The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes (Numisheet 2011)

Two-surface plasticity Model and Its Application to Springback Simulation of Automotive Advanced High Strength Steel Sheets Taejoon Parka, Dong-Yoon Seoka, Chul-Hwan Leea, Nobuyasu Nomab, Toshihiko Kuwabarab, Thomas B. Stoughtonc and Kwansoo Chunga a

Department of Materials Science and Engineering, Research Institute of Advanced Materials, Seoul National University, Daehak-dong, Gwanak-gu, Seoul 151-744, Republic of Korea b Division of Advanced Mechanical Systems Engineering, Institute of Engineering, Tokyo University of Agriculture and Technology, 2-24-16, Nakacho, Koganei-shi, Tokyo 184-8588, JAPAN c General Motors Global Research and Development Center MC 480-106-244, Warren, MI, 48090-9055, USA

Abstract. A two-surface isotropic-kinematic hardening law was developed based on a two-surface plasticity model previously proposed by Lee et al., (2007, Int. J. Plast. 23, 1189-1212). In order to properly represent the Bauschinger and transient behaviors as well as permanent softening during reverse loading with various pre-strains, both the inner yield surface and the outer bounding surface expand (isotropic hardening) and translate (kinematic hardening) in this twosurface model. As for the permanent softening, both the isotropic hardening and the kinematic hardening evolution of the outer bounding surface were modified by introducing softening parameters. The numerical formulation was also developed based on the incremental plasticity theory and the developed constitutive law was implemented into the commercial finite element program, ABAQUS/Explicit and ABAQUS/Standard using the user-defined material subroutines. In this work, a dual phase (DP) steel was considered as an advanced high strength steel sheet and uni-axial tension tests and uni-axial tension-compression-tension tests were performed for the characterization of the material property. For a validation purpose, the developed two-surface plasticity model was applied to the 2-D draw bending test proposed as a benchmark problem of the NUMISHEET 2011 conference and successfully validated with experiments. Keywords: Isotropic-kinematic hardening, Constitutive behavior, Bauschinger effect, Transient behavior, Permanent softening, Advanced high strength steels (AHSS), Spring-back PACS: 62.20.F-

INTRODUCTION Major efforts has been made to replace conventional steel sheets with light weight sheets and advanced high strength steel sheets in the automotive industry. However, there are several technical obstacles to overcome for their automotive applications such as inferior formability and larger spring-back. One of the effective ways to resolve such drawbacks is to optimize sheet forming processes utilizing computational methods, which can properly predict failure and spring-back in the process design stage. Since the spring-back of advanced high strength steel sheets is generally larger than that of conventional steel sheets and its computational prediction needs more sophisticated material characterization and numerical formulations. In order to properly account for mechanical properties of advanced high strength steel sheets, a constitutive model was developed based on a two-surface plasticity model previously proposed by Lee et al. [1]. Both the inner yield surface and the outer bounding surface expand (isotropic hardening) and translate (kinematic hardening) in this two-surface model. As for the permanent softening, both the isotropic hardening and the kinematic hardening evolution of the outer bounding surface were modified by introducing softening parameters. In this work, a dual

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1175-1183 (2011); doi: 10.1063/1.3623736 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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phase (DP) steel was considered as an advanced high strength steel (AHSS) sheet. Besides uni-axial tension tests, disk compression tests and uni-axial tension-compression-tension tests with various pre-strains were performed for the characterization of the DP sheet. For a validation purpose, the developed model was applied to the 2-D draw bending test proposed as a benchmark problem of the NUMISHEET 2011 conference.

THEORY In order to represent the Bauschinger and transient behavior during reverse loading, the combined type isotropic– kinematic hardening constitutive law based on the modified Chaboche model (Chung et al. [2], Chaboche [3]) is given by M f (   iso 0

(1)

where  is the Cauchy stress,  is the back-stress for the kinematic hardening and the effective stress (related to the isotropic hardening),  iso , is the size of the yield surface as a function of the accumulative effective strain,

 ( d  ) . Note here that the effective strain increment, d  , is defined to be a conjugate value of the effective

stress,  iso , under the following work equivalence principle:

dwiso  (   ) : d p   iso d

(2)

In the two-surface plasticity model, the back-stress for the kinematic hardening of the (inner) yield surface consists of its relative kinematic motion,  , (with respect to the bounding surface kinematic hardening) and the back-stress evolution for the kinematic hardening of the (outer) bounding surface; i.e.,

  

(3)

where  and are the back-stress for kinematic hardening of the yield surface and the bounding surface, respectively. As for the evolution of  , the following evolution rule is applied:   (   )      d d   h         iso    iso   iso

  f

1 M

(4)

( )

where  is the difference between the size of the bounding surface, iso , and the size of the inner yield surface,  iso , while d  is defined in Eq. (2). Note that the parameter h controls the rate of the relative kinematic motion. As for isotropic hardening of the inner yield surface to describe the expansion of the yield surface size, d iso is obtained from simple tension/compression tests. As for the description of the bounding surface, the evolution of the isotropic hardening is defined as, d iso  Kbound nbound ( 0,bound  )nbound 1 d 

(5)

which is the Swift type law. As for the kinematic hardening of the bounding surface, h   (   )  d  h2,bound  1,bound     d  h2,bound   iso  

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(6)

In order to describe the permanent softening behavior, the hardening evolution of the bounding surface was modified: s

 iso  * *  iso bound    iso ( , n )     * s bound h1,bound  h1,bound  kine ( , n* )

(7)

bound bound where iso and  kine are the softening parameters of the bounding surface for isotropic hardening and kinematic

hardening, respectively. Here,  * is the accumulative effective strain measured during the n* -th (current) reverse 0.0  iso ( * , n* )  1.0 , 0.0  kine ( * , n* )  1.0 , iso ( *  0.0, n* )  1.0 loading and while and

kine ( *  0.0, n* )  1.0 . With   1.0 in Eq. (7) for all reverse loading, reverse loading curves without permanent softening are recovered.

MATERIAL CHARACTERIZATION In this work, a dual phase (DP) steel, DP980, produced by Arcelor-Mittal and supplied by GM was considered. Uni-axial tensile, uni-axial T-C-T and disk compression tests were performed for the characterization of the DP980 sheet.

Uni-axial Tensile Behavior Uni-axial tension tests were carried out based on ASTM E 8M standard procedure with specimens prepared by a wire-cutting process along the rolling direction. Tensile grip speed was 0.05 mm/sec, which corresponds to approximately 0.001/s in the engineering strain rage considering the 50 mm gauge length. For anisotropy, uni-axial tensile tests along the rolling, 45ȋ off and transverse directions were performed to measure the hardening curves and R-values based on ASTM E 8M and ASTM E 517, respectively. The measured thicknesses, Young’s moduli (E), 0.2% offset yield strength (YS), ultimate tensile strength (UTS), uniform and total elongations as well as plastic strain ratios (R-values), averaged for three tests per each, are summarized respectively in Table 1 and the engineering stress-engineering strain curves for each direction are comparatively shown in Fig. 1.

Engineering stress(MPa)

1200

DP980-550

1000 800 600 400 RD 45D TD

200 0 0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Engineering strain

FIGURE 1. Uni-axial tension curves of the DP980 steel

Thickness (mm) 1.188

TABLE 1). Elastic and plastic properties of the DP980 steel %Elongation Dir. E YS UTS (MPa) (MPa) (degree) (GPa) Uniform Total 0º 45º 90º

204.095 212.998 208.755

833.69 788.12 751.00

1177

1104.70 1098.33 1098.24

7.11 5.31 5.73

11.53 9.69 9.77

R-value 0.985 0.835 1.121

Uni-axial Tension-Unloading-Tension Behavior In order to measure the reductions of Young’s modulus according to the equivalent plastic strain increase, uniaxial tension-loading-tension tests were performed. Tensile tests were performed following the ASTM E 8M standard procedure at a constant tensile speed of 0.05 mm/s, which corresponds to approximately 0.001/s in the engineering strain rate with the gauge length of 50 mm. Using the specimens prepared along the rolling direction, Young’s moduli during loading and unloading for each pre-strain increment, 1% and 3% in engineering strain, were measured as shown in Fig. 2. The measured Young’s moduli were fit to the following equation (Yoshida et al., [4]). E  E0  ( E0  ESAT )(1  eCSAT  )

(8)

Here, E0 is the initial Young’s modulus measured from the initial true stress-true strain curves of the uni-axial tension tests, ESAT is the saturated Young’s modulus when the effective strain,  , goes to infinity and CSAT is the convergent speed factor. Determined parameters, E0 , ESAT and CSAT are 204.095 GPa, 176.337 GPa and 67.743, respectively.

Engineering stress (MPa)

1200

DP980-550

1000 800 600 400 200 0 0.00

1% pre-strain interval 3% pre-strain interval

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Engineering strain

FIGURE 2. Uni-axial tension-unloading-tension curves of the DP980 steel

Disk Compression Tests The Rb value (the ratio of transverse to rolling direction strains in balanced biaxial tension) was obtained from the disk compression tests. Assuming that hydrostatic pressure has no influence on yielding, through-thickness disk compression tests were performed on a Gleeble Tester 3500. Several specimens, with a diameter of 6.0 mm were compressed through the sheet normal direction. For each specimen and each material, compression was performed with increments of 4kN from 20kN to 40kN. After each compression increment, diameters in the rolling and transverse directions were measured. Based on these measurements, assuming incompressibility Rb value and the 0.2% offset yield stress under the balanced biaxial stress condition,  b , were calculated as 1.1657 and 713.199 MPa, respectively.

Uni-axial Tension-Compression-Tension Behavior In-plane tension-compression-tension tests were performed using the testing apparatus developed by Kuwabara et al. [5]. Since buckling of the specimen is a major problem while carrying out compression tests on thin samples, this testing apparatus uses specially designed comb-shaped dies for applying a small blank holding pressure of 5 MPa to a sheet specimen to prevent buckling of the specimen. In order to prevent the specimen from galling the dies, it was lubricated on both sides with Vaseline and Teflon sheets. A run without the sample is carried out to measure the friction in the die assembly, which is later compensated in the results by subtracting (or adding) it from measured load values. Tests were carried out under a displacement controlled mode and two different speeds were used: in the elastic regions, the test speed was 0.5 mm/min whereas, in the plastic region, it was 1.5 mm/min. As shown in Fig. 3, using the specimens prepared along the rolling direction, T-C-T tests were carried out for the DP980 steel under the following true strain histories:

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(a) 0.02 ˧ 0 ˧ 0.02, 0.04 ˧ 0 ˧ 0.04 and 0.06 ˧ 0 ˧ 0.06 (b) 0.04 ˧ -0.04 ˧ 0.04 and 0.06 ˧ -0.06 ˧ 0.06

DP980-550

1000

True Stress (MPa)

True Stress (MPa)

1000 500 0 -500 0.02 pre-strain 0.04 pre-strain 0.06 pre-strain

-1000

DP980-550

500 0 -500 -1000

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

-0.06 -0.04 -0.02

True Strain

0.00

0.02

0.04

0.06

True Strain

(a)

(b)

FIGURE 3. Uni-axial tension-compression-tension curves of the DP980 steel Experiment Fitting (Swift type) Fitting (Voce type) Fitting (Combined type)

1500

DP980-550

True stress (MPa)

1000 500 0 -500 -1000 -1500 -0.06 -0.04 -0.02

0.00

0.02

0.04

0.06

True strain

FIGURE 4. Calibrated and measured uni-axial tension-compression-tension curves of the DP980 steel

From the measured uni-axial tension-compression curves, at each unloading position, the effective yield stress ( ) is calculated first as

 iso 

 f r

(9)

2

Here,  f is the yield stresses at the start of unloading and  r is the initial yield stress during reverse loading. Then, the measured effective stresses,  iso , at each unloading position were fit to the following Swift type hardening law as summarized in Table 2:

 iso  Kinner ( 0,inner  )n

inner

(10)

After the characterization of the isotropic hardening,  iso , of the (inner) yield surface, the isotropic hardening and kinematic hardening motions of the bounding surface were calibrated before the characterization of the kinematic hardening motion of the inner surface. Under the uni-axial tension/compression in the rolling direction, the following relation is derived from Eq. (6): d  h1,bound  h2,bound d

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(11)

As shown in Fig. 4, the DP980 steel does not show the permanent offset (softening) during the first reverse loading of the uni-axial tension-compression-tension test, compared to the reverse loading curves calculated from the Voce type, Swift type and combined type hardening coefficients. However, the DP980 steel shows permanent softening during the second reverse loading. In summary, for the DP980 steel, no permanent softening was assumed for the first reverse loading, while permanent softening was assumed for the second loading. After the characterization of the effective stresses,  iso , of the (inner) yield surface, the effective size of the bounding surface,  , for each unloading position was fit to the following Swift type hardening law: total

nbound total   Kbound ( 0,total bound  )

(12)

For the uni-axial deformation, the following relationship is derived from Eqs. (5) and (6):

  iso  iso (0) iso  iso (0) 

(13)

where iso (0) is the initial effective size of the bounding surface. Also, d   d iso d  d  (1  )d , 0 

1

(14)

Here, the parameter, , determines the ratio of the isotropic hardening increment with respect to the effective size of the bounding surface increment, d iso  d  , and the ratio of the kinematic hardening increment with respect to the effective size of the bounding surface increment, d  (1  )d  . Assuming the parameter,

, to be a specific

constant, the isotropic hardening, iso , and the kinematic back-stress movement, , of the bounding surface can be determined from the measured effective size of the bounding surface,  . In this work, the parameter, assumed to be 0.5 for simplicity; i.e., iso  iso (0) d iso  (0) 0.5d 

, was

(15)

 (0) d  0.5d 

where (0)  iso (0) and (0)  0 . The isotropic hardening of the bounding surface was fit to the Swift type hardening law as summarized in Table 2. Now, the general solution of Eq. (11) for the kinematic hardening becomes,

( )  e h

2,bound

(  0* )

 h  *  e 2,bound (  0 ) ! h1,bound ( )d  ( 0* )  *   0 

(16)

when h2,bound is a constant as assumed here. Then, h1,bound is determined by the measured kinematic hardening, , and the assumed constant h2,bound . In order to describe the change of the parameter, h1,bound , the following equation was used: h1,bound ( ,  n* , n* )  a1 g1 (1  e d1 ) b1ec1 (  0 ) b1ec1 (  1 ) b1ec1 (  2 ) *

*

*

b1ec1 (  n ) *

(17)

Here,  0* and  n* denote the initial effective stain and the effective strain at the n* -th reverse loading, respectively. If the saturation rate parameter, c1 , is large enough and the reverse loading number, n* is zero, Eq. (17) becomes h1,bound ( )  a1 g1 (1  e d1 )

1180

(18)

After the characterization of parameters for the kinematic movement of the bounding surface, the following softening parameter was introduced only to the kinematic hardening of the bounding surface in order to describe the permanent softening of the second reverse loading curves: bound kine ( n* , n* )  (1.0  a4 ) a4 eb 

* 4 n

(19)

Finally, in order to control the different rate of the relative kinematic hardening motion,  , of the inner surface during initial loading and reverse loading, the following effective strain dependent values of h was used: h( n* , n* )  h0 (hsat  h0 )(1  eCh ( n  n1 ) ) *

*

(20)

The measured parameters were summarized in Table 2, while the measured and calibrated true stress-true strain curves during the uni-axial tension-compression-tension curves are compared in Fig. 5.

Yld2000-2d Yield function The eight anisotropic coefficients of the plane stress yield stress function Yld2000-2d [6] were determined based on eight mechanical measurements  0 ,  45 ,  90 , R0 , R45 , R90 ,  b and Rb are listed in Table 3.

K (MPa) 803.842 a1 (MPa)

 iso h1,bound

40881.6

bound kine

TABLE 2). Elastic and plastic properties of the DP980 steel  0,bound 0 Kbound (MPa) n diso 0.2225 0.4953 1019.37 0.000012996 g1 (MPa) d1 b1 (MPa) c1 (MPa) h2,bound 100000 500 21941.0 10.4831 a4 b4 0.2057

n*  0 h

500 (constant)

# c11

# c22

0.8199

1.3428

h0

Ch

h0

161.5685

63.4993

32.8226

186.7992

0.9822

0.1141

0.1853

1.054

185

Ch

23.2201

TABLE 3). Anisotropic coefficients of Yld2000-2d ## ## ## ## c11 c12 c21 c22

Experiments Two-yield surf.

15.1693

# c66

## c66

1.060

0.9365

Experiments Two-yield surf.

1000

1000

True Stress (MPa)

True Stress (MPa)

34.5485 n* " 2 hsat

n*  1 hsat

nbound

0.0436

500 0 -500

500 0 -500 -1000

-1000 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

-0.06 -0.04 -0.02

0.00

0.02

0.04

0.06

True Strain

True Strain

(a)

(b)

FIGURE 5. Comparison of the calibrated and measured uni-axial tension-compression-tension curves of the DP980 steel

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NUMERICAL SIMULATION OF THE 2-D DRAW BENDING TEST For validation purposes, the developed constitutive model was applied for the simulation of the 2-D draw bending test proposed as a benchmark problem of the NUMISHEET 2011 conference. Two different specimens were prepared for the 2-D draw bending tests: one with base material (without pre-strain) and the other with prestretching by 0.04 in true strain. A schematic view of tools and their dimensions are shown in Fig. 6 and Table 4. The hardening model was incorporated into the commercial finite element programs, ABAQUS/Explicit and ABAQUS/Standard, utilizing the characterized mechanical properties of the DP980 steel sheet. Forming process was simulated using the explicit finite element code, ABAQUS/Explicit, with the user-defined subroutines VUMAT, while springback behavior was simulated using the implicit finite element code, ABAQUS/Standard, with the userdefined subroutines UMAT. The 4-node three-dimensional rigid body element (R3D4) was used for tools and the 4node shell reduced volume integration element (S4R) with 9 integration points through thickness was employed for the blank. The element size of the blank was approximately 1.0mm Ý 1.0mm throughout the whole blank and only the quarter of the whole blank was simulated for the computational efficiency. Blank holding force of 300 kgf (2.94 kN) was maintained throughout the test and the friction coefficient was assumed to be a constant of 0.1. The simulation punch speed was 100mm/sec, which is 100 times higher than the actual speed. Fig. 7 shows the measured and simulated spring-back profiles. The simulation results reasonably well represent the spring-back behavior of the DP980 grade advanced high strength steel sheet. Plane of symmetry

W1 G1 Thickness

W3

R1 Holder

Punch

Holder

Die

Sheet Die

R2

W4 Stroke

W2

(a) W

Rolling direction

L

(b) Lgrip Grip Position

Lgrip Rolling direction

L0

Grip Position

Cutting

Grip Position

W0

Grip displacement

Rolling direction

L

Grip Position

Cutting

(c) FIGURE 6. Schematic view of specimens and tools and their dimensions for the 2-D draw bending test: (a) tools, (b) base specimens (without pre-strain) and (c) 4% pre-strained specimens

Parameters Dimensions Parameters Dimensions

W1 50.0 L 300.0

TABLE 4). Dimensions for the 2-D draw bending test (unit: mm) W2 W3 W4 R1 R2 G1 54.0 89.0 89.0 5.0 7.0 2.0 W L0 W0 Lgrip Grip displacement 30.0 300.0 30.0 34.705 9.411

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Stroke 70.0 L 240.0

Experiments (w/o pre-strain) Simulation

50

50 40 Z coord.(mm)

Z coord.(mm)

40 30 20

30 20

10

10

0

0 0

10

20

30

40

50

60

70

80

90

100

Experiments (with 4% pre-strain) Simulation

0

10

20

X coord.(mm)

30

40

50

60

70

80

90

100

X coord.(mm)

(a)

(b)

FIGURE 7. Measured and simulated spring-back profiles for the 2-D draw bending test: (a) without pre-strain, (b) with 4% prestrain

CONCLUSION A two-surface isotropic-kinematic hardening law was developed to account for the springback of automotive advanced high strength steel sheets. A dual phase (DP) steel sheet, DP980 steel, produced by Arcelor-Mittal was considered as an advanced high strength steel sheet in this work. Besides uni-axial tension, disk compression and uni-axial T-C tests, T-C-T tests were performed with various pre-stains for the characterization of the DP980 sheet. The numerical formulation was also developed based on the incremental plasticity theory and the developed constitutive law was implemented into the commercial finite element program, ABAQUS/Explicit and ABAQUS/Standard using the user-defined material subroutines. For a validation purpose, the developed two-surface plasticity model was applied to the 2-D draw bending test proposed as a benchmark problem of the NUMISHEET 2011 conference and successfully validated with experiments.

ACKNOWLEDGMENTS The work was performed under the collaboration between GM and SNU. This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (R11-2005-065)/the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2010-220-D00037)/the Engineering Research Institute at Seoul National University. The authors are also grateful to Prof. Myoung-Gyu Lee at POSTECH, Dr. Kanghwan Ahn and Dr. Kyung-Hwan Chung at POSCO for performing the disk compression tests, uni-axial tension tests and 2-D draw bend tests.

REFERENCES 1. 2. 3. 4. 5. 6.

M.-G. Lee, D. Kim, C. Kim, M. L. Wenner, R. H. Wagoner, K. Chung., K., Int. J. Plast. 23 (7), 1189-1212 (2007) K. Chung, M.-G. Lee, D. Kim, C. Kim, M. L. Wenner, F. Barlat, Int. J. Plast. 21 (5), 861-882 (2005) J. L. Chaboche, Int. J. Plast. 2 (2), 149-188 (1986) F. Yoshida, T. Uemori, Int. J. Plast. 18, 661-686 (2002) T. Kuwabara, Y. Kumano, J. Ziegelheim, I. Kurosaki, Int. J. Plast. 25, 1769-1776 F. Barlat, J. C. Brem, J. W. Yoon, K. Chung, R. E. Dick, S-H Choi, F. Pourboghrat, E. Chu, D. J. Lege, Int. J. Plast. 19 (9), 1297-1319 (2003)

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Paradigm Change: Alternate Approaches to Constitutive and Necking Models for Sheet Metal Forming Thomas B. Stoughtona and Jeong Whan Yoonb a

General Motors Global Research and Development Center MC 480-106-244, Warren, MI, 48090-9055, USA b Faculty of Engineering and Industrial Sciences, Swinburne University of Technology PO Box 218, Hawthorn, VIC 3122, Australia Abstract. This paper reviews recent work proposing paradigm changes for the currently popular approach to constitutive and failure modeling, focusing on the use of non-associated flow rules to enable greater flexibility to capture the anisotropic yield and flow behavior of metals using less complex functions than those needed under associated flow to achieve that same level of fidelity to experiment, and on the use of stress-based metrics to more reliably predict necking limits under complex conditions of non-linear forming. The paper discusses motivating factors and benefits in favor of both associated and non-associated flow models for metal forming, including experimental, theoretical, and practical aspects. This review is followed by a discussion of the topic of the forming limits, the limitations of strain analysis, the evidence in favor of stress analysis, the effects of curvature, bending/unbending cycles, triaxial stress conditions, and the motivation for the development of a new type of forming limit diagram based on the effective plastic strain or equivalent plastic work in combination with a directional parameter that accounts for the current stress condition. Keywords: Non-Associated Flow Rule, Plastic Dilatancy, Pressure Sensitivity, Distortional Hardening, Necking Limits. PACS: 62.20.F-,62.20.mm

INTRODUCTION This paper reviews recent work that proposes paradigm changes for constitutive and necking models in sheet metal forming. The first section reviews the motivating factors and benefits in favor of both associated and nonassociated flow models for metal forming in the view of experimental, theoretical, and practical aspects. This section completes with a review of three examples of non-associated flow, one allowing capture of the most common documented anisotropies of the material response in both plastic flow and yield behavior using quadratic function, another a fully anisotropic treatment of pressure-dependent yielding in the absence of plastic dilatancy, and finally, one allowing treatment of four degrees of anisotropic hardening. The topic of constitutive models will be followed by a section reviewing a paradigm change of metal forming limits from strain to stress metrics. This section will review the topic of the forming limits, the limitations of strain analysis, the evidence in favor of stress analysis, and motivation for the development of a new type of forming limit diagram based on the effective plastic strain or equivalent plastic work in combination with a directional parameter that accounts for the current stress condition. Particular attention is paid to common misunderstanding in the application of stress metrics, which is contributing to the continued use of strain metrics for the analysis of formability under the violated conditions. The section will also review the role of stress and strain gradients through the sheet thickness, the effect of non-zero normal stress, and cyclic bending and unbending, all of which affect not only the interpretation of the stress and strain conditions during forming, but also the interpretation of the forming limit curve itself.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 15-34 (2011); doi: 10.1063/1.3623589 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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PARADIGM CHANGE IN CONSTITUTIVE MODEL FLOW RULES The associated flow rule (AFR) is most commonly used as a basis in constitutive models for metals for several reasons. One reason is simplicity; By equating the plastic potential, which is used to define the direction of the plastic strain rate, to the yield function, one need only define a single function, which ostensibly is simpler and easier to calibrate. Another reason is theoretical; Bishop and Hill (1951) derived a proof that the AFR applies to a material whose deformation is restricted to simple slip mechanisms characterized by reaching a critical shear stress on a slip plane known as the Schmid criterion, which is understood to be the primary mechanism of plastic deformation. A third reason is physical and numerical stability; Drucker (1951) proved that use of the associated flow rule guarantees positive plastic work and uniqueness of solution to the equations of classical plasticity, and further established a class of stable metals, known as non-energetic materials in Drucker (1959). Additional questions of stability under non-AFR have been raised by Sandler and Rubin (1987), who derived an indeterminate (non-unique) solution to the elastic-plastic wave equation under non-AFR, which they argued proved that non-AFR is an intrinsically unusable concept in plasticity modeling. While the AFR is well established in plasticity modeling of metals, this popularity is brought under serious challenge by the experimental work of Richmond (1980), and Spitzig and Richmond (1984), who studied pressure sensitivity of yielding and plastic dilatancy of steel and aluminum alloys. They found a small but finite pressure sensitivity of the yield behavior for all steel and aluminum alloys that they studied. Under the AFR, this pressure sensitivity would be expected to result in an associated small but finite plastic dilatancy, or permanent volume change. However, they observed no plastic dilatancy in their experiments within their experimental resolution, which was more than 10 times smaller than the amount of dilatancy that was expected under the AFR. This led them to conclude that the AFR does not generally apply to metals, a conclusion that has not yet had much influence on the selection of constitutive modeling for metals. The experimental evidence against the AFR raises doubts about the significance of the theoretical arguments that suggest it is necessary. One possibility that undermines Bishop and Hill’s theoretical foundation for the AFR, for example, is that it does not consider the effect of dislocation interactions or other effects that could lead to deviations from the Schmid criterion. Qin and Bassani (1992), and Bassani et al.(2001) describe dislocation interactions and structures that lead to non-associated flow effects. Also, Bulatov et al. (1999) describes details on a specific dislocation mechanism derived from first principles that predicts the magnitude of the AFR violations in Richmond’s work. Stoughton and Yoon (2006) derived a set of four conditions that apply to both AFR and non-AFR models in response to the stability issues raised by Drucker. The constraints, which are described later in this paper, ensure uniqueness and positive-definite plastic work when applied to a class of non-AFR models. The conditions include generalizations of convexity requirements applied individually to the plastic potential and yield functions, a requirement of positive work hardening (for uniqueness under applied stress changes), and a fourth requirement that is a generalization of Drucker’s postulate. The nature of that latter requirement, which is automatically satisfied under the AFR, can also be satisfied under a class of non-AFR. This work proves that satisfaction of Drucker’s postulate is a sufficient, but not necessary condition of uniqueness of solution and positive-definite plastic work. Stoughton and Yoon (2006) also discuss the intrinsic energetic properties of non-associated flow, showing that the characteristics of an energetic material are in a different class from questions of uniqueness and positive plastic work. It is pointed out that the amount of work that can be productively used in a closed strain cycle comes from the stored elastic energy, is a small fracture of that energy, is uniquely defined in terms of the load state and the size and direction of the strain cycle, and is only released during the first cycle since the entire strain cycle is within the yield function after the initial loading. In those terms, its nature is not comparable to the type of instabilities that would occur if the plasticity equations had non-unique solutions or permitted negative plastic work. Stoughton and Yoon (2008) also reviewed the derivation of the indeterminate solution to the elastic-plastic wave equations that was derived by Sandler and Rubin (1987), showing that this solution does not explain the equations of motion unless the plastic wave speed is strictly constant. They then showed that the curvatures of the yield function, plastic potential, and stress-strain response that are typically used to characterized metal deformation, generally work to prevent the plastic modulus and plastic wave speed from reaching a static value at any time during plastic deformation.

16

While the interpretation of experiments, such as the implications of Spitzig and Richmond’s work, and the theoretical arguments in favor of either the AFR or the non-AFR, will continue to be debated for some time, the third factor that plays a role in an engineer’s selection of material model are simplicity, the practical issues of accuracy, complexity, calibration, and cost. As implied earlier, there may be a wrong assumption that non-AFR is intrinsically more complex, involves more parameters that require calibration, and leads to higher computational cost due to the fact that there are two functions involved in the constitutive equations, instead of one. However, it can be argued the other way, that with the constraint of the AFR, it becomes a greater challenge to simultaneously capture the anisotropy of both yielding and plastic flow with a single function, which often appear to be unrelated, forcing one to introduce functions of increasing complexity. The most simple model for anisotropic metals under the AFR is the quadratic yield function proposed by Hill (1948). The limitations of this yield function under the AFR are well known. Bramley and Mellor (1966) were one of the first to show that this model is not able to predict the in-plane anisotropy of the yield behavior of steels when the parameters were calibrated from the experimental r values. They noted that if one ignores the in-plane anisotropy and restricts the model to consider only normal anisotropy, or average r value in the plane of the sheet, then the model gives an acceptable fit to the ratio of the yield in equal biaxial tension and the average yield in uniaxial tension for the four steel alloys that the considered in their study. With the state of the art of material modeling of anisotropic materials in its infancy, Bramley and Mellor proposed using the special case of the normal anisotropic version of Hill’s quadratic function for steels. Pierce (1968) expanded on Bramley and Mellor’s idea by applying Hill’s normal anisotropic model to nine metals with average r values ranging broadly from 0.18 to 3.80. He found that the normal anisotropic model did not correlate well for materials with average r value less than one. This paper may have been the first to refer to this discrepancy between experiment and the quadratic model as the ―anomalous yield‖ behavior in equal-biaxial tension associated with metals with r values less than one. Not surprisingly, researchers for aluminum alloy sheets played a leading role in advancing constitutive models for anisotropic materials in the next several decades. Even today, the dominance of the use of Hill’s normal anisotropic model, and the primacy of the consideration of r value in decisions to use more advanced models, can probably trace its origins to the influence of the work of Bramley and Mellor (1966) and Pierce (1968), respectively. However, the role of the AFR was not considered in these earlier works. If non-AFR had been considered, the conclusions would have been that Hill’s fully anisotropic quadratic function has the capability to exactly match all the experimental r values and yield stress at 0, 45, and 90 degrees to the rolling direction, as well as the yield stress in equal-biaxial tension, regardless of the r value. In that case, at least within the context of the data presented, the conclusion would have been that Hill’s fully anisotropic model was good choice of function to use under non-AFR for the 4 steel alloys in Bramley and Mellor’s study and all 9 metals in Pearce’s study, including the zinc alloy with average r equal to 0.18. These conclusions were reported in Stoughton (2002a), who proposed a simple non-AFR model based on two independent quadratic functions of the form proposed by Hill. This model will be reviewed in the next subsection, followed by a series of developments including a pressure-sensitive anisotropic yield function that is consistent with Spitzig and Richmond’s work published by Stoughton and Yoon (2004), and a simple, but powerful distortional hardening model that takes advantage of the quadratic form to analytically incorporate the anisotropy in the strain hardening functions at 0, 45, and 90 degrees to the rolling direction, as well as in equalbiaxial tension. The final subsection concludes with a discussion of potential directions in material model development, including higher order functions and kinematic models, which introduce a higher level of flexibility with minimal complexity under non-AFR.

Fully Anisotropic Non-Associated Flow Rule The yield criterion for an anisotropic metal under isotropic strain hardening is defined as follows,

 y  11 ,  22 ,  23    f  p 

17

(1)

where σf(εp) is the strain hardening function along the rolling direction of the coil, in the normalization adopted in the yield function. Using the model proposed by Hill(1948) with normalization to the rolling direction along the 1axis, Stoughton (2002a) defined a non-AFR model with yield function parameters as follows 2  y   112   y 22  2 y 11 22  2  y 122

   y   0    90 

2

1   2   y    0  2    45  

2

2 2 1    0    0       y  1   2    90    B     2      0     B  

(2)

where σ0, σ45, and σ90, are the initial yield stress in uniaxial tension along the rolling, diagonal, and transverse directions and σB, is the initial yield stress in equal biaxial tension. To generalize this model using the flow rule,

 p E ij( p )   p  ij ,

(3)

the plastic potential is given by an identical quadratic form with different coefficients under a non-AFR. The definition is the same as that used by industry in most calibrations of Hill’s model under the AFR, 2  p   112   p 22  2 p 11 22  2  p 122

p 

1  1 / r90 1  1 / r0

p 

1  1/ r  1/ r

1 1  1 / r0

(4)



90 1  2r45   p   0 2  1  1 / r0 

where r0, r45, and r90, are the r values in uniaxial tension along the rolling, diagonal, and transverse directions. As noted in Stoughton and Yoon (2004), it is convenient for expressing the constitutive equations to use lower case symbols p and y to define the gradient tensor components of the plastic potential and yield functions respectively,

 p11  1 p   p 22   p  p12 

  11   p 22     p 22   p 11   2  p 12   

 y11  1 y   y 22   y  y12 

  11   y 22     y 22   y 11   2  y 12   

(5)

and use upper case P and S to define the tensor product of the elastic matrix C and, respectively, the gradient of the plastic potential p and the total strain e=[E11,E22,2E12] ,

S  C : e

P  C:p

(6)

where

1  E  C  1 1  2  0 0

 0  1  2 1   0

18

(7)

and E and ν are Young’s Modulus and Poisson Ratio, respectively. This leads to the following definitions for this particular model

1   p  11   p   p  22  E   p  p  22   p   11  P 2  1 -  p   1   p 12  





S 

E 1 - 2





 E11  E 22      E 22  E11  .  1   E12   

(8)

Then the rate of change of the effective plastic strain is given by,

 p 



MAX 0, y11S11  y 22 S 22  y12 S12 h  y11 P11  y 22 P22  y12 P12



(9)

where h is the derivative of the stress-strain relation along the rolling direction, which is used in the yield condition defined in Eq. 1. The constitutive law relating the stress rates to the total strain rates is then given by,

 11   S11   P11      S    P   22   22   22  p  12   S12   P12 

(10)

It should be noted that the only difference in computational costs using this non-AFR is the three extra calculations given in Eq. 4, since all the subsequent calculations leading to the solution in Eq. 10 are identical to those used in an AFR in the case p=y. This small extra computation cost is offset by the advantage that the model now exactly matches the initial uniaxial and equal biaxial yield stresses and the uniaxial strain ratios at 0, 45 and 90 degrees.

Non-AFR with Pressure Sensitive Yielding One of the motivations for non-AFR discussed in the introduction was the experimental evidence published by Spitzig and Richmond’s of pressure-sensitive yield behavior in the absence of plastic dilatancy. In the general full stress version of the non-AFR model in the previous section, the plastic potential and yield functions are both invariant under application of hydrostatic stress. As a result, the model constrains the plastic dilatancy to zero, in agreement with the experimental results, but also results in zero pressure sensitivity. In order to obtain a physically meaningful description of the pressure sensitivity, we consider the analysis reported in Spitzig, Sober, and Richmond (1975). This study reported the yield behavior for a high strength steel subjected to combined tension with hydrostratic pressures from 0 to 8 GPa. They found the best fit to the data is described in terms of the J2 and I1 invariants for an isotropic material in the following form,

3J 2 1  I1    f  p 

(11)

where α is a small material constant of inverse stress such that αI1 is on the order of a few percent. This is the term that gives rise to a small pressure sensitivity. This yield function also predicts a strength differential effect (SDE) between tension and compression, which agreed with experiment. Stoughton and Yoon (2004) generalized this yield criterion for full anisotropy in both the invariant contributions for plane stress conditions to obtain the following component of a non-AFR for plane-stress conditions,

 y  ij 1  1 11   2 22    f  p 

19

(12)

where 2  y  1 112   2 22  2 y 11 22  2  y  122

1 

C 0  T0 C 02  T02

 1 1    1   1T0

2    

2

C 90  T90 2 C 90  T902

T  1   2   0 T 1   T 90 2 90  

T 1 1  y   1   2   0 2  TB 1   1   2 TB 

  

2

2

   

2T  2T 2T    2  T45 4TB   1  2T    y   0  0  0  0  1 T0    2  T45 TB  T45 TB 2 T T 45    B 1  12  1   2 T45  2TB  

1  12  1   2 T45 2 1   1   2 TB 2

(13)

where T0, T45, and T90, are the initial yield stress in uniaxial tension along the rolling, diagonal, and transverse directions, TB, is the initial yield stress in equal biaxial tension, and C0, and C90, are the initial yield stress in uniaxial compression along the rolling and transverse directions. Note that while the equations for some of the 6 constants of this model are complex, they are all explicit functions of easily measured 6 experimental yield stresses. Although we could also generalize the plastic potential, there may be no need for metals characterized by zero plastic dilatancy. Consequently, the plastic potential described by Eq. 4 was adopted in Stoughton and Yoon (2004). While the material parameter definitions for the yield function are more complex, this does not affect the computational speed. Consequently, the ability to account for SDE and pressure sensitivity in the yield stress may make the modest increase in complexity of this constitutive law worth the effort. In this case the gradient of the yield function y becomes

 1 11   y 22   1  1   1 11   2 22   y 2 22   y 11    2  y  y  2  y 12   0   

(14)

Surprisingly, no other changes or additional computations are required to find the solution using Eqs. 6-10.

Non-AFR with Distortional Hardening One of the disadvantages of a yield criterion in which the coefficients of the yield function are constants is that it is not possible to capture the anisotropy in the rate of hardening that is actually observed in most metals. While the rate of hardening in uniaxial tension in the plane of the sheet appears to be nearly isotropic to a good approximation, this is not generally true in comparison of the rate of hardening in uniaxial to equal biaxial tension. Consequently, a noticeable increase in accuracy can be obtained over the range of biaxial stress conditions that occur in metal forming processes by allowing for distortional hardening, as is noted in Stoughton and Yoon (2009).

20

One way to capture this differential hardening is to treat the coefficients of the yield function as explicit functions of the plastic strain, as described by Zamiri and Pourboghrat (2007). However, for high exponent yield functions, the fidelity of the predicted hardening is affected by the interpolation function used to define the evolution of the coefficients. Furthermore, it may be difficult to closely match the hardening behavior and control the shape of the yield function without excessive discretization of the interpolation function when using higher exponent functions for fully anisotropic materials. This difficulty is a result of the convoluted relationship between the coefficients and the hardening behavior that do not allow analytical forms to directly control the shape of the stress-strain response. The dependence on interpolation function does not occur for a quadratic yield function because the coefficients can be defined explcitly in terms of the stress-strain relations at the four critical experimental conditions, i.e., at 0, 45, and 90 degrees in uniaxial tension as well as in equal biaxial tension. In this case, no parameter interpolation functions are required. The accuracy of the distortional hardening is limited only by the accuracy of the analytic relationship between stress and strain along the directions of the model calibration. Such a simple approach to distortional hardening would not be attractive under an AFR because, not only would one loose correlation of the predicted plastic strain rates, which as mentioned earlier, occurs under isotropic hardening models, but the distortion of the yield surface would introduce a deformation dependence in the direction of plastic strain under linear loading conditions, which is not typically observed. However, in a non-AFR model, the plastic potential can remain as defined in Eq. 4, allowing the model to exactly match the primary experimental stress-strain responses in uniaxial tension at 0, 45, and 90 degrees and in equal biaxial tension, without affecting the direction of the plastic strain components. The first step in the construction the distortional hardening model described in Stoughton and Yoon (2009) is to redefine the yield criterion in Eq. 1 as follows:

        122 4 2 f y   2 11  2 22  11   22   11 222  2 12  1          B  p   45  p  90 p   0 p

(15)

where the four functions σx(εp) are the explicit stress-strain relations that are experimentally determined along the directions x=0, 45, 90 in uniaxial tension, and in equal biaxial tension for x=B. [We discuss calibration of these relations later in this subsection.] Although perhaps not obvious, Eq. 15 is mathematically equivalent to Eqs. 1 and 2 in the limiting case when all four stress-strain relations are equal to the function on the right hand side of Eq. 1. Unlike most other models, the yield function above is dimensionless, and although it also defines an implicit yield surface for a given value of εp, and the yield function serves the same purpose as the yield stress in the constitutive equations, there is no explicit scalar function that can be called the yield stress. Due to the dimensionless character of the yield function, the gradient tensor of the yield function y has dimensions of inverse stress.

2 11   22     22   0    22  1  1  1      y 2   11   2 2 22   11   2 0  2  11       0  p   90  p   45  p   B  p      8 12  2 12  0 0 1

(16)

The only other major change to the model is the definition of the function h, in Eq. 9 which is now defined by requiring the yield function fy be invariant under changes to εp during plastic deformation in the same manner as is done in conventional models. This constraint leads to the following definition, which is again dimensionless,

         122 4 2 h  2  2 11 h0  p   2 22 h90  p  11   22   11 222 hB  p   2 12 h45  p  (17)     0  p    90  p   B  p   45  p    

21

where the four dimensionless functions hx(εp) for X=[0,45,90,B] are given by

h X  p  

1

 X  p 

d X  p  d p

.

(18)

The yield criterion given by Eq. 15 is defined in a way that the stresses in the four stress-strain responses are equal to the actual true stresses in uniaxial and equal-biaxial tensions. In other words, they do not need to be converted to an effective stress to account for anisotropy, as is commonly the case when comparing experiment to predictions of anisotropic models. However, as discussed in more detail in Stoughton and Yoon (2009), the effective plastic strain in these four functions is not equivalent to the measured plastic strain, so that calibration of the functions used in the model is necessary. For calibration, the measured plastic strain must be scaled using a formula derived from the flow rule and plastic potential, Eqs. 3 and 4. The scale factor for true strain in uniaxial tension at 0, 45 and 90 degrees and for the sum of the principal true strains in equal biaxial tension are given, respectively, by

1

1 2

1   p  2 p   p 

p

1   p  2 p .

(19)

This model was compared in Stoughton and Yoon (2009) to conventional isotropic models for three aluminum and two steel alloys for which equal-biaxial and uniaxial tension data was available in 15 degree increments in the plane of the sheet. Using the additional uniaxial data at 15, 30, 60, and 75 degrees, it is possible to obtain an estimate of the error between experiment and this model, as well as other popular models that do not consider anisotropic hardening. As expected, with calibration of the isotropic hardening models to the initial stress, the root mean square error of most of the models studied, including the model described in Barlat et al. (2003) and the isotropic hardening non-AFR model described in a previous subsection of this paper, was found to be under 1%. The small error in these models at the onset of yielding arises from differences in the behavior at uniaxial tension angles of 15, 30, 60, and 75 degrees, whose data was not used in the parameter definitions. However, what is more interesting is that as the plastic strain increases to 10% strain or higher, the error for all the isotropic hardening models increased quickly to levels comparable to the error found using von Mises and Hill quadratic model under the AFR, even for the aluminum allows using the most sophisticated high exponent yield functions. In contrast, the error for the distortional hardening model is found to remain static over all levels of plastic strain at under 1%. This result suggests that unless distortional hardening is included in the constitutive model, there may not be as much benefit to use advanced constitutive models with isotropic hardening for analysis of problems with a large mixture of biaxial stress conditions.

Future Developments Using Non-AFR We close this section with some comments about the fertile ground for research in the application and extension of non-AFR models. We have considered above only quadratic forms of the plastic potential and the deviatoric component of the yield function. However, it is well known that the curvature of the yield function in plane strain conditions is relatively flat for aluminum, leading to higher exponent functions. It appears to be more convenient to use a simpler higher exponent function to describe the yielding combined under non-AFR with a lower exponent or quadratic function for the plastic potential. Furthermore, a quadratic plastic potential has advantages because there is an explicit analytic expression for the effective plastic strain, as well as the relation between the stress and plastic strain tensor directions. The higher convexity of plastic potentials also means that the solution is less sensitive to small variations in stress near the plane strain condition, allowing more easily converged solutions even using low convexity high exponent yield functions, including those with zero curvature and vertices. These more sophisticated non-AFR models may have numerical and perhaps physical advantages over traditional higher exponent functions under AFR. In addition, the expansion of non-AFR into the realm of kinematic hardening is also wide open with opportunities adding another dimension to consider in coupling the kinematic properties of the yield function and plastic potential (for example, Taherizadeh et al.(2010)). All considered, the flexibility of non-AFR models to capture a higher degree of complex material behaviors is one of its most attractive features. However, while the authors have

22

addressed the issue in Stoughton and Yoon (2006, 2008) for the class of non-AFR model in which the above examples are included, the issue of stability generally needs to be addressed when expanding non-AFR beyond the class of models considered in this and previous papers. In this class of scalar hardening models, the following set of constraints were proven to be necessary and sufficient to ensure desired characteristics of plastic stability, including unique solution and positive changes to the plastic work,

 11  P :  22   0  12 

 11  Y :  22   0  12 

h0

hP:y 0

(20)

The first, third and fourth constraints are necessary to ensure positive increments to the plastic work. The third and fourth constraints are also necessary to avoid non-uniqueness or singularity under stress-rate or strain-rate boundary conditions, respectively. The second constraint is the least important of the four, and was shown to be only necessary to ensure that the second order work under proportional loading is positive definite, a characteristic which may be desired to avoid the possibility if yield point elongation, for example. Further details of the meaning of these constraints can be found in Stoughton and Yoon (2006). The first two models described in the previous subsections satisfy all four constraints, but it is important to consider that these stability conditions may come into play with any new variant of non-AFR. For example, an interesting violation of the third condition was found in application of the distortional hardening law described in the previous subsection. This violation arises from the stress-dependent weighting function of the biaxial stress-strain term in the explicit definition of h in Eq. 17, which becomes negative between uniaxial and pure shear stress conditions. It was found in the empirical fits for all three aluminum alloys considered in Stoughton and Yoon (2009), with the relative hardening rate differences between uniaxial and equal biaxial tension, that the h parameter falls below zero in pure shear loading at a strain of 0.449 to 0.600, depending on the alloy. This occurs even though the strain-hardening in all four stress-strain relations is positive. While h=0 leads to non-uniqueness in stress-controlled processes at this critical strain level, the fourth condition, which would lead to non-uniqueness in strain-controlled processes, was not observed in the data because the produce P:y was sufficiently large for all stress conditions. It is interesting that the third stability conditions remain satisfied up to these very high strains for aluminum, which suggests the possibility that they may be physically associated with a strain localization mechanism and metal failure. So in this case, the violation of the third stability may not be evident of a flaw or limitation of the application of this non-AFR, but could be representative of a physical instability or characteristic of these aluminum alloys. And, while there is no direct connection in the topics of non-AFR and necking models in this paper, the question of the role of instability on localization mechanism is a good segue into the topic of forming limits.

PARADIGM CHANGE IN FORMING LIMIT MODELS Continuing along the common thread of paradigm change, we now review developments in modeling failure in sheet metal forming. Many practitioners of the metal forming community remain faithful to the idea that strain metrics are useful for formability assessment, while many others who are aware of at least some of the limitations of strain metrics are nevertheless reluctant to use stress metrics because of concerns about their viability. In this paper we will consider a broad set of these concerns and point to evidence in the literature that may help to resolve the issues. Finally, a new type of forming limit diagram is described that has advantages of both stress and strain metrics. Perhaps the most serious limitation of the strain FLC is shown in Fig 1A. However, it may not be immediately obvious how serious the problem is because it is often argued that the complex pattern of strain FLCs shown here do not apply when strain paths are linear. But that argument is seriously misinterpreting what the data in Fig 1A means. The FLC’s in this figure are in actuality all characterization of the strain FLC for linear paths. To understand this meaning, consider, for example, the following three FLC’s in the figure: 1) the BLACK FLC with FLDo at a longitudinal strain of about 0.19 for the zero-prestrained condition, 2) the BLUE FLC with a cusp close to the horizontal axis at a transverse strain of about 0.13, and 3) the RED FLC with a cusp just below the horizontal axis.

23

The set of all three of these curves define the ―single‖ FLC for a linear strain path corresponding to uniaxial strain along the transverse direction. What this means is the strain FLC for linear strain path is not static, but is an intrinsically dynamic limit. This is not semantic difference. It has serious consequences on the use and utility of the strain FLD. It means that the strain FLC for the as-received condition gives absolutely no clue about the margin of safety of a given process, regardless of whether the path is linear or non-linear. Once the implications of this data are understood, it cannot be rationally argued that the strain FLD can be used in conditions where the strain path is ―almost‖ linear. In other words, the only point on the original FLC, the conventional FLC for the as-received material, is the endpoint of the single strain path on which a linear strain path sets its initial course. In view of that, few measures are more limited in application than the strain-FLC for materials in the as-received condition, Experimental FLC's of 2008 T4 Al After Prestrain Stress-Based FLC's of 2008 T4 Al Before and After Selected Prestrains

0.50

500 0.40

450 400

0.20

0.10

P1

U2

350

Major True Stress (MPa)

LongitudinalTrue Strain

0.30

E2 300 250 Yield Surface after E2 prestrain

200 150

P4 Initial Yield Surface

100

-0.30

-0.20

-0.10

0.00 0.00

0.10

0.20

50

0.30

U4

0 0

-0.10

50

100

150

200

250

300

350

400

450

500

Minor True Stress (MPa)

TransverseTrue Strain

FIGURE 1. A) Net-strain FLC’s of 2008 T4 AA reported by Graf and Hosford (1993) for the as-received and for 12 prestrain conditons. B) Corresponding stress FLC(s) for the as-received and 5 prestrain conditions selected from shown in (A) that result in prestress to points U2, P1, E2, P4, and U4 as calculated and reported in Stoughton (2000). The solid black curve shows the expansion of the initial yield surface to E2, which is the starting point for plastic flow in the second strain increment for that case.

It continues to be argued explicitly or implicitly that strain paths in most metal forming applications and particularly in the first draw die, are sufficiently linear that the strain FLC for the as-received condition can be used without serious risk of error in formability assessment. In fact, with the exception of axisymmetric and uniform sectioned parts, studies of industrial applications show that most of the strains in automotive applications that reach levels of plastic strain that would be considered at risk, also include a substantial nonlinear component in the strain history. Nonlinear strain paths occur in every region of late contact with 3D tooling surfaces, as well as in areas in which the metal flows from one region of the tool geometry to another. While nature tends to take the shortest and therefore linear path, the complex 3D geometries we attempt to form metal to, do not allow metal to be stretched to high strains by linear paths, except in very rare circumstances.. Arrieux, Bedrin, and Boivin (1982) were among the first to publish evidence that the dynamic nature of the strain FLC for steel arises from a single path-independent stress limit in combination with the non-unique relationship between the loading stress and net plastic strain tensor components. They were among the first to propose using the stress FLD in formability analysis. This discovery has been shown by others to apply to all metals, including the 2008 T4 AA using in Graf and Hosford (1993) shown in Fig 1A and B. The stresses reported here are calculated using the method described in Stoughton (2000) based on the Graf’s reported material properties and either following a linear loading condition from zero strain to the as-received FLC, or linear loading to the prestrain, followed by stress unloading and reloading in different directions to the final net strain on the corresponding strain FLC. The final stress conditions on each of these strain FLCs appears to be consistent with a single stress FLC, within experimental uncertainty. Based on these and similar experimental studies, the stress condition appears to be the casual factor in the initiation of necking. More importantly, it support a simple practical solution to the

24

ambiguity of strain forming limits in manufacturing, as originally proposed by Arrieux, et al.; As long as the stress condition is maintained below this single stress FLC, the forming process will be safe from necking. So the stress FLC satisfies the need of industry for a simple diagram to clearly separate safe and non-safe forming conditions, much as the strain FLC is currently misused today. Although this FLC is apparently path-independent, the stresses calculated in the finite element (FE) analysis are path dependent variables. Therefore, it is necessary to monitor the stress state at each step of the computation and determine if it is below or above the stress-based FLC. For this purpose, it is convenient to use a single parameter to monitor the formability margin as described in Stoughton and Yoon (2005). The formability parameter γC, defined in terms of critical stress ratios, as well as an alternate formability parameter γ’ defined in terms of critical effective plastic strain ratios proposed by Zeng et al. (2008) are defined as follows,

C 

 y  ij   Y  FLC  1 

 FLC  pij 

 C 

p11  p22

 p  ij

p  FLC

e1FLC



(21)

p11  p22  p11  p22 

 p11  p22 2  p122  p11  p22 2  p122

where σY(ε) is the stress-strain relation, εp is the effective plastic strain, σy(σij) is the yield function, σp(σij) is the plastic potential, β is the direction of the current increment to the plastic strain, and e 1FLC is the major strain at the point on the strain FLC in the direction defined by β in the strain FLD. In Eq. 21, γC is a single stress scaling parameter that represents the degree of formability according to the criteria γC <1 (safe) or γC >1 (neck). In the next subsection, when we review the effect of curvature, we will show that this determination must be made at each integration point, and that necking occurs only if the γ C values at all integration points simultaneously exceed the value of 1. Despite the strong experimental evidence supporting a path-independent stress FLC, among of the factors that have slowed the acceptance of stress-based metrics are the decades of theoretical work and development of models to explain the strain FLC. These theoretical works include bifurcation analysis conducted by pioneers in the field of plasticity, including Hill (1952), Storen and Rice (1975), Hutchinsen and Neale (1977), as well as necking models proposed by Marciniak and Kuczynski(1967). This strong theoretical basis for the strain FLC has contributed to overconfidence in strain analysis, and perhaps a lack of appreciation for its limitations. In contrast to what is perceived by some to be a proof of the validity of a strain forming limit criterion, careful study of these models show that they are in reality a proof of the validity of a path-independent stress forming limit criterion. As pointed out in Stoughton and Zhu (2004) in a review of these bifurcation models, at some point in the derivation of the strain limit, the authors of these models imposed an assumption of linear strain. Prior to imposing this constraint, the equations defining the instability condition were shown by Stoughton and Zhu to be actually in the form of a relationship between the stress variables. These relations were shown to have no dependence on a history variable that isn’t uniquely defined by the current stress state. Furthermore, Zeng et al. (2008) reported the results of a study of bilinear strain paths using MK Analysis for a model with isotropic hardening, predicting the pattern of behaviors such as seen in Fig 1A. However, they also found no difference in the stress condition within the matrix when the strain rate inside the defect reached the critical level associated with onset of necking. This work shows that MK Analysis also naturally leads to a path-independent stress limit. In considering the application of the stress FLC, it is important to understand that although it is possible in principle to measure stresses using X-rays or in specially equipped tube hydro-forming experiments, such stress limits would actually not be helpful for assessing formability in finite element simulation. This is because our material models are not sophisticated enough to predict the stresses with sufficient accuracy to compare them to actual measured forming limit stresses. Seemingly, that fact may discourage some from using the stress FLC for formability analysis. But in fact, this is also not a justified position. To understand why this is so, for the sake of argument, let us refer to the ―stresses‖ we calculate using any given model as ―process-control-variables‖ or PCV’s. We would like to think of these PCVs as the real stresses, but in fact, their actual function in the metal forming

25

simulation is to define the force equilibrium conditions with sufficient accuracy that we can predict the strain history and strain distributions in the part. If we think of these PCVs as strain-control parameters used in the forming simulation and admit that the correlation with real stresses need not be perfect to obtain good prediction of the strains and strain history, we clearly put ourselves at risk when we use them to predict springback, since then we rely on the accuracy of the material model to allow us to treat these PCVs as the real stress tensors. Now the question is, ―Do we also put ourselves at risk in using these PCVs to predict forming severity?‖ The answer to the latter question is perhaps surprising, ―No‖. To demonstrate that, Stoughton (2002b) calculated the limit conditions for several sets of PCVs calculated from a variety of divergent material models, including models developed by von Mises, Logan-Hosford (1980), and Barlat (1997), and the non-AFR proposed by Stoughton (2000), all calculated based on the same strain FLC’s shown in Fig 1. As expected, the shape and location of the FLCs defined in terms of these sets of PCVs in the space of the PCVs was shown to be significantly different for each material model. However, it was also shown that the PCV-space FLC’s collapsed to the same limit condition for linear and nonlinear strain paths, such as is shown in Fig 1B using Hill’s model. What this means is that for formability analysis, it does not matter if the PCVs are real stresses or proxies for the real stress; As long as we translate the strain FLC to the space of the PCVs using the same material model as we use in simulation, we can use these variables to deal with the path dependent nature of strain metrics. However, for convenience, we can continue to refer to these variables as stresses.

Role of Curvature One of the primary factors that cause confusion in understanding forming limits is the role of the stress and strain gradients through the sheet thickness. These gradients are intrinsic to curved sheet and therefore critical to understanding and applying forming limit criterion based on stress or strain. For example, stretching a 1 mm thick sheet over a 2 mm radius will introduce a difference in the true strain between the top and bottom side of the sheet of up to ln(1.5)=0.405, depending on the amount of in-plane tension that thins the metal. That strain difference is on the order of the FLDo value of most steels and twice the limit of aluminum. So this raises the question, ―What layer do you use to define the stress (or strain) that will be compared the stress (or strain FLC) in the formability analysis?‖ When industry first started to implement the FLD in the 1960’s, it was quickly discovered that strains measured most conveniently on the convex side of the sheet were commonly found to be well above the FLC with no sign of necking. Remarkably, without any experimental evidence to justify the decision, the metal forming industry adopted the approach of using the membrane strains in making comparison to the strain FLC. This assumption has continued unchecked in both physical tryout and analysis of numerical simulations for nearly two decades, and continues to be the dominant practice used in industry today. Unfortunately, the assumption is wrong, and the truth has serious consequences in both the interpretation of forming limits and their application in analysis. Tharrett conducted a series of simple bending under tension tests at General Motors on strips of different thickness of steel, aluminum, and brass and different punch tip radii with the objective to determine what strains through the thickness are the cause of necking. He discovered that necking initiated not when the membrane strains exceeded the strain FLC, as was previously thought, but much later in the forming process, when the strains on the concave side of the sheet rose to the level of the FLC. While the tests were limited to plane strain conditions, the results were confirmed in all materials and tooling geometry. The details of the experiments for steel were later published by Tharrett and Stoughton (2003), and the results for one test geometry are shown in Fig 2. There are two necks observed in this specimen on either side of the center punch tip radius at the location where the strains on the concave side, shown by the enlarged circles, rose to the level of the FLC for this material. Considering the importance of stress metrics, Stoughton (2008) noted that Tharrett’s results are also understood to apply to the stress conditions, so that this important factor can be applied to both linear and nonlinear deformation processes. In other words, for a neck to initiate, the stress on all layers through the thickness must exceed the stress FLC. This generalization has interesting consequences because often the stress field is more complex than the strain field, due for example, to a history of cycling bending/unbending. Furthermore, it is important to note that use of membrane values will result in overly conservative predictions of necking on curved sheet. This mistake will undermine correlation with experiment, but also, because the level of the conservative estimate is proportional to the strain gradient through the thickness, it will result in a proportional bias in the safety margin towards regions of higher curvature, while providing no additional margin of safely in regions of zero curvature or through-thickness

26

stress gradient. Since failures most often occur away from curved areas of the product for this very reason, the bias of using membrane strains in formability assessment provides no real benefit to producing robust processes. There is another insidious effect of Tharrett’s results, which is on the interpretation or definition of the experimental FLC which is used as the basis of both strain and stress analysis. Many forming limit curves are based on data obtained directly from the Nakajima Test where the necking strains are measured on the convex side of the sheet stretched over a 200 mm diameter hemispherical punch. In view of Tharrett’s results, the limit strains should be measured on the concave side. But by using the strains on the convex side, the shape of the FLC is biased in a complex manner that affects both the major and minor strain of each point. Furthermore, it introduces a thicknesses effect on the FLC that is not real. The bias on these strain measurement becomes even more severe if sub-size tooling smaller than a 200mm diameter punch is used to generate the FLCs. The effect of using this biased FLC generated from the convex surface strains without compensation for Tharrett’s effect will proportionally undermine correlation with reality and may exacerbate the controversy over correlation between prediction and tryout.

FIGURE 2. A) Sum of the principal strains for a 50 wide strip of 1008 AK steel stretch-bent over a punch wedge with a ¼ inch radius to the depth at which onset of necking occurs, as reported in Tharrett and Stoughton (2003). The forming limit is characterized as a simple limit on the sum of the principals because the minor strain was less than or equal to zero at all points along the strip in a region of the FLD characterized by a limit on thinning strain for this metal. The FLC and FLDo was obtained from standard FLD tests independent of the stretch-bend test.

Role of Cyclic Bending/Unbending In the late 1990’s, Chrysler engineers brought attention to the observation that strains in the die wall within the ―bead affected zone‖ (BAZ), i.e. metal that has undergone cyclic bending/unbending through drawbeads prior to drawing into the die wall, were frequently found to be higher than the FLC with no sign of necking. Since the FLC is used in decisions to approve the die and forming process as production ready, the engineers struggled with resolving the question whether it was really necessary to continue to work the process to reduce strains further in the BAZ, especially since no problems with formability were observed in tryout arising from these high strains. In order to come to an understanding and solution to this problem, the US Auto-Steel Partnership launched the Enhanced

27

Forming Limit Project, which was completed in 2003 and published in a Technology Report (2003) and associated documents. The project involved an extensive experimental program involving 6 sheet steels and generation of about 70 FLC’s for the as-received materials and after a large number of prestrain conditions obtained entirely through cyclic bending and unbending. The cyclic prestrain was obtained in a large draw die that pulled 12 inch wide strips of metal into a vertical channel die wall to a depth sufficient to expose a 12 inch square of the material in the die wall to uniform cyclic deformation. The amount of net strain from the cyclic deformation was varied from a few percent up to 40% through a variable die radius (3, 6, and 12 mm) and zero, one, or two insertable square or round beads of variable radii bolted in slots located near to the die opening. Another insertable bead, located more than 12 inches outboard of the two inner bead locations, was used as an option to add backforce, which increases the stretch by cyclic deformation from the inner beads and die radius through the exponential effect of the product of friction and angle of contact with the bead and die radii. After the prestrain, the material from the 12 inch square region in the die wall was cut and stretched in a Marciniak Test to obtain the FLC, defined in terms of the net strain, including the prestrain through cyclic near-plane-strain conditions. These FLCs were compared to the measured FLC for the as-received materials. This test geometry and process is also familiar to the global industrial community because it was also used in Benchmark #3 in the Numisheet 2005 Benchmark Study as described by Stoughton et al. (2005). Forming Limit Diagram Forming Limit Diagram AKDQ-CR 0.7 mm

Major Strain (%)

AKDQ-CR 0.7 mm - Prestrain Condition # 403

Major Strain (%)

100

100

DFLC0 = + 10.4 %

90

90

80 80

70 70

60 60

50 50

40 40

30 30

As-received

After prestrain

As-received Theoretical FLC0: 33.7 % Experimental FLC0: 32.3 % DFLC0: -1.4%

-20

-10

Experimental FLC0:

0

10

20

30

20

42.7 %

Back tension ratio: 0.000 Square double beads: 4 mm Penetration: 50 %

10 0

-30

Experimental FLC0: 32.3 %

20

10 0

-20

-15

-10

-5

0

5

10

15

20

Minor Strain (%)

Minor Strain (%)

FIGURE 3. TypicalKeeler-Brazier results from theofUS LimitsFLC offorCyclic Deformation (2003). Data Experimental as-received material prediction FLC Auto-Steel Partnership Study on Forming Experimental FLC after Experimental (A) for the as-received AKDQFLCsteel and (B) the same material after passing through a double set prestrain of square 4mm radius beads at Not necked necked 50% penetration andNotacross a 6 mm die radius, resulting in a net major (engineering) prestrain of 14.8% in plane-strain In the neck field In the neck field conditions. Necked Necked

An example of the experimental results that illustrates the objective of the study is shown in Fig 3B for an AKDQ cold-rolled steel. Figure 3A shows the FLC for the as-received material, with FLDo of 32.3% engineering strain. Figure 3B shows the FLC after prestrain in cyclic bending/unbending process using two square beads with 4mm radii at half penetration. The cyclic prestrain results in a net engineering stretch of 14.8%. (Recall that because this net principal strain is from cyclic deformation, the effective plastic strain (EPS) and strain hardening on the surface layers is much higher, and in this 2 square bead geometry is expected to be in excess of 100%.) Remarkably, despite the excessive amount of plastic work on the surface layers and the high net principal strain under plane-strain conditions, the FLC for the net strain limit is shifted upward, with a new FLDo at 42.7% engineering strain. This confirms the original observations made by Chrysler engineers that the effect is real and significant. The absolute shift of +10.4% is about 70% of the engineering prestrain of 14.8% (0.72 if using true strain metrics). It is as though 70% of the prestrain causes no damage to the material with respect to necking. This behavior was found in all 6 steels, including a DP 600 alloy, although the shift was found to average to a smaller percentage, 60% over all bead and process configurations.

28

The primary purpose of the A-SP study was to improve the interpretation of forming severity in die tryout. For this purpose, the A-SP developed a practical solution for die tryout, leading to techniques to estimate the component of the net strain due to cyclic bending/unbending by measuring the change in thickness on either side of the impact line (boundary of the BAZ). The measured principal strains within the BAZ are then corrected by subtracting 40% of the cyclic strain from the measured major strain before plotting the data on the conventional as-received FLC (for example, before plotting in Fig 3A). With this technique, it is not necessary to measure or use other FLCs such as the one shown in Fig 3B. In addition to the application of the effect of enhanced formability from cyclic bending in physical tryout, the effect also impacts the interpretation of results from simulation. However, since the most significant impact is in areas subjected to the largest history of cyclic deformations, which are in regions where the metal pulled through bead geometries, the effect is not as important to consider in simulation as it is in physical tryout if drawbeads are approximated by a line force model, as is often done in industrial applications for reasons of efficiency. Nevertheless, whether or not physical beads are included in the tool geometry used in simulation, the effect of enhanced formability from cyclic bending impacts other areas in which the metal bends and unbends over tool radii. Although the resulting shifts in the FLC are usually smaller than what occurs in the BAZ because they usually stem from only one cycle of bending and unbending, the shift in the forming limit can be enough to lead to mistakes in formability assessment in numerical simulation, especially for metal drawing across a sharp feature radius. Therefore, it is generally important to account for this effect in numerical simulation and as explained earlier, also to account for it using stress metrics to handle nonlinear deformation processes. Stoughton and Green (2005) analyzed the A-SP Project database and found that depending on the use of kinematic and/or distortional hardening models, the effect can be described with a simple stress limit. However, an easier method of implementing the stress analysis using isotropic hardening models is to apply the correction to the accumulated effective plastic strain (EPS) computed in the simulation in the same manner as proposed by the A-SP method for physical tryout. To show that this is effective, Fig. 4A shows the calculated stress for the data points shown in Fig. 3A that were classified as necked, in order to define the stress FLC for the as-received condition. That FLC is reproduced in Fig. 4B, which overlays the calculated stress conditions for the necked data for the prestrained condition shown in Fig 3B. However, to obtain the correlation of the prestrained data with the as-received FLC, the effective plastic strain from the prestrain was reduced by a factor of 0.60, from a value of 0.170 to a value of 0.102.

Stress FLD data for As-Received FLC

Cyclic Prestrain to EPS 0.170 Annealed to 0.102

700 700

600 600

500

Major True Stress

Major True Stress

500

400

As-Received Fit

300

400

As-Received FLC

300

Initial Yield

Initial Yield 200

Necked

200

Necked

Isotropic Yield

100

100

0

0

Annealed Yield Prestrain Path

0

100

200

300

400

500

600

700

0

Minor True Stress

100

200

300

400

500

600

700

Minor True Stress

FIGURE 4. Stress conditions calculated from the measured strain conditions in areas classified as necked in the test results shown in Fig 3, where A) shows the as-received forming limit behavior, and B) shows results based on an ―annealed‖ or reduced EPS from prestrain by a factor of 0.60 from a value of 0.170 to a value 0.102. All calculations are based on Hill’s isotropic hardening model for a material with normal anisotropy.

29

To apply this correction to simulation with isotropic hardening, it is necessary to keep a separate record of the contributions to the effective plastic strain, EPS, from compression, EPSC, and tension, EPST, at each integration point. The smallest of these two components is equal to half the contribution from cyclic bending and unbending, and the difference is the non-cyclic component. The cyclic contribution is then reduced by the empirical factor fEFL representing the ―annealing‖ of the ―Enhanced Formability Effect‖ of cyclic deformation. And finally, this annealed value is added to the non-cyclic component to obtain an annealed or corrected value of the EPS for the integration point, resulting in the following relation,

EPS  max EPS T , EPS C   2 f EFL  1 min EPS T , EPS C  .

(22)

This modified value that can be used to correct or calculate the forming stress components by projecting the predicted stresses onto the ―annealed‖ yield surface defined by the stress-strain relation at this value of EPS. This reduction is an empirical correction, similar to the one proposed by the A-SP for the strain FLD. Results would be different, and a correction may not be necessary at all, if a kinematic model is used in the analysis. While it is not common practice to consider lab tests involving cycles of compression and tension to measure forming limits or provide data to calibrate strain FLC’s, care should be taken to consider the beneficial effect of compression on formability to avoid serious mischaracterization of forming limits and formability prediction.

Role of Through-Thickness and Hydrostatic Stress Another area of interest in forming limits is the role of through-thickness and hydrostatic stress, which has resulted in several recent papers regarding the impact of these variables on strain limits. Since stress metrics have more general application to nonlinear processes, we will focus in this paper on the effect of these variables on stress limits, and will only mention the consequences on strain limits as it helps to understand the effect on stress. The through-thickness and hydrostatic stress does not arise in simulation based on shell elements because by assumption, the through-thickness stress is zero, and whatever effect there is to the hydrostatic component would be build into the definition of the forming limit for plane-stress conditions. To understand the impact of non-zero through-thickness and more generally, hydrostatic stress, we first imagine that we have employed MK Analysis to define the strain limit for any specific (linear or nonlinear) strain path under plane-stress conditions. If this is a linear path, we expect to obtain a point on the conventional FLC such as shown in Fig 3A and if it’s nonlinear, we would expect to obtain a point on an FLC with a shape like one of those in Fig 1A. Similarly, we expect this same analysis will result in a single stress FLC, regardless of whether the strain path was linear or not, as has been confirmed in a numerical study by Zeng et al. (2008). So the question is, ―What is the effect on this calculated necking limit if the through-thickness stress was nonzero in our MK Analysis?‖ To obtain the answer to this question, it is helpful to first ask, ―What is the effect on this calculated necking limit if we add a hydrostatic stress to the deformation history in our calculation?‖ We can answer this question without actually doing the MK Analysis, through the following stream of logic. If the material model is based on a pressure-insensitive constitutive relation, as we most often use in modeling metal deformation, then the answer is simple. Adding a constant stress to all three normal stress components has no effect in either the matrix or defect of the MK Analysis on yielding, on the direction of plastic flow, and as well as on the stress-strain relation. Consequently, the strain history within the defect, which is used to define the onset of necking, would be unaffected by hydrostatic pressure. It follows that the strain FLC obtained using MK Analysis under hydrostatic pressure will be identical to the strain FLC obtained under plane-stress conditions. Perhaps surprisingly, the independence of the strain FLC with respect to hydrostatic pressure has an important consequence on the application of the conventional stress FLC as previously described in the literature and as shown in Fig 1B and Fig 4. The FLC’s shown in these figures as functions of the in-plane principal stresses do not apply under conditions when the through-thickness stress is non-zero. The reason for this is that if the triaxial stress condition (σ1, σ2, σ3=0) is a point of necking instability on the conventional stress FLC, then, as a result of the invariance with respect to hydrostatic stress, all stress states (σ1+c, σ2+c, σ3=c) will also be unstable against necking. So as a result, all points in the entire (σ1, σ2) space used in the conventional stress FLC could be safe or necked,

30

depending on the magnitude of σ3. This ambiguity in the 2D stess FLD can be eliminated using the two in-plane deviatoric components of stress, but this dramatically changes the shape of the stress FLC. A simpler solution, proposed in Stoughton (2008) and Stoughton and Yoon (2011), which retains the shape of the conventional stress FLC for plane-stress conditions, is to simply subtract the current value of σ3 from all three principal stresses to obtain a plane-stress equivalent state, (σ1-σ3, σ2-σ3, 0). Plotting this data in modified stress FLC in the 2D space, (σ1σ3, σ2-σ3) accounts for the effects of both the invariance of the necking limit under changes in the hydrostatic stress and the dependence on the through-thickness stress component. Under the plane stress condition, Stoughton and Yoon (2011) combined the necking limit with a model for the fracture limit in the principal stress space by employing a stress-based forming limit curve (FLC) and the maximum shear stress (MSS) criterion. A new metal failure criterion for in-plane isotropic metals is described, based on and validated by a set of critical experiments. This criterion also takes into consideration of the stress distribution through the thickness of the sheet metal to identify the mode of failure, including localized necking prior to fracture, surface cracking, and through-thickness fracture, with or without a preceding neck.

The Polar Effective Plastic Strain Diagram One of the difficulties of the stress FLD is due the reduction of the slope of the true stress-strain relation. Due to this effect, larger changes in strain occur at stress levels close to the necking limit compared to at stress levels further below the limit stress. This makes it difficult to visually see or quantify the margin of safety without a magnifying glass or overlay of the contours of equivalent strain in the stress FLD. To remedy this difficulty, we will consider using the effective plastic strain as one of the metrics to assess formability following a modification of the idea proposed by Zeng et al. (2008). Although effective plastic strain is described as a type of strain, it is not directly linked to the principal or tensor components of the strain tensor. It is however, uniquely linked to the stress tensor through the yield function and stress-strain relation, and therefore falls under the category of a stress metric. To understand the idea proposed by Zeng et al.(2008), it is helpful to begin with the equations for translating the strain FLC into the stress FLC. In the translation of the principal strains to principal stress for a point (e 1, e2) on the conventional (as-received) strain FLC, for a material model defined by a in-plane isotropic plastic potential, σp(σ1,σ2), yield function, σy(σ1,σ2), and monotonic stress-strain relation, σY(εp), we can use the following set of relations for calculating the corresponding point (σ1,σ2) on the stress FLC,



e2 e1

p  1 

    

e1   e2  p 1,  

,

 Y  p   y 1,  

(23)

 2   1

where β is the strain ratio, α is the stress ratio, and α(β) is the inverse function or solution to the following explicit relation,

 p  1 ,  2 



 2  p  1 ,  2   1

.

(24)

 1 , 2 1, 

The definitions of the function α(β) and εp have more simple analytic forms than those given above in the case of a quadratic plastic potential, but the above equations are more general and apply to both quadratic and non-quadratic forms of σp(σ1, σ2). Generalization of this translation to anisotropic material models, including anisotropic forming limit data, is described in Stoughton and Yoon (2005).

31

It is useful to point out that that the accounting of nonlinear paths comes into play at only two places in Eq. 23. The first is in the definition of β, which for non-linear strain paths is more generally defined in terms of the ratio of the current strain rates. Nonlinear paths also play a role in the definition of the effective plastic strain ε p, which is more generally defined by the time integral of a function of the strain rates, rather than the net strains as shown in Eq. 23. This integration is also further complicated by the fact that α and β, which is used to define α, are also changing in time. No other relation in Eq. 23 depends explicitly on deformation history. What this means is that the forming limit for linear and nonlinear deformations can also be characterized as a simple limit on the accumulated effective plastic strain, εp , as a function of β, or as a function of α. The idea of an FLD based on the variables (εp ,β) was proposed by Zeng et al. (2008) as a better way to account for formability. As seen in Eq. 23, it is mathematically equivalent to the stress FLC for models with positive work hardening, in other words, as long as the function is monotonic, since each point in (εp,β)-space maps to a unique value in the (σ1,σ2)-space through these equations. Although it is equivalent, the new metric has several important practical advantages; First it scales with the magnitude of strain, so it is easier to visualize safety margins for conditions that are near to the necking limit. Second, it does not depend on the stress-strain relation at all. In one sense, that means it is less complex to use; But this independence with respect to the stress-strain relation has an even bigger advantage in that the forming limit criterion can be extended to material models that allow for hardening stagnations, or in which there is softening in the relation between true stress and true strain, such as in models in which temperature effects are considered. If for any reason, the stress-strain relation, σY(εp) is not monotonic, the effective plastic strain may continue to be a good metric for formability under nonlinear conditions, whereas the saturating or decreasing stress variables that arise from a non-monotonic stress-strain relation, would no longer be practical for assessment of formability. Although monotonic stress-strain relations ensure stability and uniqueness as discussed in the third requirement for stability given in Eq. 3, it is not necessary to forbid hardening behaviors that might lead to physical instabilities such as necking. So in these cases, the effective plastic strain is an attractive solution. Experimental FLC's in Conventional Strain Diagram

Experimental FLC's in Polar EPS Diagram

0.50

0.50

0.45 0.40

0.40

0.35

0.30

EPS*COS(θ)

LongitudinalTrue Strain

0.30

0.20

0.25

0.20

0.15

0.10 0.10

-0.30

-0.20

-0.10

0.00 0.00

0.05 0.10

0.20

0.30 -0.30

-0.20

-0.10

0.00 0.00

0.10

0.20

0.30

-0.10 TransverseTrue Strain

EPS*SIN(θ)

FIGURE 5. A) Strain FLC’s of the 5 prestrain and as-received condition corresponding to the 6 Stress FLC’s shown in Fig 1B. B) Calculated EPS FLC’s shown in the PEPS Diagram with the direction given by the angle defined in Eq.2 4. Note that the scatter in the EPS strain is clearly on the order of the experimental uncertainty, which is more difficult to argue from Fig 1B.

Another alternate and still equivalent solution to using the (εp, β) variables as proposed by Zeng et al. (2008) may be more compelling to industrial engineering applications. In this generalization, we propose to plot the data in a polar diagram of the εp variable with the angle defined as the arctangent of the ratio of the principal strain rates,

  tan 1 e1 , e2  .

32

(25)

In a Cartesian equivalent system, the variables of the proposed diagram become (v1, v2)= (εpcos(θ), εpsin(θ)). The reason the polar diagram is compelling is seen in comparison of the shape of the conventional strain-FLD and the polar diagram in Fig 5. As is the case for the as-received condition in the strain-FLD, all the FLC’s in the polar diagram have cusps or low points in the plane-strain condition. Furthermore, since the effective plastic strain is actually a stress metric and the radial direction in this diagram uniquely defines the stress state, it is not surprising that we see in Fig 5B that the FLCs for these nonlinear strain paths are insensitive to the remarkably different strain histories. Interestingly, the shape of the FLC in the polar diagram is very similar to the shape of the strain FLC for the as-received condition. Most importantly, the radial directions in the polar diagram that correspond to uniaxial, plane-strain, equal-biaxial, etc. are parallel to the corresponding directions in the strain FLC. As long as engineers understand what this means, and do not make the mistake of plotting net strain tensor component in the polar diagram or plot EPS values without consideration of strain path, the polar diagram could be a convenient tool for more reliable formability assessments in manufacturing. For this reason, we propose to refer to it as the PEPS FLD, in reference to its polar nature and its radial variable defined by the effective plastic strain.

SUMMARY In this paper we review recent work that raises questions about the applicability of associated flow rules and use of strain metrics in the assessment of necking limits. We describe some simple models based on non-associated flow that provide less complex solutions to capture a wide range of initial and evolving material anisotropy. We also review discussions of stability under associated and non-associated flow models. For formability, we discuss the importance of strain path, through-thickness stress-gradients and strain-gradients caused by curvature, cyclic bending/unbending effects, and through-thickness or normal stress effects, including practical solutions to deal with these effects through stress analysis. Finally, we proposed a new ―stress-based‖ forming limit criterion based on a polar diagram of the effective plastic strain with the direction defined by the arctangent of the ratio of the current plastic strain rates, both of which are defined by the current stress condition, or the most recent stress condition while the material was undergoing plastic deformation. This diagram, which we refer to as the PEPS FLD in reference to its polar nature and its radial variable defined by the effective plastic strain, has advantages over the previously proposed Stress FLDs based on the principal stresses or in case of non-zero normal stress, the difference between the in-plane and through-thickness principal stresses. The primary advantages of the PEPS FLD over other diagrams based on variables with stress dimensions is the lack of dependence on the stress-strain relation and any saturation or softening that may be included in the material model. These advantages are shared by the model described by Zeng et al. (2008) in which the limit on the effective plastic strain is characterized by an unfamiliar parameter, the ratio of the plastic strain rates. The primary distinction and potential appeal of the proposed PEPS FLD is that the FLC is similar in shape to the strain FLC, and the directions in the diagram corresponding to uniaxial, plane-strain, equal-biaxial, etc. are parallel to the corresponding directions in the conventional strain FLD.

ACKNOWLEDGMENTS The authors would like to thank Daniel Green from University of Windsor, Xinhai Zhu from Livermore Software Technology Corporation, Ming F. Shi from United States Steel Corporation, Cedric Xia from Ford Motor Company, John Siekirk from Chrysler, Ching-Kuo Hsiung, Siguang Xu and Lumin Geng from General Motors, all for stimulating discussion and contributions to our understanding metal forming limits. The authors would also like to thank John Bassani of University of Pennsylvania, John Hutchinson from Harvard, Rich Becker from US Army Research Laboratory, Robert Wagoner from the Ohio State University, Rebecca Brannon from the University of Utah, and especially Owen Richmond (deceased) formerly from Alcoa Technical Center, for stimulating discussion and contributions to our understanding of the issues regarding non-associated flow. Finally, the authors are particularly indebted to the reviewers of our papers for their thorough and critical reviews to challenge our understanding and improve our presentation of the topics covered in this and prior related papers.

REFERENCES Arrieux, R., Bedrin, C., Boivin, M., 1982. Determination of an intrinsic forming limit stress diagram for isotropic metal sheets. In: Proceedings of the 12th Biennial Congress of the IDDRG, pp. 61–71. Auto-Steel Partnership, Enhanced Formability Project Report, 2003, www.a-sp.org, Publications.

33

Barlat, et al., 1997,―Yield function development for aluminum alloy sheet,‖ J. Mech. Phys. Solids, 45, p. 1727. Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dicke, R.E., Legea, D.J., Pourboghrat, F., Choif, S.H., Chuh, E., 2003. Plane stress yield function for aluminum alloy sheets—Part 1: Theory. Int. J. Plasticity 19, 1297. Bassani, J.L., Ito, K., Vitek, V., 2001. Complex macroscopic plastic flow arising from non-planar dislocation core structures. Mater. Sci. Eng. A319-321, 97–101. Bishop, J.F.W., Hill, R., 1951. A theory of plastic distortion of a polycrystalline aggregate under combined stresses. Philosophical Magazine 42, 414. Bramley, A.N., Mellor, P.B., 1966. Plastic Flow in Stabilized Sheet Steels. Int. J. Mech. Sci. 8, 101–114. Bulatov, V.V., Richmond, O., Glazov, M.V., 1999. An atomistic dislocation mechanism of pressure-dependent plastic flow in aluminum. Acta Mater. 47, 3507–3514. Drucker, D.C., 1951. A more fundamental approach to plastic stress–strain relations. In: Proc. of the First US National Congress of Applied Mechanics, ASME, New York, pp. 487–491. Drucker, D.C., 1959. A definition of stable inelastic material. ASME, J. Appl. Mech. 26, 101–106. Graf, A., Hosford, W.F., 1993. Effect of changing strain paths on forming limit diagrams of aluminum 2008-T 4. Metall. Trans. A24, 2503–2512. Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic metals. Proc. Roy. Soc. London A 193, 281–297. Hill, R., 1952. On discontinuous plastic states with special reference to localized necking in thin sheets. J. Mech. Phys. Solids 1, 19–30. Hutchinson, J. and Neale, K., 1977, Sheet necking-II:time-independent behavior, Proceedings of the Symposium of Mechanics of Sheet Metal Forming, Edit by D.P. Koistinen and N.M. Wang, 127-153. Logan, R. and Hosford, W. F., 1980, Int J. of Mech. Sci.,22, p. 419. Marciniak, Z. and Kuczynski, K., 1967, Limit strains in the processes of stretch forming sheet steel, J. Mech. Phys. Solids 1, 609– 620. Pearce, R., 1968. Some aspects of anisotropic plasticity in sheet metals. Int. J. Mech. Sci. 10, 995–1004. Qin, Q., Bassani, J.L., 1992. Non-associated plastic flow in single crystals. J. Mech. Phys. Solids 40, 835. Richmond, O., 1980. Plastic dilatancy in metals. In: Lee, E.H., Mallet, R.L. (Eds.), Plasticity of Metals at Finite Strain: Theory, Experiment and Computation. p. 343. Sandler, I.S., Rubin, D., 1987. The consequences of non-associated plasticity in dynamic problems. In: Desai, C.S. (Ed.), Constitutive Laws for Engineering Materials: Theory and Applications. Elsevier Sci. Pub.Co., Inc., Amsterdam, pp. 345–353. Spitzig, W.A., Richmond, O., 1984. The effect of pressure on the flow stress of metals. Acta Metall. 32,457. Spitzig, W.A., Sober, R.J., Richmond, O., 1975. Pressure dependence of yielding and associated volume expansion in tempered martinsite. Acta Metall. 23, 885–983. Storen, S., Rice, J.R., 1975. Localized necking in thin sheets. J. Mech. Phys. Solids 23, 421–441. Stoughton, T.B., 2000. A general forming limit criterion for sheet metal forming. Int. J. Mech. Sci. 42, 1–42. Stoughton, T.B., 2002a, A non-associated flow rule for sheet metal forming. Int. J. Plasticity 18, 687–714. Stoughton T.B., 2002b,The influence of the material model on the stress-based forming limit criterion. SAE 2002-01-0157. Stoughton, T.B., Yoon, J.W., 2004. A pressure-sensitive yield criterion under a non-associated flow rule for sheet metal forming. Int. J. Plasticity 20, 705–731. Stoughton, T.B., Zhu, X., 2004. Review of theoretical models of the strain-based FLD and their relevance to the stress-based FLD. Int. J. Plasticity 20, 1463–1486. Stoughton, T.B., Yoon, J.W., 2005. Sheet metal formability analysis for anisotropic materials under non-proportional loading. Int. J. Mech. Sci. 47, 1972-2002. Stoughton, T.B., Green, D., and Iadicola, M., 2005, Specification for BM3: Two-stage channel/cup draw, Numisheet 2005 Conference Benchmark Study, AIP Conference Proceedings 778 B, pp. 905-918. Stoughton, T.B., Yoon, J.W., 2006. Review of Drucker’s postulate and the issue of plastic stability in metal forming. Int. J. Plasticity 22, 391–433. Stoughton, T.B., 2008. Generalized Metal Failure Criterion,. Numisheet 2008 Conference Proceedings, Ed. P. Hora, 241-246. Stoughton, T.B., Yoon, J.W., 2008. On the existence of indeterminate solutions to the equations of motion under non-associated flow. Int. J. Plasticity 24, 583–613. Stoughton, T.B., Yoon, J.W., 2009. Anisotropic hardening and non-associated flow in proportional loading of sheet metals. Int. J. Plasticity 25, 1777-1817. Stoughton, T.B., Yoon, J.W., 2011. A new approach for failure criterion for sheet metals. Int. J. Plasticity 27, 440-459. Taherizadeh, A., Green, D.E., Ghaei, A., Yoon, J.W.,2010. A Non-associated Constitutive Model with Mixed Iso-Kinematic Hardening for Finite Element Simulation of Sheet Metal Forming. Int. J. of Plasticity, 25, 288-309. Tharrett, M., Stoughton, T., 2003. Stretch-bend forming limits of 1008 AK steel. SAE 2003-01-1157. Zamiri, A., Pourboghrat, F., 2007, Characterization and development of an evolutionary yield function for the superconducting niobium sheet, Int. J. of Solids and Structures, 44, 8627-8647. Zeng, D. Chappuis, L., Xia, Z. C., and Zhu, X. 2008, A Path Independent Forming Limit Criterion for Sheet Metal Forming Simulations, SAE 2008-01-1445.

34

Zero Failure Production Methods Based on a Process Integrated Virtual Control P. Horaa*, J. Heingärtnera, N. Manopuloa, L. Tonga a

IVP, ETH Zurich,Tannenstrasse 3, 8092 Zurich, Switzerland

Abstract. Although the virtual methods are nowadays fully established as a widely used tool in the planning and optimization of forming processes they are still completely omitted for a direct, “intelligent” process control in the later production of the parts. The paper presents a proposal for a Process-Integrated-Virtual-Control (PIVC) considering the real process perturbations induced by deviations of material properties as well as by further time dependent process parameters like the tool temperature. For the detection of the material deviations an in-line eddy-current measurement method and the appropriate evaluation method for the definition of the stochastic yield curves will be presented. The paper closes with a virtually taught control system modifying the blank-holder forces in dependency of thermal conditions and material deviations. The goal of this PIVC coupled to in-line process controls is to achieve a Zero Failure production even under alternative time dependent process conditions Keywords: Forming technology, zero defect production, modelling of perturbation effects, virtual robustness control, intelligent process. PACS: 46.80; 46.65.+g, 46.55.+d, 46.50.+a

INTRODUCTION To establish a ZD-production1 became in last years a more and more challenging task. The reasons are different: x The complexity of the parts increases continuously. The requested reduction of weight can be only achieved by the use of new light weight and high strength materials. Especially the widely used TRIP and HSS show significant temperature dependency and a high spring-back potential and are not easy to control. x To achieve crash resistance, new fabrication methods like press-hardening have been introduced. The PHprocess requires additional thermal parameters. x The requirements of quality as well as the requirements of close tolerances continuously increase. x New and more complex assembly methods will have to be applied if body in white with a complex material mix has to be realized.

Methods for an In-Process-Integrated Virtual-Control The new virtual based strategy to achieve the Zero Failure productions goals bases on three pillars as described in Table 1. Their interaction is plotted in FIGURE 1 left. For their realization following concepts have been applied:

1

ZD „Zero Defects“ synonym for „Zero Failure”

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 35-47 (2011); doi: 10.1063/1.3623590 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

35

TABLE 1.

FIGURE 1.

“Pillars” of the Zero Defect Production Control

Technical concepts

Process Integrated Virtual Control (PIVC)

Mapping of the system behavior based on virtual tryouts under stochastic conditions. Description with metamodels

In-line material control

In-line, non-destructive measurement of constitutive parameters (eddy current methods)

“Intelligent” Tool and Press Equipment

In-line control of process parameters with sensors

Modules of the Virtually Controlled Zero-Defect Production Strategy.

x Process Integrated Virtual Control The goal of the PIVC is to establish a virtual process control based on the FEM-sensitivity analysis of the process. To achieve the needed very fast response for the on-line process regulation the system behavior will be described with metamodels. The models can base on the FEA - especially on their sensitivity analysis - as well as be taught directly on the process (“Self Teaching Neuronal Networks”). x “Intelligent” Tool and Press Equipment Until now the deep drawing processes were not directly controlled. If any changes of the process settings are needed, they are done by the operating staff based on the check of some specific parameters like the draw in of the sheet and an optical quality control. For those reasons it is not surprising that for an identical problem different “correction strategies” will be applied. The goal of the “Intelligent Tool” is to detect the critical parameters automatically and so to allow a close loop regulation of the process. Methods for the realization of such principals are discussed by Annen et al., [1]. x On-line Material Control A further significant process parameter may be the non-constancy of the material properties. They can be essential as well for the possible influence on the process robustness as for the on-line actions to correct the process parameters. The applicability of the non-destructive measurement methods of the relevant parameters was recently presented by Heingärtner et al., [2]. The paper will show how the connection among those components can be realized and how they are coupled. The theoretical background is described more profoundly in the different papers of the “Zero Failure Production Methods” conference proceedings, [3].

DETECTION OF SIGNIFICANT PERTURBATION PARAMETERS AND A PROPOSAL FOR THEIR APPROPRIATE NUMERICAL MODELLING As mentioned above the process unreliability may be induced by fluctuation of different process parameters. One of the significant sources of uncertainty in the process is the fluctuation of the material properties, [4, 5]. For those reasons it is indispensable to detect this behavior. In the next section a distortion free monitoring method will be presented.

Eddy-current method for the detection of material changes The applied eddy-current measurement method (EC) bases on the detection of the magnetic response at different frequencies. The relation to the physical value will be described by a correlation function (Metamodel) between the classically evaluated tensile test data and the eddy current signals, FIGURE 2.

36

FIGURE 2.

Eddy current measurement equipment and the principal of the model.

With the eddy-current system realized by Heingärtner [2] and Born [6], the material properties can be measured with a frequency of about 24 to 28 blanks per minute. FIGURE 3 shows the blank cutting unit, where the ECmeasurement was installed. Per coil usually about 1’500 blanks will be evaluated. The diagram in FIGURE 3 demonstrates the detected changes of Rp0,2, Rm and Ag-values as measured by Born, [6].

FIGURE 3.

On-line measurement system installed in the blank cutting unit.

In FIGURE 4 those data are re-plotted as distribution functions.

FIGURE 4.

Material DC06. Distribution of material properties as measured on about 6’000 specimens of different coils.

37

For the behavior of the material and its modelling the relation between the values are significant. FIGURE 5 shows the correlation among the values Rp0,2 and Rm as well the influence of the hardening ratio Rm/Rp0,2 on the development of the uniform elongation Ag.

FIGURE 5.

Material DC06. Correlations between the parameters Rp0,2, Rm and Ag.

For the investigated material DC06 the distribution functions in FIGURE 4 prove that the variation of the mechanical properties is very small inside a single coil. Under the consideration of the correlations of FIGURE 5 the deviations became even smaller. If different coils are considered, FIGURE 6, the changes of the properties are more evident. Nonetheless the variations for the investigated material remain moderate.

a) b) FIGURE 6.

Material DC06. Changes over different coils a) Correlation Rm-Rp0,2. b) Uniform strain distribution

Generation of Yield curves based on the Rp0,2, Rm and Ag Data The stochastic yield curves for the investigated material DC06 will be described based on a “slope fixed”, material specific Hockett-Sherby model k HS f

V sat  (V sat  V 0, 2 ) exp(  M const M n )

(1)

In contrast to the normal H-S-approximation [7] the parameter Mconst is considered as a predefined material constant. As demonstrated in FIGURE 7 Mconst influences, independently of the other parameters, the yield curve slope in the extrapolated area. The idea to keep this parameter constant bases on the assumption, that for a specific material the hardening type will not change significantly.

38

HC220YD 1200 exp M = 0.5 1000

M = 1.0 M = 2.0

kf [MPa]

800

600

400

200

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

 [-]

FIGURE 7.

Material HC220YD. Determination of a material specific constant M.

The value of Mconst can be defined by a “full” Hockett-Sherby approximation for all 4 parameters using the combined tensile-bulge yield curve. If the bulge date are not available, the diffuse necking condition

dk f

kf

dM

(2) can be applied as an additional constrain condition. Both methods are described in [9] and [10]. For the material DC06 Mconst= 1.70 was identified. Because the EC system delivers the Rp0,2, Rm and Ag-values and not directly the yield curves, the H-S parameters have to be generated based on those data. This can be done in two different ways. Method 1: A direct way is to specify the stochastic yield curve in dependency of the data Rp0,2, Rm and Ag using eq.1. In this approach the parameters V0,2, Vsat and n of eq. 1 will be specified as follows: x Eq. strain at diffused necking

M DN

ln(1  Ag )

(3)

R p 0, 2

(4)

x Yield stress V0,2

V 0,2 x Stress DN at strain Ag

V DN

R m exp(M DN )

(5)

x Parameter sat and n of the HS-equation will be evaluated using the both conditions k HS f (M DN )  V DN ˆ 0 wk HS f (M DN

)

 V DN ˆ 0 wM This procedure was used to generate the stochastic yield curves in FIGURE 11.

(6) (7)

Method 2: An alternative method for the description of the stochastic influence is to overload a mean yield curve with a deviation function 'k stoch . In this case the specific yield curve can be written as f

k HS f

39

~ kf

k HS  'k stoch f f

(8)

where 'k stoch can be described with a linear interpolation of the deviated values V0,2 (=Rp0,2) and DN corresponf ding to Rm · § § M gl  M · ¸  'V gl ¨ M ¸ 'V 0, 2 ¨ ¨ M gl ¸ ¨ M gl ¸ ¹ ¹ © © 'V 0,2 (1 - m)  'V gl m

'k stoch f

(9)

The value m is restricted to the range 0 d m d 1 and will be assumed m=1 in the extrapolated range.

Impact of strain rate and temperature Industrial deep drawing processes achieve strain rates up to 10; after a short production time the tools warm up to 50qC and the temperature of the sheet reaches locally 80qC ore more. For those reasons it should be checked if even for the normal “cold” deep drawing processes the temperature and the strain rate influence are not relevant either [8]. This is definitely the case for special materials like the TRIP steel. Their hardening has then to be described by more complex hardening laws [10], [11], [12].

kf

f (M , M , T )  'k f (TRIP)

(10)

But even for the investigated DC06 material the temperature influence is of importance, s. FIGURE 12, and will be taken into consideration in the present study.

Impact of yield locus shape changes One of the recent yield locus models is the so called YLD2000 description proposed by Barlat et al. [13]

I where the functions

I 'I ' ' 2V a

(11)

I ' and I ' ' are defined as I'

X 1'  X 2'

a

a

I ' ' 2 X 2''  X 1''  2 X 1''  X 2''

(12)

a

Both linear transformations of YLD2000 are given with

X'

C' ˜ s

C' ˜ T ˜ 

L' ˜ 

(13.1)

X ''

C '' ˜ s

C '' ˜ T ˜ V

L'' ˜ V

(13.2)

where L’ and L’’ can be defined as ªLc º 2/3 0 0º « 11 » ª c » « 1 / 3 « L12 0 0»» ªD1 º « » « » « c » « 0  1 / 3 0» «D 2 » « L21 » «Lc » « 0 2 / 3 0» ««D »» ¬ 7¼ « 22 » « 0 1»¼ « L c » «¬ 0 ¬ 66 ¼

ª L cc º « 11 » cc » « L12 « » cc » « L21 « L cc » « 22 » « L cc » ¬ 66 ¼

40

8 2 ª 2 2 « 1 4 4 4 1« « 4 4 4 1 9« 2 2 « 2 8 «¬ 0 0 0 0

0º ª«D 3 º» 0»» «D 4 » « » 0» «D 5 » » 0» «D » « 6» 9»¼ «D » ¬ 8¼

(14)

As specified in Table 4, for the material HC220YD, the yield stresses and so the yield locus increase non- homogenously.

Experimental stress and r-values at different strain levels for the material HC220YD.

TABLE 2.

strain

V0

V45

V90

Vb

r0

r45

r90

rb

0.00 0.05 0.10 0.15 0.20

339 407 440 465

326 388 421 446

343 410 444 468

366 454 502 532

1.28 1.28 1.28 1.28

2.08 2.08 2.08 2.08

1.86 1.86 1.86 1.86

1.0 1.0 1.0 1.0

For the description of this type of anisotropic hardening Hora et al. [9] proposed to describe the Dk-parameters as linear functions of the equivalent strain:

D k (H ) D k0  D k' H

(14)

Based on the data of Table 2 the parameters of eq. 14 get the values specified below. The corresponding linear functions are given in FIGURE 8; the yield loci are plotted in FIGURE 9.

FIGURE 8.

FIGURE 9.

k

D k0

D k'

1

0.9631

-0.3039

2

1.0801

0.2157

3

1.0323

-0.9148

4

0.9372

-0.1600

5

0.9608

-0.1342

6

0.8470

-0.8739

7

1.0877

0.0033

8

0.9886

0.7053

Material HC220YD. Dk-parameters of the YLD2000 function, if evaluated under the constraint of linear strain dependency

Material HC220YD. Yield locus evolution under the consideration of the linear changes of the Dk-parameters as in eq. 14.

41

How significant the continuous yield locus change with an increasing Vb/VY-ratio is, demonstrates the comparison of the thickness distributions in FIGURE 10 calculated in Case A) without and in Case B) with consideration of the anisotropic yield locus evolution. Case A: isotropic hardening

Case B: Anisotropic hardening

tmin = 0.47

tmin = 0.59

FIGURE 10.

Influence of the YLD2000 description on the strain distribution without (left) and with the consideration of the anisotropic yield locus change as given in FIGURE 9.

MODELLING OF THE REAL PROCESS BEHAVIOR UNDER CONSIDERATION OF PERTURBATIONS Besides the correct “deterministic” modeling, as mentioned earlier in this paper, the real production process is perturbed by the fluctuation of different process parameters. The influence of those changes on the process behavior will be demonstrated again on the cross deep drawing part. For the investigation following parameters have been considered: x x x x

blank specific (stochastic) changes of hardening behaviour (yield curve) thermal influences on hardening thermal influences on friction dependency of the blank holder force FN on above parameters

FIGURE 11 shows the yield curve evaluated by the method 1 given by eq. 3 to 7 for a single coil. The light lines in the plot correspond to the slope wk f / wI of the hardening curve and indicate at the cross point the position of the uniform strain Ig. FIGURE 12 specifies the impact of temperature increases on the yield curve. FIGURE 13 describes the temperature dependency of the friction coefficient for one specific lubricant.

42

FIGURE 11.

Case

V0,2

Vsat

Mconst

n

RT max RT min

142 134

579 540

1.7 1.7

0.542 0.537

DC06 - Yield curve distribution evaluated by the EC-method for one coil.

Case

V0,2

Vsat

Mconst

n

RT mean heated 60qC heated 100qC

117 110 105.05

566 491 449.55

1.7 1.7 1.7

0.562 0.576 0.5747

FIGURE 12.

FIGURE 13.

DC06 Thermal influences on the yield curve.

Temperature dependency of the friction coefficient P as given by Krauer [12].

43

Sensitivity ranking of the process parameters The sensitivity analysis in FIGURE 14 demonstrates that moderate material fluctuations, as seen in FIGURE 11 for the DC06 grades, will have no significant impact on the process behavior. This may be the most surprising result of the process analysis based on the real stochastic data. For those reasons – for the prediction of the system behavior in FIGURE 16 - the yield locus will be considered constant. The major influence comes clearly from the friction, which is in turn temperature dependent.

Parameter AF-Sigma

Min.

Yield Curve DC06 Rp0,2 deviations Yield Locus R0 R90 Biaxial factor Friction P Pressure

kf_mean at RT -10 MPa Banabic 1.45 1.5 0.9 0.05 3 MPa

FIGURE 14.

Max 10 MPa 1.70 2.8 1.2 0.25 15 MPa

Sensitivity analysis of the process investigated for the cross deep drawing part.

FIGURE 15 demonstrates this behaviour for the extremes of the stochastic yield curves as well as for two thermal cases. The strong temperature sensitivity shows, that the self-heating effect, induced by plastic work, will have a significant influence on the process behaviour. Already a warming up to 60qC, which is not unrealistic under continuous production conditions, will provoke necking2.

Case RT max

Case RT min

2

For the temperature dependent description of the yield curves the data of FIGURE 12 have been used.

44

Case Temp. 60q

Case Temp. 100q

FIGURE 15.

DC06 material. Deep drawing behaviour under consideration of stochastic and thermal influences of the yield curve as given in figures 16 and 17.

ZERO FAILURE CONCEPTS BASED ON PROCESS INTEGRATED VIRTUAL CONTROL The goal of the PIVC is the detection of admissible parameter fields and their integration in an in-line process control.

Temperature dependent process control In the simulation below the influence of the parameters P(T), kf(T) and FN(T) have been investigated. FIGURE 16 visualizes the impact of increasing temperature on the admissible blank holder force. The changes are driven by the thermal dependency of the friction, FIGURE 13, and by the thermal change of the hardening as specified in FIGURE 12. The comparison of the both diagrams in FIGURE 16 proves that the temperature dependency of the yield curve significantly strengthen the T-dependency of the process as a whole, s. Fig. 16 b.

a) FIGURE 16.

b) Influence of the temperature changes on the admissable FN-settings. a) Consideration of the P(T)-influence. b) Consideration of both P(T) and kf(T) influences.

45

Material dependent process control The GUI showed in FIGURE 17 is directly coupled to the eddy-current measurement system. Based on a “virtual” knowledge of the deep drawing behavior, predicted in the same way as for the thermal dependency and mapped with a metamodel the system can immediately detect if the specific blank can be formed without necking.

FIGURE 17.

GUI for the visualization of the characteristic parameters coupled with the analysis for the applicability for a specific part.

If the material reaches critical values, a ZD-production can be achieved only after an adaption of the process parameters. One possibility for their modification is demonstrated in FIGURE 16 by the adaption of the blank holder forces. Another approach is to influence the friction by increasing the amount of lubricant.

FIGURE 18.

Layout of an integrated “Zero-Defect-Process Control”.

46

Integration of the control system in a production line The introduced modules are the pillars of an integrated process control unit as shown in FIGURE 18. The “intelligent” process monitoring makes it then possible to avoid part defects induced by material fluctuations or long-term heating effects. In the demonstrated example, the process modification was done by adaption of the blank holder forces. For complex parts the integral BH-force have to be split into single column forces. The basic idea of the approach stays the same. The last “pillar” of the concept, the sensors of the “intelligent” tool, has not been discussed. The detection of temperature changes can be done with commercial sensors. In many cases it is helpful to control the process over the draw-in behavior of the sheet, too. This can be done directly in the tool or in-directly following the forming step. For the direct measurement different mechanical and laser sensors are under development. The indirect measurements base usually on optical evaluation methods.

ACKNOWLEDGMENTS The presented paper bases on results which have been realized in close cooperation with Daimler AG and the steel supplier VoestAlpine. We would like to thank all of those partners for their support, especially Prof. K. Roll and Dr. D. Hortig, who have promoted this project and Ms. A. Neumann for her support with the experiments.

REFERENCES 1.

2.

3. 4. 5. 6. 7.

8.

9.

10. 11.

12. 13.

Ch. Annen , F. Schwarz, M. Wahl, P. Hora, “Virtual layout of adaptive deep drawing processes”, in Zero Failure Production Methods, Conference Proceedings of the Forming Technology Forum FTF-2011, edited by P. Hora, 2011, ETH Zürich. Heingärtner J., Hora P, “Application of the eddy-current methods in industrial car body production”, in Zero Failure Production Methods, Conference Proceedings of the Forming Technology Forum FTF-2011, edited by P. Hora, 2011, ETH Zürich. P. Hora, “Zero Failure Production Methods”, Conference Proceedings of the Forming Technology Forum FTF2011, edited by P. Hora, 2011, ETH Zürich. K. Grossenbacher, “Virtuelle Planung der Prozessrobustheit in der Blechumformung“, Ph.D. Thesis, ETH Zurich, Nr. 17470, VDI-Verlag, VDI Rh. 2, Nr. 667, 2008. M.H.A. Bonte, “Optimization Strategies for Metal Forming Processes”, Print Partners Ipskamp, ISBN 978-90365-2523-7, Enshede, 2007. M. Born, "Integration des Wirbelstromsystems zur Erfassung der Werkstoffkennwerte in der Produktion“, M.Sc. Thesis, ETH Zurich, 2009, ETH Zürich. J.E. Hockett and O.D. Sherby, Large strain deformation of polycrystalline metals at low homologous temperatures. J. Mech. Phys. Solid, 23-2 (1975) pp. 87-98. P. Larour, "Strain rate sensitivity of automotive sheet steels: influence of plastic strain, strain rate, temperature, microstructure, bake hardening and pre-strain", Ph.D. Thesis, RWTH Aachen, Publishe in Berichte aus dem Institut für Eisenhüttenwesen, Shaker Verlag, Band 1/2010, 2010. P. Hora, B. Hochholdinger, A. Mutrux, L. Tong: “Modelling of anisotropic hardening behaviour based on Barlat 2000 yield locus description”, in Constitutive Modelling of Kinematic and Anisotropic Hardening Effects for Ductile Materials, Conference Proceedings of the Forming Technology Forum FTF-2009, edited by P. Hora, 2009, ETH Zürich, pp. 21-29. P. Hora: "Modellierung des Kaltverfestigungsverhaltens bei metallischen Werkstoffen“, Conference Proceedings MEFORM 2011, edited by R. Kawalla, Freiberg, 2011. A. Hänsel: "Nicht-isothermes Werkstoffmodell für die FE-Simulation von Blechumformprozessen mit metastabilen austenitischen CrNi-Stählen", Ph.D. Thesis, ETH Zurich, Nr. 12672, published in VDI Reihe 2 Nr. 491. J. Krauer, "Erweitere Werkstoffmodelle zur Beschreibung des thermischen Umformverhaltens metastabiler Stähle", Ph.D. Thesis, ETH Zurich, Nr. 19070; published in VDI Reihe Nr. 676. F. Barlat F., J.C. Brem, J.W. Woo, K. Chung, R.E. Dick, D.J. Lege, F. Purboghrat, S.-H. Choi, E. Chu: "Plane stress yield function for aluminium alloy sheets – part I – theory“, Int. J. of Plasticity, 19(2003), pp. 1297-1319.

47

New Forming Technologies for Autobody at POSCO Hong-Woo Leea, Myung-Hwan Chaa, Byung-Keun Choia, Gyo-Sung Kima, Sung-Ho Parkb a

Products Application Research Center, POSCO, 180-1 Songdo-dong, Yeonsu-gu, Incheon 406-840, Korea Steel Technology Strategy Department, POSCO, 892, Daechi4-dong, Gangnam-gu, Seoul, 135-777, Korea

b

Abstract. As development of car body with light weight and enhanced safety became one of the hottest issues in the auto industry, the advanced high strength steels have been broadly applied to various automotive parts over the last few years. Corresponding to this trend, POSCO has developed various types of cold and hot rolled AHSSs such as DP, TRIP, CP and FB, and continues to develop new types of steel in order to satisfy the requirement of OEMs more extensively. To provide optimal technical supports to customers, POSCO has developed more advanced EVI concept, which includes the concept of CE/SE, VA/VE, VI and PP as well as the conventional EVI strategy. To realize this concept, POSCO not only supports customers with material data, process guideline, and evaluation of formability, weld-ability, paint-ability and performance but also provides parts or sub-assemblies which demand highly advanced technologies. Especially, to accelerate adoption of AHSS in autobody, POSCO has tried to come up with optimal solutions to AHSS forming. Application of conventional forming technologies has been restricted more and more by relatively low formability of AHSS with high tensile-strength. To overcome the limitation in the forming, POSCO has recently developed new forming technologies such as hydro-forming, hot press forming, roll forming and form forming. In this paper, tailored strength HPF, hydroformed torsion beam axle and multi-directional roll forming are introduced as examples of new forming technologies. Keywords: Hydro-forming, Hot Press Forming, Roll forming, Form forming, EVI

INTRODUCTION Steel, typically used in vehicle structures, enabled the economic mass production of millions of units over the past several decades [1]. It has been argued during last decade that steel would be replaced as body structure materials by more expensive light materials such as Al, Ti and CFRP (Carbon Fiber Reinforced Plastics) for weight reduction. However, the majority of vehicle bodies are still manufactured from sheet steels. This means that steel products are able to provide not only the excellent physical properties (in strength, formability, weld-ability and corrosion resistance) but also the cost-effectiveness. However, application of conventional steels to vehicle bodies has been restricted more and more by the regulations of environment and safety. As is well known, weight reduction and safety enhancement are the most important goals of automotive industry. However, generally, the increase of strength for certain metallurgy is accompanied with the loss of ductility. To overcome above mentioned problem, steel makers have been developing new grades of high performance steel called as advanced high strength steel(AHSS). DP, TRIP, FB, CP steels are the recent examples of AHSS. These steels have expanded in application for weight reduction and improvement of safety. The principal difference between conventional HSS and AHSS is their microstructure[2]. By appropriate balance of microstructure, AHSS can have their own unique mechanical properties. For example, some types of AHSS have a higher formability than conventional HSS within same strength range and other types show very good stretch flange-ability with high strength. As mentioned above, good performance of AHSS enables to increase in the use of ultra high strength steels for autobody with allowing weight reduction. In order to expand utilization of AHSS more and more, various steel application technologies, i.e. integrated steel solutions are highly needed. Those include not only evaluation of mechanical properties but also alternative forming technologies such as hydroforming, hot press forming, roll forming and form forming. However, application solutions of AHSS do not mean a radical change compared to solutions of conventional HSS. But, in order to secure

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 48-53 (2011); doi: 10.1063/1.3623591 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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target performance, several process requirements should be considered in addition to conventional forming technologies. The present paper describes the current status of the forming technologies at POSCO.

CUSTOMER SUPPORT TECHNOLOGIES Auto makers continuously pursue not only high performance and quality but also low costs of production. As these technological strategies, auto makers devote their best efforts to apply the optimal material such as AHSS and other light weight material in each part of a new car model. Also, various new part manufacturing processes have been adopted for weight reduction, cost saving and safety enhancement of car body. Tailor-welded blank (TWB) is now widely accepted, and hydro-forming and hot press forming are extended to various applications for auto part manufacture. Steel makers are able to be involved in the early stage of car designing by contributing materials data as well as developed materials for new car project or phase shift. Moreover, to intensify this supporting activity, steel makers often propose alternative designs for individual parts and develop the new concept for the whole body structure. The typical example of this supporting activity, ULSAB (Ultra Light Steel Auto Body) as shown in Fig.1 and ULSABAVC (advance vehicle concepts), which are designed and tested by WAS (WorldAutoSteel) in collaboration with a consortium of worldwide steel companies[3]. ULSAB/-AVC proved to be lightweight, structurally sound, safe, executable and affordable. One of the major contributors to the success of the ULSAB/-AVC was a group of new steel types and grades called AHSS and new forming technologies such as TWB and hydro-forming. FSV (Future Steel Vehicle) program [4] is recently progressing. This program focuses on the achievement of future safety requirements coupled with the demonstration of low CO2 emissions, and affordability of a steel intensive vehicle body using optimized design, AHSS usage and advanced power train technologies. The phase 2 results for electric vehicle design is shown in Fig.1 Even individual companies have attempted to develop their own concept models utilizing their high performance steels together with new ideas of design. These include NSB (New Steel Body Concept) by TKS, ABC (Arcelor Body Concept) by Arcelor and ATLAS (Advanced Technology for Lightweight Auto- bodies in Steel) by Salzgitter. POSCO also developing electric vehicle body by applying newly developed steel and forming technologies. The above supporting process by steel suppliers is called EVI which covers the activities involved in the vehicle concept design and tooling/ prototyping stages. POSCO has developed a more advanced EVI concept to serve auto makers better. In POSCO EVI program, the supportive activities cover beyond the tooling/prototyping stage, being responsible for part manufacturing and sub-assembling too. Advanced EVI program consists of the following four strategies: CE (Concurrent Engineering) / SE (Simultaneous Engineering), VA (Value Analysis) / VE (Value Engineering), VI (Value Innovation) and PP (Part Proposal), as shown in Fig.2.

(a) ULSAB body[3]

(b) FSV Phase 2 design for electric vehicle[4]

FIGURE 1. Concept car models utilizing high performance steels together with new design ideas.

FIGURE 2. Advanced EVI concept of POSCO

49

CE/SE means that a workflow of POSCO carries out a number of tasks in parallel with customers’ process. In order to establish win-win strategy with auto makers, POSCO provides the optimal steel solutions concurrently and simultaneously in each stage from styling to mass production. On the base of VA/VE strategy, POSCO proposes the best fit materials from cost viewpoint including degradation, thinning, logistics, packing and so on. From this, customers can obtain optimal material information with weight reduction and cost saving or performance improvement. VI is the POSCO EVI activity to reduce or remove unnecessary values to save the prime cost in order to strengthen customers’ competitiveness. VI activity doesn’t need high technology because it aims to maximize the efficiencies of the properties on what customers already have or use. For example, through cost saving program such as substituting GI for EG product, POSCO gives customers maximum profit without modifying the existing process. In PP activity, POSCO supports customers with new forming technologies including new design concept, prototyping and manufacturing. In order to implement CE/SE, VA/VE, VI and PP programs, various steel application technologies are highly needed. POSCO has developed integrated steel solutions which are fully utilized through whole car manufacturing process. To realize this concept, POSCO not only supports customers with material data, process guideline, and evaluation of formability, weld-ability, paint-ability and performance but also provides parts or sub-assemblies which demand highly advanced technologies. Especially, to accelerate adoption of AHSS in autobody, POSCO have tried to come up with optimal solutions to AHSS forming.

NEW FORMING TECHNOLOGIES As development of car body with light weight and enhanced safety became one of the hottest issues in the auto industry, the advanced high strength steels have been broadly applied to various automotive parts over the last few years. Corresponding to this trend, POSCO has developed various types of cold and hot rolled AHSSs such as DP, TRIP, CP and FB, and continues to develop new types of steel in order to satisfy the requirement of OEMs more extensively. Meanwhile, application of conventional forming technologies has been restricted more and more by relatively low formability of AHSS with high tensile-strength. To overcome the limitation in the forming, POSCO has recently developed new forming technologies such as hydro-forming, hot press forming, roll forming and form forming. From this point of view, POSCO’s technical activities will be shown in the following sections.

Tailored Strength Hot Press Forming As a way to acquire high strength and good formability in manufacturing of automotive parts, an advanced forming technique, often called hot press forming (HPF) was developed. During this process, sheet metal is heated to austenizing temperature, formed rapidly and then quenched in a tool. Soft austenite transforms into very hard martensite due to high hardenability of HPF material, so HPF can be used to fabricate complex and crash-resistant parts with ultra high strength. Recently, two innovative approaches, named tailored strength HPF in this article which has two different graded strength levels with 1500MPa and 600MPa in the formed parts, have been implemented. The main purpose of these approaches is to enhance the ductility in some localized areas on the hot press formed part in order to obtain both anti-intrusion by harder strength area and energy absorption with the deflection of ductile strength zone at the same time during crash accidents.

(a) (b) (c) FIGURE 3. Conceptual definition of tailored HPF for (a) MS-HPF part, (b) process conditions and (c) TWB-HPF part.

50

The first approach, named Multi-Strength HPF (MS-HPF) defined in Fig. 3(a), is carried out by using different heating temperature path of a monolithic blank in the special furnace before forming and quenching. The harder strength can be given by conventional condition of HPF temperature and resulted from martensitic phase transformation. However, lower heating temperature history leads to diffusional phase transformation, so we can get a local area of ductile strength. The second approach, named TWB-HPF described in Fig. 3(b), can be obtained by originally different hardenability of two materials at the same temperature condition. Tailor Welded Blank (TWB) sheet used for this process is prepared by laser welding of HSLA material and HPF material. Ductile strength of TWB-HPF is provided due to low hardenability of HSLA material. The characteristic of mechanical properties and different microstructure shown in Fig. 4 for MS- and TWB-HPF result in tailored strength of hot press formed parts. Our newly designed special oven for MS-HPF was already installed and suitable partner material for TWB sheet was also designed to get lower grade strength even after rapid quenching. We have studied optimization of process parameters and evaluated the performance of tailored strength HPF parts to apply automotive components to our customers’ vehicles to increase crash performance and improve safety.

(a) (b) FIGURE 4. Hardness distribution and miscrostructure after heat treatment for (a) MS-HPF and (b) TWB-HPF.

Tube Hydroforming The main goals of Tube hydroforming are to reduce the part weight and improve structural strength of the part at the same time. Therefore the choice of the material which is used for the hydroforming application is essential and depends on several parameters. Formability of the material, material cost, requirement of part performance and weld quality are only a few of them. High strength hydroformed structural parts including engine cradles, suspension parts, etc. can help achieve the high goals of the automotive industry. Parts can be designed with thinner wall thicknesses therefore, the part will be lighter and at the same time the physical properties of AHSS grades like DP590 or DP780 will increase the stiffness of the part. The Torsion Beam Axle (TBA) developed by POSCO combines the advantages of hydroforming such as high geometrical repeatability, increased wall thickness at the feeding area due to axial feeding, and the material properties of DP780. Fig.5 shows the TBA and the connection nodes with other suspension components. To achieve the requested part performance, the choice of the material thickness for this TBA design was very important because roll stiffness and maximum stress of the part depend on the wall thickness of the part. Simulations were carried out to determine the correlation between the wall thickness, roll stiffness and maximum stress of the part. From the simulation results, it was observed that the higher the TBA wall thickness, the higher the roll stiffness will be and the lower the maximum stress will be. This leads to a conflict between roll stiffness, maximum stress and wall thickness. Choosing a wall thickness to satisfy the allowable stress means that the roll stiffness would be too high and likewise, choosing a wall thickness to achieve the required roll stiffness would mean that maximum stress would be too high. Fig.6 shows the stress contour at rolling condition of the TBA and it can be seen that the areas where the highest stress occurred are located in the axial feeding zone during hydroforming. To satisfy the requested roll stiffness and the allowable stress of the TBA, 3 major criterions had to be met: • The original tube thickness could not change in the middle section of the TBA. • Due to axial feeding during hydroforming, the wall thickness needed to be increased in the area where the highest stress occurred.

51

• The original design needed to be modified and a bead designed and implemented into the affected area. Based on the above design concept, the final part and manufacturing process were designed by the FEA simulation. The final design and hydroforming tool of TBA are shown in Fig.7 and Fig.8 respectively.

FIGURE 5. Torsion beam axle(TBA)

FIGURE 6. Stress contour of TBA

Bead

Green color indicates increase of wall thickness due to axial feeding FIGURE 7. Hydroformed TBA design to satisfy the required performance.

FIGURE 8. Hydroforming Tool of TBA

Roll Forming Roll forming is a process producing long-shaped profile from sheet metal through roll sets located serially. It has an advantage of high productivity such as high speed production and little scrap amount. Especially, the technology is being magnified recently in the automotive industry because of relative suitability for forming of UHSS which has a tensile strength over than 1 GPa as its application is getting increased. Roll design has been dependent on designer’s experience more than simulation because of long calculation time required in addition to low accuracy of simulation. So POSCO has placed more emphasis on shortening of analysis time to practical level which leads to a prompt technical assistance to customers. For the achievement, POSCO uses new developed analysis software designed for roll forming only as well as various common analysis tools. One of major restrictions in roll forming process is that it can produce only uniform sectional product. To overcome this limitation, POSCO has been developing MD-RF(Multi-Directional Roll Forming) technology. MDRF is a process which can produce variable sectional profile by linear and rotational movement of roll stands. It is expected by developing this process to provide more flexibility in designing automotive structure and extend application of roll forming process. For the research of MD-RF technology, which is expected to be one of POSCO future EVI ways, POSCO has set up an in-house test facility and developed analysis software designed for MD-RF only.

(a) Roll Forming Simulation (b) MD-RF Simulation FIGURE 9 POSCO EVI activities for roll forming and MD-RF.

52

CONCLUDING REMARK Steel makers have made a lot of efforts to develop advanced high strength steels for the weight reduction. But, AHSS is not easy to be applied by conventional forming technology due to relatively poor formability. To overcome the limitation in the forming, POSCO has recently developed new forming technologies such as hot press forming, hydro-forming, roll forming and form forming. Hot press forming is a forming technique to acquire high strength and good formability by heating and quenching in manufacturing of automotive parts. Recently, POSCO has developed MS- and TWB-HPF to enhance the ductility in some localized areas on the hot press formed part in order to obtain both anti-intrusion by harder strength area and energy absorption with the deflection of ductile strength zone. Hydroforming is able to reduce the part weight and improve structural strength of the part at the same time. In the hydroforming, AHSS such as DP590 or DP780 is generally applied on hydroformed parts as other forming techniques. In this paper, a torsion beam axle is introduced as an example for AHSS and a self-designed part. The torsion beam axle (TBA) developed by POSCO combines the advantages of hydroforming such as high geometrical repeatability, increased wall thickness at the feeding area due to axial feeding, and the material properties of DP780. AHSS, especially steels with tensile strength more than 1 GPa are difficult to be formed by stamping. For such steels with poor formability, it is useful to make a part by bending. Roll forming and form forming which have studied in POSCO, are the typical forming techniques by bending. MD-RF is a special process which can produce variable sectional profile by linear and rotational movement of roll stands.

REFERENCES 1. P. Prasad and J. E. Belwafa, Vehicle Crashworthiness and Occupant Protection, AISI, 2004. 2. WorldAutoSteel, Advanced High Strength Steel (AHSS) Application Guidelines, Online at www.worldautosteel.org, 2009. 3. S.H. Park, “Application of ULSAB Technology to manufacture High Safety Low Weight Vehicle”, KATECH Seminar, Korea, 1999, p.83. 4. Future Steel Vehicle Phase 2, Report by World Auto Steel, 2011.

53

Drawability Prediction Method using Continuous Texture Evolution Model Toshiharu Morimotoa and Jun Yanagimotob a

Graduate student,the university of Tokyo,4-6-1 Komaba Meguro-ku,Tokyo,153-8505,Japan b Professor,the university of Tokyo,4-6-1 Komaba Meguro-ku,Tokyo,153-8505,Japan

Abstract. Drawability is one of steel strip properties which control press forming. Many predicted method for the Lankford value have been proposed. First, we predict recystallization texture based on the idea that total amount of microscopic crystal slips is proportional to accumulated dislocation density in grain boundaries. Next, we can predict Lankford value of ultra low carbon strips and ferritic stainless steel strips using Sachs model. Our method is very practical to use in hot and cold steel rolling industry. Keywords: drawability,Lankford value,recrystallization texture,Sachs model PACS: 81

INTRODUCTION Drawability is one of steel strip properties which control press forming. Drawablity of a steel strip is affected by density of {111} preferred orientations. Thus, to increase {111} preferred orientations, an ultra low-carbon steel strip was developed, which are generally used for automobile bodies. An ultra low-carbon steel strip has as low carbon and nitrogen contents as possible. Cold rolled by more than 80% reduction ratio and annealed above recrystallization temperature, an ultra low-carbon steel has superior deep drawability. While, being much alloyed with about 16 wt% chrome, a ferritic stainless-steel strip has less {111} preferred orientations and lower drawability than an ultra-low carbon strip does. It is capable of estimating Lankford value from measured orientations of final cold rolled and annealed strips[1]. But, few research have been reported to estimate drawability of steel strips considered hot rolling, cold rolling and annealing pattern because it’s difficult to predict recrystallization texture. We made a mathematical modeling to predict rolling texture and transformation texture[2][3]. Furthermore, we developed a method to predict recystallization texture combined with our rolling texture prediction model[4].This method was based on the idea that total amount of crystal slips is proportional to accumulated strains in grain boundaries. By this method, we can predict recrystallization texture of an ultra low-carbon strip and a ferritic stainless-steel strip precisely considering hot and cold rolling condition. This method is very practical for use in the hot and cold steel rolling industry. Finally, we calculated Lankford value using our deformation texture prediction model easily[3].

RECRYSTALLIZATION FORMATION MODEL Many mechanism for the formation of the recrystallization texture in steels have been proposed, including the oriented nucleation model[5], the oriented grain growth model[6] .In other models, such as the dislocation passed model[7], the dislocation build-up model[8], the crystal rotation model[9], the dislocation twist area model[10] and the encroachment model[11], recrystallized grains encroach on deformed non-recrystallized grains. However, no definite conclusions have been reached regarding the validity of each model. In the dislocation twist area model[12], it was predicted that the recrystallization texture of ultra low-carbon steels was inherited from the cold rolling texture. Dillamore proposed a prediction model using the Taylor factor of the pre cold rolling texture[13]. However, the dislocation twist area model was a different type of model because crystal slips and rotations that occurred during cold rolling were taken into account. In this study, we developed a new prediction method expanded The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 57-62 (2011); doi: 10.1063/1.3623592 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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from the dislocation twist theory. First, we predicted the rolling texture using the crystal plasticity by applying Lee’s elastic-plastic decomposition rule[14]. Second, we predicted the recrystallization texture from the predicted cold rolling texture. There are two classical models of deformation texture prediction. One is the Taylor model[15], in which macro strains are equal to micro strains, and the other is the Sachs model[16], in which macro stresses are equal to micro stresses. The prediction accuracy of the steel deformation texture has been reported to be good for the Taylor model[17]. Since nucleation during the recrystallization of polycrystals is generally not only in the transition bands but also in the grain boundaries[18], within which the range of orientations mostly exists. Our method of using the total numbers of crystal slip to predict the recrystallization and grain growth texture has not been previously reported. Our method has many advantageous features. For example, even though it is based on simple assumptions, it enables us to predict the recrystallization texture accurately. Moreover, it can predict the recrystallization texture of polycrystals with few parameters and in a short time. Furthermore, we explain the mechanism of recrystallization texture formation used by our prediction method. By the way, it was too trouble to calculate Lankford value from measured orientations, using the way to determine the minimum Taylor factor[1]. Thus, we used the Sachs deformation texture prediction model[16] to calculate Lankford value more easily.

PREDICTION ACCURACY OF RECRYSTALLIZATION TEXTURE Rolling Texture Analysis Model Here, J , x , a and b are the slip rate, the position, the unit vector normal to the slipping plane and the unit vector in the slipping direction, respectively. Thus, the plastic strain rate D P

P

and plastic spin W

P

can be defined using

Equations (1) and (2), respectively. The plastic strain rate D is calculated using the Orowan calculation model[3]. A b.c.c. crystal lattice has 12 slip systems of the {110}<111> type, 12 slip systems of the{112}<111> type and 24 slip systems of the{123}<111> type. Strain equilibrium, the shear stress of each slip system and the rate-dependent rule are expressed using Equations.(5)-(7), respectively. Equation (5) implies that the sum of the slip values J for each slip system in a crystal is equal to the sum of the micro strains dž . Equation (6) gives the relationship between micro stress ǔ and the resolved shear stress W acting on the slip plane. Equation (7) is the Asaro rate-dependent rule and m is a material parameter[19]. Here, J0 , W y and dt are the reference slip strain rate, the critical shear stress and the time increment, respectively. The critical shear stress W y is determined from the equivalent flow stress V , which is calculated using the macro-micro combined model for microstructure evolution [2] using Equations(8). Rotation tensor R is calculated using Equation (9). I is unit tensor.

G Dij

p

W ij

P

where  Pij

Qij dH i G  W j G G  dJ j         

Wy

1 J a i b j  a j bi Pij J .G  (1) 2 1 J a i b j  a j bi Qij J .G  (2) 2 1 ai b j  a j bi .G  (3) 2 1 a i b j  a j bi . G (4) 2 ¦ Pij dJ j .G G    (5)

¦P V ij

i

.         (6)

J0 * dt * W j / W y

1/ m

* sign(W j ) G G  (7)

V / TaylorFactor                         (8)

58

Rij where

: ij

I ij  dt: ij  1 / 2dt 2 : ij ˜ : ij            )

Wij  Wij and ƺ is lattice rotation spin and W is total spin. P

Prediction Accuracy of Rolling Texture Obtained the Taylor model with the Asaro rate-dependent rule (hereafter referred to as the Taylor model), Figures 1 and 2 respectively show the predicted and observed cold rolling textures of ultra low-carbon steel and hot rolling texture of ferritic stainless-steel. Here, J0 is 100 and m is 0.05. The prediction accuracy for ultra low-

Observed Analyzed

Cold strip

carbon cold rolled texture and ferritic stainless-steel hot rolled texture is high. We consider that because ferritic stainless-steels hardly recrystallize during hot rolling in the finisher, we can accurately simulate hot rolling texture of ferritic stainless-steel strip using the Taylor model.

Observed Analyzed

Hot strip

FIGURE 1. Texture prediction accuracy of ultra low carbon cold strip.

FIGURE 2. Texture prediction accuracy of ferritic stainless-steel hot strip.

Drawability Prediction of Ultra Low-Carbon Steel and Ferritic Stainless-Steel Here we suggested together the conventional assumptions in the above model. The recrystallized grains have a range of orientations in the deformed state that satisfy the preferential orientation model. Nucleation occurs from piled-up dislocations at grain boundaries. The total number of microscopic slips is estimated as the total number of displaced dislocations. The magnitude of slips of crystals is calculated from the Schmid factor. In other words, crystals with a low Schmid factors slip a longer distance than crystals with a high Schmid factors. Needless to say, the Taylor factor is approximately inversed to the Schmid factor. We introduce one further assumption here. The total number of microscopic slips is estimated as the total number of dislocations piled up at grain boundary[20]. This might be a self-evident assumption. Figure 3 shows the case when we selected crystals with the highest slip value as those having the preferential orientation. We can predict the recrystallization texture of cold rolled and annealed ultra low-carbon steel strip precisely because this analysis simulated that the gamma fiber structure encroached the alpha fiber structure. Figure 4 shows the case when we selected crystals with the lowest slip value as those having preferential orientation. We can predict the recrystallization texture of hot rolled and annealed ferritic stainless-steel strips precisely because this analysis simulated the existing bow out into the grains which have the

59

Cold strip after annealing Analyzed Observed

higher stored energy otherwise nuclei develop from subgrains close to that grain boundaries. Figure 5 shows the accuracy of predicting the texture of cold rolled and annealed ferritic stainless-steel strip whose grains were classified as the highest number of slips from the predicted texture of the hot rolled and annealed ferritic stainlesssteel strip in Fig. 4 by continuous analysis. We believe that it is possible to analyze the continuous change in the texture from hot rolled strips to cold rolled and annealed strips by considering both hot rolling and cold rolling conditions.

Observed Analyzed

Hot strip after annealing

FIGURE 3. Texture prediction accuracy of ultra low carbon annealing strip.

Observed Observed Analyzed

Cold strip after annealing

Analyzed

Cold strip

FIGURE 4. Texture prediction accuracy of ferritic stainless-steel annealing strip.

FIGURE 5. Texture prediction accuracy of cold rolled and annealed ferritic stainless-steel.

60

DISCUSSION According to the recrystallization textures of ultra low-carbon steel and ferritic stainless-steel whose orientations were classified by total number of slips using the Taylor model, the orientation of the recrystallization grains is the same as that of the rolled grains. In other words, this result proves that the nucleation preferred grain model is correct. However, why is the rolled texture classified by the total number of slips using the Taylor model the same as the observed recrystallization texture of ultra low-carbon steel and ferritic stainless-steel? The orientation of ultra low-carbon steel cold rolled at reduction ratio of 80% or of ferritic stainless-steel hot rolled in the nonrecrystallized region is considered to be almost stable. Two recrystallization mechanisms of ultra low-carbon steel have been proposed[11]. In mechanism A, the gamma fiber grains with dislocations undergo self-recrystallization. In mechanism B, the gamma fiber grains with fewer dislocations make encroach on alpha fiber grains, which have more dislocations. The final texture of cold rolled and annealed texture of ultra low-carbon steel is determined by mechanism B. As gamma fiber grains have a less stable orientation than alpha fiber grains, subgrains are formed at grain boundaries and gamma fiber grains initially undergo self-recrystallization. Next, alpha fiber grains that are elongated in the rolling direction and located next to gamma fiber grains are encroached upon by the gamma fiber grains. For the above-mentioned reasons, a texture whose orientations classified by the total number of slips with the highest order using the Taylor model become a final recrystallization texture of cold rolled and annealed ultra low-carbon steel strip. In contrast, as hot rolled and annealed ferritic stainless-steel strip recrystallizes by mechanism A, the orientation classified by the total number of slips with the lowest order using the Taylor model is the same as that in the observed recrystallization texture.. Mechanism A has been argued to contradict to mechanism B by many researchers, but we consider the both mechanisms are valid. Thus, mechanism A should be referred to as recrystallization and mechanism B should be referred to as grain growth, which means that coalescence and shrinkage occurs between recrystallized grains and nonrecrystallized grains. However, more investigation is required to clarify these differences.

LANKFORD VALUE CALCULATION MODEL We estimated accuracy of our Lankford value prediction method. First, we measured crystal orientation of ultra low-carbon cold rolled and annealed strip by Electron Back Scatter Diffraction Patterns, whose cold reduction was 80% and heated at 800 degrees for 90 seconds. Next, we predicted texture of a 15% stretched ultra low-carbon strip and Lankford value using Sachs deformation prediction model with the Asaro rate-dependent[3],[16]. Here, J0 is 1.0 ™ 10-3 and m is 0.05. Figure 6 shows difference between calculated and measured Lankford values in longitudinal, diagonal and cross direction. Prediction accuracy was good because grains of stretched strip were controlled by the law of large numbers. And Figure 7 shows measured texture, predicted texture using Sachs model and predicted texture using Taylor model of 15% stretched cold rolled and annealed ultra low carbon strip. Our method makes us predict Lankford value of steel strips easily.

Lankford value,r

2.5 2.0 1.5 1.0

Calculated

0.5

Measured

0.0 0

15

30

45

60

75

90

Angle from rolling direction,/° FIGURE 6. Lankford value prediction accuracy of ultra low carbon strip using Sachs deformation prediction model.

61

FIGURE 7. Measued texture, predicted texture using Sachs model and predicted texture using Taylor model streched by 15 % at thickness center of ultra low carbon cold rolled and annealed strip.

CONCLUSION The prediction accuracy for the textures of cold rolled ultra low-carbon strip and hot rolled ferritic stainless-steel strip using the Taylor model is high. Moreover, the prediction accuracy of the recrystallization texture classified by the total number of slips using the Taylor model is also high. The reason for this is that ultra low-carbon steels and ferritic stainless-steels recrystallize in accordance with the oriented nucleation model. The hot rolled and annealed texture is classified by the lowest total number of slips because nuclei develop from subgrains close to the grain boundaries. In contrast, the cold rolled and annealed texture satisfies the highest total number of slips because gamma fiber grains encroach on alpha fiber grains in the latter recrystallization period. Next we can predict Lankford value of ultra low carbon strips and ferritic stainless steel using the Sachs model. Though the Taylor model and the Sachs model are classical deformation texture analysis methods, it is practical to use in hot and cold steel rolling industry.

REFERENCES 1. H.Inoue and N.Inakazu:J. Jpn. Inst. Met.,58 (1994),pp.892-898. 2. J.Yanagimoto:Modeling and Simulation in Materials Science and Engineering,10(2002),pp.R1-R24. 3. T.Morimoto,F.Yoshida,Y.Kusumoto and A.Osamu:ISIJ Int.,50 (2010),pp.1683-1688. 4. T.Morimoto,F.Yoshida,Y.Kusumoto and A.Yanagida:Current advances in materials and processes,24(2011),pp.393. 5. W.G.Burgers and T.J.Tiedema:Acta Metallurgica,1(1953),pp.234-238. 6. J.J.Jonas and L.S.Toth:Scripta Metallurgica,27(1992),pp.1575-1580. 7. B.F.Decker and D.Harker:J.Appl.Phys.,22(1951),pp.900-904. 8. C.G.Dunn and P.K.Koh:Trans. AIME,206(1956),pp.1017-1024. 9. T.Urabe and J.J.Jonas:ISIJ international,34(1994),pp.435-442. 10. O. Akisue:J. Jpn. Inst. Met.40(1976)206-210. 11. D.Vanderschueren,N.Yoshinaga and K.Koyama:ISIJ international,39(1996),pp.1046-1054. 12. O. Akisue:Tetsu-to-Hagane,72(1986),pp.1320-1327. 13. I.L,Dillamore and H.Katoh,Metals Sci.,8(1974),pp.21-27. 14. E.H.Lee:J.Appl.Mech.,36(1969),pp.1-6. 15. G.I.Taylor:Roy.Soc.Proc.,A,116(1927),pp.16-39. 16. G.Sachs:Z.Ver.Deut.Ing.72(1928),pp.734-736. 17. K.Sekine:Recrystallization texture and their application to structural control,ISIJ,Tokyo(1999),pp.164-168. 18. I.L.Dillamore,P.L.Morris,C,J,E.Smith and W.B.Hutchinson:Proc.R.Soc.Lond A,329(1972),pp.405-420. 19. R.J.Asaro and A.Needleman:Acta metal,33(1985),pp.923-953. 20. W.B.Hutchinson:Metal Science,8(1974),pp.185-196.

62

A General Yield Function within the Framework of Linear Transformations of Stress Tensors for the Description of Plastic-strain-induced Anisotropy Shun-lai Zanga ,Myoung-gyu Leeb School of Mechanical Engineering, Xi’an Jiaotong University, No.28, Xianning Road, Xi’an, Shaanxi, P.R. China b Graduate Institute of Ferrous Technology, Pohang University of Science and Technology, San 31, Hyoja-dong, Nam-gu, Pohang, Geongbuk 790-784, Republic of Korea a

Abstract. A general yield function based on linear transformations of stress tensors for orthotropic sheet metals is presented to describe plastic strain-induced anisotropy. The subsequent anisotropy is considered by introducing a set of tensorial state variables in the linear transformations of stress tensors, and defining the tensorial state variables evolved with plastic deformation. The evolution of the tensorial state variables is based on eigen decomposition. The eigenvectors are normalized and used to represent the rotation of orthotropy axes of polycrystalline materials which relates to the texture rotation. The eigenvalues are used to capture the local reorientation of the grain preferred directions. Taking the anisotropic yield function Yld2000-2d as an example, the evolution of eigenvalues and eigenvectors of linear transformation tensors is investigated for AA3003-O aluminum alloy sheet. Based on the observations, a modified Yld2000-2d yield function incorporating with isotropic hardening model was developed to predict the anisotropy and subsequent yield surfaces in uni-axial tensions to confirm the validity of current yield function. Keywords: Anisotropic material; Yield function; Linear transformations; Plastic strain-induced anisotropy PACS: 83.60.La; 81.40.Jj

INTRODUCTION In the last two decades, the so-called isotropic plasticity equivalent (IPE) theory generalized by Karafillis[1] has gained some popularity in developing new anisotropic yield functions. In the IPE theory, plastic anisotropy is represented by one or more linear transformation tensors on the stress tensor, further the principal values of these linear transformation tensors are substituted into an isotropic yield function to get a new anisotropic yield function. Within the IPE framework, many anisotropic yield functions have been proposed to describe initial anisotropy of metallic sheets with phenomenological constitutive models[2-6], such as Yld2000-2d and Yld2004-13p/18p yield functions by Barlat[3,7] and Bron's anisotropic yield function [4]. The material coefficients of these anisotropic yield functions are usually constant and identified from the experimental initial tensile, shear or bi-axial yield stresses and/or r-values. However, Hu pointed out that it is difficult to get exact initial yield stress since the anisotropy feature at initial yield state is too sensitive to the definition of yield point[6]. Therefore, the plastic strain-induced anisotropy seems to be more important in relation to the constitutive models. Wagoner and Suh assumed that the coefficients characterizing anisotropy do not change with plastic deformation, while the stress exponent M associated with the shape of yield surface in most anisotropic yield functions, is a function of equivalent plastic strain[8-9]. But, since changing exponent M is not a general concept, it is difficult to apply this method to other anisotropic yield functions[6]. To obtain a general model which can cover plastic strain-induced anisotropy, a dynamic yield function was recently proposed, which assumes that the material coefficients of anisotropic yield function are not constant and identified from the current flow stresses and r-values[6]. However, the dynamic yield function is not easy to be implemented for non-quadratic yield functions because iterative solution procedures of non-linear is inevitably involved at each increment of plastic deformation. In addition, only successive strainhardening process can be considered, which means that the plastic deformation is continuous without abrupt

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 63-70 (2011); doi: 10.1063/1.3623593 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

63

loading-path change and the stress path of loading-unloading-reloading is not included as well. Recently, Wang also presented a yield function to consider the plastic strain-induced anisotropy by assuming the material parameters are polynomial functions of equivalent plastic strain [10]. In this paper, a general yield function theory based on linear transformations of stress tensors is firstly presented to consider plastic strain-induced anisotropy. In order to understand the evolution of subsequent anisotropy, the eigen decompositions for the linear transformation tensors of anisotropic yield function Yld2000-2d of the AA3003O sheet at different equivalent plastic strains are carried out. Thereafter, the evolution of eigenvalues and eigenvectors is investigated and discussed. Based on the observations, a simple evolution law was proposed and applied to predict the flow stresses and r-values in uni-axial tensions to validate the accuracy of current general yield function.

THEORY According to the classical plasticity theory, an isotropic stress yield function can be expressed with three invariants of the stress tensor [11]. Although many sets of three invariants have been introduced, the principal values are usually adopted to represent the isotropic yield function.

f ( ij )  F  1 , 2 , 3  where f is an isotropic stress yield function with respect to the Cauchy stress tensor of stress tensor  ij , and F is an isotropic function of its arguments, i.e.,

F  I , II , III   F  1 , 2 , 3 

(1)

 ij ,  i

are principal values

(2)

where (I,II,III) are permutations of (1,2,3). When pressure-independent plastic deformation is considered, the general formulation of the yield function can be expressed as a function of the three invariants of the deviatoric stress tensor,

g (sij )  G  S1 , S2 , S3 

(3)

where g is an isotropic stress yield function in the deviatoric stress space, G is an isotropic function of its arguments as well, Si represents the principal values of the deviatoric stress tensor sij . To formulate an anisotropic yield function, a reference frame associated with the material symmetry axes must be defined. In the isotropic plasticity equivalent approach for pressure-independent polycrystalline materials, a stress tensor sij is introduced as a linear transformation of deviatoric stress tensor sij , i.e.,

s = Cs = CT = L

(4)

where C and L are 4th-order tensors, which contain the anisotropy coefficients expressed in the material frame, T is a 4th-order tensor as well, which transforms the Cauchy stress tensor to its deviator. Substituting the principal values Si of the linearly transformed stress tensor s into Eq. (3), an anisotropic yield function  can be obtained,

  sij   g  sij   G  S1 , S2 , S3  Note that tensor s .



(5)

is an anisotropic yield function of s , while g is still an isotropic yield function with respect to stress

64

Considering the symmetry of the stress tensor s and the assumed symmetry of s , C can be represented as a 6  6 matrix. The symmetry of the material and the zero trace of s further reduce the number of independent coefficients in matrix C ; for instance, 9 for an orthotropic material. Therefore, C can be written as,

0 C12 C  21 0 C C32 C =  31 0  0  0 0  0  0

C13

0

0

C23 0

0 0

0 0

0

C44

0

0 0

0 0

C55 0

0

0 0 0 0 C66 

(6)

From a mathematical point of view, the linear transformation tensor C makes an initial isotropic yield surface homogeneously deform to an anisotropic yield surface. In most of phenomenological constitutive models, the matrix C is assumed to be constant. Recently, the importance of the description of plastic strain-induced anisotropy has been recognized and emphasized[6,10,12]. Within the isotropy plasticity equivalent framework, the matrix C should be evolved with plastic deformation in order to consider plastic strain-induced anisotropy. Similarly as deformation kinematics, the eigen decomposition can also be applied to the matrix C ,

C = VAV 1

(7)

where V includes the eigenvectors representing the rotation of orthotropy axes of polycrystalline materials, and A contains the eigenvalues which can be used to capture the local reorientation of the grain preferred directions. If we divide the plastic deformation into N steps, then the matrix C at k-step can be written as,

Ck = Vk Ak Vk1

(8)

In Eq. (8), A k is

1k  0 0 Ak   0 0   0

0

0

0

0



0

0

0

0



0

0

0

0

k 4



0

0

0

0

5k

0

0

0

0

k 2

k 3

0

0 0 0 0 6k 

(9)

The matrix C at (k+1)-step can be written as,

Ck 1 = Vk 1Ak 1Vk11

(10)

In Eq.(9), considering the evolution of A k , its current eigenvalues can be defined as

ik 1  ik  i

65

(11)

Since the local reorientation of the grain preferred directions usually influences the intensity of texture, the k eigenvalues i could be used to capture texture reorientation by defining proper evolution laws of i , i.e.,

i  i   p ,  p ,

(12)

In terms of the texture rotation, if the incremental rotation between k-step and (k+1)-step is could be expressed as

Q , then the Vk 1

Vk 1  QT Vk Q

(13)

To consider the influence of plastic deformation on texture rotation, deformation as well;

Q should be related to the plastic

Q  Q   p ,  p ,

(14)

Considering the physical aspects of texture evolution, and then defining proper evolution laws of Eq.(12) and (14), the plastic strain-induced anisotropy is expected to be accurately described.

EVOLUTION OF LINEAR TRANSFORMATION TENSOR In this section, taking the anisotropic yield function Yld2000-2d as an example, the evolution of eigenvectors and eigenvalues of linear transformation tensors is investigated. For this yield function, two linear transformation tensors are used to represent the anisotropy in plane stress condition, and total eight anisotropy coefficients, 1  8 , are usually adopted to transform the Cauchy stress to stress tensor sij , as shown in Eq. (15)[11].

1 0 0

 C =  0  2 0  0 0  7 

and

4 5   3 1 C =  2 3  2 5 3  0

2 6  2 4 4 4   6 0

0

0 3 8 

(15)

where these eight anisotropy coefficients are usually calculated from the measured experimental data,  0 ,  45 ,  90 ,  b , r0 , r45 , r90 and rb . Here   and r are yield stresses and r-values along different angles to rolling direction,  b and rb are bi-axial yield stress and r-value, respectively. To describe plastic strain-induced anisotropy, the subsequent yield surfaces should evolve with plastic deformation. It implies that these experimental data at the same subsequent hardening state should be determined to re-calculate the anisotropy coefficients. Regarding the experimental data, there are two methods to determine the experimental data in relation to a subsequent yield state. One method assumes that the anisotropy of yield function is a function of plastic work, which indicates that the experimental data at the same plastic work should be used to determine the subsequent yield surface. The other is based on the so-called principle of equivalent strain-hardening work, which can well explain the hardening behavior of perfect plastic body in which no further hardening occurs we may still have the response of the increment of plastic strain [6]. Here, the first order plastic work principle is used that can be defined as

dw   ij d ijp    d p

(16)

In this work, the aluminum alloy 3003-O in literature [6] was used as experimental data, and the Eqs. (17) - (19) were adopted to fit all experimental stress-strain curves and r-values as suggested by Hu [6].

   k e  p  ,    k e  n

66

n

(17)

 b  kb  be   bp  ,  b  kb  be  nb

1 n

1 nw

r  n k  

nw

nb

(18)

1 n 1 nw 1 1  nw n n  nw kw    n k   nw  

   

1

(19)

where k , n , k w , nw are the constants of material parameters of tensile tests at angle  to rolling direction, kb , nb material parameters of equibiaxial tension, and  e ,  be are the limit elastic strains of corresponding experiments. All the material parameters are listed in Table 1. Base on the above plastic work principle, the material properties of aluminum 3003-O at different hardening states are calculated as listed in Table 2. [6]

TABLE 1. Material parameters of aluminum 3003-O . Directions: Rolling Transverse Regarding the strain:  wp0  0p  90p  wp90 k values 195.0 240.0 183.0 240.0 n values 0.214 0.217 0.215 0.227 TABLE 2. Material properties of aluminum 3003-O at different hardening states. Directions: Rolling Transverse Diagonal Equibiaxial Plastic strain: (MPa) (MPa) (MPa) (MPa) Yield stress (0.2%) 51.58 48.1 47.1 49.98 0.0077 72.32 68.26 67.71 75.56 0.0120 78.23 73.91 73.49 81.04 0.0159 82.42 77.91 77.58 84.81 0.0195 85.77 81.10 80.85 87.76 0.0402 99.03 93.73 93.80 99.09 0.0740 112.33 106.38 106.83 110.07 0.1087 121.75 115.34 116.09 117.67 0.1446 129.3 122.53 123.52 123.68 0.1818 135.71 128.63 129.84 128.73 0.2203 141.34 133.99 135.39 133.13

Diagonal

Equibiaxial

 45p

 wp45

 bp

187.0 0.222

232.0 0.229

171.3 0.17

r0

r90

r45

0.704 0.677 0.672 0.668 0.665 0.655 0.646 0.641 0.637 0.633 0.631

0.662 0.576 0.559 0.548 0.54 0.511 0.488 0.473 0.463 0.455 0.449

0.842 0.769 0.754 0.745 0.737 0.712 0.690 0.677 0.667 0.660 0.654

As shown in Table 2, four stresses and three r-values are available for AA3003-O aluminum alloy sheet. With these data, only seven anisotropy coefficients can be identified. Therefore, 3   6 is assumed for the anisotropic yield function Yld2000-2d. Fig. 1 shows the evolution of eigenvalues of linear transformation tensors with plastic deformation. It can be seen from Fig. 1(a), the eigenvalues of linear transformation tensor C gradually saturate to the constants after large plastic deformation. However, for tensor C , the obvious abrupt changes exist for all eigenvalues when plastic deformation is small. It might result from that the deformation at early stage is quite unstable due to the fact that the early plastic range may not be completely plastic, or may be elastic-plastic transient regions. Fig. 2 shows the evolution of eigenvectors of linear transformation tensor C with plastic deformation. Here, the evolution of eigenvectors is respectively represented by the angles between current eigenvector and the initial one as shown in Fig. 2(a), and the angles between current eigenvector and previous one as shown in Fig. 2(b). Supposed n1k , n 2k , and n 3k are the current eigenvectors of linear transformation tensor C , n1k 1 , n 2k 1 , and n3k 1 are the 1 2 3 previous eigenvectors, n 0 , n 0 , and n 0 are initial one, the angles used here are respectively defined as

 ni  ni k 0  nik  ni0 

i  arcos 

   

and

 ni  ni di  arcos  i k k i 1  n k  n k 1 

   

(20)

Fig. 2(a) indicates that the eigenvector evolved with plastic deformation. When the traditional initial yield stresses and r-values, i.e., assuming the equivalent plastic strain is zero, are selected to determine the initial eigenvectors, the

67

subsequent eigenvectors have abrupt changes. However, when the eigenvectors which correspond to equivalent plastic strain of 0.0077 and 0.0120 are used to represent the initial state, the evolution of subsequent eigenvectors is continuous and almost linear. Fig. 2(b) shows that the eigenvectors have a rapid change rate at small plastic deformation, and after that the rates of  i to equivalent plastic strain are almost constant. It implies that the evolution of directions of eigenvectors has a linear relation to equivalent plastic strain.

(a) Linear transformation tensor C (b) Linear transformation tensor C FIGURE 1. Evolution of eigenvalues of linear transformation tensors with plastic deformation.

(a) Angles between current eigenvectors and initial one (b) Angles between current eigenvectors and previous one FIGURE 2. Evolution of eigenvectors of linear transformation tensor C with plastic deformation.

PREDICTION OF THE ANISOTROPY AND SUBSEQUENT YIELD SURFACES As shown in Fig. 1, the eigenvalues of linear transformation tensors tend to saturate towards constants after large deformation. Two possible reasons result in such behavior. One is that only monotonic loading paths are considered in the present work. In such cases, the evolution of anisotropy and hardening behavior is continuous, and thus no abrupt changes exist. It implies that the evolution of the eigenvalues should be continuous too. The other reason is that, for most polycrystalline metallic materials, the hardening rate decreases with plastic deformation. Therefore, it can be inferred that the change rate of anisotropy and hardening behavior should decrease with plastic deformation as well. In this work, for simplicity the eigenvectors of linear transformation tensor C are assumed to be fixed during plastic deformation as a first attempt and Eq. (21) is used to describe the evolution of eigenvalues.



i  i0  Qi 1  eb 

68

i

p



(21)

where Qi and bi are material parameters,  equivalent plastic strain, the plus-minus denotes that the eigenvalues increase or decrease. This sign might be related to the plastic strain rate when strain path changes exist. In our case, for simplicity, the signs are directly defined based on the evolution of eigenvalues as shown in Fig. 1. The important part of the current general yield function is to define the initial eigenvalues and eigenvectors of linear transformation tensors. As mentioned above, the definitions of initial yield stresses and r-values might be very difficult and inaccurate with traditional methods. To overcome it, the early stable anisotropy at small plastic deformation should be chosen to represent the initial anisotropy. In this work, the experimental data from equivalent plastic strain of 0.0077 was used to fit the initial eigenvalues and eigenvectors. For the other material parameters of anisotropy-evolved yield function Yld2000-2d, inverse identification is carried out by optimization using uni-axial tensile tests with a developed Matlab toolbox SMAT. A cost function is defined in the least square sense and minimized, starting from an initial guess of parameters. Finally, the identified material parameters are summarized in Table 3. The initial eigenvectors are shown in Eq. (22). p

TABLE 3. Material parameters of anisotropy-evolved yield function Yld2000-2d for aluminum 3003-O. Transformation tensors: b Q Q b 0 0 0

C C

3

Q3

b3

0.9033

0.0507

17.435

1.0351

0.1005

17.006

1.0204

0.0268

16.796

0.9524

0.0

0.0

1.1922

0.0

0.0

1.2522

0.0

0.0

1

1

1

2

2

2

0

0.8402 0.6546  V0 =  0.5423 0.7560 0  0 0 1.0 

(22)

Fig. 3 is the simulated flow stresses and r-values compared with experimental data. It is found that the stressstrain and r-value curves predicted by the current yield function are in good agreement with the experimental curves in all three directions, especially, for the evolution of r-values. For the case of constant material coefficients of yield function, r-values are usually kept when isotropic hardening model is adopted. However, the current method gives a good correlations with simplified evolution equation, Eq. (21).

(a) Cauchy stress (b) R-value FIGURE 3. Predicted Cauchy stresses and r-values in uni-axial tensions.

When a hardening state is selected, its corresponding yield surface could be predicted by the current yield function. Choosing the hardening states presented in Table 2, the initial and subsequent yield surfaces are plotted as shown in Fig. 4 when the shear stress is zero. For comparisons, the yield surfaces calculated by the constant and variable material coefficients of yield function are expressed in the same graph, marked with ‘anisotropy-unevolved’ and ‘anisotropy-evolved’, respectively. The results show that the subsequent yield surfaces predicted by the two methods are rather different, especially, for the bi-axial tensile stress states.

69

FIGURE 4. Initial and subsequent yield surfaces with anisotropy-unevolved and anisotropy-evolved yield functions.

CONCLUSIONS A general yield function based on linear transformations of stress tensors for orthotropic sheet metals is presented to model plastic strain-induced anisotropy. The subsequent anisotropy is considered by defining proper evolution laws for the eigenvalues and eigenvectors of linear transformation tensors. Taking the anisotropic yield function Yld2000-2d and AA3003-O aluminum alloy sheet, the evolution of eigenvalues and eigenvectors of linear transformation tensors is investigated. To confirm the validity of the present method, the flow stresses and r-values of uni-axial tensions are predicted by a modified Yld2000-2d yield function which can cover plastic strain-induced anisotropy under successively strain-hardening process. The results show that the present model can accurately describe the anisotropy and hardening behavior, especially, the evolution of r-values. As a future work, the model will be applied to a real sheet metal forming process to verify the importance of the plastic induced anisotropy.

ACKNOWLEDGMENTS Shun-lai Zang would like acknowledge financial support by the Specialized Research Fund for the Doctoral Program of Higher Education (No.200806981025), and by the National Natural Science Foundation of China (No.11002105) and by the Opening Project of Key Laboratory of Testing Technology for Manufacturing Process (Southwest University of Science and Technology), Ministry of Education (No.10ZXZK03).

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

A.P. Karafillis and M.C. Boyce, J. Mech. Phys. Solids 41, 1859-1886 (1993). D. Banabic et al, Int. J. Plast. 21, 493-512 (2005). F. Barlat et al, Int. J. Plast. 19, 1297-1319 (2003). F. Bron and J. Besson, Int. J. Plast. 20, 937-963 (2004). O. Cazacu et al, Int. J. Plast. 22, 1171-1194 (2006). W. Hu et al, Int. J. Plast. 23, 620-639 (2007). F. Barlat et al, Int. J. Plast. 21, 1009-1039 (2005). R.H. Wagoner, Metallurgical and Materials Transactions A 11, 165-175 (1980). Y.S. Suh et al, Int. J. Plast. 12, 417-438 (1996). H.B. Wang et al, Comput. Mater. Sci. 47, 12-22 (2009). F. Barlat et al, Int. J. Plast. 23, 876-896 (2007). J.H. Hahm and K.H. Kim, Int. J. Plast. 24, 1097-1127 (2008).

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Experimental Investigations on Anisotropic Evolution of 304 Stainless Sheets under Tensile Pre-Strains Lai Tenga and Cheng Guob School of Mechanical Engineering, Xi’an Jiaotong university, No.28 Xianning West Road.Xi’an.Shaanxi, Xi’an, 710049, China Abstract. The anisotropic evolution of cold rolled 304 stainless steel sheets under pre-strains is investigated experimentally. Uni-axial tensile yield stress and r-value are measured in experiments to represent the anisotropy. The tensile pre-strains under plane stress are achieved by cutting large specimens into small one at different angles to rolling direction. Then the uni-axial tensile tests are performed on the small specimens to investigate the anisotropic evolution. It is found that the yield stress increases with the increase of the pre-strains and decreases with the increase of the angles. However, the changes of r-value are hardly affected by the pre-strains, the small changes of r-value show that the material may remember the rolling direction even after the pre-strains. The sigmoidal shape can be observed in the tensile curves, and its shape depends on the pre-strains and angles. The change of hardening rate can be divided into three stages, and is the most significant at 90° to the rolling direction in the three stages, at the same time the pre-strains cause noncoincidence of the hardening rate curves at the same angle. Moreover, the hardening rate depends on the directions of tensile loading. Second derivative of the hardening rate also can be divided into three stages, and the differences of three stages may mainly be controlled by the different volume fraction of martensite. Keywords: SUS304, austenitic stainless steel, martensitic transformation, anisotropic hardening. PACS: 81.40.Ef, 81.30.Kf.

INTRODUCTION The 304 stainless steel sheets are widely used as structural components in machines and structures because of good toughness, ductility and resistance to corrosion. The anisotropy of cold rolled materials is a well-known phenomenon and is of practical interest for the formability of sheets. So a great deal of efforts has been made to understand the yield and hardening behavior of cold rolled anisotropic materials experimentally [1, 2] as well as theoretically [3, 4]. Austenitic phase in 304 stainless steels is metastable, and it is known to develop the straininduced martensitic transformation caused by lower stability of austenitic phase when it is subjected to plastic deformation. Many reports refer to the response of these evaluations in terms of the effects of the martensite contents in the cold rolled 304 stainless steel [5, 6]. Type 304 steel is one of the typical austenitic stainless steels and exhibits remarkable work hardening [7], thus pre-strained type 304 steel will be expected to behave in different manner from unpre-strained one. This paper utilized the method of Kim and Yin [8]. In their previous work, tensile deformation was applied at some angles to the rolling direction. Kim and Yin suggested that the orientations of the orthotropy axes are altered by the tensile pre-strain at an angle to the rolling direction. In this study, the change of yield stress, r-value, true stress-strain and hardening rate will be presented for the better description of the hardening behavior. The main purpose of this work was to report the experimental observations and to attempt to understand the observed plastic yield and hardening behavior. Moreover, the work did not pay attention to the change of the orthotropy axes and microscopic structure.

EXPERIMENTAL SCHEME An experimental method for the study of anisotropy of cold rolled 304 stainless steel sheets is to perform tensile tests at angles to the rolling direction. The initial anisotropy is enhanced by stretching 12 full size sheets along the The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 71-77 (2011); doi: 10.1063/1.3623594 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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rolling direction by 6%, 11%, 16% and 21% tensile pre-strains. After the pre-strains, uni-axial tensile tests are performed at every 22.5º interval to rolling direction for 5 small specimens (FIGURE 1). The full size sheets were loaded different pre-strains with the help of huge chucks, and digital image correlation system (XJTUDIC) and photogrammetry system (XJTUSM) were used to confirm homogeneous deformed zone. Center lines were drawn in the huge chucks and full size sheets in order to ensure stretch axis coincided with center axis of the full size sheets, and many spots were corroded at the back of the full size sheets, the photogrammetry system (XJTUSM) is used to measure strain after spring back (FIGURE 2).

FIGURE 1. Schematic illustration of experimental procedure.

(a) Digital image correlation system (XJTUDIC). (b) Photogrammetry system (XJTUSM). FIGURE 2. The huge chucks, full size sheet and strain measurement system.

(a) Before pre-strain (b) After pre-strain FIGURE 3. Strain measurement of the full size sheet before and after the pre-strain.

FIGURE 3 shows the strain measurement before and after the pre-strain, and the homogeneous deformation can be observed. However, the strain is negligible before the pre-strain. So it could cut the small specimens from the deformed zone to investigate the anisotropic evolution. The true stress, longitudinal and transverse strain (  l and  w ) have been measured in tensile tests, and yield points have been measured at 0.2% offset strain. The incremental plastic strain ratio ( r ) is defined by

r Here,

 tp

d wp d wp   . d t p d lp  d wp

(1)

means the through-thickness plastic strain. The r-value is determined from the longitudinal and

72

transverse strains under the assumption of plastic incompressibility. Measurement noise in the strain channels and uncertainties in the elastic properties of the material tend to disrupt the estimation of r-value in the small strain range. Therefore r-value has been measured at a sufficiently large longitudinal strain up to 10%, 20% [8-11].

EXPERIMENTAL RESULTS AND DISCUSSION The strain rate is about   4.78  10 for the full size sheets and   3  10 for the small specimens at room temperature. The following figures show the results of tensile tests. Figure 4 shows the variations of the uni-axial yield stress for the small specimens applied at 0º, 22.5º, 45º, 67.5º and 90º to the rolling direction. The yield stress of the small specimens increases with the increase of the pre-strains and decreases with the increase of the angles. In contrast to the variations of the yield stress, the changes of r-value are found to be hardly affected by the pre-strains in Figure 5, the small changes of r-value show that the material may remember the rolling direction even after the pre-strains. 4

3

FIGURE 4. Uni-axial yield stress (at 0.2% offset strain) for the small specimens applied at Ψ=0°，22.5°，45°，67.5°，90°, respectively.

(a) Longitudinal plastic strains

 lp =10%

(b) Longitudinal plastic strains

 lp =20%

FIGURE 5. Incremental r-value distributions at 10% and 20% of longitudinal plastic strains  l for the pre-strain of 6%, 11%, p 16% and 21%: (a), (b) Longitudinal plastic strains  l =10%, 20%, respectively. p

It was mentioned in many literature that deformation could induce martensitic transformation from austenite in the austenitic stainless steels [12, 13, 14]. The volume fraction of transformed martensite increased with the increase of deformation, and this was explained from transformation kinetics in [15]. Some researchers have shown that martensitic transformation can increase the strength and decrease the toughness of the austenitic stainless steels, and the results of tests are in accordance with the previously reported results (FIGURE 6). For further study change of the hardening rate, true stress-strain and strain-hardening rate curves for each pre-

73

strain condition need to be examined in detail. Figure 6 shows the true stress-strain curves of the small specimens. As mentioned above, for the different pre-strains, it can be observed that strength increases and elongation decreases with the increase of the pre-strains, it may mainly due to the pre-strains martensite in its matrix, and the effect is similar to that of martensite in the dual-phase steel. The yield strengths are slightly different at different angles to rolling direction in the same pre-strains, while the hardening rate is essentially similar in each direction except for the rolling direction. The sigmoidal shape can be observed in the tensile curves, and the larger the pre-strain is, the more obvious its trend is. Moreover, such trend is more obvious at other directions than at the rolling direction, for it may transfer more mastentics at other directions [16, 17, 18].

(a) Small specimens after 6% pre-strain

(b) Small specimens after 11% pre-strain

(c) Small specimens after 16% pre-strain (d) Small specimens after 21% pre-strain FIGURE 6. True stress-strain of the small specimens at angles to the rolling direction for different pre-strains: (a), (b), (c), (d) after 6%, 11%, 16%, 21% pre-strains along the rolling direction, respectively.

FIGURE 7 shows the hardening rate curves of the small specimens, and it cuts off the curves for better expression. The curves can be divided into three stages in the FIGURE 7. In the first stage, the hardening rate decreases with the increase of true strain, but the initial trends are similar to that of other materials, for the content of transformed martensite in the austenitic matrix may not result in difference of initial hardening rate. In the second stage, the hardening rate exhibits an opposite trend compared to the first stage. In the final stage, the hardening rate decreases with the increase of true strain again. The change of hardening rate is the most significant at 90° to the rolling direction in the three stages, at the same time the pre-strains can cause noncoincidence of the hardening rate curves at the same angle. Moreover, the hardening rate depends on the directions of tensile loading. Due to the different hardening rate in each of loading directions, the yield stress depends on the offset plastic strain level. For further study, second derivative of the hardening rate has been gotten from FIGURE 7. The pre-strains have significant effects on initial and final stage, but it makes very little difference in intermediate stage. The change s of the three stages can be explained by martensitic phase transformation, for the volume fraction of transformed martensite increases with the increase of the pre-strains when the full size sheets undergo the different pre-strains. Moreover, the differences of initial and final stage in FIGURE 8 may mainly controlled by the martensite which was

74

induced by the previous pre-strains. However, the curves trend to be uniform in the intermediate stage, this is because the volume fraction of martensite may be same after the following deformation.

(a) Small specimens along Ψ=0°

(b) Small specimens along Ψ=22.5°

(c) Small specimens along Ψ=45°

(d) Small specimens along Ψ=67.5°

(e) Small specimens along Ψ=90° FIGURE 7. Strain-hardening rate at angles to the rolling direction for different pre-strains: (a), (b), (c), (d), (e) the small specimens applied at Ψ=0°, 22.5°, 45°, 67.5°, 90°, respectively.

For further study on microstructure evolution of 304 stainless steel under the pre-strains and the tensile loading, in the next stage, light optical microscope will be used to determine microscopic structural change, and X-ray

75

diffraction will be used to measure the relative amounts of different phases formed during pre-prestrain and tensile loading in cold rolled 304 stainless steel sheets.

(a) Small specimens along Ψ=0°

(b) Small specimens along Ψ=22.5°

(c) Small specimens along Ψ=45°

(d) Small specimens along Ψ=67.5°

(e) Small specimens along Ψ=90° FIGURE 8. Strain-second derivative of hardening rate at angles to the rolling direction for different pre-strains: (a), (b), (c), (d), (e) the small specimens applied at Ψ=0°, 22.5°, 45°, 67.5°, 90°, respectively.

76

CONCLUSION The yield stress increases with the increase of the pre-strains and decreases with the increase of the angles. The change of r-value is affected by the pre-strains. The sigmoidal shape can be observed in the tensile curves. The changes of the hardening rate and second derivative of the hardening rate curves can be divided into three stages.

ACKNOWLEDGMENTS Lai Teng is grateful to Dr. Zang for his valuable advice and comments and to Professor Liang at Xi’an Jiaotong University for his experimental devices and encouragement during the progress of this work. Support from Yang Zhang and Xiang Guo of Research Institute of Mould and advanced forming technique with the experiment process and from Hua Zhang with the specimen preparation are greatly appreciated.

REFERENCES 1. Mallick, K., Sanabta, S.K., Kumar, A., “An experimental study of the evolution of yield loci for anisotropic materials subjected to finite shear deformation”, J. Eng. Mat. Tech. Trans. ASME 113, 1991, pp. 192. 2. Truong Qui, H.P., Lippmann, H., “Plastic spin and evolution of an anisotropic yield condition”, Int. J.Mech.Sci. 43, 2001, pp. 1969–1983. 3. Hill, R., “Continuum micro-mechanics of elastoplastic polycrystals”, J. Mech. Phys. Solids 13, 1965, pp. 89–101. 4. Berveiller, M., Zaoui, A., “An extension of the self-consistent scheme to plastically-flowing polycrystals”, J.Mech. Phys. Solids 26, 1979, pp. 325–344. 5. T. Yamaski, S. Yamamoto, M. Hiro, NDT&E Int. 29, 263 (1996). 6. D.Q. Sullivan, M. Cottrell, I. Meszaros, NDT&E Int. 37,265 (2004). 7. Schuster G, Altstetter C. “Fatigue of annealed and cold worked stable and unstable stainless steels”, Metall Trans A 14A,1983, pp. 2077-2084. 8. K. H. Kim, J. J. Yin, “Evolution of anisotropy under plane stress”, Journal of the Mechanics and Physics of Solids 45, 1997, pp. 841-851. 9. J.H. Hahm, K.H. Kim, “Anisotropic work hardening of steel sheets under plane stress”, International Journal of Plasticity 24, 2008, pp. 1097-1127. 10. Magnus Harrysson, Matti Ristinmaa, “Description of evolving anisotropy at large strains”, Mechanics of Materials 39, 2007, pp. 267-282. 11. Z. J. Li, G. Winther, N. Hansen, “Anisotropy of plastic deformation in rolled aluminum”, Materials Science and Engineering A 387-389, 2004, pp. 199-202. 12. P.L. Mongonon, G. Thomas, “Structure and properties of thermal-mechanically treated 304 stainless steel”, Metall Trans 1, 1970, pp. 1587-1594. 13. F.D. Fischer, Q.P. Sun, K. Tanaka, “Transformation Induced Plasticity”, Appl. Mech. Rev 49, 1996, pp. 317-364. 14. D. G. Park, D. W. Kim, C. S. Angani, “Measurement of the magnetic moment in a cold worked 304 stainless steel using HTS SQUID”, Journal of Magnetism and Magnetic Material 320, 2008, pp. 571-574. 15. Smaga M, Walther F, Eifler D, “Deformation Induced Martensitic Transformation in Metastable Austenitic Steels”, Materials Science and Engineering A, 2008, pp.483-484. 16. K. H. Lo, D. Zeng, C. T. Kwok, “Effects of sensitisation-induced martensitic transformation on the tensile behaviour of 304 austenitic stanless steel”, Materials Science and Engineering A 528, 2011, pp. 1003-1007. 17. K. Spencer,M. Ve´ron, K. Yu-Zhang, “The strain induced martensite transformation in sustenitic stainless steels Part1Influence of temperature and strain history”, Materials Science and Technology 25(1), 2009, pp. 7-17. 18. XUE Zong-yu, ZHOU Sheng, WEI Xi cheng, “Influence of Pre-transformed Martensite on Work-Hardening Behavior of SUS 304 Metastable Austenitic Stainless Steel”, International Journal of Iron and Steel Research 17(3), 2010. pp. 51-55.

77

([SHULPHQWDO&KDUDFWHUL]DWLRQDQG&RQVWLWXWLYH0RGHOLQJRI 7$90HFKDQLFDO%HKDYLRULQ3ODQH6WUDLQ6WDWH DW5RRP7HPSHUDWXUH G. Gillesa,b, V. Tuninettia, M. Ben Bettaïeba, O. Cazacuc, A.M. Habrakena,b and L. Duchênea,b a

Department ArGEnCo, Division MS²F, University of Liège, Chemin des Chevreuils 1, 4000 Liège, Belgium b Fonds de la Recherche Scientifique-FNRS, Belgium c Department of Mechanical and Aerospace Engineering, University of Florida REEF, 1350 N Poquito Road, Shalimar, FL 32579, USA $EVWUDFWThis paper presents an experimental and theoretical study of the quasi-static behavior of TA6V titanium alloy in plane strain state. In order to quantify the anisotropy of the material, tests were carried out at room temperature on specimens cut out from a sheet along three loading directions. The initial yield locus is described by the phenomenological CPB06ex3 criterion and Voce’s type isotropic hardening is used. Finite element simulations are performed and compared with the experiments. .H\ZRUGVTA6V titanium alloy, material anisotropy, strength differential effect, anisotropic yield criterion,  3$&646.35.+z, 62.20.F-

,1752'8&7,21 TA6V titanium alloy displays mechanical properties for which industrial sectors such as aerospace, biomedical or transportation have shown a high interest. Indeed, owing to its high-strength-to-weight ratio, good formability and biocompatibility, it is used in the design of aircraft engines and structures, surgical implants, valves, connecting rods, etc. [1,2]. Many investigations have already been conducted in order to characterize and model the mechanical behavior of TA6V [3,4,5]. This hexagonal-closed packed (hcp) material exhibits both a pronounced anisotropy as well as a strength asymmetry in tension and compression. Due to these specific characteristics, classical plasticity models (Hill 1948, J 2 -flow theory) are not adapted to describe the mechanical response of this alloy [6]. New phenomenological yield criteria [7,8] have been proposed to take into account both the tension-compression asymmetry and the anisotropy of hcp materials. Recently, an experimental study on the quasi-static mechanical behavior of TA6V in tension and compression has been performed [9]. The yield criterion CPB06ex3 [8] was used in order to describe yielding. However, tensile and compressive tests are not enough to identify the material parameters with accuracy. In this study, quasi-static plane strain tests in three different directions at room temperature have been performed on 0.6 mm thick samples. A description of the methodology used to analyze the experimental results and inspired by [10] is presented. The observed behavior is next modeled with CPB06ex3 yield criterion and an isotropic hardening. The model identification is improved by taking into account the onset of plasticity for the plane strain loading. The ability of the modeling to predict the material behavior is then examined by comparing the experimental and simulation stress-strain curves in plane strain tension until large strains, as well as the experimental and numerical results obtained from a simple shear test.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 78-85 (2011); doi: 10.1063/1.3623595 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

78

(;3(5,0(17$/&+$5$&7(5,=$7,21 0DWHULDO The investigated TA6V is a commercial alloy produced by TIMET (France). The material is in sheet form with a 0.6 mm thickness. It presents an average grain size of 11 µm for the hcp α-phase and 1 µm for the bcc β-phase, respectively. The chemical composition of the sheet is given in 7$%/(.

Top Bottom %DO EDODQFH

7L Bal. Bal.

7$%/(Chemical composition of the TA6V sheet alloy [in %].

$O 6.22 6.27

9 3.93 4.00

2 0.19 0.20

1 0.006 0.006

& 0.008 0.009

+ 0.01 0.0086

)H 0.16 0.16

&KDUDFWHUL]DWLRQRIWKH$QLVRWURSLF%HKDYLRULQ3ODQH6WUDLQ The experimental device used in this work is shown in ),*85( . The machine includes a vertical and a horizontal actuators (denoted V1 and V2 in ),*85( , respectively) which enable to carry out successive or simultaneous plane strain and simple shear tests. This biaxial machine was developed and validated at the Materials and Structures Mechanics Laboratory of the University of Liège [11].

9

9

),*85(Biaxial machine: the white arrows indicate the displacement of the actuators.

In order to characterize the material anisotropy in plane strain tensile state, tests were performed at a velocity of the actuators of 5x10-3 mm/s (corresponding to an initial strain rate of 1.67x10-3 s-1 when neglecting any slip of the specimen under the grips) and room temperature along three in-plane directions, namely the rolling (RD), transverse (TD) and 45° directions. Each test was duplicated four times in order to check the experimental reproducibility. The specimens are rectangular, with notches at mid-length (see ),*85(  D ). The overall length and width are respectively 120 mm and 30 mm, while the gage zone is 25 mm wide and 3 mm high.

79

(a)

(b)

),*85(D Geometry and dimensions of the plane strain specimen. E Configuration of the specimen inside the grips. ),*85( E shows the configuration of a specimen inside the grips of the biaxial machine. A stochastic pattern is previously painted on the sample’s surface in order to determine the strain field of the gage zone by using the optical measurement system Aramis®. ),*85( shows the typical distribution of the axial (x1 direction) and transversal (x2 direction) strains along the width of the gage zone for different stages of the deformation process. It can be distinguished two different areas: - a homogeneous central zonein plane strain state (the transversal strain is low compared to the axial strain); - a heterogeneous zone at the edges of the specimen.



),*85(Evolution of the axial and transversal strain fields along the width of the gage zone

Since the strain field is not uniform, the edge effects must be taken into account in the stress computation. The methodology used in this work is similar to the one proposed by [10]. The homogeneous strain field zone is first identified at the different stages of the deformation. Theoretically, the strain distribution is uniform when: ∂ε11 (1) =0 ∂x2

80

where ε11 denotes the axial strain. In practice, a range ξ is defined at each stage and the condition (1) is expressed as: ∂ε −ξ ≤ 11 ≤ ξ (2) ∂x2 In this study, the boundary ξ was chosen as the average deviation of the derivative from zero at a 5 mm wide central zone of the specimen. The homogeneous strain field width WH is then computed as follows:

WH = x2max − x2min

(3)

with x2 verifying (2), i.e.:

§ ∂ε · § ∂ε · x2min = x2 ¨ 11 = −ξ ¸ , x2max = x2 ¨ 11 = ξ ¸ (4) ∂ ∂ x x © 2 ¹ © 2 ¹ Based on a numerical analysis of a plane strain tensile test for different materials and specimen geometries, [10] showed that the axial stress in the homogeneous zone can be computed by: F F σ 11 = α T + (1 − α ) T (5) WT tH WH t H where FT is the force measured by the load cell of the machine and WT is the total width of the gage zone, while t H is the actual thickness of the homogeneous area computed thanks to the initial thickness t0 and the average strain at the specimen’s center ε 11c as follows:

t H = t0 exp ( −ε11c )

(6)

It has to be noted that, in equation (6), the volume conservation is assumed. The coefficient α in equation (5) is a correction factor which was assessed at 0.98 in [10]. ),*85( shows the average axial stress-axial strain curves along RD, TD and 45°-direction obtained with the methodology described above, while 7$%/( lists the plane strain yield stress in each orientation. The latter were determined using the work-equivalence principle [12] and the experimental results in tension and compression reported in [9]. It can be observed that the material displays moderate in-plane anisotropy in plane strain, the strongest behavior being in TD.

),*85( Plane strain test results along the rolling (RD), transverse (TD) and 45° directions. 7$%/( Plane strain yield stresses determined by the work-equivalence principle 

/RDGLQJGLUHFWLRQ>GHJUHHV@ 0 45 90

81

03D@ 1139 1185 1199

02'(/,1* ,QLWLDO
(

F = ¦ G Σ1( ) , Σ(2 ) , Σ(3 ) , k ( ) , a i =1

i

i

i

i

)

(7)

where

G ( Σ1 , Σ 2 , Σ 3 , k , a ) = ( Σ1 − k Σ1 ) + ( Σ 2 − k Σ 2 ) + ( Σ3 − k Σ3 ) a

a

a

(8)

(i )

In Eqs. (7) and (8), the material parameters k , i = 1, 2,3 , allow for the description of strength differential (SD) effects, a is the degree of homogeneity, while Σ1(i ) , Σ (2i ) , Σ(3i ) are the eigenvalues of the transformed tensors:

Ȉ( i ) = &(i ) : V

(9)

(i)

where V is the stress deviator and & , i = 1, 2,3 , are fourth-order symmetric and orthotropic tensors which are represented in Voigt notations by:

ªC11(i ) « (i ) «C12 « (i ) «C (i ) & = « 13 « 0 « 0 « «¬ 0

C12(i )

C13(i )

(i )

(i )

C22

C23

(i )

(i )

C23

C33

0

0

0

0

0

0

(i )

0

0

C44

0

0

0

0

C55(i )

0

0

0

0

0 º » 0 » » 0 » » 0 » 0 » » i C66( ) »¼

(10)

The symbol “:” in (9) denotes the double contracted product between fourth- and second-order tensors, i.e. ( & : V )ij = Cijkl skl .

+DUGHQLQJ The investigation of the mechanical behavior of TA6V in tension and compression [9] has shown that, although more accurate results can be obtained with distortional hardening, only a few discrepancies are observed when considering an isotropic hardening. In order to assess “the degree of anisotropy” in plane strain, Voce’s type evolution law is used:

(

σ = R0 + RSAT 1 − exp ( −CR ε p )

)

(11)

where σ and ε p are respectively the equivalent stress and plastic strain associated to CPB06ex3 yield criterion (8) while R0 , RSAT and CR are the initial yield stress in tension along RD, the asymptotic value and the rate of the isotropic hardening, respectively.

3DUDPHWHU,GHQWLILFDWLRQ The classical simulated annealing (CSA) method [13] was used to fit the SD parameters and anisotropy coefficients in CPB06ex3 yield criterion. The error function presented in [9] was improved to take into account the experimental data in plane strain tests. The results of the identification are given in 7$%/(.

82

7$%/(Initial yield function CPB06ex3 coefficients for TA6V using tensile, compressive and plane strain experimental data.

k -0.4024 k′ 0.0909 k ′′ -0.8845 a 

C 1.0 C′

C -2.1321 C′

C -4.2088 C′

C  1.3359 ′ C 

C  -3.0723 ′ C 

C  1.5532 ′ C 

C  -3.5393 ′ C 

1.0 C′′

0.6184 C′′

-0.6446 C′′

2.6200 ′′ C 

-3.7190 ′′ C 

2.0367 ′′ C 

2.6680 ′′ C 

1.0

-3.0046

-1.0884

1.1286

-1.0194

0.8479

-4.3452

The hardening parameters in (11) were fitted on the experimental stress-strain curve in tension along RD by using the solver in Excel. The optimized parameters are listed in 7$%/(. Moreover, additional simple shear tests, which will not be described in this paper, were performed thanks to the biaxial machine in order to determine other points on the yield surface. 7$%/(Isotropic hardening parameters.

R0 >03D@ 971

R6$7 >03D@ 212

C R >@ 12.6

5HVXOWV ),*85( displays the theoretical yield surface for TA6V in the biaxial plane ( σ 3 = 0 ) according to Von Mises and CPB06ex3 criteria in comparison with the experimental data corresponding to the plastic work levels associated to the onset of plasticity ( W p = 1.95 MPa) and a plastic strain of 5% in tension along RD ( W p = 51.3 MPa). It is

worth noting that, since only one component of the stress tensor can be determined from the plane strain tensile tests, the condition imposed on the shape of the yield locus is in this case to be tangent to a straight line. It can be noticed that the initial anisotropy and tension-compression asymmetry exhibited by the material are well predicted by CPB06ex3 yield criterion. In addition, it has to be noted that, when observing the location of the experimental points at both plastic work levels, the hardening is slightly anisotropic.

),*85(Initial yield locus according to Von Mises criterion (blue line) and CPB06ex3 criterion (red line) in comparison with experimental data in tension, compression, simple shear and plane strain (symbols and straight lines) at two different plastic work levels.

Plane strain tensile tests were simulated by using a single 8-node 3D brick element with one integration point. ),*85(  compares the experimental stress-strain curves in RD and TD with the results of the simulations.

83

Although the onset of plasticity is pretty well captured (see ),*85(  E ), the flow stress for plastic strain is underestimated in both loading directions. Nevertheless, the work-hardening rate is in quite good agreement after about a strain of 2%.

(a)

(b)

),*85((a) Comparison between experimental and simulated stress-strain curves in plane strain tension for the rolling and transverse directions; (b) zoom at the onset of plasticity.

In order to test the phenomenological modeling, a simple shear test (),*85(D ) was performed on the biaxial machine and next simulated by using the finite element mesh shown in ),*85(E . During the experiment, the applied force was measured by the load cell of the horizontal actuator and the shear strain at the center of the specimen was computed thanks to Aramis®. ),*85(F compares the experimental and numerical curves. It can be observed that, although the results do not perfectly match, they are however in good agreement.

(a)

(b)

(c)

),*85((a) Simple shear test diagram; (b) finite element mesh; (c) force applied to the specimen as a function of the shear strain at its center and comparison with the simulated curve.

&21&/86,21 In addition to a previous study on TA6V in tension and compression, plane strain tests were carried out in order to obtain new data on the quasi-static mechanical behavior of the alloy. From the experimental results, it can be observed that the material displays a moderate anisotropy in plane strain state. The initial yielding was described by using CPB06ex3 yield criterion adapted to hcp metals and the hardening was assumed to be isotropic, even if the experiments show a slightly anisotropic hardening. Finite element simulations of the plane strain tests and a simple shear test were performed in order to examine the ability of the model to predict the material behavior. The experimental and numerical results are not in good agreements in plane strain. However the simulations show that efforts must be focused on the description of the hardening. In order to improve the modeling, hardening depending on the strain path is required.

84

$&.12:/('*0(176 The authors acknowledge the Belgian National Fund for Scientific Research F.R.S.-FNRS and the Belgian Federal Science Policy Office (Contract P6/24) for their support.

5()(5(1&(6 1. R. Boyer, G. Welsch and E.W. Collings, Materials Properties Handbook: Titanium Alloys, Materials Park, OH: ASM International, 1994. 2. G. Lütjering and J.C. Williams, Titanium, Berlin Heidelberg: Springer, 2007. 3. A.S. Khan, R. Kazmi and B. Farrokh, Int. J. Plast. , 931-950 (2007). 4. A. Majorell, S. Srivatsa and R.C. Picu, Mater. Sci. Eng. A , 297-305, 2002. 5. R.C. Picu and A. Majorell, Mater. Sci. Eng. A , 306-316, 2002. 6. T. Kuwabara, C. Katami, M. Kikuchi, T. Shindo and T. Ohwue, Proceedings of the 7th International Conference on Numerical Methods in Industrial Forming Processes, Toyohashi, Japan, 2001, pp. 781-787. 7. O. Cazacu, B. Plunkett and F. Barlat, Int. J. Plast. , 1171-1194 (2006). 8. B. Plunkett, O. Cazacu and F. Barlat, Int. J. Plasticity , 847-866 (2008). 9. G. Gilles, W. Hammami, V. Libertiaux, O. Cazacu, J.H. Yoon, T. Kuwabara, A.M. Habraken and L. Duchêne, Int. J. Solids Struct. , 1277-1289 (2011). 10. P. Florès, V. Tuninetti, G. Gilles, P. Gonry, L. Duchêne and A.M. Habraken, J. Mater. Process. Tech. , 1772-1779 (2010). 11. P. Florès, “Development of Experimental Equipment and Identification Procedures for Sheet Metal Constitutive Laws”, Ph.D. Thesis, University of Liege, 2005. 12. R. Hill, J. Mech. Phys. Solids , 23-33 (1987). 13. E. Aarts and J. Korst, Simulated annealing and Boltzmann machines: a stochastic approach to combinatorial optimization and neural computing, New York: John Wiley & Sons, Inc., 1989.

85

Anisotropic yield function capable of predicting eight ears J.H. Yoona, O. Cazacua* a

Department of Mechanical and Aerospace Engineering, University of Florida, REEF, Shalimar, FL 32579, USA.

Abstract Deep drawing of a cylindrical cup from a rolled sheet is one of the typical forming operations where the effect of this anisotropy is most evident. Indeed, it is well documented in the literature that the number of ears and the shape of the earing pattern correlate with the r-values profile. For the strongly textured aluminum alloy AA 5042 (Numisheet Benchmark 2011), the experimental r-value distribution has two minima between the rolling and transverse direction data provided for this show that the r-value along the transverse direction (TD) is five times larger than the value corresponding to the rolling direction. Therefore, it is expected that there are more that the earing profile has more than four ears. The main objective of this paper is to assess whether a new form of CPB06ex2 yield function (Plunkett et al. (2008)) tailored for metals with no tension-compression asymmetry is capable of predicting more than four ears for this material.

Keywords: Anisotropic yield function; AA5042; Orthotropy; Deep drawing.

INTRODUCTION The main objective of this paper is to assess the ability of the form of the CPB06ex2 yield function for metals with cubic crystal structure to predict more than four ears in single-stage cup drawing for a strongly anisotropic aluminum. The paper is organized as follows. We begin by presenting the anisotropic yield function that will be used in the FE simulations along with the identification procedure used for determination of the anisotropy coefficients based on rvalues and tensile flow stresses. Comparison between simulation results and data are presented for two sets of anisotropy coefficients and different friction conditions. Irrespective of the sets of parameters used and friction conditions, the CPB06ex2 yield function predicts the eight ears.

ANISOTROPIC YIELD FUNCTION Cazacu et al. (2006) proposed a pressure-insensitive yield criterion that accounts for material anisotropy as well as yielding asymmetry between tension and compression associated with deformation twinning or non-Schmidt effects at single crystal level. This criterion can be applied to represent plastic behavior of various materials with FCC, BCC, and HCP structures and is defined as G ( 1 , 2 ,3 ,k , a) =  61  k 61   6 2  k 6 2   63  k 63 , a

a

a

(1)

To improve representation of the anisotropy, linear transformations are incorporated into the CPB06 criterion. The orthotropic yield criterion with two linear transformations, CPB06ex2 (Plunkett et al., 2008), is of the form:

F  6, 6 c, k,k c ,a = G ( 1 , 2 ,3 ,k , a) + G ( 1c ,c2 ,3c ,k c,a) =  61  k 61   6 2  k 6 2   63  k 63   61c  k c61c   6c2  k c6c2   6c3  k c6c3 a

a

a

a

a

a

(2)

where a is the degree of homogeneity, and k and k are material parameters describing strength differential effects; 61 , 6 2 , 6 3 and 61c , 6c2 , 6c3 are the principal values of the tensors  and c , respectively. These tensors are defined as The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 86-91 (2011); doi: 10.1063/1.3623596 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

86

c Cc : s

 C:s ,

(3)

where s indicates the stress deviator while C and C’ are orthotropic fourth-order symmetric tensors. With respect to the axes of orthotropy, anisotropic coefficients are represented by

C

ªC11 «C « 12 «C13 « « 0 « 0 « ¬« 0

C12

C13

0

0

C 22

C23

0

0

C23 0

C33 0

0

0 0

0

0

C 44 0

0

0

0

C55 0

c ªC11 «C c « 12 «C13 c « « 0 « 0 « ¬« 0

0 º 0 »» 0 » », Cc 0 » 0 » » C66 ¼»

c C12

c C13

0

0

c C 22 c C 23

c C23 c C33

0

0 0

0

0

0 c C 44

0

0

0

0 c C55

0

0

0

0

0 º 0 »» 0 » ». 0 » 0 » » c ¼» C66

(4)

Note that when C = C and k = kƍ, CPB06ex2 yield criterion reduces to the CPB06 yield criterion (see Eq. (2)). For materials with cubic crystal structure for which the yield in tension is equal to the yield in compression, the effective stress according to CPB06ex2 becomes:

V eff



1

a

+  2 +  3 + 1c + c2 + c3 a

a

a

a



a 1/a

(5)

c , are related to the in-plane properties of the sheet. Without The parameters Cij , Cijc , with i, j = 1…3 and C66 , C66

c = 1. The remaining Cij , Cijc with i, j = 1…3 (10 coefficients) and the loss in generality, we can set C11 = 1 and C11 c can be determined based on the experimental normalized flow stress values shear coefficients C66 and C66

VT

V T / V 0 and Lankford coefficients rT (T represents the angle between the uniaxial loading direction)

experimental biaxial tension and experimental rb. To simplify the equations, we introduce the following notations:

1  2 C11 - C12 - C13 , 3 1  2 =  2C12 -C22 -C23 , 3

1 =

3 =

1 =

1  -C11 + 2C12 - C13 3

(6)

1  -C12 +2C22 -C23 3 1  3 =  -C13 +2C23 -C33 3

2 =

1  2C13 - C23 - C33 , 3

From Eq. (3), the only non-zero components of the transformed stress tenor 6are

 xx = 1V xx + 1V yy ,  yy =  2V xx +  2V yy ,  zz =  3V xx +  3V yy and  xy = C 66  xy .

(7)

Similarly, the components of the transformed tensor c are expressed in terms of the Cauchy stress components by relations similar to (7) using notations similar to (6) for defining )1c , )c2 , )c3 in terms of the anisotropy coefficients Cijc . The principal values of the transformed tensors  and c are given by

1§ 1 = ¨  xx + yy + 2©



xx

2 · - yy + 4  2xy ¸ , ¹

2 1§  2 = ¨  xx + yy    xx - yy + 4  2xy 2© and 3 =  zz c3 = czz .

·, ¸ ¹

1§ 1c = ¨ cxx +cyy + 2©

 c

2 · -cyy + 4 cxy2 ¸ ¹

1§ c2 = ¨ cxx +cyy  2©

 c

2 · -cyy + 4 cxy2 ¸ ¹

xx

xx

(8)

Let denote by VT the yield stress in a direction at angle T with the rolling direction x. According to the criterion, for cubic metals with no tension-compression asymmetry (see (5)):

87

1



­°  a +  2 a + 3 a + 1c a + c2 a + c3 a ½° a 0 ® 1 a a a a a a ¾ ¯° 1 +  2 + 3 + 1c + c2 + c3 ¿°

(9)

where V0 is the tensile yield stress in the rolling direction (i.e. for  = 0$ ) and

§  + cos 2 +  1 + 2 sin 2 · 1 ¨ 1 2 ¸ 1 = 2 ¨¨ +    - cos 2 +   - sin 2 2 + 4 C 2sin 2 cos 2 ¸¸ 1 2 1 2 66 © ¹ 2 2 §  + cos  +  1 + 2 sin  · 1  1 2 ¸ 2 = ¨ 2 ¨¨     - cos 2 +   - sin 2 2 + 4 C 2sin 2 cos 2 ¸¸ 1 2 1 2 66 © ¹ 2 2  3 =  3cos + 3sin  ,

(10)

and )1 , ) 2 , ) 3 and <1 , < 2 , < 3 are given by (6). Relations similar to (10) express A1c , Ac2 , Ac3 in terms of the angle T and the anisotropy coefficients Cijc , respectively. For biaxial stress conditions, yielding occurs when

 xy = 0 and  xx =  yy =  b , where 1

a a a a a a ­° ½° a 1 +  2 + 3 + 1c + c2 + c3  b = 0 ® a a a a a a ¾ °¯ 1  <1 +  2  <2 + 3  <3 + 1c  <1c + c2  <2c + c3  <3c °¿

(11)

Furthermore, we assume that the plastic potential coincides with the yield function. Let denote by rT the strain ratio (width to thickness strain ratios) under uniaxial tensile loading in a direction at angle  with the rolling direction in the plane (x-y). Hence:

sin 2 r = -

w eff w w -sin(2) eff +cos 2 eff w xx w xy w yy w eff w eff + w xx w yy

(12)

where V eff is given by (5). In particular, according to the criterion, the strain ratio in the rolling direction is

r0 = where

1 +1c  2 +c2

a-1 1 = 1a-11 +(-1)a ( a-1 2  2 + 3  3 )

(13) a-1 a a  2 = 1a-1 (1 +1 ) + (-1) a ( a-1 2  2 + 3  3 +  2 +  3 )

with similar expressions for 1c and c2 . The strain ratio in the transverse direction (  = 90$ ) is given by:

r 90 = -

 3 +c3  4 +c4

(14)

with a a-1 a-1  3 = - a-1 2  2 + (-1) (1 1 +  3  3 )

a a-1 a-1 a a  4 = - a-1 2 ( 2 + 2 ) + (-1) (1 1 +  3  3 + 1 +  3 )

with similar expressions for c3 and c4 . The in-plane shear coefficients may be determined using theoretical expression of the yield stress in pure shear in the sheet plane.

88

1

 xy = 0 2

­° 1 a + 2 a + 3 a + 1c a + c2 a + c3 a ½° a ® ¾ a a C66 + Cc66 ¯° ¿°

(15)

c of the new yield In conclusion, using the above equations, the 10 coefficients Cij , Cijc , with i, j = 1…3 and C66 , C66 criterion (5) can be determined by minimizing an error function of the form th § VT i F = ¦Ki ¨ 1 data ¨ i VT i © n

2

· ¸ + ¸ ¹

m

§ ¨ ©

¦ J j ¨1j

2

r thj · § thb ·  w ¨1- data ¸ data ¸ ¸ rj ¹ © b ¹

2

(16)

In the above equation, “n” (respectively, m) represents the number of experimental directional flow stresses (respectively, r-ratios) available, V b denotes the biaxial flow stress; the superscript indicates whether the corresponding value is experimental or calculated with the criterion (Eq.5), while Ki , J j , and w are weight factors. For 3-D stress conditions, four additional parameters C44 ,C55 ,C44 c ,C55 c which are associated with out-of-plane anisotropic properties are involved in the expression of the anisotropic criterion (5). Those coefficients may be determined based on out-of-plane data such as yield stress for simple shear in the y-z and z-x planes and uniaxial tension at 45º between y and z, and between z and x. These properties can not easily be measured on plate material. However these data points can be determined by the use of crystal plasticity calculations. If texture data are not available, an alternative is to approximate the remaining parameters by setting them to their isotropic values. Experimental Characterization of AA5042

The experimental data provided for the benchmark on AA5042 include the stress-strain curves of AA5042 calibrated with a Voce type law, the experimental r-value distribution, and the biaxial r-value, rb = 0.991.

V T H p AT  BT exp CT H p

(17)

In the identification routine are used the flow stresses and r-values in the 7 orientations and the biaxial flow stress corresponding to the same level of accumulated plastic work Wp of 35.52 MPa calculated by integrating the biaxial flow curve with respect to the effective plastic strain from 0 to 0.1. The homogeneity parameter is set to: a 10 . Fig. 1 shows comparisons between experimental and theoretical variations of the normalized yield stresses (yield stresses normalized by the biaxial yield stress) and r-values using the yield function given by Eq. (5). Two sets of anisotropic coefficients (Set 1, Set 2) were determined by minimizing the error function of Eq. (16) (see Table 2). For Set 1 equal weights were given to all experimental data, while for Set 2, the r-value at 15° from the rolling direction was given double the weight. Note that the theoretical predictions are in excellent agreement with the experimental results for both sets of coefficients. 1.5 Experiment CPB06ex2 (Set 1) CPB06ex2 (Set 2)

0.94

1.2

0.93

r-value

Yield stress ratio

0.95

0.9

0.92

0.6

0.91

0.3

0.90

0

15

30

45

60

75

90

Experiment CPB06ex2 (Set 1) CPB06ex2 (Set 2)

0

15

30

45

60

75

Tension axis angle from rolling direction (Deg.)

Tension axis angle from rolling direction (Deg.)

(a)

(b)

89

90

FIGURE 1. Experimental results in comparison with theoretical predictions using CPB06ex2 for AA5042: (a) normalized flow stress at plastic work of 35.52 MPa; (b) r-values directionality.

TABLE 1. Anisotropy coefficients of CPB06ex2 yield function for AA5042 at the plastic work of 35.52 MPa Set

1

2

Anisotropy coefficients of the yield function, Eq.(5) at Wp = 35.52 MPa (a =10) C12

C13

C22

C23

C33

C66

0.0787

-0.6757

0.7452

0.8157

0.1041

-1.2274

C12'

C13'

C22'

C23'

C33'

C66'

-0.3142

0.9887

0.6077

0.617

-0.2642

1.3858

C12

C13

C22

C23

C33

C66

0.00011

-0.5999

0.9926

0.8037

0.0867

1.2343

C12'

C13'

C22'

C23'

C33'

C66'

0.0684

-0.0790

1.3981

0.1279

-1.1592

-1.3359

TABLE 2. Tool dimensions for the cup-drawing test of AA5042 sheet. Blank radius (Rb) Die opening radius (Rd)

38.062 mm 23.368 mm

Punch radius (Rp)

22.86 mm

Die-profile radius (rd)

1.905 mm

Punch-profile radius (rp)

2.229 mm

Initial blank thickness (t0)

0.208 mm

Finite element (FE) simulation of cup-drawing test for AA5042 was performed with ABAQUS/Explicit adopting the CPB06ex2 using a user material subroutine (UMAT). Table 2 shows the tool dimensions used in the cupdrawing test. In the FE analysis, the tools were considered as rigid-bodies and 15630 solid elements (C3D8R) were used to mesh the initial blank. Sliding contact conditions were assumed between the tools and the sheet (ABAQUS built-in surface-to-surface formulation, with hard contact). The applied blank holding force was of 8.9 kN and the Coulomb friction coefficient between the blank and the tools was of 0.05. Fig. 2 shows the earing profile for the completely drawn cup using CPB06ex2 yield criterion with both sets of anisotropy coefficients. Note that the earing profile appears to be a mirror image of the experimental r-value variation with (T+90°). Both analysis results predict eight ears and a similar earing profile. However, there is a big difference in cup height around 75° from the rolling direction since the theoretical predictions of the r-value at 15° varies largely between the two sets of coefficients. To investigate the effect of friction on the earing profile, simulations were also performed for frictionless conditions (P=0). As the friction coefficients decreases, cup height at 75° from the rolling direction decreases largely because the tangential friction force during the cup-drawing test interrupts material flow. The earing profile at the final depends on the frictional conditions (see Fig.3).

90

21.0 Set 1.

Cup height(mm)

20.5 20.0 19.5

Set 2.

19.0 18.5

0

15

30

45

60

75

90

Angle from rolling direction(Deg.)

FIGURE 2. Earing profiles of fully drawn cup for AA5042 corresponding to yield criterion (5) for two different sets of anisotropy coefficients. 21.0

Cup height(mm)

20.5

P=0.05

20.0 19.5 P=0.0

19.0 18.5

0

15

30

45

60

75

90

Angle from rolling direction(Deg.)

FIGURE 3. Frictional effect on earing profiles for AA5042 corresponding to yield criterion (5) for anisotropy coefficients (Set 1).

CONCLUSION In this paper, it was assessed the ability of a recently proposed yield function CPB06ex2 (Plunkett et al. (2008) to describe the anisotropy of metals with no strength-differential effects, and to predict more than four ears. To this end, CPB06ex2 yield criterion was applied to a strongly anisotropic aluminum alloy, AA5042. It is worth pointing out that the anisotropic criterion in conjunction with isotropic hardening predicts 8 ears. The earing profile at the final stage is sensitive to the friction conditions.

REFERENCES 1. B. Plunkett, O. Cazacu, and F. Barlat. Int. J. Plasticity 24, 847-866 (2008). 2. O. Cazacu, B. Plunkett, F. Barlat, Int. J. Plasticity 22, 1171–1194 (2006).

91

Forming Limit Diagrams of Zircaloy-4 and Zirlo Sheets for Stamping Process of Spacer Grids of Nuclear Fuel Rod Yunmi Seoa, Hong Chul Hyuna, Hyungyil Leea and Naksoo Kima a

Sogang University, Department of Mechanical Engineering, Seoul 121-742, Korea

Abstracts We investigated the theoretical forming limit models for Zircaloy-4 and Zirlo used for spacer grid of nuclear fuel rods. Tensile tests were performed to obtain stress-strain curves and anisotropic coefficients, such as r-values. The experimental forming limit diagrams (FLD) for two materials were obtained by dome stretching tests following the specification of NUMISHEET 96. Theoretical FLD depends on forming limit model and yield criterion. To obtain the right hand side of FLD, we applied the forming limit models (Swift’s diffuse necking, Marciniak-Kuczynski damage defect, Storen-Rice’s vertex theory) to Zircaloy-4 and Zirlo sheets. Hill’s local necking theory was adopted for the left side of FLD. To consider the anisotropy of sheets, the yield criteria of Hill (1948) and Hosford (1979) were applied. Comparing the predicted curves with the experimental data, we found that the FLD for Zircaloy-4 can be described by the Swift model with the Hill 48 yield criterion, while the FLD for Zirlo can be explained by the Storen-Rice model and the Hosford yield criterion (a = 8). Keywords: Zircaloy-4 Sheet, Zirlo Sheet, Forming Limit Diagram, Yield Criterion PACS: 81.70.-q; 62.20.F- ; 81.20.Hy; 62.20.-x

INTRODUCTION Formability is the capacity of a material to deform plastically without necking and tearing. The forming limit diagram (FLD) developed by Keeler [1] and Goodwin [2] is commonly used as a judgment criterion of formability. The vertical axis in FLD represents the major stain and the horizontal axis the minor strain. If the major / minor strain components of a deformed sheet are below the forming limit curve (FLC) in FLD, the sheet will be safe from necking and tearing. The state of strain depends on the loading-path so that the strain-based FLC is valid only for the case of proportional loading [3]. For this reason, in multi-step stamping process, strains far below the conventional FLC are sometimes found to neck while in other areas strains far above the curve are safe from necking. Nevertheless, the convenience of measuring strain rather than stress makes the strain-based FLC be used in judgment of the formability. Various models for theoretical forming limit based on material properties were proposed; diffuse necking model by Swift [4], localized necking mechanism by Hill [5], initial imperfection model by Marciniak and Kuczynski [6], vertex theory by Storen and Rice [7]. Swift proposed that in the right hand side of FLD, or the first quarter of the stain plane, instability will occur when the load reaches a maximum. Hill introduced a general criterion for localized necking in thin sheets under plane stress states. His analysis predicts the localized necking in only the negative minor strain region. Marciniak and Kuczynski suggested an initial imperfection model (so-called M-K model) for localized necking to occur in biaxial stretching. However, the M-K analysis proves to be very sensitive to the choice of shape and size of the initial imperfection. In addition, it does not describe the limit strain of equi-biaxial mode. For these reasons, Storen and Rice explained localized necking on the RHS of FLD using vertex theory. Factors influencing the forming limit were also investigated: strain-rate sensitivity, anisotropy, and yield criteria. Spacer grids for supporting the nuclear fuel rod are formed through stamping processes. Materials recently used for spacer grids such as Zircaloy-4 and Zirlo have not been sufficiently investigated in their formability. More detailed studies are necessary for the estimation of formability. In this work, we performed tensile tests to obtain stress-strain curves and anisotropic coefficients of Zircaloy-4 and Zirlo. Then we constructed the theoretical FLDs of two materials using several forming limit models (Swift's diffuse necking, Hill's localized necking, M-K theory, and

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 92-98 (2011); doi: 10.1063/1.3623597 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

92

S-R vertex theory) with yield criteria. Comparing the predicted forming limit curves with experimental data, we tried to investigate the validity of theoretical forming limit models.

THEORETICAL ANALYSIS General metallic crystal has orientations of metal crystal lattice; and hence materials behavior depends on the orientation. The anisotropic ductile metals should be described with the anisotropy yield criteria. The forming limit diagram is largely affected by yield condition. To obtain the forming limit diagram, we used the anisotropy yield criteria (Hill’ 48 [8] and Hosford’ 79 [9]) considering crystal structure. The relationship between the forming limit diagram and yield criteria was further studied.

Hill’48 Yield Criterion Hill formulated a yield criterion in 1948; the theory assumed that the tensile and compressive yield strengths are equal for given direction. If the plane stress condition were assumed, Hill’s 48 yield condition is given as

f V

F

F  V 2  V 3  G V 3  V 1  H  V 1  V 2 2

2

1§ 1 1 1 · ¨ ¸; G   2 ¨ V y 2 2 V y 32 V y12 ¸ © ¹

2

1

(1)

1§ 1 1 1 · ¨ ¸; H   2 ¨ V y 32 V y12 V y 2 2 ¸ © ¹

1§ 1 1 1 · ¨ ¸   2 ¨ V y12 V y 22 V y 32 ¸ © ¹

where Vyi is the yield strength and subscript i denotes the coordinate direction. F, G and H are constants related to the anisotropy of material. Hill’48 yield condition describes the behavior of steels well, but it does not explain the behavior of metals having anisotropy coefficient less than unity. Further it assumes the isotropic hardening which is generally not useful in situations where components are subject to cyclic loading. It does not account for the Bauschinger effect that is observed in most metals. Although Hill’s yield criterion cannot describe the Bauschinger effect, it is widely used due to its simplicity.

Hosford Yield Criterion Under the plane stress state, Hosford suggested a yield condition in the form of Eq. (2)

f V

F V 2  V 3 a  G V 3  V 1 a  H V1  V 2

a

1

(2)

where a is the parameter which characterizes the shape of the yield surface. F, G and H are coefficients related to the anisotropy of sheet material. The parameter a can be determined from the crystallographic theory. Considering the crystallographic structure, plastic deformation is generated by the slip. The slip occurs along the slip system having the greatest planar density. Metals with FCC or BCC or HCP crystal structures have a number of slip systems, 12 or 24 or 6, respectively. Metals with FCC or BCC crystal structures have a relatively large number of slip systems. These metals are quite ductile because extensive plastic deformation is normally possible along the various systems. Conversely, formability of HCP metals, having few active slip systems, is relatively poor compared to other metal structures. Mostly a = 6 is used for the BCC metals, a = 8 for the FCC metals. If a = 2, the yield condition is equal to Hill 48 yield condition. The main advantage of this yield criterion is that suitable choice of a ensures a good fitting of experimental data.

TABLE 1. Material properties for predicting the FLD E (GPa) K (MPa) n

r0

r45

r90

r

Zircaloy-4

110

1030

0.169

1.528

2.506

2.520

2.265

Zirlo

84

830

0.163

2.247

2.223

1.846

2.135

93

EXPERIMENTAL WORKS Experiments for Material Properties The tensile tests were carried out to obtain the material properties and the anisotropy coefficients. The materials used in this study were Zircaloy-4 and Zirlo sheets. Specimens with a gauge length of 50 mm and a thickness of 0.5 mm following KS B5 specifications were used for the tensile tests (Fig. 1). The tests were conducted in the INSTRON 5882, and a constant cross-head speed of 4.8 mm/min was employed in all cases. An advanced video extensometer is used to measure the displacement. Table 1 shows the material properties; E is the Young’s modulus, n is the strain hardening exponent and K is the strength coefficient of materials. We first derive the true stress-strain curve up to necking point with the load-displacement curve from tensile test. To evaluate the anisotropy of sheets, r-value test was performed. The samples were prepared along three different directions namely 0o, 45o and 90o to the rolling direction (T) of the sheets. Each specimen was imprinted grid with 2mm square using a silk-screen process. Values of 50 and 20 mm were used for longitudinal and transverse gage length, respectively. The strain of thickness direction is calculated according to the plastic incompressibility assumption

Hl  H w  Ht

(3)

0

where Hl, Hw and Ht refer to the strain in the length, width, and thickness directions, respectively. The anisotropy coefficients r is defined as rT { H w / H t

(4)

Anisotropy coefficients were measured before necking (elongation = 15%) using the deformation amount of grid. In this work, a normal anisotropy is assumed. The normal anisotropy is calculated by the equation:

r

r0  2r45  r90 4

(5)

FIGURE 1. Plat specimen for tensile test

FIGURE 2. Specimens after dome stretching test

Dome Stretching Test In order to estimate the formability of the sheets, the dome stretching test was performed. In the test, a series of sheet metal specimens with different widths were stretched by hemispherical punch, referred to the specification of NUMISHEET 96, until fracture occurs. The samples of varying widths are used, all of which are 200mm long, and with each width of 25, 50, 75, 100, 125, 150, 175 and 200mm. All of the tests were first done without any lubricant. Then lubricants were applied to further tests with width of 135, 150 and 200mm. Figure 2 shows deformed shape of samples after the tests. The punch speed was 1 mm / min and applied binder force was 980 kN. To measure the deformations of specimens, circle grids of 0.9mm diameter were printed. After dome stretching tests, the geometry of deformed samples were scanned using ATOS, and the strain distributions were measured using ARGUS from ISO 12004 (Figs. 3-4). The forming limit diagrams obtained in this way are plotted in Fig. 5. The smaller the width of specimens is, the lower the minor strain is. With better lubrication, the failure occurred near the top of the dome.

94

H1

FIGURE 3. 3D White light scanning systems (ATOS)

0.713 0.623 0.445 0.267 0.178 0.000

FIGURE 4. Major strain distributions of Zircaloy-4 specimens were analyzed by ARGUS after dome stretching test

RESULTS AND DISCUSSIONS In the right hand sides of forming limit diagram, or the first quarter of the strain plane, the major strain H1 is always higher than minor strain H2. The wrinkling occurs in the left region of the line for uniaxial tension corresponding to U= - (r +1) / r. We compared forming limit curves derived from forming limit models with experimental data. The power law stress-strain relation is used: V = KH n.

Swift Model When the instability occurs in biaxial stretching of sheet metal, the local area is decreased evenly, which is called the diffuse necking. Physically the local necking cannot form under stretching conditions when the strain ratio is positive. For this reason, we obtained the forming limit curves using Swift diffuse necking model [Eq. (6)]. The limit strain depends on the yield criteria. 2

H1

§ wf · § wf ·§ wf · ¸ V2 ¨ ¸¨ ¸ V w © 1¹ © wV 2 ¹© wV 1 ¹ n ; H 2 2 2 § wf · § wf · V1 ¨ V  ¸ ¸ 2¨ © wV1 ¹ © wV 2 ¹

V1 ¨

2

§ wf · § wf · § wf · ¸  V1 ¨ ¸¨ ¸ V w © 1¹ © wV 1 ¹ © wV 2 ¹ n 2 2 § wf · § wf · V1 ¨ V  ¸ ¸ 2¨ © wV1 ¹ © wV 2 ¹

V2 ¨

(6)

Here f is the yield surface function, and V1 and V2 stand for major stress and minor stress, respectively. With Hosford criterion, the curve of a = 6 is similar to that of a = 8. The forming limit curves obtained for yield criteria are drawn from Fig. 5. 0.8

0.8

Zircaloy

Swift with Hill Swift with Hosford (a = 6) Storen-Rice (a = 6) Storen-Rice (a = 8) Experimental data

0.6

Major Strain

Major Strain

0.6

0.4

0.2

0.4

0.2 Hill model

Hill model

0.0 -0.6

Swift with Hill Swift with Hosford (a = 6) Storen-Rice (a = 6) Storen-Rice (a = 8) Experimental data

Zirlo

-0.4

-0.2

0.0

0.2

0.4

0.0 -0.6

0.6

Minor Strain

-0.4

-0.2

0.0

0.2

Minor Strain

(a) (b) FIGURE 5. Comparison of FLCs with experimental data for (a) Zircaloy and (b) Zirlo

95

0.4

0.6

Hill Model In sheet, the local necking occurs after diffuse necking. Diffuse necking of sheet involves contraction in both the lateral and width directions, while local necking thins the sheet without further contraction of width. Swift’s and Hill’s theories are used to calculate the forming limit strains on the left and the right side of FLD, respectively. Hill suggests the following expressions for the limit strain sets with negative strain ratio: wf wV 1

H1

wf wf  wV 1 wV 2

n;

H2

wf wV 2

wf wf  wV 1 wV 2

n

(7)

It is obvious that

H1  H 2

n

(8)

According to Eq. (8), the FLD based on Hill’s model is not dependent on the yield criteria, but dependent on the strain hardening coefficient.

M-K Model The one known as M–K model assumes the existence of an initial imperfection. It is composed of two regions; a is a homogeneous region, and b an imperfection region. The initial imperfection factor is defined as the thickness ratio of two regions. When the deformation is concentrated in the imperfection region which is in the biaxial mode, the imperfection region is under the plane strain condition. We assume the homogeneous region has proportional deformation path. We can obtain the limit strains when the constitutive equation including strain-rate sensitivity is substituted into force equilibrium equation. The constitutive equation is given by the formula.

Ve

(9)

K H nH m

The groove force balance must be maintained. Therefore,

V a1ta exp  H a 3 V b1tb exp  H b3 1

Ia

 H e , a  d H e, a

m

n

§ Ea · m ¨ ¸ dH a2 © Ua ¹

(10) m

n§E · exp  H b3  H a 3  H e,b  d H e,b ¨ b ¸ d H bm2 Ib © Ub ¹

f

0.8

0.8 Zirlo, f = 0.99, m = 0

Experimental data

Zircaloy, f = 0.99, m =0

0.6

0.4 Hosford 79 (a = 6) Hill' 48 (a = 2) a=8

0.2

0.0 -0.6

Experimental data

0.6

Major strain

Major strain

(11)

-0.4

-0.2

0.0

0.2

0.4

0.4 Hosford 79 (a = 6) Hill' 48 (a = 2)

0.2

0.0 -0.6

0.6

Hosford 79 (a = 8)

-0.4

-0.2

0.0

0.2

Minor Strain

Minor Strain

(a) (b) FIGURE 6. Comparison of FLCs with experimental data for (a) Zircaloy and (b) Zirlo

96

0.4

0.6

where subscript a and b is the homogenious and imperfection region, I (ೆVe / V1) and E (ೆHe / H1) represent ratio of effective stress to major stress and ratio of effective strain to major strain, respectively. The procedure is to (1) assume the valuses of initial strain increment (dHa1 ) and strain ratio (Ua) inside groove, (2) compute He,a , dHa2, Ia and Ea. (3) find dHb1 by calculating Ub, Db, Ib, Eb and dHe, b using dHb2 = dHa2 and (4) repeat the calculation until dHb1 satisfies Eq. (11). Generally the necking has occurred when the increment of the equivalent plastic strain in groove region exceeds ten times the increment in homogenious region. When the criterion is satisfied, the strain in the homogenious region is used. In the M-K model the limit strains are affected by the initial imperfection and strainrate sensitivity. Using Newton-Raphson method, the assumed increment inside the groove (dHb1) was made equal to 0.001 and a convergence = 10-6 was used throughout this work. Figure 6 shows forming limit curves with yield criteria, when f = 0.99 and m = 0 were used. The variables in Eq. (11) are varied in yield criterion.

Storen-Rice Model The vertex theory was first proposed by Storen and Rice. They used a bifurcation analysis to describe the behavior of material that arises from a vertex on the yield surface. According to this theory with Hosford’79 yield criterion, the limit strains on the negative and positive strain ratio region are evaluated from the equation. N1 ªD a  2  AN 2 º f a D N ¬ ¼  3 , H 2  0; H1 AD1 AB

H 2 ! 0; H1

N1 ªD a  2  N 2 º f a D N ¬ ¼  3 D1 BC

(12)

If we assume the strain ratio and major strain, we can obtain minor strain. The variables of Eq. (13) are given by

H2 ; f D { D sgn V V e ; A 1  U ; B 1  DU ; C H1

2E  1 ; U 2E

D

a 2

N1

1  r0 ;

D1

BC ªD a  2  r90  r0D a  2 1  D ¬

N2



r90 1  D

(13)

Cn  1

; N3



a 1

a 2 º

¼

where a represents different yield criteria; a = 6 for the BCC and a = 8 for the FCC material. Figure 5 shows the forming limit curve calculated using the above formula for different values of a. Mainly the state of sheet is biaxial mode during the stamping process. In Figure 7, we magnify this region to show the region of positive minor strain in detail. Comparing the predicted curves with the experimental data, we found that the FLD for Zircaloy-4 can be described by the Swift model with the Hill’s criterion, while the FLD for Zirlo can be explained by the Storen-Rice model and the Hosford criterion (a = 8).

Zirlo

Zircaloy Swift with Hill 48 Swift with Hosford S-R with Hosford (a = 6) S-R with Hosford (a = 8)

0.1

0.0 -0.1

0.2

Major Strain

Major Strain

0.2

Experimental data Hill model

0.0

0.1

0.2

Swift with Hill 48 S-R with Hosford (a = 6) S-R with Hosford (a = 8)

0.1

0.0 -0.1

0.3

Minor Strain

Swift with Hosford

Experimental data Hill model

0.0

0.1

0.2

Minor Strain

(a) (b) FIGURE 7. Magnification of the H2 > 0 region shown in Fig. 8: FLCs for (a) Zircaloy-4 and (b) Zirlo

97

0.3

CONCLUSIONS In this study we obtained theoretical forming limit diagram in order to predict the occurrence of local necking during the stamping process of fuel rod spacer grid. For Zircaloy-4 sheet, the predicted forming-limit strains based on Swift model with the Hill's criterion are in good agreement with the experimental results. For Zirlo sheet, the experimental FLC has a better agreement with the curve predicted by the Storen-Rice model and the Hosford yield criterion having the exponent a is 6. It would be beneficial to formability prediction in stamping process of fuel rod spacer grid.

ACKNOWLEDGMENTS This research was supported by National Nuclear R&D Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2010-0020425).

REFERENCES 1. S. P. Keeler and W. A. Backofen, ASM Transaction Quarterly 56, 25-48 (1963). 2. G. M. Goodwin, “Application of Strain Analysis to Sheet Metal Forming Problems in the Press Shop” in SAE Automotive Engineering Congress, Detroit, 1968, pp. 380-387. 3. T. B. Stoughton, International Journal of Mechanical Sciences 42, 1-27 (2000). 4. H. W. Swift, Journal of the Mechanics and Physics of Solids 1, 1-16 (1952). 5. R. Hill, Journal of the Mechanics and Physics of Solids 1, 19-30 (1952). 6. Z. Marciniak and K. Kuczynski, International Journal of Mechanical Sciences 9, 609-620 (1967). 7. S. Storen and J. R. Rice, Journal of the Mechanics and Physics of Solids 23, 421-441 (1975). 8. R. Hill, Proceedings of Royal Society of London A 193, 281-297 (1948). 9. W. F. Hosford, “On Yield Loci of Anisotropic Cubic Metals” in Seventh North American Metalworking Research Conference, Dearborn, Michigan, 1979, pp. 191-196.

98

Evaluation of Experimental Forming Limit Curves and Investigation of Strain Rate Sensitivity for the Start of Local Necking Wolfram Volka and Pavel Horab a

Prof. Dr.-Ing. Wolfram Volk, Institute of Metal Forming and Casting (utg), Technische Universität München, Walther-Meißner Str. 4, 85748 Garching near Munich, Germany

b

Prof. Dr.-Ing.Pavel Hora, Institute of Virtual Manufacturing, ETH Zürich, Tannenstr. 3, 8092 Zurich, Switzerland Abstract. The failure prediction in sheet metal forming is typically realized by evaluating the so called forming limit curves (FLC). Up to now, the FLC determination was performed with failed specimens of Nakajima or Marciniak test setups. Standard methods determine the failure by considering the occurrence of cracking and do not consider the possibility of time continuous recording of the Nakajima test. Consequently forming limit curves which have been evaluated in such way are often “laboratory dependent” and deviate for identical materials significantly. This contribution presents an algorithm for a fully automatic and time-dependent determination of the beginning plastic instability based on physical effects. The algorithm is based on the evaluation of the strain distribution based on the displacement field which is evaluated by optical measurement and treated as a mesh of a finite element calculation. The critical deformation states are then defined by 2D-consideration of the strain distribution and their time derivatives using a numerical evaluation procedure for detecting the beginning of the localization. The effectiveness of the proposed algorithm will be presented for different materials used for the Numisheet’08 Benchmark-1 with Nakajima test. Additionally the effect of strain rate sensitivity on the beginning instability of local necking is discussed. It can be shown that the strain rate sensitivity is from major importance and should not be neglected for the forming simulation of sheet metal materials. Keywords: Sheet Metal Forming, Forming Limit Curves, Strain Rate Sensitivity. PACS: 62.20 Mechanical Properties of Solids

INTRODUCTION The prediction and evaluation of failure is one of the main tasks in sheet metal forming simulation in industrial practice. The current standard is a so-called a posteriori interpretation. The results of the simulation are strains and stresses and the calculated strains are compared in a post processing step with theoretical or experimental reference curves, the so-called Forming Limit Curves (FLC). The curves of experimental methods are normally obtained with Nakajima or Marciniak [1] tests and have to be accomplished under laboratory-like conditions. The ISO standardization is given in [1]. The established evaluation method of the ISO standardization is the intersection line method. Therein the failed specimen is analysed and the strains of beginning instability are calculated with a mathematical approach using the 2nd derivative of the thinning distribution. This method is robust but sometimes inaccurate because the strains of start of necking are re-calculated. With the development of photogrammetric equipment it is nowadays possible to record the whole experiment and identify directly the start of necking, see Figure 1. This procedure is called the time continuous evaluation method. Feldmann & Schatz [2] proposed to analyse directly the strain gradient around the necking zone. Volk [3] suggested the evaluation of the thinning gradient around the necking zone over time. This method is mechanically motivated, and the identification of the start of necking was realized by a so-called frequency diagram. Based on this approach Eberle et al. [4] and Eberle [5] made further examinations and additional proposals. Volk & Hora [6] summarized the ideas of [3], [4] and [5] in combination with a simple and fully automated evaluation algorithm. Merklein et al. [7] proposed the application of the 2nd derivative of the strain rate in an evaluation field around the necking zone. All of these time continuous evaluation methods are based on experimental series, where gridded samples are being deformed and simultaneously The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 99-106 (2011); doi: 10.1063/1.3623598 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

99

the position of the intersection of the grid lines is being recorded with photogrammetric methods (figure 1). Typical boundary conditions are punch velocity of 1 mm/s and recording frequency of 10 pictures per second.

FIGURE 1. Principle sketch and experimental specimen of Nakajima test equipment with camera system

In the present paper the basic ideas of [6] are used for the identification of the start of necking. With this evaluation method some additional investigations are presented. It will be shown, that for a 5182-Aluminum alloy, it is possible to see the local instabilities of the Portevin-Le-Chatelier effect beside the identification of the start of necking. Finally some basic assessments of the effect of strain rate sensitivity on the beginning instability are presented.

EVALUATION OF STRAINS AND STRAINRATE DISTRIBUTION Input Data and Finite Element Approach The only required input data for the new method is a full set of global coordinates of the grid line crosspoints, which can be handled like global coordinates of a finite element mesh, see Figure 2 and [3].

FIGURE 2. Interpretation of measurement grid as finite element mesh

Following this idea all interesting physical quantities (strains, strain rate etc.) can be calculated by using well established methods of FE theory. If the test data is obtained with a regular grid pattern the crosspoints of the grid can be directly used as the vertices of the FE mesh. If crosspoints of the optical measurement are missing the coordinates of the voids can be calculated by suitable mathematical methods, e.g. [3] or [4]. If stochastic spray structures are applied for the optical measurement an additional preparation step to generate the grid pattern before starting the evaluation process is necessary.

100

Calculation of Strains For the determination of the forming limit strains a Total Lagrange description is used, where the initial mesh is taken as reference configuration Xk and the current picture as present configuration xk. The strains are evaluated using classical FEM 4-node membrane element theory, described for example by [8]. Following this the linear Cauchyian strain tensor can be described with:

1 2

 ij  ( H ij  ( H ij )T )

(1)

where

H ij 

ui N N  J 1 i ( x j  X j )  J 1 i u j x j  

(2)

Therein ui is the displacement vector, J-1 is the inverse of the Jacobian and Ni the typical isoparametric functions [8]. The plane principal strains result from the characteristic equation with:

det( ij   ij )  0 1 2 1  2  ( 11   22  ( 11   22 ) 2  4 122 2

 2  ( 11   22  ( 11   22 ) 2  4 122

(3)

Ultimately the Henckyian strain gauge is applied due to large strain components because of the Total Lagrange description:

1  ln(1   1 )  2  ln(1   2 )  3  (1   2 )

(4)

With these strain components, the points in the FLC are determined.

Calculation of Strain Rates For the determination of the strain rates and especially of the thinning rate, it is necessary to take the numerical derivative of the deformation gradient with respect to time into account. The determination of the deformation gradient tensor on the basis of the displacement gradient is performed as follows:

Fij  ( I  H ij )

(5)

Subsequently, with the derivative of the deformation gradient, the velocity gradient tensor is figured out and the deformation rate tensor can be computed

Lij 

Fij

1 1 1 T Fij  Fij Fij ; Dij  ( Lij  Lij ) t 2

101

(6)

Finally, the plane principal components of the deformation rate are evaluated by computing the eigenvalues D1 and D2 of the deformation rate tensor Dij. To calculate the necessary thinning rate, the deviatoric character of D is used:

 : D3  ( D1  D2 )

(8)

The thinning rate is the main physical quantity to identify the picture with beginning instability and calculating the strain values for the entries of the FLC. The numerical derivative of the deformation gradient will be very un-accurate, if no smoothing over the time steps is applied. For those reasons in [3] a quadratic interpolation over time with typically seven time history points is proposed for the deformation gradient Fij(t), see Figure 3.

FIGURE 3. Determination of

Fij (t ) by using quadratic least square interpolation with seven time history points

Detection of the Forming Limit States From a continuum mechanical point of view, the plastic instability is the local concentration of the remaining plastic deformation in small (shear-) bands and a fall-back of the other areas in the elastic range, Figure 4.

FIGURE 4. Development of the localized necking by the reduction of the plastic zone to the shear band(s).

Hence, two effects can be noticed, namely the strain inside this band increases on the one hand and on the other hand it has to remain roughly constant outside. This is displayed in Figure 4. It is remarkable in that figure that the characteristics of thinning over time seem to grow relatively continuously, which can cause problems for an automatic detection of beginning plastic instability. In [3] – [6] it has been shown that the thinning rate is a suitable physical quantity to identify the start of necking for the time continuous evaluation method. The motivation can be also seen in Figure 4. The concentration of the remaining plastic deformation in small shear bands will lead to high thinning rates where the thinning rates outside the shear bands will stay nearly constant. The algorithm to detect the start of necking proposed by Volk & Hora [6] makes use of this effect. In a first step a sufficient number of grid points are identified which are sure in the necking area. Then, only for these points the mean value of thinning rates are examined over time. In Figure 5 the typical behavior is shown.

102

FIGURE 5. Development of the localized necking by the reduction of the plastic zone to the shear band(s) [6].

At the beginning one can see the stable nearly homogeneous deformation and the localized necking as instable deformation at the end until the crack occurs. The stable and instable areas are fitted with two linear curves by using the least square method. The picture which is next to the intersection of these two straight lines will be defined as beginning instability, see Figure 5. This method was applied to the different materials of the Numisheet 2008 Benchmark problems [9] in two different laboratories. HC220YD, 0.8 mm

HC220YD, 1.6 mm

HC260LAD, 1.1 mm

FIGURE 6. Detection of the FLC. Comparison of different materials, laboratories and mesh sizes [6].

Visualization of Local Instabilities In addition to the evaluation of Forming Limit Curves the analysis of the thinning rates deliver interesting results. It is well known that some Aluminum alloys tends to transient local instabilities due to crystalline stick slip effects, the so-called Portevin-Le-Chatelier effect [10]. For the visualization of the effect a so-called two color plot is applied. In the evaluation field of 20 mm x 20 mm and a grid size of 1 mm the 25 elements with the highest thinning rate are marked with red rectangles. All other elements are displayed with blue diamonds. Therewith the concentration of plastic deformation in each time step can be visualized. The material is an AA5182 alloy from Numisheet 2008 Benchmark [9]. In Figure 7 the two color plots of a Nakajima 100 mm specimen with a punch velocity of 1 mm/s are displayed. The time step is set to zero for a punch height of 3 mm before the specimen failed. Interestingly one can see the jump of the highest thinning rates within the evaluation field for the different time steps. Finally, one of the transient localizations ends in the local necking and causes the failure of the Nakajima specimen.

103

FIGURE 7. Visualization of transient local instabilities (Portevin-Le-Chatelier effect) for AA5182

EFFECT OF STRAIN RATE SENSITIVITY ON START OF LOCAL NECKING The strain rate sensitivity of materials influences significantly the material behavior for strain values larger the tensile strain. The explanation is the non-homogeneous deformation which causes also different strain rates. If a material has a positive strain rate sensitivity like most of the steel materials this leads to a regularization of the postcritical behavior for strains larger than the tensile strain. This effect can be easily examined by an artificial boundary value problem with the FE method, see Figure 8 and Volk & Charvet [11].

FIGURE 8. Boundary value problem of ideal plain strain test

The problem is now, that for simulations without strain rate sensitivity the finite elements become numerically instable before reaching strain values of the FLC. It can be seen that without strain rate sensitivity the failure prediction is too conservative in comparison to the experimental FLC, and feasible geometry proposals would have been rejected, see Figure 9.

104

FIGURE 9. Simulation results without strain rate sensitivity

Even with a small strain rate sensitivity which can be lower than measured material parameters this effect nearly vanishes, see Figure 10.

FIGURE 10. Simulation results with strain rate sensitivity

It is recommended especially for dynamic explicit codes to always perform a simulation with small strain rate sensitivity if no accurate experimental values exist. In [11] it has been shown, that this strategy can be necessary for the simulation of complex geometries in automotive industry.

105

FIGURE 11. Simulation results without (left) and with (right) strain rate sensitivity for an industrial application .

Additional assessments on strain rate sensitivity in combination with temperature dependence on the yield curve and material failure can be found in Hora [12].

CONCLUSIONS AND OUTLOOK The prediction of material failure is an essential task in sheet metal forming simulation. In the present paper the discussion of new approaches for the evaluation of experimental FLC are discussed. It has been shown that the thinning rates are the interesting physical quantity for the assessment of beginning instabilities. Beside an accurate determination of experimental FLC this concept can be used to demonstrate the appearance of transient local instabilities for an Aluminium alloy AA5182. It is emphasized, that the proposed method of detecting the start of necking is not restricted to any loading or strain path. The problem of non-proportional loadings with non-linear strain paths can also be evaluated with the presented algorithm. Finally the importance of the strain rate on start of necking and beginning material instability has been proven with a simple numerical boundary value problem.

REFERENCES 1. ISO copyright office, “Metallic materials -Sheet and strip – Determination of forming limit curves – Part 2: Determination of forming limit curves in laboratory” in ISO/DIS 12004-2, 2006. 2. Feldmann, P. and Schatz M., “Effective Evaluation of FLC-Tests with the Optical In-Process Strain Analysis System AUTOGRID®” in Proceedings of FLC-Zurich 06, edited by P. Hora, Zurich, 2006. 3. Volk, W.,“New Experimental and Numerical Approach in the Evaluation of the FLD with the FE-Method” in Proceedings of FLC-Zurich 06, edited by P. Hora, Zurich, 2006. 4. B. Eberle, W. Volk and P. Hora, “Automatic approach in the evaluation of the experimental FLC with a full 2d approach based on a time depending method” in NUMISHEET’08 - Proceedings of the 7th Int. Conf. and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, Part A, pages 279-284, Switzerland, ISBN 978-3-909386-80-2. 5. B. Eberle, “Entwicklung eines robusten numerischen Verfahrens zur automatischen Bestimmung experimenteller FLCKurven“, Master thesis, Inst. of Virtual Manufacturing, ETH-Zurich, 2008. 6. Volk, W. and Hora P.,“New Algorithm for a Robust User-Independent Evaluation of Beginning Instability for the Experimental FLC Determination” in Int. J. Mater. Form., DOI 10.1007/s12289-010.1012-9, 2010. 7. M. Merklein, A. Kuppert, S. Mütze and A. Geffert, “New Time Dependent Method for Determination of Forming Limit Curves Applied to SZBS800” in Proc. of 50th Conference of IDDRG 2010 edited by R. Kolleck, Verlag der Technischen Universität Graz, p. 489-498, Graz 2010. 8. K.-J. Bathe, “Finite Element Procedures”, Springer Verlag (1995), ISBN 0133014584. 9. W. Volk, “Benchmark 1 – Virtual Forming Limit Curve”, NUMISHEET’08 - Benchmark Proceedings of the 7th Int. Conf. and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, Part B, pages 3-42, Zurich 2008, ISBN 9783-909386-80-2. 10. A. Portevin and H. Le Chatelier, “Heat treatment of aluminium-copper alloys” in Transactions of the American Society of Steel Treating, Vol. 5, p. 457-478, 1924. 11. W. Volk and P. Charvet, “Virtual Engineering and Planning Process in Sheet Metal Forming” in Proceedings of the 7th European LS-Dyna Conference, Salzburg 2009. 12. P. Hora, “Modellierung des Kaltverfestigungsverhaltens von metallischen Werkstoffen” in Proceedings of MEFORM Conference, Freiberg 2011.

106

1XPHULFDO9DOLGDWLRQRI$QDO\WLFDO%LD[LDO7UXH6WUHVV±7UXH 6WUDLQ&XUYHVIURPWKH%XOJH7HVW M. Vucetica, A. Bouguechaa, I. Peshekhodova, T. Götzea, T. Huininka, H. Friebeb, T. Möllerb and B.-A. Behrensa a

Institute of Metal Forming and Metal-Forming Machines (IFUM), Leibniz Universität Hannover An der Universität 2, 30823 Garbsen, Germany [email protected] b

GOM mbH Mittelweg 7-8, 38106 Braunschweig, Germany E-mail: [email protected], Web page: www.gom.com $EVWUDFW The present investigation deals with the validation of the experimentally obtained biaxial true stress - true strain curves of the HCT 780 C sheet material from the bulge test with the help of the FEA. Furthermore the investigation handles the consideration of the bending influence via the blank curvature evaluation with an optical measurement system Gom ARAMIS. .H\ZRUGVmaterial characterization, bulge test, yield curve, FEA, optical strain measurement. 3$&602.70.Dh, 81.70.-q

,1752'8&7,21 Experimentally determined yield curves from tensile tests are useful only for values within the range of uniform elongation, due to the reason of the uniaxial deformation behavior of tensile specimens under tension load. Beyond the point of the uniform elongation a common approach is to extrapolate the yield curve using mathematical models after e.g. Ludwik, Nadai, Swift or Gosh. Due to used mathematical algorithms flow curve values are extrapolated differently with the result of varying curve shapes. A more accurate and reliable extrapolation of the yield curve beyond the uniform elongation is archived by using the bulge test which enables an experimental determination of the biaxial true stress – true strain curve at much larger strain values (figure 1). This paper discusses and analyzes the latest improvements in the determination of the biaxial yield curve utilizing the bulge test. 

 ),*85( Flow curve from the tensile, its extrapolations Ludwik, Gosch and a biaxial stress strain curve of the bulge test for the HCT 780 C 1.5 mm

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 107-114 (2011); doi: 10.1063/1.3623599 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

107

(;3(5,0(17$/,19(67,*$7,212)7+(%8/*(7(67 The bulge test is carried out with a tool, which consists of a blank holder, die and a fluid chamber. A circular blank is completely clamped with the lock bead between the die and blank holder. A bulge is formed by pressing a fluid against the blank until final fracture occurs. When the pressure of the fluid p, the radius of curvature , the local thickness t and the local strains  from a bulge test are know, it is possible to determine the corresponding biaxial membrane stress using the equations given in figure 2 [3, 4, 5].

),*85( Test principle of the bulge test [5]

The bulge test at the Institute of Metal Forming and Metal Forming Machine (IFUM) is realized in a modular tool system in a hydraulic press. The hydraulic press can apply a blank holder force up to 2500 kN. The diameter d = 100 mm or d = 200 mm of the die can be modularly exchanged. A lock bead is realized to avoid any material flow during the test. Furthermore the tool is equipped with a measurement system in the middle of the fluid chamber which enables to measure the pressure of the fluid during the test. The pressure increase is realized with a hydraulic unit with a maximum pressure of 280 bars [5].

),*85( Modular tool system for the bulge test with a optical measurement system Gom ARAMIS

For the measurement of the curvature radius and the strain values the measuring system ARAMIS [6] was used. A typical sensor (two cameras and a light source) is shown in figure 3. To protect the measuring system from splashing oil at failure, a glass plate is in front of the lenses of the optical measuring system. The calibration of the system was done with inserted protection. A stochastic pattern was applied to the specimen's surface and synchronized stereo images of the pattern are recorded at different loading stages. Figure 4 shows the undeformed and the deformed situation for a specimen. In this case the image of the undeformed situation of the left camera is divided into a large amount of subsets (facets). The center of each facet is shown as a cross, five of the facets are shown as quadrangle. For each face the

108

corresponding area is calculated for the right camera also for all loading stages. Based on the well known geometry of the optical setup for each center of a facet (each cross in figure 4) a 3D point is calculated for all loading stages and for each point following result values are e.g. derived: 3D coordinates, displacements and speeds Surface strain tensor (strain x, -y, xy, major, minor, thinning) and strain rate Using an optical measurement system typically all results are derived from the surface and are typically directly used for the calculation of the yield curve.

 ),*85( Typical measured images: left side full image size, right side enlarged local area with mathematical subsets (facets) for the calculation of surface point.

&203$5,6212)(;3(5,0(17$/5(68/76 By defining a general rule to calculate biaxial true stress – true strain curves from bulge tests by a IDDRG working group a large number of influences were discussed. Comparing the results of two different test departments for the same material with the current state of the rule still some differences can be recognized. In figure 5 the results for HCT780C with an initial thickness of 1.5 mm are shown. Both departments have performed three similar tests to show their reproducibility. The continuous curves where calculated from a test device with a die diameter of 100 mm by Voest Alpine, the dashed curves with 200 mm were carried out at the IFUM. At a true strain value of 0.53 for the tests with 100 mm show true stress values around 1100 MPa, the tests with 200mm true stress values around 1065 MPa. This is a difference of approx. 3%.

),*85( True stress true strain curves for the HCT780C for different blank holder diameter (continuous =100 mm, dashed = 200 mm), (a) image with complete curve, (b) image with reduced range for the y-axis to point out the differences

For the calculation, the simplified equation for biaxial membrane stress was used as given in figure 2. An obvious difference between these two tests was the die diameter. This formula was defined based on the assumptions that the thickness is much smaller than the diameter. The results of the optical measuring system (strains, curvature and thickness) were calculated from the surface (outer layer) of the specimen and directly used for the simplified formula. If these assumptions are not valid, the influence of the material thickness has to be taken into account in more detail:

109

x

Strains in the material are a mixture of stretching and bending influences. In thickness direction the strains of the outer layer are bigger than in the middle of the material or as the average over the thickness. This means that o the average strains are overestimated (true strain too large) o the actual thickness is underestimated (thickness strain too small = true stress too large) x The radius for the surface curvature Us is bigger than the radius of the middle layer Um (true stress too large) x The pressure affects are only at the inner surface of the material [9] (true stress too large) So typically in the standard procedure the calculated true stress values will be overestimated. All these single influences are leading to higher values. The compensation of the thickness influence is an obvious idea and can be used to explain the higher values for the smaller die diameter in figure 5.

)(02'(/62)7+(%8/*(7(676 To verify the experimental results and to better understand the stress and strain distribution in the blank, finite element simulations of the bulge test were carried out. Two 3D models were set up as shown in figure 6 with the help of the HyperMesh pre-processor from Altair. Similar to the experiment, two die diameters d = 100 mm and d = 200 mm were investigated while the other geometry and material parameters were kept constant. To reduce the computational costs, the die was modelled as a discrete rigid body. The deformable blank was meshed with linear full-integration elements with four elements along the z axis. Linear reduced-integration elements, which are usually used for metal plasticity problems, tend to underpredict bending stiffness due to hourglassing and were therefore not used for the present study [7]. Linear full integration elements are free from this drawback. Though leading to an overly stiff material behaviour in bending, these elements were found to be more suitable for the present investigation. According to the test set-up, the implemented boundary condition at the outer blank edge as well as the corresponding symmetry boundary conditions along the x and y axes were applied as shown in figure 6 (a). To simulate sheet bulging, an equally distributed pressure p was gradually applied to the blank bottom surface corresponding to the test conditions as presented exemplarily for d = 200 mm in figure 6 (b). The constant friction coefficient of 0.15 was assumed between the die and blank. The sheet material was modelled with the von Mises yield criterion, the associated flow rule, and isotropic hardening with the material data determined from the uniaxial tensile test carried out at Voest Alpine. Beyond the uniform elongation, the hardening curve was extrapolated with the Gosh approach [8]. The analysis was carried out with the explicit solver LS-Dyna from Livermore Software Technology Corporation.

),*85(Finite element model of the bulge test with the applied boundary conditions

110

(;3(5,0(17$/$1'6,08/$7,215(68/76 To verify the hardening curve extrapolation from the uniaxial tensile test, the simulated dome height was compared with its experimental counterpart. The results presented in figure 7 show a good agreement between the simulation and experiment and prove a good extrapolation quality.

 ),*85(Simulated (a), (b) and experimental (c), (d) dome height

To assess the accuracy of the analytical evaluation of the biaxial stress strain curve, the so determined curve was compared with the equivalent stress vs. equivalent strain curve estimated numerically in the upper element at the pole centre. As it can be seen in figure 8, both in experiment and simulation the equivalent strain – equivalent stress curve in case of the smaller die diameter is higher than the one estimated for the larger die diameter. The steeper hardening for the smaller die diameter can be attributed to higher bending and stretching. Furthermore, the comparison of the experimental and simulated curves also shows higher discrepances between the two in the range of the equivalent plastic strain from 0 to approx. 0.15.

 ),*85(Simulated and experimental biaxial true strain – true stress curves

A further interesting issue obtained with the help of the numerical simulation is shown in figure 9. Here, the level of the equivalent stress in the top and bottom elements of the blank for two locations X and Y as a function of the test time is presented. As it can be seen, the top and bottom layer of the blank undergo different extent of plastic deformation. The discrepancies in the yield stress between the layers are more pronounced close to the die but gradually decrease with increasing pole curvature.

111

),*85(Development of the equivalent stress in the upper and bottom layer of the blank during the test.

The level of the equivalent stress in the top and the bottom layer of the blank depending on the radial distance from the pole centre for the time frame t = 1/3 ttest is presented in figure 10. The figure shows that the differences in the equivalent stress through the blank thickness are considerably higher for the die diameter d of 100 mm. From the relative radial distance of approx. 0.4 (d = 100 mm) and approx. 0.9 (d = 200 mm) to 1.0 the bending effects cause the curves to deviate further from each other.

),*85(Equivalent stress in the upper and bottom layer of the blank depending on the radial distance from the pole centre.

The results of the analytical and numerical investigations conclude that a consideration of stretching and bending influences with the aid of the material thickness compensation in the analytical calculation of the biaxial stress strain curve of bulge test is necessary.

0DWKHPDWLFDOH[WHQVLRQIRUFDOFXODWLQJELD[LDOVWUHVVVWUDLQFXUYH By using the ARAMIS system for the optical measurement a material thickness compensation can be activated for the post-processing. This eliminates the bending influence in the strain determination by calculating the strain values and coordinates directly for the middle layer of the blank (engineer strains M1m, M2m and true strains H1m, H2m). By further investigations it was approved that the real actual thickness t is similar to a simplified computed thickness tm (based on the middle layer strains) and can be calculated by the following equation

t

1

³ 1  M (t ) ˜ 1  M (t ) dt

t0

1

| tm

2

t0

1 1  M1m ˜ 1  M 2 m

(1)

Therefore only the following assumptions must be valid: x Linear strain distribution (M1(t), M2(t)) in material thickness direction x The initial material thickness (t0) has to be known x The volume of the material does not change during the deformation For the determining the radius Um* of the curvature in the middle layer a first idea was the following equation:

U m*

Us 

tm 2

(2)

However, a comparison with a real fit in the coordinates of the middle layer pointed out significant deviations. Due to the local thinning at the apex of the dome, equation (2) does not give reasonable results for the real Um for

112

higher true strain values. Figure 11 shows the principle difference between the real Um and Um*. In this figure the radius Us of the complete outer layer is printed with a constant value. In the case of a homogeneous thinning of the material (left side) equation (2) is valid. On the right side the case of a localized thinning at the apex of the dome is shown and Um is much smaller than Um* from the left side.

  ),*85( Real Um for the case of a localized thinning defined by a best fit in middle layer coordinates (right image side) in comparison to simple approach by equation (2), which is only valid for a homogenous thinning (left image side)

The equation for the biaxial membrane stress (figure 11) is based on the balance of forces introduced by the pressure on the inner surface and the tangential stresses in the material can be written as:

V ˜ 2S ˜ U ˜ t

p ˜S ˜ U 2

(3) For taking the thickness influences into account this equation can be modified by the related different layer values:

V ˜ 2S ˜ U m t m

V

t · § p ˜ S ¨ Um  m ¸ 2¹ ©

2

§ t · p ˜ U m ¨¨1  m ¸¸ © 2U m ¹ 2t m

2

(4)

This new equation was used to calculate a modified true stress true strain curve based on the same test data as used in figure 5. Figure 12 shows the change of the results for one sample of both tests (diameter 100 mm and 200 mm). Additionally the influence of a modification only for single values is shown (only strain of middle layer, radius of curvature determined for the middle layer or pressure on inner surface). As discussed above all influences are reducing the true stress values. In this case the calculation of the real radius Um of the curvature with the middle layer coordinates has the biggest influence and legitimates the additional effort. The sum of all influences compensated by equation (4) shows a significant reduction of the true stress values. At a true strain value of 0.53 for the tests with 100 mm the curve is reduced from 1100 MPa to 1035 MPa (approx. 6% reduced). For the tests with 200 mm the values at the end of the curve is reduced from 1065 MPa to 1035 MPa (approx 3% reduced). Because of the higher curvature, the compensation is stronger for the smaller die diameter.

),*85( True Stress true strain curves based on “surface data” with the overestimated values; with different compensations and the full compensation by using “middle layer data” with equation (4)

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The comparison of the new results is shown in figure 13 for both diameters. The approach of reducing the true stress values lead to a better correlation of the curves for 100 mm and 200 mm diameter (thin continuous and dashed curves). The differences are small and at a true strain level of 0.53 there is no significant difference (the variation of the repetitions for the same diameter is larger than differences for 100 mm and 200 mm).

),*85(. Comparison of true stress true strain curves for different die diameter (continuous = 100 mm, dashed = 200 mm) calculated directly from “surface data” (thick) or with the new thickness compensations by using the “middle layer data” with equation (4) (thin); (a): all three tests, (b): only one test for 100mm and one for 200mm



6800$5< This paper comprised numerical and experimental investigations of the HCT780C (CP 800) with bulge test. An influence of stretching and bending could be numerically and experimentally demonstrated. With the aid of the material thickness compensation by the analytical calculation of the biaxial stress strain curve enables a more precise and reasonable determination. The remaining differences will be a topic for future investigations

$&.12:/('*0(176 We would like to thank Voest Alpine for the support and provision of the material HCT780C and the material data. Furthermore we would like to thank the working group “bulge-test” of the German Deep Drawing Research Group.

5()(5(1&(6 1. W. Pankin, “Der hydraulische Tiefungsversuch und die Ermittlung von Fließkurven”, dissertation Hochschulblatt, 1959 Stuttgart. 2. W. Bleck, I. Kim, „Die Online-Erfassung von Fließkurven und Formänderungswegen während des hydraulischen Tiefungsversuchs an neu entwickelten Tiefziehstählen“, final report fort he German research foundation, Aachen 2001. 3. S. Keller et al, “Experimantal Procedure” in Yield Curve Determination Using the Bulge Test combined with Optical Measurement, Proceedings of IDDRG 2009 International Conference, Golden, CO, USA, pp. 319-330 4. F. Gologranc: Beitrag zur Ermittlung von Fließkurven im kontinuierlichen hydraulischen Tiefungsversuch, Berichte aus dem Institut für Umformtechnik, Universität Stuttgart, Nr. 31 (1975), Verlag W. Girardet, Essen 5. C. Sunderkötter, „Vergleichende umformtechnische Charakterisierung von Blechwerkstoffen unter mehrachsiger Beanspruchung, dissertation in PZH-Verlag, Garbsen, 2007. 6. H. Friebe: Kennwertermittlung und Verifikation von Umformsimulationen mittels optischer Messverfahren, Tagung Werkstoffprüfung 2006, "Fortschritte der Kennwertermittlung für Forschung und Praxis", Verlag Stahleisen, Düsseldorf, 2006, 239-244 7. Abaqus Analysis User’s Manual 8. M.-S. Aydin, J. Gerlach, L. Kessler, Verfahren zur Extrapolation der Fließkurve aus den Daten des Zugversuches jenseits der Gleichmaßdehnung, 8. LS-Dyna Forum 2009, Germany 9. C. Leppin, Fragestellungen und Gedanken um die Versuchsauswertung des hydraulischen Bulge Tests, AK Bulge, Düsseldorf (9 March 2011)

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Identification of constitutive equation in hierarchical multiscale modelling of cup drawing process J. Gawad∗,† , A. Van Bael∗∗,‡ , P. Eyckens∗∗,§ , G. Samaey∗ , P. Van Houtte∗∗ and D. Roose∗ ∗

Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium † Department of Applied Computer Science and Modeling, University of Science and Technology AGH, al. Mickiewicza 30, 30-059 Krakow, Poland ∗∗ Department of Metallurgy and Materials Engineering, Katholieke Universiteit Leuven, Kasteelpark Arenberg 44 bus 2450, 3001 Heverlee, Belgium ‡ Department IWT, Limburg Catholic University College (KHLim), Campus Diepenbeek, Agoralaan Gebouw B, bus 3, 3590 Diepenbeek, Belgium § Materials Innovation Institute, Mekelweg 2, 2600 GA Delft, the Netherlands Abstract. In this paper we discuss extensions to a hierarchical multi-scale model (HMS) of cold sheet forming processes. The HMS model is capable of predicting changes in plastic anisotropy due to the evolution of crystallographic textures. The ALAMEL polycrystal plasticity model is employed to predict the texture evolution during the plastic deformation. The same model acts as a multilevel model and provides "virtual experiments" for calibration of an analytical constitutive law. Plastic anisotropy is described by means of the Facet method, which is able to reproduce the plastic potential in the entire strain rate space. The paper presents new strategies for identification of the Facet expression that are focused on improving its accuracy in the parts of the plastic potential surface that are more extensively used by the macroscopic FE model and therefore need to be reproduced more accurately. In this work we also evaluate the applicability of identification methods that (1) rely exclusively on the plastic potential or (2) can take into consideration also the deviatioric stresses derived from the Facet expression. It is shown that both methods provide the Facet expressions that correctly approximate the plastic anisotropy predicted by the multilevel ALAMEL model. Keywords: hierarchical multiscale modelling, texture evolution, multilevel model, sheet forming, parameter identification PACS: 02.70.-c, 46.15.-x, 46.90.+s, 62.20.F-, 81.40.Lm

INTRODUCTION The flow of metal in sheet forming processes can greatly depend on the plastic anisotropy of the sheet. In many cases the anisotropy appears due to crystallographic texture. The textures are continuously changing as a result of the plastic deformations, leading to heterogeneous texture distributions throughout the work-piece. It is quite well recognised that the evolution of texture is mutually tied together with evolution of plastic anisotropy. Ideally, this should be taken into account in simulations of sheet forming processes with the Finite Element (FE) method, thus the development of new deformation textures should be reflected by a modification of the plastic anisotropy of the sheet. The most straightforward way to implement this into the FE simulations exploits a concept of Representative Volume Element (RVE). An RVE consists of a representative number of crystals, and it can be invoked each time a constitutive response is required by the FE simulation. The advantage of such embedded approach is that texture evolution is automatically accounted for. But unfortunately such an approach is too computationally demanding for industrial applications. In this respect, hierarchical approaches offer more feasible alternatives. Virtual mechanical tests are then performed with a crystal plasticity model on representative aggregates of crystals, and the results of these experiments are used to derive analytical constitutive laws. Since the constitutive response is much easier to calculate, the FE simulations can be much faster. For example, the paper [1] shows how the parameters of Vegter yield locus can be calibrated on the basis of virtual tests using a crystal plasticity FE model. The resulting constitutive law was then used in a simulation of a stamping process. Van Houtte et al. [2] have proposed the Facet method which can describe yield loci in stress space or plastic potentials in strain rate space. The coefficients of the Facet can be readily derived from a large number of virtual experiments on a representative number of crystals. The experiments can be conducted by means of a multilevel model, e.g. the Full Constraint (FC) Taylor [3] or the ALAMEL [4] model. Recently this formulation

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 115-122 (2011); doi: 10.1063/1.3623600 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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was extended by development of a hierarchical multi-scale (HMS) model [5, 6] that takes into account evolution of the crystallographic texture during the deformation and is able to adjust the analytical constitutive law to the updated texture. One of applications of this multi-scale model is presented in the paper by Eyckens et al. [7] at this congress. The aim of the present study is to improve a procedure for reconstruction of the constitutive law that is used in the HMS model. Since the constitutive law is a key component of the HMS, reliability of the constitutive response is essential for mutual interactions of textures and plastic deformations. The paper shall also assess whether the identification method that was used in the previous papers by the present authors provided sufficient accuracy.

HIERARCHICAL FRAMEWORK FOR PLASTIC ANISOTROPY Macroscopic constitutive laws are very common in Finite Element models of anisotropic plastic deformation. Many successful analytical macroscopic descriptions of yield loci have been proposed over the years. However, if one wants to take into account the evolution of the material at a crystal level, which is indispensable for predicting the evolution of texture, the calibration of the constitutive law becomes a challenge. The scope of the current paper is limited to the cases for which the constitutive law accounting for plastic anisotropy of textured polycrystalline material can be described by means of a plastic potential in strain rate space. For materials without strain rate sensitivity and under assumption of plastic incompressibility, the deviatoric stress tensor s derived from the plastic flow stress can be calculated for a given strain rate tensor d as: ∂ ψ(d) , (1) ∂d where ψ(d) is a plastic potential. It is convenient to express the deviatoric stresses and plastic strain rates as fivedimensional vectors. More details about this transformation from relevant second-rank tensors into five dimensional vectors can be found e.g. in [8]. The current paper adapts the convention that capital letters D and S denote fivedimensional quantities converted from appropriate tensors. Suppose that the results of "virtual experiments" are available for a set of strain rates Di , covering the entire strain rate space by a grid of almost uniform density. The multilevel model is employed to lift macroscopic boundary conditions, such as strain rates Di into the microscopic level, and then to homogenise the micro-scale response over representative clusters of grains. The results may include average Taylor factors or macroscopic stress tensors m corresponding to imposed strain rates. Let’s assume that the pairs (Dm i , Si ) form a discrete grid of the multilevel data. A model function that can accurately reproduce the output from the virtual experiments is then considered as a constitutive model of the material. If a model function correctly approximates the macroscopic response of the multilevel model, the following system of equations needs to be satisfied for every point in the grid: s=

ψ(Dm i )=C   ∂ ψ(D)  = Sm i . ∂D  m

(2a) (2b)

D=Di

where ψ(D) is the plastic potential and C selects an equipotential surface. Van Houtte et al. [2] proposed the Facet expression that utilises a homogeneous polynomial to describe the plastic potential. Such expressions contain a much larger number of coefficients than traditional phenomenological constitutive laws. The coefficients are basically derived from a large number of virtual experiments on a representative number of crystals. The Facet method has the distinct advantage over other analytical expressions based on series expansions (e.g. [8]) that it strictly guarantees convexity of the yield loci and the plastic potentials. The Facet expression defines the plastic potential ψ(D) in the strain rate space as the function: 1

ψ(D) = [Gn (D)] n ,

(3)

where n is an even natural number greater than 1 and Gn (D) is a homogeneous polynomial of degree n and given by the formula: Gn (D) =

K

∑ λk (Smk · D)n ,

(4)

k=1

where λk are parameters and Sm k are deviatoric stresses obtained by means of a multilevel model, such as the FC Taylor or the ALAMEL model. Provided that λk ≥ 0 for all k and n is a positive even number, it can be proven that the equipotential surface as defined by Equation (3) and Equation (4) is always convex [2].

116

Identification based on plastic potential The method described in this subsection follows the idea of identification algorithm as proposed in [2] and extended in work [5]. Conceptually the method relies solely on values of plastic potential, i.e. it assumes that it is sufficient to approximate plastic equipotential surface in order to derive corresponding stresses. Only Equation (2a), i.e. the first equation of the system (2), is explicitly satisfied. Given that the potential (3) is positive, one can limit the calculation to searching for a pre-defined value of the polynomial Gn (D) for every point in the grid. Such formulation has the clear advantage that the problem to solve is linear with respect to λ . For the purpose of the parameter identification, it is suitable to rephrase the Equation (4) in matrix form. The parameters λ can be determined by solving the following system of equations:  1 [W] {λ } ∼ (5) = Cn ,  n m where the components of the matrix Wi j = Dm · S are considered as the homogeneous n-th degree polynomials i j of the rate of plastic work that is dissipated per unit volume. If i = j, then the deviatoric stress vector Smj corresponds with the Dm i for which the multilevel model has been evaluated. Strain rates are scaled in order to ensure that rate of plastic work per unit volume is equal to unity. The right-hand side vector in Equation (5) selects the equipotential surface. Inasmuch as the system (5) is linear, it can be efficiently solved using the Non-Negative Least Squares (NNLS) algorithm [9], which ensures that all components of λ are either positive or zero.

Identification based on plastic potential and deviatoric stresses Although the method presented in the previous subsection is fast and robust, it has some limitations related to insufficient control over the accuracy of approximation in preselected regions of the strain rate space. This has led us to the development of a new identification method, which ensures that both zeroth and first derivative of the plastic potential are close to the results of the "virtual experiments". This means that every point in the grid of multilevel data satisfies equations (2), which gives much stronger premises on precision of the approximation. Unfortunately, resulting system of equations (2) is nonlinear, which involves a complex solution procedure. In this work we employ a Trust Region [10] algorithm to solve the system of equations (2) as a nonlinear least squares problem with linear constraints. The problem is solved by minimizing the norm of the vector F: ⎧ ⎫T w1 u(ψ(Dm ⎪ 1 , λ ) −C) ⎪ ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎨ w u(ψ(Dm , λ ) −C) ⎪ ⎬ N N F(λ ) = , (6) T m m ⎪ ⎪ w1 (S(D1 , λ ) − S1 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎩ T ⎭ m m wN (S(DN , λ ) − SN ) where the first N components represent the deviations from equipotentiality of the plastic potential surface, while the latter 5N components correspond to the difference between the stress components derived from the Facet expression and predicted by the multilevel model. As one can notice, the plastic potential and stress vectors are now considered as functions of both strain rate and vector λ , which is with the purpose of identification of parameters λ . The weighting factors wi allow one to attach greater importance to selected grid points and possibly refine the quality of the approximation at specified parts of the plastic potential surface. In most cases, the weights wi are all set to the same value 1.0. If the refinements are requested, the weights are assigned to the grid points according to their distance from some arbitrarily chosen strain rate D: A  p wi (d) = (7) 1 + dd0 where d is the angular distance between Dm i and D, while A, d0 and p are parameters. Another weighting factor u is introduced with the aim to give an option to eliminate all the components related to the zeroth derivative of the plastic potential. Weight u can take value of either zero or one. If u = 0, the explicit requirements for the equipotentiality of

117

the plastic surface is abandoned, thus the method can be considered as an opposite of those described in the previous subsection. It is also possible to assign other values to the weight u with the intention of balancing the importance of equipotentiality and stress conformance. In the general case, the vector F(λ ) comprises of M = 6N components in total. Formally, to find the minimiser minλ ∈RN ||F(λ )||2 where the function F(λ ) : RN → RM and N ≤ M, the algorithm attempts to find a minimum of:   (8) min F(λ  ) + J(λ  )(λ  − λ  ) λ ∈RN

  with subject to λ  − λ   ≤ Δ , where λ  and λ  are current and new guess on λ , respectively, J(λ  ) is Jacobi matrix and Δ is the current size of the trust region. Non-negativity constraints are imposed by the solver to the components of the vector λ . The Jacobi matrix is required to solve the nonlinear least squares problem given by equations (6) and (8). Since the size of the matrix J is M × N, the computational cost expands quickly with an increase of grid density. It is possible to simplify the problem in a way similar the one of the previous subsection, i.e. by assuming that it is sufficient to use the polynomial Gn (D, λ ) in lieu of the plastic potential (3): ⎧ 1 n ⎪ w1 u(Gn (Dm ⎪ 1 , λ ) −C ) ⎪ ⎪ . ⎪ ⎪ .. ⎪ ⎪ ⎪ ⎨ 1 m wN u(Gn (DN , λ ) −C n ) F(λ ) ≈ T m m ⎪ w1 (S(D1 , λ ) − S1 ) ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎪ ⎪ ⎩ T m wN (S(DN , λ ) − Sm N) In principle, the Jacobi matrix J can be then split into two submatrices:   U J= L

⎫T ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

.

(9)

(10)

where submatrix U is constant and can be evaluated just once, since it contains the partial derivatives of the components in the vector F that are linear with respect to λ , while the submatrix L has to be evaluated at every iteration of the minimisation algorithm. Obviously, once the weights u are set to zero, one can entirely eliminate the submatrix U.

RESULTS In this section we compare the identification methods for a number of textured materials. All deformation textures shown in the following subsections were predicted by the HMS [6] model applied to simulation of a cup drawing process. The model made use of the ALAMEL to forecast the evolution of textures. The textures were represented by discrete ODFs consisting of 5000 crystals of orientations given by Euler angles in Bunge notation [11]. For sake of simplicity, these virtual samples are taken at the same point in the cups, approx. 8 mm from the cup rim at 90◦ to rolling direction (RD). Grid of the same angular resolution is used for the calibration of the Facet expressions; values of stresses are calculated by the ALAMEL model. For fcc metals, {111} <110> slip systems are adopted, and for bcc metals, {110} <111> + {112} <111> systems. Identical critical resolved shear stresses are assumed for all slip systems. Anisotropic plastic properties are characterised by a contraction ratio q-value, defined as the absolute value of the ratio q = −d22 /d11 with the plastic strain rate in the width direction d22 < 0 and the one in the longitudinal direction d11 > 0 upon uniaxial tension. The relation between Lankford r-value and the q-value is expressed by q = r/(r + 1). The q-values as calculated from the Facet expressions are compared to those directly obtained by means of the ALAMEL model. We shall refer to identification methods as following: Method 1 is based exclusively on the plastic potential (see Equation (5)), identification Method 2 combines equipotentiality of the plastic surface with the conformance of deviatoric stresses (see Equation (6)) and Method 3 relies only on conformity with respect to the stresses, i.e. it uses Equation (6) with u set to zero.

118

a)

0.58

180

0

0.56

0

0.54

q-value

0.52 0.50 0.48 0.46 0.44 0.42

90

0.40 0

PHI2=

45

15 30 45 60 75 90 105 120 135 150 165 180

ε =0

angle to RD, (deg)

b)

0.70

0 0.65

180

0

q-value

0.60 0.55 0.50 0.45 0.40

90 0.35 0

PHI2=

45

15 30 45 60 75 90 105 120 135 150 165 180

ε = 0.63

angle to RD, (deg)

FIGURE 1. q-values (on the left) calculated for the texture (on the right): a) aluminium AA1100 in annealed state, b) deformed AA1100. The ALAMEL model (filled symbols) and the Facet: Method 1 (open squares), Method 2 (open circles) and Method 3 (open triangles). Isolines in φ2 = 45◦ ODF sections at intensities 0.7, 1.0, 1.4, 2.0, 2.8, 4.0, 5.6, 8.0, 11.0 and 16.0.

Comparison of identification methods A brief comparison of these methods for AA110 aluminium is presented in Figure 1. Two states of the material are analysed: the sheet as received (after rolling and annealing) and a virtual sample deformed in the cup drawing process to the equivalent plastic strain ε ≈ 0.63. The Facet expression of order n = 8 is used. It can be seen that the shape of q-value curve (Fig. 1a) is not perfectly reproduced by none of the methods. However, the Method 1 produces the worst approximation, and other two methods are fairly comparable. This is no longer the case for highly deformed material, as presented in Figure 1b. The shape of the q-values curve is almost ideally replicated if the Method 2 is used, while Method 3 delivers only slightly better predictions than the Method 1. However, unfavourable effects are prone to appear if the requirement for the equipotentiality of the plastic surface is discarded. Figure 2 shows a comparison of the q-values obtained for a deformed DC06 steel. It is sufficient to use sixth-order Facet expression to reflect the anisotropic plastic behaviour of this material. The upshots of identification Methods 1 and 2 guarantee very good agreement of the derived q-values with the multilevel model. However, there is an observable discrepancy between the ALAMEL predictions of q-values and those obtained from the Facet approximation if the Method 3 is applied. Irregular shape of the curve is correctly reproduced by the Facet identified by means of other two methods. Unfortunately, small inflections that appear in the curve predicted by the ALAMEL become exaggeratedly protruded by the stress-based identification procedure. Further investigation is needed to clarify the underlying distortions of the plastic surface.

Local adaptations of constitutive law An example of enhanced capabilities of new identification algorithms is shown in Fig. 3. Texture data for annealed sheet of AKDQ steel were taken from the Numisheet Benchmark 1 [12]. A φ2 = 45◦ section of the ODF is shown in Fig. 3b. Figure 3a illustrates how the accuracy of Facet approximation can be increased for selected regions of the plastic potentials. The Facet expressions of order n = 6 were identified for the same grid data by means of Method 1 and Method 2. However, the Method 2 has an advantage of possible fine tuning of the approximation. This can be done

119

0.72

180

0

0.71

0

0.70 0.69

q-value

0.68 0.67 0.66 0.65 0.64 0.63 0.62

90

0.61 0

PHI2=

45

ε = 0.03

15 30 45 60 75 90 105 120 135 150 165 180

angle to RD, (deg) a)

b)

FIGURE 2. a) q-values calculated for deformed DC06 steel: the ALAMEL model (filled symbols) and the Facet: Method 1 (open squares), Method 2 (open circles) and Method 3 (open triangles). b) φ2 = 45◦ section of the ODF for the corresponding deformation texture, isolines at intensities 0.7, 1, 1.4, 2, 2.8, 4, 5.6, 8, 11 and 16

0.52

0

q-value

180

0

0.50 0.48 0.46 0.44

90

0.42 0

15 30 45 60 75 90 105 120 135 150 165 180

PHI2=

Angle to RD (deg) a)

45

b)

FIGURE 3. a) q-values calculated by means of the ALAMEL model (filled symbols) and the Facet identified by: Method 1 (continuous line), Method 2 (open squares) and Method 2 with weighting factors wi emphasising the uniaxial tensile state at 45◦ (open triangles) or uniaxial tensile at 135◦ (open circles), b) φ2 = 45◦ section of the corresponding ODF for AKDQ steel sheet (isolines at intensities 0.8, 1.0, 1.3, 2.0, 2.5, 3.2, 5.0 and 6.4)

by adjusting the weighting factors wi for appropriate grid points. The continuous line in Figure 3a shows an outcome from the Facet expression that was calibrated solely to obtain the equipotential plastic surface. As it can be seen, the q-values are overestimated for ranges [0◦ , 75◦ ] and [160◦ , 180◦ ]. The Facet calibrated by the Method 2 performs much better, however it only partially reproduces the peaks at 50◦ and 140◦ w.r.t RD. The accuracy of the prediction can be readily improved by imposing higher weighting factors to the grid points in the vicinity of yield loci attributed to those peaks. The sigmoidal function given by the Equation (7) is used for calculations of the weights; the following values of parameters are taken: A = 1, d = 0.68 radian and p = 2. By doing so, one can obtain almost perfect agreement for the peak at 140◦ w.r.t. RD (open circles in Fig. 3a) or increase the correspondence for the peak at 50◦ (open triangles in Fig. 3a).

Evolution of textures If texture evolution is taken into account in the HMS model, the constitutive law has to be periodically adapted to new anisotropic properties of the material. Obviously, this demands an appropriate form of the analytical constitutive equation that needs to be flexible enough to reproduce the effects of texture that is much sharper than the one present in the initial material. Besides, the calibration of the law should be able to extract at least the prevailing trends in the altered anisotropic plastic properties. Figure 4 presents a comparison of q-values obtained from the multilevel ALAMEL model and derived from the Facet expressions for selected snapshots of crystallographic textures predicted by HMS model in a simulation of cup

120

0.72

a)

180

0

0.71

0

0.70

q-value

0.69 0.68 0.67 0.66 0.65

90

0.64 0

PHI2=

45

15 30 45 60 75 90 105 120 135 150 165 180

ε =0

Angle to RD, (deg)

b)

0.72

0

q-value

180

0

0.70

0.68

0.66

0.64

90 0.62 0

PHI2=

45

15 30 45 60 75 90 105 120 135 150 165 180

ε = 0.06

Angle to RD, (deg)

c)

0.76

0

0.74

180

0

0.72

q-value

0.70 0.68 0.66 0.64 0.62 0.60

90

0.58 0

PHI2=

45

15 30 45 60 75 90 105 120 135 150 165 180

ε = 0.23

Angle to RD, (deg)

a)

d)

b)

0.85

0

180

0

0.80

q-value

0.75 0.70 0.65 0.60 0.55

90 0.50 0

15 30 45 60 75 90 105 120 135 150 165 180

PHI2=

45

ε = 0.59

Angle to RD, (deg)

FIGURE 4. Left side: q-values calculated for a sequence of deformation textures by the ALAMEL model (filled squares) and by means of the Facet expression. Identification of the Facet was based on equipotentiality of the plastic surface (open squares) and on both equipotentiality and stresses (open circles). Right side: φ2 = 45◦ ODF section of corresponding textures. Evolution of the texture is predicted by the HMS model [6] for cup drawing of DC06 steel sheet, isolines at intensities 0.7, 1, 1.4, 2, 2.8, 4, 5.6, 8, 11 and 16. Equivalent plastic strain corresponding to given texture is denoted by ε.

drawing of DC06 steel. Similarly as in the previous examples, virtual measurements of the texture are taken at the same point in the cup. Throughout the cup drawing process, the texture was subjected to increasing plastic strains along a complex deformation path, including bending, unbending, tension and compression. In this study we limit our attention to identification methods based on the equipotentiality of the plastic surface and on simultaneous plastic

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equipotentiality and the accordance of derived stresses with the multilevel model. As it can be noticed, the plastic anisotropy of the material changes substantially in the course of the deformation. Gradual modification of the texture lead to immense alteration of anisotropy, as it is manifested by the shape of the q-values curve. Both Method 1 and Method 2 are capable of capturing the anisotropic behaviour for low and moderate strains, as shown in Figures 4a – 4c. Identification Method 2 generates the Facet expression that slightly better reproduces appearance of irregularities and inflections in the q-values curve, but the Method 1 also yields very accurate predictions. There are some exceptions revealed in the Fig. 4b, where for the angular range [45◦ , 75◦ ] the Method 2 clearly leads to discrepancy between multilevel prediction and q-values derived form the Facet expression. However, neither of the methods is able to replicate high peaks around the angles 45◦ and 135◦ w.r.t. RD (see Fig. 4d). Also the bottom of the curve corresponding the angle 90◦ is not captured well. Nevertheless, we attribute this effect to relatively low order of the Facet expression.

CONCLUSIONS The paper deals with a problem that is fundamental to hierarchical multiscale models of forming processes. Exactness of an approximation provided by a macroscopic constitutive law is crucial for the entire HMS model, since the law effectively replaces homogenised response of the multilevel models working at micro-scale. In this work we analysed the identification procedures that ensure conformance of the zeroth and first derivatives of the plastic potential with the multilevel model. The analysis indicates that generally it is the best to fit simultaneously the analytical constitutive law to both equipotentiality of the plastic surface and to the correspondence of stresses derived from the constitutive law with the outputs of the multilevel model. However, the improvement of accuracy is not spectacular in comparison to the calibration method based exclusively on the plastic potential. One may then also conclude that assumptions made on identifications of the Facet expression in papers [2, 5, 6] are valid. The concept of adaptively updated analytical constitutive law describing the plastic anisotropy in the FE was applied in the model. Owing to the successive reconstructions of the plastic potential, coupled to the evolution of the crystallographic texture, the mutual interactions between micro- and macro-scale can be taken into account.

ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support from the project IDO/08/09, funded by K.U.Leuven, and from the Interuniversity Attraction Poles Program from the Belgian State through the Belgian Science Policy agency, contracts IAP6/24 and IAP6/4. GS is Postdoctoral Fellow of the Research Foundation - Flanders (FWO). Research of PE affiliated to the Materials Innovation Institute (M2i) was carried out under the project number M41.2.08307a/M41.10.08307b in the framework of the Research Program of M2i (www.m2i.nl). The scientific responsibility rests with the authors.

REFERENCES 1. 2. 3. 4. 5.

M. Kraska, M. Doig, D. Tikhorimov, D. Raabe, and F. Roters, Comp. Mat. Sci. 46, 383–392 (2009). P. Van Houtte, S. K. Yerra, and A. Van Bael, Int. J. Plast. 25, 332 – 360 (2009). G. I. Taylor, J. Inst. Metals 62, 307–324 (1938). P. Van Houtte, S. Li, M. Seefeldt, and L. Delannay, Int. J. Plast. 21, 589 – 624 (2005). J. Gawad, A. Van Bael, S. K. Yerra, G. Samaey, P. Van Houtte, and D. Roose, AIP Conference Proceedings 1252, 770–777 (2010). 6. J. Gawad, A. Van Bael, P. Eyckens, P. Van Houtte, G. Samaey, and D. Roose, Steel Res. Int. 81, 1430–1433 (2010). 7. P. Eyckens, J. Gawad, Q. Xie, A. Van Bael, D. Roose, G. Samaey, and P. Van Houtte, “Anisotropic sheet forming simulations based on the ALAMEL model: Application on cup deep drawing and ironing,” in Numisheet, 2011. 8. P. Van Houtte, and A. Van Bael, Int. J. Plast. 20, 1505 – 1524 (2004). 9. C. L. Lawson, and R. J. Hanson, Solving Least Squares Problems, Prentice-Hall, 1974. 10. A. R. Conn, N. I. M. Gould, and P. L. Toint, Trust-Region Methods, Society for Industrial Mathematics, 1987. 11. H. Bunge, Texture Analysis in Materials Science, Butterworth, London, 1982. 12. J. W. Yoon, and R. E. Dick, Numisheet benchmark 1 (2011), URL http://www.numisheet2011.org/Benchmarks.

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Constitutive Modeling and Numerical Simulation of Super Plasticity Forming Processes Longchang Tong and Pavel Hora Institute of Virtual Manufacturing, ETH Zurich, 8092 Switzerland Abstract. Up to now the power function   A m is commonly used to evaluate the flow stress to describe the material behavior in the numerical simulation of Super Plastic Forming (SPF) processes with the assumption that the m-value is a material constant. However, experiments revealed that the m-value is not a constant. It varies as the strain rate changes and reaches its maximum value at the ideal super plastic strain rate. Considering this fact, a mathematic expression is presented in this work. The variation of the m-value is taken into account in this model. Comparison with the measurement verified the method and simulation examples are shown as well. Keywords: Super plastic forming, Numerical simulation, constitutive modeling. PACS: 62.20.fq

INTRODUCTION Most applications of Superplastic Forming have been found in the aircraft and aerospace manufacturingindustries. Meanwhile, it can also be adopted to produce parts with complex geometrical forms or special requirements in small quantity in other cases, for example, for the manufacturing of some medical instruments. Because most super plastic forming parts are produced in small quantity, the costs of the forming tools are one of the most important factors. Trial and error method is economically unfavorable. In contrast, the numerical simulation plays an important role for the designing of superplastic forming processes. In the last decades simulation technique and the computer science have been developed at enormous speed. Many sheet forming processes have been well simulated. Systematical information provided by the simulation has helped the engineers efficiently. The period of the developing phase for a new product has been reduced remarkably. Economical significance is considerable. In order to achieve a successful simulation, not only computer hardware and software are indispensible, also the material models play the role as the base stone for the simulation technique. Unfortunately many materials models have shown lacks of accuracy for the description of material behaviors. Obviously the accuracy of the simulation results cannot be ensured if the model used for the simulation is unable to describe the material behaviors with sufficient precision. Many works of simulation for super plastic forming can be found in the literature [1, 2]. They used generally rigid-plastic model and power function   A m to calculate the flow stress of the materials. In most works the mvalue is treated as a material constant. The m-value, also called as strain rate sensitivity, is the most important parameter for super plastic property, the higher the m-value, the better the super plastic property. Usually the super plastic materials show high m-values between 0.3 and 0.8. However, it is well known that the m-value, which is the slope of the flow stress diagram plotted in log   log   , is not a constant [3]. It is small at the low strain rate and reaches a maximum value at the ideal super plastic forming rate, then decreases again as the strain rate increases. Figure 1 shows a typical diagram for the m-value. Of course it is favorable to keep the deformation rate at the ideal strain rate and achieve the most super plastic deformation. Unfortunately it is impossible because the strain rates, as well as the strains are distributed in the real forming parts according to the geometrical forms and other forming conditions. Therefore, using the simple power function to calculate the flow stresses causes errors, especially in the cases when the deformations in different regions are remarkably different. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 123-128 (2011); doi: 10.1063/1.3623601 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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FIGURE 1. The strain typical rate sensitivity of super plastic materials [3]

MODELING FOR SUPER PLASTIC MATERIALS Based on the experimental results, a mathematical expression is derived for the computation of flow stresses in the simulation of super plastic forming processes. It is well known that the m-value is defined as

m

d (log  ) . d (log  )

(1)

Suppose the m-value can be expressed as a parabolic function of strain rate 2

 log   log    (2) m  m   mmax  mmax  mmin   log   log 0  where the parameters mmax , mmin ,  and 0 are material constants and are shown schematically in Figure 2.

FIGURE 2. Parabolic function for m-factor

Let x  log  and y  log  and Substitute (2) into (1), we get a simple differential equation

 xx dy  mmax  mmax  mmin  dx  x  x0 where

x  log  and x0  log 0 are still constant values.

124

2

  . 

(3)

The equation can be easily integrated to get the solution:

y  mmax x 

mmax  mmin  x  x 3  C . 2 3x  x0 

(4)

In order to determine the integral constant C in the solution, an additional initial value must be given. The simplest way is to measure the flow stress  0 at the strain rate 0 and set x0 and y0 into equation (4). The constant C can then be determined as

C  y0  mmax x0 

mmax  mmin  x 3

0

 x.

(5)

Substitute (5) into (4), we get the expression as

y  y0  mmax x  x0   Since y  y0  log   log  0  log

 0

mmax  mmin  x  x 3  x  x 3 . 0 2 3x  x0 

(6)

, it is easy to rewrite (6) as

   0 10  ,

(7)

mmax  mmin  x  x 3  x  x 3  can be calculated accordingly. 0 2 3x  x0  Notice x  x0   log   log 0  log(  / 0 )  x is a constant, and x  x0   log   log 0  log(  / 0 ) ,

where   mmax x  x0  

the expression can be simplified as

  mmax log(  / 0 ) 

mmax  mmin  log 3  /    x 3  .

3x 2 The natural logarithm can be used as well. Set x  ln(  / 0 ) , the expression can then be written as

 

   0 exp  mmax ln(  / 0 ) 

mmax  mmin  ln 3  /    x3  .  

3x 2

(8)

(9)

Together 5 parameters are necessary to establish a flow curve for a super plastic material in the forming conditions, namely  0 , mmin , mmax ,

0 and  . The

parameters can be evaluated by fitting the experimental data.

However, if the variation of m-value is known, it is quite directly to obtain the parameters 0 , and. The last parameter

0

mmin ,mmax .

can be determined by means of some simple tests, for example, the tensile test. 300

m = 0.6

stress [MPa]

250

m-variable

200

m = 0.45

150 100

50 0 0

0.005

0.01

0.015

strain rate FIGURE 3. Flow stress using different models

Suppose the m-value varies as shown in Figure 2, i. e. 0 ,  10 5 ,  10 3 mmin  0.3,mmax  0.6 , and the initial stress  0  4.0 MPa , the flow curve is plotted in Figure 3. As comparison the curves generated using power function

125

  A m are plotted with both the average m-value m  0.45 and maximal value m  0.6 respectively. It is seen directly that the power function is unable to describe the variation of m-value. Although the initial flow stresses at 0 , are set with the same value, remarkable deviation appears immediately as the strain rate increases. Combined with the rigid-plastic description, the model has been implemented into the finite element code StreckForm developed at IVP, ETHZ. Successful applications have verified the material model for the simulation of superplastic forming.

COMPUTATIONAL EXAMPLES Super Plastic Forming of a Titanium Alloy A super plastic forming process from the real applications was taken to test the model. The forming process of a dish made of titanium alloy Ti6-4ELI is calculated using the model presented in this work. The material parameters are listed in Table 1. The thickness was measured in different positions. Total 38 points were chosen for the measurement. The result from the simulation is compared in Figure 4 with the measured values. It is seen that the agreement between the simulation and reality is satisfactory. The maximal error is 7% and 34 points show deviation within 5%. 0.4

0.35

Thickness

0.3 0.25 0.2

Measure data

0.15

Simulation

0.1 0.05 0 1

3

5

7

9 11 13 15 17 19 21 23 25 27 29 31 33 35 37

Messure position FIGURE 4. Comparison of Simulation and measurement TABLE 1. Material parameters and initial thickness for the titanium alloy

 0 [MPa]

0 ,



mmin

mmax

t0 [mm]

3.99

10-5

10-3

0.35

0.6

0.42

Process Design of a Rectangle Box with Super Plastic Forming The most important significance of numerical simulation is not only to compute the existing processes but also to provide the engineers a virtual tool for the process design. For a new product the prototyping phase can be shorten remarkably. Meanwhile, the costs for the experiments and trial-and-error method can be reduced dramatically. Figure 5 shows a simple rectangle dish made of titanium alloy and should be produced with SPF. If no special condition was applied, the process would be also very simple. The sheet can then be blown under super plastic forming conditions into the die.

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Thickness [mm]

However, as a strict tolerance was set for the distribution of thickness on the bottom. The simple process cannot fill the requirement (Figure 5 b) as the sheet touches the die in the middle of the bottom without sufficient stretching. As a consequence, small thickness at the corners is resulted. 0.4

0.35

0.3

0.25 0.2 -10

10

30

50

70

Length [mm] a)

A rectangle dish for SPF

b) Distribution of thickness in diagonal direction with simple blow

FIGURE 5. Comparison of Simulation and measurement

Different methods can be used to reduce the thickness at the middle of bottom. One of them was suggested to use an additional pre-forming step to stretch the sheet in the middle in advance (Figure 6 a). The part is then deformed in the reverse direction to get the final shape (Figure 6 b). As it is not a standard process, much experimental work would be inevitable without the virtual tools offered by numerical simulation. With the simulation results, the parameters for the pre-forming such as the diameter and height of the dome can be determined within hours. The final process is also computed to confirm the feasibility.

a)

Preforming to stretch the middle of bottom

b)

Final form of the part

FIGURE 6. SPF with a pre-forming step

Thickness [mm]

0.45

Two steps

0.4

One step

0.35 0.3 0.25 0.2 -5

15

35

55

Length [mm] FIGURE 7. Different distributions of thickness with and without pre-forming

The thickness distribution was compared in Figure 7. It is seen that the distribution of thickness is much more homogeneous than the distribution with one simple SPF process. As a result the tolerance set has been satisfied.

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The numerical simulation can also reveal process failures if some mistakes could be made in the design. An example is shown in Figure 8. If the height of the dome in the first forming step is too large, the second forming process would fail due to the wrinkling, as shown in Figure 8.

FIGURE 8. Wrinkling in the final step

CONCLUSIONS A model for the simulation of super plastic forming was presented. Compare with power function, this model takes the variation of strain rate sensitivity into account and describe the material flow stress with more accuracy. It is very easy to be implemented in the codes for finite element simulation. The computational examples have shown the applications and verified the model as well.

ACKNOWLEDGMENTS The authors are grateful to the company FormTech GmbH, Germany for the measurement of thickness shown in Figure 5 and the collaboration of the simulation of SPF.

REFERENCES 1. N. R. Harrison et al., Influence of Friction and Die Geometry on Simulation of Superplastic Forming of Al-Mg Alloys, Advances in Superplasticity and Superplastic Forming, TMS 2004, 301-309 2. Li, Xiaomei and Soo, Steven, Numerical Simulation of the Superplastic Forming of a Dental Ridge Augmentation Membrane, Materials and Manufacturing Processes, 25: 12, 1470-1476 (2010). 3. R. Kaibyshev et al., Achieving high Strain Rate Superplasticity in an Al-Li-Mg Alloy through equal Channel angular Extrusion, Materials Science and Technology vol 21, 408-418 (2005) 4 M. Mulholland et al., Studies of Void Growth and Interaction in a Eutectic Sn-Pb Alloy, Advances in Superplasticity and Superplastic Forming, TMS 2004, 289-298 5. A. Arieli and A. Mukherjee, Factors Affecting the Maximum Ductility in a Superplastic Titanium Alloy, Materials Science and Engineering, vol 43, 47-54. (1980) 6. Yan Ma and T. G. Langdon, The Characteristics of Cavitation in Superplastic Metals and Ceramics, Metallurgical and Materials Transaction A, vol 27A, 873-878 (1996) 7 R.J. Lederich, S.M.L. Sastry, M. Hayase, T.L. Mackay, Superplastic formability testing, J. Met. 34 (1982) 16–20. 8 M. Vulcan, K. Siegert and D. Banabic, The Influence of Pulsating Strain Rates on the Superplastic Deformation Behaviour of Al-Alloy AA5083 Investigated by Means of Cone Test, Materials Science Forum Vols. 447-448 (2004) pp 139-144

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Prediction of Path Deviation in Robot Based Incremental Sheet Metal Forming by Means of a New Solid-Shell Finite Element Technology and a Finite Elastoplastic Model with Combined Hardening Yalin Kiliclara, Roman Laurischkatb, Ivaylo N. Vladimirova and Stefanie Reesea a

Institute of Applied Mechanics, RWTH Aachen University, Mies-van-der-Rohe-Str.1, 52074 Aachen, Germany b Chair of Production Systems, Ruhr-University Bochum, Universitätsstr. 150, 44780 Bochum, Germany

Abstract. The presented project deals with a robot based incremental sheet metal forming process, which is called roboforming and has been developed at the Chair of Production Systems. It is characterized by flexible shaping using a freely programmable path-synchronous movement of two industrial robots. The final shape is produced by the incremental infeed of the forming tool in depth direction and its movement along the part contour in lateral direction. However, the resulting geometries formed in roboforming deviate several millimeters from the reference geometry. This results from the compliance of the involved machine structures and the springback effects of the workpiece. The project aims to predict these deviations caused by resiliences and to carry out a compensative path planning based on this prediction. Therefore a planning tool is implemented which compensates the robots’s compliance and the springback effects of the sheet metal. The forming process is simulated by means of a finite element analysis using a material model developed at the Institute of Applied Mechanics (IFAM). It is based on the multiplicative split of the deformation gradient in the context of hyperelasticity and combines nonlinear kinematic and isotropic hardening. Low-order finite elements used to simulate thin sheet structures, such as used for the experiments, have the major problem of locking, a nonphysical stiffening effect. For an efficient finite element analysis a special solid-shell finite element formulation based on reduced integration with hourglass stabilization has been developed. To circumvent different locking effects, the enhanced assumed strain (EAS) and the assumed natural strain (ANS) concepts are included in this formulation. Having such powerful tools available we obtain more accurate geometries. Keywords: Simulation, Finite element method, Material modelling, Sheet metal forming, Combined Hardening PACS: 02.70.-c

INTRODUCTION For a cost efficient mass production, typically traditional sheet metal forming processes are used. However, for individual prototyping and small-batch manufacturing the lack of flexibility as well as high investment costs clearly point out the limits of conventional sheet metal forming for these parts. Therefore, different kinematics-based sheet metal forming processes have been developed (e.g. single-point incremental forming – SPIF and two-point incremental forming – TPIF). In most cases these methods only require simple dies or supports ([4], [5]). In this regard, a robot-based incremental sheet metal forming process (roboforming) [6] is developed at the Chair of Production Systems (LPS) at the Ruhr-University of Bochum. The roboforming principle is based on flexible shaping by means of freely programmable path-synchronous movements of two industrial 6-axis robots driving universal workpiece-independent forming tools. Figure 1 a) shows the robot cell which consists of two KUKA KR360 robots and a clamping device. The final shape is produced by the incremental inward motion of the forming tool in depth direction and its movement along the contour in lateral direction by driving either on parallel layers or on a helical path. The supporting tool holds the sheet on the backside by moving synchronously along the outer contour. The first possibility is that the supporting tool remains at the same level in thickness direction. Alternatively it may be directly opposed to the forming tool. Thereby, a predefined gap between the two hemispherical tools is achieved (Fig. 1b). Due to the kinematics of the robots and the tools’ simplicity the method of roboforming is highly flexible with respect to the geometries to be manufactured by this forming process. Therefore roboforming provides The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 129-133 (2011); doi: 10.1063/1.3623602 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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a quite interesting method for industrial prototyping and low batch size production. Using the two 6-axis kinematics it is possible to form undercuts (Fig. 1d) in a multi-step strategy. This goes beyond the forming of free-form surfaces (Fig. 1c) achieved by other forming strategies.

a)

b)

moving forming tool

c)

moving supporting tool moving forming tool

d)

moving supporting tool

FIGURE 1. a) Roboforming setup, b) basic forming strategies, c) free-form surface and d) cylinder with undercut (97° wall angle)

Nevertheless, up to now the resulting geometries formed in roboforming deviate several millimeters from the planned geometry. The reproducibility of these deviations can be shown by various experiments. Hence, the forming strategy with its forming parameters has a main influence on the process result. Depending on the material and the geometry being formed, parameters like the infeed, the forming velocity or the tool path especially affect the part’s accuracy. Additionally, the sheet is subject to springback. While these two effects emerge in the context of all incremental sheet metal forming processes, the result of the roboforming process is further influenced by the compliance of the robot. With increasing availability of fast computer hardware the finite element method is used more and more efficiently to predict and simulate material and structural behavior. First of all, the use of a purely isotropic hardening model (expansion of the yield surface) is not sufficient. Therefore, a finite strain constitutive model which combines nonlinear kinematic and isotropic hardening, based on the multiplicative split of the deformation gradient, developed in a recent work by Vladimirov et al. [7], is used in this work. The papers Choi et al. [8], Dettmer & Reese [9], Hakansson et al. [10], Menzel et al. [11], Svendsen et al. [12], Wallin et al. [13] are dealing with this subject. In order to reduce computational time, a new efficient finite element technology is developed. Nevertheless, the simulation of roboforming remains a difficult task. The computation time is extremely high for implicit simulations which is due to the complex contact behavior.

MATERIAL MODELLING The constitutive model is based on the multiplicative split     of the plastic deformation gradient into “elastic” and “inelastic” parts,     being the classical multiplicative split of F. As a result, a continuum mechanical extension of the classical rheological model of Armstrong-Frederick kinematic hardening [14] can be achieved. The Helmholtz free energy per unit undeformed volume  is additively decomposed into the three parts            . The first part  describes the macroscopic elastic material properties. The second term   corresponds to the elastic energy stored in dislocation fields due to kinematic hardening and vanishes if the kinematic hardening is zero. The third term represents elastic energy due to isotropic hardening, where  is the isotropic hardening variable. The Helmholtz free energy is a function of the elastic right CauchyGreen tensor        and the elastic part of the plastic right Cauchy-Green tensor is defined as   

     

 . Inserting this in the Clausius-Duhem inequality        results in a relation for the second Piola-Kirchhoff stress tensor S. The derivation of the material model is suitably carried out in the intermediate configuration. For the numerical implementation of the constitutive equations it is, however, more appropriate to work in the undeformed or reference configuration.

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The set of constitutive equations of the model in the reference configuration is summarized below [1]: •

Stress tensors !     "  $%   #"

•

#)"

  $*     %$ *    %

. / #) 0. /  . / 1

6 4   - 2 *  $    $   5 - 3

(2)

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FINITE ELEMENT TECHNOLOGY Recent research focuses on the large deformation version of a new eight-node solid-shell finite element based on reduced integration with hourglass stabilization [15]. The major problem of low-order finite-elements used to simulate thin structures as the sheet metal is locking, a nonphysical stiffening effect. Therefore in our recent solidshell formulation the enhanced assumed strain (EAS) as well as the assumed natural strain (ANS) concept are implemented to circumvent locking. To cure transverse shear locking the corresponding transverse shear terms evaluated at locking-free sampling points are interpolated within the element domain. The same procedure applied to the transverse normal strain cures curvature thickness locking. For the EAS concept, the derivation is based on the well-established two-field variational functional B C$ D   EI F DG HD3 KL BMN  

(5)

B6 C$ D   EI F DG HD KL  

(6)

J

J

which depends on the displacement vector u and the enhanced Green-Lagrange strain tensorD . In this enhanced strain tensor, the strain in thickness direction is enriched linearly. D3 is the compatible Green-Lagrange strain tensor. Hence, the EAS concept simplifies to a scalar equation, which leads in combination with the ANS concept to an efficient and robust element formulation. Further important key points to improve the efficiency of the hourglass stabilization are different Taylor expansions with respect to the shell director. The hourglass kernel OP depends on the material behavior in the integration point. This new finite element technology has been implemented into ABAQUS as user element subroutine (UEL).

FINITE ELEMENT SIMULATION The roboforming process is simulated by means of a finite element analysis performed in ABAQUS/Standard. In order to estimate the performance of the model, an experiment was conducted using a 220x220 mm² fully clamped DC04 metal sheet with a thickness of 0.8 mm. Both, the forming and the supporting tool have a tool tip radius of 6 mm. The Young’s Modulus is 206000 MPa and the Poisson’s ratio is 0.3. Instead of applying a simplified analytical function for the tool path, we are using a complex forming path obtained from the real-time path (optical measuring device Metris K600) of the robots. The x and y axes describe the plane of the sheet. In Fig. 2 a), the tool path is given. The z axis is used for the thickness direction as shown in Fig. 2 b).

131

FIGURE 2. a) Tool path in X-Y-View, b) Tool path in X-Z-View

The time dependent path is givenn by x(t), y(t) and z(t). These coordinates over time are a translated into a user subroutine to specify prescribed bounddary condition (DISP). In this, the influence of accelerration and velocity are be neglected. The simulation of a compllicated forming process like roboforming requires a reealistic description of the material behavior. As presented abovve, a large deformation elastoplastic material model with w combined nonlinear kinematic and isotropic hardening iss used to handle the springback effect of the sheett. It should describe the Bauschinger and ratcheting effect whiich belongs to the most characteristic phenomena of the t hardening behavior of metals. The Bauschinger effect is deffined by the fact that straining in one direction reducces the yield stress in the opposite direction, while ratcheting deenotes the progressive increase of the mean strain as a result of unsymmetrical stress cycles. Due to the fact that thhe contact behaviour is very complex, the solution tiime is dominated by the number of contact iterations which is i relatively high in this simulation. Therefore the contact c formulation is an important aspect. Here, the forming annd supporting tools are modeled as rigid bodies. The interaction between them and the deformable sheet metal is deefined by penalty surface to surface contact methodd with finite sliding. The friction coefficient is 0.05. The resultts shown in this paper are computed using an 8-node continuum-shell element with reduced integration (SC8R). The equivalent plastic strain of the deformed sheet metal for an infeed of 13.9 mm can be seen in Fig. 3 a) and b).

Figure 3. a) Equivalent plastic strain (FE-Simulation) b) Maximum infeedd

SUMM MARY AND FURTHER RESEARCH The presented work shows the applicability a of FE Analysis for an individual formiing process, operated by industrial robots to obtain an effectivve production of prototypes. Due to the fact that the computation time is still

132

very high, as well as still path deviations exist, different investigations are still to be done. With the implementation of the mentioned finite-element-technology, we expect a reduction of the CPU time. Additionally, the contact algorithms have to be improved. Having such powerful tools available we expect to obtain more accurate geometries.

REFERENCES 1. 2. 3. 4.

5. 6. 7 8 9 10 11 12 13 14 15

I. N. Vladimirov, M. P. Pietryga, S. Reese, International Journal for Numerical Methods in Engineering 75, 1–28 (2008). M. Schwarze, I. N. Vladimirov, S. Reese, Computer Methods in Applied Mechanics and Engineering, online (2010). M. Schwarze, S. Reese, International Journal for Numerical Methods in Engineering, online (2010). J. Douflou, A. Szekeres, P. Vanherck, “Force Measurements for Single Point Incremental Forming: A Experimental Study” in Advanced Materials Research, Proceedings of the 11th Int. Conference on Sheet Metal SHEMET, Nürnberg, 2005, pp. 441448. J. Jeswiet, F. Micari, G. Hirt, A. Bramley, J. Douflou, J. Allwood, “Asymmetric Single Point Forming of Sheet Metal”, Proceedings of the 55th CIRP General Assembly in Antalya, (2005). H. Meier, V. Smukala, O. Dewald, J. Zhang, “Two Point Incremental Forming with Two Moving Forming Forming Tools”, Proceedings of the 12th International Conference on Sheet Metal, Palermo, Sicily, Italy, 2007, p. 599-605. I. N. Vladimirov, “Anisotropic material modelling with application to sheet metal forming”, Dissertation, Technische Universität Braunschweig, (2009). Y. Choi, C. S. Han, J. K. Lee, R. Wagoner, International Journal of Plasticity 22, 1745-1764 (2006). W. Dettmer, S. Reese, Computer Methods in Applied Mechanics and Engineering 193, 87-116 (2004). P. Hakansson, M. Wallin, M. Ristinmaa, International Journal of Plasticity 21, 1435-1460 (2005). A. Menzel, M. Ekh, K. Runesson, P. Steinmann, International Journal of Plasticity 21, 397-434 (2005). B. Svendsen, V. Levkovitch, J. Wang, F. Reusch, S. Reese, Computers & Structures 84, 1077-1085 (2006). M. Wallin, M. Ristinmaa, International Journal of Plasticity 21, 2025-2050 (2005). Frederick, PJ Armstrong, CO, A mathematical representation of the multiaxial Bauschinger effect, C.E.G.B. Report RD/B/N731, Berkeley Nuclear Laboratories, Berkeley, U.K. (1966). M. Schwarze, S. Reese, International Journal for Numerical Methods in Engineering 80, 1322-1355 (2009).

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Earing Prediction in Cup Drawing using the BBC2008 Yield &riterion Marko Vrha, Miroslav Halilovia, Bojan Starmana, Boris Štoka, Dan-Sorin Comsab, and Dorel Banabicb, a

Laboratory for Numerical Modelling & Simulation; Faculty of Mechanical Engineering, University of Ljubljana; Ljubljana – Slovenia b Technical University of Cluj Napoca, CERTETA Research Centre, 103-105 B-dul Muncii, 400641 Cluj Napoca, Romania Abstract. The paper deals with constitutive modelling of highly anisotropic sheet metals. It presents FEM based earing predictions in cup drawing simulation of highly anisotropic aluminium alloys where more than four ears occur. For that purpose the BBC2008 yield criterion, which is a plane-stress yield criterion formulated in the form of a finite series, is used. Thus defined criterion can be expanded to retain more or less terms, depending on the amount of given experimental data. In order to use the model in sheet metal forming simulations we have implemented it in a general purpose finite element code ABAQUS/Explicit via VUMAT subroutine, considering alternatively eight or sixteen parameters (8p and 16p version). For the integration of the constitutive model the explicit NICE (Next Increment Corrects Error) integration scheme has been used. Due to the scheme effectiveness the CPU time consumption for a simulation is comparable to the time consumption of built-in constitutive models. Two aluminium alloys, namely AA5042-H2 and AA2090-T3, have been used for a validation of the model. For both alloys the parameters of the BBC2008 model have been identified with a developed numerical procedure, based on a minimization of the developed cost function. For both materials, the predictions of the BBC2008 model prove to be in very good agreement with the experimental results. The flexibility and the accuracy of the model together with the identification and integration procedure guarantee the applicability of the BBC2008 yield criterion in industrial applications. Keywords: Anisotropy, Yield criteria, BBC2008 model, Earing prediction, Numerical integration, NICE scheme, Simulation. PACS: 46.35.+z

INTRODUCTION Numerical simulation of the manufacturing processes is an important way of reducing the overall production costs and the failure rate. During the last decades, this computational instrument has extended its applicability in various industrial fields. The earing phenomenon has been noticed in the early applications of the rolled metallic sheets for the deep-drawing of cylindrical parts. The number of ears is usually four, but there are also situations when six or eight ears may occur at the upper edge of the drawn cups. Which particular evolution of the earing process will take place depends generally on the anisotropy of the sheet metal and the lubrication conditions. The development of the numerical techniques during the last three decades allows the earing prediction by finite element simulation. The quantitative analyses related to the accuracy of the numerical results show that the constitutive models used in the simulation have a significant influence on the predicted earing profile. As a consequence, many researchers have focused their interest on the investigation of the possibilities to improve the quality of the numerical predictions by adopting more realistic material models. The anisotropic yield criterion proposed by Hill in 1948 [1] was extensively used for the simulation of the cup drawing processes with the aim of analyzing the earing profile. But, since earing prediction based on the Hill 1948 yield criterion proved to be rather limited, more sophisticated yield criteria have been developed [2-14] during the last two decades. Some of these material models have been implemented in commercial finite element codes and used for a prediction of the earing profile. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 142-149 (2011); doi: 10.1063/1.3623604 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

142

As mentioned above, cylindrical cups obtained by deep-drawing usually exhibit four ears. In the case of some materials having a more specific anisotropy, six or even eight ears may develop during the drawing process. The experimental and theoretical studies have proved the existence of direct relationship between the number of ears and the variation of the anisotropy coefficient in the plane of the sheet metal. The occurrence of more than four ears can be predicted only by those yield criteria that use at least eight material parameters associated to different planar directions. In fact, such plasticity models have been developed mainly as a response to this challenge. An analysis of the number of mechanical parameters needed to correspondingly describe the anisotropy and their influence on the predicted earing profile has been performed by Soare and Banabic [15] using the Poly 8 yield criterion. The first yield criterion that uses general stress state and more than eight mechanical characteristics in the identification procedure has been developed by Barlat et al. [7]. The capability of this model to predict the occurrence of six or eight ears in the drawing of cylindrical cups has been proved by Yoon et al. in a series of papers [16-21]. Aiming at improving the accuracy of the earing predictions Yoon et al. [20] have taken in their simulations also the evolution of the anisotropy during the forming process into account. Comsa and Banabic [22] have recently proposed a simple plane-stress yield criterion (BBC 2008) defined as an extension of the BBC 2005 model [12]. As an alternative of using linear transformations of the stress tensor and absolute value function, the model is very convenient for finite element implementation. Due to the fact that in the identification procedure of the new constitutive model more than eight mechanical parameters can be used, a more accurate description of the anisotropy is possible. The capability of the BBC 2008 yield criterion to predict the occurrence of more than four ears will be proved in the next sections of this paper by performing corresponding numerical simulations of the cylindrical cup deep-drawing with two aluminium alloys considered.

BBC 2008 YIELD CRITERION The BBC 2008 yield criterion is a plane-stress criterion developed to describe the plastic behaviour of highly orthotropic sheet metals [22]. The yield surface defined by this model results from the implicit equation ) (V 11 , V 22 , V 12

V 21 , Y ) : V (V 11 , V 22 , V 12

V 21 )  Y

(1)

0

where V (V 11 , V 22 , V 12 V 21 ) t 0 is the equivalent stress (see below), Y ! 0 is the yield stress, while V 11 , V 22 , and V 12 V 21 are planar components of the stress tensor expressed in an orthonormal basis superimposed to the local axes of plastic orthotropy. One assumes that the third unit vector of the local basis is always perpendicular to the mid-surface of the sheet metal. In the particular case of the BBC 2008 yield criterion, the equivalent stress is defined as follows [22]:

V 2k w 1 L( r )

¦ ^w s

r 1

r 1

^ª¬ L

(r )

2k

 M ( r ) º¼  ª¬ L( r )  M ( r ) º¼

2k

`  w ^ª¬M sr

(r )

2k

 N ( r ) º¼  ª¬ M ( r )  N ( r ) º¼

2k

``,

A1( r )V 11  A (2r )V 22 ,

M (r )

ª¬ m1( r )V 11  m2( r )V 22 º¼  ª¬ m3( r ) V 12  V 21 º¼ ,

N (r )

ª¬ n1( r )V 11  n2( r )V 22 º¼  ª¬ n3( r ) V 12  V 21 º¼ ,

2

k , s  ` \ ^0` , w

2

2

(2)

2

(3 / 2)1/ s ! 1, A1( r ) , A (2r ) , m1( r ) , m2( r ) , m3( r ) , n1( r ) , n2( r ) , n3( r )  \.

The quantities denoted as k , A1( r ) , A (2r ) , m1( r ) , m2( r ) , m3( r ) , n1( r ) , n2( r ) and n3( r ) (r 1,! , s ) are material parameters. It is easily noticeable that the equivalent stress defined by Eq. (2) reduces to the isotropic formulation proposed by Barlat and Richmond [23] if A1( r ) A (2r ) m1( r ) m2( r ) m3( r ) n1( r ) n2( r ) n3( r ) 1/ 2, r 1,! , s. (3) Under these circumstances, the value of the integer exponent k may be adopted according to the crystallographic structure of the sheet metal, as in Barlat and Richmond’s model: k 3 for BCC alloys, and k 4 for FCC alloys. The other material parameters involved in Eq. (2) are evaluated upon a corresponding identification procedure. Their number is n p 8s , where s  ` \ ^0` is the summation limit. Let ne be the number of experimental values

describing the plastic anisotropy of the sheet metal. If ne t 8, the summation limit s must be chosen according to the constraint s d ne / 8. When ne  8, the minimum value s 1 must be adopted. In this case, the identification

143

constraints obtained from experiments should be accompanied by at least 8  ne artificial conditions involving the material parameters. For example, if ne 6, one may enforce m1(1) n1(1) and m2(1) n2(1) . Due to the expandable structure of the yield criterion, many identification strategies can be devised. Comsa and Banabic developed a numerical procedure based on the minimisation of an error-function operating only with normalized yield stresses and r-coefficients obtained from uniaxial / biaxial tensile tests [22].

IMPLEMENTATION OF THE BBC 2008 YIELD CRITERION IN ABAQUS The above presented constitutive model has been implemented in a general purpose finite element code ABAQUS via VUMAT subroutine. For the integration of the constitutive model the NICE (Next Increment Corrects Error) explicit integration scheme, developed recently by some co-authors, is used. Its task is to find a proper increment of the plastic multiplier 'O from a given total strain increment 'H ij . The basic ideas of the NICE scheme are presented in Halilovic et al. [24], whereas in Vrh et al. [25] the reader can find its theoretical background adequately elaborated. The comparison studies and the proof given in the appendix of the latter paper show, that the accuracy of the new scheme is comparable to the accuracy of the backward-Euler scheme, while it is computationally far more (up to ten times) efficient in explicit dynamics simulations. Implementation of the constitutive model via user subroutine requires integration of the constitutive model along a known strain path, which is mathematically reflected in known total strain increments 'H ij . Although deduction of the NICE integration scheme is general, its implementation for shell applications needs a particular care. Namely, in order to satisfy the zero normal stress condition during the whole integration a throughthickness strain increment has to be adequately chosen in each integration step.

