STUDIES IN APPLIED MECHANICS 45
Advanced Methods in M a t e r i a l s Processing Defects
STUDIES IN APPLIED M E C H A N I C S 20. 21. 22. 23. 24. 25. 28. 29. 31.
Micromechanics of Granular Materials (Satake and Jenkins, Editors) Plasticity. Theory and Engineering Applications (Kaliszky) Stability in the Dynamics of Metal Cutting (Chiriacescu) Stress Analysis by Boundary Element Methods (Balas, Sl&dek and Sl~dek) Advances in the Theory of Plates and Shells (Voyiadjis and Karamanlidis, Editors) Convex Models of Uncertainty in Applied Mechanics (Ben-Haim and Elishakoff) Foundations of Mechanics (Zorski, Editor) Mechanics of Composite Materials - A Unified Micromechanical Approach (Aboudi) Advances in Micromechanics of Granular Meterials (Shen, Satake, Mehrabadi, Chang and Campbell, Editors) 32. New Advances in Computational Structural Mechanics (Ladev~ze and Zienkiewicz, Editors 33. Numerical Methods for Problems in Infinite Domains (Givoli) 34. Damage in Composite Materials (Voyiadjis, Editor) 35. Mechanics of Materials and Structures (Voyiadjis, Bank and Jacobs, Editors) 36. Advanced Theories of Hypoid Gears (Wang and Ghosh) 37A. Constitutive Equations for Engineering Materials Volume 1: Elasticity and Modeling (Chen and Saleeb) 37B. Constitutive Equations for Engineering Materials Volume 2: Plasticity and Modeling (Chen) 38. Problems of Technological Plasticity (Druyanov and Nepershin) 39. Probabilistic and Convex Modelling of Acoustically Excited Structures (Elishakoff, Lin and Zhu) 40. Stability of Structures by Finite Element Methods (Waszczyszyn, Cicho5 and Radwar~ska) 41. Inelasticity and Micromechanics of Metal Matrix Composites (Voyiadjis and Ju, Editors) 42. Mechanics of Geomaterial Interfaces (Selvadurai and Boulon, Editors) 43. Materials Processing Defects (Ghosh and Predeleanu, Editors) 44. Damage and Interfacial Debonding in Composites (Voyiadjis and Allen, Editors) 45. Advanced Methods in Materials Processing Defects (Predeleanu and Gilormini, Editors) General Advisory Editor to this Series: Professor Isaac Elishakoff, Center for Applied Stochastics Research, Department of Mechanical Engineering, Florida Atlantic University, Boca Raton, FL, U.S.A.
STUDIES IN APPLIED M E C H A N I C S 45
Advanced Methods in M a t e r i a l s Processing Defects
Edited by
M. Predeleanu
a n d P. G i l o r m i n i
Laboratoire de M~canique et Technologie, ENS de Cachan, CNRS, Universit6 de Paris 6, 61 avenue de President Wilson, 94235 Cachan Cedex, France
1997 ELSEVIER Amsterdam - Lausanne - New York- Oxford - Shannon - Tokyo
ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 RO. Box 211, 1000 AE Amsterdam, The Netherlands
ISBN 0-444-82670-x 91997 Elsevier Science B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, RO. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A.- This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the copyright owner, Elsevier Science B.V., unless otherwise specified. No responsibility is assumed by the publisher f9r any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands.
Third International Conference on Materials Processing Defects J u l y 1-3, 1997, C a c h a n , F r a n c e O r g a n i z e d by: Laboratoire de Mdcanique et Technologie (ENS de Cachan, CNRS, Universitd Paris VI) Conference Chairmen: M. PREDELEANU and P. GILORMINI International Advisory Committee: L. ANAND D. BESDO P.R. DAWSON I.S. DOLTSINIS S.K. GHOSH P. HARTLEY M. HASHMI T. INOUE J. KIHARA K.W. NEALE A. NEEDLEMAN E. OI~ATE O. RICHMOND E. STEIN C.L. TUCKER III
USA Germany USA Germany Germany UK Ireland Japan Japan Canada USA Spain USA Germany USA
R.H. W A G O N E R
USA
O r g a n i z i n g C o m m i t t e e (France): J.F. AGASSANT B. BAUDELET M. BRUNET J.L. CHENOT A~ COMBESCURE J.P. CORDEBOIS J.C. GELIN
P. LADEVI~ZE G. MAEDER J. OUDIN A~ POITOU J. GIUSTI C. TEODOSIU
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vii
This volume includes the contributions to the Third International Conference on Materials Processing Defects, which follows its two predecessors held in Cachan, France (1987), and Siegburg, Germany (1992). The Conference focused on advanced methods for predicting and avoiding the occurrence of defects in manufactured products. A new feature was included, namely, the influence of the processing-induced defects on the integrity of structures. The following topics were developed: Damage modeling Damage evaluation and rupture Strain localization and instability analysis Formability characterization Prediction of shape inaccuracies Influence of defects on structural integrity The main manufacturing operations have been covered and various materials were considered: new and conventional metal alloys, ceramics, polymers and composites. It is worth noting that damage theory is used increasingly both for estimating the soundness of the formed product and for defining the workability of a given material. Moreover, new damage models were proposed. High-rate loading conditions, which occur for instance in machining or explosive forming, were also considered. Finally, numerical simulations were used extensively to predict and avoid the shape inaccuracies arising in most forming processes. We believe that this series of conferences stimulates innovative approaches in this important field with obvious economic implications and m u s t be continued. July 1997 M. Predeleanu and P. Gilormini
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CONTENTS Preface ..............................................................................................................................................
vii
DAMAGE MODELING On the dynamic cavitation in solids L. Badea and M. Predeleanu ...........................................................................................................
3
A ductile damage model including shear stress effect J.C. Boyer and C. Staub ...................................................................................................................
13
A mesoscopic approach of ductile damage during cold forming processes G. Brethenoux, P. Mazataud, E. Bourgain, M. Muzzi and J. Giusti ...............................................
23
A fully coupled elasto-plastic damage damage theory for anisotropic materials J.E Charles, Y.Y. Zhu, A.M. Habraken, S. Cescotto and M. Traversin ..........................................
33
Mathematical modelling of dynamical deforming and combined microfracture of damageable thermoelastoviscoplastic medium A.B. Kiselev .....................................................................................................................................
43
A mathematical model for the formation and development of defects in metals V.L. Kolmogorov, V.P. Fedotov and L.F. Spevak ............................................................................
51
Healing of metal microdefects after cold deformation V.L. Kolmogorov and S.V. Smirnov ................................................................................................
61
Definition of the form for kinetic equation of damage during the plastic deformation S.V. Smirnov, T.V. Domilovskaya and A.A. Bogatov .....................................................................
71
DAMAGE EVALUATION AND RUPTURE The influence of critical defect size in a ceramic of alumina elaborated by process sol-gel route N.H. Almeida Camargo, M. Murat and E. Bittencourt ...................................................................
83
Defect evolution during machining of brittle materials A. Chandra, K.E Wang, Y. Huang and G. Subhash ........................................................................
89
Modeling the influence of gradients in strength on the evolution of damage in metals P.R. Dawson, D.J. Bammann and D.A. Mosher .............................................................................
99
On the fracturing of brittle solids with microstructure I. St. Doltsinis ..................................................................................................................................
111
Fracture prediction of sheet-metal blanking process R. Hambli, A. Potiron, S. Boude and M. Reszka ............................................................................
125
Elastic-plastic finite-element modelling of metal forming with damage evolution P. Hartley, F.R. Hall, J.M. Chiou and I. PiUinger .....................................................................
135
Processing of zinc oxide varistors:sources of defects and possible measures for their elimination A.N.M. Karim, S. Begum and M.S.J. Hashmi ................................................................................
143
Analysis of metallic solid fractures by quasimolecular dynamics Y.S. Kim and J.Y. Park ....................................................................................................................
155
Damage framework for the prediction of material defects: identification of the damage material parameters by inverse technique E Lauro, T. Barfi~re, B. Bennani, P. Drazetic and J. Oudin ...........................................................
165
Damage influence in the finite element computations for large strains elastoplastic mechanical structures P. Picart, G. Piechel and J. Oudin ....................................................................................................
175
Microplasticity and tensile damage in Ti-15V-3Cr-3A1-3Sn alloy and Ti-15V-3Cr-3A1-3Sn/SiC composite W.O. Soboyejo, B. Rabeeh, Y. Li, A.B.O. Soboyejo and S.I. Rokhlin ...........................................
185
STRAIN LOCALIZATION AND INSTABILITY ANALYSIS Defects in hydraulic bulge forming of tubular components and their implication for design and control of the process M. Ahmed and M.S.J. Hashmi ........................................................................................................
197
Nimerical and experimental analysis of necking in 3D sheet forming processes using damage variable M. Brunet, S. Mguil-Touchal and F. Morestin ................................................................................
205
Localization of deformation in thin shells with application to the analysis of necking in sheet metal forming J.C. Gelin and N. Boudeau ..............................................................................................................
215
Microcrack induced bifurcation of stress-strain relations for sintered materials D.G Karr and S.A. Wimmer ............................................................................................................
225
Instability analysis for ellipsoidal bulging of sheet metal D.W.A. Rees .....................................................................................................................................
235
FORMABILITY CHARACTERIZATION Compression of a block between cylindrical dies and its application to the workability diagram S. Alexandrov, N. Chikanova and D. Vilotic ...................................................................................
247
Sheet metal formability predicted by using the new (1993) Hill's yield criterion D. Banabic ........................................................................................................................................
257
Characterization of the formability for aluminum alloy and steel sheets S. Barlat, J.C. Brem, D.J. Lege and K. Chung ................................................................................
265
Material plastic properties defects and the formability of sheet metal J.D. Bressan .....................................................................................................................................
273
xi Formability analysis based on the anisotropically extended Gurson model E. Doege, A. Bagaviev and H. Dohrmann ................................................................................
281
Rupture criteria during deep drawing of aluminum alloys J. Proubet and B. Baudelet ..............................................................................................................
289
PREDICTION OF SHAPE INACCURACIES Prediction of flange wrinkles in deep drawing J. Cao, A. Karafillis and M. Ostrowski .......................................................................................
301
Filling defects in ceramics forming process F. Chinesta, R. Torres, I. Mont6n, A. Poitou and F. Olmos ......................................................
311
Localisation of debinding zone for fluid-particle flows in metal injection molding M. Dutilly and J.C. Gelin ................................................................................................................
321
Simplified approaches for the prediction of deep-drawing ears P. Gilormini and B. Bacroix ............................................................................................................
331
Springback and 'rebound' phenomenon analysis with the software PLIAGE F. Morestin, M. Boivin, F. Bublex, X. Deng and M. E1 Mouatassim ... ..................................
341
Prediction of elastic springback defects in sheet stamping processes using finite element methods E. Ofiate and J. Rojek ......................................................................................................................
349
Creep deformation in heat treated components T.C. Tszeng and W.T. Wu ................................................................................................................
361
A study of sharskin defects of linear low density polyethylene C. Venet, J.E Agassant and B. Vergnes .............. .............................................................................
373
INFLUENCE OF DEFECTS ON STRUCTURAL INTEGRITY Influence of initial imperfection on the collapse of thin walled structures A. Combescure .................................................................................................................................
385
On modeling of laminated composite structures featuring interlaminae imperfections M. Di Sciuva, U. Icardi and L. Librescu .....................................................................................
395
Delamination, instability and failure of multilayered composites E. Stein and J. Tel3mer .....................................................................................................................
405
Statistical damage tolerance for cast iron under fatigue loadings H. Yaacoub Agha, A.S. Brranger, R. Billardon and E Hild ...........................................................
415
Author index ....................................................................................................................................
425
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DAMAGE MODELING
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Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
O n t h e d y n a m i c c a v i t a t i o n in solids L. Badea a and M. Predeleanu b aInstitute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-70700, Bucharest, Romania bLaboratoire de M6canique et Technologie/E.N.S. de Cachan / UnivSrsit6 Paris 6/61, Avenue du Pr6sident Wilson, 94235 Cachan, France
1
INTRODUCTION
The nucleation of voids or cavities represents one of the main aspects of the ductile microfracture process in solids, specially in metals and alloys, subjected to high straining as in forming or impact problems. It is established that the ductility of the materials is increased if the formation of new microcavities can be stopped or diminished. The influence of the void nucleation on the various coalescence mechanisms (as direct impingement of the voids, microstrain localisation, etc.) was determined quantitatively in recent studies [1-3]. The void nucleation sites were extensively observed, the most attention being focussed on the nucleation attached to second phase particles : void formation by particle fracture or by decohesion of the particle - matrix interface or even at grain boundaries in polycrystals [4]. This kind of nucleation was classified as heterogeneous in contrast with homogeneous nucleation observed in particle free regions [5]. In [6], it was reported, for instance, that in AISI 304 stainless steel the void density was greater than the particle density by a factor of one hundred. Many microcavities were observed in areas of high dislocation densities. The threshold stress for the homogeneous nucleation of voids is of course greater than the one for heterogeneous nucleation and is generally very high. It is worth noting that in shock-loaded materials subjected to explosive forming or high rate impact loadings the tensile mean stress can exceed 10-20 GPa. The sudden formation of voids in solids (a phenomenon named "cavitation" as in fluid mechanics) has been observed in various kinds of materials (rubbers, elastomers, metals, composites) submitted to tension loadings. The paper of Gent and Lindley [7] has drawn special attention because the experimental critical load for the occurrence of a void in a short rubber cylinder pulled in tension agrees with the theoretically deduced one. Analysing the behaviour of spherical cavities in an infinite elastic (neo-Hookean) body loaded by a hydrostatic tension, they found that there is a critical value of the load at which the void grows rapidly without bound (" cavitation instability"). Consequently, Gent and Lindley explained the cavitation phenomenon by the rapid growth (without bound) of a pre-existing microscopic void. An alternative approach for cavitation problems was proposed by Ball [8] which used the discontinuous radially symmetric solutions for the equilibrium equations of an
elastic solid sphere submitted to traction loadings on the boundary. He has shown that for a certain class of elastic materials there exists a critical value of the loading at which a non-homogeneous solution describing the formation of a central cavity bifurcates from the trivial homogeneous solution, which becomes unstable. The cavitation solution is energetically favorable. Due to scaling of the two above problems in finite elasticity, the critical load obtained by Gent and Lindley for cavitation instability is that obtained by Ball for the bifurcation of the discontinuous solutions. Equivalent results were reached by Sivaloganathan [9] and Horgan and Abeyaratne [10], who have treated the cavitation problem for the sphere by considering at its center an infinitesimal pre-existing void (the initial radius approaches zero at critical load). The study of the solutions that allows the cavitation for the full three-dimensional problem in nonlinear elasticity is given in [11]. The cavitation phenomenon has been examined also for elastic-plastic solids for symmetric loadings in [12-17] and for non-symmetric loadings in [lS]. The radially symmetric cavitation problem for rate-dependent materials has been treated in [19]. A comprehensive review of the literature on cavitation may be found in [20]. Very few studies have been concerned with the dynamical problem of cavitation and that in the context of finite elasticity, [21-22], and of viscoplasticity, [23]. This paper is concerned with the continuum micromechanics analysis of homogeneous nucleation and its objective is to deduce analytically the conditions for the sudden cavitation in elasto-plastic and viscoplastic solids subjected to high dynamic loadings. In the next section we state the formulation of this mechanical problem for a hollow sphere submitted to a symmetric traction loading in the dynamical case, for both viscoplastic and elasto-plastic materials. By considering the incompressibility of the material our problem reduces to solving some second order nonlinear differential equations. These equations proceed from the equation of motion and have as unknown the current void radius. In Section 3 we shall consider the initial void radius to be infinitesimal, and consequently, we shall take this radius as vanishing. In what follows we find an expression for the critical load and we give some theoretical results concerning the dependence of the solution on this critical load and various initial conditions. We also make some remarks concerning the theoretical results and we shall draw some conclusions for each material type, viscoplastic and elasto-plastic. 2
MECHANICAL
PROBLEM FORMULATION
Consider a hollow sphere with its center situated at the origin of a spherical coordinate system (R, O, r and denote by A and B its inner and outer radii in the undeformed configuration. The sphere is supposed to have in time only a spherically symmetric motion and therefore, the coordinates in the current deformed configuration (r, 0, ~o) are of the form
o=o,
(:)
for the time interval 0 < t < oo. The inner and outer radii in the current deformed configuration will be denoted by a and b, respectively. m
Taking into account (1), the velocity field is 0r = ~
= ~,
(2)
and consequently, the Eulerian strain rate tensor has the form Ovr
D = -b-;~ o ~ + ~(~0~ o ~ + ~ ~ ~ ) .
(3)
The Cauchy stress tensor can be written as (4)
er = a f t e r | er + aoo(eo | eo + e~ | e~).
As usual, we shall assume an additive decomposition of the strain rate tensor, D = D" + D p,
(5)
in its elastic, D ", and plastic, D p, parts. We shall suppose that the sphere is an isotropic and homogeneous material. The thermoelastic response is defined by v
(6)
D~ = 1 + E v& _ -~tr&l + ar
where E and v are the elastic constants, I is the unit tensor, T the absolute temperature and a the coefficient of linear thermal expansion. The dot denotes the material time derivative. We note that in our case, the corotational (Jaumann) time rate of the Cauchy stress tensor coincides with its material time derivative. The viscoplastic strain rate is written in the overstress form, widely used in dynamic problems, [27], F
DP=
r
~-
OF
1 >)0er,
(7)
where "r is a temperature-dependent viscosity function, r is a control function, r is the viscoplastic overstress function, Ey the material isotropic hardening-softening temperature-dependent function, and F the quasi-static yield function. We shall consider a von Mises material, F(er) = ~/3erd. era, where erd is the deviatoric part of the stress tensor. For the elastic-plastic material, we shall take Dp
. OF
= P0~'
(8)
where i5 = ~ / ] D p " D p is the equivalent plastic strain rate. Also, the yield criterion for the plastic zone will be written as, F - ay - kp n = 0,
(9)
where k > 0 and 0 < n _~ 1 are some material constants and ay is the elastic limit. Supposing the material incompressible, t r D = O, from (3) we get OUr Vr 0--7 + 2--r = O.
(10)
As consequence, we deduce that r 3 = a 3 + R 3 - A 3. The motion equations are reduced in our case to a single one, (%Trr
T
Or
2-,r
p6,=
(11)
where the density p is constant and r = a00 - a ~ . As we mentioned above, the inner surface of the hollow sphere is stress free, a~(a) = O, and a uniform nominal tensile stress is applied on the its exterior surface, arr(b) = po(-~) 2. We shall multiply the above equation by v~ and by the integration on the whole volume of the hollow sphere, taking into account the boundary conditions, we get, d [a3( 1
b)h2 ] = 2a2h(..b_) [p0
_
2
b
(12)
For the viscoplastic solid, we have (elastic strain neglected), 7 = sign(v.)Ev[1
+
r162
3'
I v. IH(I~ I
~))1.
_
r
(13)
and for the elastic-plastic solid
r = ay Jr kp" = 0,
ln~=
(14)
3E r + ~
where H the Heaviside step function, i.e. H(x) = 1 if z > 0 and it vanishes for x < 0. In the above equations we shall take r = x 1/m, with 0 < m < 1, E~ = a~p", 0 < n < 1 and r = 1. The temperature-dependent coefficients are taken as,
uy(T)
t
0
= =,(T)
ifT>T,,,
(16)
~(T), ~(T) = ~ exp V ( 1 / T - 1/V~), where r/is viscosity, Y, V > 0 are some material constants, and Tm and r/m are the melting temperature and viscosity at the melting temperature, respectively, [25]. The simplified heat conduction equation may be written as
pcoT - - d i v q + X~" D ,
(17)
where co is the specific heat, q the heat flux and X the Taylor-Quinney coefficient that takes into account the stored elastic energy (X ~- 0.9). In our spherical case the heat 0T flux, using the Fourier's law, is of the form, q - - k o g r a d T - -ko~Ter. Therefore, we have,
i)T a2h tOT " 02T a2a pco(-~-~ + --~--~-~r ) = ko-~r2 + 2X7 r 3
(18) 9
For this equation we shall consider the initial condition T(0) = To. The heat flux vanishing on both surfaces will be considered as a boundary condition, q(a) = q(b) = O.
3
CRITICAL LOAD, CAVITATION AND BIFURCATION
In this section, in order to study the cavitation phenomenon, we shall consider the initial void radius is vanishing, A = 0. 3.1
Viscoplastic solid
We shall substitute the temperature dependent coefficients (16) in the formula of r, (13), and then the motion equation (12) becomes
2 B)2 [po(t)_ sign(h)Po(t)],
pd[a3(1 - b)h 2] + Q(t) - 2a h ( ~
(19)
where Q(t) = 4a2lhl
~(
r3
)'~H(]~-]- Zu)(21n~) r '
b )2 f b r ). dr P0(t) = 2 ( ~ au(2 In 5 -r--.
(20)
(21)
For the equation (19) we shall first consider the following homogeneous initial conditions, a(0) -- 0,
h(0) -- 0.
(22)
The convergence of the integrals in (20) and (21) is better seen if we change the inte{1 we get gration variable, s = 7n3" Writing x = ~, [ 3
2
1,,
J0
ds
(23)
In this way, we see that the above integral is convergent. Also, we remark that Po(t) > 0 and Q(t) > 0 for 0 < T < Tin. On letting t -+ 0 in (23), the critical load is obtained as
2 fl
2
1
ds
Per = P0(0)= 3 ]o au(To)(-~ ln-s )" 1 - s "
(24)
R e m a r k 3.1 In the following, the temperature T will be considered as a continuous function of a and h, that is, the solution T of equation (18) depends continuously on the coefficients. By a solution a of equation (19) on the interval [0, co) we shall consider the classical solutions, that is, a E C2(0, cr f3 Ca[0, co). In the following we shall give some theoretical results. Theirs proofs can be found in [23]. The next lemma gives the behaviour of a non zero solution on a time interval.
L e m m a 8.1 /f on a time interval (tx, t2), 0 < tl < t2, a solution a of equation (19) satisfies
a(t) > O, a(t) # 0 on (t~, t2), a(tx) = 0 or a(t~) = 0, a(t2) > 0 and a(t2) # 0, sign(po(t) - sign(h)Po(t)) = sp ~ O, constant on (ta, t2),
(25)
then sign(a) = sp, and consequently, on (tl, t2) we have the following equivalences i)po(t) > Po(t) ~ a> 0 ii)po(t) < -Po(t) ~ a < O,
the case IPo(t)l < Po(t) being impossible. Concerning the solution of the problem corresponding to applied loads smaller than the critical load P~, we have P r o p o s i t i o n 3.1 If the applied load po(t) is continuous and 0 < po(t) < P~ on any closed finite time interval, then problem (19), (22} has a unique solution, that is, it has only the zero solution. R e m a r k 3.2 The above proposition also states that the problem (19), (22) can have more than one solution only if po(t) >_ P~, and consequently, for po(t) = P~ we may have a bifurcation of the solution. Let us consider now nonhomogeneous initial values for our problem, that is in the place of the conditions (22) we shall consider the following initial conditions a = ao, ei = v0,
(26)
where a0 > 0 or v0 > 0 and aovo = 0. The following proposition proves that for the above initial conditions, the problem has a solution only if the applied load exceeds the critical load, or, in other words, we can get cavitation only with applied loads greater than the critical load. P r o p o s i t i o n 3.2 Supposing t > O, we then have / } / f 0 < po(0) < P~ then i/}/fpo(0) > P~ and a(t) exists a t2 > 0 such that a(t)
that the applied load po(t) is a continuous function for problem (19) and (e6) has no classical solution, is a non-nut solution of problem (19) and (Ca), then there > 0 for t ~ (0, t2).
R e m a r k 3.3 The result of item i) in the above proposition is natural taking into account that in our model the initial radius of the hole is vanishing. The following theorem describes the local behaviour of a solution for a time tl > 0. T h e o r e m 3.1 Let 0 < tl < t2 and a(t), t E [0, t2] be a solution of problem (19) with initial conditions (22) or (26). Then i} if 0 < po(tl) < P0(tl) then there exists a neighbourhood of tl on which either
a(t) = a(t) = 0 or a(t)a(t) # O, ii} if po(ta) > Po(ta) and a(tl) > 0 or a(t~) # 0 then there is a tz, tx < t3 < t2 such that a(t) > 0 on (t~, t3), or there exists a neighbourhood of tl on which a(t)a(t) ~ O.
3.1.1
Conclusions
We remark that in the above results the connection between the applied load po(t) and Po(t) is essential. In what follows we shall make some remarks concerning the behaviour of a nonzero solution and the critical load at which the cavitation take place, plotting in a plane P - x the curves P ( z , T) =_ Po(t), in which the temperature T is considered as a parameter. In Fig. 1 we have considered a specific example to point out the shape of these curves. This example corresponds to n = 0.24 and Y = 2206 M P a and the initial and melting value of the temperature were To = 293 K and T,~ = 1673 K, respectively.
\
~\
1
I
0.8
\ , \ \ \ \ \ \ \
0.6
\
0.4
\
I
I
I
I
I
T1 = 293 K T2 = 493 K T3=693K T4 = 893 S T5 =1093 K
~
T6=1293K =1493 K
-
-
t
0"2
t~, 0
500
Ts
T4
T3
T2
T1
-
P (MPa) 1000 1500 2000 2500 3000 3500 4000 4500 5000
Fig. 1. Post-cavitation for n = 0.24 and different temperatures First, we remark that P(x, T) is a decreasing function depending on T. Also, the curves are left turning for n < 1.0 (see also Abeyaratne and Hou, [19], for the case of rate-dependent materials), and we shall assume that the temperature is an increasing function during the deformation. Suppose now that the load po(t) > 0 will remain greater than P~ if it exceeds this value at a certain time (even if it starts from a value p0(0) < Pc~), and that the problem has homogeneous initial conditions. In this case, using Proposition 3.1, the item i) of Lemma 3.1 and the item ii) of Theorem 3.1, we can prove that for po(t) > P~ a non-zero solution is an increasing function in time. Consequently, for such loads, the point (P, x) will be situated in our figures, either on the segment [0, P~], if po < P~, or in the right region of all curves, if po > P~. On the other hand, if the load po(t) > 0 assumes values less than P~ move than once, then the point (P, x) can theoveticaly be situated in the left region of the curve corresponding to the temperature T(t). In this case, from item i) of Theorem I it results that for a time to, the solution will be discontinuous in two situations: either the point is not on the segment [0, P~r] and h(to) = O, or the point lies in [0, P~] and h(to) # O. As for the points on the curves P - z, we remark from equation (19) that there are no static states (in which the sphere remains undeformed while stressed) if p(t) > O. Indeed, if the point lies on a curve P - x, replacing h(to) = 0 in (19), we get/i(t0) > 0, and consequently, in a time interval (to, t l) the solution will pass into the region to the
10
right of the curves. Suppose now that, for the problem with homogeneous initial conditions, until the time to we have only the zero-solution and at this time we perturb the solution taking a(to) = ~ > 0. Taking into account Proposition 3.2, we can conclude that: ifpo(to) < P~ then the solution will go back to zero (having a jump) and if po(to) > P~, then a(t) for t > to will be an increasing function. The obtained critical load P~, for our example, calculated from (24), was of 4767.6 MPa. Also, we remark from Fig. 1 that P~ is an decreasing function of temperature T. Other aspects of the thermal effects on the void growth are reported in [26-27]. 3.2
E l a s t o - p l a s t i c solid
In this subsection the temperature influence will not be considered. For simplicity, we shall also assume the imposed load p0 to be constant in time. In the case of po depending on time and temperature influence is taken into account, we can derive similar conclusions to those in the previous subsection. The term in (12) which contains the integral will be denoted as in the previous subsection by P0 = 2(~-) 2 f~ ~dr, and we shall write, P = (-~)2(p0 -/90). In order to study the behavior of this term we shall R3 change the variable in the integral, s = 7 , and also, we shall write x = ~-. We obtain in this way
Po(x) =
2
_ fo ~ 1 - s ds'
2 P(x) = po x2 _ -~ rio
(27)
r(s) 1- s
(28)
and we shall denote 2
P= = Po(1)= 5 fo x T(S) d3,
(29)
which will be named in the following critical load. We mention that ~" in the above formulas is obtained as the solution of (14) and (15). We also remark that 0 _< x _ 1. We shall give in the following some theoretical results. Concerning P(x) we have L e m m a 3.2 For 0 < x < 1 the integral in (PT) makes sense. Concerning the behaviour of P(x) we have 9 (i) if po > _2 x/3~--~-- then P(x) is an increasing function on [0,1], E (ii} if po > i-4";, then there ezists 0 < Xo < 1 such that P(z) is increasing on [x0,1], E (iii} if po < i-4-;, then there exists 0 < zo < 1 such that P(z) is decreasing on [z0,1]. From the item (i) of the above lemma we shall immediately get the following consequence. C o r o l l a r y 3.1 An upper bound of the critical load is given by P~ < -
21~3E--E-
X-l-v"
11 For the second order differential equation (12) we shall consider the following initial conditions a(O) = O,
a(O) =
(30)
where vo > 0 is given. We have, P r o p o s i t i o n 3.3 If po < P~, then the problem (1P) and (30) has only the null solution. If po > P~, then this problem should have without the null solution another one nonnull. P r o p o s i t i o n 3.4 Assume po = P~. Then the problem (12) and (30) has only the null E If P~ < Tu E then this problem should have without the null solution if P~ > T-4"~" solution another one non-null. 3.2.1
Conclusions
An upper bound of the critical loads is given in Corollary 3.1. Evidently, we expect the values of the real critical load to be lesser than this enough large upper bound which is valid in the case of the linear elasticity, too. Proposition 3.3 shows that cavitation is possible only if the imposed load is greater than the critical load, p0 > Per. For the imposed load less than the critical,/90 < P~, cavitation is impossible. Finally, Proposition 3.4 shows that the solution bifurcation at the critical load, p0 = P~r, is possible only if this value does not exceed a certain upper bound, Per < l+v" E E cavitation at the critical load is impossible. This fact does In the case when P~r > 1-~' not contradict the previous remarks, where po ~ P~ and we assume that p0 is constant in time.
4
REFERENCES
1. V. Tvergaard, Adv. Appl. Mech., 27 (1990), 83. 2. G. Perrin and J. B. Leblond, Int. J. Plasticity, 6 (1990), 677. 3. X.P. Xu and A. Needelman, Int. J. Plasticity, 8 (1992), 315. 4. S.H. Goods and L.M. Brown, Acta Metallurgica, 27 (1979), 1. 5. M.A. Meyers and C. Taylor Aimone, Progress in Material Science, 28 (1983), 1. 6. H.G.E. Wilsdorf, Mat. Sci. Engn, 59 (1983), 1 7. A. N. Gent and P. B. Lindley, Proc. R. Soc. Lond., A, 249 (1958), 195. 8. J. M. Ball, Phil. Trans. R. Soc. Lond., A, 306 (1982), 557. 9. J. Sivaloganathan, Arch. Rational Mech. Anal., 96 (1986), 97. 10. C. O. Horgan and R. Abeyaratne, J. Elasticity, 16 (1986), 189. 11. S. M/iller and S. J. Spector, Arch. Rational Mech. Anal., 131 (1995), 1. 12. R. F. Bishop, R. Hill and N. F. Molt, Proc. Phys. Sco. 57 (1945), 147. 13. R. Hill, The mathematical theory of plasticity, Clarendon Press, Oxford (1950). 14. D. Durban and M. Baruch, J. Appl. Mech., 46 (1976), 633.
12 15. D.-T. Chung, C. O. Horgan and R. Abeyaratne, Int. J. Solids Structures 23 (1987), 983. 16. Y. Huang, J. W. Hutchinson and V. Tvergaard, J. Mech. Phys. Solids, 39 (1991), 223. 17. V. Wvergaard, V. Huang and J. W. Hutchinson, Eur. J. Mech., A/Solids 11 (1992), 215. 18. H.-S. Hou and R. Abeyaratne, J. Mech. Phys. Solids, 40 (1992), 571. 19. R. Abeyaratne and H.-S. Hou, J. Appl. Mech., 56 (1989), 40. 20. C. O. Horgan and D. A. Polignone, Applied Mechanics Reviews, 48 (1995), 471. 21. K. A. Pericak-Spector and S. J. Spector, Arch. Rational Mech. Anal., 101 (1988), 293. 22. M.-S. Chou-Wang and C. O. Horgan, Int. J. Engn Sci., 27 (1989), 967. 23. L. Badea and M. Predeleanu, Mechanics Research Comm. 23 (1996), 461. 24. P. Perzyna, Adv. Appl. Mech., 9 (1966), 243. 25. M. M. Carroll, K. T. Kim and V. F. Nesterenko, J. Appl. Phys., 59, 6 (1986), 1962. 26. D.R. Curran and L. Seaman, Proc. Int. Seminar on high temperature fracture mechanisms and mechanics, Dourdan, 13- 15 October, 1987. 27. A. Neme and M. Predeleanu, in Structures under Shock and Impact, eds N. Jones et al., CMP (1996).
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
13
A ductile d a m a g e m o d e l including shear stress effects J.C.BOYER a and C.STAUB b a Laboratoire de M6canique des Solides, INSA Lyon. 20, av A.Einstein. 69 621 Villeurbanne Cedex France b Centre de Recherche et d'Etudes Technologiques, GIAT Industries. 2, rue Alsace-Lorraine. B.P. 1450 - 65014 Tarbes Cedex France
1. INTRODUCTION Since the early work of Gurson [ 1], ductile damage macroscopic constitutive laws based on microvoid growth have been improved, the well-known Tveggaard [2] plastic potential included some coalescence effects, Rousselier [3] formulated relations for finite deformations of plastically dilatant materials, more recently Brunhs and Schiesse [4] have developed a continuum model of elastic-plastic materials with anisotropic damage by oriented microvoids. Most of the existing plasticity-damage theories consider the mean normal stress as the main parameter controlling the microvoid growth even though the deviatoric part of the stress tensor acts on the phenomena at the microscopic or mesoscopic levels. In the first part of this work, numerical modellings of the mechanical behaviour of a microvoid with or without a free inclusion in an elastic-plastic material under several stress states are presented. Then, these predictions of the void volume fraction evolution are compared to a modified Rice and Tracey model intended for any stress states including pure shear loadings. In the second part, a yield function for a stress dependent density material is discussed and identified with the modified Rice and Tracey model.
2. FINITE ELEMENT MODELLING OF THE VOID GROWTH
2.1. Uniaxial loading In their last paper the authors [5] started to simulate the growth evolution of a spherical void in a cylindrical unit cell under uniaxial loading. Using axisymmetry only one quarter of a vertical section is analysed. A spherical void is located centrally in the cylinder such that typical void fractions and distributions are in agreement with experimental data [6]. On the outer surface normal to the cylindrical axis, the displacement is imposed as a remote uniaxial tensile or compressive loading. On the cylindrical surface, the radial components are constant. The behaviour of the matrix is supposed elastic-plastic with linear piece-wise hardening. The particle material is elastic with a ratio of 10 or 0.1 to the matrix's Young modulus. The matrixparticle interface is supposed debonded since the start of the loading. The behaviours of a cell with an inclusion and a cell without an inclusion are completely different. Under tensile loading the void volume fraction increases with effective strain (Figure 1). The phenomenon is less emphasised for a void without inclusion. The more the
14 inclusion Young modulus is high, the more the closing of the void is constraint, but the void volume fraction evolution is quite similar, that's the reason why in the next simulations, the inclusion will be considered as a rigid body. The void volume change predicted with the Rice and Tracey modified model is in very good agreement with the finite element modelling of the filled void. Under compressive loading (Figure 2), the void volume fraction for a cell without an inclusion reduces, whereas it grows for the void with an inclusion.
Figure 1 9Void volume fraction evolution under tensile loading
Figure 2 9Void volume fraction evolution under compressive loading
2.2. Void volume fraction evolution under zero mean stress loading 2.2.1 Mesh
The model used for the finite element simulation is quite similar to the one used for uniaxial loading, but to take into account local plasticity with more accuracy, the mesh is improved (Figure 3). The void volume will be integrated numerically at each step of the loading. When the loading is symmetric (Figure 4) only one quarter of the unit cell is analysed.
Figure 3 9Generalised plane strain Mesh of the unit cell 96962 nodes
Figure 4 9Axisymmetric loading Mesh of a quarter : 1886 nodes
The behaviour of the matrix and the boundary conditions were presented in section 2.1. Such a model is useful to analyse the behaviour of a cylindrical cell with a spherical void as well as to observe the growth evolution of a cylindrical void in a planar cell.
15 2.2.2 Axisymmetric loading" cylindrical shear stress state Using the axisymmetric model, the displacements dz = 2a and dr = - a are imposed.
Figure 5 : Cylindrical shear Unit cell without an inclusion
(see Figure 4)
Figure 6 : Cylindrical shear Unit cell with an inclusion
As shown in Figure 5 and Figure 6 the void volume fraction changes with the plastic strain. The rate is positive for a filled void and negative for an empty void. The behaviour of a cell under cylindrical shear strain is quite similar to the one of a cell under compressive loading. 2.2.3 Generalised plane strain : pure deviatoric stress state
Figure 7 : Pure shear Unit cell without an inclusion
Figure 8 : Pure shear Unit cell with an inclusion
16 In the principal axis, a pure deviatoric strain state can be easily simulated by imposing a displacement dr = a and a displacement dz = - a in the perpendicular direction. As for the cylindrical shear strain state, the void volume fraction decreases with effective strain for a void without an inclusion (Figure 7) whereas it grows for a cell with an inclusion (Figure 8). Shear stress induces cavity closing for empty voids, whereas for a filled void, the void volume fraction increases whatever the nature of the loading.
2.3 Principal direction of the ellipsoid Using the generalised plane strain model without an inclusion, the aim of this section is to find the evolution of the principal direction of the ellipsoid versus a representative parameter which could be used in a constitutive law.
Figure 9 : Loading history
Figure 10 : Evolution of the principal direction of the elliosoid
Figure 11 : Loading history
Figure 12 9Ellipsoid principal direction
The strain rate principal direction ranging from 45 degrees to 31 degrees was gradually decreased and increased (see Figure 9). The principal direction of the ellipsoid is quite
17 identical to the strain rate principal direction (Figure 10). Even if the principal strain state direction is changed rapidly (Figure 11), the principal direction of the ellipsoid follows it. The more a loading direction is maintained constant, the more the ellipsoid direction is closed to the imposed principal strain rate direction (Figure 12).
3.A MODIFIED RICE AND TRACEY MODEL
For the case of an empty spherical void of radius R in a remote uniform plastic strainrate field D ], Rice and Tracey [7] have developed a growth model of microvoids under arbitrary remote stress states ~j = sij +OmS~j, where s~j is the deviatoric stress tensor and O' m the mean normal stress. An approximate estimate of the rates of change in the radii of the void in the directions of the principal strain-rates for the case of isotropic linear hardening materials has the form" 5
+ 3 O"m DP]R
(1)
1L= ?D~ 4~M n
the equivalent plastic strain-rate and ~M the yield stress of the matrix material considered as incompressible. with D p
The void growth rate equation (1) can be used to give useful estimates of the change in shape and volume of an initial spherical void by means of an approximate or numerical integration for proportional loading but it has to be extended to pure shear stress states. 3.1. Void without an inclusion
Thomason [8] gives general expressions for the three integrated principal radii of the resulting 'ellipsoidal' void but the predicted apparent plastic dilational strain does not change under zero mean normal stress. Once the void is no longer spherical, the rates of change of the three radii are different as the 'ellipsoidal' void has different stiffness in the three directions of the principal plastic strain rates, some curvature correction linked to the equivalent spherical void must be introduced in the original model :
+2 (~m Dp
Rk
with the 'spherical' void radius R
4 NM
= ( R , R 2 R 3 ) '/3
(2)
If Vc is the void volume, its logarithmic rate is given as 9
v
r 3ool
(3)
For an empty void, this equation leads to a good fit with volume void fraction rates predicted by numerical modelling for tensile, compressive and shear stress-state, see Fig. n~ 2, 5 and 7 respectively. For large effective plastic strain, the real shape of the void is far from the assumed ellipsoid, these geometric differences are the main reasons of the discrepancies
18 between the values of the void volume fraction predicted with this modified Rice and Tracey model and by the finite element analysis. Nevertheless, the initial rates are in good agreement and the first stages of the plastic deformation of an initial spherical void under non-isotropic plastic strain rates induce volumetric change even under zero mean normal stress. 3.2. V o i d with an inclusion
For a void filled by an inclusion without interface strength, the void growth rate equation (1) is still a versatile basis with unilateral kinematic boundary conditions on the direction of the principal plastic strain-rates. a. Tensile stress state D p = D p and (Yl
= ~ M = 3t3m9
} E ] ~
DPei = - D p / 2 ,1~3 = 1~2 << R 1
b. Compressive stress state D p = - D p and ~,------~M-'3~m D ~ = - D p / 2 15.3 = R 2 > > R , _
_
Wc - - -5- ' ~ ' - -3
~-~
We
3
(Ym
~p = 23 ~ p
4 ~M
5
1-2
3 (~m ]Dp
17 ~p
c. Plane shear stress state D p -- - D R p , DP 3 -- 0 and r~,~ = 0 9~ 5 5%f3"-~p DP -- 2D p / ~f3 et R2 (<( RI J ~ - Vc = -3 D~ = 6 d. Cylindrical shear stress state D p / 2 = -D~ = - D p > 0 and ~m = 01 Vr 5 DP = D~ I~,>>I~ 3 = 15.2 J ~ - - =V-cD P3=
5-~p 3
(4)
(5)
(6)
(7)
These extensions of the Rice and Tracey model are also in good agreement with the numerical predictions of the void volume fraction rates until the void shape degenerates. These results are noteworthy as they predict a positive void volume fraction rate for every stress states, the deviatoric part of the stress states gives rise to the main component of the volumetric change rates of the void filled with an inclusion. Furthermore, the magnitude of these rates are always greater than the values observed for an empty void.
4. K I N E M A T I C S OF THE ELASTIC-PLASTIC TRANSFORMATION The framework of the finite elastic-plastic transformation is now widely spread and only some particular points need to be discussed for the development of a specific constitutive law including damage effects. The additive split of the total strain rate in elastic and plastic parts is consistent with the multiplicative composition of the elastic and plastic transformation through the derivatives of the transformation gradients F*, F p and the spatial gradients U , L p of the velocity field with the initial configuration as reference configuration for the plastic transformation and the virtual intermediate configuration as reference configuration for the elastic part :
19 F : F e . F p ::~ L : F F - I :
~'e(Fe)-I
+ FeFP(Fp)-1(Fe
)-1 : L r
p
(8)
As the symmetric part of the velocity gradient is the strain rate tensor, the plastic strain rate and the elastic strain rate can be identified as 9 D e = I ( L e + L e) 2
D p = I ( L P + L p)
(9)
Further decomposition for the damage induced part has been set up, see Brunhs et Schiesse [4], but in the present work, the plastic transformation is supposed to include the damage effects as the void growth is considered as a local plastic phenomenon. The first law of thermodynamics for an adiabatic process lay down the specific elastic stress power as the rate of work per unit mass of the Cauchy stress tensor o and the elastic strain rate D e in the current configuration or the rate of work per unit mass of the second Piola-Kirchhoff stress tensor S and the rate of the Green-Lagrange elastic strain tensor E e with the intermediate configuration as the reference one: S
dE *
G: D e
. . . . . p~ dt Pc
(10)
As noticed by Rousselier [3] the apparent elastic moduli are related to the density change of the material, some coupling effects are introduced in this way as Pi is the density in the intermediate configuration. This density change from the initial configuration is easily obtained if small elastic strains are assumed 9 (11)
so, the densities in the current and intermediate configuration can be considered as equal if the ratio of the mean normal stress to the elastic bulk modulus is small. For the plastic behaviour of the material, only the current density will appear in the sequel of this work and the dissipated power is considered in the current configuration, and as the numerical modelling of the void growth suggests some orthotropic behaviour in the directions of the principal plastic strain rate, their rotational motion has to be traced and the plastic rotation tensor defined by Reddy and Gultop [9] will be used for numerical purposes.
5. A SPECIFIC YIELD FUNCTION According to the results of the previous sections, void growth arises for any stress state, induces density changes, and is related to global plasticity. When such a damage takes place, the conservation of mass at the representative volume element level has to be fulfilled for any yield function and flow rule, so for finite strain transformations only specific energy balances are significant. The Von Mises yield function has to be modified with the equivalence between the shear energy coming from the deviatoric part s~j of the stress state o,j in the infinitesimal volume dV of current density Pc and the corresponding one under tensile
20 condition of an infinitesimal element of same mass with k the usual hardening parameter. The effective stress ~t of this modified J2 theory is related to the current density Pt under tensile stress state 9 ( ) sijsij _ k 2 J2 _ k 2 = 0 g (Y~j = =-2Pc Pc
~
~
.~ Pt
SijSij
(12)
Pc 2
The energetic equivalence per unit mass makes necessary the measurement of the material density during tensile test, several methods are described by Montheillet and Moussy [10]. An associated flow rule can be connected to this plastic potential and the density considered as stress dependent and the usual normality rule, with ~, the rate of the proportionality factor, the components of the plastic strain rate are:
O~ -" ~ ' ~ i j -" ~' Pe
(13)
3 P"~t m
With the definition of a consistent effective plastic strain rate D p , the energy rate balance between the dissipation power P under tensile stress state and any stress sate reduce the components of the plastic strain rate to 9
I
3 Dp -_ (~t'DP -- (~ijD~ ~ Do = -2 ~~t Sij 9~ P~
o,, 3 Pt Ooij
/
(14)
2Pc 0oij
The source of the plastic strain rate is twofold, the deviatoric stress is the usual one but the stress dependent density adds new terms to the six plastic strain rate components if the damage phenomena are fully anisotropic. As the numerical modelling of the void growth suggests some orthotropic behaviour in the directions of the principal plastic strain rate, in the sequel of this work this assumption is applied to the macroscopic density change which is to be identified with the microscopic or mesoscopic void volume fraction change. If Vc is the void volume, VM the incompressible matrix volume of density PM and V T the total volume, the void volume fraction fc, the dilational plastic strain D~k and the relative differential of the current density are expressed as :
v,=v~+vo~fo
= mV~ ~ ~ =~ D~k = -w - - .-- f o ~ and vT v~ v~
p o =(I- fc)pM~
dpc Pc
df~ 1-f~
=--~
(15)
As all the energy dissipation process takes place in the material matrix with the same effective plastic strain rate, the effective stresses in the damaged material and in the matrix material have to fulfil the specific energy balance 9 u
m
~t Dp ~M Dp " - - - - - - 2 ~ - - " ~ =:~ ~ t = (1-- ft)~ M Pt PM
(16)
In order to identify the macroscopic 'damage strain rate' and the void volume fraction changes, this part of the expression of the components of the plastic strain rate is formulated
21 with the void volume fraction and the yield stress of the sound material. From the macroscopic point of view, the dilational plastic strain rate induced by the void growth is :
(
D~k = D P . ~ M - ~ i j 8ij / 2 14
% Of,] 2(1- f~) 3o'ij
(17)
As the rates of change of the ellipsoidal void radii can be expressed in the directions of the principal strain rate with the principal components of the remote deviatoric stress tensor and the remote mean normal stress as : I~K = (aKSn + bKO'K)._ Do F p RK
(18)
O=M
The modified Rice and Tracey model leads to the following dilational plastic strain rate expression: m
D~k =(A,r~, +A2~ 2 +ABr~B).f~.Dp_ DP
(19)
O"M
For tensile stress states in each of the K directions of the principal plastic strain rate, the void volume fraction change is governed by the following three equations : 1 0fo = 2A K (~K [ (~M
fc O(~K
AK.fc O'~2]
~-------~/L1 (1 -- fc) -2(YM
(20)
In bulk forming processes, the void volume fraction is lower than 1% and the triaxiality ratio for hydrostatic tension is typically lower than 1, a first order approximation checked by numerical integration of equation (21) gives an exponential function of the triaxiality ratio. With this last assumption, the final expressions of the components of the principal plastic strain rate are : 3DP(s D ~ = 2 Nt ~ K+
2AK rl f,].f~.O'K) 3 "
(21)
This constitutive equation satisfies the strong form of the Clausius-Duhem inequality if the specific dissipated power is positive for any stress state : - ~ ~ SKSK+ A K f~[1- f,]O'K~K > 0 3Pc Nt -3 " -
(22)
As the effective strain, the effective stress and the material density are positive quantities and as the void volume fraction value is in the range 0 to 1, this condition holds if the A coefficients are positive. The modified Rice and Tracey model shows values close to two for the free directions of the principal plastic strain rate and close to zero for the inclusion
22 locked direction. So, the specific yield condition (12) and its associated flow rule are consistent with the first and the second laws of thermodynamics.
6.CONCLUSION The numerical modelling of the microvoid behaviour in an elastic-plastic unit cell and the modified Rice and Tracey model are in good agreement for the prediction of the void volume fraction rate at the early stage of the damage phenomenon for uniaxial stress states and pure shear stress states so far the real shape of the void remains casi-ellipsoidal. Both models predict a positive rate for a void initially filled by an inclusion under any stress states whereas negative rates arise for an empty void under compressive or pure shear stress states. The numerical approach and the analytical model corroborate the sensitiveness of the void volume fraction changes to the deviatoric part of the stress tensor. The shear loadings of the generalised plane strain unit cell show that the directions of the ellipsoidal void follow the directions of the principal plastic strain rate for non radial path. Such a preview enforces the problem of a stress dependent damage compared with the mean normal stress dependent damage. The proposal of a yield function formulated per unit mass seems to be a right answer which is an extension of the Von Mises theory complying with the continuity equation.
REFERENCES 1. A.L. Gurson, J. of Eng. Mat. and Tech., January (1977) 2. 2. V. Tvergaard, Int J. Fracture, 17 (1981) 389. 3. G. Rousselier, Three dimensional constitutive relations and ductile fracture, S.NematNasser (Eds), North Holland, (1980) 331. 4. O.T. Brunhs, P. Schiesse, Eur. J. Mec. A/Sol., 15 (1996) 367. 5. C. Staub, J.C. Boyer, Journal of Material Processing Technology, 60 (1996) 304. 6. G. Le Roy, J.D.Embury, G. Edwards, M.F. Ashby, Acta Metall., 29 (1981) 1509. 7. Rice and Tracey, J. Mech. Phys. Solids, 17 (1981) 210. 8. P. F.Thomason, Ductile Fracture of Metals, Pergamon Press, (1990). 9. B. D. Reddy, T. Gtiltop, Eur. J. Mech. A/Solids, 14 (1995) 499. 10. F.Montheillet, F.Moussy, Physique et M6canique de l'endommagement, Les Editions de Physique, Greco Grandes D6formations et Endommagement, (1986).
Advanced Methods in MaterialsProcessingDefects M. Predeleanuand P. Gilormini(Editors) 9 1997 Elsevier Science B.V. All rights reserved.
23
A m e s o s c o p i c approach of ductile damage during cold forming processes G. Brethenoux, P. Mazataud # , E. Bourgain +, M. Muzzi* and J. Giusti IRSID, LAMEF Department, BP 30320, 57283 Maizi+res-l+s-Metz, France
1. INTRODUCTION During cold forming processes, steels are submitted to both ductile damage and strain hardening. These two competing mechanisms determine the mechanical properties of the final product, in particular its fracture resistance and fatigue life. In the present context of constant renewal of products and improvement of their capabilities, it is essential to be able to predict the evolution of the local damage in a component, from its conception and through each transformation stage fight up to its final state. Microscopic observations of steel show how damage begins preferentially around inclusions, by means of decohesion or fragmentation (nucleation). The cavities thus created grow and finally coalesce in micro-fractures. Numerous models have been proposed to describe the cavity growth in a matrix [1,2] and, later, the plastic behavior of porous media [3,4]. This approach has been recently extended to the case of a strain hardenable matrix and of non spherical cavities [5-8]. The fact that the mathematical formulation of these models is complex and their validity is limited has led to the development of numerical methods [9-11 ]. In this paper, ductile damage is investigated by means of mesoscopic approach. This analysis, based on the finite element method, is applied to torsion and notched tensile tests and offers satisfying results. The comparison of the numerical approach with experimental observations leads us to introduce a nucleation term in our model, resulting in a noticeable improvement of its capabilities.
2. MESOSCOPIC APPROACH The mesoscopic approach consists of evaluating, with finite element analysis, the void growth in a representative cell containing a simple cavity, or an inclusion, embedded in an elasto-plastic matrix (cf. Figure 1). This analysis is here referred to as <<mesoscopic >) since the scale of the elementary cell lies between that of the subgrain (microscopic) and that of the specimen (macroscopic). The boundary conditions applied to the elementary cell guarantee the # Compagnie Bancaire, Pads, France + Soci6t6 Griset, Villers-St Paul, France Politechnico di Milano, Milano, Italy
24 periodicity conditions. The use of the finite element code ABAQUS enables the mesoscopic cell to be submitted to any loading condition, characterized by its triaxiality path. Moreover, the loading condition can always be replaced by an axisymmetric equivalent, with the same triaxiality. This allows us to use an axisymmetric mesh and to reduce the calculation time.
) / / / J
Figure 1. Example of a mesoscopic mesh evolution (simple cavity).