Treatment of zero normal stress constraint in shell applications General approach for calculation of the through-thickness strain increment is derived in Vrh et al [25]. In classical elasto-plastic plane stress applications the user of the NICE scheme can calculate the through-thickness strain increment with the following simple equation § · w) C33kl 'H kl*  ¨ )  Cijkl 'H kl* ¸ 4E ¨ ¸ wV ij © ¹ (4) 'H 33  C3333  4 2 E





where 1

w) § · V ij ¸ ¨ ; i j 3 wV ij ­°0 Y w) w) w) w * ¸ , 4 w) C (5) Cijkl E ¨  ® ii 33 , H ij p H ¨ wV ij ¸ Y wV kl wY wH eq wV ii °¯ ij ; otherwise ¨ ¸ © ¹ and ) is the plastic potential (see Eq. (13)). In the above equations derivative w) / wV 33 is required which defines the influence of the through-thickness stress on yielding. The most elegant way to overcome this enigma is upgrading of the yield criterion with a simple substitution of the stress state using the deviatoric expression ) (V 11 , V 22 , V 12 , Y ) ) (V 11  V 33 , V 22  V 33 , V 12 , Y ) (6) Note that with such a manipulation the plane stress problem solution is not influenced since V 33 0 , whereas the plastic incompressibility condition w) / wV 11  w) / wV 22  w) / wV 33 0 (7) is preserved.

Integration of BBC 2008 with NICE scheme Assuming, that in each increment the total strain increments are available from the computed increments of the displacement field (note that strain increments 'H kl* are then given as input in VUMAT subroutine for further

144

processing, whereas 'H 33 can be calculated by considering (4)), the stress state at the end of the increment must be calculated using the following system of differential and algebraic equations:



) V ij , Y





'V ij

Cijkl 'H kl  'H klp

'H ijp

w) 'O wV ij

V kl 'H eqp 'Y

wY wH eqp

(8)

0

w) wV kl 'O Y

(9)

'H eqp

The plastic deformation is constrained by the consistency condition  0 , Eq. 8, which must be respected through all the integration process during the evolution of plastic strains. According to the NICE scheme, this is achieved by expanding the consistency condition into Taylor series, where higher order differentials are neglected. The numerical scheme is thus based, provided the values of the state variables are known at the beginning of the considered increment, on imposing   d 0 (10) to be fulfilled in the considered increment, which leads to w w  V ij  Y 0 (11) wV ij wY With regard to the forward-Euler approach which uses a differential form of the consistency condition, i.e. d 0 , our approach considers the additional term  . Though this term should be zero, because it represents a function whose value should obey the consistency condition  0 , numerically this is usually not true. This small difference between the two explicit schemes, NICE and forward-Euler, is the key reason for a considerable improvement in stability of the numerical integration. The remaining equations in the above DAE system, which accomplish the integration procedure, are the evolution equations, here expressed in the incremental form (9). Considering those equations and expanded consistency condition (11) yields the increment of the plastic multiplier w Cijkl H kl  wV ij (12) 'O w V kl wV kl w w w dY  Cijkl Y wV ij wV kl wY dH eqp With the increment of the plastic multiplier 'O calculated the incremental solution of the elasto-plastic model, defined by the evolution equations (9), can be considered solved. The numerical integration procedure requires a calculation of the yield function  and its respective derivatives in every increment. For implementation purposes, the yield function can be rewritten in the following form: V 2 k V ij Vˆ V ij )  1 0 (13) 1 Y 2k Y 2k with the derivatives used in the numerical scheme being: 1 w) Vˆ w) 2k 2 k 1 ; wY wVˆ Y 2 k Y (14) s ª r r r w) w) wV⏳ wL wV wM  wV? wN  º   « » r wV ij wVˆ r 1 «¬ wL r wV ij wM  r wV ij wN  wV ij »¼ where





¦

145

wVˆ

r wL

wVˆ wN

r



ª r r 2k  w  1 wr 1 « L  M  ¬





2 k 1

ª r r 2k  w  1 ws  r « M   N  ¬





r r  L  M 

2 k 1

2 k 1 º

» ¼









r r  M   N



 



2 k 1 º

(15)

» ¼

2 k 1 2 k 1 º ½ ­ r 1 ª  r r r r  L  M  °w « L  M » ° ° ¬ ¼ ° 2k  w  1 ® ¾ ° ws  r ª M  r  N  r 2 k 1  M  r  N  r 2 k 1 º ° « »° ° ¬ ¼¿ ¯

wVˆ

r wM 







and wL w r

r wM  w

^A

M

r

r N 

 r

r

, A 2 ,  A1  A 2

1

1

wN  w  r M

r r

^M m , M m , M  m  m , m 2V ` ^N n , N n , N  n  n , n 2V `

r

1 r N

, 0`

r r 1

r r 1

r r

r

2

r r 2

r

r 2

r 2

r

1

r

r 3

1

r 3

12

12

(16)

r r m1 V 11  V 33  m2 V 22  V 33

r

n1

V11  V 33  n2 r V 22  V 33 w, w r

Equation (16) is due to simplicity written in Voigt notation, where

­° w, r w, r w, r w, r ½° , , , ® ¾ for wV wV 22 wV 33 wV12 ° ¯° 11 ¿

r 1, !, s . Let us remind that according to the explicit approach all the state variables appearing in the above equations and expressions are written at the beginning of the considered increment. Once the increment of the plastic multiplier 'O is calculated, the respective increments of the other state variables can be readily calculated using Equations (9). The integration procedure is presented in Figure 1.

FIGURE 1. Integration (implementation) of constitutive model

146

EARING PREDICTION USING THE BBC2008 YIELD CRITERION In this section a validation of the presented approach is discussed, based on two performed round cup deep drawing simulations. Due to orthotropic material model only a quarter section of the cup with the corresponding symmetric boundary conditions applied is analyzed. A total of 2560 shell elements with reduced integration (ABAQUS S4R) and 21 section points through the sheet thickness are used for the simulation. To model the elastoplastic sheet metal response the BBC2008 model is employed. Its implementation with respective parameters identified for each of the sheet metals considered is enabled via user material subroutine VUMAT. The material parameters of the equivalent stress were calculated using experimental values of the normalized yield stresses and rcoefficients in uniaxial and biaxial tension. A detailed description of the BBC2008 identification procedure is given in [22].

Earing prediction for AA5042-H2 The simulation of deep drawing of aluminium AA5042-H2 follows the one, presented in Yoon et al. [20]. The tool geometry is as follows: blank diameter is 76.07mm, die opening diameter is 46.74 mm, punch diameter is 45.72 mm, die-profile radius and punch-profile radius are 2.28 mm. The initial thickness of the blank is 0.274 mm and holder force is 10 kN. Displacement of the punch is set to be large enough to pull the whole blank into the die. The friction is considered with the Coulomb model, the coefficient of friction being 0.008 for all contact surfaces. In this 18.416 H p

eq case the Voce hardening law Y 404.16  107.17 e is assumed to model the work hardening behaviour of the sheet metal. Material data (normalized yield stresses and r-coefficients) are given in [20], whereas the identified parameters of the BBC2008 model are given in Table 1.

TABLE 1. Parameters for 16 parameters BBC2008 version for AA5042-H2

k

s

w

A1(1)

A (1) 2

m1(1)

m2(1)

m3(1)

4

2

1.224744

0.719915

0.350253

0.573905

0.606047

0.718855 x 10-4

n1(1)

n2(1)

n3(1)

l1(2)

l2(2)

m1(2)

m2(2)

m3(2)

0.306974

0.492406

0.15355 x 10-5

0.564817

0.322344

0.374385

0.468435

0.737118

n1(2)

n2(2)

n3(2)

0.320818

0.641638

0.

Figure 2a shows the final geometry of deep drawn cup, as simulated, with the equivalent plastic strain field distribution displayed, whereas in Figure 2b a comparison between the predicted ears and the experimental ones [20] is given where h(D ) denotes the height of the deep drawn cup as a function of angle D (angle from the rolling direction). Also, for the sake of comparison, the calculated ears profile with the CPB06ex2 model [20] is included.

FIGURE 2. Earing prediction for aluminium AA5042-H2, a) Simulation b) Ears profile

147

Prediction of ears profile with the BBC2008 model is in good agreement with results of CPB06ex2 model, proposed by Yoon et al. [20]. The predicted ears are in same location, but are predicted to be little more intense. Eight ears are predicted by the model and agreement with experimental results is, at least qualitatively, very good.

Earing prediction for AA2090-T3 BBC2008 is in this section validated also on aluminium AA2090-T3. The round cup drawing experiment and a simulation with Yld2004 model is presented in Yoon et al. [18]. In this case the tool geometry is as follows: blank diameter is 158.76 mm, die opening diameter is 101.48 mm, punch diameter is 97.46 mm, die-profile radius and punch-profile radius are 12.70 mm. The initial thickness of blank is 1.6 mm and holder force is 22.2 kN. The friction is considered with the Coulomb model, the coefficient of friction being 0.1 for all contact surfaces. Hardening is modelled with the following stress-strain curve Y



646 0.025  H eqp



0.227

. The identified parameters of the

BBC2008 model for 8 and 16 parameters versions are given in Table 2 and Table 3, respectively. TABLE 2. Parameters for 8 parameters BBC2008 version for AA2090-T3

k

s

w

A1(1)

A (1) 2

m1(1)

m2(1)

m3(1)

n1(1)

n2(1)

n3(1)

4

1

1.500000

0.449938

0.513218

0.630315

0.601445

0.727299

0.153818

0.479391

0.499818

TABLE 3. Parameters for 16 parameters BBC2008 version for AA2090-T3

k

s

w

A1(1)

A (1) 2

m1(1)

m2(1)

m3(1)

n1(1)

n2(1)

4

2

1.22474

0.130866

0.621742

0.783422

0.660402

0.79 x 10-4

0.110991

0.048245

n3(1)

l1(2)

l2(2)

m1(2)

m2(2)

m3(2)

n1(2)

n2(2)

n3(2)

0.307522

1.033922

-0.071963

1.13 x 10-4

0. 77 x 10-4

0.538047

0.055764

1.018603

0.778150

Figure 3a displays the simulated final geometry of deep drawn cup (for 16 parameters model) and the corresponding equivalent plastic strain distribution, whereas in Figure 3b a comparison between predicted and experimental ears is given. Also, for the sake of comparison, ears profile calculated with the Yld2004 model [18] is included.

FIGURE 3. Earing prediction for aluminium AA2090-T3, a) Simulation b) Ears profile

Predictions of the BBC2008 model are in good agreement with a prediction of the Yld2004 model and also with the experiment [18]. As expected, the 8 parameters version was unable to predict six ears, which were experimentally observed. On the contrary, the 16 parameters version predicts 6 ears and their location, and at least qualitatively, the results are in good agreement with the experiment.

148

CONCLUSION In the paper the authors have shown an approach to modelling of plastic anisotropy in numerical simulation, which includes the BBC2008 plane-stress yield criterion, an efficient procedure for its implementation and a procedure for the identification of the respective model parameters. The constitutive model is derived in a form of a finite series that can be correspondingly expanded to retain more or less terms, depending on the available experimental data. The simulations of deep drawing of a round cup, which were performed with ABAQUS/Explicit with shell S4R elements, prove the capability of the model to predict more than four ears occurrence. Further, considering a comparison with the experimentally obtained ears profiles we can conclude that the presented approach is physically objective and is able to predict complex plastic anisotropic behaviour of sheet metal. Since the BBC2008 model does not use linear transformations of the stress tensor and absolute value function in the yield criterion formulation it is very convenient for implementation in finite element programs. Due to the fact, that it is in this paper explicitly integrated, the computational efficiency of the approach should be superior in the simulation of sheet metal forming processes comparing to existing techniques.

REFERENCES 1. R. Hill, A theory of the yielding and plastic flow of anisotropic metals, Proceedings of the Royal Society of London. Series A, 193, p.281-297 (1948) 2. F. Barlat and J. Lian, Plastic behaviour and stretchability of sheet metals (Part I): A yield function for orthotropic sheet under plane stress conditions. Int. J. Plasticity 5, 51–56 (1989). 3. F. Barlat et al., A six-component yield function for anisotropic materials. Int. J. Plasticity 7, 693–712 (1991). 4. F. Barlat et al., Yielding description for solution strengthened aluminium alloys. Int. J. Plasticity 13, 185–401 (1997). 5. F. Barlat et al., Yield function development for aluminium alloy sheets. Journal of the Mechanics and Physics of Solids 45, 1727–1763 (1997) 6. F. Barlat et al., Plane stress yield function for aluminium alloy sheets – Part 1: Theory. Int. J. Plasticity 19, 297–319 (2003). 7. F. Barlat et al., Linear transformation based anisotropic yield function. Int. J. Plasticity 21, 1009-1039 (2005). 8. A.P. Karafillis and M.C. Boyce, A general anisotropic yield criterion using bounds and a transformation weighting tensor, Journal of the Mechanics and Physics of Solids 41, 1859–1886 (1993). 9. G. Ferron et al., A parametric description of orthotropic plasticity in metal sheets, Int. J. Plasticity 10, 431-449 (1994). 10. D. Banabic et al., A new yield criterion for orthotropic sheet metals under plane-stress conditions. Proceedings of the 7th Conference ‘TPR2000’, Cluj Napoca, Romania, 217–224 (2000). 11. D. Banabic et al., Non-quadratic yield criterion for orthotropic sheet metals under plane-stress conditions. International Journal of Mechanical Sciences 45, 797–811 (2003). 12. D. Banabic et al., An improved analytical description of orthotropy in metallic sheets. Int. J. Plasticity 21, 493–512 (2005). 13. D.S. Comsa and D. Banabic, Numerical simulation of sheet metal forming processes using a new yield criterion. Key Engineering Materials 344, 833–840 (2007). 14. S. Soare et al., Applications of a recently proposed anisotropic yield function to sheet forming, In: Advanced Methods in Material Forming, Ed. Banabic D., Springer, Heidelberg-Berlin, 131-149 (2007). 15. S. Soare and D. Banabic, About the mechanical data required to describe the anisotropy of thin sheets to correctly predict the earing of deep-drawn cups, International Journal of Material Forming 1, 285-288 (2008). 16. J.W. Yoon et al., Prediction of eight ears in drawn cup based on a new anisotropic yield function, In: Proc. of the NUMIFORM 2004 conference, Columbus (2004) 17. J.W. Yoon et al., Plane stress yield function for aluminium alloy sheet – Part II: FE Formulation and its implementation. Int. J. Plasticity 20, 495-522 (2004). 18 J.W. Yoon et al., Prediction of six or eight ears in a drawn cup based on a new anisotropic yield function, Int. J. Plasticity 22, 174-193 (2006). 19. J.W. Yoon et al., Analytical prediction of earing for drawn and ironed cups, NUMISHEET2008 Proceeding, Interlaken, Switzerland, 97-100 (2008). 20.J.W. Yoon et al., Earing predictions for strongly textured aluminium sheets, International Journal of Mechanical Sciences, 52, 1563-1578 (2010). 21. J.W. Yoon et al., A new analytical theory for earing generated from anisotropic plasticity, Int. J. Plasticity, (in press) (2011). 22. D.S. Comsa and D. Banabic, Plane-stress yield criterion for highly-anisotropic sheet metals, Numisheet 2008, Interlaken, Switzerland, 43-48 (2008). 23. F. Barlat and O. Richmond, Prediction of tricomponent plane stress yield surfaces and associated flow and failure behavior of strongly textured f.c.c. polycrystalline sheets, Materials Science and Engineering, 95, 15-29, (1987). 24. M. Halilovic et al, NICE-An explicit numerical scheme for efficient integration of nonlinear constitutive equations, Mathematics and Computers in Simulation 80, 294-313 (2009). 25. M. Vrh et al., Improved explicit integration in plasticity, International Journal for Numerical Methods in Engineering 81, 910-938 (2010).

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Formability study of magnesium alloy AZ31B Z.G Liua, P. Lasneb, E. Massonia a CEMEF (Centre of Material Forming), Mines-Paristech 1, rue Claude Daunesse, BP 207, 06904, Sophia-Antipolis Cedex, France E-mail: [email protected], elisabeth.massoni@ mines-paristech.fr b TRANSVALOR company 694, av du Dr Maurice Donat, 06255, Mougins Cedex, France E-mail: [email protected]

Abstract. The main aim of this paper is to study the formability of the AZ31B magnesium alloy at various temperature and strain rates. The tensile tests are performed to describe the rheological behavior of material, and the constitutive law is identified with Voce law [1], which contains a softening item. The law is proved effectiveness by fitting the equation with the experimental data. Nakazima experiments with hemispherical punch have been performed at CEMEF on a hydraulic testing machine. Six strain paths are selected by performing various sample geometries [2]. The Aramis© Optical strain measurement system has been used to obtain principle forming limit strain. The Forming Limit Diagram (FLD) is obtained by the critical point on the specimen surface at various temperatures. It is shown that the forming limit curve is higher at high temperature. Based on the Voce law model, finite element simulations of deep drawing test have been done with the commercial finite element code FORGE® in order to investigate the feasibility of hot stamping process for AZ31. In the simulation, the punch load and the thickness distributions have been studied. Meanwhile, the cross-shaped cup deep drawing simulations have been conducted with the data provided in the conference Website. The similar conclusion are obtained that the formability of AZ31 improve at high temperature and the simulation is effective in hot stamping processing. The study results are helpful for the application of the stamping technology for the magnesium alloy sheet [3]. Keywords: Formability, AZ31B Mg alloy, FLD, Finite element method. PACS: 62.20.F

1. INTRODUCTION Magnesium alloys are the lightest structural alloys at present, its density is about two-third that of aluminum alloy and one-fourth that of steel [4]. It has been widely used for structural components in the aerospace, automobile and civil industry in order to reduce fuel consumption. However the magnesium-alloy usually exhibits limited ductility at room temperature due to their hexagonal close-packed (HCP) crystal structure [5]. But the mechanical properties of magnesium-alloy can be improved at elevated temperatures. The AZ31 is one of magnesium-based alloy which include 3%Al and 1%Zn, and it has been reported that AZ31 sheets have good formability at elevated temperature between 150 and 3000C [6]. Deep drawing process is an important and wide-used process in assessment of formability of sheet metal, and it has been studied using the finite element method and experiments [4, 7, 8]. However, there are still few studies to profoundly investigate the forming limits of Mg sheet at elevated temperatures experimentally and theoretically. In the present study, the deformation behaviors of the AZ31B magnesium sheet are analyzed by tensile test at various temperatures and strain rates. The relationship between stress and strain is studied and an advanced constitutive model containing a softening item is developed to describe the deformation behavior of AZ31. The Nakazima experiments with hemisphere punch have been performed at various temperatures to obtain the experimental forming limit strains. Meanwhile, the finite element simulations of deep drawing test have been performed with the commercial finite element code FORGE®. In simulation, the punch load, thickness distribution and temperature distribution have been gotten, the forming limit strain have been obtained and compared with experiment.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 150-157 (2011); doi: 10.1063/1.3623605 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

150

2. THE MECHANICAL PROPERTY The material used in this study is a commercial AZ31B-O magnesium alloy sheet with thickness of 1.2mm. The average grain size is about 10µm. The tensile tests are selected to obtain the mechanical data of material, because it is the basic and simple test to study mechanical property of material in industry.

2.1 The tensile test The tensile specimens are made from the sheet with a gauge of 40mm length and 30mm width. The tensile tests are carried out in an Instron tensile machine between room temperature and 300 0C under three constant tensile velocities (0.04, 0.4, 4mm/s). Figure 1 shows the deformed tensile test specimens of AZ31 Magnesium alloy sheet at various temperature and velocities. It is shown that the elongation increase remarkably with increasing temperature and decreasing velocity. It is evident that the ductility is substantially enhanced.

(a) Specimens at various temperatures (v=0.4mm/s) (b) Specimens at various strain rate (T=3000C) FIGURE 1. Deformed specimens at different temperatures and strain rates

The true stress–true strain curves obtained at various temperatures and strain rates are shown in Figure 2. These curves are plotted up to appearance of the fracture phenomenon. It can be seen that the flow stress decreases with increasing temperature while the strain to fracture increases with increasing temperature, and the yield stress is higher at the lower strain rate. At room temperature, the strain hardening phenomenon is clear for each strain rate. But at higher temperatures, the flow stress is practically independent of strain, no significant work hardening is observed for any specimens. The softening phenomenon can be seen at high temperature, especially at 300 0C. 

300

200 150 100 50

400

-1

RT, 0.01S 0 -1 100 C, 0.01S 0 -1 200 C, 0.01S 0 -1 235 C, 0.01S 0 -1 300 C, 0.01S

350

True stress (MPa)

True stress (MPa)

400

-1

250 200 150 100

300

0.2

0.3

0.4

True strain

(a)

0.5

0.6

0.7

0 0.0

250 200 150 100 50

50

0.1

-1

RT, 0.001S 0 -1 100 C, 0.001S 0 -1 200 C, 0.001S 0 -1 235 C, 0.001S 0 -1 300 C, 0.001S

350

True stress (MPa)

0

100 C, 0.1S 0 -1 200 C, 0.1S 0 -1 235 C, 0.1S 0 -1 300 C, 0.1S

250

0 0.0





300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.0

0.1

True strain

 =0.1s-1

0.2

(b)  =0.01s-1 FIGURE 2. True strain-True stress curves at the various temperatures

0.3

0.4

0.5

0.6

0.7

0.8

0.9

True strain

(c)

 =0.001s-1

2.2 The Equation for flow stress The Voce Law including a softening item is used to describe the flow behavior of AZ31. The equation is following [1]:

  K m ,

K  (1  W ) K ecr  W * K sat

(1)

K ecr : Strain hardening function, K sat : saturation function, W : softening function For the AZ31 magnesium alloy sheet, it has been shown clear softening character, but the saturation stage is not evident. So, the saturation function K sat has not been considered. The power law has been used to describe the

151

strain hardening. The softening phenomenon is different at various temperatures, so the polynomial expressions which depend on temperature have been used in softening function. The strain hardening and softening expressions are following: (2) K ecr  k (   0 ) n * exp(  / T )

W  1  exp( r *  ) , with r  r0  r1 *T

(3)

Finally, the Voce law is following:

  k (   0 ) n * exp(  / T ) * exp((r0  r1T ) *  ) *  ( m m *T ) 0

(4)

1

In order to get a proper global constitutive equation to describe the flow behavior of AZ31 magnesium alloy sheet at various temperatures and strain rate, the genetic algorithm (GA) in Matlab toolbox® has been used. The objective function is following: n

O( f ) 

 (

exp i

  i )2 cal

(5)

i

n

 (

exp 2 i

)

i

n is the number of data point,  i is the stress at the data point “i”. The genetic algorithm is sensitive to the initial value of parameters, the parameter value ranges have been set by fitting the individual stress-strain curves, and then getting the maximum and minimum value of parameters. Finally, according the Voce law and the setting objective function, the global optimal fitting parameters have been obtained (table 1). Table 1. optimal fitting parameters for Voce constitutive equation Parameters

k

n



r0

r1

m0

m1

0

Optimal evaluation

235.15

0.13787

86.343

-0.999

0.0107

0.062

0.00043

1.00E-06

300

200

200

150 100 50 0 0.0

Exp, 100deg Exp, 200deg Exp, 235deg Exp, 300deg Cal, 100deg Cal, 200deg Cal, 235deg Cal, 300deg

250

stress (MPa)

250

stress (MPa)

300

Exp, 100deg Exp, 200deg Exp, 235deg Exp, 300deg Cal, 100deg Cal, 200deg Cal, 235deg Cal, 300deg

150 100 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.0

0.8

strain -1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

strain

(a)  =0.01s (b)  =0.001s-1 FIGURE 3. Comparison between calculated and experimental flow curves at various temperatures.

The comparison between calculated and experimental flow curves at various temperatures has been shown in figure 3. It can be seen that the constitutive equation can describe correctly the flow properties of AZ31, especially the softening behaviors. From the optimal fitting equation, the sensitivity of process parameters can be analyzed. The choice of this law has undoubtedly improved the modeling of magnesium at every temperature, but it is clear that there is still some dissatisfaction, especially during the phase of softening. To do so would undoubtedly go to a finer coupling taking into account the evolution of microstructure.

3. EXPERIMENTAL FORMABILITY STUDY It is known that the Nakazima and the Marciniak tests are two common experimental tests which provide information on formability of sheet material. The main difference between these tests is the shape of the punch which is respectively hemispherical or flat. The Nakazima set-up has been initially selected for a straight-forward reason that it is much simpler to perform than Marciniak test, and it does not need any driving blank.

152

3.1 Nakazima test set-up The Nakazima set-up is made of a hemispherical punch, a die, a blank-holder with drawbeads to prevent any sliding motion. A furnace has been installed close to the press so that the sheet can be transferred readily into the stamping device (figure 4). An AZ31 magnesium alloy sheet of 1.2mm thick is used for the experiment. The device and these tests have been performed at CEMEF on a tension-compression hydraulic testing machine Dartec (maximum 300mm/s punch velocity and 300kN punch load).The advantages of this set-up are to support high temperature and to allow a rapid insertion of the hot blank into the press (between the die and the blank holder) thanks to the pneumatic cylinder. In stamping test, a rectangular sheet specimen is clamped firmly, and stretched by a semi-spherical headed punch. In order to prevent wrinkle and keep the sheet surface smooth, the drawbead have been used in the blank holder zone. The blank holder plays an important role in stamping process, it keeps the material flowing smoothly and makes the blank and die contacting well. In this study, the constant blank holder force has been used.

(a) (b) Figure 4. set-up for Nakazima tests (a) and the temperature controller (b)

3.2 Experimental FLD The selected set-up has to allow varying the strain paths in order to get fracture in different strain states and strain history configurations: from a uniaxial tensile mode to pure bi-expansion mode. This test technique requires the use of different sample geometries and lubrication condition to generate all possible strain states. Each sample represents one specific strain path on the FLD [2]. In this study, the BN (Boron Nitride) lubricants have been used. Generally, two optical strain measurement systems, called ASAME© (automated strain analysis and measurement environment) and Aramis©, are used to measure forming limit strains. These measurement systems are widely used to analyze formability issues for many industrial applications. In this work, The Aramis strain measurement system has been used to obtain principal strain. Figure 5 shows the deformed samples after Nakazima tests and the strain distribution by Aramis. The samples have appeared necking or fracture phenomenon ultimately.

(a) Samples after test (b) Aramis strain measurement Figure 5. Samples after Nakazima tests and strain measurement

It has been shown that the localized necking of AZ31 is followed closely by the diffused necking at room temperature from stress-strain curves. But the plasticity is improved obviously with temperature increasing, the

153

plastic deformation can still occur after diffused necking at high temperature, and then up to localized necking. So, generally, the forming limits diagram has three types: the safe FLD, the necking FLD and the broken FLD. The broken FLD of AZ31 has been shown in figure 6. It is illustrated that the strain paths of experimental FLC have discrepancies with theoretical FLC, because the strain measurement and lubricant is complicated in experiment. And the more important reason is a sudden change of strain occurs after necking, this phenomenon is obvious on the deformation images. It must be defined a certain safety margin if the broken FLD used in the actual application. The tensile tests have demonstrated that the AZ31 sheet still have some plasticity when necking occurs, especially at high temperature. So, the necking FLC is lower than the broken FLC in the forming limit diagram. This result is also published in the reference [3]. Nakazima_experiment 0.7 0.6

Major strain

0.5 0.4

300deg 200deg

0.3

100deg

0.2 0.1 0 -0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Minor strain

Figure 6. FLD determined by Nakazima experiment

4. SIMULATION 4.1 hemisphere punch deep drawing The Nakazima stamping (hemispherical punch drawing) simulations are conducted at various temperatures. In these simulations, the geometry is just quarter model about the Z-axis due to axisymmetric deformation mode. The blank is meshed with two elements through the thickness. Smaller meshing is used in forming zone, and coarse meshing is used in flange area. A drawbead is used in order to compare with real Nakazima hot stamping test. The flow stress model of AZ31 sheet is based on Voce law with parameters identified and shown in table 1. The friction coefficient used in the simulations is obtained from reference [8] and it is assumed not to vary locally with temperature and pressure. The heat transfer coefficient is assumed to be uniform for all surfaces and the value is selected based on the results published in the literature [9]. The blank holder force also plays an important role in simulation. In this work, constant blank holder force is used. The inputs to the FE simulation are shown in table 2. Table 2. Process parameters used in simulation

Tooling setup Punch diameter (mm): 60 Blank-holder and die corner radius (mm): 10 Thickness (mm): 1.2 Friction coefficient, μ: 0.10

Mechanical properties Density (kg/dm3): 1.78 Young’s modulus, E (GPa): 45 Poisson’s ratio, ν: 0.3

154

The punch loads obtained from the simulation at various temperatures are compared with experiments in figure 7. The maximum punch load obtained at all the simulated temperature is higher than the load in experiment. It is shown that the tendency is obvious, when the temperature increase, the punch load decrease. And the results match well at small displacement. Nevertheless, the discrepancies appear especially at large displacement. Many reasons could explain these deviations. Firstly, the friction coefficient is overestimated especially at low temperature. Secondly, the behavior model well represents the flow behavior at high temperature from stress-strain curves, but there is deviation at low temperature. Thirdly, in the simulation, the material is considered isotropic. However, magnesium alloy sheets exhibit some normal anisotropy, although it is not obvious. However, the best anisotropic yield criterion that represents yielding in Mg alloy may still have to be determined. 25

35 experiment,100deg

25 20 15 10

simulation, 200deg

9

experiment, 200deg

8

Punch load (KN)

20

Punch load (KN)

Punch load (KN)

10

simulation, 100deg

30

15

10

7 6 5 4 3 2

5

5

Simulation, 300deg Experiment,300deg

1

0

0

0

0

5

10

15

20

25

30

0

5

10

Displacement (mm)

15

20

25

30

35

40

0

5

10

15

20

25

30

35

40

Displacement (mm)

Displacement (mm)

(a) T=1000C (b) T=2000C (c) T=3000C FIGURE 7. Punch load obtained from simulation and experiment at various temperatures.

Thickness distribution from experiments is available at various temperatures from Aramis system. A comparison of the simulated and experimental results along a section of blank is shown in figure 8. The tendency of thickness distribution at 2000C predicted by simulation fit well with corresponding experiment result for all three geometry samples. However, less thickening and more thinning have been observed in the simulation comparing to the experiments. Maximum thinning has been observed in punch radius zone for both simulation and experiment. It is probably because the strength of the punch radius zone in warm forming is not uniform and depends on the temperature distribution. Exp. (w=200mm)

0,8

thickness thinning

Cal. (w=200mm) 0,7

Exp. (w=125mm)

0,6

Cal. (w=125mm) Exp. (w=50mm)

0,5

Cal. (w=50mm)

0,4 0,3 0,2 0,1 0 0

10

20

30

40

50

60

70

80

90

100

surface distance (mm)

FIGURE 8. Thickness distribution obtained from simulation and experiment at 2000C.

4.2 cross-shaped cup deep drawing The AZ31 cross-shaped cup deep drawing simulations are performed using FORGE software. The stress-strain data using uniaxial tension test and the tool setting are provided by conference committee. The surface temperature of the die and the blank-holder are 2500C, the surface temperature of punch and pad are 100 0C. The punch velocity is 0.15mm/s. the blank holder force increase linearly from 1.8KN to 4.8KN, the pad force increase linearly from 0 to

155

4KN. The lubricant is barrierta L55/2. A cross-shaped cup sample and the simulation model are shown in figure 9. The simulation geometry is just quarter model about Z-axis thanks to axisymmetric deformation. The thickness of blank material is 0.5mm and meshed with two elements through the thickness. The Hill 48 anisotropic yield criterion is used to describe material anisotropy. The inputs to the FE simulation are shown in table 3. Table 3. Process parameters used in simulation

Mechanical properties Young’s modulus (GPa): 45 Poisson’s ratio, ν: 0.3 Friction coefficient, μ: 0.1 Density (kg/mm3): 1.770

Thermal properties Thermal conductivity (W/(m 0C): 96 Heat capacity (J/(kg 0C): 1000 Interface heat transfer coefficient (W/(m2 0C) : 4500

(a) A cross-shaped deep drawing cup (b) the quarter of model in FORGE FIGURE 9. A cross-shaped cup sample (a) and the quarter of simulation model (b)

The punch loads obtained from simulation are shown in figure 10(a). The punch loads have not much difference with various meshing size, and there are slight fluctuation at each punch force curve especially at high displacement. The possible reason is the temperature does not distribute uniformly during forming processing, the force distribution is fluctuated correspondingly. And other possible reasons are the boundary condition and contact which play an important role in and change during simulation. Figure 10(b) shows the temperature distribution at the punch displacement of 25mm. As the stamping is non-isothermal, the temperature distributes gradually along the wall. And similar phenomena are observed during all of processing, there are no obvious temperature changes at the punch radius and die radius zone. 16

Punch Load (KN)

14 12 10 8

Mesh=1.5mm

6

Mesh=1mm

4

Mesh=0.5mm

2 0 0

5

10

15

20

25

30

Punch displacement (mm)

(a) Punch load and displacement curves (b) Temperature distribution at deep height 25mm FIGURE 10. Punch load and displacement curves at various meshing (a) and temperature distribution of the blank (b)

Thickness distributions are available in simulation at the punch displacement of 10mm and 25mm (figure 11). It is obviously shown that the thickening occurs at flange zone and thinning at cup wall zone. The maximum thinning is observed at the die corner radius for both punch displacement which is contrary to the conventional stamping where the maximum thinning is obtained at punch radius. It is probably due the strength of the cup wall in this stamping forming is not uniform as in conventional isothermal stamping. As in figure 10(b), the initial die and blank holder temperature (2500C) are higher than the punch and pad temperature (100 0C). Therefore, the yield strength of the material at the die corner radius is lower compared to the material at punch corner caused thinning in the cup

156

wall. The thickness distribution is sensitive with the temperature, and the plasticity improves dramatically with temperature increasing.

(a) Deep height=10mm (b) Deep height=25mm FIGURE 11. Thickness distribution of the blank along the center line of X-axis

CONCLUSION AZ31 magnesium sheets exhibit higher yield stress and smaller elongation at room temperature, but its formability improve significantly with temperature increasing and strain rate decreasing. The tensile experiments have been performed to describe the flow behavior of AZ31 magnesium alloy sheet. The softening phenomenon is obvious at higher temperature. A new model containing softening items is developed based on mathematical fitting, and the genetic algorithm is used to get a global optimal equation. It is proved that this Voce law can fit the softening phenomenon well, but the discrepancies still exist and the model must be improved. The Nakazima hot stamping experiments have been performed to investigate the formability of material. The broken FLD are obtained using image correlation system. Although the strain paths of experimental FLC have discrepancies with theoretical FLC, It is clearly illustrated that the formability is better at higher temperature. The hot Nakazima simulation based on Voce law is performed with FORGE. From the comparison between simulation and experiment, the punch load predicted by simulation overestimate the experimental results, but the trends match well with experiment. The distribution of thickness thinning observed in the simulation match with experiment and more thinning are obtained in the simulation. Meanwhile, the cross-shaped cup drawing simulations have been studied using conference data. The similar conclusion are obtained that the formability of AZ31 improve at high temperature, the simulation is effective despite occurring dissatisfaction. And the study results are helpful to conduct the magnesium alloy sheet application in industry.

REFERENCES 1. A. Gavrus, E. Massoni, J.L. Chenot. An inverse analysis using a finite element model for identification of rheological parameters. Journal of Materials Processing Technology, 60 (1996) 447-454. 2. Ozturk F, Lee D. Analysis of forming limits using ductile fracture criteria. Journal of Mater Process Technology, 147 (2004) 397–404. 3. Fuh-Kuo Chen, Tyng-Bin Huang. Formability of stamping magnesium-alloy AZ31 sheets. Journal of Materials Processing Technology, 142 (2003) 643-647. 4. K.F. Zhang, D.L. Yin, D.Z. Wu. Formability of AZ31 magnesium alloy sheets at warm working conditions. International Journal of Machine Tools & Manufacture, 46 (2006) 1276–1280. 5. Yong Qi Cheng, Hui Zhang, Zhen Hua Chen, Kui Feng Xian. Flow stress equation of AZ31 magnesium alloy sheet during warm tensile deformation. Journal of materials processing technology, 208 (2008) 29-34. 6. E. Doege, K. Droder. Sheet metal forming of magnesium wrought alloys - formability and process technology. Journal of Materials Processing Technology, 115 (2001) 14-19. 7. S.H. Zhang, K. Zhang, Y.C. Xu, Z.T. Wang, Y. Xu, Z.G. Wang. Deep-drawing of magnesium alloy sheets at warm temperatures. Journal of Materials Processing Technology, 185 (2007) 147–151. 8. K. Droder. Analysis on forming of thin magnesium sheets. Ph.D.Dissertation, IFUM, University of Hanover, 1999. 9. S.L. Semiatin, E.W. Collings, V.E. Wood, T. Altan. Determination of the interface heat transfer coefficient for nonisothermal bulk forming process. Journal of Industrial Engineering, 109 (1987) 49–57.

157

Crystal Plasticity Simulation of Forming Limit Strains for Fcc Polycrystalline Sheets with Different r-values Kengo Yoshidaa and Mitsutoshi Kurodaa a

Graduate School of Science and Engineering, Yamagata University, 4-3-16 Jonan, Yonezawa, Yamagata, 992-8510 Japan

Abstract. Plastic deformation characteristics and limit strains are simulated for textured face-centered cubic polycrystalline sheets using a generalized Taylor-type crystal plasticity model. The r-values are predicted to be 1.04, 7.74 and 0.17 for the random, {111} and {001} textures, respectively. The {111} texture gives limit strains as large as the random texture, whereas the {001} texture yields limit strains evidently higher than the other two even though its r-value is extremely low. Thus, the r-value cannot act as an indicator to the stretchability of sheet metals. For the {001} texture, a superior strain-hardening ability under plane-strain stretching mode is found to be responsible for the increase in the limit strains under plane-strain and equi-biaxial stretching modes. We conclude that the enhancement of the strain-hardening ability for plane strain mode is one of the key factors for high stretchability sheets. Keywords: Forming limit, r-value, stretchability, polycrystalline sheet, crystal plasticity PACS: 46.15.-x, 81.05.Bx

INTRODUCTION Improvement of formability of polycrystalline metal sheets, such as drawability, stretchability and bendability, by controlling microstructure is an attractive and considerable challenge. Intensive researches have been conducted to clarify an essential factor affecting the formability. Lankford et al. [1] experimentally investigated the formability in automotive fender draws for various low carbon steels having almost the same work-hardening exponent (nvalue), and demonstrated that the drawing performance has good correlation with the r-value, which is a ratio of width strain to thickness strain under uniaxial tension. The positive correlation between the r-value and drawability was again experimentally observed for various sheets, such as steels, aluminum alloys, copper and brass, which have the r-value ranging from 0.6 to 1.6 [2]. In addition to the experimental studies, a numerical simulation supporting this correlation has also been reported [3]. Therefore, the r-value is widely known to be a key parameter representing the drawability of sheet materials. The r-value has close connection with the preferred crystallographic orientations of polycrystalline metal [4], [5], [6], [7], [8]. Analyses based on a crystalline plasticity model revealed that the high r-value with small in-plane variation is obtained for the {111}<110> and {111}<112> orientations [4], [5], [7]. In a fully annealed low carbon steel sheet, the {111} texture, which includes the {111}<110> and {111}<112> orientations, is generated, and this results in high r-values. On the other hand, in a fully annealed aluminum alloy sheet, texture develops around {001}<001> orientation. The r-value of this orientation is unity for the rolling and transverse directions and zero for 45 ° direction from the rolling direction. Thus, the r-value of aluminum alloy sheet is generally smaller than unity. It is known that the out of plane shear deformation could generate the shear texture characterized by the {001}<110>, {111}<110> and {111}<112> orientations in aluminum alloy sheets [9], [10], [11]. Thus, the r-value could be increased by creating the strong shear texture. Asymmetric rolling, in which circumferential velocities of

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 158-164 (2011); doi: 10.1063/1.3623606 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

158

the top and bottom rolls are different, has been developed in order to generate shear strain throughout the thickness and to improve the r-value. In the past decade, asymmetric rolling has been widely investigated [11] , [12] , [13]. It is often presumed, without evidential experimental data, that the high r-value provides good stretchability, which refers the formability in biaxial stretching mode, based on the knowledge of the significance of the r-value in the deep drawing. As is mentioned, the texture components that raise the r-value have been known, but the role of the r-value in the stretchability is not fully understood. Ayres et al. [14], for instance, concluded based on their experiments that the stretchability is not be related to the r-value. Sowerby and Duncan [15] carried out numerical studies on forming limit curves for sheets having different r-values by means of the Hill’s quadratic yield function[16] in conjunction with the Marciniak and Kuczynski (M-K) model [17], and showed that the limit strains decrease with increasing the r-value. On the other hand, Lian et al. [18] and Graf and Hosford [19] showed that the forming limit strains do not depend on the r-value, provided that a higher-order phenomenological yield function is employed in the analysis. Lian et al. [18] also demonstrated that a ratio of the major stress at plane strain stretching to that at equi-biaxial stretching on a yield surface can be an indicator that governs the limit strains in biaxial stretching range. Wu et al. [20] used a generalized Taylor-type model of polycrystalline plasticity for computations of forming limit curve of the cube texture ({001}<001>), whose r-value is rather low. It was shown that the cube texture with some orientation scatter improves limit strains near equi-biaxial stretching mode. Yoshida et al. [21] reported that the forming limit curve for the cube texture is significantly increased when the sheet is stretched along 45 ° direction relative to the rolling direction, although the r-value in this direction is almost zero. The computation carried out by Yoshida et al. [21] shows that the {001}  uvw  texture, which has quite low r-values, enhances the forming limit curve for the full range of linear strain paths. These results show that the stretchability does not have any dependence on the r-value, and that conflict with the belief that a high r-value gives high stretchability. In the present study, plastic deformation characteristics and stretchability of textured aluminum alloy sheets are simulated based on a crystal plasticity model in order to re-examine the influence of the texture and the r-value on stretchability. For this goal, attention is directed to the {111}  uvw  and {001}  uvw  textures, because these texture give extreme cases of very high and low r-values, respectively. First, the plastic deformation characteristics such as the r-values, yield stresses and yield loci of these textures are illustrated using a generalized Taylor-type polycrystalline plasticity model. Then, limits to localized necking under plane-stress condition are computed by means of the M-K-type model. In the program of investigation, how the texture and r-value relate to the limit strains are discussed, and we clarify a key factor for superior stretchability of sheet metals other than the r-value. It is noted that the present paper is a part of our recent work [22].

PROCEDURE OF NUMERICAL SIMULATION

Crystal Plasticity Model A finite strain crystal plasticity model used here is along the lines presented in Peirce et al. [23]. The slip rate

 ( ) on the  th slip system is given by a power law dependence on the resolved shear stress  ( ) , 1/ m  ( )  ( )   0 sgn( ( ) ) ( ) g

(1)

where  0 is a reference slip rate, m is a strain rate sensitivity exponent, and g ( ) is a slip system hardness. The evolution law for g ( ) is specified by g ( )  h  (  ) , 

 h h  h0 1 0 a   0n  

n 1

,

a  

t

0

 

( )

dt , dt

(2)

where  0 is the initial value of g ( ) , h0 is the initial slip hardening modulus, n is a power law hardening exponent, and t is time. An isotropic elasticity is assumed in the present applications, which is determined with the Young’s modulus E and Poisson’s ratio  . As a model for a polycrystal, a Taylor type approach [24] is adopted. The deformation in each grain is taken to be identical to the macroscopic deformation of the aggregate. Taking the volume fraction of each grain to be identical, the macroscopic stress σ and macroscopic plastic strain rate D p are respectively obtained from averaging

159

the Cauchy stress σ and plastic strain rate D p in each grain over the total number of grains. A macroscopic measure of plastic strain for a polycrystal is defined by t

 eq   (2 / 3)Dp : Dp dt .

(3)

0

Texture generation The procedure for generating textures is equivalent to one used in the previous studies [8] [21], [25]. To represent crystal orientations, we introduce Bunge’s Euler angles (1 , ,2 ) . Euler angles  and 2 are set to be   54.7° and 2  45° for the {111}  uvw  texture and   0° and 2  0° for the {001}  uvw  texture, and 1 is randomly selected from the range between 0° and 90° . A misorientation from the ideal orientation is given such that misorientation angles follow a Gaussian distribution with the standard deviation 0 . In addition to these two textures, random texture is generated for comparison purposes. The three textures constructed by 2000 grains with 0  15 ° . These three textures conceptually have in-plane isotropic property. As a matter of fact, small in-plane anisotropy arises because the textures consist of a finite number of discrete crystallographic orientations.

RESULTS In-plane distribution of r-value and 0.2% proof stress In this subsection, the r-value and 0.2% proof stress are analyzed for the random, 111  uvw  and 001  uvw  textures by means of a generalized Taylor-type ppolycrystalline plasticity model described above. Material y parameters are taken to be E /  0  1400,   0.3,,  0  0.002,, m  0.002, h0 /  0  30, n  0.2 and  0  50MPa. Prescribed quantities for uniaxial tension are  111  0 ,  ij  0 for any other i, j, L11   0 (prescribed), and L21  L31  L32  0 . The six unknown values,  11 , L22 , L33 , L23 , L13 and L32 , are solved with the rate-form constitutive relationships in each increment. The x1 -, x2 - and x3 -axes are firstly coincide with the RD, TD and ND directions. When the uniaxial tension is applied along a ° direction from the RD, the textured sheet is rotated

° about x3 -axis and the boundary conditions described above are imposed. The direction of   0°‫ޓ‬and 90° correspond to the uniaxial tension along the RD and TD, respectively. The r-value is defined by the ratio of plastic strain rates D22p / D33p at  eq  0.1, and the 0.2% proof stress, denoted by  0.2 , is defined as a tensile flow stress  11 at  eq  0.002. Figure 1 shows the in-plane distribution of r-value and 0.2% proof stress for the random, 111  uvw  and 001  uvw  textures, where values in the parenthesis are arithmetic averages of the predictions from   0 ° to 3.50

10 9

random (3.23)

8

3.25

6

{111} (7.74)

5

0.2 / 0

r-value

7 eq = 0.1

4

random (1.04)

3

{001} (0.17)

3.00

{111} (3.12) {001} (2.65)

2.75

2 1 0

0

15

30

45

60

75

2.50

90

0

15

30

45

60

75

90

 (°) (a) (b) FIGURE 1. In-plane distribution of r-value (a) and 0.2% proof stress (b).  is an angle between tensile and rolling directions. A value in parenthesis is an arithmetic average of predictions from   0 ° to 90 ° .  (°)

160

90 ° . Since the three textures have conceptually in-plane isotropy, the computed r-values do not have apparent orientation dependence. The highest r-value of 7.74 is obtained for the 111  uvw  texture, and the lowest r-value about 0.17 is predicted for the 001  uvw  texture. The 111  uvw  texture involves the 111  110  and 111  112  orientations, which are known to have high r-value, and the 001  uvw  texture consists of the 001  001  orientation rotating about ND, whose r-values are in between zero and unity. Thus, Fig. 1 is consistent with the well-known crystallographic orientation dependence of the r-value. The r-values for the 111  uvw  and 001  uvw  textures are much larger and much less than the random texture, respectively. On the other hand, the 0.2% proof stress, shown in Fig. 1(b), is the highest for the random texture and those for the other two textures are lower. Both the textures lead to softer response than the random texture under the uniaxial tension.

Sheet necking analysis

Problem formulation of sheet necking analysis Sheet necking analysis is carried out based on the Marciniak–Kuczyński (M-K)-type model [17] in conjunction with the generalized Taylor-type model. Limit strains are expressed as a forming limit curve in strain space. A sheet specimen in the x1 x2 plane with an initial inhomogeneity of a reduced thickness band under the plane-stress condition is considered (Fig.2). The RD and TD axes have an angle  I relative to the x1 and x2 directions, respectively. The same texture model is assigned to the regions both inside and outside the imperfection band. In all computations performed here, linear strain paths, L D   22  22 , (4) L11 D11 are assumed outside the band. The range of the strain ratio is taken to be -0.5  1. The other velocity gradient components outside the band are set to be L12  L21  L23  L32  L13  L31  0. g is judged byy the occurrence of a much higher principal strain rate inside the band The onset of localized necking than outside the band, 1b / 1 ! 105 , where 1b and 1 respectively denote the maximum principal values of the rate of deformation tensors inside and outside the band. Localization strains 11L for a certain strain ratio are computed for various initial band angles I with 5 ° interval. The minimum value of localization strains is defined as the forming * limit strain, denoted by 11* and  22 , and the corresponding initial band angle is defined as the critical initial band angle. The initial imperfection value, i.e., the ratio of the thickness inside the band to that outside the band, is taken to be 0.999 for all computations. Forming limit curves The material parameters used here are the same as ones in the previous section. Initial orientation of RD and TD is taken to be I  0° , and the initial band angle ranges from I  0° to 90° . Forming limit curves for the random, 111  uvw  and 001  uvw  textures are shown in Fig. 3. The forming limit strains of the 111  uvw  texture are almost the same as those for the random texture over most strain range, with a local increase at   0.25. The 111  uvw  texture exhibits extremely high r-value as shown in Fig. 1. The forming limit curve is, however,

FIGURE 2. Textured thin sheet with imperfection band initially inclined at angle I .

161

almost the same as the random texture. In the case of 001  uvw  texture, the limit strain is almost identical to that of the random texture at   -0.5, while in the other strain range, the limit strains are clearly higher than the random texture. At   1 the limit strain is about double the others. The forming limit curve of the 001  uvw  texture, exhibiting low r-value, is the highest among those three textures. The limit strains at   0 and 1 are definitely dependent on the texture, and the mechanism behind it is examined below. For the plane-strain stretching (   0), the critical initial band angle is calculated to be 0° for all textures, thereby the deformation inside the band is also the plane-strain stretching. The strain-hardening ability of the material under this strain mode is one of the most influential factors that govern the strain level at the onset of localized necking. Instantaneous hardening rates for the plane-strain stretching mode (   0) are shown in Fig. 4. Here, the instantaneous hardening rate H is obtained as a slope of the true stress-logarithmic strain curve, i.e., H  d11 / d11 , and  0 is a flow stress  11 at  eq  0.002 for each texture. The developments of hardening rate for the random and 111  uvw  textures almost coincide with each other, and this leads to the same level of the forming limit strains at   0 as is seen in Fig. 3. The instantaneous hardening rate is clearly greater for the 001  uvw  texture than for the other two after 11  0.1. The superior strain-hardening ability delays the occurrence of the localized neck. Therefore, the forming limit strain is increased for this texture. The high hardening 0.6 0.5 0.4

{001}

11*

0.3

{111}

0.2

random

0.1 0.0 -0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

22* FIGURE 3. Forming limit curve for the three textures 8

 = 0

7 6

H / 0

5

{001}

4

random

3 2 1 0 0.0

{111} 0.1

0.2

0.3

0.4

0.5

"" FIGURE 4. Development of instantaneous hardening rate under plane-strain stretching mode. H  d11 / d11 and  0 represents a flow stress  11 at  eq  0.002.

162

rate of this texture is attributed to the geometrical hardening and this topic has already been discussed in detail by Yoshida et al. [25]. Next, the limit strain for the equi-biaxial stretching is considered. For the equi-biaxial stretching mode, localization strains 11L are identical for any initial band angles, provided that the material has in-plane isotropy. The textures considered here all have in-plane isotropy. (We neglect the in-plane anisotropy arose from the finite number of discrete orientations. As a matter of fact, the variation of localization strains 11L for the initial band angles between 0° and 90° is about 0.02 for the random and 111  uvw  textures and is about 0.07 for the 001  uvw  texture.) The investigation is proceeded considering the case that the initial band angle is I  0° . When the initial band angle is I  0° , only normal components of stress and strain arise inside the band and no shear components appear. The mode of straining inside the band is almost the equi-biaxial stretching at the beginning of loading, and it gradually changes toward the plane-strain stretching mode as the strain localization proceeds. Thus, the modes of strain outside and inside the band are not identical for the equi-biaxial stretching mode, unlike for the plane-strain mode considered above. The strain hardening characteristic is examined for the following two strain paths: (i) proportional strain path for the equi-biaxial stretching (   1), (ii) non-proportional strain path in which the ratio of strain rate  is unity at the beginning of loading program and then gradually changes to zero, that is, strain path changes form equi-biaxial stretching to plane strain stretching mode. The change in  is described in Fig. 5(a). A resultant non-proportional strain path fairly coincides with the strain paths inside the band for the random and 111  uvw  textures. Developments of instantaneous hardening rate for the three textures under the proportional and non-proportional strain paths are depicted in Fig. 5(b). The results for the proportional strain path are shown by thin lines. The development of hardening rate of the 001  uvw  texture is almost the same as that of the random texture, and the hardening rate of the 111  uvw  texture is lower than the other two. Thus, no systematic correlation between the degree of the hardening rate and the forming limit strain is seen. The thick lines indicate development of hardening rate for non-proportional strain path. For the case of the random and 111  uvw  textures, the hardening rates for the non-proportional strain path are close to those for the proportional strain paths. This means that the strain hardening behaviors are almost the same for both inside and outside the band in the sheet necking analysis. On the other hand, for the 001  uvw  texture, the instantaneous hardening rate becomes greater for the non-proportional strain path than for the proportional strain path as the strain path deviates from equi-biaxial stretching at 11  0.15. Thus, for this texture, an additional hardening occurs for the non-proportional strain path. In the sheet necking analysis, this increase in hardening rate can restrain the growth of a neck. Therefore, the strain localization has not predicted for the 001  uvw  texture at the strain level where the strain localization occurred for the other two textures. The increase in the strain hardening rate consists of the superior strain hardening behavior for the plane strain stretching mode shown in Fig.4. Therefore, a key factor for the high formability is again found to be the strain 8

thick: non-proportional strain path thin: proportional strain path with  = 1

7

1.0

0.5

4 3

1

0.15

11

{001}

2

0.33

0.0 0.0

random

5

H / 0

 (=D22 / D11)

6

0 0.00

0.35

{111} 0.05

0.10

0.15

0.20

0.25

0.30

0.35

11

(a) (b) FIGURE 5. Variation of the ratio of strain rate in non-proportional strain path (a) and development of instantaneous hardening rate for proportional strain path in equi-biaxial stretching and non-proportional strain path described in (a).  0 represents a flow stress  11 at  eq  0.002 for equi-biaxial stretching mode

163

hardening ability under the plane strain mode.

CONCLUSIONS In this study the influence of the texture on the plastic deformation characteristics and stretchability has been investigated for the random, 111  uvw  and 001  uvw  textures. It has been confirmed that there is no positive correlation between the r-value and stretchability. Therefore, the high r-value is not a necessary condition for the high stretchability. The superior hardening behavior for plane strain mode yields high forming limit strains at the plane strain mode as well as at the equi-biaxial stretching mode for the 001  uvw  texture. The enhancement of the strain-hardening ability for plane strain stretching mode is a key factor for high stretchability sheets.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

W.T. Lankford, S.C. Snyder and J.A. Bauscher, Trans. ASM 42, 1197-1232 (1950). R.L. Whiteley, D.E. Wise and D.J. Blickwede, Sheet Metal Industries 38, 349-358 (1961). W.F. Hosford and R.M. Canddell, Metal Forming, Second ed., Prentice-Hall, 1993. P.H. Lequeu, P. Gilormini, F. Montheillet, B. Bacroix and J.J. Jonas, Acta Metall. 35, 1159-1174 (1987). P.H. Lequeu and J.J. Jonas, Metall. Trans. A 19, 105-120 (1988). F. Barlat and O. Richmond, Mater. Sci. Eng. 95, 15-29 (1987). J. Hu, K. Ikeda and T. Murakami, Mater. Trans. JIM 36, 1363-1371 (1995). M. Kuroda and S. Ikawa, Mater. Sci. Eng. A 385, 235-244 (2004). J.-H. Han, J.-Y. Suh, K.-K. Jee and J.-C. Lee, Mater. Sci. Eng. A 477, 107-120 (2008). O. Engler, M.-Y. Huh and C.N. Tomé, Metall. Mater. Trans. A 31, 2299-2315 (2000). K.-H. Kim and D.N. Lee, Acta Mater. 49, 2583-2595 (2001). H. Jin and D.J. Lloyd, Mater. Sci. Eng. A 465, 267-273 (2007). S.-B. Kang, B.-K. Min, H.-W. Kim, D.S. Wilkinson and J. Kang, Metall. Mater. Trans. A 36, 3141-3149 (2005). R.A. Ayres, W.G. Brazier and V.F. Sajewski, Appl. Metal Working 1, 41-49 (1979). R. Sowerby and J.L. Duncan, Int. J. Mech. Sci. 13, 217-229 (1971). R. Hill, Proc. Roy. Soc. London A193, 281-297 (1948). Z. Marciniak and K. Kuczy?ski, Int. J. Mech. Sci. 9, 609-620 (1967). J. Lian, F. Barlat and B. Baudelet, Int. J. Plasticity 5, 131-147 (1989). A. Graf and W. F. Hosford, Metall. Trans. A 24A, 2497-2501 (1993). P.D. Wu, S.R. MacEwen, D.J. Lloyd and K.W. Neale, Mater. Sci. Eng. A 364, 182-187 (2004). K. Yoshida, T. Ishizaka, M. Kuroda and S. Ikawa, Acta Mater. 55, 4499-4506 (2007). K. Yoshida and M. Kuroda, Numerical investigation on a key factor for superior stretchability for face-centered cubic polycrystalline sheets, in preparation. D. Peirce, R.J. Asaro and A. Needleman, Acta Metall. 31, 1951-1976 (1983). R. J. Asaro and A. Needleman, Acta Metall. 33, 923-953 (1985). K. Yoshida, Y. Tadano and M. Kuroda, Comp. Mater. Sci. 46, 459-468 (2009).

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Numerical Investigation into the Effect of Uniaxial and Biaxial Pre-Strain on Forming Limit Diagram of 5083 Aluminum Alloy F.Zhalehfar, S.J. Hosseinipour, S. Nourouzi and A.H. Gorji Metal Forming Research Group, Faculty of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Iran ([email protected])

Abstract. In this paper, numerical simulations are carried out to determine the forming limit diagram (FLD) of 5083 aluminum alloy. The aim is to predict the effect of strain path change on the forming limit curve (FLC) of this alloy. For this purpose, out-of-plane formability test method with hemispherical punch was simulated by using commercial finite element software, ABAQUSE 6.9. In the first stage, square blanks were modeled and then some of them were prestrained uniaxially by tension test and some others were pre-strained biaxially by stretching over the hemispherical punch. In the second stage, the formability test specimens’ models were prepared by trimming the pre-strained blanks with the longitudinal axis parallel and perpendicular to the rolling direction. For trimming, a program was written in MATLAB 7.6 which could determine the new elements and introduce their properties to the FEM model. Ductile fracture criteria were used to predict failure. Furthermore, forming limit stress diagram (FLSD) was determined. The numerical results were compared with the experimental findings. Uniaxial pre-straining increased and shifted the FLC to the left hand side of the diagram for both parallel and perpendicular to the rolling direction. Biaxial pre-straining shifted the FLC to the right hand side of the diagram for both directions, and also decreased the FLC for the specimens parallel to the rolling direction. Keywords: Aluminum alloy 5083, FEM simulation, Forming limit diagram, Pre-Strain. PACS: 81.20Hy

INTRODUCTION Non-heat treatable 5XXX series aluminum alloys are used in various industries e.g. marine, TV towers and cryogenics. Because of their properties such as corrosion resistance as well as reasonable strength, ductility and weldability, they have primary potential for lightweight structural application in automotive and aerospace industries. It is believed that identification of forming capability in these alloys will enhance the potential for such applications. Forming limit diagram (FLD) is a graph of the major strain (1) versus the minor strain (2) at the onset of localized necking. However it is a well-known method to describe the formability of sheet metals, researches indicate that the FLD is dependent of the strain path. In many sheet metal forming industries, sheets are deformed under various dies and experience different strain paths. Therefore, it is important to investigate the influence of strain path change on the FLD in various alloys. Stoughton [1] showed that the forming limit for both proportional loading and non-proportional loading can be explained from a single criterion which is based on the state of stress rather than the state of strain. Graf and Hosford [2,3] studied the influence of strain path change on forming limits of AA2008 and AA6111-T4 by using uniaxial, plane strain and biaxial pre-strains parallel and perpendicular to the rolling direction. Dariani [4] investigated the effect of non proportional loading path on forming limit of St-12 steel alloy. Kuwabara et al [5] studied the effect of abrupt strain path change on the yield surface of AA6XXXT-4 by using a biaxial tensile testing machine. Kuroda and Tvergaard [6] considered the effect of different non-proportional strain paths on the FLD of anisotropic metal sheets. Reyes and Hopperstad [7] investigated numerically the effects of non-proportional loading on the Forming limit diagrams. Chow et al. [8] extended an anisotropic damage model for predicting FLDs under non-proportional loading.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 165-172 (2011); doi: 10.1063/1.3623607 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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In this paper, numerical simulations are carried out by using commercial finite element software, ABAQUSE 6.8 and programming in MATLAB 7.6 to predict the effect of strain path change on the forming limit curve (FLC) of 5083 aluminum alloy sheet. The numerical results are compared with experiments which discussed in the previous woks [9].

FINITE ELEMENT SIMULATION Simulations were performed by using commercial finite element software ABAQUS/explicit 6.9. The out-ofplane formability test (dome test) with a hemispherical punch was modeled to determining the forming limit diagram according to the Figure 1. It consists of three main tools, hemispherical punch, blank-holder and die with draw-bead. In this method, eight specimens with different geometries, as shown in Figure 2, are required to complete a forming limit diagram.

FIGURE 1. Out-of-plane formability test setup [10].

FIGURE 2. The geometry of the FLD specimens [10].

Due to symmetry, only one-quarter of the geometry was modeled, as shown in Figure 3. The tools were assumed rigid parts and the specimens were modeled as elastoplastic and meshed by S4R shell element. The size of the element 0.0015 m was obtained by comparison of different mesh size for the specimens. Commercial 5083 aluminum alloy sheet with 1 mm thickness was examined in this research. Tensile tests, according to the ASTM B557m, were carried out at 0, 45 and 90 degree to the rolling direction. The mechanical properties of the material are summarized in Table 1. The stress-strain curve was approximated by the Hollomon potential law. The anisotropic properties were described by the Hill quadratic yield criterion. The model was assumed to be rate independent. Coulomb friction law of 0.16 was assumed for contact surfaces [11,12].

166

FIGURE 3. FEM simulation model of out-of-plane formability test. TABLE (1). Mechanical properties of the material

Young’s modulus of elasticity (GPa) Poisson’s ratio Density (kg/m3) Yield strength (Mpa) (0.2%) Strength coefficient (MPa) Strain hardening coefficient (n) Normal anisotropy ( r )

70 0.3 3 27 00 28 4 52 2 0.1 1 0.7 4

For introducing biaxial pre-strain, in the first stage square blanks with 200 mm length in each dimension were stretched by the hemispherical punch to 8.5mm and 10.8 mm stroke. Then, the stretched blanks were trimmed according to the FLD specimens shown in Figure 2, with the longitudinal axis parallel (RD) and perpendicular (TD) to the rolling direction, as shown in Figure 4. In the second stage, these specimens were tested to obtaining the FLD.

FIGURE 4. The simulation sequence for biaxial pre-straining and FLD test.

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For introducing near uniaxial pre-strains, in the first stage square blanks with 200 mm length in each dimension were notched with a special radius according to the Figure 5. These blanks were pre-strained uniaxially to 5.8 mm and 6.5 mm stroke. Then the FLD specimens were prepared by trimming these blanks according to the FLD specimens shown in Figure 2, with the longitudinal axis parallel (RD) and perpendicular (TD) to the rolling direction. In the second stage, these specimens were tested to obtaining the FLD. For trimming the pre-strained specimens, a program was written in MATLAB 7.6 which could determine the new elements and introduce their properties to the FEM model. Clift and Cockcroft ductile fracture criteria were used to predict localized necking [12-15]. The total procedure was summarized in the flowchart of Figure 6. Some experiments were carried out for comparison [9].

First stage of uniaxial tension Preparing FLD specimens by trimming Deforming the FLD specimens FIGURE 5. The simulation sequence for uniaxial pre-straining and FLD test.

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Simulation of the Pre-straining stage of the initial blanks in ABAQUS

Take a report from the elements properties of the prestrained blanks and importing to MATALB

Preparing the FLD specimens by trimming the prestrained blanks in ABAQUS

Transferring the elements properties from the prestrained blanks to the FLD specimens in MATLAB

Building a new INP file in MATLAB and executing in ABAQUS

Simulation of the dome test stage of the FLD specimens in ABAQUS

Taking a report from the elements properties at each iteration of the dome test and importing to MATALB

Calculating the Clift et al and the Cockroft and Latham fracture criteria for whole elements at each iteration

Checking the elements which satisfy the fracture criteria earlier

Recording the strains and stresses of the elements.

FIGURE 6. Description of the total simulation procedure.

RESULTS AND DISCUSSION Figure 7 shows the experimental and numerical forming limit curves in as-received condition at rolling (RD) and transverse (TD) directions. It is seen that there are an acceptable agreement between the numerical results and experimental findings. The differences could be due to differences in predicting the necking (FLCN) and fracture (FLCF) [15,16]. The FLD0 is about 0.085 at RD and it is reduced to about 0.07 at TD.

169

(b)

(a)

FIGURE 7. The FLD in as-received condition, a) at RD, b) at TD.

Figure 8 shows the forming limit curves at RD and TD for biaxial pre-strained conditions. It is seen that the FLD0 is reduced to about 0.07 at RD and also is shifted to the right hand side of the diagram. The FLD0 is not changed significantly at TD, but it is shifted to the right hand side of the diagram.

(b)

(a)

FIGURE 8. The FLD in biaxial pre-strained condition, a) at RD, b) at TD.

Figure 9 shows the forming limit curves at RD and TD for uniaxial pre-strained conditions. It is seen that for RD the FLD0 is slightly increased and shifted to the left hand side of the diagram. The FLD0 for TD is increased significantly, and also shifted to the left hand side of the diagram. These results are similar to the experimental data of Graf and Hosford [2,3]. Figure 10 shows the FLSD diagram for biaxial pre-strained conditions at RD directions. It is seen that there is not any significant change between them. This indicates that the FLSD is independent of pre-straining effect and is suitable to use for non proportional loading path.

170

(b)

(a)

FIGURE 9. The FLD in unaxial pre-strained condition, a) at RD, b) at TD.

FIGURE 10. The FLDS in biaxial pre-strained condition at RD.

CONCLUSION The effect of biaxial and uniaxial pre-strains on the forming limit diagram of commercial 5083 aluminum alloy has been investigated numerically. Simulations have been carried out at two stage; pre-straining and dome test, by using commercial finite element software, ABAQUSE 6.9. Clift and Cockcroft ductile fracture criteria have been used to predict failure. MATLAB 7.6 has been used for preparing the FLD specimens from the pre-strained blanks and also for determining the ductile fracture criteria. Acceptable agreement between the numerical results and experimental findings has been achieved. The results showed that uniaxial pre-straining increased and shifted the FLC to the left hand side of the diagram for both parallel and perpendicular to the rolling direction. Biaxial prestraining shifted the FLC to the right hand side of the diagram for both directions, and also decreased the FLC for the specimens parallel to the rolling direction.

ACKNOWLEDGEMENT The authors would like to appreciate the office of the Vice President for Research of Babol Noshirvani University of Technology for its financial support.

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REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14. 15. 16.

T. B. Stoughton, Int. J. of Mech. Sci. 42, 1-27 (2000). A. Graf and W. Hosford, Int. J. Mech. Sci. 36, 897-910 (1994). A. Graf and W. Hosford, Metall. Trans. A 24A, 65-75 (1993). B. Dariani, " Effect of non proportional loading path on forming limit of sheet metals ", Ph.D. Thesis, Amirkabir University of Tech., 1999. T. Kuwabara, M. Kuroda, V. Tvergaard and K. Nomura, Acta Materialia 48, 2071-2079 (2000). M. Kuroda1 and V. Tvergaard, Int. J. of Mech. Sci. 42, 867-887 (2000). A. Reyes and O. S. Hopperstad, 7th European LS-DYNA conference 2009. C. L. Chow, L. G. Yu, W. H. Tai and M. Y. Demeri, Int. J. Mech. Sci. 43, 471-486 (2001). F. Zhalehfar, S. J. Hosseinipour, S. Nourouzi and A.H. Gorji, “Effect of Equal Biaxial Pre-Strain on Forming Limit Diagram of AA5083” in Advances in materials and processing Technologies-AMPT 2010, edited by F. Chinesta et al., AIP Conference Proceedings 1315, American Institute of Physics, 2010, pp. 353-358. F. Ozturk and D. Lee, J. Mater. Proce. Technol. 170, 247–253 (2005). K. S. Ragavan, Metall. Trans. A 26A, 2075–2084 (1995). T. Yoshida, T. Katayama and M. Usuda, J. Mater. Proce. Technol. 50, 226-237 (1995). M. G. Cockcroft and D. J. Latham, J. Inst. Met. 96, 33-39 (1968). S. E. Clift, P. Hartley, C. E. N. Sturgess and G. W. Rowe, Int. J. Mech. Sci. 32, 1-17 (1990). H. N. Han and K. H. Kim, J. Mater. Proce. Technol. 142, 231–238 (2003). F. Ozturk and D. Lee, J. Mater. Proce. Technol. 147, 397–404 (2004).

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A Comparison between Three Numerical Criteria for Prediction the Forming Limit Diagram of St14 Steel M. Moslemi, S.J. Hosseinipour, M.E. Hosseini and A.H. Gorji Metal Forming Research Group, Faculty of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Iran ([email protected])

Abstract. In this paper, the forming limit diagram (FLD) of a low carbon steel St14 (DIN 1623) is investigated experimentally and numerically. The objective of this study is to find a numerical criterion which enables a simple and reliable determination of the FLD. For this purpose, Out-of-plane stretching test method with hemispherical punch was simulated by using commercial finite element software, ABAQUSE 6.9. One-quarter of the geometry was used due to symmetry. The material was modeled as elastoplastic and the anisotropic properties were described by the Hill quadratic yield criterion. The model was assumed to be rate independent. Coulomb friction law was defined for all contact surfaces. The simulation process was performed in two steps. In the first step the blank-holder moves down and deforms the blank into the draw-bead. Then the punch moves up at 20 increments and deforms the specimen to a specified displacement. Three numerical criteria including maximum second thickness strain derivative (CRIT1), equivalent plastic strain increment ratio (CRIT2), and total equivalent plastic strain ratio (CRIT3) were evaluated and the forming limit curve (FLC) were obtained. The predicted FLC were compared with experimental data. Unlike the CRIT2, the CRIT1 and CRIT3 were in good agreements with were experimental data. The CRIT1 criterion predicted the lower bound of the experimental results. While by using the CRIT3 criterion both the lower and upper bounds of the experimental results were predicted. Keywords: FEM simulation, Forming limit diagram, Numerical criteria, St14 steel. PACS: 81.20Hy

INTRODUCTION A forming limit diagram (FLD) defines the locus of the limit strains corresponding to various strain ratios. It is conventionally described as a curve in a plot of major strain vs. minor strain. It is a useful measure to evaluate the formability of sheet metal. As long as the principal strains are below the curve in the diagram, that region of the metal will be safe from necking and tearing [1-4]. One of the earliest experimental works on forming limit diagram is reported by Keeler and Backhofen [5]. They developed the right side of the FLD. Goodwin [6] extended this concept for the left side. Marciniak et al. [7] used an experimental method with flat cylindrical punch to determine all the strain ratios of the FLD. Nakazima et al. [8] and Hecker [9] introduced a similar test with a hemispherical punch. Generally, the experimental methods to determine FLD are categorized in two main types, in-plane stretching test using a flat cylindrical punch and out-of-plane stretching test using a hemispherical punch. Both techniques give nearly identical FLD, but the results from the hemispherical punch are slightly larger. Because of surface effect and friction, the out-of-plan test has been used much more extensively than the in-plan test [3, 10]. In practice, experimental determination of a FLD is very time consuming procedure and requires special expensive equipment. Finite element simulation is an alternative method to predict FLD. For numerical prediction of a FLD, the significant problem is the definition of the onset of localized necking under various sheet metal forming conditions. Many efforts have been spent on developing more accurate and reliable numerical procedure. For instance, Takuda et al. [11-13] utilized the ductile fracture criteria to prediction of forming limit of sheet metal with the finite element simulation of in-plane cylindrical test. Yoshida et al. [14] simulated the hemispherical punch stretching using an elastic-plastic three-dimensional finite element model. They predicted the limiting cup height and

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 173-179 (2011); doi: 10.1063/1.3623608 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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rupture location for several materials. Ozturk and Lee [10] investigated the application of the ductile fracture criteria for prediction of the forming limit diagram of sheet metal with using finite element simulation of out-of-plane stretching test. Brun et al. [15] simulated the out-of-plan stretching test and predicted the localization by introducing the idea of the second time derivative of the thickness strain. Similar analysis was carried out on the out-of-plan stretching test by Geiger and Merklein [16]. They determined the FLD base on this concept that the materials necking are related to the gradient of major strain versus time. They described that the strain gradient during the forming process changes rapidly at the moment the localization occurs. Petek et al. [17] and Pepelnjak et al. [18] introduced an alternative method based on the Brun et al. [15] and Geiger and Merklein [16] concepts, which the second time derivative of the thickness strain is related to the single node or element and defines the time and area of localized necking. They simulated the Marciniak in-plane test with this methodology. In this paper, the forming limit diagram of a low carbon steel St14 (DIN 1623) is investigated experimentally and numerically. The objective of this study is to find a numerical criterion which enables a simple and reliable determination of the FLD. Simulation is performed by using the commercial finite element software ABAQUS 6.9. Different numerical procedures are used to predict the onset of localized necking. The accuracy of the predicted FLCs are compared with experiments.

EXPERIMENTAL PROCEDURE Commercial low carbon steel St14 (DIN 1623) sheet with 0.9 mm thickness was used. The mechanical properties of the material are shown in Table 1, which were determined by tensile tests according to ASTM A370. The plastic strain ratio (r-value) was evaluated in three directions with the tensile axis at 0, 45 and 90 degree to the rolling direction of the sheet according to ASTM E 517. Out-of-plane stretching test was used to determine forming limit diagram. Figure 1 shows the tooling and geometry. It is included of a hemispherical punch, a blank-holder, and a die with draw bead. In this test, various strain ratios are generated by forming specimens with different geometries. The dimension of each specimen is shown in Figure 2. The specimens were prepared in the rolling direction of the sheet metal and were marked by circular grids with 4.4 mm diameter, which were printed on the specimens by a gelatin stamp.

Direction 0 45 90

rm

TABLE (1). Mechanical properties of St14 mild steel sheet. Yield strength Tensile strength Elongation (MPa) (MPa) (%) 201 312 45 218 333 40 206 311 40

(r0  2r45  r90 ) / 4 1.51

Punch

Die

Blank-holder FIGURE 1. Out-of-plane stretching test tools.

174

r-values 1.77 1.16 1.94

D 10 0

W 20

W 15

0

0 75 50 40

17

12

5

5

15

10

0

0

12

75

5 FIGURE 2. The specimens for out-of-plane stretching test [19].

The experiments were performed by DMG 600 kN capacity universal hydraulic press equipped with a computerized control system. The specimens were clamped by pushing the material in the draw bead, using a springloaded blank holder. The punch displacement rate was 10 mm/min and the tests were conducted up to the failure. After experiments, the circular grids on the specimens were deformed to elliptic shapes. Mayler tape was used to determine the major and minor limited strains by measuring the major and minor diameters of the ellipses that was located at the nearest distance to the failure zone.

FINITE ELEMENT SIMULATION Simulations were performed by using the commercial finite element software ABAQUS/explicit 6.9. Out-ofplane stretching test was modeled according to the experiments. One-quarter of the geometry was used due to symmetry. Figure 3 shows the simulation model of the die, punch head and blank-holder. The specimens were meshed by S4R shell element and to increase the accuracy, the mesh size was reduced from the edge to the center area of the specimens under the punch, as shown in Figure 4. The material properties which were used in the simulations are presented in Table 2. The material was modeled as elastoplastic where the stress-strain curve was approximated by Hollomon equation. The anisotropic properties were described by the Hill quadratic yield criterion. The model was assumed to be rate independent. Coulomb friction 0.16 was assumed for all contact surfaces. The simulation process was performed in two steps. In the first step the blank-holder moves down and deforms the blank into the draw bead with 40 KN force to prevent the specimen sliding between the blank holder and the die. Then the punch moves up in 20 increments and deforms the specimen to a specified displacement.

FIGURE 4. Finite element meshing of specimens

FIGURE 3. One-quarter model of out-of-plane stretching test.

TABLE (2). The material properties.

Young’s modulus of elasticity (GPa) Poisson’s ratio Density (kg/m3) Yield strength (Mpa) (0.2%) Strength coefficient (MPa) Strain hardening coefficient (n)

175

210 0.3 7850 201 548 0.23

FAILURE CRITERION In the finite element simulation approach of determining the FLD it is necessary to detect at which element on the specimen and when the localized necking occurs. For this purpose, three numerical criteria are evaluated: maximum second thickness strain derivative (CRIT1), equivalent plastic strain increment ratio (CRIT2), and total equivalent plastic strain ratio (CRIT3). For CRIT1, according to Petek et al. [17] and Pepelnjak et al. [18], the localized necking was specified by tracing the thickness strain and its first and second derivatives versus time at thinnest nodes as shown in Figure 5 When necking appears, a sharp change in the strain behavior is observed on the curve. The second derivative of thickness strain versus time represents a peak in time which assumed as the onset of localized necking. Three nodes were selected for each specimen. The limit strains have been determined by evaluating the major and minor strains for the selected nodes. For CRIT2, according to Marciniak and Kuczynski (M–K) theoretical model [20], at first the equivalent plastic strain increment ( 'H ) for the nodes in the localized necking zone (zone A) and its adjacent zone (zone B) was determined (Figure 6). Then the ratio of equivalent plastic strain increment in zone A to zone B was calculated. The localized necking time was specified when the equivalent plastic strain increment ratio reaches to 10, ( 'H A / 'H B t 10 ) [21]. The major and minor strains in Zone B are noted as the limit strains. Three nodes were selected for each zone. CRIT3 is similar to the CRIT2, except that at first the total equivalent plastic strain ( H ) for the nodes in the localized necking zone (zone A) and the node in zone C was determined (Figure 6). Then the ratio of total equivalent plastic strain in zone A to zone C was calculated. The localized necking time was specified when the total equivalent plastic strain ratio reaches to 10, ( H A / H C t 10 ). The major and minor strains in Zone B (CRIT3-B) and zone C (CRIT3-C) are noted as the limit strains.

Thickness strain First derivative

0.0 -0.5

-3

Knee zone

-1.0 -1.5 -2.0

Second derivative

0 -6 -9 -12

Sharp change in strain rate

-15

600 300 0 -300 -600 -900 -1200 0.00

Onset of necking 0.25

0.50

0.75

1.00

Simulation time / Total time

FIGURE 5. Thickness strain as a function of time.

FIGURE 6. Finite element simulation after stretching

RESULTS AND DISCUSSION Figure 7 shows the specimens after stretching tests. The failure zone appears as a groove in the deformed region of the sheet metal. Figure 8 shows the FLD that obtained from experiments for St14 steel. A fourth order polynomial curve has been used for fitting the data. As seen the FLD0 (the plane-strain state) is about 0.25, which is near to the material strain hardening coefficient.

176

FIGURE 7. The experimental specimens after stretching tests. 0.6

Major strain

0.5

0.4

0.3

0.2 0.1

Exp Data Poly Fit

0.0 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30

Minor strain

FIGURE 8. The experimental FLD of the St14 steel.

In Figure 9 the experimental specimens are compared with the simulation models. It is seen simulations are in good agreement with experiment.

FIGURE 9. Comparison of the experimental specimens and the simulation models.

177

0.6

0.6

0.5

0.5

Major strain

Major strain

In Figure 10 the numerical results of the FLC for St14 steel are compared with the experimental data. Figure 10a shows the numerical criterion CRIT1. It indicates that the overall shape of curve based on the numerical points is in good agreement with the experimental curve. But it seems the CRIT1 criterion shows the lower bond of the experimental results. This numerical approach predicted the onset of necking on the specimen. It is noted that finding the critical nodes and determination of the FLC based on CRIT1 criterion is complicated. In Figure 10b the predicted FLC based on CRIT2 criterion is compared with the experimental data. It indicates that the overall shape of curve based on the simulation is not in good agreement with the experimental curve. The numerical obtained points of the FLC based on CRIT3 criterion are shown in Figure 10c. It indicates that the overall shape of curve based on the numerical points is in good agreement with the experimental curve. The CRIT3-C criterion shows the lower bond of the experimental points and the CRIT3-B criterion shows the upper bond of the experimental points. It seems the lower line on the diagram represents the localization, whereas the upper line represents fracture limit curve. It is noted that the calculation in this criterion is simple.

0.4

0.3

0.4

0.3

0.2

0.2

0.1

0.1

Exp Data FEM CRIT1

Exp Data FEM CRIT2

0.0 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.0 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30

Minor strain

Minor strain

(a)

(b) 0.6

Major strain

0.5

0.4

0.3

0.2

Exp Data FEM CRIT3-B FEM CRIT3-C

0.1

0.0 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30

Minor strain

(c) FIGURE 10. Comparison of the experiment data with the predicted FLC based on;

a) CRIT1 criterion, b) CRIT2 criterion and c) CRIT3 criterion

CONCLUSION The forming limit diagram (FLD) of a low carbon steel St14 (DIN 1623) is investigated experimentally and numerically. Out-of-plane stretching test method was simulated by using commercial finite element software, ABAQUSE 6.9. Three numerical criteria including maximum second thickness strain derivative (CRIT1), equivalent plastic strain increment ratio (CRIT2), and total equivalent plastic strain ratio (CRIT3) were evaluated and the forming limit curve (FLC) were obtained. While the CRIT1 criterion was in good agreement with experimental data,

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it predicted the onset of necking on the specimen and its determination was complicated. The CRIT2 criterion was not in good agreement with experimental data. By using the CRIT3 criterion both the lower and upper bounds of the experimental results were predicted. Since this numerical approach for determination of the FLC is simple and the numerical curves were placed in the critical area of the experimental points, it seems it can be used for determination of the material formability.

ACKNOWLEDGEMENT The authors would like to appreciate the office of the Vice President for Research of Babol Noshirvani University of Technology for its financial support.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

T. B. Stoughton, Int. J. of Mech. Sci. 42, 1-27 (2000). T. B. Stoughton and X. Zhu, Int. J. of Plasticity 20, 1463-1486 (2004). V. M. Nandedkar and K. Narasimhan, J. of Mater. Process. Technol. 89, 24-29 (1999). P. D. Wu, A. Graf, S. R. MacEwen, D. J. Lloyd, M. Jain and K.W. Neale, Int. J. of Solids and Structures 42, 2225–2241 (2005) S.P. Keeler and W.A. Backhofen, ASM Transactions Quarterly 56, 25-48 (1964). G. M. Goodwin, SAE paper No. 680093 (1968). Z. Marciniak, K. Kuczynski and T. Pokora, Int. J. Mech. Sci. 15, 789-805 (1973). K. Nakazima, T. Kikuma and K. Asuka, Yawata Technical report 264, 678-680 (1971). S. S. Hecker, Sheet Met. Ind. 52, 671-676 (1975). F. Ozturk and D. Lee, J. of Mater. Process. Technol. 147, 397–404 (2004). H. Takuda, K. Mori, H. Fujimoto and N. Hatta, J. Mater. Process. Technol. 60, 291–296 (1996). H. Takuda, K. Mori and N. Hatta, J. Mater. Process. Technol. 95, 116–121 (1999). H. Takuda, K. Mori, N. Takakura and K. Yamaguchi, Int. J. Mech. Sci. 42, 785–798 (2000). T. Yoshida, T. Katayama and M. Usuda, J. Mater. Process. Technol. 50, 226–237 (1995). R. Brun, A. Chambard, M. Lai and P. de Luca, Proc. NUMISHEET 99, Besançon, France 393-398 (1999). M. Geiger and M. Merklein, Annals of the CIRP 52, 1-9 (2003). A. Petek, T. Pepelnjak and K. Kuzman, J. of Mech. Eng. Slovenia 51, 330-345 (2005). T. Pepelnjak, A. Petek and K. Kuzman, Advanced Material Research 6, 697-704 (2005). B. Dariani, " Effect of non proportional loading path on forming limit of sheet metals ", Ph.D. Thesis, Amirkabir University of Tech., 1999. C. Zhang, L. Leotoing, D. Guines and E. Ragneau, J. of Mater. Process. Technol. 209, 3849–3858 (2009) W. F. Hosford and R. M. Caddell, Metal Forming; Mechanics and Metallurgy, 3rd Edition, Cambridge university press, 2007.

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Possibility of Expressing Anisotropic Yield Functions of Metals using the Invariants of Stress Tensor M.H.Parsa1, K. A. Shirvanedeh2 and P. H. Matin3 1

Professor, School of Metallurgy and Materials Engineering, University College of Engineering, University of Tehran, Iran, P.O. Box 11155/4563, [email protected] 2 Graduate Student, School of Metallurgy and Materials Engineering, University College of Engineering, University of Tehran, Iran, P.O. Box 11155/4563, [email protected] 3 Assistant Professor, Department of Engineering and Aviation Sciences, University of Maryland Eastern Shore, Princess Anne, MD 21853, USA, [email protected]

Abstract. Most of existing yield function have been developed base on the curve fitting of experimental data and also some theoretical effort have been done for deriving yield functions. Framework of invariant theory prepared valuable opportunity for modeling non-linear constitutive equation that it is shown in the works of Rivlin. It has been tried using invariant theory for representation of yield function for isotropic material and it is shown that invariant theory could cover all yield functions needed for isotropic metals using a general equation. Therefore, in the present paper it has been tried to extend invariant theory for anisotropic materials. Results have shown that invariant theory have the capacity for deriving a general equation for representation of anisotropic yield function for orthotropic materials. Keywords: Yield function, invariant theory, anisotropy PACS: 62.20.fq

INTRODUCTION Development of metal forming technology accompanies of fast progress in using Finite Element Method (FEM). One of the most important parameters which affect the accuracy of FEM is the yield criterion. This criterion defined the boundary between elastic and plastic behaviors of metals which defines a closed surface in the stress space. Therefore defining the yield surface is one of the main aspects of plastic behavior of metals which should be described appropriately. One of main features of metals is anisotropic behavior due to their structures. This anisotropic behavior is very important in the case of sheet metals and it also affects the yield function. There are many anisotropic yield functions which are proposed by different researchers. First anisotropic yield function has been proposed by Hill [1] in 1948. This criterion was extension of Von Mises isotropic yield function. Simplicity of Hill’s quadratic yield function caused to make it as most popular anisotropic yield function. It has one major problem. This yield function cannot predict anomalous behaviors often observed in aluminum alloy sheets. Therefore Hill tried to improve his criterion [2-4]. Hosford [5] has been used another approach for developing phenomenological yield function that was based on polycrystal aggregate calculations. Hosford’s anisotropic yield function turns out to be a base for other developments in anisotropic yield functions. In recent two decades many different kind of yield functions have been developed by researchers. Barlat and co-workers had major role in extending Hosford yield function and they proposed series of yield functions for anisotropic sheet metals [6-9]. Majority of yield functions are proposed for orthotropic sheet metals. Most of mentioned yield functions have proposed base on the experimental observations and in order to fit the experimental results. In an attempt to express the yield function of metal based on the mathematical foundation, authors have been tried to use invariant theory. Ronald Rivlin was pioneer of using invariant theory in solid mechanics of isotropic and anisotropic materials. In 1940s, Rivlin established the basic of the finite elasticity for nonlinear continuum mechanics by invariant theory for isotropic materials [10]. Also Ericksen and Rivlin [11] for first time studied strain energy function for transversely isotropic materials. In the previous work, authors showed that invariant theory could be used for representation of yield surface of isotropic metals [12]. In the present work the ability of invariant theory for expressing yield surface for orthotropic metals are investigated. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 180-186 (2011); doi: 10.1063/1.3623609 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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Deriving framework for orthotropic sheets yield function In a Cartesian coordinate defined by x, y and z directions, an orthotropic anisotropy has been considered. In such case, the material symmetry Group S can be shown by: S={±I, S 1 , S 2 , S 3 }

(1)

Where I is Identity matrix and S 1 , S 2 , and S 3 are the reflections with respect to the basic planes consisting of (y, z), (z,x), and (x,y) directions of orthonormal frame 0f x, y and z. Two series of symmetric second order tensors are needed for defining the framework of yield surface of anisotropic media using invariant theory. In First step it is necessary to find structural tensors that account for the material internal structure symmetries. In the second step, tensors that are corresponding for plastic behaviors of material are defined. Three symmetric second order tensors have been considered as structural tensors [13]: 1 0 0 0 0 0 0 0 0 𝑀1 = 0 0 0, 𝑀2 = 0 1 0, and 𝑀3 = 0 0 0 0 0 0 0 0 0 0 0 1

(2)

Selection appropriate stress tensor is very important for representing mechanical behaviors of material such as yield function. Two stress tensors, stress tensor in the principle direction and deviatoric stress tensor, have been chosen for investigating the possibility expressing anisotropic yield function using invariant theory. These two symmetric second order tensors are as shown in equation (3) and (4). 𝜎11 𝜎𝑖𝑗 = � 0 0 ⎛ 𝜎𝚤𝚥́ = ⎜ ⎝

0 𝜎22 0

0 0 � 𝜎33

(2𝜎11 − 𝜎22 − 𝜎33 )� 3 0 0

(3)

0

(2𝜎22 − 𝜎11 − 𝜎33 )� 3 0

0

0

(4) ⎞ ⎟

(2𝜎33 − 𝜎11 − 𝜎22 )� 3⎠

Where subscripts 1~3 are referred to the directions of the Cartesian coordinates. Integrity bases are essential part for establishing new formulation of yield function using invariant theory. Most of integrity base for different kind of tensors are established and they can be found in handbooks [13]. Integrity bases of an arbitrary symmetric second order tenor A that is tried to use in yield function expressed in polynomial form considering orthotropic anisotropy should be consist of terms such trM 1 A, trM 2 A, trM 3 A, trM 1 A2, trM 2 A2, trM 3 A2, trA3. The General yield function F can be expressed using seven mentioned integrity bases [13]. The general yield function F is considered to be a scalar function of the seven scalar invariants of three structural tensors and one stress related tensor related. Therefore, the general yield function F can be expressed as shown in equation (5). F(trM 1 A, trM 2 A, trM 3 A, trM 1 A2, trM 2 A2, trM 3 A2, trA3, k)=0 (5) Various traces of arbitrary symmetric second order tenor A are presented in equations (6) to (8) that can be used to express the various invariants of the mechanical tensor A. 𝑡𝑟 𝐴 = (𝐴11 + 𝐴22 + 𝐴33 ) 2 𝑡𝑟𝐴2 = (𝐴11 + 𝐴222 + 𝐴233 ) 3 3 𝑡𝑟𝐴 = (𝐴11 + 𝐴322 + 𝐴333 )

(6) (7) (8)

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The general form of equation (5) can be expressed using a polynomial function of the seven integrity bases as shown in equation (9). This the most general polynomial forms for yield surface of orthotropic anisotropic materials. a 0 (trM 1 A)b0+ a 1 (trM 1 A2)b1+a 2 ( trA3)b2+ a 3 (trM 2 A)b3+ a 4 (trM 2 A2)b4+ a 5 (trM 3 A)b5+ a 6 ( trM 3 A2)b6 +a 7 (trM 1 A)b7(trM 1 A2)b8+ a 8 (trM 1 A)b9( trA3)b10+ a 9 (trM 1 A)b11(trM 2 A)b12+ a 10 (trM 1 A)b13(trM 2 A2)b14+ a 11 (trM 1 A)b15(trM 3 A)b16+ a 12 (trM 1 A)b17( trM 3 A2)b18 + a 13 (trM 1 A2)b19( trA3)b20+ a 14 (trM 1 A2)b21 (trM 2 A)b22+ a 15 (trM 1 A2)b23 (trM 2 A2)b24 +a 16 (trM 1 A2)b25(trM 3 A)b26+ a 17 (trM 1 A2)b27 ( trM 3 A2)b28+ a 18 ( trA3)b29 (trM 2 A)b30+ a 19 ( trA3)b31 (trM 2 A2)b32 + a 20 ( trA3)b33 (trM 3 A)b34+ a 21 ( trA3)b35 ( trM 3 A2)b36 + a 22 (trM 2 A)b37 (trM 2 A2)b38 + a 23 (trM 2 A)b39 (trM 3 A)b40+ a 24 (trM 2 A)b41 ( trM 3 A2)b42+ a 25 (trM 2 A2)b43 (trM 3 A)b44 + a 26 (trM 2 A2)b45 ( trM 3 A2)b46+ a 27 (trM 3 A)b47 ( trM 3 A2)b48=k

(9)

where a i (i=0,1 … 27) and b j (j=0,1 … 48) are material constants and k is related to yield strength of material. In special case, when mechanical tensor A is diagonal, some equalities will be arose such as (trM 1 A)2=trM 1 A2, (trM 2 A)2=trM 2 A2, and (trM 3 A)2=trM 3 A2. Therefore equation (9) could be reduced to the form presented by equation (10) a 0 (trM 1 A)b0+ a 1 (trM 2 A)b1+ a 2 (trM 3 A)b2 +a 3 ( trA3)b3+a 4 (trM 1 A)b4(trM 2 A)b5 + + a 6 (trM 3 A)b8(trM 1 A)b9 + a 7 (trM 1 A)b10( trA3)b11+ a 5 (trM 2 A)b6(trM 3 A)b7 b12 3 b13 b14 3 b15 a 8 (trM 2 A) ( trA ) + a 9 (trM 3 A) ( trA ) =k

(10)

Where a i (i=0,1 … 9) and bj (j=0,1 … 15) are limited to ninth and fifth values. Therefore, diagonalization of mechanical tensor A, reduces the number of required in comparison to the general case.

Result and discussion Selection of stress tensor have significant role on number of parameters and coefficients. It is shown that by selection of principle stress tensor, the general form of equation (9) can be reduced to equation (10) which has very fewer coefficients than equation (9). In this research, two special types of diagonal tensor have been considered for representing yield function of materials. Therefore, the equation (10) can be used for examining the effect of different parameters on the capacity of this equation for forming the closed surfaces in the stress space that could be used to represent yield surface of materials. In the first step, stress tensor in the principle form (equation (3)) has been used and general form of yield function has been derived (equation (11)). It is shown that equation (11) is combination of single and multiple terms of integrity bases of principle stresses tensor. a 0 𝛔 1 b0+ a 1 𝛔 2 b1+ a 2 𝛔 3 b2 +a 3 (𝛔 1 3+𝛔 2 3+𝛔 3 3)b3+a 4 𝛔 1 b4 𝛔 2 b5 + a 5 𝛔 2 b6 𝛔 3 b7 + a 6 𝛔 3 b8 𝛔 1 b9 + a 7 𝛔 1 b10(𝛔 1 3+𝛔 2 3+𝛔 3 3)b11+ a 8 𝛔 2 b12(𝛔 1 3+𝛔 2 3+𝛔 3 3)b13+ a 9 𝛔 3 b14(𝛔 1 3+𝛔 2 3+𝛔 3 3)b15=k

(11)

It has been tried to evaluate the resultant closed surfaces that can be established using equation (11) in the stress space. Commercial software has been used to exhibit graphical contours that can be established using different combination of single and multiple factors in stress space [14]. Calculation shows that none of the mentioned terms in equation (11) could produce proper closed-surface to representation of yield surface in stress space. Therefore, it is revealed that first chosen stress tensor couldn’t represent yield surface of materials. In the next step, it has been tried to use second stress tensor (equation (4) to construct integrity bases required for generating general yield function. Established general yield function is shown as equation (12). a 0 (2𝛔 1 - 𝛔 2 - 𝛔 3 )b0+ a 1 (2𝛔 2 - 𝛔 1 - 𝛔 3 )b1+ a 2 (2𝛔 3 - 𝛔 1 - 𝛔 2 )b2 +a 3 ((2𝛔 1 - 𝛔 2 - 𝛔 3 )3+(2𝛔 2 𝛔 3 )3+(2𝛔 3 - 𝛔 1 - 𝛔 2 )3)b3+a 4 (2𝛔 1 - 𝛔 2 - 𝛔 3 )b4 (2𝛔 2 - 𝛔 1 - 𝛔 3 )b5 + a 5 (2𝛔 2 - 𝛔 1 - 𝛔 3 )b6 (2𝛔 3 𝛔 2 )b7 + a 6 (2𝛔 3 - 𝛔 1 - 𝛔 2 )b8 (2𝛔 1 - 𝛔 2 - 𝛔 3 )b9 + a 7 (2𝛔 1 - 𝛔 2 - 𝛔 3 )b10(((2𝛔 1 - 𝛔 2 - 𝛔 3 )3+(2𝛔 2 𝛔 3 )3+(2𝛔 3 - 𝛔 1 - 𝛔 2 )3)b11+ a 8 (2𝛔 2 - 𝛔 1 - 𝛔 3 )b12(((2𝛔 1 - 𝛔 2 - 𝛔 3 )3+(2𝛔 2 - 𝛔 1 - 𝛔 3 )3+(2𝛔 3 𝛔 2 )3)b13+ a 9 (2𝛔 3 - 𝛔 1 - 𝛔 2 )b14(((2𝛔 1 - 𝛔 2 - 𝛔 3 )3+(2𝛔 2 - 𝛔 1 - 𝛔 3 )3+(2𝛔 3 - 𝛔 1 - 𝛔 2 )3)b15=k

𝛔1𝛔1𝛔1𝛔1-

(12)

This equation like equation (11) composed of single and multiple terms of integrity bases of deviatoric stress tensor. The effect of the single and multiple terms in individual or in combination with other terms have evaluated by using computational software for representing closed surface in the stress space. For evaluating the effect of elements of

182

equation (12), first the effect of single terms has been examined individually. Next the effect of combined terms has been evaluated for establishing closed surface in the stress space.

Examination the effect of terms in forming closed-surface All the exponents of equation (12) can only be integer numbers. Corresponding of a 0 (trM 1 A)b0 in equation (10) is a 0 (2x-y-z)b0 in equation (12). Established shape of surface using a 0 (trM 1 A)b0 term mainly depends on values of exponent b0. If odd numbers are attributed to b0, these cause to build of flat surfaces in stress space. If even numbers are attributed to b0, these cause to formation of two flat parallel surfaces. Evaluations shows that two other single terms of a 1 (trM 2 A)b1 and a 2 (trM 3 A)b2 have same results by this difference that the directions of formed surfaces are different and they cut each other in the stress space. In the next step the first three single terms has been investigated for possibility of making closed surface. Exponents b 0 , b 1 , and b 2 and coefficients a 0 , a 1 , and a 2 could be changed separately. Examining of exponents variations show that each exponent term that have been attributed bigger value becomes dominant term and eliminates other terms effect in the equation and it is not possible to form closed-surface. Therefore, b 0 , b 1 , and b 2 must get identical values (b 0 =b 1 =b 2 ). Further studies show that when the exponents are attributed odd numbers, no closed-surface can be built. But when even numbers are chose for exponents, it is possible to build closed-surface in the stress space. Therefore by choosing even numbers for exponents in the three first single terms (a 0 (trM 1 A)b0, a 1 (trM 2 A)b1, and a 2 (trM 3 A)b2) different cylindrical closed-surface could be achieved by changing coefficients (a 0 , a 1 , and a 2 ). Figure 1 shows closed surface variation by changing of coefficients when exponents b0, b1 and b2 are equated to 4. Curve number one, two and three show intersections of π-plane by closed-surfaces result from assumption of coefficients as a 0 =2, a 1 =1, and a 2 =1, a 0 =1, a 1 =3, and a 2 =1, a 0 =1, a 1 =1, and a 2 =4, respectively.

Figure 1. effect of changing constant coefficient in first three single terms

Fourth single term, a 3 ( trA3)b3, has different behavior from other three first single terms, so it has been examined separately. In this term like other single terms exponent b3 play major role in capacity of forming closed-surface. When b3 is attributed odd numbers it can’t form closed-surface and only several intersected flat surfaces are formed. But when b3 is attributed even numbers, this term can build closed-surfaces. By adding this single term to other first three single terms, it becomes possible to bring new capacity to equation and increase flexibility of equation on showing yield surface in stress space. Figure 2 shows this case. Curve number one shows closed-surface formed from first three single terms when b 0 , b 1 , and b 2 are equated to 6 and a 0 , a 1 and a 2 are equated to one. Curves

183

number two and three show the effect of adding fourth term to three first single terms when is b3 equated to 2 and a 3 is attributed 1 and 4, respectively. It is clear that adding of fourth term to other single term increase ability of equation in forming variety of closed-surfaces and yield surfaces of anisotropic materials.

Figure 2. effect of changing constant coefficient in forth single terms by adding to other three single terms

In the next step, the effects of multiple terms have been examined. Three first multiple terms a 4 (trM 1 A)b4(trM 2 A)b5, a 5 (trM 2 A)b6(trM 3 A)b7, and a 6 (trM 3 A)b8(trM 1 A)b9 have similar behavior individually and also in combinations. Therefore, all of them have been examined in one case. Exponents b4, b5, b6, b7, b8, and b 9 have main role in forming of closed-surface and defining suitable exponents is very important. Computational results show that b 4 , b 6 , and b 8 must be identical (b4=b6=b8) and also b5, b7, and b9 should be equated (b5=b7=b9). Consequently, only two groups of exponents must be selected to be odd or even numbers. Only when even numbers are attributed to the exponents of multiple terms, combinations of them could form closed-surfaces that stand for anisotropic yield functions of materials. Also when these three terms are added to other first three single terms, it causes that resultant equation get more ability in prediction of yield function and increase the flexibility of yield function. This case is shown in the Fig. 3. The three first curves show the intersection of the closed surface formed from the combination of first three multiple terms with the π-plane when b4, b6, and b8 are equated to 10, b5, b7 and b9 equated to 2 for curve one, b4, b6 and b8 equated to 8 and b5, b7 and b9 equated to 4 for curve two, and b4, b6, b8, b5, b7 and b9 are equated to 6 for curve three. Figure 3 shows that change in selection of exponents cause rotation of closed surface π-plane. Curves four and five show When exponents b4, b6 and b8 are equated to 11, b5 b7 and b9 are equated 1 and coefficients a 4 , a 5 and a 6 are equated 1, curve four resulted. But when exponents b4, b6 and b8 are equated 10, b5, b7 and b9 are equated 2, and coefficients a 4 , a 5 and a 6 are equated to 1 curve five resulted. In both cases the three first single terms assuming exponents of b0, b1and b2 which equated to 12 and coefficients of a 0 , a 1 and a 2 that equated to 1 are added to the multiple terms.

184

Figure 3. Effect of changing exponential coefficients in first three multiple terms and also adding of them to first three single terms.

The effect of three first multiple terms have been examined and it is seen that they have very similarity in behavior and they cause reduction in amount of coefficients. The three last multiple terms a 7 (trM 1 A)b10( trA3)b11, a 8 (trM 2 A)b12( trA3)b13, anda 9 (trM 3 A)b14( trA3)b15 have same trend mentioned for the three first multiple terms and they have similar behavior. Computational results show that exponential coefficients b10, b11, b12, b13, b14, and b15 can be divided in to two category, in first category b10, b12, and b14 that must be identical (b10=b12=b14) and second category b11, b13, and b15 that should be attributed same values (b11=b13=b15). Examinations of equation (10) show that most of exponents are depend on each other. Computational results prove that some of exponents must be equal and consequently numbers of exponents in equation (10) reduced from 15 to 5. The result can be shown in equation (13). a 0 (trM 1 A)b0+ a 1 (trM 2 A)b0+ a 2 (trM 3 A)b0 +a 3 ( trA3)b1+a 4 (trM 1 A)b2(trM 2 A)b3 + b2 b3 b2 b3 a 5 (trM 2 A) (trM 3 A) + a 6 (trM 3 A) (trM 1 A) + a 7 (trM 1 A)b4( trA3)b5+ a 8 (trM 2 A)b4( trA3)b5+ a 9 (trM 3 A)b4( trA3)b5=k

(13)

Conclusion Selection of tensor that can be used to represent the yield behavior of materials, play very important role on ability of invariant theory in expressing anisotropic yield functions. It is seen that by selection of principle stresses tensor, invariant theory couldn’t represent any closed-surface to stand for anisotropic yield functions. But by choosing deviatory stresses tensor, it is seen that invariant theory could be a reliable theory for deriving yield functions for anisotropic materials. Invariant theory generates a very general equation for expressing yield function of orthotropic materials that it is shown by equation (9). But by selecting a diagonal tensor it is possible to reform equation (9) into equation (10). This results in reduction the number of coefficients and exponents from 27 and 48 in equation (9) to 9 and 15 in equation (10), respectively. The effects of each terms of equation (10) have been analyzed in order to form closed surface and the relationship between coefficients of equation (10) have been revealed. It is observed that exponents have major role in forming closed surface. Therefore, by appropriate choice of them it is possible to build a range of closed-surface that could represent yield function of orthotropic materials. The results show that exponent of equation (10) have relationship with each other and some of them must be identical until they could form closed surface. Therefore equation (10) can be reformed into equation (13) which reduces the number of exponents from 15 to 5.

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R. Hill, A theory of the yielding and plastic flow of anisotropic metals, Proc. R. Soc. Lond. A 193, p.p 281–297, 1948 R. Hill, Theoretical plasticity of textured aggregates, Math. Proc. Cambridge Philos. Soc. 85, p.p 179–191, 1979 R. Hill, Constitutive modeling of orthotropic plasticity in sheet metals, J. Mech. Phys. Solids 38, p.p 405– 417, 1990 R. Hill, A user-friendly theory of orthotropic plasticity in sheet metals, Int. J. Mech. Sci. 15, p.p 19–25, 1993 W. F. Hosford, A generalized isotropic yield criterion, J. Appl. Mech. 39, p.p 607-609, 1972 F. Barlat, O. Richmond, Prediction of tricomponent plane stress yield surfaces and associated flow and failure behaviour of strongly textured F.C.C. polycrystalline sheets, Mat. Sci. Eng. 91, p.p 15–29, 1987 F. Barlat, J. Lian, Plastic behaviour and stretchability of sheet metals – Part I: A yield function for orthotropic sheets under plane stress condition, Int. J. Plasticity 5, p.p 51–66, 1989 F. Barlat, D.J. Lege, J.C. Brem, A six-component yield function for anisotropic materials, Int. J. Plasticity 7, p.p 693–712, 1991 F. Barlat, J.C. Brem, J.W. Yoon, K. Chung, R.E. Dick, S.H. Choi, F. Pourboghrat, E. Chu, D.J. Lege, Plane stress yield function for aluminium alloy sheets – part 1: theory, Int. J. Plasticity 19, p.p 1297–1319, 2003 R.S. Rivlin, Large elastic deformations of isotropic materials IV. Further development of the general theory, Philos. Trans. Royal Soc. London A241, p.p 379–397, 1948 J.L. Ericksen, R.S. Rivlin, Large elastic deformations of homogeneous anisotropic materials, J. Rational Mech. Anal. 3, p.p 281–301, 1954 M.H. Parsa, K.A. Shirvanedeh, P. H. Matin, An attempt to express yield functions of isotropic metals based on the invariants of stress tensor, TMS 2011 Conference J. P. Boehler, “Applications of tensor functions in solid mechanics”, Springer-Verlag, Wien-New York, 1987 Wolfram Mathematica7, 2008.

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The Effect of Anisotropy on Thin-Walled Tube Bending K. Hasanpoura*, B. Aminib, M. Poursinaa, M. Baratic a

Department of mechanical engineering, Faculty of engineering, University of Isfahan, P.O.Box 81746-73441, Isfahan, Iran. b

Department of Mechanical Engineering, Islamic Azad University, Khomeinishahr Branch, Iran. c

Department of Mechanical Engineering, K. N. Toosi, University of Technology, Iran.

*

Corresponding author: +98-3117934021, Fax: +98-3117932746, [email protected].

Abstract. Thin-walled tube bending has found many of its applications in the automobile and aerospace industries. The rotary-drawbending method which is a complex physical process with multi-factor interactive effects is one of the advanced tube forming processes with high efficiency, high forming precision, low consumption and good flexibility for bending angle changes. However it may produce a wrinkling phenomenon, over thinning and cross-section distortion if the process parameters are inappropriate. Wrinkles propagate permanently in thin-walled tube, but finally, localize in a finite zone and lead to failure. The prediction of wrinkling in thin-walled tube bending processes has been a challenging topic. In this paper, firstly, the plastic deforming behavior and wrinkling mechanism for a thin-walled tube is simulated and the results will compare with the available experimental ones. Then, the effect of anisotropy on ovalization, thickness and wrinkling of tube will be investigated using FEM. Extensive numerical results are presented showing the effects of the various kinds of materials and geometric parameters on wrinkling using anisotropic yield function. Keywords: Thin-walled tube; Anisotropy; Wrinkling.

INTRODUCTION Thin- walled tubes find wide applications in many branches of engineering. Examples include aircraft, spacecraft, cooling towers, steel silos and tanks for bulk solid and liquid storage, pressure vessels, pipelines and offshore platforms. The rotary-draw bending process of thin-walled tubes has been attracting more and more applications due to its high forming precision advantage and satisfying the increasing needs for high strength per weight ratio products. However, the inner side of thin-walled tubes may experiences a wrinkling phenomenon if the process parameters are inappropriate especially for tubes with large diameter and thin wall thickness. It is because of the compressive stress during the bending process, which will leads to the process failure or even die damage if wrinkling is severe. Possible wrinkling at two locations, before the material enters the bending die and at the bending intrados, are identified. How to predict this phenomenon rapidly and accurately is one of the urgent key problems to be solved for the development of this process at present. It is well known that the problem resolution relies heavily on experience and involves repeated trial-and-errors in practice, which spends excessive manpower, raw material, and time in designing and adjusting the process and dies, and moreover makes the production efficiency abate drastically. Even so, sometimes contented outcomes may not be obtained. Li et al. developed a complete 3D elastic-plastic FEM model of the process using ABAQUS/Explicit code based on the analysis of the forming characteristics by analytical and experimental methods, and thus the plastic deformation characteristics with small bending radius were investigated [1]. Yang et al. have proposed a method to obtain the wrinkling limit of thin-walled tube with large diameter under different loading paths by the numerical control (NC) bending process analytically [2]. Zhan et al. have established one analytical model of the mandrel (including mandrel shank and balls) and have deduced some reference formulas in order to select the mandrel parameters preliminarily, then carried out experiment to verify the analytical model [3]. Gu et al. proposed and investigated the interactive effects of the wrinkling and other two defects in the process by using the developed Explicit FE model based on the ABAQUS/Explicit code. then a strategy for selecting the positive parameters was proposed based on the understanding of the interactive effects of wrinkling, thinning and section distortion [4]. Kumar was simulated rotary draw bending process focusing more on wrinkle formation [5]. The effects of various process parameters on the

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 187-193 (2011); doi: 10.1063/1.3623610 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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formation of wrinkles have been analyzed and optimum process parameters were selected to get wrinkle-free bend. Gu et al. revealed the spring back mechanism and rule of thin-walled tube NC bending based on the numerical simulation of the whole process [6]. Yang et al. established a 3D elastic–plastic finite element model and a wrinkling energy prediction model under multi-die constraints considering the characteristics of the bending processes of aluminum alloy thin-walled tubes with large diameters[7]. Li et al. developed an energy-based wrinkling prediction model for thinwalled tube bending by using the energy principle, combined with analytical and finite element (FE) numerical methods [8]. Yan et al. developed an analytical model for thin-walled tube NC bending and then based on ABAQUS platform, a series of 3D-FE models are developed to simulate the bending process with large diameter and small bending radiuses[9]. Also, numerical study on the deformation behaviors of bending process was conducted. Kou et al. aimed at revealing the deformation behaviors of thin-walled tube in rotary draw bending under push assistant loading conditions with multiple defects. They presented an analytical description of push assistant loading conditions firstly and proposed some indices for evaluation of push assistant loading roles, then the interactive effects of push assistant loading conditions on wall thinning, cross-section deformation and wrinkling investigated extensively using 3D-FE simulation considering push assistant loading functions combined with experiment [10]. In this paper, firstly, the plastic deforming behavior and wrinkling mechanism for thin-walled tube is simulated and the results will compare with the available experimental ones. Then, the effect of anisotropy on ovalization, thickness and wrinkling of tube will be investigated using FEM. Extensive numerical results are presented showing the effects of the various kinds of materials and geometric parameters on wrinkling using anisotropic yield function. THE ROTARY-DRAW BENDING PROCESS The whole process of thin-walled tube bending includes three processes: bending tube, retracting mandrel and spring back. In the thin-walled tube rotary-draw bending process, as shown in Figure 1, both sides of the tube are subjected to various tooling’s strictly contacting force, such as bend die, clamp die, pressure die, wiper die and mandrel. The tube is clamped against the bend die; drawn by the bend die and the clamp die, the tube goes past the tangent point and rotates along the groove of the bend die to the desired bending degree and the bending radius. So the process needs precise coordination of various dies and strictly controlling of forming parameters. Among the above tooling, mandrel is positioned inside the hollow tube to provide the rigid support.

FIGURE 2. Illustration of FEM model for rotary-draw bending process

FIGURE 1.Forming principle of rotary draw bending method

Figure 2 shows FEM model for rotary-draw bending process. The four-node doubly curved thin shell is applied to describe three-dimensional deformable tube. Five integration points are selected across the thickness to describe the tube bending deformation better. The external and internal rigid tools are modeled as rigid bodies using 4-node 3D bilinear quadrilateral rigid element to describe smooth contact geometry curved faces. In this paper, the results will be compared with the experimental data of Li et al. [3]. The geometry parameters are reported in Table 1. The material is aluminum alloy LF2M, which its mechanical properties are shown in Table 2. The material model is the commonly used Swift’s power-law plastic model as in Eq. (1):

V

KH

n

188

(1)

Where K is the strength coefficient, n the work-hardening exponent [7]. TABLE 1. Geometrical parameters of simulations

Parameters Tube outside diameter(D) Wall thickness(t) Bending radius(R) Mandrel diameter(d) Nose radius(r) Extension length(e) Number of balls(n) Balls thickness(k) Space length between mandrel Shank and ball(p) Mandrel length Pressure die length Clamp die length Wiper die length

Value 38mm 1mm 57mm 35.60mm 6mm 6mm 1 12mm 15mm 153mm 250mm 115mm 120mm

In the tube bending process, five contact interfaces between tube and dies are as following: tube/mandrel (ball), tube/wiper die, tube/bend die, tube/clamp die and tube/pressure die. The classical Coulomb model has been chosen to represent the interfaces’ friction conditions:

Vf Where

P Vn

(2)

V f is the frictional stress, P the friction coefficient (0  P  0.5) and V n the stress on the contact surface.

The tube’s bending deformation depends on the contact and friction between various tube portions and different dies. According to the different contact conditions, the friction coefficient can be classified into 4 kinds: 0.05, 0.1, 0.25 and ‘‘Rough’’, in which ‘‘Rough’’ type refers to no relative slipping when nodes contact each other and suitable for the tube/clamp die friction conditions. The other friction coefficients have been assigned to the different contact interfaces as shown in Table 3. TABLE 2. Mechanical properties of tube

Material parameter Ultimate tension strength (MPa) Extensibility (%) Poisson’s ratio Initial yield stress (MPa) Hardening exponent, n Strength coefficient K Young’s modulus E (GPa) Density (kg/m3)

TABLE 3. Friction conditions in various contact interfaces

Contact interface Tube/wiper die Tube/pressure die Tube/clamp die Tube/bend die Tube/mandrel Tube/balls

190 22 0.34 90 0.262 398 56 2700

Friction coefficients 0.05 0.25 Rough 0.1 0.1 0.1

HILL ANISOTROPY CRITERIA To express the behavior of anisotropic materials, different equations have been introduced. Among these equations, quadratic equation of Hill is more popular in the papers and its constants can be easily calculated by performing tensile test. In this paper, the Hill yield function was used to study the effect of anisotropy which is as follows:

189

V2

F (V yy  V zz )2  G(V zz  V xx )2  H (V xx  V yy )2  2LV 2 yz  2MV 2 zx  2 NV 2 xy

(3)

The constants in this equation can be obtained using the following relations:

Vzzy

2 y y Vxx ; Vyy H F

2 y y Vxx ;Vxy G F

1 y y Vxx ;Vyz L

1 y y Vxx ; Vxz M

1 y Vxx N

(4)

In these formulas the yield stress in the extruding direction ( V xx ), has been selected as a reference yield stress. It must y

be noted that in this finite element model, the x axes is in the extruding direction, the z one is in the circumferential direction and the y direction represents the radial direction. A convenient method for determining the coefficients in equation (3) is conducting three tensile tests, using specimens cut from the tube at the angles 0°, 45° and 90° with respect to the extruding direction, with the additional conditions: (5) H  G 2; N=L=M Lankford coefficients are defined as follows[12]: H H 2N  F  G (6) r0 ; r90 ; r45 G F 2(G  F ) To define the anisotropic behavior of the material, the six ratios of yield stresses to the reference yield stress in extruding direction i.e. x direction ( V 0

R11

V xxy V0

V xxy ) are used. These ratios are:

1; R22

The normal anisotropy index

V yyy ; R33 V0

V zzy ;R V 0 12

3

V xyy ;R V 0 13

3

V yzy ; R23 V0

3

V xzy V0

(7)

(r ) and the planar anisotropy ('r ) are calculated using the following formulas:

1 ( r0  2 r45  r90 ) 4 (8) 1 'r (r0  2 r45  r90 ) 2 In normal anisotropy r0=r45=r90 .In this paper, the effect of normal anisotropy on the wrinkling of tube under bending is investigated. Lankford coefficients and Hill yield function values associated with this case is given in Table 4. r

TABLE 4. Normal anisotropic cases

Case 1 2 3 4 5 6 7 8 9 10

r

R11

R22

R33

R12

R13

R23

0.5 0.6 0.7 0.8 1.1 1.2 1.3 1.4 1.5 2

1 1 1 1 1 1 1 1 1 1

0.866 0.894 0.921 0.948 1.024 1.048 1.072 1.095 1.118 1.224

1 1 1 1 1 1 1 1 1 1

1.060 1.044 1.030 1.019 0.992 0.985 0.978 0.973 0.968 0.948

1.060 1.044 1.030 1.019 0.992 0.985 0.978 0.973 0.968 0.948

1.060 1.044 1.030 1.019 0.992 0.985 0.978 0.973 0.968 0.948

190

RESULTS At first the validity of results is verified comparing with ones in reference [3]. Simulations show that the initial tube circular cross section deforms to the oval, through the arc of bended tube, which is in agreement with the experiments. The parameter of ovalization in tube cross-section is defined as: Major Diameter - Minor Diameter (9) Ovalization Ratio u 100 % Initial Tube Diameter

As can be seen in Figure 3, parameter of ovalization in tube cross-section for the middle of the bending angle was used to compare the results, which indicates good agreement between the results of two researches. Von mises stress contour in bending angle of 104° is shown in Figure 4.

FIGURE 4. Von mises Stress distribution in the bending angle of 104 º

FIGURE 3. Comparison of the ovalization parameter in this research and reference [3]

Now, the normal anisotropy effect on wrinkling and tube deformation in rotary-draw bending process is studied. In Figure 5, the effect of normal anisotropy on ovalization of tube cross-section is shown. In this case, the bending angle is 90° and the ovalization of tube cross-section is calculated in angle 45°. This diagram shows that with increasing in normal anisotropy coefficient, ovalization of tube cross-section will be increased. Based on experimental findings, depending on the applied conditions, possible wrinkling at two locations, before the material enters the bending die and at the bending intrados, will be occurred as is shown in Figure 6. The simulated process in this paper, leads the tube to be wrinkled in both locations.

Ovalization Ratio at 45

o

15

10

5 0.5

1

1.5

2

r-value

FIGURE 5. Anisotropy effect on ovalization of tube crosssection

FIGURE 6. Two wrinkling locations occurred in rotary-draw bending process

191

To study of anisotropy effects on wrinkling, the regions  and II are considered. Figure 7 illustrates the node numbers 1 to 35 on the tube edge which will be entered to the intrados of bended tube.

FIGURE 7. Node numbers and definition of Y value to study of anisotropy effects on wrinkling

The Y value is considered for the study of anisotropy effects on wrinkling. According to Figure 7, for nodes were located in left hand side of line AB, the value of Y is equal to the distance of nodes from the line CD and for nodes were located in right hand of line AB, the value of Y is equal to the distance of nodes from the point O (which is the center of bending die). When the tube does not wrinkle, Y value is almost equal to 38 mm, considdering the distance of point O from axis of tube is 57mm and tube radius is 19 mm. If the tube wrinkles, the value of Y will be more or less which this deviation is a characteristic of wrinkling. In this section, all geometrical features are similar to previous case except tube thickness. For having a great wrinkle wavelength and observing wrinkles on the tube, tube thickness is considered 0.5 mm. The anisotropy effects on wrinkling are shown in the Figures 8 and 9. In Figure 8, Anisotropy coefficients are between 0.5 to 1 and Figure 9 relating to the anisotropy coefficients between 1 and 1.5. 40.5

40

40

39

Y value

Y value

39.5 39

38.5

38.5

38

38

37.5

37.5 0

r = 1.1 r=2

39.5

r = 0.5 r = 0.8

5

10

15

20

25

30

37 0

35

Node Number

5

10

15

20

25

30

35

Node Number

FIGURE 8. Wrinkling of intrados for r <1

FIGURE 9. Wrinkling of intrados for r> 1

In Figure 10, the contour of tube thickness, for four cases with normal anisotropy coefficients 0.5, 0.8, 1.1 and 2 are shown. It is seen that with increasing the normal anisotropy coefficient, the maximum value of wall thickness is increased.

192

FIGURE 10. Tube thickness contours for different r values.

CONCLUSION The thin-walled tube bending process was simulated with the aim of investigating the effect of tube anisotropy on ovalization, thickness and wrinkling of tube which is exposed to the most severe conditions. An anisotropic tube material with Hill quadratic yield criterion was employed and normal anisotropy was considered in this study. Based on this investigation, the following conclusions can be drawn: 1- The larger normal anisotropy index (r) leads to the higher the ovalization ratio. 2- The maximum value of tube wall thickness is increased by increasing of the normal anisotropy index (r), for r<1. 3- The wrinkling of tube is decreased by increasing the normal anisotropy index (r).

REFERENCES 1. H. Li and H. Yang and M. Zhan and R. J. Gu, Transactions of Nonferrous Metals Society of China 16, 613–623 (2006). 2. H. Li and H. Yang and M. Zhan and R. J. Gu, Journal of Materials Processing Technology 177, 192–196 (2006). 3. H. Li and H. Yang and M. Zhan and Zh. Sun and R. Gu, International Journal of Machine Tools & Manufacture 47, 1164–1175 (2007). 4. H. Li and H. Yang and M. Zhan and R. J. Gu, Journal of Materials Processing Technology, 502-507 (2007). 5. A. Kumar, International Journal of Modelling and Simulation 28, (2008) 6. R. J. Gu and H. Yang and M. Zhan and H. Li and H. W. Li, Computational Materials Science 42, 537–549 (2008). 7. H. Yang and J. Yan and M. Zhan and H. Li and Y. Kou, Computational Materials Science 45, 1052–1067 (2009). 8. H. Li and H. Yang and M. Zhan, Modelling and Simulation in Materials Science and Engineering 17 , (2009). 9. H. Li and H. Yang and J. Yan and M. Zhan, Computational Materials Science 45, 921–934 (2009). 10. H. Li and H. Yang and M. Zhan and Y. L. Kou, Journal of Materials Processing Technology 210, 143–158 (2010). 11. H. Z. Orban, “Analysis of deformation during structural tube hydroforming”, Ph.D. Thesis, Michigan University, 2006. 12. L. Duchene, “FEM study of metal sheets with a texture based”, Ph.D. Thesis, Liege University, 2003.

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Plastic Instability in Complex Strain Paths Predicted by Advanced Constitutive Equations Marilena.C. Butuca, Frédéric Barlata,b, José J. Gracioa and Gabriela Vinczea a

Centre for Mechanical Technology and Automation, University of Aveiro, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal b Graduate Institute of Ferrous Technology, Pohang University of Science and Technology, San 31 Hyoja-dong, Nam-gu, Pohang, Gyeongbuk 790-784, Republic of Korea Abstract. The present paper aims at predicting plastic instabilities under complex loading histories using an advanced sheet metal forming limit model. The onset of localized necking is computed using the Marciniak-Kuczinsky (MK) analysis [1] with a physically-based hardening model and the phenomenological anisotropic yield criterion Yld2000-2d [2]. The hardening model accounts for anisotropic work-hardening induced by the microstructural evolution at large strains, which was proposed by Teodosiu and Hu [3]. Simulations are carried out for linear and complex strain paths. Experimentally, two deep-drawing quality sheet metals are selected: a bake-hardening steel (BH) and a DC06 steel sheet. The validity of the model is assessed by comparing the predicted and experimental forming limits. The remarkable accuracy of the developed software to predict the forming limits under linear and non-linear strain path is obviously due to the performance of the advanced constitutive equations to describe with great detail the material behavior. The effect of strain-induced anisotropy on formability evolution under strain path changes, as predicted by the microstructural hardening model, is particularly well captured by the model. Keywords: Forming limits; Constitutive equations; Complex loading; Anisotropic; Hardening; Kinematic PACS: 07.05.Tp, 75.47.Np, 62.20.fq, 81.20.Hy, 81.40.Cd, 81.40.Ef, 83.10.Bb, 81.40.Np

INTRODUCTION The forming limit diagram (FLD) is a constructive concept for characterizing the formability of sheet metal. It was proved to be an essential tool for material selection, design and try out of the tools for deep drawing operations. The maximum admissible limiting strains strongly depend on the deformation mode, loading history and material behavior. To date, the MK method has become one of the most important tools in predicting the sheet metal formability. The selected constitutive equations keep a crucial significance on the prediction of plastic flow localization. For larger strains and for abrupt strain path changes, the evolution of the strain hardening is an issue and is the subject of an intensive recent research. Sheet metal forming processes often impose very intense forming sequences, leading to severe strain-path changes, which drastically influence the forming limits. Therefore, it was introduced the stress-based forming limit concept (FLSD), apparently independent on the strain path changes. Butuc et al.[4] showed that the independence of stress-based forming limits on strain path changes is basically originated on the plastically calculation of the stress state by assuming an isotropic hardening model, no depended on strain path. In addition, Yoshida et al. [5] showed that the work-hardening behavior after prestrain plays an important role in the path-dependence of the FLSD. The aim of the present work is a more relevant study on the prediction of plastic instabilities under complex loading histories using an advanced sheet metal forming limit model that take into account the strain path induced anisotropy. A particular attention is paid to the origin of the evolution of formability under strain-path changes involving biaxial prestrain followed by uniaxial tension and the reverse sequence.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 194-201 (2011); doi: 10.1063/1.3623611 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

194

THEORETICAL COMPUTATION OF FORMING LIMIT DIAGRAMS The simulation of plastic instability is carried out by using a MK-type analysis. This section summarizes the main steps of the theoretical computation. The model is based on the growth by plastic deformation within a thin sheet of an initial defect in the form of a groove-like, narrow band of diminished thickness. A plane stress condition is assumed throughout. The initial inclination angle of the band with respect to the minor principal axis of the stress tensor is denoted by ψ0. The initial value of the geometrical defect is characterized by the ratio e b0 / e 0a where e 0a and

e b0 are the initial thickness in the homogeneous region and in the groove respectively. The x, y, z-axes correspond to rolling, transverse and normal directions of the sheet, whereas 1 and 2 represent the principal stress and strain directions in the homogeneous region. The set of axes bound to the groove is represented by n, t, z, where t denotes the longitudinal axis. It is supposed that the material has a rigid-plastic, rate-independent behaviour. Several isotropic and anisotropic hardening models are considered to describe the hardening behaviour of material. The initial shape of the yield locus is given by the Yld2000-2d plane stress yield function [2]. Small increments of equivalent strain are imposed in the homogeneous region, the corresponding stress and strain states being computed according to the selected constitutive equations. In order to define the strain and stress states within the band, the Newton-Raphson method is applied for solving a system of two polynomial equations resulting from the yield criterion and the requirement of deformation compatibility in the longitudinal direction of the necking band. The Jacobian matrix required in the Newton-Raphson computation is based on an analytical differentiation of the constitutive equations of the model. When the effective strain increment in the band ( d b ) becomes 10 times larger than the one in the homogeneous zone ( d a ), it is considered that a plastic flow localization occurs in the band, whereas the corresponding accumulated principal strains in the homogeneous zone define the forming strain limits. For each considered strain path, the analysis is repeated for different values of ψ0 (between 0º and 90º) and the limit a point on the FLD is obtained through the minimization of the principal strain 11 in the homogeneous zone versus ψ0. The simulation of the complex strain paths involves a prestrain of the homogeneous zone followed by a more or less sharp change in strain path. Specifically, the sheet metal is unloaded to a zero-stress state after the prestrain and subsequently reloaded following the second strain path. To account for complex strain paths at large plastic strains, the prestrain will be accounted for when considering the evolution of the internal variables of the microstructural hardening model. More precisely, the initial values of the internal variables at the beginning of the second loading are taken equal to their values at the end of the first loading. In addition, it is also considered that the initial band orientation at the beginning of the second strain path is the one achieved at the end of the first deformation stage.

REVIEW OF THE SELECTED CONSTITUTIVE EQUATIONS Microstructural Hardening Model The microstructural hardening model of Teodosiu and Hu describes the hardening of the material by four internal state variables, denoted by {R, X , S, P}. The evolution laws of the internal state variables are written in a workhardening/recovery format, reflecting the dynamic processes of the production/annihilation of dislocations and of the formation/dissolution of dislocation structures. The evolution of R is given by

R  CR  Rsat  R   , R(0)=0,

(1)

with CR-the rate of the isotropic hardening, Rsat-the saturation value of R. The evolution of the tensor P, which describes the polarity of planar dislocation structures, is given by

P  CP  A  P   , P(0)=0,

(2)

where A is the current direction of the plastic strain rate tensor. The evolution of the back stress X is governed by the equation

X

X  CX  sat  ' X   X  X(0)=0,

e 

(3)

195

with CX-the saturation rate of X, Xsat-the saturation value of the norm of X,

 ' -the deviator stress tensor and  e -

the equivalent stress corresponding to the effective stress tensor σ'  X . Xsat is assumed as:

X sat  X 0  1  m  SD2  q SL , 2

where X0 is the initial value of

(4)

X sat , while q and m are material parameters.

The tensor variable S describing the directional strength of planar dislocation structures is decomposed as

S  SD A  A  SL ,

(5)

with SD = A : S : A and SL = |SL|. SD is related to the currently active slip systems and SL to the latent part of the dislocation structures. The evolution of SD is governed by the equation:

SD  CSD h hP  Ssat  SD   hX SD   ,

(6)

where CSD and Ssat are material parameters. hX and hP are 2 functions expressed as:

1 X:A hX  1   e . 2 X sat  σ ' X  : A 

(7)

 CP SD  P:A , if P:A  0  1  C C S  SD P sat hP   CP SD

1  P:AnP  1  , otherwise.    C C S SD P sat 



(8)

The evolution of S L results from the interaction between microbands generated by newly activated slip planes after a strain-path change and the preformed dislocation walls. nL

SL

SL  CSL SL  , S 

Ssat 

(9)

where CSL characterizes the rate of disorganization of the preformed dislocation structures. The yield stress is given by

 Y  Y0  R  mS ,

(10)

where Y0 is the initial yield stress, m is a material parameter and S is the norm of S, i.e. S=

S = | SL |2  SD2 .

(11)

The model involves 13 material parameters, namely Y0, X0, Ssat, Rsat, CP, CSL, CSD, CX, CR, nP, nL, m and q

The Yld2000-2d Yield Function The Yld2000-2d plane stress yield function introduces the plastic anisotropy by using two linear transformations on the Cauchy stress tensor (Barlat et al., 2003). It is expressed in terms of the deviatoric stress components (Barlat et al., 2007) as

     S       S    2 Ya ,

where



and

(12)

  are two isotropic functions defined by

   S    S '1  S '2 , a

(13)

   S    2S2"  S1"  2S1"  S2" , a

a

(14)

while S  and S  are linear transformations of the effective stress tensor s, which is defined as difference between the deviatoric part σ  of the Cauchy stress and the back stress X, i.e. s  σ  X . (15)

196

Hence (16) S   C  s, S   C  s, where C  and C  represent the linear transformations of the effective stress tensor s and contain the material anisotropy coefficients, which are computed using as input the experimental values of the stresses and anisotropy factors in tension along three directions and the balanced biaxial flow stress as well as the balanced biaxial anisotropy coefficient.

RESULTS AND DISCUSSION Materials Characterization Two metal sheets of deep-drawing quality were considered in our study, namely an aluminum-killed and titanium-treated mild steel (DC06 steel) and a bake-hardened steel (BH steel). Table 1 presents the corresponding values of the yield stress and anisotropy factor r for three orientations of the uniaxial tensile direction with respect to the rolling direction, ultimate tensile strengths, maximum uniform elongations, as well as the balanced biaxial yield stress b and the balanced biaxial anisotropy coefficient rb.. TABLE 1. The material plastic properties Material Orientation DC06 steel

Y [MPa] 136 134 136 210/208 224/218 214/211

0º 45º 90º 0º 45º 90º

BH steel

uts [MPa] 374 369 365 410 417 409

Uniform elongation

Anisotropy factor r

b

rb

29% 27% 28% 23% 21% 22%

2.6 2.1 3.2 2.2 1.5 2.7

142.2

0.8

216.56

0.869

Table 2 contains the material parameters involved in the microstructural hardening model. TABLE 2. The material parameters of the microstructural model Parameters Y0 CR Rsat CX X0 CSD

CSL

Ssat

nL

np

q

m

CP

DC06 steel

121.1

31.9

90.0

446

15.9

4

1.86

231.1

0

27.9

1.5

0.445

5.5

BH steel

202.93

37.2

73.45

160.15

2.33

4.93

5.575

229.8

0.25

24.05

1.855

0.297

3.5

Table 3 shows the values of the eight parameters of the Yld00-2d yield function, as obtained by a numerical identification from the experimental data: r0, r45, r90, rb, 0, 45, 90, b. TABLE 3. The coefficients of the Yld00-2d yield function Material 1 2 3 a DC06 steel BH steel

6 6

1.055 1.072

1.09 1.039

1.00 1.027

4

5

6

7

8

0.9 0.916

0.937 0.943

1.056 1.0423

1.069 0.982

0.926 0.837

Strain and Stress Based Forming Limits: Computation and Discussion In order to achieve the best agreement with the experimental data, the initial value of the MK geometrical defect was individually selected for each material, namely 0.996 for DC06 steel and 0.994 for BH steel. The shape of yield surface is described by the YLD00-2d yield function and the strain hardening is defined by the microstructural hardening model. Figure 1 shows the experimental and theoretical forming limits for DC06 steel under proportional (acronym LSP) and non proportional loadings involving two strain path changes, maintaining constant the first and second stress ratio (acronym ) and varying the amount of prestrain under uniaxial tension (acronym UT), biaxial stretching (acronym BS) and plane strain (acronym PS), respectively. Figure 2 shows the experimental and

197

theoretical FLDs for BH steel sheet determined under linear and non-proportional loading using a sequence of two linear strain paths, keeping constant the preliminary strain/stress ratio and the amount of prestrain and varying the subsequent stress ratio. Specifically, it was considered prestrain at several values under UT, BS and PS followed by different strain paths (acronym X) between UT and BS. A good correlation between experimental data and the predicted results for linear and complex strain paths is observed for both of the selected materials. It may be reasonable assumed that this remarkable accuracy in the FLD prediction results from the relevance of the applied constitutive models. Concerning the complex strain paths, it is observed that UT prestrain along RD shifts the whole FLD to the left, raising the strain limits in biaxial tension region and consequently increasing the slope of right-hand side of the FLD. If the first stage of strain is under BS, the subsequent FLD shifts to the right and down with respect to the original FLD, while under PS prestrain the slopes of right–hand side and left–hand side of FLD increase. Moreover, the strong effect of the amount of prestrain in the predicted forming limits under complex strain paths is clearly showed in Figure 1. It is noticed a decrease of forming limits for small values of the prestrain followed by an increase of forming limits as the prestrains increase. Such effect of decrease of forming limits is much pronounced when the second strain path is between UT and PS and it becomes softer (for BS prestrains) and even disappears (for UT prestrains) as the strain ratio characterizing the second strain path increases in the range PS-BS. 0,8 PS

Exp.Data

 =0.0

LSP

0,7 UT  =0.0

0,6

BS

UT-BS

 =1.0

BS-UT MicYLD LSP

UT-alfa=0.2

0,5

2

UT-PS UT-alfa=0.6 UT-alfa=0.7

0,4

UT-alfa=0.8 UT-alfa=0.9 UT-alfa=1.0

0,3

UT-BS

BS-UT

0,2

BS-PS BS-alfa=0.6

BS-alfa=0.7

0,1

BS-alfa=0.8

f0=0.996

BS-alfa=0.9

0 -0,5

-0,4

-0,3

-0,2

-0,1

0

0,1

1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

FIGURE1. Forming Limit Diagram under linear and complex strain paths for DC06 steel. 0,6 PS

0,5

Exp. Data

BS

 =0.0

LSP

 =1.0

7%UT-X

UT  =0.0

14%UT-X

MicYLD UT-BS

0,4

BS-UT BS-PS

2

LSP

0,3

7%UT-X 14%UT-X

21%UT-X

0,2

5%BS-X 8%BS-X

10%BS-X 15%BS-X

0,1

5%PS-X

f0=0.994

10%PS-X

0 -0,3

-0,2

-0,1

0

0,1

0,2

0,3

0,4

0,5

0,6

1 FIGURE2. Forming Limit Diagram under linear and complex strain paths for BH steel.

198

Figure 3 shows the stress-based forming limits for DC06 steel under linear and complex strain paths. The effect of the amount of prestrain in the predicted stress-based forming limits is clearly observed. Figure 4 shows the stressbased forming limits for BH steel under linear and complex strain paths involving prestrain at different values under UT, BS and PS , respectively, following by different strain paths. All the results illustrate a well defined effect of the strain path change on the predicted stress-based forming limits. This dependence of the stress-based forming limits in the strain path change is due to the strain path induced anisotropy that is taken into account by the microstructural hardening model based on the dislocation structure. It is very interesting to observe that in general the stress-based forming limits under almost all selected strain path changes are higher than the FLSD under linear strain paths. Moreover excepting few cases for small UT prestrains values the stress-based forming limits increase with the increase of prestrain value for all assumed prestrains. 600

500

MicYLD LSP UT-alfa=0.2

400

UT-PS

2

UT-alfa=0.8

300

UT-alfa=1.0 BS-UT

 =1.0

200

BS-PS BS-alfa=0.6

BS-alfa=0.7

100

f0=0.996

BS-alfa=0.8 BS-alfa=0.9

0 0

100

200

300

400

500

600

700

1 FIGURE3. Forming Limit Stress Limits under linear and complex strain paths for DC06 steel. 600

500 MicYLD

400

LSP 7%UT-X

2

14%UT-X

300

21%UT-X 5%BS-X

200

8%BS-X 10%BS-X

=1.0 100

15%BS-X

f0=0.994

5%PS-X 10%PS-X

0 0

100

200

300

400

500

600

1 FIGURE4. Forming Limit Stress Diagram under linear and complex strain paths for BH steel.

199

Figures 1 and 2 show that in both of studied cases the lowest curve on the FLD is the one obtained for BS-UT and the uppermost curve is the one obtained for UT-BS. It has been found an increase of formability under UT-BS until 25% for DC06 steel and 13% for BH steel, as well as a loss of formability under BS-UT until 33,8% for DC06 steel and 23,3% for BH steel, although these two deformation histories have the same value of the parameter introduced by Schmitt et al. [7] to characterize the strain-path changes (namely 0.269). Actually, the earlier appearance of plastic instability under the sequence BS-UT may be attributed to the fact that the second path is the uniaxial tension, which is more sensitive to strain localization. For UT-BS, the balanced biaxial stress state leads to a very stable strain path. Therefore, the occurrence of plastic flow localization is controlled primarily by the occurrence of a geometrical instability (necking in tension). This geometrical instability is captured in the MK model through its necking criterion. Figure 5 shows the evolution of the MK necking criterion for different strain paths for DC06 steel. As expected, the MK necking criterion is firstly attained for BS-UT and lastly for UT-BS, fact which explains the forming limits results. Figure 5 also shows that the choice of the MK necking criterion is practically irrelevant for the computed FLDs, as the strain-rate in the groove increases very rapidly after this condition is attained, because of the low current hardening rate. 25

BS-UT

20

BS

UT

PS

UT-BS

BS

15

d

UT-BS

PS

BS-UT

UT

b

MK Necking criterion

d a

10

5

0 0

0,1

0,2

0,3

0,4

0,5



0,6

0,7

0,8

0,9

1

a

FIGURE5. Evolution of the MK necking criterion for different strain paths for DC06 steel 1 PS

 =0.0

0,9

LSP f0=0.992

UT

BS

 =0.0

0,8

f0=0.996

 =1.0

f0=0.998

0,7

f0=0.999 UT-BS

0,6

2

f0=0.992

0,5

f0=0.996 f0=0.998

0,4

f0=0.999

0,3

BS-UT

f0=0.992

0,2 f0=0.996

f0=0.998

0,1

f0=0.999

0 -0,5

-0,4

-0,3

-0,2

-0,1

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1 FIGURE6. The Influence of the initial value of MK geometrical defect in FLD under linear and complex strain paths for DC06 steel

200

Figure 6 shows the significant influence of the initial value of MK geometrical defect in the predicted FLD under linear and complex strain paths for DC06 steel. A slight increase of initial value of the MK geometrical defect allows to a considerable increase of the predicted forming limits under UT-BS and in the right–hand side of the FLD under LSP. Such increase of the forming limits is much smaller in the left–hand side of the FLD under LSP and under BS-UT.

CONCLUSIONS An advanced model for FLD prediction based on MK theory has been used to predict the strain and stress-based forming limits for DC06 steel sheet and BH steel sheet under linear and complex strain paths. A very good agreement between experiments and theoretical results for the selected materials has been found. The great accuracy on the prediction of the forming limits is due to the ability of the microstructural hardening model of Teodosiu and Hu, which accounts for the anisotropic work-hardening induced by the microstructural evolution at large strains and of the Yld2000-2d yield function to describe in great detail the material behavior. The plastic instability prediction was performed for several complex strain paths. The strain path change clearly influences the plastic flow localization. It is notable the significant decrease/increase of formability that occurs under strain histories involving a biaxial stretching prestrain followed by uniaxial tension and a uniaxial tensile prestrain followed by biaxial stretching even if these two sequences have the same value of the Schmitt factor to characterize the strain path change. The occurrence of plastic flow localization is controlled primarily by the occurrence of a geometrical instability (necking in tension). This geometrical instability is captured in the MK model through its necking criterion. Theoretically, the occurrence of plastic flow localization is mainly connected with the accomplishment of the M-K necking criterion, which is strictly dependent on the assumed strain path. Taking into account the strain path induced anisotropy through the microstructural hardening model it was proved that the stress-based forming limits are also considerable affected by the strain path changes. In addition it is noticed that the stress-based forming limit concept is not efficient to evaluate the sheet metal formability, since under specific complex loadings (i.e BSUT) a significant increase of the stress-based forming limits while a considerable decrease of the strain-based forming limits was observed.

ACKNOWLEDGMENTS Especial thanks are expressed to Prof. Cristian Teodosiu for all support concerning the microstructural hardening model. The authors thank Laboratory of the Mechanical and Thermodynamic Properties of Materials (LPMTM, U. Paris Nord, France) for providing the material parameters of microstructural model.

REFERENCES [1] Marciniak, Z., Kuczynski, K., Pokora, T., “Influence of The Plastic Properties of A Material On The Forming Limit Diagram For Sheet Metal In Tension”, Int. J. Mech. Sci., 15, 789-805 (1973). [2] Barlat, F., Brem, J. C., Yoon, J. W., Chung, K., Dick, R. E., Lege, D. J., Pourboghrat, F., Choi, S. -H. Chu E., “Plane stress yield function for aluminum alloy sheets-part 1: theory” Int. J. Plasticity, 19, 1297-1319 (2003). [3] Teodosiu C., Hu Z., “Microstructure in the continuum modelling of plastic anisotropy” In: Roskilde. Danemark (Eds.). Proc. Of the 19th Riso International Symposium on Modelling of Structures and Mechanics of Materials from Microscale to Product , 149-168 (1998). [4] M.C. Butuc, J.J. Gracio and A. Barata da Rocha, “An experimental and theoretical analysis on the application of stressbased forming limit criterion”, Int. J. Mech. Sci., 48 (4), 414-429 (2006) [5] Yoshida, Kengo, Suzuki, Noriyuki “Forming limit stresses predicted by phenomenological plasticity theories with anisotropic work-hardening behavior” Int. J. Plasticity 24, 118-139 (2008) [6] M.C. Butuc , F. Barlat, J.J. Grácio, A. Barata da Rocha, “A new model for FLD prediction based on advanced constitutive equations”, Int. J. of Mat. Forming, DOI 10.1007/s12289-009-0667-6 (2009) [7] Schmitt, J.-H., Shen E.L., Raphanel J.L., 1994, A parameter for measuring the magnitude of a change of strain path: Validation and comparison with experiments on low carbon steel. Int. J. Plasticity 10, 535.

201

Non-proportional/Non-monotonous Deformation Modeling of an Ultra High Strength Automotive Steel Sheet Rahul K. Verma*, Yuki Ogihara**, Toshihiko Kuwabara*** and Kwansoo Chung**** *Research and Development Division, Tata Steel Limited, Jamshedpur, India **Graduate School of Mechanical Systems Engineering, Tokyo University of Agriculture and Technology ***Institute of Symbiotic Science and Technology, Division of Advanced Mechanical Systems Engineering, Tokyo University of Agriculture and Technology ****Department of Material Science and Engineering, Research Institute of Advanced Materials, Seoul National University

Abstract. In this work, as non-proportional/non-monotonous deformation experiments, two-stage and tensioncompression-tension uniaxial tests were performed, respectively, for a cold rolled ultra high strength dual phase steel sheet: DP780. Deformation behaviors under such deformation paths were found different than those of the ultra low carbon single phase steels observed by Verma et al. (Int. J. Plast. 2011, 82-101). To model the newly observed deformation behaviors, the combined type constitutive law previously proposed by Verma et al. (Int. J. Plast. 2011, 82101) was successfully applied here. Permanent softening observed during reverse loading was properly characterized into the isotropic and kinematic hardening parts of the hardening law using tension-compression-tension test data. The cross effect observed in two-stage tests was also effectively incorporated into the constitutive law. Keywords: Non-proportional deformation, reverse loading, dual phase steel, permanent softening, cross effect.

INTRODUCTION The automotive application of ultra high strength steels is driven by the need to reduce the weight and to improve the crash performance of automobile body structures. However, there are drawbacks to overcome such as lower formability and higher amount of springback for their wider automotive application. A common approach to resolve such drawbacks is to numerically predict them in advance and take corrective measures during the die design stage. However, numerical prediction has not been of acceptable accuracy, especially for non-proportional/nonmonotonous deformation conditions, attributed by the unavailability of proper constitutive models and sufficient experimental data under such deformation modes [1-4]. Verma et al. [4] performed two-stage and tension-compression uniaxial tests for an ultra low carbon interstitial free high strength (IFHS) steel sheet to model non-proportional/non-monotonous deformations. They also proposed a combined type hardening law based on the one developed by Chung et al [3] and successfully incorporated permanent softening and the cross effect observed for the IFHS steel. It was found that, during the two stage tests, the stress-strain curve showed some initial cross effect but it was soon saturated to the original monotonic curve. The same set of experiments was performed for a dual phase (DP780) steel in this work. To model newly measured data, the hardening law previously proposed in [4] was also applied along with plastic strain dependent Young’s modulus.

CONSTITUTIVE MODELLING The tension-compression-tension (TCT) and two-stage uniaxial tests performed for the DP780 steel sheet showed significant permanent softening and cross effect, which were modeled utilizing the combined type hardening law (refer to [4] for details) as briefly described here. The yield function for the combined isotropic-kinematic hardening law can be represented by

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 202-209 (2011); doi: 10.1063/1.3623612 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

202

m f (   )  V iso (H )

0

(1)

where f is an m-th order homogeneous yield function, V is the Cauchy stress, D is the back stress and  iso is the effective stress to represent the size of the yield surface. Here,  iso is obtained from the definition of the initial effective stress by replacing V with V  D , while its conjugate effective plastic strain increment, d  , is obtained from the following plastic work equivalence principle as proposed in [3]:

    ˜ d p V iso d H

(2)

where d H p is the plastic strain increment. The nonlinear kinematic hardening model was proposed in [5, 6] by adding an additional term to the linear kinematic hardening model proposed in [7], to describe the transient behavior, which was further generalized in [3] to account for anisotropy. As has been discussed elsewhere [3], in the modified Chaboche model, the back stress increment is assumed to be composed of two terms, d d1  d  ; i.e., d

§ dD1 · (   ) § dD 2 · dH ¸  ¨ ¨ dH dH ¸ V © ¹ iso © dH ¹

(3)

where d 1 is an effective back stress increment obtained from the definition of the initial effective stress by replacing V with d1 , while d2 is a non-effective incremental quantity (refer to [8] for its physical implications). If, for simplicity, the simple tension in the rolling direction of the sheet is considered as a reference stress state, Eqs. (1) and (3) along with the plastic work equivalence principle lead to dV x

dD1 B dD 2D x  dV iso

(4)

where  x and x are respectively the components of V and D in the rolling direction. In Eq. (4), the negative sign of the second term is for loading and the positive sign is for unloading (or reverse loading). On manipulating Eq. (4) and considering d H d H xp as a reference quantity, the following relationship is obtained from Eq. (4): d D1 d H xp dD 2 Dx d H xp

d D1 ( { h1 ) dH

º dV § dV · 1 ª§ d V x · «¨ p ¸ »  iso  ¨ px ¸ 2 «© d H x ¹reverse loading © d H x ¹loading » d H xp ¬ ¼

dD 2 ( { h2 )D x dH

º § dV · 1 ª§ d V x · «¨ p ¸ »  ¨ px ¸ 2 «© d H x ¹reverse loading © d H x ¹loading » ¬ ¼

(5)

where h1 and h2 are the slopes of back stress evolution to be determined from the slopes of experimental hardening curves measured by uniaxial TC tests (details are referred to [9]). The simple tension / compression tests showed that the DP780 steel sheet has asymmetric properties as well as anisotropic properties. It was found that the flow stress during compression in the transverse direction was about 3% constantly higher than the corresponding values during the tension tests. In order to account for the asymmetric and anisotropic properties, the following yield function proposed in [4] by modifying Hill’s 1948 yield function [10] was utilized:

^

a V 2x  AV x V y  BV 2y  CW 2xy

`

1/ 2

 k1V x  k 2 V y V 0

(6)

There are six coefficients which may be evaluated using two tensile yield stresses along the rolling and the transverse directions ( V Tx and V Ty , respectively), two compressive yield stresses along the rolling and the

203

transverse directions ( V xC and V Cy , respectively), tensile yield stress at 45 degrees to the rolling direction ( V 45 ) and the balanced bi-axial yield stress ( V b ). The values used for the different yield stresses have been shown later.

Incorporation of Permanent Softening and Cross Effect For incorporating permanent softening, the method discussed in [4] was used as briefly summarized here. In the combined isotropic-kinematic hardening law, total hardening is decomposed into isotropic and kinematic parts; i.e.,

V

V iso  D

(7)

where  iso is the isotropic part signifying the size change of the yield surface and  is the kinematic part representing the translation of the yield surface. To characterize the above decomposition, particularly during initial hardening, tension-compression tests were performed with various tensile pre-strains and the translation of the center position of the yield surface (or  ) and the isotropic hardening data was calculated using Eq. (7) [9]. It was proposed in [4] that, for hardening decomposition (and also permanent softening) for reverse loading, the same method, as used during forward loading, can be used to characterize the isotropic and kinematic hardening parts using tension-compression-tension tests with various pre-strains during compression. In the present case, a simplified method was adopted (as was done in [4]) to incorporate the size change of the yield surface during stress reversals by modifying the flow stress as,

V iso Current loading V iso previous loading  G

(8)

It was found that the expression for G for the current DP780 steel grade could be expressed by

\

 B F F pre

E  A 1.0  e

^

(9)

where A and B are constants and  pre is the strain before stress reversal whereas F  F pre is the strain during the current loading. As for the kinematic hardening part without permanent softening, the following expression was used:

h1,2

a  b e c H

(10)

where a, b and c are constants. It was proposed in [4], in order to account for permanent softening attributed by kinematic hardening, the hardening parameter h1 of the back stress evolution law can be modified by introducing a softening factor, E , as done in [11]; i.e., current loading

h1

E (H pre ) ˜ h1

previous loading

(0  E (H pre ) d 1)

(11)

where E is a function of the effective pre-strain with E (0) 1.0 in general. In the present case, it was found that  rH fits. As to the cross-effect is concerned, it was incorporated by modifying the isotropic hardening E p  qe part whenever there was loading direction change by 90 degrees. pre

EXPERIMENTAL METHODS The material used for this study was a cold rolled and annealed DP780 steel sheet with the thickness of 1.4 mm. Work hardening characteristics and r -values obtained from uniaxial tensile tests along 0q, 45q and 90q directions from the rolling direction of the material are listed in Table 1. Figure 1 shows the stress-strain diagram for rolling,

204

diagonal and transverse directions. It can be seen that though the initial yield strength in the transverse direction is similar with those in rolling and diagonal directions, further hardening in the transverse direction is more rapid. TABLE 1. Mechanical properties of test material (DP780) K*/ MPa V 0.2 / MPa Rolling 751 1415 Diagonal 750 1444 Transverse 730 1424 *Approximated using V K (H 0  H p ) n at H p 0.001㹼0.093 **Measured at uniaxial plastic strain H p 0.093

n*

F0 *

r -value**

0.106 0.109 0.106

0.00023 0.00023 0.00024

0.85 0.76 1.07

FIGURE 1. Stress-strain diagram for uniaxial tension tests in three directions

Tension – Compression – Tension Tests Other than the uniaxial tensile tests, set of in-plane tension-compression-tension (TCT) tests were carried out for evaluating the deformation behaviour of the sheet under cyclic loading. Tests were performed on specimens prepared along the rolling direction, using the testing apparatus as shown in Figure 2. This testing apparatus was originally designed and developed in [12]. The comb-shaped dies were developed for application of continuous in-plane stress reversal to a sheet specimen. The lower die 1 is fixed to the lower plate of the die set and the lower die 2 is on a slide rail that enables the die to move smoothly in a horizontal direction. A sheet specimen is set on the lower dies 1 and 2 and both ends of the specimen are attached to the dies by chucking plates. The upper dies are placed on the specimen so that the four positioning pins fixed to the lower dies mesh with the holes of the upper dies. Accordingly, the movement of the upper dies is synchronized with that of the lower dies. The lower die 2 is actuated in the horizontal direction by the servo-controlled hydraulic cylinder A, so that continuous in-plane stress reversals are applied to the specimen. The tooth width is 2.7 mm, the tooth length is 55.5 mm, and the gap between the teeth is 3.0 mm. Blank holding force (BHF) was applied to a specimen through the upper dies 1 and 2 in order to prevent out-of-plane buckling. The BHF per unit area of the specimen surface was approximately 1% of the yield stress of the sample; therefore, the effect of the BHF on the yielding behavior of the sample can be neglected. In order to prevent the specimen from galling the dies, it

was lubricated on both sides with Vaseline and Teflon sheets. The tests were carried out under a displacement controlled mode and two different speeds were used: in the elastic regions, the test speed was 0.5 mm/min whereas, in the plastic region, it was 1.5 mm/min. Strain was measured using a strain gauge (YFLA-2, Tokyo Sokki Kenkyujo Co., Ltd).

205

Hydraulic cylinder B

Blank holding platen Upper die 1

Upper die 2 Load cell

Specimen

Lower plate Lower die 1

(a)

Slide rail

Hydraulic cylinder A

Lower die 2

(b)

FIGURE 2. Experimental apparatus for applying in-plane stress reversals to a sheet specimen: (a) set-up of the dies and (b) an overview of the testing apparatus.

Two-Stage Uniaxial Tests In the two-stage uniaxial test, the specimen was aligned along the rolling direction in the first stage and then it was pre-strained in tension by 0.02 and 0.05 or in compression by -0.02. From the pre-stained large specimen, a small specimen along the transverse direction was cut out to measure hardening in tension and compression, respectively. Specimen dimensions are shown in Figure 3. While performing the compression test either during the first or the second stage, the same die set-up was used as described in [12], though sample dimensions were different as given in Fig. 3.

Big specimen for st 1 loading

Small sample for nd 2 loading in tension Small sample for nd 2 loading in compression

Rolling direction

FIGURE 3. Schematic diagram showing the dimensions and directions of samples used in the first and second stages. (for tension, L = 290 and W = 65, for compression, L = 190 and W = 65) (unit: mm)

VALIDATION The finite element analysis using the above model was performed using the commercial software ABAQUS Standard along with its UMAT subroutine [13]. S4R type shell elements with 9 through thickness integration points were used for simulations. Tables 2 and 3 show the values of hardening coefficients used in simulations. The yield stresses and the coefficients of the modified asymmetric yield function are given in Tables 4 and 5. As for the cross effect, the value was found to be constant, which was 50 MPa for the second stage tension and 70 MPa for the second stage compression. It is to be noted that in the case of the IFHS steel sheet used in [4], the stress-strain curve in the transverse direction during second loading saturated to the original monotonic curve signifying that even an isotropic hardening model should be sufficient to model the two stage uniaxial deformation. Unlike the IFHS steel, DP780 showed a constant gap between the original monotonic curve and the second stage curve so that an isotropic

206

model was not sufficient and the cross effect needed to be properly accounted for. This was achieved by adding a constant value to the isotropic hardening component, which was 55 MPa when the pre-strain was in tension and 70 MPa when the pre-strain was in compression.

TABLE 2. Hardening coefficients for combined hardening model.

h1 a 54704

a be

c H

B - 29505

c 26.79

\

E  Q 1 e Q 95.0

Tension

h2

R F  F pre

^

a 151.89

a  b ec H b -81.36

T iso  X (F0 Y ) Z c 32.87

X 1392.7

Y 0.09402

TABLE 3. Coefficients for softening parameters.

p  qe  r H

E

R 22.6

p 0.58

q 0.4

TABLE 4. Yield stresses along different directions Compression Tension Compression Equi-biaxial

r 5.37

Tension

(Rolling)

(Rolling)

(Transverse)

(Transverse)

tension

(Diagonal)

527

558

527

527

580

527

a 0.9717

Z 0.53083

TABLE 5. Coefficients for the asymmetric yield function. A B C k1 1.23834 1.059101 3.29659 0.028302

k2 0

Tension – Compression – Tension Tests Figure 4 shows comparison between the simulation and experimental values with different amounts of pre-strains. It can be seen that there is good agreement between the two. Whenever loading changes from tension to compression or visa-versa for each integration point, the permanent softening was activated in the simulation.

FIGURE 4. Comparison between experiments and simulations for cyclic tests with different pre-strains.

207

Two-Stage Uniaxial Tests Figure 5(a) to 4(c) show comparison between the simulation and experiment results. Figure 5(a) shows the case when the pre-strain was -0.02, followed by the second stage tension and compression. Similarly, Figures 5(b) and 5(c) show cases for pre-strain values of 0.02 and 0.05, respectively. It can be seen that, for all cases, agreement is reasonably good. However, as deformation proceeds in the transverse direction, discrepancy between the simulation and the experiment develops. This could be due to the fact that the initial yield in the transverse direction is the same with that of rolling direction, but further hardening in the rolling and transverse directions are different. Therefore, more sophisticated yield functions in which anisotropic coefficients change with deformation are expected to give even better results. This comparative study will be a part of the future work.

(a)



(b)

(c) FIGURE 5. Comparison between experiments and simulations for the second stage tests. (a) Pre-strain -0.02 (b) Pre-strain 0.02 (c) Pre strain 0.05

CONCLUSIONS In this work, two stage uniaxial and tension-compression-tension tests were performed for a DP780 steel sheet, as previously done for the IFHS steel in [4]. The performance of the combined type hardening model proposed in [4] was also evaluated for those tests. It was found that, unlike the IFHS steel, the cross effect was permanent for the DP780 sheet in nature and that the incorporation of permanent softening was very important for accurate simulation

208

of cyclic behavior. Furthermore, the model proposed in [4] was effective even for the DP780 steel sheet studied in the present work, especially considering that Young’s modulus was plastic strain dependent.

ACKNOWLEDGMENTS This work has been performed as a part of the joint project between Tata Steel, India, Tokyo University of Agriculture and Technology, Japan and Seoul National University, South Korea. The authors acknowledge the financial support from Tata Steel, India.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

R. Hill, J. Mech. Phys. Solids. 48, 1683 – 1699 (2000). T. Kuwabara, Int. J. Plast. 23, 385-419 (2007). K. Chung, M. G. Lee, D. Kim, C. Kim, M. L. Wenner and F. Barlat, Int. J. Plast. 21, 861-882 (2005a). R. K. Verma, T. Kuwabara, K. Chung and A. Haldar, Int. J. Plasticity 27, 82-101 (2011). J. L.Chaboche, Int. J. Plast. 2, 149-188 (1986). J. L. Chaboche, Int. J. Plast. 7, 661-678 (1991). H. Ziegler, Quart. Appl. Math., 17, 55 (1959). J. Kim, W. Lee, K.H. Chung, D. Kim, C. Kim, K. Okamoto, R.H. Wagoner and K. Chung, Met. Mater. Int. 17, 83-98 (2011). M. G. Lee, D. Kim, C. Kim, M. L. Wenner, R. H. Wagoner and K. Chung, Int. J. Plast. 21, 883-914 (2005). R. Hill, Proc. Royal Soc. London A 193, 281-297.(1948). K Ahn , D. Yoo, M. H. Seo, S. Park and K. Chung, Metals and Materials Int. 15, 637-647 (2009) T. Kuwabara, Y. Kumano, J. Ziegelheim, I. Kurosaki, Int. J. Plast. 25, 1759-1776 (2009). ABAQUS, 2007. User’s manual (version 6.7), Hibbit, Karlsson & Sorensen Inc., USA.

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An evolutionary yield function based on Barlat 2000 yield function for the superconducting niobium sheet Payam Darbandi and Farhang Pourboghrat Mechanical Engineering Department, Michigan State University, East Lansing, MI 48824

Abstract.

Superconducting radio frequency (SRF) niobium cavities are widely used in high-energy physics to accelerate particle beams in particle accelerators. The performance of SRF cavities is affected by the microstructure and purity of the niobium sheet, surface quality, geometry, etc. Following optimum strain paths in the forming of these cavities can significantly control these parameters. To select these strain paths, however, information about the mechanical behavior, microstructure, and formability of the niobium sheet is required. In this study the Barlat 2000 yield function has been used as a yield function for high purity niobium. Results from this study showed that, due to intrinsic behavior, it is necessary to evolve the anisotropic coefficients of Barlat’s yield function in order to properly model the plastic behavior of the niobium sheet. The accuracy of the newly developed evolutionary yield function was verified by applying it to the modeling of the hydrostatic bulging of the niobium sheet. Also, in a separate attempt crystal plasticity finite element method was use to model the behavior of the polycrystalline niobium sheet with a particular initial texture.

Keywords: crystal plasticity, Barlat 2000, niobium, evolutionary yield function

INTRODUCTION The superconducting accelerator niobium cavities are traditionally fabricated by deep drawing of subcomponents, which are later joined by electron beam welding. High purity niobium sheet is the favorite material for the fabrication of these cavities. The knowledge of the mechanical properties of the niobium sheet plays a significant role in the fabrication and treatment of these cavities. Most laboratories working on superconducting accelerator technology have strict requirements for the niobium material that they use, and have developed a quality assurance (QA) strategy for this purpose. Nevertheless, the quality control tests that they perform are limited in scope. But recently, due to the need for more powerful accelerators, new fabrication techniques for making seamless cavities have been considered. The idea of manufacturing seamless cavities would have the following benefits:  Elimination of electron beam welds  Improvement of mechanical and physical properties of the cavities  Reduction of manufacturing costs  Facilitating mass production Several attempts has been made by Kneisel and Palmieri [17], and Singer et al [29] to fabricate seamless cavities by using techniques such as tube hydroforming, explosive forming, or spinning of deep drawn or spun tubes and more recently combination of swaging and hydroforming of high purity niobium tubes. It was shown by Pagani et al. [23]that the most important issue to be considered in the manufacturing of these cavities is the optimum design of its geometry; such as the cell length, iris radius, sidewall inclination and position, cell radius, etc. Therefore, the forming processes must be carefully selected in order to produce the desired geometry after deformation. For this purpose, understanding the material behavior and developing accurate constitutive and mathematical models are important. Kneisel et al. [17] reported observing non-uniformity in the hydroformed highly pure niobium cavities, which they thought was related to the non-uniformity of the structure of the niobium tube. Several investigators, e.g. Nakamachi et al. [20], Asensio et al. [2]have shown that for good formability, the material must have the following properties: 

High rm value (mean normal anisotropy coefficients),

 

High n value (work hardening exponent), Low r value (planar anisotropy coefficient or earring parameter, which is responsible for cross section geometry and thickness variation),

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 210-217 (2011); doi: 10.1063/1.3623613 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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All the above parameters can be measured from simple tensile tests at different directions, compared with rolling direction. The ductility of the material also can be precisely determined using these tests. Asensio et al.[2] reported that texture has a strong influence on rm value and r in cold formed and annealed metallic sheets, however, n value is primarily influenced by the chemical composition and the presence of second-phase particles, and only secondarily by the grain size. Zamiri et al. [33] provided an overview of the mechanical properties and microstructure of superconducting niobium sheet. Anisotropy has a significant effect on the strain distribution in a metallic alloy sheet forming. The anisotropy of a metal in a sheet metal forming process is a combination of the initial anisotropy of the sheet before forming, which is related to the initial microstructure and texture of the sheet metal, and the deformation-induced anisotropy, which is due to the microstructure and texture evolution of the sheet metal during forming process. The initial anisotropy leads to symmetry with orthotropic properties in a sheet metal while the deformation-induced anisotropy can destroy this symmetry and make the modeling of the plastic anisotropy very complex in a forming process. Therefore, for practical purposes, it is assumed that the deformation-induced anisotropy is small and negligible. Several anisotropic yield functions for modeling the plastic deformation of metals have been proposed based on this assumption. For example, see the quadratic yield function proposed by Hill [13], and non-quadratic functions developed by Hill [14,15],Hershey[12], Hosford [16], Bassani[8], Gotoh [11], Logan and Hosford [19], Budianski[9], Barlat and Lian[3], and Barlat et al. [4-7], Zamiri and pourboghrat [34] discussed the forming properties of high purity niobium using an evolutionary yield function based on the Hill’s model. In this paper we will attempt to evaluate its forming properties by means of experimental and numerical studies. We used both crystal plasticity and Barlat2000 based on evolutionary model in order to define our numerical framework.

EVOLUTIONARY MODEL BASED ON BARLAT YLD2000-2D Barlat proposed a plane stress yield function that describes the anisotropic behavior of sheet metals. This model has several advantages such that it can define better anisotropic plasticity formulations that can guarantee the convexity, make the FE implication and application simpler and take  o ,  45 ,  90 , ro , r45 , r90 and  b into account for plane stress condition. This model uses isotropic yield functions and for anisotropy cases replace the principle stresses by principal transformed stresses obtained from several linear transformations. An isotropic yield function can be simply written as

        2 a

(1)

Where

   s1  s 2

a

and    2 s 2  s1  2s1  s 2

a

(2)

For anisotropic case, the linear transformation can be written as :

  C11  X xx      X yy   C 21  X xy      0   C11  X xx      X yy   C 21  X xy      0

C12  C 22 0

0   s xx    0   s yy     s xy  C 66

(3)

C12  C 22

0   s xx    0   s yy     s xy  C 66

(4)

0

Or using

211

We can write:

1  2   3  3 0   1 2  T  0  3 3    0 1  0  

(5)

X   C .s  C .T .  L . X   C .s  C .T .  L .

(6)

The anisotropic yield function is then given as:

   X 1  X 2 ,    2 X 2  X 1  2 X 1  X 2 a

a

a

  2 0 0  3  L11     L     1 0 0    12    3  1   L21    1  0  2     0  3      L22    7  2  L66    0 0 3   0 1  0

8 2  L11   2 2  L    1  4  4 4  12       4  4  4 1  L 21       2 8 2 2  L22  L66      0 0 0 0

(7)

(8)

0  3  0  4  0  5    0  6  9  8 

(9)

Although this yield function has been successfully applied to FCC aluminum alloys, its application to BCC metals such as high purity niobium requires some modifications. Since the anisotropy coefficients in Barlat’s equation are highly dependent on the r-values, and the r-values of the niobium sheet vary with plastic deformation and plastic work [34], we propose to use a modified version of Barlat’s yield function that will take into account the changes in the anisotropy coefficients with plastic work. Table 1 shows how anisotropy coefficients change with plastic work for the niobium sheet.

CRYSTAL PLASTICITY FINITE ELEMENT METHOD Using the optimization technique, it can be shown that during the plastic deformation of a single crystal the slip rate on any slip system could be expressed by:

sign(  )

  

 y

   exp[ (   1)] m y

   exp[ (   1 m    1] y N

212

(10)

where m and  are material parameters that control the shape of the single-crystal yield-surface, and which has been shown to have a direct correlation with the staking fault energy (SFE) of the material. For most materials m=1 and  is obtained using the SFE of the material,



is a measure of the rate of plastic work in a crystal , and 





and  y are the

resolved shear stress and critical resolved shear stress for slip system  respectively (Zamiri and Pourboghrat(2010)) The strain hardening model is expressed by the following hardening equation: N

y   h   

(11)

 1

Where 

 

is the plastic slip rate of the active slip system  , and

h

denote the components of the hardening matrix.



h is known as the self-hardening moduli, and h for (    ) is known as the latent hardening moduli. A common type of hardening matrix is then proposed in the following form:

h  h  [q  (1  q)  ]

(12) Here q is the so-called latent-hardening ratio, which is the ratio of the latent-hardening rate to the self-hardening rate of a slip system with values in the range of 1 < q < 1.4. The parameter q can be considered as 1 for coplanar slip systems 



is the Kronecker's delta function. h is an evolutionary function and 1.4 for non-coplanar slip systems and  denoting the self-hardening rate, which can be expressed as a function of either shear slips or resolved shear stress on slip systems.

h  is considered to evolve as:   y h  ho 1    s 

a

   y  (13) sign  1    s    Where h0 , a , and  s are slip system hardening parameters. h0 is the initial hardening rate,  s denotes the saturation value of slip resistance and a is the exponent that describes the shape of the function. For polycrystalline niobium the 

hardening values have been reported in Table 2. These values have been obtained using the curve fitting with the stressstrain tension curve (Figure 3). TABLE1.Barlat YLD2000-2d material model anisotropy coefficients for Nb as a function of equivalent work

Anisotropy coefficient

1 2 3 4 5 6 7 8

Logarithmic fit

0.0062 log( wp 0)  0.08506 0.0146 log(wp 0)  1.069 0.0262 log(wp 0)  0.9786

0.0092 log(wp 0)  0.9978 0.0058 log(wp 0)  1.0064 0.00378og ( wp 0)  0.8723 0.016 log(wp 0)  0.9209 0.0096 log(wp 0)  1.0729

213

TABLE2.Hardening parameters obtained from experimental observation and curve fitting. Parameter

C11

C12

C22

S

 (110)

 (112)

 (123)

value

245600

138710

29300

130

40

40

1000

ho

q

2500

1.4

RESULTS As shown above, the plastic behavior of niobium sheet is complicated. Its r-value is sensitive to the plastic strain and direction. Therefore, in developing a yield function for niobium sheet one needs to take into account the variations of the r-values. Furthermore, the yield function should have enough parameters to accurately predict the forming properties of the high purity niobium sheet. Figure 1shows the change in the shape of the Barlat’s yield function with plastic strain using evolving coefficients. The shape of the yield function is similar to a highly anisotropic material at different plastic strains. Another interesting point about the behavior of an evolutionary yield function is that the direction of the normal to the yield surface (as seen in Figure 1) at a particular strain also changes with plastic strain. To validate the proposed algorithm, the modified Barlat yield function was implemented as a user material model (UMAT) into ABAQUS finite element code. A biaxial tension test was carried out on the niobium sheet, by means of applying hydrostatic fluid pressure to bulge the sheet, and its results were compared with finite element simulations. Figure 2c shows one-half of the bulged niobium sheet with fracture occurring parallel to the rolling direction and away from the pole of the sheet. Two different forms of the yield function were used in the finite element simulation; one with constant anisotropic coefficients and the other with anisotropic coefficients evolving with plastic strain. Figures 2a and 2b show the contour of the thickness strain in the bulged sheet predicted with and without the anisotropy coefficients evolving with the plastic strain, respectively. As can be seen from Figs. 2a and 2b, the yield function with evolutionary anisotropic coefficients predicts, similar to the actual bulged sheet (Figure 2c), two regions with strain localization in the deformed part. On the other hand, the yield function with constant coefficients doesn’t show any strain localization. The material properties shown in Table 2 for the polycrystalline plasticity model were obtained by curve fitting its parameters to the unixail tensile test data. An ongoing effort is underway to simulate the bulge test using the crystal plasticity model and to compare its results with the phenomenological models and the experiment. In order to calculate the eight anisotropy coefficients of theBarlat 2000-2d model, we need the value of rb (the slope of the yield function in the balance-biaxial tension), which in this study has been obtained using the Barlat96 yield function. Another approach is to use the crystal plasticity finite element method to directly calculate the value of rb . 20%

1.5



2

21% 22%



23%

1

24% 25% 26% 27%

0.5

28% 29% 30%

0 -1.5

-1

-0.5

0

0.5

1



32% 1



1.5

33% 34% 35%

-0.5

37% 38% 39% 40%

-1

41% 42% 44% -1.5

45%

FIGURE 1-Tricomponent yield surface for Nb predicted with Yld2000-2d (Yld96 rb).

214

a- Thickness strain

b- Thickness strain Fracture

FIGURE 2-(a) Simulation using yield function with constant coefficients. (b) Simulation using evolutionary Barlat’s yield function. (c) Experiment (half of bulged specimen) at strain rate

215

FIGURE 3- Comparison of stress- strain curve obtained from experiment and Simulation using the crystal plasticity

ACKNOWLEDGEMENT The authors wish to thank the National Superconducting Cyclotron Lab (NSCL) at Michigan State University for the support of this project. The authors would also like to thank the National Science Foundation for their support of this project through the MRI-0723090 grant.

REFERENCES 1. ABAQUS manual (2001), Hibbit, Karlsson & Sorensen Inc, Providence, RI, Version 6.3 2. Asensio J., Romano G., Martnez V.J., Verdeja J.I., Pero-Sanz J.A. (2001) Optimization of hot rolled textures through cold rolling and annealing, Materials Characterization, 47, 119. 3. Barlat, F., Lian, J. (1989) Plastic behavior and stretchability of sheet metals Part I: yield function fororthotropic sheets under plane stress conditions. Int. J. Plasticity 5, 51. 4. Barlat, F., Lege, D.J., Brem, J.C. (1991) A six-component yield function for anisotropic metals. Int. J. Plasticity 7, 693. 5. Barlat, F., Chung, K., Richmond, O. (1993) Strain rate potential for metals and its application to mini-mum plastic work path calculations. Int. J. Plasticity 9, 1. 6. Barlat, F., Maeda, Y., Chung, K., Yanagawa, M., Brem, J.C., Hayashida, Y., Lege, D.J., Matsui, K.,Murtha, S.J., Hattori, S., Becker, R.C., Makosey, S. (1997) Yield function development for aluminumalloy sheet. J. Mech. Phys. Solids 45, 1727. 7. Barlat F., Brema J.C. , Yoonb J.W., Chung K., Dicke R.E., Legea D.J., Pourboghrat F., Choif S.H., Chuh E. (2004) Plane stress yield function for aluminum alloy sheets—part 1: theory, Int. J. Plasticity, 19, 1297. 8. Bassani, J.L. (1977) Yield characterization of metals with transversely isotropic plastic properties. Int. J. Mech. Sci. 19, 151. 9. Budianski, B. (1984) Anisotropic plasticity of plane-isotropic sheets. In: Dvork, G.J., Shield, R.T. (Eds.), Mechanics of Material Behavior. Elsevier, Amsterdam, The Netherlands. 10. Daniel D., Jonas J.J. (1990) Measurement and prediction of plastic anisotropic in deep drawing steels, Metallurgical Transactions A, 21A, 331. 11. Gotoh, M. (1977) A theory of plastic anisotropy based on a yield function of fourth order (plane stress state). Int. J. Mech. Sci. 19, 505.

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12. Hershey, A.V. (1954) The plasticity of an isotropic aggregate of anisotropic face centered cubic crystals. J.Appl. Mech. 30, 241. 13. Hill R. (1948) A theory of the yielding and plastic flow of anisotropic metals, Proceeding of the royal society of London. Series A, mathematical and physical sciences, 193(1033), 281. 14. Hill, R. (1979) Theoretical plasticity of textured aggregates. Math. Proc. Camb. Phil. Soc. 85, 179. 15. Hill, R. (1990) Constitutive modeling of anisotropic plasticity in sheet metals. J. Mech. Phys. Solids 38, 405. 16. Hosford, W.F. (1972) A generalized isotropic yield criterion. J. Appl. Mech. 39, 607. 17. Kneisel P., Palmieri V. (1999) Development of seamless niobium cavities for accelerator applications,proceeding of the 1999 particle accelerator conference, New York, 943. 18. Liu Y.S., Delannay L., Van Houtte P. (2002) Application of the Lamel model for simulating cold rolling texture in molybdenum sheet , Acta Materialia , 50, 1849 -1856 19. Logan R., Hosford W.F., (1980) On yield loci of anisotropic cubic metals, Int. J. Mech. Sci., 22, 419. 20. Nakamachi E., Xie C.L. (2003) Design of texture for improved formability of high- strength steel, Materials Science&Engineering, A340, 130 21. Nemat-Nasser S., Guo W. (2000) Flow stress of commercially pure niobium over a broad range of temperatures and strain rates, Materials science & Engineering, A284, 202 22. Nesterova E.V., Bacroix B., Teodosiu C. (2001) Experimental observation of microstructure evolution under strain-path changes in low-carbon IF steels, Materials science & Engineering, A309, 495. 23. Pagani C., Barni D., Bosotti A.,Pierini P. (2001) Design critical for elliptical cavities, Proceeding of the 2001 particle accelerator conference, Japan. 24. Palmieri V. (1999) New technology in superconducting cavity fabrication, IEEE Transaction on Apllied Superconductivity, 9(2), 1036 25. Park Y.B., Lee D.N., Gottstein G., (1997) Evolution of recrystallisation texture from cold rolling textures in interstitial free steel, Materials Science Technology, 3403, 289. 26. Piehler H.R., Sc. D. thesis (1967) Massachusett Institute of Technology, Cambridge. 27. Read-Hill R.E., Kaufman M.J. (1994) On evaluation the flow stress in niobium of commercial purity, Acta Matter., 43(5), 1731. 28. Singer W., Kaiser H., Singer X., Weichert G., jelesov I. (2001) Hydroforming of superconducting tesla cavities, Proceeding of the 2001 particle accelerator conference, Japan. 29. Singer W., Kaiser H., Singer X., Weichert G., jelesov I., Khabibuline T. (2001) Hydroforming of NBCU clad cavities at DESY, Proceeding of the 2001 particle accelerator conference, Japan. 30. Toth L.S., Jonas J.J., Daniel D., Ray R.K. (1990) Development of Ferrite rolling textures in low carbon steels, Metallurgical Transactions, 21A, 2985. 31. Vanderschueren D, Yoshinaga N, Kestens L. (2000) Texture and Anisotropy Control of IF High Strength Steel, IF steel Conference 2000, Pittsburgh, 247 32. Yoon J.W., Barlat F., Dick R.E., Chung K., Kang T.J. (2003) Plane stress yield function for aluminum alloy sheets-part II: FE formulation and its implementation, Int. J. Plasticity, 20, 495. 33. Zamiri A., Pourboghrat F., Jiang H., Bieler T.R., Barlat F., Brem J., Compton C., Grimm T.L. (2006), On mechanical properties of the superconducting niobium, Materials Science and Engineering A, Vol. 435-436, 658-665. Zamiri A., Pourboghrat F., Characterization and Development of an Evolutionary Yield Function for the Superconducting 34. Niobium Sheet, International journal of Solids and structures 44(2007), 8627-8647

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The stiffness and buckling behavior of panels with stampinginduced imperfections Hongqing Shen a ˈShuhui Li a,b, and Guanlong Chen b

a

Auto Body Manufacturing Technology Center, School of Mechanical Engineering, 800 Dongchuan Road, Shanghai, 200240, PR China b State Key Laboratory of Mechanical System and Vibration, 800 Dongchuan Road, Shanghai, 200240, PR China *Corresponding author: Tel.: +86 21 34206304; Fax: +86 21 34204542 ; E-mail: [email protected] Abstract. With the decrease of sheet thickness, the stiffness and buckling behavior of automobile body panel are more and more concerned. In this paper, the stiffness and buckling behavior of panels with stamping-induced imperfection is studied. Imperfections caused in drawing and springback are obtained by finite element analysis and introduced to define the initial state of the panel before indenting. The indenting procedure is simulated in ABAQUS Standard. In order to solve the buckling problem, a viscous pressure that is proportional to the relative velocity between surfaces is added artificially. For comparison, a simplified model in which imperfections are not considered is also established. Comparing the simulation results with the reference experiment, we find that the stamping-induced imperfections lead to a more severe oil canning phenomena under a buckling load, although they increase panel’s initial stiffness a bit. Moreover, among all stampinginduced imperfections, the geometrical imperfection (surface deflection) is beneficial to increase panel’s buckling load, and the material imperfections (residual stress and uneven thickness) help to decrease panel’s buckling load and can make oil canning more severe during post-buckling. Influences of blank holder force, sheet thickness and material property on panel stiffness and buckling behavior are also studied. The work of this paper is beneficial to processing optimization, material selection, and reinforcement design for automotive exterior panels. Keywords: stiffness, buckling, surface deflection, sheet metal forming PACS: 81.20.Hy

INTRODUCTION Due to fuel economics, ecological reasons and product properties, mass reduction plays a critical role in automobile design and manufacturing [1]. With the application of thinner sheets and new materials, surface stiffness and dent resistance of the automotive exterior panel are becoming more and more concerned [2-9]. In 1995, Nader Asnafi [2] derived equations to predict a double-curved panel’s stiffness and dent resistance. In 2002, Stefan Holmberg and Per Thilderkvist [6] found that high strength steel showed a much higher dent resistance and the aluminum panels had the lowest stiffness. In addition, panels stamped with lower blank holder force (BHF), in general, had a lower dent resistance but a slightly higher initial stiffness and oil-canning load. In 2004, Stefan Holmberg and Babak Nejabat [7] simulated the stiffness and dent property of an automotive exterior panel. Their simulation showed sufficient accuracy in terms of stiffness property but overestimated panels’ dent resistance. In 2010, Hongqing Shen, Shuhui Li and Guanlong Chen [8] found that the Bauschinger effect could reduce panels’ dent resistance and this effect should be considered in numerical analysis of panels’ dent resistance. The stamping effect causes thickness change, residual stress and surface defects on the panel [10, 11]. Since the buckling of shells are sensitive to imperfections, influence of the stamping-induced imperfections on panel stiffness and buckling behavior is further studied in this paper.

FE SIMULATION Problem Description FE Simulation in this paper refers to the experiment work of Stefan Holmberg and Per Thilderkvist [6]. Blanks are first pressed to a depth of 28 mm to form a double-curved panel, and then an embossment with a depth of 8 mm is formed in the opposite direction. Finally, the panel is fixed in each corner and a hemispherical punch indents onto the panel slowly. Reaction force and displacement of the indentor are recorded during the indenting procedure. Figure 1 shows the sample panel and the two evaluation points. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 221-227 (2011); doi: 10.1063/1.3623614 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

221

FIGURE 1. Evaluation points on the sample panel.

All the cases studied in this paper are listed in Table 1. Material characteristic for BSUFD and B170P1 are presented in Table 2. Rp0.2 is the yield strength, R80% is the ultimate tensile strength, n is strain-hardening exponent, k is strength coefficient, E is Young’s modulus and  is Poisson’s ratio. TABLE 1. Panels studied in this paper

Case I II III IV V VI

Material BSUFD BSUFD BSUFD B170P1 B170P1 B170P1

Thickness (mm) 1.0 1.0 1.0 1.0 0.9 0.8

Blank holer force (kN) 228 400 540 228 288 288

TABLE 2. Material characteristics

Material

Rp0.2 (MPa)

R80% (MPa)

n

k

E (GPa)



B170P1

194

362

0.228

639

210

0.3

BSUFD

137

360

0.257

538

210

0.3

FE Modeling The finite element model is shown in Figure 2. Tools are assumed to be rigid. The von Mises yield criterion and the nonlinear combined hardening model is utilized to describe the blank’s yield and hardening behavior. The linear finitemembrane-strain reduced-integration quadrilateral shell element (S4R) is adopted. Meshes are refined around the tool angles. The drawing procedure is simulated in ABAQUS Explicit. The springback and the indenting procedures are simulated in ABAQUS Standard. Simulation results of the drawing procedure are transferred to define the initial state of springback. To solve localized instability in springback, viscous forces are added to the global equilibrium equation. The dissipated energy by viscous damping is controlled to be less than 0.1% of panel’s total strain energy. To simulate the buckling phenomenon, a viscous pressure that is proportional to the relative velocity between the indentor and the panel is introduced. It is defined by where P follows:

f

f ( h, h )

P (h )h

(1) is the damping coefficient and h is the overclosure between surfaces. Damping coefficient, P , is specified as

P (h)

0, ­ ° ® P0 ( h  c0 ) /(1  K )c0 , ° P0 , ¯

for h  -c0 for - c0 d h  -K c0 for h t -K c0 (2)

222

whereߤ଴ is the value of the damping coefficient at zero overclosure and  is the fraction of the overclosure interval [-c0, 0] over which the damping coefficient is equal to ߤ଴ . In our model, ߤ଴ is defined as 0.1% of the relevant pressure between the indentor and panel. It is assumed that c0= 6 mm and =0.833. The max value of the damping force is less than 0.5 N, which is low enough and will not distort the solution.

FIGURE 2. The finite element model (unit: mm).

For comparison, the stiffness and buckling behavior of an ideal panel is also simulated. The geometry of the panel refers to the design value and the material is assumed virgin. Other aspects of the model are identical with the model, in which stamping-induced imperfection is considered.

RESULTS AND DISCUSSION Model Verification Simulation results are compared with the reference experiment [6] in Figure 3. In case I, the load-deflection curves of experiment show an apparent buckling when indenting loads approach to 190 N at Point 1 and 165 N at Point 2. Simulation considering the stamping-induced imperfection succeeds in predicting the buckling phenomenon at both points. The predicted buckling loads are 171 N at Point 1 and 193 N at Point 2. However, if the imperfections are not considered, simulation will fail to predict buc at both poilingnts. So our model which considers the imperfections in simulation is better at prediceting panels’ buckling behavior.

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a) Load-deflection curve at Point 1. b) Load-deflection curve at Point 2 FIGURE 3. Load-deflection curves of simulation and experiment. (Case I)

a) Load-deflection curve at Point 1. b) Load-deflection curve at Point 2 FIGURE 4. Load-deflection curves of panels with various BHFs (Case I, II and III)

Influence of Blank Holder Force Load-deflection curves from Case I to Case III are shown in Figure 4. When BHF increases from 288 kN to 540 kN, the initial stiffness at Point 2 increases from 89 N/mm to 109 N/mm, but the reaction force reduces more sharply after buckling occurs.A high BHF enhances the initial stiffness of the panel but lead to a more severe oil-canning phenomenon˄sharp reduction of reaction force after buckling˅. Figure 5 shows the dispalcement of the panel in the thickness direction under various BHFs (positive value means convex ). When BHF is high, a hump forms in the center of the panel and increases the curvature locally. According to Zhang’s [9] equation,

K

P fp

4 Et 2 k x k y (1  P 2 ) 3

,

(3)

where E is the Young’s modulus, t is thickness of the panel, kx and ky are the curvature in x and y direction and the Poisson’s ration, the initial stiffness is proportional to the curvature, panels’ initial stiffness.

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P is

k x k y . So a higher BHF helps to increase

a) BHF = 288 kN

b) BHF = 400 kN

c) BHF = 540kN d) Buckling mode 1 FIGURE 5. Surface defects and buckling mode

In Figure 5, the springback of the panel under 400 kN and 540 kN has a tendency to be convex in the central area. This tendency is similar to the first buckling mode of the panel under a load at point 2, shown in Figure 5 d). For an imperfection-sensitive structure, geometrical imperfection consisting of buckling modes can trigger buckling easily. So when BHF increases, although the buckling load may increase a bit, a more severe oil-canning phenomenon ୄsharp reduction of reaction force after buckling୅ take place.

Stamping-induced Imperfections The stamping induced imperfections include geometrical imperfection (surface deflection) and material imperfection (residual stress and uneven thickness).In Figure 6, load-deflections curves of panels only with geometrical imperfection are compared with those of panels with all stamping-induced imperfections. If there were only geometrical imperfection (surface deflection) on the panel, the buckling load would be 252 N at Point 1 and 268 at Point 2. In the effect of material state imperfection, the buckling load drops to 208 N at Point 1 and 235 N at Point 2. So the geometrical imperfection (surface deflection) is beneficial to increase panel’s buckling load and the material imperfections help to decrease panel’s buckling load. On the other hand, during post-buckling reaction force of panels only with geometrical imperfection does not drop so sharply as those of panels with all the stamping-induced imperfections. This indicates that the material imperfection can make the oil canning more severe during post-buckling.

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a) Load-deflection curve at Point 1. b) Load-deflection curve at Point 2. FIGURE 6. Load-deflection curves under the influence of various imperfections (Case III)

Influence of Sheet Thickness Load-deflection curves of panels of various thicknesses are compared in Figure 7. The stiffness of B170P1 and BSUFD panels are close to each other when the thickness of both panels is 1.0 mm. When sheet thickness decreases from 1.0 mm to 0.8 mm, the initial stiffness of B170P1 panel decreases from 44 N/mm to 19 N/mm at Point 1 and from 87 N/mm to 44 N/mm at Point 2, respectively. A decrease of sheet thickness weakens panels’ stiffness even high strength steel is used. This conclusion agrees with the work of other researchers [2, 6, 7, 9]. For example, according to Eq. (3), the initial stiffness of a panel is not related to the material’s the initial yield stress but related to the material’s elastic property, thickness and curvature. Moreover, a decrease of sheet thickness also reduces the buckling load sharply. In Figure 7 b), when sheet thickness decreases from 1.0 mm to 0.8 mm, the buckling load at Point 2 drops from 218 N to 114 N. In automobile engineering, buckling load less than 100 N should be prevented.

a) Load-deflection curve at Point 1. b) Load-deflection curve at Point 2 FIGURE 7. Load-deflection curves of panels with various thicknesses. (Case I, IV, V and VI)

CONCLUSIONS In this paper, the stiffness and buckling behavior of a double-curved stamping panel is studied. The following conclusions apply: z Panels with stamping imperfections may show an obvious buckling phenomenon under indenting. In order to simulate the buckling phenomenon, stamping-induced imperfections should be considered. z When BHF increases, although the buckling load may increase a bit, a more severe oil-canning phenomenon (sharp reduction of reaction force during post-buckling) can take place. z The geometrical imperfection (surface deflection) is beneficial to increase panel’s buckling load. z The material imperfections help to decrease panel’s buckling load and can make oil canning more severe during post-buckling.

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z

A decrease of sheet thickness not only weakens panels’ stiffness but also reduces the buckling load heavily.

ACKNOWLEDGEMENTS The authors acknowledge the support from Research Project of State Key Laboratory of Mechanical System and Vibration MSVMS201101) and Research Fund for the Doctoral Program of Higher Education of China (20100073110034).

  REFERENCES 1. Jeswiet. M.Geiger. U.Engel. CIRP Journal of Manufacturing Science and Technology 1 (2008) 2–17. 2. Nader Asnafi. Journal of Materials Processing Technology 49 (1995) 13-31. 3. Palmquist N. Technical Report LM 500704; 1995. 4. Chen C-H, Rastogi P, Horwath C. Automotive Body Materials–IBEC’93; 1993. 5. Lundberg M, Gunnarsson L, Thuvander A, Melander A. Swedish Institute for Metals Research; 2000. 6. Stefan Holmberg, Per Thilderkvist. Materials and Design 23 (2002) 681–691. 7. Stefan Holmberg, Babak Nejabat. Materials and Design 25 (2004) 361–368. 8. Hongqing Shen, Shuhui Li, Guanlong Chen. Material and Design 31 (2010) 870-876. 9. Yan Zhang, Xinmin Lai, Ping Zhu, Wurong Wang. Materials and Design 27 (2006) 64–68. 10. C.D.Park, W.J.Chung, B.M.Kim. Journal of Materials Processing Technology 187-188 (2007) 99-102. 11. A. Andersson. Journal of Materials Processing Technology 209 (2009) 821–837.

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Analysis, Simulation and Prediction of Cosmetic Defects on Automotive External Panel A. Le Porta, S. Thuilliera, C. Borotb, J. Charbonneauxc*, Aethra Automotive Systems a

LIMATB, Université Bretagne-Sud, BP 92116 – 56321 Lorient, France b ESI Group, 5 Parc du Golf, 13856, Aix en Provence, France c ESI Group, Parc d’Affaires SILIC, 99 rue des Solets, BP 80112, 94513 Rungis, France *Corresponding author : [email protected] Abstract. The first feeling of quality for a vehicle is linked to its perfect appearance. This has a major impact on the reputation of a car manufacturer. Cosmetic defects are thus more and more taken into account in the process design. Qualifying a part as good or bad from the cosmetic point of view is mainly subjective: the part aspect is considered acceptable if no defect is visible on the vehicle by the final customer. Cosmetic defects that appear during sheet metal forming are checked by visual inspection in light inspection rooms, stoning, or with optical or mechanical sensors or feelers. A lack of cosmetic defect prediction before part production leads to the need for corrective actions, production delays and generates additional costs. This paper first explores the objective description of what cosmetic defects are on a stamped part and where they come from. It then investigates the capability of software to predict these defects, and suggests the use of a cosmetic defects analysis tool developed within PAM-STAMP 2G for its qualitative and quantitative prediction. Keywords: cosmetic defect, stamping, flanging, sheet metal forming, automotive, stoning. PACS: 62.20.F-

INTRODUCTION In the automotive industry the surface defects on the body part are taken into account for the quality of the vehicles which are produced. The brand image is put into play. A customer has a look on it. That is why during the conception of the part the manufacturer has to predict the defects. During the production of the blank, defects have been checked by different ways (touchable and visible) in different steps of the process. It is useful to avoid any issue. In the reality, the defects can appear after some specific stages like the springback, the flanging or the hemming… Thanks to the improvement of the software this problematic is now predictable during the simulation. PAM-STAMP 2G is able to simulate the different processes and a cosmetic defects tool has been implemented for showing easily the quality surface.

STAKES OF COSMETIC DEFECTS CONTROL ON AUTOMOTIVE EXTERNAL PANELS In the automotive industry there is a ceaseless search for more resistant and lighter materials, modern design, fuel savings, safety, comfort, quality and repeatability in production and especially lower environmental impacts. AETHRA, like other automotive press shops, facing daily the challenges presented by this quest. New materials are developed, thicknesses are reduced and product designs are made more complex. For such a company to grow given the current world economy, wasting time and raw materials is economically unacceptable. In the stamping process, most of the waste is concentrated in tool tuning, which also turns out to be the most costly work for the company, hence the necessity to reduce the required work time in testing and tuning. A

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 228-236 (2011); doi: 10.1063/1.3623615 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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reliable stamping simulation software should effectively reduce trial times as well as the number of material blanks needed for testing, in addition to contributing to securing part quality. For external panels, the quality of the product’s surface is critical to customers (Fig. 1). To validate an external panel it is necessary for it to be free of tool marks and any kind of superficial defect. Furthermore, though the product must have structural hardness, it cannot show any sign of necking caused by excessive strain. When this kind of flaw is found during testing, the tuning process typically becomes very expensive and time-consuming. Although the die is already machined and assembled, the press isn’t in fact producing: personnel is spending time on analysis and careful modification, and for each test one metal blank is rejected.

FIGURE 1. Overall view of a high end commercialized car. After the body assembly, operators check the completed body in white in a light tunnel (Fig. 2) before continuing the process by the painting. All front fenders, doors, hood and trunk have been hand-feel checked in deeper at standstill in order to verify the excellent appearance of the vehicle, the proper tightening of the sheet and the nonexistence of any asperity. It is crucial to obtain a perfect quality of surface without cosmetic defect.

FIGURE 2. Detection of the defect in a light inspection room. But it is difficult to detect the cosmetic defect easily and automatically because it is essentially subjective. The surface defects which are detected by human visualization inspection (stoning, hand feel checking, light tunnel and visual checking) or by mechanical processes like the digital probe are occurred at forming and assembly processes. The appearance quality is checked after each operation: stamping, flanging, hemming, assembly, after painting and on the final vehicle. Moreover a part is considered compliant when no defect is detected on the finished vehicle. That is why, it is crucial to remove all cosmetic defects in the early operations.

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Types of defects in the automotive industry Depending on the production process, defects can be sorted in different categories. There are two main types of cosmetic defects. The first one is “slip lines”. It corresponds to marks of the blanks which are caused by the flow of the material over a tooling radius. It is easier to predict where it might occur but it is not so easy to predict its severity. The second type of defect is a geometrical modification of the surface. It can be hollows, shallows or ‘teddy bear ears’ which are located in the flattish area of the outside panel and generally adjacent to handles, edges, flanges... It appears after the stamping and trimming operation, after flanging or after hemming process. The main cause is the deformation of the blank due to a lack of stretch. The defect appears during the release of the stresses.

Localization of the defect A cosmetic defect is something very small around 10µm deep on a small area 1cm². They are quite difficult to localize but different methods exist for detecting them. They all rely on manual action where several levels of verification can be done. On all parts the surface quality is hand-feel checked. The operator strokes the part with a special glove and marks it with a pen. The rating of the defect ranges from a minor defect to a visible defect (Fig. 3).

FIGURE 3. Example of surface defects on an automotive door handle. The marker B correspond to a hump, the areas C and D corresponds to ‘teddy bear ears’ defects. For a deeper analysis a part is randomly taken from the production line and is checked by dragging the hand and by moving a stone onto the surface. The defect is highlighted by this process. If the part is scratched, there is no defect otherwise there is a defect. By the same way it is possible to visualize a defect by putting a ruler on the surface. If there is a depression between the ruler and the blank, a defect exists (Fig. 4). Another method is the lighting of the part in an inspection room (Fig. 5). In this case the defects are picked up by observing the deformations of the light patterns over the panel surface.

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FIGURE 4. Detection of the defect with a ruler. For more precision the part can be scanned with an optical or mechanical sensor, measuring shape deviations.

WHAT IS A COSMETIC DEFECT? The detection of surface defects on real parts remains very subjective because investigations are done by hand, with empirical rules for quantifying the severity of the defects. Human intervention along the process line and the different iterations to modify the manufacturing tools increase the cost of the vehicle. To reduce the cost of cosmetic defect control and corrective actions, it is important to first understand what cosmetic defects are and what causes them. UBS experimental door corner The flanging device has been designed so as to reproduce the same kind of defect that appears in the door-upper corner region at the flanging stage. This device aims to be compact, modular and easily processed. Figure 5 illustrates the flanging device set on a Zwick-Roell BUP200 testing machine. It is composed of a flanging blade, a blank-holder and a die. The geometry of the sample with a flanging height of 5 mm is defined here below; it consists of a disc with a missing quarter obtained by laser cutting. The results have been obtained using dual phase DP500 steel sheets with a thickness of 0.6 mm.

FIGURE 5. Process and geometry of the experimental door corner. The reproducibility is checked by performing three tests in the same configuration and one of them is then selected to be processed. A selection of flanged blanks has been measured on a Brown&Sharpe ® MicroXcel pfx 4.5.4 tridimensional measuring machine with an accuracy of 3µm. The blank is positioned by three holes on a dedicated setup and its surface is scanned every millimeter before and after flanging. The output is the coordinates of each scanned point. For this processing the displacement in the xy plane is assumed to be negligible in the considered area. Due to the slightly non-planar surface of the virgin sample, the output Z-coordinate corresponds to the difference between the deformed and initial values. By magnifying the Z-coordinate of the measured blank, it is observed that the defect consists of a depression near the corner radius and of two humps along the flanged edges. Figure 6 illustrates that the depression shape is not dependent on Y, contrariwise to the hump, the height of which decreases when the Y-coordinate increases. The surface defect exhibits a depth of 50µm, and a size of 60 mm along the X-axis and 30 mm along the Y-axis.

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 FIGURE 6. Profiles along X-axis for different Y values.  Flanged blanks have also been evaluated by experts in surface-defect characterization. A surface defect was effectively found in the corner area, with a low degree of severity (indeed the lowest in the scale) that has been marked as “only visible by a specialist”, but nevertheless will necessitate some modification of the geometry of the tools or process parameters to suppress it. After some treatments of the results, the profiles have been smoothed and the curvatures have been computed (Fig. 7).

FIGURE 7. Profiles and curvatures along X-axis for different Y values. The analysis of the experimental results shows that surface defects are characterized by a local inversion of the curvature developed during springback following the drawing stage, but also the flanging stage. That is why the detection is very difficult. For this simple case where the extracted shape is relatively flat and the defect is only a hump, the measure of the curvatures is easy and logical. But on some shapes where the curvatures exist and are known, the results of the curvature modification are not enough to post-treat. It is possible to show a modification of curvature but it is not a defect, it corresponds to the shape of the part. To conclude this method works on flat blank but has some limitations to use for shapes with normal curvature changes.

INNOVATIVE SOLUTION FOR SIMULATING AND MEASURING COSMETIC DEFECT WITH PAM-STAMP 2G To reduce drastically the costs induced by cosmetic defect it is necessary to predict the apparition of such defects and to evaluate the gravity of the problem. If this analysis is done at the right time, corrective action plans can be defined and tested before any real prototyping. This is the objective of the surface defect functionality that is proposed in PAM-STAMP 2G.

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Focusing on the way industrials are checking the quality of the panels before painting, PAM-STAMP 2G proposes similar tools for localizing the defects on simulation. The difference with reality is that surface defect functionality can also measure the area of defects and the depth; it proposes thus an objective tool for comparing the defects and for adapting the empirical classification of defects.

Stoning method The stoning method proposed in PAM-STAMP 2G is like a straight stone which is moved on the blank along a direction (Fig. 8). If there is no defect, a null value will be plotted. But if there is one, the depth of the defect will be measured.

FIGURE 8. Description of the stoning method. The rolling method (Fig. 9) can be compared to the stoning method. But this method has a curved stone. It is useful for a part which has known curvature. This method can be used when the defect is in a concave area of the part (e.g. the defects on the door handle).

FIGURE 9. Description of the rolling method. Depending on the geometry of the global part, stoning or rolling methods can be selected. For example, the rolling method gives better results in a concave area instead of the stoning method (Fig. 10).

FIGURE 10. Comparison between rolling and stoning methods for a concave area.

PAM-STAMP 2G Stoning method on UBS Experimental door corner The computation of this case has been performed with PAM-STAMP 2G 2011.0. The results have been analyzed after the springback stage. The arrow indicates the movement of the stone.

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FIGURE 11. Comparison between experimental and simulation results of the door corner with the stoning contour in PAM-STAMP 2G.

PAM-STAMP 2G SURFACE DEFECT SOLUTION APPLIED ON AETHRA DOOR PART AETHRA needed to secure the parts quality according to customer requests, whilst saving development time and money. Simulated results close enough to the real parts concerning splits, wrinkles, superficial defects and tool marks allowed the tools tuning with significant reduction in work hours. For this study, the sheet metal forming team has performed some tryouts on a door panel in order to find a solution on the detection of the surface defect early enough in the process.

Data set-up This project simulates the stamping of the outer door panel of a vehicle and especially the handle area. The goal is to visualize some defects around this area. After a positioning of the blank on the tool, a double action process has been performed with a high quality simulation. A springback stage ends the process in order to simulate the tools removal.

Cosmetic defects on the simulated handle area The first is to detect the defect in the real part. After some handmade inspections on this part, the defects have been localized and marked. The “teddy bear ears” and a mark on a corner of the handle are the most important defects on this case. In parallel the simulation of the part has been performed until the release of the stress after the stamping operation. The door handle has been inspected by the stoning contour on PAM-STAMP 2G and the defects appear. By doing the same actions like in reality it is possible to see the same defects in the same areas (Fig 12 and Fig 13).

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FIGURE 12. Stoning contour in PAM-STAMP 2G on the door handle

FIGURE 13. Stoning contour in PAM-STAMP 2G on the door handle

CONCLUSION After acquiring PAM-STAMP 2G, the engineering and the tool testing teams from AETHRA started a reverse engineering work aiming to set the contour scale from PAM-STAMP 2G’s analysis in order to find the defect that actually occurred during physical testing. The goal was to benefit from the lessons learned by applying them to future simulations. The results obtained in the first comparisons between simulation and testing proved to be incredibly accurate. Consequently, a software feedback was implemented for every product, resulting in a significant increase in result accuracy and reduced testing time for die release to AETHRA’s press shops. Cosmetic defect is a major issue for production of automotive external panels. Understanding what they are and where they are generated was a mandatory step for developing an objective method of prediction. PAM-STAMP 2G v2011.0 offers surface defect tools, based on the know-how of the industrial. The stoning contour is an important functionality for predicting the cosmetic defects at the right time and therefore set-up the good corrective actions without real prototyping.

ACKNOWLEDGMENTS The authors would like thank Aethra, ESI Group and UBS for their collaboration. The authors would also like to thank the Direction Générale de la Compétitivité, de l’Industrie et des Services, Région Bretagne and Conseil Général du Morbihan for their financial support as well as the continuous support of PSA during all the projects.

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REFERENCES 1. A. Le Port, “Caractérisation et prédiction des défauts d’aspect sur les ouvrants automobiles”, Ph.D. Thesis, Université de Bretagne-Sud, 2010 2. C.D. Park, W.J. Chung, B.M. Kim. A numerical and experimental study of surface detections in automobile exterior panels, Journal of Materials Processing Technology 187-188 (2007) 99-102 3. T. Ohtsuka. Prediction and Compensation of Surface Deflection using PAM-STAMP, Confidential presentation from Mitsubishi done during the Global Forum 2010 of ESI Group 4. Ricardo A. Micheletti Viana, Wellingtown Caetano Soares, Arlem Picinin Campos. Aethra Automotive Systems sets the scales of PAMSTAMP obtaining identical results as to parts in try-out, regarding propagation of cracks, wrinkles, tool marks and surface defect, Aethra Sistemas Automotivos, 2011

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Crash Analysis of Automotive Chassis Structure Considering the Strain Hardening Effect A. S. Mashalkara and R. P. Parvekara a

Department of Production Engineering, S. G. G. S. Institute of Engineering and Technology, Nanded (MS), INDIA Abstract. The work hardening during the forming process is generally ignored in crash analysis. This paper presents computational details of the effect of forming process on crash response of typical Sports Utility Vehicles (SUV). Forming results for front panels of a medium-sized SUV were calculated using Hyperform and LS Dyna. These were imported into the crash model and crash results compared with and without the forming effects. Time taken to generate the forming data by a variety of methods is quantified and the trade-off between time taken and accuracy is examined. LS-DYNA was used for both forming and crash simulations. The effect of thickness, work hardening and residual stresses on the crashworthiness results is measured; crash response is seen to be significantly affected when the effects of forming are included. The paper proposes a systematic method to transfer data from the forming analysis to crashworthiness analysis (both using LS-DYNA), allowing thickness, residual stress and plastic strain data selectively or in combination to be used to initialise the crash model. The relative effect of each of these forming parameters has been examined and the importance of, work hardening on the crash response of the stamping front rail has been identified. A significant change in energy absorption, peak force and stroke of the front rail was predicted, indicating a far stiffer response in the formed rail than would be expected based on the nominal material properties. Therefore, forming effects should be accounted for in vehicle crashworthiness predictions. Keywords: FEA, Crashworthiness, Strain Hardening

INTRODUCTION Material data derived from tensile tests is an important input to finite element crash models. However, the test samples are normally taken from the coil, not from formed parts. The forming process is likely to modify the properties of the materials due to work hardening; thickness changes and residual stresses may also be important. This could lead to errors in predictions of crash response. The degree of effect on the crash behavior depends on the type of forming process. The hyper-forming process leaves very significant residual thickness changes and work hardening, and can have a major effect on crash results [1]. Even the predicted principal mode of collapse can be altered. This is because some areas of the part are both thickened and work-hardened by the hyper-forming process, leading to a considerable increase of strength and hence increased resistance to collapse in those areas. Stamped parts are thought to show less sensitivity, because areas that are work hardened would also in general be thinned by the stamping process; the two effects might approximately cancel each other. Experimental studies on axially crushed straight box sections [2] did not show an observable effect from the forming process: any effect was within the experimental scatter, and may be less significant for the geometries tested. An analytical study based on the high strength steel materials in the SUV vehicle [3] showed that residual stamping effects have some influence on the deceleration time history but no change of deformation mode was observed. Compared with other approximations in the crash models, the effects of forming might be considered to be secondary. That conclusion is also drawn by the work reported here. However, given the drive to reduce prototypes and place an ever-increasing reliance on CAE, it would not be prudent to ignore the possibility of more significant influence in some cases.

BACKGROUND Metal forming processes result in a number of changes in the material properties of a formed component. The initial blank material is subjected to large deformation during forming which results in changes in the thickness and

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 237-242 (2011); doi: 10.1063/1.3623616 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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yield point of the material. Another result of the forming process is the residual stress distributed across the component. These material properties are important inputs to computer-aided engineering analyses of component response, for example crash or durability models. A change in these properties could have a significant effect on the results of such analyses and could lead to a sub-optimal design. By using the output data from forming simulation as the input to other types of modeling it is possible to investigate the significance of these changes to the integrity of the component.

OBJECTIVES To establish a method to take forming analysis data into the crashworthiness model (including thickness, work hardening and residual stresses)

MODEL DESCRIPTION AND METHOD Flow chart for the various steps in crash analysis (using Hyperworks and LS DYNA) followed by practitioners as a standard model is given in Figure 1. A modified flow chart which takes into consideration the stamping effects while doing the crash analysis is given in Figure 2.

FIGURE 1. Flow Chart for standard model

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FIGURE 2. flow chart for stamping data crashworthiness model

The base crash model is the SUV model developed figure3. For the purposes of illustrating the principles involved, the front, side, cross rail material data is mild steel. Stress-strain properties were derived from tensile test. The model was run in LS-DYNA 970. The panels absorbing the most energy in the base crash model (Figure3) were selected for stamping analysis.

FIGURE 3. FEA Model

Boundary Condition Chassis is symmetrical about X axis and only half front side chassis is considered. The initial impact velocity of a rigid wall is 13 km/hr , mass of wall is 1300 kg. The model was run in LS-DYNA 970.

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The side rail panels absorbing the most energy in the base crash model (Figure 4) were selected for stamping analysis.

. FIGURE 4. FEA Model

Forming analysis method The forming results are generated by using a combination of Hyperform and incremental LS DYNA stamping simulation program. The mesh from the crash model is used as input. The objective is to test the simplest method by which forming results can be generated, remove undercut conditions, or otherwise make the crash mesh more suitable for forming analysis. From an initial sheet with a thickness 3.0 mm, the sheet undergoes a forming operation, followed by a mechanical preform before the hyperforming stage. A maximum binder pressure around 10000 N is obtained. The forming limit diagram is studied and revisions made to the boundary conditions until a successful forming operation is indicated, i.e. a strain distribution in the formed panel that indicated neither splitting nor severe wrinkling tendency.

FIGURE 5. Stamping FLD for inner and outer side rail

Because the same mesh is used for stamping and crash, no data mapping is necessary. The thickness and plastic strain are simply copied from the stamping model into the crash model for each element using a program to reformat the Hyperform results into LS-DYNA input. The next release of Hyper-form will output LS-DYNA format data directly, avoiding the need for reformatting. Residual stresses have been shown by previous studies to be less significant and are not considered. The difference in stamping results arising from the more detailed mesh at a typical % thickness reduction is shown in Figure 6.

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FIGURE 6. Stamping % thickness reduction for inner and outer side rail

RESULTS Deformations of the front structure from the analyses with and without the stamping effects are compared in Figures 7 and 8. There is no significant change of deformation mode.

FIGURE 7. Deformation of front structure without stamping effects

FIGURE 8. Deformation of front structure with stamping effect

The predicted force histories in the Front Side Rail are compared in Figure 9, first with no stamping effects, then with the base method in which stamping effects are calculated. When the stamping effects are included the Front Side Rail force is reduced from 148 N to 90 N (Figure 9). The peak force carried by the Front Side Rail (Figure 9) is 20% to 39% higher; the extra energy absorbed early in the crash explains the subsequent reduction in peak force

FIGURE 9. Force in Front Side Rail

FIGURE 10. Displacement in Front Side Rail

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The peak displacement value of the front side rail without stamping effect is 130 mm and with stamping effect is 127 mm. It clearly indicates that the crashworthiness results are influenced by thickness variation, work hardening and residual stresses developed during stamping operation.

DISCUSSION Forming simulation programs are beginning to offer blank-holder surface and addendum generation functions to speed up the creation of a tool design model. While such functions are reducing the time needed to create a detailed finite element model, it is likely that several iterations would be needed to achieve a satisfactory result – for example, varying the blank-holder force, edge/draw-bead conditions, and blank-holder and addendum geometry. The total time required to analyse each panel by this method is likely to be between one to three days. In view of the time required to analyse each panel and the large number of panels, it would be impractical to carry out such analyses purely for the purpose of deriving residual stresses, thickness and plastic strain for the crash model. However, where these models already exist, they would be the preferred source of data due to the increased accuracy inherent in the method.

CONCLUSION There is potential for residual stamping effects to cause more significant differences in overall crash response, therefore, they should be included in crash models. When the thickness variation, work hardening and residual stresses effects are considered with the dynamic constitutive relation, the final displacement is relatively small compared to the one with the standard model.

ACKNOWLEDGEMENT The authors would like to express their deep appreciation to Department of Production Engineering, S. G. G. S. Institute of Engineering and Technology, Nanded (MS), INDIA for the help in the study and preparation of manuscript.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Dutton, T., Iregbu, S., Sturt, R., Kellicut, A., Cowell, B., Kavikondala, K., The Effect of Forming on the Crashworthiness of Vehicles with Hydroformed Frame Siderails, SAE Paper 1999-01-3208, 1999 Shaw, J., Watanabe, K., Steel Strength and Processing Effects on Impact Deformation for a Crash Energy Management Component, SAE Paper 2001-01-1053, 2001 Simunovic, S., Shaw, J., Aramayo, G., Steel Processing Effects on Impact Deformation of Ultra Light Steel Auto Body, SAE Paper 2001-01-1056, 2001 Zaouk, A., Marzougui, D., Kan, C., Development of a Detailed Finite Element Model, Part I: Model Development, The 1st International Conference for the International Journal of Crashworthiness Detroit, MI; 1998. Hallquist J; LS-DYNA Users Manual Version 950, Livermore Software Technology Corporation, 1999. FAST_FORM3D, Users Manual, forming Technologies Inc, 2000. Milititsky, M., Matlock, D., Speer, J., Cady, C., Mataya, M., Effects of Pre-Strain on Properties of Low-Carbon Sheets Steels Tested over a Wide Range of Strain Rates, SAE Paper 2001-01-0082, 2001. Oasys PRIMER Version 8.1 Manual, Oasys Ltd, 2001. Kaufman M., Gaines D., Kundrick K., Liu S.-D., "Integration of Chassis Frame Forming Analysis into Performance Models to More Accurately Evaluate Crashworthiness", SAE, 1998 Le Roch, Y., Beard J.Y., Duval J.L, Mouatissim E., Di Pasquale E., "Coupled Sheet Metal Forming and Fatigue Simulation", IDDRG'98, 1998 LSTC, "LS-DYNA User's Manual Version 940", Livermore Software Technology Corporation, 1997. LSTC, "LS-NIKE3D User's Manual Version 981", Livermore Software Technology Corporation, 1997. Hance B. M., Foley R. P., Matlock D. K., “Effects of Strain Path on Formability and Micro-structural Evolution in Low-carbon Sheet Steels”, SAE Paper No. 970155, 1997 Trevor Dutton, Richard Sturt, Paul Richardson and Andrew Knight; “The Effect Of Forming On Automotive Crash Results”, Society of Automotive Engineers, Inc., 2001-01-3050

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Efficient planning and numerical analysis of industrial hemming processes Igor Burchitz∗ , David Fritsche† , Göran Grundmann∗∗ and Matthias Hillmann† ∗

AutoForm Engineering B.V., Krimpen a/d Ijssel, The Netherlands † AutoForm Development GmbH, Zürich, Switzerland ∗∗ AutoForm Engineering Deutschland GmbH, Dortmund, Germany Abstract. Hemming is a forming operation used in the automotive industry to join inner and outer components during the assembly of closures. These are typically opening parts of the body-in-white with additional requirements to their visual appearance. A suitable production concept of hemming operation which satisfies quality, capacity and cost requirements is determined during hemming planning activities. A digital tool to facilitate these activities and minimize the amount of trial and error iterations in try-out phase is presented in this paper. This tool can be used to define process plan, active tool surfaces and suitable process parameters for both die hemming and roll hemming operations. In case of early feasibility studies, when the die layout of the drawing operation is still not available, 3D part geometry is used directly to develop the concept of hemming process. Advanced validation studies, aimed at process optimization and controlling defects associated with hemming, can be based on complete simulation of all forming operations. Validation and analysis of developed concepts of hemming operation is done using the standard AutoForm-Incremental solver. Submesh strategy and special algorithm for contact description between inner and outer parts were implemented to ensure that accurate simulation results can be obtained within reasonable calculation time. Performance of the new software tool for hemming planning and accuracy of simulation results are demonstrated using several simple benchmarks and a real industrial part. It is shown that the new software tool can help to secure the efficient production launch by providing adequate support in try-out phase. Keywords: hemming, process planning, automotive assembly, FEM simulation

1. INTRODUCTION Computer simulation based on the finite element method has become an essential tool used extensively during the design of new sheet metal parts. Industrial practice has shown that number of trial and error iterations can be reduced considerably by using for example stamping simulations to analyze part feasibility and to find optimum process parameters. In recent years various software solutions have been developed to help the engineers to quickly generate cost optimized process layouts and to control the lead time [1]. Growing complexity of the industrial development processes also lead to the introduction of the framework for digital process planning which was developed to ensure fast to market production of quality parts at lowest costs [2]. Despite these significant developments, comparison of physical reality with available software tools reveals that not all industrial requirements are currently met [3]. Further developments are needed to improve the coverage of stamping simulations by introducing an efficient use of the software in the analysis of hemming. Hemming is a forming operation which can be used to join inner and outer components of a part during the assembly procedure. Different technologies are available to perform this operation, i.e. die hemming, tabletop hemming and roll hemming. Throughout part design and development of production process, multiple tasks related to hemming must be performed. This includes early feasibility studies, choice of appropriate hemming technology and equipment, tool design, process planning and final validation studies. The traditional approach to accomplish these tasks is to rely on empirical knowledge and experience accumulated during previous projects. After that, during the try-out phase, necessary adaptations of tooling geometry and process conditions are made iteratively, to make sure that the final part satisfies specified requirements. To minimize the number of trial and error iterations in the try-out phase, a digital tool is needed which can provide the required and reliable information at various stages of part development. It is known that, since hemming involves plastic deformations of the material, it has a considerable influence on the quality of the final part. Therefore, over the years, several comprehensive experimental studies were conducted to better understand hemming, identify defects associated with this operation and investigate sensitivity of part’s quality to various process parameters [4–9]. Possibility of modeling of hemming processes using explicit and implicit numerical codes was investigated by various authors. Multiple studies were performed which focused on different

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 243-250 (2011); doi: 10.1063/1.3623617 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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FIGURE 1.

I: definition of hemming contour, II: roller positioning and orientation.

aspects of finite element modeling of hemming, e.g dimensionality of the model [5, 10, 11]; applicability of existing yield criteria and hardening laws [12–14]; choice of appropriate element type [5, 12]; forming limits and prediction of material failure [15–17]; description of contact conditions and modeling of inner part [5, 18]. Although analysis of various benchmarks proved that the numerical technology available today is suitable for accurate modeling of hemming processes, hemming simulations are not extensively used in the industrial environment. To facilitate the usage of such tools, this paper presents the complete solution, which can be used for planning and development of hemming operations. As will be described in the next section, the hemming solution consists of AutoForm-HemPlanner - software tool for pre- and post processing - and the AutoForm-Incremental solver. General description of the HemPlanner and possible scenarios for application of the hemming solution in the automotive industry are outlined in Section 2. Special algorithms and modifications of AutoForm-Incremental solver, which were needed to guarantee realistic calculation times and high simulation accuracy, are discussed in Section 3. Finally, results of validation of solver accuracy performed with several benchmark problems are summarized in Section 4.

2. SOLUTION FOR HEMMING PLANNING In real practice, creation of the hemming plan can be started as soon as the CAD geometry of the part after forming and hemming becomes available. These surfaces together with the hemming contour are used by the AutoFormHemPlanner as the basis for the generation of geometries of tools and their kinematics. Hemming contour is shown in Figure 1 and it is defined as the curve that outlines the entry to the area of the flange radius. This curve proved to be the most optimum curve for hemming planning. Independent of manufacturing technology, AutoForm-HemPlanner can automatically generate the hemming bed and a strip of the inner part based on the hemming contour and the surfaces of the flanged part. Generation of the active surfaces of tools for pre- and final hemming, as well as their kinematics, is based on sections. The user can define any number of sections at arbitrary positions of the hemming contour. Within each such section geometries of the flanged

FIGURE 2.

Profiles for generation of tools for conventional hemming.

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FIGURE 3.

Workflow diagram describing quick and advanced hemming.

and hemmed part are displayed to help the user to define the most suitable position and geometry of the hemming tools. These sections can also be used to view results of simulations and to make decisions if corrections of the active tool surfaces or their position are necessary. For geometry generation, the software uses mesh-based techniques which are more suitable in this situation due to their superior speed and flexibility. In case of roll hemming, the number of process steps and the shape of the roller have to be defined. Roller position can be prescribed in any section of the hemming contour, see Figure 1. Based on the industrial practice, a simple set of parameters was formulated to ensure an easy and optimum positioning of the roller with respect to the hemming bed and the target flange. Similarly, for die or tabletop hemming, the number of process steps can be specified by the user. Pre-hemming tools are generated with the help of profiles which can be defined at any section of the hemming contour. To speed-up the development, a selection of the most used parametric and free form profiles was made available, see Figure 2. Geometry of the tool for final hemming operation is based on the CAD surfaces of the final hem and it is created using user-defined supplementary profiles.

2.1. Planning and feasibility engineering In the automotive industry, the first phase of hemming activities is started soon after the concept of part’s geometry is finalized, i.e. during the stage of production and process planning, even when the information about the die layout of drawing and forming operations is not yet available. The main goal of this phase is to conduct early feasibility analysis of part’s geometry, to develop concept of the process plan and to make a decision about the technology for hemming. With the help of the new software tool, process planner can perform a combined 2D and 3D evaluation of the flange geometry which must be deformed during the hemming operation. Depending on the vehicle program and part specification requirements, certain hemming technology can be selected and the user can then design the process plan, create suitable tool geometries and kinematics. To validate the generated concept, a quick numerical analysis of the developed hemming operation can be run. This is the full scale finite element calculation and in order to make it quick, CAD geometry of the flanged part is used as a starting point of the analysis. Not considering complete history of deformation, which takes place during the drawing and forming operations, will obviously have an effect on the accuracy of simulation results. However, at this stage, when part’s geometry may still undergo multiple modifications, this simulation can provide a reasonable estimate of the material roll-in and the general information about the quality of the hem, such as tendency for flange wrinkling or splitting. "Quick hemming" approach is schematically shown in Figure 3. Its main advantage is the possibility to run multiple simulations to evaluate various concepts and to predict necessary part modifications to make it manufacturable. Results of the quick hemming simulation can also generate information which is necessary to determine the type of manufacturing equipment, space requirements and production resources. This information may include area of the active tool surfaces and their kinematics in case of die or tabletop hemming, roller shapes and trajectories in case of roll hemming. Additionally, using reaction forces calculated in the simulation, one can make decisions about, for instance, type of robots which may be required during roll hemming or forces which will be needed to clamp the inner and outer parts.

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2.2. Final validation and try-out support The concept of the hemming operation is finalized during the detailed tool and process design stage of the part development process. The main goal of this phase is to make sure that a problem-free production will be achieved and quality of the hemmed part will be within specification requirements. Typically, considerable amount of information may be available at this phase, i.e. stable CAD geometry of the product and detailed description of the drawing and forming operations. To guarantee efficient usage of available information, previously validated concept of the hemming operation can be opened in the AutoForm-HemPlanner. Tool design engineers can then append results of simulation of drawing and forming operations and can update the definition of the reference geometry to incorporate any changes introduced during the previous development stages. Advanced numerical analysis can then be used to validate the updated set-up, calculate material roll-in and evaluate occurrence of typical defects, such as for example splitting of the flange, presence of wrinkles or material overlaps in the corner areas. In this analysis, accuracy of simulation results is increased by taking into account history of deformation accumulated during all previous operations. Calculated material roll-in can become the basis for the compensation of forming tools and for any adjustments of hemming equipment which may be required to obtain high quality parts. Similarly, changes in the flange shape, tooling and process conditions can be done if the simulation detects splits or wrinkles of the material or possible collisions between for example inner part and hemming tools. Typical workflow diagram of the "advanced hemming" approach can be found in Figure 3. When hemming operation is fully verified and optimized, active surfaces of tools generated by the AutoFormHemPlanner can be exported into a CAD system to create tooling geometry of milling quality. Information which may be shared with the CAD system includes working directions, tool surfaces in mesh format, guiding curves and control profiles. In case of roll hemming, generated trajectories of rollers can be used as a basis for programming of robots, estimation of cycle time and complete simulation of the manufacturing cell. Finally, tools kinematics, calculated reaction forces and the most optimum process parameters found in the numerical analysis can be used to provide adequate support during the try-out and production phases.

3. ENHANCEMENTS OF INCREMENTAL SOLVER 3.1. Modeling of inner part, roller tool and path For the efficient simulation of hemming processes some enhancements of the standard AutoForm-Incremental solver were necessary. New types of objects, inner part and hemming roller, were introduced. In the current implementation the inner part is assumed to be rigid, which means that the outer part is the only deformable body considered in the analysis. The inner part is represented by its middle surface. It has a certain thickness, an optional offset and two contact surfaces, which are the upper and lower surfaces of the object. If the deformable outer sheet is positioned between an inner part and the hemming die, any movement tangential to the sheet surface is suppressed. Thus it is assumed that the outer sheet is always perfectly clamped, without the need for a clamping force or description of friction conditions between the inner and outer parts. Roller tool can be represented by, for example, an axisymmetric surface. The special feature of this tool is that it

FIGURE 4. Roller movement (I: pure translation, II: translation and change of orientation, III: translation, change of orientation and rotation around roller axis).

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can move along an arbitrary path and this movement may include translational and rotational parts. To describe the movement of an object along a curve in space, the orientation of the local tripod in each point of the curve is necessary. This orientation is modeled by one local direction which coincides with the axis of the roller. In addition to the translation and the change of orientation, the roller can also rotate around its own axis, see Figure 4. The roller path is discretized by a number of points which is dependent on the local curvature. Between these discrete points the translational and rotational parts of the roller movement are linearly interpolated. As a result, the time discretization of the simulation is independent of the discretization of the tool paths. It is important to note that the tangential velocity of the roller is constant throughout a process step and the contact between the roller and the deformable sheet is considered to be frictionless.

3.2. Speed-up of roll hemming simulation Roll hemming is the forming process in which the plastic deformation zone at every moment in time is small compared to the dimensions of the part. The final geometry is reached by moving the plastic forming zone through the sheet several times. This process specific feature is used to significantly speed up the simulation. Instead of solving the finite element mesh of the whole sheet, only the elements are considered that are inside of the current forming zone. These elements form the so-called sub mesh, which is at any time a subset of the whole mesh. Assembling and solving the system of equations is then done for the elements of the sub mesh only. Apparently solving the sub mesh is faster than solving the whole mesh, thus it is possible to save the computational time. For typical industrial parts, the savings were found to be between 100% and 500%. It is important to note that these savings were achieved without changing the quality of the simulation results.

4. VALIDATION OF HEMMING SOLUTION 4.1. Die hemming benchmarks A general description of some of the die hemming benchmarks used for validation of hemming solution is given in Figure 5. Flat blank with a straight flange was used in the first test. In the second test the blank had a curved outline with the radius of 257mm. Detailed description of these benchmark problems and information about materials used in the hemming experiments can be found in [19]. The complete forming process consisted of several operations, i.e. flanging the material around the die radius Rt , pre-hemming the flange to an angle of 45◦ and final hemming to make the flange parallel to the hemming die. Each benchmark was simulated following quick hemming and advanced hemming approaches. Simulations were performed with two different materials, i.e. aluminum alloy AA6016 was used in the first benchmark and bake hardening steel E220BH in the second. Table 1 provides a summary of material properties and some process parameters. Material behavior was described using BBC2005 yield function with isotropic hardening in case of quick hemming and AutoForm kinematic hardening model in case of advanced hemming. The outer blank was discretized with triangular

FIGURE 5.

Outer part outlines and schematic description of die hemming benchmarks. All dimensions are in millimeters.

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TABLE 1. Sheet metal characteristics, process and simulation parameters

Young’s modulus, GPa Poisson’s ratio Yield stress/Yield strength, MPa R0 , R45 , R90 n-value Outer/inner sheet thickness, mm Friction coefficient during pre- and final hemming Forming tool radius, Rt , [mm] Roll-in measurement height, hd , [mm] 6000

E220BH

70 0.3 102/232 0.7, 0.6, 0.65 0.252 1.2/1.0 0.15 2 3.4

210 0.3 232/369 1.65, 1.45, 1.77 0.2 0.68/0.68 0.15 0.5 2.2

20000 Experiment Simulation, advanced hemming Simulation, quick hemming Vertical force Fz, [N]

5000 Vertical force Fz, [N]

AA6016

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15000

10000

5000

1000

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final hemming

0

0 0

1

FIGURE 6.

2 3 4 5 Tool displacement, [mm]

6

7

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4

5

Benchmark #1 - vertical force during pre- and final hemming compared to experimental results reported in [19].

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25000 Average experiment Simulation, advanced hemming Simulation, quick hemming

Average experiment Simulation, advanced hemming Simulation, quick hemming

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FIGURE 7.

7

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5

6

Benchmark #2 - vertical force during pre- and final hemming.

Mindlin type of shell elements with 11 integration points through the material thickness. The level of discretization was chosen to obtain a reasonable number of elements around the forming tool radius. Contact conditions were modeled using the penalty method and Coulomb friction model. Flange height after forming, h f in Figure 5, obtained in the simulation of the first and second benchmarks was 8.4mm and 8.65mm correspondingly. Figures 6 and 7 show evolution of vertical forces during pre- and final hemming as a function of tool displacement. For comparison purposes, experimental measurements reported in [19] are also plotted in these figures. Several important observations can be made while evaluating results of simulations of both benchmarks: •

Magnitude of the vertical force calculated in the quick hemming analysis is lower than that predicted in the advanced hemming simulation. This can be explained by the fact that the quick hemming analysis is started from the undeformed CAD geometry of the flanged part.

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2

0.8

Benchmark #2

Benchmark #1 0.6 Roll-in, [mm]

Roll-in, [mm]

1.5

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FIGURE 8.

• •

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Roll-in after final hemming compared to experimental measurements reported in [19].

Difference between the quick hemming and advanced hemming forces decreases during the final hemming operation as a result of material work hardening during the previous process step. Compared to the measured forces, vertical force predicted in the advanced hemming simulations is overestimated, but its evolution corresponds well to the experimental results.

Roll-in of the material along the flange length, at the end of hemming operation, is plotted in Figure 8 for both benchmarks. It is defined as the distance d between the flanged and the hemmed parts, calculated at a certain height hd above the hemming bed, see Figure 5. Average experimental measurements including standard deviation of results are also plotted in these figures as solid bars. Simulations of the first benchmark show that the quick hemming analysis underestimates the magnitude of the material roll-in. More accurate results, although still overestimated, are obtained with the advanced hemming approach. In case of the second benchmark, both simulations predicted the roll-in evolution which is similar to reality, i.e. roll-in increases when moving from the center to the free edges of the part. The magnitude of roll-in is slightly overestimated in advanced hemming simulation, but in general results are in good correspondence with experimental measurements. Based on the analysis of these benchmarks, it is possible to conclude that the quick hemming simulation can be used to estimate the roll-in and tool forces during the early stages of part development. At the same time, the observed discrepancy of material roll-in and the mismatch of calculated and measured forces in case of advanced hemming is probably related to the rigid inner part assumption and the difference in contact conditions that are known to be very complex for die hemming operations [5].

4.2. Roll hemming of real part Finally, some results of testing of the developed hemming solution using typical automotive parts are presented here. Following the advanced hemming approach, a simulation of roll hemming of a flange of an outer part was prepared

FIGURE 9.

Prediction of flange wrinkling during pre-hemming.

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in the AutoForm-HemPlanner. The simulation set-up corresponded fully to the real process, i.e. two pre-hemming rollers with 60◦ and 30◦ orientation and the final hemming roller. Geometry of the outer part after each process step was similar to the real part surface generated by a 3D scanner. In particular, as shown in Figure 9, one of the typical hemming defects - wrinkling of the flange - was properly predicted after the simulation of the first pre-hemming step.

5. CONCLUSIONS The new software presented in this paper can be used as a reliable solution for planning, development and analysis of hemming operation in the industry. Quick and advanced hemming options make it possible to apply this tool at various stages of part development process, which are characterized by different input information and objectives. Special algorithms were developed to allow quick generation of geometry of the hemming bed, inner part, tools for pre- and final hemming and their kinematics. Several adaptations of the AutoForm-Incremental solver were necessary to make it more suitable for the numerical analysis of different hemming processes. Simulation of real part and typical die hemming benchmarks showed that the new software tool can provide adequate support to process planners and engineers and can help to design the hemming operation which satisfies quality and cost requirements.

ACKNOWLEDGMENTS AutoForm-HemPlanner was developed in cooperation with INPRO. The work of all people from INPRO who contributed to the development of HemPlanner is greatly acknowledged. Valuable support and input related to the real hemming processes received from the industrial partners is gratefully acknowledged. Their feedback was very useful to develop a user oriented and industrially applicable software. Assistance and support of AutoForm development and consulting teams are also acknowledged with great appreciation.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

M. Pietsch, and S. Wagner, “Computer aided method for an associative process layout generation,” in Proceedings of the NUMISHEET 2008, edited by P. Hora, Interlaken, Switzerland, 2008, pp. 885–892. W. Kubli, and V. Prabhakar, “Digital process planning for automotive body panels,” in Proceedings of the NUMISHEET 2008, edited by P. Hora, Interlaken, Switzerland, 2008, pp. 885–892. K. Roll, “Simulation of sheet metal forming - necessary developments in the future,” in Proceedings of the NUMISHEET 2008, edited by P. Hora, Interlaken, Switzerland, 2008, pp. 3 – 11. Y. Umehara, Journal of materials processing technology 22, 239–256 (1990). M. Sigvant, The hemming process: a numerical and experimental study, Ph.D. thesis, Chalmers University of Technology (2003). A. Müderriso˜glu, M. Murata, M. Ahmeto˜glu, G. Kinzel, and T. Altan, Journal of materials processing technology 59, 10–17 (1996). H. Livatyali, M. Müderriso˜glu, M. Ahmeto˜glu, N. Akgerman, G. Kinzel, and T. Altan, Journal of materials processing technology 98, 41–52 (2000). H. Livatyali, and S. Larris, Journal of materials processing technology 153-154, 913–919 (2004). H. Livatyali, T. Laxhuber, and T. Altan, Journal of materials processing technology 146, 20–27 (2004). E. Atzema, R. Baartman, and A. Klomp, “Finite element simulations of the hemming process,” in Proceedings of NUMIFORM 1998, edited by J. Huétink, and F. Baaijens, Enschede, The Netherlands, 1998, pp. 933–939. C. Lange, E. E. Massoni, E. Felder, V. Zwilling, and F. Debuire, “Experimental and Numerical Simulation of the Hemming Process of an Aluminum Sheet,” in Proceedings of NUMIFORM 2004, edited by S. Ghosh, J. Castro, and J. Lee, Columbus, Ohio, 2004, pp. 1001–1006. M. Svensson, and K. Mattiasson, Journal of Materials Processing Technology 128, 142–154 (2002). S. Li, X. Hu, Y. Zhao, Z. Lin, and N. Xu, Materials and Design 5, 20–27 (2010). S. Thuillier, N. Le Maoût, P. Manach, and D. Debois, Journal of Materials Processing Technology 198, 226–233 (2008). M. Liewald, and R. Schleich, Internation journal of material forming 3, 1175–1178 (2010). X. Hu, Z. Lin, S. Li, and Y. Zhao, Materials and Design 31, 1410–1416 (2010). N. Le Maoût, S. Thuillier, and P. Manach, Engineering Fracture Mechanics 76, 1202–1214 (2009). N. Le Maoût, S. Thuillier, and P. Manach, Materials and Design 31, 2725–2736 (2010). C. Lange, Etude physique et modèlisation numèrique du procèdè de sertissage de pièces de carrosserie, Ph.D. thesis, Ecole Nationale Supèrieure des Mines de Paris (2006).

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Application Limits of a Method to Predict Distortion Caused by Mechanical Joining Technologies in Car Body Construction A. Eckerta, R. Neugebauera, M. Rössingerb, M. Wahlc, F. Schulzc, A. Hofmann, B. Hecht a b

Fraunhofer IWU, Chemnitz, Germany Volkswagen AG Wolfsburg, Germany c

Audi AG Ingolstadt, Germany

Abstract. This paper deals with the determination of application limits of an already presented method [1] to predict distortion of joined car body assemblies caused by mechanical joining technologies. The investigated method is particularly developed concerning the demands coming from the application in the automotive body production. Parameter studies in simulation and experiment are presented which show the influence of pre-straining and clamping on the joints properties and dimensional accuracy of the assembly. The gained cognitions allow conclusions about the quality of prediction of dimensional accuracy using the proposed simplified model in different joining situations. Finally, the investigated method is applied to a car body assembly. Keywords: Finite Element Method, mechanical joining, simplified model, distortion, application limits, process chain

INTRODUCTION In the automotive industry reducing cars weight is the most promising solution to meet customers and governmental demands - concerning fuel consumption and emissions. Reducing weight goes along with an increased use of lightweight materials like high strength steels, aluminium and magnesium. In car body assemblies, single parts made of the same type of material as well as of different types of material need to be joined. One of the main technical challenges concerning lightweight concepts is the development of appropriate joining technologies, as former established technologies like resistance spot welding (RSW) do reach their limits. Looking at aluminum and mixed material car bodies there is a high use of mechanical joining technologies like selfpiercing riveting (SPR) and clinching. Investigations of [2, 3, 4, 7] show that mechanical joining techniques significantly cause distortion in joined assemblies and influence the dimensional accuracy and quality, customers expect (shape, gap, flush). In the past different methods to predict distortions of car body assemblies using FEM (finite element method) were presented. Industrial applicable methods have a high potential to reduce costs in the planning phase of car body tools/ devices and shorten the time between first assembly from series tools/ devices and series launch. These goals can only be achieved if applied methods are reliable and the application limits are determined. In this paper application limits of a method to predict distortion caused by mechanical joining techniques are determined.

STATE OF THE ART Different papers have been published the last years dealing with assembly distortion caused by mechanical joining technologies [2, 3, 4, 6, 7]. These investigations show the relevance of assembly distortion for car body quality. Simulative methods and practices to predict multi-station sheet panel assembly dimension were shown in [14, 15]. It needs to be mentioned that this method does not take distortion caused by the applied joining technology (welding, mechanical joining) into account. Cai et al. [4] presented the only known approach to predict local and

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 251-258 (2011); doi: 10.1063/1.3623618 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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global distortion of car body assemblies joined by SPR with tubular rivet. Using this method to predict distortion of a car body structure (door assembly) it was shown that the results are sufficient in global but insufficient in local regions of the assembly. In [1] a method, using simplified models in FEM software to predict distortion caused by SPR, was presented. This method is particularly developed to meet the demands coming from the car body production process and making process chain simulation possible. Demands coming from the planning and production process of car body assemblies are: x Consideration of effects coming from prior processes (pre-straining, reduction of blank thickness and springback/ pre-deformation of the parts to be joined) x Consideration of effective surfaces of tools for clamping and joining processes x Creation of distortion caused by different joining technologies using simplified models (without explicit simulation of the joining process) The principle of the method, considering these demands, is illustrated in Figure 1. blanks

„ blank holder“ simplified model

local joint y

die Clamping (pre-deformed blanks)

x

connector Joining

tool open (springback)

FIGURE 1. Principle of the simplified model to predict distortion caused by joining [1]

The illustrated simplified model was applied to pre-deformed and non-pre-deformed riveted specimen (steel top blank, aluminum bottom blank) with a gradual increase in complexity. It was shown that the dimensional accuracy of simple as well as complex structures can be reliably predicted in local (gap) and global regions of the assemblies. In [2] the simulation of the process chain from the single parts to the hemmed assembly, including sub-assembly processes, are performed. It is shown that the shape of the sub-assembly after joining is crucial for the final geometry of this hemmed assembly. Looking at the sub-assembly process (SPR with solid rivet) it is shown that effects coming from the stamping process after springback (stress, strain, thickness reduction) have a significant influence on the simulated joining distortion in local and global areas. The consideration of pre-straining is also suggested and proven in case of welded car body structures in [10]. After own research there are no publications in science and industry relating to the influence of pre-straining on distortion caused by mechanical joining. Scientific works like [8, 9] only investigated the influences due to the joining process, geometry of joint and especially the static, dynamic and high-dynamic structural behavior [16, 17, 18]. Furthermore in the publication of [8] appropriate methods to produce test-specimens were discussed like e.g. rolling, stretching and the Deep-Drawing Marciniak test. Especially the Marciniak test was recommended but not used due to the unavailability of adequate sizes of the test-setups. Relevant positive aspects of using the Marciniak test are [13]: x production of uniformly and plane pre-strained test-specimens x different levels of pre-deformation can be reached Target-orientated studies to the influence of clamping devices on a mechanical joint or mechanical joined car body assembly also have not been the focus of investigations in recent years. In contrast to that, in the field of welding various experimental and numerical studies [11, 12] have shown the influence of clamping on residual stresses and distortion. Especially in case of clamping in the heat affected zone [12]. In addition to that Cai [5] showed an numerical and deterministic method for fixture optimization of welded assemblies taking into consideration variations of welding guns, e.g. misalignment.

APPLICABILITY OF METHODS TO PREDICT DISTORTION OF CAR BODY ASSEMBLIES In addition to previously performed studies on the proposed simplified model, application limits need to be defined. On the one hand there are topics concerning the simplified model itself and the process of gaining input information and on the other hand there are further demands coming from the process of producing car body components.

252

x

Small local deviation: Depending on the joining technique, the used joining parameters, the parts to be joined geometry as well as the material, locally only very small distortions occur. Nevertheless these small local distortions can cause relevant global distortions. To consider small distortions in simulations these distortions need to be measureable and it needs be shown that these deviations are predictable using the proposed simplified model. x Stiffness of joints: Depending on the joining technology and the joining task, joints set next to each other can interact and influence each other [1]. Known methods using simplified models neglect this influence. It needs to be found out, if there is a change in simulative results and if these come closer to measured values of experimental joined specimen. Figure 2 shows a car body clamping device with positioned and clamped parts to be joined. Due to prior stamping processes usually the parts to be joined are pre-strained. It is a goal of this investigation to identify the interactions between joints and clamps/ pre-straining looking at changes in spot quality and distortion. For the applicability of the simplified model it is crucial to keep it as simple as possible and know the limits of the method. It is to be investigated if it’s possible to use the same simplified model in different car body relevant joining situations or if it needs to be adjusted for different joining situations. x Influence of clamps on distortion and joint quality: In welding it’s known that clamps have an influence on residual stresses and the samples distortion. The proposed simplified model is based on measurements or simulations of a specific joint with its properties and characteristic distortion of reference samples. The influence of clamps on the distortion and the quality of mechanical spot joints is investigated in experiment and simulation in this paper. x Influence of pre-straining on distortion and joint quality: Reductions in sheet thickness and prestraining in general influence the quality of the joint (behaviour under load, joint geometry). Changes in the joint quality can also have an influence on the distortion of specimen and assemblies in general. Thickness

mechanical joint clamps

FIGURE 2. Car body assembly in clamping device. Left side - pre-strained parts (thickness). Right side - clamps and joints

In the following the state of the art to these topics is described and discussed.

EXPERIMENTAL AND SIMULATIV STUDIES ON DISTORTION CAUSED BY DIFFERENT JOINING TECHNOLOGIES The aluminium (AA6181-T4, upper blank 1.5mm, lower blank 1.0mm ) sub-assembly pictured in Figure 2 is an idealized assembly which includes most of the characteristics of a typical car body assembly. It allows investigating different design variants for each process step along the process chain of car body production. In detail this assembly is described in [2]. For this assembly the joining sampling process was run through for the three joining techniques Clinching, riveting with solid rivet and riveting with semi tubular rivet. Because of the limited approachability caused by the clamping device the joining direction is given for all the technologies. The used joining parameters and the micro sections of the reference samples are shown in table 1. TABLE 1. process parameters of reference samples Joining Process SPR solid rivet Machine Rivet diameter / length blankholder diameter blankholder force/ spring stiffness die

SPR semi tubular rivet

Clinching

KerbKonus 4mm / 2.7mm 12mm

Eckold 3.3mm / 4.0mm H2 18mm

Eckold -

-/ 277N/mm

2000N/ -

1500N/ -

6.5 mm (ring diameter)

-

TOX

253

18mm

micro section

The joining techniques were applied to circular blanks with a diameter of 45mm. Circular blanks were chosen because of the identical geometrical stiffness in the different directions. The specimens (7 specimens per experiment) were measured using the optical laser triangulation technique, before and after joining. Different joining technologies cause different kinds of distortion. To compare the deviations the illustrated definition (Figure 3) is used in this paper.

out-of-plane

gap

in-plane

FIGURE 3. Definition of deviation caused by mechanical joining (example riveting with solid rivet)

x out-of-plane distortion: distortion of joined blanks in vertical direction x gap: maximum distance between upper and lower blank x in-plane-distortion: difference of maximum measured horizontal distance before and after joining The measured distortions for the three joining techniques are shown in Figure 4. It can be seen that the results differ in kind, direction and value of distortion. SPR with semi tubular rivet causes for this joining parameters and joining task the lowest out-of-plane distortion but the highest in-plane distortion. Clinched specimens have a deviation of both sheets in the direction of the punch and a negative in-plane distortion. That there is no gap in a single spot specimen looking at clinching and SPR with semi tubular rivet doesn’t mean that there is no gap in multi-spot specimen.

FIGURE 4. Comparison of distortions caused by different joining techniques

SPR with solid rivet causes by far the highest out-of plane distortion. As the values of distortion differ significantly there is a gap of approximately 0.27mm. Compared to the distortion values of a mixed material specimen (steel, aluminum) with SPR with solid rivet in [1] these measured values are very low. To get an idea about the prediction quality of the simplified model it is applied to a two-spot-specimen (rectangle 100x45mm). The preciseness of the measured values of the one-spot-specimen makes it possible to exactly calibrate the simplified model before applying it to specimens of different complexity. 0.7

Reference sample B-B

gap (in mm)

0.6

Simulation B-B

0.5

Reference sample A-A

A-A

0.4

Simulation A-A Simulation B-B stiffness

0.3

Simulation A-A stiffness

0.2

B-B

0.1 0 0

10

20

30

40

50

60

70

80

90

100

Distance

FIGURE 5. Comparison of measured and simulated distortions (gap) of a two spot specimen (SPR with solid rivet)

The measured and simulated results of section A-A and B-B show very good compliance. Due to an improved measurement technique these results even improved compared to the results presented in [1]. It can be assumed, that the joints stiffness plays the most significant role if a joint causes a high in-plane distortion. As SPR with solid rivet causes the highest in-plane distortion this joining technique and the corresponding

254

simplified model is chosen. These connections characteristics are defined by tension (1KN/mm) and shear stiffness (21KN/mm). The stiffness of the connection was determined in experiment using KS2-specimen [18]. Looking at Figure 5 it can be seen that the simulative determined gap gets reduced. For this case better results are achieved using stiff connector elements. For the further investigations stiff connector elements are used. Investigations on the modelling technique, different joining situations and different joining techniques are in progress.

INFLUENCE OF PRESTRAINING AND CLAMPING ON DISTORTION AND JOINT QUALITY Investigations concerning the influences of pre-straining and clamping on the distortion and joint quality were performed on all the three presented joining techniques. Selected results for SPR with solid rivet and clinching are presented and discussed. To compare results the following quality criteria are used: x Distortion: out-of-plane distortion is compared calculating the total of the absolute values of the deviations of upper and lower blank x Joint quality: filling of shank groove (SPR with solid rivet), undercut (Clinching), reduction of base thickness (SPR with solid rivet, Clinching) Concerning the joint quality, generated micro sections of pre-strained and clamped specimen are compared to the micro section of the reference sample.

Joining of pre-strained specimen Figure 2 shows that car body single parts are usually pre-strained due to stamping processes. In the state of the art it is shown that pre-straining of blanks has a significant influence on the joints quality. It should be emphasized that sampling processes of new joining connections are executed on simply rolled, not on pre-strained specimen. In this paper the influences of pre-straining on the joints quality (as defined above, mechanical behavior) and the caused distortion in one-spot and two-spot specimen are investigated. To get plane pre-strained specimen an up scaled Marciniak-test (punch diameter 200mm, here biaxial load) is used. One (diameter 45mm) and two spot joining specimen are cut out (lasercutting) of the plane area. The process chain of assembly simulation using pre-strained specimen is illustrated in Figure 6. Ø 45

Die

Punch

45

Blank holder

marciniak test

10

0

Plane pre-strained test specimen

Assembly simulation

FIGURE 6. Experimental setup/ process chain of assembly simulation using pre-strained specimen

As car body single parts are not uniformly pre-strained, used blanks (1.5mm, 1.0 mm) are pre-strained to 3, 5, 8.5% strain. To determine the exact values AutoGrid® was used. The maximum level (8%) causes a reduction in sheet thickness of about 16% on the thick as well as the thin sheet. For this investigation sheets with the same level of prestraining are combined (thick blank punch side, thin blank die side) and joined using riveting with solid rivet (same tools for all levels – according to real production processes). In the simulation the entire process chain is ran trough. Deep drawn parts are trimmed and joined using the simplified model approach. For the simulations with the prestrained blanks the same simplified model is applied to the non-pre-strained specimen. This matches to the industrial demand to keep the simplified model approach as simple as possible. Figure 7 shows the results for the different prestraining-levels. It can be seen that pre-straining significantly influences the distortion as well as the joint quality. Caused by the reduction of the sheet thickness (corresponding to the results described in the state of the art) there is a decrease of the joint quality compared to the reference sample.

255

measured distortion - bottom sheet

1 0.8 0.6

2

0.4

measured distortion - total

joint quality

distortion - out of plane [mm]

measured distortion - top sheet

1

1.2

simulation distortion - top sheet simulation distortion - bottom sheet simulation distortion - total

0.2 0

1

0% pre-strain

2

3

3% pre-strain

4

5% pre-strain

3

8,5% pre-strain

joint quality 1 = reference 2 = sufficient 3 = insufficient

FIGURE 7. Influence of pre-straining on distortion and joint quality/ comparison of measured and experimental results

Looking at the base thickness and the filling of the shank groove the joint quality of the maximum pre-straining level can be considered as insufficient [16]. For production processes this would go along with a change of the process parameters of the reference sample. While the joint quality gets reduced the total measured distortion increases. It can be noticed that there is a higher change in distortion of the bottom sheet (side of the die ring). Qualitatively the results of the simulation correspond to the measured values. Quantitatively there are deviations of up to -33% from the simulative to the measured results. It needs to be said that the distortion gets simulative underestimated using the simplified model. To reduce this error pre-strained specimen should be used in experiment and simulation to calibrate the simplified model.

Joining of clamped specimen Car body single parts are positioned and clamped prior to joining processes. Figure 2 shows a typical car body clamping device. It can be seen that there are joints with and without clamps located next to it. To investigate different clamping situations and determine the interactions of clamps and joints a clamping device for single spot and multi-spot specimen was build. In this paper results of a circular specimen clamped with four clamps are presented. The specimens get clinched and joined with SPR with solid rivets. Looking at Figure 8 differences in the experimental setup of clinched and riveted specimen can be seen. While clamping and joining takes place in the same plane for clinching, the parts to be joined need to pushed over the die ring, being held in position by the clamps, for SPR with solid rivet to get the groove shanks filled. Blank holder Clamping point

Joining plane = Clamping plane

Specimen Die

Joining plane

Clamping plane

Joining plane = Clamping plane

FIGURE 8. Experimental setup for clamping one-spot specimen/ differences concerning joining and clamping plane of SPR and Clinching

As Figure 9 shows the significant change of boundary conditions of the SPR process leads to a reduction of the joints quality. It can be noticed that this has a strong influence on the total distortion of the specimen. As the sheets get pushed over the die ring the measured distortion of the bottom sheet increases while the distortion of the top sheet decreases. Simulative results using the characteristic information of the reference sample are in opposite hardly influenced by the clamps. In the case of clamped, clinched specimen only slight differences can be detected comparing the clamped simulated/ measured results to the results of the simulated/ measured reference specimen. There is neither a significant change in the caused distortion nor in the joints quality. To correctly predict the distortion caused by SPR with the shown experimental setup the joining situation needs to be considered while calibrating the simplified model. It is shown that clamping processes taking place in the same plane like the joining process can be correctly simulated.

256

1

0.4 2 0.3 0.2 0.1 0

3 0

not clamped unclamped

1

clamped

0.25

1

0.2 0.15 2 0.1

joint quality

0.5

measured distortion - top sheet measured distortion - bottom sheet measured distortion - total simulation distortion - top sheet simulation distortion - bottom sheet simulation distortion - total joint quality 1 = reference section 2 = sufficient 3 = insufficient

distortion - out of plane [mm]

0.6

joint quality

distortion - out of plane [mm]

0.7

0.05 0

3 0

not clamped

1

clamped

FIGURE 9. Influence of clamping on SPR with solid rivet and clinching

Application of the simplified model to a car body assembly To underline the practical relevance of the presented results the method is applied to a car body assembly. The experimental setup is illustrated in Figure 2. The parts in the simulation are simulative deep drawn parts that are riveted (with solid rivet, reference specimen) in the car body device. The simplified model (reference sample) is implemented after clamping (clamping device and joining tongs). When the last spot is set the opening of the clamps causes another change of the assemblies’ geometry. The process and the results for global and local distortion are illustrated. The above described effects of clamping and pre-straining help to interpret the gained results. It needs to be mentioned that the clamping situation strongly differs from the above described clamping situation. Due to that, the effect on the change in distortion and joint quality will be reduced and the simulative prediction quality will be improved in this case.

Clamped sheets

Clamped/ joined sheets

Global distortion

Local distotion (gap)

FIGURE 10. Simulating local and global distortions of an CAD (including thickness reduction) part

Besides clarifying the functionality of the simplified model, simulations like this (Figure 10) can for example be used in the planning phase of car body tools and devices for optimization purposes. This makes it possible to optimize the parts construction as well as the tools and the clamping device before producing first real parts. Besides that the simulation of assembly processes offers the possibility to identify parameters with the highest significance for an optimal process result. Comparing simulative results to measured results is in progress.

CONCLUSION In this paper demands to determine application limits, of a proposed method to predict distortion caused by different mechanical joining techniques, are described. Coming from the state of the art the functionality and the results´ quality is described. Based on the existing results further demands concerning the applicability in the automotive industry are named: Æ Predicting small distortions Æ Considering the joints´ stiffness Æ Influence of clamps on distortion and joint quality Æ Influence of pre-straining It is shown that different joining technologies (Clinching, SPR) cause different kinds of distortion (in-plane, out-ofplane, gap) applying these to the same joining task (aluminum, 1.5mm/ 1mm). Compared to the deviations of a mixed material specimen in [1] these deviations are small. Using optical measurement equipment the simplified model is calibrated for the SPR with solid rivet and applied to a two spot specimen. The simulative and the measured results show very good compliance. The influence concerning joints stiffness in the simulation is also illustrated on this joining task. For this connection the simulative results of the gap get underestimated. Further investigations concerning the joints stiffness are in progress. Concerning the applicability of the influence of clamps and prestraining of the blanks is investigated. The simplified model, calibrated to the reference sample of SPR and

257

Clinching, is applied to both pre-strained and clamped specimen. Pre-strained, plane specimens are gained using a scaled up Marciniak-test. The process chain is run through in simulation and experiment. Experimental results show, that the distortion increases with increasing pre-straining while the joint quality decreases. These results are highly relevant as blanks with different pre-straining levels are joined with the same tools in car body production. Comparing measured and simulative results it can be said that the simulation (using the initial state of the simplified model – calibrated with a non-pre-strained specimen) underestimates the gap´s height. To the clamped sheet two different joining techniques are applied to show the difference of joining and clamping plane. The distortion and joint quality of clinched specimen, a process with the same level of clamping and joining plane, is hardly effected by the clamps (strongly clamped specimen – 4 clamps). In contrast to that, the joint quality and the distortion of the riveted specimen (different levels of joining and clamping plane) are strongly affected by the clamps. The total distortion and the joint quality decreases. Obviously this effect can not be shown in a simulation using the initial simplified model (calibrated with a not clamped specimen). Depending on the demanded quality of simulation results used simplified models can be recalibrated using pre-strained and clamped reference specimen, but this increases the effort. To clarify the functionality and the use of the proposed method the simplified model gets applied to a car body assembly. Local and global distortion of the assembly is shown. The prior described effects can be used to interpret the gained results. A comparison of measured and simulated results using the described assembly is published in the near future. Part of current work packages is to determine the effect of the elasticity of the robot/ joining tongs and the boundary conditions of the specimen on the final position of the tools and the distortion of the specimen/ assembly. The effects described in this paper should on the one hand be considered in the planning phase when tools and clamping devices for new assemblies are chosen. On the other hand these effects are relevant looking at the phase between first part from tool/ device and series launch when the assemblies´ quality is optimized.

REFERENCES 1. R. Neugebauer, M. Roessinger, M. Wahl, F. Schulz, A. Eckert and W. Schuetzle, “Predicting Dimensional Accuracy of Mechanically Joined Car Body Assemblies”, SheMet Conference 2011, Leuven. 2. A. Eckert, T. Altermann, M. Roessinger, M. Wahl and F. Schulz, “Beitrag zur Prozesskettensimulation im Karosseriebau“, Pam-Stamp Forum 2011, Landshut 3. R. Neugebauer, I. Israel, “Erweiterung der Anwendungsgrenzen des Vollstanznietens mit zweiteiligen Matrizen”, EFBForschungsbericht Nr. 294, 2009. 4. W. Cai, P.C. Wang and W. Yang, “Assembly dimensional prediction for self-piercing riveted aluminium panels” in International Journal of Machine Tools & Manufacture 45, 2005, pp. 695-704. 5. W. Cai, “Fixture optimization for sheet panel assembly considering welding gun variations” in Journal of Mechanical Engineering Science Vol. 222, 2006, pp 235-245. 6. X. Fan and I. Masters, “Dimensional Variation in Self-piercing Riveting”, SAE World Congress 2006, Detroit. 7. I. Masters and X. Fan, ”Predicting distortion in vehicle body assemblies”, EuroPam 2005, Potsdam. 8. O. Hahn, J.R. Kurzok, W. Voelkner and R. Mauermann, ”Einfluß von Kaltverfestigungen auf den umformtechnischen Fügeprozeß“, EFB-Forschungsbericht Nr. 127, 1999. 9. L. Han, K. W. Young, R. Hewitt, A. Chrysanthou and J. M. O’Sullivan, “The Effect of Pre-Straining on the Mechanical Behaviour of Self-Piercing Riveted Aluminium Alloy Sheet” in Advanced Materials Research Vol. 6-8, 2005, pp. 157-162. 10. M.F. Zaeh, L. Papadakis, S. Roeren, T. Hornfeck, “Use of Shell Elements for the FEM-Simulation of the Welding Process of Sheet Metal Parts” in Advanced Materials Research Vols. 6-8, 2005, S. 209-216. 11. S.R. Roeren, C. Schwenk and M. Rethmeier, “Different Approaches to Modal Clamping Conditions within a Welding Simulation”, Mathematical Modelling of Weld Phenomena Vol. 8, Verlag der Technischen Universität Graz, 2007, pp. 10931106. 12. T. Schenk, I.M. Richardson, M. Kraska and A. Ohnimus, “A study on the influence of clamping on welding distortion” in Computational Materials Science 45, 2009, pp. 999-1005. 13. Norm DIN EN ISO 12004-2; “Metallische Werkstoffe – Bleche und Bänder - Bestimmung der Grenzformänderungskurve – Teil 2: Bestimmung von Grenzformänderungskurven im Labor“. Februar 2009, Berlin: Beuth Verlag. 14. S.C. Liu and S.J. Hu, "Variation simulation for deformable sheet metal assemblies using finite element methods" in Journal of Manufactoring Science and Engineering Vol.119 (1997), pp. 368-374. 15. J. Camelio, D. Ceglarek and S.J. Hu, “Modeling Variation Propagation of Multi-Station Assembly Systems with Compliant Parts” ASME, Journal of Mechanical Design 125(4), pp. 673-681. 16. DVS 3410: Self-piercing riveting - Overview, 2007, DVS-Verlag. 17. DVS 3420: Clinching - Overview., 2002, DVS-Verlag 18. DVS 3480-1: “Testing Of Properties Of Joints - Testing Of Properties Of Mechanical And Hybrid (Mechanical/Bonded) Joints - Stiffness Analysis Of Mechanical Joints”, 2010, DVS-Verlag

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Intelligent Shimming for Deep Drawing Processes Søren Tommerup∗ , Benny Endelt† , Joachim Danckert† and Karl Brian Nielsen† ∗ †

Fibigerstræde 16, DK-9220 Aalborg, Denmark Fibigerstræde 16, DK-9220 Aalborg, Denmark

Abstract. This paper demonstrates the use of an intelligent shimming system to compensate for changes in process output due to tool wear. A new tool concept with integrated hydraulic cavities used as actuators in feedback control system is presented. By prescribing a hydraulic pressure in the individual cavities the blank-holder force distribution can be controlled during the punch stroke. By means of a sequence of numerical simulations abrasive wear is imposed to the deep drawing of a rectangular cup. The abrasive wear is modelled by changing the tool surface geometry using an algorithm based on the sliding energy density. As the tool surfaces are changed the material draw-in is significantly altered when using conventional open-loop control of the blank-holder force. A feed-back controller is presented which is capable of reducing the draw-in difference to a certain degree. Further a learning algorithm is introduced to the system, which is able to improve the response of the feed-back system significantly. Keywords: Finite Element Method, Closed loop control, iterative learning, deep drawring, tool wear PACS: 45.80.+r, 46.70.-p

INTRODUCTION The process of running-in tools is still a practical challenge in the sheet metal forming industry. Problems related to this appear e.g. when shifting tools, press or material, due to changes in lubrication, temperature, wear, etc. By using intelligent shimming systems it is possible to overcome these challenges, and this paper will demonstrate how such a system can be used to compensate for changes in the process output due to tool wear. Numerical simulations of the deep drawing of a square sink is carried out. A layered tool design with four integrated hydraulic cavities is used, where the pressure in each cavity and the total blank-holder force (BHF) can be controlled individually during the punch stroke. In this way a controlled deflection of the blank-holder plate surface is achieved similar to the conventional method of shimming. But contrary to the use of small pieces of material between the tool layers it is possible to make controlled variations to the blank-holder force distribution. A series of 30 process simulations is carried out, where the tools is subject to accelerated abrasive wear proportional to the surface energy density of the previous draw. An algorithm is presented, which relates the surface energy density from the previous draw to a translation of the surface nodes normal to the tool surface. In this way numerical removal of material can be modelled. Using the wear-updated tools the performance of the system using three different control strategies is investigated: (a) A conventional open-loop method, where the same shim configuration is used through the punch stroke d p . The shim-configuration is prescribed relative to d p as a set of process parameter curves (wi , d p ) for each parameter i. This corresponds to the conventional method of adjusting the process and leave the configuration untouched afterwards. (b) A closed loop method, where the shim configuration is changed during the draw. These adjustments are carried out by making adjustments to the cavity pressures and the BHF 40 times during the punch stroke. The adjustments are applied as corrections ui to the parameter curves (wi , d p ) used in method (a). The feed-back framework used follows the implementation used by Endelt et al [1]. (c) A learning algorithm, where the parameter curves (wi , d p ) used by the closed loop method (b), are updated from draw j to draw j + 1. The update of the parameter curves are carried out using an linear Iterative Learning Control strategy following the implementation by Endelt et al [2]. The numerical results show that the process drift using method (a) can be suppressed to a certain extent using method (b), but a significant improvement can be achieved using an iterative learning control approach (c).

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 259-266 (2011); doi: 10.1063/1.3623619 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

259

TABLE 1. Limits for the process input parameters Parameter upper limit lower limit BHF 375 Ton 5 ton Cavity 1 600 bar 0 bar Cavity 2 600 bar 0 bar Cavity 3 600 bar 0 bar Cavity 4 600 bar 0 bar

Actuator system A schematic drawing of the actuator principle is seen in Fig. 1 and the dimensions of the tool is seen in Fig. 2. It consists of a classical stacked tool design with four embedded cavities at each long side of the drawn part. By adjusting the pressure in the four cavities by means of a hydraulic fluid and the total BHF the local intensity of the BHF can be controlled during the punch stroke. By increasing the cavity pressure the BHF intensity is increased in the area below the cavity as seen in Fig. 1 (b).

(a)

(b)

FIGURE 1. A schematic drawing of the shimming system. By controlling the pressure in a number of cavities embedded in a stacked tool, a controlled deflection of the tool surfaces can be obtained. The pressure in each cavity and the total BHF can be controlled independently.

FIGURE 2.

Dimensions (mm) of the deep drawing tools.

Boundaries on the cavity pressures and the total BHF and the dimensions of the tool is taken from a physical experimental setup, and are listed in table 1. The experimental setup is implemented at Dept. of Mechanical and Manufacturing Engineering, Aalborg University. Further the forces exerted by the cavities must not exceed the total BHF to avoid opening of the tools which would lead to hydraulic fluid between the tool layers.

260

Control system A schematic drawing of two control system loops is seen in Fig. 3. The inner loop is a classical 2.-order discrete state-space feedback system for reducing the error for the current part j. The outer loop is an Iterative Learning Controller (ILC) which transfers the information of from drawing part j to part j + 1.

(a)

(b)

FIGURE 3. The implemented control system. (a) The drawing of the current part j is controlled by an online 2.-order discrete feedback controller. The knowledge from drawing part j is transfered to drawing of part j + 1 by an iterative learning algorithm. (b) The flange draw-in relative to the punch displacement is used as state variables. The draw-in is measured at four locations, at the middle of the long sides of the rectangular part. The draw-in is measured perpendicular to the long sides.

Inner Control Loop The radial flange draw-in measured at the center of the long sides is used as state variables d j . The positions are seen in Fig. 3 (b). The flange draw-in gives a good indication of the strain behaviour in the drawn geometry during the punch stroke, and by measuring directly on the deformation any influences from changes in material properties and friction behaviour is avoided. The feedback system follows the implementation by Endelt et al. [2] The feedback control error is defined as the difference between the current draw-in d j and a reference draw-in drej f e j = d j − drej f

(1)

To increase the robustness of the feedback system an non-linear scaling of the entries in the error vector is introduced   e2 |e| |e| e+ e∗ = (2) e e α At each of the 40 sample points i during the punch stroke the five plant inputs p j (four cavity pressures and the total BHF) are updated as: p j,i = w j,i + u(i−1) + Ke∗i   

(3)

ui

The feedback controller is of 2.-order thus the rightmost part of equation 3 is applied as ⎤ ⎡ e1,i ⎤ ⎡ ⎢e1,i−1 ⎥ ⎡ ⎤ k1 k2 0 0 0 0 0 0 ⎥ ⎢ Δpcav 1 ⎢ e2,i ⎥ ⎥ ⎢ 0 k1 k2 0 0 0 0 ⎢Δpcav 2 ⎥ ⎢ 0 ⎢ ⎥ ⎢e2,i−1 ⎥ ⎢ ⎥ ⎥ ⎥⎢ 0 0 0 0 k1 k2 0 0 ⎢Δpcav 3 ⎥ = ⎢ ⎥ ⎢ e3,i ⎥ ⎥ ⎣Δp ⎦ ⎢ ⎦ ⎣ 0 0 0 0 0 0 k k 1 2 ⎥ ⎢e cav 4 ⎢ kbh,1 kbh,2 kbh,1 kbh,2 kbh,1 kbh,2 kbh,1 kbh,2 3,i−1 ⎥ ΔFBH i ⎣ e ⎦ 4 4 4 4 4 4 4 4 4,i e4,i−1

(4)

Table 2 shows the gain factors k and φ used in this paper. The identification procedure follows the method used by Endelt et al. [2].

261

TABLE 2. Feedback constants for the on-line feedback controller k1 k2 kbh,1 kbh,2 φ1 φ2 φbh,1 5.90 -3.94 37.52 -17.80 0.77 -0.24 5.66

φbh,2 -2.84

Inner Control Loop A linear learning algorithm is used to transfer the knowledge from a draw j to the following j + 1. The algorithm updates the cavity pressure curves w j+1 using the ILC-gain factors seen in table 2 as 

w1,j+1

w2,j+1

w3,j+1

  w4,j+1 = w1,j

w2,j

 w4,j + GE j

w3,j

(5)

where the gain matrix G and the current error matrix Ej are defined as ⎡

φ1 ⎢φ1 + φ2 ⎢ . G=⎢ ⎢ .. ⎣φ + φ 1

2

φ1 + φ2

0 φ1 .. . φ1 + φ 2 φ1 + φ 2

··· ··· .. . φ1 + φ2 φ1 + φ2

⎤ 0 0⎥ .. ⎥ ⎥ .⎥ 0⎦

0 0 .. . φ1 φ1 + φ2

⎡

and

e11 ⎢ e21 ⎢ Ecav, j = ⎢ . ⎣ ..

e12 e22 .. .

e13 e23 .. .

⎤ e14 e24 ⎥ ⎥ .. ⎥ . ⎦

em1

em2

em3

em4

φ1

(6)

and the total blank-holder force is updated as ⎡

φbh,1 ⎢φbh,1 + φbh,2 ⎢ .. wbh,j+1 = wbh,j + ⎢ . ⎢ ⎣φ +φ bh,1

bh,2

φbh,1 + φbh,2

⎤⎡

⎤ ∑ ei,1 ⎥⎢ i ⎥ ⎢ ∑i ei,2 ⎥ ⎥⎢ . ⎥ ⎥⎣ . ⎥ . ⎦ 0 ⎦ ∑i ei,m

φbh,1 .. .

··· ··· .. .

0 0 .. .

0 0 .. .

φbh,1 + φbh,2 φbh,1 + φbh,2

φbh,1 + φbh,2 φbh,1 + φbh,2

φbh,1 φbh,1 + φbh,2

φbh,1

0

(7)

Wear Model Wear can be divided into several sub phenomena, and only abrasive wear is treated in this paper. Abrasive wear is the gradual removal of material due to the relative motion of two bodies in contact with each other. Typical tool wear problems seen in the industry stem from adhesion which will lead to very fast instant tool failure. These problems will call for immediate tool repair and production stops, and can not be avoided using the system presented in this paper. A classical method for wear analysis is the Archard (or Rabinowicz) wear equation [3]. The equation states that the volume of material removed by wear W is proportional to the product of the contact load P and the sliding distance L but inversely proportional to the surface hardness H of the wearing material. δ is a material constant. W =δ

PL H

(8)

Following the steps presented by Hoffman et al [4] the equation can be transformed to a rate form, which can be used for explicit finite element codes, where p is the surface pressure and v is the relative velocity of the two bodies in contact. pv H The wear work on a unit area is then obtained by integrating equation 9 by time w˙ = δ

Z=

 t

δ

pv dt = PL H

Using equation 10 equation 8 can be written as W=

262

δ Z H

(9)

(10)

(11)

LS-DYNA® offers several different contact algorithms, and the numerical expriments carried out in this paper uses the penalty based contact algorithm. At a fundamental level, contact occurs when a node from one body penetrates into the surface of another body. Two forces are then applied to the penetrating node: (1) a force normal to the surface of the penetrated body FN , and (2) a shear force FS parallel to the surface of the penetrated body. The friction model used in LS-DYNA® is based on a Coulomb formulation, and the frictional force is calculated and applied to the slave node at every timestep based on the relation: FS = μFN

(12)

The magnitude of the normal force FN is calculated using a penalty based method based on the penetration distance x using the relationship αKbulk A2 x (13) V where α is a scale factor, Kbulk is the material bulk modulus, A is the area of the penetrated segment and V is the volume of the element being penetrated. The energy associated with a contact Econtact can be extracted from a simulation for each surface node in a contact interface. The work W in Equation 11 is included in the contact energy which is calculated per node by integration of two components. FN =

WN = Econtact =



FN dx +



FS v dt

(14)

As the penetration distance x in general is very small, the contribution from FN in equation 14 is very small compared to the contribution by FS [5]. The rightmost part of equation 14 corresponds to equation 10 besides from the scaling factor δ /H. Thus the wear volume can be estimated using Econtact and multiplying it with a certain scale factor depending on the materials at hand.

Simulation of wear The removal of material is modelled using the surface energy density extracted from LS-DYNA® . Surface energy densities are extracted for each surface node after each simulation run Calculations are carried out in a series of steps. As seen on Fig. 4 a surface node can be connected to up to four elements, as eight node quadrilaterals are used for meshing the blankholder-plate and the draw die. For each node, a surface normal is calculated as the mean of the normals of the elements to which the node is connected.

FIGURE 4. A node on the surface of the tools can be connected to up to four different elements. The node of interest is the center node. For each of the surface elements, a surface normal can be calculated from its four nodes. For the each surface node, a surface normal can be calculated from the mean of the normals of the connected elements.

nmean =

1 N ∑ nN N i=1

, N ∈ 1, 2, 3, 4

(15)

The surface nodes are then displaced by a distance calculated by the product of the surface energy density at the node Esed , the surface normal at the node nnode and a scale factor δ corresponding to equation 9. dnode = −δ Esed nmean

263

(16)

TABLE 3.

Blank material properties used in the numerical analysis.

Rolling dir. 0◦ 45◦ 90◦

Thickness (mm) 1.00 1.00 1.00

R 2.00 1.75 1.90

n 0.25 0.25 0.25

K (MPa) 550 550 550

The updated position xupd for next simulation iteration is then found by adding the surface node displacement to the current nodal position xcur . xupd = xcur + dnode

(17)

1: Perform a numerical process simulation using LS-DYNA® . 2: Increment iteration counter i 3: if i ≤ M then 4: Extract Esed for each surface node 5: for all nodes on surface do 6: Calculate the surface normal at the current node: 7: 8: 9: 10: 11: 12: 13:

nnode = N1 ∑N i=1 nN where N ∈ {1, 2, 3, 4} Calculate the node wear dnode = −δ Esed nnode Update the node coordinates: xupd = xcur + dnode end for Extract drawin to file GOTO 1 end if End

FIGURE 5. The algorithm used for updating the tool geometry to simulate surface wear.

RESULTS The numerical simulations was carried out using LS-DYNA® . The tools were modelled as elastic having material properties corresponding to steel. The blank was modelled as a low carbon steel. Blank material properties are seen in table 3. The surfaces of the tools after the 50 simulations without feedback control is seen in Fig 6, and a symmetric wear pattern is observed. The largest surface node displacements are observed in the draw-die corners and directly below the pressure cavities, as one should expect. The contact interfaces between blank and drawdie and the blank and blank-holder plate were subject to wear. The friction was modelled as Coulomb model using a constant coefficient of friction of μ = 0.1. The coefficient of friction was set to δ = 2.5 · 10−4 which is a compromise to achieve a sufficiently high wear progression rate without promoting severe pitting of the surfaces. A draw depth of 80 mm is used and the feedback loop samples each 2 mm punch travel.

No Feedback A significant difference in the achieved flange draw-in is seen when comparing the reference draw-in using unchanged tools to the draw-ins using the tools after application of wear. The draw-in error is seen in Fig. 7. Due to the wear of especially the blank-holder plate, the draw-in is now increased compared to the reference. Since a symmetric tool is used the draw-in curves of the corresponding sides match each other.

264

FIGURE 6. The surfaces of the tools after 50 draws without feedback control. The left figure shows the drawring and the blankholder plate is seen on the right. The fringe shows the distance between the initial and final position of the surface nodes.

70

Error [mm]

4 3

Draw-in 1 and 3

2

Draw-in 2 and 4

1 0 -1

600 500

60 400

50 40

300

30

Blank-holder force

200

20 10

100

Pressure Cavity 1,2,3 and 4

0 0

10

20

30

40

50

60

70

80

Cavity pressures [bar]

80

5

Blank-holder force [Ton]

6

0 0

10

20

Punch stroke [mm]

30

40

50

60

70

80

Punch stroke [mm]

.

FIGURE 7. The draw-in error using no feedback for the process parameters. A constant total BHF of 25 Ton is applied, and all four cavity pressures are kept at 100 bar.

Feedback The draw-in error is suppressed by using on-line feedback control of the process parameters as seen in Fig. 8. The positive error means that material is drawn into the die cavity at to high rate, and thus a higher BHF is applied as are higher cavity pressures on the short sides. The controller resulting error at the end of the punch stroke is below 1 mm, but is as high as 1.75 mm during the punch stroke. The controller response shows an increase in the total BHF as well as the pressures in the long side cavities. 80 Draw-in 1 and 3

Blank-holder force [Ton]

Error [mm]

2 1 0

Draw-in 2 and 4

-1 -2 -3

600

70

500

60

Blank-holder force

400

50 40

300

Pressure Cavity 1 and 3

30

200

20 10

100

Pressure Cavity 2 and 4

0 0

10

20

30

40

50

60

70

80

Punch stroke [mm]

Cavity pressures [bar]

3

0 0

10

20

30

40

50

Punch stroke [mm]

60

70

80

.

FIGURE 8. The draw-in error using feedback control of the process parameters. The process parameters are seen on the right.

265

Feedback updated using ILC A sequence of 30 simulations was carried out using the iterative learning controller to update the initial process parameters between the simulations. As seen in Fig 9 after 30 iterations the draw-in error is reduced significantly compared to using direct feedback control without update of the initial process parameter curves. The evolution of the initial process parameters during the 30 learning iterations is seen in Fig. 10. 80

2

Blank-holder force [Ton]

Error [mm]

3 Draw-in 2 and 4

1 0 -1

Draw-in 1 and 3

-2

600

70

Blank-holder force

500

60 400

50 40

300

Pressure Cavity 1 and 3

30

200

20 10

100

Pressure Cavity 2 and 4

0 0

10

20

30

40

50

60

70

80

Cavity pressures [bar]

4

0 0

10

Punch stroke [mm]

20

30

40

50

60

70

80

Punch stroke [mm]

.

FIGURE 9. The draw-in error using Feedback control with ILC-updates of the initial controller values is seen on the left. As seen the draw-in is kept in control during the punch motion, and only minor differences is seen at the end of the punch stroke. The process parameters are seen on the right. BHF [Ton] 70 60 50 40 30 20 10 0 10 20

30 40 50 Punch stroke [mm]

0 5 10 15 20 60

(a)

70

25 80 30

Iteration

Pressure [bar] 400 350 300 250 200 150 100 50 0 0 10 20

Pressure [bar] 250 200

30 40 50 Punch stroke [mm]

60

70

150 0 5 100 10 50 15 20 0 0 25 Iteration 80 30

(b)

10

20 30 40 50 Punch stroke [mm]

60

70

0 5 10 15 20 25 Iteration 80 30

(c)

FIGURE 10. The evolution of the five process parameters: (a) The total blank-holderforce. The plateau in the blank-holder force from iteration 3 to 15 at the end of the punch stroke is due to the controller algorithm prescribing a blank-holder force below the allowed range seen in table 1. (b) The pressure in cavities 1 and 3. (c) the pressure in cavities 2 and 4.

CONCLUSIONS A wear algorithm has been developed giving results in qualitatively correspondence with wear patterns described by Jensen et al. [6]. The wear algorithm was applied to the deep drawing of a rectangular box to update the tool surfaces in order to simulated the effects of severe abrasive wear. By applying the conventional open-loop control method significant drift in the process output is observed. This drift is suppressed by applying a feedback control of the pressure in four hydraulic cavities embedded in a new tool design. Although the drift is negligible in the final stages of the deep drawing, rather large differences are seen during the punch stroke.

REFERENCES 1. B. Endelt, K. B. Nielsen, and J. Danckert, Journal of Materials Processing Technology 177, 426–429 (2006). 2. B. Endelt, and J. Danckert, International Journal of Material Forming 3, 25–28 (2010), ISSN 1960-6206, 10.1007/s12289-0100698-z. 3. J. A. Williams, Wear 225-229, 1–17 (1999). 4. H. Hoffmann, C. Hwang, and K. Ersoyl, CIRP Annals - Manufacturing Technology 54, 217–220 (2005). 5. J. D. Reid, and N. R. Hiser, International Journal of Crashworthiness 9, 65–72 (2004). 6. M. R. Jensen, F. F. Damborg, K. B. Nielsen, and J. Danckert, Journal of Materials Processing Technology 83, 98 – 105 (1998), ISSN 0924-0136.

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.

Crash Simulation of Roll Formed Parts by Damage Modelling Taking Into Account Preforming Effects Edwin T. Tilla*, Benjamin Hackla, Hermann Schauera a

*

voestalpine Stahl GmbH, Voestalpine Str. 3, Linz, A-4020, Austria

Corresponding author: Tel. +4350304159919, [email protected]

Abstract. Complex phase steels of strength levels up to 1200 MPa are suitable to roll forming. These may be applied in automotive structures for enhancing the crashworthiness, e. g. as stiffeners in doors. Even though the strain hardening of the material is low there is considerable bending formability. However ductility decreases with the strength level. Higher strength requires more focus to the structural integrity of the part during the process planning stage and with respect to the crash behavior. Nowadays numerical simulation is used as a process design tool for roll-forming in a production environment. The assessment of the stability of a roll forming process is quite challenging for AHSS grades. There are two objectives of the present work. First to provide a reliable assessment tool to the roll forming analyst for failure prediction. Second to establish simulation procedures in order to predict the part’s behavior in crash applications taking into account damage and failure. Today adequate ductile fracture models are available which can be used in forming and crash applications. These continuum models are based on failure strain curves or surfaces which depend on the stress triaxiality (e. g. Crach or GISSMO) and may additionally include the Lode angle (extended Mohr Coulomb or extended GISSMO model). A challenging task is to obtain the respective failure strain curves. In the paper the procedure is described in detail how these failure strain curves are obtained using small scale tests within voestalpine Stahl, notch tensile–, bulge and shear tests. It is shown that capturing the surface strains is not sufficient for obtaining reliable material failure parameters. The simulation tool for roll-forming at the site of voestalpine Krems is Copra® FEA RF, which is a 3D continuum finite element solver based on MSC.Marc. The simulation environment for crash applications is LSDYNA. Shell elements are used for this type of analyses. A major task is to provide results of the roll forming simulation as initial conditions for the crash model, taking over the shell thickness, the variation of the plastic strain and the damage parameter over the profile. This is realized by a python [13] interface program. Profiles are manufactured by the roll forming facility in Krems with a complexphase steel grade of 980 MPa strength. The final samples are manufactured using the profiled parts with cover plates fixed to them by spotwelds. Axial crash experiments are carried out using the inhouse horizontal crash test facility. It is observed that the component shows good folding behavior with some minor failure sites at edges where there is extensive forming during roll-forming. Simulation runs are made with LS-DYNA using the GISSMO damage model. The results match reasonably well with the experimental results. The simulation tool seems to be useful in order to assess not only the integrity of the roll-forming process but also to adequately predict the crash behavior of roll-formed components. Some suggestions are made in order to improve the simulation of the evolution of damage after initiation in the future. Keywords: AHSS, Roll forming, crash simulation, damage modelling. PACS: 62.20.M-; 62.20.mt ; 02.70.Dh; 81.20.Hy;

INTRODUCTION Two of the major trends in the automotive industry are the application of advanced high strength steels (AHSS) for structural components and the use of numerical simulation tools. The first trend is driven by the increased light weight design accompanied by the challenging CO2 emission reduction requirements. Additionally the standards and requirements for crash safety are challenging. The second trend is driven by the demanding time and cost saving requirements in the development cycles. Whereas computer simulation will never completely replace full vehicle crash tests they can be significantly reduced. The enhanced usage of simulation is accompanied by the need of

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 267-274 (2011); doi: 10.1063/1.3623620 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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improving the accuracy of the models. The need has even emerged to map results of the forming processes (thinning, strain hardening and even now damage fields) onto the most relevant structural components of the crash model. Advanced high strength steels have been applied in the automotive industry since the 1990s. Dual phase steels (DP) consist of two major components, the ferrite and martensite. The first phase contributes to the exceptional strain hardening and thus drawing capabilities. The second phase determines the strength level. For this steel grade special attention has to be paid to the following characteristics, namely the limited ductility (areas of high local strain, e. g. bending) and stretch-flangeability (deformation of the material along the sheared edges). Complex-phase (CP) steel grades are composite materials with – compared to DP steels - additional phases like bainite and tempered martensite. These steel grades complement the DP steels in that they have excellent bending forming and stretch – flangeability as well as high yield strength. These steels however have less hardening than the DP steels. In Asia these steels are also known as DP steels with high yield. These steel grades are very suitable to rollforming. voestalpine Stahl produces such steel grades for strength levels of 780, 980 and 1180 MPa [1]. In order to assess the structural integrity of roll formed parts in the design stage as well as to improve the accuracy of the crash simulation, damage models may be applied. These are the two major objectives of this paper. CrachFEM’s model [2, 3] is one of the more advanced damage models. It has been developed by MATFEM along with BMW. There are three different failure modes which are accounted for in the FEM simulation: tearing due to exceeding the forming limit using shell elements, ductile and shear failure. Tearing is relevant in sheet forming processes like deep drawing only. It has been observed that the dominant failure mode in AHSS steel sheets in crash is a shear type failure. This means that subsequent to the initiation of microvoids shearing takes places between these voids [4]. In the following section a method is shown how the ductile shear failure curve is calibrated by the results of small scale tests. The basis of a damage model is a scalar function. D (damage parameter) depends on the (equivalent) plastic strain, (Hp), and the triaxiality (K) of the stress tensor. K = –p / Veq. Veq is the equivalent stress, p is the first stress invariant. Here CrachFEM’s model applies as well as the GISSMO model [5]. In more comprehensive damage models like the extended GISSMO model [6] or the modified Mohr Coulomb Model (MMC, [7, 8]) the Lode angle, which is an expression of the third stress invariant, is another independent variable for the damage. For each time increment, 't, the incremental damage, 'D = 'Hp / Hs*, is calculated and accumulated, D = 6 'D. Failure is initiated when D becomes 1. Hs* is the failure strain curve and depends on Kand is assumed to be characteristic to the material investigated. No coupling of damage D and the constitutive equations are assumed until failure is initiated. A “fade out” function can be specified in the GISSMO model as it is implemented in LSDYNA [9] with the onset before D = 1. This means that the stress gradually decreases to zero with further straining. Also the number of integration points may be specified with D = 1 before the element is finally deleted. In the implementation of CrachFEM’s model in LS-DYNA as well as Abaqus [10] the fade out is realized by a specified plastic deformation of the element after failure initiation. In the present paper a conservative approach has been taken by assuming that there is no fade out. The element is deleted instantly when D for the first integration point of the shell element becomes 1.

CALIBRATION OF CRACHFEM’S FAILURE STRAIN CURVE In this paper CrachFEM’s approach is assumed to predict failure of the material, whose failure strain in the shear failure mode is given in (1). This model is applicable to thin sheets when the plane stress assumption, min (|1|, |2|, |3|) = 0, is valid. This means that one of the principal stresses is zero. Its calibration is described in detail below. Failure - in the shear failure mode - of CrachFEM is described by the following damage surface in strain space,

H  sinh  f 4  4   H  sinh  f 4   4

H S*

 

sinh f 4   4 

where the triaxiality  = -p / V eq

Mises

, the shear ratio  = (1 - k)

p = -1/3(1+2+3), 2max = 1- 3 and 2( V eq



V eqMises / max,

,

(1)

+ = 2(1 + 1 / 3 |k|), - = 2(1 – 2 /3|k|),

Mises 2

+

) = (1-3)2 + (1- 2)2 + (2- 3)2. The only calibration parameters

-

are k, f,  ,  . To calibrate this model the following tests are performed: notched tensile tests (uniaxial tension (R2) and plane strain (R5) sample) and the bulge test (biaxial tension). Shear tests were also carried out but for the complex phase steel grade the optical strain measurement system (Aramis [11]) presently is not able to deliver appropriate measurements at the time of failure. In Figure 1 the measurements as well as the finally calibrated damage surfaces are given. It is common to provide the failure surface as a function of the triaxiality  although it is

268

more simple to do the calibration in the shear ratio  space. In principle the calibration could be done by numerical procedures like the least squares method but due to the lack of a fourth measurement set the fit is carried out by common sense. The finally identified strain failure curve is shown in Figure 1 together with the calibration parameters.

FIGURE 1. CrachFEM shear failure curve for HCT980C. -space left, -space right.

How are the appropriate data obtained? In the following different methods are described to determine the fracture strain S*. One possibility is to use optical measurement systems, like Aramis, to measure the surface strain. Due to the plane stress assumption one tends to assume that the surface value represents the strain through the thickness of the sheet. Another method is to measure the thickness, at the thinnest point, of the cracked sample. Knowing the stress state, respectively the strain ratio, it is possible to calculate an averaged fracture strain. When the strain is homogeneous with respect to the thickness of the material, the thickness measurement should provide the exact fracture strain. If not than it should provide more accurate results than the surface measurements. To compare both methods 3D FEM simulation for the HCT980C notched tensile tests are carried out, using Abaqus/Standard [10]. According to the CRACH failure model, the time at failure is identified and the real fracture strain is compared to the surface and the thickness measurement. The results are shown in Figure 2.

FIGURE 2. Surface vs. thickness vs. true fracture strain for plane strain (R5, left) and uniaxial strain (R2, right) notched tensile test for HCT980C, initial thickness of the sheet: 1.6 mm.

It can be seen that the surface measurement is substantially less than the real fracture strain (middle), whereas the thickness measurement is closer to it. In Figure 3 the distribution of equivalent strain is displayed for these two

269

simulated samples. It can easily be seen that the strain is not homogeneous through the thickness, which explains the results shown above. Thus the preferred method for obtaining the fracture strain is thickness measurement instead of surface measurement. Note that for shear tests the strain is almost homogeneous with respect to the thickness and therefore surface measurements should in principle provide good results. Figure 3 shows plots of the variation of the equivalent plastic strain. For the plane strain notched tensile test failure indeed starts in the middle of the sample width with strain ratio almost exactly plane strain. This is different to the notched specimen for the uniaxial tension test. The specimen has an exact uniaxial tension state at the edge of the notch and usually it is assumed that failure starts there. The FEM simulation shows that failure starts somewhat inside the material and not at the edge of the notch. The strain ratio is not known in advance but can just be provided from FEM simulations. Here for convenience the theoretical strain ratio ( = 1/3) is adopted.

FIGURE 3. eqp for plane strain (left) & uniaxial tension (right) sample and steel grade HCT980C.

ROLL FORMING SIMULATION AND DAMAGE EVALUATION voestalpine Krems, a company manufacturing roll formed products within the voestalpine group, has been applying simulation software to their tool and roll form process design for some years. They use COPRA® FEA RF [12], with MSC.Marc as finite element solver which uses an implicit approach. The element type used is a 3d volume element with good bending performance. Friction between rolls and sheet is also taken into account. With this software the roll forming process of two hat profiles with differing radii, 4 and 1 mm, is simulated, with one layer of the 3d volume element through the thickness. The profiles are displayed in Figure 4. The static true stress – plastic strain curve for HCT980C is seen in Figure 6.

FIGURE 4. Hat profiles simulated and manufactured by voestalpine Krems, left H36042, right H36043, steel grade HCT980C.

A challenging task is to use the roll forming simulation results and prepare the necessary (scalar) initial fields, namely the equivalent plastic strain and the damage of the profile for the subsequent crash simulation with shell elements. Also the thinning has to be taken into account. The following procedure is used to accomplish this task which has been realized by an interface python program after a representative 3d model with one element in the length direction of the profile has been saved in the format of the target code (here LS-DYNA):

270

- read in the geometrical data and compute the thickness of each element; - read in the complete history of the principal stresses and principal plastic strains in the nodes; - compute the plastic strain as well as the damage variables for the shell integration points (5 equidistant locations through the thickness) for each shell element; this is done by interpolation from the 3d nodal values; - write out node and element data as well as the initial fields for the shell element model with the specified length. At this stage the damage result may be evaluated for the roll forming processes. Figure 5 shows the equivalent plastic strains of the end of the forming with respect to the failure strains of the material in the stress triaxiality space. As expected, the profile with the small radius exhibits larger strains than the profile with the small radius. As the damage evaluation of roll forming processes will become more important in the future – with the increased use of advanced high strength steels – post processing features should be developed in order to assess more easily the likeliness of the occurrence of failures in the process but also make sure that enough deformation capability exists for a subsequent crash. 2.0

2.0

HCT980C 1.6mm H36043

HCT980C 1.6mm; H36042 1.5

failure strain HS*[]

failure strain HS*[]

1.5

1.0

0.5

1.0

0.5

0.0

0.0 -0.6

-0.4

-0.2

0.0

0.2

0.4

-0.6

0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

stress triaxiality K []

stress triaxiality K []

FIGURE 5. Damage assessment of roll forming process, left H36042 rmin = 4 mm, right H36043, rmin = 1 mm.

CRASH SIMULATION voestalpine Krems, besides delivering the roll forming simulation results, also manufactured hat profiles of the two different geometries. These profiles are then cut in length (360 mm) and cover plates are attached to the hat profiles by spot welds. The grade for these plates is H340LAD which is a micro alloyed steel and the thickness is 1.5 mm. The assembled components are then axially crashed by voestalpine’s inhouse horizontal crash simulator. The mass of the impactor is 283 kg and its initial velocity is 30 km/h. 1200

H340LAD_1.5mm

1400

1000

0 /s 1 /s 20 /s 200 /s

1000 800

HCT980_1.6mm

stress [MPa]

stress [MPa]

1200

600

800 0 /s 1 /s 20 /s 190 /s 493 /s

600 400

400

200

200 0 0.0

0.2

0.4

0.6

0.8

1.0

0 0.0

strain []

0.2

0.4

0.6

0.8

1.0

strain []

FIGURE 6. Strain rate yield curves for the the grades HCT980C and H340LAD. The model for the crash simulation is built using the preprocessor of LS-DYNA [9], using the input data from the interface program as described above. The impactor and its counterpart are modeled by rigid shell elements. The spot weld utility and the single contact feature are applied. The strain rate dependent yield curves used for the

271

respective shell parts are displayed in Figure 6. Isotropic hardening, material model 24 is specified. The length of an element in the longitudinal direction of the component is 3mm (see Figure 7 left). The damage model is applied using the GISSMO feature available in LS-DYNA version R5 of 971 in the *MAT_ADD_EROSION block. Here the failure curve (see Figure 1) is specified. As has already been mentioned in the introduction section the fade out utility is not used in the present work. This means that the stress loss is instant when D becomes one in any of the shell integration points. Also it is assumed that the element is deleted when the first shell integration point of the respective element is damaged up to D = 1. This is a conservative approach.

FIGURE 7. Model (left), deformed model after the impact (middle), damaged model in undeformed configuration and fotograph of a crashed sample (H36042 rmin = 4mm). 500

13_29.9km/h LS-DYNA w damage 30km/h LS-DYNA wo damage 30km/h

Force [kN]

400

300

200

100

0 0

20

40

60

80

100

120

displacement [mm] FIGURE 8. Force displacement curves for component with H36042 profile (rmin = 4 mm).

Both simulation and experimental results are shown in Figures 7 and 8 for the component with the 4 mm radius. First of all it can be seen that the global behavior of the crash sample match reasonably well. The force displacement curve captures the overall behavior of a crash component with the peak force, the average force throughout the crash event and the final displacement after springback. The influence of the damage included in the crash simulation with respect to the overall response, e. g. the final displacement, is small, in the order of 6 or 7 %, see Figure 8. This is due to the fact that there are small cracks only. This is reported in the documentation of the crash experiments. The simulation shows a few elements that have been deleted during the crash event, see Fig. 7, third picture from the left. No objections can be made here as to the structural integrity of the examined crashed component.

272

Some of the results of the component with the small radius are shown in Fig. 9. There are some more cracks, but still small ones, in comparison to the former sample. But still the overall structural integrity of the component is intact even for this sample with severe bending due to the roll forming (minimum radius 1 mm for 1.6 mm thick sheet!). It can be observed that there are some extensive edge cracks in the simulation (see first two pictures from the left in Fig. 9). This is due to the conservative approach taken as stated before. A closer look reveals that there is a substantial influence of the effect of preforming, when the second and third picture in Fig. 9 is compared.

FIGURE 9. Deformed model after the impact (left), damaged model in undeformed configuration with and without taking into account the predeformation effects, and fotographs of two crashed samples (H36043 rmin = 1 mm).

SUMMARY AND CONCLUSION In the present paper a modeling approach is outlined for crash simulation of roll formed components. The GISSMO damage model is used with LS-DYNA and a shell model. The failure strain curve is obtained using the proposed functions in the CrachFEM ductile shear damage model by using the results of a set of small scale tests. It is pointed out that failure strains should be obtained using thickness measurements. Surface strain measurements may lead to failure strains which severely underestimate the real failure strain. The results of the roll forming simulation are used as initial fields for the crash simulation, namely the thinning, the strain hardening (equivalent plastic strains) and the damage parameter. The results show reasonably good correlation between experimental and simulation results. It has to be observed that the damage model assumptions taken here are conservative with a tendency of overpredicting the damage effects of the investigated samples. The “fade out” feature in the GISSMO model as implemented in the LS-DYNA code is not applied here. This feature along with a more realistic assumption of deleting the elements after a substantial portion of integration points have reached the ultimate damage value of 1, should help in obtaining more accurate results in the future. The most critical strain ratio with respect to failure are plane strain and pure shear. The outlined procedure for obtaining failure strains for the plane strain regime, as shown in the relevant section, is based on flat notch tensile tests. However, as is shown by careful 3d finite element analyses, failure for these samples start in the middle of the thickness. The folding mechanism in axially crashed components show that significant bending is involved. This means that failure would start from the surface of the sample. Therefore it would be reasonable to complement the identification procedure for the failure strains including the fade out parameters using tests where substantial bending is taking place. Up to now no useful results have been obtained for samples in the pure shear deformation regime for complex-phase steels. So work is going on in this area in order to have more comprehensive data available that can be used for the failure strain curve calibration.

273

ACKNOWLEDGMENTS We gratefully acknowledge the work that has been carried out by our colleages, Franz Koller and Philipp Biegel from voestalpine Krems GmbH, especially carrying out the roll forming finite element analyses as well as doing coordination for production of the rollformed profiles.

REFERENCES 1. T. Hebesberger, A. Pichler, H. Pauli and S. Ritsche, „Dual-Phase and Complex-Phase Steels: AHSS Material for a Wide Range of Applications“ in Steels for Cars and Trucks 2008, edited by B. Fuchsbauer and H. J. Wiehland, Düsseldorf: Verlag Stahleisen GmbH, 2008, pp. 456 – 463. 2. H. Gese, H. Werner, H. Hooputra, H. Dell and A. Heath, “CrachFEM - A Comprehensive Failure Model for Metallic Structures in Sheet Metal Forming and Crash Simulation” in Europam 2004, Paris, France, 2004. 3. H. Dell, H. Gese and G. Oberhofer, “CrachFEM – A Comprehensive Approach to the Prediction of Sheet Metal Failure” in NUMIFORM ’07, Materials Processing and Design: Modelling Simulation and Applications, edited by J. M. A. Cesar de Sa and A. D. Santos, 2007, pp. 165 – 170. 4. D. Teirlinck, F. Zok, J. D. Embury and M. F. Ashby, “Fracture mechanism maps in stress space”, Acta Metallurgica 36, pp. 1213-1228, 1988. 5. F. Neukamm, M. Feucht, A. Haufe and K. Roll, “On Closing the Constitutive Gap Between Forming and Crash Simulation”, 10th Int. LS-DYNA Users Conference, Dearborn, MI, USA, June 2008. 6. M. Baran, S. D. Wölkerling, M. Feucht, F. Neukamm and D. Weichert, „An Extension of the GISSMO Damage Model Based on Lode Angle Dependence”, 9th LS-DYNA Anwenderforum, Bamberg, 2010. 7. L. Xue, “Damage Accumulation and Fracture Initiation in Uncracked Ductile Solids Subject to Triaxial Loading”, Int. J. Solids & Structures, 44, pp. 5163–5181, 2007. 8. Y. Bai and T. Wierzbicki, “Application of extended Mohr–Coulomb criterion to ductile fracture”, International Journal of Fracture 161, 1-20, 2010. 9. LS-DYNA, Dynamore, Stuttgart, www.dynamore.com. 10. Abaqus, Dassault Systemes Austria GmbH, Vienna, www.simulia.com. 11. Aramis, gom mbH, Braunschweig, www.gom.com. 12. COPRA® FEA RF, data M Sheet Metal Solutions GmbH, Oberlaindern, Germany, www.datam.de. 13. M. Löwis and N. Fischbeck, “Python 2 Einführung und Referenz der objektorientierten Skriptsprache”, Addison-Wesley, München, 2. Auflage, 2001.

274

Numerical and Experimental Applications of TWIP Steel in Automotive Parts J. W. Lee1*, J. H. Lim1, J. B. Choi1 and P. Y. Oh1 1

Products Application Research Center Metal Forming Research Group, POSCO Global R&D Center 180-1 Songdo-Dong Yeonsu-Gu, Inchon, 406-840, Korea

Abstract. Modern automotive design has been faced with the weight reduction problem to meet the CO2 emissions standard while achieving high safety and compact design. Such being the case, most car makers want to use the ultra high strength steels (UHSS). But there are several problems when such steels are used, due to presumed lack of formability. Since the disadvantage such as above, it has been suggested that UHSS need special forming methods or it should be used only limited process like simple bending, by many automotive research institutes. To overcome these shortcomings, Twinning Induced Plasticity (TWIP) steel for improved strength and formability has been developed by steel making company, including POSCO, Korea. Because of its characteristics, it is expected to be widely used in automotive parts. This paper aims at finding out several ways how to make effective use of TWIP steel in automotive parts. Especially, comprising about from 15 to 18% manganese and from 1.5 to 2% aluminum which was developed by POSCO for application of the automotive parts will be considered. Keywords: 980TWIP Applications, Automotive Parts, Formability, Forming Simulation. PACS: 46.70.-p

INTRODUCTION Automobile companies (OEMs) are constantly developing new models. Better than the past cars, superior design and performance than its competitors to develop a fine car is being studied extensively. In recent days the automotive industry is extremely influenced by an increasing demand for better fuel efficiency and reduction of emissions which result in greenhouse gas induced global warming [1, 2]. In order to reduce CO2 emissions from vehicles, it should have light weight, but it is difficult to ensure the safety of passengers. Because of these reasons, adoption rate of high strength steels at automotive industry has been growing. For this reason, many steel makers are focusing on developing a variety of high strength steel especially [3, 4, 5]. In this trend, POSCO have been developing High Mn austenitic steel, so called TWIP steel for automotive applications which has an excellent combination of strength and ductility since 1994 [6, 7]. Attempts have recently made to develop a new grade TWIP steel with better performance and processing. It has been announced its mechanical properties and performance for several times [8, 9, 10]. In this study, we'll introduce the process for developing new automotive part using TWIP steel. The TWIP steel can be used for the auto parts which are following characteristics required. First, some parts have complicated shape, therefore it is difficult to apply high strength steel to them because of its formability. Most of these parts inevitably need a variety of reinforcement to ensure the rigidity and that gives a bad affect to weight reduction of vehicle. At this point if we apply the TWIP steel to maintain or improve the rigidity of the part but it has the advantage of reduced weight. In addition, some parts divided into several components due to lack of material formability. Those components need each forming process and welding process for completed part. In this case, we can have the advantage that save tooling and assembly costs by TWIP steel application. Finally, it can be applied to the parts to absorb the crash energy for the safety of passengers during a vehicle crash. The TWIP steels can have good performance of crash energy absorption, because of its superior elongation compared with the same strength material. In this study, new automotive part has been developed using TWIP steel. To this end, the forming simulation was carried out and then actual part was fabricated based on its results. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 275-280 (2011); doi: 10.1063/1.3623621 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

275

PROCEDURE FOR DEVELOPMENT OF THE NEW AUTOMOTIVE PART APPLYING THE TWIP STEEL The object part to apply is one of the cross member in rear floor and it consist of three component. The material of each component has 440MPa tensile strength and 1.6mm thickness. In this step, we checked the possibility of forming by computational analysis, since the purpose is to integrate this part by applying TWIP steel. We used the PAM-STAMP 2G Professional for forming simulation, and material properties for analysis were used not general constitutive equation but experimental data from tensile and FLC test.

The New Part Design The existing part, such as the following FIGURE 1 consists of three components of the assembly. To produce this part, it needs each tooling sets for which one is symmetry components at both ends and the other is middle component. Thus, we designed a new part which is combined three components using the TWIP steel that has high formability. This design gives process cost saving because it does not require welding process to combine each component, and it can be made by only one tooling set. We attempt to use 980TWIP 1.2t for newly designed part to replace the existing part is used 440MPa grade 1.6t. It has advantage that is strength combined with lightness, compared with the 440MPa grade 1.6t, and therefore weight reduction can be expected.

FIGURE 1. An automotive part consisted several components.

Forming Simulation for New Part As shown in FIGURE 2 (a), forming simulation was carried out based on the design plan described above. Considering the influence of gravity, holding and stamping process was conducted in order. The blank information for simulation is listed in TABLE 1). As mentioned in the published literature, it is not appropriate to generalize through the constitutive equation for common steels, because the TWIP Steel has somewhat different behavior compared with them. Accordingly, we use raw data for flow curve obtained from tensile test to improve the accuracy of the result. TABLE 1). The blank information of 980TWIP.

Thickness Young’s modulus Poission’s ratio Density Lankford value Flow curve FLC

r0 = 0.69

276

1.2 mm 188 GPa 0.3 7.73 g / mm 6 r45 = 1.01 Experimental data Experimental data

r90 = 1.20

There were no particular details about the strain distribution in blank after completing the holding step as shown in FIGURE 2 (b). The optimal blank size and blank holding force was able to determine through parametric study and punch stroke to achieve the final part was measured with an approximately 106mm. At the same time, we have solved problem about cracks occurring in some area, through modification of tooling surface without affecting the shape of the part. As shown in the FIGURE 3(a), the strain distributions of blank does not exceed forming limit curve of 980TWIP after forming simulation was completed, in other words, those are in safe area.. However, wrinkles are likely to occur in the area marked in red and orange color, as shown in the FIGURE 3 (b). These sites are not included in the final part and they will be eliminated through trimming process. After trimming, if the flanging process for joining with other components is completed, and then the final part will be obtained. The FE analysis for this task was omitted because it does not give large deformation on the completed part. We obtained the result that it is possible to make the part without any rupture from forming limit diagram analysis, however, as shown in FIGURE 3 (c), local thinning was predicted to more than 30%. The strain distribution at this area closes to forming limit curve for 980TWIP. It is likely to be failure, when the actual part is formed by tooling. Therefore it is necessary to control the corner radius of the die and punch in order to reduce the risk of rupture where local thinning was predicted.

Die

Blank Blank holder Punch

(a)

(b)

FIGURE 2. (a) The schematic diagram for forming simulation, (b) Major strain of blank after holding process.

(a)

(b)

277

(c) FIGURE 3. Result of forming simulation. (a) The FLD of 980TWIP after stamping process, (b) Wrinkle trend analysis, (c) Thinning analysis (over 30%).

Manufacture of prototypes and formability compared with other Steel grades Based on the results of the previous section, the actual part was fabricated. We determined the forming condition for 980TWIP by the results obtained from forming simulation and obtained final product through several trial test. The FIGURE 4 (a) shows the first real attempt for forming and the blank deformation was observed according to changes in stroke, in order to determine the forming conditions. The FIGURE 4 (b) shows the result of finally obtained forming simulation, and based on this the forming conditions for real part fabrication were derived. In the first attempt to observe the deformation behavior, the blank was used more sufficient size than determined it by FE analysis and the blank holding force and die cushion pressure was applied less than result of FE analysis to prevent rupture during forming.

Before 100mm to final stroke

Before 80mm to final stroke

Before 60mm to final stroke

Before 40mm to final stroke

Before 20mm to final stroke (a)

(b)

FIGURE 4. Comparison of the blank deformations as the punch stokes changes. (a) The first attempt for the real stamping, (b) Forming simulation result.

278

Next, the forming test for various steel grades, as shown by TABLE 2), was performed to compare the formability with different steel grades. The 390R and 590DP grade were expected to be impossible to get intact product because they have considerably lower elongation than 980TWIP and as shown in the FIGURE 5. (b) - (c) below, it has been confirmed. The CQ grade was expected to be success because of better elongation, but the rupture has occurred, likewise other grades mentioned just before. As can be seen in the FIGURE 5 (a) - (c), for all the cracks seems to be occurring at the same area where was the local thinning occurred as shown in the result of forming simulation, FIGURE 3 (c). Eventually, it has been confirmed that all grades used in the experiment except 980TWIP were not proper to produce the newly designed part. The FIGURE 6 (a), (b) shows the existing part applied 440E grade 1.6t and newly designed part for TWIP980 1.2t each of them. Those are final product passed through the trimming and piercing process. TABLE 2). The information of 980TWIP and some steel grade for comparison of formability Thickness (mm) Yield stress (MPa) Tensile Stress (MPa) Elongation (%) 1.2 216 328 45 1.2 300 419 37 1.2 381 641 28 1.2 482 996 54

Steel Grade CQ 390R 590DP 980TWIP

(a)

(b)

(c)

(d) FIGURE 5. The forming results of various steel grades. (a) CQ, (b) 390R, (c) 590DP, (d) 980TWIP.

(a)

279

(b) FIGURE 6. Comparison of the shape of the final part. (a) The existing part using 440MPa grade 1.6t, (b) The newly designed part using 980TWIP 1.2t.

CONCLUSION As development of car body with light weight and enhanced safety became one of the hottest issues in the auto industry, the advanced high strength steels with new forming technologies have been broadly applied to various automotive parts over the last few years. Corresponding to this trend, POSCO has developed innovative steel solutions in order to satisfy the requirement of OEMs more extensively. The TWIP steel with the reduced content of Mn was developed for application to automotives. This steel will be suitable for application to auto industry since a great reduction in the weight can be obtained. As mentioned in the present study, it is possible that an automotive part which consists of multiple components can be integrated by TWIP steel on the basis of superior strength and formability that could not be found in the other steel grades. That integrated part can give automotive company a great benefit of cost savings and weight reduction. New automotive part made in this study which compared with the existing parts can be omitted the welding process between each component and can be reduced the tooling costs needed to produce. In addition, we can get impressive weight reduction approximately 25% because the new part has thinned but strengthened than existing. It has still some challenges remain applying TWIP steel to automotive parts. Currently, many a researcher makes an effort to resolve these problems and most have already reached the completion stage. Going by this trend, it is expected that the TWIP steel will be applied to a variety of automotive parts.