The first application of this approach is the validation of analytical damage models using the same hypotheses : no inclusion in the cavity, simplified model of material behavior and restricted loading conditions. For example, the figure 2 presents the results obtained by Mazataud & Me Dowell [12]. They have compared, for different constant values of triaxiality, the void growth estimated with the mesoscopic analysis and the Gologanu model [7] which is a new formulation of the Gurson model taking into account the changes in cavity shape. The results show a good agreement between the two models (cf. Figure 2-a). They have obtained the same results while comparing void growth with the mesoscopic analysis and a modified Gurson-Tvergaard formulation, taking into account the strain hardening of the metallic matrix (cf. Figure 2-b). However, the most important advantage of the mesoscopic model is its capability to take into account very easily the different phenomena and parameters of ductile damage that cannot be introduced into analytical formulations. The mesoscopic model has thus enabled us to study the influence of numerous parameters such as strain hardening of the matrix, presence of an inclusion in the cavity, shape of the inclusion, cohesion and friction conditions at the interface between the inclusion and the matrix...
25 0.0008
;
0.0008
e
...
T-4/3 0.0006
-
0.0004
-
." T = I
..""
T =4/3 0.0006
-
~ r~
o
lm
r~
0.0004-o ~,,
o
0.0002 -
0.0002 ...
- ......
-
.......................
T= 1/3 0.0000
'
0.0
I
0.2
'
i
0.4
'
I
0.6
T=I/3 0.0000
'
0.8
' o.o
I 0.2
~
I 0.4
'
i 0.6
' 0.8
E~ Figure 2. Void growth comparison, for constant triaxiality, between the mesoscopic analysis (solid lines) and the Gurson-Tvergaard model (a) and the Gologanu model (b)(dashed lines).
3. APPLICATION TO TORSION TEST The torsion test is a good method to study ductile damage because it corresponds to a constant triaxiality equal to zero and is independent of the constitutive behavior of the material. Moreover, analytical models are not able to predict ductile fracture during torsion tests because they do not take into account the part played by inclusions in the cavity deformation. This effect is all the more important when the triaxiality is low. For this application, we consider 5 different as rolled bar and wire steel grades : FR5, 20MnB5, FR38, C45E and C80D. These materials are very pure in terms of inclusion content and the total inclusion volume fraction (v0) can be a priori considered to equal the cementite volume fraction (estimated from the carbon content). 3.1.
Ductile
damage
mechanisms
A mesoscopic simulation of the torsion test has been carried out for the 5 steel grades. In each case, the results of the simulation bring to the fore the void growth and coalescence mechanisms (cf. Figure 3, Figure 4) 9 - the metallic matrix begins to be strained and void grows slowly along the axial direction (principal loading direction) ; - the metallic zone between two inclusions is locally under tensile conditions and thus becomes more and more narrow ; - in the final stage, the strip has become so narrow that necking occurs. At this point, the void grows in a catastrophic way along the radial direction. This corresponds to the beginning of ductile fracture.
26
Figure 3. Evolution of the mesoscopic mesh during torsion simulation (3D view of the axisymmetric analysis - only the boundary of the rigid inclusion is figured). 0.1
I
I
I
I , t
0.08 *~,,4 O')
. . . . . . . .
~ . . . . . . . . . .
i
i
i
:
i
,!
0.04
: ', : : 1 ............................
: ,
........
, : " ......... ',
: : " .......... :
: ', ' ........
I : I : 1-:----I ',
',
0.02
........
, ..........
r ........
: ' ' ..........
: ' .i........
I ', I ', [',~"
0.06
i
O
'. . . .
, ..........
'
. . . . .
~
. . . . . . . . .
"
. . . . . . . . .
' .........
"
. . . . . . . . . .
'. . . . . . . . . .
'
0
0.2
0.4 0.6 0.8 1 total equivalent strain
i
I
~
"
',
!
l:
i
li
1.2
11.4
fracture strain Figure 4. Void growth in the mesoscopic cell during torsion test simulation.
3.2. Comparison with the experimental results Mesoscopic fracture strains can be estimated very precisely using void growth curves. These deformation values are then compared to experimental values derived from the number of revolutions 9 2~NR = r:'_ V3 L
(1)
where R and L are respectively the radius and length of the specimen. The results of the comparison (of. Table 1) reveal a very good correlation between the mesoscopic model and the experimental results for the 5 steel grades. This indicates that the model can give a good account of the ductile damage mechanisms that occur during the torsion test.
27 Table 1 Prediction of torsion fracture strain with the mesoscopic model Steel Grade
Initialporosity v0(%) (equal to the cementite volume)
FR5 20MnB5 FR38 C45E C80D
Fracture Strain Mesoscopic model
0,525 3,0 5,7 7,2 12,0
2,55 1,38 1,1 0,95 0,57
Experiment 2,78 1,42 0,82 0,74 0,57
4. APPLICATION TO NOTCHED TENSILE TEST Then the mesoscopic model has been applied to axisymmetric notched tensile tests. The specimens utilized (AE2, AE4, AE10) present different notch radii (2, 4 and 10 mm respectively). They are commonly used to characterize ductile damage at different triaxiality levels (cf. Figure 5). The triaxiality path of these specimens depends noticeably on the material's constitutive behavior and must be estimated with a numerical simulation of the tensile test. In this application, 3 different steel grades for bars, used in cold forging, were studied : 27CrMo4, 27CrMo4 with sulfur and 42CrMo4.
1.4
i
i
i
1.2 ;=
1 0.8 0.6
....""
....-"""
ae2 ael0 ae4 . . . . .
r~ ///"""
0.4 t ' " 0
I
0.2
I
i
t
0.4 0.6 0.8 equivalent plastic strain
I
1
1.2
Figure 5. Triaxiality path for the different tensile specimens used.
28 4.1. Ductile damage mechanisms Mesoscopic simulations have been carried out for each steel grade and specimen. The ductile damage mechanisms are almost identical to those found in the torsion test. Nevertheless, it is interesting to note that, during the tensile test, a complete decohesion appears between the metallic matrix and the inclusion from the beginning of the deformation. This cavity first seems to grow in all the directions, but preferentially along the axial direction ; the occurrence of coalescence corresponds to a change in the cavity shape, growing this time along the radial direction (cf. Figure 6, Figure 7). Therefore, the global growth of the cavity appears like a relatively continuous phenomenon. This smooth evolution, in comparison with the one observed in torsion test (cf. figure 4), makes it less easy to determine precisely the <
) fracture strain.
Figure 6. Evolution of the mesoscopic mesh during tensile test simulation. 0.05 0.04 .i..-i 6~
.............. i ................ i................ i ................ i.....
0.03
O
0.02
..............
0.01
..............
0
i
i
!
i i
: ' ................
: ' ................
: -,' ...............
: , ....
I I--
: : ; ................ : ',
: : 1 ............................... ', ,
: :
: : T"q-" ', ',
I I
0.2 0.4 0.6 total equivalent strain
I I
0.8 l fracture strain
Figure 7. Void growth in the mesoscopic cell during tensile test simulation.
29
4.2. Characterization of the ductile damage propensity Contrary to the case of the torsion test, these steel grades contain different types of inclusions and the total inclusion volume can no longer be considered to equal the cementite volume. In this case, the analysis method involves iterations on the initial porosity vo in the mesoscopic model, in order to fit as close as possible the experimental results. The final value of v0 represents the ductile damage propensity. It should be an intrinsic parameter of the material, independent of the loading conditions. The experimental fracture strains cannot be directly determined since the strain state becomes heterogeneous from the beginning of the necking. They are estimated from the values of experimental fracture elongation by the means of a numerical simulation of the tensile test. Then, a value of v0 is fitted for each steel grade using the mesoscopic model, in order to predict the experimental fracture strain for all the triaxiality levels (cf. Table 2). Table 2 Prediction of fracture strain (%) using the mesoscopic model Steel Grade
Initial porosity
AE2
v0(%)
Exper.
Mesos.
Exper.
Mesos. Exper.
Mesos.
78 54 64
57 53 57
95 75 78
84 76 83
135 116 132
27CrMo4 0,14 27CrMo4 + S 0,21 42CrMo4 0,14
AE4
AE10
121 100 106
The comparison of the numerical and experimental fracture strains shows a mean error of about +10%. These discrepancies, in part explained by experimental uncertainties, reveal some limits of the model in its present formulation. First, the model seems to overestimate the dependence of ductile damage on triaxiality. Thus, for some steel grade, such as 22MnCrB5, and contrary to the results obtained previously (cf. Table 2), it is not possible to find any value of v0 giving a reasonable agreement between experimental and calculated fracture strain in the case where nucleation is not considered (cf. Table 3). The results show a very large dispersion in the values of v0. Moreover, the simulation made for the AE2 specimen leads to a value of v0 too small to be significant. This tendency has already been observed for some analytical void growth models. Moreover, microscopic observations reveal very few cavities near the fracture zone ; this seems to indicate that there is a nucleation threshold which cannot be neglected in the models of ductile damage. Table 3 Ductile damage propensity calculated with mesoscopic model (22MnCrB5) Initial porosity v0 (%)
AE2 AE4 AE10
0,004 0,018 0,10
Fracture strain Mesoscopic
Experiment
0,83 0,97 1,10
0,87 0,97 1,13
30 5. NUCLEATION MODEL In order to validate the previous nucleation hypothesis, we have developed an experimental procedure to characterize the nucleation mechanisms. These observations will then enable us to formulate an appropriate model. The steel grade we consider here is 22MnCrB5 (cf. w 5.1. Experimental characterization of nucleation The observations have been made on modified compression specimens which present, on their equatorial part, sufficient strain and triaxiality levels to reveal ductile damage (cf. Figure 8). Four compressions have been made at different strain levels, in order to follow ductile damage evolution with strain level. The equatorial parts of the specimens were examined in the SEM, and the relative density variations were measured following the double weighing method [ 13, 14]. The results (cf. Figure 9) show that the relative density decreases with the deformation because of void growth in the material. Nevertheless, this decrease appears only aider about 10% strain (corresponding to 25% compression ratio). Thus, this curve confirms the assumption of the existence of a nucleation threshold. The SEM observations tend to support this assumption and reveal that the nucleation appearance depends on the shape and nature of the inclusions. Moreover, cavities have been observed in parts of the specimen where the triaxiality is negative (compressive loading). In this case, a nucleation threshold is effective and seems to depend either on the local strain level or on a critical stress value.
Compression Rate (%) 20 40 60 -5.0E-04
specimen Q
~9 -1.5E-03 --
I
balance Figure 8. Preparation of modified compression specimen for ductile damage observations.
~ -2.5E-03 - - o
~ -3.5E-03
j
9S6riel AS6fie2 I
Figure 9. Density variation measured on modified compression specimens (22MnCrB5).
80
31 5.2. Nucleation formulation in mesoscopic analysis
The experimental characterizations of nucleation lead to the introduction of a nucleation term in void growth models. For this first application, a critical strain formulation has been used. Thus, the mesoscopic simulation has been modified in order to take into account the cohesion between the matrix and the inclusion : when the equivalent strain of the cell reaches a critical value, the interface is released and can grow freely (cf. Figure 11). On the base of the relative density variations, the new formulation of mesoscopic analysis has been applied to steel grade 22MnCrB5. The integration of nucleation in the mesoscopic model allows us to determine a value of the ductile damage, independent of the loading conditions (cf. Figure 10). The prediction of fracture strain is then obtained with less than 10% error. 0.05
I
I
I
I
I
il
'
la
I
0.04 |
m 0.03o 0.02 -
.v..i r.~
ae2 ael0 ae4 . . . . .
0.01 0
~ '
~; --"-4" .... ---I---
0
I
, .
[ ,,
-
~t
Ii
_
I
0.2 0.4 0.6 0.8 1 1.2 1.4 total equivalent strain
Figure 10. Prediction of fracture strain using the mesoscopic model with nucleation (v0=0.1%).
Figure 11. Introduction of nucleation in the mesoscopic analysis.
All the steel grades used for tensile test application have ferrite-perlite structures ; but on the contrary to the one of 27CrMo4 and 42CrMo4, the perlite of 22MnCrB5 is globular. It is possible that the annealing has improved the cohesion between the metallic matrix and the carbides. This could explain why the nucleation has to be taken into account in the case of 22MnCrB5. These results reveal once more the power of the mesoscopic approach in the study of ductile damage ; in future we intend to test the different nucleation criteria proposed in the literature [ 15-24].
6. CONCLUSION The mesoscopic model avoids restrictive assumptions usually used in analytical models and appears to give a good account of ductile damage phenomena (nucleation, growth and coalescence) for all kinds of loading conditions. It represents a very efficient tool for validating analytical models, for understanding the mechanisms and the part of the ductile damage
32 parameters, and for developing complete analytical formulations. The use of both the mesoscopic approach and the experimental characterization of damage has confirmed the importance of taking into account a nucleation critical level in ductile damage studies. The first integration of such a criterion in our mesoscopic model has allowed us to improve its predictive capabilities. Henceforth, it is essential to strengthen this approach by determining more precisely the respective effects of strain and critical stress (nucleation) and triaxiality (growth) on the ductile damage evolution. Experimental investigations (SEM, density balance) will contribute to this goal and to the validation of the method. At the same time, the comprehension of the damage phenomena will be the basis for the formulation of analytical nucleation criteria and void growth models, while their validation is ensured by mesoscopic simulations. Another line of investigation opened by the mesoscopic approach lies in taking into consideration different populations of inclusions, particularly in the study of coalescence.
ACKNOWLEDGMENTS Authors wish to thank UNIMETAL and ASCOMETAL companies for their collaboration and the supply of materials.
REFERENCES
1. Mc Clintock, ASME, J. Appl. Mech., Vol. 35 (1968). 2. Rice J.R., Tracey D.M., J. Mech. Phys. Sol., Vol. 17, pp. 201-217 (1969). 3. Gurson A.L., ASME, J. Eng. Mat. Tech., Vol. 99, pp. 2-15 (1977). 4. Cocks A.C.F., J. Mech. Phys. Sol., Vol. 37, pp. 693-715 (1989). 5. Gologanu M., Leblond J.B., Devaux J., J. Mech. Phys. Sol., Vol. 41, pp. 1723-1754 (1993). 6. Gologanu M., Leblond J.B., Devaux J., J. Eng. Mat. Tech., Vol. 116, pp. 290-293 (1994). 7. Gologanu M., Leblond J.B., Devaux J., Comp. Mat. Mod., ASME, Vol. 294, pp. 223-244 (1994). 8. Perrin G., PhD Thesis, Ecole Polytechnique (1992). 9. Leblond J.B., Gologanu M., Devaux J., 4th Int. Syrup. Plasticity, Baltimore (1993). 10. Hom, Me Meeking, J. AppL Mech,, Vol. 56 (1989). 11. Koplik, Needleman, lnt. J. SoL Struc., Vol. 24, 8 (1988). 12. Mazataud P., Me DoweU, lnt. Syrup. IneL Defor., Honolulu (1995). 13. Ratcliffe R.T., Brit. J. AppL Phys., Vol 16, pp1193-1196 (1965). 14. Moussy F., Roesch L., ECSC Final Report, n~ 7210-MA/307 (1982). 15. Gurland J., Plateau J., Trans. ASME, Vol 56, pp. 442 (1963). 16. Tanaka K., Moil T., Nakamura T., Philos. Mag., Vol. 21, pp. 267 (1970). 17. Tanaka K., Moil T., Nakamura T., Trans. Iron Steellnst. Japan, Vol. 11, pp. 383 (1971). 18. Argon A.S., J. Eng. Mat. Tech. ASME, Vol. 18, pp. 60 (1976). 19. Argon A.S., Im J., Met. Trans., Vol. 6A, pp.839 (1975). 20. Argon A.S., Im J., Safoglu R.,Met. Trans., Vol. 6A, pp.825 (1975). 21. Beremin F.M., Met. Trans., Vol. 12A, pp. 723 (1981). 22. Gilormini P., Montheillet F., J. Mec. Th. Appl., Vol. 3, pp. 563 (1984). 23. Gilormini P., Montheillet F., J. Mech. Phys. Solids, Vol. 34, pp. 97 (1986). 24. Ashby M.F., Ebeling R., Trans. AIME, Vol. 236, pp. 1396 (1966).
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
33
A fully coupled elasto-plastic damage theory for anisotropic materials J-F. Charles a, Y.Y. Zhu b, A-M. Habraken ", S. Cescotto" and M.Traversin c. a MSM Department, University of Liege, Belgium b ANSYS Inc., Houston, USA c R&D Cockerill-Sambre group In this paper, an elastoplastic energy-based anisotropic damage model at finite strains for ductile fracture is described. A calibration method is presented as well as the measured parameters for two steel sheets with different mechanical properties; one steel being a quasi isotropic one and the other being a classical IF steel. Two numerical examples are also given to illustrate the potential applicability of the model. I. INTRODUCTION It is well known that most structural materials exhibit some degree of anisotropy. In sheet forming, the effect of anisotropy on the deformation characteristics may be appreciable and important. Recent experimental evidence indicates that structural failures are often associated with the development of anisotropy damage even if the initial material properties are isotropic : microvoids or microcracks extend more on planes normal to the maximum principal stress axis. An analysis without taking into account the damage-induced material anisotropy may therefore yield questionable results. The model presented in this paper was originally proposed by Cordebois and Sidoroff [1] and Cordebois [2]. It is extended by Zhu [3] with some special considerations: 9 three major anisotropies are taken into account, including anisotropic elasticity, anisotropic plasticity and anisotropic damage" 9 the generalised damage effect tensor M proposed by Chow and Wang [4] is used" 9 a new damage characteristic tensor J based on the hypothesis of damage energy equivalence is proposed" 9 microcrack opening and closing mechanism is considered thanks to different effects of tensile and compressive states.
34 2. THE ANISOTROPIC DAMAGE
2.1 Effective stress and damage effect tensor M(D_D_) One basic assumption of most isotropic and anisotropic models of continuum damage mechanics is that damage can be viewed as a macroscopic state variable which translates the average microscopic damage growth in the sense of effective stress. Therefore there exists a damage effect tensor M_(D) applied to the stress tensor o which defines the effective stress tensor ~ [5-6]: ~ - M(D)~
(1)
where the damage effect tensor M(D_) is a second-order tensor depending on the damage tensor D. Chow and Wang have proposed a formulation of the tensor M(D) " . 1
1
I
1
1
1 DI'I-D2'I-D3'x/(1
M - diag
1
D2)(1-D3)'
(2)
1
L~/(7 - D 1-~-1--D3)' ~-0 ~ D2S(1 -- D1)
with .
~'--
.
.
.
.
11 ~ ( ~ 2 2 , (~33 ~ ~ 2 3 ~ t~31 ~ ~ 1 2
~m I:~m
[
I~11 ~ 1~22 ~ O'33 ~ G'23 ~ I ~ 3 1 , O'12
]
(3)
2.2 Hypothesis of energy equivalence
Instead of the conventional postulate of strain or stress equivalence, an hypothesis of energy equivalence is used. It states that the complementary elastic energy for a damaged material has the same form as that of a fictitious undamaged material except that the stress is replaced by the effective stress in the energy formulation. We(U, D) - We (~,D)
(4)
I~T -1 1 oT~-I o r ~ _ Co ~ - ~ _ _oo
(5)
m
where Co and Co are the virgin and the damage elastic material stiffness tensors respectively. So, the effective elastic strain vector is 9 me
- M-lg--e
(6)
35 3. GENERAL THERMODYNAMIC ANALYSIS 3.1 State variables In table 1, the internal variables to be used in thermodynamic analysis are listed, with their associated thermodynamic forces. The general structure of the constitutive equations is furnished by the thermodynamic theory of irreversible processes. Here after, isothermal condition is assumed. Table 1 The state variables and their associated thermodynamic forces State variables
Associated thermodynamic forces
Elastic strain e_e
Cauchy stress o Plastic hardening threshold R
Accumulated plastic strain a Damage variable D - (D 1, D2, D 3) Overall damage 13
Damage energy released rate Y = (Y~,Y3, Y3) Damage strengthening threshold B
3.2 Thermodynamic potential In the present model, the Helmholtz free energy takes the following form [7] 9
,v(o_, ~, ~, ~) - wo (o_o,p) + Vp r + Vd (~)
r
where W e (g-e, D) is the elastic strain energy, ~p (or) the free energy due to plastic hardening and ~d (13) the free energy due to damage hardening. Following the rules of thermodynamics of irreversible processes, the associated thermodynamic forces are given by the derivatives of the Helmholtz free energy. 3.3 The dissipation power According to the second law of thermodynamics, the total dissipation power is 9 r
o_~ - R a - Y D -
Sl3 -> 0
(8)
Thanks to the hypothesis of independence of energy dissipations between plastic flow and damage processes the above equation can be splitted into two parts such as 9 ak
-Rd>O (9)
- rD-
ejO>_ o
36 These two equations define the existence of a plastic dissipative potential and a damage dissipative potential, i.e.
.,(_~,_~,R): o and F.(_r,~): o
(10)
In case the criteria Fp=0 and Fd=0 are satisfied, the actual values of o,R, Y, B make the dissipation power a stationary value. If we introduce Lagrange multipliers ~p and ~'d, the dissipation power can be written" ~ = oe_"p - R & - Y I ) - B[~- ~,pFp - ~dFd
(11)
Its derivatives give the evolution law of the state variables. 4. FULLY COUPLED ANISOTROPIC ELASTO-PLASTIC MODEL
4.1 Anisotropic plastic yield surface In the material characterisation for large plastic deformations, Hill's yield criterion has been chosen. Its expression in stress has the following form"
F, (~, D, R)= F,(~, R)= ~ - R 0 -
R(~) = o
(12)
where 1% is the initial strain hardening threshold. We assume a linear plastic hardening. The effective equivalent stress ~ is"
(13) The effective plastic characteristic H is given by" ___= M(D)HM(D)
(14)
The positive definite tensor H for orthotropic materials is represented by a 6x6 matrix in the material coordinate system [8].
n
_
G+H
-H
-G
0
0
0
-H
H+F
-F
0
0
0
-G
-F
F+G
0
0
0
0
0
0
N
0
0
0
0
0
0
L
0
0
0
0
0
0
M
_
I
_
(15)
37 where F,G,H,L,N,M are parameters characterising the current state of plastic anisotropy. For a strain-hardening material, the uniaxial yield stress varies with increasing plastic deformation, and therefore the anisotropic parameters should also vary, since they are functions of the current yield stress (see [9]).
4.2 Damage evolution law and damage surface In a similar way to the arguments leading to plastic dissipative potential, on can assume that there exists a surface Fd=O, which separates the damaging domain from the undamaging one. A damage criterion in a quadratic homogeneous function of the damage energy released rate Y was proposed as [ 1]"
Fd-'Yeq-n0- B(~)- 0
(16)
where the equivalent damage energy released rate Y~q is defined by 9
Yeq- E.~1yT jy
1
(17)
in which J is the damage characteristic tensor. Normally, J should be a fourth order tensor as H. However, since we work on the principal coordinate system of damage, it can be treated like a second order tensor. The purpose of introducing a damage characteristic tensor J, like the introduction of plastic characteristic tensor H in the theory of plasticity, is to take account of the anisotropic nature of damage growth. The damage characteristic tensor J, which is a extension of the formulation due to [8, 10], is proposed on the basis of the damage energy equivalence.
J-2
J~lJ2
J2
~Jx/~lJ~ J~~2J~
(18)
Jxf~2J3 / J3
J
In the case of damage hardening materials, the equivalent damage energy released rate increases with the total damage growth, and hence, the anisotropic parameters in the above equation should also vary. The change in the equivalent energy released rate in any component depends on the total amount of damage work done in that component. For an equivalent variation, the damage work done in each component should be the same. For the case of a linear damage hardening as shown in figure 1, the damage work in component 1 is :
Wdl - - ~ D
+
'
2Dr1
)
(,9)
where Dtl is the slopes of Y~-D~ curves and Y~o the initial equivalent damage energy released rate corresponding to direction 1.
38 Similarly the damage work done in terms of equivalent damage energy released rate Yeq is : 1
w~- ~ t (~o~-~o ~)
~o~
t r,
1
~mage work.
=-D om
nt dmmtp
vtri~lc Figure 1 .Equating damage work.
By equating eqns (28) and (29), we have
:
,
':LYI j
(21)
(,~,1/D, Xu --,~o~)+,~1o~
Similarly, we obtain equations for J2 and J3 Obviously, if component 1 coincides with the reference component, Bo=Ylo, B(I3)=Y1-Y10, Jl=l and
Y12
,J3
J2:a2:(Dt2///D 1
y -j2
: a3 : (Dt~Dt1)(y12- Y102) + Y302
(22)
tl
5. CALIBRATION OF THE MODEL We perform three series of tests to determine according to [11], respectively, the elastic parameters, the Hill' s parameters and the damage parameters. The dimensions of the samples are shown on the figure 2. P15 3O0
220
1
//~m
"'1 100
I
T
Figure 2. Dimensions of the samples (for damage parameter determination R= 150 and D=3 mm otherwise R=m and D=0).
39
5.1 T h e elastic p a r a m e t e r s
We use the orthotropic relation : 1
-
E1 ~11
-- V21
f
V12
-- V13
E1 1
-- V23
E 1
E2
E2
E 2
~33
-- V31
-- V32
1
~12
E3
E3
E3
0
0
E;22
_
01
I
(~11
(~22
(23)
(Y33 L(~12
2G12
_
where the axis 11, 22 and 33 represents respectively the rolling axis, the transversal axis and the sheet-normal axis. To determine the elastic parameters, tensile tests are made with samples cut with an angle ot between the longitudinal axis of the piece and the rolling axis of the sheet. Tests performed with cz=0, 45 and 90 ~ allow the determination of El, E2, v12, v13, v23, G12. The missing parameters are taken as follows : E1 + E2 9 E 3 = "-------7-- G23
E3 " - G13 - G12 ; v31 - v13 g ,
v32 = v23
E3
E2
(24)
5.2 T h e Hill's p a r a m e t e r s
As we assume a bilinear material behaviour, the parameters (yield stresses and plastic tangent moduli) a r e : (Yly,(yEy,(Yay,(~12y , Et1,EtE,Eta, Et12. The tensile stresses and the Lankford coefficients are measured at seven angle ct [ 12]. Knowing the Hill's criterion, f(cy) = F(oz2-Cy33)2 + G(cyll. cy33)2 + H(Cyll-OZ2)2 + 2Nc~2 + 2L~3 + 2Mcr~l = 2cYt2
we can calculate the ~(o0 --
rH (or) =
OH
(25)
and rn values, the H index meaning according to Hill criterion :
2~t2 (H +G) + (F-G) sin4(oQ + (2N-2H-G) sin2(o0 eos2(o0 H - ( F + G + 4 H - 2 N ) sin2(~) cos2(~) F sin2(ot) + G cos2(ot)
(26)
(27)
Then we consider the functional : d~ = ~ i=1,7
(1- rl) [ O'n(~
" o ~ ( o q ) ]z + rl [r.(oq) - r~xp(Oq)]2 ~t
(28)
40 where Ot is the average of stress measurements Oexp and rcxp, the experimental stress and Lankford coefficients. Therefore, we determine the parameters F,G,H,N that minimise the functional 9 for two states A and B, with rl chosen equal to 0.5. The state A is taken when a zero tensile plastic strain is applied ; the state B is relative to a ten percent tensile strain for the tests performed for an angle equals to zero degree. For the other angles, we use the equivalent energy principle as described for damage hardening in 4.2 to determine the tensile strain value to adopt. Form this two states, we calculate the following stresses" O lyA ,(~2yA,(~12yA,l~3y A
, i~1B ,1~.2B,o12 B ,o 3B ," then we can determine the tangent moduli. The missing parameters are taken as follows 9E t13
-- E t23 -
Et12; o t13
-- G t23 -- ~ t12 9
5.3 The damage parameters. The parameters are : Ylo, Y2o, Y30,Dtl ,Dt2 ,Dt3" We perform loading-unloading cycles at angles equal to 0 and 90 degree. The Y~0, Dtl, Y2o, Dt2 parameters are determine directly by measuring the initial elastic moduli and their evolution according to the tensile strain. The Y30 and Dt3 parameters are calculated using the equivalent energy principle.
6. N U M E R I C A L SIMULATION We have chosen to simulate the classical Nakazima's tests achieved for determining the forming limit curve of each steel (FLC). The mechanical properties of the two selected steels are
Table 2 The material parameters
1 2
1 2
1 2
E1 (N/mm:)
E2 (N/mm:)
E3 (N/mm:)
G12 (N/ram:)
~12
~13
~23
210000 157000
203500 158000
207000 157700
83000 53600
0.28 0.36
0.32 0.34
0.30 0.34
13'yl
13'y2
~y3
Oy12
EG
Etl
Et2
(N/mm=) (N/ram~)
(N/ram~) (N/mm~)
(N/mm~)
(N/mm=) (N/mm~) (N/mm~)
266 108
277 104
284 110
161 65
1233 1773
i107 1719
Ylo (N/ram:)
Y20 (N/mm:)
Y30 (N/mm:)
Dtl (N/mm:)
Dt2 (N/mm:)
Dt3 (N/mm:)
0.34 0.075
0.38 0.068
0.39 0.077
14.65 20.55
13.45 20.55
14.05 3.53
1512 2484.5
Et12
459 650
41 This test has been selected because, on one hand, in the forming process, any drawing is prevented by the clamping conditions, on the other hand, using special lubrification conditions and plastic film, the friction is severely reduced and closed to zero. At this time of the work, we have only performed the simulation of the Nakazima's test which applied bi-axial tensile condition. The Lankford coefficients of the two materials are, respectively, closed to 1 for the isotropic steel and closed to 2 for the IF steel. The Coulomb friction coefficient is estimated at 0.05. The distributions of equivalent damage and stress for the sheet after a punch displacement of 30 mm are shown on the next figures.
Fig 3. Equivalent damage for material 1.
Fig 5. Equivalent damage for material 2.
Fig 4. Equivalent stress for material 1.
Fig 6. Equivalent stress for material 2.
If the Lankford's coefficient is reduced form 2 to 1, the peak damage and stress move towards the center of the sheet and the peak values are increased. The experimental Nakazima's test show the same results but we have to go on the simulations until the punch displacement at rupture. We will be able to calibrate a rupture criteria depending on damage variable which should be validated thanks to the other points of the forming limit curves.
42 7. CONCLUSION An energy-based anisotropic elasto-plastic damage model has been presented in this paper to characterise progressive damage. Comparison of results for isotropic and anisotropic steels illustrates significant differences in structural response. Thus in sheet forming problems where anisotropic conditions are invariably encountered, it is essential to adopt the anisotropic damage model. The obtained results with the simulation are in a good accordance with the physical interpretation of the forming process. 8. ACKNOWLEDGMENTS The authors are pleased to acknowledge the support of their work provided by the Rrgion Wallone and R&D Cockerill-Sambre group (RDCS).
REFERENCES
1. J. P. Cordebois, F. Sidoroff, Damage Induced Elastic Anisotropy, Euromech 115, Villard de Lans, France, 1979. 2. J. P. Cordebois, Crit~res d'Instabilit~ Plastique et Endommagement Ductile en Grandes Drformations, Applications ~ l'Emboutissage, doctoral thesis, Paris, 1983. 3 Y.Y. Zhu, A Fully Coupled Elasto-Visco-Plastic Damage Theory for Anisotropic Materials, Int. J. Solids Struct. 32, 1607-1641 ; 1995. 4. C. L. Chow, J. Wang, An Anisotropic Theory of Elasticity for Continuum Damage Mechanics, Int. J. Fract. 33, 3-16, 1987. 5 J.L. Chaboche, Continuous Damage Mechanics--A Tool to Describe Phenomena Before Crack Initiation, Nucl. Engng Des. 64, 233-247, 1981. 6 J.L. Chaboche, Anisotropic Creep Damage in the Framework of Continuum Damage Mechanics, Nucl. Engng. Des. 79, 309-319, 1984. 7 T.J. Lu, C.L. Chow, On Constitutive Equations of Inelastic Solids with Anisotropic Damage, J. Theory Appl. Fract. Mech. 14,187-218, 1990. 8 R. Hill, The Mathematic Theory of Plasticity, Oxford, 1950. 9. S. Valliappan, P. Boonlaulohr, I. K. Lee, Non-linear Analysis for Anisotropic Materials, Int. J. Numero. Meth. Engng 10, 597-606, 1976. 10.H. Lee, K. E. Peng, J. Wang, An Anisotropic Damage Criterion for Deformation Instability and its Application to Forming Limit Analysis of Metal Plates, Engng Fract. Mech. 21,1031-1054, 1985. 1I.E. Estevez, A.-M. Habraken, Calibration Method of an Anisotropic Elastoplastic Model Coupled with Damage, Nuphymat, 1996. 12.P. Noat, P. Montmitonnet, Y. Chastel, R. Shahani, Anisotropic 3-D Modelling of Hot Rolling and Plane Strain Compression of Al Alloys, Simulation of Materials Processing, pp. 959-964, 1995.
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini(Editors) 9 1997 Elsevier Science B.V. All rights reserved.
43
Mathematical modelling of dynamical deforming and combined microfracture of damageable thermoelastoviscoplastic medium A.B. Kiselev Mechanics and Mathematics Faculty, Gory, Moscow 119899, Russia*
M.V.Lomonosov State University, Vorobievy
Thermomechanical processes, which proceed in deformable solids under intensive dynamical loading, consist of mechanical, thermal and structural ones which correlate between themselves. Structural processes in materials include formation and motion of the orders in crystal of metals, phase transition, split bonds between molecules in polymers, accumulation of microstructural damage etc. These processes are shown themselves on microlevel as irreversible deformations, thermal isolation and fracture. Dynamic fracture is a complicated multistage process, including an appearance, development and confluence of microdefects and a formation of embryonic micro-cracks, their grow up to the break-up of the bodies with division into separate parts. Three basic types of dynamic fracture can be singled out: ductile, brittle and the mechanism of adiabatic shear failure. Ductile fracture, observed under normal conditions in metals, solid fuels and explosions, are characterized by the appearance and development of dispersed spherical micropores under plastic deformation. A large number of orientared, coin-type micro-cracks, capable of growing in the process of deformation are formed in the brittle fracture of the material. Fracture of this type can be observed in berilium, concrete, mineral rocks and certain types of steel. The mechanism of shear failure is observed under high speeds of deformation, for example, when a "plug" is forced out from the target. In this case the resulting shear is concentrated in thin layers with a thickness of up to several tens of a micron, are positioned along surfaces with maximum tangent stresses. This leads to the development of intensive plastic flow. Mechanics of continuous fracture began its development from fundamental papers [1, 2]. In this papers for the first time one scalar damageable parameter was introduced. Later on some successful attempts to introduce tensor of damage were undertaked [3], which don't stop up to now [4]. Introduction of damageable parametrs into the system of internal parameters describing the condition of material and application of thermodynamical principles of continua mechanics makes possible to construct the thermodynamically correct and boundary connected models of solids [5-11].
*Russian Basic Research Foundation is acknowledged for partial financial support.
44 New bound model of damageable thermoelastoviscoplastic medium which makes possible to describe deformation and microfracture both under the action of expanding stresses (separational fracture) and under the shear deformation (shear fracture) are consider in this paper.
1. T H E R M O M E C H A N I C A L COMBINED FRACTURE
MODEL OF DEFORMATION OF SOLIDS
AND
~i5
Introduce next parameters: aij, ~i~ and are components of the stress tensor, elastic P = 0); F, U and s are specific and plastic deformations respectively (~i1 = ~i~j + ~iPj, ~kk (per mass unit) free energy, specific internal energy and specific entropy respectively; T is absolute temperature; ~"is heat flow; p is density; w, a are scalar internal structure parameters. Physical meaning of these structure parameters consist in the next statements: w = W k k is the first invariant of damage tensor wij which is characterizes volume density of micropores in material; ~ = ( ~ i j ~ i j ) 1/2 is the second invariant of deviator of tensor wij: olij ~--- W i j - - W ~ i j . We shall call w - a parameter of damage under tension and a a parameter of damage under shear (w varies from 0 for undamaged material to 1 for complete fracture; a = 0 for initial undeformed condition). Turn to the equation for internal energy expressed in the form of influx heat equation 1
~f
--
.
- - O ' i j g i j --
P
1
(1)
-divg P
Here and further the dot over a symbol indicates material derivative with respect to time. The second low of thermodynamics expressed in the form of the Clausius - Duhame inequality, is
> -
d
v(y)
(2)
From internal energy U we pass to the free energy F = U - T s and then inequality (2) gets us, with the help of equation (1), to the next inequality 1
.
1 (gradT
_ Ts - F > 0
(3)
Because F = F ( e e j , e i ~ , w , ~ , T ) , we get next inequality from (3): OF
OF
1
.p
OF.
OF
OF .p Or
1 ~ gradT - - ~ > 0 p T -
As it's shown in [5], it follows from (4) that
(4)
45
OF
O'ij -- P ~ e ' Oeij
OF
.S =
(5)
OT
Using (5), we lead (4) to the next inequality
(6)
d = dM + dF + dT >__0,
d = PT, 7 is the production of entropy and the next disignations are introduced here
OF
OF.
d M -- ( a i j -- p ~ _ p j )~i
OF.
dR = - - p ( - ~ w w + - ' ~ a ) ,
dT = --
~ gradT T
'
(7)
dM is mechanical dissipation, dF is dissipation of continuum fracture, dT is thermal dissipation. Turn again to the equation (1) which is written over with the help of (5) in the next form Th =
OF
.p
(o'ij - p ~ 6 ~ j ) e i j
OF.
OF
- -~ww - -'~&
1 pdiv(
(8)
1.1. T h e m a i n a s s u m p t i o n s Next simplifical assumptions were done: e e 1. Elastic deformations are small: eijeij << 1. 2 Free energy F doesn't depend on cumulative plastic deformations: ~ 9
Oeii
= 0. This
hypothesis means that cumulative plastic deformations don't change elastic characteristics of material [7]. 3. Dissipation function d _> 0 (6) may be written as sum of three nonnegative addends
(7): d M >_ 0,
dF >_ 0,
d T >_ 0
(9)
Note that for Fourier low of heat ~' = -e;gradT termal dissipation has the form of Fourier inequality dT = a(gradT) 2 / T >_ 0; hypothesis dM >_ 0 is Plank inequality 9 The inequalities (9) will used for formulation of kinetic equations for internal parameters of state ~iPj, w and a. 4. For dissipation of continuum fracture we assume that
OF -p~-~- = A&,
OF -Pffw-w = A&,
(10)
where A >_ 0, A >_ 0 - parameters of material. Note that equations (10) are the consequenses of Onzager theory under the condition A = const and A = const. 5. Volume modulus K, shear modulus # and dynamical viscosity 77 are the next function of parameter of damage w for damaged material: K
= K o ( l - w),
# = #o(I - w),
~ = ~/o(I - w)
46 Here K0, #0 and 770 - parameters for nondamaged material (they can be the functions of temperature, pressure and other parameters [7]). 6. Kinetic equations for parameters w and a have next form: =
=
Here a -- akk/3 - the first invariant of the stresses tensor O'ij and r second invariant of the deviator of stresses tensor 7"ij = aij - aSij. Introduce next thermodynamic potential
(TijT"ij) 112 - the
=
e
G = F-
~O'ijCij
(11)
Derivation of (11) on the time with regard of (5) gives 1
= ----O'ij~i~ -- T s "4-
OF.
"4-
OF
(12)
&
Because G = G(o'ij,w, a , T ) , we get from (12) r
OG = - P oaij,
s =
OG OT'
OG Ow
OF Ow'
OG Oa
OF Oa
(13)
Using the hypothesis 1 - 5 we get (M
1
1
- 3 K a ~ k + -'4~ O'ij O'ij + -3 a V O ' k k ( T - To) + A - p G = 2#36K-------~
I
O~
~odw+ A
0
I
Cda+
0
+Go(T),
(14)
where a v - the coefficient of volume extension. If we introduce specific heat under constant stresses c~, then we get next equation of heat conduction from equation (8) [7]" pc,,~' + a v & T
= ,,:ls "v -3t- Ad,,2 + A& 2 - d i v ~ "
(15)
"
Using the equations (13), (14) and the hypothesis 5 - 6 we get {M
o, ~kk = ~ o + a v ( T - T o ) + -~
Ot .
~adw,
eij = 2#---~ -
.+
~a~j da
0
(16)
0 !
Here ei~ - deviator of elastic strain, a ' = a/(1 - w ) , rij = vij/(1 - w ) . Let us assume that viscoplastic deformations are described by kinetic equations of Perzina type [12]"
2,/0
r'
H(r'-
~
Yo)
(17)
47 Here ~-' = r / ( 1 - w ) . The statical yield strength Iio depends on tempetature, pressure and other variables of state [7] , Y = Yo + 277oV~C.,ij,.,ij /3 ;v ~v is dynamical yield strength, H(x) is function of Heaviside. Now we define concreteness kinetic equations for damaged parameters w and c~: :
:
-
&= r
-
= C(r'-
~-.)H(v' - 7.)
(18)
Here B, a . , C, r. - parameters of material. 1.2. The system of constitutive equations of the m o d e l Thus full system of constitutive equations for model of damageable thermoelastoviscoplastic medium consists of the next equations: Od
a' e k k = -~o +
~ v
A / O~ ( T - To) + ~ --~adw ,
r'j + A ] eiJ
=
0
27/0
r'
d.; = c2(w,o ) = B(a' - a.) H(a' - a.),
0r
2#0
0-~ij d~' 0
'
6 = r
pc~,7' + o~v&T = 7"ijii~ + A&2 + A&2 -- dive,
c~, r) = C ( ' r ' - 7.) H ( T ' - r.),
~=-x
gradT
(19)
This model generalizes model of elastoviscoplastic flow [12] and takes into account the accumulation of damage in area of intensive tension and in area of intensive shear, the effects of the processes of deformation and accumulation of micro-structural damage, the thermal effects.
1.3. Criterion of beginning of destruction Criterion of beginnig of destruction (the origin of cracks - new free surface in material) - is the entropy criterion of breaking specific dissipation (as in models [7, 8])" t.
D = f _1(dM + dF + dT)dt = D , P
(20)
o
Here t, is fracture time, D, is parameter of material (maximum of dissipation), which is defined experementally [7, 8]. Criterion (20) may be referred to the class of entropy criteria of failure. Such a criterion makes it possible to describe, in principle, the process of failure, using the mechanism of cumulative mierostructural damages occurring, for instance, at break-off failure in tension waves (in this case a decisive contribution to (6) and (20) is made by the term Ad; 2 the power of continual failure dissipation along with the power of ...v Use can also be made of the mechanism of shear mechanical dissipation dM = "r,;r
48
which is the case, for example, in problems of punching plate targets of a finite thickness with a flat-face striker. In his particular case, narrow zones of intensive abiabatic shear are known to develope in target in places of stress concentration. The work of plastic deformations converts almost completely to the heat that, because of high local deformation velocities, has no time to extend over any significant distance from the zones of developed plastic deformations. As a result, the temperature in the zones rises and great thermal gradients occur which cause an additional plastic flow and a further concentration of local plastic deformations and eventually forces a "plug" out of the target. At shear failure, a decisive contribution to (6) and (20) is made by the terms dM= rij~i~, dT = and A&2.
--qgrTdT
2. E X A M P L E S Now we consider examples of formulation of special problems of deforming and fracture of solids which make possible to estimate capabilities of the model. 2.1. F l a t c o l l i s i o n o f t w o p l a t e s w i t h s p a l l a t i o n d e s t r a c t i o n in a plate t a r g e t Consider the problem of flat collision of two thin plates which is studied experimentally very well. This problem is most often used for calculation of constants of the solids models by means of comparison of results of physical and numerical experiments of the problem of flat collision of two plates with spallation destraction in a plate target [7, 8]. Since the thickness of the plates are small as compared with the size and the characteristic time of the process is the time of several runs of elastic wave across the whole thickness of the plate target, the problem may be solved using a one-dimensional mathematical formulation (a uniaxial deformed state) and an adiabatic approximation (div~ = 0). In this case, the equations of mass, momentum and internal energy are written on the Cartesian coordinate system Oxyz (the x - axis is perpendicular to the plate surface) as follows:
/5
- = -i,
p
1 a ( f + a)
7) = -
p
pc~T + a~&T = 23--'rgl~ + AdJ2 + A&2
Ox
Here v = v. is the velocity, i = i~. = 0--7, o. i p = i ~ , r = Txx the rest of designations is identical to the ones introduced above. Besides, we take into account of the following
eyy'' = L-P:: = -~/2,
ryy = rzz = -r~=/2, since rkk = 0 and gkk "p
5-'= K 0 ( i r'
arT2 Y0
e' - ~-~0 (1 - ~7[)
=
0. Constitutive equations are given by
3 ( 1 - ~ 1 co) '
H(I
r'
2
I- 5Y0),
=
+
i _
i~
~"
+ iv,
+
~
AC 1-w
& sign T,
49
~ "- C ( ~
& = B(a~ - a,) H(a~ - a,),
[TI[ -- T,) H ( ~
[Tll -- T,)
Spallation destraction take place in the cross-section x = x*, where is realized criterion of destruction (20).
2.2. O n e - d i m e n s i o n a l s p h e r i c a l l y s y m m e t r i c p r o b l e m Now consider one-dimensional spherically symmetric problem in adiabatic approximation. In view of the spherical symmetry, the equations of the laws of conservation of mass, change of momentum and heat influx are written as PP -- -g~ - 2go ,
piJ = ~Oa~ + 2 ~a~, r - ao
pc~T + av&T = r~g~ + 2rogPO+ A&2 + A&2
Here r is the distance from the center of symmetry; v is the velocity of radial motion; ar and a0 = ar are the components of the stress tensor decomposed into the spherical and deviator parts a = (a~ + 2a0)/3 and a~ = a + r~, a0 = a + r0, respectively; gr =~'7 ov and go = vr__are the deformation rates that are representable as sums of elastic and inelastic deformation rates, the inelastic part consisting of the deformation rates due to viscous and plastic effects: g~ = e~ + g~, g0 = e~ + g~; it is also assumed that g~ + 2g~ = 0. The system of constitutive equations are given by ~
~
BA + 2go 2-~r~ o9I &t = Ko(g~+2go-avT-3(l~_w----------~&) ' g; = -----~---+
3 +-~o1-w T{ Yo ~ = ~ o ( 1 - i T~.;--------~) ;_ H(lv'-r;l-go),
T; ~ = ~o(1-i,r
u
o
1-wAC & sign(r;_r~O )
~
Yo H(lr:-r;l-Yo),
u
o
2.3. O n e - d i m e n s i o n a l cylindrically s y m m e t r i c p r o b l e m Consider one-dimensional cylindrically symmetric problem in adiabatic approximation. In this case the equations of the laws of conservation of mass, change of momentum and heat influx are written as
[)
- = -g,- - go p '
piJ =
O~Tr
-~r
+
6rr - - ~ 0
r
pc~T + a , & T = 2T~g~ + 2rOgPo + T~g~ + 7og~ + A& 2 + A& 2 In one-dimensional cylindrically simmetric case e = (a~ + a o + e z ) / 3 and a~ = a + r~, "
r. -t- ra -l- r. = O.
9
9
50 The system of constitutive equations are given by
BA
~' = go ( ~ + ~o - . , T - 3(1 - ~ )
~ = ~ + ~ +~ 3 +5-~,o + =
TIr (1 -
'
~ = ---Y-
+ ~foo + ~ - ~
r,
AC ~ 2~-'~+ r" 1~-~ ~-'
/-2 Y0 ~ H(T ' -
5.,=B(a'-a.)H(a'-a.),
5.,)
i 2 Y0)
~;
/~y0
~ = ~-~0( 1 - y ~ - ~ - 7 ) H ( r ' -
& = C(v' - v.)H(v' - r.),
~_ Y0),
v' = V/2(r~2 + ,'r;rb + r; 2)
REFERENCES
1. L.M.Kachanov, Izv. Akad. Nauk SSSR. Section of Eng. Sciences, 8 (1958) 26 (in Russian). 2. Yu.N.Rabotnov, Fatigue of Structural Components, Moscow, 1966 (in Russian). 3. A.A.II'yushin, Mechanics of Solids, 3 (1967) 21. 4. S.Murakami and Yu.N.Radaev, Mechanics of Solids, 4 (1996) 93. 5. B.D. Coleman and H.E. Gurtin, J. Chem. Phys., 2 (1967) 597. 6. V.I.Kondaurov and L.V.Nikitin, Theoretical Bases of Geomaterials' Rheology, Moscow, 1990 (in Russian). 7. A.B. Kiselev and M.V. Yumashev, J. Appl. Mech. Tech. Phys., 5 (1990) 116. 8. A.B. Kiselev and M.V. Yumashev, J. Appl. Mech. Tech. Phys., 6 (1992) 126. 9. A.B. Kiselev and M.V. Yumashev, Moscow Univ. Mechanics Bulletin, 1 (1994) 14. 10. A.B.Kiselev, Fourth Int. Conf. of Biaxial/Multiaxial Fatigue (St. Germain en Laye, France, May 31 - June 3, 1994), 2 (1994) 183. 11. A.B.Kiselev, M.V.Yumashev and A.S.Zelensky, Advances in Fracture Resistance in Materials, Eduted by V.V.Panasyuk etc., New Delhi, India, 2 (1996) 281. 12. P.Perzina, Quart. Appl. Math., 3 (1963) 321.
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
51
A Mathematical Model for the Formation and Development of Defects in Metals V.L.Kolmogorov, V.P.Fedotov, L.F.Spevak Institute of Engineering Science of the Russian Academy of Sciences (Ural Branch), 91 Pervomaiskaya St., GSP-207, 620219, Ekaterinburg, Russia SUMMARY To simulate metal forming processes, the formation and development of defects in metals, one has to solve relevant boundary value problems. The progress in the theory of plasticity is obvious (for example, the slip-line method, the finite element method, etc.), yet it retains too many unsolved problems to be applied to attain these ends. A mathematical model for the formation and development of continuity defects in metals under deformation cannot be constructed within the theory of plasticity alone (or any other section of continuum mechanics) because of the fundamental axiom of continuity. The proposed mathematical model of continuity defect formation deals with a new boundary value problem, in which the classical problem of plasticity is supplemented with a kinetic ordinary differential equation for a scalar functional depending on the stress-strain state and temperature histories. This kinetic ordinary differential equation is written for each material particle. The functional is called "metal damage", ~', caused by microdiscontinuities. Here we present a new technique for solving rather general boundary value problems, which can be characterized by the following: microdamage and macrofragmentation; the anisotropy of the materials handled; the heredity of their properties and compressibility; finite deformations; nonisothermal flow; rapid flow with inertial forces; nonstationary state; movable boundaries; changeable and nonclassic boundary conditions etc. 1. INTRODUCTION The proposed mathematical model for the formation and development of continuity defects in metals deals with the corresponding general boundary value problem, which consists of the classic problem of the plasticity theory and the condition of fracture for each material particle. This fracture condition (theory) was published in English in [ 1], then in [2-4] and in [5]. The solution for the corresponding general boundary value problem is presented here. Recent papers and books, for example [6 - 9], speak of the state of affairs in the calculation of the stress-strain state in metal forming with the application of the finite element method (FEM). Naturally, the calculation is approximate, and, notwithstanding the progress in computer hardware, the matter of topical importance is to find a better approximate solution method. To calculate the stress-strain state in metal forming (and, on the whole, in deformation mechanics) is to solve the boundary value problems of mechanics. Continuum mechanics
52 equations may be schematically divided into three types: kinematic equations, dynamic equations and constitutive relations. The characteristic feature of the majority of the works [69] is the fact that they satisfy kinematic equations exactly, satisfy constitutive relations, but, strictly speaking, they do not satisfy dynamic equations. In fact, the solution is constructed in velocities and displacements, e.g., by Lagrange's, Jourdain's, Markov's principles or Galerkin's method. Then, by the found flow kinematics (for of Markov's principle, by kinematics and mean normal stress), by means of canstitutive relations, the stress tensor field is calculated. Since direct variational methods are used for the solution, the obtained stress fields do not satisfy continuum dynamic equations, namely, the differential equations of equilibrium (or motion if the flow has mass inertial forces), the stresses do not satisfy the boundary conditions in stresses exactly ("softer" satisfiability). Certainly, as the number of variation parameters grows and/or if more suitable coordinate functions are used, the discrepancy in the satisfaction must decrease. Even solutions for very complicated problems by the FEM (as in [6-9]) often seem verisimilar. Man has accumulated some experience in estimating body forming, and the visualisation of kinematic solution results (displacements, velocity fields and even strain distribution) creates the impression of safety. However, there is little physical notion of stress fields, and paper authors seldom bring their results to the analysis of the stress state obtained. We do not state that this approximate solution method is worse than the one described in the present paper, but we are of the opinion that the latter method deserves discussion. The solution in the form of kinematic fields (of velocities, displacements etc.) satisfying all the kinematic equations and in the form of stress fields satisfying all the dynamic equations is viewed as an alternative method. Since the solution is still approximate, the "softening" falls on the constitutive relations. It should be noted that the constitutive relations are always approximate and found f~om experiments, where there are experimental errors, therefore this way of "softening" seems more preferable. The idea of simultaneous variation of the stress and strain states with the "softening" of constitutive relations alone is not new. It was proposed independently in [10,11] and developed in [ 12-17]. 2. THE CORRECT FORMULATION OF THE GENERAL BOUNDARY VALUE PROBLEM
The boundary value problem consists in the integration of an extended set of continuum mechanics equations with respect to the variables describing flow kinematics, stress, microdamage and macrol~agmentation for certain boundary and initial conditions. Some of these equations (constitutive relations) and the boundary conditions are defined fiom experiments for a specific class of problems, particularly, those of metal forming mechanics; however, any formulation ought to be correct. The correct formulation is as follows. Let a material body of volume V undergo finite plastic deformation. Let the constitutive relations be given V A / e V . They can be given in any form (e.g., they describe plastic deformation in volume Vp, which is part of V, and elastic deformation in volume V e, which is the rest of the volume etc.). We suppose that, under conditions of developed forming, the material possesses rheonom properties, the constitutive relations are known functionals of the deformation history, temperature, 0 and density, ,o etc. However, at every
53 specified time, t , including the one under study, they turn into some known tensor functions, together with their inverse functions s o. = s o. ( e k l ) , e o. = e o. ( s kl ); (1) cr = o'(9~),
r = ~(o').
(2)
..
Here, s ~J and e/j are stress deviator components and strain rate deviator components; cr and are mean normal stress and the rate of relative volume change respectively; ekl and s kl form a set of deviator components which appear in functions (1) as arguments. Among the arguments in functions (1) and (2) there may be any characteristics of the stress and strain e..
9
states (e.g., derivatives s v , o', microdamage, ~ etc.), but in (1) and (2) there are arguments that are principal ones for the following reasoning. The adopted coordinates are Lagrangian ones. Let functions (1) and (2) satisfy the conditions
3
~"/o e~l~_~ > 0,
(3)
C70-/C7~ > 0. (4) SO constitutive relations at any specified time, t (functions (1) and (2)) ought to be differentiable with respect to the mentioned arguments and have inverse functions; conditions (3) and (4) ought to be fiflfilled for the functions, as they express the known properties of metal viscosity. Let the solution be sought in the flow velocity fields, o i and temperature fields, O, which are continuous in coordinates, and in the stress fields, o" 0, for which the surface stresses, fi
_ cr iJnj are continuous on any surface inside V. Here n is a unit normal to the surface
S. Suppose the body of volume V under deformation is bounded by an external surface, S, where the boundary conditions are generalized at any specified time, t as follows
VM eS
f i _ fi(vj,...) ' vi - v ; ( f J,...),
(5)
o7'/ jl;:j > 0; VM e S
(6)
00 - 2 - ~ - - (p,(M, O).
(7)
Here f.;, v?, ~. are known functions, wita f.;, V,.* be~-g reversible; 2 ~ heat ~o.au~ti~ity coefficient. Relation (6) shows the viscous properties o f th e environment. Finally, suppose that V M ~V distributed mass forces, g/* are given. Suppose the following initial conditions are given (at t = 0) for time integration:
V M e V,
vi- v ~
o ' 0 " - o "0", P = Po,
g / - ~o,
0-0o.
(8)
On the fight, marked by zero, there are known coordinate functions. Thus we have formulated the boundary value problem of mechanics of solids under deformation. Let us solve it.
54 3. A N A P P R O X I M A T E S O L U T I O N F O R THE G E N E R A L B O U N D A R Y V A L U E PROBLEM
Consider an arbitrary, but specified of time, t . The space integration of the boundary value problem can be replaced by the equivalent task of solving the following variational equation for the principle of virtual velocities and stresses
eo'..
d"
~"
o-"
0
+ Ir 0
6{f[ I s'J (e)ae + I eij(s)ds + I o-(~)ar z 0
0
:/" -I[I f/av + I v ? a f ] a s }
+ p(W i
-
gi* )Vi" ]dV
-
(9)
vl
SO
= o.
0
l
The variation is isochronous only with respect to the virtual quantities cr 0'', v i marked in (9) by a prime. The summation is made over the indices i and j appearing in the upper limits of the integrals and in the expressions under the integral sign. Naturally, the constitutive relations ought to be such that the functional J in Eq. (9) (curly brackets) were differentiable. Besides !
the continuity in V and on S, the virtual o i ought to satisfy all the kinematic conditions. For example, if the material is incompressible, velocity vector is given on S v and its normal component is given on S S ( Sv k.) S S L) S f - S ), then VM
~ V dive'=
O; V M
~ Sv h = h ,9" V M ~ S s v ~
(10)
' -- V*v.
!
Besides the continuity of the surface stresses f i ' = cr 0 n7 in V and on S, the virtual o-0" ought to satisfy the following conditions V M ~ V, V;cr#" + p (g.i _ w J ) = 0 , V M e ST, cr #"nj = f.~.
cr 0 ' - cr J;';} (11) !
Here w j is the acceleration of material particles. Note that the virtual o-0" and o i satisfy all the equations of continuum mechanics (which are linear in this case, and this simplifies the practical implementation of the principle of virtual velocities and messes), except for the constitutive relations. The functional J calculated for any virtual stress-strain state, even entirely different from the actual one, all other factors being equal (i.e., invariable quantities), is not negative and becomes zero on reaching the absolute minimum at the actual state being the space solution for the boundary value problem with specified time. The value of J calculated for some virtual state described by the fields o i and cr/j expresses a discrepancy in their satisfying the constitutive relations. The temperature part of the problem is also solved on the basis of the variational principle. The solution of the heat conduction differential equation is equivalent to the finding of the extremum of a functional t
!
55
)2 oO'~o'dO+ IcP 0 tdO]dV +--I I~o,(M,O) doriS} - 0 (12) v z o o ' 2s0 on the virtual temperature fields (continuous in V and on S), with the variation with respect to
/9'. Here V is Laplacian operator, C is mass heat capacity factor. The approximate solution at an arbitrary time, t will be sought by using the principles (9) and (12) in the following form n m l o~cr~O"( y ) ; 0 ' = ~-'~CkOk(y). (13) vi' - Y'~agivki(Y); frO"- Z'-O" k=l k=l k=l Here, y are Lagrangian coordinates; aki , b~ and Ck are variable coefficients (with specified t and, generally speaking, functions of time);
Oki(Y), cr~'(y) and Ok(y ) are known
suitable coordinate functions (in the fight hand part there is no ~mmation over the repetitive indices
i, j). The suitable functions are selected so that o i,! ~
On v - 67x
account
that
P-Po
t where p 0 and p
..
and
0 !
are virtual.
detllOX~o/dyJll/detllCTXk/dYtll , /l il/l I[ / II
are the initial and current material density,
co-
t,
x i~ and x k are
the initial and current Eulefian coordinates of the particle, yJ and yl are their Lagrangian coordinates, the variational equation (9) and (12) are reduced to the integration of a set of
or .a y
equa o,
respectto ak, - ak, tt), t g =
(t), ck - ck(t)
view of the initial conditions (8). The question of the unique existence of the solution is discussed for each stage of the proposed solution algorithm separately. According to the theorem of functional analysis, conditions have been found that suffice to provide the unique existence of the extreme problem (9) in the Odich-Sobolev space
W lLM [18]. When the problem is solved
numerically, the properly chosen virtual state allows one to obtain the closest possible solution to the variational problem at any time. In the second stage, the problem solution is reduced to a set of ordinary differential equations where the conditions for the unique existence of the solution are known. Note that, for different times (different stages of elasto-plastic deformation), the chosen virtual states give different deviations from the exact solution. Therefore one can speak only of the unique existence of the approximate solution found with an accuracy up to the virtual state, with the convergence of the general problem solution being established numerically. There is one more variable, ~ which is still to be defined, and it is discussed in the next section.
4. ON THE FRAGMENTATION OF DEFORMABLE BODIES In the above-described solution for the boundary value problem, it was assumed (as usual) that the material volume V remains continuous in the process of deformation, that it does not become divided into parts, and that no macroscopic holes or macrocracks appear in it, i.e.,
56 that there is no considerations and mechanics is valid) time tp). This time
macroscopic fragmentation of the body under deformation. The the boundary value problem solution are valid (in as far as continuum until macroscopic fragmentation begins, i.e., discontinuity appears (at the can also be viewed as the be~nning of a new stage, i.e., a new solution for
a new boundary value problem, because on the newly-formed surfaces there appear additional boundary conditions requiring a new boundary value problem statement. The second stage continues until the next new surfaces appear and so on. The time tp when fragmentation starts and the time of fil_rther macrodiscontintfities can be determined by means of the fracture theory [8], supplied with some new statements. According to this theory, in every material particle of the body under deformation, accumulation of damage, ~ takes place. By the time t , ~t(t) is calculated by some kinematic relations. To this end, firstly, we solve the appropriate boundary value problem. Secondly, we specify (or find from special experiments) plastic characteristics of the body under deformation. Damage, ~r is calculated for every material particle. For this purpose, on the particle motion path, separate sections of monotonic deformation, are specified where the components of the particle strain rate tensor do not change the sign. We indicate the time when the tensor components change the sign as tl, t2,... , tn_1 (transition through zero of at least one component). In the first section (t o _< t < t 1), damage is determined as
H(t)
d ~br1
~/(t) = ~l(t),
d t - A? [kl(t) , k2(t)] ' ~rl(t~ = 0;
in the second section (t 1 ~ t < t 2 ), [ / / ( t ) - [ ~ l ( t l ) ] al + [ ~ 2 ( t ) ] a2,
d ~2
d---T =
H(t) A p [k,(t), ko(t)] ' ~ 2 ( t l ) = 0; K
x
-
-
--
in the n-th section (t~_ 1 _< t
< t),
d ~'n
/~(t)
d t - Ap [Klt/r--~tx,K2t)J--:t ~1' ~n(tn-1)= 0.
~ t ) = ~~'~",__~
(14)
Firstly, here H - H ( t ) i s shear strain rate intensity; k 1 = kl(t) and k 2 - k 2 ( t ) are dimensionless independent invariants of the stress tensor [k 1 = O'/T, k 2 = 2 ( 0 - 2 2 - o " 3 3 ) / ( O - l l - o " 3 3 ) - 1 where o" is the mean normal stress and T is tangential stress intensity; o'11 >_ 0"22 > 0-33 are principal normal messes]. Secondly, we have plastic characteristics of the body under deformation, which have been found from oxpo mo ts:
- A
valuos of~o ~ ~ o n
Ikl,
rofors to
va uos
~ = ~ (g~, ge) m ~o i - ~ so~on of monotom~ Ooformauo~
aro
57
By e eof ao e('--'p), (t) = ~ (tp) = 1,
(15)
and the body is saturated with microdamage (not appearing in the boundary value problem solution), the material becomes brittle and ready to form a macrocrack (body fragmentation onset). Condition (15) is the end of solving the boundary value problem within the adopted statement and the be~nning of a new stage, i. e., a new solution. How can we find tp, the macrompture spot and formulate boundary conditions on the new surface? Damage is calculated by the given algorithm simultaneously with the time integration of ordinary differential equations, allowing one (as is described above) to obtain an approximate solution to the boundary value problem. At every time t , we can solve the problem of seeking the x coordinates of the point where there is maximum damage
max [ ~/(t)]. x~V
(16)
The time t = tp will be determined when, according to (24), the maximum value of IF reaches unity, i.e.,
max [ g/(tp)] =1. x~V
(17)
At the same time, the point (or points) where a macrocrack appears can be found. How will the unlace of the macrocrack be oriented?::We can assume that, if plastic deformation precedes fracture, then, at the time t = tp, the crack will be oriented along the sites of maximum tangential stresses. The crack will have finite dimensions owing to the continuous change of ~ in the volume V. The dimensions can be calculated by solving a new boundary value problem and calculating the stress-strain state around the crack. On the spots of maximum tangential mess, the tangential and normal messes will, respectively, be as follows
l
~'n = 2 (O'l l -- 0"33 ); 1
1
0_33). j
If, when t = tp, at the point with ~ = ~max = 1
(18) , o-n > 0, then there appears a crack with
"edges" free from surface stresses, i.e., ~'n = ~ = 0. Impact unloading takes place at the edges by the value A r n = rn, Ao-n = o n . (19) If, when t - t p, at the point with ~ - ~'max- 1, we have O"n < 0 , then there appears a shear crack with edges not free from surface stresses. Impact unloading takes place at the edges by the value
I
(20)
if between the shear crack edges there is assumed to be Coulomb friction (fl is friction factor). In the subsequent solution, the shear crack edges should be viewed as surfaces with sliding friction.
58 CONCLUSION The problem solution by the proposed method can be divided into two stages: (i) space integration for any specified time up to some parameters and (ii) time integration of a set of ordinary differential equations (including the kinetic one) for these parameters. To solve the problem in stage (i), a variational principle is proposed, which develops the well-known classical principles of Jourdain, Castigliano and Bio. The field of velocities of material particles, the stress tensor field and the temperature field are varied independently at a specified time. This method allows one to find the solution satisfying all the equations of continuum mechanics exactly, with the constitutive relations being fulfilled approximately. The "soitening" of the constitutive relations, which are always approximate, is more preferable as compared with the most familiar methods when the Newtonian mechanics equations are "soitened", whereas the constitutive relations are fulfilled exactly. To substantiate the solution method, we have proved the equivalence of the application of the variational method to differential equations of continuum mechanics and the unique existence of the solution, etc. The use of the functional minimum condition results in a set of algebraic and ordinary differential equations for variable parameters depending on time (stage ii). The discussion and the solution to the boundary value problem in terms of classical continuum mechanics are valid up to the time of discontinuity tp, i.e., the onset of macroscopic fragmentation. At this time a defect appears. Impulse relief takes place on its boundary. A boundary condition is formulated according to a certain rule, and one can start solving the boundary value problem in the next stage of deformation up to the formation of the next defect etc. The instant of fragmentation onset, tp and the instants of further macroruptures can be determined by the proposed kinetic equation for the fimctional ~ . Some applications of the mathematical model for the formation and development of metal continuity defects under deformation are given. For dynamic processes, impact problems are discussed for materials with different properties under different kinematic conditions, some of the problems having known solutions. This has offered appropriate correlations. For simplicity, the application of the mathematical model for the appearance and development of continuity defects in metal under deformation is illustrated by test examples of thin elastic and elasto-plastic bars impacting a rigid obstacle. The elastic problem was solved at the following mechanical characteristics of the material: /9 = 8 0 0 0 k g / m 3,
E = 200000MPa,
Ap = 0.2 exp(-2cr / T), a - 1.2exp(l + 0.24cr / T), bar
length l = 0.1m. The virtual state was given in the form of Fourier series, the solution of the variational problem resulted in a series segment coinciding with the exact solution segment. Fig. 1 shows the values of damage, ~ cumulated along the bar at the instant when the first fracture takes place at an impact velocity of 250m/s. Fig.2,3 show fracture time tp and fracture point Xp in the bar as dependent on impact velocity ~. Note that the origin of coordinate x = 0 is in the point of contact between the bar and the obstacle. Fig.4 shows fracture point displacement before and after fragmentation at an impact velocity of 140m/s, rip. ~XOM tp = 0 . 0 0 0 0 4 8 S, Xp -- 0 . 0 3 m. As is seen from the figure, as a result of
59 unloading, the rupture edges move along different paths after fracture. The elastoplastic problem was solved with the following constraint equation for the plastic region cr = 2000x/-~, the rest parameters being the same as in the elastic problem. The virtual state was given in the difference form based on the representation of the bar as 10 linear finite elements. Tab. 1 shows the average values of damage, Ip' for each element at the instant when the first fracture takes place at an impact velocity of 300m/s, t p - - 0 . 0 0 0 0 1 5 s . The fracture is seen to occur in Element 1. 6 tp• 103 5.8 5.6 5.4 5.2 5 4.8 4.6 4.4 100
0.8 0.6 0.4 0.2 0
0.02
0.04
0.06
0.08
0.1
x ----~
Figure 1. Damage along the elastic bar.
u(t,x)•
, 150
200
-10 \ -
250
V~
v -----
250
~
3 2 1 0
0"07f~
0.011 100
200
Figure 2. The instant of the first fracture.
0.09~Xp
t I
150
Figure 3. The point of the first fracture.
1
2
3
/txl05-t 4
5
6
7
Figure 4. The displacement of the fracture point.
Table 1 Damage along the elastoplastic bar dement no. damage,
1
2
3
4
5
6
7
8
9
10
1.012 0.213 0.027 0.006 0.002 0.001 0.001 0.001 0.002 0.004
The proposed method for solving boundary value problems provides an approach to solving the problem of motion stability, which can be described by continuum mechanics equations. The method presented here within the mathematical model for defect formation is a matter of great siL-mificance in itself. It can be successfi~y applied to calculate stress-strain states and temperature fields in metal forming.
60
Acknowledgement We are grateful to Mrs E.E.Verstakova for assistance in the preparation of this paper. REFERENCES 1. V.L.Kolmogorov. Model of metal fracture in cold deformation and ductility restoration by annealing. Materials Processing Defects, S.K.Ghosh and M.Predeleanu (Editors), 1995, Elsevier Science B.V. 2. V.G.Burdukovsk% V.L.Kolmogorov, B.A.Migachev. Prediction of resources of materials of machine and construction elements in the process of manufacture and exploitation. I.J. of Materials Processing Technology, 55 (1995), 292-295. 3. V.L.Kolmogorov. Friction and wear model for a heavily loaded sliding pair. Part I. Metal damage and fracture model. I.J. Wear 194 (1996) 71-79. 4. V.L.Kolmogorov, V.V.Kharlamov, A.M.Kurilov. Friction and wear model for a heavily loaded sliding pair. Part II. Application to an unlubricated journal bearing. The Journal of Wear 197 (1996) 9-16. 5. V.L.Kolmogorov, S.V.Smirnov Healing of metal microdefects after cold deformation (an article in the present volume). 6. J.-L.Chenot, T.Coupez, L.Fourment. Recent progresses in finite element simulation of the forging process. Proceedings of the Fourth International Conference on Computational Plasticity: fundamentals and applications, Barcelona, 3-6 April 1995, pp. 1321-1342. 7. Zhi-Hua Zhong. Finite element procedures for contact-impact problems. Oxford University Press, 1993. 8. Jurgen Gerhardt. Numerische Simulation dreidimensionaler Umformvorgange mit Einbezug des Temperatmverhaltnes.Springer-Verhg, 1989 (German). 9. A.A.Pozdeev, P.V.Trusov, Yu.I.Nyashin. Large plastic deformations: theory, algorithms, addenda. M.: Nauka, USSIL 1986 (Russian). 10. A.Baltov. Variational theorems in the dynamic theory of viscoplasficity. Bull. de l'Acad. Pol. SoL set. SoL techn. 17, No5, 1969. 11. V.L.Kolmogorov. The Principle of possible variation of stress and strain. Mekhanika tverdogo tela. N2. 1967 (Russian). 12. V.L.Kolmogorov. Stresses, strains, fracture. Metallurgiya. USS1L 1970 (Russian). 13. E.P.Unksov. W.Johnson. V.L.Kolmogorov et al. Theory of plastic strains in metals. Mashinostroyenie. USS1L 1983 (Russian). 14. V.L.Kolmogorov. Mechanics of metal forming. Metallurgiya. USSR, 1986 (Russian). 15. V.L.Kolmogorov and 1LE.Lapovok. The calculation of stress-deformed state under nonisothermic plastic flow - the example of parallelepiped setting. Computers and structures. Vol. 44 No. 1/2. 1992 (English). 16. E.P.Unksov. W.Johnson. V.L.Kolmogorov et al, edited by E.P.Unksov, A.G.Ovchinnikov. Theory of forging and stamping. Mashinostroyenie. Russia. 1992 (Russian). 17. V.P.Fedotov. Variational solutions for elastic-plastic problems. Conf. Modem Problems of Plastic Metal Forming. Bulgaria, Vama. 1990 (Russian). 18. 1LA.Adams. Sobolev spaces. Academicpress, New-York - San-Francisko - London. 1975.-315p.
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
61
Healing of Metal Microdefects after Cold D e f o r m a t i o n V.L.Kolmogorov, S.V.Smirnov Institute of Engineering Science of the Russian Academy of Sciences (Ural Branch), 91 Pervomaiskaya st., GSP-207, 620219, Ekaterinburg, Russia
I. Introduction As is known, cold plastic deformation of metal (rolling, die-forging etc.) from the first stages is accompanied by microscopic defects in continuity (we call these defects and the phenomenon "microdamage" or simply "damage"). As deformation accumulates, the development of damage can result in the appearance of macroscopic defects or even in the division of the body under deformation into parts, i.e., in defective products. Definitely, this is inadmissible. Macroscopic defects can be easily revealed (external ones by visual observation, internal ones by introscopy), but correction either is impossible or requires that the defective bulk of the metal should be removed. One of the methods for avoiding macrodamage is multistage deformation with intermediate annealing at the end of every stage. This provides metal dishardening and, above all, the restoration of metal plasticity (i.e., the ability of metal to be deformed without fracture). The amount of deformation in a separate stage can be established intuitively, from one's practical experience. This paper gives specific rules for it. As distinct from macrodefects, microdiscontinuities are harder to detect under service conditions. Now industry lacks means for checking microflaws, therefore all metal products must have microflaws, which can lower the efficiency of machine parts. They have been ascertained to influence fatigue life [2]. Therefore it is urgent to study the mechanisms of eliminating (or healing) microflaws, i.e., the mechanisms of metal plasticity margin restoration by heat treatment and the ways of making it more efficient. This is the subject matter of the present paper. Together with their helpers, the authors have done the work complemented with the phenomenological theory of metal fracture developed by them. The principal features of the theory were briefly discussed in [ 1-3] published in English. The reader can familiarize himself with the details of the theory by reading [4-6] in Russian. In [2] it was shown that the abovementioned theory of metal fracture proved to be general enough and applicable to the description of the fatigue life of machine parts. In brief, the ideas of the fracture theory are as follows. The physical picture phenomenologically described by the fracture theory consists in the fact that plastic deformation is always accompanied by microfracture, i.e., by the formation of flaws (point, linear, plane and three-dimensional). As deformation accumulates this damage develops to fracture by way of the appearance of new defects and the growth of those previously formed. This can be presented mathematically as follows.
62 For a material particle of a body being deformed, the increment of this damage (A~) is assumed to be proportional to the increment of deformation and inversely proportional to plasticity margin. The increment of deformation HAt (H is shear strain rate magnitude, At is the increment of time) results from the calculation of the stress-strain state; plasticity margin, Ap is a constitutive relation (function) determined by experiment. Plasticity margin (function Ap = Ap (| of temperature | and two dimensionless invariants of the stress tensor kl,k2) is determined by the method worked out within this theory. If the increment of time approaches zero, this assumption brings us to the differential equation d~
H(t) m
dt
Ap [0(t),kl (t),k2 (t)]
(1)
Here the arguments of the function Ap (| result from the calculation of the stressstrain state and temperature. Equation (1) gives good results (at the time of macrofractureq~=l) if deformation develops monotonically. It develops monotonically if no component of the strain rate tensor changes its sign. If a material particle undergoes nonmonotonic deformation (and it is a more common case), then the damage accumulated by the time t must be calculated in a different way: n
V(t) = 2 V ai
(2)
1
where n is the number of monotonic deformation stages the particle has gone through by the time t; ~Pi is the value calculated for the i-th stage of monotonic deformation (i -- 1...n) in accordance with relation (1); ai is an exponent that is taken as a mean value for the conditions corresponding to the i-the stage of monotonic deformation (it has been found that a i > 1, and this formally reflects damage development retardation upon the change in the deformation direction). It has been found that a - a(kl,k2) , and it is the second (afterAp) constitutive relation in the fracture theory presented here. This version of the fracture theory gives good results, whereas some familiar theories of fatigue can only be viewed as particular cases. (In some cases, the mathematical model (1) can be made more accurate, but it has to be done at the cost of introducing some nonlinearity [6].) It was shown in [2] that the above-mentioned model of fracture can be viewed as generalized known conditions of fracture from low-cycle fatigue, creep, etc. There is an unconventional way of "healing" steel (metal) "tired" of deformation. The restoration of metal plasticity is based on diffusion or on the transfer of the substance into pores and microcracks and the transfer of these discontinuities to the surface of the body. This idea of using heating to maintain the serviceability of machine parts was discussed in [3] with a heavily loaded sliding bearing as an example. In this paper the problem of metal plasticity restoration by heating is discussed in detail within the above-mentioned theory of fracture described in [1-3,6].
63
2. A model for damage history in annealing As was mentioned above, damage accumulates in metals under loading of all kinds (plastic forming, machine part operation, etc.). In the experiments described below, to initiate the onset of damage, active plastic deformation was used. This proved to be an easier method, besides, flaws from plastic deformation are large, as compared with those from other kinds of loading, and therefore they are harder to heal. To work out the technology of manufacturing cold-deformed products with annealing, it is necessary to give a mathematical description (within the above-mentioned model) of how the restoration of plasticity margin (or the reduction of microdamage) proceeds under annealing. The following technique has been developed for solving this problem. Tests are performed on metal, with the plasticity Ap(kl,k2) being already known. Test specimens undergo different amounts of plastic deformation, each specimen being deformed to different amounts of damage, Wi (for example, by tension not to fracture, but to different amounts of shear strain A 1 = 2~f3 ln(d 0 / dl) where d o and d 1 are diameters of specimens before and after tension; W1 = A1 / A p is determined by a plasticity diagram for every specimen, provided that the process of tension is monotonic). Thereafter all the specimens undergo annealing in accordance with the chosen regime (0 is temperature and t is annealing duration). Plasticity restoration (or damage decrease by the value AV) takes place in annealing. After annealing, once again, all the specimens undergo plastic deformation in the same direction, but this time to fracture. Then A 2 is determined and calculated W2 = A 2 / A p . The second plastic deformation plays an auxiliary part in the determination of AW. Evidently, the total value for the specimens is ~Ij = ~IJl - A~t j + t ~ 2 =
1,
since the specimens have been brought to fracture by the second deformation. This allows one to determine the unknown decrease in damage resulting from annealing in accordance with the chosen regime AW - ~1 + ~2 - 1,
(3)
and to predict permanent damage Wp = W1 - AW which has not been healed by heat treatment. The results on damage decrease by annealing of some steels and a titanium alloy are presented in Fig.1 as an example. The annealing regimes: for steels, Oranges from 500 to 750~ and t ranges from 5 to 300 minutes; for the titanium alloy, | = 680~ and t = 60 min. The dependence presented here was found in other investigations to apply to various other metals and alloys, therefore we can draw general conclusions [6].
64 The restoration of plastic properties (or decrease in microdamage AW) under annealing by recrystallization depends on W~ to a great extent. Separated by two critical values of damage W, and ~**, there are three intervals of W~with different rates of restoration. If damage resulting from deformation is 0 < W1 < W,, then complete healing takes place on annealing, since Aq~=q~, in Fig.1. If q~, < qJ~ < q~**, then damage is not completely removed by normal annealing. AW continues to increase as W~ grows, but at a lower rate. When qJ~ > q~,, AW decreases, and it goes to zero when q~l = 1. The damage - time history for the steel of grade 20 at a temperature of 600~ is shown in Fig.2 as an example. The lower curve at ~1=0 illustrates that the annealing of a blank to be deformed, for example, of hot rolled metal, can lead to higher plasticity due to the healing of microdamages that appear in the stage of hot rolling. The curves have three distinctive parts: AB is rapid exponential decrease in damage; BC is considerable deceleration (and even stopping) of metal plasticity restoration; CD is further acceleration of the process.
/'x
f
Aq~
i
0.4
0.2
0
0.2
0.4
0.6
0.8
qJ 1
Figure. 1. Decrease in metal microdamage owing to recrystallization annealing (for fixed time and fixed temperature) for the steels 12X18H10T (1) and CT3Kn (2) and the titanium alloy BT1-0 (3) with damageW1 after plastic deformation.
Assume that the material damage, ~I-/1 resulting from plastic deformation changes exponentially (in Fig.2 within 0.5 h.), i. e., it may be written as follows q~(t) = ~Jl exp(-[3t). Here 13> 0 is the index of the exponent steepness.
(4)
65 3. I n v e s t i g a t i o n
results,
discussion,
recommendations
The restoration of plasticity margin in metals after cold deformation has been studied for a number of years in co-operation with researchers from some institutes, mostly with Prof. A.A.Bogatov (Ural State Technical University, Ekaterinburg). Fig.3 shows diagrams illustrating the degree of damage healing for some alloys in coordinates (~[,~p). By the amount of permanent damage, ~gp (caused by cold deformation resulting in calculated damage Wp and further recrystallization annealing), one can estimate the completeness of the healing of flaws appearing in the preloading stage. (Mind that, if q~p= 0, the healing is complete, whereas if q-'p=W1, all the flaws remain in the metal.) Healability is seen to be different for different alloys. However, the values of q-',,W** are similar for different alloys. This fact enables one to state that, if machine part damage (in the most unsafe place) < q~, = 0.2-0.3 in cold plastic deformation or in service, then the margin of plasticity can be restored by conventional recrystallization annealing. P 0.7, A
B
D
C
0.5 0.3 "-------~
G
0
~
e
I
0.1 ' ~ , ~ :'
2 i '~ " ~ ~ ' ~ D . . . . . . . . ~ ,
'
I
I
I
-0.1
~-,
0
:40
u
:80
~
~
120
.~
.
T
: 160
a ,w
t, mln
Figure.2. Damage history W during heat tretment (at 600 0 C). The grade of steel is 20.
Generally speaking, in this equation, the exponent is a function of heat treatment and the third constitutive relation of the fracture theory: In Fig.3a, the behaviour of q~p for steels is noteworthy. Thus, for example, carbon steels tend to show lower permanent damage as carbon content increases. Electrone microscopy has made it possible to reveal that this can be attributed to a smaller size of the flaws and, consequently, to more intensive healing by annealing. The healing of deformation damage by means of heating is connected with the change in the dislocation structure, decrease or disappearance of microdiscontinuities. Depending on the temperature of heating and the amount of deformation, healing can follow the pattern observed in polygonization, recrystallization and other phenomena. Investigations have shown that recrystallization annealing results in the healing of subgrain-size microdiscontinuities (i.e., under 2-5 ~t) by
66 intensive surface diffusion of voids when they are crossed by the moving intergrain boundary of the grain being recrystallized.
q., P
0.8
0.8
0.6
0.6
0.4
0.4
1//
0.2 O0
0.2
0.4
0.6
bo
4
0.2 0 0
0'.8
0.2
0.4
0.6
0.8
q-'I
Figure.3. Permanent damage after plastic deformation of alloys (with damage W1) and recrystallization annealing. 1-3 - carbon steels with 0.2%, 0.5%, 0.7% carbon content respectively; 4 - titanium cx-alloy of the system Ti-A1-Mn; 5 - W-Ni-Fe alloy; 6 - aluminium alloy with 7% of rare-earth metals.
The rate of damage healing can be increased by the activation of other healing mechanisms, for example, by the repeated passing of the grain boundary through a microflaw during cyclic heat treatment with phase recrystallization. This enables one to increase the value of W, limiting the range of damage, which can be completely removed by recrystallization annealing (Fig.4). Naturally, by increasing W,, heat cycling in annealing allows for fewer cycles in making colddeformed products due to a greater amount of deformation between annealings. However, the initialization of the mechanism of metal leaking into flaws (Laplace flow) due to local plastic creep seems to be more promising. This mechanism can be realized by treating materials with discontinuities in a gasostat (HIP process). q., t9
0.8 0.6 0.4 0.2 0
~ 0
0.2
2 0.4
0.6
0.8
q~l
Figure.4. Diagrams of permanent damage for the steel 40X after single annealing with phase recrystallization (1) and after heat-cycling annealing with phase recrystallization (2).
67 Fig.5 (line 1) shows the result of studying permanent damage of the nickel alloy 3H698 after treatment in a gasostat for four hours at 1100 o C in the argon environment under a pressure of 180 MPa. This kind of treatment is seen to result in the complete healing of damage caused by preceding deformation. Triaxial uniform compression does not cause any alteration of the specimen's geometry after treatment, and yet it initiates the development of metal microcreep in internal discontinuities. ~.J.,
8325
P
0.8
o
xx)'
0.6
%
8315
4 o
0.4
3
o
o
o
8295
0.2
. 9o
0
o n
8305
^
x x X'~x
•
x x
x
x
1,2
a
a ,,,,,,,~x ox
o
9
9g o
~
8285
u
A
m
-0.2
0
0.2
0.4
0.6
0.8
~
8275 1
0
0.2
0.4
0.6
0.8
I
Figure.5. Diagrams of permanent damage for the nickel alloy 3 H 6 9 8 : 1 - annealing in a gasostat; 2 - annealing under uniaxial compression; 3 - conventional annealing Figure.6. Density of nickel alloy samples: 1 -after deformation (without annealing); 2 - after deformation and conventional Complete damage healing also proved to be attainable by heating specimens under uniaxial compressive stress. To this end, samples of the same nickel alloy having undergone different amounts of cold tensile deformation, were heated up to 1100~ and subjected to initial compression with a force running 80 per cent of the yield stress at the above-mentioned temperature (30MPa). They were held at constant temperature to complete relaxation (2.5min). As distinct from HIP, uniaxial compression proved to result in the rise of undesirable permanent creep. Although the amount of permanent strain was small, it proved to be sufficient for complete damage healing (Fig.5, line 2). For comparison, Fig.5 (line 3) shows a diagram of permanent damage after conventional annealing (1100 ~ C, 1-hour holding, air cooling). The validity of the above-mentioned results, which have proved high effectiveness of heat treament after deformation under compressive stresses, has been proved. To this end, we have studied the change in the density of the speciments after deformation to different amounts of damage and after conventional heat tretment in a gasostat with uniaxial compression has been studied. It follows from Fig.6 that heat treatment under compressive stresses leads to complete healing of metal damaged because of deformation up to strains preceding fracture (with ~P, = 1). Note that, if conventional annealing is used, the area of damage to be healed must not exceed 0.3. The initiation of the gasostat healing of microdamaged metal that has undergone cold plastic deformation or reached the limit of its fatigue life in a machine or a construction is now applicable only in particular cases. Gasostatic treatment is expensive and underproductive. Besides, gasostats are unsafe because of high potential energy saved in
68 compressed gas. Instead of gasostats, fluid-metal hydrostats can be used. They can be fairly efficient and safe owing to low compressibility of metals. Pressure can be forced electromagnetically. Therefore the development of such apparatuses seems to have prospects. They are being designed by the Institute of Engineering Science, Russian Academy of Sciences (Urals Branch). Timely heat treatment offers better deformability of metal in multipass processes of metal forming. Fig.7 shows an example of a successful technological solution to the problem of how to reduce cracking of drawn tungsten wire for filament lamps. Predicted damage (q~ = 0.6) supported by metallographic examination has shown that in 0.8mm finished wire there are microcracks that cannot be healed by annealing and cause delamination of heating elements under deformation. In Fig.7 the intermediate annealing of 1.4mm wire, with damage not exceeding W = 0.6, is seen to heal the deformation defects. The practical application of this innovation has resulted in significantly reduced wire scrappage caused by delamination in heating element coiling. As far as we know, the restoration of the in-use fatigue life of machine parts by heat treatment is not adopted in practice in Russia. However, this phenomenon has been known for a long time. Thus, in [7] there is a description of a favourable result concerning the restoration of the mechanical and physical properties of rails after service by heat treatment. There is no doubt that it entails some technological difficulties. For example, it is necessary to maintain the condition of the surface during heat treatment. However, for some machine parts, including those used in forging and stamping, this way of prolonging service life seems feasible. Thus, the operational lifetime of dies can be increased after 25-30 per cent exhaustion by tempering causing the dispersal of dislocation clusters and the healing of submicroscopic flaws.
~3
P,
0
0.4
microcracks ,_7
_.
0.2
tm
t~ t" o ~
rc -
o f
0
"278
o
~
"2?4
~ .6
176
,
2 ,.~ "'J/ d, mm
Figure.7. The change in the damage of tungsten wire for electric lamps (1-conventional technology, 2-proposed technology).
Acknowledgement We are grateful to Mrs E.E.Verstakova for assistance in the preparation of this paper.
69
References 1. V.L.Kolmogorov. Model of metal fracture in cold deformation and ductility restoration by annealing. In: Materials Processing Defects, S.K.Ghosh and M.Predeleanu (eds.), Elsevier Science B.V, 1995 2. V.G.Burdukovsky, V.L.Kolmogorov, B.A.Migachev. Prediction of resources of materials of machine and construction elements in the process of manufacture and exploitation. I.J. of Materials Processing Technology, 55 (1995), 292-295. 3. V.L.Kolmogorov. Friction and wear model for a heavily loaded sliding pair. Part I. Metal damage and fracture model. I.J. Wear 194 (1996), 71-79. 4. V.L.Kolmogorov. Stresses, strains, fracture. M: Metallurgiya, 1970, 232 p. (in Russian). 5.V.L.Kolmogorov, A.A.Bogatov, B.A.Migachev et al. Plasticity and fracture. M: Metallurgiya, 1977, 336 p. (in Russian). 6. A.A.Bogatov, O.I.Mizhiritsky, S.V.Smirnov. Metal plasticity margin in metal forming. M.: Metallurgiya, 1984. (in Russian). 7. V.S.Ivanova, S.E.Gurevich, I.M.Kopiev et al.Fatigue and brittleness of metallic matrials. I: Nauka, 1968, 216 p. (in Russian).
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Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
71
Defimtion of the form for kinetic equation of damage during the plastic deformation S.V.Smimov ~, T.V.Domilovskaya" and A.A.Bogatovb "Institute of Engineering Science of the Russian Academy of Sciences (Ural Branch), 91, Pervomaiskaya st., GSP-207, 620219, Ekaterinburg, Russia bUral State Technical University, 19, Mira st., 620002, Ekaterinburg, Russia
1.1ntroduction According to contemporary ideas the damage of metals is not a simply catastrophic phenomenon, but a multi-stage process of appearing and development of microdefects, which is called in mechanics a processes of damage accumulation (plastic loosening, fracture, cracking, etc.) Purely brittle fracture is possible only in non-metallic materials with a great quota of covalent part in inter-nuclear link. Historically, the problem of fracture during plastic deformation was first considered from the position of technological interests based on empirical criteria. This approach allowed to solve some easy applied tasks and held back investigations of general problems of metal fracture during the complex stress strain state. Progress has been made in ideas of mechanics of fracture of metals during plastic deformation connected with appearance of dissipated fracture kinetic theories. As it was shown in the Novozlfilov work [ 1], the plastic deformation should be accompanied by a residual change of deformed metal volume (plastic loosening): ~ = tx].,,
(1)
where a - plastic loosening module, L - loading path which, for the large plastic deformations could be taken as a shear strain AA= ~/(2e~j e~j), where e~j - components of deformations increment tensor. Another expression of fracture processes during the plastic deformation within theories of dissipated fracture is the kinetic equations of damage in a form, where the accumulated damage is defined by kinetic equation: do/dA = f(sl,s 2),
(2)
where r represents damage of material, sl,s2 - thermomachanic parameters of deformation, depending on the loading conditions. Relations between the value of dissipated fracture with
72 plastic deformation[ 1] allow to connect the damage accumulation (cracking) with shear strain A and to insert a term ~F named the reserve of plasticity expenditure stage [2]. This value has been initiated by linear summarization principle and became a base for the majority of phenomenology theories of metals fracture Ap dA ~F= oJ" Ap ' where Ap - plasticity, could be defined as a shear strain rate, accumulated in material till the fracture point, under the monotonic deformation processes with the given constant values of thermomeehanieal parameters. Damage is a scalar parameter, equal to 0 for undeformed material and corresponding to 1 in a point of macro-fracture. Later Bogatov [3] offered to describe accumulation of plastic loosening by the more common power function e = bA a and considered it's increment as de = a(A)dA,
(4)
where ct(A) = baA(al) - intensity of plastic loosening coefficient, b,a - empirical coefficients. The limit value of plastic loosening e" in a point of macro fracture cracking appearance depends on physical-mechanical nature of the deformed material and thermomechanical conditions of loading processes. The parameter e* for constant conditions of loading in a point of fracture, estimated by value of accumulated shear strain Ap, could be defined by a
e* = b.A/p. Relation de/s* = do was identified as a damage increment. Then kinetic equation (2) could be described as dto/dA = e,A/e"
(5)
Consideration of equation(5) allows to assume that defining the form of kinetic equation of damage under constant deformation conditions can be reduced to investigation of dependence e,A and 6" of loading conditions and the value of accumulated deformation. It should be emphasized, that in literature there is no agreement in opinions on the form of kinetic equation and different authors base their choices on hypotheses fragments of published data on metal-physical investigations. As postulated in linear models of damage accumulation, the e,A depends only on momentary loading conditions [2]. Some investigators assume, that even during the monotone simple deformation, with constant external loading parameters, relationship e,A has a power [3, 4, 5] or exponential form. It is accepted in [6] that change of damage depends on accumulated damage. There are some inconsistent data in literature on the dependence of limit level of plastic loosening of loading conditions [3,5,7]. It is usually accepted in theoretical consideration of damage accumulation, that the parameter e* depends only on the nature of deformed metal.
73 The purpose of the present work is to carry out experimental investigations for basing the choice of the form for kinetic equation of deformation and cheek out hypothesis of dependence of the limit parameter r* on stress state.
2. Materials and experimental methods Relative change of density p during the deformation was considered as plastic loosening g=Ap/p0. Density definition was carried out by triple hydrostatic method. Every specimen was subjected to five times weighing to reduce a statistical error. A resulting value was defined as an arithmetic mean. Specimens were made of 6 different kinds of steel, brass and molybdenum. The surface of all specimens was treated by grinding and polishing before experimental tests. A ZDMU-30 universal testing machine was used for testing specimens of steel 45; an original testing machine equipped with a controlled hydrostatic pressure chamber [3] was used for the test specimens. The liquid used in the chamber is castor oil, which is neutral to deformed metal. The dimensions of the specimens before and after tests were measured with instrumental microscope. Five general series of experiments were carried out. Shear strain rate A calculated by formulas [2,3] was used as a measure of plastic deformation: - for tension of cylinder specimens in a minimal cross section A = 2~/31n(d0/d~); - for tension of flat specimens with a width and thickness ratio >_0.6 A = 2In(to/h); - for torsion of cylinder specimens
A = tg(p, where do, d~ - diameter of specimen in a minimal cross section before and after the test; to, t~ thickness of flat specimen in its minimal section before and after the test; q) - twisting angle of line, brought on a surface of specimen parallel to its longitudinal axis. Stress state coefficient k and coefficient of Lode-Nadai go were used to characterize a stress state of the deformed metal: k = o/T; go = (2022 - 011 - 033)/( O l l
- 033),
where 0 = (0~ + 022 + o33)/3 - average normal stress; T = ~/~S~S~ - tangent stress intensity; 0~1,022, o33 - principal normal stresses; saj - components of stress deviator. The above mentioned values are invariant parameters, so they allow to compare stress state of deformed materials with different levels of strength properties.
74 The separate effect of stress state coefficients k, go was investigated by conducting the tests in a chamber with controlled hydrostatic pressure. In this case the coefficient !~ does not change and the current value of coefficient k could be determined from the decision of Davidenkov and Spiridonova k=(1 + 3/4 d/R)q3 - pq3/a,
(6)
where d - diameter of specimen in its minimal cross section; R - radius of longitudinal profile of deformed specimen in area of deformation localization, as - intensity stress. It is obvious from equation (6), that when p is not equal to 0, as a result of changes os due to deformational hardening of material a coefficient k is not constant even within the uniform deformation stage. To define density the areas with length 2d0 containing the localized deformation zone in the center were cut off from the tested specimens. It is obvious that the density that was defined during hydrostatic weighting is the average within the volume of an investigated specimen. So changes of density correspond to shear strain rate and coefficient of stress state that are also average in deformation path and volume V 1 vj'k(V)dV, kv =-V
A v = V1V~ A(V)dV
(7)
Deformation resistance as used for calculation of coefficient k was investigated by inverse extrapolation of Lode method for multi-stage tension of specimens. For analytical description of deformation resistance Ludwick model was used a. = a,o + aAb,
(8)
where a~0 ,a ,b - has been defined by statistic treatment of experimental results. Tests were carried out using a universal testing machine ZDMU-10t with velocity of loading 6 ram/rain.
3. Simple loading The purpose of the first one was to establish the form of equation (1). Obviously, to define correctly whether the S,A value depends on accumulated deformation, it is necessary to carry out experiments with changing conditions of stress state during the loading process. So during the first series of experiments a preliminary loading was carried out by tension of cylinder or flat specimens in a range of uniform deformation. In this case it is possible to cut some area of effective part of specimens, where stress state coefficients are constant during deformation (k-0.58, p~ = -1 - for tension of cylinder specimen; k=- +1, !~ = 0 - for tension of flat specimen), and use them to define density. Comparing the results of specimens tested and the value of strain rate accumulated during the process of uniform tension allows to determine a form of equation (1) for constant quantities of stress state conditions. Experimental results of test performed are shown in fig. 1. It is obvious, that under constant parameters of stress state during the deformation process, relation between increment of
75 density and accumulated shear strain is linear. This behaviour is characteristic for tension of cylinder ( ~ . = -1) as well as for fiat shaped specimens ( ~ . = 0). As it was mentioned above, the experimental data obtained by a number of scientists suggests non-linear relation between deformation and physical characteristics of damage. These published data were subjected to analysis and the following were observed. Most of the investigated specimens had the areas with non-uniform stress-strain state (for example, tested in a non-uniform range of deformation or during drawing of wire). It is necessary to point out, that in the first case during the deformation a change of stress state coefficient according to relationship is observed (6). This causes a growth of plastic loosening intensity proportionally with variations of stress state coefficient or by the power law. In the second case non-liner relation "s-A" could be connected with signveriable type of deformation which reduces an intensity of damage accumulation. An explanation of non-linearity during the torsion of cylinder specimen is not so obvious. Similar data were observed in [5]. But in our opinion these results could not serve as a basis for definition of the form of equation (1) because of tension stress appearing in deformation hardening material, made specimens under the conditions of the big angles of torsion. The level of this stress grows with deformation development and results in the growth of coefficient of stress state (investigation results of A.V.Konovalov) and damage accumulation intensity which show themselves as non-linear decrease of density. E;
.
104 I. ~
(a)
4;
-8
,.6
-16
~
3
(b)
~ o
-8 "100
0.1
0.2
"240
0.3 A
0.05
0.1
0.15 '
0.2
A
|
i
lO~ ' ~ -2
o
-10
~ ~ o
4
(C)
' { ~ - ~ , ~
(d)
-4
o
i
.201
-6
5
xX'x
-10
4% . . . . 0'.1'
0'.2
0'.3
o'.,~
'
0
0.1
0.2
0.3
0.4 A
Figure 1. The plastic loosening e under simple loading: 1 - carbon steel 0.2%C , ~ = -1; low-alloyed structural steel (0.12%C, l%Cr, 0.3%Mo, 0.2%V), ~ = - 1 , 3 - brass molybdenum, ~ = -1; 4 - leaded brass, ~ = -1; 5 - carbon steel 0.45%C, !~ = 0; 6 - carbon steel 0.45%C, !~ =-1 _
76 4. E f f e c t o f s t r e s s s t a t e
The purpose of the second one was to define a functional relation of value S,A and limit quantity s* of the parameters of stress state. For definition of form for the relation between plastic loosening intensity and the stress state coefficient k, some specimens made of carbon steel 0.20%C were subjected to tension tests in uniform deformation range in a chamber with a controlled hydrostatic pressure. There is no reason to average k and A within the volume, which decrease statistic error. Some deviation of relation from the linear one shown at fig.2 connected with the variation of coefficient k because of the deformational hardening according to formula (6). So the average value of the plastic loosening intensity s was defined as ds/dA = (Ap/po)/AA,
(9)
and associated with the value of integrated mean coefficient of stress state k, calculated by formula (7).
in (S,A)
.
.
.
.
.
.
.
.
.
1
-7 -8
-3
-2
-1
0
k
Figure 2. Changes of plastic loosing intensity under the stress state coefficient k, p~ = -1 1 - carbon steel 0.45%C, 2 - carbon steel 0.10%C, 3 - carbon steel 0.20%C; 4 - carbon steel 0.24%C; 5 - alloyed structural steel (0.4%C, 0.8%Cr, 0.8%Ni, 0.2%Mo, 0.3%Si) Results were obtained in semi-logarithmic coordinates and that allowed to describe the required relationship in exponential manner: ds/dA = am exp(azk), where a~ and a2 - empirical coefficients, to be defined by least square method.
(10)
77 Specimens of the test materials were pulled or twisted till failure. The parts of specimens with length L=do, cut from the deformed specimens were subjected to hydrostatic weighting. An average in volume coefficient of stress state was calculated from formula (7). As it is obvious from fig.2, for all the tested materials a relationship between the plastic loosening intensity and the coefficient of stress state described correctly by the exponential form (10). It should be pointed out that the growth of tension stress during the deformation calls an increasing of plastic loosening while the growth of compressive ones are decreasing it. The direct experimental determination of the limiting value of plastic loosening ~* corresponding to the moment of micro-crack appearing is very complicated since failure starts in locally undetermined volume dimensions of which do not allow to carry out the density determining. That is why in this experiments the value e* has been calculated using an above mentioned relationship (10). It is known from the experimental tests, that in tension tests the crack of plastic metals appears in a middle area of minimal cross section of specimen neck. So the value 8 in the moment of failure, accounted for this moment taking in consideration the history of stress state coefficient was taken as Ap
(11)
e*= ~ k(h)dA, 0
where relationship k(A) could be determined by formula (7). In
s,
, , ~
-3.5
1
-4
-5.5
-'
-0.5
0
0.5
k
Figure 3. Changes of limited value of plastic loosening e* under the different stress state coefficient, Ix~ = -1:1 - carbon steel 0.45%C; 2 - carbon steel 0.10%C; 3 - carbon steel 0.20%C; 4 - carbon steel 0.24%C; 5 - alloyed structural steel (0.4%C, 0.8%Cr, 0.8%Ni, 0.2%Mo, 0.3%Si) Calculated value of plastic loosening accumulation in a central area of specimen neck till the failure moment is shown in fig. 3. Results of e* calculations by formula (11) are also shown there. Obviously, the value ~, is not constant for specimens pulled apart under different hydrostatic pressure. Variation of e* in comparison with integral mean value of stress state
78 coefficient during the tension process is shown at fig.3 in semi-logarithm coordinates and has an exponential character s* = bleXp(bzk),
(12)
where b~ and b2 - empirical coefficients. The result of investigations of other materials shown at fig. 3 are also reinforce a validity of relationship (12). Decreasing of limiting plastic loosening with increasing of tension stress contribution (k is increasing) corresponds to existing ideas about the fracture of metals. Actually, the rate of viscous fracture associated with body defects of continuity is decreasing with a concentration of tension stress and brittle fracture rate associated with appearing and spreading of the increasing fiat cracks. Obviously in limiting ease of the brittle fracture value s* will tend to zero.
5. Two-stage loading In third series of tests the cylindrical specimens made of 0.20%C carbon steel were subjected to tensile tests under atmospheric pressure ( pl = 0.1 Mpa) till different shear strain rates, then specimens were placed to the controlled hydrostatic pressure chamber with a pressure of working liquid p2 = 200, 500, 800 MPa and subjected to following tension. In the fourth series of tests the specimens were subjected to tension i n t o the chamber under hydrostatic pressure pl = 200, 500, 800 MPa till shear strain rate A~ = 0.14 and 0.35, then tension was continued under atmspheric pressure p2 = 0.1 MPa. Changes of specimens density under this conditions goesnon monotonic (see riga and fig.5). t; 10 -4
",,,,.
-2
........
1
~..,...
"~
-4
"%-.% '%
X "~.%
~ 4
'-,.. -6
~176 ~ ~ %,, "%~163
-8 -10
(a)
0
.
.
.
. 0.2
.
.
.
.
. . 0.4
.
0.6 A
Figure 4. Changes of plastic loosening intenity under two-stage tension: 1 - p = O. 1 MPa; 2 - p = 500 MPa; 3 - pl = 0.1 MPa, A1 = 0.14, P2 = 500 MPa; 4 - Pl = 0.1 MPa, A1 = 0.3 5, P2 = 500 MPa
79 It was observed that for two-stage variation of stress state the intensity of plastic loosening changes not immediatly but within certain period of adaptation of metal behaviors to the changed conditions. The rate of adaptation depends of the magnitude of damage accumulated at the first stage of deformation. Similar phenomena were observed under the conditions of changing direction of deformation. 10 "4
,~- ,-,,~,...~..
'
%
-2
(b)
"",,t.,.. .
i--?,~t h~'',,a
=
-4 -6
-8 -10
0
0.2
0.4
0.6 A
Figure 5. Changes of plastic loosening intenity under two-stage tension: 1 - p = 0.1 MPa; 2 - p = 500 MPa; 3 - pl =500 MPa, A~ = 0.14, p2 = 0.1 MPa
6. Practical calculations Comparison of the calculated results shows that non of the known theories of metal fracture could describe the observed experimental relationships correctly, but the best agreement reaches by using the formulas of Bogatov [3], which could be recommended for practical calculations under the multi-stage loading: 1/ai AAi co i = (o~i_l + )ai,
Api
where i - number of loading stage, AAi - increment of the shear strain rate at loading stage, Api- plasticity of metal under k and la, at i-stage of loading, ai - empiric index of damage accumulation intensity. Values of k and la~ could be defined by solving the plasticity test. Relationship between a, Ap and k and !~ to be defined experimentally [2,3]. Calculation and analysis of damage accumulated in metal allow to optimize technology processes of plastic treatment. One of number of practical examples may be given. At the Pervouralsk Pipe-Making Plant in Pervouralsk (Russia) pipes of carbon steel 0.45%C for poles has been produced by cold rolling. Existing equipment did not allow to satisfy the demand for this type of product. To increase a volume of production it was offered to be produce at
80
automated triple drawing line. One of the major questions stated for engineers was a question of damage of pipes during drawing process because the manufacturing line design did not suppose the intermediary annealing. Theoretical calculations allowed to choose an optimal drawing parameters, when the level of residual damage was not dangerous (fig.6). Experimental and then industrial tests of theoretical results show a validity of prediction.