REFERENCES 1. H. Hayashi, Proc. 4th Steel Symposium on Manufacturing and Application of Automotive Steels, KIMM, 2004, pp. 16. 2. Y. M. You, Proc. 50th Steel Symposium on Manufacturing and Application of Automotive Steels, KIMM, 2009, pp. 59. 3. H. W. Lee, G. S. Kim and S. H. Park, The 10th International Conference on Numerical Methods in Industrial Forming Processes, 2010. 4. S. H. Park, G. S. Kim, H. W. Lee, 2009 International Symposium on Automobile Steel, 2009. 5. O. J. Kwon, K. Y. Lee, G. S. Kim and K. G. Chin, Materials Science Forum, Vols. 638-642, 2010, pp 136-141. 6. Y. G. Kim, T. W. Kim and S. B. Hong, Proc. of ISATA, Aachen, 1993, pp. 269 7. S. K. Kim, J. W. Cho, W. J. Kwak, G. Kim and O. Kwon, Proc. of METEC 2007, Dusseldorf, 2007. 8. C. Scott, N. Guelton, S. Allain, M. Faral, Materials Sci. & Tec., Organized by James R. Fekete and Roger Pradhan, 2005, pp.127. 9. D. Cornette, P.Cugy, A. Hildenbrand, and M. Bouzekri, SAE international, 2005-01-1327. 10. J. K. Kim, C. Lei, H. S. Kim, S. K. Kim, B. C. De Cooman, Metallurgical and materials transactions. A, Physical metallurgy and materials science, vol. 40, no13, 2009, pp. 3147-3158.

280

Texture Control of Aluminum, Iron, and Magnesium Alloy Sheets to Increase Their Plastic Strain Ratios Dong Nyung Lee and Heung Nam Han Department of Materials Science and Engineering and RIAM, Seoul National University, Seoul 151-744, Korea Abstract. It is known that the limiting drawing ratio of sheet metals is proportional to their plastic strain ratios, and the plastic strain ratios of fcc and bcc metal sheets increase with increasing <111>//ND component in their textures. Conventional cold rolling and subsequent annealing of fcc metals cannot give rise to the <111>//ND component. Specifically, the cold rolling texture of polycrystalline fcc metals is characterized by the fiber connecting the {112}<111>, {123}<634>, and{011}<211> orientations in the Euler space, which is often called the -fiber. The density of each component in the fiber depends on the stacking fault energy of metals. The {112}<111> and {123}<634> textured Al alloy sheets evolve the {001}<100> texture, when recrystallized. The low plastic strain ratios of the Al alloy sheets are attributed to the {001}<100> texture. The <111>//ND texture can be obtained in shear deformed fcc sheets. Bcc steels develop the <111>//ND texture when cold rolled and recrystallized. However, the density of <111>//ND depends on the content of dissolved interstitial elements such as carbon and nitrogen. The density of the <111>//ND component decreases with increasing concentration of the dissolved interstitial elements. For a given steel, the density of the <111>//ND component can vary with varying thermomechanical treatment. Magnesium alloy sheets are subjected to sheet forming processes at temperatures of 200°C or higher because of their basal plane texture, or the <0002>//ND orientation. Many studies have been made to alleviate the component so that the magnesium alloy sheets can have better formability. In this article, the above issues are briefly reviewed and discussed. Keywords: Aluminum Alloys, Steels, Magnesium Alloys, Textures, Plastic Strain Ratios. PACS: 81.40.Gh

INTRODUCTION It is well known that the limiting drawing ratio (LDR) of sheet metals is proportional to their plastic strain ratios, or the Lankford parameters (R value)1,2. The R values of fcc and bcc metal sheets are known to increase with increasing <111>//ND component, where ND stands for the sheet normal direction, in their textures. Conventional cold rolling and subsequent annealing of fcc metals cannot give rise to the <111>//ND component. Specifically, the cold rolling texture of polycrystalline fcc metals is characterized by the fiber connecting the {112}<111>, {123}<634>, and{011}<211> orientations in the Euler space, which is often called the -fiber. The density of each component in the fiber depends on the stacking fault energy of metals. The density of the {011}<211> component is the highest in low stacking-fault-energy metals such as brass and silver, and those of the {112}<111> and{123}<634> components are the highest in medium and high stacking-fault-energy alloy sheets. The {112}<111> and {123}<634> textured sheets evolve the {001}<100> texture, when recrystallized. The {001}<100> orientation is responsible for reducing the R values of Al alloy sheets. The <111>//ND texture can be obtained in shear deformed fcc sheets. The shear deformation of sheets can be achieved by asymmetric rolling. However, the main component in the shear deformation texture is {001}<110>, which deteriorates the R value like the {001}<100> component. Recently, a study is successfully being made of reducing the {001}<110> component. Bcc steels develop the <111> texture when cold rolled and recrystallized. For a given steel, a way of increasing the density of the <111>//ND component has been developed. Magnesium alloy sheets are subjected to sheet forming processes at temperatures of 200°C or higher because of their basal plane texture, or the <0002>//ND texture. Many studies have been made to alleviate the component so that the magnesium alloy sheets can have better formability. In this article, the above issues are briefly reviewed and discussed.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 283-290 (2011); doi: 10.1063/1.3623622 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

283

LIMITING DRAWING RATIO, PLASTIC STRAIN RATIO AND TEXTURE The LDR-R relation can be well described by the following relation.1,2

ln( LDR ) K

( 2  2 a R )1 / a 2[1  ( R a /( a 1)  R) /(1  R 1 /( a 1) ) a ]1 / a

(1)

where  is a deformation efficiency to account for frictional and bending work and a | 6 for bcc metals and a | 8 for fcc metals. The plastic strain ratio (the R value) is related to the sheet texture. For sheet metals of cubic structure, Lee and Oh proposed the following equation.3

R

with

dH w

dH w

6(dH w ) j V j

dH t

6(dH t ) j V j

³ dH ³ dH v

v

6 (n ˜ w ) ( i ) (b ˜ w ) ( i ) (n ˜ l ) (i ) (b ˜ l ) (i ) and dH t

w

( g ) f ( g )dg

t

( g ) f ( g )dg

(2)

6 (n ˜ t ) ( i ) (b ˜ t ) (i ) (n ˜ l ) (i ) (b ˜ l ) (i )

ͣͣ͟ ͣ ͩ͢͟ ͧ͢͟ ͥ͢͟ ͣ͢͟ ͢ ͩ͟͡ ͧ͟͡ ͥ͟͡ ͣ͟͡ ͡

ͩ͢͟

ά͢͡͡ήͭͯ͢͢͡ άͣ͢͢ήͭͯ͢͢͢ άͣͤ͢ήͭͧͤͥͯ ά͢͡͡ήͭͯ͢͡͡

ͧ͢͟ ͥ͢͟ ΃͞ΧΒΝΦΖ

΃͞ΧΒΝΦΖ

where Vj is volume fractions of texture components j, dw and dt are plastic strains along the width and thickness directions, respectively, n and b are unit vectors along the slip plane normal and the slip direction, respectively, and w, t, and l are unit vectors along the width, thickness, and tensile directions of a sheet specimen. The superscript i and f(g) indicate slip systems and the orientation distribution function, respectively. For a sheet having texture (hkl)[uvw] where (h2+k2+l2)1/2=1 and (u2+v2+w2)1/2=1, l =[l1l2l3] = [{u cosD + (lv - kw) sinD}{v cosD + (hw - lu) sinD}{w cosD + (ku - hv) sinD}], t =[t1t2t3] = [h k l], w =[w1w2w3] = [{-u sinD + (lv - kw) cosD}{- v sinD + (hw - lu) cosD}{-w sinD + (ku - hv) cosD}] with D being the angle between the rolling direction (RD) and the tensile direction. The R - D relations calculated for various sheet orientations are shown in Fig.1. The restricted slip systems were assumed to be activated. It can be seen that the <111> and <001> orientations gives the highest and lowest R values, respectively. The recrystallization (Rex) texture of cold rolled bcc steel sheets is known to be approximated by the {111}<112> orientation. However, the density of the orientation depends on the cold rolling reduction and the concentration of solute elements, especially interstitial elements such as carbon and nitrogen. As the cold rolling reduction increases and the concentration of interstitial elements decreases, the density of the Rex texture tends to increase. Segregation of the interstitial elements in dislocations is likely to reduce the directionality of stress field due to dislocations generated during deformation and the driving force for Rex, which in turn reduce the density of the Rex texture. The directionality of the stress field controls the Rex texture (SERM model for Rex texture).

ͣ͢͟ ͢ ͩ͟͡

ά͢͢͢ήͭͯ͢͢͡

ͧ͟͡

ά͢͢͢ήͭͣͯ͢͢

ͥ͟͡

άͣ͢͢ήͭͯ͢͢͡

ͣ͟͡ ͡

͡ ͢͡ ͣ͡ ͤ͡ ͥ͡ ͦ͡ ͧ͡ ͨ͡ ͩ͡ ͪ͡ ͲΟΘΝΖ͑ΥΠ͑ΣΠΝΝΚΟΘ͑ΕΚΣΖΔΥΚΠΟ͑͝ΕΖΘ͟

͡

͢͡ ͣ͡ ͤ͡ ͥ͡ ͦ͡ ͧ͡ ͨ͡ ͩ͡ ͪ͡ ͲΟΘΝΖ͑ΥΠ͑ΣΠΝΝΚΟΘ͑ΕΚΣΖΔΥΚΠΟ͑͝ΕΖΘ͟

FIGURE 1. Calculated R-value as a function of angle to RD.

284

TAYLOR FACTOR AND TEXTURE CONTROL The Taylor factor M is defined as the following expression.4

M

¦ dJ

(k )

/ dH ij

V ij / W c

(1)

The Taylor factor indicates the sum of shear strains dJ on active slip systems k to achieve a plastic strain dij of a crystal. During cold deformation, about l-10% of the energy spent is stored in the crystal lattice5 in the form of crystal defects such as dislocations, dislocation cell walls, stacking faults6,7. Therefore, the energy stored in deformed crystals is expected to be proportional to their Taylor factors. Various attempts have been made by nany researchers to evaluate the stored energy by measuring the broadening of X-ray diffraction lines8-12 and neutron diffraction.13 The experimental results are in qualitative agreement with what expected from the Taylor factor. It should be noted that most cold-rolling reductions in the experiments were high enough to cause Rex. If the reduction is low enough not to cause Rex, the results can be against expectation because the Taylor factor can be an index of plastic anisotropy – the extent to which strength ij of a crystal varies with orientation, for a given shear strength c. An increase in the Taylor factor of a grain indicates that the grain is increasingly difficult to undergo plastic deformation. When high Taylor-factor grains neighbor low Taylor-factor grains, the low Taylorfactor grains can undergo more plastic deformation to assume an external deformation of the aggregate. In this case the low Taylor-factor grains have higher dislocation densities or higher stored energy than the high Taylor-factor grains, which grow at the expense of the low Taylor-factor grains during annealing, resulting in the orientation of the high Taylor-factor grains after annealing.14,15

SERM MODEL FOR RECRYSTALLIZATION TEXTURE The strain-energy-release-maximization (SERM) model for Rex texture was advanced by Lee16 and later elaborated.17 The model is briefly described here. The most important driving force for Rex is the stored energy due to dislocations generated during fabrication. Dislocations assume a stable array as shown in Fig. 2(a) during recovery. The long-range stress field plays a dominant role in the evolution of Rex textures. Fig. 2(b) shows the principal stress distributions for an edge dislocation array. It can be seen that the absolute maximum stress direction (AMSD) is parallel to the Burgers vector of dislocations that are responsible for the long-range stress field. When a volume of V in the stress field is replaced by a stress free single crystal of the volume V, the strain energy of the system is released. The strain-energy release is maximized when AMSD in the stressed matrix is along the minimum Young’s modulus direction (MYMD) of the stress free crystal. This and other conditions that maximize the strainenergy release determine Rex textures. For multiple slip, AMSD in plastically deformed metals can be calculated by the sum of active slip directions of the same sense taking their activities into account as shown in Fig. 3(a). The activities are equivalent to shear strains on the active slip systems. Priorities for determining Rex textures are as follows17. Priority #1. When AMSD in the deformed matrix is crystallographically the same as MYMD of the matrix material, the deformation texture remains unchanged after Rex. Priority #2. When AMSD differs from MYMD, the Rex texture is determined such that AMSD in the deformed matrix is parallel to MYMD in the recrystallized grain, together with one common axis of rotation between the deformed and recrystallized states. The common axis can be ND, TD or other directions (e.g. <110> for bcc metals). 1G

100 horizontal edge dislocations with inter-dislocation spacing being 10b b Burgers vector G shear modulus (a )

( b)

FIGURE 2. (a) Schematic drawing of dislocation array after recovery. Horizontal array gives rise to long- range stress field, and vertical array gives rise to short-range stress field. (b) Calculated principal stress distribution.18

285

b

b (1) J (1)

(2)

b

B

S

b (2) J ( 2 ) b (1) J (1)  b (2) J ( 2 ) // AMSD

(1)

A

( b)

(a )

C

FIGURE 3. (a) AMSD for active slip systems i, whose Burgers vector and activities are b and J , respectively. (b) Principal stress directions A (AMSD), B and C in deformed matrix. S is one of slip directions that is closest to 90° with A. Once A and S are known, C and B are obtained by A × S = C and C × A = B. (i )

(i )

Priority #3: When the above two conditions are not met, we obtain three principal stress directions A (AMSD), B and C as explained in Fig. 3(b). Once A, B, and C are known, the Rex texture is determined such that A, B, and C in the deformed matrix are parallel to three <100> directions in Rexed grains, when MYMDs are <100> in cubic system. Let the unit vectors of A, B, and C be a [a1 a2 a3], b [b1 b2 b3] and c [c1 c2 c3], where ai, bi and ci are respectively direction cosines of the unit vectors, a, b, and c, referred to the crystal coordinate system. If the unit vectors a, b, and c are set to be parallel to [100], [010] and [001] after Rex, components of the unit vectors are direction cosines relating the deformed and Rexed crystal coordinate systems. If the deformation orientation is expressed as (hkl)[uvw], theRex orientation (hr kr lr)[ur vr wr] can be calculated using Eq. 2.

ªa1 «b « 1 «¬ c1

a2 b2 c2

a3 º ªh º b3 »» ««k »» cc »¼ «¬ l »¼

ªhr º ªa1 «k » , « b « r» « 1 «¬ l r »¼ «¬ c1

a2 b2 c2

a3 º ª u º b3 »» «« v »» cc »¼ «¬ w»¼

ª ur º «v » « r» «¬ wr »¼

(2)

TEXTURE CONTROL OF ALUMINUM ALLOY SHEETS Al alloy sheets are considered as one of the high potential substitutes for steel sheets considering weight reduction of automobiles. However, Al alloy sheets have drawbacks such as higher prices and the inferior deep drawability. The inferior deep drawability is mainly due to low R values of Al alloy sheets made by the conventional rolling and thermal processes. The low plastic strain ratios of Al alloy sheets are due to their textures. The rolling texture of Al alloys is characterized by the  fiber. The Rex texture of Al alloy sheets is well known to be {001}<100>. The 40°<111> orientation relationship between {123}<634> and {001}<100>, which has been taken as a proof of the oriented growth theory19 has made one believe that {123}<634> is responsible for {001}<100>.20 The oriented growth theory is claimed to be associated with grain boundary mobility anisotropy. However, many data indicate that {112}<111> is responsible for {001}<100>.21-26 Rex is likely to occur first in high strain energy regions. The energy stored in highly deformed crystals is proportional to the Taylor factor. The Taylor factors calculated for major components in the rolling texture are 2.45 for {001}<100>,26 2.45 for {110}<112>,26 2.45 for {110}<001>27, 3.24 for {123}<634>26, and 3.64 for {112}<111>.26 The measured stored energies for 99.99% aluminum crystals channel die deformed to a strain of 1.5 showed that the {112}<111> region had higher energy than the {123}<634> region.28 The Taylor factors and the measured stored energies indicate that the driving force for Rex is higher in the {112}<111> grains than in the {123}<634> grains. Therefore, the {112}<111> component in the rolling texture is likely to be more responsible for the {001}<100> Rex texture even when the {123}<634> component has higher density than {112}<111>. The {112}<111> to {001}<100> texture transition was first explained by the SERM model,16 and elaborated later.26 Thus the conventionally processed Al alloy sheets lack the <111>//ND texture. On the other hand, the ideal shear deformation texture of fcc metals consists of the {111}<112> and {001}<110> orientations.29 The deformation texture does not change much even after annealing.30 The shear textured aluminum sheets can be made feasibly by asymmetric rolling, in which the circumferential velocities of work rolls are different, imposes shear deformation and in turn shear deformation textures to sheets through the thickness.29-32 One example is shown in Fig. 4, in which the density of the {001}<110> component is much higher than that of the {111}<112> component. In order to increase the plastic strain ratio, it is desirable that the density of the {001}<110> component should be reduced as

286

&

&

{111}  112 !

& &

& {111}  110 !

&

{001}  110 !

FIGURE 4. Measured (111) pole figures of aluminum sheets asymmetrically rolled by 50% at upper-roll-radius to lower-rollradius ratio of 1.5. (left) Top surface layer; (center) center layer; (right) bottom surface layer.29

much as possible. Since the {001}<110> component has smaller Taylor factor than the {111}<112> component33, we may be able to enhance the {111}<112> component using the concept described in the Taylor factor section. That is, the low Taylor-factor grains can have higher dislocation densities or higher stored energy than the high Taylor-factor grains at low rolling reductions. Therefore, the high Taylor factor grains grow at the expense of the low Taylor-factor grains during annealing, resulting in increases in the <111>//ND component and the plastic strain ratio after annealing. This concept led Kim et al.34 to obtain about 1.3 in the R value of aluminum alloy (1050) sheets.

TEXTURE CONTROL OF STEEL SHEETS Commercially available interstitial free (IF) steel sheets have well developed <111> //ND texture, or  fiber. The steel sheets are made by cold rolling and subsequent Rex. The important components in the rolling texture of IF steels are located along the <110>//RD, or  fiber, and  fiber. The rolling texture turns to  fiber after Rex. The evolution of the  fiber in IF steel has been well explained by the SERM model.35 Recently POSCO developed a copper-bearing bake hardening (CBBH) steel, which can be characterized by an intentionally added Cu content for solid solution hardening, and very low C concentration to reduce strain aging. The characteristics eventually lower the Rex temperature, leading to energy saving and low production cost.36 However, it has a little lower density of  fiber component compared with IF steel because of a little higher dissolved C content due to lack of Ti and Nb, scavengers of C and N. Oh et al.37 developed a two-step rolling-annealing process to increase the  fiber component in the Rex texture of the CBBH steel. The two-step process comprises the first rolling by a low reduction in thickness and subsequent annealing at 780°C, followed by the second rolling by a high reduction and subsequent annealing at 780°C. The first rolling process aims at seeding the  fiber oriented grains, so that they can grow at the expense of differently oriented grains developed in the second rolling process. In this way the density of  fiber component in the Rex texture of the CBBH steel much increases compared with that in the conventional one-step rolling-annealing process. The enhanced density of the  fiber component in the first rolling is attributed to the fact

FIGURE 5. Volume fractions of  fiber in textures of () first rolled-annealed and () second rolled-annealed steel sheets, respectively.37 10A-74A indicates that hot-rolled sheet is cold-rolled by 10% in first rolling reduction and annealed at 780 °C , followed by 74% in second rolling reduction and annealing at 780 °C.

287

that the Taylor factor of the <110>//RD component is smaller than that of the {111}<110> component.35 The <110>//RD component can have higher stored energy during the cold-rolling, resulting in its diminution and an enhanced density of the <111>//ND component during annealing. This <111>//ND component can act as nuclei during Rex in the second step process. The cold-rolling reduction in the second step must be high enough to cause Rex. The cold-rolling reductions in the first and second steps should be carefully controlled. For given the initial and final sheet thicknesses, an increase in the first-rolling-reduction brings about a decreased in the second-step-rollingreduction, resulting in a decrease in the driving force for Rex in the second-step-annealing, which in turn gives rise to a decrease in the <111>//ND component as shown in Fig. 5, in which the 20A-71A sheet shows the highest volume fraction of the <111>//ND component among the five differently processed sheets.

TEXTURE CONTROL OF MAGNESIUM SHEETS The use of Mg alloy in structural applications has great potential for the light weighting of transportation vehicles. Unlike aluminum alloys and steels, which have cubic crystal structures, magnesium alloys have a hexagonal crystal structure. This limits easy dislocation motion to the close-packed direction on the basal planes, which provides only three independent slip systems of the five systems required for arbitrary deformation to occur without a volume change. Hence non-basal slip and twinning are required to acquire the five independent slip systems, and both require much higher critical resolved shear stresses than the basal slip. To activate non-basal slip, the deformation temperature is increased. A number of methods have been suggested to improve the ductility of Mg alloy sheets at room temperature. The methods are mostly related to decreases in the basal plane orientation, the (0002) orientation, and are classified into chemical, mechanical, and partial melting-solidification methods. An addition of rare earth metal in Mg alloy38-40 belongs to the chemical method. The mechanical methods use an imposition of shear deformation to Mg alloys by asymmetric rolling such as differential-diameter rolling, differential speed rolling41-46, cross-roll rolling,47 equal channel angular pressing, equal channel angular rolling,48,49 etc. The partial meltingsolidification method uses defocused laser scanning. Since the mechanical methods have been so widely addressed, the present discussion will be focused on the laser scanning treatment.50

Laser Scanning Treatment Ha et al.50 suggested a method, in which high energy is supplied to both surfaces of a 1.05 mm-thick AZ31BH24 Mg alloy sheet through a laser scanning treatment combined with a defocusing technique to weaken the (0002) basal plane texture on the surface regions. An Yb fiber laser in continuous mode was used to scan both surfaces of the sheet. The laser power was 200W and the laser scanning speed was 10, 15, and 20 m/min.

100 m

FIGURE 6. EBSD orientation map and (0002) pole figures of base material (BM), heat affected zone (HAZ), and columnar grain zone (CGZ).50

288

FIGURE 7. Engineering stress–strain curves (room temperature, strain rate of 3.3×104/s) for specimens laser-scanned at power of 200W under various laser scanning speeds.50

Fig. 6 shows the electron backscatter diffraction (EBSD) orientation map and the (0002) pole figures of the BM, HAZ, and CGZ. The BM has the (0001) texture, and the HAZ has a little dispersed basal-plane texture, which might develop during grain growth under stress. The fact that the dispersion is not large implies that the elastic anisotropy in the Mg alloy is weak. Since the elastic constants of the alloy are not available, the stiffness components51 (C11 = 59.74, C33 = 61.7, C44 = 16.39, C12 = 26.24, C13 = 21.7 GPa) of Mg at 298 K have been used to calculate the maximum and minimum Young’s moduli (Emax and Emin) and Zener’s anisotropy factor A = 2C44/(C11-C12). From the stiffness data, the compliance components are calculated to be S11 = 0.02202, S33 = 0.01971, S44 = 0.06101, S12 = 0.007914, and S13 = -0.004960 GPa-1. Young’s modulus E of hexagonal crystal is expressed as 2

4

2

2

1/E = S11 (1 - a13 )2 + S 33 a13 + ( S 44 + 2 S13 ) a13 (1- a13 )

(3)

where a13 is cosine of the angle between a given direction and the hexagonal symmetry axis. Emax, Emin , and A are calculated to be 45.41 GPa, 42.83 GPa, and 0.98, respectively. When A is unity, the elastic properties are isotropic. These values indicate that Mg is approximated to be elastically isotropic. The texture of the CGZ is nearly random. In general, the columnar grains obtained during solidification tend to grow along their minimum Young’s modulus directions.52 The random orientation of the CGZ is attributed to the elastic isotropy of Mg alloy. The random orientation and coarse grain size of the CGZ give rise to higher ductility and lower strength as shown in Fig. 7.

CONCLUSIONS The R value of Al alloy sheets can be increased to about 1.3 by two-step asymmetric rolling and subsequent annealing. The <111>//ND of low C steel sheets can be increased by two-step rolling and subsequent annealing. Mg alloy sheets at room temperature can yield higher ductility and lower strength by a partial melting-solidification method like the defocused laser scanning treatment because the melted and solidified CGZ has almost random orientation.

ACKNOWLEDGMENTS This study was supported by National Research Foundation of Korea grant funded by the Ministry of Education, Science and Technology (2009-0083038) and POSCO research program.

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Influence of Transformation-induced Plasticity on Formability of TRIP Steels Hai Yan Yua, Shi Jin Yuanb, Zhe Sunc a,c, School of Automotive Engineering,Tongji University,4800 Cao’an Road, Shanghai, 201804, China b School of Software Engineering, Tongji University, 4800Cao’an Road, Shanghai, 201804, China Abstract. A micromechanical flow stress model for the transformation-induced plasticity (TRIP) steel is proposed on the basis of continuum mechanics. In the model, TRIP effect and degradation of elastic modulus are considered. TRIP effect is introduced by regarding the volume fraction of retained austenites as varying with plastic strain. Degradation of elastic modulus is evaluated by an empirical expression. Both the contribution of individual phases and that of interaction between constituent phases to the overall stress are taken into account. The proposed model is introduced into LSDYNA software to simulate the cup forming and U-channel springback. Cup drawing and U-channel bending experiments are provided. Comparison shows that formability results simulated with TRIP effect are closer to the experimental ones. Keywords: TRIP steels, flow stress, formability, springback PACS: 81.05.U-

INTRODUCTION With increasing fuel cost and worsening environment, light weighting is urgent for the sustainable development of automotive industries. Since adopting light materials is an effective method, the application of advanced high strength steels has increased significantly during the last decades. Low-alloy transformation induced plasticity (TRIP) steels are well known to combine high strength with good ductility [1-3] and have been regarded as promising lightweight materials for automotive industry. The remarkable characteristic of TRIP steels is that plastic deformation can result in martensite transformation which can induce extra plasticity in the meantime [4-5]. The transformation is accompanied by a volume expansion and by the increase of the strain-hardenability during plastic deformation. This delays the onset of necking and ultimately leads to a higher elongation. So the transformation-induced plasticity effect is an important contributor for the good ductility and favorable strain-hardenability of TRIP steels and should be considered in the flow stress model. From a micromechanical point of view, TRIP steels may be considered as multiphase composite material. Papatrianatfillou et al. [6] determined the effective properties and overall behavior of TRIP steels by using homogenization techniques for nonlinear composites. Jacques et al [7] investigated the mechanical behavior of TRIPassisted multiphase steels based on three different microstructures generated from the same steel grade. The mechanisms responsible for the work-hardenability and the resulting balance between strength and resistance to plastic localization are investigated at different length scales. Delannay et al. [8] presented an incremental mean-field model to predict transformation induced plasticity. The partitioning of strain between softer and harder constituents is computed based on an elastic-plastic Mori-Tanaka approach that accounts for the progressive transformation of austenite into martensite. The latter transformation is predicted using an energy-balance criterion that is formulated at the level of individual austenite grains. Uthaisangsuk[9] proposed a flow stress model from the point of damage and failure .The partitioning of carbon in microstructure is taken into account for the flow curve description of each individual phase and representative volume elements (RVE) are used within the framework of continuum mechanics. During recently years, some researches have pointed out that TRIP steels are of outstanding degradation of elastic modulus during plastic deformation. This phenomenon has been named as the inelastic strain recovery behavior [5]. Elastic modulus is a key parameter of flow stress and springback simulation considering the degradation of elastic

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 291-298 (2011); doi: 10.1063/1.3623623 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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modulus is more precise than the common simulation. It should also be taken into account in the flow stress model of TRIP steels. The aim of this work is to introduce the transformation-induced effect and the degradation of elastic modulus into in the flow stress modeling for TRIP steels. TRIP steels are regarded as composite materials with several phases. Contribution of each constituent phases and their interaction to the overall stress are analyzed on the basis of continuum mechanical theory. The TRIP effect is modeled by regarding the volume fraction of retained austenites as a variable of effective plastic strain. The expression describing the volume fraction of retained austenites and the one characterizing the degradation of elastic modulus are retracted from the author’s previous work [10-11] since they are not the emphasis of this work.

1. FLOW STRESS MODEL V

VdH

V

FIGURE 1. Representative Volume Element model

FIGURE 2 Interaction between the constituent phases which labels the

Assumption

(1) For simplicity, TRIP steels are assumed to be composed by four phases, namely, ferrites, bainites, retained austenites and martensites. Before plastic deformation the volume fraction of martensites is assumed to be zero. As shown in Fig.1, the four phases are named 1st, 2nd, 3rd and 4th phase, respectively. (2) Each constituent phase is of the same elastic modulus. (3) According to its content, ferrite is regarded as the matrix phase and the other three phases are the inclusions. Interaction between each two phases is taken into account. It has been shown in Fig.2. (4) Elastic modulus changes with plastic strain during plastic deformation. According to the continuum mechanics, one can use the representative volume element (RVE) model to analysis the deformation behavior of the constituent phases. As shown in Fig.1, the RVE model is in uniaxial tensile deformation under uniform stressV. If the RVE model is stretched by a strain increment, i.e. from strain H to H+dH, then each constituent phase will get a strain increment, namely, dH1, dH2, dH3 and dH4, respectively. According to the energy balance theory, the work (VdH) done by the applied stress (V) is equal to the sum of the increase of stored inner energy (dU) during the deformation from H to H+dH and the heat (dQ) emitted to the surroundings. It can be expressed as

dU =V dH -dQ.

(1)

The stored energy U in Eq. (1) is given by 4

U

4

¦ f U  ¦U i

i 1

i

int,i

(2)

.

i 1

Where, fi and Ui indicate the volume fraction of the i-th phase and the energy stored in the i-th phase, respectively. Uint,i indicate the interaction energy between the i-th phase and other phases. The shape of the grains is simplified to be spherical according to the Eshelby’s inclusion model [12] and Mori-Tanaka theory [13]. Then a general formula of Uint,i for uniaxial tensile deformation can be, 4 2 1 (3) U int,i Af i f j H i  H j . ¦ j 1, j z i 2





292

Where, Hi and Hj are the strain of i-th and j-th phase, respectively. A is a function of elastic constants. For multiphase materials, the following relations can be assumed, 4

¦ fiH i .

H

(4)

i 1

4

Q

¦ fi Qi .

(5)

i 1

Where Qi is the heat emitted from the i-th phase to the surroundings. The differential form of Eq. (2), Eq. (4) and Eq. (5) can be expressed as, 4 wU w  fiU i int,i  d H dH . ¦ ¦ w w H H i 1 i 1 4 w  f iH i dH = ¦ dH . wH i 1 4 w  f i Qi dQ = ¦ dH . wH i 1 4

dU

(6)

(7)

(8)

As it is discussed in the above assumptions, the volume fraction of retained austenites and martensites, f3 and f4, are varying with plastic strain considering the TRIP effect. And the volume fraction of ferrites and bainites, f1 and f2 are constant during plastic deformation according to some investigations [2]. The sum of f3 and f4 is regarded as a constant C during plastic deformation. So the equation (6) can be rewritten as, 4

¦ i 1

w  f iU i dH wH

4

¦f i 1

i

wU i wf dH  U 3  U 4 3 dH . wH wH

(9)

Taking Eq. (7) into account, one can develop the first item in the right side of Eq. (1) to be,

V dH

4 wH i wf dH  V ¦ H i i dH . wH wH i 1

4

V ¦ fi i 1

(10)

Taking Eq. (8) into account, one can rewrite the second item in the right side of Eq. (1) as, 4 wQi wf dH  ¦ Qi i dH . wH wH i 1

4

dQ

¦f

i

i 1

(11)

Substituting Eq. (9), Eq. (10) and Eq. (11) into Eq. (1), one can obtain.

fi

wU i  wH

4 2 wA 1 fi f j  H i  H j  ¦ Af i f j  H i  H j  wH j 1, j zi 1, j z i 2 4

j

¦

2 w  fi f j wf wQ wf 1 A H i  H j  f iV  VH i i  fi i  Qi i ¦ wH wH wH wH j 1, j z i 2

(12)

4

0.

The derivation of the volume fraction of retained austenites and martensites to strain is close to zero since the change of volume fraction is small. So Eq. (12) can be rewritten as,

w U i  Qi  wH

2 wA 1 f j H i  H j  ¦ wH j 1, j z i 2 4

4

¦

j 1, j z i

Af j  H i  H j V .

(13)

The first term in Eq. (13) is the summation of the stored energy with the heat emitted to the surroundings, which is equal to the work done on it by external stress. With the flow stress of the i-th phase Vi=Vi (Hi), one can obtain

U i  Qi

³

H

0

V i  H i dH i .

(14)

Substituting Eq. (14) into Eq. (13), one can obtain 4 2 wA 1 f j H i  H j  ¦ Af j  H i  H j V . wH j 1, j zi 1, j z i 2 4

V i H i  j

¦

293

(15)

Eq. (15) is the flow stress model considering both the transformation-induced plasticity effect and the variation of elastic modulus during plastic deformation. The first item in Eq. (15) is the contribution of the constituent phases to the overall flow stress. The second item defines the alteration induced by the variation of elastic modulus with plastic strain. The third item is the contribution of interaction among the constituent phases to the overall stress. Two important parameters need to be determined before one can use the equation (15) to calculate flow stress. The first parameter is the relation between the volume fraction of retained austenites and plastic strain, i.e. f3=f (H). One can measure the volume fraction of retained austenites by X-ray diffraction as plastic strain advances. Then an empirical expression can be summarized from the experimental data. Some references [2, 11] have given the detail information. In this research, the expression from reference [11] is used. This expression has been tested by experimental data.

f3

f 3,0 exp  cf3,0H p .

(16)

Where, f3 and f3,0 are the volume fraction of the retained austenites and the initial value before deformation, respectively. The material parameter c is a constant. Under uniaxial tension, it is equal to 24.4. The parameter H p is the equivalent plastic strain. The second parameter in Eq.(15) is the relation between the elastic modulus E and the plastic strain H, i.e.E =g (H). It can be determined from the uniaxial tension testing unloaded under different plastic strain [10].

E 156.7  45.4 exp  6.7H p . GPa

Where the meaning of

Hp

(17)

is same with the one in Eq. (16).

2. FORMING SIMULATION 2.1 Forming simulation The proposed model Eq. (15) is introduced into LS-Dyna software to predict the formability of cup drawing and U-channel springback. During springback simulation, explicit forming is switched to implicit springback simulation. Seven integration points through the thickness both for forming and springback are used. The corners of the cup and U-channel are meshed with seven elements. Dimensions of the tools for cup and U-channel are shown in Fig.3. Blank outline for cup is a circular plate. As shown in Fig.4, blanks with radius(R) 90mm, 100mm and 105mm are drawn to cups with depth (h) 50mm, 65mm and 78mm, respectively. Blank outline for U-channel is a rectangle with 350mm in length and 35mm in width.

(a)Cup forming (b) U-channel forming FIGURE 3. Dimensions of the tools for cup and U-channel forming simulation

2.2 Simulation results Simulated cup thickness is shown in Fig.4 (a) ~ (f). Form Fig.4(a) and Fig.4(b) one can find that the thickness thinning trend is almost the same. The thickness range considering TRIP effect is 0.87~1.24mm and that simulated without TRIP effect is 0.85~1.26mm. As is well known, the minimum thickness means trend of crack and the maximum thickness means trend of wrinkling. From this point, one can conclude that the simulation with TRIP effect can give more perferable formability than that neglecting the TRIP effect. The same conclusion can be drawn from the other two pairs, Fig.4(c) vs. Fig.4 (d), Fig.4 (e) vs. Fig.4 (f), respectively.

294

(a)With TRIP effect, R=105mm,h=78mm

(b) Without TRIP effect, R =105mm,h=78mm

(c)With TRIP effect, R =100mm,h=65mm

(d) Without TRIP effect, R =100mm,h=65mm

(e)With TRIP effect, R =90mm,h=50mm (f) Without TRIP effect, R =90mm,h=50mm FIGURE 4 Simulated cupthickness when blank radius is 90mm,100mm and 105mm

a b FIGURE 5 Simulated U-channel outline after springback (a) without TRIP effect and no degradation of elastic modulus ; (b) both TRIP effect and degradation of elastic modulus The simulated U-channel outline after springback is shown in Fig.5. It shows a larger flange angle derivation with both TRIP effect and degradation of elastic modulus that the one neglecting these two factors does. Between the two factors of TRIP effect

295

and degradation of elastic modulus, the latter plays a more important role. The elastic strain increases with the decrease of elastic modulus under the same stress level. So once unloading the accumulated elastic strain will recover by which the flange displays an angel derivation after springback. The larger degradation of elastic modulus is the larger springback is.

3. EXPERIMENTS AND DISCUSSION 3.1 Cup drawing experiments The investigated TRIP steel in this work is an uncoated cold-rolled sheet. Its thickness is 1.0mm. The chemical composition of the tested sheets is shown in table 1. TABLE 1 Chemical composition of TRIP steel (mass %) C 0.11

Mn 1.55

Si 1.2

P 0.008

S 0.0069

Al 0.031

(a)R=90mm,h=50mm (b)R=100,h=65mm (c)R=105mm,h=78mm FIGURE 6 Deep drawn cups when blank radius is 90mm,100mm and 105mm

FIGURE 7 Calculated and measured thickness of the cups

The stamped cups are shown in Fig.6.The thickness is measured along the cup symmetry section. The measured data and the simulated ones are shown in Fig.7.In the figure, the solid lines are the simulated thickness distribution and the points with marks „Ʒ i are the measured data. The zero point of the horizontal axis begins from the bottom center of the cup. It shows that the simulated data agrees with the experimental results. The error lies in several aspects. First, the precision of the material model and material parameters are the main error sources. Second, the simulation is carried out in perfect material and process conditions, such as constant temperature, constant velocity, and uniformly-distributed lubrication oil and so on. And this perfect state is impossible for the experimental TRIP steel sheets. Third, finite element analysis has its own error when the continuous material is replaced by discrete meshes. Fourth, the experimental procedure is involved with the operators and instruments. So the personal error and instrument error also contribute to the overall error.

296

3.2 U-channel springback

Angles (q)

FIGURE 8 U-channel parts

T1

T2

Types of Angle FIGURE 9 Comparison of simulated results and experimental values

Six U-channel parts are stamped on a hydraulic press with a digital control of the blank holder force. Both of the two sides of the blank are oil lubricated and the other parameters are accordance with those used in simulation. The U-channel is taken from the tools carefully and springback angles are measured according to the method used in Numisheet’93. As shown in Fig.8, there are some differences after they are taken outside of the die although all of them have been stamped under almost the same condition. So springback angle has to be evaluated by the mean value of the six parts. All of the simulated and experimental angles have been listed in Fig.9. Compared with other simulation methods, springback angles simulated both with TRIP effect and with varied elastic modulus are closer to the experimental value. Compared with the simulation without TRIP effect and with constant elastic modulus, the simulation both with TRIP effect and with varied elastic modulus could be of about one third of the overall error. This shows that the TRIP effect and the degradation of elastic modulus need to be considered in order to obtain a more precise springback prediction. It needs to be remarked that the anisotropy of elastic modulus hasn’t been considered in this paper, which could be another important source to improve springback simulation precision.

CONCLUSIONS ୄ1୅GA micromechanical flow stress model for TRIP steels is proposed on the basis of the continuum mechanics and Mori-Tanaka mean field theory. In the model, TRIP effect and degradation of elastic modulus are considered. ୄ2୅GThickness simulated with the proposed flow stress model agrees well with the experimental results for cup forming. Thickness range simulated without the TRIP effect is wider than that simulated with the proposed model .It indicates that TRIP effect is an important property and should be considered in the formability analysis of TRIP steels. ˄3˅GSpringback angles simulated with TRIP effect and with varied elastic modulus are larger than those simulated without TRIP effect and with constant elastic modulus. And the simulated profile for U-channel with TRIP effect and with varied elastic modulus is closer to the experimental result than that without TRIP effect and with constant elastic modulus. The degradation of elastic modulus on springback needs to be considered so as to get a more precise springback simulation.

297

ACKNOWLEDGEMENTS This work has been together supported by National Natural Science Foundation of China (No.50705067) and by Shanghai Municipal Natural Science Foundation (No. 11ZR1439900).

REFERENCES N. Tsuchida and Y. Tomota, Mater. Sci. Eng. A 285, 345-352(2000). K.I .Sugimoto, U. Noboru, M. Kobayashi, et al., ISIJ Int.32, 1311-1318(1992). W.J. Dan, W.G. Zhang, S.H. Li, et al., Comput. Mater. Sci.40, 101-107(2007). A. Andersson, J. Mater. Eng.Perfor.16,301-307(2007). R.M. Cleveland, A.K. Gosh, Int. J. Plasticity 18,769-785(2002). I. Papatriantafillou, M. Agoras, N.Aravas, et al. Computer Methods in Appl. Mech. Engin. 195, 5094-5114(2006). P.J. Jacques, Q. Furnémont, F. Lani, Acta Materialia 55, 3681-3693(2007). L.Delannay, P.Jacques, T. Pardoen, Int. J Solids and Structures 45, 1825-1843(2008). V. Uthaisangsuk, U.Prahl, W.Bleck, et al., Engineering Fracture Mechanics78, 469-486(2011). Yu Haiyan, Bao Li; Gao Yunkai, Journal of Mechanical Engineering 46, 46-51(2010). H.Y. Yu, Y.K. Gao, and D.J. Meng, Mater. Sci. Eng. A 441,331-335(2006). J.D. Eshelby,ĀThe determination of the elastic field of an ellipsoidal inclusion and related problemsā,in Proceedings of the royal society of London Series A, Mathematical and Physical Sciences, 241(1957), p.376-396. [13] T. Mori and K. Tanaka, Acta Metall. Mater.21,571-574(1973).

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

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5754 Aluminum Sheet Deformed Along Bi-Linear Strain Paths M.A. Iadicolaa, L. Hub, A.D. Rollettb and T. Foeckea a

Metallurgy Division, National Institute of Standards & Technology, Gaithersburg, MD 20899 USA Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213 USA

b

Abstract. Sheet specimens of aluminum alloy 5754 were deformed along a series of bi-linear, equal-biaxial and uniaxial, strain paths while simultaneously measuring stress-strain behavior. Using the measured crystallographic texture before and after deformation, the VPSC model that incorporates texture evolution was used to simulate the flow stress and hardening behavior. Including latent hardening of multiple slip planes allowed the model to explain the decrease in flow stress when changing from equal-biaxial to uniaxial deformation. However, the model did not capture the details of the drop in flow stress nor the magnitude of the plastic hardening after the change in deformation mode. This is likely due to room temperature recovery between the two steps of testing. Keywords: 5754 Aluminum, bi-linear straining, crystal plasticity, strain path change, texture. PACS: 81.70.Bt

INTRODUCTION Metal forming simulations performed in strain space, using a forming limit diagram or other representation, suffer from a number of fundamental limitations. Since strain is not an intrinsic state variable, there is no single value that can universally represent the point of failure, and thus the limits of usable plastic strain are strong functions of the strain path [1]. Work has recently begun to shift from a strain-based to a stress-based modeling scheme, which removes path dependence, as stress is an intrinsic state variable, and the point of failure of a component can be identified by a single equivalent stress value. Stress-based forming limits are thought to be more applicable to some advanced alloys being considered for automotive applications [2], such as dual-phase and transformation-induced plasticity (TRIP) steels, in which flow stresses are high enough that the failure mode changes from ductile failure to decohesion, which is better described by a fracture stress than a strain to failure [3]. Transitioning from multi-axial strain space to multi-axial stress space requires a robust multi-axial constitutive relation that can accurately describe the evolution of the yield surface along any strain path. This requires information about how the yield surface evolves in stress space, which is difficult to measure, but is starting to be reported in the literature [5-10]. However, to develop an understanding that can predict the mechanical response, the yield-surface data must be tied to the structural characteristics of the sheet metal. A structural characteristic that shows promise is the initial and evolving crystallographic texture, which can be tied to flow stress via a crystal plasticity formalism that treats the sheet as an interacting aggregate of single crystal grains [11]. A recent study found reasonable agreement between the measured and predicted yield surfaces using a variety of crystal plasticity based analyses [12] for multi-axial behavior for the as-received material deformed through monotonic linear strain paths. This work is extended in the present study to determine if these modeling methods can be used to capture evolving flow stresses measured in sheet deformed along two sequential linear strain paths: equal-biaxial followed by uniaxial tension.

MATERIAL The material used in this investigation was industrially processed AA5754-O annealed sheet, nominally 1 mm thick, with chemical composition within nominal values. The grain diameter in the rolling plane is approximately 40 The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 299-306 (2011); doi: 10.1063/1.3623624 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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μm. The crystallographic texture of the as-received (AR) material is measured using an X-ray diffraction (XRD) technique on a conventional diffractometer with quarter Euler cradle, Cu KȻ radiation, and a multiwire area detector. The texture measured for the material will be reported elsewhere. The alloy derives its strength from the solid solution strengthening, with magnesium atoms interacting with dislocations and inhibiting dynamic recovery processes during straining. At room temperature, this alloy is known to exhibit yield point behavior and the Portevin-Le Chatelier (PLC) effect in its mechanical response [13]. This effect is also seen in the data discussed in this work.

EXPERIMENTAL PROCEDURES AND RESULTS Samples were tested through linear and bi-linear strain paths, as shown in Figure 2. As-received samples were strained in equal-biaxial and uniaxial tension to near failure. Other samples were strained in two steps: Step 1, equal-biaxial tension to prescribed strain levels and then in Step 2, strained in uniaxial tension in one of two orthogonal directions, RD or TD.

Figure 1: Linear strain paths plotted along true-strain axes: equal-biaxial (dotted), uniaxial RD (solid), and uniaxial TD (dashed). Example of two step forming shown in grey where Step 1 is equal-biaxial straining to approximately 10 % truestrain followed by Step 2 uniaxial straining in the TD.

Figure 2. Measured equal-biaxial true-stress true-strain data RD (solid circles) and TD (open circles), measured using XRD for two experiments in each direction [6].

A modified form of the Marciniak cup test is used to deform the as-received samples in equal-biaxial tension to near failure [4]. There is no direct method to calculate stress from the measured punch force, due to unquantifiable friction; therefore, a method of stress determination through X-ray diffraction measurements, carried out during holds in the deformation, was performed on the surface of the sample above the centerline of the ram [6]. This method requires calibration of the effective X-ray elastic constant (XEC) [6-7], which was accomplished using uniaxial testing. Plastic strains for all of the biaxial samples were measured using a calibrated compact biaxial extensometer mounted slightly off the center of the specimen [6]. Figure 2 is a plot of the measured equal-biaxial true-stress true-strain data for two experiments in each direction: RD and TD. The uncertainties shown for each data point are based on the larger of either the systematic error or the quality of the XRD measurement scan. The repeat tests showed good agreement when considering the uncertainties of the measurements.

300

Figure 3a: AR uniaxial true-stress vs. true-strain plots for RD. In both plots, solid lines are a representative test results for continuously strained samples and data points are data from XRD stress measurement during extension holds using compact testing frame and measured effective XEC [6].

Figure 3b: AR uniaxial true-stress vs. true-strain plots for TD.

Uniaxial specimens in the RD and TD were machined to the ASTM E-8 sub-size geometry for sheet metals. Some samples were tested by standard methods using an 88 kN electro-mechanical testing machine with calibrated load cell, axial extensometer, and transverse extensometer. Three repeat tests in each orientation (RD and TD) showed excellent agreement. The results of a single representative test, in each direction (RD and TD), are shown in Figure 3 as solid lines. The yield point phenomena (in TD) and serrated yielding associated with the PLC effect (in RD and TD) are visible in the hardening behavior. Four samples were tested using a compact testing frame, also with a calibrated load cell and axial extensometer, with the addition of the XRD stress determination. In Figure 4, the results of two of these interrupted XRD samples (one RD and one TD, shown as circles) were compared with those of the continuously strained standard test results, (solid lines) and show excellent agreement, typically within the serrated yield stress variation. The uncertainties of the measurements are approximately twice the height of the data points shown. The effective XECs used in these measurements were partially verified using similar tests with frequent elastic unload-reload cycles during plastic deformation. The variation of effective XEC, 1/2 S2, was found to be minimal in the range of testing [7]. In the Step 1 bi-linear strain path tests, the Marciniak method and biaxial extensometer described above were used to strain as-received samples near to target levels of 5 %, 10 %, 15 %, and 20 % true-strain in each direction (along the dotted path in Figure 2), resulting in equivalent strains of approximately 10 %, 20 %, 30 %, and 40 % true-strain, respectively. In the remainder of this paper, we will describe results in terms of the true strain measured in the direction of interest, not in terms of the equivalent strain. After achieving the prescribed strains, the Marciniak samples were unloaded and trimmed as shown in Figure 8 for use in multiple studies including crystallographic texture development [14], surface roughening [15,16], XEC determination [7], and the bi-linear strain path results shown here. The uniaxial samples for Step 2 were mechanically machined from the Marciniak strained specimens in the RD and TD to a shape similar to the ASTM E-8 sub-size geometry (with shorter end tabs). No effort was made to prevent room temperature recovery of the samples after unloading from equal-biaxial straining. The samples were stored at room temperature for approximately 6 months before Step 2 machining and testing.

301

Figure 4a: Step 2 uniaxial true-stress vs. true-strain plots after various levels of equal-biaxial straining in RD, with initial elastic loading shown as dotted lines. In both plots, the Voce fit of the AR uniaxial data (from Figure 4) and the AR equalbiaxial data (from Figure 3) are shown as dashed lines for comparison.

Figure 4b: Step 2 uniaxial true-stress vs. true-strain plots after various levels of equal-biaxial straining in TD, with initial elastic loading shown as dotted lines.

In the Step 2 tests, the biaxial strained and trimmed samples were tested using the compact uniaxial frame with the calibrated axial extensometer, calibrated load cell, and the XRD system for stress determination similar to the AR uniaxial effective XEC tests described above. The testing cycle included plastic deformation followed by elastic unload-reloads (again with stress levels kept above 80 % of the AR initial yield stress) followed by further plastic straining. This permitted determination of effective XEC after the biaxial deformation. The results showed minimal variation of effective XEC and therefore a single value was used in each direction (RD and TD) [7]. From the load cell and axial extensometer measurements, during these same tests, the Step 2 continuous mechanical responses were derived and shown in Figure 4 for RD and TD, solid and dashed lines respectively. In Figure 4, the unloadreload portions have been removed and the initial elastic loading is shown as dotted lines. Dashed lines show the Voce fit to the AR uniaxial data (from Figure 3) and the AR equal-biaxial data (from Figure 2) for comparison of the AR monotonic material response to the bi-linear response. As expected, the Step 2 maximum uniform elongation decreases with increased Step 1 strain, but the entire response is more of interest in this paper. The crystal plasticity model results for the two Step process are compared to these data, through a combination of the AR equal-biaxial data up to the Step 1 strain level and the plastic hardening portion of the uniaxial data (solid lines in Figure 4) for straining during Step 2.

CRYSTAL PLASTICITY CONSTITUTIVE ANALYSIS The plasticity model used in this simulation is the Visco-Plastic Self-Consistent (VPSC) model developed by C. N. Tomé and R. A. Lebensohn at Los Alamos National Laboratory [11]. The single-crystal constitutive behavior is given by

Dijg

¦

N s

J? 1 0

mijs

W

s n o

m V , i, j,k s kl

g K lk

1,3

(1)

Dg: strain rate tensor g: stress tensor. The superscript g indicates that the tensors represent the state in the grain as opposed to macroscopic properties N: number of slip systems J?0 : reference shear rate

W os : critical resolved shear stress (CRSS) of the slip system s

K: the inverse of the rate sensitivity exponent.

s

ms: Schmid tensor of slip system s defined as m ij the slip direction and n is the slip plane normal.

302

1 2

(bi n j  b j n i )

, where b is

The VPSC model deals with the interaction of grains by introducing a homogeneous effective medium (HEM). It treats each grain as a viscoplastic ellipsoidal inclusion embedded in and interacting with the effective medium. The stress and strain in the grains can vary as a function of orientation, depending on the stiffness of the interaction. The properties of the HEM are not known a priori and are adjusted self-consistently to coincide with the average of all inclusions forming the aggregate. As a result, the stress equilibrium and deformation compatibility conditions are satisfied in an average sense. Metals generally strengthen with plastic deformation. This strain hardening behavior is described by the evolution of the CRSS on each slip system in the plasticity model,

?D W

¦E

N

?E h J

1 DE

(2)

is the hardening rate on the slip system , and it is related to the shear rate on all the slip systems by a where hardening modulus matrix hDE . Several mechanisms contribute to the increase in strength, and those considered in this work include primary work-hardening, latent hardening and kinematic hardening. In the latent hardening theory, non-active systems harden differently from the active systems. Kinematic hardening is the softening of a slip system upon strain reversal. Incorporating anisotropic hardening behaviors, Equation 2 can be rewritten as,

¦E

N

?D W

?E q hE J

1 DE

(3)

is the primary work-hardening rate, and is a hardening coefficient matrix that incorporates latent where hardening and kinematic hardening behaviors. This equation reduces to the isotropic hardening if the hardening coefficients equal one everywhere. The increase of the CRSS from primary work-hardening is given by a Voce hardening law,

§

§

©

©

W W 0  W1¨ ¨1  exp¨J

h0 ·· ¸¸ ¸ W 0 ¹¹

(4)

where  is the accumulated shear on the slip system, 0, (0+1), and h0 are the initial CRSS, the back-extrapolated CRSS, and the initial hardening rate respectively. The initial texture was incorporated into the plasticity simulation using a set of 24,939 weighted orientations derived from the measured texture. The experimental stress-strain curves used to identify these parameters include AR RD uniaxial tension, AR TD uniaxial tension, AR equal-biaxial in the RD and TD, and the eight Step 2 uniaxial tension curves (Figure 4) with varying levels of Step 1 straining (four in each orientation). A single set of strain hardening parameters was identified to be the best match to all of these stress-strain curves simultaneously.

MODEL RESULTS AND DISCUSSION Previous plasticity simulation work was carried out to predict the stress-strain curves and texture evolution for five different monotonic strain paths, including uniaxial tension in the RD and TD, plane-strain in the RD and TD, and equal-biaxial stretch [12]. In this work, the isotropic hardening behavior described by Equation 4 was imposed in the VPSC simulation, and an optimal set of strain hardening parameters (isotropic row in Table 2) was determined to be a best fit to the experimental data. Using these material parameters, an initial trial of the stress-strain curve calculation for the strain-path change tests on the same sheet aluminum alloy was performed using the isotropic hardening only. Figure 5 shows two representative plots of the comparison between the VPSC predictions and the experimental curves for the case of 5% strain in Step I; the other, larger Step I strains showed very similar results. The model predicted a slight increase of the flow stress upon the strain path change from Step 1 to Step 2, whereas the experiment shows a much larger decrease. The model also was not able to capture the high hardening rate at the initial portion of Step 2 plastic deformation seen in the experimental curves. Strain path changes often involve activations of formerly latent slip systems, and possibly slip system reversals. The large discrepancy between the isotropic hardening model prediction and the experimental data demonstrate that anisotropic hardening behaviors must be added to the analysis. Plastic deformation occurs on the {111}-<110> slip

303

systems in FCC materials. Considering both the forward and backward slips (i.e., two slip systems on the same slip plane but along opposite slip directions), there are in total 24 slip systems. Since the latent hardening is considered to be induced by the interaction between the mobile dislocation and the forest dislocations, the slip systems are divided into four coplanar groups associated with four (111) planes: (111), (1-11), (11-1), & (-111). The different types of interactions between the slip systems are categorized into three groups:

qDE qDE qDE

qself 1 , if Ȼ and ȼ are coplanar but not in a forward and backward slipping pair; qlatent z 1 , if Ȼ and ȼ are non-coplanar; qkinematic  1 , if Ȼ and ȼ are in a forward and backward slipping pair.

The fitted material parameters are shown in the bottom row of Table 2. Table 2. Strain hardening parameters for VPSC model fitted to AR multi-axial behavior (top row, from [12]), or all the available AR uniaxial, AR equal-biaxial, and bi-linear strain path data (bottom row). Hardening 0 [MPa] 1 [MPa] H0 [MPa] qlatent qkinematic Isotropic 33 96 310 1 1 Latent 33 85 330 0.5 1

Figure 5a. Comparison of the experimental stress-strain data (AR equal-biaxial, dots, and Step 2 uniaxial, crosses) and the VPSC predictions (solid lines) with isotropic hardening only. Plots are combined results for Step 1 equal-biaxial and Step 2 uniaxial tension in RD after 5 % Step 1 straining.

Figure 5b. Comparison of the experimental stress-strain data (AR equal-biaxial, dots, and Step 2 uniaxial, crosses) and the VPSC predictions (solid lines) with isotropic hardening only. Plots are combined results for Step 1 equal-biaxial and Step 2 uniaxial tension in TD after 5 % Step 1 straining.

(b)

(a)

304

(c)

(d)

(e)

(f)

(g)

(h)

Figure 6. Comparison of the experimental stress-strain data (AR equal-biaxial, dots, and Step 2 uniaxial, crosses) and the VPSC predictions (solid lines) incorporating latent hardening. Plots are combined results for Step 1 equal-biaxial and Step 2 uniaxial tension in RD after (a) 5 %, (b) 10 %, (c) 15 %, and (d) 20 % Step 1 straining, and in TD after (e) 5 %, (f) 10 %, (g) 15 %, and (h) 20 % Step 1 straining.

The slip system activities were investigated to determine whether kinematic hardening behavior is relevant in this simulation. The model simulated a uniaxial tension in the RD after a 15 % equal-biaxial Step 1 straining. The calculated shear strains on slip systems that reverse during the strain path change were recorded and normalized by the total strain in the grain. An average of only 0.31 % of such normalized shear strains was found. Varying the Step 1 strain levels was also analyzed, but the effect was negligible. We concluded that kinematic hardening does not play a role in this particular type of strain path change, and the kinematic hardening coefficient qkinematic is therefore set to 1. Latent hardening is included in the plasticity modeling to capture possible anisotropic hardening behaviors. Three values of latent hardening coefficient qlatent were tested in the simulation: 0.5, 1, and 1.5. For a set value of latent hardening coefficient, the rest of the material parameters were iterated to find the closest match between the predictions and the experimental data. A statistical best fit was found when qlatent = 0.5. The calculated stress-strain curves are shown in the Figure 6. By incorporating the latent hardening, the model successfully predicted a flow

305

stress decrease upon the strain path change. However, taking latent hardening into account, the simulation still does not fully capture the transient softening upon the strain path change. Literature studies [17,18] have reported that deformed Al-Mg alloys, similar to AA5754, could recover at room temperature. Magnesium atoms pin dislocations, due to their high diffusivity during deformation, which severely retards dynamic recovery. This leads to a high stored energy, providing a large driving force for static recovery. It is possible that static recovery occurred in the Step 1 strained samples during the time interval before they were tested in Step 2, which along with anisotropic hardening behaviors, would also contribute to the increase in the hardening rate and decrease in the initial yield strength in the second loading. Taking account of this recovery effect may explain the transient behaviors observed during the strain path change. In addition, with the amount of recovery identified, material parameters can be determined more accurately from the measured data. These are the topics of continuing work.

CONCLUSIONS The stress-strain response was measured in samples of aluminum alloy 5754 sheet subjected to various bi-linear strain paths, consisting of equal-biaxial tension followed by uniaxial tension. Simulations of the strain path change effect were performed using the Visco-Plastic Self-Consistent (VPSC) crystal plasticity model. It was found that incorporating latent hardening behavior improved the predictions of the mechanical response over simple isotropic hardening. Addition of kinematic hardening was found to not significantly change the results. The large hardening rate seen in the uniaxial response after equal-biaxial straining, potentially a result of dynamic recovery, could not be effectively simulated using any of the hardening models.

REFERENCES 1. W.F. Hosford and J.L. Duncan, JOM 51 (11) (1999) 39-44. 2. T.D. Stoughton and X. Zhu, International Journal of Plasticity 20 (2004) 1463-1486. 3. K.S. Choi, W.N. Liu, X. Sun, M.A. Khaleel, Acta Materialia 57 (2009) 2592–2604. 4. T. Foecke, M. A. Iadicola, A. Lin and S. W. Banovic: Metallurgical and Materials Transactions A 38 (2007) 306-313. 5. M.G. Stout, P.L. Martin, D.E. Helling, G.R. Canova, International Journal of Plasticity 1 (1985) 163-174. 6. M.A. Iadicola, T. Foecke, and S.W. Banovic: International Journal of Plasticity 24 (2008) 2084-2101. 7. M.A. Iadicola and T.H. Gnaeupel-Herold: Advances in X-ray Analysis 51 (2008) 131-137. 8. T. Kuwabara, International Journal Of Plasticity 23 (3) (2007) 385-419. 9. K.Yoshida, T. Kuwabara , M Kuroda, International Journal Of Plasticity 23 (3) (2007) 361-384. 10. M.O.Andar, T. Kuwabara, S. Yonemura, A. Uenishi, ISIJ International 50 (4) (2010) 613-619. 11. R.A. Lebensohn and C.N. Tomé, Acta Metallurgica 41 (9) (1993) 2611-2624. 12. L. Hu, A.D. Rollett, M. Iadicola, T. Foecke, and S. Banovic, Metallurgical and Materials Transactions (submitted 2010). 13. J. Kang, D.S.Wilkinson, M. Jain, J.D. Embury, A.J. Beaudoin, S. Kim, R. Mishira, and A.V. Sachdev, Acta Materialia 54 (2006) 209–218. 14. S.W. Banovic, M.A. Iadicola, and T. Foecke, Metallurgical and Materials Transactions A 39A (2008) 2246-2258. 15. M.R. Stoudt, J.B., Hubbard, M.A., Iadicola, and S.W. Banovic, Metall. Materials Trans. A 40A (2009) 1611-1622. 16. J.B. Hubbard, M.R. Stoudt, and A. Possolo, Materials Science and Technology (2010 Submitted). 17. Go, J. (2001) Recovery and recrystallization behavior of AA5754 and IF-Boron steel during annealing. PhD thesis. University of British Columbia. 18. H. Halim, D.S. Wilkinson and M. Niewczas, Acta Materialia 55 (2007) 4151-4160.

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A Dislocation Density Based Crystal Plasticity Model of 3D Complex Sheet Metal Forming Haiming Zhang1, Xianghuai Dong1*, Qian Wang1, Zhiying Chen2 1

National Die & Mold CAD Engineering Research Center, Shanghai Jiao Tong University, 1954 Hua Shan Road, Shanghai 200030, China 2 School of Mechanical Engineering, Shanghai Dianji University, 690 Jiang Chuan Road, Shanghai 200240, China Abstract. A dislocation density based crystal plasticity model is used to simulate sheet metal forming processes under different temperature. In the present model, dislocation densities in each slip system are taken as internal state variables. An extended Kocks-Mecking-Estrin evolution law is adopted to describe the athermal storage, and the strain rate and temperature sensitive annihilation, of dislocation densities. The critical resolved stress on a specified slip system is related with dislocation densities in all slip systems, the corresponding mesoscopic coefficients of interaction between different slip systems are determined by recent dislocation dynamics simulations. On the basis of thermally activated dislocation motion, this physically based model is capable of describing the softening resulted from dynamic recovery. This model incorporated with a hyperelastic constitutive law is then implemented into an explicit finite element code, to make it deal with the strong nonlinear problems more efficiently. An iteration-free semi-implicit method, which is more suitable for explicit FEM, is adopted as the integration algorithm of the model. This model is used to investigate the deformation characteristics of warm forming of Al-Mg alloy under different temperatures. The simulation results show that the model is robust and efficient, has the potential to be used in the practical engineering simulation. It also confirms that the micromechaincs models can be used to simulate conventional sheet metal forming. Keywords: Dislocation density, Crystal plasticity, Physically based model, Sheet metal forming

1 INTRODUCTION In the last two decades, crystal plasticity encompasses a large number of now wide-spread models accounting for the anisotropic deformation of crystalline material. Considerable progresses have been witnessed in the areas related to heterogeneous plastic deformation, lattice rotation and texture evolution when metals are subjected to large deformation, or to solve engineering problems like metal forming. In the majority of crystal plasticity models, the flow rule is considered as a power law of strain and the hardening law is considered as a power law of strain and strain rate [1]. These models are still pure phenomenological, the “material constants” are only fit parameters that do not relate to the micro-structural state of the material. One of the limitations of these conventional crystal plasticity models is that they are not fully capable of predicting properties of the material beyond the available experimental data. In contrast with the phenomenological models, the physically based material models, which consider the plastic flow as the result of the movement of dislocations in a continuum way, have the advantage that the prediction of material behavior for deformations out of the available experimental data is more certain. Furthermore, the physical significance of model parameters is very helpful with regard to the discussion of model prediction and real material behavior [2]. In this work, we couple the dislocation density based material model with crystal plasticity within the finite element framework, to study heterogeneous plasticity behavior and dislocation microstructure evolution during plastic deformation. This physically based theory describes plastic flow as the result of the movement of dislocations (namely slip) in a continuum way. As a result, the model keeps the advantages of crystal plasticity for studying heterogeneous plasticity, and is able to study dislocation microstructure evolution at the same time. Generally, to formulate the dislocation density based crystal plasticity, two sets of equations are necessary ingredients. One is the evolution equation for dislocation density, which describes the competition of storage and annihilation of dislocation, another is the Taylor-type equation which describes the work hardening. _______________ Corresponding author: phone/fax +86 21 62813435 Email address: [email protected]

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 307-313 (2011); doi: 10.1063/1.3623625 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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2 THEORY 2.1 Finite Deformation Crystal Plasticity Model In the present work, the kinematics of crystal plasticity model is based on the hyperelastic law. The multiplicative decomposition of the total deformation gradient F can be expressed as

F F eF p

(1)

Fe

Fp

corresponds to elastic distortion and super-imposed rigid body rotations of lattice, is the plastic where contribution arising solely from dislocation motion in the intermediate (undeformed) configuration. The multiplicative decomposition of deformation gradient leads to an additive decomposition of the velocity gradient as

 1 Le  F eF p F p 1F e 1 L FF

with

F p

(2)

 ¦D JD SD0 F p

(3)

where SD0 is the Schmid resolved tensor expressed in the intermediate configuration. The second Piola-Kirchhoff stress is defined by the elastic Lagrangian strain tensor as follows

T ƒ

1 eT e F F  I 2

(4)

ƒ is the fourth order anisotropic elastic tensor, for the cubic crystals, it can be specified in terms of three independent elastic constants E11 , E12 and E44 in the local coordinate systems [3]. To complete the constitutive equations, the Cauchy stress tensor relationship as

V in the current configuration is related to the Kirchhoff stress T through the 1

V det  F F eTF eT

(5)

In the intermediate configuration, which possesses the invariable slip direction mD0 and slip plane normal nD0 , D the resolved shear stress W can be expressed as

WD

SD0 :  F eT F e ˜ T

(6)

D

W is the driving force that prompts the massive dislocation lines to glide on the specified slip plane, and this D

consecutive process results in the irrecoverable plastic deformation. In other words, W is a bridge between crystal plasticity and dislocation density. Details in the derivation of the crystal plasticity model adopted in this work can be found in [4-5].

2.2 Dislocation Density Based Material Model Plastic deformation in the range of strain rates and temperatures where diffusion and creep are not dominant, is basically due to the glide of dislocations, and the motion of dislocations is impeded by both short-range and longrange obstacles. The short-range obstacles can be overcame by the help of thermal activation, however, the longrange obstacles are athermal, which can only be surmounted by the assit of the externally applied force. Based on the thermally activated mechanisms of dislocation motion at moderate rate, the relationship between plastic slip rate and dislocation density can be defined by the Orowan equation combined with the Arrhenius type equation [6]

JD

­ § W D  W aD ° 'G0 ¨ J0 exp ® 1 Wˆ0 ° k BT ¨© ¯

308

p ·q ½

¸ ¸ ¹

° ¾ sign W D ° ¿

(7)

Eq.(7) has a strong linkage to the thermally activated dislocation glide processes. W D is the stress applied on the dislocations, that is, the resolved shear defined in Eq.(6). W D is the long-range resistance, which describes the work a

hardening, includes the elastic-stress field due to grain boundaries, far-field forest of dislocation, and other structural defects. Wˆ is the mechanical stress required to cross the barrier without any thermal activation. Finally, W D can be 0

a

expressed as follow

W aD

FP T b

¦ E ADE U E



P T W P0 0

(8)

the first term is due to the elastic stress field of forest dislocation, and the second term is the approximation of all other long-range resistance, ADE is the matrix describes the anisotropic interactions between slip systems. Besides the Taylor-type equation Eq.(8), which combines work hardening with dislocation density, another necessity to dislocation density based models is the evolution equation for dislocation density. The evolution equation is in a differential form, which is determined by the competition between deformation induced dislocations multiplication and dislocations annihilation due to dynamic recovery, that is [7]

d UD

d U D  d U D

(9)

Multiplication of dislocations is generally assumed to be originated from the Frank-Reed type sources or from the grain boundaries, and experiments have shown that dislocations multiplication is an athermal process. As a result, for the polycrystalline material, dislocations multiplication can be expressed as

d U D

§ 1 1 · D D  ¨ ¸ dJ , L ¨ bDgrain bLD ¸ © ¹

K

¦ E MDE ˜ U E

(10)

where Dgrain is the average grain size, LD is the mean free path that the dislocations travel before being stored, which is proportional to the average spacing between the dislocations. Annihilation of mobile dislocations with opposite Burgers vectors is due to dynamic recovery, which explains the saturation of strain hardening. Detailed mechanism for dislocation annihilation is less well understood compared to the multiplication mechanisms of dislocations. In this work, the cross slip of screw dislocations is viewed as the only annihilation mechanism [8], In order to cross slip, the dislocation must overcome an energy barrier, therefore this phenomenon is temperature and strain rate dependent. In FCC metals, the most common cross slip mechanism is compact cross slip, i.e. Friedel-Escaig mechanism. Finally, the annihilation term is generally assumed to follow the first order kinetics, i.e. to be linear with the density of dislocations as follows

d U D

f D (JD , T ) U D ˜ d J D

(11)

where f D (JD , T ) describes the slip rate dependence and the thermal activated mechanism of cross slip. Ignoring the slip rate dependence, we write f D (J D , T ) as the Arrhenius type equation to describe the probability that the cross slip attempt is successful, as follows [9]

f D (JD , T )

ª 'G § W D ·º yc P0 exp «  cs ¨1  E ¸» P T ¹ »¼ «¬ k BT ©

(12)

3 NUMERICAL IMPLEMENTATION The expensive computational cost is always a major deterrent for the widespread using of crystal plasticity models in the practical engineering problems. Besides, the physically based models have more internal variables needed to be updated. Consequently, an efficient integration algorithm is necessity. In the present work, a forward gradient method is adopted, and the similar method was also adopted by Pierce et.al in the hypoelastic frame [10].

309

For the incremental FEM solvers, it is assumed the variables at time t are all known, and the solution dependent variables (SDVs) at time t  't are needed to be updated. Here, we use 'J D

't u JtD 't as the primary unknowns,

D

and the specified dislocation densities are updated directly by Eq.(9). Jt  't is approximated by using the first order Taylor's series expression as

§ · wJD wW D wJD wW aD wU E 't ¨ JtD  T1 'J E  T 2 'J E ¸ wW D wJ E wW aD wU E wJ E © ¹

'J D the partial derivative

wJD wW D

,

wJD wW aD

,

wW aD wU E

wU E

and

(13)

can be derived from Eq.(7), Eq.(8) and Eq.(9), respectively.

wJ E

By integrating Eq.(3), the inelastic deformation gradient at time t  't can be obtained as





Ftp't | 1  ¦ D 'J D SD0 Ftp

So, W D can be approximated as

(14)

¼»



W D | SD0 : ª A T  ¦ E 'J E CE º ¬«

(15)

with

A = Ftp T FtT't Ft 't Ftp 1 , T As a result,

wW D wJ E



1 1 ƒ   A  I , CD = ƒ  ASD0  SD0 T AT 2 2



(16)

can be obtained from Eq.(15), and a system of explicit simultaneous equations with 'J E

as unknown quantities can be established as

ª D «G DE  'G0 I D ˜ 'tJt « Wˆ0 k BT «¬

§ ·º D :  AC E E ˜ T S T b T FP d  U 1 2 0 ¨ ¸ » ˜ 'J E ADE  E ¨¨ sign W D ¸ G dJ ¸» 2 ¦ E ADG ˜ U t © ¹ »¼









D D D 'G0 I D T1'tJt S 0 : AT  W t D 'tJt  k T Wˆ sign W D B

0

t



(17)

where

ID

§ W D  W aD pq ¨1  ¨ Wˆ0 ©

p · q 1

¸ ¸ ¹

W D  W aD Wˆ0

p 1 (18)

The above integration procedure does not have any iteration, and no update is needed for the Schmid tensor and grain orientation matrix, because both are invariable in this procedure. Consequently, the whole procedure is concise and efficient, it is implemented into ABAQUS/Explicit by the user subroutine VUMAT.

310

4 MODEL VERIFICATION AND NUMERICAL APPLICATION 4.1 Material Parameters Identification Theoretically, the ideal physically based Models should not have to be parameter fitted, that is, all the needed parameters are the material intrinsic constants. However, due to the less understanding of some aspects of plastic deformation mechanism, the present dislocation density based models still have some empirical or fitted parameters. In this paper, the partial material constants are presented in Table 1. TABLE 1 Material parameters

Well-Defined Parameters b (Aluminum) k (Boltzmann constant) G0 p q

Assumed Parameter yc P0 Gcs

2.86h10-7mm 1.38065h10-23 J/K 0.20­b3 0.5 1.5

J0 Wˆ0

£

10b 18.0 0.78 eV 1.0h10-8s-1 30MPa 250

The recent discrete dislocation dynamic (DDD) simulation [11] provides the referent values for the parameters of dislocation interaction and dislocation evolution, e.g. ADE , M DE and K . Due to the difference of predefined conditions, a proportional constant D is introduced in Eq.(8), this constant is also used to scale LD . In this paper, F 2~3 combines with the parameters provided by DDD simulations shows good numerical performance. The temperature dependence of elastic constants is described by the MTS model, which is written as

Eij T and the shear modulus can be obtained from

P T

Eij0 



sij



exp tij / T  1

E44 T  E11 T  E12 T / 2

(19)

(20)

0 0 0 For aluminum and aluminum alloy, E11 , E12 , E44 , s11 , s12 , s44 , t11 , t12 and t 44 are 114.22GPa, 61.95GPa, 31.61GPa, 10.11GPa, 2.05GPa, 2.56GPa and 258.4K, 293.6K, 168.0K [12].

4.2 Verification of the Model In order to verify the dislocation density based crystal plasticity, a series of the uniaxial tensile simulations under different strain rates and temperatures are conducted, the polycrystal Al-Mg alloy AA 5754-O is chosen as the tested material, and the simulation results are compared with experiments from [13]. The initial texture is represented as mixture of the cube texture and random texture, with ratio of four to one, which is observed from experiment. The rest material parameters are set as: Dgrain

U0

25 P m , W 0

36MPa , the initial dislocation density

1011 m 2 .

The results are illustrated in Figure 1. It can be found that the simulation results are in good agreement with these from experiments. With increasing temperature, the flow stress becomes lower and the strain rate dependent becomes more obvious. With increasing strain rate, the stress becomes higher. It is concluded that the presented model is able to regenerate the stress-strain curves under a moderate large range of strain rate and temperature.

311

4.3 Numerical Application The presented dislocation density based crystal plasticity model is used to simulate the 3D sheet metal forming of cylinder deep drawing under different temperatures, to investigate the warm forming of the Al-Mg alloy AA 5754-O. The parameters are the same as the preceding sections. The blank with the thickness 1.2 mm is meshed by 3280 nodes combined with 1602 C3D8R elements, and every integration point is treated as polycrystal aggregate represented by 60 grains, which are random read from the texture file. The punch stoke is 55mm, and the total simulation time is 0.11s. The total increments are about 120000, and the cost time is about 7 hours with the computer configurations of Intel Core 2.33GHz CPU and 3.00GB memory.

(a) (b) FIGURE 1. True stress - strain curves of experiment and simulation, under different strain rates and temperatures. From the top to down, the corresponding temperature is 373K, 448k, 498K and 523K. (a) - strain rate 0.02 s-1; (b) - strain rate 0.1s-1.

FIGURE 2. Punch force - displacement curves

(a) (b) FIGURE 3. Dislocation density distribution maps (the unit is mm-2). (a)-373K, (b)- 448K, (c)- 498K

(c)

Figure 2. and Figure 3. are the “punch force - punch stroke” curves and the dislocation distribution maps under three different temperatures, e.g. 373K, 448K and 498K, respectively. With the increasing of temperature, the

312

annihilation of dislocations increases, the resistance to deformation decreases. As a result, the punch force will decrease as temperature increases, which is shown in Figure 2. During the drawing progress, the material in the flange, which bears the associated deformation of both tension and compression, must deforms heavily. The force necessary to deform the material is imposed by the punch on the bottom of the cup, mainly near the punch radius. Though failure often emerges in the area near the punch radius, the plastic deformation of the material in this area is less than the flange. It can be confirmed from Figure 3, which shows that the flange has the maximum dislocation density and the material near the punch radius has relatively low dislocation density. From Figure 3, it can also find that as the temperature increases, the dislocation density decreases. Besides, the higher the temperature of forming, the more uniform dislocation density distribution of the forming components. This phenomenon is mainly due to the physical fact that the increasing of temperature can accelerate the strain hardening saturating.

5 CONCLUSION In this paper, a dislocation density based crystal plasticity model is developed, and a semi-implicit integration algorithm is adopted to implement the material model into ABAQUS/Explicit by the user’s subroutine VUMAT. Because of coupling the micromechanical phenomena, i.e. dislocation density, to the macroscopic continuum quantities, i.e. strain and stress, the presented model is a type of multiscale modeling of material behavior, which is capable of investigating the traditional metal forming from the microscopic mechanism of plastic deformation. The comparison of experimental results and simulation results of uniaxial tension shows that the presented model has a relative strong applicability and generality in a moderate large range of strain rate and temperature. The 3D deep drawing simulation of a typical Al-Mg alloy AA 5754-O sheet shows that the model is robust and efficient, and the results reappear the typical deformation characteristics of deep drawing, it is concluded that the presented model has the potential to be used in the practical engineering simulation. It should be noted that, though many workers are striving to build up the physically based models from the micromechanic, however, due to the less understanding of some microscopic mechanism of plastic deformation, there are inevitably some empirical or fitted parameters in the presented dislocation density based crystal plasticity model,.

ACKNOWLEDGMENTS The authors would like to gratefully acknowledge the support of National Natural Science Foundation of China, Grant No. 50821003, 50835002 and 50975174.

REFERENCES 1. J. Hrder, A crystallographic model for the study of local deformation processes in polycrystals. Int. J. Plasticity,15, pp. 605624. 2. B. Berisha,P. Horaa, A. Wahlena, L.Tonga, A combined isotropic-kinematic hardening model for the simulation of warm forming and subsequent loading at room temperature . Int. J. Plasticity, 26, pp. 126-140. 3. U.F. Kocks, C.N. Tome, H.R. Wenk, Texture and Anisotropy, Cambridge, Cambridge University Press, 2000. 4. S.R. Kalidindi, C.A. Bronkhorst, L. Anand, Crystallographic texture evolution in bulk deformation process of fcc metals. J. Mech. Phys. Solids. 40, pp. 537-569. 5. A.M. Cuitino, M. Ortiz, Computational modeling of single crystals. Modell. Simul. Mater. Sci. Engrg. 1, pp. 225-263. 6. U.F. Kocks, H. Mecking, Physics and phenomenology of strain hardening: the FCC case. Prog. Mater. Sci. 48, pp. 171-273. 7. E. Nes. Modelling of work hardening and stress saturation in FCC metal. Prog. Mater. Sci. 41, pp. 129-193. 8. Y. Estrin, H. Meckling A unified phenomenological description of work hardening and creep based on one-parameter models. Acta Metall. 32, pp. 57-70. 9. W.D. Nix, J.C. Gibeling, D.A. Hughes, Time-dependent deformation of metals. Metall. Trans. A. 16, pp. 1985-2215. 10. D. Pierce, R. Asaro, A. Needleman, An analysis of nonuniform and localized deformation in ductile single crystals. Acta Metall. 30, pp. 1087-1119. 11. L. Kubin, B. Devincre, T. Hoc, Modeling dislocation storage rates and mean free paths in face-centered cubic crystals. Acta Mater. 56, pp. 6040-6049. 12. Y.P. Varshni. Temperature dependence of the elastic constant. Phys. Rev.B. 15, pp. 3952-3957. 13. A.H. van den Boogaard, “Thermally enhanced forming of aluminum sheet modelling and experiments”, Ph.D. Thesis, University Twente, 2002.

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Prediction of Yield Loci for a Magnesium Alloy Sheet using Crystal-Plasticity Finite-Element Method Takayuki HAMAa, b, Hitoshi FUJIMOTOa, and Hirohiko TAKUDAa a

Department of Energy Science and Technology, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501, Japan b VCAD System Research Program, RIKEN, 2-1, Hirosawa, Wako, Saitama 351-0198, Japan Abstract. This paper presents the prediction of yield loci for an AZ31 magnesium alloy sheet using a rate-dependent crystal-plasticity finite-element method. A differential work-hardening behavior was clearly observed; the contour of plastic work was initially rather flattened in the vicinity of equi-biaxial tension, but thereafter severely bulged. The variation of the relative activity of each family of slip systems was examined to investigate the mechanism of the differential work-hardening behavior. During uniaxial tension, the work hardening was determined mainly by the basal slip in the very beginning, while by both the prismatic slip and the basal slip in the subsequent deformation. On the other hand, during equi-biaxial tension, the activity of the basal slip systems was predominant throughout the deformation, while the relative activity of the prismatic slip systems was smaller than that in the uniaxial tension. We concluded that this difference in the relative activities of the slip systems depending on the biaxial-stress ratio eventually resulted in the differential work-hardening behavior of the contour of plastic work. The mechanism that the activity of the prismatic slip systems decreased as the biaxial-stress ratio approached to one was also discussed using a simple analytical model. Keywords: Magnesium alloy sheet, Yield locus, Crystal plasticity, Finite-element method. PACS: 46.70.-p

INTRODUCTION Magnesium (hereafter Mg) alloys are the lightest materials used for structural components and have recently been drawing much attention because of increasing demand for lightweight materials in automobile and electrical industries [1, 2]. Conventionally, components made of Mg alloys are mainly manufactured by die casting and thixoforming [1, 3, 4, 5]. The press forming of Mg alloy sheets has recently attracted attention because it can further expand the use of Mg alloys for structural components. There are a lot of studies on the press forming of Mg alloy sheets [3, 4, 5, 6]. To increase the number of applications of the press forming, it is crucial to understand mechanical properties of Mg alloy sheets. Because Mg alloys are typical hexagonal close-packed (HCP) materials, mechanical properties of Mg alloy sheets are significantly different from those of the conventional structural sheet metals that have cubic structures. Some examples are as follows: stress-strain curves of rolled Mg alloy sheets are notably different between tension and compression [7, 8], a stress-strain curve obtained under compression followed by tension exhibits a sigmoidal shape [7], and Mg alloy sheets show much stronger inelastic deformation during unloading than mild steel and aluminum alloy sheets [9, 10, 11]. Generally it is understood that these characteristic deformations are owing to the activity of direction-dependent {10 1 2} twinning. This is because a strong basal texture with most of the c-axes aligned in the thickness direction is initially formed in rolled Mg alloy sheets. Recently Andar et al. [12] measured experimentally the contours of plastic work in a rolled AZ31 Mg alloy sheet in the first quadrant of the stress space. The measured contours showed the differential work-hardening behavior significantly; the ratio of equibiaxial stress to uniaxial stress increased with the progress of plastic deformation in a range of plastic strain 0.001 ≤ ε p ≤ 0.008 . Tadano [13] also showed a qualitatively similar result in pure Mg using a crystal-plasticity finite-element analysis. It should be emphasized that such differential work-hardening arises in the first quadrant of the stress space where the {10 1 2} twinning is difficult to be activated, indicating that the The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 314-321 (2011); doi: 10.1063/1.3623626 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

314

differential work-hardening may not be primarily due to the activity of twinning, but due to other factors. To the best of our knowledge, causes of this behavior have not been understood yet. This paper is concerned with a prediction of contours of plastic work using a crystal-plasticity finite-element method in a rolled Mg alloy sheet. The mechanism of the differential work-hardening in the first quadrant of the stress space was examined in detail from a viewpoint of mesoscopic deformation.

FINITE-ELEMENT FORMULATION Virtual Work A crystal-plasticity finite-element method code developed in this study uses an updated Lagrangian rate formulation. Assuming that all the rate-form relations are preserved from time t to t+Δt, where Δt is a small time increment, an incremental form of the principle of virtual velocity is written in the form [14, 15] Δt ∫

V

{( σ

J

}

− 2σ ⋅ D ) : δD + ( L ⋅ σ ) : δL dV = Δt ∫ f ⋅ δvdS + ∫ f ⋅ δvdS − ∫ σ : δLdV S1

S1

V

(1)

where V and S denote, respectively, the domain occupied by the body and its boundary at time t. S1 is the part of the boundary S on which the nominal surface traction f is prescribed. δv is the virtual velocity, σ is the Cauchy stress tensor, σ J is the Jaumann rate of the Cauchy stress tensor, L is the velocity gradient tensor, and D is the strain rate tensor, which is the symmetric part of L. The second and third terms of the right-hand side remain in order to cancel the nonequilibrium between external and internal forces arising from the explicit time-marching algorithm.

Crystal-Plasticity Model The crystal-plasticity model employed in this study follows the formulations developed in the literature [11, 16, α 17, 18]. The crystalline slip is assumed to follow Schmid’s law. The slip rate γ ( ) of the α slip system is assumed to be given by a visco-plastic power law as

γ (α ) τ (α ) = (α ) γ0 τY where τ (

α)

1 m

( )),

sign τ (

α

τY(α ) = ∑ qαβ h γ ( β )

(2)

β

and τ Y( ) are Schmid’s resolved shear stress and the current strength of the α slip system, respectively, α

γ0 is a reference strain rate, m is the rate sensitivity exponent, and qαβ is the self (α = β) and latent (α ≠ β) hardening moduli. h is the rate of hardening, for which three different laws are used depending on the family of slip and twinning systems h = h0 ⎛ τ h = h0 ⎜ 1 − 0 ⎝ τ∞

(3)

⎞ ⎛ h0γ ⎞ ⎟ exp ⎜ − ⎟ ⎠ ⎝ τ∞ ⎠

⎧h0 ⎪ n −1 h=⎨ ⎛ γ ⎞ h ⎜ ⎟ ⎪ 0 ⎩ ⎝ γ ref ⎠

(4)

for γ ≤ γ ref for γ > γ ref

315

(5)

with τ Y( ) = τ 0 at the beginning, and α

γ = ∑ ∫ γ (α ) dt.

(6)

α

In this study, three families of slip systems–basal slip , prismatic slip , and pyramidal-2 slip – and one family of {10 1 2} twinning systems are used to model the mechanical behavior of a rolled AZ31 Mg alloy sheet. There are three basal, three prismatic, six pyramidal-2, and six twinning systems. The linear hardening (eq. (3)) is assumed for the basal slip, and Voce hardening (eq. (4)) is assumed for the prismatic slip and pyramidal-2 slip. Twinning is assumed to have a polar character in which each system is activated only by tension of the c-axis. Because shear strain that arises in a grain due to twinning can be assumed to be similar to the shear caused by an activated slip system [11, 19], the shear strain induced by twinning is also calculated using eq. (2). The combined hardening (eq.(5)) is assumed for the twinning, with γref denoting the shear strain due to twinning arising in a grain twined entirely. For the sake of simplicity, the lattice rotation due to twinning and the so-called untwinning is not taken into consideration. Because the deformation considered in this study is at most plastic strain of 0.01 in the first quadrant of the stress space and the activation of twinning is negligible as explained below, the above simplification may not affect the simulation result. The above rate-dependent crystal-plasticity model is incorporated into each Gauss point in the static finiteelement method [11, 16, 20]. The rate tangent modulus method [17] is used for the explicit time integration of the constitutive model. To prevent excessive increase of non-equilibrium between external and internal forces, the generalized rmin-strategy is employed. The plastic work Wp is given in the form ns

W p = ∫ σ :D p dt, with D p = ∑ γ (α ) α =1

(

1 (α ) s ⊗ m (α ) + m (α ) ⊗ s (α ) 2

)

(7)

where Dp is the plastic strain rate tensor, the unit vectors s(α) and m(α) are the slip direction and slip plane normal, respectively, and ns is the number of slip and twinning systems in a grain. A contour of plastic work was calculated using the following procedure proposed in the literature [12]. A uniaxial tensile test in RD was selected as a reference condition. The uniaxial stress σRD and the plastic work Wp given by eq.(7) are calculated at various p which is given by integrating the plastic strain rate DP. Then the biaxial stresses σRD and uniaxial plastic strains ε RD σTD obtained for various preset biaxial-stress ratios σTD/σRD are determined at the same plastic work. Consequently p by plotting the obtained stresses in the principal stress the contours of plastic work can be obtained for various ε RD p space. The contour of plastic work was calculated up to ε RD =1.0% in this study. It should be noted that the plastic work Wp may not be zero even during the elastic range in the stress-strain curve because of the rate-dependent model of crystal-plasticity.

MATERIAL MODELLING A finite-element model used in the present study was a cube which had seven uniform eight-node isoparametric brick elements using selective reduced integration in each direction, as shown in Fig. 1 (a). The total number of elements was 343. In this study, the same initial crystallographic orientation was assigned to all eight Gauss integration points in an element, thus the present model had 343 initial crystallographic orientations. Figure 1 (b) shows the initial (0001) pole figure used in the simulation. The initial crystallographic orientations were artificially created using a procedure proposed by Mayama et al. [20] to simulate the rolling texture of Mg alloy sheets [7, 8]. In the simulation, the x-, y-, and z-axes in Fig. 1 (a) were defined to be the rolling direction (RD), transverse direction (TD), and normal direction (ND), respectively, thus the majority of c-axes tended to align in the z direction (ND). The following boundary conditions were used to simulate the contour of plastic work: the planes x = 0, y = 0, and z = 0 were fixed in the x, y, and z directions, respectively, due to the assumption of plane symmetries, and the small displacement increments were given to the planes x = l and y = l such that pseudo-proportional stress paths were achieved.

316

RD (b)

(a)

l

TD

z

y

x

l

l

FIGURE 1. Material modelling used in the simulation. (a) Finite-element model and (b) (0001) pole figure. TABLE 1. Latent hardening parameters qαβ used.

300

Prismatic Pyramidal-2 Twinning 0.5 0.5 0.5 0.2 0.2 0.5 1.0 0.2 0.25 1.0 0.2 0.25

200 True stress /MPa

Basal Prismatic Pyramidal-2 Twinning

Basal 0.2 0.2 1.0 1.0

TABLE 2. Calibrated material parameters in equations (3), (4), and (5) used in the present simulation.

τ0 τ∞ h0 n

Basal 10 20 -

Prismatic Pyramidal-2 Twinning 84 160 48 260 360 680 750 150 10

Simulation

100 0 -100 Experiment

-200 -300 -0.1

-0.05

0

0.05

0.1

Logarithmic strain ε FIGURE 2. Stress-strain curves obtained by uniaxial tensile and compressive tests.

From an experiment of a uniaxial tensile test of the AZ31 Mg alloy sheet with a thickness of 0.8 mm, Young’s modulus E = 42 GPa and Poisson ratio ν = 0.3 were assumed [6, 11]. The rate sensitivity exponent was set to m = 0.02 and the reference strain rate to γ0 = 0.001 s-1. The choice of m has been explained in the literature [11]. The self-hardening parameters were set to 1, while the latent-hardening parameters qαβ shown in Table 1 were adopted based on the literature [11, 16]. γref in equation (5) was 0.027. Other parameters used in equations (2), (3), (4), and (5) were identified by trial and error to achieve a reasonable fit with experimental stress–strain curves obtained by the uniaxial tensile and compressive tests of the AZ31 Mg alloy sheet, as shown in Fig. 2. The determined parameters are shown in Table 2.

RESULTS AND DISCUSSION Contour of plastic work To examine a linearity of stress path during deformations in the simulations, Fig. 3 shows the evolutions of σRD and σTD for various preset biaxial-stress ratios. Clearly fairly-linear stress paths are achieved in the simulations regardless of the stress ratios, indicating that the simulation procedure adopted in this study is acceptable to calculate a contour of plastic work. p p . From the beginning of plastic deformation to ε RD Figure 4 shows the contours of plastic work at various ε RD

317

300

300 250

250

200

200

σTD /MPa

σTD /MPa

p ε RD

150 100

0.08% 0.14% 0.16% 0.2% 0.24% 0.3% 0.45% 0.64% 1.0%

150 100

50

50

0

0 0

50

100

150

200

250

300

0

σRD /MPa

50

100

150

200

σRD /MPa

250

300

FIGURE 4. Contours of plastic work obtained using p simulation. The numbers shown in the legend are ε RD .

FIGURE 3. Stress paths during deformation up to p ε RD = 1.0%.

=0.16 %, the contour is flattened in the vicinity of σTD/σRD =1. Subsequently, in the uniaxial plastic-strain range p ≤ 0.3 %, the work-hardening rates in the vicinity of σTD/σRD =1 become larger than those in the 0.16 % ≤ ε RD vicinity of σTD/σRD =0 and the contour bulges severely. Then the work-hardening rates in the vicinity of σTD/σRD =1 p p higher than 0.3%. Clearly the shapes of the contours change as ε RD increases, exhibiting become small again at ε RD the differential work-hardening behavior in the simulation result. The trends observed in the simulation results are qualitatively in good agreement with those of the experimental results reported by Andar et al. [12], demonstrating that the present simulation results are qualitatively acceptable. Because the definition of plastic strain is different between the experiment and the simulation, a quantitative comparison between the experiment and the simulation is not conducted here.

Relative Activity of Slip Systems In the following, a variation of the relative activity of each family of slip and twinning systems is used to investigate the mechanism of the differential work-hardening behavior. The relative activity is the relative contribution of each family of slip and twinning systems to plastic deformation. The relative activity of each family of slip and twinning systems i is given in the form [11, 21]

∑ ri =

∑

(

p k a family of slip and twinning systems , i

∑ p

(

∑

j all families of slip and twinning systems

)

)

Δγ ( Δγ (

p,k )

p, j )

(8)

where the numerator is the plastic strain increment contributed by the family of slip or twinning systems i, summed over all grains. The denominator is the plastic strain increment contributed by all the families, summed over all the grains. The variations of the relative activity as a function of the plastic work obtained from uniaxial and equi-biaxial p is also shown in Fig. 5 (a). The tension are shown in Fig. 5. The correlation between the plastic work and ε RD variation of the relative activity during the uniaxial tension (σTD/σRD =0) (Fig. 5 (a)) is as follows. The activity of the basal slip systems is predominant at the very beginning of plastic deformation. Shortly after that the prismatic slip systems start to be activated rapidly and at the same time the relative activity of the basal slip systems decreases. During the middle stage of plastic deformation, both the prismatic slip and the basal slip systems are activated but the relative activity of the prismatic slip systems is more predominant. The relative activities of the pyramidal-2 slip

318

1

Relative activity

0.8

0.008

0.6

0.006

(b)

Prismatic

0.4

0.004 Basal

0.2

0.002

Uniaxial plastic strain

0.01

Relative activity

(a)

1 0.8 Basal

0.6 0.4

Pyramidal

0.2

Prismatic

Twinning

0

0 0

0.2

0.4 0.6 0.8 Plastic work / MJ*m-3

0

1

0

0.2

0.4

0.6

0.8

1

Plastic work / MJ*m-3

FIGURE 5. Variations of relative activities as a function of plastic work. (a) Uniaxial tension, and (b) equi-biaxial tension.

and twinning are negligible throughout the deformation simulated in this study. This result shows that the work hardening during the uniaxial tension is determined mainly by the basal slip at the very beginning, while by the prismatic slip and the basal slip after that. The variation of relative activity during the equi-biaxial tension (σTD/σRD =1) (Fig. 5 (b)) is different from that of the uniaxial tension as below. The activity of the basal slip systems is predominant at the beginning as well as the uniaxial tension, but the strain range where this trend is kept is larger than that in the uniaxial tension. Subsequently the prismatic slip and pyramidal-2 slip systems start to be activated gradually. Because the relative activities of the prismatic slip and pyramidal-2 slip systems are much smaller than those in the uniaxial tension during the middle stage, the relative activity of the basal slip systems is kept being predominant. Interestingly the relative activity of the pyramidal-2 slip systems is larger than that of the prismatic slip systems. The relative activity of the twinning systems is negligible throughout the deformation in this study. From this result, the mechanism of the work hardening during the equi-biaxial tension may be explained as follows. The stresses are small in a small plasticstrain range because only the basal slip systems are activated at the beginning, while after that the work-hardening rate increases during the middle stage because not only the basal slip systems but also, to some extent, the pyramidal-2 slip and prismatic slip systems start to be activated gradually. The overall transitions of relative activity are qualitatively similar to those of the uniaxial tension, but the strain range where only the basal slip is activated at the beginning is larger and the increases in the relative activities of the prismatic slip and pyramidal-2 slip are more gradual than the uniaxial tension, thus emphasizing the change in the work-hardening rate from the initial to middle stages. Summarizing the above results, we conclude that the differential work-hardening in a rolled Mg alloy sheet is due to the facts that the slip systems that govern the work hardening are notably different depending on the biaxial stress ratio and that the variation of the relative activity is also different depending on the biaxial stress ratio.

Mechanism of Differential Work-Hardening p Figure 6 shows the correlations between the relative activities and the biaxial stress ratio at ε RD =0.2 % and 1.0 %. In the both results, the relative activity of the prismatic slip systems is the largest at σTD/σRD =0 (uniaxial tension). This decreases as the biaxial stress ratio σTD/σRD increases, whereas the relative activities of other slip systems, i.e. p p the basal slip systems for ε RD =0.2 % and the basal slip and pyramidal-2 slip systems for ε RD =1.0 %, increase with σTD/σRD. It is presumed from these results that one of the main reasons of the differential work-hardening behavior is the relative activity of the prismatic slip systems decreases as the biaxial stress ratio increases. A mechanism of the change in the activity of the prismatic slip systems is examined using the following simple analytical model. The directions of c-axes in rolled Mg alloy sheets in fact exhibit some variation from the sheet normal direction as shown in Fig. 1 (b). However, to simplify the model, an idealized HCP crystal structure whose c-axis trends exactly in the sheet normal direction is considered as shown in Fig. 7. Referring to the x-y coordinate system in Fig. 7, the unit vectors s and m of an arbitrary prismatic slip can be given as

⎧cos θ ⎫ ⎧cos ϕ ⎫ m=⎨ ⎬, s = ⎨ ⎬. ⎩sin θ ⎭ ⎩sin ϕ ⎭

319

(9)

1

(b)

0.8

Basal Prismatic Pyramidal Twinning

0.6 0.4

1

Basal Prismatic Pyramidal Twinning

0.8 Relative activity

Relative activity

(a)

0.2 0

0.6 0.4 0.2 0

0

0.2

0.4 0.6 σTD/σRD

0.8

1

0

0.2

0.4

0.6 σTD/σRD

0.8

1

p p FIGURE 6. Correlations between relative activity and biaxial stress ratio. (a) ε RD =0.2 %, and (b) ε RD =1.0 %.

σTD

σRD

y

ϕ

σTD

m

θ

m

σRD

ϕ − θ = 90D s

s

x

FIGURE 7. A simple analytical model of HCP crystal structure on which biaxial stresses are acting.

Assuming that the biaxial stresses σRD and σTD are acting on the crystal structure in the x and y directions respectively, the resolved shear stress of the prismatic slip τpris can be depicted in the form

τ pris = m ⋅ σ ⋅ s ⎡σ τ pris = {cos θ sin θ } ⎢ RD

τ pris

0 ⎤ ⎧cos ϕ ⎫ ⎬ ⎥⎨ 0 σ ⎣ TD ⎦ ⎩sin ϕ ⎭ = (σ RD − σ TD ) sin ϕ cos ϕ .

(10)

Equation (10) shows that τpris becomes smaller as the biaxial stress ratio σTD/σRD approaches to one, yielding the activity of the prismatic slip systems decreases. The similar trend would arise also in real rolled Mg alloy sheets with some variation of the c-axes from the sheet normal direction. Because the prismatic slip systems are more difficult to be activated for the equi-biaxial tension (σTD/σRD =1) than the uniaxial tension (σTD/σRD =0), the strain range where only the basal slip is activated at the beginning is larger for σTD/σRD =1 (Fig. 5 (b)) than σTD/σRD =0. Furthermore, from the same reason, the relative activity of the prismatic slip systems increases gradually for σTD/σRD =1, but is still smaller during the middle stage than σTD/σRD =0. The relative activities of the basal slip and pyramidal-2 slip systems also change with the change of the activity of the prismatic slip systems. We conclude that such changes in the activities of the slip systems eventually result in the differential work-hardening behavior.

320

CONCLUSIONS The contour of plastic work for an AZ31 magnesium alloy sheet was predicted using a rate-dependent crystalplasticity finite-element method. The mechanism of the differential work-hardening behavior was examined in detail using the relative activity of each family of slip systems. The results obtained in this study are as follows. (1) The contour of plastic work obtained with the simulation exhibits clearly a differential work-hardening behavior; the contour is initially rather flattened in the vicinity of equi-biaxial tension, but severely bulges at the middle stage. This trend is qualitatively in good agreement with the experimental result reported in the literature. (2) The variation of the relative activities during the uniaxial tension shows that the work hardening under the uniaxial tension is determined solely by the basal slip in the very beginning, while by both the prismatic slip and the basal slip in the following deformation. (3) From the variation of the relative activity during the equi-biaxial tension, the mechanism of the work hardening under the equi-biaxial tension may be explained as follows. The stresses are small in a small plastic-strain range because the deformation during which only the basal slip systems are activated in the beginning is kept longer than that in the uniaxial tension. Then the work-hardening rate increases during the middle stage because not only the basal slip systems but also, to some extent, the pyramidal-2 slip and prismatic slip systems start to be activated gradually. Nevertheless, the activity of the basal slip systems is kept being predominant throughout the deformation because the relative activities of the prismatic slip and pyramidal-2 slip systems are smaller than those in the uniaxial tension. (4) The activity of the prismatic slip systems decreases as the biaxial stress ratio increases. A simple analysis reveals that the prismatic slip systems are more difficult to be activated for the equi-biaxial tension than the uniaxial tension because the two biaxial stresses are cancelled with each other. From the same reason, the activity of the prismatic slip systems increases gradually during the middle stage for the equi-biaxial tension, but is still smaller than that of the uniaxial tension. We conclude that such changes in the activities of the slip systems eventually result in the differential work-hardening behavior of the contour of plastic work.

REFERENCES 1. Japan Society of Technology of Plasticity, Magnesium Processing Technology, Tokyo: Corona Publishing Co., Ltd., 2004, pp. 12-33. (in Japanese) 2. B.L. Mordike and T. Ebert, Mater. Sci. Eng. A 302, 37-45 (2001). 3. H.J. Kim, S.C. Choi, K.T. Lee and H.Y. Kim, Mater. Trans. 49, 1112-1119 (2008). 4. J. Kaneko and M. Sugamata, J. JILM 54, 484-492 (2004). (in Japanese) 5. Y.S. Lee, M.C. Kim, S.W. Kim, Y.N. Kwon, S.W. Choi and J.H. Lee, J. Mater. Process. Technol. 187-188, 103-107 (2007). 6. T. Hama, Y. Kariyazaki, K. Ochi, H. Fujimoto and H. Takuda, Mater. Trans. 51, 685-693 (2010). 7. X.Y. Lou, M. Li, R.K. Boger, S.R. Agnew and R.H. Wagoner, Int. J. Plast. 23, 44-86 (2007). 8. Y. Chino, J.S. Lee, K. Sassa, A. Kamiya and M. Mabuchi, Mater. Lett. 60, 173-176 (2006). 9. C.H. Cáceres, T. Sumitomo and M. Veidt, Acta Mater. 51, 6211-6218 (2003). 10. G.E. Mann, T. Sumitomo, C.H. Cáceres and J.R. Griffiths, Mater. Sci. Eng. A 456, 138-146 (2007). 11. T. Hama, H. Takuda, Int. J. Plast. in press (2010). 12. M. O. Andar, D. Steglich, T. Kuwabara, Proceedings of AMPT 2010, 75-80 (2010). 13. Y. Tadano, Int. J. Mech. Sci. 52, 257-265 (2010). 14. M. Kawka, A. Makinouchi, J. Mater. Process. Technol. 50, 105-115 (1995). 15. T. Hama, T. Nagata, C. Teodosiu, A. Makinouchi, H. Takuda, Int. J. Mech. Sci. 50, 175-192 (2008). 16. S. Graff, W. Brocks and D. Steglich, Int. J. Plast. 23, 1957-1978 (2007). 17. D. Pierce, R.J. Asaro and A. Needleman, Acta Metall. 31, 1951-1976 (1983). 18. R.J. Asaro, A. Needleman, Acta Metall. 33, 923-953 (1985). 19. P. Van Houtte, Acta Metall. 26, 591-604 (1978). 20. T. Mayama, K. Aizawa, Y. Tadano and M. Kuroda, Comput. Mater. Sci. 47, 448-455 (2009). 21. C. N. Tomé, R.A. Lebensohn and U.F. Kocks, Acta Metall. Mater. 39, 2667-2680 (1991).

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3D Crystal Plasticity Modelling of Complex Microstructures in Extruded Products S. Dumoulina, J. Friisa, S. Gouttebrozea, B. Holmedalb, K. Marthinsenb a

b

SINTEF Materials and Chemistry, NO-7465 Trondheim, Norway Department of Materials Science and Engineering, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway Abstract. The current study relates to the modelling of plastic anisotropy in aluminium alloy AA6063-W using a ratedependent crystal plasticity finite element approach. A virtual microstructure, in which grains are explicitly resolved, was first generated based on experimental data using a grain growth algorithm. The texture was obtained by sampling the experimental ODF. The microstructure was then meshed using three different methods and appropriate boundary conditions were used in order to simulate uniaxial tensile testing in different directions. The results obtained from the different meshes were finally compared with experimental true stress-true strain curves and discussed. Keywords: Finite Element Method; Texture; Microstructure; Anisotropy; Aluminium. PACS: 02.70.Dh; 81.40.-z; 62.20.F

INTRODUCTION In the extrusion process of 6xxx aluminium alloys the microstructure and texture strongly evolves with the deformation. This leads to strong anisotropy in all mechanical properties, and especially in strain-hardening [1,2]. Furthermore, through-thickness variations in microstructure and texture may also occur during processing. The material thus exhibits an even more complex anisotropic behaviour [3]. The source of this anisotropy is therefore attributed to these through-thickness heterogeneities but has not yet been clearly identified. Attempts in modelling the anisotropic behaviour of extruded alloys using phenomenological approaches, based on advanced anisotropic yield criteria, have usually proven successful when no through-thickness heterogeneity is present [2]. However, difficulties appear when through-thickness variations are present [1,3]. In these approaches, the complex microstructure is considered as homogenous which is suspected to be a too crude approximation. A composite shell element based modelling approach has also been suggested; each integration point of the shell element was given different material parameters, or initial yield surfaces, identified using polycrystalline – Taylortype and visco-plastic self consistent (VPSC) – models and experimental textures [3]. Nevertheless, no improvements were obtained. This was attributed to the inability of shell elements in reproducing through-thickness variations in stresses and strains, and the use of brick elements was therefore recommended. An alternative to the approach described above is the use of crystal plasticity combined with the finite element method (CP-FEM). CP-FEM has proven good capability in predicting strain heterogeneity and anisotropy in polycrystalline materials. Furthermore it has regained interest over the last ten to fifteen years owing to the increase in computational power. This approach explicitly accounts for both microstructure and texture. Evolution of texture and thereby anisotropic hardening are inherently part of the approach [4,5,6]. The current study thus aims at describing plastic anisotropy in AA6063-W by use of CP-FEM. The next section introduces the material, its microstructure and the uniaxial tensile curves. The third section describes the modelling approach: building of the microstructure, meshing, single crystal model, calibration of model parameters and modelling of tensile tests. Then, before the conclusion, the results are presented including comparisons between experiments and simulations.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 322-329 (2011); doi: 10.1063/1.3623627 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

322

MATERIAL DATA The material studied here is the heat-treatable wrought alloy AA6063, containing magnesium and silicon, solution heat treated to W temper, i.e. alloying elements are in solid solution. It is widely used as extruded-andformed part in automotive applications [3]. The material was received as 3-mm thick strip profiles. Figure 1 shows the through-thickness microstructure with micrographs at different positions along the thickness together with corresponding pole figures. It clearly appears that both microstructure and texture vary across the thickness.

(a) (b) FIGURE 1. (a) Micrographs of the microstructure at different positions through the sample thickness with (b) corresponding pole figures [3].

FIGURE 2. True stress-true strain curves until necking for uniaxial tensile tests.

Uniaxial tensile tests were carried out along three different directions, 0°, 45° and 90° with respect to the extrusion direction (ED), at an initial strain rate of 10-3 s-1. The specimens, with a gauge length of 35 mm and a width of 24 mm, were cut out from the strip. The true stress-true strain curves, shown in Fig. 2, were obtained using digital image correlation (DIC) [7]. Note that the accuracy of DIC for these tests was about 0.5% strain thus not

323

allowing computing accurately stresses at small strains. Portevin-Le Châtelier (PLC) bands were observed during the tests as attested by the serrations on the curves. Anisotropy in strength, elongation to necking and hardening is observed.

MODELLING The modelling approach consists of three steps: i) generation of a microstructure and texture, ii) building of the FE model and iii) run of FE simulations.

Generation of Microstructure and Texture First, a representative volume element (RVE) of the material was determined based on micrographs of the microstructure; its volume was estimated to be 1.5×1.5×1.5 mm3. Then a 3D voxel periodic microstructure of volume 1.5×1.5×1.5 mm3 was generated using a Potts Monte Carlo [8] grain growth algorithm based on [9,10,11]. Nuclei are initially positioned in space in order to obtain a microstructure similar to the experimental one (see Fig. 1). The RVE contained then 364 grains. Figure 3.a shows cross-sections of the generated microstructure. Variations in grain size are obtained through the thickness although they are not as large as in the experiments. The texture for the RVE was obtained by sampling a representative set of 364 orientations from the experimental texture using the approach proposed in [12]. Figure 3.b shows selected sections from the global, i.e. throughthickness, experimental and reconstructed orientation distribution functions (ODF). The experimental ODF reveals a sharp rotated cube texture with some Goss component that is smoothened in the reconstructed one due to the low number of orientations. Finally, the reconstructed texture is randomly assigned to the grains. Surface

ND

Experimental ODF

TD ED

ED Surface

ND

Reconstructed ODF TD

(a) (b) FIGURE 3. (a) Cross-sections of the microstructure generated; (b) selected sections from experimental and reconstructed orientation distribution functions.

Finite Element Model The 3D voxel microstructure is converted into a finite element model using three different methods. The first one consists in mapping the microstructure onto the integration points of an N q N q N solid element mesh with N

324

equals to 10 or 20 (hereafter called RVE-IPN); the grain boundaries are not explicitly resolved and several grains can then share the same element. The second method is similar to the previous one, only that the mapping is done onto the elements, i.e. all integration points of each element have the same initial grain orientation (hereafter called RVE-ELN); the grain boundaries are then resolved in a stair-like fashion (Figure 4.a). The last method consists in meshing the microstructure based on the grain boundaries using tetrahedron elements (hereafter called RVE-GB) using the CGAL library [13]; the grains are thus explicitly resolved (Figure 4.b) and about 45390 elements are generated. For the latter, note that some grains may not be convex since they are not convex in the generated microstructure. In all RVEs each grain has a unique initial orientation. Note that 8-node brick elements with 8 integration points are used for the two first RVEs and 4-node tetrahedron elements with 1 integration point are used for the third RVE.

ND

TD ED

(a) (b) FIGURE 4. (a) Element-based mesh (RVE-ELN) and (b) grain boundary-based mesh (RVE-GB).

In all simulations, periodic boundary conditions (e.g. [14]) are applied along the extrusion (ED) and transverse (TD) directions. In the normal direction (ND), the plane representing the surface of the material is unconstrained; however symmetric boundary conditions are applied in the opposite plane representing the mid-thickness of the material. Appropriate nodal boundary conditions are applied in order to avoid any rigid body rotation during the simulations. Tensile conditions are generated by imposing a velocity in the required direction. While this is straightforward for the 0° and 90° directions, this is not the case for the 45° direction. For the latter the texture was then rotated about the ND. Mass scaling was used in all simulations; kinetic effects, which were monitored, were negligible.

Crystal Plasticity Model The crystal plasticity model is defined within the framework of finite deformations, i.e. based on the multiplicative decomposition of the total deformation gradient into elastic and slip-induced plastic parts. It follows a classical approach as e.g. in [15,16]. For FCC metals, slip occurs on the twelve \111^ 110 slip systems. The constitutive model is as follows. Shearing rate on slip system D is defined by the power-law D



D 0 D c

1 m

sgn   D

(1)

where 0 is the reference shearing rate,  cD the slip resistance, or critical resolved shear stress, and m the microscopic strain rate sensitivity. The slip resistance is a function of the plastic straining

WcD

n

T  * ¦ qDE J E

(2)

E 1

where T defines the hardening rate for a given accumulated plastic strain, qDE is the latent hardening matrix, being equal to 1 for self-hardening and q for latent hardening, and * is the accumulated plastic shear strain. The work-

325

hardening rate T is defined as the derivative with respect to * of a modified Voce hardening model in the form (e.g. [17]) N

§

§ Tk

©

© Wk

W c  * W 0  ¦W k ¨¨1  exp ¨  k 1

·· *¸¸ ¸ ¹¹

(3)

where W 0 is the initial critical resolved shear stress, assumed to be the same for all slip systems, while W k and T k are the hardening rate and saturated value, respectively, of hardening term k. In the current work, only two terms are used. Note that this model has proven good capability in describing strain-hardening in FCC metals and has furthermore physical grounds [17]. The model has been implemented into the explicit non-linear finite element code LS-DYNA [18] via a user material subroutine developed for solid elements [19]. The model parameters were calibrated against the experimental curve for the 90° direction with the design optimisation tool LS-OPT [20] using RVE-IP10 described in the previous section. In order to improve the efficiency of the calibration m , 0 and q were taken from the literature (e.g. [16]). These parameters were then fixed in the calibration while the others were free. The values obtained after calibration are given in Table 1. TABLE 1. Model parameters used in the simulations. q m 0 (/s) W 0 (MPa) 0.001

0.01

1.4

12.6

T1 (MPa)

W 1 (MPa)

T 2 (MPa)

W 2 (MPa)

41.4

25.6

148.7

13.9

RESULTS AND DISCUSSION Figure 5 shows the comparison between experiments and simulations using RVE-IP10 for the three directions of interest i.e. 0°, 45° and 90°. A perfect fit is obtained at 90° thus validating the parameter identification. However, even though the model predicts an anisotropic material response, the trend observed experimentally is not reproduced. The simulations give similar strain-hardening in both 0° and 90° directions while strain-hardening is much higher in 0° direction in the experiments up to a strain of about 0.08-0.09. The maximum difference in flow strength between the experiment and the simulation in 0° direction is 18% at a strain of about 0.06. Furthermore, in 45° direction the strain-hardening obtained in the simulation is higher than in the experiment leading to differences in strength up to 14% at a strain of about 0.095.

FIGURE 5. Comparison between experimental and numerical (RVE-IP10, RVE-IP20) true stress-true strain curves.

In order to check whether mesh sensitivity has an effect, all simulations were rerun using a finer mesh, RVEIP20 (Figure 5). While a finer mesh does not affect the results in neither 0° nor 90° directions, an minor improvement is observed in 45° direction: the maximum difference in flow strength between the experiment and the simulation is now about 11% i.e. an improvement of about 2%.

326

Next, simulations were run using different meshes, i.e. based on elements (RVE-EL20) and grain boundaries (RVE-GB). Figure 6 shows the true stress-true strain curves obtained for the different meshes, including the one based on integration points using a fine mesh (RVE-IP20). Meshes based on integration points (RVE-IP20) and elements (RVE-EL20) give identical results with differences less than 0.7%. However, simulations run with a mesh based on grain boundaries (RVE-GB) show a slightly higher strain-hardening; maximum differences between RVEGB and RVE-EL20 are about 3% for all directions. Finally, in order to assess whether accounting for the microstructure, i.e. grain size and shape, has any influence on the material response, simulations were run using the Taylor full constraint model. The Taylor model used here was built within the framework described previously, i.e. based on finite deformations and a rate-dependent formulation. The results are included in Fig. 6. The Taylor model gives results identical to those obtained with RVEGB.

(a)

(b)

(c) FIGURE 6. Comparison between true stress-true strain curves obtained experimentally and using different RVEs, for each direction: (a) 0°, (b) 45° and (c) 90°.

RVE-IP20, RVE-EL20 and RVE-GB give similar results at the macro-scale, i.e. true stress-true strain curves, with RVE-GB generating slightly higher strain-hardening. In order to assess whether these meshes also provide similar results at the micro-scale, i.e. at the grain level, the plastic strain in each RVE in uniaxial tensile testing in 90° direction is plotted in Fig. 7. While RVE-IP20 and RVE-EL20 give almost identical results, RVE-GB generates about 10% lower plastic strain. This figure also emphasises the main difference between CP-FEM and high grainmatrix interaction models such as Taylor, i.e. CP-FEM provides inter- and intra-granular resolution. The above analysis thus implies that, in the current study, the microstructure does not play any role in the anisotropy of the material while texture is of prime importance.

327

ND TD ED (a)

(b)

(c)

FIGURE 7. Plastic strain field in (a) RVE-IP20, (b) RVE-EL20 and (b) RVE-GB, for tensile testing in 90° direction.

As suggested in [3], the anisotropic hardening of AA6063-W is mainly attributed to the ED-rotated cube texture. It is suggested that the microstructure has little if no effect since AA6063-W has equi-axed grains with a size, even though varying through the thickness, expected to be above any value giving strength contribution. In materials having symmetric cube texture, uniaxial tensile responses in 0° and 90° directions are identical while higher strainhardening is observed in the 45° direction [21]. In the current study, 0° and 45° directions show similar strainhardening while higher strain-hardening is observed in the 0° direction which is attributed to the ED-rotated texture. Furthermore, different width contractions were observed through the thickness of the specimens, depending on the tensile direction, i.e. plastic strain ratios (R-values) varied through the thickness [3]. This effect was attributed to heterogeneities in both texture and microstructure across the thickness. Variations in R-values through the thickness will lead to strain incompatibilities and thereby change the stress state in the sheet which in turn may affect the anisotropy. In general, the contribution from these through-thickness variations on anisotropic hardening is difficult to assess due to the strong correlations between the different variables involved. Based on the above analysis, weaknesses in the modelling approach can be identified. The main drawback resides in texture modelling i.e. the through-thickness texture variation was not accounted for in the current modelling. A second source of errors related to texture lies in the ODF measurement: the latter was obtained based on data from X-ray diffraction which, for heterogeneous microstructures, may lead to overestimation of some texture components. Furthermore, in the modelling approach, an homogeneous distribution of dislocations was assumed: the initial slip resistance W 0 was identical for all slip systems, which is reasonable for annealed materials but not for work-hardened materials, i.e. anisotropy due to prior deformation (extrusion) was ignored. This may however have a significant role as shown in [22]. Finally, the size of the RVE investigated in this study was not validated; this may also affect the response of the material.

CONCLUSION In the current study modelling of plastic anisotropy in aluminium alloy AA6063-W was investigated using a multiscale approach with a strong experimental background. A RVE 3D periodic microstructure was generated based on micrographs of the microstructure using a Potts Monte Carlo grain growth algorithm. The texture for the RVE was obtained by sampling the experimental ODF. This microstructural and textural RVE was then meshed using three different methods. A rate-dependent CP-FEM modelling approach was then used together with a modified Voce hardening model. Simulations of uniaxial tensile tests in three different directions were then performed. Comparison between numerical and experimental results revealed that the modelling approach had difficulties in reproducing the anisotropic material response observed experimentally. Comparison with simulations performed with the Taylor FC model showed that, for the approach chosen here, the microstructure had no effect on the anisotropy and all meshes gave similar responses. These difficulties were attributed to various sources relating to heterogeneities in texture and material properties: both variation of texture through the thickness and heterogeneity in dislocation distributions, or material history, were not accounted for.

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Ongoing work aims at accounting for through-thickness texture variation using experimental ODF at different positions across the thickness (see Fig. 1.b). In addition, the effect of the size of the RVE on the material response will also be investigated.

ACKNOWLEDGMENTS The authors would like to thank Dr. S.K. Fjeldbo for providing all experimental data. The financial support of this work from the “Through Process Modelling” Hydro-Fund project is gratefully acknowledged.

REFERENCES 1. K. Pedersen, O.-G. Lademo, T. Berstad, T. Furu and O.S. Hopperstad, J. Mater. Process. Technol. 200, 77-93 (2008). 2. D. Achani, O.S. Hopperstad and O.-G. Lademo, J. Mater. Process. Technol. 209, 4750-4764 (2009). 3. S.K. Fjeldbo, “Studies of the influence of strong through thickness variation on the plastic floe properties of extruded AA6063-W aluminium strips and tubes, with application to the modelling and simulation of tube hydroforming of T-shapes”, Ph.D. Thesis, Norwegian University of Science and Technology, 2008. 4. Z. Zhao, S. Kuchnicki, R. Radovitzky and A. Cuitiño, Acta Mater. 55, 2361-2373 (2007). 5. K. Inal, R.K. Mishra and O. Cazacu, Int. J. Solids Structures 47, 2223-2233 (2010). 6. F. Roters, P. Eisenlohr, L. Hantcherli, D.D. Tjahjanto, T.R. Bieler and D. Raabe, Acta Mater. 58, 1152-1211 (2010). 7. S. Dumoulin, L. Tabourot, C. Chappuis, P. Vacher and R. Arrieux, J. Mater. Process. Technol. 133, 79-83 (2003). 8. R.B. Potts, Proc. Cambridge Philos. Soc. 48, 106-109 (1952). 9. A. Brahme, M.H. Alvi, D. Saylor, J. Friday and A.D. Rollett, Scr. Mater. 55, 75-80 (2006). 10. A. Brahme, J. Fridy, H. Weiland and A.D. Rollett, Modell. Simul. Mater. Sci. Eng. 17, 015005 (2009). 11. E. Fjeldberg and K. Marthinsen, Comput. Mater. Sci. 48, 267-281 (2009). 12. P. Eisenlohr and F. Roters, Comput. Mater. Sci. 42, 670-678 (2008). 13. The CGAL project homepage, www.cgal.org (2010). 14. A. Prakash, S.M. Weygand and H. Riedel, Comput. Mater. Sci. 45, 744-750 (2009). 15. D. Peirce, R.J. Asaro and A. Needleman, Acta Metall. 30, 1087-1119 (1982). 16. S.R. Kalidindi, C.A. Bronkhorst and L. Anand, J. Mech. Phys. Solids 40, 537-569 (1992). 17. Y. Estrin, J. Mater. Process. Technol. 80-81, 33-39 (1998). 18. J.O. Hallquist, LS-DYNA keyword user’s manual, Version 971, California: Livermore Software Technology Corporation (2007). 19. S. Dumoulin, O.S. Hopperstad and T. Berstad, Comput. Mater. Sci. 46, 785-799 (2009). 20. N. Stander, W. Roux, T. Goel, T. Eggleston and K. Craig, LS-OPT user’s manual, Version 4.1, California: Livermore Software Technology Corporation (2010). 21. A.B. Lopes, F. Barlat, J.J. Gracio, J.F. Ferreira Duarte and E.F. Rauch, Int. J. Plast. 19, 1-22 (2003). 22. S.R. Kalidindi and S.E. Schoenfeld, Mater. Sci. Eng. A 293, 120-129 (2000).

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Anisotropic Sheet Forming Simulations Based on the ALAMEL Model: Application on Cup Deep Drawing and Ironing P. Eyckensa,b*, J. Gawadc,d, Q. Xiea,b, A. Van Baela,e, D. Roosec, G. Samaeyc, J. Moermanf, H. Vegterf and P. Van Houttea a

Materials innovation institute, Mekelweg 2, 2600 GA Delft, the Netherlands Department of Metallurgy and Materials Engineering (MTM), Katholieke Universiteit Leuven, Kasteelpark Arenberg 44, 3001 Heverlee, Belgium c Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium d Department of Applied Computer Science and Modeling, AGH University of Science and Technology, al. Mickiewicza 30, Krakow, 30-059, Poland e Department IWT, Limburg Catholic University College (KHLim), Campus Diepenbeek, Agoralaan Gebouw B, bus 3, 3590 Diepenbeek, Belgium f Tata Steel Research, Development & Technology, IJmuiden Technology Centre, PO Box 10.000, 1970 CA, IJmuiden, The Netherlands *Corresponding author: tel (+32) 16 321305, fax (+32) 16 321990, [email protected] b

Abstract. The grain interaction ALAMEL model [1] allows predicting the evolution of the crystallographic texture and the accompanying evolution in plastic anisotropy. A FE constitutive law, based on this multilevel model, is presented and assessed for a cup deep drawing process followed by an ironing process. A Numisheet2011 benchmark (BM-1) is used for the application. The FE material model makes use of the Facet plastic potential [2] for a relatively fast evaluation of the yield locus. A multi-scale approach [3] has been recently developed in order to adaptively update the constitutive law by accommodating it to the evolution of the crystallographic texture. The identification procedure of the Facet coefficients, which describe instantaneous plastic anisotropy, is accomplished through virtual testing by means of the ALAMEL model, as described in more detail in the accompanying conference paper [4]. Texture evolution during deformation is included explicitly by re-identification of Facet coefficients in the course of the FE simulation. The focus of this paper lies on the texture-induced anisotropy and the resulting earing profile during both stages of the forming process. For the considered AKDQ steel material, it is seen that texture evolution during deep drawing is such that the anisotropic plastic flow evolves towards a more isotropic flow in the course of deformation. Texture evolution only slightly influences the obtained cup height for this material. The ironing step enlarges the earing height. Keywords: Anisotropy, Texture Evolution, Crystal Plasticity, Multi-Scale Modeling, Finite Element Method. PACS: 46.70.De, 81.40.Ef, 83.50.-v, 62.20.F-

INTRODUCTION It is well recognized that the crystallographic texture of sheet metal largely determines the anisotropic characteristics of plastic flow in pressing operations. For axisymmetric deep drawing, it may lead to a macroscopic shape effect called earing. In terms of plastic anisotropy properties, not only the directional anisotropy in the flow (often characterized by r-value) may contribute significantly to earing, but also the directionality in flow stress [5]. It is even the case for yield strength differences as low as 2MPa [5], which are close to the measurement accuracy for many tensile test devices. Besides phenomenological yield loci, which rely on experimental measurements, another valuable approach consists of multi-level modeling of the texture-induced anisotropy. This last approach will be followed in this paper, by adopting a scheme that considers not only the initial texture, but also the deformation textures obtained by the ALAMEL multilevel model, as proposed in [3]. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 330-336 (2011); doi: 10.1063/1.3623628 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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DESCRIPTION OF EXPERIMENT AND SIMULATIONS Experimental set-up In the axisymmetric cup drawing and ironing experiment of the NumiSheet2011 Benchmark-1 (BM-1), a circular blank (38.062mm radius) is deep drawn under the action of a punch with external radius of 22.860mm and fillet radius of 2.229mm. Punch speed is set to 70.0mm/s. A constant blank holding force of 8.9 kN is applied throughout the process. The die has an orifice with a radius of 23.368 and a fillet radius of 1.905mm. After the circular cup is formed, the punch passes through an ironing ring with internal radius of 23.025mm and an entrance angle of 8°. Two materials are described in the benchmark, one of which (AKDQ steel with initial thickness of 0.229mm) is reported on in this paper. For more details on the experimental set-up, the reader is referred to the BM-1 description of these proceedings.

Finite Element Model The forming process is modeled by explicit Finite Element (FE) simulations, using the commercial FE software Abaqus. The blank mesh consists of three layers of first-order solid brick elements with reduced integration. Exploiting the orthorhombic sample symmetry of the initial sheet produced by rolling, only one quarter of the blank is considered in the FE model, which is divided in 3375 solid elements in total. The initial mesh can be seen in Figure 2(a). All tools are modeled as analytic rigid surfaces. Punch speed is set to 5000.0mm/s in the explicit simulation (time scaling) as a compromise between calculation time efficiency and approximation of the quasi-static forming conditions. The adopted kinematic contact algorithm avoids any penetration of blank nodes into the (analytically defined) tool surfaces (‘hard contact’). Coulomb friction is assumed between all contacting surfaces (friction coefficient: 0.05).

Multi-scale Model The multi-scale modeling scheme adopted comprises a macro-scale description of the plastic anisotropy in the form of a Facet plastic potential expressed in strain rate space [2]. As shown in [2], its convexity can be assured. A polynomial of order 6 is currently chosen. It is furthermore assumed that the material is plastically incompressible and rate-insensitive. The Facet potential, being an analytic expression, is relatively efficient to evaluate whenever required in the course of the FE simulation (compared to a multilevel model). Starting from a measured initial texture (or calculated deformation texture), the grain-interaction ALAMEL model [1] is invoked in order to identify the coefficients of the Facet potential. This identification procedure is detailed further in [2,4]. The ALAMEL model is also adopted to make predictions of deformation textures after a pre-determined amount of straining (an interval of 0.1 in equivalent strain is currently chosen). Since this scheme uses the local deformation history (i.e. as obtained from a FE integration point), the texture evolves differently for sections of the sheet that are strained differently. Also the plastic anisotropic properties will then evolve accordingly. Isotropic hardening of the sheet is furthermore assumed (besides the textural hardening described above) by specification of an equivalent stress-strain relationship. The multi-scale modeling scheme thus considers the mutual effect between the local history of plastic deformation and the local plastic anisotropy. More details on the adopted multi-scale models can be found in the accompanying conference proceedings [4] and in papers [6,7].

Material Data The sheet considered is an aluminum-killed drawing quality (AKDQ) steel of 0.229mm thickness. Table (1) mentions all the material data used: density , Young’s modulus E, Poisson coefficient  (isotropic elasticity is assumed), and the parameters K, 0 and n of the isotropic Swift hardening law: V eq

K  H  H 0 , in which  and  n

are the equivalent stress and strain, respectively. The Swift parameters are derived from uniaxial tensile tests along the rolling direction. The parameter 0 is set in order to let the initial yield stress of the material law equal the experimental yield stress Y=315.7MPa.

331

 [kg/m³] 7800.0

TABLE (1). Material data used for AKDQ steel E  K 0 [/] [GPa] [/] [MPa] 210.0 0.3 599.7 9.89E-3

n [/] 0.139

It should be pointed out that all other mechanical test data that is available from the benchmark (tensile test data in other directions, equal biaxial tension test data, disk compression test data) is not required to obtain the results shown in this paper (and so they have not been utilized). The (initial) anisotropy of the sheet is derived exclusively from measurement of the crystallographic texture, which has been provided in the BM-1 description and is shown in Figure 1.

FIGURE 1. The 1=0°, 2=45° and 1=90°sections of the ODF measured on the as-received sheet, assuming orthorhombic sample symmetry. Texture index: 1.93.

Figure 1 shows three sections of the initial ODF in which the main texture components of ferritic steels are found. The 2=45° section contains the -fibre as well as the -fibre. The -fibre contains all orientations having a <110> direction parallel to the rolling direction RD, such as the H (001)[110], J (114)[1-10], I (112)[1-10] and E (111)[110] components. The -fibre consists of all orientations with a <111> direction parallel to the normal direction ND, such as E (111)[1-10], E’ (111)[0-11], F (111)[1-21] and F’ (111)[-1-12]. This particular steel texture consists of the complete -fibre and partial -fibre, having a similar intensity. In addition, also a relatively strong fibre from rotated H-component to Cube (001)[100] is present.

RESULTS Deformed shape The deformed meshes after respectively deep drawing and ironing stages are shown in Figure 2(b-c), while the corresponding cup profiles are shown in Figure 3. After drawing a profile with 4 ears, each at about 45° with respect to the rolling direction, is obtained. The ironing process obviously increases the average cup height. Besides this, the earing height is also more pronounced: it increases from about 0.35mm after drawing to 0.65mm after ironing. The consideration of texture evolution is seen to have only a very limited effect on the earing behaviour. It leads to a local change in cup height of about 0.1mm at most after the ironing stage. After the drawing stage, the difference is neglectable. In a recent study using the same modelling approach for the deep drawing process [8], it is seen that texture evolution does not improve earing predictions for a low carbon steel and Mn-Fe-alloyed aluminium, both

332

having 4 ears, while a noticeable improvement of accounting for texture evolution is seen for an IF steel with 6 ears and a commercially pure aluminium sheet with 8 ears.

(a)

(b)

(c)

A90 B90 A45 B45

TD RD

B0 A0

FIGURE 2. (a) mesh of one quarter of blank (undeformed state). Deformed mesh after (b) deep drawing stage and (c) ironing stage. Some specific elements are highlighted and labeled.

Cup height [mm] After ironing

After drawing Angle to RD FIGURE 3. Prediction of cup height profiles without texture evolution (full line) and with texture evolution (dashed line).

Texture evolution The Figures 4-5 show the evolution of texture, as predicted by the ALAMEL model, for the labeled elements of Figure 2(a). In each case, texture is shown for the central of three solid elements used in the sheet thickness direction. It can be seen that, after the deep drawing process, the original fibrous texture is mostly broken up into individual texture components with a particular spread around them, close to the ideal components I (112)[1-10], E’ (111)[011] and Cube (001)[100]. Although these components are quite similar for all different locations, relative intensities and the shape of the texture spread around ideal orientations are clearly distinct for different positions with respect to the original RD. The texture evolution is very similar for the corresponding elements from A- and B-zones that are located on the same angular position, while larger qualitative differences are seen between the three different angular positions under consideration. The ironing process appears to decrease the orientations near Cube in favour of components near the ideal Iorientation. Also the E’-component is seen to be strengthened. In addition, a change in shape of textural spread around E’ within the shown sections of ODF is systematically seen during the ironing process. The ‘vertically’ oriented spread at the end of deep drawing (cf. for instance the dotted lines in Fig. 4(a)) appears to be rotated after ironing (cf. for instance the dotted lines in Fig. 4(b)). It implies that that the ironing causes these orientations near the E’-orientation to rotate (on average) towards the perfect -fibre, i.e. aligning a {111} plane with the sheet plane. The texture evolution during ironing is more pronounced as the location is closer to the cup rim (compare Fig. 4(b)

333

to Fig. 5b)). Sheet thickness increases towards the cup rim in the intermediate state after deep drawing, and so significantly more plastic straining takes place during the ironing stage for the A-elements compared to the Belements.

(a)

(b)

(c)

(d)

(e)

(f)

FIGURE 4. 2=45°-sections of the calculated ODFs in (a-b) A0-, (c-d) A45- and (e-f) A90-elements. (a),(c),(e): at  = 0.5 (approximately at the end of drawing stage). (b),(d),(f): final textures (after the ironing stage). ODF intensity lines: 1.0, 1.4, 2.0, 2.8, 4.0, 5.6, 8.0, 11.0, 16.0, 22.0.

(a)

(b)

(c)

(d)

(e)

(f)

FIGURE 5. 2=45°-sections of the calculated ODFs in (a-b) B0-, (c-d) B45- and (e-f) B90-elements. (a),(c),(e): at  = 0.5 (approximately at the end of drawing stage). (b),(d),(f): final textures (after the ironing stage). ODF intensity lines: 1.0, 1.4, 2.0, 2.8, 4.0, 5.6, 8.0, 11.0, 16.0, 22.0.

334

Evolution of Mechanical Properties Figure 6 presents evolution of (a part of) the anisotropic plastic behavior in the form of q-values. It is defined as the ratio of plastic contraction in the sheet width direction per unit tensile elongation in uniaxial deformation. The qvalue is bounded in the interval [0;1] and is related to the r-value (bounded in [0;]) by q=r/(r+1). Reported values are calculated by application of the Full Constraints (FC) Taylor model on the experimentally measured texture (for  = 0) and textures predicted by the ALAMEL model (for  > 0). Both the local radial (‘q0°’) as the local circumferential (‘q90°’) directions are considered as the deformation axes.

q-value

q-value

(a)

drawing



ironing

drawing

(b)

ironing 

FIGURE 6. Evolution of FC Taylor-based q-value predictions for uniaxial tension/compression along the local radial direction (‘q0°’) and along the local circumferential direction (‘q90°’), for different elements (in (a): ‘A0’,’A45’ and ’A90’, in (b): ‘B0’,’B45’ and ’B90’,), cf Fig. 2(a). The horizontal axis shows the (local) equivalent plastic strain . Transition from drawing to ironing stage occurs in both (a-b) at 0.5 <  < 0.6.

For all the considered elements, the evolution of q90° is seen to converge to about 0.5 (isotropic behavior) at the end of deep drawing. Near the rim of the blank, the local anisotropic yielding during a deep drawing process is actually described very well by q90°, as the deformation is approximately uniaxial compression1 along the local circumferential direction. So as deformation progresses, the texture is evolving in such a way that the anisotropy associated to the current deep drawing mode of deformation, is diminishing. This explains the rather limited effect of the incorporation of texture evolution on the earing profile for this material (Figure 3). On the other hand, the q0°values do not converge but show a decreasing trend during the deep drawing process. Other directions besides 0° and 90° have also been studied; only q90°-values from different locations showed the tendency to converge to each other. So it appears that the properties of anisotropic flow that are not ‘active’ during the deep drawing generally do not evolve towards isotropy in the course of deformation. The change of deformation mode occurring in the transition from drawing to ironing results in a noticeable change in evolution of the q-values. An effect of the intermediate deformation textures is seen. For instance the q0°values evolve somewhat differently for the elements A0, A45 and A90 during ironing. Correspondingly, the change in cup height as a result of texture evolution after ironing (cf. Figure 3) depends on the angle with respect to RD.

CONCLUSIONS This paper demonstrated the feasibility of a multi-scale modeling approach on the subsequent processes of deep drawing and ironing. Simulation results for the investigated AKDQ steel sheet show a 4-earing pattern after drawing that is hardly affected by consideration of texture evolution. After an ironing step, the earing profile is more pronounced and slightly more dependent on deformation texture. A detailed analysis of the deformation texture predicted by the ALAMEL model and corresponding evolution in flow properties leads to the conclusion that the 1

The uniaxial tension and compression deformation modes along a particular direction have the same q-value for the FC Taylor model with critical resolved shear stresses that are independent on the sense of plastic slip [1].

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plastic flow during deep drawing becomes increasingly isotropic. This increase of isotropy refers exclusively to the active deformation mode during deep drawing. The simulated cup height after deep drawing complies with previous findings of the same authors [8]. In this paper, earing profiles for 4-eared materials appear to have only marginal benefits from consideration of texture evolution, while clear improvements are on the other hand seen for materials that show either 6 or 8 ears, i.e. materials exhibiting more complex anisotropic behavior. Future research will be focused on those materials, more in particular by assessment of possible further improvements through increase of the mesh density in the FE model, and/or refinements in the identification procedure of the Facet plastic potential in the multi-scale model.

ACKNOWLEDGMENTS This research was carried out under the project number M41.2.08307a/M41.10.08307b in the framework of the Research Program of the Materials innovation institute M2i (www.m2i.nl). The authors gratefully acknowledge the financial support from the project IDO/08/09, funded by K.U.Leuven, and from the Interuniversity Attraction Poles Program from the Belgian State through the Belgian Science Policy agency, contracts IAP6/24 and IAP6/4. GS is Postdoctoral Fellow of the Research Foundation – Flanders (FWO).

REFERENCES 1. Van Houtte, P., Li, S., Seefeldt, M., Delannay, L., 2005. Deformation texture prediction: from the Taylor model to the advanced Lamel model. Int. J. Plasticity 21, 589-624. 2. Van Houtte, P., Yerra, S.K., Van Bael, A., 2009. The Facet method: A hierarchical multilevel modelling scheme for anisotropic convex plastic potentials. Int. J. Plasticity 25, 332-360. 3. Gawad, J., Van Bael, A., Yerra, S., Samaey, G., Van Houtte, P. and Roose, D. A coupled multiscale model of texture evolution and plastic anisotropy. In Barlat, F. (Ed.), Moon, Y. (Ed.), Lee, M. (Ed.), AIP Conference Proceedings: Vol. 1252 (1). NUMIFORM 2010. Pohang, Korea, pp. 770-777. 4. Gawad, J. et al.: Proc. Numisheet 2011 conference (submitted). 5. Mulder, J. and Vegter, H.,2011. An analytical approach for earing in cylindrical deep drawing based on uniaxial tensile test results. Proc. Esaform2011 (accepted). 6. Gawad J., Van Bael A., Eyckens P., Van Houtte P., Samaey G., Roose D., Effect of texture evolution in cup drawing predictions by multiscale model. Steel Research International, Supplement Metal Forming, 81/9 (2010) 1430-1433. 7. Van Bael A., Eyckens P., Gawad J., Samaey G., Roose D., Van Houtte P., Evolution of crystallographic texture and mechanical anisotropy during cup drawing. Steel Research International, Supplement Metal Forming, 81/9 (2010), 1392-1395. 8. Van Bael, A., Gawad, J, Eyckens, P., Samaey, G., Roose, D., Van Houtte, P., Modelling texture anisotropy and its evolution in sheet forming processes, Proc. ICTP2011 (accepted).

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Microstructure Evolution and Mechanical Properties of Al/Al-Mg/Al composite sheet metals Jaehyung Cho∗ , Su-Hyeon Kim∗ , Hyoung-Wook Kim∗ , Cha-Yong Lim∗ , Eun-Young Kim† and Shi-Hoon Choi† ∗

Korea Institute of Materials Science, 797 Changwondaero, Seongsan-gu, Changwon, Gyeongnam, 642-831, South Korea † Department of Materials Science and Metallurgical Engineering, Sunchon National University, Sunchon, 540-742, South Korea

Abstract. Two different types of aluminum alloys of AA1050 and AA5182 were used to manufacture Al/Al-Mg/Al composite sheet metals by roll bonding technology at room temperature. The composite sheet metals were annealed at 400 o C and carried out uniaxial tension tests to investigate mechanical properties. Macroscopic mechanical properties are strongly dependent on the volume (or thickness) fraction of two component layers. Microstructure and texture evolution were also investigated during roll bonding process. The AA1050 sheets located in the outer layer mainly consist of shear texture components and the AA5182 sheet located in the center layer consists of plane strain texture components. With differential speeds of the top and bottom rolls, roll bonding was also carried out. Elongation along the RD and TD was improved at a speed difference of approximately 10% − 20%. Keywords: cladding, asymmetric rolling, texture and microstructure, aluminum alloys PACS: 62

INTRODUCTION One of the most frequently used clad processes is a roll bonding. During roll bonding, surface of the sheets should be cleaned and degreased, and then be carried out scratch brushing with fine wires. The surface layers on the metals usually contain non metallic films such as oxide films, which often prevent virgin metals from being tightly bonded. Scratches on the surface increase the contact area and increase bonding strength. In addition to the surface preparation, rolling temperature and speed, reduction in area, contact friction, and geometry of sheets and rolls would also play an important role in successful fabrication of the clad materials [1, 2, 3, 4, 5, 6]. There exists some critical or threshold reduction in area to break surface oxide films, below which no bonding occurs. The threshold reduction is associated with threshold strain and strength [7]. The former indicated bonding of base metals occurs when strain is greater than the threshold value. The latter implies that the bonded interface would not be separated when bonded strength is larger than the threshold. Increased reduction in area usually resulted in increased bonding strength. Elevated temperature of sheets at the roll entry also helps to make further reduction in area and increases bonding strength. It is necessary to optimize reduction in area, and annealing time and temperature to avoid undesirable intermetallic compound layer or properties [8, 9, 10, 11]. Roll bonding of dissimilar materials is a naturally asymmetric rolling process, because their plastic responses were different [12, 13]. Sometimes, the shearing effect was intentionally used to fabricate clad materials [14, 15]. The rolling process was designed with a hard material contacting with a high speed roll, and a soft materials contacting with a slow speed roll. The asymmetrical bonding made a uniform metal flow of both metals [6]. This is because cross shear zone occurred and decreased friction hill during asymmetric rolling. Our main goal is to better understand cladding process and the resultant mechanical properties, and to finally fabricate high strength structural clads with enhanced surface treatment capability. The clad system of interest is AA1050/AA5182/AA1050, which seems to contain both high strength and elongation, and good anodizing properties. It is known that shear deformation is favorable to enhancement of the mechanical properties of aluminum alloys [16, 17, 18, 19]. Therefore, asymmetric rolling process was applied to roll bonding process in order to fabricate three-layer clads of Al/Al-Mg/Al with enhanced mechanical properties. Variation of microstructure and mechanical properties was investigated with speed ratio.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 337-342 (2011); doi: 10.1063/1.3623629 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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TABLE 1. Chemical composition of clad materials Si

Fe

Cu

Mn

Mg

Cr

Zn

Ti

AA5182

0.08

0.18

0.05

0.45

4.68

AA1050

0.04

0.37

0.07

-

-

Al

0.02

0.03

0.01

Bal.

-

0.01

-

Bal.

EXPERIMENTS Commercial aluminum alloys of AA1050 and AA5182 were used for fabrication of cladding composite. The thicknesses of the AA1050 and AA5182 sheets were 0.5 mm and 2 mm, respectively. Chemical composition of them was given in Table 1. Sample preparation of the aluminum sheets was carried out by degreasing with acetone, and followed by surface scratch-brushing using the wire brush. The diameter of the brusher is 200 mm, and each stainless steel brusher wire is 0.3 mm in diameter. Rotational speed of the brusher was 300 rpm (revolution per minute) and sample feeding speed was 0.2 mpm (meter per minute). The clad materials were stacked in a sequence of top AA1050/center AA5182/bottom AA1050, and they were clamped with steel wire to avoid their separation during roll bonding. The top and bottom sheets of AA1050 imply that they contact the top and bottom rolls, respectively. The AA5182 sheet was located between the AA1050 sheets. The diameter of the rolling mill was 280 mm and the rotational speeds of the top and bottom rolls were separately operated. Overall reduction in area was 50% per pass at the room temperature. The velocity of the top roll Vtop was held constant at 5 mpm, and that of the bottom roll Vbot was varied intentionally at 5, 5.5, 6, 6.5, and 7 mpm, and thus the speed ratio of the top to bottom rolls was varied from 1:1 to 1:1.4. Texture and microstructure were characterized using XRD (X-ray diffraction), OM (optical microscope), and EBSD (electron backscatter diffraction)/SEM. Macroscopic texture measurements were carried out by XRD on the surface region of the bonded sheets. The AA1050 sheet contacting the top roll was measured both sides, the one contacting the top roll, and the other contacting the AA5182. The center AA5182 sheet was measured on the side contacting the top AA1050 sheet. The XRD system used was Xpert Pro (PANalytical) with Cu-Kα radiation operated at 40 kV and 30 mA. Three incomplete (111), (200), and (220) pole figures were collected on a 5◦ grid up to a sample tilt of 70◦ . The obtained data were analyzed using the WIMV method with an automated conditional ghost correction [20, 21], assuming cubic crystal and triclinic sample symmetries. Microstructure and microtexture analyses were carried out using an automated high-resolution EBSD (JEOL7001F) with HKL Channel5 and a generalized EBSD data analysis code, REDS [22]. EBSD samples were mechanically polished and then electropolished using a solution of ethanol (60 ml), and perchloric acid (10 ml) at a voltage of 10 V and a temperature of −15◦ C to −20◦ C. Samples observed by OM were prepared by mechanical polishing followed by subsequent electro-chemical etching with a solution of fluoroboric acid (5 ml), and distilled water (95 ml). Annealing heat treatments were carried out with clad samples before tensile tests. Tensile specimens with a gauge length of 12.5 mm and a width of 3 mm were machined according to the ASTM E8 standard, with the tensile axis at 0◦ , and 90◦ from the rolling direction (RD). The tensile tests were conducted on a standard universal testing machine (Instron 4206) at ambient temperature with a head speed of 1.25 mm/min.

RESULTS AND DISCUSSION Figure 1 displays schematic sequence of roll bonding, heating and mechanical tests of AA1050/AA5182/AA1050 clads. Texture and microstructure evolution of as-rolled clad sheets during clad were investigated using both XRD and EBSD. Mechanical properties were measured through tensile tests after annealing at elevated temperature of 400◦ C for 1 hr. Initial microstructure and texture of AA1050 and AA5182 sheets were presented in Fig. 2. The AA1050 sheet was an as-rolled state and elongated grains were observed. The complete (111) pole figures showed the typical fcc rolling fiber or plane strain compression (PSC) texture, α(Goss{110}001 ∼ Brass{110}112) and β (Brass{110}112 ∼ S{123}634 ∼ Copper{112}111). The complete (111) pole figures of AA1050 were computed from both EBSD and XRD experiments based on the cubic crystal and triclinic sample symmetries. There existed some minor difference between pole figures computed from EBSD and XRD. The EBSD results usually provide more localized information, and both α and β fibers looked well-developed. While, the XRD results provide more bulky information, and the Copper texture component looked stronger than others in the AA1050 sheet. The overall microstructure of the AA5182 sheet showed equiaxed grain shapes and revealed that it was fully-recrystallized. The (111) pole figure computed from

338

FIGURE 1.

Schematic sequence of clad fabrication and heat treatments followed by mechanical tests.

FIGURE 2. Initial microstructure (inverse pole figure maps) and texture (111 pole figures) of AA1050 and AA5182 sheets. Contours for XRD PFs: 1, 2, 3, 5, 7, 10.

EBSD showed both Cube{100}001 and retained Brass type texture. The (111) pole figure by XRD clearly presented strong Cube texture component. Some minor components also existed. Figure 3 illustrated bent-shaped clad sheets and optical micrographs through thickness. From the optical micrographs, the outer layers of AA1050 were distinguished from the center (core) layer of AA5182. The elongated grains in the as-rolled AA1050 sheets were much more elongated during roll bonding. The equiaxed grains in the AA5182 sheets were also elongated. It is unclear to what extent the speed ratio affected the microstructure from the optical micrographs. The thickness of each layer had some variation from position to position. Figure 4 exhibited EBSD results of roll bonded clads (outer layer of AA1050) with speed ratio. The Outer region is the surface contacting rolls and the Inner is the surface contacting AA5182. The Outer region experience more deformation and grains in the Outer region were more refined than those inside the sheets. The metal flow near Inner region revealed inhomogeneous deformation due to hard materials of AA5182. Overall, elongated grains looked similar to the optical micrographs found in Fig. 3. The grain size of AA5182 layer looks greater than those of AA1050 layers shown in Fig. 4. Grain size and aspect ratio were summarized in Table 2. Figure 5 showed (111) pole figures computed from EBSD and XRD results. The (111) pole figures of the core

339

FIGURE 3. Optical micrographs of clads fabricated by asymmetric roll bonding.

FIGURE 4.

Inverse pole figure maps of clads fabricated by asymmetric roll bonding. TABLE 2. ratio

Grain size (GS) and aspect ratio (AR) with speed Ini.

1:1.0

1:1.05

1:1.10

1:1.20

AA5182 (GS)

6.4

7.8

6.1

6.0

7.7

AA5182 (AR)

1.3

2.4

2.3

2.2

2.5

AA1050 (GS)

3.0

1.2

1.3

1.4

1.5

AA1050 (AR)

2.5

2.8

2.7

2.7

2.7

340

FIGURE 5. (111) pole figures of clads fabricated by asymmetric roll bonding. Contours for EBSD PFs: 0.92, 1.42, 1.92, 2.42, 2.92, 3.42, 3.92, 4.42, 4.92, 5.42, 5.92, 6.42, 6.92, 7.42. Contours for XRD PFs: 1, 2, 3, 5, 7, 10.

FIGURE 6.

Variation of mechanical properties of clad sheets along the RD and TD.

AA5182 and outer layer AA1050 sheets were computed from EBSD maps measured from the whole thickness. In addition, XRD results taken from the top surface were also presented for the outer layer of AA1050. The core AA5182 sheet showed well-developed PSC texture of α and β fibers. Overall intensities looked similar to each speed ratio. The outer layer of AA1050 sheets displayed shearing texture, major rotated cube component, {100}011 and minor {111}//ND. The minor {111}//ND components found in the XRD were weaker than those in EBSD results. Results of the uniaxial tension tests were presented in Fig. 6. Variation of the tensile and yield strength was negligible with velocity ratio. Tensile and uniform elongation have some peak around speed ratio of 1.15. These trends were found both the RD and TD samples.

341

SUMMARY AA1050/AA5182/AA1050 clad sheets were fabricated with various rolling conditions at room temperature. Initial microstructure of AA1050 was as-rolled, and that of AA5182 was fully recrystallized. The outer layer of AA1050 sheets experienced more shear deformation than the core of AA5182. Plane strain compression texture was observed in the core, and rotated cube type texture was found in the outer layer. The variation of texture and microstructure with speed ratio from 1.0 to 1.2 was not evident during cladding. It was found that elongation was improved at a speed ratio of about 1.1 to 1.2.

ACKNOWLEDGMENTS The authors would like to thank the financial supports of Korea Institute of Materials Science (KIMS).

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

Y.-M. Hwang, H.-H. Hsu, and Y.-L. Hwang, International Journal of Mechanical Sciences 42, 2417–2437 (2000). H. Manesh, and A. Taheri, Mechanics of Materials 37, 531–542 (2005). H. Manesh, and A. Taheri, Journal of Materials Processing Technology 166, 163–172 (2005). H. Yan, Trans. Nonferrous Met. Soc. China 16, 84–90 (2006). M. Nezhad, and A. Ardakani, Materials and Design 30, 1103–1109 (2009). X. Li, G. Zu, and Q. Deng, Light Metals 2011 "(TMS2011)" 1, 615–619 (2011). X. Zhang, T. Yang, S. Castagne, C. Gu, and J. Wang, Materials and Design 32, 2239–2245 (2011). H. Manesh, and A. Taheri, Materials and Design 24, 617–622 (2003). H. D. Manesh, and A. K. Taheri, Journal of Alloys and Compounds 361, 138–143 (2003). J. Lee, D. Bae, W. Chung, K. Kim, J. Lee, and Y. Cho, Journal of Materials Processing Technology 188, 546–549 (2007). J. S. Yoon, S. H. Lee, and M. S. Kim, Journal of Materials Processing Technology 111, 85–89 (2001). A. Segawa, and T. Kawanami, Journal of Materials Processing Technology 47, 375–384 (1995). A. Segawa, and T. Kawanami, Journal of Materials Processing Technology 47, 544–551 (1995). N. Bay, H. Bjerregaard, S. Petersen, and C. dos Santos, Journal of Materials Processing Technology 45, 1–6 (1994). Y. Hwang, T. Chen, and H. Hsu, International Journal of Mechanical Science 38, 443–460 (1996). T. Sakai, S. Hamada, and Y. Saito, Scripta mater. 44, 2569–2573 (2001). H. Jin, and D. Lloyd, Materials Science and Engineering A 399, 358–367 (2005). S. B. Kang, B. K. Min, and W. D. S. J. K. Kim, H. W., Metall. Mater. Trans. A 36, 3141–3149 (2005). H. Jin, and D. Lloyd, Materials Science and Engineering A 465, 267–273 (2007). S. Matthies, and G. Vinel, Phys. Stat. Sol.(b) 112, k111–k114 (1982). S. Matthies, S. Vinel, and K. Helming, Standard Distribution in Texture Analysis, Akademie-Verlag, Berlin, 1987. J. H. Cho, A. D. Rollett, and K. H. Oh, Metall. Mater. Trans. A 36A, 3427–3438 (2005).

342

Influence of Hydrostatic Pressure on FLDs for AZ31B Sheets H. Wanga, Y. Wub, P.D. Wua, K.W. Nealec a

Department of Mechanical Engineering, McMaster University, Hamilton, Ontario L8S 4L7, Canada b

State Key Laboratory for Geomechanics and Deep Underground Engineering China University of Mining and Technology, Xuzhou, Jiangsu 221008, China

c

Faculty of Engineering, University of Sherbrooke, Sherbrooke, Quebec J1K 2R1, Canada

Abstract. Sheet metal formability in terms of the Forming Limit Diagram (FLD) for magnesium alloy AZ31B sheets is studied by using the recently developed Elastic Visco-Plastic Self-Consistent (EVPSC) model, in conjunction with the classical M-K approach. The effect of superimposed hydrostatic pressure on FLDs is numerically assessed. It is found that the superimposed pressure delays the initiation of necking for any strain path. The increment of formability becomes significant when the pressure is larger than about half the initial tensile yield stress of the sheet metal. The difference in predicted FLDs between that with a superimposed hydrostatic pressure and that with a stress component normal to the sheet plane is also discussed. Keywords: crystal plasticity; hydrostatic pressure; formability; magnesium alloy PACS: 62.20.fk

1. INTRODUCTION Conventionally rolled or extruded magnesium alloys inevitably result in a dominant basal texture and thus show low formability due to the limited number of plastic deformation modes available at room temperature. However, recent experimental works have revealed that the formability of magnesium sheets can be significantly improved through texture optimization by mainly re-orientating the basal plane through various shear processes [1-5]. In a recently study, Wang et al. [6, 7] investigated the influence of basal texture on the uniform strain under uniaxial tension and the limit strain under biaxial stretching. This study suggested that formability can be significantly improved by controlling texture even without grain refinement. Another possible way to increase formability for magnesium alloys is by superimposing hydrostatic pressure. It has been generally accepted that the superimposed hydrostatic pressure significantly enhances ductility for various monolithic metals and composites [8-11]. Based on the classic isotropic rate-independent plasticity theory, Wu et al. [12] have studied the effects of superimposed hydrostatic pressure on the sheet metal formability in terms of the FLD. It has been observed that the superimposed hydrostatic pressure increases sheet metal limit strains for any strain path. In the present study, the influence of superimposed pressure on FLDs is numerically studied for the magnesium alloy sheet AZ31B. All simulations are based on the recently developed Elastic Visco-Plastic SelfConsistent (EVPSC) model [13], in conjunction with the M-K approach developed by Marciniak and Kuczynski [14].

2. CONSTITUTIVE MODEL The elastic constitutive equation for a crystal is:

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 343-350 (2011); doi: 10.1063/1.3623630 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

343

’*



L : d e  tr (d e )

(1) ’*

where L is the fourth order elastic stiffness tensor, d is the elastic strain rate tensor and  is the Jaumann rate of e

the Cauchy stress  based on the lattice spin tensor L through the crystal elastic constants Cij .

w e . The single crystal elastic anisotropy is included in

FIGURE 1. Plastic deformation modes for hexagonal structure: (a) basal slip systems, (b) prismatic slip systems, (c) pyramidal slip systems, and (d) tensile twin.

Plastic deformation of a crystal is assumed to be due to crystallographic slip and twinning on systems ( s D , nD ) . Here, s D and nD are the slip/twinning direction and the direction normal to the slip/twinning plane for system D , respectively. In the present study we consider slip in the Basal ( {0001}  1120 ! ), Prismatic ( {10 1 0}  11 2 0 ! ) and Pyramidal ( {1 1 22}  1 1 23 ! ) slip systems, and twinning in the

{10 1 2}  1 011 ! tensile twin system (see Fig. 1). The following equation gives the grain (crystal) level plastic strain rate:

dp where

PD

J0

WD J0 ¦ P D D W cr D

1 1 m

WD W crD

(2)

is a reference value for the slip/twinning rate, m is the slip/twinning rate sensitivity,

( s D nD  nD s D ) 2 is the Schmid tensor for system D , and W D

 : P D and W crD are the resolved shear

stress and critical resolved shear stress (CRSS) for system D , respectively. The self-consistent approach is applied to obtain the response of a polycrystal comprised of many grains. In a self-consistent approach each grain is treated as an ellipsoidal inclusion embedded in a Homogeneous Effective Medium (HEM), which is an aggregate of all the grains. Interactions between each grain and the HEM are described using the Eshelby inclusion formalism. During each deformation step, the single crystal constitutive rule (which describes the grain-level response) and the self-consistency criteria are solved simultaneously. This ensures that the grain-level stresses and strain rates are consistent with the boundary conditions imposed on the HEM. To date, various self-consistent schemes have been proposed. The Elastic Visco-Plastic Self-Consistent (EVPSC) model for polycrystals recently developed by Wang et al. [13] is a completely general elastic-viscoplastic, fully anisotropic, self-consistent polycrystal model, applicable to large strains and to any crystal symmetry.Very recently, Wang et al. [15-17] evaluated various linearization self-consistent schemes employed in self-consistent modeling by applying them to the large strain behavior of magnesium alloys under different deformation processes, including torsion. It was found that the Affine self-consistent scheme gave the best overall performance among the self-consistent approaches examined. Therefore, the EVPSC model with the Affine self-consistent scheme is applied in the present study. For a detailed description of the EVPSC model we refer to Wang et al. [13].

344

3. PROBLEM FORMULATION AND METHOD OF SOLUTION Following the numerical procedure developed by Wu et al. [18], the EVPSC model, in conjunction with the M-K approach, has been implemented into a numerical code for constructing FLDs by Wang et al. [7]. The basic assumption of the M-K approach is the existence of material imperfections, which are in the form of a groove on the surface of the sheet. Marciniak and Kuczynski [14] showed that a slight intrinsic inhomogeneity in load bearing capacity throughout a deforming sheet can lead to unstable growth of strain in the region of imperfection, and subsequently cause localised necking and failure. The influence of the superimposed hydrostatic pressure on formability is studied by considering a nonproportional history using combinations of two linear deformation paths. The first loading path is to apply hydrostatic pressure to materials both outside and inside the imperfection (3) V11 V 22 V 33  p until the desired level of superimposed hydrostatic pressure p is reached. It is noted that the deformations inside and outside the band remain very small during this almost purely elastic loading path even when p is very high. In the subsequent linear strain path, the deformation outside the band is assumed to be such that

D22 D11 where H22

D22 and H11

H22 H11

D12

0, W12

0

(4)

D11 are the (principal) logarithmic strain rates and the Wij values are components of the

spin tensor. It is further assumed that D13 condition V 33

U,

D 23 W13 W23

0 . The thickness strain rate D33 is specified by the

0 ( V 33 { p ). For the orthotropic textures considered here, these boundary conditions imply that

V 13 V 23

0. By enforcing the compatibility at the band interface, the equilibrium balance on each side of the interface, and the assumption V 33 0 inside the band, the deformation inside the band

the average stress components

can be completely determined. The onset of sheet necking is defined by the occurrence of a much higher maximum b b 4 principal logarithmic strain rate inside the band than outside, taken here as the condition H D11 t 10 , where H

represents the strain rate inside the band. The corresponding principal logarithmic strains H11 and H 22 outside the band are the limit strains. The entire FLD of a sheet is determined by repeating the procedure for different strain paths outside the band as prescribed by the strain ratio U . A detailed description on how superimposed hydrostatic pressure is included in the FLD calculation can be found in Wu et al. [12]. *

*

4. RESULTS AND DISCUSSION The material studied in the present paper is the magnesium alloy AZ31B sheet; its mechanical behavior at room temperature has been experimentally studied by Jain and Agnew [19]. The initial crystallographic texture of the sheet is represented by 2160 orientations. The measured texture shows that the grains tend to have their basal planes oriented perpendicular to the normal direction of the sheet. The reference slip/twinning rate, J0 , and the rate sensitivity, m , are prescribed to be the same for all slip/twinning systems: J0

0.001s 1 and m 0.05 , respectively. The room temperature elastic constants of the magnesium single crystal are assumed to be C11 58.0 , C12 25.0 , C13 20.8 , C33 61.2 and C44 16.6 (unit of GPa) [20]. Values of the hardening parameters for

each mode are estimated by fitting numerical simulations of uniaxial tension and compression along the rolling direction (RD) to the corresponding experimental flow curves. The uniaxial tension and compression true stresstrue strain curves along the RD are presented in Fig. 2. The characteristic S-shape of the compressive flow curve clearly reveals the importance of twinning in compression. The macroscopic yield stresses for uniaxial compression and tension are about 100 MPa and 160 MPa, respectively, showing the strong tension/compression asymmetry associated with twinning. It is clear that the EVPSC model with the Affine scheme fits the experimental curves quite well. The values of these parameters will be used in the subsequent simulations. In addition, the initial imperfection in the M-K approach is assumed to be f 0 0.99 .

345

500

V 11

Compression 400

300

Tension

200 Experiment Fitting

100

0

0

0.05

0.1

0.15

0.2

H 11p

0.25

FIGURE 2. True stress and plastic strain curves under uniaxial tension/compression along the RD. The experimental data are taken from Jain and Agnew [19]. 600

V 11

500

p 0MPa

40 400

80 120 160

300

200

100

0

0

0.1

0.2

0.3

H11

0.4

FIGURE 3. Uniaxial tensile stress responses under superimposed hydrostatic pressure represented in terms of V 11 .

We first consider in-plane plane strain tension ( U Figure 3 presents the true stress

V 11

and true strain

0 in Eq. 4) under a superimposed hydrostatic pressure p.

H11 curves for the material point outside the imperfection band.

346

The calculations are stopped at necking; i.e., when H b D11 t 10 4 . It is found that the superimposed hydrostatic pressure p lowers the

V 11 ~ H11

curve vertically.

600

V 11

120

40

500

400

p 0MPa

80

160

300

200

100

0

0

0.1

0.2

0.3

H11

0.4

FIGURE 4. Uniaxial tensile stress responses under superimposed hydrostatic pressure represented in terms of

V 11 V 11  p .

0.7

H*

0.6

0.5

0.4

0.3

0.2

U U U

0.1

0

0

40

80

120

  

160

200

p (MPa )

FIGURE 5. Effect of superimposed hydrostatic pressure p on major limit strains for strain paths uniaxial tension U plane plane strain tension

U

0 and equi-biaxial tension U 1 .

347

0.5 , in-

The V 11 ~ H11 curves in Fig. 3 are converted into the V 11 V 11  p ~ H11 curves in Fig. 4. The symbols represent the necking points. It is clear that the superimposed hydrostatic pressure p does not affect work-hardening but lowers the V 11 ~ H11 curve vertically by the amount of p. However, it is also clear that the superimposed hydrostatic pressure increases the limit strain.

0.8

H

* 11

0.6

200

0.4

160 120 80 40 0.2

p= 0MPa

-0.4

-0.2

0

0

0.2

H

0.4

* 22 FIGURE 6. Effect of superimposed hydrostatic pressure p on FLDs.

Figure 5 shows the effect of superimposed hydrostatic pressure on the predicted major limit strains under uniaxial tension ( U 0.5 ), in-plane plane strain tension ( U 0 ) and equi-biaxial tension ( U 1 ). It is found that the major limit strain increases monotonically with the superimposed hydrostatic pressure for all three strain paths. Figure 6 presents the predicted full FLDs under various superimposed pressures. It is clear that the superimposed hydrostatic pressure increases the sheet metal limit strain for any strain path. Finally, it is worth mentioning that we have analyzed the effect of hydrostatic pressure on FLDs. It is noted that the effect of the stress component V 33 (normal to the sheet plane) on FLDs has been reported by Banabic and Soare [21]. They showed that the effect of the normal stress on FLDs was similar to that of a hydrostatic pressure. We have numerically assessed the effect of the normal stress on FLDs for the AZ31B sheet. Figure 7 shows the influence of the normal stress/pressure on the predicted FLDs. It is found that, in the range of p d 200 MPa , the effect of the normal stress on FLDs is very similar to that of the hydrostatic pressure. Figure 8 presents effects of superimposed hydrostatic pressure and normal pressure/stress on the major limit strain for in-plane plane strain tension U 0 . It is demonstrated that the difference in the predicted FLDs between superimposed hydrostatic pressure and normal pressure is negligible even at pressures up to 320 MPa.

348

0.8

H 11* 0.6

200

0.4

160 120 80 40 0.2

-0.4

0

-0.2

 V 33

0

0MPa

0.2

FIGURE 7. Effect normal pressure

 V 33

0.4

* H 22

p on FLDs.

0.8

H* 0.6

0.4

p  V 33

0.2

0

0

100

200

300

 V 33 ( p)(MPa )

400

FIGURE 8. Effect of superimposed pressure and normal stress on major limit strain for in-plane plane strain tension૟ U = 0.

349

4. CONCLUSIONS The effect of superimposed hydrostatic pressure has been included in the analysis of the forming limit diagram for the magnesium alloy AZ31B sheet. All the simulations have been based on the recently developed elastic-viscoplastic self-consistent (EVPSC) model, in conjunction with the M-K approach. It has been numerically demonstrated that the superimposed hydrostatic pressure increases sheet metal limit strains for any strain path. It has been also found that, in the range considered, the effect of the normal stress on FLDs is very similar to that of a hydrostatic pressure.

ACKNOWLEDGEMENTS H. Wang and P.D. Wu were supported by funding from the NSERC Magnesium Strategic Research Network. More information on the Network can be found at www.MagNET.ubc.ca. Y. Wu was supported by the National Basic Research Program of China (2007CB209400) and the 111 Project of China (B07028).

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

T. Mukai, M. Yamanoi, H. Watanabe and K. Higashi, Scripta Mater., 45, 89-94 (2001). S. R. Agnew, J. A. Horton, T. M. Lillo and D. W. Brown, Scripta Mater., 50, 377-381 (2004). X. S. Huang, K. Suzuki, A. Watazu, I. Shigematsu and N. Saito, Mater. Sci. Eng., A488, 214-220 (2008). Y. Chino, K. Sassa and M. Mabuchi, Scripta Mater., 59, 399-402 (2008). B. Song, G. S. Huang, H. C. Li, L. Zhang, G. J. Huang and F. S. Pan, J. Alloy Compd.,489, 475-481 (2010). H. Wang, P. D. Wu and M. A. Gharghouri, Mater. Sci. Eng., A527, 3588-3594 (2010). H. Wang, P. D. Wu, K. P. Boyle and K. W. Neale, Int. J. Solids Struct., 48, 1000-1010 (2011). P. W. Bridgman, Studies in Large Plastic Flow and Fracture - With Special Emphasis on the Effects of Hydrostatic Pressure. McGraw-Hill, New York. 1952. W. A. Spitzig and O. Richmond, Acta Metall., 32, 457-463 (1984). A. S. Kao, H. A. Kuhn, O. Richmond and W. A. Spitzig, Metall. Trans.,20A,1735-1741 (1989). J. J. Lewandowski and P. Lowhaphandu, Int. Mater. Rev., 43, 145-187 (1998). P. D. Wu, J. D. Embury, D. J. Lloyd, Y. Huang and K. W. Neale. Int. J. Plasticity, 25, 1711-1725 (2009). H. Wang, P. D. Wu, C. N. Tomé and Y. Huang, J. Mech. Phys. Solids, 58, 594-612 (2010). Z. Marciniak and K. Kuczynski, Int. J. Mech. Sci., 9, 609-620 (1967). H. Wang, B. Raeisinia, P. D. Wu, S. R. Agnew and C. N. Tomé, Int. J. Solids Struct., 47, 2905-2917 (2010). H. Wang, P. D. Wu and K. W. Neale. J. Zhejiang Univ.-SC. A, 11,744-755 (2010). H. Wang, Y. Wu, P. D. Wu and K. W. Neale. CMC-Comput. Mater. Con., 19, 255-284 (2010). P. D. Wu, K. W. Neale and E. Van der Giessen, P. R. Soc. London, A453, 1831-1848 (1997). A. Jain and S. R. Agnew, Mater. Sci. Eng., A462, 29-36 (2007). G. Simmons and H. Wang, Single crystal elastic constants and calculated polycrystal properties. Cambridge (MA): MIT Press, 137 (1971). D. Banabic and S. Soare, On the effect of the normal pressure upon the forming limit strains. In: Hora, P. (Ed.), Numisheet 2008 - Proceedings of the 7th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, 199-204 (2008).

350

Application of a Reduced Texture Methodology to Model the Plasticity of Anisotropic Extruded Aluminum Sheets Meng Luoa, Gilles Rousselierb and Dirk Mohra,c a

Impact and Crashworthiness Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge MA, USA b MINES ParisTech, Centre des Matériaux, CNRS UMR 7633, BP 87, 91003 Evry Cedex, France c Ecole Polytechnique ParisTech, Solid Mechanics Laboratory, CNRS UMR 7649, 91120 Palaiseau, France

Abstract. A recently developed Reduced Texture Methodology (RTM) featuring (i) a significantly reduced number of crystallographic orientations, (ii) a special experiment-based parameter calibration procedure, and (iii) reasonable computational time for industrial applications is adopted to model the anisotropic plastic behavior of a 2mm-thick extruded aluminum 6260-T6 sheet. Firstly, the full-thickness sheet is modeled with twelve crystallographic orientations, and the model parameters are identified through an optimization procedure based on uniaxial tensile tests with seven different material orientations. The calibrated model describes well the stress-strain curves and Lankford ratios for all directions, while the optimized grain orientations are in good agreement with EBSD measurements. However, the EBSD results also reveal that the present sheet exhibits a strong heterogeneity through the thickness as far as crystallographic orientations and grain sizes are concerned. To account for this heterogeneity, eight grain orientations are selected out of the total twelve for the full-thickness sheet to model the 0.7mm-thick central layer of the sheet based on the EBSD measurements. It is found that the reduced eight-grain model provides good predictions of the macroscopic responses in uniaxial tensile tests on reduced-thickness specimens, even without further calibration. A combined calibration is also performed to determine the final set of parameters which provide excellent modeling for both the full-thickness sheet (twelve-grain model) and its central layer (eight-grain model). Keywords: Polycrystalline model, anisotropy, aluminum extrusion, heterogeneity, finite element method, sheet forming PACS: 62.20.-x; 46.35.+z; 61.66.Dk

INTRODUCTION Extensive research has been conducted in the field of phenomenological plasticity modeling of anisotropic metals. As a result, the initial anisotropy of their plastic response can be described with good accuracy using advanced macroscopic models, in particular, with the use of linearly transformed stress tensors in isotropic yield functions, e.g. Refs.1, 2. However, phenomenological plasticity models cannot capture the complex evolution of anisotropy throughout loading3 or the cross-hardening behavior in non-proportional loading4, unless more sophisticated rotational or distortional yield surfaces are employed5. Polycrystalline metal plasticity models have intrinsic advantages in modeling the anisotropy and complex hardening behaviors under multi-axial and multi-path loadings. The main limitation of the polycrystalline models is the large computational effort involved for structural analysis and material model parameter identification through inverse analysis. The tremendous CPU times required for a single simulation limit the number of parameters that can be calibrated and the number of iterations for optimization. As a result, the accuracy of polycrystalline models at the macroscopic level is often compromised, despite their sound mechanism-based constitutive description at the microscopic level. Even with advanced selfconsistent models and an experimentally-measured anisotropic texture, it can be difficult to obtain a good evaluation of flow stresses and directions simultaneously3. This is a huge drawback for the accurate finite element analysis of sheet metal forming processes. Reasonable CPU time can be achieved through a significant reduction of the number of crystallographic orientations6, 7. Rousselier et al.8, 9 have used a small number of crystallographic orientations with a self-consistent polycrystalline model. Their parameter calibration procedure (which includes the texture

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 351-359 (2011); doi: 10.1063/1.3623631 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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parameters) relies on a comprehensive experimental database to suppress irregularities in the predicted stress-strain response associated with the limited number of crystallographic orientations. With this approach, they managed to obtain an accurate description of both the flow stresses and strain rates of a highly anisotropic 2090-T3 aluminum sheet; the CPU time of their model on a deep drawing FE simulation is only twice as long as that of an advanced phenomenological model8. In this paper, a Reduced Texture Methodology (RTM) is applied to model the anisotropic plastic behavior of a 6260-T6 aluminum extrusion of 2mm wall thickness. Our objective is not the modeling of the actual initial texture and its evolution, but the accurate description of the mechanical response at the macroscopic scale. One particular challenge in this study is the through-thickness heterogeneity present in the extruded aluminum sheets. Our experimental data base therefore contains results for both full-thickness and reduced-thickness specimens 10. It is shown that reduced texture methodology allows for a complete identification of all model parameters. The resulting RTM based polycrystal plasticity model is able to describe the plastic behavior of both full-thickness and reducedthickness specimens with good accuracy.

POLYCRYSTALLINE PLASTICITY MODELING In the following, , ⃗,    represent a scalar, a vector, a second order tensor and a fourth order tensor, respectively. A dot denotes differentiation with respect to time. Upper-case symbols (Σ,  , , … ) and lower-case symbols (, , … ) correspond to mechanical variables at the macroscopic and microscopic scales, respectively. All the following equations are written in a material embedded co-rotational frame8. Due to the length limit, only key ingredients of the constitutive equations are presented in this section. Readers are referred to Refs.8, 9 for details on the model formulation.

The Self-consistent Polycrystalline Model The Reduced Texture Methodology (RTM) considers a very small number of crystallographic orientations only. These orientations are determined from mechanical experiments through inverse analysis instead of crystallographic measurements. The RTM framework can be applied to any polycrystalline model. In this paper, a particular viscoplastic self-consistent model is employed. Each of the  “grains” of the model represent a set of physical grains with close orientations. In each “grain” g of volume fraction  , the stress  and strain  =  +  are assumed to be homogeneous. In the case of isotropic elasticity, with Young modulus  and Poisson’s ratio , we have 



Σ=

  with 







 = 

 = 1

(1)



 

(2)

1+  1 − 1 ⊗ 1 ∶ Σ  

(3)



 ≡   = 

A so-called -rule11, 12 which captures the elasto-plastic accommodation of intergranular deformations is adopted in the present work: 

 = Σ +   −   with = 



 

$

̇ = ̇ − !:  " ̇ " where " ̇ " = # ̇ : ̇ %

(4) (5)

where  is an intermediate deviatoric strain tensor which evolves non-linearly. The scalar parameter of the initial elastic accommodation  is close to the elastic shear modulus & =  /2(1 + ) . The fourth-order tensor ! is

352

proposed by Sai12 for anisotropic materials. With Voigt the notation, ! has 10 independent elements !'* , as it has orthotropic symmetry and it is a linear relation between two deviatoric tensors (see details in ref. 8 ).

The Single Crystal Formulation This section briefly presents the constitutive equations at the level of a single crystal. The single crystal model relates the slip rate -.̇ and the resolved shear stress 0. of each slip system (s = 1 to M). The orientation tensor 3. attributed to the slip system number 4 is defined as 3. = 56⃗7 ⊗ 8⃗7 + 8⃗. ⊗ 6⃗. 9/ 2

(6)

8⃗. and 6⃗. being the “slip plane” normal vector and the “slip direction” vector in this plane, respectively. The socalled resolved shear stress is given by 0. =  : 37

(7)

In the case of the fcc crystallographic structures, the ; = 12 glides considered in this paper are the octahedral systems {111}(100). A phenomenological viscoplastic model11 is used as the constitutive equations for each slip system: D

|0. − <. | − A. E C F B.

(8)

I.J [1 − exp(−LJ )] + H$ [1 − exp(−L$ N. )]

(9)

-̇. = ̇. sign(0. − <. )

with

̇7 = Max >0, @

K

A. = G. + H 

J

Ȯ . = -̇. − O. ̇.

with <. = PO.

(10)

For each slip system, two scalar internal variables, A. for isotropic hardening and O. for kinematic hardening, are introduced. Visco-plastic flow reaches the rate-independent limit for large . and small B. . The non-linear isotropic hardening law makes use of an interaction matrix I to represent self-hardening (diagonal terms) and latent hardening (off-diagonal terms. According to Ref.8, the 12×12 I matrix depends on only six parameters ℎ to ℎR . In addition, it is assumed here that G. ≡ GS , B. ≡ B and . ≡  are identical for all octahedral slip systems. Eq. (10) is the Armstrong-Frederick non-linear kinematic hardening model with two parameters P and . The grain plastic strain rate corresponds to the sum of the slip on the ; slip systems:

̇ = 

K

3. -̇.

.

(11)

EXPERIMENTAL PROGRAM The present aluminum sheet is extracted from the flanges of an aluminum alloy 6260-T6 extrusion, which has a wall thickness of about 2mm and maximum width of 70mm. The uniaxial tension experiments for seven different material orientations are included in the experimental database for calibration. Figure 1a shows the geometry and dimensions of the dog-bone shaped uniaxial tensile specimens, which differs from the ASTM-E8 specimen due to the limited width of the extruded material. In order to calibrate kinematic hardening, four compression-tension tests are also performed on specimens along the transverse direction (O = 90o ). The geometry of the compression-tension specimen is shown in Fig. 2b. Its special shape is designed for use with an anti-buckling device which helps maintain the in-plane compression up to 10% strain. A first loading in compression is applied up to maximum engineering strains of -0.8%, -2.6%, -4.2% and -8.5% before loading rate reversal, see Fig. 8.

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The heterogeneity of the present sheet is investigated using Electron Backscatter Diffraction (EBSD). As shown in Fig. 2, the grain size is larger near the surfaces and the “green” orientations almost disappear in the central layer. To address the through-thickness heterogeneity, a new set of 7 uniaxial tension tests are performed with a reduced thickness specimen. The test procedure and the outer profile of the specimen (Fig.1a) are the same as in fullthickness experiments, but the thickness of the specimen has been reduced to 0.7mm symmetrically. All experiments are carried out under quasi-static condition and at room temperature, The surface strain fields are all measured using a Digital Image Correlation (DIC) system (VIC2D, Correlated Solutions, SC).

MODEL CALIBRATION AND DISCUSSIONS Numerical Implementation and Parameter Calibration Procedure The polycrystalline plasticity model presented in this paper has been implemented as a user material subroutine (Zmat) in the F.E code Zset/ZeBuLoN. The Zset/ZeBuLoN package has two features which make it an ideal tool for parameter calibration:

(a) (b) (c) FIGURE 1. Specimen used for the model calibration: (a) uniaxial tensile specimen; (b) compression-tension specimen; (c) biaxial specimen with a reduced thickness gage section.

FIGURE 2. Grain orientations measured by EBSD. X and Z are the extrusion direction and the thickness direction respectively.

x A simulator which can do material point simulations x An optimizer that minimizes the user-defined objective function (usually the mean squared error between experimental data and model predictions) For the present study, a special procedure is designed using the two above mentioned building-blocks to calibrate the model parameters: i) Prepare the input files for the simulator corresponding to all calibration experiments. Each input file contains a loading table obtained from experimental measurements. The simulator is fed with the input files and then generates output files which will be compared with experimental data. Particularly, for the uniaxial tension/compression tests, the loading table contains time history of the axial strain  (T), as well as all the non-axial stress components U$$ (T) = U%% (T) = U$ (T) = U$% (T) = U% (T) = 0, and then the simulator will calculate U (T), $$ (T) and %% (T). ii) Define the objective function for the optimizer; for uniaxial tension, the calculated history of axial stress U (T) and the width strain $$ (T) are compared with experimental measurements (stress by load cell, strain

354

by DIC). Each test consists of about 150 time points, and the mean squared error is calculated using the stress and strain errors for all these time points.. iii) Specify the parameters to be calibrated. Assuming rate-independency, the present self-consistent polycrystalline model has 23 material parameters, including 7 slip hardening parameters ( AS , H, L, H$ , L$ , P,  ), 5 latent hardening parameters ( ℎ$ to ℎR ) and 11 localization parameters ( , ! , !$ , !% , !$ , !$% , !% , !%$ , !VV , !WW , !RR ) . In addition, the reduced texture methodology include the texture parameters in the calibration. The grain orientation is defined using the Euler angles (X , Y, X$ ). Due to orthotropic symmetry of the material, the texture is composed of texture components with four symmetric grain orientations ( X , Y, X$ ), (−X , Y, −X$ ), (−X , −Y, −X$ ) and (X , −Y, X$ ) and the same volume fraction . Hence, the reduced texture grain number can be N = 4, 8, 12, 16 …, grouped into 1, 2, 3, 4 … texture components. The first calibration has been performed with N = 12 grains, which corresponds to 3 texture components with 11 parameters (9 Euler angles, 2 volume fractions). The total number of parameters is thus 23+11=34. Last but not the least, one has to specify the initial values and bounds of the parameters, which largely depend on experience but could make huge difference in the CPU time for calibration. iv) Choose an optimization algorithm and the convergence criterion. A gradient-driven Levenberg-Marquardt algorithm is employed.

Calibration I Following the procedure outlined in the last section, the ‘Calibration I’ is carried out using only uniaxial tensile experiments on full-thickness specimens. The 12-grain model (N = 12) is chosen for the full-thickness material, and the kinematic hardening is deactivated (P =  = 0) for ‘Calibration I’ as the reversed loading tests are not included. In addition to 7 physical tests (every 15o in material orientations) presented in section 3, 12 pseudo-experimental tensile tests are also generated for this calibration to avoid possible irregularities of the yield surface due to reduced texture. These pseudo experiments are uniaxial tensile tests with intermediate material orientations every 5o (O = 5\ , 10\ , 20\ , 25\ , … , 85\ ), and obtained by interpolation of the A values. Namely, these pseudo experiments have the same axial strain as the closest physical tests, while the width strains are calculated using the interpolated A values. The width strain data $$ (T) of the pseudo experiments are then included in our experimental database for calibration. It is noteworthy that the actual texture was unknown while the ‘Calibration I’ was performed, so the cold rolling texture components _ (X = 60\ , Y = 32\ , X$ = 65\ ,  = 0.1), Brass (X = 35\ , Y = 40\ , X$ = 0\ , $ = 0.75 ) and Copper ( X = 90\ , Y = 35\ , X$ = 45\ ,  = 0.1 ) are chosen as the initial values of the texture parameters. The calibrated parameters are summarized in Table 1, and the performance of the calibrated 12-grain model is demonstrated in Fig. 3. Clearly, the model accurately captures the anisotropy of the present aluminum sheet in both yield stress and Lankford ratio. Meanwhile, it is even more interesting to compare the calibrated texture with the experimentally measured one. Figure 4a shows this comparison in the form of the {111} pole figures. The black poles are random selected from the EBSD measurements for full-thickness sheet, while the colored poles correspond to the calibrated texture components given in Table 1. The red and purple poles represent the red and purple orientations in Fig. 2, and the green poles roughly correspond to the more scattered green orientations. Note that the calibrated texture agrees qualitatively with the physical texture (which was not known a priori). TABLE 1. Texture and hardening parameters calibrated with full-thickness tensile tests (Calibration I) Texture Parameters jk (o) l(o) jm (o) o “red” component 174.943 81.574 -85.811 0.1094 “purple” component 196.628 107.402 173.532 0.0898 “green” component 125.725 134.213 -81.441 0.0508 Hardening Parameters GS (MPa) H(MPa) H$ (MPa) L L$ ℎ$ ℎ% 68.67 26.13 2.326 1.177 4.206 0.0104 0.1905 (MPa) ℎV ℎW ℎR ! !$ !% 0.0629 0.3915 0.0484 25188.4 484.47 62.77 4.688 !$ !$% !% !%$ !VV !WW !RR 23.64 2.704 8.058 94.13 1.083 125.32 270.51

355

(a) (b) (c) FIGURE 3. Results of Calibration I: (a-b) axial stress-strain curves; (c) width versus thickness strain curves.

(a) (b) FIGURE 4. Comparison of the measured and calibrated texture in the form of {111} pole figures of the present aluminum extrusion (XD = extrusion direction, TD transverse direction). Black dots are measured texture, and colored dots are calibrated ones: (a) full-thickness sheet modeled with a 12-grain model; (c) central-layer sheet represented by an 8-grain model.

The RTM is also used to model the differences in mechanical properties of the full-thickness and central layer sheet material. The black poles in Fig.4b depict the texture of the central layer (0.7mm) material. By analyzing Fig. 2 and comparing Fig.4a with Fig.4b, it is assumed that the central layer reduced texture can be obtained approximately by suppressing the “green” texture component and increasing proportionally the volume fraction of the remaining two components. Namely, an 8-grain model is formed using only “red” and “purple” components in Table 1, and it is then used to predict the material behavior in the reduced-thickness uniaxial tensile tests described in section 2. As shown in Fig. 5a, the 8-grain model can predict the slightly lower yield stress of the central-layer material. Regarding the strain ratios, the 8-grain model can qualitatively capture the difference of the central-layer, but discrepancies are observed in the cases of O = 30\ , 45\ , 60\ and 90\ .

(a) (b) FIGURE 5. Prediction of the anisotropic behavior of the central-layer sheet with the 8-grain model from Calibration I: (a) measured (dots) and predicted (solid lines) axial yield stress at 3% axial strain for 7 material orientations. Red: full-thickness; Blue: central-layer; (b) width versus thickness strain curves from reduced-thickness experiments and the 8-grain model prediction

356

Calibration II On the basis of ‘Calibration I’, the 7 uniaxial tensile tests with reduced-thickness specimens and 4 compressiontension tests are included in the experimental database for ‘Calibration II’. In this further calibration, the two kinematic hardening parameters P and  are involved. Moreover, instead of selecting two grain components from the full-thickness model, the texture parameters of the 8-grain model are directly calibrated from experiments. Therefore, the outcome of the ‘Calibration II’ includes one 12-grain model and one 8-grain model, which share the same set of hardening parameters at the crystal level. The calibrated parameters of ‘Calibration II’ are given in Table 2. It can be seen that the Euler angles are still close to the ones in Table 1, but the volume fractions of the 8-grain model (central-layer) have changed a lot. Regarding the hardening parameters, as the kinematic hardening comes into play, several hardening parameters are changed too, e.g. H, H$ , L$ . TABLE 2. Texture and hardening parameters calibrated with full-thickness/reduced-thickness tensile tests and additional compression/tension tests(Calibration II) 12-grain model jk (o) l(o) jm (o) o “red” component 176.634 79.748 -82.137 0.0933 “purple” component 197.003 107.503 176.078 0.1035 “green” component 129.234 128.459 -78.029 0.0532 8-grain model jk (o) l(o) jm (o) o “red” component 174.693 86.817 -89.576 0.0996 “purple” component 194.888 110.856 171.482 0.1504 Hardening Parameters GS (MPa) H(MPa) H$ (MPa) L L$ P  ℎ$ 68.55 12.38 2.913 6.424 24.992 91.842 2.346 0.0109 (MPa) ℎ% ℎV ℎW ℎR ! !$ !% 0.0886 0.0697 0.9106 0.0421 22766.4 319.40 65.19 4.146 !$ !$% !% !%$ !VV !WW !RR 22.76 2.746 7.524 119.00 1.183 86.54 199.17

Figure 6 demonstrates the validity of the models calibrated from ‘Calibration II’. The anisotropy of the fullthickness and reduced-thickness specimens is captured with good accuracy by the calibrated 12-grain and 8-grain models, respectively. The greatest improvement appears in the strain ratios predictions for the central-layer material (Fig. 6c) compared to the results in Fig. 5b. Regarding the predicted textures shown in Fig. 7, the ‘red component’ for the central-layer material (Fig.7b) is more concentrated, which is in better accordance with the EBSD measurements. The axial stress-strain responses in the four compression-tension tests are shown in Fig. 8, with the markers representing the experimental data and lines depicting the model predictions. The agreement of simulation and experimental results improves greatly with ‘Calibration II’, in which kinematic hardening is activated. As the compression-tension experiments are performed with only full-thickness specimens, all the predictions are from the 12-grain models.

(a) (b) (c) FIGURE 6. Results of Calibration II: (a) yield stress at 3% axial strain. Dots are experimental data, and solid curves are model prediction. Red is for full-thickness and blue represents central-layer sheet; (b) width versus thickness strain curves for fullthickness sheet; (c) width versus thickness strain curves for central-layer sheet.

357

(a) (b) FIGURE 7. Comparison of the measured and calibrated texture of Calibration II in the form of {111} pole figures (XD = extrusion direction, TD transverse direction): (a) full-thickness sheet modeled with a 12-grain model; (c) central-layer sheet represented by an 8-grain model calibrated from Calibration II.

(a) (b) FIGURE 8. Comparison of the measured and predicted axial stress-strain responses in compression-tension tests: (a) 12-grain model given by ‘Calibration I’; (c) 12-grain model given by ‘Calibration II’.

SUMMARY A reduced texture methodology is applied to obtain a computationally-efficient polycrystal plasticity model of a 2mm-thick 6260-T6 aluminum extrusion. Both the texture parameters and hardening parameters are identified through inverse analysis using a comprehensive database that includes the results from uniaxial tension and compression-tension experiments. Specimens are extracted from different material orientations to study anisotropy, and the through-thickness heterogeneity is investigated by testing both full-thickness and reduced-thickness specimens. It is demonstrated that the RTM based self-consistent polycrystal plasticity model can describe both the anisotropy in the flow stresses and the Lankford ratios.

ACKNOWLEDGMENTS The authors are grateful to Dr. Héripré of Ecole Polytechnique ParisTech, Solid Mechanics Laboratory, for the EBSD measurements on the aluminum extrusion. Volkswagen (Germany), particularly Dr. L. Greve, is also gratefully acknowledged for providing the extruded aluminum profiles. Thanks are due to MINES ParisTech, Centre des Matériaux for the educational license of Zset/ZeBuLoN.

REFERENCES 1. 2. 3. 4. 5. 6.

F. Barlat, J. C. Brem, J. W. Yoon, K. Chung, R. E. Dick, D. J. Lege, F. Pourboghrat, S. H. Choi and E. Chu, International Journal of Plasticity 19 (9), 1297-1319 (2003). A. P. Karafillis and M. C. Boyce, Journal of the Mechanics and Physics of Solids 41 (12), 1859-1886 (1993). A. B. Lopes, F. Barlat, J. J. Gracio, J. F. Ferreira Duarte and E. F. Rauch, International Journal of Plasticity 19 (1), 1-22 (2003). F. Barlat, J. M. Ferreira Duarte, J. J. Gracio, A. B. Lopes and E. F. Rauch, International Journal of Plasticity 19 (8), 1215-1244 (2003). A. S. Khan, A. Pandey and T. Stoughton, International Journal of Plasticity 26 (10), 1432-1441 (2010). D. Raabe and F. Roters, International Journal of Plasticity 20 (3), 339-361 (2004).

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7. 8. 9. 10. 11. 12.

Z. Zhao, F. Roters, W. Mao and D. Raabe, Advanced Engineering Materials 3 (12), 984-990 (2001). G. Rousselier, F. Barlat and J. W. Yoon, International Journal of Plasticity 25 (12), 2383-2409 (2009). G. Rousselier, F. Barlat and J. W. Yoon, International Journal of Plasticity 26 (7), 1029-1049 (2010). M. Dunand, A. Maertens, M. Luo and D. Mohr, Submitted for publication (2011). G. Cailletaud, International Journal of Plasticity 8 (1), 55-73 (1992). K. Sai, G. Cailletaud and S. Forest, Mechanics of Materials 38 (3), 203-217 (2006).

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Effect of Off-J-fiber Texture Components on R-value Anisotropy and Cup-Drawing Behavior in High-Strength Steel Sheets Eun-Young Kima, Shi-Hoon Choia and Jeong Whan Yoonb a

Department of Materials Science and Metallurgical Engineering, Sunchon National University, 540-742, Korea Faculty of Engineering and Industrial Sciences, Swinburne University of Technology, P.O. Box 218, Hawthorn, VIC 3122, Australia.

b

Abstract. A crystal plasticity finite element method (CPFEM) was used to capture the effect of off-J-fiber texture components on R-value anisotropy and cup-drawing behavior in high-strength steel (HSS) sheets. Macrotexture analysis using XRD was conducted to capture the texture components typical in HSS sheets. The orientation distribution function around a single ideal component with the Gaussian standard functions was calculated to create the J-fiber texture and offJ-fiber texture components. For the polycrystal model, the material behavior was described using crystal plasticity theory ,in which each integration point in the element was considered to be a discrete orientation of polycrystalline HSS sheets. The R-value anisotropy and cup-drawing behavior of HSS sheets containing the J-fiber texture and off-J-fiber texture components were simulated using CPFEM.

Keywords: Crystal plasticity FEM; Cup-drawing; High-strength steel; Macrotexture; R-value PACS: 62

INTRODUCTION High strength steel (HSS) sheets for application in the automotive industry are required to have good deep drawability and stretchability during press forming operations. The anisotropic properties, such as the R-value (plastic strain ratio) and yield stress, affect earing behavior in deep-drawn parts and shape fixability in stretched parts. The dominant metallurgical parameter that affects the anisotropic properties of HSS sheets is known as the crystallographic texture. Finite element modeling (FEM) technology is one of the most powerful tools for the design of new parts in the automotive industry. However, to analyze and design specific materials, FEM users need appropriate data and constitutive models. The yield surface and the stress-strain curve are important material property data for the FEM of forming operations. The constitutive equations proposed to describe the plastic yielding of anisotropic materials are divided into phenomenological and crystallographic approaches [1-7]. In the phenomenological approach, the plastic behavior of metals is assumed to be well described by an analytical yield function. Non-quadratic yield functions that describe plastic anisotropy were proposed by Barlat et al. [1,2]. In the crystallographic approach [4-6], the material is assumed to be a polycrystal consisting of many single crystals (grains). Plastic flow is assumed to occur only by crystallographic slip on given slip systems in each crystal. This approach can be used to take crystallographic texture evolution into account during plastic deformation. Inal et al. [5] simulated the earing of aluminum alloys using CPFEM with a model involving only the flange. Zhao et al. [6] used CPFEM to investigate the influence of various texture components and their combination on the earing behavior of cup- drawn aluminum samples. The study found that the ear height and profile can be minimized by an optimized combination of certain texture components including their scatter width. In the present work, the Gaussian standard function was first used to generate texture components typical in annealed HSS sheets. The R-value anisotropy and cup-drawing behavior of HSS sheets containing the three typical texture components were simulated using CPFEM.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 360-364 (2011); doi: 10.1063/1.3623632 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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FIGURE 1. ODF in 3-D Euler space and M2=45qsection of annealed HSS sheets.

REPRESENTATION OF TEXTURE COMPONENTS Hot rolling, cold rolling and annealing conditions all contribute to the development of crystallographic texture in HSS sheets. The main texture component in hot-rolled HSS sheets can be characterized by {001}<110>, which was transformed from the cube texture component in the austenite region. Cold- rolling deformation enhanced the intensity of D-fiber (<110>//RD) and J-fiber (<111>//ND) texture components simultaneously. After annealing, the D-fiber texture components decreased in intensity and the J-fiber texture components increased in intensity of the orientation distribution function (ODF). The relative intensity of these components strongly depends on the processing conditions and material composition. Macrotexture was quantitatively examined using X-ray diffraction (XRD). The (110), (200) and (211) incomplete pole figures were measured to a maximum tilt angle of 70q using the Schulz back-reflection method with Mo-KD radiation. After background and defocusing corrections, ODFs were calculated from the incomplete pole figures using the WIMV method. Figs. 1(a) and (b) show the ODF distribution in 3-D Euler space and a M2=45q section of the annealed HSS sheets, as observed by X-ray diffraction. Texture components typical in HSS sheets can be represented on the M2 = 45q section of ODF. From these results, the initial texture of annealed HSS sheets was characterized as J-fiber texture and off-J-fiber texture components (~{554}<225>). ODF around a single ideal component can be mathematically described using the “Gaussian standard function,” [8]

~) f (S, Z

N(S)e

, N(S) 1 /(I 0 (S)  I1 (S)),

S

1 x cos t e cos kt dt S 0 ln 2 /[2 sin 2 (b / 4)] for b d 2S. Ik (x )

S

~ S cos Z

³

(1)

~ Z ~ (g o , g ) ~ ) expresses the relative density of an orientation g rotated through an orientation distance Z where f (S, Z k o from the single ideal orientation g k . b (=1.665Z ; Z=Bunge's notation) and Ik(x) are the half-width (FWHM, full width at half-maximum) of the bell-shaped curve and the modified Bessel function, respectively. In anisotropic materials, typical values of b are in the range of 16.65q~33.3q (Z: 10q~20q). An intermediate value of the range, b=25q (Z=15q), was selected in the present study to describe the distribution of the typical texture components of HSS sheets. Therefore, the texture consisted of a Gaussian distribution of orientations around J-fiber and off-J-fiber texture components. Fig. 2 shows the ODF in 3-D Euler space and a M2=45q section of texture components typical in HSS sheets. Both 50 and 80% volume fractions of the off-J-fiber texture component were considered. The ODFs were used to generate discrete orientations for polycrystal modeling, and the effects of J-fiber and off-J-fiber texture components on R-value anisotropy and cup-drawing behavior were considered.

361

FIGURE 2. ODF in 3-D Euer space and M2=45qsection of texture components typical in HSS sheets.

TABLE 1. Microscopic hardening coefficients determined using the fitting procedure.

W Do (MPa)

W sat (MPa)

ho (MPa)

a

91

310

2800

5

PLANAR ANISOTROPY AND CUP-DRAWING R-value anisotropy and cup-drawing behavior were simulated using the finite element code, ABAQUS with the material model programmed based on a continuum crystal plasticity theory. A rate-dependent constitutive relationship was implemented into user material subroutine UMAT in ABAQUS [9]. The model was fundamentally based on a multiplicative decomposition of the deformation gradient, F, into a plastic part characterized by shearing rates on an active slip system, as well as a portion that accounts for the rotation and elastic distortion of the crystal lattice. This formulation leads to additive decomposition of the velocity gradient into elastic and plastic parts,

L

Le  Lp

(2)

with the plastic part determined by slip rates, J D , on slip planes with normals, m D , and slip directions, s D N

Lp

¦ J s

D D

… mD

(3)

D 1

The summation represents all of the slip systems, N. The plastic part of the velocity gradient is decomposed further into symmetric and antisymmetric parts ( Lp D p   p ) to yield the formula Dp

p

1 2

1 2

N

¦ J

D

(s D … mD  mD … s D )

D 1

N

¦ J

D

N

¦ J P D

D

D 1

(s D … m D  m D … s D )

D 1

N

¦ J W D

D

D 1

where D p is the plastic part of the rate of deformation tensor and  p is the plastic spin. As described in [4], the Jaumann rate of Kirchhoff stress can be expressed as

362

(4)

N

ˆ K : D  ¦ J D R D

(5)

D 1

where K is a fourth-order tensor based on the anisotropic elastic modulus, C. D is the rate of deformation tensor (symmetric part of the velocity gradient), and R D is a tensor that depends on the current slip plane normal and direction, the applied stress and the elastic modulus. For materials having cubic crystal symmetry, 3 independent elastic constants of pure Fe are used in the present work: C11=231.4 MPa, C12=134.7 MPa, C44 =116.4 MPa. For rate-dependent materials, shear rates are given explicitly in terms of the resolved shear stress on the active slip system and the resistance of the active slip system to shear. For these simulations, this dependence is given by J D

J oD

WD WDo

1/ m

sign (WD )

(6)

Self- and latent-hardening are readily accounted for by a suitable evolution of the reference W Do values in the constitutive law in Eq. (6). The present work employs a microscopic hardening law [4] for this purpose, as follows: W Do

N

¦H E

DE

J E

D, E 1......( Ns  N t )

H

DE

§ WD · q h o ¨¨1  o ¸¸ © W sat ¹

a

DE

(7)

where H DE is a hardening matrix introduced to account for interaction between the slip systems. q DE accounts for the hardening rate of the slip system, D, and is due to the slip activity on system, E. Here, it is assumed that the selfhardening term (diagonal term of q DE ) equals the latent-hardening term (off-diagonal term of q DE ), i.e. ( q DE =1). The fitting simulation was carried out by varying the CRSS values and microscopic hardening parameters (ho, Wsat and a) until agreement was achieved between the predicted and measured uniaxial loading curves. In the present study, two slip systems ({110}<111>, {112}<111>) were considered. CRSS values and microscopic hardening parameters were determined by fitting the experimental stress-strain curve. The set of parameters listed in Table 1 was used in the theoretical simulation.

FIGURE 3. (a) Finite element mesh used in CPFEM and (b) calculated R-value anisotropy of texture components typical in HSS sheets.

In order to account for the planar anisotropy of the material, a CPFEM simulation was performed using 50 elements (5×5×2), as shown in Fig. 3(a), with 3-D eight-node brick elements, type C3D8R. Each integration point had 20 orientations mapped into it. The R-value anisotropy of the typical texture components, calculated with the CPFEM, is shown in Fig. 3(b). The J-fiber texture component (b=25q, VF=80%, random=20%) exhibited relatively weak directionality of the R-value. However, the off-J-fiber texture component containing the lower volume fraction (b=25q, VF=80%, random=20%) exhibited strong directionality of the R-value. The R-value directionality of the off-J-fiber texture component containing the lower volume fraction (b=25q, VF=50%, random=50%) decreased. In the cup-drawing simulation, blanks 0.7mm in thickness and 80mm in diameter were drawn into a 40mm-diameter cup (drawing ratio 2:1).

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FIGURE 4. (a) Geometry for cup-drawing simulation and (b) simulated drawn cup

for texture components typical in HSS sheets. Fig. 4(a) shows the geometry of the tools used in the simulation. Due to the orthorhombic sample symmetry, only a quarter of the blank was used to simulate the cup-drawing process. The FE simulation was performed using 1098 elements with 3-D 8-node brick elements, type C3D8R. The friction coefficient between the blank and the tools was prescribed as P=0.02. Fig. 4(b) shows the deformed shape obtained at the final stage of the cup-drawing simulation for the 3 texture components of HSS sheets. It is obvious that the initial texture components significantly affected the formation of ears in the drawn cup. Fig. 5 shows the simulated earing profiles of the 3 typical texture components. The J-fiber texture component (b=25q, VF=80%) did not produce ears in the drawn cup. The first type of off-J-fiber texture component (b=25q, VF=80%) produced 6-fold ear, oriented at about 0q and 90q in the rolling direction. The second type of off-J-fiber texture component (b=25q, VF=50%) produced almost the same earing profile as the first type of off-J-fiber texture component, except for the absolute value of the earing height. A comparison between R-value anisotropy and earing profiles showed that the R-value anisotropy was strongly dependent on the earing profiles.

FIGURE 5. Effect of texture components on the earing profiles.

REFERENCES 1. F. Barlat, J. C. Brem, J. W. Yoon, K. Chung, R. E. Dick, D. J. Lege, F. Pourboghrat, S. -H. Choi, E. Chu, Int. J. Plast. 19 1297-1319 (2003). 2. F. Barlat, H. Aretz, J. W. Yoon, M. E. Karabin, J. C. Brem, R. E. Dick, Int. J. Plast. 21 1009-1039 (2005). 3. J. W. Yoon, F. Barlat, R. E. Dick, M. E. Karabin, Int. J. Plast. 22 174-193 (2006). 4. S. -H. Choi, S. H. Han and K. G. Chin, Acta Mater. 51 1947-1958 (2009). 5. K. Inal, P. D. Wu and K. W. Neal, Int. J. Plast. 16 635-648 (2000). 6. D. Raabe and F. Roters, Int. J. Plast. 20 339-361 (2004). 7. Z. Zhao, W. Mao, F. Roters and D. Raabe, Acta Mater. 52 1003-1012 (2004). 8. S. -H.Choi, J. H. Cho, F. Barlat, K. Chung, J. W. Kwon, and K. H. Oh, Metall Trans 30A 377-386 (1999). 9. ABAQUS User’s Manual, version 6.1. Hibbit, Karlsson & Sorenson, Providence, R.I., 2000.

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QuickForm Simulation System of Sheet Metal Forming with Trial-and-error Contact Algorithm Bao Yidonga Liu Leib Lu Yub and Chen Wenlianga a

College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, 29 Yudao Street, Nanjing, 210016, China b Shanghai Aircraft Manufacturing CO. Ltd., 3115 Changzhong Road, Shanghai, 200436, China

Abstract. The QuickForm numerical simulation system has been self-developed specially for sheet metal forming with complex geometries. In order to resolve the convergence problem of static implicit method due to the change of contact state, the uncoupling solution method of the non-linear differential equations is used in QucikForm system. The bending effect and drawing effect during forming procedure are separated into two independent processes in QuickForm system. During bending effect solution, the contact state between the blank nodes and rigid tools should be determined by accurate contact algorithm. A trial-and-error contact algorithm is used in the bending effect calculation process of QuickForm system. The contact procedure is carried out before the solving of bending effect FEM equations, and the contact conditions are treated as the boundary conditions of FEM equations. Numerical simulation results of complex part demonstrate the higher efficiency and accuracy of this contact algorithm and the proposed special purpose program. Keywords: Uncoupling algorithm; Sheet metal forming; Contact algorithm PACS: 81.20.Hy

Sheet metal forming process is very common in industry. With the development of non-linear theory, finite element method and computer hardware, the stamping simulation technology of sheet metal forming has been gradually utilized in industry. It becomes a useful tool to reduce the development period and reduce production costs. The sheet metal forming is a very complicated process and is influenced by many factors, such as die shape, material property, blank shape and size, boundary conditions, friction and lubrication between die and blank, etc. Although metal forming simulation technology has been developed many years, many research works still focus on this area. Nowadays, although dynamic explicit software takes major position in the current stamping simulation of autobody panel, the static implicit incremental algorithm is the most suitable simulation method for sheet metal forming theoretically since it is a quasi-static problem. So many scholars still carried on this research works to solve the problems of hard convergence and low calculation efficiency of static implicit algorithm. A novel program named QuickForm based on large-step static implicit algorithm is proposed specially for sheet metal forming simulation. This algorithm considers the bending effect and drawing effect separately so that the fast iterative algorithm can be used and convergence can be improved. In addition, due to the use of static implicit algorithm, there is no limit for the adaptive refinement level of mesh and the min. mesh could be 0.5mm. The large mesh and small mesh can coexist. Due to the above two factors, the calculation efficiency of quickform program is enhanced significantly. However, the large-step static implicit algorithm adopted much larger incremental step compared to the dynamic explicit algorithm, so the contact algorithm should be specially design for the large incremental step feature of largestep static implicit algorithm to ensure calculation accuracy. In this work, we first review the basic formulations of large step static implicit algorithm in quickform system. Then a special trial-and-error contact algorithm is proposed for QuickForm program. Finally, DynaForm based on Ls-dyna and quickform systems as CAE platform of explicit and implicit solution are adopted to simulate forming process of a typical complex part. The change of thickness of blanks is compared and strain state is compared with the FLD of material: this is done for the areas of the wrinkling and failure.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 367-373 (2011); doi: 10.1063/1.3623633 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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LARGE-STEP STATIC IMPLICIT ALGORITHM Based on uncoupling solution, large-step static implicit algorithm is a specially designed new simulation algorithm which specializes in sheet metal forming. It uses the uncoupling thought to separate the two processes of bending effect and membrane drawing effect during sheet forming and solve them respectively. As mentioned above, dynamic explicit algorithm cannot completely satisfy the requirements during die design and process parameter phase due to its limitation. Therefore, Quickform system adopts the static implicit algorithm, of which every time incremental step needs equilibrium iteration and every time of equilibrium iteration needs to solve a system of linear equations. The dimension of linear equations depends on the number of node degree of freedom. There are two solutions for system of linear equations: direct method and iteration method. For large problems, it requires large storage to solve system of linear equations with direct method, which will result in a rapid decline of solving efficiency. And when using iteration method, the stiffness factor matrix of linear equations is required to be a well-conditioned matrix. Since shell element is usually used by sheet metal forming simulation, the membrane stiffness for shell element stiffness matrix should be much greater than bending stiffness. Therefore, the stiffness factor matrix for linear equations in each iteration step is usually an ill-conditioned matrix. This is just the important reason why the traditional static implicit algorithm is not able to converge or even diverge. In order to solve this problem, the new large-step static implicit algorithm is brought up. This algorithm consists of two modules: Intermediate configuration prediction module and stress equilibrium iteration module, as shown in Fig. 1. Intermediate configuration prediction module: Before incremental step iteration, an approximate method named pre-stressed membrane element is used to achieve the intermediate configuration of sheet in this incremental step. The sheet shape of intermediate configuration depends on the current tooling position. In the contact zones this form is defined by the tools, and in the other zones it is determined by the pre-stressed membrane element.

(a) Intermediate configuration prediction

(b) stress equilibrium iteration

FIGURE 1. Two modules of uncoupling solution algorithm.

Stress equilibrium iteration module: After the intermediate configuration of sheet is determined, the nonlinear material flow within this intermediate configuration is carried out, which means that the sheet nodes can only slide on this intermediate configuration. Based on uncoupling solution, this new algorithm has two significant advantages. Firstly, there are only two displacement degrees of freedom per node during nonlinear material flow solution. In the most general shell element, each node has six degrees of freedom. Secondly, since bending stiffness and membrane drawing stiffness are solved by uncoupling solution, the stiffness matrix of linear equations in each iteration step is well-conditioned. The iteration solution use few storage space and with high convergence efficiency. In each equilibrium iteration step, the node converted coordinate system can be determined by the current configuration, as shown in the Fig.2. The linear equation is solved in the node converted coordinate system. As shown in the figure, these converted coordinate systems differ from node to node. So the element stiffness matrix only needs to be transformed to the node converted coordinate system respectively, there is no transformation into the global coordinate system. ª K Mnn  u0  KGnn  u0 K Mnt  u0  KGnt  u0 º ­° 'u ½° « »® n ¾ «¬ K Mtn  u0  KGtn  u0 K Mtt  u0  KGtt  u0 ¼» ¯° 'ut ¿°

°­  Rn  u0 °½ ® ¾ °¯ Ft  Rt  u0 ¿°

(1)

where, u0 is current displacement, u is displacement increment, R is residual force, Ft is tangential friction force as external loads, KM is the membrane component of tangential stiffness, KG is its geometrical component or

368

the component of the stiffness arising from the initial stresses. The aim of equilibrium iteration is to reduce the residual forces towards 0, so the element stiffness matrix can be transformed freely into the components along nodal tangential direction t and normal direction n. Two uncoupling equations can be obtained: °­ K Mtt  u0 'ut Ft  Rt  u0 ® °¯ K Gnn  u0 'un  Rn  u0

(2)

So far only the geometrical effect of the bending deformation has been taken into account, the actual bending effect from the influence of the non-uniform stress through the shell thickness is neglected. Two bending effects occur in deep stretch-drawing process: 1) The intermediate configuration of sheet in which the material flows is affected to a minor extent by the bending. Unlike the situation in a pure bending process, in a three-dimensional sheet metal forming this effect is negligible. Since a small alteration of intermediate configuration has no influence on the material flow itself, this bending effect is minor influence on the material flow. n t



FIGURE 2. Node converted coordinate system.

2) The drawing resistance, when drawing the sheet over radius region, caused by the bending when entering the radius region and the reverse bending when leaving this region. This is an important effect particularly in pure deepdrawing process. It must be taken into account fully in the uncoupling algorithm. If the bending drawing resistance is ignored, the equation by the principle of virtual displacement is obtained:

G umT F

³ GH

T

A

V tdA G umT ³ BT V tdA

(3)

A

Where, um represents the virtual tangential displacements of the element nodes at the mid- thickness of the shell, F represents the node forces, A is the element surface, t the shell thickness,  the strain tensor,  the stress tensor and B the displacement-strain relationship of the element. If the bending drawing resistance is to be taken into account, then the shell bending energy must be included. Assuming the Kirchoff hypothesis, the virtual displacement along node tangential direction at a distance z from the mid-thickness is:

Gu  z

 I  zN G um

(4)

So the principle of virtual displacement takes the following form:

G umT F

G umT ³ ³  I  zN B  z V  z tdzdA T

(5)

A z

Where I is the unit tensor and  is the curvature tensor at the nodes, which is re-determined from the current geometry at each iteration. The history-dependent material data are now determined through the thickness at the integration points and stored, so that the stress values through the thickness are available. The degrees of freedom are now still the two tangential displacements per node. As a result, the computing costs for the solution of the linear equations, which comprise by far the greatest part of the overall computing costs, are not increased. However, the element and material routines for the integration through the thickness must be called up several times, so that there is a slight increase in the total computing costs.

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CONTACT ALGORITHMS A special trial-and-error contact algorithm is adopted for QuickForm program. Contact procedure is carried out two tasks: searching of the nodes being in contact and computation of the contact forces. At the beginning of each time increment step, the punch makes a relative large travel, then contact searching will be done, and a series of corrections are carried out to maintain the balance of all the forces as well as the contact requirements. The same sequence of operations is made in each iteration step. This procedure is repeated until convergence is obtained. Fig. 3 illustrates the contact iteration procedure.

Travel step Geometry compatibility Punch

Punch

Iterations

Contact force compatibility Punch

Punch

FIGURE 3. Contact Iteration process.

The proposed contact algorithm and the solving of FEM equations are relatively independent, and the contact conditions just act as the boundary conditions of the FEM equations and are not introduced into the element stiffness matrix, so the contact algorithm will not add much more calculation time of QuickForm program.

Contact Searching The tool surfaces are discretized into many triangle or quadrilateral elements for 3D parts. The interaction between the sheet and rigid tool can be treated as a problem of unilateral frictional contact. The tool surfaces are treated as master surfaces and a sheet surface is treated as a slave surface. A penetration distance function Dp can be defined for each slave node since the contact conditions are considered on nodal level Dp(X(s)) = (X(s) – X(m)) n (X(m))

(6)

where X(m) is the closet projection point of the slave node X(s) on the mater surface, as shown in Fig.4, and n is the unit vector of outward normal of the mater surface defined at the point X(m). Impenetrability condition (Dp  0) is checked for all slave nodes. Contact searching is carried out in two steps: the closest master node of the slave node is searched firstly during global searching step, and then the projection point of this slave node X(s) on the master surface is obtained in the adjacent elements of the closest master node during local searching step. n X(m)

Slave node

X(s) Master surface FIGURE 4. Definition of the penetration distance function.

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Trial-and-error Processing During the forming process, the status of slave nodes is changing from time to time. However, slave nodes should satisfied two criteria: slave nodes are not permitted to penetrate into the master surfaces, and the tensile force cannot occur between slave nodes and master surfaces. Only intermediate configuration prediction module of QuickForm program needs contact algorithm to determine sheet shape. This module used an approximate method named pre-stressed membrane element to achieve the sheet intermediate configuration in each incremental step. The sheet intermediate shape depends on the current tooling position. The sheet form is defined by contact tool surface in the contact zones, and it is determined by the prestressed membrane element in the other zones. The primary purpose of this module of QuickForm program is obtain a deformed sheet shape by approximate method, so the contact relation between sheet nodes and tool nodes become a simplified contact state compared to traditional static implicit and dynamic explicit algorithms. According to the contact state between slave nodes and mater surfaces, slave nodes are classified into two catalogues: free node and sticking node. If the slave nodes is checked to be outside the tool surface, these slave nodes without penetration are defined the free nodes. On the other hand, when slave node is checked to be inside the punch (or die) surface, it will be projected onto the master tool surface, and this slave node is assumed the sticking node. These sticking nodes will be constrained as fixed nodes of FEM equations. This penetration slave node will be moved back to the master surface along slave nodal normal. Because the position of penetration slave node is adjusted is not the reasonable position, the corresponding contact force maybe inadequate, thus some necessary adjustments must be made. If the normal contact force (the component of the external force R along the normal direction to tool surface) is tensile force, the node should be released and changed into a free node, as shown in Fig.3.

NUMERICAL RESULTS AND DISCUSSION In this chapter, a typical complex fender part from the automotive industry is presented to demonstrate the effective and accuracy of above-mentioned trial-error contact algorithm of QuickForm system. The simulations are carried out by QuickForm program and Dynaform5.6 based on LS-DYNA solver, respectively. Computer conditions used in simulations: CPU is Intel Pentium 2.4GHz and memory is 2.0 Gb. The mechanical properties of material and the process variables for simulation are as follows: initial sheet thickness, t = 1.0 mm; Young’s modulus, E = 207,000 MPa; Poisson’s ratio,  = 0.28; Average Lankford value, r = 1.6; Stress-strain curve,  = 648.0(0.0013+p)0.22 (MPa); Yield stress, s = 178.0 MPa; Friction coefficient, f = 0.15.

Binder Sheet Punch FIGURE 5. Finite element models of fender.

The tools setup and finite element models of fender part are shown in Fig. 3. The initial blank mesh includes 4,142 nodes and 8,033 triangle elements used in Quickform program, and 4,140 nodes and 4,025 quadrangle elements used in LS-DYNA solver. Because of the complexity of fender part and the binder is not in a plane, the forming process type is single action and two movements are considered. Firstly, the die moves downwards until binder clamps the sheet on the die; after that, the die continues to move downwards to punch and the bent sheet with pressure applied on the binder to clamp sheet. Finally, the final deformed part is obtained. The comparison of final deformed mesh shapes between QuickForm and LS-DYNA are shown in Fig. 6. The final deformed shape and the adaptive refinement situations from two different solvers are looks approximately similar. The final blank node number of QuickForm is increased 35,356 from 4,142, on the other hand, the final blank node number of LS-DYNA is 38,933. QuickForm’s node number is little less than LS-DYNA’s node number. These results show that this proposed trial-error contact algorithm of QuickForm program have enough ability to

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deal with complicated contact circumstance of real complex parts, furthermore, the contact algorithm also meet the requirements of contact solution for large-step static implicit algorithm of QuickForm system.

LS-DYNA result

QuickForm result

FIGURE 6. Comparison of final blank mesh between LS-DYNA and QuickForm. 0.706 mm

0.643 mm

1.13 mm 1.19 mm

LS-DYNA result

QuickForm result

FIGURE 7. Comparison of final thickness distribution between LS-DYNA and QuickForm.

The thickness distributions of final blank in QuickForm and LS-DYNA are shown in Fig. 7. The maximum and minimum value of t is 1.13 mm and 0.706 mm in the final deformed blank of QuickForm, and those values are 1.19 mm and 0.643 mm for LS-DYNA. It could be seen from Fig. 7 that the position of the maximum value of t is the same approximately, and the position of the minimum value of t has small difference. It shows that the trend of the wrinkling and failure predicted by QuickForm and LS-DYNA is coincident. QuickForm and LS-DYNA are the static implicit and dynamically explicit solutions, respectively, which is the main reason for the difference in the results. The precision of the implicit solution is superior to that of the explicit solution and economizes in the time. However, the changing contact conditions lead to some problems, including that of convergence. In addition, the disadvantage of the dynamically explicit solution is that a large amount of calculating time is needed because of the time interval given when the quasi-static problems are resolved in real time. As far as fender part is concerned, the shortest time for LS-DYNA is about 1.5 h, but the longest time for QuickForm is only about 0.5 h. The latter is thus superior to the former with respect to the calculating time.

CONCLUSIONS The numerical simulation program named QuickForm has been self-developed specially for sheet metal forming. A trial-error contact algorithm as one of most important process is adopted into the QuickForm program. This contact algorithm is relatively independent of the FEM equations calculation, and the contact conditions just act as the boundary constraint in the FEM equations, so the main advantages of this contact algorithm is that contact

372

conditions could not increase the total number of degree of freedom. The numerical results have demonstrated that the present contact algorithm is effective and acceptable. Comparison results of thickness distribution and strain state of FLD have proved that the self-developed QuickForm program with trial-error contact algorithm is an efficient and accuracy simulation tool for sheet metal forming and could be a suitable fast simulation tool considering stamping parameters during the initial die design and optimization stage.

ACKNOWLEDGMENTS The Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars of State Education Ministry and the National Nature Foundation (project number: 50805075) and these supports are gratefully acknowledged.

REFERENCES 1. 2. 3. 4. 5. 6.

W. L. Xu, Y. Y. Yang, Z. R. Wang and Z. Q. Lin, Journal of materials processing technology 100, 219-223 (2000). Y. M. Kong, Z. E. Ma, L. Y. Liu and F. Mercier, Journal of materials processing technology 120, 1-5 (2002). M. Harnau, A. Konyukhov and K. Schweizerhof, Computers & Structures 83, 1804-1823 (2005). W. Kubli and J. Reissner, Journal of materials processing technology 50, 292-305 (1995). Y. D. Bao, G. K. Du, et al., Journal of nanjing university of aeronautics & astronautics 41, 606-610 (2009). Y. D. Bao, J. W. Chen, et al., NUMIFORM2010, AIP Conference Proceedings 1252, American Institute of Physics, Melville, NY, 2010, pp. 433-438.

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1HZ6ROLG6KHOO)LQLWH(OHPHQW%DVHGRQ($6DQG$16 &RQFHSWVIRU6KHHW0HWDO)RUPLQJ A. Ben Bettaieb a, L.Duchêne a,b, L.Zhang a, A.M.Habraken a,b a

ArGEnCo Department, MS²F Division, University of Liège, Chemin des Chevreuils1, 4000 Liège, Belgium. b F.R.S.-FNRS Fonds de la Recherche Scientifique, Belgium 

 $EVWUDFW We present in this paper a new eight node solid-shell finite element called SSH3D (Three Dimensional SolidShell). This element is currently implemented in the frame of the in-house research code called LAGAMINE. Here the Enhanced Assumed Strain (EAS) technique based on the Hu-Washizu variational principle, described in [1] and [2], is used to cure the volumetric locking occurring when the material shows nearly incompressible behavior and Poisson’s thickness locking caused by the high aspect ratio of the finite element. In the proposed element, the EAS technique can be combined with the Assumed Natural Strain (ANS) [3-5] concept to treat shear locking caused by the transverse shear strain and curvature thickness locking caused by the transverse normal strain. Different schemes for the ANS concept are implemented while the number of integration points and the number of EAS modes are element parameters (the element uses four integration points in the plane of the element and at least two integration points through the thickness direction in a single element layer). These features must be adjusted by the user according to the studied process (geometry and loading) so as to avoid locking and limit the calculation time. This element was successfully tested and the numerical remedies were verified using several kinds of patch tests. A double sided contact problem is modeled in order to investigate the performance and accuracy of the developed element and to validate the suggested approach. .H\ZRUGVSolid-Shell element, Enhanced Assumed Strain, Assumed Natural Strain, locking free, sheet metal forming. 3$&602.70.DH, 81.20.HY.

,1752'8&7,21 Solid-shell elementsform a class of finite element models intermediate between thin shell and conventional solid elements. They can be referred to as the finite element models that have shell kinematics assumption and posses no rotational degrees of freedom (DOFs). They have the same node and DOFs configuration then standard eight-node brick solid elements but account for shell-like behavior in the thickness direction. They are useful for modeling shelllike portions of a 3D structure without the need to connect solid element nodes to shell nodes. Despite that, solid-shell elements have also serious drawbacks. Among these drawbacks, their computational cost is higher than the conventional shell elements, especially when they are used to model layered structures. Another problem appears in high element aspect ratios, they can suffer from the ill conditioning of the element stiffness matrix, this phenomenon is called locking phenomenon. Displacement based solid elements are known to have poor performances in bending-dominated situation, such as in thin-walled structures. In the element described in this paper, the Enhanced Assumed Strain (EAS) method and the Natural Assumed Strain technique are combined to produce a new eight node solid-shell element, where the numerical integration scheme consists of four integration points in the plane of the element and an arbitrary userdefined number of integration points through the thickness direction. Different schemes for the Assumed Natural Strain (ANS) technique were implemented while the number of EAS modes can be adjusted. The features must be tuned by the user according to the studied process so as to avoid locking and limit the calculation time. ($6$1'$16(/(0(176 (QKDQFHG$VVXPHG6WUDLQ(OHPHQW($6  The EAS formulation was first introduced by [1]. With this technique various types of locking can be avoided, for instance volumetric locking or transverse shear locking. Starting point for the EAS method is the three field HuWashizu variational principles. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 374-381 (2011); doi: 10.1063/1.3623634 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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The crucial point in the EAS formulations is the enlargement of the displacement related strain field H (see(1)), adding a new field of internal variables, here called D DOFs. The EAS method has been widely adapted for 2D, 3D and solid shell formulations [6]. With the enhanced assumed strain method the total strain field is assumed to consist of the displacement based field and an assumed additive strain field. This enhanced strain field contains modes that are not present in the displacement interpolation. H H com  H EAS (1) com EAS In(1), H and H represent the compatible strain field and the enhancing (EAS) strain, respectively. H com ’ s u B(r , s, t ).U (2) Similarly to the displacement field u, which is interpolated (in the element domain) by the standard FEM straindisplacement B matrix (2), the enhanced strain field is interpolated by the G matrix (3) at the element level, the enhanced part of the strain field H EAS being added to the standard strain field (1). j0 H EAS G (r , s, t ).D .F0T .M (r , s, t ).D (3) j ( r , s, t ) In equations (2) and(3), r, s, t represent the intrinsic coordinates of the element, U is a vector containing the nodal displacements, j0 is the jacobian determinant at the center of the element, while j (r , s, t ) is the jacobian determinant at integration point. Finally, according to(2), the enhancing matrix M is expressed in the intrinsic reference frame (due to the transformation F0T ) as: ªr «0 « «0 M « «0 «0 « ¬«0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 rs rt 0 0 0 0 0 0 0 rst

0

s 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 rs st 0 0 0 0 0 0 t 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 rt st 0 0 0

0 0

rst 0 0 rst

0 0 r s 0 0 0 0 rt st 0 0 0 0 0 0 0 0 0 0 rs 0 0 0 0 0 0 r t 0 0 0 0 rs st 0 0 0 0 0 0 0 0 0 rt 0

0 0

0 0

0 0

0 0 0 0 0 0 s t 0 0 0 0 rs rt 0 0 0 0 0 0 0 0 st

0

0

0

0

0º 0 »» 0» » rst 0 0 » 0 rst 0 » » 0 0 rst ¼» 0

0

0 0

0 0

(4)

In(4), non-zero terms in the three upper-rows are applied for enhancing the additional modes of the normal strains, non zero terms in the three lower-rows are applied for enhancing the additional modes of the shear strains. As chosen in (4), the M-matrix expands the compatible strain field up to the complete tri-linear field (EAS30). It means that the EAS element has 30 potential additional modes which should be selected according to each problem. $VVXPHG1DWXUDO6WUDLQ(OHPHQW$16  The principal idea of the ANS method is to choose a certain interpolation for the transverse shear strains instead of deriving the strains directly from the classical interpolation of the displacement, as shown in FIGURE 1 and FIGURE 2. Nodaldisplacements U

Strainattheintegrationpoints(IP)

B

ª¬H com º¼ IP

ª¬’s u º¼ IP

B(rIP , sIP , tIP ).U

 ),*85( Strain interpolation without ANS method For the ANS method, the classical interpolation of the displacement field should be defined in order to construct the B-matrix from the shape functions of the element (see ),*85(and ),*85(). Additionally, the interpolation for the strain components is required to construct the BANS matrix from the sampling points and the corresponding shape functions f (seeFIGURE 2). Nodaldisplacements

U

B

Strainatthesamplingpoints(SP)

ª¬H com º¼ SP

B(rSP , sSP , tSP ).U

Linearinterpolation ofthestrain components

B ANS

Strainattheintegrationpoints

ª¬H

ANS

º¼ IP



),*85(Strain interpolation with ANS method

375



f ª¬H com º¼ , ª¬H com º¼ ... SP1 SP 2

The ANS method was originally prepared to relieve the shear locking problem that typically arises as the thickness of the shell is reduced. In the present solid-shell element we use the ANS method to treat the shear locking caused by the transverse shear strain components ( H13 and H 23 ) and the curvature thickness locking caused by the transverse normal strain ( H 33 ) in the present solid-shell element. This technique is implemented in the SSH3D solid-shell in three versions which can be selected by the user. In this first ANS version, the transverse shear strains E13 and E23 are assumed to be interpolated through the use of certain sampling points as follow: 1 .[(1  s ).E13com ( A)  (1  s ).E13com (C )] E13ANS (5) 2 1 com .[(1  r ).E23com ( D)  (1  r ).E23 ( B)] E23ANS (6) 2 Where E13com ( A) , E13com (C ) , E13com ( B) and E13com ( D) are the natural shear strains at points A,C,B and D situated on the mid-surface of the solid element (see FIGURE 3), respectively

 ),*85( Sampling points for ANS interpolation for transverse shear strain (A, B, C, D) [3, 7, 8] According to [5], in the second version, a bilinear interpolation of the transverse normal strains sampled at the four corners E,F,G,H of the element midsurface (FIGURE 4) is applied. For the compatible transverse shear strains E13com , E23com a linear interpolation (8) and (9) evaluated at the four mid-points A, B, C, D of the element edges ( at t=0) is imposed. 1 .[(1  r ).(1  s ).E33com ( E )  (1  r ).(1  s ).E33com ( F )  (1  r ).(1  s ).E33com (G )  (1  r ).(1  s ).E33com ( H )] E33ANS (7) 4 1 .[(1  s ).E13com ( A)  (1  s ).E13com (C )] E13ANS (8) 2 1 com .[(1  r ).E23com ( D)  (1  r ).E23 ( B)] E23ANS (9) 2

 ),*85( Sampling points for ANS interpolation for transverse shear strain (A, B, C, D) and for transverse normal strain (E, F, G, H) [5, 9] In the third ANS version, to avoid the transverse shear strain we work with four sampling points defining the root plane of the considered parasitic shear term [10]. E13com and E23com are assumed to be interpolated at the sampling points (A,B,C,D) and (E,F,G,H) see FIGURE 5. In order to alleviate the curvature thickness locking, we evaluate the compatible transverse normal strain E33com at the sampling points I, J, K and L, the corners of the element midplane, depicted in FIGURE 5, and interpolated them by the use of bilinear function (12). 1 .[(1  s ).(1  t ).E13com ( A)  (1  s ).(1  t ).E13com ( B)  (1  s ).(1  t ).E13com (C )  (1  s).(1  t ).E13com ( D)] (10) E13ANS 4 1 com com .[(1  r ).(1  t ).E23com ( E )  (1  r ).(1  t ).E23com ( F )  (1  r ).(1  t ).E23 (G )  (1  r ).(1  t ).E23 ( H )] E23ANS (11) 4

376

E33ANS

1 .[(1  r ).(1  s ).E33com ( I )  (1  r ).(1  s ).E33com ( J )  (1  r ).(1  s ).E33com ( K )  (1  r ).(1  s).E33com ( L)] 4

(12)

 ),*85( Sampling points for ANS interpolation for transverse shear strain (A… H) and for transverse normal strain (I, J, K, L) [10, 11] The interpolation is performed on the strain component of E ANS expressed isoparametric space, so the relationship between the isoparametric (r, s, t) strain to the physical (X, Y, Z) strain H is: j0 H ANS .F0T .E ANS (13) j ( r , s, t ) Where F0 is equal to the transformation matrix in(3). In the combined EAS-ANS element (1) becomes:

H

H ANS  H EAS

(14) The numerical accuracy of the above-mentioned procedure was assessed in the numerical results presented in the next section. 180(5,&$/5(68/762)7+(62/,'6+(//(/(0(17 In order to evaluate the performance of the present SSH3D Solid-Shell element, various tests are presented in this section. First, three tests out of the literature were performed. Then, a sheet metal forming process is modeled to assess the efficiency and accuracy of the formulation under plastic anisotropy, and double-sided contact. To make the formulation more efficient, we propose, for all following tests: x To use the total number of EAS modes (30  DOFs) to remedy the Poisson-thickness locking to improve the in-plane bending behavior. x To apply the ANS interpolation on the compatible transverse shear strain components to treat shear locking and on the compatible transverse normal strain for the curvature thickness locking [5] (third ANS version corresponding to FIGURE 5 and (10), (11), (12)). x In order to reduce the CPU time, the tangent stiffness matrix and the residual force vector are evaluated using only two integration points along the thickness direction in each element. This corresponds to classical full scheme with 2x2x2 integration points. The following examples are calculated using the in-house finite element analysis code (LAGAMINE), developed by Prof S.Cescotto (see [12]). 3LQFKHG+HPLVSKHUHZLWKDQƒ+ROH This geometrically non-linear example is often cited as benchmark for shell elements [13]. In this test, the structure consists of a thin hemispherical shell with two inward and two outward forces located at every 90° in the equatorial plane. For symmetry reason, only one quadrant of the structure has to be modeled (see FIGURE 6). The material and geometric properties are: E= 6.825x107, = 0.3, R=10, h= 0.04 Where E and  are the Young’s modulus and the Poisson’s ratio, R is the radius and h the thickness of the shell. The load factor is taken to increase from =0.0 until the final value =100.0. To assess the convergence behavior the results are compared to shell elements proposed in [14, 15]. The load deflection curves presented in FIGURE 7 are computed using a refined mesh with 8x8 and 16x16 elements. Our solution agrees very well with those provided in [14, 15].

377

 ),*85( Pinched hemispherical shell problem. Geometry, loading and meshing (16x16x1case)  3.5

6 5 absolute displacement

absolute displacement

3 2.5 2 1.5

Simoetal.(1990) 256elem(OX) S4E6P5256elem(OX)

1

S4E6P564elem(OX) 0.5

SSH3D256(OX) 0

10

20

30

40

50

60

70

80

3 Simoetal.(1990) 256elem(OY)

2

S4E6P5256elem(OY) S4E6P564elem(OY)

1

SSH3D64(OX)

0

4

SSH3D64(OY) SSH3D256(OY)

0 90

100

0

10

20

30

40

loadfactor

50

60

70

80

90

100

loadfactor

(a)

(b)

 ),*85( Pinched hemisphere with 18° hole. Displacement along OX direction (a) and OY direction (b). &DQWLOHYHU3ODWH2XWRI3ODQH%HQGLQJ This problem has been previously investigated in [5, 14, 15]. It permits to show the superior performance of the present element formulation for the out of plane bending behavior. An elastic cantilever plate with length L=10 and width w=1 is clamped on one end and subjected to two point loads F/2 at the free end. The thickness of the plate is taken as h=0.1(see FIGURE 8). We consider three different meshes (10, 12 and 16) along the length direction to model this problem. The material properties are defined as E=107 and we use four different values of  (0.0; 0.3; 0.49; 0.499) to test the performance of the present solid-shell element. The external load F is increased up to a value of 40., where  is a geometrical factor, function of the thickness and defined as =103xh3 (in the case,  =1). F/2

Clamped F/2

h w

L

),*85(Out of plane bending: geometry, mesh and boundary conditions A comparison of the tip deflection of the plate for different meshes and Poisson’s ratios obtained using the present element, the HCiS12 solid-shell element [14] and the theoretical solution [15] is shown in FIGURE 9 and FIGURE 10. To verify the coarse-mesh accuracy, The proposed element shows the same accurate results and insensitivity to mesh configuration as for the element reported in [14] (see FIGURE 9). In FIGURE 10, a set of numerical results with different Poisson’s ratio a 16x1x1 mesh is presented. The load level is the same as before. These results show that Poisson’s ratio has no effect on the bending behavior of the SSH3D element for this case as well as for the HCiS12 solid-shell element [14].

378

35

Loadlevel

30 25

theoriticalsolution

HCiS12,10elements

HCiS12,16elements

HCiS12,20elements

SSH3D,10elements

SSH3D,12elements

40 35 30 Loadlevel

40

SSH3D,16elements

20 15

25 20

HCiS12,Nu=0.0

HCiS12,Nu=0.3

HCiS12,Nu=0.49

HCiS12,Nu=0.499

SSH3D,Nu=0.0

SSH3D,Nu=0.3

SSH3D,Nu=0.49

SSH3D,Nu=0.499

15

10

10

5

5 0

0 0

1

2

3

4

5

6

7

0

8

1

2

),*85(Out of plane bending benchmark load versus displacements curve for different elements, meshes and theoretical solution (=0.3) 

3

4

5

6

7

8

Verticaltipdisplacement

Verticaltipdisplacement

),*85(Out of plane bending benchmark- influence of Poisson’s ratio on the deflection (16x1x1 mesh) 

&DQWLOHYHU%HDP,Q3ODQH%HQGLQJ In this example, an elastic clamped beam, which is discretised with 10 elements, is loaded by a transverse force F=.1000 on its free edge where  is a load factor [5, 14-16]. Elastic constitutive parameters are E=107 and  =0.3. The beam length is L=1.0, which the width w and the thickness hare 0.1. In order to demonstrate the performance of the membrane interpolation and it’s insensitivity to mesh distortion two meshes are considered as presented in FIGURE 11.

(a) (b) ),*85(In plane bending: initial and deformed configuration for the regular (a) and distorted (b) meshes (with 10 elements). A plot of the load-deflection curves for the tip displacement of nodes A and B for the two meshes is shown in FIGURE 12. These results show that the present solid-shell formulation and the element presented in [5] have a similar performance for both meshes, and the same insensitivity to the mesh distortion. 1000

pointAregularmesh(SSH3D)

absolutedisplacement

900

pointBregular mesh(SSH3D) Point B

800

pointAdistortmesh(SSH3D)

700

pointBdistortmesh(SSH3D)

600

PointAregularmesh(VuQuoc [5]) PointAdistortmesh(VuQuoc [5])

500

Point A

PointBregularmesh(VuQuoc [5])

400

PointBdistortmesh(VuQuoc [5])

300 200 100 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

loadfactor

),*85(In-plane bending: load deflection curve for both regular/ distorted mesh.

6LPXODWLRQRID&\OLQGULFDO&XS'HHS'UDZQSURFHVVZLWKD+HPLVSKHULFDO3XQFK In this section, the benchmark test B-1 of the NUMISHEET 99 conference [17] is carried out using the SSH3D solid-shell element with 2 elements through the thickness direction. For this propose we consider a cylindrical cup drawing with a hemispherical punch. The blank is an anisotropic sheet metal made of DDQ mild steel described by the hardening behavior modeled by the equivalent stress-equivalent strain power law: V k .H n (15)

379

Where k and n are the strength coefficient and the hardening exponent respectively. The hardening coefficients have been computed in three directions; however, the 0 degree direction is used as the reference for the hardening curve. The remaining material parameters are given as follows: Young’s modulus E= 200 GPa; Poisson’s ratio =0.3; k=558.46 MPa and n=0.2145. The parameters of Hill’s yield function describing the anisotropy of the material are F=0.655, G=0.7771, H=1.2229, N=2.3844, L=2.3844 and M=2.3844. The geometry dimensions are given in FIGURE 13. Only a quarter section of the specimen was analyzed due to the orthotropic material symmetry. Diameter of the hemispherical punch:100 mm Outer diameter of the drawing die: 220 mm Inner diameter of the drawing die (punch opening):102.4 mm Blank thickness: 1.0 mm Draw radius: 5 mm Initial diameter of the blank: 200 mm ),*85( Tool geometry and set up The coulomb friction coefficient used to model the contact between the steel sheet and the tools is assumed to have a value of =0.15. The forming process is finished when the punch travel reaches the value of 85 mm. The deformed configuration and the draw-in definition are shown in FIGURE 14.

 ),*85( Deformed sheet and definition of the flange draw-in. In order to validate the finite element solution, the analysis of the results is summarized and compared to the experimental resulted reports in [17]. As shown in FIGURE 15 and FIGURE 16, the present result is in good agreement with the experiment (the experimental results are presented by B1E-01,02,03,05) . 40

B1E01

35

B1E02

drawin(mm)

30 25

B1E03

20

B1E05

15 10

SSH3DSolidShell element

5 0 0

45

90

section,anglevsRD

),*85( Flange Draw-In at 0°, 45° and 90° from the rolling direction (A, B and C direction defined in FIGURE 14)

380

160000

B1E01

punchforce(N)

140000 120000

B1E02

100000 80000

B1E03

60000 40000

B1E05

20000 0 0

10

20

30

40

50

60

70

80

90

SSH3DSolidShell element

punchtravel(mm)

),*85( Punch force as a function of punch travel

$&.12:/('*0(176 The authors thank the Walloon Region (Winnomat2 Dinosaure project). The Belgian scientific Research Fund F.R.S-FNRS which finances A.M.H and L.D and the Belgian Science Policy (interuniversity attraction poles program IAP P6/24) are thanked for their financial support. The authors would also thank their industrial partner for this Winnomat project, namely AC&CS, subsidiary of CRM group (previously AMLR, subsidiary of ArcelorMittal).

5()(5(1&(6 1. 2. 3. 4. 5. 6. 7. 8.

9. 10. 11. 12. 13. 14. 15. 16. 17.

Simo, J.C. and M.S. Rifai, A Class of Mixed Assumed Strain Methods and the Method of Incompatible Modes. International Journal for Numerical Methods in Engineering, 1990. 29(8): p. 1595-1638. Andelfinger, U. and E. Ramm, EAS-elements for two-dimensional, three-dimensional, plate and shell structures and their equivalence to HR-elements. International journal for numerical methods in engineering, 1993. 36(8): p. 1311-1337. Dvorkin, E.N. and K.-J. Bathe, A continuum mechanics based four-node shell element for general nonlinear analysis. Engineering computations, 1984. 1: p. 77-88. Valente, R.A.F., et al., On the use of an enhanced transverse shear strain shell element for problems involving large rotations. Computational Mechanics, 2003. 30(4): p. 286-296. Vu-Quoc, L. and X.G. Tan, Optimal solid shells for non-linear analyses of multilayer composites. I. Statics. Computer Methods in Applied Mechanics and Engineering, 2003. 192(9-10): p. 975-1016. De Sousa, R.J.A., et al., On the use of a reduced enhanced solid-shell (RESS) element for sheet forming simulations. International Journal of Plasticity, 2007. 23(3): p. 490-515. Hauptmann, R., K. Schweizerhof, and S. Doll, Extension of the 'solid-shell' concept for application to large elastic and large elastoplastic deformations. International Journal for Numerical Methods in Engineering, 2000. 49(9): p. 1121-1141. Nguyen, N.H., Development of solid-shell elements for large deformation simulation and springback prediction, in ALTAS. 2009, University of Liège.Belguim.: http://bictel.ulg.ac.be/ETD-db/collection/available/ULgetd-11012009164626/. Klinkel, S., F. Gruttmann, and W. Wagner, A robust non-linear solid shell element based on a mixed variational formulation. Computer Methods in Applied Mechanics and Engineering, 2006. 195(1-3): p. 179-201. Schwarze, M. and S. Reese, A reduced integration solid-shell finite element based on the EAS and the ANS conceptGeometrically linear problems. International Journal for Numerical Methods in Engineering, 2009. 80(10): p. 1322-1355. Cardoso, R.P.R., et al., Enhanced assumed strain (EAS) and assumed natural strain (ANS) methods for one-point quadrature solid-shell elements. International Journal for Numerical Methods in Engineering, 2008. 75(2): p. 156-187. Cescotto, S. and H. Grober, Calibration and application of an elastic viscoplastic constitutive equation for steels in hotrolling conditions. Engineering Computations, 1985. 2(2): p. 101-106. Macneal, R.H. and R.L. Harder, A proposed standard set of problems to test finite element accuracy. Vol. 1. 1985, Amsterdam, Netherlands: Elsevier. Valente, R.A.F., Developments on Shell and Solid-shell finite elements technology in nonlinear continuum mechanics., in Faculty of engineerring. 2004, university of Porto: Port. p. 298. Simo, J.C., M.S. Rifai, and D.D. Fox, On a stress resultant geometrically exact shell model. IV, Variable thickness shells with through-the thickness stretching. Vol. 81. 1990, United Kingdom: Elsevier. Simo, J.C., D.D. Fox, and M.S. Rifai, On a stress resultant geometrically exact shell model. III, Computational aspects of the nonlinear theory. Vol. 79. 1990, United Kingdom: Elsevier. 50. NUMISHEET 1999 Organization Commitee. In Proceedings of the fifth international conference and workshop on numerical simulation of 3D sheet forming processes, 1999. Besançon-France.

381

Development and Application of Non-linear Friction Models for Metal Forming Simulation Ninshu Ma and Nobuhiko Sugitomo

JSOL Corporation, 2-2-4 Tosabori, Nishiku, Osaka 550-0001, Japan

Abstract.

Friction has a significant effect on the formability of stamping parts. A constant friction coefficient between stamping tools and blank is often employed in the metal forming simulating. In this presented work, several non-linear friction models which considers of the change of friction coefficient with contact pressure, sliding velocity, sliding distance, frictional work, plastic strain and temperature were developed using LS-DYNA customized friction subroutine. The validity was verified by numerical friction testing models and deep drawing model. Keywords: Friction model, Contact pressure, Sliding velocity, Sliding distance, Friction work, Stamping, Simulation PACS: 73.40.Jn, 62.20.fq, 81.20.Hy

INTRODUCTION The formability of stamping parts is determined by material properties, process design, tool shape and friction between contact surfaces. The effects of the material properties, tool shape and forming conditions on stamping formability were investigated by many researchers based on experimental approaches. Concerning with the prediction of formability, simulation technology is widely employed in both research and product development [1]. However, the friction phenomena in metal forming process are very difficult to evaluate qualitatively. There are two kinds of friction coefficient to describe the friction phenomena for a contact interface. One is viscous friction coefficient and another is Coulomb friction coefficient. The viscous friction coefficient is defined by the shear strength  s , which represents the limit of the resistance stress at the contact interface. If the applied shear stress is less than  s , the interface may be in sticking state. Sliding occurs if the applied shear stress is equal to  s . The reference value of

 s is the yield strength in shear which is approximated by,  (1) s  Y 3

where,  Y is the yield strength of the material. However, this value can not be exactly determined because the interface includes the lubrication oil and contacted materials. The most common friction coefficient is based on Coulomb friction model in which the shear strength of the contact interface is given as a linear function of the applied normal pressure, p , as follows,

s    p

(2)

Where  is the coefficient of Coulomb friction. Generally, the friction coefficient  is assumed to be constant. Actually, the friction coefficient  is not a constant and is dependent on many variables such as pressure, velocity, sliding distance, temperature, lubrication conditions and surface conditions (coating and roughness) etc [2-8]. It is quite difficult to express by a theoretical equation. In this presented paper, the friction coefficient  is expressed by the function of contact pressure p , sliding velocity v , sliding distance d , friction work

w , strain  and temperature T based on both the measured data by

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 382-389 (2011); doi: 10.1063/1.3623635 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

382

experiments and some published research results [2-8], respectively, which is schematically given by following equation, (3)    ( p, v, d , w,  ,T ) To meet the request from the industrial users, the table among the friction coefficient and these variables are developed and programmed using LS-DYNA customized friction interface program. The friction phenomena are confirmed by numerical experiments of friction testing models and a deep drawing model. The expected results are obtained.

MEASUREMENT OF FRICTION COEFFICIENT Experiment Conditions To measure the effects of velocity v and sliding distance d on the friction coefficient  , friction experiments were performed. FIGURE 1 shows the image of the friction experiment device [4].

Normal Force Clamp 48

v

p

R4

Picture R4

Test piece Rigid

Roller FIGURE 1. Schematic showing of friction experiment device [4]

The experimental conditions for the measurement of the friction coefficient  are shown in TABLE 1. The material and lubrication oil used in the experiments are SUS304L/t0.95mm and press oil S-478, respectively. The viscosity of press oil S-478 is 58mm2/s. The friction experiments were performed with two velocity levels and the change of the friction coefficient with the sliding distance was measured. TABLE 1 Conditions for the friction coefficient measurement

Velocity

v1  1.0mm / sec v2  10.0mm / sec

Friction coefficient measurement  (v1, d )  (v2, d )

Material: SUS304L/t0.95mm Lubrication oil: Press oil S-478 (viscosity=58mm2/s) Pressure=3.0MPa

383

Measured Results FIGURE 2 shows the change of the friction coefficient with the sliding distance measured in the experiments. The friction coefficient increases with the sliding distance if the distance is short and it becomes small after a certain distance. If the sliding distance is smaller than about 20mm, the friction coefficient at the higher velocity (10mm/s) is larger than that at the lower velocity (1mm/s). If the sliding distance is larger than 20mm, the friction coefficient at the higher velocity (10mm/s) becomes smaller than that at the lower velocity (1mm/s). The influence of velocity and sliding distance on the friction coefficient is complicated and their relation is not linear as measured in the experiments. 0.20 0.18

Friction Coefficient

0.16 0.14 0.12 0.10

v=1.00mm/sec, p=3.0MPa

0.08 v=10.0mm/sec, p=3.0MPa

0.06 0.04 0.02 0.00 0

10

20

30

40

50

Distance (mm)

FIGURE 2. Measured friction coefficient and its change with velocity and sliding distance

DEVELOPMENT OF NON LINEAR FRICTON MODELS FOR SIMULATION USING LS-DYNA CUSTOMIZED FRICTION SUBROUTINE The following non linear friction models were proposed to describe the changes of the friction coefficient with the various parameters for the practical metal forming simulation. (1) Model-V (fr-V) Non linear curve between friction coefficient and sliding velocity is proposed as shown in FIGURE 3. The friction coefficient decreases usually with the increase of the sliding velocity. If the sliding velocity is less than a certain minimum velocity given in the input curve, a constant friction at this minimum velocity is employed in the simulation. If the sliding velocity is larger than a certain maximum velocity given in the input curve, a constant friction at this maximum velocity is used. (2) Model-P(fr-P) Non linear curve between friction coefficient and contact pressure is proposed as shown in FIGURE 4. The effect of the contact pressure on the friction coefficient is not simply decreasing or increasing. If the contact pressure is lower than the smallest pressure in the input curve, the friction coefficient takes the value at the smallest pressure In the same way, the friction coefficient is determined if the contact pressure is higher than the largest pressure in the input curve. (3) Model-D (fr-D) Non linear curve between friction coefficient and sliding distance is proposed as schematically shown in FIGURE 5. The experimental results in the FIGURE 2 shows more complicated change of the friction coefficient with sliding distance. (4) Model-W (fr-W)

384

Non linear curve between friction coefficient and friction work density is schematically shown in FIGURE 6 which is based on the results done by Hashimoto [7]. (5) Model-E (fr-E) Non linear curve between friction coefficient and plastic strain is proposed as schematically shown in FIGURE 7. In metal forming, the large plastic deformation always occurs in the material. The large plastic strain will change the surface roughness and other surface qualities such as micro structure change and temperature increase. These may make the friction coefficient change. (6) Model-T (fr-T) Non linear curve between friction coefficient and temperature is proposed as schematically shown in FIGURE 8. If the temperature changes, the viscosity of lubrication oil will change and it may induce the change in the friction coefficient. (7) Model-PV (fr-PV) Non linear table among friction coefficient, contact pressure and sliding velocity is proposed as shown in FIGURE 9. The friction coefficient at a certain pressure and a velocity (marked by  ) can be interpolated from the table defined in FIGURE 9. There are upper bound and lower bounder for the friction coefficient in order to avoid mis-input. FIGURES 3-8 present the schematic tendencies with various parameters which can be defined freely based on measured results for the practical metal forming simulation. (8) Model-PW (fr-PW) Non linear function among friction coefficient, contact pressure and friction work density given by equation (3) which were proposed by Hashimoto [6].

 ( p, w)  A0  A1w  A2 p  A3 wp  A4 w2  A5 p 2  A6 w2 p  A7 wp 2  A8 w3  A9 p 3

(3)

Where, the parameters A0-A9 can be determined by friction experiments.

 (v)

 ( p)

v(mm / s) FIGURE 3. Friction coefficient vs. velocity

p(MPa)

FIGURE 4. Friction coefficient vs. pressure

 (w)

 (d )

w(MPa  mm)

d (mm) FIGURE 5. Friction coefficient vs. distance

FIGURE 6. Friction coefficient vs.friction work

385

 ( p )

 (T )



p

T

FIGURE 7. Friction coefficient vs. plastic strain

FIGURE 8. Friction coefficient vs.Temperature

 max

 (v, p)  (v, p)

p1 p2 p3

 min

V (mm / s) FIGURE 9. Friction coefficient vs. both pressure and velocity

VERFICATION OF NON LINEAR FRICTON MODELS To verify whether the non friction model works as accurate as the theory, a friction test piece shown in FIGURE 10 was modeled and friction force was simulated using the customized LS-DYNA program developed by authors. The geometry of test piece, velocity curve, testing pressures and input friction coefficient curves are also shown in the figure. The velocity is changing from 0.0 to 1000mm/sec at four levels of pressures (p=5.0, 10.0, 15.0, 20.0MPa), respectively. The input friction coefficient decreases with the increase of velocity and pressure. FIGURE 11 is verified the results describing the change of the friction coefficient with the velocity. The marks □, ○, △ and ◇ in the figure are the input points and solid lines are the reproduced friction coefficient defined by the ratio of friction force and normal force for the contacted interface between the friction holding tools and blank. The good agreement was obtained and the accuracy of the developed program was verified. =0.15 R5 1000

V

p 0.1 Time (s)

0× L20

0 W1

mm 1.0 ×t

p=5MPa p=10MPa p=15MPa p=20MPa

friction coefficient

Velocity (mm)

50

=0.14

0

p=5MPa

=0.13

p=10MPa

=0.12

p=20MPa

=0.11

p=40MPa

v(mm/s)

1000

FIGURE 10. Simple testing model and friction coefficient input for verification

386

forcecoef to normal force Ratio of friction Friction

0.16 frc-p5-v

0.15

frc-p10-v frc-p15-v

0.14

frc-p20-v 0.13

frc-p5-input frc-p10-input

0.12

frc-p15-input frc-p20-input

0.11 0.10 0

250

500 750 1000 Sliding velocity (mm/s)

1250

1500

FIGURE 11. Reproduced friction coefficient changing with velocity at various pressures

APPLICATION OF NON LINEAR FRICTION MODEL TO SQUARE CUP DEEP DRAWING SIMULATION FIGURE 12 shows the simulation model of a square cup deep drawing. FIUGURE 13 shows the detail sizes of die, punch and holder which are the same as those used in NUMISHEET benchmark [9]. The blank material is a mild steel sheet SPCC. The surface of blank is as cold rolled state without surface coating. The length, width and thickness of the blank sheet are 150mm, 150mm and 0.80mm, respectively. The blank holding force (BHF) and punch stroke are 19.6KN and 40mm, respectively. Two cases of simulation were performed in order to compare the difference of the developed non linear friction model with a constant friction coefficient as shown in TABLE 2. If the measured friction coefficient [7] used for drawing simulation is plotted by graph, FIGURE 14 gives its schematic change with the friction work at the different pressures. FIGURE 15 and FIGURE 16 shows the thickness distribution of the deep drawn square cup using a constant friction coefficient and non linear friction model, respectively. The min. thickness occurred at the position contacted with punch corner. It is not affected by the friction coefficient. The maximum thickness is about 0.96mm and 1.01mm at the edge of flange, respectively, if a constant friction coefficient and non linear friction model are employed. This is because the draw-in displacement increased especially for large punch stroke when the change of the friction coefficient with the pressure and friction work are considered as shown in FIGURE 17.

Holder 43

Punch 70

7 R8

48

2

7 R8

Stoke

R5

Holder 43

R5 2

Die FIGURE 12. Simulation model of square cup drawing

Die

FIGURE 13. Tool sizes of square cup deep drawing

Blank material

TABLE 2 Forming conditions in simulation Non linear friction model Eq. (3) Constant friction coefficient

Cold Rolled SPCC(SPCC-CR)

FIGURE 15

  0.14

387

48

0.250

Friction Coefficient

Cold Rolling Steel

fric-CR　(p=10MPa d=0～4cm) fric-CR (p=20MPa d=0～4cm) fric-CR　(p=30MPa　d=0～4cm fric-CR　(p=40MPa d=0～4cm) fric-CR　(p=50MPa d=0～4cm)

0.200 0.150 0.100 0.050 0.000 0

1

2

3

4

5

6

7

8

9

10

Friction work (KN/cm) FIGURE 14. Friction coefficient vs. pressure and friction work

Thickness (mm)

Thickness (mm)

Thickness (mm)

tmin

tmin tmax

DX

tmin=0.70 tmax=0.96

tmax

DX

A

tmin=0.70 tmax=0.96

FIGURE 15. Thickness using constant friction coefficient

tmin=0.71 tmax=1.01 FIGURE 16. Thickness using non linear friction model

Draw-in displacement (mm)

35 SPCC/t0.8-CR

30

Constant friction coef.

25

nonlinear friction coef.

20 A

15 10 5 0 0

10

20 Punch stroke (mm)

30

FIGURE 17. Draw-in displacement at point A with punch stroke

388

40

SUMMARY (1) (2) (3) (4)

Non linear friction coefficient was measured by simple experiments. Several non-linear friction models were proposed and programmed into LS-DYNA customized subroutine. The non linear friction models were verified by numerical experiments. A Non linear friction model was applied to a square cup deep drawing, the simulation gave reasonable results compared those with the constant friction coefficient

ACKNOWLEDGMENTS The authors gratefully appreciate Prof. T. Naka of Yuge National College of Maritime Technology for the measurement of the friction coefficient. The authors would also like to acknowledge Mr. T. Kyuno and S. Tamura of Aida Engineering for the providing of experimental samples.

REFERENCES 1. N. Ma and N. Sugitomo, “Dynamic Explicit FEM and Simulation on Sheet Metal Forming”, Journal of Japan Society of Technology Plasticity, Vol.47, No.540 (2006-1), pp.29-34. 2. B. Fredriksson, “Finite element solution of surface non-linearities in structural mechanics with special emphasis to contact and fracture mechanics problems”, Computer Structures, 6 (1976), pp.281-290. 3. E. Nakamachi and J. Kumada, “Derivation of the Friction Law for a coated sheet by experiments and sheet forming Finite Element Simulation”, Trans. of Japan Society of Mechanical Engineers, Vol.58, No.551 (1992), pp.1228-1233. 4. T. Naka, “ Effect of temperature and forming speed on deep drawability of 5083 aluminum alloy sheet”, Metal and Materials, 4-4 (1998), pp.464-466. 5. T. William and J. Carter, “A model for Friction in Metal Forming”, Journal of Engineering Material and Technology, ASME, Vol.116, No.2 (1994), pp.8-13. 6. K. Hashimoto, E. Isogai, T. Yoshida, Y. Kuriyama and K. Ito, “Finite Element Analysis of Sheet Metal Forming Taking Account of Nonlinear Friction Model”, Journal of Japan Society of Technology Plasticity, Vol.49, No.573 (2008-10), pp.6569. 7. K. Hashimoto, “Sliding Characteristics of Surface-coated Steel Sheets and Influence of those on Press Formability”, Doctor thesis, Northeast University, Japan, Oct., 2002. 8. Y. Tamai, Y. Yamazaki, A. Yoshitake and T. Imura, “Improvement deep drawablity of steel sheet by motion control of servo press”, Journal of Japan Society of Technology Plasticity, Vol.51, No.592 (2010-5), pp.68-72. 9. Y. Hirose, H. Uto and K. Kazuma, Benchmark B1 for Square Cup Deep Drawing”, Proc., NUMISHEET’93, Japan, 1993, pp.377-378.

389

Die Deformation Measurement System during Sheet Metal Forming J. Funada1, S. Takahashi1, and H. Fukiharu2* 1

Dept. of Mechanical Engineering, College of Indusrtial Technology, Nihon University, 1-2-1 Izumi-cho, Narashino, Chiba, 275-8575, Japan 2 Advanced Simulation Technology of Mechanics Co.,Ltd, RIKEN 2-1, Hirosawa, Wako-shi, Saitama, 351-0198, Japan

Abstract:In order to reduce affection to the earth environment, it is necessary to lighten the vehicles. For this purpose, high tensile steels are applied. Because of high strength, high forming force is required for producing automotive sheet metal parts. In this situation, since the dies are elastic, they are deformed during forming parts. For reducing die developing period, sheet metal forming simulation is widely applied. In the numerical simulation, rigid dies are usually used for shortening computing time. It means that the forming conditions in the actual forming and the simulation are different. It will make large errors in the results between actual forming and simulation. It can be said that if contact pressure between dies and a sheet metal in the simulation can be reproduced in the actual forming, the differences of forming results between them can also been reduced. The basic idea is to estimate die shape which can produce the same distribution as computed from simulation with rigid dies. In this study, die deformation analyses with Finite Element Method as basic technologies are evaluated. For example, simple shape and actual dies elastic contact problems were investigated. The contact width between simple shape dies was investigated. The computed solutions were in good agreement with experimental results. The one case of the actual dies in two cases was also investigated. Bending force was applied to the blank holder with a mechanical press machine. The methodology shown with applying inductive displacement sensor for measuring die deformation during applying force was also proposed. Keywords: elastic deformation, elastic contact, finite element method, sheet metal forming, die deformation.

1. INTRODUCTION Recently, the application of high tensile steels and Aluminum Alloy has developed to advance the automotive lightening. The high forming force is required for forming high tensile steels. According to the previous simulation studies[1㹼4], dies wererigid and analyzed simply. However, the die is deformed, the pressure distribution (pressure distribution) is changed, and the formability of the productis influenced in the practical situation. Therefore if we could design the die which is same pressure distribution (pressure distribution) as the simulation study, we consider that we obtain formation products closer the results of the simulation. In this study, we suggested measurement system of die deformation to gain the results of die forming simulation following the technical issues. We decided some requirement to analyze both of the measurement experiment of die deformation using die of experimentwith the system and the simulation of die deformation. Additionally, we report that the comparison and discussion of two outcomes such as the measurement result of die deformation and the result of die deformationfrom the simulation.

2. PRECISION INSPECTION OF THE SIMULATION CONTACT AREA Elastic contact analyses of forming dies are useful for investigating die deformation during. In order to validate the accuracy of elastic contact analyses, dies with cylindrical contact were investigated. Elastic contact width obtained from experiments and numerical simulation results were compared.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 390-397 (2011); doi: 10.1063/1.3623636 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

390

2.1. Measurement Results of Contact Area of Experiment Dies In order to investigate the accuracy of elastic contact areas using numerical simulation, the dies with cylindrical contact were applied. The dimensions of dies are shown in Figure 1. The testing apparatus is indicated in Figure 2. For measuring contact area between upper and lower dies, Prescale (Fujifilm Business Supply Co. , LTD.) was applied. The load on the upper die was 1, 7, 20 (kN). The holding period of the upper dies with the load was 10(s). The contact areas are shown in Table (1). The width of contact area was measured as indicated in Figure 3. Loadcell

Die

Figure 1. Dimensions of experimental dies

Figure 2. Experimental apparatus

Table (1). Measurement results of contact area Force

Experiment number

(kN)

1

0.6

0.6

Contact width (S/2)(mm)

㻜㻚㻢

L

㻝 S/2(mm)

3

L

S

2

0.6

㻣 S/2(mm)

㻝㻚㻜㻌 1.0

1.0

1.1

1.5

1.6

1.5

㻞㻜 S/2(mm)

㻝㻚㻡㻌

S

L L

Figure 3. Width of contact area

2.2. Measurement Results of Contact Area in Elastic Contact Simulation In order to make shorter computing time, the quarter model is employed. The number of node and element is 28,539 and 113,790 respectively. Young’s modulus and Poisson’s ratio are 210GPa and 0.28 respectively. The discretization of the model is shown in Figure 4. The contact areas at each load are indicated in Figure 5.

Figure 4. Quarter model of dies

391

a) 1kN

b) 7kN

c) 20kN

Figure 5. Contact stress at contact area for each load

2.3. Comparison between Experiment and Analysis The relationship between contact width and force is shown in Figure 6. They are good agreement with each other. 25

Analysis value

20

Experimental value

Force [kN]

15

10

5

0 0.0

0.5

1.0

1.5

2.0

Contact length [mm] Figure 6. Relationship between contact width and force

3. MEASUREMENT SYSTEM OF THE DIE DEFORMATION In order to evaluate the deformation of the dies during forming, the measuring system indicated in Figure 7 was developed. Inductive displacement sensors were applied for measuring the distance to the measurement object. The die deformation is evaluated by the following equation (1).

 ( A  A' ) (C  C ' )  Bend ( B  B' )    2  

(1)

Where A, B, C, A’, B’,C’ are indicated in Figure 7. The bottom surface of the sensors and the measurement object are not parall, if makes some error in the mesurement results. Therefore, the distance between the sensor and measurement object was measured. The error of the distance was ±1.0%, even if the angle between the bottom of the sensor and the surface of the measurement object is 2.5°.

392

Die

Inductive displacement sensor

Jig

A

C

B

Measurement object

A䇻

C䇻 B䇻 Figure 7. Calculation method of bend

4 DEFORMATION MEASUREMENT OF THE BLANK HOLDER 4.1. Experiment Method and Experiment Results For obtaining validation data of die deformation analysis, blank holder as shown in Figure 8 was used. The shim (17×70×1.2mm) is located as shown in Figure 9. It makes bending moment in the blank holder when the press slide is located in the bottom dead center. The die set of this experiment is shown in Figure 10. The measurement system of die deformation is indicated Figure 11. The results of distance between sensors and the mesurment object are shown in Figure 12. The first stable distance around 1(s) is due to the blank holder pressure.

4.2. Validation of the Analysis Results The bend of the blank holder was calculated and compared with the preceding experiments. The analysis conditions were Young’s modulus 210GPa, Poisson’s ratio 0.3, coefficient of friction 0.15, cushion force 147kN, forming force 1470kN. Figure 13 shows the model for deformation analysis of the blank holder. The number of element and node are 88,010 and 21,402 respectively. The shims are also located in the center of the blank holder.

t=26.4mm t=26.4mm Figure 8. Blank holder

393

Shim

Upper die

Blank holder

Shim

Upperdie die Lower

Figure 9. Location of shim

Figure 10. Die set

2

1

5.2 5.2

3

Displacement(mm) (mm) Distance

Blank holder der

䐡Right

55

4.8 4.8

䐟Left

4.6 4.6

Right Center Left

䐠Center

4.4 4.4

Jig Inductive displacement l sensor

4.2 4.2

Measurement object

Figure 11. Measurement system of die deformation

00

1 1

2 2

3 3

4 4

5

Time (s) Time(s) Figure 12. Distance to measurement object at each device

Figure 13. Element of blank holder model

394

6 6

5. THE IMPROVEMENT OF THE PERFERMANCE OF DEFORMATION MEASUREMENT If the dies are deformed, inductive displacement sensors are also shown in Figure 14 (a). In this case, the distance between the sensors and the measurement object is charged significantly. It makes some error in the measurement results. Therefore, a rod is inserted between the measurement object and jig as shown in Figure 14 (b). The distance from the center of the sensor to the measurement object is not changed. It makes better results of measuring bend blank holder.

Needle Rod Rod Jig Measurement object

(a)Before modification (b)After modification Figure 14. Improvement of jig for distance measurement

6. DEFORMATION MEASUREMENT OF THE DIE FOR AUTOMOBILE PARTS 6.1. Experimental Results of Bending Dies

180

180

125

The deforming die of automobile parts had various types of contact surface rather than common die used in the previous study. Then a further simulation was examined with another die shown in Figure15. Facilitating the deformation of die, two shims (151×61×2mm) were set at the center of the die (Figure 16). The center part of a riser supported the upper and lower dies were removed. In order to deform the die compulsorily, the upper die was moved under the bottom dead center point in the range from 1.2mm to 2.5mm, and bend was measured with three inductive displacement sensors such as the measurement of the blank holder deformation (Figure 17). The results at took down the range between 2.5mm is indicates in Figure 18 and Figure 19.

146

450

85

450 450

(a) Lower die

(b) Upper die Figure 15. Shape of dies for automotive parts

395

Shim Shim

1

Jig

3’

2’

1’

2

Die

3

Measurement object Inductive displacement sensor Figure 16. Location of Shim

(Evaluation points on upper die are located back side of the die)

6 5.8 5.6 5.4 5.2 5 4.8 4.6 4.4 4.2 4

3

1

Distance(mm)

Distance(mm)

Figure 17. Measurement of die deformation

2

0

2

4

6

8

10

12

2’

6 5.8 5.6 5.4 5.2 5 4.8 4.6 4.4 4.2 4

3’

0

14

1’

2

4

6

8

10

12

14

Time(s)

Time(s) Figure 18. Distance to measurement object at each device of lower die

Figure 19. Distance to measurement object at each device of upper die

6.2. Validation of the Analysis Results The analysis model is shown in Figure 20. The number of element and node is 1,313,415 and 272,202 respectively. Young’s modulus, Poisson’s ratio and friction coefficient are 210GPa, 0.3, and 0.1, respectively. The computing time was 15min. The error in bend of the upper die was 3.8% and that of the lower die was 43%. Table (2). Bend from simulation and experimental result

a) Experimental results Die Upper Lower

Bend(mm) 0.158 -0.147   

b) Simulation Die Upper Lower

Bend(mm) 0.152 -0.258

396

Figure 20. FEM model of automotive parts dies

7. CONCLUSIONS 1. The computed contact width of two dies at cylindrical shape was in good agreement with the experiment results. 2. The system for measuring die deformation during forming was developed. In order to improve the performance of the system, contact condition between the jig and the measurement object, was modified from surface to line. 3. The deformation analysis with elastic contact for automotive parts forming dies can compute bend in dies accurately. In some results, it had large error. It needs more investigation on this matter.

REFERENCES 1. K.Roll: Simulation of sheet metal forming necessary developments in the future, NUMISHEET, 2008, pp3-11 2. N. Maand Y. Umezu and Y. Watanabe and T. Ogawa: Springback prediction by yoshida-uemori model and compensation of tool surface using jstamp, NUMISHEET, 2008, pp473-478 3. P. Akerstrom and M. Oldenburg: Numerical simulation of a thermo-mechanical sheet metal forming experiment, NUMISHEET, 2008, pp569-574 4. S. Bonino and D. Xu: Sheet metal transfer simulation using flexible bodies, NUMISHEET, 2008, pp909-914

397

A Study on the Development of Dedicated FEA Tool for Roll Forming of Arbitrary Cross Sections Wan-gi Chaa, Hong-lae Kima and Naksoo Kimb a

b

Sogang university,1, Sinsu-dong, Mapo-gu, Seoul, 121-742, Korea. Sogang university, Mechanical engineering, 1, Sinsu-dong, Mapo-gu, Seoul, 121-742, Korea.

Abstract. In the cold roll forming process, the sheet metal strip is gradually and successively bent into a desired profile. Occurrences of spring back, twist, and bow are major defects in roll formed profiles. We developed a dedicated analysis program based on the elastic-plastic finite element method to simulate roll forming processes with arbitrary shaped rolls. We applied the augmented Lagrangian technique in contact algorithm. Simplified cross sections with different flange heights produced via three passes of roll forming have been experimented to be measured in the amounts of twist and bow occurred after roll forming. The performance of the developed FEA tool has been verified with measurements in terms of accuracy. Considering that the forming defects reflect all the deformation models, it is concluded that the developed tool could yield quite reasonable results. Keywords: Roll Forming, Spring Back, Elastic-Plastic FEM, Augmented Lagrangian Technique. PACS: 46.15.-x, 47.11.Fg.

INTRODUCTION Roll forming is one of the most productive processes for forming products with constant cross sections. However, twist and bow defects due to spring back are major drawbacks for the roll forming process. In order to predict these defects or for more efficient process design, FEM analysis programs have been continually developed. Kiuchi et al. [1] developed a numerical simulation for designing objects with circular cross section. Also, Kim et al. [2] developed a rigid-plastic based three-dimensional finite element method roll forming program for general roll-pass design. Hong et al. [3] developed a roll forming program with kinematically steady-state condition integrated, which allowed for identification of factors affecting roll forming. Kim et al. [4] conducted buckling analysis with elastic finite element method after calculating the material deformation and the amount of longitudinal strain with rigid-plastic finite element analysis using the roll forming program SHAPE-RF. Oh et al. [5] suggested a robust design method for the forming process of slide rail using SHAPE-RF. Besides rigid-plastic deformation of material behavior, elastic-plastic finite element method-based programs were developed to study roll forming process [6, 7]. Brunet et al. [8] developed elastic-plastic finite element method code, which combined 2D crosssection analysis and 3D analysis, to study optimal design of roll-forming process. There was also a study by Zhang et al. [9] which used elastic-plastic spline finite strip method to better analyze tube shaped roll forming products. Li [10] had studied how residual stresses affect the roll formed square hollow section. He conducted roll formed square channel and analyzed residual stresses using X-rays diffraction method. Meanwhile, we simplified the effect of residual stresses on the product by measuring shape error. Most researches had studied roll forming by verifying an example which roll formed U-channel of symmetry cross section. However, Ona et al. [11] conducted research on methods to correct twist appearing on different asymmetrical U-channels manufactured by roll forming process. Based on the researches, we chose a U-channel of arbitrary cross section, in which twist and bending can be shown clearly and conducted roll forming experiment. We developed elastic-plastic finite element method [12] which was based on program for analysis and augmented Lagrangian technique [13] which was used to calculate contact forces on the rolls. We quantifiably defined an index of twist and bowing. And through the roll forming simulation, we checked the performance of the developed program by comparing the amounts of twist and bowing in each result. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 398-405 (2011); doi: 10.1063/1.3623637 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

398

THEORY To better predict the experimental values, we developed elastic-plastic finite element method-based program for analysis. [11] Also, the augmented Lagrangian technique [12] was used for contact algorithm to enhance the accuracy and astringency of roll contact calculation. Roll forming is a forming process to assess bending deformation to a thin and wide blank sheet. Due to the characteristics of roll forming, massive deformation rarely occurs in contact region of blank and rolls. Most plastic deformation occurs in bending zone. When the blank enters between two rolls, confined contact areas frequently get into trouble in numerical convergence. Especially, when the blank passes through parallelized rolls with limited gap, it is not easy to obtain proper reactions or stable contact points of rolls due to the oscillations of solutions. The penalty method of using a single fixed penalty also frequently fail to obtain converged solutions in the case of passing through a gap of parallelized roll surfaces. At the same time, there are Lagrangian method and augmented Lagrangian method for computing precise reactions on contact nodes. In order to overcome the weakness of Lagrangian method, the augmented Lagrangian method is employed for reinforcing the weak convergence. However, it still has the extra Lagrangian multiplier variables in the solution of equations. It causes much more computing time than the penalty. The way of overcoming this limitation is the augmented Lagrangian technique. Lagrange multipliers remain as element level variables and do not enter the global structural solution. The technique only involves the conventional displacement variables as it does with the penalty method, and reduces the computing time which solves the global equations. In this technique instead of direct obtaining the Lagrange multipliers, the reaction forces are found as in Eq. (1).

r

i 1

   Ct p

i

(1)

Ct is a penalization parameter and has constant value. When the equilibrium state is obtained, the Lagrange multiplier λ, is updated as follows in Eq. (2).

r

new



old

 Ct p

equi

(2)

MODELING OF THE ROLL-FORMING PROCESS Blank and Rolls TABLE 1. Initial width, Arbitrary Flange Heights and Relative Ratio. Case No. Initial width(mm) H, Variable Flange Heights(mm) 1 154.4 50 2 141.4 37 3 128.4 24 TABLE 2. Material Properties. Young’s Modulus Poisson’s Ratio Initial Yield Stress Flow Stress

h, Fixed Height(mm)

Flange

24

Rf (H/h) 2.08 1.54 1.00

210 GPa 0.3 214.6 MPa



  615.2 0.02747   p



0.293

MPa

The example presented in this study was an arbitrary U-channel forming process that is composed of three passes, and each roll stand has a bending angle of 40°, 70°, and 90°, respectively as shown in Fig. 1. The distance of roll stands was 350 mm. Geometries of rolls at each stand are given in Fig. 2. The thickness of the initial blanks was 0.8 mm, and the length was 2,300 mm. The initial width of blanks were 154.4 mm, 141.4 mm, and 128.1 mm for case No. 1, 2, and 3, respectively, as shown in Table 1. The height of one flange was fixed, but others were variable for checking the effect of the variable ratio of two flange heights. The values of the flange heights were shown in Table 1. Material properties were found by tension test, as shown in Table 2. We used the Swift’s isotropic strain hardening law for material behavior information.

399

FIGURE 1. Section profile.

(a)

(b)

(c) FIGURE 2. Description of the roll set at (a) 40°, (b) 70°, (c) 90°.

400

Experiment Method Three roll formed products were shown in Table 1. To eliminate the influence of the rolls such as scratch and satisfy the condition of continuous processing, 1,000 mm was left as the reference length by cutting off the front and the end parts. Specific locations of the leftover products were assigned to be measured since we anticipated different degrees of twist and bow because of spring back and the residual stress. Measurements of Twist and Bowing There are two measuring methods for twist and bow. They are shown in Fig. 3. The method for measuring twist is shown in Fig. 3 (a). We first define four vertexes that form the base side of the product. If any twist occurs in a roll formed product, it is obvious that all the four vertexes cannot lie in a same plane. A plane can be defined with the three vertexes, and we measure the vertical distance from the plane and the other vertex left. If the length of this vertical distance is big, we define it as a high degree of twist; while the length is close to zero, we define it as a small degree of twist. In the actual measurement, we placed the product on a flat surface, and measured the distance between the flat bottom surface and the vertex after making sure that the self-weight was irrelevant. The method for measuring bow is shown in Fig. 3 (b). The measuring position is at center in the length of the product. The vertexes on either side can form a straight line, and from the line, the longest vertical distance made by bow is measured. In the actual measurement, we placed the product on a flat surface, and measured for the largest height difference. We decided that the larger the distance the larger degree of bow, and the smaller the distance the smaller degree of bow. In other words, the straightness of the product is higher when the largest distance is smaller. By using these two methods, we can more clearly show the degree of defects, compared to using strain distribution.

(a)

(b) FIGURE 3. Measurement methods, (a) twist measurement; the distance between one vertex and a plane made by other three vertexes, (b) bow measurement; the longest vertical distance between the base side and the line made by two vertexes on either end.

401

Simulation Simulation Set-up With the assumption that the blank went through deformation with no slip condition in roll forming process, the friction coefficient was set at 0, and the blank went through the rolls with constant speed. There were three kinds of the blank by different widths which we got ready to simulate. 8 node brick elements were used to generate the mesh for the 2,300 mm long blank. We used one element layer in the thickness direction. We made elements finer where bending deformation is predicted, and in other area, we made coarse mesh for the finite element analysis mesh. We assumed rolls were rigid surface since the deformation of roll was very small in contact area. Roll surfaces were treated as analytical surfaces.

FIGURE 4. Velocity boundary conditions on the blank.

Velocity Boundary Conditions In the simulations, two different velocity boundary conditions were used. First, in front-section, we assigned 20 mm/s as a velocity boundary condition in the x-axis direction throughout the entire simulation. Second, the nodes on the center line of the blank in longitudinal direction were assigned as fixed boundary conditions ( Vx = Vy = 0 ) as shown in Fig. 4. However, after the blank passed the last rolls, the fixed boundary condition was released to free nodes since it was hard to converge for implicit solving without constraints in all directions. Spring Back Analysis As mentioned previously, we cut off both end parts of the product to leave 1,000 mm as reference length, and measured the spring back of the leftover. In the analysis, we used the previously completed analysis results to delete the elements from the either end in the length direction to leave 1,000 mm elements, and then we performed spring back analysis. With the spring back analysis, we first completed all the passes, and applied elastic analysis on the residual stresses that are left on the elements. Simulation Sequence The procedure of the roll forming of the arbitrary U-channel simulation can be summarized as follows: i. Pull the blank through the 40° and the 70° roll sets until the 90° roll set is reached. ii. Stop the blank, and bend the front end to 90° by moving side roll inwards. iii. Restart pulling the blank again through the all roll sets until the whole blank leaves the 90° roll set and is free in the air.

402

SIMULATION RESULTS AND COMPARISONS WITH EPERIMENTAL MEASUREMENTS

(a) (b) (c) FIGURE 5. Influence of the spring back measured at specific locations; (a) twist measurements, (b) bowing measurements at fixed flange side, (c) bowing measurements at variable flange side.

In Fig. 5(a) which shows twist measurements in different cases, the tendency is that the twist measurements of experiment are decreased by reducing Rf. The numerical analysis results in twist generally show the lower values than the experimental results, the largest difference was about 35%. The simulation results show the similar tendency as experimental measurements. The case 3 is the U-channel which has the symmetric cross-section because the value of Rf is 1.00. And then it usually does not appear the twist which is a defect of spring back in symmetric condition. However, due to asymmetric geometries of rolls, the twist defect appears in the case 3. The Fig. 5(b) and (c) show that the bowing measurements on both sides in different cases. The values of bowing measurements at fixed flange side in experiments are increased as the case number increasing, while the results at the variable flange side have almost constant value. If the value of Rf is not 1.00, the ratio of bowing measurements at fixed to at variable is large. If the value of Rf is 1.00, both of the bowing measurements are almost same. Although some errors of the numerical analysis results exist, we can still conclude that they are close to the experimental results and deserve to follow the trend of experimental measurements. The compared results of the both shows that the spring back analysis results after cutting the materials were not much far away from the experimental values. It should be noted that the simulation results are quite reliable, considering the defects occur as results of springback and release of the residual stresses, which are regarded as the gradient of the degree of deformation. Fig. 6 shows the deformed shape of blank on 3-pass with von Mises stress. Residual stresses distribution of before and after cutting from the analysis results are displayed in Fig. 7. The displayed residual stresses are von Mises stress. With the elastic-plastic analysis, we find that the overall stress level is lower for the spring back analysis result after cutting the elements when compared with the results just after process completion. Meanwhile, the stress distribution is also found to change. These values cannot be compared directly with the experiment results. However, if the analysis results are within an acceptable range compared with the experimental values of spring back analysis after cutting, then indirectly, the residual stress analysis results are quite reasonable as well. With these analyses, we can predict from the roll forming processes to the after-process designs such as cutting or twist correction.

403

FIGURE 6. Distribution of von Mises stress mapped on the deformed strip during the 3-pass roll forming process.

FIGURE 7. Distribution of residual von Mises stress mapped on the roll formed product after cutting of case 1.

SUMMARY AND CONCLUSIONS In this study, we first analyzed the arbitrary U-channel with elastic-plastic finite element method and augmented Lagrangian technique which are optimized for the roll forming process, and then we compared the results with the experimental results. Also, we compared the spring back analyses results derived from the previously acquired results with the experimental results. There were no big differences in the results when the two defined methods were used for the comparisons. Based on these results, we indirectly confirmed the trustworthiness of calculated residual stress distribution. Further research may be required on whether these analyses can be generally applied on roll formed products with various cross sections.

404

ACKNOWLEDGMENTS This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (No. 2010-0023152) and sogang university research program with a Grant No. 201010042.01.

REFERENCES 1. M. Kiuchi, K. Abe and R. Onodera, CIRP Annals-Manufacturing Technology 44, Issue 1, 239-242 (1995). 2. N. Kim and S.-I. Oh,CIRP Annals-Manufacturing Technology 48, Issue 1, 235-238 (1999). 3. S. Hong, S. Lee and N. Kim, JMPT 113, 774-778 (2001). 4. Y. Kim, J. Kim, Y. Jeoung and N. Kim, Trans. of the KSME(A) 27, 644-652 (2003). 5. M. Oh, M. K. Lee and N. Kim, JMST 24(12) 2545-2553 (2010). 6. G. Nefussi, L. Proslier and P. Gilormini, Int. J. Mech. Sci. 40, 15-25 (1998). 7. Z.-W. Han, C. Liu, W.-P. Lu and L.-Q. Ren, JMPT 127, 382-387 (2002). 8. M. Brunet, S. Mguil and P. Pol, JMPT 80-81, 213-219 (1998). 9. L.-L. Zhang, N.-L. Tan and C. Liu, J. Shanghai Jiaotong Univ. (Sci.) 15, 70-75 (2010). 10. S.H. Li, G.Zeng, Y.F.Ma, Y.J.Guo and X.M.Lai, Thin-WalledStructures, 47, 505-513 (2009). 11. H. Ona and T. Jimma, JMPT 8, 273-291 (1983). 12. J. C. Simo and T.J.R. Hughes, “Integration Algorithms for Plasticity and Viscoplasticity,” in Computational Inelasticity, edited by J. E. Marsden, New York:Springer, 1998, pp.120-122. 13. O. A. Baba, B. Radi and J.-C. Gelin, Mathematical and Computer Modelling 32, 1171-1179 (2000).

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Cutting Strategy for the Numerical Simulation of Multi-Stage Forming of Thick Metal Sheets Fernando Rastellini12* , G. D´ıaz13 , Albert Forgas1 ,Gerardo Socorro1 1

Quantech ATZ S.A., Gran Capit´an 2-4 NEXUS I, Barcelona, 08034, Spain. 2 CIMNE, Gran Capit´an s/n, Barcelona, 08034, Spain. 3 Technical University of Catalonia, Jordi Girona 3, Barcelona, 08034, Spain. * Corresponding author: +34.932.047.083, Fax +34.932.047.256, [email protected]

ABSTRACT The cutting stage is an essential step in the process of multi-stage forming of metal sheets. In this work, a robust cutting strategy is thoughtfully described. The cutting strategy involves the following steps: the reading of the cutting line geometry and parameters; the projection of the cutting lines over the blank to cut; the regeneration of the FEM mesh in the vicinity of the projected lines, the removal of the finite elements that were cut off; the smoothing of simulation variables at nodal and gauss-point levels. To validate the proposed procedure, four industrial examples are given including 3D geometries discretized with isoparametric hexahedra and BST shell elements. The results show that the whole process of multi-stage forming of thick metal sheets can be successfully simulated employing this cutting strategy. KEYWORDS: Blanking, Trimming, Laser cutting, Metal forming, Thick sheets, FEM.

1

INTRODUCTION

The tasks of cutting in metal sheet forming processes, are essential tasks if wanted to achieve a good final result. The main motivation of the work presented in this article is to perform the whole numerical simulation of a multi-stage forming process that could include one or more intermediate cutting stages. The numerical execution of these cuts can involve a high degree of complexity, especially in the case of thick sheets and when the cuts to be done are laser type. The cutting step is given different names depending on if done before or after the forming: blanking, trimming, piercing, notching, perforating, etc. • Blanking: It is an operation of cutting a flat shape from a sheet metal. The part punched out is called the blank and the remaining sheet is the scrap. In the blanking operation, the blank size is equal to the die size and the clearance is added to the punch. • Piercing: It is a cutting operation by which various shaped holes are made in the sheet metal. In this operation, the hole is the required operation and the material punched out is the scrap. In the piercing operation, the hole size is equal to the punch size and the clearance is added to the die. • Trimming: Trimming operation is done to remove the flash or unwanted excess material from the previously formed components. Special dies called trimming dies are used for this operation. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 406-412 (2011); doi: 10.1063/1.3623638 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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• Perforating: This is an operation of producing number of holes evenly spaced in regular pattern on a sheet metal. The industry makes cuts of different types depending on the geometry of the final shape and the technology available. The cuts are made using a cutting tool moving in a certain direction, or employing a laser cutting machine to perform the trimming of complex geometries. When creating a model for numerical simulation, we can find that the geometry of the blank is discretized with shell or solid elements depending on the thickness of the blank and accuracy of results desired. The main objective of this work is to propose an efficient strategy to perform a cutting operation that could had sufficient flexibility to take into account different typologies: different meshes, different cutting projection types, etc.. The cutting operation is considered as a part of a multi-stage stamping, with several operations before and after the cutting steps. This procedure has been implemented in STAMPACK v625 [3], which is an explicit FEM code, specialized in metal forming processes. An outline of the remaining part of the paper is as follows: in the Section 2, we present and describe the cutting strategy. Later, in Section 3, industrial examples including the cutting operation are shown. These examples cover several types of meshes: solid and shell elements, and two cutting projection types: vector and laser. Finally the conclusions and acknowledgments are shown in Section 4 and Section 5 respectively. 2

CUTTING STRATEGY In the following, we describe the steps required to numerically perform the cutting operation: • Step 1: Projection of the cutting line. The cutting line is projected taking into account the projection type: vector or laser. If vector type, it is used a direction vector that indicates the movement of the cutting tool. And in the case of laser type, the projection is performed in such a way that each point of the cutting line is perpendicular to the surface of the sheet. To prevent that the projection is made on unwanted areas of geometry, it is used an additional parameter that takes into account the depth of the cutting. • Step 2: Identification of the area affected by the cutting. We define a bandwidth, measured in number of elements on either side of the cutting line to define the area over which will be performed the remeshing or smoothing of the FEM mesh. • Step 3: Remeshing of the affected area. This step is performed by moving, adding or removing nodes and elements of the area affected by the cutting. The algorithm can allow or avoid the inclusion of wedge elements. Note that the remeshing is done while maintaining the topology of the mesh in the different layers of elements constituting the sheet thickness. This allow the mesh to retain the condition of semi-structured in the thickness direction, keeping always the same number of layers. Another important aspect to consider is that the boundaries between elements with different materials must be maintained, and at the same time, should be taken into account the nodes with a particular condition applied to them, since these nodes define also a frontier that must be respected by the cutting and can not be moved from their location (eg. the application point of a concentrated load). • Step 4: Deletion of the mesh that was cut off. Once the cutting is performed, we need to determine which elements and nodes are to be removed from the geometry, in terms of the cutting type: ”hole”, ”trim” or ”releaf”. For the ”trim” case, we remove the part of geometry which contains the longest part of the original sheet border, and vice versa for the ”hole” case. In the case of ”releaf”, all parts are left as product of the cutting. 407

Another option to indicate which part of the geometry should be removed, it is to specify a reference node and indicating whether it is ”to keep” or ”to remove”. • Step 5: Interpolation of Gaussian variables (from original to new mesh). This step is very important because its aim is to transfer all information from the original mesh to the new mesh with the least possible loss of accuracy (usually the interpolation/extrapolation methods involves a smoothing effect on the variables). There are many methods to transfer this database, but in this paper we use an interpolation method directly from the original cloud of Gauss points to the new cloud of Gauss points. We believe this direct method involves the least error due to interpolation. • Step 6: Interpolation of nodal variables (from original to new mesh). For the interpolation of nodal values, it is used the same method employed to interpolate Gauss points database, but this time taking into account that we pass the information from the original cloud of nodes to the new cloud of nodes. • Step 7: Modification of the degrees of freedom (DOF) in the new mesh. The conditions and restrictions that are applied to the nodes are of different types: boundary conditions, applied forces or velocities, multi-nodal constraints, etc. The restrictions and conditions applied to the degrees of freedom of the old mesh, must be conveniently transferred to the new mesh in such a way that keep producing the same effect. 3

NUMERICAL SIMULATION OF INDUSTRIAL EXAMPLES

In this section, we present the results obtained using the cutting algorithm. To this end, the following layout was used in Fig.1 to Fig.4 to easy the description of each case: • On top of each figure, it is shown the cutting lines employed to perform each cut. • In all figures, the parts are shown before and after cutting. In turn, we incorporate the plastic strain field and the smoothing obtained during the cutting strategy. • Finally, we show a zoom of these figures, in where we can see a detail of the mesh of the blank, before and after the cutting. In the Fig.1(a), several cutting lines are shown, which are employed to cut an already conformed geometry. The Figs.1(b) and (c) depicts the blank before and after the cutting operation, (see Section 2); It is also shown the field of plastic strains and the interpolated field (of plastic strains) obtained during the cutting process, respectively. Finally, we make a zoom of these FEM meshes (Figs. 1(d) to (e)), to show the detail of the blank, before and after the cut. Now, focusing on the Fig. 2, we have performed two numerical simulations for the car part example, which corresponds to the benchmark 3 of NUMISHEET 2011 [5]. On one hand, we have used shell elements (EBST 3-node triangular shell element (total Lagrangian) [1]). (See Figs. 2 (a) to (e)). And on the other hand, we use solid elements (8-node hexahedral solid 3D element, total Lagrangian [2]). (See Figs. 2 (f) to (i)). The Fig. 2 (f) is the same blank geometry but modeled with hexahedra, before cutting. The plastic strain fields, as well as, the geometries employed (shell and solid) are shown in Figs. 2 (b) and (f). The results of the cutting operation are given in the Figs. 2 (c) and (g), respectively. The projection method employed in this example is laser type. As mention before, the laser cutting is done by normal projection of the cutting line, to the surface of the blank.

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Figure 1: Cutting process of a thick plate

The Fig. 3 shows two different projections of the cutting line that can be used: vector and laser types. For the same cutting line (ellipse type, see Fig. 3 (a)), the cut algorithm described in the Section 2 is capable of performing different types of cuts. The Figs. 3 (c) and (e) show the blanks obtained using different projections (vector and laser, respectively). The Figs. 3 (d) and (f) are the results obtained, in perspective, on the plane xz.

409

Figure 2: Cutting of automotive member

410

Figure 3: Cutting projections: vector and laser types

The Figs. 4 (b), (e) and (g) belong to the same stamping process, but with two different cutting strategies. On one hand, the Figs. 4 (a), (b) and (c) show a circular cutting line, which is used to cut the center of the blank. In turn, the Figs.4 (d), (e) and (f) belongs to a suspension member of a car, but with the mesh already cut. And finally, in the next stage of forming, we use multiple cutting lines. (See Figs. 4 (g) and (h)). 4

CONCLUSIONS

A new cutting strategy was developed in the explicit code STAMPACK v625 [3]. This allows the modeling of a multi-step forming process including complex cutting operations. This procedure has been developed for both shell and solid geometries, what allows the cutting of blank sheets with a considerable thick geometry. The cutting scheme involves three main phases: a) cutting line projection following a specific criterion, b) local remeshing in the neighborhood area, and c) interpolation of internal variables, from the old to the new FEM mesh. The numerically simulated industrial examples show that the performed cuts exhibit adequate mesh quality what allows continuing with the subsequent operations in a multi-stage forming process. It was also validated the versatility of this methodology for both vector and laser type cutting, showing remarkable results in all cases. 411

Figure 4: Member for the suspension of an automobile

5

ACKNOWLEDGEMENTS

The industrial examples of the Figs.1,3 and 4 have been provided by the customers of Quantech. The mechanical response of the industrial example of the Figs.2, corresponding to the benchmark of NUMISHEET 2011 [5], was performed by CIMNE[4]. The authors wish to thank all these support and contribution.

REFERENCES

[1] Fernando G. Flores and Eugenio O˜ nate(2001). A basic thin shell with only traslational DOFs for large strain plasticity. Int. J. Num. Meth. Eng. 51,(57-83). [2] O˜ nate Eugenio. Calculo de estructuras por el m´etodo de los elementos finitos. An´ alisis est´atico lineal. Cimne 2a edici´on. [3] Stampack v6.2.5. A finite element software for simulation of sheet stamping process. Quantech ATZ(2011). Gran Capit´an, 2-4 - Edificio NEXUS 08034 Barcelona, Spain. Tel.: + 34 - 932.047.083 Fax: + 34 - 932.047.256, Email: [email protected]. [4] CIMNE. Centro Internacional de m´etodos num´ericos en ingenier´ıa. Edificio C1, Campus Norte UPC. Gran Capit´an, s/n. 08034, Barcelona, Espa˜ na. [5] NUMISHEET 2011. The 8th International Conference and workshop on numerical simulation of 3D sheet metal forming processes. Seoul Korea, 2011. 412

Recent Innovations in Hyperworks® Stamping Simulation Technology Subir Roya, Lionel Morancayb, Erwan Beauchesneb, Hariharasudhan Palaniswamya a

b

Altair Engineering Inc., 1820 East Big Beaver Road, Troy, MI 48083, USA Altair Development France, 2 Rue de la Renaissance, 92184Antony Cedex, France

Abstract. HyperForm® and RADIOSS® which are components of the Altair® HyperWorks® software suite have been used for stamping simulation for the last fifteen years. Many features specialized for explicit stamping simulation to accurately capture the material behavior and contact phenomena have been developed over the years. Over the recent years several new innovative enhancements have been made to HyperForm and RADIOSS based on industry needs such as, analysis with fine mesh from the start of analysis for accurate spring back prediction without sacrifice of performance obtained with mesh adaptivity; high degree of scalability, reliability, quality and performance of the RADIOSS solver across multiple processors; modeling of thick sheet metal with solid elements; thermal forming for advanced high strength steels; tool stress analysis, and optimization of stamping processes and tool structures. This paper illustrates the benefits of the most recently implemented novel features in HyperForm and RADIOSS using real life examples.

Keywords: HyperForm, RADIOSS, HyperWorks, stamping simulation, hot forming PACS: Replace this text with PACS numbers; choose from this list: http://www.aip.org/pacs/index.html

INTRODUCTION Over the last two decades stamping simulation has radically replaced the need for soft tooling and physical prototyping resulting in significant cost savings and reduction in lead time for stamped products. HyperForm, a highly automated and customized environment for setting up finite element models for stamping simulation and RADIOSS, a high performance general purpose finite element solver are components of the Altair HyperWorks [1] software suite which have been used for stamping simulation with great success for the last fifteen years. Many features specialized for explicit stamping simulation to accurately capture the material behavior and contact phenomena have been developed over the years. As finite element simulation gained large scale acceptance as a validation tool for stamping processes, further demands to enhance the user experience and the simulation technology were encountered in order to improve the performance and scalability of stamping simulation as well as to expand into more advanced forming processes. In the recent years, several innovative enhancements have been made to HyperForm and RADIOSS to address various industrial needs. In the following we discuss the background information on the most significant features that have been implemented recently and illustrate their accuracy, performance and benefits via examples.

IMPROVING PERFORMANCE OF EXPLICIT METHOD Stability of the Explicit Method The discretized equation of motion can be written as

Mu  Cu  Ku

Fext

(1)

where M, C and K correspond to the mass, damping and stiffness matrices, u is the displacement vector and Fext is the external force vector. The mass matrix M results from assembling the elementary mass matrix

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 413-420 (2011); doi: 10.1063/1.3623639 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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³ U  ) !  ) ! dV

Me

t

:

(2)

e

where U is density and ) denotes shape function. Using the central difference scheme which corresponds to Newmark scheme with J=1/2 and E=0 and assuming there is no damping , the current acceleration is calculated at each time step n+1 by solving int int (3) Mun 1 Fnext Kun 1 1  Fn 1 , with Fn 1 The use of a lumped mass matrix avoids solving simultaneous equations for accelerations in (3). It is proved [2] that stability is insured if

't 

2

Z max

, with

Zmax the maximum eigen frequency of the system

(4)

Commercial explicit codes generally provide a way of raising the time step by adding diagonal mass to the nodes. Several drawbacks are inherent to this method such as, non conservation of global translational momentum and modification of the low frequencies of the system.

Adaptive Meshing Adaptive mesh refinement has been implemented in RADIOSS for shell elements, which is a better approach to mapping of state variables than global remeshing. Several refinement criteria are available, taking into account the geometry of contacting tools through the curvature, and the angle between neighbouring elements. A nonconforming adaptive mesh shown in Fig. 1(a) is maintained by imposing a kinematical constraint at non conforming nodes. The amount of penetration as shown in Fig. 1(b) is considered as a criterion for refinement which avoids the need for backtracking as if contact had been detected too late. The maximum number of refinement levels is input by the user, so that the full structure of the mesh is built from the start time with tracking of contact of all nodes.

FIGURE 1 (a) A non-conforming adaptive mesh (b) Penetration amount as a criterion for mesh adaptivity

Dynamic Condensation Dynamic condensation, a technique for mesh condensation to improve performance, is available in RADIOSS through adaptive meshing and its related data organization as discussed in the following. Let the reduced set of nodes of the condensed mesh be (m) and the nodes coming from the complete mesh be (n). Lumped mass M m at the reduced set of nodes is computed by considering the coarse mesh. Forces computed at condensed nodes are mapped to nodes (m) using the shape functions motion at the reduced set of nodes becomes

M m um

Im

of the coarse mesh so that the equation of

Fm  ¦ ) m ( n ) Fn

(5)

n

Forces computed at the condensed nodes are still considered as inertia forces such that

M n un  um Fn , with um

¦) m

414

m

(n)um

(6)

The lumped mass M n at condensed nodes is set big enough, so that the time step can be computed by only considering the nodes of the coarse mesh. M n is not an absolute mass but it is only relative to the mean

m . Therefore, increasing M n does not affect global momentum. displacement u

Advanced Mass Scaling Olovsson et al. [3] proposed to add additional terms to the lumped mass matrix, aiming at decreasing the highest frequencies of the system, while affecting the lower frequencies as little as possible. In RADIOSS, we consider a modified mass matrix M' M  / (7) where M is the lumped mass matrix and / is assembled from constant elementary matrices such that the rigid translational acceleration of an element is not modified.

u

M ' 1 F

(8) Solution of equation (8) is approximated using a conjugate gradient iterative resolution method within a tolerance. Contrary to dynamic condensation, matrix inversion is needed with advanced mass scaling. However, it is still able to work with a very fine mesh without sacrificing performance.

Validation Studies To study the effect of the different speed-up options in RADIOSS, the NUMISHEET2008 S-rail model shown in Fig. 2 is considered. To establish the baseline results, a starting blank mesh which is fine enough to capture the smallest feature on the formed part is used. RADIOSS results are correlated with experimental data for blank drawin at specified locations along the boundary of the formed part.

(a)

(b)

FIGURE 2 (a) S-rail for NUMISHEET2008 benchmark (b) Specified cross-sections for draw-in comparison

FIGURE 3. Comparison of deviation of draw-in between 4 different commercial stamping simulation software and experimental data. The leftmost bar plots correspond to RADIOSS results.

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Once the baseline is determined with acceptable accuracy as shown in Fig. 3, different options in RADIOSS to improve performance such as mesh adaptivity, dynamic condensation, and advanced mass scaling are turned on. Commonly used results quantities such as distribution of thinning, effective plastic strain and stress tensor are compared to determine the limits of the speed-up parameters before the results start to degrade with respect to the baseline. Table 1 shows a comparison of the CPU time for the different speed-up options without loss of accuracy. TABLE 1. Performance gains resulting from difference speed-up options in RADIOSS Simulation Time (s)

Option Baseline (fine mesh from start) Mesh adaptivity Dynamic condensation Advanced mass scaling

5493 599 157 119

Speed-up (with respect to baseline) 1 9.2 35 46

THERMAL FORMING Hill orthotropic material for temperature dependent yield stress is available in RADIOSS for both shells and solids. In addition to temperature, the yield stress is also dependent on plastic strain and strain rate which are input as tabulated data. Thermal conduction through the contact interfaces is considered by setting a thermal interface between the sheet metal and tooling and by providing the heat transfer coefficient. Thermal expansion coefficient is assumed to be constant. Convection with air is modelled using heat transfer coefficient and radiation is modelled via Stefan-Boltzmann constant and emissivity. To validate the thermal forming implementation in RADIOSS, the NUMISHEET2008 benchmark on continuous press hardening of a hot formed automotive B-pillar is studied. The setup is a single action draw with a die moving down on a stationary punch as shown in Fig. 4(a). At the beginning of the die movement, the temperature of the blank is 810oC and the tooling is 75oC. All the material data and thermal coefficients as provided by NUMISHEET2008 benchmark committee are used. The results for temperature distribution are shown in Fig. 4(b). The comparison of simulation results with experimental data for thickness along specified sections are shown in Fig. 5 and Table 2.

(a)

(b)

FIGURE 4 (a) NUMISHEET2008 automotive B-pillar benchmark model for hot forming simulation (b) RADIOSS results for temperature and thickness distribution

416

(a)

(b)

FIGURE 5 (a) NUMISHEET2008 specified cross-sections for thickness comparison (b) Measured and experimental thickness distribution along Section 1b TABLE 2. Number of points on each cross-section with a thickness deviation more than 5% from experimental data. Results from 5 other participating software are shown as A-E. Software A B C D E RADIOSS

Section 1a 3 4 4 2 1 2

Section 1b 2 -

Section 2a 3 1 3 3 2 1

Section 2b 1 1 1

Section 3a 1 -

Section 3b 1 -

As a second example to validate thermal forming simulation in RADIOSS, the NUMISHEET2011 benchmark for deep drawing of a cross-shaped cup is studied. The process consists of modelling first the heat transfer between a heated blank made from a magnesium alloy and the hardened tool steel. Once the blank reaches a steady state it undergoes a thermo-mechanical deep drawing. The input conditions have been specified by NUMISHETT2011. The results of simulation are shown in Figs. 6 and 7.

(a)

(b)

FIGURE 6 (a) NUMISHEET2011 hot forming simulation benchmark model (b) RADIOSS results for steady state temperature distribution

417

(a)

(b)

FIGURE 7 (a) RADIOSS results for temperature distribution after forming (b) RADIOSS results for per cent thinning distribution after forming

For thicker materials especially to capture possible ironing phenomena during thermal forming, solid elements are available in RADIOSS. In addition to the thermal Hill orthotropic material, to account for martensite formation in TRansformation Induced Plasticity (TRIP) steels two specialized material models have been implemented in RADIOSS. The first one is an empirical model developed by Ugine & Alz (now under the Arcelor Group) to describe the martensite volume fraction. The second one is a non-isothermal material model from Hänsel et al. [4] and modified by Avendaño et al. [5]. These models have been validated in an earlier study by Morancay et al. [6].

YOSHIDA UEMORI MATERIAL LAW To capture phenomena resulting from cyclic plasticity such as Bauschinger effect and work hardening stagnation, the Yoshida-Uemori material law [7] has been implemented in RADIOSS. This law is based on a yield surface which moves within a bounding surface during plastic deformation. The yield surface never changes its size whereas the bounding surface can change both size and location. The plastic strain dependency of the Young’s modulus is also taken into account. To validate the implementation of the Yoshida-Uemori law in RADIOSS, the NUMISHEET2011 benchmark regarding the effect of pre-strain on 2D draw-bending is considered. The material parameters for DP780 were derived iteratively by correlating the supplied experimental data with output from RADIOSS simulation of a uniaxial tensile-compression-tensile test as shown in Fig. 8.

FIGURE 8. Fitting of RADIOSS results to uni-axial tensile-compression-tensile test data with 8% and 10% pre-strains to derive Yoshida-Uemori material parameters

The simulation of the 2D draw-bending deformation is made up of 3 stages, first a tensile stretch to create a prescribed amount of pre-strain, followed by draw-bending and finally springback. Displacements resulting from springback as predicted by RADIOSS without pre-strain and with a pre-strain of 8% are shown in Fig. 9.

418

(a)

(b)

FIGURE 9. Springback displacements after draw-bending deformation with (a) no pre-strain (b) 8% pre-strain

ENHANCING USER EXPERIENCE FOR MODEL SETUP So far this study has focussed on the new developments pertaining to stamping simulation solver technology. However, as the solver technology advances, it becomes more challenging to keep the user experience for setting up the finite element models simple and intuitive so that all new features can be conveniently accessed and fully utilized. It has to be kept in mind, most often the direct beneficiaries of stamping simulation are the tooling and process designers who are often involved with day-to-day real world manufacturing issues rather than the nuances in numerical simulation. In HyperForm, the modelling environment for stamping simulation, a strong emphasis is put on process automation to create the most efficient workflow with minimal mouse-clicks to set up conventional stamping processes as shown in Fig. 10(a). On the other hand, a step-by-step approach is used to capture expert knowledge to set up non-conventional stamping process setups. During the setup, all the steps are recorded and can be saved under a user template which can be later re-used for a similar setup without having to go through the steps anymore. Aside from single stage model setup, significant enhancement has been made to simplify the setup of complex multi-stage forming processes via an object oriented and browser driven approach that assembles the different forming stages as shown in Fig. 10(b) and automatically transfers results from one stage to the next without user-intervention.

(a)

(b)

Figure 10 (a) Automatic setup of a conventional single action draw process (b) Object oriented and browser driven setup for multi-stage forming as an assembly of single stage processes

419

CONCLUSION In this study recent innovations in HyperWorks stamping simulation technology are discussed. Several options to improve the performance of the explicit solver RADIOSS without loss of accuracy are presented. The simulation results are validated against experimental data wherever available. Most examples used for validation studies are taken from NUMISHEET benchmark problems. Process automation driven setup of conventional stamping processes and an innovative approach to capture expert knowledge to standardize model setup for non-conventional stamping processes in HyperForm is discussed. A unique object oriented and browser approach to set up complex multi-stage forming processes is also presented.

REFERENCES 1. 2. 3. 4.

5.

6.

7.

HyperWorks 10.0 Help Manual, 2009, Altair Engineering. T.J.R. Hughes, The finite element method: linear static and dynamic finite element analysis, Prentice Hall Engelwood Cliffs New Jersey 0763, 1987. L. Olovsson, K. Simonsson, “Iterative Solution Technique in Selective Mass scaling” in Communications in Numerical Methods in Engineering, 2006, Vol. 22, pp.77-82. A.H.C. Hänsel, P. Hora, J. Reissner, “Model for the Kinetics of Strain-induced Martensitic Phase Transformation at Nonisothermal Conditions for the Simulation of Sheet Metal Forming Processes with Metastable Austenitic Stainless Steels” in Sim.Mat.Proc: Theory, Methods and Appl., ISBN 90 5410970 X, edited by Huétink and Baaijens, Rotterdam, 1998, pp.373378. J.L. Avendaño, J.I. Fernández, E. Schedin, J. Kajberg, L. Staubwasser, “Implementation of a Non-isothermal Material Model for Austenitic Stainless Steels in Deep Drawing Simulation” in Proc. of 8th International Conference on Technology of Plasticity, Verona, 2005. L. Morancay, G. Rios, S. Mamouri, E. Beauchesne, S. Roy, “Recent Advances in Hot Sheet Metal Forming using RADIOSS Explicit Finite Element Code” in Proceedings of 2nd International Conference on Hot Sheet Metal Forming of HighPerformance Steel, Lulea, 2009. F. Yoshida, T. Uemori, “A model of large-strain cyclic plasticity describing the Bauschinger effect and workhardening stagnation” in International Journal of Plasticity, 2002, Vol. 18, pp. 661–686.

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A Fully Automated and Robust Method to Incorporate Stamping Data in Crash, NVH and Durability Analysis Hariharasudhan Palaniswamya, Narayan Kanthadaib, Subir Roya, Erwan Beauchesnec a Altair Engineering Inc.1820 E Big Beaver Road, Troy, MI 48083, USA Altair Engineering India Pvt. Ltd, Mercury 2B Block, 5th Floor,Prestige Tech Park, Sarjapur Marathhalli Outer Ring Road, Bangalore 560103, India c Altair Development France, 2 Rue de la Renaissance, 92184Anthony Cedex, France

b

Abstract. Crash, NVH (Noise, Vibration, Harshness), and durability analysis are commonly deployed in structural CAE analysis for mechanical design of components especially in the automotive industry. Components manufactured by stamping constitute a major portion of the automotive structure. In CAE analysis they are modeled at a nominal state with uniform thickness and no residual stresses and strains. However, in reality the stamped components have non-uniformly distributed thickness and residual stresses and strains resulting from stamping. It is essential to consider the stamping information in CAE analysis to accurately model the behavior of the sheet metal structures under different loading conditions. Especially with the current emphasis on weight reduction by replacing conventional steels with aluminum and advanced high strength steels it is imperative to avoid over design. Considering this growing need in industry, a highly automated and robust method has been integrated within Altair Hyperworks® to initialize sheet metal components in CAE models with stamping data. This paper demonstrates this new feature and the influence of stamping data for a full car frontal crash analysis. Keywords: RADIOSS, HyperForm, HyperMesh, HyperWorks, stamping, crash, results initializer PACS: Replace this text with PACS numbers; choose from this list: http://www.aip.org/pacs/index.html

INTRODUCTION Crash, NVH (Noise, Vibration, Harshness), and durability analysis are commonly deployed in structural CAE analysis for mechanical design of components especially in the automotive industry. In automotive, sheet metal components constitute a majority of its structure. In CAE analysis the sheet metal components are modeled at a nominal state with uniform thickness and no residual stresses and strains. However in reality these components are manufactured by sheet metal forming process that introduces non uniform strain hardening and change in its thickness. The amount strain hardening and thickness change depends on the part geometry and its material properties. This strain hardening significantly increases the yield strength of the material especially for advanced high strength (AHSS) steels that are being used increasingly in automotive structures to reduce weight. Effect of the formability results on the crashworthiness of the automotive structure has been widely studied [1-7] by several automotive OEMs and suppliers. It was observed that including formability results into crash simulation improved the correlations with the experiments. Several methods to include the crash result have also been presented [5]. Most of these methods suggest using the incremental stamping simulation results by mapping them on to the crash models. Incremental stamping results for individual sheet metal parts are usually supplied by process development engineers. Sheet metal forming process development by itself is a design process that involves several iterations of draw development by die engineering based on the part geometry to establish a process sequence using incremental forming simulation for each of the component. This is significantly time consuming exercise considering it to be repeated over several parts. Usually these results are available at latter stages of the product development cycle. This limits the use of stamping history from incremental stamping simulation for CAE analysis The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 421-428 (2011); doi: 10.1063/1.3623640 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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for crash worthiness, NVH and durability during earlier stages of the product design cycle. Also a significant effort is needed for mapping various formats and management of these data, as the manufacturing of automotive body structure gets distributed to several suppliers and each supplier uses their own preferred stamping simulation software. An alternative to incremental stamping simulation, one step stamping simulation is commonly used by product design engineers to check for the manufacturing feasibility of the designed part. Contrary to incremental solution, one-step solutions are quick that can be setup easily by the product design engineers and do not require draw die development. Also the data transfer to CAE models is much simpler without any need for mapping and can be handled locally on product designers’ desktop. In this study, a fully automated and robust method to incorporate stamping data into crash, NVH and durability analysis inside Altair’s CAE pre-processing tool HyperMesh® using industry proven and robust one step solver RADIOSS® is discussed. The developed process is solver neutral thus can be used for CAE models built for commonly used crash solvers LSDYNA, RADIOSS and PAMCRASH. In this paper, the developed tool is used to add stamping history on a full vehicle model for frontal crash and side crash, respectively. The crash worthiness results with and without adding stamping results from RADIOSS one step solver are discussed. Also ease of use of the developed tool is presented.

RADIOSS RADIOSS is a multi-physics solver from Altair Hyperworks software with good accuracy, scalability and repeatability that has been widely used for CAE analysis in the industry for the last fifteen years [8]. One step stamping simulation is another capability of the RADIOSS solver which is seamlessly interfaced with HyperForm®, a highly automated environment for modeling of one step and incremental sheet metal stamping simulation. HyperForm has been widely used in the industry with RADIOSS one step solver for analyzing the stamping feasibility of stamped parts, blank size estimation, and blank nesting etc. RADIOSS one step solver works on the basis of geometric mapping and minimum energy principle to flatten the sheet metal part to a flat sheet. The strains, stress and the thickness change are estimated based on the material properties from this one step non linear analysis. The part geometry, thickness and material properties are the input to the one step analysis. It should be noted that, contrary to incremental analysis in the one step analysis, it is not necessary to have a draw die development in order to consider the effect of addendum and its influence on material flow. RADIOSS one step solver, as an alternate allows user the option to provide edge force on the part free edges to mimic the addendum and control the material flow. Also additional options of defining friction, the effect of curvature (bending and unbending) are available. These additional options of RADIOSS one step solver have taken one step results closer to the incremental results. Judicious combination of these additional options of RADIOSS one step solver of have helped users over years to take the one step results closer to Incremental stamping. An example of structural part (Numisheet 2005 benchmark channel part), comparing the RADIOSS one step and incremental forming results are shown in Fig. 1. It is observed that the plastic strain distribution is close between the incremental and one-step results with some additional localized peak strain at the edges for the one-step results because of the absence of a developed addendum.

a) Incremental stamping simulation b) One-step stamping simulation FIGURE 1. Comparison of effective plastic strain predicted by incremental and one step simulation for Numisheet 2005 channel part.

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HYPERMESH RESULTS INITIALIZER HyperMesh is a general purpose pre-processing environment for CAE model building under Altair HyperWorks. HyperMesh interfaces with several commonly used FEA solvers including RADIOSS, LSDYNA, PAMCRASH, ABAQUS, NASTRAN etc. Results initailizer is a new tool developed under HyperMesh version 11.0 environment to seamlessly integrate stamping results from RADIOSS one step solver to structural CAE models with the user remaining in one familiar modeling environment. The process flow for the stamping results initialization process to CAE models in HyperMesh is shown in Fig. 2. Rapid developments in hardware technology and parallelization of FE codes have significantly reduced the computation time over last decade. This has allowed CAE engineers to incorporate as much as possible the vehicle components from CAD into the vehicle CAE model with fine mesh to capture the smallest features in these components for better correlation with experiments. In this development, the mesh of sheet metal structure in the crash, NVH or durability CAE model is used directly for the one step analysis rather than with mesh additional mesh refinement. The results from one step analysis are directly initialized one to one on sheet metal components in the CAE model rather than mapping thus significantly reducing the results initializing time and data management. The developed tool guides the user through the series of steps in the process with an adjacent browser as shown in Fig. 3. Each step is explained in the following section.

FIGURE 2. Schematic illustrating the process involved in results initialization using HyperMesh.

FIGURE 3. Results initialization user interface for parts selection in HyperMesh

Step 1: Selection of Parts and Model Preparation In first step the user is prompted to select parts for initializing with stamping results. This can be accomplished by selecting parts from the Graphical user interface (GUI) of HyperMesh. Once the parts are selected, the user has the choice to fill the holes in the parts. An example application of hole filling feature implemented is shown Fig. 4. In stamping process the parts are formed without holes and the holes are pierced in successive stages. Similar process is emulated with this option thereby one step analysis is performed on the part with holes filled and then the results

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are initialized and then the holes are removed to get back the original CAE mesh topology. Internal studies indicated that presence of holes in the part results in concentration of stress and strain near the hole. Hence, hole filling before one step analysis is recommended.

FIGURE 4. Automatic hole filling option in the HyperMesh results initializer tool.

Step 2: Model Setup The material and sheet thickness information are extracted from the CAE model and converted to equivalent material for one step forming analysis. The user has the option to set the blank holder force level. Also additional options to optimize the blank holder force based on plastic strain and FLC are available.

Step 3: Initialization process Once the user chooses to initialize the selected parts, these selected parts will be automatically checked for excessive undercuts before submitting to the one step solver without user interaction. After the one step analysis is completed, the results are automatically read into HyperMesh for visualization. Additionally, the ascii files that contain the results in the respective solver format are attached as include files to the CAE model in HyperMesh.

Step 4: Results review The implemented tool allows users to review all the stamping relevant results such as thinning, plastic strain, forming limit diagram (FLD), equivalent and principal stresses for the selected parts as shown in Fig. 5. Based on the results, the implemented tool allows user the freedom to switch back to any previous steps and repeat them with different blank holder force or with new parts.

Step 5: Cutoff This is an optional step. In some parts, one step simulation may some times predict higher strains than in reality. This is due to the absence of real tool with addendum or the presence of certain unfilled areas. These unrealistic strains can be saturated or filtered to a user defined upper limit using this step. Accordingly thickness change is also modified. As described this tool integrates preprocessing, solving and post processing of one step stamping analysis in one environment followed by management of the stamping data for the CAE model with minimum user interaction. Thereby it enables the CAE engineer to remain focused on the full car crash model building and analyses rather than spending significantly large amount of time and effort in setting up manually the one step analysis and managing the data transfer between the one step and crash models.

FIGURE 5. Visualization of one step results

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Considering the need that this initialization may need to be done for several parts in a full car model outside the HyperMesh environment, a batch process has also been developed that follows the same process with no user interaction. As a minimum, the user is required to provide CAE model, the list of the components and blank holder force for each of them.

APPLICATION The developed results initializer in HyperMesh is used to initialize sheet metal components in full car crash CAE model for frontal and side impact. Frontal crash and side impact CAE analyses were conducted with and without initializing the sheet metal components with stamping results. Crash worthiness results were compared for both frontal and side crash in this study to illustrate the significance of including stamping results and the need for the developed tool.

Case study 1: Frontal crash Public domain ford Taurus model [9] available from National Crash Analysis Center is used in the Frontal crash NCAP study. About 370 sheet metal parts in the full car model as shown in the Fig. 6 are selected for initialization with stamping results. The developed results initialization tool discussed earlier is used to initialize the selected components with plastic strain and thickness distribution resulting from stamping. Fig. 6 also shows the initial plastic strain in the components due to addition of the stamping history. Frontal crash analysis was conducted on the Ford Taurus model with and without the stamping history to study the effect of including the stamping results on the crash worthiness of the model. The deformed shape of the engine compartment structure which takes the predominant energy from the frontal crash due to its deformation is shown in Fig. 7. Additional longitudinal cross section view along the driver seat is provided in Fig. 8 highlighting the difference in the deformed shape of the front passenger and engine compartment between the two CAE models. It could be observed for this Ford Taurus model, the addition of stamping results didn’t change the deformation mode during the frontal crash. However, appreciable difference in the amount of deformation of the structure that takes majority of crash energy was observed. Less deformation was observed in the model initialized with stamping history compared to the model without stamping history. This indicates better crash worthiness for the CAE model initialized with stamping results compared to without stamping results. Components initialized with stamping strains have higher yield strength due to the prestrains and the strain hardening behavior of the sheet material thus requiring more energy to plastically deform them compared to the component without stamping strains. This means the CAE model initialized with stamping strain shows stiffer response compared to the model without.

a)

Ford Taurus CAE Model

b) Sheet metal parts for initialization

c) Initial plastic strain distribution

FIGURE 6. Sheet metal parts in Ford Taurus frontal crash CAE model selected for initializing with stamping history and distribution of initial plastic strain from stamping results initialization in the selected parts.

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a)

CAE model without results initialization

b) CAE model with stamping results initialization

FIGURE 7. Deformed shape comparison of the loading structures in front engine and passenger compartment after full frontal crash with and without considering stamping history in the CAE model

FIGURE 8. Cross section view comparing deformed shape of the front passenger and engine compartment after full frontal crash with and without considering stamping history in the CAE model

Case study 2: Side impact A side impact crash analysis model’s sheet metals parts are initialized with stamping history for this study to analyze the effect of pre-strains on the crash worthiness of the model as shown in Fig. 9. The developed tools are used to initialize approximately 470 parts in this side impact crash analysis model. Fig. 9 also shows the initial plastic strain in the components due to addition of the stamping history. The deformed shape of the B-pillar which takes the predominant energy from the side crash due to its deformation is shown in Fig. 10. Additional longitudinal cross section view along the driver seat is provided in Fig. 11 highlighting the difference in the deformed shapes between the two CAE models. Some differences in the deformation mode could be observed during the side crash between the two CAE also models. Also appreciably less deformation is observed in the model initialized with stamping results compared to the one without. As explained earlier, these differences attributed to higher energy required to

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plastically deform them compared to the component without stamping strains. Thus the CAE model initialized with stamping strain shows stiffer response compared to the model without initializing with stamping strains.

a) Side crash analysis CAE Model

b) Sheet metal parts for initialization c) Initial plastic strain distribution

FIGURE 9. Sheet metal parts in side impact CAE model selected for initializing with stamping history and distribution of initial plastic strain from stamping results initialization in the selected parts.

a)

CAE model without results initialization

b) CAE model with stamping results initialization

FIGURE 10. Deformed shape comparison of the loading structure B-pillar with and without considering stamping history in side impact CAE model

FIGURE 11. Cross section view comparing deformed shape of the front passenger and engine compartment after side impact with and without considering stamping history in side impact CAE model

CONCLUSION In this study, a fully automated and robust tool for initializing sheet metal parts in CAE models with stamping history using RADIOSS one step solver and HyperMesh has been presented. This tool integrates and automates preprocessing, solving, post processing of one step stamping analysis, and data management in one user environment thereby enabling the CAE engineer to remain focused on the CAE model building and analyses rather than on stamping. The developed tool is used to study the effect of including stamping history on the crash worthiness of frontal crash and side impact of full vehicle model. In both frontal and side crash model considered in this study, inclusion of stamping results did not significantly change the overall deformation modes. However noticeable difference was observed in the deformation of load bearing structures. In both analyses, CAE model with stamping results showed a stiffer response and better crash worthiness as pre-strained components absorb more energy to

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plastically deform them compared to the components without stamping strains. The study illustrates the need for considering stamping history in crash CAE models.

REFERENCES 1. K.Takashina, K. Ueda, T. Ohtsuka, “Investigation of accuracy improvement in crashworthiness simulation with Pre-simulation of Metal Forming”, Proceedings of 7th European LS-DYNA Conference, 2009, www.dynalook.com 2. K. Wolf, R. Schilling, J. Lutjens, M. Hunkel, T. Waalmersperger, Jankowski, D. Sihling, K.Wiegand, M.Heuse, “Coupled FEM calculations – A CAE Tool to Improve Crash relevant Automotive Body Components by Local Hardening”, Proceedings of 7th European LS-DYNA Conference, 2009, www.dynalook.com. 3. C.S. Bottcher, S.Frick, “Consideration of Manufacturing Effects to Improve Crash Simulation Accuracy”, Proceedings of 4th European LS-DYNA Conference, 2006, www.dynalook.com. 4. H.Kim, S.Hong, S.Hong, H.Huh, “The Evaluation of Crashworthiness of Vehicles with forming effect”, Proceedings of 4th European LS-DYNA Conference, 2006, www.dynalook.com. 5. J. Cafolla, R.W.Hall, D.P.Norman, I.J. McGregor, “Forming to Crash” Simulation in Full Vehicle Models”, Proceedings of 4th European LS-DYNA Conference, 2006, www.dynalook.com. 6. N. Vallino, “Coupling between Stamping results and Crash simulation”, Proceeding of European HyperWorks Technology Conference, 2010 7. T. Holene, “One step forming simulation with HyperForm” Proceeding of European HyperWorks Technology Conference, 2009 8. L.Morancy, G.Rios, S.Mamouri, E. Beauchesne, S. Roy, “Recent Advances in Hot Sheet Metal Forming using RADIOSS Explicit Finite Element Code”, Proceedings of 2nd International Conference on Hot Sheet Metal Forming of High Performance Steel, Lulea, 2009 9. http://www.ncac.gwu.edu/vml/models.html

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$SSOLFDWLRQRI7KLFNQHVV*UDGLHQW%DVHG1HFNLQJ&ULWHULRQWR ,QYHVWLJDWHWKH,QIOXHQFHRI:HOG&RQILJXUDWLRQRQWKH)RUPLQJ /LPLWRI7DLORU:HOGHG%ODQNV R. Ganesh Narayanana, K. Narasimhanb a

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Department of Mechanical Engineering, IIT Guwahati, Guwahati 781 039, India Department of Metallurgical Engineering and Materials Science, IIT Bombay, Powai, Mumbai 400 076, India

$EVWUDFW The application of thickness gradient based necking criterion to understand the effect of weld conditions on the forming limit of TWB is explored in this paper. The weld conditions considered for the analysis are weld-base material property combination, weld orientation and weld location. Limiting Dome Height (LDH) test is simulated using PAM STAMP 2G, an elastic-plastic finite element code and limit strains of un-welded, welded blanks are predicted using thickness gradient based necking criterion. Transverse weld has insignificant effect on the limit strains when it is placed at the centre of the sheet. In the case of transverse weld placed at critical locations of the sheet, decrease in limit strains is observed. In other cases, weld properties show negligible influence on the forming limit strains of transverse welded blanks. Longitudinal weld has significant effect on the limit strains of TWB depending on the weld properties and strain path. However, the overall FLC of longitudinal welded blank is by and large similar to that of unwelded blank. Some of the simulation results are found to be consistent with the available experimental observations qualitatively. .H\ZRUGV Tailor welded blanks, Necking criterion, Limit strain, Sheet forming 3$&6 81.20.Hy

,1752'8&7,21 The forming behavior of TWB is affected by thickness of sheets welded, strength of the sheets welded, weld orientation, and weld location. The important properties of the weld, say strain hardening exponent and yield strength, is expected to have significant effect on the forming behavior of TWB in comparison to that of un-welded sheet depending on the weld orientation and location. Most of the existing studies performed in TWB details the effect of thickness ratio, weld orientation, weld location etc. on the forming behavior of TWB [1-4]. The influence of TWB parameters is actually the compounding effect of those parameters on the forming behavior. For example, the impact of thickness ratio on the forming limit strain is the synergistic effect of thickness difference of the blanks welded and the weld conditions (like weld region, weld location, orientation etc.). This holds good for any other parameter involved with respect to welded blanks. The individual effect or sensitivity of parameters, mainly weld conditions, on the TWB forming behavior is not separately addressed in the literature. Hence, it is worth to study the influence of just the weld conditions on the TWB forming behavior, say forming limit strains. In recent past, there is a trend in predicting the forming limit of welded blanks by using different limit strain theories. For example, Ganesh and Narasimhan [5] predicted the forming limit of TWBs made of IF steel using thickness gradient based criterion and validated the results with experimental observation. The predicted results are consistent with the experimental FLCs. Similarly Jie et al. [6] predicted the forming limit of TWBs made of 5754-O material using a localized necking criterion and the predictions are coinciding well with the experimental FLCs. It is found from the literature that application of necking criteria in understanding the influence of varied weld properties or conditions on the forming behavior is not explored much. In this work, the application of thickness gradient based necking criterion to analyze the effect of just the weld conditions viz., weld tensile properties, weld orientation, weld location, on the forming limit strains (or FLC) during

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 431-438 (2011); doi: 10.1063/1.3623641 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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TWB forming is investigated. Limit Dome Height (LDH) test is simulated to achieve this. In order to avoid the effect of other parameters and to study just the effect of weld conditions, blanks of same thickness and quality are considered for the present analyses.

0(7+2'2/2*< 0RGHOLQJ6LPXODWLRQRI/'+WHVWLQJRI7:% The LDH test simulation was performed using PAM STAMP 2G, a commercial FE code. The methodology of constructing the tools and welded blanks required for LDH test simulation is described elsewhere [7]. In FE simulations, the weld region is modeled as a single entity without heat affected zone. The base material and weld zone model were comprised of quadrilateral shell elements of the Belytschko–Tsay formulation, with five through-thickness integration points. A uniform mesh size of 1 mm was used in the weld region and base metal as this has been reported to validly predict the forming limit of TWBs acceptably [5]. From the previous study, it was found that the mesh size should be 2-5 mm to follow this FLC criterion, which is equivalent of what is followed in experimental determination of FLC. After remeshing during forming simulation, thickness average of few elements was considered to apply the forming limit criterion. Hill’s 1948 isotropic hardening yield criterion was used as the plasticity model. The standard eight different strain paths (25×200 mm to 100×200 mm to 200×200 mm) were simulated to predict the Forming Limit Curve (FLC) of un-welded blank in LDH test. In the case of welded blanks, only three different strain paths, viz., tensile (25×200 mm), near plane strain (100×200 mm) and bi-axial stretching (200×200 mm) conditions were simulated to obtain the forming limit strains. Limit strain was obtained using a novel thickness gradient based necking criterion which is presented in a later section. Friction coefficient ‘µ’ was assumed to be 0.12, and was kept constant through out the study. An optimal (BHF) (~ 100 kN) was chosen such that the blank neither draws-in nor tears near drawbeads during forming simulation.

:HOG3URSHUWLHV/RFDWLRQDQG2ULHQWDWLRQ The base metal properties considered for the analysis represent a typical Steel grade material and are given in Table 1. The values of ‘n’ and ‘K’ were obtained from the literature [8]. Weld conditions, viz., (a) Weld orientation, (b) Combination of mechanical properties of weld and base metal, and (c) Weld location were considered for studying the effect on limit strains developed during TWB forming. Longitudinal orientation and transverse orientation are considered for the analyses. The combination of mechanical properties of weld and base material is quantified by (a) ratio of strain hardening exponent of the weld (nw) to base material (nb) i.e., nr = nw / nb and (b) ratio of yield strength of the weld (YSw) to base material (YSb) i.e., YSr = YSw / YSb. The values of these parameters were chosen such that YSr > 1, failure should not occur in the weld zone and this is the case with most of the steel TWBs. The ‘nr’ and ‘YSr’ values considered were: nr = 0.56, 0.83, 1.1, 1.3, 1.7 and YSr = 3, 4, 5. All other properties of the weld zone were assumed to be same as that of base metal. Initially, weld region was placed at the blank centre (for both longitudinal and transverse weld orientations) and its width was kept constant as 5 mm throughout the analysis. The limit strains of TWB will be compared to that of un-welded sheet. In the case of transverse weld orientation, the effect of weld location was also studied by placing the weld at some critical offset locations from the centre. For each weld location, weld properties (nw, YSw) were changed to investigate their influence. The FLCs of un-welded and welded sheets (with changing weld locations) were compared. 7$%/(Base material properties assumed for the simulations [8]

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)DLOXUH/LPLW6WUDLQ&ULWHULRQ A novel thickness gradient-based necking criterion was used to obtain FLC. This criterion basically perceives the localized necking as the critical, local thickness gradient developed in the sheet [9]. A local critical thickness gradient (Rc) must exist at the onset of a local visible neck. In the simulations, the thickness gradient developing in the adjoining elements was monitored and when this gradient (Ractual) drops below Rc, then it was concluded that the local neck has initiated in the element where thickness is lower. The major and minor strain in the adjoining thicker element was then reported as the forming limit strain of that particular deformation condition. It was found from the earlier experiments that necking occurs in the sheet metal when the thickness gradient (ratio of thickness of neighboring elements or circles) falls below 0.92 [10]. Hence in the present FE simulations performed, element pairs where the thickness ratio equals or falls below 0.92 were considered as necked elements. It was observed that the necking criterion is satisfied between few element pairs at the same punch travel. Hence the major strains in all the thicker elements of such element pairs were recorded. The largest major strain and the corresponding minor strain of such elements were treated as the forming limit strain. This procedure of obtaining the limit strain was followed for both un-welded and welded blanks.

5(68/76$1'',6&866,21 /LPLW6WUDLQVRI7:%LQ7UDQVYHUVH:HOG2ULHQWDWLRQ Figure 1 shows the predicted Forming Limit Curve (FLC) of un-welded sheet and TWB with transverse and longitudinal weld orientations, when weld is placed at the centre. The limit strains of the TWB for the different strain paths, for the various weld conditions (properties, orientations) are shown within ellipses/circles. Limit strains of TWB are shown for three strains paths – tensile, near plane strain and bi-axial stretching, only. The TWB limit strains are grouped as ellipses within which limit strains vary with the ‘nr’ and ‘YSr’ values. It is observed that, in the case of transverse weld orientation, weld has insignificant effect on the limit strains developed in TWB. All the three strain paths – tensile, near plane strain and bi-axial stretching conditions of TWB have limit strains similar to that of un-welded sheet. The possible reasons behind this behavior are: (i) less severe ‘YSr’ and ‘nr’ values, (ii) smaller weld width, and (iii) weld located in non-critical region. Since weld exhibits higher yield strength in comparison with the base metal (more than 3 times that of the base metal), it is conceivable that the weld is not contributing to the deformation. Also failure occurs in the base metal, weld being at centre of the sheet, and the failure location is near punch nose region as in LDH testing of un-welded sheet. This failure location is at 30-40 mm from the weld zone. Since weld is placed at the noncritical location in TWB, the effect is insignificant. In order to analyze further, the weld width and weld-base metal property combinations are chosen as before and the weld was placed in the critical regions or severely strained regions (near punch nose region), i.e., at some offset from the centre. Here critical regions are considered as regions of the sheet where peak strains and strong thickness gradient develop. Tensile, near plane strain and bi-axial stretching conditions are studied in this case. Once again the weld zone properties are changed as described earlier and the limit strains are predicted as before. In all the three strain paths, weld is placed at two different critical locations from the centre of the blank. In the case of tensile strain path, the weld is placed at 20 mm and 35 mm from the centre of the blank. In the case of near plane strain condition, the weld is located at 10 mm and 25 mm from the centre. In bi-axial stretching mode, the weld is located at 15 mm and 30 mm from the centre of the blank. Figure 2 shows the effect of weld when it is placed at severely strained regions for tensile, near plane strain and bi-axial stretching conditions. In the case of near plane strain stretching condition, limit strains have decreased significantly and are nearly zero (~ 0.005). This limit strain value is very low and is unexpected. When placed at different locations (10 mm and 25 mm from the centre), the effect is same. Failure occurs at the interface of weld and the base metal, when weld is at an offset position. It is observed that the weld zone, which had an initial thickness of 1.2 mm before deformation, has a thickness of 1.18 mm when limit strains are measured. So it is clear that weld is not deforming much but it just changes the limit strains of the TWB. In the case of bi-axial stretching conditions, decrease in limit strain is observed when compared to that of TWB with weld at centre of the sheet. But the decrease in limit strains in this case is not to the extent as observed in near plane strain condition. The different weld locations, 15 mm and 30 mm

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from the centre of the blank have nearly equal limit strains like in near plane strain condition. Since yield strength of the weld (YSw) is higher than the base metal, it behaves like a rigid body and the deformation is insignificant in the weld region. Since weld is harder, it concentrates strain or deformation in neighboring regions at a faster rate as compared to that of un-welded blank (or centre weld case) and hence the limit strains of TWB are much lower than that of un-welded sheet. The above argument is supported by observing the thickness distribution under different conditions.

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Figure 3 compares the thickness distribution of TWB with weld located at centre to weld located at critical offset location from the centre of the blank. TWB with weld at centre exhibits lesser thinning when compared to that of TWB with weld at critical location. Though thickness distribution is same in the base metal region, considerable difference is observed near the weld and failure region. This is because of strong thickness gradients developing near the weld region. Also in the case of bi-axial stretching condition, the difference in thinning is lesser when compared to that of plane strain condition. This is the reason why not much of limit strain difference is seen in this case. Also, decrease in limit strains is not to the extent as observed in near plane strain condition. It is quite clear from the discussion that not only the weld orientation and weld properties affect the limit strains, but weld location also plays a vital role in affecting the limit strains. In the case of tensile strain path, weld is located at 20 mm and 35 mm from the centre of the blank. It is observed that limit strains have increased when weld is placed at 20 mm from the centre when compared to that of un-welded blank (or weld at centre case). When the weld is located at 35 mm from the centre, the limit strains are decreasing as expected (shown in Fig. 2). In the case of TWB with weld at 20 mm from the centre, the failure occurs nearly 20 mm from the weld region, but in between the weld and clamping region. This behavior of increase in limit strains of TWB to that of un-welded blank is due to the reduction in effective area of deformation (X) (see Fig. 4). In the case of 20 mm weld offset, the effective area of deformation is reduced, unlike previous cases. After a particular punch travel, strain distribution will be non-uniform and also concentrated on the effective deformation area. This will generate a local tensile mode of deformation in the small effective deformation area. The rigid weld during the deformation process can

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just locally stretch the region, because of which both major and minor strain increases significantly till the limit strain is reached. In order to demonstrate the existence of tensile mode of deformation, major strain is monitored with increasing punch travel till failure is reached. As mentioned earlier, the thickness gradient criterion will be satisfied between two successive elements. The major strain and corresponding minor strain in the thicker element will be identified as forming limit strains.

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Limit strain of un-welded blank in tensile 0.4 strain path 0.3

Tensile stretching – weld at 20mm

FLC of un-welded blank

0.2

Tensile stretching – weld at 35mm

Bi-axial stretching – weld at 15mm and 30mm

0.1 Near plane strain – weld at 10mm and 25mm 0 -0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Minor strain ),*85(Limit strains of transverse TWB with weld at critical locations of the blank Figure 4 shows the variation of major strain with punch travel in thinner and thicker elements where thickness criterion is satisfied. This is shown for both TWB with weld at centre and at 20 mm offset. It can be understood from the figure that tensile mode of deformation exists at the effective deformation area (X) in 20 mm weld offset case. As a result the major strain is increasing and non-linear. It is higher when weld is placed at 20 mm offset as compared to weld at centre case. Though the change is major strain in the thinner element finally reaches approximately same peak major strain, the change in thicker element is considerable. Also, one can clearly observe the increase in limiting major strain in weld at 20 mm offset case, when compared to that of weld at centre. Not only limiting major strain, but the entire change in major strain is different in these two cases, with weld at 20 mm offset case showing higher strains till failure. This resulted in increasing forming limit strains in the case of weld at 20 mm offset location. In whole, when weld is at center, limit strains of TWB are same as that of un-welded blanks. TWB limit strains are found to decrease when weld region is placed at critical locations of the blank. This particular inference on the weld location effect will be significant during TWB product design. It is difficult to locate the weld line position in actual TWB parts before forming. This depends on the TWB forming characteristics, structural design and TWB part compatibility with its mating sheet parts. Since weld is not deforming, irrespective of weld location, the influence of weld properties (nr, YSr) on the forming limit strains of TWB with transverse weld is found to be insignificant.

435

1.21

Plane strain More difference

1.19 1.17

Thickness (mm)

1.15 1.13 1.11 Near plane strain - w eld at 25mm from centre Near plane strain - w eld at centre Bi-axial stretching - w eld is at 30 mm from centre Bi-axial stretching - w eld at centre

1.09 1.07

Bi-axial - Less difference

1.05 1.03 0

25

50

75

100

125

150

175

200

225

Distance from one end of sam ple (m m )

),*85( Comparison of thickness distribution of TWBs with weld at centre to weld at critical locations (Weld properties: nr = 0.83 and YSr = 3) 0.6 Weld Weld Weld Weld

0.5

at at at at

center - Thinner element (2) center - thicker element (4) 20 mm of f set - thinner element (1) 20 mm of f set - thicker element (3)

1 2 3

Major strain

0.4 Thinner element 

0.3

Necked region

;

4 Thicker element 

0.2

Specimen geometric center

0.1 0 0

5

10

15

20

25

30

35

Form ing he ight (m m )

),*85( Variation of major strain with forming height at necked region

436

40

/LPLWVWUDLQVRI7:%LQORQJLWXGLQDOZHOGRULHQWDWLRQ In the case of longitudinal weld orientation, as shown in Fig. 1, limit strains are considerably different when compared to that of un-welded sheet in all the three strain paths. Weld is placed only at the centre location. In this case, failure has initiated in the weld and propagated to the base metal and hence the weld conditions affect limit strains of TWB significantly. As shown in Fig. 1, in the case of tensile and near plane strain conditions, the limit strains of TWB are higher or lower than that of un-welded blank depending on the weld properties. In the case of bi-axial stretching condition, the limit strains are lower than that of un-welded blank. The increase or decrease of limit strains of TWBs in comparison to that of un-welded blank depends on the base metal-weld property combination. It was hypothesized that the increase in limit strain observed for a particular weld-base metal property combination may not only be in tensile and plane strain deformation mode, but could also be occurring for all other strain paths in between. If this is the case, then one can expect a higher FLC for TWBs when compared to that of un-welded blank for that weld-base metal combination. In order to verify this hypothesis, few weld-base metal property combinations, nr = 1.67, YSr = 3; nr = 0.56, YSr = 5; nr = 1.11, YSr = 5, are selected for analyses randomly. All the strain paths are simulated using LDH set up for these weld-base metal property combinations and FLCs obtained are compared to that of un-welded blank. Figure 5 compares the FLCs of TWB with that of un-welded blank. It is quite clear from the figure that the overall FLCs obtained for TWBs are similar to that of un-welded blank. Negligible difference is observed in all the cases. So the increase in limit strains observed for some weld-base metal property combinations is only in few strain paths.

0.5

Major strain

0.4 Un-welded blank

0.3

FLC of un-welded blank

nr1.67ysr3 nr0.56ysr5

0.2

nr1.11ysr5

0.1

0 -0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Minor strain

),*85(Comparison of FLCs of TWB having longitudinal weld with un-welded blank

&21&/86,216 The following are the conclusions obtained from the present work. x Simulation results show that transverse weld located at the centre of the blank has no effect on the limit strains developed in the forming of TWB. The limit strains of TWB are nearly equivalent to that of un-welded blank. x Weld location has significant effect on the limit strains of TWB in transverse weld orientation. Decrease in limit strains is observed when the weld is placed at the critical locations of the blank, where strain has tendency to concentrate. In the case of TWB with weld at 20 mm offset in tensile strain path, anomalous behavior of increase in

437

x x

limit strain is witnessed. Weld tensile properties, viz., yield strength and strain hardening exponent, are found to show insignificant effect on the forming limit strains of transverse welded blanks. In longitudinal weld orientation, the limit strains are lower or higher than that of un-welded blank depending on the weld properties and strain path. Though increase and decrease of limit strains is seen in few selected strain paths because of weld zone properties, the overall forming limit is same for un-welded and longitudinal welded blanks.  The relative location of failure with respect to weld region plays a vital role in deciding the forming limit strains of TWB. In transverse weld orientation, drastic decrease in limit strain is witnessed when failure location is very near to the weld region. Increase in limiting major strain is also witnessed in a particular case because of the change in failure location. In TWBs with longitudinal weld orientation, the location of elements between which thickness gradient criterion is satisfied, i.e., in the weld region or in base material, decides the change in forming limit strains.

5()(5(1&(6 Y. Choi, Y. Heo, H. Y. Kim, and D. Seo, Journal of Materials Processing Technology , 1-7 (2000) L. C. Chan, C. H. Chang, S. M. Chan, T. C. Lee, and C. L. Chow, Journal of Manufacturing Science and Engineering  , 743-751 (2005) 3. A. Reis, P. Teixeira, J. Ferreira Duarte, A. Santos, A. Barata da Rocha, and A. A. Fernandes, Computers and Structures , 1435-1442 (2004) 4. R. W. Davies, M. T. Smith, H. E. Oliver, M. A. Khaleel, and S. G. Pitman, Metallurgical and Materials Transactions A $, 2755-2763 (2000) 5. R. Ganesh Narayanan, and K. Narasimhan, Journal of Strain Analysis for Engineering Design  , 551-563 (2008) 6. M. Jie, C. H. Cheng, L. C. Chan, C. L. Chow, and C. Y. Tang, Journal of Engineering Materials and Technology , 151-158 (2007) 7. R. Ganesh Narayanan, and K. Narasimhan, International Journal of Forming Processes  , 154-178 (2007) 8. K. Abdullah, P. M. Wild, J. J. Jeswiet, and A. Ghasempoor, Journal of Materials Processing Technology , 91-97 (2001) 9. V. M. Nandedkar, “Formability Studies on a Deep Drawing Quality Steel”, PhD Thesis, IIT Bombay, India, 2000 10. S. Chaudhary, and K. Narasimhan, “Prediction and Validation of Forming Limit Strains” in The 5th International conference and workshop on Numerical simulation of 3D sheet forming process, edited by D. Y. Yong et al., Jeju Island, Korea, 2002, pp. 481-486 1. 2.

438

Shear Fracture of Dual Phase AHSS in the Process of Stamping: Macroscopic Failure Mode and Micro-level Metallographical Observation Wurong Wanga , Xicheng Weia, Jun Yangb, Gang Shib a

School of Materials Science and Engineering, Shanghai University (Room 607, Rixin Building, No. 149 Yanchang Road, Shanghai 200072, China) b SAIC Motor Company (No.201 Anyan Rd. Jiading Shanghai 201804, China) Abstract. Due to its excellent strength and formability combinations, dual phase (DP) steels offer the potential to improve the vehicle crashworthiness performance without increasing car body weight and have been increasingly used into new vehicles. However, a new type of crack mode termed as shear fracture is accompanied with the application of these high strength DP steel sheets. With the cup drawing experiment to identify the limit drawing ratio (LDR) of three DP AHSS with strength level from 600 MPa to 1000 MPa, the study compared and categorized the macroscopic failure mode of these three types of materials. The metallographical observation along the direction of crack was conducted for the DP steels to discover the micro-level propagation mechanism of the fracture. Keywords: DP AHSS; Shear fracture; Stamping. PACS: 81.05.Bx; 81.20.Hy; 81.40.Lm; 83.85.Ei.

INTRODUCTION The application of high-strength steel (AHSS) has been one of the major ways to reduce the vehicle weight and fuel consumption, while improving vehicle crashworthiness. Dual Phase (DP) steel sheets, as the most common type of AHSS, has been increasingly used into vehicles and served as structural parts due to the combination of special mechanical properties such as high tensile strength, high work hardening rate at early stages of plastic deformation as well as very good ductility. One of key issues associated with DP AHSS is the springback occurring in the process of stamping. To reduce springback, small radii are often recommended [1]. However, a new type of crack mode which is termed as “shear fracture” is periodically seen along small bending radii in the stamping with DP AHSS [2-4]. Shear fracture is termed because the fracture cracks with limited localized necking, and cracks on alternating 45 planes, through thickness. Due to the nature of limited necking in shear fracture of DP, the traditional Forming Limit Diagram (FLD) proves insufficient to predict such fractures because the theory based FLD is mainly derived on the basis of concentrated necking. A significant effort has been made to address this challenge. Microstructure changes was taken into account to model stress and strain relationship for DP steel to analyze its strain hardening properties [5]. Sun et al. [6] suggested considering plastic strain localization, resulting from the incompatible deformation between the harder martensite phase and the softer ferrite matrix, to predict ductile failure of dual phase steels. Wierzbicki et al. [7] developed a Modified Mohr–Coulomb (MMC) ductile fracture criterion and applied it to analyze the failure behavior of a Dual Phase (DP) steel sheet during a simple quasi-static stretch-bending operation. In our previous study [8], different hardening formulas and yield functions were compared in the numerical prediction of onset crack for DP1000 sheet metal drawing. The investigation showed that a Swift and Hockett-Sherby combined formula was in good agreement with the flow curve of the tensile test and Batlat-89 yield model successfully predicted the onset shear crack of DP AHSS.

Corresponding Author: Wurong Wang; Email: [email protected]; Tel: +86 21 56331377; Fax: +86 21 56331466

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 439-445 (2011); doi: 10.1063/1.3623642 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

439

DP steels contain two major phases, ferrite and martensite. Such ferrite–martensite structure is called dual phase microstructure in which, soft ferritic network provides good ductility; while, hard particles and martensitic phase play the load-bearing role [9]. Many researches have compared the martensite morphology produced by some variations of two basic heat treatments, the quenching process from austenite region, or the intercritical annealing [10-12]. Martensite morphology and volume has a significant influence in the void nucleation and crack propagation. G. Avramovic-Cingara et al. [13] investigated the effect of martensite morphology and distribution in a ferrite matrix on the mechanical properties and the damage accumulation in uniaxial tension of two different automotivegrade dual phase DP600 steels. And they found that voids nucleation occurred by martensite cracking, separation of adjacent martensite regions, or by decohesion at the ferrite/martensite interface. In the SEM observation of DP780 fracture by 3-point bending, H.C. Shih et al. [14] revealed that (i) ductile fracture initiated from void nucleation and growth in ferrite phases and (ii) zigzag crack path bypassing martensitic grains. However, According to Kim and Thomas [15], cleavage fracture occurred at ferrite in a coarse martensite structure, whereas voids initiated at ferrite– martensite interfaces in a fine martensite structure. The cause why different damage mechanisms of particular dual phase steels were reported may be due to their chemical compositions, heat treatment history, and differences in their final microstructure. In the present work, three commercial DP steel sheets of strength level from 600MPa to 1000MPa are studied. Firstly, their chemical compositions are analyzed. Martensite distribution and volume are also identified. Then cup drawing experiments are conducted for these three types of materials to identify their limit drawing ratios (LDR). Macroscopic failure modes are determined for the LDR failure parts. Finally with the help of scanning electron microscope (SEM), the fracture propagation mechanism are observed and compared with each other.

MATERIALS AND EXPERIMENTAL PROCEDURE The three commercial DP AHSS sheets, 1.7 mm thick DP600, 1.2 mm thick DP800 and 1.0 mm thick DP1000 were studied in the as-received condition. The chemical compositions of the three steel are shown in Table 1. With increasing strength level, steels contain more C, Si and P. Steel DP600 DP800 DP1000

C 0.110 0.134 0.161

TABLE 1. The chemical compositions of steels in wt.% Si Mn P Cr Ni Al 0.193 1.400 0.005 0.170 0.017 0.0457 0.202 1.490 0.034 0.016 0.037 0.0461 0.501 1.480 0.037 0.021 0.036 0.060

Co 0.0066 0.0186 0.0189

Fe balanced balanced balanced

To observe the martensite distribution and volume of three DP steels, three samples were cut from the original sheets by electrical discharge machining (EDM). Then the samples were mechanically polished and etched in a 4% Nital solution for 10-15 seconds. The revealed microstructure in JSM-6700F SEM shows a dark ferrite phase with embedded white martensite. From FIGURE 1, one can see that the martensite phase in DP600 and DP800 is in island shape; while in DP1000, the martensite phase shows a continuous mesh shape. For the particular DP600, DP800 and DP1000 sheets in this study, the martensite volume is approximately 18%, 32% and 50%, respectively. As mentioned in introduction, the martensite distribution and volume play a dominant role in load bearing and will have a significant effect on the fracture propagation.

(a)



(b)



FIGURE 1. Microstructure of three DP steels.

(a) DP600, (b) DP800, (c) DP1000

440

(c)



LDR is an indicator of material formability, defined as the ratio of the maximum blank diameter that can be safely drawn into a cup without flange to the punch diameter. A punch of I32 mm diameter recommended by the International Deep-Drawing Research Group (IDDRG) was selected in this study. FIGURE 2 shows the press and the die tools. A group of DP1000 circular blanks which were prepared by EDM are shown in FIGURE 2. The diameter of the critical level that is safely drawn is the maximum diameter Dmax , and the next level is identified as the onset crack level. The LDR is then calculated by the ratio of Dmax to 32.

FIGURE 2. The press and die tools.

FIGURE 3. The group of DP1000 circular blanks.



RESULTS AND DISCUSSION LDR and Macroscopic Failure Mode Through the LDR cup drawing test, the maximum blank diameter that can be safely drawn into the cup without flange for 1.7 mm thick DP600, 1.2 mm thick DP800 and 1.0 mm thick DP1000 is 70 mm, 68.75 mm and 66.25 mm respectively, which gives a LDR of 2.19, 2.15, and 2.07 for the three materials. FIGURE 4 shows the crack level for each material. It can be seen that all three cups crack near the punch radius. In view of drawing depth, three DP steels crack at the initial stage of the cup drawing, which indicates they crack under stretch. Focusing on the fracture morphology, they all underwent shear fracture as they cracked on alternating 45 planes through thickness and there were very limited necking phenomena. Therefore, shear fracture is not only observed on DP800 and above steel grade, but also observed on DP600 in the deep drawing process. Moreover, DP600 only cracked with latitudinal fracture (square marked in FIGURE 4 (a)). However, not only latitudinal fracture (square

441

marked in FIGURE 4 (b) and (c)), but also longitudinal fracture (circular marked in FIGURE 4 (b) and (c)) were found on DP800 and above DP grade. This indicates that higher the DP strength, the more brittle the crack initiates.

(a)

(b)

(c)



FIGURE 4. The LDR crack level.

(a) DP600, (b) DP800, (c) DP1000

Micro-level Metallographical Observation To identify the surface morphology of the latitudinal fracture and the longitudinal fracture, samples were cut from the cracked cup and observed by SEM. The SEM result of the latitudinal fracture and the longitudinal fracture is shown in FIGURE 5 and FIGURE 6, respectively. By observation, both the latitudinal fracture and the longitudinal fracture are of mainly ductile fracture with dimple appearance. Through FIGURE 5, one can see that the higher the strength of the DP sheets, the smaller diameter and the shallower depth of the dimples. This corresponds well with formability of three DP steels. Moreover, the dimples in FIGURE 6 (a) is smaller and shallower than those in FIGURE 5 (b), as is the case in FIGURE 6 (b) and FIGURE 5 (c), which tells ductility of the latitudinal fractures is better than that of the longitudinal fracture.

(a)



(b)



(c)

FIGURE 5. Surface morphology of the latitudinal fracture.

(a) DP600, (b) DP800, (c) DP1000

(a)



(b)

FIGURE 6. Surface morphology of the longitudinal fracture.

(a) DP800, (b) DP1000

442





The fracture propagation mechanism is analyzed through the longitudinal fractures in DP800 and DP1000 cracked cups as the longitudinal fracture provides a relatively smooth plane for SEM sampling. Two samples were cut from the locations of the longitudinal fracture. Then the samples were also mechanically polished and etched in a 4% Nital solution for 10-15 seconds. FIGURE 7 shows an x32 image of the fracture. The SEM result of the fracture end for DP800 and DP1000 is shown in FIGURE 8 and FIGURE 9, respectively. It can be seen from FIGURE 8 that the fractures of DP800 propagate along the ferrite–martensite interfaces. Concerning the DP1000 fractures, there are two propagation modes that are (i) propagation along the ferrite–martensite interfaces (marked as A in FIGURE 9) and (ii) propagation across the martensite grains (marked as B in FIGURE 9). However, it is difficult to determine where and how fracture is initiated. Potential causes can be (i) inside the ferrite phases (void nucleation and growth), (ii) at the ferrite-martensite interface (interfacial debonding), (iii) inside the Martensite particles (cleavage fracture) and (iv) separation between two martensite particles [16].

FIGURE 7. The zoom-in of the longitudinal fracture.



END

FIGURE 8. The SEM result of the longitudinal fracture (DP800)

443



A

B

END

FIGURE 9. The SEM result of the longitudinal fracture (DP1000)



The difference in the fracture propagation mode of DP800 and DP1000 is majorly due to the ferrite/martensite volume and their distribution in each sheet metal. For DP800, soft ferrite is still the dominant phase and hard martensite is the second phase and distributes in island shape. Therefore, DP 800 has a better ductility and higher LDR than DP1000. The finding that the DP800 fractures propagate along the ferrite–martensite interfaces is likely related to the void nucleation along the interfaces. While for DP1000, the volume of ferrite and martensite is balanced and hard martensite is in continuous mesh shape. Thus, the fracture must bypass the martensite grain and forms a mixed propagation mode.

CONCLUSION Three commercial DP steel sheets of strength level from 600 MPa to 1000 MPa were studied in this work. The study focused on the recognition of their macroscopic failure mode and understanding of their micro-level fracture propagation mechanism. Major conclusions can be drawn as follows. (1) For the particular DP600, DP800 and DP1000 sheets in this study, the martensite volume is approximately 18%, 32% and 50%, respectively. The martensite phase in DP600 and DP800 is in island shape; while in DP1000, the martensite shows a continuous mesh shape. (2) A LDR of 2.19, 2.15, and 2.07 has been achieved for the three materials. And they all encounter consistent shear crack. Not only latitudinal but also longitudinal cracks are observed on DP800 and above steel grades. (3) SEM result shows the fractures of the three DP sheets are ductile failures. DP800 sheets crack and propagate along the ferrite–martensite interfaces. As for the DP1000 fractures, there is a mixed propagation mode that contains (i) propagation along the ferrite–martensite interfaces and (ii) propagation across the martensite grains.

444

ACKNOWLEDGMENTS The joint support from National Science Foundation of China (Grand No. 51005144), Innovation Program of Shanghai Municipal Education Commission and Shanghai Automotive Industry Science and Technology Development Foundation (Grand Nos. 0910 and 1009) is greatly acknowledged. The author also wishes to thank the guidance from Dr. Z Cedric Xia through several communications and the sheet materials provided by SSAB, Sweden.

REFERENCES 1. P. Chen, M. Koç, M.L. Wenner, Trans ASME J Manu Sci Eng 130, 0410061-8 (2008). 2. D. Zeng, Z.C. Xia, H.C. Shih, M. Shi, SAE Paper No. 2009-01-1172 (2009). 3. Wagoner RH, Kim JH, Sung JH. Int J Mater Form 2, 359-362 (2009). 4. A.W. Hudgins, D.K. Matlock, J.G. Speer, C.J. Van Tyne, J Mater Proc Technol 210,741-750 (2010). 5. F.G. Caballero, M.J. Santofimia, C. García-Mateo, J. Chao, C. García de Andrés, Mater Des 30, 2077-2083 (2009). 6. X. Sun, K.S. Choi, W.N. Liu, M.A. Khaleel, Int J Plasticity 25, 1888-1909 (2009). 7. M. Luo, T. Wierzbicki, Int. J. Solids Struct 47, 3084–3102 (2010). 8. W.R. Wang, C.W. He, Z.H. Zhao, X.C. Wei, Mater Des 32, 3320-3327 (2011). 9. T. Baudin, C. Quesnea, J. Jura, R. Penelle, Mater Charac 47, 365-373 (2001). 10. H.S. Lee, B. Hwang, S. Lee, C.G. Lee, S.J. Kim, Metall. Trans. 35A, 2371-2382 (2004). 11. X.J. He, N. Terao, A. Berghezan, Metal Sci. 18, 367-373 (1984). 12. D.A. Korzekwa, R.D. Lawson, D.K. Matlock, G. Krauss, Scripta Metall. 14, 1023-1028 (1980). 13. G. Avramovic-Cingara, Y. Ososkova, M.K. Jain, D.S. Wilkinson, Mat Sci Eng 516, 7-16 (2009). 14. H.C. Shih, M. F. Shi, “An Experimental Study on Shear Fracture of Advanced High Strength Steels,” in 2009 AISI Great Design in Steel Seminar, May 2009, Livonia, MI. 15. N.J. Kim, G. Thomas, Metall Trans. 12A, 483-489, (1981). 16. Z.C. Xia (private communication).

445

'HWHUPLQLQJ([SHULPHQWDO3DUDPHWHUVIRU7KHUPDO 0HFKDQLFDO)RUPLQJ6LPXODWLRQFRQVLGHULQJ0DUWHQVLWH )RUPDWLRQLQ$XVWHQLWLF6WDLQOHVV6WHHO Philipp Schmida, Mathias Liewaldb a Forschungsgesellschaft Umformtechnik mbH Stuttgart, Kornbergstraße 23, 70176 Stuttgart, Germany Institue for Metal Forming Technology, University of Stuttgart, Holzgartenstraße 17, 70174 Stuttgart, Germany

b

$EVWUDFW The forming behavior of metastable austenitic stainless steel is mainly dominated by the temperaturedependent TRIP effect (transformation induced plasticity). Of course, the high dependency of material properties on the temperature level during forming means the temperature must be considered during the FE analysis. The strain-induced formation of '-martensite from austenite can be represented by using finite element programs utilizing suitable models such as the Haensel-model. This paper discusses the determination of parameters for a completely thermal-mechanical forming simulation in LS-DYNA based on the material model of Haensel. The measurement of the martensite evolution in non-isothermal tensile tests was performed with metastable austenitic stainless steel EN 1.4301 at different rolling directions between 0° and 90 °. This allows an estimation of the influence of the rolling direction to the martensite formation. Of specific importance is the accuracy of the martensite content measured by magnetic induction methods (Feritscope). The observation of different factors, such as stress dependence of the magnetisation, blank thickness and numerous calibration curves discloses a substantial important influence on the parameter determination for the material models. The parameters obtained for use of Haensel model and temperature-dependent friction coefficients are used to simulate forming process of a real component and to validate its implementation in the commercial code LS-DYNA. .H\ZRUGVstainless steel, martensite formation, Haensel model, TRIP, thermal-mechanical forming simulation 3$&664.60.Ej; 64.60.My; 64.60.qe

,1752'8&7,21 The trend of utilizing FEA for sheet metal forming objectives of austenitic stainless steel has been increased in the last few years. Temperature and the martensite content must be taken into account, in respect to forming history [1]. Highly temperature-dependent TRIP-effect (Transformation Induced Plasticity Effect) hereby plays an important role with regard to material modelling and model parameter determination. There are many efforts spent during last decades on modelling strain-induced martensite formation. One of the most discussed models is the nonisothermal model by Haensel [1]. It is important to determine the actual martensite and temperature evolution in austenitic stainless steel accurately, independent from preciseness of the used model. Measurements of the martensite and temperature evolution for determination of model parameters during tensile tests in austenitic steel as proposed by Haensel have been conducted by various authors [4, 5, 6]. Typically, the martensite content is measured by magnetic induction methods (feritscope), but it is challenging to calculate the hardening behaviour influencing ’-martensite content. This paper validates the parameter determination process including implementation of Haensel material model into LS-DYNA code and shows the possibility to put thermalmechanical forming simulations of austenitic stainless steel into practice.

0(7$67$%/(67((/6$1'0$7(5,$/02'(/ Modelling of metastable austenitic stainless steel is mainly based on feasibility and alloy composition [2] or on the effect of martensite formation at shear band crossings [3]. As a high temperature dependence of martensite formation is proven by Angel [7], attempts by Tsuta [8] include influence of raising temperature level. Various

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 446-452 (2011); doi: 10.1063/1.3623643 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

446

complex matrix or thermodynamically-bases material models for strain induced martensite formation are known from Hallberg [9] or Galleé [11]. Incremental descriptions for non-isothermal forming operations were formulated by Haensel [1] and Springhub [12]. Haensel model for that reason until now is one of the most discussed models and showed sufficient correlation with real components in [13, 4, and 5]. Schedin [14] implemented the Haensel model into commercial code of LS-DYNA, but commented found misinterpretation of yield surface based on von Mises theory and the lack of temperature-dependent forming limit curves. Attempts to clarify this issue and to overcome found weaknesses were conducted by Krauer [5]. Parameter determination was done by Hansel’s approach of using non-isothermal tensile tests in this work with the aim of validating the martensite and temperature-measurement methods, parameter calculation, implementation in LS-Dyna and the model itself. The Haensel model is separated into two equations, one describing the martensite formation rate, the other describes occurring work hardening.

dVM d

Q

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The work hardening is based on the above calculated martensite content Vm

k ges. f

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HS

@



 BHS  AHS ˜ exp  m ˜  n ˜ (K1  K 2T)  kfo ˜ VM '

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(;3(5,0(176 6KHHWPDWHULDO The metastable austenitic stainless steel EN 1.4301 with sheet metal thickness of 0.8 mm was supplied by ThyssenKrupp Nirosta / Germany. Its chemical composition is shown in table 1. The Md30-temperature can be calculated as 15.8°C using the formula of Nohara [10]. 7$%/( chemical composition of the investigated steel grade EN 1.4301 [%] PDWHULDO& (1

0.036

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3

6

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1

0.39

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8.01

0.051

1RQLVRWKHUPDOWHQVLOHWHVWLQJRYHUYLHZ The procedure was adopted by Haensel at different test velocities of  ୙0,01 s-1,  ୙0,005 s-1 and  ୙0,002 s-1 to keep temperature level low and the martensite evolution relatively high. Tests were performed according to DIN ISO 10002 with a minimum of 5 valid test runs per velocity and rolling direction. Strain, temperature and martensite content were continuously measured and recorded during the uniaxial tensile test. In order to determine martensite evolution, a Feritscope MP30 (Fischer Comp. / Germany) was used. It was directly assembled to specimen surface and fixed with an elastic cord. One must ensure, that function of lateral contraction gauge as used in common tensile test machines (here Roell & Korthaus RKM 100/ 20) was not influenced by the elastic cord. Temperature itself was measured by a non-contact pyrometer. Emissivity factor fluctuates during the tensile test because of roughening effects of the surface. Calibration with thermocouples before and after the test was used to adjust this effect by linear interpolation.

447

Measurement and calculation of the martensite evolution Measuring procedure of the martensite content basically turned out to play an important key role in parameter determination procedure, because of its direct influence on the hardening behaviour. To calculate the actual martensite content, three main effects must be included in the procedure as follows: (1) Calibration factor between Fertiscope indication and the true ’-martensite content. To simplify the determination, Fischer [15] recommends a constant factor of 1.7, while calibration curves are recommended by Greisert [16] and other authors, see figure 1. Because Greisert used x-ray diffractometer and pre-electro-polished surface of test specimen to eliminate residual martensite influences, his calibration curve is used in this work. It must be noted, that no data is available for Feritscope values below 2%, which leads to unknown failure in this relevant range. 30

Greisert Hünicke Peterson Hecker Fischer

Feritscope %

25 20 15 10 5 0 0

10

20

30

40

50

60

'-martensite; %

),*85( Calibration curves of different authors in comparison to constant factor 1.7 by Fischer

(2) Villari-effect which describes the stress level influence on the magnetic behaviour of steel. The correction factor  can be calculated as



VM VM0

(3) 

0

In equation 3, VM is the indicated martensite content during the test and VM the measured martensite content value after load relief. all tests were carried out at 0°, 45°, 90° rolling directions, calculated Villari factors show positive values for 0° and 45° and 90° neutral. A precise determination is difficult because of the limited Feritscope measurement accuracy. The Villari correction factor for high stress levels was extrapolated using the existing data.

Villari correction 

1,02 1,01

0° 45° 90°

1,00 0,99 0,98 150

200

250

300

350

400

2

stress[N/mm ]

 ),*85( Calculated Villari-effect correction factors with consideration of rolling direction

448

(3) Blank thickness influences the measurement results and can be corrected by correction tables of Fischer or the equation calculated by Haensel [1]:

1 a0 





a 1  In VM ˜ s 2 (4) s

Where a0 equals 0.10435 and a1 = 0.01976. The compensated martensite content can be calculated using equation (5). It must be noted here, that blank thickness is not a constant factor and that thickness reduces during the test from 0.79 mm to approximately 0.65mm.

VM

VMV ˜

(5) 

A further point which was not taken into consideration is the residual ferromagnetic phase. In as-delivered condition, EN 1.4301 grade has 0.2-1% magnetic indication, usually caused by pre-formed martensite during cold rolling operations, but presence of residual ferrite can not be excluded either.

7HQVLOHWHVWUHVXOWV Tensile test are performed under the conditions described above. Figure 3 (a) shows the flow curve of EN 1.4301 in the rolling direction at different velocities. The velocity indirectly influences the temperature development (figure 3 (b)) because of lower heat conduction und convection loss, coupled with a faster test. Change in temperature mainly influences the elongation at fracture and tensile strength within this temperature range. Maximum temperature was measured at 32°C in the zone without necking immediately before fracture. (a) flow curves

(b) temperature development

1200

35

30

800

tem perature [°C]

yield stress [N/mm 2]

1000

 = 0,01 1/s 600

25

 = 0,005 1/s  = 0,002 1/s

400

 = 0,01 1/s

20

 = 0,005 1/s

200

 = 0,002 1/s 15

0 0

10

20

30 strain [%]

40

50

60

0

20

strain [%]

40

60

),*85( Flow curve of EN 1.4301 in rolling direction and measured temperature development

Low test temperatures promoted high martensite formation. Figure 4 shows the final calculated martensite evolution for EN 1.4301 at different rolling directions over the strain. Martensite formation starts at a certain martensite start temperature Ms. This temperature must not be reached at the same strain level; in this case it differs between 16.8% for  = 0.01 to 20.8% for  = 0.002. The total maximum martensite content is approximately the same at the 0° and 45° rolling direction, but on average 1.05 times higher at 90°. Anisotropy of martensite formation is low, but cannot be excluded.

449

(b) 45° direction of rolling

25 20 15 10 5 0 0

20 40 strain [%]

(c) 90° direction of rolling

35

calc. martensite content [%]

 = 0,01 1/s  = 0,005 1/s  = 0,002 1/s

30

calc. martensite content [%]

calc. martensite content [%]

(a) 0° direction of rolling 35

 = 0,01 1/s  = 0,005 1/s  = 0,002 1/s

30 25 20 15 10 5

35

25 20 15 10

0

60

 = 0,01 1/s  = 0,005 1/s  = 0,002 1/s

30

5 0

0

20 40 strain [%]

60

0

20 40 strain [%]

60

),*85( Calculated martensite evolution in EN 1.4301 for strain rates of 0.01, 0.005 and 0.002 1/s and rolling directions

'HWHUPLQLQJWKHSDUDPHWHUVRIWKH+DHQVHOPRGHO Six parameters A, B, Q, p, C, D of the Haensel material model mentioned above were determined using the least square method and fitting the model parameter to the data. The martensite formation rate dVm/d can be calculated from the results in figure 4 (from its derivation). The strain , the temperature T and the martensite content Vm must be adapted. It became apparent that a proper fit of the model to the measured data cannot be found if the complete strain range was implicated. The model was only fit to the measured data in a range between approximately 18 and 45% strain. Comparison of measured martensite formation rate and the model fit is show in figure 5. Due to complexity of Haensel model, the variation of starting values, solver adjustments, chosen measured values and range influence the result of the least square method operations. As noted by other authors [4, 5], it can still not be determined whether the found result is a global or local maximum, which can lead to many correct parameter combinations.

martensite forming rate

0,25 0,20 0,15 0,01 model

0,10

0,002 model 0,05

0,01 measured 0,002 measured

0,00 0

2

4

6

8

martensite content [%]

),*85( Comparison between model fit and measured martensite formation rate

However, the model parameter could be calculated for EN 1.4301 and all three rolling directions as shown in table 1. Heinemann [13] attempted to minimize the number of parameters and proposed to set the B, C, D and Q constant. This work also attempted to fit a curve on the measured values using the parameters from Heinemann. A satisfying correlation could not be found for any test series; in addition, constant parameters could not be determined in any case. It must be stated that the constant factors determined by Heinemann are not valid for any austenitic stainless steel grades or types.

450

7$%/( Parameter for Haensel material model in reference to rolling direction of sheet A

B

C

D

p

Q

0°

8.986

0.340

-45.012

0.1452

4.413

1384.35

45°

1.929

0.148

-47.870

0.0641

8.443

1376.15

90°

5.591

0.249

-47.870

0.0641

5.544

1375.27

7$%/( Parameter for Haensel hardening model in reference to rolling direction of sheet AHS

BHS

m

n

k

dkf -’

0°

334.9

994.6

4.3243

0.962

0

702.3

45°

352.1

1703.0

1.2937

0.9041

0

320.9

90°

279.0

979.6

5.324

1.149

0

857.8

6,08/$7,215(68/76 The parameters determined were used for the Haensel-model implementation (mat113) in LS-DYNA and the tensile test was simulated under adiabatic conditions. Former attempts done by Hochholdinger [17] or Krauer [18] show good correlations between trial and simulation. Figure 6 shows the distribution of the martensitic content along the specimen - the qualitative correlation is sufficient, but because of the disregarded temperature loss, the total martensitic contents did not match well. Because temperature is an important input parameter to the model, temperature development in tools must be simulated as well. Therefore, thermal boundary conditions (heat conductance, heat radiation, convection and heat transfer) must be know and must be incorporated.

60 start martensite 1.4301 0º 0,01 1.4301 0º 0,005 1.4301 0º 0,002

martensite content [%].

50 40

fracture

30 20 10 0 0

2

4

6 8 10 12 14 16 18 20 22 24 measurement position [cm]

),*85( Simulated tensile test and measured martensite contents along the specimen

In further simulations a circular cup was simulated with the TRIP model in LS-Dyna and with regard of the thermal development in the forming tool. It was possible to show the influence of higher temperatures on the martensite content during start-up processes, which varies in a forming process relevant range.

451

',6&866,21 An accurate simulation of the TRIP-effect during sheet metal forming processes according to state of the art looks possible, but challenging. The performed measurements of the ’-martensite content in austenitic stainless steel by magnetic methods proved to be influenced by various factors which are not adequate as of yet. The influence of rolling direction on the martensite evolution looks fairly low, but cannot be excluded. Determining parameters for the Haensel model is difficult because of its high complexity. The calibration of measured temperature and martensite distributions make parameter identification unstable. Even different parameter combinations result from the rolling direction; the influence of marginal alloy variations has not yet been investigated and will be a part of further investigations. First tests showed a high influence of alloy variations on the martensite evolution and are part of current investigations. Further FEM-simulation will be planned with the inclusion of thermal boundary conditions and especially the heat development, which influences the whole process and is needed as an input parameter for forming simulations with regard to the TRIP-effect. It is no longer sufficient to measure the temperature in real process and set this temperature as a constant tool temperature; a fully thermal-mechanical forming simulation will be done in future work.

$&.12:/('*0(176 This work was supported by the research consortium “Heat Development in Thin Sheets of Stainless Steel during Forming Operations“ of the Forschungsgesellschaft Umformtechnik mbH in Stuttgart / Germany in collaboration with the Institute for Forming Technology (IFU) at University Stuttgart from May 2009 to October 2011.

5()(5(1&(6 1. A. Haensel, „Nichtisothermes Werkstoffmodell für die FE-Simulation von Blechumformprozessen mit metastabilen austenitischen CrNi-Stählen“, Dr.-Ing. Thesis, ETH Zürich, 1998. 2. DC. Ludwigson, JA Berger: “Plastic behaviour of metastable austenitic stainless steels”, Journal of the Iron and Steel Institute, 1969 3. G.B. Olsen and Cohen, Kinetics of Strain Induced Martensitic Nucleation,Metallurgical Transactiona 6A, p.791-794, 1975 4. Next Generation Vehicle, “Stainless Steel Automotive Engineering Guide Lines”, p.30, 2007 5. J. Krauer, „Erweiterte Werkstoffmodelle zur Beschreibung des thermischen Umformverhaltens metastabiler Stähle“, Dr.-Ing. Thesis, ETH Zürich 2010 6. E. Schedin and J. Kajberg, Real Behaviour and FEA Modelling of Stainless Steels, Proceedings of the FLC, Zürich, 2006 7. T. Angel, “Formation of Martensite in Austenitic Stainless Steels, Effect of deformation, Temperature and Composition”, Journal of the Iron and Steel Institute, p.165, 1954 8. T. Tsuta, JAC. Ramirez, Y. Mitani, K. Osakada: Flow Stress and Phase Transformation Analyses in the Austenitic Stainless Steel under Cold, JSME Int. Journal, Vol. 35, No. 2, 1992 9. H. Hallberg, Paul Håkansson, Matti Ristinmaa, “Thermo-mechanically coupled model of diffusionless phase transformation in austenitic steel“, International Journal of Solids and Structures 47, p.1580–1591, 2010 10. K. Nohara, Y. Ono, N. Ohashi, Tetsu-to-Hagane 63 (1977) p. 5 11. S. Gallée, P. Pilvin, “Deep drawing simulation of a metastable austenitic stainless steel using a two-phase model”, Journal of Materials Processing Technology 210, p.835–843, 2010 12. B. Springhub, „Semi-analytische Betrachtung des Tiefziehens rotationssymmetrischer Bauteile unter Berücksichtigung der Martensite evolution“, Dr.-Ing. Thesis, IFUM Universität Hannover, 2006 13. G. Heinemann, “Virtuelle Bestimmung des Verfestigungsverhaltens von Bändern und Blechen durch verformungsinduzierte Martensitbildung bei metastabilen rostfreien austenitischen Stählen“, Dr.-Ing. Thesis, ETH Zürich, 2004 14. E. Schedin, D. Hilding, “Finite Element Simulation of the TRIP-effect in Austenitic Stainless Steel”, SAE International 04M3, 2003 15. H. Fischer GmbH, FERITSCOPE® FMP30 Bedienungsanleitung, 2008 16. C. Greisert, “Strain-induced formation of martensite during forming and springback behaviour of annealed and hard coldrolled stainless steel grades EN 1.4301 and EN 1.4318”, Dr.-Ing. Thesis, 2004 17. Hochholdinger, “Simulation of Austenitic Stainless Steel in LS-DYNA”, 7.Anwenderformum LS-DYNA, Frankenthal, 2007 18. P. Hora, J.Krauer, Thermoactive process modelling of deep drawing with austenitic stainless steel, IDDRG international conference, p.17-28, Golden Co USA, 2009

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Efficient and Robust Prediction of Localized Necking in Sheet Metals Holger Aretz1and Olaf Engler Hydro Aluminium Rolled Products GmbH, Research & Development, Georg-von-Boeselager-Str. 21, D-53117 Bonn, Germany Abstract. The recently proposed Critical Specific Tension (CST) model is jointly used with the well-known Marciniak and Kuczy´nski (M-K) model to predict localized necking in anisotropic sheet metals in the regime of negative and positive minor in-plane strains, respectively. A significantly simplified method is presented to calculate the critical tensile stress required in the CST model, without the need of iterative computations. In the present work the CST/M-K model is used along with a rate-independent phenomenological elasto-plastic constitutive model as well as the known visco-plastic self-consistent (VPSC) crystal plasticity model developed by Tomé and Lebensohn. A comparison between experimental data and the limit strains predicted by means of the phenomenological constitutive model reveals a very good agreement. In order to validate the correctness of the non-trivial computational implementation of the VPSC-based CST/M-K model the predicted necking strains are compared with results obtained by using the phenomenological constitutive model. It is shown that the results of both approaches are in good agreement. Keywords: Sheet metal forming, Formability, Localized necking, Limit strains PACS: 81.05.Bx, 81.20.Hy, 81.40.Ef, 81.40.Lm, 83.10.Ff, 83.10.Gr, 83.50.Uv, 81.70.Bt, 83.60.La, 83.80.Ab, 89.20.Bb, 89.20.Kk

INTRODUCTION The most popular model to predict localized necking in sheet metals is the one proposed by Marciniak and Kuczy´nski [14], referred to as the M-K model in the following. The M-K model assumes a pre-existing initial imperfection in form of a narrow band or groove whose orientation is assumed of being normal to the major stress axis. The original M-K model was intended to predict necking strains in the regime of positve minor in-plane strains. Due to the presence of the imperfection the strain will localize there. In other words the M-K model assumes that a neck develops normal to the major stress axis. In order to predict necking strains in the regime of negative minor in-plane strains Hutchinson and Neale [9, 10] extended the M-K model. It is well-known that for negative minor in-plane strains the assumption of a neck that is normal to the major stress axis is not valid. For example, in uniaxial tension of sheet specimens a neck develops at an angle that is inclined to the tensile direction. In the Hutchinson-Neale extension [9, 10] different orientations of the imperfection are tested and the one leading to the most conservative (i.e. lowest) limit strain is finally selected. Obviously, anticipating an acceptable resolution of imperfection orientations to be tested is a task left to the user and is a compromise between accuracy and computational effort. In a typical computational implementation of the M-K model the considered (imperfect) sheet is incrementally stretched for different imposed strain-modes, usually ranging from uniaxial over plane-strain to equibiaxial tension. Necking occurs when the strain increment inside the imperfection becomes several times larger than in the region outside, see e.g. [2]. The in-plane strain at necking associated with the region outside the imperfection constitute a single point of the forming limit curve (FLC), which is the central part of the well-known forming limit diagram (FLD). The imposed strain increments used in an M-K analysis must be small in order to capture the instant of necking accurately. Usually, the strain increments are of the order 1 · 10−4 to 1 · 10−3 . For most metallic materials this results in a large number of increments until necking is detected. In each increment the field quantities inside the imperfection must be calculated, which is achieved by incrementally solving the force equilibrium equation in the direction normal to the imperfection. Due to the non-linearity of the governing boundary value problem the associated numerical solution process is costly, especially in the regime of negative minor in-plane strains. In particular, the solution process becomes very time demanding when computationally expensive texture-based crystal plasticity models are being used

1

Corresponding author; e-mail: [email protected]

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 453-460 (2011); doi: 10.1063/1.3623644 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

453

to describe the mechanical response of the material at hand. In this contribution the recently proposed critical specific tension (CST) model is jointly used with the M-K model to predict localized necking in anisotropic sheet metals [3] in the regime of negative and positive minor in-plane strains, respectively. The CST model avoids all of the previously mentioned pitfalls of the M-K model and is, therefore, very appealing. In the aforementioned reference it was already demonstrated that the predictions of the CST model agree very well with predictions of the M-K model. The critical tensile stress required in the CST model can be easily computed using the major strain at plane-strain tension predicted by the M-K model; compared to the original paper [3] a significantly simplified procedure will be presented without the need of iterative computations. In the present work the combined CST/M-K approach is applied to predict localized necking of different materials. Phenomenological constitutive models as well as the known visco-plastic self-consistent (VPSC) crystal plasticity model developed by Lebensohn and Tomé [12] are utilized to describe the material behavior.

MODELLING OF LOCALIZED NECKING In the present work the M-K model [14] is combined with an extended version of Hill’s localized necking model [6]. Due to space limitations only the extended Hill model will be described in the following. For further details the reader is referred to [2, 3]. In Hill’s model of localized necking rigid-plastic material behavior is assumed, see [6]. An equivalent, but more general, derivation of Hill’s model can be found in the book of Marciniak et al. [15, p. 68]. Their derivation is actually independent of the material behavior, i.e. it holds for rigid-plastic as well as for elastic-plastic materials, no matter whether the response is rate-dependent or rate-independent. Therefore, the derivation of Marciniak et al. [15, p. 68] is followed in the present paper. Accordingly, necking is assumed to set in when the major force per unit width, f1 , attains or passes a maximum, i.e. d f1 = d(σ11 · t) = d σ11 · t + σ11 · dt ≤ 0

⇔

d σ11 + σ11 · d ε33 ≤ 0

(1)

with t being the current sheet thickness, σ11 the 11-component of the tensor of the Cauchy stress and d ε33 = dt/t the thickness strain increment. Due to the fact that necking is assumed to coincide with the maximum of f1 , Hill’s criterion for localized necking can be referred to as the Maximum Specific Tension (MST) criterion. The motivation for extending the MST model stems from the following idea, see also [3]. The MST model can only be applied to negative minor in-plane strains while the Marciniak and Kuczy´nski [14] (M-K) model can be applied to both the regime of negative and positive minor in-plane strains. Since the MST model is significantly more efficient than the M-K model it is appealing to combine the MST model with the M-K model, especially when computationally demanding material models are being used (e.g. texture-based crystal plasticity models). However, for plane-strain tension the M-K model generally predicts lower limit strains than the MST model does, which is due to the imperfection used in the M-K model. From a geometrical perspective there will always be a ‘gap’ between the limit strains predicted by both models. This shows that a combination of the M-K model and the MST model is not straightforward. For this reason the MST model was slightly modified based on the following assumption: Necking sets in when f1 attains a critical value, which is not necessarily coincident with the maximum of f1 . Accordingly, the extended MST criterion reads as follows: d f1 = d(σ11 · t) = d σ11 · t + σ11 · dt ≤ f1∗ f1∗

⇔

∗ d σ11 + σ11 · d ε33 ≤ σ11

∗ , σ11

(2)

The critical values are denoted as and respectively. In the following the inequalities (2) are referred to as the Critical Specific Tension (CST) model. ∗ can be considered as material dependent parameters. Thus, one may think of The critical values f1∗ and σ11 determining them experimentally. This is, however, not in the scope of the present work. It is recalled that the motivation for the CST model was to combine it with the M-K model. By means of a numerical procedure the ∗ can be adapted so that the predictions of the CST model and the M-K model agree for planecritical stress σ11 ∗ is used for the other strain-modes being considered. An iterative strain tension. Thereafter, the same value of σ11 ∗ was proposed in [3]. Meanwhile, it was realized that such an iterative procedure can be procedure to calculate σ11 entirely avoided. The new procedure works as follows. Provided that the prediction of the M-K model at plane-strain ∗ for the CST model can be readily computed from tension is available, the critical stress σ11 plane-strain  ∗ σ11 := d σ11 + σ11 · d ε33 (3) M-K model

454

The bracketed quantities are to be computed by the M-K model for plane-strain tension. In the computational implementation differentials are replaced by finite differences, see [3]:  plane-strain ∗ k+1 k+1 k+1 σ11 := Δσ11 + σ11 · Δε33 ,

with

k+1 k+1 k Δσ11 := σ11 − σ11 ,

k = 0, 1, 2, . . .

(4)

M-K model

The superscripts k and k + 1 are used to label consecutive time steps during the solution process. The combination of the CST model with the M-K model will be referred to as the CST/M-K model in the remainder of this paper. For more details it is referred to [3].

CONSTITUTIVE MODELLING The same rate-independent elastic-plastic CST/M-K model as in [3] was used in the present work. For a detailed description it is referred to [3]. In addition to the elastic-plastic model, in the present work the visco-plastic self-consistent (VPSC) model (version 7) developed by Tomé and Lebensohn [12, 17] is used in combination with the CST/M-K model to predict localized necking. Further details of the VPSC model may be found in the book [11, pp. 466-509]. Compared to the elasticplastic approach used in [3] there are no conceptional differences when the VPSC model is used. This is due to the fact that both material models repond in the same way, i.e. a strain increment is prescribed/imposed and the corresponding stress tensor is returned. Hence, the numerical implementation of the VPSC-based CST/M-K model follows exactly the same principles as for the elastic-plastic version described in [3]. For the VPSC-based necking prediction the prescribed strain increment is 5 · 10−4 , which is somewhat larger than the value used in the elastic-plastic model described in [3]. Signorelli et al. [16] implemented the VPSC model in an M-K model including the HutchinsonNeale extension for the regime of negative minor in-plane strains. The VPSC model is a rate-dependent plasticity model. Since the rate-dependence can not be entirely switched off a comparison with results obtained by means of a rate-independent elastic-plastic model is a priori difficult and a perfect agreement can not be expected. As a workaround the rate-dependence of the VPSC model can be reduced to a low level by using an appropriate value for the parameter that controls the rate-sensitivity of the mechanical response. However, due to reasons of numerical stability the rate-sensitivity parameter can not be arbitrarily chosen, see below. Furthermore, it is known that the limit strain for plane-strain tension predicted by the M-K model is shifted to larger values when the rate-sensitivity has a positive sign (see e.g. [15]). Hence, in order to obtain the same limit strains for plane-strain tension different imperfection sizes must be used in the VPSC-based and in the elastic-plastic M-K model. In general, in order to obtain comparable plane-strain predictions the imperfection in the VPSC-based M-K model must be larger in size than for a rate-independent model. Since the VPSC code was not designed for the present purpose some minor but important modifications had to be employed which are described in detail in [13]. The major modification refers to all history-dependent variables (hardening, texture). In the M-K model the force equilibrium between the imperfection and the surrounding matrix must be established. Due to the non-linearity of the governing equation an iterative procedure is required, preferably using Newton’s method for non-linear equations. However, unless appropriate modifications are made the VPSC code updates the history-dependent variables each time it is called. Thus, the major modification being made was to store the history variables at the beginning and the end of the current time increment. Before entering a Newton iteration the history variables are initialized by their values at the beginning of the time increment. An update of the history variables is only employed when convergence of the force equilibrium iterations is achieved. Moreover, one has to ensure that the input data is read only at the very beginning of the considered strain-mode, otherwise the history variables will always be overwritten by their initial values. It is finally noted that in the present work solely plane-stress problems are considered. The VPSC code [17] can handle plane-stress problems by properly specifying the boundary conditions. In particular, the user can specify a zero stress in the sheet normal direction, which was done in the present work.

455

EXAMPLE NO. 1: ALUMINIUM ALLOY AA6016-T4 Objectives The objective of this application example is to demonstrate that the predictions of the elastic-plastic CST/M-K model are in close agreement with the Hutchinson-Neale extension of the M-K model. For this purpose, the simulation results of Butuc et al. [5] were used as a reference. The same example was already considered in [3], but in the said reference ∗ -value was computed by means of an iterative procedure. The new method outlined above is used in the present the σ11 work.

Elastic-Plastic CST/M-K Model The material data used in the present work correspond to the aluminium alloy AA6016-T4 considered in the work of Butuc et al. [5]. Isotropic hardening is assumed. The reference strain-hardening curve is given as follows [5]: Yref (ε¯ ) = 318.103 − 191.103 · exp(−8.706 · ε¯ ) MPa

(5)

According to [5] the directional r-values, uniaxial yield stresses as well as the equibiaxial yield stress are as follows: Uniaxial tension, 0◦ : yield stress 127 MPa, r-value 0.80. • Uniaxial tension, 45◦ : yield stress 127 MPa, r-value 0.43. • Uniaxial tension, 90◦ : yield stress 114 MPa, r-value 0.61. • Equibiaxial tension: 123.44 MPa. •

In order to account for plastic anisotropy the yield function ‘Yld96’ [4] was used, in accordance with [5]. The calibration of Yld96 involved the anisotropy data listed above. The parameters of the yield function were adopted from [5] and read as follows: c1 = 1.0474, c2 = 0.7752, c3 = 1.0720, c4 = 1.0000, c5 = 1.0000, c6 = 0.9288, αx0 = αx1 = 2.0, αy0 = αy1 = 3.5, αz0 = 1.0, αz1 = 1.05, m = 8. They correspond to Yref = 127 MPa. With these parameters the yield function Yld96 fits the input data listed above exactly. A Young’s modulus of E = 70 GPa and a Poisson’s ratio of ν = 0.33 were used, along with the assumption of isotropic elasticity. Furthermore, the M-K imperfection used in the present work had a value of 0.998, in agreement with [5]. Linear strain-paths were assumed in the necking predictions.

Results The calculated forming limit curve using the elastic-plastic CST/M-K model is shown in Fig. 1. One may notice the good agreement of the predictions of the M-K model of Butuc et al. [5] (including the Hutchinson-Neale extension) ∗ = 0.65977 · 10−2 MPa was calculated. and the present elastic-plastic CST/M-K model. In the CST/M-K model σ11 The total computation time took 10 s for 10 points of the forming limit curve. The good agreement of the computed limit strains with the experimental data shown in Fig. 1 is remarkable.

EXAMPLE NO. 2: ALUMINIUM ALLOY AA6016-T4 WITH PRE-STRAINS Objectives The objective of this application example is to demonstrate that the elastic-plastic CST/M-K model can be applied to non-linear strain-paths as well, which is a fundamental requirement for the general applicability of necking models. The same material as in the previous section (i.e. AA6016-T4 according to Butuc et al. [5]) is considered. Only qualitative results are presented, i.e. a comparison with experimental data is not given.

456

0.4

Log. major in-plane strain [-]

0.35 0.3 0.25 0.2 0.15 0.1 0.05

Elastic-Plastic CST/M-K Model Rigid-Plastic Simulation Butuc et al. Experiment Butuc et al.

0 -0.2

-0.1

0

0.1

0.2

0.3

0.4

Log. minor in-plane strain [-]

FIGURE 1. Predicted forming limit curve for AA6016-T4 using the elastic-plastic CST/M-K model. For comparison, the curve predicted by Butuc et al. [5] is shown as well.

Elastic-Plastic CST/M-K Model Exactly the same material data that was used in the first application example on AA6016-T4 is used here. The value of the M-K imperfection was again 0.998. Three cases were generated involving the following types of pre-straining: 1. Case 1: No pre-strains. 2. Case 2: Pre-deformation under equibiaxial straining with a pre-strain of 0.05. 3. Case 3: Pre-deformation under uniaxial tension with a pre-strain of 0.1.

Results Figure 2 shows the predicted forming limit curves (FLC) for the generated cases described above. Compared to case 1 the minimum of the FLC corresponding to case 2 is shifted to the lower right, which is typical for predeformation under equibiaxial straining. In contrast, the minimum of the FLC corresponding to case 3 is shifted to the upper left, which is typical for pre-straining under uniaxial tension. The results confirm that the present model can be used for forming limit strain prediction under linear as well as non-linear strain-paths.

EXAMPLE NO. 3: ISOTROPIC ALUMINIUM Objectives The primary objective of the application example was to validate the non-trivial computational implementation of the VPSC-based CST/M-K model. For this purpose the intensively validated elastic-plastic CST/M-K model described in [3] (and also used in the previous application examples of the present paper) was applied whose predicted limit strains serve as a reference.

VPSC-Based CST/M-K Model The imperfection factor used in the VPSC-based M-K model was 0.992. The ‘secant’ inclusion-matrix interaction type was used along with a rate sensitivity parameter of n = 80 (see [17] for details). Note that larger values for

457

0.4

Log. major in-plane strain [-]

0.35 0.3 0.25 0.2 0.15 0.1 Case 1 Case 2 Case 3

0.05 0 -0.2

-0.1

0

0.1

0.2

0.3

0.4

Log. minor in-plane strain [-]

FIGURE 2. Predicted forming limits curves for AA6016-T4 using the elastic-plastic CST/M-K model. Different types of prestraining are imposed.

the rate-sensitivity parameter resulted in an error termination. The hardening on the slip systems is described by the following extended Voce approach [17]:

τ (Γ) = τ0 + (τ1 + θ1 · Γ) · (1 − exp(−Γ · |θ0 /τ1 |)) MPa

(6)

Herein, Γ denotes the accumulated shear strain in a grain and τ the shear yield stress. The hardening parameters used in the present work are as follows [13]: τ0 = 14.67, τ1 = 12.67, θ0 = 62.00, θ1 = 8.78 (units omitted). The considered material is isotropic aluminium with fcc lattice structure. Accordingly, a random orientation distribution of the grains was used. For a parametric study the number of grains was varied as follows: in the first case 100 and in the second case 500 grains were used. The crystallographic texture was not updated, i.e. the initial texture was used throughout deformation. This ensures that the predictions of the VPSC-based model are comparable with the predictions of the elastic-plastic model that uses isotropic hardening, see below.

Elastic-Plastic CST/M-K Model In the elastic-plastic CST/M-K model the considered material is isotropic aluminium with a Young’s modulus of E = 70 GPa and a Poisson’s ratio of ν = 0.33. The yield surface is described by means of the non-quadratic plane-stress yield function ‘Yld2003’ [1]. Accordingly, the equivalent stress is defined as follows: 1/m   1 |σ1 |m + |σ2 |m + |σ1 − σ2 |m σ¯ := (7) 2 The generalized principal stresses σ1 , σ2 , σ1 , σ2 are defined as follows:  a8 σ11 + a1 σ22 (a2 σ11 − a3 σ22 )2 σ1 ± + (a4 σ12 )2 := σ2 2 4  σ11 + σ22 (a5 σ11 − a6 σ22 )2 σ1 ± + (a7 σ12 )2 :=  σ2 2 4 The dimensionless parameters a1 , . . . , a8 can be calibrated to experimental data. In the present work a1 = . . . = a8 = 1 is used, which reduces the yield function to the one proposed by Hosford [7]. The latter one describes the yield surface of isotropic fcc metals accurately. Furthermore, according to the recommendations concerning fcc materials given in [8] the exponent was set to m = 8.

458

1

200

VPSC (100 Orientations) VPSC (500 Orientations) Elastic-Plastic

0.8 Log. Major Strain [-]

True Stress [MPa]

150

100

0.6

0.4

50 0.2 VPSC (100 Orientations) VPSC (500 Orientations) Elastic-Plastic

0 0

0.2

0.4

0.6

0.8

0 1

-0.4

FIGURE 3. curves.

-0.2

0

0.2

0.4

Log. Minor Strain [-]

True Strain [-]

Left: comparison of macroscopic strain-hardening curves under uniaxial tension. Right: computed forming limit

In the elastic-plastic model isotropic strain-hardening was used. The utilized reference hardening curve is analogous to that implemented in the VPSC code and is given as follows: Yref (ε¯ ) = A + (B +C · ε¯ ) · (1 − exp(−D/B · ε¯ )) MPa

(8)

When the elastic-plastic CST/M-K model is compared with its VPSC-based counterpart it is very important that the same hardening curve is used in both approaches. An important difference between both models is the following: the VPSC model uses a hardening curve in form of a shear stress that depends on the accumulated shear strain for each slip system, while the elastic-plastic model uses a macroscopic hardening description. The transformation of both hardening models is not straightforward. Thus, in addition to the prediction of forming limit strains the VPSCbased model was also used to compute a stress-strain curve for uniaxial tension. Thereafter, the material-dependent parameters A, B, C, D of the macroscopic hardening description used in the elastic-plastic model were calibrated by means of a least squares solver so that (8) agrees with the hardening curve predicted by the VPSC model. They read as follows: A = 43.7844, B = 37.9214, C = 76.7851, D = 538.94 (units omitted). The imperfection factor used in the elastic-plastic M-K model was 0.996, which represents a somewhat smaller imperfection than the one used in the VPSC-based model, see above.

Results In Fig. 3 the macroscopic hardening curves predicted by the VPSC-based model are compared with the macroscopic description (8) that was utilized in the elasto-plastic model. As one may see the hardening curves are practically coincident. Hence, the fundamental requirement of equal hardening curves in both models is satisfied. Furthermore, there is no notable difference between the prediction that used 100 grains and the one which used 500 grains. Figure 3 also shows the predicted forming limit curves. As can be seen the difference between the prediction that used 100 grains and the one that used 500 grains is negligible. Furthermore, the differences between the VPSCbased model and the elastic-plastic model are small for positive minor in-plane strains, whereas larger differences can be observed for the regime of negative minor in-plane strains. The most significant difference can be observed for uniaxial tension. However, the agreement between both approaches is very promising. The agreement at plane-strain tension can be improved by adapting the imperfection being used in the M-K model, but this was not attempted in the present work. The computation time consumed by the elastic-plastic model was 12 s (for 30 points of the forming limit curve) while the VPSC-based model required 24 min and 50 min (for 10 points of the forming limit curve) using 100 and 500 grains, respectively. All three simulations were run on the same computer using the same compiler options. The

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high computation times associated with the VPSC-based model indicate that the use of an efficient necking model ∗ -values are as follows: (i) σ ∗ = 0.16238 · 10−2 MPa for the VPSC-based can be very beneficial. The calculated σ11 11 ∗ ∗ = model with 100 grains, (ii) σ11 = 0.15783 · 10−2 MPa for the VPSC-based model with 500 grains and (iii) σ11 −2 0.21491 · 10 MPa for the elastic-plastic model.

CONCLUSIONS The combined CST/M-K model is a very efficient tool to predict forming limit curves for sheet metals. This becomes very beneficial when the utilized material model is time consuming, like the VPSC model used in the present work. The results in the present paper represent a snapshot of ongoing development work. At present the VPSC code is being transferred from FORTRAN 77 to Fortran 90/95 programming language, mainly to make programming safer (by using Module Procedures available in Fortran 90/95), to simplify maintenance and to reduce its execution time.

ACKNOWLEDGMENTS Parts of the presented VPSC-based CST/M-K model were implemented by G. Liu during his MSc thesis, see [13].

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

H. Aretz: Applications of a new plane stress yield function to orthotropic steel and aluminium sheet metals, Modelling and Simulation in Materials Science and Engineering 12 (2004) 491-509 H. Aretz: Numerical analysis of diffuse and localized necking in orthotropic sheet metals, International Journal of Plasticity 23 (2007) 798-840 H. Aretz: An extension of Hill’s localized necking model, International Journal of Engineering Science 48 (2010) 312-331 F. Barlat, Y. Maeda, K. Chung, M. Yanagawa, J.C. Brem, Y. Hayashida, D.J. Lege, K. Matsui, S.J. Murtha, S. Hattori, R.C. Becker, S. Makosey: Yield function development for aluminium alloy sheets, Journal of the Mechanics and Physics of Solids 45 (1997) 1727-1763 M.C. Butuc, J.J. Gracio, A. Barata da Rocha: A theoretical study on forming limit diagrams prediction, Journal of Materials Processing Technology 142 (2003) 714-724 R. Hill: On discontinous plastic states with special reference to localized necking in thin sheets, Journal of the Mechanics and Physics of Solids 1 (1952) 19-30 W.F. Hosford: Yield locus of randomly oriented fcc metals according to the Bishop-Hill theory, Metallurgical Transactions 4 (1973) 1416-1417 W.F. Hosford, R.M. Caddell: Metal Forming – Mechanics and Metallurgy, 2nd edition, Prentice Hall, 1993 J.W. Hutchinson, K.W. Neale: Sheet necking-II. Time independent behavior, In: D.P. Koistinen, N.-M. Wang (Eds.), Mechanics of Sheet Metal Forming, Plenum Publ. Corp, pp. 127-153 J.W. Hutchinson, K.W. Neale: Sheet necking-III. Strain-rate effects, In: D.P. Koistinen, N.-M. Wang (Eds.), Mechanics of Sheet Metal Forming, Plenum Publ. Corp, pp. 269-285 U.F. Kocks, C.N. Tomé, H.-R. Wenk: Texture and Anisotropy, Cambridge University Press, 1998 R.A. Lebensohn, C.N. Tomé: A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: application to zirconium alloys, Acta Metall. Mater. 41 (1993) 2611-2624 G. Liu: Formability simulation from texture data by means of the VPSC polycrystal plasticity code, Master thesis, RWTH Aachen University, 2010 Z. Marciniak, K. Kuczy´nski: Limit strains in the process of stretch-forming sheet metal, International Journal of Mechanical Sciences 9 (1967) 609-620 Z. Marciniak, J.L. Duncan, S.J. Hu: Mechanics of Sheet Metal Forming, 2nd ed., Butterworth-Heinemann, 2002 J.W. Signorelli, M.A. Bertinetti: On the role of constitutive model in the forming limit of FCC sheet metal with cube orientations, International Journal of Mechanical Sciences 51 (2009) 473-480 C.N. Tomé, R.A. Lebensohn: Manual for Code Visco-Plastic Self-Consistent (VPSC) Version 7a, updated July 2006

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Fracture prediction in hydraulic bulging of AISI 304 austenitic steel sheets based on a modified ductile fracture criterion Y. Xua, H.W. Songa*, S.H. Zhanga, M. Chenga a

Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, China

Abstract. The demand for weight reduction in modern vehicle construction has resulted in an increase in the application of hydroforming processes for the manufacture of automotive lightweight components. This trend led to the research of evaluation on formability of the sheet or tube hydroforming to be noted, particularly the prediction of fracture. In this study, a new proposed approach based on damage theory for fracture prediction considering the deformation history was introduced. And the modified ductile fracture criterion was applied to predict the failure for hydraulic bulging of AISI 304 austenitic steel sheets. The material parameters in terms of the function of strain rate in the failure criterion were determined from the equivalent fracture strains corresponding tensile tests under different stress conditions. Then, in the finite element simulation the effect of strain rates and their distribution as well during practical sheet metal forming process was considered. The hydraulic bulging tests were carried out to identify the fracture behavior predicted from FE analysis. A comparison between the prediction and experimental results showed that the proposed approach with a modified ductile fracture criteria can give better fracture predictions than traditional ways. Keywords: Hydraulic bulging; Austenitic steel; Ductile fracture criteria; Deformation history PACS: 62.20.mm; 81.20.hy; 83.50.uv

INTRODUCTION To save energy and reduce exhaust gas, hydroforming technologies as a modern advanced technology have got wide applications for manufacturing lightweight structural parts in automotive, aeronautic and aerospace industries [1, 2]. By using different blanks, hydroforming technologies can be divided into three types: tube hydroforming, shell hydroforming and sheet hydroforming. For the past two decades, a lot of research works have been carried out in key technologies and solution about the main defects such as wrinkling, buckling and bursting during the hydroforming process. Basically, buckling failure, which prevails when the axial compressive stress on a part exceeds the strength of the material, takes place during the beginning stage of the forming process and causes localized necking due to excessive tensile forces. Wrinkling failure, on the other hand, occurs during both the initial and intermediate stages of forming and can sometimes be eliminated by increasing the internal pressure during the final stage of the hydroforming process. In contrast with buckling and wrinkling, bursting failure is an irrecoverable failure mode. Hence, in order to obtain sound hydroformed products, it is necessary to study the effects of the process parameters on bursting failure in the hydroforming processes[3]. Among the various forming limit criteria suggested for sheet metal forming processes, FLD (forming limit diagram) has been most widely utilized following the pioneering work of Keeler and Backofen [4]. FLD, however, has a strong strain path dependent nature, such that it needs a tremendous amount of data acquisition to depict various in-plane strain modes involved in sheet forming processes[5]. As alternatives to FLD, ductile fracture criteria and continuum damage models have also been used[6]. With recent advances in numerical analysis, particularly in the finite element method, the use of ductile fracture criteria to predict the occurrence of fracture has become popular. Moreover, in order to make the prediction close to the practical situation for various materials and process conditions, the ductile fracture criteria have been constantly modified. In the present study, a new approach was proposed for extending the applications of existing ductile fracture criteria to sheet metal forming

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 461-468 (2011); doi: 10.1063/1.3623645 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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with the effect of deformation history considered. And then its applications in hydraulic bulging of AISI 304 austenitic steel sheets were discussed.

NEW APPORACH FOR FRACTURE PREDICITON WITH A MODIFIED DUCTILE The Background of Ductile Fracture The idea of ductile fracture criteria to predict failure in a material has been applied for the past fifty years since Freudenthal [7] derived a generalized plastic work criterion based on von Misses stress and strain. There have been many ductile fracture criteria proposed over the years, several of which are based on the void growth relation reported by Rice and Tracey [8]. Cockroft and Latham [9] observed that ductile fractures in metal forming processes tend to occur in the region of largest tensile stress, and they proposed a criterion that is based on the total plastic work per unit volume. Brozzo et al. [10] created a criterion which is a modified version of Cockcroft and Latham and takes hydrostatic pressure into account. Oyane et al. [11] have derived a ductile fracture criterion from the equations of plasticity theory for porous materials. Clift et al. [12] have carried out some plane strain experiments and some finite element calculations on an aluminum alloy. It was found that the total plastic-work criterion was the successful criterion for the prediction of site of fracture. Fracture criteria based on continuum damage mechanics have also been proposed by Lemaitre [13]. Takuda et al. [14] applied various fracture criteria to predict fracture in axisymmetric deep drawing of aluminum alloys and mild steel sheets. Hambli [15] simulated damage evolution and fracture failure during bulk metal forming including stamping and extrusion using damage fracture criterion based on continuum damage mechanics.

New Proposal for Fracture prediction Fracture criteria for metal forming processes as mentioned above usually involve a function of the form:

C where f is a certain function of

Vh V

Vh

³ f(V

)dH

and C is an integration when

(1)

H

is the equivalent strain at fracture. It is

considered that a fracture occurs when the C value exceeds the critical value. Although a wide variety of fracture criteria exist, it is still difficult to find one suitable for all forming processes and it should be noted that all the aforementioned fracture criteria were proposed and used for predicting the fractures in cold metal forming with no consideration of effect of deformation parameters, i.e. temperature and strain rate on the accumulative fracture strain. However, the inhomogeneity of deformation leads to varied strain rates at different areas of the workpiece. In other words, there has been a deformation history which has an influence on the occurrence of the facture in the process of deformation and the critical value C considered to be characteristic of material constant was inaccurate to predict the forming process that has a complicated deformation history. Fracture criteria in terms of the influence of temperature and strain rates on deformation history were firstly proposed by the author in 2007 [16, 17] and used for fracture prediction in Non-isothermal forming for magnesium alloys sheet and hot forming of titanium alloy. In our early work [16], only the deformation temperature history was considered for sheet metal forming, and there is a lack of the influence of the strain rates in the criterion. Recently, Kim et al [3] applied the approach for magnesium alloy sheet forming and considered both temperature and strain rate effects. However, just the influence of deformation history on the critical fracture strain that was a function of Zenerholloman parameter was considered in their criteria without the effect of deformation history on the process of damage evolution. In the proposed approach by considering the effect of deformation history on the process of damage evolution, two hypothesizes are proposed: (1) the strain rate is regarded constant when a small part of material is undergoing a finite strain in a short interval; (2) the contribution of the finite strain to final fracture can be expressed with a parameter 'D which can be calculated with the following equation:

'D

f(

Vh ) ˜ 'H / C (T , H ) V

462

(2)

H

f (V h / V ) are the average value of the strain rate and f function in the incremental step that the 'H , C (T , H ) is the critical value of fracture at the deformation condition with temperature T and strain rate H . The integral form of expression (2) can be written as: Hf ª V º (3) D ³ « f ( h ) / C (T , H )»dH 0 ¬ V ¼ where f and C (H ) have the same meanings as in equations (1) and (2); D is named fracture danger index which can where

and

material experienced a finite strain

be regarded as the accumulation on the contribution of finite strain at different deformation conditions to final fracture. When fracture danger index D exceeds 1, a fracture is implied. Then equations (3) is the proposed modified fracture criterion corresponding to the traditional one shown in equations (1).

MATERIALS AND RELATED PARAMETERS Materials Austenitic stainless steel is one kind of widely used engineering materials, due to their high corrosion resistance and versatile mechanical properties. These metastable alloys belong to a kind of typical transformation induced plasticity (TRIP) steels [18] whose high strength and excellent ductility are expected for their strain-induced martensitic transformation [19-21]. They can usually be manufactured by cold working including cold rolling [22], drawing [23] and hydroforming [24]. Therefore prediction on the forming limit for the process of hydraulic bulging of AISI 304 austenitic steel sheets was carried out.

Determination of the Material Parameters in the modified criterion In the present study, the hydraulic bulge tests were carried out at room temperature without considering little internal thermal lift. Since the applied stress and fracture strain of AISI 304 austenitic steel are strongly depended on strain rates, the present work can be considered as an isothermal process and just the influence of the strain rates on accumulated damage value was considered. In addition, the modified Oyane’s ductile fracture criterion is based on damage evolution where fracture occurs when the damage level, D reaches unity. According to the approach in preceding section, f (V h / V ) can be expressed as following:

f (V h / V )

Vh  A(H ) V

where A(H ) is a correction factor including the influence of the strain rate Therefore, equation (3) can be rewritten as follows:

Vh  A(H )) / C (H )]dH V where C (H ) is the critical damage value with strain rate H . D

³

Hf

0

[(

(4)

H . (5)

To determine the material parameters in ductile fracture criterion, experiments of different stress triaxialities [25] or uniaxial tension combined with numerical simulations [26] were generally carried out. However, in the present study considering that it is inaccurate to determine the material parameters from the results of numerical simulation, since when fracture occurs can not be estimated by the commercial finite element software before inducing the ductile fracture criterion in them. So a threshold value of some deformation parameter needs to be pre-set. It is considered that fracture occurs if the simulation results surpass the threshold value, which has inaccuracy compared with the real process. In consequence, two groups of mechanical experiments of different stress state for AISI 304 stainless steel were investigated to determine the modified parameters A(H ) and C (H ) in the present research. Data measurement and collection need to be as simple as possible. Bar notched specimens are widely used to investigate tensile fracture under different stress trixialities. According to analysis by Bridgman [28], relationship between stress trixiality and geometry of specimens is as following:

463

Vh V

§ a 2  2aR  r 2 1  ln¨¨ 3 2aR ©

· ¸¸ ¹

(6)

This ratio is 1/3 at the notch root, r=a, while for r=0 the ratio achieves the maximum value:

Vh V where

V

is equivalent stress and

§ a · 1  ln¨¨1  0 ¸¸ 3 2 R0 ¹ ©

(7)

V h is the hydrostatic pressure or the mean normal stress, a and R are the radius of

the minimum cross-section and the radius of the circumferential, notch, a0 is the initial value of a. The geometries of smooth and notched bar tensile specimens which were selected for the present study are shown in Fig. 1. The stress triaxiality calculated using the initial geometry for both smooth and notched bar tensile specimens were 0.33 and 0.68 respectively. However, it should be noted that the change of the radius of curvature of the neck due to the deformation was not considered in calculating stress and strain components using the Bridgman solution.

FIGURE 1. Smooth and notched bar specimens

A series of tensile tests has been carried out at room temperature under various strain rates within the range for metal forming. Fig. 2 shows the true stress-strain curves of smooth and notched specimens respectively. Fracture strain of AISI 304 stainless steel at various strain rates for different shape specimens was shown in Fig. 3. According to the results of tensile tests, it is obviously noted that the deformation behavior of AISI 304 stainless steel is strain rates-dependent. Fracture strain grew down with strain rates increasing. 1200

1200

(a)

1000

1000

32

800 4 76 5

600 400 200 0 0.0

0.1

0.2

0.3

0.4

0.5

1 1--0.001/s 2--0.003/s 3--0.005/s 4--0.008/s 5--0.01/s 6--0.05/s 7--0.1/s 0.6

True strain

True stress

True stress /MPa

1400

(b)

1

800 7

600

6

5

400 200 0 0.00

0.05

0.10

0.15

True strain

FIGURE 2. The true stress-strain curves at various strain rates for (a) smooth specimens, (b) notched specimens

464

32 4 1--0.001/s 2--0.003/s 3--0.005/s 4--0.008/s 5--0.01/s 6--0.05/s 7--0.1/s 0.20 0.25

0.72

smooth specimens notched specimens

0.5

calculated value fitted curve

0.68 0.64

0.4

C

Fracture strain

0.6

0.3

0.60 0.56

0.2

0.52

0.1 0.00

0.02

0.04

0.06

0.08

0.10

Strain rates FIGURE 3. Fracture strain at various strain rates for different specimens

0.00

0.02

0.04

0.06

0.08

0.10

Strain rates FIGURE 4. Calculated C- H relation through tensile tests and equation (5)

Combined with equation (5) and the data from the tensile tests, the modified material parameters

A(H ) and

C (H ) could be calculated. Fig. 4 shows the relation of C values with strain rates obtained. From curve fitting, the material parameters can be expressed as: (8) A 0.05402 exp( H 0.00645)  0.17955 C 0.44453  0.12951 exp( H 0.25367)  0.09115 exp( H 0.00438)  0.07327 exp( H 0.00437)

(9)

FAILURE PREDICTION IN SHEET HYDRAULIC BULGING Sheet Hydraulic Bulge Test Hydraulic bulge tests were carried out on rectangular specimens of dimension 200 mmÝ200 mmÝ0.46 mm. In this experiment, the die set was installed in a hydraulic driven type universal sheet forming testing machine. Firstly, the sheet metal was clamped at its edges by a blank holder. When the lower bar began to rise and push the lower chamber, oil used as a pressure carrying medium in the lower chamber was pressurized. Then the sheet was bulged into the cavity of the upper die. The structure of hydraulic bulge test setup was shown in Fig. 5.

FIGURE 5. Tools used in hydraulic bulge test

Fracture Prediction Based on the Modified Fracture Criterion In order to numerically predict the bursting which appeared in the previous experimental tests, the currently developed fracture criteria were implemented into the user subroutine of finite element code MSC. Marc, and the damage values were evaluated at each step to monitor the fracture initiation. In addition, the sheet blank was meshed by four-node 3-D shell elements. The die and blank holder were defined rigid. Due to the symmetry of the model, 1/4 part was applied as shown in Fig. 6. Six loading paths were applied in the simulation as shown in Fig. 7. The maximum hydraulic pressure was uniform, while the loading time was different for the six loading paths. In other

465

words, the loading rates varied. The results of simulation using the modified fracture criterion are compared with the traditional one in Fig. 8. It can be seen that the predicted expanding height using the modified fracture criterion is more close to the experimental results than using traditional one. And it is also obviously noted that the limit expanding height of bulged sheet decreases with increase of loading rate by using the modified fracture criterion. However, the limit expanding height rarely fluctuates for different loading paths in the simulation by using tradition ways. From the results of tensile tests, it is known that it has a great influence of strain rates on the deformation behavior of AISI 304 stainless steel. That is to say, the modified fracture criterion makes the fracture prediction more accurate and close to the real situation. In addition, as shown in Fig. 9, the simulation results based on the modified fracture criterion show that the fracture position occurs near the dome top of the bugled sheet instead of at the center of the dome top, which agreed well with the experimental observation.

FIGURE 6. Finite element model in simulation

Expanding height /mm

Hydraulic pressure /MPa

18 16 14 12 10

Path 1 Path 2 Path 3 Path 4 Path 5 Path 6

8 6 4 2 0

2

4

6

8

10

200 400 600 800 1000

Time /s

FIGURE 7. Loading paths applied in the simulation 36 prediciton by new approach prediciton by traditional approach 34 experimental result

32 30 28 26 24 22 -2

0

2

4

6

8

10

12

14

16

18

Loading rate /MPa*s-1 FIGURE 8. Comparison of the limit expanding height of bulged sheet obtained from the simulation

466

FIGURE 9. Comparison of fracture predicted by finite element simulation coupled (a) with the modified fracture criterion, (b) without the traditional fracture criterion and (c) the experimentally observed fracture near the dome top of bulged specimen

CONCLUSIONS (1) A new approach for fracture prediction based on damage theory by considering the influence of strain rate history is proposed. (2) Uniaxial tensile testes for AISI 304 steel specimens under different stress conditions are carried out at different strain rates, and the material parameters (modification factor A and critical damage value C) in the proposed fracture criterion are determined to be function of strain rate. (3) The application of the new fracture criterion in the fracture prediction of hydraulic bulging of AISI steel sheets shows a good agreement with the experimental results, which validates the proposed method.

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F. Dohmann and C. Hartl, J. Mater. Process. Technol 60. 669-676 (1996). M. Koc and T. Altan, J. Mater. Process. Technol 108. 384-393 (2001). M. Koc and T. Altan, Int. J. Mach. Tools Manuf. 42:123-138 (2002) H. K. Kim and W. J. Kim, Mech. Mater 42. 293-303 (2010). T. B. Stoughton and X. Zhu, Int. J. Plasticity 20, 1463-1486 (2004). J. Kim et al., Int. J. Adv. Manuf. Technol 22. 357ˀ 362 (2003). A.M. Freudenthal, The Inelastic Behaviour of Engineering Materials and Structures, Publisher City: John Wiley & Sons, New York, 1950. J. R. Rice and D.M. Tracey, J. Mech. Phys. Solids 17. 201–217 (1969). M. G. Cockroft and D.J. Latham, J. Inst. Met 96. 33-39 (1968). P. Brozzo et al., Proceedings of the Seventh Biennial Conference of IDDRG, Amsterdam, The Netherlands, 1972. M. Oyane et al., J. Mech. Work. Tech 4. 65 (1980). S. E. Clift et al., Int. J. Mech. Sci 32. 1-17 (1990). J. Lemaitre, Trans. ASME: J. Engng. Mater. Technol 107. 83–89 (1985). H. Takuda et al., J. Mater. Process. Technol 95. 116-121 (1999). R. Hambli and D. Badie-Levet, Comput. Methods Appl. Mech. Engrg 186. 109-120 (2000). W. T. Zheng et al., Proc. IMechE, B: J. Eng. Manu 221. 981-986 (2007).

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17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

S. H. Zhang et al., Proceedings of the Plasticity of Conventional and Emerging Materials: Theory and Applications, Alaska, 2007. F. D. Fischer et al., Int. J. Plasticity 16. 723–748 (2000). S. S. Hecker et al., Metall. Trans 13A. 619 (1982). M. R. Rocha et al., Mater. Sci. Eng. A 517. 281–285 (2009). Y. Xu et al., Mater. Lett 65. 1545–1547 (2011). S. Gallée and P. Pilvin, J. Mater. Process. Technol 210. 835–843 (2010). M. Ahmetoglu et al., J. Mater. Process. Technol 98. 25–33 (2000). D. Raabe, Acta. Mater 45. 1137-1151 (1997). M. S. Mirza et al., J. Mater. Sci 31. 453-461 (1996). P.W. Bridgman, Studies in Large Plastic Flow and Fracture, Publisher City: McGraw-Hill, New York, 1952.

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Identification of Deformation Mechanisms Responsible for Failure in Incremental Forming using a Damage Based Fracture Model Rajiv Malhotra, Liang Xue, Jian Cao* and Ted Belytschko

Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA Abstract. Single Point Incremental forming (SPIF) has generated significant interest recently due to its increased formability and greater process flexibility. However, the complicated deformation mechanisms involved in SPIF have prevented conclusive identification of the primary mechanisms responsible for failure. This work successfully predicts the forming forces and occurrence of failure in SPIF using explicit FEA with a damage based fracture model in which failure envelope depends on the hydrostatic pressure and the Lode angle. Furthermore it is shown that through-thethickness shear is primarily responsible for failure in SPIF. Simulations are also performed to form the same component using SPIF and using a conventional punch and die, and a comparison is made between the dominant mechanisms of failure in the two processes. Furthermore, it is shown that reduction of tool-sheet friction can delay fracture in SPIF and the mechanism behind this effect is discussed as well. Keywords: Incremental Forming, Fracture, Through-the-thickness shear, Friction PACS: 81.20.Hy

INTRODUCTION Single Point Incremental Forming (SPIF) is a method of forming sheet metal into a desired shape without using a conventional die and punch set and has envisioned considerable interest due to its greater process flexibility, higher formability and significant cost savings for low batch volume production. However, a significant challenge in this process has been the prediction of fracture during the forming process and a better understanding of the physical mechanisms governing fracture in SPIF is of significant importance to the engineering community. In the recent past Silva et al. [1] compared a membrane based analytical solution of the stress and strain fields with a shell based simulation in LS-DYNA and found good agreement between the principal stresses. Malhotra et al. [2] investigated the use of various material models to simulate SPIF using FEA and showed that a damage based material model can predict forces in the process quite well. Cerro at al. [3] simulated SPIF of a pyramid with a 75° wall angle with shell elements and obtained a 5% difference between the maximum values of the measured and calculated tool z forces. Ambrogio et.al [4] simulated SPIF with one linear brick element through the thickness of the sheet and reported that a compressive mean stress was observed in the local zone of contact near the tool. Van Bael et al. [5] extended a Marciniak-Kuczyisnki type of localized necking prediction model with three different material models and showed that while the forming limit predictions were higher than that for monotonic loading, their models still underestimated the forming limits obtained experimentally in SPIF. What has been lacking in most of these efforts has been the lack of a means to accurately predict occurrence of fracture in SPIF. This work deals with the prediction of fracture and localization in SPIF using a damage based material model in FEA in which damage accumulation is a function of the hydrostatic pressure and shear (via the Lode angle). SPIF experiments were performed to form a cone and a funnel shape and the measured forces and fracture locations are found to match well with corresponding FEA predictions. The same material model is used to simulate the forming of the cone shape using a larger punch and die and the differences between conventional forming and SPIF in terms of the mechanisms governing fracture are discussed. Moreover, it is shown that a reduction in friction can increase the fracture depth in SPIF and the mechanisms responsible for this effect are discussed as well.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 469-476 (2011); doi: 10.1063/1.3623646 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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MATERIAL MODEL In the present work, the mechanical response of the blank material is modeled using the damage plasticity model proposed by Xue [6]. The constitutive relationship consists of a damage coupled yield function, the evolution laws for the plastic strain and a reference fracture envelope. In particular, the stress-based fracture envelope is chosen. The damage coupled yield function is expressed as shown in Eqn. 1

) V eq  w( D )V M d 0

(1)

where eq is the equivalent stress, M is the matrix resistance and w is a scalar weakening factor used to describe the material deterioration. The factor w is related nonlinearly to the damage variable D as w(D)=1-D where  is a material constant. In Eqn. 1 the undamaged matrix resistance M is expressed as a function of the plastic strain p using the Swift hardening relationship i.e. M=y0 (1+p/0 )n where y0 is the initial yield stress, 0 is the initial yield strain and n is the hardening exponent. The associated flow rule is enforced. The damage accumulation is described using a power law function of the plastic strain p and a limiting fracture strain f and the evolution of damage is expressed in the rate form as x

D

§Hp m¨ ¨H © f

· ¸ ¸ ¹

m 1

x

Hp Hf

(2)

where m is a material constant and f is a function of the hydrostatic pressure and the Lode angle of the current stress state. The fracture envelope is given in the stress space envelope Mf [7], which takes the form of a modified Tresca type of pyramid as Mf=f0(1+kpp)1/n(3/2cosL)1/n where f0 is the fracture stress under uniaxial tension, kp is a material constant related to pressure sensitivity, p = - (kk /3) is the hydrostatic pressure and L is the Lode angle. The limiting fracture strain f is then obtained from the inverse of the matrix stress as shown in Eqn. 3. An element completely loses its load carrying capacity and is removed when the value of the damage variable D reaches 1.0. Additionally, closed form solutions for the onset of the diffused necking, i.e. a maximum power criterion, which is a three dimensional generalization of the one-dimenional Considere condition [8], and the onset of localized necking, i.e. occurrence of local shear bands, in a Hadamard-Hill sense [9] are included in the material model as well. 1/ n ½ ­§ V ·1 / n ª 3 º ° °¨ f 0 ¸ H f ( p,T L ) H 0 ®¨ (1  k p p )  1¾ « » ¸ 2 cosT L ¼ °¿ °¯© V y 0 ¹ ¬

(3)

EXPERIMENTS AND FINITE ELEMENT ANALYSIS SPIF experiments were performed to form a 70° wall angle cone and a variable angle funnel (Figs. 1a and 1b) on 1 mm thick AA5052 sheet with tool of diameter 9.525 mm, feed rate 150 mm/min, using a helical toolpath with incremental depths of 1.0 mm and 0.5 mm respectively. To examine the effect of reduction in tool sheet friction coefficient, experiments were also performed for SPIF of the funnel with a Teflon sheet wrapped around the tool and Teflon lubricant was used on the sheet in addition to the regular lubricant. The Z forces on the tool were measured throughout the SPIF process using a Kistler dynamometer mounted below the fixture and the tool tip depths at which fracture occurred were recorded in each case. FEA was performed in LS-DYNA to simulate SPIF of the cone and the funnel shape with coefficient of friction  = 0.10 at the tool sheet interface (Fig. 1c), SPIF of the funnel with  = 0.0 (corresponding to the reduced friction funnel experiment) and for the 70° wall angle cone formed with a larger punch (Fig. 1d) using four linear brick elements through the thickness of the sheet. The tools were modeled as rigid bodies and the fixture was modeled using a top and a bottom clamping plate with coefficient of friction of 0.15 between the blank and the fixture. A user subroutine was coded to implement the aforementioned material model in LS-DYNA. The Young’s Modulus E, yield stress y0, hardening coefficient n and initial yield strain 0 for the blank material were obtained from tensile tests (Table 1). The material constants f0, kp , m and  cannot be obtained using tensile tests and were calibrated manually by matching the tool Z forces from 3 trial simulations with those from experiments for SPIF of the 70° cone (Table 1). To show the quality of calibration of the damage model parameters, the tool Z forces from experiments and simulation for the 70° cone are compared in Fig. 2a. Furthermore, the

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maximum thinning just before fracture was measured to be 64% from experiments and 64.1% from simulation. The fracture depths measured from experiments and predicted by corresponding simulations for SPIF are shown in Table 2. The predicted and experimental tool Z forces for SPIF of the funnel with the regular lubricant are shown in Fig. 2b. Both predicted tool Z forces and the fracture locations in SPIF are found to match quite well with experiments. For the simulation of forming the cone with the large punch the predicted fracture depth was 14.5 mm, i.e. the formed depth in SPIF was greater than in the punch forming case. The deformation mechanics in SPIF and the large punch case were examined by looking at the deformation history from FEA, of four elements through the thickness of the sheet at four sections A, B, C, and D, along the wall of the formed component (Fig. 2c). Section D is the zone where the through-the-thickness crack first appears. At each of these sections the four elements through the thickness are labeled ‘1’ to ‘4’ where ‘1’ is the element on the side of the sheet which is in contact with the tool (i.e. the inner side) and ‘4’ is the element on the other side of the sheet (i.e. the outer side). 25 mm

50

30

70°

SPIF tool Top clamp (green)

20 mm

Blank (Red)

Large punch Blank holder (green)

Blank (Red)

Bottom clamp (yellow) Bottom die (yellow)

85°

(a) (b) (c) (d) FIGURE 1. (a) CAD model of 70° cone (b) CAD model of variable angle funnel (c) FEA model for SPIF (d) FEA model for 70° cone formed with large punch TABLE (1). Calibrated material parameters.  (kg/m3) 2680

E (GPa) 68.6

 0.3

y0 (MPa) 117

n 0.22

0 0.0045

fo(MPa) 490

kp (Mpa-1) 0.0001

m 2.0

 3.0

TABLE (2). Comparison of SPIF tool-tip depths at fracture measured from experiments and predicted by FEA Component Experiment (in mm) FEA (in mm) 70° cone 16.9 16.1 Funnel (regular lubricant) 15.02 15.03 Funnel (regular lubricant & Teflon) 15.56 15.45

Section A Section B Section C Section D

(a) (b) (c) FIGURE 2. (a) Comparison of tool z-forces for 70° cone (b) Comparison of tool z-forces for variable angle funnel with =0.10 (c) Sections along component wall at which deformation history is examined

Material Instability and Failure This section examines the reasons for the increase in the formed depth in SPIF as compared to the punch forming case in terms of occurrence of material localization. Figures 3a and 3b show that the plastic strain p for SPIF is always higher at any section, than it is for the punch case. Moreover, the plastic strain before failure in SPIF is about

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20% higher than that before failure in the punch case. The reasons for this trend can be found by comparing the occurrence of diffused and localized necking in both cases (Figs. 4 and 5). In Figs. 4 and 5, for the localization flag contour plots, the localization flag has a value of 1.0 (blue colour) in regions when the material is in a diffused necking state and a value of 3.0 (red colour) when the material is in a localized necking state (i.e. shear bands have formed). Furthermore the regions with diffused and localized necking are marked as D and L respectively.

(a) (b) FIGURE 3. Evolution of plastic strain along sections A,B,C,D for 70° cone formed using (a) SPIF (b) Large punch

FIGURE 4. For SPIF (a) at onset of diffused necking (b) at onset of localized necking (c) just before fracture

FIGURE 5. For Punch forming case (a) at onset of diffused necking (b) at onset of localized necking (c) just before fracture

Figures 4a and 5a show that in SPIF diffused necking starts out at an earlier Z depth (5.6 mm) as compared to the punch case (8.6 mm). In both cases the material deformed after this point is in a diffused necking state for the next 5 mm of Z depth before localized necking starts (Figs. 4b and 5b). Once localized necking begins the component fractures very quickly in the punch case (within 1.5 mm Z depth) as compared to SPIF where material for about 6 mm in Z depth undergoes localized necking before fracture. Since deformation in the case of the punch forming is global in nature, after initial localized necking occurs this unstable material is still being actively stretched by the

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punch resulting in rapid growth of the shear bands which leads to rapid fracture. In contrast, in SPIF the tool is constantly moving and while the newly deformed regions of the blank in the tool contact zone are becoming unstable, the previously formed unstable region is not being actively deformed. Therefore the shear bands already present in the previously formed unstable regions do not grow as quickly leading to a larger region which is in a state of localized necking before fracture occurs. The effect of this previously formed material becoming unstable enough to undergo localized necking but not reaching the point of fracture, is that the some of the deformation of newly formed material in the subsequent passes of the tool is taken up by this previously formed localized region. This allows the newly formed material to undergo greater amount of plastic strain and reach a greater Z depth without fracture. This effect is also seen by examining the plastic strain contours for both cases which show that in the punch case (Fig. 5c) the plastic strain becomes concentrated very quickly into the shear bands in the localized necking region before fracture. However, in SPIF (Fig. 4c) the plastic strain is distributed over a larger localized necking region and has greater magnitude before fracture, as compared to the punch case. Further evidence for this is obtained by observing that in the punch case (Fig. 3b) the plastic strain is very highly concentrated at sections C and D. In fact at section D after a certain point (at which point this section is in a state of localized necking) the plastic strain rate increases dramatically. No such effect can be seen in SPIF (Fig. 3a) where the increase in plastic strain from sections A to D is fairly regular. The occurrence of material localization was further confirmed by the observation of regions along the circumference of the formed sheet on the outer surface showing distinctive bands that indicated material localization in these regions (Fig. 6). At this point it is useful to keep in mind that occurrence of diffused and subsequent localized necking is influenced by damage accumulation as well as by material weakening [8,9]. As will be shown in the next section, through-the-thickness shear plays a significant role in damage evolution in SPIF and therefore indirectly influences occurrence of material instability as well as the resulting effects mentioned above.

FIGURE 6. Regions along the outer surface of SPIF components indicating material localization

Deformation Mechanisms affecting Damage Evolution

(a) (b) FIGURE 7. evolution of damage index (D) along sections A,B,C,D for 70° cone formed using (a) SPIF (b) Large punch

In SPIF the damage index D (Fig. 7a) evolves faster for element 4 as compared to element 1. On the other hand in the case of the punch forming (Fig. 7b) the difference in D between elements 1 and 4 is negligible. Therefore, in

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SPIF element 4 is the first element to be removed and the crack propagates from outside to the inside of the sheet, whereas for the large punch all the elements through the thickness of the sheet are removed simultaneously. In SPIF the hydrostatic pressure (Fig. 8a) is negative on the inner side (element 1) and positive on the outer side of the sheet (element 4). This is due to local bending of the sheet around the small hemispherical SPIF tool. When the tool has passed over a particular region, that region undergoes local springback as a result of which the hydrostatic pressure becomes negative on the inner side and positive on the outer side of the sheet. This means that for SPIF the term (1+kpp)1/n in Eqn. 3 is greater than 1.0 for element 4 and lesser than 1.0 for element 1 which, in the absence of a significant shear effect, should cause the limiting fracture strain f to be higher for element 4 than for element 1. For the punch forming case (Fig. 8b), since the material is continuously being stretched without any local springback, the hydrostatic pressures are always equally negative on either side of the sheet. Therefore the aforementioned pressure term will also be equally lesser than 1.0 on either side of the sheet, and will not cause a difference in f between the two sides of the sheet. To pinpoint the effect of shear on f the product term [(1+kp.p). (3/2.cos L)](1/n) in Eqn. 3 is split into a pressure term (1+kpp)1/n and a Lode angle term (3/2.cos L) (1/n) and the evolution of all these terms at section D is plotted against p (Fig. 9). In SPIF (Fig. 9a), inspite of the pressure term following the aforementioned trends, the product term and therefore f is much lower for element 4 due to the Lode angle term being much lower for element 4. Therefore, in SPIF the effect of shear on f dominates over the effect of the hydrostatic pressure. In the case of the punch the the Lode angle terms are similar on either side of the sheet (Fig. 9b) resulting in the product term and therefore the fracture strain f being similar on either side as well. In SPIF since p is higher (Fig. 3a) and f is lower (Fig. 9a) on the outer side of the sheet, damage is higher on the outer side (from Eqn. 2) causing the crack to begin on the outer side of the sheet. In the case of the punch since the difference between p and f on either side of the sheet is negligible, so is the difference in damage accumulation and the crack begins simultaneously through the thickness of the sheet. Therefore in SPIF the dominant mechanisms governing the evolution of damage and the nature of fracture are local bending (causing higher p for element 4) and through-thethickness shear (causing lower f for element4). In contrast, in the case of the large punch the deformation is dominated by uniaxial tension along the component wall and failure is by a continuous circumferential crack.

(a) (b) FIGURE 8. Evolution of Hydrostatic Pressure along sections A,B,C,D for 70° cone formed using (a) SPIF (b) Large punch

A further effect of shear on fracture in SPIF can be seen by examining the deformation mechanisms responsible for the increase in fracture depth for SPIF of the funnel when the friction at the tool sheet interface is reduced. Figures 10a and 10b show that when  =0.0, as compared to when  =0.10, the reduction in damage accumulation is much larger for element 1 than it is for element 4. Therefore the increase in final fracture depth of the component is mainly because element 1 takes longer to be removed when  is reduced. Figure 10d shows that the reduction in plastic strain when  =0.0 is again much larger for element 1 than it is for element 4 which by itself implies larger reduction in damage accumulation for element 1 than for element 4 (Eqn. 2). Comparing the tensile strains along the component wall (Fig. 10e) and the shear strains along and perpendicular to the tool motion (Figs. 10f and 10g respectively) shows that the reduction in shear strain along the toolpath for element 1 is not just greater than that for element 4, it is also the most significant reduction among these three major components of the strain. Therefore the reduction in shear along the toolpath motion, is the most significant factor responsible for the significantly greater reduction in plastic strain for element 1 as compared to element 4 (Fig. 10d), when  is reduced.

474

(a) (b) FIGURE 9. Comparison of product term, pressure term and Lode angle term for 70° cone with (a) SPIF (b) larger punch

(a)

(b)

(c)

(d)

(e) (f) (g) FIGURE 10. Comparison of (a) Damage evolution (b) Reduction in damage evolution (c) Plastic strain (d) Reduction in plastic strain (e) Tensile strain along component wall (f) Through-the-thickness shear along component wall (g) Through-the-thickness shear along tool motion, between  =0.10 and  =0.0 case, for elements 1 and 4.

.

(a) (b) FIGURE 11. Comparison of Pressure, Lode angle and Product terms of Eqn .3 between  =0.10 and  =0.0 cases for (a) Element 4 (b) Element 1.

Comparing the individual components of the fracture envelope f at section D for the  =0.10 and  =0.0 cases (Fig. 11) shows that for both elements 1 and 4 the pressure term does not differ significantly with a reduction in  and for element 4 the Lode angle terms and therefore the product terms are quite similar irrespective of the value of

475

 (Fig. 11b). However, for element 1 (Fig. 11a) the Lode angle term is slightly reduced with reduced , resulting in a reduction in the product term. Therefore the reduced through-the-thickness shear due to reduced friction reduces f and therefore tends to accelerate damage accumulation on the inner side of the sheet. Since damage accumulation depends on the ratio p/ f (Eqn. 4) and the reduction in p is much larger (Fig. 10b) than the reduction in f (Fig. 11a), the net effect is a larger reduction in damage accumulation for element 1 as compared to element 4 when  is reduced (Figs. 10a and 10b). Physically this means that reduced tool sheet friction causes the crack to initiate on the outer side of the sheet at pretty much the same time but the crack takes longer to propagate all the way to the inner side and cause a complete through-the-thickness crack. This results in the observed increase in fracture depth.

SUMMARY AND CONCLUSIONS This work uses a damage based fracture and localization prediction model in which damage evolution is dependent on the hydrostatic pressure as well the shear (via the Lode angle), to simulate SPIF of a cone and a funnel shape as well as forming of the same cone with a larger punch and die. Good agreement between experiments and simulations is obtained for tool forces and fracture locations in SPIF. It is shown that in SPIF, the evolution of damage is controlled by local bending around the tool as well as by through-the-thickness shear whereas in the punch case it is influenced mainly by tension along the component wall. The evolution of damage also affects occurrence of diffused and localized necking [8,9]. In SPIF the local nature of the deformation causes diffused and localized necking to develop faster than in the punch case, which is characterized by a more global deformation. However, this very feature of SPIF also prevents the localized region of the sheet from going to failure very quickly. As a result a significant portion of the sheet undergoes localized necking before failure allowing this region to take some of the plastic strain in subsequent passes of the tool. As a resulting the plastic strain and the Z depth before before fracture is greater in SPIF than in the punch forming case. In terms of failure in SPIF, any combination of operational parameters that can reduce the rate of damage accumulation and therefore delay localization, or increase the region of the sheet in a state of localized necking before fracture, can increase formability. One such operational parameter is friction at the tool sheet interface. This work shows both experimentally and via simulations for SPIF of a funnel shape that a reduction in friction reduces damage accumulation on the inner side of the sheet, primarily via a reduction in through-the-thickness shear and consequently in plastic strain. The result is an increase in fracture depth.

ACKNOWLEDGMENTS The authors gratefully acknowledge the support provided by the National Science Foundation Grant #CMMI0758607, for this work.

REFERENCES 1. 2.

3. 4.

5.

6. 7. 8. 9.

M.B. Silva, M. Skjoedt, N. Bay, P.A.F. Martins, “Revisiting single-point incremental forming and formability/failure diagrams by means of finite elements and experimentation”, Journal of strain analysis 44, 221-234 (2009). R. Malhotra, Y. Huang, L. Xue, J. Cao, T. Belytschko, “An Investigation on the Accuracy of Numerical Simulations for Single Point Incremental Forming with Continuum Elements” in Proceedings of the 10th International Conference on Numerical Methods in Industrial Forming Processes NUMIFORM 2010, AIP Conference Proceedings 1252, 221-227 (2010). I. Cerro, E. Maidagan, J. Arana, A. Rivero, P.P. Rodríguez, “Theoretical and experimental analysis of the dieless incremental sheet forming process” in Journal of Materials Processing Technology 177(1-3), 404-408, (2006). G. Ambrogio, L. Filice, L. Fratini, F. Micari, “Process Mechanics Analysis in Single Point Incremental Forming” in Proceedings of the 8th International Conference on Numerical Methods in Industrial Forming Processes NUMIFORM 2004, AIP Conference Proceedings 712, 922-927 (2004). A. Van Bael, P. Eyckens, S. He, C. Bouffioux, C. Henrard, A.M. Habraken, J. Dulfou, P. Van Houtte, “Forming limit predictions for single point incremental sheet metal forming” in Proceedings of the 10th ESAFORM Conference on Material Forming, AIP Conference Proceedings 907, 309-314 (2007). L. Xue,“Damage accumulation and fracture initiation in uncracked ductile solids subjected to triaxial loading” in International Journal Solids and Structures 44(16), 5163-5181 (2007). L. Xue “Stress based fracture envelope for damage plastic solids” in Engineering Fracture Mechanics 76(3), 419-438, (2009). L. Xue, “Localization conditions and diffused necking for damage plastic solids” in Engineering Fracture Mechanics 77, 1275-1297, (2010). L. Xue, T. Belytschko, “Fast methods for determining instabilities of elastic-plastic damage models through closed-form expressions” in Internationl Journal for Numerical Methods in Engineering 84, 1490-1518, (2010).

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Prediction of Fracture in Deep Drawing Process, Using Ductile Damage Criteria Hassan Nejatbakhsha , Mohamad Khataeib*, Mehrdad Poursinac a,

Faculty of Engineering, University of shahrekord, 8118634141 shahrekord, Iran b* Department of Mechanical Eng., Isfahan University of Technology, Iran. c Faculty of Engineering, University of Isfahan, 8174473441 Isfahan, Iran

Abstract. In the deep drawing process, determination of the drawing depth and prediction of the time and the place that fracture occurs has been one of the important case studies which engineers tend to take into account. Because of a drastic reduction in design and manufacturing expenditures, numerical methods are extended to calculate the drawing depth during the process. In this paper, ductile damage model in which the stress triaxiality and equivalent strain are the most effective parameters on the damage growth and fracture of the material is used to predict fracture. For prediction the place and time of ductile fracture, according to ductile damage criteria, the fracture strain for various stress triaxiality values should be determined. To obtain the parameters of ductile damage model for St12 steel, some tensile tests have been performed on the notched specimens. Numerical simulation of deep drawing was performed using commercial finite element ABAQUS. Results obtained from simulation are in good agreement with the experimental ones and emphasize that using ductile damage model is appropriate to anticipate the place and time of the fracture during the deep drawing process. Keywords: Fracture; Ductile damage; Deep drawing; Experiment; St 12 steel. PACS: 62.20.me

INTRODUCTION Exceeding the usages of finite element software in forming processes simulations, several failure and damage models have been implemented in finite element software and researchers have investigated intensely about different damage models as well. Rice and Tracey [1] have shown that fracture of the ductile metals is strongly dependent on hydrostatic stress by studying growth of spherical voids. Hancock and Mackenzie [2] performed experiments on notched specimens and observed that ductile damage depends severely on stress triaxiality value applied to the material. Johnson and Cook [3] in 1985 presented a damage model that depending to stress triaxiality, strain rate and temperature by performing practical experiments. Their offered model included some material constants which had to be determined via specific experiments. Many studies were carried out defining the parameters of Johnson and Cook’s model. One of the most appropriate approaches was presented by Holmquist [4] in 1991 which was based on investigating the effects of each parameter separately on materials failure. In 1998, Rule [5] reported a method for specifying the constants of the model with high values of strain rates. Following that, Juthras [6] presented an almost perfect and accurate technique for Johnson and Cook’s model in 2008, using torsion tests. After that, several investigations were carried out to find an equation to relate fracture strain to stress triaxiality for various materials. Hooputra et al. [9] from BMW R&D center calculated the damage parameter according to an equation for stress triaxiality. With the aim of their suggested equation, it is possible to evaluate damage growth numerically and predict the probable arias of fracture. In the present paper, the tension tests were carried out initially on notched specimens and constant values of Hooputra model for St12 steel were obtained. Following that, deep drawing process was modeled and simulated commercial finite element ABAQUS, applying ductile damage model for predicting the failure zones in the simulation. Finally, real deep drawing tests were performed and simulation results from Hooputra ductile damage model were compared with experimental results.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 477-483 (2011); doi: 10.1063/1.3623647 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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DUCTILE DAMAGE MECHANIC Ductile damage is a proceeding physical phenomenon which causes the fracture of the materials. This phenomenon occurs in three steps which are void nucleation, void growth and void coalescence. To analysis the growth of ductile damage in every step of loading, a parameter is defined as ductile damage parameter (D) which is the ratio of the void area in intersection to the total area of the intersection. Obviously this parameter is always between zero and one values (0
DnG =

δ SD δS

(1)

Where is void area in intersection and is total area of the intersection. In ductile damage model it is possible to predict fracture at every material point by numerical evaluation of damage parameter (D). Therefore fracture strain of the material would have a direct relation with damage parameter and its growing rate. In ductile damage model the fracture strain is assumed to be affected by stress triaxiality which is applied to the material [9]. ε f = f (η ) (2) In which εf is fracture strain and η is the stress triaxiality applied to the material and is calculated as:

η=

σm σ eq

(3)

And

σm =

σm =

[

1 (σ 1 + σ 2 + σ 3 ) 3

1 (σ 1 − σ 2 )2 + (σ 2 − σ 3 )2 + (σ 1 − σ 3 )2 2

(4)

]

(5)

Where σ1 , σ and σ are principal stresses. Hooputra equation defines the fracture strain of a material as a function of stress triaxiality [9]:

ε f = D1 exp(−Cη ) + D2 exp (C η ) coefficients D1 , D2 and C which are materials

(6)

The equation has three property and should be obtained by experiments. After obtaining the above coefficients for the specified material, the damage parameter at each material point can be calculated according to the following equation [10]: εf

D=

dε f

∫ ε (η ) 0

(7)

f

Having D for every element, the numerical evaluation of growing rate can be reached. By reaching the D value to a critical value (D ), the material is fractured in that element and does not withstand any stress practically. The value of D is different for each material; however it is usually between 0.2 and 0.5 for ductile metals [8].

EXPERIMENT To obtain the constant coefficients of equation (6) and to calibrate this equation for St12 steel, three samples were cut from a steel sheet in identical direction (rolling direction), Fig. 1. To have different states of stress, these samples were given three notches in center with different radius, while the intersection area of each sample is similar.

478

FIGURE 1. Geometry of test specimen

These three samples were under gone to tensile test and were pulled till fracture occurs. Because the stress and strain in the center of the samples are local, fracture occurs in the center. The stress triaxiality and fracture strain were calculated for using the equation (7), which are improved Bridgeman’s equations for plain-strain state of sheet samples [11]

η=

t ⎞⎤ 3⎡ ⎛ 1 + 2 Ln⎜1 + 0 ⎟⎥ ⎢ 3 ⎣ ⎝ 4 R ⎠⎦

(8)

⎛t 2 Ln⎜ 0 3 ⎜⎝ t f

(9)

εf =

⎞ ⎟ ⎟ ⎠

Where t the initial ligament thickness of the specimen, t is the fracture ligament thickness of the specimen and R is the local radius of a neck. This above formula defines the average strain through the cross section, Fig. 2.

(a)

(b)

(c)

(d)

FIGURE 2. Plane strain flat specimens with different notches (a) R=4mm; (b) R=6mm; (c) R=12mm and (d) fractured specimen. (The specimens have the same thickness (1.6 mm) of the minimum cross section)

After performed experiments, the value of εf and η is available for each sample and fracture strain presented in Table1. By using these values, the constant coefficients of equation (6) for St12 steel will obtain. The fracture diagram of this material in terms of εf and η is shown in figure (3). TABLE (1). Experimental result for calibration Johnson-cook equation.

Notch radius(mm) 4 6 12

Fracture strain(εf ) 0.7016 1.2557 1.6121

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Stress triaxiality (η) 0.6874 0.6518 0.6152

FIGU URE 3. Fracturre locus of St122 steel in stress triaxiality t and equivalent e strainn plain using joohnson-cook eq quation

NUMERIICAL SIMU ULATION In this section, the deep d drawing process simuulated by ABA AQUS, the duuctile damagee model of Ho ooputra was h pushes the applied acccording to preevious sectionn data. In the simulated deeep drawing prrocess, a cylinndrical punch material intto the die while a holder wiith 20 KN forcce prevents an ny shrinkage on o the sheet. T The punch, diee and holder were assum med to be rigidd and the sheeet was consideered a deformaable and axis--symmetric. T The simulated process and the positionn of sheet, punnch and die is illustrated in figure 4.

FIGURE 4. Deep D drawing process p simulattion

Two sim mulations withh different sizzes for die annd punch’s rad dius were carrried out. In thhe first simulaation the die and punch’’s radius weree 6 and 8mm, respectively. Figure 5 show ws the ductile damage paraameter distribu ution for the deep drawnn specimen. Itt can be obseerved that the damage param meter is at itss maximum vvalue at the bo ottom of the vessel whicch indicates thhe critical zone for imminennt failure to occcur. Consideering the figurre 5, the maximum value off damage paraameter is 0.13 which has a great margin with critical value and prescribes p thaat the ductile damage moddel predicts faailure never occur in this ssimulation. Th he produced part, by die and punch 6 and 8mm sizes, respecttively and 35mm depth off drawing, and force-displaacement are ved in the parrt. demonstratted in figure 6 while no failuure or rupturee can be observ In the second s simulaation, the sizees of punch annd die were assigned a smalller, equal to 3 and 4mm, respectively. r Figure 7 eluucidates the damage d param meter distribution in such con nditions and with w 25mm deepth of drawin ng.

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FIGURE 5. Damage parameter distribution

(a)

(b)

FIGURE 6. a)The produced part without any failure or rupture b) Force-Displacement diagram for simulation and experimental

FIGURE 7. Damage parameter distributions with 4mm die radius

In these conditions, the damage parameter has reached to its critical value in side areas which is 0.45 and indicates that the failure may occur in the area. The simulated part is illustrated in figure 8 applying eliminating elements technique and predicting the imminent failure zones, alongside with experimental specimen. Obviously, the predicted failure zone by ductile damage model has satisfactory agreement with experimental failed zones.

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The damage parameter of ductile damage for critical element of model respect to depth of drawing is demonstrated in figure 9. It can be observed that the damage parameter has increased by displacing, exceeding to its critical value. Following that, the damage parameter has been remained constant due to applying eliminating elements technique and the considered element has been discarded off calculations.

FIGURE 8. Simulated part and experimental specimen

FIGURE 9. The damage parameter curvature respect to depth of drawing for an element of specimen

CONCLUSION Using damage models, the time and location of failure can be satisfactorily predicted in forming processes. Comparing the experimental results with simulation outcomes and reasonable agreement between theoretical and practical outputs proves the considerable capabilities of Hooputra model of ductile damage in predicting time and areas of failure in forming processes. In addition, the accuracy of results in determining the constant parameters of the Hooputra model by running experiments over notched specimen, this method can be considered as a proper approach in determining the model parameters.

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REFERENCES 1 J.R. Rice and D.M. Tracey, “On the ductile enlargement of voids in triaxial stress fields”, J. Mech. Phys. Solids, Vol. 17, pp 201-217, 1969 . 2 J.W. Hancock, A.C. Mackenzie, “On the mechanisms of ductile failure in high strength steels subjected to multi-axial stressstates”, J. Mech. Phys. Solids, Vol. 24, pp 147-169, 1976. 3 G.R. Johnson, W.H. Cook, “Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures”, Engineering Fracture Mechanics Vol. 21, No. 1, pp. 31-48, 1985. 4 T.J. Holmquist, G.R. Johnson, “Determination of constants and comparison of results for various constitutive models”, Journal de physique IV, Colloque C3, suppl. au Journal de physique III, Vol. 1, pp. 853-860, 1991. 5 W.R. Rule, S.E. Jones, “A revised form for the Johnson-Cook strength model”, International Journal of Impact Engineering, Vol. 21, No. 8, pp. 609-624, 1998. 6 M. Jutras, "Improvement of The characterization Method of The Johnson-cook Model", Ph.D. Thesis, Faculte des Sciences et de Genie Universite Laval Quebec, 2008. 7 J. Lemaitre, D. Rodrigue, "A course on damage mechanics", Springer, Verlag Berlin, 1992 8 J. Lemaitre, D. Krajcinovic, “Continuum Damage Mechanics Theory and Applications”, CISM Lectures, Springer, Berlin Heidelberg New York, 1987 . 9 H. Hooputra, H. Gese, H. Dell, H. Werner, “A Comprehensive Failure Model for Crashworthiness Simulation of Aluminium Extrusions”, Int. Journal of Crashworthiness, Vol. 9, No. 5, pp. 449−463 2004. 10 T.Wierzbicki, Y.Bao, Young Lee, Y.Bai, “Calibration and Evaluation of Seven Fracture Models”, International Journal of Mechanical Sciences. Vol. 47, 2005, pp 719–743. 11 Yuanli Bai, Xiaoqing Teng, Tomasz Wierzbicki, On the Application of Stress Triaxiality Formula for PlaneStrain Fracture Testing, Journal of Engineering Materials and Technology Vol. 131, 2009, pp 557-566. 12 M. Khataei, M. Poursina, M. Kadkhodaei, “A Study on Fracture Locus of St12 Steel and Implementation Ductile Damage Criteria” in Numiform-2010, edited by F. Barlat et al., AIP Conference Proceedings 620, American Institute of Physics, Pohang, 2010, pp. 107.

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Ductile Fracture of AHSS Sheets under Multi-axial Loading: Experiments and Modeling M. Dunanda,b and D. Mohra,b a

Impact and Crashworthiness Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA b Solid Mechanics Laboratory (CNRS UMR 7649), Ecole Polytechnique, Palaiseau, France Abstract. Fracture experiments on TRIP-assisted steel sheets covering a wide range of stress states (from shear to equibiaxial tension) are performed to create a comprehensive experimental database to calibrate and evaluate the shearmodified Gurson model (Nielsen and Tvergaard, 2010) and the Modified Mohr-Coulomb (MMC) fracture model (Bai and Wierzbicki, 2010). The experimental program includes notched tensile tests as well as fracture experiments on butterfly-shaped specimens under combined tension and shear loading. Both phenomenological fracture models are physics-inspired and take the effect of the first and third stress tensor invariants into account in predicting the onset of ductile fracture. The MMC model is based on the assumption that the initiation of fracture is determined by a critical stress state, while the shear-modified Gurson model assumes void growth as the governing mechanism. The model accuracy is quantified based on the predictions of the displacements to fracture for experiments which have not been used for calibration. It is found that the MMC model predictions agree well with all experiments (less than 4% error), while less accurate predictions are observed for the shear-modified Gurson model. A comparison of plots of the strain to fracture as a function of the stress triaxiality and the normalized third invariant reveals significant differences between the two models except within the vicinity of stress states that have been used for calibration. Keywords: Ductile Fracture, Shear-Modified Gurson model, Modified Mohr-Coulomb model, Stress triaxiality, Lode angle. PACS: 62.20.fk, 62.20.M, 62.20.fq

INTRODUCTION Gurson type of models have been developed to study ductile fracture when void nucleation and growth is the predominant mechanism at the mesoscale. A shortcoming of those models, namely their inability to predict fracture under shear-dominated loading conditions, has been addressed recently (e.g. [1]) by introducing a dependency of the damage evolution on the third stress invariant. As an alternative to Gurson type of models, damage indicators have been proposed that can be used in conjunction standard incompressible plasticity models. Unlike in Gurson models, it is assumed that the evolution of damage has no effect on the effective stress-strain response of the material before fracture occurs. Recently, Bai and Wierzbicki [2] transformed the Mohr-Coulomb fracture criterion into the space of stress triaxiality, Lode angle and equivalent plastic strain, defining the so-called Modified Mohr-Coulomb (MMC) fracture criterion. This conference paper is an abridged version of Ref. [3]. In this work, the fracture prediction capabilities of the shear-modified Gurson model by Nielsen and Tvergaard [4] and the MMC fracture model are investigated for a representative advanced high strength steel sheet material. The results from an extensive experimental program [5,6] are used to calibrate and validate the fracture models. The program includes fracture experiments under nine different loading conditions thereby covering a wide range of stress triaxialities and Lode angles. The models are evaluated based on their ability to predict the fracture displacements in experiments that have not been used for calibration.

EXPERIMENTAL PROGRAM Nine distinct experiments have been carried out to characterize the fracture response.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 484-491 (2011); doi: 10.1063/1.3623648 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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(a) (b) (c) FIGURE 1. Fracture specimens: (a) Tensile specimen with 20mm notch radius; (b) tensile specimen with a central hole; (c) Butterfly specimen for bi-axial fracture testing

All specimens are extracted from 1.5mm thick TRIP780 steel sheets provided by POSCO (Korea). The plastic behavior of this TRIP steel under monotonic loading conditions has been investigated over a wide range of stress states using a combined tension and shear testing technique for sheet materials [7]. For monotonic loading conditions, the plastic behavior of the TRIP780 material is modeled using a standard plasticity theory with: (1) an anisotropic quadratic Hill48 yield function, (2) non-associated quadratic anisotropic flow rule, and (3) an isotropic hardening model. The strain hardening response of the material is modeled by a piecewise-linear stress-strain curve (determined from biaxial punch experiments [5]).

Tension experiments Tensile experiments are performed on flat specimens with circular cutouts and on a flat specimen with a central hole (Figs. 1a and 1b). The specimen loading axis is always oriented along the rolling direction. All notched specimens are 20mm wide and feature a b=10mm wide notched gage section. Specimens with three different notched radii are prepared: R=20mm, R=10mm and R=6.67mm. A forth type of specimen featuring an 8mm diameter circular hole at the center is also used. All experiments are carried out under displacement control at a constant crosshead velocity of 0.5mm/min. Digital Image Correlation (DIC) is used to measure the relative displacement of specimen boundaries and the displacement field at the center of the gage section. The instant of the onset of fracture is defined through the first discontinuity in the measured surface displacement field.

(a) (b) FIGURE 2. Force displacement curves for tensile specimens with (a) a 20mm notch, (b) a central hole. Experimental results depicted in black, numerical results using the MMC model in red and the Nieslsen-Tvergaard model in blue. Vertical dashed line show displacements to fracture.

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(a) (b) FIGURE 3. Force displacement curves for the butterfly experiments. (a):  = 90°. (b):  = 0°. Experimental results depicted in black, numerical results using the MMC model in red and the Nieslsen-Tvergaard model in blue. Vertical dashed line show displacements to fracture.

The measured force-displacement curves of the 20mm notched tensile and central hole specimens are depicted in Fig. 2 with black dots. For all four geometries, a force maximum is reached before fracture occurs. It is associated with the onset of through-the-thickness necking at the specimen center for the notched geometries and in the two specimen ligaments for the central hole geometry. In addition, a circular punch experiment is performed: a sheet specimen is clamped on a circular die and subsequently loaded through a hemispherical punch. The applied force increases monotonically until a sharp drop in the force level is observed at the instant of the onset of fracture.

Combined tension-shear experiments Bi-axial fracture experiments are carried out in a dual actuator system that permits to apply a combination of normal and transverse loads to the boundaries of a butterfly-shaped specimen (Fig. 1c). It features a gage section of reduced thickness which is bounded by shoulders of clothoid shape. The butterfly specimen has been designed such that fracture initiates at the center of the gage section, remote from the lateral free edges [6]. The ratio of the applied vertical force F and the horizontal force F , is characterized by the biaxial loading angle  (where tan  = F F ). Four different loading conditions are investigated:  = 0° (pure shear),  = 25° (shear-dominated),  = 63° (tension dominated) and  = 90° (transverse plane strain tension). All experiments are performed under force control to enforce a constant loading angle . During the experiments for  = 63° and 90°, the control mode is switched to displacement control shortly before the onset of localized necking; the actuator velocities are chosen such that  remains approximately constant. Displacements of the specimen boundaries (both translations and possible rotations) are recorded by Digital Image Correlation (DIC) throughout the experiments. For each experiment, we measure two force-displacement curves: (1) the tangential force versus horizontal displacement curve, and (2) the axial force versus vertical displacement curve (depicted with black dots in Fig. 3 for the loadings  = 90° and 0°). The fracture displacement is defined as the relative displacement (either horizontal or vertical) of the specimen boundaries at which a sudden drop of force occurs.

Range of stress states covered In almost all experiments described above, plastic deformation localizes within the gage section prior to fracture. A finite element simulation is performed of each experiment to obtain the stress and strain history at the location of fracture initiation. All specimens are meshed with reduced-integration eight-node 3D solid elements. Details on the FE meshes and discretization errors are given in [5, 6]. The displacements measured by DIC are directly imposed as boundary conditions in the numerical model. Figure 4 provides an overview on the stress states reached in different specimens as obtained from hybrid experimental-numerical analysis. The experimental program covers the entire

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FIGURE 4. Average stress state in each experiment at the location where the highest equivalent plastic strain is reached. Experiments used to calibrate the MMC model are depicted with red stars, while experiments used to calibrate the NielsenTvergaard model are depicted with blue crosses. The black dashed line show the relation between stress triaxiality and third stress invariant in case of plane stress condition.

range of positive stress triaxialities accessible in sheet materials (0    23), and the full range of normalized third stress invariants ( 1   1). We define the stress triaxiality and normalized third invariant as: =

tr  3

=

27 det  2 

(1)

where  is the deviatoric part of the Cauchy stress tensor  and the Mises equivalent tensile stress.

FRACTURE MODELING Phenomenological approach (uncoupled model) Damage is measured through the scalar variable  (damage indicator) and its increase   0 is defined through the increment of the equivalent plastic strain  with respect to a stress-state dependent reference failure strain , dD =

 ()

(2)

Assuming an initial value of  = 0 for the material in its undeformed configuration, it is postulated that fracture initiates as  = 1. Note that it is assumed that the elasto-plastic material behavior is not affected by the progressive internal damage of the material (uncoupled model). We make use of the so-called MMC [2] reference failure strain. This particular weighting function is considered as it includes parameters that can adjust the damage accumulation function for low and high stress triaxialities as well as the Lode angle parameter: .

( )! 1 + ' ( )! 1 ( )! (  ( )  ) sec  (, ) =   + " 1"# $ $%& cos  " + '  + sin  ""*3 6 6 3 6 2 3 3

' /

(3)

with   ( ) = 4

1 

for  0$ for < 0

and

 = 1

2 acos

!

(4)

The material parameters ' ,  ,  and  are calibrated based on four fracture experiments: the three butterfly fracture experiments (at  = 0°,  = 63° and  = 90°) and the punch test. These four experiments cover a wide range of positive stress triaxialities and the complete range of third stress invariants ( 1   1). The calibration

487

consists in finding a set of parameters [' ,  ,  ,  ] such that the model describes correctly the onset of fracture in all calibration experiments.

Shear-modified Gurson model We make use of a recent extension by Nielsen and Tvergaard [4] which includes all features of the original Gurson model as well as the effect of the third invariant on the damage accumulation. The material state is described by the void volume fraction f, the macroscopic plastic strain tensor 8: , and the deformation resistance ; of the (undamaged) matrix material. The macroscopic yield surface is given by >(, ;, ?) = 0, where 3 E  A (B' ? C ) >(, ;, ?) = @ A + 2B' ? C cosh @ B 2 ; ;

(5)

In the Nielsen-Tvergaard model, the empirical modification permitting to predict fracture under shear loading conditions affects only the differential equation describing the evolution of the void volume fraction, which reads: ?G = (1 ?)8G  : I + JGK + ;L ?MN ()

: 8G  OK

(6)

where the first term accounts for the growth of voids due to hydrostatic stress. The second term accounts for the nucleation of voids. The third term has been introduced to account for material deterioration due to shear loading. ;L is a model parameter and MN () a stress-state dependent weighting function:

MN () = M( )P()

where

M( ) = 1



and

1 for  < '  ) P() = for '     $ R('  ) 0 for  >  Q S (

(7)

It is emphasized that the original physical meaning of f is lost due to the addition of the empirical third term in Eq. (6). In other words, ? can no longer be seen as the void volume fraction and is interpreted as an empirical damage parameter. The reader is refered to [4] for a complete overview of the constitutive equations. The fourteen material parameters of the Nielsen-Tvergaard model are determined from four experiments: (1) uniaxial tension, (2) punch experiment: (3) notched tension with U = 20VV, and (4) shear-dominated butterfly experiment ( = 0°). A Monte-Carlo based inverse method is used to identify most model parameters.

Comparison of model predictions and experimental results The fracture models are evaluated by comparing the predicted and experimentally-measured displacements to fracture for nine distinct experiments. FE simulations are run of all experiments with the measured displacements imposed as boundary conditions. The predicted displacement to fracture corresponds to the applied displacement at the instant when  = 1 for the MMC model and ? = 0.9?X for the modified Gurson model. Figure 5 depicts the ratios of the predicted and measured displacements to fracture for all nine experiments. Error bars are included to represent the experimental scatter on the measured fracture displacements (when applicable). Recall that four fracture experiments (three butterfly fracture experiments and the punch test) have been used to identify the parameters of the MMC model, and three for the Nielsen-Tvergaard model. It is found that the MMC model (in red in Fig. 5) predicts the displacements to fracture with great accuracy for all experiments. The relative error is smaller than 2% for eight of nine experiments. The largest relative error of 3.2% is observed for the tensile specimen with a central hole. Fracture displacements predicted by the Nielsen-Tvergaard model are shown in blue in Fig. 5. For the three experiments used for calibration, the fracture displacements are predicted with less than 0.5% error. The fracture displacement of the 10mm notched specimen is very accurately predicted, with a relative error of 0.2% which is less than the experimental scatter. For the two other full thickness tensile experiments (6.67mm notch and specimen with a central hole), the relative error affecting the predicted displacement to fracture is less than 10% (respectively 6.9% and 4.5%). However, the shear-modified Gurson model is clearly inaccurate for three butterfly experiments (more than 10% error): it underestimates the displacement to fracture by 11% for the shear-dominated combined loading

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FIGURE 5. Ratio of predicted and experimental fracture displacements. Experiments used for calibration have their name underlined

( = 25°) and by 17.7% for tension-dominated combined loading ( = 63°); conversely, the fracture displacement is overestimated by 10.6% for the transverse plane strain tension experiment ( = 90°).

DISCUSSION Effect of damage on the elasto-plastic response can be neglected The accumulation of damage affects the elasto-plastic material response prior to fracture when using the Gurson model. Conversely, the predictions of the standard plasticity model which is used in conjunction with the MMC model remain unaffected by the damage accumulation. The comparison of the force-displacement curves predicted by the standard plasticity model and the Nielsen-Tvergaard model (respectively in red and blue lines in Figs. 2 and 3) show that both models predict the same global response of the specimens at the early stages of loading. Before nucleation occurs, material damage is negligible and the Gurson yield function, given in Eq. (5), reduces to a standard quadratic yield function. The predicted force-displacement curves slightly differ only when material damage becomes significant.. It is interesting to see that the Nielsen-Tvergaard model does not always predict the force prior to fracture with greater accuracy; for tension with a central hole (Fig. 2b) and the butterfly experiment at  = 0° (Fig. 3b), the uncoupled plasticity model predicts the force-displacement curves more accurately than the Gurson model. Based on the above observations, it is tentatively concluded that the modeling of the effect of damage on the elasto-plastic behavior does not yield any improvement of the accuracy of force-displacement curve predictions.

Modeling of shear-induced material deterioration inaccurate According to the Nielsen-Tvergaard model, the shear-induced damage rate for low stress triaxialities reads fGYZ[\] = ;L ?M()

: 8G :

(8)

where ? is the damage parameter, ;L is a material parameter and M = 1  measures the deviatoric state of stress. G The quadratic dependence of ?^_`b on the third stress invariant (as proposed in [1]) is a purely empirical assumption. Among the above experiments, the results from shear-dominated butterfly experiments ( = 0° and  = 25°) can be used to assess the validity of this choice. In these experiments, fracture occurs at stress triaxialities close to zero, while the corresponding third stress invariants are significantly different (Fig. 4). Recall that the experiment for  = 0° has been used for calibration; however, the inability of the Nielsen-Tvergaard model to predict the fracture displacement for  = 25° suggests that the empirical choice of the quadratic relationship G and needs to be revisited. between ?^_`b

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(a)

(b)

(c) (d) FIGURE 6. Fracture loci for proportional loading conditions in the (, ,  ) plane of the MMC fracture criterion (in red) and Nieslen-Tvergaard model (in blue). Cross sections at constant stress triaxiality: (a)  = 0.1, (b)  = 0.5; and constant third invariants: (c) = 0, (d) = 1.

Fracture locus for proportional loading Proportional loadings are characterized by constant stress triaxiality g and normalized third invariant j throughout loading. In this specific case, the equivalent plastic strain to fracture, or fracture strain k l , can be represented as a function of g and j to create the so-called fracture locus in the (g, j, k m ) space. Figures 6a and 6b depict the fracture strains for proportional loading as a function of the third stress invariant, at a fixed stress triaxiality ( = 0.1 in Fig. 6a,  = 0.5 in Fig. 6b), while Figs. 6c and 6d depicts the fracture strains for proportional loading as a function of the stress triaxiality, at fixed third invariants ( = 0 in Fig. 6c, = 1 in Fig. 6d). The MMC fracture strain decreases in stress triaxiality and exhibits a U-shaped dependence on the third invariant, resulting in a continuous fracture locus shaped as a half-tube in the (g, j, k m ) space [2]. Note that the MMC fracture locus is asymmetric with respect to j = 0 and that, at a fixed stress triaxiality, the minimum ductility does not correspond to the generalized shear stress state (characterized by j = 0). The Nielsen-Tvergaard fracture locus shows a more complex dependence on the stress state. For low stress triaxialities (0 <  < g' ), the fracture locus is a U-shaped function of the third invariant. The fracture strain is smallest for generalized shear stress states (j = 0). For axisymmetric loading states (j = ±1), there is no shear-induced damage and the fracture strain increases asymptotically as the stress triaxiality decreases to 0. For intermediate stress triaxialities (g' <  < g ), the fracture strain increases in stress triaxiality for loadings states with third invariants close to 0, as the shear-induced damage slows down. For high stress triaxialities (g > g ), the shear damage term in Eq. 6 is inactive: the fracture strain

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decays almost exponentially in stress triaxiality and is independent of the third invariant. Note that the NielsenTvergaard fracture locus is symmetric with respect to j = 0. The calibrated MMC and Nielsen-Tvergaard fracture loci are remarkably different over the entire range of stress states considered, except within the vicinity of the experimental data that have been used to calibrate the NielsenTvergaard model: for  p 0.5 and p 0.7 (average stress state in the 20mm notch tensile experiment) and for  p 0.7 and p 1 (average stress state in the punch experiment), the two fracture loci intersect.

CONCLUSIONS The predictive capabilities of the shear-modified Gurson model by Nielsen and Tvergaard and the phenomenological MMC fracture criterion are evaluated over a wide range of stress states (shear to equibiaxial tension). The MMC model can predict the onset of fracture with great accuracy for all experiments. The predictions of the shear-modified Gurson model are found to be less accurate. It is shown that the Lode angle dependence of the shear-induced damage mechanism needs to be modified further to improve the accuracy of the Nielsen-Tvergaard model. Aside from the small differences in the prediction accuracies, it is surprising to see that two fundamentally different fracture models (stress-based criterion versus void growth model) are both able to predict the fracture displacement over a wide range of stress states. This observation suggests that the underlying physical assumptions are less important than the models’ mathematical flexibility to be fitted to a wide range of experimental data. Another key fundamental difference between the MMC model and the Nielsen-Tvergaard model lies in the coupling of plasticity and damage. The MMC model makes use of a damage indicator function which has no effect on the elasto-plastic behavior, while damage reduces the load carrying capacity and changes the shape of the yield surface in the case of the shear-modified Gurson model. However for the TRIP780 material, both models show equally good predictions of the force-displacement curves. The MMC model is recommended for practical applications because of the greater computational stability of uncoupled damage models, the smaller number of parameters to be identified and the ease of their identification based on experiments. However, it is important to identify the MMC model parameters based on experiments that cover a wide range of stress states.

ACKNOWLEDGMENTS POSCO Steel (Korea) is thanked for providing the material. The financial support of the AHSS Fracture Consortium at MIT (AISI, Hyundai, Nissan, Posco, ThyssenKrupp, Volkswagen) is gratefully acknowledged.

REFERENCES 1. 2. 3. 4. 5. 6. 7.

K. Nahshon and J. Hutchinson, Eur. J. Mech. A-Solids 27, 1-17 (2008). Y. Bai and T. Wierzbicki, Int. J. Fract. 161, 1-20 (2010). M. Dunand and D. Mohr, J. Mech. Phys. Solids 59, 1374-1394 (2011). K. Nielsen and V. Tvergaard, Eng. Fract. Mech. 77, 1031-1047 (2010). M. Dunand and D. Mohr, Int. J. Solids. Struct. 47, 1130-1143 (2010). M. Dunand and D. Mohr, Eng. Fract. Mech, In Revision. D. Mohr, M. Dunand, K.H. Kim, Int. J. Plasticity 26, 939-956 (2010).

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An Ellipsoidal Void Model for the Evaluation of Ductile Fracture in Sheet Metal Forming Kazutake Komoria* *Department of Integrated Mechanical Engineering, School of Engineering, Daido University 10-3, Takiharu-town, Minami-ward, Nagoya-city, Aichi-prefecture, 457-8530, Japan Abstract. We propose a void model that can be used for the evaluation of ductile fracture in the simulation of metal forming processes. This void model is an extension of the Thomason model of void coalescence, which is based on the internal necking of intervoid matrix ligaments. The validity of the new void model for bulk metal forming processes was confirmed by experiments and simulations. In this paper, we confirm the validity of the new void model in the simulation of a bore expanding test, which is one of the material tests used in sheet metal forming. Keywords: Ductile fracture, Thomason model, Ellipsoidal void, Multi-scale simulation. PACS: 46.35.+z, 46.50.+a, 62.20.fk.

INTRODUCTION Ductile fracture [1], which occurs when a material is subjected to a large plastic deformation, is a troublesome problem in metal-forming processes. Many fracture criteria for various materials have been proposed. However, no fracture criterion that is applicable to all metal-forming processes has been found. Fracture criteria which are widely used for metal-forming processes are fracture criteria proposed by Freudenthal [2], Cockcroft and Latham [3], Brozzo et al. [4], and Oyane [5]. All these fracture criteria are expressed in terms of a definite integral of an integrand that is a function of stress and strain components, while, according to Lemaitre [6], the damage variable D is a function of stress and strain components. Since stress and strain are defined macroscopically, each fracture criterion is derived from a macroscopic point of view. However, since ductile fracture occurs through nucleation, growth, and coalescence of voids, it is a microscopic phenomenon. Hence, it is difficult to improve the accuracy of microscopic ductile fracture predictions using a macroscopic fracture criterion. We have been investigating the prediction of ductile fracture in metal-forming processes from a microscopic point of view. In our previous study, using a void model that we had previously proposed, we simulated the inner fracture defects in drawing of carbon steels for machine structural use [7] and high-carbon steel wire rods [8], while we simulated the blanking process of sheet metals [9]. Thomason proposed a model of void coalescence based on the internal necking of intervoid matrix ligaments [10]. The model was derived from the upper bound method, in which the material is assumed to fracture when the energy required to coalesce voids by internal necking is less than the energy required to deform the material homogeneously. The model of void coalescence is useful, since it has definite physical meaning. Our void model is derived from the Thomason model. Since the Thomason model assumes that voids are rectangular, our void model assumes that voids are parallelogrammic. However, parallelogrammic voids are unrealistic in shape. We recently proposed an improved void model [11] in which voids are assumed to be ellipsoidal. Ellipsoidal voids are more realistic in shape and thus the ellipsoidal void model is an improvement on the parallelogrammic void model. In the present study, we use the ellipsoidal void model in the simulation of a bore expanding test, which is one of the material tests used in sheet metal forming. The simulation results are compared with the experimental results. This comparison confirms the validity of the ellipsoidal void model.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 492-497 (2011); doi: 10.1063/1.3623649 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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SIMULATION METHOD The simulation method is described in our previous paper [9] in which parallelogrammic voids are assumed. The present study assumes ellipsoidal voids. Hence, we describe the void shape in the simulation in detail below.

Outline Figure 1 shows the outline of the whole simulation. Our simulation is a multiscale simulation. The macroscopic simulation models the deformation of the material by the rigid-plastic finite-element method, while the microscopic simulation evaluates the fracture of the material using our void model. Since the deformation of the material after the material fracture is not required in this study, the simulation is performed until the material fractures. Macroscopic Simulation (Simulate Deformation of Material byRigid-Plastic Finite-Element Method) DisplacementGradient Rate VoidVolumeFraction Rate

Determine Whether Material Fractures

Microscopic Simulation (EvaluateFracture of Material by our Void Model) FIGURE 1. Outline of whole simulation.

Outline of Macroscopic Simulation The simulation of the deformation of the material is performed using the conventional axisymmetric rigid-plastic finite-element method [12]. The yield function proposed by Gurson [13] is adopted: §V 3 V ijc V ijc ˜ 2  2 f cosh¨¨ kk 2 VM © 2V M

)

· ¸ 1  f 2 ¸ ¹

0,

(1)

where V M is the tensile yield stress of the matrix and f is the void volume fraction of the material. Since the yield function ) is not a function of the second power of stress, it is not easy to perform a rigid-plastic simulation using Equation (1). Hence, cosh x is approximated to be 1  x 2 2 [14]. The approximated yield function )c used in this study is )c

3 V ijc V ijc f ˜  2 V M2 4

§V 2 ˜ ¨¨ kk2 ©VM

· ¸  1  f 2 ¸ ¹

0.

(2)

On the other hand, the following evolution equation, which denotes the change in the void volume fraction, is assumed: f

1  f Hkk  A

V kk  B H , 3V

(3)

where V is the equivalent stress, H is the equivalent strain rate and A and B are material constants. Here, x in Equation (3) denotes Macauley’s bracket. In other words, x is equal to x when x is positive, while x is equal to zero when x is negative. The first and second terms of the right-hand side of Equation (3) denote void

493

growth and void nucleation, respectively. The second term indicates that a void nucleates when the stress triaxiality V kk 3V is larger than B .

Outline of Microscopic Simulation The outline of the microscopic simulation in each step, from the calculations of the void volume fraction and the average deformation gradient to the determination of whether or not the material fractures, is shown below: (1) The void volume fraction f and the average deformation gradient wx wX are calculated from the void volume fraction rate f and the displacement gradient rate wu wX , which are calculated by the macroscopic rigidplastic finite-element simulation. (2) The void configuration and the void shape are calculated. (3) The ratio of the energy-dissipation rate of internal necking to the energy-dissipation rate of homogeneous deformation E is calculated. (4) Whether or not the material fractures is determined. For simplicity, the deformation gradient of the material in the macroscopic simulation is assumed to be identical with the deformation gradient of the void in the microscopic simulation. In this study, the experiment of the bore expanding test using a prestrained material is performed. The void before the bore expanding test for a non-prestrained material is circular. However the void before the bore expanding test for a prestrained material is not circular but ellipsoidal, so that the deformation gradient of the void coincides with that of the material.

Evaluation of Material Fracture Figure 2 shows the coordinates in the macroscopic simulation and the microscopic simulation. In the simulation of blanking [9], the cylindrical coordinates r and z are used in the macroscopic simulation due to the axisymmetric deformation. Furthermore, the cylindrical coordinates r and z are used in the microscopic simulation, since the normal stress in the circumferential direction V T becomes the intermediate principal stress. In this study, the cylindrical coordinates r and z are used in the macroscopic simulation due to the axisymmetric deformation. However, the cylindrical coordinates T and z are used in the microscopic simulation, since the normal stress in the circumferential direction V T becomes the maximum principal stress [15].

z

0

z

Macroscopic Simulation

r

0

Microscopic Simulation

T

FIGURE 2. Coordinates in macroscopic simulation and microscopic simulation.

Figure 3 shows the two neighboring voids and the velocity field. Figure 3(a) shows the two neighboring voids. The surfaces of the velocity discontinuity shown by broken lines are assumed. Here, v denotes the material velocity in the direction of the maximum principal stress. T1 and T 2 denote the angles between the direction of the maximum principal stress and the direction of the surface of the velocity discontinuity. 2l1 and 2l2 denote the lengths of the surface of the velocity discontinuity. L denotes the distance between two neighboring voids. Figure 3(b) shows the velocity field denoted as a hodograph. Here, 'v1 and 'v2 denote the amounts of velocity discontinuity. First, the rate of energy dissipation on the surface of the velocity discontinuity when two neighboring voids coalesce by internal necking, i.e., the energy dissipation rate of internal necking, is expressed as

494

k ˜ 'v1 ˜ 2l2  k ˜ 'v2 ˜ 2l1

l1 sin T 2  l2 sin T1 . sin T1  T 2

4kv

(4)

Next, the rate of energy dissipation by homogeneous plastic deformation of the material, i.e., the energy dissipation rate of homogeneous deformation, is expressed as L ˜ 2 k ˜ 2v

4kvL.

(5)

Since void coalescence is assumed to occur when the energy dissipation rate of internal necking is less than the energy dissipation rate of homogeneous deformation, the criterion of void coalescence is expressed as Lt

l1 sin T 2  l2 sin T1 . sin T1  T 2

(6)

The ratio of the energy dissipation rate of internal necking to the energy dissipation rate of homogeneous deformation E is defined as E

l1 sin T 2  l2 sin T1 . L sin T1  T 2

(7)

When E is less than one, the criterion of void coalescence is satisfied. Moreover, the material is assumed to fracture when E is less than one.

L v

v

ȟ v1 T2

T 2 T1

ȟ v2

T1

0

l2 l1 v

v

(a) Two neighboring voids (b) Velocity field FIGURE 3. Two neighboring voids and velocity field.

SIMULATION RESULT Experiment and Simulation of Tensile Test The tensile test was performed prior to the bore expanding test. The uniaxial tensile specimen was made of JIS SPCC, which is equivalent to ISO CR1. The thickness of the material was 1 mm. The following stress-strain relationship was obtained in the experiment using the uniaxial tensile specimen, and applied in the simulation:

VM

460 ˜ H M  0.005

0.14

MPa ,

(8)

where H M is the strain of the matrix. The simulation using the uniaxial tensile specimen was performed, in which plane stress is assumed. The material constants A and B in Equation (3) were selected such that the reduction in thickness of the material calculated in the simulation was almost equal to that obtained in the experiment. Here, B is assumed to be zero to simplify the simulation.

495

A

0.06, B

(9)

0

Experiment and Simulation of Bore Expanding Test Table 1 shows the experimental condition for the bore expanding test. To produce a prestrained sheet metal, a thick sheet metal was rolled to obtain a thin sheet metal with a thickness equal to the thickness of a non-prestrained sheet metal. Two rolling directions were used in the laboratory: rolling in the rolling direction of the as-rolled sheet and rolling in the width direction of the as-rolled sheet. Thickness of as-rolled sheet (mm) 1 2 3

TABLE 1. Experimental condition. Thickness in bore expanding test (mm) 1 1 1

Magnitude of prestrain 0 0.69 1.10

Figure 4 shows the illustration and photograph of the specimen in the bore expanding test. The outer and inner diameters of the specimen were 85 mm and 9 mm, respectively. The die for the bore expanding test was shaped like a cone with a corner angle of 45 degrees. The bore expanding ratio is defined as d d 0  1 , where d is the fractured bore diameter and d 0 is the pierced bore diameter.

Beforetest

After test

d0

d

(a) Illustration (b) Photograph FIGURE 4. Illustration and photograph of specimen.

Figure 5 shows the finite element meshes of the specimen before and after the bore expanding test in the case of no prestrain. Axisymmetry of the specimen was assumed.

(a) Before bore expanding test (b) After bore expanding test FIGURE 5. Finite element meshes of specimen in case of no prestrain.

Figure 6 shows the relationship between the prestrain and the bore expanding ratio. In the experiment, the bore expanding ratio for the specimen rolled in the rolling direction of the as-rolled sheet is almost the same as that rolled in the width direction of the as-rolled sheet. As the prestrain increases, the bore expanding ratio decreases. Although the bore expanding ratio calculated from the simulation is slightly larger than that obtained in the experiment, the simulation result agrees with the experimental result qualitatively.

496

Bore expanding ratio

2.5 2.0

Experiment (rolling direction in laboratory = rolling direction of as-rolled sheet) Experiment (rolling direction in laboratory = width direction of as-rolled sheet) Simulation

1.5 1.0 0.5 0.0 0.0

0.3

0.6

0.9

1.2

Prestrain FIGURE 6. Relationship between prestrain and bore expanding ratio.

CONCLUSION We performed a simulation of the bore expanding test to confirm the validity of the new model of void coalescence. Since the simulation results and the experimental results generally agree, the method for considering the prestrain in the new model was proved to be effective.

REFERENCES B. Dodd and Y. Bai, Ductile Fracture and Ductility, London: Academic Press, 1987. A. M. Freudenthal, The Inelastic Behavior of Engineering Materials and Structures, New York: John Wiley & Sons, 1950. M. G. Cockcroft and D. J. Latham, Journal of the Institute of Metals 96, 33-39 (1968). P. Brozzo, B. De Luca and R. Rendina, Proceedings of the Seventh Biennial Congress of the International Deep Drawing Research Group, (1972). 5. M. Oyane, Bulletin of JSME 15, 1507-1513 (1972). 6. J. Lemaitre, A Course on Damage Mechanics, 2nd Edition, Berlin: Springer-Verlag, 1996. 7. K. Komori, Acta Materialia 54, 4351-4364 (2006). 8. K. Komori, Theoretical and Applied Fracture Mechanics 50, 157-166 (2008). 9. K. Komori, Materials Science and Engineering A 421, 226-237 (2006). 10. P. F. Thomason, Journal of the Institute of Metals 96, 360-365 (1968). 11 K. Komori, AIP Conference Proceedings 1252, 97-102 (2010). 12. S. Kobayashi, S.-I. Oh and T. Altan, Metal Forming and the Finite-Element Method, New York: Oxford University Press, 1989. 13. A. L. Gurson, Transactions of the ASME Journal of Engineering Materials and Technology 99, 2-15 (1977). 14. Y. Tomita, Numerical Elasticity and Plasticity, Tokyo: Yokendo, 1990 (in Japanese). 15. Y. K. Ko, J. S. Lee, H. Huh, H. K. Kim and S. H. Park, Journal of Materials Processing Technology 187-188, 358-362 (2007). 1. 2. 3. 4.

497

Formability Prediction of Advanced High Strength Steel with a New Ductile Fracture Criterion Yanshan Loua, Sungjun Lima, Jeehyang Huha, and Hoon Huha a

School of Mechanical, Aerospace and Systems Engineering, KAIST, 335, Gwahanno, Daedoek Science Town, Daejeon, 305-701, Republic of Korea

Abstract. A ductile fracture criterion is newly proposed to accurately predict forming limit diagrams (FLD) of sheet metals. The new ductile fracture criterion is based on the effect of the non-dimensional stress triaxiality, the stress concentration factor and the effective plastic strain on the nucleation, growth and coalescence of voids. The new ductile fracture criterion has been applied to estimate the formability of four kind advanced high strength steels (AHSS): DP780, DP980, TRIP590, and TWIP980. FLDs predicted are compared with experimental results and those predicted by other ductile fracture criteria. The comparison demonstrates that FLDs predicted by the new ductile fracture criterion are in better agreement with experimental FLDs than those predicted by other ductile fracture criteria. The better agreement of FLDs predicted by the new ductile fracture criterion is because conventional ductile fracture criteria were proposed for fracture prediction in bulk metal forming while the new one is proposed to predict the onset of fracture in sheet metal forming processes. Keywords: Ductile fracture criterion, Advanced high strength steel, Forming limit diagram, Sheet metal forming. PACS: 46.50.+d

INTRODUCTION A forming limit diagram (FLD), first introduced by Keeler and Backofen [1] and Goodwin [2], is the most popular tool to characterize the formability of sheet metals. An FLD can be constructed by experimental methods such as hemispherical punch-stretch tests and Marciniak cup tests [3]. Experimental methods, however, requires intensive effort and tremendous time. For more efficient construction of FLDs, many analytical models were proposed based on different hypotheses such as the Hill’s localized necking model [4], the Swift’s diffuse necking model [5], the Marciniak-Kuczynski model [6], the vertex theory [7, 8] and the modified maximum force criterion (MMFC) [9]. Recently, various ductile fracture criteria [10-15] were utilized in finite element analysis (FEA) to predict fracture initiation in sheet metal forming processes [16-21]. The problem of these conventional ductile fracture criteria is that these criteria were proposed for the fracture prediction in bulk metal forming processes. Conventional ductile fracture criteria could not describe the formability of sheet metals with higher accuracy [22, 23]. It is because conventional ductile fracture criteria were derived based on the mechanism of void growth while the fracture in sheet metal forming processes is the combination of the void growth inside sheet metals and necking outside [24]. A new ductile fracture criterion is proposed based on the effect of the non-dimensional stress concentration factor, the stress triaxiality and the effective plastic strain on nucleation, growth and coalescence of voids. The new ductile fracture criterion is then applied to predict FLDs of AHSS steels: DP780, DP980, TRIP590 and TWIP980. The AHSS steels are characterized by the Swift’s strain hardening rule and R-values obtained from uniaxial tensile tests and the FLDs from hemispherical punch-stretch tests. The Yld89 anisotropic yield function is employed to construct the yield surfaces. FLDs of these AHSS steels are predicted by the new ductile fracture criterion and conventional ones and compared with the experimental results in order to evaluate the performance of the new ductile fracture criterion.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 498-505 (2011); doi: 10.1063/1.3623650 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

498

THEORETICAL BASIS Ductile Fracture Criteria The forming limit of metal forming depends severely on the deformation history. The history of plastic deformation is the main concern in ductile fracture criteria. Generally, ductile fracture criteria have a form of



f

f  σ d   C

0

(1)

where f denotes a weight function considering the stress state, σ is a stress state,  is the equivalent plastic strain,  f is the equivalent plastic strain at fracture and C is the material constant. Review of Conventional Ductile Fracture Criteria Various ductile fracture criteria have been proposed empirically as well as theoretically based on different hypotheses. Cockcroft and Latham [10] postulated that fracture was controlled by the maximum principal tensile stress integrated over the plastic strain as below:



f

0

 *d   C1

(2)

Brozzo et al. [11] proposed a modified criterion in which the effect of hydrostatic stress was introduced such as:



f

0

2 * d   C2 3  *   m 

(3)

Oh et al. [12] introduced the effect of the non-dimensional stress concentration factor into ductile fracture criteria as below:



f

0

* d   C3 

(4)

Oyane et al. [13] derived a ductile fracture criterion from the equations of the plasticity theory for porous materials in a form of



f

0

m   C4 d   C5   

(5)

Clift et al. [14] substituted the equivalent stress for the maximum stress of the Cockcroft–Latham criterion:



f

0

 d   C6

(6)

Ko et al. [15] proposed another modification of the Cockcroft-Latham criterion to consider the influence of the hydrostatic stress in a form of



f

0

 * 1  3 m d   C7  

 x , when x  0 x    x , when x  0

499

(7)

1.3

0.70

(a)

(b)

0.65 1.2

0.60 Stress triaxiality

Stress concentration factor

Plane strain

1.1

1.0

Equibiaxial tension

0.55 Plane strain

0.50 0.45 0.40 0.35

0.9 -0.75 -0.50 -0.25

Uniaxial tension

Equibiaxial tension

Uniaxial tension 0.00

0.25

0.50

0.75

1.00

0.30 -0.75 -0.50 -0.25

1.25

Strain Path 

0.00

0.25

0.50

0.75

1.00

1.25

Strain Path 

FIGURE 1. Effect of strain paths on weight functions in ductile fracture criteria: (a) the stress concentration factor; (b) the stress triaxiality.

In Eqs. (2)-(7),  * is the maximum principal stress,  m is the mean or hydrostatic stress,  is the effective stress, and C1  C7 are the material constants in ductile fracture criteria. New Ductile Fracture Criterion for Accurate Prediction of FLDs The ductile fracture criteria reviewed above were proposed to predict the onset of fracture in various bulk metal forming processes such as extrusion, rolling, upsetting drawing and hub-hole expanding [10-15]. These conventional ductile fracture criteria cannot predict FLDs accurately especially in the biaxial stress region [22, 23]. Ductile fracture is preceded by severe plastic deformation involving nucleation, growth and coalescence of voids in metals and alloys. These steps are influenced mainly by three factors: the hydrostatic stress, the maximum stress and the equivalent plastic strain. Three factors should be considered in ductile fracture criteria in order to accurately predict fracture initiation of sheet metals. The hydrostatic stress is normally considered in terms of stress triaxiality    m  while the effect of the maximum stress is introduced by the stress concentration factor    *  . The stress concentration factor and the stress triaxiality are functions the strain path   d  2 d 1 , as presented in FIGURE 1 (a) and (b). The stress triaxiality increases monotonously as the strain path shifts from uniaxial tension to balanced biaxial tension. The stress concentration factor  increases as the strain path moves from uniaxial tension to plane strain tension and then decreases when the strain path shifts to balanced biaxial tension from plane strain tension. The stress triaxiality and the stress concentration factor are utilized as the weight function in the new ductile fracture criterion. To adjust the effect of the stress triaxiality and the stress concentration factor on the accumulated damage of materials, a material constant of an exponent is added to each of these factors. The new ductile fracture criterion is proposed to have a form of



f

0

C8

  *   3 m  9     d   C10      C

(8)

where C8, C9 and C10 are material constants. The materials constants are evaluated from three experiments: the uniaxial tensile test; the plane strain test; the balanced biaxial tensile test. These three kinds of experiments are carried out to measure the limit strains in the uniaxial tension, plane strain and balanced biaxial tension conditions. From the limit strain in the uniaxial tension, the material constant C10 is obtained as below:

C10   u 0

(9)

where  u 0 indicates the limit strain in uniaxial tension. In the plane strain condition, Eq. (8) is reduced to a form of

500



 psC 1  ps 1   ps  8



C9

 ps 0  C10

(10)

Here  ps ,  ps    2  1  and  ps 0 denote the stress concentration factor, stress ratio and the limit strain in the plane strain conditions. In the balanced biaxial tension d 1  d  2 , the new ductile fracture criterion has a form of



eqC eq 1   eq  8



C9 1

 eq 0  C10

(11)

where eq ,  eq    2  1  and  eq 0 are the stress concentration factor, stress ratio and the limit strain in the balanced biaxial tension conditions. The material constants in the new ductile fracture criterion are calculated by solving Eqs. (9), (10) and (11). Yld89 Yield Function Barlat and Lian [25] generalized the Hosford79 non-quadratic anisotropic yield function [26, 27] to consider the shear stress which has a form of

a K1  K 2  a K1  K 2  c 2 K 2 m

m

m

 2 m

(12)

where

K1 

 xx  h yy 2

  xx  h yy  2 2 , K2     p  xy 2   2

(13)

where c, h and p are anisotropic parameters. The exponent m is suggested to be 6 for BCC metals and 8 for FCC metals. The anisotropic parameters c and h are easily obtained as below:

c  2a 

R0 R90 R0 1  R90 ,h  1  R0 1  R90 1  R0 R90

(14)

Whereas the anisotropic parameter p is evaluated by the Lankford coefficient in the diagonal direction R45 by solving an implicit equation as below: 2 2    1 h  1 h    m 1 m 1 R45  a  K1  K 2   2 K 2 1  h     a  K1  K 2   2 K 2 1  h      2  2     2   1 h  m 1 m 1 2  c 2 K 2 1  h   a  K1  K 2   K 2 1  h    p2     4   2 2     1 h  m 1 m 1 1  h  2  a  K1  K 2   K 2 1  h    p   2c  2 K 2    p2   0   4  4   

(15)

where

1 h K1  , K 2  4

1  h 

2

4

501

 4 p2 (16)

TABLE 1. Mechanical properties of AHSS steels (Unit: MPa). Material

Thickness

K

0

n

R

u0

 ps 0

 eq 0

DP780 (1.0t)

1.0 mm

0.002 0.002

0.179 0.109

0.20

0.105

0.32

1.2 mm

1429 1442

0.930

DP980 (1.2t)

0.658

0.12

0.07

0.19

TRIP590 (1.2t)

1.2 mm

1192

0.015

0.285

0.995

0.39

0.24

0.40

TWIP980 (1.4t)

1.4 mm

2537

0.120

0.726

0.812

0.58

0.46

0.48

TABLE 2. Anisotropic parameters in the Yld89 yield functions of AHSS steels. Material DP780 (1.0t) DP980 (1.2t) TRIP590 (1.2t) TWIP980 (1.4t)

a

c

h

p

1.0363 1.2063 1.0025 1.1038

0.9637 0.7937 0.9975 0.8962

1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000

yy [MPa]

2000

Stress [MPa]

1750

1000 750

1500

500

1250

250

m 6 6 6 6

DP780 (1.0t) DP980 (1.2t) TRIP590 (1.2t) TWIP980 (1.4t)

1000 750

-250 DP780 (1.0t) DP980 (1.2t) TRIP590 (1.2t) TWIP980 (1.4t)

500 250

-500 -750

(a) 0 0.0

0.1

0.2

0.3

0.4

0.5

(b) -1000 -1000 -750 -500 -250

0

250

500

750 1000 xx [MPa]

Plastic strain

FIGURE 2. (a) Stress-strain curves of AHSS steels; (b) yield surfaces of AHSS steels.

MATERIAL PROPERTIES The new ductile fracture criterion is applied to predict FLDs of four AHSS sheets: DP780, DP980, TRIP590 and n TWIP980. The materials are characterized by the Swift’s strain hardening law (   K   0    ) and the R-values obtained by uniaxial tensile tests. The coefficients in the Swift’s strain hardening law and the R-values are summarized in Table 1. The true stress - true strain curves in the rolling direction are fitted with the Swift’s strain hardening law and shown in FIGURE 2 (a). The anisotropic parameters of the Yld89 yield function are calculated with Eqs. (14), (15) and (16) as presented in Table 2. The yield surfaces at yielding are compared in FIGURE 2 (b). Hemispherical punch-stretch tests were conducted to construct FLDs of AHSS steels. Different specimen sizes and lubrication conditions are applied to obtain various strain paths [28]. The width of arc-shaped specimens is selected as 20 mm, 40 mm, 60 mm, 80 mm, and 100 mm while the dimension of square specimens is 200 mm × 200 mm. Teflon and Plasticine are used for the arc-shaped specimens as lubricants while square-shaped specimens are lubricated as dry (D), Teflon (T), Teflon + Vaseline (T+V), and Teflon + Plasticine (T+P). Specimens tested are shown in FIGURE 3 (a), (b), (c) and (d) for DP780, DP980, TRIP590 and TWIP980, respectively. FLDs are constructed in the true strain space by measuring the deformation of circular grids from fractured specimens. The limit strains in the uniaxial, plane strain and balanced biaxial tension conditions are listed in Table 1.

502

FIGURE 3. Fractured specimens of hemispherical punch-stretch tests. (a) DP780; (b) DP980; (c) TRIP590; (d) TWIP980. TABLE 3. Material constants in ductile fracture criteria. C2 C3 C1 Material DP780 (1.0t)

145.20

0.18

0.16

C4

C5

C6

C7

C8

C9

C10

-0.03

0.06

140.19

0.37

12.86

-1.51

0.20

DP980 (1.2t)

103.27

0.11

0.10

0.01

0.04

100.13

0.23

12.29

-1.11

0.12

TRIP590 (1.2t)

241.19

0.37

0.34

0.13

0.18

231.67

0.78

8.36

-0.93

0.39

TWIP980 (1.4t)

716.79

0.62

0.57

1.11

0.84

686.28

1.31

4.63

-0.61

0.58

APPLICATION TO FLD PREDICTION OF AHSS SHEETS After careful experiments, material constants from C1 to C7 are calculated with the corresponding equation for each ductile fracture criterion after appropriate experiments of the uniaxial and plane strain tension tests. Material constants C8, C9 and C10 in the new ductile fracture criterion are calculated with Eqs. (9), (10) and (11) after careful experiments of the uniaxial, the plane strain and the equibiaxial tension tests. The material constants calculated are presented in Table 3. These material constants calculated are utilized to predict FLDs of AHSS sheets. The FLDs predicted by various ductile fracture criteria are compared with experimental results in FIGURE 4 (a), (b), (c) and (d) for DP780, DP980, TRIP590 and TWIP980, respectively. For FLD prediction in the left hand side, the Cockcroft, Brozzo, Oh, Clift and Ko-Huh criteria underestimate the limit strain in the uniaxial tension conditions for DP780, DP980 and TRIP590. For TWIP980, the limit strain predicted by the Cockcroft, Oh, Oyane-Sato, Clift and new ductile fracture criteria are close to experimental results while the Brozzo and Ko-Huh criteria overestimate the formability in the left hand side of FLDs. The formability in the left hand side of FLDs can be predicted by the Oyane-Sato and new ductile fracture criteria with higher accuracy than others because these two ductile fracture criteria has more material constants than other ductile fracture criteria.

503

0.4

0.4

Major strain

0.3

Fracture Necking Safe

0.3

0.2

0.2 Cockcroft Brozzo Oh Oyane-Sato Clift Ko-Huh New criterion

0.1

(a) DP780 -0.2

-0.1

Major strain Fracture Necking Safe

0.0

0.1

0.2

0.1

(b) DP980 0.4 -0.2

0.3

Cockcroft Brozzo Oh Oyane-Sato Clift Ko-Huh New criterion

-0.1

0.0

0.1

0.2

0.6

0.5

0.3

0.4

Minor strain

Minor strain

0.7

Major strain Fracture Necking Safe

Major strain Fracture 0.6 Safe 0.5

0.4 0.4 0.3 0.3 Cockcroft Brozzo Oh Oyane-Sato Clift Ko-Huh New criterion

0.2

0.1 (c) TRIP590 -0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

Cockcroft Brozzo Oh Oyane-Sato Clift Ko-Huh New criterion

0.2 0.1 (d) TWIP980 0.6-0.3

Minor strain

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Minor strain

FIGURE 4. Comparison of FLDs predicted by ductile fracture criteria with experimental results. (a) DP780; (b) DP980; (c) TRIP590; (d) TWIP980.

For prediction of the limit strain in the plane strain condition, the Oyane-Sato and the new ductile fracture criteria can accurately describe it because of the larger number of material constants in these two ductile fracture criteria. Other ductile fracture criteria cannot accurately estimate the formability in the plane strain condition because there is only one material constant in these ductile fracture criteria. The only material constant in these ductile fracture criteria are calculated simply by the limit strain in two tests: the uniaxail and the plane strain tension tests. Consequently, based on ductile fracture criteria with only one material constant, the predicted limit strain in the uniaxial tension is greater than the experimental results if the limit strain predicted in plane strain condition is less than the experimental data and vice versa. For prediction of FLDs in the right hand side, conclusions are obviously obtained that FLDs can be accurate predicted by the new ductile fracture criterion while conventional ductile fracture criteria are too conservative in this stress region. It is because the conventional ductile fracture criteria were proposed to predict the onset of fracture in bulk metal forming processes while the new ductile fracture criterion is developed to accurately predict FLDs of sheet metals. In the new ductile fracture criterion, more material constants are introduced to consider the effect of stress triaxiality and the stress concentration factor on the nucleation, growth and coalescence of voids. More material constants make the new ductile fracture criterion more flexible than conventional ductile fracture criteria. For the accurate prediction of FLDs in both the left and right hand sides, the new ductile fracture criterion is recommended for sheet metals. FLDs predicted by the Oyane-Sato ductile fracture criterion are close to the experimental FLDs in

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the left hand side. Moreover, only two tests are required to calculate the material constants in the Oyane-Sato ductile fracture criterion. The Oyane-Sato ductile fracture criterion is qualified if the fracture initiation is expected to occur in FLDs of the left hand side.

CONCLUSIONS A ductile fracture criterion is newly proposed to accurately predict FLDs of sheet metals. The new ductile fracture criterion considers the effect of the stress triaxiality, the stress concentration factor and the plastic strain on the nucleation, growth and coalescence of voids in ductile fracture. The new ductile fracture criterion is applied to predict FLDs of four AHSS steels: DP780, DP980, TWIP590 and TWIP980. FLDs predicted by the new ductile fracture criterion are compared with experimental results and those predicted by conventional ductile fracture criteria. The comparison demonstrates that conventional ductile fracture criteria cannot predict FLDs of sheet metals especially in the right hand side of FLDs. If the onset of fracture is expected to occur in the left hand side of FLDs, the Oyane-Sato ductile fracture criterion is suggested to be utilized. The new ductile fracture criterion is recommended to be utilized to predict fracture initiation from the uniaxial tension to balanced biaxial tension.

REFERENCES 1. S. P. Keeler and W. A. Backofen, Trans. ASM 56, 25-48 (1963). 2. G. M. Goodwin, Trans. Soc. Automot. 25, 1413-1424 (1968). 3. ISO 12004-2, Metallic material – Sheet and strip – Determination of forming limit curves – Part 2: Determination of forminglimit curves in the laboratory, First Edition, (2008). 4. R. Hill, J. Mech. Phys. Solids 1, 19-30 (1952). 5. H. W. Swift, J. Mech. Phys. Solids 1, 1-18 (1952). 6. Z. Marciniak and K. Kuczynski, Int. J. Mech. Sci. 9, 609-620 (1968). 7. S. Storen and J. R. Rice, J. Mech. Phys. Solids 23, 421-441 (1975). 8. X. Zhu, K. Weinmann and A. Chandra, J. Eng. Mater. Technol. 123, 329-333 (2001). 9. P. Hora, L. Tong and J. Reissner, Proc. Numisheet’96 Conf. Dearborn, Michigan, USA, 1996, pp. 252-256. 10. M. G. Cockcroft and D. J. Latham, J. Inst. Met. 96, 33-39 (1968). 11. P. Brozzo, B. Deluca and R. Rendina, Proc. 7th Biennial Conf. IDDRG on Sheet Metal Forming and Formability, 1972. 12. S. I. Oh, C. C. Chen and S. Kobayashi, J. Eng. Ind. 101, 36-44 (1979). 13. M. Oyane, T. Sato, K. Okimoto and S. Shima, J. Mech. Work. Technol. 4, 65-81 (1980). 14. S. E. Clift, P. Hartley, C. E. N. Sturgess and G. W. Rowe, Int. J. Mech. Sci. 32, 1-17 (1990). 15. Y. K. Ko, J. S. Lee, H. Huh, H. K. Kim and S. –H. Park, J. Mater. Process. Technol. 187/188, 358-362 (2007). 16. H. Takuta, K. Mori and N. Hatta, J. Mater. Process. Technol. 95, 116-121 (1999). 17. H. Takuda, K. Mori, H. Fujimoto and N. Hatta, J. Mater. Process. Technol. 92/93, 433-438 (1999). 18. H. Takuda, K. Mori, N. Takakura and K. Yamaguchi, Int. J. Mech. Sci. 42, 785-798 (2000). 19. F. Ozturk and D. Lee, J. Mater. Process. Technol. 147, 397-404 (2004). 20. H. S. Liu, Y. Y. Yang, Z. Q. Yu, Z. Z. Sun and Y. Z. Wang, J. Mater. Process. Technol. 209, 5443-5447 (2009). 21. J. S. Chen, X. B. Zhou and J. Chen, J. Mater. Process. Technol. 210, 315-322 (2010). 22. Y. S. Lou, H. Huh, Y. K. Ko and J. W. Ha, Proc. KSAE Spring Conf. Pusan, Korea, 2010, pp. 40-45. 23. Y. S. Lou, S. B. Kim, Y. K. Ko and H. Huh, Proc. KSTP Fall Conf. Jeju, Korea, 2010, pp. 298-303. 24. T. B. Stoughton and J. W. Yoon, Int. J. Plasticity 27, 440-459 (2011). 25. F. Barlat and K. Lian, Int. J. Plasticity 5, 51-66 (1989). 26. W. F. Hosford, Proc. 7th North American Metalworking Conf., Sme, Dearborn, MI 191-197 (1979). 27. R. W. Logan and W. F. Hosford, Int. J. Mech. Sci. 22, 419-430 (1980). 28. H. Huh, C. H. Lee and J. W. Chung, J. Kor. Soc. Technol. Plast. 7, 81-93 (1998).

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Numerical Modeling for Hole-Edge Cracking of Advanced High-Strength Steels (AHSS) Components in the Static Bend Test Hyunok Kim1*, William Mohr1, Yu-Ping Yang1, Paul Zelenak1, and Menachem Kimchi1 1

Edison Welding Institute, 1250 Arthur E. Adams Drive, Columbus, Ohio 43221-3585, U.S. *Corresponding author: 1-614-688-5239, 1-614-688-5001 and [email protected]

Abstract. Numerical modeling of local formability, such as hole-edge cracking and shear fracture in bending of AHSS, is one of the challenging issues for simulation engineers for prediction and evaluation of stamping and crash performance of materials. This is because continuum-mechanics-based finite element method (FEM) modeling requires additional input data, “failure criteria” to predict the local formability limit of materials, in addition to the material flow stress data input for simulation. This paper presents a numerical modeling approach for predicting hole-edge failures during static bend tests of AHSS structures. A local-strain-based failure criterion and a stress-triaxiality-based failure criterion were developed and implemented in LS-DYNA simulation code to predict hole-edge failures in component bend tests. The holes were prepared using two different methods: mechanical punching and water-jet cutting. In the component bend tests, the water-jet trimmed hole showed delayed fracture at the hole-edges, while the mechanical punched hole showed early fracture as the bending angle increased. In comparing the numerical modeling and test results, the load-displacement curve, the displacement at the onset of cracking, and the final crack shape/length were used. Both failure criteria also enable the numerical model to differentiate between the local formability limit of mechanical-punched and water-jettrimmed holes. The failure criteria and static bend test developed here are useful to evaluate the local formability limit at a structural component level for automotive crash tests. Keywords: Failure analysis, Hole-edge failure, Component bend test. PACS: NUMI_ABS215

INTRODUCTION Hole-edge cracking has been observed during plastic deformations in stamping of advanced high-strength steels (AHSS) and vehicle crash tests. Due to the complex nature of edge fracture behavior, finite-element (FE) modeling with conventional material model/data has had only limited success for predicting edge cracking. There are several important factors which influence on hole-edge cracking, including:    

Sheet material properties Edge quality of hole from punching Pre-strains on the edge caused by trimming or punching Burr formation (up or down with respect to a moving punch).

STATE OF THE ART Research is actively ongoing in the area of hole-edge cracking prediction. Two different approaches include more complicated modeling techniques by either correlation of FE simulation results of blanking/piercing with stamping [1], or conducting numerical modeling to explicitly consider the volume fraction of martensite and ferrite in the dual-phase microstructure [2, 3]. However, these approaches may be limited for the direct use in industry forming simulations, due to the higher computation cost and unknown parameters that the FE model cannot take into account, such as hole-edge imperfections (e.g., micro-cracks remained on hole-edge after punching) and microstructure characteristics (banding width and micro-void areas, etc.). The stress-triaxiality-based failure model to predict the failures in deep-drawing TRIP 700 material and bending the automotive structural component [4]. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 506-513 (2011); doi: 10.1063/1.3623651 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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Compared to these complicated models, simpler prediction models were recently introduced [5, 6, 7]. A regression model was developed to handle all these unknown parameters in the hole-tensile test and hole-expansion test (HET) [5]. A simple ductile damage model was used to predict edge cracking in the bow-tie tensile test and HET [6]. A stress-strain based damage model was used to predict the local shear fracture during draw-bending of AHSS [7]. To account for macro-crack formation in FE simulations of HET, recently an inverse calibration method based on a damage model utilizing a triaxiality-dependent fracture criterion and hardening behavior was developed [8]. Therefore, for a given sheet material (including material properties and microstructure) and trimming process (punching, water-jet cutting, and laser cutting), a unique number or a small range of critical damage values that is the energy consumed to failure can be determined. With this background, it is desirable to develop a simple and practical failure model for industry applications for FE modeling of stamping and vehicle crash tests.

OBJECTIVE The primary objective of this study was to predict the hole-edge failures for AHSS structural components. The specific objectives are to:  Develop a component bend test (i.e., the structure design and test conditions) to evaluate the hole-edge failures  Predict hole-edge failure by using FE simulations with reliable failure criteria.

THE FAILURE CRITERIA FOR HOLE-EDGE CRACKING The hole-edge failure criteria were determined by using different local formability testing (LFT) methods and FEM model. Fig. 1 illustrates the flow chart to determine the failure map and its application for component bend simulations. Four different LFT methods such as a tensile, hole-tensile and a plane-strain shear and the hole expansion tests were used to characterize the effects of material properties and trimming methods on the hole-edge failures. Experimental results were analyzed by using FEM simulations to calculate the local strain, damage value and stress-triaxiality (i.e., a ratio of equivalent stress and mean stress). From experimental and simulation models, two different failure criteria: a local-strain-based damage model and a stress-triaxiality-based failure model were determined. The failure models were used to design the component bend test and predict the hole-edge failures observed in the component bend test.

Figure 1. Flow chart to determine the failure map

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The failure map was constructed with the local strains and different stress triaxiality calculated from FE simulations of various local-formability tests. The detailed procedure to construct the failure map is available in [4]. The CDV for TRIP 780 material was determined for different trimming conditions. Furthermore, two different failure maps were constructed for the punched-hole and water-jet-cut-hole cases. Both failure criteria were implemented for LS-DYNA simulations with MAT_224 as shown in Fig. 2. In the stress-traxiality based failure criteria, the failure strain varies with the stress triaxiality, while the local-strain based damage model has a constant failure strain as shown in Fig. 2. When the strain of FE model reaches this failure strain, the element at failure strains starts eroding in LS-DYNA, which emulates the crack initiation and propagation in experiments.

Figure 2. Failure map and criteria defined for FEM simulations for bending of the structural part

DESIGN OF THE COMPONENT BEND TEST A component bend test was developed by conducting preliminary FE simulations of various hat-shaped structure designs by bench-marking other designs available in literature [9, 10]. Different locations and sizes of hole and spot welds were investigated to obtain reproducible hole-edge failures in the test.

Figure 3. Schematic of the component structure and a bending configuration

Fig. 4 illustrates the procedure to optimize key design parameters in the hat-structure geometry and bending configuration. A series of spot weld models were defined on the flange of a lower channel as shown in Fig. 5. A weld-pitch of 25-mm was used and individual weld nugget model was defined to have 6-mm diagonal length and 1.65-mm thickness based on previous welding simulation experiences.

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Figure 4. A flow chart of the component design using FEA

Figure 5. FE model of a lower channel part and a spot weld model

EXPERIMENTAL RESULTS Component bending tests were conducted at the MTS machine (300-kN capacity) and Fig. 6 shows the testing setup. During the test, the high-resolution camera recorded the hole-edge failure as the upper ram moved down and the structure is bent.

Figure 6. A 3-pt. bend test for AHSS structures

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The component has a rectangular opening with four corner holes that were trimmed by punching or water-jet trimming. The lower hat-shaped channel was bent by using a press-brake machine. Finally, a top plate and a lower channel section were spot welded (e.g., 6-mm spot weld nugget diameter). The same TRIP 780 material was used for the entire component structure. During the test, the load-displacement curve was measured and synchronized with a high-resolution video camera monitoring the hole-edge region. Testing was conducted up to 120-mm stroke. Test results were summarized in Table 1. The punched structure samples showed the edge cracking in a lower stroke (e.g. 30~40-mm) compared to the water-jet structure samples and this resulted that the punched samples gave 5-10 mm larger crack length compared to the water jet samples at the final stroke (i.e. 120-mm). Fig. 7 shows both punched and water jet samples tested. The overall bend angles were similar between the punched and the water-jettrimmed structures. However, the punched case showed more severe cracks at two trimmed holes compared to the water-jet case. Table 1. Summary of experimental results

Sample No. WJ-1 WJ-2 WJ-3 P-4 P-5 P-6

Stroke at the start of cracking 87 mm 95 mm 100 mm 65 mm 60 mm 57 mm

Maximum Force 19.41 kN 19.27 kN 19.35 kN 19.31 kN 19.33 kN 19.23 kN

Figure 7. Tested component structures

FEM RESULTS Fig. 8 shows the FE model for 3-point bend testing. A commercial FEM code, LS-DYNA was used for simulations. The fine mesh (2-mm) was used for the areas near the hole-edges and the remaining areas were meshed with 5-mm elements. The rollers were meshed with 4-mm elements uniformly. When the local strain reaches the limit strain, the element erosion is initiated as shown in Fig. 9, and they are propagated to neighboring elements.

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Figure 8. Testing configuration of a component bend test

Figure 9. FEM predictions of the initiation of the hole-edge cracking (left) and the final fractured structure with the effective stress contour (right)

FEM simulations were conducted with the local strain-based damage model and the stress-traxiality-based failure map (Fig. 2). Compared to the test results of the punched case, the local-strain-based damage model gave better correlations than the stress triaxiality based failure map in terms of severity of cracking (Fig. 10) as well as the critical stroke at the onset of hole-edge cracking as shown Fig. 11.

Figure 10. Comparison of edge-failure shapes between the experiment and FEM predictions with different failure criteria

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Figure 11. Comparison of the critical stroke at the onset of hole-edge cracking

The stress triaxiality and local strain were compared for two different failure criteria of punched and water-jet cases (Fig. 12). The local strain value (0.74) from the ductile failure of a normal tensile test showed no edge failure and the local strain exceeded the boundary of failure strain limit that were determined by other failure criteria.

Figure 12. FEM predictions of the stress triaxiality and local strain with the different failure criteria for the punched case

DISCUSSSIONS In this study, different trimming methods such as a mechanical punching and a water-jet cutting were used to make the hole of specimens remain different residual stresses and damages at the hole-edge and this difference can result in different behaviour of hole-edge failure in secondary forming and crash tests. A local-strain based failure model was determined by using the CDV for different trimmed edge quality. This model was evaluated for FE simulations of various tests and it showed the better predictions of hole-edge failures compared to the complex failure map that was established by considering the stress triaxiality. Therefore, to be able to reliably apply the complex stress triaxility based model for the large scale crash model, a further study on stress triaxiality based failure model should be followed as a future work.

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CONCLUSIONS The following conclusions can be summarized from this study.  A new methodology was able to evaluate the quality of hole edge with different trimming methods and also quantify the damage/local strain remained from those trimming processes.  The punched hole showed the hole-edge failure at a lower strain compared to water-jet cutting.  The local-strain based failure criterion gave better correlations with experiments than the stress-triaxiality based failure criterion. A failure map based on the stress-triaxiality and local strain is useful to predict the hole-edge failure.

REFERENCES 1. R. Wiedenmann, P. Sartkulvanich, T. Altan, Finite element analysis on the effect of sheared edge quality in blanking upon hole expansion of advanced high strength steel, International Deep Drawing Research Group (IDDRG) 2009 International Conference, Golden, CO, United States. 2. X. Sun, K.S. Choi, Effects of AHSS Microstructure Inhomogeneity on Strain Localization, Shear Fracture Technical Progress Review, Auto/Steel Partnership, Sept., 10, 2008, Southfield, MI, United States. 3. X. Wu, C. Xie, H. Bahmanpour, Flange shear affected zone study, Shear Fracture Technical Progress Review, Auto/Steel Partnership, Sept, 10, 2008, Southfield, Michigan 4. D.Z. Sun, F. Andrieux, M. Feucht, Damage modeling of a TRIP steel for integrated simulation from deep drawing to crash, 7th European LS-DYNA Conference, May 14-15, 2009, Salzburg, Austria. 5. K. Watanabe, M. Tachibana, K. Koyanagi, K. Motomura, Simple Prediction Method for the Edge Fracture of Steel Sheet during Vehicle Collision (1st report) – Evaluation of Fracture Limit from the Edge using Small-Sized Test Pieces, LS-DYNA Conference. 2006. 6. J.S. Lee, Y.K. Ko, H. Huh, H.K. Kim, S.H. Park, Evaluation of Hole Flangeability of Steel Sheet with Respect to the Hole Processing Condition, Engineering Plasticity and its Applications 340-341, 2007, pp. 665-670. 7. H. Kim, A.R. Bandar, Y. Yang, J.H. Sung, R.H. Wagoner, Failure Analysis of Advanced High Strength Steels (AHSS) During Draw Bending, International Deep Drawing Research Group, IDDRG 2009 International Conference, Golden, CO, United States. 8. K. Chung, N. Ma, T. Park, D. Kim, D. Yoo, C. Kim, A modified damage model for advanced high strength steel sheets, International Journal of Plasticity, Article In Press, 2011 9. N. Kojima, K. Fukui, N. Mizui, Improvement in Bending Crashworthiness of Hat-Shape Columns by using High Strength Steel Sheets, 40th Mechanical Working and Steel Processing Conference, 1998, Pittsburgh, PA, United States. 10. K. Koyanagi, K. Motomura, Simple Prediction Method for the Edge Fracture of Steel Sheet during Vehicle Collision (2nd report), LS-DYNA Conference 2006, Anwenderforum, Ulm.

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A new strategy for stiffness evaluation of sheet metal parts Q. Caia, W. Volkb, A. Düsterc, E. Ranka a

Chair for Computation in Engineering, Technische Universität München, Arcisstraße 21, 80290 Munich, Germany b Institute of Metal forming and Casting, Technische Universität München, Walther-Meißner-Straße 4, 85748 Garching, Germany c Numerical Structural Analysis with Application in Ship Technology, Technische Universität Hamburg-Harburg, Schwarzenbergstraße 95c, 21073, Hamburg, Germany

Abstract. In the automotive industry, surfaces of styling models are shaped very often in physical models. For example, in the styling process of a car body important design work is realized by clay models and the resulting geometry information typically comes from optical scans. The scanned data is given in the form of point clouds which is then utilized in the virtual planning process for engineering work, e.g. to evaluate the load-carrying capacity. This is an important measure for the stiffness of the car body panels. In this contribution, the following two issues are discussed: what is the suitable geometric representation of the stiffness of the car body and how it is computed if only discrete point clouds exist. In the first part, the suitable geometric representation is identified by constructing continuous CAD models with different geometric parameters, e.g. Gaussian curvature and mean curvature. The stiffness of models is then computed in LS-DYNA and the influence of different geometric parameters is presented based on the simulation result. In the second part, the point clouds from scanned data, rather than continuous CAD models, are directly utilized to estimate the Gaussian curvature, which is normally derived from continuous surfaces. The discrete Gauss-Bonnet algorithm is applied to estimate the Gaussian curvature of the point clouds and the sensitivity of the algorithm with respect to the mesh quality is analyzed. In this way, the stiffness evaluation process in an early stage can be accelerated since the transformation from discrete data to continuous CAD data is labor-intensive. The discrete Gauss-Bonnet algorithm is finally applied to a sheet metal model of the BMW 3 series. Keywords: Stiffness evaluation, Gaussian curvature, point clouds, Gauss-Bonnet theorem, Automotive engineering. PACS: Geometric mechanics, 02.40.Yy

INTRODUCTION During the virtual planning process in the automotive industry, it is necessary to judge the structural stability of a car body on the basis of early styling models in order to assess its feasibility of production. Despite the tremendous advancements in computer based geometric modeling, these styling models are very often built physically with clay. In the virtual planning process, measuring and predicting the bending resistance of these clay models is a functional relevant issue. Especially for flat geometries like roofs, front hoods or trunk lids, it is difficult to meet the requirement of an adequate bending resistance because the curvature of the parts is an essential factor between suitable car package and sufficient bending resistance. The bending resistance, which is also called stiffness, is directly related to the buckling strength of flat geometries. An insufficient buckling strength often leads to hollows which is a main problem in the sheet metal production of car body panels. Currently only basic approaches exist to predict the stiffness and often big differences between predicted and experimental obtained values occur. This can lead to expensive modifications in the maturation phase of the product development process. Therefore it is desirable to have an accurate method for predicting the stiffness of flat sheet metal parts in the virtual world. In the first part of this contribution, Gaussian curvature is proved to be most competitive one for evaluating the stiffness of the surface, which also coincides with the statement in the literature. In the literature, points with zero Gaussian curvatures are considered to be parabolic or planar surface points which lie on the locally flat surface (e.g., The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 525-532 (2011); doi: 10.1063/1.3623653 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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plane, cylinder, etc.). And locally flat surfaces in particular have a worse load-carrying capacity than non-flat surfaces (e.g., sphere, saddle, etc.), see [10]. Hence, it is desirable to compute the Gaussian curvature of car body panels during the virtual-planning process, in order to check the load-carrying capacity of panels before they are put into the production. On the other hand, Gaussian curvature, the product of two principal curvatures at each point, can be computed directly if the point is located on a C2-continuous surface. Unfortunately, the first-hand geometric information of the models comes from scanned styling models, often given in the discrete form of point clouds. Therefore, there are two ways to evaluate the load-carrying capacity of a car body model with discrete data: one way is to transform these data into continuous CAD data and to compute the Gaussian curvature analytically in Figure 1(a); the other way is to estimate the Gaussian curvature directly from discrete point clouds in Figure 1(b).

a) Based on CAD data b) Directly from data clouds FIGURE 1. Gaussian curvature computation in two different ways

As Figure 1(a) indicates, the transformation from discrete point clouds to continuous CAD data, a typical laborintensive procedure, is nested inside the evaluation loop, which means that we have to perform it for each model. As styling models are frequently modified during the early design phase, a more efficient process is required, as is shown in Figure 1(b). It demonstrates that if we could estimate the Gaussian curvature directly from discrete point clouds, the transformation procedure could be carried out after we find a qualified styling model, which would avoid a repetition of the time-consuming work. A number of approaches have been proposed to analyze the Gaussian curvature from discretized surfaces. Meyer [5] outlines a method estimating the curvature of surface directly from a triangular mesh. The main idea is based on the discrete Gauss-Bonnet theorem, which can estimate the Gaussian curvature directly from a discretized surface without knowing the principal curvatures of the surface. Taubin [4] proposes a procedure estimating the tensor of curvature of a surface at the vertices of a polyhedral approximation. The Gaussian curvature can be computed after the principal curvatures have been calculated. Hamann [6] presents the parabolic fitting method based on local analytic surface approximation. The main idea is to use a rotated local quadratic surface to approximate the current surface and to compute from this the curvature. In addition, different methods of Gaussian curvature estimation have been also compared in the treatment of discrete surfaces [3]. Evgeni [1] states the comparison of the accuracy of different estimation methods using several typical 3D models. These methods were the Gauss-Bonnet method, the Parabolic fitting method, the Taubin's method, and the Watanabe and Belyaev method. According to the mean error of the Gaussian curvature's value, the Gauss-Bonnet scheme is demonstrated to give the best accuracy among these algorithms. Thus, the algorithm based on Gauss-Bonnet theorem was evaluated [7] in order to investigate its sensitivity with respect to the mesh quality of the model. When the experiments are performed with a variety of mesh models, the mesh-processing software MeshLab [11] is used.

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THE OPTIMAL GEOMETRIC REPRESENTATION OF THE STIFFNESS We compute the stiffness of the sheet metal form with LS-DYNA and compare the result to the experimental one in order to velidate the simulation result. The metal sheet for the experiment and the corresponding mesh model for the numerical simulation are shown in Figure 2. The material for the model is set with Young’s modulus 195GPa and prescribed displacements are enforced on the punch. Figure 3 shows the measured load on punch versus prescribed displacements graph and corresponding test points on the model are marked in Figure 2(left).

FIGURE 2. The car roof for the experiment (left) and corresponding mesh model for the numerical simulation (right)

FIGURE 3. The measured load versus prescribed displacements and corresponding test points

We can observe in Figure 3 that the simulation result has to be filtered due to the numerical noise. Comparing the loading path in the simulation (the upper red curve) to the experimental one (black curve), we see that simulation results coincide well with the experimental result for different test points on the sheet metal form, which indicates that the numerical simulation is reliable in the following curvature-stiffness analysis. To determine the most relevant geometric parameter to the stiffness, we employ two curvature related parameters – mean curvature and Gaussian curvature – to evaluate the stiffness of simple geometries. The geometric model is a part of an ellipsoid with two principle radii (correspond to principle curvatures) r1 and r2, see Figure 4. Different groups of geometry have the same projection area but different combination of r1 and r2. Exactly the same material property and load steps are applied for simple geometries as in the simulation for the car roof model.

FIGURE 4. The geometric model cut from an ellipsoid (left and middle), corresponding mesh in LS-DYNA (right)

The stiffness of different groups of geometric models is then computed by LS-DYNA and the result is shown in Figure 5. Two graphs demonstrate the stiffness versus Gaussian curvature and the mean curvature respectively. Point 1 has the principle radius 0.25m-1, 1m-1 and point 2 has the principle curvature 0.5m-1, 0.5m-1, which means that two points have the same Gaussian curvature but different mean curvatures. It is obvious that the stiffness at point 1 and

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point 2 is very close regardless of the difference in the mean curvature. In general, the stiffness increases monotonically while the Gaussian curvature increases. The mean curvature also increases but without monotonic tendency which is an important property for the representative geometric parameter. Although point 2 and point 3 have a close value of mean curvature, the stiffness has a considerable discrepancy since they have different Gaussian curvatures. Therefore, Gaussian curvature is considered as a better parameter of the surface from which the stiffness of the model can be evaluated.

Point 2 (0.5m-1, 0.5m-1)

Point 1 (0.25m-1, 1m-1)

Point 2 (0.5m-1, 0.5m-1)

Point 1 (0.25m-1, 1m-1)

Point 3 (1m-1, 0.1m-1)

Point 3 (1m-1, 0.1m-1)

FIGURE 5. Stiffness versus Gaussian curvature (left) and the mean curvature (right)

EVALUATION OF THE ALGORITHM WITH DIFFERENT MESH QUALITIES A spherical surface model that has a similar radius to the sheet metal model in the automotive industry is used in the evaluation. The prototype is a part of the sphere with a radius of 5000 mm, which is called an “umbrella” structure, illustrated in Figure 6. In the spherical coordinate system, the surface of “umbrella” structure models can be simply represented by two angle parameters  and  , namely a longitudinal and a latitudinal angle. According to the definition, the principal curvatures at the vertex v are: 1 1 (1) 1   2    2  104 (mm1 ) r 5000 Thus, the exact Gaussian curvature is: (2) K  1   2  4 108 (mm2 ) In order to get the discrete data in the form of point clouds, we retain a certain number of vertices on the sphere surface and create several discrete facets to approximate the smooth surface. Figure 6 shows the discretized surface model schematically. We can apply the algorithm to this model to obtain an estimated value of Gaussian curvature. During the application of the algorithm, the discrepancies between the estimation and the exact value of the different models can demonstrate how the algorithm behaves with respect to different meshes and can thus help us to evaluate the sensitivity of the algorithm.

FIGURE 6. “Umbrella” structure in prototype and mesh model (schematic)

FIGURE 7. Voronoi area A for non-obtuse elements(left), Voronoi area A1+A2 for an obtuse element(right)

Basic formulation of Gauss-Bonnet Theorem Let us suppose M is a compact two-dimensional Riemannian manifold with boundary M . Then the GaussBonnet theorem states: (3)  KdA   kg ds  2 (M ) M

M

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Here, K is the Gaussian curvature of M, dA is the element of area of the surface, k g is the geodesic curvature of

M , ds is the line element along the boundary of M and  is the Euler characteristic of M [12]. In ''umbrella'' structure models, the integral of the geodesic curvature on the piecewise linear boundaries is zero. Therefore the term k g ds is equal to zero and accordingly the Gaussian curvature over the discrete surface A



M

around a certain vertex can be calculated using the following equation [5]:  KdA  KA  2    i  2  i A

i

(4)

i

A is a properly defined area around the corresponding vertex, which has a direct influence on the result. Since the Gaussian curvature K is assumed to be constant within the area A, the integration is simply equal to the product of K and A. For non-obtuse elements, i.e. i  90 , we simply use the Voronoi area, which is composed of the surface area formed by using the circum-center of the triangles. Because the circum-center is the intersection of the perpendicular bisectors, the external angle  i is equal to  i in Figure 7(left), which is easier for computing. Thus, the Gaussian curvature at each vertex can be computed by means of the following equation. K

1   2   i  A i 

(5)

From the equation above we can see that the calculation of A is an important issue. However, if there is an obtuse triangle among the one-ring neighboring elements, the Voronoi region is extended beyond the one-ring triangle elements as depicted in Figure 7(middle). Since the Voronoi region is no longer valid in this case, we should define the area in a different way for the obtuse element. We draw two perpendicular bisecting lines through the edges containing the obtuse angle and take the two intersection points lying on the edge opposite to the obtuse angle. The area A1+A2 is considered as an effective area for this obtuse element as in Figure 7(right). Compared to other dividing schemes for obtuse elements [5], it has an indirect connection to the Voronoi region. We extend the obtuse triangle by mirroring it over the edge that is opposite the obtuse angle. If we connect the two vertices containing obtuse angles, it is easy to prove that two recreated triangles are non-obtuse. Considering each non-obtuse triangle separately, we found that area A1 and A2 can be regarded as the Voronoi region in each new triangle respectively. A similar approach can be also found in [2].

Sensitivity of Gauss-Bonnet Theorem to different parameters of mesh A good initial triangle mesh is a prerequisite for accurate results in estimating the curvature. In this section, several parameters of the models were taken into consideration during the evaluation of the Gauss-Bonnet algorithm: the latitudinal and longitudinal angle of triangle elements as well as the angle distribution around a vertex. These parameters are all related to the mesh quality, which could influence the accuracy of the algorithm. The influence of latitudinal angles  We choose a constant longitudinal angle  10 and vary the latitudinal angle in a range of 0    90 . Corresponding models can be seen in Figure 8.

FIGURE 8 The Meshes in different quantities of 

FIGURE 9. Comparison between the numerical and exact Gaussian curvature with varying 

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We can see result in Figure 9 that the degree of error is generally tends to increase in line with the increase in the number of elements, which means that the angle in the triangle elements decreases. Furthermore it can be observed that the graph oscillates significantly when the angle  is smaller than approximately 2 , which means that the accuracy of the algorithm is affected by the latitudinal angle of triangle elements when it exceeds a critical value. This critical value should be chosen based on the tolerance in the evaluation, which is 2 in this case. The influence of longitudinal angles To see the effect of longitudinal angles, we retain the latitudinal angle   36 and change the longitudinal angle  . The meshes in different quantities of  are presented in Figure 10. Obviously, the estimation of the curvature of the smooth spherical surface becomes worse when we increase the angle  , as depicted in Figure 11.

FIGURE 10. The Meshes in different quantities of 

FIGURE 11. Comparison between the numerical and exact Gaussian curvature with varying 

It can also be observed that when the mesh is too coarse, such as   30 , the error, which is bigger than 20%, is not acceptable anymore. The influence of the inequality of elements We keep  10 ,   45 constant and change the distribution of angles around the vertex. In order to discuss the effect of inequality and the order of elements, non-obtuse triangles are used for the estimation. To quantify the inequality of the elements in the mesh, we introduce a frequently used coefficient in economics, called the Gini coefficient. The Gini coefficient [13] is a measure of statistical dispersion, most prominently used as a measure of inequality of wealth or income disribution. The Gini coefficient, which indicates how evenly the variable is spread, is applied here to reflect the distribution of element angles around the top vertex. It is defined as a ratio between 0 and 1. A value of 0 corresponds to an equal distribution. In our case, it means that the angle related to the top vertex is the same for all the elements. By contrast, 1 corresponds to a perfect inequality. The interpretation is that there is only one element that covers the whole area around the top vertex. The attractiveness of the Gini coefficient is that it has an intuitive geometric interpretation [9]. That is, the Gini coefficient can be defined geometrically as the ratio of two areas in a unit box as shown in Figure 12. A is the area between the curve called the Lorenz curve and the 45 line in the box. B is the area under the Lorenz curve. The Lorenz curve is the graphical representation of the cumulative distribution of a probability distribution [8].

FIGURE 12. Discrete Lorenz curve

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In Figure 12, the Gini coefficient is then computed by the following equation: A (6) G A B Since we consider meshes with a limited number of elements, the Lorenz curve is not smooth but it consists of piecewise continuous lines. The area under the Lorenz curve can thus be calculated with the following formula 1 n1 (7) B   ( xi 1  xi )( yi 1  yi ) 2 i 0 We can distribute the angles in a different way to get a range of Gini coefficients. With the same longitudinal angle and the number of elements, the angle around the top vertex can be dispersed in a different way as shown in Figure 13.

FIGURE 13. Meshes with different Gini coefficients

FIGURE 14. Comparison between the numerical and exact Gaussian curvature with different inequality

Figure 14 depicts the effect of inequality on the estimation result. Generally speaking, the error of Gaussian curvature estimation tends to increase in line with the increase in the Gini coefficient. However, according to the value to the scale of the y axis, one can also see that this change is not very significant compared to the exact value (up to approximately 0.75%).

INDUSTRIAL APPLICATION In order to see whether it is feasible to apply the Gauss-Bonnet algorithm to the metal sheet of a car body, the door model of BMW 3 series is investigated using the proposed method. To compare our estimation to the exact value, we create a discretized surface from a smooth CAD surface.

FIGURE 15. Gaussian curvatures of the sheet metal model of the BMW 3 series

In Figure 15, the maximum value of the Gaussian curvature is assigned to the color blue, zero Gaussian curvature to the color green and the minimum value to the color red. Comparing the estimated result in Figure 15 (left) to the analytical result in Figure 15 (right), we can observe that the most critical region, which has the Gaussian curvature value zero, is colored green. These areas have a relatively low bending resistance and consequently a low buckling strength which is of a great importance to car body designers. The estimated result is accurate enough to substitute the analytical result in design process of the automotive industry to assess the feasibility of the production.

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CONCLUSION In this paper, we presented numerical simulations, demonstrating that the Gaussian curvature is a better geometric parameter to evaluate the bending resistance than the mean curvature. We employed the Gauss-Bonnet algorithm to estimate the Gaussian curvature of surfaces based on point clouds. To further analyze the effect of mesh quality on the algorithm, we considered a wide variety of “umbrella” structure models that represent discrete spherical surfaces with radii of 5000mm, close to the value of the sheet metal model in industry. The numerical investigation of the algorithm was performed by comparing the estimates to the analytical Gaussian curvature values. The investigation of spherical surface models has shown that when there are only non-obtuse triangles in the meshes, the accuracy of the estimation is influenced by the latitudinal angle  , the longitudinal angle  , of the spherical coordinate and the distribution of the elements. When   2 , the estimation value of Gaussian curvature is highly oscillatory resulting in inaccurate results. The degree of error increases in line with the increase in the longitudinal angle  . It is therefore advisable to avoid having excessively distorted elements in triangle meshes. The inequality of the element distribution was quantified by the Gini coefficient used in economics. It was shown that when the Gini coefficient increases (i.e. the elements become more uneven), the error of estimation also increases. However, compared to the longitudinal and latitudinal angles, the distribution of elements has less impact on the results. We applied the Gauss-Bonnet theorem to estimate Gaussian curvatures of the sheet metal model from BMW 3 series. The comparison between the estimated result and the analytical results from CAD continuous data showed that the estimation of Gaussian curvature based on the Gauss-Bonnet theorem is accurate enough to indicate zero Gaussian curvature surfaces which demonstrates a region with weak load-carrying capacity. Therefore, the use of the Gauss-Bonnet theorem in Gaussian curvature estimation is feasible in the automotive industry.

ACKNOWLEDGMENTS The first author gratefully acknowledges financial support of Munich Centre of Advanced Computing (MAC) and International Graduate School of Science and Engineering (IGSSE) at Technical University of Munich.

REFERENCES 1. M. Evgeni, S. Pctavian, R. Ehud, A Comparison of Gaussian and Mean Curvature Estimation Methods on Triangular Meshes, IEEE International Conference on Robotics and Automation, 2003, pp. 1021-1026. 2. N. Dyn, K. Hormann, S-J. Kim, D. Levin, Optimizing 3D triangulations using discrete curvature analysis, Press Innovations In Applied Mathematics Series archive Mathematical Methods for Curves and Surfaces, 135-146. 3. A. M. McIvor, R. J. Valkenburg, A comparison of local surface geometry estimation methods, Machine Vision and Applications 10, 1997, pp. 17-26. 4. G. Taubin, Estimating the tensor of curvature of a surface from a polyhedral approximation, The Fifth International Conference on Computer Vision, 1995, 902-907. 5. M. Meyer, M. Desbrun, P. Schroeder, A. H. Barr, Discrete Differential Geometry Operators for Triangulated 2-Manifolds, Visualization and mathematics, 2002, 3.34-57. 6. B. Hamann, Curvature approximation for triangulated surfaces, Computing Supplements 8, 1993, 139-153. 7. Q. Cai, "Computation of Gaussian curvature based on data clouds", Master Thesis, Technische Universität München, 2008. 8. http://en.wikipedia.org/wiki/Lorenz curve 9. K. Xu, How Has the Literature on Ginis Index Evolved in the Past 80 Years?, Department of Economics, Dalhousie University, Canada, 2004. 10. M. Farshad, Design and analysis of shell structures, Springer, 1st edition, 1992, 4-5. 11. P. Cignoni, MeshLab supported by EPOCH (EU Network of Excellence, IST-2002-507382), Visual Computing Lab of the Italian National Research Council, 2005. 12. http://en.wikipedia.org/wiki/Gauss-Bonnet theorem 13. http://en.wikipedia.org/wiki/Gini coefficient

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A Novel Method of Modeling the Deformation Resistance for Clad Sheet Hu Jian-lianga*, Yi You-pinga, Xie Man-tangb a

School of Mechanical and Electrical Engineering, Central South University, Lushan South Road, Hunan Province, Changsha 410083, China b

Dalian Design & Research Institute, First Heavy Industries Company Ltd, Northeast Street Road, Liaoning Province, Dalian 116600, China

Abstract˖Because of the excellent thermal conductivity ,the clad sheet (3003/4004/3003) of aluminum alloy is extensively used in various heat exchangers, such as radiator, motorcar air conditioning, evaporator, and so on. The deformation resistance model plays an important role in designing the process parameters of hot continuous rolling. However, the complex behaviors of the plastic deformation of the clad sheet make the modeling very difficult. In this work, a novel method for modeling the deformation resistance of clad sheet was proposed by combining the finite element analysis with experiments. The deformation resistance model of aluminum 3003 and 4004 was proposed through hot compression test on the Gleeble-1500 thermo-simulation machine. And the deformation resistance model of clad sheet was proposed through finite element analysis using DEFORM-2D software. The relationship between cladding ratio and the deformation resistance was discussed in detail. The results of hot compression simulation demonstrate that the cladding ratio has great effects on the resistance of the clad sheet. Taking the cladding ratio into consideration, the mathematical model of the deformation resistance for clad sheet has been proved to have perfect forecasting precision of different cladding ratio. Therefore, the presented model can be used to predict the rolling force of clad sheet during the hot continuous rolling process. Keywords: deformation resistance model; clad sheet; hot compression test; finite element analysis; cladding ratio PACS: 60

INTRODUCTION The clad sheet (3003/4004/3003) of aluminum alloy is widely used in various industries due to various good properties such as high electrical conductivity, high corrosion resistivity, and high strength [1]. The production of clad sheet by rolling process, which is a more efficient and economical approach compared with others, has become an increasingly important subject of study [2-5]. However, the plastic deformation behaviors of the clad sheet are complex during the hot continuous rolling process, which leads to the difficulty of modeling. So far, a limited number of paper has been published [6-12].In this paper, the deformation resistance model of Al 3003 and 4004 were firstly established through hot compression tests. And the novel deformation resistance model for clad sheet was proposed by finite element analysis based on the experimental results.

EXPERIMENTAL PROCEDURE The stress-strain relationship of Aluminum 3003 and 4004 were investigated by hot compression tests. The experiments were performed on the Gleeble-1500 thermo-simulation machine under five different temperatures (200ć, 300ć, 400ć, 450ć, 500ଇ) and six different strain rates (0.1 s-1, 1 s-1, 5 s-1, 10 s-1, 20 s-1, 50 s-1).

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 533-540 (2011); doi: 10.1063/1.3623654 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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Specimens are cylindrical with a diameter of 8 mm and a height of 12 mm. Each specimen was heated to the deformation temperature at a rate of 10 ଇ/s by thermo-coupled feedback-controlled AC current, and held for 5 min at isothermal conditions before compression tests. The reduction in height is 60% at the end of the compression tests. In order to minimize the frictions between the specimens and die during hot deformation, the flat ends of the specimen were entrapped the lubricant of graphite mixed with machine oil.

SIMULATION METHOD The compression simulation model was created and operated on DEFROM-2D software. The finite element

analysis model is rectangular. The cladding ratio is set as 20%,. the thickness of the Al 4004 cover layers being 3 mm, and the thickness of the Al 3003 core layer 9 mm, as shown in Figure 1. The simulation has five different temperatures (200ć, 300ć, 400ć, 450ć, 500ଇ) and six different strain rates (0.1 s-1, 1 s-1, 5 s-1, 10 s-1, 20 s-1, 50 s-1).

FIGURE 1.

Quadrate Specimen and Simulation Model

In order to obtain the critical flow stress of clad sheet with different cladding ratio(13.3%, 16.7%, 20%, 23.3%, 26.7%, 30%, 33.3%), the simulations were also performed at 300 ଇ, with strain rate of 10 s-1 and a reduction of 20%.

RESULTS AND DISCUSSION Resistance Models of Aluminum 3003 and 4004 Based on the Zuzin’s flow stress model [13], the deformation resistance models of aluminum 3003 and 4004 were obtained by the hot compression tests are written as follows: x

V 3003

H

H H · § 98.706 exp[ 1.176(T  1)]( ) 0.350.213T ¨1.015( ) 0.087 0.072T  (1.015  1) ¸ 10 0.2 0 .2 ¹ ©

(1)

x

H H · § ) 0.0610.084T ¨1.04( ) 0.1910.087T  (1.04  1) ¸ 10 0.2 0.2 ¹ ©

V 4004 147.049 exp[0.525(T  1)]( Where

V

is true stress(MPa);

H

H is true strain; T =t/300, t is temperatureć); H is strain rate(S-1).

534

(2)

Resistance Model of Clad Sheet with a 20% Cladding Ratio The flow stress curve of clad sheet at different strain rate and temperature were obtained through simulation as shown in Figure 2. When the temperature is 20ć the flow stress decreases slowly and comes to a level with the increase of strain. The reason is that the resistance of aluminum 3003 is smaller than that of aluminum 4004, aluminum 3003 has larger deformation, so the true stress decreases at the beginning. With the compression going on, the deformation between aluminum 3003 and 4004 comes to a balance, which corresponds to a steady stage of true stress. Due to the friction between aluminum 3003 and 4004, the lateral flow of aluminum is stocked, which make the compressive stress increase, and the flow stress grows with the stain when the temperature is higher. 160

200ć

200ć

140

120 100

True Stress(MPa)

True Stress(MPa)

140

300ć

80

400ć 450ć 500ć

60 40

120

300ć 100

400ć 450ć 500ć

80

60

40

20 0.0

20

0.2

0.4

0.6

0.8

1.0

0.0

0.2

˄a˅ H =0.1s-1

0.8

1.0

180

160 140 120

300ć

100

400ć 450ć 500ć

80

200ć

160

200ć

True Stress(MPa)

True Stress(MPa)

0.6

˄b˅ H =1s-1

180

60

140

300ć

120

400ć 450ć 500ć

100 80 60 40

40 20 0.0

0.2

0.4

0.6

0.8

20 0.0

1.0

0.2

True Strain

0.4

0.6

0.8

1.0

True Strain

˄c˅ H =5s-1

˄d˅ H =10s-1

180

200

200ć

140

180

True Stress(MPa)

160

True Stress(MPa)

0.4

True Strain

True Strain

300ć

120

400ć 450ć 500ć

100 80 60 40

200ć

160 140

300ć

120

400ć 450ć 500ć

100 80 60 40

20 0.0

0.2

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0.6

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20 0.0

1.0

True Strain

0.4

0.6

0.8

1.0

True Strain

˄e˅ H =20s-1 FIGURE 2.

0.2

˄f˅ H =50s-1

Flow Stress Curve under Different Strain Rate and Temperature

According to the simulation results, the deformation resistance model of clad sheet with 20% cladding ratio was given in equation (3).

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x

V

H H H · § 129.529 exp[0.856(T  1)]( ) 0.0430.029T ¨ 0.972( ) 0.030.033T  (0.972  1) ¸ 10 0.2 0.2 ¹ ©

(3)

Revised Resistance Model of Clad Sheet with Different Cladding Ratio In order to analyze the effect of cladding ratio on deformation resistance model, the deformation resistance model was modified to equation (4). x

V

Where,

V 0 exp[0.856(T  1)](

H

10

H H · § ) 0.0430.029T ¨ 0.972( ) 0.030.033T  (0.972  1) ¸a 0.2 0.2 ¹ ©

(4)

V 0 is the critical stress; b is the cladding ratio, a is the regression coefficient which is related to cladding

ratio. The critical stress of clad sheet has relation with aluminum 3003 and 4004, so the V 0 can be described as follow:

V0

cV 3003  (1  c)V 4004

(5)

Where, c is a coefficient that connects to cladding ratio. Table 1 is simulation results of critical stress at different cladding ratio. Figure 3 is drawn using the data in table 1, which shows a good linear relation between coefficient c and cladding ratio b. So the equation can be expressed as follow:

c TABLE (1). Cladding ratio /% Critical stress /MPa

0.31  1.16b

(6)

Critical Stress at Different Cladding Ratio

13.3

16.7

20

23.3

26.7

30

33.3

124.609

122.704

120.720

119.000

117.010

115.190

113.415

0.75

data point fit line

0.70 0.65

c

0.60 0.55 0.50 0.45 0.10

0.15

0.20

0.25

0.30

0.35

b FIGURE 3.

Relationship between Coefficient c and Cladding Ratio b

Combining the equation (4) to (6), the revised resistance model by critical stress can be expressed as

536

x ­ °V V 0 exp[0.856(T  1)]( H ) 0.0430.029T §¨ 0.972( H ) 0.030.033T  (0.972  1) H ·¸ u a ® 10 0.2 0.2 ¹ © ° ¯V 0 (0.31  1.16b)V 3003  (0.69  1.16b)V 4004

(7)

Figure 4 indicates that the revised resistance model is more close to the simulation results than regression model. However, the flow stress of revised model is a bit lower than simulation curve at high temperature. What’s more, the revised model almost have the same precision at different strain rates, so the strain rate will not be taken into consideration in the following revising resistance model progress. simulation curve regression model revised model by critial stress

140

180

ć

ć

160

100

True Stress(MPa)

120 True Stress(MPa)

simulation curve regression model revised model by critial stress

ć

80 ć ć ć

60 40

140

ć 120 100

ć ć

80

ć

60

20 0.0

40

0.2

0.4

0.6

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1.0

0.2

(a) Strain Rate of 0.1S-1 FIGURE 4.

0.4

0.6

0.8

1.0

True Strain

True Strain

(b) Strain Rate of 5S-1

Flow Stress Curve of Simulation, Regression Model and Revised Model by Critical Stress at Strain Rate of 0.1 S-1,5S-1 while Cladding Ratio is 20%

Figure 5 shows connections of revised deformation resistance model and simulation data of different cladding ratio at the strain rate of 1S-1. The revised model can describe the flow stress of different cladding ratio. But the revised model has a little offset to the simulation data at the temperature of 450ଇ and 500ଇ, and the offset increases with the increase of the cladding ratio. simulation curve revised model by critial stress

160

simulation curve revised model by critial stress

160

ć

140

True Stress(MPa)

True Stress(MPa)

ć

140 120 ć

100 ć ć ć

80 60 40 20 0.0

0.2

0.4

0.6

0.8

120 ć

100 ć

80

ć ć

60 40 20 0.0

1.0

(a)

0.2

0.4

0.6

True Strain

True Strain

Cladding Ratio is 13.3%

(b)

537

Cladding Ratio is 16.7%

0.8

1.0

simulation curve revised model by critial stress

160 140

ć

140

True Stress(MPa)

True Stress(MPa)

simulation curve revised model by critial stress

160 ć

120 ć

100 ć

80

ć ć

60 40 20 0.0

0.2

0.4

0.6

0.8

120 ć

100 ć

80

ć ć

60 40 20 0.0

1.0

0.2

0.4

True Strain

(c)

Cladding Ratio is 20.0%

ć

140

ć

100 ć ć ć

80 60 40

0.2

0.4

0.6

0.8

ć

120 ć

100 80

ć

60

ć ć

40 20 0.0

1.0

0.2

0.4

True Strain

(e) FIGURE 5.

1.0

simulation curve revised model by critial stress

140

120

20 0.0

0.8

Cladding Ratio is 23.3%

160

True Stress(MPa)

True Stress(MPa)

(d)

simulation curve revised model by critial stress

160

0.6

True Strain

0.6

0.8

1.0

True Strain

Cladding Ratio is 26.7%

(f)

Cladding Ratio is 30.0%

Flow Stress Curve of Simulation and Revised Model by Critical Stress at Strain Rate of 1S-1 of different Cladding Ratio

Table 2 is the difference between revised deformation resistance model by critical stress and simulation data. Minus means the model is bellow the simulation data. TABLE (2).

Difference Value Between Revised Model by Critical Stress and Simulation Data (MPa) Cladding ratio %

13.3

16.7

20.0

23.3

26.7

30.0

200ଇ

0

0

0

0

0

0

300ଇ

0

0

0

0

0

0

400ଇ

0

0

0

-1.1

-1.8

-2.5

450ଇ

-0.8

-2.1

-3.1

-3.9

-5.3

-6.2

500ଇ

-3.2

-4.0

-5.2

-5.7

-7.0

-7.8

From the figure 5, it can be seen that the curve of revised model could fit well with the simulation data if the model curves of 400ଇ, 450ଇ and 500ଇ make an upward movement. This means, there exists a factor that the equation (7) should be multiplied by to make the model fit the simulation curve. Figure 6 is the curve of factors at different simulation conditions, which show a fine linear fit between factor a and cladding ratio b.

538

ć ć ć ćILWOLQH ćILWOLQH ćILWOLQH

1.16 1.14 1.12

factor a

1.10 1.08 1.06 1.04 1.02 1.00 0.10

0.15

0.20

0.25

0.30

0.35

cladding ratio b

FIGURE 6.

The Relation between Factor a and Cladding Ratio b

When the temperature is 400ć,the equation is a 0.9517  0.268b

˄8˅

0.9707  0.364b

˄9˅

When the temperature is 450ć, the equation is a

When the temperature is 500ć, the equation is a 1  0.483b

˄10˅

When T= t/300ˈthe equation (8) to (10) can be describe as one equation: a

(0.761  0.142T )  (0.577  0.632T )b

˄11˅

As a conclusion, the deformation resistance of clad sheet can be revised as follow: x

V

V 0 exp[ 0.856(T  1)](

V0

H 10

H H · § ) 0.043 0.029T ¨ 0.972( ) 0.03 0.033T  (0.972  1) ¸u a 0. 2 0.2 ¹ ©

(0.31  1.16b)V 3003  (0.69  1.16b)V 4004

˄12˅

when t˘400ć when 400ć  t  500ć

a 1 a (0.761  0.142T )  (0.577  0.632T )b

In order to verify the model, the curves of simulation data, regression model and the revised model with the cladding ratio being 30% and strain rate of 20S-1can be shown in figure 7, which reveals that the revised model fits the simulation data better than the regression model. 200

True Stress(MPa)

180

ć

simulation curve regression model revised model

160 140

ć

120 ć

100

ć ć

80 60 0.0

0.2

0.4

0.6

0.8

1.0

True Strain FIGURE 7.

The Curves of Simulation Data, Regression Model and the Revised Model

539

CONCLUSIONS In this work, for the first time, the mathematical model for the deformation resistance of clad sheet was obtained through hot compression test and simulation. The following conclusions are achieved: 1.

In the compression simulation of clad sheet, the deformation resistance increases smoothly since the friction

2.

The results of hot compression simulation demonstrate that the cladding ratio of clad sheet has great effects on

between layers prevents the lateral flow of the clad sheet. the resistance of the clad sheet. 3.

With attention to fine agreement between the predicted results obtained from the revised model and the simulation results, the model developed in this work is useable for the simulation of rolling of symmetrical clad sheet.

ACKNOWLEDGMENTS This work was supported by Dalian Design & Research Institute, First Heavy Industries Company Ltd.

REFERENCES 1. Aluminium Brazing Handbook, Nihon keikinzoku yousetsu kouzou kyoukai, Japan, 1992. 2. Y.M. Hwang, H.H. Hsu, H.J. Lee, Analysis of sandwich sheet rolling by stream function method, Int. J. Mech. Sci. 37 (3) (1995) 297-315. 3. J.S. Yoon, M.-S. Kim, Effect of cold rolling condition on the brazing characteristic for Al 4343/3003/4343 brazing sheet, J. Kor. Inst. Met.Mater. 37 (1999) 992-999. 4. Y.M. Hwang, H.H. Hsu, H.J. Lee, Analysis of plastic instability during sandwich sheet rolling, Int. J. Mech. Tools Manuf. 36 (1)(1996) 47-62. 5. Z.C. Lin, T.G. Huang, Different degree of reduction and sliding phenomenon study for three-dimensional hot rolling with sandwich flat strip, Int. J. Mech. Sci. 42 (2000) 1983-2012. 6. R. Hawkins, J.C. Wright, Observation on the deformation properties of sandwich materials, Int. J. Mech. Sci. 14 (1972) 875-883. 7. A.A. Afonja, D.H. Sansome, A theoretical analysis of the sandwich rolling process, Int. J. Mech. Sci. 15 (1973) 1-14. 8. A.G. Atkins, A.S. Weinstein, The deformation of sandwich materials, Int. J. Mech. Sci. 12 (1970) 641-657. 9. L.R. Vaidyanath, M.G. Nicholas, D.R. Milner, Pressure welding by rolling, Brit. Weld. J. 6 (1959) 13-28. 10. S.C. Pan, M.N. Huang,Analysis of asymmetrical cold and hot bond rolling of unbounded clad sheet under constant shear friction, Journal of Materials Processing Technology, 177, (1-3), 2006, 114-120 11. Manabu Kiuchi, Ken Shintani, Yeong-Maw Hwang, Mathematical Simulation of Clad Sheet Rolling and Sandtvich Sheet Holling, CIRP Annals - Manufacturing Technology, 41(1), 1992, 289-292 12. Frank Nowicke Jr., Antonios Zavaliangos, Harry C. Rogers, The effect of roll and clad sheet geometry on the necking instability during rolling of clad sheet metals, International Journal of Mechanical Sciences, 48(8), 2006, 868-877 13. Zuzin W I, M Ya Browman and Melikov A F. Metallurgy, Moscow, 1964

540

Analytic Study on Pure Bending of Metal Sheets Jeerachai Supasuthakula, Peter D. Hodgsona, and Chunhui Yangb a

Centre for Material and Fibre Innovation, Deakin University, Waurn Ponds, VIC 3217, Australia b

School of Engineering, Deakin University, Waurn Ponds, VIC 3217, Australia

Abstract. In this work, analytical models of pure bending are developed to simulate a particular type of bend test and to determine possible errors arising from approximations used in analyzing experimental data. Analytical models proposed for steels include a theoretical solution of pure bending and a series of finite element models, based on the von Mises yield function, are subjected to different stress and strain conditions. The results show that for steel sheets the difference between measured and calculated results of the moment-curvature behaviour is small and the numerical results from the finite element models indicate that experimental results obtained from the test are acceptable in the range of the pure bending operation. Further for magnesium alloys, which exhibit unsymmetrical yielding, the algorithm of the yield function with a linear isotropic hardening model is implemented by programming a user subroutine in Abaqus for bending simulations of magnesium. The simulations using the proposed user subroutine extract better results than those using the von Mises yield function. Keywords: Pure bending, metal sheets, theoretical solution, finite element modelling, and yield function. PACS: 46.15.-x

INTRODUCTION Sheet-metal bending is a widely-used sheet forming operation. Although bending is technically a simple operation, it causes a complex stress history and material flow in a sheet. For isotropic materials Ludwik [1] firstly established a theory of the plastic bending of a non-strain-hardening sheet subjected to only bending moments. Hill [2] extended Ludwik’s work by defining a plane-strain condition for the case where the radius of curvature of a bent sheet is large compared to the sheet thickness and the through-thickness stress is negligible. He then considered the movement of the neutral surface for small radius bends. Marciniak et al [3] suggested that the large radius bend approximations are suitable for a bent sheet having a curvature greater than five times the curvature in elastic bending. Lange [4] proposed a theoretical model under elastic-perfectly plastic deformation to determine the total bending moment. In reality, sheet metals, generally manufactured by cold rolling, have anisotropic mechanical properties varying with direction. Tan et al [5] presented a model of the plastic, plane-strain bending of anisotropic sheet metals incorporating the Bauschinger effect to consider such effects on material thinning. A model of the elastic-plastic bending of anisotropic sheets was introduced by Chakrabarty et al [6] to determine a momentcurvature relation. They accounted for the elastic-plastic bending, and showed a relation of the bending moment to the curvature. However, this model did not consider deformation of fibers away from the middle surface as Marciniak et al did where they considered the movement of the neutral surface. For magnesium alloy sheets which exhibit asymmetrical yielding, an analytical model of bending with tension was proposed by Lee et al [7], based on the classical beam theory, linear strain hardening, and observations from experimental stress-strain tests. For numerical solutions of sheet metal forming of magnesium alloys, the finite element method has been applied widely. Palumbo et al [8] developed a fully coupled thermo-mechanical finite element model for the stretch-bending operation and studied the bend radius and temperatures. Kim et al [9] proposed a hardening law based on the orthotropic yield criterion of Cazacu et al [10] for shell-element models on three-point bending of magnesium alloy sheets. Lee et al [11] proposed an asymmetric hardening model for magnesium alloys using the modified anisotropic Drucker-Prager yield criterion. Lee et al [12] extended their own

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 541-548 (2011); doi: 10.1063/1.3623655 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

541

work to prediction of springback of magnesium alloy AZ31B using shell-element models of the draw bend test. However, when the thickness stress plays an important role on forming operations, solid-element models using such yield function should be considered. When considering the yield criteria for magnesium alloys, the yield function of Cazacu and Barlat [13] is attractive because the yield function is derived from the test and includes the third stress invariant J c3 describing the difference between compressive and tensile stresses. In the current work, considering the elastic-plastic bending and the deformation of fibers through the thickness, a theoretical bending model for isotropic materials, such as stainless steels, is firstly developed. We deal with two loading conditions: plane strain and uniaxial tension, and study which condition is more suitable for the modelling of pure bending. Secondly, a series of finite element models for stainless steels are developed, and the bending condition of the apparatus is studied to see whether or not these two conditions in the test rig can be approximated to pure bending. Thirdly, an algorithm for the yield function of Cazacu and Barlatis is implemented in the user subroutine UMAT for Abaqus [14, 15], and it is further employed in the finite element model to simulate forming operations of magnesium alloy sheets without heat involved. Lastly, the results from both the analytical modelling and experimental work are compared and verified.

THEORETICAL SOLUTION OF PURE BENDING OF A SHEET OF ISOTROPIC MATERIAL For a thin sheet which is bent by end couples, the maximum bend radius in the sheet is not less than three or four times the sheet thickness. According to the classic bending theory, a plane normal section in the sheet will remain plane and normal and converge on the centre of curvature [3]. In addition, the material is assumed as homogeneous and isotropic, and the sheet thickness is assumed to be constant throughout the process. In this study, let the 1-, 2and 3- directions be the circumferential, width and radial directions, respectively. It is assumed that the neutral surface is located at the mid-surface of the sheet and the axial strain of a fiber is,

H1 H b | y / R,

(1)

where Hb is the bending strain, y is the distance measured from the middle surface, and R is the radius of curvature at the middle surface. In elastic bending, the relation of axial stress to axial strain is

V1

E cH1 ,

(2)

where Ec is the modulus of elasticity of bending in the circumferential direction. Hooke’s law under uniaxial-tension condition (V1; V2 = 0; V3 = 0) indicates that

Ec

E,

(3)

In the plastic bending, assuming uniaxial deformation, the process is

 1p ;  p2

 1 / 2  1p ;  3p

 1 / 2  1p .

(4)

Under such loading condition, the von Mises yield function is

f = Vef  Vy(Hef) = V1  Vy(Hef),

(5)

and the effective strain derived from Drucker’s postulate is

Hef =  1p . For a cold-worked material with a work-hardening behaviour, the following equation can be used to the data:

542

(6)

Vy = Ks (H0 + Hef)n,

(7)

where Ks is the strength coefficient, H0 is the pre-strain constant and n is the strain hardening index. The axial stress can thus be expressed as follows:

V1 = Ks (H0 +  1p )n.

(8)

In the case of elastic-plastic bending, for unit width, the equilibrium condition becomes

M

t 0 /2 · § ye n ¨ 2 ³ E y / R y dy  ³ K s > 0  y / R   y / E @ y dy ¸ , ¸ ¨0 ye ¹ ©

(9)

where ye is the distance to the interface of elastic and plastic zones, ye = R Vy / Ec. Considering the pure bending of a sheet under plane-strain condition (V1; V2 = V1 / 2; V3 = 0), Hooke’s law indicates that

Ec = E / (1  Q2).

(10)

In the plastic bending under the plane-strain condition, the deformation process is

 1p ;  p2

0 ;  3p

 1p .

(11)

Under this loading condition, the von Mises yield function is

f = Vef  Vy = (31/2 / 2) V1  Vy,

(12)

and the effective strain derived from Drucker’s postulate is

Hef = (2 / 31/2)  1p .

(13)

V1 = (2 / 31/2) Ks [H0 + (2 / 31/2) (H1   1e )]n,

(14)

The axial stress is therefore

In the case of elastic-plastic bending, the equilibrium condition is

M

t 0 /2 § ye 2 ¨ ³ Ec y / R y dy  ³ 2 / 3 K s  0  2 / 3 ¨0 ye ©



>



where the distance ye = (2 / 31/2) Vy R / Ec.

^y / R  2 / 3 

y

/ E c

`@

n

· y dy ¸ , ¸ ¹

(15)

FINITE ELEMENT MODELLING ON PURE BENDING OF METAL SHEETS In the study, two-dimensional finite element models were developed by using the commercial finite element analysis package–Abaqus. The bend equipment prototype for pure bending designed by Weiss et al [16] was used to generate the experimental data as shown in Fig. 1. The bend test can be performed in a standard tensile testing

543

machine to produce appropriate results of nearly pure bending. Bending and reverse bending can be performed when the crosshead of the tensile testing machine is displaced either upwards or downwards.

Fixed Point

Reference Point of Clamp

Clamp

Reference Point of Ram Ram

(a)

(b)

U1

FIGURE 1. (a) Bending apparatus and (b) finite element modelling with translation of the ram [7]

In the 2-D finite element models for both isotropic and anisotropic materials, since the bending operation is symmetric about a plane along the centreline of the sheet, only half of the sheet was modelled, as shown in Fig 1 (b). Two finite element models using two types of solid elements: plane-strain and plane-stress elements, to model the sheet are devised, respectively. The clamp and ram are modelled as discrete rigid parts whilst the surfaces of the clamp that contact the top and bottom surfaces of the strip are modelled as discretized rigid surfaces. As for interactions between two parts, the regions in which the surfaces of the sheet and clamp contact each other are fused using tie constraints, and a connection between two points of the clamp and ram is modelled using a joint type connector. In the bend tests of isotropic materials, a commercial stainless steel sheet of 200mm length, 19.84mm width and 5mm thickness was used. Its elastic properties are Q of 0.3 and E of 190 GPa, and its plastic data is obtained from fitting Eq. (7) with the true stress-strain curve, so Ks = 690 MPa, H0 = 0.0009, and n = 0.095. The von Mises yield function was employed. In the bend tests of anisotropic materials, Magnesium alloy strip (AZ31B) of 150mm length, 18mm width and 2mm thickness was used in the bending experiment. Its elastic properties are defined by a Poisson ratio of 0.35 and Young’s modulus of 45 GPa. Fitting the stress-strain equation to the true stress-strain curve derived from a tensile test gave the following parameters: Ks = 450 u 106 Pa, H0 = 0.002, and n = 0.14. The yield function of Cazacu and Barlat that describes the asymmetry in yielding was then used for the magnesium alloy:

f where

Jc

3/2 2

 cJ c3



1/ 3

 y ,

(16)

J c2 is the second stress invariant, J c3 is the third stress invariant, c is the material parameter of

unsymmetrical yielding and Wy is the yield stress in pure shear. In addition to the yield function of Cazacu and Barlat, Hooke’s law and Drucker’s postulate were applied. The implicit integration technique, called the radial return method [14], was utilized and improved to solve the constitutive equations. For plastic data in the user subroutine UMAT [15], since linear hardening is defined, only parameter H is specified; a suitable value of H is 2200 MPa. Other parameters that need to be specified in the user subroutine are the initial tensile stress and initial compressive stress; the initial tensile stress of AZ31B is 210 MPa and the initial compressive stress is 120 MPa.

544

RESULTS AND DISCUSSION The proposed theoretical solution of the bending of isotropic materials based on the von Mises yield criterion leads to a relation of the bending moment to curvature. Using the proposed solutions, the values of bending moment and curvature were predicted. Finite element modelling of the bending operation was also devised to study two loading modes: plane strain and plane stress, and two bending approaches  by applying a rotation to a reference point of the clamp and applying a translation to the ram making the clamp rotate. For the validation of the theoretical and simulation models, experimental work was also employed. In Fig. 2, the theoretical solution under plane-strain yields higher results than the experimental data, but under uniaxial-tension conditions the curve is lower. When considering the stress state of both solutions, the intermediate principal stress V2 is likely to have an important effect. Under uniaxial tension V2 is zero and the bending moment is significantly underestimated. The solution under plane strain provides results closer to the experimental data. In the experiments the width to thickness ratio of the test-piece is 9; the greater it is, the more closely the bend process would be expected to be plane strain.

FIGURE 2. Moment-curvature diagram from theoretical solutions and experimental data

To obtain better simulation results other conditions may be considered. For two-dimensional problems without regard to heat transfer, Abaqus offers two major kinds of solid elements  plane-strain elements and plane-stress elements – which characterize the condition of deformation for the process. When the plane-strain condition is defined, the simulation probably overestimates results; hence, the plane-stress condition may be more appropriate for this problem. Fig. 3 shows a difference between simulation results under the plane-strain and plane-stress conditions. Besides selecting proper elements for a sheet, the characteristics of loading in the bending operation are examined. Two simulations whose loadings are different: the one with applying a translation at the ram, similar to the experiment, as shown in Fig. 1 (b), and the other with applying a rotation at the clamp at its reference point (called pure-bending model) , produce close results.

545

FIGURE 3. Moment-curvature diagram from simulations under plane-strain and plane-stress conditions

To investigate the magnitude of errors of the experimental data, simulations are performed at further displacement of crosshead. In Fig. 4, bending moments obtained from the simulation having translation of the clamp begin to drop at a curvature of about 15.708 /m or at a displacement of the crosshead of 0.095 m. This indicates that for the experiments in this work, where the curvature did not exceed about 5 /m, either mode of analysis will give similar results.

FIGURE 4. Moment-curvature diagram from simulations under different loadings for more severe bend

The user subroutine was firstly examined by reducing the form of the yield function of Cazacu and Barlat into the form of the von Mises yield function by assuming the tensile and compressive strengths are equal. Results obtained from the user subroutine were compared with those from the model following the definition of linear kinematic hardening available in Abaqus and from the model containing pairs of plastic data of stresses and strains.

546

FIGURE 5. Moment-curvature diagram from simulations and experiment for stainless steel sheet

Fig. 5 shows results of the relationship between moment and curvature from all the three models. In Fig. 5, under the plane strain condition, the simulation results are higher than the experimental result, and the model using the linear kinematic hardening in Abaqus predicted the same result as the model using linear isotropic hardening in the proposed user subroutine. This verifies the user subroutine is reliable. In the bending of AZ31B magnesium alloy sheet, a variation in the material constant c for the yield function of Cazacu and Barlat is studied, as shown in Fig. 6. When the initial compressive stress is less than the initial tensile stress, the moment-curvature result is lower and closer to the experimental result. This shows that the yield function of Cazacu and Barlat improves the simulation results.

FIGURE 6. Moment-curvature diagram from simulations and experiment for AZ31B magnesium alloy sheet

547

CONCLUSIONS In this paper, a theoretical solution and a series of finite element models for the pure bending of metal sheets have been developed for steels and magnesium. For steels, the theoretical solution of pure bending under the planestrain condition gives better agreement to the experiment than that under the uniaxial-tension condition but overestimates the moment by an appreciable amount. A finite element model of the process using a plane-stress model gives a better agreement with experimental results. It reveals that the numerical model developed by using the planestress condition and the von Mises yield criterion is apparently an optimal choice for bending simulations of stainless steel sheet. For the finite element modelling of magnesium alloys, the user subroutine UMAT for Abaqus/Standard has been developed, verified and implemented. When the material constant c for the asymmetry in yielding is incorporated, the simulations using the yield function of Cazacu and Barlat provides a better agreement to the experimental data than the simulations using the von Mises yield function approximately. It therefore shows the differential strength effect on magnesium alloys. In the experimental work, the mechanism of the bending apparatus is considered to ensure that the apparatus produces proper results of pure bending. The simulation under the translation of the crosshead, representing the experiment, predicts results that are different to those from the simulation under the rotation at the end of strip, representing the pure bending. It shows an increase of the error in the bending progression of experiment. The error can be neglected at an admissible value to ensure the experiment operates as pure bending.

ACKNOWLEDGEMENTS Authors would like to acknowledge the financial support from the ARC Centre of Excellence for the Design of Light Metals and the Australian Laureate Fellowship for Peter D. Hodgson and the kind assistance of Dr Matthias Weiss in the bending experiments and analysis. The valuable discussions with Prof John L. Duncan are also appreciated.

REFERENCES P. Ludwik: Technologische Studie über Blechbiegung (Technische Blätter, Praha 1904). R. Hill, in: The Mathematical Theory of Plasticity, Chapter, XI, Oxford University Press (1950), pp. 287-293. Z. Marciniak, J.L. Duncan and S.J. Hu, in: Mechanics of Sheet Metal Forming, Butterworth-Heinemann (2002), pp. 82-92. K. Lange, in: Handbook of Metal Forming, chapter, 19, McGraw-Hill (1985). Z. Tan, B. Persson and C. Magnusson: International Journal of Mechanical Sciences, Vol. 37 (1995), pp. 405-421. J. Chakrabarty, W.B. Lee and K.C. Chan: International Journal of Mechanical Sciences, Vol. 43 (2001), pp. 1871-1880.\ M.G. Lee, J.H. Kim, K. Chung, S.J. Kim, R.H. Wagoner and H.Y. Kim: International Journal of Plasticity, Vol. 25 (2009), pp. 399-419. 8. G. Palumbo, D. Sorgente and L. Tricario: Journal of Materials Processing Technology, Vol. 201 (2008), pp. 183-188. 9. J. Kim, H. Ryou, D. Kim, Da. Kim, W. Lee, S.H. Hong and K. Chung: International Journal of Mechanical Sciences, Vol. 50 (2008), pp. 1510-1518. 10. O. Cazacu, B. Plunkett and F. Barlat: International Journal of Plasticity, Vol. 22 (2006), pp. 1171-1194. 11. M.G. Lee, R.H. Wagoner, J.K. Lee, K. Chung and H.Y. Kim: International Journal of Plasticity, Vol. 24 (2008), pp. 545-582. 12. M.G. Lee, S.J. Kim, R.H. Wagoner, K. Chung and H.Y. Kim: International Journal of Plasticity, Vol. 25 (2009), pp. 70-104. 13. O. Cazacu and F. Barlat: International Journal of Plasticity, Vol. 20 (2004), pp. 2027-2045. 14. Dunne F. and Petrinic N., in: Introduction to Computational Plasticity, Chapter, 5, Oxford University Press (2005), pp. 146153. 15. DS Simulia, Abaqus V6.9 User Manual (2009). 16. M. Weiss, H. Wolfkamp, B.F. Rolfe, P.D. Hodgson and E. Hemmerich: Measurement of Bending Properties in Strip for Roll Forming (IDDRG International Conference, Colorado 2009). 1. 2. 3. 4. 5. 6. 7.

548

$6KRS)ORRU$SSURDFKWR6SULQJEDFN3UHGLFWLRQ K.B. Nagur Aziza , Zakaria, Aa, K. Khairul Za’ima , M. Achmeda and M.A.L. Mohd Safuana a



Institute of Product Design & Manufacturing, Universiti Kuala Lumpur, 119 Jalan 7/91, Taman Shamelin Perkasa, 3.5 miles Cheras, 56100 Kuala Lumpur, Malaysia.

$EVWUDFW This paper presents a new methodology called a shop floor approach to springback prediction. It begins with the development of a springback model based on die and stamped part historical data by Multiple Regression (MR) technique. This model is then used to validate a numerical springback model using Finite Element Analysis (FEA). Three different pressed parts were selected from industry for this study. They represent three different levels of springback severity namely high, medium and small. Significant factors affecting springback such as die radius, die angle, die depth, punch radius and die width were used as independent or predictor variables meanwhile shape deviation refer springback response between stamped part and die surface were used as dependent or criterion variable. Initial results show that the multiple regression models for the three parts are linear with a prediction error of less than 9%. However a combined part1 and part3 regression model improves the prediction accuracy by 7% compared to values predicted by individual regression model. An on-going work is being conducted to validate the model by using commercial FEA software. .H\ZRUGVShop Floor Approach, Multiple Regression, Springback. 3$&6 80.81.05.Bx

,1752'8&7,21 Though springback phenomenon issues have been around in automotive manufacturer industries for decades, its entire phenomenon is never really understood. Tool geometry is one of the contributing factors to springback occurrences [1]. While geometry correction on finished tool is physically compensated being the standard practice in the current industries, it is really costly and time-consuming job [2]. The influences of geometric parameters namely die radius, sector angle and numbers of elements on springback are observed in previous research [3]. These are the key parameters as once experimented and also published [4]. In most cases springback is greatly dependent on geometric parameters at minimal load condition and decreases with more compression depth. Attempts have been made to mitigate these geometrical defects on diverse springback parameters via materials, part design, tool design and processes. Kim et al., [5] proposed an analytical model to predict bend allowance. Naceur et al., [6-8] used Response Surface Method (RSM) to predict punch radius and die radius. Narayanasamy and Padmanabhan [9] investigate punch travel, strain hardening exponent, punch radius, punch velocity and width of the sheet with comparing using regression and neural network modelling. Eggertsen and Mattiasson, [10-11] proposed a material model using Finite Element Method (FEM). Marretta et al., [12] studied the blank holder force (BHF), lubricant conditions and strain-hardening of the material using FEM, RSM and Monte Carlo Simulation (MCS) method. Iwata et al., [13] employed the FEM for bend radius study. Garcia-Romeu et al., [14] analyze bend angle and bend radius based on an experimental work. Most researches of springback were responses towards specific approach for specific springback parameters whereby the using of the experimental and analysis part design are quite simple and arbitrarily modeled. There are also less concentration and consideration to all prerequisites process that need to be in-place prior to the actual implementation and adoption as a whole to ensure its success. As in the industries, all of these parameters or variables happen simultaneously, interact and affecting each other. Therefore, it be seen, the proposed approaches not accurately represented in simulation [15] and reality formulations [16], difficult to establish the cause of discrepancy between the magnitudes of springback obtained in simulation and reality [17] especially for the products with complicated geometry or in cases with different materials. The discrepancies of the springback cannot be

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 549-556 (2011); doi: 10.1063/1.3623656 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

549

explained on the basis of variability of the input parameters or numerical factors alone [18] especially when the product geometry is complicated [19]. Thus, in this paper, the techniques called a shop floor approach by using an analytical Multiple Regression (MR) technique were develop for an initial springback model by examining the sensitivity of springback input parameters on existing tooling geometry. The same process is currently being simulated by using Finite Element Analysis (FEA) software. The springback predicted from both methods were then compared. An actual tools, processes and products studied to understand the phenomenon and to capture the actual variables involve.

0XOWLSOH5HJUHVVLRQ05 0RGHO In this paper, the multiple regressions (MR) analysis is first carried out to examine and to identify the significant factors affecting springback. MR is a statistical technique of data analysis that may be appropriate whenever a dependent or criterion variable is to be examined in relationship to any other independent or predictor variables [20]. The selection of an actual part design (stamped part) and tool design (tool surface) data from the stamping production line process that relates to the input variables to the response variable should be able to predict the overall response of the process. To stabilize the response value of springback, a measurement of correlation value (R, R2 and Adjusted R2)between the measured value and predicted value was calculated and is analyzed for each single run. This correlation value is a measure of how good the dependent variable can be predicted by knowing the independent variable. It also gives the most useful measure of the success of the proposed MR model. Once the process is performed, the values of the actual tool/die surface data are transforming in the mathematical equation to predict the response of the springback. With this, MR model analysis [21] with mathematical expressions is written as:



Y =  + ß1 X1 + ß2 X2 +……………+ ßn Xn or; Y =  + ißi Xi

(1) (1a)

where; Y is dependent variable (predicted value) i is independent variables index (i = 1, 2....n)  is intercept (constant) ßi is regression coefficient Xi is independent variable n is number of independent variables The strength of the correlations between the proportion of variance in the values of the dependent variability in Y that is explained by independent variables in X is given by coefficient of determination value5 (or 56TXDUH ; 

R2 = SSR/SST = 1 - SSE/SST

(2)



SSR, SSE and SSTare regression, residual (error) and total sum of square. These R2 is a measure of the regression model as whole and percentage of response variable variation that is explained by its relationship with one or more predictor variables. If there is a closer or perfect fit R2 value is to one (1), the better the model R2=1 describes the data as linear relationship. If there is no linear relationship, R2 value will equal to 0 or R2=0. Sometimes, the adjusted coefficient of determination R2 value (DGMXVWHG5  or 5DGM) is used whenever R2 is biased upward. This R2adj mathematical expression is given as;    (3)   R2adj=1- {SSE/(n-k)} / {SST/(n-1)}   SSE, SST and k are the residual (error), the total sum of squares and constraint. R2adj used as an indicator of an adequacy of the model, as it takes into relation of deviations and degrees of freedom. Basically, the significant of R2adj value is the same as R2 value whereby the closer to one (1) the better the model.

550

In this study, die parameters shown in Table are used as independent variables:

1R3DU 1 2 3 4 5

7$%/( Die Parameters

DPHWHUV1RWDWLRQV8QLWV

/HJHQG

Die radius (dr) in mm Die angle (da) in ø Die depth (dd) in mm Punch radius (pr) in mm Die width (dw) in mm

0(7+2'2/2*< 3DUWDQG7RRO'HVLJQ'DWD Three (3) types of industrial stamping part of automotive Rear Floor Assembly-Body in White (BIW) components (see Table 1) were selected from the stamping production line as sample parts study. These selected sample parts encapsulates a comparatively complex shape design channel structure with different draw and surface depths as well as draw bead, addendum and reinforced features and also with the different range degrees of springback deviation along the parts length. The material used is High Strength steel (HSS) and its properties are shown in Table 2 The characteristics of the selected stamped parts as shown in Table 2. 7$%/(. Selected Parts Dimension

3DUW 1R

3DUWV1DPH

7\SHVRI%HQGLQJ 'HJUHHRI 3URFHVV 6SULQJEDFN

1

Side Member

U-Bend (S-Rail)

High

2

External Rear Floor

V-Bend

Medium

3

Sill Rear Floor

U-Bend

Low

6WDPSHG3DUW 'LPHQVLRQV

7$%/(Material Properties

Type of Material Thickness Friction Coefficient Poisson Ratio Young Modulus Yield Strength Tensile Strength

HSS Dual Phase (DP 600) 1.2mm 0.15 0.3 207GPa 400MPa 632MPa

For each part, 4 samples were randomly taken from the production line of 1,500 pieces.

551

%ODQN'LPHQVLRQV

6SULQJEDFN3UHGLFWLRQ7HFKQLTXH The Shop Floor Approach The comparison of approaches between the proposed shop floor [22-23], industrial and current academic studies for springback prediction and solution is shown in Table 4. 7$%/(Comparison between Industrial, Academic and Shop Floor Approach for Springback Prediction & Solution

3URFHVV

Industrial Approach

Current Academic Approach

Shop Floor Approach

0RGHOLQJ

6LPXODWLRQ 

7RRO  0DQXIDFWXULQJ

Part and tool design

Functionality & Formability defects prediction

Machining process matched with desired part shape

Springback detected

Springback compensation technique

BIW components part production

N/A

N/A

N/A

N/A

Machining of tools of ‘similar’ part

N/A

N/A

New ‘similar’ BIW components part production

Experimental (Lab) Modeling on: - Material - Process - Tool

Analytical (ANN, GA, DOE) & Numerical (FEA) analysis for springback prediction

MR model based on digitized BIW components part production

Comparing MR springback model with FEA springback model

3UHVV7U\  2XW

7RRO  0RGLILFDWLRQ

0DVV  3URGXFWLRQ

Basically, this shop floor approach consists of two (2) main stages namely MR analysis stage followed by verification and validation stage via FEA. Figure 1 illustrates the proposed of shop floor approach. Process

Input

05 $QDO\VLV

Digitized Die Surface

`

fMR(x)  fFEA(x) y=fFEA(x) Output Digitized Die Surface

Digitized Stamped Part vs Die Surface

Output y=fMR(x)

VW6WDJH (MR Analysis) QG6WDJH (FEA Validation)

Springback Response

)($ 9DOLGDWLRQ

Input

Process ),*85( Shop floor Approach

552

5HILQHG 6SULQJEDFN 0RGHO

In the first stage, actual data from automotive BIW digitized stamped part and die are analyzed by analytical MR technique to measure springback response as well as identification of significant factors contributing to springback model; y=fMR(x1, x2,...xn). Meanwhile, the second stage of numerical simulation approach based on Finite Element Analysis (FEA) technique will simulate stamped part springback response model; y=fFEA(x1, x2,...xn) that is generated from digitized die surface. By doing so, the springback model from the first stage; y=fMR(x1, x2,...xn) is verified, validated against the springback response model from the second stage; y=fFEA(x1, x2,...xn). Finally, this refined springback model where fMR(x)fFEA(x) will be used as guide for tool and part designer to mitigate the springback phenomenon.

6SULQJEDFN0HDVXUHPHQW3URFHGXUHV The input geometry from die parameters is obtained along with the stamped part profiles through dice or cross sections [24-26] process as shown in Figure 2. Ten (10) dice sections (A, B, C, D, E, F, G, H, I, J) of each stamped and die surface are selected for springback deviations measurements, as shown in Figure 2 & 3.

),*85( Dice Section Shape x 3 parts

),*85( Dice Section Process

The shape deviations that were captured in stage 1 over the whole length of stamped parts and die were used to measure the springback response. For this purpose, the digitized data of stamped part and die surface were obtained by using optical scanners FARO Arm and ATOZ. The two surface data were than superimposed using GEOMAGIC software and part geometry deviation at all section were measured. Finally, MR analysis was done to obtain the relationship between different input parameters and the response variable. Minitab 14 software was used to analyze this relationship. This relationship analysis explains data ability to represent the parameters and predict the springback response approximately. Figure 4a shows the superimpose technique and Figure 4b illustrates how the springback response value is calculated. Springback Response (Measured) = øA - øB

(a)

(4)

(b)

),*85( (a) Superimposed part & die surfaces (b) Springback response (measured) calculation

By using the same digitized die surface model and actual blank shape, springback values predicted by Hill hardening model were obtained by means of commercial FEA software Hyperform. In this case, we first verified the most significant factors affecting springback by using virtual DOE. Having obtained the data, the MR analysis was repeated to generate a new MR model i.e. y’ = fFEA(x). The final springback model wills a compensated MR model produced by FEA.

553

5(68/76 ',6&866,21 The results from the analysis show the springback responses can be predicted by four linear regression equations. L

3DUW yP1 = - 307 + 6.80dr-L + 2.32pr -L + 0.393da-L + 0.503dd - 0.0902dw + 5.05dr-R- 0.527pr -R + 0.804da-R where; L=left side and R=right side, therefore; R2 = 97.4%, R2(adj) = 96.3% and R2(pred) = 94.98%

LL 3DUW yP2= - 304 - 0.0637dr-L + 0.311pr -L - 1.11da-L+ 0.360dd + 0.514dw - 0.307dr-R + 2.36da-R therefore; R2 = 99.4%, R2(adj) = 99.2% and R2(pred) = 98.96% LLL 3DUW yP3 = - 259 – 1.55dr-L + 5.30pr -L + 0.0027da-L + 1.28dd + 0.066dw + 3.54dr-R – 28.7pr -R - 0.00083da-R therefore; R2 = 95.4%, R2(adj) = 93.9% and R2(pred) = 90.98% LY &RPELQDWLRQRI3DUWDQG yP1+3 = 51.0 + 1.18dr-L + 0.532pr -L - 0.186da-L - 0.0349dd - 0.114dw + 0.275dr-R + 0.188pr -R - 0.178da-R therefore; R2 = 93.2%, R2(adj) = 92.1% and R2(pred) = 90.29% The equation coefficients are given in Table 5 7DEOHCoefficients Value Coefficients Constant

dr-L

pr-L

da-L

dd

dw

dr-R

pr-R

da-R

-307 -304 -259 51.0

6.80 -0.0637 -1.55 1.18

2.32 0.311 5.30 0.532

0.393 -1.11 0.0027 -0.186

0.503 0.360 1.28 -0.0349

-0.0902 0.514 0.066 -0.114

5.05 -0.307 3.54 0.275

-0.527 nil -28.7 0.188

0.804 2.63 -0.00083 -0.178

Parts 1 2 3 1+3

From the MR analysis, it is shown that the parts classified by the industry as high, medium and low degrees of springback can be represented by three different models. Part2 model appears to be the best among the three. By virtue of having the same U profile, a combined MR model for Part1 and Part3 improves the prediction accuracy slightly as shown in Table 7. Part2 model can’t be combined with other two simply because of different input parameters i.e missing pr-R value. Results on FEA will be added here

554

0RGHOV&RPSDULVRQZLWK$FWXDO0HDVXUHG9DOXHV A comparison of measured and predicted springback value using MR model are presented in Table 7. 7$%/( Comparison of measured and predicted springback values Springback Degrees Dice Sections

Part 1

Measured Part Part 2 3

Part 1+3

Error (+/- 10%)

Part 1

MR Technique Part Part 2 3

Part 1+3

Part 1

Part 2

Part 3

Part 1+3

A

10.39

6.62

2.33

8.72

9.44

6.35

2.18

8.07

-9%

-4%

-6%

-7%

B

9.63

13.17

5.00

8.32

8.80

12.73

4.74

7.77

-9%

-3%

-5%

-7%

C

10.32

12.81

5.78

9.63

9.44

12.73

5.56

9.06

-9%

-1%

-4%

-6%

D

11.42

13.27

4.78

9.32

10.65

13.24

4.63

9.06

-7%

0%

-3%

-3%

E

9.32

12.62

5.78

10.39

8.80

12.73

5.60

10.24

-5%

1%

-3%

-1%

F

14.31

6.28

4.57

5.00

14.24

6.35

4.50

4.97

0%

1%

-1%

-1%

G

10.53

6.17

5.42

5.00

10.65

6.35

5.56

4.97

1%

3%

3%

-1%

H

8.88

12.35

4.51

2.56

9.44

12.73

4.63

2.61

6%

3%

3%

2%

I

8.27

12.58

4.57

2.56

8.80

13.24

4.74

2.61

7%

5%

4%

2%

J

9.99

0.99

2.13

4.72

10.65

1.05

2.24

4.99

7%

6%

5%

6%

&21&/86,21

 The accuracy of spring back prediction depends largely on the part and die design. This is indicated by the MR technique adopted in this study. Parts with similar base profile but of different complexity can be represented by a common MR model. This paper has also demonstrated the feasibility of springback model validation by the ‘compensated MR model’ instead of actual material model as proposed by others [13, 17].

$&.12:/('*0(176 This project is part of the Technofund (TF0608C073-17) project of Computationally Optimized Fuel-Efficient Concept Car (COFEC²), under the Ministry of Science, Technology and Innovation (MOSTI) of Malaysia in collaboration of Perusahan Automobil Nasional Holdings Sdn Bhd (PROTON). We would like to express our sincere gratitude for the cooperation of all those concerned in this project who assisted with valuable help and support.

5()(5(1&(6 1. T. de Souza and B. Rolfe, “Multivariate Modeling Of Variability In Sheet Metal Forming”, Journal of Material Process Technology, vol. 203, 2008, pp. 1–12. 2. T. Meinders, I.A. Burchitz, M.H.A. Bonte, and R.A. Lingbeek, “Numerical Product Design: Springback Prediction, Compensation and Optimization,” International Journal of Machine Tools & Manufacture, vol.48, 2008, pp. 499-514. 3. S.K. Panthi, N. Ramakrishnan, M. Ahmed, S.S. Singh, and M.D. Goel, “Finite Element Analysis of Sheet Metal Bending Process to Predict the Springback,” Journal of Materials and Design, vol. 31, 2010, pp. 657-662. 4. S.K. Panthi, N. Ramakrishnan, K.K. Pathak, and J.S. Chouhan, “An Analysis of Springback in Sheet Metal Bending Using Finite Element Method (FEM),” Journal of Materials Processing Technology, vol. 186, 2007, pp. 120-124. 5. H. Kim, N. Nargundkar, and T. Altan, “Prediction of Bend Allowance and Springback in Air Bending,”Journal of Manufacturing Science and Engineering, vol. 129, 2007, pp. 342-351. 6. H. Naceur, and S. Bezzina, “Finite Element Procedures for the Design of Sheet Forming Parameters,” The Arabian Journal for Science and Engineering, vol. 31, 2006, pp. 117-138.

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7. H. Naceur, Y.Q. Guo, and S. Ben-Elechi, “Response Surface Methodology for Design of Sheet Forming Parameters to Control Springback Effects,” Journal of Computers and Structures, vol. 84, 2006, pp. 1651-1663. 8. H. Naceur, S. Ben Elechi, and J.L. Batoz, “On the Design of Sheet Metal Forming Parameters for Springback Compensation,” in Proceedings of the 8th International Conference on Computational Plasticity, 2005, pp. 1-4. 9. R. Narayanasamy, and P. Padmanabhan, “Comparison of Regression and Artificial Neural Network Model for the Prediction of Springback during Air Bending Process of Interstitial Free Steel Sheet,” Journal of Intelligent Manufacturing, 2010, pp. 1-8. 10. P.A. Eggertsen, and K. Mattiasson, “On the Modelling of the Bending–Unbending Behaviour for Accurate Springback Predictions,” International Journal of Mechanical Sciences, vol. 51, 2009, pp. 547–563. 11. P.A. Eggertsen, and K. Mattiasson, “Material Modelling for Accurate Springback Prediction,” International Journal of Material Forming, vol. 2 (1), 2009, pp. 793–796. 12. L. Marretta, G. Ingarao, and R. DiLorenzo, “Design of Sheet Stamping Operations to Control Springback and Thinning: A Multi-Objective Stochastic Optimization Approach,” International Journal of Mechanical Sciences, vol. 52, 2010, pp. 914-927. 13. N. Iwata, A. Murata, Y. Yogo, H. Tsutamori, M. Niihara, and H. Ishikara, “Highly Accurate Numerical Prediction of Springback Shape of Stamped Thick Metal Sheet,” R&D review of Toyota CDRL, vol. l41(3), 2007, pp. 47-52. 14. M.L. Garcia-Romeu, J. Ciurana, and I. Ferrer, “Springback Determination of Sheet Metals in an Air Bending Process Based on an Experimental Work,” Journal of Materials Processing Technology, vol. 191, 2007, pp. 174-177. 15. H.S. Cheng, J. Cao, and Z.C. Xia, “An Accelerated Springback Compensation Method,” International Journal of Mechanical Sciences, vol. 49, 2007, pp. 267-279. 16. I. Burchitz, “Springback: Improvement of its Predictability,” Literature Study Report, Netherlands Institute for Metals Research, 2005. 17. X. Li, Y. Yang, Y. Wang, J. Bao, and S. Li, “Effect of the Material-Hardening Mode on the Springback Simulation Accuracy of V-Free Bending,” Journal of Material Process Technology, vol. 123, 2002, pp. 209-211. 18. T. Kuwabara, Y. Asano, S. Ikeda, and H. Hayashi, “Forming the Future: Global Trends in Sheet Metal Forming,” Editors IDDRG, 2004, pp. 55-64. 19. S. Chatti, “Effect of the Elasticity Formulation in Finite Strain on Springback Prediction,” Journal of Computers and Structures, vol. 88, 2010, pp. 796-805. 20. D.E. Berger, “Introduction to Multiple Regression,” Lectures Notes, Claremont University, 2003. 21. L.O. Michael, “Multiple Linear Regression Analysis Using Microsoft Excel,” Chemistry Department, Oregon State University, 1996. 22. K.B. Nagur Aziz, Z. Ahmad, K. Khairul Za’im, M. Achmed, and A.L. Mohd Safuan, “Shop Floor Approach for Springback Prediction and Solution,” presented at the PROTON-MOSTI Technofund (TF0608C073-17) project on of Computationally Optimized Fuel-Efficient Concept Car (COFEC²) Symposium, Putrajaya Malaysia, 2011. 23. K.B. Nagur Aziz, A.Zakaria, K. Khairul Za’im, M. Achmed, and M.A.L. Mohd Safuan, “Technical Paper on Springback Prediction and Solution,” presented at the PROTON-MOSTI Technofund (TF0608C073-17) project workshop on Computationally Optimized Fuel-Efficient Concept Car (COFEC²), Putrajaya Malaysia, 2010. 24. A. Andersson, “Numerical and Experimental Evaluation of Springback in Advanced High Strength Steel,” Journal of Materials Engineering and Performance, vol. 16(3), 2007, pp. 301–307. 25. S.A. Asgari, M. Pereira, B.F. Rolfe, M. Dingle, and P.D. Hodgson, “Statistical Analysis of Finite Element Modeling in Sheet Metal Forming and Springback Analysis,” Journal of Materials Processing Technology. vol. 203, 2007, pp. 129–136. 26. J.H. Song, H. Huh, and S.H. Kim, “A Simulation-Based Design Parameter Study in the Stamping Process of an Automotive Member,” Journal of Materials Processing Technology, vol. 189, 2007, 450–458.

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An Experimental Analysis Device for Obtaining Skid Line Limit Diagrams Chatchai Wanintradula, Gopinath T. Gurumurthya, L. M. Smitha, Changqing Dub, Lumin Gengc, D.J. Zhouc, Ching-Kuo Hsiungc, Jizhou Chend, Chao Fenge a

Oakland University, 248 DHE, Rochester, MI, 48309 USA b Chrysler Group LLC, Auburn Hills, MI 48326, USA c General Motors Corporation, Warren Technical Center, Warren, MI 48090, USA d Ford Motor Company, Dearborn, MI 48127, USA e Beijing Jiaotong University, Beijing, Haidian District, PRC Abstract. A novel design for a machine intended to measure directly various in-plane and contact normal forces acting upon a sheet metal specimen during a stretch-bend-draw process is proposed, in order to gain insight into the formation of skid line defects in sheet metal. The new machine, called a Stretch-Bend-Draw Simulator (SBDS) is designed specifically to be integrated into a typical laboratory tensile testing machine, thereby making it accessible to those researchers lacking the resources to acquire expensive additional tooling. As the strip of sheet metal is pulled over a round tool radius, the SBDS is shown to be capable of collecting pulling force, back force, tool normal force, and the corresponding draw bead clamping force. Analysis of the force data in conjunction with visual observations of the actual pulled specimens allows researchers to ascertain the conditions under which so-called skid lines arise. Experimental results, including a newly unveiled Skid Line Limit Diagram (SLLD), are provided and discussed. The SBDS appears to be a promising new electro-mechanical laboratory device for improving researchers’ knowledge of the physical phenomena associated with skid lines in sheet metal products created in stamping dies. Keywords: Sheet Metal Forming, Bending-under-Tension, Skid Line, Surface Distortion PACS: 81.70.Bt

INTRODUCTION Customer enthusiasm for products featuring stamped sheet metal parts is based upon both performance and appearance. Research and the resulting powerful computer simulation models introduced over the past two decades have essentially eliminated major concerns about how to design a stamping die without introducing splits in the metal. Huang et al. [1], for example, implemented a procedure coupling a response surface methodology with a commercially available finite element program, resulting in an optimized design for a deep drawn cup. In conformance with many other researchers’ findings, they further verified that the finite element model is, indeed, a powerful tool for predicting the development of sheet metal splits during the forming process. Three significant challenges, however, still exist for formability engineers. Two of these challenges involve springback prediction and crash behavior. The main difficulty with springback is the deficiency in material modeling capability as it relates to how it affects the final geometry of a product. The work of Ma et al. [2] represents one of many examples which could be brought forth. In Ma’s study, a computer subroutine of a specialized material model was integrated into a commercial program. The subroutine updated the Young’s modulus change with respect to plastic strain within the context of a springback simulation for two stamping parts. Tool surface compensation was introduced in order to determine the best die design. K.M Shao et. al. [3] studied on cyclic loading by employed cyclic three-point bending test. Material parameter is identified that account for hardening behavior subjected to cyclic loading, as sheet metal undergo in stamping, which predicted springback very accurate. The structural integrity of a stamped sheet metal product is also of concern to those interested in evaluating its crash worthiness. If the thickness of the sheet metal at a particular location in the product is below some critical value, then the ability of the part to absorb appropriately energy during a crash may be compromised. Additionally, if the material properties and strain distribution within the product at the end of the simulated stamping process are

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 557-564 (2011); doi: 10.1063/1.3623657 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

557

not accurately brought into the crash simulation, then the crash simulation may yield poor predictions of the actual behavior. Abedrabbo et al. [4], for example, employed rate sensitive constitutive models for crash test analysis of hydroformed tubes. Their numerical modeling allowed them to study with good accuracy the effects of material models and other factors on energy absorption in crash testing of tube structures. Uncontrolled springback behavior and/or poorly understood material behavior both affect the performance of the sheet metal product; the appearance of the sheet metal, however, does not affect such performance. Yet, it is true that even some of the most seemingly minor surface distortions on painted outer automotive panels, for example, can dissuade a customer from making a purchase. For this reason, formability engineers take the issue of minor surface distortions in stamped sheet metal parts very seriously

OBJECTIVE The purpose of this research is to investigate the utility of a completely new machine called the “Stretch-BendDraw Simulator” (SBDS), which may be employed for the purpose of more fully understanding the mechanics of skid line formation in sheet metal products formed in stamping dies. Offered in this paper is a presentation of the machine design, experimental methodology, and example results. Figure 1(a) illustrates a simplified drawing of two common surface distortions observed in sheet metal panels: a kink, and a skid line. A kink features local curvature of the same direction on adjacent sides of the sheet metal, while a skid line features local curvature of different directions on adjacent sides. For the purpose of promoting clarity in future technical communications among researchers, the following definition of a skid line is put forth in accordance with the United States Automotive Manufacturing Partnership [5] Definition of skid line is a visually observable surface distortion band on the non-contact side of the portion of a stamped sheet metal panel which has been pulled over a radius, as example in Fig. 1(b). Among the different types of surface defects discussed, the skid line is arguably the most difficult to prevent. A minor skid line may often appear more pronounced when it is painted, thereby leading to dramatic reductions in product quality perception. To circumvent the problems associated with skid lines, trial and error methods have been implemented in industry with varying degrees of success. The need for a quantitative, experimental assessment of the mechanical boundary conditions associated with the onset of skid line formation would help engineers and researchers understand how to manage effectively skid line formation in sheet metal. To satisfy this need, a laboratory machine for creating skid lines must be accessible. A laboratory machine which can be used to simulate the mechanical boundary conditions, while measuring the associated forces in the work piece, leading up to skid line formation is not available. This lack of availability hinders growth in sheet metal surface quality analysis technology.

PRELIMINARY ANALYSIS OF SKID LINE As the punch tool is driven into the die cavity, the bending/unbending action varies with respect to work piece region. For the geometry shown in Fig. 2(a), the deformation mode in region 1 is dominated by stretching. The material point then moves into region 3, which upon entry, features an additional unbend action. This translated to design challenge in fig. 2(b).This difference in deformation is believed by the authors to be intimately coupled to the

FIGURE 1. (a) Illustrations of kink and skid line surface distortions in sheet metal (b) Skid Line on Sheet Metal (Features have been exaggerated for illustrative purposes.)

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FIGURE 2. (a)Two-dimensional drawing of rigid punch tool and work piece. (b) Design challenge

skid line effect, but it is not clear how it is so coupled. This presumption is linked to what is called the enhanced forming limit effect in which the strain space formability of regions undergoing bend/unbend action is greater than those undergoing only membrane action [6]. A definitive explanation of the physical mechanisms accounting for the enhanced formability phenomenon has not yet been fully disseminated. Likewise, a full explanation for skid line formation has not yet been offered in literature. The experimental analysis using the SBDS is expected to serve as a vital tool for providing reliable experimental data to those interested in searching for the exact mechanism for skid line formation. Based upon personal correspondence, it has been observed by the authors that numerous numerical model and/or experimental analysis studies on skid lines have been conducted within the confines of private industrial settings. Reporting of this technology in the journals has been found to be relatively narrow in scope and lacking in direct experimental measurement of forces acting on the work piece. For example, Mori et al. [7] conducted finite element simulations of multi-stage forming of a wheel disk in order to better understand the mechanics of what the authors call a kink (Mori calls it a “shock line.”). Finite element simulations in conjunction with experimental verification indicated to Mori that an increase in the tooling radius over which the work piece is bent will tend to reduce the prominence of the shock line upon subsequent unbending. Mori’s work, although helpful, did not involve the stretching effects during the later stages of the forming process. The absence of the stretching effect renders the utility of Mori’s work limited in scope when more complex stamping processes are considered. It should be noted that Kokubo et al. [8] did consider stretch effects in later stages of the forming process, but the reporting is not available in the journals to the authors’ knowledge. Y.H. Moon [9] works included stretching in their analytical model. Even their work focus on prediction of sidewall curl, similar apparatus (bending-under-tension) had been employed. This sidewall curl phenomenon is also what the authors call a kink. In-depth analysis of the mechanics of skid line formation is planned by the authors for future studies. In order to provide a glimpse at the phenomena associated with skid line formation, some preliminary finite element analysis results are conducted, but result is not included, as it is limited in scope and it appears to be contrary to observed behavior. Therefore more additional explorations are required in order to render such analysis more complete. For example, instead of isotropic hardening model, Y. You [10], calculated draw bead restraining force applying kinematic hardening material model which consider Bauschchinger effect, and be able to obtain better result. The fact that these finite element model results do not agree with the observed experimental results is exactly why the SBDS has been introduced; the experimental data produced through SBDS experiments are to be used help refine finite element and analytical models.

DESIGN OF STRETCH-BEND-DRAW SIMULATOR Figure 2(b) presents design challenge for this investigation. A strip of sheet metal workpiece must be stretch-bent over a round tool with a minimal amount of draw (relative tangential motion) occurring. Then the workpiece must be drawn over the round tool under the action of a pre-defined stretch condition. Four forces must be measured in real

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FIGURE 3. (a) The SBDS and data acquisition system integrated into a tensile test machine (b) Front view of SBDS

time along the process: pulling force, back force, clamping force, and contact normal force. The entire machine must be integrated into a standard tensile machine typically found in most material properties laboratories. There are some previous works on bending-under-tension apparatus, but there were usually focus on measure normal contact pressure, or friction. S.K. Kernosky et. al.[11] has developed force transducer to measure shoulder force which sheet metal pull over. Similarly, JL Andreasen et al. [12] has developed apparatus which can simulate bending-undertension process, the front tension, back tension and torque on the tool pin can be measured directly, thus enable to accurate measurement of friction. While finding in their works are very useful information, some parameters from stamping process, tool radius and wrap angle, for example, which are believed by the authors to effect formation of skid line are still missing. Since tensile test machines is most available equipment, a major design constraint of having to pull the sheet metal specimen vertically was self-imposed. A picture of the SBDS while integrated into a 50 kN tensile test machine is provided in Fig. 3(a). A front view of the SBDS with key component labeled is shown in Fig. 3(b) including the draw bead blocks from new Angle Draw Bead Simulator (ADBS) introduced by Smith et al. [13]. A sheet metal strip (A) is clamped into a tensile test jaw grip (B), aligned adjacent to round tool (C), and slid through male/female draw bead blocks (D, E). Tool (C) is positioned with respect to rotation point (G). Tool (C) may be translated by threaded shaft (F), and/or rotated (H). A wrap angle may be introduced through rotation (J). The sled slides freely along tracks (K). Sled translation positioning (L) may be achieved using threaded shaft (M). Threaded shaft (N) drives the draw bead blocks together, clamping the specimen. A clamping force is measured using Load Cell (Q). Back force is measured using Load Cell (R). Tool contact force is measured using Load Cell (P). Pulling force is measured using Load Cell (B) from tensile test machine (not shown).

EXPERIMENTAL METHODOLOGY Test Matrix The SBDS can accommodate a wide variety of material types and thicknesses. For this study, galvanized, Bake Hardenable (BH) 210 Steel was considered. A test matrix similar to that shown in Table 1 was employed. It should be noted that the back stress corresponding to the onset of a skid line is not known a priori. Therefore, a trial “net” of back stresses must be cast with the hope of enveloping the back stress associated with the onset of a skid line. To this end, five back stress values are recommended. Eight replications were carried out for each case and each sample was then evaluated for a skid line severity ranking from 1-5, which 5 is most severe skid line.

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TABLE 1. Typical Test Matrix for Ranking BH210, galvanized, 15Û:UDS$QJOH Back Stress as % Tool Radius (mm) UTS 4 8 16 60 70 80 90 100

32

Procedure The following procedure is proposed in order to reproduce the condition happened in stamping die. 1. Install the SBDS into the tensile test machine, taking care to tighten sufficiently all joints and contact surfaces. Make note of the tool radius and wrap angle setting. 2. Shear each sheet metal specimen to the appropriate width and length. Depending upon the quality of the machining, some de-burring may be necessary. Label each specimen with a permanent marker or etching tool. (Current Study: 46 mm x 404 mm) 3. Apply lubrication to the tooling and a trial specimen based upon pre-defined specifications. (Current Study: Liberal Application of Mill Oil) 4. Allowing for only elastic bending, slide the specimen into the tensile test machine jaw grips, and through the male/female draw beads. The work piece should not yet be in contact with the round tool. 5. Lock the specimen into the tensile test machine jaw grips, making certain that the it is vertically oriented. 6. Adjust the position of the draw bead until the trial specimen is nearly perpendicular with bead block and tangent with round tool. Lock the female bead at this position. 7. Close the draw beads to a controlled, pre-defined gap (or load). (At this point, the specimen is still not yet in contact with the round tool at point A.) 8. Drive the round tool such that the portion of trial specimen between the jaw grips and the round tool is vertically oriented. Lock the round tool in this position. 9. Re-adjust the position of the draw beads such that the trial specimen is perpendicular to the face of the draw beads (ie. such that the sharp corner edge of the draw bead does not scrape the trial specimen.) 10. Remove the trial specimen. Repeat steps above for new specimen, but do not change the position of the beads. 11. Activate electronic data acquisition instruments. Deploy the cross head motion of the tensile test machine in order to pull the metal work piece (upward) over the round tool.

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12. Note the nature of the defect (ie. fracture, neck, scratch, kink, skid line) in the specimen in the region which was pulled across the round tool. Modify the draw bead geometry and gap (of force) in order to adjust the pulling force appropriately. This process may take several hours in order to ascertain a reasonable first guess at the lowest and highest pulling force setting for a given material, tool radius, and wrap angle. 13. Conduct the necessary tests in accordance with the test matrices previously defined. 14. Assess the severity of the skid line through either qualitative or quantitative means. For this study, the authors provided a ranking based upon qualitative tactile and visual analysis for each replication.

FIGURE 4. Skid Line Limit Diagram (SLLD). R and t are tool radius and sheet thickness, respectively. The %UTS on the abscissa axis refers to the engineering restraining stress as a percentage of the UTS.

FIGURE 5. Example of raw measured data for force results for BH210 material. Experimental uncertainty across all tests ranged from 100 N – 400 N.

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FIGURE 6. Example force results for BH210 material. r = 4 mm. This specimen had a skid line severity ranking of 5 .

Example Set of Results A selected portion of the authors evaluated the pulled specimens with the naked eye under the illumination of a set of long, florescent lights. Each specimen was lightly coated with oil in order to elucidate any skid line that may be present. Skid line severity forms were then completed. The data were then processed (averaged) and cast into the form of what shall be called a Skid Line Limit Diagram (SLLD) as shown in Fig.4. The SLLD is complimented with the quantitative information of forces from the load cells. An example set of force results for 0.74 mm BH210 material is shown in Fig.5. The uncertainty in these force measurements is approximately 200 N. For the example results shown in Fig.5, the UTS was 358 MPa, and K and n were 588 Mpa and 0.19, respectively, where hardening  =   . The uncertainty in the %UTS at the most is assumed to be governed by the power law hardening curve:  estimated to be 3%. (This percentage is based upon the uncertainty in the measured forces, which has been determined to be, at most, 400 N, where the ratio of 400 N to a UTS force of 13,500 N leads to 0.03.) From the SLLD, it appears that it is possible to prevent skid line formation if restraining force (restraining stress) is sufficiently large. This trend seems to be in contrast to that indicated in the finite element analysis study. Equally as significant, is the fact that there exists no meaningful standard for interpreting predicted thickness gradients in finite element models. Therefore, the ground for much more work connecting finite element model predictions with the SLLD is fertile. Using needle-tip hand calipers, the thickness with respect to the length of a deformed specimen (with skid line severity ranking of 5) was measured and recorded. The resulting variation of thickness plot is shown in Fig. 6. These results are in qualitative agreement with those from the finite element analysis study. The information gleaned from Fig. 4-6 provides a solid foundation upon which additional research in the area of skid line mechanics may be performed.

CONCLUSION A Stretch-Bend-Draw Simulator (SBDS) has been created and employed for the production of multiple sheet metal specimens for skid line severity assessment. From the SBDS tests, a skid line limit diagram (SLLD) may be derived in order to help guide the design of sheet metal stamping dies. This milestone answers the need for reliable experimental data related to skid line formation. From the SBDS data, researchers can improve the reliability of their analytical or finite element skid line formation models. Additional work in this emerging research area will include

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experimental analysis of a more broad range of material types, much more in-depth finite element analysis, and explorations into the analytical mechanics of skid line formation.

ACKNOWLEDGEMENT The authors wish to acknowledge the contribution of Dr. Thomas Stoughton and Dr. C. T. Wang, both of General Motors Corporation, for their input into the early phase of the SBDS design. Also, the authors wish to thank the USAMP Auto/Steel Partnership for the research grant which made this work possible.

REFERENCES 1. Y. Huang, Z.Y. Lo, and R. Du, Minimization of The Thickness Variation in Multi-step Sheet Metal Stamping, Journal of Materials Processing Technology, 177, 2006, pp. 84-86. 2. N. Ma, Y. Umezu, Y. Watanabe, and T. Ogawa, Springback Prediction by Yoshida-Uemori Model and Compensation of Tool Surface Using JStamp, Numisheet, Interlaken, Switzerland, 2008 3. K.M. Zhao and J.K. Lee, Generation of Cyclic Stress Strain Curve for Sheet Metals, Journal of Engineering Materials and Technology, 123, 2001, pp 391-397 4. N. Abedrabbo, R. Mayer, M.J. Worswick, and I. van Riemsdijk, Crash Response of Hydrformed High Strength Steel Tubes, Numisheet, Interlaken, Switzerland, 2008 5. United States Automotive Manfacturing Partnership, Southfield, MI, U.S.A., Design of Skid Line Simulation Machine for Sheet Metal Surface Quality Analysis, Executive Summary, 2010 6. United States Automotive Manfacturing Partnership, Southfield, MI, U.S.A., Enhanced Forming Limit Diagram, Technology Report, January, Committee : Enhanced Forming Limit Diagram Project Team, 2003 7. K. Mori, Y. Abe, and O. Ebihara, Prevention of Shock Lines in Multi-stage Sheet Metal Forming, International Journal of Machine Tools & Manufacture, 43, 2003, pp. 1279–1285 8. I. Kokubo, S. Nomura, and Y. Yutori, Press forming problems with reduced gage steel sheet for autobody panels (shock lines and pimples), Research and Development Kobe Steel Engineering Reports, 33, 1983, pp. 70–73. 9. Y.H. Moon, D.W. Kim, and C.J. Van Tyne, Analytical Models for Prediction of Sidewall Curl During Stretch-Bend Sheet Metal Forming, International Journal of Mechanical Sciences, 50, 2008, pp 666-675 10. Y. You, Calculation of Drawbead Restraining Forces with the Bauschinger Effect, Journal of Engineering Manufacture, 212, 1998, pp 549-553 11. S. K. Kernosky, K. J. Weinmann, and J. R. Michler, Development of a Die Shoulder Force Transducer for Sheet Metal Forming Research, Journal of Manufacturing Science and Engineering, 120, 1998, pp 42-49 12. J. L. Andreasen, D. D. Olsson, K. Chodnikiewicz, and N. Bay, Bending under tension test with direct friction measurement, Journal of Engineering Manufacture, 220, 2006, Part B, pp 73-80 13. L.M. Smith, Y.J. Zhou, D.J. Zhou, and C. Wanintradul, A New Experiment Test Apparatus for Angle Binder Draw Bead Simulations, Journal of Materials Processing Technology, 209, 2009, pp. 4942-4948

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Biaxial Tensile Test of Cold Rolled IF Steel Sheet for Large Plastic Strain Range Ryotaro Enatsu a and Toshihiko Kuwabarab a

Department of Mechanical Systems Engineering, Graduate School of Engineering, Tokyo University of Agriculture and Technology, 2-24-16, Nakacho, Koganei-shi, Tokyo 184-8588, JAPAN b Division of Advanced Mechanical Systems Engineering, Institute of Engineering, Tokyo University of Agriculture and Technology, 2-24-16, Nakacho, Koganei-shi, Tokyo 184-8588, JAPAN Abstract. Deformation behavior of cold rolled IF steel sheet (SPCE) under biaxial tension has been investigated for large plastic strain range over 15 %. The test material was bent and TIG welded to form a tubular specimen with an outer diameter of 46.2mm and wall thickness of 0.8mm. The tubular specimens have been subjected to linear stress paths in the first quadrant of stress space with the use of a servo-controlled tension-internal pressure testing machine developed by one of the authors [T. Kuwabara, K. Yoshida, K. Narihara, S. Takahashi, Anisotropic plastic deformation of extruded aluminum alloy tube under axial forces and internal pressure, Int. J. Plasticity 21, 101-117 (2005)]. Moreover, biaxial tensile tests using a cruciform specimen have also been carried out to more precisely measure the deformation behavior for a small strain range following initial yielding. True stress-true plastic strain curves, contours of plastic work in stress space and the directions of plastic strain rates have been measured and compared with those calculated using selected yield functions: the von Mises, Hill’s quadratic and Yld2000-2d [Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Lege, D.J., Pourboghrat, F., Choi, S.H., Chu, E., Plane stress yield function for aluminum alloy sheets - Part 1: Theory. Int. J. Plasticity 19, 1297-1319 (2003)]. The plastic deformation behavior up to a work equivalent plastic strain of  0p  0.19 has been successfully measured. It is found that the test material exhibits differential hardening and that the Yld2000-2d yield function with an exponent of six most closely predicts the contours of plastic work and the directions of plastic strain rates. Keywords: IF steel, biaxial tensile test, yield function, differential hardening PACS: 46.35.+z, 83.60.-a, 83.85.-c

INTRODUCTION In sheet metal forming simulations the effect of yield functions on the predictive accuracy of material deformation behavior is significant [1-6]. One of the authors developed a biaxial tensile testing method for sheet metals using a cruciform specimen [7] and successfully determined appropriate anisotropic yield functions for sheet metals for industrial use, such as cold rolled interstitial free steel [7-9], high strength steel [10], austenitic stainless steel [11] , aluminum alloys [12] and pure titanium [13]. The larger the work hardening exponent (n-value) of the sample, the larger the maximum plastic strain that is applicable to the cruciform specimen; however, it is 9 % at most even for the austenitic stainless steel sheet [11], the n-value of which is over 0.5. One of the authors investigated the work hardening behavior of pure titanium sheet using biaxial tensile tests [14]. In order to measure the biaxial stress-strain curves of the sample for large plastic strain range, tubular specimens were fabricated by bending it and welding its edges together and combined internal pressure and axial load were applied to them using the servo-controlled tube bulge testing machine [15]. Consequently, the maximum plastic strain of 8.5 % could be applied to the specimen, although it was only 0.2 % when using a cruciform specimen. In this study we apply the same tube bulge testing method as that in [14], as well as the biaxial tensile test using a cruciform specimen, to cold rolled interstitial free steel sheet. Several linear stress paths are applied to the specimens to measure the biaxial true stress-true plastic strain curves, contours of plastic work in stress space and the directions of plastic strain rates. They are compared with those calculated using selected yield functions to determine the most appropriate yield functions to the test material. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 565-570 (2011); doi: 10.1063/1.3623658 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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EXPERIMENTAL METHOD Test Material The test material used in this study was 0.8mm thick cold rolled steel sheet (JIS SPCE). The work hardening characteristics and r-values in different directions to the rolling direction (RD) are listed in Table 1. Table 1 Mechanical properties of test material Tensile direction / deg 0 45 90

 0.2 /MPa

c* /MPa

n*

*

r -value**

148 158 153

610 622 614

0.308 0.308 0.318

0.0080 0.0092 0.0101

2.06 1.63 2.35

*Parameters for Swift’s hardening law,   c(   p ) , for the strain range of 0.002   p  0.09. **Measured at a uniaxial nominal strain of 0.10.