O" 0.2
-"
-4.0
0.15 0.1
f -8.0
0.05
II
III
IY
Figure 6. Changes of calculated damage co (1) and plactic loosening ~ (2)under drawing of 0.45%C carbon steel pipe: I - III - number of drawing pass; IY- annealing
REFERENCES 1. V.V. Novozl~ov. Applied Mechanic and Mathamatic, v.29, No 4 (1965), 681-690 (in Russia) 2. V.L.Kolmogorov. Stress, Strain, Fracture, M., Metallurgy, 1970, 230 p.(in Russia) 3. A.A.Bogatov, O.I.Mizhiritsky, S.V.Smimov. Reserve of metal plasticity during the treatment of metals by pressure, M., Metallurgy, 1984, 144 p (in Russia) 4. V.A.Ogorudnikov. Estimation of the deformability of metals during the treatment by pressure, Kiev, Vishya Shkola, 1983, 175 p (in Russia) 5. Z.J.Luo,W.H.Ji, N.G.Guo at al. Journal of Materials Processing Technology, N_o 30 (1992), 31-43. 6. B.A. Migachiev. Metals (1994), 3, 52-55 (in Russia) 7. H.Sekiguchi, K.Osakada, H.Hayashi. Journal of Institute of Metals v.101 (1973), 167-174
DAMAGE EVALUATION AND R U P T U R E
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Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
83
The influence of critical defect size in a ceramic of alumina elaborated by process sol-gel route. N.H.Almeida Camargoa; M. Murat b and E. Bittencourt a. aFaculdade de Engenharia de Joinville - UDESC, Campus Universithrio-Bairro Born Retiro 89.203-100 - Joinville - S.C.- Brasil blNSA de Lyon, Laboratoire GEMPPM, URA CNRS n ~ 341, 20, Av. A. Einstein 69621 Villeurbanne. Abstract: Optimizing the conditions of growth of a-alumina during the thermal transformation of a transition alumina prepared by the sol-gel process allows us to obtain a hot-pressed aalumina ceramic with low critical defect size, which presents mechanical characteristics significantly better than those of alumina ceramics obtained from conventional powders.
1. INTRODUCTION The research program has elaborated a ceramic by the process sol-gel route using as precursor organo - mineral a blend of pseudo-boehmite and acetate of aluminum (powder DISPERAL P3 from CONDEA CHEMIE, RFA). The filiation sol-gel obtained was dryed resulting a xerogel, then alumina of transition by thermal processing and a-alumina at T~ and final ceramics by hot-pressing under mechanical charge at T2. The temperature T~ was varied from 1100~ until 1500~ The ceramic resulting hot-pressing at T: = 1550~ during 30rain presented low critical defect size, and mechanical properties significantly superior than of the ceramics of alumina obtained from conventional powders. Observations in scanning electron microscopy (SEM) on the powder processed thermicaUy at T~ = 1300~ put in obviousness a elementary crystals growth constituted of aggregates of t~-alumina and the hot pressed product presented an homogeneous micro structure, with dimensions of elementary grain of alumina not exceeding some microns. The micro structure of the product transformed into corundum at T1 = 1500~ and then hot pressed at T2 = 1550~ put clearly in obviousness the grains growth detected by a significant decreasing mechanical properties, related by an increase of critical defects size.
2. EXPERIMENTAL PROCEDURE
The presem research works on the elaboration of a ceramic of pure alumina prepared by the process sol-gel route, the alumina was optimized characteristic. A blend of pseudo-boehmite and acetate of aluminum (powder Disperal P3 of Condea Chemistry., RFA),
84
transition, which considered in this case a mixture of phases eta and theta. This powder presents itself in the form of aggregates of elementary crystals with size, d <100nm. The obtained powder iscooled at ambient temperature, then placed in an furnace of hot-pressing to under go its first its transformation in alpha alumina by isothermal treatment under vacuum during three hours in a given T~ without application of the mechanical charge. Then the furnace atmosphere was controlled using argon and the mechanical charge (40 MPa) was progressing applied on the material during a linear increasing temperature until T2 > T] to realize the hot-pressing. In the present case T2 = 1550~ with isothermal maintenance of the charge during 30 minutes to avoid a too pronounced growth of grains of a-alumina, what would be deleterious to the obtaition of good characteristic and mechanical properties. A preliminary study of alumina of transition behavior prepared according to previously described protocol had leaned to choose l l00~ as temperature T~, after isothermal processing under void for three hours at this temperature, the pure alumina of transition was totally transformed into corundum [1-2] The surface grains of corundum showed then a very strong increasing, the classic micro structure microporous vermicular (the elementary crystals having a transverse diameter in the order 100 to 200nm) observed by others authors having used others precursor types [3-7]. The mechanical properties of ceramic obtained after hot-pressing at 1550~ ffigures, la and lb) are similar to those of ceramic of alumina prepared by conventional hot-pressing powders (flexural strength generally in order to 350-400 MPa although recently have been obtained values in the order to 560 MPa for samples prepared by conventional process with very fine commercial powders of alpha alumina. In the present case, the microstructure fracture of the material after hot pressed comprises very fine particles (some hundred of nm) dispersed around crystals aggregates of alumina with well superior dimension (some microns)
700 600 E
~ ' 500
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-4
-
o. r~
400
-
300
-
etO
-3
O b-
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I000
I
1100
'
i
'
I ......
' .......
I
1200 1300 1400 Temperature T1 (Oc)
'
I
1500
2
'
1600
Figure 1" Variation, according to the temperature TI, the 4 flexural point strength (curve a) and toughness ( curve b).
85 We tried to improve this micro structure varying the parameter T, from 1100~ until 1500~ This had put in obviousness the existence of a temperature Tt (in the occurrence 1300~ where the ceramic resulted of hot pressing at 1550~ present case maximum value of the flexural 4 point strength fracture (op = 684 MPa, fig. l a) although the value of the report d/dth (apparent density and theoretical density) is only 96%. For Tl = 1300~ was observed equally (fig. 1b) a maximum value of the toughness Ktc (4,12 MPa re'a), and the hardness Hv (15,1 GPa ) but especially a minimum value of the dimension of the critical defect size (x = 1 l~tm, fig. 2) calculated from the relation K,c Y'.CrR.(Tt.X)]/2 with Y chosen arbitrarily equal to unit. =
100 ..,90
8o ~'70 ~
60
N 5o "= r.) 4 0 30 ~ 20 100 ~
~
1000
'
I
1100
'
I
'
I
'
I
1200 1300 1400 Temperature T 1(~
J
I
1500
'"
1600
Figure 2 9Variation, according to the temperature Tt in relation of critical defect size. The observation of the powder obtained at T, = 1300~ (and even that already obtained at 1200~ put in obviousness a growth of elementary crystals constituting alpha alumina aggregates (figure 3a), and the product hot pressed presented a homogeneous micro structure (figure 3b), the elementary grain dimension of alumina not exceeded some jam. For values of T, over 1300~ an incontest process of pre-densification (natural sintering) of powders, associated with the transformation of the alumina of transition. This phenomenon is perfectly put in obviousness by thermal dilatation curves (figs. 4 and 5), powders after processed at Tt ( notable decreasing of the retraction at high temperature when T, was increased). This natural sintering, associate to a exagered growth of alumina, limits the previous action of the hot-pressing. The micro structure of the product transformed into corundum at T, = 1500~ then hot-pressed at T2 = 1550~ put clearly in obviousness this increasing of grains, but the aspect presented by crystals is not favorable to characteristics of the material that presents an important porosity (d/dth = 80,5%) with lair value of the flexural strength fracture (173 MPa) and the toughness (2,83 MPa.m ~'2, figures l a and 1b). The value of critical defect size tbr this material was raised (90~tm).
86
Figure 3a: Microstructure of the powder of alumina sol-gel Tt = 1300~ (SEM)
Figure 3b: Microstructure sol-gel pure alumina strength fracture by a couple of temperature T~/T2= 1300~176 (SEM)
9,
A
Temp6rature (~
0 /
200
400
600
800
1
1400
1600
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........
E
re 9149149 9oo-ei,.'..i" 'l~.~,dMl~:~dp~oQ
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Figure 4: Thermal dilatation curve of the x6rogel powder of pure alumina treated at T~ = 1100~ (curve OABC: linear increasing of the temperature; curve OF: isothermal return at 1500~ A direct thermal processing at 1550~ to realize the hot-pressing without using a constant sintering temperature T~ conducted to the obtainment of a material with large grains (10~tm), high K[c (4,1 MPa.m ~/2) but of small C~R(134 MPa), what results the very high value of the critical defect size (309~tm, figure 2).
87 0 t
A d(~
-5 .1
Temperature (~
o o~0~ o o6~0 o o o 8 ~ o o d ~ 0 ~ ) o o g g ~ a o o ~ 4 ~ d ~ C 1600
~176176 ~
fie O0
E
-10 -15 -20 --25 --30 --
Figure 5" Thermal dilatation curve of the xerogel powder of pure alumina treated to T1 = 1300~ (curve OAC: linear increasing of temperature; curve OF: isothermal return at 1500~
3. CONCLUSION The optimization of the temperature formation of corundum at T~ = 1300~ from a alumina sol-gel behaved, after hot-pressing of this corundum, to a ceramics presenting mechanical characteristic clearly superior than of ceramic prepared from powders of conventional alumina. This result can be interpreted as a better control condition of formation of the alpha alumina, from a powder of alumina of transition prepared itself by thermal processing of a xerogel of alumina [10], and the homogenization of the micro structure (figure 3a and 3b), and diminution of the critical defect size (figure 2).
REFERENCES 1. M. Murat, F. Mignard, F. Hue et G. Fantozzi, Elaboration des poudres c~ramiques composites "Alumine/Whiskers SiC par proc6d6 sol-gel. L'ind. C6rm., n ~ 892, (1994), p. 236-238. 2. M. Murat, N.H.A. Camargo et F. Sorrentino, Stabilit6 thermique d'une alumine de transition: effet de l'incorporation pr6alable d'une poudre nanom6trique de carbure de silicium. C.R. Acad. Sci., Pads, 318, s6rie II, (1994), p. 611-614. 3. G. Varhegyi, J. Fekete et M. Gemessi, Reaction kinetics and mechanism of ot-A1203 formation. Compte Rendu du 36me Congr6s International de I'I.C.S.O.B.A., Nice. (1973), Sedal Editeur, p. 575-584. 4. F.W. Dynys, J.M. Halloran, Alpha Alumina formation in alum-derived Gamma alumina. J. Amer. Ceram. Soc., 65, (1982), p. 442-448.
88 5. F.W. Dynys, M. Ljungberg et J.W. Halloran, Micro structural transformations in alumina gels. in Bitter Ceramics trough Chemistry, Materials Research Society, vol. 32, (1994), p. 321-326. 6. J.E. Blendell, H.K. Bowen et R.L. Coble, High purity alumina by controlled precipitation from Aluminum sulfate solutions. Ceram. Bull., 63, (1984), p. 797-801. 7. S. Rajendran, Production of ultrafine alpha alumina powders and fabrication of fine grained strong ceramics. J. of Mat. Sic., 29, (1994), p. 5664-5672. 8. L.M. Sheppard, Enhancing performance of ceramic composites. Ceram. Bull., 71, (1992), p. 617-631. 9. J. Zhao, L.C. Stearns, M.P. Hamer, H.M. Chart and G.A Miller, Mechanical behavior of almnina-silicon carbide nanocomposites. J. Amer. Ceram. Soc., 76, (1993), p. 503-510. 10. N. H. Almeida Camargo et M. Murat Caract6ristiques m6caniques d'une c6ramique d'alumine prdparde par le procdd~ sol-gel: influence des conditions de formation de l'almnine alpha. C.R. Acad. Sci., 319, s6rie II, (1994), p. 893-897.
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
89
Defect evolution during machining of brittle materials A. Chandra, K. E Wang, Y. Huang and G. Subhash Department of Mechanical Engineering and Engineering Mechanics Michigan Technological University, Houghton, M149931 ABSTRACT A simple stress based defect evolution model is developed to assess the influence of various process paramters on material removal rate (MRR) and induced damage during ceramic grinding processes. Model predictions for normal and lateral damage zones under normal indentations are first compared to fracture models as well as experimental observations on pyrex glass. The proposed model is then extended to simulate oblique indentation events depicting abrasive gritworkpiece interactions during ceramic grinding. It is also easily extendable to real grinding situations involving multiple interacting abrasive grits. Process design options for reducing induced damage in the finished part, and increasing MRR are considered next. In particular, the potential of a new design avenue involving intermittent unloading is investigated. For pyrex glass, it is observed that intermittent unloading can facilitate significant increase in force per abrasive grit without increasing the associated surface and sub-surface fragmentation in the finished part. This design feature may enable significant increase in MRR, while maintaining a very low level of process induced damage in the finished product. 1. INTRODUCTION Currently, precision ceramic components are being used ever increasingly in various engineering applications. Compared to metals, however, most ceramics (e.g., glass, SiC, Si3N4, ZrO 2, CBN, sapphire, spinel, etc.) are also much harder and much more susceptible to brittle fracture. These advanced ceramic components, in most cases, require finishing operations to achieve surfaces of required geometry, tolerance and finish, while maintaining the desired level of strength. Finishing operations, e.g., grinding, typically induces surface and sub-surface damages in the finished part, that can severely degrade its strength and useful life under service conditions. Hence, very expensive free abrasive polishing is necessitated for such precision ceramic parts. Accordingly, finishing operations typically account for 25-80% of the total cost of many precision ceramic components. It is observed (e.g., Eckert and Weatherall 1990, Office of Technology Assessment 1988) that process induced damage and high finishing cost pose two significant barriers against commercial success of precision structural ceramic components. Jahanmir et al (1992) provide a review and assessment of current practices and research needs in precision finishing of ceramic components. Brittle grinding makes the finished surface susceptible to induced surface and sub-surface damages and the strength of finished parts have been observed to be reduced by 30-60% (Jahanmir et al 1993). Accordingly, the present work focuses on first developing a simple stress based model that will be capable of depicting defect evolution at high strain rates (103 - 10S/s) that are representative of a machining process. The abrasive grit - workpiece interaction in grinding is modeled as an indentation event subject to normal and tangential loading. Model simulations for
90 normal and lateral damage under normal indentations are compared against fracture models, and verified against experimental observations on pyrex glass. The validated model is then utilized to further explore the "process design space", and identify potential design parameters for process improvements. Intermittent unloading during chip formation is identified as a keydesign variable, and its effects are investigated in detail. Following introduction, existing literature on scratch tests and indentation fracture mechanics models for brittle materials are briefly reviewed in the background section. This is followed by development of the proposed model, and its validation against experimental observations from static and high strain rate normal indentations on pyrex glass specimens. The proposed model is then utilized, in particular, to investigate the effects of intermittent unloading on defect evolution and material removal mechanisms in ceramic machining processes. 2. DEFECT EVOLUTION MODEL AND VALIDATION A single chip, in a ceramic grinding process, is typically formed in 10-100 ~ts. This time gets even shorter for modem high speed grinding at 60,000-150,000 rpm (Kovach et al 1996, Ashley 1995, O'Connor 1995). As a result, strain rates approach 103-105/s. Accordingly, the present work focuses on depicting defect evolution in brittle ceramics under high loading and strain rates. It is important to note here, that most process induced damage in ceramic grinding is observed (e.g., Jahanmir et al 1992, 1993, Allor et al 1993, Xu and Jahanmir 1994, 1995a, b, Xu et al 1995) to be less than 1 mm in size. Typical dilational wave speeds are about 104 m/s in ceramics and 23x103 rn/s in glass. In view of this, the time needed for the wave to propagate 1 mm is expected to be <1 Its, which is only a fraction of the imposed pulse-time. Accordingly, as a first attempt, elastodynamic effects are neglected in this paper. The present model assumes that micro-defects always exist in brittle materials. Details of the damage model is presented in Chandra et al (1997). 2.1. Estimation of Damage Zones: A simple stress based model of the form:
Iol ->
(1)
is proposed for estimations of different damage zones induced by a single grit indenting on a workpiece. In Eq. (1), I~1 is a suitable norm of the induced stress field due to the action of the abrasive grit on the workpiece. As warranted, I~1 may be identified with magnitudes of appropriate stress components, principal stresses, or other invariants. 6f(g) is the rate dependent failure strength of the workpiece material, and admits a failure strength of the material that has been observed experimentally at the apprpriate strain rate regime. For a completely elastic solution under a concentrated load, a stress singularity exists at the point of contact between the abrasive grit and the workpiece. Hence, the damage zone will initiate there, and propagate outward as long as 16[ is greater than the intrinsic rate dependent failure strength 6f(~) of the material. This finite zone, is then identified as the corresponding damage zone. In the present work, we focus on damage evolutions normal to the finished surface (median and radial damage), and parallel to the finished surface (lateral damage). The normal damage zone represents the effects of median and radial cracking (and induced secondary cracking), while the lateral damage zone represents the extent of lateral cracking. It is interesting to note, that normal
91 damage is typically left behind in the finished part, and is responsible for its strength degradation. The lateral damage, on the other hand, facilitates material removal (Cook and Pharr 1990). Thus, they may be benevolent, and aid in enhancing the material removal rate (MRR). Details of the damage model is presented in Chandra et al (1997). The experimental scheme aims at delineating the effects of various material and process parameters influencing the material removal and damage evolution mechanisms during ceramic grinding processes. A set of static normal indentation tests at peak loads of 5-100 N is carried out first~ A modified split Hopkinson pressure bar (also called Kolsky bar) facility with a momentum trap is used for indentation experiments involving high loading rates (Subhash and Nemat-Nasser 1993, Koeppel and Subhash 1997, Koeppel et al 1997). 2.2. Model Predictions and Validations for Normal Indentations: For the present work, tensile failure strength of pyrex glass is assumed to be 20% of its compressive failure strength. For model simulations, the tensile failure strength is taken to be 200
MPa for nominal strain rates less than 103/s, and is assumed to increase linearly to 400 MPa for a strain rate of 104/s. For simulations using a fracture model (e.g., Lawn et al 1980, Evans and Marshall 1981, Chiang et al 1982a,b), Klc of pyrex glass (typically in the range of 2.7 - 3.3 MPam 1/2) is assumed to be 3 MPa-m 1/2. Fig. 3 shows comparisons of simulated results with experimental observations for fully unloaded pyrex glass specimens that have experienced different peak indentation loads. Predictions of normal damage zone depths (shown in Fig. 2a) based on the proposed simple stress-based model agree quite well with the median crack size estimates obtained from a fracture model (e.g., Lawn et al 1980, Evans and Marshall 1981, Chiang et al 1982a,b, Marshall 1984, Ritter et al 1984, 1985, Hu and Chandra 1993). The normal damage depths were measured experimentally by polishing a window on the side of the glass specimens. Two sets of experimental data are shown in Fig. 2a. It may be observed that model predictions also agree well with the experimental observations. Fig. 2b shows the comparisons of simulated surface traces of normal damage with experimental observations. In this case, the model seems to over-predict the surface trace extensions. The model assumes the surface traces of median cracks. Experimentally observed surface traces may be due to median or radial cracks, and this may explain the discrepancy between model predictions and experimental observations. Lateral damage occurs due to unloading, and Fig. 2c shows the comparisons of lateral damage zone predictions with experimental observations under fully unloaded configurations. It may be observed that model predictions for lateral damage zone sizes agree very well with the experimental observations at fully unloaded configurations. Upon indentation loading, normal damage is initiated. The normal damage zone grows bigger in size and penetrates deeper with increasing load, finally reaching its maximum depth at the peak load. At this instant, no lateral damage is observed, and the surface trace of normal damage is also usually very small. Thus, the normal damage zone evolves as a series of elongated ellipses (with major axis oriented along the loading direction) during the loading phase. Upon unloading from the peak indentation load, the depth of normal damage remains almost constant, however, its surface trace gradually extends, approaching a circular shape under complete unloading. The evolution of normal damage in soda-lime glass during a complete loading-unloading cycle is shown in Fig. 3. The model predictions are then compared to experimental observations of
92 Marshall and Lawn (1979) on the evolution of a median crack in annealed soda-lime glass slabs (H = 5.5 GPa, Kie = 0.75MPa-m 1/2) during a complete loading-unloading cycle. For the damage model, tensile failure strength of soda-lime glass is taken to be 50 MPa. At the peak load of 90N, the damage model predicts a normal damage zone depth of 0.3 mm, while the fracture model predicts 0.34 mm. The experimental data (Marshall and Lawn 1979) ranges from 0.214 mm to 0.363 mm. Thus, the model predictions are within the range of experimental observations. It is particularly interesting to note the non-self-similar nature of normal damage growth under unloading. Typically, fracture models assume a half-penny shaped crack front, and its predictions are restricted to self-similar circular crack-fronts. A damage, model, however, can fully capture such effects. Upon unloading, the simulation results predict extensions in surface traces of the normal damage zone, while its depth of penetration is held constant. Such a trend is also evidenced by crack arrest marks observed experimentally by Marshall and Lawn (1979). Their experimental data (Fig.6 in Marshall and Lawn 1979) shows a fully unloaded aspect ratio (depth/ surface trace) of about 0.5, while the present simulations predict 0.52. Similar to experimental observations, the proposed stress-based model also predicts lateral damage initiation only upon unloading beyond a certain threshold (that depends on the tensile failure strength of the material). With continued unloading, the lateral damage zone propagates parallel to the free surface, reaching its maximum size at the fully unloaded configuration. 3. DESIGN SPACE EXPLORATION Utilizing the validated model, several issues pertaining to process design of a ceramic grinding operation are investigated in this section. In Figs. 2 and 3, the normal force on the indenter is correlated with various damage zone estimates. In view of these results, it may be observed that Peak Force/Abrasive Grit is a key parameter effecting defect evolution during a ceramic grinding process (Bifano et al 1991, Subramanian et al 1996, Kovach et al 1996). In the present work, we focus on identifying additional variables, that may significantly effect the product quality and economy (through Material Removal Rate) in ceramic grinding processes. 3.1. Potential of Intermittent Unloading: It is particularly interesting to note from the above investigations, that detrimental normal damage nucleates and propagates in depth under loading, while benevolent lateral damage is primarily facilitated by unloading. Upon unloading, the normal damage zone has also been observed to propagate in the transverse direection with extending surface traces, while maintaining constant depth of penetration. This indicates that process induced damage (left behind in the finished part) in a ceramic grinding process is typically induced as the abrasive grit penetrates the workpiece and reaches its full indentation load. By contrast, the material removal mechanism (of lateral damage) is initiated only upon unloading from the peak load, and material is removed as the indenter gradually leaves contact with the workpiece. Thus, the loading
produces the undesired product damage and strength degradation, while unloading provides the desired material removal. In a conventional grinding process, however, loading always precedes unloading. As Force/Grit is increased gradually from zero, normal damage is initiated first, and propagates to its maximum depth at peak load. Only later, upon unloading, lateral damage responsible for material removal is initiated. Thus, in a traditional grinding operations on brittle materials, detrimental normal damage is usually fully developed, even before material removal mechanism of lateral damage has had a chance to initiate. It is important to note here, that one does not necessarily need to wait for unloading from the
93 peak pulse load to initiate lateral damage. Unloading, from a threshold load (dependent on the failure strength of the material) is only required to induce lateral damage in the material. Accordingly, the proposed intermittent unloading technique attempts to achieve the following: 1. Initiate the material removal mechanism of lateral damage early in the process, before normal damage is fully developed~ 2. Utilize the shielding effects of lateral damge to retard and potentially eliminate normal damage from the finished product. This shielding effect is shown schematically in Fig. 4. It is interesting to note that the lateral crack (or damage), initiated by the intermittent unloading, acts as a barrier against further penetration of normal crack (or damage) upon reloading. Given a fixed load Pint from which unloading has occured, this shielding effect is effective until a threshold Pshield is exceeded in reloading. The effect of intermittent unloading from an indenter load of Pint, is shown in Fig. 5. In order to reduce process induced damage in the finished part, an additional intermittent unloading step from Pint is introduced within a typical load-pulse during normal indentation prior to reaching the peak indenter load of Pmax. The load pulse is shown as inset in Fig. 5. It is observed that intermittent unloading establizes a lateral damage zone before normal damage becomes fully developed. The size of this lateral zone depends on the magnitude of Pint- Upon reloading beyond Pint, the normal damage zone would normally attempt to grow in size, and penetrate to greater depth. Unlike a traditional case, however, the pre-existing lateral crack or damage (generated from prior unloading) now provides a shielding effect, and essentially annihilates any singularilty associated with median/radial crack or normal damage. As a result, the normal damage zone cannot penetrate to greater depth until (under continued increase in load) it's lateral size has grown bigger than the size of the lateral damage zone induced by Pint- The corresponding indenter load at this instant is denoted as Pshield. Thus, upon reloading followed by intermittent unloading from Pint, the normal damage zone is inhibited from penetrating to greater depth so long as the indenter load does not exceed Pshield" Fig. 5 shows a typical case, where unloading from a Pint = 75N resulted in a Pshield exceeding 250N. Fig. 6 graphically presents the effects of intermittent unloading, and resulting interactions of lateral and evolving normal damage zones in the load range Pint < P < Pshield- Defining AP = Pshield--Pint' the workpiece can carry an additional indenter load AP beyond Pint without any further increase in normal damage depth. Upon exceeding Pshield, the normal damage penetrates to higher (as predicted for monotonous loading) depth, and the shielding effect of intermittent unloading is lost. The present investigation shows that the effect of intermittent unloading remains the same over a wide range of indentation loads (0.1N to 300N), and results in a Pshield that is about 3*Pint. This offers a potential avenue for increasing Force/Grit (and associated MRR) in grinding of ceramics, without any associated increase in process induced damage in the finished product.
3.2. Oblique Indentation Events: Both normal and tangential loads are significant in real life grinding processes. Accordingly, implications of intermittent unloading under more realistic oblique indentation events are investigated in this section. Single- and multi-grit scratch tests show that normal load (Pn) / tangential load (Pt) ratios typically vary from about 1 for sharp well dressed wheels to about 2 for
94 worn and loaded wheels (e.g., Hockey and Rice 1979, Jahanmir 1993). Accordingly, oblique indentation simulations are conducted with Pn/Pt = 0.5, 1.0 and 2.0, at fixed resultant loads (R). As a first approximation, it is assumed that Pn/Pt remains fixed throughout the chip formation cycle. Fig. 7 shows the contours of normal damage zones for oblique indentations for different ratios of Pn/Pt. As expected, the normal damage zone is skewed. It is interesting to note, that the fundamental nature of defect evolution remains the same as that for normal indentation. Normal damage is initiated and propagated under loading, while lateral damage is initiated and propagated under unloading. The qualitative nature of the normal damage evolution also remains the same. Fig. 8 shows the effects of intermittent unloading for oblique indentation events. Again, the qualitative nature of the shielding characteristics remains the same as that observed under normal indentation. For a given Rint, the actual value of AR = Rshield- R int is now dependent on the Pn/Pt ratio. It is observed that (for 0.1N < Rint < 300N), AR varies from 0.26*Rin t to 1.05*Rint as Pn/Pt is varied from 0.5 (corresponding to single point cutting) to 2.0 (worn grinding wheel). For a sharp grinding wheel (Pn/Pt = 1.0), a AR of 0.55*Rin t is observed. Thus, the benevolent effects of intermittent unloading, observed under normal indentations, seems to carry forward to realistic grinding situations. 4. DISCUSSION AND CONCLUSION A very simple stress based model is proposed in the present work to represent defect evolution during grinding of brittle materials such as glasses and ceramics. For grinding, the zone of interest is usually restricted to a depth of less than 1 mm from the free surface. A strain rate dependent static model is a reasonable approximation of the essentially dynamic event, as long as the loadpulse time is much larger than the time taken by the dilatational wave to travel the depth of interest (usually <1 mm) in the workpiece material. Thus, the proposed model is valid for a wide range of brittle glasses and ceramics at moderately high grinding wheel velocities. Due to inherent advantages of transparency and well characterized properties, pyrex glass is chosen as a demonstration material. The model predictions are first validated against experimental observations for static as well as dynamic (load-pulse durations of 10-100 ~ts) normal indentation events. Strain rate effects are accounted for via a rate dependent failure strength of the material. Normal indentation with a standard Vickers indenter is chosen to isolate the effects of different variables, and due to the existing literature on static tests under such configurations. The correlation between model predictions and experimental observations (Figs. 2 and 3) establishes the validity of the proposed model. Such a validated model may be utilized to ensure scalability and migratability of experimental data and varied circumstances. The damage model is then utilized to investigate the potential of a new design avenue involving intermittent unloading. It is observed that a combination of Force/Grit and intermittent unloading may be utilized to effectively and efficiently minimize any residual damage in the finished product, while ensuring a high material removal rate. This can be carried out under brittle fracture mode of material removal, and intermittent unloading from a Force/Grit of Pint can offer very effective shielding against further process induced residual damage. This, in turn, can facilitate higher MRR without any further increase in process induced residual damage. The proposed stress based defect evolution model is also extended to realistic oblique indentation events, and the effectiveness of intermittent unloading under such conditions is investigated. It is observed that fundamental observations remain the same as those under normal
95
indentation, while the quantitative values for different threshold values are altered due to obliqueness of the indentation events. It can also be extended to represent cutting actions of multiple interacting abrasive grits. Such simulations, as well as experimental work on single- and multi-grit oblique indentations and scratch tests are currently in progress. ACKNOWLEDGEMENTS The authors gratefully acknowledge the financial support provided under Grant No. D M I 9610454 of the U.S. National Science Foundation. They are also thankful to Mr. Brian J. Koeppel and Mr. Richard Anton for their kind help with the indentation experiments. REFERENCES Allor, R. L., Whalen, T. J., Baer, J. R. and Kumar, K. V., 1993, Machining of Silicon Nitride: Experimental Determination of Process/Property relationships, Machining of Advanced Materials, NIST Pub. 847, pp. 223234. Ashley, S., 1995, High-Speed Machining Goes Mainstream, Mech. Eng., ASME, May 1995, pp. 56-61. Bifano, T. G., Dow, T. A., and Scattergood, R. O., 1991, "Ductile Regime Grinding: A New Technology for Machining Brittle Materials," ASME J. of Engineering for Industry, vol. 113, pp. 184-189. Chandra, A., Wang, K. P., Huang, Y. and Subhash, G., 1997, Role of Unloading in Machining of Brittle Materials, J. Mfg. Sc. Eng., ASME (submitted). Chiang, S. S., Marshall, D. B. and Evans, A. G., 1982, The Response of Solids to Elastic/Plastic Indentation: (a) Stresses and Residual Stresses, and (b) Fracture Initiation, J. Appl. Phys., vol. 53, pp. 298-317. Cook, R. F. and Pharr, G. M., 1990, Direct Observation and Analysis of Indentation Cracking in Glasses and Ceramics, J. Am. Ceram. Soc., vol. 73(4), pp. 787-817. Doyle, E. D. and Dean, S. K., 1980, "The fine grinding of glass and ceramics using conventional wheels," The Science of Ceramic Machining and Surface Finishing n (B. J. Hockey and R. W. Rice, eds.), NBS Special publication 562, pp. 93. Eckert, C. and Weatherall, J., 1990, Advanced ceramics: 90's global business outlook, Ceramics Industry, April issue, pp. 53-57 Espinosa, H.D., Raiser, G., Clifton, R.J., and Ortiz, M., 1992, "Experimental Observations and Numerical Modeling of Inelasticity in Dynamically Loaded Ceramics," J. Hard Mater. 3, [3-4] 285-313. Evans, A. G. and Marshall, D. B., 1981, "Wear mechanisms in ceramics," Fundamentals of Friction and Wear of Materials, (D. A. Rigney, ed.), ASME, New York, USA, pp. 439. Gioia, G., and Ortiz, M., 1996, The Two-dimensional Structure of Dynamic Boundary Layers and Shear Bands in Thermoviscoplastic Solids, J. Mech. Phys. Solids, vol. 44(2), pp. 251Grady,D.E., and Lipkin,J., 1980, Criteria for Impulsive Rock Fracture," Geophys. Res. Lett, 7, 255-258 Hu, K. X. and Chandra, A., 1993, "A fracture mechanics approach to modeling strength degradation in ceramic grinding processes, J. Eng. Ind. ASME, 115, pp. 73-84. Jahanmir, S., Ives, L. K., Ruff, A. W., and Peterson, M. B., 1992, "Ceramic Machining: Assessment of Current Practice and research Needs in the United States," NIST Special Publication # 834. King, R. I. and Hahn, R. S. (eds.), 1986, Handbook of modern grinding technology, Chapman and Hall, New York, USA. Koeppel, B. J. and Subhash, G., 1997, "A Novel Technique to Determine Dynamic Indentation of Materials," Experimental Techniques (in press). Koeppel, B. J., Subhash, G. and Chandra, A., 1997, "Dynamic Indentation Hardness of Metals and its Implications to High Speed Machining (submitted). Komanduri, R. and Maas, D. (eds.), 1985, "Proceedings of Milton C. Shaw Grinding Symposium," ASME, PED16, New York, USA. Kovach, J. A., Laurich, Ziegler, K. R., Malkin, S., Sunderland, J. E., Guo, C., Zhu, B., Ganesan, M., 1996, Development of advanced grinding technology for structural ceramics, Proc. NAMRC, pp. 51-56. Lankford, J., 1981, Mechanisms responsible for Strain Rate Dependent Compressive Strength in ceramic Materials, J. Am. Ceram. Soc., vol. 64, c33-c34.
96
Lankford, J. and Blanchard, C.R., 1991, Fragmentation of Brittle Materials at High rates of Loading, J. Mater. Sci, 26, 3067-3072. Lankford, J., 1989, Dynamic Compressive Fracture in Fiber-reinforced Ceramic Matrix Composites, Mater. Sci. Eng. A 107, 261-268. Lankford, J. and Blanchard, C.R., 1989, "Response of Whisker-Reinforced Ceramic Matrix Composites to dynamic Compressive Loading," Mater. Sci. Eng. A107, 261-268. Lawn, B. R. and Evans, A. G., 1980, "Elastic-plastic indentation damage in ceramics: the median/radial crack system," J. Am. Ceram. Soc., Vol., 63, no 9/10, pp. 574-581. Longy, F. and Cagnoux, J., 1989, Plasticity and Microcracking in Shock Loaded Alumin, J. Am. Ceram. Soc., 72 [ 16], 971-979. Malkin, S., 1984, "Grinding of metals: theory and applications," J. App. Metalworking, vol. 3, no. 2., pp. 95. Malkin, S., 1989, Grinding Technology: Theory and Applications of Machining and Abrasives, Ellis Horwood Pub, Chichester, UK. Malkin, S. and Anderson, R. B., 1974 "Thermal aspects of grinding: part 1-energy partition," ASME J. Eng. Ind., vol. 96, pp. 1177-1182 Malkin, S. and Ritter, J. E., 1989, "Grinding mechanisms and strength degradation for ceramics," ASME J. Eng. Ind., vol. 111, pp. 167-174. Marshall, D. B., 1984, "Geometrical effects in elastic-plastic indentation," J. Am. Ceram. Soc., vol. 67, no. 1, pp. 57-60. Marshall, D. B. and lawn, B. R., 1979, Residual effects in sharp contact cracking, J. Mat. Sc., vol. 14, pp. 20012012. Nemat-Nasser, S., Isaacs, J. B. and Stan'ett, J. E., 1991, Hopkinson Techniques for Dynamic Recovery Experiments, Proc. R. Soc. Lond., A435, pp. 371-391. O'Connor, L., 1995, Machining with Super-fast Spindles, Mech. Eng., ASME, May 1995, pp. 62-64. Office of Technology Assessment, 1988, Advanced Materials by Design, U. S. Congress, OTA-E351, U. S. Govt. Printing Office, Washington, DC. Ravichandran, G. and Subhash, G., 1995, A Micromechanical Model for High Strain Rate Behavior of Ceramics, International Journal of Solids and Structures, 32 [ 17/18] 2627-2646. Subhash, G. and Nemat-Nasser, S., 1993, Uniaxial Stress Behavior of Y-TZP, Journal of Material Science, 25 59495952. Subramanian, K. and Keat, P. P., 1985, "Parametric study on grindability of structural and electronic ceramics - part 1," Machining of Ceramic Materials "and Components, (K. Subramanian and R. K0manduri, eds.), PED-vol. 17, ASME, New York, pp. 25. Subramanian, K., Ramnath, S. and Redington, P. D., 1996, Norton Co. report. Suresh, S., Nakamura, T., Yeshurun, Y., Yang, K.-H., and Duffy, J., 1990, Tensile Fracture Toughness of Ceramic Materials: effects of Dynamic Loading and Elevated Temperatures," J. Am. Ceram. Soc., 73 [8] 2457-2466. Tennery, V. J., 1994, ORNL Foreign Trip Report, ORNL,/b'TR-5170dated October 25, 1994 Tennery, V. J., 1993, ORNL Foreign Trip Report dated October 22, 1993. Xu, H. H. K. and Jahanmir, S., 1995a, Microfracture and Material Removal in Scratching of Alumina, J. Mat. Sc., vol. 30, pp. 2235-2247. Xu, H. H. K. and Jahanmir, S., 1995b, Scratching and Grinding Of a Machinable Glass-Ceramic with Weak Interfaces and rising T-Curve, J. Am. Ceram. Soc., vol. 78(2), pp. 497-500. Xu, H. H. K. and Jahanmir, S., 1994, Simple Technique for Observing Subsurface Damage in Machining of Ceramics, J. Am. Ceram. Sot., vol. 77(5), pp. 1388-90. Xu, H. H. K. and Jahanmir, S. and Wang, Y., 1995, Effect of Grain Size on Scratch Interactions and Material Removal in Alumina, J. Am. Ceram. Soc., vol. 78(4), pp. 881-891. Yoffe, E. H., 1982, Elastic Stress Fields Caused by Indenting Brittle Materials, Philos. Mag., A, vol. 46, pp. 617628.
97
0.~
. . . .
I
. . . .
I
. . . .
I
. . . .
o.,o~ /
~O.OJ
9 .
I
/
9 I
0.00 0
. . . .
5O
I
I
. . . .
100
Figure 1: Schematic of Indentation Crack Systems.
I
I
. . . .
1.,50
. . . .
I
. . . .
~,,.,~.~
I
. . . .
I
I
2O0
.~
. . . .
250
300
IndentationLoad(N)
(a)
,•
0.60
....
I ....
t ....
I ....
I ....
I .... .
0.50
0.5
~meI ru~nmuB
D~l%ml~ ~
0.40
~l~a
Ke..~L
IS~q~
!
i
9
11.30
9
9
Z 0.211
i
0,0
~
-0,5
-1,0
0.0
!
9
0.5
DIISe
i
1,0
I
0.00
Figure 3: Comparisons of Model Predictions to Experimental Observations (Fig. 6 in Marshall and Lawn 1979).
Normal_ \X ~,
,~
0.25
~
0.20
I
I
200
250
300
9 9
0.15
~. 0
/
o,o
F ~ I
0.05
/ / /
. _
I
150
e~
Dama,,r \"
Lateral Datnago Zone Slze upo~t Unloadi,g
I
100
(b)
P
/
1
50
IndentationLoad(N)
0.30
l
Model
0.10
.,..
t
0.00
.... 0
/ .... 50
I .... 100
I .... 150
I .... 200
I .... 250
300
IndentationLoad(N)
(c) Figure 4: Schematic of Interactions of Lateral and Evolving Normal Damage Zones upon Reloading after Unloading.
Figure 2: Comparisons of Model Predictions to Experimental Observations under Fully Unloaded Configuration: (a) Depth of Normal Damage Zone, (b) Surface Trace of Normal Damage Zone, (c) Size of Lateral Damage Zone.
98
0.35
....
,~
0.30
b~
0.25
~
0.21)
I ....
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I ....
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N oA5 0,1
Z o.lo ~ -~ 0.05 0.00
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,
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1
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0.'t
,
,
0,2
(a)
Figure 5" Effect of Intermittent Unloading from Pint on Depth of Penetration of Normal Damage upon Subsequent Loading tO Pmax.
700
,
0.0
. . . .
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~,00 0,1
.~ 300 ga
200 100
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.
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9
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300
9
0.1
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,
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(b)
Pint (N) Figure 6: Variation of AP = Pshield - Pint with Pint.
3.50
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. . . .
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. . . .
I
. . . .
"1
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50
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200
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Rint (N) Figure 8: Variation of AR -- Rshield - Rint with Rint for Pn/Pt=0.5, 1.0, 2.0.
300
Figure 7: Contours of Normal Damage Zone under Oblique Indentation at R=200N, (a) Prd Pt=0.5, (b) Pn/Pt=l.0, (c) Pn/Pt=2.0.
,
,
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini(Editors) 9 1997 Elsevier Science B.V. All rights reserved.
99
Modeling the Influence of Gradients in Strength on the Evolution of Damage in Metals P.R. Dawson a, D.J. Bammann b, and b.A. Mosher b a Cornell University, Ithaca, New York, USA b Sandia National Laboratories, Livermore, California, USA
Abstract The growth of voids during the ductile extension of a notched bar is examined using numerical simulation. The evolution of the strength is assumed to depend not only on the strength itself, but also on the spatial gradients of its distribution. Through the dependence of the void growth on the material strength, the gradients in strength impact the void fraction by limiting the strength in zones of highest stress.
1.0 I N T R O D U C T I O N The concept of continuum damage was first introduced by Kachanov [1] to describe the effects of an isotropic distribution of spherical voids on plastic flow. In this approach, a state variable, termed damage, is introduced to describe the degradation of the material due to the presence of the voids. The damage variable is introduced into the flow rule in such a manner as to effectively concentrate the stress, thereby enhancing the plastic flow. This approach has been utilized and extended by many authors since its introduction (see [2] for a review). The basic difference between many of these models results from the proposed time evolution of the damage. This evolution is generally based upon either analytical or numerical micromechanical simulations of the ductile growth of voids in an inelastic matrix, the differences stemming from the varying assumptions in the micromechanical modeling. In all models the damage growth rate has a strong dependence upon the stress triaxiality, or the ratio of the tensile pressure to the material strength or hardness. The model used in the following analyses is an internal state variable model with one scalar state variable introduced to represent the hardening associated with the dislocation density and another scalar variable introduced to account for the softening associated with the presence of microdefects, such as voids and commonly referred to as damage. Many dislocation based plasticity models are based upon the Taylor assumption that the flow stress is proportional to the square root of the forest dislocation density. In fact, we can introduce a state variable, ~, which represents the hardening of the material to the
100 increased dislocation density as a -
2Gbx/~
(1)
Where, p represents the density of dislocations, G is the shear modulus and b is the Burger vector of the active dislocation systems. These types of models, coupled with time evolution equations to predict the creation, multiplication and annihilation of dislocations have been used successfully to describe many observed material phenomena. State variable models based upon dislocation density have been shown to accurately predict observed material response such as creep, stress relaxation, and other rate and temperature history effects. In this paper, we introduce a nonlocal term into a well established model by adding a spatial gradient in the evolution of the hardening. One physical reason for introducing such a term is an attempt to describe the coalescence of two voids. Several approaches may be pursued in attempting to model this phenomena. For example, one could reason that two voids in close proximity tend to grow towards each other, and therefore a force of attraction between the two voids could be introduced through a gradient of void density or damage. This would lead to a non standard or negative diffusion term in the evolution of the damage. We have investigated this approach but concluded that neither the results nor the physical motivation behind this type of model are very satisfying. In actuality, the voids are not attracted to each other. Instead, the material between two large voids in close proximity experiences a large stress concentration. This results in an increased density of dislocations and a significantly reduced dimension for the dislocation substructures which form. Dislocation spacing will finally reduce to a level at which the repulsive force between dislocations becomes important. This can be modeled at a macroscopic length scale by the introduction of a gradient in the dislocation density in the substructure, or therefore, the hardness. The net result is the spatial spreading of the hardness resulting in a lower magnitude of strength between the two voids. Since the damage growth depends upon the ratio of the tensile pressure to the strength or hardness of the material, the evolution of damage will be enhanced simulating the coalescence of the voids. Therefore, the introduction of a spatial gradient in the evolution of hardening should result in a concentration of damage. Another motivation for introduction of a spatial gradient term in the evolution of the dislocation density is discussed in a proposal by Bammann and Aifantis [3] in which they require the state variables associated with dislocation density to satisfy complete balance laws. This leads naturally to the appearance of a spatial gradient in the equation describing the evolution of the dislocation density. Coupled with the relationship between the dislocation density and the hardness given in Eql, we therefore include a spatial gradient in the temporal evolution of the strength, kappa. The inclusion of this gradient term introduces a length scale into the continuum. This length scale is critical when addressing problems which occur over very small distances, where the assumptions associated with local continuum mechanics are less appropriate. Gradient theories can be related to the more classic nonlocal theories in which a material property at a point is assumed to depend not only on the current state at that point, but also on the state of the surrounding neighborhood. This formulation results in an integral expression in which the effect of the neighborhood is incorporated by an appropriate choice of the kernel in the integral. Expanding this kernel in a Taylor series results in a series of increasing spatial
101 gradients which can be truncated to any level of approximation required. The importance of the inclusion of this term and the resulting length scale which is introduced into the continuum on problems associated with localizations of porosity and strain are some of the issues we will address in this paper. A review of nonlocal and gradient type models of plasticity can be found in [4].
2.0 P L A S T I C F L O W 2.1 B a l a n c e Laws The motion of the body is governed by the equations of equilibrium
V.tr=0
(2)
where inertia and body forces are neglected and by conservation of mass
v.
=0
(3)
where the material is assumed to be incompressible. Here, cr is the Cauchy stress, and u is the velocity field. Boundary conditions consist of imposed velocity or imposed tractions. 2.2 Y i e l d C o n d i t i o n and F l o w Law The plastic flow of the metal at fixed state is described by two constitutive equations, the yield condition and the flow law. We assume isotropy, and write the yield condition simply as
a' = T'(O, D', n, '~)
(4)
and the flow law as D' -
3D'
(5)
,7'
Here, ,7' is the deviatoric part of the Cauchy stress, D ' is the deviatoric part of the deformation rate, T' is the flow stress, 0 is the temperature, n is a scalar state variable describing the strength, and 9 is a scalar state variable describing the damage level, a' and D' are effective values of tr' and D', respectively. (a' - [3tr' 9tr'] 89," D' - [~D2,. D'] 89 A number of models have been proposed that fit into this form, and here we employ a variation of the model proposed by Hart [5]. In this model the flow stress is the sum of two contributions, given by
r
~'~ + ~-'P,
(6)
where
D') 1/M a'-
aoOX
and
(8)
102 The flow stress increases with straining due to an increase in the dislocation density since the dislocations act as barriers. The hardness, a, quantifies the barrier strength in an average (isotropic) sense. The stress in the frictional (first) element is usually much smaller than that in the plastic (second) element except at high strain rates or low temperatures. Here, G is the elastic shear modulus and the variables a* and D* depend on the temperature and hardness as shown, with ao, Q~o, fo, Qo, m, M , and 7 being model parameters.
3.0 E V O L U T I O N
OF STRENGTH
AND DAMAGE
Through the course of a deformation history the microstructure evolves. This evolution is quantified by the state variables, whose changing values subsequently alter the mechanical properties. For metals considered here, there are two features of the state modeled: the strength, a, and the void fraction, (I). Of principal interest is the influence of gradients in the strength, which appear in the evolution equation for the strength, on the rates of void growth. 3.1 Strain H a r d e n i n g The evolution equation proposed by Hart has been modified to include the dependence on the gradient of the strength variable, a, as per the earlier discussion. Using e~ as the model parameter to control the magnitude of the gradient contribution gives DR = e ~ D V 2 ~ + Con Dt
(9)
D'
where Co, m ~, and n are model parameters in the original hardening expression. 3.2 D u c t i l e Void G r o w t h The damage evolution equations proposed by Lee and Dawson [6] consist of equations for two regimes of loading. For low levels of triaxiality, the evolution equations are drawn from analyses of voids growing under macroscopic hydrostatic stress. For higher levels of triaxiality, the evolution equations are derived from the growth of voids in a cylindrical volume with various levels of effective deformation rate in addition to the macroscopic mean stress. It is this latter regime that is most applicable to forming operations because the mean stress typically is less than a factor of three times the strength (the former equations dominate when the mean stress is more that several times larger than the strength). The appropriate void growth equations for the low triaxiality case are D(~ _ ~ , r Dt 1 - (I)exp cl
D~;
~* -
f o mr) exp c 2 -~(-~ ROrJ
Here, Dr, ~r, ~r, Cl, and c2 are model parameters.
103 4.0 F I N I T E E L E M E N T I M P L E M E N T A T I O N Constitutive equations having gradient terms in the evolution of state typically have a different mathematical character than evolution equations with only local dependencies (the equations are parabolic partial differential equations instead of hyperbolic ordinary differential equations). Consequently, the numerical solution of the evolution equations with gradient terms involves a somewhat different approach than can be taken with a strictly local theory. For nonlocal theories, Galerkin finite element formulations are effective for handling both the spatial and temporal dimensions using the same methodologies employed in solving transient heat conduction problems. The complete solution for the motion and evolution of state involves integrating the evolution equations together with determining the velocity (or displacement) field from the balance laws. The computation of the velocity field follows precisely the same approach used with a strictly local form of the constitutive theory. A weighted residual on the equilibrium equations is constrained by conservation of mass via a consistent penalty formulation. Because the gradient terms in the evolution of state are restricted to the evolution equations and do not appear in the yield condition, the solution procedure for the velocity field is unaffected by the nonlocal effects. As constructed, the velocity field at an instant depends on the state, but not on the gradients in state or strain. The motion of the body and the evolution of state are coupled, but for the purposes of simulation are separable. The technique for determining the velocity field may be found in existing references [7]. 4.1 S t r e n g t h evolution The modification of the material strength with deformation is embodied in the constitutive model through the evolution of the strength, as described by Eq 9. The numerical integration of Eq 9 is based on a weighted residual formed over the domain as
R,~-/aq~,~[-~tt-~'~D'V2~-c~
( G ) m' ( ~ ) ~ D ' ]
d~
(11)
where ~ is a set of scalar weighting functions and ~t is the volume of the body. Following standard finite element practices [8], the continuity requirements on the trial functions used to represent the dislocation density are shifted to the weighting functions through integration by parts, assuming sufficient interelement continuity. This gives, after application of the divergence theorem
where F is the boundary of ~t, and F 1 is that portion of the boundary where the dislocation density gradient is known c~D'V~, n = ]
(13)
The remainder of the surface is denoted F 2 with the boundary condition that ~=~
(14)
104 This latter condition is not encountered in the applications discussed later, however. We next introduce into the weak form of the evolution equation the temporal discretization for the strength as +
-
-
2
k-
~t
(15)
and finite element interpolation for the trial and weight functions as = [N]{K:)
~
= [g]{s)
(16)
where [Y] are the interpolation functions, {K~) are nodal values of the strength, and {S) are nodal values of the weights. We require that R~ vanish for arbitrary values of the weights equation, giving a matrix equation for the nodal values of the strength at the end of each time interval based on the values at the beginning of the interval and the rate of strain hardening during the interval.
4.2 Damage (Void) Evolution The rate of growth of damage may depend on the gradients of damage in the same manner that the strength evolution depends on it gradients. While we have explored this possibility to some extent, here we limit our attention to the class of models exhibiting only local dependencies. Ductile void growth is described by Eq 10
(I))]dR
(17)
where ~r is a set of scalar weighting functions and / ~ - ~ooexp(c2a---~m)D'
(18)
Again, we introduce into the weak form of the evolution equation the temporal discretization for the void volume as _
+
_ 2
(19)
and finite element interpolation for the trial and weight functions as ' (I)- [N]{@}
~r
[N]{S)
(20)
where {@} are nodal values of the strength, and {8} are nodal values of the weights. As with the formulation for the evolution of the strength, we require the residual vanish for arbitrary values of the weights equation, giving a matrix equation for the nodal values of the void volume at the end of each step in terms of values at the start of the step.
105 Table 1: Material Parameters for the Simplified Hart's Model for Aluminum. ao fo (s -1) (s -1) 9.64 x 1059. 2.12 x 1019
G (GPa) 24.
~o (MPa) 64.0
Qo/n (K)
Q'o/R (K)
Co
")'
1.45 x 104
1.45 x 104
6.19 x 10 -9
0.15
M
m
Tt
m I
7.8
5.0
4.5
3.5
Table 2: Material Parameters for the Damage Model for Aluminum. Cl (-) 1.6
c2 Dr ( - ) ( s -1) 24.0 1.0
0r (K) 373.0
ar (MPa) 64.0
5.0 A P P L I C A T I O N Void growth during the extension of a notched rod was studied previously [9] using the theory presented here, with the exception of the gradient term appearing in the evolution equation for the strength (Eq 9). We return to that application to demonstrate the strong influence that gradients in the strength have on the void growth. The axisymmetric specimen shown in Fig 1 is pulled in tension with a fixed crosshead speed of 2.1167 x 10-Sm/s for a total of 200 s. The initial strength is 64 MPa and the initial void fraction is 0.0011. The temperature initially is 373 K, and because the test is conducted very slowly, isothermal conditions are assumed to persist throughout the test. Parameters for both the plastic flow model and the void growth model are given in Tables 1 and 2. To illustrate the impact of the gradient term in the hardening, simulations performed with two different values of e~ are compared to the original formulation in which no dependence on the gradients appears. Shown in Figs 2 and 3 are the strength and void fraction corresponding to the case of e~ of zero at the end of the loading. The strength is elevated by the largest amount in a band across the smallest cross section where the straining is most intense. The peak value of ~ occurs within this band and is approximately 128 MPa. The void fraction exhibits the greatest rise along the centerline and at the minimum cross section, as reported previously [9]. The void fraction reaches a maximum value of about 0.01 in this case. The effects of the gradient term in the evolution of ~ are shown for two values of e~: 0.0001 and 0.0003. Shown in Figs 4 and 5 are the strength and void fraction distributions corresponding to the end of the loading for the case of e~ of 0.0001. Note that the while the strength distribution is qualitatively similar to the case in which e~ is zero, the peak
106
t
t
t ~ / A
A B C D
12.2870 3.0607 6.1087 1.8732
cm cm cm r
Figure 1: Schematic of notched tensile specimen.
Figure 2: Distribution of strength in the notched zone at the end of loading for e~ of 0.(] Peak value- 128 MPa.
107
Figure 3: Distribution of void fraction in the notched zone at the end of loading for e~ of 0.0. Peak value- 0.010. value is lower (118 MPa) and the gradients near the minimum cross section are weaker. The corresponding void fraction distribution shows maximum values more than double (~max - 0.024) those predicted in the zero e~ case. The differences in the responses between e~ of 0.0 and 0.0001 are accentuated for c~ of 0.0003. These appear in Figs 6 and 7 for the strength and void fraction, respectively. The strength gradients are suppressed near the minimum cross section and the peak value is the lowest of the three cases considered. The void fraction, on the other hand, is approximately three time the result obtained using no gradient term in the evolution ((I)max : 0.036). The influence of gradients in the evolution of the strength can be seen to have a pronounced effect on the growth of voids. This occurs here because the peak value of the mean stress occurs at the same spatial location as the highest strength. Such an occurrence is clearly not unusual, as it implies that both the mean stress and the deviatoric stress are high at the same spatial location. It has a strong effect because the void growth rate is quite sensitive to the ratio of the mean stress to the strength. The effect of the gradient term is to diffuse the peak values of strength, which results in higher values of this ratio and therefore higher void growth rates.
6.0 S U M M A R Y The goal of this paper is to demonstrate the substantial impact that gradient terms in the evolution of the strength can have on the growth of voids. In the notched zone of the tensile specimen, the increases in the strength during deformation are less when the gradient dependency is present, which acts to elevate the ratio of mean stress to strength and accentuates the growth of voids. Due to the exponential dependence of the growth
108
Figure 4: Distribution of strength in the notched zone at the end of loading for e~ of 0.0001. Peak value- 118 MPa.
Figure 5: Distribution of void fraction in the notched zone at the end of loading for e~ of 0.0001. Peak value- 0.024.
109
Figure 6: Distribution of strength in the notched zone at the end of loading for e~ of 0.0003. Peak value - 115 MPa.
Figure 7: Distribution of void fraction in the notched zone at the end of loading for c~ of 0.0003. Peak value- 0.036.
110 rate on this ratio, the void fraction is sensitive to the small variations in the ratio that are introduced through the influence that gradients in strength have on its own evolution.
ACKN OWLED G EMENT S The authors with to thank Dr. Ashish Kumar for his assistance in completing the simulations of the notched tensile specimen.
REFERENCES
1. L.M. Kachanov, IZV. Akad.Nauk. SSSR O.T.N.Tekh. Nauk.,8(1958)26. 2. Y.-S. Lee, Ph.D. dissertation, Cornell Univ., 1991. 3. D.J. Bammann and E.C. Aifantis, Acta Mech., 45(1982)91. 4. E.C. Aifantis, Comp. Mat. Mod., ASME AD-42(1994)199. 5. Y.-S. Lee and P.R. Dawson, Mech. Mater., 15(1993)21. 6. E.W. Hart, J. Engr. Mat. Tech., 98(1976)193. 7. Y.-S. Lee and P.R. Dawson, Mech. Mater., 15(1993)35. 8. G.M. Eggert and P.R. Dawson, Int. J. Mech. Sci., 29(1987)95 9. P.R. Dawson, Y.-S. Lee, and A. Kumar, J. Mat. Proc. Tech., 32(1992)119. 10. O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, 4th Edition, McGrawHill, London, 1989.
Advanced Methodsin MaterialsProcessingDefects M. Predeleanuand P. Gilormini(Editors) 91997ElsevierScienceB.V. All rights reserved.
111
On the fracturing of brittle solids with microstructure Ioannis St. Doltsinis ~ aInstitute for Computer Applications, University of Stuttgart Pfaffenwaldring 27, D-70569 Stuttgart The present account deals with theoretical issues in modelling brittle microcracking solids. In this connection, a constitutive framework is presented first for the elastic fracturing continuum. Then, the consistency of the parameters accounting for fracturing is elucidated by considering an elastic solid with cracks. Thereby, the configuration of the crack system becomes significant. The description of progressive fracturing relies on the combination of constitutive description and fracture criterion with the structural morphology of the material. The latter defines a network of potential microcracks, and determines the evolution of fracturing. 1. I N T R O D U C T I O N The paper aims at a constitutive description of brittle microcracking materials accounting for structural characteristics [1] resulting from the manufacturing process. A constitutive framework is developed first on the continuum level. Thereby, the material is assumed elastic undergoing damage by progressive fracturing, [2]. The formalism becomes more specific on the background of a continuous formation and extension of distributed microcracks. At this stage, the parameters accounting for the appearance of the crack system remain rather abstract. In order to explore the nature of the above parameters which affect the elastic properties, the constitution of a solid containing a deterministic crack system is investigated. For this purpose, structural relations between interior microscopic stress and strain, and boundary quantities provide a basis for the introduction of a homogeneous material macroscopically equivalent to the elastic solid with defects in the form of cavities. Specification for cracks is performed on the assumption of either noninteracting or interacting systems. This development follows mainly the approach of [3], [4] but our formalism and the reasoning to the interacting system are different. In particular, a stiffness is assigned here to the interacting crack system. The macroscopic stress-strain relation for the elastic solid containing a deterministic crack array being established, we deal next with the issue of the progress of fracturing in the material under loading. In this connection, the stability of the system is examined, a condition for the formation of new cracks is postulated, and an algorithmic procedure is indicated, following the development of cracks within a defined structure. The above forms the basis for a computational material model. The pattern of potential cracks is supplied by the morphology of the material structure. In studies on ceramics and plasmasprayed ceramic coatings in [5], the grain boundaries were assumed to form the potential
112 fracturing network. 2. D E S C R I P T I O N OF F R A C T U R I N G C O N T I N U U M 2.1. C o n s t i t u t i v e f r a m e w o r k In developing a constitutive framework on the continuum level of brittle microcracking solids, the material is considered elastic and homogeneous. The influence of evolving microcrack patterns is assumed to be accounted for by a number of parameters collected in the vector array D which in this context characterizes the actual state of the crack system. At D - const., the functional dependence
(i)
a(e, D) = ~(D)e establishes an elastic stress-strain relationship for the material.
Here, the 6 x 1 vec-
t~r arra~s a -- {axx cr~ azz V~ax~ V/-2~r~zV~rxz } an~l ~ -- {~xx ~y ~zz x/r2~xyx / ~ z v/-2~xz} represent the local stress and strain state respectively. Since the description ultimately addresses progressive fracturing, the parameter array D appears explicitly in (1) accounting for the current pattern of the relevant material structure, cf. [1]. Differentiation of (1) yields the relation
0r
0r
&= ~
i+ ~
b = a(D)i + 4-
(2)
between the time rates of the constitutive variables a,e and D. In (2), the elasticity matrix ~(D) determines the instantaneous response as long as the variables D remain stationary. The additional term accounts for D ~ 0 during progressive fracturing, cf [2]
4-= [i)a/i)D]D = k(D)e
(3)
The definition of a condition limiting elastic states either in stress or strain space, r
_ 0
or
~(e,D)< 0
(4)
and of an evolution law for the microcracking parameters b = r
r162
(5)
complete the constitutive framework for the material. The energy difference between loading and elastic unloading of the unit volume element amounts to G=
/
t
1
at~dt - "~ate
(6)
o
Differentiating, and with reference to (1), (2), (3) we obtain the time rate 1
t"
1 t
r = - ~ e r = - ~ a k(D)a
(7)
113
2.2. Microcracking Changes of the elasticity matrix ~(D) are associated here with irreversible fracturing processes, and therefore (7) supplies the rate of energy dissipation in the volume element. If an area A is collectively assigned to the system of distributed cracks in the unit element, the energy consumption during extension is 27dA, with 7 denoting the specific surface energy of the material. The equality dG = 27 G = d--A
(8)
expresses the balance between the energy release rate G and the specific surface energy 7 during the course of a stable progress of microcracking. In order to examine the stability of the stress state, we determine the energy release rate G for a virtual extension of the crack system at constant stress, and define the criterion
o = marx( ot
a
) -27_<0
(9)
Analogously for the strain ~ = max (2 et [-dn(D)dA] e) - 27 < 0 Critical states satisfy the equality in (9) resp. (10). The maximum condition for the energy release rate refers to the variation of the parameters D which may be not uniquely specified by the variation of the collective crack area A. In order to expound on the argument we consider more closely the energy release rate in (10), and express it with reference to (7) as
letan(D) I t dr a=-~ dA e=-~e~-~ The scalar product in (11) attains a maximum when strain e, i.e. for dT d~(D) dA= dA e = - , ~ e
(11)
dr/dA
is directed along the given
(12)
/,From the last equality in (12) it can be concluded that e is an eigenvector of the matrix d n ( D ) / d A and )~ the associated eigenvalue. The latter is requested to attain a maximum as well. Neither maximum condition, however, necessarily determines a unique variation of the microcracking parameters D. As a consequence, a single variation of the collective area A, the decisive parameter in the energy criterion, may be associated to different variations of D entering the elasticity matrix n(D). With (12), the criterion (10) assumes the form ~(e, A) = - )~et e - 2 7 2
<0
(13)
and is considered a function of the strain e and the collective crack area A in conformity with the foregoing remarks. We avoid here the standard discussion on loading/unloading
114 conditions based on (13). We merely remark that during stable microcracking the equality sign is applicable in (13), and determines A to = ~tc
(14)
Assuming a functional dependence A(A) in (14) and forming the time rate we obtain ( d A ) -1 ct~
(15)
for the evolution of the collective crack area in the unit volume element. Use of (15) in (12) yields + = --A/[e = 2A2 ~
eet~
(16)
for the stress decrement. In (16), we introduced the abbreviation 1 e = ~etec
(17)
which identifies the direction of the strain vector c. It is seen from (14) that an increasing intensity of the strain e diminishes A, whilst the crack area A is supposed to grow. Therefore, dA/dA < 0 in (15), (16). Equation (16) completes the determination of the stress decrement + in (2). Differentiation of the fracturing function (13) with respect to the strain yields, Oct
(18)
=%c
and reveals that +, (16), (12), is directed along the interior normal of the surface ~(c, A) = 0 in strain space. It thus satisfies the material stability requirements, cf. [2]. At this stage, the modified elasticity matrix ~(D) can be determined neither directly nor via its dependence on the microcracking parameters D since an evolution law (5) for their update is lacking. 3. E L A S T I C S O L I D W I T H C R A C K S In order to explore the consistency of the parameters D accounting for microcrack patterns in the elastic solid, we refer to a representative volume element V of the material composed of an elastic matrix with volume VM and comprising a number n of cavity defects with collective volume Vc such that V = VM + Vc.
3.1. Interior and boundary quantities Relationships between stresses am(x) and strains era(x), x = {Xl x2 x3}, in the interior of the volume element, and tractions t = {tl t2 t3} respectively displacements u = {Ul u2 u3} on the surface S bounding the volume V are established by the virtual work principle. The static equivalence between t and am is expressed by n
S
V
VM
i = l Si
115 In (19), the standard volume integral referring to a continuum is first represented as the sum over the matrix, VM, and the remainder Vc = E~=I Vci. The latter contributions are replaced by surface integrals accounting for the tractions on n inclusions or for loading on cavity surfaces. In the case of traction-free cavities or cracks, the respective term in (19) vanishes. Alternatively to the above gradual derivation, a direct statement of the last expression in (19) may be referred to matrix plus cavities, ti denoting local reactions in case of loaded cavity surfaces. For virtual displacements u = x~
(20)
compatible with a virtual strain field e = const., it follows from (19) n
J X'
J..dV = J
S
V
(21)
a.dV +
VM
i=l Si"
In (20), (21) the matrix
X=
xx 0 0 0 x2 0 0 0 x3
~/~ o ~/~ ] 19/ ~ ~/v~ o o ~/~ ~,/~
J
(22)
conforms with the definition of the 6 x 1 strain vector. The kinematic compatibility between c,~ and u is expressed by
J 'udS: i S
n
at~:'dV = f ~-t~:,,.,dV+ ~ f([tudS)i
V
VM
(23)
i=l Si
where the sum of the surface integrals in the last expression extends over the n cavities, respectively inclusion boundaries. For virtual surface tractions, t, = N~
(24)
associated with virtual stresses ~ = const., we obtain
fN'udS:i
VM
(25) i=1 Si
between displacements and strains. In (24), (25), the matrix
N=
nl 0 0
0 0 n2 0 0 n3
n~/Vf 0 "~/~'1 ~ , / ~ ,,3/~ o o n~/~ ,,,,Iv1
J
(26)
specifies operations with the normal n = {nx n2 n3} on boundary surfaces in conformity with the definition of the 6 x i stress vector. We point out the identical structure of N and the matrix X in (22).
116
3.2. Equivalent homogeneous material Next, we consider tractions t which, applied to the external boundary of the volume element, would have produced a constant stress a in a homogeneous material, whilst a,~(x) in the actual one. Application of (21) to either case leads to the equality aV = / a ~ d V
/a,~dV
=
V
(27)
VM
Accordingly, the macroscopically homogeneous stress a is obtained as the volume average of the microscopic stress field am. Thereby, any self-equilibrating system of internal stresses is seen not to contribute to the macroscopic average. For cavities which are free of tractions, the volume integral in (27) extends over the compact matrix material only. For the kinematics, displacements u compatible with a constant strain e in a continuum element are applied to the outer surface of the material volume element inducing strains era(x) in the actual material and displacements ui along internal surfaces. Applying (25) to either case we obtain the equality eV
= /e~dV + ~/(NtudS)i VM
(28)
i=l Si
The averaging process for the macroscopic strain e in (28) eliminates fluctuations of the microscopic strain e,~. The matrix material is assumed elastic, obeying a local stress-strain relation am = JCoe~ + rm
(29)
The original elasticity matrix is denoted by too and r,,, refers to self- equilibrating initial stresses. With (29) the first integral on the right-hand side in (28) is transformed to, J emdV = ~-x / [a~ - Tin] dV = tc=laV VM
(30)
VM
and (27) has been used for the stresses. Substitution of (30) in (28) contributes to the establishment of a relation between macroscopic strain e and stress a in the form r :
/~o"10" ~- ~0
(31)
Here,
1 ~/(NtudS)i ~U -- V i = l s i
represents the contribution of cavities or inclusions to the macroscopic strain.
(32)
117 3.3. M i c r o c r a c k s For a completion of the constitutive relation (31), the strain cc must be expressed in terms of the macroscopic stress a. This will be performed in the following for microcracks as the material defects of interest here. For a crack with area Ai, (Si = 2Ai), and defining the displacement discontinuity across the crack by the crack opening displacement vector bi = {bx b2b3}~, the single contribution to the strain ec in (32) reads,
f (NtudS)i= f Si
n
(NtbdA)i : ( N t < b > A)i
and
eo = V ~-~(Nt < b > A)i
(33)
i=1
Ai
requiring evaluation of the integral only over the area of the crack. The last expression applies to flat cracks with unique normal ni, and < bi > denotes the average crack displacement vector over the crack area Ai. For a single isolated crack, the approximate relation < bi > = Bi < ti >
(34)
is proposed in [3] between the average crack opening displacement < bi > and < ti >, the average of a nonuniform traction, respectively a traction vector ti uniformly applied to the crack faces. The matrix Bi, the crack compliance, depends on crack shape and size, the geometry of the sample and on the properties of the elastic medium. In the anisotropic case, orientation is also of relevance. Expressions for Bi were given in [31, [41 for representative crack geometries. As an elementary example we consider a rectilinear crack in the plane, and with reference to the local n, s-system (normal and tangential to the crack line), eq. (34) assumes the form
b, > i
B,,B,,
i < t, >
For the isotropic material, normal and shear modes are uncoupled, and the off-diagonal terms in the matrix Bi vanish. Otherwise B,n = B,, for the elastic medium, and Bi is a symmetric matrix. In the infinite plane, B,n = B,, = 7rli/2E" (crack length Ii, elastic modulus Eo, Poisson's ratio to, E'o = Eo for plane stress, E'o = E o / ( 1 - U2o) for plane strain). In this case, Bi = (Trli/2E~o)I is an isotropic matrix invariant to coordinate rotations, I denoting the identity operator. The average tractions along the crack faces defined by the local stresses, are specified here as to compensate the action of the applied loading at the same location in a crack-free solid. Thus,
1 f(NadA)i
< ti > = ~iAi
= Ni < ai >
and
ti = Nia
(ai = const. = a)
(36)
where ai denotes the stress along the plane of the virtual crack defined by the normal vector ni respectively the matrix Ni. It accounts for the presence of any other defects but for the cracks. For a uniformly distributed stress we have along each single crack ai = a, the macroscopically applied stress.
118 If interactions can be neglected, relation (34) for the single isolated crack can be applied to each member of a crack system. With (36), (34) and (33), the contribution (32) of the cracks to the strain then becomes ec' =
NtBNA)i a = N t l 3 N a
"~
,
N = { N i } , B =-v
1 [AiBiJ
(37)
In the last expression in (37), the definition of the matrix arrays N and 13 substitutes the summation sign. We notice that expression (37) for the strain ee establishes a symmetric constitutive relation as by (31). It reads e = n / i + N t I ] N ] = n-1 a
and
g-1
=
go- 1 _[_N t l ] N
(38)
Representation of the crack system in (38) is based on the orientation Ni and the compliance Bi of the individual cracks, which thus define the parameters D in (1). 3.4.
Interacting
crack
system
For interacting cracks, we assign a stiffness to the crack system introduced by the relations < t~ > = CII < bl > +C12 < b2 > + . . . + C~. < b . > < t2 > = C21 < bl > -~-C22 < b2 > + . . . + C2n < b . > (39) < t, >=
Cnl
<
bl > +Cn2 < b2 > + . . . + C , , < b , >
A short-hand form of (39) reads t = Cb
with
t = {< ti > } , b = {< bi >} and C = [Vii]
(40)
where the symbols t, b and C denote hypervectors and hypermatrix arrays respectively. and the hypermatrix. In (39), each coefficient matrix Cij determines the average traction < ti > induced Mong the plane of the i th crack by an average displacement discontinuity < bj > imposed on the j th crack, whilst all other cracks are kept closed. Interpreting (36), determination of the above contribution to < ti > requires the stress distribution resulting from opening the j th crack. For this purpose, we assume that the stress Vrj can be represented as o'j = a~tj where a~ denotes a 6 x 3 array formed from unit solutions for the isolated j th crack and t] is the uniform, respectively average traction along the crack face. In the two-dimensional case for instance, the crack-induced stress aj may be represented as
crj = pjapj + qj%j
(41)
where pj = n j tt j ,
,
q j - - s jtt j ,
(42)
119 denote the traction components along the normal direction n and the tangential direction sj of the i th crack. The stress fields apj, aqj are those arising in the elastic plane from unit normal and shear tractions pj = 1 and qj = 1 respectively at the j th isolated crack. Combining (41) with (42) yields aj - [apnt-t - aqS t] J tj' -- ajtj''
(43)
which defines a~. Then, with (34) aj =
a~t] = a]B~"1 < bj >
(44)
and from (36) the single contribution to < ti > follows 1
< ti >j---- Aii [Ni f a;dAi]B~" < bj >
(45)
Ai
~,From (45) the coefficient matrix Cij is deduced as, Cij = ~i1 [Ni ] a~dAi] B~-' = (~ijS~-1
(Cii = B~-x , Cii - = I)
(46)
Ai
and essentially requires averaging over the i th crack of the unit solution for the j th crack constituting a~. The stiffness matrix of the crack system may be written as C = (~B -1 with (3
=
[Cij], B---- [BiJ
(47)
The particular constitution of the diagonal elements of the hypermatrices in (47) is shown in (46). Ultimately, the tractions applied to the crack faces must compensate the action of the external loading on the specimen. For a uniformly distributed stress a they axe given by t = N a and solving (40) for the crack opening displacements yields b = C - 1 N a = B(~ - 1 N a
(48)
With b = { < bi > } from (48), the contribution of the crack system to the overall strain is determined by (32), respectively (33). In matrix notation, e.c =
Nt]3(~-SNa
(49)
which replaces the uncoupled expression (37). The constitutive relation following from
(31) i~ e -- [too1 + N t I 3 ( ~ - 1 N
]a
= n-xa
with
~-1
= ~o'+ N~t]e-XN
(50)
_
and is now non-symmetric because of the interaction matrix C. Thereby, mutual positions of the cracks appearing in the interaction matrix enter as additional parameters in the array D in (1).
120 4. E V O L U T I O N OF M I C R O C R A C K I N G
4.1. Stability of crack system The question arises now, whether at a given state of loading the crack system can undergo changes by the appearance of additional cracks. In this connection, the following considerations do not deal with continuous crack formation and extension, but rather with an increasing number of discrete crack entering the system instantaneously with the full dimension. A modification of the crack system with nl to one with n2 cracks (n$ > nl ) at constant applied stress a causes a change in strain e. The strain difference emanates exclusively from the part ev, the contribution of the crack system, cf. (31), and determines the energy release in the material volume, Vvrt [ec$ - eel ] = ~ E t[ < b~ - b, >i Ai 2 i=1
(51)
In (51), the energy release rate has been alternatively expressed for the discrete system in terms of the tractions ti = Nia along the n$ crack areas and the change in average crack opening displacement < b2 - bl >i. For the additionally introduced (n2 - n l ) cracks < b~ - b ~ > i = < b~ >i. Comparison of the energy release from (51) to the surface energy required for the creation of the new cracks states that the change will not appear as long as, n2
y~ 2
[~o~ _ ~o,] _ 2 ~ ~jAj < 0 ~
(52)
where the second term represents the surface energy for the extension of the system from n l to n2 cracks. If crack interaction can be neglected, the strain cv is determined by (37). The difference for the two configurations of the crack system reads, f-C$- f-C1 ~ ' ~
L nl
J
the summation extending over the trial cracks (n2 - n l ) only. This is a consequence of the non-interaction assumption stating that the new cracks do not affect the displacement discontinuity along the existing cracks. Using (53) and for 7j = 7 = const., the stability criterion (52) becomes, Aj
(NtBNA)j a - 2~' _< 0
(54)
L nI
which is the counterpart of the continuous criterion (8) for a certain virtual extension of the crack system. Alternatively, (53) could be derived from (8) in conjunction with (31), (37) for the present virtual crack extension. The same result is obtained directly in terms of microscopic quantitites, using the energy expression on the right-hand side of (51). The contribution of interacting cracks to the strain is given by (49), and yields
Aj
N ~ [B~C~ ~ - B, C; ~] N ~ - 2~ < 0
(55)
121 instead of (54). Interaction necessitates consideration of the complete system in the two configurations, the existing and the one virtually extended by the (ng - n l ) cracks.
4.2. Progressive fracturing The above theory relies entirely on deterministic crack systems requiring specification of orientation, shape, size and mutual position of cracks. Nevertheless, the data may be generated at random within the sample. Accordingly, examination for possible appearance of microcracks can be performed at random, in principle. If the structural morphology of the material favours any particular crack formation, the task of exploring the evolution of fracturing is simplified. As an example, if fracturing preferably occurs along grain boundaries, a network of potential cracks is defined by the grain structure of the material. Expressions (54) and (55) introduce microscopic morphological parameters in the respective continuum forms. For an algorithmic treatment of progressive fracturing we preferably refer to the microscopic variables. We consider a state of fracturing (n cracks) stable under the given conditions, and modify the loading. It is assumed that a new single crack is thereby formed if the elastic energy release balances the surface energy. Using the expression on the right-hand side of (51), (56)
t i < b, - b ~ >i Ai = 27jAj 2 i=l
and 2vjAj denotes the surface energy for the j th trial crack. In the non-interacting case, the introduction of the new crack does not affect the response of the existing system. With reference to (34) for constant loading
>iCj = 0 ,
>j==Bj
(57)
It follows from (56) that at a given state of uniform macroscopic stress a the j th crack is considered formed if, tjt < bj > = tjtBjtj > 47j
(tj -- Nja)
(58)
and variation of the index j -- n + 1, m supplies the cracks forming by the performed modification of the loading. Since, however, the criterion for crack formation does not depend here on the state of fracturing, the crack pattern associated to a given macroscopic stress a can be obtained at once. In the case of radial loading (a = tat) the equality sign in (58) specifies the instant tj when the j th crack forms 4-yj t~ = att(NtBN)jat
(59)
The above equation may be deduced also directly from (54). Within a defined network of potential cracks, (59) determines a priori the sequence of fracturing patterns during the course of the radial loading. If the interactions in the system can not be neglected, the determination of the energy release in (56) involves all cracks, the existing and the trial one, and requires a complete solution of the matrix equation (40) for the displacement discontinuities in the presence
122 of the surplus crack at constant applied load. For the system of n + 1 cracks, equation (30) can be presented in the form (60) In (60), the index n refers to the existing array, the index j to the new crack. Depending on the number (m - n) of the trials to be undertaken, an exploration of the extended crack pattern may become cumbersome. A simplified treatment disregards the influence of the single new crack on the system. Then the upper row in (60) supplies the average crack opening displacements bn = {< bl > } , ( i = 1, n), as
(6:)
bn = C n-In t n = C n-1n N n a
Consequently, the difference < b~ - bl >i vanishes for all i = 1, n. The lower row in (60) yields, bj = C~ 1 [tj - Cjnbn] = Bj [Nj - ajnbn]
and
< b~ - bl >j--< bj > - bj
(62)
which completely accounts for the influence of the existing system on the trial crack. It follows from (56) that at a given state of uniform macroscopic stress a the j th crack is considered formed if tj> < =bj t
ttBj [tj - Cjnbn] >_ 47j
(63)
The criterion (63) differs from (58) by the interaction term Cj~b~. If the transition from n to n + 1 implies a radial stress variation (a~+x = fa~), the equality sign in (63) determines the multiplier
f~ = min (47j(a~NjBj[Nj- Cj.C:=:N.]a.)-')
(64)
for the formation of the next among the (m -- n) candidates in the network. Expression (64) helps to indicate explicitly the reference stress an prior to the progress of fracturing. Its evaluation is conveniently performed with the quantities in (63) determined for an and scaled by the factor f to the actual ones. We remark that the assumption of a negligible influence of the extension on the existing system at least necessitates a small number of additional cracks, preferably a single one in each incremental modification of the loading. 4.3. M i c r o c r a c k i n g n e t w o r k For a simulation of microcracking in the material it is proposed to consider a sample representative of the structure defining a network of potential cracks. Such a network may be provided by the grain boundaries. Information on the morphology of the structure, respectively the grains must be available in a form suitable for numerical processing. For this purpose, artificial microstructures with specific characteristics are generated by a computer algorithm [5]. In two dimensions, the procedure starts from a regular hexagonal lattice. The cells, representing grains, are subjected to statistical distortions by displacing triple points. This usually leads to only moderate fluctuations in grain size, and in order to obtMn larger variations, several adjacent grains are merged together. Thereby, some triple points are eliminated at random and new grain boundaries are created there.
123 Apart from grains, other elements may appear in the microstructure as well. Plasmasprayed ceramic coatings, for instance, exhibit a lamellar structure as a result of the manufacturing process, and pores. Lamellae are introduced in the computer generated microstructure by the definition of polynomial curves attracting triple points in the vicinity. Pores are randomly placed by activating a virtual void growth process at triple points in the sample accounting for the fact that they preferably appear at lamella~ interfaces. The resulting microstructure is specified by a record containing the topological and geometrical data of the facets. An algorithm pursuing crack formation in the defined network in accordance to Section 4.2 can be given by the following instructions:
Loading loop Advance state of applied stress a. Determine tractions tj = Nja and average crack opening displacements < bj > an by eq. (62) for each of the (m - n) unbroken facets.
Facet loop Calculate "energy release" expression t~ < bj >. Allocate a crack if tjt < bj > >_ 47j.
End facets Update cracks n and facets ( m - n). Determine strain state e and elasticity matrix n for actual crack pattern.
End loading REFERENCES
1. J.R. Rice, Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms, in A.S. Argon (ed.), Constitutive Equations in Plasticity, The MIT Press, Cambridge, Massachusetts, and London, England (1974). 2. J.W. Dougill, On stable progressively fracturing solids, Journal of Applied Mathematics and Physics (ZAMP) 27 (1976) 423-437. 3. M. Kachanov, Elastic solids with many cracks: a simple method of analysis, Int. J. Solids Structures, 23/1 (1987) 23-43. 4. M. Kachanov, Effective elastic properties of cracked solids: critical review of some basic concepts, in V.C.L.Li (ed.), Micromechanical Modelling of Quasi-brittle Behaviour, Applied Mech. Rev., 45 (1992) 304-355. 5. I.St. Doltsinis and R. Handel, Modelling the behaviour and failure analysis of brittle microcracking materials, Proceedings, ECCOMAS 96, John Wiley, 1996.
This Page Intentionally Left Blank
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
Fracture
prediction
125
of sheet-metal blanking process
Ridha HAMBLI 9Alain POTIRON 9Serge BOUDE and Marian RESZKA ENSAM CER-ANGERS; Laboratoire de G6nie M6canique et C.A.O 2 Boulevard du Ronceray - B.P. 3525 49035 ANGERS CEDEX
1. I N T R O D U C T I O N In industrial processes, the making of thin mechanical parts which requires costly tools and machines, is widely used. A modern way to decrease the development's costs is to implement a numerical simulation. In the case of sheet-metal forming, the process involves complex sollicitations of the material and many physical phenomena, such as hardening and damaging, may occur leading to modifications of the material's behaviour. In some processes as blanking, shearing and punching, the rupture of the sheet is wanted. Consequently, during the numerical simulation, a mechanical behaviour model will necessarily account for damaging and will include several failure criteria. This allows for a more realistic outlining of the industrial process from its starting point up to the final breaking of the part. Moreover, prediction of the spreading damage defaults inside the material and the final geometrical shape of the blanking part, will be well predicted. The aim of this work is to provide a quite general finite element model allowing for the numerical studies of structures, subjected to damage and ductile fracture. In order to meet this goal, the best suited models describing the whole blanking process, will finally be used. 2. NUMERICAL MODELLIZATION OF DAMAGE AND DUCTILE FRACTURE Inspection of the last studies in the field of industrial process simulation and despite the increasing progress reached in numerical computing simulation, there appears to be a lack in the prediction of the blanking process including damage and ductile failure phenomena. Recently, Clift and al (1990) [1] carded out a comparative study by means of finite element models including failure of the material, in order to investigate the metal forming problems. The fracture has been numerically predicted by means of several criteria described in literature. Comparing the numerical and experimental results they concluded that some
126 literature. Comparing the numerical and experimental results they concluded that some criterion which are well suited to simulating the behaviour of the material for some particular process and geometry, cannot efficiently be used in other technological cases. In conclusion, there is not a universal failure criterion which allows for the description of any industrial metal forming process.
2.1 Ductile fracture criteria In order to predict the possibility of a structure to undergoing rupture, numerous authors have proposed their own criterion. In the isotropic case, these failure criteria are scalar functions invol'~ing stresses and/or strains, depending on some physical and mechanical parameters. These are usually identified by performing rheological tests on specimens. Particularly the failure is represented by a mathemetical function which is supposed to represent the physical behaviour of the material, and occurs when it reaches a critical value Cc. The mathematical formulations are often written in the form of: If
fO f ( o, eeq ) deeq - Cc < O, there is no failure.
(l-a)
If
f ~ f ( O, eeq ) deeq- Cc >_O, the failure occurs.
(l-b)
In the above expressions, e is the total strain, eeq is the equivalent strain defined by means of plastic part eeq
-
V3-
Different failure criteria which can be applied to predict the ultimate elasto-plastic behaviour of structures subjected to external forces, have been detailed in [2]. It is shown that these criteria are not suitable for describing the gradual material-degradation initiating the first cracks. Consequently, theoretical formulations have been developed by means of elastoplastic laws coupled with continuous damage, overcoming the aforementioned inefficiencies. With this in mind, we are reminded of the work of Lemailre and Chaboche [3].
2.2. Lemaitre behaviour law coupled with damage [3] As a matter of simplification, the damage will be considered as isotropic and depending on a scalar value D. In the unidimensional case and for an elemental domain, D is defined as the ratio between the default and apparent areas. As a result, the material damage state corresponds to the following situations: a- If: D = 0 then there is no damage. b- If 0 _ D _
127 By means of the classical constitutive elasto-plastic laws, Lem~tre and Chaboche propose that the behaviour has to be coupled with the material damaged state. It follows that the corresponding relationship defining the general problem is constituted by: 1- The additive strain decomposition in an elastic strain eel and plastic strain epl: et = eel+epl
(2)
2- The elastic Hooke's law coupled with damage: 0 = (l-D) [Del] g
(3)
[Del] is the isotropic elastic matrix lbr Hooke's material. 3- The yield function f, defining the region of purely elastic response, coupled with damage: f - oeq- (Oel + o0)( 1 -D)
(4)
in which Oeq is the Von Mises equivalent stress, Oel is the yield stress and o0(epl) is the hardening stress function.
4- The flow rule:
/;pl = i 0_..~f 00
(5)
~pl is the plastic strain-rate and Z. is the plastic strain-rate multiplier deduced from the plastic- consistency condition. where
5- The incremental damage law: OD=
Dc
(6)
ER- 8D e R and eD are the strain values corresponding respectively to damage initiation and rupture. The scalar function Rv is written in the form of:
Rv-3(1
+ v ) + 3 ( 1 - 2v) [OHt2 ~ooq! o H is the hydrostatic component of stresses.
(7)
2.3. Algorithm implementation In the classic case, the constitutive aforementioned laws are integrated by means of an incremental procedure. Considering a time increment [tn, tn+ 1], an elastic prediction of stresses is carried out at time tn+ 1 in the following manner:. .T On+ 1 = On + A o
(8)
128 ()T means "test" and the stress increment takes the value: Ao = (1 - Dn) [Dell Aet Dn is the damage value at the beginning of the increment. The yield function f is computed: a- If: f < 0, the predicted value is right and the true stress tensor corresponds to the testing one:
(9)
O n + 1 -- ofT+ 1
b- If: f >__0 the predicted stress value falls into a non permissible region and a plastic correction is needed. The corresponding algorithm is the two-step scheme elastic-prediction plastic-correction, involving the orthogonal projection P(oT+I)
on the yield surface.
This has been implemented into the F.E. code ABAQUS [4] by means of the user's routine UMAT. 2.4. C rack initiation and propagation During the structural analysis, it is supposed that cracks initiates at some points of the material where the damage value reaches a critical value Dc. Consequently, the damaging parameter D is set to a value DR near 1. Therefore, it can be seen from (3) that the material
stiffness value falls to zero and in the finite element modellization, the stiffness of the elements belonging to that region is negligible. In this way, the failure-of the finite element occurs in the vicinity of the critical points and the crack grows and follows the damaged material domain, characterized in the mesh by the elements with the lower values of the stiffness. 3. N U M E R I C A L
TEST
A fiat plate subjected to axial loading has been studied as a testing model for the numerical algorithm which has been developed. The EE. model includes just one rectangular eightnodes element. Clamped on one side, it is subjected to a prescribed displacement U on the other side. Geometry and boundary conditions are depicted on figure 1. The material's elastic parameters are taken to be the Young'modulus E = 210000 Mpa and the Poisson's ratio v = 0.3. For isotropic hardening, the equation is:
eq~ o0 = oel + K (epl
with the values of:
oel =200 Mpa ; K = 480 Mpa ; n = 0.406. Accounting for Lemaitre damaging law (6), the material parameters take the following values: e D = 0 . - e R = 1. - O c = 0 . 3 5
129
I
~
U 9 "1
t
/
b
~-H--~ Figure 1- Tensile test of a quadrangular finite element with eight nodes. Performing all the aforementionned steps of the numerical algorithm leads to the following conclusions, depicted on figures 2a-b. Accounting for the material damage D, or not, results in two different curves drawn on figure 2a. These curves correspond to the equivalent stress evolution Oeq versus equivalent total strain et- If the damage is taken into account, the failure of the structure can be predicted. Otherwise, the structure would never fail. 800
r~va~ aress(lVl~
700
~7
0,8
J
J
(:J30
j f
500
0,6
300
0,4
200 o2 1130 .
o
.
.
.
i
.
0,2
.
.
.
i
0,4
.
.
.
.
,
0,6
-a-
.
.
.
.
;
0,8
.
.
.
.
,
1
,
9
,
9
i
1,2
0
,
0
,
02
,
,
,
0,4
0,6
,
I
,
i
0,8
-b-
Figure-2- Equivalent stress and damage evolution versus equivalent strain. As it is shown on figure 2b, when the equivalent strain exceeds the value eR, the damage increases rapidly inducing a decreasing stress due to the reduced stiffness of the material, as it can be seen on figure 2a. Taking these results into account, we can conclude that the algorithms we have implemented, allow for a good prediction of the material behaviour and damage, leading to the complete rupture of the structure without computing divergence. In the following, the blanking process simulation of a XC 60 sheet-metal will be described by employing the previous numerical method.
130 4. S H E E T - M E T A L B L A N K I N G S I M U L A T I O N
Among the industrial processes dealing with large elasto-plastic deformations, the sheetmetal blanking simulation is one of the most hard to perform, due to the difficulties arising from a right description for the damage evolution, the crack initiation and its propagation throughout the material. In former works and especially [5], the complete material failure of the sheet could not be attained and consequently, the numerical results don't match experiments. In order to overcome these difficulties, we propose the following approach. 4.1. Modellization of damage and failure mechanisms The sheet-metal blanking process on press, has been investigated by many researchers. Recently, we have pointed out by means of an analytical stud)" and experimental tests, that the physical mechanisms leading to the complete failure of the sheet material can be described as lbllows: Firstly, a crack initiates at the cutting edges A and B of the t ~ l s (figure 3) due to the
penetration of the punch into the sheet. Secondly, this crack goes on the region where eeq overpass a critical value and cut progressively the material fibres one after another.
~~n~h
-
fibres
~/~~
~ Sheet -a- initial
-b- crack initiation
-e- crack propagation (fibres cutting)
Figure-3- Rupture of the sheet from [2]. Experiments on technical devices equipped with electrical gauges and a force transducer, were performed on a 4000 KN hydraulic press and we have implemented different failure modellizations in the algorithm. Having analysed the results, it was found that the model of LemMtre was the more appropriated to describe the progressive degradation of the material, the complete rupture of the sheet being reached in a realistic manner.
131
4.2- Numerical blanking simulation of a XC 60 steel sheet The axisymetric blanking operation of circular parts has been choosen as experimental work. All the geometrical parameters are shown on figure 4-a. Accurate experiments performed on the sheet material have given the following values: Young's modulus E = 200000 M p a , Poisson's ratio v =0.3 The hardening law is characterized by the values of: Oel = 250 Mpa , K = 1045 Mpa, n = 0.194. The damage coefficients a~sociated with Lemai'tre's model are: eD = 0. - eR = 0.8. - Dc = 0.37 The meshing of the model involving 1400 quadrangular four nodes axisymetric element, is depicted on figure 4-b.
4.3.
Results
The computations corresponding to different steps of the punch penetration, figures 5-a and 5-c, show the crack propagation inside the mesh.
132
ABA@L0 Punch
! -a- punch penetration = 40%
,2
-b- punch penetration = 42%
numerical profi!
i ~ -V ~/i
I
,-~,1
!
/experimental profil ~
I\
I ! I
r
-~- punch penetration = 45%
-d- Complete rupture of the sheet (punch penetration = 65%)
Figure-5- Numerical prediction of the rupture in the sheet Despite all non-linearities arising from contacts between the sheet and the tools and the elasto-plastic behaviour of the material, the results enlighten the robustness and the fiability of the algorithm. A Coulomb's law has been choosen in the contacts. It can be seen that the distorsion of the mesh has no influence on the results accuracy because the distorted dements vanish during computation. From the several failure criteria which have been implemented in the F.E. code, particularly the Gurson's criterion, it has been found that the better results are deduced from the Lemaitre and Chaboche approach. Consequently the sheet-metal blanking process simulation would be best predicted using a constitutive damaging law coupled wiyh elasto-plasticity.
133 As the optimal choice of the press and corresponding tools is always an industrial goal, we have computed the punch force vs. the punch displacement, occuring all along the blanking process. On figure 6, in the case of an optimal clearance - 10% of the sheet thickness, two CUl-Ves are drawn. They are respectively, the numerical prediction depending on all aforementionned models and algorithm, and the experimental results. 350
Fax
lii
s ~
300I j~,
250-
a-
Ip--=,ql-,,-o ~
200.
r
Punch
Die 50-
an~rmmme/~ O~ 0
,
I
I
I
I
I
I
I
i
10
20
30
40
50
03
70
80
Figure 6 Punching forces vs. punch travel
Figure 7 D a m a g i n g map
It can be viewed that the more realistic description corresponds to a damaging material model and consequently it can be concluded that the blanking process would necessary account for damage.The failure of the sheet material is obtained for a punch penetration of about 65% of the sheet thickness. The resulting computed damaging field is given on figure 7, on which it can be observed that the damage is restricted in the neighborhood of the crack line and mainly apparent in the gap between punch and die. 5. C O N C L U S I O N The numerical studies which have been presented in this paper, are based on robust and efficient algorithms. The experiments have shown the computed results' accuracy. It has been shown that the damaging and ductile failure phenomenon occuring during a sheet-metal blanking operation must be accounted for. Then the correct description of the general elastoplastic behaviour of the material is achieved. The crack initiation and propagation can accurately be predicted without computational divergence. From the moment of the crack initiation to the complete rupture of the sheet-part, all along the process simulation, experimental and numerical results are always in good agreement. These algorithms and constitutive laws seem also to be siuted in the case of extrusion and plastic forming.
134
REFERENCES [1]- C L I F F S.E., H A R T L E Y P., S T U R G E S S C.E.N. and R O W E G.W. " Fracture prediction in plastic deformation process" Int. J. Mech. Sci. Vol. 32, n ~ - 1990 - p: 1-17 [2] H A M B L I Ridha " Etude exp6rimentale, num6rique et th6orique du d6coupage des t61es en vue de l'optimisation du proc6d6 " - Th~zsc de Doctorat - E N S A M d ' A N G E R S - 15 Oct. 1996 [3]- L E M A I T R E J . , C H A B O C H E J.L. " M6canique des mat6riaux solides " - Dunod - 2/~me 6dition - 1988 [4]- A B A Q U S
- HKS
9 Manuel d'utilisation - version 5 . 4 - 1
[5]- H O M S I M., W R O N S K I M. et R O E L A N D T J.M. " Mod61isation num6rique de la coupe " - S T R U C O M E - 1994 - p: 677 - 690 [6]- P O P A T P.B., G H O S H A. and K I S H O R E N . N " Finite element analysis of the blanking process" Jour. of Mech~W-6rking Techn. p" 2 6 9 - 282
- 1989.
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
135
Elastic-plastic f i n i t e - e l e m e n t modelling of metal forming with d a m a g e evolution E Hartley, ER. Hall, J.M. Chiou and I. Pillinger Solid Mechanics & Process Modelling Research Group, School of Manufacturing and Mechanical Engineering, The University of Birmingham, Edgbaston, Birmingham, B15 2TI', UK.
ABSTRACT A model for damage accumulation proposed by Lemaitre has been incorporated in an elastic-plastic finite-element simulation of the plane-strain side pressing of an aluminium rod. Predictions of failure site agree closely with experimental observations. The level of deformation at which failure is predicted shows a small difference to experiment. Further predictions of failure were undertaken with variations on the basic Lemaitre model, (i) with no damage accumulation permitted for compressive triaxiality, and (ii) with an exponential dependence on triaxiality. When combined with a suitable fracture criterion the latter model showed very close correlation with experiment. 1. INTRODUCTION The recent advances in computer simulation techniques combined with models for damage evolution has provided the opportunity to analyse in detail material behaviour in forming processes and how the evolution of damage affects the integrity of a formed product. There is however considerable debate on the most effective damage model to use. In this paper the Lemaitre damage model is used [1,2] together with modifications proposed by Chiou [3]. Each of three different models are incorporated in simulations of the plane-strain side pressing of a circular section workpiece, see figure 1. This type of test, investigated initially by Jain and Kobayashi [4], has proved very useful in assessing finite--element models incorporating ductile fracture criteria [5] or damage models [6]. This is largely due to the consistency with which fracture initiation sites can be identified and related to the initial geometry of the workpiece. The circular section workpiece used here will always develop a fracture at the centre of the cross-section. 2. FINITE ELEMENT FORMULATION The elastic-plastic thermo-mechanical finite-element code (epfep3) used for the modelling employs an updated-Lagrangian approach with a large displacement finite-strain formulation [7,8]. The material is assumed to be isotropic and the constitutive expression is derived from the Prandtl-Reuss flow rule and von Mises' yield criterion. A radial return algorithm [3,9] is used for return mapping to the yield surface.
136
Figure 1. Plane strain side pressing of a circular section workpiece. 3. DAMAGE MODELS Damage models are used to represent the effect of plastic deformation and progressive deterioration of the material. Often damage models may be used in conjunction with some local criterion to predict ductile fracture. Numerous damage models have been proposed, for example those of Rice and Tracey [10], Gurson [11], Tvergaard [12], Rousselier [13] and Lemaitre [1,2]. In this paper the focus is on the incorporation of the Lemaitre model in an elastic-plastic finite-element formulation. The damage evolution equation is shown below (this is referred to here as the L model). /)=
D~ 2 e,~ - e~ [-5(1 + v)+ 3(1- 2v)(
)Z]pZ/Mp
(11
where v is Poisson's ratio, M is the Ramberg-Osgood hardening exponent, ER and ED are the uniaxial rupture strain and uniaxial damage threshold, p is the generalised plastic strain and Dc is the critical damage value Two variations on Lemaitre's model are also investigated. In the first case the effect of hydrostatic compression is neglected (referred to as the LS model), giving the conditions below. 15 = e,~D~ 2 + v) + 3(1 - 2V)(O~q ot~ )2 ]p2/M p - e~9 [3(1
when on O,q
>0
/)
when~
<0
0
(2)
O eq
A further modification (referred to as the P model) based on the observation that damage evolution is affected exponentially by the triaxiality ratio is represented by the expressions below. 2
~
15 = et exp(1.5x)p ~ p
(3)
137 where c~ is a constant obtained by the integration of this expression between D = 0 and D = Dc when P = 0 and P = ER respectively, x is the triaxiality ratio OH/O'eq where OH is the hydrostatic stress and O'eq is the equivalent (or generalised stress). Here the critical damage value is not a constant and a fracture criterion has been developed based on the assumption that the fracture strain can be related to the triaxiality ratio [3] in the form, ~,~ = 0. 316 exp(-0. 705
~
Cleq
)
(4)
The three models are represented in figure 2 (L model), figure 3 (LS model) and figure 4 (P model).
o E a
a
0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02
~
I
L"NI t~
== d d
d
"f'~
r
c~ Plastic Strain
d 9
I
~
'CN
~ Triaxiallty
~
O' d o' d ~,
l
"-"
d
~ ~ d d
Figure 2. Three--dimensional representation of the Lemaitre model.
138
0.180.16 a~ 0 . 1 4 o 0.12 E a 0.I
d
d ~.
d
0.08
0.06
~:~1";
~
d
Plastic Strain
c~ d d o
o
d
Triaxialily
...: o "
c~ ,~. d d ~ '
,
,
~
~ C~ o
o,
Q o
d
(:5
Figure 3. Three-dimensional representation of the LS model.
0.22 0.18 0.16 m a~ 0.14 a E a
0.12 0.I
(5
~d 0.04
d
r
(:5
d ""
9 a'O . , I
" ~, . _ c~ ~
Trlaxlality
. ~
, -r ~, o,
,-~
d
d
d
Figure 4. Three-dimensional representation of the P model.
Plastic strain
139
4. FINITE E L E M E N T RESULTS The material properties of 2024-T351 aluminium alloy were obtained from compression and tension tests [3] and can be represented by the equation below. 1
1
o = Oy + Ke ~ = 370 + 562e "'~4(MPa)
(5)
Young's modulus is 75.05 GPa and Poisson's ratio 0.33. All the experimental tests were performed slowly at room temperature. A central fracture was found to occur at 14.7% reduction in height. One quarter of the cross-section was modelled using 365 elements. The friction factor (m) was evaluated from ring tests [3] to be 0.37. Comparisons of grid distortions from the experiments and those from the finite--element simulation showed a very good correlation [3]. For brevity the grid distortions are not included here. Illustrated in figures 5, 6 and 7 are the predicted values for generalised stress, generalised strain, triaxiality and damage contours for the L, LS and P models respectively.
Figure 5. Finite element results using Lemaitre's model at 13.70% reduction.
140
Figure 5. Finite element results using the LS model at 15.18% reduction.
5. DISCUSSION The finite-element results shown in figures 4 to 6 show clearly the most critical area containing the greatest stress, strain and damage is at the centre of the workpiece and in what appears to be a region of shear close to the tool-workpiece contact comer. With the L model the highest positive triaxiality ratio is located at the region midway between the workpiece centre and the flattened surface. In this region the plastic strain is relatively low. In the analyses with the LS and P models the greatest triaxiality appears near the outer curved surface. Of importance is not only the fracture initiation site which many damage models can identify, but also the level of deformation at which failure occurs. The finite element results incorporating the L model
141
indicated fracture at 13.7% reduction and the LS model indicated 15.18% reduction, both are reasonably close to the experimental observation of 14.7% reduction. With the P model, two predictions are given for fracture initiation, one at 11.8% (based on the fracture criterion proposed here) and one at 14.56% (based on Lemaitre's critical damage data Dc = 0.23).
Figure 6. Finite element results using the P model at 14.56% reduction.
142 6. CONCLUSIONS The LS model is more representative of damage evolution in compression as an allowance is made for reduced damage in compression has been made. The finite element results with the LS and P models are similar, and if the P model is to be developed further the fracture criterion will need some modification to predict fracture initiation with improved accuracy. Further work in this area should concentrate on the load path dependency of damage and fracture initiation, improvements to the model descriptions and the accurate assessment of microstructural damage in materials. ACKNOWLEDGEMENTS The authors are grateful for the support provided for J.M. Chiou by the Aeronautical Industry Development Centre in Taiwan to enable him to undertake this research. REFERENCES 1. 2. 3. 4.
5. 6. 7. 8. 9. 10. 11. 12. 13.
J. Lemaitre, Trans. A.S.M.E.J. Eng. Mat. Techn., 107 (1985) 83. J. Lemaitre, Comput. Meths. Appl. Mech. Eng., 51 (1985) 31. J.M. Chiou, A study of ductile damage in metal forming, Ph.D. Thesis, The University of Birmingham (1996). S.C. Jain and S. Kobayashi, Deformation and fracture of an aluminium alloy in plane-strain side pressing, Proceedings of the 1lth Int. M.T.D.R. Conf., Birmingham, Ed. S.A. Tobias (1970) 1137. S.E. Cliff, P. Hartley, C.E.N. Sturgess and G.W. Rowe, Int. J. Mech. Sci., 32 (1990) 1. Y.Y. Zhu, S. Cescotto and A.M. Habraken, J. Mat. Proc. Techn., 37 (1993) 295. G.W. Rowe, C.E.N. Sturgess, P. Hartley and I. Pillinger, Finite element plasticity and metal forming analysis, Cambridge University Press (1991). I. Pillinger, The elastic-plastic finite-element method, in Numerical Modelling of Material Deformation Processes, Eds. P. Hartley, I. Pillinger and C.E.N. Sturgess, Springer-Verlag (1992) 225. J.M. Chiou, F.R. Hall, P. Hartley and I. Pillinger, Elastic-plastic finite element modeliing of material damage in metal forming, Proceedings of the 4th Int. Conf. on Computational Plasticity, Barcelona, Eds. D.R.J. Owen and E. Onate, Pineridge Press (1995) 1411. J.R. Rice and D.M. Tracey, J. Mech. Phys. Solids, 17 (1969) 201. A.L. Gurson, Trans. A.S.M.E.J. Eng. Mat. Techn., 99 (1977) 2. V. Tvergaard, Int. J. Fract., 18 (1982) 237. G. Rousselier, Finite deformation constitutive equations including ductile fracture damage, in Three Dimensional Constitutive Relations and Ductile Fracture, Ed. S. Nemat-Nasser, North-Holland (1981) 331.
Advanced Methods in Materials Processing Defects M. Predeleanuand P. Gilormini(Editors) 9 1997Elsevier ScienceB.V. All rights reserved.
143
Processing o f zinc oxide varistors: sources o f defects and possible measures for their elimination A N M Karim, S Begum and M S J Hashmi School of Mechanical and Manufacturing Engineering Dublin City University, Dublin 9, IRELAND
The processing of ZnO varistors primarily follows the conventional ceramic route. A number of dopants are added to zinc oxide powder to prepare the constituent material and some additional finishing operations are performed fgr securing the varistor action. Thus there involved a large number of processing operations and there are naturally multiple sources from which defects can originate. To qualify a varistor performance several parameters are commonly investigated of which energy absorption capability of the device is a significant one. In this respect flaws and defects developed in the varistor body during the processing have detrimental effects and can considerably restrict the energy absorption capability. This paper briefly outlines the probable sources of defects of the metal-oxide varistor device and their possible solutions mainly with sintering orientation, passivation thickness and electrode area.