Biaxial Tensile Test Using a Cruciform Specimen In order to determine an appropriate anisotropic yield function for the sample that is able to reproduce its elasticplastic deformation behavior for small strain range, biaxial tensile tests were performed using the cruciform specimen shown in Fig. 1. The geometry of the specimen is similar to that used in [7-9]. Each arm of the specimen has seven slits, 60 mm in length and 0.2 mm in width, at 7.5 mm intervals, so as to exclude the geometric constraint on the deformation of the 60 mm × 60 mm square gauge section. The slits were fabricated by laser-machining. Hereafter, the RD and transverse directions (TD) of the material are defined as the x- and y-axes, respectively. Biaxial tensile forces (Fx, Fy) were applied to the cruciform specimen using a servo-controlled biaxial tensile testing machine [7]. The nominal tensile stress components ( N x ,  N y ) were in fixed proportions during each test:  N x : N y  1:0, 2:1, 4:3, 1:1, 3:4, 1:2 and 0:1. Biaxial total strain components (  x ,  y ) were measured using four uniaxial strain gauges (Tokyo Sokki Kenkyujo, YFLA-2) mounted on the centerlines of the specimens at (x, y)=(±10 mm, 0). True stress components ( x ,  y ) were determined by dividing (Fx, Fy) by the current cross-sectional area of the gauge section, which was determined from the measured values of the plastic strain components ( xp ,  yp ) with an assumption of constant volume. xy was assumed to be zero, as  x and  y were measured on the centerlines of the specimen. For  N x : N y  1:0 and 0:1, standard uniaxial tensile specimens (JIS 13 B-type) were used. The equivalent plastic strain rate was (1 ~ 4)×10-4 s-1. According to the finite element analysis of the cruciform specimen and the strain measurement position as shown in Fig. 1, using selected isotropic (von Mises) and anisotropic yield functions as material models, the error of stress

Units in mm Rolling direction Strain gauge

Slit width: 0.2mm

Fig. 1 Cruciform specimen for biaxial tensile test

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measurement is estimated to be less than 2 % [16, 17]. In order to evaluate the work hardening behavior of the sample under biaxial tension, the notion of the contour of plastic work in stress space was introduced [18, 19]. The stress-strain curve obtained from a uniaxial tensile test along RD was selected as a reference datum for work hardening; the uniaxial true stresses  0 and the plastic work per unit volume W0 corresponding to particular values of offset logarithmic plastic strains  0p were determined. The uniaxial true stress  90 obtained from a tensile test in TD and the biaxial true stress components ( x ,  y ) obtained from biaxial tensile tests were then determined at the same plastic work as W0. The stress points ( 0 , 0) , (0,  90 ) and ( x ,  y ) thus plotted in the principal stress space comprise a contour of plastic work corresponding to a particular value of  0p . When  0p is taken as sufficiently small, the corresponding work contour can be practically viewed as a yield locus.

Biaxial Tube Bulge Test In the biaxial tensile tests of the sample using a cruciform specimen, fracture occurs at an early stage of plastic deformation, 0.025   0p  0.045, in one of the arms of the cruciform specimen. In an effort to observe the plastic deformation behavior of the sample over a much larger plastic strain range, we fabricated tubular specimens by bending the sample and welding its edges together to carry out biaxial tensile tests of the sample by applying combined tension-internal pressure to the tubular specimens; the diameter was 46.2 mm and the length 210 mm with the rolling direction of the sheet being taken to be parallel to the axial direction of the tube. To avoid changes to the mechanical properties of the sample, thermal stress relief was not applied to the tubular specimens. Linear stress paths,  x : y  2:1, 4:3, 1:1, 3:4, and 1:2, were applied to the tubular specimens using the electrically servo-controlled tension-internal pressure testing machine developed in [15], see Fig. 2a.

Strain gauges

R

(a)

(b)

Fig. 2 Experimental apparatus. (a) Electrically servo-controlled tension-internal pressure testing machine. (b) Arrangement of strain gauges and spherometer.

The axial and circumferential membrane strains,   and  , were measured using strain gauges designed to measure large plastic deformations (Tokyo Sokki Kenkyujo Co., YFLA-2). These were mounted on the centerline of the specimen as shown in Fig. 2b.   and  were assumed to be uniform in the circumferential direction of the specimen, and were evaluated at the mid wall using the following equations,

     S  ln   ln

R R  (t / 2)

,

(1)

D exp( S )  t Dt ,  ln 0 D0  t0 D0  t0

where   S and  S are the axial and circumferential true strains measured on the outer surface of the specimen, D0 and t0 are the initial outer diameter and wall thickness of the specimen, and D and t are the deformed outer diameter and wall thickness measured at the center of the specimen. The current wall thickness t was determined under the

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assumption of constant volume and neglecting elastic strains as D  D   ( D  t )t     0 0 0 2  2   exp(  ) 2

t

  . 

(2)

The maximum strain to which the strain gauge could maintain contact to a specimen was ten to fifteen percent, varying according to the strain ratio given to the specimen. Therefore, we changed the strain gauges every ten to fifteen percent strain measurement. The axial and circumferential stresses,   and   , were calculated as the values at the mid wall, using the following equations based on the equilibrium requirements at the center of the specimen: P ( D / 2  t ) 2  T ,  ( D  t )t ( R  t )( D  2t ) Dt P     , (2 R  t ) t 2 R  t

 

(3)

where T and P are measured values of the axial load and the internal pressure, respectively. Regarding the closed-loop electrical feedback circuit used for controlling the true stress components,   and   , see [15]. The measured values of T, P,   ,  and R were inputs to a personal computer and   and   were calculated from equation (3). These values of   and   were compared with the command values of   and   . Based on the discrepancy, two PI-controllers then output command signals for controlling T and P every 0.01s. The command signals for T and P were then compared with the measured values, and the servo-valves were controlled according to the difference between the measured and command values of T and P so as to activate the hydraulic cylinders and intensifier, such that the desired values of   and   are applied to the specimen. The resolution in the measurements for the axial loads was 7 N, and that for the internal pressure was 1 kPa. Resolution in the measurements for strain components was 6 106 . The outputs of T, P,   ,  ,   ,   and R were recorded every 0.3s with A/D data conversion and a personal computer, and were saved on a disk.

RESULTS AND DISCUSSION Fig. 3a compares the true stress-true plastic strain curves in the transverse direction of the test material measured using a tubular specimen and a cruciform specimen for equibiaxial tension,   :   1:1. It is found that both curves almost coincide with each other when the curve obtained from the tubular specimen is horizontally offset approximately by 0.2 % plastic strain. The necessity of the offset is possibly caused by the work hardening of the tubular specimen due to the pre-bending in fabricating the tube. The Bauschinger effect of the pre-bent sample may also have some influence on it. Fig. 3b compares the measured stress points comprising contours of plastic work for different levels of  0p . The same procedure as shown in Fig. 3a for offsetting the true stress-true plastic strain curves were adopted to determine the work contours for tubular specimens. The work contours measured using both specimens are in good agreement with each other. Thus, the validity of the biaxial tensile test data using the tubular specimens have been verified. Fig. 4 shows the measured stress points that comprise the contour of plastic work. In Fig. 4b all stress values comprising a work contour are normalized by 0 corresponding to a specific value of  0p . Also depicted in the figures are the theoretical yield loci based on the von Mises [21], Hill’s quadratic (Hill ‘48) [22], and the Yld20002d yield functions [23]. The unknown parameters of Hill’s quadratic yield function were determined using r0, r45, r90 and 0/0, and those of the Yld2000-2d yield function were determined using r0, r45, r90 and rb, and 0/0, 45/0, 90/0 and b/0, where r and  are the r-value and tensile flow stress measured at  from the RD, respectively, and rb and b are the ratio of plastic strain rates, d  yp / d  xp , and the flow stress, respectively, at equibiaxial tension,  N x : N y  1:1. The values of r0, r45 and r90 used were the same as those in Table 1 both for Hill ’48 and the Yld2000-2d yield functions. The values of 45/0, 90/0, b/0 and rb used for determining the Yld2000-2d yield functions were selected from those for the work contour for the  0p referred to in the figures. It was found in Fig. 4a by calculating the standard deviation of the theoretical yield loci from the measured work contours that the Yld2000-2d with an exponent of M  6 has closer agreement with the experimental data than that

568

300

p

0 Cruciform 0.015 0.02 0.025 0.03 0.035 0.04 0.045 Tubular 0.015 0.02 0.025 0.03 0.035 0.04 0.045

400 200

 /MPa

True stress  /MPa

500

300 200

100

x :y =1:1 (y =)

100

Tubular specimen Cruciform specimen

0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

0 0

100

Logarithmic plastic strain 

(a)

200

 /MPa

p

300

(b)

Fig. 3 Comparison of biaxial stress test data obtained using tubular specimens and cruciform specimens: (a) true stress-true plastic strain curves in the transverse direction of the test material for equibiaxial tension,   :   1:1, and (b) measured stress points comprising contours of plastic work for different levels of  0p . 500

1.5

p

 /MPa

400 300 200 100 0 0

100

200

300

 /MPa

400

p

0 = 0.002 0.01 0.08 0.131 0.19 von Mises HIll'48 Yld2000-2d p (0 = 0.131) M=4 M=6

1.0

 / 0

0 = 0.002 0.01 0.02 0.03 0.04 0.06 0.08 0.10 0.131 0.16 0.19 Yld2000-2d M=4 M=6

0.5

0.0 0.0

500

0.5

 / 0

1.0

1.5

(a) (b) Fig. 4 Results of biaxial tensile tests using tubular specimens: (a) measured stress points comprising contours of plastic work and (b) the directions of plastic strain rates compared with those calculated using the selected yield functions.

with M  4 for  0p  0.13. On the other hand, the Yld2000-2d with M  4 shows better agreement with the experimental data than M  6 only for  0p  0.19. In order to validate the normality flow rule for the yield functions, the directions of plastic strain rates,  , were measured at  0p  0.002, 0.01, 0.131 and 0.19. As shown in Fig. 5a,  is almost constant for the strain range of 0.002   0p  0.19. Judging from the standard deviations of the measured  from those predicted by the selected yield functions as shown in Fig. 5b, the Yld2000-2d with M  6 has the closest agreement with the measured data for the strain range of 0.002   0p  0.19. From the results of Fig. 5, it is concluded that the Yld2000-2d with M  6 is the most appropriate material model for the test material at least for linear stress paths.

CONCLUSIONS The biaxial true stress-strain curves of cold rolled interstitial free steel sheet (JIS, SPCE) were successfully measured by bending it and welding both edges together to fabricate a tubular specimen and apply combined axial force and internal pressure to the specimen. Contours of plastic work in stress space and the directions of plastic strain rates were determined for liner stress paths up to a work equivalent plastic strain of 0.19. It was found that the

569

12 p

0 0.002 0.01 0.131 0.19

90

Yld2000-2d p (M=6; 0 =0.002)

Yld2000-2d p (M=4, 0 =0.002)

45

 y

von Mises



0



Hill '48

-45

0

15

Standard deviation  / °

Direction of plastic strain rate  / °

135

30

45

x

60 75 Loading direction  / °

90

10 8 6 4 von Mises Hill'48 Yld2000-2d M=4 Yld2000-2d M=6

2 0 0.00

0.04

0.08

0.12

0.16

0.20

p

Logarithmic plastic strain 0

(a) (b) Fig. 5 Validation of normality flow rule. (a) Measured directions of plastic strain rates compared with those calculated using selected yield functions. (b) Standard deviations of the measured directions of plastic strain rates from those predicted using the selected yield functions.

test material exhibited differential hardening and that the Yld2000-2d yield function with an exponent of 6 most closely predicted the work hardening behavior of the material.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

11. 12. 13. 14.

15. 16. 17. 18. 19. 20. 21. 22. 23.

J.-W. Yoon, F. Barlat, R.E. Dick, and M.E. Karabin, Int. J. Plasticity 22, 174-193 (2006). T. Kuwabara, K. Hashimoto, E. Iizuka and J.-W. Yoon, J. Mater. Processing Technol. 211, 475–481 (2011). T. Moriya, T. Kuwabara, S. Kimura and S. Takahashi, Steel Research Int. 81, 1384-1387 (2010). K. Hashimoto, T. Kuwabara, E. Iizuka and J.-W. Yoon, Tetsu-toHagané 96, 557-563 (2010). (in Japanese) T. Kuwabara, Int. J. Plasticity 23, 385-419 (2007). D. Banabic, F. Barlat, O. Cazacu and T. Kuwabara, Int. J. Mater. Form. 3, 165-189 (2010). T. Kuwabara, S. Ikeda and T. Kuroda, J. Mater. Process Technol. 80-81, 517-523 (1998). T. Kuwabara, M. Kuroda, V. Tvergaard and K. Nomura, Acta Mater. 48, 2071-2079 (2000). T. Kuwabara, A. Van Bael and E. Iizuka, Acta Mater. 50, 3717-3729 (2002). R. Saito, T. Kuwabara and E. Iizuka, “Material modeling of 980MPa dual phase steel sheet based on biaxial tensile test and in -plane stress reversal test” in Proc. IDDRG 2010, edited by R. Kolleck, Verlag der Technischen Universität Graz, Graz, Austria, 2010, pp. 669-678. S. Murakoso and T. Kuwabara, J. Solid Mech. Mater. Eng. 3 1330-1339 (2009). T. Kuwabara, M. Umemura, K. Yoshida, M. Kuroda and S. Hirano, J. JILM 56, 323-328 (2006). (in Japanese) M. Ishiki, T. Kuwabara and Y. Hayashida, Int. J. Mater. Form. DOI10.1007/s12289-010-1024-5. M. Ishiki, T. Kuwabara, M. Yamaguchi, Y. Maeda and Y. Hayashida., “Differential Work Hardening Behavior of Pure Tita nium Sheet under Biaxial Loading” in Proc. Numisheet 2008, edited by P. Hora, Institute of Virtual Manufacturing, ETH, Zurich, Switzerland, 2008, pp. 161-166. T. Kuwabara, K. Yoshida, K. Narihara, S. Takahashi, Anisotropic plastic deformation of extruded aluminum alloy tube under axial forces and internal pressure, Int. J. Plasticity 21, 101-117 (2005). Y. Hanabusa, H. Takizawa and T. Kuwabara, Steel Research Int. 81, 1376-1379 (2010). Y. Hanabusa, H. Takizawa and T. Kuwabara, J. JSTP 52, 282-287 (2011). (in Japanese) R. Hill, S.S. Hecker, and M.G.. Stout, Int. J. Solids Struct. 31, 2999-3021 (1994). R. Hill and J.W. Hutchinson, J. Appl. Mech. 59, S1-S9 (1992). Abaqus Analysis, User’s Manual, Version 6.6, Dassault Systemes, 2006. R. Von Mises, Göttingen Nachrichten, math.-phys. Klasse, 582-592 (1913). R. Hill, Proc. Roy. Soc. London A193, 281-297 (1948). F. Barlat, J.C. Brem, J.W. Yoon, K Chung, R.E. Dick, D.J. Lege, F. Pourboghrat, S.H. Choi and E. Chu, Int. J. Plasticity 19, 1297-1319 (2003).

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Evaluation of Constitutive Models for Springback Prediction in U-draw/bending of DP and TRIP Steel Sheets Jeong Yeon Lee, Myoung-Gyu Lee and Frédéric Barlat Graduate Institute of Ferrous Technology (GIFT), Pohang University of Science and Technology (POSTECH), San 31, Hyoja-dong, Nam-gu, Pohang, 790-784, Republic of Korea Abstract. U-draw/bending experiments and simulations were performed to investigate the characteristics of springback in sheet metals. The finite element method is frequently used for the simulation of springback, but the predictions are strongly influenced by the constitutive models such as yield criteria and hardening laws. In the present study, springback of DP and TRIP steel sheets after U-draw/bending was predicted with a finite element analysis. Various yield functions were considered, namely, the isotropic von Mises and anisotropic Hill models. For strain hardening, isotropic, non-linear kinematic and combined isotropic-kinematic hardening models were considered. In order to characterize the isotropic hardening behavior, both uniaxial and balanced biaxial (hydraulic bulge) tension tests were carried out. For the characterization of the combined isotropic-kinematic hardening model, forward-reverse simple shear tests were conducted. The springback predictions were greatly influenced by the choice of the hardening model but slightly affected by the choice of the yield criterion. The kinematic hardening and combined isotropic-kinematic hardening models provided satisfactory predictions for DP590 and TRIP590 sheets, respectively. Due to an extended measurable strain range, the use of the flow curves from the hydraulic bulge test led to more reliable results than those of uniaxial tension. Keywords: Springback, U-draw/bending, Finite element simulation, AHSS, Kinematic hardening model, Hydraulic bulge test, Simple shear test PACS: 62.20.fq

INTRODUCTION In sheet metal forming processes, springback is the elastically-driven shape change, which is frequently observed after the material is unloaded. In general, higher residual stresses and lower Young’s moduli increase the range of elastic deformation after forming and cause larger springback. Since the automotive industry attempts to utilize Advanced High Strength Steels (AHSS) for manufacturing their products, springback is becoming a significant issue. Finite Element (FE) simulations provide a cost-effective and reliable way of springback predictions for complex geometries and advanced constitutive models. In FE simulations of springback, the constitutive model strongly affects the accuracy of the result. Therefore, the influence of the constitutive models on simulation results has been a common interest in many related studies [1-3]. The role of hardening is particularly significant when the material experiences forward-reverse loading during forming [1, 3]. In the present study, U-draw/bending (Numisheet’93 benchmark problem [4]) simulations were conducted to predict springback for two AHSS, a dual-phase (DP) steel and a transformation induced plasticity (TRIP) steel, both with a nominal tensile strength of 590 MPa. The aim is to assess the influence of the constitutive model on springback simulations. A constitutive model consists of a yield function and its associated flow rule, and a hardening law. In this work, the von Mises and Hill 1948 yield functions [5] were associated with isotropic, Chaboche-type nonlinear kinematic [6] and combined isotropic-kinematic hardening models [6]. The stress-strain behavior was characterized in two different ways, namely: (1) Balanced biaxial tension flow curve determined from the hydraulic bulge test as an alternative to the uniaxial tension flow curve to characterize isotropic hardening. (2) Forward-reverse simple shear curves in order to characterize the combined isotropic-kinematic hardening model.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 571-578 (2011); doi: 10.1063/1.3623659 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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EXPERIMENTS Two grades of advanced high strength steel sheets, DP590 and TRIP590 (both 1.2 mm thick) produced by POSCO, were selected for this study. Mechanical tests were conducted in order to characterize the stress-strain behavior and U-draw/bending tests were carried out to assess springback as described below. All tests were duplicated in order to confirm the reproducibility.

Uniaxial Tensile Test Uniaxial tension tests were conducted with the Zwick/Roell tensile testing machine on ASTM E8 standard specimens for three loading directions, 0ȋ, 45ȋ and 90ȋ from the rolling direction (RD). Two mechanical extensometers were attached to the specimen during the test in order to measure the gauge length elongation and the gauge width reduction. The crosshead strain rate was controlled to be 1×10-3 /s for all tests. The resulting mechanical properties such as Young’s modulus (E), yield stress (σ ) and r-value (or plastic strain ratio, r ) are shown in Table 1.

Hydraulic Bulge Test Hydraulic bulge tests were conducted with the Erichsen bulge/FLC tester model 161. The velocity of the punch, which compresses the hydraulic oil, was set to 12 mm/min. After the tests, it was confirmed that this velocity resulted in a thickness strain rate of about 3×10-3 /s. The oil pressure was recorded by a pressure sensor installed in the machine. The oil temperature was maintained between 15 ȋC and 35 ȋC. A mechanical measurement device was placed on the top of the specimen allowing in-plane elongation and curvature measurements using an extensometer and a LVDT (Linear Variable Differential Transformer), respectively. From these measurements, the membrane stress and thickness strain were calculated as described in the literature [7]. The yield stress in balanced biaxial tension (σ ) was calculated based on the work-equivalence principle by comparing the bulge and uniaxial tension flow curves. The averaged value of flow stress ratio σ /σ is listed in Table 1.

Forward-reverse Simple Shear Test A device which imposes deformation to a rectangular sample from the parallel displacement of two rigid grips [8] was used for the simple shear tests. The specimens were cut into rectangular shapes of 50 mm (in the RD) × 10 mm (in the transverse direction, TD) and inserted into the grips of the simple shear device. An appropriate amount of torque, 28 Nm, was applied through a set of screws on the grips in order to prevent sliding. The constant grip velocity was set to give a shear strain rate of 2×10-3 /s to the sample. The axial force signal was recorded from the load cell and the displacement of the grip was measured by an extensometer. This displacement was assumed to be suitable to calculate the shear strain and to determine the simple shear flow curves. The shear axis was always in the RD. Forward-reverse simple shear tests were conducted for three pre-strains (5 %, 10 % and 15 % engineering shear strains) in addition to monotonic simple shear experiments.

U-draw/bending Test U-draw/bending tests were performed with a forming machine developed in POSCO. The tool geometry follows the Numisheet’93 benchmark problem [4] and is illustrated in Figure 1(a). The material was cut into rectangular shape specimens with the dimension of 350 mm × 45 mm (length in the RD × width in the TD). A lubricant (BW90EG) was applied on both sides of the specimens. It was measured that this lubricant gives a friction coefficient about μ=0.14. The punch stroke of the U-draw/bending test was fixed to 70 mm. A blank holding force of 25.7 kN was applied to constrain plastic flow.

DP590 TRIP590

E [GPa] 191 189

TABLE 1. Mechanical properties of DP590 and TRIP590 sheets. σ [MPa] σ [MPa] σ [MPa] r r 411 406 391 0.91 0.84 432 442 421 1.34 1.16

572

r 1.12 1.30

σ /σ 1.02 1.06

θ2

40mm Punch 50

Holder 55

R5

R5

35mm

Holder

ρ

6.5

Blank

53

θ1

Die

Die

15mm

350

(a)

Unit: mm

(b)

FIGURE 1. (a) Geometry of U-draw/bending tool and (b) the measurement of springback angles and sidewall curl radius.

After the tests, the edges of the specimens were scanned to a file by 3D digitizing system (Surveyor DS-4060, manufactured by Laser Design Inc.). Then, the springback angles and sidewall curl radii were measured according to the definitions illustrated in Figure 1(b).

CONSTITUTIVE MODELS Yield Criteria In plane stress, the anisotropic Hill 1948 yield function [5] is expressed as

Fσ + Gσ  + H σ − σ + 2Nσ  = σ .

(1)

Here, x and y correspond to the RD and TD, respectively. σ represents the effective stress. F, G, H and N are the yield function coefficients, which are given in Table 2 for DP590 and TRIP590. Note that the coefficients in Table 2 are valid only when RD uniaxial tension is the reference stress state. The yield stress in the RD (σ ) and r-values in three different loading directions (r , r , r ), as listed in Table 1, were used to calculate these coefficients.

Hardening Laws The hardening behaviors of the two materials under monotonic loading were determined using uniaxial tension and hydraulic bulge tests. These experimental curves were fitted by the Hollomon, σ  = K(ε ) , and Swift, σ = K(ε + ε ) , laws as shown in Figure 2. The coefficients for the hardening laws are given in Table 3. The uniaxial flow curve or, alternatively, the balanced biaxial flow curve can be used as a reference for isotropic hardening.

TABLE 2. Coefficients for the Hill1948 yield function (reference: uniaxial tensile stress state). F G H N DP590 0.463 0.556 0.444 1.263 TRIP590 0.440 0.427 0.573 1.441 TABLE 3. Coefficients of the hardening laws for DP590 (Hollomon) and TRIP590 (Swift). Uniaxial tension (RD) Balanced biaxial tension K [MPa] n K [MPa] n ε ε DP590 934.19 0.14 976.48 0.15 TRIP590 1113.81 0.020 0.24 1218.20 0.016 0.24

573

True stress or membrane stress [MPa]

True stress or membrane stress [MPa]

1200

DP590

1000 800 600 400 200 0 0.0

Uniaxial tension (RD) Fit of uniaxial tension Bulge Fit of bulge

1200

TRIP590

1000 800 600 400

Uniaxial tension (RD) Fit of uniaxial tension Bulge Fit of bulge

200 0 0.0

0.1 0.2 0.3 0.4 0.5 Plastic logarithmic strain or thickness strain

0.1 0.2 0.3 0.4 0.5 Plastic logarithmic strain or thickness strain

(a)

(b)

FIGURE 2. Fitting of hardening curves for (a) DP590 and (b) TRIP590.

According to the Chaboche nonlinear kinematic hardening model [6], the back stress increment is given as a function of effective plastic strain, 2 (2) d = cd − γdε , 3 where c and γ are the material coefficients which are considered constant in this study. In pure kinematic hardening, the size of yield surface does not change and strain hardening is achieved only by the back stress evolution. The parameters for pure kinematic hardening are listed in Table 4. In the combined isotropic-kinematic hardening model, the yield surface can expand and translate in stress space during plastic deformation. Therefore, an additional evolution rule for isotropic hardening is introduced as,  = σ + Q(1 − exp(−b ∙ ε )). σ

(3)

The subscript ‘iso’ was given to the effective stress in order to emphasize its contribution to isotropic hardening. This Voce-type isotropic hardening is used, rather than power law hardening, in conjunction with equation (2) for combined isotropic-kinematic hardening. A set of coefficients (see Table 5) for the combined isotropic-kinematic hardening model was determined so that it can reproduce the forward-reverse simple shear curves as shown in Figure 3. Note that a linear combination of several back stress components (d = d + d + ⋯ ) was used [6].

Shear stress [MPa]

600

TRIP590

Monotonic

400 200 Prestrain J=5% Prestrain J=10% Prestrain J=15%

0 -200 -400 -600 -0.4

Experiment IH-KH -0.2

0.0 0.2 Shear strain

0.4

0.6

FIGURE 3. Shear stress-strain curves predicted by the combined isotropic-kinematic hardening model (dashed) compared to the experimental curves (solid).

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TABLE 4. Coefficients for pure kinematic hardening model (top: uniaxial tension flow curve, bottom: bulge flow curve as reference). c [MPa] c [MPa] c [MPa] γ γ

γ DP590 1231 2.63 5159 23.13 11511 127.80 TRIP590 2021 2.43 4861 25.17 -

DP590 TRIP590

c [MPa] 889 757

c [MPa] 3173 2506

γ 1.61 0.36

c [MPa] 7827 4893

γ

13.57 6.36

γ 51.09 33.06-

TABLE 5. Coefficients for combined isotropic-kinematic hardening model (reference: uniaxial tension flow curve). Q [MPa] b c [MPa] c [MPa] γ γ

9.83 3388 47.60 TRIP590 1200 0.50 3101

In order to compare the forward and reverse stress-strain behavior from the different hardening models, single element FE simulations were performed for uniaxial tension-compression. The corresponding stress-strain curves are shown in Figure 4.

SPRINGBACK PREDICTIONS Finite Element Modeling A FE model of the U-draw/bending process was constructed in Abaqus/Standard (implicit code) version 6.8 based on the tool geometry shown in Figure 1(a). Only a quarter of the whole model was considered because of the geometrical and material symmetries. A blank holding force of 6.4 kN was applied on the quarter geometry, and the friction coefficient was μ=0.14. The tools, namely, the punch, die and blank holder, were modeled as analytical rigid surfaces while the blank was meshed with shell elements with reduced integration (S4R). Plane stress conditions were assumed in the quarter model. It was found from a series of parametric study that the use of 1 mm × 1.5 mm (length × width) mesh and 9 integration points through the thickness was reliable for the current geometry.

Influence of the Yield Function In order to observe the influence of the yield function, springback was predicted with two different models (von Mises and Hill 1948) and isotropic hardening, characterized with the uniaxial tension flow curve. The results are not shown in this paper, but elsewhere [9]. These two yield functions predicted very similar springback for DP590 and TRIP590, which is probably due to the isotropic yielding behavior of these materials. 1000

TRIP590

750

True stress [MPa]

500 250 0 Kinematic

-250

Isotropic-kinematic -500 Isotropic -750 -1000 -0.10

-0.05

0.00 0.05 True strain

0.10

FIGURE 4. Stress-strain curves under tension-compression for different hardening models, obtained from FE simulations.

575

100

DP590 BHF: 25.7kN

Experiment VM + IH uniaxial VM + IH bulge VM + KH uniaxial VM + KH bulge

Y [mm]

75

TRIP590 BHF: 25.7kN

Experiment VM + IH uniaxial VM + IH bulge VM + KH uniaxial VM + KH bulge

75

Y [mm]

100

50

25

50

25

0

0 0

25

50

75 X [mm]

100

125

150

0

25

50

(a)

75 X [mm]

100

125

150

(b)

FIGURE 5. Springback predicted by different hardening models and different reference curves for (a) DP590 and (b) TRIP590.

Influence of the Hardening Laws Figure 5 shows springback predictions with two different hardening models, the isotropic and Chaboche kinematic hardening models. Each hardening model is characterized by either the uniaxial tension (indicated as ‘uniaxial’) or bulge (indicated as ‘bulge’) flow curves. The von Mises yield criterion was selected for all the simulations shown in Figure 5. For DP590, the choice of the reference hardening curve (‘uniaxial’ or ‘bulge’) seems to have little effect on the springback predictions. This might be related to the similarity of two reference curves shown in Figure 2(a). However, the choice of the reference curve certainly affects the predictions for TRIP590. This is likely due to the fact that the bulge flow curve is much higher than the uniaxial flow curve for TRIP590, Figure 2(b), leading to higher residual stresses. This results in larger predicted springback with the bulge flow curve. The choice of the hardening model, either isotropic or kinematic, has more influence on springback than the choice of either the yield function or the reference hardening curve. The isotropic hardening model tends to overestimate springback while the kinematic hardening model tends to underestimate it. This is mainly because of the difference in residual stresses obtained through forward-reverse bending in the sidewall region. The kinematic hardening model shows early re-yielding (or Bauschinger effect) which makes the final residual stress much lower than that by the isotropic hardening model. Therefore, a smaller springback is expected with kinematic hardening compared to isotropic hardening.

TRIP590 BHF: 25.7kN

Experiment VM + IH VM + KH VM + IH-KH

75

Y [mm]

von Mises equivalent stress [MPa]

100

50

25

0 0

25

50

75 X [mm]

100

125

150

(a)

1000 800

Region I Region II Region III

TRIP590 VM+uniaxial IH IH-KH

600 400

KH 200 0 0.00

0.05

0.10 0.15 0.20 Equivalent plastic strain

0.25

(b)

FIGURE 6. (a) Springback predicted by different hardening models and (b) Von Mises equivalent stress-strain curves obtained from finite elements in Regions I (punch corner), II (sidewall) and III (die corner).

576

0

Angle [ ] / Radius of curvature [mm]

180 160

Numbers: prediction error with IH-KH model TRIP590 -3.0%

140 120

-0.1%

Experiment VM+ IH VM+ KH VM+ IH-KH

100 -1.2%

80 60 40 20 0

T1

U

T2

FIGURE 7. Springback angles and radii of curvature predicted by three different hardening models for TRIP590.

In terms of accuracy, springback of DP590 is well predicted by kinematic hardening with roughly a 1 % error for springback angles and a 3% error for sidewall curvature. For TRIP590, the springback angles are also well predicted by kinematic hardening within an error smaller than 2 %. However, the sidewall curvature is not well captured using either the reference behavior from uniaxial tension (17 % error) or bulge (9 % error) tests. Since the prediction of the curl radius was not satisfactory for TRIP590, the combined isotropic-kinematic hardening model was employed for that material. Figure 6(a) shows the springback profiles predicted with three different hardening models, and Figure 7 shows the corresponding springback angles and sidewall curl radii. The von Mises yield criterion and reference hardening curve in uniaxial tension were consistently used for these simulations. It was found that the springback predictions were much improved using the combined isotropickinematic hardening model comparing to isotropic or by kinematic hardening. This result can be explained in terms of residual stresses in the sidewall at the end of forming. The equivalent stress-strain curves were extracted from the finite elements at three regions (I: in contact with punch corner; II: in contact with die corner; III: sidewall) as shown in Figure 6(b). Only monotonic loading occurs in Regions I and III, for which any hardening model leads to the same stress-strain response. If the load is reversed, as in Region II, the kinematic hardening and combined hardening models are characterized by early reverse yielding (or Bauschinger effect), which makes the final residual stress much smaller than that made with isotropic hardening. It is worth noting that the residual stress predicted from combined isotropic-kinematic hardening is between those from isotropic and kinematic hardening. Considering the fact that springback is proportional to the residual stress, the springback obtained from combined hardening is intermediate between the springbacks obtained from the other two models. Since this explanation is valid only in the sidewall region, the differences between the hardening models are significant only for the sidewall curl radius, as shown in Figure 7.

DISCUSSION In this study, the materials were characterized using bulge and forward-reverse simple shear flow curves. Based on the experimental and simulated results presented above, a few comments about these characterization methods are given below. It is common to use the uniaxial tension flow curve to characterize the isotropic hardening behavior. The finite element simulation in Figure 6(b) showed that the plastic equivalent strain during U-draw/bending can exceed 20 %. It is difficult to obtain the flow curves up to this level of plastic strain in uniaxial tension tests for some AHSS sheets. For this reason, the uniaxial tension flow curves are extrapolated using power or exponential laws. Generally the hydraulic bulge test provides the flow curve with a much extended strain range. For example, for the DP590 and TRIP590 used in this work, the maximum strains were in the range 35 - 40 %. The use of the bulge flow curve can, therefore, reduce a possible error coming from extrapolations if the hardening curves are similar in both uniaxial tension and balanced biaxial tension such as for DP590 (Figure 2(a)). When the flow curves are quite different like for TRIP590 (Figure 2(b)), there is no clear answer to which flow curve is more suitable for U-draw/bending (which is often regarded as plane strain problem, i.e., between uniaxial and balanced biaxial tension).

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In order to characterize the material with combined isotropic-kinematic hardening, it is necessary to characterize the material behavior under non-monotonic loading. This is usually achieved with uniaxial tension-compression or three-point bending tests [1, 3]. Uniaxial tension-compression test is a straight-forward procedure to obtain the model parameters [1]. However, the maximum strain achievable in compression is limited due to elastic or plastic buckling. In contrast, the three-point bending test has no such a limitation and the experimental procedure is rather simpler. However, additional FE simulations are necessary for the three-point bending and the model parameters are indirectly determined through numerical iteration schemes [3]. In the forward-reverse simple shear test, as proposed in this research, the maximum strain achieved is much larger than in uniaxial tension-compression. This also requires additional FE simulations for simple shear but simpler than for three-point bending because a single element simulation is sufficient in this case. It was not verified in this work that the forward-reverse simple shear behavior allows a good prediction of forward-reverse bending. However, the model coefficients obtained using forward-reverse simple shear were shown to predict a reasonable uniaxial tension-compression behavior.

CONCLUSIONS The springback in the U-draw/bending process was predicted for two AHSS using various yield functions, von Mises and Hill 1948, and hardening laws, isotropic, kinematic and combined isotropic-kinematic hardening models. The mechanical properties of the two AHSS sheets, DP590 and TRIP590, were investigated using uniaxial tension, hydraulic bulge and simple shear tests. The measured properties were used in the different constitutive models in order to simulate springback. A comparative analysis leads to the following conclusions: (1) For DP590 and TRIP590, the yield criterion has a minor effect on springback while the hardening model plays a major role. (2) Isotropic hardening tends to overestimate springback while kinematic hardening tends to underestimate it, especially regarding the radius of curvature in the sidewall region. This tendency is directly related to the difference in residual stress levels produced by these models at the end of forming. (3) The springback predictions also depend on the choice of the reference hardening curve, i.e., uniaxial tension or bulge flow curves, but not as much as the hardening model. (4) The springback predictions for DP590 were satisfactory within an acceptable error using the kinematic hardening model. In contrast, the predictions for TRIP590 were satisfactory only with the combined isotropickinematic hardening model.

ACKNOWLEDGEMENTS This research was supported by POSCO and by WCU (World Class University) program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (R32-10147). Experiments were conducted with the help of Ms. Le Xu, Mr. Hyun Jin Kim, Mr. Jin Woo Lee and Mr. Seok Nyeon Kim from Materials Mechanics Laboratory, GIFT, POSTECH.

REFERENCES 1. M. G. Lee, D. Kim, C. Kim, M. L. Wenner, K. Chung, Int. J. Plasticity 21, 915-953 (2005) 2. R. H. Wagoner, J. F. Wang, M. Li, “Springback” in ASM Handbook volume 14B Metal Working: Sheet Forming, ASM, Material Park, OH, 2006, pp. 733-755. 3. P. A. Eggertsen, K. Mattiasson, Int. J. Mech. Sci 51, 547-563 (2009). 4. L. Taylor, J. Cao, A. P. Karafillis, M. C. Boyce, J. Mater. Proc. Technol. 50, 168-179 (1995). 5. R. Hill, Proc. Roy. Soc. Lond. 193, 281-297 (1948). 6. J. L. Chaboche, Int. J. Plasticity 2, 149-188 (1986). 7. M. G. Lee, D. Kim, C. Kim, M. L. Wenner, R. H. Wagoner, K. Chung, Int. J. Plasticity 21, 883-914 (2005). 8. E.F. Rauch, “Non-linear phenomena in materials science II” in Solid State Phenomena volume 23-24, edited by G. Martin, L. Kubin, 1992, pp. 317-334. 9. J. Y. Lee, Master thesis, Pohang University of Science and Technology, 2011.

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Simulation and Analysis of Finite Volume of Hot Forging Process of Nut M. Maarefdousta and M. Hosseynib a

Islamic Azad University, Gonabad Branch, Iran Islamic Azad University, Gonabad Branch, Iran

b

Abstract. In this study the forging operations of nut has been modeled. This nut is a part which is manufactured with the help of hot forging. The aim of this research is utilizing computers in designing forming process, and in particular, modeling of hot forging in the nut and to inquire the stresses appeared on the mold. For this purpose Pro/Engineer software for modeling, and SuperForge2004 software for analyzing the process have been used. This part is formed in two stages. To enrich the results coming out of the use of the software, the findings achieved from the modeling of the first stage are compared with its analytic dissolving. In the second stage modeling of metal forming the effect of rake on increasing the stresses imposed to the die mold is studied. The aim of this research is to correct the molds and the volume of the raw materials so that we can produce high qualified parts in spite of raw material low volume and low pressure on the molds. Keywords: Hot Forging, Simulation, FVM, Nut. PACS: 81.20.HY

INTRODUCTION Forging process is one of the popular methods in metal forming. Many of the industrial parts are produced by this method because of its high strength and production rate of parts. Parameters such as temperature, geometry of raw material and die are effective in reducing production cost and increasing part quality. If the design and manufacturing of parts are on basis of experience that results in time waste and high cost. The prediction of material flow can be achieved completely by computer simulation. Main parameters in computer simulation are filling the die completely without leaving any defect, reducing material loss and stress in die and increasing die life. In order to reduce the cost of forging process and make it competent with other production methods, it is essential to optimize the design of part and die. Moreover, considering required forces, energy and time in such a way to attain an accurate analysis and design could be highly important. Therefore, using of the accurate computer simulation can be an effective method for reaching these goals. Forging of complex parts is usually performed in several steps. Design of step numbers, step types and intermediate steps (pre-forms) along with the design of final die, forging parameters and process conditions are the most complex tasks in forging simulation. Generally, application of computer to design the required steps based on analytical and experimental data has been considered excessively by other researches previously. However, finite volume methods (FVM) are accepted especially today as a powerful method to be used widely to analyze various bulk metal forming processes including forging. As a result of extensive research on FVM different commercial software have been obtained with the capability of simulating different metal forming processes. Using these tools, one can build a virtual model of dies and defines different materials characteristics in order to simulate the forging process precisely and to get valuable information about how the material is being formed in the die. Thus it can be guaranteed that the parts have desired shapes before making the real die. In addition, using the results of performed numerical analysis many of the important design parameters such as forming forces, stresses on the part and die and also plastic energy are determined. Because of the functions and capabilities of this method in simulating the process, the manufactures know this method as the so-called virtual production. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 581-587 (2011); doi: 10.1063/1.3623660 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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During the last decade use of simulation software has grown remarkably in the forging industry. Most of the software presented in this field are formed based on the Finite Element Method. METALFORM in 1993 [1] DIEFORGE in 1996 [2] and QForm in 2001 [3] are among the software which are developed based on this method. Finite Volume Method are developed to analyze numerically fluid dynamic (CFD), and then it was utilized in the same field because the huge power in analyzing transient problems and large deformation. In 1997, M. Wheel used this method for analyzing bending deformation of thick and thin plates [4]. In 1998, W.J. Ferguson, employed this method for analyzing creep problems [5]. In 2002, K. Zarrabi, stated the way of using this method in elastic analysis of pressure vessel with axis metric geometrical figures[6]. In 2002, this method was utilized by G.A. Taylor and his colleague for modeling the welding Phenomenon [7]. The purpose of the present studying was using computer in designing metal forming and specifically simulation of hot forging of nut and investigating the stress created on die. To do so, we have used pro/Engineer software to modeling and MSC. Super forge 2004 for procedural analysis.

MATERIALS AND METHOD Simulation depicts an exact behavior of how metal flows inside the die. This leads us to investigate problems such as filling of the die, the possibility of creating defects in the work piece and figure of material flow in work piece. The results of simulation of forging processes including the figures of stress , strain, material flow and the way of temperature distribution, one used in the estimation of final product properties. Generally, the problems which are used mostly in simulation are as follow: 1) Ensuring that the die is completely full without any defects in the finisher. 2) Reduction of wastes. 3) Increasing the dies’ life time using optimization of material flow in the die (Reduction of wear), 4) Reduction of the load on the die and preventing from losing the dies. Nut is a work piece which is produced via hot forging method. The Primary row work piece is cylindrical. Deformation phases include blocker and finisher phases. In figure [1], the processes of deformation for this work piece are shown. Figure [2] shows the way of locating the work piece between dies. The aim of simulation of the process of deformation for this die is designing the geometrical figures of dies, and determining the appropriate volume of the primary work piece. The lack of accurate designing of die can lead to several difficulties such as creating cracks in the work piece and making defects in it. If the volume of the primary work piece is less than the accurate volume, the die can’t be filled. If the volume of work piece to creation of flash, increasing the stress on the die and decreasing their lifetimes, wasting the raw material and a lot of energy. In order to avoid such kinds of problems and defects and lack of estimating unpredicted problems, simulation of hot forging was done for foots of the deformation steps.

a)

b)

c)

FIGURE 1. Different processes of forming of nut a) Primary billet b) geometrical figure of billet after the first forming phase c) geometrical figure of billet after the second forming phase

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a)

b)

FIGURE 2. Locating methods of billet and dies, a) The first phase of forming, b) The second phase of forming

TABLE 1. The characteristics of used crank press Crank radius(R)

Rod length(L)

Revolution

70(mm)

520(mm)

90(rpm)

TABLE 2. The used input data Die Material Billet Material Din 1.1191 Din 1.2765

Die Temperature

200

C

Press kind Crankshaft press

Ambient Temperature

25C

RESULTS AND DISCUSSION Analytical Solution for the first step of deformation In this step, a raw cylindrical work piece is located between 2 parallel plates and is pressed from one plate. This deformation is called upsetting. This process is used widely in screws, gears, pins, and nails. In order to do analytical solution for this problem we use slab method. In this method, stress and load are directly calculated via integration of partial deferential of equilibrium equation. Using this method, accurate answers are found for the distribution of stress in the problem of upsetting. The hypothesis are as follow: 1) Symmetrical Deformation (including plane strain and axial symmetric) 2) Avoiding weight and inertial force. 3) column’s coefficient of friction is fixed. 4) Yield stress is function of strain and strain rate and temperature. According to figure [3] a cylinder is pressed with height of H and beam of R. Supposing that the coefficient of friction is fix and writing the balance equation on partial deferential in the direction of “r” and omitting the 2nd order sentences, The following equation is obtained: 2  Pr dr  h  dr  h r dr  hrd r  0 (1) In axis symmetric

   r

(2)

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And so, we have,

   r if

(3)

 r ,  z are the main strain, with substitution in Trsca equation, we have:

1  2  Y   r  P  Y 

(4)

d  r   dP From the equations (1), (2), and (3)we can exclude:

dP 2 dr  P h

(5)

Via integration of the following relation: ‫ا‬

 r ( R)  0  P  Y

(6)

Contact pressure function in beam direction is extracted like the following formula:

 2  P  Y exp  ( R  r )  h 

(7)

In the following section, the obtained curve is designed and the results of software are compared with it.

Simulation of the first step of deformation process and its comparison with analytical Solution Considering the geometrical figure of raw work piece and figure of work piece after the first step of deformation, simulation of this step using triangular elements in ax symmetric state was done specifications of press and entry data to the software are shown in table [1] and [2]. Since the reflection of the contact pressure on the die are put on the work piece. Maximum of the contact pressure are considered as the output of the software. To converge the results of the software, with increasing the number of elements, diagram sensitive mesh are designed. Figure [4]. This figure shows that with increasing the number of elements the answers of the problems are converged into the analytical solution. This diagram also shows if the number of elements are more than a specific number, it doesn’t have any special impact on the accuracy of the responses, and it only multiplies the time of simulation. After the responses are converged, contact strains are extracted in eleven points on the beam direction and were compared with analytical solution. Figure [5]. This figure shows a good correspondence between the result of software and analytical solution. To find the percentageof error in each point, the following relation is used: (8) PP error  100 P In this relation, P is the contact pressure and ‘P’ is the contact Pressure obtained from software analysis. The rate of errors in each point are shown in figure [6] and maximum and minimum and average error of software solution are also shown in table [3]. The amount of error shows that the results obtained from the software are closed to analytical solution with a perfect accuracy. TABLE 3. Maximum, Minimum and average software error in terms of percent Erorr Max Erorr Min 1.05%

0.05%

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Erorr Average 0.41%

Maximumcontact pressure(MPa)

70 60 50 40 Superforge

30

Analytical

20 10 0 0

500

1000

1500

2000

2500

3000

No. of elements

FIGURE 3. under pressure cylindrical and forces put on differential particles

FIGURE 4. Diagram of sensitivity to Mesh for the maximum stress in the first phase of formation

59

1.2

58 Analytical Superforge

57

1 0.8

55

error%

P(MPa)

56

54 53

0.6 0.4

52

0.2

51 50

0 0

1

2

3

4

5

6

7

8

9

10

11

0

1

2

3

4

r(mm)

5

6

7

8

9

10

11

r(mm)

FIGURE 5. Comparison of analytical solution and super forge solution

FIGURE 6. calculation of software error

Simulation of the Second Deformation Process This step of deformation process is simulated by the elements of hexagonal and in the 3 dimensional environments. In this step the contact stress is considered between die and work piece. To examine the pressure on the dies. In figure [8] with breaking the elements, diagram mesh sensitive and convergence of responses were investigated. The purpose of this simulation was to make the appropriate volume of the raw material clear that while filling the die and forming a work piece which is faultless, no flash is produced. The existence of the flash wastes the raw material and also increases the contact stress between work piece and die and finally the wear of the die or break of it. Figure [9] shows the way of flash forming and figure of contact stress. With observing the existed standards in designing the die and presented limitation and with the help of trial and error, the appropriate optimized figure of dies and the accurate volume of the raw material are recognized. Analysis shows that with doing these changes, not only the flashes are removed, but also contact stress on the dies are decreased up to 3 times. In figure [10] a die after the process corrector and removing the flashes and contract pressures are shown.

585

Maximum contact pressure(MPa)

460 450 440 430 420 410 400 0

500

1000

1500

2000

No. of elements

FIGURE 7. Contact pressure on the work piece surface as a result of software in the first phase of formation

FIGURE 8. Mesh sensitive diagram of Simulation in the Second formation Process

a)

b)

FIGURE 9. Distribution of Contact pressure with flash, a) real work piece b) simulated work piece

a)

b)

FIGURE 10. Distribution of Contact pressure without flash on nut, a) real work piece b) Simulated work piece

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CONCLUDING REMARKS In the current study, hot forging process for nut has been simulated and analyzed by Super Forge software. To valid the results obtained from software, the responses obtained from the simulation of first step were compared with its analytical situation. This comparison showed a good correspondence between analytical results and software results. The simulation of the 2nd step also shows that the accurate designing of dies and determining the raw material leads to the remove of flashes and reduction of stress on the dies.

REFERENCES 1. F. Kazerooni , M. Kazerooni, “ Finite Element Simulation of Forging Process”, production and manufacturing engineering conference,Tehran, 1993. 2. J. C Choi , B. M Kim, S. W Kim, “Computer–aided forging-die design for two-dimentional cross sectional components”,Journal of Material Processing Technology, 1996, pp124-132. 3. Biba N., Stebounov S.,Lishiny A., “Cost effective implementationof forging simulation”, Journal of Material Processing Technology, 2001, pp34-39. 4. M. A. Wheel, “A finite volume method for analysing the bending deformation of thick and thin plates”, Copmuter Method in Applied Mechanics and Engineering, 1997, pp199-208. 5. W. J. Ferguson, “The control volume finite element numerical solution technique applied to creep of soft woods”, Int. J. Solid Structures, 1998, Vol 35, No.13, pp1325-1338. 6. K. Zarrabi, “A finite volume element formulation for solution of elastic axisymmetric pressurized components”, Int. J. of pressure vessels and piping, 2000, pp 479-484. 7. G. A. Taylor, M. Hughes, N, Stursevich, K, Pericleous, “Finite volume method applied to the computational modelling of welding phenomena”, Applied Mathematical Modelling, 2002, pp. 309-320. 8. W. J. Slagter, C. L. Florie, and A. J. Venis, “Advances in 3-D forging process modeling”, 2000. 9. K, Lange, “Handbook of metal forming”, 1985.

587

Numerical Simulation Study on Hot Forming Process of Large TC4 Alloy Cylindrical Component Xu Qinghuaa, Zhou Wanlin a a

College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, Jiangsu China Abstract. Titanium alloys have outstanding performance on corrosion resistance and intensity, which are now partly used in the construction of the platform instead of steels. Based on the rigid-plastic finite element method(FEM) and modified iterative Newton-Raphson method, the deformation region of equivalent strain, equivalent stress, thickness distribution and load during the hot forming of large TC4 alloy cylindrical component were analyzed by numerical simulation under different processing conditions. This simulation was carried out by FE analysis software MSC Marc. The results show that the deformation temperature and punch velocity have important influence on the equivalent strain, equivalent stress and thickness distribution, which have greater effects on the load of TC4 alloy sheet hot forming process than friction. At last, this article achieved optimized technological parameters for the experiment and the simulated results agreed well with the experimental results. Keywords: TC4 alloy; hot forming; finite element method(FEM); numerical simulation PACS: 81.20.Wk, 81.05.-t, 83.50.-v

INTRODUCTION TC4 alloy is widely used in aerospace industries due to its lightweight property, high specific mechanical strength, creep resistance, corrosion resistance and excellent welding property[1]. Apart from the aerospace industry, titanium alloys have led to widespread non-aerospace applications due to good corrosion resistance to many highly corrosive environments. The attention has been paid to the high strength, unique corrosion resistance and nonmagnetic properties of titanium. It result in rather a wide application of titanium in chemical, marine and industrial applications[2]. At present, the use of titanium alloys in engineering structures for construction of the platform has become one of new titanium applications. The large cylindrical component is divided into six equal pieces and formed by mould pressing. Due to the component is made up of TC4 alloy sheet which has big spring-back and is easy to crack when formed at room temperature, hot forming process is usually used for the formation of titanium alloy workpiece. Recently, progress has been made by using the FEM to deal with the hot forming process. With the development of engineering technology, FEM can be used to simulate the metal forming process and gain optimized technological parameters for experiment[3-10].

FINITE ELEMENT ANALYSIS MODEL Geometry Model A 3-D model of hot forming process is established by FE software MSC Marc. Before the forming process, the sheet is put on the die and the temperature and punch velocity is then used to form the sheet in the deformation region. Two forming moulds are needed: punch and die. Fig.1 shows the 3-D FEM model of forming process. TC4 alloy cylindrical component is 10.0mm in thickness, 1780mm in upper diameter, 1760mm in bottom diameter and 1350mm in height. According to the constant volume principle in this model, the sheet is 10.4mm in thickness,

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 588-593 (2011); doi: 10.1063/1.3623661 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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1350mm in length and more than 890mm in width of the deformation region. The punch velocity is between 0.1mm/s and 2.5mm/s. A total of 7200 thin shell elements describe the model.

FIGURE 1. Geometry model and FEM meshes: (a) side view and (b) oblique view

Friction Model The Marc code uses different models for the friction. The classical Coulomb friction formulation:

Ffr

P Fn

Here Ffr is the tangential force between two bodies in contact, Fn is the relative normal reaction and Coulomb friction coefficient. In the model,

P

(1)

P

is the

is 0.1, 0.2 and 0.3.

Constitutive Equation The material of cylindrical component is TC4 alloy, whose mechanical behavior at high temperature is characterized by Backofen’s equation:

x

V

xm

kH

(2)

Here V is real effective stress, H is real effective strain rate, k is a constant of the material, and m is strain rate sensitivity exponent. When the deforming temperature is 650ଇ, 700ଇ and 750ଇ, k is 43.33, 29.94, 21.06 and m is 0.105, 0.115, 0.125, respectively.

Boundary Conditions As shown in Fig.2a, two edges of the sheet are fixed during the mould pressing process. In order to avoid another two edges of the sheet concaving along z direction, the displacement in z direction is set as zero as shown in Fig.2b.

FIGURE 2. Geometry model and FEM meshes: (a) side view and (b) oblique view

589

RESULTS AND DISCUSSIONS Strain and Stress Fig.3 shows the distribution of equivalent strain, equivalent stress under the condition of the deformation temperature for 700ଇ, the punch velocity for 0.1mm/s and the friction coefficient for 0.2 after unloading. As shown in Fig.3, the distribution of both equivalent strain and stress is uneven. And the maximum strain appears in circle angle area while the maximum stress appears in the middle of the component.

FIGURE 3. The distribution of (a) the equivalent strain and (b) stress

Deformation temperature has important influence on hot forming process of metal materials. With the increase of the deformation temperature, the softening effect, material flow and metal forming performance improves. Table 1 shows the influence of deformation temperature on equivalent strain and stress in the middle of the component at the last step before unloading. As shown in table.1, when the deformation temperature increases, the maximum equivalent strain and the maximum and minimum equivalent stress decreases while the minimum equivalent strain increases. The difference value between the maximum and minimum equivalent strain decreases, which is the same for the equivalent stress. And the non-uniformity of deformation distribution in the component reduces. Therefore, during the hot forming process of titanium alloy, the internal deformation of the component tends to be uniform when the deformation temperature increases appropriately. TABLE 1. The influence of the deformation temperature on the equivalent strain and stress Deformation Equivalent Strain/mm Equivalent Stress/MPa Temperature/ଇ Max Min Max Min 650 0.0542 0.0316 31.84 30.77 700 0.0528 0.0339 21.47 20.72 750 0.0513 0.0348 14.67 14.14 Punch Velocity: 0.1 mm/s, Friction Coefficient: 0.2

Table 2 shows the influence of the punch velocity on the equivalent strain and stress in the middle of the component at the last step before unloading. With the increase of the punch velocity, the maximum equivalent strain and stress increases while the minimum equivalent strain and stress decreases. The difference value between the maximum and minimum equivalent strain increases, which is the same for the equivalent stress. And the nonuniformity of the deformation distribution in the component intensifies. TC4 alloy has strong sensitivity to strain rate, when the punch velocity increases, the forming force will increase that results in less deformation time and material can't fully recrystallize. With the increase of the equivalent stress, the material hardening is greater than dynamic recrystallization softening effect. Considering the production efficiency, the reduction of the punch velocity has a positive effect on saving the production of energy consumption and improving mold life. In Fig.4, when the punch velocity increases, the maximum residual strain and stress increases and the minimum residual strain decreases after unloading. TABLE 2. The influence of the punch velocity on the equivalent strain and stress Punch Equivalent Strain/mm Equivalent Stress/MPa Max Min Max Min Velocity/mm 0.1 0.0528 0.0339 21.47 20.72 0.5 0.0664 0.0238 21.97 20.09 2.5 0.0981 0.0157 22.99 19.31 Deformation Temperature: 700ଇ, Friction Coefficient: 0.2

590

FIGURE 4. The influence of the punch velocity on (a) the residual strain and (b) stress(Deformation Temperature: 700ଇ, Friction Coefficient: 0.2)

Thickness Distribution During the hot forming process, the processing parameters have important influence on thickness distribution. In this work, the influence of the deformation temperature and punch velocity is focused on.

FIGURE 5. The influence of (a) the deformation temperature and (b) punch velocity on the thickness distribution(a, Punch Velocity: 0.1mm/s, Friction Coefficient: 0.2; b, Deformation Temperature: 700ć, Friction Coefficient: 0.2)

FIGURE 6. The middle and edge thickness distribution(Deformation Temperature: 700ć, Punch Velocity: 0.1mm/s, Friction Coefficient: 0.2)

Fig.5 shows the influence of the temperature and punch velocity on thickness distribution in the middle of the component after unloading. When the deformation temperature changes from 650ć to 750ć the sheet thickness

591

will be closer to 10.0mm. In molding process, with the increase of the deformation temperature, the material softening degree increases and promotes material flow uniformly. when the punch velocity increases from 0.1mm/s to 2.5mm/s, the thickness distribution will be more uneven and far away from 10.0mm. With the increase of punch velocity the equivalent strain and stress increases and the non-uniformity of the deformation distribution and material flow resistance in the component intensifies. Therefore, an excessively high punch velocity is not advised. In this case, according to the calculating result, 0.1mm/s is considered appropriate. In Fig.6, it shows the middle and edge thickness distribution is basically the same.

Load-Stroke Curve In forming process, the component load directly affects the choice of the equipment size. The load-stroke curve is the macroscopic mechanical performance of the change of the material internal organization property in certain extent, it is necessary to study on the load-stroke under different deformation process conditions. Fig.7 shows the deforming force increases when the stroke increases in the whole forming process. For TC4 alloy, the increase of the strain rate will cause the deformation resistance and the load increases remarkably. At the beginning of the deformation, the sheet is less deformation and need small deformation force while the contact area of the sheet and mould is small. With the increase of the deformation degree, the contact area of the sheet and mould increases gradually and the deforming force increases slowly. In the later deformation, the deforming force has risen sharply due to the metal deform sharply. And the contact area between the sheet and mould increases while the friction increases and the metal flow is difficult. These reasons prompt the increase of deforming force. The rapid increase of the load has the disadvantageous aspects with the mould and equipment. In Fig.7, the deformation temperature and punch velocity have greater effects on the load of TC4 alloy sheet hot forming process than friction.

FIGURE 7. The influence on load made by (a) the deformation temperature, (b) punch velocity and (c) friction(a, Punch Velocity: 0.1 mm/s, Friction Coefficient: 0.2; b, Deformation Temperature: 700ଇ, Friction Coefficient: 0.2; c, Deformation Temperature: 750ଇ, Punch Velocity: 0.1 mm/s)

592

EXPERIMENTAL STUDY In order to verify the validity of simulated results, the corresponding experiments were done. The experimental condition of hot forming process is that: the deformation temperature is 750ଇ, the punch velocity is 0.1mm/s. Comparison of the simulated and experimental thickness distribution at the same condition. The simulated value shows the thickness distribution is between 9.91mm and 10.03mm. The experimental value shows the thickness distribution is between 9.90mm and 10.04mm. The simulated results agree well with the experimental results.

CONCLUSIONS 1. Numerical simulation results show that the distribution of both equivalent strain and stress is uneven. And the maximum strain appears in circle angle area while the maximum stress appears in the middle of the component. 2. Deformation temperature and punch velocity have important influence on the equivalent strain, equivalent stress and thickness distribution. When the deformation temperature increases, the maximum equivalent strain and the maximum and minimum equivalent stress decreases while the minimum equivalent strain increases. And the non-uniformity of deformation distribution in the component reduces. With the increase of punch velocity, the maximum equivalent strain and stress increases while the minimum equivalent strain decreases. And the nonuniformity of deformation distribution in the component intensifies. The maximum residual strain and stress increases after unloading. 3. During the hot forming of large TC4 alloy cylindrical component the deforming force increases when the stroke increases in the whole forming process. The deformation temperature and punch velocity have greater effects on the load of TC4 alloy sheet hot forming process than friction.

REFERENCES 1. ZHAO Yong-qing, XI Zheng-ping, QIU Heng-lei. Current situation of titanium alloy materials used for national aviation [J]. Journal of Aeronautical Materials, 1005-5053 (2003) S0-0215-05. 4. I. Gurrappa. Characterization of titanium alloy Ti-6Al-4V for chemical, marine and industrial applications [J]. Materials Characterization, 51 (2003) 131– 139. 3. WANG Ming-wei, ZHANG Li-wen, PEI Ji-bin, LI Chen-hui, ZHANG Fan-yun. Effect of temperature on vacuum hot bulge forming of BT20 titanium alloy cylindrical workpiece [J]. Trans. Nonferrous Met. Soc. China, 17(2007) 957-962. 4. Zhao Bing, Li Zhiqiang, Hou Hongliang, Liao Jinhua, Bai Bingzhe. Three dimensional FEM simulation of titanium hollow blade forming process [J]. Rare Metal Materials and Engineering, 2010, 39(6): 0963-0968. 5. Z.M. Hu, T.A. Dean. Aspects of forging of titanium alloys and the production of blade forms [J]. Journal of Mechanicals Processing Technology, 111(2001) 10-9. 6. LUO Jiao, LI Miao-quan. FE-based coupling simulation of Ti60 alloy in isothermal upsetting process [J]. Trans. Nonferrous Met. Soc. China, 20(2010) 849-856. 7. Cheng Lv, Liwen Zhang, Zhengjun Mu, Qingan Tai, Quying Zheng. 3D FEM simulation of the multi-stage forging process of a gas turbine compressor blade [J]. Journal of Materials Processing Technology, 198(2008) 463-470. 8. ZHANG Shi-hong, SUN Cheng, WANG Zhong-tang. Finite element simulation on press forging of magnesium alloy AZ31 sheets [J]. Trans. Nonferrous Met. Soc. China, 18(2008) S269-S272. 9. YU Min, LUO Ying-she, PENG Xiang-hua, QIN Yin-hui. Numerical simulation and its application of rheological forming of titanium alloy vane disk [J]. Trans. Nonferrous Met. Soc. China, 16(2006) S671-S675. 10. LV Cheng, ZHANG Li-wen. Numerical simulation on forging process of TC4 alloy mounting parts [J]. Trans. Nonferrous Met. Soc. China, 16(2006) 1386-1390.

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601

Numerical Prediction of Microstructure and Mechanical Properties During the Hot Stamping Process Dongbin Kan, Lizhong Liu, Ping Hu , Ning Ma, Guozhe ShenˈXiaoqiang Han, Liang Ying State Key Laboratory of Structural Analysis for Industrial Equipment, School of Automotive Engineering, Dalian University of Technology, Dalian 116024, P.R.China Abstract. Numerical simulation and prediction of microstructures and mechanical properties of products is very important in product development of hot stamping parts. With this method we can easily design changes of hot stamping products’ properties prior to the manufacturing stage and this offers noticeable time and cost savings. In the present work, the hot stamping process of a U-channel with 22MnB5 boron steels is simulated by using a coupled thermo-mechanical FEM program. Then with the temperature evolution results obtained from the simulation, a model is applied to predict the microstructure evolution during the hot stamping process and mechanical properties of this U-channel. The model consists of a phase transformation model and a mechanical properties prediction model. The phase transformation model which is proposed by Li et al is used to predict the austenite decomposition into ferrite, pearlite, and bainite during the cooling process. The diffusionless austenite-martensite transformation is modeled using the Koistinen and Marburger relation. The mechanical properties prediction model is applied to predict the products’ hardness distribution. The numerical simulation is evaluated by comparing simulation results with the U-channel hot stamping experiment. The numerically obtained temperature history is basically in agreement with corresponding experimental observation. The evaluation indicates the feasibility of this set of methods to be used to guide the optimization of hot stamping process parameters and the design of hot stamping tools. Keywords: Hot stamping, Numerical simulation, 22MnB5 steel, Phase transformations, Hardness estimation. PACS: 81.05.Bx, 81.20.Hy

1. INTRODUCTION In order to reduce the weight of automobiles and improve the crash safety, more and more high-strength or ultrahigh-strength steel components are used on the vehicles. Conventional cold stamping of high-strength steel leads to disadvantages such as low formability and sever springback, so hot stamping technology is consequently developed for quenchenable boron steels such as 22MnB5. However, there are many process parameters in the hot stamping process and the tools need to be designed with cooling system, these make the design of hot stamping process and tools really complex. And this problem will become much knottier when we design hot stamping products with tailored properties. Numerical simulation and prediction of microstructures and mechanical properties of products can solve these problems mentioned above. Through this method, we can simply modify hot stamping technical process parameters and the design of tools before the manufacturing stage until the mechanical properties of the hot stamping products meet our expectation. In other words, the hot stamping process can be controlled precisely and quantitatively. This will greatly reduce the development costs and shorten the research cycle time. In this paper, the hot stamping process of a U-channel with 22MnB5 boron steels was simulated by using FE program LS-DYNA. Then with the temperature evolution results obtained from the simulation, models proposed by Li et al [1] were applied to predict the microstructure evolution during the hot stamping process and final

Corresponding author, email address: [email protected]

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 602-609 (2011); doi: 10.1063/1.3623663 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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mechanical properties of this U-channel. The hot stamping experiment was designed to evaluate the prediction method.

2. EXPERIMENTAL SET-UP The material studied is a low alloyed boron steel called 22MnB5 with a thickness of 1.6mm. The chemical composition of this material is given in Table 1. The tests were performed in a hydraulic press machine. The tool consists of a punch and a die, both with cooling systems, Fig. 1. In the experiment, the blank was heated in a furnace to 950ć, and held for 5 minutes. After having a homogeneous austenitic microstructure, the blank was transferred to the water cooled tools where stamping and quenching took place simultaneously. After deformation, the hot blank remained in the tool to be cooled for 8s. Figure 2 shows the dimensions of the rectangular blank and the location of the temperature measurement point. The temperature evolution of the blank during the whole process was measured at the point on the sheet surface, using type K thermo-couple. C (%) 0.22–0.25

TABLE 1. Chemical composition of the 22MnB5 steel given in wt.% B (%) Si (%) Mn (%) Cr (%) P (%) S (%) 0.002-0.005 0.2–0.3 1.2–1.4 0.11-0.2 0–0.02 0–0.005

Al (%) 0.02-0.05

FIGURE 1. Illustration of hot stamping tools with cooling system.

FIGURE 2. Dimensions of the blank and location of the temperature measurement point, all dimensions are given in mm.

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3. FE MODELING The FE-code LS-DYNA has been used in this simulation. The modeled punch, die and blank as shown in Fig. 3 are corresponding to the experimental set-up. Only half of the blank and tools is modeled and symmetry constraints are applied. The simulation model consists of 9760 Belytschko-Tsay shell elements, with 5 through thickness integration points. Set TSHELL to 1 in the keyword *control_shellˈso that in the thermal calculations of the blank, the shells will be treated as 12-node brick elements to accurately calculate the through thickness temperature gradient, which is very important in the simulation of hot stamping process [2,3]. Material model 106 [3] which is a thermal elastic visco-plastic material model is used. Stress versus strain data is taken from the Numisheet 2008 BM03 benchmark specification [4]. Fig. 4 shows stress-strain curves for 22MnB5 steel for different temperatures from 500ć to 800ć at a strain rate of 0.1s-1. Data for Young’s modulus, Poisson’s ratios, thermal conductivity and heat capacity of the blank as a function of temperature are taken from the literature [5]. And the heat capacity and thermal conductivity for the material used in the tools are extracted from [6].

FIGURE 3. FE-model of blank and tools.

FIGURE 4. Stress–strain curves at temperatures of 800–550ć at a strain rate of 0.1 s1.

A convection heat transfer coefficient of 8 Wm-2K-1 is used and the radiation heat transfer coefficient is set to107 Wm-2K-1. Thermal boundary conditions are turned off for areas in contact with tools. The contact heat transfer coefficient which has a strong dependency on the contact pressure is a most critical parameter controlling cooling of

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the blank. In this work, the contact heat transfer coefficient is defined as a function of contact pressure, and the data given in the Numisheet 2008 BM03 benchmark specification [4] is used. The frictional coefficients is defined as a function of temperature, the data is extract from [7]. According to the experiment, temperature of the blank at the beginning of the die movement is set to 810ć, and the tools are held at 75ć. The punch is given a velocity of 70mm/s which is the actual punch velocity in the experiment. After the stamping procedure, the blank is held in the tools for 8s for the quenching.

4. MODELING OF MICROSTRUCTURAL EVOLUTION The numerical prediction model of microstructure and mechanical properties consists of a phase transformation model to predict the austenite decomposition into ferrite, pearlite, bainite and martensite during the cooling process, and a mechanical properties prediction model to predict the products’ final hardness distribution.

4.1 Phase Transformation Model The diffusion controlled transformations of austenite decomposition into ferrite, pearlite, and bainite for the isothermal condition is described by a set of reaction kinetics equations developed by Li et al [1]. The austenite to ferrite reaction equation is given as:

20.41G  Ae3  T exp  27500 RT X F 0.4 1 X F 1  X F 3

dX F dt where

0.4 X F

(1)

FC exp 1.00  6.31C  1.78Mn  0.31Si  1.12 Ni  2.70Cr  4.06 Mo

FC

For the pearlite reaction, the equation is stated as:

20.32G  Ae1  T exp  27500 RT X P 0.41 3

dX P dt

 XP

1  X P

0.4 X P

PC

where

(2)



exp 4.25  4.12C  4.36Mn  0.44Si  1.71Ni  3.33Cr  5.19 Mo

PC



And the bainite reaction equation is expressed as:

dX B dt where

BC

20.29G  Bs  T exp  27500 RT X B 0.4 1 2

 XB

1  X B

0.4 X B

BC

(3)

exp  10.23  10.18C  0.85Mn  0.55 Ni  0.90Cr  0.36 Mo

In Equation (1)-(3), G is the ASTM grain size number for the austenite; Ae3 , Ae1are the critical temperatures from the Fe–C equilibrium diagram; Bs is the bainite start temperature; R is the universal gas constant, T is the current temperature; XF , XP and XB are the volume fraction of ferrite, pearlite and bainite, respectively. The diffusionless austenite-martensite transformation is modeled using the Koistinen and Marburger relation [8], which is expressed as

Xm

X J (1- e-D ( Ms -T ) )

(4)

where Xm is the volume fraction of martensite, X¤ is the volume fraction of austenite available for the reaction, ¢is a constant coefficient set to 0.011, Ms is the martensite start temperature, and T is the current temperature.

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4.2 Mechanical Properties Prediction Model The mechanical properties prediction model is given as [1]:

Hv

X M HvM  X B HvB   X F  X P HvF  P

(5)

where XM, XB, XF and XP are the volume fraction of martensite, bainite, ferrite, and pearlite, respectively; and HvM, HvB and HvF+P are the Vickers hardness of martensite, bainite, and the mixture of ferrite and pearlite, respectively. For the calculation of HvM, HvB and HvF+P, empirically based expressions suggested by Maynier et al. [9] are used:

HvM HvB

127  949C  27 Si  11Mn  8 Ni  16Cr  21log Vr

323  185C  330 Si  153Mn  65 Ni  144Cr  191Mo 

 89  53C  55Si  22Mn  10 Ni  20Cr  33Mo log Vr HvF  P

42  223C  53Si  30 Mn  12.6 Ni  7Cr  19 Mo 

10  19Si  4 Ni  8Cr  130V log Vr

(6)

(7)

(8)

where Vr is the cooling rate at 70Wć (ć/s).

4.3 Prediction Process The kinetics equations are solved by the RUNGE_KUTTA method. With the application of the additivity rule, Equation (1)-(3) can be used to computing phase transformations under continuous cooling conditions. A fourth order polynomial is used to fit the temperature history curves obtained from the simulation. And then the coefficients of the polynomial are used as initial data in the prediction program. Fig. 5 shows a schematic structure for the prediction program.

FIGURE 5. Schematic structure for the prediction program.

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5. RESULTS AND DISCUSSION 5.1 Temperature History The measured and calculated temperature history of the point (Fig. 2) is compared in Fig. 6. In the experiment, the hot blank is transferred from the furnace to the tools manually. The blank is air cooled to 810ć when the stamping begins. As it is illustrated in Fig. 6, the calculated temperature history is in good agreement with the measured data. The cooling rate of the point is higher than 150ć/s. However, deviation appears in the last few seconds of the quenching process. The main reasons can be summarized as follows: 1) The tools were held at 75ć in the simulation, which is not in correspondence with the actual situation. 2) The phase transformation latent heat is not considered in the simulation.

FIGURE 6. Measured and calculated temperatures for the point defined in Fig. 2.

FIGURE 7. Calculated temperature distribution of the blank after the stamping process.

Figure 7 shows the calculated temperature distribution of the blank just after the stamping process. As we can see from the picture, the lowest temperature is found at the lower fillet of the U-channel. The blank material at this area keeps in contact with the fillet of the punch during the whole stamping process, almost no relative sliding is observed. Thus the contact heat transfer is high efficient because of the high contact pressure. The highest temperature value can be observed in the bottom and flange area, because these areas are the last ones which come in contact with the tools. Figure 8 shows the temperature evolution of four representative nodes (Fig.7) during the whole process. The average cooling rate (between 800 and 300ć) is the highest around the lower fillet of the U-

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channel, about 340ć/s. While the average cooling rate is the lowest at the bottom area and side wall, about 120ć/s. These can be also attributed to the contact conditions at these areas.

FIGURE 8. Temperature evolution of four representative nodes illustrated in Fig.7 during the whole process.

5.2 Microstructure and Hardness The hot stamped U-channel can be divided into five parts: the flange area, the side wall, the bottom area, the upper fillet and the lower fillet. Prediction of microstructures and hardness is conducted by the developed program for selected points from each area. Location of selected points and prediction results are shown in the unfolding drawing of the U-channel FE model, illustrated as Fig.9. The cooling rates at these points are always high enough to drive the martensite transformation, so all the predicted values of martensite volume fraction are above 95%, and the values of hardness above 490. While at the upper and lower fillet, the values of martensite volume fraction and hardness are a little higher than those at other areas, because the cooling rate is higher at the fillet than at other areas.

. FIGURE 9. Location of selected points, predicted Martensite volume fraction (%) and Vickers Hardness(within parenthesis).

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6. CONCLUSIONS The simulation of hot stamping process of a U-channel component with 22MnB5 boron steels was conducted by using FE program LS-DYNA. In addition, a program was developed to predict the final microstructure and mechanical properties. The temperature history of the blank from calculated and measured results are basically in agreement, which indicates that this set of simulation is feasible. According to the simulation, the lower fillet of blank and the fillet of punch keep in contact with high pressure during the stamping process. As a result, the cooling rate is higher at this area, and the predicted values of martensite volume fraction and hardness are a little higher than those at other areas.

ACKNOWLEDGMENTS This work was funded by the Key Project of the National Natural Science Foundation of China (No. 10932003), “973” National Basic Research Project of China (No. 2010CB832700), “863” Project of China (No. 2009AA04Z101) and the Fundamental Research Funds for the Central Universities(No. 893324, DUT11ZD202). These supports are gratefully acknowledged. Many thanks are due to the referees for their valuable comments.

REFERENCE 1. M. Li, et al., "A computational model for the prediction of steel hardenability," Metallurgical and Materials Transactions B, 1998, pp. 661-672. 2. G. Bergman and M. Oldenburg, "A finite element model for thermomechanical analysis of sheet metal forming," International Journal for Numerical Methods in Engineering, 2004, pp. 1167-1186. 3. J. O. Hallquist, "LS-DYNA Keyword User’s Manual - Version 971," Livermore Software Technology Corporation, 2007. 4. Nunmisheet 2008, The Numisheet Benchmark Study, Benchmark Problem BM03, Interlaken, Switzerland, 2008. 5. A. Shapiro, "Finite Element Modeling of Hot Stamping," Steel Research International, 2009, pp. 658-664. 6. P. Akerstrom, "Modelling and simulation of hot stamping," PhD. Thesis, Lulea University of technology, 2006. 7. T. Stöhr, M. Merklein and J. Lechler, "Determination of frictional and thermal characteristics for hot stamping with respect to a numerical process design," 1st International Conference on Hot Sheet Metal Forming of High-Performance Steel, Kassel, Germany, 2008, pp. 293–300. 8. D. P. Koistinen and R. E. Marburger, "A general equation prescribing the extent of the austenite-martensite transformation in pure iron-carbon alloys and plain carbon steels," Acta Metallurgica, 1959, pp. 59-60. 9. P. Maynier, J. Dollet and P. Bastien, "Hardenability Concepts with Applications to Steels," D.V. Doane and J.S. Kirkaldy, eds., AIME, New York, NY, 1978, pp. 518-44.

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Simulation of Tailored Tempering with a ThermoMechanical-Metallurgical Model in AutoFormplus S. Ertürka, M. Sestera, M. Seliga, P. Feuserb, K. Rollb a

AutoForm Development GmbH, Technoparkstrasse 1, CH-8005 Zurich, Switzerland b Daimler AG, Bela-Barenyi-Str. 1, 71063 Sindelfingen, Germany

Abstract. For automotive applications, the hot stamping of ultra-high-strength steels such as 22MnB5 is a wellestablished process providing significant reduction of fuel consumption and improving the component strength and geometrical accuracy due to reduced springback. Tailored tempering is a special type of hot stamping, in which different areas of the component experience different cooling histories leading to different final properties. The potential of manufacturing-optimised components consisting of high-strength and high-ductility regions in harmony with an enhanced crash performance makes tailored tempering very attractive compared with other conventional hot stamping processes. The optimisation of this process, where deformation and cooling take place simultaneously, requires a complete understanding in terms of material behaviour, formability, heat transfer and phase transformation kinetics. To this end, a thermo-mechanical-metallurgical model has been implemented in AutoFormplus in order to capture the material behaviour during the forming and quenching processes. Both radiation and convection are taken into account to describe heat transfer to ambient. Moreover, latent heat is considered and its effect on simulation is discussed. A guideline for parameter identification strategy has been developed and validated by separate experiments. The simulation results of tailored tempering of a B-pillar are presented together with measured tensile strength and elongation at fracture. Keywords: Tailored tempering, phase transformation, high-strength steels PACS: 81.05.Bx; 81.40.Ef

INTRODUCTION The number of automobile structural components produced from ultra-high-strength steels such as 22MnB5 is continuously growing due to the need to reduce the fuel consumption together with improved safety requirements. In hot stamping of 22MnB5, high initial temperatures are used together with high cooling rates leading to a martensitic phase transformation and thus high strength. Besides high strength, the components produced by conventional hot stamping exhibit a high geometrical accuracy due to reduced springback [1]. The benefits of hot stamping mentioned above have been extended to obtain improved crashworthiness by a special type of hot stamping, i.e. tailored tempering where forming and phase transformation can be optimised. The material properties of the parts can be selectively adjusted by introducing different localised temperatures within a tool. This affects the cooling rate, therefore phase transformation. Consequently, combining areas with high strength and areas with high ductility is possible via tailored tempering. This favours production of structural components with improved properties such as an optimised crash resistance [2].

SPECIAL FEATURES OF TAILORED TEMPERING In the simulation of a conventional direct hot-forming process, temperature field computation is necessary only for the simulation of material flow at elevated temperatures in order to ensure that the part can be produced without splits. The design of a conventional hot-forming process excluding quenching can be accomplished easily with the hot-forming functionality of AutoForm-Incrementalplus. On the other hand, the planning of tailored tempering processes places much higher demands on simulation software because now the quenching process itself has to be modelled in detail. Tailored tempering with heated tools and with cooled tools provides the possibility of adjusting The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 610-617 (2011); doi: 10.1063/1.3623664 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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zones with very high strength and zones with enhanced ductility in the same part, with a gradual transition of properties between these zones. A simulation model that is capable of predicting final part properties must, on the one hand, provide the complete temperature history of a material point with sufficient accuracy and, on the other hand, employ a phase transformation model. Such a model for the formation of martensite and bainite, for generation of latent heat and for prediction of final properties, has been implemented in AutoForm-ThermoSolverplus and is described in the next section.

THERMO-MECHANICAL-METALLURGICAL ANALYSIS In order to predict final part properties with sufficient accuracy, a complete modelling of the relevant phenomena with a coupled thermo-mechanical-metallurgical description is essential. The phenomena observed in tailored tempering can be described by thermal (i.e. heat transfer between hot sheet, tools and environment), mechanical (i.e. plastic deformation of sheet) and metallurgical descriptions (i.e. phase transformation of sheet resulting from cooling). Figure 1 illustrates the interaction between thermal, mechanical and microstructural fields, which is adapted in AutoForm-ThermoSolverplus. A more detailed discussion on the interaction can be found in [3]. Contact

Mechanical

Phase transformation

Thermal

Heat generation

Metallurgical

Latent heat

Mechanical properties

FIGURE 1. Interaction between thermal, mechanical and metallurgical fields in AutoForm-ThermoSolverplus

Mechanical Effects Since metals show temperature-dependent yielding in the temperature regime of tailored tempering, a coupled temperature and displacement analysis is required. Moreover, the strain rate effect on plasticity cannot be neglected for any hot-forming process, since it is pronounced at elevated temperatures. In AutoFormplus, hardening curves can depend on strain, temperature and strain rate. Details regarding temperature- and strain-rate-dependent hardening and the strategies to determine model parameters are explained in [4]. The hardening description can be combined either with the quadratic Hill yield criterion or with the non-quadratic BBC yield criterion. However, for hotforming applications a simple isotropic description is usually sufficient.

Thermal Effects Both mechanical and metallurgical effects strongly depend on temperature. This makes accurate temperature calculation crucial. Temperature calculation according to Newton's cooling law is achieved by taking thermal energy balance into account. For a given volume, the summation of all heat flux applied to the body and the rate of energy generated within the body has to be equal to the internal energy rate.

qtool

Punch

qtool qtool

q amb

qtool

HTCtool (Tblank  Ttool )

q amb

effective HTCamb (Tblank

effective HTCamb

 Tamb )

radiation convection HTCamb  HTCamb

Die

effective HTCamb

Ambient heat transfer coefficient

Blank holder

Heat transfer coefficient [mW/mm2K]

100 90

Effectiveradiation HTCamb Radiation Convection convection

80 70

HTCamb

60 50 40 30 20 10 0 0

plus

FIGURE 2. Left: Heat fluxes between sheet, tools and ambient in AutoForm-ThermoSolver of heat transfer to ambient

611

200

400

600

800

Temperature

Temperature [°C]

1000

Right: Temperature dependency

The generated energy considered in a simulation of AutoForm-ThermoSolverplus can be any arbitrary combination of generated energy due to plastic deformation, phase transformation and friction. The modelling of heat flux is summarised in Figure 2. Heat conduction within the blank can be neglected in a hot-forming simulation but should be taken into account for tailored tempering due to a longer process time. Heat transfer to ambient can be defined as a function of temperature when both convection and radiation contribute to the heat transfer to ambient. Radiation plays a significant role especially at elevated temperatures and causes a strong deviation from the linearity of heat transfer. Heat transfer to tools is modelled based only on convection. The linear heat transfer coefficient regarding heat transfer to tools can be a function of contact pressure as described in [5].

Metallurgical Effects The base material 22MnB5 exhibits a pro-eutectoid ferritic-pearlitic microstructure with a hardness of about 170 HV10. However, as seen in the continuous cooling transformation (CCT) diagram [6] (Figure 3) after quenching with a cooling rate exceeding the critical cooling rate, the component might have a martensitic microstructure with a hardness of about 475 HV10. 900 C 700

A+

Ms 0.2

0.7

228

163

10 8 6 5 3 278

474

475 470

Hardness HV10

100

429 417

200

A+M

30 25 20

Cooling rate K/s

300

100 80

Temperature

B

500 400

A+P

A+

600

50%

F

Source: Arcelor

2h

1h

15 mi n

in 5m

2m

1m

in in

0

A: Austenite F: Ferrite P: Pearlite B: Bainite M: Martensite

Time

FIGURE 3. Continuous cooling transformation (CCT) diagram of 22MnB5

The phase transformation model implemented in AutoForm-ThermoSolverplus is the product of a co-operation with GMS GmbH [7]. The complexities of phase transformation and its kinetics are adapted and captured especially for 22MnB5. This enables the calculation of phase fraction and the resulting material properties, which are hardness, tensile strength and elongation at fracture respectively. Phase transformation and its kinetics are mainly derived from chemical composition. 22MnB5 gains special characteristics due to the presence of carbon (C), manganese (Mn) and boron (B). To this end, the chemical composition of 22MnB5 defined with respect to C, Mn and B is essential input because the corresponding phase transformation temperatures such as Ms are calculated based on chemical composition. Moreover, the energy generated during phase transformation, called latent heat, is also computed by the phase transformation model and taken into account in the temperature calculation of AutoForm-ThermoSolverplus.

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IDENTIFICATION OF ADDITIONAL MODEL PARAMETERS FOR TAILORED TEMPERING Since material flow in tailored tempering takes place in the same temperature regime compared to conventional hot-forming of 22MnB5, mechanical material characterisation as described in the Numisheet 2008 Benchmark 3 “Continuous press hardening” [8] is sufficient, and the corresponding material data preparation is performed as explained in [4]. No additional mechanical material data is required for simulation of tailored tempering. In order to determine the effect of latent heat and the effect of heat transfer to ambient, cooling curves of 22MnB5 sheets and austenitic stainless steel 1.4310 sheets subjected to air cooling were measured [9]. The latent heat was calculated as 75.8 kJ/kg based on the different cooling curves of 1.4310 and 22MnB5. Moreover, a strong temperature dependency of heat transfer to ambient was demonstrated in [9].

VALIDATION In co-operation with Daimler AG, an experimental setup was designed as shown in Figure 4. The designed tool consists of four modules, one pair cooled by cooling fluid and the other pair heated by heating cartridges. Austenised blanks at 950°C were transferred from the furnace to the tool and subjected to contact pressure during closure for controlled cooling. All tests were performed at the Lehrstuhl für Fertigungstechnologie (Prof. M. Merklein) of the University of Erlangen-Nuremberg in Germany. The nominal tool temperature was varied up to 550°C. Apart from tool temperature, contact pressure, transfer time, holding time and blank thickness were chosen as process parameters. During the experiments, the blank temperature was measured by a thermocouple with a diameter of 1.0 mm, which was connected to a small hole in the edge of the blank. In addition, the tool temperature was recorded simultaneously at four measuring points close to the surface of the contact plate. The measured tool surface temperatures are about 25°C higher than nominal values. All experiments were performed with a Schenck-Trebel RM400, which makes it possible to log the loaddisplacement path of the machine. Tensile tests were performed on the specimens prepared from the blank subjected to cooling experiments for the purpose of validation of resulting mechanical properties.

upper mounting plate

bores for water-cooling contact plate pressure pin

COOLED

HEATED

Experimental tool • partially cooled (15°C) • partially heated (variable heating up to 550°C) bores for heating • tests performed with Schenckdevices Trebel RM400 • tool material: 1.1730 positioning bolt

base plate

disc spin support

Measuring of • tool temperature at 4 points • specimen temperature at 1 point • load-displacement path of machine P1 P2

specimen

lower mounting plate

contact plate

FIGURE 4. Experimental tool setup for validation

Simulations of the experiments described before were performed with a pre-release version of AutoFormThermoSolverplus in order to validate the thermo-mechanical-metallurgical model. The material parameters used for the simulations are summarised in Table 1. Including a pressure-dependent heat transfer coefficient between the tool and the blank as described in [5] had only a minor effect on the results.

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Density [kg/m3] 7951.07

TABLE 1. Input data of 22MnB5 for the simulations Heat capacity Thermal conductivity Latent heat Chemical composition [J/(kg*K)] [W/(m*K)] [kJ/kg] [wt%] 550 32 75.8 C: 0.217; Mn: 1.162; B: 0.0029

Table 2 shows the heat transfer coefficients used for the simulations. For the tools, a constant temperature Ttool=Tnominal+25°C was prescribed. TABLE 2. Heat transfer data for the simulations Temperature HTC tool [°C] [mW/(mm2*K)] 1050 20 2100 250 2625 500 3150 750 3500 950

Pressure [MPa] 0 1 2 5 20

HTC amb [mW/(mm2*K)] 20 26.71 41.87 68.39 100

For two experiments with a nominal tool temperature of 400°C and 500°C, the measured and computed temperature curves are shown in Figure 5. A temperature rise due to latent heat is observed after removal, since bainite transformation is not completed at this time. The measured temperatures are described well by the simulation. On the other hand, if latent heat is neglected in the simulation, the temperature during air cooling is underestimated.

1000 Experiments

900

AutoForm^plus with Latent Heat AutoForm^plus without Latent Heat

Temperature [°C]

800

coole d

700

heate d

Nominal tool temp. 500° 500°C

600 500 400

Nominal tool temp. 400° 400°C 300

Transfer

Quenching

Removal

Air cooling

200 0

10

20

30

40

50

60

70

80

Time [s] FIGURE 5. Temperature predictions together with the experimental result for a nominal tool temperature of 400°C and 500°C

Figure 6 shows the computed and measured hardness results for a nominal tool temperature of 500°C. The hardness results are plotted over the distance from the intersection line between the hot and cooled tools. From the hardness measurements, the width of the transition region between hard and soft blank can be defined as 15-20 mm. The width of the transition region is captured well by the simulation results taking heat conduction into account. The tools are idealised to have a constant and homogeneous temperature. In reality, there is a temperature gradient between the tools. This can explain why the hardness between s=0 and s=5mm is not captured exactly by the

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simulation. However, the magnitude of the hardness distribution curve is calculated sufficiently in order to evaluate the dimension of the transition zone. 600

coole d

550

Measured line 1

heate d s

Measured line 2 Measured line 3

Hardness [HV10]

500

AutoForm^plus with Conductivity AutoForm^plus without Conductivity

450 400 350 300 250 200 -10

-5

0

5

10

15

20

25

30

Distance s [mm]

FIGURE 6. Hardness predictions with and without heat conduction in the blank together with the experimental results for a tool temperature of 500°C

Figure 7 shows computed and measured tensile strength results in a measurement series where the temperature of the heated tool was varied keeping all other process parameters constant. The tensile test specimens were taken from the soft zone of the blank, far away from the transition zone. 1550

coole d

Tensile strength [MPa]

1400

heate d

1250 1100 950 800

Experiments AutoForm^plus with Latent Heat

650 500 150

AutoForm^plus without Latent Heat

200

250

300

350

400

450

500

550

600

Tool temperature [°C]

FIGURE 7. Tensile strength predictions with and without latent heat effect together with the experimental results for different tool temperatures

Tensile strength as a product of phase transformation is predicted successfully by the simulations with latent heat for all temperatures. Neglecting the effect of latent heat leads to deviation from the experiments in the case of warmer tool temperatures, i.e. more than 350°C, where bainitic transformation dominates.

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TAILORED TEMPERING RESULTS OF A B-PILLAR Basic knowledge concerning the process window and resulting material properties as a function of the relevant process parameters was developed by tests using a planar and partially tempered tool. The simulation model of AutoForm-ThermoSolverplus was calibrated and validated by measurements from these tests described in the previous section. With the objective of transferring and applying the findings to the production of a real partially press-hardened body-in-white component and in order to review the quality of simulation results under these conditions, a tool for a B-pillar was manufactured by Daimler and tests were performed in Sindelfingen, Germany. The B-pillar comprises three zones of different material characteristics. For the upper area, a tensile strength of about 1500 MPa and a corresponding martensitic microstructure was targeted. The middle area of the part was to have a tensile strength of 850 MPa, and the lower section a tensile strength of 750 MPa combined with an elongation at fracture of about 15%. Different mechanical properties can be adjusted and controlled by the cooling rate and the tool temperature. The process parameters and the part geometry are shown in Figure 7. Using this configuration, a small batch of partially press-hardened parts was produced. In order to validate the simulation results with a focus on the mechanical properties, several A30 tensile specimens (Z1, Z2, Z3 and Z4) were extracted from different areas of the manufactured components.

Tool

Process parameters 500°C 450°C 25°C

Z1

Part 750 MPa 850 MPa 1500 MPa

Z2 Z4

Z3

material: oven temperature: oven time: tool temperature: transfer time: thickness: contact force: closure time:

22MnB5 + Zn 910 °C 320 s variable 9s 1.50 mm 8000 kN 25 s

FIGURE 8. Tailored tempering of a B-pillar with process parameters

The model implemented was applied to simulate the production of a partially hardened B-pillar as described before. The final material properties with respect to tensile strength and elongation at fracture obtained from simulation with and without latent heat effects and corresponding test are compared in Table 3. TABLE 3. Simulation with and without latent heat and test results of final material properties Measured (n=3)

Z1 Z2 Z3 Z4

759±1.4 838±10.0 1441±24.0 1501±6.4

Tensile strength [MPa] AutoFormAutoFormThermoSolverplus ThermoSolverplus with latent heat without latent heat 767 830 869 906 1430 1450 1430 1450

Measured (n=3)

13.0±0.2 10.1±1.2 7.4±0.5 8.0±0.1

Elongation at fracture A30 [%] AutoFormAutoFormThermoSolverplus ThermoSolverplus with latent heat without latent heat 14.1 11.8 10.3 9.0 6.3 6.3 6.3 6.3

By comparing measured and calculated data for tensile strength and elongation at fracture with and without consideration of latent heat, a statement can be established: The disregard of latent heat leads to raised strength values and in the same context to an elongation at fracture that is too low. Under this condition, the elevated specimen’s cooling rate can be named as the reason for this effect. Simulation with latent heat improves the quality of the calculated results significantly. Both tensile strength and elongation at fracture can be calculated quite accurately in AutoForm-ThermoSolverplus. Thus, the consideration of latent heat is indispensible for proper

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simulation of the considered tailored tempering process, where the bainite transformation is used to define the mechanical properties of the produced component.

CONCLUSION In order to capture the relevant phenomena in tailored tempering, a coupled thermo-mechanical-metallurgical description is employed in AutoForm-ThermoSolverplus. The implementation was validated via hybrid studies of experiments and simulations. The predicted final part properties of a B-pillar show satisfactory agreement with results from tensile tests. The additional CPU time for the combined thermo-mechanical-metallurgical model is about 5% for this example. Concerning industrial application, the further development of hot-forming processes is a very important step with regard to process and product quality. On the one hand, this development enables engineers to perform a profound simulation of tailored tempering. On the other hand, the presented feature contributes to a better understanding of conventional hot-forming processes, where the temperature history of produced parts and the related phase transformation as well as resulting mechanical properties are aspects of paramount importance for feasibility studies and for process design.

REFERENCES 1. Merklein, M.; Lechler, J.: Investigation of the thermo-mechanical properties of hot stamping steels. In: Journal of Materials Processing Technology 177 (2006), pp. 452–455 2. Feuser, P.; Schweiker, T.: Tailored Tempered Parts – Presshärtbauteile mit maßgeschneiderten Eigenschaften. In: Merklein, M.: Tagungsband zum 4. Erlanger Workshop Warmblechumformung, Erlangen, Germany (2009) 3. Karbasian, H.; Tekkaya, A.E.: A review on hot stamping. In: Journal of Materials Processing Technology 210 (2010), pp. 2103–2118 4. Sester, M.; Krasovskyy, A.; Kubli, W.: Material data for advanced yield surface and hardening models in AutoForm. In: Proceedings of International Deep Drawing Research Group (2009), pp. 207–218 5. Hoff, C.: Untersuchung der Prozesseinflussgrößen beim Presshärten des höchstfesten Vergütungsstahls 22MnB5. Dissertation, LFT, University of Erlangen-Nuremberg (2007) 6. Lechler, J.: Beschreibung und Modellierung des Werkstoffverhaltens von presshärtbaren Bor-Manganstählen. Dissertation, LFT, University of Erlangen-Nuremberg (2009) 7. Gesellschaft für metallurgische Systeme mbH (GMS) 8. Oberpriller, B.; Burkhardt, L.; Griesbach, B.: Benchmark 3 - Continuous Press Hardening. In: Proceedings of Numisheet (2008) 9. Ertürk, S.; Sester, M.; Selig, M.; Feuser, P.; Roll, K.: Simulation of Tailored Tempering Using AutoFormplus. In: Proceedings of International Conference on Hot Sheet Metal of High-Performance Steel (2011)

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Simulation of the Press Hardening Process and Prediction of the Final Mechanical Material Properties Bernd Hochholdingera, Pavel Horaa, Hannes Grassb and Arnulf Lippb a

Institute of Virtual Manufacturing, ETH Zurich, Tannenstrasse 3, 8092 Zurich, Switzerland b BMW Group, 80788 Munich, Germany

Abstract. Press hardening is a well-established production process in the automotive industry today. The actual trend of this process technology points towards the manufacturing of parts with tailored properties. Since the knowledge of the mechanical properties of a structural part after forming and quenching is essential for the evaluation of for example the crash performance, an accurate as possible virtual assessment of the production process is more than ever necessary. In order to achieve this, the definition of reliable input parameters and boundary conditions for the thermo-mechanically coupled simulation of the process steps is required. One of the most important input parameters, especially regarding the final properties of the quenched material, is the contact heat transfer coefficient (IHTC). The CHTC depends on the effective pressure or the gap distance between part and tool. The CHTC at different contact pressures and gap distances is determined through inverse parameter identification. Furthermore a simulation strategy for the subsequent steps of the press hardening process as well as adequate modeling approaches for part and tools are discussed. For the prediction of the yield curves of the material after press hardening a phenomenological model is presented. This model requires the knowledge of the microstructure within the part. By post processing the nodal temperature history with a CCT diagram the quantitative distribution of the phase fractions martensite, bainite, ferrite and pearlite after press hardening is determined. The model itself is based on a Hockett-Sherby approach with the Hockett-Sherby parameters being defined in function of the phase fractions and a characteristic cooling rate. Keywords: press hardening, thermo-mechanically coupled simulation, 22MnB5, final part properties PACS: 81.40.Gh, 81.40.Cd, 02.70.Bf, 81.40.-z

INTRODUCTION Today the press hardening process is well established for the manufacturing of body-in-white parts in the automotive industry. The number of parts in a new car generation that are manufactured by press hardening is still increasing. In general two variants of the press hardening process have to be distinguished. In the direct press hardening process (fig. 1a) the part is formed and quenched in one process step whereas in the indirect process (fig. 1b) the part is first classically cold-formed and afterwards heated and quenched in a separate operation.

(a) austenitization

forming and quenching

timming

treatment of surface

(b) cold-forming

trimming

austenitization

quenching

treatment of surface

FIGURE 1. Principal process steps for the direct press hardening process (a) and the indirect press hardening process (b) as proposed by the steel company voestalpine AG.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 618-625 (2011); doi: 10.1063/1.3623665 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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The steel that is typically used for press hardening is the boron alloyed steel 22MnB5. It is available from different steel suppliers that offer 22MnB5 with different coatings. Depending on the actual chemical composition of the coating, the supplied 22MnB5 is either suited for the direct or the indirect process variant. With the upcoming of press hardening in the automotive industry the focus lay on the production of structural parts with ultra-high strength. Recently the press hardening process is enhanced to produce parts with tailored properties. Still the generation of ultra-high strength properties is the principal objective of press hardening but at the same time it is desired to get areas within a part that have lower strength but increased ductility. There are several ways to achieve this. A standard approach to realize different mechanical properties within a part is to use tailored blanks by joining two or sheets made of different steel. Another way to increase the ductility is to anneal the part after quenching as for example discussed by Laumann [1]. Instead of using different materials or annealing the part, the hardening process itself can be modified to generate different mechanical properties. In the so called tailored tempering process, as for example discussed by Lenze et al. [2], the tools are partially heated. This results in a locally reduced cooling rate. Instead of a solely martensitic microstructure, as it is the case for cooling rates faster than approx. 30 K/s, a mixture of ferrite, pearlite, bainite and martensite forms that has a lower strength but higher ductility than pure martensite. Another way to locally reduce the cooling rate is to insert grooves in the tool surface. Depending on the size and depth of the groove the heat flux from the hot part to the cold tools at the location of the groove is orders of magnitude smaller than in areas where part and tool are in direct contact. By the insertion of grooves the tool surfaces are not continuous anymore and cannot be used for forming a part. Therefore this approach may only be used in the indirect process variant, where the part is formed in a separate operation before the hardening takes place. For the virtual assessment of the press hardening process, thermo-mechanically coupled simulations have to be conducted, which require a variety of input parameters. Depending on which process variant is followed – direct or indirect press hardening – the relevance of the input parameters regarding the process simulation varies. As shown by Burkhardt [3], for the simulation of hot forming as part of the direct press hardening process, the accurate definition of the yield stress depending on strain, strain rate and temperature as well as temperature dependent friction are of major importance for a reliable prediction of the feasibility of the forming step and the geometrical properties, like sheet thickness and draw-in, of the part after forming. An experimental procedure for the determination of the yield stress and a comparison of the applicability of different mathematical models is given in Eriksson [4] and Hochholdinger [5]. Regarding the prediction of the final mechanical properties after the direct or the indirect process, the accurate prediction of the local temperature history within the blank is of major importance. The temperature history depends on various boundary conditions like thermal convection and radiation to the environment, but mainly on the timedependent contact situation between blank and tool. Especially for the generation of tailored properties by the insertion of grooves in the tool surfaces, the correct modeling of the interfacial heat flux is mandatory. Therefore the focus of this work regarding the determination of input parameters is set to the determination of the contact heat transfer coefficient as described in the following chapter.

DETERMINATION OF THE CONTACT HEAT TRANSFER COEFFICIENT Depending on the actual contact situation during the hardening process the effective heat flux from the blank/part to the tools may vary by several orders of magnitude. The input parameter that determines the heat flux due to contact is the contact heat transfer coefficient (CHTC) h. In general the following two contact situations have to be distinguished: 1. Part and tool are in direct contact. The CHTC depends on the effective area in contact. Since the effort to model the effective contact area accurately with finite elements would be much too high, the CHTC is defined as a function of contact pressure h(p). It is obvious that for high contact pressures the effective contact area is larger than for small pressure values. 2. Part and tool are not in direct contact, but there is a small gap between the contact partners. Since part and tool are close to each other, there is heat transfer due to radiation and convection of the fluid in the gap. In this case CHTC is a function of the gap distance h(lgap). With the gap between part and blank getting larger, the heat flux due to the interaction between the contact partners is decreasing. For gaps with lgap>lgap,max the heat loss due to convection and radiation to the environment is dominant and the heat flux between the contact partners can be neglected.

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Experimental Setup In order to determine the CHTC for the contact situations described above, simple experiments at the Chair of Manufacturing Technology of the University of Erlangen-Nuremberg have been conducted. The principle experimental setup is shown in fig. 2a.

(a)

upper tool blank

(b)

lower tool

FIGURE 2. Experimental setup for determining the CHTC (a) and experimental and simulated temperature over time curves for an applied contact pressure of 30 MPa (b).

After heating up the blank it is quenched between upper and lower tool. In the closed configuration the closing force of the press can be adjusted such that the blank is clamped with a predefined pressure. In the experiments pressure values of 0, 5, 10, 20, 30 and 40 MPa have been used. Alternatively the blank can be placed on adjustable pins on the lower tool and the upper tool can be positioned at a specified distance away from the upper side of the blank. Like that the second contact configuration with a small gap between blank and tools is realized. In the experiments gap distances were set to 0.1, 0.3, 0.5, 1.0 and 2.0 mm. The temperature is measured through thermocouples in the middle of the blank thickness and in the upper and lower tool 1 mm underneath the tool surfaces. In order to ensure the same initial temperature of the tools when conducting a series of consecutive experiments, upper and lower tool can be water-cooled. Further details on the experimental setup can be found in Hoff [6].

Inverse Identification of the CHTC For the determination of the CHTC based on the measured temperature over time curves different approaches can be employed. A fairly simple, analytical approach is to use Newton’s law of cooling as e.g. done by Lechler [7]. When using Newton’s law of cooling it is assumed that the blank has a constant temperature over its thickness and that the tool temperature is constant over time. Especially for the test configurations with blank and tool being in direct contact, the measured tool temperatures showed that the assumption of constant tool temperatures is not appropriate (see fig. 2b). Instead of trying to employ an analytical formula, an inverse approach for the identification of the CHTC is chosen. For this purpose a simple simulation model of the experiment is set up, where the measured temperatures in the upper and lower tool are applied as boundary conditions at the nodes 1 mm underneath the tool surfaces. For each of the tested configurations with a specific pressure or gap distance applied, an optimization run was conducted. The objective of these optimization runs is defined as the mean square error (MSE) between simulated and measured blank temperature over time. In order to avoid a falsification of the results by the release of latent heat during the cooling process, only the temperature history prior to the occurrence of phase change was considered for the evaluation of the MSE. The effects of phase change could be included in the optimization process by specifying the latent heat and the phase change temperature as additional optimization variables. Since such an approach led to unstable results, solely the CHTC was defined as a variable within the optimization procedure. The dashed curve in fig. 2b shows the resulting blank temperature over time using the “optimal” CHTC for the configuration with applied contact pressure equal to 30 MPa. In fig. 3 the normalized values of the determined CHTCs over gap distance and pressure are shown.

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FIGURE 3. Normalized values of the (CHTC) over gap distance (left) and contact pressure (right).

SIMULATION OF THE PRESS HARDENING PROCESS Depending on which answers and results are expected from the simulation, the model setup as well as the numerical strategy for the virtual assessment might differ significantly. If just the feasibility of the forming phase during the direct press hardening process is of interest, it is in most cases sufficient to simulate gravity, closing and hot forming of the blank without consideration of heat conduction or cooling within the tools. As can be seen from the results submitted for Benchmark 3 of the Numisheet 2008 Conference (Oberpriller [8]) the sheet thickness and the position of local necking after hot forming can be fairly well predicted by the use of finite element models that are quite similar to the ones employed for standard cold-forming simulations. If on the other hand the objective of the simulation is the prediction of the mechanical properties after hardening, the process steps that have to be considered as well as the requirements regarding the FE model are more comprehensive. In the following two chapters these issues are discussed.

Simulation of the Process Steps during Press Hardening In order to be able to predict the properties of the final part correctly all process steps that influence the thermal history of the part, have to be considered in the simulation. Fig. 4 shows the process steps for the indirect press hardening process.

heating

transfer

gravity

closing

quenching

FIGURE 4. Simulated process steps during the indirect press hardening process.

All process steps are simulated using a weak coupling of the mechanical and the thermal problem. Like that it is possible to use different time integration schemes with different time step sizes for both problems. Table 1 shows the chosen settings for the thermo-mechanical simulation of the process steps. The starting point for the coupled simulation is the trimmed part after cold-forming. During heating from room temperature up to an austenitization temperature of approx. 900 °C the volume of the part increases due to thermal expansion. The correct definition of the temperature dependent coefficient of thermal expansion (CTE) is essential to get the right dimensions after heating. The FE code LS-DYNA® that is used for the simulations utilizes the tangent CTE in contrast to the secant CTE that is e.g. used by the FE program Abaqus®. After heating the part and holding it at a constant temperature for several minutes, it is assumed that residual stresses and the accumulated plastic strain that evolved during cold-forming are cancelled out. During transfer from the oven to the press the part

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cools due to radiation and convection to the surroundings. The temperature drop depends on the effective duration of transfer, usually 3-8 seconds, and the values provided for the convection heat transfer coefficient and the emissivity of the blank surface. Reasonable values for these input parameters are stated in Shapiro [9]. In the subsequent gravity simulation the part is put on the lower tool. In this step there is a heat flux on the lower side of the part due to local contact with the cold tool and on the upper side due to convection and radiation. For the closing step a separate simulation is setup, since a different time integration scheme is used for the solution of the mechanical problem (see table 1). The experience has shown that the use of an explicit time integration scheme is more robust in handling the changing contact situations than the dynamic implicit Newmark integration. During the quenching step the part is tightly clamped between upper and lower tool. As discussed above, the effective heat flux depends on the local pressure or on the gap distance at the part-to-tool interface. Either the explicit or the implicit time integration may be used. If the explicit approach is chosen the application of mass and time scaling is recommended to reduce the computation time. A detailed description of the recommended settings for the explicit time integration scheme can be found in Lorenz [10]. To reduce the cycle times in serial production, the duration that the tools are held closed should be as short as possible. Accordingly the tools are usually opened when the part is still well above room temperature. In order to get the correct geometry at room temperature, it might therefore be necessary to conduct an additional simulation of opening the tools and cooling down of the part. The more or less free thermal contraction during that cooling phase on air might lead to geometrical distortion of the part. TABLE 1. Applied time integration schemes for the thermo-mechanically coupled simulation of the separate process steps. Process Step Mechanical Problem Thermal Problem heating static implicit implicit transfer static implicit implicit gravity dynamic implicit implicit closing explicit implicit quenching explicit or dynamic implicit implicit

FE-Model for the Simulation Press Hardening In general there are several approaches to model blank/part and the tools in a forming simulation. In standard cold-forming simulations of deep drawing processes, the tools are usually modeled as rigid contact surfaces providing the reference geometry for the part to be formed. For the simulation of press hardening, as discussed in this paper, it is as well assumed that the tools do not deform and therefore can be considered as rigid for the mechanical part of the problem. For the thermal problem the assumption of tools having a constant temperature over time is in most cases not adequate. When just the hot forming phase of the direct process has to be evaluated then the error of “thermally rigid” tools is probably acceptable as shown by Lorenz [10]. However if the final mechanical properties of the part are to be predicted, which requires an accurate simulation of quenching and cooling, then the tools have to be modeled with continuum or at least boundary elements. Only if heat conduction within the tools is considered in the simulation, the time dependent distribution of the tool surface temperatures can be captured, which is a prerequisite for the correct evaluation of the heat flux. TABLE 2. Recommended modeling of tools and blank for the thermo-mechanically coupled simulation of press hardening. Process Step Mechanical Problem Thermal Problem modeling of the tools rigid: no stresses, no strains thermal tetra- or hexahedron modeling of the blank/part elasto-viscoplastic shell 12 node thick thermal shell

For the structural (mechanical) problem, the blank/part is modeled with the same fully integrated shell elements that are usually employed for the simulation of cold-forming. Of course the material model that is used for the simulation of press hardening has to be capable of considering the temperature dependency of the mechanical input parameters, like e.g. Youngs modulus E(T) and yield stress Vy(H, dH/dt, T). For the thermal part the blank is modeled with a 12 node thick thermal shell element (red nodes in fig. 5b) as proposed by Bergman and Oldenburg [11]. The shell element is defined by the 4 nodes of the mechanical shell (black nodes in fig. 5b). The nodes defining the upper and lower surface of the thick shell are generated internally by LS-DYNA. The shell uses linear shape functions in plane and quadratic ones in the thickness direction. This offers several advantages. Like that it is possible to apply and model different boundary conditions on the upper and lower side of the shell, as for example during the gravity step (see fig. 4), and to accurately capture the resulting temperature gradient through the thickness.

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(a)

(b)

FIGURE 5. Solid model of the tools (a) and thick thermal shell element used for modeling the blank (b).

PREDICTION OF THE FINAL MECHANICAL MATERIAL PROPERTIES For the prediction of the mechanical properties after press hardening a post processing procedure was developed that automatically evaluates the temperature history of all nodes in the quenched part over all process steps after heating. The CCT diagram shown in figure 6 is used to evaluate the phase fractions that result after hardening at the nodes of the blank. Fast cooling rates result in a purely martensitic microstructure with a yield stress of more than 1500 MPa and Vickers hardness above 480. For intermediate cooling rates the austenite changes to a mixture of martensite, bainite, ferrite and pearlite. Rather slow cooling rates result in a microstructure that consist of ferrite and pearlite which is quite similar to the unquenched state of the 22MnB5.

FIGURE 6. CCT diagram of 22MnB5 from Naderi [12] for an austenitization temperature of 900 °C.

For the prediction of the yield curves after hardening a simple model based on the well-known Hockett-Sherby yield curve approximation is developed.

V y H pl Ax , dT  ( Ax , dT  Bx , dT ) exp M x , dT H plN

(1)

Where Ax,dT is the saturation stress, Bx,dT is the initial yield stress, Mx,dT is the saturation rate and N the hardening exponent. As indicated by the subscripts, the Hockett-Sherby parameters A, B and M are defined as functions of the phase fractions x and a characteristic cooling rate dT/dt at 700 °C (973 Kelvin), while the same hardening exponent N, which is unequal to one, is used for all configurations, independently of the specific microstructure or cooling rate. As can be seen from equation (2) each of the parameters Ax,dT, Bx,dT and Mx,dT is defined as linear combination of the phase fractions martensite xmart, bainite xbain, ferrite and pealite xferr-perl and the natural logarithm of the cooling rate at 973 K. As inspiration for this approach served the mixture rule developed by Maynier et al. [13], which uses the chemical composition of the steel to predict the Vickers hardness. The coefficients ax, bx and mx in equation (2) are determined by multi-linear regression. For this purpose tension tests of specimens with different cooling rates have been conducted. The resulting stress-strain curves are fitted employing the mentioned Hockett-Sherby approximation. The phase fractions for each tension test specimen are evaluated with the CCT diagram shown in fig. 6.

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Ax , dT Bx , dT M x , dT

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Based on this model the following post processing is developed, that allows predicting the yield curves in the part after press hardening: 1. Extract temperature history from all process steps starting with the transfer simulation until the end of the hardening process. 2. Evaluate the resulting microstructure with the CCT diagram shown in fig. 6. 3. Employ the developed model (eq. (2)) to calculate an individual yield curve for each finite element in the simulation model. 4. Sort elements with similar yield curves in different groups. For this purpose the plastic potential of each yield curve is calculated. Calculate representative yield curve for each group of elements. 5. Output of the post-processed part including the determined yield curves. Fig. 7 shows two small sample press hardening setups and the corresponding results using the described post processing procedure. Fig. 7(a) shows a standard hardening tool. In fig. 7(b) the part after hardening is shown. The different colors represent different yield curves that are shown in fig. 7(c). The figures 7(d) to (f) show tools, the part after quenching and normalized yield curves for a setup, where grooves have been inserted in the surfaces of upper and lower tool. At the location of the grooves the cooling rates are slower than in areas where the tools and the part are in full contact. The yield curves at the location of the grooves are accordingly lower than in areas with higher cooling rates.

(a)

(b)

(c)

(d)

(e)

(f)

FIGURE 7. Figure (a) shows the tools, (b) the hardened part with areas having different properties and (c) the corresponding normalized yield curves for standard press hardening setup. In figures (d), (e) and (f) the tools, the quenched part and the corresponding normalized yield curves for a setup, where grooves have been inserted in the tool surfaces are shown.

SUMMARY AND CONCLUSIONS The intention of this work was to discuss the most relevant issues that are necessary to predict the yield curves of a part after press hardening. The preliminary for an accurate simulation is the knowledge of the input parameters for the thermo-mechanically simulation. The basic idea of press hardening is to quench a sheet metal blank or part made of hardenable steel, as for example the boron alloyed steel 22MnB5, by bringing it into contact with cold tools. The simulation of this process requires the definition of a large variety of input parameters. When the objective of the simulation is to predict the final part properties the by far most important parameter is the contact heat transfer coefficient. An inverse approach was chosen to determine the pressure and gap dependent CHTC from the experimental temperature over time curves. Furthermore a strategy for the simulation of the subsequent steps that are

624

involved in the press hardening process was presented. Since the mechanical and the thermal solution are just weakly coupled, different time integrations schemes for the mechanical and the thermal part may be used for solving the initial boundary value problem. In order to get the correct spatial and time-dependent temperature distribution the tools are modeled with volume elements. For the part a thick thermal shell is employed that allows capturing the temperature gradient over the sheet thickness and to apply different boundary conditions on top and bottom surface of the blank. For the prediction of the yield curves of the final part a phenomenological model was developed. This model requires the input of the microstructure in form of the phase fractions martensite, bainite, ferrite and pearlite. The phase fractions are determined by employing a CCT diagram from literature. The model itself is based on a Hockett-Sherby approach. The Hockett-Sherby parameters are defined in function of the phase fractions and a characteristic cooling rate. The evaluation and output is automatically done during post processing of the subsequent simulations. In order to determine the parameters for the presented model tension or compression tests of specimen that have been quenched with different cooling rates have to be conducted. As an alternative to the determination of the phase fractions by using the CCT diagram the material model developed by Åkerström and Oldenburg [14] could be used. This model employs transformation kinetics based on Kirkaldy’s rate equation to predict the phase fractions after quenching. Disadvantages of this model are the required input parameters, like the chemical composition, the austenite grain size and the activation energies of the diffusion driven transformation reactions, and the fact that the time-scaling cannot be applied to the simulation.

ACKNOWLEDGEMENTS The kind support of this work by the BMW Group is gratefully acknowledged. Also many thanks to Dr. A. Shapiro from LSTC for the implementation of many new features into the FE software LS-DYNA without which the virtual assessment of the press hardening process would not have been possible.

REFERENCES 1. T. Laumann, “Qualitative und quantitavie Bewertung der Crashtauglichkeit von höchstfesten Stählen”, Doctoral Thesis, , University of Erlangen-Nuremberg, 2009. 2. F.-J. Lenze, J. Banik, S. Sikora and T. Gerber, “Further development of manganese boron steels for the lightweight design of body in white structures”, IDDRG 2010 Conference Proceedings, edited by R. Kolleck, Institute of Tools and Forming, Graz University of Technology, 2010, pp. 17-26. 3. L. Burkhardt, “Eine Methodik zur virtuellen Beherrschung thermo-mechanischer Produktionsprozesse bei der Karosserieherstellung”, Doctoral Thesis, ETH Zurich, 2009. 4. M.C.F. Eriksson, “Modelling of Forming and Quenching of Ultra High Strength Steel Components for Vehicle Structures”, Doctoral Thesis, Luela University of Technology, 2002 5. B. Hochholdinger, H. Grass, A. Lipp, A. Wahlen and P. Hora, “Determination of Flow Curves by Stack Compression Tests and Inverse Analysis for the Simulation of Press Hardening”, Proceedings of the 7th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, Part A, edited by P. Hora, Institute of Virtual Manufacturing, ETH Zurich, 2008, pp. 633-639. 6. C. Hoff, “Untersuchung der Prozesseinflussgrössen beim Presshärten des höchstfesten Vergütungsstahls 22MnB5”, Doctoral Thesis, University of Erlangen-Nuremberg, 2007. 7. J. Lechler, “Beschreibung und Modellierung des Werkstoffverhaltens von presshärtbaren Bor-Manganstählen”, Doctoral Thesis, University of Erlangen-Nuremberg, 2008. 8. B. Oberpriller, L. Burkhardt and B. Griesbach, “Benchmark 3 – Continuous Press Hardening – Part B: Benchmark Analysis”, Proceedings of the 7th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, Part B, edited by P. Hora et al., Institute of Virtual Manufacturing, ETH Zurich, 2008, pp. 119-129. 9. A. Shapiro, “Finite Element Modeling of Hot Stamping” in steel research international, Volume 80, Issue 9, Düsseldorf: Verlag Stahleisen, 2009, pp. 658-664. 10. D. Lorenz, “Simulation of Thermo-Mechanical Forming Process with LS-DYNA”, Proceedings of the 9th LS-DYNA Forum, Bamberg, DYNAmore GmbH, 2010, pp. C-I-35-42. 11. G. Bergmann and M. Oldenburg, “A finite element model for thermo-mechanical analysis of sheet metal forming” in International Journal of Numer. Meth. Engng, Volume 59, Issue 9, 2004, pp. 1167-1186. 12. M. Naderi, “Hot Stamping of Ultra High Strength Steels”, Doctoral Thesis, RWTH Aachen University, 2007. 13. P. Maynier, B. Jungmann and J. Dollet, “Creusot-Loire system for the prediction of the mechanical properties of low alloy steel products” in Hardenability concepts with application to steels, edited by D.V. Doane and J.S. Kirkaldy, AIME, New York, 1978, pp. 518-544. 14. P. Åkerström and M. Oldenburg, “Austenite decomposition during press hardening of a boron steel – Computer simulation and test” in Journal of Materials Processing Technology, 174, 2006, pp. 399-406.

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633

Effect of Strain Rate and Temperature Gradient on Warm Formability of Aluminum Alloy Sheet R. Bagheriasla , K. Ghavam and M.J.Worswick University of Waterloo, Mechanical and Mechatronics Engineering Department 200 University Ave W, Waterloo ON Canada N2L 3G1 a [email protected] Abstract. The effect of temperature gradient and forming speed on warm formability of aluminum alloy sheet has been studied using a coupled thermal mechanical finite element model of cup deep drawing. A user-defined material model was developed using the Bergstrom temperature and strain-rate dependant hardening model and Barlat YLD2000 anisotropic yield surface, which was implemented within LS-DYNA. The stress-strain curves for AA3003 at elevated temperatures and different strain rates were used to fit the Bergstrom hardening parameters. The anisotropy parameters were considered to be non-temperature dependant. The numerical model was validated against experiments from previous work and was found to accurately predict punch force for warm deep drawing. Increases in forming speed are shown to have a negative effect on formability. It is concluded that non-isothermal warm forming can be used to improve the formability of aluminum alloy sheet. Keywords: Warm forming, Strain rate sensitivity, Formability PACS: Forming, 81.20.Hy

INTRODUCTION Aluminum alloys have high strength-to-weight ratio and good corrosion resistance and are considered good replacements for conventional steels in automotive applications. The use of aluminum alloys can improve the fuel efficiency of vehicles due to its lighter weight, but there exists several technical difficulties that limit its widespread application in vehicles. Some of these limitations are lower formability and higher springback compared to steel sheet as well as higher forming process costs. Warm forming is a promising processing route that has been considered recently to improve the formability of aluminum alloy sheet in which sheet forming is performed at an elevated forming temperature that is lower than the re-crystallization temperature [1]. In comparison to traditional stamping of aluminum alloys, warm forming has some important advantages, the most important being that the forming limit strains at elevated temperatures are increased significantly; therefore, more complex geometries can be achieved by this method and/or fewer die progressions are necessary to form a given geometry. Warm forming was studied for many years, e.g. in the 1970s and 1980s by Shehata et al. [2]. Schmoeckel et al. [3] studied the drawability of 5XXX series alloys at the elevated temperatures. Li and Gosh [4] have shown that warm forming can considerably improve the formability of aluminum alloy sheet. Van Den Boogaard and Huetink [5] and McKinley et al. [6] studied the cylindrical cup deep drawing at different gradient of temperatures and observed that the formability of Al–Mg alloy sheet can be improved by increasing the temperature in some regions of the sheet and cooling other regions. The challenge for process design is to investigate the complex interaction between mechanical and thermal effects on formability. Numerical simulation using finite element methods (FEM) is an economical approach to develop optimized forming processes and to reduce the number of experimental trials. In order to accurately model the warm forming of aluminum alloy sheet by numerical simulation, proper constitutive equations must be used. This requires an accurate hardening law, accounting for thermal softening and strain rate-sensitivity, and an appropriate anisotropic yield function. The reported constitutive fitting exercises [6,7] have identified the Bergstrom [5] or NES [8] models as appropriate hardening models for use in numerical simulation of warm forming. In addition to capturing material anisotropy in plastic region, it is necessary to also incorporate an advanced material yield surface to capture the crystallographic nature of yielding in FCC aluminum alloys. In particular this includes capturing material R-values The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 634-641 (2011); doi: 10.1063/1.3623667 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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which control thinning in textured sheet materials. The Barlat et al. [9] YLD2000 yield surface has been shown to accurately describe the anisotropic material behavior of aluminum alloy sheet [10]. Many efforts have reported the application of the NES/Bergstrom model and also the Barlat-2000 yield surface separately; however, there still exist a need to implement the Barlat YLD2000 yield surface in conjunction with the Bergstrom hardening model to accurately model aluminum alloy sheet during warm forming. The present research considers AA3003 sheet, which is an aluminum-manganese alloys. These alloys typically have manganese additions ranging from 0.05 to 1.8%, are of moderate strength (UTS of 110 to 280 MPa) and exhibit good corrosion resistance, good formability and are well suited for use at elevated temperatures. One of the first uses of these alloys was in fabrication of pots and pans and they are the major component today for heat exchangers in automobiles and power plants. Their moderate strength, however, often precludes their consideration for structural applications.

NUMERICAL SIMULATIONS Material Model Yield criterion Barlat’s YLD2000 anisotropic yield criterion [9] is expressed for plane stress state as ‫ ׎‬ൌ ȁଵᇱ െ ଶᇱ ȁୟ ൅ ȁʹଶᇱᇱ ൅ ଵᇱᇱ ȁୟ ൅ ȁʹଵᇱᇱ ൅ ଶᇱᇱ ȁୟ ൌ ʹɐ ഥୟ (1) ᇱ ᇱᇱ where ଵǡଶ and ଵǡଶ are the principal values of the linear transformations of the stress tensor;  ᇱ and  ᇱᇱ which are defined as ܺ ᇱ ൌ ‫ ܥ‬ᇱ ܵ ൌ ‫ ܥ‬ᇱ ܶߪ ൌ ‫ܮ‬ᇱ ߪ , ܺ ᇱᇱ ൌ ‫ ܥ‬ᇱᇱ ܵ ൌ ‫ ܥ‬ᇱᇱ ܶߪ ൌ ‫ܮ‬ᇱᇱ ߪ (2) Further, L' and L'' can be expressed as functions of eight independent coefficients, ߙ௜ Ǥ These independent coefficients are all that is needed to describe the anisotropy behaviour of material. For the material model to account for changes in temperature, the anisotropy coefficients must be expressed as functions of temperature, i.e.‫ ׎‬ൌ ߪതሺߪǡ ܶሻǡ though the current research considers the anisotropy coefficients to be temperature independent. Flow stress A Bergstrom hardening rule [5] was used to represent the flow stress of the material. The flow stress represents the size of the yield function during deformation. The temperature and rate dependant hardening model is written as: ߪ௙ ൌ ݃ሺܶሻ൫ߪ଴ ൅ ߙ‫ܩ‬௥௘௙ ܾඥߩ൯ (3) The term ݃ሺܶሻ is shear modulus and is expressed as ் (4) ݃ሺܶሻ ൌ ͳ െ ‫ ’š‡ ்ܥ‬ቀെ భቁ ் where ‫ ்ܥ‬and ܶଵ are fitting parameters, ‫ܩ‬௥௘௙ is the reference shear modulus, ߙ is a scaling parameter of order 1, ܾ the Burgers vector (2.857×1010 m), ߪ଴ is the strain rate independent term. The Bergstrom model is a strain and strain rate dependant model. The effects of strain and strain rate are captured in terms of the competitive processes of dislocation annihilation and accumulation which controls ɏ, the dislocation density [5]; ɏ୧ାଵ ൌ ቂ

୙బ :

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(5)

Material parameters Tensile data at different temperatures and strain rates are used to fit the Bergstrom parameters (TABLE 1). FIGURE 1 shows the predicted flow stress curves at different temperatures based on these parameters which agree well with the measured data. In order to calculate the eight D୧ anisotropy parameters, the yield stress in the longitudinal, transverse and diagonal directions (଴ ǡ ଽ଴ ǡ ସହ ) and biaxial yield stress (ୠ ሻǡas well as R values in the longitudinal, transverse and diagonal directions ( ଴ ǡ  ଽ଴ ǡ  ସହ ), are required [10]. Since biaxial data was unavailable, ୠ was set equal to (଴ ൅ ସହ ሻȀʹ [11]. The anisotropy coefficients have been fit for the studied material and are summarized in TABLE 2. The yield surfaces for different temperatures are plotted and shown in FIGURE 2. The heat transfer and thermal properties of the blank were also modelled with an isotropic thermal material model. The material thermal properties for aluminum alloys are shown in TABLE 3.

635

The Bergstrom hardening rule and Barlat yield surface were coded within a user-defined material subroutine or UMAT and linked with the LS-DYNA finite element code. TABLE (1) Bergstrom parameters ɐ଴

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198.62

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TABLE (2) Anisotropy coefficients for Barlat’s YLD2000 હ૜  હ૝ હ૞ હ૟

ͲǤͻͶͲͻͻ

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ͳǤͲʹͷͳͳͻ

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હૠ  ͲǤͻ͹ͷͳͺ

TABLE (3). Thermal properties of steel and aluminum [12] Aluminum 904 Heat Capacityሺ۸Ȁ‫܏ܓ‬۹ሻ 174 Thermal Conductivity ሺࢃȀ࢓૛ ࡷሻ 2710 Density ሺࡷࢍȀ࢓૜ ሻ

T=25qC

T=200qC

હૡ  ͳǤͲ͵Ͳͷ

Steel 450 80 7800

T=250qC

FIGURE 1. Measured stress-strain curves at different temperatures and strain rates and predicted curves using itted parameters .

FIGURE 2. Yield surfaces for AA-3003 at elevated temperatures

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Validation of UMAT against Measured Tensile Data In order to validate the UMAT implementation, a finite element model of a simple tensile test was constructed to confirm that the UMAT constitutive model could reproduce the measured tensile data. A one-half model of the tensile test sample (FIGURE 3) is used due to symmetry. A mesh with 1.0 mm elements was created as shown in FIGURE 3. The total gauge length is 56 mm, the width of the sample is 12 mm and the thickness is 0.5 mm.

FIGURE 3. Tensile test mesh model

Belytschko-Tsay plane stress shell elements are used with 5 integration points through-thickness. The simulations are time scaled by a factor of 1000 to reduce the computation time. All tensile test simulations are isothermal. The nodes of the middle of the sample (symmetry line) are constrained in the tension direction. Velocity boundary conditions corresponding equal to nominal strain rates of ͹ ൈ ͳͲିଶ ǡ ͹ ൈ ͳͲିଷ ǡ ƒ†͹ ൈ ͳͲିସ ǡare applied to the other end. FIGURE 4 compares the predicted tensile stress-strain response with measured data at 25, 200 and 250qC at different strain rates. It can be seen that the model predictions at different temperatures are quite accurate. The model underestimates the stress values at strains lower than 10%. The bottom right plot compares the material response for different strain rates at 250qC. It can be seen that the material exhibits a softer behavior at lower strain rates.

FIGURE 4. Comparison of numerical results with measured engineering stress-strain curves clockwise from top left at 0.07, 0.007 and 0.007 1/sec strain rate and different strain rates at 250qC

637

Deep Drawing at Elevated Temperatures Finite element mesh Figure 5 shows the meshes used to model the tooling elements and blank. Only one-quarter of the geometry was meshed due to the symmetry to reduce computation time. The tools were modeled with four node rigid quadrilateral elements. A uniform mesh consisting of 1.0mm by 1.0mm elements was adopted. However, a coarser mesh was used for the components of tooling which are modeled as rigid bodies. The blanks were modeled using BelytschkoTsay elements with 9 through-thickness integration points. Thermal contact The tooling was held at a constant temperature throughout the simulations. The blank temperature was determined by heat transfer from the tooling. The initial blank temperature was set at room temperature. Thermal contact is one of the least published aspects of warm forming simulations. The most commonly used thermal conductance for contact between tool steel and aluminum sheet is 1400 Ȁଶ  as published by [13].

FIGURE 5. Meshes used to model the tooling and blank

Simulations Simulations were solved using the developed UMAT within LS-971 version of the explicit dynamic finite element code LS-DYNA which employs a central difference method of dynamic explicit time integration. For coupled thermal mechanical simulations, fully implicit thermal time steps (backwards difference) are performed between mechanical time steps. Explicit simulations require a small time step, which results in computationally expensive simulations. To reduce computation time, all simulations are time-scaled by a factor of 1000. The inertial force, the force required to accelerate the blank to forming speeds, is kept below 0.1% of the total forming force to eliminate the inertial effects due to time-scaling. Tooling velocities are increased by a factor of 1000; the thermal conductivity is also increased by a time-scaling factor and, in the material model, the strain rate is divided by 1000 to correspond to the correct operative strain rate during the forming process. The die is fixed in all directions while both the blank holder and punch are free to travel along the Z-axis. The blank holder is closed first, and then the punch descends. The punch is operated under displacement control and the blank holder is under load control. A constant force is applied to the blank holder after an initial ramp-up. The punch contacts initially with the blank after which the punch is held at a fixed position to allow heat transfer to occur between the punch and sheet prior to the forming stroke.

638

Results FIGURE 6 shows measured and predicted punch force versus punch displacement for deep drawing of 203.2 mm (8”) diameter blanks formed at different tooling temperatures with a Dasco Cast lubricant [6]. The experiments and simulations used in the model validation considered a blank holder force of 40 kN, 2 mm/s punch speed and a coefficient of friction of 0.08 for this lubricant. The graph compares the punch force vs. punch displacement for the cases of: (i) tooling and blank at room temperature; (ii) warm punch (100qC) and warm dies (200qC); and, (iii) warm dies (200qC) and cold punch (10qC). The room temperature forming and the warm dies-warm punch forming resulted in premature failure of cup. Warm forming with 200°C dies and 100°C punch reduced the maximum punch load by 36%. Both the room temperature and warm tooling simulations over-predict the maximum punch force which is expected for room temperature based on the Bergstrom model’s over-prediction of stresses at low temperatures [5]. The numerical results for warm dies at 200qC and cold punch at 10qC match the experimental results well for most of the draw though after 30mm of punch displacement, the predicted punch force drop off more rapidly. This could be a result of change in friction in the experiments that was not be matched in the simulations. However, the numerical results predict the overall trends and total draw depth accurately.



FIGURE 6. Punch force versus displacement for 203.2 mm (8”) blanks formed with 40 kN BHF [6] FIGURE 7 compares the predicted shapes of the part formed at room temperature and with the part formed under non-isothermal conditions with 250qC dies and punch cooled to 10qC. Both models considered a blank holder force of 13.8 kN, 8 mm/s punch speed and a coefficient of friction of 0.08. Wrinkles were predicted in the room temperature model while non-isothermal forming resulted in the desired shape.

FIGURE 7. Predicted shape and temperature contours under room temperature (left) and non-isothermal (right) forming

639

FIGURE 8. Punch force vs. punch displacement formed with 15kN binder force with warm dies at 250qC and cold punch at 10qC and punch speed of 8mm/s FIGURE 8 compares the punch force versus punch displacement for blanks drawn under room temperature and non-isothermal warm forming conditions. The forming speed for all simulations is set to 8mm/s and a clamping force of 15kN is applied. The temperature of punch is set to 10qC during the simulations. Two levels of friction are simulated by applying COF=0.05 and COF=0.08. The numerical results showed that warm forming considerably reduces the punch force. Warm forming with 250°C dies reduced the maximum punch load by 35% when applying lower friction and 32% when applying higher friction in simulations. FIGURE 9 illustrates the effect of forming speed on punch force for three levels of punch velocity (1.6, 8 and 40 mm/s). The simulations have been performed for warm forming of 203.2mm (8") blanks while the die and blank holder temperatures are 250°C, punch is kept cold at 10°C, and the COF=0.05 is applied. A blank holder pressure of 15 kN has been applied for all three punch velocities. Increasing the punch velocity causes a significant increase in the maximum punch force which is attributed to the rate sensitivity of the warm flange regime of the cup. Increasing the punch speed from 1.6mm/s to 8 mm/s has smaller effect on calculated punch force.

FIGURE 9. Punch force vs. punch displacement for different punch velocities; BHF=15kN, dies at 250qC and punch at 10qC

640

CONCLUSION The developed UMAT using the Bergstrom hardening equation and Barlat's Yld2000 yield surface was found to accurately predict the mechanical behavior of AA3003 and reproduce the measured tensile data. Coupled thermomechanical finite element models are found to be able to predict the punch force for warm deep drawing accurately. Also friction has an important effect on the thinning. It has been shown that applying a gradient of temperature between the punch and the die improves formability. The forming speed has an important effect on punch force, although the effect is negligible when varying the forming speed at low ranges.

ACKNOWLEDGMENTS Financial support for this research from Dana Canada Corporation (Long Manufacturing), the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Ontario Research Fund is gratefully acknowledged.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Tebbe P.A. and Kridli G.T., Warm forming of aluminum alloys: an overview and future directions, Int. J. Mater. Prod. Technol., 21, 24–40, 2004. Shehata F., Painter M.J., Pearce R., Warm forming of aluminum/magnesium alloy sheet, J. Mech. Work Technol., 2, 279– 291, 1978. Schmoeckel D., Liebler B.C., Speck F.D., . Grundlagen und Modellversuche Temperaturgefu hrterStofffluß beim tiefziehen von Al-blech-realversuche, B¨ander Bleche Rohre, 36, 14–21, 1995. Li D. and Ghosh A., Tensile deformation behavior of aluminum alloys at warm forming temperatures. Mater. Sci. Eng. AStruct., 352, 279–286, 2003. Boogaard A.H.; Huetink J., Simulation of aluminum sheet forming at elevated temperatures, Computer Methods in applied mechanics and engineering, 195, 6691-6709, 2006. Mckinley J., Abedrabbo N., Worswick M.J. and Kozdras, M., Effect of Independent Die and Punch Temperature Control on the Formability of 3003 Aluminum Alloy in Warm Deep Drawing, 581-586, Interlaken-Switzerland, Numisheet 2008. Bagheriasl R., Ghavam K., Worswick M., Stress analysis of a deep drawn cup with plastic anisotropy and rate sensitivity at elevated temperatures, International conference IDDRG, Bilbao, Spain, 2011. Kurukuri S., van den Boogaard A.H., Mirox A., Holmedal B., Warm forming simulation of Al-Mg sheet, J. Mat. Processing Tech., 209, 5636-5645, 2009. Barlat F., Brem J.C., Yoon J.W. Chung K., Dick R.E, Plane stress yield function for aluminum alloy sheets-part 1: theory, Int. J. Plast., 19, 1297-1319, 2003. Yoon J., Barlat F., Dick R.E., Chung K., Kang T.J., Plane stress yield function for aluminum alloy sheets—part II: FE formulation and its implementation, Int. J. Plast., 20, 495-522, 2004. Abedrabbo N., Pourboghrat F., Carsley J., Forming of aluminum alloys at elevated temperatures ˀ Part 1: Material characterization, Int. J. Plast. 22, 314ˀ341, 2006. Hu J., Mechanics of Sheet Metal Forming, Elsevier Butterworth-Heinemann, 2002. Takuda H., Mori K., Masuda I., Abe Y., Matsuo M., Finite element simulation of warm deep drawing of aluminum alloy sheet when accounting for heat conduction, J. Mater. Process. Technol. 120, 412–418, 2002.

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Influences on the Molding in Hydroforming Using Granular Material as a Medium 0DUWLQ*UQHUD 0DULRQ0HUNOHLQD a

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Chair of Manufacturing Technology, University Erlangen-Nuremberg, Egerlandstrasse 13, 91058 Erlangen, Germany

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mm

x-coordinate FIGURE 10,PSURYHPHQWLQPROGLQJDFFRUGLQJWRLQFUHDVLQJSXQFKIRUFHV

651

90



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652

Prediction of Forming Limit Diagram for Seamed Tube Hydroforming Based on Thickness Gradient Criterion Xianfeng Chena, Zhongqin Lina,*, Zhongqi Yua, Xinping Chenb, Shuhui Lia a

Shanghai Key Laboratory of Digital Autobody Engineering, Shanghai Jiao Tong University, Shanghai 200240, China b Shanghai Baoshan Iron & Steel Co., Ltd, Shanghai 201900, China * Corresponding author. E-mail: [email protected] Tel.: +86-21-34206304 Fax: +86-21-34204542 Abstract. This study establishes the forming limit diagram (FLD) for QSTE340 seamed tube hydroforming by finite element method (FEM) simulation. FLD is commonly obtained from experiment, theoretical calculation and FEM simulation. But for tube hydroforming, both of the experimental and theoretical means are restricted in the application due to the equipment costs and the lack of authoritative theoretical knowledge. In this paper, a novel approach of predicting forming limit using thickness gradient criterion (TGC) is presented for seamed tube hydroforming. Firstly, tube bulge tests and uniaxial tensile tests are performed to obtain the stress-strain curve for tube three parts. Then one FE model for a classical tube free hydroforming and another FE model for a novel experimental apparatus by applying the lateral compression force and the internal pressure are constructed. After that, the forming limit strain is calculated based on TGC in the FEM simulation. Good agreement between the simulation and experimental results is indicated. By combining the TGC and FEM, an alternative way of predicting forming limit with enough accuracy and convenience is provided. Keywords:Forming limit diagram (FLD); Seamed tube; Tube hydroforming; Thickness gradient criterion (TGC) PACS: 81.20.Hy

INTRODUCTION In sheet metal forming operations, the sheet can be deformed only to a certain limit that is usually imposed by the onset of localized necking, which eventually leads to fracture. Forming limit diagram (FLD) is a well-known method of describing this limit and predicting the occurrence of necking during sheet metal forming. So it has been widely used to compare the formability of different materials and as the failure criterion for predicting the limit strains in the simulation of sheet metal forming. Similarly, FLD is an important tool for assessing formability of tube hydroforming, which is commonly obtained from experiment, theoretical calculation and finite element method (FEM) simulation. However, the experimental mean is restricted in the application due to the equipment costs and the different experimental methods and subjective measurement approaches. Since 2000, Davis et al. [1] proposed a tooling and experimental apparatus to establish FLD for AA6061 tube based on the free-expansion tube hydroforming. The FLD based on tube free hydroforming is an effective method of proven and authoritative. But in order to obtain the right hand side of FLD, experimental apparatus are required highly and expensive to clamp the two ends of tube when the tube is subjected to the internal pressure. Theoretically, many researchers have developed various theoretical models as an alternative to fill up the deficiency of experiment. Varma et al. [2, 3] analyzed the localized necking in aluminum tubes free hydroforming using the M-K method along with an anisotropic version of the Gurson model. Hwang et al. [4] obtained the FLD via the Swift’s diffused necking criterion and Hill’s localized necking. Kim et al. [5] also used Oyane’s ductile fracture criterion to predict the initiation of necking under combining internal pressure and independent axial feeding. The FLD for tube hydroforming based on these theoretical models are the same as or different from the FLD for sheet metal forming.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 653-660 (2011); doi: 10.1063/1.3623669 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

653

With increasing application of computational techniques, numerical predictions of FLD have become more attractive and the FEM has been selected to simulate the Nakazima and Marciniak tests for sheet metal forming. In analyzing the simulation results for the onset of necking, an appropriate failure criterion is a key to numerical determination of FLD. One of the pioneers in this research area was Brun [6] who analysed the material's thinning in order to determine the onset of necking by simulating the Nakazima test. The method is applicable to the entire range of deformation states of the FLD. Basing on the same test, Geiger and Merklein [7] considered that the gradient of major strain changed rapidly when localized necking occurred. Using the limiting dome height (LDH) test, Narasimhan [8] has predicted the onset of necking by the thickness strain gradient across neighboring regions. A novel failure criterion-thickness gradient criterion (TGC) was proposed. The original idea of TGC is closely related to the hypothesis of thickness inhomogeneity in the M-K model. Some researchers put forward the TGC, which combining the M-K model and Storen-Rice analysis to predict the dome height with good accuracy. Zhang et al. [9] simulated the Marciniak test with the commercially available finite element (FE) program ABAQUS. Based on FEM results, the equivalent plastic strain increment ratio (CRIT3) criterion was chosen to analyze the onset of localized necking. Now numerical predictions of FLD have been widely used in sheet metal forming. However, no many numerical studies on the prediction of FLD for tube hydroforming have been presented. In the field of tube hydroforming, only Kim et al. [10] predicted forming limit for tube hydroforming by means of the FEM combined with Oyane’s ductile fracture criterion based on the Hill’s quadratic plastic potential. Based on the TGC, this work predicts the FLD of seamed tube hydroforming with the FEM simulation of tube hydroforming. The critical thickness gradient in the TGC is determined by the FEM simulation of tube bulge test under plane strain state. The numerical result is compared with the experimental data to show the validity and reliability of the means of the FEM combined with the TGC and the research method adopted.

EXPERIMENTS FOR STRESS-STRAIN CURVE OF SEAMED TUBE There are three test methods for obtaining the stress-strain curve of tube, including uniaxial tensile test, ring tensile test and tube bulge test. In order to perform pertinent numerical simulations of tube hydroforming process, it has been demonstrated that material data obtained from tube bulge test were more suitable than the classical tensile test. Fuchizawa et al. [11] have developed an analytical model for tube bulging. Experimental stress-strain curves obtained by tube bulging and uniaxial tensile tests have been compared and the pertinence of tube bulge test has been confirmed especially for tube manufactured by roll forming and welding. Sokolowski et al. [12] and Strano and Altan [13] also performed tube bulge test which is proposed by Fuchizawa. Koc et al. [14] investigated different method to conduct the tube bulge test depending whether the parameters required for the calculation of the stress– strain relationship (longitudinal and circumferential radius of curvature, bulge height, internal pressure and thickness at the top of the dome) are measured on-line or off-line. In this section, tube bulge tests with two ends fixed are performed to obtain the parameters, such as circumferential and longitudinal radius of curvature, bulge height, internal pressure and thickness at the top of the dome. Then the stress-strain curve of tube can be calculated with equations.

Description of the methodology Consider a tube with two ends fixed which is subject to an internal pressure Pi , as illustrated in Fig.1. For an element at the top of the tube dome, the following equilibrium equation can be written.

FIGURE 1. Stresses on an element at the top of the tube dome

V1 V 2  U1 U 2

Pi ti

654

(1)

As shown in Fig.1, Pi is the instantaneous internal pressure, t0 and ti are the initial and the instantaneous tube wall thickness, V 1 and V 2 are stress components in the circumferential and the axial directions, H 1 and H 2 are strain components in the circumferential and the axial directions, U1 and U 2 are the instantaneous circumferential and longitudinal radius, d 0 is the initial tube radius, and A1 , A2 and A3 are the cross-sectional areas of the inner circle at the top of tube dome, the cross-sectional area of the inner circle at the guiding zone and the cross-sectional area at the top point of bugling area. In the case both tube ends were fixed in axial and radial directions, while the central region of the tube will be free to expand. Considering the point at the top of the tube dome, the axial force can be written as: Fo Pi A1  Pi A2 (2) From the cross section geometry shown in Fig.1, one obtains

t 2

A1

S ( U 1 i ) 2

A2

S( A3

Combining Eqs. (2) – (5), one can write

V2

d0  t0 ) 2 2 2SU1ti

>

Pi (2 U1  ti ) 2  (d 0  2t0 ) 2 8U1ti

Fo A3

(3) (4) (5)

@

(6)

Additionally, Eq. (1) can be rewritten as

V1

(

Pi V 2  )U1 ti U 2

(7)

Assume that the tube material obeys the Swift hardening relationship:

V

K (H 0  H ) n

(8)

where K is the strength coefficient, V is the equivalent stress, H 0 is the initial equivalent strain, H is the equivalent strain and n is the strain hardening exponent. According to Hill’s yield criteria, the equivalent stress V and the equivalent strain H can be written as:

V2

H And

V 12 

2r V 1V 2  V 22 1 r

(9)

1 r 2r H 12  H 1H 2  H 22 1 r 1  2r U1 H 1 ln

(10) (11)

(d 0  t0 ) / 2

H3

ln

ti t0

(12)

where H 3 is strain component in the radial direction. Assuming volume constancy, one can write:

H2

(H 1  H 3 )

(13)

Conducting several experimental bugle tests using different internal pressure, it is possible to obtain a set of ( V , H ) couples representing the experimental stress–strain relationship of the material during tube bulge test. These couples plotted in a V ,

H

diagram, can be fitted by means of the Swift’s law V

655

K (H 0  H ) n .

Bulge tests for base metal In this study, tube bugle tests were performed for stress-strain curve on the hydroforming machine designed in Auto Body Technology Center of Shanghai Jiaotong University. The maximum allowable working pressure of the machine is 200MPa and the maximum allowable axial force is 1000kN. Fig. 2 and Fig. 3 show the schematic of experimental apparatus and dies for tube bugle test.

FIGURE 2. Schematic of experimental apparatus for tube bugle test

FIGURE 3. Schematic of dies for tube bugle test

The tubes used in the experiments are straight seamed tubes with 58mm outside diameter, 2.5mm wall thickness and 245mm long. Additionally, circular grids with a diameter of 2.5mm are etched on the tube surface before the experiments. During bulge tests of seamed tubes, six levels of internal pressure are scheduled. The hydroformed parts at different internal pressure levels are shown in Fig.4.

FIGURE 4. Hydroformed parts at different internal pressure

After bugle tests, the corresponding U1 , U 2 , ti , H1 and H 2 for each level are measured and recorded. First, a micrometer is used to measure the bulged diameter of the tube at the pole for the instantaneous circumferential radius U1 . Then the thickness ti at a curved surface is measured with an ultrasonic thickness meter. A profile projector is used to measure the major strains e1 and minor strains e2 of the deformed grids on the tube surface. The necking limit strain is 0.199. Laser 3D scanner is used to measure the instantaneous longitudinal radius U 2 . The tube bugle tests results of ti , U1 and U 2 for different internal pressure levels Pi are listed in Table 1. TABLE 1. Experimental results of the bugle test for seamed tube

Pi /MPa

ti /mm

U1 /mm

43 45 48 50 52

2.485 2.324 2.285 2.191 2.012

28.351 39.321 30.319 30.852 32.351

U2

/mm

1825.3 910.65 481.89 387.65 167.42

Once the above experimental data are obtained, the flow stress and the H 0 , K and n values of the tubular materials can be determined numerically. Then the calculated stress–strain relation of base metal for seamed tube is fitted to the flow curve as V 680(0.040  H ) 0.145 .

Tensile tests for weld zone A seamed tube is composed of weld metal, heat affected zone (HAZ) and base metal, as illustrated in Fig.1. Only the material properties of base metal can be obtained from tube bugle test. Hence, the assumption of iso-strain in both weld metal and HAZ used by Saunders and Wagoner [15] is applied in this study. First, the determination of an appropriate method for measuring the width of weld metal and HAZ is crucial and thus micro-hardness profile is used in this work. Fig.5 depicts measuring positions along the circumference direction.

656

Micro-hardness profile of a seamed tube with 2.5mm thickness and 58 mm outer diameter is plotted in Fig.6. This profile indicates that weld metal and HAZ width is approximately 4 and 6 mm.

FIGURE 5. Measuring positions for micro-hardness test

FIGURE 6. Micro-hardness profile of a seamed tube

Then according to the widths, four weld-only specimens and four specimens of mixed metal containing weld metal and HAZ, which are cut from seamed tubes, are prepared. The two type schematic tensile test specimens are illustrated in Fig.7. The two calculated stress–strain relation for the HAZ and weld metal are fitted to the flow curve as V 699(0.023  H ) 0.126 and V 721(0.011  H ) 0.091 .

TGC BASED FLD PREDICTION The original idea of TGC is closely related to the hypothesis of thickness inhomogeneity in the M-K model. Based on many experimental observations of previous studies, the TGC can be expressed as follow: R  RC (14) When the current thickness gradient R in the metal material drops below the critical thickness gradient RC , the localized necking is considered to take place. In this section, TGC is used to predict the FLD for seamed tube hydroforming. The critical thickness gradient in the TGC is determined by the FEM simulation of tube bulge test under plane strain state. After that, the FE models for the experimental tools and methods of our innovations in previous research are constructed. Then the forming limit strain is calculated based on TGC in the FEM simulation of tube hydroforming test. The commercial FEM code LS-DYNA is used as the solver. The Hill’s yield function is adopted in the simulation. The Swift hardening rule V K (H 0  H ) n is used to characterize the flow stress.

Determination of critical thickness gradient The simulation of tube bugle test is performed to determine the critical thickness gradient of the test tube under plane strain state. During the deformation, when the major strain reaches the necking limit strain 0.199, the localized necking is assumed to take place. The current thickness gradient value is recognized as the critical thickness gradient RC in formula (14). For the critical thickness gradient RC is independent of strain path, the TGC is established for predicting the localized necking in the next step simulation of tube bugle tests for FLD of seamed tube. Accuracy of the tube bulge test simulation is tested by comparing the simulated with the experiment via apparatus in Fig.2. The hydroformed part under plane strain state is shown in Fig.8 and Fig.9 shows the simulation result under plane strain state.

FIGURE 7. Tensile test specimens cut from seamed tubes

FIGURE 8. Hydroformed part under plane strain state

After the simulation, the length of element, the thickness variation, the major strain and minor strain can be obtained in the postprocessor. The critical thickness gradient RC is defined in the following procedures: when the major strain reaches the necking limit strain, the node A with the lowest thickness value and its neighboring node B

657

along the direction of major stress are found out; then the ratio of thickness value of node A and B is divided by the length of them to gain the value of critical thickness gradient. The thickness gradient variation in the direction perpendicular to the neck which is attained in the simulation of tube bugle test is plotted in Fig.10. The thickness gradient value decreases then increases across the neck, achieves its minimum value in the necking area. And then the thickness gradient value between HAZ and weld metal decreases a little. It can be seen that the thickness gradient value between adjoining nodes (approximately 1.00mm apart) across the neck reaches a critical magnitude of 0.784, which is considered as the critical thickness gradient value for predicting FLD in the FEM simulation of tube hydroforming. The TGC can be further expressed as: R  0.784 (15)

FIGURE 9. Simulation result under plane strain state

FIGURE 10. Variation of the thickness gradient across the necking area

FE simulation for FLD prediction In order to numerically predict localized necking during tube hydroforming process of a seamed tube, the FEM simulation combined with TGC has been carried out. The FE model of tube hydroforming is set up in FE program LS-DYNA, which is composed of a rigid die-up, a die-low, two punches and a deformable seamed tube. The contact interaction is modeled using Coulomb’s law: friction coefficient for die-tube and die-punch are both 0.05. Different loading paths in the test represent different linear strain paths. For each simulation, the thickness gradient developing in the adjoining node is monitored. When the thickness gradient drops bellow RC , the localized necking is considered taking place. Then the corresponding limit major and minor under different strain paths are recorder and plotted as the forming limit curve of the seamed tube. For the left hand side of FLD, a classical tube free hydroforming tool set is used. During free expansion, the tube is subject to axial compressive force F and an internal pressure Pi . The principle schematic is shown in Fig.11 and the FE model is shown in Fig.12.

FIGURE 11. Principle schematic for the left hand side of FLD

FIGURE 12. FE model for predicting left hand side of FLD

The forming limit is determined by controlling the material flow along the linear strain paths. Based on Asnafi [16] constructing analytical models, the relationship for generating constant strain ratio paths between the internal pressure and the axial force is determined. Fig.13 shows the simulation result at one strain state.

FIGURE 13. Simulation result of E

0.16

FIGURE 14. Principle schematic of tension -tension strain states

658

In previous researches, the left hand side of FLD for tube hydrforming could be easily obtained by free expansion. But for the right hand side of FLD, the difficulty is to clamp both ends of tube in free tube hydroforming under the tension-tension strain states. In our previous studies, a novel hydroforming tool set is designed for the right hand side of FLD for tube hydroforming. The principle is to control the material flow along tension-tension strain paths by applying the lateral compression force and the internal pressure. In order to generate different forming limit states, three type pressing blocks are used. The width w is 50mm, 75mm and 100mm. The principle schematic is shown in Fig.14 and Fig.15 shows the FE model. The linear strain paths in Fig.16 are determined by FEM simulation.

FIGURE 15. FE model for predicting right hand side of FLD

FIGURE 16. Loading paths for tension-tension strain states

RESULTS AND DISCUSSIONS Many experiments have been performed for FLD of tube hydroforming. The results of the products after free hydroforming with axial feeding and internal pressure for different strain ratio are shown in Fig.17. It is known that the bursting lines occur at the base material near to HAZ and the maximum bulge height increases with the absolute value of the strain ratio. Fig.18 displays the products with lateral compression force and internal pressure.

FIGURE 17. Hydroformed parts for E

FIGURE 18. Hydroformed parts for E

 0 .2 ~ 0

0 .1 ~ 0 .4

A three dimensional image processing system is used to measure the engineering major strains e1 and minor strains e2 of the deformed grids on the tube surface. And the values of the true strain ( H 2 , H1 ) are transformed from ( e2 , e1 ) and used to construct the experimental forming limit points in Fig.21. In the FEM simulation of tube hydroforming test, the formula (15) is used as the failure criterion for predicting the forming limit. When the thickness gradient of each specimen drops below 0.784, the corresponding forming limit strain is recorded as the solid line plotted in Fig.19.

FIGURE 19. Comparison of the calculated FLD with experimental data

659

Fig.19 shows a comparison between the calculated FLD and experimental data for a seamed tube. The calculated FLD matches well with the experimental data provided by seamed tube hydroforming. In the state of plane strain, the calculated FLD is superposed on the experimental point. This is owed to the TGC is established by tube bulge test at plane strain state. In other deformation states, the calculated FLD based on TGC matches well with the experimental data. The combination of TGC and FEM is shown to be a practical method for predicting forming limit diagram for seamed tube hydroforming with enough accuracy and convenience.

CONCLUSIONS Prediction of forming limit diagram in virtual tube hydroforming was performed by means of the FEM combined with the TGC. The following conclusions are obtained. (1) The critical thickness gradient value was successfully determined in the FEM simulation of tube bulge test under plane strain state to establish the TGC, which is used as the failure criterion in the next step simulation. (2) The FLD of QSTE340 seamed tube was calculated based on TGC in the simulation of tube hydroforming test. The comparative evaluations have shown an excellent correlation between numerically and experimentally obtained FLD, which demonstrate the accuracy and validity of the TGC and the research approach adopted. (3) By combining the TGC and FEM, the FLD of seamed tube can be calculated with enough accuracy and convenience when its three parts properties are available by tube bulge test and uniaxial tensile test.

ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support of National Natural Science Foundation of China (Grant No. 50605043) and Baosteel Group the supply of test materials.

REFERENCES 1. R. Davies, G. Grant, D. Herling, M. Smith, B. Evert, S. Nykerk, and J. Shoup, “Formability investigation of aluminum extrusions under hydroforming conditions”, SAE Technical Paper, 2000, 2000–01–2675. 2. N. S. P. Varma, R. Narasimhan, A. A. Luo, and A. K. Sachdev, “An analysis of localized necking in aluminium alloy tubes during hydroforming using a continuum damage model”, International Journal of Mechanical Sciences, 2007, 49, pp. 200– 209. 3. N. S. P. Varma and R. Narasimhan, “A numerical study of the effect of loading conditions on tubular hydroforming”. Journal of Materials Processing Technology, 2008, 196, pp. 174–183. 4. Y. M. Hwang, Y. K. Lin, and H. C. Chuang, “Forming limit diagrams of tubular materials by bulge tests”, Journal of Materials Processing Technology, 2009, 209, pp. 5024-5034. 5. J. Kim, S. W. Kim, W. J. Song, and B. S. Kang, “Analytical and numerical approach to prediction to prediction of forming limit in tube hydroforming”, International Journal of Mechanical Sciences, 2005, 47, pp. 1023–1037. 6. R. Brun, A. Chambard, M. Lai, and P. DeLuca, “Actual and virtual testing techniques for a numerical definition of materials”, NUMISHEET’1999, Besancon, France, 1999, pp. 393-398. 7. M. Geiger and M. Merklein, “Determination of forming limit Diagrams-a new analysis method for characterization of materials’ formability”, Annals of the CIRP, 2003, 52, pp. 213–216. 8. K. Narasimhan, “A novel criterion for predicting forming limit strains”, In: Proceedings of NUMIFORM 2004, Ohio, USA, 2004, pp. 850–855. 9. C. S. Zhang, L. Leotoing, D. Guines, and E. Ragneau, “Theoretical and numerical study of strain rate influence on AA5083 formability”, Journal of Materials Processing Technology, 2009, 209, pp. 3849-3858. 10. J. Kim, Y. W. Kim, B. S. Kang, and S. M. Hwang, “Finite element analysis for bursting failure prediction in bulge forming of a seamed tube”, Finite Elements in Analysis and Design, 2004, 40, pp. 953–966. 11. S. Fuchizawa And M. Narazaki, “Bulge test for determining stress-strain characteristics of thin tubes”, Advanced Technology of Plasticity, 1993, 1, pp. 488–493. 12. T. Sokolowski, K. Gerke, M. Ahmetoglu, and T. Altan, “Evaluation of tube formability and materials characteristics: hydraulic bulge testing of tubes”, Journal of Materials Processing Technology, 2000, 98, pp. 34–40. 13. M. Strano and T. Altan, “An inverse energy approach to determine the flow stress of tubular materials for hydroforming applications”, Journal of Materials Processing Technology, 2004, 146, pp. 92–96. 14. M. Koc, Y. Aue-u-lan, and T. Altan, “On the characteristics of tubular materials for hydroforming-experimentation and analysis”, International Journal of Machine Tools & Manufacture, 2001, 41, pp. 761–772. 15. F. I. Saunders and R. H. Wagoner, “Forming of tailor welded blanks”, Metallurgical and Materials Transactions A, 1996, 27, pp. 2605–2616. 16. N. Asnafi, “Analytical modeling of tube hydroforming”, Thin-Walled Structures, 1999, 34, pp. 295–330.

660

Finite Element Analysis of Tube Hydroforming in NonSymmetrical Dies Abhishek V. Nulkara, Randy Gub and Pilaka Murtyc a

EASi Engineering, 1551 E. Lincoln Ave, Madison Hts, MI 48071, USA Department of Mechanical Engineering, Oakland University, Rochester, MI 48306, USA c Department of Mathematics, Physical Science, and Engineering, West Texas A&M University, Canyon, TX, 79016 b

Abstract. Tube hydroforming has been studied intensively using commercial finite element programs. A great deal of the investigations dealt with models with symmetric cross-sections. It is known that additional constraints due to symmetry may be imposed on the model so that it is properly supported. For a non-symmetric model, these constraints become invalid and the model does not have sufficient support resulting in a singular finite element system. Majority of commercial codes have a limited capability in solving models with insufficient supports. Recently, new algorithms using penalty variable and air-like contact element (ALCE) have been developed to solve positive semi-definite finite element systems such as those in contact mechanics. In this study the ALCE algorithm is first validated by comparing its result against a commercial code using a symmetric model in which a circular tube is formed to polygonal dies with symmetric shapes. Then, the study investigates the accuracy and efficiency of using ALCE in analyzing hydroforming of tubes with various cross-sections in non-symmetrical dies in 2-D finite element settings. Keywords: FEA, positive-semi-definite, singularity, hydroforming, non-symmetry, contact.

PACS: 02.70.Dh, 46.55.+d

INTRODUCTION Since the advent of hydroforming in 1950 many research articles have been published on tube hydroforming (THF). The study by Dohmann and Hartl [1] shows hydroforming of tubes with straight axis and pre-bent tubes. The article also includes discussion of failure cases like buckling, bursting, wrinkling and folding back when tubes with a straight longitudinal axis are expanded. In a comprehensive study, Koc and Altan [2] present an overall review of tube hydroforming technology. Hwang and Altan [3] discuss and compare hydraulic expansion alone and crushing process combined with preforming and hydroforming to a rectangular die. Through finite element modeling the article shows that crushing combined with hydroforming can yield better results. Koc and Altan [4] present a study in which finite element simulation of two-dimensional tube hydroforming is given. Kim and Kang [5] use a backward tracing technique and finite element method to determine the initial tube dimensions and intermediate configuration for preform design of square and rectangular dies. Rama et al. [6] develop a new two-dimensional numerical method based on membrane analogy to tube-sheet deformation with the concept of finite elements for sheet discretization. The efficiency of the method is verified using commercially available FEA packages. Keum and Kim [7] present a study showing the finite element simulation of axisymmetric tube hydroforming. Abrantes et al. [8] make an effort to establishing a basic understanding of the THF processing of aluminum and copper tubes by giving finite element simulations of THF. Some experimental results of THF of T-parts are also included in the article. Hwang and Chen [9] propose a mathematical model considering sliding friction between tube and die. The model explores the plastic deformation behavior of the tubes in a square cross-sectional die. FE simulations are given and experiments are carried out for comparison. As it appears, finite element method has become an important tool in THF for assisting in tool design, predicting the final shape, determining manufacturing parameters, and so on. In this study, we will first disclose the difficult challenge encountered in finite element simulations of THF in a two-dimensional setting. A novel contact element whose material behavior is analogous to the air is proposed to address the difficult task. The stiffness matrix is formulated based on the material model. The effects of the air-like contact element are discussed. Two numerical examples involving symmetric dies are solved using the proposed algorithm and the commercial code ANSYS [10]. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 661-668 (2011); doi: 10.1063/1.3623670 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

661

The accuracy of the proposed algorithm is compared against the ANSYS code. Finally, two numerical simulations of tube hydroformed to a triangular and pentagonal dies using the proposed algorithm are presented.

PROBLEM STATEMENT Many applications of THF [3~9] reported involve models featuring symmetric characteristics. With symmetric settings, only a portion of the geometry is needed in finite element modeling. Additional boundary conditions are imposed on nodes on the planes of symmetry. As a result of the additional symmetric boundary conditions, the model becomes well-posed even if the original full model is not. The first two models presented later are such examples. For THF, the cross-sections of the final tube are not necessarily symmetrical. In this event, the symmetric boundary conditions are no longer valid. Consequently, the finite element models do not have sufficient supports or are not constrained at all. As illustrated in FIGURE 1 a deformable object does not have sufficient support at the beginning of the static analysis. The finite element system for such a problem often suffers positive semi-definiteness as a result of singular total stiffness matrix. In this study, a methodology is proposed to tackle the challenging problem.

FIGURE 1. A deformable object with insufficient support

MATERIAL MODEL FOR AIR-LIKE CONTACT ELEMENT The air-like contact element (ALCE) closely resembles the physical behavior of the air or ideal gas, thus it is given the name. First we will look at the physics law of an ideal gas and how it evolves into a material model for ALCE.

Law of Air / Ideal Gas It is well known that the air or ideal gas is governed by the physics law below under a constant temperature. pV c , (1) where p is the pressure, V the volume of the gas, and c a constant. FIGURE 2 shows the plot of the physics law in solid line which is a hyperbolic curve. Note that the pressure and volume are both positive numbers in Eqn. (1). If the curve is first translated to the left and then reflected to the negative half plane, it becomes the base law for ALCE. The transformed curve can be described below.  p(V  V0 ) c , (2) where V0 is the original volume.

FIGURE 2. Physics law of an ideal gas and its transformed versions.

Material Description for ALCE In solid mechanics, the pressure is considered a negative bulk stress Vb, while the bulk strain Hb is 'V/V0, and 'V is the change in volume. Therefore the transformed curve is in analogy to the following. V b (1  H b ) c , (3) which has the two asymptotes given below.

662

V b 0; H b 1 . (4) The two asymptotes have the slopes of 0 and f, respectively. This material law does not serve the purpose of solving the current problem. Now the asymptotes are modified slightly as below. V b E x (1  H b ); V b EnH b , (5) where Ex and En are, respectively, a large (maximum) and a small (minimum) numbers. Note that the two lines intersects at a point very close to (Hb, Vb) = (-1, 0). Thus, we propose the following material law for ALCE through the bulk stress Vb and bulk strain Hb. (6) V b EbH b . Here, the bulk modulus Eb is defined in terms of Ex, En in the following way. En , if b t 1  e ­ ° 3 (7) Eb ®( E x  E n )(2[  3[ 2 )  E x , if  1  e ! b t 1 , ° E  , if   1 x b ¯ where e is a very small number and, Hb 1 (8) [ , 0 d [ d 1. e

The graphical representation of Eqns. (6) and (7) is given in FIGURE 3. It is seen that for easy numerical treatment, the sharp intersection of the two asymptotic lines given in Eqn. (5) is smoothened by the cubic polynomial as shown in Eqn. (7). In addition to the material law given in Eqns. (6) and (7) which governs the mechanical behavior of the ALCE, it is also assumed that the ALCE is inactive to the shear stress.

FIGURE 3. Material description for ALCE. Solid line is for Eqn (6) while the dashed line is for Eqn (7).

FINITE ELEMENT FORMULATION FOR ALCE According to the material description given above, the strain energy U of ALCE occupying a volume V is defined as follows. U

³ ³V

b dH b dV

.

(9)

V

In the following development, only triangular ALCE is considered. The finite element matrix for such an ALCE is given below.

Stiffness Matrix for ALCE Let the space between the object and the rigid barrier be filled with ACLE as seen in FIGURE 4. It is readily seen that when the area of the triangular element is decreased to smaller than e, the two objects start engaging in contact and the material behavior of ALCE enters the nonlinear zone indicated as BDG in FIGURE 3. We will now formulate the stiffness matrix of this nonlinear elastic contact element.

663

FIGURE 4. Area coordinates for ALCE.

The coordinates xi, i = 1, 2, of a generic point within the ALCE can be found as follows. xi N ij X j ,

(10)

where indicial summation convention is implied, Xj, j = 1, 2, …, 6, are the components of the element nodal coordinate vector, and Nij is the component of the shape-function matrix defined as 0 N2 0 N3 0 º ªN . (11) N ij « 1 0 N 0 N 2 0 N 3 »¼ 1 ¬

> @

The shape functions Ni are in fact the area coordinates [13]. 1 Ai Ni ai  bi x1  ci x2 . A 2A where, referring to FIGURE 4, A A1  A2  A3

(12)

a1

X3X6  X5X 4,

a2

X 5 X 2  X1 X 6 ,

a3

X1 X 4  X 3 X 2 ,

b1

X 4  X6,

b2

X6  X 2,

b3

X 2  X 4,

.

(13)

c1 X 5  X 3 , c2 X 1  X 5 , c3 X 3  X 1. Note that, as shown in FIGURE 4, the area of an ALCE, A, is positive, when the nodes are defined in the counter-clockwise direction. The shape function matrix given in Eqn. (10) is also employed for interpolating the displacements in the element. Therefore the bulk strain Hb becomes, (14) H b u i ,i N ij ,iU j ,

where ui is the component of the displacement vector at a point within the element, and Uj is the component of the element nodal displacement vector. Thus the exact strain-displacement matrix is given below. 1 (15) N ij ,i ¬b1 c1 b2 c2 b3 c3 ¼ . 2A It is seen that the bulk strain Hb is constant in the element. Recall that ALCE does not support shear stresses. Hence, the exact stiffness matrix of ALCE, when Hb t -1+e, may be formed in the usual fashion and is given below. ªb12 b1c1 b1b2 b1c2 b1b3 b1c3 º « » c12 c1b2 c1c2 c1b3 c1c3 » « tEn « b22 b2 c2 b2b3 b2c3 » (16) « N ij ,i ». 4A « c22 c2b3 c2c3 » « b32 b3c3 »» symm « «¬ c32 »¼

> @

> @

Eqn. (16) represents the stiffness matrix of ALCE at any state provided En is replaced by the bulk modulus Eb given in Eqn. (7). Note that when A = 0 stemming from a contact situation, a small value of A is used, instead. We now examine the numerical characteristics of the ALCE.

Effects of ALCE Let 3 be the discretized total potential energy of the elastic object and ALCE in FIGURE 4. Thus, 3 * 3  tAS A , where A is the area all ALCE occupy, t the thickness of the plane elastic object, and

664

(17)

1 (18) K ijU iU j  RiU i , 2 which represents a PSD quadratic functional of the elastic object due to insufficient supports. In addition, SA is the strain energy intensity of ALCE which in essence is the area under the Vb-curve in FIGURE 3. For example, when Hb d -1, SA is equal to the sum of the areas of triangle OAB, ABDC and trapezoid CDGF shown in FIGURE 3. Refer to [12] for more details. Following the usual manner, we may form the total stiffness matrix of the entire system with ALCE included. It is seen that when there is no contact, Hb t -1+e, SA has such an insignificant value that 3* | 3. However, it provides a small disturbance to the singular Kij so that the finite element system is no longer PSD. This effect is known as regularization in many ill-posed problems [14, 15]. On the other hand when the elastic object engages in contact with the rigid barrier, SA registers a very large value as indicated by the area of ABDC and CDGF in FIGURE 3. This is in analogy to the penalty function seen in common optimization problems [16]. Therefore, the current ALCE contains both the regularization and penalizing effects for the constrained minimization problem with a PSD finite element system. 3

NUMERICAL SIMULATIONS The effectiveness and accuracy of ALCE are demonstrated using a few tube hydroforming examples in the following section. For comparison purpose, two symmetric dies are included first which are solved by both ANSYS and the current ALCE algorithm. Two non-symmetric dies are solved using ALCE algorithm are given afterwards.

Symmetric Dies Using ALCE and ANSYS The detailed drawing for the first symmetric die and tube is given in FIGURE 5(a), which shows a square die with rounded corners. The outer diameter of the tube is intentionally chosen at 80 mm, slightly smaller than the square so that there is no contact at the beginning of the analysis. To ensure symmetry, both the die and the tube share the same center. FIGURE 5(b) depicts the finite element mesh for calculating the contact between the tube and the die surface using the ALCE. Due to symmetry, nodes on the Y-axis are constrained in the X-direction while nodes on the X-axis are constrained so that they are not allowed to move in the Y-direction. Nodes for the die are fixed against movement in both directions. The model contains 485 nodes and 120 8-noded elements for the tube, 15 nodes and 14 line elements for the die surface and 190 ALCE elements. Note that there are only a few ALCE shown in FIGURE 5(b); in actual calculation there are a lot more ALCE covering the entire die surface. Plane strain model and frictionless contact are assumed throughout the numerical simulations. For comparison, a finite element model is created for ANSYS. The model is shown in FIGURE 5(c) which features 845 nodes and 212 quadratic elements for the steel tube. Note that the block for the die is also meshed in the model.

(c) (b) (a) FIGURE 5. (a) A square die. Unit: mm; (b) a quarter finite element model for the proposed algorithm using ALCE; and (c) finite element model for ANSYS. TABLE 1 Material constants of steel tube and ALCE. Constant Steel ALCE E ( Ex / En ) 200 GPa (1.0×1010 Pa / 1.0×10-2 Pa) 0.3 Q ET 10 GPa 100 MPa VY e 1.0×10-8

665

The material properties for the steel tube and ALCE are given in TABLE 1, where E is the modulus of elasticity for the steel tube, Q Poisson’s ratio, ET tangential modulus and VY yield stress. The rate-independent elastoplastic model of bilinear isotropic hardening is assumed for the steel. Refer to [9, 12] for the sensitivity of Ex and En on the numerical accuracy. FIGURE 6 shows the computed distance traveled by point Q, as labeled in FIGURE 5(a), during forming. Note that the maximum distance point Q can travel is 15.26 mm which is indicated by the top horizontal portion of the curves. It is seen that point Q reaches the die surface when the calculated hydro-pressure is ph = 308 MPa using ALCE and 320 MPa using ANSYS. The maximum difference in the distance traveled between the two predictions is 0.73 mm when the hydro-pressure is near 150 MPa. The deformed shapes of the tube as calculated by ANSYS and ALCE are given in FIGURE 5(b). It is seen that the tube has more thinning on the flattened area as predicted by the algorithm using ALCE than by ANSYS. Other values of parameters for ALCE are also tested which provide similar results. Note also that the CPU times used by ANSYS and the algorithm with ALCE are, respectively, 142 seconds and 59 seconds.

(b) (a) FIGURE 6. (a) Distance traveled of point Q, r = 40 mm, T = 45q, as labeled in FIGURE 5 (a); (b) Deformed shapes of the tube at ph = 160 MPa predicted by ANSYS (shaded mesh) and ALCE (clear mesh).

FIGURE 7(a) shows the detailed drawing for the second symmetric die and tube. As indicated, the tube is in contact with the lower surface of the die. The gap size between the tube and the vertical die surfaces on both sides is set equal to achieve symmetry. FIGURE 7(b) depicts the half model finite element mesh with only a few ALCE shown for calculation using the ALCE. Note that there is no contact between the tube and the die surface in the finite element model. As usual, nodes for the die are fixed against movement in both directions, while nodes for the tube on the Y-axis are constrained in the X-direction. As a result, the model has a rigid-body-motion model in the Ydirection. The model contains 303 nodes and 60 8-noded elements for the tube, 26 nodes and 25 line elements for the die surface and 300 ALCE elements. For comparison, a finite element model is created for ANSYS. The model is shown in FIGURE 7(c) which features 2,937 nodes and 770 quadratic elements for the steel tube. Note that the block for the die is also meshed in the model. In order to achieve a converged solution, there are a couple treatments applied on the model. In addition to the usual boundary conditions, the finite element mesh for the tube is initially in contact with the lower die surface. Further, the two nodes in contact are “coupled” together so the two have the same Y-displacement. Consequently, there is no rigid-body-motion mode for the model.

(a)

(b)

(c) FIGURE 7. (a) A hexagonal die with rounded corners with radius 10 mm. Unit: mm; (b) a half finite element model for the proposed algorithm using ALCE; and (c) the finite element model for ANSYS.

The calculated deformed shapes of the tube are depicted in FIGURE 8. It is seen from FIGURE 8(a) that the tube makes contact with the three surfaces at ph = 18.5 MPa as predicted by ANSYS and at ph = 19.6 MPa by the

666

ALCE algorithm. The deformed shapes of the tube when the hydro-pressure is 100 MPa are displayed in FIGURE 8(b). The results indicated that the ALCE algorithm exhibits stiffer behavior for the tube, which may be attributed to the much fewer elements used for the tube (only one-tenth the number of elements used in the ANSYS model). Finally, the hydro-pressure for the tube to reach the complete die is ph = 244 MPa as predicted by ANSYS as opposed to ph = 268 MPa by the ALCE algorithm.

(b) (a) FIGURE 8. Deformed shapes of the tube predicted by ANSYS (shaded mesh) and by ALCE (clear mesh) (a) when the tube contacts three surfaces at ph = 18.5 MPa (by ANSYS) and ph = 19.6 MPa (by ALCE); (b) at ph = 100 MPa.

Non-Symmetric Dies Using ALCE In this section, two numerical examples of tube hydroforming with non-symmetric dies are presented. Due to the limited capability of the ANSYS code in producing converged solution involving insufficient supporting conditions, only numerical predictions from the ALCE algorithm are presented. FIGURE 9(a) shows the detailed drawing for the first non-symmetric die and tube. FIGURE 9 (b) depicts the full model finite element mesh without ALCE. Note that there is no contact between the tube and the die surface anywhere in the finite element model. As usual, fixed boundary constraints are defined for nodes for the die, while there is no constraint imposed on the tube. The model contains 600 nodes and 120 8-noded elements for the tube, 27 nodes and 27 line elements for the die surface and 647 ALCE elements (not shown). FIGURE 9 (c) shows the deformed tube at various hydro-pressures as calculated by the ALCE algorithm. The algorithm fails to produce a converged solution corresponding to the final form of the die. Apparently there is not sufficient material for the tube to be stretched to the final form of the die. Numerical simulation for two-stage forming using ALCE algorithm has not been attempted.

(a)

(c) (b) FIGURE 9. (a) A triangular die with rounded corners. Unit: mm; (b) a full finite element model for the proposed algorithm using ALCE; and (c) the calculated deformed shape for the tube using ALCE algorithm.

The second numerical example involves a pentagonal die with rounded corners as shown in FIGURE 10(a). The geometry is not symmetric about any lines. A full finite element model is shown in FIGURE 10(b) which has 600 nodes and 120 8-noded elements for the tube, 49 nodes and 49 line elements for the die surface and 939 ALCE elements (not shown). All the nodes for the die surface are constrained from any movement, while the tube is not constrained at all. FIGURE 10(c) displays the deformed shapes of the tube predicted by ALCE algorithm at two hydro-pressures. It is seen that the tube reaches the corners with larger radius earlier while leaving a large gap to the corner with smaller radius. According to the ALCE algorithm, the final form of the tube can be achieved when the hydro-pressure is 194 MPa.

667

(b) (c) (a) FIGURE 10. (a) A pentagonal die with rounded corners; (b) a full finite element model for the proposed algorithm using ALCE; and (c) the calculated deformed shape for the tube using ALCE algorithm.

CONCLUSIONS There are some tube hydroforming applications that involve non-symmetrical dies. When such dies are used, a tube would be “floating” inside the die without any supports. Finite element system of such a problem is often positive semi-definite and could not be solved by many commercial codes. A numerical algorithm employing a novel airlike contact element is presented. The nonlinear element has both regularization and penalty effects to tackle the constrained minimization problems. The algorithm is first examined using symmetric models that can be solved using ANSYS. It is found that the results predicted by the proposed algorithm are compatible with ANSYS. The proposed algorithm is further tested using two THF examples involving unsymmetrical dies. The predicted deformed shapes of the tube at various pressures are presented. It is found that the proposed algorithm is numerically efficient in handling positive semi-definite finite element systems seen in THF.

REFERENCES 1. 2. 3. 4. 5. 6. 7.

8. 9. 10. 11. 12. 13. 14. 15. 16.

F. Dohmann, Ch. Hartl, Tube hydroforming – research and practical applications, 1997, J. Mat. Proc. Technol. 71 (1997) 174- 186. M. Koc, T. Altan, An overall review of the tube hydroforming (THF) technology, J. Mater. Process. Technol. 108 (2001) 384–393. Y-M. Hwang, T. Altan, Finite element analysis of tube hydroforming processes in a rectangular die, Finite Elements in Analysis and Design 39, 1071-1082 (2002). M. Koc, T. Altan, Application of two dimensional (2D) FEA for the tube hydroforming process, Int. Journal of Machine Tools & Manufacture, 42 (2002) 1285-1295. J. Kim, B.S. Kang, Implementation of backward tracing scheme of the FEM for design of initial tubular blank in hydroforming, 2002, J. Mat. Proc. Tech. 125-126 (2002) 839-848. S.C. Rama, K. Ma, L.M. Smith, J.M. Zhang, A two-dimensional approach for simulation of hydroforming expansion of tubular cross-sections without axial feed, J. Mater. Process. Technol. 141, 420-430 (2003). Y.T. Keum, Y.S. Kim, Finite Element Simulation of Axisymmetrical Tube Hydroforming Processes, Materials Science Forum, v 437-438, pp. 387-390, 2003, Advanced Materials Processing II; Proceedings of the 2nd International Conference on Advanced Materials Processing. J.P. Abrantes, A. Szabo-Ponce, G.F. Batalha, Experimental and numerical simulation of tube hydroforming, 2005, J. Mat. Proc. Tech. 164–165 (2005) 1140–1147. Y-M. Hwang, W-C. Chen, Analysis of tube hydroforming in a square cross-sectional die, Int. Journal of Plasticity, 21 (2005) 1815-1833. ANSYS Workbench User’s Guide, Revision v12.1, ANSYS Inc., Canonsburg, PA. R. J. Gu, P. Murty, and Q. Zheng, Use of penalty variable in finite element analysis of contacting objects with insufficient supports, Computers & Structures 80 (31), 2449-2459 (2002). P. Murty, Finite Element Analysis of Positive Semi-Definite Systems, Ph.D. dissertation, Oakland University, 2001. R. D. Cook, D. S. Malkus, M. E. Plesha, R. J. Witt, Concepts and Applications of Finite Element Analysis, 4th Ed., Wiley, 2001. A.N. Tikhonov, and A.V. Arsenin, Solutions of Ill-Posed Problems, Winston, Washington, D.C., 1977. J. V. Beck, B. Blackwell, and C. R. St. Clair, Jr., Inverse Heat Conduction—Ill-Posed Problems, Wiley-Interscience, New York, 1985. Reklaitis GV, Ravindran A, and Ragsdell KM, Engineering Optimization: Methods and Applications, New York: John Wiley, 1983.

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674

A Study on the Effect of Pre-Bending Radius on the Formability of Hydroformed Steel Tubes A. Prabhu, N. Mishra and K. Narasimhana a Department of Metallurgical Engineering and Materials Science, Indian Institute of Technology - Bombay, Powai, Mumbai 400076, India

Abstract. Pre-bending is an important component of the tube hydroforming process for the manufacture of automotive components. In this paper the correlation between the bending radius and the formability of pre-bent DP steel tubes in the tube hydroforming process is discussed. The effect of the bending on the thickness variation, nature of necking and bulge height along the tube length is investigated. The analysis is carried out using the commercial FEM package PAM-TUBE. Keywords: tube hydroforming, pre-bending, simulation PACS: 81:20.Hy

INTRODUCTION Tube hydroforming (THF) is the process of applying hydraulic pressure to form parts from tubular blanks into required die shapes. Tube hydroforming has assumed significant importance in the manufacture of complex automotive parts due to various advantages it offers over conventional forming processes. THF allows greater repeatability, part consolidation, thus requiring lower tooling costs. The method also confers a higher strength to weight ratio, as well as lower springback in the formed product. The method is particularly attractive since it allows for reduction in the number of secondary operations required for manufacture [1]. The drawbacks of the method include high cycle time, higher initial capital investment and a nascent knowledge base concerning the process. FEM analysis is used to analyze the forming procedure due to the complex nature of the plastic deformation during hydroforming. Complicated structural part geometries may need a pre forming operation like pre-bending prior to the hydroforming step[2]. This step has important bearing on the final quality of the product. Various process parameters need to be carefully chosen to achieve optimum results in the bending process. In this paper, the effect of varying bending radius on the formability of steel tubes is studied. The forming limit strains are used as a measure of the formability and the necking is predicted using a thickness gradient based criteria. The effect of the bulge forming on the thickness gradient, and bulge height is also reported.

METHODOLOGY FEM Simulation Scheme In the tube bending process a straight tube is formed into the required bend radius and angle. This forming process can be carried out through various techniques such as rotary draw bending, press bending, roll bending etc. The rotary draw method is the preferred method used for thin walled tubes, since the process allows precise control of bending radii, versatility, and is cost effective [3-4]. The FEM in this study is based on the rotary draw bending process. Commercially available explicit finite element code PAM-TUBE is used to model and predict the behavior of the circular tubes. A virtual rotary draw bender is used to generate FEM based bending models for pre and post processing, and is used to interpret the simulation results.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 675-682 (2011); doi: 10.1063/1.3623672 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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Fig.1 shows a schematic setup of the bending tools. All tools are represented as rigid bodies, and Coulombic friction model with friction coefficient 0.12 is implemented. Four-node shell elements comprised the tubular blank. An isotropic, homogeneous, elastic–plastic material following the Hill 1948 yield criterion and Holloman flow n

equation, =K is used as the material model.

FIGURE 1. (a) External geometry of the bend tooling (FE mesh). (b) Internal geometry of the tooling (FE mesh).

TABLE 1. Material property data for DP-590 K Yield Strength n (MPa) (MPa) 0.2

1054

415

r

Young Modulus (GPa)

Poisson’s ratio

Thickness (mm)

0.8

210

0.33

1.4

For the present study, tubes with internal diameter of 50mm, constant bend angle of 90 degrees, and a wall thickness of 1.4mm have been considered. The tube radii considered for the present work are 200mm, 250mm and 300mm. The material considered in the present study is DP-590 steel. The relevant properties of the material are listed above in Table 1.

Forming Method and Parameters . The base tube is defined using 2D-based strain estimation. Figure.2 shows the bend tube and tooling setup required for the bent tube hydroforming which includes upper die, lower die, left piston and right piston. The computed model of the bent tube geometry obtained from the previous bending operation is imported into the forming calculation. Here a range of pressure paths and axial feed paths are used to form the bent tube into die cavities of varying bulge length to generate different strain paths on the FLD. Sample curves for these paths are represented in figure 3. For a given FLD, 4 samples are formed with provision of axial feed while forming. Another 4 samples each are formed with allowance for free expansion of tube, and with tube ends held fixed. A Coulombic friction model with coefficient 0.08 is used for bulge forming with axial feed and for free forming of the bent tubes. For bulge forming in the fixed condition, a columbic friction of 0.5 has been assumed between the tube and the die wall, to prevent material flow into the bulge region.

676

FIGURE 2. Bend tube and tooling for hydroforming.

Pressure (GPa)

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

Input 1 Input 2

0

2

4

6

8

10

Time (sec) (a) 80 Axial Feed (mm)

Input 1

60

Input 2

40 20 0 0

5

10

15

20

Time (sec) (b) FIGURE 3. (a) Pressure – Time path and (b) Axial feed path used during

Necking Criteria To predict the forming limit strains during simulation, a thickness gradient based criterion is used.[5-6] According to this criterion, the presence of a neck is perceived as development of a local critical thickness gradient across neighboring elements during forming. Such a perception of the neck is independent of the strain path, rate of forming and the type of material. The predicted onset of necking is verified using the output pressure-time curve, and a drop is observed in the output pressure at the predicted onset of necking.

677

RESULTS Results of Bending Procedure

(a)

(b)

FIGURE 4. (a) Major strain distribution post the bending operation in the tube with R = 300mm. (b) Thickness distribution in the same die post forming operation.

The bending operation leads to compressive strains and tube thickening on the inner fiber, while leading to tube thinning and tensile strains on the outer fiber. The thickness and major strain contours generated due to the bending process are depicted in figure 4. Due to the nature of the process, a greater inhomogenity in the tube thickness is observed in the case of tubes with higher curvature, as can be inferred from figure 5. The major strain also exhibits higher magnitude in the case of lower curvature. The length of the tube closer to the end has comparitively uniform thickness and are relatively free of strain.

Thickness on inner fiber (mm)

1.55 R=200 R=250

1.5

R=300 1.45

R=200 R=250 R=300

1.4

1.35 0

50

100

Position ID along length (a)

678

150

(b) FIGURE 5. (a)Thickness distribution along inner fiber of tube post rotary bending operation (b) Thickness distribution along outer fiber of tube post rotary bending operation

Results of Tube Forming

0.6 Straight Tube 0.5

Major Strain

Comparison of Forming Limit Strains

0.4 R=300 R=250

0.3 R=200 0.2

R=200 R=250 R=300 Straight Tube

0.1

0 -0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

Minor Strain FIGURE 6. Comparison of Forming Limit Curves for tubes of different radii. Only the left quadrant has been plotted

The forming limit curves for the different bulge radii have been plotted in figure 6. Due to the nature of the prestrain, all limit strains in the current battery of experiments show a strain path on the left quadrant of the FLD. A notable proportional relationship is seen to exist between the radius of curvature and the magnitude of the limit strains. The highest limit strains at the onset of necking are observed in the case of the tube with R=300mm. This can be attributed to the lower magnitudes of stress present in the bent tube prior to the hydroforming step. As expected, the tube with R=200mm shows the earliest onset of necking for different strain paths. Closer to the plain strain condition, necking is seen occur especially early in the tube with lowest radius.

679

The forming limit strains in the forming of bent tubes can be seen to be significantly lower than that in the straight tube hydroforming. In the straight tube hydroforming, forming with provision of axial feed usually shows a strain path in the drawing region, while the forming with fixed tube condition shows strain paths in the stretching region. The free forming typically is observed to produce strain paths in the plain strain region. Though all the strain paths in the case of bent tubes lie in the left quadrant, the behavior of the strain paths of the necked elements displays behavior analogous to the un-bent tube as shown in figure 7. The strain path of the necked element in the axial feed condition displays a strain path to the extreme left, with higher minor strain, while on the contrary the strain path of the necked element in the fixed condition shows it tending towards a lower magnitude of minor strain after an initial increase.

0.2

Major Strain

0.25

0.15 Axial

0.1 0.05

Fixed Free

0 -0.06

-0.04

-0.02

0

Minor Strain

FIGURE 7. Strain paths for necked elements in different bend conditions for R=300mm

Thickness Distribution and Neck Orientation

FIGURE 8. Thickness distribution plotted along the tube circumference for tubes of different radii as a function of position from the outer fiber at the onset of necking.

The thickness The time required failure does not appear to change significantly with the increase in radius. However, as evident from figure 8, failure occurs at a lower thickness in tubes with higher radius of bending. This

680

can be attributed to lower concentration of stress in tubes with larger radii. Higher curvature in the geometry shows higher accumulation of material along the inner fiber of the tube. The points of onset of necking can be identified in the figure as the points of maximum gradient, i.e at position 11 and position 26, through application of the thickness gradient based criterion discussed earlier. During forming the tube, failure is always observed to occur through necking in the axial direction on the neutral fiber as shown in fig 9. The major strain is always directed in a direction perpendicular to the direction of neck. This is in contrast to the case of necking in the un-bent tube, where the direction of necking is circumferential when formed in the fixed condition.

(a)

(b)

FIGURE 9. (a) Thickness distribution in the tube with R=300. The necked region can be percieved as the blue region with the minimum thickness. (b) The major strain distribution for the previous case. Maximum major strain during failure is present on the neutral fiber, and is directed perpendicular to the direction of necking.

Comparison of Bulge Height The bulge heights attained in table 2, were measured as the increase in radial distance of the necked element after the forming process. An increase in radius of bending, led to a relatively uniform increase the bulge height throughout all forming conditions. Abrupt high values of bulge height were obtained in samples exhibiting nonuniform bulging along the length. TABLE 2. The bulge radii as a function of process condition Axial Condition Fixed Condition

Average

Free Condition

R=200 mm

1.8

1.32

2.35

1.58

1.29

1.58

1.28

1.26

5.27

1.97

R=250 mm

2.28

1.44

2.38

1.39

1.23

2.15

2.55

2.97

2.75

2.1

R=300 mm

2.48

1.89

2.6

1.55

1.97

3.11

3.01

2.57

10.65

3.3

681

CONCLUSION The nature of the tube bending operation demonstrates a significant effect on the formability in the final tube hydroforming operation. The formability in the hydroforming process is observed to increase with an increase in the bending radius. This is intuitively explained by the greater unevenness of strain and thickness in the tubes with lower radii, after the bending operation. The formability and nature of failure in the bent tube is examined, and compared to that of the un-bent tube. The bent tubes display significantly lower formability, and the circumferential mode of necking is found to be absent in the bent tubes.

ACKNOWLEDGEMENT The authors would like to acknowledge the support from the CAR-TIFAC Project (05TI002) by the Government of India

REFERENCES 1. 2. 3. 4. 5. 6.

M. Koc, T. Altan, “An overall review of the tube hydroforming (THF), technology”, J. Mater. Process, 108 (2001) 384-393 L.Gao, S.Motsch, M.Strano, “Classification and analysis of tube hydroforming process with respect to adaptive simulations”, J. Mater. Proc. Technology, 129 (2002) 261– 267 H. Lee, C.J. Van-Tyne, D. Field, “Finite element bending analysis of oval tubes using rotary draw bender for hydroforming applications” J. Mater. Proc. Technology, 168 (2005) 327–335. R.R. Stange, “Tooling and methods for tube and pipe bending: the importance of proper equipment and proven techniques”, Tube Pipe J. (1997) 28–35, Nov/Dec. S. Kumar, P.P. Date, K. Narasimhan, “A new criterion to predict necking biaxial stretching”. J. Mater. Proc. Technology, 45, (1994) pp. 583. V. Nandedkar, “Formability Studies on a Deep Drawing Quality Steel”, Ph.D Thesis IIT-Bombay (2000).

682

A Compaarison of o Sheett Formiing Metthods foor 5754--O Alum minum Alloy A S Senthilkum mar Venk katesana, F Farhang Po ourboghrata*, John E E. Carsley yb a

Department of Mechanicaal Engineeringg, Michigan State University, East Lansinng, MI 48823, USA b Genneral Motors Research R &D Development Center, C Warren n, MI 48090, USA

Abstract. The goal off this research was to verifyy, through expeerimentation an nd numerical m modeling, thatt the sheet ming process iss a suitable alteernative to connventional form ming methods such s as stampinng for 5754-O aluminum hydroform alloy sheeets. Numerical analyses included implemenntation of a temperature-depeendent anisotroopic yield funcction (YLD 2000-2d) and corresponnding forming limit l diagram (FLD) into thee commercial FEM F code LS--Dyna as a useer material subroutinee (UMAT). Thhe temperature-d dependent mateerial model waas used in a coupled thermo-m mechanical finiite element model to assess the acccuracy of the UMAT’s U abilityy to predict bo oth forming and failure behavvior of AA575 54-O under various foorming conditioons [4]. Numeerical investigat ations proved th hat counteractin ng pressure useed in sheet hyd droforming drasticallyy improves the draw depth, irons out wrinkkles and delayss the onset of failure when ccompared to co onventional forming processes Keywords: Thermo-hyddroforming, AA5754-O aluminnum sheet, coun nteracting pressure PACS: 662.20.F

INT TRODUCT TION Sheet metal stamping s has the t advantagee of providing seamless partts in a cost-eff ffective manneer suitable for large production ruuns. Warm fo orming introdduces heat to the stamping process, whicch softens thee material thereby rrequiring lesss force to form m the part. The added heeat also makees the materiaal strain rate sensitive, leading to uniform thickness distrib bution and deeeper draws. Hydroforming H applies pressuurized, viscou us fluid to the surfaace of the sheeet metal, coun nteracting the punch force. The main ad dvantage of shheet hydroform ming over stampingg is that the pressurized flu uid supports thhe sheet metaal throughout the forming pprocess and also a forces the sheet metal to takee the shape of the punch. Thherefore, in hy ydroforming itt is not necesssary to use a feemale die, thus dram matically reduucing the tooliing cost. Therm rmo-Hydroform ming introducces heat to thee hydroformin ng process thereby aallowing all thhe advantagess of the warm m forming metthod to be com mbined with tthe advantagees of sheet hydroform ming.

EXPER RIMENTA AL AND CO OMPUTAT TIONAL OBSERVAT O TIONS Numerous exxperiments weere conducted at MSU in orrder to quantiffy the differennces between stamping, and hydrroforming proocesses. Thesee experiments [7] were con nducted primaarily using a 66” (152.4mm)) diameter cylindrical punch, butt a limited num mber of expeeriments were also conductted using a tap apered square punch. A picture of these two puunches and thee experimentaal setup is show wn in Fig. 1.

(aa) (b) (cc) FIGU URE 1. 6” diam meter cylindricall punch, (b) tapeered square pun nch, (c) stampin ng, warm formiing, hydroformiing and theermo-hydroform ming testing settup at MSU usin ng a 30-ton serv vo hydraulic pre ress. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 683-689 (2011); doi: 10.1063/1.3623673 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

683

To oobjectively andd quantitatively compare the peerformance of th he sheet formin ng processes, it was decided to o record for each form ming process thee maximum pun nch height achiievable prior to the onset of wrinkling or tearring of the sheeet. With the shape of tthe punch beingg fixed (i.e., cy ylindrical with a diameter of

D p  6" ), thee goal of the sttudy was to maaximize the

punch dispplacement ( Y p ) achievable fo or each of the pprocesses using available formiing conditions, such as the blaank holding

HF ), the blannk holding position ( BHP ) rrelative to the sheet, force ( BH s temperatu ures of the sheeet ( Ts ), punch h ( T p ) and die ( T d ),, temperature of o the fluid ( T f ) and the fluidd pressure ( P ). It should be noted that in thhis study the sh hape of the blank wass assumed to be b fixed as cirrcular but its ddiameter ( Ds ) was allowed to vary betweeen 11”-12” (2 279.4mm – 304.8mm)). In the next seection a brief summary of som me of the experrimental and nu umerical resultss obtained for each e of the processes will be presentted.

Stamping Figure 2 show ws a sample of o the punch fforce-displaceement curves for f 11.5” andd 11” blanks stamped at room tem mperature (777F) with con nstant blank hholding forcees ranging fro om 3000lb-100,000lb. The range of constant BHF was selected in a waay to avoid wrrinkling but maximize m the punch displaccement witho out tearing the sheett. It can be seeen that all the curves are siimilar in shape and the effeect of increasin ing BHF is to cause the sheet to ffail earlier. In the case of 11.5” blanks, thhe sheets faileed at a maxim mum punch dissplacement off about 1.1” for all BH HF. In the caase of 11” blaanks, the sheetts failed at lo ower punch diisplacements aas BHF was increased. i The maxximum punch displacement achievable prrior to tearing g the sheet ran nged from 0.669” to 0.98”. Numerical N results obbtained with LS-Dyna L 3D finite f element code confirm med experimental results. Stamping g (RT, Constant BHF, 11 1" Blank, AA5754)

Stamping (RT, Co onstant BHF, 11.5" Bllank, AA5754) 12000

25000

20000

8000

Punch Load (lbf) ( )

Punch Displacement (inch)

10000

BH HF=3000lb BH HF=4000lb

6000

BH HF=5000lb BH HF=6000lb BH HF=7000lb

4000

B BHF=3000lb

15000

B BHF=4500lb B BHF=5500lb 10000

B BHF=10,000lb

5000 2000

0

0 0

0.2

0.4

0.6

0.8

1

0

1.2

0.2

0.4

0.6

0.8

1

1.2

Pu unch Displacem ent (inc ch)

Punch Load (lbf)

( (a)

(b)

(c) FIGURE 2. Punch load vs. punch displlacement curvees for various co onstant blank ho olding forces obbtained from (aa) stamping m alloy sheets att room temperature (RT). (c) The T picture of a stamped 11” blank. b of 11.5” aand (b) 11” AA55754 aluminum

H Hydroforming Many sheet hydroforming h experiments were conduccted at room temperature t (R RT) and blan nk holding 684

forces (B BHF) with 111” and 11.5” AA5754 blannks. Unfortun nately, due to o the limited punch force and BHF capabilitiies of the MS SU’s 30-ton hy ydraulic servoo press, it wass not possiblee to reach the highest presssure levels required to prevent thee sheets from wrinkling at ddeepest drawss. Leaking pro oblems at fluidd pressures hiigher than 1500 psi prevented us from being able a to suppreess all the wrin nkles. To circcumvent this pproblem, finitte element simulatioons were perfoormed with LS S-Dyna 3D coode in order to o computationally determinee the optimum m pressure levels neeeded to supppress all the wrinkles w withoout tearing th he sheet. Finitte element sim mulations performed at lower preessure levels showed s that in n fact parts woould wrinkle similar s to expeerimental resuults. One of the distinct d characcteristics of thhe sheet hydrroforming is that t during thhe forming prrocess the pressurizzed fluid forcees the sheet metal m to conforrm to the shap pe of the puncch early in thee forming pro ocess, thus creating a large contaact surface to carry the form ming loads. This T large con ntact surface ddistributes thee forming loads oveer a larger suurface area, th hereby reducinng any chances of plastic deformation llocalizing at the t punch corner raadius. With coomputational design of thee optimum hy ydroforming pressure p profiile as a functiion of the punch dissplacement it is possible to form deep drrawn parts witthout wrinkling or tearing thhe sheet metall. Figure 3 show ws 11.5” blan nks hydroform med at RT, usiing various maximum fluidd pressures an nd BHF. It can be seen that withh a maximum fluid pressurre of 1200 pssi it is possib ble to draw thhe part to 1.5” without wrinklingg or tearing thhe sheet metaal (see Fig. 3cc). However, the t same part wrinkles on its side wall when w it is formed too a punch dispplacement of 2” using a higgher BHF and d pressure (seee Fig. 3b). Figgure 3a showss that even increasinng the maximuum fluid presssure to 1500 ppsi does not help h to preven nt the part from m wrinkling at a a punch displacem ment of 2.5”. Finite elemennt simulation of o hydroformiing was perforrmed with LS-Dyna 3D sofftware to find the limits of hydrooforming withh a cylindricall punch, sinc e experimental capabilitiess were limiteed to a maxim mum fluid pressure of 1500 psi, a maximum BH HF of 75,000 lbf, and a maaximum punch h force of 58,0000 lbf. Figurees 4 and 5 show thee FE simulatioon results for 11” and 11.5”” blanks hydro oformed with pressure and BHF profiles shown in Figs 4a-bb and 5a-b. Mechanical M pro operties of AA A5754 alumin num alloy sheeet used in thhe FE simulattions were obtained from previouus publicationss. Figures 4a and a 4b show the BHF annd pressure vs. v time proffiles used in the simulatio on of RT hydroform ming of an 111” blank. Figu ure 4c shows the predicted shape of the hydroformedd part at time increment t=25 ms or punch dispplacement of 1.89”. 1 Figure 4d shows the FLD contourr for this part indicating thaat part has failed byy tearing. Figuure 4e shows that t the same ppart can be hy ydroformed without w wrinklling or tearing g up to the time incrrement of t=224.5 ms or a punch displaacement of 1.85”. Figure 4f 4 shows the FLD of the same part indicatingg that it severrely thins but does d not tear. Figure 5d shoows that an 11.5” blank cann be drawn to o a higher max ximum punchh height of 1.8 89”, using the assum med pressure and BHF proffiles, without wrinkling or tearing the sh heet. This connfirms the exp perimental result shown for an 11.5” 1 hydrofo ormed blank iin Fig. 3c. Fiigure 5c show ws that the ppart will deveelop slight wrinkles at the maxim mum punch diisplacement oof 2.44”. Thesse numerical results show that on the average an 11.5” blaank can be draawn deeper th han an 11” blaank without wrinkling w or teearing, partly due to the larrger blank holding aarea. Figure 6, shows the reesults for RT hydroforming g of an 11.5” blank b using a higher BHF, as shown in Fig. 6aa. Figure 6b shows s the corrresponding prressure profile resulting in the higher BH HF. With thee modified BHF, it w was possible to t draw the sh heet metal to a maximum punch p displaceement of 2.244” without wrinkling or tearing thhe sheet, as shhown in Fig. 6c. 6 Figure 6d shows the FL LD contour forr the formed ppart. It can bee seen that the formeed part is sevverely thinned d but hasn’t faailed yet. In the t next sectio on, results forr thermo-hydrroforming process w will be presentted.

(a)

(b)

685

(c) FIGURE 3. Room tem mperature (RT) hydroformed pparts at variouss punch displaccements, fluid ppressure, and BHF; B (a) a maximum m pressure of 1500 1 psi was not n enough to prevent an 11..5” blank from m wrinkling at 2.5” punch dissplacement (BHF=25,,000 lbf (111 KN)); K (b) althou ugh smaller sidee wall wrinkless were formed, the maximum ffluid pressure of o 1500 psi was not ennough to prevennt this 11.5” blaank from wrinkkling at 2” puncch displacementt (BHF=39,0000 lbf (173 KN)); (c) with a maximum m fluid pressure of 1200 psi th his 11.5” blank was prevented d from wrinklin ng at a maximuum punch displaacement of 1.5” (BHF F=25,000 lbf (111 KN)).

Pre essure(Gpa) vs Time(ms)-RT 0.07

0.06

Pressure(Gpa)

0.05

0.04

0.03

0.02

0.01

0 2

3

4

5

6

7

8

9 10 11 12 13 3 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 2 33 34 35 Time(ms)

(a)

(b)

(c )

(d)

686

(e)

(f)

FIGURE 4. Room tem mperature (RT) simulation of ssheet hydroform ming of 11” blaanks with LS-D Dyna 3D FE so oftware; (a) BHF vs. ttime profile useed in the simulation; (b) correesponding fluid d pressure used; (c) the final sshape of the hy ydroformed part at tim me increment t= =25 ms or puncch displacemennt of 1.89” (BH HF=76,500 lbf, P=4687 psi); ((d) FLD showing that the part failedd by tearing; (e)) wrinkle-free shape s of the hyydroformed partt at time increm ment of t=24.5 m ms or punch dissplacement of 1.85” (B BHF=90,000 lbbf (400 KN), P= =4555 psi (0.03314 Gpa)); (f) FLD F of the part showing severee thinning witho out failure. Pressure(Gpa) vs Time(ms) 0.04

0.035

0.03

Pressure(Gpa)

0.025

0.02

0.015

0.01

0.005

0 2

3

4

5

6

7

8

9 10 11 12 2 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Time(ms)

(a)

(b)

(c)

(d)

FIGURE 5. Room tem mperature (RT) simulation s of shheet hydroform ming of 11.5” bllanks with LS-D Dyna 3D FE so oftware; (a) mulation; (b) coorresponding fluid pressure used; (c) slightlly wrinkled sh hape of the BHF vs. time profile ussed in the sim hydroform med part at timee increment t=32 ms or punch ddisplacement of 2.44 (BHF=112,000 lbf, P=44300 psi); (d) wrinkle-free w shape of tthe hydroformeed part at time increment of t= =25ms or puncch displacementt of 1.89” (BHF HF=118,000 lbf (524 KN), P=2900 pssi (0.02 Gpa)).

687

Pressure(Gpa) vs Tim me(ms)-RT 0.05

0.045

0.04

Pressure(Gpa)

0.035

0.03

0.025

0.02

0.015

0.01

0.005

0 2 3 4 5 6 7 8 9 10 11 12 1 13 14 15 16 17 18 19 20 21 22 2 23 24 25 26 27 28 29 30 31 32 33 34 35 Time (ms)

(a)

(b)

(c)

(d)

FIGURE 6. Room tempperature (RT) simulation of shheet hydroforming of 11.5” blanks with LS-D Dyna 3D FE so oftware; (a) BHF vs. time profile p used in th he simulation; ((b) correspondiing fluid pressure used; (c) impproved wrinkle-free shape modified B of the hyddroformed part at time incremeent of t=29.5mss or punch displlacement of 2.2 24” (BHF=1012250 lbf (580 KN N), P=4500 psi (0.031 Gpa)).

Sum mmary of Reesults The Table below b quantifi fies the resultts discussed in this reporrt for the sheeet forming processes; p stampingg and RT hydrroforming. Th he maximum punch displaccements reporrted below aree given for th hose cases where noo tearing or wrinkling occcurred. Also, for comparison, BHF an nd pressure att the maximu um punch displacem ment are reporrted. TABLE 11. BHF and presssure at the max ximum punch ddisplacement Proceess Maax. Punch Max. Punch o Increase Compared to Disp. Disp Stamping S

Increease Compared d to Stamping

111” Blank

11.5” Blank

11” Blank

11.5” Blank

Stampping

0.669”-0.98”

1.1”

-

-

RT T Hydrofoorming 77 F @ (BHF; Prressure

1.85”

2.24”

122%

104%

(90,000 lbf; 4 4550 psi)

(129,000 lbf; 4500 psi)

688

CONCLUSION Aluminum sheets are gaining importance in weight reduction applications mainly in automotive and aerospace industries. Currently there is a need to determine the forming process of aluminum sheets that can produce seamless complex parts with no wrinkles or tearing. The hydroforming process is one of the manufacturing methods that can be used to attain the desired results for various critical components. Experimental and numerical analyses were conducted to quantitatively evaluate various sheet forming processes. The experiments included studying stamping and hydroforming with constant and varying pressure loadings of AA5754-O sheets. Due to the limited capability of the experimental setup, some of the hydroforming experiments requiring high pressures were not performed. In these cases, numerical modeling was used solely to complete the study. Numerical analyses of stamping and hydroforming processes were conducted with LS-Dyna 3D FEA code with a temperature-dependent material model representing AA5754-O that was implemented into the FEA code using the UMAT option [1-5]. It was shown that the computational model was capable of accurately capturing the tearing and the anisotropic wrinkling characteristics of AA5754-O. Table 1 provides a quantitative comparison of the performance of the forming processes studied in this research based on the maximum amount of the punch displacement to draw the material into the die cavity without tearing or wrinkling the sheet. The authors believe that this is the most meaningful method to compare the performance of various sheet forming processes. For the part geometry with an 11” blank that was tested, it can be concluded that compared to stamping with hydroforming, it was possible to draw the same material up to 122% deeper than stamping because the hydrostatic pressure supports the majority of the blank and leaves very little unsupported areas that are prone to wrinkling. Furthermore, this pressure distributes strain more evenly than stamping which delays the onset of strain localization that leads to rupture. These effects are particularly important when forming parts that require sharp radii and features. These are impressive improvements in drawability when compared with stamping of aluminum sheets. It should be noted however that the results for hydroforming and were based on numerical simulations and were not all experimentally verified. In other words, it can be concluded that the sheet hydroforming process is the most ideal forming process for deep drawing of aluminum sheets with sharp radii features. With the additional drawability provided by the sheet hydroforming process, the automotive, aerospace and other industries would be able to make difficult-toform aluminum parts that cannot be stamped without product concessions such as increased die radii.

ACKNOWLEDGEMENTS The authors would like to thank General Motors for their generous support of this research project. The authors especially would like to sincerely thank Drs. Anil Sachdev, Jim Schroth and Tom Stoughton from the GM Research and Development Center for their assistance and helpful discussions in support of this research. The authors would also like to express their sincerest gratitude to NSF for its support of the thermohydroforming project through the MRI-0723090 grant.

REFERENCES 1. Abedrabbo, N., Pourboghrat, F., Carsley, J.E., 2007. Forming of AA5182-O and AA5754-O at elevated temperatures using coupled thermo-mechanical finite element models. Int. J. Plasticity 23, 841-875. 2. Abedrabbo, N., Pourboghrat, F., Carsley, J.E., 2006 Forming of aluminum alloys at elevated temperatures – part 1: material characterization. Int. J. Plasticity 22, 314-341. 3. Abedrabbo, N., Pourboghrat, F., Carsley, J.E., 2006. Forming of aluminum alloys at elevated temperatures – part 2: numerical modeling and experimental verification. Int. J. Plasticity 22, 342-373. 4. Simo, J.C., Ortiz, M., 1985. A unified approach to finite deformation elastoplasticity analysis based on the use of hyperelastic constitutive equations. Comput. Meth. Appl. Mech. Eng. 49, 221-245. 5. Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Lege, D.J., Pourboghrat, F., Choi, S.H., Chu, E., 2003. Plane stress yield function for aluminum alloy sheets – Part 1: Theory. Int. J. Plasticity 19, 1297–1319. 6. Marciniak, Z., Kuczynski, K., 1967. Limit strains in the processes of stretch-forming sheet metal. Int. J. Mech. Sci. 9 (9), 609–612. 7. Zampaloni, M.A., Pourboghrat, F., Yu, W.-R., 2004. Stamp thermo-hydroforming: a new method for processing fiberreinforced thermoplastic composite sheets. J. Thermoplastic Composite Mat. 17 31-50.

689

Cold and Warm Hydroforming of AA754-O Sheet: FE Simulations and Experiments Manan Shaha, Eren Billura, Partchapol Sartkulvanicha, John Carsleyb and Taylan Altana a

Center for Precision Forming, 339 Baker Systems, 1971 Neil Avenue, The Ohio State University, Columbus, Ohio 43210, USA b General Motors Research & Development Center, 30500 Mound Road, Warren, MI 48090 Abstract. The sheet hydroforming with punch (SHF-P) process offers great potential for low and medium volume production, especially for forming: (1) lightweight sheet materials such as aluminum (Al) and magnesium (Mg) alloys and (2) thin gage high strength steels (HSS). Mg and Al alloys are being increasingly considered for automotive applications, primarily due to their lightweight and high strength-to-weight ratios. However, there is limited experiencebased knowledge of process parameter selection and tool design for SHF-P of these materials. Thus, there is a need for a fundamental understanding of the influence of process parameters on part quality. This paper summarizes analyses of the SHF-P process of AA5754-O sheet using finite element (FE) simulations. FE simulations and preliminary experiments of SHF-P were conducted to determine the process parameters (blank holder force versus punch stroke and pot pressure versus stroke) to form a challenging shape (a cylindrical cup with a reverse bulge) successfully at room and elevated temperature (~150°C). The material properties of the sheet material were obtained from tensile tests at room temperature up to 260°C as presented by [1]. The FE model was established using PAMSTAMP 2G, Version 2009. SHF-P experiments were conducted in order to (i) evaluate the formability of the part at room and elevated temperatures and (ii) validate FE simulation results. This study shows that the SHF-P at elevated temperature can form a cup with larger cup height and better reverse bulge profile than SHF-P at room temperature. Moreover, the FE predictions of part profiles and thinning distributions matched reasonably well with the experimental results. Keywords: sheet hydroforming, AA5754-O aluminum sheet, counteracting pressure, optimal pressure profile, finite element simulation PACS: Replace this text with PACS numbers; choose from this list: http://www.aip.org/pacs/index.html

INTRODUCTION Manufacturing technology development of materials and processes for automotive body and structural applications has been focused on mass reduction to improve vehicle fuel economy and emissions. While advanced high strength steels provide significant opportunity to optimize mass with performance for the passenger compartment structure, lighter weight materials such as aluminum (Al) and magnesium (Mg) offer potential for significant mass reduction of body and closure panels. For example, it was reported that a 10% weight reduction could improve the fuel efficiency by 6-8% for an average automobile [2]. Krajewski et al. described opportunities for implementing wrought, cast and extruded Al and Mg materials throughout the automobile [3]. However, fabrication methods with such wrought materials can be challenging because they are difficult to form into complex shapes at room temperature. Since the deformation behavior of some Al sheet alloys is greatly enhanced at elevated temperatures of 150-300°C [1], development of warm forming technologies for Al and Mg sheet alloys could enable production of complex panels requiring deep draws. Warm stamping and warm hydroforming are two such processes that have received considerable attention recently [4, 5]. For the sheet hydroforming with punch (SHF-P) forming method, a sheet blank is deep drawn against a counter pressure from compressed fluid inside the pot, as presented in FIGURE 1, rather than against a female die as in conventional stamping operations. The medium in the pressure pot can be either “passive” (pressure generated due to incompressibility of the medium during forward stroke of the punch) or “active” (pressure generated by an external pump) as defined by [6]. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 690-697 (2011); doi: 10.1063/1.3623674 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

690

FIGURE 1. Schematic illustration of the SHF-P process [6]

For the warm hydroforming method, the sheet and the flange portion of the die and the blank holder are heated to the required temperature. The pressurizing fluid may be maintained at slightly higher than room temperature, while the punch is cooled. During the process, the lower temperature of the punch cools the portion of the sheet that is in contact with the punch and increases its load carrying capacity. Thus, failure caused by excessive thinning is postponed and the process yields a higher limiting draw ratio (LDR) than those obtained from deep drawing at room temperature [4, 5] The most common defects encountered during the SHF-P process are wrinkling, excessive thinning (leading to fracture) and leaking of the pressurized fluid during forming. Conventionally, the process parameters are estimated by performing trial and error experiments which requires considerable time and effort. Therefore, the present research focuses on the use of finite element (FE) simulations along with physical experiments to estimate the optimum blank holder force (BHF) vs. punch stroke and optimum pressure vs. punch stroke that is needed to form a part successfully. This preliminary study assumes isothermal conditions in the analysis of SHF-P at elevated temperatures (meaning all tooling components, blank and fluid were heated to approximately the same temperature). The benefit of a cooled punch was not investigated in this study.

OBJECTIVE / APPROACH The objective of this study was to improve the hydroforming process of aluminum and magnesium alloys using FE simulations and validated by experimentation. More specifically, the objective was to establish an efficient method for estimating the process parameters (BHF, pot pressure, punch velocity) that are used for room and elevated temperature SHF-P processes to produce defect-free parts. The following tasks were included in the approach to determine the process parameters: 1) Fluid pressure and temperature control systems of the machine setup were evaluated for accuracy and sensitivity to control pressure and temperature during a forming operation. 2) SHF-P experiments were conducted at room and elevated temperatures. 3) Formed samples were measured using a coordinate measuring machine (CMM) and height gauge measurement devices. 4) A FE model was created to determine the effects of various SHF-P process parameters and validate replacing the experimental trial and error with simulation.

EXPERIMENTAL APPARATUS The schematic of the tooling is presented in FIGURE 2. The lower die consisted of three main components including the pressure pot, lower post (fixed in the center) and blank holder. The upper die consisted of the punch and the spring setup which allowed the punch to travel to a maximum stroke of 1.5 in (38 mm). The purpose of the lower post was to create a reverse bulge at the center of the deformed part when forming a part by conventional deep drawing without using fluid pressure. However, necessary modifications of the deep drawing tooling restricted the travel distance of the punch. In these SHF-P tests, the lower post did not touch the sheet and the fluid pressure can be considered as the only active force that forms the reverse bulge into the center recess of the punch.

691

Punch

Blankholder Blank Die

Post

ri h

(a) Initial setup

Dd

(b) After Deformation

FIGURE 2. Schematic of the tooling for SHF-P experiments: (a) initial setup (b) after deformation, where ri = initial blank radius, Dd = draw depth and h = bulge height.

A Mokon heating system was used to control the forming fluid temperature. With the fill and vent valves open, fluid will circulate through the tool. The system includes a reservoir that maintains fluid at temperature and available for filling and draining the tool cavity through the fill and vent valves. Dynalene 600, a synthetic oil, was used as the fluid for building up forming pressure in the pot cavity. Prior to forming tests at room and elevated temperatures, preliminary FE simulations were conducted to establish initial test parameters (BHF, punch velocity, and pot pressure) that could be used as experimental inputs. The sheet material was AA5754-O with an initial diameter, di, and thickness, ti, of 17 in (432 mm) and 0.039 in (1.0 mm), respectively. Table 1 lists the input parameters for the room temperature experiments.

TABLE 1. Input parameters for SHF-P tests at room temperature (punch velocity=0.075 in/sec) Sample #

BHF (kip)

Pot Pressure (psi)

Punch stroke (inch)

5

5.6

300

1.35

6

5.6

300

1.13

In the preliminary FE simulations of the room temperature SHF-P process, the pot pressure was maintained at approximately 300 psi (2.07 MPa) for various BHF of 5.6, 11.25 and 33.8 kip (25, 50 and 150 kN). With the downward movement of the punch, the punch displaced oil causing an increase in counter pressure. As this pressure approached the set limit of 300 psi (2.07 MPa), the relief valve cycled to maintain a steady pot pressure. In the case of elevated temperature deformation, the maximum pot pressure input for the relief valve was set to limiting pressure levels as listed in Table 2. Initially, a pot pressure of 300 psi (2.07 MPa) and BHF of 5.6 kip (25 kN) were used similar to room temperature SHF-P, but the part consistently failed around the punch corner region. Therefore, new process conditions were used by applying a lower BHF, around 1 kip (4.45 kN), and increasing pot pressures. A set of samples were formed at a constant BHF and draw depth of 0.56 in (14 mm), but with variable pressure limits of 500, 1000 and 2000 psi (3.4, 6.9 and 13.8 MPa), to study the effect of forming pressure on the part shape. For the pressure limit of 2000 psi (13.8 MPa), the sample parts were further drawn to depths of 0.7, 1.0 and 1.5 in (17.8, 25.4 and 38.1 mm), to measure the amount draw-in that occurred during forming.

692

TABLE 2. Input parameters used to conduct warm SHF-P test at elevated temperature of about 150°C (experimental test with maximum cycle time = 20 sec) Sample # Pot Pressure (psi) BHF (kip) Punch Stroke (inch) 21 500 1 0.56 27 1000 1 0.56 28 2000 1 0.56 29 2000 1 0.70 30 2000 1 1.00 33 2000 1 1.50

EXPERIMENTAL RESULTS Table 3 compares input (entered at the control panel) and output values (BHF and pot pressure measured from sensors inside the press machine) and the measured draw depth, Dd, of the formed part. Since the output values of BHF fluctuated throughout the punch stroke, an average value of BHF was calculated. Measured flange perimeter, Ff, draw depth, Dd, and the bulge height, h, are presented in Table 3.

TABLE 3. Comparison of experimental input to the output readings and measurements at room temperature (initial flange perimeter, fi= 53.41 in) EXPERIMENTAL INPUT EXPERIMENTAL OUTPUT Pot Pot Flange Draw Bulge Sample BHF Punch BHF Pressure Pressure Perimeter, Ff Depth, Dd Height, h # (kip) stroke (in) (kip) (psi) (psi) (in) (in) (in) 5 5.6 300 1.35 6.78 304.7 49.1 1.349 0.353 6 5.6 300 1.13 6.81 307.5 50.1 1.083 0.333 Table 4 compares the input and output values (BHF and forming pressure) and the measured draw depths, D d, of the samples formed at elevated temperature. The steady BHF output obtained from the experimental results is higher than the experimental input value. This may be because the BHF (clamp load) was very small relative to the press capacity. Also, this load was measured using pressure transducers, which are placed on either side of the actuator‟s piston and the cross sectional areas of the piston areas are then considered to calculate the load. Thus, it gave only approximation for measurement of a very small load. The comparison of output values of pot pressure for samples 21, 27, and 28 indicates that a maximum of 842.5 psi (5.8 MPa) was needed to draw this part to a depth of 0.56 in (14 mm). In order to draw the part to a depth of 1.5 in (38 mm), the required pot pressure increased to 1494 psi (10.3 MPa), keeping the BHF to the same value of 2.2 kip (9.8 kN). TABLE 4. Comparison of experimental input and output readings and profile measurements, for SHF-P experiments at elevated temperature (Initial flange perimeter, Fi = 53.41 in) EXPERIMENTAL INPUT EXPERIMENTAL OUTPUT Flange Draw Sample BHF Pot Pressure Punch Stroke BHF Pot Pressure Perimeter, Depth, # (kips) (psi) (inch) (kips) (psi) Ff (in) Dd (in) 21 1 500 0.56 2.19 595.3 52.3 0.578 27 1 1000 0.56 2.21 834.6 51.9 0.570 28 1 2000 0.56 2.21 842.5 52.1 0.573 29 1 2000 0.70 2.23 977.5 51.8 0.715 30 1 2000 1.00 2.19 1140.1 50.2 1.025 33 1 2000 1.50 2.22 1494.4 47.9 1.514

The temperature of the tooling (die, blank holder and punch) was controlled by two dual channel Watlow F4 temperature controllers using resistance heaters. The fourth Watlow channel was used to monitor the oil temperature with a thermocouple positioned inside the lower die. Thermal data was collected for each component of the tooling using thermocouples that were embedded in the tooling. Due to the constraints of the machine, it was very difficult to obtain isothermal conditions within a 10C range. Results indicate that the temperatures of the blank holder, die and oil ranged from 127 to 140C, while the punch temperature was measured to be 157C.

693

NUMERICAL ANALYSIS The part geometry was designed at General Motors as representative of a challenging feature to form with aluminum by conventional stamping. The AA5754-O sheet blank was 0.039 in (1 mm) in thickness (ti) and 17 in (432 mm) in diameter (di). The FE model was created using PAMSTAMP 2G, Ver. 2009, which enabled input of the material model as a function of temperature and strain rate simultaneously. Only one quarter of the geometry was modeled since the tooling and part geometries were axisymmetric. For all temperatures, the process was modeled using the Aquadraw Module in PAMPSTAMP 2G, which allows efficient control of pressure input in the tool cavity, based on compressed fluid volume and maximum pressure limit to be applied on the blank. To create a simplified model and replicate the elevated temperature deformation, an „isothermal condition‟ was assumed. By approximation, the FE model was set up at 150C (whereas temperature readings of tools and fluid ranged from 127 to 157C). The input parameters used during the simulation are summarized in Table 5. The major difference between simulation of this process at room and elevated temperature conditions is the input of material properties. As the material at elevated temperature is strain rate sensitive, the deformation speed at different regions of the blank in the simulation should take into account the influence of strain rate. Abedrabbo et al. (2007) conducted tensile tests and applied the Fields and Backofen‟s material model (power law equation) to describe the stress-strain behavior of AA5754-O as show in Equation (1). Flow stress data was input as a function of strain, strain rate and temperature, in tabular format in PAMSTAMP 2G.

  K n m

(1)

Where: K = 503.7-0.592*T (for T = 25-93C) and 641.3-1.829*T (for T = 93-260C) n = 0.3304 -0.000529*T (for T = 25-93C) and 0.4048-0.001192*T (for T = 93-260C) m = 0.00118*exp(0.0161*T) (for T = 25-260C) TABLE 5. Input parameters for FE simulations of SHF-P. Mechanical Properties Blank material

AA5754-O

Flow stress (obtained by tensile test)

(Abedrabbo et al. 2007)

Young‟s Modulus (E)

69 GPa

Poisson‟s ratio (ν)

0.3

R0, R45, R90

1 (material assumed isotropic)

Interface Condition Friction coefficient µ (blank/ tools)

0.12

Mesh Element type

Shell (Belytschko-Tsay )

Object type Blank

Elastic, Plastic

Tools

Rigid Sheet Hydroforming with Punch (SHF-P)

Aquadraw

Activate

Bulk Modulus, K

50 GPa

Pot Pressure

Variable

BHF Punch stroke

Variable Variable

Punch Velocity

0.074 in/sec

694

COMPARISON OF FE PREDICTIONS WITH EXPERIMENTAL RESULTS The results from room temperature FE simulations have been summarized below for Sample #5 (BHF = 6.8 kip, pot pressure = 304.7 psi and punch stroke = 1.35 in). The results compared in FIGURE 3 indicate that the flange perimeter measurements, Ff from the experimental and FE predictions match very well. FIGURE 4 presents the thickness distribution along the curvilinear length of the part. Note that the maximum thinning location occurred near the punch corner region (location #4) for the part formed at room temperature.

FE Output Measured

Flange Perimeter (in)

52

1372

54

1321

52

50

1270

48

1219

46

1168

44

Initial Perimeter Perimeter

SHF-P at Room Temp Temp SHF-P at Elevated Elevated Temp Temp (Sample #33) (Sample #5)

1118

Flange Perimeter (mm)

54

50

48

46

44

Initial Perimeter Perimeter

FIGURE 3. Comparison of flange perimeter measurement and prediction between initial blank and final part after SHF-P at room and elevated temperatures. 1.10

2

1.06

Maximum Thinning Location

1

5

Thickness (mm)

1.02 2

0.98

6

1

3 9

4 5 6

7

7

8

4 3

0.94 Thickness profile along the curvilinear length for Sample 33 (BHF= 2.22 kip, Pot Pressure limit= 2000 psi, Punch Stroke= 1.50 in)

0.90 1.08

0.86 1.06 1.04

FE Output Measured

Thickness (mm)

0.82 1.02 1

0.78 0.98 0.96

0

25

0.94

50

75 100 125 150 Curvilinear Length (mm)

175

200

225

0.92 FIGURE 4. Thickness distribution along the curvilinear length for the part formed at room temperature (BHF = 6.81 kip, pot 0.9 pressure = 304.7 psi) 0.88 0

50

100 150 Curvilinear Length (mm)

200

250

Sample #33 (BHF= 1 kip (4.45 kN),Experimental pot pressure = 2000 psi (13.8 MPa) and punch stroke=1.5 in (38 mm)) was measurements with Error bar FE profile selected for FE analysis at elevated temperature. The results of SHF-P at the elevated temperature of 150C are presented in Table 4. There was only a small difference (0.5%) in the values of the final flange perimeter determined by FE simulation and experimental measurement. The BHF prediction of the FE simulation for Sample #33 indicates some fluctuation, but the average value remains close to the experimental output. The FEA input of BHF force did not work in PAMSTAMP because forming pressure is balanced by blank holder only, but in reality the punch force adds to BHF to balance the forming pressure. So in simulation, the forming pressure caused the blank holder to open up. Therefore, input clamp/blank holder stroke (instead of BHF) that was recorded in ITC data file. In putting clamp stroke provided simulation that matched experiments. Another set of tests were simulated to determine the effect of pot pressure on the formed part geometry. The formed samples are # 23, 27 and 28, for BHF ~2.2 kip (9.8 kN), cycle time = 20 sec, 0.56 in (14 mm) punch stroke and variable pressure limits of 500, 1000 and 2000 psi (3.4, 6.9 and 13.8 MPa). From Table 6, it can be inferred that with an increase in the input pot pressure limit, the maximum thinning along the part would increase. However, there is not a significant experimental difference in percent thinning between 1000 and 2000 psi (6.9 and 13.8 MPa),

695

SHF-P at Room Te Te (Sample #5)

because the physical output for the maximum pressure only attained about 840 psi (5.8 MPa), regardless of the value input at the control panel. In order to form a part at elevated temperature (ET) for a draw depth (D d) of 0.56 in (14 mm) and 2.2 kip (9.8 kN) BHF, a maximum of 592.5 psi (4.1 MPa) pot pressure should be used to achieve the least thinning percentage ~ 6.5%. TABLE 6. Percent thinning affected by the applied pot pressure while keeping the other process parameters constant (BHF ≈ 2.2 kiPs, punch stroke = 0.56 in, cycle time = 20 sec, average temperature = 150°C)

7

--

--

--

--

--

40 6.54 8.51 35 8.56

500 1000 2000

Max. location 3 thinning 4 5 6 Reverse bulge region Reverse bulge region Reverse bulge region

FEA CMM

30

Part Depth (in)

23 27 28

2

--

Max. thinning 0 (%) 1

--

Set pot pressure (psi)

--

Sample #

Max. output pot pressure (psi) --1.60 592.5 --1.40 834.6 842.5 --1.20 --1.00

25 FIGURE 5 is a comparison of the part profile measurement using the CMM and the FE results of SHF-P for --0.80 20 sample #33. For the CMM measurement of the experimental sample, a significant bend in the reverse bulge region --0.60 contraction of the part was observed. This bend in the part may 15 be due to springback and distortion from thermal after part was removed from the press and cooled to room --0.40 10 temperature. 5

--0.20

0

--0.00

7

Dd = Draw Depth

30 25 20 15 10 5

h  Bulge height

0 0

25

50

75 100 125 Radial Distance (mm)

(b)

150

175

0

--1.60

40

--1.40

35

--1.20

30

--1.00 --0.80 --0.60

20 15 10

--0.20

5

--0.00

0 0

25

FIGURE 5. (a) experimental workpiece, (b) Pam-Stamp result of quarter model, (c) comparison part profile from FEM and experiment, for the SHF-P of Sample #33 (punch stroke = 1.50 in, average temperature = 150C, pot pressure = 840 psi) 1.10 1

Thickness (mm)

1.06

5

1.04

2

1.02

2

FE Output Measured

1 6

1

3 9

4 5 6

7

7

8

4 3

5

1.00

6

7 8

2

0.98

Maximum Thinning Location

Thickness profile along the4curvilinear length for Sample 33 (BHF= 2.22 kip, Pot Pressure limit= 2000 psi, Punch Stroke= 1.50 in)

0.96 1.08

3

9

1.06

0.94 1.04 Thickness (mm)

1.02 0.92 1

0.90 0.98 0 0.96

25

50

0.94

75 100 125 150 Curvilinear Length (mm)

175

200

225

0.92 FIGURE 6. Thickness profile along the curvilinear length measured by the mechanical indicator and compared with FE results 0.9 for Sample #33 (punch 0.88 stroke = 1.50 in, average temperature = 150C, pot pressure = 840 psi) 0

50

100 150 Curvilinear Length (mm)

Experimental measurements with Error bar

696

200

FE profile

250

3

FEA CMM

(c)

1.08

2

25

--0.40

200

1

--

6

--

5

--

--

--

--

4

FEA CMM

35 Part Depth (mm)

3

200

--

2

175

Part Depth (in)

1

150

--

0

40

75 100 125 Radial Distance (in)

--

50

--

25

--

0

--

(a)

50

75 1 Radial Dis

For comparison of the thickness profile generated from the FE simulations, the mechanical indicator (Mitutoyo) was used to measure thickness along the curvilinear length of the formed sample. FIGURE 6 presents the comparison of the thickness profile of Sample #33 formed at elevated temperature. The thickness profile for the FE simulation matches fairly well with the measurements. FE results predict that maximum thinning occurred near the reverse bulge region (position 8) along the curvilinear length.

CONCLUSIONS FE modeling of the SHF-P process at room and elevated temperatures with the assumption of isothermal conditions was completed using PAMSTAMP 2G Ver. 2009. Simulation results compared favorably with experimental measurements. The experimental values of the process parameters (i.e. fluid pressure) were input into the FE model to emulate similar conditions. The final flange perimeter (Ff) predictions for both room and elevated temperature were within ± 0.5% of the experimental results. For room temperature, the part profile prediction matched the experimental measurement, whereas for the elevated temperature case, there was some difference in the part profile in the reverse bulge region. This difference could be caused by artifacts of springback and thermal distortion as the part cooled to room temperature or by thermal gradients within the part during the forming process. The maximum thinning for the room temperature drawn part was predicted to occur near the punch corner region. For elevated temperature, however, the maximum thinning location occurred within the reverse bulge region of the part. Simulation of thickness strain or thinning distribution across a curvilinear length of the part matched reasonably well with the experimental measurements. This study also shows that the SHF-P at elevated temperature can form a cup with larger cup height and better reverse bulge profile than SHF-P at room temperature. Since the tooling (blank holder, lower die, punch and fluid) temperatures during the actual forming process were not isothermal, ranging from 127 to 157C, elevated temperature FE simulation was attempted using a nonisothermal model that considered heat transfer. In this non-isothermal model, the initial temperature conditions and the heat transfer coefficients were the required inputs of the FE model, but this exceeded the capability of the FE code. The current software used for this study did not allow modeling of the hydroforming process (using the Aquadraw module) and heat transfer simultaneously. There was also difficulty to define the heat transfer condition between the fluid and the blank under non-isothermal conditions. ESI Group (USA) is improving the capability of the PAMSTAMP code in order to model the warm SHF-P process. Future research in the continuation of this study would include additional experimentation and FE analysis at higher forming temperatures (250C) and warm forming of other materials (e.g. Mg alloys) using solid dies. Furthermore, non-isothermal conditions will be modeled to study the benefit of using a punch that is cooler than the remaining tooling environment to enhance warm forming of metal sheet.

REFERENCES 1 N. Abedrabbo, F. Pourboghrat and J. Carsley, " Forming of AA5182-O and AA5754-O at elevated temperatures using coupled thermo-mechanical finite element models", Int.J.Plast. 23 (5), 841-875 (2007). 2 R.A. Schultz, "Aluminum for Light Vehicles: an Objective Look at the next Ten to Twenty Years". In: Metal Bulletin 14th International Aluminum Conference, 1999. 3 P. E. Krajewski et al., "Automotive Aluminum and Magnesium: Innovation and Opportunities," Light Metal Age, 67, 6-13 (2009). 4 M. Koc, A. Agcayazi and J. Carsley, "An Experimental Study on Robustness and Process Capability of the Warm Hydroforming Process," Journal of Manufacturing Science and Engineering 133 (2), 021008 1-13 (2011). 5 S. Venkatesan et al., "Quantitative comparison of hydroforming and stamping of 5754-O aluminum alloy sheet at room and elevated temperature," J. Mat. Proc. Tech , (in review). 6 M. Aust, "Modified hydromechanical deep drawing", in Hydroforming of Tubes, Extrusions, and Sheet Metals, edited by K. Siegert, 2001, pp. 215-234.

697

Possibilities, Challenges and Risks in Creating Material Cards for Forming Simulations L. Kesslera, J. Gerlacha, T. Beiera , H. Grassb, I. Heinleb and A. Lippb a

ThyssenKrupp Steel Europe AG, Research and Development, Kaiser-Wilhelm-Straße 100, Duisburg, 47168, Germany b BMW AG, Knorrstraße, 147, Munich, Germany

Abstract. The fit of complex yield locus models with different experiments is often not unique and the quality of modeling can not be derived directly. To overcome this problem a set of simple material tests are proposed, one is discussed in detail. This test allows the individual estimation how realistic yield locus results are and is also adequate to calibrate the free yield locus exponent. The in literature given exponent of M=6.0 for steels can not be confirmed here. Better is a range of M=4.5 to M=5.0 in combination with a strain rate depending material hardening for some steels, but not in general. With the help of an optimization strategy in combination with an inverse model calibration the yield locus exponent, friction and material hardening are automatically determined for a mild steel grade. Finally the results of the validation experiments are confirmed in a study for a side wall frame produced with the same material grade. Keywords: FEM, yield locus models, strain rate dependency, necking, material hardening. PACS: 81.20.Hy ; 81.40.Lm; 62.20.F-

INTRODUCTION Forming simulation has developed to a fundamental part of manufacturing processes during the last two decades. Beside to be a glutton for work in the areas of die design and method planning in the tool shop phase more and more decisions are based on the forming simulation feasibility prediction when determining the general forming potential of different materials in early concept phases. In the second mentioned, the limits for component design are studied with numerical try out phases for different material types, but solely based on the material model selection. A further field of application for forming simulations can be identified in the estimation of process robustness influenced by material scatter or process parameter variations. As a consequence the demands for an even more realistic prediction of the real world forming behavior for all kind of materials are often pronounced. Several opportunities and challenges exist to model material grades for every day practice. Nevertheless the key issue for industrial application has to be the adequate interpretation of results when comparing different material grades.

Material models in sheet metal forming simulations The application of material models for industrial simulations has to be done with respect to some important issues and conditions. The total setup of the numerical simulation in tool and die engineering has to be a compromise of cost, reliability of the results; the exactness wished for the phase of engineering and of course affords to introduce a new material in an existing data base. Components of the material models for forming simulations are isotropic material hardening given by a yield curve, a yield locus formulation in plane stress condition and partly a strain rate dependent material behavior. For special applications like spring back prediction an isotropic-kinematic hardening is sometimes used, but not discussed in this paper. The material hardening formulations of Swift or Ludwig have been applied for a long time. Due to the development of new steel as well as aluminum grades several formulations or mixture functions are used today. All findings in this paper are based on a hardening extrapolation by means of the ThyssenKrupp Extrapolation Method /1/. Here a hydraulic bulge test is recalculated in combination with a selected yield locus. This resulting hardening

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 701-708 (2011); doi: 10.1063/1.3623675 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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TABLE 1. Overview for available yield locus models and the necessary input parameters Level Number of Independent Input values from tensile test parameter exponent R00 R45 R90 V00 V45 V90 I 4 X X X X Hill ´48 II 4 X X X X Hill ´90 II 4 X X X X Barlat ´89 III 8 a X X X X X X Barlat 2000 III 8 M X X X X X X Banabic 2005

Yield locus model

Additional test(s) rB VB X X X X

X X

curve is valid for all yield locus models which have the same ratio of the biaxial stress point to the uniaxial stress value. Looking back, the yield locus model of Hill ´48 /2/ is for steel grades the most used formulation in the European industrial forming simulation practice. One of the advantages is the easy calibration by data of tensile tests in three angles to rolling direction. Because of quite some simplifications of the model, some material characteristic values of stresses Lankford coefficients are not always expressed in an acceptable tolerance. This of course is material specific. With the models of Hill ´90 /3/ and Barlat ´89 /4/ proposals for non-quadratic and by this, more flexible yield loci exist since two decades. In addition to this, even more complex models have been published and implemented in commercial software; for example Barlat 2000 /5/ and Banabic 2005 (BBC 2005) /6/. In Table 1 a selection of yield locus models, the required input parameters and a ranking of the model fitting complexity is given. Only the model of Hill ´48 can solely be calibrated with the tensile test data. All other models need at least an additional point in biaxial stress space or shear stress point. The with level II marked models allow a well defined fitting, when having this additional stress point in biaxial stress space. In contrast to this the level III models have one independent parameter that allows a shaping of the yield locus without interacting or influencing the biaxial stress point. A calibration of this parameter (exponent of the yield locus) can be done by using the shear stress point information or an assumption according to crystallographic considerations. The shear point information is seldom accurate enough to allow a definite selection of the exponent. Even when this method is possible for some materials, others exist which do not allow the final shape fitting with this extra point. An example is given in /7/. Some authors have already suggested a constant exponent (M=6, a=6) for steel grades and of M=8 for aluminum /5, 6, 8/.

Experimental testing for material models calibration The classical procedure for yield locus calibration is the combination of different experiments. All methods base on the tensile test to gather information for uniaxial hardening and the material anisotropy. The Lankford coefficient is usually determined in a strain range of 10 – 20 % strain. Most stress information for the yield locus calibration is used at the individual yield stress or at a constant defined plastic work. This data is later combined with different strategies. Among the additional experiments the hydraulic bulge test or stack compression test are favorite experiments for the biaxial yield information. Both are up to date not standardized. In general the yield locus experiments can be accomplished with a shear test and a plane strain tension test in different directions. An outcome of research work published in /9/ was the finding that the additional experiments do not always allow the final fitting of the independent exponents. One of the major draw backs has been analyzed to be the exactness of the measurement, the impact of strain rate and the identification of stress combinations in the planes strain tension test /10/. In Fig. 1 a) the process sketch is given for creating material cards for industrial forming simulations. The traditional way with Hill ´48 has mostly concentrated on the straight forward strategy. The individual measurement are directly transformed to a material card and used in praxis. Even making a validation step did not offer too many opportunities to modify the material cards with only limited free parameters. The more complex models need first of all more input parameters and have in a second step also more chances to calibrate the numerical response without changing the fundamental test data like hardening and anisotropy. This possibility is accompanied with an unknown risk of having unrealistic effects in areas, which are not directly covered by the measured values. An aim of this process is nevertheless to find one unique material card which can be applied for all kind of forming conditions in practice. As small changes in the yield locus description can lead to a complete change of the forecast for formability (Fig. 1b)) an additional validation step is necessary to assure the material card quality. The challenge is to find a set of experiments that allow the target-oriented checking or fitting of the free yield locus parameters in combination with the chosen material hardening. In a first step the validation should be done with a focus on the pure material behavior. Most of boundary conditions should be eliminated. A risk in this validation process is the insensitivity of

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some experiments regarding the expected answers of material modeling /9/. A finalizing step might be the testing of material cards inside the complete numerical environment.

FIGURE 1. a) Different strategies to fit material models for forming simulation. B) Impact of a yield locus calibration with different models on the feasibility result for a door inner with mild steel

A MATERIAL MODEL VALIDATION PROCESS In the following some selected examples are presented to demonstrate the challenges and risks in creating material cards for forming simulations. A special emphasis is laid on the yield locus description with level III models and the comparison to especially developed experiments.

Challenges in Yield Locus Calibration Using Different Models For all the material discussed here, the data of uniaxial tensile tests as well as the hydraulic bulge test with circular blanks has been available. This data has been prepared for the modeling according to /1/. A special interpretation of the biaxial anisotropy rb has not been done. The value was assumed to 1.0 and the impact of this value is only sporadically checked. In Fig. 2 the plots of different yield loci are illustrated exemplarily for three different steel grades. The DC 06 is an interstitial free mild steel with extra deep drawing quality and relative high Lankford coefficients. The HX260YD is a high strength IF-steel with an average Lankford coefficient rm = 1.7. Finally the HX420LAD is a cold rolled microalloyed high strength steel grade. To generate the yield loci plots the Lankford coefficients and individual initial yield strengths in 0°, 45° and 90° to rolling direction and the biaxial stress point from the bulge test (average factor to rolling direction) are considered. A basis for the exponent of the Barlat ´89 fitting is the biaxial stress point. Thus the factor may vary for each material grade. Here we have a range of 3.0 to 4.0. In addition to this the literature given constant value of 6.0 is applied in the model of Banabic 2005. A use of the Barlat 2000 model would give the same tendency in the yield locus plot. To discuss the influence of the free parameter M or parameter a the value of 5.0 is also investigated. It is getting obvious, that a direct determination of a yield locus exponent requires exact experimental measurements to identify the differences in the plane strain areas. In Table 2 the differences in transverse direction are calculated in comparison to the Banabic 2005 model with M=6.0. Choosing an average hardening curve (not only the longitudinal stress strain curve) would result in other values for the differences in plane strain for both principle directions. Even knowing the biaxial stress point, a material card can not be identified directly with only these measurements. Furthermore the selection of an exponent is hardly able by additional experiments for all the steels shown here. A maximum difference of only 4% might bear a risk in calibrating a yield locus only with this information.

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FIGURE 2. Yield locus model calibration using tensile and hydraulic bulge tests for a) DC 06, b) HX260YD and c) HX420LAD TABLE 2. Stress differences calculated for yield locus models in transverse direction plane strain area Material 1) BBC 2005 M=6 to Barlat ´89 2) BBC 2005 M=6 to M=5 DC 06 10 MPa 4 MPA HX260YD 17 MPa 7 MPa HX420LAD 45 MPa 10 MPa

Validation of Material Cards To overcome the above mentioned challenges of individually identifying the adequate yield locus for each material special experiments have been developed in a common work of BMW Group and ThyssenKrupp Steel Europe. Three different experiments have been identified, referenced as Yield Locus Identification Tools (YLIT 1, 2 or 3) /10/. The general idea of the experiments consists of the individual test of each orientation to rolling direction separately. Furthermore in contrary to the normal yield locus calibration, the onset of necking is part of the material card identification procedure. In Fig. 3 one general tool setup is shown (Fig. 3a)) as well as the corresponding specimen with the onset of localized necking (Fig. 3b)). The advantage of the experimental setup is a continuous recoding of all specimen positions with strain analyzing systems.

FIGURE 3. a) Experimental setup for the yield locus identification test 1, b) Specimen with onset of necking

In the following the example of the high strength IF-steel is discussed in detail. This steel shows differences in the yield locus mostly in plane strain area, whereas the stress fit in all other directions is in good agreement with the experimental measurements. The Lankford coefficients in 0° and 90° to rolling direction differed by around 0.7. As a result the in Fig. 4 illustrated position of rupture and the ultimate drawing depth at failure are quite different for both directions. A well balanced material card for forming simulation should allow the prediction of the right position of the onset of necking as well as the corresponding drawing depth in parallel. The criterion for an onset of necking is always defined by one forming limit curve.

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FIGURE 4. Specimen of the YLIT 1 experiments for a HX260YD and its analysis of strain distribution

In Fig. 5 simulation results are illustrated for three different material cards with the simulation program AutoForm. The same tendencies have also been detected when using the code LS-Dyna in combination with the yield locus of Barlat 2000 and a-values of 5.0 and 6.0. The strong impact of the yield locus models becomes obvious when looking at the final drawing depth when reaching the forming limit curve. The Barlat ´89 generates a compromise of underestimating (TD) and overestimating (RD) the maximum punch travel at onset of necking (FLC). In contrast to this the calibrated model of BBC 2005 makes a better proposal for a drawing in rolling direction, but are completely underestimating the possible forming potential in transverse direction.

FIGURE 5. Influence of a yield locus fit in a simple validation experiment for HX260YD

FIGURE 6. Influence of a yield locus fit to strain path of neck location and position of neck from pole

Interpreting the strain path of the necked element, first one touching the FLC, the large deviation in punch travel is getting clearer. Only the model of Barlat ´89 predicts the location of failure correctly (Fig. 6 right). This is one reason for a correct forecast of the strain path. With the model setups for Banabic 2005 the onset of necking is shifted to the pole of the specimen. By doing this the critical strain path does not follow the experimental results (Fig. 6, left). A much too early onset of necking is predicted in transverse direction. If we estimate, that a validation of the yield locus is only done by additional test in longitudinal direction, a better in the meaning of more realistic result, would have been found for the BBC 2005 models (figure 5, middle and right). Only assuring other loading conditions with respect to rolling direction or further experiments can reduce the risk of misinterpreting validation or modeling results. This is extremely important for industrial applications, where a material model has to fulfill all practical relevant forming situations with reliable forecast of manufacturing. A tuning to one limited forming condition is not sufficient. With respect to the position of failure and drawing depth the often proposed exponent of M=6.0 calculated the most conservative results. This is also valid for other steel grades like DX 54, DC 06, HX420LAD looked at in this work. A similar result has been published by Vegter in /12/.

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CHANCES FOR AN OPTIMIZATION USING INVERSE METHODS According to the presented results (Fig. 6), a potential of determining the yield locus exponent by an inverse approach, is recognizable. The exponent is derived from the lowest possible deviation between the simulation results and the measured data. The described task can be formulated as an optimization (minimization) problem. Additionally, it also is necessary to answer the question as to whether friction and the strain rate dependency of the material have a strong influence on the simulation results. For the presented investigation, the Barlat 2000 yield locus is deployed. As depicted (Fig. 7a), each optimization step comprises a calibration of the yield locus parameter depending on the selected yield locus exponent a. After the yield locus parameters are determined, the input files for the simulation are generated, the forming simulation is performed and the results are evaluated. Subsequently, the simulation results are prepared for the comparison with the measured data. Finally the value of the objective function is computed on the basis of a specimen in rolling and in transverse direction. Aside from the drawing depth, also the predicted strain state is compared with measured data. Thereby, the strain state of a material point is taken, which is located within the later developing necking zone. However, the measured strain state refers to a drawing depth, which does not show localized necking. For treating this multiple objective problem a desirability function is applied /13/. The best prediction of the strain state and the drawing depth is obtained by an exponent between 4.8 and 5.0. This range is derived from Fig. 7 b) by considering the best fitness value related to each yield locus exponent, see also Fig. 8. Friction does not play a crucial role for the prediction of the measured quantities of the investigated YLIT-experiment and satisfying results are only obtained under the consideration of the strain rate effect /13/. In this optimization, the optimizer automatically focused on a setup considering the strain rate sensitivity of the material.

FIGURE 7. a) Optimization principle for yield locus parameters and b) result for an optimization of a DX 54 material with a yield locus formulation Barlat 2000 in combination with LS-Dyna

FIGURE 8. Drawing depth reaching the forming limit curve and selected results for yield locus optimization variations (DX 54)

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INDUSTRIAL APPLICATON EXAMPLE As an industrial example, a side frame has been chosen, which consists of a DX54 steel grade. The process conditions and the geometry of the investigated forming operation are identical to the real production process in the press shop. Thereby, a simulation on the basis of a Hill 48 yield locus, without consideration of the strain rate effect, is taken as a reference. This model had acceptable, but a little bit too robust results in the yield locus identification tests (compare Fig. 8, SR=off). The simulation, performed under application of this yield locus, predicts a stable production process, which is in accordance to the real process. Figure 9 shows the difference of the predicted thinning between the reference and simulation results obtained from the Barlat 2000 yield loci. Areas which are stronger affected by the change of the yield locus are highlighted with a circle, in general nearly all areas show more thinning with the Barlat 2000 formulation. The calculation with Barlat 2000 and a=5.0 is done in two ways, with and without strain rate sensitivity for the material hardening. Figure 9 a) shows, in very good agreement to Fig. 8, a slightly higher thinning in the comparison to Hill ´48 without strain rate sensitivity. In contrast to this, the model calibration with identical values (a=5), but without strain rate sensitivity estimates a total unstable forming process with a multiple areas with unrealistic thinning, Fig. 9 b). This formulation has already been very conservative for the calibration experiment with the same material. A change of the exponent to a=4.0 or a=6.0 with strain rate sensitive hardening results finally in a different pronunciation of a possible thinning near the deck lit. Again this correlates well with the findings of Fig. 8 where all Barlat 2000 yield locus calibration had been similar or more critical in the feasibility study of the validation experiment. > ' 0.08 mm

Side frame made of DX 54 Initial thickness = 0.75 mm Hill ´48 minus (-)

Alternative model

Difference thinning / mm to Hill ´48 0.10

equal (=)

0.06

Result a) to d)

0.02

Reference simulation with Hill ´48 (SR=off) Here, comparison of other model results to the result of Hill ´48

-0.02

a) Barlat 2000, a=5.0, SR=on

b) Barlat 2000, a=5.0, SR=off

c) Barlat 2000, a=4.0, SR=on

d) Barlat 2000, a=6.0, SR=on

-0.06 -0.10

FIGURE 9. Difference of thinning for a side frame simulation with DX 54 in comparison to the reference results with Hill ´48

DISCUSSION AND CONCLUSION In principle the today available yield locus models for forming simulation offer a lot of variety and flexibility to shape the individual setup to the material needs. These models are quite sensitive to small changes and need extreme care when trying to adapt them in industrial practice. The real challenge can be identified in the well balanced fit of these models to the tool shop needs. To do so, additional experiments are needed. In the following discussion we separate the modeling in two principle ways. The so called conventional way is based on a fixed hardening estimation, for example Swift for the extrapolation, and a separate modeling of the yield locus with Hill ´48 or Barlat ´89 models. This way is fully based on uniaxial tensile tests and by this well defined but will not be discussed here.

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In our studies we focused on a second way. Here more experiments are used and the hardening is in correspondence with the tensile test and for the extrapolation with the hydraulic bulge test. The data of the hydraulic bulge test is in parallel also applied to calibrate the biaxial yield point of the yield locus. This method often results in lower, less pronounced hardening of the yield curve when comparing it to the prior mentioned conventional method. As a conclusion we can summarize our findings as follows for materials with a tensile strength below 500 MPa: x Yield locus exponents greater than 5.0 can not be advised so far for steel grades, even though using a strain rate sensitive material hardening formulation based on tensile test results. x In all our studies the assumption of using an exponent of a=6.0 or M=6.0 for steel has not been proven for all materials we looked at, generally it did not produce the best quality with respect and comparison to the real forming process experiments. x The exponent for Barlat 2000 or Banabic 2005 yield locus can hardly be identified when having only uniaxial tensile and hydraulic bulge tests available. The influence of the exponent on the forming prediction can be of extreme importance. Applying a constant, general valid exponent for all steel grades appears to be not possible. For each material the exponent has to be identified individually, the concentration on only one direction for a calibration might lead to unrealistic results (Fig. 5). x Using a well designed set of validation experiments allows the critical examination of the forming forecast of each material model; we consider a final determination of the exponent with this method. x An optimization strategy can give valuable information for critical parameters, allows the overall material model fit for different directions and includes the prediction of failure as well. With respect to all the above findings and results a direct use of complex models can bear a recognizable risk to underestimate the real material formability. The challenge of the future can be seen in the individual modeling of each material grade to the die design and forming simulations needs. A clear benefit of the more complex models is of course the ability to shape the individual quality of the formability forecast with respect to an overestimation or underestimation of the material potential. Here the models of Hill ´48 and Barlat ´89 were more limited.

REFERENCES 1. J. Gerlach, L. Kessler, „A consistent approach for simplifying the material modelling in forming simulations aiming on cold rolled steel grades”. In Steel Grips 4, 2006, pp. 255-260 2. R. Hill, “A theory of yielding and the plastic flow of metals“in Proc. Roy. Soc. A193, London 1948, pp. 281-297 3. R. Hill, “Constitutive Modelling of Orthotropic Plasticity in Sheet Metals” in Journal of the Mechanics and Physics of Solids, Vol. 38, No. 3, 1990, pp 405–417 4. F. Barlat and J. Lian, “Plastic behaviour and stretchability of sheet metals.” in Part I: A yield function for orthotropic sheets under plane stress conditions. International Journal of Plasticity 5, 1989, pp. 51-66 5. F. Barlat et. al. , “Plane stress yield function for aluminium alloy sheets”, in Part I: Theory. International Journal of Plasticity 19, 2003, pp. 1297-1319 6. D. Banabic, D. S. Cosma, M. Sester, M.. Selig, W. Kubli, K. Mattiasson and M. Sigvant, " Influence of onstitutive equations on the accuracy of prediction in sheet metal forming simulation ", In: Proc. Numisheet 2008, , Interlaken, Switzerland, 2008, pp. 37-42. 7. T. Beier, J. Gerlach, H. Grass, I. Heinle, L. Kessler and A. Lipp, “Identification and validation of yield locus parameters with respect to industrial forming simulation needs” in Proceedings of ICTP 2011 (accepted paper). Aachen 2011. 8. W. F. Hosford, “On the crystallographic basis of yield criteria” in Textures and Microstructures, Overseas Publisher Association, Amsterdam 1996, pp. 479-493 9. T. Beier, J. Gerlach, L. Kessler and M. Linnepe, "The Impact of Advanced Material Simulation Parameters in Press Shop Operations Using Mild Steel Grades", in SAE Int. J. Mater. Manuf. V119-5, published by SAE International, 2010. 10. M.-S. Aydın, A. Canpolat, J. Gerlach, L. Kessler and A. E. Tekkaya, “On the Study of Constitutive Parameter Identification of Advanced Yield Criteria”, in Proceedings of SheMet, Leuven, 2011, pp. 11. I. Heinle, L. Kessler, T. Beier, H. Grass, H., J. Meinhardt, A. Lipp and T. Bäck, “Challenges in validation of material modelling for forming simulations”, in: Proc. AutoMetForm, November 24-26, Freiberg, Germany, 2010, pp. 83-99 12. H. Vegter, C. H. ten Horn and Y. G. An, “Modelling of the plastic material behaviour in advanced sheet metal forming simulation”. In: Proc. Automotive sheet metal forming, Jamshedpur, India, 2007, pp. 7-18 13. T. Bäck, I. Heinle and L. Kessler, “Evolutionary strategies for identification and validation of material model parameters for forming simulations”. Accepted paper in: Proc. Genetic and Evolutionary Computation Conference, July 12-16, Dublin, 2011

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,QIOXHQFHRI'LVVLSDWHG)RUPLQJ(QHUJ\RQ)ORZ&XUYHVRI $XVWHQLWLF6WDLQOHVV6WHHO Rainer Steinheimera, Bernd Engela a

University of Siegen, Paul-Bonatz-Strasse 9-11, Siegen, D-57076, Germany

$EVWUDFW Finite element (FE) simulations are widely used to design sheet metal forming processes. Flow curves and forming limit curves of the semi-finished goods are required for these computations. Mostly flow curves are obtained by conversions of stress-strain caracteristics from uniaxial tensile tests. In these calculations, uniform strain and stress within the gauge length is postulated until reaching elongation without necking. This precondition is true only if specimens remain homogenous during the test procedure. Effects from dissipated mechanical energy and heat flow on the results of uniaxial tensile tests were examined with specimen made of austenitic stainless steels with practical experiments and FE simulations. .H\ZRUGVaustenitic stainless steel, dissipated energy, flow curves. 3$&6: 62.20.fk

*(1(5$7,212))/2:&859(6 Mechanical data of sheet metals mostly are obtained by uniaxial tensile tests. At present the valid common standard is the EN ISO 6892-1. In appendix D specimens geometries and tolerances for flat bar specimen made from sheet metal are defined. Concurrently the DIN 50125:1991-04 is a standard for the shape and the dimensions of tensile specimens. For the Experiments, specimens type H with width of 12,5 mm and an initial parallel length of 75 mm were used. The cross section A0 is depending from the wall thicknesses of the sheet. For FE-simulation programs flow curves are necessary for modeling the forming processes. For this a mathematical function is adapted to the experimental generated relation between true strengths kf and true strains M. With the aid of this function the extrapolation of the flow curve is also possible. For most of the low alloyed steels the function of Ludwig (Equation 1) can be used. (1) k f a ˜M n Here are two parameters, the factor a and the hardening exponent n. These parameters can be calculated from the true strain M and the technical stress Rm at maximum tensile force Fmax. The criterion of instability is valid at maximum tensile force only (Equation 2).

wk f wM

(2)

kf

The gradient of flow curves at maximum tensile force is equivalent to the value of their flow strain kf. This value is taken from the experimental data. The values of the parameters are given subsequently by Equations (3) and (4). (3) n M Fmax

a

Rm ˜ e n

(4)

3/$6720(&+$1,&$/02'(/ The tensile forces F during uniaxial tensile tests can be calculated by multiplying the yield stress kf with the actual specimens cross sectional area A (Equation (5)).

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 709-716 (2011); doi: 10.1063/1.3623676 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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F

kf ˜ A

(5)

The change of the initial cross sectional area AB0B during the test can be calculated using the law of volume constancy, expressed by Equation (6). (6) A A0 ˜ e M So, the force can be calculated using Equation (7). This expression was also used by el-Magd [1].

F

k f ˜ A0 ˜ e M

(7)

Strains will differ, if the flow curve is not constant in respect to the gauge length. This is modelled with a differing factor a(l). The local strains will be higher in regions with lower strength and vice versa. However, tensile force is constant in respect of the specimens length. Differentiating the tensile force with respect to the specimens length will give a value of zero (Equation (8)).

wF wl

0

w(M (l ) n ) ˜ A0 ˜ a(l ) ˜ e M (l ) ) wl

(8)

This leads to the subsequent differential equation (Equation (9)).

· wM (l ) § n wa(l ) 1 ˜¨  1¸ 0 ˜  wl ¨© M (l ) ¸¹ wl a(l )

(9)

This equation was also used to discuss the influences of geometrical defects in the initial measuring length on the determination of the hardening exponent [2]. The variables of this differential equation can be separated and the equation is integrable. The solution uses the boundary condition, that the highest strain MBmax B is located at the place of the lowest strengths, that is represented by the lowest factor aminB. The result is shown in Equation (10).

M (l ) n ˜ e M ( l )

amin n ˜ M max ˜ e Mmax a (l )

(10)

Equation (10) links the distribution of the local strength a(l) to the resulting local true strains M(l). These strains cannot be calculated directly. It is possible to prove, that only the true strain at the smallest strength aminB reaches the value of the strain hardening exponent at the state of maximum tensile force (Equation (11). (11) M max n This condition enables a scaling of Equation 10 (Equation (12)). The graph of the equation is diagrammed for different hardening exponents in fig. 1. n

a(l ) § n · n M (l ) (12) ¨ ¸ ˜ M (l ) ˜ e amin © e ¹ The graphs reach their minimum value at M(aB) = n. If the area with the lowest strength reaches this strain, all other places have smaller strains. The strain distribution of specimen depends also from the hardening exponent. For materials with smaller hardening exponents, the differences in respect of the strains are bigger. Marciniak and Kuczynski came to a similar result [2].

),*85( Relation between the ratio of prefactor a to the minimum prefactor amin and the scaled distribution of the true strains for different hardening exponents n

The Ludwik equation does not describe the flow curves of austenitic stainless steel quite well. So, calculations with this plasto-mechanical model will not give satisfying quantitative results. However, the qualitative information

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makes sense. The distribution of strain is broader with greater differences in strength. The strain distribution of specimen depends also from the hardening exponent. For materials with smaller hardening exponents, the differences in respect of the strains are bigger too. Different flow curves in one specimen could be caused by fluctuations in the material, but they definitely will result for austenitic stainless steels, if the temperature will not remain constant concerning the initial gauge length during the test procedure.

7+(50$/&21',7,216 During uniaxial tensile tests mechanical work brought into the specimens. This work is the integral of the product of drawing force and the specimens elongation. This work is mostly dissipated into heat within the specimen. Consequentially, the temperature of the specimens rises. Assuming a uniform elongation concerning the parallel length, this area is heated up even. Due to the larger cross sectional area in the Zone between parallel length and the chuck heads, elongation is decreased and heat capacity is increased in this area. This results in lower temperatures in and heat flow from the parallel length to this area. Thus temperature is non-uniform in the initial measuring length. Furthermore, heat is emitted to the ambient air by convection. Figure 2 shows the heat flows in the specimen schematically. The convectional heat flow is biggest in the area with the highest temperature. So convection tends to equalize the temperature profile. This leads to lower peak values of temperature and a more uniform temperature distribution within the initial measuring length as well.

),*85(Heat flow in specimens for uniaxial tensile tests (schematical)

Dissipation of forming energy as well as heat conduction to the clamping areas are proportional to the cross sectional area. Convective heat transfer to the surrounding just acts on the specimens surface, resulting in a lower surface temperature. So, a temperature profile is also built up concerning the cross section of the specimen. Hence, the influence of convection falls and both peak temperature and thermal inhomogeneities in the initial measuring length rise with increasing thicknesses of the specimens.

7(03(5$785('(3(1'(17)/2:&859(6 Flow curves of metals are generally temperature dependent. To quantify the temperature dependency, uniaxial tensile tests were carried out under almost isothermal conditions in a temperate chamber. The experimental setup is shown in Fig. 3a, the gained curves of true stresses vs. true strains in Fig.3b.

),*85(Temperature chamber a, True stresses vs. true strains for different temperatures b

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The chamber is heated with hot air from a heat gun to achieve a high convection. Temperature sensors fixed on the specimens surface measured maximum temperature differences of 4°C. The influence of different temperatures becomes evident from true strains of ca. 0,1 on. From then, the slopes of the curves depend on temperatures. The flow stresses are higher at lower temperatures. The temperature range for the experiments was chosen so that the maximum temperature is higher than the temperature of the specimens under adiabatic conditions. The curves were linearised for use in FE-Simulations. The used flow curve is given by equation 13, the parameters are listed in Tab. 1.

kf

A  B ˜M C

(13)

7$%/(Coefficients for the flow curves used in FE-Simulations

7+(50$//<&283/(')(6,08/$7,216 Thermally coupled FE-Simulations were performed using the software PAM-Stamp of the ESI GmbH. A scheme of the model is shown in Fig. 4.

),*85(Scheme of the thermally coupled FE-Simulation model (PAMSTAMP)

Dissipation of forming energy is incorporated into the model, assuming that 90% is dissipated into heat [3]. Values for heat capacity, heat conductance and convective heat transfer are given in Table 2 [4]. 7$%/(Thermal properties used in FE-Simulations

The ends of the specimen were drawn with constant velocity of 0,01 mm/s. Displacements of both marked knot lines were recorded as well as the resulting tensile force characteristics. These quantities enable for the calculation of flow curves. The clamping areas were held at constant temperature of 20°C and act as heat sinks. The flow curves used as input compared with the flow curves calculated from the results of the FE-Simulations under different conditions are shown in Fig. 5.

712

),*85(Input flow curves and flow curves gained from Simulation results under different thermal conditions

Flow curve 1 was simulated conventionally without thermal coupling. Flow curve 2 was generated with simulations that include dissipated forming energy. Conduction of heat and convective heat transfer to the ambient was added to the model to generate flow curve 3. Flow Curve 1 is very close to the input flow curve for 20°C until maximum tensile force. This is a evidence, that the simulations are interpretable until maximum force is reached. Flow curves gained by simulations including thermal coupling – curves 2 and 3 - start at the input curve for 20°C and approach the input curve for 50°C. Their elongation without necking is lower. The peak temperature at maximum tensile force is higher, if convection is not considered. In this case, the temperature gradation is more inhomogeneous as well. This is caused by holding the temperature of the clamping areas on a constant level of 20°C.

35$&7,&$/(;3(5,0(176 For first tests, specimen made of X10 CrNi 188 (material number 1.4310) were used. The specimens temperature gradations were recorded by an infrared picture camera (fig. 6a). A typical plot of the chronological temperature progression is shown in fig. 6b.

),*85(Temperature graduation measured by an infrared picture camera a, plot of the temporal temperature progression b

The temperature starts homogeneous concerning the measuring length and rises during test progress. A temperature profile develops caused by faster heating of the specimens center compared to the margins of the measuring length, from which heat is led to the clamping areas. A significant temperature maximum builds up near the center. The center itself was marked and so the sharp maximum is due to a different emission coefficient. This result corresponds qualitatively to the results of the FE-simulations.

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For the test serious, sheets of X5 CrNi 1810 (material number 1.4301) were used in thicknesses of 0,96 mm and 0,2 mm. Traverse velocities were varied between 10 mm/s and 0,01 mm/s and held constant during the respective test. Different convective conditions were achieved by wrapping isolation around the specimen (reduced convection) or blowing with pressurized air (forced convection). The elongations without necking and the maximum tensile forces under different convective conditions are diagrammed toward the drawing velocities for specimens with thicknesses of 0,96 mm in fig. 7

),*85(Tensile strength and uniform strains in dependency of the traverse velocity under different convective conditions

With increasing convection, both uniform elongation Hgl and tensile strength Rm rise with lower drawing velocities. The increase of tensile strength results from higher heat loss at forced convection and consequential lower peak temperatures. The increase of the uniform elongation is due to a more homogeneous temperature concerning the measuring length. The influence of convective conditions is less at 10 mm/s the highest traverse velocity. Heat, that is led out from the specimens surface has to be conducted from the internal sections. Thus, temperature of these sections is higher than on the surface. These temperature distributions concerning the cross section is depending from the specimens thicknesses as well as from drawing velocities. The temperatures of inner sections and on the surface will be more equally distributed with slower drawing velocities and for thinner specimens. So the minimal tensile velocity of 0,2 mm/s will lead to an likewise lower temperature and a homogeneous temperature distribution concerning the thickness as well. The results of tensile tests from specimens with thicknesses of 0,2 mm are shown in fig. X. For these specimens, the ratio of surface to cross sectional area is increased by a factor of 5 compared to the specimen with thicknesses of 0,96 mm.

),*85(Tensile strength and uniform strains in dependency of the traverse velocity under natural and forced convection

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Again proportional elongations and tensile strength are higher with forced convection. For velocities less than 1 mm/s with forced convection, the influence of velocity under forced convection is small. These conditions lead to almost isothermal behavior. With natural convection, tensile strength is nearly constant whereas proportional elongations increase slightly with lower velocities. To achieve nearly isothermal conditions, velocities have to be slower than 0,1 mm/s. The greatest differences are found at a velocity of 1 mm/s. The uniform elongation reaches a value of ca. 74 %, whereas the uniform elongation of specimens with wall thicknesses of 0,96 mm just reach ca. 58 %. So it can be estimated, that the experiments are not real isothermal even at slow drawing velocities. The dotted line represents the estimated development of the proportional elongations with differing drawing velocities under – almost - isothermal conditions. This dependency is explicitly less than the dependency found with conventional conditions.

&21&/86,216 The determination of mechanical properties of stainless steel with uniaxial tensile tests was investigated under different convective conditions. A significant influence of the tensile velocity was found. The results are interpreted the way that the flow curves are nearly independent from the forming speed. The reasons are different temperature distributions, which develop in the measuring length of the specimens due to heating from dissipated forming work and time dependent heat emission from heat flow and convection. In the examined velocity range, the velocity dependence of the flow curves under nearly isothermal conditions seem to be small. Heat conduction is the reason for temperature inhomogeneity concerning the measuring length. Discussions of limits have the result that both infinite heat conductivity and a conductivity of zero will lead to uniform temperatures in the initial measuring length. The temperatures will be different, but this should not have an effect on the uniform elongation. However higher conductivity will result in lower tensile strength. Heat emission to the ambient by convection lowers and homogenizes the temperature within the measuring length. The ratio of heat flow from conduction to heat emission by convection is substantial for the temperature profile that develops in the gauge length. Because convection just acts on the surface, specimen from sheets with less thickness will be more homogeneous. This will result in bigger uniform elongations. Mechanical properties determinate by uniaxial tensile tests therefore are dependent on thermal characteristics as heat capacity and heat conductivity as well as from the specimen geometry. The flow curves of the material X5 CrNi 1810 are dependent on the temperature in a temperature range, reached by dissipation of forming energy. Temperature distributions will lead to strain distributions concerning the measuring length even before maximum tensile force is reached therefore. Figure 9 shows the strain distribution of a specimen. The test has been stopped at an averaged strain of 35 % clear before maximum force was reached. The uniform strain under these drawing conditions is expected at a value of ca. 37 % (see fig. 7). The strains are measured with the system AutoGrid® of Vialux GmbH, Chemnitz, Germany.

),*85(Strain distribution in the measuring length before maximum tensile force is reached

The maximum strain found near the middle of the specimen has a value of ca. 40%. Near the ends of the actual gauge length, the strains are lower. This is qualitatively consistent to the results of the plasto-mechanical model, the

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FE-Simulations and the theory described above. So there will be no really uniform strain at all in uniaxial tensile tests with this austenitic stainless steel under conventional conditions. It seems to be necessary to separate the influences of temperature distribution and forming speed for the modeling of flow curves used in FE-Simulations. To get more realistic results, FE-Simulations of could forming processes with austenitic stainless steels should be thermally coupled. Heat flow to tools should be included also.

287/22.21)857+(5,19(67,*$7,216 Further investigations are planned on different subjects. The deviations of flow curves generated under conventional conditions from those obtained under isothermal conditions will be investigated for different materials. This is necessary especially to estimate for which materials thermally coupled Simulations are needed. In many sheet forming processes the forming force is not applied in the actual forming zone. It is transmitted from the application zone to the forming zone. The material in the transmission zone is already formed and warmed up by dissipated forming energy. If there is tensile stress in this zone, it will be prone to failure by cracks. Typical processes are deep-drawing and the rotary draw bending of tubes. For those processes cooling of the transmission zone and where applicable combined with warming of the forming zone could enlarge forming limits. Cooling of the transmission zone leads to higher transmittable forming forces and warming of the forming zone will result in lower required forming forces. First FE-Simulations show, that the application of a heated blank holder zone (T=90°C) in combination with a isothermal punch (T=20°C) can enlarge the draw depth for ca. 30 %. The investigations will be performed with FE-simulations and practical experiments as well.

5()(5(1&(6 1. E. El-Magd, „Auswirkungen kleiner Formabweichungen des Probestabs auf die Nenndehnung bei Höchstlast im Zugversuch” Materialprüf. 15.Nr. 3, (1973), pp. 90 – 95. 2. B. Engel, R. Steinheimer, ³Obtaining more precise flow curves from unaxial tensile tests for FE-simulations”, in Tagungsband "9th International Conference on Technology of Plasticity ICTP 2008", 7.-11. September 2008, Gyeongju, Korea ISBN 978-89-5708-151-.8. 3. Z. Marciniak, K. Kuczynski, “Limit Strains in the Process of Strech-forming Sheet Metall” in Int. J. Mech. Sci., Vol. 9, (1967), pp 609-620, Pergamon Press Ltd. E. El-Magd, „Auswirkungen kleiner Formabweichungen des Probestabs auf die Nenndehnung bei Höchstlast im Zugversuch” Materialprüf. 15.Nr. 3, (1973), pp. 90 – 95. 4. R. Kopp and H. Wiegels, “Einführung in die Umformtechnik” 1. Auflage 1998, Verlag der Augustinus Buchhandlung, Aachen, Germany, p 62, ISBN 3-86073-665-5. 5. N. N., “Merkblatt 821 Edelstahl Rostfrei – Eigenschaften” 3. Auflage 2003, p. 16, Herausgeber Informationsstelle edelstahl Rostfrei 40013 Düsseldorf, Germany.

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On The Experimental Verification Of The Numerical Simulation Based Inferences From Strain Distribution In Axisymmetric Drawn Sheet Metal Parts Navneet Varmaa, Manoj Ukhandeb and P. P. Datea a : Department of Mechanical Engineering, IIT Bombay Powai, Mumbai 400076. b : R & D, Bharat Forge Ltd., Pune Abstract : Studies based on numerical simulations of axisymmetric parts showed that the scaleability of strain distribution as described above was limited to only about 60% geometric upscaling in conventional forming of a cylindrical part and to about 90% upscaling in hydroforming. These findings may be criticized as they amount to verification of the scaleability of the differential equations being solved. In this context experiments were performed on cylindrical sheet metal cups drawn to three different diameters keeping the draw ratio as well as the R/t ratio at the die entry and the punch nose constant. The meridional strain distribution was determined using GOM automatic strain measuring system. The strain distribution was fitted into a Fourier series and the variation of the Fourier coefficients with punch travel plotted for the three sizes investigated. This variation in the major Fourier coefficients was found to run parallel indicating scaleability of the strain distribution. An attempt was made to predict the strain distribution in the larger cup based on that in the smallest one. During the course of the simulation based studies, it was noticed that the sine coefficients show a sudden increase at imminent failure. This was borne out by the measured strain distribution using a uniaxial tensile sample and that from the deep drawn cups that failed during the experimentation.

1. INTRODUCTION Geometric scaleability of thickness strain distribution in sheet metal forming is a challenge due to involvement of non-deterministic, non-scaleable effects like friction, symmetry of product shape etc.. However, if it were indeed possible to predict the thickness strain distribution in a large sized product based on that of a substantially scaled down model, it would lead to enormous savings in prototyping cost and the cost of product development as a whole. Even FEM simulations would be much more economical compared to those for a full scale model. Studies on strain distribution and different approaches used to extract deformation related information have been reported in the literature in the past [1-4]. Effect of geometric scaling in the context of forming of microparts vis-à-vis that of macroparts was discussed by Vollertsen et. al [5] and Picart et. al [6] to name a few. Variables influencing sheet formability in biaxial deformation were documented by Keeler [7] and many others, while Danckert et. al [8] and many others brought out the mechanics of a more uniform strain distribution and better formability by hydroforming. However, there has been very little work reported on the topic presented in this paper. Initial work [9] revealed the potential of the present methodology, which was carried forward to investigate practical applications [10,11].

2. APPROACH. In view of this, simulations were carried out on deep drawing of cylindrical and rectangular cups, and hydroforming of cylindrical cups. As expected, the process of hydroforming of axisymmetric components showed maximum scaleability of the thickness strain distribution due to a negligible presence of friction, followed by conventional drawing of cylindrical cups, which are axisymmmetric. Minimum scaleability of strain distribution was shown by the square cups, being symmetric, but not axisymmetric [10,11].

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 717-724 (2011); doi: 10.1063/1.3623677 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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The spatial thickness strain distribution profiles obtained in a section encompassing the minimum deformed and the maximum deformed regions in a sheet metal blank were fitted into a Fourier series. TABLE 1: Dimensions of drawing specimens, tool geometry and test conditions

Scaling factor

1

1.1

1.2

1.3

1.4

1.5

1.6

(mm)

50

55

60

65

70

75

80

Blank thickness (mm)

0.8

0.88

0.96

1.04

1.12

1.2

1.28

Blank dia

105

115.5

126

136.5

147

157.5

168

42.62

46.88

51.15

55.41

59.67

63.93

68.19

Punch diameter (mm)

48

52.8

57.6

62.4

67.2

72

76.8

Die corner radius (rcd) (mm)

4.5

4.95

5.4

5.85

6.3

6.75

7.2

Punch corner radius (rcp) (mm)

3.5

3.85

4.2

4.55

4.9

5.25

5.6

BH Force

12.5

13.75

15

16.25

17.5

18.75

20

5

5.5

6

6.5

7

7.5

8

Die diameter

Cup height

(mm) (mm)

(KN)

Punch velocity (mm/ms)

TABLE 2: Dimensions of drawing specimen and tools for square cup

Scaling factor

1

Blank size

(mm)

40×40

Blank thickness

(mm)

0.8

Cup height

(mm)

25

Die size

(mm)

22×22

Punch size

(mm)

20×20

Die corner radius (mm)

3

Punch corner radius (mm)

2.5

BH Force

(KN)

10

(mm/ms)

5

Punch velocity

In case of a cylindrical cup it would be a diametral section while in a rectangular component it would be along a diagonal of the dawn cup. The thickness strain distribution was determined from one end of the flange to the other end along the cut section. This spatial range was mapped onto an interval - to + radians and a Fourier fit obtained.

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If the Fourier coefficients showed scaleability, i.e., showed the same nature of variation over two dimensional scales, the thickness strain distribution would become predictable. Simulations were performed using dimensions at different scaling factors as enumerated in the table below. As seen from the tables, the r/t ratio (radius of bend to sheet thickness ratio) was maintained constant at all geometric scales in an effort to minimize non-scaleable effects.

3. RESULTS AND DISCUSSION Typical variation of the Fourier coefficients characterizing thickness strain distributions at different dimensional scales in cylindrical vis-à-vis a square component is shown in Fig. 1 a and b respectively. Fig 1a shows the normalized fourier coefficient a2 (coefficient of cos (2x)) for a cylindrical cup, while Fig. 1b shows the behaviour of normalized a2 for a rectangular cup. Normalisation of Fourier coefficients is performed using the corresponding coefficients for thickness strain distributions in a reference geometry (i.e., scaling factor of 1). Hence, ai (normalized) = ai (scale ‘s’)/ai (scale 1) which also stands for the scaling factor of that Fourier coefficient with respect to that of the reference geometry. Fig. 1a and b both show two regions of variation, the initial one showing random variation of the coefficient upto a certain punch travel followed by a steady variation indicating scaleability. The initial portion of the stroke refers to the process of wrapping the sheet around the punch where frictional effects would dominate. Later, deformation progresses steadily with Fourier coefficients at different scales running parallel, indicating scaleability. The zone dominated by friction is seen to be about 14mm punch travel for a square shape (Fig. 1b) similar to that for an axisymmetric cylindrical shape (Fig. 1a). In contrast, in Fig. 1c the coefficient a2 for hydroforming does not show the initial ‘disturbed’ region characteristic of drawing in the presence of friction. In fact, due to a virtual absence of friction in the initial part the values of a2 are seen to be similar regardless of geometric upscaling. The strain distribution of an upscaled geometry calculated from the upscaled Fourier coefficients correlated reasonably well with that from simulation (Fig. 2). At 60% upscaling a maximum error of 7% between the Fourier calculation (Fourier fit of the order 5) and the simulation based strain calculation was obtained [11] whereas experimentally, a reasonably accurate prediction of strain distribution could be made using upscaled Fourier coefficients to a geometric scaling factor of 2 : 1. This procedure may be criticized on the grounds that a simulation result on scaleability would reflect the scaleability of the differential equations being solved. Hence the scaleability(or otherwise) as inferred from the differential equations might not be the true representation of the actual process. In view of this, experiments were performed on drawing aluminium blanks of 2mm thickness and different diameters to cylindrical cups. To this end, cups were fully drawn to different heights (h/d ratios) maintaining same draw ratio (D/d = constant = 2), constant thickness (=2mm) and constant r/t at the punch nose and the die entry. Circular blanks of diameter 50, 100, 150 and 180mm were considered, implying a total scaling factor of 3.6, i.e., 360 %. Five punch travels were considered for each draw. Repeatability was ensured using three sets of samples per condition. The meridional strain distribution was measured using the ARGUS measuring system of GOM. Strain distributions were compared for cups drawn to constant D/d, h/d and r/t ratios. A typical safe drawn cup is shown in Fig. 3(a) and the corresponding true thickness strain distribution in Fig. 3(b). A failed drawn cup and its strain distribution are given in Fig. 4(a,b). The sharp peak and steep strain gradients characterize flow localization due to necking, calling for a Fourier fit of n = 50. Table 3 shows the Fourier coefficients for different h/d ratios for two different scales – 1: 1 and 2 : 1. Higher ratios of upscaling led to incomplete data due to premature failure of the cups. The true thickness strain distribution in successfully formed cups could be adequately described by a relatively low order Fourier series. Hence, an and bn determined upto n = 10 or so was adequate to capture the features of the strain distribution profile. Wherever a strain peak due to necking was observed, the sharp radius of

719

(a)

(b)

(c) FIGURE 1. Variation of Fourier coefficient a2 characterising true thickness strain distribution for cups geometrically upscaled up to different levels (a) meridional section along a diameter, conventionally drawn cylindrical cup and (b) diagonal section of a square cup. (c) meridional section along a diameter, hydroformed cylindrical cup

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FIGURE 2. A correlation between the strain calculated from Fourier coefficients (Fourier fit of order 5) and the strain predicted from simulation for 60% dimensional upscaling of a cylindrical cup

(a)

(b)

FIGURE 3 : (a) Drawn cup to h/d = 0.48 and (b) the corresponding true thickness strain distribution obtained using ARGUS system of GOM TABLE 3 : Fourier coefficients for meridional strain distributions of cups drawn to different h/d ratios, reference scale (scale 1) and scale 2: 1 (scale 2)

Scale 1 h/d 0.48 0.56 0.64 0.72

a0 -0.0787 -0.0897 -0.0959 -0.1139

a1 -0.0854 -0.0852 -0.0875 -0.0773

a2 0.1070 0.1323 0.1324 0.1522

a3 0.0287 0.0076 0.0122 -0.0017

a4 0.0003 0.0065 0.0152 0.0165

a5 -0.0112 -0.0015 -0.0199 -0.0222

b1 0.0070 0.0099 0.0041 -0.0072

b2 -0.0066 -0.0219 -0.0085 0.0056

b3 0.0057 -0.0017 0.0069 -0.0038

b4 -0.0023 -0.0031 -0.0059 0.0005

b5 -0.0020 0.0108 0.0017 -0.0007

-0.0279 -0.0324 -0.0430 -0.0355 -0.0527

-0.0025 -0.0003 -0.0060 -0.0035 -0.0019

0.0013 -0.0147 0.0180 0.0113 -0.0037

-0.0002 0.0064 -0.0116 0.0006 -0.0063

-0.0005 0.0056 0.0002 -0.0007 0.0057

-0.0020 0.0066 0.0009 -0.0116 0.0094

Scale 2 0.48 0.56 0.64 0.72 0.8

-0.0488 -0.0520 -0.0579 -0.0761 -0.0676

-0.0523 -0.0663 -0.0590 -0.0461 -0.049

0.1371 0.1473 0.1484 0.1372 0.1543

-0.0164 -0.0370 -0.0322 -0.0213 -0.0388

-0.0315 -0.0091 -0.0080 -0.0111 -0.0048

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(a)

(b)

FIGURE. 4. (a) Drawn cup of diameter 75mm. and (b) spatial distribution of true thickness strain in three directions of anisotropy measured using ARGUS system of GOM.

(a)

(b)

(c)

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(d) FIGURE 5 : Cosine coefficients for two scaling factors (a) a1 (b) a2 (c) a3 (d) a5

(a)

(b) FIGURE 6. Variation in the sine coefficients with punch travel for two different scales, namely 1: 1 and 2:1 (a) coefficient b1 and (b) coefficient b5 0.3

0.25

0.2 sine coefficients

b5 0.15

0.1

b1 b2

0.05

b4 0 b3 -0.05

2

4

6

8 10 12 punch travel(mm)

14

16

FIGURE 7. A sudden increase in sine coefficients on flow localisation due to necking under uniaxial tension (a) simulation (b) experiment

curvature at the peak could be captured only by higher order Fourier coefficients. Hence, n = 50 or so was necessary to represent the strain distribution to a reasonable accuracy. A pictorial representation of the

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18

variation of the normalised Fourier coefficients (also scaling factors for the respective coefficient) is given in Fig. 5(a-d) for cosine coefficients while Fig. 6 (a,b) shows typical variation of the sine coefficients. It would therefore appear that the cosine coefficients are by and large scaleable while the sine coefficients are not, leading to an interpretation that whereas the scaleable effects could be represented by the cosine coefficients, the non-scaleable ones are represented by the sine coefficients. Naturally non-scaleable events like localisation would then cause the sine coefficients to rise indicating imminent failure. This was supported by a comparison of the behaviour of the coefficients using simulated thickness strain distribution with those determined from experiments (Fig. 7a,b). A Fourier series of order 5 (n=5) was adequate to fit the strain distribution in uniaxial tension. In experimental results on biaxial deformation however, n=50 was necessary and higher order Fourier coefficients increased near the draw depth of failure to capture the sharpness of the strain peak.

4. CONCLUSIONS 1.

2.

3.

4. 5.

Scaleability of the thickness strain distribution is confirmed experimentally. A reasonably accurate prediction of the strain distribution may be obtained for a part geometrically scaled to double its dimensions, by virtue of the scaleability exhibited by the Fourier coefficients. Product symmetry and friction play a significant role in the scaleability of the thickness strain distribution. Hence predictability of strain distribution using scaleability of the Fourier coefficients is higher for highly symmetric parts (eg., axisymmetric geometries) and diminishes with decreasing symmetry. The Fourier coefficients describing the thickness strain distribution in hydroforming shows absence of the initial arbitrary variation (observed in conventional forming) on account of uncontrollable frictional effects. A much higher order of Fourier series is required to fit high strain gradients and sharp radii of curvature in a thickness strain distribution characteristic of localization processes. The degree of upscaling up to which predictability of strain distribution is sufficiently accurate needs to be enhanced for the procedure illustrated here to be of widespread practical use.

5. REFERENCES 1. S. G. Desai and P. P. Date, “On the quantification of strain distribution in drawn sheet metal products”, J. Mater. Processing Technol., 177(2006)439-443 2. Date P.P., Padmanabhan K.A., “On the Formability of 3.15 mm Thick Low-Carbon Steel Sheets”, Journal of Materials Processing Technology, Vol.35, 1992, pp. 171-175 3. Rees D.W., Power R.K., “Process Signatures and Finite Elements in Sheet Metal Forming”, journal of material processing technology, vol.77, pp. 139-141. 4. Schedin E., Melander A., On the Strain Distribution During the Stretch Forming of Low and High Strength Sheet Steels”, Journal of Mechanical Working Technology, Vol.15, 1987, pp. 181-201. 5. Kals R., Vollertsen .F., Geiger M., “Scaling Effect in Sheet Metal Forming”, Proceedings of the International Conference on Sheet Metal Forming, 1996, pp.65-75 6. Michel J.F., Picart P., “Size Effect in the Constitutive Behavior for Brass in Sheet metal forming”, journal of materials processing technology, vol.141, 2003, pp. 439-440. 7. Keeler S.P., “Die Design and Lubrication, Sheet Metal Formability Part-5”, Sheet Metal Industries, 1971, pp. 511-517. 8. Langa L.H., Wang Z.R., Kang D.C., Yuan S.J., Zhang S.H., Danckert J., Nielsen K.B., “Hydroforming highlights: sheet hydroforming and tube hydroforming”, Journal of Materials Processing Technology, Vol.151, 2004, pp. 165-167. 9. Vamsi Krishna Y., “Study of Strain Distribution in Forming of Sheet Metal Parts of a Single Family”, M.Tech. Project, IIT Bombay, 2004. 10. M. Ukhande and P. P. Date, A strain distribution based failure criterion in drawing of sheet metal, Proc. IDDRG 2007 Conference, 21-23 May 2007, Gyor, Hungary. 11. M. Ukhande, “Effect of geometric scaling on strain distribution in deep drawn sheet metal parts”, M. Tech Dissertation, 2006, IIT Bombay.

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Modeling of Sandwich Sheets with Metallic Foam H. Mataa, R. Natal Jorgeb, A. Santosc, A.A. Fernandesd, R.A.F. Valentee M.P.L. Parentef a, b, d, f

IDMEC-FEUP Faculdade de Engenharia da Universidade do Porto Rua Dr. Roberto Frias 4200-465 Porto {dem09005, rnatal, aaf, mparente}@fe.up.pt, web: http://www.fe.up.pt c

INEGI Faculdade de Engenharia da Universidade do Porto Rua Dr. Roberto Frias 4200-465 Porto [email protected], web: http://www.fe.up.pt e

Universidade de Aveiro Campus Universitário de Santiago 3810-193 Aveiro e-mail: [email protected], web: http://www.ua.pt

Abstract: World-wide vehicles safety experts agree that significant further reductions in fatalities and injuries can be achieved as a result of the use of new lightweight and energy absorbing materials. On this work, the authors present the development and evaluation of an innovative system able to perform reliable panels of sandwich sheets with metallic foam cores for industrial applications. The mathematical model used to describe the behavior of sandwich shells with metal cores foam is presented and some numerical examples are presented. In order to validate those results mechanical experiments are carried out. Using the crushable foam constitutive model, available on ABAQUS, a set of different mechanical tests were simulated. There are two variants of this model available on ABAQUS: the volumetric hardening model and the isotropic hardening model. As a first approximation we chose the isotropic hardening variant. The isotropic hardening model available uses a yield surface that is an ellipse centered at the origin in the p–q stress plane. Based on this constitutive model for the foam, numerical simulations of the tensile and bulge test will be conducted. The numerical results will be validated using the data obtained from the experimental results. Keywords: Sandwich Structures, Numerical Model, Metal Foam.

INTRODUCTION Despite significant improvements in car safety in the last 25 years, the actual number of deaths and wounded arising from automobile accidents, in addition to all social and economic costs, remains unacceptable. Here, the passive safety systems still have great potential for development as a way to reduce deaths and injuries. On the other hand, from an environmental point of view, the use of materials optimized in terms of energy absorption, with a reduced weight, has a direct impact on the thermal efficiency, and consumption of the engines, thus emitting less greenhouse gases for the atmosphere. Within this framework, it makes sense the study and

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 725-732 (2011); doi: 10.1063/1.3623678 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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development of this new composite material, formed by two sheets of aluminum separated by a core of aluminum [1-2]. The composite structure used on this work is formed by two sheets of aluminum with 1mm thickness, separated by a aluminum foam core with a thickness of 8mm. Since the specific characteristics of this composite material are mainly due to the behavior of its core, namely the energy absorbing during impact and vibration isolation properties [3 -4], the foam will be studied in detail on this work. The structure of the foam corresponds to a three-dimensional arrangement of cells, which can be divided into two gr oups: open and closed cells. On a metal foam with open cells, the individual pores are open. When the individual pores are closed, the metal foam belongs to the closed cells group [5 -6].

FIGURE 1. Panels with aluminum foam core.

CONSTITUTIVE MODEL To describe the plastic behavior of the metal foam in this work, the constitutive model proposed by Deshpande was used [7]. This model was chosen due to its capability to describe the behavior of porous metals and due to fact that the yield surface used on this model depends only on the plastic Poisson's ratio.

Yield Surface Metal foams have an approximately linear elastic behavior for small strains. Metallic foams plastic deformation occurs when there is a change in volume, unlike solid metals. For metal foams the yield criterion can be formulated as follows: (1)

σˆ ≥ σ y with 1

σˆ 2 =

2

[(σ e 2 + α 2σ m 2 ]

(2)

(1 + (α / 3) ) 1 σ m = − (σ 1 + σ 2 + σ 3 ) 3

σe ≡

(3)

3 ′ σ ij : σ ij′ 2

Where the equivalent stress is given by σˆ , and

σe

is the von Mises equivalent stress.

726

(4)

σm

is the mean stress

and is defined as

1 3

σ m = σ kk . The shape of the yield surface is defined by α

and

σy

is the yield stress of the

material [9]. Considering the stress plane σ m , versus stress equivalent σ e , from Figure 2 is possible to obtain the points 1, 3, 6, 8 that define the yield surface. These points were obtained by varying the pressure p, and the uniaxial load [7-8].

FIGURE 2. Definition of the yield surface Deshpande model

Deshpande Model Based on the experimental results of the definition of the yield surface previously presented, but now considering a reference σ m → p, σ e → q, the model of Deshpande [9] can be summarized as follows: Elastic law:

σ = De : εe Yield Surface:

φ=

1 α  1+   3

2

 q 2 + α 2 p 2  − σ y (ε p ) = 0

Plastic evolution law:

εɺ p = γɺ N Elastic evolution law:

εɺ e = εɺ − γɺ N Evolution of the equivalent plastic strain:

εɺ p = γɺ

2 N:N 3

727

EXPERIMENTAL TESTS Uniaxial Compression Test To validate the numerical model previously presented, uniaxial compression tests of samples obtained from the panels were performed . Figure 3 illustrates the experimental procedure, from which it was possible to record force /displacement values for this material, which are then compared with the numerical values. This mechanical test is one of the tests commonly used to characterize the mechanical behavior of metallic foams.

FIGURE 3. Uniaxial compression test.

Tensile Test

C

W

Tensile tests were also performed in order to evaluated the mechanical behavior of this composite material under traction. To this end specimens were cut from the panel with the dimension shown on Figure 4. G=50 mm, L W=12.5 mm, B A B R=20 mm, L=200 mm, A=75 mm, B=50 mm, T ≈20 mm, C R G T- Thickness [mm] FIGURE 4. Specimens used on the tensile test, according to ASTM E 8M -04.

The following Figure shows de experimental setup used to conduct the tensile tests.

728

FIGURE 5. Experimental tensile test.

NUMERICAL SIMULATIONS Numerical simulations of the same experimental tests were performed. Table 1 shows the properties used for the aluminum sheets. For the aluminum sheets an elasto-plastic behavior was assumed. For the metallic foam core an elasto-plastic behavior was assumed, based on the model previously defined. Table 2 shows the properties used to describe the metal foam behavior. The stress-strain curve obtained by the uniaxial compression test is also used to describe the metallic foam behavior [7 , 10-11].

TABLE 1. Plate Properties

Mechanical Behavior

E

υ

Elastic

70 GPa

0.33

Plastic

Yield Stress

Plastic Strain

110 MPa

0

250 MPa

0.08

TABLE 2. Metal foam Properties

Mechanical Behavior

E

υ

Elastic

0.354 GPa

0.33

Plastic

Compression Yield Stress Ratio

υp

1.71

0.013

The numerical simulation were conduct using the software ABAQUS. The finite elements models are representations of the actual specimens. Only 1/8 of each specimen was represented numerically in order to minimize the computational effort, and imp rove the quality of the approximation given by the finite element method.

729

PRESENTATION AND DISCUSSION DIS CUSSION OF RESULTS Figure 6a shows the spciments before and after the compression test. It can be observed the high volume change by compression, suffered by the foam. Figure 6b shows the deformed mesh obtained for the numerical simulation of the compression test.

a) Final result of experimental compression test.

b) Final numerical result.

FIGURE 6. Compression test

Figure 7 shows a comparison between the force/displacement curves obtained from the experimental and numerical compression tests. F (N) 8000

6000

4000

Experimental test 2000

Numerical test 0

0

0,5

1

1,5

2

2,5

3

d(mm)

FIGURE 7. Force/displacement curves for the uniaxial compression test.

It is possible to observe in Figure 7 that a good agreement between the numerical results obtained using the Deshpande model and experimental results. It is also possible to observe the numerical instabilities obtained. For the numerical test, the metal core foam assumes an importing role. After a small domain of elastic behavior, the foam quickly reaches its yield value. This can be physically interpreted as the beginning of the collapse of the cells that constitute the foam. This process is almost constant with some small variations in strength, and is visible in Figure 7. After all the cells have collapsed, the numerical specimen starts behaving as a solid, resulting on a rapid increase on the applied force for small increments of displacements. The approximation of the numerical curve to the behavior of the experimental results validates the use of the Deshpande constitutive model, used to describe the difficult behavior of plastic porous materials. Figure 8 shows the speciments after the tensile tests experiments. The obtained numerical and experimental results for the tensile tests are shown in Figure 9. Observing the force/displacement curves of Figure 9, they appear to be similar to a typical curve of a tensile test for an homogeneous solid metal, with an

730

initial elastic part and a second plastic part. The high forces that leads to the rupture of the speciments are mainlly supported by two aluminum sheets, since the rupture of metal foam occurs for much lower forces than those achieved during the experimental tensile test.

FIGURE 9. Point’s force/displacement for the tensile test FIGURE 8. Point’s force/displacement for the tensile test

CONCULUSIONS Comparing the numerical and experimental curves obtained for the uniaxial compression test, it is possible to conclude that the model of Deshpande can be used to describe the plastic behavior associated with the metal foam. Comparing the numerical results with the experimental data for the compression test, one can see that the values are at the same order of magnitude, as it is visible in Figure 7 , although there is a deviation. For the tensile test a good agreement between the numerical and experimental results was obtained. It is also still possible to conc lude that the aluminum sheets are the main responsible for the high force values achieved experimentally, due to the foam’s low ability to withstand tensile forces.

Finally, and as a way to improve the numerical results, additional experimental studies would be needed in order to obtain the properties of the sheets separated from the foam, and an approach to study the anisotropy associated with both materials that co nstitute these panels.

ACKNOWLEDGEMENT

The authors acknowledge the funding provided by Ministério da Ciência, Tecnologia e Ensino Superior – Fundação para a Ciência e a Tecnologia (Portugal) and by FEDER/FSE, under grant PTDC / EME – TME / 098050 /2008.

REFERENCES [1] [2]

MF Ashby et al.; Metal foams: a design guide, Oxford, Butterworth-Heinemann; 2000. industry”; Mat and Design; 18:217-220; 1997). Baumeister et al.; “Aluminium foams for transport industry”;

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Topin, F., et al., Experimental Analysis of Multiphase Flow in Metallic foam: Flow Laws, Heat Transfer and Convective Boiling; Advanced material Engineering; 2006. 8(9): p. 890-899. [4] Banhart, J., Manufacture; Characterization and application of cellular metals and metal foams; Progress in materials Science; 2001. 46: p. 559-632. [5] Mata H. et al, Study of sandwich shells with metallic foam cores Int J Mater Form Vol. 3 Suppl 1:903– 906 DOI 10.1007/s12289-010-0914-x, Springer-Verlag France, 2010; [6] Mata H. et al, FEM analysis of Sandwich Shells with Metallic Foam Cores, Key Engineering Materials Vol. 473, Trans Tech Publications, Switzerland, 2011; [7] Deshpande, V. S., and N. A. Fleck; Isotropic Constitutive Model for Metallic Foams; Journal of the Mechanics and Physics of Solids; pp. 1253–1276; 2000. [8] A.G. Hanssen et al., Validation of constitutive models applicable to aluminium foams; International Journal of Mechanical Sciences 44 (2002) 359–406; 2001. [9] Neves M. P. Rui, Modelação Constitutiva do Comportamento Mecânico de Espumas Metálicas, 2009. [10] R.E. Miller / International Journal of Mechanical Sciences 42 (2000) 729}754. [11] T.M McCormack et al., Failure of sandwich beams with metallic foam cores, International Journal of Solids and Structures 38 (2001) 4901-4920; 2000.

[3]

732

5ROO)RUPLQJRI9LEUDWLRQ'DPSLQJ&RPSRVLWH6KHHWV Bernd Engela and Johannes Buhla a

University of Siegen, Chair of Forming Technologies, Paul Bonatz 9-11, 57076 Siegen (Germany)

$EVWUDFW Vibrations damping three-layer sandwich sheets often need to be formed. Failure modes during and after the forming process are described in this paper. Material parameters, especially the maximal shear stress, are determined to calibrate numerical calculations. With plastomechanical preliminary design the shear force between the cover sheets is calculated. Exactly calculation of the roll forming process is achieved by FE-Methods. .H\ZRUGVSheet metal forming, Application of simulation, Multilayer sheet, viscoelastic interlayer, failure modes, displacement, delamination 3$&662.20.F

$33/,&$7,212)9,%5$7,21'$03,1*&20326,7(6+((76 For many applications there are claims concerning the metal sheet, which cannot be reached with a single sheet. To achieve new mechanical, thermal and acoustic properties with one sheet laminates with different properties were consolidated. (1) Sheet Metal Laminates (SML) with vibrations damping qualities can be made of steel sheets enclosing a viscoelastic plasticcorelayer. (2) The vibration energy of the oscillating coversheet is converted to heat. In the automotive lightweight constructions, sandwich sheets with noise-absorbent behavior are highly useful in the engine bay. (3), (4), (5) Accessorily, noise damping components made of sandwich sheets have a higher security against cracking. Due to the relocatability of the outer and the inner sheet, leakage of, e.g., an oil pan (s. fig. 1) in contact with the subsoil or in the case of a crash is unlikely for composites in contrast to a comparable one layer sheet.

)DLOXUHPRGHVGXULQJDQGDIWHUWKHIRUPLQJSURFHVVHV

 ),*85( Oil pan made of three layer sandwich (1) (a); failure modes of a sandwich with viscoelastic-plastic interlayer: displacement (b), delamination (c), crinkling (d)

Under load, during and after the forming process special effects occur. (6) As failure modes, mainly displacement and delamination of the cover sheets of the viscoelastic-plastic interlayer occur (see Figure 1). (7) Buckling at the inside surface is detected especially by vibration-damping composite sheets with long non-deformed legs. (8) The cover sheets themselves usually do not crack. Plastics and adhesives on the other hand creep under load. Because of the residual stresses in the cover sheets or due to temperature influences, the composite delaminates often after hours or days, respectively. (9)

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 733-741 (2011); doi: 10.1063/1.3623679 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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(;3(5,0(17$/7(676$1'7+(,56,08/$7,21)25'(7(50,1,1* 0$7(5,$/3523(57,(6 Mechanical properties of laminates were determined with experimental tests (see. fig. 2).

),*85( Uniaxial tensile test (a), tensile shear test (b); pilot project: v-die bending (c) and swing-folding (d)

([SHULPHQWDOUHVXOWV With resistance strain gauges the elasticity and with uniaxial tensile tests the plastic behaviors of the metal sheets were determined. Three uniaxial tensile tests carried out under the same conditions are shown in figure 3 (cover sheet material: DC04, DC06, 1.4301 and 1.4640). The isotropic hardening is described with the yield stress and the plastic strain curve for numerical calculations. For the plastomechanical preliminary design the yield strength (Rp0,2), tensile strength (Rm), and uniform elongation (φgl) of composites with cover sheets are read out. In fig. 3 the uniaxial tensile test (a) and the tensile shear test (b) are shown in fig. 3.

  ),*85( Uniaxial tensile tests of four metal sheet materials (a), tensile shear tests compared with the simulation for three velocities (b) (10)

734



1XPHULFDOFDOFXODWLRQRIWHQVLOHVKHDUWHVWV In figure 4 the experimental and numerical tensile shear tests of the adhesive film are shown. (11) The shear stress  increases approximately linear until the maximum  . Force and displacement ∆ are gauged. The shear stress is calculated out of the force and the initial overlapping length. For three velocities, the shear stress is shown over the displacement (fig. 3b). With high velocity, the tolerable displacement becomes lower, but the shear stress   ) gets higher. In literature, several investigations higher. So the specific stiffness of the adhesive (

=  ∆ determined the stress-distribution depending on the overlapping lengths (12 S. S. 403-481).

 ),*85( Tensile shear tests compared with simulation (a), geometry of tensile shear test (b)

Delamination and displacement of laminated sheet metal can be analyzed with finite elements. There are different modes of delamination. G. Alfano and M. A. Crisfield e.g. established an interaction model “mixed-mode” which summarized many fracture criteria proposed in literature. (13) To describe the elasticity of the adhesive, the traction/relative displacement law is used. In this paper, the damage law with nominal stress of the normal mode and the both other directions is chosen. Numerical results of the uniaxial tensile shear tests fit to the experiments. Special attention for forming sandwich sheets is laid on the failure of adhesive bond. As seen in fig. 4 the elastic and damage behavior is calculated exactly, but the maximal shear force and displacement depart up to 9%.

3/$6720(&+$1,&$/35(/,0,1$5<'(6,*1 With plastomechanical calculations the displacement of the metal layers is described. (14) As described by the tensile shear tests, the metal sheets can slide on each other under shear force. The adhesive layer is considered as viscoelastic. This shear force affects the bending behavior of the shear fixed layers. The metal layers itself remain shear fixed. (15) (16)

735

),*85( K-factor, part of transmitted shear force: none (a), partly (b), full (c), bending slat with superposed force (d)



With reference to the calculation of the angel of delamination αdelam which does not include the cohesive force (10), the influence of the adhesive layer on the individual sheet metal layer is described by a k-factor. Special attention must be paid to the position of the neutral axes Rn of each layer (17) (see figure 6). Part (b) of figure 6 shows the outer cover sheet of a symmetrical shear-fixed sandwich. In figure 5 two theoretical modes, a sandwich with an adhesive which cannot transmit any (a) and the whole (c) shear force are shown. A realistic situation for viscoelastic interlayer’s are displayed in part b. So the bending moment  of both cover sheets are superposed with the force ü . The compressive strain and elongation of the cover sheets are determined k-factor.

 ),*85( Principle of k-factor for each layer (a-e), geometric of a sandwich with different layer-thicknesses (f)

The strain distribution of sandwiches with many metal layers, which may have different thicknesses, can be described by the k-factor. For each metal layer with the thickness si, the neutral axis is computed dependent on the thickness of the sandwich s for the shear fixed mode:

 =  + ( + 1) ∗  =

 2

 −1 

(1) (2)

Therefore, the strain is defined step-by-step. A linear strain distribution (s. fig. 7) according to ludwig is assumed and can be described as follows:

 =

  _

 = –

  ∗ (1 +  ) <  < ∗ (1 −  ) 2 2

(3)

The young's modulus definite the material responses till a distance yf to the bending axis. As seen in fig. 7 the plastic behavior kf of the sheet material is described linearly (Eq. 4).

 = , + !

!=

736

"1 + #$ % − , , #$ − &

(4)

 ),*85( Stress and strain distribution of a cover sheet (a); influence of the overlapping length to the neutral axis (b)

Force is defined as integral of stress over the cross-section. (18) In contrary to Euler–Bernoulli beam theory (19) the forces of the compressive and strained zone differ. 