1. I N T R O D U C T I O N A zinc oxide varistor is an electronic ceramic material which possesses a non-linear current (I) - voltage (V) characteristics with a symmetrical sharp breakdown ~1 - similar to that of a zener diode. But unlike a diode, a varistor can limit overvoltages equally in both the polarities, thus giving rise to an I-V characteristics which is analogous to the two back-to-back diodes. This has enabled it to provide an excellent transient suppression performance. The state-of-the-art of the metal-oxide varistor is represented by a broad range of products manufactured 3 to meet the need of the present day transient voltage suppression. The products cover from arresters for power systems to the low power and low energy application such as integrated circuits, automotive systems and other modem electrical and electronic circuits. There are several critical application parameters some of which are associated with the various regions of the I-V curve. These parameters serve functions in the design and operation of a surge protector. The most desirable device should have a high value of non-linear coefficient or a low value of protective level, a low value of leakage current, a long varistor life and high energy absorption capability. Processing parameters of varistors, though basically follow the conventional ceramic fabrication route, vary to a certain extent depending upon the configuration of the device. The
144 present study is concerned with the cylindrical varistor discs applied for arrester application. The manufacturing route of the varistor disc is presented by the block diagram in Figure 1.
I Zincoxide I
I Additiveoxido I
I
I
I Mixing and Milling I Spray drying I Calcining I Milling I Spray drying I Compaction I Sintering I Lapping I Ultrasonic cleaning I Electroding ! Passivation I Testing
Figure 1. The fabrication procedure of metal-oxide arrester block The application of zinc oxide varistor is selected on the basis of some critical parameters such as non-linear coefficient, nominal voltage, leakage current, energy absorption capability etc. The material and processing parameters such as the green and the fired body, homogeneity, grain size, porosity, varistor chemistry and sintering parameters are identified to affect the energy absorption capability remarkably 49. The life of varistors is largely dependent on the leakage current and the energy absorption capability
145
1.1 E n e r g y A b s o r p t i o n C a p a b i l i t y
Commercial arresters are rated conservatively according to the lowest level of energy at which failures initiate. The failures occur randomly in a lot starting from a low level of energy sustaining up to a high level. The minimum level of energy at which failure starts restricts the enhanced rating of the arrester block. A typical failure distributionS~ is presented below. 100 "
f
80-
60 " II I
E~ 4 0
-
rj
20-
.e,-----
_____._------
9
250
300
350
Energy
400
450
(J.cnl 3)
Figure 2. A typical failure pattern of arrester block in energy test It is obvious that few earlier failures occurring at the lower level of energy absorption capability considerably restrict the rating of the varistor. Prevention of the initial failures would allow enhanced rating of the discs. In that case it will be possible to apply the arrester blocks for more demanding applications or their volume could be reduced proportionately. In other words the same lot will improve the system functional reliability. 2. S I N T E R I N G
ORIENTATION
AND ORIGIN OF FAILURE
The orientation of the cylindrical arrester discs or blocks during sintering is shown in Figure 3. It is observed that the face of a disc remaining in contact with the liner material (bottom face) is not physically as good as the top face. As a result more material needs to be ground off from this face. But even then this face does not become functionally as good as the top face. In the experiment the origin of failure was tracked and it was found that the distribution of failure origin was not same for both the faces. A remarkably higher number of failures were found to
146 Top face
Liner , , ~
J
Bottom
face
I
Figure 3. Sintering orientation and identification of different section of an arrester block originate from the bottom face during the test for the energy absorption capability. The percentage share of failures according to the destruction mark on the face of a disk is shown in Figure 4.
Figure 4. Frequency of failure origin as observed in the test for energy absorption capability This uneven distribution of failure has been correlated by the variation of physical properties of varistors as affected by the sintering orientation. Tensile strength and density gradient of the sintered discs were evaluated.
147
2.1 Sintering Orientation and Tensile Strength The influence of sintering orientation on the mechanical strength of the arrester blocks was evaluated by the diametral compression test. Measurement was conducted on five arrester blocks having a nominal diameter of 32 mm and a height of 34 mm. They were sectioned perpendicular to the axis by a diamond cutter into three equal disc shaped pieces. Load was applied on the disc specimen by an Instron machine with a cross-head speed of 1 mm/min. The breaking load was recorded and the tensile strength of the specimen was calculated. In Figure 5 the tensile strength for the three sections is plotted.
50-
t
40~0 30t~ o~..q t~ tD
t--,
20-
10-
1"
Bottom
!
Middle
Top
Section of arrester disc
Figure 5 Variation in tensile strength due to sintering orientation It is clear that the bottom section, the face which remains in contact with the liner material during sintering is relatively weaker. The strength of this section is about 40 percent less than that observed in the case of the middle or the top section. The top and the middle sections appear to have similar strength. It is clear that at least up to two-thirds of the height of the 34 mm tall disc is not affected by the contact with the liner material. Though it may not be possible from this investigation to quantify how far the effect of contact reaches, it is clear that the bottom section is weaker. This lower mechanical strength of the bottom part of the disc can only be attributed to the influence of sintering orientation.
2.2 Sintering Orientation and Density Gradient To further investigate the effect of the sintering orientation, the density gradient of a fired disc was evaluated. A piece cut from an arrester block was sectioned as shown in Figure 6. The
148 small parallelepiped shaped specimens taken from the eighteen grid points were measured for density. I
/
" .....
~ ......
!
h i 9 ......
:. . . . . .
; ......
I I I i
r
Figure 6: Sample preparation for measuring the density gradient Each specimen was polished with the fine grinding paper to make the surface sufficiently smooth. These were then washed in deionized water using ultrasonic washing bath so that all the loose debris were removed from the specimens. They were then dried in an oven at 125 ~ for more than one hour to expel the trapped water or moisture content completely from the open pores. Weighing was performed in a high precision laboratory type balance by adopting Archimedes' principle. It should be mentioned here that the specimens were wrapped with water-tight masking tape to prevent water from being entered into the open pores during weighing in water. Care was taken so that there could not be any air bubble trapped inside the wrapping. In Table 1 the density of the specimens are given according to the grid points in terms of radius and height. With the grid point densities the contour lines indicating the constant-density were drawn as shown in Figure 7. Table 1 Density (gm/cc) of varistor material at different grid points Radial grid(r) 1 2 3
5.36 5.56 5.59
Grid number along the height (h) 3 4 5 5.55 5.49 5.55 5.57 5.52 5.57 5.56 5.58 5.59 5.58 5.62 5.59
6 5.53 5.61 5.59
2.3 Alternative Sintering Approach This investigation includes the scope of alternative sintering configuration. The evaluation is based on the frequency of regrinding and the energy absorption capability.The objective of this study was to evaluate the feasibility of alternative liner support and sintering orientation of arrester blocks. The sintered discs were characterized to evaluate the effect of the new method.
149
~/ )I~~.. r~~ 5.585
5.5
5.60
~
\ 5.58
5.50
"
I ,
I
Radius Figure 7: Density gradient as a function of sintering orientation
In addition, to enhance the process capability in terms of the performance the foreseeable advantages are (i) reducing the problem arising from regrinding (ii) minimizing the level of bismuth contamination from the liner material due to contact (iii) increasing the scope of repetitive use of the liner material (iv) lowering the allowance of block height for regrinding and (v) better geometry of the disc. The orientation of a varistor disc in conventional sintering operation has already been shown in Figure 3. The modified arrangement for this experiment is presented in Figure 8. It should be mentioned here that for the horizontal sintering, the Vee-groove supports were made from the fired arrester discs. To prevent sticking of the discs during the sintering operation the supports were covered by spreading spinel powder. Spinel is known to have an inhibiting effect on the grain growth ~ and its selection was attributed to keep the dimensional elongation along the contact to a minimum. But the dry powder poured on the surface did not stick to it due to the inclination of the surface. To ensure proper adhesion of the dry liner power with the inclined surface, it was necessary to lightly wet the supports by spraying water. The Vee-groove support facilitates to keep the edges free from any physical contact during the sintering process. Improved faces with uniform edges thus achieved were helpful in
150 reducing the frequency of regrinding operation. Energy absorption capability was also found to be consistent having initial failure at a higher level of energy
Green
disc -
" -
.........
::
Fired
, .
t
:
iI
/ ~i
Liner_
I
Sap.~er
I (a) Front view
(b) Side view
Figure 8" Horizontal or Vee-groove sintering orientation . 3. P A S S I V A T I O N
THICKNESS
Arrester blocks are passivated with glass material to prevent dielectric breakdown when subjected to stresses due to transient electrical surges. The material property and thickness of the coating are very critical in this respect. In this study the effect of glass thickness was evaluated with standard glass material and glassing procedure. Passivation of an arrester block is very critical to prevent the failure by flashover. Thinner passivation leads to more flashover while the thicker passivation results in failure due to problem with heat transfer, usually accompanied by pinholes. Experiment was conducted on 32 mm diameter discs by depositing glass coating of 110, 220 and 340 Ixm. The glass thickness plays an important role in varistor performance. The influence of glass thickness on the varistor performance was assessed by evaluating the energy absorption capability. The plot of the energy absorption capability of the varistors with different glass thicknesses is presented in Figure 9 with the mean energy along with standard error. The standard amount of glass taken as the minimum level in the experiment had shown inferior performance and the varistors started to fail at a very low energy level. Nevertheless, the thickest coating did not exhibit the best performance. The coating of glass acts as an insulator and restricts heat transfer. The effect is more obvious in the second and the third shot in the energy test. The heavier coating acts as higher insulator and does not dissipate heat as effectively as the lower thickness. Thus, the high temperature in the ceramics makes it more vulnerable to failure. This feature is supported by the fact that the discs with the thicker coating failed through the ceramic and most of the discs failed by electrical puncture (number of pinhole (PH) increased substantially) as presented in Table 2. Here interface indicates the zone where ceramic and glass coating meet and FO indicates flashover.
151
t.)
I=
550 500 450 400 350 300 250 200 150 100
i
I
i
110
I
i
220
340
(31ass thickness 0xm) Figure 9 Effect of passivation thickness on the energy absorption capability It was also noticed that for the best cell, 47% of the discs survived after 400 J.cm 3. The mode of failure with various glass thickness was also categorised. It is clear from the figure that the increase of glass thickness has shifted the failure. More than 85% of the varistors failed through the ceramic coated with the thicker passivation coating compared to 27% in the case of the thinner coating. Table 2 Failure mode with different glass coatings during "strength to destruction" test Glass thickness (~tm) 110 220 340
4. E F F E C T
Sample Intersize face 8 15 3 15 14
PH 3 4 8
FO
Failure mode Rupture Interface+ FO
Interface+ PH
PH+FO
-
-
3
-
-
4
3
-
1 1
-
2
2
-
2
OF MARGIN ON ELECTRODE
The margin on electrode as shown in Figure 10 appears to be advantageous in one respect but harmful in the other. This experiment was undertaken to identify the effect of margin on the energy absorption capability. The experiment was conducted by covering the top and the bottom face of an arrester block with aluminium deposited by arc-spray method. Three options were selected (i) control leaving a margin on electrode on both the faces (ii) one face electroded with margin and the other face fully electroded leaving no margin and (iii) both the faces were fully electroded. The three categories of arrester discs were identified by CONT, OFFE and BFFE respectively.
152
(a) Top view
(b) Side view
Figure 10 Electrode with margin as shown on the face of an arrester disc
4.1 Evaluation of Energy Absorption Capability The energy absorption capability was evaluated by the standard testing procedure. The performance of the three cells is demonstrated in Figure 11.
100% -80%
o
-
CONT(F) - - ~ OFFE (F) BFFE (V)
--o- CONT(S) -'- OFFE (S) "- BFFE (S)
60%"
..
40%-r~ 20% 0%200
250
300
350
400
450
500
Energy absorption capability (J.cm "3) Figure 11 Effect of margin on electrode on the Energy absorption capability The sample size was 17, 16 and 15 respectively for CONT, OFFE and BFFE. It should be mentioned here that the test for energy was initiated with a charging voltage of 24 KV which was equivalent to a level of 140 J.cm "3. With the increment of 1.2 KV for every subsequent cycle testing was continued up to 49.9 KV, maximum limit of the generator. But out of 48 discs only 18 failed while the remaining 30 discs survived. Energy absorption capability of the survivor discs was computed on the basis of the data obtained at the last cycle of test at the charging voltage of 49.9 KV of the generator. In the legend the letters 'F' and 'S' in parenthesis stand respectively for the failed and survived disc. From the percentage of discs survived after the maximum possible energy injection for the three cells it is apparent that the full face electrode is conducive to energy absorption capability. However, unsatisfactory results with full face
153 electrode may not be unlikely when the quality of passivation and its thickness are not properly maintained. 5. C O N C L U S I O N S Defects generated in arrester blocks due to their sintering orientation have significant effect than normally perceived. The adverse effects arising from the bottom face cannot be avoided by simply removing more materials through grinding. Investigation revealed that there was a considerable density gradient in the sintered body which is influenced by the sintering orientation resulting in the lowest density at the central part of the bottom face. Horizontal sintering on the Vee-groove support was found to be advantageous in minimizing the density gradient and delaying the earlier failures in energy test. Passivation thickness was found to greatly influence the energy absorption capability of varistors. Neither too thin nor too thick passivation was found to be favourable for superior varistor performance. Thus there is scope of optimizing the amount of glass to be deposited on the side of the arrester block for passivation. Full-face electrode was found to improve the energy absorption capability. However, if the passivation is not proper in terms of the quality and thickness this method may result in poor performance. In such a condition full electrode may result in a preferential path of current flow causing earlier failure and thus lead to a wrong perception about the effect of full-face electrode.
References 1.
M Matsuoka, "Non-ohmic Properties of Zinc Oxide Ceramics", Jpn. J. Appl. Phys., Vol. 10, No. 6, pp 736-46, 1971 Tapan K. Gupta, "Application of Zinc Oxide Varistors", J. Am. Ceram. Soc. 73(7), pp 18171840, 1990 3. "Transient Voltage Suppression Devices 1995", a Databook published by Harris Semiconductor, Melbourne, FL 32902, USA 4. K. Eda, A. Iga and M. Matsuoka, "Current Creep in Non-ohmic ZnO Ceramics " Jpn. J. Appl. Phys., 18 [5], pp. 997-98, 1979. 5. E. Saksaug, J. Burke and J. Kresge, "IEEE Trans. on Powder Del. Vol.4 No. 4, Oct., 1989. 6. W.N. Lawless and T. K. Gupta, "Thermal Properties of Pure and Varistor Zinc Oxide at Low Temperature", J. Appl. Phys., 60 [2], pp. 607-11, 1986. 7. J. Ozawa, et al, IEEE Transac. on Powder App. and Sys.", VoI.PAS102,No. 5, May, 1993. 8. N. Amiji, et al., Advanced Cearmic Materials, 113], pp. 232-36, 1986. 9. K. Eda "Destruction Mechanism of ZnO Varistors Due to High Currents", J. Appl. Phys., 56(10), 15 Nov., pp. 2948-55, 1984. 10. A N M Karim, "Effect of Compaction Parameters and Sintering Configurations on the Performance of ZnO Varistor" PhD thesis,Dublin City University, Ireland, 1996. 11. E. Olsson, G. Dunlop and R. Osterland, "Development of Functional Microstructure during Sintering of a ZnO Varistor Material", J. Am. Ceram. Soc., 76 [ 1], pp-65-71, (1993) 12. S Begum, "Powder Processing Parameters and Their Influence on the Electrical Performance of ZnO Varistor", PhD thesis, Dublin City University, Ireland, 1996.
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Advanced Methodsin Materials ProcessingDefects M. Predeleanuand P. Gilormini(Editors) 91997Elsevier ScienceB.V. All rights reserved.
155
Analysis of Metallic Solid Fractures by Quasimolecular D y n a m i c s Youngsuk KIM a and Junyoung PARK b
aprofessor, Department of Mechanical Engineering, Kyungpook National University, Tae-Gu, 702-701, Korea bGraduate Student, Department of Mechanical Engineering, Kyungpook National University, Tae-Gu, 702-701, Korea
Abstract
Recently, quasimolecular dynamics has been successfully used to simulate the deformation characteristics of solid materials in actual size material. In quasimolecular dynamics, which is an attempt to bridge the gap between atomistic and continuum simulations, molecules are aggregated into large units, called quasimolecules, to evaluate the large scale material behavior. In this paper, a numerical approach using quasimolecular dynamics has been performed to simulate the crack initiation and propagation behavior of a Cu-plate subjected to uniform bending. The bending simulation of the Cu-plate has clarified the effects of aspect ratio and the existence of surface imperfection upon the fracture behavior of the specimen. 1. INTRODUCTION An investigation on fracture behavior related with micro-crack growth of industrial materials is of interest, for example, to engineers, mathematicians, and geologists. Recently, many simulation researches on the microscopic level have been successfully performed in order to investigate the micro-crack growth or fracture behaviour of solid materials in view of atomistic or molecular level using molecular dynamics(MD)[1]. These studies could be possible due to the recent development in computer that deals with a large amount of data with high speed CPU. Nevertheless, the material size that can be analysed is limited because the material of actual size is composed of an astronomic numbers of molecule. Recently, Greenspan's quasimolecular dynamics(QMD) approaches make this possible, in that molecules are aggregated into large units, called virtual quasimolecules[2,3]. Moreover, a few investigations of QMD to industrial applications have been performed[2,3,5,6]. In this paper, the fracture behavior of Cu-plate subjected to uniform bending is investigated numerically within the framework of Greenspan's QMD method. Moreover, the effects of aspect ratio and the existence of surface imperfection upon the fracture
156
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o o o o o o o o . . . . . .
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---" --
Potential Function " Force Function
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00000000000000081
1
40
Figure 1. The initial arrangement for the thick specimen of Cu-plate under bending
Figure 2. Potential and force function
behavior are clarified. 2. BASIC ANALYSIS Let us consider a rectangular Cu-plate which is about 8• The 6-12 Lennard-Jones potential function for two copper atoms r A apart is given by [3]. ~(r)=-
1"398068x10-1~
r6
+
1"55104x10-~
rl z
(erg)
( 1)
From equation (1), the force F interacting between two atoms is derived as follows F ( r ) = --
8.38840 x 10 - 2 r7 +
1.866125 x 10 rl 3
(dyn)
(2)
The minimum of ~ ( r ) results when F(r)=0, at r=2.46/~. We assume that the Cu-plate is composed of the structure of Faced-Centered Cubic, and also the distance between two neighbors is 0.2cm on the plane (1 1 1). Then, the resulting arrangement is shown in Figure 1. The number N* of atoms in the plate is, approximately, N* = 8 x 108 . 19.9186 x 108 - 30.41123707 x 1016 2.46 2.13
(3)
A quasimolecular mass m from total mass is given by m = N* x 1.0542 x 10 z~ / 4698 = 6.824079 x 10 - 19
(g)
(4)
in which we assumed that the total mass is distributed over the 4698 qusimolecules. The total energy E* of the system of atoms is, E*=
3 x (30.41123707x 1016) x (-3.15045 x 10-13) = - 2 . 8 7 4 2 7 2 5 x 105(erg)
(5)
157 Now assume that the force F between two quasimolecules is given by G H F(R) -- ~ + R5
(6)
in which R is measured in centimeters. From the condition, that F(0.2)=0, and total energy E equals E*, equation (6) takes the specific form F(R) = -
3.262974315 R:~ +
0.130518972 R5
(7)
The force acting between two molecules which are seperated by a distance of 4.92 A, i.e. twice the mean distance 2.46A between two molecules, can be negligible due to its very small value comparative to weight. We can also apply this property to quasimolecules. Now, we introduce a normalizing constant a such that at a distance 0.4era the force between quasimolecules is small relative to their weight. If we define "small relative to weight" to mean 0.1% of the effect of weight, then we must have a]-3.262974315.(0:~3 + 0 "~_~5_~972 II <
(0.001)" 980m
which results in our choice a = 1.25X10~~ of each quasimolecule is m
(8)
The dynamical equation for the motion
d2Ri -~0) [( 3.262974315 0.130518972) Rii dt 2 - (1.25 • 10 ~, ~o).~ + (Ro)5 ~
(9)
in which R; is a position vector of the point Pi, Rii is a position vector with initial point Pi and terminal point Pi- The summation is taken over the neighbors of Pi. From equation (5), and introducing the computationally convenient transformations R*=4R, TZ=10tz, equation (9) reduces to d2R~dve -- ~-]'[( -
1"530099184+0"979263473) R i * ' ( R (R~)5 ~ j ) 3 --:TRii
(10)
Now, for solving eqn (10) numerically, we adopt the leap-frog method, a kind of the Verlet method which is frequently used in the MD simulation. For positive time step At, let tk=kAt, where k=0,1,2, 9..... . Here, we assume that the number of particles is N. Also, at tk let Pi having mass mi be located at ri,k, have velocity Vi,k, and acceleration a~,k. Then the leap-flog formulas, which relate position, velocity and acceleration are expressed as follows
$,.~/2 =
$;.o+ (z/t) 2
"
-~,. /m, .o
Vi.k+1/2-- V i.~-l/2+(At) F i . k / m i , k = 1,2... ri.~,+l= ri.k+(At) Pi.k+l/2 , k = 0 , 1 , 2 , . .
(11)
158
,L,9
120"
h
h=a H, a=0.025
A'
9 " "iiiiiiiii'i!'!i'iii'!i'"''"" 9 9 9 oe*
e
.
9
"!i'!!!!! !!'!i!!!'!! i!'!i i!'!i!" Figure 3. The initial imperfection of the specimen on the upper convex surface
Figure 4. Displacement boundary condition of quasimolecule at both ends
In order to simulate the crack initiation and propagation behavior in QMD it is necessary to break the quasimolecular bond when the force acting between two neighboring quasimolecules reaches a certain limit. In this study we introduce the separating criterion of quasimolecular bond recommended by Ashurst and Hoover[4], there the quasimolecular bond is broken when the distance between two neighborhood quasimolecules reaches the distance R* at which dF/dR* first becomes negative. From eqn (10), then, R*=l.06667(Figure 2). 3. NUMERICAL SIMULATION MODELS Figure 1 shows the initial arrangement of quasimolecules for a relatively thick specimen, (called Model I case). In order to generate naturally the crack initiation and propagation under tensile strain a geometrical imperfection of the specimen is introduced as a V-notch shape on the upper convex surface of the specimen, as shown in Figure 3. The size of the specimen of Cu-plate is a 8cm X 19.9186cm which gives an aspect ratio of 1:2.5. To compute the position change and the force for every quasimolecule, a constant-displacement boundary condition at the both ends was applied. During pure bending simulation, the arc length of neutral axis of the specimen is kept at a constant value and the quasimolecules of both ends are assigned to turn by unit degree per unit time as shown in Figure 4. The boundary conditions for bending simulation are given by
=
cos ~0 cosAO - s i n AO][ ~ ] -
[ s i n AO
'
1 I AO
AO -
l AO
" sinAO "
]
(12)
cosAO
where i means the half length of the neutral line. The Q matrix represents rotation transformation of the coordinate axis and the D matrix is introduced to take into consideration that the neutral line of the specimen has a curvature in proportion to
159 turning angle. The turning angle 0 is measured from the vertical line to quasimolecules on the end line. A time step AT is taken as 0.1 and degree step A O is kept at 0.01. So, dO /dY=0.1 (deg/time). 4. N U M E R I C A L RESULTS AND DISCUSSION Figure 5 shows the initial state and arrangement of the quasimolecules of the thick specimen with V-notch on the upper surface(Model Case I). The deformation behaviors and crack propagation tendency during bending operation are shown in Figures 6 to 10. Their figures correspond to turning angles of 0 =17~176176 ~ and 22 ~, respectively. Until the specimen bends to 0 =17 ~ (Figure 6), the specimen seems to deform uniformly with pure bending, in which generally the upper convex surface is under
Figure 5. The initial state of thick specimen with notch before bending
Figure 6. Deformed state of thick specimen with notch at 0 =17 ~
Figure 7. Deformed state of thick specimen with notch at 0 =18 ~
Figure 8. Deformed state of thick specimen with notch at 0 =19 ~
Figure 9. Deformed state of thick specimen with notch at 0 =21 ~
Figure 10. Deformed state of thick specimen with notch at 0 =22 ~
160 tensile stress and the lower concave surface under compressive stress. However, when the turning angle reaches /9 =18 ~ (Figure 7), a visible crack generated from the notch occurs due to the high tensile stress acting on the upper convex surface, at which the bond of quasimolecules are firstly broken. The crack generated from the notch propagates deep into the specimen as the specimen continues to bend. Figure 8 for the deformed state of /9 =19 ~ illustrates that the crack propagation extends deeply into the center of the specimen. As can be seen in the figure, we can see a more realistic crack propagation pattern in that firstly the front of the crack may propagate into two directions, an almost vertical direction to the neutral axis and is an inclined direction to the vertical direction. This crack propagation into two directions ceases as the bending of the specimen proceeds, and the crack propagation along a major direction becomes predominant as the bending proceeds. This pattern of crack propagation is repeated during the bending operation. Therefore the rough fractured surface will appear. This crack propagation tendency is similar with the actual fracture phenomena of the ductile materials subjected to bending stress. Figure 9 shows the deformed state at /9 =21 ~ In this case, the crack proceeds almost up to the neutral axis of the specimen in the major direction, also there we can observe the rough fractured surface due to the stepped propagation pattern as discussed above. Moreover at the lower concave surface of the specimen some dense quasimolecules area subjected to high compressive stress seems to appear. This dense area of quasimolecules at the concave side of the specimen may contributes to the development of plastic hinge(compression instability) there. Figure 10 shows the deformed state of /9 =22 ~ , at which plastic hinge on the concave side is fully developed. High energy is accumulated at the plastic hinge area as expected from
Figure 11. Deformed state of thin specimen with notch at • =40~
Figure 12. Deformed state of thin specimen with notch at 0 =45 ~
eqn (1) because the density of molecules is high there. Next in order to clarify the effect of the thickness upon the fracture behavior of the specimen, simulation for a relative thin specimen of aspect ratio 0.32:2.5(called Model II case) was performed under the same boundary condition with case I. Figures 11,~,12 show the deformed states of /9 =40 ~ and /9 =45 ~ respectively. In Figure 11 the comparatively thin plate bents uniformly until /9 =40 ~ without any crack propagation around the notch area and any dense quasimolecules area at concave side. This deformation characteristic of uniform bending of thin plate is proven everywhere by actual phenomena. From the comparison between the deformed state of thin material at Figure 11 and that of thick material, it is certain that in spite of the
161
Figure 13. Deformed state of thick specimen without notch at 0=22 ~
Figure 14. Deformed state of thick specimen without notch at 0=24 ~
Figure 15. The force distribution along y axis position of 0 =17 ~
Figure 16. The force distribution along y axis position of 0=18 ~
Figure 17. The force distribution along y axis position of 0 =19 ~
Figure 18. The force distribution along y axis position of 0=21 ~
existence of the same magnitude of imperfection, the thin material deformes more uniformly than the thick material. This could be explained as follows. In pure bending, the magnitude of tensile stress acting on the upper fiber of the specimen is almost proportional to the thickness of the specimen, and therefore the tensile stress acting on the upper surface for the thin specimen is lower than that for the thick specimen. The similar analogy could be introduced in the case of the concave side. This makes the thin material more deform uniformly without any fracture occurring at
162 convex side and any dense quasimolecules at concave side. The fully developed V-shape fracture is observed in Figure 12 for the deformed state of 0 =45 ~ With comparison to Figure 10 of thick specimen it is certain that the fracture of the specimen develops at the front of the V-notch only in a direction perpendicular to the neutral axis. Moreover the fractured surface for the thin specimen is more fine and has a regular V-shape. Figures 13~,14 illustrate the deformed states at 0 =22 ~ and 24 ~ in the case of no surface imperfection for the thick specimen. These case (Model III case) was chosen to investigate the effect of the existence of the surface imperfection, V-notch, upon fracture behaviors of the specimen. Figure 13 shows no signs of fracture occurring on both sides of the specimen until the specimen bends to 0 =22 ~ However, in Figure 14 for the case of 0 =24 ~ some dense quasimolecular area in shear band-like form is developed at the concave side. As the specimen continues to bend this process becomes more remarkable. Therefore, it can be concluded that the specimen with no notch is apt to produce the compression instability at the concave side but not tensile fracture at convex side in the bending operation. Figures 15~-18 illustrate the longitudinal force distributions interacting between two neighborhood quasimolecules along the y-axis positions of the quasimolecules with respect to the turning angle in Model I case. In these figures X-coordinate represents a magnitude of longitudinal force and Y-coordinate represents the y-axis position of quasimolecules. The measurement of the interacting forces was performed at the vicinity of the notch area because of stress propagation velocity. Here, the forces at the quasimolecules locating between the 2755th and the 2794th molecules along A-A section are measured as shown in Figure 5. In case of Figure 15 for bending angle of 0 =17 ~ the longitudinal forces of each quasimolecule are distributed almost uniformly with proportion to the y-distance of each quasimolecule from the neutral axis, center line of the specimen. This assures that t h e uniform bending deformation preserves at 0 =17 ~ as discussed in the deformed state at Figure 6. However, when the crack on the convex side propagates deeply into the center of the specimen as shown in Figure 7---10 according to increase of the bending angle, the longitudinal forces which were acting on the fracture occurring area are released rapidly. Therefore, the magnitude of the force acting on the fractured area finally approaches to zero as can be seen in Figure 16, Figure 17 and Figure 18 corresponding to 0 =18 ~ = 19~ and 0 =21 o, respectively. In Figure 16 for a bending angle of 0 =18 ~ the proportional force distribution like Figure 15 is firstly broken and the force acting at the upper convex surface diminishes. Next the area where the force approaches zero becomes wider in Figure 17 for the case 0 =19 ~. The state shown in Figure 18 for 0 =21 ~ depicts that the force acting on the fractured area at the convex side becomes almost zero and therefore the fractured area transmits no more force under bending operation. Moreover, we could found the compressive unstable states at the concave side which
163 implies the possibility of the occurrence of a plastic hinge as discussed in Figure 9. 5. CONCLUSIONS In general the numerical simulations of quasimolecular dynamics show that the investigation of material behavior is faster and more simple than any other method, for instance, FEM using the continuum model or MD using the discrete model. Actually, we can compute the bending operation for a 8cm• model within 25 minutes with a 133MHz PC. On the contrary, it is impossible to compute this model by molecular dynamics in reasonable speed because of the limit of computer capacity. This is a best merit in quasimolecular dynamics. According to the results of the quasimolecular simulation for bending operation we could clarified the fracture behavior and crack propagation pattern of the specimen. The investigations on the effect of the specimen size make clear that the thick plate reaches the material fracture more early than the thin plate under the same magnitude of surface imperfection. Moreover, comparison between notch specimen and no notch specimen reveals that the specimen without notch continues to deform without denoting any fracture at the convex side until 0 =24 ~ However, at the concave side a dense quasimolecular area like a shear band localization is observed. From this study we can conclude that the bending operation is very sensitive to the imperfection at the outer surface and as the thickness of the specimen thickens the tendency becomes more remarkable. REFERENCES
[1] [2] [3] [4] [5] [6]
Celis B.de, Argon A.S., and Yip S., J.Appl.Phys., 54, 4864(1983) Greenspan D., Comput.Struct., 22, 1055(1986) Greenspan D., J.Phys.Chem. Solids, 50, No.12, 1245(1989) Ashurst W.T. and Hoover W.G., Phys.Rev., B14, 1465(1976) D.K. Choi and H.K. Ryu, Proc. KSME Spring Conf., A,102 (1996) Y.S. Kim and J.Y. Park, Proc. the 47th JSTP Conference ,457(1996)
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Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
165
D a m a g e f r a m e w o r k for the prediction of material defects: identification of the d a m a g e material parameters b y inverse technique F. Lauro a, T. Barribre a, B. Bennani a, P. Drazetic a and J. Oudin a alndustrial and Human Automatic Control and Mechanical Engineering Laboratory, CNRS Research Unit, MECAMAT, CSMA, University of Valenciennes and Hainaut-Cambresis, Le Mont Houy, B.P. 311, 59304 Valenciennes Cedex, France.
1. I N T R O D U C T I O N The finite element method is a precious tool to analysis the crashworthiness during the design phases of vehicles, trains and trucks or the formability of metal sheet by stamping [ 1-2]. Under the large dynamic loading involved, large strain and strain rate values appear, leading to consequent damage in metallic parts such as carbon steel or aluminium alloy. The evolution of the micro-structural damage in the range of 20 s-1 to 500 s-1, ---the strain rate values generally observed during crash analysis or stamping processm, is similar to that observed for moderate strain rate values. Ductile fracture consequently occurs in the polycrystalline materials. The micro-structural damage process is characterized essentially by the decohesion at the interface between the matrix and inclusions or second phase particles. This is followed by the growth of voids leading to a macroscopic fracture responsible for the ductile fracture [3-4]. The high speed loading increases the material ductility by delaying the striction instabilitie occurrences. Nevertheless, the damage process remains very similar to that described for moderate strain rate [5-6]. Successful results have been obtained for damage prediction in the metal forming processes by using a specific elastoplastic potential available for porous materials [7-9]. This previous damage model has been implemented in the explicit finite element code for non-linear dynamic analysis PAM-SOLID TM. New elasto-viscoplastic porous material law has been developed in the case of convected shell elements [ 10] using the evolution of the porosity with growth, nucleation and coalescence of the microvoid volume fraction [ 11-12]. Accurate prediction of the damage evolution using the previous damage model implies those material parameters of this model are well defined. These damage material parameters are essentially the initial porosity, the geometrical form of the microvoids, the inclusion volume fraction, the mean nucleation strain and the value of the microvoid volume fraction at coalescence onset. Due to their microscale, the identification of these parameters by direct observation needs expensive measurement machines, and complex measurement techniques Which challenge the role of numerical simulations. To determine the damage material parameters of the proposed damage model, an original identification technique is developed. This identification technique is based on the inverse method. The variation due to straining of a macroscopic variable strongly dependent on the material parameter values, is measured experimentally. An optimisation technique is then used to determine the values of the material parameters which correlate the experimental variation of the macroscopic variable and its numerical prediction using the finite element method. The damage material parameters are identified by using a tensile test of a notched bar and measuring the variation of the inner radius of the specimen during straining as experimental setup. A numerical sensitivity analysis showed the strong dependence of this geometrical variable
166 on the damage material parameters. The optimisation of the damage material parameters is undertaken by interfacing the optimiser and finite element software. The optimiser is based on a classical conjugated gradient method [ 13]. In this paper, the damage model formulations are initially described. The identification technique of the damage material parameters by inverse method is then defined. The experimental set-up and optimisation technique are described. Finally, the application of the identification technique for a mild steel is shown. 2. CONSTITUTIVE DAMAGE MODEL The plastic flow of the porous elasto-viscoplastic material is described by the Gurson modified by Tvergaard potential defined by O2f + 2qlf*C~ ( 3 q2 ~M Om) - ( 1+ q3 f*2 ) with Om >0 f~evp = ~--~M or
Oef
(
)
~evp = ~--~M+ 2qlf* - 1+ q3 f*2 with ~rn < 0
(la)
(lb)
in which Gef is the yon Mises effective stress defined in its deviatoric forms by O'er=(3/2 sij:sij) 1/2 with sij the deviatoric stress tensor, ql, q2, q3 are the material parameters dependent on the microvoid form, ~M is the elasto-viscoplastic flow stress, ~m is the mean stress and f* is the specific coalescence function. The evolution of the micro-structural damage is represented by the current void volume fraction f defined by f = Va - Vm
(2)
Va where Va, Vm are, respectively, the elementary apparent volume of the material and the corresponding elementary volume of the matrix. The rate of increase of the microvoid volume fraction is given by t" = t'n + t'g
(3)
in which i'g and i~n are, respectively, the growth and nucleation rate of microvoids. The rate of increase of the microvoid volume fraction, due to the growth of existing microvoids, is determined by requiring that the matrix material must be plastically incompressible. The plastic incompressibility relation leads to the following expression _
VA - (1- f)trDP
(4)
in which trDP is the first invariant of the macroscopic plastic strain rate tensor. Assuming plastic strain controlled microvoid nucleation, the nucleation microvoid volume fraction rate is expressed by
167
l'n =
fN 2(, SN SN 2 . ~ e
(~M = AII~M
(5)
where fN is the nucleated microvoid volume fraction, SN is the Gaussian standard deviation, 8N is the mean effective plastic strain for nucleation, s is the effective plastic strain and A1 is interpreted as the volume fraction of particles converted to microvoids per unit plastic strain. The coalescence of neighbouring microvoids leading to the rapid loss of the stress carrying capacity is described using the specific function f* into the microvoided material potential as shown previously. The coalescence phenomena occurs for a determined level of microvoid volume fraction as described f* = fc + fu - fc ( f _ fc) when f > fc
fF-fc
(6a)
or f* = f when f < fc (6b) where fc is the critical microvoid volume fraction at coalescence onset, fF is the microvoid volume fraction at ductile fracture occurrence which corresponds to the complete vanish of the carrying stress capacity where the von Mises effective stress is equal to zero and fu is the corresponding value of the coalescence function fu=F(fF). The effective plastic strain rate 8M is computed from the equivalence between the plastic power dissipated into the material and into the corresponding matrix, o:D p ~:M = (1- f) O"M
(7)
in which r is the Cauchy stress tensor, (YM is the elasto-viscoplastic flow stress and DP is the macroscopic plastic strain rate tensor defined in the case of the associated plasticity by Dp = ~ (~"2evp . b~
The viscoplastic multiplier ~, is deduced from the consistency condition
(8)
~'~evp=0 and
~evp = 0 leading to solve
~)~'2evp "t~= 0 ~evp = ~evp = c ) ~ e v p . 6 + c)~evp "(~M + J~GM J~ " The plastic multiplier is then expressed by
(9)
168
(lOa)
--O~evp .ce.O~'~evp - ~'2evp O(~M O~-evp[ O(I OG OGM OEM A3 - t)f (1-
f)
OO,evp.I+A4 ] 06
6: O~evp 06 M and A 4 = A1A 3. with A 3 _- ( 1 - f)6
(10b)
in which C e is the isotropic material tensor and I is the second order identity tensor. The previous constitutive damage model is implemented in the three-dimensional, finite element, explicit, code PAM-SOLID TM as a new constitutive material law. The implementation of the constitutive damage model modified the stress calculation algorithm. The stress calculation is based on an elastic prediction and plastic correction schema. 3. I D E N T I F I C A T I O N OF THE MATERIAL PARAMETERS The material parameters of the previous damage model are quantified with difficulty by direct experimental measurements. Thus, an original inverse identification technique is used. This technique is based on the determination of the material parameters rninirnising the cost function representative of the correlation between the partial macroscopic response of a mechanical test, obtained by experimentation and numerical simulation. The cost function expressed by the least square approximation is given by
nb_point [Zi
sim
(~)-Zi
exp
2 ]
Q(ct)
i_-,
(11)
[ sim
where o~ is the material parameters, Z i and Z ex~ are the simulated and experimental macroscopic responses and nb-point is the number of experimental points of the experimental response. An optirniser is used to find the material parameters minimising this cost function. An iterative method using the cost function value as criterion and a convergence method taking the information about gradients and second derivatives (Hessien) of the cost function into account are the most efficient. The optimiser OPTB2L was developed with a conjugated gradient method and a Davidon Fletcher Powel or BFGS method. Two criteria are used to stop the optimisation. A global convergence on the cost function defined by Z[ im ( e ) -
Z exp
Z~XP
___131
and a stability criterion of the material parameters defined by
(12)
169 o~k _ ~k-1 ---132 ~k-1
(13)
where [31 and [32 are values given by user and k is the iteration index of the minimisation algorithm. The optimiser OPTB2L is linked with the finite element code to automate the identification of the material parameters (Figure 1). The development of this tool on a modular concept allows an interfacibility with all the finite element codes. User interface ) Optimisation ~ control
~ Initial parameters and zexp
Optimiser OPTB2L
l
~fi
Interface ~ x l Finite element creation of the ~ - ~ code nite element file) I
Data analysis L zsim(o0 ~ r
Fig 1. Flowchart of the identification by inverse method. The previous identification technique is used to determine the material parameters of the damage model. The tensile test of a thin notched specimen is used as the mechanical test to measure the macroscopic response Z. This set-up is the variation of the inner radius of the specimen in function of the elongation. The specimen and the experimental set up are shown on figure 2. The corresponding finite element modeling of the specimen is defined to ensure the independence of the variation of the inner radius beside the modeling (Figure 3).
170
Fig 2. Geometrical description of the thin notched specimen and position of the two extensometers on the tensile machine.
Fig 3. Finite element mesh, boundary conditions and loading for one quarter of the notched tensile specimen.
171 The numerical sensitivity analysis of the material parameters on the variation of the inner radius is undertaken by increasing each parameter by 10% and calculating the norm defined by
norm
--
width(t~ i + 10%)- width(~ i ) X (Xi width(oq) 10%
(14)
This sensitivity analysis is performed with standard elastoplastic material defined by ~M = t~y + C e~ with t~y=290 MPa the elastic yield stress, n=0.22 the hardening coefficient and C=770 MPa the plastic consistence, eM the effective plastic strain. The material damage parameters are typical for a mild steel, ql'-1.5, q2=1 and q3=2.25 for the elasto-viscoplastic potential, f0= 10 .5 for the initial void volume fraction, fN=0.04, SN=0.1 and eN=0.2 for the nucleation and finally fc=0.15 fF=0.25 for the coalescence. The sensitivity analysis results are shown on figure 4.
Fig 4. Sensitivity of the material damage parameters on the width evolution in the inner radius in function of the elongation of the notched specimen. The material parameters identification is divided into two stages. The initial stage corresponds to the straining up until the beginning of the coalescence phase and the second stage corresponds to the coalescence phase. The most important parameters for each stage are the behaviour law parameters C and n, the nucleation parameters eN and fN for the first one and the coalescence parameters fc and fF for the second one. The identification of the material damage parameters for a E24 steel is performed. The experimental flow stress is deduced from the tensile test of an unotched specimen (Figure 5). Due to the form of the experimental flow stress, the flow stress is described by successive tangent modulus (Table 1).
172
300-
r~
200-
r~
0~:
100-
I I 0.1 0.2 Effective plastic strain
I 0.3
Fig. 5. Experimental flow stress. Table. 1 Flow stress for a E24 steel Et 192.5 5240 225 235.5 OM
1374 248.9
1184 269
845.8 292.1
557 319.9
The ductile rupture of the specimen appears for a 0.25 effective plastic strain. The last tangent modulus is then extended up to the value of the effective plastic strain equal to 2. The numerical simulation of the tensile test of the notched bar is performed by using the experimental flow stress and the reference damage parameters. The corresponding evolution of the inner radius of the specimen in function of the elongation is compared with the experimental one. The important gap between the two curves is essentially due to the appearance of striction in the tensile specimen. The only determination of the damage parameters will be unable to correct the curve. Hence the first optimisation is undertaken on the last tangent modulus with the reference damage parameters. The value of the new tangent modulus is Et=530 MPa and is found with 18 solver calls. With approximately a variation of 5% of the tangent modulus, the correlation between the two curves is better (Figure 6). A second optimisation is made by adding the most influent damage parameters; the nucleation parameters fN and eN and the void form parameter ql. The optimisation is achieved after 292 solver calls and the values are fN=0.03372, eN=0.335 and q l = 1.295. The correlation between the experimental and computed curves is very good. However, a slight divergence appears at the end of the tensile test which is mainly due to the deterioration of the finite elements in the necking zone and the non determination of the other damage parameters. The experimental tensile test was performed until the ductile rupture of the specimen. It has been impossible to experimentally record the propagation of the cracks corresponding to the coalescence phase. The tensile test of the notched specimen is not sufficient to identify the coalescence parameters.
173 To predict the ductile fracture of the notched specimen, the value of fc can be taken at the value of the porosity at the end of the numerical simulation which is equal to fc=0.14. The value of fF is very close of fc and can be taken to fF=0.15. The evolution of the'porosity is presented in figure 6. The material damage parameters are identified and the simulation gives a good correlation with the experimental results.
Fig 6. Experimental-numerical comparison of the width evolution in the inner radius of the notch in function of the tensile specimen elongation for different optimisations.
Fig 7. Microvoid volume fraction evolution in the tensile specimen for the elongation equals to (a) 6 mm and (b) 9 mm.
174 4. C O N C L U S I O N The improvement of the numerical simulation for crashes or forming processes under dynamic loading takes the prediction of the ductile rupture into account. The present paper proposes a coupled damage-mechanical model for strain rate dependant voided material. The microvoid nucleation, growth and coalescence are modeled. The ductile fracture is predicted at the end of the damage process. This model has been implemented in the explicit finite element code for non-linear dynamic analysis PAM-SOLID TM for elasto-viscoplastic porous material for the convected coordinate shell elements. The numerous material damage parameters must be defined to make the damage model available. An inverse method has been developed by coupling an optimiser and the finite element code. An experimental tensile test of the notched specimen is used to determine the variation of the inner radius of the specimen in function of the elongation. This measurement is compared with that obtained by numerical simulation and the material damage parameters are identified by minimising the gap between these two measurements. An application is made with a E24 steel. The material damage parameters are identified accurately. The influence of the damage evolution on the variation of the inner radius is shown and ductile rupture is well predicted. REFERENCES
1.
E. Haug and D. Ulrich, Second European Cars/Trucks Simulation Symposium, West Germany, 1989. 2. E. Haug, J. Clinckemaillie, X. Ni, PAM '95, (1995) 443. 3. F. Montheillet et F. Moussy, Physique et MEcanique de l'Endommagement, Editions de Physique, Paris, 1986. 4. Y. Ravalard, J. Oudin et J.C. Gelin, Physique et MEcanique de la Mise en Forme des M6taux, F. Moussy et P. Frangois Eds, Presses du CNRS, Paris, (1990) 248. G. Pluvinage, 6~me Colloque MEcanique et Mat&iaux de Tarbes, 1987. 6. M. Lacomme, A. Froger, J.P. Ansart et R. Dormeval, Congr6s DYMAT (1988). 7. B. Bennani, P. Picart and J. Oudin, Engng. Comp., 10 (1993) 409. 8. B. Bennani, P. Picart and J. Oudin, Int. J. Dam. Mech., 2 (1993) 118. 9. L. Lazzarotto, P. Picart and J. Oudin, Comp. Mat. Sci., 5 (1996) 167. 10. T. Belytschko and C.S. Tsay, Comp. Meth. Appl. Mech. and Engng., 42 (1984) 225. 11. A.L. Gurson, J. Eng. Mat. Tech., 2 (1977) 95. 12. V. Tvergaard, Int. J. Solids. Structures, 18 (1982) 237. 13. P. Trompette, F. Fleury et C. Knoft-Lenoir, Optimisation des structures, approche de l'ingdnieur, Institut pour la promotion des sciences de ringdnieur, Pads. .
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
175
D a m a g e i n f l u e n c e in the finite e l e m e n t c o m p u t a t i o n s for l a r g e s t r a i n s elasto-plastic m e c h a n i c a l s t r u c t u r e s P. Picart, G. Piechel, and J. Oudin Industrial and H u m a n Automatic Control and Mechanical Engineering Laboratory, CNRS Research Unit 1775, MECAMAT, University of Valenciennes, BP 311, 59304 Valenciennes Cedex, France
1. INTRODUCTION The efficient development and optimisation of cold forging technology need more and more use of the finite element analysis. The up-to-date modelling of large strain mechanical problems requires to take into account the constitution of the material microstructures. The initial microstructures of engineering materials such as steels and aluminium alloys are often more or less microvoided, or inclusions content. A realistic prediction of the damage evolution during the forging sequence allows us to avoid defects initiation and control the final physical caracteristics of the mechanical parts. The microvoids come from heterogeneities, second-phase particles (inclusions) and precipitates existing within the material. Up to now, a lot of microscopic observations have shown three main phases to describe the microvoid evolution [1]. First, new microvoids may appear by nucleation that mainly occurs either by decohesion at inclusionmatrix interfaces, or by inclusions or matrix fractures. Second, as the strenghtening of the materials continues the macroscopic effective plastic strain increases, the growth of existing microvoids is observed. Third, the coalescence of microvoids may appear and leads to the ductile rupture of the material. The prediction of the evolution of the microvoid volume fraction in the material has been previously performed with a constitutive model which involves a set of eight supplementary material parameters [1]; three parameters for the nucleation, three parameters for the growth and two for the coalescence. The nucleation rate has been found to be mainly related to the effective plastic strain in the matrix or the effective isotropic yield stress and macroscopic hydrostatic stress. The damage goes on with the growth of the existing microvoids. The material matrix is considered as incompressible. So the growth phase is related to the apparent volume changing. The ultimate step during the microvoid increase is the coalescence of microvoids leading to ductile rupture. Most available computations have been concerned with a restrictive set of
176 material parameters and consequently, a set of finite element computations is achieved in the large strain non linear framework to enhance the sensitivity of the material parameters in regard to computed results. The sensitivity is measured by the relative variation of either local mechanical variables such the microvoid volume fraction, the effective plastic strain and the hydrostatic stress, or global mechanical variables such displacement of free boundaries and surface forces in comparison with microvoid free material. For that aim, a tensile test of a single axisymmetric Q4 element is proposed, because the hydrostatic stress is always positive and permits the immediate activation of the microvoid evolution. Significant results are obtained and discussed [2]. It is clearly shown that the initial microvoid volume fraction, the effective plastic strain at incipient nucleation and the potential nucleated microvoid volume fraction are the most sensitive material parameters. The consequences of parameters variations on global and local mechanical variables have been evaluated and permits the characterisation accuracy to be fixed To complete these previous analysis, a new set of finite element computations is presented to analyse the influence of the initial microvoid volume fraction density and distribution in the matrix. So a tensile test of a patch of one hundred axisymmetric Q4 elements is proposed, because various initial microvoid volume fraction values can be used for each finite element. The influence of the initial density and distribution of the microvoids is measured by relative variation of the local and global mechanical variables. Finally, a specific forging test for the preform design of a H-shaped cross sectionnal axisymmetric workpiece is proposed to complete and validate the previous results but also to enhance the actual data base related to cold forging processes.
2. CONSTITUTIVE MODEL In ABAQUS code, the damage evolution for elasto-plastic structures including large strains follows two phases; firstly, the nucleation of new microvoids, either by decohesion of the matrix/particle interface, or by second phase particles fracture, that involve three material parameters and second, the growth of existing microvoids that involve three other material parameters. The nucleation rate is mainly controlled by the effective plastic strain and therefore, depends on the mean effective plastic strain value of the matrix at incipient nucleation EN, the Gaussian standard deviation of the normal distribution of inclusions SN and the nucleated microvoid volume fraction related to the inclusions volume fraction fN. The growing rate depends on three material parameters, two parameters which are used in the Gurson's elasto-plastic potential for porous material [3], ql and q2, the third one is the initial microvoid volume fraction in the material matrix fi. A bibliographic analysis of the research works about the damage material parameters characterization [2] allows us to define for the finite element
177 computations a set of reference values for each of them; s ql = 1.5 and q2 = 1.0.
=
0.2,
SN
=
0.1, fN = 0.04,
3. FINITE ELEMENT COMPUTATIONS 3.1. Introduction A typical stress-strain relationship used in the computations refers to a 1070 annealed steel behaviour at room temperature: for the elastic behaviour, the Young's modulus is equal to 200 GPa and the Poisson's ratio is equal to 0.3; for the
plastic behaviour, the yield stress of the matrix is equal to
650{l+EM0"18} in
N . m m -2, the effective plastic strain being in the range of 0.05-0.7. 3.2. Tensile test To analyse the sensitivity of the initial microvoid volume fraction distribution a set of finite element computations are achieved for a tensile test of a patch of one h u n d r e d axisymmetric Q4 elements. Initially, height and radius of the workpiece are fixed to 1 mm, then a prescribed axial displacement from 0 to 1 mm is applied to the upper nodes, which corresponds to a 100% axial_elongation of the workpiece in order to obtain a range of moderate effective plastic strain from 0 to 0.69 (Fig. 1). To focus the analysis on the initial microvoid volume fraction distribution, a first set of calculations are achieved as reference, with an h o m o g e n e o u s distribution of the initial microvoid volume fraction in the workpiece for three different values of the initial porosity fi respectively, 0., 0.001 and 0.01. Then, three random distributions are tested and compared to the reference results.
Qeomet~: a=b=lmm
Hxed displacements: Uz=lmm
Figure 1. Tensile test of a patch of 100 axisymmetric Q4 elements, dimensions and boundary conditions.
178 For the first set of computations, the initial microvoid volume fraction is the same for all elements. The final values of the mechanical variables for 100% axial_elongation are given in Table 1 as reference. Table 1 Reference mechanical values for the tensile test of a patch of 100 axisymmetric Q4 elements, ,,
fi Initial porosity
r Effective plastic strain
am Hydrostatic stress (N.mm -2)
f Final porosity ,
0 0.001 0.01
374.1 372.6 359.5
0.6674 0.6667 0.6598
Ur Radial (mm) displacement
,,
0.0623 0.06504 0.08367
0.2808 0.2803 0.2765
The second set of finite element computations involved, refers to three random distributions of the initial porosity. For each one, 10 elements have an initial porosity and the other are pore free elements (Fig. 2). All distributions are tested for two different values of the initial microvoid volume fraction fi, 0.001 and 0.01. The mechanical variable values are presented and compared to the reference values in Table 2.
mmm mmmmmmmmmmmmmmmm mmm mm mmmmmmm mmmmmmmm mmmmmmm mmm mmmm mmm mnmmmmmmm mmmmmmmmm
BI-
~ ~
~
I m
I I I i
m
LI I (A)
(B)
{ I/
(C)
Figure 2. Tensile test of a patch of 100 axisymmetric Q4 elements for random distributions of the initial porosity. For the smallest value of the initial porosity, all computations give very similar final distributions and numerical values compared to the reference test. Therefore, the distributions stay quasi-homogeneous.