  − ∗ 2 2



⎛ ü = ' + * = - ⎜ 3 & 4 + 3 5 4 − 3 & 4 − =0

=0

=

(5)

  −7 + ∗8 2 2

3

⎞

5 4⎟

=

⎝

⎠

  7  8 ∗ ! ü> ≈ - @ ∗ ,> + 2 B A>

(6) ü_ = !CD"ü> %

 $E =  $E ∗ (Dü + 2 ∗ F ∗ G ∗ A ) ∗ -

(7)

Therefore, the maximal superposed force is used for dimensioning of the sandwich. By laying down the force which the adhesive transmits, the equilibrium of forces can be established and transposed to the overlapping length. With this simplified formulation, the plastomechanical preliminary design is done.

ü_ =  $E   7 8 ∗ ! B  @ ∗ , + 2 A  − 2 ∗ F ∗ G ∗ A  $E   ∆ = G ∗ 7H7− 8 ∗ ( + 1)H + H7+ 8 ∗ ( + 1)H8 2 2

(8)

(9)

Dü =

(10)

To minimize the calculated displacement, following options could be found: x x x x x x

Keep the required overlapping length, respectively the side length Choose an adhesive depending on to the shear force which is calculated Enlarge the numbers of metal layers Reduce the thicknesses of the outer layers Choose a material for outer layers with low yield strength Choose a small hardening coefficient

3,/27352-(&76$1'7+(,56,08/$7,21 With the simulation of the basic- and pilot tests, material properties and interactions were calibrate, to predict the suitability of a vibration-damping composite metal sheet for forming. The material was performed with bilinear, quadrilateral elements. The reduced integration plane stress space with hourglass control (CPS4R) was used. The

737

penalty method approximates hard pressure-overclosure behavior. The penetration distance is proportional to the contact force. An isotropic friction is chosen with μIJKL = 0,1.

 ),*85( Pilot project: Experiment v-die bending (a), v-die bending with cover sheet material DC06 (b)

The v-die bending process of a symmetrical sandwich with a length of 2L is considered in fig. 4. The displacement of both cover sheets is determent for tow zones, the bending area and the adjacent area. By the punch radius r and the bending angel α, the bending area is defined. The adhesive layer transmits shear force of the adjacent area into the bending area. The stress distributions of cover sheets out of DC06 with a thickness of 0,7mm in each care are shown. For distances t=[1,2,3,…]mm from the bending axis the displacement is measured.

 ),*85( Numerical results of die-bending, displacement of cover sheets for the bending area (a) and the adjacent area (b) to bending axis

Numerical results of die-bending of vibration damping sandwich sheets (DC06) are shown in fig. 9. The displacements Δl for the sandwich and tow single sheets (oK), which are located frictionless on each other are plotted. The displacement increases in the bending area (a). Beyond the bending area the displacement of the sandwich decreases of up to the edge. This decrease characterizes the transmitted force of the adhesive between the slide legs. Therefore, the displacement of unbonded sheets (ok) increase further. The slide legs are comparable with the overlapping length in the tensile shear test.

738

  ),*85( Numerical results of elongations of the external fibers (bending axis at distance t=40mm)

With a punch radius of r=8,5mm sandwich sheets (marked with a small arrowhead Å) and tow single sheets without adhesive and were formed. The absolute strains of the upper and under fiber of sheets without adhesive show similar values. So the neutral axis for unbonded sheets can be assumed in the middle of the current layer. The strains of the upper and under layers differ because of the varied bending radius. In contrast to the unbonded layers, the sandwich shows remarkable differences. There is a local maximum at the bending axis (L=40mm) and the bending area is bigger. For symmetrical sandwiches with tow metal layers of the same material, a shear fixed mode with ki=-1 and ka=1 can approximately be achieved. The strain of the upper fiber of the under layer decreased. The higher friction, the less is the maximum of strain. After forming, the maximum remains, but all in all the strain is higher. This is lead back to the shear force of the adhesive, which is as large as it can form the metal sheets. Calculate by plastomechanical preliminary design the “constant strain over bending angle” is shown in fig. 10, too. The influence of the side length on the displacement of the edges is shown in fig. 10 for the side lengths L=[40,35,30,25]mm. Predicted by plastomechanical preliminary design, the displacement decreases by increasing side length. Results fit by trend. With fig. 11b following conclusion can be made: the center of area between the under and upper fiber in the strain-displacement diagram designs the k-factor and therefore the position of neutral axes.





),*85( Numerical results of die-bending; displacement Δl over bending radius α for variation of side length L (a), elongation of the external fiber of the both layers with bending axis at distance t=40mm (b)

Inaccuracies of preliminary design like are ascribed to the: x Constant thickness during and after the bending process

739

x x x x

Negligence of elastic behavior of the cover sheets Linearization of the material properties Linearization of the elongation distribution Constant strain over bending angle

The normal plastic strain component in bending direction depends on the thickness of the metal layers. The thinner the inner cover sheet the meaner the plastic strain and the displacement of the edges (see no. 2-4). On the other hand the inner layer tends to buckling. To get the lowest normal stress in bending direction, the thickness should increase see configuration no.4. Also the spring-back of the undamaged sandwich depends on the layerconfiguration.

52//)250,1*2)9,%5$7,21'$03,1*&20326,7(6+((76 First, the roll forming process was calculated for pilot geometries. There is a 90-degree v-profile of DC06 for the cover sheets and with the inner radius of r=1mm and wide of band b=97mm. The thickness of the sandwich is s=1,5mm. Because of the long side length, no measurable delaminations and displacements of the cover sheets occur. In fig. 12a the flower pattern for the complex guiding profile with a shear fixed adhesive is shown. By FEmethods the geometry of sheets and viscoleastic adhesive is calculated with the material properties displayed in fig. 4. The calculation concurs with the experiment. Exactly the same failure-modes occur and both edges of the profile are damaged. Displacement and delamination occur on radius one and tow (b). If the adhesive fails, delaminates occurs based on the different spring back behavior. Different spring back of the cover sheets occurs, due to the varied bending radii; the outer radius is larger by half the composite thickness. (10) As a result of fixed edges, buckling occur on straight sides (see. fig. 12b).

 ),*85( Roll-forming of vibration-damping composite sheets, required flower pattern (a), damage of the adhesive (b), damage status of the adhesive

With the plastomechanical preliminary calculations, the first step of the flower pattern is damaged. This is verified by the FEM- calculations (c). Maximal shear stress is passed and the failure of the adhesive bond reaches 100% on the edges. This status (c) can be understood with the fig. 4 “tensile shear tests”.

&21&/86,216 At the Chair of Forming Technologies, University of Siegen vibrations damping composite sheets were investigated regarding their forming limits. The prediction of the roll forming process of this sandwich sheets could be achieved. During forming, failure modes are delamination and displacement of the cover sheets. The mechanical properties of the metal sheets were determined by uniaxial tensile tests. Tensile shear tests were carried out under variation of the shear velocity. The numerical calculations were calibrated with the tensile shear test. A plastomechanical

740

preliminary design was developed. The causes of displacement were identified by calculations. Based on these calculations, opportunities for forming without failure were identified. Exactly prediction of a guiding profile geometry which is produced by a roll forming process requires FEMethods. Plastomechanical preliminary design shows sufficient but improvable deliverables. Especially the elastic behavior of the unformed sides should be improved. Further improvable inaccuracies are in addition to the strain distribution over the thickness and the bending angle, the material description. On the base of further investigations, these inaccuracies should be minimized by more precisely assumptions. Instead of symmetrical sandwiches, significant improvements can be achieved by e.g. reducing the thickness of the outer metal layer. With increasing the number of sheets the total thickness of the composite can be achieved. On the other hand thin inner layers tend to buckling. The buckling tendency should be investigated further.

$&.12:/('*0(176 The present research was a part of Zentrales Innovationsprogramm Mittelstand (ZIM), Bundesministerium für Wirtschaft und Technologie (BMWi).

5()(5(1&(6 1. &DUUDGz$HWDO Metal/polymer/metal hybrid systems: towards potential formability applications. Composite Structures. Jg. 93, 2011, Vol. H. 2. 2. -DQGHO%HUQG0HXWKHQXQG$OPXWK6LJUXQ Coil Coating. Wiesbaden : Vieweg & Sohn Verlag, 2008. ISBN 978-3-83480338-2. 3. (5RRV.0DLOH Werkstoffkunde für Ingenieure. Berlin Heidelberg : Springer, 2008. ISBN: 978-3-540-22034-3. 4. DQWLSKRQ antiphon® MPMTM - vibration damping sandwich material. [Online] S-670 40 ÅMOTFORS Schweden, 04 20, 2010. [Cited: 04 21, 2010.] www.antiphon.se. 5. 7K\VVHQ.UXSS 6WHHO Bondal® Körperschalldämpfender Verbundwerkstoff. [Online] 10 1, 2009. [Cited: 04 21, 2010.] www.thyssenkrupp-steel.com. 6. 3HWHUV 6DVFKD Handbuch für Technisches Produktdesign: VERBUNDWERKSTOFFE. Berlin Heidelberg : Springer-Verlag, 2006. ISBN 13 978-3-540-21416-8. 7. 1XW]PDQQ'U,QJ0DUF Umformung von Mehrschichtverbundblechen für Leichtbauteile im Fahrzeugbau. Aachen : Shaker Verlag, 2008. 8. .H‰OHU/XW] Simulation der Umformung organisch beschichteter Feinbleche und Verbundwerkstoffe mit der FEM. Aachen : Shaker, 1997. ISBN 3-8265-2256-7. 9. +HOOLQJHU9 New Possibilities for Improved Bending of Vibration Damping Laminated Sheets. CIRP annals: manufacturing technology. Berichte der Internationalen Forschungsgemeinschaft für Mechanische Produktionstechnik, 1999, Vol. 48/1/199. 10. Metal Forming of Vibration-Damping Composite Sheets. (QJHO %HUQG %XKO -RKDQQHV DOI: 10.1002/srin.201000205, Weinheim : 2011 WILEY-VCH Verlag GmbH & Co. KGaA, 2011, Vol. steel research international. 11. ',1 1RUPHQDXVVFKXVV 0DWHULDOSUIXQJ 103  LP Strukturklebstoffe - Bestimmung des Scherverhaltens struktureller Klebungen, Teil 2: Scherprüfung für dicke Fügeteile (ISO 11003-2:2001, modifiziert). Berlin : Beuth Verlag GmbH, 2004. 12. +DEHQLFKW*HUG Kleben. Heidelberg : Springer-Verlag, 2009. ISBN 978-3-540-85264-3. 13. * $OIDQR 0 $ &ULVILHOG Finite element interface models for the delamination analysis of laminated composites: mechanical and computational issues. International Journal for Numerical Methods in Engineering. 50, 2001, Vol. 7, Pages: 1701–1736. 14. 0LFKLKLUR 7$.,*8&+, )XVDKLWR <26+,'$ Deformation Characteristics and Delamination Strength of Adhesively Bonded Aluminium Alloy Sheet under Plastic Bending. JSME International Journal. Series A, Vol. 46, 2003, Vol. No. 1. 15. )ULW]HQ3URI'U,QJ&3 Technische Mechanik II (Elastomechanik). Siegen : UNI SIEGEN, 2006. 16. .UHXOLWVFK+HULEHUW92(67$/3,1(6WDKO/LQ]*PE+ Formgebung von Blechen und Bändern durch Biegen. Wien : Springer, 1995. ISBN 3-211-82595-9. 17. 0 :HLVV % ) 5ROIH 0 'LQJOH - / 'XQFDQ Elastic Bending of Steel-Polymer-Steel (SPS) Laminates to a Constant Curvature. JOURNAL OF APPLIED MECHANICS. 2006, Vol. Vol. 73. 18. 9',%XFK Handbuch Umformtechnik. Berlin Heidelberg : Springer, 2007. ISBN 978-3-540-23441-8 . 19. /DQJH . Umformtechnik: Blechbearbeitung, Handbuch für die Industrie und Wissenschaft. Berlin Heidelberg : SpringerVerlag, 1990.

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Validation of Uniaxial Data Beyond Uniform Elongation Mark A. Iadicolaa a

Material Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899-8553 USA Abstract. Uniaxial testing is one of the simplest methods used to determining mechanical properties for sheet metals, but the classic test does not determine properties to sufficiently high strain levels for use in models of very large deformations such as forming operations. This limitation is primarily due to the classical test requiring uniform deformation in the gage length. Full-field optical techniques are now being used to extend the applicable strain range of uniaxial testing to strains beyond uniform elongation and well into diffuse necking, but assumptions must be made as to the multi-axial state of stress in this regime. In order to determine if such assumptions are correct, uniaxial experiments are performed combined with a unique combination of digital image correlation to measure plastic strains and stress tensor measurement through X-ray diffraction methods. The combination of metrology permits determination of the true stress-strain response to strains four times uniform elongation and the associated multi-axial stresses at the highest strain levels. Keywords: X-ray diffraction; Digital Image Correlation; Mechanical Testing; Mechanical Properties; Plasticity PACS: 61.05.cp; 62.20.-x; 62.20.f; 62.20.m; 81.70.-q

INTRODUCTION Uniaxial tensile testing is probably the most common method used to determine the mechanical properties of sheet metal materials. The data developed is used in numerical modeling of various sheet metal forming processes, but it is well known that the application of uniaxial testing in this manner has some severe limitations. The tests results are only applicable up to uniform elongation in the gauge section [1]. The strain level at that point is known to be well below strains achievable in multi-axial strain states such as equal-biaxial tension. Real forming processes involve these multi-axial strain states, thus when modeling these processes some assumptions must be made to extrapolate the uniaxial data beyond the applicable range of the test or the use of other testing methods [2,3]. Modeling of uniaxial tests beyond the uniform elongation through inverse-methods has been promising, but requires an assumption of an appropriate constitutive law with the correct parameters that can be fit based on the available experimental data [4]. Multi-axial test methods that can achieve these higher strain levels do exist (e.g. shear, bulge, tube extension/inflation), but they are often difficult to perform and require the assumption of a constitutive law or bending of the as-received sheet. The result is a blending of the actual mechanical behavior with certain experimental limitations and complications. There has long been a desire to extend the applicability of uniaxial tensile testing to beyond uniform elongation when strain localization (i.e. diffuse necking) starts. Bridgeman [7] Davidenkov [6] initially described how this might be done for a cylindrical specimen and then extended it to wide sheets with plane-strain necking [7]. This and similar work assumes a known symmetric strain field in the diffuse neck that can be described by the curvature of the outside edges of the specimen. In addition, the stresses were then calculated from the applied strain field based on an assumed constitutive law. This was extended by Aronofsky [8] to a flat sheet by machining two samples one cylindrical (to determine a material properties) and one a bar with rectangular cross-section (to determine the strain field) and comparing the results. Many researchers have begun to extend the uniaxial tensile tests using a combination of these results and the use of full-field strain measurement methods in the diffuse neck during uniaxial testing, but it is unclear if all the assumptions are valid especially on the multi-axial state of stress that occurs in the necked area that has not been measured and must be inferred from the strain field and the applied load. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 742-749 (2011); doi: 10.1063/1.3623680 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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The purpose of this study is to measure simultaneously and in situ the strain field and state of stress within diffuse neck, and use this data to validate various theories. Either result will enable correct application of relatively simple uniaxial testing to a broad range of numerical modeling. This experiment requires the ability to measure stress and strain at a localized spot on a tensile specimen during plastic deformation. A combination of full-field strain measurement and X-ray diffraction (XRD) is chosen to accomplish the measurements near the diffuse neck. XRD is used to measure the interatomic lattice spacing of the specimen of interest while holding displacement with the specimen under applied load. The system determines the stress tensor from the measured elastic lattice strain tensor just inside the surface of the specimen at the location of the beam spot. This spot can be sized and positioned to measure key locations such as the multi-axial stress at a feature (e.g. a diffuse neck). A key point to understand is that the applied stress results in an elastic strain in the lattice, plastic deformation through dislocation motion does change the macroscopic-scale strain, but the average local lattice spacing within the crystal lattice is still determined by the elastic behavior of the lattice. The details of the XRD measurements are discussed below. Simultaneous to the XRD measurements, a two-camera digital image correlation (DIC) system and processing software are used to perform 3D displacement measurements that are used to determine the surface strain field of the specimen. The method requires application of a surface pattern to track and clear line of sight for both cameras, therefore the XRD measurements in this study are made on the opposite side of the specimen from the DIC measurements. Since "localized plastic deformation" can describe multiple behaviors, it is useful to define the terms used in this study. There are three significant strain localizations: Lüders banding, diffuse necking, and localized necking. Figure 1 shows the load displacement plot for a uniaxial test with exemplary strain fields (each for a different range of first principal strain) showing different regions of behavior. Some steels are known to exhibit Lüders banding, where a portion (or band) of the gauge length has a sudden localized increase in strain accompanied by a sudden drop in load. This band expands along the gauge length (Fig. 1 strain field B) with additional specimen elongation at approximately the same load level until the entire gauge length is at the higher strain state. After this behavior is complete, a nearly homogeneous strain field is seen in the specimen gauge section (Fig. 1 strain field C). Upon further straining, once the maximum load is reached, the strain field in the gauge section is no longer homogenous [1]. At this point, a portion of the gauge length begins to strain more than the remainder (Fig. 1 strain field D), resulting in an area described as a diffuse neck that is generally symmetric about the longitudinal axis of the specimen across the width. Additional loading will result in a portion of the diffuse neck straining further than the remainder of the diffuse neck at an angle to the longitudinal axis, and is referred to as localized necking (Fig. 1 strain field E). Localized necking is the precursor to ductile fracture. This work is concerned with the measurement of stresses and strains between initial diffuse necking to onset of localized necking. The behavior in this range has been seen to control the ductile fracture behavior of the material [8], but is potentially more useful in extension of the uniaxial stress-strain curve for use in numerical modeling of sheet metal forming processes. 0.04

B

ε1 0 0.06

C

ε1 0 0.30

D

ε1 0 0.55

E

ε1

0.35

FIGURE 1. Plot of engineering stress and strain for HSLA specimen (on left) and related first principal strain maps for various types of localized straining (labeled B-E).

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EXPERIMENT Specimen The material of interest here is a high strength low alloy (HSLA) steel sheet approximately 1 mm thick. The sheet is fine grained and zinc coated for use in automotive sheet metal forming. The zinc coating on the specimen surface is removed (with a solution of HCl and water) before testing. The mechanical response of the material is known to be anisotropic, with different properties in the rolling direction (RD) of the sheet as compared to the direction transverse to the rolling direction (TD). The specimen is machined to the ASTM-E8 standard subsize specimen for sheet metal (Fig. 2) with RD in the x-axis direction. The edges of the gauge section are then sanded to create a slight hourglass taper within the gauge section, but still within the deviations acceptable for E8 requirements. This directed the development of the diffuse neck toward the center section of the gauge length within the field of view of a full-field strain mapping system. The precise location of the initial diffuse neck centerline did not perfectly match this location, but was within one specimen width of the smallest initial gage width. 1.0

25.4

y

y local 12.7

fixed

z

x

6.0 average R 6.35

DIC neck

XRD

extension

>100 FIGURE 2. Nominal specimen dimensions (in mm). The areas used for strain averaging are shown in dashed lines.

Experimental Setup A portable uniaxial testing frame with manual screw loading is used to deform the sample. The specimen end tabs are held in plate style grips, with extension imposed on the positive x-axis end with the other grip held fixed (Fig. 2). The grips are seated in a guide frame to prevent initial misalignment. The load (P) is measured using a load cell that is calibrated using a force ring prior to the experiment. As mentioned above, strain is measured using a commercial two-camera DIC system mounted under the uniaxial testing frame (Fig. 3b, positive z-direction in Fig. 2) with the bisector of the cameras perpendicular to the specimen positive z-axis surface with an approximate total field of view of 30 mm. The field of view is offset slightly to the positive x-direction to compensate for the motion of the specimen gauge section toward that direction during extension. The XRD head is located above the specimen (Fig. 3a) in the negative z-axis direction (Fig. 2). The orientation of the XRD head and its translations in the x and y directions was calibrated to the DIC system orientation to permit the spatial mapping from one system to the other system.

y Specimen

X-ray Head

x

z ~33o

Specimen

x

Grip

Grip Light

y Left Camera

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Right Camera

(a)

(b)

FIGURE 3. Images of the experimental setup (a) upper view with XRD and (b) lower view with DIC.

Strain Measurement The DIC system was calibrated using a manufacturer recommended method using a known length along the specimen surface. This process, in addition to a coordinate transformation, is used to put the DIC and XRD systems into a common specimen coordinate system and align the resulting measurement positions. Assessments of the noise level in the DIC set up showed a measurement uncertainty of ± 0.001 strain. The strain measure used by the DIC system is the Hencky strain (or Natural strain). This is the same as the true strain, but includes the higher order terms when calculated from the displacement field. For the remainder of the paper we will treat the measured Hencky strain values as equal to the true strain values, as defined below, even though they may differ by a very small amount, due to the higher order terms. During each image pair acquisition, a digital sample from the load cell signal is taken to ensure the strain and load measurements are synchronized. The surface strain measurement orientations follow the axes shown in Fig. 2, and are denoted by subcripts xx, yy, and xy. These data are averaged over three different lengths, thus defining three different strain measures. The areas that are averaged over are sketched and labeled in Figure 2. The area labeled "average" will result in a strain measure similar to an extensometer with axial gauge length of approximately 15 mm, and will be denoted with the superscript "avg". The smaller area labeled "neck" is located after the test is complete to be centered on the final diffuse neck location with an axial length of approximately 2.5 mm and the full specimen width, and the related strain values are denoted with the superscript "neck". The third area shown as a small circle approximately 1 mm in diameter labeled as "local" is located at the same x and y location as the X-ray diffraction measurement on the upper surface of the specimen, and the related strain measurements are denoted with the superscript "local".

Stress Measurement The use of XRD to measure residual stress in cold worked materials is well known [9]. In this work, a similar method is used to determine the yield stress at extension hold points during plastic deformation (similar to [10] but measuring the full stress tensor). Residual stresses in the sheet material will effect the measurements until the material is initially yielded through the specimen thickness [11]. The governing equation is,

dψφ − do do

= 12 S2 {σ xx cos 2 φ + σ xy sin(2φ ) + σ yy sin 2 φ − σ zz } sin 2 ψ

(1)

+ 12 S2 {σ xz cos φ + σ yz sin φ} sin(2ψ ) + 12 S2σ zz + S1{σ xx + σ yy + σ zz }, where dψφ are the measured lattice spacing, do is the stress-free lattice spacing, and ½ S2 and S1 are the effective Xray elastic constants. These constants are sensitive to material composition, lattice direction, and other microscopic effects. € These constants will be calibrated as part of the experimental procedure using the specimen being tested, rather than relying on values from the literature for similar materials. Eqn. 1 has six unknown stresses (σij) and the unknown stress-free lattice spacing (do). A sufficient number of measurements at various combinations of angles are required to solve the resulting system of equations. For the measurements shown here, nine tilt angles (± 28.0º, ± 21.87º, ± 11.8º, ± 6.87º, 0º) and four in-plane rotations (-135º, -45º, 0º, 90º) are used. The system also has two detectors positioned symmetrically about the X-ray source beam. This results in 72 independent measurements of dψφ for each XRD scan. Since the imposed deformation is in the plane of the specimen (thin compared to the other dimensions), it is assumed that σ33=0. This assumption allows determination of do from the system of equations for each XRD scan. The system of equations can then be solved for the remaining five σij values. This is done using the data for all 72 measurement angles simultaneously. This is an over-determined problem resulting in the least squares solution and an estimate of the associated uncertainty. These uncertainties were found to be < 6 MPa in magnitude for all of the scans discussed in this work. An XRD scan of a stress-free iron powder sample is taken using the same angles to verify the system performance at the start of each day of testing. These powder scans typically result in an average magnitude of stress < 2 MPa with no single component > 17 MPa. The scans of the specimen are corrected by subtracting the powder scan data to reduce the effect of system bias, but as a conservative estimate for the systematic uncertainty the value of ± 17 MPa is still used. Since the systematic uncertainty (± 17

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MPa) is greater than the uncertainties associated with each scan (± 6 MPa) the systematic uncertainty will be assumed here for all the XRD data shown. The X-ray beam was reduced by an aperture to a spot size of 1 mm in diameter. This spot size was chosen based on the spatial variation seen the strain fields of preliminary tests. At each hold in extension, the XRD spot was repositioned in the specimen xy-plane based on the strain field measured. After focusing an XRD scan is performed taking approximately 900 s. The stresses measured in this manner are based on the current state of the sample at the point being measured and are appropriately described as true stresses. It is assumed that the orientation of the XRD axis to the specimen axes (shown in Fig. 2) remains constant throughout the experiment. This is not exactly correct when portions of the specimen rotate slightly due to a rotation in the principal strain directions such as during Lüders banding or localized necking, but these rotations are minor and the behaviors in these ranges of plastic deformation that are not the main focus for this study. Similar to the strain measurements, the cross-sectional area and stretch ratio used to determine the stress can take multiple forms. An "average" axial stress is defined using the minimum initial cross-sectional area in the gauge section and the average stretch ratio based on the εavg measured by DIC. A second axial stress measure with superscript "neck" is used based on the same initial cross-section, but using the εneck values to determine the stretch ratio. The stress values with superscript "XRD" are measured by the X-ray diffraction system after the calibration procedure described below.

Stress & Strain Definitions The standard definition of engineering stress (s) and engineering strain (e) are used where P is the load, Ao is the initial cross-sectional area, and lambda is the stretch ratio of the current length (L) over the initial length (Lo),

P and e = λ −1. Ao

(2)

P P = λ and ε = ln( λ), A€ Ao

(3)

s=

Volume conservation is assumed during the plastic deformation allowing us to describe true stress and true strain as,

σ=

€ where A is the current cross-sectional area. The definition of the variables Ao, L, and Lo can result in different values of stress and strain whenever the strain or stress fields are not homogeneous. In the regime where homogeneous deformation is expected, between the end of € Lüders development and diffuse necking, the data should result in consistent values of stress and strain. Care will be taken to define the method of calculation for each change of € variable definition. After development of any localized strain, the stress field will similarly be distorted, and it is well accepted that a diffuse neck results in a multi-axial state of stress [6,7]. Similar assumptions are made here which are later verified based on the measured data. Recall, since no stresses are applied on surfaces perpendicular to the z-axis and the specimen is much thinner than it is wide, the stress through the thickness is assumed to be zero. Although no stress is applied on the surfaces normal to the y-axis, it is assumed that there could exist nonzero stress values in the plane away from the specimen width edges as a result of any strain localization. Other common assumption such as symmetry of the strain field [6,7], and hardening or constitutive laws will not be needed, but will be measured for verification of these assumptions. It is often desirable to reduce multi-axial stress states to uniaxial equivalents, therefore the measured multi-axial stress and strain data will be converted to two commonly used uniaxial equivalent measures, the Von Mises and Tresca equivalents. Von Mises equivalent stress and strain are defined as,

σ yVM =

(σ1 − σ 2 ) 2 + (σ 2 − σ 3 ) 2 + (σ 3 − σ1 ) 2 2 2 2 2 VM and ε = (ε + ε 2 + ε 3 ) 2 3 1

(4)

where σi and εi are the principal true stress and principal true strain, respectively. The Tresca equivalent stress and strain are defined using the same components as Tr (5) σTr = ε i max y = (σ1 − σ 3 ) and ε

€ of the sheet material, but is just a preliminary analysis € The experiment shown here does not consider the anisotropy of the experimental method. Subsequent work will look at the material behavior in other orientations such as TD.

€

€

746

Procedure After machining, the specimen is stripped of the zinc coating in the gauge section. The surface for DIC is painted with a light coating of matte white paint followed by a speckled coat of matte black paint. The surface used for XRD was not coated. Once the specimen was clamped in the grips, the reference DIC images were taken. The extension is applied by a manually screw drive. At selected points the extension was held and an XRD scan of the stress was taken (including positioning and auto focusing). DIC data was acquired during the extensions and at the hold point before and after XRD scan. The extension was continued through Lüders banding into the plastic hardening regime with holds at various levels of strain to perform XRD measurements. Figure 4a plots the “local” engineering strain and “local” engineering stress as a continuous line for the entire test with diamond symbols at the points of XRD scans. Load drops occur after each halt in loading and could be related to the variable speed of manual extension or some backlash in the loading screw. A series of elastic unloading/reloading cycles are used for XRD calibration, with one being a full unloading at the middle of the experiment. The DIC data was processed during extension holds to determine the appropriate location for the XRD measurements. During diffuse necking the position of the XRD measurement was moved into the high strain region near the longitudinal centerline of the specimen, but not the exact location of the final localized neck since this is not known a priori. The test was halted when a developed localized neck was visible to the eye.

RESULTS As shown in Fig. 4a at approximately elocal = 0.10 strain, the specimen is elastically unloaded and reloaded with frequent hold points for XRD measurements to be used in calibration of the X-ray elastic constant. The XRD locations are held fixed during these elastic unload/reload cycles so that the correction factor determined is associated with a consistent value εlocalxx. This procedure is repeated three more times with the elastic unload points selected to be once just after the start of diffuse necking (between elocal = 0.18 strain and 0.20 strain) and two more times before localized necking began. A correction factor is determined for the X-ray elastic constant determined from each elastic unload/reload. A maximum variation of 10 % was seen in these correction factors, with an average correction factor of 1.11. As a simple correction that includes these variations, the X-ray elastic constant correction factor values for all stress values measured here are interpolated from these measurements based on the current maximum local strain level at the location of the XRD measurement. A similar variation was seen in an aluminum alloy [12], and further analysis of the variation seen here is planned in future work.

(a)

(b)

FIGURE 4. Engineering stress vs. engineering strain plot (a) with XRD scan points shown, and true stress vs. true strain plot (b) with XRD data raw (default elastic X-ray elastic constant) and corrected (calibrated X-ray elastic constant).

Figure 4b plots the stress and strain results for only the points of XRD scanning with the elastic unload/reload data points removed. The continuous line is for the calculated local stress from the load cell assuming no multi-axial stress components, and the points are the uncalibrated (system default elastic constants) and calibrated X-ray measured stresses in both the x and y directions. The calibrated correction factor has a substantial effect for the higher values of stress. From here on the XRD stress data discussed are the calibrated values.

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As described above the specific location of the diffuse neck along the gauge length is not known a priori, thus the position of the XRD measurement is initially selected to be near to the minimum specimen width and as the deformation develops the position is moved based on the current strain field. The location associated with εlocal is the 1 mm diameter spot on the positive z-face of the specimen centered at the same x and y location as the XRD on the negative z-face. And it is this strain value at which the XRD data is plotted below. During Lüders banding that point was kept near the smallest cross-section with the exception of one scan that was taken at the edge of band (see Fig. 4b at approximately 0.04 strain) that showed a distinctly non-zero σ22 component. After diffuse necking began, the stress measurements were performed at specific locations in areas of high strain. These locations were found to be < 0.7 mm from the eventual center of the fully developed neck. Based on the final strain profile across the neck and the microscope images of another specimen thickness across the neck at the start of localized necking, the extent of the neck in the axial direction seems to be between one to two specimen widths. It is difficult to predict the exact location of the diffuse neck without biasing the gauge width, with a gage length having a more hourglass shape than is permitted by the E8 specimen shape. Such a geometry might bias the resulting neck shape and stresses as well, similar to notched bar sample geometries that are used to purposefully create a triaxial state of stress. The σXRD values show (Fig. 5a) that the yy component is nominally zero below approximately 0.35 strain, with the exception of one scan specifically taken on the edge of a Lüders band that did show σXRDyy = 40±17 MPa. It is interesting to note that the diffuse neck begins to develop after approximately εlocalxx = 0.17, well before the σyy component develops. Another interesting feature in the σXRDyy data is an increased slope at approximately εlocalxx = 0.47. Figure 5b plots the other three measured components of stress after calibration, these are seen to be quite small with the exception of the last point near localized necking (εlocalxx = 0.53). The values seem to vary randomly with the exception of the σXRDxz and σXRDxy components that increase in magnitude after approximately εlocalxx = 0.35 which is consistent with the increase in σXRDyy. These shear components of stress do not seem to show the same rapid increase after εlocalxx = 0.47, but the existence of the distinctly nonzero value of σXRDxz could be the result of a through thickness stress variation in σXRDxx at these strain levels.

(a)

(b)

FIGURE 5. True stress (XRD) vs. true strain (local) for principal direction (a) and XRD shear stress components (b).

Figure 6 compares true stress-true strain calculated using the average, neck, and XRD/local definitions, also plotted are the Von Mises and Tresca equivalents. Since the location of the XRD data is not exactly the center of the final localized neck the “local” strain never achieves the highest strains that are measured by “neck” strain measure. For the region where there is data for all of the definitions (0 strain to 0.22 strain), there is good agreement. In the “local” strain range from 0.22 strain to 0.47 strain, there is good agreement between the “neck” data and the Tresca equivalent data, but the σXRDxx data and the Von Mises equivalent tend to be greater than the “neck” data. This is not surprising since the σXRDyy values become distinctly positive starting at 0.35 strain which is where the deviation between the “neck” and σXRDxx deviate. The maximum true stress value occurs approximately at 0.47 strain for all the measures based on the XRD data. The “neck” measure places the maximum closer to 0.53 strain. The general agreement between the “neck” and Tresca equivalents for the majority of the strain range shown and the fact that the Von Mises equivalent results are generally higher in stress, suggests that the “neck” method of extending the true stress - true strain data into the diffuse neck region is valid, but is likely a lower bound on the expected behavior.

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FIGURE 6. True stress vs. true strain for the various definitions including the effective Von Mises and Tresca.

CONCLUSIONS The experimental design is sufficient to develop the data required to verify common theories of the mechanical behavior within a diffuse neck. The XRD method has sufficient resolution to measure the multi-axial stress tensor at a small spot on the specimen. This combined with full field DIC and proper alignment of the various coordinate axes, permits determination of the local stress-strain behavior that can be compared to the theoretical stress-strain behavior. For this HSLA sample, the typical assumptions using the average strain in the neck and the overall load to calculate the effective mechanical response seems to be a lower bound to the behavior, but also could over-predict the maximum obtainable strain before localized necking.

REFERENCES 1. A. Nadai, Theory of Flow and Fracture in Solids, McGraw-Hill, NY,1950, pp. 70-88. 2. Y.H. Wang et. al., Experimental Techniques March/April, 54-59 (2010). 3. D. Zeng, Z. Xia, Proceddings of SAE International 2010-01-0981 4. Y. Ling, AMP Journal of Technology, 5,37-48 (1996) 5. E.E. Cabezas, D.J. Celentano, Finite Elements in Analysis and Design, 40, 555-575 (2004) 6. N.N. Davidenkov, N.I. Spiridonova, Proceedings of ASTM, 46, 1147-1158 (1946) 7. P.W. Bridgman, Studies in Large Plastic Flow and Fracture, McGraw-Hill, NY,1952, pp. 9-37 8. J. Aronofsky, Journal of Applied Mechanics, March, 75-84 (1951) 9. I.C. Noyan and J.B. Cohen, Residual Stress: Measurement by Diffraction and Interpretation, Springer-Verlag,1987 10. M.A. Iadicola, T. Foecke, and S.W. Banovic: International Journal of Plasticity 24 (2008) 2084-2101. 11. T. Foecke, M. A. Iadicola, A. Lin and S. W. Banovic: Metallurgical and Materials Transactions A 38 (2007) 306-313. 12. M.A. Iadicola and T.H. Gnaeupel-Herold: Advances in X-ray Analysis 51 (2008) 131-137.

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Characterization of Initial Anisotropy of Sheet Metals Employing Inhomogeneous Strain Fields Alper Günera, Celal Soyarslana, Alexander Brosiusa, A. Erman Tekkayaa a

Institute of Forming Technology and Lightweight Construction TU Dortmund Baroper Str. 301 44227 Dortmund Germany

Abstract. The realistic finite element simulation of sheet metal forming processes necessitates the accurate characterization of plastic behavior of the sheet. For this purpose different experiments and specimen geometries are used in the field of sheet metal characterization to supply information to the increasing number of sophisticated constitutive models demanding many parameters. This work proposes an inverse procedure that reduces the number of needed experiments by utilizing the distribution of strains in the identification of the initial yield locus of sheet metals. The procedure includes the optical measurement of strains on a flat specimen with a varying cross-section. The geometry of the specimen allows obtaining a distribution of strains rather than a uniform deformation. The measured deformation field is compared with the numerical results which use the YLD2000-2D yield criterion and the difference is tried to be minimized in an inverse framework using the Levenberg-Marquardt algorithm. The proposed method is applied to the commercial aluminum alloy AA6016 by varying the specimen orientation with respect to the rolling direction. The results show that the use of the strain distribution in the objective function is a crucial point in identification of the yield locus. The inversely obtained yield loci are in accordance with the conventionally obtained yield stresses and r-values and it is showed that the proposed specimen and the method can successfully characterize the initial anisotropy of the sheet metals in the first quadrant by using one tensile experiment and one layer compression test. Keywords: Sheet Metal, Finite Element, Yield Locus, Inverse Method

INTRODUCTION The fact that the sheets show a certain texture due to the rolling operation in the production stage makes it difficult to assume isotropy in the numerical simulations of sheet metal forming processes. Instead an orthotropic symmetry of the sheet is assumed, in which the rolling, transverse and normal directions are considered to be the lines of symmetry. The Hill´48 model [1] is a widely used quadratic yield criterion, which is utilized especially to model anisotropic steel sheets. With the reduction of the model to plane stress state, the needed 3 material parameters are obtained from the uniaxial tensile tests at 0°, 45° and 90° to the rolling direction. Further invesatigations on the sheet metal behaviour in different stress states led to development of this model as Hill´90 [2], which enabled the consideration of the equi-biaxial stress state as an additional material parameter. Meanwhile, Barlat et al. proposed different sets of yield functions. Yld91 [3] accepts the uniaxial yield stress values at 0°, 45° and 90°, together with the equi-biaxial yield stress as material parameters. This model is followed by Yld94, Yld96 and Yld2000-2d. Among them Yld2000-2d has been used extensively in the last years and is also implemented in the commercial code LS-Dyna [4]. This non-quadratic yield function uses yield stresses and strain ratios (r-value) in three uniaxial cases, 0°, 45° and 90° and one equi-biaxial case to define the yield locus, adding up to 8 material parameters totally. Banabic et al. proposed another model, BBC2000, which accepts 7 material parameters [5]. The model is then improved to BBC2005 with 8 material parameters, which can also be derived from the Yld2000-2d model [6, 7]. The yield function description proposed by Vegter and van den Boogaard utilizes quadratic Bezier functions that are defined by the help of the different states of stresses [8]. This model requires 17 free material parameters if the experiments are performed at 0°, 45° and 90°. The consequence of these complex material models is the necessity of an extensive analysis of the sheet under various stress states, such as uniaxial tensile test, shear test, biaxial tension test, plane strain tensile test, layer compressions test, etc [9]. Increasing flexibility of the yield functions is desired on one hand, and on the other hand the number and complexity of the material tests are tried to The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 750-757 (2011); doi: 10.1063/1.3623681 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

750

be kept at reasonable levels. Since these tests are not standardized completely, it is difficult to transfer and compare the results of different laboratories. Another disadvantage of the conventional testing methods is the assumption of uniform stresses and strains in the considered plastic zone of the specimens. Analytical calculation of the stresses and strains is only possible with this assumption and it is not satisfied for all specimen geometries. On the other hand inverse approach can identify the material parameters with an iterative manner in which for each iteration a finite element model is used [10]. Inverse methods enable the usage of unusual test set-ups or even the forming process itself [11, 12]. These methods compare conventionally experimental measures like tool forces and tool displacements [13, 14]. Today, by the help of the optical measurement systems the strains on the sheets surfaces can be measured directly. The utilization of optical measurement systems in the inverse parameter identification together with finite element simulations has been in use in the field of characterization of sheet metals [15]. However, the usage of complex yield functions has not been studied thoroughly. In the present paper a specimen with a varying cross-section is introduced, which enables an inhomogeneous distribution of strains. These strains are captured with an optical measurement system. At the same material points the strains are also identified in the finite element simulations and the differences are minimized with an optimization algorithm, in which the material parameters are varied in a certain manner.

MATERIAL MODEL AND ITS IMPLEMENTATION Assuming a , b , and c as three second order tensors, together with the Einstein's summation convention on repeated indices, c a x b represents the single contraction product with [c]ik [a]ij [b] jk . This product preserves its structure in between vectors and matrices also. d d [a]ij [b]ij where d is a scalar.

a : b represents the double contraction product with

sym [l ] represent the spatial rate of deformation tensor, where l w t F x F 1 w X v denotes the spatial velocity gradient, with v w t x. The kinematics of the current framework relies on the additive decomposition of the symmetric part of the rotated rate of deformation tensor, which preserves validity in the context of metal plasticity due to infinitesimal elastic strains compared to the larger plastic strains, ˆ D ˆe D ˆ p. (1) D t e p ˆ ˆ ˆ with D and D denoting the elastic and plastic rotated rate of deformation with D R x d x R . Following integrations are defined in the corotated coordinates e p p ˆ dt , Ee E ³D (2) ³Dˆ dt , E ³Dˆ dt. Let d

The Cauchy stress tensor, T , in this corotational form gives ˆ R t x T x R, (3) T e e e ˆ ˆ where a hypoelastic relation w t T C : D , is assumed with C denoting the elastic constitutive tensor. Due to ˆ , q ) is postulated in the corotational framework. Let V a denote the ˆp ) ˆ p (T anisotropy, the yield function ) equivalent stress and V 0 denote the initial yield stress in a certain material direction, we start by defining the following generic form of the yield function is assumed where nonlinear isotropic hardening is taken into account ˆ , q ) V  >V  q @ . ˆ p (T ) (4) 0 where q q (D ) represents the expansion of the yield locus due to isotropic hardening where hardening strain like variable. We select a saturation type dependence which reads q KD  ª¬V f  V 0 º¼ ª¬1  exp > GD @º¼ .

is the isotropic (5)

with K , V and G denote material parameters. For the anisotropic yield function YLD2000-2D, the equivalent stress V is defined as a degree one homogeneous function of T as follows f





1/ a

a a a · §1 X 1c  X 2c  2 X 2cc  X 1cc  2 X 1cc  X 2cc ¸ . ¨ ©2 ¹ where X ic and X icc , i 1, 2,3, respectively denote the principal values of tensors Xc

V

(6)

Lc x T and Xcc

Lc and Lcc are defined in terms of material parameters D i , i 1,...,8 as proposed in [4].

751

Lcc x T where

ˆ p and w D Normality postulate and plastic work identity gives the following evolutionary equations for D t ˆ p Jw ˆ ) ˆ , q ), w D J, ˆ p (T D t

T

(7)

where J denotes the plastic multiplier. In solution of local integration algorithms in a strain driven nature of FEM, it is typical to assume that for a typical ˆ , E p ,D and the solution at tn 1 is sought for time step 't tn 1  tn the solution at tn is known as T n n n

^Tˆ

n 1

,E

p n 1

^

`

`

, D n 1 in the current corotational setting. We follow an elastic predictor plastic/damage corrector type

operator split, where the elastic prediction is carried out assuming the trial values, where the trial stress and yield surface expansion respectively read, ˆ trial Ce : ªE  E p , trial º , ½ T n 1 n 1 ¬ n 1 ¼ ° (8) ¾, f trial 0 trial ª º ª º ª º qntrial K D V V GD     1 exp . 1 n 1 n 1 ¼ ¼ ° ¬ ¼¬ ¬ ¿ Within the time step, the elastic or plastic character of the status is checked by inserting the trial stresses into the ˆ trial , q trial . Once ) ˆ p , trial d 0 is satisfied, an elastic state at t is defined and the trial ˆ p , trial ) ˆp T yield function, )



n 1

n 1

n 1



n 1

n1

values does not require any correction. Otherwise, plastic flow is due. During flow, together with an implicit backward Euler integration and utilizing 'J n 1 : 'tJn 1 , the return mapping in the residual form becomes, with a time dependent form, ˆ p ,½ R np1 Enp1  E np  'J n 1w Tˆ ) n 1 ° n 1 (9) ¾, ) p ˆ ˆ Rn 1 'J n 1  ) (Tn 1 , qn 1 ). °¿ We recollect this nonlinear equation set in R n1 , where the solution is sought for the unknown vector y n1 using the Newton method ˆ °½ ­°R p ½° °­ T R n 1 ® )n 1 ¾ and y n 1 ® n 1 ¾ . (10) ¯° Rn 1 ¿° ¯°'J n 1 ¿° Using the iteration index as

k , the solution algorithm reads, A (nk)1 x 'y (nk)1 R (nk)1 , ½° ¾, y (nk11) y (nk)1  T ( k ) 'y (nk)1 .°¿

(11)

which will run until R (nk)1 d TOL . T ( k )   0,1@ is a suitably selected line-search parameter. Where the explicit definition of the matrix case.

A which is derived following the linearization is given as follows for the rate independent





1

ª Ce , 1  'J ( k ) w 2ˆ ˆ ) ˆ p ,( k ) ˆ p ,( k ) º w Tˆ ) n 1 Tn1 … Tn1 n 1 n 1 n 1 (k ) « » . (12) A n 1 ,( ) p k p(k ) » « ˆ ˆ w ) w ) n 1 n 1 ¼ 'J n1 » Tˆ n1 ¬« This ends the Backward Euler return mapping scheme for the plastic correction which falls into a general closest point projection type algorithm, see e.g. [16]. This framework has been implemented as a VUMAT subroutine for ABAQUS-Explicit.

PROPOSED SPECIMEN AND FINITE ELEMENT MODEL The needed inhomogeneous deformation field was realized with a specimen having a varying cross-section as shown in Figure 1.a. The specimen can simply be loaded under tension with a tensile test machine. The geometry was modeled in ABAQUS-Explicit using 720 fully integrated shell elements (Element Type S4 in ABAQUS) as presented in the figure. The symmetry of the problem enables the modeling of one half of the geometry. The maximum tool displacement is given as 4 mm as in the experiments. After this point, the occurring strains go beyond the limits of the uniaxial tension test and therefore the flow curve of material should be extrapolated. This is prevented in this study.

752

FIGURE 1. (a) The geometry of the specimen and the finite element model of the problem. (b) Specimen with stochastic pattern on it. The strain state of the specimen can be analyzed with the strain diagram shown in Figure 2. The diagrams present the major and minor strain of the elements predicted by the simulations for the material AA6016 in rolling direction at the maximum tool displacement of 4 mm.

FIGURE 2. Strain diagram generated from the finite element simulations of AA6016 for isotropic and anisotropic cases When the problem is simulated with von Mises yield condition for an isotropic case, it is seen that the deformation is between uniaxial tension and plane strain tension states. There are elements directly on the uniaxial tension line which are located on the symmetry line of specimen. The deformation state changes significantly when the material is simulated with YLD2000-2D with the correct material parameters of AA6016. The points are shifted in the direction of plane strain tension state. This is because of the Lankford`s coefficients (r-values) of the material which are all lower than 1.0. This means that in uniaxial tension case the plastic strain in the thickness direction is larger than the strains in transverse direction. This effect in uniaxial tension test can also be observed in this specimen geometry. As compared to the isotropic case, anisotropic aluminum does not flow much in the width direction and therefore the observed deformation shifts to the plane strain tension. These diagrams point out two issues about the specimen. The specimen can reflect the plastic behavior of anisotropic materials with its strain distribution on the surface. Secondly, at any instance of the test there exist material points with different strain hardening, meaning that they lie on different yield loci. These affects can be detected with an optical strain measurement system using a stochastic pattern on the specimen (Figure 1.b) and this cloud of information will be used to characterize the initial anisotropy of the material.

753

STRUCTURE OF INVERSE PARAMETER IDENTIFICATION For inverse parameter identification it is crucial to compare the experimental and numerical results in an automated manner. However, the discretization of the optical measurement system is different than the one used in finite element simulations, meaning that the location, dimension and the number of elements are different. Therefore a mapping algorithm is needed to obtain the strains at the exact locations of the experimental measurement points. This is performed in two steps. Firstly, the enclosing finite element of each experimental measurement point is found according to the procedure described by Jayadevan and Narasimhan [17]. In this step the local coordinates of the experimental point is also determined in its surrounding element. Secondly, the nodal values of the finite element are interpolated to the desired coordinates of the point with the following interpolation function:

u

1 1 1 1 1  [ 1  K u1  1  [ 1  K u2  1  [ 1  K u3  1  [ 1  K u4 4 4 4 4

(13)

where ȟ and Ș are the local coordinates and the ui’s are the nodal values of the major and minor strains. This procedure is applied at definite tool displacement increments. In this case the 4 mm tool displacement is divided into 8 intervals and for each 0.5 mm of tool displacement the strains from the simulations are obtained. The tool force values at these increments are also recorded from the simulations and experiments. With the known tool forces and strain values at the same material points for each tool displacement increment, the objective function can now be formed. In conventional manner, the YLD2000-2D parameters are defined with the uniaxial tension tests in 0°, 45°, 90° and equi-biaxial tests. It is shown in the previous section that the deformation state of the proposed specimen lies between uniaxial tension and plane strain tension states, meaning that the region around the equi-biaxial state is not covered. Therefore, the information of the biaxial stress state should be included explicitly in the objective function. The objective function should include the major and minor strain differences, tool force differences and equi-biaxial point differences between the simulations and experiments. As these values are totally different in their order of magnitudes the values are normalized according to the initial contribution of the strains to the objective function, with all the parameters of the YLD2000-2D model set to 1.0 for the isotropic case which is the initial guess. With the formed objective function, the problem is non-linear least-square type and the function is minimized using the Levenberg-Marquardt algorithm [18, 19]. With the introduction of a scalar controlling the magnitude and direction of the iterations, the algorithm calculates a search direction that is between the Gauss-Newton direction and steepest descent direction and therefore eliminates the disadvantages of the Gauss-Newton method especially in the proximity of the optimum solution. In order to automate the optimization procedure a MATLAB code is written which starts the ABAQUS simulations, reads the outputs by calling a Python script, compares the results with the experimental measurements, builds and minimizes the objective function.

CHARACTERIZATION OF INITIAL ANISOTROPY In order to check the results of the inverse scheme, firstly the conventional characterization of AA6016 with 1 mm thickness was performed with uniaxial tension tests and a layer compression test. The results of these tests together with the calculated YLD2000-2D parameters can be seen in Table 1. ı0 112.5 Į1 0.979

TABLE 1. Material parameters of AA6016 (All stresses in MPa) ı45 ı90 ıb r0 r45 r90 rb ı0 107.4 110.0 115 0.85 0.48 0.77 1.0 112.5 Į2 Į3 Į4 Į5 Į6 Į7 Į8 K 0.998 0.885 1.008 1.001 0.965 0.953 1.242 308.9

ı’ 223 į 20.14

The main concern in this study is the applicability of the method to identify the initial anisotropy of the material. Therefore it is assumed that the flow curve of the material is known and solely the parameters of the yield condition are varied throughout the iterations. This assumption allows also to construct a relation between these parameters and one parameter turns out to be dependent to others. Therefore instead of 8 Į’s, only 7 of them are varied. The yield function exponent a was taken to be equal to 8 as recommended for the FCC materials. Two strategies are followed in the characterization procedure as analyzed in [20]. In the first one, only the specimen in 45° orientation is used and in the second one, the specimens in the 0° and 90° orientations are used simultaneously. In the second

754

case, two simulations are run one after another and the objective function is formed in the same way as before but this time containing the differences of both of the cases. For both strategies the initial guess is the isotropic case, where all the material parameters are equal to 1.0.

FIGURE 3. Comparison of analytically and inversely obtained yield loci The obtained yield loci with the two cases can be seen in Figure 3. The white lines represent the analytically fit YLD2000-2D yield loci with the parameters listed in Table 1. The difference in the absolute values of the yield surfaces is less than 5%. In the second and third quadrants of the stress space there is a one to one match between the analytical and inversely obtained results. The only remarkable dissimilarity is observed in the plane strain regions where the slopes are different. Therefore the predicted plane strain tension points on the yield loci are different. When the results of the both strategies are compared, there is not a significant difference. The both cases can predict the initial anisotropy of material. It should be noted that this methodology does not take the r-values or the yield stress of the material at different orientations as input. Nevertheless the yield stress in 90° or the slopes of the yield locus at the axes, which eventually reflect the r-values, are predicted correctly.

FIGURE 4. Experimental and numerical tool force – displacement diagrams for three orientations The force-displacement curves of the experiments and simulations can be seen in Figure 4. The case with the 45° specimen predicts the forces with a maximum error of 10%. The deviations are even lower with the case of specimens in 0° and 90°. Therefore not only the strains, equi-biaxial point and the r-values but also the predicted tool forces are in good agreement with the experiments.

755

EXPERIMENTAL VERIFICATION In order to verify the obtained Yld2000-2D parameters a scaled car hood geometry is used (Figure 5.a). This industry-oriented geometry is interesting for verification purposes since the material in the side walls flow mainly through the drawbeads, where a complex deformation pattern is observed. AA6016 with 1 mm thickness from the same charge as in the characterization section was used in the experiments. The yield locus of the material with respect to two different yield conditions can be seen in Figure 5.b. The main difference between the two yield loci occurs in the vicinity of the equi-biaxial stress state. Since Hill´48 model does not take this stress state into account, the predictions are unsatisfactory. The strains on the surface of the car hood were measured with an optical measurement system and the measurements are compared with the numerical results along the two sections in Figure 6. The section 1 goes across the workpiece diagonally and the section 2 is along the half height of the rear side wall.

FIGURE 5. (a) Scaled car hood geometry and the analyzed sections (b) Yield locus of AA6016 according to Hill´48 and Yld2000-2D For comparison the results with Hill´48 material model are also included, since this model is still widely in use. Since the verification geometry is large in dimensions, the strains are measured stepwise and afterwards the measurements are combined. The gaps in the experimental results in Figure 6 are due to the gaps between these subprojects. The Hill´48 model apparently oversimplifies the plastic behavior of the AA6016 material. Especially in the corner regions of the geometry, the predicted thicknesses with Hill´48 are 12% lower than the measured thicknesses. As a general tendency the results with the Yld2000-2D model are in good agreement with the experiments. In the regions where moderate deformations occur, the both models predict similar thickness distributions with Yld20002D having slightly larger thickness values.

FIGURE 6. The comparison of the simulation results and the experimental measurements for the two selected sections.

756

CONCLUSION The presented work analyses the applicability of the utilization of strain distributions in the identification of the initial anisotropy. It is showed that in addition to the conventionally used integral measures like force and displacement, utilization of an optical measurement system provides the needed information for an inverse parameter identification of yield functions. If the material properties change significantly along different orientations, it is shown that multiple experiments in different directions can also be used simultaneously, of course leading to longer computation times. Another crucial issue is the flexibility of the used yield condition. Therefore the YLD2000-2D model is implemented in this study. If the same study were performed with Hill´48 for instance, the obtained solution would not be the same, since this model oversimplifies the problem and therefore it is too robust to satisfy changing yield stresses and R-Values along different directions. Since the presented framework can function with all yield loci without limitation, more flexible yield functions with more material parameters should also be tested. The assumption of negligible kinematic hardening is also an important aspect of the study. For this reason the current method is not suitable for materials that show a pronounced anisotropic hardening behavior. In that case the obtained solution would be the best compromise between the initial yield locus and subsequent yield loci. The proposed method should therefore be further developed to include combined isotropic-kinematic hardening models.

ACKNOWLEDGMENTS The authors would like to thank the German Research Foundation (DFG) for their financial support of the work reported here, which is acquired within the scope of the research project PAK250. The support of the second author has been provided by DFG under Contract No. TR 73. The authors would also like to thank Dr.-Ing. Stefan Keller and Hydro Aluminum Rolled Products GmbH for supplying the car-hood toolings.

REFERENCES R. Hill, Proc. Roy. Soc. A 193, 281-297 (1948). R. Hill, J. Mech. Phys. Solids 38, 405-417 (1990). F. Barlat, D. J. Lege, J. C. Brem, Int. J. Plast. 7, 693-712 (1991). F. Barlat, J. C. Brem, J. W. Yoon, K. Chung, R. E. Dick, D. J. Lege, F. Pourboghrat, S. H. Choi, E. Chu, Int. J. Plast. 19, 1297-1319 (2003). 5. D. Banabic, T. Kuwabara, T. Balan, D. S. Comsa, D. Julean, Int. J. Mech. Sci. 45, 797-811 (2003). 6. D. Banabic, H. Aretz, D. S. Comsa, L. Paraianu, Int. J. Plast. 21, 493-512 (2005). 7. D. Banabic, D. S. Comsa, M. Sester, M. Selig, W. Kubli, K. Mattiasson, M. Sifvant, Proceedings of the 7th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes (Numisheet 2008), edited by P. Hora, Institute of Virtual Manufacturing, ETH Zurich, 2008, pp. 37-42. 8. H. Vegter, A. H. van den Boogaard, Int. J. Plast. 22, 557-580 (2006). 9. T. Kuwabara, Int. J. Plast. 23, 385-419 (2007). 10. K. Kavanagh, R. Clough, Int. J. Solids Struct. 7, 11-23 (1971). 11. J. Unger, M. Stiemer, A. Brosius, B. Svendsen, H. Blum, M. Kleiner, Int. J. Solids. Struct. 45, 442-459 (2008). 12. M. Kleiner, A. Brosius, CIRP Annals. Manuf. Technol. 55, 267-270 (2006). 13. J. C. Gelin, O. Ghouati, Annals. CIRP 44, 189-192 (1995). 14. J. C. Gelin, O. Ghouati, J. Mater. Process. Tech. 45, 435-440 (1994). 15. J. Kajberg, G. Lindkvist, Int. J. Solids Struct. 41, 3439-3459 (2004). 16. J.C. Simo, T.J.R Hughes, Computational Inelasticity, Springer (1998). 17. K. R. Jayadevan, R. Narasimhan, Comput. Struct. 57, 915-927 (1995). 18. K. Levenberg, Q. Appl. Math. 2, 164–168 (1944). 19. D. Marquardt, SIAM J. Appl. Math. 11, 431–441 (1963). 20. A. Güner, Q. Yin, C. Soyarslan, A. Brosius, A. E. Tekkaya, Int. J. Mater. Form., (online first) DOI 10.1007/s12289-010-1009-4

1. 2. 3. 4.

(2010).

757

Fracture of 1045 steel under complex loading history Yuanli Bai ([email protected]) Department of Mechanical, Materials and Aerospace Engineering University of Central Florida, Orlando, FL 32816, USA

Abstract. In the paper, a 3D fracture locus of 1045 steel under proportional loading conditions was calibrated using round, plane strain, and tubular specimens. The fracture locus includes both the stress triaxiality and Lode angle dependence. As an extension to the conventional linear damage evolution assumption, a new form of ductile fracture model considering the loading history effect was proposed, in which two weighting functions were introduced to the damage indicator calculation. One considers the non-linear damage evolution under proportional loading, the other accounts for the effect of change in loading directions. A round of comprehensive fracture tests on 1045 steel was conducted to validate the proposed fracture model. These tests include monotonic loading tests for 3D fracture locus calibration and other tests with complex loading histories, for example two-stage-tension test, compression-tension test and torsion-tension test.

1 1.1

Introduction Ductile fractures under complex loading conditions

The effect of loading history on metal sheets necking has been recognized for many years [1-3]. The strain-based forming limit diagram (FLD) is not unique under non-proportional loading conditions. The effect of loading history on FLD was also studied by many researchers [3-7]. However, the effect of loading history on fracture is still not well noted by the community of fracture mechanics. Both the physically-based fracture models (for example Tvergaard and Needleman [8]) and empirical fracture models [9-11] are usually calibrated and validated by tests under monotonic loading conditions. A fracture model calibrated only from monotonic loading usually fails to predict fracture under complex loading conditions. For example, it was pointed by Johnson and Cook [10] that the total damage is always less than unity at fracture (up to 40%) when the material is subjected to torsion followed by tension. Recently, the effect of strain reversal on fracture was studied for 2024-T351 aluminum alloy [12-14], which is a common phenomenon in mechanics. Consider for example three-point bending of a solid section beam. Initially, there are positive strains on the tensile side and negative strains on the compressive side. As the crack propagates towards the center of the beam, it clearly passes through a region that was initially under compression. This interesting phenomenon has been described by Bao and Wierzbicki [12, 13], who also introduced a modified form of damage function to account for the effect of pre-compression. Another practical example in which the strain rate ratio changes sign is an axially compressed prismatic tube [14]. The objective of the present paper is to study the loading history effect on fracture through a round of comprehensive experimental study. Firstly, the 3D fracture locus of 1045 steel is calibrated under monotonic loading conditions. Secondly, tests with complex loading conditions are designed and conducted. These tests includes two-stage-tension test, compression-tension test, and torsion-tension test. A new form of ductile fracture model is proposed and validated based on the experimental results.

1.2

Characterization of the stress state

For isotropic materials,  1 ,  2 and  3 denote three principal stresses. It is shown that the stress state can be uniquely described by two parameters, stress triaxiality  and normalized Lode angle parameter  [15], which are defined as follows.  (1) = m.  6 2 (2)  = 1 = 1  arccos  .





The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 758-764 (2011); doi: 10.1063/1.3623682 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

758

where  m is the mean stress,  is the equivalent stress,  is defined as normalized third deviatroic stress 3

invariant,  =  J 3  . Here J 3 = s1s2 s3 is the third deviatoric stress invariant. The parameter  can be further    

related to the Lode angle  through  = cos(3 ) . The range of  is 1    1 . All loading conditions can be characterized by the above defined set of parameters ( , ) . In this paper, the ductile fracture model will be reformulated in such a coordinate system.

2 2.1

Constitutive modeling A plasticity model considering combined hardening rule

In order to simulate a test under complex loading paths, an isotropic hardening rule is not enough, so the kinematic hardening of the material has to be introduced. According to the work of Armstrong and Frederick and Chaboche [16, 17], the evolution of the back stress is given by 2 (3) d [ Sback ] = C[d  p ]  [ Sback ]d  p , 3 where C and  are material constants and Sback is the back stress tensor. The isotropic hardening takes the form of the asymmetric plasticity model proposed by the author [15], which assumes that a isotropic yield surface is controlled by both hydrostatic pressure and Lode angle parameter. The following isotropic yield surface was proposed,    m 1   (4)  yld = f  p ,[ ]  [ Sback ] =  iso  p 1  c (   0 )   cs  (cax  cs )     , m  1    where  and cax are two parameters defined by





=

 

 cos( / 6)  1  1 = 6.4641sec(   / 6)  1 , 1  cos( / 6)  cos(   / 6) 

(5)

for   0 for   0

c t cax =  c c

(6)

The range of  is 0    1 , in which  = 0 is corresponding to plane strain or generalized shear condition, and  = 1 is corresponding to axial symmetry. The parameter m is a non-negative integer. There are four t

s

c

material constants, ct , cs , cc and m , need to be calibrated. The values of c , c , and c are relative, and at least one of them is equal to unity. This depends on which type of reference test is used to calibrate the isotropic strain hardening function  iso ( p ) . The isotropic hardening rule is a specific definition used by Chaboche and Lee [17, 18].

 iso  

p

=





  st 1  e  b

p

,

(7)

where   is the first yield stress, and the stabilized value is     st . Here the material is assumed to follow a combined hardening rule, so Eq. (7) will be used. It is shown in Ref. [19] that the material of 1045 steel has neither pressure dependence nor Lode angle dependence on plasticity, so the following parameters of plasticity are assumed: c = 0.0 ,  = 0.72 ,

cs = ct = cc = 1.0 , and

m = 6.

In this particular case, the proposed plasticity model reduces to a combined

isotropic/kinematic hardening model with von Mises type of flow. Khan and Jackson[20] proposed a method of finding parameters of both isotropic hardening and kinematic hardening of materials. Here, a new type of compression-tension test on notched round specimens with optical measurement was developed to calibration these two types of hardening [21]. Aslo, finite element model using 4-node axisymmetric elements (CAX4R) is built in ABAQUS to simulate the notched round bar subjected to compression-tension loading. Since the experimental data of hardening are only up to  p = 0.152 that is far below the maximum equivalent plastic strains occurring in the tests, the hardening parameters are adjusted around the experimentally calibrated values at each iterate run. Finally, a very good correlation of the force-displacement curves between experiments and numerical simulation is achieved, as shown in Fig. 1. The finally calibrated parameters of the plasticity model is listed in Table. 1. A comparison of two hardening curves between the initial calibration and final adjustment is shown in Fig. 2. This group of parameters will be used for all the remaining tests. 759

Fig. 1: Comparison of force-displacement curves under compression-tension loading.

Table 1:

Fig. 2: Measured isotropic hardening and kinematic hardening of 1045 steel

Calibrated parameters of the plasticity model of 1045 steel





 st

b

C

754 MPa

500 MPa

0.4

3500 MPa

c

0.3 

cs

ct

c

m

0.0

-0.72

1.0

1.0

1.0

6

E 2.2e+5MPa

2.2

c

 23

A new form of ductile fracture model considering the loading history effect

A new parameter  is introduced to describe the stress state of non-proportionality,



S   back =   Sback ,  Sback

(8)

which denotes the difference between the directions of current stress and back stress. It can be proven that the range of  is 0    2 . A new parameter  , defined by p

p

0

0

  10    d  p = 10  

  Sback d  p ,

(9)

is introduced to measure the accumulated amount of non-proportionality during the loading process, For convenience, a coefficient 10 is used to make the range of coefficient  vary around unity. It is shown that the parameter  can distinguish different stages of a non-proportional loading process [21]. If the two stress state parameters(  and  ) change under a loading path or the parameter  increases away from zero, then the loading is called non-proportional loading. Because the fracture locus ˆ f ( ,  ) is defined and calibrated only under proportional loading conditions, additionally designed experiments with non-proportional loadings are necessary to calibrate and verify the loading history effect on fracture initiation. The proposed ductile fracture model can be written in an incremental form (Eq. (10)), which is easy for the implementation in the material subroutine. It is assumed that material points will fail when D = Dc = 1 . c   d p   k (10) dD =  cg D  c g  1  ch D 1  2 . ˆ f  ,   e g 1   Two weighting functions were introduced in Eq. (10) to the calculation of damage indicator ( D ). One function using parameter cg considers the non-linear damage evolution under proportional loading, the other function,





using another set of parameters ( ch , 1 ,  2 , k ), accounts for the effect of change in the loading directions. In this paper, the 3D fracture locus ˆ f  ,   is defined as [15]

760

 1 ˆ() ˆ()  f   f   ˆ (f )   2  12 ˆ (f )  ˆ (f )   ˆ (f) 2 1 D  D  D   D  D  D  1   D1e 2  D5e 6  D3e 4   2  D1e 2  D5e 6   D33e 4 . 2 2 

ˆ f  ,   = 







(11)



d testss with both mo onotonic and non-proportioonal loading conditions c In next sectioon, carefully designed will be ussed to calibratte and verify the t newly propposed ductile fracture modeel.

3

Experim mental and d simulatio n results

3.1 3D frracture locu us calibratiion for 1045 steel A round of fracture f tests consisting off seven types of specimens was perform med to calibraate the 3D fracture locus of 1045 steel. A co ollection of thhese tests is shown in Fig g. 3, which iincludes smoo oth round specimenn and notchedd round specim men in tensionn, flat specimeens with different radii of ggrooves in ten nsion, tube specimenn in tension annd tube specim men in torsionn. Details abou ut test data and FE simulatiion results can n be found in Ref. [[21, 22]. The fracture testss discussed inn this section n are either under u the loadding condition n of axial symmetryy tension (  = 1 ) or plane strain s tension (  = 0 , or callled generalizeed shear). It iss found that th he fracture locus of 1045 steel is strongly s depen ndent on the L Lode angle parrameter (  ) even e the its pllasticity does not n [22]. The data poinnts from the numerical n sim mulations will be used to co onstruct the 3D D fracture loccus. Since nder the loadinng condition of axial symm metric compreession (  = 1 ), the 3D there is nno data pointss available un fracture llocus is assum med to be sym mmetric, ˆ (f  ) = ˆ (f  ) , or D1 = D5 and D2 = D6 . The caalibrated paraameters of the fractuure locus (Eqq. (11)) is lissted in Table 2. A 3D geometrical rep presentation oof the fracturee locus is illustratedd in Fig. 4. Thhe error of every data pointt is also marked in Fig. 4. It is shown thaat the 3D fracture locus agrees w well with all the t data pointts. Since all tthe data pointts were obtained under moonotonic load dings, this fracture llocus will be used as a reference for thhe correction of loading hiistory effect uunder compleex loading conditionns. Table 2: 2 Calibrated pparameters of the fracture lo ocus of 1045 ssteel

D1 0.77121

D2 1.6968

D3

D4 1.9454

0.51887

Fiig. 3: Specimenns used to calibrrate the fracturee locus of 1045 steel.

D5

D6

0.7121

1.6 6968

Fig. 4: Calibrated 3D fractture locus of 1045 1 steel from monotonic m loaading conditio ons.

3.2 Load ding history y effect on ffracture of 1045 steel The fracture of o 1045 steel has been welll calibrated in n Section 3.1 under u monotoonic loading conditions. In this seection, three types t of tests with complexx loading con nditions will be b used to callibrate and vaalidate the proposedd correction foor the loading history effecct, Eq. (10). Totally, T there are a five materrial parameterrs, cg , ch h changing nootches, reversal loading 1 ,  2 , aand k , to be calibrated. Thhose tests incllude two stagees tension with of comprression-tensionn, and the torssion followed by tension. A collection off specimens iss shown in Fig g. 5.

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Fig. 5: List of o specimens for three typees of tests with h complex loading historiess: (a) two-stag ge-tension, (b) compression-tension rreverse loading, (c) torsion-tension tests.

In the first tyype of test, sm mooth round sspecimens werre subjected to pre-tensile lloading. One specimen was pulleed always to fracture. f The other four weere denoted reespectively as group A andd group B baseed on two different amounts of pre-tension. p The specimenss were re-macchined to intro oduce notchess, then they were w tested under tennsion to fractuure. The damaage calculatedd without histo ory effect correction is shoown in Fig. 6. Note that i found thatt the first weeighting functtion (nonlineaar accumulatiion under  = 0 inn this type off test, so it is proportioonal loading condition) is negligible n exceept for the preemature fractu ure on specime men 4. So, the parameter p matical singulaarity. cg is sett as cg = 0.0001 . A very smaall value is useed in order to avoid mathem

Fig. 6: Com mparison of damage d accum mulation for th he two-stage-teension tests w without loading g history correection (case 1, for pure tenssion; case 2&3 3, group A; caase 4&5, grouup B).

The second type of test is the comppression-tension reversal loading l test. Different am mounts of pre-comppression weree applied, then n specimens w were subjected to tension until u crack. T The third type of test is torsion-teension test. Tube specimen ns were subjeccted to differeent amounts of o pre-torsionn, then they were w failed under tennsion again. All A the historiees of the param meters (  p ,  ,  , and  ) at the fract cture initiation n sits were obtained from the finnite element simulations. Without loaading history correction, tthe calculated d damage wn in Fig. 7 and 8 as cirrcles. It is fo ound that the fracture locuus calibrated from the indicates D are show c in monotonnic loading coonditions can not predict thhese two typees of fracture ( Dc  1 ). It iss found that changes loading ddirections (orr the parameteer  ) increasses the ductillity of 1045 steel s because the values of o damage indicatorr at fracture are a greater thaan one. Hencee we chose k = 1 in Eq.((10). A Matlaab code was written to calculate the damages accumulation n and optimizze the model parameters: p ch , 1 , and  2 . The objecttive in the 762

optimization is the make all the values of damage accumulation at fracture Dc as close to unity as possible. Finally, a combination of these parameters is achieved. The calibrated parameters of loading history effect is listed in Table 3.

Fig. 7: Correction of the loading history effect on fracture, notched round bars of 1045 steel subjected to compression followed by tension.

Fig. 8: Correction of the loading history effect on fracture, tubing specimens of 1045 steel subjected to torsion followed by tension

Comparisons of the total damages at fracture between without and with correction of loading history effect are shown in Fig. 7 and 8. Now, the total damages with correction at fracture are much closer to unity for all cases (diamond plots). If the damage is assumed to occur at Dc = 1 (as is proposed by the model), then the predicted displacements to fracture will be very close to the fracture displacement in all tests. Therefore, one can see that the proposed loading history functions works well for both the compression-tension loading and the torsion-tension loading. In other words, the test data validated the proposed ductile fracture model under complex loading history. Table 3: Calibrated parameters of the loading history effect on fracture of 1045 steel

4

cg

ch

1

2

k

0.0001

5.5

2.0

2.0

-1

Conclusion

As an extension to the conventional linear damage evolution assumption, a new form of ductile fracture model accounting for the loading history effect was proposed in this paper. A round of comprehensive fracture tests on 1045 steel was conducted to validate the proposed fracture model. Firstly, the combined isotropic/kinematic hardening plasticity model of the material was calibrated using compression-tension tests. Secondly, the 3D fracture locus of 1045 steel was calibrated using smooth/notched round bars, flat-grooved specimens, and tubular specimens. Thirdly, the proposed correction of loading history effect is studied and calibrated by three groups of tests: two-stage-tension test, compression-tension test and torsion-tension test. Two opposite trends of loading history effect on ductile fracture were observed. Changes in loading direction can either decrease the material ductile (for example, 2024-T351 aluminum and OFHC copper) or increase the material ductility (for example 1045 steel). Using the calibrated fracture locus and the loading history correction, the proposed model well predicts ductile fractures of all tests under complex loading conditions. A more comprehensive study on the 3D fracture locus and the loading history effect on ductile fracture are undergoing to further confirm the present findings.

Acknowledgement The work was done while the author at MIT. Thanks due to Professor Tomasz Wierzbicki of MIT for 763

his advising. Partial financial supports from the NSF/Sandia alliance program and the AHSS MIT industry consortium are greatly appreciated. Thanks due to the continuous support of HyperMesh program from the Altair Company. References [1] Muschenborn, W., and Sonne, H.M. Influence of the strain path on the forming limits of sheet metal. Arch. Eisenhuttenwes, 46:597-602, 1975. [2] Graf, Alejandro and Hosford, William. The influence of strain-path changes on forming limit diagrams of A1 6111 T4. International Journal of Mechanical Sciences, 36(10):897--910, 1994. [3] Stoughton, Thomas B. A general forming limit criterion for sheet metal forming. International Journal of Mechanical Sciences, 42(1):1--27, 2000. [4] Atkins, A. G., and Mai, Y. W. Elastic and Plastic Fracture. Chichester:Ellis Horwood, 1985. [5] Cao, Jian and Yao, Hong and Karafillis, Apostolos and Boyce, Mary C. Prediction of localized thinning in sheet metal using a general anisotropic yield criterion. International Journal of Plasticity, 16(9):1105--1129, 2000. [6] Chow, C. L. and Yu, L. G. and Tai, W. H. and Demeri, M. Y. Prediction of forming limit diagrams for AL6111-T4 under non-proportional loading. International Journal of Mechanical Sciences, 43(2):471--486, 2001. [7] Yuanli Bai and Tomasz Wierzbicki. Forming severity concept for predicting sheet necking under complex loading histories. International Journal of Mechanical Sciences, 50(6):1012 - 1022, 2008. [8] Tvergaard, V. and Needleman, A. Analysis of the cup-cone fracture in a round tensile bar. Acta Materialia, 32:157--169, 1984. [9] Wilkins, M L. and Streit, R D. and Reaugh, J E. Cumulative -strain-damage model of ductile fracture: simulation and prediction of engineering fracture tests, UCRL-53058. Technical report, Lawrence Livermore Laboratory, Livermore, California, 1980. [10] Johnson, Gordon R. and Cook, William H. Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Engineering Fracture Mechanics, 21(1):31-48, 1985. [11] Hooputra, H. and Gese, H. and Dell, H. and Werner, H. A comprehensive failure model for crashworthiness simulation of aluminium extrusions. International Journal of Crashworthiness, 9:449-464, 2004. [12] Bao, Yingbin. Prediction of ductile crack formation in uncracked bodies. PhD thesis, Massachusetts Institute of Technology, 2003. [13] Bao, Yingbin and Treitler, Roland. Ductile crack formation on notched Al2024-T351 bars under compression-tension loading. Materials Science and Engineering A, 384(1-2):385--394, 2004. [14] Bai, Yuanli and Bao, Yingbin and Wierzbicki, Tomasz. Fracture of prismatic aluminum tubes under reverse straining. International Journal of Impact Engineering, 32(5):671--701, 2006. [15] Yuanli Bai and Tomasz Wierzbicki. A new model of metal plasticity and fracture with pressure and Lode dependence. International Journal of Plasticity, 24(6):1071 - 1096, 2008. [16] Armstrong, P.L., and Frederick, C.O. A mathematical representation of the multiaxial Bauschinger effect, G.E.G.B. Report RD/B/N 731. 1966. [17] Chaboche, J.L. Viscoplastic constitutive equations for the description of cyclic and anisotropic behavior of metals. Bulletin de l'Academie Poloanise des Sciences. Serie des Sciences Techniques, 25:33, 1977. [18] Lee, D., and Zavenl, Jr. F. A generalized strain rate dependent constitutive equation for anisotropic metals. Acta Metallurgica, 29:1771, 1978. [19] Bai, Yuanli and Teng, Xiaoqing and Wierzbicki, Tomasz. Derivation and application of stress triaxiality formula for plane strain fracture testing. Presented at ASME conference of Applied Mechanics of Materials, Austin, TX, June 3-7, 2007. [20] Khan, Akhtar S. and Jackson, Kamili M. On the evolution of isotropic and kinematic hardening with finite plastic deformation Part I: compression/tension loading of OFHC copper cylinders. International Journal of Plasticity, 15(12):1265--1275, 1999. [21] Bai, Yuanli. Effect of Loading History on Necking and Fracture. PhD thesis, Massachusetts Institute of Technology, 2008. [22] Yuanli Bai and Xiaoqing Teng and Tomasz Wierzbicki. On the Application of Stress Triaxiality Formula for Plane Strain Fracture Testing. Journal of Engineering Materials and Technology, 131(2):021002, 2009.

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Effect of Seam Welding on Forming Limits of IF-Steel Sheet D. Sutariya1, H.Raval1, K. Kalaivani2, K. Hariharan2, A. Prabhu3, K. Narasimhan3* 1

Department of Mechanical Engineering, Sardar Vallabhbhai National Institute of Technology, Ichhanath, Surat, Gujarat. 395007, India 2 Advanced Engineering, Ashok Leyland Ltd, Chennai, India 3 Department of Metallurgical Engineering and Materials Science, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India *Corresponding author Phone +91-22-2576-7630, Fax 91-22-2572 3480, Email [email protected] Abstract. Laser welding is the most commonly used process for producing Tailor Welded Blanks (TWB). Although laser welding is best suited for TWB applications, it is an expensive process. In this work an alternate cheap process, seam welding, is used to produce welded sheets of IF-steel of same grade and thickness. The effect of this welding on forming limit strains is explored in this work. Forming behavior is characterized by Forming Limit Diagram (FLD). The FLDs of welded blanks are compared with that of un-welded blanks. The effect of longitudinal, transverse and 450 weld orientation on formability is studied. Weld location includes both centered and offset weld location in the transverse weld orientation. Numerical simulations are carried out by considering weld as a zone. Experimental results are compared with the predictions carried out by FE method. Keywords: Forming limit strain, Tailor welded blank (TWB), Weld property, Formability for different weld orientations, FEM simulations. PACS: 81.20.Hy

INTRODUCTION Tailor welded blanks comprise sheets of different thickness, strength, coating etc. welded in a single plane prior to stamping. TWBs offer several benefits such as reduced manufacturing cost, reduced scrap, weight reduction etc. [1] and are therefore currently widely used in automotive industries .They are usually fabricated by a laser welding process, which creates a narrow weld zone. The formability of TWBs is determined by (a) thickness and material combination of blanks welded, and (b) by the weld conditions like weld orientation and weld location. The compounding effect of thickness ratio, weld line movement and weld orientation on the forming limit strain has been studied by Chang et al [2]. An inverse relationship between the thickness ratio and TWB formability has been reported in the research on laser welded TWBs by Chang et al [3]. Wonoh Lee et al [4] studied the friction stir welded (FSW) TWBs of aluminum alloys, dual phase steel and magnesium alloy and investigated formability performance by comparing LDH value of base and TWB sample. A lowering in the LDH was reported in the TWBs in comparison to the base metal regardless of the stretching mode. Some of the researcher studied the effects of weld orientation and weld location on forming behavior of laser welded and friction stir welded blank. Garware et al [5] studied the tensile and fatigue behavior of FSW TWBs of aluminum alloy 5754 for different weld orientations with respect to loading direction, observing that both tensile and fatigue failure in the tailor welded blank occurred on the thin gage side irrespective of the weld orientation with respect to the loading direction. Panda et al [6] studied the effect of different weld location and orientation on laser welded AHSS blanks, reporting that the formability LDH and failure location strongly depends upon the weld location. LDH increases when the weld is positioned at the extreme positions away from the center because of more uniform strain distribution on deform dome. Ganesh et al [7] have also studied the effect of weld configuration on The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 765-771 (2011); doi: 10.1063/1.3623683 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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the forming limit of laser welded blanks and compared with that of un-welded sheet, concluding negligible influence on the FLC when the weld was located at center in both longitudinal and transverse direction. They also reported that the FLD of TWB is lower than the FLD of un-welded sheet. Results show that FLD is much lower at the plan strain condition for the transverse weld configuration than the un-welded sheet as well as longitudinal TWB configuration. In this paper the effect of weld configuration on forming limit strain has been experimentally investigated. TWBs were fabricated by seam welding with same materials and 1.2 mm of thickness. FE simulation of LDH test was carried out using PAM STAMP 2G commercial FEM code to construct a FLD. Limiting Dome Height tests (LDH) were conducted for samples of different widths to get a complete FLD and the strain data was recorded by GOM using Argus software.

MATERIAL PROPERTIES IF-steel was used in present study. The tensile properties of un-welded and welded sheets are shown in Table 1. All these parameters were evaluated by tensile testing of standard ASTM geometry in Instron machine. The ASTM standard procedure was followed to evaluate above parameter [8]. The longitudinal weld configuration was considered to evaluate mechanical properties of TWBs and 5.2 mm width of specimen was tested. The table 2 shows tensile parameters of welded sheet. The engineering stress strain curves of un-welded and weld only sheets are shown in Fig. 1. TABLE 1. Mechanical properties of un-welded sheet Rolling direction 0 45 90 Average

Longitudinal

ys (MPa) 138.84 154 163 151.94

UTS (MPa) 281.04 294.63 292.70 289.45

n 0.3018 0.2894 0.2768 0.2893

R 1.8428 1.7142 2.7045 -

TABLE 2. Tensile parameter of weld only specimen ys (MPa) n 312.35 0.124

Weld only Base metal

FIGURE 1. Engineering Stress-Strain curve of un-welded sheet and the weld region.

766

K (Avg.)

545.6

K (MPa) 614.5

METHODOLOGY Weld Condition Weld condition, namely weld orientation and weld location, were considered for the study of influence on limit strains in TWBs. Longitudinal, 450 and transverse weld orientations were studied. The longitudinal weld orientation represents a weld parallel to major straining direction, 450 weld orientation represents a weld is 450 to the major straining direction, and transverse weld orientation represents a weld normal to the major straining direction. In the case of longitudinal-welded blanks, the weld zone was kept at the geometric centre of the sample (Fig. 2). In transverse-welded blanks, the weld zone was located at two different positions including one at the geometric centre and others at 20mm offset (Fig. 2).

(a) (b) (c) FIGURE 2. Schematic diagram of weld configuration: (a) Longitudinal (b) Transverse (c) 45 degree

Out of Plane Formability Test To study the effect of weld configuration on FLC of TWBs, limit dome height (LDH) test was performed. Geometry of standard LDH tooling set up is shown in Fig 3. LDH test was conducted on double acting hydraulic press with standard hemispherical punch at constant speed of 1 mm/sec and 4 tons of binder force was applied to avoid draw in or failure in clamping region. Circle grid was marked on all samples using screen printing. All the samples were deformed up to necking. Eight different width samples were tested to obtain different strain paths. GOM system and ARGUS software was used for post deformation analysis, and recording major and minor strains further used to plot the forming limit diagram.

Figure 3. Geometry of out of plane test

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Simulation The PAM STAMP 2G FE code was used to simulate LDH test in order to predict the influence of the weld conditions. During the simulation, the weld region is modeled as a single entity without separate HAZ. Different blank widths ranging from 25mm to 200 mm in steps of 25 mm were simulated to achieve different strain paths. Mesh size of 1mm was kept constant in the weld zone and base metal. Hollowman’s power law was used to describe the strain-hardening behavior. Hill’s 1948 isotropic hardening yield criterion was used as the plasticity model. Friction coefficient was assumed to be 0.12 and 150 kN binder force was chosen and kept constant throughout the study. Limit strains were obtained using the thickness gradient based necking criterion described by Nandedkar [9].

RESULTS AND DISCUSSION Effect of Weld Orientation Fig. 4 shows the experimental FLD, the effect of weld on forming limit strain of TWBs under different weld orientations like longitudinal, transverse, 45 degree with weld at the geometric center and comparison with that of un-welded sheet. It is observed that the FLC of the transverse welded blank is slightly lower than that of the base metal. In this orientation, the lower influence of weld is attributed to the fact that weld zone is not participating in deformation as failure occurs at 20 – 30 mm away from the weld zone. As shown in Fig. 4, in case of longitudinal weld at center, FLD of welded blank is lower than that of un-welded sheet. The higher deviation in limit strain is found in the drawing (in tensile) region and failure occurs normal to the weld zone in punch nose region of TWBs. The strain path in drawing region approaches tensile strain path which is witnessed during standard tensile testing, where in reduced ductility of longitudinal welded blank is observed in comparison to that of un-welded blank. In the plane strain condition limit strains are almost the same for the both cases of longitudinal and transverse weld. In case of 45 degree weld orientation, FLC is significantly lower than the other weld orientation in plane strain and stretching mode.

Figure 4. Experimental FLC of welded and un-welded sheets

Effect of Weld Location in Case of Transverse Weld Orientation In the transverse weld orientation, both weld at center and weld at an offset by 20 mm was studied. Fig 5 shows the forming limit diagram of both the cases and compares it to that of un-welded blank.

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Figure 5. Experimental FLC of transverse weld at center and 20 mm offset

In the case of transverse weld at 20 mm offset position (Fig 5), the experimental FLC is considerably lower than that of un-welded blank and weld at center in plain strain and bi- axial strain condition. This is because of failure has occurred in weld zone for plain strain (100 x 200) and bi- axial (200x200) conditions.

Comparison of Experimental and Predicted FLCs Fig. 6 (a) – 6 (c) shows the comparison between experimental and predicted FLCs of un-welded sheet and TWBs for varied weld orientation. In the case of base metal (Fig. 6-a) predicted limit strains are in good agreement with experimental limit strain values in drawing and plan strain condition, but in case of stretching region of FLD limit strains are deviated. This is possible due to the assumed constant material properties throughout the simulation.

(a)

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(b)

(c) Figure 6. Comparison between experimental and predicted FLCs: (a) base metal (b) longitudinal (c) transverse

As shown in Fig. 6 (b) – 6 (c), predicted FLCs are significantly lower than experimental FLCs. Modeling of weld zone plays vital role in FEA analysis. During the simulation weld zone is modeled by considering uniform thickness and weld width while it is not the case in experiment. Failure observed near weld zone and weld zone itself in FEM analysis. Above discussed reasons significantly lower the predicted FLC as compared to experimental FLC.

CONCLUSION In this work, effect of weld configuration on FLDs is studied in seam welded TWB sheets of interstitial free steel. Following conclusions are obtained from present study. (1) Forming limit strains of transverse welded sheet shows negligible effect as necking has occurred away from the weld zone and weld has not participated significantly in deformation. (2) In case of 450 oriented welded sheet, deviation was observed in bi- axial stretching. (3) LDH test shows that the forming limit strain of longitudinal and transverse TWBs are almost same when weld is placed at center. But in case of transverse weld at 20 mm off set FLC is considerably lower especially near plan strain condition. (4) Predicted limit strains of un-welded sheet coincide well with experiments. In case of welded sheets FLCs shows similar trend when compared to experimental FLCs, but are significantly lower.

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REFERENCES 1. Auto/Steel Partnership, 1995, “Tailor Welded Blank Design and Manufacturing Manual” Technical Report 2. Chan, L. C., Chang, C. H., Chan, S. M., Lee, T. C., and Chow, C. L. “Formability analysis of tailor welded blanks of different thickness ratios” J. Mfg Sci. Engng, 2005, 127(4), 743–751. 3. Chang S.M., Chang L.C. and Lee, T. C. “Tailor welded blank of different thickness ratios effect on forming limit diagram” J. Mat. Proc. Tech. 132, 2003, 95-101. 4. Wonoh Lee, Kyung-Hwan Chung, Daeyong Kim, Junehyung Kim, Chongmin Kim, Kazutaka Okamoto, R.H. Wagoner, Kwansoo Chung “Experimental and numerical study on formability of friction stir welded TWB sheets based on hemispherical dome stretch tests” International Journal of Plasticity 25 (2009) 1626–1654 5. M. Garware, G.T. Kridli, and P.K. Mallick “Tensile and Fatigue Behavior of Friction-Stir Welded Tailor-Welded blank of Aluminum Alloy 5754” Journal of Materials Engineering and Performance, Volume 19(8) November 2010—1161 6. S.K. Panda, J. Li, V. H. Baltazar Hernandez, Y. Zhou, F. Goodwin “Effect of Weld Location,Orientation, and Strain Path on Forming Behavior of AHSS Tailor Welded Blanks” j. Engg. Mat. & Tech. Oct 2010, Vol 132 7. R Ganesh Narayanan1, K Narasimhan “Influence of the weld conditions on the forming-limit strains of tailor-welded blanks” J. Strain Analysis 2008 Vol. 43 p- 217-227. 8. ASTM E646-98 ”Testing method for tensile strain hardening exponents (n-values) of metallic materials” In Annual book of ASTM standards, section 3, vol. 03.01, 2000 (ASTM International, West Conshohocken, Pennsylvania). 9. Nandedkar V. M. “Formability Studies on a Deep Drawing Quality Steel” PhD Thesis, IIT Bombay, 2000.

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3UHGLFWLRQRI)RUPLQJ/LPLWVRIWKH0XOWLOD\HU0HWDOOLF6KHHW T. Oya1*, C. Jeong2,and J. Yanagimoto2 a

b

Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo, 153-8505, Japan

$EVWUDFW In this paper, the forming limit of a multilayer metallic steel sheet is numerically investigated. Multilayer metallic sheets are an emerging sheet metal that is designed to achieve both high strength and enough formability, which is seriously demanded especially from the automotive industry. Unfortunately, its mechanism of enhanced ductility, fracture behavior, and forming limits have not been sufficiently studied. Therefore, a new stress-strain curve formula is constructed so as to precisely predict the forming limit that is influenced by the unique characteristics of the multilayer metallic sheets. Tensile and stretching experiment on a 15-layer multilayer metallic sheet is conducted, then the results are used to verify the effectiveness of the presented stress-strain expression, and the predicted forming limits are compared to the experimental value, resulted in a good agreement. .H\ZRUGVMultilayer metallic steel sheet, sheet metal, forming limit, ductility enhancement. 3$&6 81.05.Bx, 83.10.Gr, 83.50.Uv.

,1752'8&7,21 The use of high-strength steels is now becoming inevitable in various industrial fields such as the automotive, aerospace, and other heavy industries. High-strength steels are expected to produce lightweight structures without ruining the rigidity and the performance. However, some hurdles, such as poor shape-freezing property and difficulty in predicting forming limits, have been an obstacle to broaden the application field of the high-strength steels. One of the most troublesome properties would be its low ductility in terms of press forming. Recently, multilayer metallic steel sheets are emerging as a next generation steel to urge the utilization of the high-strength steel. This material consists of alternate hard material layer and soft material layer to achieve the high strength and enough formability. Due to the strong interfaces, the ductile soft layers restrain the onset of necking in the hard layers, leading to a sufficient elongation as a sheet metal. However, the forming limit prediction of multilayered material is difficult because of its unique characteristics of layered geometry. Therefore, mechanism of enhanced ductility, fracture behavior, and forming limit prediction should be rigorously explored in order to promote the application of the multilayer metallic sheet. Syn et al. [1] studied the effect of layering on tensile ductility using a composite made of a particle-reinforced aluminum metal matrix composite (MMC) and a highly ductile Mg -9%Li alloy. Due to the enhanced tensile ductility, the laminate exhibited higher ductility than that of the constituent aluminum MMC. Semiatin and Piehler [2] studied on the forming limit of clad laminates based upon the rule of mixture. Manesh and Taheri[3] evaluated the formability of an aluminum-clad steel sheet by the Erichsen cupping test to determine the optimum conditions of heat treatment in the production process. Mori and Kurimoto [4] evaluated the formability of a stainless-steel aluminum clad sheet by tensile and bending tests. Yoshida and Hino [5] examined the forming limit of stainless steel -clad aluminum sheets under plane stress condition. They prepared two- and three-ply laminate and conducted punch-stretching experiments. Their model based upon Hill’s localized necking theory and the M-K theory predicted forming limits. Hino et al. [6] investigated the springback behavior by draw-bending experiments using two-ply sheet metal laminates. Ohashi et al. [7] produced a 35-layer laminated metal made of ultrahigh-carbon steel (UHCS) and brass, and they investigated its ductility and fracture characteristics upon bending. Carreño et al. [8] created a 7layer laminated steel composed of ultrahigh-carbon steel and mild steel, which was used in impact tests to investigate the fracture mechanism. Inoue et al. [9] have already confirmed the effectiveness of the layering concept by using a brittle martensitic steel and a ductile steel. The layered material they created exhibited a good strength-ductility balance and no

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 772-779 (2011); doi: 10.1063/1.3623684 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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delamination was observed during the tensile test. Nambu et al. [10] used a layered material consisting of highstrength martensitic steel and an austenitic ductile steel to investigate the transition under a large deformation. Nambu et al. [11] also investigated the effect of the strength of the interfacial bonding of multilayered steel composites, where a marked enhancement of tensile ductility was observed with increased bonding strength. In these studies the fundamental characteristics of multilayer metallic steel sheets have been revealed. In Oya et al. [12], the bending formability of multilayer metallic sheets were evaluated through V- and hemming-bending. Monolithic martensitic steel specimens fractured in hemming-bending, on the other hand, the same martensitic steel was successfully bent even in hemming-bending when it was sandwiched by the ductile steel layers. Oya et al. [13] conducted finite element analyses on multilayer metallic sheets; V-bending was modelled and springback angles were used to verify the results of the simulation with the experiments, and good agreements were obtained. In these studies, the stress-strain curves of the constituent brittle materials were predicted by the rule of mixtures because tensile test on a monolithic brittle metal cannot provide an entire curve of the brittle metal layer in a multilayer metallic sheet. Yanagimoto et al. [14] presented a theoretical explanation on why multilayer metallic sheet can undergo a large strain more than the original fracture limit of the constituent brittle metal. Geometrical restriction by the soft material layers would induce a delay of diffuse necking in the hard material layers, leading to the improved elongation of a multilayer metallic sheet. Chen et al. [15] conducted a numerical study on the effect of cladding on necking and fracture in sheet metals. They reported that cladding by a ductile layer enhances the overall work hardening, leading to the increase in the uniform strain that slows down the stress development causing the void nucleation delay. Liu et al. [16] reported that the tensile plasticity of cold-rolled Cu could be enhanced by a layered structure stacked alternately by thicker Cu and thinner Al layers. They explained that a strong interface can delay the development of premature strain localization and local necking in the ductile Cu layer, and crack formation also can be prevented. In this paper, a new stress-strain formula is proposed to predict the formability of the multilayer metallic sheet. The formula is capable of handling the variation of both strength and work-hardening property; therefore, it would be useful to characterize the flow curve of the multilayer metallic sheet in which the deterioration in the workhardenability can be observed. Numerical simulations were conducted to investigate the performance of this expression by using the experiment of a tensile test.

08/7,/$<(50(7$//,&6+((7 Test materials used in this paper is same type as those the authors have used in the previous studies [12]-[14]; therefore, its processing and characteristics are briefly described. A multilayer metallic steel sheet consists of soft material and hard material; they are alternately laminated with soft layer placed at both outermost layers, then the edges are welded. The laminated sheet is subjected to hot and cold rolling with appropriate heat treatment, reducing its thickness into 1mm. Finally the sheet is quenched to obtain high strength. The multilayer metallic sheets fabricated by this process have sufficiently high bonding strength in each interface; thus, no delamination or crack was seen when the sheet was subjected to severe bending. Owing to this strong interface, the constituent hard material can break the fracture limit even if it is a brittle material.

0HFKDQLVPRIGXFWLOLW\HQKDQFHPHQW Prediction of forming limits of multilayer metallic sheets is very difficult because the sheet consists of two different materials, namely hard steel and soft steel. Generally, hard steel fractures earlier than the soft material; however, a multilayer metallic sheet can be elongated beyond the fracture limit of the constituent hard material. This eye-catching phenomenon is called ductility enhancement that has been studied theoretically and experimentally by the authors [14]. Here, the mechanism of ductility enhancement is briefly described. When a multilayer metallic sheet is subjected to tensile elongation, it elongates uniformly at the early elongation phase. The hard layer material probably attain to its maximum load ahead of the soft layer. Then, the hard layer initiates diffuse necking, but the layers are geometrically restricted by the soft layers through strongly bonded interfaces; therefore, the abrupt decrease in the thickness of the hard layers would be suppressed. The sheet can be elongated as if it is a monolithic sheet, but void may be generated in the hard layers, which would cause deterioration in strength and ductility. This situation would continue as long as the work-hardening capacity remains enough in the soft layers. When the load in the soft material attains to its maximum value, the soft layer also initiates necking; then the sheet initiates fracture immediately because the hard layer cannot sustain the load increase. )LJXUH illustrates this mechanism schematically.

773

When it comes to mathematical modeling and numerical simulation, handling this mechanism directly is difficult. If one would like to model a multilayer metallic sheet perfectly based upon the geometrical and mechanical aspects, the computing time will be considerable. Furthermore, it is very difficult to predict the forming limit of a multilayer metallic sheet by such a bottom-up manner. Therefore, in this paper, the authors propose a simple but effective way to deal with the formability of a multilayer metallic sheet by using a modified power-law-like hardening rule. By defining a new work-hardening formula, formability evaluation of a multilayer metallic sheet will become easy and handy. This is also convinient to conduct a full-sized forming simulation using a multilayer metallic sheet and to discuss the forming limit theoretically. The concept is shown in )LJXUH . Because of the deterioration in the hard layers, a multilayer metallic sheet would lose its strength and ductility as strain increases. This phenomenon may be involved in the flow curve. Thus, in order to handle the formability of the multilayer metallic sheets, the conventional n-power law formula is not sufficient because its parameters are constant. The detailed explanation follows in the next section.

),*85(. Schematic illustration of the elongation property of a multilayer metallic sheet. After the point when the maximum load is achieved in hard layer, uniform deformation in hard layer changes to local deformation. However, the soft layers would suppress its decrease in thickness until the maximum load in soft layer is achieved.

),*85(. Concept of the presented flow curve in which a decrease in work-hardening affects the curve. The work-hardening of the multilayer metallic sheet would be decreased as strain increases due to the weakening in hard layer.

774

)/2:&859((;35(66,21)2508/7,/$<(50(7$//,&6+((76 The conventional and empirical n-power law expression is

V(H ) KH n ,

(1)

where K is a strength coefficient and n is a work-hardening exponent. This is no more than an experimental formulation; however, the exponent n can be interpreted as a press-formability parameter of a sheet metal. On the other hand, the coefficient K is an adjuster to let the stress value be equal to the designated stress value when the variable strain is 1. This expression can represent a single property of hardening; therefore, it is not enough to represent the hardening curve of the multilayer metallic sheet because of the above-mentioned reason. A material of which work-hardening is dependent on strain is not uncommon. Hasegawa et al. [17] presented a good estimation of stress-strain relationship by defining new constitutive formulae in which the work-hardening exponent n is dependent on strain. They conducted FE simulations, where the obtained load-elongation curve was compared with that of the experiment. The parameters were optimized to obtain a good fitting for the specimens such as annealed steel and prestrained high-strength steel; and the results showed an excellent agreement between the analytical and experimental curves even in the post-uniform elongation range. Joun et al. [18] presented a method to acquire true stress-strain curves over a large range of strain. They used a reference true stress-strain curve that satisfy the maximum load condition, then the curve was iteratively modified with the finite element solutions to decrease the difference in the tensile forces between the tensile test and the analyzed results. The authors propose a new stress-strain expression based upon the n-power law; the new expression can cover a large strain range, and can satisfy the maximum load condition, and can exhibit a gradual decrease in the workhardening exponent n with respect to strain. The work-hardening expression presented in this paper is

V(H ) Ke DH H ne

 EH

,

(2)

where K and n are same as those of the n-power law;  and  are parameters to control the rate of decrease in the H attenuating function e . Analogically, positive  and  are the attenuation constants that represent the attenuation rate of an evanescent wave in a near-field wave domain. Thus, as strain increases, the exponent of strain attenuates, which means that the work-hardening of this material gradually decreases as strain increases. The parameter  controls the magnitude of the adjuster K, and the parameter  controls the attenuation rate of the work-hardening. There are four parameters; K, n, , and . Two of them, K and n, can be replaced by  and  by using the maximum load condition. Namely, the use of the Swift’s diffuse necking criteria [19] dVH dH VH , and of VH = Vleads to the parameter elimination. After some calculations, the parameter K and n can be obtained as functions of  and . In this process, the common non-zero part is eliminated, and then the uniform strain Hu is substituted into the variable H, leading to

n

e EH u (1 D )H u . 1  EH u ln H u

(3)

Similarly, the strength coefficient K is obtained by using a true stress at the necking point. Thus K is

K

e

DH u

Vu H u ne

 EH u

,

(4)

where Equation (3) is used for n in this expression. Now, the independent parameters of the presented expression are  and . The user can attempt to fit the curve onto the experimental values by using the remaining  and , and the obtained curve always satisfies the diffuse necking condition dVH dH VH and VH = V. Moreover, the remaining parameters  and  can be determined by using the Hill’s localized necking theory [20] in the tensile axis. Namely, by solving the simultaneous equations

775

1 dV(H f ) V(H f ) dH

1 dV(H u ) 1, V(H u ) dH

1 2

,

(5)

where Hu is a uniform elongation and Hf is a fracture strain. If one use the formula with undetermined  and , these parameters are used for curve fitting, and the obtained curve precisely pass through the diffuse necking point in the stress-stain curve. And if one use the formula with determined  and , there are no room for more fitting; however, the curve mathematically satisfy the Swift’s diffuse necking criterion and the Hill’s localized necking criterion.

180(5,&$/,19(67,*$7,216 The proposed stress-strain formula is evaluated by the fitting error and the predicted uniform and limit strains. These analyses are conducted by numerical calculations using the experimental data by means of the least squares method and Newton method. For simplicity, elastic strain was not separated from the total strain because it was an ignorable small value compared to the uniform and fracture strains. The sample data is a result of the tensile test conducted on a 15-layer multilayer metallic sheet. The constituent hard layer is martensitic Type-420J2 and the constituent soft layer is austenitic Type-301 that is located at the outermost layers. In the tensile test, the gauge length was 50mm and the sheet thickness was 1mm, and the tensile speed was 10mm/min. The tensile strength was about 1.2GPa, and the true uniform strain and fracture strain was 0.213 and 0.298, respectively. The monolithic Type420J2 was too brittle to be formed alone; however, the multilayer metallic sheet including this metal was successfully formed due to the enhanced ductility. Three patterns of expressions were adopted for the numerical analyses. The case 1 adopted the conventional npower law formula. The case 2 adopted the flow curve in which the reference point at the maximum load was used to modify the strength coefficient and the hardening exponent. This technique can be seen in [18]. In case 3, the presented formula was tested. This case was further divided into 2 patterns. In the case 3-1, both parameters  and  are used for the least squares fitting. In the case 3-2, the parameters  and  were determined by solving the simultaneous equations of Swift's criterion and Hill's criterion as explained above. The fitting error was evaluated by the root mean square residual (RMSR). This value means the averaged error of all strain range; therefore, the smaller is a better fitting. Next, uniform and limit strains were calculated to be compared with the experimental value. Fitting comparison can be seen in )LJXUH , and the numerical results are shown in 7DEOH . Detailed comparisons are below. In the case 1, the n-power law of (TXDWLRQ (1) was used to fit the experimental data. The best fit was obtained by the least squares method, and the result is shown in )LJXUH as a blue curve. The fitting parameters were determined as K = 2175 and n = 0.235. Then, in the case 2, the flow curve using a reference point was used for fitting. In this formula, the hardening exponent is chosen as a true strain at the diffuse necking point, and the strength coefficient is calculated to make the true stress at the necking point be equal to the experimental value. Both parameters can be determined this way; therefore, this expression does not require fitting procedures any more. The result is represented in )LJXUH as a green curve. The fitting parameters were K = 2117 and n = 0.213, which is of course same as the true uniform strain. Finally, the evaluations of cases 3-1 and 3-2 are presented. In the case 3-1, the parameters  and  are determined as  = -0.714 and  = -2.693. And in the case 3-2, the obtained parameters were  = 0.735 and  = 0.10. The obtained curves are seen in )LJXUH. From )LJXUH and 7DEOH , the fitting results of case 1, 2, and 3-1 brought relatively good predictions, especially the proposed curve (case 3-1) offered the best fit; and the case 3-2 resulted in the worst. In terms of characteristic strains prediction, the case 1 yielded a small error in the uniform strain prediction and overestimated the limit strain. The others matched the experimental value of the uniform strain; however, only the case 3-2 predicted the limit strain precisely. Note that the case 3-1 could not yield a limit strain value because of some numerically ill situation. The case 1 curve is no more than a fitted curve with two parameters. The case 2 curve has a physical meaning that it satisfy the diffuse necking criteria, but there is no fitting ability. The case 3-1 curve has the best ability of fitting, and it also satisfies the diffuse necking criteria. The case 3-2 curve has not fitting ability; however, it has much physical meaning because it satisfy both the diffuse necking and localized necking criterion. In other words, the case 1 is a geometrical fitting model, and the case 2 is a half physical fitting model, and the case 3-1 is a geometrical fitting model with a half physical meaning, and the case 3-2 is a full physical model. In terms of forming limit prediction, most important part of a flow curve is at and after the uniform strain point. Namely, the latter part of the flow curve plays significant role in characterizing the hardening property of a material. Therefore, a curve in which as accurate curve as possible is represented is required to predict the forming limit. In

776

)LJXUH, numerical results of the forming limit using the curve of the case 1, 2, and 3-2 are presented; this figure is combined with the experimental data that was obtained from a punch stretching experiment. In this paper, only the negative strain ratio region was prepared for this comparison; thus, the second quadrant of FLD is presented. Full FLD comparison will be a future work. Note that the numerical predictions were conducted by using the above-used tensile test result; however, the strain measurement for this half FLD was conducted on the different sheet. Although the chemical compositions and heat treatment history were the same, some differences between the limit strains can be seen. The test sheet was 200mm in the edge length, and the hemispherical punch was 50mm in radius. Forming speed was 10mm/min. From the figure, the proposed formula presented a reasonable predictions; on the other hand, two conventional n-power law based methods yielded overestimations. This result may be apparent because of the discussion presented above. Namely, the presented curve is expected to represent the hardening characteristics embedded in a flow curve very accurately, and this favorable feature of the proposed formula led to the relatively good prediction of forming limits. 7$%/(. Comparison of the fitting errors and characteristic strains. Errors are evaluated by root mean square residual (RMSR), and the strains are true values. Experiment Case 1 Case 2 Case 3-1 Case 3-2 Fitting error (RMSR) – 85.7 93.5 71.03 115.6 Uniform strain 0.213 0.235 0.213 0.213 0.213 Limit strain 0.298 0.470 0.426 NA 0.298

),*85(. Comparison of the fitted curves and the experimental result on a tensile test.

777

),*85(. Fracture limit predictions along with the experimental results.

&21&/86,21 In this paper, the forming limit of a multilayer metallic steel sheet has been numerically investigated. A new formula to represent the work-hardening property has been presented and investigated by comparing with another methods. These investigations were conducted using the tensile and stretching experiments. The presented expression can satisfy the instability criterion such as diffuse necking and localized necking; therefore, the presented expression is significantly advantageous when it comes to predicting the forming limits. Needless to say, there is much material in which work-hardening deterioration can be observed, and the presented formula would be effective for such materials. In future, the positive strain ratio region of a FLD should be explored by adopting the S-R theory [21].

$&.12:/('*0(176 The present research was conducted as part of the LISM (Layer-Integrated Steels and Metals) Project funded by the Ministry of Education, Culture, Sports, Science and Technology of Japan.

5()(5(1&(6 1. C. K. Syn, D. R. Lesuer, O. D. Sherb, Enhancing tensile ductility of a particulate- reinforced aluminum metal matrix composite by lamination with Mg–9%Li alloy. Materials Science and Engineering A , 201–207(1996) 2. S. L. Semiatin, H. R. Piehler, Deformation of sandwich sheet materials in uniaxial tension, Metallurgical Transactions $, 85-96(1979). 3. H. D. Manesh and A. K. Taheri, A. K, Bond strength and formability of an aluminum-clad steel sheet, Journal of Alloys and Compounds , 138-143(2003). 4. T. Mori and S. Kurimoto, Press-formability of stainless steel and aluminum clad sheet, Journal of Materials Processing Technology , 242-253(1996). 5. F. Yoshida and R. Hino, Forming limit of stainless steel-clad aluminium sheets under plane stress condition, Journal of Materials Processing Technology , 66-71(1997). 6. R. Hino, Y. Goto, F. Yoshida, Springback of sheet metal laminates in draw-bending, Journal of Materials Processing Technology , 341-347(2003). 7. Y. Ohashi, J. Wolfenstine, R. Koch, O. D. Sherby, Fracture behavior of a laminated steel-brass composite in bend tests, Materials Science and Engineering A , 37-44(1992).

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8. F. Carreño, J. Chao, M. Pozuelo, O. A. Ruano, Microstructure and fracture properties of an ultrahigh carbon steel-mild steel laminated composite, Scripta Materialia , 1135-1140(2003). 9. J. Inoue, S. Nambu, Y. Ishimoto, T. Koseki, Fracture elongation of brittle/ductile multilayered steel composites with a strong interface, Scripta Materialia , 1055-1058(2008). 10. S. Nambu, M. Michiuchi, Y. Ishimoto, K. Asakura, J. Inoue, T. Koseki, Transition in deformation behavior of martensitic steel during large deformation under uniaxial tensile loading, Scripta Materialia , 221-224(2009). 11. S. Nambu, M. Michiuchi, J. Inoue, T. Koseki, Effect of interfacial bonding strength on tensile ductility of multilayered steel composites, Composites Science and Technology , 1936-1941(2009). 12. T. Oya, N. Tiesler, S. Kawanishi, J. Yanagimoto, T. Koseki, Experimental and numerical analysis of layer-integrated steel sheets upon bending, Journal of Materials Processing Technology , 1926-1933(2010). 13. T. Oya, J. Yanagimoto, Finite element analysis of layer-integrated steel sheets undergoing bending, AIP Conference Proceedings (NUMIFORM2010) , 283-289(2010). 14. J. Yanagimoto, T. Oya, S. Kawanishi, N. Tiesler, T. Koseki, Enhancement of bending formability of brittle sheet metal in multilayer metallic sheets, CIRP Annals - Manufacturing Technology (1), pp.287-290(2010). 15. X. X. Chen, P. D. Wu, D. J. Lloyd, J. D. Embury, Y. Huang, Enhanced Ductility in Sheet Metals Produced by Cladding a Ductile Layer, Journal of Applied Mechanics , 041015-1–041015-7(2010). 16. H. S. Liu, B, Zhang, G. P. Zhang, Delaying premature local necking of high-strength Cu: A potential way to ehance plasticity, Scripta Materialia , 13–16(2011). 17. K. Hasegawa, Z. Chen, K. Nishimura, and K. Ikeda, Estimation of stress-strain relationship in post-uniform elongation range of tensile test, Journal of the Japan Society for Technology of Plasticity (in Japanese) , 565, 143-147(2008). 18. M. Joun, J. G. Eom, and M. C. Lee, A new method for acquiring true stress-strain curves over a large range of strain using a tensile test and finite element method, Mechanics of Materials , 586-593(2008). 19. H. W. Swift, Plastic instability under plane stress, Journal of the Mechanics and Physics of Solids , 1-18(1952). 20. R. Hill, On discontinuous plastic states, with special reference to localized necking in thin sheets, Journal of Mechanics and Physics of Solids , 19-30(1952). 21. S. Storen and J. R. Rice, Localized necking in thin sheets, Journal of Mechanics and Physics of Solids , 421-441(1975).

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&RQWDFW3UHVVXUH(IIHFWRQ)ULFWLRQDO&KDUDFWHULVWLFVRI6WHHO 6KHHWIRU$XWRERG\ S.S.Han a, and D.J.Kimb a

Department of Mechanical Engineering, Kumoh National Institute of Technology, Kumi 730-701, Korea b Automotive Steel Research Group, POSCO, Kwangyang 545-711, Korea

$EVWUDFWThe high strength steel (HSS) is widely used in auto body part due to its advantage of weight reduction. The usage of HSS extends the range of contact pressure than that of mild steel’s and makes it is not disregardable fact that the effect of contact pressure on frictional characteristics of steel sheet. To investigate the influence of contact pressure on frictional behavior of steel sheet, the flat type friction test with high strength bare steel sheet was conducted under various contact pressures. According to the test result, the relationship between contact pressure and friction coefficient shows U shape. When the contact pressure is lower than 10MPa, the friction coefficient was slightly decreased as contact pressure was increased. However the amount of decrement was very small. Above 10MPa contact pressure the friction coefficient was increased as the contact pressure was increased and the amount of increment of friction coefficient was not negligible. This study shows that the effect of contact pressure on frictional behavior of steel sheet is very big, especially on HSS stamping which has the wide range of contact pressure. .H\ZRUGVHigh strength steel, Flat type friction test, Bare steel sheet, Oil pocket 3$&662.20.Qp

,1752'8&7,21 Frictional behavior at the tool and workpiece interface during stamping process greatly influence on the strain distribution and formability. Frictional characteristic is interface property and it depends on surface roughness, oiling, properties of coated layer etc. Many of research are focused on that topic[1~5]. However it is also influenced on stamping process parameter such as sliding velocity between tool and work piece, contact pressure. Some of research results showed that the value of friction coefficient is decreased as contact pressure is increased [6,7] and the others showed opposite tendency - friction coefficient is increased as contact pressure is increased [8]. Even both results showed different tendency on the effect of contact pressure on frictional behavior of steel sheet, it is clear that contact pressure is change the value of friction coefficient.

G

(a) SPCEN

(b) DP590  ),*85( Distribution of maximum normal contact pressure over time for cross member stamping The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 780-783 (2011); doi: 10.1063/1.3623685 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

780

For the weight reduction of autobody and the improvement of crashworthiness of car, the usage of high strength steel is increasing dramatically. The high strength of steel sheet results very high contact pressure between tool and workpiece during stamping compared to that of mild steel’s. As shown in the figure 1, the maximum normal contact pressure of mild steel SPCEN stamping was about 280MPa and that of high strength sheet DP590’s was about 500Mpa which was 1.8 times bigger than that of mild steel’s. Since the range of contact pressure fluctuation of high strength steel when stamping is much higher than that of mild steel’s, the stamping process of high strength steel is more easily influenced by the change of friction properties with respect to contact pressure. In this paper the friction test of high strength bare steel was conducted with flat type friction test under various contact pressure to investigate the influence of contact pressure on frictional behavior of steel sheet for autobody.

)5,&7,217(67 To analyze the change of frictional behavior of steel sheet due to contact pressure contact variations, friction test with the flat type friction tester was conducted (Figure.2). In the test, bare steel DP 590 1.2t were used. Since the maximum clamping load of friction tester was about 1,600kgf, to increase the contact pressure the area of testing tool should be reduced. Friction testing tool of contact area size 20mm*20mm were used in order to get high contact pressure. Although the contact pressure can be increased by reducing contact area of testing tool, but too small size can be problem in terms of repeatability, so this size(20mm*20mm) were adopted. This friction test was done with the same oiling condition in the stamping line. The oil used in the test was P340, currently used in steel maker as rust preventing oil. The intervals of test load are 200~300 kg and test was repeated three times with same loading condition.

FIGURE 2 Flat type friction tester and tool shape for friction test

5(68/7$1'',6&866,21 Figure 3 shows friction test results of DP 590 bare steel with respect to contact pressure. The friction test results varied much from each contact pressure status. According to the test result, the relationship between contact pressure and friction coefficient shows U shape. When the contact pressure was relatively low – less than 10 MPa, the friction coefficient slightly decreased by increasing the contact pressure. However the amount of decrement was very small. Above 10MPa contact pressure the friction coefficient was increased as the contact pressure was increased and the amount of increment of friction coefficient was not negligible. This study shows that the effect of contact pressure on frictional behavior of steel sheet is very big, especially on HSS stamping which has the wide range of contact pressure. The test result of bare steel sheet fits well with the result of J. M. Lanzona [6] and J. W. Shim [7]; decrease in friction coefficient with respect to the increase in contact pressure, under the condition of low contact pressure. With contact pressure relatively high, the tendency of friction coefficient with respect to contact pressure also fits well with H. Kim [8] ; increase in contact pressure leads to increase in friction coefficient.

781

Friction Coefficient

0.25 0.2 0.15 0.1 0.05 0 0

10 20 30 40 Contact Pressure(MPa)

50

),*85( Friction coefficient with respect to contact pressure(DP590 Bare)

53URILOH 3.0 [um]

[um]

53URILOH 5.0 4.0

2.0

3.0

1.0 2.0 1.0

0.0

0.0

-1.0

-1.0

-2.0 -2.0

-3.0

-3.0 -4.0

-4.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

[mm]

[mm]

(a) Original

(b) 10MPa

53URILOH

53URILOH [um]

[um]

3.0 2.0

3.0 2.0 1.0

1.0

0.0

0.0 -1.0

-1.0 -2.0

-2.0

-3.0

-3.0

-4.0 -5.0

-4.0 0.0

0.5

1.0

1.5

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3.0

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4.5

0.5

1.0

1.5

2.0

2.5

3.0

(c) 20 MPa

(d) 40MPa

),*85( Surface roughness profile for each testing contact pressure

782

3.5

4.0

4.5 [mm]

[mm]

The cause for the change in friction coefficient with respect to contact pressure mentioned above can be explained in the change in asperities and bearing area due to contact pressure. Figure 4 shows change in asperities of DP590 bare steel sheet for each contact pressure. When the tool contacts sheet steel, the peak of asperities of steel sheet makes first contact and they are deformed flat. This deformation of asperities leads to formation of enclosed oil pockets which improve the lubrication of the interface between tool and workpiece and this causes steel sheet`s friction coefficient to decrease(Figure 4 (b)). However, when the contact pressure is over the limit, the area of flat surface caused by contact pressure increases which decrease the proportion of the area of oil pocket to the total contact area and increase the proportion of real contact area between tool and workpiece. In conclusion, the friction coefficient decrease as contact pressure increase(Figure 4 (c, d) ) When stamping the high strength steel sheet of figure 1(b), the maximum contact pressure between tool surface and steel sheet shows several hundred MPa, but the contact pressure done in this test was lower than the maximum. Theoretically as contact pressure increases, the friction coefficient increases to reach is maximum column friction coefficient, but proof has not been done because of the lack of capacity of the test rig. Further proof will be done after compensating the capacity of test rig.

&21&/86,216 These following conclusions were achieved by the test; how contact pressure effects the autobody steel`s frictional characteristics 1) In the case of bare steel, as contact pressure increases the friction coefficient increase after some decrement. 2) When the contact pressure over the limit, as pressure increased, also the friction coefficient increased considerably. 3) The change in friction coefficient with respect to contact pressure can be explained by deformation of asperities, formation of oil pocket and change in the bearing area. 4) To conclude the test results made, it is appropriate to estimate steel sheet`s friction properties by considering the range of contact pressure done on steel sheet stamping.

5()(5(1&(6 1. 2. 3. 4. 5. 6. 7. 8.

S. S. Han, K. C. Park, Transactions of Materials Processing , 2009, v.17, No.5, pp. 401~406 B. H. Lee, Y. T. Keum, Transactions of Materials Processing, 2002, v.11, No.4, pp. 349~354 P. Skarpelos, J. W. Morris, Jr., WEAR, 1997, v. 212, pp. 165~172 A. Azushima, M. Sakuramoto, Annals of the CIRP, 2006, v. 55 W. T. Kim, D. H. Lee, M. S. Suh, Y. H. Moon, Transactions of materials processing, 2005, v.14, No.8, pp.668~674 J. M. Lanzona, M. J. Cardew-Hall ,P. D. Hodgson, J of Mate. Process. Tech. 1998, v. 80–81, pp.251–256 J. W. Shim, Y. T. Keum, Transactions of Materials Processing, 2007, v.16, no.5, pp.381~385 H. Kim, S.Han, Q.Yanc, T.Altan , CIRP Annals - Manufacturing Technology, 2008, v.57, pp.299~304

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A Multi-scale Friction Model for Sheet Metal Forming Simulations J.Hol∗ , M.V. Cid Alfaro† , T. Meinders∗∗ and J. Huétink∗∗ ∗ Materials innovation institute (M2i), P.O. box 5008, 2600 GA Delft, the Netherlands Tata Steel Research, Development & Technology, P.O. box 10000, 1970 CA IJmuiden, the Netherlands ∗∗ University of Twente, Faculty of Engineering Technology, chair of Forming Technology, P.O. box 217, 7500 AE Enschede, the Netherlands †

Abstract. This paper presents a multi-scale friction model for large-scale forming simulations based on the surface changes on the micro-scale. The surface texture of a material changes when it is subjected to normal loading and stretching. Consequently, the frictional behavior between contacting surfaces, caused by the adhesion and ploughing effect between contacting asperities, will change when the surface texture changes. A friction model has been developed which accounts for the change of the surface texture on the micro-scale. Statistical parameters have been introduced to make a fast and efficient translation from micro to macro modeling. The flattening models are validated by means of FE simulations on micro-scale and the implementation of the advanced macroscopic friction model in FE codes is discussed. Keywords: friction, ploughing, adhesion, flattening, roughening, coefficient of friction PACS: 46.15.-x, 46.35.+z, 46.55.+d

INTRODUCTION Finite Element (FE) simulations of sheet metal products are everyday practice in the automotive industry. The usability of such a code largely depends on the accuracy of the numerical models in these codes. An accurate forming analysis can only be made if, among others, the material behavior and friction conditions are modeled accurately. For material models, significant improvements have been made in the last decades, but in the majority of simulations still a simple Coulomb friction model is used. Consequently, it is still cumbersome to predict the draw-in and springback of a blank during forming processes correctly. In the past, research has been carried out on contact and friction phenomena on different length scales. This paper focuses on the phenomena which play a role in the boundary lubrication regime, which is the most common condition during sheet metal forming. On the microscopic level, friction is due to the adhesion and ploughing effect between contacting asperities [1, 2, 3, 4]. The real area of contact, defined as the area summation of contacting asperities, plays an important role in characterizing friction. The real area of contact depends on the roughness of both tool and workpiece, where the roughness of the workpiece is liable to changes due to flattening and roughening mechanisms. The main flattening mechanisms during sheet metal forming are flattening due to normal loading [4, 5, 6, 7, 8] and flattening due to stretching [4, 9, 10]. Flattening increases the real area of contact, resulting in a higher coefficient of friction. An advanced friction model is proposed which couples the most important friction mechanisms. Based on statistical parameters a fast and efficient translation from micro- to macro modeling is included. A newly developed flattening model, including work hardening effects, has been proposed to describe the increase of real contact area due to normal loading. Asperity flattening due to stretching has been described by the flattening model proposed by Westeneng [4] and the influence of ploughing and adhesion on the coefficient of friction has been described by the friction model of Challen & Oxley [2, 4]. A brief overview of the friction model is presented and the translation from micro to macro modeling is outlined. Next, the theoretical background of the models used to describe the various friction mechanisms are briefly discussed. In the following section, the flattening models are validated by means of FE simulations on the micro-scale. Finally, the implementation of the advanced macroscopic friction model into FE codes is discussed.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 784-791 (2011); doi: 10.1063/1.3623686 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

784

210 Measured Gauss fit Weibull fit Fourier fit B-spline fit

probability [mm− 1]

168

126

84

42

0

FIGURE 1.

-4

-2

0 2 height [mm]

4

6 ×10−3

Measured surface texture (a, left) and corresponding asperity height distribution (b, right)

THEORETICAL BACKGROUND Numerical framework A numerical friction framework has been developed to couple the various micro friction models. The friction model starts with defining the process variables and material characteristics. Process variables are the nominal contact pressure and strain in the material. Significant material characteristics are the hardness of the asperities and the surface properties of the tool and workpiece material. Once the input parameters are known, the real area of contact is calculated based on the models accounting for flattening due to normal loading and flattening due to stretching. The amount of indentation of the harder tool asperities into the softer workpiece asperities can be calculated if the real area of contact is known. After that, shear stresses due to ploughing and adhesion effects between asperities and the coefficient of friction are calculated. It is noted that in reality flattening due to normal loading and flattening due to stretching will appear simultaneously during sheet metal forming, as well as the combination between flattening and sliding. Nevertheless, it has been assumed that the various mechanisms act independently of each other in this research.

Characterization rough surfaces Friction models encompassing micro-mechanisms are generally regarded as too cumbersome to be used in largescale FE simulations. Therefore, translation techniques are necessary to translate microscopic contact behavior to macroscopic contact behavior. Using stochastic methods, rough surfaces are described on the micro-scale by their statistical parameters (asperity density, mean radius of asperities and the asperity height distribution). Assuming that the surface height distribution on the micro-scale represents the surface texture on the macro-scale, it is possible to describe contact problems that occur during large-scale FE analyses of sheet metal forming processes [4]. Figure 1a shows a 2 dimensional roughness measurement of an electrical discharged textured (EDT) steel material. The location of asperities and the asperity density can be obtained by using the nine-point summit rule [11, 12]. Summits are points with a local surface height higher than their 8 neighboring points. Once the location of the asperities is known and assuming that asperities are spherically tipped, the radius of the asperities is related to the local curvature at the surface. The curvature κ is defined as the second order derivative of the function, which can be obtained by the second order finite difference method [12]. The expressions given in Equation 1 can be used to obtain the radius β in two perpendicular directions and the equivalent radius. The expressions are based on the three point definition of a summit curvature in which zx,y represents the local surface height at the asperity location (x, y).

κ|| = β||−1 =

zx−1,y − 2zx,y + zx+1,y dx2

κ⊥ = β⊥−1 =

zx,y−1 − 2zx,y + zx,y+1 dy2

785

−1 κeq = βeq =

κ|| + κ⊥ 2

(1)

When using stochastic methods, only the mean radius of asperities is of interest:

β=

1 nasp

nasp

−1 ∑ κeq,i

(2)

i

The histogram of all local asperities is called the asperity height distribution (Figure 1b). To describe the histogram a continuous function is desirable to eliminate the need for integrating discrete functions during the solution procedure of the friction model. Various methods exist to describe discrete signals by continuous functions. The Gauss distribution function can be used if it is assumed that the surface height distribution is symmetric and approximates a normal distribution function. However, the initial surface height distribution is usually asymmetric and will become even more asymmetric if there is flattening of contacting- and rising of non-contacting asperities. The asymmetric Weibull distribution function is a more flexible criterion but can only approximate smooth surface height distributions. A more advanced method to describe discrete signals can be achieved by using a Fourier series or by using B-splines. A Fourier series makes it possible to describe non-smooth asymmetric distribution functions from which the accuracy of the evaluation depends on the number of expansions used. Using B-splines, non-smooth asymmetric distribution functions can be evaluated from which the accuracy depends on the number of lines used to construct the curve. In Figure 1b, the asperity height distribution corresponding to the measured surface roughness (Figure 1a) is evaluated by a Gaussian, Weibull, Fourier and B-spline function. For the Fourier and B-spline function 10 Fourier expansions and 10 cubic lines were used, respectively. As can be seen, using a Gaussian or a Weibull function a large error in representing the histogram is introduced. A better evaluation can be obtained by using a Fourier series or a B-spline function, noting that the error could be reduced further by using more Fourier expansions or more lines to construct the B-spline. An advantage of the B-spline function, compared to the Fourier series, is that the derivative at the end points approaches zero, which can have a stabilizing effect in the friction algorithm. The Fourier series tends to oscillate towards the end points of the distribution when large tails are present, which represents unrealistic behavior and has a destabilizing effect on the friction algorithm. The computation time to solve the friction model, using the various distribution functions, will be discussed in the final section of this paper.

Flattening models Two flattening mechanisms have been implemented in the friction model to calculate the real area of contact of the workpiece: flattening due to normal loading and flattening due to stretching. A non-linear plastic load model has been developed inspired by the ideal-plastic load model proposed by Westeneng [4, 13]. Besides an ideal-plastic load model, Westeneng proposed an ideal-plastic stretching model [4, 13] which has been used in this research. A brief explanation of the normal loading model is given in this paper, a detailed derivation of the model will be published in an upcoming paper. Westeneng assumed the tool as rigid and perfectly flat, which indents into a soft and rough workpiece material. This assumption is valid since the difference in hardness and length scales between the tool and workpiece material is significant in the case of sheet metal forming processes. The asperities of the rough surface are modeled by bars which can represent arbitrarily shaped asperities, Figure 2. Westeneng introduced 3 stochastic variables as presented in Figure 2: The normalized surface height distribution function of the asperities of the rough surface φ (z), the uniform rise of the non-contacting surface U (based on volume conservation) and the separation between the tool surface and the mean plane of the asperities of the rough surface d. Contact between a flat hard smooth surface and a soft rough surface is assumed without sliding and bulk deformation. Only plastic deformation of asperities is assumed including work-hardening effects. Using the normalized surface height distribution φ (z), expressions to obtain the amount of flattening of contacting asperities d and the rise of non-contacting asperities U can be obtained by energy and volume conservation laws: Pnom =

B (γ + ηβ ) ρω

U (1 − α ) =

∞

(z − d) φ (z) dz

d−U

with

α=

∞

φ (z) dz

(3)

d−U

ρ represents the asperity density, ω can be regarded as an external energy factor while γ and β can be regarded as internal energy factors. ω , as well as γ and β are variables which depend on the statistical parameters U (the constant rise of asperities) and d (the separation between the tool surface and the mean plane of the asperities of the rough

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Rise non-contacting Surface U

Crushed asperities

∞ Tool surface

d z

Mean plane

φ (z) Workpiece asperities U

FIGURE 2.

A rough soft surface flattened by a smooth rigid surface

surface). In addition, ω is a function of the normal forces acting on the asperities FN (z). It should be noted that an equal rise of asperities has been assumed in the derivation of ω , γ and β , which corresponds to the experimental results of Pullen & Williamson [8]. η represents the persistence parameter which describes the amount of energy required to lift up the non-contacting asperities. A value of η = 0 means that no energy is required to rise the asperities, a value of η = 1 implies that a maximum amount of energy is required to rise the asperities. Known parameters in Equation 3 are Pnom , the nominal contact pressure (input parameter), and B, a hardness parameter. Since non-linear plasticity is assumed, the hardness H of the softer material can be described by H = BσY (with B=2.8 for steel materials). The yield strength σy can be described by a flow rule which analytically describes the relation between the strain in the material and the yield strength of the material. A typical phenomenological model is the Nadai or Swift relation. The Nadai hardening function defines the yield stress σy as a function of the equivalent plastic strain ε p and the strain at the initial yield stress ε0 : ⎧   λ +d −z ⎪ ⎪ ln for z > d ⎪ ⎪ ⎪ λ ⎪ ⎨ λ +d −z (4) σy = C (ε p + ε0 )n with εp = for d −U ≤ z < d ln ⎪ λ  ⎪  ⎪ ⎪ λ +U ⎪ ⎪ for z ≤ d −U ⎩ ln λ in which C and n represents specific material characteristics and ε p the logarithmic strain in the asperities. The strain in the asperities is defined as the amount of flattening or rise of asperities relative to the initial height of the asperities λ . In this respect, a definition for the strain can be derived for 1) asperities in contact with the indenter, 2) asperities which will come into contact due to the rise of asperities and 3) asperities which will not come into contact with the indenter (Equation 4). The model described above is based on a normal loading case without additional bulk strain. To account for flattening due to stretching, the model has to be adapted. The change of the fraction of the real contact area as a function of the nominal strain can be presented as:

dαSi l = φ dSi−1 −USi−1 dε E

(5)

with i the iteration number. The subscript S is used for variables that become strain dependent. The contact area ratio is updated incrementally by: αSi = αSi−1 + dαSi (6) The initial values αS0 , dS0 and US0 are obtained from the model without bulk strain. To calculate the change of αS , the value of US and dS needs to be solved simultaneously while ε is incrementally increased. Based on volume conservation and the definition of the fraction of real contact area (Equation 7) US and dS can be obtained.

αS =

∞

φ (z) dz

US (1 − αS ) =

dS −US

∞ dS −US

787

(z − dS ) φ (z) dz

(7)

0.2

0.13

0.58 Fraction of real contact area (-)

Fraction of real contact area (-)

0.17

0.7 FEM: elastic ideal-plastic FEM: elastic nonlinear-plastic Analytical solution

λ =∞

0.1

λ =4.0Rt → 2.0Rt → Rt 0.067 0.033 0

0

FIGURE 3.

FEM: elastic ideal-plastic FEM: elastic nonlinear-plastic Analytical solution: ideal-plastic

0.47 2500 asp/mm2 0.35

5000 asp/mm2 9000 asp/mm2

0.23 0.12

20

40 60 80 Nominal contact pressure (MPa)

0

100

0

0.02

0.04 0.06 Logarithmic strain (-)

0.08

0.1

Development real contact area of analysis 1 (a, left) and development contact area of analysis 2 (b, right)

Shear stresses The model of Challen & Oxley [2, 3] takes the combining effect of ploughing and adhesion between a wedge-shaped asperity and a flat surface into account. Westeneng [4] extended the model of Challen & Oxley to describe friction conditions between a flat workpiece material and multiple tool asperities. He assumed that the flattened peaks of the asperities are soft and perfectly flat and the surface of the tool material is rough and rigid. The difference in hardness between the tool and workpiece material and the difference in length scales between the two surfaces is significant in the case of a sheet metal forming process. Therefore, it is valid to make a subdivision in two length scales using a rigid tool and a soft workpiece. The extended version of Challen & Oxley’s model [4] has been implemented in the friction model to describe friction conditions between the tool and workpiece material: Fw = ρt αS Anom

smax

μasp πωβt H φt (s) ds

(8)

δ

with ω the amount of indentation, βt the mean radius of the tool asperities and μasp the coefficient of friction at single asperity scale [3]. ρt represents the asperity density of the tool surface, Anom the nominal contact area and φt the normalized surface height distribution function of the tool surface. The bounds of the integral are described by smax , the maximum height of the tool asperities, and δ , the separation between the workpiece surface and the mean plane of the tool asperities. Since the normal force is known (input parameter), the coefficient of friction can finally be obtained by: Fw μ= (9) FN

VALIDATION The newly developed non-linear load model as well as the ideal-plastic strain model of Westeneng have been validated by means of FE simulations on a two-dimensional rough surface. In the first analysis, a two-dimensional rough surface of 4mm long was deformed by a perfectly flat and rigid tool. The second analysis was focused on flattening a rough surface by a normal load including a bulk strain in the underlying material. Three simulations have been executed for each analysis using different roughness profiles. The roughness profiles equal three roughness measurements of DC04 low-carbon steel. The surface was modeled by 4 node 2D plane-strain elements. The yield surface was described by the Von Mises yield criterion using the Nadai hardening relation to describe work-hardening effects. The surface height distribution used for the analytical model corresponds to the roughness distribution of the FE simulation. The development of the real area of contact has been tracked during the simulation and compared with the analytical solution. Results shown in Figure 3 are the mean values of the three simulations performed per analysis case. The non-linear load model uses the Nadai hardening relation to account for work hardening effects of the flattened asperities. An unknown parameter has been introduced during the derivation of this model: the initial height of

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asperities λ (see Equation 4). Calculations have been performed using different values for this parameter: 1.0, 2.0 and 4.0 times the Rt value (Figure 3a). The Rt value represents the maximum peak to valley distance between asperities. The amount of strain build up in the asperities will be lower when using a higher value for the initial height λ . From the results, it can be concluded that the exact development of real contact area can be found when using an initial height of 2.0 times the Rt value. Increasing the initial height of asperities to infinity represents ideal plastic material behavior, since the strain build up in the material approaches zero. As shown in Figure 3a, the analytical solution describes the results of the ideal-plastic FE simulation well when using a value of λ = ∞. Combined normal loading and stretching the underlying bulk material decreases the effective hardness [9]. A lower hardness results in an increase of the real area of contact. Both the analytical and the FE results of analysis 2, where a rough surface has been flattened by a nominal load and a bulk strain has been applied to the underlying material, are presented in Figure 3b. As for analysis 1, results shown are the mean values of three simulations performed per analysis case. It can be concluded that work-hardening effects have a large influence on the flattening behavior of the asperities. The strain model developed by Westeneng uses the asperity density to obtain the flattening behavior of asperities. A line profile has been used in this section to generate the FE models. The asperity density of this profile has been obtained by the 3-point summit rule (1D version of the 9-point summit rule for surfaces). By taking the square of the asperity density found by the 3-point summit rule, a density of 2500 asp/mm2 was found. Results obtained by the analytical strain model, using an asperity density of 2500, 5000 and 9000 asp/mm2 , are shown in Figure 3b. The trend of the graphs corresponds well, but the development of the real area of contact is significantly higher compared to the ideal-plastic FE results if an asperity density of 2500 asp/mm2 is used. If a higher value of the asperity density is used the amount of real area of contact will be lower. Using an asperity density of 5000 asp/mm2 it is possible to describe the results of the elastic ideal-plastic FE solution well.

IMPLEMENTATION The developed friction model can be implemented in FE codes using two different strategies. One strategy is to implement the code of the friction model into the contact algorithm of the FE code. The friction model is called if a node of the workpiece comes in contact with the tool, resulting a friction coefficient belonging to that specific node. This strategy returns a specific coefficient of friction for each node in contact. Another strategy is to initialize the coefficient of friction for a predefined range of process variables. Process variables are the nominal contact pressure and the bulk strain in the workpiece material. Since a rough guess can be made about the range of these variables a matrix can be constructed including friction values for all possible combinations. Figure 4a shows for example a friction matrix for a nominal contact pressure in between 0 and 100 MPa and a bulk strain in between 0 and 150%. The time required to construct this matrix, using various functions to evaluate the asperity height distribution, is shown in Figure 4b. The Gauss and Weibull distribution functions are the two fastest methods. However, the applicability of these methods is restricted by the relatively simple shape of the functions. Fourier series or B-splines could be used if more complex shapes are desired. The time required to build a friction matrix using Fourier series increases when increasing the number of Fourier expansions. The time required to construct a friction matrix using B-splines depends on the order of the B-spline (in this case cubic lines where used) instead of the number of lines used to construct the B-spline. Concerning the flexibility of the B-spline function and the numerical stability of the friction algorithm, the B-spline function is favorable in describing complex distributions. The Weibull distribution function is favorable in case of normally distributed distributions. For the first strategy the friction routine is called for every node in contact each step of the simulation. Hence, this strategy can result in a huge increase of the simulation time. For the second strategy, the friction matrix has to be constructed only once. When the friction matrix is constructed it can be used during a FE simulation to find nodal friction values. Since only an interpolation scheme is required to find the nodal friction values, this method will hardly increase the simulation time. Difficulties with the second approach are however expected when it is desired to take history dependency of the deformed asperities into account. Loading/unloading and straining/unstraining of the workpiece material will take place during sheet metal forming which requires the use of history dependency. For the first strategy, friction values are calculated per step per node which makes it more straightforward to take the flattening history of previous steps into account.

789

30 Nr. expansions: 5 - 10 - 15 - 20

25

Nr. lines: 5 - 10 - 15 - 20

Time [s]

20 15 10 5 0

FIGURE 4.

Gauss

Weibull

Fourier

B-spline

Friction matrix (a, left) and evaluation time using different distribution functions (b, right)

APPLICATION A cross-die product is used to test the numerical performance of the developed friction model in a large-scale FE simulation (Figure 5a). Due to symmetry only a quarter of the workpiece was modeled. The workpiece was meshed with 9000 triangular discrete Kirchhoff shell elements using 3 integration points in plane and 5 integration points in thickness direction. The coefficient of friction used in the contact algorithm was calculated on the basis of the friction model presented in this paper. Simulations have been performed using both implementation strategies discussed in the previous section. A distribution of friction coefficients can be observed from the results presented in Figure 5a. Values of the friction coefficients are found ranging from 0.13 to 0.19. The gray area represents the non-contacting area. It can be observed from Figure 5a that higher values of the coefficient of friction occur at regions where high strains occur (region A, B and C). Region A is purely stretched, region B is compressed which causes thickening of the material and region C is stretched over the die radius. Overall it can be concluded that the distribution of the coefficients of friction lies within the range of expectation. The draw-in pattern of the simulation is compared to the simulation result in which the standard Coulomb friction model has been used with a friction coefficient of 0.13. It can be observed that the draw-in significantly deviates from the draw-in obtained with the Coulomb friction model. This is logical since the maximum obtained friction coefficient by the developed friction model is much higher than the fixed value of 0.13. When a fixed value of 0.19 is used, which is the maximum value found when using the developed friction model, failure of the cross-die will occur. As mentioned before, simulations have been performed using two different strategies to couple the friction model to the FE code. Figure 5a shows the results obtained by using a friction matrix (the second strategy proposed). Results match perfectly with these obtained when using the first strategy (direct implementation of the friction algorithm in the contact algorithm of the FE code). However the simulation time grows significantly when using the first strategy. An increase of 3 times the simulation time required to perform a Coulomb based simulation was observed. On the other hand, an increase of less than 1% was observed when using the second strategy.

CONCLUSIONS A friction model that can be used in large-scale FE simulations is presented. The friction model includes two flattening mechanisms to determine the real area of contact at a microscopic level. The real area of contact is used to determine the influence of ploughing and adhesion effects between contacting asperities on the coefficient of friction. A statistical approach is adapted to translate the microscopic models to a macroscopic level. The friction model has been validated by means of FE simulations at a micro-scale. A good comparison was found between the FE simulations and the results obtained by the newly developed non-linear loading model. It has been shown that the non-linear load model can be used to describe both ideal-plastic and non-linear plastic material behavior. If a nominal strain is applied to the bulk material, the effect of work-hardening becomes significant. The ideal-plastic strain model is able to describe the trend of the FE results. However, it was not possible to accurately describe the

790

120

0.19 A

y (mm)

90

60

Draw-in Coulomb 0.13 Draw-in friction model

30

C 0.13

0

0

B

FIGURE 5.

30

60 x (mm)

90

120

Development coefficient of friction cross die (a, left) and amount of draw-in (b, right)

ideal-plastic FE results, as well as the influence of work-hardening effects. Other, more advanced, models are required to accurately describe the influence of bulk straining on the flattening behavior of asperities. The friction model could be directly implemented into FE codes or being coupled to the FE code by using a friction matrix. The first strategy requires to calculate nodal friction values each step of the simulation, which can result in a huge increase of the simulation time. When a friction matrix is used, only an interpolation scheme is required to find nodal friction values during the FE simulation which will result in a lower increase of the simulation time. The friction model has been applied to a full-scale sheet metal forming simulation to test the numerical performance and feasibility of the developed friction model. The results look very promising. The modest increase in simulation time shows the feasibility of the friction model in large scale sheet metal forming simulations.

ACKNOWLEDGMENTS This research was carried out under the project number MC1.07289 in the framework of the Research Program of the Materials innovation institute M2i (www.m2i.nl).

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

W. Wilson, Americal Society of Mechanical Engineers 10, 13–23 (1988). J. Challen, and P. Oxley, Wear 53, 229–243 (1979). J. Challen, and P. Oxley, International Journal of Mechanical Siences 26, 403–418 (1984). J. Westeneng, Modelling of contact and friction in deep drawing processes, Ph.D. thesis, University of Twente (2001). J. Greenwood, and J. Williamson, Proceedings of the Royal Society of London. Series A, Mathematical and Physical sciences 295, 300–319 (1966). Y. Zhao, and L. Chang, Journal of Tribology 123, 857–864 (2001). D. M. Y. Zhao, and L. Chang, Journal of Tribology 122, 86–93 (2000). J. Pullen, and J. Williamson, Proceedings of the Royal Society of London. Series A, Mathematical and Physical sciences 327, 159–173 (1972). W. Wilson, and S. Sheu, International Journal of Mechanical Science 30, 475–489 (1988). M. Sutcliffe, International Journal of Mechanical Science 30, 847–868 (1988). J. Greenwood, Proceedings of the Royal Society of London. Series A, Mathematical and Physical sciences 393, 133–157 (1984). M. de Rooij, Tribological aspects of unlubricated deepdrawing processes, Ph.D. thesis, University of Twente (1998). J. Hol, M. C. Alfaro, M. de Rooij, and T. Meinders, WEAR (2010, doi:10.1016/j.wear.2011.04.004).

791

Pre-Forming Effects on AHSS Edge Cracking Xiaoming Chen a , Ke Chenb and Lorenzo Smithc a

United States Steel Corporation, Automotive Center, 5850 New King Court, Troy, MI, USA b College of Mechanical Engineering, Tongji University, Shanghai, China c School of Engineering and Computer Science, Oakland University, Auburn Hills, MI, USA

Abstract. Edge cracking in advanced high strength steels (AHSS) is a significant failure mode in many sheet metal stamping processes. Edge pre-forming into a wave (or scallop) shape is a common technique used in conventional steels to gather material in high edge stretch regions in preparation for the subsequent edge stretch process. The pre-forms designed for mild steels do not always apply to AHSS because the properties of AHSS can differ greatly from those of conventional steels. This work has studied the effects of pre-forming on AHSS edge cracking. Experiments have been conducted to stretch pre-formed steel strips to failure. Strain distributions of pre-forms with various levels of stretch have been measured using digital image correlation (DIC) technology. Finite element analyses have been performed and compared with the experimental results. Different failure criteria have also been evaluated for use in this type of application. Keywords: Edge crack, Sheet metal forming, DIC, Finite element analysis

INTRODUCTION AHSS have the advantages of combining high strength with enhanced formability and high work hardening capability, which provide an opportunity to make complex geometric automotive parts with strengths far exceeding those of conventional high strength grades of steels. The high strength components provide an opportunity to improve vehicle crash and structural performance while reducing the overall weight of the vehicle. Therefore, AHSS are increasingly used in the automotive industry to meet future fuel economy and safety requirements. However, AHSS also have disadvantages and some forming issues have arisen in the stamping processes. Edge cracking, stretch bending fracture in a tight radius (shear fracture), and springback are some of the major manufacturing issues for AHSS, and these issues are more severe for AHSS than for the conventional high strength steels. Due to the high strength and multi-phase microstructure of AHSS, the edge cracking and shear fracture limits are lower than the conventional forming limit curve. The higher strength and significant Bauschinger effects cause more severe springback after forming. These issues cannot be predicted using the conventional simulation technology and often involve significant trial and error procedures in the tool and stamping process designs. Thus, a computer aided tool would be very useful to shorten the lead time and to reduce the cost of tool and die development processes. In the last several years, significant research has been conducted to solve these AHSS manufacturing issues. Many publications can be found on better understanding of the mechanisms. In the study of edge cracking, hole expansion and tension tests are usually used. Several experiments [1-3] were conducted to investigate the effects of different shearing methods, process parameters, and sheared edge characters on AHSS edge cracking. Steel microstructures were also found to play an important role in the edge cracking process [4,8]. Various simulation techniques were used to integrate the sheared deformation into hole expansion analyses to provide more accurate results. Different failure criteria were used to improve the AHSS edge cracking predictability [5,6]. In the AHSS shear fracture studies, various experiments were conducted to generate the failures for different stretch bending radii and stretch tensions, and empirical shear fracture limits were proposed [7,8]. Alternative failure criteria were also used in simulations to improve the fracture predictions [9]. In springback studies, prediction accuracy was significantly improved by using the kinematic/isotropic hardening material models for AHSS and upgrading simulation technology [10-12]. In flanging of conventional steels, a stretched edge usually forms a wave (or scallop) shape in the previous stage for gathering material. However, this is not applicable for AHSS stamping in many cases. Figure 1 shows an example of the pre-forming, trimming and then flanging processes of dual phase (DP) 780 steel using the procedure employed in forming conventional steels. The edge cracked in the pre-forming stage and the part could not The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 792-799 (2011); doi: 10.1063/1.3623687 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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withstand flanging in the next stage. AHSS pre-forming and stretching are complicated forming processes. The edge experiences bending, unbending and stretching deformations, which involve all three AHSS issues discussed above. Better understanding of this process is important for future AHSS applications. In this work, experimental and simulation studies were conducted on pre-forming and stretching of two AHSS (DP780 and 590R). The mechanical properties of both steels are shown in Table (1). The DIC technique was used to measure the deformation history of both forming stages. Finite element analysis was also conducted to simulate those processes. The objective is to develop a simulation technique for optimal design of AHSS edge stretching processes. TABLE(1). Mechanical Properties Material

590R DP780

Yield Strength (MPa) 440 497

UTS (MPa)

Total Elong (%) 22.9 15.0

636 856

n value

r ave

0.134 0.122

0.77 0.84

FIGURE 1. Edge cracking in pre-forming of DP780

EXPERIMENTS Experiment Setup A laboratory experiment was designed to simulate the pre-forming and edge stretching of a steel strip. Figure 2 shows the tool used for pre-forming. Using this tool, a pre-formed wave can be created from a straight steel strip. Both one and two waves were made, with an example of a one wave pre-form shown in Figure 3.

FIGURE 2. Tools for pre-forming

FIGURE 3. Pre-formed strip

A pre-formed strip was then pulled in a tension test machine as shown in Figure 4. Pulling stopped when fracture occurred. Figure 5 shows a force-displacement curve for DP780 steel. This experiment was designed to use DIC for strain measurement. The steel samples were sprayed with paint to make a speckle surface for DIC strain measurements. For ce- Di spl acement Cur ve

For ce ( KN)

16 12 DP780 TD DP780 RD

8 4 0 0

10

20

30

40

Di spl acement ( mm)

FIGURE 4. Stretching of pre-formed strip

FIGURE 5. Force-displacement curves

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Strain Distributions The strain distributions were measured using DIC technology. Figure 6 shows the color maps of true major and minor strain (Hencky strain) distributions on the top surface of a pre-formed sample. The maximum strain is about 7% on the peak of the pre-form. Figures 7 and 8 show the distributions of strains at fracture for DP780 and 590R, respectively. It can be seen that the DIC technique can obtain the strain distributions clearly and capture the strains at fracture.

(a)

(b)

FIGURE 6. Major (a) and minor (b) strain distributions on a pre-formed sample of DP780

(a)

(b)

FIGURE 7. Major (a) and minor (b) strain distributions at fracture (DP780 transverse)

(a)

(b)

FIGURE 8. Major (a) and minor (b) strain distributions at fracture (590R transverse)

Another advantage using DIC is that the deformation history can be recorded, which provides very useful information for multi-stage forming analyses as in the current case. The above strain mapping figures show that the major and minor strain distributions are reasonably uniform along the width of the specimens. Therefore, the deformation history along the width can be represented in one section. Figures 9 to 12 show the major strains along the longitudinal direction of the specimens. Strains are measured on the top surfaces at various steps. The first step is pre-forming and bending is the dominant deformation. High tension is found at the peak and compression at the valleys near the peak. The specimen is then stretched in the tension machine. As the specimen is stretching,

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unbending occurs at the pre-formed region and tension strain is increasing along the whole specimen. In the specimen tested in the transverse direction, strains reach the failure limit and fracture occurs in the pre-formed regions as show in Figures 9 and 10 for DP780 and 590R steels, respectively. In the specimen tested in the rolling direction, fracture occurs outside the pre-formed regions and fracture strains are significantly higher as shown in Figures 11 and 12 for DP780 and 590R steels, respectively.

0. 08

Fracture

Maj or St r ai n

0. 07 0. 06

Pr ef or m 5mm 10mm 15mm 17. 5mm

0. 05 0. 04 0. 03 0. 02 0. 01 0

FIGURE 9. Major strain of DP780 along transverse direction

0. 3

Fracture

Maj or St r ai n

0. 25

Pr ef or m 5mm 10mm 15mm 20mm 25mm 30mm 35mm 38mm

0. 2 0. 15 0. 1 0. 05 0 - 0. 05

FIGURE 10. Major strain of 590R in transverse direction

795

Fracture

0. 18 0. 16 Pr ef or m 5mm 10mm 15mm 20mm 25mm 30mm 37. 5mm

Maj or St r ai n

0. 14 0. 12 0. 1 0. 08 0. 06 0. 04 0. 02 0

FIGURE 11. Major strain of DP780 in the rolling direction.

Fracture

0. 4 Pr ef or m 5mm 10mm 15mm 20mm 25mm 30mm 35mm 40mm 45mm

0. 35 Maj or St r ai n

0. 3 0. 25 0. 2 0. 15 0. 1 0. 05 0 - 0. 05

FIGURE 12. Major strain of 590R in the rolling direction

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Fracture Three types of tensile failures were observed in the tests. Figure 13 shows those for DP780 steel. Type A is a milled edge without machining damage at the edge. Fracture initiated from the center of a specimen. Thinning and necking are observed. Type B is a sheared edge. Minor necking and thinning are observed around the fracture. In Type C, a sheared edge specimen is pre-formed then pulled. Fracture occurs at the pre-formed region. Fracture initiates at the edge and shear fracture is observed with minimum thinning and necking. Figure 14 is the comparison of the fracture strains for the three types. There is a critical strain value for sheared edge pre-forming and stretching. If a material does not fail at this value, failure will occur outside the pre-formed region, and the failure strain will be significantly higher. This critical strain value depends on the pre-forming geometry, edge damage severity, steel grade and material orientation.

0.50

0. 39 0.31

True Strain

0.40 0.30

0.26

0.30 0.15 0.11

0.20 0.10

0s he ar ed 78 0s he ar +p f 59 0m il le d 59 0s he ar ed 59 0s he ar +p f

78

78

0m il l ed

0.00

FIGURE 13. Three types of fractures of DP780

FIGURE 14. Comparison of the fracture strain for all types

SIMULATION Finite element analyses were conducted to simulate the pre-forming and stretching tests. Figure 15 shows the model. Full integration shell elements and solid elements were respectively used and run on Software packages of LS-Dyna and ABAQUS. Only half of the blank width was used due to the symmetry.

FIGURE 15. Finite element model for simulation

Result Comparison Various material models were used in the simulations. Figure 16 shows the comparison of the cases including von Mises, Hill’s and Barlat yield criteria with the isotropic hardening law, Hill’s yield criterion with modified Yoshida isotropic/kinematic hardening laws, and Johnson-Cook’s fracture model[13]. It can be seen that all models can catch the bending and unbending deformations in the pre-forming and stretching processes. The strain

797

distribution trends can be predicted. It seems that the results are not sensitive to the different material models and no significant difference is found in the strain prediction. Though the trend can be predicted, it is difficult to obtain a good agreement with experiments in magnitude. The four fracture cases shown in Section 2 were simulated and the results were compared to the experimental data. Only the DP780 transverse case shows a reasonably good agreement with the experimental data, in which the fracture strain is at the lowest level. Figure 17 shows the discrepancies between the strain distributions. The simulations predicted more severe bending/unbending strains in the pre-formed regions. The results using explicit methods and solid elements over-predict, while shell elements under-predict the strain level. The results using implicit methods and shell elements over-predict the bending/unbending effects. Therefore, more studies are needed to improve the prediction of bending and unbending deformation for pre-forming.

FIGURE 16. Result comparison of various material models

FIGURE 17. Discrepancy of simulations in the pre-form region.

Forming Limit Figure 18 shows the forming limit diagram (FLD) strain mapping on the major-minor strain plane when fracture initiates. It can be seen that all strain points at the edge are along the uniaxial tension line and far below the limit curve at failure. The comparison of different forming limits and strain paths is shown in Figure 19. The strain paths at four different locations are shown in the figure. The edge point at the peak (EP) of the pre-forming moves up and down along the uniaxial tension path during bending and unbending. The edge point at the valley (EV) moves along the uniaxial compression line. The center point at the peak (CP) is in plane strain condition and moves along the major strain axis. The center point at the valley (CV) moves along the minor strain axis. The FLD is drawn based on the sheet thickness and n value. It lies far above the failure strain point. The milled edge failure is shown in a horizontal line at 0.3 major strain. It closes to the intersection of the FLD and the uniaxial tension line. The sheared edge failure line is at 0.15 major strain. The failure strain of a sheared edge with pre-forming is at 0.11 major strain. This line can be used to predict failure in this case. 0.3

Major Strain

Milled edge

0.2

Sheared edge

FLC

Sheared edge+preform

EP

0.1

EV CV CP

0 -0.1

-0.05

0

0.05

Minor Strain

FIGURE 18. FLD strain mapping at fracture initiation.

FIGURE 19. Comparison of different forming limit and strain paths.

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CONCLUSIONS AND FUTURE WORK • • • • • •

Pre-forming is important in the prevention of edge cracking of AHSS in flanging and edge stretching processes. The conventional approach needs to be modified for proper pre-forming of AHSS. Surface strain distributions of experimental pre-forming and stretching can be measured with DIC technique, which provides full field and history of strain distribution for pre-form design. There is a critical strain value for pre-forming and edge stretching which is a boundary of failure occurring in or out of the pre-formed region. This strain value depends on the pre-formed geometry, edge damage severity, steel grade and material orientation. The conventional FLD cannot be used to predict edge failure in pre-forming and post stretching since the failure limit is much lower due to sheared edge and stretch bending deterioration. In this case, the limit strain can be measured with the pre-form and stretch testing. A newly designed pre-forming tool will be used to test the critical strain values of different pre-formed geometries and various AHSS. A computer assisted technology is important for optimal designs of pre-forming. Simulation technologies need to be developed to accurately predict the bending and unbending effects of pre-forming and subsequent stretching.

ACKNOWLEDGMENTS The authors would like to acknowledge the tooling engineers of United States Steel Corporation Automotive Center, for inspirational discussions on the AHSS edge cracking issues. The material in this paper is intended for general information only. Any use of it in relation to specific applications should be based on independent examination and verification of its unrestricted availability for such use, and determination of suitability for the application by professionally qualified personnel. No license under any United States Steel Corporation patents or other proprietary interest is implied by the publication of this paper. Those making use of or relying upon this material assume all risks and liabilities arising from such use or reliance.

REFERENCE 1. A. Konieczny and T. Henderson “On Formability Limitations in Stamping Involving Sheared Edge Stretching ", SAE technical paper 2007-01-0340. 2. A J. Chintamani and S. Sriram, “Sheared Edge Characterization of Steel Products used for Closure Panel Applications”, SAE technical paper 2006-01-1589. 3. Jian Wang, Todd Link, Mattew Merwin, “AHSS Edge Formability in Sheared Edge Tension”, International Conference on New Developments in Advanced High-Strength Sheet Steels, 2008. 4. C. Chiriac, “Sheared Edge Performance of DP780 Steels -Conventional vs. Improved Bending and Flanging (IBF) Steels”, Materials Science and Technology (MS&T) 2010 , Houston, Texas. 5. X. M. Chen , C.Du, X. Wu , X. Zhu ,and S-D. Liu, “Sheet Metal Shearing and Edge Characterization of Dual Phase Sheets”, IDDRG International Conference, June 2009. 6. M. F. Shi and X. Chen, “Stretch Flange-ability Limits of Advanced High Strength Steels”, SAE technical paper 2007-01-1693. 7. Hua-Chu Shih and Ming F. Shi, " Experimental Study On Shear Fracture Of Advanced High Strength Steels", MSEC-ICMP 2008-72046. 8. Hudgins A.W. Matlocck D.K. Speer J.G., “Shear failure in bending of advanced high strength steels” Proceedings, Interational Deep Drawing Reasearch Group , Conference, pp. 53-64, Golden, CO, 2009. 9. Xiaoming Chen, Meng Luo, et.al.,“AHSS Shear Fracture Predictions Based on a Recently Developed Fracture Criterion”, SAE technical paper, 2010-01-0988. 10. Ming F.Shi, Xinhai Zhu, Cedric Xia, Thomas Stoughton, “Determination of Non-linear Isotropic/Kinematic Hardening Constitutive Parameters for AHSS using Tension and Compression Tests”, Numisheet 2008. 11. Xiaoming Chen, et. al, “Springback Prediction Improvement Using New Simulation Technologies”, SAE technical paper 2009-01-9881. 12. Ke Chen, Jianping Lin, Maokang Lu, Liying Wang, “Advanced High Strength Steel Sheet Forming and Springback Simulation”, Advanced Materials Research 2010, 97-101, 200-203. 13. LS-Dyna Keyword User Manual .

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Material Modeling of 6000 Series Aluminum Alloy Sheets with Different Density Cube Textures and Effect on the Accuracy of Finite Element Simulation Daisaku Yanagaa, Toshihiko Kuwabarab, Naoyuki Uemac and Mineo Asanoc a

Department of Mechanical Systems Engineering, Graduate School of Engineering, Tokyo University of Agriculture and Technology, 2-24-16, Nakacho, Koganei-shi, Tokyo 184-8588, JAPAN b Division of Advanced Mechanical Systems Engineering, Institute of Engineering, Tokyo University of Agriculture and Technology, 2-24-16, Nakacho, Koganei-shi, Tokyo 184-8588, JAPAN c Research & Development Center, Sumitomo Light Metal Industries, Ltd., 3-1-12, Chitose, Minato-ku, Nagoya 4558670, JAPAN Abstract. Biaxial tensile tests of 6000 series aluminum alloy sheet with different density cube textures were carried out using cruciform specimens similar to that developed by one of the authors [Kuwabara, T. et al., J. Material Process. Technol., 80/81(1998), 517-523.]. The specimens are loaded under linear stress paths in a servo-controlled biaxial tensile testing machine. Plastic orthotropy remained coaxial with the principal stresses throughout every experiment. Successive contours of plastic work in stress space and the directions of plastic strain rates were precisely measured and compared with those calculated using selected yield functions. The Yld2000-2d yield functions with exponents of 12 and 6 [Barlat, F. et al., Int. J. Plasticity 19 (2003), 1297-1319] are capable of reproducing the general trends of the work contours and the directions of plastic strain rates observed for test materials with high and low cube textures, respectively. Hydraulic bulge tests were also conducted and the variation of thickness strain along the meridian direction of the bulged specimen was compared with that calculated using finite element analysis (FEA) based on the Yld2000-2d yield functions with exponents of 12 and 6. The differences of cube texture cause significant differences in the strain distributions of the bulged specimens, and the FEA results calculated using the Yld2000-2d yield functions show good agreement with the measurement results. Keywords: aluminum alloy sheet, biaxial tensile test, yield function, hydraulic bulge test, finite element analysis. PACS: 46.35.+z, 83.60.-a, 83.85.-c

INTRODUCTION The crystallographic textures of rolled sheet metals have significant effects on the plastic deformation behavior [1-3]. In sheet metal forming simulations based on the phenomenological theory of plasticity, the difference in the plastic deformation characteristics of real sheet metals caused by difference in crystallographic texture is expected to be reproduced using appropriate yield functions [4-8]. Therefore, validation of this notion is crucial to deepen our knowledge of the effects and limitations of yield functions on the accuracy of forming simulations based on the phenomenological theory of plasticity. This study aims to clarify the accuracy of yield functions to reproduce differences in plastic deformation behavior caused by different crystallographic textures using two kinds of 6016 aluminum alloy sheets with the same chemical compositions but different cube textures. An appropriate yield function for each test material was firstly determined by biaxial tensile tests with linear stress paths using cruciform specimens. Hydraulic bulge tests were then conducted to apply large biaxial plastic deformation to the test materials and the thickness strain distribution along meridian directions of the bulged specimen was measured to investigate the effect of crystallographic texture difference on the deformation behavior of the aluminum alloys. Finally, finite element analyses (FEA) of the hydraulic bulging tests were conducted using selected yield functions determined from the biaxial tensile tests. The calculated thickness strains were compared to the measured thickness strains to reveal the effect of the yield functions on the accuracy of the FEA for the hydraulic bulge test. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 800-806 (2011); doi: 10.1063/1.3623688 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

800

EXPERIMENTAL METHOD Test Material Two types of 6016 aluminum alloy sheet (Al-1.0Si-0.5Mg-0.1Mn (mass%), 0.9mm thick) were prepared with different intensities of the cube orientation: the intensities of the cube orientation at the 1/4 thickness position were 133 and 21 (at random), respectively. The former is referred to as HC (High Cube) and the latter as LC (Low Cube). The work hardening characteristics and r-values in different directions to the rolling direction (RD) are listed in Table 1. In order to reduce the effect of age hardening on the experimental results as much as possible, both materials were heat treated a sufficiently long period ago prior to the experiments. Table 1 Mechanical properties of test material Tensile direction / deg

 0.2 /MPa

c* /MPa

n*

*

r -value**

High Cube (HC)

0 45 90

169 152 163

494 469 485

0.24 0.28 0.25

0.008 0.014 0.007

0.54 0.13 0.55

Low Cube (LC)

0 45 90

152 146 145

474 469 466

0.25 0.27 0.26

0.007 0.009 0.007

0.80 0.26 0.70

*Parameters for Swift’s hardening law,   c(   p ) , for the strain range of 0.002   p   Bp , where  Bp is the logarithmic plastic strain at the maximum load. **Measured at a uniaxial nominal strain of 0.10.

Biaxial Tensile Test In order to determine an appropriate anisotropic yield function that is able to reproduce the elastic-plastic deformation behavior of the material, biaxial tensile tests were performed using the cruciform specimen shown in Fig. 1. The geometry of the specimen is similar to that used by Kuwabara et al. [2, 9, 10]. Each arm of the specimen has seven slits, 30 mm long and 0.2 mm wide, at 3.75 mm intervals, so as to exclude the geometric constraint on the deformation of the 30×30 mm2 square gauge section. The slits were fabricated by laser-machining. Hereafter, the RD and transverse directions (TD) of the material are defined as the x- and y-axes, respectively. Biaxial tensile forces (Fx, Fy) were applied to the cruciform specimen using a servo-controlled biaxial tensile testing machine [9]. The nominal tensile stress components ( N x ,  N y ) were in fixed proportions during each test:  N x : N y  1:0, 4:1, 2:1, 4:3, 1:1, 3:4, 1:2, 1:4 and 0:1. The biaxial total strain components (  x ,  y ) were measured using four uniaxial strain gauges (Tokyo Sokki Kenkyujo, YFLA-2) mounted on the centerlines of the specimens at

Strain gauge

Fig. 1 Cruciform specimen for biaxial tensile test

801

(x, y)=(±10 mm, 0). The true stress components ( x ,  y ) were determined by dividing (Fx, Fy) by the current crosssectional area of the gauge section, which was determined from measurements of the plastic strain components ( xp ,  yp ) with an assumption of constant volume. xy was assumed to be zero, because  x and  y were measured on the centerlines of the specimen. For  N x : N y  1:0 and 0:1, standard uniaxial tensile specimens (JIS 13 B-type) were used. The equivalent plastic strain rate was 1 to 4×10-4 s-1. According to the FEA of the cruciform specimen and the strain measurement position as shown in Fig. 1, the error of the stress measurement was estimated to be less than 2 % using selected isotropic (von Mises) and anisotropic yield functions as material models, [11, 12]. To evaluate the work hardening behavior of a material under biaxial tension, the concept of the contour of plastic work in the stress space was introduced [13, 14]. The stress-strain curve obtained from a uniaxial tensile test along the RD of the material was selected as a reference datum for work hardening; the uniaxial true stresses  0 and the plastic work per unit volume W0 corresponding to particular values of offset logarithmic plastic strains  0p were determined. The uniaxial true stresses  90 obtained from a tensile test in the TD and the biaxial true stress components ( x ,  y ) obtained from biaxial tensile tests were then determined at the same plastic work as W0. The stress points ( 0 , 0) , (0,  90 ) and ( x ,  y ) thus plotted in the principal stress space comprise a contour of plastic work corresponding to a particular value of  0p . For a sufficiently small value of  0p the corresponding work contour can be practically viewed as a yield locus.

Hydraulic Bulge Test Hydraulic bulge tests were performed in order to quantitatively evaluate the effect of the crystallographic texture difference on the deformation behavior during biaxial stretch forming. Fig. 2 shows the experimental apparatus used for the hydraulic bulge test. The diameter of the die opening was 150 mm and the blank diameter was 220 mm. The normal strain components along the rolling (RD) and transverse (TD) directions of the original sheet were measured using strain gauges attached at a distance of 5 mm from the center of the blank. The radius of curvature at the top of the bulged specimen was measured along the direction 45º from the RD using a spherometer, as shown in Fig. 2, the gauge length of which was 40 mm. The spherometer can move in the vertical direction and rotate about an axis normal to the plane of Fig. 2, so that it is always in contact with the bulged specimen at three points. The hydraulic pressure was controlled during each test so that the equivalent plastic strain rate was kept approximately constant (  0.001 s-1). Samples were bulged up to a height of 38 mm. They had an array of grids of 10 mm square for the measurement of the radial and circumferential plastic strains ( rp ,  p ) , along the meridian directions 0, 45 and 90 from the RD. After the bulge test, the thickness strains  zp along the meridian lines were determined as  zp   rp  p . Blank holding force

Spherometer

Pressure

(a)

P 150 150 (b)

190 Fig. 2 Experimental apparatus for the hydraulic bulge test.

Fig. 3 Schematic illustration of FEM model. (a) Tool. (b) Initial mesh division of a blank

802

FINITE ELEMENT SIMULATION OF HYDRAULIC BULGE TEST Material forming simulations of the hydraulic bulge test were carried out using Abaqus/Standard Ver.6.9-1 [15]. 오류! 참조 원본을 찾을 수 없습니다. Fig. 3 shows the finite element mesh used for the analysis. One quarter of a circular blank was analyzed because of the orthotropic anisotropy of the material. The blank diameter was 190 mm and the nodal displacement along the periphery of the blank was assumed to be zero, because the radial position of the draw-bead on the experimental die was 95 mm. 4-node shell elements, S4R, were used. The increment of element division was 2.5º in the circumferential direction and 1 mm in the radial direction in the range of radial coordinates 30 mm  R  95mm. Mesh division was automatically performed in the range of R  30 mm. A Surface-to-Surface contact condition of the blank to the die was selected with a blank holding force of 30 kN and a coefficient of friction of 0.3.

RESULTS AND DISCUSSION

1.2

1.2

1.0

1.0

0.8

0.8

0.6 0.4 0.2

von Mises Hill '48 Yld2000-2d M=6 p (0 =0.002) M = 12 p (0 =0.040)

High Cube p 0 0.002 0.010 0.020 0.030 0.040

0.0 0.0

0.2

y / 0

y / 0

Fig. 4 shows the measured stress points that comprise the contours of plastic work. All stress values comprising a work contour are normalized by 0 corresponding to a specific value of  0p . The theoretical yield loci based on the von Mises [16], Hill’s quadratic (Hill ‘48) [17], and the Yld2000-2d yield functions [18] are also depicted in the figure. The unknown parameters of the Hill's quadratic yield function were determined using r0, r45, r90 and 0/0, and those of the Yld2000-2d yield function were determined using r0, r45, r90 and rb, and 0/0, 45/0, 90/0 and b/0, where r and 

0.4

0.6

x / 0

0.6

Low Cube p 0 0.002 0.010 0.020 0.030 0.045

0.4 0.2

0.8

1.0

0.0 0.0

1.2

0.2

von Mises Hill '48 Yld2000-2d M=4 p (0 =0.002) M=6 p (0 =0.045)

0.4

(a)

0.6

0.8

x / 0

1.0

1.2

(b)

135

Direction of plastic strain rate  / °

Direction of plastic strain rate  / °

Fig. 4 Measured stress points comprising contours of plastic work, compared with theoretical yield loci. Each symbol corresponds to a contour of plastic work for a particular value of  0p .

High Cube

y

90

 

45

x p

0

0.002 0.010 0.020 0.030 0.040

0

-45

0

15

30

45

60

von Mises Hill '48 Yld2000-2d M=6 p (0 =0.002) M = 12 p (0 =0.040)

Loading direction  / °

75

90

(a)

135 Low Cube

y 90

 

45

x p

0

0.002 0.010 0.020 0.030 0.045

0

-45

0

15

30

45

60

von Mises Hill '48 Yld2000-2d M=4 p (0 =0.002) M=6 p (0 =0.045)

Loading direction  / °

75

90

(b)

Fig. 5 Comparison of the directions of measured plastic strain rates with those of the local outward vectors normal to yield loci.

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are the r-value and tensile flow stress measured at  from the RD, respectively, and rb and b are the ratio of plastic strain rates, d  yp / d  xp , and the flow stress at equibiaxial tension,  N x : N y  1:1, respectively. The values of r0, r45 and r90 used were the same as those in Table 1, both for the Hill ’48 and Yld2000-2d yield functions. The values of 45/0, 90/0, b/0 and rb used to determine the Yld2000-2d yield functions correspond to those comprising the work contours for the  0p as referred to in the figure. We find that the variation in shape of the work contours is small for a strain range of 0.002   0p  0.010, while for  0p  0.010 the work contours show a tendency of approaching to the Yld2000-2d yield function with higher exponents with the increase of  0p . The Yld2000-2d yield function with an exponent of M  6 for HC and 4 for LC had the closest agreement with the measured work contour for  0p  0.002, while the M that provided the closest agreement with the measured work contour at a highest strain (  0p  0.040 for HC and  0p  0.045 for LC) was 12 for HC and 6 for LC, respectively. In order to validate the normality flow rule for the yield functions, the directions  of plastic strain rates were measured at each linear stress path, and the results are shown in Fig. 5. The Yld2000-2d yield function provided the closest agreement with the measurement. From the results of Figs. 4 and 5, it is concluded that the Yld2000-2d yield function is capable of reproducing the plastic deformation behavior of the material more accurately than other yield functions. Fig. 6 shows the measured thickness strain along the meridian lines, 0, 45 and 90º directions from RD, of hydraulic bulge specimens with a bulge height of 38 mm, compared with those calculated using FEA with selected yield functions. In Fig. 6a, the equivalent stress-equivalent plastic strain curve used in the FEA was determined by approximating the measured uniaxial  x -  xp curve using Swift’s power law as shown in Table 1, while in Fig. 6b, it was determined by approximating the  b -  zp curve obtained from the hydraulic bulge test. Comparison of the experimental results for HC and LC shows that  zp at the top of the bulged specimen is larger for LC than for HC, and  zp at the periphery (near die profile) is larger for HC than for LC. This difference in thickness strain distribution between HC and LC can be explained from the difference in the shapes of the work contours, as shown in Fig. 4. The difference in flow stresses between the top (equibiaxial tension) and the periphery 0.4

0.3

Yld2000-2d

0.2

von Mises Hill '48 0° 45° 90° Yld2000-2d ( M = 12 ) 0° 45° 90°

0.1

High Cube Swift's power law: Uniaxial 0.0

0.3

10

20

30

40

50

60

70

80

90

0.1

Low Cube Swift's power law: Uniaxial

100

0

10

Initial radial coordinate R / mm

20

30

40

0.3

0.2 Yld2000-2d

Experimental von Mises 0° Hill '48 45° 0° 90° 45° 90° Yld2000-2d ( M = 12 ) 0° 45° 90°

10

20

30

40

50

60

70

80

90

Hill '48

p

High Cube Swift's power law: Biaxial 0

70

80

90

100

(a2)

0.1

0.0

60

0.4

Logarithmic plastic strain |z |

p

Logarithmic plastic strain |z |

Hill '48

50

Initial radial coordinate R / mm

(a1) 0.4

von Mises Hill '48 0° 45° 90° Yld2000-2d (M=6) 0° 45° 90°

Yld2000-2d

0.2

0.0

0

Experimental 0° 45° 90°

Hill '48

p

Experimental 0° 45° 90°

Logarithmic plastic strain |z |

Hill '48

p

Logarithmic plastic strain |z |

0.4

100

0.3

0.2 Yld2000-2d

Experimental von Mises 0° Hill '48 45° 0° 90° 45° 90° Yld2000-2d (M=6) 0° 45° 90°

0.1

Low Cube Swift's power law: Biaxial 0.0

0

10

20

30

40

50

60

70

80

90

100

Initial radial coordinate R / mm

Initial radial coordinate R / mm

(b1) (b2) Fig. 6 Measured thickness strain along the meridian lines of hydraulic bulge specimens, compared with those calculated using finite element analysis with selected yield functions.

804

(plane strain tension) is smaller in HC than in LC; therefore, the plane strain elongation at the periphery is accelerated more in HC than in LC, and the reduction in thickness at the periphery becomes accordingly larger in HC than in LC. Thus, it is concluded that the difference in crystallographic texture causes the relative difference in the magnitudes of flow stresses between equibiaxial tension and plane strain tension and, consequently, causes the difference in the thickness distribution along the radial direction for the two materials. Comparison of Fig. 6 (a) and (b), it is clear that the calculated results obtained using the Yld2000-2d yield functions in Fig. 6 (b1) and (b2) provide the closer agreement with the measurement than those in Fig. 6 (a1) and (a2). Thus, it is recommended to carry out the biaxial stress test that closely reproduces the real stress state to be analyzed in order to determine the work hardening equation used in the FEA. Comparison of the experimental and calculated results indicates that the Yld2000-2d yield functions are capable of reproducing the experimental tendency with good accuracy for both materials, and have the closer agreement with the experimental results than other yield functions, regardless of the work hardening laws used in the FEA. In contrast, the von Mises and Hill’s quadratic yield functions overestimate and underestimate the reduction in thickness at the top and periphery, respectively, which results in a large deviation from the experimental results. The results in Fig. 6 are therefore in accordance with those in Fig. 4 and Fig. 5. Thus, it is concluded that the biaxial tensile testing with cruciform specimens is an effective method to determine appropriate anisotropic yield functions for given sheet metals to improve the predictive accuracy of FEA for stretch forming.

CONCLUSIONS Biaxial tensile tests of 6000 series aluminum alloy sheets with different density cube textures (HC and LC) were conducted using cruciform specimens. The difference in crystallographic texture was clearly detected as the difference in the plastic deformation behavior in the biaxial tensile tests; the shapes of the work contours and the directions of the plastic strain rates were in good agreement with those calculated using the Yld2000-2d yield functions. FEA of the hydraulic bulge test showed that the Yld2000-2d yield function, which had better reproducibility of the biaxial tensile test results than other yield functions, also resulted in closer agreement with the hydraulic bulge test results for both materials. It is concluded that the biaxial tensile testing with cruciform specimens is an effective experimental method to determine appropriate yield functions for given sheet metals and is useful to improve the predictive accuracy of FEA for aluminum alloy sheet forming processes.

ACKNOWLEDGMENT The authors are grateful to Professor Jeong Whan Yoon of the Swinburne University of Technology for kindly providing the subroutine program for the Yld2000-2d yield function used in the FEA.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

F. Barlat, Mater. Sci. Engng. 91, 55-72 (1987) T. Kuwabara, A. Van Bael and E. Iizuka, Acta Mater. 50, 3717-3729 (2002). K. Yoshida, M. Kuroda, and S. Ikawa, Acta Mater. 55, , 4499-4506 (2007)． J.-W. Yoon, F. Barlat, R.E. Dick, K. Chung and T.J. Kang, Int. J. Plasticity 20, 495-522 (2004). J.-W. Yoon, F. Barlat, R.E. Dick, and M.E. Karabin, Int. J. Plasticity 22, 174-193 (2006). T. Kuwabara, Int. J. Plasticity 23, 385-41 (2007). D. Banabic, F. Barlat, O. Cazacu and T. Kuwabara, Int. J. Mater. Form. 3, 165-189 (2010). T. Kuwabara, K. Hashimoto, E. Iizuka and J.-W. Yoon, J. Mater. Processing Technol., 211, 475–481 (2011). T. Kuwabara, S. Ikeda and T. Kuroda, J. Mater. Process Technol. 80-81 517-523 (1998). T. Kuwabara, M. Kuroda, V. Tvergaard and K. Nomura, Acta Mater. 48, 2071-2079 (2000). Y. Hanabusa, H. Takizawa and T. Kuwabara, Steel Research Int. 81, 1376-1379 (2010). Y. Hanabusa, H. Takizawa and T. Kuwabara, J. JSTP 52, 282-287 (2011). (in Japanese) R. Hill, S.S. Hecker, and M.G.. Stout, Int. J. Solids Struct. 31, 2999-3021 (1994). R. Hill and J.W. Hutchinson, J. Appl. Mech. 59, S1-S9 (1992). Abaqus Analysis, User’s Manual, Version 6.9, Dassault Systemes, 2009. R. Von Mises, Göttingen Nachrichten, math.-phys. Klasse, 582-592 (1913). R. Hill, Proc. Roy. Soc. London A193, 281-297 (1948). F. Barlat, J.C. Brem, J.W. Yoon, K Chung, R.E. Dick, D.J. Lege, F. Pourboghrat, S.H. Choi and E. Chu, Int. J. Plasticity 19,

805

1297-1319 (2003).

806

A User-friendly 3D Yield Function for Steel Sheets and Its Application Fusahito Yoshidaa, Shohei Tamuraa, Takeshi Uemorib and Hiroshi Hamasakia a

Department of Mechanical Science & Engineering, Hiroshima University, 1-4-1, Kagamiyama, Higashi-Hiroshima, 739-8527, Japan b Department of Mechanical Engineering, Kindai University, 1, Umenobe, Higashi-Hiroshima, 739-2116, Japan

Abstract. A 6th-order polynomial type 3D yield function, which has a high flexibility of describing anisotropic behavior of steel sheets is proposed. A user-friendly scheme of material parameter identification for the model, where only limited experimental data of r-values and yield stresses in three directions ( r0 , r45 , r90 and V 0 , V 45 , V 90 ) plus equi-biaxial yield stress ( V x

Vy

V b ) are needed, is presented. The accuracy of this yield function is verified by comparing the

numerical predictions with the corresponding experimental data on planar anisotropy of r-values and flow stress directionality, as well as the shape of yield loci, on several types of steel sheets (high r-value IF steel and high strength steel sheets of 440-980MPa TS levels). The advantage of this model is demonstrated by showing a numerical simulation of hole-expansion test. Keywords: Yield function, anisotropy, sheet metal forming

PACS: 62.20.fq INTRODUCTION For accurate numerical simulation of sheet metal forming and springback, the use of plasticity models that properly describe material behavior is of vital importance. For material modeling, two items should be considered, one is the yield function to express the sheet anisotropy, and the other is the kinematic hardening law to describe the Bauschinger effect and cyclic workhardening. The present authors have already proposed an advanced kinematic hardening law (Yoshida-Uemori model [1-3]), and the accuracy of this model have been verified by many researchers (e.g., Eggertsen & Mattiasson [4], Ghaei & Green [5]). To describe planar r-value anisotropy and flow stress directionality, many types of anisotropic yield functions have been proposed in the past (e.g., Hill [6-8], Gotoh [9], Barlat et al. [10-12], Banabic [13], et al., Vegter & Boogaard [14] etc.). However, still classical Hill48 quadratic function [6] is widely used for FE simulation in the sheet metal forming industry, because its material parameters are easily determined from uniaxial tension experiments in three directions (0, 45 and 90o with respect to rolling direction) of a sheet, and furthermore, its 3D expression is clearly presented. Although most of FE simulations of sheet metal forming are performed with shell elements on the assumption of plane stress condition, in some cases of forming, we have to take into account the 3D stress condition, e.g., sheet forging, bottoming, ironing etc.. The present paper begins with the summary of some existing yield functions, and points out the advantages of these models and also their drawbacks, and then proposes a new 3D yield function of 6th-order polynomial. A userfriendly scheme of material parameter identification for the model is also presented, where only limited experimental data of r-values and yield stresses in three directions ( r0 , r45 , r90 and V 0 , V 45 , V 90 ) plus equi-biaxial yield stress ( V x V y V b ) are needed. The accuracy of this yield function is verified by comparing the numerical predictions with the corresponding experimental data on planar anisotropy of r-values and flow stress directionality, as well as the shape of yield loci, on several types of steel sheets (high r-value IF steel and high strength steel sheets of 440-980MPa TS levels). The advantage of this model is demonstrated by showing a numerical simulation of holeexpansion test. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 807-814 (2011); doi: 10.1063/1.3623689 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

807

SUMMARY OF SOME EXISTING YIELD FUNCTIONS Polynomial Type Yield Functions The most popular anisotropic yield function is Hill48 model [6], of which plane stress description is given by the following quadratic function:

IH

A1V x2  A2V xV y  A3V y2  3 A4W xy2

V 02 ,

(1)

where, A1 ~ A4 are anisotropic parameters which are all unity for the von Mises isotropic material. They can be determined by using r-values r0 , r45 and r90 in three direction with respect to rolling direction of a sheet, as follows:

A1

A1( r )

1, A2

A2( r )

2r0 , A3 1  r0

A3( r )

r0 (1  r90 ) , A4 r90 (1  r0 )

A4( r )

(r0  r90 )(1  2r45 ) . 3r0 (1  r0 )

(2)

Alternatively, if we use the normalized yield stress values V 0 , V 45 and V 90 in the three directions, as well as the equibiaxial flow stress V b : V 0 1 , V 45 V 45 / V 0 , V 90 V 90 / V 0 and V b V b / V 0 , these parameters are determined as

A1

A1(V )

1, A2

A2(V )

1

1

V

2 90



1

V

2 b

, A3

A3(V )

1

V

2 90

, A4

A4(V )

1§ 4 1 · ¨ 2  2 ¸. 3 © V 45 V b ¹

(3)

Hereafter, we call Hill48 model when using parameters A1( r ) ~ A4( r ) ’Hill48-r’, and when using parameters

A1(V ) ~ A4(V ) ’Hill48-V’. As already well known that, Hill48-r predicts the planer r-value anisotropy reasonably well for most of sheet metals, but it describes the flow stress directionality too strongly. Furthermore, the shape of the yield loci calculated by Hill48-r do not fit the experimental observations, e.g., for high r-value sheets it overestimates the equi-biaxial yield stress V b , and for low r-value sheets (e.g., aluminum sheets) underestimates it (so-called ‘anomalous’) . Hill48-V well predicts the stress directionality, as well as the shape of yield loci for steel sheets, but it poorly predicts the r-value planar anisotropy. For aluminum sheets, the shapes of yield loci are very different from its predictions. Despite such drawbacks, it has an advantage over the other models that its material parameters are explicitly identified by using just three r-values (in Hill48-r), and its 3D expression is clearly presented. The quadratic ellipse function always guarantees the convexity of the yield locus. To overcome the above problems in Hill48 model, Gotoh [9] proposed a 4th-order polynomial type yield criterion. It can be expressed by the following equation by using nine material parameters B1 ~ B9 which are all unity for the von Mises isotropic material.

IG

B1V x4  2 B2V x3V y  3B3V x2V y2  2 B4V xV y3  B5V y4  6  B6V x2  B7V xV y  B8V y2 W xy2  9 B9W xy4

(4)

V 64 .

Here, material parameters B1 ~ B5 are determined by using three direction r-values r0 , r45 and r90 , as well as V 90 and V b , explicitly, as follows:

B1

A1( r )

1, B2

A2( r )

2r0 , B5 1  r0

1

V 904

, B4

2 B5 r90 , B3 1  r90

· 1§ 1  B1  2 B2  2 B4  B5 ¸ . ¨ 3 © V b4 ¹

(5)

To identify the rest of four material parameters B6 ~ B9 , we need r45 and V 45 , and two more experimental data. Gotoh recommended the use of r22.5 and V 22.5 which are obtained from a uniaxial tension experiment in 22.5o direction. However, when using such a set of material parameters identified by Gotoh’s explicit approach, the overall characterization of the flow stress directionality, V D , and planar r-value anisotropy, rD , can show unbalanced big errors. To overcome this problem, Hu [15] proposed to use the following equations:

808

V 22.5

(V 0  V 45 ) / 2 , V 67.5

(V 45  V 90 ) / 2

(6)

instead of r22.5 and V 22.5 . Furthermore, Hu [15] recently gave a 3D expression for Gotoh’s yield function. The convexity of this yield locus is not generally guaranteed, however, we can check it analytically (see Soare et al. [16]). Soare et al. [16] recently discuss four, six and eight-order polynomial yield functions, where they proposed a scheme of material parameter identification based on the numerical optimization technique which includes the constraint to satisfy the convexity of the yield loci. This approach is sophisticated, however, for the optimization, we need a lot of additional material data other than r0 , r45 , r90 , V 0 , V 45 , V 90 and V b .

Yield Functions Based on Linear Transformation of Stress Tensors Different from the polynomial type, several anisotropic yield functions have been proposed based on the linear transformation of stress tensors (Barlat & Lian [10], Barlat et al. [11], Banabic et al. [12]). Among these, the simplest plane stress model is Yld89 [10], as follows:

IB 89 K1

a K1  K 2

M

 a K1  K 2

1 V x  hV y , K 2 2

M

  2  a 2 K1

M

2V 0M , (7)

2 1 2 V x  hV y   pV x ,  4

where exponent M takes a value of 6 for BCC or 8 for FCC metals. Three anisotropic parameters a, h and p are determined by using three of the six material data of r0 , r45 , r90 , V 45 , V 90 and V b . Hereafter, when the parameters are identified from r0 , r45 and r90 , we call Yld89 model, ‘Yld89-r’. In contrast, when using and

V 45 , V 90

V b , we call it ‘‘Yld89-V’. This model has been extended to have a higher flexibility of description of yield loci,

e.g., Barlat et al’s Yld2000-2d [11] has a form:

IB

I c  I cc 2V a

Ic

X 1c  X 2c , I cc

a

X c C 's

Lc ,

X ''

a a 2 X 2cc  X 1cc  2 X 1cc  X 2cc ,

(8)

C '' s = Lcc,

where V and s are the stress tensor and its deviator, and C ' and C '' are linear transformation matrices representing the anisotropy, where their components are all unity for a isotropic material. By using principal values of thus obtained X ' and X '' ,

X 1' , X 2' , X 1'' and X 2'' , two functions I ' and I '' are defined. Since I ' and I '' are

real-valued convex functions, the yield function and it has been verified that

IB

IB

is also real-valued convex. This approach is very sophisticated

has a high capability of expressing the anisotropy of sheet metals (e.g. Barlat et al.

[12]). Although a 3D formulation has also discussed (Barlat et al. [12]), its expression by the stress components is rather complex because it is only obtained from a calculation scheme for 3D principal stress values.

Plane Stress Yield Function by Interpolation of Biaxial Stress States Vegter & Boogaard [14] proposed an anisotropic plane stress yield function based on the interpolation by second-order Bezier curves. In this model, six reference stress points are selected in the principal stress plane V 1 , V 2 , which are two uniaxial stress points, equi-biaxial stress point, two plane-strain stress points, and two pure shear stress points. Based on the normality rule, giving plastic strain ratios U

H2 / H1 at these reference stress points,

the V 1  V 2 yield locus is defined. For full description of the yield locus, we need a set of data of the reference stress points and the corresponding plastic strain ratios (outward normals) in several principal directions defined by the angle between the first principal stress V 1 and the rolling direction, T . For that purpose, Fourier series are used, where their parameters are determined from the V 1  V 2 yield points and the corresponding outward normals

809

for several choices of T -directions. An advantage of this model is a high flexibility of description of the yield locus. On the other hand, it has a drawback in the material parameter identification, i.e., we have to conduct unconventional biaxial experiments by using specimens cut from a sheet in several different directions T . Furthermore, its extension to 3D model is difficult.

PROPOSITION OF SIXTH-ORDER POLYNOMIAL YIELD FUNCTION Yield Function and Its Material Parameter Identification For the plane stress state, we propose the following anisotropic yield function of a 6th-order polynomial.

IY

C1V x6  3C2V x5V y  6C3V x4V y2  7C4V x3V y3  6C5V x2V y4  3C6V xV y5  C7V y6

9  C8V x4  2C9V x3V y  3C10V x2V y2  2C11V xV y3  C12V y4 W xy2 27  C13V x2  C14V xV y  C15V y2 W xy4  27C16W xy6

(9)

V 06 .

Here, C1 ~ C16 are anisotropic parameters which are all unity for the von Mises isotropic material. Parameters C1 ~ C7 can be determined explicitly by using two uniaxial flow stresses, V 0 and V 90 , three biaxial stresses: equibiaxial stress V b ; plane strain stresses V a and V c , as schematically shown in FIGURE 1, as well as r-values r0 and r90 , as follows:

A1( r )

1, C2

B2

A2( r )

2r0 , C7 1  r0

1

B1

C3

16 § 1 · 1 § 1 · 8 § 1 · 7 1 7 35 ¨ ¸ ¨ ¸ ¨ ¸  C 2  C6  C7  , 9 © V a6 ¹ 3 © V b6 ¹ 9 © V c6 ¹ 4 2 12 24

V 906

, C6

2C7 r90 , 1  r90

C1

C4

16 § 1 · 5 § 1 · 16 § 1 · 3 3 45 45 ¨ ¸ ¨ ¸ ¨ ¸  C2  C6  C7  , 7 © V a6 ¹ 7 © V b6 ¹ 7 © V c6 ¹ 2 2 28 28

C5

8 § 1 · 1 § 1 · 16 § 1 · 1 7 35 7 ¨ 6 ¸  ¨ 6 ¸  ¨ 6 ¸  C2  C6  C7  . 9 © Va ¹ 3 © Vb ¹ 9 © Vc ¹ 2 4 24 12

(10)

For shear-normal stress coupling parameters C8 ~ C12 , we assume that they can be determined using Gotoh’s parameters B1 ~ B5 , by introducing a coefficient [ as follows:

FIGURE 1. Five reference stress points for determining (V x , V y ) yield locus: uniaxial stresses V 0 and V 90 ; equi-biaxial stress V b ; plane strain stresses V a and V c .

810

Ci 7

[ Bi , i 1, ",5 .

(11)

For a prescribed value of [ , the rest of four coupling parameters C13 ~ C16 are determined explicitly so as to satisfy three flow stresses, V 22.5 , V 45 and V 67.5 , as well as r45 . Coefficient [ is identified by minimizing the difference between experimental and numerical results for overall rD -values. One of the advantages of this model is a high flexibility of description of the yield locus. Especially, in the present model we can directly use the plane-strain yield points V a and V c , similarly to Vegter’s model. The convexity of this yield locus is not generally guaranteed (refer to Soare et al. [16]), however, it will be confirmed numerically by checking the Hessian matrix w 2I / wV i wV j is semi-positive definite. In the present model, from the assumption of Eq. (10), if the biaxial (V x , V y ) yield loci of 4th- and 6th-order polynomial are convex, the convexity of (V x , V y ,W xy ) yield loci in the (V x , V y ) plane for given shear stress

W xy

is always guaranteed, as

shown in FIGURE 2. The plane-stress model Eq. (9) is easily extended to 3D version as follows:

C1 V x  V z  3C2 V x  V z V y  V z  6C3 V x  V z V y  V z

IY

6

5

4

2

7C4 V x  V z V y  V z  6C5 V x  V z V y  V z  3C6 V x  V z V y  V z  C7 V y  V z 3

3

^

4

2

5

9 C8 V x  V z  2C9 V x  V z V y  V z  3C10 V x  V z V y  V z 4

3

2

2C11 V x  V z V y  V z  C12 V y  V z 3

^

4

` W

2 xy

 W yz2  W zx2

27 C13 V x  V z  C14 V x  V z V y  V z  C15 V y  V z 2

27C16 W xy2  W yz2  W zx2

3

2

` W

2 xy

6

2

(12)

 W yz2  W zx2 . 2

V 06

Here note that the expressions for the normal-shear coupling terms are different from those of Soare et al’s 3D polynomial yield function [16]. 1.5

1

Vy㻌V

0.5

0

-0.5

-1 Experiment

-1.5 -1.5

-1

-0.5

0

0.5

1

1.5

Vx㻌㻛V FIGURE 2. (V x , V y ,W xy ) yield loci in the (V x , V y ) plane for several values of shear stress W xy on 590MPa HSS. Experimental

0.43, r45 1.41, r90 0.61, V 45 0.936, V 90 1.047 and V b 1.145 are estimated from the calculation by Yld89-V ( P 0 .in Eq. (13)).

data are from Hashimoto et al. [17]. r0 stresses V a

1.094 and V c

811

1.000 . Plane strain

User Friendly Scheme of Anisotropic Parameter Identification For users’ convenience, we will offer a scheme of anisotropic parameters identification, in which all the parameters are determined using only r0 , r45 , r90 , V 0 , V 45 , V 90 and V b . As mentioned above, besides these seven material data, theoretically we need some additional data, two uniaxial yield stresses, V 22.5 and V 67.5 , two planestrain yield stresses, V a and V c , and furthermore some of rD -values. Instead of conducting biaxial stretching experiment, we will estimate these values by using simple anisotropic yield functions, e.g., Hill48 or Yld89, as follows. Since uniaxial yield stresses V D in D-directions can be estimated by Hill48-V or Yld89-V, and rD -values by Hill48-r or Yld89-r reasonably well (see FIGURE 3), we may use such estimations for material parameter identification. Note that the values of V 22.5 and V 67.5 calculated by Hill48-V and Yld89-V are almost the same as Hu’s assumption given by Eq. (6). For the prediction of two plane-strain yield stresses,

Va

and V c , we may use

either Hill48-V, Yld89-Vpredictions, or their combinations:

Va

PV aHill 48  1  P V aYld 89 , V c

PV cHill 48  1  P V cYld 89 ,

(13)

where P is a weighting coefficient, which may depend on materials. For example, the calculated yield locus on 590 MPa HSS is compared with the experimental result (after Hashimoto et al. [17]) in FIGURE 2, where V a and V c are estimated by Yld89-V ( P

0 in Eq. (13)).

Verification of the Yield Function for Steel Sheets To verify the proposed yield function, uniaxial tension experiments in various directions were conducted on several types of steel sheets such as a high r-value IF steel, high strength steel sheets (HSS) of 440-980MPa tensile strength levels (eighteen different sheets), where data of the flow stress directionality V D and the r-value planar anisotropy rD were obtained. Moreover, (V x , V y ) yield loci were obtained by performing biaxial stretching experiments using a cruciform specimen. Besides our own experimental data, previously published data on steel sheets by Kuwabara et al. [18, 19] and Hu [20] were also checked. The calculated results of V D and rD , as well as the yield loci, agree reasonably well with the corresponding experimental results. For example, FIGURE 3 shows the flow stress directionality V D and the r-value planar anisotropy rD , predicted by the present model (indicated as ‘Yoshida-St model’) and the corresponding experimental results on 590 MPa HSS (experimental data are from Hashimoto et al. [17] which are for the same sheet as shown in FIGURE 2). 1.15

1.5

Normalized flow stress VD

1.1 1.05

Lankford value rD

Experiment Hill'48-r Hill'48-V Yld2000-2d (a=6) Yoshida-St

1 0.95 0.9

1

0.5

Experiment Hill'48-r Hill'48-V Yld2000-2d (a=6) Yoshida-St

0.85 0.8

0 0

15

30

45

60

75

0

90

15

Tensile angle from R.D. D [deg.]

FIGURE 3. Flow stress directionality

VD

30

45

60

75

90

Tensile angle from R.D. D [deg.]

(left) and r-value planar anisotropy rD (right) on 590 MPa HSS. Experimental data are from Hashimoto et al. [17].

812

3

1.06 1.04

Lankford value rD

Normalized flow stress VD

1.08

1.02 1 0.98 Experiment Hill'48-r Gotoh Yoshida-St

0.96 0.94

0

15

30

45

60

75

2.5

2

1.5

1

90

Experiment Hill'48-r Gotoh Yoshida-St 0

Tension angle from R.D. D[deg] FIGURE 4. Flow stress directionality

VD

15

30

45

60

75

90

Tension angle from R.D. D[deg]

(left) and r-value planar anisotropy rD (right) on high r-valued IF steel. Experimental

1.5

y

Normalized yield stress in T.D. V /V

0

data are from Kitayama et al. [21].

1

0.5 Experiment Hill'48-r Gotoh Yoshida-St 0 0

0.5

1

1.5

Normalized yield stress in R.D. Vx㻌/V0

FIGURE 5. Biaxial yield locus of high r-valued IF steel. Experimental data are from Kitayama et al. [21].

For the present material parameter identification, experimental data of employed, together with the estimation for

Va

r0 , r45 , r90 , V 0 , V 45 , V 90 and V b are only

and V c by Yld89-V. Both for V D and rD , the predictions are agree

with the experimental results very well. In FIGURE 4, the flow stress directionality V D and the r-value planar anisotropy rD predicted by the present model and the corresponding experimental results on a high r- value IF steel sheet [21]. The results on the biaxial yield locus are also shown in FIGURE 5. In these figures, calculated results by Hill48-r and Gotoh’s 4th-order yield function are also indicated. It is seen from this, the calculated yield locus by Gotoh’s model is too distorted, in contrast, the newly proposed Yoshida-St model simulates the locus reasonably well.

APPLICATION: SIMULATION OF HOLE EXPANSION Recently, Hashimoto et al. [17] pointed out that the numerical simulation of hole expansion is strongly influenced by a choice of yield functions, especially for the strain localization characterization. We performed FE simulation of hole expansion on 590HSS using the proposed model, as well as some other models such as Hill48-V,

813

Gotoh and Yld2000. The simulation results were compared with the experimental observations, and consequently, it was found that the present model can well simulate the strain localization characterization (details will be presented at the conference).

CONCLUDING REMARKS A 3D yield function of 6th-order polynomial type to describe the anisotropic behavior of steel sheets, YoshidaSt model, has been proposed. The advantages of this model are summarized as follows: 1.

2.

The model has a high flexibility of describing the planar r-value anisotropy, the flow stress directionality and the shape of yield locus. Specifically for the description of (V x , V y ) yield locus, it offers a freedom to give two plane-strain stress points, V a and V c , besides two uniaxial stresses V 0 , V 90 and the biaxial stress V b . A user-friendly scheme of material parameter identification has been presented. It includes an idea of using a simple yield function, such as Hill48 or Yld89, for the estimations of V a and V c as well as V D and rD . Consequently, for the parameter identification, we need only limited experimental data of r-values and yield stresses in three directions ( r0 , r45 , r90 and V 0 , V 45 , V 90 ) plus equi-biaxial yield stress ( V b ).

3. 4.

Once all the material parameters for plane stress (2D) yield function are determined, no additional parameters are needed for 3D version of the model. The validity of the model has been confirmed by comparing the numerical simulations of the planar r-value anisotropy, rD , and the flow stress directionality, V D , with the experimental data of several types of steel sheets (a high r-value IF steel and high strength steel sheets of 440-980MPa TS levels, totally more than twenty types of sheets). The yield loci calculated by the model well capture the shape of several types of steel sheets. Furthermore, FE simulation result of hole expansion showed that it is useful for accurate numerical simulation of sheet metal forming.

REFERENCES 1. F. Yoshida, T. Uemori, and K. Fujiwara, Int. J. of Plasticity 18, 2002, pp. 633-659. 2. F. Yoshida and T. Uemori, Int. J. of Plasticity 18, 2002, pp. 661-686. 3. F. Yoshida and T. Uemori, Int. J. of Mechanical Sciences 45, 2003, pp. 1687-1702. 4. P. A. Eggertsen and K. Mattiasson, Int. J. of Mechanical Sciences 52, 2010, pp. 804-812. 5. A. Ghaei and D. E. Green, Computational Material Science, 48, 2010, pp. 195-205. 6. R. Hill, Proc. Royal Soc. London A 193, 1948, pp.281-297 7. R. Hill, J. Mech. Phys. Solids 38, 1990, pp. 405-417. 8. R. Hill, J. Mech. Phys. Solids 42, 1994, pp. 1803-1816. 9. M. Gotoh, Int. J. Mech. Sci. 19, 1977, pp.505-512. 10. F. Barlat and J. Lian, Int. J. Plasticity 5, 1989, pp. 51-66. 11. F. Barlat, J. C. Brem, J. W. Yoon, K. Chung, R. E. Dick, D. J. Lege, F. Pourboghrat, S.-H.. Choi and E. Chu, Int. J. Plasticity 19, 2003, pp.1297-1319. 12. F. Barlat, J. W. Yoon and O. Cazacu, Int. J. Plasticity 23, 2007, pp.876-896. 13. D. Banabic, H. Aretz, D. S. Comsa and L. Paraianu, Int. J. Plasticity 21, 2005, pp.493-512. 14. H. Vegter and A. H. van den Boogaard, Int. J. Plasticity 22(2006), pp.557-580. 15. W. Hu, Int. J. Plasticity 23 (2007), pp.620-639. 16. S. Soare, Int. J. Plasticity 24(2008), pp.915-944. 17. K. Hashimoto, T. Kuwabara, E. Iizuka and J. W. Yoon, Tetsu-to-Hagane 96, 2010, pp.27-33. 18. T. Kuwabara, A. Van Bael and E. Iizuka, Acta Materialia 50, 2002, pp.3717-3729. 19. T. Kuwabara, S. Ikeda and K. Kuroda, J. Mat. Process. Technology 80-81, 1998, pp.517-523. 20. W. Hu, Int. J. Plasticity 21 (2005), pp.1771-1796. 21. K. Kitayama, T. Kobayashi, T. Uemori and F. Yoshida, Tetsu-to-Hagane 97, 2011, pp.221-229.

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Determination of the Forming Limit Diagram of an UltraThin Ferritic Stainless Steel Sheet Hyuk Jong Bonga, Frédéric Barlata, and Myoung-Gyu Leea a

Materials Mechanics Laboratory, Graduate Institute of Ferrous Technology, Pohang University of Science and Technology, Pohang, Gyeongbuk, 790-784, Korea Abstract. The forming limit diagram (FLD) of a 98μm-thick ferritic stainless steel sheet sample was determined using a modified Marciniak test and the conventional Nakazima test (ASTM E2218-2). Wrinkling and undesired fracture were not observed in the modified Marciniak test. However, for some strain paths, wrinkling of the sheet specimens and undesired fracture at the upper die radius and the draw bead locations occurred in the Nakazima test. In spite of the problems observed with the Nakazima test, similar FLDs were obtained with the two methods. Keywords: Forming limit diagram, modified Marciniak test, Nakazima test, ultra-thin ferritic stainless steel sheets PACS: 60.20.F-

INTRODUCTION During recent decades, there has been a world-wide surge of efforts to replace internal combustion engines with more fuel-efficient and environmentally friendly technologies. Particularly, fuel cell technology has received great interest because of its characteristic high energy conversion efficiency and zero pollutant emissions. In the development of commercial fuel cells using metallic bipolar plates, a key obstacle has been the mass of the bipolar plates, which represents roughly 60% of the total mass cells1. Therefore, it is necessary to reduce the mass of the bipolar plates by using metallic sheet thinner than 0.1 mm. While reducing mass, fuel cell efficiency must be maintained and requires forming deep micro-channels in the bipolar plate. However, it has been an engineering challenge to form ultra-thin (<0.1 mm) steel sheet for this application. The problem is compounded because there are no standard practices to characterize formability of ultra-thin metals and corresponding limited formability information, such as forming limit diagrams (FLDs), are available. The FLD describes the formability of a material in strain space with respect to necking. The Nakazima test and modified Marciniak test are two common methods used to determine the FLD of a material. The Nakazima test2 applies to sheet thicknesses ranging from 0.5 mm to 3.3 mm, and no standardized methods have been developed for sheet materials with thickness below 0.5 mm. Despite the lack of standard methods, several researchers have attempted to determine the FLD of thinner sheet materials. Some investigations were carried out using a hydraulic bulge test3-4. However, this test method is unable to define the left side (drawing region) of the FLD, and furthermore, it does not provide information for the plane strain state. Therefore, some researchers conducted uniaxial tension as a complement to the bulge tests and to determine the left side of FLD5-6. However, a significant issue arises from the fact that the limit strains on the left side of the FLD are determined by an in-plane technique, whereas those on the right side are determined by an out-of-plane technique. It has been sufficiently demonstrated experimentally that the limit strains determined by in-plane tests are always slightly lower than those obtained with out-of-plane tests7-8. Thus, it is important to develop a unified test method that can be applied to the entire FLD and settles all of the above issues. The Marciniak test was first proposed by Marciniak et al.9. When they conducted the test for the first time, the driving blank did not have a hole. In 1973, this test was markedly improved 10 with the introduction of a circular hole in the driving blank to allow free expansion of the specimen blank and the use of two elliptical punches with different aspect ratios to enable different strain paths. The modified Marciniak test may be the best test method to overcome the limitations as previously described and offers the following advantages: (1) the FLD is determined

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 815-820 (2011); doi: 10.1063/1.3623690 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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using a single tooling geometry and equipment; (2) the entire region (drawing+stretching) of the FLD can be determined; and (3) the test method is straightforward and reproducible. In the last decades, several researchers used the Marciniak test with significant modifications7-8, 11-16. Tadros and Mellor17 conducted Marciniak tests to determine the right hand side of the FLD for steel, aluminum, and brass materials, and also to investigate the influence of the surface roughness development on the FLD. Kleemola and Kumpulainen5, 18 determined the entire range of the FLD using sheet strips of varying widths to obtain different strain paths as in the Nakazima test. Raghavan8 developed a systematic way of selecting the blank dimensions from blank geometries classified into four groups described in the next section. In this study, the FLD of a POSCO-proprietary 98 μm thick ferritic stainless steel, developed for the bipolar separator application in the proton exchange membrane (PEM) fuel cell, was determined by the modified Marciniak and Nakazima tests.

EXPERIMENTAL PROCEDURES Materials The tensile properties of the steel were determined by uniaxial tensile test in accordance with ASTM E8. Uniaxial specimens were tested at orientations 0°, 45°, and 90° relative to the rolling direction (RD). Table 1 summarizes the mechanical properties of the ultra-thin steel sheet. Orientation relative to RD 0° 45° 90°

TABLE 1. Tensile properties of the ultra-thin ferritic stainless steel. Yield Ultimate Uniform Total Work Stress Tensile Elongation Elongation Hardening (%) (%) (MPa) Strength Exponent (MPa) 455 577 19.4 26.3 0.213 464 580 19.7 28.2 0.223 465 572 19.8 32.2 0.226

r-value

1.79 1.82 2.51

Modified Marciniak Test Figure 1 shows a schematic of the modified Marciniak test used in this study. Several key characteristics of the modified Marciniak test can be observed. First, a flat-bottomed cylindrical punch is used instead of the hemispherical punch used in the Nakazima test (Fig. 3). Second, a driving blank with a circular hole at its center is placed between punch and blank. The driving blank must have greater formability than the specimen blank and it deforms together with the specimen blank during the test. The driving blank prevents circumferential fracture of the specimen blank along the punch shoulder during the test. In addition, the hole in the center of the driving blank assures free expansion of the specimen blank which leads to strain concentration and ultimate fracture in the center region of the specimen blank not in contact with the driving blank. One additional advantages of using a driving blank with thin sheet in the Marciniak test is that, it is easier to apply a uniform blank holding force (BHF) along the draw bead line because of the increased thickness provided by the driving blank. Moreover, the bending effect and the friction between specimen blank and tooling are eliminated because the flat surface in the central area of the specimen blank is maintained during the test. Finally, the central flat surface also facilitates strain measurements.

FIGURE 1. Tool geometry for the modified Marciniak test (Dimensions in millimeters).

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FIGURE 2. Schematic geometry of the specimen blanks and driving blanks used in the modified Marciniak test.

Type 1 1 2 2 3 3 4

TABLE 2. Dimensions of specimen and driving blanks. W R D Strain Ratio (Minor/Major Strain) 20 90 N/A Negative 45 77.5 N/A Negative 100 N/A 44 Negative 120 N/A 44 Negative 125 37.5 44 ~Plane strain 135 32.5 44 Positive 200 N/A 33 ~Balanced Biaxial

The dimensions of the specimen and the driving blanks are shown in Fig. 2 and Table 2. These dimensions were taken from the work of Raghavan8. The driving blank was made from an uncoated cold-rolled EDDQ steel with a thickness of 1.2 mm. The specimens for this test fall into 4 types. Type 1 was used for the negative minor strain region and no driving blank was used. In this case, the strain localization was similar to that in an uniaxial tensile test. Type 2 blanks were used for the near-plane-strain tension mode. Type 3 and Type 4 specimens were used to determine the limit strains in the biaxial strain and balanced biaxial modes, respectively. All specimens and driving blanks were oriented such that the major strain was parallel to the transverse sheet direction (TD) in consideration of the direction of the ridging phenomenon that occurred in the ultra-thin sheet steel. Friction conditions at the punch-driving blank and the driving blank-specimen blank interfaces are keys to the success of this test. The friction between the punch and the driving blank must be minimal, whereas that between the driving blank and the specimen blank should be high. To meet these objectives, Teflon film and tallow (an animal fat-based lubricant) were used as lubricants between the punch and the driving blank. In contrast, no lubricant was applied between the driving blank and the specimen blank. For each strain path, at least three tests were conducted. Tests were conducted at room temperature with a punch speed of 15 mm/min and a holding force of 150~200 kN. Square grids (2mm u 2mm) were silk printed onto the specimen blanks to eliminate any surface damage of the ultrathin sheet material.

Nakazima Test In addition to the modified Marciniak test, the Nakazima test was conducted to find which of the two methods are best suited to determine the FLDs of the ultra-thin sheet material. Figure 3 shows schematic of the Nakazima test apparatus. In this test method a hemispherical punch is used instead of the flat-bottom cylindrical punch used in the modified Marciniak test. The Nakazima tests were carried out with the same conditions to the modified Marciniak test with respect to punch speed, grid size, grid shape, and geometry of specimens. Consistent test conditions are essential for fair comparison of the two methodologies. A driving blank is not used in the Nakazima test and fracture initiated at the die radius and draw bead regions for some test conditions. Figure 4 shows some examples of this type of fracture. This experimental difficulty is the major problem of the Nakazima test when it is applied to ultra-thin sheets. In addition, wrinkling occurred in several of the specimen geometries. Wrinkling of the specimen may cause errors in the limit strain measurements and its effect on the final FLD requires investigation.

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FIGURE 3. Tool geometry for the Nakazima test.

FIGURE 4. Undesired fractures in the Nakazima test.

RESULTS AND DISCUSIONS Figure 5 shows tested specimens from the modified Marciniak and the Nakazima tests. Wrinkling of the specimen did not occur in the modified Marciniak test but it was observed in the Nakazima test. Figure 6 shows the FLDs determined by the modified Marciniak and the Nakazima tests. Despite wrinkling in the Nakazima test, the FLDs determined by both methods are similar. In addition, the trends of both FLDs are similar. Therefore, it is likely that wrinkling does not affect the final level and shape of the FLD, and both test methods can be applied for the FLD determination of ultra-thin sheet. However, it is much easier to prevent undesired fracture at the draw bead with the modified Marciniak test because of the buffer action of the driving blank. This advantage, together with the elimination of the bending and friction effects between the specimen blank and tooling, indicates that the modified Marciniak test is a more robust method to determine the FLD of ultra-thin sheet (<0.5 mm thick) materials.

FIGURE 5. Fractured samples from (a) the modified Marciniak test and (b) the Nakazima test.

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FIGURE 6. FLDs of the ultra-thin ferritic stainless steel sheet determined by (a) the modified Marciniak test and (b) the Nakazima test.

FIGURE 7. Comparison of forming limit curves (FLCs) of the ultra-thin sheet by the modified Marciniak and Nakazima tests.

Figure 7 shows a direct comparison of FLCs of the ultra thin ferritic stainless steel determined by the modified Marciniak and Nakazima tests. Generally, the forming limit strains determined by the Nakazima test are 5~10% higher than those determined by the modified Marciniak test7-8, 19-20. There have been several explanations for this observation, including the influence of the sheet curvature 19, the bending effect 19, and the delay of strain localization by friction7-8, 19. However, even if the limit strains determined by the Nakazima test are slightly higher than those determined by the modified Marciniak test, the difference is not as large as in previously published results. A possible source for these differences can be attributed to the strain measurement methods on curved surfaces in the Nakazima test in which the strain of the convex surface could be higher than that at other positions such as the mid-plane. In previous studies the sheet thickness was greater than 0.5 mm and the FLD was more likely to be affected by the curvature and bending effects. However, the sheet material investigated in this study is ultra-thin, therefore bending and curvature effects are expected to be negligible.

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SUMMARY The forming limit diagram of an ultra-thin ferritic stainless steel sheet sample was determined by the Nakazima and modified Marciniak test methods. Both of test methods were able to cover the entire strain rage of the FLD. However, the ultra-thin sheet specimens produced wrinkles or undesired fractures at the upper die radius and bead locations in the Nakazima test. Wrinkling or undesired fracture did not occur in the modified Marciniak test which turned out to be a robust technique. Almost identical FLDs were produced by each test methods. This result indicates that wrinkling in the Nakazima test does not significantly affect the final FLD results. Overall, the modified Marciniak test is a better method to determine the FLD of ultra-thin sheet materials because of the experimental difficulties associated with the Nakazima test. Moreover, the modified Marciniak test eliminates most of the friction, curvature and bending effects.

ACKNOWLEDGMENTS This research was supported by POSCO and by the World Class University (WCU) program through the National Research Foundation (Grant #R32-10147).

REFERENCES 1. Y. Liu and L. Hua, Journal of Power Sources 195 (11), 3529-3535 (2010). 2. K. Nakazima, T. Kikuma and K. Hasuka, Yawata Technical Report 264, 141-154 (1968). 3. A. Diehl, S. Geissdörfer and U. Engel, Proceedings of the 2nd ICOMM, Clemson University: Proceedings of the 2nd International Conference on Micromanufacturing, 156-161 (2007). 4. R. Mahmudi, Journal of Materials Processing Tech. 37 (1-4), 203-216 (1993). 5. H. J. Kleemola and J. O. Kumpulainen, Journal of Mechanical Working Technology 3 (3-4), 289-302 (1980). 6. H. B. Campos, M. C. Butuc, J. J. Grácio, J. E. Rocha and J. M. F. Duarte, Journal of Materials Processing Technology 179 (13), 56-60 (2006). 7. M. M. Moshksar and S. Mansorzadeh, Journal of Materials Processing Technology 141 (1), 138-142 (2003). 8. K. S. Raghavan, Metallurgical and Materials Transactions A 26 (8), 2075-2084 (1995). 9. Z. Marciniak and K. Kuczyński, International Journal of Mechanical Sciences 9 (9) (1967). 10. Z. Marciniak, K. Kuczyński and T. Pokora, International Journal of Mechanical Sciences 15 (10), 789-800 (1973). 11. T. Foecke, S. W. Banovic and R. J. Fields, JOM 53 (2), 27-30 (2001). 12. F. Stachowicz, Journal of Mechanical Working Technology 19 (3), 305-317 (1989). 13. K. Yamaguchi, N. Takakura and S. Imatani, Journal of Materials Processing Tech. 48 (1-4), 27-34 (1995). 14. T. Pepelnjak and B. Barisic, Journal of Strain Analysis for Engineering Design 44 (6), 459-472 (2009). 15. C. Guangnan, S. Huan, H. Shiguang and B. Baudelet, Materials Science and Engineering A 128 (1), 33-38 (1990). 16. G. Palumbo, D. Sorgente and L. Tricarico, International Journal of Machine Tools and Manufacture 48 (14), 1535-1545 (2008). 17. A. K. Tadros and P. B. Mellor, International Journal of Mechanical Sciences 20 (2), 121-133 (1978). 18. H. J. Kleemola and J. O. Kumpulainen, Journal of Mechanical Working Technology 3 (3-4), 303-311 (1980). 19. A. R. Ragab and B. Baudelet, Journal of Mechanical Working Technology 6 (4), 267-276 (1982). 20. E. Hsu, J. E. Carsley and R. Verma, Journal of Materials Engineering and Performance 17 (3), 288-296 (2008).

820

Simulation Analysis on Backward Spinning of Extra thinwalled Shielding Can with High Ratio of Diameter/Thickness Xinhe Lia, Xiahui Hea*, Xinquan Yanga a

School of Electrical and Mechanical Engineering, Central South University, Changsha ,410083,China

Abstract. A new FEM simplified elastoplastic mechanical model is constructed for the backward spinning process of the extra thin-walled shielding can with high ratio of diameter/thickness. The feed mode of the roller adopts the pulsing schedule, which is closer to the actual working condition. Additionally, this new FEM simplified model is significant saving in computing time. In this paper, the metal plastic flow field and the distributions of stress and strain of the deformation zone are studied by the large-scale FEM analysis software MSC.Marc. The causes of the defects of the workpieces are interpreted and the dividing point of the flowing direction of the metal is obtained, accordingly. And the theoretical guidance for the actual processing of the extra thin-walled shield can with high ratio of diameter/thickness is provided. The C-276 shielding can with 0.42mm thickness and 561.5mm diameter is successfully trial-produced. It proves that above mentioned analysis is reasonable. Keywords: backward spinning, elastoplastic FEM, simulation analysis, shielding can PACS: 81.20.Hy, 83.50.UV

INTRODUCTION Shielding can is one of the important parts of caned motor pump in the Third-generation nuclear power plants AP1000. Guan Rui [1] analyzed the process of the shielding can. Since the shielding can works in the high temperature, high pressure and corrosive media, it is necessary to find a suitable plastic processing method to manufacture the cylindrical workpieces with high-precision, extra-thin-wall and high ratio of diameter/thickness that is more than 1000[2]- [4]. Spinning process is a kind of successive partial plastic deformation of point-by-point processing technology. It has unparalleled advantages in machining high-precision thin-walled cylindrical parts. Therefore, spinning plays an important role in the aviation, aerospace, weapons, vehicles and other metal precision plastic processing technology fields [5]. In recent years, some scholars try to use the finite element method to analyze power spinning. The issue on spinning is complex, especially for large parts, so it is required for simplification in the finite element simulation analysis. Li Yuqiang [6] studied Multi-pass deep drawing spinning by means of a four-node FEM model. Mori, KenIchiro [7] proposed an axisymmetric model to simulate the shear spinning, saving computing time largely. This paper built a new FEM simplified model for backward spinning the tube with extra-thin-wall ,high ratio of diameter/thickness by the large scale software MSC.Marc, studied the metal plastic flow field in the deformation zone, obtained the distribution of stress and strain, provided the theoretical guidance for processing the tube with extra-thin-wall ,high ratio diameter/thickness in practice.

FINITE ELEMENT MODEL The Power Spinning Physical Model of the Tube The backward spinning of the tube is shown in Fig.1, the cylindrical blank is placed outside the mandrel. Since the teeth fixed the blank in one end, in the chuck there is no relative movement between the mandrel and the blank. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 823-830 (2011); doi: 10.1063/1.3623691 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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This end of blank is fixed. The other end can be freely stretched along the axis. In spinning, a plastic deformation is produced in the small contact zone between the blank and roller as result of the spinning force. Mei Ying [8] thinked that the resistance is very large in circumferential direction, the extruded metal flows along the axis where the resistance is least. The material flowing direction and the feed direction of the roller are opposite in backward spinning. The working stroke of the roller is equal to the length of the blank.

FIGURE 1. Power backward spinning.

A Simplified Mechanical Model As result of the complex condition and the restriction of the computing power of the computers, it is difficult to analyze the large parts in FEM. The process of backward spinning is often simplified. Taking the characteristics of axial symmetry of the tube into consideration, an axial symmetry model is built by taking out a cross-section from the axis of the workpiece, shown in Fig.2. In the past, the roller feeds along the axis in linear way in simplified model. However, those results prove that the reducing rate and the stress are not entirely consistent with actual situation. The process of roller acting on blank is shown in Fig.3. Therefore, we present a new way of the roller feeding. The feed mode of the roller adopts the pulsing schedule, which is closer to the actual working condition. The roller press the metal down, then lift up and move forward for a feed rate f. The path of the roller is shown in Fig.4.

Elastoplastic Model In the spinning, the unrecoverable plastic deformation and some of elastic deformation occurred in the blank. According to The Prandt l-Reuss elastic-plastic theory stated by Shuisheng Xie [9], the total strain increment is composed of the elastic strain increment

 ij     e ij

 ije

and plastic strain increment

 ijp

 ij

, namely:

p ij

According to stress-strain relationship, by use of V • Mises flowing rule and consistency condition, elastic-plastic material constitutive relation can be got, namely: ep  ij  Dijkl  kl

Where, ep ijkl

D

 ij

is the stress increment.

p ep  D  Dijkl Dijkl e ijkl

De

Dp

, is the stiffness matrix. ijkl 、 ijkl are elastic and plastic matrix tensor, which have a relationship with the material elastic modulus, plastic modulus, shear modulus, Poisson’s ratio, yield stress and other parameters . In this study, the material of the blank is Hastelloy-C276. Its material properties and geometric parameters are shown in Table 1. The true stress-strain curve of Hastelloy-C276 is shown in Fig.5.

2

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FIGURE 2. Simplified mechanical model.

FIGURE 4. The path of the roller.

FIGURE 3. The process of roller on section.

Table.1 Properities of the material Density

Modulus of elasticity

Poisson ratio

ρ/(kg/m3) 8890

Initial yield

Tangential modulus

limit

E/Mpa

μ

σ/Mpa

G/Mpa

205000

0.307

343.5

79000

FIGURE 5. True stress-strain curve.

825

3

Meshing Due to the high hardness, the deformation of the mandrel and roller is very small. To analyze conveniently, the mandrel and roller are set as the rigid bodies that are not meshed. According the real processing model and the element description provided by MSC.Marc, the blank is meshed in a suitable mesh density . The element type with four-node quadrilateral that can show the dynamic analysis is selected. Because of the deformation of the blank is very large in spinning, it is easy for element to produce a greater distortion, which make the calculation hard to continue. Therefore, the adaptive mesh is adopted in calculation. The deforming state of tube spinning is basically steady. In order to reduce the time of simulation, we can simulate the process for only one part of the blank, shown in the Fig.6

Contact In spinning, three parts of the blank are constrained: the fixed end of the blank, the inner surface and the area of blank contact with the roller. The Coulomb friction model is applied to the contact conditions. Both the friction coefficients of the roller and blank, the mandrel and blank are set to 0.05.

FIGURE 6. Axisymmetric meshing model

Boundary Condition In practice, the blank rotates with the mandrel, the roller feeds along the axis. After the axisymmetric model built, the constraints are imposed as follows:1.The mandrel is fixed completely;2.The end of the blank away from the initial position is fixed;3.The roller has a pulsing feed movement along axis as shown in Fig.4.

Process Parameter The process parameters are very important for the tube spinning. The reasonable parameters are the important conditions that ensure the success of spinning. The main parameters in spinning are forming angle of the roller, radius of the roller corner, spindle speed, reduction rate, feed rate, and so on, shown in Table 2.

Parameter External diameter of blank/mm Thickness of blank/mm Length of blank/mm Diameter of mandrel /mm Diameter of roller/mm Forming angle of roller/ º Fillet radius of roller/mm Spindle speed/r﹒min-1 Reducing rate/ % Feed rate/mm﹒r-1

4

Table.2 Process parameter in backward spinning simulation Sign D t L d D0 α r n ψ f

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Value 565 0.8 1000 563.2 160 25 1.2 300 30 1.5

ANALYSIS ON RESULTS Analysis on Plastic Flow Field in The Deformation Zone Fig.7 is the nodal displacement vector in the deformation zone at some time, which is based on the calculating result of FEM. We can see that the blank flowing field has a significant partition. As the Fig.7 depicted, the hidden line is the dividing line. The material on the dividing line is only extruded. The material in front of the dividing line is of the trend flowing forward. Meanwhile, the material behind the dividing line has the trend flowing backward. The material in the already forming zone has been basically stable flow. The export speed of the material and the feed rate are related to the reduction rate. In order to ensure material flowing smoothly, good lubrication should be kept between the mandrel and the blank, the roller and the blank.

FIGURE 7. The nodal displacement vector diagram of the backward spinning mechanical model

The Distribution of The Ttress and Strain in Deformation Zone In the spinning, the diameter expansion, the diameter reduction, and the uplift are related to the distribution of the stress and strain in deformation zone. So in order to explain the mechanism of the backward spinning of extra- thinwall tube with high diameter/thickness, the distribution of the stress and strain in deformation zone is required to research. We extract the data of the stable stage to analyze the distibution of stress and strain. The axial distribution of three-dimensional stress Figure.8.a, Fig.8.b, and Fig.8.c describe the axial distribution of the stress of the material in surface, middle and bottom (the bottom point of roller in horizontal axis is set to zero. The roller feeding direction is set to positive direction). The figure shows that the metal in deformation zone is in the tri-axial compressive stress state, and reaches the maximum. In the surface, the ratio of the radial compressive stress to the axial compressive stress to the circumferential compressive stress isσ r:σ z:σ θ ≈ 1 : 0.76 : 0.85. In the middle, it is about 1 : 0.47 : 0.69. At the bottom, it is 1 : 0.33 : 0.56. In Fig.8.a, we can see that there is a intersection point in front of the roller. The point is about 1.7mm away from the roller. At this point, the stresses in three directions are equal. According the metal plastic forming theory, we can derive that there is only spherical tensor of stress here without deviatoric tensor of stress. This means that there is no shape change and plastic deformation. So here is the cut-off point of the direction of material flowing. By analyzing the Fig.8.b, we can know that the cut-off point appears in front of the roller. This point is 2.5mm away from the roller. The cut-off points distribute with a linear downward trend.

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5

(a) The distribution of the surface stress (b) The distribution of the middle stress (c) The distribution of the bottom stress FIGURE 8. The axial distribution of the stress in deformation zone.

The radial distribution of the tri-axial stress Figure.9.a, Fig.9.b, and Fig.9.c respectively describe the radial distribution of tri-axial stress in deformation zone, front the roller, and behind the roller (The zero in the horizontal axis is the inner wall of the blank). Through the Fig.9.a, we can know that the metal that is 2mm in front of the roller is still in tri-axial compressive stress state. But the stress is significantly reduced relative to the area exerted by the roller. What’s more, the axial compressive stress is the biggest of the three dimensional stress in the surface, and the radial compressive stress is the smallest. However, the radial compressive is the biggest of the three dimensional stress, regardless in the middle or at the bottom, and the axial stress is the smallest. This is because the roller feed toward the fixed end along the axis, causing the metal in front of the roller extruded. Then the upper material has a trend accumulating forward because of the obstacles behind the material. The middle and lower material has a trend flowing backward because the end behind the deformation zone is free. It is also consistent with the law of least resistance that metal flowing follows. Figure.9.b shows that the compressive stress is the smallest in the inner wall where the workpiece touches the mandrel in radial direction. The compressive stress is the largest in the outer wall where the workpiece touches the roller in radial direction. In Fig.9.c, we can see that the tri-stress reduce in turn from the bottom to the surface, axial and radial stress tends to zero. This is because the forming end is free in axial direction in backward spinning.

(a) (b) (c) (a)The distribution of the stress of 2mm in front of the roller (b)The distribution of the stress under the roller (c)The distribution of the stress of 4mm behind the roller FIGURE 9. The radial distribution of the stress in deformation zone

The radial distribution of the tri-axial strain Figure.10.a, Fig.10.b and Fig.10.c respectively describes the axial distributions of the tri-axial strain in surface, in middle, and at the bottom (the bottom point of roller in horizontal axis is set to zero. The roller feeding direction is set to positive direction).This figure shows that the metal deformation at the roller is larger than at the mandrel. The deformation closed to the mandrel is only about half the surface. The deformation in front of the roller is small as well as the circumferential deformation. So it can be seen that the flow of the metal is very small in circumference and in front of the roller. The metal contacting with the roller has a greater radial compressive strain, occurs circumferential and axial elongation under the action of the roller. Behind the roller, the tri-axial strain is basically stable. The metal has radial compression, axial elongation and a bit extension in circumference. The diameter

6

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expanding occurs when the total elongation is greater than the total compression in circumference, otherwise necking. The Fig.10.a shows that there is a small axial compression in front of the roller, a little radial and circumferential elongation, which is the cause of the metal accumulation.

(a) The axial distribution of the surface strain(b)The axial distribution of the middle strain (c)The distribution of the bottom strain FIGURE 10. The axial distribution of the strain in deformation zone

. Production With the analysis above, the theoretical guidance for the actual processing of the extra thin-walled shield can with high ratio of diameter/thickness is provided. We adjust the process parameter reasonably. The C-276 shielding can with 0.42mm thickness and 561.5mm diameter is successfully trial-produced. It is shown in Fig.11.It proves that above mentioned analysis is reasonable.

FIGURE 11. The product of shielding can

CONCLUSION A new FEM model is constructed. Through elasto-plastic FEM analysis, obtained the distribution of the plastic flowing field of the extra-thin-wall tube. Its trend is consistent with the backward spinning process in practice. The metal contacting with the roller is in the tri-axial compressive stress state, which reaches the maximum. In the radial, the compression close to the inner wall is the smallest, while it is the largest close to the outer wall. The tri-axial stress is significantly reduced in front of the roller, and the axial stress is the largest. The tri-axial stress becomes zero gradually because the end is free behind the roller. There is a dividing line in deformation zone. The stress in the dividing line is spheric stress (the third principal stress is equivalent). The deformation closed to the roller is larger than the deformation closed to the mandrel. The deformation closed to the mandrel is only about half the surface. The deformation in front of the roller and in circumference is very small. It means that the flow toward the circumference and forward the roller is very small. Then the uplift and diameter expanding is not very large.

829

7

In order to make the material flow more smoothly, we can consider that the appropriate field is introduced to improve the distribution of the spheric strain and accelerate the flow of material along the axis. Therefore, the trend of the material accumulation is restrained.

ACKNOWLEDGEMENT The authors would like to thank the Major State Basic Research Development Program for its financial support.

REFERENCES 1. Guan Rui, Gao yong-jun, The brief analysis of the main pump of the AP1000 reactor, Equipment Manufacturing. 2008,1(1), pp.49-53. 2. ASTM, 2000 Annual Book of ASTM Standards, V01.01,2000,USA. 3. T.Toyooka,Y.Kawabata. Development of Vltra Fine Grain Steel Tube with High Strength and Excellent Fonnability,

Proceedings of International Joint Symposium on Innorative Tubes and Tubular Products for Antomotire and Electric Application , Osaka ,2000. 4. Y.Koyama,T.Toyooka.High Strength and High Elongation Tubular Products ‘History Steel Tube’ With Good Bendability, Kawasaki Steel Technical Report, (43) 2000, pp.55-57. 5. Zhao Yunhao, Li Yanli. The technology and Application of the Spinning, Beijing: Mechanical Industry Press, 2007. 6. LI Yu-qiang,CUI Zhen-shan,RUAN Xue-yu, Effects of parameters on the distributions of stress and strain of multiprocess draw-spinning, Journal of Placity Engeering.2004,pp.17-20. 7. Mori Ken-Ichiro, Nonaka, Takayuki, Simplified three-dimensional finite element simulation of shear spinning process based on axisymmetric modeling, NAMRC 32, June 1, 2004. 8. Mei Ying, Rui-Qing Li, Zhang Chenai et al., The simulation of the tube’s power backward spinning and the analysis of the spinning force, 2007,pp.65-68. 9. Xie Shuisheng, Li lei. The simulation technique and application of the FEM of the metal forming, Beijing: Science Press,2008.

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Characterization of ironing in the hole-flanging process of an aluminum alloy A. Kacema, A. Krichena and P.Y. Manachb a

University of Sfax, National Engineering School of Sfax, LGPMM, BP 1173, 3038 Sfax, Tunisia b Université de Bretagne-Sud, LIMATB, Rue de Saint Maudé, 56321 Lorient, France

Abstract. Hole-flanging is a sheet metal forming process which is often used to produce a flange around holes. Many industrial applications require a longer flange that can be used for example to increase bearing surface or to increase the number of threads that will fit in a tapped hole. Ironing is usually used during such process to increase the flange height. Due to the thinning of the flange caused by edge stretching, some difficulties arise when trying to set a clearance value for which the process is performed with ironing. Therefore, the occurrence of ironing needs to be more clarified. It is also important to quantify the effect of ironing intensity in the flange geometry. In the present work, FE simulations and experiments are performed to investigate hole-flanging with and without ironing. For this purpose, hole-flanging is carried out by varying the clearance between the punch and the die while keeping constant the other parameters. The aim is to identify the occurrence of ironing by the determination of a limit between the hole-flanging with ironing and the hole-flanging without ironing. Attention is focused on studying the effect of the hole-flanging conditions on the punch load, the forming kinematics and the final shape. The experiments are carried out to check the FE results on an aluminum alloy. Keywords: hole-flanging, ironing, metal forming, finite elements.

1. INTRODUCTION Hole-flanging is a sheet metal forming process which is often applied to produce a flange around holes. In the conventional hole-flanging process, a blank-holding force is set to clamp firmly the sheet between the blank-holder and the die. Then, a punch forces the periphery of the hole into the die as shown in Fig. 1. Depending on the clearance between the punch and the die, hole-flanging can be performed with or without ironing.

FIGURE 1. Conventional hole-flanging process with ironing (right part) and without ironing (left part).

In order to have ironing during the hole-flanging, the process is performed by setting a low value of the clearance between the punch and the die (C). In this case, the metal is squeezed between the punch and the die as shown in The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 831-838 (2011); doi: 10.1063/1.3623692 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

831

Fig.1. In practice, to control the hole-flanging conditions, the clearance-thickness ratio (Rc) which is defined as the ratio of the clearance C to the workpiece thickness (t) is considered. Some authors have performed ironing in the hole flanging process to succeed the forming of flanges for two-ply sheet metal [1] and thick sheet metal [2]. Recently, Kacem et al. [3] have performed ironing in the hole flanging process to obtain long flanges. They have identified the occurrence of ironing by using experiments and finite element (FE) model based on the von Mises yield criterion. Others techniques were proposed to produce a long flange such as the cold extrusion [4,5], the Maslennikov’s technique [6,7] and the multistage incremental forming [8]. In the present work, the conventional hole-flanging process with and without ironing was used. Hole-flanging conditions were carried out by varying the clearance between the punch and the die (C) and keeping the other parameters unchanged. Efforts were focused on the characterization of ironing by studying the effect of the clearance thickness ratio (Rc) on the flange shape and punch load. FE model based on von Mises yield criterion was used. To evaluate the effect of the normal anisotropy of the sheet in the predicted results, FE model based on Hill’s 1948 yield criterion was also used. In parallel, experiments were carried out to check the FE results.

2. MATERIAL AND TECHNIQUES 2.1. Material behaviour A 2 mm thick commercial aluminium alloy sheet was considered for this work. For the FE analysis, the aluminium alloy was assumed to have an elastic plastic behaviour and to obey either the von Mises or Hill’s 1948 yield criterion with an isotropic hardening rule. The yield function is given by: f  J 2 ( )  R (1) J2 denotes the second stress invariant that is defined as follows in the Von Mises yield criterion:

J 2 ( ) 

3 S:S 2

(2)

and for Hill’s 1948 yield criterion:

J 2 ( ) 

3 S: H :S 2

(3)

Where S is the deviatoric part of the Cauchy stress tensor (  ) and H is the fourth order Hill tensor which contains

the six coefficients: F, G, H, L, M and N. It was assumed that the sheet is a transversely isotropic material. Therefore, F=G=

1 r 1

, H = 1 – G and N =

2r  1 r 1

Where r is the normal anisotropy. Owing to a lack of available data on the mechanical behaviour in the sheet thickness, L and M are kept equal to their value in case of isotropic behaviour (i.e. L = M = 1.5). R denotes the isotropic hardening function that is chosen with a saturation form of Voce type: R  R 0  Q (1  exp(b )) (4) Where R0, Q and b are the initial yield stress, the hardening saturation and the rate at which saturation is achieved, respectively. The evolution of the plastic strain rate is obtained by using the normality rule:

d P  d

f  d n 

(5)

Where n gives the direction of the plastic strain increment and dλ, called the plastic multiplier, determines the magnitude of the plastic strain increment. The material parameters of the implemented work hardening models (R0, Q and b) were identified with the experimental stress-strain curve obtained from a monotonic tensile test in the rolling direction (RD) as shown in Fig. 2-a. The tensile test was carried out on rectangular sample of dimensions 150 x 20 x 2 mm3 under displacement

832

control at a constant rate of 5 mm/min. The Cauchy stress was obtained by a load cell of a 50 kN capacity. The logarithmic strain was measured by a field measuring system by 3D image correlation (Aramis). The Young’s modulus (E=68 GPa) was determined from the experimental stress-strain curve. The poisson’s ratio was taken equal to 0.28. To determine the material anisotropy, tensile tests were also performed at 45° and 90° to the RD and the measurement of both transverse and longitudinal strains (see Fig.2-b) leads to a normal anisotropy coefficient 0.61. The values of the model parameters (R0 = 65 MPa, Q = 53 MPa and b = 750) were obtained by the best fit to the experimental stress-strain curve as shown in Fig. 2-a. b -0.03

120 Cauchy stress (MPa)

Plastic thickness strain (ε33p) -0.02 -0.01

0

0

100 Experiment 80

Model

-0.01 D 0°/R

60

D 45°/R 90

°/

RD

40

r0 = 0.45

20

r45 = 0.55

-0.02

r90 = 0.89

Plastic transverse strain (ε22p)

a

0 0

0.01

0.02 0.03 Logarithmic strain

0.04

0.05

-0.03

FIGURE 2. (a) Experimental and numerical stress-strain curve in the rolling direction. (b) Calculation of the plastic anisotropy coefficient by measuring the slop of ε22p = f(ε33p).

2.2. Hole-flanging parameters and Experiments The workpiece is an aluminium washer. Conventional hole-flanging tools including die, truncated pilot-punch and blank-holder were used. The nomenclature and the values of the tool geometry parameters are illustrated in Fig. 3. A range of clearance-thickness ratio (Rc) values between 0.41 and 1.52 was studied, the other geometrical parameters being kept constant. Rc was controlled by keeping the radius of the punch (Rp) constant and changing the radius of the die (Rd).

FIGURE 3. Tool geometry parameters and nomenclature of the conventional hole-flanging process.

The experimental procedure is as follow. After centering the workpiece on the die, it was clamped between the die and the blank-holder in such a way to prevent its upward motion [9]. After that, the hole-flanging process was

833

carried out with a universal tension-compression testing machine of maximum load 50 kN. It was performed under displacement control at a constant low rate of the punch equal to 5 mm/min. The outputs of the displacement gauge and of the load cell were continuously recorded in a data acquisition system.

2.3. FE modelling A FE model was developed to perform numerical analysis on the hole-flanging process. The analyses were performed using the standard version of the software ABAQUS. As shown in Fig. 3, the geometry of the tools, the workpiece and the loading are axisymmetric. Moreover, as the metal sheet was assumed to be isotropic or transversely isotropic, the material properties are also axisymmetric. Therefore, a 2D axisymmetric models were adopted. The tools (die, punch and blank-holder) were modelled as rigid surfaces. The workpiece was modelled as a deformable body with an elastic-plastic material using the material properties specified in section (Material behaviour). A 3-nodes linear axisymmetric triangle element (CAX3) and 4-nodes bilinear axisymmetric quadrilateral element with selective reduced integration (CAX4R) were used. The choice of a reduced integration element was made to speed up the computation as this element uses only one integration point per element instead of four. An automatic mesh generator program was used to generate the initial FE mesh as shown in Fig. 4. Due to the large mesh distortions, remeshing techniques were used in order to compute an accurate solution. The adaptive remeshing process involves the iterative generation of multiple dissimilar meshes to determine a single, optimized mesh that is used throughout the analysis. The mesh size can be refined or coarsened according to an error indicator. The equivalent plastic strain error indicator was chosen to decide if the workpiece needs to be remeshed. A typical obtained FE mesh is shown in Fig. 4. Remeshing was applied for each case studied in this work. Coulomb’s friction law was used to solve the frictional effect of the tool-workpiece interfaces. To describe reasonable contact conditions between the tools and the workpiece, the friction coefficient was set to 0.2. FE simulations were performed in two steps. In the first step, the die and the blank-holder were rigidly clamped and the punch was constrained to move in the axial direction. In the second step, the blank-holder was removed to analyse the final shape after spring-back phenomenon.

FIGURE 4. FE model (a) initial mesh (b) mesh after 3 adaptive remeshing iterations.

3. RESULTS 3.1. Flange shape For all considered experimental conditions, the finished products are obtained free from necking and fracture. Some typical examples are shown in Fig. 5-a for different values of Rc. To illustrate the shape of the flange, a sectional view of each finished product is plotted in Fig. 5-b. Fig. 5-c and 5-d show the final deformed shapes obtained from FE simulations based on the von Mises and Hill’s 1948 yield criterion, respectively.

834

FIGURE 5. (a) finished products, (b) sectional view for different Rc values, (c) and (d) FE deformed shape

Both experimental and numerical results show that whatever the value of Rc, thinning of the flange occurs and a cylindrical boring was obtained at the inner side of the flange as shown in Fig. 5. For high value of Rc, the outer side of the flange was formed with a partial contact between the flange and the die. This leads to an outer curved shape and a non-uniform thickness distribution along the flange as shown in Fig. 5-b, c and d. For low value of Rc, a full contact between the flange and the die was established. The metal is squeezed between the punch and the die. Consequently, ironing occurs. This leads to a cylindrical shape with a uniform thickness distribution along the flange. In the next section, a critical value of Rc limiting the two hole-flanging conditions was determined numerically and experimentally.

3.2. Occurrence of ironing The squeezing of metal between the punch and the die makes the thickness at the flange extremity (te) (see Fig. 5-c) as the most suitable parameter to determine the critical value (Rcc) of Rc for which ironing starts. Fig. 6 shows the evolution of te with Rc obtained from the experiments and the models based on the von Mises yield criterion and Hill yield criterion. Workpiece thickness

2

Hole-flanging with ironing

Hole-flanging without ironing

te (mm)

1.5

1

Hill48 von Mises experiment

0.5 Rcc Hill

Rcc von Mises

0 0

0.2

0.4

0.6 0.8 1 1.2 Clearance-thickness ratio (Rc)

1.4

1.6

FIGURE 6. The thickness at the flange extremity (te) versus Rc.

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The critical value Rcc was determined by assuming that ironing occurs when te becomes equal to the clearance (C) and a full contact between the flange and the die is established. It reveals that the critical value Rcc predicted numerically by using the von Mises yield criterion is about 0.68. However, Rcc is about 0.63 for the model based on Hill yield criterion. The critical value can also be determined analytically as the geometrical configuration for Rcc is in agreement with the assumptions proposed by Asnafi [10] relating te and t by: 1 /(1 r )

 te  Ri    t  R d  t e / 2 

(6)

Taking into account that te = C and Rd = Rp + te in this case, and by using eq. (5), we obtain the following relationship:

 Ri R cc    R  0 .5 R cc  p

1 /(1 r )

  t 

(7)

Therefore, Rcc can be determined analytically for isotropy ( r =1) and transverse isotropy ( r =0.61) assumptions by resolving the following equation:

0.5 t R cc

2 r

1 r

 R p R cc

 Ri  0

(8)

The obtained values of Rcc, that are in agreement with that predicted by the FE models, are 0.67 and 0.61, respectively. For hole-flanging without ironing (i.e. Rc higher than Rcc), thinning leads to a constant value of te. It is about 1.35 mm (i.e. 32% t) and 1.25 mm (i.e. 37% t) for the model based on von Mises yield criterion and Hill yield criterion, respectively. The difference of the thinning between the two models was expected since the value of the normal anisotropy was lower than the unity leading to a lower resistance to thinning. In Fig. 6 the experimental values of te are superimposed. Comparison between experimental and numerical results shows that the model based on Hill yield criterion provides the closest agreement with the experimental results. However, the model based on von Mises yield criterion overestimates thinning by about 7 %.

3.3. Flange geometry From an industrial point of view, it is important to quantify the effect of ironing on the flange geometry. For that purpose, the characteristic dimensions of the flange are analysed as a function of hole-flanging conditions. In particular, the geometrical evolution of both the inner and the outer profile of the flange are investigated versus Rc. 3.3.1. Inner profile

The straight profile length (hs) and the curved profile length (hc) depicted in Fig. 5-c are the main characteristic dimensions of the inner profile. The evolution of hs and hc predicted by the model based on von Mises yield criterion with Rc are plotted in Fig. 7-a.

hS

6

b

6 Hill48 5

von Mises experiment

4 4

h (mm)

(hS) , (hc) (mm)

5

a

Rcc Hill Rcc von Mises

Rcc Hill Rcc von Mises

7

3 hC

2

3

Hill48

2

experiment

von Mises

1

1 Hole-flanging with ironing

0 0

0.2

0.4

Hole-flanging without ironing

0.6 0.8 1 1.2 Clearance-thickness ratio (Rc)

1.4

Hole-flanging with ironing

0 1.6

0

0.2

0.4

Hole-flanging without ironing

0.6 0.8 1 1.2 Clearance-thickness ratio (Rc)

1.4

1.6

FIGURE 7. (a) The straight profile length (hs) and the curved profile length (hc) versus Rc (b) The flange height (h) versus Rc.

836

It can be seen that for higher values of Rc, hs and hc are almost similar. When Rc decreases hs increases while hc decreases. hs increases more sharply for low values of Rc. Namely, hs becomes 13 times longer than hc for Rc equal to 0.41. For this value of Rc, hs is 3.4 times higher than the workpiece thickness that seems to be advantageous in the product design from an industrial point of view. The predicted values of hs and hc obtained by the model based on Hill yield criterion are superimposed in Fig. 7-a. It can be seen that both models gives very similar results, which means that these values are highly conditioned by the geometry of the tools rather than the material behaviour. In Fig. 7-a, the experimental values of hs and hc are superimposed. A good agreement between experimental and numerical values is observed. The average difference is about 6% and 8% of the numerical value for hs and hc, respectively. 3.3.2. Outer profile

The flange height (h) which is depicted in Fig. 5-c is the main characteristic dimension of the outer profile. The evolution of h with Rc obtained from the experiments and the model based on von Mises yield criterion and Hill yield criterion is plotted in Fig. 7-b. Compared to the characteristic dimensions of the inner profile, the transition between the two hole-flanging conditions is more pronounced for the flange height. When hole-flanging was performed without ironing, the portion of the workpiece subjected to the deflexion (see Fig. 3) decreases with Rc. As a consequence, a slight decrease of h is observed. In this case, the values predicted by the model based on Hill yield criterion are higher than that predicted by the model based on von Mises yield criterion. The difference is about 4 %. This is a consequence of the larger thinning marked by the model based on Hill’s 1948 yield criterion. The occurrence of ironing causes a sharp increase of h. In this case, the predicted values of h from both models are similar. Numerical results overestimate h by about 12% but, globally, a good agreement between experimental and numerical results is observed.

3.4. Punch load From an industrial point of view, it is also important to investigate the effect of ironing on the maximum punch load (Fmax). The evolution of Fmax with Rc obtained from the experiments and the model based on von Mises yield criterion and Hill yield criterion is plotted in Fig 8-a. 6

a

5

b

Hill48 von Mises

5 Punch load (kN)

experiment 4 3

Rcc Hill Rcc von Mises

Maximum punch load (kN)

6

2

Hill48 von Mises experiment

1 Hole-flanging with ironing

0 0

0.2

0.4

1.4

3

Rc=0.41

2

Rc=1.52

1

Hole-flanging without ironing

0.6 0.8 1 1.2 Clearance-thickness ratio (Rc)

4

0 1.6

0

2

4

6

8 10 Punch travel (mm)

12

14

16

FIGURE 8. (a) Evolution of the maximum punch load as a function of clearance-thickness ratio (b) Punch load versus punch travel for Rc equal to 1.52 and 0.41.

It can be observed that as Rc decreases, Fmax increases. Fmax increases more sharply when the hole-flanging is performed with ironing. The values predicted by the model based on Hill yield criterion are lower than that obtained by the model based on von Mises yield criterion. Comparison between experimental and numerical results shows a good agreement. However, this quality of agreement is less when the relationship between punch load and punch travel is analysed.

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The relationship between punch load and punch travel obtained from the experiments and the model based on von Mises yield criterion and Hill yield criterion is plotted in Fig. 8-b for high and low values of Rc corresponding to the hole-flanging without ironing and the hole-flanging with ironing, respectively. It was found that the change in Rc does not affect the punch load-punch travel relationship at the beginning of the process. After this stage, the punch load increases with decreasing Rc. After reaching the maximum punch load (Fmax) and when the process was performed with ironing, the punch load increases again to a secondary maximum characterizing the ironing process that occurs at this stage as shown in the work of Kacem et al. [3]. The predicted punch load by the model based on Hill yield criterion is lower than that predicted by the model based on von Mises yield criterion. The difference is more significant when the process was performed with ironing. The experimental punch load-punch travel relationships are superimposed in Fig. 8-b. Globally, a good agreement between experimental and numerical results is observed. However, these results can be better predicted by taking into account the transverse anisotropy and the damage of the sheet. This will be done in future works.

4. CONCLUSION FE simulations and experiments were carried out to characterize ironing in the hole-flanging process. It was found that thinning occurs usually in the hole-flanging process. Ironing starts from a critical value of the clearancethickness ratio that depends on the anisotropy of the sheet. When ironing occurs, a long flange with a straight profile is obtained. The comparison between FE simulations and experiments shows that the developed FE model can be an appropriate tool to predict the punch load and the final shape. However, this model can be improved by taking into account the transverse anisotropy and the damage of the sheet.

REFERENCES 1. T. Kumagai, H. Saiki, Y. Meng, Hole flanging with ironing of two-ply thick sheet metals 1999. Journal of Materials Processing Technology, 89-90, 1999, pp. 51– 57. 2. T. Kumagai and H. Saiki, Deformation analysis of hole-flanging with ironing of thick sheet metals. Metals and Materials 4, 1998, pp. 711 – 714. 3. A. Kacem, A. Krichen, P.Y. Manach, Occurrence and effect of ironing in the hole-flanging process. Submitted to Journal of Materials Processing Technology. 4. H.S. Lin, C.Y. Lee, C.H. Wu, Hole flanging with cold extrusion on sheet metals by FE simulation. International Journal of Machine Tools and Manufacture 47, 2007, pp. 168 – 174. 5. H.S. Lin, C.W. Tung, An investigation of cold extruding hollow flanged parts from sheet metals. International Journal of Machine Tools and Manufacture 47, 2007, pp. 2133 – 2139. 6. S. Thiruvarudchelvan and M. Tan, Investigations into collar drawing using urethane pads. Journal of Materials Processing Technology 191, 2007, pp. 87–91. 7. S. Thiruvarudchelvan, The potential role of flexible tools in metal forming. Journal of Materials Processing Technology 122, 2002, pp. 293 – 300. 8. Z. Cui, L. Gao, Studies on hole-flanging process using multistage incremental forming. CIRP Journal of Manufacturing Science and Technology 2, 2010, 124–128 . 9. A. Krichen, A. Kacem, M. Hbaieb, Blank-holding effect on the hole-flanging process of sheet aluminium alloy. Journal of Materials Processing Technology 211, 2011, pp. 619–626. 10. N. Asnafi, On stretch and shrink flanging of sheet aluminium by fluid forming, Journal of Materials Processing Technology 96, 1999, pp. 198 – 214.

838

Surface Defects in Sheet Metal Forming: a Simulative Laboratory Device and Comparison with FE Analysis Sandrine Thuilliera,b, Alban Le Porta,b and Pierre-Yves Manacha,b a

Université de Bretagne-Sud, LIMATB, rue de Saint-Maudé BP 92116 – 56321 Lorient Cedex, France b Université Européenne de Bretagne, Rennes, France

Abstract. Surface defects are small concave imperfections that can develop during forming on outer convex panels of automotive parts like doors. They occur during springback steps, after drawing in the vicinity of bending over a curved line and flanging/hemming in the vicinity of the upper corner of a door. They can alter significantly the final quality of the automobile and it is of primary importance to deal with them as early as possible in the design of the forming tools. The aim of this work is to reproduce at the laboratory scale such a defect, in the case of the flanging along a curved edge, made of two orthogonal straight part of length 50 mm and joint by a curved line. A dedicated device has been designed and steel samples were tested. Each sample was measured initially (after laser cutting) and after flanging, with a 3D measuring machine. 2D profiles were extracted and the curvature was calculated. Surface defects were defined between points where the curvature sign changed. Isovalues of surface defect depth could then be plotted, thus displaying also the spatial geometry on the part surface. An experimental database has been created on the influence of process parameters like the flanging height and the flanging radius. Numerical simulations have been performed with the finite element code Abaqus to predict the occurrence of such surface defects and to analyze stress and strain distribution within the defect area. Keywords: Forming, Springback, Surface Defects, Experiments, Numerical Simulation. PACS: 62.20.F-

INTRODUCTION Surface defects or deflections, or also cosmetic defects, occur on outer panels of automobiles after forming and springback steps. They are small amplitude depressions over a generally curved surface (typical radius of curvature for automotive opening parts are 1 m) and there is a change of the curvature sign compared to the main curvature of the panel. These defects are difficult to visualize on the naked sheet but may alter significantly the vehicle quality after painting. They therefore demand a dedicated inspection, in order to validate the tools and process parameters in the validation step. To spare time and tedious trials, these defects should be predicted numerically, in the early stages of the virtual mechanical design. Such surface defects have typical depth (measured along the sheet normal) of a few tenths of micrometers and an in-plane size of several tenths of millimeters. Therefore, their numerical prediction, with discretized geometries for the blank and the tools, should be validated in a first step. Fukumura et al. [1] analyzed the area near the door handle embossment, which is prone to defect called “teddy bear ears”. They extracted 2D profiles, according to a given direction, from a digitalization of the deformed part with a 3D measuring machine and calculated the curvature of these profiles. From two distinct peaks in the curvature distribution, they identified surface defects. No quantified evaluation was performed but a comparison with finite element prediction, obtained after a forming step within an explicit framework and an implicit step for springback. They concluded that finite element analysis can predict the occurrence of surface defect. Park et al. [2] worked on a similar geometry, and obtained experimental values and predicted ones, but also proposed a post-treatment from the curvature sign, in order to define a defect depth. They obtained depth values of 0.05 up to 0.1 mm, and numerical predictions tended to underestimate the defect depth. They used shell elements of maximum size 1 mm and an arc-length method to overcome instabilities such as wrinkles. Andersson [3] tested the influence of the yield criterion (Hill’s 1948 or Barlat’s 1991) and hardening, either isotropic or mixed, on the numerical prediction of defect localization and curvature, on similar The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 839-846 (2011); doi: 10.1063/1.3623693 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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geometries as in [1,2]. They found good correlation but curvature measurement does not lead to a direct equivalence in terms of depth and in-plane size. To enlarge the type of defects investigated, Le Port et al. [4,5] developed a device to flange samples of reduced dimensions, compared to automotive opening parts, in order to reproduce defects that develop near the upper corner of an automotive door, after forming, flanging and springback [6]. Fig. 1 shows the geometry of the sample, the outer radius of which is 55 mm. Flanging is performed along the edge composed of the two linear parts and the connecting radius. Specialized technicians from PSA Peugeot Citroën have analyzed such samples and marked the area highlighted in Fig. 1a as a defect of small amplitude. A digitalization of the outer surface was performed with a tridimensional measuring machine (Fig. 1b) and profiles along a given direction were extracted and post-treated in order to derive the curvature.

(a) (b) FIGURE 1. Visualization of a surface defect after flanging and springback (a) The outer radius of the sample is 55 mm. Two areas were highlighted by a specialized technician from industry as surface defects (b) Overall geometry of the defect nearest to the corner.

This article is dedicated to the numerical simulation of the corresponding forming process, i.e. flanging and springback steps, in order to compare experimental and numerical values of the defect dimensions. An experimental database of surface defects after flanging was obtained with a dedicated device [5], and the influence of some process and design parameters like the flange height, the flanging radius and the corner radius was investigated. A method was developed, base on curvature sign, to determine the defect dimensions. A dual phase steel DP500 was chosen and its mechanical behavior was characterized under different strain paths in order to determine the material parameters to be used as input of the numerical simulation [7]. After recalling these preliminary results, the following section describes the numerical simulation of the process with the finite element code Abaqus, in the case of a reference configuration. Both load evolution during flanging and defect dimensions were analyzed. Finally, the influence of the flange height, the flanging radius and the corner radius on the output was also investigated numerically.

MATERIAL PROPERTIES Thin sheets of dual phase steel DP500 were used in this study, of 0.6 mm thickness. This material is not currently used for automotive opening parts but was chosen as a prospective material. The mechanical behavior was characterized in uniaxial and biaxial tension, and in simple shear [7]. Its mechanical properties are given in Table 1. TABLE 1. Mechanical properties of DP500 steel determined in uniaxial tension. Rp0.2% stands for the conventional initial yield stress and Rm stands for the ultimate tensile strength. E (GPa) Rp0.2% (MPa) Rm (MPa) r0 r45 r90 191 345 515 0.87 1.04 1.03

The mechanical behavior was modeled with Hill’s 1948 anisotropic yield criterion coupled to isotropic hardening with a Swift law. Inverse identification was performed and Hill’s parameters F, G and N as well as hardening parameters Y0, K and n for Swift law were adjusted in order to give a correct prediction of both the stress level and

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the anisotropy coefficients. Results are presented in figure 2 and material parameters are given in Table 2. For Hill’s parameters, values derived directly from the anisotropy coefficients were also calculated (Table 2) and it can be seen that the gap between these values and the ones obtained by inverse identification does not exceed 12% for G (7% and 4% respectively for F and N). Fig. 2 shows that the stress level in loading is well predicted but as this material exhibited a pronounced Bauschinger effect, the model could not describe the reverse loading behavior.

FIGURE 2. Comparison of experimental and predicted stress-strain curves for DP500 steel. The model used was Hill’1948 anisotropic yield criterion and isotropic hardening. TABLE 2. Material parameters for DP500 steel corresponding to Hill’s 1948 yield criterion and isotropic hardening (Swift law). For Hill’s parameters F,G and N, values directly calculated from r-values are given in brackets. Y0 (MPa) K (MPa) n F G N 259 833 0.18 0.48 (0.45) 0.48 (0.53) 1.58 (1.52)

FINITE ELEMENT MODEL FOR THE REFERENCE CONFIGURATION The general shape of the sample is illustrated in figure 3. The outer radius is 55 mm and the two straight edges were linked by a corner radius equal to 4 mm. The flanging radius was 0.5 mm, the blade radius 2 mm and the flanging height 3 mm. This height was decreased in the corner, in order to avoid rupture during flanging. The blank was meshed with 8 node elements with reduced integration and also some 6 node elements at the interfaces between fine and coarser meshes. In the area under the blankholder, elements had a characteristic length of 1 mm and this size was decreased down to 0.1 mm in the bent area. Six layers of elements were used in the thickness. The three tools, i.e. die, blankholder and flanging blade, were undeformable parts discretized with smaller elements than the blank. Flanging step was performed within the explicit framework of the finite element code Abaqus, to avoid any convergence problems due to friction between the blank and the tools. A constant blankholder force of 19 kN was applied during this step. The flanging blade had an imposed displacement of 8 mm, such that its initial and final velocities were null. The step time was fixed to 0.5 ms. In order to represent a quasi-static process, kinetic energy was checked to remain below the critical threshold of 10% of the total internal energy, over the whole step. Springback was simulated within the implicit framework, by removing the contacts with the tools. Boundary conditions correspond to the position of the blank on the measuring device, and are applied on the border of the drilled holes (see Fig. 3 for the hole positions): one node is fixed in the three directions (hole n°2), another one is fixed along y and z directions (hole n°1) and a third one (hole n°3) is fixed along z direction.

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Samples were laser cut so that the rolling direction was inclined at 45° with respect to the global directions in the sheet plane. Contact with Coulomb friction was considered. Two different friction coefficients were introduced, in order to reproduce the load evolution during flanging: the friction coefficient fg = 0.5 corresponded to the interaction of all the blank/tool surfaces except for the blade radius, whereas the friction coefficient for this area was decreased down to fl = 0.3 (Fig. 4a). It should be emphasized that no lubricant was used during the experiments.

(a) (b) FIGURE 3. Initial mesh of the blank (a) The whole blank was meshed mainly with 8 node element with reduced integration. The three holes were machined in order to fix the sample on the measuring machine. These holes were also used to impose boundary conditions during springback. Starting from the bottom and turning counterclockwise: holes n° 1, 2 and 3. (b) The mesh was refined in the most deformed area. The global frame is in black and the material frame used for the anisotropy directions is in red.

Load evolution is shown in figure 4b. It exhibited a maximum value of approximately 5 kN for a blade displacement of 2 mm and after the load decrease, a load stagnation corresponding to the sliding of the blade along the bent edge. The coefficient fg was adjusted in order to reproduce this plateau level. It can be seen in figure 4b that the model gives a close description of the load evolution.

(a) (b) FIGURE 4. (a) Definition of the two areas with different friction coefficient: fg = 0.5 was related for all blank-surface interactions except for the blade radius, where fl = 0.3. “lame” stands for blade, “presseur” for blankholder, “matrice” for die and “flan” for blank (b) Experimental and simulated load (F)-displacement (d) curves.

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The corresponding surface defect geometry is shown in figure 5a. A method based on curvature calculation, along 2D profiles oriented in a given direction, was used both for experimental and numerical results [5]. From the deformed mesh, a set of nodes was obtained and profiles along X direction were extracted and fitted by splines. Then, the curvature was calculated and its changes of sign were detected: indeed, as depressions corresponded to a positive curvature, a straight line was defined in-between two consecutive “change-of-sign” (from negative to positive and then positive to negative) points and the distance between this line and the profile corresponded to the defect depth. If no inversion was found, the depth was fixed to a null value. Fig. 5a shows that the defect was highly localized in the corner area and its depth reached a rather low value, of 14 µm at the maximum. For comparison’s sake, the experimental value corresponding to this case was 20 µm. Though the depth was underestimated, it can be concluded that the defect was correctly predicted by finite element simulation.

(a) (b) FIGURE 5. Surface defect geometry (a) in the reference configuration (b) for a constant flanging height and a corner radius of 10 mm.

INFLUENCE OF NUMERICAL, PROCESS AND DESIGN PARAMETERS Mesh size The aim was to investigate the stability of the numerical prediction with respect to the mesh size in the sheet plane and in the thickness. Starting from the previous model with an in-plane element size of 1 mm and 6 layers in the thickness (Fig. 5a), element size was changed from 0.5 mm (Fig. 6a) up to 2 mm (Fig. 6b) and the number of element layers was changed from 3 (Fig. 6c) up to 9 (Fig. 6d). It can be seen that the defect localization was not affected by these changes of mesh. However, decreasing the in-plane element size led to a more localized defect in the corner. The depth was also influenced and reached 14 µm for the two smallest element sizes. The opposite influence on the defect localization was found concerning the number of element layers in the thickness. The depth varied in-between 10 and 14 µm, which was still lower than the experimental value of 20 µm. A model with shell elements having 5 integration points in the thickness was also performed. The mesh was identical to the surface mesh of figure 3. Friction coefficients were modified in order to give a correct description of the load evolution (fg = 0.4 and fl = 0.2, cf. Fig. 4b). However, the defect depth was largely underestimated, with a value of 3 µm. Analysis of the stress and strain states in the defect area showed that after springback, a stress gradient of bending type was obtained through the thickness, that could lead to a depression [8]; the gradient was much higher with solid than with shell elements.

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Flanging radius and height The flanging radius was also changed, it was increased from 0.5 mm, which was a typical value used for steels, up to 2 mm. Numerical prediction of the load evolution was obtained for two flanging height (3 and 5 mm) and plotted in Fig. 7a. The scale on the vertical axis was the same as in Fig. 4a; it can be shown that increasing the flanging radius led to a significant load decrease from 5 kN to 3 kN for the maximum value. Concerning the defect depth, a numerical value of 40 µm was obtained, which was the same order of magnitude than the experimental value. Therefore, the numerical model was able to predict the influence of the flanging radius on both the load evolution (except for the load stagnation at the end of the process, for a flanging height of 3 mm) and the defect depth. The increase of the depth with increasing the flanging radius could come from springback intensity. Indeed, when this radius increased, the plastic strain in the bent area decreased and therefore, the intensity of springback was increased. Flanging height was changed from 3 up to 5 mm, with a reduced value in the corner to avoid rupture. The experimental tendency of an increase of the defect depth with increasing the flanging height (20, 25 and 25 µm respectively for 3, 4 and 5 mm) was also obtained numerically, though with lower values, i.e. 14, 16 and 20 µm. (a) In-plane element size: 0.5 mm

(b) In-plane element size: 2 mm

(c) 3 element layers

(d) 9 element layers

FIGURE 6. Surface defect geometry as a function of the mesh size: the element size was changed in the sheet plane (a), (b) and also the number of layers in the thickness (c), (d).

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Corner radius The corner radius that linked the two straight edges was increased up to 10 mm, value to be compared to 4 mm in the reference configuration. Samples with a constant flanging height of 4 mm were used, which led experimentally to a rupture in the bent zone for a corner radius of 4 mm but not for 10 mm and to a higher maximum force in flanging. Moreover, the defect depth was 45 µm. This increase of the depth was also obtained numerically, but it was underestimated to 30 µm (Fig. 5b for the defect depth and Fig. 7b for the load).

(a)

(b)

FIGURE 7. Numerical prediction of load evolution during flanging and comparison with experimental values (a) influence of the flanging radius (b) influence of the corner radius.

RESULTS AND DISCUSSION The previous results concerning the influence of the mesh size on the surface defect geometry were summarized in Table 3. It shows that whatever the mesh size, a surface defect was detected in the corner area, with smaller dimensions than experimental ones. TABLE 3. Surface defect dimensions, as a function of the mesh size and comparison with experiments, for the reference configuration. For solid elements, the first line gives the number of element layers in the thickness and the second one the in-plane element size (in mm).

Depth (µm) In-plane size (mm2)

3 1 10 15x15

Numerical value Solid 6 0.5 1 2 14 14 10 15x15 15x15 15x15

Exp. Shell 9 1 12 15x15

3 10x5

20 20x25

Concerning process (flanging height and radius) and design (corner radius) parameters, the results were summarized in Table 4. It can be seen that the numerical model gave the same tendency, with an increase of the defect depth when both the flanging height and the flanging radius increased, and also the corner radius. The predicted in-plane size was less affected by a change of these parameters than the experimental values.

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TABLE 4. Surface defect dimensions, as a function of the flanging height h (r stands for reduced height), the flanging radius Rt and the corner radius Rc, and comparison with experiments.

Num. Exp.

Depth (µm) In-plane size (mm2) Depth (µm) In-plane size (mm2)

h3r 14 15x15 20 20x25

h4r 16 15x15 25 20x25

h5r 20 20x15 25 20x25

h3r Rt = 0.5 mm Rt = 2 mm 14 50 15x15 20x15 20 40 20x25 35x33

Rc = 4 mm 45 30x35

h4 Rc = 10 mm 30 15x15 60 30x35

CONCLUSION A surface defect occurring after flanging and springback on automotive door corner was reproduced at the laboratory scale on planar surfaces and an experimental database on load evolution and surface defect dimensions was obtained. The aim of this study was to perform the numerical simulation of the process, in order to test the reliability of the simulated defect size when compared to experiments. A 3D model was mainly used and the mesh size was changed, in order to highlight its influence on the defect dimensions. It can be concluded that a surface defect was detected in the corner radius and that its dimensions were globally underestimated, when measured with the same method as in experiments. The flanging height, the flanging radius and the corner radius were changed and their influence on the load evolution during flanging and the defect size were correctly reproduced.

ACKNOWLEDGMENTS The authors are indebted to the Conseil Régional de Bretagne, Conseil Général du Morbihan and DGCIS for their financial support. They also would like to thank PSA Peugeot Citroën for supplying the material and fruitful collaboration on the subject during the EMOA project (http://web.univ-ubs.fr/limatb/emoa/).

REFERENCES 1. M. Fukumura, Y. Yamasaki, D. Inage and T. Kujita, “Finite element simulation of surface defects in the automobile door outer panel” in Materials Processing and Design: Modeling, Simulation and Applications, Proceedings of Numiform, 2004, pp. 1917-1922 2. C. Park, W. Chung and B. Kim, J. Mater. Process. Technol. 187-188, 99-102 (2007) 3. A. Andersson, J. Mater. Process. Technol. 209, 821-837 (2009) 4. A. Le Port, S. Thuillier, P.Y Manach and J. Garabed, “Experimental study of surface defects on automotive doors during flanging and their numerical prediction” in EPJ web of conferences, 14th International Conference on Experimental Mechanics, 6, 15001, Poitiers (France) 5. A. Le Port, “Caractérisation et prédiction des défauts d’aspect sur les ouvrants automobiles”, Ph.D. Thesis, Université de Bretagne-Sud, 2010 6. A. Le Port, S. Thuillier and P.Y Manach, Int. J. Mater. Form. 3, 215-223 (2010) 7. S.L. Zang, S. Thuillier, A. Le Port and P.Y. Manach, Int. J. Mech. Sciences 53, 338-347 (2011) 8. A. Le Port, S. Thuillier and P.Y Manach, submitted in J. Mater. Process. Technol. (2011)

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Studies in Cup Drawing Behavior of Polymer Laminated Sheet Metal M. Elnagmia, M. Jaina, M. Bruhisa and K. Nielsenb a

Department of Mechanical Engineering, McMaster University, Hamilton, L8S4M1, Canada b 3M Canada Company, London, Ontario, Canada

Abstract. Axisymmetric deep drawing behavior of a polymer laminated sheet metal (PLSM) is investigated using an axisymmetric cup drawing test. PLSMs are of interest as a replacement for painted finishes for automotive applications as they have the potential to achieve good quality long lasting and aesthetically appealing surfaces on stamped parts. However, there is limited understanding of PLSMs in automotive deep drawing situations to produce complex 3-D parts. The tests are carried out using well-controlled, laboratory-based, dual-action, servo-hydraulic forming presses under blank-holder force and punch displacement control conditions. An optical strain mapping system is used to measure the surface strains (and to construct 3D strain maps) from the film side of the deformed samples for a range of forming conditions. Deep drawing characteristics such as punch load versus punch displacement traces, strain distribution along the cup profile, flange wrinkling and fracture characteristics are experimentally assessed for stainless steel-plastic film laminated sheet materials. Also the effect of lamination pressure on wrinkling and delamination is investigated for a decorative pressure sensitive adhesive film affixed to the stainless steel sheet.

Keywords: Polymer Laminated Sheet Metal (PLSM), Deep drawing, Wrinkling, Strain. PACS: 40-46.70.-p

INTRODUCTION Polymer laminated sheet metals (PLSM) are of interest as a replacement for painted finishes for automotive applications as they have the potential to achieve good quality long lasting and aesthetically appealing surfaces on stamped parts. The environmental benefits include reduction or elimination of paint lines in automotive manufacturing plants and reduced use of polluting lubricants in stamping lines. Most research activities on polymer coated metals focus on corrosion protection or corrosive delamination. Less attention is given to the industrial forming of coated metals by means of deep-drawing (1-3). Some recent papers deal with loss of adhesion and recovery of polymer coated steel (4) and laminated steel sheets sandwiched with a polymer core (5). There is virtually no history of the use of PLSMs in automotive stamping processes to produce complex 3-D parts. In this paper, deep drawing characteristics such as punch load versus punch displacement traces, strain distribution along the cup profile, flange wrinkling and fracture characteristics were experimentally assessed for several different metal-polymer film laminates. Good quality cups were produced from different laminate combinations under optimized process conditions.

Experiments 1.

Materials

The materials used in this study consisted of as-received bare 304SS sheet with a gauge thickness of 0.6 mm and various polymer films and adhesive system [film A, film B, film C and Metallic film. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 847-854 (2011); doi: 10.1063/1.3623694 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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2.

Lamination conditions and test specimen geometry

Blanks of 4 different diameters (3.5, 4.5, 5.5 and 6.5 inches) were utilized for axisymmetric deep-drawing experiments. All specimens were cut by a water jet machine and all sharp edges were deburred and polished by using a rotary grinder. The steel specimens were laminated with film A, film B, film C and metallic films using the roll laminator available at McMaster which provided air pressure for lamination. All specimens were laminated at room temperature under the same conditions. The sheet metal substrate was passed only once through the gap between the rollers and kept parallel to the initial rolling direction of the sheet metals. The gap between the rolls was set at 80% of the combined thickness of sheet metal and the film layer. All of the specimens were held for 72 hours of dwell time after lamination for stabilizing the deformed polymer layer.

3.

Grid imprinting for strain measurement

Bare 304SS specimens were electrochemically etched with a periodic grid pattern of solid circles prior to testing as shown in Fig. 1a. The center-to-center spacing of the solid circles was kept at 2 mm. The pattern was applied using a grid stencil placed on the metal surface. An absorbing cotton pad was soaked with an electrolytic solution (Electrolyte 53NC) which was placed on top of the grid. Any exposed surfaces in contact with the solution became oxidized when current (15-20 amps) was passed through the sheet material which produced the dot pattern (see Fig. 1a). For laminated specimens, regular dots were imprinted by first applying a precise plastic mask with periodic circular holes (a distance of 3 mm center-to-center) to the polymer side of the specimen and then applying ink through the holes in the mask using an air brush (Fig. 1b). During the deep drawing process, the dot pattern is distorted as it deforms with the laminate. The change in separation distance between the circular dots is used by the ARGUS optical system to measure the strain distribution on the specimen at the end of the process (Fig. 2). After the test, the ARGUS camera is utilized to take a series of photographs of the deep drawing specimen from different angles and point of view. The software then calculates the relative distance between the centers of dots and strain over the deformed region by comparing the deformed and grid initial spaces.

(a) A solid regular dots electrochemically etched prior to the testing.

(b) A solid regular dots imprinted on the film specimen prior to testing.

Figure 1. Regular dots for a (a) bare 304 SS specimen and (b) film laminated specimen.

Figure 2. Optical strain measurement system (ARGUS).

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4.

Experimental procedure

A schematic of the conventional deep drawing process for sheet metal is shown in Fig. 3. It is to be note that film free side of the laminate is in contact with the punch during deep drawing. The deep drawing for all different laminated specimens (A, B, C and metallic films) were performed on a MTS mechanical testing machine fitted with two servo-controlled actuators as shown in Fig. 4. The capacity of the MTS test system was limited to 25,000 lbf force and a maximum stroke of 6 inch for both actuators. The upper actuator system allowed a close control of the clamping force during the forming process, while the lower actuator system, attached to the punch, allowed its controlled displacement. The specimens were clamped up to clamping load of 14,000 lbf based on an optimized blank holder force profile as discussed later. The drawing experiments were performed at a speed 0.01 inch per second (or 15 mm/min) up to the chosen cup height. Punch load versus displacement data was continuously recorded during the test.

Figure 3. A schematic of cup deep drawing process.

Die

Cup Blank

holde r Punch Figure 4. A photograph of MTS axisymmetric cup drawing system in an ‘open’ position to illustrate various components. The die is moved down to clamp the blank and then the punch is moved upwards to draw the blank.

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Results and discussion 1.

Modification of cup drawing system

In an effort to limit wrinkling and wall friction of deformed specimens, modifications were made to the drawing die. The clearance between die and punch was increased to reduce the friction between the laminated film and die. The comparisons of deformed specimens before and after modification are shown in Fig. 5.

(a)

(b)

(c)

(d)

Figure 5. Photographs of cups, (a) metallic film with 304SS substrate before modification, (b) after modification, (c) 9000J film with 304SS substrate before modification, and (d) after modification .

2.

Clamping Process Optimization for Cup Drawing

In order to produce a defect free deep drawn cup, an optimization strategy for the cup drawing process was developed. During the cup drawing test the laminate was clamped between the blank and die. Typically, a constant blank holder force is applied during the cup drawing test process which involves competition between wrinkling of the flange region and tearing at the punch profile radius. An excessive blank holder force will reduce the material flow into the die and result in fracture or tearing at the punch profile radius while not enough blank holder force will lead to wrinkling of the flange. An optimization procedure for blank holder force involved linearly increasing the blank holder force before the punch reached the maximum force. Fig. 6 shows the process history of punch and blank holder for cup drawing test. Referring to the thick solid black line for blank holder force and the thin solid black line for the punch force, one observes that blank holder force is initially held constant (segment AB of the thick black line) at about 14000 lbf to suppress wrinkling and still allow flange to drawing into the die cavity. However, once the punch displacement is initiated and the punch force starts to rise rapidly, the blank holder force is further increased (segment BC of the thick black line). This change in blank holder force part-way through the cup drawing process enabled further suppression of the wrinkles in the flange while still allowing for the material to flow into the die. The blank holder force was then held constant while the punch load dropped for the remaining stroke of the punch. The optimized blank holder force profile resulted in improved cup drawing to different heights as shown in Fig. 7.

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Figure 6. Time history of various process parameters during cup drawing.

Figure 7. SS304/film A cups drawn to different heights.

3.

Wrinkling and tearing of laminated films during cup drawing

Wrinkles often formed in the flange area and along the cup wall during the test and subsequently over time. Different methods to suppress wrinkle formation in the film after cup were explored. Three solutions are proposed and tested to stop wrinkles after cup drawing test. First, the film at the edge of the flange is cut away from the laminate before the cup drawing test as shown in Fig. 8a. When wrinkles in the film were observed after removal of the specimen from the test system, their advancing front along the cup wall was cut with a knife as shown in figure 8b. Second, the clearance between punch and die was reduced at the top of the cup wall to increase friction. This resulted in film undergoing large shear deformation thereby ‘fusing’ to the substrate which suppressed the onset and

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development of wrinkles. Third, the cup, once drawn and removed from the test system, is machined to remove the flange and adjacent regions of the cup wall (see Fig. 9). After two months, in storage for these cups, the wrinkling tendency of different films on 304SS substrate of cup drawing has been evaluated as shown in Fig. 10. The 304SS/film A laminates showed that delamination started at the machined edge of the cup drawing (see Fig. 10a). Micro-wrinkles were observed on the cup wall close to the edge in 304SS/Metallic film cups over time (see Fig. 10b), on the other hand, no changes in 304SS/film B and 304SS/film C cups were observed over time as shown in Figs. 10c and 10d. It therefore, appears than film B and film C are more suited to a cup drawing operation.

(a)

(b)

High friction at cup wall

Cup redrawing of laminated Magnification of cup(c) wall sheet shows a high friction where applied on wall cup Figure 8. (a) Part of film has been removed before cup drawing, (b) Film has been cut in the cup wall region to stop wrinkles after cup drawing, (c) the clearance between punch and die is changed to increase friction at the cup wall.

304SS/ film Bfilm SS304/10BDC

304SS/ film film A SS304/9000J

304SS/Metallicfilm film SS304/Metallic

304SS/film C SS304/FRA045J

Figure 9. Final cups obtained using different plastic films on 304SS substrate, wrinkled flange region has been removed by machining.

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Figure 10. Surface appearances of drawn and machined cups after two months, (a) 304SS/film A (delamination), (b) 304SS/Metallic film (micro-wrinkles), (c, d) 304SS/film B and 304SS/film C.

4.

Strain distribution measured over formed cups

Once the cup drawing test were done for different blank diameters with periodic grids, a strain map was produced using the ARGUS system, A section along the cup profile was taken from the top of the cup wall to its bottom to obtain the distribution of major and minor strains (see Fig. 11). Fig. 12 shows a plot of the major and minor strains along the cup line profile of Fig. 11. These profiles are typical of what was observed in cup drawing.

(a)

(b)

Figure 11. Example of ARGUS results for a cup with 304SS/film A, (a) deformed and pattern, and (b) major strain distribution over the cup surface.

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0.4

Major Strain Minor Strain

0.3

0.2

True Strain

0.1

0.0 0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

-0.1

-0.2

-0.3

-0.4 Surface distance (mm)

Figure 12. Major and minor strain profile along the section line shown in Figure 11.

Conclusions The research presented in this paper offers a new blank holder force control strategy which can be useful in controlling the material flow and suppress film wrinkling to produce defect free deep drawn parts from polymer laminated sheet metals. Deep drawn cups from different 304SS/film laminates have been successfully obtained. A range of strategies have been explored to stop wrinkles after cup drawing test. Further investigation is continuing on aspects of formability (in other words limiting draw ratio determination) and wrinkle suppression during cup drawing of 304SS/film laminates. Also an existing cup redrawing system has been modified to achieve further gains in deep drawability.

ACKNOWLEDGMENTS The authors gratefully acknowledge the financial supports of Ontario Centre of Excellence (OCE), and Natural Science and Engineering Research Council (NSERC) of Canada as well as material support from 3M of Canada.

REFERENCES 1. D.F. Chang, J.E.Wang, Analysis of draw-redraw processes, Int. J. Mech. Sci.,1998, 40, 793–804. 2. L. Liu, J. Wang. Modeling springback of metal–polymer–metal laminates, J. Manuf. Sci. Eng. , 2004,126,599–604. 3. J.M. Lee, D.C. Ko, K.S. Lee, B.M. Kim, Identification of the bulk behavior of coatings by nano-indentation test and FEanalysis and its application to forming analysis of the coated steel sheet, J. Mater. Proc. Technol., 2007, 187–188, 309–313. 4. A. Fedorov, , R. Van Tijum, , W.P. Vellinga, , J.T.M. De Hosson, Work of adhesion in laser-induced delamination along polymer–metal interfaces. J. Appl. Phys., 2007, 101, 043520. 5. H.S. Cheng, J. Cao, H. Yao, S.D. Liu, B. Kinsey, Wrinkling behavior of laminated steel sheets, Journal of Materials Processing Technology, 2004, 151, 133–140.

854

$XWR%RG\6WDPSLQJ5REXVW'HVLJQ$SSURDFKDQG$SSOLFDWLRQ Lin Zhong-qina,b , Hou Boa,b , Li Shu-huia,b , Yu Zhong-qia,b and Zhao Yi-xia,b a

Shanghai Key Laboratory of Digital Auto-body Engineering, 800 Dong chuan Road, Shanghai, China School of Mechanical Engineering, Shanghai JiaoTong University, 800 Dong chuan Road, Shanghai, China

b

$EVWUDFW The uncertainties in the auto body stamping process affect the stamping quality significantly. The current stamping simulation and optimization may lead to design failure without considering these uncertainties. A systematic stamping robust design approach is proposed. The variation behavior of the local auto sheet properties is analyzed, and the parameters are obtained. A stochastic analysis method is developed, where FE-simulation, uniform design, response surface methodology (RSM) and Monte-Carlo simulation are used. The robust optimization model composed of mean and standard deviation is constructed and solved. The proposed approach is verified through feature box and deck lid inner panel forming process. The developed approach offers rapid and reliable results for engineers to deal with potential stamping problems during the early phase of product and tooling design, saving more time and resources. .H\ZRUGVStamping Robust design Local auto sheet Stochastic analysis Optimization 3$&602.50.Fz, 02.60.Pn,81.05.Bx, 81.20.Hy

,1752'8&7,21 In complex auto body stamping process, traditional trial and error experimental campaigns have been progressively replaced by FE-simulation and optimization method. However, many uncertainties exist in the stamping process, such as the variation of raw material properties form coil-to-coil and batch-to-batch, die and press setup conditions, long-term tool wear, etc. Current practice of stamping simulation and optimization seldom take these uncertainties into account. The stamping operation under these design results may be less robust, and result in increased reject rate in the form of split or wrinkled panels as well as panels out of dimensions. The present trend is, therefore, increasingly leading to the use of stochastic FE analysis trying to reflect the manufacturing reality including the uncertainty. A survey to the recent literature suggests that the challenges faced presently are: The high computational cost required by directly integrating FE-simulation and design of experiment (DOE), although the advanced Latin hypercube design were used to reduce the number of runs compared with Monte-Carlo simulation (MCS)[1,2]. Advanced sampling techniques [3] and response surface (meta-model) based method [4] are developed to reduce the computational cost without substantially reducing accuracy. Several attempts have been made for optimization under uncertainty [5,6], where the focus has been placed not on the robust optimization of the process but on the definition of “stochastically feasible regions” in the parameter space. As discussed above, consideration of the stochastic nature of stamping process in FE-simulation and optimization is now receiving increasing attention. In this study, it is aimed to develop a simple, easy operating stamping robust design approach. The technology roadmap and the implementation process are introduced. The key technologies including meta-model based stochastic analysis and robust optimization strategy are introduced. Robust design for two kinds of auto body part is performed to demonstrate the accuracy and validity of the approach.

0(7+2'2/2*<)2567$03,1*52%867'(6,*1 The basic concept for stamping robust design can be illustrated in Figure. 1, including several essential factors such as the stamping quality target(specification) y0, design factors x, noise factors z and stamping quality index y. y0 is the expected quality goal in the stamping production with tolerance of [y-,y+]. x represent the controllable process parameters, such as blank holder force (BHF), die fillet radius and drawbead, etc. z are the parameters that influence the system but are difficult, expensive or impossible to control, such as the variation of raw material properties, die and press setup conditions, etc. y is the output of the design results, such as the split index, which is

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 855-862 (2011); doi: 10.1063/1.3623695 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

855

influenced by the design variables and noised factors as a function of x and z. The objective of stamping robust design is to minimize the sensitivity of stamping quality index y to noise factors z (material scatters, lubrication, etc) by regulating the controllable factors x (BHF, drawbead, etc), aiming at the lowest manufacturing cost. The implementation process specially developed for stamping robustness is shown in Figure. 2 and is categorized into three main modules. (I) System design, which includes three steps as follows. ¾ Define stamping quality index y. As shown in Figure. 3, the distance between point A and forming limit curve (FLC) is set as cracking index d1 to evaluate the possibility of cracking. Similarly, the distance between point B and wrinkle limit curve(WLC) is set as wrinkling index d2 to evaluate the extent of wrinkling ¾ Define stamping quality target y0. In mass production, the quality targets are usually reflected by sigma level or reject rate. According to the statistical theory, a 3-process implies 27 products of ten thousand products are defective, meanwhile 2.75-process needs to be meet for quality specification of 0.3% reject rate. ¾ Screening key factors. In order to reduce the complexity of the problem, sensitivity analysis is performed to screen the dominant controllable factors x and the dominant noise factors z in support of the local steel sheet statistical variation property database. The sensitivity analysis method is described in [7].

+

f(xi)

x

Controllable factors x

Start -

ଠx ଠx

,6\VWHPGHVLJQ Define quality index Define quality target

Local steel sheet property variation statistical database

Screening key factors

y=y(x,z)

Input

Quality Indices

y0,ଠy-,ଠy+

,,4XDOLW\LQGH[VWRFKDVWLFDQDO\VLV

Output

Noise factors z

RSM modeling y=f(x,z)

f(zi)

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Accuracy test Pass Monte-Carlo simulation

z ),*85( The basic concept for stamping robust design.

Statistical results of the stamping quality index˄, , etc)

,,,5REXVWRSWLPL]DWLRQVWUDWHJ\ RSM modeling of the  and  of the stamping quality index Fail

WLC

B

Accuracy test

FLC

d2

Pass Optimization for statistical result of the stamping quality index (Determine the robust optimal process)

d1 Verification in small-batch experiments

A Fail

Reject rate test Pass

Minor strain

End

),*85( Illustration of the stamping quality index in FLD),*85( Implementation flowchart

856

(II) Meta-model based stochastic analysis. FE-simulation, DOE and response surface methodology (RSM) are used to construct meta-model, which approximates the relationship between the quality index (d1, d2) and inputs (x, z). Significance of the meta-models is judged by ANOVA. The determination coefficient R2 is checked for model accuracy. If the accuracy test failed, new DOE and surrogate model will be adopted to improve accuracy. In this study, uniform design (UD) and polynomial model are proved to be practical tools for constructing meta-model with enough accuracy. Based on the meta-model and probability distribution of the noise factors z, inexpensive MCS are performed to generate the exact statistical distribution of product quality(d1, d2), such as mean , standard deviation (S.D.) , etc. (III) Robust optimization strategy. A combined stochastic analysis and RSM approach is used to calculate the mean, S.D. values of stamping quality index as a function of inputs x, namely, (x) and (x). UD is applied to generate DOE matrix. Based on the quality specification y0, the robust optimization model can be formulated, where the mean and S.D. are involved. Levenberg-Marquardt algorithm is used to solve the robust optimal design parameters. Small batch experiments using the robust optimal parameters are necessary for verification. If the reject rate is close to or below the expected value, the approach is valid. Otherwise, we must start over from the first step.

67$7,67,&$/'$7$%$6(2)/2&$/67((/6+((763523(57,(6 The variability of raw material impacts on the statistical characteristic of auto body stamping quality. Precisely characterization of material properties variation is crucial for stamping stochastic analysis. In this study, a considerable amount of test data from a local steel sheet producer has been analyzed. The measured material properties include total elongation, tensile strength b, yield strength s, hardening exponent n, average anisotropy coefficients R, etc. The test samples are obtained from head to tail of different steel coils and batches (more than 40 thousand steel coils), which covers different strength grades, coatings and strengthening mechanisms including mild steels, bake hardening steels and dual phase steels, etc. The probability distribution and statistical parameters of DP590 and DC06 material properties are characterized, and their mean, S.D. values are shown in Table 1 and Table 2. Similarly, 5 series, 22 kinds of local auto sheets are statistically analyzed, and the corresponding statistical database is developed which provide a foundation for stamping robust design. 7$%/(The statistical datasheet for DP590 sheet E0SD  3DUDPHWHUV V0SD  Mean 412 676 S.D. 21.1 19.6 Range 372-436 639-708

7$%/(The statistical datasheet for DC06 sheet 3DUDPHWHUV V0SD nYDOXH RYDOXH Mean 142 0.240 2.197 S.D. 10.8 0.006 0.1771 Range 125-165 0.23-0.26 1.9-2.5

&$6(678'<52%867'(6,*1)25'(&./,',11(53$1(/67$03,1* A deck lid inner panel is selected as the focal part to demonstrate the robust design approach. Figure 4 shows the base design for the drawbead development, where it consists of six drawbead segments. As shown in Figure 5, three Split-Prone Zones(SPZ) and a Wrinkle-Prone Zone(WPZ) are identified though detailed examination of formability

DB2

SPZ-1

DB3 DB5

WPZ

DB1

DB4

SPZ-2

DB6

SPZ-3

),*85( Drawbead development),*85( Formability results

857

results. A slight change of process parameters or material properties may lead to split in SPZ-1, SPZ-2 and SPZ-3, severe wrinkle in WPZ. A cold-rolled local steel sheet (DC06) of 0.8mm is used. Four drawbead segments (DB1, DB2, DB3, DB4) in Figure 4 and the blank-holder pressure (BHP) are the main control factors. As a matter of experience, the value range of BHP is set as 0.65~0.8Mpa, and that of drawbead strength for DB1 to DB4 are set as 0.2~0.35, where 0.2 and 0.35 represent the weak and medium drawbead strength respectively. Sensitivity analysis shows that BHP, DB1 and DB2 are the dominant controllable factors. The considered noise factors include material properties (s, n, R), thickness t and lubrication. The range of t (±0.08mm) is specified by the steel suppliers, and a Gaussian distribution is assumed. The statistical parameters of DC06 sheet is listed in Table 2. A Gaussian distribution of static friction coefficient f within 0.06 and 0.10 is assumed. Three dominant noise factors screened out are: n, R and f.

0HWDPRGHOEDVHGVWRFKDVWLFDQDO\VLV UD is applied to generate DOE matrix. A part of the UD (6 out of 60 runs) is shown in Table 3, where x1-x6 represent the screened three control factors (BHP, DB1 and DB2) and three noise factors (n, R and f) respectively. The cracking index d1 and wrinkling index d2 are shown in the right column of Table 3, which are engineering strain results generated by FE-simulation according to the corresponding combination of inputs. 7$%/(UD matrix with six factors at six equal levels and FE-simulation results ,QIOXHQFHIDFWRUVDQGOHYHOV VLPXODWLRQUHVXOWV 5XQ x x x x x x d d 1 0.65 0.35 0.26 0.255 2.2 0.084 13.3 6.7 2 0.8 0.29 0.23 0.255 2.35 0.084 14.4 5.9 3 0.65 0.32 0.26 0.23 1.9 0.084 3.5 8.4 4 0.65 0.2 0.2 0.235 2.05 0.067 1.8 7.9 5 0.74 0.32 0.2 0.255 2.2 0.06 19.9 7.8 60

0.8

0.23

0.29

0.255

2.05

0.076

11.6

6.7

Based on the UD matrix in Table 3, polynomial meta-models for d1 and d2 are fitted, and the coefficients are determined with Levenberg–Marquardt algorithm. ANOVA indicated that the two models are significant at 95% confidence level. A high proportion of variability (R2>0.95) imply that the two models are reliable. Utilizing the meta-models, MCS can be performed cost-effectively to determine uncertainty propagation. The principle of meta-model based stochastic analysis is shown in Fig. 6, and the main steps are explained below: Step 1: 6HWXSPHWDPRGHOV, y=f(x1, x2, ..., xq). Meta-models are used, where y represents stamping quality index, x1, x2, ..., xq represent the input control factors and noise factors. Step 2: *HQHUDWHDV HWRIUDQGRPLQSXWV, x1i, x2i, ..., xqi (i=1,2,…,n). According to the statistics of noise factors listed in table 2, a sequence of random numbers conformed to their statistical distribution are generated. Step 3: (YDOXDWHWKHUHVSRQVH. Record the response of meta-models as yi and repeat steps 2 and 3 for i=1 to n. Step 4: $QDO\]HWKHUHVXOWV. The recorded yi can be analyzed using histograms, and the corresponding mean, S.D. value, probability density function (pdf) and other statistical parameters can be obtained.

),*85( Principal of uncertainty propagation),*85( Meta-model based stochastic analysis for d1

858

A set of random numbers of noise factors (x4~x6) are generated with the control factors (x1~x3) at a constant level which is assigned based on experience. The d1 and d2 are calculated by the fitted polynomial meta-models. Figure 7 shows the histograms, fitted probability density curves, the mean and S.D. values of d1. Assuming that the part is rejected when d1 is less than 5%, the probabilities of cracking are obtained as 85.6%. It is indicate that the current design is not feasible. The BHP and drawbead strength must be adjusted until a specified reject rate is achieved.

5REXVWRSWLPL]DWLRQVWUDWHJ\ In some cases it is not enough only to predict the stamping quality scatters, but it is also necessary to determine the robust optimal solution automatically. Robust optimization strategy is developed to improved design efficiency. The meta-model based stochastic analysis and RSM are combined to construct meta-model between control factors and the mean, S.D. values. UD is applied to generate DOE matrix. A part of the UD matrix (6 out of 48 runs) is shown in Table 4, where x1~x3 represent the BHP, drawbead strength of DB1 and DB2 respectively. The mean, S.D. values of the d1 (1, 1) and d2 (2, 2) are also shown in the right column of Table 4, which are generated by the meta-model based stochastic analysis under the corresponding combination of control factors. 7$%/(UD matrix with three factors at six equal level and stochastic analysis results ,QIOXHQFHIDFWRUV &UDFNLQJLQGH[ :ULQNOLQJLQGH[ 5XQ x x x     1 0.74 0.26 0.32 -1.102 5.732 7.935 0.714 2 0.65 0.32 0.35 1.687 4.149 6.983 1.321 3 0.8 0.32 0.29 -2.144 9.051 6.433 0.833 4 0.77 0.35 0.2 2.148 9.566 7.574 0.626 5 0.74 0.23 0.29 2.08 6.98 7.803 0.592 48

0.77

0.29

0.2

3.682

6.732

8.559

0.787

Based on the UD matrix in Table 4, polynomial meta-models are fitted, and the coefficients are determined with Levenberg–Marquardt algorithm. The fitted equations for 1 and 1 are: 1

-6445.31x13  16455.77 x23  11081.34 x33  6346.86 x12 - 24174.88 x22  9474.25 x33 - 28175.3 x1 x2 - 25100.19 x1 x3  38427.94 x2 x3  14169.59 x1 x1 x2  14804.18 x1 x1 x3  14111.06 x2 x2 x1  26.23 x2 x2 x3  6830 x3 x3 x1  15271x3 x3 x2  2427.36 x1

(1)

 15640.92 x2  10247.04 x3  2819.21 1

14220.67 x13  16258.05 x23  87918.33 x33  15275.64 x12 - 7040.99 x22  37538.63 x33  1120.03x1 x2  2641.51x1 x3  635.89 x2 x3  1107.64 x1 x1 x2  808.71x1 x1 x3  884.19 x2 x2 x1  458.21x2 x2 x3  2707.68 x3 x3 x1  1691.09 x3 x3 x2  6987.29 x1

(2)

 916.23 x2  7735.5 x3  4827.1x14  14316.5 x24  79148.7 x34  777.98

Similarly, the mean and S.D. values of wrinkling index (2, 2) are calculated as the function of control factors. ANOVA indicated that the meta-models are significant at 95% confidence level. A high proportion of variability (R2>0.99) imply that the four models are reliable. The quality specifications in the deck lid inner panel stamping include: (1). Minimize the wrinkling extent in WPZ; (2). Reject rate (mainly caused by split) less than 0.01%. The mean and S.D. values of wrinkling index (2, 2) should be minimized, as shown in Eq.(3). According to the statistical theory, reject rate of 0.01% equates to 3.72 level. It’s assumed that the part is rejected when d1 is less than 5%, thus the constraint function can be written as Eq. (4). By combining Eq. (3) and Eq. (4), the robust optimization model is formulated as follows.

 P2  V 2

(3)

P1  3.72V 1 ! 5%

(4)

Minimum

s.t.

The robust optimal solution is obtained as 0.65Mpa for BHP, 0.26 for drawbead strength of DB1 and 0.228 for

859

DB2. The results of robust optimization are compared with that of traditional deterministic optimization [8]. As shown in Figure 8(a), by the robust optimization, the probability of split in the production of this panel is reduced significantly from 19.1% for deterministic optimum to 0.01%. As shown in Figure 8(b), by the robust optimization, although the mean of wrinkling index increases, the probability of wrinkling failure is reduced from 0.0583% for deterministic optimum to 0.0348%, as a result of the reduction of S.D. value.

(a) (b) ),*85( Comparison of the robust and deterministic optimization results for (a) d1 and (b) d2

&$6(678'<52%867'(6,*1)25)($785(%2;'5$:,1*352&(66 A feature-box drawing die is developed to cover the typical strain state and process characteristics in auto-body stamping. The detail of the feature-box part is shown in Figure 9, which reflects the typical geometrical characteristics of the auto body product with complex shapes. It is selected as the focal part to demonstrate the robust design approach with small batch experiments in lab. A cold rolled dual phase steel sheet (DP590) of 1.0mm is used. The BHF is the main control factor with range of 20~30t. The considered noise factors include material properties (s, b), thickness t and lubrication. The range of t (±0.11mm) is specified by the steel suppliers, and a Gaussian distribution is assumed. The statistical parameters of DP590 sheet is listed in Table 1. A Gaussian distribution of static friction coefficient f within 0.1 and 0.15 is assumed.

Short side wall

Long side wall

Large convex

Concave

Small convex

),*85( The geometrical characteristics of feature-box part A detailed examination of the formability results reveals that severe deformations are mostly concentrated on the short side wall (Split-prone) and the flange (Wrinkle-prone) in Figure 9. UD of five factors with five equal levels is applied to generate DOE matrix U55(55). Based on the UD matrix, polynomial meta-models are fitted to predict the relationship between stamping quality index (d1, d2) with the control and noise factors. The fitted equations are: d1 1178.603x1 -1.99 x2  0.7 x3  2.626 x4 - 9033.908 x5  478.343 x12  7.8E-04x22  2.6E-03x32  0.066 x42  26812.485 x52  0.974 x1 x2  0.949 x1 x3  8.393x1 x4  2333.536 x1 x5  5.7E-03x2 x3  0.012 x2 x4  0.447 x2 x5  0.031x3 x4  2.057 x3 x5  67.509 x4 x5  462.075

860

(5)

d2

0.169 x1  3.6E-04 x2  8.6E-04 x3  2.7E-03 x4  0.201x5  0.094 x12  3.473 x22  6.811x32  2.388 x42  0.814 x52  1.3E-04 x1 x2  4.74 x1 x3  1.4E-04 x1 x4  0.019 x1 x5  3.099 x2 x3

(6)

 3.508 x2 x4  2.5E-04 x2 x5  1.109 x3 x4  8.132 x3 x5  0.015 x4 x5  0.514

where x1-x6 represent t, s, b, BHF and f respectively. ANOVA indicated that two meta-models are significant at 95% confidence level. A high proportion of variability (R2>0.9) imply that the two models are reliable. Based on the statistical distribution and the meta-models, MCS are performed to determine the variation of stamping quality. As shown in Table 5, the BHF range is divided into 11 levels with interval of 1t. The mean, S.D. values of the d1 (1, 1) and d2 (2, 2) are also shown in the right column of Table 5, which are generated by the meta-model based stochastic analysis under the corresponding level of BHF. 7$%/(6tochastic analysis results for different BHF levels &UDFNLQJLQGH[  :ULQNOLQJLQGH[ %+)W     20 37.7 3.1 25 0.134 21 37.2 3.3 24.9 0.149 22 36.8 3.1 24.8 0.165 23 35.7 4 24.7 0.18 24 34.7 4.5 24.6 0.196 30

26.3

7.9

23.8

0.289

Based on the data in Table 5, polynomial meta-models are fitted to predict the relationship between the mean, S.D. values of the stamping quality index with control factor (BHF), and the coefficients are determined with Levenberg–Marquardt algorithm. The fitted equations are:

2

1

21.31  2.144 x  0.065 x 2

1

5.788  0.574 x  0.022 x

2

25.748  0.021x  0.029 x

(7)

2

(8) 2

0.165  0.015 x  0.000019 x

(9) 2

(10)

The quality specifications in the feature-box stamping include: (1). Minimize the wrinkling extent in the flange; (2). Reject rate caused by split less than 0.01%. It’s assumed that the part is rejected when d1 is less than 10%, thus the robust optimization model is formulated as follows. The robust optimal solution is obtained as 26.7t for BHF.

 P2  V 2 P1  3.72V 1 ! 10%

Minimum

s.t.

(11) (12)

In order to evaluate the influence of lubrication and material properties variation on the feature-box stamping results, small batch of experiments are carried out under different BHF levels in Table 5, as shown in Figure 10.

),*85( A part of the experimental stamping feature-box with DP590 sheet

861

Severe wrinkling has been found when the BHF level is lower than 26t, and split parts are often found when BHF level is higher than 28t. With the perturbations existing in the material properties and lubrication conditions, more than 100 pieces of the parts are stamped under BHF level of 27t. No split part has been found and only slight wrinkling occurs at the flange. Thus the robust optimal solution is determined as 27t from the experimental results. The predicted robust optimal solution matches the experimental data well, which demonstrate the accuracy and validity of the proposed robust design approach.

&21&/86,21 A methodology of stamping robust design is developed, which helps to predict the production robustness and to improve the design reliability for auto body stamping. It is demonstrated that the proposed methodology is feasible and effective for the stamping process parameters involved. The robust design results match well with the specifications and experimental results, which demonstrates the accuracy and validity of the approach. By using the meta-model based stochastic analysis, the exact statistical distribution of product quality caused by stamping uncertainties can be predicted with the less number of FE-simulations. The robust optimization allows engineers to determine the robust optimal process automatically, avoiding long term trial and error procedures. The method is simple, appropriate for industrial use. Using the developed approach, engineers can solve potential stamping problems months before production start-up - a real resources and money saver for manufacturers. In future the tolerance design will be integrated in order to perform robust optimization from the economical point of view. For example, in some case the quality specification is less strict, and it is reasonable to use cheaper material with higher dispersion of properties. The correlation of input variables needs to be considered because uncertainty of some material parameters is clearly correlated, e.g. yield strength and ultimate tensile strength.

$&.12:/('*0(176 The authors greatly acknowledge the support from National High-Tech Research and Development Plan (No.2007AA04Z145) and State Key Laboratory Mechanical System and Vibration (Shanghai Jiao Tong University) (No.MSV-MS-2008-12).

5()(5(1&(6 1. 2. 3. 4. 5. 6. 7. 8.

R. Moshfegh, L. Nilsson and M. Larsson, Struct. Multidiscip. O , 587-600 (2008). P. Chen and M. Koc, J. Mater. Process. Tech , 189-198 (2007). D. Roos, in: Proceedings of the NUMISHEET’08, (2008), pp. 835-845. H. Müllerschön, W. Roux, D. Lorenz and K. Roll, in: Proceedings of the NUMISHEET’08, (2008), pp. 819-828. G. Gantar and K. Kuzman, J. Mater. Process. Tech , 237-243 (2005). N. Manopulo, J. Heingärtner and P. Hora, in: Proceedings of the NUMISHEET’08, (2008), pp. 871-876. Z.Q. Lin, S.H. Li, et al., Auto body stamping simulation, Beijing: China Machine Press, 2004, pp. 263-266. Z.Q. Lin, S.H. Li, et al., Auto body lean forming technology, Beijing: China Machine Press, 2009, pp. 249-252.

862

Accurate Die Design for Automotive Panel Stamping Considering the Compensation Related with Die Deflection and Blank Thinning Jun Chena, Dongkai Xua, Guodong Xiaa, Xifeng Lia, Jieshi Chena, Jian Zhangb, Wei Yanb and Yue Li b a

Department of Plasticity Technol., Shanghai Jiao Tong University, 1954 Huashan Road, Shanghai, 200030, China Technical Center, FAW Tool & Die Manufacturing Co. Ltd, 5519 Dongfeng Avenue, Changchun, 130011, China

b

Abstract. In order to improve assembly accuracy, automotive body panels have to be fabricated with higher dimensional and surface quality requirements, therefore the die faces should be designed more accurately to consider more relevant factors. In the presented study, we proposed algorithms to realize the following functions: through forming process simulation, the thinning distribution on the deformed blank was extracted as first kind of compensation; through die structural CAE analysis which automatically mapped the boundary contact forces onto the die surfaces from process simulation results, the die deflection was calculated as second kind of compensation. These two quantitative contributions were added together to compensate the die face. The proposed methodologies were programmed and integrated with LS-Dyna and HyperWorks, and also integrated with Autoform and CATIA linear CAE functionalities separately. In addition, a software toolkit to calculate the contacting ratio was also developed to evaluate the effectiveness of die face compensation. The second toolkit developed was verified by an automotive structural part forming die design, through die compensation and geometric optimization, the predicted contact ratio between the die face and formed blank was improved a lot, and the first toolkit was testified by a fender drawing die design. It shows that the die face compensation can be realized and integrated seamlessly between CAD model, process simulation model and die structural CAE model with the help of data I/O tools developed by the authors. Keywords: Automotive body panel, die design, geometric compensation, process simulation, structural analysis

INTRODUCTION The design and manufacture of large automobile panel is one of the key technologies in automobile industry. The dynamic fitting accuracy between punch and die is reflected by matching ratio, and it guarantees the operation stability of stamping under mass production conditions. Traditionally, the matching ratio is measured and improved through physical die try-out, which directly results in high labor cost and severely prolongs the development duration. With the application of accurate design technology, the compensation amount may be obtained in early design stage based on studying the influencing factors of matching-ratio. The die face is modified to meet the requirement of fitting accuracy, which can reduce both labor cost and development lead time and improve the quality of stamping parts. The deformation of press and tooling system has been already documented in literatures [1-3]. Tool-surface deformation is a kind of local deformation mainly caused by thickening of the blank and other factors. Traditionally, this kind of deformation is less than 0.1mm [4]. However, if a larger stamping die is not properly designed with isostiffness, or mounted on the workbench of the press which has larger functional hole, larger elastic deformation may occur on the die structure as shown in Figure 1 where the die gap during home position was measured as 0.27mm. In such kind of condition, forming quality was absolutely affected. Traditionally, stamping process simulation ignores The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 863-870 (2011); doi: 10.1063/1.3623696 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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the elastic deformation of the tooling system, as the die, punch, and blank holder are assumed to be rigid. Obviously, if elastic deformation is to be calculated, the assumption of rigid tools has to be abandoned. Here the straight forward approach would be to discretized the tools by a sufficiently accurate full 3D finite element model. Additionally, the machine stiffness may be considered to the model for completeness. Obviously, this will lead to enormously increasing calculation time especially for complicated stamping parts [5]. In order to analyze elastic deformation of stamping dies, a simple and effective method for die structure analysis based on the sheet metal formability simulation was proposed by one of the author and others, and the interaction force between forming blank and die faces was mapped onto die faces to serve as load boundary conditions for die structure analysis [6].