179 For the upper value, the mechanical variable distributions are similar but less h o m o g e n e o u s . In order to validate these observations, finite element computations are performed with 20 initial porous elements for various random distributions and led to the same conclusions. Table 2 Numerical results for random and reference distributions ,
,
,
, ,,,
fi
Cp
0.001 (A) 0.001 (B) 0.001 (C) Reference 0.01 (A) 0.01 (B) 0.01 (C) Reference
0.666-0.671 0.665-0.671 0.666-0.67 0.6667 0.646-0.694 0.643-0.695 0.645-0.69 0.6598
,,
,
......
~ .....
r~m
f 0.0621-0.0649 0.0622-0.0649 0.0622-0.0649 0.06504 0.0606-0.0898 0.0599-0.0884 0.0600-0.0889 0.08367
369-379 370-377 372-375 372.6 326-428 340-406 353-387 359.5
Ur 0.2806-0.2815 0.2809-0.2813 0.2809-0.2813 0.2803 0.2760-0.2848 0.2785-0.2833 0.2790-0.2830 0.2765 L
To complete these conclusions, the initial porosity of the matrix is located in a specific part of the workpiece. In that case, the mechanical behaviour is strongly affected and lead fastly to ductile rupture, even for small values of initial microvoid volume fraction (Fig. 3).
Jim mmmmnl mmmmmm i U l immmummm! iummmuunl Ill ill mR iiumi
mime mmmmmmmmm| m mmmmmm mmmmm mmm| mmmmmmmmm|
|
liltlttltt
IRUimmnmml immimi|mDi liUlnilinl IlimeiNIml immiiHmiml immmmimmml mmmmmmmmmm |mmmmmmmml |mmmmmmmml
Ilttt
IIII]
IIIIIIIIII IIIIIIIIII
IIIIIIIIII (a)
(b)
Figure 3. Tensile test of a patch of 100 axisymmetric Q4 elements, final meshes for (a) random and (b) localized distributions of the initial porosity.
180 In order to perform realistic predictions of forging sequence for elasto-plastic damage material by finite element simulation, an accurate characterisation of initial porosity distribution isn't necessary for small value of the initial porosity. Against, the occurrence of initial localized porosity must be verified.
3.3. Preform design in H-shaped cross sectionnal axisymmetric forging It is now suggested to show up the influence of damage on mechanical variables through the preform design in H-shaped cross sectionnal axisymmetric forging. Initially the workpiece is 11.13 mm high and diameter from 4.8 to 6 ram. The material involved is a 1070 annealed steel. Due to symmetry, only a quater of the specimen is used for the finite element mesh and contained 260 axisymmetric Q4 elements with four integration points (Fig. 4). Computations are achieved for 4.817 mm prescribed displacement of the upper die. During the analysis, frictionless contact conditions are assumed at die/workpiece boundaries. For that specific example the comparison is based on two cases; a first microvoided material with fi - 0.001 (case 1) and a second microvoided material with fi = 0.01 (case 2). The other related microvoided material parameters are CN = 0.2, SN = 0.1 and fN = 0.04.
+
+
Figure 4. Preform design of H-shaped cross sectionnal axisymmetric forging, initial mesh and geometry (a), deformed meshes for intermediate positions, (b) 32.6% and (c) 86.5% of height reduction.
181
The final h y d r o s t a t i c stress d i s t r i b u t i o n s are v e r y similar for e a c h case (Fig. 5), b u t t h e final m i n i m u m a n d m a x i m u m v a l u e s are r a t h e r d i f f e r e n t s . So, the m a x i m u m v a l u e of the c o m p r e s s i v e stress is r e s p e c t i v e l y 3000 M P a for the small initial p o r o s i t y case 1 a n d 2380 M P a for case 2.
(Ym (N.mm-2) A B C D E F G H I J K L
-291 -538 -785 -1030 -1270 -1520 -1770 -2010 -2260 -2510 -2750 -3000
(a) ~m
(N.mm-2) A B
C D E F G H I J K L
+127 -100 -328 -556 -784 -1010 -1240 -1460 -1690 -1920 -2150 -2~80
Co) Figure 5. Final h y d r o s t a t i c stress distributions for case 1 (a) a n d case 2 (b). The effective plastic strain v a l u e s a n d d i s t r i b u t i o n are n e a r l y similar for e a c h case (Fig. 6). At a h e i g h t r e d u c t i o n of 50.6%, the h i g h e s t v a l u e s are in the d i a g o n a l p a r t of the w o r k p i e c e , the r a n g e is 0.625-0.775.
182
A B C D E F G H I J K L
+0.225 +0.275 +0.325 +0.375 +0.425 +0.475 +0.525 +0.575 +0.625 +0.675 +0.725 +0.775
7
Figure 6. Effective plastic strain distribution for a height reduction of 50.6%. For the maximum final form maximum
two cases, the damage evolution appears fastly in the vicinity of the radius of the deformed workpiece in a tension zone (Figs 7 et 8). The of the microvoid volume fraction distributions are similar and the damage located in the same part of the workpiece.
f A C D
+3.60E-03 + 1.39E-02 +2.43E-02 +3.46E-02 +4.50E-02
Figure 7. Microvoid volume fraction distribution in case 1 for a height reduction of 86.5%.
183
f A
DE
+4.63E-03 +1.75E-02 +3.05E-02 +4.35E-02 +5.65E-02
Figure 8. Microvoid volume fraction distribution in case 2 for a height reduction of 86.5%. The final m a x i m u m values of the microvoid volume fraction are respectively 0.045, for case 1 ( fi = 0.001 ) and 0.0565 for case 2 ( fi = 0.01 ). Noticed that for an initial ratio of 10, the final ratio is 1.25. For case 1, the evolution curves of the nucleation and g r o w t h parts of the microvoid volume fraction in the most damage finite element versus the axial prescribed dsiplacement of the u p p e r die are represented on figure 9 . I
I
I
I
.05-
.04
Void fraction Void fraction due to nucleation Void fraction due to growth
.03
.02
.00 I I I I I I ""1" - 4. I -4.8-4.4-4.0-3.6-3.2-2.8-2.4-2.0-1.6-1.2
I~
-.8
-.4
Figure 9. Evolution curves of the nucleation and growth parts of the microvoid volume fraction in the most damage finite element versus the axial prescribed displacement of the upper die in case 1.
184 4. CONCLUSIONS A set of finite element computations are achieved with ABAQUS code on elasto-plastic structures with microvoided material. Firstly, a tensile test is focussed on the microvoid volume fraction evolution for various random initial distributions of the porosity and two values of the initial microvoid volume fraction fi, 0.001 and 0.01. It is clearly shown that to perform realistic predictions of forging sequence for elasto-plastic damage material by finite element simulation, an accurate characterisation of the initial porosity distribution isn't necessary for small values of the initial porosity. Against, the occurrence of an localized initial porosity must be verified. Then the analysis of a specific example of cold forging process, preform design of a H-shaped cross sectionnal axisymmetric workpiece, shows up the importance to take into account damage for the optimisation and the development of such problems. Indeed, the influence of the damage material parameters is analysed and discussed in regard to local mechanical variables, notably, the microvoid volume fraction, the effective plastic strain and the hydrostatic stress.
ACKNOWLEDGMENTS The authors are grateful to CNRS, and the French Ministry of research and Technical for supports to achieve computations.
REFERENCES
1. B. Bennani, P. Picart and J. Oudin, International Journal of damage Mechanics, 2 (1993) 119. 2 L. Lazzarotto, P. Picart and J. Oudin, International Journal of damage Mechanics, 5 (1996) 259. 3 R. Becker, A. Needleman, O. Richmond and V. Tvergaard, J. Mech. Phys. Solids., 36, 3 (1988) 317.
10
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
185
Microplasticity and Tensile Damage in Ti-15V-3Cr-3AI-3Sn Alloy and Ti15V-3Cr-3AI-3Sn/SiC Composite W.O. Soboyejo+, B. Rabeeh +, Y. Li+, A.B.O. Soboyejo x and S. Rokhlin* + Department of Materials Science and Engineering, The Ohio State University, 2041 College Road, Columbus, OH 43210-1179 x Department of Aeronautical Engineering, Applied Mechanics and Aviation, The Ohio State University, 155 West Woodruff Avenue, Columbus, OH 43210 * Department of Industrial, Welding and Systems Engineering, The Ohio State University, 190 W. 19th Ave. Columbus OH 43210 ABSTRACT Microscopic evidence of plastic flow at stresses below the bulk yield stress is presented for a met~tstable 13 titanium alloy (Ti-15V-3Cr-3A1-3Sn) and model Ti-15V-3Cr3A1-3Sn/SiC composite deformed to failure under monotonic loading at room-temperature. Microplasticity is shown to initiate in Ti-15V-3Cr-3A1-3Sn (Ti-15-3) alloy at stress levels between 5 and 10% of the bulk yield stress. Evidence of microplasticity is obtained via scanning electron microscopy examination of the deformed surfaces of smooth specimens that are loaded in incremental stages to failure. In the case of the monolithic Ti-15-3 alloy, deformation is shown to occur by a wide range of mechanisms at room temperature. These include: grain boundary sliding, grain boundary flow and bulk flow mechanisms. A wider range of matrix damage is observed in the model Ti-15-3/SiC composite deformed to failure under monotonic loading. These include all the damage components observed in the matrix and additional damage shear localization/slip band formation which is presumed to occur as a result of constraint effects in the composite. Plasticity in the Ti-15-3/SiC composite is also shown to involve early interfacial debonding, fiber fracture and multiple crack coalescence stages prior to the onset of catastrophic failure. The potential implications of the above results are also discussed. 1.
INTRODUCTION
It is generally accepted that the elastic deformation of metallic materials occurs by bond stretching at low stress levels. It is also commonly accepted that the onset of bulk plastic deformation occurs at stresses beyond the bulk yield stress due to the onset of dislocation motion. However, recent work [ 1] has shown that localized plasticity may initiate at stress levels that are significantly below the bulk yield stress. Such localized plasticity, which henceforth will be described as microplasticity, may occur in favorably oriented grains by a range of deformation mechanisms. Similarly, composites may exhibit a range of microplastic deformation phenomena in the so-called "elastic" regime. There is, therefore, a need for careful studies of microplasticity in metallic alloys and their composites. The current paper presents recent evidence of microplasticity in a metastable 13 titanium alloy (Ti-15V-3Cr-3A1-3Sn). Evidence of microplasticity and tensile damage mechanisms are also presented for a symmetric eight ply [0/9012s Ti-15V-3Cr-3A1-3Sn composite reinforced with silicon carbide fibers. Microplasticity in the monolithic alloy is shown to occur at room-temperature by a wide range of mechanisms that include: grain boundary sliding, grain boundary flow and bulk flow. A wider range of mechanisms are
186 observed in the composite presumably as a result of local constraint effects due to the composite architecture. These include: all the mechanisms identified previously in the monolithic alloy, slip band and subgrain formation, interfacial debonding and composite cracking phenomena. The potential implications of the results are. discussed for the modeling of elasticity and plasticity phenomena. 2.
MATERIALS
The monolithic titanium alloy (Ti-15V-3Cr-3A1-3Sn) was supplied by the NASA Lewis Research Center. It was produced by the rolling of foil at Textron Specialty Materials, Lowell, MA. The as-rolled material had a metastable 13 microstructure with an average grain size of 180 ktm. Two types of heat treatment were used to control the monolithic alloy microstructure. The first type of heat treatment involved annealing at 540~ for different durations (10, 50, and 100 hours) followed by air cooling. This heat treatment resulted in the transformation of the metastable 13(as-received) structure to a Widmanst~itten colony structure (ct+13 phase field). The second type of heat treatment involved annealing at 815oc for the same durations (10, 50 and 100 hours) followed by air cooling. Recreystallization of the asreceived structure occurred in the 13phase field at 815 oC. The symmetric eight ply [0/9012s Ti-15V-3Cr-3AI-3Sn/SiC (SCS-6)composite material that was used in this study was supplied by Textron Specialty Metals, Lowell, MA. It was produced by the foil/fiber/foil technique via hot pressing at 982~ for 2h. A slightly non-uniform distribution of SiC fibers was produced in the [0/9012s composite due to the effects of fiber "swimming" during composite processing. Large (--500 l.tm average diameter) 13 grains are also observed in the as-received composite. The layered interface has a highly complex microstructure [2, 3] that consists predominantly of titanium carbides (TiC and Ti2C). Two sets of heat treatments were used to control the matrix and interfacial microstructure. The first set of heat treatments involved annealing for different durations (10h, 50h and 100h) at 540~ (below the 13 solvus of approx. 800~ for Ti-15-3). The heat treatment at 540~ resulted in a Widmanst~itten colony microstructure with small circular t~ grains in a matrix of 13 9The changes in the matrix microstructure occurred without significant coarsening of the interfacial structure. Some slip bands were also observed in the Widmanst~tten matrix microstructure. It is presumed that these were induced as a result of matrix yielding due to residual stresses in the composite. The second set of heat treatments were carried out at 815oc. Annealing durations of 10, 50 and 100h were employed. These heat treatments were designed to promote significant coarsening of the interfacial microstructure without inducing significant alteration of the matrix microstructure. However, unlike the four-ply unidirectional Ti-15-3/SiC (SCS-9) composite that was examined in previous studies [ 14], the degree of coarsening of interfacial microstructure was observed to be limited in the [0/90] 2s Ti-15-3/SiC (SCS-6)composite that was examined in this study. Further details on the complex structure of the layered interfacial region are provided in Ref. 3. The coarse grained (500 ktm average grain size) as-received microstructure is retained after annealing in the 13 phase field. 3.
EXPERIMENTAL
Tensile tests were performed at room temperature on two sets of specimens. The first set of tensile specimens were 125 mm long with rectangular cross sections (1.7 mm x 10
187 mm). The second set were smooth 125 mm long tensile specimens with rectangular cross sections (2 mm x 12 mm). A servohydraulic test machine was employed in the mechanical testing. The first set of tensile specimens were loaded continuously to failure at a strain rate of 5x10 -4 sec-1. Strain was measured with contact extensometer with a gauge length of 25.4 mm. A second set of tensile tests were conducted on smooth specimens to study the deformation and cracking phenomena associated with damage under monotonic loading. The specimens were loaded in incremental steps of 0.1 CYtrrs, to various fractions of the ultimate tensile stress, Ctrrs, determined from the first set of tests. Damage phenomena associated with the different incremental monotonic loading steps were then identified by ex-situ scanning electron microscopy examination of the sides (of the gauge sections) of the deformed tensile specimens. In this way, the sequence of damage was identified for the two microstructural conditions that were employed. 4.
RESULTS AND DISCUSSION
4.1. Stress-Strain Behavior Monolithic Ti- 15-3 Alloy Tensile properties of monolithic Ti-15-3 alloy at room temperature under the asreceived and the heat treated conditions are summarized in Table 1. Characteristic stressstrain plots of the as-received material and materials annealed at 540oc for 50 hours are presented in Fig. 1. The stress-strain plots of 540oc materials was almost identical after annealing for 10, 50 or 100 hours. This is consistent with the similar microstructures of the materials annealed at 540~ The material strength of 540~ was also higher than as-received material strength, presumably as a result of phase transformation from metaste.ble 13 phase (as-received) to o~+13 phase (Widmanst~itten) microstructures. Annealing at 540oc was also associated with smaller plastic strains to fracture as shown in Table 1. The stressstrain plots of materials annealed at 815~ for 50 hours is also presented in Fig. 1. The strength of 815~ Ti-15-3 alloy was also similar to that of the as-received material, and the strengths of materials heat treated at 815~ decreased with increasing annealing duration (Table 1). The stress-strain response of monolithic Ti-15-3 alloy at different heat treatments was identical conventional stress-strain with distinguishable elastic and plastic rejoins. The higher strengths obtained after the 540oc heat treatments are attributed to the cx+13 phase Widmanst~itten structure. Similarly, the improved ductility of the material annealed at 815~ is consistent with the results obtained from previous studies of titanium alloys. Furthermore, both 540oc and 815~ heat treated materials have different yield stresses, although the yield strains were almost identical in the heat treatment conditions that were examined (Table 1). Ti- 15-3/SIC Composite Tensile properties of the as-received and heat treated Ti-15-3/SIC composite materials are summarized in Table 2. Characteristic stress-strain plots are also presented in Fig. 2 for 540oc/50h/AC and 815oc/50h/AC materials compared to the as-received material. The stress-strain characteristics of the as-received and 815~ materials were almost identical, consistent with the similar composite microstructures in these two conditions. The composite strength were also higher after annealing at 540oc for 50h which resulted in the
188
Table 1 S u m m a r y of Tensile Properties of Ti- 15-3 Allo~ Modulus (E) Yield Stress/Strain Condition Heat Treatment [GPa] OYield Strain [MPa] % AR 733 801 0.1 1044 1125 0.1 540~ C/10h/AC 957 1205 0.1 540~C/50h/AC 1320 1008 0.1 540~C/100h/AC 774 809 0.1 815~ 732 806 0.1 815~C/50h/AC 847 755 0.1 815~ AR = As-Received f = Failure UTS= Ultimate Tensile Stress
Ultimate Tensile Stress/Strain Ours Strain [MPa] % 812 0.2 1234 0.4 1223 0.2 1208 0.3 826 0.2 798 0.3 783 0.2
Failure Stress/Strain Strain of % [MPa] 723 1.40 1224 0.50 1223 0.16 1197 0.38 744 0.90 718 2.34 704 1.30
Table 2 S u m m a r y of Tensile Properties of Ti-15-3/SiC Composites Condition Heat Modulus of Elasticity Deformation Stress Treatment [GPa] [MPa] E1 E2 E3 E4 Ol o2 o3 (Failure) As-Received 106 88 80 54 211 472 616.6 540o C/50h/AC 133 116 97 87 124 345 800 815oc/50h/AC 151 104 85 58 57 377 611
1500
OUTS
Strain to Failure %
856 1028 891
0.11 0.10 0.11
OUTS
[MVa]
1200 ~--
As-Received
~,
540C/50h
As-received 540C/50h
"
815C/50h
815C/50h
900 1000 t_...a
t......a
r~
r~
ra~
600
ra~
500:
300
0 ~
0.00
0 ~
0.01
0.02
0.03
0.000
Strain Fig.1.
Stress-Strain Behavior of Ti- 15-3 Alloys.
0.003
0.006
0.009
0.012
Strain Fig.2.
Stress-Strain B e h a v i o r of Ti- 15-3/SIC Composites.
189 transformed c~+13 Widmanst~itten microstructure. Such annealing was associated with lower ductility than that of the as-received and 815~ materials (Table 2 and Fig. 2). However, unlike monolithic Ti-15-3 alloy, the composite does not have a distinguishable elastic or plastic range of deformation. The stress-strain plots (Fig. 2) revealed similar stress-strain characteristics in the composites annealed at 540oc and 815oc prior to tensile loading. The stress-strain response was approximately linear at room temperature until a critical stress was reached. Average values of this critical stress, Crl, with the slope (modulus) are presented in Table 2. Non-linear behavior ensued beyond this critical stress. In all cases, the curves exhibit almost four distinct critical stresses with different slopes (modulus). The ultimate tensile stresses of 540~ annealed material are higher than that of as-received and 815 ~ annealed material, and the plastic strain remained almost the same (unlike monolithic Ti-15-3 alloy). The non-linear stress-strain response of titanium metal matrix composite was due to the underlying damage mechanisms which will be discussed later. 4.2. Damage Mechanisms Monolithic Ti- 15-3 Alloy Local evidence of damage initiation was observed early in the deformation sequence, i.e., prior to bulk yielding across the gauge of the Ti-15-3 alloy specimens annealed at 540oc/50h/AC. Damage initiation in the specimens occurred by microcracking along grain boundaries (GB) at very low stresses. Local evidence of microplasticity was also observed early in deformation sequence at 0.3 crtrrs (Fig. 3a). The microplasticity manifested itself in the form of localized flow of material. This is illustrated in Fig. 3b in which surface damage features are observed to flow with increasing load. Note also that the flow results in the widening of the region between the boundaries. However, flow in this microplasticity regime is associated with linear stress-strain characteristics. Beyond the initial regime of microplasticity, fracture initiated rapidly from grain boundaries. Catastrophic failure occurred by ductile dimpled fracture mechanisms along prior 13grain boundaries (Fig. 3c). Defects observed in the Widmanst~itten structure were observed to flow across the grains during deformation in the microplastic regime. Such large scale (micrometer levels) evidence of flow is clearly inconsistent with current elasticity and plasticity theories. However, the mechanisms of such flow processes are not fully understood at present. The stress and strain levels associated with the observed deformation and cracking phenomena are summarized schematically in Fig. 4 for Ti-15-3 alloy annealed at 540oc for 50 hours. Unlike the specimens annealed at 540oc/50h/AC, the specimens annealed at 815~ doesn't exhibit early damage initiation. However, some grain boundary sliding were observed under monotonic loading at 0.40trrs, follow by initiation of small voids along grain boundaries.. Subsequent damage is associated with slip band formation along small o~ precipitates, nucleation of voids along the ~ precipitates on these slip bands and surface roughening due to the intersection of slip bands. Microvoid linkage/coalescence then occurs predominantly along slip bands. Localized plastic flow and further surface roughening are observed just before the onset of catastrophic failure, which occurred by intergranular/ transgranular ductile dimpled fracture and secondary cracking between the slip bands.
190
Fig. 3. Damage Mechanisms of Ti-15-3 Alloy Specimens Annealed at 540oc/50h/AC and Deformed under Monotonic Loading at Room Temperature. (a). Crack Nucleation from Grain Boundary and Localized Plastic Flow at 0.3 Gtrrs; (b). Localized Plastic Flow and Bulk from Grain Boundary at 0.5 ~UTS; (c). Ductile Dimpled Fracture at 1.0 GUTS.
191
Fig. 4. Summary of Stress-Strain Behavior with Underlying Damage Mechanisms of Ti- 15-3 Alloy Annealed at 540~ and Deformed to Failure under Monotonic Loading.
Ti- 15-3/SIC Composite The damage phenomena associated with the different regions of the stress-strain curves at room temperature are presented in Fig. 5 for 540oc/50h/AC. Note that damage initiation is described arbitrary to include all the damage events (interfacial cracking, slip band formation and sub-grains) prior to matrix cracking, while damage propagation/evolution is considered here to involve all the subsequent stages of damage (multiple crack growth, fiber fracture and crack coalescence) after the onset of matrix cracking. Local evidence of plasticity (damage initiation) was observed early in the deformation sequence, i.e., prior to bulk yielding across the gauge of the specimens annealed at 540oc. This form of local plasticity, which is referred to subsequently as microplasticity, occurred at the very low stress (approx. 0.1 ~trrs), and is illustrated in Fig. 5a. The microplasticity manifested itself in the form of slip bands, which were observed to nucleate from the fiber/matrix interface. Similar slip band initiation mechanisms have been observed in previous studies by Majumdar et al. [4, 5] on 0 and 90 degree Ti-15-3/SCS-6 composites. Note that the stress-strain behavior of the composite is still approximately linear in the microplasticity regime, as is typically observed in conventional monolithic materials. Beyond the initial regime of microplasticity, crack initiation occurred in the outer 900 plies (Fig. 5b), followed by slip band intersection in the regions between the 0 and 900 plies. These slip bands appeared to have been initiated by the localization of strain in the vicinity of interfacial/reaction zone cracks, as reported by Majumdar et al. in Refs. [4, 5] for 0 and 900 composites. Debonding was observed to occur at the region between the titanium carbide interface and the Ti-15-3 matrix. Subsequent matrix crack initiation occurred by the
192 extension of interfacial crack into the matrix at higher stresses of 0.6 C trrs (Fig. 5c). Transgranular matrix cracks were nucleated by the extension of interfacial cracks into the matrix. This occurred initially in the inner plies, while slip band activity was still dominated in the inner plies. Intergranular matrix crack growth/coalescence was observed at the boundaries between the large [3 grains in outer plies. The initiation of matrix crack growth preceded final fracture via the coalescence of matrix and fiber cracks in the mode I direction. Catastrophic failure occurred by ductile dimpled fracture and cleavage/quasi-cleavage fracture of SCS-6 fibers in 540~ microstructure (Fig. 5d). The stress and strain levels associated with the observed deformation and cracking phenomena are summarized schematically in Fig. 6. The damage phenomena associated with the different regions of the stress-strain curves at room temperature for the specimens annealed at 815 oC/50h/AC are similar to those observed in the specimens annealed at 540oc/50h/AC. However, unlike the specimens annealed below beta transus, slip bands were not observed in these specimens. Instead, matrix microplasticity occurred by formation of sub-grain structure. Fiber fracture was also found to precede catastrophic failure, which occurred in the mode I direction during monotonic loading at room temperature. Catastrophic failure occurred by ductile dimpled fracture in the matrix, and cleavage/quasi-cleavage fracture of SCS-6 fibers in composites annealed at 815~ as those observed in the specimens annealed at 540oc/50h/AC. 5. Implications It is apparent from the above results and discussion that the observed "linear elastic" behavior (Figs. 1 and 2) is clearly associated with plasticity phenomena at stress levels well below the bulk yield stress levels. The local scale of some of the observed plastic flow processes (e.g. grain boundary sliding and bulk flow in Ti-15-3 alloy, or formation of slip bands and crack initiation in T-15-3/SiC composite) is also considerable greater than the submicroscopic ~ levels of flow that are typically associated with conventional dislocation glide/climb. Such non-linearities have been measured in previous studies on other material systems [6-8] using strain gauges that can detect strains that are low as 10-8. Such small strain changes and associated non-linear stress-strain behavior may be importance in the design of high tolerance components such as the lenses in the Hubble telescope. It also gives a insight light on damage initiation mechanisms of fatigue, especially at stresses well below the bulk yield stresses of materials. However, more work is needed to understand the mechanisms of plastic flow at such low stresses. Nevertheless, it is apparent from the observed microplasticity phenomena that the so-called linear elastic response of materials may correspond to linear elastic behavior. This is in spite of micro-scale of the observed plasticity phenomena. 6. Summary The stress-strain response of Ti-15-3 alloy and Ti-15-3/SiC composite has been studied under monotonic loading. The damage mechanisms in these two materials under monotonic loading were investigated via incremental techniques. Strong evidence of microplasticity is observed at stresses well below the bulk yield stress levels (at -30% of the bulk yield stress for Ti-15-3 alloy and at -10% of the bulk yield stress for Ti-15-3/SiC composite). Microplasticity in Ti-15-3 alloy with 13 structure occurs initially via grain boundary sliding and slip band formation/intersection, while microplasticity in Ti-15-3 alloy with a model Widmanst~itten ct+~ microstructure occurs by bulk or grain boundary flow mechanisms that are not well understood at present. Final fracture in the model 13and
193
Fig. 5. Damage Mechanisms of Ti-15-3/SIC Composite Specimens Annealed at 540oc/50h/AC and Deformed under Monotonic Loading at Room Temperature. (a). Slip Bands Initiation in Outer 900 Ply with Debonding at 0.2 CUTS; (b). Matrix Crack Nucleation from Interface in Outer Ply at 0.3 CUTS; (C). Matrix Crack Nucleation from Interface with Debonding in Inner Ply at 0.6 ~UTS; (d). Ductile Dimpled Matrix Fracture at 1.0 ~UTS.
194 Matrix Crack Coalescence r~ r~
r~
ire Debonding And PS 7ormation in Inner Ply
Inner Ply
VlatrixCrack Nucle in Inner Plies at 0.6 Band Formation in~
.j~"~,,"/" Matrix Cracking in
Strain Fig. 6. Summary of Stress-Strain Behavior with Underlying Damage Mechanisms of Ti-153/SIC Composite Annealed at 540~ and Deformed under Monotenic Loading.
Widmanst~itten ~+13 structures occurs by classical ductile fracture mechanisms. In Ti-153/SIC composites, a wider range of microplasticity mechanisms are observed. These mechanisms include: all the mechanisms identified previously in the monolithic alloy, slip band and subgrain formation, interfacial debonding and composite cracking phenomena. ACKNOWLEDGMENTS The research was supported by the Division of Mechanics and Materials of the National Science Foundation. The authors are grateful to the Program Monitors, Dr. William A. Spitzig and Dr. Oscar Dillon, for their encouragement and support. REFERENCES 1. B.M. Rabeeh, S.I. Rokhlin and W.O. Soboyejo, Scripta Materialia, 35 (1996) 1429. 2. B.A. Lerch, T.P. Gabb, and R.A. Mckay, A Heat Treatment Study of SiC/Ti-15-3 Composite System, NASA Technical Report No. 2970 (1990). 3. J. Shyue, W.O. Soboyejo and H.L. Fraser, Scripta Materialia, 33 (1995) 1695. 4. B.S. Majumdar, G.M. Newaz and J.R. Ellis, Metall. Trans. A24 (1993) 1597. 5. B.S. Majumdar, G. Newaz, Phil. Mag., A66 (1992) 187. 6. A. Pusk~ir, Microplasticity and Failure of Metallic Materials, Elsevier Publishing, First Edition, 1989. 7. J.F. Bell, The Physics of Large Deformation of Crystalline Solids, Springer-Verlag, 1968. 8. C.W. Marchal and R.E. Maringer, Dimensional Instability, Pergamon, 1977.
STRAIN LOCALIZATION AND INSTABILITY ANALYSIS
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Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
197
Defects in hydraulic bulge forming of tubular components and their implication for design and control of the process
M. Ahmed and M.S.J. Hashmi School of Mechanical and Manufacturing Engineering Dublin City University, Dublin 9, Ireland
ABSTRACT Bulge forming process is used to manufacture various types of tubular components from straight cylindrical tubes. Products include components like T-branches, X-branches, angle branches etc. The bulge forming process is accomplished by the application of hydrostatic pressure to the tubular blanks placed in a die bearing the shape of the desired product and at the same time pushing the ends of the tube towards the bulging zone. The deformation in the process is very complex. The loads acting in the deformation zone are multidimensional and the ratio of the loads in different directions may not be constant during the forming process. Therefore, the process is very prone to instability and thereby defects. Two most common defects are the rupture of the bulge and the buckling in the main branch. This paper presents finite element simulation of bulge forming of T-branches leading to the point of failure. Simulation results show that depending on the dominance of the axial compressive load or the pressure load the process terminates by buckling or by rupture respectively. By manipulating the loading pattern it is possible to have more deformation and thereby a longer T-branch.
1. INTRODUCTION Conventional process of manufacturing tubular components involves several operations. An alternative way of manufacturing this type of products is bulge forming which gives nearnet shape products. In this process a straight cylindrical tube is placed in a die bearing the shape of the designed component and a hydrostatic pressure is applied at the inner side of the tube. Application of only internal pressure thins the bulged area rather quickly and leads to rupture. To delay the rupture additional material from other non-bulging areas of the blank is pushed into the bulging region. In the case of bulging from straight cylindrical tubes, this is done by compressing the tube ends. Simultaneous application of the above two types of loading help to obtain large deformation in the process. However, a critical balance of these loading are essential for a stable process. Prominence of any one type of loading would lead to a defective product.
198 Different kinds of defects may develop in bulge forming. But the most frequent ones are rupture and buckling. Rupture defects occur when the pressure loading is high compared to the axial compressive load. Buckling, on the other hand, develops when the axial compressive load is dominant over the pressure loading. Theoretically[I,2], if the structure could be so loaded that the axial strain is equal and opposite to the meriodinal strain, then the strain in the thickness direction would remain unchanged. This implies that under such an ideal condition the bulging of the structure can be continued indefinitely. But in reality such loading or more specifically straining is impossible specially during the forming of asymmetric products like Tjoints or X-joints. There are different analytical studies on bulge forming of asymmetric products. Hashmi[3] presented a theoretical model to predict thickness distribution in a hydraulically bulged T- branch. Hashmi, in a different presentation[4], also developed analytical method to predict thickness in X-joints. Filho et al.[5] analysed the equilibrium of forces at different zones of T-joint formed by pressing elastomer and then solved the resulting equations simultaneously to obtain the total axial load. Ahmed and Hashmi[6] theoretically determined the axial load required to form T-branches by internal hydraulic pressure and axial compressive load. There are no theoretical studies in predicting the formability of T-branches which would be helpful to produce defect free products. Hutchinson[7], however, experimentally established a formability zone for forming T-joints from a certain blank material and geometry. Analysis by numerical methods could help identify suitable loading process that would give maximum deformation without onset of any instability leading to defects. This paper is an effort towards that objective. It presents finite element simulation of bulge forming ofT-joints for different loading pattern. It is seen that by changing the loading pattern, it is possible to maximise deformation in the tube but it is not possible to delay the instability indefinitely. The simulations were done using LS-DYNA3D commercial FE package.
2. FINITE ELEMENT MODEL In this work, bulge forming of a commercially pure copper cylindrical tube of 24.12 mm diameter and 107mm length into a T-branch was simulated. The tube thickness was 1.37 mm. The diameter of the T-branch was same as the tube diameter. The forming load was internal hydraulic pressure and axial compressive load. The discretized model is shown in Figure 1. One half of the problem was modelled because of the symmetry. Both the die and the tube were discretized with Belytschko-Tsay shell elements[8]. The interface between the die and the tube was modelled by automatic surface to surface contact algorithm [8]. Three different loading pattern were simulated. Table 1 presents the loading pattern against simulation time. The load values vary linearly with simulation time. Henceforth, the loading patterns are termed as Loading I, Loading II and Loading III. The die is modelled as a rigid body. The tube material is modelled as piece-wise linear elasto-plastic material. Different parameters for the tube material are as below: Young's modulus = 3.66x103 N/mm z', Yield stress=l 8.3 N/mm 2", Poisson's ratio=0.3 Plastic failure strain = 1.0 ; Density=8900 Kg/m 3
199 Table 1" Pattern of loading used for different simulation cases Pattern Load Type Simulation Time, See. 0.0025
0.01
Pressure, N/mm 2
0.0 0.0
Displacement, mm Pressure, N/mm 2
0.0 0.0
30.0
29.0 30.0
Displacement, mm Pressure, N/mm 2
0.0 0.0
30.0
29.0 30.0
Displacement, mm
0.0
0.0
29.0
30.0
Loading I
Loading II
Loading III
lrt
~,~X@f/lll/llll//l/Ib I~f~"JlH//II#Ill/If lIlIIA x
l~N'/ ~711Hlil/il1~ ~~_ ...~I /II II ! ! / ! ! !!! !/I , ~111 IIIIilll Ii III 111 i ~ ~ I III II I II Ill Ill Ill I
Figure 1" Discretized finite element model of the bulge forming problem.
3. SIMULATION RESULTS AND DISCUSSIONS Figures 2, 3 and 4 show the distribution of X- coordinate displacement for the three loading situations at equally acceptable state of deformation. Simulation beyond that point
200 branch height = 8.58 mm
Figure 2: T-branch height developed by Loading I branch height = 16.47 mm
Figure 3 T-branch height developed by Loading II branch height = 9.60 mm
Figure 4: T-branch height developed by Loading III
201 leads to instability and defect. Maximum deformation in the X- direction gives the maximum Tbranch height. It can be seen from the figures that the maximum T-branch height is obtained for loading II as in Figure 3. Maximum T-branch height in the case of Loading I and Loading III are only about half of that by Loading II. Figures 5, 6 and 7 respectively show the failed mode due to the three loading situations. For loading I the structure has buckled at the main branch . In loading II, the structure has buckled at the main branch and ruptured at the T-branch.. For loading III, the bulge formed Tbranch has ruptured. The tube was simulated to fail at plastic strain value of 1.0. Figure 8 presents the process of straining in the structure for different loading situations. The figure shows maximum equivalent plastic strains developed at the mid- surface of the shells along with the pressure and displacement loading at different times of the simulation. In general the maximum strains developed at different stages of simulation were located in the areas where the tube ultimately failed as can be seen in figures 5, 6, and 7. In loading I where the pressure and axial compressive load were simultaneously increased, the strain increased linearly and after the loading reached half of the intended values, the strain started increasing at a faster rate and ultimately the tube buckled in the main p~rt. The maximum pressure that could be applied in this pattern of loading was 19.2 N/mm 2 against 30 N/mm 2 intended pressure. The corresponding axial displacement was 18.56 mm against an intended value of 29 mm. Because of buckling failure in loading I, it was thought that if the pressure was applied at a faster rate compared to the displacement the deformation, would be higher. In loading II, full intended pressure was applied within a quarter of the intended simulation time of 0.01 seconds. The displacement loading was linearly increased throughout the simulation period. From the figure, it can be seen that the strain in the tube increased at a faster pace until the full application of pressure load. After that the strain increases very slowly until the onset of instability. The specimen failed by both buckling at the main branch and rupture at the Tbranch. However, buckling initiated in the specimen much before the rupture. In this loading the full intended pressure was applied and the tube ends were pushed by 23.2 mm. Since buckling was the predominant failure mode in loading II, it was thought that the axial displacement loading may be delayed and applied at a slower rate in order to achieve more deformation. Loading III was patterned in that fashion. After initial straining by the pressure load, the strain in the tube virtually did not increase for a while. Then the strain started increasing and eventually ruptured at the T- Branch. In this loading, the full pressure could be applied but the tube ends could be compressed inwards by only 5.8 mm. Comparison between the three loading patterns reveal some interesting aspects of the bulge forming. In loading I the structure has started straining at a much faster rate after the loads are only about 50% of the intended load. Since the structure eventually failed by buckling, it is thought that an appreciated pressure loading at that stage might delay the instability. In loading II when the pressure was full applied, the strain in the structure was about 0.49. In loading III, on the other hand, when the full pressure was applied the strain in the tube was 0.77. The only difference between the last two loading patterns is the axial displacement load. It appears that the axial displacement load in loading II has actually kept the strain low in the structure by relieving some stress caused by the pressure load. This phenomenon has enabled more deformation in Loading II. Looking at the results by all three loading patterns, it is apparent that there is an optimum loading pattern that would give maximum deformation of the structure which has to be obtained by more trial simulations
202
Figure 5" Buckling mode of failure by Loading I
Figure 6: Buckling and rupture failure by Loading II
Figure 7 Rupture failure by Loading III
203 Loading I 20 18 16 14
pressure, N/sq.mm.
12
- - - I - - disp. mm.
10 8
strainXlO
6
0.003
0.0044
0.006
Simulation time, seconds Loading II
251 / 30
A
A pressure,N Isq .mm.
20
--In
disp. ram.
"-
strain XIO
15 10 5
m
0
--
, 0
0.002 0.0028 0.005
0.007
0.008
Simulation time, seconds Loading III 30 25 i---r
20 15
pressure, N/sq.mm. disp. rnrn
10 -"
6
strain XlO
0 0
0.0016
0.0024
0.0032
0.004
simulation time, seconds
Figure 8" Straining of the bulged tubes for different loading pattern
204 4. CONCLUSION Finite element simulation of the bulge forming of a T-branch from straight cylindrical tube was carried out. It was found from the simulations that the structure fails either by buckling at the main branch or by rupture at the bulged T-branch. Certain loading conditions give more deformation than others but eventually fails by any of the above mentioned modes. By trial, an optimum deformation mode may be obtained. However, the theoretical possibility of indefinite deformation is not worth pursuing as innumerable trials are possible. But at the same time, this study does not rule out that possibility. REFERENCES
.
.
.
Tirosh J., Neuberger, A. and Shirzly, A. "On tube expansion by internal fluid pressure with additional compressive stress"; Int. J. Mech. Sci. Vol. 38 Nos 8-9, PP. 839-851, 1996. Thiruvarudchelvan, S. and Lua, A.C. "Bulge forming of tubes with axial compressive force proportional to the hydraulic pressure"; J. Mater. Shaping Technol. Vol.9, PP. 133-142;1991. Hashmi, M.S.J. "Radial thickness distribution around a hydraulically bulge formed annealed copper T-joint: experimental and theoretical predictions" 22nd Int. M.T.D.R. conf. pp 507-516; 1981. Hashmi, M.S.J. "Forming of tubular components from straight tubing using combined axial load and internal pressure: theory and experiment" Proc. of int. conf. on dev. on drawing of metals, Metals Society, pp 146-155; 1983. Filho, L.A.M., Menezes, J.C. and AI-Qureshi, H.A. "Analysis of unconventional tee forming on metal tubes" J. Mat. Proc. Technol. Vol. 45 pp 383-388, 1994. Ahmed, M. and Hashmi, M.S.J. "Estimation of machine parameters for hydraulic bulge forming of tubular components" accepted for publication in J. Mat. Proc. Tech. Vol. 64 (in press). Hutchinson, M.I. "Bulge forming of tubular components" PhD thesis, Sheffield City Polytechnic, 1988. Hallquist, J.O. "LS-DYNA3D theoretical manual" ; Livermore Sot~ware Technology Corporation, Livermore, California, 1993.
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
205
NUMERICAL A N D EXPERIMENTAL ANALYSIS OF NECKING IN 3D. SHEET FORMING PROCESSES USING DAMAGE VARIABLE
M. Brunet a, S. Mguil-Touchal ~ and F. Morestin ~ aLaboratoire de M6canique des Solides, Institut National des Sciences Appliqu6es, Bat. 304, 69621, Villeurbanne, France
Abstract: As fracture in metal forming is mainly due to the development of ductile damage and to represent the failure of anisotropic sheet-metals, an extension of the Gurson-Tvergaard model is presented and implemented in the context of plane-stress for shell elements. Moreover, a new necking criterion is proposed within the framework of the modified force maximum criterion which includes the determination of local necking for arbitrary strain paths as well as for anisotropic materials. The difference of orientations between the principal stresses axes and the orthotropic axes are taken account. The paper closes with a numerical and experimental study of the necking of rectangular strips in several Marciniack's experiments. 1. INTRODUCTION There are several ways to achieve analysis of necking occurrence in sheet-metal forming. One way consists to carded out a conventional F.E. simulation and by postprocessing the F.E. results, in using a theoretical or experimental necking criterion, to detect the zones where risks of necking can occur. It is the approach we have employed in [1] and [2] introducing the concept of Forming Limit Stress Surfaces for anisotropic sheets. It has been found that the Forming Limit Stress Diagrams are much more intrinsic that the conventional F. L. Strain Diagrams strongly influenced by the strain path which may vary significantly from the direct strain path in the case of complex sheet-metal forming processes. If experimental F.L.D. are not available , the strains of elements calculated in every steps by F.E. analysis are compared with the necking limits obtained by formulas based on plastic-instability theories such that the StorSn-Rice criterion [3]. However, many Limiting Dome Height (L.D.H.) tests [4] on steel-sheets have shown that theoretical formulas give smaller heights than the measured values, except for aluminum alloys. On the other-hand, a large number of macroscopic fracture criteria for failure which occurs after necking have been evaluated by Doege and co-workers [5],[6], consisting of products, integrals and sums of macroscopic stresses and strains. To determine the values of these criteria at the onset of failure, both experiments and F.E. simulations are needed. When applying these criteria, it was found that the main factor affecting the accuracy is the mode in which failure takes place, mainly under deep-drawing or under stretching conditions. The equivalent Mises stress was judged best for the prediction of both deep-drawing and stretch-drawing cracks but the locus of maximum equivalent stress does not necessarily coincides with the locus of failure in the sheet [6]. Moreover, the thickness distribution may also indicate the
206 wrong locus of failure since this parameter is operation dependant and there is no material dependant critical sheet thickness reduction. Since deformation after necking up to fracture consists of sheet thinning within the neck together with complementary tensile stretching perpendicular to the neck and no straining along the neck, m many forming operations, the onset of necking is considered to be limiting. Also there is a need in the simulation process to achieve better localization of the onset of necking. This can be expected by the coupled approach where the damage process is incorporated into the constitutive rdations. Many investigations have shown that ductile fracture involves four successive damage processes which are the nucleation of voids from inclusions, void growth, void coalescence and cracking propagation. One constitutive equation to account for these processes is the Gurson-Tvergaard model [7],[8], which was derived in an attempt to model a porous isotropic plastic material containing randomly disposed voids. As suggested by Doege and co-workers [9], we have extended the Gurson-Tvergaard model to anisotropic matrix behavior and implemented with our simplified triangular shell element suitable for simulating sheet-metal forming processes [10]. After localization by the iso-value curves of the porosity which acts as an macroscopic internal damage variable, the onset of necking may be found numerically by mathematical considerations due to the fact that the strain state gradually drifts to plane strain after the onset of load instability. 2. DAMAGE MODEL
The proposed damage model is based on an extension of the Gurson-Tvergaard model [7],[8] in which microvoid volume fraction evolution in the constitutive matrix is described. The proposed yield condition takes the form : 2
2
*
*2
- q / % + 2ql f cosh(-3q2p/2%) - (1 +qaf ) = 0
(1)
with O-m= -p the hydrostatic pressure. f is the state variable available for microvoided material, called microvoid volume fraction or porosity and defined by : f - (VA - VM) / VA (2) with V A elementary apparent volume of material and V u the corresponding matrix one. In relation (1) q is the effective stress of the macroscopic Cauchy stress tensor owhich, instead of the original Mises stress is replaced by the orthotropic Hi11'48 [11] effective stress or by any others general quadratic or non-quadratic loci descriptions as proposed by Hill [12],[13] or Barlat and Lian [14] : qZ = o,T M o" (3) where for plane-stress condition and in the orthotropic axes x and y : yy
and
M =
-
F+H
0]
(4)
The anisotropic plastic behavior of the metal is described by the anisotropy parameters G,H,F and N which are defined in terms of the Lankford parameters ro,rgo,and r4s as : H=ro/(1 +ro) G= 1-H F=ro/[rgo(1 +ro)] N=(rgo+ro)(2r4s+ 1)/I2r9o(1 +ro)] (5) The Lankford parameters are determined by three experiments in the various directions as pointed out by their different indices. This model is often preferred for industrial applications with steel sheets. If H = G = F = I / 2 and N=3/2 Eq.(4) abridges to standard Mises isotropy. Due to the Hill function and the associated flow rule not being isotropic, the direct Eulerian constitutive law based on this criterion is not
207 objective. In order to assure the objectivity, the rotating frame formalism is applied. The axes of orthotropy of the Hill criterion can be updated by a rotation which can be chosen as the material spin rate to (co-rotational stress rate) or from the polar decomposition F = R U (Green-Nagdi stress rate). Since the elastic strain are assumed to be small and from practical sheet forming applications, the differences between these different rotations are very small. Tvergaard [8] introduced the constant ql, qz and qa=q z as coefficients of the void volume fraction and pressure terms in order to make the predictions of the Gurson model agree with numerical studies of ordered voided materials in plane strain tensile fields, typically 9 q~= 1.5 q2= 1 and qa=2.25 instead of qx=qz=qa 1 in the original Gurson model. The flow rule is derived from the yield potential Eq.(1), the presence of the hydrostatic pressure in the yield function results in non-deviatoric plastic strains: =
o~ [ o ~ op o~ o q ] dep = dx ~-6 = dx ~~ ~6 + ~-~ ~-6 The hardening of the fully dense matrix material is described through
(6)
O'y-h(~P). The
evolution of ~P is assumed to be governed by the equivalent plastic work relation 9 (l-f) o ' y d ~ p = tr T d c p (7) The damage model takes into account the three main phases of damage evolution. The microvoid volume fraction is given by 9 df = df N + df G + df c
(8)
Considering a random distribution of second phase particles, microvoid volume fraction increment due to nucleation is expressed by [15]" df~
_
SN fn 2~n expf_(~:P-~:n)z 1 22S IJ d~P
(9)
The normal distribution of the nucleation strain has a mean value e~ , standard deviation SN and nucleates voids with volume fraction fN. For steel-sheet metals the possible values are fN=0.04 , 0.01 -< SN -~ 0.1 and 0.3 ~- e N ~- 0.7 [15]. Growth of existing voids is based on the apparent volume change and the law of conservation of mass and is expressed as " dfG = (l-f) (dePxx+ dePy + deP3) (10) Finally, the modification of the yield condition to account for coalescence and final material failure, is introduced through the function f (f) specified by [16]" f = fcR + K ( f - fR) i f f ~ fcR and f = f iff
(12)
at ductile rupture and f r is the porosity at final failure. It is worth noticing that if the damage evolution takes place in small areas as in the case of sheet metal forming, large values of porosity up to 20% can be only found on a micro level.
208 3. C O M P U T A T I O N A L ASPECTS
3.1. Explicit solution procedure The three node simplified shell element 110] with only translational degree of freedom but with bending capability is adopted for the spatial discretization of the sheet. A large numbers of analysis have shown that sheet forming processes can be analyzed successfully by both the implicit static method and explicit dynamic procedure if the latter is run at a relatively low speed (---10 m/s). With the use of lumped mass matrix, the advantages of the explicit dynamic algorithm is that the stiffness matrix does not need to be formed and the contact conditions are modelled accurately in a simple manner because of the requirements of small time steps. Moreover the material behavior can be complex which is the case with internal damage variable leading to softening of the material.