Figure 1. Elastic deformation measurement of larger stamping die

In this paper, two important factors which greatly affect the dynamic fitting accuracy of stamping dies are analyzed. The algorithms for die surface compensation and matching ratio optimization technique based on FEM simulation were proposed. The proposed methodologies were programmed and integrated with LS-Dyna and HyperWorks, and also integrated with Autoform and CATIA linear CAE functionalities separately. In addition, a software tool to calculate the contacting ratio was also developed to evaluate the effectiveness of die face compensation. The first toolkit developed was verified by an automotive structural part forming die design, through die compensation and geometric optimization, the predicted contact ratio between the die face and formed blank was improved a lot, and the second toolkit was testified by a fender drawing die design. It shows that the die face compensation can be realized and integrated seamlessly between CAD model, process simulation model and die structural CAE model with the help of data I/O tools developed by one of the authors and others.

METHOD OF CALCULATING MATCHING RATIO Definition of matching ratio Matching ratio is an important index of die surface quality. Traditionally, matching ratio is estimated during the stage of die trial. Figure 2 shows traditional calculation and optimization flow chart for matching ratio. Matching ratio is estimated after the completion of tools manufacturing and first try-out, which definitely adds the production cost and increase development period. On stamping parts, flat regions are generally evaluated, and those regions with large curvature variation, fillet and addendum surface are often excluded. As shown in Figure 3, the regions enclosed by red lines are used to estimate matching ratio. The stamping part shown in Fig.2 is taken as an example, and the definition of matching ratio (MR) is described as follows. N

M

i 1

i 1

MR   Smi /  St j 100% Where

(1)

Smi is the area contacting with die face in ith element of the blank St j is the area of jth element in the blank N is the amount of elements which contact with die surface within the trimming lines M is the total amount of elements within the trimming line

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Completion of tools manufacturing Paint red lead powers on die working faces

Stamping process testing Examine contacting areas Modify die surface design Estimate matching ratio Fail Pass

Figure 3. Sketch of regions for estimating stamping die matching ratio

Qualified die

Figure 2. Flow chart for Traditional calculation and optimization for stamping die matching ratio

Procedure of die surface compensation and matching-ratio optimization The die surface compensation mainly derives from blank thickness reduction and elastic deformation of die structure. The calculation of the two aspects mentioned above can be concurrently processed after stamping process simulation. From formability simulation results, node coordinates, element information and thickness distribution of the stamping part are extracted. Therefore, thickness reduction induced compensation amount at each contacting node in three directions of x, y and z is calculated by area weighting factor method. The contacting forces between the blank and die structure is generated also from formability simulation results, which are exerted on the outer nodes of the boundary 3D solid elements on the die structure. The elastic deformation of die structure is computed. A new load mapping algorithm is the improvement of that in the reference [7]. The proposed block mapping methodology has less severe requirements for CPU and memory when the industrial case is a complicated stamping part. The total compensation on die face is furthermore processed by linear superposition of the above two compensation amounts. The matching ratio may also be calculated in the compensation regions. The original die surface is modified by calculated compensation amount, and the optimal die face can be obtained within several iterative steps. Massive nodes searching by sub-region Since discretization of die face even at regions with small geometric features usually results in several hundred thousand nodes, when the load mapping on different nodes is handled sequentially, all the elements on the blank have to be searched to find the nearest element which likely interact with the selected boundary node on the die face. At the same time, compensation induced by sheet metal thinning has also to be implemented by the similar searching processes. If each searching process is realized to go through all nodes, it will be quite time-consuming for the complicated case. Thus a new algorithm is proposed to search the node on die face close to node on stamping part, and the flow chart is shown in Figure 4. In this way, the searching efficiency can be dramatically improved. Divide A into n2 regions

Obtain boundary A

Aij（i j=1~n）

Calculate nodal parameter near to P

Determine the central point Pij of Aij

Search the nodes near to P in the searching regions

Select searching regions Aij, A(i+1)j, Ai(j+1), A(i+1)(j+1)

Figure 4. Flow chart of massive nodes searching by sub-region

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Select any node P from boundary B

COMPENSATION ALGORITHM OF DIE FACE Compensation amount induced by thickness reduction Since the element compensation was first calculated as the thickness following the element normal n as shown in Figure 5, ti is the thickness of element Bi, which vertex is V1, V2 and V3, and T is the matching gap between punch and die. e2, e2 and n denote edge vectors and normal vector of the triangle element, respectively. Then the elemental compensation vector c can be represented as follows.

n V3 e2 V1

e1

V2

Figure 5. Compensation related parameters in an element

 e1  V3  V1  e  V V 2 1  2   n e e  1 2  n c  (T  ti ) n 

(2)

Compensation amount for thickness reduction at a specific node can be represented by area weighting factor method as follows: K

K

i 1

i 1

c j   cij  Si /  Si

(3)

Where Cij is the elemental compensation along j direction at ith element Sj is the area of the triangular element K is the number of the elements which share the node

Compensation amount induced by elastic deformation of the die structure Since the strategy for die face modification is implemented on the die, and no modification on the punch, the elastic deformation induced compensation amount on die surface consists of 2 parts, one is the direct deformation of the die face itself u d (u x ,u y ,u z ) , and another is the compensation caused by the punch deflection v p (v x ,v y ,v z ) , while v p (v x ,v y ,v z ) can be calculated by finding the nearest 3 nodes around the specific node on the die outer surface. Compensation amount induced by elastic deformation of the die structure The total compensation δ at specific node on the die surface can be added together as follows:

 j =c j  u j  v j

(4)

Where j =x, y, z

Software tool development

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The software system was developed by C++, and the main interface is shown in Figure 6, it can be run as a stand-alone system. There are 2 versions, one integrated LS-Dyna for formability simulation and HyperWorks for die structure CAE modeling, another integrated Autoform for formability simulation and CATIA for linear CAE analysis.

Data reading Thinning induced compensation Die deformation induced compensation Superposition of 2 kinds of compensation Matching ratio calculation

Figure 6. Main interface of the developed software

CASE STUDY A fender is selected for case study, the initial thickness of the blank is 0.7mm, material is Sigaudi-t99 with yield strength as 272MPa. The friction coefficient and binder force are 0.14 and 1200KN, respectively. During stamping simulation by Autoform, die surfaces are regarded as rigid. The simulation results indicate there are no cracks and wrinkles on the forming part. Thickness distribution is shown in Figure 7, and thinning amount at each node was extracted and converted to three translational values which would be applied as compensation on die surface.

Figure 7. Contour of thickness distribution Using the exported results from Autoform, the nodal pressure file was imported into CATIA GAS module, and the distributed boundary forces were then mapped to the boundary nodes of die structure which is discretized by 3D tetrahedral elements. Die structure CAE modeling was conducted using CATIA linear CAE functionality, and the deflection of die structure is indicated in Figure 8. The maximum and minimum values are 0.223mm and 0.153mm, located at point A and point B respectively.

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Figure 8. Deflection contour on two symmetric punch geometries mounted at the lower workbench Boundary nodal deflection induced compensation (EL Comp) as shown in Figure 9 and thinning induced translational compensation (TH Comp) as shown in Figure 10 were added together to generate the compensation for die face. The elastic deformation of punch surface was also mapped on die face. In Figure 9, it shows that larger EL Comp is distributed in red regions, covering 0.0719mm~0.0799mm, while smaller EL Comp is distributed in blue regions, covering 0 mm~0.0080mm. Figure 10 shows that larger TH Comp is distributed in red regions, covering 0.1186mm~0.1318mm, while smaller TH Comp is distributed in blue regions, covering 0mm~0.0132mm. The two compensation values were further superposed. Maximum value in Z direction of total compensation amount on die face is 0.1562mm.

Figure 9. Compensation value on die face by elastic deformation

Figure10. Compensation value on die face by thickness reduction

The cloud point data at compensated region was used to calculate the matching ratio of the region. As shown in Figure 11, the region enclosed by red lines is defined as compensation region. When the threshold value of contact clearance is 0.05mm, 57.14% of matching ratio is obtained.

Compensation region

Figure 11. Compensated region surrounded by red lines

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The second case is an automotive floor panel, and the sheet metal is commonly used DC04 with initial blank thickness as 0.8mm. LS-Dyna was used to simulate drawing process, frictional coefficient is 0.09. In order to get accurate simulation result, real drawbead geometry was used for simulation, and the formability simulation result is shown in Figure 12.

Figure 12. Formability simulation result by LS-Dyna The boundary forces extracted were automatically mapped onto the die surface, punch surface and binder surface using the proposed algorithm mentioned above. HyperWorks was used to conduct die structure CAE analysis, and the die structure deformation in Z direction is shown in Figure 13, maximum deflection on the punch is 0.04536mm, while the maximum deflection on the die is 0.02153mm. In such process condition, the initial matching ratio is 66.58%.

(a) Elastic deflection on the punch (b) Elastic deformation on the die Figure 13. Die structure deflection in Z direction The proposed method was used to generate EL Comp value and TH Comp value, while the threshold contacting clearance is 0.05mm. After the compensation was added on die surface, the matching ratio was improved to 75.91%. Figure 14 shows the comparison before (left) and after (right) compensation. Less green points were displayed in the right picture.

Figure 14. Comparison of contact before and after compensation

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5. Conclusion Two important factors affecting the dynamic fitting accuracy of stamping dies, namely, thickness reduction of the deformed blanked and elastic deflection of die structure were taken into account for die surface compensation, and compensation algorithms of die face based on FEM simulation were studied. The definition of matching ratio is given. A new algorithm of searching massive nodes by sub-region was proposed, which can improve calculation efficiency. The compensation methods induced by thickness reduction and elastic deformation were analyzed, and were programmed and integrated with LS-Dyna and HyperWorks, and also integrated with Autoform and CATIA linear CAE functionality respectively. Furthermore, the software tool to calculate the matching ratio was also developed to evaluate the effectiveness of die face compensation. The aim of die surface compensation and matching-ratio optimization was achieved before the tryout of stamping dies.

ACKNOWLEDGMENTS The research was funded by the Ministry of Industrialization & Information Technology, China through the National Science & Technology Key Special Program with Grant #2009ZX04013-31, and co-funded by the National Natural Science Foundation of China through Grant #50821003 and the Doctoral Program of Higher Education of China through Grant #20090073120058.

REFERENCES 1. M. Firat. Materials and Design, 28(4):1304-1310 (2007) 2. R.Struck, S.Kulp, B.-A. Behrens, R. Krimm. SIMVEC-Numerical Analysis and Simulation in Vehicle Engineering 2008, 2031:185-199 (2008) 3. R.X.Gao, S.Sah, N.Mahayotsanun. CIRP Annals - Manufacturing Technology, 59(1): 399-402 (2010) 4. R.A. Lingbeek, T. Meinders. Proceedings of the 9th International Conference on Numerical Methods in Industrial Forming Processes, 908:723-728 (2007) th 5. A.Haufe, K. Roll, P. Bogon. Proceedings of the 7 International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, Interlaken, Switzerland, 743-748 (2008) 6. G.B. Zhang, J.Chen, Y.C.Tang, L.Chen, A.Wang, H.Xiao, L.Shi, H.M. Jiang. International Journal of Materials and Product Technology, 38(2-3):184-197 (2010) 7. G.B. Zhang, J.Chen, A.Wang. Chinese Journal of Mechanical Engineering, 44(8):174-179 (2008)

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Numerical Simulations of Forming Aluminum Beverage Can End Shells Jing Hana, Koetsu Yamazakib, Takashi Hasegawaa, Ryouiti Itoha, Sadao Nishiyamaa a

Universal Can Corporation, 1500 Suganuma, Oyama-Cho, Sunto, Shizuoka, 410-1392 Japan b

Kanazawa University, Kakuma-machi, Kanazawa, Ishikawa, 920-1192 Japan

Abstract. Forming simulations of can end shell have been implemented and compared with the experimental observations of the shell actual forming process. The influences of the loads applied to tools, the clearances between tools, the shapes of the tool profiles and the positions of tools, on the shell forming quality, have then been investigated numerically. The design optimization method based on the numerical simulations has been applied to search optimum design points, in order to reduce the amount of thinning subjected to the constraints of the shell geometric shape and the suppression of wrinkles. The optimization results show that the amount of thinning can be reduced up to 4% by optimizing the forming route, adjusting the clearances and the loads, and modifying the tool shapes. Keywords: Shell forming process, Thinning, Numerical analyses, Process optimum design.

INTRODUCTION Two-piece aluminum cans (Fig.1(a)) have been used as beverage containers for several decades due to their many merits, such as lightweight, high productivity and high recyclability, and hence environmental friendliness. A two-piece can is double seamed at the end and the body. The end is formed from a coil of coated aluminum sheet. First, the sheet is fed through a shell press to be formed into the basic end shell, and a sealing compound is then applied to the inside curl of the shell on a compound liner. The lined shell is formed and scored into a finished end shell in a conversion press through multi-forming processes, and the tab is also formed and attached to the end shell in the same press. Many researchers have investigated end shell forming and performance experimentally, analytically and numerically [1-6]. Can makers have continued research and development to pursue lighter can ends by reducing the end size and gauge while still keeping enough buckling strength by changing the shape of end shell profile. Efforts have been done to optimize the profile of the end shell to minimize the weight subject to the constraints of internal pressure resistance and other design requirements [5]. However, to form the end shell into a new geometrical shape, the optimum designs of the dies and forming process are also expected. Forming the sheet into the basic end shell involves three processes, namely, blanking, shell forming and curling. The purpose of the blanking operation is to cut the sheet into a circular blank, while that of the curling operation is to deform the curl of the shell into the shape required for seaming with the can body. The shell before curling is called the “uncurled end shell” (Fig.1(b)), and its forming process is investigated in this paper. Though a number of shell shapes and forming methods have been developed in the past years [2], this paper focuses on the uncurled shell of the MF206 end, that is, one of the common 206 diameter ends (outside diameter of the double seam is 2+6/16 = 2.375 inches, approximately 60 mm). The profile and nomenclature of the shell are shown in Fig. 2. After filling the beverage into the can body, the seaming panel radius of the end is placed in contact with the corresponding part of the can body. Then, the curl is double seamed with the flange of the can body by a seamer. The shape and thickness of the shell play an important role in the seaming quality and structural performance. Three geometrical dimensions The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 871-878 (2011); doi: 10.1063/1.3623697 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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of the shell, namely, the unit depth H1, the lip height H2 and the panel depth H3 are adopted for a quick check of the shell forming results. In this paper, the shell forming simulation is performed using the finite element (FE) method and is compared to experimental observations. The influences of the load applied to the tools, the clearance between the tools, the shape of tool profile and the initial position of the tool are then investigated numerically. Because a thinning reduction may increase the buckle pressure resistance of the shell, the required initial sheet thickness may be reduced, and lighter weight may then be expected to save materials. The design optimization method based on the numerical simulations is applied to search for optimum design points, to minimize the amount of thinning subject to the constraints of the geometrical shape of shell and the wrinkle suppression.

End Double Seam Body

(b) An uncurled end shell

(a) A 2-piece can

FIGURE 1. Photos of 2-piece aluminum beverage can and shell

Seaming panel Curl Seaming panel radius H2 Panel

Panel radius Chuck

H1

H3

Panel wall

Countersink radius FIGURE 2. Cross section of uncurled end shell

SHELL FORMING PROCESS SIMULATION The axisymmetric cross section of the tooling system utilized to form the end shell at its initial position is illustrated in Fig. 3. The system consists of three upper tools and three lower tools. The three upper tools are the draw die, the upper piston and the die center. Three lower tools are the lower piston, the die core ring and the panel punch. During shell forming process, constant loads of P1, P2, P3 and P4 are applied to the upper piston, the die center, the lower piston and the panel punch, respectively. The movement of the draw die is controlled by the shell press. The shell is formed by moving the draw die down to the bottom dead center, then up to the top dead center. The end shell is formed inside of a single-action shell press within one stroke, and its forming process is invisible. Therefore, shell forming simulations using the commercial FE code, MSC.MARC are performed, based on an axisymmetric model. All tools are assumed to be rigid. The cross section of the blank is discritized into axisymmetric elements with a four-node quadrilateral cross section. The material model used for the aluminum blank is an elasto-plastic von Mises material with isotropic hardening. A Young’s modulus E = 68.6 GPa, Poisson’s

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ratio ν = 0.33 and yielding stress σ0 = 300 GPa are assumed. The friction coefficient is assumed to be a constant value of 0.05. The initial thickness of the blank is T0 = 0.260 mm. Figure 4 shows the history curve of the minimum thickness during the forming process. The sheet profiles at the main hinges of the curve are displayed in the same figure, and the thinnest locations are indicated by arrows. It is observed that the minimum thickness decreases sharply between hinges NA and NB when the tip of the die center begins to apply a load to the sheet, and 4% thinning occurs at the panel radius. The minimum thickness remains almost constant but decreases slightly when the die center begins to draw the sheet into the die core ring cavity. The thickness decrease ratio increases again between hinges NC and ND, and another 3% thinning occurs at the contact region with the outside of the die center tip. After the sheet lip slides out of the clamp between the draw die and the lower piston, the minimum thickness remains constant until the die reaches bottom dead center. The minimum thickness increases somewhat between hinges NE and NF then drastically decreases when the panel punch rises to form the panel and the panel radius. Between NF and NG, another 4% thinning occurs in the panel wall. The crosssectional shape of the shell profile obtained by the sheet forming simulation is compared to sectioned shells formed by the shell press. The simulated and actual shell profiles matched each other well. Hence, the shell forming simulation can be used for further study of the shell forming process. Wrinkling is also one of important factors for evaluating the formed quality of the end. There are several ways to predict the wrinkle from the simulation results. This paper uses the Circumferential Plastic Strain (CPS) to predict the occurrence of wrinkling at the seaming panel radius of the shell. If the minimum CPS ε0 is smaller than its threshold value, wrinkles may appear. The shell formed by the shell press does not show wrinkles, so it is confirmed that the minimum value of the CPS at the seaming panel radius region, ε = −0.17, does not reach the critical limit to cause wrinkling. P2

Upper piston Die center

Draw die

P1

Aluminum Sheet

P4

P3

Panel punch

Lower piston Die core ring

FIGURE 3. Axisymmetric cross section of the tooling system

INFLUENCE FACTORS ON SHELL FORMING QUALITY The loads applied to the tools, profiles of the tool upmost surface, clearances between tools and the forming route (Fig.5) are considered to investigate the influences on the forming quality, such as the resulting shape of the shell profile, thinning and wrinkling. From the simulation results, it is concluded that the loads applied to the tools influence the geometrical shape of the shell profile and the minimum CPS more than the minimum thickness of the shell. The influence of the panel punch radius is also investigated, because the panel punch causes the minimum thickness to decrease sharply when the die center begins to apply the load to the sheet and when the panel punch rises to form the panel radius. In Models MD1 and MD2, as shown in Table 1, the cross section profile of panel punch corner is changed from a circular curve to elliptic curve. The panel diameter is required to remain constant to ensure enough space for the panel design of the end after the shell forming operation. Therefore, the length of the horizontal axis of the elliptic curve remains the same as the diameter of the original circular curve. The shape change

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ratio AR is defined as the ratio of the vertical axis length to the horizontal axis length of the elliptic curve. The simulation results are shown in Fig.6. As shown in Table 1, Model ME1 is calculated to study the influence of the change ΔLS in longitudinal clearance between the die center and the upper piston, while Model MF1 is used to understand the influence of the change ΔRS in the horizontal clearance between the draw die and the die core ring. The simulation results are also shown in Fig.6. To reduce further the amount of thinning, innovation is required in the shell forming process. The minimum thickness decreases sharply when the die center begins to apply the load to the sheet (from NA to NB in Fig.4) and before the sheet escapes the clamp by the draw die and the lower piston (from NC to ND in Fig.4). Therefore, new forming routes are tried by changing the longitudinal position of the die center and the panel punch. The die center of Models MG1 and MG2 in Table 1, are moved up distances of ΔLA = 3 mm and ΔLA = 6.25 mm at the initial stage, respectively, to delay drawing the sheet into the die core ring cavity. The panel punch of Models MH1 in Table 1 is moved down a distance of ΔLB = −6.5 mm at the initial stage to delay the contact between the metal and the panel punch. The shell forming simulation results, as shown in Fig.6, show that the minimum thickness increases up to 4%, the lip height decreases approximately 6% and the minimum CPS decreases approximately 2% for the Model MG2.

0.265 0.260

NA

M inimum thickness (mm)

0.255 0.250

NB 0.245

NC

NF

0.240

NE ND

0.235 0.230

NG

0.225 0.220 0

2

4

6

8

10

12

Stroke (mm) Draw die movement (mm) FIGURE 4. History of the minimum thickness and location

874

14

16

ΔLS ΔRS ΔLA ΔLB Elliptic curve Circular curve

FIGURE 5. Changes in the tooling system

TABLE 1. Changes in the tooling system models ΔLS Model No. AR (mm) M0 1.00 0 MD1 1.30 MD2 0.68 ME1 0.035 MF1 MG1 MG2 MH1

ΔRS (mm) 0

ΔLB (mm) 0

ΔLA (mm) 0

−0.02 3.00 6.25

− 6.5

10.0% Unit depth Lip height Panel depth M inimum thickness M inimum circumferential plastic strain

8.0%

Rate of change

6.0% 4.0% 2.0% 0.0% -2.0% -4.0% -6.0% -8.0% MD1

MD2

ME1

MF1

MG1

M odel No. FIGURE 6. Influence on forming quality

875

MG2

MH1

OPTIMUM DESIGN OF FORMING PROCESS On the basis of the shell forming simulation results, it is observed that the amount of thinning can be reduced by selecting the proper forming route. Meanwhile, the geometric shape of shell profile can remain constant by designing the die shape, the clearance and the loads applied to the tools. The deign optimization method is applied to achieve the optimum design for forming the shell. The objective of design optimization is to minimize the amount of shell thinning, which is equivalent to maximizing the minimum thickness of the shell. The unit depth H1, the lip height H2 and the panel depth H3 are restricted within prescribed limits to ensure the shape of shell profile for the following shell forming process, the seaming operation and to maintain the required buckling pressure resistance. To avoid wrinkling, the minimum CPS ε0 is also considered to be larger than an allowable minimum bound. The design variables may be selected from the forming conditions, the die shape and the positions. The problem is then posed as: Find design variables: X = {xi } , i = 1, . . . , n (n: the number of design variables) that (1) maximize f = Tmin (X), (2) subject to g1 = H1min / H1 (X) − 1 ≤ 0 , g2 = H1 (X) / H1max − 1 ≤ 0, g3 = H2min / H2 (X) − 1 ≤ 0, g4 = H2 (X) / H2max − 1 ≤ 0, g5 = H3min / H3 (X) − 1 ≤ 0, g6 = H3 (X) / H3max − 1 ≤ 0, (3) g7 = εmin /ε (X) − 1 ≤ 0,

xi ≤ xi ≤ xi , i = 1,..., n L

U

(4)

where H1min and H1max, H2min and H2max, H3min and H3max are the allowable upper and lower bounds of the unit depth H1, the lip height H2 and the panel depth H3, respectively. The allowable lower minus bound of the CPS is εmin, and U

L

xi and xi are the upper and lower bounds of design variable i, respectively. The Response Surface Approximation (RSA) method [7], based on the FE analyses, is used to solve the optimization problem. First, the design variables are selected and the design space is defined. Then, an orthogonal array in the design-of-experiment is used to arrange the design points, and the FE code is utilized to implement the simulation of the forming process. The RSA technique is then applied to generate an approximate response surface in terms of the selected design variables. The numerical optimization program is then used to perform the optimization calculation based on the response surfaces. If it is necessary to move the design space according to judgment based on the optimization results, then the above steps are repeated. As a numerical example, the vertical moving distance of the die center at the initial stage ΔLA, the change in the longitudinal clearance ΔLS between the die center and the upper piston, the rate of change in the load applied to the panel punch AP4, and the shape change ratio ΑR of the panel punch are selected as the design variables. The design space in the first design cycle is defined as: 0 ≤ ΔL A ≤ 7.0mm, − 0.2mm ≤ ΔL S ≤ −0.1mm, 0.95 ≤ AP 4 ≤ 1.05, 0.5 ≤ AR ≤ 1.5. The sheet forming simulations are performed for 9 design points that are arranged using the orthogonal array L9. The numerical results at each design point are shown in Fig.7. The relative influence ratios [8] of each design variable are shown in Fig.8, which shows that the design variable ΔLA influences the minimum thickness, the lip height and the minimum CPS strongly. The design variable AP mainly influences the panel depth, and the design variable AR mainly influences the unit depth and panel depth. The design variable ΔLS is used to finely adjust the optimum point. On the basis of the numerical simulations, the response surfaces of the minimum thickness, the unit depth, the lip height, the panel depth and the minimum CPS are constructed in terms of the four design variables, and the approximated optimization problem is then solved by the mathematical programming. The optimum values for the design variables at the optimum point are obtained as ΔLA = 6.45 mm, ΔLS = −0.10 mm, AP4 = 0.95, ΑR = 1.50. Although three design variables reach the lower or upper bounds, the design space of AP4 and AR is not changed to ensure a safe design, whereas the upper bound of ΔLS is moved to −0.05 mm in the second design cycle. The final optimum value of ΔLS = −0.07 mm is calculated based on three new design points. The forming simulation results at the optimum design point show that the minimum thickness Tmin = 0.240 mm, changes in H1, H2 and H3 are within 0.03 mm, 0.02 mm and 0.02 mm, respectively, and ε0 is the same as that of baseline. The history curve of the minimum thickness at the optimum point is compared to that of baseline, as shown in Fig. 9. The amount of thinning is reduced from 11.9% to 7.7%.

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15%

Unit depth Lip height Panel depth Minimum thickness minimum circumferential plastic strain

Rate of change

10% 5% 0% -5% -10% 1

2

3

4 5 6 Design point

7

8

9

FIGURE 7. Simulation results of design points

Relative influence ratio

2.0

Unit depth Lip height Panel depth Minimum thickness Minimum circumferential plastic strain

1.5 1.0 0.5 0.0

ΔLA LA

ΔLS LS AAP P4 Design variable variable

AAR R

Minimum thickness (mm)

FIGURE 8. Relative influence ratio

0.27

Model M0 Optimum Model

0.26 0.25 0.24 0.23 0.22 0

5

10

15

Draw die movement (mm)

20

FIGURE 9. Minimum thickness changing curves

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CONCLUSIONS Simulations of the forming of end shells have been conducted to obtain a better understanding of the forming process. The simulation results agree well with experimental observations on the actual shell forming process. The influences of the loads applied to the tools, the clearances between the tools, the shapes of the tool profiles and the positions of the tools have also been investigated. The design optimization method based on the numerical simulations has been applied to search the optimum design points to reduce the amount of thinning subjected to the constraints of the shell geometric shape and the suppression of wrinkles. The optimization results show that the amount of thinning can be reduced from 11.9% to 7.7% by optimizing the forming route, adjusting the longitudinal clearance between the die center and upper piston, adjusting the load applied to the panel punch and modifying the shape of the panel punch corner. Furthermore, it was shown that the optimum design method based on shell forming simulations may be performed to develop new tooling systems and new shell profiles.

REFERENCES 1 Trageser, A. B., Dick, R. E., Aluminum can design using finite element methods, Proceedings of the SME Can Manufacturing Technical Symposium, Chicago, 1988. 2 Biondich, S.C., Dick, R.E., A new reformed shell forming process, Transactions of the North American Manufacturing Research Institution of the Society of Manufacturing Engineers (NAMRI/SME), 18, 1990, pp.33-40. 3 MacEwen S.R., Perrin, R., Green, D., Makinde, A., Neale, K., The science of modeling can forming and performance, Morris, J.G., Merchant, H.D., Westerman, E.J., Morris, P.L. (eds.), Aluminum Alloys for Packaging, the Minerals, Metals & Materials Society, 1993, pp. 85-101. 4 Chu, E.W., Biondich, S.C., An understanding of coining beverage can ends. J Mater Process Manuf Sci, 2, 1994, pp.285-303. 5 Yamazaki, K., Itoh, R., Watanabe, M., Han, J., Nishiyama, S., Applications of structural optimization techniques in light weighting of aluminum beverage can ends, J Food Eng, 81, 2007, pp.341–346. 6 Han, J., Nishiyama, S., Yamazaki, K., Itoh R., Ergonomic design of beverage can lift tabs based on numerical evaluations of fingertip discomfort. Appl Ergon. 39(2), 2008, pp.150-157. 7 Myeres, R.H., Montgomery, D.C., Response Surface Methodology-Process and Product Optimization Using Designed Experiments, John Wiley & Sons, New York, 1995. 8 Han, J., Itoh, R. Nishiyama, S., Yamazaki, K., Application of structure optimization technique to aluminum beverage bottle design, Struct Multidiscip O, 29(4), 2005, pp.304-311.

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New Tooling System for Forming Aluminum Beverage Can End Shell Koetsu Yamazakia, Takayasu Otsukaa, Jing Hanb, Takashi Hasegawab, Taketo Shirasawab a

b

Kanazawa University, Kakuma-machi, Kanazawa, Ishikawa, 920-1192 Japan

Universal Can Corporation, 1500 Suganuma, Oyama-Cho, Sunto, Shizuoka, 410-1392 Japan

Abstract. This paper proposes a new tooling system for forming shells of aluminum beverage can ends. At first, forming process of a conversional tooling system has been simulated using three-dimensional finite element models. Simulation results have been confirmed to be consistent with those of axisymmetric models, so simulations for further study have been performed using axisymmetric models to save computational time. A comparison shows that thinning of the shell formed by the proposed tooling system has been improved about 3.6%. Influences of the tool upmost surface profiles and tool initial positions in the new tooling system have been investigated and the design optimization method based on the numerical simulations has been then applied to search optimum design points, in order to minimize thinning subjected to the constraints of the geometrical dimensions of the shell. At last, the performance of the shell subjected to internal pressure has been confirmed to meet design requirements. Keywords: Optimum design, Sheet metal forming, Finite element analyses, Thinning reduction.

INTRODUCTION For developing aluminum beverage containers, the Finite Element (FE) method and structure optimization method based on numerical simulations have been applied to predict performance, to simulate the can forming process, operating processing and to obtain optimum solutions [1-10]. Sheet metal forming simulation technology has been continuously developed and computer’s capacity and processing speed have been greatly improved recent years, which make it more realistic to design and optimize dies and the forming processes based on numerical simulations. Figure 1(a) shows a basic end shell of a two-piece aluminum beverage can. The nomenclature and three kinds of basic geometric dimensions of the shell, the unit depth H1, the lip height H2 and the panel depth H3, are shown in Fig. 1(b). The thickness and the dimensions of shell play an important role in seaming quality as well as structural performance. In our previous study for the end shell development, structural optimization technology based on numerical simulations has been applied to minimize the weight of end shells subject to constraints of the buckling strength, the panel growth suppression and the other design requirements [9]. Numerical simulations and optimum design have also been performed for the shell forming process [10]. This paper proposes a new tooling system for forming the end shell. At first, forming process of a conventional tooling system is simulated using three-dimensional finite element models. The new tooling system is then proposed based on the conventional tooling system. Influences of the tool upmost surface profiles and tool initial position of the new tooling system are investigated and the design optimization method based on the numerical simulations are then applied to search optimum design points, in order to minimize thinning subjected to the constraints of the geometrical dimensions of the end shell. At last, the performance of the shell subjected to internal pressure is confirmed to meet design requirements.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 879-886 (2011); doi: 10.1063/1.3623698 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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Seaming panel Curl Seaming panel radius H2 H1 Panel radius Chuck

Panel wall

Panel

H3

Countersink radius (a) An end shell

(b) Cross section of end shell

FIGURE 1. Aluminum beverage can end shell

SHELL FORMING PROCESS UTILIZING CONVENTIOANL TOOLING SYSTEM The axisymmetric cross section of conventional tooling system utilized to form the end shell at its initial position is illustrated in Fig. 2. The system consists of three upper tools and three lower tools. The three upper tools are the draw die, the upper piston and the die center. Three lower tools are the lower piston, the die core ring and the panel punch. During the shell forming process, loads of P1, P2, P3, P4, are applied constantly to the upper piston, the die center, the lower piston and the panel punch, respectively. Movement of the draw die is controlled by the shell press. The shell is formed by moving the draw die down to bottom dead center and then up to top dead center. Shell forming simulations using the commercial FE code, LS-DYNA were performed, based on three dimensional models. All tools were assumed to be rigid. A quarter of the circular blank was modeled with quadrilateral shell elements. The material model used for the blank is an elasto-plastic von Mises material with isotropic hardening. Young’s modulus E = 68.6 GPa, Poisson’s ratio  = 0.33 and the yielding stress 0 = 0.3 GPa, mass density 2.7e-6 kg/mm3 were assumed. The friction coefficient between die surfaces and the blank was assumed to be 0.05, on the basis of reference studies and influence studies of friction condition. The initial thickness of the blank is T0 = 0.260 mm. Shell forming is finished within about 0.06 sec. Figure 3 shows the forming simulation results of the base model of the conventional tooling system, Model CTS (Conventional Tooling System). The simulation starts as shown in Fig. 3(a). With the draw die traveling downward, the upper piston, the die center and the lower piston move down simultaneously. The sheet flange clamped by the draw die and the lower piston gradually slides into the clearance between the die core ring and the draw die, and at the same time the center of the sheet is drawn into the die core ring cavity by the die center as shown in Fig. 3(b). After reaching bottom dead center as shown in Fig. 3(c), the die center is lifted up by the draw die, and the panel punch rises to form the panel wall as shown in Fig. 3(d). After the panel of the shell is clamped tightly by the die center and the panel punch, the upper piston is lifted up by the shell as shown in Fig. 3(e). The shell forming is finished after the upper piston is lifted up by the draw die as shown in Fig. 3(f). Figure 4 shows the thickness reduction distribution of Model CTS, in which minus values mean the thickness increase. As shown in Fig.4(a), at bottom dead center, the minimum thickness of the shell is located at contact area with the outer surface of the die center tip. Figure 4(b) shows that the location of the minimum thickness of the shell in its final profile is at the contact area with the inner surface of the die center tip. It is observed that the thickness of the panel has almost no change. The thickness becomes thinner than its initial value at portion of panel radius, and reaches its minimum value Tmin at the panel wall. The thickness returns to its initial value and then decreases again at the portion from the center of the countersink radius to the outer part of the countersink radius. At the portion from the chuck wall to the edge of the shell, the thickness becomes thicker than its initial value. The maximum thickness reduction at bottom dead center is about 7%, and final maximum thickness reduction in the panel wall is about 11%.

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P2

Upper piston Die center

Draw die

P1

Aluminum Sheet

P4

Panel punch

Lower piston Die core ring

P3

FIGURE 2. Axisymmetric cross section of the tooling system

(a) Time = 0 ms

(b) Time = 8 ms

(d) Time = 38 ms

(c) Bottom dead center

(e) Time = 48 ms

(f) Final profile

FIGURE 3. Forming process of Model CTS

7% 3% - 6% - 4% - 8% -12% -15%

11% 7% 2% - 2% - 7% -11% -16%

(a) Profile when reaching bottom dead center

(b) Final profile

FIGURE 4. Thickness reduction distribution of Model CTS

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It is confirmed that the above simulation results agree well with simulation results based on an axisymmetric model as well as experimental observations [10]. Because computational time of the axisymmetric model is much shorter than that of the three-dimensional one, the axisymmetric model is used for further study.

SHELL FORMING PROCESS UTILIZING A NEW TOOLING SYSTEM As observed above, since the tip of the die center is one of the key factors to cause thinning during shell forming, the tip is removed in the new tooling system proposed to improve further forming quality. Figure 5 illustrates the upmost cross-sectional surfaces and initial position of dies in the new tooling system. To form an end shell that is equivalent or more to one formed by the conversional tooling system, the initial position of the upper piston and load P1 in the new tooling system were subordinately adjusted. The base model of the new tooling system is referred as Model NTS (New Tooling System). Figure 6 shows the shell forming simulation results of the new tooling system, Model NTS. It is obvious that movement of dies in Model NTS is similar to that in Model CTS until reaching bottom dead center. However, the countersink radius of the shell in Model NTS is then formed freely without any restraint from the die center. The forming process is finished after the upper piston is lifted up by the draw die. Figure 7 shows the thickness distribution of Model NTS after forming process. The thinnest portion is located around the panel radius of the shell and is improved about 3.6%, as compared with that of Model CTS. Influence factors, such as loads applied to the tools, profile of the tool upmost surface and clearance between tools were investigated to understand sensitivities on the formed results. Figure 8 shows simulation results of models used to study the upmost surface profiles of the panel punch and the die center, the up moving distance of the die center at its initial position 'LA. As shown in Fig.5, AR is the axis length ratio of the vertical axis to horizontal axis of the curve at the panel punch elliptical corner. If AR = 1, the curve is a circular curve, otherwise, the curve is an elliptic curve. Similarly, CR is the axis length ratio of the vertical axis to the horizontal axis of the curve at the die center corner. The length of horizontal axes of the curves is kept the same as those of Model NTS. The rates of change in the unit depth H1, the lip height H2, the panel depth H3, and the minimum thickness Tmin, as compared with Model NTS in which AR = CR = 1, 'LA = 0, were obtained when changing AR, CR and 'LA. It is observed that the minimum thickness of the shell increases about 2.3%, 1.0%, 0.5% when AR = 1.5, 'LA = 7.0, CR = 0.75, respectively. Moreover, it is found that 'LA and CR give relatively large influence on the lip height. The panel depth increases almost 2% when CR = 0.75, and the unit depth decreases about 1% when 'LA = 7.0.

Upper piston Die center

Panel punch

Draw die

CR AR

Sheet Lower piston

Die core ring

FIGURE 5. Upmost surfaces of dies in new tooling system

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(a) Time = 0 ms

(b) Time = 10 ms

(c) Bottom dead center

(d) Time = 42 ms

(e) Time = 52 ms

(f) Final profile

FIGURE 6. Forming process of Model NTS

5

Unit depth Lip height Panel depth Minimum thickness

4 66

Rate of change [%]

3

Thickness [mm] Thickness [mm]

Shell profile profile [mm] [mm] Shell

0.290 0.29 0.280 0.28 0.270 0.27 0.260 0.26

33

0.250 0.25 0.240 0.24 0.230 0.23

Thickness 00 distribution

0.220 0.22

-3 -3

Shell profile 20

25 30 Radial direction [mm]

2 1 0 -1 -2 -3 -4

35

-5 0.5

AR

1.5

3.5

'LA

7.0

0.75

CR

1.25

FIGURE 7. Thickness distribution of Model NTS FIGURE 8. Influences on forming quality

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OPTIMUM DESIGN OF NEW TOOLING SYSTEM On the basis of the shell forming simulation results, it is observed that thinning can be improved, meanwhile, the geometric dimensions of the shell can be kept within a range by designing the die shape and by selecting proper forming route. The deign optimization method is applied to perform optimum design for forming the end shell. The objective of the optimization is to minimize thinning of the shell due to the forming process, which is equivalent to maximize the minimum thickness. The unit depth H1, the lip height H2 and the panel depth H3 are restricted within the prescribed limits, to ensure the shape of the shell profile for the following shell forming process, the seaming operation and for keeping required buckling strength. The design variables may be selected from the forming conditions, the die shape and position. The problem is then posed as: Find design variables: X = {xi } , i = 1, . . . , n (n: the number of design variables) maximize f = Tmin (X), subject to g1 = H1min / H1 (X)  1  0 , g2 = H1 (X) / H1max  1  0, g3 = H2min / H2 (X)  1  0, g4 = H2 (X) / H2max  1  0, g5 = H3min / H3 (X)  1  0, g6 = H3 (X) / H3max  1  0,

xi d xi d xi , i L

U

1,..., n

(1) (2)

(3) (4)

where H1min and H1max, H2min and H2max, H3min and H3max are the allowable upper and lower bounds of the unit depth H1, the lip height H2, and the panel depth H3, respectively. xi

U

L

and xi are the upper and lower bounds of design

variable i, respectively. The Response Surface Approximation (RSA) method [11] based on the FE analyses is used to solve the optimization problem. At first, select the design variables and define the design space. Then, an orthogonal array in the design-of-experiment is used to arrange the design points and the FE code is utilized to implement the simulation of forming process. The RSA technique is then applied to construct an approximate response surface in terms of the selected design variables. The numerical optimization algorithm is then used to perform the optimization calculation based on the response surfaces. If it is necessary to change the design space, according to judgment based on the optimization results, then the above steps are repeated. As a numerical example, the axis length ratio of the panel punch corner AR, the increase of up moving distance of the die center at its initial position 'LA, the axis length ratio of the die center corner CR, are selected as the design variables. The panel punch is moved down to be at the same horizontal level as the die core ring. The lower and upper bounds of the design variables are defined to be 2.0 d AR d 3.0 , 0.0 mm d 'LA d 10.0 mm, 0.75 d C R d 1.25 , and the rates of changes in shape dimensions of the shell are restricted from 0.7% to +0.7% for H1, from 2.5% to +16.4% for H2, from 0.9% to +3.6% for H3. Each design variable has three equal design levels and 27 design points are arranged using the orthogonal array L27. The shell forming simulations are carried out for each design point, and rates of changes in the minimum thickness and dimensions of the shell are calculated as shown in Fig. 9. We can observe that the minimum thickness at Design point 8 (AR = 2.0, 'LA = 10.0, CR = 1.0) increases almost 4% and changes in geometric dimensions are within the range of restrictions, hence Design point 8 is considered as a candidate of optimum solutions. Based on the simulation results, the response surfaces of the minimum thickness, the unit depth, the lip height and the panel depth are then constructed in terms of the design variables, and the approximated optimization problem is then solved by the mathematical programming method. However, no solution better than Design point 8 is obtained. Therefore, Design point 8 is the optimum design point, and is referred as Model NTOS (New Tooling Optimum System). Figures 10 and 11 show the thickness distribution and the forming process of Model NTOS.

CONFIRMATION OF SHELL PERFORMANCE Due to different tooling system, hence, different forming processes, countersink radius profile and thinning location of the shell formed by the new tooling system are different from those of conventional shells. Both of shell

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shape and thickness are critical to the shell performance under internal pressure, so a simplified buckling strength analysis model under internal pressure is created as shown in Fig. 12(a) to confirm that the new shell’s buckle assistant meets design requirement of the end. Figure 12(b) shows a deformed shell when buckling, simulated by the FE code, MSC.MARC. The shell deforming simulation results indicate that the buckle pressure resistance of Model NTOS is 3.9% larger than that of Model NTS, and 5.5% larger than Model CTS. 8 Lip height

Panel depth

Minimum thickness

4

0

-4

-8 0

3

6

9

12

15

18

21

24

27

Design point

FIGURE 9. Numerical simulation results of each design point. 0.290 0.29 0.280 0.28 0.270 0.27

0

0.260 0.26 0.250 0.25

Shell profile [mm] Shell profile [mm]

3

Thickness䇭[mm] Thickness [mm]

Rate of change[%]

Unit depth

-3 Thickness distribution

0.240 0.24 0.230 0.23

Shell profile

0.220 0.22

-6 25 30 35 25 30 35 Radial direction direction [mm] [mm] Radial FIGURE 10. Thickness distributionof optimum design Model NTOS 20 20

(a) Initial (b) Bottom dead center (c) Final profile FIGURE 11. Forming process simulation of optimum design Model NTOS

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(a) Analysis model (b) Deformed shell when buckling FIGURE 12. Buckling strength analysis

CONCLUSIONS The shell forming simulation of the conventional tooling system has been implemented based on threedimensional models. Simulation results of three-dimensional models and axisymmetric models have been confirmed to be consistent with each other, and simulations for further study have been performed based on axisymmetric models to save computational time. A new tooling system was proposed and thinning improvement of about 3.6% has been achieved. The influences of the shape of the tool upmost surfaces and the tool initial position have been then investigated and the design optimization method based on the numerical simulations have been applied to search an optimum design points, in order to reduce the thinning subjected to the constraints of the geometric dimensions of the shell. The optimization results show that thinning can be improved up to 4%. Finally, the buckle pressure resistance of the shell formed by the new tooling system has been confirmed to be larger than that of the shell formed by the conventional tooling system.

REFERENCES [1] Trageser, A. B. and Dick, R. E., “Aluminum Can Design Using Finite Element Methods,” Procedings of the SME Can Manufacturing Technical Symposium, Chicago, 1988 . [2] Biondich, S.C. and Dick, R.E., “A New Reformed Shell Forming Process,” Transactions of the North American Manufacturing Research Institution of the Society of Manufacturing Engineers (NAMRI/SME), 18, 1990, pp.33-40. [3] Boers, S.H.A., Schreurs, P.J.G. and Geers, M.G.D., “Operator-split Damage-plasticity Applied to Groove Forming in Food Can Lids,” International Journal of Solids and Structures, 42, 2005, pp.4154-4178. [4] Han, J., Itoh, R. Nishiyama, S. and Yamazaki, K., “Application of Structure Optimization Technique to Aluminum Beverage Bottle Design,” Structural and Multidisciplinary Optimization, 29(4), 2005, pp.304-311. [5] Han, J., Yamazaki, K. and Nishiyama, S., “Optimization of the Crushing Characteristics of Triangulated Aluminum Beverage Cans,” Structural and Multidisciplinary Optimization, 28(1), 2004, pp.47-54. [6] Han, J., Yamazaki, K., Itoh R. and Nishiyama, S., “Multi-Objective Optimization of a Two-Piece Aluminum Beverage Bottle Considering Tactile Sensation of Heat and Embossing Formability,” Structural and Multidisciplinary Optimization, 32(2), 2006, pp.141-151. [7] Han, J., Nishiyama, S., Yamazaki, K. & Itoh R., “Ergonomic Design of Beverage Can Lift Tabs Based on Numerical Evaluations of Fingertip Discomfort,” Applied Ergonomics, 39(2), 2008, pp.150-157. [8] Chihara, T., Yamazaki, K., Itoh, R., Han, J., Evaluation of drinking ease relative to the opening diameter and beverage type of aluminum beverage bottles, Journal of Food Engineering, 95(2), 2009, pp. 264–271. [9] Yamazaki, K., Itoh, R., Watanabe, M., Han, J. & Nishiyama, S., “Applications of Structural Optimization Techniques in Light Weighting of Aluminum Beverage Can Ends,” Journal of Food Engineering, 81, 2007, pp.341–346. [10] Han, J., Yamazaki, K., Hasegawa, T., Itoh, R. and Nishiyama, S., Numerical Simulations of Forming Aluminum Beverage Can End Shells,” Proceeding of the 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes (Numisheet 2011), Seoul, Korea, Paper No.083, 2011. [11] Myeres, R.H. and Montgomery, D.C, Response Surface Methodology-Process and Product Optimization Using Designed Experiments, John Wiley & Sons, New York, 1995.

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Investigation on Twisting and Side Wall Opening Occurring in Curved Hat Channel Products Made of High Strength Steel Sheets Masato Takamuraa , Ayako Fukuib, Hiroshi Yanoc, Takayuki Hamad, Hideyuki Sunagaa, Akitake Makinouchia, and Motoo Asakawae a

VCAD System Research Program, RIKEN, 2-1 Hirosawa, Wako-shi, Saitama, 351-0198, Japan Graduate School of Fundamental Science and Engineering, Waseda University, Tokyo, Japan c UNIPRES Corporation, Yokohama, Japan d Department of Energy Science and Technology, Kyoto University, Kyoto, Japan e Department of Applied Mechanics and Aerospace Engineering, Waseda University, Tokyo, Japan b

Abstract. High strength steel sheets are becoming increasingly important for the weight reduction of automotive bodies to meet the requirements for reduced environmental impact. However, dimensional defects resulting from springback are serious issues, and effective methods of predicting and reducing such defects are necessary. In this study, we numerically and experimentally analyzed the mechanisms of dimensional inaccuracies caused by springback occurring in curved hat channel deep drawing products. The analysis was based on the static explicit FEM software “TP-STRUCT” (the solver part is known as “STAMP3D”). The results of the experiments and simulations similarly show that the twist angle is positive (right-hand system) when the drawing height is relatively large. We calculated the twist torque around the longitudinal axis using the stress distributions obtained by FE analysis. Through the investigation of twist torque and its transition during the drawing and die removal processes, we found that the negative torque generated by side wall opening occurring in the die removal process is the dominant factor of the positive twist. Knowing such mechanisms of twist in cases with a relatively large drawing height, we attempted to explore methods of reducing side wall opening by giving the side wall a stepped shape with the eventual aim of reducing twist. Consequently, we concluded that the stepped shape on the side wall has marked effects of reducing side wall opening, mainly through the elimination of bendingunbending effects on die shoulders, which was verified by observing the stress distribution obtained by FE analysis. Keywords: High strength steel, Hat channel, Twist, Torque, Side wall opening, FEM PACS: 62.20.F-

INTRODUCTION High strength steel sheets are increasingly applied in automotive body parts as a means of reducing weight. However, the twisting caused by the significant springback typically observed in channel-shaped products made of high strength steel sheets has become a serious issue. Therefore, it is crucial to establish a method of reducing the amount of twist by studying the mechanism by which it is generated. For this purpose, several studies to clarify the causes of twist through experiments [1, 2] or by evaluating twist torque at the bottom dead point through numerical procedures [2, 3] have been carried out. In this study, we are concerned with the twist occurring in curved hat channel products, and propose a numerical procedure to analyze the source of twist. Furthermore, we attempt to explore methods of reducing side wall opening by giving the side wall a stepped shape with the eventual aim of reducing twist, by determining the mechanisms of the twist through the proposed procedure.

STATIC-EXPLICIT FEM CODE In the “TP-STRUCT” (the solver part is known as “STAMP3D [4, 5]”) code, the updated Lagrangian rate formulation is employed to describe the finite deformation problem. Hill’s quadratic yield function and the The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 887-894 (2011); doi: 10.1063/1.3623699 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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associated flow rule are used. In this code, a static-explicit approach is applied. Assuming that the relations described by rate-type equilibrium equations and constitutive equations are preserved from time t to t  't , where 't is a small time increment, all rate quantities are simply replaced by the incremental quantities, and the corotational stress increment ' J and the strain increment 'd can be written as

' J

 J ˜ 't , 'd

D ˜ 't ,

(1)

where  J and D are the Jaumann derivative of the Cauchy stress and the strain rate tensor, respectively. To prevent large nonequilibrated forces arising owing to the explicit time integration, the “r-min” method proposed by Yamada et al. [6] is employed to impose a limitation on the size of the time step.

INVESTIGATION OF MECHANISMS OF TWIST Experimental Procedure The geometry of the curved hat channel model employed in this study is schematically shown in Fig. 1. The model had a curved area and two straight areas adjoining both sides of the curved area. The curved area had a radius of 400 mm with a central angle of 36 degrees, and the straight areas had a length of 150 mm. The sheet was drawn at six different drawing heights H ranging from 10 mm to 60 mm as shown in Fig. 2. The punch and die shoulder radii were 6 mm. The blank holding force was 1,200 kN. Dual-phase high strength steel sheets with a tensile strength of 980 MPa and a thickness of 1.6 mm were used. 36 deg.

(60)

R6 R400

150

60

FIGURE 1. Geometry of curved hat channel model

H = 10mm 20mm 30mm 40mm 50mm 60mm FIGURE 2. Schematic view of formed panels with various drawing heights H

Analysis Procedure FE Analysis Model Eight-node brick elements with selective reduced integration were used for the sheet model. The model had four elements through thickness, which was empirically sufficient for the prediction of bending and springback [7]. Only half of the channel shape was simulated owing to its symmetric geometry. The stress-strain behavior was approximated using Swift’s equation. To simplify the model, kinematic hardening was not taken into consideration. The mechanical properties of the sheet material are shown in Table 1. TABLE 1. Mechanical properties used in simulations of curved hat channel Swift law approximation Yield Young’s Stress modulus Q C/MPa n H0 E/GPa V0/MPa 212 708 0.3 1411 0.10 0.00001

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Sec 0 Sec 2 Sec 4 Sec 6 Sec 8

Sec 1 Sec 3 Sec 5 Sec 7

FIGURE 3. Definition of cross sections

Measurement cross section

Reference cross section

I Twist angle FIGURE 4. Definition of twist

Evaluation of Twist Torque We evaluated the twist angles and twist torques of the models during the deformation by the procedure proposed in the literature [2]. The procedure is briefly explained below. We defined nine cross sections perpendicular to the longitudinal direction (Fig. 3). The twist angles I of the punch bottom between the cross section on the plane of symmetry (Sec 0 in Fig. 3) and other cross sections were calculated (Fig. 4). The twist torque of each cross section is calculated as

T

¦W

LC ,i

˜ Li ˜ Ai

(2)

i

where i is the index of the quadrilateral domain generated by cutting the brick elements by the cross section, WLC,i is the stress component in the local coordinate system shown in Fig. 5 with L and C denoting the longitudinal and circumferential directions, respectively. The radial direction is defined to be that of the vector from the centroid of the cross section to a point on the material. Li is the distance between the quadrilateral domain and the centroid, and Ai is the cross-sectional area of the quadrilateral domain (Fig. 6). In addition, the distributions of torque at a quadrilateral domain over the sheet width are also examined. For detail, the reader is referred to the literature [2]. After the sheet was drawn, the die was first removed, followed by the punch to clearly examine the transition of twist torque. We focused our investigation on three stages during the process of tool removal; at the bottom dead point, after die removal, and after punch removal. We assumed the right-hand screw direction with respect to the longitudinal axis as the positive direction for both twist and torque.

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ªV x W yx W zx º « » «W xy V y W zy » «W xz W yz V z » ¬ ¼

Centroid

ª V L W RL W CL º «W » « LR V R W CR » «¬W LC W RC V C »¼

C : Circumferential direction R : Radial direction L : Longitudinal direction

FIGURE 5. Definition of twist

Li Ai

i

Centroid

ª V L W RL W CL º «W » « LR V R W CR » ¬«W LC W RC V C »¼

FIGURE 6. Physical quantities used in the Calculation of twist torque

Results and Discussion

Twist Angle Ideg.

Figure 7 shows the relationship between drawing height and twist angle obtained by the simulation and experiment. The twist angle is negative at the drawing heights of 10 and 20 mm, and increases with the drawing height. Eventually it is positive for drawing heights of 40 mm and above. It is negligible at the drawing height of 30 mm. The results obtained by the simulation and the experiment show a similar tendency. It is presumed that the mechanism that determines the twist angle is different depending on the drawing height. In this study we focus on the result at the drawing height of 60 mm and the mechanism that determine the twist angle is examined using the twist torques at the three stages during the process of tool removal. 12.0 8.0

Experimental results

4.0 0.0 -4.0

Numerical results

-8.0

-12.0 0

10

20

30

40

50

60

Drawing height H /mm FIGURE 7. Relation between drawing height and twist angle

Figure 8 shows the twist torque distribution along the longitudinal direction of the model at the drawing height of 60 mm. The twist torque is positive at the bottom dead point, then becomes negative after the die is removed, and finally becomes negligible after the punch is removed. This result indicates that factors that generate twist may be different during between the die removal and punch removal; thus the resultant twist may be determined by the net effect of multiple factors

890

Twist torque T /N㺃m

1500 Bottom dead point

1000 500 After punch removal

0 -500 After die removal

-1000 0

50 100 150 200 250 300 Length along longitudinal direction from symmetry plane to measurement section L /mm

FIGURE 8. Twist torque distribution (drawing height H=60mm)

From the above-mentioned results, there are two main factors that are considered to cause the twist as follows. The first factor is the shear stresses that arise during the drawing process. Since the flange regions of the curved area undergo in-plane tensile and compressive deformation during the drawing process, shear deformation may arise in the flange and side wall regions. The twist torque can be roughly explained in conjunction with the in-plane shear stress component WLW, where W denotes the width direction of the sheet along the deformed geometry. Figure 9 shows the distribution of the shear stress component WLW at the bottom dead point. Large shear stresses are clearly observed at the drawing height of 60 mm, suggesting that the twist torque is positive owing to the in-plane shear deformation, which is the source of negative twist. Distribution of WLW /MPa 100

-100 H= 20 mm

300

In plane tensile and compressive deformation 100

100 300

H= 60 mm -100

FIGURE 9. Distribution of shear stress

W LW

at the bottom dead point

Stretching deformation

Shrinking deformation FIGURE 10. In-plane deformation due to sidewall opening

The second factor is the opening of the side wall (and curl) caused by springback. Because of the geometrically necessary constraint imposed by the side wall opening, the flange region that undergoes shrinking deformation requires in-plane compression, whereas the flange region that undergoes stretching deformation requires in-plane

891

tension, as schematically shown in Fig. 10. These in-plane deformations have the opposite effect to the first factor and give rise to negative torque during the process of side wall opening due to die removal (Fig. 11). Consequently, the positive twist caused by negative torque was observed during the punch removal process where a large side wall opening arose. Because the amount of side wall opening increases with the drawing height, the negative torque owing to the opening of the side wall may be predominant at the drawing height of 60 mm rather than the positive torque owing to the shear stresses, yielding the resultant twist positive. Distribution of WLW /MPa

100

-100

H= 20 mm

-300

100

-100 100

300 -300 -100

-100 H= 60 mm

FIGURE 11. Distribution of shear stress after die removal (drawing height H=60mm)

REDUCTION OF SIDE WALL OPENING Conditions for experiments and numerical analyses The above results suggest that, for deep drawing products as in the case of the drawing height of 60 mm in the above results, the suppression of the side wall opening may be effective to reduce the twist. We propose here a new method of reducing side wall opening by giving the side wall a stepped shape. In the following, the validity of the proposed method is examined in detail using a two-dimensional draw bending. Figure 12 shows the geometry of the tools with a two-dimensional hat shape used in this study. Experiments were carried out with two different shapes; (i) with stepped shapes in the middle of the vertical walls and (ii) without steps. The clearance between the die and the punch was 1.76mm. The sheet was drawn to a punch stroke of 60 mm. The same dual-phase steel sheet was used as the blank material. The blank was 150mm (direction perpendicular to this paper with respect to Fig. 12) x 227mm (drawn direction). The blank holding force and pad force were 93kN and 500kN, respectively. 60 BH

Clearance

Punch R2 5

BH

R6 Punch R2 Die R 6

60 30

Pad

Die

FIGURE 12. Geometry of two-dimensional hat channel product and tools

892

E/ GPa 212

TABLE 2. Mechanical properties used in simulations of curved hat channel Swift and Chaboche law approximation V0/ C/ Cx/ Q n MPa H0 MPa MPa 708 0.3 1337 0.103 0.0021 37

xsat /MPa 44

Four-node degenerated shell elements with five integration points through the thickness direction are used in numerical simulations. Only a quarter part of the model was analyzed owing to symmetrical geometry. Kinematic hardening is taken into account in this simulation and a stress-strain behavior was described by Swift’s isotropic hardening combined with Chaboche’s kinematic hardening. Table 2 shows the material parameters used.

Experimental and numerical results The numerical and experimental results of the side wall opening 'W, the angle of the flange 'Tf and the curvature of wall curl  are shown in Tables 3. Figure 13 shows the definitions of 'W, 'Tf and . Negative values of  indicate an inward wall curl. Qualitative tendencies coincided with each other, verifying the validity of the numerical results. ԥW

ԥW

'W

a 5

N

1

U

U1

5

U2 U3

a

FIGURE 13. Definition of springback

TABLE 3. Experimental and numerical results of springback Experimental result Numerical result (i) With step (ii) Without step (i) With step (ii) Without step 13.48 16.03 9.28 15.41 ԥW (mm) 22.15 22.85 16.01 25.49 ԥTf (deg) 0.0113 0.0091 N1/mm) 0.0051 -0.0033 0.0084 0.0087 N1/mm) -0.0077 -0.010 N1/mm)

Discussions Both 'W and 'Tf are smaller in model (i) with steps than in model (ii) without steps, as shown in Tables 3 and 4, respectively. It is considered that these tendencies are caused by the relaxation of the bending-unbending effect on the die shoulder at the initial stage of the drawing owing to the larger clearance between the die and the punch in model (i). The above mechanism can be observed by comparing the sheet behavior at the drawing height (dh) of 25mm, as shown in Fig. 14. Figure 15 shows a comparison of the distributions of the vertical component of normal stress in the thickness direction at the bottom dead point (dh = 60mm). The stress distribution generated by typical bending-unbending deformation, which causes wall curl, can be observed in model (ii), whereas it is apparently eliminated in model (i), implying that the relaxation of the bending-unbending behavior causes the reduction in wall curl, resulting in smaller 'W, 'Tf and 3. These results clearly show that the sidewall opening can be suppressed using the proposed method in the two-dimensional draw bending process. Because a basic mechanism of occurring sidewall opening is the same in curved hat channel products, a similar effect of suppression of the sidewall opening is expected in deep-drawing curved-hat-channel products. The application of the method to the above curved hat channel product will be our future work.

893

On the other hand, another procedure should be proposed to suppress twist occurring in shallow-drawing curvedhat-channel products because the shear stresses rather than the side wall opening may be a predominant factor for the twist as explained above. This will also be our future work. Area without bending deformation

dh=25mm

ཛ h=30 mm

ཝ Original

FIGURE 14. Deformation profile at drawing height of 25 mm

dh= 60mm 48.8 1000

ཛ

500 0 -500

-1000

punch side -0.8

-0.4

die side 0

0.4

Stress V /MPa

Stress V /MPa

1000

punch side

die side

500 0

49.7

ཝ

-500

-1000

0.8

Position /mm

-0.8

-0.4

0

0.4

Position /mm

0.8

FIGURE 15. Stress distribution at drawing height of 60 mm

CONLUSIONS The mechanism of twist occurring in curved hat channel drawing was examined in detail by finite element simulation. The results obtained in this study were as follows. 1) The mechanisms of twist occurring during the unloading process can be explained by the following two factors: a) Positive torque induced by the in-plane tensile and compressive deformation of the flange regions during the drawing process, which is a source of negative twist. b) Negative torque generated by the side wall opening and curl during the die removal process, which is a source of positive twist. 2) The resultant twist can be determined by the net effect of the above two factors. The negative torque has a greater effect on the behavior at the drawing height of 60 mm, resulting in positive twist. Aiming at reducing twist at larger drawing height, we proposed a new method of reducing side wall opening by giving the side wall a stepped shape. The validity of the proposed method was examined in two-dimensional hat bending with stepped shapes on vertical walls. We concluded that the stepped shape on side walls has marked effects of reducing side wall opening, mainly through the elimination of bending-unbending effects on the die shoulder, which was verified by observing the stress distribution obtained by FE analysis.

REFERENCES 1. T. Yamano and J. Iwaya, Proceedings of the 56th Japanese joint Conference for the Technology of Plasticity, Okinawa, 2005, pp. 137-138 (in Japanese). 2. M. Takamura, A. Fukui, T. Hama, Y. Miyoshi, M. Sakata, H. Sunaga, A. Makinouchi and M. Asakawa, Steel Research Int., 81-9 (2010), pp. 821-824. 3. K. Ito, T. Kondo, G. Uemura and N. Mori, Proceedings of Numiform’07, Porto, 2007, pp. 1477-1482. 4. M. Banu, M. Takamura, T. Hama, O. Naidim, C. Teodosiu and A. Makinouchi, J. Material Process. Technol., 173-2 (2006), pp. 178-184. 5. M. Banu, T. Hama, J.L. Alves, O. Naidim, Steel Research International, 79 (2008), Special Edition, vol. 1, pp. 186-193. 6. Y. Yamada, N. Yoshimura, T. Sakurai, Int. J. Mech. Sci., 10 (1968), pp. 343-354. 7. M.C. Oliveira, J.L. Alves, B.M. Chaparro, L.F. Menezes, Int. J. Plasticity, 23 (2007), pp. 516-543.

894

Design of Channel Type Indirect Blank Holder for Prevention of Wrinkling and Fracture in Hot Stamping Process Hong-seok Choia, Se-yoon Hab, Seung-hoon Chab, Chung-gil kangc and Byung-min Kimc† a

Precision Manufacturing Systems Division, Pusan National University, Busan, 609-735, Korea Department of Stamping Tool Process Planning, Shin Young Co., Ltd., Yeongcheon, Kyeongbuk, 770-140, Korea c School of Mechanical Engineering, Pusan National University, Busan, 609-735, Korea

b

Abstract. The hot stamping process has been used in the automotive industry to reduce the weight of the body-in-white and to increase passenger safety via improved crashworthiness. In this study, a new form die with a simple structure that can prevent defects such as wrinkle and fracture is proposed for the manufacture of hot stamped components. The wrinkling at the flange cannot be eliminated when using a conventional form die. It is known that the initiation of wrinkling is influenced by many factors such as the mechanical properties of the sheet material, geometry of the sheet and tool, and other process parameters, including the blank holding force (BHF) and the contact conditions. In this research, channel type indirect blank holder (CIBH) is introduced to replace general blank holder for manufacturing the hot stamped center pillar. First, we investigate the tension force acting on the blank according to the channel shapes. We determine the appropriate range by comparing the tension force with the upper and lower BHFs in a conventional stamping process. We then use FE-analysis to study the influence of the slope angle and corner radius of the channel on the formability. Finally, the center pillar is manufactured using the form die with the selected channel. Keywords: Channel type indirect blank holder (CIBH), Hot stamping, Tension force, Slope angle, Corner radius, Center pillar, Form die. PACS: Replace this text with PACS numbers; choose from this list: http://www.aip.org/pacs/index.html

INTRODUCTION The use of hot stamping has allowed automotive manufacturers to produce components that are lighter and stronger than those made from conventional cold-stamped parts [1-2]. Overall reduction in the mass of a vehicle helps improve the fuel economy. The use of strong materials such as rapidly cooled boron steel (22MnB5), which has a tensile strength of more than 1,500 MPa, also helps improve the crashworthiness of vehicles by preventing intrusion into the passenger compartment. Hot stamping is performed at elevated temperatures in order to increase the formability. However, it is not feasible to use a blank holder to prevent wrinkling at the flange and side wall of the product because of the large temperature gradient between the blank and the die, which leads to the formation of crack at that position [3]. Cha et al. investigated the effect of stamping methods on the formability of the front side member by using ultra high strength steel of 980 MPa grade and claimed that the form-draw method delivers a good performance in terms of reducing springback, wrinkling, and fracture [4]. Appropriate choice of the stamping method is important for manufacturing reliable hot stamping components without defects. However, there are very few published reports on the forming method and its effectiveness. Keraucsch and Schönbach investigated the influence of the blank geometry, blank holder distance, tool velocity, and blank position on thinning, by using the hot-forming benchmark model proposed in the Numisheet 2008 [5]. Stoehr et al. determined the frictional and thermal characteristics by using a deep-drawing die, with a distance plate between the die and the blank holder to prevent rapid cooling in that region [6]. Kim et al. manufactured a bumper impact beam by using a three-piece tool composed of a die, a pad, and a punch to verify the validity of the The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 895-902 (2011); doi: 10.1063/1.3623700 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

895

developed simulation skill [7]. According to the ongoing research on hot stamping, hot stamped components are usually produced by using a draw die, which has a small gap between the blank holder and the die. Consequently, the structure of the hot stamping tool, which has complicated cooling channels, becomes highly complex; this structural complexity results in the gap effect. In this study, a new form die with a simple structure that can prevent defects such as wrinkle and fracture is proposed for the manufacture of hot stamped components. The wrinkling at the flange cannot be eliminated when using a conventional form die. In this research, channel type indirect blank holder(CIBH), which make it possible to eliminate the wrinkling without fracture, is introduced for manufacturing the hot stamped center pillar. We investigate the tension force acting on the blank according to the channel shapes such as slope angle and corner radius. We determine the appropriate range by comparing the tension force with the upper and lower BHFs in a conventional stamping process. Furthermore, available forming condition without fracture is selected and FEanalysis is conducted to investigate formability. The center pillar is manufactured using the form die with the selected channel.

HOT STAMPED CENTER PILLAR USING FORM DIE Hot Stamped Center Pillar Manufactured by Conventional form die Fig. 1 shows a hot stamped center pillar manufactured using a conventional form die. In this process, forming is achieved by the upper die after the blank sheet has been clamped by the upper pad [8]. Folding caused by severe wrinkling during the forming operation occurs at the flange of the central region, as shown in region A. On the other hand, region B, where there is a high risk of crack formation due to the biaxial tensile mode, is formed normally without fracture. This is because hot stamping is performed without applying a BHF to the sheet in the flange area; BHF interrupts material flow into the drawn wall. Hence, it is important to prevent wrinkling at the flange for the successful production of a hot-stamped center pillar using a form die.

FIGURE 1. Hot-stamped center pillar formed using conventional form die.

Channel Type Indirect Blank Holder (CIBH) in Form Die When we use a form die for the hot-stamping process, it is essential to prevent folding at the flange caused by wrinkling during the forming process. To control the formation of wrinkle and fracture, a CIBH is introduced on both sides of the form die. Fig. 2 shows the stamping procedure for a form die with the channel. The tool is composed of a die, a pad, and a punch with the channel, similar to a conventional form die. At the first stage, the pad is moved downwards to clamp the blank. At the second stage, the blank is formed into the required shape. In the forming process, if the tool surface at the channel is curved, a contact pressure, p, is generated. When the blank slides along the surface of the channel, a friction force, μp, is generated, where μ is the friction coefficient. A BHF can be applied indirectly to the flange by the friction force caused by sliding along the radius and slope of the channel. Consequently, wrinkling is reduced to a significant extent during the forming operation.

FIGURE 2. Schematic of stamping procedure performed using form die with channel.

896

MECHANISM OF INDIRECT BHF GENERATION ACCORDING TO CIBH Calculation of Unit Tension Force at Blank Sliding Curved Surface It is necessary to calculate the tension force generated at the blank during the stamping operation in order to estimate the indirect BHF, FIBHF, depending on the channel shape. The unit tension force, T, which is transmitted through the blank, is defined as:

T  t

(1)

where, σ and t denote principal stress and initial material thickness, respectively. Duncan et al. investigated the tension, strain, and thickness distribution of elements sliding on a curved surface in a two-dimensional stamping process [9]. Fig. 3 shows an equilibrium diagram of the unit force acting on an element sliding over a curved surface. From the equilibrium conditions shown in Fig. 3, if the tension at a point, α, in the section is known, the tension at some other point, β, can be calculated as follows:  dT  ∫T T ∫0 d T

(2)

The unit tension force at β is obtained from the tension force at α, friction coefficient, μ, and angle, θαβ, i.e.,

T  T exp(  )

(3)

where θαβ is the angle between the two points. This implies that it is possible to determine variables such as the corner radius of the punch and die, BHF, and thickness distribution in the stamping process if either the BHF or the strain at some point is specified.

dθ

μp



 dθ pR

dθ μpR dθ

Td

θ

FIGURE 3. Equilibrium diagram of force acting on an element sliding on a curved surface.

Determination of Indirect Upper and Lower BHFs The range of the indirect upper and lower BHFs (FIUBHF, FILBHF) must be known because a very large clamping force may result in fracture, while a very small force would be ineffective in preventing wrinkling. Consequently, the unit tension force acting on the blank during hot stamping should be between FILBHF and FIUBHF:

897

FILBHF  T  FIUBHF

(4)

In this study, FIUBHF and FILBHF are estimated on the basis of the conventional stamping model, and the estimated values are used to determine the channel shape. Fig. 4 shows a schematic of the conventional stamping process and the tension forces acting on the blank. As can be seen in Fig. 4, it is possible to determine the tension at each point of the blank using Eq. (3). If the strain at point O is specified, the unit tension force, TO, in the case of hot stamping would be

TO   t  C ( ) m t o exp(  o )

(5)

where σ is the effective stress of the blank material; C, the material constant; ε , the effective strain rate; m, the strain-rate sensitivity exponent; to, the thickness of the blank; and εo, the initial strain. εo can be calculated by using the formula

o 

y

(6)



where ρ and y denote the bending radius of the curvature and the distance from the middle surface, respectively. Note that ρ between the points O and A can be assumed to be large when the punch face is flat, as can be seen in Fig. 4. Therefore, εo is close to zero and is assumed to be 0.005, which causes the yield in the blank.

F

F

clearance μF

TD

Punch TC

θ CD

μF

TD

μF

μF

F θ AB

To

TB Die

TA

TA,B,C and D denotes tension force at each point.

FIGURE 4. Tension forces acting on blank in conventional stamping process.

On the other hand, because the flow stress of the blank material (22MnB5) in the hot stamping process is dependent on temperature and strain rate, a rough estimate of the flow stress can be obtained by using the following equation, on the basis of the results reported by Eriksson et al. [10]; in this case, temperature should be less than 700 °C, and the strain should be in the range 0.005 to 0.05:

  171( ) 0.1036 [ MPa]

(7)

where ε denotes the average effective strain rate, which is determined to be 0.04 s-1 by FE-analysis. FIUBHF can be calculated using the above equations with respect to the material properties and the tension force acting on the blank. FIUBHF is defined as a unit force when occurring the necking at the wall of the deformed blank:

FIUBHF 

UTS  t 0 [ N / mm] 2 exp( CD )

(8)

where UTS is the ultimate tensile stress of the material, and θCD is π/2 if the clearance between the punch and the die is assumed to be narrow. The material thickness, t0, is assumed to be constant. The UTS of 22MnB5 at a temperature of 700 °C and a strain rate of 0.04 s-1 is determined to be 220 MPa. The friction coefficient at the elevated temperature, μ, is assumed to be 0.4 [11].

898

The outer side of the channel has a large curvature, as shown in Fig. 5, because of the complicated shape of the center pillar. This curvature leads to wrinkling. FILBHF should be just sufficient to prevent wrinkling, and experience shows that the pressure on the flange should be 2% of the average flow stress of the sheet so that wrinkling is prevented [12]. Consequently, FILBHF per unit length can be expressed as

FILBHF  0.02 

YS  UTS  L [ N / mm] 2

(9)

where YS and L denote the yield stress of the material and the width of the flange in Fig. 5, respectively. The width of the flange can be roughly estimated from the spread of the final product. On the basis of Eqs. (8) and (9) and the parameters determined, FIUBHF and FILBHF are calculated to be 176 N/mm and 93 N/mm, respectively. Consequently, the unit tension force, i.e., the indirect BHF generated in accordance with the CIBH, should be in the following range:

93 [ N / mm]  TO  176 [ N / mm]

(10)

: Blank : Die face

C

L

Punch Die face Channel

FIGURE 5. Die face having large curvature (C) on the outer side of the channel.

Calculation of Tension Forces for Determining Channel Shape Fig. 6 shows a schematic of the deformation of the blank during the penetration of the die. The shape parameters considered in this type of blank holder are the slope angle, θ, and the corner radius, R. These variables, which determine the shape of the channel, influence material flow into the die cavity during the stamping operation. The height of the holder (H), one of the parameters that determine the channel shape, is fixed and is determined along the die face of the product. The initial blank shape is one of the main factors that influence formability. However, it is difficult to take into account the influence of the process variables because the optimal blank shape depends on these variables. In this study, the initial blank is chosen on the basis of the results of a previous research [3]. The tension force acting on the blank is maximized at point B, i.e., the maximum tensions are TiB and TjB when the forming depths are Hi and Hj, respectively. Furthermore, the tension force varies with the main parameters influencing the channel shape, R and θ. In this study, when most of the flange enters the die cavity, the tension force at point B is calculated using Eqs. (2), (3), and (4).

Die θj θi

TiB Tio

TiA

R TjB

Hi

Tjo

Hj

TjA

Channel

Punch

θ

FIGURE 6. Schematic drawing of deformation of blank during penetration of die.

899

Fig. 7 shows the tension force diagram in terms of R and θ. The maximum tension force increases with a decrease in R and θ. This implies that the tension force increases with the contact area between the blank and the tool, and this in turn leads to an increase in the friction area between the blank and the tool. Consequently, the appropriate range for R and θ can be obtained by calculating the tension force, as shown in Fig. 7. Material usage increases

Corner radius, R (mm)

28

Tension force (N/mm)

24 20

Safety zone

16 180.0

170.0

12 176.0 8 4 90

150.0 160.0 170.0 176.0 180.0 190.0 200.0 210.0 220.0 230.0 240.0

Selected condition R16 - 

190.0

95

100

105

110

o

115

120

Slope angle,  ( ) FIGURE 7. Tension force diagram in terms of corner radius, R, and slope angle, θ, of CIBH.

FE-ANAYSIS OF HOT STAMPING FOR A CENTER PILLAR FE-model FE-analysis is conducted to validate the indirect BHF using the calculated tension force and to determine the channel shape via a more accurate analysis of hot stamping. The dynamic explicit code LS-DYNA is used to analyze the hot stamping simulation. FE-analysis is performed where both forming and heat transfer are simultaneously solved. *MAT_106, which is an elastic-viscoplastic material with thermal effects, is used as a model of the 22MnB5. Flow curves in dependency of strain, temperature, and strain rate are obtained by a tensile test at an elevated temperature. A contact heat transfer coefficient based on the contact pressure is used in the simulation [13]. Fig. 8 shows the FE-model composed of a die, a pad, and a punch with a channel and FE-analysis conditions. The pad force is fixed to 15 ton. The heating and transfer procedures are not taken into account, and hence, the initial blank temperature just before stamping is set to 750 °C, by taking into account the temperature decrease during the transfer.

Upper pad

Pad

Upper Die die

Blank Blank

Channel Lower punch Punch with CIBH

FIGURE 8. FE-analysis model and applied conditions for hot stamping of center pillar.

900

FE-analysis results and discussion Fig. 9 shows the deformed shapes by FE-analysis for evaluating the formability under the condition of selected CIBH. It is known from Figs. 9(a)–(c) that the flange area is smoothed and subsequently flattened as the forming operation progresses. The maximum thinning at the side wall is 25.5%.

FIGURE 9. FE-analysis reveals deformed shapes during hot stamping operation for a selected CIBH. (a) Before 30 mm from LDC; (b) Before 10 mm from LDC; (c) After completion of hot stamping; (d) Thinning distribution of hot stamped center pillar.

VALIDATION OF CIBH THROUGH HOT STAMPING EXPERIMENT Hot stamping experiment using form die with CIBH A hot stamping experiment using a form die with a channel, which is machined to achieve θ and R values of 99° and 16 mm, respectively, is performed to validate the effectiveness of this type of blank holder. A hydraulic press with a 1,000-ton capacity is used in the experiment to maintain the tools in a closed state for quenching. Fig. 10 shows the die set, which is composed of a pad, a die, and a punch with a channel. The blank is heated to 900 °C in a furnace and then transferred to the press for about 10 s. The temperature of the blank at the moment of stamping is approxi-mately 740–770 °C, similar to that in the analysis. The forming time is approximately 10 s. The blank is rapidly cooled for 20 s when the forming process is complete. Furnace

Punch Channel

FIGURE 10. Schematic of stamping performed using conventional form die.

Experimental result Fig. 11 shows the center pillar manufactured by hot stamping using a form die with CIBH with θ = 99° and R = 16 mm. Although severe wrinkling and folding occur at the flange of the center pillar manufactured by a conventional form die, as can be seen in Fig 1, wrinkling is not observed because of the effect of the channel. This

901

type of blank holder provides an indirect BHF resulting from the friction to the flange. Thus, severe wrinkling can be suppressed during the forming process.

FIGURE 11. Center pillar manufactured by hot stamping using form die with CIPH; marked number indicates location of tensile test specimen.

CONCLUSIONS In this study, we have proposed a new conceptual blank holder that can eliminate severe wrinkling; in this holder, there is no direct contact between the blank and the tool, and hence, fractures resulting from the drastic temperature gradient are prevented. This channel type indirect blank holder (CIBH) has been proposed for hot stamping of the center pillar. We calculated the tension force, performed an FE analysis, and carried out experiments with a form die with this CIBH. The following conclusions could be drawn: (1) During hot stamping with a conventional form die, folding occurs at the flange because the tool is not equipped with a blank holder that can prevent wrinkling. (2) An indirect BHF can be generated at the flange because the friction force leads to sliding along the radius and slope of the channel. This indirect BHF is obtained by comparing the calculated tension force with FIUBHF and FILBHF. (3) The design parameters used to determine the shape of the channel are the slope angle (θ) and corner radius (R). Selected channel shape for successful hot stamping of the center pillar is θ of 99° and R of 16 mm. Hot stamping using a form die with the selected CIBH can be employed for the formation of a high strength center pillar without wrinkles and fractures.

ACKNOWLEDGMENTS This research was financially supported by the Ministry of Knowledge Economy(MKE) and Korea Industrial Technology Foundation (KOTEF) through the Human Resource Training Project for Strategic Technology.

REFERENCES 1. 2. 3. 4. 5.

L. Vaissiere, J. P. Laurent and A. Reinhardt, SAE Paper No. 2002-01-2048 (2002). M. Merklein and J. Lechler, SAE Paper No. 2008-01-0853 (2008). H. S. Choi, C. G. Kang, S. Y. Ha, D. H. Lee and B. M. Kim, KSAE 2009 Annual Conference, 2439-2445 (2009). S. H. Cha, S. K. Lee, D. C. Ko and B. M. Kim, J. Korea Society for Technology of Plasticity 18, 1, 39-44 (2009). M. Kerausch and T. Schönbach, Proc. of the 1st Int’l Conference on Hot Sheet Metal Forming of High Performance Steel, 105-112 (2008). 6. T. Stoehr, M. Merklein and J. Lechler, Proc. of the 1st Int’l Conference on Hot Sheet Metal Forming of High Performance Steel, 293-300 (2008). 7. H. G. Kim, H. S. Son and S. H. Park, POSCO Technical Report 10, 1, 115-123 (2006). 8. Auto/Steel Partnership, “Die Sketch,” in Advanced High-Strength Steel Applications, Southfield, Michigan: Auto/Steel Partnership, 2010, pp. 231-239. 9. J. Duncan, S. J. Hu and Z. Marciniak, Mechanics of Sheet Metal Forming. 2nd edn. Butterworth-Heinemann. Massachusetts, 2002. 10. M. Ericksson, M. Oldenburg, M. C. Somani and L. P. Karjalainen, Modeling and Simulation in Materials Science and Engineering 10, 277-294 (2002). 11. M. Geiger, M. Merklein and J. Lechler, Proc. Eng. Res. Devel. 2, 269~276 (2008). 12. Z. Marciniak and J. L. Duncan, Mechanics of Sheet Metal Forming. 1st edn. Edward Arnold. Great Britain (1992). 13. K. J. Nam, H. S. Choi, D. H. Lee and B. M. Kim, KSAE 2009 Annual Conference, 2435-2438 (2009).

902

,QIOXHQFHRI9DULDEOH%ODQN+ROGLQJ)RUFHRQWKH'UDZLQJ %HKDYLRURI6WHHO6KHHWV Nemisha Mishraa, Dilip Sutariyab ,K.Narasimhana a

Department of Metallurgical Engineering and Materials Science, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India b Department of Mechanical Engineering, Sardar Vallabhbhai National Institute of Technology, Ichhanath, Surat, Gujarat. 395007, India

$EVWUDFW Blank holding force is one of the important factors affecting sheet metal forming that needs to be controlled for maximizing the formability of sheet metal. Too high a blank holding force leads to early splitting and too low a blank holding force gives rise to wrinkling. Formability can be improved to a limited extent by application of constant blank holding force. In the present paper, numerical simulations are carried out using commercial FEM software PAMSTAMP-2G to study different patterns of varying blank holding force on formability. .H\ZRUGV: Variable blank holder force, Formability, FEM simulation . 3$&681:20.Hy

,1752'8&7,21 Sheet metal forming is the process of deformation of flat sheets into differently shaped parts without causing any instability like wrinkling or excessive localized thinning. Enormous efforts have gone into understanding the process in order to improve the formability of sheet metal parts. The formability can be quantified in terms of maximum height of deformed cup without any failures. Formability can also be compared by the thickness of sheet metal parts at a particular stage of deformation [1]. There are several ways to prevent a work piece from defects including use of blank holder, variation of the shape of the tool piece or drawbead geometry [2, 3, 4] Blank holder controls the amount of metal flow into the die cavity. So, variation in blank holding force will affect the formability. In recent years, control and/or optimization of linear and non-linear schemes of variable BHF profiles have been studied extensively [5, 6]. Blank holding force has been varied in various fashion to improve the formability. Thiruvarudchelvan et al [7] proposed friction actuated blank holder. This involves generation of BHF using an annular urethane pad. The BHF is then roughly proportional to the punch force throughout the process. Optimization of blank holding force can also be done by closed loop control which will take the thickness of blank as input and then accordingly apply force [1, 8]. Along with optimum blank holder profile and closed loop control, vibration of tool is also a method of increasing formability. [9]. In this paper, the effect of various blank holding force profiles on deep drawing behavior of DP steels have been studied

0(7+2'2/2*< ([SHULPHQWDO6HWXS The tool setup used for studying the formability is standard LDR geometry.The simulations were performed on blanks of 115 mm diameter and of 0.8mm thickness. This particular diameter was chosen because all the smaller diameters formed into full cups without getting failed (on application of constant BHF). For 115 mm the maximum

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 903-910 (2011); doi: 10.1063/1.3623701 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

903

cup height of a full formed cup is 54mm. However, the maximum cup height obtained by applying constant BHF is 41mm. This paper aims in improving the maximum cup height for 115 mm diameter blanks.

0DWHULDOSURSHUWLHV Pertinent properties of the drawing quality steel considered for the present work is shown in Table 1.  7$%/(Material properties for DP steel E Poisson ratio 210

0.3

1.06

n

K(Mpa)

Thickness

0.183

94

0.8

)DLOXUHFULWHULD Criteria for necking This study employs the thickness gradient criterion to predict the forming limit strains, which states that localized necking during deformation takes places when there is a critical local thickness gradient in the sheet independent of strain path, rate of forming and the type of sheet metal (i.e. the material properties) being formed. The thickness gradient can be expressed as equation 1 (1) It has been established experimentally that when this ratio is smaller than 0.92 sheets is said to be necked. [10] Criteria for wrinkling At any level of deformation, the blank is said to have failed by wrinkling if the wrinkle height of flange exceeds 0.15 mm. This height was measured as difference in the height of flange above die and the punch stroke. [1]

 ())(&72)&2167$17%/$1.+2/',1*)25&( To study the effect of blank holding forces, the maximum cup height obtained at different constant blank holding forces was measured. The blank holding force was varied from 1kN to 15kN in steps of 1kN and then from 20 kN to 100kN in steps of 10kN. Figure 1 shows the variation of maximum cup height with blank holding force. In this analysis the cup height to failure is either due to wrinkling or necking, whichever occurs earlier.

904

  

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%+)N1 ),*85(Effect of BHF on Formability 

%/$1.+2/',1*)25&()250$%,/,7<:,1'2: The authors in [11] have stated a novel concept of BHF formability window based on the safe and failure zones of forming limit curve. It gives, for a particular punch stroke, the type of failure obtained on forming at a given magnitude of blank holding force. To study the effect of variable blank holding force. Figure 2 shows blank holding force formability window for DP steel blanks of 115mm diameter. Also, to test the effect of variable blank holding force on formability, we ran the simulation along the lower line which depicts minimum blank holding force below which wrinkling occurs and with an offset of 5 kN and 10kN as shown in figure 3. It was observed that when we ran the simulations along minimum blank holding force line the blank wrinkled at 23 mm. And when blank holding force was offset with 5 and 10 kN, cup height of unfailed cup was 41 mm. 

7HDULQJ%+)



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3XQFKVWURNHPP ),*85(Experimental blank holding force formability window for DP steel

905



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%+)N1

     





3XQFKVWURNH





),*85( Variable blank holding force profiles tested according to the wrinkling BHF values obtained in blank holding force formability window

$33/,&$7,212)9$5,$%/(%/$1.+2/',1*)25&(9%+) 21 )250$%,/,7< To study the effect of variable blank holding force, three types of BHF profiles were used viz. linearly increasing, linearly decreasing and pulsating.

/LQHDUO\,QFUHDVLQJ9%+) The linearly increasing profiles studied in this paper are categorized on the basis of the initial blank holding force, 1kN, 10kN and 20kN with different slopes as shown in Table 2. It was observed that initial BHF of 1kN was too low and the cups formed failed by wrinkling at early stages. For initial BHF of 10kN, larger cup depths were obtained with cup heights ranging from 39-42mm.When the initial BHF was increased to 20kN, blanks started tearing at initial stage. For initial BHFs of 1kN and 10kN, increase in slope played a very important role, the maximum cup height increased with increase in slope. For initial BHF of 20kN there was very minimal effect of slope. Since the maximum cup height obtained was highest for initial BHF 10kN, we compared the thickness gradient with the cups formed at constant 13kN BHF (which was the constant BHF at which maximum cup height of 41mm was obtained). It can be observed that the thickness at the punch stroke of 30mm was best for a slope of 1.08 . As the slope was increased, the thickness decreased and minimum thickness was for slope of 0.18.The comparison is depicted in figure 4. 7$%/( Maximum cup height obtained for linearly increasing BHF profile Initial BHF Slope Maximum Initial BHF Slope Maximum value Cup height value Cup height 0.163 12 0.18 39 0.345 14 0.36 40 1kN 0.527 15 10kN 0.54 40 0.709 16 0.72 40 0.89 20 0.9 41 1.08 42

906

Initial BHF value

20kN

Slope

Maximum Cup height

0.18 0.36 0.54 0.72

20 20 20 20



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'LVWDQFHIURPFHQWHULQWKHXQGHIRUPHGEODQNPP (b) ),*85((a)Comparing the thickness distribution obtained on application of constant BHF of 13kN and variable blank holding force of linearly increasing profile and initial BHf of 10kN (b) zoomed view of Figure 6(a)

/LQHDUO\'HFUHDVLQJ9%+) The categorization of linearly decreasing profiles was also based on the initial BHF. The profiles studied were initiated at 10kN and 15kN as shown in table 3. For linearly decreasing profiles, slope had negligible effect on failure height. The cups failed at 24 mm and 38 mm irrespective of slope for initial BHFs of 15 kN and 10 kN

907

7$%/( Maximum cup height obtained for linearly decreasing BHF profile Initial BHF value Slope Maximum Cup Initial BHF value height 0.09 24 15kN 0.18 24 10kN 0.254 24

Slope

Maximum Cup height 38 38 38

0.054 0.09 0.163

3XOVDWLQJ9%+) In order to study the effect of variable blank holding force, pulsating profiles were also studied. The study comprised of varying the frequency and the amplitude. Effect of frequency: The tooling geometry and amplitude were made constant. All the simulations were run with constant mean BHF of 10kN and with a variation of 5kN. The various frequencies tested were 0.2, 0.1, 0.05 and 0.033 cycles/mm (Table 4) each with both initial blank holding force of 5kN and 15kN. 7$%/( Maximum cup height of pulsating BHF profiles with mean BHF of 10kN and amplitude of 5kN VBHF profile

Initial BHF

Frequency

Maximum cup Height

I

15kN

0.2

23

II

15kN

0.1

23

III

15kN

0.05

23

IV

15kN

0.03

22

V

10kN

0.2

35

VI

10kN

0.1

36

VII

10kN

0.05

36

VIII

10kN

0.03

36

When the initial BHF was 15kN, the blank cracked at all the frequencies and the frequency of 0.2 cycles/mm showing the maximum cup height. While, when the initial BHF had lower value, all the blanks failed by wrinkling and the frequency of 0.033 cycles/mm showed the minimum cup height. Effect of amplitude: The effect of amplitude on formability was studied for a mean BHF of 10kN and variations of 3, 5 and 10kN (table 5). It was observed that a variation of 3kN gave the maximum cup height of 40mm while with other two amplitudes of 5 and 10kN the blank failed at 23 and 20 mm. The formability of blanks obtained on applying constant BHF of 13kN and by applying a pulsating BHF of 3kN variations on 10kN mean BHF was compared by comparing thickness at punch stroke of 30mm. (Figure 5) It can be seen that at the same punch stoke the thickness of pulsating profile is better than constant BHF 7$%/( Maximum cup height of pulsating BHF profiles with mean BHF of 10kN and frequency of 0.05 cycles/sec

VBHF profile IX X XI

Mean BHF 10kN 10kN 10kN

Amplitude 3kN 5kN 10kN

Frequency 0.05 0.05 0.05

908

Maximum Cup height 40 23 20



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'LVWDQFHIURPFHQWHULQWKHXQGHIRUPHGEODQNPP )LJXUHComparing the thickness distribution obtained on application of constant BHF of 13kN and pulsating profile IX

&21&/86,21 In this paper, the effect of variable blank holding force on deep drawing behavior of DP steel sheets of 115 mm diameter and 0. 8 mm thickness was studied. The following conclusions can be drawn from the present work: 1) The maximum cup height of 42mm was obtained at constant blank holding force of 13kN. 2) Both linearly increasing and decreasing profiles of variable blank holding force were tested. It was observed that initial BHF of 10kN gave similar cup height but a slight improvement in thickness distribution as compared to constant blank holding force of 13kN. The initial BHF of 1kN and 20kN failed by wrinkling and tearing respectively. 3) When linearly decreasing profiles were tested, it was found that irrespective of initial BHF, the cups failed early but wrinkling. 4) Pulsating blank holding force can have ambivalent effect on formability. It is observed that amplitude variation is a greater determinant of formability than frequency. Pulsating profile IX shows a better thickness distribution as compared to constant BHF of 13kN.

5()(5(1&(6 1.Sy-Wei Lo and Tsu-Chang Yang, Closed-Loop Control of the Blank Holding Force in Sheet Metal Forming with a New Embedded-Type Displacement Sensor, International Journal of Advanced Manufacturing Technology, 24(78), pp 553-559 2. N. Kawai , T. Mori, H. Hayashi, F. Kondoh , Effects of Punch Cross-section on Deep-Drawability of Square Shell of Aluminum Sheet, ASME Journal of Engineering Industry,1987 ,109, pp 355–361 3. S.B Park., Y,Choi, B.M.Kim, J.C .Choi, A Cad-Cam System For Deep Drawing Dies In A Simple-Action Press, Journal of Material Process Technolgy, 1999, 87, pp 258–265 4. K.J. Weinmann, J.R Michler, V.D Rao, A.R. Kashani, Development of a Computer-Controlled Drawbead Simulator for Sheet Metal Forming, 1994, Annals of CIRP, 43(1), pp 257–261 5. L.Wang, L.C.Chan, T.C.Lee, Process Modeling Of Controlled Forming With Time Variant Blank Holder Force P Using Rsm Method, (2007) International Journal Of Machine Tools And Manufacture, 47 (12-13), pp 1929-1940 6. M.A.Ahmetoglu, G.Kinzel, T .Altan, Forming of Aluminum Alloys - Application of Computer Simulations and Blank Holding Force Control, Journal of Material Process Technology, 71 (1), pp. 147-151

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7. S. Thiruvarudchelvan, W.G. Lewis, Deep Drawing with Blank Holder Force Approximately Proportional to the Punch Force, ASME Journal of Industrial Engineering, 112 , 1990, pp 278-285. 8. W. R. Wang, G. L. Chen, Z. Q. Lin, S. H. Li, Determination of Optimal Blank Holder Force Trajectories for Segmented Binders of Step Rectangle Box Using PID Closed-Loop FEM Simulation, International Journal of Advanced Manufuring Technology, (2007) 32: 1074–1082 9.T.Jimma,,Y. Kasuga,, N.Iwaki, O.Miyazawa,E.Mori,K. Ito, H.Hatano,An application of ultrasonic vibration to the deep drawing process ,Journal of Materials Processing Technology, 1998, 80-81, pp. 406-412. 10.Sujith Kumar, P.P. Date, K. Narasimhan, A New Criterion To Predict Necking Failure under Biaxial Stretching, Journal Material Process Technology, 1994, 45,pp 83.

910

Three Dimensional Forming Simulation of the Shielded Slot Plate for the MCFC Using a Ductile Fracture Criterion C. H. LEEa, D. Y. YANGa,*, S. R. LEEb, I. G. CHANGb, T. W. LEEb a

School of Mechanical Engineering & Aerospace System, KAIST Doosan Heavy Industries & Construction Co., Fuel Cell Development Center * Corresponding author, School of Mechanical Engineering & Aerospace System, KAIST Tel.: +82-42-350-3214, E-mail address: [email protected] b

Abstract. The shielded slot plate, which has a sheared corrugated trapezoidal pattern, is a component of the metallic bipolar plate for the molten carbonate fuel cell (MCFC). In order to increase the efficiency of the fuel cell, the unit cell of the shielded slot plate should have a relatively large upper area. Additionally, defects from the forming process should be minimized. In order to simulate the slitting process, whereby sheared corrugated patterns are formed, ductile fracture criteria based on the histories of stress and strain are employed. The user material subroutine VUMAT is employed for implementation of the material and ductile fracture criteria in the commercial FEM software ABAQUS. The variables of the ductile fracture criteria were determined by comparing the simulation results and the experimental results of the tension test and the shearing test. Parametric studies were conducted to determine the critical value of the ductile fracture criterion. Employing these ductile fracture criteria, the three dimensional forming process of the shielded slot plate was numerically simulated. The effects of the slitting process in the forming process of the shielded slot plate were analyzed through a FEM simulation and experimental studies. Finally, experiments involving microscopic and macroscopic observations were conducted to verify the numerical simulations of the 3-step forming process. Keywords: MCFC, Shielded slot plate, Ductile fracture criterion, Slitting PACS: 46.05.+b

INTRODUCTION Molten carbonate fuel cell (MCFC) is a promising source of a renewable energy system [1, 2, 3]. A MCFC is composed of bipolar plates and membrane electrode assemblies (MEA). The bipolar plate of the MCFC is made from stainless steel due to manufacturability and cost considerations. The bipolar plates of the MCFC are hence referred to as metallic bipolar plates. The Metallic bipolar plates of MCFCs are composed of a center plate, an anode shielded slot plate and a cathode shielded slot plate, as shown in Fig. 1(a). The unit cell of the shielded slot plates is trapezoidal in shape, as shown in Fig. 1(b). In order to increase the efficiency of the MCFC, the contact area between the shielded slot plate and the electrode should be relatively large. The open sheared corrugated unit cell should have relatively large upper area without defects from the forming process. In order to realize these features, appropriate simulations should be carried out in advance. However, FEM simulation of the shielded slot plate is hard work, because the forming process of the shielded slot plate includes a slitting process, which makes the unit cell sheared. Numerical studies on the shearing process have been carried out since the 1990s. Employing the ductile fracture criterion, Taupin et al. [4] simulated the shearing process. Wisselink et al. [5] used an arbitrary Lagrangian Eulerian formulation to simulate the slitting process. Ghosh et al. [6] used a failure criterion to simulate the shear-slitting process for aluminum alloys. In this work, the slitting process is considered in the FEM simulation of the shielded slot plate employing the ductile fracture criterion. First, the ductile fracture criterion was determined from an analysis of the uni-axial tension test. Applying this ductile fracture criterion, the uni-axial tension test was simulated and the results were compared with the experimental results. The ductile fracture criterion of the shearing process was determined by comparing the simulation results and experimental results of the shearing process. Employing the ductile fracture criterion, the

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 911-918 (2011); doi: 10.1063/1.3623702 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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multi-step forming process of the unit cell of the shielded slot plate was investigated. In addition, the effects of the slitting process of the shielded slot plate were analyzed through the FEM simulation and experimental results.

(a) (b) FIGURE 1. Schematic figure of MCFC metallic bipolar plate and the shielded slot plate

PROPERTIES OF THE SHIELDED SLOT PLATE Material The metal sheets used for the shielded slot plates are cold rolled stainless sheets (KS-D3698 manufactured by POSCO). Table 1 presents the chemical composition of the material. This stainless steel incorporates more Cr and Ni than other austenitic stainless steel sheets in order to increase corrosion resistance. The uni-axial tensile tests were performed to study the mechanical properties of the sheets in the quasi-static condition. The Young’s modulus was determined to be 150GPa. The yield strength of the material was determined to be 230MPa. The fracture strain was determined to be 0.69. The flow stress is given in (1). The isotropic hardening and von-Mises yield criterion were employed in the user material subroutine VUMAT in ABAQUS/Explicit.

  1199.7(0.0232   )0.447 Table 1. Chemical composition of the material (wt. %) Cr C Si Mn P 25.50

0.0434

0.5

1.35

0.025

(1)

S

Ni

Cu

Mo

N

0.0007

19.60

0.05

0.06

0.0324

Unit Cell of the Shielded Slot Plate The metallic bipolar plate of the MCFC is composed of a center plate, an anode shielded slot plate, and a cathode shielded slot plate. The center-plate separates the anode gases and the cathode gases. The shielded slot plates support the center-plate and the MEA as presented in Fig. 1(a). The shielded slot plate makes space through which the reactive gases flow. The catalyst of the MCFC is placed on the shielded slot plate. For these reasons, the unit cell of the shielded slot plate is trapezoidal in shape. The slitting process is necessary to form the sheared trapezoidal shape. Two additional forming processes, the performing process and the final forming process can be included after the first slitting process in order to form a high aspect ratio pattern. The punch and die set for the experiment are shown in Fig. 2.

FIGURE 2. Experiment for shielded slot plates and the punch and die set

912

DETERMINATION OF THE DUCTILE FRACTURE CRITERION Ductile Fracture Criterion Ductile fracture is characterized by tearing of ductile materials due to significant plastic deformation. There have been many experimental and numerical studies to establish ductile fracture criteria [7, 8, 9, 10]. In numerical studies, ductile fracture occurs when the damage value of an element exceeds a certain critical damage value. It is assumed that as soon as an element exceeds a certain critical damage value, the element starts to fracture and is removed from the analysis domain. In the present work, the Cockroft-Latham criterion presented in (2) was adopted. In (2), d is the damage value,  f is the equivalent strain at which fracture occurs, σmax is the maximum principal stress,  is the equivalent stress, and C is the critical value of the ductile fracture criterion. If the damage value exceeds the critical value, the ductile fracture occurs. Cockroft-Latham criterion is implemented to commercial FEM software user material subroutine VUMAT in ABAQUS/Explicit. Fig. 3 shows the flow chart of the user material subroutine VUMAT employing the ductile fracture criterion and the element deletion method. The critical value of the ductile fracture criteria was determined by comparing the simulation results and the experimental results of the tension test and the 2D shearing test. d 

f

0

Check yield criterion and update stress and strain

ABAQUS/Explicit User defined Subroutine

max d  C 

Update state variable, damage factors

(2)

Damage prediction

yes

No

End and return to the ABAQUS/Explicit

Delete the corresponding element

FIGURE 3. Flow chart of user material subroutine VUMAT employing the ductile fracture criterion

Simulation of the Uni-axial Tensile Test For the ductile fracture, determination of the critical value for the ductile fracture criteria is important, because the critical value is related with the crack initiation. In the uni-axial tensile test, the critical value can be determined analytically. The maximum principal stress is equal to the equivalent stress and the equivalent strain is equal to the strain in the tensile direction. Thus, the critical value which is the material constant, C1 in (2) is equal to the fracture strain of the uni-axial tensile test. Using this ductile fracture criterion, the uni-axial tensile test with an ASTM sub-size specimen was simulated and the results were compared with the experimental results. The simulation model of tensile test specimen is 6mm in width and 0.3mm in thickness, as shown in Fig. 4(a). The size of the specimen is the same as the specimen used in the experiment. The user material subroutine VUMAT is employed for implementation of the material and the ductile fracture criterion in the commercial FEM software ABAQUS/Explicit. The material was modeled as 14,469 three dimensional shell elements (S4R). (a) Î Local deformation after necking Thickness(t)

Effective stress

w=4.26mm

(b) Effective strain

tmin=0.209mm

(c) Damage value

f

I  0

m d 

Cockcroft-Latham criterion

(d)

(e)

FIGURE 4. FEM model of uni-axial tensile test and simulation result

913

The material shows local necking before fracture as presented in Fig. 4 (a). The width in the deformed state is 2.09mm in the simulation result and 2.01mm in the experimental result. The plastic strain and effective stress are concentrated on the center after necking as shown in Figs. 4 (b), (c), and (d). The damage value of the CockroftLatham criterion is also concentrated on the center as shown in Fig. 4(e). The load-displacement curves of the simulation and the experiment are presented in Fig. 4(f). From these results, the simulation model can predict the necking and fracture of the uni-axial tension test precisely.

Simulation of the Shearing Process Employing the aforementioned Cockroft-Latham ductile fracture criterion, the shearing process was simulated and the results were compared with the experimental results. 21,825 two dimensional plain strain elements (CPE4R) are used for the modeling of the material. The center region of the material where severe plastic deformation occurs is modeled as 20,000 4 node plain strain elements, as shown in Fig. 5. The user material subroutine VUMAT was employed for the simulation of the 2D shearing process. The arbitrary Lagrangian Eulerian(ALE) option in ABAQUS/Explicit, which combines the features of a Lagrangian analysis and an Eulerian analysis and avoids excessive element distortion and contact surface penetration [6], was used for the center region. The thickness of the material is 0.3mm. The radius of the punch and the die is 10% of the thickness, respectively. The clearance between the punch and the die is 10% of the thickness. The contact condition was modeled as Coulomb friction with μ=0.1. The critical value of the Cockroft-Latham criterion determined from the uni-axial tensile test was used. As shown in Fig. 6, the edge in the simulated results contains four distinct zones: rollover, burnish, fracture, and burr. The rollover is the result of the elastic and plastic deformation at the beginning of the shearing process. As the punch penetrates the sheet further, the shearing results in the severe deformation at the center and the vertical burnish zone. The damage value increases during the shearing process. When the damage value of the ductile fracture criterion exceeds the critical value, fracture initiates.

FIGURE 5. FEM modeling of the shearing process

FIGURE 6. Simulation result of the shearing process

Experimental Results Experiments on the shearing process were conducted through the die set shown in Fig. 3. The center part of the material after the first slitting process was compared with the simulation results of the shearing process. The experimental conditions corresponded with the simulation condition. Fig. 7(a) shows the experimental result and the SEM image. Fig. 7(b) is the SEM images of the surface of the part and the scrap. Fig. 7(c) shows the SEM images of the fracture surface. The deformed geometry contains four distinct zone including rollover, burnish, fracture and burr, in agreement with the simulation results shown in Fig. 7(b). However, the lengths of the rollover, burnish and fracture are different with the simulation results. In particular, the lengths of the burnish and the fracture show large discrepancies with the simulation results. This is attributed to their strong dependence on the initiation of cracks.

914

In the uni-axial tensile test, the maximum principal stress is equal to the equivalent stress before necking. The strain in the tensile direction is equal to the equivalent strain. In contrast, the maximum principal stress and equivalent stress are different in the shearing process, because compressive and shear deformation mainly occur in the shearing process. Thus, the ductile fracture criterion obtained from the uni-axial tensile test cannot be applied to the shearing process.

(a)

(b)

(c) FIGURE 7. Experimental result of shearing process

Correction of the Critical Value of the Cockroft-Latham Criterion In order to obtain the critical damage value of the Cockroft-Latham ductile fracture criterion, parametric studies of the shearing process in relation to the critical value of the Cockroft-Latham criterion have been conducted. By comparing the experimental results and the simulation results, the critical value of ductile fracture criterion is determined. The lengths of the shear area and the fracture area in the simulation results were compared with the experimental results, because they show a large discrepancy for each critical value. In the experiments, the ratio of the length of the shear area to the thickness ranged from 0.43 to 0.48. The ratio of the length of the fracture area to the thickness ranged from 0.19 to 0.32. In the simulation of the shearing process, as the critical value of the Cockroft-Latham ductile fracture criterion increases, the length of the shear area rises and the length of the fracture area decreases, as shown in Fig. 8(a). In the case of the critical damage value of 2, the simulated lengths of the shear area and the fracture area show good agreements with the experimental results. The experimental result is presented in Fig. 8(b). The simulated result of the shearing process for a critical value of 2 is given in Fig. 8(c). The experimental results and simulation results show good agreement when the critical value of the ductile fracture criterion is 2.

(a)

(b) FIGURE 8. Simulation result of the shearing process

915

(c)

FEM SIMULATION OF THE SHIELDED SLOT PLATE FEM Modeling The forming process of the shielded slot plate is composed of a three-step forming process including the slitting process, the preforming process, and the final forming process as shown in Fig. 9(a). Due to the large size of the problem, a quarter model is used. Fig. 9(b) presents the FEM modeling of the slitting process. The sheet is constrained with the top die and the bottom die, as shown in Fig. 9(b). In FEM modeling, the die and the punch of each forming process are placed on the sheet as presented in Fig. 9(b). The top die, the bottom die, the dies of each forming step, and the punches of each forming step are modeled as rigid surfaces. The center region of the sheet is meshed more densely than the other regions in order to account for localized deformation and stresses, as shown in Fig. 9(b). 74,880 brick elements (C3D8R) are used in the modeling of the center region of the sheet. The Arbitrary Lagrangian Eulerian(ALE) option in ABAQUS/Explicit was used for the center region. The thickness of the material is 0.3mm, and the radius of each punch and die is 10% of the thickness. The clearance between the first forming punch and the top die is 10% of the thickness. The user material subroutine VUMAT is used for implementation of the material and the ductile fracture criterion in the commercial FEM software ABAQUS/Explicit. The critical value of the Cockroft-Latham ductile fracture criterion, which is determined from a comparison of the shearing process, is used. The multi-step forming process is simulated by adjusting the contact condition. In the slitting process, the slitting die and the punch are in a contact condition with the sheet. In the final forming process, the final forming die and punch are in a contact condition with the deformed sheet.

(a) (b) FIGURE 9. FEM modeling for the three-step forming simulation of the unit cell

Simulation Results of the Forming Process of the Shielded Slot Plate The multi-step forming process of the shielded slot plate can be simulated successfully by employing the ductile fracture criterion. Fig. 10(a) presents the FEM simulation result of the slitting process. Fig. 10(b) presents the simulation result of the section A-A’. The top die and the bottom die are not shown in Fig. 10(b). The deformed shape contains four distinct zones including rollover, shear, fracture and burr. The lengths of each zone show relatively small difference with the simulation result of the shearing process. However, in the slope of the trapezoidal shape, the deformed shape is different from the center of the trapezoidal shape. In the slope of the trapezoidal shape, the length of the fracture zone is relatively long. Fig. 11(a) is the simulation result of the three-step forming process of the shielded slot plate. The deformed shape after the final forming process is shown in Fig. 11(b). Fig. 11(c) is a sectional view of the simulation result. In the simulation result, the effective strain is more concentrated at the upper round region than at the lower round region. Significant deformation is induced at the upper round region of the trapezoidal shape. The minimum thickness is shown in the upper round region. The ratio of the minimum thickness to the initial thickness is 0.77. An unusual observation is the non-uniform length of the rupture zone at the section B-B’, which is presented in Fig. 11(c). The length of the rupture zone is relatively small around point B. However, the length of the rupture zone increases around point B’. The different lengths of the rupture zone are induced from the slitting process. At the slope of the trapezoidal shape, the sheet is sheared by the slope of the slitting punch. The material starts to be

916

sheared more easily in this part than in other parts, because the stresses in that part are induced from both the tension along the punch and shearing.GThe length of the fracture zone is longer than the other parts. G

(a) FIGURE 10. Simulation results of the slitting process

(b)

FIGURE 11. Simulation results of the three-step forming simulation

FIGURE 12. Experimental results of the shielded slot plate

RESULTS AND DISCUSSION The experiments were conducted using the die set shown in Fig. 3. Fig. 12(a) shows the three-step forming process result. The lower region of the sheets undergoes only the slitting process, the middle region undergoes the

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slitting and preforming process, and the upper region is the final result of the experiment. The holes in the left and right side were made for alignment in the multi-step forming process. Fig. 12(b) presents the SEM images of the experimental result for each step. Fig. 12(c) is the sectional view of the experimental result. In the experimental result, the minimum thickness is shown in the upper round region and the ratio of the minimum thickness to the initial thickness is 0.79. Fig. 12(d) shows SEM images presenting details of the section C-C’ shown in Fig. 12(c). The relative length of the rupture zone around point C’ is longer than that of point C. The experimental results and the simulation results are in good agreement. In the center part where the sheet contacts with the punch parallel deforms identically with deformation observed in the shearing process. However, the sheet that contacts with the slope of the punch shows non-uniform fracture surface. The deformation of the center part depends mainly on the shearing. The deformation at the slope of the slitting process depends mainly on the tensile deformation along the slope and the shearing deformation. In order to reduce the non-uniform fracture area and length of the fracture area, the inclined surface of the punch should be minimized. In the performing step, a preform with a modified shape [11] is used in order to increase the formability. The shielded slot plate can be formed without fracture. In the FEM simulation including the slitting process, it is found that the deformation is concentrated on the upper round region of the trapezoidal shape. For a more uniform strain distribution, optimization of the forming process is necessary.

CONCLUSION FEM simulations employing a ductile fracture criterion are conducted to predict the deformed geometry of a unit cell of the shielded slot plate. The user material subroutine VUMAT is used for implementation of the material and ductile fracture criterion in the commercial FEM software ABAQUS/Explicit. The critical value of the ductile fracture criterion is defined from a comparison of the simulation and experiment results of the shearing process. In the three dimensional simulation of the unit cell, the ALE option in ABAQUS/Explicit is used. In the simulation results, non-uniform fracture surface is found on the surface of the part. In the upper round region, the ratio of the minimum thickness to the initial thickness is 0.77. The experiments were conducted to verify the simulation results. The result of the experiments showed good agreement with the simulation results.

ACKNOWLEDGMENTS This work was supported by the New & Renewable Energy of the Korea Insitute of Energy Technology Evaluation and Planning(KETEP) grant funded by the Korea government Ministry of Knowledge Economy (No. 2008-N-FC12-J-04-1000)

REFERENCES 1. G. Durante, S. Vegni, P. Capobianco and F. Golgovici, Journal of Power Sources, vol. 152, pp. 204-209 2. C. Yuh, J. Colpetzer, K. Dickson, M. Farooque and G. Xu, Journal of Materials Engineering and Performance, vol. 15(4), 2006, pp. 457-462 3. S. Fragini, Journal of Power Sources, vol. 182, 2008, pp. 462-468 4. E. Taupin, J. Breitling, W. T. Wu and T. Altan, Journal of Materials Processing technology, vol. 59, 1996, pp. 68-78 5. H. H. Wisselink and J. Huetink, Journal of Materials Processing technology, vol. 148, 2004, pp. 328-341 6. S. Ghosh, M. Li and An. Khadke, Journal of Materials Processing technology, vol. 167, 2008, pp. 91-102 7. M. G. Cockroft and D. J. Latham, Journal of Institute of Metals, 1968, pp. 33-39 8. M Oyane, T. Sato, K. Okimoto and S. Shima, Journal of Mechanical Working Technology, vol. 4, 1980, pp. 65-81 9. V. Tvergaard, Advanced in Applied Mechanics, vol. 27, 1990, pp. 83–147. 10. R. Hambli and M. Reszka, International journal of Mechanical Sciences, vol. 44, 2002, pp. 1349-1361 11. C. H. Lee, S. M. Ryu, D. Y. Yang, D. W. Kang, I. G. Chang, and T. W. Lee, “Analysis of Multi-step Forming of Metallic Bipolar Plate for MCFC Using Various Shapes of Preforms” in NUMIFORM 2010, edited by F. Barlat et al., AIP Conference Proceedings 1252, Americal Institutie of Physics, 2010, pp. 296-303

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Saving Material with Systematic Process Designs M. Kerauscha a

AutoForm Engineering Deutschland GmbH, Ingolstädter Str. 102, Pfaffenhofen, D-85276, Germany

Abstract. Global competition is forcing the stamping industry to further increase quality, to shorten time-to-market and to reduce total cost. Continuous balancing between these classical time-cost-quality targets throughout the product development cycle is required to ensure future economical success. In today's industrial practice, die layout standards are typically assumed to implicitly ensure the balancing of company specific time-cost-quality targets. Although die layout standards are a very successful approach, there are two methodical disadvantages. First, the capabilities for tool design have to be continuously adapted to technological innovations; e.g. to take advantage of the full forming capability of new materials. Secondly, the great variety of die design aspects have to be reduced to a generic rule or guideline; e.g. binder shape, draw-in conditions or the use of drawbeads. Therefore, it is important to not overlook cost or quality opportunities when applying die design standards. This paper describes a systematic workflow with focus on minimizing material consumption. The starting point of the investigation is a full process plan for a typical structural part. All requirements are definedaccording to a predefined set of die design standards with industrial relevance are fulfilled. In a first step binder and addendum geometry is systematically checked for material saving potentials. In a second step, blank shape and draw-in are adjusted to meet thinning, wrinkling and springback targets for a minimum blank solution. Finally the identified die layout is validated with respect to production robustness versus splits, wrinkles and springback. For all three steps the applied methodology is based on finite element simulation combined with a stochastical variation of input variables. With the proposed workflow a well-balanced (time-cost-quality) production process assuring minimal material consumption can be achieved. Keywords: process design, sensitivity analysis, material saving, process robustness. PACS: 89.20.Bb

INTRODUCTION The motivation for the automotive industry to find material cost optimized solutions is mainly driven by two aspects. On the one hand, the price for both steel and aluminum has increased in the past two years in the order of 30-40% [1, 2]. On the other hand, more and more high-strength steels and light weight materials are used in automotive body structures to reduce weight while at the same time improving crash safety. Since advanced sheet materials are more expensive than classical deep drawing steels, this trend also increases the amount of material cost per body-in-white. Both trends, rising material prices and the increasing usage of advanced sheet material, have negative influence on the value chain of sheet metal products. But the biggest impact will be felt by stamping suppliers, as material represents their highest cost and typically they have few options to spread higher costs across production [3]. Therefore competition is forcing stamping industry to minimize material cost while classical functional and quality targets of stamping products still have to be ensured by die layout and process engineers. Balancing between time-cost-quality targets is already an important task during the development process of automotive body panels and will also be a sophisticated future challenge. Simulation and digital process planning, in particular, plays a decisive role in balancing and decision making process [4];depending on the specific body panel (skin, structural, inner, etc.) and the targeted market for the vehicle program. Luxury cars for example, which are produced in small numbers, part quality with perfect finish and fit is much more important than material usage. The opposite with high-end middle-class or even low-end car model, where the material cost is the dominant factor. Every millimeter saved in blank size and each scrap part avoided represents significant cash savings.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 919-924 (2011); doi: 10.1063/1.3623703 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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COMPREHENSIVE DIGITAL PROCESS PLANNING Two examples point out the importance of proper balancing for the decision making process during die layout engineering. Material cost versus tooling cost: is it preferable to avoid a cam for a cutting operation by a tool cutting in working direction due to an optimized tipping for cutting but which requires a slightly larger blank? Assuming the cam adds 100k$ to the overall tools cost and the tool cutting solution requires a blank which is 0.25$ more expensive. In this case the cam should be avoided only if the planned production volume is less than 400 thousand cars. Although this example is strongly simplified, it shows that general design rules like "avoiding cams helps save money" can lead to counterproductive results. A conclusion from this example could be that material cost should be the sole focus for mass volume cars and everything else remains secondary. But what if a radical blank size reduction that saves $0.50 per blank but causes 10% rejected parts due to robustness problems? That means for a production volume of 1 million parts there is a saving potential of 500k$, but just theoretical. Assuming a price of 3$ per blank the savings are reduced by 300k$, because of the higher rejection rate. If additional quality assurance measures, production down time and manual rework exceed 200k$, the conservative solution for material utilization becomes more cost-effective. Decisions similar to the examples above have to be taken upfront during process engineering. At that time there is typically a lack of information regarding the impact of different process variants on production and tool costs as well as tryout and production lead time. Today either design guidelines or processes solutions from the past are used as reference to assist the die layout engineer in the decision making process. As a consequence, innovative solutions with additional material saving potential but without production experience involve an unforeseeable risk. Comprehensive Digital Process Planning (CDPP) can create the required transparency during the engineering workflow to improve the information basis for balancing between the interacting goals regarding function, quality, lead time and cost [4, 5]. Thus, comprehensive digital process planning is a key factor for true cost saving. Without the ability to calculate the production risk of unprecedented stamping solutions early during process engineering, innovative solutions which offer material saving potential will hardly be realized. Recognizing whether a new die layout will result in an overall saving will support creative solutions with less material consumption.

FIGURE 1. (a) Improvements to 'fit' may drive material cost from optimum, (b) Reducing tooling cost may affect quality or ramp-up time adversely [5]

PROCESS DESIGN The goal is to introduce a CDPP-based workflow which improves the balancing during process engineering. In particular this workflow should assist draw die engineers to evaluate new material saving solutions with respect to their performance during production. As a representative example a typical structural part, a hinge reinforcement of a rear door, is used (Figure 2). The starting point is a full process plan according to a set of typical die design and blank size standards (Table 1) following commonly used parameters in industry. The manufacturing process consists of a single action drawing operation, a subsequent cutting, two flanging operations and a final cut including

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separation of the parts. The trapezoid blank is cut from a 955 mm coil with a progressive feed of 460 mm; the blank material H220BH has an initial thickness of 1.5 mm.

FIGURE 2. (a) draw die geometry according to typical die design standards, (b) blank geometry TABLE 1. Die design and blank size standards applied to the investigated example.

Design Parameter Addendum: Wall angle Addendum: Die radius Addendum: Distance between left and right part Binder: Distance between die entry and drawbead center line Blank: Distance between drawn-in blank and drawbead center line

Abbr.

Value

DW rD ldf ldb lmin

10° 7 mm 20 mm 20 mm 15 mm

With respect to typical quality targets for thinning, wrinkling and springback the die layout delivers acceptable results. In a conventional development workflow, the simulation-based process engineering is completed and the proposed tool geometry would be released for milling (Figure 3). A reason for relying on die layout standards, similar to those used for reinforcement part, is to implicitly ensure a stable production. Since die layout standards are general rules, they normally make sure that a certain safety margin is included in the proposed solution.

FIGURE 3. Conventional workflow and additional steps for CDPP-based material saving.

During draw die development it is valid to ask the question, if there is still unexploited potential for material saving due to the safety margin of general design guidelines. A robustness check can help to answer this question. The robustness check is a stochastical analysis that reveals the impact of production noise on the proposed die layout [6, 7]. Multiple simulations are conducted with a variation of noise parameters reflecting the existing but noninfluenceable scatter during serial production (Table 2). The variation or the noise parameters follow a normal distribution defined by center value, standard deviation and minimum and maximum value. As a result of robustness checks the process capability index Cpk can be calculated. The Cpk value is a statistical measure that predicts how many parts will be produced out of specification due to cracks, wrinkles or springback. If the design guideline serves its purpose of ensuring a stable production the predicted defect rate should be zero. Typically this will not the case; beside being statistically improbable this would be a clear indication that the process design is over-engineered. If an instable process is detected due to the proposed CDPP-based workflow in Figure 3, then ans adaption of the proposed die layout is required. If Cpk is between 1.0 and 1.33 up to 0.135% of the

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production results are likely outside of the quality specification limit. In this case the proposed process design has the potential for a stable production, but there is normally no further potential for material saving measures, since small modification already could lead to undesired results. A Cpk greater than 1.33 indicates that there is a sufficient process window for a robust production. In this case a systematic check of the proposed die layout regarding material saving potential should follow. TABLE 2. Noise parameters for stochastical investigation of a robust production.

Noise Parameter Tensile Strength Yield Strength (depended on V0) Material

Process

Anisotropy (depended on r0) Initial Thickness Position (in x-direction) Lubrication Binder Force

Abbr.

Center

StdDev

Min

Max

'V0 [MPa] 'Rm [MPa] r0 [-] r45 [-] r90 [-] t0 [mm] x [mm] P [-] FB [kN]

0 0 2.03 1.62 2.51 1.5 0 0.15 3500

8.3 8.3 0.135 0.108 0.167 0.05 0.33 0.005 117

-25 -25 1.62 1.3 2.01 1.35 -1 0.135 3150

25 25 2.44 1.94 3.01 1.65 1 0.165 3850

The material consumption during drawing operations is influenced by the developed section length of the addendum and binder surface, as well as the resulting drawing depth. For this example the wall angle and die radius are successively minimized until the quality limit for cracks is reached. In a second step the blank dimensions and the drawbeads are adapted according to modified addendum. According to the design rule that distance between drawn-in blank and drawbead center line should be minimum 15 mm; to check for material saving potential the blank shape is further reduced until the quality limit for wrinkles is reached. As final step of the proposed workflow again a robustness check is required, since safety margins of the die design standards intentionally beyond the limit. In case the robustness check indicates an instable production; a further loop is required to indentify a more moderate material saving solution. As soon as robustness is validated the conventional workflow continues. Finally, the new die layout with less material consumption can be released for milling.

MATERIAL SAVING According to the CDPP-based workflow in Figure 3 an initial robustness analysis is applied to the proposed manufacturing process for the hinge reinforcement part with respect to the design standards of Table 1 (conventional solution). Scatter in material properties and process noise is modeled as variable input parameter according the values summarized in Table 2. 90 random samples of the noise parameters are calculated to ensure a adequate level of confidence for the later statistical assessment; 15 per independent noise parameter. The upper left and right picture in Figure 4 shows the result of the robustness check for the conventional solution. Except a small area in the wall of an embossing, which may cause up a defect rate of up to 2.22%, the drawing process is validated as robust. This critical area was overlooked during classical process design. In other words the safety margin included in the general design guideline underestimates the risk of cracks for this example; problems due to cracks will occur in production; also the robustness in terms of springback indicates two instable areas in the flanges for a geometric tolerance of ± 0.5 mm. Three robustness issues have been detected for the conventional solution. Although this could be interpretated that there is not much potential for further process optimization, the CDPP-based workflow with focus on material saving is proceeded. The goal is to optimize the material consumption and to improve process robustness in the wall of the embossing. Since the scatter in springback of the two flanges is geometry based, a prevention without changing the part geometry is not expected. Thus, the goal is to end up with maximal same variation in springback results for the flange area. In a first modification loop, addendum and binder geometry are checked for material saving potential. Since the developed length of the surrounding addendum wall is less than 3 mm in the longitudinal direction of the part and maximum 12 mm on the smaller side, the saving potential due to a steeper wall angle is marginal. For this example the die radius offers the chance to effectively reduce the material consumption due to two effects. Firstly the drawing depth is reduced by the same amount 'rD the die radius is reduced; secondly the developed length is reduced by approximately 0.5*S*'rD (valid for a 90° angle). For this example the initial die radius of 7 mm is minimized to 5 mm and accordingly the drawing depth is adapted as well as the drawbead center line.

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FIGURE 4. Comparison between initial and final robustness check. Left: Robustness versus cracks after drawing operation. Right: Robustness versus springback after final cutting operation.

In a second step the blank shape is adapted to the modified draw die geometry. A variance from the design guideline; the target draw-in limit is defined as minimum 10 mm distance to either the entry of the die radius respective 10 mm distance to the cutting line. The second restriction is applied in areas, where the part is developed onto the binder. After several conventional simulation loops a revised process that meets thinning, wrinkling and springback criteria was identified. The resulting trapezoid blank shape has a width of 960 mm (before 955 mm) and a progressive feed of 400 mm (460 mm). In Figure 4 the results of the final robustness check for the modified solution (lower) are compared with the initial robustness check of the conventional process (upper). The stability problem in the wall of the embossing could be solved through an improvement in draw-in. As expected, the springback issues could not be eliminated without compensation, however the overall result regarding springback stability has improved; for one of the two flanges the springback variation due to scatter in material properties and production noise is now acceptable.

CONCLUSION This paper describes a systematic workflow with focus on minimizing material consumption. With respect to the example process, a hinge reinforcement, the blank volume was reduced from 659 103 mm3 to 576 103 mm3 (-12.6%). Compared to the starting point of the investigation an overall improvement in process robustness was achieved. Although the standard solution passes all quality criteria of a defined set of die design guidelines, process instabilities were identified by a stochastical FE-analysis. The safety margin included in the general design guideline underestimates the risk of cracks for this example. As a consequence problems due to cracks and springback will occur later in production. Due to a robustness check during simulation based process design, instabilities can be detected. Countermeasures to ensure a robust production can be taken in time. For the example process binder and addendum geometry were systematically checked for material saving potentials. Subsequent the blank shape and draw-in were adjusted to meet thinning, wrinkling and springback targets for a minimum blank solution. Finally the identified die layout was validated with respect to production robustness versus splits, wrinkles and springback. With the proposed workflow a well-balanced (time-cost-quality) production process assuring minimal material consumption can be achieved.

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REFERENCES 1. www.meps.co.uk. 2. www.westmetall.de. 3. R. G. Matthews, "Steel-Price Increases Creep Into Supply Chain", in The Wall Street Journal, Dow Jones & Company, 3 February 2011. 4. W. Kubli and V. Prabhakar, "Digital Process Planning for Automotive Body Panels", in Numisheet 2008, edited by P.Hora, Proceedings of the 7th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, Institute of Virtual Manufacturing, Zurich, Switzerland, 2008, pp. 885-892. 5. D. Banabic, "Numerical Simulation of the Sheet Metal Forming Processes", in Sheet Metal Forming Processes, Heidelberg: Springer-Verlag, 2010, pp. 213-295. 6. M. Tisza, Z. Lukács and G. G. Gál, "Integrated Process Simulation and Die-Design in Sheet Metal Forming", in Proceedings of IDDRG 2007 International Conference, Gyor, 2007. 7. A. Emrich, M. Liewald and G. Ruf, "Stochastic Analysis in FE-Simultaion of Sheet Metal Forming as a Key Enabler for Robust Production Processes", in Zero Failure Production Methods, edited by P. Hora, Proceedings of 4th Forming Technology Forum Zurich 2011, Institute of Virtual Manufacturing, Zurich, Switzerland, 2011, pp. 31-36.

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PROCESS MODELING AND OPTIMIZATION

Robust Design of Sheet Metal Forming Process Based on Kriging Metamodel Yanmin Xie School of Mechanical Engineering, Southwest Jiao tong University, Chengdu, China Abstract. Nowadays, sheet metal forming processes design is not a trivial task due to the complex issues to be taken into account (conflicting design goals, complex shapes forming and so on). Optimization methods have also been widely applied in sheet metal forming. Therefore, proper design methods to reduce time and costs have to be developed mostly based on computer aided procedures. At the same time, the existence of variations during manufacturing processes significantly may influence final product quality, rendering non-robust optimal solutions. In this paper, a small size of design of experiments is conducted to investigate how a stochastic behavior of noise factors affects drawing quality. The finite element software (LS_DYNA) is used to simulate the complex sheet metal stamping processes. The Kriging metamodel is adopted to map the relation between input process parameters and part quality. Robust design models for sheet metal forming process integrate adaptive importance sampling with Kriging model, in order to minimize impact of the variations and achieve reliable process parameters. In the adaptive sample, an improved criterion is used to provide direction in which additional training samples can be added to better the Kriging model. Nonlinear functions as test functions and a square stamping example (NUMISHEET’93) are employed to verify the proposed method. Final results indicate application feasibility of the aforesaid method proposed for multi-response robust design. Keywords: Robust design; Kriging model; Sheet Forming; Optimization. PACS: 46. Continuum mechanics of solids

INTRODUCTION In engineering optimization a direct coupling between analysis models and optimization routines may be very inefficient, as during optimization a large number of iterative calls to possibly time-consuming analysis models may be necessary. In those situations it is preferred to uncouple analysis and optimization through the use of so called metamodels or surrogate models: fast to evaluate approximations for objective and constraint functions. These metamodels may also be helpful in large-scale multidisciplinary optimization (MDO), to exchange relevant information between the disciplines involved and to study the effect of design changes on coupled sub-problems. Among the various metamodel techniques for optimization, response surface methodology (RSM) and Kriging attract the most attention. Of the most widely used approximation methods, response surface methods (RSM) have been used successfully in many deterministic engineering design problems [1, 3]. As an alternative approximation method to RSM, Kriging has not been used for the engineering design applications until Sacks et al. [4] introduced it into the literature. Kriging models can accurately approximate an unknown system even with high nonlinearity. Kriging originates from the field of geostatistics as a method to predict the responses for spatially correlated data, such as the thickness of ore layers from a limited number of experimental data [5]. The theory of Kriging was developed from the work of its inventor, Danie G. Krige and further developed by Matheron [6]. Giunta [7] utilized this method for the multidisciplinary design optimization of a High Speed Civil Transportation Aircraft, in which he compared RSM with a Kriging method. An extension of the Kriging method was developed by Osio and Amon [8] for a numerical optimization problem of the thermal design of an embedded electronic package. Booker et al. [9] solved a helicopter rotor structural design problem using an approximation method based on Kriging. More recently, Sakata et al. [10] developed an optimization method using Kriging approximation, which was applied to the optimization of a stiffened cylinder for an eigenfrequency problem. Polynomial and interpolating models were compared by Giunta [11]. Simpson [12] analyzed experiment designs and approximate models. Park [13] used a

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 927-934 (2011); doi: 10.1063/1.3623704 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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Kriging method to obtain the shape optimization of the plate-fin type heat sink. Bang [14] optimized a jaw structure using a Kriging interpolation method. A Kriging-based metamodeling technique is used to minimize the risk of failure in a sheet metal forming process [15]. Although conventional deterministic optimization model can find an optimum from mathematic point of view, variations caused by noise factors from material, assembly and machining during manufacturing are not considered. Li [16] developed a CAE-based six sigma robust design method. Statistical techniques and dual response surface approximate model were integrated to perform robust design of deep drawing process of a rectangular cup. Tang [17] developed a robust design by adaptive importance sampling in a cup deep drawing. In this paper, a high nonlinear function is employed for proving the Kriging model. At the same time, a practical engineering problem is also robust optimized by an adaptive Kriging method.

KRIGING MODELING APPROACH The basic approach to the construction of a Kriging model is to model the response stochastic process [4, 18]:

y ( x) as the realization of a

Y ( x)  f T ( x)  Z ( x)

(1)

It is the combination of a ‘global’ model f ( x)  and a random process T

corrections to the ‘global’ model. Where f ( x)   f1 ( x),

, f m ( x)  and    1 , T

z ( x) which allows for ‘local’ ,  m  , m denotes the number T

Y ( x) is the unknown function of interest, f ( x) is a known function of x ,  is the regression coefficient vector, and z ( x) is assumed to be a stationary process with zero mean and

of basis functions in the regression model, covariance

cov( xi , v j )   2 R( xi , v j ) i, j  1, 2,

,n

(2)

R(.,.) is a correlation function and  2 is the process variance, xi and x j present different two variables and cov( xi , x j ) is the covariance of these two variables. Where n denotes the number of experiments(sampled points),

For a certain experimental design such as central composite design, a set of experimental points are generated as

x  x1 , x2 ,

, xn , xi R p

(3)

Where p is the number of design variables. Thus, function to be modeled are given as:

Y   y1 ( x), y2 ( x),

, yn ( x)

From these outputs the unknown parameters ˆ  ( F R F ) F R Y T

1

1

T

1

x is a n  p design matrix. The resulting outputs from the (4)

 and 

2

can be estimated

1 n

ˆ 2  (Y F ˆ )T R-1 (Y F ˆ )

(5)

Where F is a vector including the value of f ( x) evaluated at each of the experiment points and R is the correlation matrix, which is composed of the correlation function evaluated at each possible combination of the experimental points R ( x1 , xn ) 

R ( x1 , x2 )   (6) R    R( x , x )  R ( xn , xn )  n 1  However, before calculating ˆ and ˆ , first the unknown parameters of the correlation function have to be estimated. Using maximum likelihood estimation, they result from the minimization of 1 (7) 2(n ln ˆ 2  ln det R) Which is a function of only the correlation parameters and the response data. For the estimation of these parameters, the best linear unbiased prediction of the response is 2

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yˆ( x0 )  f T ( x0 )  r T ( x0 )R 1 (Y F  ) Where r

T

(8)

( x0 ) is a vector representing the correlation between an unknown set of point x0 and all known

experimental points

r T ( x0 )  R( x0 , x1 ), R( x0 , x2 ),

The second term r

T

, R( x0 , xn )

(9)

( x0 )R (Y F  ) is in fact an interpolation of the residuals of the regression model. Thus, 1

all response data will be exactly predicted. In all the literature concerning the method of Sacks et al. [4], a correlation binary function of the type R( xi , x j )  R( xi x j ) is generally selected, and a product correlation rule is used for mathematical convenience p

R( xi , x j )   R( xik x kj )

(10)

k 1

Where

xik and x kj denote kth component of the experiments. The correlation function R( xi , x j ) is specified by

the user, and several correlation functions exist in the literature. Among them, however, the following function is mostly used 2  p R( xi , x j )  exp   k xik x kj   k 1 

(11)

Which permits control of both the range of influence and the smoothness of the approximation function. In this paper, the correlation function given in Eq. (11) is implemented for building metamodels. This is the paragraph spacing that occurs when you use the Enter key.

VALIDATION METHODS OF A KRIGING MODEL After the approximation model is constructed, it must be validated to ensure sufficient accuracy. The additional validation points are used to compute error. Where error is defined as the difference between the predicted and actual values, such as Root Mean Square Error (RMSE), Empirical Integrated Squared Error (EISE) [16], Mean Absolute Percentage Error (MAPE), and so on. (1) The Root Mean Square Error (RMSE): The RMSE is: H

  yˆ ( x) y ( x)  i

RMSE=

2

i

i 1

(12)

H

Where H is the number of additional sample points,

yi ( x) is the actual value, and yˆi ( x) is the predicted value.

(2)The Empirical Integrated Squared Error (EISE):

EISE 

1 H

H

 ( yˆ ( x) y ( x)) i

2

(13)

i

i 1

(3)The Mean Absolute Percentage Error (MAPE):

MAPE 

1 H

H

 i 1

yˆi ( x) yi ( x) yi ( x)  

0 ! ! #0.01

 "

yi ( x)

0

yi ( x)  0

(14)

NUMERICAL STUDY OF KRIGING To illustrate the practical performance of Kriging it is applied to the two-variable analytical test function [19]:

y  2  4 x1  4 x2 x12 x2 2  2sin(2 x1 ) sin(2 x2 ) The test function is shown in figure 1.

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0.5 $ x1 , x2 $ 3.5

(15)

For the Kriging model, a global linear regression model and a Gaussian random process are chosen. 2  2 R( xi , x j )  exp  k xik x kj   k 1 

f ( x)  (1 x1 x2 x12 x1 x2 x2 2 )T

Which depends on the optimization of the parameter,

(16)

k .

It is shown that differences between the predictions and true functional values are predominantly caused by the experimental design used for the training. This can be illustrated by training the same Kriging model on a

32 full

42 full factorial design and 52 full factorial design. Results are presented in Table 1. The Kriging 2 approximate surface of the test function is shown in figure 2 for 5 full factorial design. The Values for MAPE,

factorial design,

EISE and MSE, calculated on a regular 31 × 31 grid in design space, are also reported in Table 1. TABLE 1. Comparison of Kriging results with different design levels.

Factor

3 42 52

2

1

H

2

MAPE (%)

EISE

RMSE

961

40

160

10

1.16

1.20

961

160

2560

9.77

1.01

1.09

961

1.25

0.88

0.71

0.0066

0.16

From the results, the Kriging model matches the true function well, and the model with 2

2

5 full factorial design

2

with factor levels captures the function more adequately than 3 and 4 .

y

y

½ experiment design

x2

x2

x1

Figure 1. the original surface of the tested function

x1

Figure 2. the approximate surface of the function

APPLICATION OF A KRIGING MODEL IN SQUARE DRAWING In the optimization of sheet metal forming, large computational efforts are needed to calculate the value of optimization function and the engineering constraints to check both the feasibility and optimality. The Kriging metamodel can replace the original model, thus reducing the computational burden of evaluating numerous designs. An adaptive optimization based on Kriging is proposed in order to find the global optimum value, as shown in figure 3. Deep-drawing of a square cup is an international benchmark at the NUMISHEET’93[20]. The material parameters are taken as follows. The thickness of a mild steel plate is 0.78mm. The Young’s modulus E is 206 GPa, the friction coefficient is 0.144 and the blank holding force (BHF) is 19.6KN. The anisotropy coefficient r0 is 1.79,

r45

is

1.51,

r90

is

  565.32(0.007117   p )

2.27. 0.2589

The

mean

true

stress-true

strain

relationship

expression

. Geometric parameters for the cup drawing are shown in figure 4.

930

is

Determine the optimization model

DOE for key factors

PUNCH

Construct Kriging metamodel Add the modified design the samples by the adaptive sample method

2

Robust optimization based on Kriging model, Obtain new design, and its MSE

2

170

74 70

2

70 74

Run LS-DYNA to get corresponding objective values

2

170

DIE 170

No

Is convergence criterion satisfied?

70

50

7 PUNCH

Yes

50 43 BLANK HOLDER

R8 STROKE MIN.50

Obtain robust design

Stop

Figure 3. Procedure of robust optimization based on Kriging model with an adaptive sample method

R5 2

48

Figure 4. The tool geometry of square cup forming (mm)

Figure 5 shows the draw-in direction, which is one forth of the forming. In the direction OA the distribution of thickness strain is illustrated in Figure 6, which shows a good agreement of simulation with reported experiment results in [20]. Initial blank edge C

DY

D D

B

Deformed blank edge

Y DX

X

A

0

Figure 6. Thickness strain along OA

Figure 5. Definition of draw-in direction

Design of Experiment In this study, the design of experiment method - a powerful tool for parameter design of performance characteristics - is used. Three controllable factors are chosen, which are the bank holding force ( BHF ), punch radius ( R1 ), and friction coefficient ( % 3 ). The each parameter is designed to three levels according to the NUMISHEET experiment, denoted 1, 2 and 3, as shown in table 2. So an array with 27 rows is used for the simulation. The multi-objective quality characteristics of the deep-drawing process are defined as following. The first and second objective functions are in term of forming limit diagram (FLD). Principal strains obtained for a particular

931

TABLE 2. Control factors and their various levels.

%3

R1

BHF

Level

1 2 3 processes are compared with the forming limit curve (FLC) as demonstrated in figure7. Two FLC in the principal plan of logarithmic strains proposed by Hillman are given 20 40 60

7 8 9

0.10 0.15 0.20

1  &s ( 2 ) 1  &w ( 2 ) (17) Where &s and &w are the FLC functions which control the rupture and wrinkling respectively and defined as the safety FLC by

's ( 2 )  &s ( 2 ) s

'w ( 2 )  &w ( 2 )  s

(18)

Where s is called safety tolerance, taken from the true FLC, and this tolerance is constant during the optimization process and defined by user. Thus, the objective functions are defined for each element as following. (1) The tolerance between the actual strain  1 and the rupture criterion in FLC for a given strain  2 expressed by

f1  d1 ( 2 ) 

1 sum e 1 's  2e  for 1e $ 's  2e      sum e1 

(19)

(2) The tolerance between the actual strain  1 and the wrinkle criterion in FLC for a given strain  2 expressed by:

f 2  d 2 ( 2 ) 

1 sum 'w  2e  1e  for 1e ( 'w  2e      sum e1 

(20)

(3) The third objective function is based on thickness variation between initial and final state.

1 sum te t0  f3    sum e1  t0 

4

(21)

Where t0 , te are initial and final thickness, which is principal stretch along thickness. The minimization of this

Major strain e1

objective function lets to attenuate thickness variations, so it can improve feasibility of the work piece.

s

'w  2  &w   2 

&s   2 

wrinkle criteria d2  2 

Safety domain

d1   2 

's  2 

rupture criteria

s

Minor strain e2 Figure 7. Illustration of de criteria in FLD

Optimization of Square Forming Making use of the above validated FEM model, the finite element analysis is implemented for each row of the design of experiment. The multiple responses in different parameters are obtained. The Kriging model is trained by the obtained data through the standard Kriging procedure. For the Kiging model, a quadratic polynomial and a

932

Gaussian random function are selected. The objective functions for the sheet forming are obtained. The objective prediction from the trained Kriging for f1 is illustrated in figure 8. x1 is BHF ; x2 is R1 ; x3 is % 3 . The objective prediction from the trained Kriging for

f 2 is illustrated in figure 9.

f2

f1

x3

x3

x1 (KN)

x1 (KN)

Figure 8. The first object under x1 and x3 in Kriging metamodel

Response prediction from the trained Kriging for

Figure 9. The second objeuer x1 and x3 in Kriging metamodel

f 3 is also illustrated in this method. In the square drawing, the

first and second functions are selected as constrain functions, the third function is selected as the robust optimization function, as follows:

Min )1  % f3  )2   f3 S .t. Where

f1 $ 0

)1 and )2

f2 $ 0

(22)

are weighting factors for mean and variance, respectively.

%f

3

and  f3 are the mean and

variance of the objective function. For deterministic optimization, the problem is formulated as:

Min

f3

S .t.

f1 $ 0

f2 $ 0

(23)

The drawing process is robust optimized, and the optimum parameters are obtained. They are 33.54, 8.87, and 0.146 for BHF , R1 and % 3 . However, the results for the deterministic optimization are 30.02, 7.83, and 0.101. Figure 10 and Figure 11 show the wrinkle constrain distribution under robust and deterministic optimum solutions. Deterministic optimization has the lower optimum, but its variance is larger. However, the variance of the adaptive robust optimization is much less, so it is much more robust in feasibility probability calculations.

Figure 10. Wrinkle constraint distribution of deterministic optimization

Figure 11. Wrinkle constraint distribution of robust optimization by an adaptive method

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CONCLUSION This paper investigated an adaptive robust optimization based on Kriging. In order to prove accuracy and efficiency of the Kriging method, a nonlinear function is implemented. At the same time, a square drawing is also robust optimized successfully. The Kriging metamodel effectively replaces the function of numerical simulation during optimization iteration, and enhances the efficiency of optimization. The final results indicate the proposed method is feasible for the robust design of sheet metal forming process. In the future work, a dual-Kriging model should be constructed to further improve the Kriging model.

ACKNOWLEDGMENTS The research is supported by the National Natural Science Foundation of China (NSFC51005193), supported by Fundamental Research Funds for the Central Universities (SWJTU09CX016), and supported by youth funds of Southwest Jiao Tong University.

REFERENCES 1. G. Venter, R.T. Haftka, J.H. Starnes, Construction of response surfaces for design optimization applications, In: Proceedings of the 6th AIAA/NASA/ISSMO symposium on multidisciplinary analysis and optimization. Bellevue (Washington), September 4-6, 1996, Part 1, pp. 548-564. 2. O. Balabanov, Development of approximations for HSCT wing bending material weight using response surface methodology [D]. Virginia Polytechnic Institute and State University, Blacksburg (VA); 1997. 3. G. Ingarao, R. D. Lorenzo, Optimization methods for complex sheet metal stamping computer aided engineering, Struct. Multidisc. Optim. 2010, 42(3), pp. 459-480. 4. J. Sacks, S.B. Schiller, W.J. Welch, Design for computer experiment. Technometrics 1989, 31 (1), pp.41ˀ 47. 5. N. A.. C. Cressie, Statistics for Spatial Data, J. Wiley: New York, 1993. 6. G. Matheron. Principles of geostatistics. Economic Geology, 1963, 58, pp. 1246-1266. 7. A.A. Giunta, Aircraft multidisciplinary optimization using design of experiments theory and response surface modeling methods. Ph.D. dissertation, Virginia Polytechnic Institute and State University, Blacksburg (Virginia); 1997. 8. J.O. Osio, C.H. Amon, An engineering design methodology with multistage Bayesian surrogates and optimal sampling, Res Eng Des 1996; 8(4), pp. 189-206. 9. A.J. Booker, J.E. Dennis, P.D. Frank, et al. A rigorous framework for optimization of expensive functions by surrogates. Struct Optimization, 1999, 17(1), pp. 1-13. 10. A. Sakata, F. Ashida, M. Zako. Structural optimization using Kriging approximation. Comput Meth Appl Mech Eng 2003, 192(7ˀ 8), pp.923-39. 11. A.A. Giunta, L.T. Watson. A comparison of approximation modeling techniques: polynomial versus interpolating models, AIAA-98-4758. 1998, 36(1), pp.275-286. 12. T. W. Simpson, J.D. Peplinski, P.N. Koch, Metamodels for computer-based engineering design: survey and recommendations. Engineering with computers. 2000, 17(2), pp. 129-150. 13. K. Park, P.K. Oh, H.J. Lim. The application of the CFD and Kriging method to an optimization of heat sink. International Journal of Heat and Mass Transfer, 2006, 49(19-20), pp.3439-3447. 14. Il-Kwon Bang, Dong-Seop Han, Geun-Jo Han, et al. Structural optimization for a jaw using iterative Kriging metamodels. Journal of Mechanical Science and Technology, 2008, 22(9), pp.1651-1659. 15. H. Wang, Enying Li, G. Y. Li. Parallel boundary and best neighbor searching sampling algorithm for drawbead design optimization in sheet metal forming. Struct Multidisc Optim, 2010, 41(2), pp.309-24. 16. Y.Q. Li, Z.S. Cui, X.Y. Ruan, et al. Application of six sigma robust optimization in sheet metal forming. 2005 AIP Conf Proc, 778, pp. 819–824. 17. Y.C. Tang, J. Chen. Robust design of sheet metal forming process based on adaptive importance sampling. Struct Multidisc Optim, 2009, 39(5), pp.531-544. 18 H. Wang, Enying Li, G. Y. Li, et al. A metamodel optimization methodology based on multi-level fuzzy clustering space reduction strategy and its applications. Computers & Industrial Engineering, 2008, 55(2), pp. 503-532. 19. J. J. M. Rijpkema, L. F. P. Etman, A. J. G. Schoofs. Use of design sensitivity information in response surface and Kriging metamodels. Optimization and Engineering, 2001, 2(4), pp. 469-484. 20. Joachim Danckert. Experimental investigation of a square-cup deep-drawing process. Journal of Materials Processing Technology, 1995, 50(1-4), pp.375~384.

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A Numerical Model for the Search of the Optimum Capacitance in Electromagnetic Metal Forming Ruben Otin, Roger Mendez, and Oscar Fruitos CIMNE - International Center For Numerical Methods in Engineering Parque Mediterráneo de la Tecnología, edificio C3, despacho 206 E-08860 Castelldefels (Barcelona, Spain) tel.: +34 93 413 41 79, e-mail: [email protected] Abstract. Electromagnetic forming (EMF) is a high velocity forming technique that uses electromagnetic forces to shape metallic workpieces. In this work we present a numerical model that is able to compute the current flowing trough the coil, the Lorentz force acting on the workpiece and the optimum capacitance of the EMF process. The main advantage of our approach is that it provides an explicit relation between the capacitance of the capacitor bank and the frequency of the discharge, which is a key parameter in the design of an electromagnetic forming system. This method is computationally efficient because it only requires solving the time-harmonic Maxwell equations for a few frequencies to have completely characterized the EMF system. The approach can be very useful for estimating the order of magnitude of some parameters, for experimentation on modeling conditions, for coil design or for modeling complex geometries. Moreover, it can be easily included in a sequential coupling strategy without the worry of numerical instabilities. Keywords: Electromagnetic forming, optimum capacitance, optimum frequency, numerical analysis, finite element method

INTRODUCTION Electromagnetic forming (EMF) is a high velocity forming technique that uses electromagnetic forces to shape metallic workpieces. The process starts when a capacitor bank is discharged through a coil. The transient electric current which flows through the coil generates a time-varying magnetic field around it. The time-varying magnetic field induces electric currents in any nearby conductive material. These induced currents flow in the opposite direction to the primary currents and, therefore, a repulsive force arises between the coil and the conductive material. If this repulsive force is strong enough to stress the workpiece beyond its yield point then it can shape it with the help of a die or a mandrel. This technique present several advantages: no tool marks are produced on the surfaces of the workpieces, no lubricant is required, improved formability, less wrinkling, controlled springback, reduced number of operations and lower energy cost. But, in order to design sophisticated EMF systems and control their performance, it is necessary to advance in the development of theoretical and numerical models of the EMF process. This is the objective of the present work. More specifically, we focus our attention on the numerical analysis of the electromagnetic part of the EMF process. In this paper we present a numerical model that characterizes an EMF process in frequency domain. The input data required are the geometry and material properties of the system coil-workpiece and the electrical parameters of the capacitor bank. With these data and the time-harmonic Maxwell’s equations we are able to calculate the optimum capacitance, the current flowing trough the coil and the electromagnetic forces acting on the workpiece.

FREQUENCY-DOMAIN ELECTROMAGNETIC MODEL The electromagnetic model presented in this work starts by solving the time-harmonic Maxwell’s equations for a given frequency interval. For each frequency ω we compute the electromagnetic fields E(ω) and H(ω) inside a volume containing the coil and the workpiece. With E(ω) and H(ω) we calculate the inductance Lcw (ω) and the resistance Rcw (ω) of the system coil-workpiece. With Lcw (ω) and Rcw (ω) we compute the intensity I(t) flowing through the coil. Finally, with I(t) and the fields E(ω) and H(ω), we obtain the Lorentz force exerted on the workpiece and an estimation of the optimum capacitance of the EMF process. In the following it is described each step in more detail.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 935-942 (2011); doi: 10.1063/1.3623705 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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Time-harmonic Maxwell’s equations We start solving numerically the time-harmonic Maxwell’s equations inside a volume containing the coil and the workpiece. For that, we employ the in-house code called ERMES, which implements in C++ the finite element formulation explained in [1]. The electromagnetic fields E(ω) and H(ω) obtained at this step will be used to compute the inductance and the resistance of the system coil-workpiece and the Lorentz force acting on the workpiece. Although, in theory, we need all the frequencies ω ∈ [0, ∞) to fully characterized an EMF system, in practice, we have observed in the literature that the frequencies of interest lay in the interval 0.5 <

δ < 1.5 τ

where τ is the thickness of the workpiece and δ is the skin depth. The skin depth is defined as  2 δ= ω μσ

(1)

(2)

being μ the magnetic permeability of the workpiece and σ its electrical conductivity.

Intensity through the coil The next step is to obtain the electrical current I(t) flowing through the coil. We assume that the coil, workpiece and capacitor bank form a RLC circuit with resistance R = Rcb + Rcw , inductance L = Lcb + Lcw and capacitance C = Ccb , where the subscript cb denotes the parameters of capacitor bank and the subscript cw denotes the parameters of system coil-workpiece. The parameters Rcb , Lcb and Ccb are given data, while Rcw and Lcw are obtained from [2] Lcw (ω) =

1 |In |2

Rcw (ω) =

1 |In |2

 υ

 υ

μ|H(ω)|2 dυ,

(3)

σ |E(ω)|2 dυ

(4)

where υ is a volume containing the coil and the workpiece and In is the current injected into the system through the input terminals. We can imagine the coil and the workpiece inside the volume υ with only its input terminals protruding and consider the set as a generic, two-terminal, linear, passive electromagnetic system operating at low frequencies. The current I(t) in the RLC circuit formed by the capacitor bank and the system coil-workpiece is given by the expression V (5) exp (−γ t) sin (ω0 t) I(t) = ω0 L being

 ω0 =

 2 1 R − LC 2L

and

(6)

R . (7) 2L The inductance L and the resistance R are evaluated at the frequency ω0 . The parameter V in (5) is a given data that represents the value of the initial voltage at the terminals the capacitor bank. V is related with the energy of the discharge U by 1 U = CV 2 . (8) 2 Equation (5) shows that the intensity I(t) is determined by the values of ω0 , Ccb , V , L(ω0 ) and R(ω0 ). As mentioned above, the capacitance Ccb and the voltage V are given data, the inductance L(ω0 ) and the resistance R(ω0 ) are obtained γ=

936

with the help of (3) and (4), but, on the other hand, the value of ω0 is unknown. This frequency is univocally related with the capacitance Ccb for a given set of coil and workpiece. The frequency ω0 is the solution of the equation 4L(ω) −Ccb = C(ω) −Ccb = 0, 4ω 2 L(ω)2 + R(ω)2

(9)

where the relation of the left hand side comes from reordering expression (6). Once ω0 is known, we can evaluate L(ω0 ) and R(ω0 ) and calculate I(t) with (5). In practice, we obtain the capacitance C(ω) for every frequency after computing L(ω) and R(ω) with (3) and (4). Then, when Ccb is fixed, we look for the capacitance C which accomplish C(ω) = Ccb to obtain ω0 .

Lorentz force on the workpiece If we assume that the dimensions of the system coil-workpiece are small compared with the wavelength of the prescribed fields (quasi-static regime), the workpiece is linear, isotropic, homogeneous and non-magnetic and the modulus of the velocity at any point in the workpiece is always much less than 107 m/s then, the force per unit volume f(r,t) acting on the workpiece can be expressed as [3, 4, 5] f(r,t) = J(r,t) × B(r,t) = σ μ ( E(r,t) × H(r,t) ) ,

(10)

where r is a point inside the workpiece, J = σ E is the induced current density and B = μH is the magnetic flux density. Under the above assumptions, the force acting on the workpiece can also be expressed as a magnetic pressure applied on its surface  1  (11) P(r0 ,t) = μ |H(r0 ,t)|2 − |H(rτ ,t)|2 , 2 where r0 is a point placed on the workpiece surface nearest to the coil and rτ is a point placed on the opposite side. Assuming quasi-static regime and linear materials, the field H(r,t) of (10) and (11) is related with the field H(r, ω) calculated with the time-harmonic Maxwell’s equations by means of the inverse Fourier transform H(r,t) =

1 2π

 ∞ −∞

Hn (r, ω) I(ω) eiωt dω,

(12)

√ where i = −1 is the imaginary unit, Hn (r, ω) is the magnetic field per unit intensity at the frequency ω and I(ω) is the Fourier transform of the intensity I(t) flowing through the coil (equation (5))   1 1 1 , (13) I(ω) = − 2 ω + ω0 − iγ ω − ω0 − iγ being ω0 the frequency given in equation (6) and γ the constant defined in (7). The magnetic field per unit intensity Hn (r, ω) is defined by H(r, ω) , (14) Hn (r, ω) = In being H(r, ω) and In the magnetic field and intensity appearing in (3). To relate E(r,t) with E(r, ω) we make an equivalent development to the one shown above, being E(r,t) =

1 2π

 ∞ −∞

En (r, ω) I(ω) eiωt dω.

(15)

Optimum frequency and capacitance The frequency at which the discharge current oscillates is a key parameter in the design of an electromagnetic forming system. In [6, 7, 8] it is shown that, for a fixed energy U and a given set of capacitor bank, coil and workpiece, there exist a frequency at which the maximum deformation of the workpiece is achieved. The use of this optimum frequency saves energy and prevents the premature wearing of the coil.

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To modify the frequency of the discharge in an EMF system we change the capacitance Ccb of the capacitor bank. The relationship between Ccb and ω is described in equation (9). This relation is biunivocal and, therefore, the search for the optimum frequency ωop is equivalent to the search for the optimum capacitance Cop . To obtain Cop we can compute the deformation of the workpiece for all the available values of Ccb . Thereafter, we select the Ccb which produces the maximum deformation. This is the approach followed in [6, 7, 8]. To reduce the number of simulations we can make an initial estimation Cie and calculate the deformation produced by the capacitances Ccb with a value around Cie . In fact, if Cie is close enough to Cop , we can find Cop with only three coupled electro-mechanical simulations. To obtain Cie , or equivalently ωie , we look for the frequency which produces the maximum momentum P in the first n semi-periods, where a semi-period is half a period T /2 = π/ω. That is, we look for the frequency which makes maximum the quantity ΔPn =



nπ ω

0

Ftot · dt,

(16)

where Ftot is the total magnetic force acting on the workpiece and ΔPn is the momentum produced by this force in the first n semi-periods. The number n must be selected with care, if we use a high n then the frequency ωie will be too high and if n is too low then ωie will be too low. The number of semi-periods we select is the minimum natural number n which accomplish 2Lω∞ , (17) n≥ πR where ω∞ is the frequency which makes maximum the quantity ΔP∞ and L and R are the inductance and the resistance evaluated at ω∞ . The quantity ΔP∞ is obtained when n → ∞ in (16). The expression (17) comes from reordering (nπ/ω) ≥ (1/γ), where γ is defined in (7). The minimum natural number which satisfies the inequality (17) represents the minimum number of semi-periods required to release more than 80% of the total momentum ΔP∞ . In summary, we first locate the frequency ω∞ which makes maximum the quantity ΔP∞ . Second, we compute n with (17). Finally, ωie is the frequency which makes maximum ΔPn , being n the natural number calculated in the second step. All this process is performed neglecting the workpiece deformation. We do not require any additional simulation. We are using the data obtained in the initial frequency sweep of our electromagnetic model. It takes only a few seconds to compute all the integrals and obtain ωie .

APPLICATION EXAMPLE In this section we apply the electromagnetic model explained above to the EMF process presented in [7]. This process consists in the expansion of a cylindrical tube by a solenoidal coil. In [7] is analyzed the tube bulging process under different working conditions, here, we will find the magnetic pressure acting on the workpiece when the capacitance is Ccb = 160 μF and Ccb = 800 μF, the coil length is  = 200 mm and the initial charging energy is U = 2 kJ. Also, we will compute Cie for all the cases analyzed in [7]. The coil is made of d = 2.0 mm diameter copper wire. The outer diameter of the coil is Dc = 37.0 mm. The separation between each loop is p = 3.0 mm. To compute the magnetic pressure we used a coil of length  = 200 mm (N = 68 turns). We assume an electrical conductivity for copper of σ = 58e6 S/m. The coil is approximated by coaxial loop currents, concentric with the workpiece and placed inside it. Thus, the problem is considered axis symmetric. The workpiece is a cylindrical tube made of annealed aluminum A1050TD with an outer diameter Dwp = 40.0 mm and a thickness of τ = 1.0 mm. We assume an electrical conductivity for the workpiece of σ = 36e6 S/m. We also assume that ε = ε0 and μ = μ0 for workpiece and coil. In [7] is said that Lcb and Rcb are less than 1.0 μH and 2.0 mΩ respectively. But, in [5], where it is used an EMF set-up similar to that used in [7], it is reached to the conclusion that these quantities do not take into account the inductance Lcon and the resistance Rcon of the wires connecting the capacitor bank with the coil. In [5] is found that Lcb + Lcon = 2.5 μH and Rcb + Rcon = 15.0 mΩ are more realistic values for the electrical parameters external to the system coil-workpiece. Then, in this work, we consider R(ω) = Rcw (ω) + 15.0 mΩ and L(ω) = Lcw (ω) + 2.5 μH as the total resistance and inductance of the RLC circuit formed by the capacitor bank and the system coil-workpiece. We computed the fields E(r, ω) and H(r, ω) with the finite element tool ERMES. The problem was driven by a current density uniformly distributed in the volume of the coil wire. Then, inside the volume of the coil, equation (4) was replaced by  |J|2 1 dυ (18) Rcw (ω) = 2 |In | υ σ

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where J is the imposed current density and σ is the conductivity of the coil. The values of Hn , L, R and C obtained with ERMES as a function of the frequency f = ω/2π are shown in figures 1 and 2.

Figure 1. Left: Modulus of the normalized magnetic field on the workpiece surface nearest to the coil |Hn (r0 , ω)| and on the opposite side |Hn (rτ , ω)| as a function of the frequency f = ω/2π. Right: Total inductance L of the RLC circuit formed by the capacitor bank and the system coil-workpiece as a function of the frequency f .

Figure 2. Left: Total resistance R of the RLC circuit formed by the capacitor bank and the system coil-workpiece as a function of the frequency f . Right: Capacitance C of the RLC circuit as a function of the frequency f .

We computed the intensity flowing trough the coil of length  = 200 mm (N = 68 turns) for the capacitances Ccb = 160 μF and Ccb = 800 μF, being the initial charging energy of U = 2 kJ (see right graph of Fig. 3). For the case of Ccb = 160 μF we found that f0 = ω0 /2π = 4.6 kHz, L(ω0 ) = 6.9 μH and R(ω0 ) = 110.1 mΩ. For the case of Ccb = 800 μF we found that f0 = 1.7 kHz, L(ω0 ) = 8.3 μH and R(ω0 ) = 102.8 mΩ. To obtain these values we only need to find f0 in the right graph of Fig. 2 and use this frequency in L(ω) and R(ω). In Fig. 3 is also shown the maximum current intensities measured in [7] and the maximum current intensities computed in this work for several capacitances Ccb and initial charging energy U = 1 kJ. Once the intensity was known we used equations (13), (12) and (11) to compute the magnetic pressure acting on the workpiece. In Fig. 4 we show the magnetic pressure calculated in this work compared with the magnetic pressure calculated in [7]. We had to average the magnetic pressure over the surfaces of the workpiece to compare our results with those provided by [7]. The differences shown in Fig. 4 can be attributed to the features of each electromagnetic model. In [7] is considered the movement of the workpiece, assumed a uniform current density along all the the length of the coil and an exponential decay of the magnetic field from the inner surface to the outer surface of the metallic

939

Figure 3. Left: Maximum current intensities measured in [7] compared with the maximum current intensities computed in this work for several capacitances Ccb . The initial charging energy is U = 1 kJ. Right: Intensity flowing trough the coil of length  = 200 mm for the capacitances Ccb = 160 μF and Ccb = 800 μF, being the initial charging energy U = 2 kJ.

tube. On the other hand, we neglected the workpiece deformation, considered a more realistic coil geometry and calculated numerically the magnetic field on the surfaces of the metallic tube.

Figure 4. Magnetic pressure on the workpiece for the  = 200 mm coil and initial charging energy U = 2 kJ. The magnetic pressure calculated in [7] is compared with the magnetic pressure calculated in this work. Left: Ccb = 160 μF. Right: Ccb = 800 μF.

In [7] is analyzed the expansion of a tube by a solenoidal coil under different working conditions. They used several coils with lengths  = {100, 200, 300, 400, 500} mm. They computed for each coil the bulge height with the capacitance varying from Ccb = 20 μF to Ccb = 1600 μF. They obtained the optimum capacitance Cop for each coil length and concluded that the optimum frequency satisfy δ = 0.9 τ in all the cases. We analyzed the same problem with the method expalined in the previous section. The number of semi-periods (17) was n = 2 in all the cases. Then, the initial estimation Cie is the capacitance which makes maximum the quantity ΔP2 defined in (16). In Fig. 5 we show the momentum ΔP2 as a function of the capacitance Ccb for the coil lengths  = 200 mm and  = 500 mm. The initial charging energy in both cases is U = 2 kJ. In Table 1 we show the values of Cie obtained in this work compared with the optimum capacitances Cop calculated in [7]. In [7] is also analyzed a coil with  = 100 mm and U = 1 kJ. They measure the bulge height for the capacitances Ccb = {24, 50, 100, 200, 400, 800, 1600} μF and they found that the maximum height was in Ccb = 200 μF. They also calculated numerically the optimum capacitance and they obtained a value of Cop = 310 μF. We calculated the initial estimation and we found that Cie = 296 μF.

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Figure 5. Momentum ΔP2 as a function of the capacitance Ccb for the coil lengths  = 200 mm and  = 500 mm. The initial charging energy is U = 2 kJ. The initial estimation Cie is the capacitance which makes maximum the quantity ΔP2 . The values of Cie are shown in Table 1. Table 1. Optimum capacitance (Cop ) calculated in [7] and initial estimation (Cie ) calculated in this work for different coil lengths ()  (mm)

100

200

300

400

500

Cop (μF) Cie (μF)

310 296

160 161

100 108

70 83

40 67

CONCLUSION In this paper we have presented a numerical model for the simulation of electromagnetic forming processes. This method is computationally efficient because it only requires to solve the time-harmonic Maxwell equations for a few frequencies to have completely characterized an EMF system. The approach can be very useful for estimating the order of magnitude of some parameters, for experimentation on modeling conditions, for coil design or for modeling complex geometries. Moreover, it can be easily included in a sequential coupling strategy without the worry of numerical instabilities. The method also allows us to estimate the optimum frequency and capacitance at which it is attained the maximum workpiece deformation and it offers a new insight into the physics of EMF.

ACKNOWLEDGMENTS This work is part of the project SICEM (SImulación multifísica para el diseño de Conformado ElectroMagnético) founded by the Spanish Ministry of Education and Science through the National R+D Plan 2004-2007, reference DPI2006-15677-C02-01.

REFERENCES 1. 2. 3. 4.

R. Otin, Electromagnetics 30, 190–204 (2010). J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, Inc., 3rd edition, 1999. J. A. Stratton, Electromagnetic Theory, McGraw-Hill, 1941. T. E. Manea, M. D. Verweij, and H. Blok, The 27th General Assembly of the International Union of Radio Science (URSI 2002), Maastricht, The Netherlands, 17-24 August (2002).

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5. T. E. Motoasca, Electrodynamics in Deformable Solids for Electromagnetic Forming, Ph.D. thesis, Delft University of Technology (2003). 6. Y. U. Haiping, and L. I. Chunfeng, Journal of Materials Processing Technology 209, 1053–1059 (2009). 7. H. Zhang, M. Murata, and H. Suzuki, Journal of Materials Processing Technology 48, 113–121 (1995). 8. J. Jablonski, and R. Winkler, International Journal of Mechanical Sciences 20, 315–325 (1978).

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Prediction Algorithm of Blank Shape for Rubber Forming of Aircraft Sheet Metal Part Zhu Minghuaa,b Xu Ying Qianga Le Yua and Chen Wenliangb a

Shanghai Aircraft Manufacturing CO. Ltd., 3115 Changzhong Road, Shanghai, 200436, China

b

College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, 29 Yudao Street, Nanjing, 210016, China Abstract. The aircraft sheet metal part is the important component of aircraft structures and the rubber forming is one of the main forming method for aircraft sheet metal part, the prediction of initial blank sheet shape is a difficult problem for rubber forming. A fast prediction method for initial blank outline of rubber forming based on one-step inverse forming is proposed. Firstly, the designed sheet metal part is treated as the final configuration. Then the quick reverse simulation is carried out to obtain the initial blank outline by the one-step inverse analysis. Finally, the outline from the above method is applied into real rubber forming manufacture. The experimental results are well demonstrated that this method is an effective way to predict the initial blank shape for rubber forming process. Keywords: Rubber forming; One-step inverse analysis; Sheet metal part; Blank sheet PACS: 81.20.Hy

INTRODUCTION Rubber forming process is widely used in aerospace components manufacturing. Aircraft sheet metal parts as an important part of the aircraft structure, influencing the accuracy of aircraft shape and structural bearing capacity directly, its manufacturing process has become an important part of the overall performance of the aircraft. Rubber forming is one of the main methods of aircraft sheet metal forming. Compared with the conventional stamping production, rubber forming method can be used to create more complex shape parts, to improve the accuracy of mould sticking and improve part surface quality, to better solve the conflicts among parts strength, stiffness, size and precision. The parts using high pressure rubber forming can be applied directly to assembly without nap or a few nap, it is suitable for medium and small batch production and trial productionૡ As the same as other sheet metal forming, the initial blank design of rubber forming is a very important issue. Tests show that reasonable initial blank shape can not only reduce the workload on trimming after forming, but also to improve the strain distribution of materials in forming process, improve the quality of forming parts. The development of sheet metal parts is a process of developing the complex 3D geometric shape into flat blank. Currently used in practice or seen reports of the sheet metal develop methods are mainly empirical formula method, slip line method, potential field simulation, the geometric mapping method, analytical method and one-step inverse analysis. One-step inverse analysis is a kind of numerical simulation method to solving initial blank shape. Sheet metal forming has nonlinear problems with geometry, materials and boundary conditions at the same time, it generally need numerical simulation with increment theory. But in the aircraft sheet metal parts manufacturing process, the mould and initial blank are generally generated parallel to the design of parts design, then we find the initial blank without mold information provided, and one-step inverse analysis can solve this problem well. This paper discussed the basic principle of rubber forming and the classification of rubber forming parts firstly, and then deduced the basic theory and formula of one-step inverse analysis, At last, take a typical rubber forming sheet metal part – the rib of flat empennage on large aircraft as an example, predicting the initial blank sheet shape by one-step inverse analysis based on simulated calculation, then compared the outline of initial blank with the

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 943-947 (2011); doi: 10.1063/1.3623706 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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experimental value, so we can point that the prediction of rubber forming initial blank by one-step inverse analysis is effective enough.

Finite element formulation of one-step inverse analysis with bending effects C

n, z q

h

p

xq uq

up

n,z 0

q

Z

0

h0

0

p0

O

x

C0

0 q

Y

X

FIGURE 1. Shell geometry and kinematics

In the one-step inverse analysis, only two configurations are considered: the initial flat blank C0 and the final 3D workpiece C. Using the Kirchhoff assumption, the initial and final position vectors of a material point can be expressed on C, as shown in Fig.1: 0 0 0   xq  x p  u p  z n    xq  x p  zn

(1)

where up is the displacement of the point p on the mid-surface of the sheet, n0 and n are the normals of the midsurface at p0 and p. The thickness stretch is supposed to be constant through the thickness. So, the inverse Cauchy-Green left tensor [B]-1 takes the following form:

B

1

 F

T

F

1

(2)

The eigenvalue calculation of [B]-1 gives two principal plane stretches λ1 , λ2 and their direction transformation matrix [M]. Then the thickness stretch λ3 is calculated by the incompressibility assumption. Neglecting the transverse shear effects of the thin sheet, the element internal force vector in the global coordinates is obtained:

F  [T ]  [B ] e int

T

T

Ve

m

 z[ Bb ]T  dzdA

(3)

where [Bm] is membrane matrix and [Bb] is bending matrix. These resultant forces can be obtained by numerical integration through the thickness at the element central using five or seven Gauss integration points. In the one-step inverse analysis, only the initial C0 and the final configuration C are compared. Therefore, the elastic-plastic deformation is assumed to be independent on the loading path and a total constitutive law is obtained,

 xx    2 (2  r )  yy   3 (1  2r )  xy   

r 1  r  r  1 r   0 0

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0 0 1 

 xx     yy   xy   

(4)

where  and  are the total equivalent strain and equivalent stress. The mean planar isotropic coefficient r is obtained from the three anisotropic coefficients with the relation, r = (r0 + 2r45 + r90)/4. This operation is performed for each numerical integration point through the thickness. Since the one-step inverse analysis cannot deal with contact problems depending on the loading history, the tool actions (punch, die and draw-bead) are simply represented by some external forces at the final configuration. The consideration of the tool actions will give the external force vector, leading to the following non-linear equilibrium system.

{R(U ,V )}   ({Fexte }  {Finte })  0

(5)

e

NUMERICAL RESULTS AND DISCUSSION In this chapter, a rubber forming part-the rib of flat empennage is presented. It is a typical rubber forming part of the large aircraft structures. Its geometrical model shown in figure 2.

Figure 2. Geometrical model of the flat empennage rib

The material of the flat empennage rib is aluminum sheet 2024-0. The mechanical properties of this material and the process variables for simulation are as follows: initial sheet thickness, t = 1.016mm; Young’s modulus, E = 7.24e4 MPa; Poisson’s ratio, = 0.33; Average Lankford value, r = 0.653; Stress-strain curve, σ = 548.9(0.001+εp)0.23 MPa; Yield stress, σs = 131.0 MPa. Computer conditions used in simulations: CPU is Intel Pentium 2.4GHz and memory is 2.0 Gb. ‫ٻ‬Firstly, take the rib of flat empennage as the final shape in the procedure of one-step inverse analysis, and then discrete it into finite element mesh. At last the respective total number of the triangle and the quadrilateral elements is 4 707 and the number of nodes is 4 799, just as shown in figure 3.

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Figure 3. Final geometry meshes

Due to the geometric model of the rib of flat empennage was established in the fuselage coordinates, it needs to set the direction of rubber forming automatically before the simulated computation by one-step inverse analysis. After calculating all the elements’ average normal, making the component rotate 125.523 degrees and -88.255 degrees along the Z axis and Y axis respectively in global coordinate, the position of the component will be rotated as shown in figure 3. After this operation, the rib’s finite element model has arrived at a proper place and satisfies the requirements of one - step inverse analysis. In addition, in order to prevent translation of the rigid body in the process of calculating and ensuring the accurate initial heralded position of the sheet metal after calculating, it must apply the X displacement constraint at the nodes which is rowed at the bottom of the rib finite element model. After a few seconds of the quickly computation, it gets the finite element mesh as shown in figure 4. According to the topology relation of the finite element mesh, we can get the outline of initial blank quickly.

Figure 4. The initial blank

Finally , we can compare the outline of initial blank which obtained by one-step inverse analysis to the outline which mold designers adopts by repeatedly adjusting and trimming in process of practical production. The comparison results as shown in figure 5 below.

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‫ٻ‬

‫ٻ‬ Figure 5. The comparison of initial blank’s outline

By the comparison of figure 5, we can see the contour line shape of the initial blank which we get by the onestep inverse analysis is close to‫ٻ‬the actual initial blank. Besides that the error is bigger at the right head positions of the figure 5, the rest space is consistent with the experimental value. Basing on this result, we can see that the prediction method for initial blank sheet of rubber forming based on one-step inverse analysis can indicate the outline of initial blank accurately and quickly, this outline is a must for rubber forming, in other way this method can greatly reduce the workload when mould designers adjust and trim the mold in process of practical production. What must be emphasized is that the above simulated process of initial blank predicted only need 5 seconds, if using dynamic explicit incremental FEM to make the same simulation the time will be extended hundreds of times.

CONCLUSIONS A fast prediction method for initial blank shape of aircraft sheet metal rubber forming based on one step inverse analysis method was established in this paper, take the sheet metal parts as final configuration, one step inverse analysis was adopted to predicting the initial blank. This method was applied to the actual rubber forming successfully, accurately predicted the initial blank shape of a typical aircraft sheet metal part which was forming by rubber forming; experimental comparison results show that the prediction method in this paper is effectiveness. This method also has guiding significance for actual rubber forming production practice. It can not only reduce the workload after forming and improve the forming parts’ quality, but also save large amount of materials and artificial waste and reduce the cost of mould making at last.

ACKNOWLEDGMENTS The Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars of State Education Ministry and the National Nature Foundation (project number: 50805075) and these supports are gratefully acknowledged.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

Z. R. Han, N. Wu and Q. X. Zhang, Journal of Plasticity Engineering 17, 98-102 (2010). L. Chen, Y. B. Zhang and S. L. Li, Machine Tool & Hydraulics 38, 82-84,63 (2010). Z. R. Han and Y. Z. Liu, Journal of Plasticity Engineering 16, 6-11 (2009). Z. R. Han, H. Tao and L. Zhu, Acta Aeronautica et Astronautica Sinica 26, 125-128 (2005). J. L. Batoz, Y. Q. Guo and F. Mercier, Engineering Computations 15, 864-892 (1998). C. H. Lee and H. Huh, Journal of materials processing technology 82, 145-155 (1998). J. Lan, X. H. Dong and Z. G. Li, Journal of materials processing technology 170, 624-631 (2005). Y. D. Bao and W. L. Chen, International Journal of Modern Physics B 22, 6057-6063 (2008). Y. D. Bao and H. Huh, Journal of materials processing technology 187-188, 108-112 (2007).

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Systematic Process Improvement of Sheet Metal Forming Processes Bart Carleer and Michael Stippak AutoForm Engineering Germany, Emil-Figge-Str. 76-80, D-44227 Dortmund, Germany Abstract. The design of a forming process of sheet metal forming parts is a complex issue. Many boundary conditions must be fulfilled and many considerations must be made to come to a successful solution. Elimination wrinkles and splits very often need contrary measures. Many times the approach to come to a successful solution is an iterative process which is also dependent on the person who deals with the job. Generally this job has been solved with help of simulation software. AutoForm developed a methodology, systematic process improvement, to systematically approach this job. The systematic process improvement is a standardized way to effectively design forming processes. This systematical approach reduces the number of loops, gives transparency of the different solution statements and makes it easier to pass the work to a colleague. As a result the development of a forming process can be done faster, more reliable and less dependent on individuals. The systematic process improvement will be illustrated on the design of forming process of an automotive part. Keywords: stochastic analysis, process optimization

INTRODUCTION The setup of a sheet metal forming process becomes more and more complex. This increasing complexity has been caused by the continuously increasing geometrical complexity as well as the increasing application of high strength steels. Furthermore, the highly competitive market situation forces one to deliver high quality output under the consideration of an always decreasing lead time. As a result less time is available to develop a technically mature process plan. On the other hand because of the new materials only limited experience exists of processing these materials. One of the consequences we are facing today is that despite of powerful software tools to support the process definition many problems need to be solved during tool tryout. Many of those problems originate from a not optimal elaborated forming process. In order to overcome the problems in tryout and to increase the competitiveness one must improve the process maturity. Sheet metal forming simulation software has widely been used to improve the forming process. Generally two different methods have been applied, manual optimization and automatic optimization [1]. Manual Optimization During manual optimization the user iteratively improves the process. Subsequent simulations are started and evaluated in order to come to a satisfying final result. The effectiveness of manual optimization strongly depends on the experience and skills of the engineer. So, large differences can be distinguished between the beginner and the expert. Both engineers apply the identical iterative approach: the trial-and-error approach. With this approach decisions have been take based on experience and gut feeling. These decisions have been validated in the simulation software and one stops until a satisfactory result has been obtained. This satisfactory result is a result that fulfills the minimal requirements; no more effort has been made to look for an even better result or identify a necessary process window [2]. Automatic Optimization The temptation for the automatic optimization is very large. One defines in advance the desired result by means of a suitable object function. Then the software automatically looks for all kind of parameter variations in order to approximate the desired result as good as possible. The evaluation skill of the engineer are hardly used, the The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 954-961 (2011); doi: 10.1063/1.3623708 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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optimized setup has been taken for granted. In case this optimized setup not exactly matches the expectations the user hasn’t the understanding why the desired result hasn’t been achieved neither he understands what needs to be done more to further improve. It is a kind of black-box approach [3]. As shortly described above both optimization methods have their pros and cons. In this paper we would like to introduce a third approach making use of the advantages of both methods and avoiding the disadvantages as much as possible. This new approach to improve the forming process and increase the process maturity has been called systematic process improvement. The starting point for the systematic process improvement is the following requirements: x The method must be user-level independent, i.e. the method can be applied by both beginner as well as expert. x The method must be reproducible, i.e. the trial-and-error component must be eliminated as much as possible. In case some best practice or engineering guessing has been applied it must at least be retraceable. x Manual subsequent execution of simulation must be avoided since this is very time consuming and generally not very effective; certain automation must be ensured. x The method must not only give a result that fulfills the desired results but must indicate the best possible result given the defined boundary conditions. x The evaluation skill of the engineer must be used. The human being is very well able to interpret various different results and weight their effect. Implementing these ‘soft’ skills into an algorithm is very challenging and case sensitive. In the next section the systematic process improvement methodology will be introduced whereas in the subsequent section this method will be illustrated with an example.

SYSTEMATIC PROCESS IMPROVEMENT In the simulation tool the forming process has been defined in a way as will be realized in production. In the early phase of the engineering work the forming process still shows many failures like splits, wrinkles, skid lines etc.. In order to overcome these failures the engineer has to decide which parameters must be modified. In our case the engineer will not manually execute various simulations but he enters the first phase of the systematic process improvement, the preparation phase.

Preparation Phase During the preparation phase the parameters which might solve the forming problems have been identified. In most cases one can select the most promising parameters out of a set of typical parameters. The selected parameters are called design variables and one has to define a range of variation for these variables. Also here it is very well possible to define pre-selected values for every design variable. E.g. the drawbead restraining force varies from 0.15 (weak) to 0.60 (severe). This first selection can very well be standardized within a company making use of the experience of the experts. Typical design variables are: x Geometrical part features e.g. radii, addendum geometry x Drawbead restraining force x Blank shape x Blank position Once the set of design variable has been defined the user starts the calculation, which brings us to the next phase.

Calculation Phase The calculation phase is highly automated. Starting the calculation is just pushing one button and the software automatically starts several simulations each with a unique set of parameter settings. The number of simulations depends on the number of design variables defined. After all simulations have been finished the calculation phase has ended and one goes to the next phase the evaluation phase.

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Evaluation Phase In the evaluation phase, the critical area and the area of risk have been identified based on the result on the basic simulation. Several evaluation zones have been defined depending on various result variables and their corresponding limit. Those areas in the part exceeding this limit are called critical and are allocate to a zone. Typical results to be evaluated are splits and wrinkles but also springback. So, at the end of the evaluation phase several evaluation zones depending on various result variables are defined. In the next phase, the identification phase, the most effective design variables to overcome each problem will be identified.

Identification Phase The identification phase is the heart of the systematic process improvement methodology; this phase makes full use of the whole set of simulation [4]. In order to evaluate the set of simulation some clever evaluation methodology is required, no one will evaluate many simulations individually. For every zone the most effective design variable has been identified. Figure 1(a) shows the Pareto chart that has been used to identify this most effective design variable. The bar height indicates the amount of influence the corresponding design variable has on the result in the selected zone, the higher the bar the more influence. In order to identify the optimized value of a design variable the scatter plot has been used (Fig. 1(b)). The scatter plot represents the correlation between a result variable and a design variable value. The red circles represent the various simulations. The formation of the circles indicates how and how strong the result variable and design variable are correlated. The black horizontal line indicates the limiting value for the particular result value. Three different categories for the design variable can be distinguished; these categories are defined by the defined limit. x Unconditionally fulfilled, all design variable values in this category produce result that fulfill the defined limit x Conditionally fulfilled, design variable values in the category produce results that both fulfill and violate the defined limit. So, fulfilling the desired criterion also depends on other design variables x Unconditionally violated, all design variable values in this category produce result that violated the defined limit

FIGURE 1. (a) Pareto chart showing the influence of every design variable on the result, (b) scatter plot showing the correlation between design variable and result.

This design variable categorization gives an indication of the process window each design variable has in order to achieve good results. Now for every zone, for all design variables with significant influence, the design variable can be categorized. All these values are put in a so called identification matrix representing the design variable zone combination. After everything has been consolidated the next phase can be entered, the implication phase.

Implication Phase In the implication phase the final conclusion with respect to the best possible design variable values will be drawn. For each design variable has been defined which value gives the desired result. So at the end, as a logical consequence the best possible design variable values come out of the implication phase. We shortly described the five phases of the systematic process improvement methodology. When following this methodology our requirements for the optimization procedure have been fulfilled. The methodology will be illustrated on a concrete example in the next section. The example is the stamping of a b-pillar upper cover.

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EXAMPLE OF SYSTEMATIC PROCESS IMPROVEMENT The five phases will be illustrated on the forming of the b-pillar upper cover, usage with courtesy of the NGVconsortium. Figure 2(a) shows the geometry of the b-pillar upper cover. The process sequence showing the subsequent operations in order to produce the part can be seen in Fig. 2(b), drawing – trimming – cam-trimming forming.

FIGURE 2. (a) part geometry (b) process plan, process sequence to produce the b-pillar upper cover

The first critical step in the forming operations is the drawing operation. In this stage of the process development we will focus on the drawing operation. At a first result evaluation both splits and wrinkles occur. The process needs to be optimized in order to overcome these failures. So, we will enter the first phase of systematic process improvement, the preparation phase.

Preparation Phase B-Pillar Upper Cover The simulation just analyzed we call the basic simulation is our basis for the preparation phase. The engineer has to define which parameters can be used to overcome the identified problems like the splits and wrinkles. The selection of parameters can be based on own experience but that will make the procedure very user dependent. Alternatively we recommend selecting the parameters from a (company) standard set of parameters or alternatively shortly consult an experienced colleague. Figure 3 shows the parameters that have been selected to optimize the forming process. The design variables are the three drawbead restraining forces and the blank geometry. The blank geometry has been defined as three independent design variables being the blank size in x-direction both left side and right side of the blank as well as the blank size y-direction. Keep in mind that the lower side of the blank is the line of symmetry. Once the design variables have been defined the user can start the calculation.

FIGURE 3. Definition of design variables, the parameters to be varied in order to optimize the process.

Calculation Phase B-Pillar Upper Cover The calculation phase is highly automated; the software automatically starts several simulations each with a unique set of parameter settings. In this case with six independent design variables, 140 simulations have been executed.

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Evaluation Phase B-Pillar Upper Cover In the evaluation phase, the critical areas have been identified based on the result of the basic simulation. Several evaluation zones have been defined depending on various result variables and their corresponding limit. The first result we will look at is the maximum failure, a maximum failure value of 1 means the point is at the FLC, a maximum failure value of 0,7 means the point is still 30% below the FLC. Generally one works with a 20% safety margin so the limit for maximum failure is set on 0.80. This evaluation gives four critical zones (Fig. 4(a)). The next result to be evaluated is the wrinkling criterion which is based on the ratio of minor and major strain. In this investigation we use a limit of 0.04. This evaluation results in one other critical zone (Fig. 4(b)). These two result variables have also already been evaluated during the initial evaluation which was the initiation of the systematic process improvement. In case more critical issues must be analyzed these can simply be introduced in the evaluation phases. Like in the case of the b-pillar upper cover, it can be seen that the draw-in can become critical, the draw-in exceeds the developed flange (Fig. 4(c)). So, we defined an additional evaluation issue, draw-in to our analysis. Adding critical issues later is a major advantage compared to the automatic optimization. At the automatic optimization one has to define an objective function in advance where after the optimization algorithm looks for the best setting based on the objective function. In case other issues pop-up, the objective function needs to be redefined and the optimization run needs to be repeated. In case of systematic process improvement one can just add an evaluation zone after having executed all simulation without re-running the analysis.

FIGURE 4. (a) Result evaluation splits (b) Result evaluation wrinkling tendency (c) Draw-in compared with flange development

So, six critical issues have been found for our part. In the next phase we will identify which design variables should be used, and what value they should have in order to solve these critical issues.

Identification Phase B-Pillar Upper Cover For every critical issue we need to identify which design variables have significant influence followed by the categorization of the design variable values. The result of this identification and categorization will be added in the matrix representing the parameter zone combination. In the remainder of this section for ever type of issue, max failure, wrinkling and draw-in, one example will be given how to identify the design variable and categorize the design variable values. Identification Phase B-Pillar Upper Cover - Max Failure Zone 1 Figure 5(a) shows the Pareto chart for zone 1. As can be seen two design variables have significant influence on the failure behavior, the blank size in x-direction right side and drawbead 2. For those two design variable the scatter plot will be analyzed and the design variable values will be categorized. It can already be concluded that the other design variables don’t influence the failure result in zone 1. Figure 5(b) shows the scatter plot for the blank size in x-direction right side which is the variable having the most influence on failure behavior in zone 1. The black horizontal line indicates the critical limit for max failure of 0.8. It can be seen that for values less than 0.32 all results unconditionally fulfill the defined limit, all other values only conditionally fulfill the defined limit.

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Figure 5(c) shows the scatter plot for drawbead 2. Again the critical limit has been indicated with the horizontal black line. In case the drawbead restraining is less than 0.26 the process unconditionally fulfills the failure limit. In case of values greater than 0.26 the process only conditionally fulfills the limit.

FIGURE 5. (a) Pareto chart for zone 1 identifying the most effective design variables (b) Scatter plot for zone 1 showing the correlation between max failure and blank size in x-direction right side (c) Scatter plot for zone 1 showing the correlation between max failure and drawbead 2.

Table 1 shows the values for identification matrix with the consolidated data for zone 1. The three values indicate the amount of influence in % and the limits between the categories unconditionally fulfilled – conditionally fulfilled and conditionally fulfilled – unconditionally violated. For easy evaluation the highest influence value per zone has been printed in bold. For the other failure zones the identical procedure will be followed in order to fill the identification matrix. TABLE 1. Identification matrix with categorization of the design variables with respect to zone 1.

Failure Zone 1

Bead 1

Bead 2

Bead 3

Blnk XL

Blnk XR

Blnk Y

-

25%, 0.26, max

-

-

45%, 0.32, max

-

Identification Phase B-Pillar Upper Cover - Wrinkling Zone 5 Figure 6(a) shows the Pareto chart for zone 5, three design variables have significant influence on the wrinkling tendency. These three variables are drawbead 1, drawbead 3 and the blank size in x-direction left side. For two design variables with the greatest influence the scatter plot analysis will be illustrated. Generally all design variables having in sum at least 80% influence will be categorized. It can already be concluded that the design variables blank size in y-direction, blank size x-direction right side and drawbead 2 don’t influence the wrinkling result in zone 5. Figure 6(b) shows the scatter plot for drawbead 1, the variable having the most influence on wrinkling tendency in zone 5. The black horizontal line indicates the critical limit for the wrinkling criterion of 0.04. It can be seen that for values less than 0.10 all results violate the defined limit, for values between 0.10 and 0.42 the limit has been fulfilled conditionally. For values greater than 0.42 the limit has been unconditionally fulfilled. Figure 6(c) shows the scatter plot for drawbead 3. The critical limit has been indicated with the horizontal black line. In this case all drawbead restraining values only conditionally fulfill the limit. The only conclusion that can be drawn is that the higher the value the higher the probability the limit will be respected.

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FIGURE 6. (a) Pareto chart for zone 5 identifying the most effective design variables (b) Scatter plot for zone 5 showing the correlation between wrinkling criterion and drawbead 1 (c) Scatter plot for zone 5 showing the correlation between wrinkling criterion and drawbead 3

Identification Phase B-Pillar Upper Cover – Draw-in Figure 7(a) shows the Pareto chart for the draw-in feasibility. The two design variables the blank size in xdirection right side and the blank size in y-direction have the most influence. The design variables drawbead 3 and drawbead 1 have minor influence whereas drawbead 2 and the blank size in x-direction left side have no significant influence. Figure 7(b) shows the scatter plot for the blank size in x-direction right side. The result variable draw-in feasibility has a value of 1 in case the draw-in criterion has been fulfilled; in case of not fulfilling the draw-in criterion the draw-in feasibility becomes less than 1. It can be seen that for values less than 0.29 all results unconditionally violated the draw-in criterion. For values greater than 0.29 the draw-in criterion has been fulfilled conditionally. Figure 7(c) shows the scatter plot for the blank size in y-direction. It can be seen that for values less than 0.29 all results unconditionally violated the draw-in criterion. For values greater than 0.29 the draw-in criterion has been fulfilled conditionally.

FIGURE 7. (a) Pareto chart for draw-in feasibility identifying the most effective design variables (b) Scatter plot for draw-in showing the correlation between draw-in and blank size in x-direction right side (c) Scatter plot for draw-in showing the correlation between draw-in and blank size in y-direction

For all critical issues the most effective measures have been identified and the design variable values have been categorized. All date have been added to the identification matrix. Now, we can step to the next phase and define the best possible value for every design variable in the implication phase.

Implication Phase B-Pillar Upper Cover In the implication phase the final conclusion with respect to the best possible design variable values will be drawn. Where in the identification phase the matrix has been filled row-wise, in the implication phase the matrix will be evaluated column wise. In this paper we will shortly discuss the determination of the best value of two design variables. Firstly, we will look at the column of drawbead 1 and secondly we will have a look at the column of blank size in x-direction right side.

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Bead1 We see that with respect to wrinkling zone 5 this design variable has the highest influence. Values greater than 0.42 result in conditionally fulfilling the limit, whereas values less than 0.10 result in unconditionally violating the limit. Also for failure zone 2 and 3, drawbead 1 has significant influence. In both cases values less than 0.13 result in unconditionally fulfilling the limit and greater values result in conditionally fulfilling the limit. As so often, failure and wrinkling give contrary recommendation. In this case the issue with the highest influence has been counted for. So, wrinkling is privileged for drawbead 1 resulting in a best value of 0.45 Blanksize in x-direction right size This design variable has in three failure zones the highest influence. For values less than 0.32 this will unconditionally fulfill the limit, for greater values the limit has been fulfilled conditionally. With respect to the draw-in again contra dictionary conclusions must be drawn, values less than 0.29 unconditionally violate the limit. Greater values conditionally fulfill the limit. Including both issues, results in a very small process window between 0.29 and 0.32 for this design variable. In this case we chose the best value of 0.32 for the blank size in x-direction right side. In this way for every design variable the best value can be identified. The last row in table 2 shows the best value of each design variable. It is recommended to execute a confirmation simulation based on the best value found in the identification matrix in order to be sure that nothing has been overlooked. Evaluation the confirmation simulation with the best values from table 2 showed that all critical areas were solved and s successful process has been defined. TABLE 2. Identification matrix with categorization of the design variables.

Failure Zone 1 Zone 2 Zone 3 Zone 4 Wrinkling Zone 5 Draw-in global Best Value

Bead 1

Bead 2

Bead 3

Blnk XL

Blnk XR

Blnk Y

14%, 0.13, max 29%, 0.13, max -

25%, 0.26, max -

-

72%, 0.38, max

45%, 0.32, max 69%, 0.57, max 65%, 0.41, max -

15%, min, max

37%. 0.42, 0.10

-

24%, max, min

13%, max, min

-

-

0.45

0

0.45

0.30

17%, max, 0.29 0.32

14%, max, 0.29 0.32

CONCLUSIONS In this paper we introduced the systematic process improvement as a methodology to optimize sheet metal forming processes. This methodology consists of five well defined phases: preparation, calculation, evaluation, identification and implication. Systematic process improvement can be characterized as highly standardized. It also reduces the trial-and-error approach of the manual optimization as well as that is avoids the black box approach of the automatic optimization. The methodology can be used in a reproducible way by both the expert and the beginner. Based on the prepared identification matrix the best value in order to achieve the desired result can be easily determined.

REFERENCES 1. P Hora, “New approaches for the virtual system planning of complex, narrow tolerated forming systems” in New Developments in Sheet Metal Forming, edited by Klaus Siegert, Frankfurt: Mat info Werkstoff-Informationsgesellschaft, 2004,pp. 153-174. 2. A. Emrich, G. Ruf and M. Liewald, “Simulative validation of robust forming processes” in New Developments in Sheet Metal Forming, edited by Mathias Liewald, Frankfurt: Mat info Werkstoff-Informationsgesellschaft, 2010, pp. 359-380 3. D. Wei, Z. Cui and J. Chen, “Robust optimization of a stamping process using polynomial chaos expansion,” in Numisheet 2008, edited by Pavel Hora, Zurich: Institute of Virtual Manufacturing, 2008, pp. 865-869. 4. B. Carleer, T. Zwickl, A.H. van den Boogoord, E. Atzema, “One day tryout, identifying influence and sensitivity in stamping,” in Numisheet 2008, edited by Pavel Hora, Zurich: Institute of Virtual Manufacturing, 2008, pp. 829-834.

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Accelerating the FE-Simulation of Roll Forming Processes with the Aid of specific Process’s Properties Ahmad Abrassa, Mahmut Özelb, Peter Grocheb a, b

Institute for Production Engineering and Forming Machines, Darmstadt University of Technology, Petersenstrasse 30, 64287 Darmstadt, Germany a [email protected]

Abstract. Roll forming is an effective and economical sheet forming process that is well-established in industry for the manufacturing of large quantities of profile-shaped products. In cold-roll forming, a metal sheet is fed through successive pairs of forming rolls until it is formed into the desired cross-sectional profile. The deformation of the sheet is complex. For this reason, the theoretical analysis is very difficult, especially, if the strain distribution and the occurring forces are to be determined [1]. The design of roll forming processes depends upon a large number of variables, which mainly relies upon experience based knowledge [2]. In order to overcome the challenges and to optimize these processes, FE-simulations are used. The simulation of these processes is time-consuming. The main objective of this work is to accelerate the simulation of roll forming processes by taking advantage of their steady state properties. These properties allow the transformation of points on the sheet metal according to a mathematical function. This transformation function is determined with the help of the finite element method and then the next forming steps are computed, based on the generated function. With the aid of this developed method, the computational time can be reduced effectively. The details of the FE-model and new numerical algorithms will be described. Furthermore, the results of numerical simulations with and without the application of the developed method will be compared regarding computational time and numerical results. Keywords: Cold roll forming, FE-simulation, U-channel PACS: 46.15.-x, 47.11.Fg

INTRODUCTION The design of pass-schedules and roll profiles mainly relies upon experience based knowledge. The crucial first step in this process is to establish the number of passes and a good profile flower which depends on numerous factors (section depth, material thickness). A lot of researches has been attempted to analyze the roll forming process. Nowadays it is very complicated to find the correct process parameter and geometry parameters. In [3], Kiuchi has analyzed the roll forming process by using two shape functions. The deformation is defined by the fact, that the dissipated energy during the forming becomes minimal. In this work, the main drawback is that the influence of rolls geometry and abrupt change of the run cannot be considered. Nefussi und Glormini have presented the coons patch technique for the simulation of roll forming process. They have used kinematic approximation to determinate the optimized form and longitudinal strain of sheet metal during the forming [4]. In their study, rigid plastic material behavior was implemented. In [2], they have integrated elastic plastic material behavior. The development and the analysis of the roll forming simulation, with the help of PAM-Stamp, have been carried out in [5]. The numerical and experimental results were compared. The resulted divergence amounts to about 10%. To optimize the simulation, adaptive mesher was used in the simulation. The authors have pointed out, that the simulation of roll forming is time-consuming. For that reason, the simulations are not very efficient. In [1], the finite strip method has been developed. The finite strip method is a special case of the finite element method and considers the elastic plastic material behavior. In this method, the element is treated as strip. In each strip, a B-spline function is integrated to compute the longitudinal strain. Indeed, because of the oversimplification the results quality changes for the worse in comparison to finite element method. Hong [6] has developed the combined 2D/3D Analyses, which is established in [7]. In his work, the assumption of uniform strain-rate over the cross section was adopted. The numerical results were evaluated using the experiments conducted by Damm [10]. Bui has accomplished many simulations and has compared the numerical results with the

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 962-969 (2011); doi: 10.1063/1.3623709 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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results from experiments [8]. Different parameter are considered (longitudinal strains, roll velocity). Only one FEM Model was considered without any changing in FEM parameter. In [9], Boman make used of the LagrangeEulersche formulation to reduce the computational time for this process. However, within the scope of his work, no comparison with the experiments or with the Lagrange formulation has been performed to validate the results. The researches have exhibited, that the reduction of the computational time of roll forming processes was aim at numerous work using different new methods. In this work, accelerating the simulation of roll forming processes will be achieved by taking advantage of their steady state properties. Using these properties, the deformation of points on the sheet metal is determined with the help of a mathematical function, which will be established by means of the finite element method.

STEADY STATE PROCESSES The steady state property refers to the condition where the state variables of a process at any single point in the system do not change over the time. Mathematically, this statement is expressed as:

 

=0

(1)

Where S represents the state variables, which determine the behavior of material during the process. Here the elastic and plastic strain are considered as state variables, because these variables specify the change of geometry, which can be used with the material law to compute resulting stresses in the elements. The definition (1) states, that the behavior of elements during steady state process could be predicted using numerical function, which is valid only in defined time range.

FIGURE 1. Changes of state variables of three material particles with regard to the position

For example, imagine that three material particles (P 1, P2, P3) move in x-direction with constant velocity v. it is supposed, that the process is steady state process. Here the condition 1 will be satisfied. S t11, St12 and St13 represent the behavior of these particles at time t1. St21, St22, St23 are the behavior of these particles at time t2. As in figure 1, we observe these particles at two times (t1, t2). If one determines a numerical function, which can describe the change of state variables during the process, one will get the solution at any time without any time dissipation. In this case, the solution is defined as following:

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At any time t

  = ( +  )

(2)

Where X is the initial configuration of certain particle in coordinate system. In this example, the change of state variables over the time is set to zero with reference to a fix point in the coordinate system. Consequently, if the above-mentioned condition (1) is fulfilled, the process can be defined as steady state process. The behavior of material particles can be predicted by making use of steady state properties. By roll forming processes, the above-mentioned condition is satisfied. In figure 2, a FE-model for the simulation of a roll forming process is depicted. The model consists of metal sheet and rolls. Through the FE-Simulation, the sheet metal will be deformed into a U-profile. To check the steady state condition, we will study the displacements of three nodes, which are located on the sheet (figure 2). The displacement and the plastic strain of the nodes in y- and z-direction are presented in figure 3. It is obvious, that the gradient of the three curves over the distance lie close together.

FIGURE 2. FE-model of roll forming process

FIGURE 3. The displacement and the plastic strain of three defined nodes on the sheet metal during the simulation

In this case, the strain can be considered as state variable. With the aid of the strain, the stress will be computed. In the next section, the details of new algorithms are described.

NEW EFFICIENT ALGORITHMS This Algorithm consists of two parts. The first part (Contact Algorithm) will be implemented in the FE-Software (Marc Mentat) as subroutine. The next part (Approximation Algorithm) is external software, which computes and changes the numerical results regarding the new efficient algorithms.

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The flow diagram is depicted in figure 5. In the first step, the FE-model is established. Afterwards the FE-simulation will begin and continue, until two conditions are satisfied. The first is that the contact between the sheet metal and roll station occurs. The second condition confirms that the contact forces are computed at the contact surface. If the mentioned conditions are satisfied, the numerical results produced from finite element simulation are enough to compute the new required state variables for the next time steps. Generally, the required results from finite element simulation are the displacements and strains of all nodes at the deformation zone. The principle of this computation is to find certain analytical function, which can describe the behavior of sheet metal at any time within a defined range. It means that the computed analytical function is valid as long as the work piece and rolls do not come in contact one more time. If new contact occurs, the simulation must be started to compute the numerical results in the new deformation zone. As above-mentioned, every row of nodes takes over the same behavior, therefore this behavior can be represented as spline function.

FIGURE 4. Spline function

In all intervals Ii= [x , x  ] , the spline interpolation S(x) is described with following cubic polynomials P (x) (figure. 4):

S(x) = P (x) = a ∗ x  + b ∗ x  + c ∗ x + d

,

x ϵ [x , x  ]

Where (x , y ) is the sampling point to be interpolated through spline S(x) . Additionally the first and second derivative of S(x) must be continuous in (x , y ).

FIGURE 5. The flow diagram of the developed algorithms

For each required variables, a spline function will be generated to compute the necessary variable at the next time steps. Afterward the sheet metal is translated to the new position corresponding to the new computed time steps. After that, the numerical results will be computed using spline function, until the sheet metal comes in contact with the next roll station. In this case, the finite element simulation continues to compute the new state variables required

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at every contacted roll station. One repeats the last steps in the case, that the two above-mentioned conditions are satisfied. In the next section, the numerical result with and without applying the developed algorithms will be introduced.

IMPLEMENTATION AND VALIDATION OF THE DEVELOPED ALGORITHMS For the validation of the developed algorithms, the simulation of an U-profile will be carried out. Figure 6 illustrates the considered final channel section of the roll forming process used in experimental work. The material used in the experimental work is ZStE500. The Poisson’s ratio was specified as 0.3 and the Young’s module 210000 N/mm2. The flow stress-strain curve is obtained from the experiments.

FIGURE 6. Suggested final channel section

With the help of the software RIM, the flower pattern was generated. This Software considered different methods to calculate the required parameters for roll forming process. The suggested profile will be produced in six forming steps (figure. 7). The width of the sheet metal is also computed with the aid of the RIM-Software. For this calculation, the constant radius method is used.

Step 1 2 3 4 5 6

Angle 13 29 47 65 80 90

FIGURE 7. Generated flower pattern and bending angle at each forming step

In order to validate the new algorithm, only three rolling stations are considered. The FE-model of the roll forming process has been build up by means of the Marc Mentat Software. In figure 8, the FE-model of the sheet metal is shown. Due to symmetry properties of the profile, only half of sheet metal is depicted with its boundary conditions. The nodes in symmetry plane are fixed in y-direction. Additionally, the nodes at the bottom of the symmetry plane are fixed in z-direction. In contrast to the experiments, the rolls are moved in the simulation. Only one node must be fixed in x-direction The generated mesh is shown in figure 8. There are two elements in thickness direction. In width direction the region of bending edge is meshed finer than in the other regions. Here the element width is approx. 0.8 mm, while the element width amounts to approx. 1.9mm in other zones. In figure 8-b, the arrangement and geometry of rolls are shown. In this finite element simulation, the rolls are modeled as rigid bodies. The distance between them is 500mm and the friction between the rolls and sheet metal is neglected. The rolls move with constant velocity (v=5m/sec). The simulation is carried out using the implicit solver Marc.

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(a)

(b)

Figure 8. Finite element model a) Boundary condition b) Rolls geometry and process parameter

In figure 9-10, the numerical results of the finite element simulation are depicted. The equivalent stress is shown in figure 9. One can notice, that the distribution of computed stress with and without applying the developed method are reasonably similar. The equivalent stress increases at the contact zones because of contact forces. In order to validate the developed method, the equivalent stress for three points (figure 9) is evaluated. As shown in figure 9, the deviation between both results is small and it amounts to about 5%.

Figure 9. Equivalent stress a) Applying the developed method b) Without applying the developed method

As shown in figure 10, the distributions of the longitudinal strain with and without applying the developed method are the same over the computed steps. The value of longitudinal strain rises from -0.04 to 0.023 at the edge. It is obvious from the diagram in figure 10 in reference to the observed three points , that the deviation of plastic strain between the computed result with and without the developed method is about 5%.

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Figure 10. Strain in the longitudinal direction a) Applying the developed method b) Without applying the developed method

The next diagram proves that the computational time, with the help of the developed method can be efficiently reduced. Without applying the developed method, the conventional simulation requires about 28 h for three roll stations. In contrast, the simulation with the developed method needs about 12 h. Consequently, the computational time will be reduced more than 50% with the aid of the developed method.

Figure 11. Computational time with and without applying the developed method

SUMMARY With the aid of the developed method, the computational time can obviously be reduced for the simulation of roll forming processes. The saved time depends upon the number of contact surfaces. In this paper, an algorithm, which is required to accelerate the simulation of roll forming, is introduced. In the future, the application of this algorithm will be accomplished automatically in order to simulate profiles that are more complex. Thereby, the generated numerical results will be validated according to experiments. Additionally, the relative error, which arises by applying the new algorithm, must be computed.

ACKNOWLEDGMENTS The authors thank the Deutsche Forschungsgemeinschaft (DFG) for funding the project “Entwicklung eines Verfahrens zur beschleunigten Simulation von Walzprofilierprozessen“.

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REFERENCES C. Liu, Y. Zhou and W. Lu, Numerical simulation of roll forming by B-spline finite strip method. J. Mater. Process. Technol. 60, 1996 , pp. 215–218 2. G. Nefussi, L. Proslier and P. Gilormini, A simulation of cold roll forming for elastoplastic materials. Int. J. Mech. Sci. 40 1, 1998, pp. 15–25 3. M. Kiuchi, K. Abe and R. Onodera, Computerized Numerical Simulation of Roll-Forming Process, CIRP Annals Manufacturing Technology, Volume 44, Issue 1, 1995 , Pages 239-242 4. G. Nefussi, P. Gilormini, A simplified method for the simulation of cold roll forming, Int. J. Mech. Sci. 35, 1993, pp. 867–878 5. F. Heislitz, H. Livatyali, M. A. Ahmetoglu, G. L. Kinzel and T. Altan, Simulation of roll forming process with the 3-D FEM code PAM-STAMP, J. Mater. Process. Technol. 59, 1996 ,pp. 59–67 6. S. Hong, S. Lee and N. Kim, A parametric study on forming length in roll forming, J. Mater. Process. Technol. 113, 2001, pp. 774–778 7. M. Brunet, B. Lay and Pol, P., Computer aided design of roll-forming of channel sections, J. Mater. Process. Technol. 60, 1996 , pp. 209–214 8. V. Q. Bui and J. P. Ponthot, Numerical simulation of cold roll-forming processes ,Journal of Materials Processing Technology, Volume 202, Issues 1-3, 20, 2008, pp. 275-282 9. R. Boman, L. Papeleux, Q. V. Bui and J. P. Ponthot, Application of the Arbitrary Lagrangian Eulerian formulation to the numerical simulation of cold roll forming process, Journal of Materials Processing Technology, Volume 177, Issues 1-3, Proceedings of the 11th International Conference on Metal Forming 2006, Pages 621-625, ISSN 0924-0136 10. K. Damm, Determination of longitudinal strains in roll forming of standard section in a multi-stand machine, Institute for Production Engineering and Forming Machines, Germany, 1989 1.

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Optimization of Cooling Simulation in Press Hardening M. Medricky1, G. Klawitter1, R. Struck1, M. Vucetic2, B. -A. Behrens2 1 2

Volkswagen AG, Berliner Ring 2, Wolfsburg 38436, Germany

Institute of Metal Forming and Metal-Forming Machines, Leibniz Universität Hannover, An der Universität 2, Garbsen 30823, Germany

Abstract. This paper compares methods used for integration of cooling system in the press hardening simulation. For this purpose an experimental tool was developed and constructed. This tool makes it possible to compare and validate different simulation methods applicable for cooling systems and resolves the question of their importance. The simplified shape of the tool has been chosen to break down the problem to less numerous parameters limited to thermodynamics and fluid dynamics. Such a simplified problem can better address the parameters like heat transfer coefficient in channels depending on channel geometry and flow speed of cooling water, and can easily relate them to the reality. A temperature distribution on the tool surface for different geometries of channels is presented under chosen flow speeds of cooling medium and heating power of integrated heating elements and compared to the results from both FEM and CFD simulation methods. Resultant heat transfer coefficients for different flow speeds of water and cooling channels geometries will be calculated through the inverse calculation, compared to the analytical calculation. The conclusion should reveal whether it is better, for a given geometry, to use a constant analytically calculated heat transfer coefficient, constant coefficient from the inverse calculation, or variable heat transfer coefficient from CFD simulation. Keywords: Cooling channel, Heat transfer coefficient, Simulation methods, CFD, Hot forming, Press hardening. PACS: 47.27.te

INTRODUCTION Press hardening is a manufacturing method used to produce steel parts of exceptionally high strength. It is achieved through rapid cooling (quenching) of hot formed sheet within a press. The quenching takes place between two forming tools (punch and die) inside of which the cooling channels are designed. Advantages of the parts are today exploited mainly in car body structures by increasing the safety while keeping or even decreasing the weight. Their production is a very complicated process embracing many fields of physics like mechanics, thermomechanics, thermodynamics, and fluid dynamics comprising distinctively high number of parameters which influence the final properties. Due to this complexity, a significant effort is needed to produce a desired part. Developing time and costs reduction is therefore very desirable. One possibility to reduce time and costs is to predict the part properties already before the part is produced. This can be done by means of a simulation tool. The accuracy of the existing simulation tools is limited which is partially due to an exclusion of cooling system. To include a cooling system in the simulation, 3D modeling tools need to be created. The main complication comes first when the boundary conditions are to be applied. This study compares two types of boundary conditions applicable to the walls of cooling channels, CFD computed heat transfer coefficient on the channel walls and constant convective boundary conditions analytically determined from Gnielinski formula. Their advantages and disadvantages will be clarified and a method for finding an optimal convective heat transfer coefficient in order to get reality close results will be presented. The optimized heat transfer coefficient will be proposed for two different flow speeds and two different geometries.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 970-977 (2011); doi: 10.1063/1.3623710 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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COOLING SIMULATION - BOUNDARY CONDITIONS Cooling simulation within press hardening can be approached in several ways. The easiest way to simulate the cooling is through assignment of constant temperature boundary conditions (BCs) so called Dirichlet conditions, the second possibility is heat flux BCs so called Neumann conditions, and the third convection BCs are called Robin conditions, which combine the first two BCs while a temperature must be defined as well as a convective heat transfer coefficient.1 In all these cases, the BCs are applied on the wall segments of cooling channels modeled inside of the 3D model. By Dirichlet BCs and Neumann BCs, no fluid properties and type of flow must be taken into consideration. The Robin convection BC needs to be analytically calculated with use of special formulas for water in turbulent flow. Dittus Boelter formula is usually picked when smooth straight channels are used, thereby, no channel roughness needs to be determined. 2 The channel roughness can be on the other hand considered through the following Gnielinski formula3 where the letter f represents Darcy – Weisbach friction factor. § k · ª ( f / 8)(Re− 1000) Pr º h = ¨ ¸« » 0 .5 2/ 3 © D ¹ ¬1 + 12.7( f / 8) (Pr − 1) ¼

(1)

These formulas consider an ideal case of fluid flowing through a straight cooling channel where no turbulences caused by a curvature of the channels are taken into account. Anyway, convective heat transfer coefficient (HTC) with Robin’s boundary conditions and constant predetermined temperature is very frequently used in simulations optimizing the cooling systems.4 5 A method with use of so called Bulkflow elements can be used as well. This uses 1D calculation of water heating along the cooling channels with constant heat transfer coefficient.6 Another way for the determination of HTC is with use of computational fluid dynamics (CFD). The CFD uses knowledge of fluid dynamics based on Navier Stokes equation for calculation of flow characteristics approximating a real flow of fluid.7 The HTC is in this way calculated with respect to local characteristics like viscosity, speed, and turbulence of the flow, which are in turn based on actual temperature of a medium and geometry of the cooling channels. In following, the focus will be aimed at the reliability of the convective BCs within FEM calculation and CFD calculated convective BCs and question what BC is the best choice for a 3D cooling simulation will be answered.

EXPERIMENTAL TOOL For determination of boundary conditions in press hardening simulation and better understanding of cooling simulation in relation to reality, a simplified press hardening tool was designed and constructed. During designing, a main focus was aimed at simplicity which should reduce a number of possible influencing factors to a minimum. This should in turn make it possible to simulate the real problem without complications and to accomplish the experiments with high precision. The experimental tool shaped as a simple block is composed of four main parts as shown in Fig.1. A ground and insulating plate serve for fixing and thermal isolation of the tool from the ground respectively. Inside of the heating plate, 16 heating elements were inserted to heat the tool. Fourth part designed for cooling the system is a plate which includes milled channels several meters long. The cooling plate is fixed with bolts on the perimeter to the heating plate. Temperature was measured at eight points along the cooling channels from upper, middle, and lower side (green points in Fig.2). Flow speed of water in the channels was measured at the channel entrance. The input and output temperatures of the cooling water were besides for the cooling power calculation used for comparison with the CFD 1

Cheng, A. and D. T. Cheng (2005). “Heritage and early history of the boundary element method, Engineering Analysis with Boundary Elements,” 29, 268–302. 2 Dittus P. W., and Boelter L. M. K., “Heat Transfer in Automobile Radiators of the Tubular Type,” Univ. Calif. Pub. Eng., (2/13): 443-461, 1930; reprinted in Int. Comm. Heat Mass Transfer, (12): 3-22, 1985. 3 Gnielinski V., “New equations for Heat and Mass Transfer in Turbulent Pipe and Channel Flow,” Int. Chem. Eng., (16):359-368, 1976. 4 Steinbeiss H., So H., Michelitsch T., Hoffmann H., “Method for optimizing the cooling design of hot stamping tools,” Production Engineering, Volume 1, Number 2, October 2007, pp. 149-155(7). 5 Saifullah A. B. M., Masood S. H. “Finite Element Thermal Analysis of Conformal Cooling Channels in Injection Moulding,”5th Australasian Congress on Applied Mechanics, ACAM 2007, 10-12 December, 2007. 6 Lorenz D., Haufe A,“Simulation von Warmumformprozessen mit LS-DYNA.“ In: 2. Erlanger Workshop Warmblech – umformung, 2007. 7 Anderson, John D. (1995), “Computational Fluid Dynamics: The Basics With Applications,” Science/Engineering/Math, McGraw-Hill.

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simulation. Additionally, the temperature distribution along the flat surface was measured by a thermocamera. Positions of the temperature sensors and of heating elements are shown in Fig.1. Two different geometries of cooling channels were considered. Both channels have the same length and diameter. They differ in the number of curves only. The channel with geometry 2 has three times more 180° curves as the channel with geometry 1 as shown in Fig.2.

Water flow speed sensor Temperature sensor

Temperature measuring points 3 2

Water outlet

Cooling plate

1 Water inlet

Heating plate Water temperature sensors

Insulating plate Ground plate

Heating elements

FIGURE 1: Experimental tool with description

The same covered area

Geometry 1

Geometry 2

The same diameter and flow speed

The same length

Temperature measuring points

FIGURE 2: Cooling plates geometries

Turbulences caused by the more curved geometry destroy the boundary layer in the channels and thus higher HTC should be observed. Calculation of curved cooling channels was studied by Butuzov et. al.,8 anyway, this theory is applicable on higher curvature radii only.

EVALUATION OF RESULTS First of all the heating of the tool without cooling were simulated and compared to reality in order to determine the correct amount of energy entering the system. Afterwards, the result affecting parameters were detected and optimized through an inverse simulation. With this optimized simulation model, several calculation results were analyzed and compared to experimental results for two different geometries (Fig.2) and two different flow speeds. For each case an optimal convective HTC in the cooling channels was determined through an inverse modeling.

8

Ito H., “Friction Factors for Turbulent Flow in Curved Pipes,” J. Basic Eng., (81): 123-134, 1959.

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Heating experiment – inverse simulation To check whether the heating power was well measured, a heating simulation was compared to the heating in reality. An input energy corresponding to the experiment was set as a constant heat flux on the heating elements for 20 minutes. During this experiment, there was no cooling water flowing through the cooling channels. The temperatures were measured above, next to, and below the channel in total at eight points illustrated in Fig.2. These points were exactly localized in the simulation through a global coordinate system.

FIGURE 3: Comparison of heating experiment and simulation – top left thermocamera picture; top right simulation with LSDyna; bottom left measured temperatures; bottom right simulated temperatures

Figure 3 visually compares a simulation with an experiment of temperature evolution with time during 20 minutes heating and consequently 40 minutes cooling on the air. The scale is in this case not as important as the compared temperature distribution, which is in good accordance with both thermocamera pictures and graphs. All the points distributed along the tool respond accordingly and attain the same final temperature as in reality. To compare each sensor value is in this case not as necessary as the fact whether the curves together follow the same shape. We can say that results are after inverse simulation in very good accordance with reality (also when comparing the individual sensors). The slight discrepancies were caused by an incorrect setup of boundary conditions causing the heat dissipation through the tool walls to the air by convection, radiation, conduction to the ground, and conduction between the tool parts. These parameters have been allocated and corrected through the inverse simulation. These optimized parameters were used in the following heating - cooling simulations.

Cooling experiment – inverse simulation The cooling experiment was accomplished with several geometries of channels and speeds of flowing water. In this paper, as an example, results of the experiment with only two geometries (see Fig.2) and two flow speeds of water are presented. The temperatures at the eight points along cooling channel were recorded in quasi static state. This state was achieved at the time when the measured temperature change did not exceed 0,01 K in 30 minutes but maximal after about 7 hours of contemporaneous cooling and heating due to limited experimental possibilities. Pictures with letter (a) are made with the thermocamera after achieving of steady state. Letter (b) represents the ANSYS simulation results where the HTC in the channels were calculated with use of CFD simulation after setting up the inlet water temperature and flow speed. Figures with letter (c) show the result from the simulation where a Robin’s BCs were used and the convective HTC was determined with use of Gnielinski formula and DarcyWeisbach friction factor. The water temperature was in this case held constant at the real inlet value. Behind letter (d) is the same simulation as in (c) but with averaged water temperature.

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Geometry 1 – 4,1 l/min Visual comparison of the results from the experiment and simulation with CFD and FEM software for geometry 1 and flow speed of four liters per minute is presented in the following section.

FIGURE 4: Visual comparison of cooling experiment with geometry 1 and 4,1 l/min flow speed in cooling channels (a) Thermocamera, (b) CFD – ANSYS, (c) FEM – LS-Dyna Gnielinski HTC = 2576 W/m2K, constant water temperature 20,1°C, (d) same as (c) but with averaged water temperature 31°C

The thermocamera visualization reveals that the 4,1 l/min water flow is not sufficient for uniform cooling of the tool surface and a given 8kW heating power. In this case difference of 30K is between the water inlet and outlet. This effect can be very well approximated with CFD simulation as seen in Fig.5(b) where the x axis represents experimental results and y axis simulation results. The result (c) shows relative big deviations due to constant water temperature. The first three points at the water inlet correspond to the real values, but, due to water heating, the further points show higher discrepancies. The case (d) where the constant temperature corresponds to the average temperature between the water inlet and outlet (30K) gives a standard deviation of 8,5 as seen in Fig.5(d). This means that when we know the temperature difference between the inlet and outlet, we can use this method with relatively high precision. But this is usually not the case. All we know at the beginning is the flow speed and initial temperature of the cooling water.

b) Standard deviation = 2,8 c) Standard deviation = 13 d) Standard deviation = 8,5 FIGURE 5: Comparison of simulation results with experimental results and standard deviations for a flow speed 4,1 l/min

For determination of a convective HTC, such that no temperature at the outlet needs to be known while the result still well approximates the reality, a second inverse simulation has been done. Figure 6 illustrates a result from an FEM simulation in LS-Dyna where the convective HTC in channels was set to such a value that the temperatures in the eight points along the tool achieved the smallest possible standard deviation with respect to experimental values. The temperature was set to the same real inlet value as in the case c)

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FIGURE 6: FEM – LS-Dyna – optimized convective HTC through inverse modeling, standard deviation = 8,2

Geometry 1 - 16,6 l/min Similar procedure was used in a second experiment with the same geometry and 16,6 l/min flow speed.

FIGURE 7: Visual comparison of cooling experiment with geometry 1 and 16,6 l/min flow speed (a) Thermocamera, (b) CFD – ANSYS, (c) FEM – LS-Dyna Gnielinski HTC = 6533 W/m2K, constant water temperature 18,2°C, (d) same as (c) but with averaged water temperature 21,4 °C.

For the four times higher flow speed than in the previous example, we can see a better temperature distribution on the tool surface. This can be again very well simulated with use of CFD simulation which, as Fig.8 illustrates, has achieved 1,8 points of standard deviation.

b) Standard deviation = 1,9 c) Standard deviation = 5,3 d) Standard deviation = 3,5 FIGURE 8: Comparison of simulation results with experimental results and standard deviations for a flow speed 16,6 l/min

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FIGURE 9: FEM – LS-Dyna – optimized convective HTC through inverse modeling, standard deviation = 3,2

Simulation with the optimized convective HTC results again in a deviation better than the method (c) and (d) but is still worse than the CFD simulation. Geometry 2 – 4,1 l/min

FIGURE 10: Visual comparison of cooling experiment with geometry 2 and 4,1 l/min flow speed (a) Thermocamera, (b) CFD – ANSYS, (c) FEM – LS-Dyna Gnielinski HTC = 2576 W/m2K, constant water temperature 20,6°C, (d) same as (c) but with averaged water temperature 29,1 °C

b) Standard deviation = 4,7 c) Standard deviation = 10,4 d) Standard deviation = 6,5 FIGURE 11: Comparison of simulation results with experimental results and standard deviations for a flow speed 16,6 l/min

The CFD simulation again matches visually very well the reality. Applied convective boundary conditions with constant inlet temperature give again small discrepancies. These can be, like in the case of previous geometry, removed by applying averaged inlet-outlet temperature (d). When using an optimized convective BC, we get lower standard deviation than with (d). (6,4 except for 6,5)

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FIGURE 12. FEM – LS-Dyna – optimized convective HTC through inverse modeling, standard deviation = 6,4.

ACKNOWLEDGMENTS This study compares cooling simulations with different approaches for convective boundary conditions (BCs) in the cooling channels to real measurements on an experimental cooling tool. It presents a new alternative with use of optimized convective heat transfer coefficient (HTC). The CFD simulation shows the best results. Although it can very well account for turbulences caused by a curvature in geometries and water heating, its complexity inevitably results in excessive calculation times. FEM Simulation with use of Robin’s convective BCs with HTC from Gnielinski formula and constant water temperature equal to inlet temperature has delivered the worst results from the three compared simulations. Knowing an outlet water temperature, the second FEM simulation with constant average water temperature has proven to be a very good method for cooling channels calculation. The new alternative of calculation is based on optimized convective HTC such that no temperature at the outlet needs to be known. This method has shown second best results out of the four possibilities right after the CFD simulation with approximately 300 times shorter calculation time. A decreasing difference in standard deviations among the four methods when increasing the flow speed had proven the result to be very affected by water heating. As a next step, an analysis comparing different lengths of cooling channels with respect to the input energies and resultant optimized convective HTC should be done in order to allocate the robustness. The presented alternative of convective HTC calculation has a potential in the hot forming processes where the water heating continuously changes along the whole cooling cycle and thus the inlet-outlet temperature difference is never constant, therefore, it will be tested on the real cycling unsteady state process.

REFERENCES 1. Cheng, A. and D. T. Cheng (2005). “Heritage and early history of the boundary element method, Engineering Analysis with Boundary Elements,” 29, 268–302. 2. Dittus P. W., and Boelter L. M. K., “Heat Transfer in Automobile Radiators of the Tubular Type,” Univ. Calif. Pub. Eng., (2/13): 443-461, 1930; reprinted in Int. Comm. Heat Mass Transfer, (12): 3-22, 1985. 3. Gnielinski V., “New equations for Heat and Mass Transfer in Turbulent Pipe and Channel Flow,” Int. Chem. Eng., (16):359-368, 1976. 4. Steinbeiss H., So H., Michelitsch T., Hoffmann H., “Method for optimizing the cooling design of hot stamping tools,” Production Engineering, Volume 1, Number 2, October 2007, pp. 149-155(7). 5. Saifullah A. B. M., Masood S. H. “Finite Element Thermal Analysis of Conformal Cooling Channels in Injection Moulding,”5th Australasian Congress on Applied Mechanics, ACAM 2007, 10-12 December, 2007. 6. Lorenz D., Haufe A,“Simulation von Warmumformprozessen mit LS-DYNA.“ In: 2. Erlanger Workshop Warmblech – umformung, 2007. 7. Anderson, John D. (1995), “Computational Fluid Dynamics: The Basics With Applications,” Science/Engineering/Math, McGraw-Hill. 8. Ito H., “Friction Factors for Turbulent Flow in Curved Pipes,” J. Basic Eng., (81): 123-134, 1959.

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Efficient Robust Optimization of Metal Forming Processes using a Sequential Metamodel Based Strategy J.H. Wiebenga∗ , G. Klaseboer† and A.H. van den Boogaard∗∗ ∗

Materials innovation institute (M2i), P.O. Box 5008, 2600 GA, Delft, The Netherlands † Philips Consumer Lifestyle, P.O. Box 201, 9200 AE, Drachten, The Netherlands ∗∗ University of Twente, P.O. Box 217, 7500 AE, Enschede, The Netherlands

Abstract. The coupling of Finite Element (FE) simulations to mathematical optimization techniques has contributed significantly to product improvements and cost reductions in the metal forming industries. The next challenge is to bridge the gap between deterministic optimization techniques and the industrial need for robustness. This paper introduces a new and generally applicable structured methodology for modeling and solving robust optimization problems. Stochastic design variables or noise variables are taken into account explicitly in the optimization procedure. The metamodel-based strategy is combined with a sequential improvement algorithm to efficiently increase the accuracy of the objective function prediction. This is only done at regions of interest containing the optimal robust design. Application of the methodology to an industrial V-bending process resulted in valuable process insights and an improved robust process design. Moreover, a significant improvement of the robustness (> 2σ ) was obtained by minimizing the deteriorating effects of several noise variables. The robust optimization results demonstrate the general applicability of the robust optimization strategy and underline the importance of including uncertainty and robustness explicitly in the numerical optimization procedure. Keywords: Metal Forming Processes, FEM, Optimization, Uncertainty, Robustness, Sequential Optimization PACS: 02.60.Pn

INTRODUCTION When dealing with the numerical optimization of metal forming processes using computationally expensive FE simulations, arriving at an optimal process is only one side of the coin. The other challenge is to optimize towards robust metal forming processes. More specifically, the goal is to improve the quality of a product or process by minimizing the deteriorating effects of uncertain parameters. In fact, very often an obtained deterministic optimum lies at the boundary of one or more constraints that constitutes a sharp border between acceptable products and waste. Variations in material properties, lubrication or process settings will now lead to a product rejection rate of approximately 50% in practice. To avoid such waste, uncertainty has to be taken into account explicitly in any numerical optimization strategy. Apart from control variables (x), the process is influenced by stochastic variables or noise variables (z); see the process–diagram in Figure 1(a). The input variation is subsequently translated to the response ( f ) which will now also display a probability distribution instead of just a deterministic value. Process robustness is related to the amount of variation in the process response, i.e. the narrower the response distribution, the more robust the manufacturing process. Robustness is thus a measure of the reproducibility of products that result from applying the process. An overview of the most important developments in the field of robust optimization can be found in [1] and [2]. In this paper, a robust optimization strategy is proposed that bridges the gap between deterministic optimization techniques and the industrial need for robustness. In the first section of this paper, the strategy that has been developed will be introduced. The modeling step and sensitivity analysis will be discussed in detail in the second section, after which the robust optimization procedure is described in detail. The paper continues with the introduction of a sequential optimization algorithm. The final section demonstrates the applicability of the strategy through the robust optimization of an industrial V-bending process.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 978-985 (2011); doi: 10.1063/1.3623711 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

978

A ROBUST OPTIMIZATION STRATEGY

Control variables

Modeling

2

Sensitivity Analysis

3

Design Of Experiment

4

Run FE simulations

5

Fit metamodels

6

Validate metamodels

7

Robust Optimization

8

Validate Optimum

9

Accuracy OK

xn

x1 ... Input

1

Process

f

Response

Sequential Improvement

The proposed robust optimization strategy consists of 10 steps. A flowchart is presented in Figure 1(b). This flowchart serves as a guide through the process of mathematically modeling and solving the robust optimization problem in a structured way. The strategy was specifically developed for solving optimization problems including time–consuming FE simulations and was implemented in the optimization software O PT F ORM developed at the University of Twente [3]. The software is applicable to any metal forming process and is suitable for use in conjunction with commercially available FE packages.

...

z1

zn

10

Noise variables

No

Yes

Robust Optimum

(a) FIGURE 1.

(b)

(a) Process-diagram and (b) flowchart of the robust optimization strategy

MODELING AND SENSITIVITY ANALYSIS The first step (1) in the robust optimization strategy is to model the optimization problem under consideration. Both control and noise variables have to be selected and ranges have to be quantified. For the latter type of variables, a normal distribution is assumed. The stochastic variables can now be expressed by a mean value μz and a corresponding variance σz2 . The same holds for the objective function f and constraints g since they can be influenced by the noise variables. Control variables x can be treated the same as in a deterministic problem where its ranges are bounded by Lower Bounds (lb) and Upper Bounds (ub). A typical variance–based robust optimization formulation is given by: find x min μ f + k f σ f s.t. LSL ≤ μg ± kg σg ≤ USL lbx ≤ x ≤ ubx

(1)

z ∼ N(μz , σz2 ) In Equation 1, the objective function is to minimize the weighted sum of both the mean μ f and standard deviation σ f of the response, i.e. both the location and the width of the response distribution are minimized simultaneously. A similar type of formulation is used for describing the constraints. Note that using the weighted sum formulation for a constraint can also be interpreted as a reliability constraint that ensures a 3σ reliability with respect to a certain Lower Specification Limit (LSL) and Upper Specification Limit (USL) for kg = 3. If g is assumed to be normally distributed, then one can subsequently calculate the scrap rate.

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Before proceeding with the actual robust optimization, a sensitivity analysis is performed in step (2). Screening techniques are applied to reduce the size and complexity of the optimization problem. This reduction is possible based on the Pareto effect stating that 80 % of the consequences stem from 20 % of the causes. In the context of optimization, screening techniques for variable reduction are used to determine the control and noise variables that have the most effect on the objective function and implicit constraint(s).

ROBUST OPTIMIZATION AND VALIDATION Solving the robust optimization problem is performed in the following steps. A (3) Design Of Experiment (DOE) is performed based on a full factorial design combined with a space filling Latin Hypercube Design (LHD). After having run the FE simulations (4) corresponding to the settings specified by the DOE, a metamodel is fitted in the combined control–noise variable space using both Response Surface Methodology (RSM) and Kriging metamodeling techniques (5). In the robust optimization strategy, the computationally expensive non-linear FE simulations are thus replaced by a surrogate model. Metamodel validation (6) is performed using ANalysis Of VAriance (ANOVA) techniques [4]. To estimate the performance of the approximate model, leave–one–out cross–validation is used. Each DOE point is selected once as the validation data, and the remaining DOE points as the training data. The level of fit of each metamodel is calculated and used to select the most accurate metamodel with respect to the FE-model response. From the single metamodel, two models for the response mean and variance are subsequently extracted using Monte Carlo Analyses (MCA) of 10.000 points each. The latter step can be performed very efficiently since the MCA is executed onto the metamodel. Both models of the mean and variance can now be used for robust optimization (7) by applying a Genetic Algorithm (GA). Initially, a minimum number of DOE points is chosen in step (3) to limit the required number of expensive function evaluations. The design engineer must be aware of the fact that the solution of the approximated problem is only an estimate of the true robust optimum. To obtain an accurate and reliable solution, the metamodels have to be validated (8). This is done by performing approximately 100 FE simulations in the robust optimum using Latin Hypercube Sampling (LHS). The resulting coarse estimation of the robustness criterion is used to validate the accuracy of the metamodel prediction (9). If the accuracy of the robustness prediction is not sufficient according to the design engineer, a sequential improvement step can be applied to update the metamodel successively (10).

SEQUENTIAL ROBUST OPTIMIZATION The goal of sequential optimization is to increase the accuracy of the objective function prediction at regions of interest containing the optimal design. This is achieved by adding new DOE points to the original set. An update algorithm is required to select the location of the next infill point. The algorithm applied in this work will be outlined in the following subsections. After evaluating the new infill point, the metamodels are updated and validated taking into account the additional response. A robust optimization procedure is started and the new prediction of the robust optimum is determined. Instead of validating this optimum using FE simulations, a stopping criterion is evaluated. The sequential improvement strategy continues to add DOE points until the stopping criterion is fulfilled. As a final check, the design engineer can decide to validate the accuracy of the newly predicted robust optimum.

Expected Improvement Criterion In [5, 6], an update algorithm is proposed based on the principle of Expected Improvement (EI). The proposed Efficient Global Optimization (EGO) strategy adds additional sample points by maximizing the EI. The Improvement is defined as:  0 if y(x) > fmin I= (2) fmin − y(x) otherwise The improvement is calculated by taking the difference between the objective function value y(x) with respect to the minimum feasible sampled (or tested) response of the true objective function value fmin if y(x) < fmin . In addition to a

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mean value prediction y(x ˆ 0 ) at an untried setting x0 , the EI algorithm makes use of the prediction error s(x ˆ 0 ) provided by the metamodel. In other words, the predictor y(x ˆ 0 ) represents a realization of a stochastic process Y in which the randomness is governed by the uncertainty s(x ˆ 0 ) about the true objective function; see e.g. [7, 8]. The so–called posterior distribution at x0 can be modeled as a normal distribution with Y ∼ N(y(x ˆ 0 ), sˆ2 (x0 )). Replacing y(x) in Equation 2 by the random variable Y will result in an expression of the improvement that is also random variable. As shown in [6], the expected value of the improvement can now be expressed in a closed form given by:     fmin −yˆ  ( fmin − y)Φ + sˆφ fminsˆ−yˆ if sˆ > 0 ˆ s ˆ E(I) = (3) 0 if sˆ = 0 with φ and Φ the probability density and cumulative distribution function of the standard normal distribution respectively. For notational simplicity, the dependence on x is omitted here. The first term in Equation 3 contributes to the EI if yˆ is smaller than fmin . The second term contributes to the EI at locations of high uncertainty about whether yˆ will be better than fmin . The criterion is thus capable of searching both locally (first term) and globally (second term).  Maximizing the EI finally provides the coordinates of the infill point x .

Extension of the Expected Improvement Criterion In the case of robust sequential optimization, the goal is to find a suitable location of the infill point which requires   the determination of an optimal setting for both the control variables x and noise variables z . Focusing on the EI algorithm, several difficulties arise if noise variables are taken into account. Firstly, the goal is to find an infill point that is most promising with respect to the robustness criterion instead of the metamodel itself. Secondly, the prediction uncertainty or suitable error estimation sˆ associated with respect to the objective function is not accounted for. Only a few studies have been published that consider sequential update strategies in which noise variables are taken into account; see e.g. [9, 10, 11]. The robust sequential optimization algorithm that is applied in this work is proposed  in [12]. The search for an infill point in the control-noise space is divided into two steps. First, x is determined in the   control space after which z is identified in the noise space. Determining x requires the evaluation of the robustness criterion by application of a MCA onto the metamodel. As a result, the majority of the sampling points that are incorporated in the prediction are untested. That means that an uncertainty sˆ about the predictor yˆ at each point (x, zi ) remains present. As a result, two random variables are obtained in the same probability space. Each point (x, zi ) is now a realization of the stochastic noise variable Z and the posterior error distribution in this point. The stochastic process Y can now be written as (Y |Z) ∼ N(y(Z), ˆ sˆ2 (Z)). To evaluate the prediction uncertainty with respect to the robustness criterion, the conditional variance formula is used: var(Y ) = E(s(Z) ˆ 2 ) + σY2

(4)

The presence of prediction uncertainty adds an additional term to the prediction variance of the stochastic process Y which equals the expected value of s(Z) ˆ 2 over the noise space. Returning back to the definition of the EI, a revision has to be done. This is because the definition of the current best solution of the robustness criterion can now also be seen as a random process with (Y ∗ |Z) ∼ N(yˆ∗ (Z), sˆ∗ (Z)2 ). The EI can now be written as: E(I) =

 b a

(yˆ∗ − y)( fY (y) − fY ∗ (y))dy

(5)

with fY (y) and fY ∗ (y) the probability density function of the stochastic process Y and current best solution Y ∗ . The integration bounds a and b depend on the intersection points of fY (y) and fY ∗ (y) which can be calculated analytically. As a first step, the current best solution is calculated by evaluating the robustness criterion for all settings of the control variable x = xl that are part of the set of DOE points. Subsequently, the EI can be maximized by evaluating Equation 5 for a grid of points x representing the candidate infill points. In exploring the noise space, the deteriorating effects of the noise variables are of special interest. Maximizing the product of the error prediction and the probability   of occurrence results in the optimal noise variable setting, i.e. z = arg max(s(x ˆ , z)2 fZ (z)). Together, these optimal   settings (x , z ) define the infill point at which a new FE simulation is performed. The resulting FE response is added to the set of responses and the metamodel is updated. The update sequence continues until a threshold value for the EI or a maximum number of runs is reached.

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APPLICATION TO A V-BENDING PROCESS The proposed new robust optimization strategy will now be applied to optimize an industrial V-bending process. The goal of this optimization study is to gain more insight into the production process. With that, the process can be optimized such that products are produced within specification in a robust way. Figure 2(a) shows the 2D FE-model of half of the part, MSC Marc has been used as FE-code. One simulation takes about 7 minutes to perform. Both the die and punch are modeled to be non-rigid such that the real process is represented more realistically.

α D

L

β R

σy

M

92◦ ≤ θT ≤ 96◦

θM = 90◦ ± 1.2◦ (a) FIGURE 2.

(b)

(c)

(a) 2D FE-model, (b) definition of control and noise variables and (c) constraints on the flange shape

Modeling and Robust Optimization A sensitivity analysis is first performed to reduce the size and complexity of the optimization problem. As a result, the number of variables has been reduced to four control variables and two noise variables. The remaining set of variables is shown in Figure 2(b). The control variables are the radius of the die (R), the final distance (D) between the flange of the die and punch (if no deformation of the tooling would occur), the dimensions of the punch (L) and the angle of the die (α ) and the punch (β ). Since α and β must be equal, both angles can be described by a single control variable. As with the process in practice, the material thickness (M) and yield stress (σy ) are considered uncertain. To ensure a correct performance of the final product, constraints on the flange shape of the product are prescribed. The flange shape is defined by a transition angle (θT ) and a main angle (θM ) spanned up by the marked line segments; see Figure 2(c). The constraint on the main angle is stricter since this angle is most critical with respect to the performance of the final product. In the current V-bending process, active steering of D is required to obtain products that satisfy the requirements for both angles. The goal of the robust optimization study is to achieve an 3σ process for both angles without the need to adjust D. The main angle is taken into account as the objective function f while satisfying ±3σ constraints on the transition angle: min |(μθM − 90)| + 3σθM s.t. 92 ≤ μθT − 3σθT μθT + 3σθT ≤ 96 92 ≤ α = β ≤ 93 1 ≤ R ≤ 1.3 4≤L≤5 0.4 ≤ D ≤ 0.65 M ∼ N(0.51, 0.012 )

σy ∼ N(350, 6.662 )

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(6)

Results and Evaluation

98 96 94 92 90 88 0.56 0.54 0.52

θM

θM

As a basis for the metamodel, an initial DOE of 300 points is constructed in the combined 6D control-noise variable space. The FE simulations are performed on four parallel processors reducing the total calculation time to approximately 9 hours so that all simulations can be performed overnight. Using ANOVA, a second–order Kriging model resulted in the most accurate fit for both f and g. Impressions of the metamodels are shown in Figure 3. Figure 3(a) and 3(b) show the metamodels of the main angle as a function of the distance D, and the material thickness M and yield stress σy respectively. For visualization purposes, the remaining variables are set to their current process settings as shown in the first column of Table 1. Evaluating the shape of the metamodel, an increase of the main angle is observed for large D since the material is only bent and not yet flattened. Note that for small D, the main angle approaches the tooling angle caused by the flattening of the material. In between these extreme settings, an area can be observed in which both the influence of the distance D and both noise variables is small. For this specific setting, this could potentially yield a process which is robust with respect to noise. Moreover, by looking at the slope of the metamodel in the noise direction it can be seen that the standard deviation increases for increasing D. A similar behavior is observed for the transition angle; see Figures 3(c) and 3(d). Note that especially the lower constraint on the transition angle (presented by the lower plane) significantly decreases the feasible area also excluding the robust area.

96 94 92

0.6 0.5 0.5 M 0.48 0.4 D

0.7

90 370 0.8 360 0.7 350 0.6 340 0.5 σy 330 0.4 D

0.8

(a)

(b)

0.6 0.5 0.5 M 0.48 0.4 D

102 100 98 96 94 92 90 370 360 0.8 350 0.7 0.6 340 0.5 σy 330 0.4 D

θT

θT

100 98 96 94 92 90 0.56 0.54 0.52

0.7

0.8

(c) FIGURE 3.

98

(d)

Kriging metamodel impressions of the (a,b) main and (c,d) transition angle

The metamodels are subsequently used for optimization, solving Formulation (6). The initial optimization approach did not result in a 3σ process. This is mainly caused by a non-robust behavior of the transition angle in the feasible design space. Therefore, the 3σ constraint on the transition angle is slightly relaxed to a 2σ constraint. The optimal process settings for the revised optimization problem are given in the second column of Table 1. The results of the optimized process are shown in Figure 4 in which the main and transition angle are plotted as a function of D. The vertical bars represent the ±3σ bounds caused by the influence of the noise variables. The optimal distance D is found to be 0.55 mm which corresponds to a mean value of the main angle of 90.39◦ and a standard deviation of 0.1◦ ; see Table 2. Note that the modified 2σ requirement for the moment matching constraint on g has been satisfied. In the case of deterministic optimization, neglecting the influence of the noise variables will result in an optimum that lies at the boundary of the lower constraint on θT for D = 0.52 mm. The variation in M and σy will in this case lead to a high number of violations of the constraint in the real V-bending process. From this, it can be concluded that the effect of adding noise is significant if one compares the deterministic to the robust optimal settings for D.

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TABLE 1. Current process settings, process settings after robust optimization and after sequential robust optimization Parameter

Current Settings

α, β R L D

92.9◦ 1.14 mm 4.9 mm 0.55 mm

1.16 mm 4.8 mm 0.55 mm

104 102 100 98 96 LSL 94 92 USL 90 88 0.65 0.4

θT

96 94 92 90 88 0.4

Sequential Robust Optimization

92.4◦

92 1.15 mm 5 mm adaptive

θM

98

Robust Optimization

◦

0.45

0.5

0.55

0.6

(a) FIGURE 4.

D

LSL USL

0.45

0.5

0.55

(b)

0.6

D

0.65

Optimal process results for the (a) main and (b) transition angle as a function of the distance D

Validation and Sequential Robust Optimization The prediction accuracy in the robust optimum is evaluated by a first validation step from which the results are presented in Table 2. The FE simulation based prediction of the mean value of both the objective function and the constraint show a deviation in comparison to the metamodel based predictions. Next to that, a small increase of the standard deviation is obtained for both responses. From this it can be concluded that the initial metamodels approximate the standard deviation rather well. The prediction of the mean value in the vicinity or the robust optimum is less accurate which leaves room for sequential optimization. It must be noted here that drawing firm conclusions about the exact prediction accuracy is risky due to the limited number of FE simulations. However, further increasing the number of expensive function evaluations would decrease the efficiency of the strategy. A trade-off has to be found. Using 100 FE simulations in the validation step will provide the design engineer with a feeling of the approximation accuracy of the metamodel in the robust optimum. TABLE 2. Response of the main and transition angle in the robust optimum Response type

μθ M σθM μθM + 3σθM μθ T σθT μθT + 2σθT μθT − 2σθT

First Validation Step Metamodel FE simulations 90.39 0.10 90.69 93.33 0.66 94.65 92.01

90.69 0.13 91.08 93.77 0.78 95.33 92.21

Second Validation Step Metamodel FE simulations 90.47 0.10 90.77 93.77 0.85 95.47 92.07

90.53 0.12 90.89 93.83 0.82 95.47 92.19

Executing the sequential improvement procedure resulted in an additional 32 DOE points, mainly clustering around the current robust optimum for varying tooling angles α and β . The settings of the robust optimum after sequential optimization are given in the third column of Table 1. The optimal setting of the tooling angle has increased where the other three control variables show negligible changes. Visualization of the metamodels shows that the location of the constrained robust optimum with respect to the tooling angle is highly sensitive for small changes in the shape of the constraint function. The combination of this phenomenon with an increased prediction accuracy of the constraint explains the shift of the robust optimum.

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The results of the second validation step are given in Table 2. The metamodel based and FE simulation based responses show an increase of the prediction accuracy of both the objective function and constraint in comparison to the first validation step. This shows that the sequential optimization step is an efficient way to increase the reliability of the robustness measure in the vicinity of the robust optimum. Application of the robust optimization strategy to the V-bending process resulted in a significant improvement of the robustness (> 2σ ). This is achieved by improving the process design for robustness. The deteriorating effects of the noise variables are minimized by changing the tooling angle and determining the optimal process depth setting (D). This will make the active steering of the depth setting in the current process redundant. Finally, valuable process insights are obtained, among others by visualization of the metamodels used in the robust optimization strategy.

CONCLUSIONS AND FUTURE WORK The robust optimization strategy presented in this paper allows for modeling and solving robust optimization problems in the metal forming industries. Uncertainties such as material variation and process settings, are taken into account explicitly. The sequential update algorithm is based on an expected improvement measure and has proven to increase the accuracy of the objective function prediction at regions of interest in an efficient way. The developed strategy goes beyond just using deterministic strategies and algorithms, it assists metal forming industries to achieve robust and reliable mass production processes. The challenge remains to balance the number of FE simulations spent on the robustness evaluation and the reliability of the robustness measurements themselves. Performing the validation step that is incorporated in the robust optimization strategy, is therefore absolutely mandatory. Focusing on the prediction accuracy of the initial metamodel in the V-bending study, the first validation step pointed out that the response prediction was quite accurate. This suggests that there is room for decreasing the number of initial DOE points. Locally increasing the prediction accuracy of the initial metamodel can subsequently be done by applying the sequential improvement algorithm. This will automatically increase the efficiency of the robust optimization strategy. More generally, future work will focus on including manufacturing variability, process robustness and reliability during optimization in an efficient manner.

ACKNOWLEDGMENTS This research was carried out under the project number M22.1.08303 in the framework of the Research Program of the Materials innovation institute (www.m2i.nl).

REFERENCES H.-G. Beyer and B. Sendhoff. Robust optimization - A comprehensive survey. Journal of Computer Methods in Applied Mechanics and Engineering, 196:3190–3218, 2007. 2. G.-J. Park, T.-H. Lee, K.-H. Lee, and K.-H. Hwang. Robust design: an overview. Journal of the American Institute of Aeronautics and Astronautics, 44:181–191, 2006. 3. M. H. A. Bonte. Optimisation strategies for metal forming processes. Phd thesis, University of Twente, 2007. 4. R. H. Myers and D. C. Montgomery. Response Surface Methodology. John Wiley & sons, 2002. ISBN 0-471-41255-4. 5. M. Schonlau. Computer Experiments and Global Optimization. Phd thesis, University of Waterloo, 1997. 6. D.R. Jones, M. Schonlau, and Welch W.J. Efficient global optimization of expensive black-box functions. Journal of Global Optimization, 13:455–492, 1998. 7. J. Sacks, W.J. Welch, T.J. Mitchell, and H.P. Wynn. Design and analysis of computer experiments. Statistical Science, 4: 409–423, 1989. ISSN 0883-4237. 8. T.J. Santner, B.J. Williams, and W.I. Notz. The Design and Analysis of Computer Experiments. Springer Verlag, 2003. ISBN 0-387-95420-1. 9. B.J. Williams, T.J. Santner, and W.I. Notz. Sequential design of computer experiments to minimize integrated response functions. Statistica Sinica, 10:1133–1152, 2000. 10. D. Huang. Experimental Planning and Sequential Kriging Optimization using Variable Fidelity Data. Phd thesis, Ohio State University, 2005. 11. J. S. Lehman. Sequential Design of Computer Experiments for Robust Parameter Design. Phd thesis, Ohio State University, 2002. 12. F. Jurecka. Robust Design Optimization Based on Metamodeling Techniques. Phd thesis, Technischen Universität München, 2007. 1.

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Review of Approximate Analyses of Sheet Forming Processes Matthias Weissb , Bernard Rolfea*, Chunhui Yanga, Tim de Souzab, Peter Hodgsonb b

a School of Engineering, Deakin University, Waurn Ponds, Victoria, Australia. 3217 Institute for Technology Research and Innovation. Deakin University, Waurn Ponds, Victoria, Australia. 3217 * corresponding author: [email protected]

Abstract. Approximate models are often used for the following purposes: • in on-line control systems of metal forming processes where calculation speed is critical; • to obtain quick, quantitative information on the magnitude of the main variables in the early stages of process design; • to illustrate the role of the major variables in the process; • as an initial check on numerical modelling; and • as a basis for quick calculations on processes in teaching and training packages. The models often share many similarities; for example, an arbitrary geometric assumption of deformation giving a simplified strain distribution, simple material property descriptions - such as an elastic, perfectly plastic law - and mathematical short cuts such as a linear approximation of a polynomial expression. In many cases, the output differs significantly from experiment and performance or efficiency factors are developed by experience to tune the models. In recent years, analytical models have been widely used at Deakin University in the design of experiments and equipment and as a pre-cursor to more detailed numerical analyses. Examples that are reviewed in this paper include deformation of sandwich material having a weak, elastic core, load prediction in deep drawing, bending of strip (particularly of ageing steel where kinking may occur), process analysis of low-pressure hydroforming of tubing, analysis of the rejection rates in stamping, and the determination of constitutive models by an inverse method applied to bending tests. Keywords: Stamping, hydroforming, roll forming, analytical modelling, laminates PACS: 81.05.Bx, 83.50.Uv, 81.40.Lm, 81.20.Hy

INTRODUCTION Metal forming studies have followed several traditional approaches; one, the mathematical, continuum mechanics style, another using extensive numerical modelling, mainly finite element techniques and thirdly the method that can be described as engineering plasticity. The latter was pioneered by, to name just a few, Johnson and Mellor, Marciniak, Fukui, Sachs, Popov and Backofen. This follows more the approach of strength of materials for elastic materials in that it uses approximations and simplifying assumptions to reduce the problem to formulae that allow rapid calculation. The object of engineering plasticity has been not only to provide numbers for process design, but also to identify the governing variables so that students can obtain in their minds an understanding of the limits of a metal forming operation. Two very clever, but at the same time essentially simple examples of models that have been extensively used over the years can be mentioned. The Bland and Ford approximation [1] for the roll force in slab rolling is;  = 

(1)

where F is the roll force per unit width, R, the roll radius and, ¨h, the reduction. The analysis is based on the assumption that the stress on the deformation length,, is the yield stress, Y. In the early control algorithms for

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 986-996 (2011); doi: 10.1063/1.3623712 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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automatic gauge control in strip mills, a tuned version of this equation was used and in teaching this is a useful starting point as it shows how roll radius affects the roll force. Another model still widely used for predicting springback in sheet metal forming is the Gardiner equation [2];  =

 

(2)

where, ¨ș, is the springback, E, is Youngs modulus, R, the bend radius and, t, the sheet thickness. The assumption is that the sheet is fully plastic before unloading and that there is no axial tension; the equation presented is also simplified mathematically. At Deakin University, closed form approximations have been used in the initial stages of study of a number of metal forming studies and these are reviewed briefly here. It should perhaps be mentioned that there is a recommended approach to all these problems that follows the order given below: x Geometry: describe the deformation of the material in a simple and approximate form which will lead to defining the strain distribution. x Equilibrium: write down the equilibrium equations that apply without, as yet, specifying the material behaviour. x Material behaviour: describe material behaviour by a simple constitutive law. In this review, the different processes studied are presented briefly. In all cases, the above approach was used and, for brevity, only the simplifying assumptions are described.

DEEP DRAWING Many analytical models have been used for deep drawing of a circular cup [3]; for a rigid, perfectly plastic material a typical model can be summarized in the equation, 0   = 2 1.13 0    +  () ()  0

(3)

where the variables are illustrated in Figure 1 and, 0 , and,  , are the outer blank radius and cup radius respectively. The cup wall angle, ș, can be found from trigonometry.

(b)

(a)

FIGURE 1. (a) Schematic diagram of the deep drawing of a cylindrical cup (dimensions in millimeters) and (b) Geometry of deep-GUDZQFXSIRUGHWHUPLQDWLRQRIWKHFRQWDFWDQJOHș

This model was extended to include strain hardening and the details are given in Ref. 4. Constant thickness deformation was assumed to exist throughout and at any instant in the process, the average yield stress,  , was taken as the effective stress for an element on the flange at a radius that divided the flange into two equal areas. The effective strain for this radius can be readily determined from the initial and current blank dimensions. To test this model, commercial quality cold rolled sheet steel was deep drawn, and the experimental and theoretical punch displacement curves are shown in Figure 2. This also shows the results of finite element analysis of the process using Abaqus/Standard; the results using two models, shell elements (SAX1) and solid elements (CAX4R), are also shown in this diagram.

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FIGURE 2. Punch displacement-force diagram.

The closed form solution gives slightly better prediction of the maximum drawing force than the finite element analyses; all the models under-estimate the punch force in the final stages of the process. The advantage of this approximate model [5] is that the effect of friction, strain hardening and tool geometry on the outcome of the drawing operation can assessed quickly and this is valuable both in process design and in teaching.

ASSESSMENT OF REJECTION RATES The inherent variability of the incoming sheet and the die conditions in a stamping operation can influence the quality of the final product and cause excessive rejects. A single finite element simulation is successful at predicting the forming behaviour and springback of a stamping process; however, it lacks the ability to assess the influence of the true stochastic nature of the process. In this work [6], a probabilistic analytical model has been developed to investigate the effect of variation in material and process conditions on the shape stability or springback for a simple stretch forming process. The stamping process is simplified to the two-dimensional stretching of a thin sheet over a gently curved form block, as shown in Figure 3, the applied tension in the sidewall is used to simulate various process operating windows. The rejection criterion was based on an arbitrarily assigned acceptable limit of springback which was measured by the change in the radius of curvature; a simple model [3] gives the change in curvature as: (1) = (1)

|!1 !"1 |" 1

(4)



where |!1 !"1 |" 1 is the plastic modulus of the material for the mid-surface principal strain, "1 , at the point of unloading.

Geometrical Parameter Semi-width of the former, a Radius of curvature (punch face), U 0 Punch corner radius, Rp Initial sheet thickness, t 0 Crown height, h Wrap angle, T

FIGURE 3. Two-dimensional stretch forming operation.

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Value 250 mm 1m 50 mm 1.5 mm 31.25 mm 77.85º

To assess the sensitivity of the sheet metal process to variation in input parameters, a multivariate probabilistic modelling approach was adopted. Probabilistic modelling is primarily based on the application of statistics for probability assessment of uncontrollable events. A multivariate probabilistic model permits the introduction of multiple noise parameters for analysis, where each noise parameter has a univariate normal distribution. For example, the variability in yield stress in an incoming coil is represented by a normal distribution of yield stress in individual blanks assessed from information in the literature on steel variability. We denote the n-dimensional jointnormal distribution with a mean vector # and a covariance matrix Ȧ, hence the probability density function is: $( |# , %) =

{ (12)(

# )& ' 1 (

# )}

*(2) |'|

(5)

A full-factorial DOE was performed for five noise parameters (yield stress, n-value, Young’s Modulus, sheet thickness and friction co-efficient) at five equi-spaced levels to generate an appropriate output response space. For each DOE run, the probability of the inputs occurring was calculated using Equation (5) and, in turn, the likelihood of the output is determined. Springback histograms for various operating windows, assessed by analysing various sidewall tensions, are represented in Figure 4. The reject rate for each process setup state and, subsequently, its robustness can be assessed.

FIGURE 4. Springback distributions for a range of sidewall tensions and reject rate given an upper specification limit.

From Figure 4, it may be seen that for the variability parameters selected, the rejection rate is very sensitive to side-wall tension (indicative of binder pressure). For a tension of 600 kN/m the reject rate is 89%; this drops to less than 1% if the tension is increased to 750 kN/m. Stochastic modelling is well suited to analytical modelling methods, primarily due to extremely short solution times compared to Monte Carlo finite element techniques which often involve running hundreds of individual simulations. Analytical models of this type could be used by manufacturers during initial feasibility and try-out stages to assess the ‘true’ performance of their stamping processes.

BENDING OF METAL/POLYMER LAMINATES Material in which a thin polymeric sheet is sandwiched between two sheets of steel or aluminium has been used in the automotive industry and found to be effective in reducing weight and noise transmission. To improve the understanding of forming this sheet, a bending model was created [7]. It was assumed that the moment due to bending of the core would be negligible and the polymer was modelled as a material having a zero Young’s Modulus, but a finite elastic shear modulus. In bending the sheet in a circular die, the outer sheet is stretched and the inner one is compressed (Figure 5a). Figure 5b shows a schematic illustration of the shear stress acting at the interface AB due to the shear deformation of the interlayer and the resulting tension in the outer cover sheet.

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(a)

(b)

FIGURE 5. a) An element of the laminate material bent at a constant radius of curvature. b) Force equilibrium diagram of both the cover sheet and the interlayer.

The force equilibrium within the cover sheets was determined which led to: &(, ) =

2 /( + 4) 

4

!5 6 !

(6)

With the tension force &, the shear angle , - , the plane strain modulus, the radius of curvature , half the !5 thickness of the cover sheets and the interlayer  and 4 respectively and the rate of change of shear strain . (The ! shear modulus is included in determining the shear strain [7]). Assuming that the bending moment on the interlayer is small relative to the cover sheets and can be neglected, the bending moment on the laminate is composed of the moment to bend each cover sheet plus the moment due to the tensions on each cover sheet. 7 8  = 79 + 78: + 2( + 4)&(, )

(7)

Equation (6) and (7) were applied to investigate the bending moment in a metal/polymer laminate that is bent to a circular arc in a frictionless die (Figure 6a). (b)

(a)

FIGURE 6. a) Bending of the metal/polymer laminate material to a constant radius ofҏ ҏIRUDIULFWLRQOHVVFODPSHGGLHE 7KH moment diagram of the metal/polymer laminate bent to a circular arc for different values of the bend radius.

The results showed that the moment in the laminate is non-uniform along the bent strip (Figure 6b), even though the curvature is constant. This is due to tension and compression forces introduced in the cover sheets by the shear reaction force of the interlayer material. In a second approach [8] the model was extended to the elastic plane strain deformation of a strip that overhangs the die as shown in Figure 7.

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FIGURE 7. Schematic of a homogeneous metal strip and a metal/polymer/metal laminated sheet clamped between dies of circular profile showing the difference in the shape of the free standing strip (Gullwing defect).

The schematic diagram compares the free-standing shape of a homogeneous steel strip clamped between two dies of circular profile and that of the metal/polymer laminate. In both cases, the strip is tangent to the die surface at the edge and while the overhanging steel strip is straight, in the laminate there is a reversal of curvature at the tangent point (reverse bend). The model developed was applied to study the influence of process parameters on the shear deformation of the core, the tension forces within the cover sheets and the final shape of metal/polymer laminated strip. The investigation of the deformation of the free bending sheet outside the tooling showed that the reverse bend defect is due to tension within the cover sheets, which is a result of the core shear stress transferred at the interface between the core and the cover sheet layer. An additional benefit of the model is that it permitted the calculation of the stress at the interface; in practical forming operations, this stress can exceed the bond strength leading to delamination.

LOW PRESSURE TUBE HYDROFORMING (LPTH) A new technology called low pressure tube hydroforming (LPTH) has gained increasing attention in the automotive the industry for the forming of high strength steels. In this process a fluid filled and pressurized tube is formed to the required shape using a moving die (Figure 8) leading to a significant reduction of the internal pressure and die closing force required to form the part compared to the conventional hydroforming process. To improve the understanding of the process and to evaluate the die closing force and internal tube pressures needed to form a simple, wrinkle free part, a simplified analytical model was developed [9] for the low pressure tube hydroforming of a rectangular tube shown in Figure 8.

FIGURE 8. Schematic of low pressure hydroforming and the part shape analysed.

The model assumed a symmetrical deformation of the tube with negligible change in tube perimeter and a constant and uniform wall thickness. A rigid, perfect plastic material model was used and the tension in the tube wall was assumed constant around the corner radii. It was found that the die closing force consists of two major components: one is the compressive force in the tube wall to deform the corner radius and the other is the force due to the internal pressure that is transmitted through the tube wall contacting the upper die. The relation developed for the die closing force is given by.

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 = 2 ; 1.165 + 2&

(8)

With the internal tube pressure, , the compressive force in the tube wall, &, and the displacement during forming, . The model developed further showed that the circumferential compressive force in the tube wall required to form the material into the die corners increases as the corner radius decreases. The comparison with experimental and numerical results, Figure 9, showed sufficient agreement for initial process design.

FIGURE 9. Die closing forces for the forming of a pressurized tube (10 MPa) at 10 mm die displacement.

INVERSE MODELLING Researchers at Deakin University recently developed a free bending test for the investigation of the material behavior of metal strip at forming strains close to yield [10]. The test is of particular interest in studying the roll forming process where small changes in yield can lead to major forming defects. It was found that for some materials there is a significant difference between the elastic plastic transition in bending calculated from the yield point measured in the tensile test and that observed in actual bending; this was ascribed to residual stresses in the material introduced by manufacturing processes such as skin passing and roller leveling. It was concluded that for forming processes where bending is the major deformation mode, such as the roll forming process, material data based on the bending test may lead to improved accuracy in CAD and FEA based design packages [11, 12]. A typical moment curvature diagram obtained in the free bending test is shown in Figure 10a. (a)

(b)

FIGURE 10. a) Moment curvature diagram determined in the pure bending test. b) Numerical model of the bending test.

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While the Young’s modulus and the bending yield stress can be determined by simple analysis of the elastic portion of the curve, a direct conversion of the moment curvature diagram in the elastic plastic region into a plastic true stress strain curve is not possible by a simple analysis. An inverse routine was therefore developed that is based on a numerical model of the bending test (Figure 10b); the output is a true stress strain law that if used in a simple bending model will reproduce the experimental moment curvature characteristic [13]. This is a simplification in that it neglects residual stresses in the material and other non-uniformities, but this is justified as many existing modeling packages for bending analyses also ignore these effects. Since the material properties for the elastic region (Yield Point and the Young`s Modulus) are directly measurable from the bending test data the inverse routine is limited to the plastic region. The constitutive equation used here is: D

 = @ + AB(" )C , for  > @

(9)

where the parameters B and D are the variables to be found. (B is the strength coefficient and D the plastic strain hardening index). A schematic of the inverse modelling system is shown in Figure 11.

FIGURE 11. Schematic of the inverse model used showing the seven main steps.

At the start, an initial guess of the best values of tensile parameters is given to the optimization routine, (step 4) and a moment curvature diagram is numerically predicted using the FEA model of the bending test (step 5) and the estimated tensile curve. A root-mean-squared error is used to calculate the difference between the estimated and experimental moment-curvature curves (step 6) and an optimized set of material parameters (B and D) is found using a Matlab optimization command. Using the optimized tensile input, a second iteration (steps 4-7) is started; this process is repeated until the difference between the estimated and experimental moment-curvature characteristic is below a value set by the operator. The true stress strain curve generated can be used in numerical models of the roll forming process. Initial studies have indicated that for some conditions, improved model accuracy can be achieved if material input based on the bending test is used instead of conventional tensile data [14]. This method of obtaining a simple material law from the bending test is not a closed-form analysis, but it is included in this work as the outcome, i.e. the material equation, can be used for simple, approximate analyses of processes of the type given in other examples.

UNSTABLE BENDING OF SHEET A technological defect in free bending of aged steel sheet is the formation of creases or kinks due to an instability in bending; it manifests as sharp bends across the width when uncoiling sheet and as creases in roll forming. It may also appear when bending a sheet over a die radius; instead of the sheet curving smoothly, facets appear that are separated by sharp bends. The phenomenon is, in some respects, analogous to the onset of diffuse necking in a tensile test. Diffuse necking starts when the load reaches a maximum allowing one portion of the test-piece to strain in an accelerated manner to failure, while material on either side unloads elastically. Similarly it may be postulated that in bending, the process will be stable if the moment curvature characteristic has a positive slope and unstable if

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the slope is negative [15]. To study this, an analysis was required for the moment in free bending a strip having the stress strain characteristic shown in Figure 12; this shows that the elastic plastic transition occurs at an upper yield stress, 9 , followed by deformation at a plastic flow stress, $ ; the degree of ageing can be represented by, F=

9 $

(10)

³

Stress

³u ³f

E

¦

Strain

FIGURE 12. Assumed stress strain relation in aged sheet. G

Assuming a linear strain distribution in bending, i.e. "4 = , the stress distribution through the thickness is 

shown in Figure 13, where the elastic plastic transition is at a distance, G = 

¦E

9

Plastic

, from the mid-surface.

³I

t/2

\H

Elastic

³X

¦

³

t/2 Plastic

FIGURE 13. Assumed strain and stress distribution in bending of aged sheet.

The equilibrium equation for this stress distribution is [3], 7 = 2H

G

0

/2 2 G !G + 2 H $ G !G  G

(11)

where M is the moment per unit width. From this, we obtain,

2 2F 7 = $ J + G2  4 3

1L

(12)

The moment curvature diagram calculated for different values of F is shown in Figure 14. This is for a typical case of aged carbon steel in which, t = 1 mm, Young’s modulus E = 200 GPa and the flow stress is, $ = 180 7@. For no ageing, the upper yield stress, 9 , is equal to the flow stress, i.e. F = 1, and the curve has a positive slope throughout and stable bending is expected. For aged material with an upper yield stress of, 9 = 270 7@ , i.e. F = 1.5 , the slope of the curve is zero after yielding and this would be the start of unstable bending. For a heavily aged material with an upper yield stress of, 9 = 324 7@ , i.e. F = 1.8 , the slope of the curve beyond elastic bending is negative and the bending would be unstable. From this, it was proposed that the criterion for the onset of unstable bending or kinking is, F O 1.5

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(13)

FIGURE 14. Theoretical moment–curvature diagram in bending aged materials without tension.

Experiment shows that if the upper yield stress is determined from a bending test rather than from a tensile test and the flow stress is obtained by back extrapolation of the true stress strain curve, this criterion is reasonable [15,16,17].

CONCLUSION The approximate and simplified models presented have been useful in the initial study of a wide range of manufacturing processes, both for the initial assessment of the magnitude of variables in process design and also to assist students in understanding the factors governing common processes. They are not intended to replace more accurate numerical or analytical models, but rather to provide additional tools for industrial metal formers and for students entering this field.

ACKNOWLEDGMENTS The authors thank those students whose work is included here, specifically Daniel Hurst, J Supasuthakul and Chetan Nikhare and also Emeritus Professor J L Duncan for his input.

REFERENCES 1. Bland, D. R., Ford, H., the calculation of roll force and torque in cold strip with tensions. Proc. Inst. Mech. Engrs 159, 144 (1948). 2. Gardiner, F,. J. ‘The spring back of metals’, Tram. Am. SOC. mech. Engrs 1943 65,817. 3. Marciniak, Z., J.L. Duncan, and S.J. Hu, Mechanics of Sheet Metal Forming. 2nd ed. 2002: Butterworth-Heinemann. p. 211. 4. Supasuthakul, J., P.D. Hodgson, M. Weiss, and C. Yang, Analytical Solutions and Finite Element Modelling of Deep Drawing Process for Cylindrical Metal Cups. Key Engineering Materials, 2010. 443: pp. 104-109. 5. Supasuthakul, J., Modelling of Bending-Dominated Forming Processes of Metal Sheets, PhD Thesis. Deakin University, Victoria. 2011. 6. de Souza, T. and B.F. Rolfe, Multivariate Modelling of Variability in Sheet Metal Forming. Journal of Materials Processing Technology, 2008. 203(1-3): pp. 1-12. 7. M. Weiss, B.F. Rolfe, M. Dingle, J.L. Duncan, Elastic Bending of Steel-Polymer-Steel Laminates (SPS) to a Constant Curvature, Journal of Applied Mechanics (ASME), 2006. 73 (4): p. 574 - 579. 8. Weiss, M., An investigation on the forming behaviour of metal/polymer laminates, in Faculty of Science and Technology. 2006, Deakin University: Geelong. p. 214. 9. C. Nikhare, M. Weiss and P. D. Hodgson, Die closing force in low pressure tube hydroforming, Journal of Materials Processing Technology, 210 (2010) 2238-2244. 10. Weiss, M., et al., Measurement of bending properties in strip for roll forming, in International Deep Drawing Research Group, D.K.M. B.S. Levy, C.J. Van Tyne, Editor. 2009: Golden, Colorado, USA. p. 521-532.

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11. Weiss, M., Y. Chunhui, and B.F. Rolfe, The effect of skin passing on Bending and Tension: An experimental Investigation, Steel Research International, 2010. 81(9): p. 789-792. 12. Weiss, M., et al. The effect of skin passing on the material behavior of metal strip in pure bending and tension in NUMIFORM 2010: Proceedings of the 10th International Conference on Numerical Methods in Industrial Forming Processes Dedicated to Professor O. C. Zienkiewicz (1921-2009). AIP Conference Proceedings. 2010. Pohang: American Institute of Physics 13. Weiss, M., B.F. Rolfe, and P.D. Hodgson, Comparison of bending and tensile properties for automotive grade strip steel, in International Deep Drawing Research Group. 2010: Graz, Austria. 14. Henning, R., et al., Understanding the shape defects in roll forming using a novel material characterising method, in Symposium of Plasticity 2011. 2011: The Casa Magna Marriott Puerto Vallarta Resort & Spa. 15. Ding, S.C. and J.L. Duncan, Instability in bending-under-tension of aged steel sheet. International Journal of Mechanical Sciences, 2004. 46(10): p. 1471-1480. 16. J. L. Duncan, M. Sue-Chu, X. J. Wang, Instability in plastic bending of thin sheet, in: C. J. (Ed.) Mechanical behaviour of materials -IV, Pergamon press: Stockholm, Sweden, 1993. 17. Weiss, M., et al., Effect of thermal treatment on the bending properties of pre-strained carbon steel. Materials Science and Engineering: A, 2011. 528(13-14): p. 4528-4536.

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Sensitivity Analysis For Determination The Optimal Blank Shape For Hydromechanical Deep Drawing Of Steel Cups S. A. Zamania and M. Hossinzadeb a

Faculty of Mechanical Engineering, Sama Technical and Vocational Training College, , Islamic Azad University, Ayatollah Amoli Branch, Amol, Iran b Department of Mechanical Engineering, Ayatollah Amoli Branch, Islamic Azad University, Amol, Iran Abstract. In the deep drawing of cups, the earing defects are caused by planar anisotropy in the sheet. Several methods have been developed to predicting the optimal blank to avoid the formation of ears. In the deep drawing processes, the use of an optimal blank shape not only saves the material but also it may reduce the occurrence of defects e.g. wrinkling and tearing. In addition, finding the desirable flange contour, eliminate the trimming step in the deep drawing of parts with flange. In this study, a systematic method based on sensitivity analysis has been used to find the optimal blank to obtain a desirable flange contour for a cylindrical cup by hydromechanical deep drawing process. With the aid of a wellknown dynamic explicit analysis code ABAQUS; optimum initial blank shape has been obtained. With the predicted optimal blank, both computer simulation and experiment are performed and the effect of blank shape on the some parameters such as thickness distribution and blank holder force during the forming process were investigated. It is shown that formed cup with an optimum blank shape has a better thickness distribution in the both rolled and perpendicular of rolled direction. And it was shown that the blank holder force of modified blank is higher than circular blank. Keywords: Sheet Hydroforming, Optimal Blank Shape, Sensitivity Analysis. PACS: 81.20.Hy

INTRODUCTION A problem that has attracted some attention in the recent years is optimal blank shape design for the deep drawing processes. The optimum blank shape leads to the prevention of tearing, uniform thickness distribution and to the reduction of the press load during drawing. If the blank shape is designed optimally, the formability will be increased and the final product will require the least amount of trimming at the end of the process. Therefore, it is desirable to design the blank shape with a uniform flange of its periphery after deep drawing. However, because its deformation during the forming process is very complicated, it is not easy to design the optimum blank shape even by the skilled labor based on the experience of many years. Among the methods, the slip line field method was used by Gloeckl [1], the geometric mapping method was used by Sowerby [2], Toh used the trial-and-error method based on the FE method [3], Iseki used the inverse method [4], ideal forming was studied by Barlat [5], backward tracing was utilized by Kim [6] are notable. In this study, a systematic method based on sensitivity analysis that was used by Shim [7], has been used to find the optimal blank for the hydromechanical deep drawing of cylindrical cups. Fig. 1 shows the schematic illustration of hydromechanical deep drawing.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 997-1003 (2011); doi: 10.1063/1.3623713 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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Figure 1. Schematic of hydromechanical deep drawing set up.

OPTIMIZATION PROCEDURE The overall procedure of obtaining the optimal blank to produce an acceptable final product is shown in the flowchart of Fig. 2. In this procedure, an initial blank shape is selected and a corresponding mesh is produced by the finite element code, ABAQUS. This blank was deformed and the obtained flange contour of the formed part is compared with the required target contour. If the flange contour is different from target contour, depending on its deviation, the initial blank are corrected and a new blank shape is produced and then the modified blank is simulated. It is noticeable that the element shape and element number is kept constant during the optimization procedure. This procedure is repeated until the deviation between flange contour and target contour, shape error, falls below a user defined value ‘‘TSE”.

Figure 2. Flowchart for optimal blank design.

In order to quantify the deviation between the flange and the target contours, a geometrical measure namely, Target Shape Error (TSE), is used. Target shape error, expressed in mm, and is defined by the following equation:

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TSE =

1 n ⎛ x init − x final ⎞ ⎟ x inter − x final ∑f ⎜ n 1 ⎜⎝ x init − x inter ⎟⎠

(1)

Where    

  

 

   "

   

    h  !

     

 

#   $%&' ()'(*+,'(

-

(2)

For further description, Let (xinit, yinit) represents the position vector of the nodes located at the outermost of the flange of meshed blank before deformation and (xfinal, yfinal) represent the position of those nodes after deformation. If (xfinal, yfinal) does not lie on the target contour, (xtarget, ytarget), the position of the material point before deformation, (xinit, yinit), should be moved in order to make the deformed contour coincide with the target contour. Thus, at the end of each simulation, the straight line between initial location (xinit, yinit) and final location (xfinal, yfinal) of nodes is computed. This line intersects the target contour at a point (xinter, yinter), as illustrated in the Fig. 3 with two different cases.

Figure 3. Computation of deviation of nodal points.

Based on these three points, the corrected position of the border nodes (xmod, ymod), defining the new blank shape, is computed, as followed:

xmod = xinit + (xinter - xfinal ) , ymod = yinit + (yinter - yfinal )

(3)

By this new modified blank the process is simulated and the optimization producer that mentioned above, continued. TSE is evaluated at the end of each simulation and when it reaches a value less than ‘‘δ”, a value predetermined by the user for a required accuracy in the flange shape, the iterative procedure is stopped because the optimal blank shape for the part has already been obtained.

FINITE ELEMENT SIMULATION The finite element code ABAQUS/Explicit 6.7 is used to simulate the process and the computational results are analyzed via ABAQUS/CAE. To investigation of the blank shape, 3D simulation of the die set and the blank are conducted. The blank is simulated as deformable object and other component tool are treats as discrete rigid. The working material is described using an elastic–plastic model and is meshed using four-node shell elements (S4R) and the Hill 48 criterion was used to model the blank. The punch, blank-holder and die were modelled surface elements R3D4. The Coulomb friction model is used for the contact analysis. The friction coefficient between the blank and die, between blank and blank holder and between blank and punch are considered 0.05, 0.05 and 0.1. To model the oil, uniformly distributed pressure is applied on the blank. The applied material is St14 steel sheet with 1 mm thickness. The properties of the blank have been obtained from tensile test and are shown in Table 1.

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Table 1. Properties of the blank. Young's modul, E (GPa) 210

Strain hardening exponen, n

Strength coeffici, K (MPa)

0.24

610

Yield stress, σ (MPa)

Poisson’s ratio

201

0.3

Density, ρ (Kg/m3) 7800

The true stress-strain curve along the rolled direction, obtained from tensile test, is shown in the Fig. 4.

Figure 4. True stress-strain curve of St14 obtained from tensile test.

Anisotropy values of the used material (St14) obtained from tensile test is shown in the Table 2. Table (2). Anisotropy values of the used material (St14). r45 r90 r0 Δr r 1.77

1.16

1.95

0.7

1.5

RESULTS OF SIMULATION In order to investigation of the blank shape on deep drawing parameters, deep drawing of a cylindrical cup has been chosen as an example. With the predicted optimal blank, both computer simulation and experiment are performed. Fig. 5 and Fig.6 shows the initial blank, simulated and experimental drawn cup at the first and final step. As it is seen, after final modification step, a drawn cup with rounded flange obtained.

(b) (a) (c) Figure 5. first step a)initial blank b)simulated drawn cup c)experimental drawn cup.

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(a) (b) (c) Figure 6. final step a)modified blank b)simulated drawn cup c)experimental drawn cup.

EFFECT OF BLANK SHAPE ON THICKNESS DISTRIBUTION In order to find the effect of the initial blank shape on the deep drawing process, thickness distribution of the formed parts were compared by each other. To obtain the thickness distribution of the formed cups, the cups were cut from one of their symmetry plane and the thickness was measured through the plane perpendicular to that plane. This is presented in Fig. 7.

Figure 7. The cut part for measuring the thickness distribution.

Fig. 8 shows the simulation and experimental result of comparison of the thickness distribution between circular and final modified initial blank along the rolled direction.

(a)

(b) Figure 8. Comparison of thickness distribution between cups formed by circular blank and final modified blank along the rolled direction a) simulation b) experimental.

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As it can be seen, the optimal blank has a better thickness distribution than circular blank. It means that the maximum thinning in the formed cup by modified blank is less than formed cup by circle blank. This is because of decreasing the flange area of the optimal blank that causes lower friction force. And so, simulation and experimental comparison of the thickness distribution of the drawn cup along the perpendicular of the rolled direction are shown in the Fig. 9.

b a Figure 9. Comparison of thickness distribution between cups formed by circular blank and final modified blank along the perpendicular the rolled direction a) simulation b) experimental.

As it can be seen, the optimal blank has a better thickness distribution than circular blank. It means that the maximum thinning in the formed cup by modified blank is less than formed cup by circle blank. This is because of decreasing the flange area of the optimal blank that causes lower friction force.

EFFECT OF BLANK SHAPE ON BLANK HOLDER FORCE In order to better investigation of the effect of the initial blank shape on the deep drawing parameters, the blank holder force of both circle and modified blank drawn cup compared with each other during the forming. Fig 10 shows this comparison.

Figure 10. Comparison of blank holder force between cups formed by circular blank and final modified blank during the forming (simulation).

As it can be seen in the figure, forming by modified blank has a higher blank holder force. It means that when initial blank modified to obtaining cups with desirable flange contour, it changes the material flowing and in some case (like this case study), the flange tend to wrinkling or increasing the thickness. This is remarkable in thickness distribution along the perpendicular of the rolled direction (Fig.9).

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CONCLUSION In this paper, a prediction model was established for finding the optimal initial blank of a cylindrical cup with desirable flange contour. Simulated results showed that cylindrical cups with desirable flange obtain by using the modified initial blank shape. The result of experiments conducted on the blanks of modified shaped agreed closely with the simulated results. Also, it was showed that thickness distribution of drawn cups along the rolled direction of modified blank is better than circular blank. And the blank holder force of modified blank is higher than circular blank.

REFERENCES 1. 2. 3. 4. 5. 6. 7.

H. Gloeckl, K. Lange, proceedings of the 11th NAMRC 243-251 (1983). R. Sowerby, J. L. Duncan, E. Chu, International Journal Mechanical Science 28, 415-430 (1986). C. H. Toh, S. Kobayashi, International Journal Mechanical Science 25, 15-32 (1985). H. Iseki, R. Sowerby, JSME Int. A 38, 473-479 (1995). F. Barlat, K. Chung, O. Richmond, Metal. Mater. Trans. A 25, 1209-1216 (1994 ). S. D. Kim, M. H. Park,S. G. Kim, D. G. Seo, Journal of Materials Processing Technology 75, 94 -99 (1998). H. Shim, K. Son, K. Kim, Journal of Materials Processing Technology 104, 191-199 (2000).

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2SWLPL]DWLRQRIWKH%ODQN+ROGHU)RUFH8VLQJWKH1HXUDO 1HWZRUN$OJRULWKP Albut A., Ciubotaru V, Radu C. and Olaru I. University “Vasile Alecsandri” from Bacau, Calea Mreti, No. 157, Bacau, 600115, ROMANIA $EVWUDFW In case of sheet metal forming the main dimensional errors are caused by the springback phenomena. The present work deals with numerical simulation related to draw bending and springback of U - shaped parts. The current paper is trying to prove out the important role of the blank holder force variation during the forming process. The Dynaform 5.6 software was used to simulate the forming process, in which the blank holder force varies linearly in four steps between 20 and 50 kN. The factorial simulations test plan was made using the Design Experts 7.0 software and 72 simulations were necessarily to cover completely the variation domain. The part obtained after each simulation is analyzed and measured to quantify the errors caused by springback. Parameters as: angle between flange and sidewall, angle between sidewall and part bottom, chamfer radius between part bottom and sidewall or chamfer radius between sidewall and flange are recorded in a data base. The initial simulations plan together with the generated data base is loaded in a neural network software called NeuroSolution 5. The presented optimization method is a good method to reduce the springback effect. The inconvenient of this method is the large number of simulations tests that must be done and the large amount of data necessarily as input for the NeuroSolution software. .H\ZRUGVspringback, blank holder, neural network 3$&662.20.fg.

,1752'8&7,21 During the last few decades the steel industry has been forced to improve the quality of its products as a result of tension on the steel market and increased customer demands. One of the main aspects in improving the quality of a product is knowledge of the production process and particularly of the material behaviour during the deep drawing process and after the part is removed from the forming tools. In deep drawing process , the quality of the final part is affected by the amount of metal flow into the die cavity. Excessive metal flow will cause wrinkles in the part, while insufficient metal flow will result in tears or splits. The blank holder plays a key role in regulating the metal flow by exerting a predefined blank holder force (BHF) profile. When selected properly, this BHF profile can eliminate wrinkles and delay fracture in the drawn part [1,2]. Usually, in deep drawing, a constant BHF is applied over the punch stroke. During the drawing process, the state of stress in the deforming material changes significantly. Consequently, the process conditions that reduce wrinkling and fracture also change. To take into account these changes, it is reasonable that the BHF should also be modified to increase the formability of the drawn part [3]. A good variable BHF profile is usually determined by conducting time-consuming FEM simulations or experimental trials [4]. Springback is an important problem in sheet metal forming process because of its inconsistency. Moreover it is root of the main dimensional error. There are many factors which affect springback, such as mechanical properties of the material, friction conditions, and process parameters and so on. Therefore, workpiece measurement aiming to check whether it complies with technical requirements implies higher cost in comparison with the case of higher thickness sheets. Numerous fundamental studies have been conducted over the years in an attempt to obtain a basic understanding of springback behaviour. These research activities have been extended from elasticity to plasticity, from small curvature bending to large curvature bending, from pure bending to forming using different BHF. The stress distribution in a sheet metal bending part before unloading decides the magnitude and direction of springback of the part after unloading.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1004-1009 (2011); doi: 10.1063/1.3623714 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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0$7(5,$/3523(57,(6 Simulation of the forming processes based on finite element method using Dynaform 5.6 software requires as input data the mechanical properties of the used materials. Determination accuracy of the mechanical properties has an important influence on simulation results. To obtain the mechanical properties of the base materials and of the welding line, tensile tests were performed on a universal mechanical testing machine, equipped with Hottinger force cells of 5 tf and a Hottinger – Baldwin electronic measurement system for PCs – type Spider 8. The data acquisition, processing and visualisation were performed using Catman Express software. The measurement of specific strains for determination of stress - strain curves was performed using an Epsilon extensometer for a strain rate of 1.6 · 10-3s-1 and 3.3·10-3s-1. Engineering strain/stress data were converted to true stress/strain according to:  = ln(1+e)

(1)

 = S(1+e)

(2)

where  is the true strain,  is the true stress, e is the engineering strain, S is the engineering stress. The specimens were cut as a function of the rolling direction being achieved sets of specimens corresponding to the directions of 0º and 90º and were worked by milling and grinding in order to obtain the prescribed dimensions. The reference length of the specimen was equal to 50 mm. To obtain a good accuracy of the results, 3 specimens were tested for each determination. The results from the tensile test for E220 steel are shown in figure 1 and the mechanical properties are presented in table 1.

),*85( Real stress-strain curves for E220 steel considering the rolling direction and strain rate 7$%/(Mechanical properties of the E220 steel Orientation toward rolling direction 0o 90o

Young modulus (MPa) 399281 417416

Yield strain (Mpa) 189.8333 296.4825

Uniform elongation Total elongation (%) (%) 18.6246 34.3841 20.1708 31.835

strain hardening exponent (r) 2.2342 2.1941

plastic strain ratio (n) 0,1948 0,1899

To determine the Forming Limit Diagrams (FLC) of the E220 steel after performing the Marciniak test, an image analyses technique have been used. In order to measure the occurred strains, a high resolution camera is used to record images during the test. For this method, a random pattern was necessary. A 5 frames per second recording rate was used for the entire test. By using specialized software the recorded series of images are scanned and then an automatic analogy between the first image and all the other images is made. The result of this analogy is the

1005

Major deformations

measurement of the displacement of an initial chosen point on the sample surface. The FLC obtained using the image analysis technique are depicted in figure 2.

Minor deformations

),*85( FLCs obtained by using image analysis technique

6,08/$7,210(7+2'2/2*< The objective of this work is determine the influence of the BHF variation during the forming process on the amplitude of the springback phenomena. For the numerical simulation it was considered the workpiece in figure 3. We performed simulations with the following reference values: friction coefficient P=0.125, material steel and sheet thickness of 0.8 mm, control pattern parameters{F1, F2, F3, F4} in the 20-50kN range. The stripes are 350 mm long and 30 mm wide. The forming time was splited in 4 equal periods, during the first period the BHF is was F1, for the second period F2, for the third period F3 and for the fourth period F4.

),*85( Part considered for simulation

The factorial simulations test plan was made using the Design Experts 7.0, Design of Experiments (DOE) software and 72 simulations were necessarily to cover completely the variation domain .DOE approach is able to increase the accuracy of performance estimates and to reduce the number of required tests. DOE, one tool in the lean engineering and manufacturing toolbox, is a very useful statistical tool when the number of factors is large and there are interdependencies between the factors. By varying the values of all factors simultaneously, a statistically validated estimate of the results for every possible combination of the factors is possible. This analysis approach determines not just the primary effects of each factor but also the interactions between the factors. DOE also requires far fewer experimental (time in the test cell) iterations than the traditional one-factor-at-a-time approach and makes it possible to quickly identify the optimal values for all factors in precisely the right combination. In table 2 is presented a sequence from the experimental plan necessary to cover the combination of the fourth blank holder forces.

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7$%/(Design of experiments )))) 1

40

50

40

40

2

50

40

40

50

3

20

20

50

40

4

50

50

30

40

5   

50

30

30

50

70

40

50

30

30

71

30

30

50

30

72

30

40

20

50

The material was modelled as elastic-plastic, where elasticity is considered isotropic and plasticity is modelled as anisotropic using Hill quadratic anisotropic yield criterion. In this simulation, fine mesh technology was applied to interested regions such as the punch profile radius and the tailor-welded strip model to improve the accuracy of the analysis for the springback. The geometrical model is presented in figure 4. The sheet was modelled as deformable body with 400 shell elements on one row with 5 integration points through the thickness. The tools (punch, die and blankholder) were modelled as analytical rigid because they have the advantage of reduced calculus efforts and a good contact behaviour. Rigid body movements are controlled by reference points. The working conditions and parameters are: friction coefficient  = 0.125; punch displacement in the negative sense of Y axis 50 mm; drawing speed: 18mm/s. In order to decrease the calculation time needed for simulation the following simplifications were applied: a symmetrical model with respect to the longest axis was used; a condition of symmetry with respect to the longest axis (axis of symmetry) was imposed for the blank; the die components were considered as no deformable; this condition eliminated the need of their discretization in finite elements but it imposed to associate a symbolic weight equal to 25kg for the blank holder; the die was considered as fixed, but for the punch and blank holder component a condition of displacement on Y axis was imposed.

),*85(The geometrical model

Springback parameters that were observed during the simulation tests are presented in figure 5: x T1 – sidewall angle between the bottom of the part and part sidewall; x T2 – flange angle between real profile and theoretical profile.

2

1 ),*85(Springback parameters),*85(Overlap of the simulations results

1007

$33/,&$7,212)7+(237,0,=$7,210(7+2' The parameters which need optimization in order to reduce the effects of springback are the blank holder forces F1, F2, F3 and F4. These parameters will be modified according to the experimental plane elaborated on the Design Expert software. All this parameters must be optimized for E220 steel and the part presented before. The first step to implement a neural network is to generate the training data. The ANN model will correctly solve the problem if it learns from a representative crowd of peculiar examples. If the data used in the training set is not representative for the input data class, we can expect poor performance with the test data set, even though performance can be excellent with the training data set. In the present work, the 72 values of the process parameters established according to the fractional factorial experiment design were used as input data for the neural network and the values of the springback parameters resulted from the simulations were used as their associated targets. Neural networks must first be trained in order to be able then to generalize. The learning process assumes the connections weights adjustment. The most common method to change the weights connected to the net hidden neuron layers is the backpropagation algorithm that is: after the propagation of an input through the network, the error is calculated and it is propagated back through the net while the weights are adjusted in order to make the error smaller. In order to study the applicability of the neural network to control the springback phenomenon in the case of U shaped part, a three-layer neural network (the initial, intermediary and final layer), was used with a sigmoid activation function between the input and hidden layers and a linear activation function between the hidden and the output layers. The number of the neurons within the hidden layer must be chosen so that the square mean error (MSE) to the end of the training process to be minimum. 7$%/( Hidden layer neurons and their MSE Number of neurons within the hidden layer

Training MSE

Cross validation MSE

3

0.046271941

0.000573588

4 5

0.044984524 0.048337562

0.000191369 0.000302192

0.35 0.3

Training MSE Cross Validation MSE

MSE

0.25 0.2 0.15 0.1 0.05 0 1

90

179

268

357 446 535 Epoci de antrenare

624

713

802

891

),*85( MSE/Training epochs

Analyzing the above diagram it is possible to conclude that the chosen ANN model leads to an adequate variation of the learning curve; as a consequence, the model will be used to the next functional phase of the neural network.

5(68/76 After training is over, the test for a network performance, is to stipulate correct outputs for that it has not yet seen. In the case of the present analyses, a data set of 25% from the total inputs was given to the network. In the presented case the error between the output values of the network and the real data obtained by simulation is less the 0.2 %. In conclusion there is a good concordance between the desired outputs and the stipulated ones could be observed; as a consequence the neural network is validated.

1008

In order to find the best set of parameters that lead to the diminishing of the springback intensity, the above validated network will be tested for different combinations of process parameters. The neural network should stipulate the outputs for different inputs without have defined the target values of these inputs. This step emphasizes the main advantage of the neural network method – the shortest time of calculation (the time required to calculate the outputs is about few seconds). Good results, with respect to the ideal geometry of the parts were obtained for the following set of process parameters: F1 = 21.3kN; F2 = 20.8kN; F3 = 48.4kN and F4 = 41.2 kN. To validate the neural network method, a simulation has been performed using as input data this set of parameters and the obtained results were compared with the nominal geometry of the part. The shape of the obtained part is presented in figure 8.

),*85( The final part obtained with optimized blank holder force

&21&/86,216 In consequence with the optimisation of the variable blank holder force, the shape error of the part, caused by the springback phenomena, is considerable diminished as follows: 89.5% error reduction for T1 angle between bottom of the part and part sidewall, 92.3% for flange angle T2, 97.7% for radius between sidewall and part bottom and 94.3% for radius between flange and part sidewall. By applying the Neural Networks method the deviations of the geometrical parameters of the part can be significantly reduced, using les time. The presented method is a valuable algorithm for designers of the deep drawing tools.

$&.12:/('*0(17 This project has been funded by the Romanian Ministry of Education, Research, Youth and Sports, Project TE_256, contract no. 17 from 10.08.2010.

5()(5(1&(6 1. F. Pourboghrat, E. Chu, Springback in plane strain stretch/draw sheet forming, Int. J. Mech. Sci. 36 (1995) 327–341. 2. F. Paulsen, T. Welo, Application of numerical simulation in the bending of aluminum-alloy profiles, J. Mater. Process. Technol. 58 (1996) 274–285. 3. T.X. Yu, W. Johnson, Influence of axial force on the elastic–plastic bending and springback of a beam, J. Mech. Work. Technol. 6 (1982) 5–21. 4. T.X. Yu, W. Johnson, The large elastic–plastic deflection with springback of a circular plate subjected to circumferential moments, ASME J. Appl. Mech. 49 (1982) 507–515. 5. J.C. Gerdeen, J.L. Duncann, Springback in sheet metal forming, AISI Report No. 1201-465, 1986.

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An Optimum Design Index of the Bottle with the Vacuum Insulation Structure Takuya HORIUCHIa, Tatsuya TANAKAa, Yutaka IMAIDAa, Keiji NAKAIb and Koji UTSUMIb a

Doshisha University (Kyotanabe-City Kyoto 610-0394, Japan) b Tiger Corporation (Kadoma-City Osaka 571-8571, Japan)

Abstract. In this paper, we present an optimum design index of the bottle with a vacuum insulation structure. Thinner wall design is required to produce lighter bottles. When the wall thickness is too thin, the bottles crushed external pressure. Therefore it is necessary to provide the optimum design index of the bottle. We showed the factors that may affect on the deformation of bottles. We though the factors are classified into shape and material of the bottle. The factors in shape are length L, diameter D and thickness t of the bottles. And the factors for material are Young's modulus and yield stress. The influence of each factor the critical deformation of bottles was verified by using FEM simulation. The nonlinear structural analysis LS-DYNA of the analytical software was applied. The analytical model simplified the base of the external cylinder is hollow cylinder model with shell element. Material properties for stainless steel (sus304), commercially pure titanium (Ti) and titanium alloy (Ti-6Al-4V). These each analytical model was loaded an external pressure by time steps. The pressure when the analytical model was transformed then was obtained. The result shows that the bottle’s strength has the definite relation from its shape and greatly influences the material rigidity. Keywords: Stainless steel, Container, Thin cylinder, vacuum.

INTRODUCTION The vacuum insulation structure is the container that has the dual structure and the vacuum layer between inner and external cylinder as shown FIGURE 1. The container has ability to keep the temperature inside of it by vacuum layer. Therefore the bottles with the vacuum insulation structure had been often-used to carry such as drink generically. But demand of these bottles decrease by the invention of plastic bottles and became widely used these that were lighter and more useful. The plastic bottles have a quality for end up in a landfill after use. The more popular the plastic bottles became, the more they generate waste. The mass production of the plastic bottles led to the mass consumption in resource. However, in recent years, people are raising awareness about environmental problems. Therefore attention is focused on bring own bottle again. The bottles with the vacuum insulation structure such water bottles at one time have been changing from the stainless-steel bottles with a request in recent. There we challenge to increase the usefulness of the bottles. As part of an attack is the thinner board thickness of the bottle. It is come out of the improvement of lighter of the bottle and convenience of carrying it. There,a problem caused by thinner it is strength of the bottle’s external cylinder. The external cylinder is always subjected to external pressure by the vacuum layer. As the board thickness is too thinner, the external cylinder is crushed by the pressure. The optimum shape of the bottle is needed to design its optimum board thickness. The mostly design of the bottles has been determined by technical know-how. However, it is insufficient to design the optimum bottles only in technical know-how. Therefore it is necessary to provide the theoretical optimum design index of the stainless-steel bottle. This study is intended to provide the theoretical optimum design index of the stainless-steel bottle. At first, we researched the factors in which it influence problems that the external cylinder of the bottle is crushed by the external pressure. The influenced factors were classified into shape and material of the bottle. The factors in shape are the height, the diameter and the thickness of the bottle. And the factors in material are such as Young's modulus The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1010-1015 (2011); doi: 10.1063/1.3623715 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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and yield stress in mechanical characteristic. This study verified the relationship between these factors and strength of bottles by external pressure in using FEM simulation.

FIGURE 1. Cross-section view of bottles with the vacuum insulation structure

ANALYSIS In this study, the strength of crushed bottles by pressure was researched by the finite element method simulation. This simulation was created the analytical models of the simplified base of the external cylinder. And the external pressures that linearly increase by time steps loaded the analytical models-wide. This section describes the analysis soft and the analysis condition in the finite element method.

Analytical Model Used an analytical software is the nonlinear structural analysis LS-DYNA ver.971 made by Livermore Software Technology Corp. The analytical model simplified the base of the external cylinder is hollow cylinder model with only shell element as shown FIGURE 2. The central coordinate is the center point of the cylinder bottom. The all shell elements on cylindrical surface is square and the size is 5 mm x 5 mm. The shell is the complete integral shell. The dimensions of the analytical model were shown TABLE 1. Where, D is diameter of the analytical model and L is Length of it. The models were divided into three types of different diameter, and each type has three types of different length. A total of nine models were used to investigate the strength of the bottles by pressure. And the thickness t of each analytical model was the four types of 0.1 mm, 0.15 mm, 0.2 mm and 0.25 mm.

FIGURE 2. Analytical model

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TABLE 1. Dimensions of the analytical model Model Name D [mm] a b 1 50 c a b 2 60 c a 3 67 b c

L [mm] 100 150 200 160 200 240 115 150 200

Boundary Condition The restraint condition on the boundary condition were had some nodes of the analytical model as shown FIGURE 3. Their nodes on the circumference of both edge faces of the model were defined a moving constraint in the X, Y and Z axial direction. The analytical model had the initial imperfection to start point for crushed of it in. In initial imperfection, a node of the coordinates of point was (x=D/2, y=0, z=L/2) had -0.01mm at X axial displacement. The fluid in the analytical model was presupposed the atmosphere. The volume elasticity K of the fluid was 1.4 x 105 Pa and the mass density ρa of it was 1.293 kg/m3. Material dates of the analysis conditions were stainless steel (sus304), commercially pure titanium (Ti) and titanium alloy (Ti-6Al-4V) as shown TABLE 2, TABLE 3, and TABLE 4. This study had each material date in FEM simulation and the physical metallurgy of each material date was used an elastic perfectly plastic solid.

FIGURE 3. Restraint condition and initial imperfection TABLE 2. Mechanical characteristic dates of material in analysis condition Metal Mass density [g/cm3] Young’s modulus [GPa] Yield stress [MPa] Sus304 8.03 197 205 Ti 4.51 106 272 Ti-6Al-4V 4.42 113 888

Poisson ratio 0.30 0.34 0.33

TABLE 4. Chemical component of stainless steel, mass [wt%] Metal C (max) Si (max) Mn (max) P (max) S (max) sus304 0.08 1.00 2.00 0.045 0.030

Ni (min/max) 8.00 / 10.50

Cr (min/max) 18.00 / 20.00

Fe Bal.

TABLE 4. Chemical component of titanium materials, mass [%] Metal N (max) C (max) H (max) Fe (max) O (max) Ti 0.03 0.08 0.010 0.15 0.15 Ti-6Al-4V 0.05 0.08 0.015 0.30 0.20

Al (min/max) 0.00 5.50 / 6.75

V (min/max) 0.00 3.50 / 4.50

Ti Bal. Bal.

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RESULTS AND DISCUSSION Results of Output Dates in Analysis At first, the procedure that leads the output dates to the results in analysis is described. FIGURE 4 is line out the procedure. FIGURE 4 (a) shows a relationship between pressure loading with the analytical model and elapsed time in analysis. FIGURE 4 (b) shows a relationship displacement of the measurement nodes and elapsed time in analysis. Where, the measurement nodes are outputted the displacement dates on the analytical model in shown FIGURE 4 (c)-before crushed. The coordinate of their nodes are circumferential nodes in the middle of analytical model: (x, y, z=L/2). FIGURE 4 (c) shows both before and after crushed analytical models. As will be noted form FIGURE 4, if pressure loading will approach a particular value and the analytical model is crushed behavior drastically. The analytical models of all types are shown this crushed phenomenon. Tc is a time when the analytical model is crushed by pressure in FIDURE 4 (b). The pressure when elapsed time is Tc is defined a collapsing pressure Pc in shown FIGURE 4 (a). Below sections see the analysis results that examine strength of bottles through the lens of a collapsing pressure Pc

(a)

(b)

(c) FIGURE 4. Lead procedure to work up collapsing pressure Pc. In analysis relationship (a): pressure loading or (b): displacement, and elapsed time. (c) is crushed behavior of analytical model

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Relationship between Strength and Shape of the Bottles The examination is a relationship between strength and shape of the bottles. Each collapsing pressure Pc of the all analytical models in TABLE 1is obtained following in the preceding section. When, material dates of analysis condition is stainless steel (sus304). Each plot in FIGURE 5 shows the analysis results that relationship between collapsing pressure Pc and thickness of models t. 1 atm is a maximum value that real bottles with the vacuum insulation structure are loaded by external pressure. FIGURE 5 suggests that increasing thickness have considerable influence a collapsing pressure. These results have trend that the less a value of diameter become, the more the pressure increasing value.

FIGURE 5. Each collapsing pressure of all models

In the strength of bottles, define a critical thickness tc that bottles can maintain their shape, if bottles are loaded the determinate external pressure. When, a value of the pressure is 1 atm. Finding each critical thickness of all models from results in FIGURE 5, its strength is indicated a relationship their shape that is made L, D and tc as shown FIGURE 6. As will be note from FIGURE 6, its strength has a constant relationship to the shape. The trend of this results shows a similar relationship L, D and t of thin cylinder shape [1][2]. Therefore, it is said that L, D and t of shape factors decide strength of bottles.

FIGURE 6. Influence on strength by different shape of bottles

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Strength Influenced by Different Material of the Bottles The examination is what the different material has influence a strength of the bottles. The critical thickness tc is found from the analysis results used commercially pure titanium (Ti) and titanium alloy (Ti-6Al-4V) of material date by similar process. FIGURE 7 shows results that compares the critical thickness in the case of sus304, Ti and Ti6Al-4V of analysis condition. This result shows that analytical model used sus304 of material date has most strength in these material dates. However different material date has influence of its strength, FIGURE 7 shows the similar trend of its strength on shape of bottles in each material. These strengths change a value in concert with Young’s modulus of mechanical characteristic in each material. And these results don’t show that other mechanical characteristics have influence on the strength. Therefore, it is said that Young’s modulus has a major influence on strength of bottles. In these, we will be able to think that bending rigidity has an influence on the strength.

FIGURE 7. Influence on strength by different material dates of bottles

CONCLUSION Here is that result obtained from this study in analysis. Strength of bottles loaded by external pressure is determined by shape of them, the strength-relevant shape bears a constant relation to length L, diameter D and thickness t on their shape factors. Strength of bottles influenced by different material is determined by Young’s modulus of mechanical characteristic. Increasing value of Young’s modulus, the strength escalates. In these, we will be able to think that bending rigidity has an influence on the strength. These parameters prove to be beneficial in deciding strength of bottles. However, if design used only them is too theoretical in bottles. To design the optimum bottles, it is said that other factors will be needed to consider such as forming process of bottles.

REFERENCES 1. X. Wang, Jun Xiao and Y.C. Zhang, International Journal of Pressure Vessels and Piping 81 (2004), pp.907–912 2. D F. Windenburg, Mechanical Engineering, “Vessels under External Pressure”, 1937, pp.625-632

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Development of a New Technology for Trimming of Dual Phase Steels Andrey M. Ilinicha, Sergey F. Golovashchenkob, Lorenzo M. Smitha a

Department of Mechanical Engineering, Oakland University 2200 N. Squirrel Road, Rochester, MI 48309, USA b Manufacturing Research Department, Ford Research & Advanced Engineering 2101 Village Rd., Dearborn, MI 48121, USA Abstract. This paper describes the development and investigation of a new technology for trimming of dual phase steels and provides a comparison with other trimming processes. This technology utilizes an elastic support of the offal, sharp upper trim knife and angled lower trim knife. Such a configuration allows for a high quality trimmed surface with little or no burr and without slivers over a wide range of cutting clearances. Keywords: Dual phase steel, trimming, shearing, burr, sliver, total elongation, hole expansion, edge stretching, sheared edge quality. PACS: 46.15.-x, 46.50.+a

INTRODUCTION Modern product design and manufacturing often utilizes a wide variety of materials. Where once low carbon steel dominated, a variety of new materials such as advanced high-strength steels (AHSS) and aluminum alloys are now being utilized. Although such alternative materials may provide a variety of benefits in product manufacturing and design, they also present difficulties when subjected to manufacturing processes originally designed for low carbon steel. Dual phase (DP) steel is an extremely popular subset of the AHSS family. It provides a favorable balance of strength and formability, thereby allowing it to be processed by conventional means, in spite of its relative high strength. DP steel is a multi phase material as are most other AHSS. Two of its main phases are ferrite and martensite. The minimum ultimate tensile strength of different grades varies between 450 and 1000 MPa. Stronger grades are less ductile and have a larger volume fraction of martensite. Increased strength of this material and presence of martensite particles lead to difficulties in trimming operations. Trimming dies that have been designed in accordance with guidelines developed for the mild steel often lack sufficient stiffness for cutting DP steels. Thus, the cutting clearance can easily drift out of specifications due to elastic deflection of the die. Also, the actual width of this specification window is usually smaller than that for the mild steel, since thinner gages are used and the width is typically expressed as a percent of the material thickness. Following mild steel guidelines often results in inappropriate die material and heat/surface treatment selection from wear resistance prospective, since hard martensite particles accelerate the wear rate. Improper cutting clearance or worn cutting edges result in burr appearance which is a main defect in trimming operations. Burrs not only decrease the panel appeal and geometrical accuracy, but they also diminish its stampability since the burr height is a primary factor influencing trimmed edges' ability to stretch [0]. Therefore, a manual deburring operation is frequently required which drives up cycle time and cost of the part. The authors’ previous research highlighted two technologies for trimming of DP steels [0]: the conventional trimming and dulled upper trim edge with bottom support (DUBS) trimming. Conventional trimming process, shown in Fig. 1a, is most widely used in production. In this process, the blank is initially clamped to the lower shearing knife and then the upper shearing knife moves down to trim the panel. It was shown that this technology provides a good quality trimmed surface up to a cutting clearance of 15%. Starting at approximately 20% clearance a visible burr appears on the sheared edge of the blank. Further increase of the clearance leads to an increase in the

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1016-1026 (2011); doi: 10.1063/1.3623716 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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burr size. It also was shown that the total elongation (TE) along the sheared edge of the samples experiences a substantial drop after 20% clearance. Moreover, this drop was more severe for samples that had been cut perpendicular to the rolling direction of the blank (transverse direction). Additionally, the conventional technology is known to generate very undesirable hair-like particles that are commonly referenced to as slivers. These particles get torn away from the offal during trimming because of its bending [0]. After being generated, they get imprinted into parts and/or tooling which causes damage to their surfaces. The sliver problem has been observed in both aluminum and DP steel applications. Fig. 2 shows a picture of a DP steel sliver observed in this study.

a) Clamping pad Sharp lower trim steel

b) Sharp upper Clamping trim steel pad Blank Cutting clearance

Dull upper trim steel R 0.01mm

Sharp Elastic lower pad trim steel

Fig. 1. Schematics of the trimming processes: a) conventional trimming process; b) dulled upper trim edge with bottom support trimming process.

Fig. 2. Slivers generated by the conventional technology.

The DUBS technology (Fig. 1b) was initially suggested for trimming of aluminum alloys. It introduced a significant improvement over the conventional process in quality of the sheared surface by resolving the issues of slivers, burrs, and splits [0-0]. The idea of this technology is to use a polyurethane pad to support the offal. The offal support eliminates workpiece bending and promotes similar strain distributions around the upper and lower shearing edges. In order to initiate crack propagation from the lower shearing edge, a small radius is fabricated on the upper edge lowering the maximum strain level in the area. Previous studies of this technology [0] confirmed that the DUBS process eliminates burrs, but for a more narrow clearance range relative to that for aluminum. For DP steels, burrless trim was observed for clearances of up to 40%, while for aluminum it is in excess of 80%. Also, it was shown that restriction of offal rotation by the elastic support leads to complete elimination of slivers. In terms of the TE along the trimmed edge, the DUBS trimming technology demonstrated relatively stable values of the TE that are almost independent of the trim clearance, but with significant and almost constant difference between different blank orientations with respect to the rolling direction. Compared to the conventional technology in this regard, this process did not appear advantageous. In particular, it showed a substantially better elongation only in the longitudinal direction and at cutting clearances of more than 25% of the material thickness. For the same large clearances and the transverse direction, it only marginally outperformed the conventional technology. For clearances under 25% which are of more practical interest, the DUBS process showed less TE with a particularly large difference in transverse direction. Since the DUBS trimming technology with the offal support did not solve all problems for DP steels as it did for aluminum, a new technology that would provide burrless and sliverless trim and that would relieve strict requirements and tolerances for die manufacturing is desired. In this paper, the authors document efforts to modify the DUBS process. The aim is to fix its known problems, while maintaining the benefits that include burr absence and suppressed sliver generation.

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FINITE ELEMENT ANALYSIS Finite element analysis was employed throughout this work to help understand the trimming process under different boundary conditions, to predict probable crack initiation areas, and to help develop preliminary tool design. The finite element modeling approach employed is described in more detail in the literature [0, 0]. The simulation was carried out using an internally developed 2D plane strain code [0]. The main features of this code include socalled mild contact that does not allow penetration and a calculation of a cumulative damage variable for each element on every time step using a fracture criterion first introduced by Kolmogorov [0] and later modified by Bogatov [0]. The mild contact algorithm helped prevent a contact problem that was observed using commercially available codes in which there was a trade off between the amount of penetration and the level of deformation/distortion of the blank’s outer layer of elements (elements adjacent to the blank surface). This problem is especially severe for the current application due to the large contact forces and small tool radii that make stiffness-based contact algorithms impractical. The damage variable that was used takes on values ranging from 0 – intact, to 1 – fully damaged. It was calculated as follows: t

\

H

³H 0

dt

(1)

fr

where H is the effective strain rate at a given integration point within an element at a given time t,

H fr

is an

effective fracture strain limit, which is determined experimentally, and is a function of hydrostatic pressure and Lode’s stress parameter, vV . Lode’s stress parameter has the following formula:

vV where:

V1, V 2

and

V3

2V 2  V 1  V 3 V 1 V 3

(2)

are the first, second, and third principal stresses, respectively.

The effective fracture strain limit

H fr

is a strain at which fracture occurs under a monotonic load with a given

level of hydrostatic pressure and Lode’s stress parameter. Since the trimming process for this study is a plane strain process, vV is omitted because its value is zero. The method to experimentally obtain H fr V 0 V y curve is





described in [0]. The resultant curve that was used in the current analysis is shown in Fig. 3.

6

§ V 0 ·¸4 ¨ ¸ © Vy ¹

H fr ¨

2

0 1

 0.5

0

0.5

1

V0 Vy

Fig. 3. A diagram of maximum strain before fracture initiation for DP500 as a function of hydrostatic pressure for plane strain conditions.

Vo

and

Vy

are the mean stress and yield stress, respectively.

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EXPERIMENTAL TECHNIQUE In this study, 0.65 mm DP500 steel is the primary material of interest. This material is attractive for exterior automotive panel applications as it enables weight savings through the use of a thinner gage sheet. Some work has been completed with DP600 which is a stronger grade DP steel used mostly for automotive structural components. Preliminary results for this material are in agreement with those obtained for DP500; details will be reported in a future publication. The sequence of the experiment is shown in Fig. 4. Specimens of 12.7 mm width were trimmed from a 300mm x 75mm strip. One side of each sample was trimmed under various conditions in an experimental die (Fig. 5). This side is shown by dashed lines in Fig. 4. In all cases, trimming was performed at a right angle (i.e. direction of the upper tool movement, shown in Fig. 1 by arrows, is normal to the blank). The other side (shown by solid lines in Fig. 4) was trimmed with a high quality manual shear, and then ground first by a medium and then by fine sand paper. This was done in order to make the surface quality as high as possible so that its influence on the TE of the sample would be minimal. The grinding technique replaced EDMing (i.e. cutting by a wire in electric discharge machining) of "Half-a-dog bone” shaped samples used in previous studies 1 [0]. After the grinding, samples were stretched parallel to the trimming line to a failure. Experiments were repeated 10 times for each trimming condition, and the results were averaged.

75

Scrap Part Scrap Part Scrap Part

Trimmed surface

Part Ground surface 12.7

Fig. 4. Sequence of the experiment.

A picture of the trimming die system for this study is shown in Fig. 5. The dies were secured to upper and lower plates, each of which were approximately 432 mm in length and 280 mm in width. The upper plate was translated vertically through the action of the force from four nitrogen cylinders. Hardness of the trimming tools was 60 HRC.

Fig. 5. The experimental tool.

In order to analyze the trimmed surface quality, cross-sections of samples were mounted in epoxy, then polished and etched for 15-30 s by 4% nital acid. After etching, samples were washed in water and dried by compressed air. 1 It should be noted that tensile testing of straight samples and grinding technique is only suitable for materials that exhibit substantial work hardening (have substantial n-value). Otherwise cracks tend to initiate on the grips boundary

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To analyze crack propagation, partial trimming was employed. The trimming process was interrupted prior to final separation. Partially sheared samples were then polished and analyzed. Both trimming and tensile tests were conducted using Instron testing machine, Model 1125. For the trimming experiments, a ram speed of 0.5mm/s was prescribed; a lower speed of 0.1mm/s was used for partial trimming.

DEVELOPMENT OF A NEW TECHNOLOGY The first step in searching for a way to improve the TE along the sheared edge of the DUBS trimming technology involved an attempt to find a possible explanation of the difference in TE between the DUBS process and the conventional technology at small clearances. After studying closely the cross-section of the samples that were trimmed by the DUBS technology, a small step was observed in a region where the burnished area meets the ductile fracture area on all samples regardless of the clearance (Fig. 6).

5%

15%

30%

Fig. 6. Step-like defects on cross-section of samples that were trimmed by the dulled upper trim edge with bottom support technology. Material DP500 thickness 0.65 mm. Cutting clearance left to right were: 5, 15, 30% of the material thickness.

This step might have been one of the causes of the difference in TE since it could serve as a stress concentrator and, therefore, was a likely site for crack initiation. To check this hypothesis an experiment was performed where, after trimming, samples were stretched along the sheared edge, stopping before failure at different amounts of strain. After this process, the sheared surface was studied under a microscope. In this experiment, it was observed for most cases that cracks started in the region of the step (Fig. 7)

Fig. 7. A source of crack initiation in stretching along the sheared edge for the dulled upper trim edge with bottom support technology.

One possible method to eliminate the step feature would involve the use of a sharper upper trim knife. This, however, re-introduces the aforementioned burr problem since conditions around the upper and lower shearing edges would be the same due to the symmetry of the loading and crack could initiate from the upper shearing edge, leaving a burr. An experiment confirmed this conclusion showing that while usage of the sharp upper knife improves the TE for small cutting clearances, it make it worse for large ones, where burrs of varying heights appear in this case. In order to leave the burr on the scrap side, it is important to impose crack initiation and propagation from the lower shearing edge. To achieve this goal, a difference in stress-strain conditions should be created between the edges in a way that material ductility is exhausted most quickly on the lower edge. To accomplish this, an angled lower knife (Fig. 8) was introduced. Basic research of this technology has been published in [0]. The angle dimension of the knife was picked somewhat arbitrarily. Its influence on the performance has not been studied, and

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is an area of the future work. The only consideration in this study was trimming edge wear (Smaller angles mean less structural strength of the knife, hence, the edge would be more susceptible to wear). Based on this, the angle of 85 degrees was chosen for study in this work. In addition to 85°, the 75° case was considered, but wear of the trimming edge (for 75°) was significantly more intense as anticipated.

Clamping pad Sharp lower trim steel, 85o Lower post

Sharp upper trim steel

Blank

Elastic pad

Fig. 8. Angled lower trim edge with bottom support trimming technology.

FEA was performed as a first step in evaluation of the angled lower trim edge with bottom support (ALBS) technology. Fig. 9 and Fig. 80 show distribution of accumulated damage for cutting clearances of 10% and 30% respectively. Both figures demonstrate that the most probable area of crack initiation is at the lower trimming edge.

Fig. 9. Distribution of accumulated damage for the angled lower trim edge with bottom support trimming technology and the cutting clearance of 10%.

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Fig. 80. Distribution of accumulated damage for the angled lower trim edge with bottom support trimming technology and the cutting clearance of 30%.

RESULTS Three distinct failure modes were observed in the tensile tests (Fig. 91). Mode 1 – typical fracture observed in standard tensile tests, when fracture is initially localized as shear bands, and then separation happens with typical angles defined by material anisotropy Mode 2 – slow opening of the crack perpendicular to the tensile direction Mode 2+1 – combination of modes 2 and 1, when fracture starts from mode 2, but then finalized in mode 1 since the material is close to its limit of elongation.

Mode 1 Mode 2+1 Mode 2 Fig. 91. Stretching of trimmed samples failure modes.

It is worth mentioning that in all cases, the crack initiated from the sheared edge and not from the ground edge. Analyzing the experimental data, it is observed that a clear link between the mode of fracture and final elongation exists. Mode 1 and 2 failure specimens proved to have the greatest and least stretch, respectively, prior to fracture; Mode 2+1 specimens featured intermediate results. These results are consistent with the conventional understanding that formability is compromised when brittle-like modes of failure prevail.

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Fig. 102 shows cross-sections of samples that were trimmed by the ALBS trimming technology. It can be seen that this technology provides high sheared surface quality with no visible burrs for clearances of up to 40% of the material thickness and without the step-like defect featured in the DUBS process. The ALBS technology also provided a significant improvement over the DUBS technology with respect to rollover height, which is approximately the same as that for the conventional technology at the same clearances.

10%

15%

20%

30%

40%

50%

Fig. 102. Cross-section of samples trimmed by the angled lower trim edge with bottom support trimming technology. Material DP500 thickness 0.65mm. Cutting clearance left to right were: 10, 15, 20, 30, 40, 50% of the material thickness.

Partial trimming photos shown in Fig. 113 confirmed modeling on place of crack initiation. For all studied clearances cracks started from the lower trimming edge.

10%

20%

40%

Fig. 113. Cross-section of partially trimmed sample. The angled lower trim edge with bottom support trimming technology. Material DP500 thickness 0.65mm. Cutting clearance left to right were: 10, 20, 40% of the material thickness.

Fig. 102 and Fig. 113 also confirm that the ALBS technology, just as the DUBS technology, does not have sliver problem due to the offal rotation prevention. Results of the tensile tests for the ALBS process are shown in Fig. 124. Bars on the graph show the range of the TE values obtained for each set of experimental parameters. As seen from the figure, this technology provides a relatively high level of TE when the rolling direction is parallel to the sheared edge (longitudinal direction) for clearances of up to 40%. Unfortunately, the ALBS technology inherited almost constant and large difference in TE between longitudinal and transverse directions from the DUBS technology. For clearances of more than 40%, TE decreases with increasing clearance. This decrease is associated with a change in the separation mechanism that

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results in a beak-like shape of the trimmed surface (see the rightmost photo in

Fig. 102). 40

Total Elongation, %

35 30 25 20 15 T-Dir

10

L-Dir

5 0 0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

Clearance between shearing edges/t

Fig. 124. Influence of the cutting clearance and rolling direction on total elongation along trimmed surface. The angled lower trim edge with bottom support trimming technology. Material DP500 thickness 0.65mm. Error bars represent the full range of the data values for each condition.

Fig. 15 shows a failure mode distribution in the tensile tests obtained for the ALBS technology. While the longitudinal samples featured mostly ductile failures (mode 1) for clearances less than 45%, the transverse samples featured only brittle failures (modes 2 and 2+1). This difference in failure mode distribution between longitudinal and transverse directions reflects the difference in the TE just as it did for the DUBS technology. Also noteworthy is a shift in the prevailing failure modes at large clearances (45% and 53%) which is in agreement with the results shown in Fig. 124.

9

8

8

7

7

Frequency

10

9

Frequency

10

6 5 4 3 2 1

5 4 3 2

12.2%

20.4% 28.5%

Clearance

36.6%

1 0

Mode 2 Mode 2+1 Mode 1

0

a)

6

44.8%

12.2% 20.4% 28.5%

b)

52.9%

Clearance

36.6%

Mode 2 Mode 2+1 Mode 1 44.8%

52.9%

Fig. 15. Frequency of failure modes: a) – longitudinal direction, b) – transverse direction. The angled lower trim edge with bottom support trimming technology. Material DP500 thickness 0.65mm.

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COMPARISON WITH OTHER TRIMMING TECHNOLOGIES

40

40

35

35

30

30

Total Elongation, %

Total Elongation, %

The proposed ALBS technology looks beneficial with respect to the sheared surface quality. It does not lead to burrs or slivers for cutting clearances of up to 40%, and promotes significantly smaller rollover than the DUBS technology. Fig. 16 shows a comparison of the TE between different trimming technologies separately for longitudinal and transverse directions. From this comparison it is clear that, while better than the DUBS process, the ALBS technology features a lower TE than that of the conventional technology at small clearances in the transverse direction. The reason for this is not clear since both (the ALBS and the conventional) processes provide approximately the same sheared surface quality in this case. Nevertheless, the ALBS technology appears beneficial for the longitudinal direction, where it shows the same or better amount of TE along the trimmed edge when compared to the conventional process for all but one clearance. An exception can be seen at a clearance of 12% for which the ALBS process underperformed both other technologies. The reason for this is also not clear.

25 20 Conventional trimming technology

15

85deg lower trim steel + support technology dull upper trim steel + support technology

10

25 20 15 10

Conventional trimming technology 85deg lower trim steel + support technology

5 0 0.0%

5

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

0 0.0%

Clearance between shearing edges/t

dull upper trim steel + support technology

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

Clearance between shearing edges/t

Fig. 16. Comparison of the total elongation along the trimmed surface between different trimming technologies. Material DP500 thickness 0.65mm.

Overall, it seems that the presented technology can fully substitute the DUBS process in the trimming of DP steels. The most beneficial area of its application would be if no additional stretching is imposed on the sheared edge after the trimming. Since the TE of the sheared edge is not a concern in this case, the ALBS technology is the best choice, as it leads to burrless and sliverless trim in a wider clearance window. Another beneficial area would be in the case where the blank's edge is stretched after trimming, but the stretch is mostly applied in one direction (as in stretch bending for example). In this case, it seems to be a good idea to position the blank in a way that the stretch is applied along the sheared edges that are parallel to the rolling direction.

CONCLUSIONS Introduced in this study, is a new trimming technology which features a sharp upper knife, angled lower knife, and lower elastic support. General observations are offered as follows: The ALBS trimming technology is clearly better than the DUBS technology in trimming of DP steels. It provides better sheared surface quality and better TE along the trimmed edge. The ALBS technology inherits both positive and, unfortunately, negative sides from the DUBS process. On the positive side there is suppressed burr generation and the offal rotation restriction that eliminates sliver generation. On the negative side there is a significant and almost constant difference in TE between longitudinal and transverse oriented specimens (with respect to the rolling direction). Lower TE is observed when compared to the TE of the conventional technology at small clearances in transverse direction. Improving this condition and making possible modification of the presented technology is a subject of future work. Compared with the conventional process, the ALBS technology demonstrates equivalent or better sheared surface quality depending on the cutting clearance. At clearances of less than 20% the quality is about the same but for clearances of 20% and greater the quality is better due to absence of burr. Also, this technology has another very

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important advantage over the conventional one: it does not generate slivers. In terms of the TE along the trimmed edge, it shows at least the same elongation in longitudinal direction with just a minor exception. In transverse direction, the proposed ALBS process shows less TE at cutting clearances of less than approximately 26-29% but improves at larger clearances. Trimming at angles other than 90 degrees needs to be studied to more fully understand the potential of this technology. This will be included in future work.

REFERENCES 1. D. A. Smith, Die design handbook, Society of Manufacturing Engineers, Dearborn, 1990. 2. A. M. Ilinich, S. F. Golovashchenko, L. M. Smith, “Material anisotropy and trimming method effects on total elongation in DP500 sheet steel”, J. of Mat. Proc. Tech., 2011, 211(3), pp. 441-449. 3. S. F. Golovashchenko, “A study on trimming of aluminum autobody sheet and development of a new robust process eliminating burrs and slivers,” Int. J. of Mech. Sci., 2006, 48(12), pp. 1384-1400. 4. S. F. Golovashchenko, and M. Blodget, “Apparatus for trimming metal,” US Patent 7197970, 2007. 5. S. F. Golovashchenko, “Analysis of trimming of aluminum closure panels,” J. of Materials Engineering and Performance, 2007, 16(2), pp. 213-219. 6. N. M. Bessonov, S. F. Golovashchenko, and V. A. Volpert, “Numerical Modelling of Contact Elastic-Plastic Flows,” Math. Model. Nat. Phenom., 2009, 4(1), pp. 44-87. 7. S. F. Golovashchenko, and A. M. Ilinich, “Analysis of Trimming Processes for Advanced High Strength Steels”, 2009, SAE Paper 2009-01-1175. 8. V. L. Kolmogorov, Stresses, Strains, Fracture, Metallurgy, Moscow, USSR, 1970. 9. A. A. Bogatov, O. I. Mizhiritski, S. V. Smirnov, Resource of material ductility in metal forming, Metallurgy, Moscow, USSR, 1984. 10. S. F. Golovashchenko, and A. M. Ilinich, “Analysis of Trimming Processes for Advanced High Strength Steels”, 2008, SAE Paper 2008-01-1446.

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Study on Edge Thickening Flow Forming Using the Finite Elements Analysis YOUNG JIN KIMa and JIN SUNG PARKb , CHONGDU CHOc a

Dept. of Mechanical Engineering, Inha University Graduate School, Incheon, 402-751, Republic of Korea Dept. of Mechanical Engineering, Incheon University Graduate School, Incheon,406-772, Republic of Korea c Dept. of Mechanical Engineering, Inha University, Incheon, 402-751, Republic of Korea

b

This study is to examine the forming features of flow stress property and the incremental forming method with increasing the thickness of material. Recently, the optimized forming method is widely studied through the finite element analysis to optimize forming process conditions in many different forming fields. The optimal forming method should be adopted to meet geometric requirements as the reduction in volume per unit length of material such as forging, rolling, spinning etc. However conventional studies have not dealt with issue regarding volume per unit length. For the study we use the finite element method and model a gear part of an automotive engine flywheel as the study model, which is a weld assembly of a plate and a gear with respective different thickness. In simulation of the present study, a optimized forming condition for gear machining, considering the thickness of the outer edge of flywheel is studied using the finite elements analysis for the increasing thickness of the forming method. It is concluded from the study that forming method to increase the thickness per unit length for gear machining is reasonable using the finite elements analysis and forming test.

Keywords: Flow stress, Flow forming, Flywheel, Edge thickening, Forming roller PACS: 46. Continuum mechanics of solids

INTRODUCTION Recently, many plastic forming methods such as cold and hot forging, hydro forming, deep drawing, extrusion, rolling and spinning have been researched and developed to manufacture the required geometry using reducing the volume per unit length and extreme plastic deformation of material and flow stress in the metal plastic forming techniques. [1-4]. However, researches are not profound about against forming method such as increase volume per unit length. Thus, this study will be considerate flow forming which using flow stress of material by finite elements analysis. This study analyzed a flywheel of auto engines which consists of a plate and a ring gear, as shown in Fig. 1, and is widely applied with the assembling method of the ring gear on the external face of the plate through welding individual parts manufactured separately. In general, 2.0- to 3.5-mm-thick plates are manufactured with stamping and gears are manufactured with the machining of 10- to 15-mm-wide and 7- to 12-mm-thick rims. we analyzed the forming technique for increasing the thickness to a dimension which allows machining by using the flow stress of the material, so we can avoid manufacturing a separate individual part, machining the ring gear rim, or weldassembling. For this study, we calculated plastic analysis using DEFORM, common finite elements analysis program. The material of the gear part of the flywheel that we examined in this study is AISI 1035. We analyzed through a tensile test this material, based on properties like flow stress. In other to deduce the final forming of a rectangular cross-section rim shape to enable gear machining, we applied the incremental forming technique, and the formed roller cross section shape was designed and analyzed in steps. And, formed shapes and metal flows were analyzed by flow forming analysis of deformable body. Also, forming load and stress distribution of forming roller were predicted by elastic analysis.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1027-1033 (2011); doi: 10.1063/1.3623717 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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FIGURE 1. This Welded flywheel

MATERIAL MECHANICAL PROPERTY TEST As a material for flow forming that facilitates an increase in volume per unit length, AISI 1035 carbon steel, widely used as a gear material of flywheel for auto engines, was used. We obtained the mechanical characteristics, flow stress, and work-hardening index of the material by executing a tensile test to measure the material resistance and work-hardening characteristics through forming.

FIGURE 2. Tensile test specimen

The tensile test specimen that we used for the study is shown in Fig. 2, the specimen standard according to the ASTM tensile specimen standard. [7-8] Table 1 and Fig. 3 show the results of the tensile test. The material for the forming roller was AISI 52100 (bearing steel), widely used in the rolling process, and Metal Handbook was used for its basic material characteristics as a reference. [10] TABLE 1. AISI 1035 mechanical property Property 177 GPa 0.28 406 MPa 603 MPa 22.5 % 0.116

Property Fields Elasticity modules Poisson ration Yield stress Tensile stress Elongation n-value

V

^[_UZ ˜ H WUXX]

(1)

FLOW FORMING FEA Finite elements modeling In order to enable the machining of the flywheel gear part, the technique of forming a 10-mm-wide and 8-mmthick rim by forming a 2.5-mm-thick plate, as shown in Fig. 4, was used, and the model was formed as the size of

1028

the welded flywheel. The model of the weld-type flywheel shown in Fig. 4a was assembled by welding the wheel plate and ring gear, and was designed to form the rim of the rectangular cross-section. The gear was machined via flow forming, instead of the individual parts of the plate and ring gear, as shown in Fig. 4b.

FIGURE 3. Tensile test specimen

(a) Welded flywheel model (b) Flow formed flywheel FIGURE 4. Forming model description

The modeling for the finite element analysis was performed only for the actually formed area, and the invariant volume during the forming was considered in the determination of the plate size before the forming. For the FE model, full modeling was done because even though the model can be assumed to be axis-symmetrical, the deformation behavior according to the rotational angular velocity of the model during the forming cannot be described as axis-symmetrical. The elements of the FE model were defined as 3D Tetra elements, and 23,739 nodes and 89,560 elements were used for the analysis. All the elements, except the deformable body, were defined as rigid elements. The forming rollers were defined as the elastic models for the stress analysis after the flow forming analysis.

Flow forming roller design

In the final forming of the ring of the rectangular cross-section that can be gear-machined, the forming cannot be done at once; hence, the incremental forming technique was applied, and the roller cross-section shape was used for each step. The first forming step involved setting a total of four steps. Fig. 5 shows the shape of the forming roller cross-section by step.

FIGURE 5. Forming roller profile

The groove depth setting of the forming roller in each step was defined by a quadratic function that assumed the increase in the forming load and the reduction of the rim thickness increment. Eq.(2) expresses the reduction of the forming thickness of the rim in each processing step, and Eq.(3) expresses the groove depth of the forming roller

1029

according to the process thickness obtained from Eq.(2). In the definition of the groove depth of the forming roller, it was assumed that there was no volume change in each process step.

› O— P

› W  › Y › › —  Y W  —  ›W Y s s k WO— P  k y O— P Y

(2) (3)

Boundary condition for the FEA For the forming analysis, the rigid element ring that could be constrained was applied under the assumption that there was no deformation in the inward radial direction of the deformable body, and the forming roller and the forming support roller were applied in each process step. Fig. 6 shows the forming analysis and conditions.

FIGURE 6. Boundary condition for FEA

For the analysis conditions, a rotational angular velocity of 37.7 rad/s was applied to the inner rigid ring for the deformation in the tangential direction of the deformable body. For the forming roller, a forming speed of 3 mm/s toward the radius center was applied. To consider the roller rotation according to the relative rotational force, it was assumed that there was no friction between the deformable body and the forming roller.

FIGURE 7. Step Increment description Fig. 7 shows the forming analysis conditions in each forming step. For the elastic analysis for the stress analysis of the forming roller after the flow forming analysis, the calculated forming load and distribution in each flow forming step was applied to the forming roller.

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FEA RESULTS for FLOW FORMING Formed shape according to FEA

Fig. 8 shows the sectional shape of the model in each forming step as a result of the flow-forming finite element analysis for the forming of a rim with a rectangular section to enable the machining of a flywheel gear for an automobile engine, which is the model of this study. As shown in the results, it was possible to analytically calculate the forming technique that increased the volume per unit length through the designed sectional shape of the forming roller and the incremental forming.

Figure 8. Formed shape results by each step Fig. 9 shows the metal flow of the model according to the flow forming. It shows that not only the flow to the directions of the radius and thickness of the section, but also the flow to the rotational direction, were generated. Furthermore, the metal flow allowed the forming without folds via excessive volume change.

Figure 9. Formed metal flow by each step

Forming load and roller stress

Fig. 10 shows the forming load and load distribution of the forming roller in each step of the flow analysis. It can be seen that the forming load increased as the forming steps progressed. It is supposed that as shown in Equation (1), the forming load increased due to the work hardening according to the plasticity of the material and the demand for a relatively high volume change in the progress of the forming steps

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Figure 10. Formed load result Furthermore, Fig. 11 shows the load distribution calculated from the formation analyses for the elastic analysis of the forming roller.

Figure 11. Formed load distribution

Figure 12. Forming roller stress distribution The stress analysis results for the forming roller, as calculated from pre analyzed load distribution, are shown in Fig. 12. As can be seen from the figure, excessive stress was generated on the forming roller in each flow forming step. In particular, the higher stress in the second forming roller than in the forming load can be regarded as the effect of the stress concentration according to the forming groove depth.

CONCLUSIONS

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This study presented a forming technique that increases the thickness per unit length to make a rim shape with a rectangular section, which allows gear processing through the forming of a outer surface on a fly wheel plate for automobile engines, from the analysis results using material flow stress analysis, forming roller design, incremental forming technique, forming conditions, and finite element analysis conditions. It was found that the flow stress characteristics of the material increased the forming load in the forming process, which generated excessive stress distribution to the forming roller. More studies are needed to minimize the load and stress of the forming roller based on stable forming, through the optimization of the forming process conditions such as the flow stress characteristics, according to the strain rate of the material and the optimized design of the forming roller, as well as through flow forming tests.

REFERENCES 1. C.C. Wong, T.A. Dean, J. Lin, International Journal of Machine Tools & Manufacture, (2003), 43, pp. 1419-1435 2. C.C. Wong, A. Danno, K.K. Tong, Journal of Materials Processing Technology, (2008), 08, pp. 53-62 3. M. Jie, C.H Cheng, L.C Chan, International Journal of Mechanical Science, (2009), 51, pp. 269-275 4. C.C. Wong, T.A. Dean, J. Lin, Journal of Material Processing Technology, (2004), pp. 60-66 5. J.N. Park, J.E, Keh, B.M. Kim, Journal of Korea Society of Precision Engineering, (2002), 19, pp. 142-149 6. N.K. O, Y.S. Ho, P.H. Jin, Korea Society of Machine Tool Engineers, (2006), pp. 119-124 7. ASTM E8-01, ASTM STANDARDS (2002), Volume 03.01 8. ASTM E517-00, ATSM STANDARDS 2002, Volume 03.01 9. ASTM E646-00, ATSM STANDARDS 2002, Volume 03.01 10. N.S. Kim, H.J. Kim, ‘Plastic Forming and Analysis; Moonwoodand 1st ED, 2002

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A Study on Forming Characteristics of Roll Forming Process with High Strength Steel ByeongDon Jooa, HyunJong Leea, DongKyu Kimb and YoungHoon Moona a

School of Mechanical Engineering/ Engineering Research Center for Net Shape and Die Manufacturing, Pusan National University, Busan, 609-735, Korea b Sungwoo Hitech Technical Institute Advanced Technology Team, Yerim-ri, Jeonggwan-myeon, Gijang-gun, Busan, 940-150, Korea Abstract. Roll forming is a kind of sheet metal forming process used to manufacture long sheet metal products with constant cross section. Recently, roll forming technology draws attentions of automotive industries due to its various advantages, such as high production speed, reduced tooling cost and improved quality. In automotive industries, roll formed automotive parts used as structural components in vehicle body frame or sub frame and high strength steel becomes more common to improve safety and fuel efficiency. However, when roll forming process is performed with high strength steel, rolling forming defects, such as spring back, buckling and scratch should be considered more carefully. In this study, efforts to avoid roll forming defects and to optimize forming parameters were performed. FE analysis was performed with high strength steels using commercially available simulation software, COPRA-RFTM and SHAPE-RFTM. Forming characteristics were analyzed and roll flower model and proper roll-pass sequences were suggested by analyzing longitudinal strain and deformation behavior. This study provided considerable experience about roll forming process design that using high strength steel. Keywords: Roll forming, High strength steel, Roll flower, FEM PACS: 81. 20. Hy

INTRODUCTION Roll forming is a kind of sheet metal forming process and is used to manufacture long sheet metal products with a constant cross-section thickness [1,2]. In this process, an originally plane sheet is transformed onto a desired profile through the action of a series of pairs of rotating rolls, as illustrated in Fig. 1. Roll forming has many advantages such as high production speed, reduced tooling cost and improved quality [3,4] and it draws attentions of automotive industries. Various automotive parts such as bumpers, door beams, frame rails and roof bows are formed with roll forming process. Recently, high strength steel becomes more common to improve safety and fuel efficiency in automotive industry [5]. However, when roll forming process is performed with high strength steel, rolling forming defects, such as spring back, buckling and scratch should be considered more carefully [6]. Because, with using high strength steel, the difficulty to control the deformation behavior include longitudinal stretching, longitudinal bending, transverse bending and shear is increased. In this study, efforts to understand forming characteristics of roll forming process were performed with SPFC590, SPFC780 and SPFC1180. Roll flower pattern for sill side was suggested and roll-pass sequences were designed with SPFC590. With the model, forming characteristics according to the material were analyzed by evaluating longitudinal strain and deformation behaviors. Also, optimal roll flower deign and proper roll-pass sequences were suggested for each material.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1034-1040 (2011); doi: 10.1063/1.3624622 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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FIGURE 1. Typical roll-stands.

ROLL FORMING SIMULATION USING FINITE ELEMENT ANALYSIS A finite element analysis for the process using COPRA-RFTM and SHAPE-RFTM was performed. Table 1 shows the material properties and the analysis conditions. For the efficient implementation, steady-state, rigid plastic finite element method was used. The section profile of sill side which used in this study is shown in Fig. 2. Roll flower pattern and roll profiles were designed for SPFC590 which is widely using in automotive industries and the results are shown in Fig. 3. As the number of roll pass is directly related with production cost and quality of roll formed part, optimized pass number of 11 was determined. Additional analysis was preformed with various roll flower pattern and roll profiles. TABLE 1. Finite element analysis conditions.

FEM Program Material Initial Strip Thickness(mm) Initial Strain Roll velocity(m/s) BCC

COPRA-RF SHAPE-RF SPFC590, SPFC780, SPFC1180 1.2 0.01 0.1 Full

FIGURE 2. Section profile of sill side.

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FIGURE 3. Roll flower pattern and roll profiles designed for SPFC590.

FORMING CHARACTERISTICS ACCORDING TO YIELD STRENGTH Finite element simulations are used in the current study to evaluate the yield strength influence on longitudinal strain and deformation behavior. The number of forming steps was 11 pass, input material strip width was 194.31mm and part weight was 1.83kg. Fig. 4 shows the longitudinal strain due to bending from an original flat strip to a final shape and Fig. 5 shows longitudinal strain distribution of the exit section for pass No. 11. When yield strength was 590Mpa, strain distribution was stable and longitudinal strain at final pass was even lower than 0.06%. When the yield strength increase to 780Mpa, maximum longitudinal strain at left side was 0.011% and at right side was 0.022%. The range of longitudinal strain is not higher than 0.06%. However considering over bending process using for correcting spring back, the range of longitudinal strain will increase to 0.06%. Therefore additional roll passes between pass 8 ~ 9 and 9 ~ 10 are recommended. As shown in Fig. 4 (d), when high strength steel, SPFC1180, was used, the process was unstable. Longitudinal strain was rapidly increased and longitudinal strain at measuring point 6 was higher than 0.06%. As the range of longitudinal strain at final pass was 0.088%. Defects such as edge wave and twist are expected. Fig. 5 shows defects at exit of roll pass No. 2 with SPFC1180. After pass No. 2, edge wave was occurred and sheet was moved to left side, 2mm. In general, the main reason why wave occurs is high amount of deformation concentrated on local region. And the main reasons why roll form sections twist are non symmetrical sections, roll adjustment with uneven pressure and side roll stands with excessive pressure on one side. In this case, optimal process design is needed.

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0.04

Longitudinal strain

0.03 0.02 0.01 0.00 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06

mesuring point 1 2 3 4 5 6

1

2

3

4

5

(a)

Longitudinal strain

Longitudinal strain

0.01 0.00

-0.04 -0.05 -0.06

9

10 11

0.03

0.02

-0.03

8

0.04

0.03

-0.02

7

(b)

0.04

-0.01

6

Roll pass

mesuring point 1 2 3 4 5 6

1

2

3

0.02 0.01 0.00 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06

4

5

6

7

Roll pass

8

9

10 11

mesuring point 1 2 3 4 5 6

1

2

3

4

5

6

7

Roll pass

(c)

8

9

10 11

(d)

FIGURE 4. Longitudinal strain according to yield strength; (a) measuring point, (b) SPFC 590, (c) SPFC 780 and (d) SPFC 1180

Longitudinal strain

0.04 0.02 0.00 -0.02 -0.04

SPFC590 SPFC780 SPFC1180

-0.06 0

20

40

Node number

FIGURE 5. Longitudinal strain distribution after final pass

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60

80

FIGURE 6. Defects at exit of roll pass No. 2 (SPFC 1180)

OPTIMIZATION OF ROLL FORMING PROCESS To optimize the roll flower deign and proper roll-pass sequences for high strength steel roll forming, finite element analysis was performed at various conditions. To minimize deviation of longitudinal strain which caused in left section and right section, target section profile of sill side was rotated 11°, counter-clockwise direction. With that profile, roll flower pattern and roll profiles were designed and optimized roll flower pattern is shown in Fig. 7. Fig. 8 and Fig. 9 show longitudinal strain on roll passes and longitudinal strain distribution after final pass with optimized roll profiles. Comparing with Fig. 4 (d) and Fig. 5, longitudinal strain was remarkably decreased and strain distribution was more uniform. By rotating section profile, non symmetrical section can be formed with high strength steel, without increasing the number of roll passes.

FIGURE 7. Modified roll flower pattern

1038

0.07

Longitudinal strain

0.06 0.05 0.04 0.03

mesuring point 1 2 3 4 5 6

0.02 0.01 0.00 -0.01 -0.02 -0.03

1

2

3

4

5

6

7

Roll pass

8

9

10 11

FIGURE 8. Longitudinal strain on roll passes with modified roll profiles

Longitudinal strain

0.04 0.02 0.00 -0.02 -0.04 before optimization after optimization

-0.06 0

20

40

60

80

Node number

FIGURE 9. Longitudinal strain distribution after final pass with modified roll profiles

CONCLUSIONS In this study, finite element simulation of roll forming process was performed using commercially available simulation software, COPRA-RFTM and SHAPE-RFTM to analyze forming characteristics due to yield strength. Roll flower pattern and roll profile were designed for sill side which has non symmetrical section and forming characteristics were analyzed with comparing longitudinal strain. Considering defects such as wave and twist, optimal flower pattern and roll profile were redesigned and the results shows that by rotating section profile, non symmetrical section can be formed with high strength steel, without increasing the number of roll passes.

ACKNOWLEDGMENTS This work was partially supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-521-d00206) and grants-in-aid for the National Core Research Center Program from MOST/KOSEF (No. R15-2006-022-03003-0).

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REFERENCES 1. 1. 3. 4. 5. 6.

A. Alsamhan, P. Hartely and I. Pillinger, J. Mater. Process. Technol. 142, 102-111 (2003). M. Brunet, S. Mguil and P. Pol, J. Mater. Process. Technol. 80-81, 213-219 (1998). G. ZENG, X. M. LAI, Z. Q. YU and Z. Q. LI, J. Iron. Steel Res. Int. 16, 32-37 (2009). Q. V. Bui and J. P. Ponthot, J. Mater. Process. Technol. 202, 275-282 (2008). K. Sweeney and U. Grunewald, J. Mater. Process. Technol. 132, 9-15 (2003). M. Lindgren, J. Mater. Process. Technol. 186, 77-81 (2007).

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Modeling and Analysis of Reverse Drawing Process of Steel Sheet and Comparison with Simple Drawing Process M. Maarefdousta , M. Kadkhodayanb, M. Goharimaneshc a

Islamic Azad University, Gonabad Branch, Iran b Ferdowsi University, Mashhad, Iran c Iran University of Science and Technology, Tehran, Iran Abstract. In this paper, using finite element analysis, an analytical comparison process of deep drawing steel sheet in two ways, ordinarily and reversely is considered. The results show that, under plane strain, steel sheet is caused suffering throat in the ordinary method, when achieves stretching deformation factor m 0.162 (Ratio of final 13mm Initial width 80mm ), after suffering the effects of tensile deformation mechanism of activation failure. Whereas the same conditions of the process to reverse the stretching method, a whole sheet steel deformation without failure gives the amount of force needed to punch, per the same amounts of deformation of Reverse process of stretching is less than ordinary process .Changes in the stress of plate, displacement curve sheet, sheet strain changes for the two stretching modes and simple inverse elasticity are compared. Keywords: Simulation, Reverse drawing, Stretching, Finite Element Method. PACS: 81.20.HY

INTRODUCTION Deep drawing of sheet forming processes is most widely in the transformation of sheet metal and is used of its conversion to the hollow parts. In this process, change of thickness is very small with previous pieces, usually drawn almost level with the level of initial match sheet. The process for producing such types of metal containers, pressurized tanks, aircraft parts and tinned cans is used in figure [1]. However to achieve the desired shaping factor in this method (or the maximum amount of stretch) considering the possibility of activating any of the mechanisms of disability, including failure (Rupturing), creating throat (Necking), wrinkles (Wrinkling) and reversible High spring (Large spring back) is severely restricted. Usually, the limited maximum force deep drawing using a stretching step cannot be deepen enough for most applications achieved. But through a multi-stage deep drawing can be reduced in diameter and depth or height of the wall is increased. Two methods of multi-stage deep drawing called multi-stage direct tension and stretch is seen in the reverse in figure [2]. One common solution to increase the process of shaping metal deep drawing, stretching the standard design process has two stages, so that the second stretch in the first stage direction occurs (inverse elasticity). The process of shaping sheet with two stages of non-stop, we can towards shaping the appropriate amount (maximum amount of tensile stretching more than a simple process) is reached, figure [3]. The reverse design process for alloys with tensile strain high hardness is very complex and requires precise analytical modeling [1]. Many researchers have done Analyzing to draw a simple process using finite element method. Esche and colleagues have used numerical and experimental studies on this stretch sheet metal. They wrote a program usinf finite element, model a strain axial symmetry deep drawing of a stage model and the strains obtained from numerical methods and experimental were compared with each others and got good results [2]. Dick and his colleagues plan components are limited to an analysis of the cup deep drawing process and wrote it with the experimental results were compared [3]. Baptista and colleagues analyzed the simulated multi-stage deep drawing process with the method of IVR (Incremental Volumetric Remapping) were performed [4]. Here, simulation and analysis reverse stretching sheet steel DP600 aided conventional finite element software ABAQUS was carried out and are compared. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1043-1049 (2011); doi: 10.1063/1.3623718 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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FIGURE 1. Deep drawing process

FIGURE 2. Multi-draw and Reverse drawing

FIGURE 3. Geometry of perform the deep-drawing in two stages

MATERIALS AND METHOD Finite element model To evaluate the deep drawing process in two ways, a Simple tension and tension strain condition is reversed with the sheet, thick steel plate deformation, Width, primary and the final width Mechanical specifications. Table [1]

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shows its specifications. Defining the two models and two-dimensional finite element analysis assuming rigid structure to form, punch and plate holder, in two cases stretching process is simulated. (Here, the normal stretching process of analyzing is intended steel sheets, Thickness 1 mm and the width is 80 mm. During an initial stage of deformation plate width of 80 mm sheet width reaches 13 mm. Problem analysis of a model with two-dimensional plane strain conditions can be used. Strain is not in the direction perpendicular to the page that the correct conditional length of sheet is high in this direction. According to the symmetry of the geometry model, the middle page is only half of the model components. And punch card is an Instant at assumed rigid. Geometrical model is shown in Figure[4]. Swift model to analyze the issue of the characteristics used in Table [2].

FIGURE 4. Modeling of simple deep-drawing TABLE 1. The characteristics of used crank press (steel)DP600

E Q

TABLE 2. Swift law with isotropic hardening parameters Swift Law

210 (Gpa) 0.33

Y0

308.3

H0

0.00082

n C

0.132 720.2

B) Modeling and analysis of reverse drawing of steel sheets First Figure [5] which is modeled in ABAQUS software environment took place in the process and performance is indicated. To compare the two stretching modes and simple inverse thickness 1 mm and 80 mm width equally for the two modes is considered. As in Figure [6], it is observed in the two-dimensional model assuming rigid structure to form, punch and do maintenance and process sheet deformation is predicted in two stages. the first step by moving from top to bottom U shaped punch plate width of Width Reduced, and secondly by moving the second punch from the bottom up (L shape) of the plate width Will be reduced. Note that the inverse stretching process often form the first stage as the preservative in the second stage stretching(Blank Holder) and punch the first stage (U-shape punch) as the template in the second act, and should define the boundary conditions for each component considered above.

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FIGURE 5. Modeling of reverse deep-drawing

FIGURE 6. Mesh Sensitivity diagram

DEFINED BOUNDARY CONDITIONS AND PROCEDURE Considering the complexity of the problem in reverse stretching technique, with three independent definition included on the sheet side of the middle point, of the left half of sheets, point side of the middle right side of the left half of sheet, and the four points for mold, holding, first and second punch, shaping process follows six steps is simulated. First stage: first contacting, second stage: conservative force, the third step: started moving punch and contacting U-shape, the fourth stage: moving punch U shape, the fifth stage: the second contact and beginning the process of punch stretching. sixth stage of the reverse: moving the second punch. Here, first by two sheet stuck around the plate is gripped and then sheet is drawn into the chamber by first punch and when the first punch reaches its stroke, the second starts to move upwards during its stroke. During this process, defects may occur because of rupture of metal that are affected by factors such as geometry, material properties, friction form force and tension on the retention plate. The force applied by the sheet stuck to the sheet is a deterrent to produce preventing flow of material and produce the main deterrent force of friction. Many researchers have done research to control force of sheet stuck to improve the forming. Sheet stuck force using criteria based on the rupture plate is found by trial and error. Here the work is done by trial and error. We try to force and the 200 kilo Newton optimum force plate was achieved. Also, researchers have found that due to limitations stretch ratio increased retention and reduce the force plate can be improved during the sheet forming process. But nowadays they try to use the vibrating plate Lists. It is noteworthy that the role of process reversed tension and punch in the form of first and second stage is different form the first stage, i.e. as a preservative in the second phase as a template and punch in the second stage they play a role in defining the boundary conditions is very important.

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Friction coefficient of 0.09 is considered. Figure [6] is plotted To achieve optimum mesh pieces intended point and curve number of elements in the form of Von Mises stress. Optimal mesh number 800 is considered. Here are two stages to achieve deformation plate deformation factor is predicted. In the first stage the initial sheet width of 80 mm to 42 mm width and 42 mm width of the second stage of the width reaches 13 mm. Sheet forming process with Von Mises tension total length of the process is shown in figure [7].

First stage( first punch travel)

Second stage( first punch travel)

Fourth stage(second punch travel)

Third stage( first punch travel)

Sixth stage( second punch travel) Fifth stage(second punch travel) FIGURE 7. Von Misses stress during process

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RESULTS AND DISCUSSION Maximum amount of tension is applied to the mandrel (i.e. the reaction plate) in the two methods were compared in Figure (8). As we know, at ordinary method the amount of required force is increased whereas to reverse tension method. In the ordinary stretching method, before completing the stretching process (reaching a sheet to width of 13mm), With tensions reaching to maximum value limit of tensile strength sheets, and thus cause of activation mechanism throat, sheet after suffering nearly (19 mm)In the area stretching between the mold and punch will suffer failure. But the reverse stretch deformation method is completed and reaches without disability, within two stages stretching sheet standard. Displacement curve for plate stretching mode is in the ordinary and reverse stretching is shown in figure[9]. Punch force curve (1) for two methods of stretching and a simple inverse elasticity in Figure [10] Shows that the punch forces are required (1) in the simple tension higher than its value in the method of stretching that is reversed. Strain variation for plate stretching mode in ordinary and reverse forms of tension is shown in figure [11 and 12]. Strain rate changes on the sheet for both ordinary tension and tension in the reverse shows that in the ordinary method of stretching, maximum of plastic strain is equal to 119 Percent per maximum stress and at the latter, maximum plastic strain is equal to 46 Percent, per maximum stress.

FIGURE 8. Comparison Von Mises stress between bending and reverse bending

FIGURE 9. Comparison displacement maximum between bending and reverse bending

FIGURE 10. Compare punch force(1) between drawing and reverse drawing

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FIGURE 11. Strain simple drawing and

FIGURE 12. Strain reverse drawing

CONCLUDING REMARKS In this study, a simple drawing operation, drawing steel sheets was simulated and analyzed by finite element method. analytical Comparison between the process shows that the method of reverse traction force less punch card is required for shaping and stress caused tension is lower on the piece in reverse mode. Also if the sheet in two stages (drawing reversed) will be formed the shaping becomes better and easier and phenomena are able to being prevented such as the throat in pieces.

REFERENCES 1- Z. Keran et al, “Finite element approach to analysis of axisymmetric reverse drawing process”, ISSN 0543-5846 ,2006, 341346. 2- Sven K. Esche, Mustafa A. Ahmetoglu, Gary L. Kinzel, Taylan Altan. “Numerical and experimental investigation of redrawing of sheet metals”, Journal of Materials Processing Technology, 2000. 3- R. E. Dick, J. W. Yoon, F. Barlat, “Convolute Cut-Edge Design for an Earless Cup in cup Drawing”, NUMISHEET 2005, Eds. L.M. smith, F. Pourbograth, J.W Yoon, T.B. Stoughton, 2005, 713-718. 4- A. Baptista, J. L. Alves, M. Oliveira, D.M. Rodrigues, L. F. Menezes, “Application of the Incremental Volumetric Remapping Method in the Simulation of Multi-Step Deep Drawing Processes”, NUMISHEET 2005, Eds. L.M. smith, F. Pourbograth, J.W Yoon, T.B. Stoughton, 2005, 173-178 5- METAL FORMING HANDBOOK, Professor Taylan Altan.

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Springback Control of Sheet Metal Forming Based on High Dimension Model Representation and Genetic Algorithm Tang Longa Wang Hua Cai Yonga Mao Lichena and Li Guangyaoa* a

State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, China, *E-mail: [email protected] Abstract Springback is related to multi-factors in the process of metal forming. In order to construct an accurate metamodel between technical parameters and springback, a general set of quantitative model assessment and analysis tool, termed high dimension model representations (HDMR), is applied to building metamodel. Genetic algorithm is also integrated for optimization based on metamodel. Compared with widely used metamodeling techniques, the most remarkable advantage of this method is its capacity to dramatically reduce sampling effort for learning the input-output behavior from exponential growth to polynomial level. In this work, the blank holding forces (BHFs) and corresponding key time are design variables. The final springback is well controlled by the HDMR-based metamodeling technique. Keyword: Springback; Blank holding force; Metamodel; HDMR; Optimization PACS: 02.60.Pn

INTRODUCTION Springback caused by a combination of elastic-plastic bending and stretch deformation of the blank exists widely in sheet forming. It is very necessary to predict and reduce springback so that the final workpiece can be controlled. Currently, the finite element (FE) method is used frequently to predict springback in research and industry. However, for most engineering problems including strongly nonlinear geometric and material behaviors, the computational cost of one time FE simulation is expensive. This is particularly serious in optimization procedure where objective function is evaluated by FE simulation for many times. Development of metamodeling techniques supplies a strong tool for optimization problem based on FE simulation. Metamodeling technique is to construct a response surface approximating expensive analysis with a few samples in the design space. In optimization process, getting help from metamodel of objective function, the efficiency could be enhanced obviously. In present, there are several metamodeling technologies widely used, such as polynomial, radial basis function (RBF), support vector regression (SVR) and Kriging. Recently, Wang, G.G and Shan, S [1] reviewed the applications of metamodeling techniques in the context of engineering design and optimization. Chen et al [2] summarized the pros and cons of the design of experiment methods and approximation models. However, with the increasing of complexity and dimensionality, metamodeling techniques confront two major challenges. First, the size of sample points should be increase exponentially as the number of design variables increases. Second, it is difficult to give the explicit correlation relationships amongst design variables by popular metamodeling techniques. Therefore, a new high-dimension model representation (HDMR) is suggested in this paper. The most remarkable advantage of this method is its capacity to exploit correlation relationships among variables of the underlying function. Furthermore, HDMR can reduce corresponding computational cost from exponential growth to polynomial level. Thus, the essence of the assigned problem could be presented efficiently. In this paper, the relationship between Blank Holder Forces (BHFs) changing with time and springback will be built by HDMR. Additionally, GA is employed to search the optimal BHFs minimizing springback.

DESCRIPTION OF SPRINGBACK CONTROL PROBLEM DEFINITION OF OBJECTIVE FUNCTION

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1050-1055 (2011); doi: 10.1063/1.3623719 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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Here, m test points on the blank are employed to measure springback after forming. The objective function is based on the spatial location variation of each test point between the states before and after springback and is given out by following: 1 n (1) f d



n i1

i

Where is the displacement of the th test point after springback. The minimization of objective function (1) lets to attenuate springback.

GENETIC ALGORITHM The GA is used to optimize constructed metamodel in this work. The GA was initially proposed by J.Holland[3]. The GA is an intelligent global searching algorithm based on Darwin’s theory of evolution. The remarkable feature of the GA is to operate on individuals directly without limiting continuity and differentiability of the objective function. The optimizing process starts with a population of random individuals and searches for the optimum by updating generations. The GA can acquire searching space automatically and adjust the direction of search adaptively and has robust ability to search optimum. For these reasons, GA is applied to optimize the HDMR based metamodel.

HDMR APPROACH BASIC THEORY OF HDMR A HDMR represents the mapping between input variables x  R n and the output f (x ) . A general form of HDMR is shown as follows: n

f ( x )  f 0   f i ( xi )  i 1

f

1 i  j  n

ij

( xi , x j )   

f

i1i 2 il 1 i1  il  n

( xi1 , xi2 , xil )   f12n ( x1 , x2 ,, xn )

(2)

where (non-coupling term) gives the contribution of the is a constant; the component function variable acting independently upon the output; the component function , (the first order coupling term) describes the related contribution of the variables and upon the output. The subsequent terms reflect the effect of increasing number of variables acting together upon the output. Considering the efficiency of calculation, Cut-HDMR, which express the by a superposition of its values on lines, planes and hyper-planes passing through a cut center listed as refer to reference [4].

, is introduced. The component functions of Cut-HDMR are

HDMR-GA TECHNOLOGY With regards to most engineering problems, high order coupling terms are not sensitive to the output, so only non-coupling terms and the first order coupling terms are considered here: n (3) f ( x)  f  f (x )  f (x , x ) 0

 i 1

The procedure of the algorithm is shown by Fig.1.

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i

i



1 i  j  n

ij

i

j

Fig.1. Flowchart of HDMR-GA

BENCHMARK PROBLEM DESCRIPTION Blank Holder Forces (BHFs) is one of the most important factors in sheet forming which affect the springback significantly. There are two different methods to exert BHFs. The first way is to keep the BHFs constant during the forming. The other way is to load BHFs changing with time in several stages, which makes it feasible that material flow can be controlled more subtly. So in this work, the value of BHFs and corresponding key time are design variables and a mathematic model between them and the springback will be constructed by HDMR. In consequence, based on the mathematic model, GA can be performed to control springback effectively by proper BHFs. Here, the process of forming is divided to three phrases by time. The three values of BHFs and two key nodes of time are considered as variables and the BHFs are shown in Fig2.

F1 BHF F2 F3 t1

t2 Fig.2 BHFs 1052

time

The geometry data, material data and other related data we adopt entirely come from Numisheet2011_Benchmark3. The aim of the process parameters design is to obtain a deep-drawn product without tearing and wrinkling whose shape accuracy can be controlled in a specified range at each measuring point after springback through adjusting the BHFs. As needs to be illustrated specially, the coordinates of constrained points for springback provided by BM3 are fixed and the coordinates of FE nodes after forming are uncertain so that there is probably no node on the standard constrained points after forming. Therefore, in order to simply calculation, the nearest nodes to the standard constrained points are used as constrained points for springback. Similarly, the measuring points for springback are defined as the two end nodes nearest to the cutting line after trimming. The constraints of the design variables are shown in Table 1, and the objective function is presented as formulation (1) in which means the displacement of the ith measuring point in z direction after springback. All of the tools are modeled as grid shell elements, and quadrilateral shell elements are applied in the blank. Considering the balance of the accuracy and efficiency of the FE simulation, the size of the grid is set to 5mm. Table.1. Design variables and their constraints Design variables F1(kN) F2(kN) F3(kN) t1(ms) t2(ms)

Upper bound 1200 1200 1200 19 28

Lower bound 500 500 500 10 21

OPTIMIZATION AND RESULTS HDMR-GA is used to solve the problem described in benchmark 4.1. The time spent for simulation computation of benchmark 4.1 for once is about 0.42h. The optimization result is shown by Table.2. Table.3 and Table.4 give the location of the 20 measuring points before and after springback. In order to show the springback more visually, the location of a y-section line before and after springback is described by Fig. 3(a) and Fig. 3(b) is comparison of shape of the product before and after springback. Table.2. Optimization result Design variables

optima

F1(kN) F2(kN) F3(kN) t1(ms) t2(ms)

640.4 850.2 849.9 14.5 24.499

value of objective function 4.0335mm

Table.3. Location of the 20measuring points before springback node x y z node x 1 -94.8865 572.1666 -82.3039 11 96.9708 2 -94.8866 438.0919 -82.1962 12 96.9482 3 -94.8866 308.8994 -82.0812 13 96.9256 4 -97.9632 179.2295 -82.2166 14 91.8212 5 -112.7771 50.0234 -82.0748 15 37.7299 6 -127.5814 -79.0984 -82.1519 16 53.7272 7 -142.8585 -212.3454 -82.1210 17 97.2443 8 -120.9167 -341.1614 -82.1052 18 139.9460 9 -38.7312 -469.2339 -82.0223 19 146.8390 10 -19.8866 -600.7599 -82.1088 20 146.8161 Table.4. Location of the 20measuring points after springback node x y z node 1 -97.1016 574.2717 -80.6907 11 2 -97.5189 440.1871 -80.8133 12 3 -97.4499 310.9991 -80.8629 13 4 -100.6278 181.3457 -80.8805 14 1053

x 98.7256 98.3933 97.2278 91.3053

number of simulation 70

y 571.1995 444.7449 312.2393 178.7823 50.2014 -79.7452 -210.9158 -339.6282 -470.4977 -602.0728

z -81.9640 -81.9069 -81.8873 -82.0335 -81.9805 -81.7965 -81.6815 -80.7606 -81.2098 -81.5452

y 572.5413 443.1034 313.6212 180.1919

z -81.2091 -80.7540 -80.9042 -81.3215

5 6 7 8 9 10

-115.3400 -130.3435 -146.1281 -124.8966 -43.7075 -24.8159

52.1328 -77.0124 -210.2377 -339.4230 -468.0544 -599.7194

-80.6127 -80.0648 -79.0146 -77.5730 -76.2747 -74.9940

15 16 17 18 19 20

38.1767 54.7047 98.1386 140.7057 147.0820 147.4170

51.2057 -78.4589 -209.5896 -338.3440 -469.4388 -601.1196

-79.8083 -77.3022 -74.5141 -70.7521 -70.3150 -68.8163

40 before springback after springback 20

0

-20

-40

-60

-80

-100 -40

-20

0

20

40

60

80

100

120

140

160

Fig.3(a). Comparison of a y-section line before and after springback

Fig.3(b). Comparison of the product before and after springback

CONCLUSION In the present paper, the springback control problem of sheet forming is modeled and optimized by HDMRGA method. The most remarkable advantage of HDMR, especially for simulation based problem, is its ability of reducing corresponding computational cost from exponential growth to polynomial level when the input parameters are numerous and coupling with each other. Thus, the ultimate purpose of this work is to supply a new idea for sheet forming optimization based CAE. The accuracy of HDMR based metamodel, and the ability of searching global optima should be improved through future study.

ACKNOWLEDGMENT This work is supported by the National 973 Program of China under the grant number 2010CB328005; the Project of National Science Foundation of China (NSFC) under the grant number 10902037; the National High Technology Research and Development Program (“863” Program) of China under the grant number 2009AA044501; Program for Changjiang Scholars and Innovative Research Team in University and the State Key Laboratory of Advanced technology for vehicle design and manufacture.

REFERENCES 1. Wang, G.G; Shan, S; Review of Metamodeling Techniques in Support of Engineering Design Optimization.

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Journal of Mechanical Design. 2007.129(4):370-380 2. Chen, V.C.P; Tsui, K; Barton, R.R; Meckesheimer, M; A Review on Design, Modeling and Applications of Computer Experiments. 2006.38(4):273-291 3. J.Holland; Adaptation in Natural and Artificial Systems. 1992 4. Rabitz H; Alis OF; General foundations of high-dimensional model representations. Journal of Mathematical Chemistry. 1999.25:197-233 5. Simpson TW; Peplinski JD; Koch PN, et al. Metamodels for computer-based engineering design: survey and recommendations. Engineering with Computer. 2001.17(2):129-150 6. Cheng, L; Zhang, W; Lu, BC, et al. International Conference on Advances in Materials and Manufacturing Processes. Materials Processing Technologies.2011.154-155:902-906 7. Shan, SQ; Wang, G.G; Metamodeling for high Dimensional Simulation-Based Design Problems. Journal of Mechanical Design 2010.132: 051009-1 8. Wang, H; Li, EY; Li, GY; Adaptive MLS-HDMR metamodeling techniques for high dimensional problems. Expert Systems with Applications 2011 9. Tang, L; Li, GY; Wang, H; Kriging-HDMR metamodeling technique for nonlinear problems. Chinese Journal of Theoretical and Applied Mechanics 2011

1055

Hybrid Die Compensation Method to Accommodate Springback Error in Sheet Metal Agus D. Anggonoa, Waluyo A. Siswantob, Badrul Omarb a

Department of Mechanical Engineering, Universitas Muhammadiyah Surakarta Jl. A. Yani PO.BOX I Pabelan 57162 Sukoharjo, Indonesia

b

Faculty of Mechanical and Manufacturing, Universiti Tun Hussein Onn Malaysia Parit Raja, Batu Pahat 86400 Johor, Malaysia

Abstract. Springback phenomenon theoretically occurs for anything kind of deep drawing or sheet metal forming process. In the real manufacture, it is a well known and avoided characteristic of every process that involves plastic deformations on elastic metals. To improve the accuracy of the formed parts, the die surfaces are required to be optimized and compensate therefore after springback the geometry falls at the expected shape. The proposed procedure combined two methods of die compensation; Displacement Adjustment Method (DAM) and Spring-forward method (SFM) of Karafillis and Boyce (K&B) called Hybrid Method (HM). Both are based on iteratively comparing the deformed shape with the target. According to the advantages of each method DA and K&B to solve the springback compensation problem, these methods are applied alternately. This paper presents and discusses the proposed framework in accommodating springback in sheet metal forming. Keywords: Springback, Hybrid Method, Die Compensation, Optimization. PACS: 82.20.Wt

INTRODUCTION The role of modeling and simulation in engineering and in manufacturing industry has been continuously increasing. Due to the recent developments, both in software and hardware, modeling and simulation have become an everyday tool in engineering practice, especially in sheet metal forming. Sheet metal forming is a widely used mass production process, e.g. in the packaging industry, airplane parts and automotive industry. Springback occurrence theoretically exists in whatever kind of sheet metal forming or stamping process. That is a well known characteristic of every process that involves plastic deformation. Elastic recovery of blank material which happens during unloading can cause dimension deviation. Many efforts have been done to eliminate the springback. Most of them focus on optimization and configuration of draw-beads, radius of curvature, draw-beads forces and blank holder force [1,2,3]. The other way is reducing springback effect by modifying the tooling surfaces at geometry to compensate springback [4,5]. Generally, there are two methods for surface modifying to compensate springback, the Displacement Adjustment method (DAM) and the Spring-Forward methods (SFM) [6]. The DAM is a strictly geometrical method and very simple way, to move the surface nodes defining the die surface in the direction opposite to the springback error. The displacement vectors at each node are used to adjust the trial die design until the target part shape is achieved. The DAM has been proven to be the most successful method but one of the problems of this DAM is that it could not converge to the curved features in the side-wall area of dies. The SFM has a more physical approach, based on the internal stresses that cause springback and computing the constraint forces to maintain equilibrium following forming. Gan and Wagoner [7] implemented the DAM and SFM to compensate die tool successfully. DAM converges faster than SF but SF iterations show steadily decreasing error. In both DAM and SFM, the iterations are successfully applied to U-channel bending. The die surface shapes produced by the two methods are similar, and resulting from springback shapes are nearly identical. Siswanto and Omar [8] successfully apply SFM to compensate surface tool and develop “Automatic Mesh Generator” to generate new compensated mesh. Most of successfully method is applied to simple model, that is, if a method is successful and efficient for simple geometries, it should have a good chance being capable of solving for complex geometries. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1056-1063 (2011); doi: 10.1063/1.3623720 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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In this work, the combination of DAM and SFM algorithm is proposed to be coupled to solve a difficult sheet metal forming processes that involve of springback error compensation. THE HYBRID COMPENSATION METHOD Hybrid method is the combination between DAM and SFM. Hybrid method as a model for a forming process, a horizontal part is bent downwards plastically. The compensation method of DA is directly based on the spring-back displacement, due to the occurrence of primarily elastic recovery of the part. Simple equation of forming process could be written as following:

Pf  Ps  Pp  Pu

0

(1)

where Pf is loaded blank sheet, Pe is the elastic recovery of the part and Pu is the unloaded sheet part. Pp connected to value that unloading is not purely linear behavior as known Bauschinger effect. Bauschinger effect is a phenomenon when metal is stretched in tension and unload, the unloading is not linear. If Pd is the desired part and the tolerance of springback is , the compensation is to obtain a Pu close to the desired shape then the relationship is defined as

| Pf  Ps  Pd |d G

(2)

Regard the reference or desired surface geometry R, given as a collection of n points in R3 and the springback geometry S.

R {rj | rj  R 3 },0  j  n

(3)

{S j | S j  R },0  j  n

(4)

S

3

The compensated geometry surface C is now calculated as follows:

C

R  a( S  R) ,

(5)

where a is the overbending factor. a(S – R) is called the shape modification field . The iterative application of DAM is conducted until the springback shape within the desired target. The first compensated surface C is now referred to as C1, and with this new surface a FE simulation is carried out. The resulting springback mesh S1 is now used to modify surface C1, delivering the second compensated geometry C2. The iterative formulation of equation (3) in iteration i is:

C i 1

R i  a( S i  R)

(6)

By applied iterative to the formula, the compensation process does not need to guess an overbending factor because the tool geometry converges to the optimal shape iteratively. The factor a is still present in the formula to gain control over the amount of compensation that is applied per iteration. A value of 1.0 is recommended. The compensation method of SF applied inverse residual stresses that are recorded during unloading to the formed part [9]. The shape of the blank in the FE model is described by the displacement vector ul1, where the material stress-strain response causes the development of internal force vector Il1. The tooling is set identical to the desired shape in order to obtain ul1, so ul1= udes. Springback occur upon removal all contact forces Fl1 = Il1, and the new displacement vector uul1 of blank is:

U ul1 (U ul1  I ul1 ) U ul  U sb1 (U ul1 , I l1 )

(7)

where usb1 is the displacement increment vector during springback. Also springback can be considered as the additional deformation described by usb1 which occur due to the application of –Fl1 on blank in the fully loaded state and configuration resulting in a zero net load applied to blank.

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As shown in FIGURE 1, HM is the combination between DAM and SFM become an integrated algorithm. However, for all method, the first step of deformation a flat sheet to the target is the same. A flat sheet is stretched to the target, and the blank is allowed the spring back and then delivering the springback mesh. Both original mesh and formed mesh are compared. The displacement of the node in the blank during springback can be calculated directly and the error (y ) is used to compensate die shape. The second step in the HM is using the algorithm of spring forward which applied the internal stress to the formed part. A new trial die surface is obtained then used to form blank sheet and internal forces are recorded by FE during simulation. A new formed shape is then checked again, and iteration will continue until the target shape is attained within a desired part.

FIGURE 1. Illustration of Hybrid Method

MESH GENERATOR After the springback can be predicted accurately then the results used to modify the previous tooling surface to produce a target part shape. Punch and die surface must be generated with an accurate and constant gap. Tool surface created by using the translation process is to compensate a surface with the normal direction from the surface. Generating upper die surface based on the node surrounded by four elements translated perpendicular to the elements as illustrated in FIGURE 2. The calculation of the node translation vector NA, which move the node A to A’, is based on its normal translation vectors (N1, N2, N3, and N4) from surrounding (dependent) elements. The opposite directions of vectors (N1, N2, N3, and N4) are used in case of generating lower die surface.

FIGURE 2. Node normal translation from around four elements

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As all normal translation vectors have the same magnitude (half of the part thickness), the direction of vector TA (represented with its unit vector ݁ோಲ ) is the same with the resultant of normal translation vectors T1, T2, T3 and T4. The direction of vector TA is shown as:

eTA

RA | RA |

(8)

N1  N 2  N 3  ...N n

(9)

where

RA

Res ( N i )

The subscript n in Eq. (9) denotes the number of the contributing normal vectors to the vector NA. The location of the vector NA can be stated by direction cosines angle .

D

cos 1 (eN i .eRA )

(10)

where eNi is unit vector of Ni The translator vector NA of node A can be obtained

NA

| Ti | ER cos D A

(11)

In order to create a constant clearance between the part and the tool surface, all normal vectors are set to a constant t. (12) | N1 |  | N 2 |  | N 3 |  | N 4 |  | N i | t The translation Eq. (11) can be generalized for the whole of nodes,

( N A )i

t (e R A ) i cos D

(13)

where i represents the node number from the part after springback or springforward. This equation is only dependent to the surrounding elements. In a global coordinate system, the location of the new generated node i, A’ is obtained by a general vector addition of vector A and vector NA. (14) ( A' ) i ( A) i  ( N A ) i Since the normal translation vectors at a node rely on its surrounding elements, each node must recognize its surrounding (dependent) elements. THE SIMPLE ONE-ELEMEN TEST In order to test and to investigate the usefulness of the proposed Hybrid Method, a two dimensional shell part was conducted to analyze. A one rectangular shell element was used in this test as illustrated in FIGURE 3. and the finite element software in this test is IMPACT dynamic finite element program as processor. This software is selected because of the freeware, availability and open-source software. The pre and post processor is used GiD software. Two nodes (1 and 2) are suppressed in both x and y-directions, and the others (3 and 4) are pulled in the xdirection. The blank sheet material in this test is HX260LAD micro-alloyed steel grade with high yield strength for cold forming according to DIN EN 10346 / DIN EN 10143 and the properties can be seen in TABLE 1. 10 mm

5 mm

1

3

2

4

V

V

FIGURE 3. Model setup for one-element analysis.

Properties Value

TABLE 1. Material properties of HX260LAD. Young’s modulus Poisson ratio Yield stress (MPa) (MPa) 210 0.3 176

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Thickness (mm) 1.0

Simulations with three different methods were performed, i.e. displacement adjustment, spring-forward and hybrid method. The material is stretched in the x-direction until reach 13 mm total lengths as a target in this forming. By using velocity of 0.1 m/s the time requirement is 130 ms. Velocity is then set over time as [0,0.1,130,0.1,130.1,OFF,132,OFF] that mean velocity set 0.1 at time 0 until at time 130 and OFF at time 130.1. The value changes are linear and the condition is deactivated by using OFF. Points 3 and 4 displacements are recorded during loading and unloading. The points were not matched with the target dimension and giving deviation value (x) 0.2046 mm, therefore, the next displacement should be 13.2046 mm. After unloading at the second stretch is a gift the x-dimension of 12.99803 mm or giving error 0.00197 mm. Then the stretch is continued to the third with the total length of 13.20657 mm, and the final length is 13.009452 mm as shown in FIGURE 4. The deviation is bigger than the second step so the iteration should be stopped at step two. When SFM is applied to the same problem, reduced error is achieved in same 3 cycles but getting bigger error value 0.0174247 mm than DAM. SFM simulation continued to step 4 by applying 0.2279354 kN force, but did not reduce error to an acceptable value. HM method minimized error closely to DAM at the step 3 of simulation and the error value of -0.0057352 mm is smaller than DAM. The complete result's value could be seen in TABLE 2.

Deviation,mm

0.25 0.2

DAM

0.15

SFM

0.1

HM

0.05 0 0.05 1

2

3

4

0.1 0.15 0.2 0.25

LoadingStep

FIGURE 4. Dimensional errors of DAM, SFM and HM compared TABLE 2. Deviation values data of DAM, SFM and HM for one element simulation. Step Deviation (mm) DAM SFM HM 1 0.2046307 0.2046307 0.2046307 2 0.0019652 0.0275207 0.0019652 3 -0.0099452 0.0174246 -0.0057352 4 -0.2105108 COMPENSATING THE NUMISHEET PRODUCT The successful implementation of HM on the simple one-element model encouraged attempts to another model of sheet metal forming process, the benchmark problem of Numisheet ’93 2D model. This problem was selected because of its availability in the assembly which is consisting of a punch, holder, die and blank sheet. In this analysis, the main focus observation is focused on the blank sheet hence the blank is designed as a deformable part and the others are a contact elements only as shown in FIGURE 5.(b). The ideal springforward shape for single element can be calculated in an analytical or numerical manner. By applying all the boundary conditions and stress – strain state, springforward is able to calculate because the biggest problem in HM is springforward calculation. The deformable blank part is meshed become 800 elements as a half side. Punch stroke maximum is about 76 mm in –y-direction from the blank position, punch velocity is set to 4 m/s from time 0 – 35 ms. At the punch position of 76 mm velocity became 0, and then the fix boundary condition is applied to all nodes in the mirror line of blank or the center position until a time of 35.1 ms. During this loading, the

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internal force is recorded and to show the springback effect. All loads are released by using OFF at the boundary conditions. The friction coefficient is taken 0.3 as reported for the contact between the punch and blank sheets and also between holder and blank. The punch position at the maximum stroke or loading step is shown in FIGURE 6. Due to the highly nonlinear nature of the springback problem, it is difficult to evaluate the convergence rate of a compensation method on arbitrary part geometry. Punch Holder

Blank

Die

(a)

(b)

FIGURE 5. (a). Geometry of 2D assembly model in mm, (b). Meshing all parts

(a)

(b)

FIGURE 6. Displacements of punch and blank. (a). Total displacements. (b). y-direction of displacements.

The actual finite element mesh is refined five times to get the smooth results. Punch and blank are symmetric with respect to the YZ plane. There are four steps in the forming simulation. Step 1, the movement is initiated by moving the holder to the blank. Step 2, the punch moves down to deform the blank until a maximum deformation for about 76 mm. Step 3, the clamps are added to the symmetric edge of blank. Step 4, all the contacts are removed and allow the blank to springback due to plasticity. As shown in FIGURE 7. (a), to know the movements of blank holder and punch, two points are subjected to be observed by using tracking point during the forming process. Binder pressed the blank sheet and the punch moved down to bend the blank, after that both binder and punch are released at the same time. Springback phenomenon could be seen during the unloading process, and this step is described by plotting of point movement in the blank as shown in FIGURE 7. (b).

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FIGURE 8. is the result of track points in the die compensation by using the hybrid method. In the first iteration step is applying the displacement adjustment algorithm. The second iteration, by applying springforward algorithm, the sheet was steadily moved to the target surface, and the third iteration is use displacement adjustment again, the shape of bending region was closed to the desired target. 250

Binder

200

Y, mm

150

Punch

Unloading

100 50 0 50

0

10

20

100

30

40

50

Time, ms

(a) 1 0 1

0

10

20

30

40

50

Node 10

Y , mm

2

10 per. Mov. Avg. (Node 10)

3

Springback

4 5 6 7

Time, ms

(b) FIGURE 7. (a) Point travel position at the binder and punch during forming. (b) Movement point of blank sheet show the springback error.

Target Iteration 2 Iteration 3 Iteration 1

FIGURE 8. Track of points in the die compensation of Hybrid Method.

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CONCLUSION The underlying scope of the present work has been to outline finite element technique for predicting springback deformation compensation in sheet metal forming. A method for the die compensation of sheet metal forming dies due to springback has been successfully developed. The springback method or displacement adjustment method (DAM) is an iterative technique based on comparing a target part shape with a formed and unloaded part shape simulated using finite elements. In contrast, the spring-forward method (SFM) in the sheet metal forming applied the inverted stress which using the FE simulated forces for a similar iterative procedure. In this proposed framework, it combined these two methods which have been successful in simulate die compensation in simple one element model and 2D forming model. ACKNOWLEDGMENTS The authors would like to thanks to Universiti Tun Hussein Onn Malaysia (UTHM). This work has been conducted under Advance Dynamic and Vehicle Safety (AdVeS) research group. Part of the work has been supported by Fundamental Research Grant Scheme (FRGS) Vot No.0557 and Vot. No. 0748. REFERENCES 1. W.A. Siswanto, B. Omar, A.D. Anggono, A.A. Mathew, Proceeding of National Conference on Advanced Manufacturing and Materials Engineering 2010, UTHM, Johor, Malaysia, May 2010, pp. 197-203. 2. S. Yoshihara, K. Manabe, H. Nishimura, Journal of Materials Processing Technology, 170, pp 579-585, June 2005. 3. L. Maretta, G. Ingarao, R. Di Lorenzo, International Journal of Mechanical Sciences, 52, pp 914-927, March 2010. 4. A.P. Karafillis, M.C. Boyce, Journal of Materials Process Technology, 32, pp. 499-508, 1992. 5. T. Mainders, I.A. Burchitz, M.H.A. Bonte, R.A. Lingbeek, International Journal of Machine Tools and Manufacture, 48, pp. 499-514, 2008. 6. R.Lingbeek, J.Huetink, S.Ohnimus, M.Petzoldt, J.Weiher, Journal of Processing Technology, 169, pp. 115-125, April 2005. 7. W.Gan, R.H. Wagoner, International Journal of Mechanical Sciences, 46, pp. 1097-1113, June 2004. 8. W. A. Siswanto, B. Omar, International Journal of Material Forming, 2, pp 797-800, April 2009. 9. A.P. Karafillis, M.C. Boyce, International Journal of Machine Tools and Manufacture, 36, pp. 503-526, 1996.

1063

A COMPREHENISVE ANALYSIS OF BENCHMARK 4: PRE-STRAIN EFFECT ON SPRINGBACK OF 2D DRAW BENDING

P.-A. Eggertsena,b, Kjell Mattiassona,b, Mats Larssonc,d a

Div. of Material and Computational Mechanics, Dept. of Applied Mechanics Chalmers University of Technology, SE-412 96 Göteborg, Sweden b Volvo Cars Safety Centre, SE-405 31 Göteborg, Sweden c Dept Press, C15-1 TMNKM, Saab Automobile, SE-461 80 Trollhättan, Sweden d Department of Engineering Science, University West, SE-461 86 Trollhättan, Sweden

Abstract. In order to be able to form high strength steels with low ductility, multi-step forming processes are becoming more common. Benchmark 4 of the NUMISHEET 2011 conference is an attempt to imitate such a process. A DP780 steel sheet with 1.4 mm thickness is considered. In order to understand the pre-strain effect on subsequent forming and springback, a 2D draw-bending is considered. Two cases are studied: one without prestrain and one with 8% pre-stretching. The draw-bending model is identical to the “U-bend” problem of the NUMISHEET’93 conference. The purpose of the benchmark problem is to evaluate the capability of modern FEmethods to simulate the forming and springback of these kinds of problems. The authors of this article have previously made exhaustive studies on material modeling in applications to sheet metal forming and springback problems, [1],[2],[3]. Models for kinematic hardening, anisotropic yield conditions, and elastic stiffness reduction have been investigated. Also procedures for material characterization have been studied. The material model that mainly has been used in the current study is based on the Banabic BBC2005 yield criterion, and a modified version of the Yoshida-Uemori model for cyclic hardening. This model, like a number of other models, has been implemented as User Subroutines in LS-DYNA. The effects of various aspects of material modeling will be demonstrated in connection to the current benchmark problems. The provided material data for the current benchmark problem are not complete in all respects. In order to be able to perform the current simulations, the authors have been forced to introduce a few additional assumptions. The effects of these assumptions will also be discussed.

Keywords: Multi-stage forming, springback, material modeling, pre-straining PACS: Replace this text with PACS numbers; choose from this list: http://www.aip.org/pacs/index.html

INTRODUCTION Multi-step forming of sheet metal materials is a kind of process that is becoming more and more common in the industry. In terms of numerical simulations, very little has been done in this field. Benchmark 4 of the NUMISHEET 2011 conference is an attempt to imitate such a process. In order to understand the pre-strain effect on subsequent forming and springback, a 2D draw-bending is considered. Two cases are studied: one without pre-strain and one with 8% pre-stretching before the draw-bending process. The draw-bending model is identical to the “U-bend” problem of the NUMISHEET’93 conference. The purpose of the benchmark problem is to evaluate the capability of modern FE-methods to simulate the forming and springback of these kinds of problems. The current paper starts with a brief description of the material models used in the present study. The following section describes the material parameter identification processes, and in the final section the analyses of the drawbending problem are accounted for.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1064-1071 (2011); doi: 10.1063/1.3623721 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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MATERIAL MODELING The authors of the present paper have earlier shown the importance of the modeling of the material behavior for successful springback predictions [1],[2],[3]. In the previous work it is demonstrated that there are three material model constituents, which are important for a proper description of the material behavior. These are: the yield surface, the hardening law, and, finally, the modeling of the elastic stiffness degradation with plastic strain. Below follows a brief description on how these constituents are chosen for the present problem.

Yield Surface To accurately account for the anisotropic behavior, the eight parameter yield criterion by Banabic and Aretz [6] is chosen: M M M ª1 σ = « Γ + Ψ + Γ − Ψ + 2Λ ¬2

(

1

·º M ¸» ¹¼

(1)

The functions Γ, Ψ and Λ are defined as Lσ xx + Kσ yy

Γ=

2

Ψ=

Λ=

( Nσ

xx

− Pσ yy )

2

+ Q 2σ xyσ xy

4

( Rσ

xx

− Sσ yy ) 4

(2)

2

+ T 2σ xyσ xy

where the parameters L,K,N,P,Q,R,S and T are identified from uniaxial- and equibiaxial-test data.

Kinematic Hardening Description To account for the material behavior at cyclic loading, two different kinematic hardening laws are considered: the classical hardening law by Arnstrong and Frederick (A-F) [7] and a modified version of the Yoshida-Uemori (YU) [8] hardening law. The back-stress evolution of the A-F law is described by:

ı −Į ª º − Į » ε p Į = C x «α sat σ ¬ ¼

(3)

where Cx and Įsat are material parameters. The A-F hardening law is able to account for the Bauschinger effect and the transient behavior during reverse loading. However, the important permanent softening effect (see Refs. [1][3][8]) is not accounted for. To also consider the permanent softening effect, a modified version of Y-U hardening law is also used in this study. In this modified version of the Y-U model, the plastic hardening curve is provided as input data, instead of being a part of the parameter identification procedure as in the original formulation. This also results in that the number of unknown material parameters is reduced from seven to four. Below follows a brief description of this modified version. For a full description the reader is referred to earlier work by the authors [1], [3]. The Y-U model includes both translation and expansion of the bounding surface, while the active yield surface only evolves kinematically. The evolution of the back-stress is expressed as:

Į = Į * + ȕ with

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(4)

§R · Į * = C x ⋅ ¨ ⋅ ( ı - Į ) − Į* ¸ ⋅ ε p ©Y ¹ b § · ȕ = k ⋅ ¨ ⋅ ( ı - ȕ ) − ȕ ¸ ⋅ ε p ©Y + R ¹

(5)

where Į* is the relative kinematic motion of the yield surface with respect to the bounding surface, ȕ is the centre of the bounding surface, Y is the size of the yield surface, Cx, b and k are material parameters, and R is the isotropic hardening of the bounding surface. In the current modified version, the isotropic hardening is described as:

R (ε p ) = H (ε p ) − β (ε p )

(6)

where H (ε p ) is the given plastic hardening curve, and β (ε p ) is the uniaxial backstress of the bounding surface. Besides the three already mentioned material parameters, the Y-U model also includes a fourth material parameter, h. This parameter affects the amount of cyclic hardening. The larger value of the parameter h the smaller the prediction of cyclic hardening.

Elastic stiffness The amount of springback during unloading depends to a great extent on the elastic stiffness of the material. It has been shown that during plastic deformation, the elastic stiffness decreases [1][3][9]. In this work, an analytical expression proposed by Yoshida et al. [8], was used to describe the elastic stiffness degradation with plastic strain:

(

Eu = E0 − ( E0 − Esat ) ⋅ 1 − e −ξ ⋅ε

p

)

(7)

where E0 is the initial Young’s modulus, Esat is a value that the unloading modulus saturates towards, and ȟ is a material parameter.

MATERIAL CHARACTERIZATION The material used in this study is a DP780 material with a thickness of 1.4 mm. The material characterization is based on a number of experiments. Below follows a brief description of these experiments, and on how the various material parameters are determined from these experiments.

Uniaxial Tension Tests Tension tests were performed in the rolling, transverse and diagonal directions. The basic experimental data from the tension test in the rolling direction are presented in TABLE 1. The uniaxial tension tests also form the basis for description of the anisotropic behavior of the material. Unfortunately, only the material coefficients for the Barlat Yld2000 and the Hill’48 yield criteria are provided in the benchmark material data sheet. However, the Banabic BBC2005 model (used in our simulations) and the YLD2000 model are identical, although their coefficients have different meanings. It has therefore been possible to transform the YLD2000 coefficients to BBC2005 coefficients. The recalculated coefficients for the BBC205 criterion are presented in TABLE 2. The exponent M (a in Yld2000) has the same meaning for both criteria and is set to 6. TABLE 1. Uniaxial tension test data

Direction Rolling direction

E [GPa] 198.8

YS [MPa] 527.0

UTS [MPa] 831.5

R-value 0.781

Poisson’s ratio 0.3 (assumed)

TABLE 2. Material constants for the Banabic 8-parameter (BBC2005) yield criterion

Sample DP780

L 1.029

K 0.925

N 1.043

P 1.062

Q 1.032

R 0.956

S 0.995

1066

T 1.004

M 6

The uniaxial tension test in the rolling direction also forms the basis for the plastic hardening curve. However, due to the occurrence of diffuse necking, plastic hardening data is only provided up to 10% plastic strain. The maximum plastic strain level that can be anticipated in the springback examples is much higher than that. This means that the plastic hardening curve has to be extended up to higher values of effective plastic strain by extrapolation. This introduces a significant source of error in the simulations. It would have been desirable, if the experimental hardening curve was extended by means of data from a bulge test or a shear test. In our simulations the hardening curve is extended by a simple extrapolation of Hollomon type: H (ε p ) = K ⋅ ε p n

(8)

where H is the effective stress, and K and n are material parameters. Both these parameters are determined based on the slope at the end of the experimental tension test curve, such that K = 122.4 and n = 0.125. An illustration of the extrapolation is given in FIGURE 1a. The resulting hardening curve can be seen in FIGURE 1b, where the first part of the curve is the tension test data and the second part is based on the extrapolation described above. In the numerical simulations this hardening curve is given directly in the input data.

(a) FIGURE 1. (a) Extrapolation of the isotropic hardening curve. (b) Resulting isotropic hardening curve with one part from uniaxial tension test in the rolling direction and one part from Hollomon extrapolation.

In-Plane Tension-Compression-Tension Tests In order to characterize the cyclic behavior of the material, in-plane tension-compression-tension tests were performed. The test specimens were prepared in the rolling direction, using the testing apparatus described in Kuwabara et al. [4], for different amounts of pre-strains: 0.02, 0.04, 0.06, 0.08 and 0.1. The identification procedure is performed with the optimization code LS-OPT, such that a stress-strain relationship from the constitutive driver becomes as close to the experimental values as possible. In the optimization procedure we have chosen to use two of the experimental stress-strain curves as target curves: the curves corresponding to 0.04 and 0.1 plastic pre-strain. The reason for using two target curves is to assure that the obtained parameter set-up covers a wide range of plastic strain levels. By using only one target curve, there is a risk that the optimal solution only is suitable for that particular strain level. A detailed discussion on this matter can be found in an earlier work by the authors [5]. The obtained material parameters are provided in TABLE 3. In FIGURE 2 the performance of the kinematic hardening models with the obtained material parameters is shown. TABLE 3. Kinematic hardening parameters for the two considered hardening laws

Hardening law A-F Y-U

Cx 37.6 137.9

Įsat 327.5 -

b 165.7

k 29.1

h 0.99

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(b)

(a) FIGURE 2. Verification of predicted uniaxial tension-compression-tension data for: (a) the modified Yoshida-Uemori kinematic hardening model and, (b) the Armstrong-Frederick hardening law, with material parameters according to TABLE 3. The red curves represent predicted values and the blue curves are experimental data.

Elastic Stiffness Degradation The stiffness degradation as a function of plastic strain is measured from stress-strain relationships at five different levels of pre-straining. Based on this data, the material coefficients in Eq. 7 are determined by a least square fitting of the analytical expression to the experimental data points. In TABLE 4 the obtained parameters are listed. An illustration of the obtained analytical curve can be seen in FIGURE 3. In the figure the experimental data points are illustrated as red “stars”. TABLE 4. Material constants describing the elastic stiffness degradation according to Eq. 7

Sample DP780

E0 [GPa] 198.8

Esat [GPa] 167.0

ȟ 87.0

FIGURE 3. Unloading modulus curve according to Eq. 7 with parameters presented in TABLE 4. The stars represents experimental data.

SPRINGBACK The aim of this paper is to discuss benchmark problem 4 of this conference. The benchmark considers 2D-draw bending of the DP780-steel described earlier in this paper. Two different cases are compared: in the first one the virgin base material is used in the draw bending process, and in the second case the material is pre-tensioned up to 8% engineering strain. An illustration of the experimental set-up is given in FIGURE 4a, and the dimensions of the tools are provided in TABLE 5. The applied blank holding force is 2.94 kN. In FIGURE 6 and TABLE 7 the dimensions of the specimens used in the draw-bending example are defined.

Numerical Simulations The numerical simulations were performed by means of the LS-DYNA code. For the forming simulations an explicit time integration scheme was used. The sheet material was modeled by fully integrated quadratic shell

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(b)

elements. The forming tools were modeled by shell elements and were assumed to be completely rigid. The friction coefficient between all contact surfaces was assumed to follow the simple Coulomb frictional law with a constant friction coefficient equal to 0.1. The importance of the mesh size in springback simulations has been demonstrated in previous investigations. In this study a mesh convergence analysis was carried out. The result from the mesh sensitivity study is shown in FIGURE 5 and TABLE 6. As can be seen, the mesh that is called “mesh 3”, 0.469x0.469 mm, seems to be the best choice. Therefore, this mesh density is used for both the considered cases. The pre-straining of the material is modeled such that the nodes in one end of the material are locked in the length direction, while the nodes in the other end are given a prescribed displacement, corresponding to an axial strain level of 8% engineering strain. After the pre-straining, a springback simulation is performed so that the specimen becomes stress-free. The pre-strained specimen is then used in a forming operation, and a subsequent springback simulation. For the springback simulations implicit time integration is used. The unloading method for the springback simulation is the so-called “one step unloading” method, which means that all the forming tools are removed before the springback calculation takes place.

(a)

(b)

FIGURE 4. (a) Schematic view of the tools of the 2-D draw bending test. (b) Definition of the various springback measures. TABLE 5. Dimensions for the 2D draw-bending test

Parameters Dimensions

W1 50.0

W2 54.0

W3 89.0

W4 89.0

R1 5.0

R2 7.0

G1 2.0

Stroke 71.8

(a) FIGURE 5: Illustration of the mesh sensitivity. (a) springback profile, (b) zoom-in of the specimen tips in figure (a).

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(b)

TABLE 6: Mesh-sizes in the mesh sensitivity study (FIGURE 5)

Mesh Mesh 1 Mesh 2 Mesh 3 Mesh 4

Size [mm x mm] 1.875 x 1.875 0.938 x 0.938 0.469 x 0.469 0.234 x 0.234

FIGURE 6. Schematic view of specimens and their dimensions for: (a) the base material case (without pre-strain), (b) the prestrained case. TABLE 7. Dimensions of the specimens in FIGURE 6.

Parameters Dimensions

Base material L W 360.0 30.0

Pre-strained specimen (8% engineering strain) L0 W0 Lgrip Grip displacement L 360.0 30.0 25.0 24.3 324.0

The springback profiles for the two considered kinematic hardening laws are displayed in FIGURE 7. FIGURE 7a shows the results for the base material, while in FIGURE 7b the corresponding results for the pre-strained material are displayed. Obviously the A-F hardening model yields more springback than the Y-U model. The reason for this can be deduced from the determination of the kinematic hardening parameters, and from the stress-strain relationship for the A-F hardening law at high cyclic strain levels (see FIGURE 2b). As can be seen in FIGURE 2b the predicted stresses at reverse loading are significantly higher than the experimental ones. This means that the numerical simulations of the draw-bend test, using the A-F hardening law, will also over-predict the stresses in the side-wall of the formed part. Therefore, also the bending moment will be over-predicted, and as a consequence of this also the resulting springback. Comparing FIGURE 7a and FIGURE 7b, we note that the difference between the results for the two considered hardening laws is larger for the base material case than for the pre-strained case

(a) FIGURE 7. Springback profiles for (a) the base material, (b) the pre-strained material.

Since the Y-U hardening model fits experimental data best, it can be anticipated to result in the most accurate springback predictions. The resulting springback profiles for the Y-U model applied to the base material and to the pre-strained material, respectively, are shown in FIGURE 8a. It can be observed that the pre-strained material results in considerably more springback. The reason for this can be seen in FIGURE 8b. The figure shows the through

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(b)

thickness stress distribution at a point in the middle of the side wall, after the forming phase, and before the springback phase. As can be seen the stresses are much higher for the pre-strained case, which in turn results in higher side-wall curl. The reason for the higher stresses in the pre-strained case can easily be understood, if one considers the pre-strained material just as a material with a higher hardening curve than the base material. In TABLE 8 the values of the springback measurements illustrated in FIGURE 4b are shown for both the base material and for the pre-strained material, for the two considered kinematic hardening laws.

(a)

(b)

FIGURE 8. Y-U hardening model. (a) Springback profiles for the base material and the pre-strained material. (b) Longitudinal stress distribution through the thickness, for the base material, and for the pre-strained material, in a point in the middle of the sidewall. TABLE 8. Springback measurements for all considered cases

Hardening law Y-U A-F

Without pre-strain ș1 ș2 ȡ 97.2 81.5 655 99.4 80.0 542

With pre-strain ș1 ș2 ȡ 105.5 79.5 525 107.2 78.0 466

CONCLUSIONS Benchmark 4 of the NUMISHEET 2011 conference has been studied. It has been shown that the springback deformation after a draw-bend operation, such as the one presented in the benchmark, is very sensitive to the mesh size and the choice of kinematic hardening model. In this paper both the A-F hardening law and the Y-U hardening law has been used. Since the Y-U hardening model fits experimental cyclic stress-strain data best, it can be anticipated to yield the most accurate springback predictions. When it comes to the effect of pre-straining of the material before the draw-bend operation, it is shown that the resulting springback for the pre-strained case becomes higher than for the base material case. This is a result of the prediction of higher stresses in the side-wall after the forming stage, resulting in larger side-wall curl, for this case. It should finally once again be pointed out that the numerical results obtained in this study are highly dependent on the appearance of the hardening curve, which we, unfortunately, to a great extent have had to guess. The errors originating from this assumption may perhaps be of the same order as the differences in results obtained with the two kinematic hardening laws.

REFERENCES 1. P.A. Eggertsen and K. Mattiasson, Int. J. of Mech. Sci. 51, 547-563 (2009). 2. P.A. Eggertsen and K. Mattiasson, Int. J. of Mech. Sci. 52, 804-818 (2010). 3. P.A Eggertsen, "Prediction of springback in sheet metal forming – with emphasis on material modeling", licentiate Thesis, Chalmers University of Technology, 2009. 4. Kuwabara et al, Int. J. of Plast.. 25, 1759-1776 (2009). 5. P.A. Eggertsen and K. Mattiasson, Int. J. of Mtrl. Forming (2011). 6. D. Banabic et al, Int. J. of Plast, 21, 493-512 (2005). 7. P.J. Armstrong and C.O. Frederick, G.E.G.B report RD/B/N 731 (1966). 8. F. Yoshida and T. Uemori, Int. J. of Plast, 18, 661-686 (2002).

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Evaluation of Surface Deflection in Automobile Exterior Panel by Curvature Based Method W. J. Chunga, W. S. Kima, J. H. Kima, J. H. Seob and T. C. Jungb a

Dept. of Product Design and manufacturing, Seoul National Univ. of Science and Technology, 172, Gongneung-Dong, Nowon-Gu, Seoul, 139-743, KOREA b Advanced Die & Mold manufacturing Eng. Team, Hyundai Motor Co., 772-1, Jangduk-Dong, Hwaseong-Si, Gyeonggi-Do, 445-706, KOREA

Abstract. Surface deflections in exterior panel have a great effect on the external appearance of automobiles. Usually, they are occurred on large flat panels containing sudden shape changes and shows very small amplitude. Traditional methods for detecting surface deflections on stamped automobile panels are scratching the surface with a wheatstone with oil, rubbing the surface with technician’s fingers, or reflecting light on the surface. All of these methods rely on a technician’s sense. In this study, a curvature based method was proposed to detect and quantify the surface deflection. The measure of surface deflection was defined as the variation of curvature difference between tool and blank. The degree of surface deflection can be predicted by the correlation between the proposed measure and visual inspection. Today’s simulation code has enough accuracy to predict surface deflection by using the proposed method. The appropriate simulation method to find the surface deflections accurately were studied. As a verification example, a shallow rectangular drawing with rectangular embossing was studied. The corresponding experiment was carried out. The simulation and experiment were compared and investigated to prove the effectiveness of the proposed method and the effectiveness of prediction. Keywords: automobile panel, surface deflection, curvature, visualization of surface deflection PACS: 02.70.Dh

1. INTRODUCTION Recently, the importance of cosmetic appearance of automobiles is rapidly increasing due to innovative exterior design. Surface deflection is one of the major defects in automobile’s appearance. These defects mainly appear in area where sudden shape changes occur on the large flat exterior panels such as the corner area around embossing. Not only measuring but correcting surface deflection is very difficult because of their very small amplitude about r30~300Pm. Fig. 1 shows an example of surface deflections around the handle portion of the exterior door panel. From the figure, the surface deflection means the distance (Fig.1 (b)) between the ideal curve and the section curve with a typical length being from several decades of mm to several hundreds of mm. The ideal curve is the virtual curve assuming no surface deflections and the section curve is the curve obtained from the panel.

D

D

(a) Experimental panel containing surface deflections (b) Definition of surface deflection FIGURE 1. An example of surface deflections around the handle portion of the exterior door panel.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1072-1077 (2011); doi: 10.1063/1.3623722 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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Traditional methods for detecting surface deflections on stamped automobile panels are scratching the surface with an oiled wheatstone, rubbing the surface with technician’s fingers, or reflecting light on the surface [1]. Since all of these methods rely on a technician’s sense, the results may vary depending on the person. Nowadays, due to the difficulties of detecting and correcting the surface deflections, much of the lead time is spent resolving these problems. Several studies have been conducted to understand the mechanisms of surface deflections. Yoshida, et al. [2] have investigated the effects of the tensile stress and additional transverse tensile stress on surface deflections by simplified the handle portion of the exterior door panel. Matsui, et al. [3] have studied the initiation and growth of surface deflections under uniaxial tension and biaxial tension using the Yoshida buckling test. A variety of methods have been used to detect surface deflections visually: reflected screen, hologram interferometry and image shearing interferometry and other techniques. Liu, et al. [4] evaluated the surface deflections of pressed automobile panels using an optical reflection method. Zhenzhong and Yuying [5] demonstrate experimentally that the surface deflection is identified well by squaring the curvature difference between tool and panel. Sawata et al. [6] applied a finite element method using to predict surface deflections of the handle portion of the exterior door panels. Anderson [7] carried out forming and consecutive springback analysis on a simplified doublecurved panel by dynamic explicit FEM. By using a dedicated visualization system, the simulation results were compared with experimental results. Park, Chung and Kim [8] developed the method which can predict the locations and amplitudes by using the curvature analysis. In this paper, prediction algorithm and efficient simulation method are suggested to evaluate the location of surface deflection and to quantify degrees of surface deflection. It is proposed that the degree of surface deflection is predicted by the variation of the curvature difference between tool and blank along the scanning direction. Since the prediction of surface deflection is based on the blank shape from forming analysis and subsequent springback analysis, the appropriate simulation method to get the more accurate blank shape is explored. To verify the accuracy of the system, simulations and experiments were done on the simplified embossing problem. The effectiveness is discussed by comparing simulations and experimental results.

2. EVALUATION OF SURFACE DEFLECTION USING CURVATURE ANALYSIS It is reported experimentally that the curvature variation of formed panel correlates highly with visual inspection [1]. This principle is very effective in flat panels with low curvatures since blank shows high curvatures only in the regions with surface deflections after springback. However, in today’s recent innovative exterior design, the surface with high curvature becomes common. Therefore, it becomes difficult to predict the surface deflections only from the curvature of panel. In this study, a new analysis method based on the curvature was proposed for the quantitative measurement of surface deflections. It is suggested that the variation of curvature difference between tool and blank after springback is used to determine where the surface deflections are and to quantify the degree of surface deflection. Since the tool is very similar to the shape which we want to get, the curvature difference between tool and blank after springback reflects the deviation from the ideal curve. It should be noted that curvature difference is proportional to the bending moment in the springback, which is the main cause of the growth of surface deflection. Thus, it is thought that the curvature difference between tool and blank after springback has to do with the degree of surface deflection. Even in the drastic surface with high curvature, the present measure can take into account surface deflection since the curvature difference between tool and blank is utilized. Fig. 2 shows the procedures of curvature analysis to predict surface deflection. Firstly, the possible region should be defined by user. Next, scanning direction is required to generate sections. From the tool and blank, equally spaced parallel sections are extracted along scanning direction around the possible regions of surface deflections. For calculation efficiency and contour convenience, the sections are converted to the piecewise linear curves by sampling evenly and regularly spaced points along the sections. Next, curvatures are calculated. Since the surface profiles extracted from both the simulation and experiment contain a considerable amount of noise, local differentiation is not adequate for calculating the curvature. Here, curvature is calculated by finding the least square circle. A circle is sought that approximate given point

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s ( xi , z i ) most closely. Let ( a, b) be the center of the circle and r be the radius. Then we minimize the function G given by N

G (a, b, r )

¦ (( x

i

 a ) 2  ( z i  b) 2  r 2 ) 2

(1)

i 1

Finding r by using the algorithm in the reference [9], curvature can be calculated along the section. The span for the least square circle can be chosen from the result of the dial gage type curvature apparatus [1]. It is known that in general a long span tends to be insensitive to curvature and a short span tends to be sensitive to noise. Using noise filtering techniques is helpful for eliminating the noise of the curve before calculating curvature. Next, curvature difference between tool and blank after springback is obtained. Finally, curvature difference is visualized in a possible region by contouring. Curvature difference may be plotted in the graph. Since there may be oscillation in the calculated curvatures along the section, some oscillations are common in the contour or in the graph. From the contour plot, the location of surface deflection can be identified. The surface deflection can be quantified by defining the variation of curvature difference by using the graph of the curvature difference along the candidate sections. The degree of surface deflection can be predicted by using the high correlation between the variation of curvature difference and the visual inspection. The construction of correlation database is mandatory for the effective prediction of surface deflection. +02761(611.#0&$.#0-/'5*

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FIGURE 2. Flowchart of curvature based method

3. SIMULATION FOR THE PREDICTION OF SURFACE DEFLECTION The surface deflection grows as a small wrinkle with a wave section due to surplus material and compressive stress during forming and subsequent springback. To predict surface deflections accurately, simulation conditions should be appropriate to capture the growth of surface deflections. We investigated the influence of the simulation parameters such as blank mesh size, tool mesh size, time increment and the partition of mesh. To predict surface deflections accurately, the area where the surface deflections were predicted in the panel should be meshed finely and uniformly enough to describe the growth of surface deflection. Since a half wave length of surface deflection exceeds 10 mm in automobile exterior panel, 1 mm blank mesh size is sufficient enough to describe the mechanical behavior during the growth of surface deflection. Tool mesh also has an important role in the accuracy of prediction of surface deflection. Since the curvature of most of outer skin can be predicted within 105 /mm with 5 mm tool mesh size, we choose the tool mesh size of 5 mm. It is found that smaller time increment also improves the accuracy of surface deflection prediction. Especially, at the end of the forming analysis, time increment should be much smaller. If the entire blank is meshed with 1 mm mesh size, the number of finite elements is too large to solve with ordinary computer in most of engineering company. Thus, only regions with the possibility of surface deflections should be meshed with fine and uniform mesh. In reality, the regions with the possibility of surface deflections are small compared to the entire region. Therefore, it is expected that surface deflection can be predicted with reasonable time using appropriate partitioning of blank and tool mesh.

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4. APPLICATION Surface Deflection Analysis in the shallow rectangular drawing with rectangular embossing As an example, a shallow rectangular drawing with rectangular embossing is considered. Fig. 3 shows a quarter geometry of a rectangular drawing. The height of drawing is 30 mm and the height of inner embossing (H) is 10 mm. The draft angle (D) is 550. The corner radius (RC) is 15 mm. In the horizontal direction, the radius of curvature of tool is 5,000 mm. In the vertical direction, the curvature varies to maintain the same drawing height. The blank holding force of 140 ton is applied. The material data of SPRC340 steel sheet used in this study is listed in Table 1. Uniform mesh size of 1 mm was applied in the region where the blank contacts with punch head. Fig. 4 (a) shows the mesh after forming. Due to this partitioning, the simulation time is 9 minutes and 12seconds using 3 GHz PC with quad core. The simulation time with the whole uniform mesh of 1 mm is 16 minutes and 38 seconds. Fig. 4 (b) shows the contour of curvature difference between tool and blank along the horizontal direction. Since the curvature is the largest along the horizontal direction, the scanning was chosen as horizontal. Along the section AA’, parallel to and 35 mm away from the horizontal symmetry line, the largest value of curvature difference was found near the embossing corner. The analysis is performed with Autoform R2 due to the less noise in the curvature of blank.

FIGURE 3. Geometry of upper die TABLE 1. SPRC340 Steel Sheet Material Data [ V Material Parameter Strength Coefficient Offset Strain

H 0

n

]

Value 663.46 MPa

(K )

0.015141

Strain Hardening Exponent Average Rankford Value Thickness

K H 0  H p

n

0.26628

R

1.61 0.7 mm

(a) (b) FIGURE 4. (a) final mesh after springback, (b) Contour of curvature difference between tool and panel

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Experiment and Discussion Fig. 5 shows the upper and lower dies used in the experiment. To minimize the human error, the machining of die was finished using a very fine tool pitch of 0.3 mm and the polishing was minimized. Fig. 6 shows the formed blank. The blank holding force of 140 ton was applied using a die cushion. Along the section AA’, where the simulation predicts the highest value of curvature difference, the tool and panel were measured by using the coordinate measuring machine with resolution of 1 Pm . The curvature difference between tool and CAD surface was limited under 10-5 mm-1, which is much smaller than the curvature difference of a typical surface deflection. Thus, it is thought that the accuracy of tool is sufficient enough to access the accuracy of the simulation. Then, the curvatures of tool and panel were computed by the proposed algorithm and the curvature difference was obtained. In Fig.7, the curvature differences from simulation and experiment were compared. It is found that the curvature difference from the experiment shows more noise than the one from the simulation. In terms of the location of maximum and minimum, the experiment shows a small deviation to the simulation. Maximum value and minimum value from the experiment are in good agreement with the ones from the simulation, respectively. Also, the variation of curvature difference from the experiment, 1133x10-6 mm-1 is very similar to the one from the simulation, 1080x10-6 mm-1. Thus, it is demonstrated that the simulation can be effectively used is to identify the location of surface deflection and to quantify the surface deflection using the variation of curvature difference.

FIGURE 5. Upper and lower die

FIGURE 6. Blank after forming

FIGURE 7. Comparison of curvature difference between simulation and experiment

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5. CONCLUSION In this study, surface deflections that are crucial to the surface qualities of the automotive exterior panels are studied. A new prediction method based on the curvature analysis has been proposed not only to detect surface deflections but also to quantify the degree of deflections. This method is based on the curvature obtained by the least square fitting and the measure of surface deflection is defined as the variation of curvature difference between the tool section curve and the blank section curve after springback. The proposed measure is suggested to assess the degree of surface deflection by the high correlation between the measure and visual inspection. For the prediction of surface deflection, the appropriate simulation method is developed not only to predict the surface deflection accurately but also to save the computation time. In order to verify the effectiveness of the proposed method, a shallow rectangular drawing with a rectangular embossing was simulated. The corresponding experiment was performed. On the section where the curvature difference between tool and blank has a largest value, the variation of curvature difference from the simulation shows good agreement with the experimental one. In terms of the locations of maximum and minimum of curvature difference, simulation is also in good agreement with experiment. The variation of surface deflection also shows a good correspondence between simulation and experiment. Finally, it is concluded that the proposed method is effectively used to predict surface deflection. In addition, it is shown that the simulation could be used to predict surface deflection with reasonable accuracy.

ACKNOWLEDGMENTS This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government(MEST)" (NRF-2010-0025432)

REFERENCES ‫ډڌ‬The Japan Sheet Metal Forming Research. Handbook of Ease or Difficulty in press forming. National Center of Manufacturing Science. Japan, 1993. 2. M. Yoshida and T. Shimomura , Journal of the Japan Society for Technology of Plasticity, 24, 1983, pp.1260-1267. 3. M. Matsui and N. Iwata, Journal of the Japan Society for Technology of Plasticity, 24, 1983, pp.1268-1273. 4. L. Liu, T. Sawata, and M. Sakamoto, Journal of Material Processing Technology, 103, 2000, pp.280-287. 5. S. Zhenzhong and Y. Yuying, J. of Materials Processing Technology, 180, 2006, 53-59. 6. T. Sawata and L. Liu, Simulation of Materials Processing: Theory, Methods and Applications, edited by J. Huetink, F. P. T. Baaijens, Balkema, 1998, pp. 863-868. 7. A. Anderson, NUMISHEET 2005 Conference Proceedings, edited by L. M. Smith, F. Pourboghrat, J. W. Yoon, T. B. Stoughton , 2005, pp.113-118. 8. C. D. Park, W. J. Chung, B. M. Kim, J. of Materials Processing Technology, 187-188, 2007, 99-102. 9. C. O. Lee, K. S. Park, B. C. Park & Y. W. Lee, “Algorithms for Stylus Instruments to Measure Aspheric Surfaces”, Research Report Series 2004-1, KAIST, 2004.

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Prediction of the Spring-back Calculated with Numerical Simulations for the Household Industry Mihael Volka, Mihael Deželakb, Blaž Nardina, Stanko Stepišnikb a

Gorenje Orodjarna d.o.o., Partizanska 12b, Velenje, 3320, Slovenia EMO Orodjarna d.o.o., Bežigrajska cesta 10, 3000 Celje, Slovenia

b

Abstract. Within industry, spring-back is a problem which has to be considered during the manufacturing of tools. Over the last few years this has become an even bigger problem because of more complex drawn-products and the discovery of new materials. The shapes of products depend significantly on the spring-back effect, therefore an accurate prediction of the spring-back effect plays a significant role during a product's development process. Spring-back in sheet-metal forming can be described as the change in sheet-metal's shape compared with the shapes of the tools after the forming process. Using numerical computer analyses, spring-back is a difficult problem to solve, especially when the tolerances of the component parts are very narrow. This research work presents the effects of different holding systems on the spring-back effect in sheet metal deep-drawing processes. A conventional holding system and a segmented holding system with distributed Blank holder forces (BHF) were evaluated. Analyses were carried out using computer finite element simulations, using the Pam-Stamp software package for calculating. The results were compared and the conclusions given. Keywords: Sheet metal forming, numerical simulations, segmented holding system, spring-back

INTRODUCTION Spring-back is a very complex problem. Many authors are of the opinion that spring-back is one of the most difficult problems during sheet-metal production. Spring-back in sheet-metal forming can be described as the change in sheet-metal's shape compared with the shapes of the tools after the forming process [1]. We separate the following types of spring-back when considering the geometry of a product: angular change, sidewall curl, and twist (Figure 1). The deep-drawing process of sheet metals is a combination of angular changes and sidewall curls and usually occurs in the walls of the products, whilst the effect of torsion spring-back deviation (twist) is visible throughout the entire area of the product.

FIGURE 1. Different types of spring-back effect Industry solves these problems mainly by compensating the shape of the die but this procedure takes a lot of the time, and money [3]. This method is also inaccurate. Many repeated trail-loops are needed to assure tolerant dimensions of the products. These loops are more repetitive when there is less experience with the part's geometry and material. By using computer FEM analyses, the numbers of these loops and the time used for trails can be significant reduced [2]. However, even with today's computer knowledge, the spring-back is hard to calculate, and results are inaccurate. Several optimization and intelligent methods have been used to prove the accuracy of the The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1078-1085 (2011); doi: 10.1063/1.3623723 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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results [5, 6]. Spring-back can also be theoretically calculated [7], but this method is slow, unreliable, and only proven for simple models. Papaleux (et. al) investigated the impact of parameters on spring-back appearance, and proved that the BHF has a big influence on spring-back. The more-stretched deep-drawing sheet metal products are less sensitive to spring-back [8]. Using the segmented holding system, the stretching control is improved, as described in the work of [9], but this work presents different holding system and concentrates on sheet-metal deepdrawing, and doesn’t include spring-back analyses. The presented research work shows the prediction of spring-back, calculated using computer FEM simulations for different holding systems. The goal of this research work was to reduce those dimensional errors which occur due to the spring-back effect. Spring-back was calculated for a conventional holding system and for a newlydeveloped segmented holding system with segment inserts. With using this holding system, the deep-drawing process is more controlled, the technological window is bigger, and the process more stable.

EXPERIMENTAL PROCEDURE AND SPRING-BACK MODELING A sheet metal part from the household industry was used for this experimental evaluation of spring-back. This is the Front Panel (FP) of a cooking device. The shape of the FP can be clearly seen on the front side of the device, therefore the tolerance requirements are very strict. Scratches, wrinkles or cracks are disallowed. Dimensional errors should be also minimal due to mounting problems. The figure below shows the tool for the drawing operation of the FP. This tool is designed for presses that could separately control die cushion axes and can produce distributive holding forces. The cushion system and its control are presented in the work of [10].

(b)

(a) (c) FIGURE 2. a) Front panel deep-drawing tool and the FP part b) version 1 c) version 2 Several versions of the FP can be made (different cutting operations and the same drawing shape) using this tool. FPs can also be from different initial sheet-materials. Therefore, compensation of the die geometry is limited and spring-back should be compensated mainly by changing the holding-force. A newly-developed segmented holding system was used for this purpose. This holding system consists of a blankholder plate as used for holding within the intermediate areas and for guiding the segments’ inserts (Figure 3). The blank is directly in contact with the segments’ inserts, which allows the system to have better control over the deep-drawing process.

FIGURE 3. Holding system with segments’ inserts

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By using this holding system, all the holding forces of each segment can be separately controlled. For this system, distributed BHFs were computed and optimized for each segment, using fuzzy logic rules. The main criterion for the optimization was the forming limit-curve. From the results of the optimizations, it was proven that the quality of the drawing parts were better. The technological window became slightly bigger, which can be seen from Figure 4. In this study, two types of the front panel were calculated for one material. DC04 sheet material was used for the blank and material characteristics were taken based on manufacturer’s recommendations. The BHFs for calculation were chosen from a technological window. BHFs were chosen near the wrinkling boundary, near the cracking boundary, and in the middle of the technological window regarding good parts – optimal settings.

FIGURE 4. Schema – technological window FEM spring-back prediction results from the forming simulation. A number of numerical parameters during each simulation step transformation can have a significant impact on the final forecast changes in geometry after the withdrawal of forming forces, and the latter were not varied using the default value, as we focused only on the very important technological parameter – BHF. For the above-mentioned experiment, we presented using the Pam-Stamp program as a virtual model, by which we tried to make the best model for the real deep-drawing process. Material characterization was performed with yield-function of the Hill48 and Krupkowsky Law with isotropic hardening. An explicit algorithm was used for deep-drawing process modeling and an implicit algorithm for the spring-back effect. When measuring spring-back we had to remove all rigid body motion, so that only the spring-back phenomenon would be analyzed, and for this reason three locked nodes of the model were chosen during implicit spring-back simulation. Firstly, the spring-back for a conventional holding system was calculated. Some points from the technological window of the deep-drawing processes were taken. Similar values of the BHFs were used for the segmented holding system but, because of the wider technological window the computed BHFs were also higher (up to 660kN). These values are the sum of all segment inserts' BHFs, and their configurations are listed in Table 1. TABLE 1. Configurations of BHFs for segmented holding system. Name/Segment number S260 S290 S400 S440 S480 S560 S660

1 40kN 40kN 30kN 40kN 50kN 40kN 50kN

2 10kN 20kN 40kN 40kN 40kN 60kN 60kN

3 10kN 20kN 40kN 40kN 40kN 60kN 60kN

4 10kN 20kN 40kN 40kN 40kN 60kN 60kN

5 40kN 40kN 30kN 40kN 50kN 40kN 50kN

6 40kN 40kN 30kN 40kN 50kN 40kN 60kN

7 10kN 20kN 40kN 40kN 40kN 60kN 60kN

8 10kN 20kN 40kN 40kN 40kN 60kN 60kN

9 10kN 20kN 40kN 40kN 40kN 60kN 60kN

10 40kN 40kN 30kN 40kN 50kN 40kN 50kN

BH 40kN 10kN 40kN 40kN 40kN 40kN 100kN

30 measured points were chosen for clarification of the results. These points were also the measured points of the tool customer. Whole FP parts were used for calculating but, because of the symmetry, measured points were chosen

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for only one-half of the model. Furthermore, for the simplified results presentation, the part is divided into three colored areas (flange – marked with yellow, walls – gray, and the visual area of the FP part - green).

FIGURE 5. Points for deviation measuring due spring-back

RESULTS The following diagrams contain the results of spring-back FEM analyses for different holding systems and BHFs. Figure 6 presents the results for the drawing part, Figure 7 presents the results for the final part of version 1 and Figure 8 results for final part of version 2. As mentioned, the FP part was also divided into 3 areas; Flange (Figure 6a, 7a and 8a), Walls (Figure 6b, 6c, 7b, 7c, 8b and 8c) and the visual area of the FP part (Figure 6d, 7d and 8d). The diagram shows the average values of deviations. All deviations are absolute values (millimeters) no matter in which directions are the variations.

(a)

(b)

(c) (d) FIGURE 6. Deviation of the drawing part in measured points for a) Flange, b) and c) Walls and d) Visual areas

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(a)

(b)

(c) (d) FIGURE 7. Deviation of the final part – version 1 in measured points for a) Flange, b) and c) Walls and d) Visual areas

(a)

(b)

(c) (d) FIGURE 8. Deviation of the final part – version 2 in measured points for a) Flange, b) and c) Walls and d) Visual areas

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The flange is a less important part of the FP, therefore its influence is also less important. However, the deviations within this area were biggest. By optimization BHF, the deviations decreased by 30% but a lot of the measured points were still outside the tolerances (±0,5mm). In general, the deviations of walls increased by increasing the BHF due to sheet-metal thinning. This was especially true for the higher-side of the FP part, where the thinning is maximal. The most important area is the central area of the model which is visible to the customers. Here the tolerances and quality of the surface are the most important. Using the segmented holding system, almost all the measured points fell into tolerances or just above the tolerances. The improvements were for approximately 35%. The improvement can also be seen from Figure 8, which presents the actual measured values of the chosen points for each area. These results are for only one case (version 2) and for both used holding systems. BH200 and BH400 profiles presents displacements for conventional holding system, and S660 and S400 results for segmented holding system.

FIGURE 9. Deviation of the final part – version 2 in measured points for Flange, Walls and Visual areas (relative values)

Figure 10 presents the displacement due spring-back on the computer model. On these figures are results for a conventional holding system (Figure10a) and results for the segmented holding system (Figure 10b).

(b)

(a)

FIGURE 10. Displacements due spring-back of FP version 2; a) conventional and b) segmented holding system The presented results show that the segmented holding system is more reliable for controlling spring-back, and the deviations are generally smaller. The only exceptions were configurations S260 and S290 whose results were even worsen than all the results obtained from the conventional holding system because the applied BHFs were too small or not optimally distributed.

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Because of huge geometrical deviation a comparison between simulation results and measured values for S260 configuration was done. 56 points were chosen across the whole area of the FP drawing part and results of both research methods are presented on Figure 11.

FIGURE 11. Simulation and experimental results for S260 stamping part Comparison of simulation and experimental results show some deviation. The main reason could be in using relatively simple material-model (Hill 48), which is good enough for forming simulations, but has some disadvantages in spring-back modeling. This yield criterion is today wide used and for this reason, accuracy of spring-back simulations is still limited. In spite of deviation, spring-back tendency is well seen and this data are very useful in forming process design.

CONCLUSION This paper presents the prediction of spring-back, as calculated using numerical simulations of an industrial case. A front panel tool from the household industry was used for demonstration. More versions of the front panels for this cooking device can be made using this tool but only two versions and one drawing part are presented in this study. The main advantage of this FP tool is its integrated segmented holding system. The wider technological window can be achieved when using this system. The goal of this research work was to decrease dimensional errors due spring-back effect. Presented results show that the main goal was achieved. Using the segmented holding system with optimized BHFs it is possible to decrease deviations due spring-back for more than 30% for all measured areas, only the side walls are exceptions. In general, the deviations of walls increased by increasing the BHFs due to sheet-metal thinning. The most important area is the central area of the FP part which is visible to the customers of the cooking devices. For these areas almost all measured points fell into tolerances or just above. However, the simulation results show that for the conventional holding system the influence of increasing the BHFs on the spring-back is smaller, therefore controlling the springback with the BHF is less useful for this holding system. But for the presented segmented holding system the improvement is bigger and spring-back control more reliable.

ACKNOWLEDGMENTS Operation part financed by the European Union, European Social Fund. This work was also part of the 7th Framework programme, research and development project, Self-Learning sheet metal forming system (LearnFORM), Grant agreement N° NMP2-SL-2009-228346.

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REFERENCES 1. Y. Tongxi, Plasticity Bending Theory and Application, Beijing Science Press, 1992. 2. F. Cimolin, R. Vadori and C. Canute, Springback Compensation in Deep Drawing Applications, Meccanica, 2008, Vol. 43, pp. 101-113. 3. W. Gan and R. H. Wagoner, Die Design Method for Sheet Springback, International Journal of Mechanical Sciences, 2004, pp. 1097-1113. 4. S. K. Panthi, Finite Element Analysis of Sheet Metal Bending Process to Predict the Springback, Material and Design, 2010, pp. 657-662. 5. W. Liu, Springback Prediction for Sheet Metal Forming Based on GA-ANN technology, Journal of Materials Processing Technology, 2007, Vol. 187-188, pp. 227-231. 6. H. Liu, Adaptive Multiple Scale Meshless Simulation on Springback Analysis in Sheet Metal Forming, Engineering Analysis with Boundary Elements, 2011, Vol. 35, pp. 436-451. 7. P. Xue, T. X. Yu and E. Chu, Theoretical prediction of the springback of metal sheets after a double-curvature forming operation, Journal of Materials Processing Technology, 1999, Vol. 89-99, pp. 65-71. 8. L. Papeleux and J. P. Ponthot, Finite Element Simulation of Springback in Sheet Metal Forming, Journal of Materials Processing Technology, 2002, Vol. 125-126, pp. 785-791. 9. H. Palaniswamy, Optimal Programming of Multi-point Cushion Systems for Sheet Metal Forming, Annals of CIRP, 2006, Vol. 55/1/2006. 10. R., Neugebauer, Influence on Materials Flow in Deep Drawing Using Individual Controllable Draw Pins and Smooth Blankholder Design, 1997, SAE paper no 970989

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Springback Simulation and Compensation for High Strength Parts Using JSTAMP Terumasa Shindo, Nobuhiko Sugitomo and Ninshu Ma JSOL Corporation, 2-2-4 Tosabori, Nishiku, Osaka 550-0001, Japan Abstract. The stamping parts made from high strength steel have a large springback which is difficult to control. With the development of simulation technology, the springback can be accurately predicted using advanced kinematic material models and CAE systems. In this paper, a stamping process for a pillar part made from several classes of high strength steel was simulated using a Yoshida-Uemori kinematic material model and the springback was well predicted. To obtain the desired part shape, CAD surfaces of the stamping tools were compensated by a CAE system JSTAMP. After applying the compensation 2 or 3 times, the dimension accuracy of the simulation for the part shape achieved was about 0.5 mm. The compensated CAD surfaces of the stamping tools were directly exported from JSTAMP to CAM for machining. The effectiveness of the compensation was verified by an experiment using the compensated tools. Keywords: Sheet metal forming, Springback, Yoshida-Uemori kinematic model, Tool compensation, JSTAMP PACS: 01.50.hv, 02.60.Cb, 07.05.Tp

INTRODUCTION Recently high strength steel sheets have been applied to automotive body parts in order to reduce the weight of the body and improve the collision safety. However, the strength enhancement of steel sheets tends to increase the springback when the stamped parts are removed from the stamping tools. This phenomenon causes shape deviations and makes it difficult to ensure the dimension accuracy. Accordingly it is important to control the springback, so there are lots of countermeasures such as the improvement of forming process design and conditions [1-2], the use of lock beads [3] and the die shape compensation [4-6]. In any of these cases, computer simulation has already been necessary to predict the springback and verify the effectiveness of the countermeasures. Since the countermeasures are based on the springback prediction, in particular the accuracy of the springback simulation is vitally important, and CAE systems are required to have the capability to apply and verify the countermeasures easily. To meet the needs, the accuracy of the springback prediction has been significantly improved [7-8], and the sheet metal forming simulation system JSTAMP has been providing the advanced capabilities [9-12]. The objective of this paper is to simulate a drawing process for a pillar part using two material models, an isotropic hardening model and a Yoshida-Uemori kinematic hardening model, in order to investigate the impact of the material models before applying the springback compensation. The CAE system JSTAMP is used to compensate CAD surfaces of the stamping tools based on the springback prediction, and an experiment using the compensated tools is performed to verify the effectiveness.

SPRINGBACK SIMULATION Since the countermeasures such as the die shape compensation are based on the springback prediction, the accuracy of the springback simulation is extremely important. For that reason, the Yoshida-Uemori kinematic hardening model which describes the transient Bauschinger effect as well as the workhardening stagnation has been proposed [7-8]. The Yoshida-Uemori kinematic hardening model is the most reliable and also the most expensive in terms of computation time for the springback prediction of high strength steel sheets. To save some computation

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1086-1091 (2011); doi: 10.1063/1.3623724 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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time, isotropic hardening models can be used depending on the radius/thickness ratio and the steel grade as mentioned in [13]. A drawing process for a pillar part made from several classes of high strength steel, SPC440, SPC590DP and SPCN780 are shown in TABLE 1, was simulated using LS-DYNA and JTAMP applications software. The thickness of the steel was 1.0 mm. Two material models, the isotropic hardening model (the standard LS-DYNA material model Mat 37) and the Yoshida-Uemori kinematic hardening model (the user implemented LS-DYNA material model Mat 47 [14]), were employed to investigate which model was better for the iterative springback compensation loops.

Steel Grade SPC440 SPC590DP SPCN780Y

Yield Stress [MPa] 314 373 457

TABLE 1. Material properties. Tensile Strength Elongation [MPa] [%] 478 35.0 606 33.3 833 23.3

n-value 0.199 0.212 0.170

FIGURE 1 shows the shape correspondence between the part shape using the isotropic hardening model and the part shape using the Yoshida-Uemori kinematic hardening model. In the case of the part made from 440MPa steel, the shape correspondence within a tolerance of 0.5mm was about 90.5% of the part area and the shape correspondence within a tolerance of 1.0mm was about 99.6% of the part area respectively. FIGURE 2 shows the comparison of the sections using these two material models, the differences increased according to the strength enhancement of steel sheets. These results suggest that it would be possible to employ the isotropic hardening model instead of the Yoshida-Uemori model in the case of 440MPa steel in order to save on the computation time. In fact, the Yoshida-Uemori kinematic hardening model required about 1.7 times more computation time than that of the isotropic hardening model. To make the compensation iteration loops faster, it is economical to use the isotropic hardening model for the part made from 440MP steel. On the other hand, it is necessary to employ the YoshidaUemori kinematic hardening model to simulate the stamping process for the part made from 590MPa or higher strength steel. 100

Shape Correspondence (%)

90 80 70 60 50 40

Shape Tolerance +/- 1.0mm Shape Tolerance +/- 0.5mm

30 20 10 0 SPC440

SPC590DP

SPCN780Y

FIGURE 1. Shape correspondence between the part shape using IH model and the part shape using YU model.

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SPC590DP

SPC440

SPCN780Y

Sec 1

Sec 2

Sec 3

䇷 section using IH model 䇷 section using YU model

FIGURE 2. Comparison of the sections using IH model and the sections using YU model.

SPRINGBACK COMPENSATION The major algorithms for springback compensation known as the ‘Spring Forward Method [5]’ and the ‘Displacement Adjustment Method [6]’ were implemented in JSTAMP [11]. The iterative springback compensation scheme is shown in FIGURE 3, and part shape is possible to get closer to the desired shape by performing the iterative springback compensation loops until the shape tolerance is satisfied.

Original CAD surfaces

Compensated

Drawing simulation

CAD surfaces

Iterative compensation loops

Springback simulation

Check shape deviation

No

Die Shape Compensation 1st Compensation㸸Spring forward method or Displacement adjustment method

Yes 2nd Compensation㸸Displacement adjustment method Nth Compensation㸸Displacement adjustment method Compensated CAD surfaces for machining

FIGURE 3. Iterative springback compensation scheme in JSTAMP.

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The Spring Forward Method compensates stamping tools directly according to springback or springforward displacement. The Displacement Adjustment Method compensates stamping tools according to the shape deviation between the springback geometry and the reference or desired geometry. These algorithms in JSTAMP have the following unique characteristics: (1) JSTAMP is able to create the compensated CAD surfaces easily. (2) There is very little difference between the original fillet radius and the compensated one. JSTAMP never causes undesired behavior in this area. (3) Even when the upper and the lower geometries of the stamping tools are different, JSTAMP can create these compensated tools while maintaining the clearance. These characteristics make it possible to directly export the compensated CAD surfaces from JSTAMP to CAM for machining (see FIGURE 4). In addition, the application and the verification of the compensation can easily be conducted in JSTAMP. It is beneficial to use JSTAMP in order to reduce product development time and manufacturing costs.

Conventional Process

New Process using JSTAMP

CAE JSTAMP

Back to CAD

CAD

CAM

CAD

Forward to CAM directly

CAM

Back to CAD

CAE

Integration of SB Evaluation and Compensation FIGURE 4. New compensation process using JSTAMP.

VERIFICATION OF SPRINGBACK SIMULATION AND COMPENSATION To verify the springback simulation and the compensation, stamping processes for the pillar part were simulated using LS-DYNA and JSTAMP applications software. The processes were two-stage forming, the drawing process followed by a trimming operation, and a bending process followed by a final trimming and piercing operations. The material was 440MPa high strength steel which was modeled using the isotropic hardening model (the standard LSDYNA material model Mat 37), and other properties are listed in TABLE 1. FIGURE 5 shows the shape deviation before and after applying the compensation in the first stage. The shape deviation with original tools indicates the part was seriously twisted, but after applying the compensation 2 times, the dimension accuracy for the part shape achieved was about 0.5 mm on 99.5% of the part area. In the bending stage, a maximum shape deviation of 2.6 mm was measured before applying the compensation. After applying the compensation 3 times, the deviation was reduced sufficiently, and the dimension accuracy achieved was about 0.5 mm on 88.9% of the part area.

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The shape Deviation with 1st compensated tools is +/- 0.5 mm on 80.3% of the part area

The shape Deviation with original tools is +/- 0.5 mm on 17.8% of the part area

The shape Deviation with 2nd compensated tools is +/- 0.5 mm on 99.5% of the part area

FIGURE 5. Shape deviation between the springback geometry and the desired geometry during compensation.

The compensated CAD surfaces of the stamping tools were directly exported from JSTAMP to CAM to manufacture the actual stamping tools. To verify the effectiveness of the compensation, an experiment was carried out by using the tools. FIGURE 6 shows the differences between the experimental measurement points and the simulated ones. The differences were within 0.5 mm at 19 out of 23 points, this means the compensation capabilities in JSTAMP are useful to develop the actual stamping tools.

2.5

Differences (mm)

2 1.5 1 0.5 0 -0.5 -1 -1.5 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 Measurement Point

FIGURE 6. Differences between the experimental measurement points and the simulated measurement points.

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CONCLUSIONS It is necessary to employ the Yoshida-Uemori kinematic hardening model to simulate the stamping process for the pillar part made from 590MPa or higher strength steel. In contrast, in the case of the part made from 440MPa steel, it would be possible to employ the isotropic hardening model instead of the Yoshida-Uemori model in order to save on the computation time. The compensation capabilities in JSTAMP are powerful methods to compensate the stamping tools, and the compensated CAD surfaces of the stamping tools can be directly exported from JSTAMP to CAM for machining. The effectiveness of the compensation using JSTAMP was verified by the experiment.

REFERENCES 1. Y. Morishita, T. Kado, M. Iwatani and S. Abe, Development of sheet metal forming technique by multi-stage action (2nd report) Die shape determination in draw bending process using the moving punch, Bulletin of the Hiroshima Prefectural Technology Research Institute, West Region Industrial Research Center (52), 2009, pp. 44-47. 2. T. Yoshida, T. Katayama, K. Hashimoto and Y. Kuriyama, Shape Control Techniques for High Strength Steel in Sheet Metal Forming, Shinnittetsu Giho, No. 378, 2003, pp. 25–29. 3. J. R. Fekete, J. N. Hall, D. J. Meuleman and M. Rupp, Progress in implementation of advanced high-strength steels into vehicle structures, Iron & Steel Technology, Vol. 5, No. 10, 2008, pp. 55–64. 4. M. Clarke, J. He and X. Zhu, A Real World Approach for using LS-DYNA to Achieve True Springback Compensation on AHSS Components During Forming, 7th EUROPEAN LS-DYNA CONFERENCE 2009, Salzburg and Bad Reichenhall, pp. 72. 5. A. P. Karafillis and M. C. Boyce, Tooling And Binder Design for Sheet Metal Forming Processes Compensating Springback Error, Int. J. Mach. Tools. Manuf., Vol. 36, 1996, pp. 503–526. 6. R. A. Lingbeek, J. Huetink, S. Ohnimus and J. Weiher, Iterative Springback Compensation of Numisheet Benchmark #1, AIP Conf. Proc., Vol. 778, 2005, pp. 328-333. 7. F. Yoshida and T. Uemori, A model of large-strain cyclic plasticity describing the bauschinger effect and wrokdhardening stagnation, International Journal of Plasticity, 18, 2002, pp. 661-686. 8. F. Yoshida and T. Uemori, A model of large-strain cyclic plasticity and its application to springback simulation, International Journal of Mechanical Sciences, Vol. 45, 2003, pp. 1687-1702. 9. N. Ma and N. Sugitomo, Dynamic Explicit FEM and Simulation on Sheet Metal Forming, Journal of Japan Society of Technology Plasticity, Vol.47, No.540 (2006-1), pp.29-34. 10. N. Ma, Y. Umezu and Y. Watanabe, The Development and Applications of Simulation System JSTAMP/NV for Tube HydroForming, NUMIFORM2007, Proc. of Porto, June 17-21, 2007, pp.1471-1476. 11. N. Ma, Y.Umezu, Y.Watanabe and T.Ogawa, Springback Prediction by Yoshida-Uemori Kinematic Model and Compensation of Tool Surface using JSTAMP, NUMISHEET2008, Interlaken, Switzerland, Sept. 1-5, 2008, pp.473-478. 12. Y. Kato, Y. Umezu and Y. Watanabe, Recent Developments in JSTAMP/NV for the Best Stamping Simulation Environment, 11th International LS-DYNA Users Conference, June 6-8, 2010. 13. L. Kessler, J. Gerlach and M.-S. Aydin, Springback simulation with complex hardening material models, 7th German LSDYNA Forum, Bamberg, 30th September 2008. 14. N. Ma, Y. Umezu, T.Hamada and Y. Watanabe, Development of Various KinematicHardening Material Subroutine for LSDYNA and Its Applications, Proc. of JSTP JointConference, pp. 175-176, Nov. 2005

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An Automatic Spring-back Compensation Die Design Method Based on Genetic Algorithm and IsotropicKinematic Hardening Laws Xiao-Xing Lia, Jing Hua, Kwansoo Chungb, Guo-Feng Zhoua, and Rao Yaoa a School of Mechanical Engineering and Automation, Beihang University, Beijing 100-191, China Department of Materials Science and Engineering, Research Institute of Advanced Materials, Seoul National University, Seoul 151-742, Republic of Korea

b

Abstract. We present the study of a finite element method for die contour to compensate spring-back in sheet metal forming process, which is based on the genetic algorithm and isotropic-kinematic hardening laws. The Chaboche type combined isotropic-kinematic hardening law was formulated and used to account for the Bauschinger and transient behavior in the finite element analysis. Using a S shape stretch bending process as an example, it was demonstrated that the new method optimizes the die profile effectively. The good performance of the die profile optimized utilizing the new method was also experimentally verified , confirming that the new method might be more effective in cost reduction than common design practices in practical applications. Keywords: FEM, GA. Tool, Optimization PACS: 07.05.Wr

INTRODUCTIION Many parts used in the aviation industry are made of sheet metals such as wing-fuselage fusion profiles and complicate saddle panels. During the forming process of those sheet parts, various defects can be introduced . While defects like slip lines, coarse grains, bulging and excessive thinning can be controlled by multi-step forming or restricting stretch forming velocity and maximum deformation [1,2,3], efforts to effectively control spring-back or to minimize the adverse impact of spring-back still need further technical improvement. The amount of spring-back is affected by both material properties and forming process parameters. In general, the greater the strength, the lower the elastic modulus of a material, the greater the spring-back is [4]. The bending curvature and the die gap also have a positive effect on the spring-back. Many research works have been focused on optimizing forming process parameters in order to minimize spring-back [5,6]. One of important steps in controlling spring-back in the forming process of sheet metals is to design the die profile properly to compensate spring-back. In the past, spring-back compensation methods in sheet metal forming processes were based on die designers’ experiences, mainly involving (experimental) trial and error to achieve ultimate target part shapes. With the advent of computation technology, sheet metal forming process can be analyzed prior to experiments using the finite element method (FEM). FEM became the basis of spring-back compensation method as demonstrated by Karalifills and Boyce [7] in their work to develop a spring-forward arithmetic to compensate spring-back. The displacement adjustment (DA) method was also proposed, which has been proven to be successful in compensating spring-back mainly by adjusting the surfaces of tools in the opposite direction to that of spring-back [8,9]. However, the DA method could only applied to simple forming processes and two dimensional products [10]. One of drawbacks of the initial DA method was that the shape modification field can be defined only on the nodes of blank-meshes. A method to overcome the limitation was proposed recently, which applied shape modification to any meshes, including generally large and topologically different tool meshes, or even analytically defined CAD geometries [11]. Another method called the inverse displacement method [12,13] is also available, which is very similar to the DA method, with difference mainly onadjustment directions. The genetic algorithm (GA) [14] applied in the new springback compensation method is a directed search algorithm based on the mechanism of biological evolution, which The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1092-1099 (2011); doi: 10.1063/1.3623725 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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has been widely used in business, scientific and engineering fields. The algorithm is a stochastic optimization technique, employing the concepts of natural selection and genetics. Therefore, the method can effectively identify the global minimum, especially for highly nonlinear problems with multiple local minima [15,16,17]. Optimization of plastic deformation processes such as sheet metal forming processes are highly nonlinear and there are a few works [18,19,20], which employed the GA method for optimization, but works to control spring-back are rare In the present study, a new alternate method to design die profiles by compensating spring-back in the process design stage is developed. The method is also one of numerical trial-error methods but based on both the genetic algorithm and the finite element analysis. The method can be easily extended to optimize other process parameters in addition to the die profileThe method is also automatic for industrial applications. The main goal of the present work is to develop a die shape design method to compensate spring-back in sheet metal forming, which is robust and automatic for industrial applications. In this method, an initial die profile is assumed randomly for a prescribed sheet part and FEM is applied to simulate forming, using the initial die, and also to predict the final part shape after spring-back. During FEM processing, the modified Chaboche type combined isotropic-kinematic hardening law is used to improve the simulation accurate. Then, the precision of the final shape is evaluated provided by GA automatically. The procedure is repeated until predetermined accuracy reaches. The optimized die profile is exported and transferred into a CAD file, which could be used as a numerical control (NC) code for tool production. With this GA method, expensive prototype tests can be avoided and the die design optimization process might be faster and more cost effective than previous practices.

FLOW FORMULATION OF THE CHABOCHE MOD(/AND NUMERICAL IMPLEMENTATION PROCEDURES The yield surface is described by m

(1)  f (V  D )  V iso 0 Where f is the yield function and  is the Cauchy stress, while  is the back-stress by which the current yield stress surface is translated from an initial position, therefore, Eq. (1) leads to m  2 wV iso wf wf dV  dD  mV iso (V  D ) ˜ dH P 0 w ( V  D ) w ( V  D ) w H  (2) In the Chaboche model, the back-stress increment is composed of two terms, dD dD 1  dD 2 to differentiate the transient hardening behavior during loading and unloading (or reverse loading). dD 1 dD 1 (V  D ) dD 1 dH ) (V  D ) ( V iso V iso dH   (3)   Consider the plastic strain increment as dH P

 dO

dD 2

dO

(

dD 2 d H )D dH

wV iso w (V  D )

dH

 wV iso w(V  D )

wV iso wV iso ˜D ˜ C ˜ dH  d D 1  dD 2 w (V  D ) w (V  D ) wV iso wV iso  hiso ˜C ˜ w (V  D ) w (V  D )

 Then consider linear isotropic elastic constitutive law as e  dV C ˜ dH

C ˜ ( d H  dH P )



(4)



(5)



(6)

 (7) Where C is standing for the constitutive Jacobian matrix in the UMAT, and it is updated during the strain increment, but in the VUMAT, C is a constant matrix without any updating at all. For a strain increment dH prescribed at every time increment, Eqs. (5) and (6) determine the plastic strain increment and then the back-stress and Cauchy stress are updated by the Jaumann stress increments as shown in Eqs. (3), (4) and (7), respectively. The numerical scheme to solve non-linear boundary value problems of sheet forming processes commonly consists of three steps at every process increment [21]. At the first step, the discrete strain increments are calculated

1093

from trial displacement increments and then at the second step, the stresses and other variables such as plastic strains, back stresses are updated from the discrete strain increments using the elasto-plastic constitutive equation. Finally, the tolerance is checked whether the static or dynamic principle of momentum is satisfied. Iteration continues until the principle is satisfied within the tolerance. In the stress update scheme at the second step, the updated stress is initially assumed to be elastic for a given discrete strain increment 'H . Therefore, V T V n C ˜ 'H  n 1 (8) where the superscript ‘T’ stands for a trial state and the subscript denotes the process time step. Also, preserving the plastic quantities as the previous values, T

 H n 1 H n , D n1 D n If the following yield condition is satisfied with the trial values for a prescribed elastic tolerance Tole, T

1

(9)

T

f m (V nT1  D nT1 )  V iso (H n1 )  Tol e

 (10) the process at the step n + 1 is considered elastic. If the above condition on yielding is violated, the step is considered elasto-plastic and the trial elastic stress state is taken as an initial value for the solution of the plastic corrector problem. Thus the plastic strain increments are obtained such that the following new trial stress stays on the new yield surface: T P  V V  C ˜ 'H (11) The condition that the new stress stays on the new yield surface leads to the consistency requirement. To effectively solve the non-linear equation, several yield surfaces were introduced between the trial elastic stress and the initial stress V n and the solution was obtained progressively as schematically shown in Fig. 1[22].

OPTIMIZATION METHOD BY GENETIC ALGORITHM The GA method is a stochastic optimization technique, which is based on the concepts of natural selection and genetics. In the GA method, independent variables to be optimized are called chromosomes and a set of chromosomes is called an individual, while a set of individuals is called a population. The GA optimization method involves the selection of the best fit individual harnessed with optimal chromosomes out of a population, which evolves positively every generation. Therefore, the GA method consists of three procedures: a procedure to produce a new improved population out of a current population, a way to evaluate the performance of an individual for every population and a criterion to terminate the calculation, as schematically illustrated in Fig.2. The first population is generated with random numbers assigned to every individual within the value domains of variables (chromosomes). Next, FEM simulations are performed and the FEM performance of every individual is evaluated according to an objective function (which is supposed to be minimal for the best fit in general) to rank. Then, a new population is generated by transforming the current population by means of three operators: the crossover and mutation of individuals as well as selection by the elitist strategy. These procedures are iterated until a criterion to terminate calculations is met. The size of the population is one of important input parameters to be determined. If the population size N is too small, calculation is fast but accuracy might be lost. if it is too large, however, calculation may waste computational resources [23] without improving accuracy. Usually, the recommended number of individuals is 5~10 times the number of variables [24]. To evaluate the FEM performance of an individual, an object function is defined, which is dependent on the difference between the target sheet shape and the current simulated sheet shape after spring-back. As the performance of an individual is better, its value of the object function is supposed to be smaller. The GA method proceeds until the object function value becomes smaller than a prescribed value (called the tolerance) for an individual. Another practical way to terminate the calculation is to prescribe the number of reproduction (which is equivalent to the number of improvements by reproduction). In the reproduction of a new generation, the crossover operation produces a new pair of individuals from a pair of current individuals. There are the discrete crossover and the arithmetical crossover. In this study, the arithmetical crossover [23] was adopted because the discrete crossover just exchanges positions and the search band of the arithmetical crossover is larger. For the I-th generation, consider the total N individuals: q1 , q2 , q3 , }, qN , I

1094

I

I

I

while each individual qiI contains n chromosomes: qiI (ZiI1 , ZiI2 , } , ZinI ) and (i = 1,…, N). The crossover operation is performed as follows; Step-1: Prescribe the crossover probability number Pc, which is usually between the boundaries of 0.4 and 0.9 [23]. I Step-2: Generate N/2 pairs of individuals by randomly pairing two individuals, qxI (ZxI1 , Z xI 2 , }, Z xjI , }, Z xn ) and

q yI

(Z yI1 , Z yI 2 ,}, Z yjI ,}, Z ynI ) .

Step-3: For each pair, generate random arrays RhI

(rhI1 , rhI2 , }, rhnI ) (h=1,…, N/2) with the range from 0.0 to 1.0.

I

Step-4: For the h-th individual pair, if rhj is lower than Pc, the following operation takes place for corresponding chromosomes of q x and q y , to generate a pair of new individuals, qx I 1 I

q y I 1

I

(ZxI1 , ZxI 2 ,}, ZxjI 1 ,}, ZxnI ) and

(Z yI1 , Z yI 2 ,}, Z yjI 1 ,}, Z ynI ) , where ZxjI 1 rhjI ZxjI  (1  rhjI )Z yjI

(12)

Z yjI 1 rhjI Z yjI  (1  rhjI )ZxjI

(13)

I

Step-5: Else (if rhj is equal or higher than Pc), corresponding chromosomes are preserved without modification. After the crossover is performed, the mutation operation is carried out. When GA approaches the optimal solution, the mutation operation accelerates the optimization procedure. In this study, the uniform mutation method is applied. For this operation, a mutation probability number Pm is prescribed by a user between 0.001 and 0.1 [23,24]. Then, random arrays, K iI 1 ( kiI11 , kiI21 , }, kijI 1 , } , kinI 1 ) (i = 1,…, N), are generated for each individual,

qiI 1 , using the value from 1.0 to 1.0. If kijI 1 (1  j  n) is between the mutation probability Pm and Pm, the following operation takes place for the corresponding chromosome:  ZijI 1 LB j  kijI (UB j  LB j )

 

(14)

and  qiI 1

(ZiI11 , ZiI21 , } , ZijI 1 , } , ZinI 1 )





(15)

Here, LBj and UBj are the low and upper bounds of the chromosome value, which are also prescribed by a user. After a new population is produced by the crossover and mutation operations, the FEM performance of individuals are evaluated considering the object function. Then, the elitist strategy [26] was further adopted in this work, in which the worst 10% performers (as individuals) are replaced by the top 10% performers obtained in the previous population. The elitist strategy not only represents the survival principle of the fittest but also avoids trapping in the local minimum. From a probabilistic point of view, these three operators tend to improve the population and increase the performance of individuals in the next iteration of the optimization process. The crossover and mutation probability numbers influence the convergence speed of GA but not affecting the optimization result [27].

APPLICATION TO THE STRETCH FORMING PROCESS To execute the GA method, the general mathematics software MATLAB [28] was used in this study, which provides a platform for multi-language programming to call external executable files. An optimization procedure based on the MATLAB GA tool was developed here. For efficiency, the stretch forming was simulated using the explicit code, ABAQUS/explicit, with the user-defined subroutine VUMAT, while spring-back was simulated using the implicit code, ABAQUS/standard, with the user-defined subroutine UMAT, after forming simulation results were imported from the explicit code to the implicit code. Geometrical features of die profiles are described by a series of points in finite element analysis. A common way is to divide (or disperse) the surface of the die along its characteristic profile lines by applying meshing software, and then the data for all the coordinates of nodes are obtained. An S type die surface divided according to this method (using the ABAQUS/Explicit code) is seen in Fig.3, in which only a half of the die is considered by symmetry (in Fig. 2(b)). Note that optimizing the die profile implies optimizing every nodal position of the die,

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which is numerically cumbersome. Therefore, smooth and continuous function is utilized to define the die surface, while their independent variables which describe the die shape are optimized. The half target sheet part shape considered in this work was defined as ­° 125 2  x 2 0 d x d 63.5 z ® (16) °¯ 235 2  x 2 63.5 d x d 279  300 d y d 300

where x, y, z are the coordinates of the target sheet shape, as illustrated in Fig. 4. For the blank material, an aluminum alloy 2A12-O [30] sheet whose dimension was 1400 mm ×600 mm with the thickness of 1.2 mm. For the hardening behavior of the aluminum alloy sheet, the Chaboche type combined isotropic-kinematic hardening law was assumed, while Mises yield criterion was applied to describe the anisotropic yield stress. The mechanical properties are summarized in Table 1. The initial set up of the stretch bending process is shown in Fig. 3(a). The length of the blank sheet between two clamps was 1,200 mm (aligned along the rolling direction of the sheet). Considering the symmetry of the die and the blank sheet, a half of the whole was considered in the FEM analysis as shown in Fig. 3(b). Here, the die and clamp were meshed using the rigid element (R3D4), The sheet was meshed using shell elements (S4R) with 9 integration points through thickness. The sheet and the clamp were tied together. The 20# engine oil was used as a lubricant and the friction coefficient between the sheet and die was assumed to be 0.1. In the forming step the lower die moved up for 150 mm and the upper die moved down for 150 mm, respectively, then, during the post-stretching step, two clamps moved inwardly by 118.000 mm, simultaneously with the upper die and lower die moved up by 168mm together. Note that the post-stretching processes were added in this work since it was proven to be effective to control spring-back in the previous works [30,31,32,33]. However, it was not enough to remove the spring-back completely so that the die shape was further optimized in this work to remove the spring-back completely by compensating the spring-back. In order to calculate the spring-back, the half part of the S type die shape was separated by different curvatures, the half of the die profile shape was defined as ­ 125 2  (125 x / b ) 2 1  z °® °¯ 235  (235 x / b2 )  300 d y d 300 2

0 d x d 63.5 2

63.5 d x d 279



(17) Here, x, y, z are the coordinates of the die profile with two variables (chromosome b1, b2) to be optimized using the GA method. The variable b1 ranges between 115 mm and 125 mm, variable b2 ranges between 225 mm and 235 mm. The variation of the die shape with respect to the variables is shown in Fig. 4. After one generation iteration, the part with the curvature of 235 will be moved along the red line to be joint with the other part, and then the next generation iteration will begin. The following object function was used in this work to evaluate the FEM performance of the individual:  f max z ( m )  Z ( m )  (18) where z(m) is the z-coordinate (of the m-th node) in a simulated (using the individual data) sheet shape after spring-back, while Z(m) is the z coordinate of the target shape obtained from Eq. (16), considering the simulated x coordinate of the m-th node. In this work, there are two variables and the size of population, which is the number of individuals in a population, was chosen to be 16. Also, the probability numbers of crossover and mutation were selected as 0.8 and 0.1within the boundaries, respectively. Calculations were terminated after 20 generations well after a converged solution was obtained. Fig. 5 shows the evolution of the optimum solution (individual) of each population (generation) in terms of the two variable values. It was confirmed that the converged solution was obtained after the fourth generation: the ultimate solution was b1 = 123.5mm and b2=232.0 with the object function value 0.4073. The performance of the optimized die profile was experimentally verified. After the coordinates of die nodes were generated from Eq. (17) with the optimum b values, the shape of the whole final die profile which was used as the digital model to manufacture the die was obtained using the surface reconfiguration tool of CATIA [34]. The sheet contour of the manufactured part after spring-back was compared with that of the optimized simulation part

1096

and the target part sheet as shown in Fig.6, which confirmed good agreement between the manufactured part and the target part with less than 5.7 mm in the z-value deviation and the relative error of 0.79%.

SUMMARY Geometric distortion of sheet parts incurred by spring-back in sheet forming is one of major defects to control to improve the precision and the performance of sheet parts. In order to control the spring-back, process parameters are optimized together with the die shape. In the present work, a die shape design method to compensate spring-back in sheet metal forming was developed based on the finite element analysis and the genetic algorithm, which is robust and automatic for industrial applications. The method was applied for a four-stage stretch bending process of an aluminum alloy sheet and it was demonstrated that the new method optimized the die profile effectively. The performance of the die profile optimized utilizing the new method was experimentally verified and it was confirmed that the optimum die profile removed spring-back completely along with pre- and post-stretching stages added to control the spring-back; therefore, suggesting that the new die design method might be cost effective in practical applications.

REFERENCES 1. M. Wan, X. B. Zhou, X. X. Li, Acta Areo. Astro. Sinic 20, 326 (1999) 2. Z. H. You, Y. S. Huang, Wu, R. H., Avia. Engi. Main. 6, 45 (2001). 3. M. Wan, X. B. Zhou, X. X. Li, J. Beihang Uuiv. 24, 654 (1998). 4. N. M. Wang, the 13th Biennial Congress IDDRG, Melbourne, Australia (1984). 5. G. Gasper, P. Tomaz, K. Karl, J. Mater. Process. Tech. 130-131, 54 (2002). 6. R. Bahloul, S. Ben-Elechi, A. Potiron, J. Mater. Process. Tech. 173, 101 (2006). 7. A. P. Karafillis, M. C. Boyce, Int. J. Mach. Tool. Manu. 36, 503 (1996). 8. R. H. Wagoner, Proc. ESAFORM (V. Brucato), p. 7, Salerno, Italy (2003). 9. W. Gan, R. H. Wagoner, Int. J. Mech. Sci. 46, 1097 (2004). 10. W. Gan, R.H. Wagoner, K. Mao, S. Price, F. Rasouli, J. Mater. Process. Tech. 126, 360 (2004). 11. R. Lingbeek, J. Huetink, J. Weiher, J. Mater. Process. Tech. 169,115 (2005). 12. Q. Zhang, Z. R. Wang, T. A. Dean, J. Mater. Process. Tech. 194, 74 (2007). 13. W. P. Li, W. D. Bian, S. M. Nie, Forg. Stamp. Tech. 32, 86(2007). 14. J. H. Holland, Ph. D. Thesis, University of Michigan Press, Ann Arbor, USA (1975). 15. A. Pandit, X. Lu, C. Wang, G.S. Kassab, Am. J. Physiol.-Heart C. 288, 2581 (2005). 16. S. Vigdergauz, Int. J. Solids Struct. 38, 8593(2001). 17. C. L. Karr, I. Yakushin, K. Nicolosi, Eng. Appl. Artif. Intel. 13, 625 (2000). 18. W. Liu, Y. Y. Yang, J. Mater. Process. Tech. 208, 499( (2008). 19. Z. H. YouୈY.S. Huang, R. H. Wu, Avia. Engi. Main. 6, 45 (2001). 20. Z. Michialewicz, Genetic algorithms +data structures=evolution programs. Springer, USA(1996). 21. K.Chung, O.Richmond: Int. J. Plasticity, Vol. 9(1993), p.907 22. J.W.Yoon, D.Y.Yang, K.Chung: Comput. Methods Appl. Mech. Eng., Vol. 174(1999), p.23 23. Y. J. Lei, Ph. D. Thesis, p.60, The press of xidian university, China (2005). 24. Y. P. Huang, C. H. Huang, Fuzzy set syst. 87, 265 (1997). 25. G. Rudolph, Ieee T. Neural. Networ. 5, 96 (1994). 26. Y. Ledoux, P. Sebastian, S. Samper, J. Mater. Process. Tech. 210, 474(2010). 27. MATLAB, 2008. User’s manual for version 7.6.0, The Mathworks Inc. 28. ABAQUS, 2008. User’s manual for version 6.8.1, Dassault Systemes Simulia Corp. 29. GB/T 3880-1997, Aluminium and aluminium alloy rolled sheet. 30. A.Parris, Ph. D. Thesis, Massachusetts Institute of Technology, USA (1996). 31. C.H.Jin, X.B.Zhou, et a.l, Mater. Sci. Tech.-Lond. 12, 394 (2004). 32. A.H.Clausen, O.S.Hopperstad, J. Mater. Process. Tech. 102, 241 (2000). 33. Y. M. Zhang, X. B. Zhou, J. Beihang Uuiv. 33, 201 (2007). 34. CATIA, 2006. User’s manual for version5.17, Dassault Systemes Corp

1097

Table 1ˊThe material parameter of 2A12-O (E-Electric Module, v-Poisson’s Ratio, s-Yield Strength, b- Ultimate tensile strength)

v

E Gpa 72.65

0.3

s

b

MPa

MPa

88.3

194.8 Fig.1 A schematic view of the progressive update of the new stress

Population P ୄi+1 Evaluating with Object function

Ending Condition

i୅

GA Tool: 1୅ Crossover 2୅ Mutation 3୅ Selection

N

Y Profile Nodes Fig.2. Flow chart of the genetic algorithm.

(b)

(a)

Upper die

Upper die

y

600

Sheet 600 o

F

Clamp

Sheet

E

D

x

Lower die Lowe die Clamp

1098

Fig3 (a) A schematic view of the S stretch bending process (unit: mm) (b) meshes in the die and the sheet (only a half of the whole was considered by symmetry, unit: mm)

Target Surface b1=120

b2=235

b2=230

b1=125

Fig4 The target sheet part shape and the die shape with variables (two chromosomes, b1 and b2, unit: mm).

10 236

Best b1 value Best b2 value

124.5

8

6 234

4

124.0

Best b2 vaule

Best b1 vaule

125.0

2 232

123.5 0 0

5

10

15

20

Generation

Fig.5. The evolution of the chromosome value (b1 and b2).

Object shape Simulation shape Experiment shape

Fig. 6. Comparison between the object shape, the final shape of simulation and experiment.

1099

+LJK)LGHOLW\6SULQJEDFN6LPXODWLRQDQG&RPSHQVDWLRQZLWK 5REXVW)RUPLQJ3URFHVV'HVLJQ Intaek Leea , B.D. Carleerb, and S.Haagec a

b

AutoForm Japan K.K. ,Level 9 Orix Shinbashi Building 5-8-11Shinbashi,Minato-ku,Tokyo, 105-0004, Japan AutoForm Engineering Deutschland GmbH,Technologiezentrum Emil Figge Strasse 76-80, Dortmund, D-44227, Deutschland c AutoForm Engineering Deutschland GmbH, Marktstrasse 46, Ravensburg, D-88212, Deutschland $EVWUDFWFor the efficient virtual try-out loop, geometric change and bending angles have been compensated during last 20 years. This approach was based on some restrictions like pure bending, plane strain state and isotropic behavior. For more complex forming processes, this has been applied without reviewing this limitation. Analytical force consideration to reduce the amount of springback is also another idea to compensate geometrical displacement adjustment efficiently. In other view, the springback prediction accuracy is also one major topic with various material model developments. All these topics are absolutely of high importance in order to increase springback accuracy and effective compensation to help reduce the trials efforts. But the focus should not only be on these advances issues since the basics must be right as well. Based on our long term experiences in simulation and several project outcomes covered 20 different parts during last 2 years, we will present our experience and investigations in order to geometrically compensate forming process. It is emphasized that we did not just simulate the springback but the compensated surfaces where brought into the real tools. So, the experiences are not only based on numerical analysis but of all this part a physical tryout has been performed as well. The experiences are consolidated in a set of principles of robust springback compensation. These principles will be illustrated and explained at an example part, a B-pillar upper reinforcement. Indeed, it is can be seen that that compensation is a straight forward activity. However, our experiences have shown that it is only a straight forward activity in case certain ‘boundary conditions are fulfilled. We will discuss some of this boundary condition using the Bpillar upper example. Through this study, basic requirements for successful springabck compensation, the full scope of simulation range, set-up the nominal springback simulation and robustness of forming process will be clearly described to help understanding practical meanings of springback accuracy and strategic compensation. .H\ZRUGVSpringback Accuracy, Compensation, Material behaviors, Virtual & real try-out, Finite element method, Robustness, Optimal forming process  3$&6=

 ,1752'8&7,21 Over the past decades, stamping simulation has being dramatically evolved and adapted in standard engineering work process to get optimum process with its accurate prediction in crack, wrinkle and even surface defects into micro level. In the competition, shorter deadlines, advanced materials and challenging geometries are just come to the many challenges faced by sheet metal forming industry. Delivering a dimensionally stable part has become a complex issue. One major challenge is to neutralize the springback behavior by geometrical compensation. Thinkable and efficient engineering process for dimensional accuracy is the intensive use of simulation from product design stage to production engineering stage. For the practical use of springback simulation, the physical match to reality, as like a high fidelity modeling technique to physical meaningful engineering is most important element for engineering work process chain and wide visional-comprehensive engineering. Springback analysis and compensation is nothing new. Already last century by means of simple graphs and formulas bending angles have been compensated [1]. This approach was based on some restrictions like pure bending, plane strain state and isotropic behavior. For more complex forming processes not fulfilling the boundary conditions finite element simulations have been applied.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1100-1107 (2011); doi: 10.1063/1.3623726 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

1100

The approximation of these simulation models is sufficient to yield good forming simulation results as well as springback values. However, in order to increase the springback accuracy, the models must be amplified at various points [2]. One of the main topics which has been investigated is the material behavior since the springback is strongly affected by material properties in terms hardening behavior, anisotropy and effects like kinematic hardening [3], [4]. All these topics are absolutely of high importance in order to increase springback accuracy. But the focus should not only be on these advanced issues since the basics must be right as well. In the paper we will present our experiences and investigations in order to geometrically compensate forming process. These experiences have been gained during the last two years. During this time AutoForm together with many of their customers, compensated more than 20 different parts. Keep in mind that we didn’t just simulate the springback but the compensated surfaces where brought into the real tools. So, the experiences are not only based on numerical analysis but of all this part a physical tryout has been performed as well. The experiences are consolidated in a set of principles of robust springback compensation. These principles will be illustrated and explained at an example part, a B-pillar upper reinforcement. In the next section we will show how the B-pillar upper reinforcement will be compensated. Indeed, you will see that that compensation is a straight forward activity. However, our experiences have shown that it is only a straight forward activity in case certain boundary conditions are fulfilled. In section 3 we will discuss some of this boundary condition using the B-pillar upper example. This section will show you that springback compensation is a ‘Virtual high fidelity modeling in focus of multiple real engineering issues through product design to production engineering’.

 63,1*%$&.&203(16$7,212)$%3,//$5833(55(,1)25&(0(17 In this section we will introduce the B-pillar upper reinforcement. We will have a look at the process plan of producing this part and we will have a closer look at the springback process. After understanding the process we will show how this part has been compensated in order to get a part fulfilling the geometrical requirements.

 6RPHGHILQLWLRQVIRUPLQJSURFHVVDQGVSULQJEDFNPHDVXUHPHQWV The process plan in order to produce this part exists of three operations. Figure 1 shows a schematic overview of the process plan. In operation 20 the part has been drawn, segmented trimming and piercing of the part takes place in operation 30 and 40. In operation 50 the flange will be formed. After this last forming the spingback will be evaluated.

(a) Process planning and definition

(b) Constraint for Springback

),*85(Forming processes and springback measurements

(c) Sample points

In order to evaluate the springback in a repeatable way, we define a so called constrained springback. The constrained springback represents the springback evaluation in a fixture. It is very important to assure that the clamping concept doesn’t influence the springback behavior in a negative way by distorting the part and assembly forces are not exceeding a certain forces. One should avoid over-determined clamping. The part position has been defined by two pilots positioned in the hole and the slot of the part and the three clamps. With these pilots and clamps the 6 degrees-of-freedom of the rigid body motion have been suppressed (Figure 1(b)).

1101

In this clamped position the springback values have been evaluated. We distinguish two different springback values. The material displacement represents the amount of springback of a material point; the distance from reference represents the difference of the part after springback and the reference geometry. In this paper we will evaluate and discuss the springback results in a set of well defined sample points (M0 – M10). The sample points are positioned at the edge of part, starting at left flange M0 following the part edge going around the corner ending at the tip of the flange M10. Figure 1(c) shows the positions of the 11 sample points.

6SULQJEDFNDQG&RPSHQVDWLRQUHVXOWV Springback reduction to help the reduction of troubles in production line and springback compensation loop already begins in the product design stage. Some geometrical stiffness affects the amount of springabcak, as known as that a relatively flat structure is more prone to springback with the in-plane shrinkage and its reduction with reinforcement ribs. Part stiffness related with thickness, radii can also improve the dimensional stability. To understand this geometrical stiffness effect balanced with stress distribution formed by the manufacturing process is not an easy and trivial task. Part designer designs his part with thousands of geometrical parameters decided by not only manufacturing but also structural performance of final product. To do this partially integrated engineering process between manufacturing and performance of the part in very early stage, analytical methodology of directional stress release is developed to understand this geometrical stiffness and stress distribution by forming processes. This directional stress release would be directly used in collaboration part design stage so that part designer could get an idea to reduce the springback and understand the springback mechanism of his part during the forming processes. Directional and 3 dimensional selective stress release method is applied to this B-pillar reinforcement as part designer and press engineer like to know the springback mechanism. Since it is the decision of engineering sense, operation 20 draw and operation 50 flange forming stages are investigated where major stress direction is reference for directional selection for stress release.

(a)Nominal Draw (b)Nominal Flange (c)X-dir Draw (d) X-dir Flange

(e) Y-dir Draw (f) Y-dir Flange

),*85( Springaback sensitivity on the forming process and directional stress distribution

Figure 2 presents the springback displacement from the reference geometry and table 1 presents the comparison values on measuring points which are feely defined apart area from center of part, because of its large values on outside edges. 4 points are selected on draw die and 8 points are selected on flange die. These measuring points are depicted on figure 2 (a) and (b) with diamond marks, and the clockwise numbering from right upper side is used. 7$%/(. Springback values from sensitivity models &DVH2SHUD WLRQ 1 2 3 Nominal Draw 2.243 1.511 -0.754 Flange 1.267 1.078 -0.494 X-dir Draw 3.113 1.998 -1.120 Flange 1.364 0.823 -0.342 Y-dir Draw 0.756 0.591 0.511 Flange 0.920 0.806 -0.602

0HDVXULQJSRLQWV 4 5 -1.794 -0.260 -0.011 -2.740 -0.318 0.072 0.005 -0.258 0.034

6

7

8

0.091

-0.125

-0.249

0.154

-0.314

-0.137

0.047

0.129

0.024

The amount of springback being compared on the draw and flange stage along with each global direction X and Y, Y direction having a large portion of crossed forming radii is effective stress control for draw operation. The reduction of springback is drastically improved to nominal draw operation, but it does not so much affect the amount

1102

of springback for flange operation. This implies the compensation of draw die also not influential idea for final part shape. The same way is also applied to operation 50 flange forming as the three different local zones and direction are defined to know the sensitivity of springback in flange operation only. First model deals with entire area of flange operation and global Y direction. The 2nd model has selected local flange radii area for stress release and stress alignment vector direction is defined along with the cross sectional flange radius. 3rd model has been defined as whole flange surface; because of balance issue with in-plane stretching, stress distribution and direction vector is the identical to 2nd model.

(a) Global Y direction and whole area (b) Local flanges radii area and direction (c) Local flange whole area ),*85(Springback sensitivity on the flange operation

Figure 3 shows the improvement of springback values with local sensitive area in flange radii and further improvement with extended area to whole flange shape. Comparing these three results, we came to the conclusion that the flange radii area is critical and stress distribution of flat area of flange is influential area as well.  7$%/(. Springback values improvement from sensitive local area selection models

&DVH2SHUDW Global 1st Local 2nd Local 3rd

LRQ

Flange Flange Flange

1 1.444 -0.190 -0.0318

2 -0.223 -0.812 -0.614

3 0.103 -0.644 -0.675

0HDVXULQJSRLQWV 4 5 -0.030 0.003 -0.252 0.048 -0.259 0.047

6 0.228 0.058 0.055

7 0.001 0.163 0.169

8 -0.371 0.069 0.065

In next step, springback compensation strategy is studied. The springback values in the sample points of the basis simulation are shown in Figure 4(a). As can be seen the first sample point only shows a small springback value which is also not very surprising since it is situated relatively close to a clamping point. From the third to the eighth sample point we find negative values meaning the flange moves down. The last three sample points on the flange of the B-pillar upper reinforcement have positive value, the flange springs up. In order to evaluate the geometrical accuracy of the part, the part after springback has been compared with the wanted part geometry also called the reference geometry. The geometry of the forming tools of the basis coincides with the geometry of the final product. So, if we evaluate the distance from reference, these values for the basis match well with the material displacement. Only points 7 and 8, at the bottom radius of the flange, show a deviation of around 0,1 mm meaning that the radius is not formed exactly. This effect can be caused by intermediate springback effects in the various operations (OP30/OP40) and can be neglected in many cases.

(a) Basis springback values

(b) First compensation

(c) First and second compensation

),*85( Springback results of the nominal and several compensation loop

1103

During geometrical compensation the active tool geometries are compensated for the amount of springback. So, we ‘extract’ the amount of springback from the tool geometry in the hope that the part springs into the correct geometry. One compensation loop of the B-pillar upper reinforcement has been executed. The drawing operation of the part has been compensated as well as the forming operation of the flange. Figure 4(b) shows the springback values of the basis simulation as well as of the first iteration. Let’s first compare the material displacements. It can be seen that the material displacement hasn’t changed that much. So, the amount of springback before and after compensation remains almost identical. Comparing the distance from reference of the basis simulation and first iteration clear differences can be distinguished. The distance from reference of the first iteration shows values smaller than 0,2mm meaning that the deviation from the wanted geometry as smaller than 0.2mm. So, the first compensation loop already improved fulfilling our geometrical tolerances considerably. In case of this effective convergence behavior of the geometrical compensation, these surfaces will already be used for milling the actual tool surfaces. Nevertheless for illustration purposes, a second compensation loop based on the remaining deviation has been executed. The springback results of this second iteration have been added to the graph of springback results as can be seen in Figure 4(c). One can see that the distance from reference has been reduced; the maximum deviation is now less than 0,1mm. One can also see that the material displacement of the second iteration again remains almost unchanged. In this section we have shown the springback analysis and the compensation results of the B-pillar upper reinforcement. Indeed, you have seen that the compensation is a straight forward activity. However, our experiences have shown that it is only a straight forward activity in case certain ‘boundary conditions’ are fulfilled. In the next section we will discuss some of this boundary condition, or requirements for successful springback compensation. I would like to repeat that the focus is not on a numerical exercise but that we want to be better and faster at the physical tryout.

 5(48,5(0(176)2568&&(66)8/635,1*%$&.&203(16$7,21 Several requirements in order to successfully compensate for springback will be discussed at this section. You just saw in the last section how well the compensation worked; in two iteration loops the final geometry matched the reference geometry. So, in this discussion the basis simulation is defined as the reference and the springback results will be compared with this basis. The first requirement says you have to simulate all operations in order to form the part; you have to execute a so called full process simulation. Next, you need to have a proper basic simulation. A proper basic simulation means that no splits and wrinkles must be present anymore and that you simulate with the final blank shape. The last inevitable requirement discussed is that you need a robust process, if you process is not stable and repeatable your springback cannot be predicted. The next three sections will illustrate these three requirements on our B-pillar upper reinforcement.

)XOOSURFHVVVLPXODWLRQ Generally you need more than a drawing operation in order to give the part its final geometry. Also in case of our B-pillar upper reinforcement subsequent operations like trimming, piercing and reforming are needed. Since the final geometry needs to fulfill the geometrical requirements the geometry in that state must be evaluated; and the sprinback values at that state are needed to effectively compensate the part. Figure 5 shows sprinback values of the sample points at two different stages, first our basis at the end of the last operation, second the same points at the end of drawing. In order to make a fair and reproducible comparison the same constraints after drawing are applied as at the end of the process. You can clearly see that the springback values strongly differ. It is not only a translation of the points, the sample points of the flange (points 9 – 11) show opposite springback behavior. Where the points of the basis spring up, these points spring down at the end of drawing.

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),*85(Springback results of the basis simulation at the end of the forming process and at the end of drawing

So, if we would have compensated our tool surface with the values at the end of drawing we would not be successful. Especially the flange area would be far away from the required geometry.

3URSHUEDVLFVLPXODWLRQ Before starting to compensate the active surface the forming process must be improved until no defects like splits and wrinkles are present anymore. In case the tryout team has to fix the last details; the compensated surface are of no value anymore. In the next to sub-sections we will show how springback values are dramatically influenced by only small defects like a split or a wrinkle. 3.2.1 Only a small split Figure 6 shows an intermediate result in the process development of our B-pillar upper reinforcement. You can see one split in the part, the split is not even in the final part geometry; it is in a trimmed off area.

(a) Formability plot showing split

(b) Amount of springback values

),*85( An intermediate results showing a split and its influence in springback

Very often these kind of small defects are shifted to the tryout team. In order to avoid this small split the tryout team takes several measures. The measures like modifying the blank holder force or blank geometry result in changing process parameters which on this turn result in (slightly) changing results. So, what looked like a harmless split actually destroys the cumbersome elaborated process set-up In our simulation we will show how this small split will significantly influence the springback results. Figure 5(b) shows springback values of the basis simulation and springback result of a simulation with a split. In order to only study the influence of a split and not comparing two totally different processes the following approached has been followed. We took the basis simulation and forced a split 7mm before bottom stroke by cutting the split at the position as indicated in Figure 6(a). You can clearly distinguish that the split has an influence on the springback values. Around the corner of the part (points 5, 6) the springback is reduced with 0.2 mm which is in this case 50%. So, using the values of the part with the split will result in different compensated active surfaces.

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3.2.2 Only a small wrinkle An identical procedure has been followed in order to illustrate the effect of only one small wrinkle in the simulation. Figure 7(a) shows the position of the wrinkle which is visible as a small ‘bump’ on the part. Again we took the basis simulation and ‘formed’ the wrinkle at the end of the forming process. This methodology results in an almost identical strain and stress distribution only a local geometrical deflection is present.

(a) Small wrinkle on the part area

(b) Amount of springback valuses

 ),*85( The B-PLR UPR reinforcement with a small wrinkle and its influence in springback

Figure 7(b) shows the spingback values of our sample points for both the basis simulation and the simulation with the forced wrinkle. You can see that this geometrical feature stabilizes the part considerably; the springback values are much smaller than those of the basis simulation. So, if the compensation has been executed on the smaller values the compensation would underestimate the amount of springback. So again, using the ‘wrinkle results’ would lead to incorrect compensation. The tryout team will eliminate this wrinkle and creating a different basis for compensation as we used for our compensation. Additionally, these kind of small ‘accepted’ wrinkles result in not completely closed tools during forming. The not completely closing of the tools results in a process change because of the change contact pressures. The springback behavior will in the case change even more dramatically.

5REXVWSURFHVV The numerical compensation is based on a simulation. Even if we secured that this result is fulfilling all feasibility requirements; we don’t know for sure how representative these springback values are. In real life we have to deal with tolerances for many parameters. This makes that these parameters can be slightly different from our basis simulation. In order to evaluate how representative our springback results are we will execute a robustness analysis. The robustness analysis shows how sensitive the process reacts on noise. In case the variation of the output, like springback is only limited, we speak about a robust process; which on its turn assures that our basis simulation gives representative values for springback. In case the process is not robust the springback result might differ a lot from our basis simulation. Figure 8(a) shows the process capability Cp of our drawing process. The red spot indicates high variation of our failure value indicating a considerable risk of splitting. In case this splitting occurs during tryout, the tryout time will eliminate this split probably by smoothen the geometry.

(a) Process capability with the respect to split (b) Locally smoothened tool geometry to prevent splitting ),*85( Process capability with the respect to failure of the B-PLR UPR reinforcement

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Figure 8(b) shows the positions where the geometry will be smoothened. This modified geometry will result again in different strain and stress values in our B-pillar upper reinforcement which on it turn will result in different springback values. Figure 9 shows the springback values of the basis simulation and the smoothened or ‘grinded’ simulation. Smoothening the geometry leads to larger springback values of the corner. The values increase from -0.49mm to 0.75mm and from -0.32mm to -0.71mm for point 5 and point 6 respectively. In the flange (point 11) we see a reduction of the springback value from 0.94mm to 0.52mm. So, these different springback results will again result in different compensations.

),*85( Springback results of the basis simulation and a comparable simulation with a forced wrinkle

In order to avoid this kind unforeseen differences between the basis simulation and the tryout result it is strongly recommended to secure that the process in robust. Only if the process delivers repeatable results simulation and tryout will match, and only in that case the numerical compensated surfaces will deliver a geometry fulfilling the geometrical tolerances.

 6800$5<$1'287/22. In this paper we presented our experiences and investigations in order to geometrically compensate forming processes. Our experiences have been illustrated and at an example part, a B-pillar upper reinforcement. We showed that springback compensation is indeed a straight forward activity. But it will only lead to satisfying results during tryout if certain requirements are fulfilled. The requirements illustrated here are: z You need a full process simulation meaning all operations must be simulated z You need a proper basic simulation, you have to eliminate all spits and wrinkles and don’t leave anything to fix by the tryout team. z You have to assure that your process is robust, meaning it delivers repeatable result again don’t leave anything to fix by the tryout team These requirements make springback compensation an elaborated topic, it is more than just compensating active surfaces. The wide range of engineering processes with focus on springback analysis and compensation would be more practical and applicable with this standardized simulation procedures along the whole processes chain from part design to production engineering.

5()(5(1&(6 1. Lange K.; Umformtechnik: Handbuch für Industrie und Wirtschaft, Ausg. 3; 1990 2. Roll K.; Simulation of sheet metal forming – necessary developments in the future; Numisheet 2008, S. 3-11, Interlaken, Switzerland, 2008 3. Yoshida F.; Uemori T.; Abe S.; Hino R.; A model of large-strain cyclic plasticity and its numerical simulation applications to springback prediction and compensation; Numisheet 2008, S. 19-24, Interlaken, Switzerland, 2008 4. Kubli W.; Krasovsky A.; Sester M.; Advanced modeling of reverse loading effects for sheet metal forming processes; Numisheet 2008, S. 479-484, Interlaken, Switzerland, 2008 5. Dorel Banabic.; Sheet metal forming processes, Constitutive modeling and numerical simulation; ISBN 978-3-540-88112-4; Springer Heidelberg Dordrecht London New York, 2010

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Development of a New Gradient Based Strain-Criterion for Prediction of Bendability in Quality Assurance and FEA Dipl.-Ing. Ralf Denningera,b*, Prof. Dr.-Ing. Mathias Liewald MBAa and Dr.-Ing. Manfred Sindelb a

Institute for Metal Forming Technology – Holzgartenstr. 17, 70174 Stuttgart, GERMANY Webpage: www.ifu-stuttgart.de b Audi AG – Postfach 1144, 74148 Neckarsulm, GERMANY Webpage: www.audi.de

Abstract. Numerical simulation systems are more and more used in process development of car bodies. Nowadays, also the hemming process is optimised in FEA. Thus, the analysing of process robustness calls for a failure criterion for the specific bending and hemming load condition. For that purpose the experimental determination of bendability under various pre-load conditions that occur in real production, e.g. during deep drawing in press shop, is content of this contribution. Using these experimental results, a new approach for a strain-gradient based failure criterion for bending operations is presented to optimise bendability prediction. The bending-strain-gradient approach can be used both in production related departments of quality assurance as well as for simulative process design or process validation for vehicle manufacturing planning.

Keywords: Sheet metal forming, Sheet metal bending, bending and hemming, material characterisation PACS: 81.20.Hy

INTRODUCTION Due to the legal framework conditions with regard to energy efficiency and pollutant emissions of automobiles prescribed within the EU within the last few years, European automobile manufacturers have been increasingly required to make stepwise reductions in fuel consumption of their produced car fleets. To meet the requirements made of them, large automobile manufacturers have increasingly relied on the use of modern, light-weight construction designs for car bodies. The use of light materials linked with weight reduction in the construction of car bodies results in new technological challenges within the manufacturing processes, nevertheless, automobile manufacturer expects a constant or even improved component quality without any increase in manufacturing costs. To fulfil such goals, deeper knowledge of materials behaviour regarding its processability and limits due to failure along entire process chain of modern car body manufacturing looks indispensable today. This contribution discusses material failure caused by incipient surface cracks on bending edges during predominant bending processes such as those that occur, e.g. when flanging and hemming vehicle body part along its break line. Main thrust hereby is a comprehensive examination of the local changes in shape during the bending process, as well as the introduction of a new approach to evaluate the predictability of bending failure. The effects of material pre-strain on the observed bending characteristics due to previous deep drawing processes are considered within the scope of this examination. Today established trials and criteria of the characteristics of sheet metals regarding their bending/ hemming characteristics, e.g. three-point bending test (Figure 1), provide parameters for a sufficient description of material with regards to its bendability in 80% of all cases. Some of the evaluation models that build on these trials are based on determination of geometrical parameters of bending specimen (punch stroke f, position of supporting rolls c, p). Established Bending factor diagram [1, 2] is of special interest here, which indicates the critical inside radius of a sheet thickness for a defined (remaining) bending angle, depending on the level of pre-stretching. Additional approaches are based on an examination of critical strain value at the bending edges, according to the forming limit curve, (FLC) depicted in the forming limit diagram FLD as a bending limit curve (BLC) within the area of in plane major / minor strain [3, 4, 7]. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1108-1114 (2011); doi: 10.1063/1.3623727 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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FIGURE 1. Schematic representation of (left side:) the Tree Point Bending Test Specimen and (right side:) the three-point bending test compliant to the standard (DIN EN ISO 7438) [5]

This contribution sheds equal light on both of these influences and also contains a scientifically-oriented estimation of how a new approach in the evaluation of bending operations on sheet metal components can be derived from the knowledge gained. Goal of this contribution for that very reason is the publication of an approach to the evaluation of bending that can be easily applied to the various bending operations of thin sheets within the field of vehicle body construction, independent of the geometrical parameters of workpiece.

Consideration of the local forming processes within the three-point bending test Experimental Examination of the Bendability Characteristics of Thin-Sheet Aluminium Materials using Three-Point Bending Test To clearly present the requirements made of the evaluation criteria in the bending trial, various bending samples are analysed in a study, each using an identical bending angle but each with significantly different bending characteristics. The study is comprised of bending samples made of aluminium alloy (AA6016, 1.5 mm sheet thickness) with five different levels of uniaxial pre-stress (0%, 5%, 10%, 15%, 20%). All samples were bent in the three-point bending test (corresponding to Figure 1) at a defined bending radius along a pre-determined bending path f between 2 and 12 mm corresponding to residual bending angle α (Formula 1). Using such three-point bending trial compliant to DIN EN ISO 7438, the bending path and angle are geometrically independent and can be converted according to the formula 1 [5]. A bending path (punch stroke) can be assigned to a certain bending angle and to a unique geometrical state or a bending sample shape. In this contribution the bending path is to be understood as the geometrical description of bent workpiece.  ( p * c  p 2  ( f  c)2  c 2 * ( f  c)   2 2   p ( f c )    

 angle ( f punchstroke )  2  arcsin 

(1)

The differences in the geometrical characteristics (bending angle / sample shape) are infinitesimally small after bending for the variously pre-treated trial samples. However, both the condition of surface of bent edge (Figure 2) as well as amount of bending deformation (Figure 3) does show significant differences. The analysis of surface condition at bending edge describes/ classifies the material failure due to incipient surface cracks on the drawn side. The sample in Figure 2 shows the different stages from incipient cracking to deep cracks.

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FIGURE 2. Cracks on the outer side surface of the sample during increasing bending strokes in bending test

The deformation analysis of bending operation provides the explanation of the respective condition of the surface. Various geometrically identical samples are contrasted in Figure 3 from left (not pre-strained) to right with uniaxial pre-strain increasing up to 20%. The distribution of colour overlapping corresponds to the main distribution in the major strain [8].

0% pre-strain

10% pre-strain

15 % pre-strain

20% pre-strain

FIGURE 3. bending samples with different level of pre stretching

Amount of minor strain can be ignored in this contribution because during bending process the drawing side is subjected to plane elongation (φ2=0) [2, 8]. Such analysis shows extent to which pre-strained samples tend to have higher rates of deformation within a lower range. In the following, the area of a bending sample in which bending deformation occurs is referred to as the "bending affected zone“. This expression was chosen to correspond to "shear affected zone“ from contribution [6]. Aiming to quantify width of bending affected zone of various pre-strained bending samples, a deformation analysis was evaluated in the following using line cuts vertical to the bending edge. Figure 4 shows the example for the bending path of 10 mm (punch stroke), with the representation of line-cut of the major strain distribution over its length. Contrasting the various pre-stressed primary samples concretises the phenomenon of deformation localisation and also allows quantifiable analysis of the line cuts. Characteristic deformation curves of the outer fibres of respectively samples can be derived, depending on the uniaxial pre-strain of bending samples. Deformations become increasingly concentrated on the bending edge with increasing pre-strain. This is apparent both in an increase in local bending deformation as well as in a reduction of bending affected zone. In the evaluation (Figure 4) special focus should be placed on the relative differences of deformations introduced on edge. The contrast of the various pre-strained samples indicates rates of deformation that are higher by as much as 60% at identical bending angles. Significant increase of bending deformation at prestrained component respectively sample edges gives evidence to such observation.

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bending stroke = 10mm

major strain [-]

0.4

Line Cut

0 % pre-strain 5 % pre-strain 10 % pre-strain

0.3

15 % pre-strain 20 % pre-strain

0.2 0.1 0 -6

-4

-2

0 2 width of line cut [mm]

4

6

FIGURE 4. Representation of amount of deformation / definition of the bending affected zone

Mathematical description of bending localisation

A modified form of the Gaussian bell shaped curve has proven to be a suitable mathematical formula for explicit description of deformation curves of line cuts observed. Formula 2 describes the course of the deformation.

y(x)  a  e

     

 x e   b 

2     

 c 

(2)

The shape of the Gaussian formula is described in this case (with consideration of the bending operation – separate from the pre-strain parameter c ) by parameters a and b of Formula 2. Parameter a stands for height of curve and therefore the amount of the maximum measured major strain, parameter b describes width of bell curve, and therefore the width of the bending affected zone. Evaluations of the experimental examinations according to Formula 2 show a characteristic bending behaviour of aluminium material, which allows the distribution function to be mathematically formulated independent of former loads (bending path and amount of pre-strain) [8]. Novel, elongation-based approach to the evaluation of material edges dominated by bending The basic idea for a new approach to the evaluation of bendability of sheet metal materials in the construction of vehicle bodies derives from the knowledge gained in experimental examinations. The idea of this criterion is based on a characterisation that is independent of any geometry and pre-strain, comprising an evaluation approach that corresponds to the damage in the bending affected zone, i.e. the local condition of the surface or a later growth of crack on the edge during bending [8]. In contrast to initial criteria mentioned [1, 2, 3] novel approach does not assume that large deformations or sharp hemming geometries necessarily result in material failure. Rather, this criterion supposes that kind of a critical strain gradient is required to initiate (incipient) surface cracks. In order to simplify the critical elongation gradients of the bending evaluation for the application, a factor is introduced from the major strain at the bending edge and the bending affected zone. Compliant to formula 3, the bending evaluation factor (BEF) is comprised of the quotients of the square of the major strain a² sustained during the bending process, and the width of the bending affected zone represented by the variable b.

 bending 2

a2 Bending Evaluation Factor (BEF)   width of the bending affected zone b

(3)

Using this novel approach to the evaluation of bending, all cases of bending of sheet metal can be mapped, characterised and compared with each other independent of bending shape, geometry and “heritage” linked with a bundle of meaningful parameter.

1111

Application of BEF to real material edges The BEF can conceivably be used both during simulation as a design criterion as well as for quality assurance purposes in production area. The usage of the novel criterion does only require a forming analysis comparable to common evaluations. Usage of the BEF in FEM Simulation and Comparison with bending strain criterion The evaluation of simulation results of with different criterion is shown in the following section. The evaluation of the forming analyse, ether in FEM simulation or in quality assurance includes several line analysis along hemming edge. The edge geometry is analysed by regarding strain values during bending load. In Figure 5, the results of the strain analyses are compared to the novel BEF criteria.

risk of fracture

a) Region of Interest b) Evaluation by pure bending strain c) Evaluation by BEF FIGURE 5 Comparison of common criteria and the novel Bending Evaluation Factor

Regarding the statement of the results, there are significant deviations between the simple strain criterion and the BEF. By using the bending strain criterion (Figure 5.b), wide areas of the hemming edge are quantifies with high risk of fracture. In contrast, the novel BEF criterion (Figure 5.c) separates the truly critical areas more clearly. Independent of the actual limiting fracture-value for each criteria, there are significant differences in its sensitivity. Especially for the red highlighted area, where cracks appear on real parts, the advantages of the EBF became obvious. While there are very few characteristic differences in the major strain values (Figure 6 - blue line). The BEF (green line) clear separates the critical and non-critical areas along the edge very precisely. 0,6

1 major strain

0,5

bending evaluation factor high risk of fracture

0,8

0,4 0,6 0,3 0,4 0,2 0,2

0,1

0

0 0

10

20

30

40

50

No. of Line Cut

FIGURE 6. Comparison of common criteria and the novel Bending Evaluation Factor

This analyse shows, that the novel BEF reflects reality more precisely than a simple bending strain analyse.

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Evaluation of BEF results Simulation of forming process or deformation analysis of real specimen thereby delivers a line cut of the bending edge to be examined. The characteristic values for the bending evaluation factor can be calculated from the line cut according to the described procedure, and be compared to a critical limit value defined for particular material. Ratio between the BEF and the limit value can thereby assume values between 0 (not critical) and 1 (high risk of failure). This quotient represents a kind of "crack probability" of a bending edge, equivalent to the "residual bendability" (Formula 4). To generate characteristic limit value for a material and calibrate a defined blank surface quality the three-point-bending-test is used. Therefore only very few specimens are needed, because this criterion is independent of strain path.

Evalutaion Ratio 

BEF Limit Value

(4)

An example for the application of the evaluation ratio has been carried out on a real sheet metal component. Multiple line cuts were generated at several positions along a fairly straight slope of edge of component, and elongation curves were measured / assessed according to the BEF criteria. 0.4 Gauß'sche Interpolation 95% Vorhersagegrenzen Messwerte

0.35

major strain [-]

0.3 0.25 0.2 0.15 0.1 0.05 0 0

2

4

6 BAZ [mm]

8

10

12

FIGURE 7. Evaluation/ Interpolation procedure of measured values (left) from one of the line cuts (right) during flanking operation

A particular line cut-was analyzed in more detail in Figure 7. compliant to the analysis of bending edges explained above. This takes into account that this deformation analysis comprises both deep-drawing operations as well as trimming of component. Consequently, general form of used Gaussian formula was adjusted in such a manner that bending effect now can be assessed independently of pre-stress conditions. This is done by offsetting the x-axes of the curve from their origins by parameter e and the y axis taking into account the pre-stressing or the predominant deformation value c within the immediate area.

TABLE 1. Parameter for evaluation ratio applied to Aluminium material AA6016, sheet thickness 1,25 Value System of Units 0.2862 [-] 0.0489 [-] 0.3351 [-] 3.426 [ mm ] 1.454 [-] 0.0056 [-] 0.008 [-]

Function / Variable a pure bending strain c pre-strain a+c total major strain e offset from center b “length” of bending affected zone a² / b Bending Evaluation Factor cV characteristic fracture value for this material - gained from experiments

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From the analysis of the line cut, a value of 0.286 is obtained for the (pure) bending deformation, and a value of 1.45 for the bending affected zone, whereby a BEF of 0.0056 results for the quotient. Measured against the comparative samples, the BEF of 0.0056 is within a range that can still be considered uncritical, compared to the characteristic material fracture value of 0.008.

Summary As is clear from the examinations at hand, the further developments of existing approaches as well as the development of new approaches are both required for the clear evaluation of the bending behaviour of sheet metal materials. The studies introduced in this paper have shown values for local differences in deformation that are higher by up to 60% on geometrically comparable samples. These significant differences have been consciously taken into consideration in the new bending evaluation factor (BEF) and allow the expected edge quality to be predicted. In this approach, influences of geometry and pre-strain on the bending edge are taken into consideration solely in an indirect way, by means of the influence of the characteristic deformation distributions on the bending edge. The application of the new bending evaluation factor has been used to successfully locate the critical bending areas on the hemmed outer car body part. (Even more precisely then with any other existing post processing criterion)

REFERENCES 1. R. Schleich et al., Beitrag zur Qualifizierung von Falzprozessen im Automobilbau, UTF Science, 2008 2. F. Ostermann, Anwendungstechnologie Aluminium, 2., neu überarbeite und aktualisierte Auflage, Seite 502f, Springer Verlag 2007 3. R. Schleich et al., Beitrag zur Verbesserung der simulativen Absicherung von biegedominierten Umformprozessen im Automobilbau, Blech Rohre Profile 2008 4. R. Schleich et al., Beitrag zur Charakterisierung von Versagensmechanismen in der Umformtechnik, wt Werkstattstechnik online, 2008, Heft 10 5. N.N., Metallische Werkstoffe - Biegeversuch, DIN EN ISO 7438, 2005 6. R. Schleich, Erweiterte Methoden zur Charakterisierung der Umformbarkeit von hochfesten Stahlblechwerkstoffen, Internationales Stuttgarter Symposium, 2009 7. M. Liewald, Aktuelle Tendenzen in der Forschung auf dem Gebiet der Blechumformung am Institut für Umformtechnik der Universität Stuttgart, Internationale Konferenz „Neue Entwicklungen in der Blechumformung“, 2010 8. R. Denninger, Verbesserte Vorhersage der Biegbarkeit von Blechwerkstoffen, wt Werkstattstechnik online, Jahrgang 100(2010) H.10

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Effects of Blank Curvature and Tool Conditions on the Spring Back of Thin Sheet Panel Formed through Local Embossing and Edge L-Bending Keecheol Park, Jongyoun Park and Jaebok Nam Product Application Research Center, POSCO, 180-1 Songdo-dong, Yeonsu-gu, Incheon 406-840, Korea Abstract. Due to the application of thinner sheet steels, the stamped panels in the forming process, generally, are severely distorted. The wavy shape of embossed panel finally converted to residual stress embedded in the panel at final forming (edge L-bending) and it is known as the cause of twisting and oil canning of spring backed panel. Another important source of stamped shape deviation is the curvature of blank. The effects of blank curvature on the shape defects (panel curvature and twisting) after stamping were investigated from defective panel analysis, model experiment and stamping simulation. And the effect of tool conditions (BHF and bead height change) on spring backed shape of real TV bottom chassis were studied. The initial curvature of blank was remained in the flat area of stamped panels as width directional curvature. It converted from length direction curvature of blank. The curvature of initial blank reduced the wavy shape after local emboss forming, but twisting after edge L-bending was increased at large blank curvature cases. The effects of emboss forming conditions, the forming heights and blank holding force were studied and it was found that the wavy shape of stamped sheet was rapidly changed although the forming conditions altered very small amount. Keywords: Blank, Stamping, Embossing, L-Bending, Spring Back, Curvature, Twisting, Oil Canning, Residual Stress PACS: 89.20.Bb

INTRODUCTION In the digital display products such as LCD and PDP TV and monitors, steel chassis are applied in order to sustain parts and maintain stiffness of display product. Due to the strict quality requirements on these display products, very small geometric tolerances are allowed and sheet steel parts should have sufficient stiffness. A representative stamped part in the display products, called bottom chassis, has many protrusions in the panel for parts assembly and strengthening, as shown in Fig.1 (a). It is made through local stretching (embossing) and edge Lbending successively. A typical example of tool is illustrated in Fig.1 (b).

i–››–”G ŠˆšššVG l”‰–šš

(a) Typical Bottom Chassis of LCD TV (b) An example of typical forming tool FIGURE 1. Two typical blank feeding methods applied in the stamping digital display parts. One is coil feeding line (a) and the other is sheet feeding line (b)

Because of the forming characteristics, typical shape defects such as oil canning and twisting those are allied to uneven distribution of deformation and spring back, are easily come out [1]. As shown in Fig.2, oil canning is a kind of wave formed on the flat surface and twisting is deformation of a corner of panel normal to a plane made by three remaining corners. Curvature of panel is un-flat surface of stamped panel. The residual stress appeared due to uneven distribution of emboss forming in plane sheet is known as the source of shape defects [1]. As thinner gauge

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1115-1120 (2011); doi: 10.1063/1.3623728 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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sheet steels are applied in digital display parts, the surface of stamped panels in the forming process, generally, are severely distorted. From experiment and numerical simulations, it was verified that the distorted or wavy shape of panel finally converted to residual stress embedded in the panel at final stage of forming (edge L-bending) and it is the cause of twisting and oil canning of spring backed panel [2]. {žš›•Ž

v“GŠˆ•••Ž

Ⱦ jœ™ˆ›œ™Œ

FIGURE 2. Three typical shape defects of such as oil canning, twisting and curvature.

In addition to the residual stress generated during stamping, the initial blank curvature also influence the geometry of panel and shape defects such as panel curvature, oil-canning and twisting as depicted in Fig.2. Because the digital display parts are generally retain large portion of surface as original blank and curvature change during forming and edge L-bending produce bending stress which constitute a part of residual stresses, three shape defects could be affected from blank curvature. The effects of blank curvature on the shape defects of panel were analyzed from defective sample analysis, model experiment and forming simulation. And another influencing factor on stamped shape was tool conditions. The effects of bead forming heights and blank holding force were studied.

SOURCE OF CURVED BLANK AND PANEL CURVATURE In the case of stamping lines where coil feeding is adapted as shown in Fig.3 (a), there is big curvature change along the length of sheet. Though feeder levelers at the entrance of press are equipped, it is generally known that it is not perfect for leveling the curvature change within the coil. But in the case of sheet blank feeding cases (Fig.3 (b)), generally, the leveler in the cut to length line is relatively perfect for leveling curvature in the coil if leveler operated properly. But in digital display part stamping industry, coil feeding is dominant process. Therefore much more consideration should be applied on the feed leveler operation and coiling process in the sheet processing lines.. j–“G š“››•ŽG “•Œ

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The major source of curvature variation in the coil is coiling [3]. The schematic illustration of sheet deformation in the coil is shown in Fig.4. In order to verify the deformation of sheet in the innermost part (tail) of coil, strips (divided as 7 strips along width) curvature after slitting (before recoiling) in three positions of coil (head, middle, tail) were measured and summarized as in Fig.5. It was found that at the innermost of coils, large plastic deformation occurred. By repeated coiling and uncoiling along the process lines, the head and tail sheets in the coil are subjected to plastic deformation and those must have large curvature. ΄ΜΖΝΡ͑ΓΠΨ͑ΠΗ ͢Ξ͑ͽΖΟΘΥΙ͙͑ΞΞ͚͑͑

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1116

By considering the thickness distribution along width of sheet (Fig.6 (a)) and coiling tension, circumferential stress within coil is computed as Fig.6 (b). The model[3] used for analyzing stress within the coil treats coiling as lap by lap and consider coiling of one lap as non-uniform radial stress applied on the hollow cylinder. The nonuniform radial stress is obtained from coil outer radius distribution along width and coiling tension. Circumferential stress concentrated in the center of innermost part of coil. By adding the bending stress on the circumferential stress and considering the yield stress of sheet (210MPa), sheet located less than 400mm radius in the coil affected the plastic deformation and curvature change occurred. ͢͟͢͡

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A geometric dimensional defects occurred in curved blank case is shown in Fig.7. If flat blank is applied, there is no spec. out panels but dimensional problem appeared when stamping width directional curved blank. The quantity of dimensional problem can be computed by simple geometrical analysis and blank curvature (0.084m-1). $



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(a) Blank with curvature (0.084m-1) (b) Stamped part and section deformation (0.4mm spec. out for 400·WU3mm tol.). FIGURE 7. Blank with width directional curvature (a) and stamped bottom chassis of LCD TV and section A-A (b).

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A-Case: 0.31m-1 (Length), 0.10 m-1 (Width) B-Case: -0.06m-1 (Length), 0.10 m-1 (Width)

(a) 32” size stamped panel and width dir. curvature

(b) Curvature of EDM cut specimens

FIGURE 8. Comparison of curvature in the stamped panel and curvature of EDM cut specimens.

Another curvature defect of stamped panel case is shown in Fig.8. B-case has small panel curvature within tolerance but A-case has larger than allowable. In order to find the cause of two cases, cut samples for verifying blank shape were made from none emboss formed area. From the curvature level of cut samples, the panel curvature can be roughly computed from simple elastic relations as " N C  Q u N L 0.10m 1  0.29 u 0.31m 1 0.19m 1 " and this value is nearly the same as panel curvature of A-case (0.17m-1). Panel curvature of B-case was estimated as "0.10m1  0.29u 0.06m1 0.08m1 " and this is also nearly the same as measured panel curvature. Therefore it is possible to say that blank curvature is the major influencing factors on the panel curvature and dimensional accuracy.

1117

CURVATURE OF BLANK AND PANEL TWISTING In order to verify the effect of blank curvature on spring back after forming, laboratory tests simulating bottom chassis of LCD TV by using the material shown in Table 1. Curved blank (curvature range: 0~2m-1) along length direction of specimen were made by roller leveler. 240โ180 size blanks were used for stamping small embosses and followed by edge L-bending process as shown in Fig. 10. Embosses and beads heights are 2mm. TABLE 1. Mechanical properties of tested specimens. Specimen Thickness (mm) Yield Stress (MPa) 0.5t 0.493 238 0.6t 0.591 227 0.8t 0.781 218

Tensile Stress (MPa) 326 325 329

U-El. (%) 21.7 22.9 21.4

T-El. (%) 38.1 39.5 37.5

Without emboss forming, curved blank were edge L-bended for studying the effect of blank curvature only on twisting. When the curvature is small there is no twisting, but when curvature increases twisting occurred as shown in Fig.9. In this simulative test, the stiffness for twisting is large due to small size of panel. If the size of blank is increased or if there is in plane residual stress within the panel, the blank curvature will be more sensitive to twisting of L-bended panels. After edge L-bending, length directional curvature were reduced to nearly zero and width directional curvature appeared by elastic relation ship between orthogonal curvatures. In Fig.9 the computed width directional curvature (gutter) is nearly the same with measured values, especially when curvature level is small. ~‹›G‹™UG jœ™UGOnœ››Œ™P

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Curved blank, embossed panel and edge L-bended panel made from 0.6t sheet are shown in Fig.10. It could be seen from figure that embossed panels made from curved blanks have more uniform curvature than flat blank. The deformation of nine points and twisting were measured after hanging specimens in order to eliminate gravity effects as the same as previous report [2]. Y[WŸX_W i“ˆ•’

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The height deviation (distortion) and twisting after edge L-bending for 0.5, 0.6 and 0.8t blanks are summarized in Fig.11. The height deviation of embossed blank is significantly reduced when curved blank cases. But twisting is increased when curved blank was applied cases. The curvature of blank adds bending stress in the panel and it prevents embossed panel from distortion but it increased twisting after edge L-bending. The level of height

1118

deviation is the same for 0.5~0.8t sheet but twisting amount is reverse relation with sheet thickness. For 0.5t, 0.6t and 0.8t sheet cases the twisting level occurred is about 11, 5 and 0.1mm respectively. Because the in-plane residual stress occurred due to stamping in the sheet is about the same though thickness of sheet is different but the twisting stiffness is increased when thickened sheet case. Therefore, twisting reduced dramatically as thicker sheet is applied.

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The effect of blank curvature and thickness on twisting was analyzed according to previous procedure [2] by small blanks (240×180) such as flat blank (0.60t), curved blank (0.60t) with 1.0m-1 curvature and thinner flat blank (0.57t). As shown in Fig.12, curved blank (5.7mm twisting) showed 18% larger spring back deformation (twisting) than flat blank (6.7mm twisting) case. And in the case of 5% thinner blank case, twisting increased 1.8%. This result is the same trend with the experimental results.

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(a) Flat blank 0.60t (b) Curved (1.0m-1) blank 0.60t (c) Flat blank 0.57t FIGURE 12. Twisting of blank (240×180, YS=207MPa) with and without curvature (emboss h.=2mm,Ɇ =0.14, BHF=30ton).

STAMPING TOOL CONDITIONS ON SPRING BACK OF PANEL In order to investigate the effects of representative forming tool conditions on spring back shape of stamped panel, a typical bottom chassis of 940mm diagonal length size was analyzed by varying blank holder force and depth of frame bead. Selected model for analysis and a simulation result of spring back shape are shown in Fig.13. Twisting and bending are measured by hanging the specimen and computed by Pam-Stamp software [4]. The material was modeled by isotropic hardening and normal anisotropic Hill 48 law. iomd[WW›–• {žš›•Ž

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ɍ =535MPa(0.005637+ȿ p)0.2037 (ravg. = 1.37, Ɇ =0.14).

1119

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(a) Effects of BHF (b) Effects of frame bead depth FIGURE 14. Effect of blank holding force and frame bead depth on spring backed panel.

In the case of 6.0mm frame bead height, spring backed shape according to blank holding forces are shown in Fig.14 (a). As increasing BHF, the bending in stamped part is reduced but twisting is increased. In large BHF case, amount of flange draw in is reduced and therefore residual stress level also lowered, but with reduced bending the tendency of twisting is increased due to the increased in plane residual stress by flattening (or reduced bending). Frame bead height effect on stamped shape is shown in Fig.14 (b). Dramatic and complicated change in spring backed shape was predicted as frame bead depth differed. But the trend in bending and twisting is contrary to each other as in the case of BHF effect. From the simulation results, it could be possible to say that in real stamping situations the effect of forming conditions such as BHF, friction and emboss forming depth are too big to easily adjust or control the shape of stamped panel if the tool conditions are not robustly established. TABLE 2. Comparison between computed and measured panel shape at frame bead depth is 5.7mm case. Data Twisting Bending Computed 15.0 10.0 Measured 14.5 8.9

In the case of 5.7mm frame bead depth, the computed and measured twisting and bending of stamped panel were compared in Table 2. The real stamped shape was predicted about 10% error range. Therefore stamping simulation could be applied to design robust tool conditions for press formed parts of digital display equipments.

CONCLUSION The effects of blank curvature on the shape defects (panel curvature and twisting) after stamping were analyzed from defective sample analysis, model experiment and stamping simulation. And the effect of tool conditions (BHF and bead height change) on spring backed shape of real TV bottom chassis were investigated. The initial curvature of blank was remained in the flat area of stamped panels as width direction curvature. It converted from length direction curvature of blank. The curvature of blank reduced the wavy shape after local emboss forming, but twisting after edge L-bending was increased at large blank curvature cases. Therefore, exact curvature control of blank is important for improving shape quality of stamped panels. The effects of emboss forming conditions, the forming heights and blank holding force, were studied and it was found that the wavy shape of stamped sheet was rapidly changed although the forming conditions altered very small amount. But stamping simulation could be applied to find robust tool conditions for press formed parts of digital display equipments.

REFERENCES 1. K.C.Park, “Effect of Material and Forming Conditions on the Shape Defects of Sheet Panel Formed Through Local Embossing and Edge Bending” in Advanced Technology of Plasticity 2008, edited by D.Y.Yang et al., Proceedings of 9th International Conference on Technology of Plasticity (ICTP2008), Korean Society for Technology of Plasticity, Gyeongju, Korea, 2008, pp. 558-563. 2. K.C.Park and J.Y.Park,” Analysis on the Cause of Twisting Defects Occurred in Sheet Panel Formed Through Local Embossing and Edge Bending” in NUMIFORM 2010, edited by F. Barlat et al., Proceedings of the 10th International Conference on Numerical Methods in Industrial Forming Processes, POSTECH, Pohang, Korea, 2010, pp.882-887. 3. S.Yanagi, S.Hattori and Y.Maeda, “Analysis Model for Deformation of Coil of Thin Strip under Coiling Process”, Journal of JSTP, Vol.39, No.444, 1998, pp.51-55. 4. ESI Goup, PAM-STAMP 2G & PAM-TUBE 2G User’s Guide, ESI Group, 2009.

1120

A Springback Compensation Method for Complex-Shaped Flange Components in Fluid-Cell Forming Process Xiaoqiang Lia, Dongsheng Lia, Weijun Yanga a

School of Mechanical Engineering and Automation, Beihang University, No. 37 XueYuan Road, HaiDian District, Beijing, 100191, PRC Abstract. The fluid-cell forming process (FFP), one of the main forming technologies in aeronautical manufactories, suffers seriously from the product quality problem due to springback. A springback compensation method is proposed particularly for complex-shaped flange parts. This method is derived from the classic displacement adjustment method, but differs in the way to predict the springback amount and calculate the adjusting vectors. Based on discussion on the relationship between the springback angles and the geometrical parameters of flange parts, a springback distribution function (SDF) is proposed to predict springback angles of curved flanges. The springback angles are mapped to nodal adjusting vectors of the die mesh and then the optimized die is obtained. The above procedures are applied to two fuselage flange parts and the experimental result shows that this method is valid.

Keywords: Fluid-cell forming process, Flange components, Springback compensation, Springback distribution function PACS: Replace this text with PACS numbers; choose from this list: http://www.aip.org/pacs/index.html

INTRODUCTION FFP is a kind of flexible sheet metal forming technologies [1, 2]. The low cost and satisfactory product quality makes the process popular in the aeronautical manufacturing industry of small quantities and large varieties. In FFP, there are a large number of products which can be sorted in flanges components. As the deformation is close to pure bending in the flange parts, the geometric errors due to springback seriously affect the product quality. Therefore springback compensationୈthe most ideal approach to reduce springback errors for sheet metal forming [3], is desirable in the tool design of FFP. A lot of works has been done on springback compensation with FEM in sheet metal forming processes. Karafillis and Boyce proposed a method denoted as “Spring-forward” which was capable for compensation of 3D geometries [4]. W. Gan and R.H. Wagoner developed “displacement adjustment” (DA) method [3]. Compared with "Springforward" method, DA method keeps several advantages in convergence, correction and generality [3]. In fact, DA methods cannot be directly applied to complex-shaped parts. Because the compensation vectors computed by FEM are defined on the nodes of the blank mesh only, not those of the die mesh [5]. Then, several improved algorithms were further developed to make the DA method available for the arbitrary 3D components [5-7]. Their work extends practical application of the springback compensation technology. However, for these compensation methods with FEM, it takes a long time to analyze the forming and springback procedures and perform the iterative optimization. For FFP, it is limited to apply these methods to the practical manufacture, because the large number of product varieties leads to an unacceptable huge period for analysis work. So if there is a new tool design method which utilizes the compensation method without long time of analysis work, it will be a great promotion for the product quality and efficiency of FFP. In this study, a compensation method is introduced, which offers a quick and accurate solution of springback compensation for complex-shaped flange parts in FFP.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1121-1128 (2011); doi: 10.1063/1.3623729 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

1121

PART GEOMETRY More than 90% of the parts formed with FFP in aeronautic manufacture can be sorted in flanges components. These parts are designed as the sub-load-bearing structures and used for the connection between outer shapes and inner fuselage components. They are composed of a web plate, several flanges and other features, i.e. holes, reinforcing posts. The two typical parts are shown in figure 1(a). Web plate

Hole

Web plate Straight flange

Convex flange

Straight flange Straight flange

Straight flange

Curved flangeflange Straight

Curved flange

Concave flange

Curved flange

(a) Typical flange parts (b) The flange shapes FIGURE 1.Typical flange parts and shape

The flanges can be categorized into two kinds: straight flanges and curved flanges, as depicted in figure 1(b). The curved flanges can be sorted as convex flange and concave flange. The main parameters of a curved flange are illustrated in figure 2. L0 is the length of the flange curve and R is its radius. h is the flange height, D is the flange angle and r is the bending radius. For a curved flange, R is variable at different location of the flange curve. Web plate L0

Bending curve

R r

¢ h Curved flange

FIGURE 2. The schematics of a curved flange.

EXPERIMENT INVESTIGATION A series of flange experiments is designed to investigate the influence of flange geometry on the springback.

Material, Die and Blank Material is Al 2024 sheets in W temper and its thickness is 1.016mm. There are two groups of curve flange die and one groups of straight flange die. One side of curve flange die is convex flange and the other side is concave flange. Die geometry and parameters are shown in figure 3(a) and table 1. R1 is curvature radius of concave flange and R2 is curvature radius of convex flange. Radius of die is 3 mm which is often used in tooling design in factory. There are different dimension of blank for curve flange considering where the effect of different flange line length on forming is considered and the wrinkle must be avoided in order to measure springback. Blank geometry and parameters are shown in figure 3(b) and table 2. ( ) h

45Ϩ R2

R2

R1

R1

30Ϩ

h ©

 (a) Die geometry and parameter

(b) Blank geometry and parameter FIGURE 3. Die and blank

TABLE 1. Parameters of flange dies Die ID R1/mm(concave flange)

1 150

2 150

R2/mm (convex flange)

250

200

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1 45

2 30

h /mm

15

15

1122

One straight flange dies (Die ID is 3) has two types of tooling radius which shown in figure 4. Different combination of die and blank is shown in table 3. In each combination there are three times experiments. R5

R3

(a) Section (b) 3D geometry FIGURE 4. Straight flange die TABLE 3. Experiment combination Experiment ID Die ID Blank ID

1 1 2

2 2 1

3 3 -

Experiment Process Experiment is done in 45000 ton fluid cell forming press in Shanghai Aircraft Manufacturing Factory. Forming pressure is 28 Mpa. Manufactured die and blank are shown in figure 5.

(a) Die

(b) Blank

FIGURE 5. Manufactured die and blank

Measurement Method Springback angles in 9 sections of curve flange or in 5 sections of straight flange which located uniformly are measured by goniometer. Springback angles in different places of straight and curve flange are shown in figure 6-7.

FIGURE 6. Spring back data of straight flange

(a)

(b) (c) FIGURE 7. Spring back data of curve flange

1123

(d)

From experimental result we can find that: (1) The springback of straight flange is larger than one of curve flange; (2) The springback angles of convex flanges is larger that those of the concave flanges at the same radius of curvature; (3) The springback of curve flange is relative to curvature radius of bending curve. The larger curvature radius, the larger springback angle; (4) The springback of curve flange is different along the bending curve. The more close to the edge, the larger the springback angle; (5) The springback is relative to the length of bending curve. The shorter the length of the bending curve, the larger the springback angle.

SPRINGBACK PREDICTION Springback Distribution Function According to conclusion obtained through experiment, a spring back distribution function is given to predict the spring back of complex flange. T N is the springback angle of the node N, belonging to the flange curve. T s is the springback angle of the straight flange with the same bending radius and the same material as curve flange. This value represents the effect of material property and the fillet radius. Here spring back distribution function

K T N T and it represents the distribution of spring back angle along bending line. K can be expressed as such: s

Te T N ˜ T s Te

K

(1)

where Te is the springback angle at the edge of the flange, which is the maximal along the bending curve. K is expressed as the product of two function as follows:

K g ( R, L0 )

g ( R, L0 ) ˜ f (l , L0 , R)

(2)

Te indicates the relationship between the maximal angle of curved flanges and one of the straight Ts

flange with the same material and radii. g ( R, L0 ) increases as R increases and it cannot be bigger than one because curve flange inclines to straight flange when R becomes bigger. g ( R , L0 ) increases as L0 decreases and its expression is, g ( R, L0 )

(1 

bL a )(1  0 ) R SR

(3)

f (l , L0 , R ) T N T e represents the distribution rule for springback angles along the bending curve. l is

distance from curve point to the edge of bending line and represents the location of curve point. The spring back in the center of bending line is smallest and the one in the edge of bending line is biggest for equal curvature flange. The spring back increases when R increases or L0 decrease for variable curvature flange. A quadratic function is chosen to describe function f . f (l , L0 , R)

T L T 4e (1  o )(l  0 ) 2  o Te Te 2 L20

1124

(4)

f (l , L0 , R )

4e(1  Oo )(

l 1 2  )  Oo L0 2

(5)

Where To is the minimal springback angle, right in the middle of the bending curve for single-curvature flanges. Here,

Oo ( R, L0 )

Oo ( R, L0 ) 1 

To Te

(6)

c dL0  R SR

(7)

Equations (2-7) compose the spring back distribution function as follows: ­ ° K g ( R, L0 ) ˜ f (l , L0 , R) ° bL a ° g ( R, L0 ) (1  )(1  0 ) °° SR R ® l ° f (l , L0 , R ) 4e(1  Oo )(  1 ) 2  Oo ° L0 2 ° dL c ° Oo ( R, L0 ) 1   0 °¯ R SR

(8)

Coefficients in the function can be fitted by basic straight and curve flange spring back data. e flange and e

1 for concave

0.1 for convex flange because a quadratic function cannot describe the distribution law of spring

back in convex flange well. For flange with 3mm radius and Al 2024 in W temper with 1.016mm thickness the coefficient of spring back distribution function is as follows. TABLE 4. Parameter of spring back distribution function for flange with 3mm radius and Al 2024 in W temper with 1.016mm thickness Flange type

Straight flange

Concave flange

Convex flange

Parameter

Ts

a

b

e

c

d

a

b

e

c

d

value

3°

62.7

0.06

1

0.65

0.017

11.6

0.02

0.1

0.43

0.011

SDF calibrated by one material, one thickness, one temper and one flange radius can be used to predict the springback of complex flange with same material, thickness, temper and flange radius. Bending angle with 90 degree is considered in SDF and the influence of different bending angle on springback is considered in following springback compensation process.

Validation Springback angles predicted by SDF given are shown in figure 8. For concave flange, the trend and value between prediction and experiment are consistent. The trend of convex flange of No.2 die is not predicted well because wrinkles happen and influence the measurement of springback.

1125

(a)

(b)

(c)

(d)

FIGURE 8. Prediction of SDF

SPRINGBACK COMPENSATION METHOD The principle of the compensation method can be expressed as: JK JK JK JK C R  a( S  R) (9) JK JK JK where C is the compensated shape; R is the reference shape; S is the shape after springback; a is the JG compensation coefficient. Here, use V to describe the adjusting vectors. JG JG JG V a( S  R) (10) JG V is calculated from spring back which is predicted by springback distribution. Following algorithm is developed to calculate the adjusting vectors of die nodes from the springback angles predicted with the SDF. G For a single node P in the tool, Q p is the adjusting vector. G G (11) Q p 'Q
lPR

T pc

(13) (14)

D

Tp

(15) ˜ OT s 90 lPR is the bending length, 'z is the height from the node P to the web plate. T pc is the compensation angle, T p is the springback angle, a is the compensation coefficient. These parameters are shown in figure 9(a). The bending curve is meshed and curve nodes are obtained. The springback angle of each node can be calculated according to the SDF. The die node P is projected as the node N between two nodes N1 and N2 on the curve. Then the T p can be calculated from T N 1 and T N 2 with the interpolation method. D Section of Flange parts

T pc

N

Tp

D

Section of flange parts

lPR

G

Qp 'z

Compensated Reference shape After springback

G

Q uni

P

G e3

P

G e2

I3 I2

I4 I1

G e4 G e1

(a) Flange section (b) The adjusting vector of the node P. Figure 9. Flange section and adjusting vectors

1126

G The adjusting direction Q p is calculated according to the elements containing the node P in figure 9 (b). Then the G unit vector Q uni can be obtained. G 1 n G Qp Ii ei ¦ n i 1 I ¦ i i 1 (16) G G Qp Q uni G |Q p | (17) G ei is the uniform normal of the element that contains the node P. Ii is the angle at the node P in this element, as the weighting factor.

G

For the node P in the die mesh, the adjustment Q p can be calculated. Then the new die mesh is obtained.

APPLICATION VERIFICATION The method is applied to two flange parts of one regional passenger airplane. The material of parts is 2024 in temper T3 with 1.016mm thickness. The fillet radius is 3mm. Parts are formed using compensated tooling. Forming process is “Blanking in O temper, Drilling location hole, deburring, quenching, forming in W temper and natural aging”.

CASE 1 The part geometry is shown in figure 10(a) and its size is about 540mm h 150mm. Compensated tooling manufactured and formed parts are shown in figure 10 (b-c).

(a) Part CAD model

(b) springback compensated tooling

(c) formed parts

Figure 10. Part CAD model, springback compensated tooling and parts 92 10

11

12

13

Measurement of No.1 Part Measurement of No.2 Part Design

R

9

14

Bending AngleT

8

91

90

89

1

2

3

4

5

(a) Measurement locations

6

7

88

2

4

6 8 10 12 Measurement Points

14

(b) Comparison between measure and design of bending angle

Figure 11. Measurement locations and result 

Measurement locations are shown in figure 11(a). It is shown that the angle error is below 0.5e in the whole flange area excluding the edge such as points of No.1, 7, 8, 14 in figure 11(b). In these points angle measure error is large because the flange height is very small.

CASE 2 The part geometry is shown in figure 12 and its size is about 540mm™150mm. Compensated tooling manufactured and formed parts are shown in figure 12 (b-c).

1127

(a) Part CAD model

(b) springback compensated tooling

(c) formed parts

Figure 12. Part CAD model, springback compensated tooling and parts

90.5

o

91.5

Measurement of No.1 Part Measurement of No.2 Part Design

Bending Angle /

Bending Angle /

R

91.0

91.0

90.0

7 6 5 1

2

8

9

10

11

Measurement of No.1 Part Measurement of No.2 Part Design

90.5

89.5

12 13

89.0

3

(a) Measurement locations

4

1

2 3 Measurement Points

4

90.0

5 6 7 8 9 10 11 12 13 14 Measurement Points

(b) Comparison between measure and design of bending angle Figure 13. Measurement locations and result 

It is shown that the angle error is below 1e in the whole flange area. The angle error of part2 is more than the one of part1 mainly because large deformation after quenching of part2 leads to forced location of blank on the die. The solution of this problem is to size the blank using multi-roller sizing machine after quenching.

CONCLUSIONS A springback compensation method is presented for complex-shaped flange parts in FFP. Based on the discussion of springback of flange parts, the springback distribution function is proposed as the substitution of the large analysis work for springback prediction. The application to two fuselage parts show the validity of this method. Future work will also include the study of different function forms in order to obtain more approximate description of the springback distribution.

ACKNOWLEDGMENTS The authors would like to thank the National Natural Science Foundation of China (50905008), the China Postdoctoral Science Foundation funded project (201003038), the Fundamental Research Funds for the Central Universities (YWF-10-01-B08).

REFERENCES 1. G. Sala, Materials & Design, 22, 289-299 (2001). 2. H. Palaniswamy, G. Ngaile, and T. Altan, Journal of Materials Processing Technology, 146, 28-34 (2004). 3. W. Gan, R. H. Wagoner, International Journal of Mechanical Sciences, 46, 1097-1113(2004). 4. A. P.Karafillis, M. C. Boyce, International Journal of Machine & tools manufacture, 36, 503-522 (1996). 5. J. Weiher, B. Rietman, K. Kose, NUMIFORM2004 Proceedings, 1011-1015, (2004). 6. R. Lingbeek, J. Huetink, S. Ohnimus, et al. Journal of Materials Processing Technology, 169, 115-125 (2005). 7. G. Z. Shen, P. Hu, X. K. Zhang, X. B. Chen, and X.D. Li, AIP Conf. Proc., 778, 334-339 (2005) 8. N. Asnafi, Journal of Materials Processing Technology, 96, 198-214 (1999).

1128

6SULQJEDFN$QDO\VLVRI8EHQGLQJZLWK%RWWRPLQJ Takayuki Ogawaa and Fusahito Yoshidab a

Industry Solution Department, ESI Japan, Ltd., 6-14-1 Nishi-Shinjuku Shinjuku-ku, Tokyo, 160-0023, Japan b Department of Mechanical Science & Engineering, Hiroshima University, 1-4-1 Kagamiyama Higashi-Hiroshima, 739-8527, Japan $EVWUDFW The effect of bottoming on the reduction of springback was investigated by performing U-shaped bending experiments and corresponding 3D FE simulation on 590MPa level high strength steel sheet. From experiments using three punch-die sets, each have different gaps between punch and die (0, 5 and 10% less than the sheet thickness) at punch corner, it was clarified that springback decreases with increasing bottoming load to some extent but a certain amount of springback remains even under a higher load. From 3D FE simulation of the bottoming on U-shaped bending, it was found that bending stresses at punch R-corner are much reduced by bottoming, but these stresses around the end of R-corner cannot be eliminated. This is a reason why complete elimination of springback by bottoming on U-bending is so difficult. Therefore, it would be recommended in actual press forming operations to apply a certain amount of bottoming load, but it should not be too large, for reduction of springback. Another important conclusion, found in the present study, is that an appropriate choice of material model is essential for accurate FE simulation of bottoming. Furthermore, deformation of punch/die slightly affects the springback. The best combination is the use of Yoshida-Uemori kinematic hardening law for material model and 3D deformable solid model for tools. .H\ZRUGVSpringback, Bottoming, FE analysis, Yoshida-Uemori material model, Solid element 3$&662.20.fg

,1752'8&7,21 Recently, high strength steel sheets (HSSs) have been increasingly used for car body structures in order to reduce car body weight and improve crash safety. Since HSSs show significantly large springback, its reduction is a key technology in stamping. One of effective and popular techniques for it is to reduce bending moment by bottoming, where a compressive load is applied normally to the sheet at the final stage of stamping [1]. For example, the effect of bottoming in three-point air bending on the reduction of springback was reported by some researchers in the past [2-3]. On U-shaped channels, which are widely used for body frames and chassis members of automobiles, the bottoming effect was also discussed by Nagai [4] and Iwaya et al.[5], however, their conclusion is that bottoming induced bending-unbending plays a major role for the reduction of springback, but the reduction of bending moment by bottoming has a minor effect. It seems that, in their experiments, no counter pad was installed, and as a result, the bottom part of a sheet was slightly curved during forming, and it was straightened by bottoming. Contrary to the above, in the present work, we examined a possibility of reducing bending moment (and consequently the springback) in a U-shaped channel by bottoming both experimentally and numerically. For that purpose, we used a specifically designed tool set consisting of a punch, a die and a counter pad, where forming gaps (e.g., gaps between die and punch) at bending corners were set smaller (10% maximum) than the sheet thickness. In the experiment, the effect of bottoming loads on the springback was examined for three different conditions of forming gap. Furthermore, in order to investigate the die-sheet contact and the stress distribution at several stages of bottoming, 3D finite element simulation of U-bending followed by bottoming was conducted. In this, the improvement of springback prediction from two points of view, one is material modeling and the other is the consideration of deformation of tools, is discussed. Finally, a guideline for bottoming in real stamping operation to reduce springback is presented.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1129-1136 (2011); doi: 10.1063/1.3623730 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

1129

(;3(5,0(17$/ )RUPLQJ7RROVDQG([SHULPHQWDO&RQGLWLRQVRI8EHQGLQJ U-shaped bending experiments were performed on 590MPa class HSS of which yield strength was 482MPa, tensile strength 635MPa and average Lankford value 1.03. Dimensions of blank sheet were 80mm in length, 18mm in width and 1.02mm in average thickness. The tool set consisted of a die, a punch and a counter pad, as shown in ),*85( The forming gap between the punch and die were 1.02mm, which was equal to thickness of blank sheet, except the area of bending corner. To examine the effect of forming gap, three different die-corner radii, Rd = 4.02, 4.07 and 4.12mm were chosen, so that the gaps were 0, 5 and 10% smaller than the sheet thickness, respectively. The counter pressure to the sheet was kept constant (1.86MPa) during the forming by using a pad which was attached to the punch with two coil springs, as shown in ),*85( . The punch/pad travel was limited by the bottoming block. Hereafter, the punch position when the pad surface first contact with the bottoming block (the punch travel is 10.0mm) is called ‘the bottom’. Punch 28.0

.0 0 6

Punch

R3

Blank Sheet

.0 0 R2 1

Pad Coil Spring

.0 0 4

Die

Rd

30.06 Die

Die Holder

Bottom Block

20.0

(Width = 20.0)

 ),*85( Experimental set-up for U-bending and dimensions of tools.

([SHULPHQWDO5HVXOWV The bent angle of the sheet after springback was measured, as shown in ),*85( . The effect of bottoming load on the springback, when using three different die-corner radii (Rd=4.02, 4.07 and 4.12mm), is summarized in ),*85( . From this figure, it is found that the bent angle decreases with increasing punch load, rapidly at the early stage of bottoming, but it soon approaches a certain value (around 92.5o). It should be noted that the bottoming effect is saturated at a load of around 30kN, and the bent angle after springback cannot be reduced further even by a much higher load. 96 Rd4.02 Bent Angle [degree]

95

Bent Angle

Rd4.07 Rd4.12

94 93 92 91 90 0

20

40

60

80

100

120

140

Final Punch Load [KN]

),*85( Decrease of bent angle of sheet after springback with increasing punch load.

1130

')(6,08/$7,212)635,1*%$&.$)7(58%(1' )(0RGHO Usually tools are modeled by the rigid shell element, and the deformation of tools is neglected. However, as already pointed out (e.g., Tsutamori & Yoshida [6], Roll [7]), the elastic deformation of tools influences the numerical results of stamping. Hence, in the present work, we employed both the rigid shell element and the deformable solid element, as shown in ),*85(  D  andE , respectively, to investigate the effect of elastic deformation of tools on the springback simulation. For modeling of the blank sheet, hexahedron solid elements were used, where the length of the element was 0.2mm so that the sheet has five FE layers through thickness direction.

6ROLG3XQFK Blank Sheet (Solid)

Punch

Symmetry

Blank Sheet Symmetry Die Pad Spring

Bottoming Plate

6ROLG'LH

D Rigid shell tools.

PAD (Shell)

E Deformable solid tools.

),*85( U-bending models using solid blank sheet with rigid shell or deformable solid tool.

0DWHULDO0RGHOVIRU')(6LPXODWLRQ It is already well known that Bauschinger effect strongly influences the springback behavior of HSS. Although there are many papers on the improvement of FE simulation of springback by using kinematic hardening models (e.g., Eggertsen & Mattiasson [8], Ghaei and Green [9]), most of the FE simulations were done by using shell element and the pressure acting normally to the sheet surface was neglected. Our interest here is whether the Bauschinger effect of a material also affects the springback in the case of bottoming. To clarify this, in the present work, Yoshida-Uemori kinematic hardening model [10-12] (hereafter, we call it ‘Y-U model’), which describes the Bauschinger effect and cyclic hardening characteristic accurately, was used. This model is constructed in the framework of two-surface modeling, , where the yield surface moves within the boundary surface(refer to [10-12]). In the present work, FE code of PAM-STAMP2G V2009 was employed. In standard version of PAMSTAMP2G V2009, we can use Y-U model only with shell element, but not solid elements, therefore, we have developed our own 3D code of Y-U model for solid elements and implemented it using the user subroutine. The framework for implementation of Y-U model into the dynamic explicit FEM is described as follows. Assuming von Mises type yield function, the subsequent yield function is expressed by: f

3 S   : S    Y 2 2

0

(1)

Here, S denotes stress deviator,  is the back stress deviator. In FE simulation, the strain increment 'H is obtained every time step by the dynamic explicit scheme of the main routine. Using the return mapping algorithm, the plastic ~ strain increment 'Hp is calculated by the following equations. When Eq. (2) is assumed, the elastic predictor try㻿  can be expressed by Eq. (3). Since the yield criterion must be satisfied, the plastic strain increment 'Hp can be calculated as a value which satisfies Eq. (4). ~ S try ~ S

S

(2)

~ S  2G'H c

(3)

1131

3 2





try ~



S  'H p  :



try ~

S  'H p   Y 2

0

(4)

2Gn p  A

(5)

§ a 3 a * : S~ · ~ ¸ S  m§¨ b S~   ·¸ A C¨  ¨ Y 2 * Y 2 ¸ ¹ ©Y ¹ ©

(6)

Here, G denotes shear modulus of elasticity, and n p is the unit normal vector to the yield surface, which shows the direction of plastic strain increment. * , * ,  denote the relative kinematic motion of the yield surface with respect to the bounding surface, it’s size and the centre of the bounding surface, respectively. Y, a, b, C and m are material parameters. 7$%/(Material parameters of Y-U model for blank sheet [Unit: MPa]. Y 270

a0 200

b 150

Rsat 180

m 12

C1 1,200

C2 300

h 0.45

E 230,000

E0 172,000

 352.6

800 600

Stress [MPa]

400 200 0 -0.05

-200 0

0.05

0.1

0.15

0.2

-400 Y-U EXP

-600 -800

Strain

),*85( Stress-Strain curves measured by experiment and calculated by Y-U model.

A set of material parameters of Y-U model for the present sheet are listed in 7DEOH . A comparison of stressstrain curves measured from uniaxial tension and cyclic straining tests with the curves calculated by Y-U model is shown in ),*85( . Y-U model well captures deformation behaviors, not only uniaxial tension but also cyclic stress-strain responses characterized by the Bauschinger effect and cyclic hardening, as well as the work hardening stagnation. In the present work, simulations with the isotropic hardening material model (hereafter we call it ‘IH model’) were also carried out. Both results are compared with experimental results to examine the influence of the Bauschinger effect on springback in the case of bottoming.

,QIOXHQFHRI0DWHULDO0RGHOVRQ$FFXUDF\RI6SULQJEDFN6LPXODWLRQ The relationship between bent angle after springback and punch load are investigated by 3D FE simulation with rigid shell tools. In the simulation, same three types of die, Rd=4.02, 4.07 and 4.12mm, were used as in experimental test. The punch loads were measured when the punch stops at the bottom, and also 0.01, 0.02, 0.03, 0.04, 0.08 and 0.12mm short of the bottom. Springback simulation was done by removing the punch and bent angles were measured when the blank sheet motion had been converged. ),*85( shows the comparison of experimentally obtained bent angles with the simulation results by Y-U and IH models. Obviously, the results by Y-U model are much closer to the experimental results, compared to the results by IH model. From this, it would be concluded that, in the case of bottoming, the Bauschinger effect should be taken into account in the springback analysis, and for that, an appropriate selection of material model is of vital importance.

1132

96 EXP Rd4.02_IH Rd4.07_IH Rd4.12_IH Rd4.02_Y-㹓 Rd4.07_Y-U Rd4.12_Y-㹓

Bent angle [degree]

95 94 93 92 91 90 89 0

20

40

60

80

100

120

140

Final Punch Load [KN]

),*85( Comparison of experimentally obtained bent angles after springback with FE simulation results by Y-U and IH model.

,QIOXHQFHRI(ODVWLF7RRO'HIRUPDWLRQRQ$FFXUDF\RI6SULQJEDFN6LPXODWLRQ In the bottoming, the punch load increases drastically, and especially for the cases of Rd=4.07 and 4.12mm, where the forming gaps are smaller than the sheet thickness, large compressive loads act directly on the die/punchcorners. For such a case, it is very interesting to know how the elastic deformation of tools affects on the accuracy of springback prediction. To examine it, FE simulation with deformable tools was conducted for the case of Rd=4.12mm, using Y-U model. 95

Experiments

Bent Angle (degree)

Y-U_Rigid_Tool

94

Y-U_Deformable_Tool

93 92 91 90 0

20

40

60 80 Punch Load (KN)

100

120

140

),*85( FE simulation results of bent angle after springback using rigid and deformable tools, together with experimental results.

),*85( shows the comparison of experimental results with 3D FE simulation results using the rigid shell tools and the deformable solid tools. The results by deformable tools are slightly closer to the experimental results compared to the case of using the rigid tools, however the difference in the numerical results by the two tool models is not so large. Then it would be concluded that the elastic deformation of tools has secondly effect on springback prediction for a given punch load. ),*85( shows a contour of deformation of the punch and die when the punch reaches its bottom level. This figure is a view on the symmetry section. When the die with Rd=4.12mm is used, the gap between the punch and die

1133

is 0.1mm smaller than the initial blank sheet thickness. However, the maximum value of tool deformation is much smaller than it. Although the elastic deformation of tool is rather small, a numerical result of punch load for a given punch travel is influenced by the deformation of tools significantly. ),*85(D DQG E show the increase of punch load with punch travel calculated by FEM with deformable and rigid tools. Both results from rigid and deformable tools show similar tendency to increase punch load up to the position of 0.05mm short from the bottom. At just the bottom, the punch load reaches 94.6KN when using rigid tools, however it reaches only 33.4KN when using deformable tools. From this, it would be concluded that the use of deformable tools in FE simulation is preferable, and it is essential for the prediction of punch load for a given punch travel.

Displacement

0.018~0.019 0.016~0.018

0.004~0.006 0.006~0.0064

),*85( Deformation of tools at the bottom of punch travel calculated by FEM with deformable tools.

100

Punch Load [KN]

100 Deformable

80

80

Rigid 60

60

40

40

20

20 0

0 -10

-8

-6

-4

-2

0

-0.5

Punch Stroke [mm]

(a) From -10mm to 0mm.

(b)

-0.4 -0.3 -0.2 -0.1 Punch Stroke [mm]

0

From -0.5mm to 0mm.

),*85( Increase of punch load with punch travel calculated by FE simulation with deformable and rigid tools.

',6&866,2121%27720,1*())(&7 ),*85(6D ±F show major stress distributions on the center cross section of blank sheet before springback, calculated by FE simulation with deformable tools. ),*85(D is the result at a punch load of 5.5KN (0.1mm shorter of the bottom), E at 33.4KN (bottom) and F at 90.3KN (0.05mm further from the bottom). ),*85(6 D ± F  show the contact areas of tool surfaces with blank sheet at the same punch loads as shown in ),*85(6 D F .

1134

0.6~0.8

-0.2~0.0

-0.2~0.0

-0.6~-0.4 -0.4~-0.2

-0.4~-0.2

-0.2~0.0 -0.4~-0.2

-0.6~-0.4 0.6~0.8

-0.2~0.0

0.6~0.8

0.0~0.2

0.8~

0.8~

0.6~0.8

D Punch Load = 5.5KN (Bottom-0.1mm䠅

E Punch Load = 33.4KN F Punch Load = 90.3KN (Bottom䠅 (Bottom+0.05mm䠅  ),*85( Major stress distributions before springback at the section of blank sheet for three levels of punch load.

Punch

Punch

No contact

Die D Punch Load = 5.5KN (Bottom-0.1mm䠅

Punch

Contact with Blank sheet

Die

Die

E Punch Load = 33.4KN (Bottom䠅

F Punch Load = 90.3KN (Bottom+0.05mm䠅

),*85( Contact areas of tool surface with blank sheet for three levels of punch load.

When punch load is relatively small, i.e. 5.5KN, where the punch stays still 0.1mm short of the bottom, the punch- and die-corners contact only partially with the blank sheet. In this case, since a force acting normally to the blank sheet is rather small, the bending stresses (high compressive stresses at the inside and high tension stresses at outside of the bent sheet) still remain. When a punch load increases to 33.4KN at the bottom, the punch- and diecorners fully contact with the blank sheet, and consequently, the compressive stress appears at the outside of bent sheet. As a result, the bending moment, which is the driving force of springback, is reduced there. However, it should be noted that even if the punch go further, e.g., the case of ),*85(F where a punch load of 90.3KN (punch goes 0.05mm further from the bottom), the bending stress still remains around the end of bent corner. This is a reason why the effect of bottoming on the reduction of springback is limited to some extent.

&21&/8',1*5(0$5.6 The effect of bottoming on the reduction of springback has been investigated by performing U-shaped bending experiment and the corresponding 3D FE simulation. In addition, from the viewpoint of accurate springback simulation, specifically for the case including the bottoming, the effects of material models and the consideration of tools’ deformation have been examined. The present findings are summarized as follows:

1135

1.

2.

3. 4.

Springback is reduced with increasing bottoming load to some extent. However, zero-springback is not possible even under an extremely high load. In the present work, we examined a case that the forming gap at the die/punch corner is as narrow as 10% of the initial sheet thickness, but still a certain level of springback remained. From 3D FE simulation, it was found that bending stresses remain around the end of bent corner of the sheet even when a large bottoming load is applied. This is a reason why the effect of bottoming on reduction of springback is limited. From the above, it would be recommended in actual press forming operations to apply a certain amount of bottoming load, but it should not be too large, for reduction of springback. The Bauschinger effect of a material influences the springback behavior greatly. In addition, the elastic deformation of tools affect the calculation of bottoming load for a given punch travel. Therefore, it would be concluded that, for an accurate 3D FE simulation, the best combination is the use of an advanced kinematic hardening model (Y-U model) together with the deformable solid elements for tools.

5()(5(1&(6 1. Y. Nagai, Journal of the JSTP, Japan, 38-442, 1997, pp. 967-971. 2. H. Ogawa, T. Gotoh and A. Makinouchi, Journal of the JSTP, Japan, 39-445, 1998, pp. 121-125. 3. T. Ogawa, A. Hirahara and F. Yoshida, Proceedings of the 10th International Conference on Numerical Methods in Industrial Forming Proceses (NUMIFORM 2010), 2010, pp.919-926. 4. Y. Nagai, Journal of the JSTP, Japan, 29-324, 1988, pp. 69-74. 5 J. Iwaya, F. Tanaka, H. Shirasawa and M. Miyahara, Journal of the JSTP, Japan, 35-404, 1994, pp. 1122-1127. 6. H. Tsutamori, F. Yoshida, Journal of the JSTP, Japan, 46-532, 2005, pp. 1961-1964. 7. K. Roll, Proceedings of the 7th International Conference NUMISHEET, 2008, pp.3-11. 8. P. A. Eggertsen and K. Mattiasson, Int. J. of Mechanical Sciences 52, 2010, pp. 804-812. 9. A. Ghaei and D. E. Green, Computational Material Science, 48, 2010, pp. 195-205. 10. F. Yoshida, T. Uemori, and K. Fujiwara, Int. J. of Plasticity, 2002, pp. 633-659. 11. F. Yoshida and T. Uemori, Int. J. of Plasticity, 2002, pp. 661-686. 12. F. Yoshida and T. Uemori, Int. J. of Mechanical Sciences 45, 2003, pp. 1687-1702.

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A Study on the Effect of Input Parameters on Springback Prediction Accuracy Y.S. Hana, W.H. Yanga*, K.Y. Choib, and B.H. Kima a

Hyundai Motor Company, 700 YangChung-Dong, Buk-Ku, Ulsan, Korea b HanKook ESI, 660-6 DeungChon-Dong, GangSeo-Ku, Seoul, Korea

Abstract. In this study, it is considered the input parameters in springback simulation affect factors to use member part by Taguchi’s method into six-sigma tool on the basis of experiment for acquiring much more accurate springback prediction in Pamstamp2G. The best combination of input parameters for higher springback prediction accuracy is determined to the fender part as the one is applied for member part. The cracks and wrinkles in drawing and flanging operation must be removed for predicting the higher springback in accuracy. The compensation of springback on the basis of simulation is carried out. It is concluded that 95% of accuracy for springback prediction in dimension is secured as comparing with tryout panel. Keywords: springback, Taguchi’s method, six-sigma

PACS: 89

INTRODUCTION On demanding the light weight for improving fuel efficiency, the use of high strength steel sheet has been recently increased. However, in case of manufacturing press product with high strength steel sheet, it is not so easy to satisfy the accuracy of product because the amount of springback is much more occurred than conventional strength steel sheet. In order to solve this problem, it has been necessary procedure to simulate entire forming process and springback analysis prior to manufacturing press die. Especially, it is very significant to improve the springback accuracy for solving springback issue in advance. Even though the large number of research about input parameter has been carried out in order to enhance the springback prediction accuracy, the actual input parameter study under real die system is insufficient.1)~3) The purpose of this research is to check the current level of the springback prediction accuracy using Taguchi method in six-sigma under actual condition of die system, to select the optimized input parameters by analyzing the amount of effect about input parameter at springback simulation, and to improve the springback prediction accuracy through the detailed analysis by conditional case of the most effective parameter.# #

INPUT DATA OPTIMIZATION FOR SPRINGBACK ANALYSIS Effective Parameter Analysis Using Six-Sigma After selecting four input parameters affected to prediction accuracy of springback analysis in order to optimize input data for it, the orthogonal array table (L82 [(7)]) based on Taguchi method in six-sigma for four input data is set up. The amount of effect for each parameter in orthogonal array is analyzed and then, the input parameter for springback analysis is optimized as selecting main factor and defining the optimal input data consisted of it.

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1137-1142 (2011); doi: 10.1063/1.3623731 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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Selection of Simulation Part for Whole Process For comparing results of springback analysis and product, the member front side rear for simulation part is selected as following reasons. First, the behavior of springback is enormous. Second, the boundary conditions for springback is simple and it is easy to measure the flush and distortion of whole part. Finally, the solving time is shorter than a fender part, so the results could be checked quickly. The springback boundary conditions and measuring locations are shown in Figure 1.

FIGURE1. The springback boundary conditions and measuring location of member

The tensile strength, thickness, and size of member part are 440MPa, 1.6mm, 1,100mm × 570mm respectively. The type of press is single action. The blank holding force and cushion stroke are 60tons and 170mm. The whole process consists of four operations which are OP10 (draw process), OP20 (trim process), OP30 (cam or restrike process), and OP40 (pierce process). Selected Input Parameters The most effective input data and detailed level for springback analysis based on the experiences of HMC(Hyundai Motor Company)and recommendations of ESI company are selected as following; The adaptive mesh level is 2 and 3 for OP10, and 3 and 4 for OP30. Gauss integration point number through the thickness is 5 and 7. As selecting the four input data and the level of two on applying cam offset (adding 10% of the thickness) or not, the orthogonal array consisted of total eight parameters based on Taguchi method in six-sigma is chosen. The detail contents of orthogonal array (L82 [(7)]) are shown in Table 1. TABLE1. Orthogonal array table by Taguchi method in six-sigma

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Comparison of Results For the exact setting of the blank sheet, the gage is installed at OP10 die. The measuring for 35 points of flush and 18 points of distortion are carried out with CMM (Co-ordinate Measurement Machine). The detail measuring locations are shown in Figure 2.

FIGURE2. The measuring locations (flush and distortion) of springback

Accuracy of Springback Analysis on Each Factor The accuracy of springback analysis is judged by a percentage of the number of OK which means the tolerance between analysis results and the measuring data of panel indicates within ± 0.5mm. The highest effective parameter for springback prediction accuracy is factor(8) of eight factors and its accuracy is 74%. The springback prediction accuracy at each factor is shown in Figure 3.

FIGURE3. Springback prediction accuracy at each factor

Analysis of The Amount of Effect at Each Factor The highest effective factor is OP10 adaptive mesh level(OP10 AML) and its amount of effect is 39.6%. The other factors are cam offset, Gauss integration point number through the thickness(GIP) and OP30 adaptive mesh level in order. Their amounts of effect are 35.2%, 22.2%, and 1.2% respectively. Therefore, the optimal input conditions for springback prediction accuracy are optimized when OP10 AML and OP30 AML is higher, GIP is greater, and the cam offset is applied. These conditions are consistent with factor(8). The results of analyzing the amount of effect at each factor are shown in Table 2.

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TABLE2. The amount of effect at each factor

IMPRONEMENT OF SPRINGBACK ANALYSIS ACCURACY Analysis Considering Detailed Conditions about Major Factor In order to improve the accuracy of springback analysis, the analysis considering detailed conditions about major factor is carried out. As selecting the level four for OP10 AML which is the highest effective input data of 4 input conditions, the level 5 for OP30 AML which is less effective than OP10 but it is correlated with OP10 AML, the input value 5, 11, 48 (max.) for GIP and the cam offset, and 5%, 15% for panel thickness, these analyses for case studies are performed in detail. In addition, the amount of effect for cam speed and friction which is not included in main four input data is analyzed. The Effect of Cam Offset The springback prediction accuracy of cam offset reaches the highest rate in case of 10% among 5%, 10%, and 15% of the panel thickness and there’s not quite different about solving time. The effect of cam offset to springback prediction accuracy is shown in Figure 4.

FIGURE4. The effect of cam offset to springback prediction accuracy

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The Effect of Gauss Integration Point through the Thickness The springback prediction accuracy corresponding to GIP level is not quite effective and the solving time incase of GIP 48(maximal number of input value at Pamstamp 2G) is two times shorter than GIP 7 approximately. The springback prediction accuracy is the same rate, 74% in case of GIP 5 and GIP 7 within ± 0.5mm error, but the accuracy of GIP 7 is stable than GIP 5 because the standard deviation in case of GIP 7(0.6562) is lower than GIP 5(0.6582). The effect of GIP to springback prediction accuracy is shown in Figure 5

FIGURE5. The effect of GIP to springback prediction accuracy

The Effect of Cam Speed Generally, it is occurred that the springback prediction accuracy is improved or the solving time could be taken long according to lower cam speed. The cam speed, 5m/s would be appropriate as considering both the springback prediction accuracy and the solving time. The effect of cam stamping velocity to springback prediction accuracy is shown in Figure 6.

FIGURE6. The effect of cam stamping velocity to springback prediction accuracy

The Effect of Cam Friction Overall, the springback analysis is carried out as reducing the friction from 0.1 to 0.05 for inducing springback behavior larger because the amount of springback on simulation tends to occur less than real panel. As a result, the

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springback prediction accuracy is 81% and 92% within ±0.5mm and ± 1.0mm tolerance respectively and its input condition are shown in Table 3. TABLE3. The input condition of springback prediction OP10 Mesh Size GIP (AML) 421(3) 7

Cam Offset 10%

OP30 Mesh Size (AML) 10.50.25(3)

Cam Friction 0.05

CONCLUSION The amount of effect for major input data of springback analysis is analyzed by using Taguchi method in sixsigma, then the results between simulation and T/O of member part in vehicle are compared. The conclusions are arranged as following; (1) The amount of springback of the simulation tends to occur less than the real panel. It is required that the input data should be modified. (2) Generally, the smaller final mesh size is, the higher springback prediction accuracy would be. (3) It is considered that GIP(Gauss integration point number through the thickness) 7 and the cam stamping velocity 5m/s would be the most appropriate in respect to the springback prediction accuracy and the solving time efficiency.

REFERENCES 1. H. Yao, S.-D.Liu, C.Du, Y.Hu, “Techniques to Improve Springback Prediction Accuracy Using Dynamic Explicit FEA Codes”, SAE Technical paper,2002, No.2002-01-0159. 2. K. Li, R. H. Wagner, “Simulation of Springback, Simulation of Material Processing : Theory, Methods and Applications”, Hu etink and Baaijens (eds), 1998, pp. 21~31. 3. K. Mattiasson, P. Thilderkvist, “Simulation of Springback in sheet metal forming, Simulation of Material Processing : Theory, Methods and Applications”, Shen & Dawson (eds), 1995, pp. 115~124.

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Analytical Solution on the Failure of Strips Under Bending and Tension Y. Bai 1* ,

T. Wierzbicki 2 and

([email protected], 1*

2

[email protected],

M. Luo 2

[email protected])

Department of Mechanical, Materials and Aerospace Engineering University of Central Florida, Orlando, FL 32816 Department of Mechanical Engineering, MIT, Cambridge, MA 02139

Abstract. The analysis of stretch-bending of a strip is important in understanding failure initiation in the sheet metal forming process. Most of the current work in this area has been purely experimental or numerical simulation. In the paper, a procedure for constructing an analytical solution for failure of strips under bending and tension was developed. One important aspect of the derivation was the consideration of the logarithmic strain instead of engineer strain, as was done in the existing solutions found in the literatures. The other important aspect is the assumption of tension/bending damage coupling. The input data of the analytical solution are the geometry of the tool (die radius), sheet thickness, parameters of the plasticity hardening rule, magnitudes of fracture strains under uni-axial tension and plane strain, and the tension/bending damage coupling coefficient. The present closed-form solution was validated by test results of the present authors as well as data from the open literature. A good correlation is obtained between the measured and predicted radius to thickness ratio ( R / t ) to fracture under a given pre-tension. The results of the paper will be useful in the design of stamping and deep drawing operation.

1 Introduction As the increasing need for energy saving in various industries, application of advanced high strength steel (AHSS) has drawn a lot of attention from researchers. One of the big problems is how to accurately predict necking and fracture in sheet metal forming. While gaining high strength, AHSS sheets are more susceptible to wrinkling, spring back and fracture during press shop operations. A systematic study of the mechanics of the stretch-bending problem had begun in 1960s and 1970s. Marciniak and Duncan [1] emphasized the importance of using logarithmic strains or force generated in deep drawing processes. They developed special equipment for bending and stretching a metal strip which is known as a Modified Duncan-Shabel apparatus [2]. Lots of research on stretch-bending of metal strips have been conducted in recent years [2-4]. Most of them were focusing on experimental studies. Walp et al [2] performed a series of tests on metal strips cut from four different high strength steels to determine a combination of the strip tension and punch radius causing fracture. This empirical fracture locus was constructed using the stress formulation. For each normalized punch radius, a unique side wall stress to fracture was determined. More details were summarized in Section 3.1. The press shop experience indicates that fracture may occur either on the die radius or at the sidewall on leaving the die or else at the tangency point between the sidewall and punch [2]. Examples of these fracture locations from Luo and Wierzbicki[5] is shown in Fig. 1. Bai and Wierzbicki analyzed the stress and strain states and demonstrated predicting fracture of AHSS strip under combined bending and tension using Modified Mohr-Coulomb Criterion (MMC) using finite element simulation [6, 7]. The application of MMC fracture model in predicting strip bending failure was further confirmed by Luo and Wierzbicki[5, 8]. Most of current work in this area was purely experimental or numerical simulation. In the paper, an The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1143-1150 (2011); doi: 10.1063/1.3623732 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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analyticall solution for ffailure of stripp under bendinng and tensionn is developed.. One importannt aspect of thhe derivation is the connsideration of the logarithmiic strain insteaad of engineerr strain. The oother importannt aspect is thee assumption of tensionn/bending dam mage coupling. A closed-forrm solution is derived to preedict the nomiinal wall stress to fracture under givven R / t ratioo (die radius vv.s. strip thickkness) of stretcch-bending prrocess. The innput data of thhe analytical solution aare the geom metry of the toool (die radiuus), sheet thicckness, param meters of the pplasticity harddening rule, magnituddes of fracture strains under uni-axial tension and plane strain tensionn, and the coeffficient of tenssion/bending damage ccoupling. The solution is verrified by the teest data from tthe literatures [2, 5].

Fig. 1: Fractured DP P780 specimenss after SFS testss with four diffe ferent lower die radii [5].

2

Fig. 2: A sketch of thee drawing process with puncch and die

Analytiical solution n of local sttrain underr combined bending/teension

A sketch of drrawing processs is shown inn Fig. 2. The die d is stationarry and the punnch moves dow wnwards. A closed-form solution w was derived bby the presentt authors for the punch foorce P and ppunch displaceement  as functions of instantaneoous wrapping angle  [6]..  2M   P=  T  sin  . R  d 

(1)

 =  g   R p  Rd  1  sin    tann    R p  Rd  1  cos   ,

(2)

moment at a beending plastic hinge. The foollowing inputt parameters where, M  =  bt iss the fully plasstic bending m 4 are enteriing the solution: Punch radiuus R p , Die raadius Rd , Gapp between the ppunch and thee die g , Gaugge thickness 2

of the striip t , strip widdth b , Pre-tennsion force T , Plane strain yield stress  = 2  .  y 3

The above sollution has beeen validated byy finite element solutions [66], but it will not tell us thee location to T fracture aand the side waall stress correesponding to ffracture. Bending and re-bennding is a planne strain probllem, so local analysis oof stress statees and strain under strip bbending with tension is a necessary steep to predict the fracture initiation.. The magnituude of true staains (or logaritthmic strains) in the bendinng and re-bendding process were w already derived bby the presentt authors in R Ref. [6]. The m main results aare summarizeed again for thhe easy of reaading in the followingg subsection foor both rigid-pperfectly plastiic and work haardening mateerials.

2 2.1

Rigid--perfectly plastic p mateerial

A segment off a bent strip with a definittion of geomeetrical parameeters involvedd is shown in Fig. 3. It is assumed tthat the bent sstrip fully conffirms to the raadius of die/puunch, which is denoted by R . The initial thickness of the strip is denoted byy t . It is reccognized thatt tension in thhe strip can oovercome friction and mayy reduce the thickness. This effect can be accounnted for in ann iterative wayy, however inn the present derivation, the change of thickness is not taken innto account.

1144

Fig. 3: A skketch of a stripp's bending undder tension W When a tensille force is appplied, there woould be a shifft of neutral axxis. Under thee plane strain assumption, the maxim mal equivalentt strain under stretch-bendinng is obtained,, 

 max

   2  R / t 1 = ln   1 3  R / t  1T      2

for

T 1

(3)

i clear that thhe maximum where T = T = T is the normaalized tensile fforce. From thee above simplle analysis, it is N   t equivalennt strain to fraacture under sstretch-bending for a material with a connstant yield sttress is controolled by two parameterrs, the radius tto thickness raatio R / t and the normalizeed tensile forcee T .

2 2.2

Work k hardeningg material

T above derrivation is exteended to the sttrain hardeninng material witth the Swift foormula, The

 = H    y  , n

(4)

where H is the stresss amplitude, n is the harddening exponeent, and  y iis the initial yyield strain. Inn the present derivationn, a simply isootropic hardenning plasticityy model is connsidered. A M Matlab program m was writtenn to evaluate the shift of neutral axiis. Compared to the rigid-pplastic materiaal, a joint effeect of three m main input parrameters i.e. be approximated by the folloowing equatioon, R / t , T and n can b 

 max

   2  R / t 1 0.6 = ln   1   0.1  0.7T  n  . 1 3  R / t  1 T     2 

(5)

The aboove formula is applicable to predict fracturre at either punnch or die raddius.

3 3 3.1

Analytiical solution n for the modified Dun ncan-Shabeel tests Modiffied Duncan-Shabel teests

Walp et al. [2]] performed a series of draw W wing tests on 440mm wide strrips using the Modified Dunncan-Shabel apparatuss, sketched in Fig. 4(a). The die radius w was kept consttant at Rd = 1 15mm , while five different punch radii were usedd ranging from m one to ten m millimeters. Inn each tests, tthe restrainingg force (pre-tennsion force T ) was kept constant, and measuredd was the punnch force from m which the w wall stress wass calculated. T The maximum draw depth m . Tests weree performed fo for four differeent materials including DP6600, TRIP800,, CP800 (Com mplex Phase) was 90mm and DP8000. A typical result of the aabove test serries showing tthe dependencce of the wall stress on the normalized punch raddius, R = R / t , is shown in Fig. 4(b). Thee solid curve iin this figure separates s testss with fracturee from those without fr fracture. One cconclusion froom the above ttest is that therre is a clear booundary betweeen the bendinng dominate region (onn the left) andd the tension ddominate regioon (on the righht). For exampple, the criticall normalized ppunch radius ( R / t raatio) in the casse of DP600 w was equal to 4.0.

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Fig. 4: A tyype of modifieed Duncan-Shhabel test to caalibrate drawinng limit of AH HSS, after Walpp et al. [2] T diagram oof wall stress v.s. The v the R / t ratio can connsist of two reggions. The firsst region is whhere the side wall stresss is less than the initial yieeld stress. If sttrips fracture aat this region, it means that the material iis too brittle for sheet forming. Thiss kind of experrimental data usually u don't sshow in the litterature. The ssecond region is where the side walll stress is greater than the initial yield stress. In thiss situation, theere could be strong coupliing between tension annd bending. T The test resultss shown in Figg. 4(b) belong to the second region, whichh is also the m main focus of this paperr. The solutionn of these two regions will bbe analyzed inn Section 3.4 aand 3.5.

3 3.2

Curvee fitting of engineering e g stress v.s.. true strain n

The results reeported in the Fig. 4(b) aree the engineerring stresses w T which are the average stripp wall stress corresponnding to fractuure. Accordingg empirical fracture modelss, the strain thhat controls thhe fracture inittiation is the true strain or logarithm mic strain. Too obtain analyytical solutionn for fracturee in modified Duncan-Shabbel tests, an expressioon of the enginneering stress versus true staain is needed for convertingg the fracture ttensile strain bback to strip wall stresss. I a uni-axial tensile test (for examplee, dog-bone sspecimens tennsile test) forr calibrating tthe material In stress-straain curve, the force P andd displacemennt d of a certtain gauge lenngth L are reecorded. The engineering stress annd engineeringg strain are defined as

S=

F A

and E = d . The truue strain can be calculateed from the L

engineerinng strain by  = ln 1  E  . It should be noted that thhe strain transfformation form mula is valid only before necking. The peak streess in the meaasured engineeering stress is usually calledd the ultimatee tensile strenggth, Suts , of the materrial. It is propposed to use tthe Voce equaation to descriibe the relatioon between enngineering streess and true stain.

S = Suts  A Ae B where 

p pl

is the plasttic strain, and

pl

= Sutss  Ae



 B   yiield

,

(6)

During the currve fitting to gget the two coonstants, A  yield is the yield strain. D

and B , thhe experimenttal data beforee necking are used. Taking an example oof DP780 steeel reported in Ref. R [5], the yield straain is   = 0.0 f gives ttwo material 008 , and the ultimate tensile strength iss Suts = 871MP Pa . Curving fitting constants, A = 188MP mparison betweeen curve fittinng and tested results is show wn in Fig. 5. Pa and B = 557.65 . A com Good corrrelation is achhieved using E Eq. (6).

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Fig. 5: Curve fitting of engineering stress versus true strain for DP780 steel

3.3

Fig. 6 Assumption of damage contribution under combined bending and tension

Assumption of damage accumulation under combined bending and tension

In the stretch-bending tests, especially when the tensile stress (or side wall stress) is greater than yield stress, material damage comes from both bending and tensile loads. The loading conditions at die or punch radii are either plane strain tension or plane strain compression, while the loading condition at the side wall are uni-axial tension. Besides the bending tension coupling, the loading history of materials at fracture location could also be very complex. To predict this kind of fracture using analytical solution is very challenging. It is proposed to decouple the tensile damage and bending damage using the following assumption. Denote the fracture strain under uni-axial tension as  fT , and the fracture strain under plane strain tension as  fB . In a combined loading condition, it is assumed that failure will occur if the following criterion is satisfied, M

M

 B  T  (7)  B    T  = 1,    f   f  where  B is the amount of equivalent strain contributed by the pure bending without axial tension,  T is the amount of equivalent strain from axial tension, and M is the power coefficient describing the tension/bending coupling. A plot of Eq. (7) for different values of M is shown in Fig. 6. The M is a critical parameter to be determined from test results.

3.4

Solution of region 1: wall stress less than yield stress

In the absence of axial tension ( T = 0 and side wall stress equals to zero), according to Eq. (3), the equivalent strain introduced by pure bending is   (8)   2 R / t 1 ln   3 R /t  1  2 

B=

If  B   fB , then a strip will fail under pure bending only. This gives the cut-off value of the diagram of wall stress versus the R / t ratio.

 R / t cutoff

3 B    1 2e 2 f  =  3 . 2 B  e 2 f  1 

(9)

When the axial tensile stress (side wall stress) is less than initial yield stress, the main effect of tension is to shift neutral axes in the bending process, in which the tension doesn't produce much damage accumulation itself. If the fracture initiation is at bending phase of die or punch radii, the material was subjected to monotonic loading condition. General speaking, the equivalent strain to fracture is higher in plane strain compression than that of plane strain tension, so fracture usually occurs at the tensile side of a strip. From Eq. (5), one can get the following fracture criterion.

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

where

T

 max

   2  R / t 1 0.6 B = ln   1   0.1  0.7T  n  =  f , 3  R / t  1 1 T      2

is the normalized tension, which can be approximated by

T=

S S yield

(10)

. Solving Eq. (10), one can get the

corresponding local side wall stress at fracture for a given R / t ratio. If a strip crack at a location just leaving the die radius, then it becomes a reverse loading condition because materials were subjected to bending and re-bending. It was found that reverse loading can either increase the material ductility or reduce the material ductility [9]. A complicated damage accumulation rule was proposed by Bai [9, 10] for fracture prediction under complex loading history, but it was mainly designed for numerical solutions, which can not be applied to analytical solution directly. So, it is hard to get an explicit solution for this region. This is still an open question.

3.5

Solution of region 2: wall stress greater than yield stress

As the R / t ratio increases, the side wall stress corresponding to fracture increases as well. When the tensile stress (side wall stress) becomes greater than the initial yield stress. There would be significant amount of damage contributing from tension. According to damage assumption described in Session 3.3, the tension damage and bending damage are assumed to couple as Eq. (7), so the tensile strain causing strip to failure becomes, 1

   B M  M (11)  T =  fT 1   B   .    f     For a given R / t ratio, the amount bending damage can be evaluated according to the bending strain introduced by pure bending (without axial tension, T = 0 in Eq. (3)), which yields,   2  R / t 1  (12) B  = ln  . 3 R /t  1   2 Substituting Eq. (11) into the hardening law, Eq. (6), the corresponding wall stress at fracture can be obtained,

S wall =  S =  Suts   Ae

1 M M      B    B Tf 1     B     f  

.

(13)

It should be noted that the S wall is the engineering stress corresponding to true tensile strain at fracture. Here a coefficient  is introduced because there could be some difference between local stress and average cross-section stress of a strip. Substituting Eq. (12) into Eq. (13), the solution of wall stress at fracture versus a given R / t ratio is,

S wall

   =  Suts   A exp  B fT   

1 M M          2  R / t 1     1   ln    . B 1   3 f  R / t          2      

(14)

The inputs of this solution include three sets of parameters. The first set is ( Suts , A , B ), which can be obtained from the engineering stress versus true strain curve. The second set is the (  f , T

 Bf ),

which are the equivalent

strains to fracture for uni-axial tension and pure bending respectively. The third set is (  , M ). The parameter  counts for the difference between local fracture stress and average engineering wall stress at fracture. Its typical range is 0.95  1.0 . The parameter M is the coefficient of bending/tension fracture coupling, which is to be determined by test results.

4 Validation using experimental data

1148

I this sectionn, two experiimental data ppublished in the literature will be usedd to verify thhe analytical In solution dderived in Secction 3 for the strip fracture at modified D Duncan-Shabeel tests. One m material is the D DP600 steel reported iin Ref. [2]. Thhe other one iss the DP780 stteel extensively studied by Luo L et. al [5, 88].

4 4.1

DP6000 steel

Several materrials were studdied by Walp et. al to calibbrate their sheear fracture in forming [2]. The DP600 S steel was taken as an example materrial to check thhe analytical ssolution. The first f set of parrameter, ( Suts , A , B ), is determineed from the pplasticity and hardening h prooperties. The uultimate tensille stress is Suuts = 647 MPa . The initial yield streess is S yield = 3396MPa . Acccording to Eq. (6), the param meter A = Sutts  S yield = 2511MPa . The paarameter B is determiined by matchhing the hardenning exponentt n = 0.17 , w which gives B = 32 . T second seet of parameterrs, (  f , The T

 Bf ) needs to be ccalibrated from m the tensile frracture tests. A According to

the total elongation e at fracture (0.266 for uni-axial tension [2]), the uni-axial tensile fracturre strain can bbe estimated from the finite elementt simulation, w which gives  Tf = 0.75 . The bending fractture strain is assumed a to bee decreasing from tenssile fracture sttrain in an expponential function of e 1.5 , where the  is the stress triaxiality [99]. Then, we get  Bf = 0.52 . It shoulld be noted thhat these two pparameters arre based on esstimations sincce no accuratee calibration proceduree was conductted.

Figg. 7: Dependence of the wall strress at fracture on o the norrmalized punch radius for DP6000 steel. The fraacture sttrain under the pplane strain tennsion was estimaated from m the test resultts [2].

Fig. 8 Comparisoon between anaalytical solutioon and experiiments for DP780 steel

R Regarding thee third set off parameter (  , M ), takke  = 0.98 as a simple approximatioon. The key parameterr is the bendinng/tension fraccture couplingg coefficient M , which is tto be determinned from the tthe modified Duncan-S Shabel tests reesults. It is fouund that M = 0.44 gives a nice cure fittiing for all thee test results. IIllustrated in Fig. 7 is tthe plot of thee side wall streess at fracture versus given ratio of punchh radius to striip thickness. The T test data point in this t figure weere taken from m the solid cuurve of Fig. 4(b). A very ggood data corrrelation was aachieved for DP600 steel.

4 4.2

DP7800 steel

Luo et al. syystematically studied the D L DP780 steel from materiaal fracture loocus characterrization and stretch-beending tests too finite elemeent simulationns [5, 8]. It was w shown that using the Modified M Mohhr-Coulomb (MMC) ccriterion, the fracture in tthe stretch-beending tests ccan be accurrately predicteed in the finnite element simulation[5]. In this paper, the em mphasis is putt on the analytical solutionn. The plasticcity and strainn hardening p in propertiess are determinned from curvve fitting to enngineering streess versus truue strain curvee, which was presented Section 33.2. Those threee constants are a Suts = 871M and B = 57.65 . The calibrated MM MC fracture M MPa , A = 188MPa locus givves  Tf = 1.09955 and  Bf = 0..36 . Take  = 0.98 as ann approximatiion again. It is found thaat M = 0.33 makes a nnice cure fittinng for all the fo four test resultss, see Fig. 8.

1149

The fracture strain under plane strain tension,  Bf , is used to determine the cut-off value of the R / t ratio (see Eq. (9)), which corresponds to the pure bending fracture (without axial tension). The result is ( R / t )cutoff = 0.87 for DP780 steel. The region where wall stress is less than yield stress is plotted as a dashed line because of no explicit analytical solution available in this small region.

5 Conclusion In this paper, the authors tried to analyze the stretch-bending test from three aspects. The global response of punch force versus punch displacement was obtained through full plastic bending hinge method. The local stretch-bending analysis gave the shift of neutral axis and magnitude of maximum equivalent strain (logarithmic strain). Using the damage accumulation assumption of combined bending and tension, described in the Section 3.3, an analytical solution was derived for the modified Duncan-Shabel tests. Test data from the literatures were used to verify the analytical solution. One is the DP600 steel reported by Walp et al [2]. Since no fracture strain data available for this material, some parameters were based on estimation. Good correlation was achieved in predicting the wall stress to fracture versus given strips R / t ratios. The other material is the DP780 steel investigated by Luo et al [5, 8], where the fracture test data are sufficient. It has been shown that by adjusting the bending/tension fracture coupling coefficient M , the analytical solution gives very good prediction of wall stress to fracture for given strips R / t ratios, which validated the analytical solution again. Work is in progress to demonstrate the applicability of the present methodology to other representative of metal sheets.

Acknowledgement The authors gratefully acknowledge partial financial support from the AHSS MIT/Industry Consortium.

References [1] Marciniak, Z. and Duncan, J.L. Mechanics of Sheet Metal Forming. Edward Arnold, A division of Hodder & Stoughton, 1992. [2] Walp, Matthew S., and Wurm, Astrid, and Siekirk, John F. III, and Desai, Ajit K. Shear fracture in advanced high strength steels. SAE technical papers 2006-01-1433, 2006. [3] Damborg, F.F. and Wagoner, R. H. and Danckert, J. and Matlock, D. K. Stretch-Bend Formability. [4] Wagoner, Robert and Kim, Ji and Sung, Ji. Formability of advanced high strength steels. International Journal of Material Forming, 2:359-362, 2009. 10.1007/s12289-009-0644-0. [5] Meng Luo and Tomasz Wierzbicki. Numerical failure analysis of a stretch-bending test on dual-phase steel sheets using a phenomenological fracture model. International Journal of Solids and Structures, 47(22-23):3084 - 3102, 2010. [6] Bai, Y. and Wierzbicki, T. Predicting Fracture of AHSS Sheets on the Punch and Die Radii and Sidewall. Proceedings of Numisheet 2008, Interlaken, Switzerland, Sept. 1-5, 2008. [7] Bai, Yuanli and Wierzbicki, Tomasz. Application of extended Mohr-Coulomb criterion to ductile fracture. International Journal of Fracture, 161:1-20, 2010. 10.1007/s10704-009-9422-8. [8] Luo, Meng, and Chen, Xiaoming, and Shi, Ming F., and Shih Hua-Chu. Numerical Analysis of AHSS Fracture in a Stretch-bending Test. NUMIFORM 2010, Proceedings of the 10th International Conference edited by F. Barlat, Y.H. Moon, and M.G. Lee, 2010. [9] Bai, Yuanli. Effect of Loading History on Necking and Fracture. PhD thesis, Massachusetts Institute of Technology, 2008. [10] Bai, Y. Fracture of 1045 steel under complex loading history. Proceedings of Numisheet 2011, Seoul, Korea, August 21-26, 2011.

1150

An Improved Numerical Integration Method for Springback Predictions R. Ibrahima, L.M. Smitha and Sergey F. Golovashchenkoc a,b

Department of Mechanical Engineering,118 Dodge Hall, Oakland University, Rochester,MI 48309, USA c Ford Motor Company, Dearborn, MI 48121, U SA

Abstract. In this investigation, the focus is on the springback of steel sheets in V-die air bending. A full replication to a numerical integration algorithm presented rigorously in [1] to predict the springback in air bending was performed and confirmed successfully. Algorithm alteration and extensions were proposed here. The altered approach used in solving the moment equation numerically resulted in springback values much closer to the trend presented by the experimental data, Although investigation here extended to use a more realistic work-hardening model, the differences in the springback values obtained by both hardening models were almost negligible. The algorithm was extended to be applied on thin sheets down to 0.8 mm. Results show that this extension is possible as verified by FEA and other published experiments on TRIP steel sheets. Keywords: Springback; V-die bending; Sheet metal forming; numerical integration. PACS: 81.20.Hy

INTRODUCTION The aim of this investigation is to examine and extend the detailed model developed in [1] for the springback and fracture in v-die air bending of thick stainless sheets. The recognized application presented in [1] was to provide a consistent mathematical model, which can be used by the Technical Customer Service or Application Support at material suppliers for quick estimation of the:    

smallest bending radius (fracture); springback; least die width (fracture); and maximum required bending force.

However, the second and fourth topics are coupled in [1] analysis and were consequently coupled in this investigation. The way by which the maximum required bending force for a given bending angle was used in the previously published algorithm was improved and implemented in a computer programming code. The improvement resulted in better controlling the iteration of the algorithm; the maximum required force was obtained as a result instead of being used as a target. The target in the improved code was the bending angle. Consequently, the springback angle was highly correlating the experimental results and mathematically raises the confidence in use of this algorithm. Because in the air bending the bend geometry is produced by three points bending, the inner sheet radius at the end of air bending (before unloading), Ri could be classified as one of the most significant parameters in v-die bending [1], this radius is also closely related to the bending moment prior to unloading and, thereby, the springback and the required bending force. Another aim of the author study was to investigate how large the error would actually be, if the shift in the position of the neutral axis and the thinning were neglected, this point was not addressed in this investigation, a significant ratio in this regards would be Ri /t0, those topics are well addressed in many references like [3]. One important limitation of the study [1] and in this investigation as well, that the die radius has to be less or equal to the blank bending radius before unloading, otherwise the multi-point bending case, friction and membrane strains would invalidate the results. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1151-1158 (2011); doi: 10.1063/1.3623733 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

1151

The author study can be consulted for the derivation of the moment formula. Efforts have been made in order to avoid ambiguity in the symbols and equations used in this investigation, after all the author study will again have to be consulted for further explanations of unclear symbols and/or equations. The moment equation (4) represents the balance between the internal moment and the elastic moment after unloading calculated at the punch which owns the max moment value along the blank. This equation is used for the two work-hardening models that are covered in this investigation, namely linear hardening and nonlinear hardening (Ludwik model).

THEORETICAL TREATMENT As mentioned in the introduction the derivation of the moment balance will not be presented here and the focus will be only on its integral. Other formulas will be used should the need arise.

Fundamentals  Plain strain prevails everywhere during the bending about z axis (εz = 0), width/thickness ratio of the sheet is large (σy = 0) where y is the thickness direction and x is the length direction.  The neutral surface is always in the centre of the sheet. The thickness does not change significantly during bending.  True strain is almost equal to the engineering strain (εx = ex = y/ R0) where R0 is the sheet radius at the end of bending and before springback. Fig. 1 displays the important notations for the angles and radii that are used in the study before and after loading/unloading (springback).

FIG. 1. Air bending: the sheet before and after unloading.

To facilitate the analysis [1], the sheet will be treated as a cantilever beam that is rigidly supported at the symmetry axis and loaded by the force Q (Fig. 2). The bending line y(x), is a function that represent the deflection of

1152

the beam, it depends on the magnitude of the force Q, or the moment M(x), and the assumed cross section shape. It also depends on the beam length or the half of the die width, w/2 in Fig. 2, etc. As also seen in Fig. 2, the radius of curvature increases from the canter (at the origin point (0,0) toward the beam edge where it theoretically becomes a very high value. During the analysis, an algorithm will be presented that accurately predicts R0 along the x axis (Fig. 2). At any point along y(x), the curvature of the bent beam can be calculated by, [2], 2 1  d 2 y    dy      / 1   R0  dx 2    dx  

3/ 2

(1)

FIG. 2. The bent sheet is treated as a cantilever beam rigidly supported at the symmetric axis and loaded by the force Q.

Linear hardening model Assuming that the material behaviour is described best by

  E

in the elastic region

(2)

and

   0 e  K e e

in the plastic region

where σ =stress, E = young's modulus, ε = strain,

(3)

σe = effective stress, and εe = total effective strain. *

The moment balance equation that will take the following form (4) where  x = the circumferential strain at the elastic-plastic interface (theoretical yielding), ν = Poisson's ratio.

M

max

   2    



* x

R

 0

0

E 1 

2

  y2 dy      R0  

t0 / 2



* x



R

0

2  3

0e



The radius at which the yielding occurs is given by [5], and can be rearranged into

1153

4K 3

e

 y  ydy  (4) R0   

 



R0*  3  t 0  E  K e  / 4  1   2   0e



(5)

*

If R0 > R0 then the deformation is elastic in the entire cross section and the moment will be

 t0 / 2 E y2  E 1 t03  M max  2    dy    2 2  R0 12  0 1  R0  1 

(6)

The load Q (/ unit width) was estimated according to [4]

Q  (1  4  t 0 / w)  t 02  Rm / 2 w 

(7)

where Rm is the ultimate tensile strength of the selected sheet. This value was used by the author in the first iteration to get the bending angle then was modified after that if the desired bending angle not obtained. Using this value to control the algorithm iterations was modified. The desired bending angle was used directly to control the interactions and a significant improvement was noticed, see below. The procedure used to calculate the variation of R0 along x axis of the sheet (toward the die edge) was used to judge whether the deformation is elastic in the entire section (R0 < R0*) or elastic/plastic. According to [1], the variation took the pattern of the two equations in two different regions. Upon replication of that work a discontinuity was detected as appearing in Fig. 3: M ate rial 904L, t0 = 20.9 m m , σ=327+1873ε E= 200,000 M pa, w = 150 m m Q (pe r unit w idth = 1394.5) N/m m

1/R0i

0< x < 45 mm

1/R0i 75 > x > 45 mm

0.0008 0.0007

y = -1E-05x + 0.0007 1/R0i (mm-1)

0.0006 0.0005 0.0004 0.0003 0.0002

y = -8E-06x + 0.0006

0.0001

x (m m )

0 0

20

40

60

80

FIG. 3. The variation of curvature 1/R0 with x axis as reproduced in the current investigation using the presented integration of Eqs (4), Material 904L, thickness = 20.9 mm with die width w = 150 mm. Q is calcu. by Eq. (7)

Revision was performed in order to determine the difference source. An integration of Eq. (4) revealed that some terms are missing in the presented integration. The correct result of the integral must be



M max    R0 * x



2

 2 * E  8 2 K e  t 03  2   x  Ke    0 e   0 .2887   0 e  t 0  2 9 9 R0 3   3  1  

(8)

which yields exact results found in [1] as in Fig. 4. Rf which represents the radius after unloading was calculated recognizing that the unloading is pure elastic recovery.

1154

FIG. 4. The variation of curvature 1/R0 with x as reproduced in the current investigation using Eqs (8), Material 904L, thickness = 20.9 mm with die width w = 150 mm. Q is calcu. by Eq. (7). (Linear hardening)

Determination of springback angle Eqs. relating radii before and after unloading can only be solved numerically [1] to find deflection y(x) in order to get the final profile of the sheet metal before/after unloading. This profile will directly give the springback angle in the form of αf-α0 (Fig. 1). Steps of the numerical solution are inspired by [5], this numerical integration yielded much better correlation of springback angles than the correlation presented in [1] The Euler-Bernoulli equation is presented by 2  d 2 y    dy    2  / 1     dx    dx  

3/ 2



M EI

(9)

where M=moment, E=Young’s modulus, I = moment of inertia. In the author’s study and this investigation the right hand side is in the form of F(x). In Eq. (16) set dy/dx = tanθ leads to



 d (tan  )  2  / 1  tan      dx



3/ 2



 F ( x) , simplifying leads to, (sec  ) 2 d

dx

/(sec ) 

2 3/ 2

 F ( x)

which can be written as

cos  d = F(x)dx

(10)

eq. (10) was then numerically integrated to get the angle θ which represent =Ф0/2 and Фf/2 in Fig. 1 for the before/after unloading cases respectively. The two angles used in the calculations of springback are related to Ф0/2 and Фf/2 by the following two Eqs.

 0  180  2 0 / 2  2  arctan(dy / dx) (before unloading)

(11)

 f  180  2 f / 2  2  arctan( dy / dx ) (after unloading)

(12)

and

In order to validate comparison with the experimental data, the author in [1] had to apply an iterative procedure in the following manner[1]: 1.

A certain load was selected and used to calculate dy/dx prior to unloading (dy/dx = tanΦ0/2, see Fig. 2)

1155

2.

Assume that load was insufficient to bend this sheet in the experimental part (  0  180  2 0 / 2, Fig .3 ).

3. 4.

Then the load was increased until the selected bending angle or dy/dx was obtained. The iteration over the moment equation was then used to finally get dy/dx, y(x) and αf. The used load was then compared to the load calculated by Eq. (7)

Noticing points 1-4 above and how α0 was considered fixed between the theoretical and experimental parts, this is actually more like feeding the procedure with a specific α0 instead of relying on the one that corresponds to the force Q calculated by Eq. (7). Alternatively in this investigation the program was fed with the specific α0 then by the way of iterating over α0 itself and changing Q value (another level of iteration was then introduced over that described in [1] over the x axis).

Non-Linear hardening Assuming that the material behaviour is described best by

  0e  Ke e n , in the plastic region

(13)

where n =strain-hardening exponent. Same line of analysis as above come up to

M

max

   x*  R 0 E   2      0 1  

2

n   t 0 / 2 2 2Ke  2   y y2  dy          *  R 3 0 e R0 3 3    R0   x 0

  

n

  ydy  

   

(14)

Integration of Eq. (14) yields n

n

3 2 2 E 1 * 2  t02 * 4 Ke  2   1         Mmax  R   R x e x 0 0 0  3 n  2  3   R  3 12 R0 3  4    0 









 t0 n2 * n2     x R0   2    





(15)

The radius at which the yielding R0 is given as a function of  e by *

*

 



R0*  3  t 0  E / 4  1   2   e* where



(16)

 e* = the effective stress at the elastic-plastic interface and was calculated by

   0e * e

 *   K e   e   E 

n

(17)

RESULTS In [7] springback behavior of cold rolled transformation induced plasticity (TRIP) steels in air v-bending process was reported. The improved algorithm presented here was able to reproduce the reported results for the TRIP steel with accuracy over 97%. Only data for a constant modulus was used for the comparison. However, the extension of algorithm presented here can be performed and will include the variation in modulus into account. One example for T1.45 as mentioned in [7] is to use the equation E  201 105 GPa (18) Such an equations needs to be introduced to the moment equation like Eq. (8) above and included into the iterative algorithm to converge to specific bending angle, hence calculate the springback more accurately.

1156

LS-Dyna commercial software was used to achieve independent comparison basis for the thin sheets. Shell elements with full integration formulation were used for the blank. Symmetry was utilized and plane strain conditions were imposed, Fig (5). Implicit solver was used for springback analysis using Dynain file approach. The dynein file approach was preferred over the seamless springback approach, see [8] because convergence was some time hard to achive and better stabilization control was generally required. Large springback and flexible parts are considered difficult springback problems even for simple geometry as in this case. The contact stiffness was reduced below the recommended values for metal forming by LS-Dyna, [6]. By selection of suitable contact stiffness scale and reasonable mass scaling, error down to 5% was obtained as compared to the improved algorithm presented here.

FIG. 5. LS-Dyna analysis for Material is 904L, with die width = 300 mm.

The implicit analysis by LS-Dyna deals with the stresses in formed material which is the same starting point of calculations as presented in this paper, however the convergence by the algorithm is guaranteed and proved accurate. Fig. 6 displays the normalized springback obtained theoretically, (α0/αf)theoretical vs. the normalized springback found experimentally (α0/αf)experimental as noticed by the author in a specific range of bending angle α0 (87o≤α0≤86o) for material 253 MA . The author states that the pattern shown may initiate the idea of having a correction factor that is able to bring all the theoretical values to coincide with the experimental values.

FIG. 6. Normalized springback obtained theoretically, (α0/αf)theoretical, vs. normalized springback found experimentally (α0/αf)experimental as noticed by the study author Material is 253 MA. The thick line represents (α0/αf)theoretical = (α0/αf)experimental.

1157

Fig 7 displays the corresponding normalized data; for a different material however; associated with the trend lines as obtained in this investigation. This figure displays the results obtained by both linear and non-linear models, they show very good correlation.

FIG. 7. Normalized springback obtained theoretically, (α0/αf)theoretical, vs. normalized springback found experimentally (α0/αf)experimental as per this investigation, Material is 904L MA. The thick line represents (α0/αf)theoretical = (α0/αf)experimental.

CONCLUSIONS The following can be concluded:   

A partially analytical model (including numerical integration) is correctly reproduced from [1], and by employing improved accurate numerical solution, the model is able to predict the springback values of the sheet metals for both thin and thick sheets. The model gives results that are comparable to those reported by implicit solvers, the convergence in the presented model is guaranteed and easy to implement. Non-linear hardening (Ludwik model) does not add a significant value to the springback results. Linear model does then maintain simpler analysis and does achieve enough accuracy in such applications.

REFERENCES 1. N. Asnafi, “Springback and fracture in v-die air bending of thick stainless steel sheets”, Materials and Design 21(2000) 217236. 2. Johnson W, Mellor PB. Engineering plasticity. London, Great Britain: van Nostrand Reinhold Company Ltd, 1973:124-128. 3. William F. Hosford, Robert M. Caddell Metal Forming.Cambridge University Press, ISBN: 9780521881210. 4. Weinmann KJ, Shippel RJ. Effect of tool and workpiece geometries upon bending forces and springback in 908 v-die bending of HSLA steel plate. Proceedings of the Sixth North American Manufacturing Research Conference (NAMRC)., held in Gainsville-Fl. Dearborn Michigan: SME, 1978:220-227. 5. Bisshopp, K. E., and Drucker, D. C., “Large Deflections of Cantilever Beams,” Quarterly of Applied Mathematics, Vol. 3, 1945, pp. 272-275. 6. H. V. Gajjar, A. H. Gandhi and H. K. Raval, “Finite Element Analysis of Sheet Metal Air-bending Using Hyperform LSDYNA,” Int. J. Mech.Sys. Sci. Eng., vol. 1, no. 2, pp. 117-122, 2008. 7. Fei, D., Hodgson, P., 2006. Experimental and numerical studies of springback in air v-bending process for cold rolled TRIP steels. Nuclear Engineering and Design, 236(18):1847-1851. 8. Maker, B.N. and Zhu, X, “Input Parameters for Metal Forming Simulation Using LS-DYNA,” April, 2000, available at www.feainformation.com/forming_parameters2.pdf or by email from [email protected].

1158

Numerical Modeling for Springback Predictions by Considering the Variations of Elastic Modulus in Stamping Advanced High-Strength Steels (AHSS) Hyunok Kim1* and Menachem Kimchi1 1

Edison Welding Institute, 1250 Arthur E. Adams Drive, Columbus, Ohio 43221-3585, U.S. *Corresponding author: 1-614-688-5239, 1-614-688-5001 and [email protected]

Abstract. This paper presents a numerical modeling approach for predicting springback by considering the variations of elastic modulus on springback in stamping AHSS. Various stamping tests and finite-element method (FEM) simulation codes were used in this study. The cyclic loading-unloading tensile tests were conducted to determine the variations of elastic modulus for dual-phase (DP) 780 sheet steel. The biaxial bulge test was used to obtain plastic flow stress data. The non-linear reduction of elastic modulus for increasing the plastic strain was formulated by using the Yoshida model that was implemented in FEM simulations for springback. To understand the effects of material properties on springback, experiments were conducted with a simple geometry such as U-shape bending and the more complex geometry such as the curved flanging and S-rail stamping. Different measurement methods were used to confirm the final part geometry. Two different commercial FEM codes, LS-DYNA and DEFORM, were used to compare the experiments. The variable elastic modulus improved springback predictions in U-shape bending and curved flanging tests compared to FEM with the constant elastic modulus. However, in S-rail stamping tests, both FEM models with the isotropic hardening model showed limitations in predicting the sidewall curl of the S-rail part after springback. To consider the kinematic hardening and Bauschinger effects that result from material bending-unbending in S-rail stamping, the Yoshida model was used for FEM simulation of S-rail stamping and springback. The FEM predictions showed good improvement in correlating with experiments. Keywords: Springback, Stamping, Advanced high strength steel. PACS: NUMI_ABS214

INTRODUCTION Auto industry experiences the increase of AHSS applications for automotive structure to improve the crashworthiness without corresponding weight increases. AHSS is known to have unique elastic and plastic material behaviors. This becomes more difficult for stamping engineers and tooling designers to reduce or compensate for springback in their production and tooling design. Therefore, the reliable prediction and practical compensation of springback are important to allow production of the desirable stamped parts. Authors conducted the springback analyses using a simple V-die bend test and FEM [1]. S-rail stamping study was conducted and the springback of Srail parts was conducted using FEM with an isotropic hardening model and a variable elastic modulus [2]. In this paper, the new results of the S-rail stamping and springback are presented by considering a kinematic hardening model and a variable elastic modulus. The objective of this study was to improve the prediction of springback in FE simulation of stamping AHSS parts by considering the variation of elastic modulus, kinematic hardening, and the Bauschinger effect.

APPROACH The following three different testing methods were used in this study to evaluate springback (Fig. 1):   

U-Bend Test (UBT) Curved-Flanging Test (CFT) S-Rail Stamping Test (SST)

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1159-1166 (2011); doi: 10.1063/1.3623734 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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c) Final part shapes predicted by FEM

Figure 1. Schematic of various forming tests with S-shaped tooling and the final shape predicted by FEM.

Two different commercial FEM codes, LS-DYNA and DEFORM-3D, were used to analyze the experimental results and predict springback. As shown in Fig. 1, various stamping tests were conducted with an S-rail die at a 160-ton hydraulic press with the CNC-controlled cushion system. Final blank shapes and their locations at a S-rail die for different testing methods were determined to avoid any cracking and wrinkling using preliminary stamping trials and FEA.

RESULTS Characterization of material properties A dual phase (DP) 780 material was used in this study. The elastic and plastic material properties of DP780 were determined by using the ASTM standard tensile test and biaxial bulge test. Three samples were tested for each condition. Table 1 shows the sheet material properties that were obtained from the tensile test in both rolling and transverse directions. Table 1. Summary of ASTM tensile test results. Sheet Material DP 780 (t0=1 mm)

Tensile Testing Direction RD TD

0.2% Yield (MPa) 538.6 531.7

UTS (MPa) 865.5 872.4

Total Elongation (%) 19.3 15.1

A larger range of true stress-strain data for DP 780 was obtained by conducting the viscous pressure bulge (VPB) test, because the tensile test gave a relatively small range of data. As compared in Fig. 2, while the tensile test reaches a maximum strain of 0.11 for DP 780, the VPB test reaches higher maximum strains, up to 0.37, which are often observed in stamping processes.

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Figure 2. True stress-strain data obtained from the VPB test and tensile test for DP 780.

The cyclic loading-unloading tensile test was conducted to quantify the change of elastic modulus of DP 780 materials as the strain increased. The elastic loading and unloading modulus were obtained by calculating the slopes of apparent elastic loading and unloading curves, as shown in Fig. 3.

Loading

Unloading

Figure 3. True stress-strain curves in the loading and unloading tensile test for DP 780 material.

The apparent elastic unloading modulus is more relevant to springback than the apparent elastic loading modulus because springback takes place when the workpiece is released from the punch or die. The apparent elastic modulus of DP 780 was found to decrease about 28% from the initial elastic properties to the elastic modulus at 0.11 strain (Fig. 4). The variation of elastic modulus during unloading was used to determine the mathematical forms of the variable elastic modulus model, the “Yoshida-Uemori model”, as shown in Fig. 4. Three coefficients (E0, EA, and ζ) in the Yoshida model were determined by fitting the outputs of the model with the measured data at a 97% confidence level.

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Figure 4. Yoshida model vs. experimental data for variable elastic modulus of DP 780.

Evaluation of Springback in the UBT and FEA The UBT was conducted for DP 780 material with the S-rail die and the blank holder. The specimen size of 40 × 185 mm was determined by considering the straight-side edges of punch as well as the easiness to measure the bending angle of specimen side-wings. Two different blank holder forces (BHF), 2 and 8 tons, were used to investigate the stretching effect on springback. Most of the tests were conducted with no lubricant. All the measurement of springback angle of tested samples was done by using an X-Y table and a high-resolution camera. Measurements were made for all the tested samples (e.g., three samples for each condition). As the BHF increased, the springback was found to decrease. FEM analyses for the UBT were conducted by using two different commercial FEM codes, DEFORM-3D and LS-DYNA. The same input parameters of material properties, friction, and the same element size were used in both FEM codes. The coefficient of friction was defined to be 0.25 that is reasonable input for a dry friction condition used in experiments. LS-DYNA springback simulations were run using both a variable (see Fig. 4) and constant value (207 GPa) for the elastic modulus. The isotropic hardning model was used in simulation. Two LS-DYNA simulations considered both the variable elastic modulus and the constant elastic modulus. Two different springback angles in the channel opening angle (Springback 1) and the flange angle (Springback 2) were compared between FEM predictions and experiments. The comparison of springback in UBT at a BHF of 2-tons is given in Fig. 5. In the BHF 2-ton, LS-DYNA with variable elastic modulus showed the best correlations with experiments, while DEFORM-3D and LS-DYNA with the constant elastic modulus showed similar predictions and both results gave about 4 to 5 degrees error with respect to experiments.

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Figure 5. Comparison of springback in U-shaped stretch bending of DP 780 at a BHF of 2 tons.

Evaluation of Springback in the CFT and FEA The springback was investigated with the CFT. In the CFT, non-uniform stress and strain conditions cause more severe dimensional errors such as variations of channel opening and side-wall curls, as well as twisting of the drawn channel. The effects of CFT were investigated with tests conducted at two different BHF levels. The CFTs were conducted for DP 780 with an S-rail die by placing the rectangular sample (60 × 185 mm) with respect to the S-rail punch. Most of the experiments were conducted without lubricant. The six different angles on the tested specimens which were measured are shown in Fig. 6.

Figure 6. Locations of the measurements for various angles after springback.

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The mid-section profile was difficult to accurately measure using the camera system or the manual-type protractor. Therefore, the side-edge profiles of tested samples were measured and the average value of measured angles was used to quantify springback. Final bending angles of FEM predictions and experiments at 8-ton BHF are compared in Fig. 7. The variable elastic modulus gave better predictions of the flange angles, βin and βout. However, for the channel angles (өin and өout) and the shoulder angles (αin and αout), it is not clear which elastic modulus model gave better predictions compared to experiments. This may be caused by possible measurement errors for these angles. It was more difficult to accurately measure the channel angles and the shoulder angles compared to the flange angles, because of the sidewall curl and twisting.

Figure 7. Comparison of the final bending angles of curved flanging between FEM predictions and experiments at a BHF of 8 tons.

Evaluation of Springback in the SST and FEA In the SST, the distributions of stress and strain on the part are more complicated than for UBT and CFT. Because of complex geometry and stress conditions, three different types of springback, defined twisting of the channel, channel opening, and sidewall curl, occur in S-rail stamping. Experiments were conducted with DP 780 (t0=1.0 mm) materials. Fig. 8 shows the stamping test setup and the CAD models of the initial blank geometry and the designed stamping geometry. FEM simulations for S-rail stamping were conducted by using a commercial FEM code, LS-DYNA. Two different elastic modulus models; a constant (207 GPa) and the variable (as given in Fig. 4); two different hardening models; the isotropic hardening and kinematic hardening models; were used for DP 780 in the FEM simulations. Shell elements with a 2-mm element size were used for the sheet blank model. The coefficient of friction was input as 0.125 to consider the lubrication at the tool-workpiece interface. As shown in Fig. 8, the FEM model was carefully set up by considering the relative positions of the initial sheet blank to other punches and dies, which were measured in the experiments.

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Figure 8. S-rail tooling and the initial blank and final stamping geometries, and FEM model

In the test, the final depth of stamping was determined to be 20 mm and BHF was limited to 2 tons to avoid any cracking of sheet material. A pressure pad force on the top of the hat surface was measured to be increased from 10 to 44 kN as the stroke increase up to 20 mm during stamping. The initial blanks were lubricated with a stamping lubricant before the test. The stamped part was scanned by the 3-D white light scanning method to obtain a 3-D CAD model from the actual stamped part. The isotropic hardening model did not provide accurate results of springback regardless of elastic modulus models; a constant or variable. The detailed results has been published in [2]. Therefore, in this study, the YoshidaUemori model [3] was used to consider the effects of the kinematic hardening and the Bauschinger effect on springback. This model was implemented by using an advanced material model (*MAT_125) that is available in LS-DYNA [4]. The detailed parameters for DP 780 were provided by a steel supplier [5], as shown in Table 2. Two different sections on the stamping were compared for springback between the actual and FEM predicted final part geometry as shown in Fig. 9. Two reference points at the top flat sectional profile were fixed to compare the multiple profiles together. TABLE 2. Parameters for the Yoshida-Uemori model for DP 780 Variable CB Y SC K Value 453 MPa 291 MPa 513 MPa 62 Variable EA COE IOP C1 Value 149 GPa 54 0.0 0.052

RSAT 700 MPa C2 0.955

SB 449 MPa

H 0.95

Figure 9: Two selected sections for springback comparison

Fig. 10 compares springback results for two different sections. New simulation results (A and B) with the Yoshida-Uemori model showed better correlations with experiments (Exp) in comparison to the isotropic hardening model results (C and D). The variable elastic modulus (A) gave better predictions of the channel opening (θ) and the flange angle (β), while the constant elastic modulus showed slightly better correlation with experiment for the dieshoulder angle (α).

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Figure 10: Comparison of FEM predictions and experiment

CONCLUSIONS The following conclusions can be drawn from this project:  As the plastic strain increases, the elastic modulus significantly decreased up to 28% for DP 780 from the initial values.  The springback decreased with increasing BHF from 2 to 8-tons, as confirmed in the CFT and the UBT results. However, the side-wall curl was increased for increasing the BHF.  The isotropic hardening models with constant and variable elastic modulus showed limitations in SST to predict the channel opening and the sidewall curl compared with experiments.  The Yoshida-Uemori model with the consideration of kinematic hardening and the Bauschinger effects showed better correlations with springback measurements in S-rail stamping compared to the isotropic hardening models regardless of the elastic modulus models.  To predict the springback of the relative simple part geometry; V-die or U-die bending; the constant elastic modulus and the isotropic hardening model can be reasonable input data of FEM. However, to predict the spingback of the complicate stamping geometry that encounters the repeated bending and unbending, it is recommended to define the non-linear hardening behaviors and elastic modulus change in the material input data for reliable FEM simulations.

ACKNOWLEDGEMENTS This project was conducted with support from the State of Ohio, Department of Development, and Thomas Edison Program. The authors would like to thank Dr. Taylan Altan, Director of Engineering Research Center for Net-Shape Manufacturing (ERC/NSM), the Ohio State University and Dr. Boel Wadman, Project Manager of Swerea-IVF, for their technical supports and advices on this research program.

REFERENCES 1. Kim, H., Kimchi, M., Altan, T. Control of springback in bending and flanging advanced high-strength steels (AHSS), International Automotive Body Congress (IABC), 4-5 November 2009, Troy, Michigan, USA. 2. Kim, H., Kimchi, M., Kardes, N., Demiralp, Y., Mete, O., Altan, T., Predictions of springback in the S-rail stamping of AHSS using FEM with the variable elastic modulus, IABC, 3-4 November 2010, Troy, Michigan, USA. 3. Yoshida, F., Uemori, T., A model of large-strain cyclic plasticity describing the Bauschinger effect and workhardening stagnation, International Journal of Plasticity 18, 2002, pp. 661-686. 4. LSTC, LS-DYNA Keywords User’s Manual, Vol. 2, Material Models, Version 971,Rev 5.0, 2010, pp. 479-482. 5. Chen, X., Shi, M., Zhu, X., Du, C., Xia, Z.C., Xu, S., Wang, C.T., Springback prediction improvement using new simulation technologies, SAE Paper 2009-01-0981, 2009.

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Free Surface Roughness Prediction in Bending Under Tension Sergei Alexandrova, Ken-ichi Manabeb and Tsuyoshi Furushimab a

A.Yu.Ishlinskii Institute for Problems in Mechanics of the Russian Academy of Sciences, 101-1 Prospect Vernadskogo, 119526 Moscow, Russia b Department of Mechanical Engineering, Tokyo Metropolitan University, 1-1 Minami-osawa Hachioji, 192-0397 Tokyo, Japan

Abstract. The paper presents the first step of research aimed at determining the specific form of a new empirical equation for free surface roughness evolution. In contrast to conventional empirical equations for free surface roughness evolution, the new equation introduces a dependence of the increment of roughness parameters on two kinematic variables. Nevertheless, it can be conveniently represented by a curve in appropriate space. The experimental method chosen is the process of plane strain bending of a sheet stressed by tensions applied to the ends. When the convex surface of the sheet is considered, one point of the aforementioned curve can be obtained. The theoretical part of the approach proposed consists of an analytic solution for an idealized process and simple geometric considerations. Keywords: Free Surface Roughness, Empirical Equation, Bending Under Tension, Semi-Analytic Method. PACS: 46

INTRODUCTION The roughness has a great impact on the quality of surface after metal forming processes. Yet, it may have an effect on the overall deformation in course of such processes. The effect depends on the ratio of roughness parameters to a characteristic length of the process and, therefore, is more pronounced in microforming processes. A typical micro-forming process where the free surface roughness plays a crucial role is die-less drawing of microtubes. Theoretical methods for predicting the evolution of free surface roughness include empirical equations, for example [1, 2] and numerical methods, for example [3, 4]. The empirical equation proposed in [1] has been successfully used in applications [5, 6]. The empirical equation given in [2] generalizes the equation from [1] and is based on an analysis of several independent experimental results [7-13]. In contrast to the original equation, the new equation introduces a dependence of the increment of roughness parameters on two kinematic variables. Nevertheless, it can be conveniently represented by a curve in appropriate space. A general numerical method for predicting the free surface roughness evolution in the case of plane strain deformation has been proposed in [14]. This method relates the solution of a boundary value problem of a phenomenological theory of plasticity and the finite element solution for a representative element of material in the vicinity of the free surface. Both experimental and numerical methods can be adopted to find the specific shape of the aforementioned curve. When any plane strain process in which infinitesimal fibers normal to the free surface are subject to compression is considered, just one point of the curve can be found. In the present paper, the convex surface of the sheet subject to bending under tension is chosen for this purpose. An advantage of this process over the pure bending process is that much larger strain can be achieved. On the other hand, the theoretical analysis based on conventional assumptions is practically the same for both processes. In particular, a unified approach for the analysis of pure bending proposed in [15] has been successfully used for several material models in [16-19] and then extended to bending under tension in [20]. These solutions are semi-analytical. In particular, a numerical treatment is usually necessary for integrating and solving transcendental equations. However, in the case under consideration the experimental technique is based on displacement control. Therefore, one additional relation to connect kinematic variables is known in advance. As a

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1167-1174 (2011); doi: 10.1063/1.3623735 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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result, for any isotropic incompressible material the complete system of equations used to relate the measurable input variables and roughness parameters is obtained from simple geometric considerations without solving the stress equations. This means that the analysis is independent on the particular hardening law.

GENERAL APPROACH The empirical equation for predicting the evolution of free surface roughness proposed in [2] can be written in the form

'R ag where

§ Hn · ¨ H eq ¸¸ © ¹

H eq : ¨

(1)

'R is the increment of any roughness parameter, Ra or Rz , ag is an average grain size, H eq is the

equivalent strain,

Hn

is the logarithmic strain normal to a free surface, and : is a function of its argument. This

function should be determined either from experiment or from numerical simulation for several ratios

H n H eq . The

equivalent strain is defined by t

³[

H eq

eq

dt , [ eq

0

where

[eq

is the equivalent strain rate,

[ij

2 [ij[ij 3

(2)

are the components of the strain rate tensor, and t is the time. The

integral in (2) is taken over the actual strain path. Some general features on the function :

H

n

H eq

found in [2]

with the use of experimental data are illustrated in Figure 1. It has been hypothesized that the function has two local maximums and one local minimum. The local maximums are located in the vicinity of the points and

H n H eq

 3 2 . The local minimum should be near the point H n H eq

shape of the curve are from (2) that addition,

H n H eq

Hn  0

E1 ! E

3 2

0 . The restrictions imposed on the

0 d H n H eq d 1 and J 1 ! J in the range 0 t H n H eq t 1 . It is seen

3 2 in the case of plane strain deformation, if the material is incompressible. If, in

then equation (1) can be rewritten in the form

'R ag Thus

in the range

H n H eq

§

H eq : ¨¨  ©

3· ¸ H eq : ps , : ps 2 ¸¹

§ 3· : ¨¨  ¸¸ © 2 ¹

(3)

'R is proportional to the equivalent strain and the coefficient of proportionality is ag : ps . Since the

average grain size is known, it is sufficient to make one experiment from which the roughness coefficients should be found to determine the value of : ps and thus one point of the curve shown in Figure 1. An alternative approach is to determine the evolution of roughness parameters numerically, for example with the use of the method proposed in [14]. It is seen from Figure 1 that results of three or four different experiments (or numerical solutions) should provide sufficient information to propose an analytic representation on the function of variation of its argument, 1 t H n surfaces under the condition

:  H n H eq in the entire range

H eq t 1 . The present paper deals only with plane strain deformation of free

H n  0 . One of possible tests in this case is plain strain bending under tension. The

roughness parameters should be measured on the convex free surface of the sheet.

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FIGURE 1. Typical shape of the curve representing the function :.

BENDING UNDER TENSION An idealized process of bending under tension is illustrated in Figure 2. The initial shape is rectangular A1B1C1D1. Any intermediate shape is sector ABCD with AB and CD being circular arcs whereas AD and BC are straight lines. Such transformation of the rectangular into the sector is possible if a force F and a bending moment M are applied at BC and AD and, in addition, a uniform pressure p is applied over CD. It is obvious that F and p are not independent. However, this relation is not important for the purpose of the present paper. The pure bending process is obtained as a particular case when F 0 and p 0 . In this particular case, a unified approach for analysis of the process for quite an arbitrary material model has been proposed in [15]. This approach has been used in [16-19] and then extended to the process of bending under tension in [20]. A similar approach has been proposed in [21]. The starting point of the approach proposed and developed in [15, 20] is the following transformation equations between Eulerian Cartesian coordinates xy and Lagrangian coordinates ]K (see Figure 2)

]

x H

a



s s cos  2aK  , 2 a a

a)

K= L/H C1

y

]

y H

a

b)

K= L/H

 =0

y B

=0

=1

T H=h

p

2L

T O

(4)

M r F C

B1

 =1

s sin  2aK a2



x

O1

O

rCD

x

h

rAB

D F A1

D1

M

A

K= L/H

K= L/H

FIGURE 2. Geometry of the problem and coordinate systems (a) initial shape, (b) intermediate shape.

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where H is the initial thickness of the sheet, s is a function of a, a is a function of the time, and a

0 at t

0 . The

dependence of a on t is immaterial for rate-independent materials. The function s  a should be found with the use

of constitutive equations and the equilibrium equations. It is possible to verify by inspection that equations (4) correspond to the transformation of rectangular A1B1C1D1 into sector ABCD if

1 4

s at a

(5)

0 . It has been shown in [15] that the radii of curves AB and CD are given by

rAB H

s , a

sa a

rCD H

(6)

respectively. Moreover, the principal components of the strain tensor are determined from the equations

H]

1  ln ¬ª 4 ] a  s ¼º 2

HK

(7)

The equivalent strain can be found from these equations in the form

H eq Since

]

1 ln ª 4 ] a  s º¼ 3 ¬

(8)

0 on surface AB throughout the process of deformation, it follows from (8) that

H eq ]

0

H AB

ln  4 s 3

(9)

It is obvious that this value of the equivalent strain is involved in (3). When F is prescribed, for example F 0 in the pure bending process, numerical integration and solution to

transcendental equations are required to find the function s  a . Once this function has been determined, the values of rAB , rCD and

H AB

can be found from (6) and (9) with no difficulty. However, the process of bending under

tension can be performed under displacement control. In this case the function s  a can be determined from geometric relations independently of the constitutive model. In particular, the experimental set-up used in the present study is shown in Figure 3. In order to relate it to the idealized process illustrated in Figure 2, it is necessary to assume that the concave surface of the sheet is a circular arc. Then, the geometric scheme of the real process can be represented by a diagram shown in Figure 4. Here L and l do not change with the time. Therefore, points C1 and C2 are fixed in space. On the other hand, Z is a prescribed function of the time. The value of Z can be measured at any stage of the process. U is a function of the time and U o f as t o 0 . It is obvious that U rCD . Therefore, it follows from (6) that

U H

sa a

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(10)

Using elementary geometric relations (Figure 4) it is possible to arrive at

2l 2L

FIGURE 3. Experimental set-up.

O

2G

U

2J C1

C2

Z

2l 2L FIGURE 4. Idealized scheme of the process.

U

L , tan G sin J

l 2 § J G · 2 , 2 L sin ¨ ¸ sin J Z   L  l Z  L cot J © 2 ¹

(11)

It follows from these equations that

tan J  tan  J  G tan J tan  J  G  1 where

J G

tan J l L ª¬1   Z L tan J º¼

(12)

º » » ¼

(13)

should be excluded by means of

J G

ª sin J 2arcsin « « 2 ¬

2

l· §Z· § ¨ ¸  ¨1  ¸ © L¹ © L¹

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2

Then, equation (12) can be solved numerically for

J

sa a Since the function

J Z

. Combining (10) and (11)1 gives

L H sin J

(14)

has been determined, this equation provides a relation between s and a at any value of Z.

One more relation is required to find the functions s  Z and a  Z . It follows from (7) that

HK ] HK

1

where

HKCD

HK

ln  UJ L (Figure 4). Using (11)

CD

is the value of the strain

HKCD

1 ln ª 4  s  a º¼ 2 ¬

(15)

on surface CD (Figure 3). On the other hand, this strain can be calculated as

§ J · ln ¨ ¸ © sin J ¹

HKCD

(16)

Comparing (15) and (16) gives the second relation between s and a in the form

§ J · 4s  a ¨ ¸ © sin J ¹

2

(17)

Excluding s  a between (14) and (17) leads to

a

J H

(18)

2 L

Then, using (17) 2

s

1§ J · J H ¨ ¸  4 © sin J ¹ 2 L

It is seen from this solution that (5) is satisfied as

H AB

Since the function

J Z

(19)

J o 0 . Substituting (19) into (9) gives

2 1 ª§ J · Hº  ln «¨ 2 J » ¸ L ¼» 3 ¬«© sin J ¹

has been determined, equation (20) provides the dependence of

(20)

H AB

on Z L .

ROUGHNESS EVOLUTION A bending under tension test was performed with the use of the experimental set-up shown in Figure 3 on an A1050 aluminum alloy sheet 0.7mm thick. The length of the specimen used for the current work was 120mm and

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the width was 20mm. The spans between the inner and outer supports had lengths of 10mm and 100mm, respectively. An average grain size in the specimens after this treatment was of the order of 25Pm. The initial arithmetic mean roughness was Ra | 0.35P m . Using the geometric parameters of the test the dependence of H AB on Z L has been found by means of (20). It is illustrated in Figure 5. Roughness parameters have been measured at several stages of the process (several Z L - values) with the use of a standard technique. Then, using the dependence of

H AB

on Z L (see Figure 5) the variation of the roughness parameters with

In particular, the variation of Ra with

H AB

H AB

has been obtained.

is depicted in Figure 6. It is seen from this figure that this dependence is

linear, which is in agreement with (3). Using this dependence and (3) it is possible to find that ag : ps | 9.5P m .

0.38 and one point of the curve shown in Figure 1 has been determined.

0.14 0.12

HAB

0.1 0.08 0.06 0.04 0.02 0 0

0.05

0.1

0.15

0.2

0.25

0.3

Z/L FIGURE 5. Variation of HAB with Z/L.

2

Arithmetical mean roughness Ra [Pm]

Thus : ps

1.5

1

0.5

0 0

0.05

0.1

0.15

Equivalent strain H$% FIGURE 6. Variation of the arithmetic mean roughness with strain.

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CONCLUSIONS A theoretical/experimental procedure based on the plane strain bending under tension test and its analytical treatment has been proposed and completed to determine one point of the curve representing the empirical equation for free surface roughness evolution proposed in [2]. The theoretical treatment is quite simple and reduces to numerical solution of equation (12). Another point of the curve can be in general found from the same test considering the concave surface of the sheet because it is also traction free. In this case H n ! 0 and therefore the argument of the function :

H

n

H eq

is

H n H eq

3 2 . Still another point can be obtained by measuring the

evolution of roughness parameters on the sides of the specimen where made (see Figure 1) the function

H n H eq | 0 . According to the assumptions

:  H n H eq attains local maximums at H n H eq

r 3 2 and a local minimum

H n H eq | 0 . Therefore, the values of the function at three aforementioned points allow one to propose a rather accurate specific dependence of : on H n H eq in the entire range of H n H eq values, 1 d H n H eq d 1 . Other at

possible tests for determining the function

:  H n H eq have been proposed in [2].

ACKNOWLEDGMENTS The first author acknowledges support from the Russian Foundation for Basic Research (Project 10-08-00083).

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