3.2. Integration of constitutive equations It is known that one of the best algorithm for integrating constitutive equations is the Backward Euler or implicit scheme [17]. However, in case of plane-stress condition, the out of plane component of strain is not defined kinematically and must be added as an extra unknown in the local Newton iteration scheme. This fact and the presence of "cosh" terms in the yield function and flow rule may lead to numerical difficulties when the damage variable increases rapidly. The authors have chosen a substepping scheme on the modified Euler algorithm which incorporates error control where the details can be found in [18]. This approach is suitable with explicit dynamic analysis since it takes advantage of the small time step required by the overall stability limit. Then on each substep, the following set of incremental forms of equations are used to compute the stress increments : dtr = do-% D de p (13) where d~re is satisfying the multiplier dx is ncP= -dx o~/op
the elastic stress increment vector and D the elastic (3x3) matrix plane stress assumption. From the flow rule Eq.(6), the plastic eliminated with the following two equations : and A e q - - d~ O~/Oq with A c p o~/oq + A~: q o~/op = 0 (14)
Eq. (1) is used to yield : ncP=0 if Crm<- 0 and neP=klnCq if Crm>0 with kx=[3qxqzf sinh(-3qzp/2o'y)] / 2q (15) Noticing the gradient vector a so that : a = aq/a,r = q-1 M o-
and
dq = a r thr = a r du e - k 2
Ae q
(16)
where it is found that for plane-stress : k z = ktE(a~x +ayy)/[3(1-v)] + aT D a
(17)
The equivalent plastic work Eq.(7) gives the effective strain increment : de p = neq(q-ksP)/[(1-f) o'v]
(18)
Use of h'=d~v/d~ p the hardening modulus of the matrix in Eq.(17) and in Eq.(18) leads to: /Xeq = ardo"e (1-f)o'v/[(h'+k3)(q-ktP)] where k3 = kz(1-f) Cy/(q-kxP ) (19) The plastic out-of-plane strain increment can be now written as : d~:P3
=
k 1ne q -(dexx+deyy) p p
(20)
209 Notice that if the porosity f = 0 then k l = 0 and the plastic incompressibility is obtained. For the first order Euler algorithm, the stress at the end of a substep is given by : o'K+1 = o"K + doh (21) and it is the same for each internal state variable : the effective strain and porosity f where all quantifies have been evaluated at the stress state o-K. A more accurate estimate of ~K.~ and state variables may be obtained from the modified Euler scheme which gives : crK.1= o"K + (do"1+do'z)/2
(22)
where ~ z and all quantities are evaluated at the stress state o'K+~. The global error in the solution may be controlled by ensuring that the relative error for each substep is less than some specified tolerance : ,(do- 2- do])/2,/llo-K+~, < TOL
(23)
The size of each substep is continually updated during the integration procedure to satisfy Eq.(23) where TOL is a small positive number in the range 1.E-02 to 1.E-05 . 4. NECKING CRITERION
4.1. Physical considerations: The strain ratio /3=ncz/ne 1 has an evident influence on the internal damage of sheet metals. At the same level of deformation, it is generally noted that the damage increment is the greatest at plane strain. This indicates that plane strain is the most dangerous strain state at which strain localization is most likely to appear. Gradually forming of plane strain state after load instability (diffuse necking) may be the common origin of leading to strain localization at tension-tension and tension compression stress states. In both regions, the strain evolves towards the plane strain state such that Ae2=0 when the localized necking occurs. 4.2. Mathematical form: Our formulation follows the recent works of Hora and co-workers [19] but our criterion is formulated in terms of the principal stresses and their orientations in preference to the more usual components relative to the orthotropic directions. This type of decomposition is needed in particular when the deformation is viewed from the standpoint of a rectangular specimen cut from the sheet at any angle as it can be shown on Fig.(1). Moreover, the damage model may be introduced in the anisotropic constitutive equations used for the criterion in order to be consistent with the coupled F.E. analysis. Y 1
Figure 1. Orthotropic and principal stresses axes
210 On Fig.(1), x and y are the in-plane orthotropic axes and 0"1>0"2 are the principal components of stresses typically directed at an anticlockwise angle o~ to x and y respectively. At the onset of load instability (dFl-~ 0), the plane stress assumption gives : 0"1
+ de z + de s ~-0
(24)
with ~ as the major Cauchy stress. In this section, the elastic part of the strain is neglected and the internal damage up to the diffuse necking is assumed to be very small so that the plastic incompressibility leads to : dr 0~1 ~ 0"1 or 0"x replaced by 0"10"y/q if internal damage is taken account (25) Since the state of strain evolves towards the plane strain state, due to the related stress state change, there is an additional hardening effect such that: 00"1 d0`l = ~ 1
80"1
de1 + -Off d~
(26)
where the stress dependency 0"1(el,/3) has been earlier introduced by Hora and coworkers [19] with the strain ratio /3=Aez/Ae 1 . Then the necking criterion takes the form: 00`1 00`1 d # 0 e l -]- - ~ ~ 1
(27) -< 0`1
4 . 3 . Intrinsic formulation with Hi11'48 material behaviour: In order to determine explicitly the left-hand side of Eq.(27) and without loss of generality, the yield criterion Eq.(3) and (4) is written as: 2
2
2
q = a(a) 0"1 - 2 c(a) 0"1o"2 + b(tO 0`2 where:
(28)
a(o0 = [ ( F + G ) + 2 N - 2(F-G)coszo~ + (F+G+4H-2N)cosZa ]/4 b(o0 = [ ( F + G ) + 2 N + 2(F-G)cos2~ + (F+G+4H-2N)cosZo~ ]/4 c(cz) = - [(F+G)-2N - (F+G+4H-2N)cosZ~ ]/4 The change from orthotropic axes to principal stresses axes is motivated by the fact that the necking criterion Eq.(27) is written in terms of the major stress and the wish to make explicit the dependence on loading orientation. The formulation of the flow rule in terms of intrinsic variables (0"1,0"2,o0 gives the components of the plastic strain increments on the principal axes of stresses as 9 de I =
da (90"
1
de z = dA 0q
00"2
2(0"1-0"2) de12
= dX aq atr
(29)
Assuming a work-equivalent strain increment de conjugate to q, that is: q d~ = 0`1de1 + 0`zdez
(30)
211 It follows that d~t=d~ since: oq oq q = ~0`1 + ~0`2
(31)
The normal components of strain increments on the principal axes of stresses are: de 1 = de [a(o00`1 -c(~)0`2]/q de z = de [-c(o00`1 + b(o00`z]/q It is deduced that the stress ratio: 0`2 = n(~,t~) 0`1 with n(~,t3) = [a(~)t3+c(~)l/Ib(~)+t3c(~)] is dependent of the strain ratio t3=dez/de 1 and of the orientation oc. Having regard to (28) and (34), the major stress 0`1 is expressible as:
(32) (33) (34)
(35) The first term of the left-hand size of Eq.(27) can now be determined explicitly by the chain rule of derivation: o.1 = q [a(oO +b(oOnZ-2c(=)n] -1/z
00"1 0e I
00`1 0q 00`y 0~ 0q 0% 0~ ~
(36)
The first derivative which appears can be evaluated as : 00`1
- [a(~) +b(~)n2-2c(=)n] -~/2 oq If the internal damage is not coupled with the necking criterion then:
(37)
0q = 1
(38)
00`y
otherwise the corresponding derivative takes the more complicated form: aq _ [A]1/2 + 0`v [A]-1/2 B/2 0%
(39)
where: A = 1 - 2qlf cosh(-3q2P/20`y) + q3fz B = - [3qlq2P f sinh(-3q2p/20`y)]/0`y In both cases, the third term in Eq.(36) describes the material behaviour by the yield curve of the fully dense matrix material in the common form: h = d0`r/d~ (40) Next some formulae are recorded for the last derivative of Eq.(36) where the strain ratio t3 is introduced in the inverse of Eq.(32) and (33) and where Eq.(28) is used: de _ 1 ~[~'(a,t3)" ~1-~ where: /x = a(a)b(a)-c(o02 ~(a,t3) = al(a) + a2(a)13 + %(a)/32 in which: ax(o0 = a(~)b(~)2 _ b(o0c(~)2 a2(~) - 2[a(~)b(=)c(~) - c(~) 3] a3(o0 = b(o0a(~) 2 - a(o0c(~) 2
(41)
212 The second term of the left-hand side of Eq.(27) represents the additional hardening effect which increases the strain level due to the dependency of the major principal stress to the strain ratio /3. Then, using the expression of the major stress Eq.(35), we find: 0% d/3 13 [2b(~)m' - 2c(~)n'] 0/3 ~ 1 = ~ [a(~)+b(00t~Z-2c(o0f2]3/z where n' is the derivative of n such that:
(42)
on _ a(~) _ c(~)[a(~)+t3c(~)] or3 b(~) + ~c(~) [b(~) + t3c(~)]z
(43)
By a direct implementation into any F.E.-codes where the principal stresses and their orientation ~ with respect to the orthotropic axes are calculated at each time step or increment of load, the proposed criterion can be used as a stop-test for arbitrary strain paths in a general 3D. sheet metal forming analysis. Moreover, F.L.C. can be numerically evaluated by Eq.(27), with the possibility to include damage effect, for linear strain paths as well as for non-linear paths and different orthotropic axes orientations. 5. F O R M I N G
LIMIT C U R V E
As an example, the comparisons between the proposed necking criterion and our experiments have been obtained from a mild steel sheet named XES of thickness 0.67 mm with the following material properties: Elastic modulus: 198 GPa, Poisson' ratio: 0.3 and the coefficients of the true-stress and logarithmic-strain uniaxial curve are: = B(c+e) n (44) B=551.14, c=9.54E-03, n=0.2797 and with the anisotropic Lankford's coefficients: Ro=2.2 R9o=1.6 and R4s=1.9 Experimental Marciniack's tests have been carried out on rectangular strips where various widths (40---/-160ram) have been used to obtain different strain ratios. The rectangular test-pieces were cut such that the orthotropic axes may not necessarily coincide with the principal stresses axes. To avoid fracture of sheet metals on the die radius, a second blank is pasted on the test-piece with a circular hole at the center of about 25 to 40mm. Moreover, a new method of measurement of displacement field based on the digital image correlation principle has been developed in our laboratory. The precision of the measure can reach 1/60th of a pixel and then it is possible to measure swains between 5.E-05 to 0.7 on a plane surface. However, lost or alteration of the random aspect (speckle), deposed on the surface, may not guarantee the success of the measure but it is not generally the case in the Marciniack's tests. It is also found that this method with the software sp~ially developed for analyzing the successive digital images obtained by the video camera system, gives F.L.C. on a lesser level of major strain mainly at plane strain and tension-compression conditions as it can be shown on Fig.(2) by comparison with a classical method of strain measurement (grid deposit). Fig.(2) shows also the comparison of the F.L.C. calculated using the proposed criterion where the given strain ratio ~ varies from -0.5 to 1 and is imposed along a linear strain path in this example, the stress state is determined incrementally by integration until the criterion is satisfied. The calculated curves has been obtained without and with internal damage coupled. In the latter case, an initial porosity of fo = 0.004 is introduced in conjunction with the following damage parameters 9SN=0.1, eN=0.5, fN=0.04 and f~r =0.04 with K=5 .
213 MAJOR
STRAIN
0.7 0.6
...............................
i , ............................................................................................................................................................................................................
0 . 5 ........................................................ [ ] ' ~ .....................................................................................................................................................................................................................
.....................
. .........................
.
...........................
. .................
.......
.........................
0.2
..............~...........................................................................................................................................................................................................................
0.1
...........................................................................................................................................................................................................................
0
. . . .
-0.5
-0.4
-0.3
-0.2
-0.1 MINOR
9 Num.
-t- Num. w i t h damage
0
0.1
0.2
0.3
0.4
0.5
STRAIN ~
Exp. ( g r i d )
[]
Exp.
(speckle)
Figure 2. Experimental and Calculated Forming Limit Curves 6. FINITE ELEMENT ANALYSIS The first examples of the F.E. applications are with the tools and blanks geometries used in the Marciniack's tests. The rectangular sheet blanks are clamped fLrmly at the periphery by a lock bead. The F.E. analysis is done for the complete formed part and the thickness of shell elements is twice in the zone where a second blank is pasted. The proposed criterion is introduced in our F.E. code as a stop-test in order to detect the onset of necking. It has been found that the additional straining due to the coupled damage analysis is only more than 2-3% of major strain than that for the case without damage considered. However, the maximum damage variable value gives a better geometric localization than the thickness strain distribution. Moreover, for a precise evaluation of the onset of necking and its geometrical position, an adaptive mesh refinement in the critical area is performed based on the growing of damage.
Figure 3. Deformed meshes at necking (widths: 60 and 150mm)
214 Broadly speaking, the enrichment process is based on a simple side splitting operation, where-by nodes are generated in the middle of those sides that belong to elements with damage variable exceeding a given value for each remeshing procedure. Fig.(3) shows the deformed meshes and refinements at necking for blank widths of 60 and 150ram and where the orthotropic axes coincide with the principal strains axes. For the width 60mm (tension-compression state), the punch stroke at necking is 21.7mm and e 1=0.511, ez=-0.248 where the experimental values are e1=0.562, e z=-0.253. For the width 150mm (tension-tension state), the punch stroke at necking is 23.9mm and et=0.356, ez=0.322 where the experimental values are e, =0.339, e~=0.310. In both cases, the experimental values are those measured by ~e digital linage correlation method. Others examples but with the orthotropic axes making different orientations with respect to the principal strains axes show the same level of agreement. 7. CONCLUSION The method presented in this paper shows how the Gurson-Tvergaard's model can be extended to anisotropic sheets. A better localization of the onset of necking is expected by a coupled damage analysis of 3D. sheet forming processes. Moreover, a necking criterion is proposed based on the fact that the strain state becomes a plane strain state in the neck localized by the damage variable. The criterion is written in an intrinsic form where the angle of the orthotropic axes with respect to the principal stresses axes is introduced. The given criterion is as well applicable to linear strain paths as well as to non-linear paths. REFERENCES
1. Brunet, M., Materials Processing Defects, Ed. by Ghosh,S.K. and Predeleanu,M. Elsevier Ed.,(1995) 235. 2. Brunet,M. Arrieux,R. and Nguyen-Nhat, T., NUMIFORM'95, Ed. by Shen,S.F. and Dawson,P. Balkena,A.A. ed., (1995) 669. 3. Storen,S. and Rice,J.R., J. Mech. Phys. Solids, Vol. 123, (1975) 421. 4. Yoshida,T. Katayama,T. Usuda,M., J. of Mat. Proc. Tech., Vol. 50, (1995) 226. 5. EI-Dsoki,T. Doege,E. Groche,P. , NUMISHEET'91 VDI Beritch 894, VDI-Verlag ed. (1991) 301. 6. Doege,E. EI-Dsoki,T., J. of Mat. Proc. Tech., Vol. 32, N* 1-2, (1992) 127. 7. Gurson,A.L., J. Eng. Mat. Tech. Vol. 99, (1977) 2. 8. Tvergaard,V., Int. J. Fract. Mech. Vol. 17, (1981) 389. 9. Doege,E. E1-Dsoki,T. Seibert, D., J. of Mat. Proc. Tech., Vol. 50, (1995) 197. 10. Brunet,M. Sabourin,F., J. of Mat. Proc. Tech., Vol. 50, (1995) 238. 11. Hill, R. , Proc. Roy. Soc. London, A193, (1948) 281. 12. Hill, R. , Math. Proc. Cambridge Phil. Soc., 85, (1979) 179. 13. Hill, R. , J. Mech. Phys. Solids. Vol. 38, (1990) 405. 14. Barlat, F. and Lian,J. , Int. J. Plasticity, 5., (1989), 51. 15. Chu,C.C. Needleman,A., J. Eng. Mat. Tech. Vol. 102, (1980) 249. 16. Tvergaard,V. and Neexlleman,A. , Acta Metall. , Vol, 32, (1984) 157. 17. Aravas,N., Int. J. Nun. Meth. in Eng. Vol. 24, (1987) 1395. 18. Sloan,S.W., Int. J. Num. Meth. in Eng. Vol. 24, (1987) 893. 19. Hora,P., Tong,L. and Reissner,J. , Proc. of NUMISHEET'96, Ed. by Wagonner and al.,Dearbom, U.S.A., (1996)252.
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
215
L o c a l i z a t i o n of d e f o r m a t i o n in thin shells with application to the analysis of n e c k i n g in sheet m e t a l f o r m i n g J.C. Gelin and N. Boudeau Applied Mechanics Laboratory, UMR CNRS 6604, University of Franche-Comt6 24 Rue de 1' Epitaphe, 25030 Besanqon Cedex, France
Necking occurs quite naturally in stamping or deep drawing processes due to the development of instability associated to material behavior and boundary conditions. Our study is focalized on the discussion and comparison of necking phenomenon considered as a problem of instability of the local equilibrium state for sound material or a problem associated to the development of anisotropic damage leading to the instability. Results for localized necking predictions obtained from finite element simulations of sheet metal forming processes are given illustrating the proposed approaches. 1. INTRODUCTION During sheet metal forming processes, the main causes of fracture result from plastic instabilities associated with localization phenomena commonly called necking. The plastic instabilities are related to the mechanical conditions associated with the loss of stability of the equilibrium problem. This problem can be treated as a local problem by using the singular perturbations technique [1], the linear stability analysis [2][3], the Hill's stability theory [4] or on the well known MK [5] model. In the first case, the perturbation of the equations characterizing the equilibrium state leads to a linear system where the non trivial solution corresponds to a nonlinear algebraic equation, in the second case the linearization of the local problem leads to a polynomial equation where the unknown is the instability parameter, in the third and fourth case the integration of constitutive and equilibrium equations along a given strain path without or with a small initial thickness defect permit to detect the level of strains at necking as a function of the defect. As some of the instability analyses are based on the introduction of a thickness or material defect, a way to predict sheet necking can be to introduce the development of damage in the material behavior law. This damage results from the apparition of microvoids or microcracks growing under large straining conditions and leads to macroscopic discontinuities in the sheet metal. From observations at the microscopic level, the main cause of internal damage in sheet metal forming is the ductile fracture mechanism occurring in crystallographic materials [6]. Ductile fracture mechanism results from the decohesion of the matrix material around inclusions or second phase particles, then the microcavities or microvoids growth due to large straining and they tend to form macroscopic cavities or cracks leading to macroscopic fracture. The approaches proposed in the paper are mainly based on the linear stability analysis and are developed taking into account the initial or induced anisotropic material behavior. The initial anisotropic behavior of the sheet metal is described by the quadratic Hill's model [7]. On the other hand the induced anisotropic behavior is based on a polycrystalline law [8]. Such a
216 behavior law allows to take into account the crystalline structure, the evolution of the orthotropy axes during straining and the induced anisotropy without making kinematic assumptions. In order to describe the ductile damage, an extension of the damage model proposed by Gelin and Predeleanu [9][10] accounting for anisotropic plastic behavior is proposed. The linear perturbation technique is then used for the equilibrium equations which are defined at the local level. The necking analysis is included as a post-processing capability of a finite element software for modeling sheet metal forming processes. The proposed approaches are applied to different necking characterization tests as Nakazima's tests, Marciniak's tests and square cup deep drawing tests. 2. THE LINEAR PERTURBATION TECHNIQUE The mechanical state of a material point is defined by the following equations 9the plastic behavior and the hardening laws, the mechanical equilibrium equations, the p l a s t i c incompressibility (or the evolution of the void volume fraction in the case of a damaged material) and the compatibility equations for the strain rate components. The mechanical equilibrium, the plastic incompressibility and the strain rate compatibility are given respectively in equations (1),(2) and (3) : div (h (~) = 0
(1)
trace D = 1~/( 1 - f)
(2)
D11,22 + D22,11 = 2 D12,12
(3)
where D stands for the plastic strain rate, f for the void volume fraction and c for the Cauchy stress tensor. The behavior law and the hardening law being related to the constitutive approach, the dimension of the equation set describing the mechanical state depends on this approach. However the resulting equation set can be written with the following formal expression : A(u) = 0
(4)
Let note that equation (4) represents the equilibrium at the unit volume in case of a phenomenological approach as it represents the equilibrium at the grain level in case of a microscopic approach. Let u ~ be a solution of equation (4). To study the stability of the solution u ~ in the sense of the linear perturbation technique, the perturbed solution u is considered : u = u ~ + 8u
(5)
with 8u = 8u 0. exp(rlt), exp (i ~ x.n)
(6)
where 8u ~ is the amplitude of the perturbation, rl is the temporal part of the perturbation and its spatial part. The vector x represents the spatial localization where instability is supposed to occur in a global frame (el,e2) related to the sheet. The vector n is the normal to the direction (A) along which instability could develop itself. The perturbation of the initial set (4) following by a linearization leads to the problem below : A(u~
8u ~ = 0
(7)
217 where ~ is the angle between the x- and n-vectors (figure 1). The condition to have a perturbation is that equation (7) has a non trivial solution for 15u~ That is satisfied when 9 det [A(u~
=0
(8)
It can be shown that this determinant can be reduced to a low degree polynomial form from which the instability parameter can be calculated easily. The study of equations (5) and (6) shows that it is the sign of the perturbation parameter 11 which determines the stability or the instability of the solution. Finally, the instability criterion is 9 Re(rl) > 0
(9)
X2 n
e2tJ w
} el
x=O
Xl
Figure 1. Representation of the x- and n-vectors 3. MATERIAL BEHAVIOR 3.1 Anisotropic elasto-plastic behavior
We first consider a phenomenological anisotropic behavior law. Sheets metal for deepdrawing being generally obtained by cold rolling, the orthotropic behavior can be accurately described using the Hill's criterion [7] (at least for Lankford coefficients greater than 1). Then the plastic behavior is given by the following equation : E
D = -
t~
H'~
(10)
where m
o=H'o'o
(11)
H represents the Hill's tensor, ~ the Cauchy stress tensor and D the strain rate tensor. ~ is the equivalent stress defined by (11) and ~: the equivalent strain rate. The hardening law can be a Swift law with sensitivity to the strain rate 9 <3o(~:' ~) = k.(a ~ +
~2)n.(~)m
(12)
Sheets for metal forming generally are not strain rate dependent. However, it is needed for the establishment of the instabilitycriterion.At last,the strain rate sensitivityparameter m will
218 be imposed to be equal to zero. According to section 2, the set of equations (1) to (3) and (10) to (12) is of dimension 9. The equation (8) leads in this case to a third degree polynom with the following expression" 1"12(C 2112 + C 1 1] + C0) = 0
(13)
The roots of this polynom are easily calculated and the instability occurs when the criterion (9) is satisfied for at least one of the non-zero two roots. 3.2 Anisotropic damaged elasto-plastic behavior In order to establish the damage model for the initially orthotropic material, an extension of the damage model introduced by Gelin and Predeleanu [9][10] is proposed. In that case the plastic potential is defined on the following form: (tro'~'] , ( o , 0 o, o0) = , , / o : H ' o - o 0 1-[3Ln(0O)exp~,~-o0)j
(14)
where H is the Hill's tensor and 0 d is the volume change associated with damage evolution (the plastic flow is assumed to be isochoric). 0 d is related to the void volume fraction by: 0 d = (1
f0)/(1-f)
(15)
The plastic strain rate tensor is given by the normality relation as: D=
~[Ljo:H:o H!_a. + L?d ~exp(~-~0)l tro ]
(16)
The plastic potential defined in relation (16) is closed from the form obtained in [ 11 ] and used to predict defects occurrence in various metal forming processes[ 12] [13]. The case 0 d =1 or f=0 corresponds to the Hill's case whereas f= 1 corresponds to the complete degradation of the material. In the physical reality f=l is never reach and the load carrying capacity of the material decreases considerably for small values of the void volume fraction (from f=0.05 to f=0.2). The hardening law is the same as given by equation (12) and the stability analysis can be conducted in the same manner. 3.3 Crystalline plasticity behavior law As sheet metals for deep-drawing are polycrystalline materials then a phenomenological behavior law can not model well what is happening at the microscopic level. It can not take into account the induced anisotropy which is important for the large deformation encountered in deep-drawing. This induced anisotropy is due in particular to the rotation of the grains in response to the external loading. It is then interesting to use more physical models for the material behavior and actually a lot of works are done is this sense. In our approach, the material behavior is modeled using the Taylor's model [14]. This implies some adaptation of the linear perturbation technique proposed in section 2 to the new constitutive equations. With this microscopic approach for the material behavior, the plastic law and the hardening law are defined at the grain scale. The plastic law is given by the following equation 9 Ns
DP = ~'i,(cx)P a (X
(17)
219 where N s is the number of slip systems, j,(a) is the slip rate on the slip system oc and p(a) the symmetric part of the tensor defined by the tensor product of the slip direction s (a) and the normal to the slip plane m (cx) : p(a) = (sCa)|
(18)
The hardening law is given below"
''m
sgn(x(a))
(19)
I where i'0 is the reference slip rate, ,~(a) is the resolved shear stress on the slip system o~ and X(ca) the critical resolved shear stress on theslip system o~. The resolved shear stress on the slip system a is defined as following" x(a) = P(a) 91:
(20)
with x the macroscopic Kirchhoff stress tensor. Then the equations set defined by the equations (1), (2), (3), (17) to (20) is of dimension 7+N s. The application of the linear perturbation technique to this set leads to a polynom whose expression is given below 9 rl
Ns
(Clrl + C 0) = 0
(21)
The above equation allows to calculate the instability parameter 1"1 and then to determine the stability or instability of the considered grain. The passage from the monocrystal to the polycrystal is done using the Taylor's theory and the perturbed stress state at the grain level is obtained by the maximization of the plastic power [8]. 4. RESULTS 4.1 The modeling of necking by post-processing finite element results The numerical results presented in the following subsections are carried out with the explicit version of the code Polyform developed in our laboratory. The finite element approach used is based on 3D shell elements with the zero normal stress constraint and an assumed strain interpolation in the reference element frame to avoid shear locking due to the shear strain components [ 15]. Each finite element result at each integration point of the sheet metal corresponds to a mechanical equilibrium state described by 3D-strain and -stress states. According to the linear perturbation technique presented in section 2, the method adopted consists to perturb the equilibrium state at each integration sample [16]. The matrix A defined in section 2 needs some adaptation to the available finite element results i.e. the logarithm strain tensor, the incremental logarithmic strain tensor and the Cauchy stress tensor. These adaptations are easily accomplished using the determinant properties. Finally the new problem is the following: det A(A~ij,~ij,~,TI,V) = 0 where ~ and crij are respectively the equivalent strain and the Cauchy stresses at the increment I and A~ij the increment of the logarithmic strain between
220 the increments I and I+i where i represents the frequency of storage of the finite element results. Its value will depend on the complexity of the problem and will insure that the strain path has not a too large variation between these two instants. 4.2 Nakazima's test
The Nakazima's test is a very classical test used to build FLD. It consists to draw bands of metal cut in a circular flange with an hemispherical punch in a cylindrical die. The variation of the band width allows variations of the strain path. A complete circular flange corresponds to balanced biaxial stretching. The flange considered in the present case is 20 mm of width that corresponds to a negative strain path and the material parameters are : E=210000 MPa, v=0.3, k=536.22 MPa, n=0.2315, e0=0.0001 and f0=0. The necking is generally observed in the middle of the wall. The instable zones shown in figure 2a corresponds effectively to this localization. The thinning contours for the same depth of drawing are shown in figure 2b and do not reveal a critical thinning. The thinning is smooth across the sheet and its does not let appear a risk of rupture which is however present and well detected by the perturbation technique. The post-processor predicts necking for a drawing depth of 27 mm and an equivalent strain of 0.203. The experimental works detect the phenomenon for a depth of 30 mm for an equivalent strain of 0.205 [17]. The predictions are then in a very good agreement with the experimental measures. The same calculations conducted with the damage model combined with the linear stability analysis leads to the same strains at necking with a void volume fraction equal to 0.3.10 -4.
Figure 2. Numerical results (a) instable zones in the middle of the wall in agreement with experimental observations (b) thinning contours which do not show a critical thinning in the sheet and then do not reveal the risk of rupture. 4.3 Marciniak's test
The Marciniak's test is another classical test to build FLD. Its particularity is that it does not depend on the friction between tools and sheet using two identical flanges drawed together by a cylindrical punch with a hole in its centre, in a cylindrical die. That provides also the risk of prematurated rupture near the die radius. As the same manner of the Nakazima test, the flanges are cut in an initial circular one and the variation in the width allows a variation of the strain path. In the example presented, the strain path is still negative. The material parameters are : E=198000MPa, v=0.3, k=521.22 MPa, n=0.224, e0=0.00511, f0-0. During experiments necking occurs in the middle of the flat zones of the drawed part. The instable zones are effectively observed in this zone (figure 3).The prediction with the perturbation technique of
221 the principal strain at necking are (0.555;-0.258). The measured ones and the ones predicted by the Stress Limit Curve concept [18] were respectively (0.567;-0.274) and (0.559;-0.277). The agreement between the different results is still convincing. The result obtained applying the damage model combined with the linear stability analysis is about the same as previously mentioned and leads to a final void volume fraction f0- 0.63.10 4.
Figure 3. Instable zones predicted by the linear perturbation technique. The localization of necking in the middle of the flat part of the specimen is in good agreement with experimental observations
4.4 Numerical investigations in the square deep drawing The square cup deep-drawing is simulated with different blankholder forces, friction coefficients and punch radii, in reference with experimental works presented in [ 19].
Influence of the blankholder force: For the lower force (1250 N), instability is detected under the blankholder for 32 mm of depth as wrinkles are observed under the blankholder for 33 mm. Then instability is detected in the comer under the punch radius for a depth of 34 mm. For the higher force (2600 N), instability is detected in the corner under the punch radius for 13.25 mm depth as necking is observed in the same zone for 15 mm depth. The proposed approach seems then to be able to detect any kind of instability 9wrinkles and necking. More, the instability prediction and the instability observations are in good accordance as for the localization as the depth. Let note that the predicted instable zones corresponding to wrinkles are zones in compression (principal stress of opposite sign) as the ones corresponding to necking are in stretching zones (principal stress of same sign and positive). More the instability parameter rl is in a range of at least the unity when it corresponds to wrinkles as it is equal to 0.01 to 0.1 when it corresponds to necking. Influence of the friction: Experimental works have shown the importance of the friction between tools and sheet for the success of a deep-drawing process. Generally, an important friction on large contact zone leads to a late necking occurrence as on very localized zone, it leads to prematurated necking. For the study of the friction influence on necking phenomenon the square cup deep-drawing is carried out with two friction coefficients between punch and sheet : 0.162 and 0.3. In this case, we are in presence of localized contact zone in the comer. For the lower friction coefficient, instability (necking) is predicted for a 16 mm depth as for the higher one, it is predicted for 12 mm depth. The numerical results are then very convincing for the good behavior of the method in front of the process parameters.
222
Influence of the punch radius: The square cup deep-drawing is again simulated with different punch radius. This time the friction coefficient is constant and equal to 0.162. The main idea is to modify the wideness of the contact zone without changing the friction coefficient and then find again the results above. The results are still very encouraging. For the smaller radius (5 mm), necking is predicted for 12 mm as for the other one (8 mm) the phenomenon is detected for 16 mm depth. 4.5 Influence of induced anisotropy In order to studY the influence of the induced anisotropy, a Taylor texture program developed in our laboratory [8] has been used with different linear or non linear strain paths. The results of the Taylor texture program in terms of necking are analyzed by the linear stability analysis. Then Forming Limit Diagrams are built and the textures at necking are analyzed through the { 111 } and { 100 } pole figures. The influence of the crystalline structure is also studied. The studied material is an aluminum alloy with fcc structure. Its initial texture is a typical cold rolling texture. The FLD obtained for linear and non linear strain path are shown in figure 4. The experimental trends are found by the approach : i) a second path is plane strain leads to prematurated necking, ii) a sequence thinning-expansion leads to higher level of strain at necking, iii) a sequence expansion-thinning leads to higher level of strain at necking, and then the phenomenon becomes more prematurated. This is not really observable on the presented FLD for a lake in numerical points, but it has been well validated in the case of an initial isotropic texture.
EI 0.40
g-,I
1
~An
0.40 isotropic material
0.30
0.30
0.20
0.20
0.10
0.10
o.0o (a)
4
0.oc -0.24 -0.12 0.00 0.12 Linear strain paths
0.2482
024 )
Ooln21 0~0 pO~ me am t
024
Figure 4. FLD obtained in linear strain path (a) and non linear one (b) with the second strain path in plane strain sequence uniaxial tensile-balanced biaxial stretching. The study of the texture at necking for linear strain paths and non linear ones (figure 5) shows texture of cube-type which seems in agreement with other authors [20]. First we insist on the fact that the texture has an important evolution which reveals the importance of the induced anisotropy for the strain developed in deep-drawing. Let note that the cube-type texture is stronger when one of the strain path is an expansion one and that it is weaker when
223 none of the strain path is an expansion one. It seems then that the cube-type texture is stable when one of the strain path is an expansion one and we can ask us if there is a relation between this texture stability observed numerically and the stability of the dislocation cells in case of expansion strain path observed experimentally in [21 ]. Xl
X1
3
Strain path 1
Xl
3
Strain path 2
Strain path 4
" path 3
Strain path 5
Figure 5. Pole figures { 100} obtained for linear and non-linear strain paths 5. CONCLUDING REMARKS Firstly, it is shown that the increasing of the void volume fraction in the case of sheet metal forming cannot be considered as a sufficient parameter for indicating necking. Secondly the linear perturbation technique adapted to the post-processing of finite element results is a real interesting tool as it allows to detect the necking zones. The critical zones, the strain level and the critical punch displacement are in very good agreement with the experimental results. The numerical investigations carried out in this paper reveal the good behavior of the necking prediction post-processor in front of the process parameters: This criterion will be interesting to pilot an adaptive remeshing during finite element simulation to let appear the necking band and wrinkles naturally. Concerning damage evolution it seems that one have to consider the damage development after the detection of an incipient necking predicted from standard instability analyses. Finally the linear perturbation technique applied to a polycrystalline law reveals the importance of the anisotropy induced by the large deformations present in deep-drawing. The influence of the crystalline structure is studied. The study of the texture at necking in case of fcc material reveals a characteristic texture : a texture of recrystallization of cube-type. This aspect let appear the existence of a physical necking criterion and the necessity to introduced more physical behavior law to simulate the deep-drawing processes References [ 1] J.R. Rice, The localization of plastic deformation, in Theoretical and Applied Mechanics, Ed. W.T. Koiter, North-Holland (1976).
224 [2] D.Dudzinski, A. Molinari, Modelisation et prevision des instabilites plastiques en emboutissage, In Physique et Mecanique de la Mise en Forme des Metaux, Ed. by E Moussy and P. Franc~osi, Presses du CNRS, (1990) 444-460. [3] C. Fressengeas, A. Molinari, Instability and localization of plastic flow in shear at high strain rates, J. Mech. Phys. Solids, Vol 35, N ~ 2, (1987) 185-211. [4] R.Hill, On discontinuous plastic states, with special refernce to localized necking in thin sheet, J. Mech. Phys. Solids, Vol 1, (1952) 19-30. [5] Z.. Marciniak, K. Kuczynski, Limit strains in the processes of stretch-forming sheet metal, Int. J. Mech. Sci., vol 9, (1967) 609-620. [6] F. Barlat, B. Baudelet, Influence de 1' endommagement sur la limite de formage, In Physique et Mecanique de la Mise en Forme des Metaux, Ed. by E Moussy and P. Franc~osi, Presses du CNRS, (1990) 461-470. [7] R. Hill, The mathematical theory of plasticity, Clarendon Press, Oxford, (1950). [8] N. Boudeau, S. Salhi, J.C. Gelln, Prediction of te localized necking in sheet metal forming based on crystalline plasticity computations, Submitted to Eur. J. Mech. A/Solids, (1996). [9] J.C. Gelin, M. Predeleanu, Finite strain elasto-plasticity including damage- Applications to metal forming processes, 3rd Int. Conf. on Numer. Meths. in Industrial Forming Processes, Fort-Collins, USA, Ed. by E.G. Thompson et al., Balkema, (1989) 151-157. [10] J.C. Gelin, M. Predeleanu, Recent advances in damage mechanics: modelling and computational aspects, 4th Int. Conf. on Numerical Methods in Industrial Forming Processes, Ed by J.L. Chenot et al., A.A. Balkema, (1992) 89-98. [ 11] A.L. Gurson, Continuum theory of ductile fracture by void nucleation and growth: Part IYield criteria and flow rules for porous ductile media, J. Eng. Mat. Tech., 99, (1977) 2-25. [12] N. Aravas, The analysis of void growth that leads to central bursts during extrusion, J. Mech. Phys. Solids, 34-1, (1986) 55-79. [13] J.C. Gelin, Theoretical and numerical modelling of isotropic and anisotropic ductile damage in metal forming processes, in Materials Processing Defects, Ed. by S.K. Ghosh and M. Predeleanu, Elsevier, (1995) 123-140. [14] G.I. Taylor, Plastic strain in metals, J. Inst. Met., Vol 62, (1938) 307-325. [15] L. Boubakar, L. Boulmane, J.C. Gelin, Improved finite element modelling of the stamping of anisotropic elasto-plastic sheets metal, Eng. Comp., Vol 13, n~ 1996, 143-171. [16] N. Boudeau, J.C. Gelin, Prediction of the localized necking in 3D sheet metal forming processes from FE simulations, J. Mater. Process. Tech., Vo145, (1994) 229-235. [ 17] A. Bodin, Direction de la Recherche, Renault, Internal Technical Note, (1991). [18] M. Brunet, R. Arrieux, T. Nguyen Nhat, Necking prediction using forming limit stress surfaces in 3D sheet metal forming simulation, Proceedings of Numiform'95, Ed. by S.F. Shen and P.R. Dawson, Balkema, (1995) 669-674. [ 19] Danckert J., Experimental investigation of a square cup deefap drawing process, 2nd Int. Conf. Numerical Simulation of 3D Sheet Metal Processes. Verification of Simulation with Experiment, Numisheet'93, Tokyo, (1993) 353-359. [20] Y. Zhou, K. Neale, Texture evolution during the biaxial stretching of fcc sheet metals, Textures and Microstructures, Vol 22, (1993). [21] E Ronde Oustau, B. Baudelet, Microstructure and strain path in deep-drawing, Acta. Metall., Vol 25, (1977) 1523-1529.
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
225
Microcrack induced bifurcation of stress-strain relations for sintered materials* D. G. Karra and S. A. Wimmera aDepartment of Naval Architecture and Marine Engineering, University of Michigan, 2600 Draper Road, Ann Arbor, Michigan, U.S.A, 48109-2145 The presence of cavities and cracks, which are inherent in sintered materials, are known to have effect on the mechanical properties of materials. Changes in microstructure, such as growth of microcracks from pores, cause changes in the overall compliance of the material. In order to study these effects, we develop three dimensional constitutive equations for porous, brittle solids based on the damage mechanics of elastic materials containing spherical pores and cracks. For homogeneous deformation modes, the microcrack growth causes nonlinearities in the stress-strain relations. Failure criteria for discontinuous bifurcations, loss of uniqueness, and localizations are examined for the developed nonlinear constitutive relations. A study of the computed compressive strength of such materials is presented in which moderate levels of lateral stresses are applied for biaxial stress states. 1.0 INTRODUCTION Sintering involves the thermal bonding of particles or powders to form a predominantly solid structure. Various processes such as die compaction, isostatic pressing, and rolling are used to influence atomic level bonding mechanisms and to attain the desired shaping and physical properties of the finished product [ 1]. Because of successful application of a variety of sintered materials and sintering processes, there has been wide and steadily growing industrial usage of sintered machine elements and structural components [2]. Sintering of ceramics, metals, and polymers is now also being achieved through the process of selective laser sintering in which computer controlled lasers are used to create objects directly from computer product models [3]. For the optimal usage of these materials and processes, distinct disciplines such as computer science, materials science, and controls theory, have merged considerably with solid mechanics and mechanical design. The green states, or initial structures of the powders prior to sintering, are initially porous and pressing is often used to compact the powder and shape the parts. Unavoidable defects thus exist even before sintering while the thermal sintering processes themselves may densify the compacted powder and reduce pore size. Defects can however be amplified by the sintering process [1]. There is thus an obviously strong linkage of process parameters to mechanical properties such as strength, ductility, and toughness of the final component. These changes in properties are of primary concern for industrial applications and require an understanding of *Research supportprovidedby the UniversityResearch Grant and ScholarshipProgramof the Ship Structure Committee is gratefullyacknowledged.
226 the influence of microstructural changes on the deformation response of sintered elements when subjected to operational stresses. Porosity, an inherent part of sintering, is known to strongly influence the strength of sintered materials. One effect is that porosity reduces the load bearing area of the material, thus increasing the mean stress acting on the material between the pores. Another effect is that the pores or flaws within the parent material act as stress concentration sources leading to microcrack nucleation and growth for brittle materials and local plastic zones for more ductile materials. The microstructures are often extremely complex three dimensional networks of pores and defects of various shapes. Despite the diversity of the final microstructure, the concept of equivalent ellipses has been used by several investigators as an attempt to quantify the stress concentrations for idealized pore geometry [4]. In the following, we use spherical pores as the source for the microcracking and examine the influence of the growth of the microcracks on the macroscale stability of the material. Our analysis therefore presumes that the grains of the material are small enough, such that strength is controlled by flaw size rather than by grain size effects [ 1]. Brittle sintered materials often have higher compressive strengths than tensile strengths. Applications of sintered materials to compressed structural elements are therefore often favored over tensile parts. Emphasis in the following analyses of microcracked induce failure therefore is on compressive stress states. When brittle solids are subjected to compression, microcracks nucleate and grow from the pre-existing pores as the stresses are increased. One approach to analyzing the growth of microcracks has been presented by Sammis and Ashby [5] for axial cracks emanating from spherical pores. Their micromechanical analysis for the growth of cracks from holes yields a relationship between the applied stresses and the length of the axial cracks. This relationship is dependent upon the fracture toughness of the material and the initial porosity; the analysis indicates that stable crack growth in which crack length increases requires an increase in applied stress. The presence of microcracks affects the volume average elastic properties of the material [6]. The formation and evolution of the microcrack damage cause a progressive loss of stiffness of the constitutive response of the material. Continuumdamage theory is used here to establish nonlinear stress-strain relations which reflect the macroscale effects of the nucleation and growth of microcracks. Discontinuous bifurcations, loss of uniqueness, and localization criteria are examined in the following sections. Hill [7] established a single necessary condition for any type of bifurcation or loss of uniqueness. Localization criteria, initially developed for shear band formation in plastic materials, have been used often for the analysis of various brittle materials such as rock, ice, concrete, and composites (see for example Ortiz [8], Wimmer and Karr [9], and Hild, et al. [ 10]). Neilson and Schreyer [ 11] associate localization with the loss of strong ellipticity which, for non-symmetrical acoustic tensors, precedes the classical discontinuous bifurcation condition. This paper focuses on establishing the compressive strength characteristics of brittle materials considering these failure mechanisms. We examine three-dimensional brittle solids idealized to contain multiple pores with microcracks as depicted in Figures l a and lb. The initial voids are assumed of radius, a, with vertical penny crack growth parallel to the xl-axis, the axis of the maximum principal compressive stress. The x3-axis is assumed normal to the plane of the crack; the stress (~33 is therefore the least compressive principal stress. The geometry depicted in Figure lb is an enlargement of a single pore plus crack. We approximate this combination of pore and cracks as an spheroid for the purpose of examining the influence of the
227 cavities on the constitutive response of the material. The spheroid has a major axis of length 2(l+a) and minor axes of 2a.
~11 xI
+ + +
:Z I
%3
x3 2a
_4_
-4// /....--
./
(a)
(b)
FIGURE 1. Crack Growth from spherical holes, a) Representative element with distributed holes, axially orientated cracks, and applied stress, b) Hole and crack geometry. 2.0 STRESS-STRAIN RELATIONS Kachanov, et al. [12] studied the effective elastic properties of solids with cavities of various shapes. The elastic potential function, f, defines the effective stress-strain relations by the equation: Df Eij "- ~Gij
(1)
The function f consists of two parts, 1 f = f o + A f = ~--[(1 ~o
+
Vo)(GlkGkl)-Vo(Gkk )2]--
4" A f
(2)
in which Af represents the change in the potential due to cavities. In equation (2), E o and v o are the Young's modulus and Poisson's ratio in the absence of any cavities, respectively. The change in elastic potential due to spheroidal cavities is [12]"
228 Vcav m f -~ ~ X
2VE o
{Al(~kk)2+ A2(~lk~kl) + [A3(~kk)~ij + A4(~ik~kj)]nini + (A5~imnmnknknl~lj)}
(3)
The porosity parameter, p, is defined as the ratio of the volume of the cavity to the total volume: p = Vcav/V. The coefficients A i are given in [12] in terms of the components of the Eshelby tensor. Substituting into equations (2) and (1) yields the constitutive relation in terms of principal stresses and strains:
I 33_1
•
1 + p~(A 1 + A 2 + A 3 +
A5)
-Vo + P~(AI + ~A3)
- Vo + P~'
1+
1 + p~,(A 1 +
1)
~A3 A2)
m
- Vo + P~',(A1 +~A3) 2.1
Damage
- v o + p~,A 1
-Vo + P~(AI + ~A3)] .
Vo+P~A
33]
u
1 + p~,(.A 1 +
(4)
1 A2)
evolution
The crack extension is dependent upon the applied stresses. Sammis and Ashby [5] considered penny-shaped cracks forming in the Xl-X2 plane when subjected to principal stresses ~11 and a33. The stress component ~11 is the most compressive (negative) principal stress and, ~33 = a~ll, is the least compressive principal stress. The intermediate principal stress, ~22 = a'~ll, is assumed not to influence the crack growth. In addition, it is here assumed that shear stresses also do not influence crack growth. The stress intensity at the poles of the hole is given as:
(5)
K! = -F(a, L ) ~ l l ~ ' a g
In equation(5), L is the nondimensional crack length: L = l/a = ~ - 1. Stable crack growth occurs when the stress intensity becomes equal to the fracture toughness, KIC, hence the nondimensional stress is defined as follows:
Oll~ Ktc
=
1 F(a,L)
(6)
229 For non-interacting cracks, the function, F, was determined [5] as"
(o,62(131.8a)_ cx)
F(cx, L) = ~'L
(7)
Combining equations (6) and (7) yields our damage evolution equation:
fill ~
KIC
=
1 ,,]-L(0.62(1-1.8o0 a)
(8)
~3
The damage evolution equation requires increasing stress to increase the crack length provided a is positive. For negative a (lateral tension), a limiting stress is eventually reached after which an increase in crack length is achieved with decreasing applied stress; locally unstable crack growth is then predicted using equation (8).
2.2 Tangent compliance and tangent stiffness The stress-strain relations given by equation (4) together with the damage evolution in equation (8) provide our description of the constitutive relation of the damaged material. The time derivative of equation (4) gives the incremental constitutive relation:
dEij - Cijkldffkl -b ~--,Cijkldff mn OGmn
(9)
The partial derivatives required in equation (9) are determined using equations (4) and by applying the product rule using equation (8). Inverting equation (9) yields the rate form of the constitutive equation involving the fourth order tangent stiffness tensor, D.
(~ij = Dijkls
(10)
3.0 BIAXIAL COMPRESSIVE FAILURE We examine the compressive failure states in this section with failure defined by either localizaton or general bifurcation. The criteria for the initiation of localization are always satisfied either aider the general bifurcation or at the same point as the general bifurcation.
230 3.1 F a i l u r e C r i t e r i a The general bifurcation [7] criterion is given by:
tU ~U = 0
(11)
which is a necessary condition for any loss of uniqueness. The general bifurcation criterion is also associated with the loss of positive definiteness of the symmetric portion of the tangent stiffness tensor, D s. Equation (11) is first satisfied when:
det(D s) = 0
(12)
A second criterion we employ is that referred to as the loss of strong ellipticity. Neilson and Schreyer [13] used this criterion as a necessary condition for localization. This associates the loss of positive definiteness of the symmetric part of the acoustic tensor with a localized deformation mode. Let M be a vector representing the orientation of the relative velocity of regions on opposite sides of a localized deformation region. The discontinuity occurs across parallel planes with normal vector N. A necessary condition for a general bifurcation with a kinematically compatible mode is [13,14]: M 9
QS* M
= 0
(13)
In equation (13), QS is the symmetric portion of the acoustic tensor: Q = N 9D 9N. As loading is increased from zero, QS is positive definite so that equation (13) is first satisfied when:
det(Q s) = 0
(14)
3.2 C o m p r e s s i v e s t r e n g t h The stress-strain relations for the model are calculated for various porosity levels and lateral stress confinement conditions. The numerical approach is to treat the crack length as a control parameter and determine the corresponding axial stress from equation (8). The components of the tangent compliance tensor can then be determined in a straight forward manner. The value of the determinates of the symmetric tensors D s and QS are then calculated at points along the constitutive path. When criteria given by equations (12) and (14) are satisfied, the constitutive equations suffer a loss of uniqueness which we define as indicating failure of the material. Shown in Figure 2 are the axial stress versus axial strain paths with a porosity ofp = 0.002. The stress-strain curves are shown for lateral stress ratios of a = 0.0, -0.01, and -0.02. The intermediate principal stress value is zero. The general bifurcation criterion is the first failure condition to occur for all lateral stress ratios. Post failure paths are also shown, but these paths are solutions for the homogeneous deformation in which conditions for localization have been satisfied.
231 The failure points of the figure are indicated for the general bifurcation, the localization criterion, equation (14), occurs just slightly after the general bifurcation point. The bifurcations of the constitutive path occur at stress levels for which the crack growth is locally stable. This is shown in Figure 3. We note t h a t for lateral tension, the axial stress versus crack length curve has a peak stress, however before this point is reached, the bifurcation conditions are reached. Thus localization occurs prior to the loss of local stability of the individual cracks. A portion of the post-bifurcation homogeneous paths for stress versus crack length, L, are also indicated in the figure for reference. Lateral tension increases the crack growth and leads to localization at stress levels below the uniaxial strength levels. Initial porosity highly influences material response both with regard to initial compliance and with regard to the onset of localization. In Figure 4 the material failure results for uniaxial loading are shown for various porosities. For very low porosities, the compressive stresses are very sensitive to changes in porosity. The theory presented here considers only stress-strain bifurcations as failure criteria for brittle materials. Other failure modes such as plastic yielding would of course limit material strengths. Some of the combined effects of porosity and lateral stress are shown in Figure 5 for conditions of plane stress. The axial failure stresses are plotted versus lateral stress ratio for several porosity levels. Higher porosities decrease critical stresses. The failure stresses increase as lateral tensile stresses are decreased. Lateral compressive stresses ( a ' > 0) also increase failure stress, but at a much reduced rate. For these cases, ~33 is zero and the failure mode is associated with deformation in the x3-direction.
100
9
o
9
9
80
~ I
40 20
/ / I
20
a =- 0.01
a =- 0.02 I
I
40 60 EllEo~r~" KIc
I
80
100
FIGURE 2. Nondimensional stress-strain curves, for biaxial loading for various lateral stress ratios, a, for p = 0.002. Failure points for the general bifurcation are indicated.
232 100
/a
80
=0.0
•l-~
a =-0.01
ffl~ 40 I 20
a =-0.02
0
0
I
I
I
I
I
I
I
1
2
3
4
5
6
7
8
L
FIGURE 3. Crack extension, L, versus normalized stress for various ratios of lateral stress, a, for p = 0.002. Failure points for the general bifurcation are indicated.
140 120 L~! ,.100
ffl
80
I
60 40 20 0
I
0
I
I
I
0.002 0.004 0.006 0.008
0.01
P
FIGURE 4. Nondimensional stress at the general bifurcation versus porosity for uniaxial loading. The general bifurcation criterion and the loss of strong ellipticity criterion occur almost simultaneously. The shear band orientation vector, N, is the normal to the shear band plane, and can be described by using two angles, r and 0. The angle, r is measured from the xl-axds to the normal vector, N; while 0, is measured from the x3-axJs. For all porosities and lateral tensile stress ratios investigated, r varied from 44 ~ to 50 ~ while 0 varied from 0 ~ to 5 ~
233 14o
p = 0.001
12o 100
ff]~
80
i
60
i
p = 0.005
'
p=0.01
40
20 0 -0.1
I
-0.05
0
0.05
0.1
FIGURE 5. Nondimensional stress at the general bifurcation versus lateral stress ratios for various porosities. 4.0 CONCLUSIONS The results presented here are similar in some regards to the two dimensional analysis of cracks emanating from holes reported in a previous study [9].In the two dimensional case, there was considerable dependence of the failure stress on the geometry of the hole pattern, an effect not addressed in this study. Lateral compression tended to stabilize the material and eliminate localization for the two dimensional case; no deformation mode out of the two dimensional plane was considered. The three dimensional analysis shows much less sensitivity of the failure strength to lateral confinement. A comparable sensitivity to lateral stress requires both (r22 and ~33 to be compressive. Our model neglects any influence of the intermediate principle stress and shear stresses on the determination of the stress intensity at the crack tip and on the damage evolution equation. An interesting extension of the analysis presented would be to include these effects. We have also idealized the net geometry of the pore and crack as simply a spheroid. A more accurate approximation of the geometry could also be included in the stress-strain relations, by allowing the two minor axes of an ellipsoid to be of different lengths. The present model does however provide a very useful description of several characteristics of porous brittle material failure. REFERENCES 1. R. German, Sintering Theory and Practice, John Wiley and Sons, Inc. New York, NY, 1996.
234 2. B. Kubicki, Sintered Machine Elements, Ellis Horwood Limited, Hertfordshire, 1995. 3. U. Lakshminarayan and H. Marcus, Ceramic Composites Fabricated by Selective Laser Sintering, Matl Manufacturing Processes, Vol. 9, No. 5, (1994) 921. 4. B. Kubicki, Stress Concentration at Pores in Sintered Materials, Powder Metall., Vol. 38, No. 4 (1995) 295. 5. C.G. Sammis and M. F. Ashby, The Failure of Brittle Porous Solids Under Compressive Stress States, Acta Metall., Vol. 34 (1986) 511. 6. S. Nemat-Nasser, and M. Hori, Micromechanics: Overall Properties of heterogeneous Materials, North-Holland, New York, 1993. 7. R. Hill, A General Theory of Uniqueness and Stability in Elastic-plastic Solids, J. Mech. Phys. Solids, Vol. 6, (1958) 103. 8. M. Ortiz, An Analytical Study of the Localized Failure Modes of Concrete, Mech. Matl., Vol. 6, (1987) 159. 9. S.A. Wimmer and D.G. Karr, Compressive Failure of Microcracked Porous Brittle Solids, Mech. Matl., Vol. 22, (1996) 265. 10. F. Hild, P.L. Larsson, and F.A. Leckie, Localization due to Damage in Fiberreinforced Composites, Int. J. Solids Structures, Vol. 29, (1992) 3221. 11. M.K. Neilsen, and H.L. Schreyer, The Initiation of Bifurcation and Localization in Damaging Materials, Proc. of Ninth Conf., Eng. Mech. Div., ASCE, College Station, Texas May 24-27, 1992. 12. M. Kachanov, I. Tsukrov, and B. Shafiro, Effective Moduli of Solids with Cavities of Various Shapes, Appl Mech Rev., Vol. 47, (1994) S151. 13. M.K. Neilsen, and H.L. Schreyer, Bifurcations in Elastic-plastic Materials, Int. J. Solids Structures, Vol. 30, (1993) 521. 14. N.S. Ottosen, and K. Runesson, Properties of Discontinuous Bifurcation Solutions in Elasto-plasticity, Int. J. Solids Structures, Vol. 27, (1991) 401. 15. D. Bigoni, and T. Hueckel, Uniqueness and Localization-I. Associative and Non-associative Elasto-plasticity, Int. J. Solids Structures, Vol. 28, (1991) 197.
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
235
I n s t a b i l i t y A n a l y s i s for E l l i p s o i d a l B u l g i n g of S h e e t M e t a l D.W.A. Rees Dept of Manufacturing & Engineering Systems, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom.
This paper presents a theory of instability for ellipsoidal bulging of orthotropic sheet metals. It provides the limiting strains and peak pressure for a pole failure when an instability condition applies. Predictions to the pressure versus bulge height curves are compared with experimental results for four sheet materials: stainless steel, brass and carbon steels with and without zinc cladding. A general approach is adopted in terms of the die axis length ratio, the sheet's r-values, the Hollomon hardening constants and any inclination of the die's minor axis to the material's rolling direction. The theory of instability given requires equivalence in the flow behaviour for all test conditions reported in [1]. From this a sub-tangent is derived to provide the equivalent instability strain which is converted into (i) the ultimate pressure and (ii) the limiting, inplane surface strains. The latter show that greater strains are possible from forming under pressure than from in-plane stretching. Thus, the attainment of a peak pressure is a desirable condition for forming since very large strains can be achieved with the correct material. The analysis presented a has wider application to any part formed from thin sheet materials without contacting dies, e.g. by fluid pressure or controlled explosion. It may be applied to determine the critical pressure and the limiting strains for any ellipsoidal shape defined by its two principal radii of curvature.
I. INTRODUCTION The circular bulge test has long been used for assessing a sheet material's formability under in-plane biaxial tension. The limiting formability is conveniently identified with a pressure maximum for which theoretical solutions have been found [2,3]. These solutions, which apply to bulging isotropic sheet, identify the subtangent to the equivalent stress-strain curve (~-versus ~ with an instability point corresponding to the maximum pressure. When coupled to a suitable description of the equivalent flow curve, it is possible to determine the equivalent plastic strain, ~P, at instability. Swift [2] gave this strain as one root to a cubic equation. For this Swift used the hardening law now attributed to him:
236 ~/% = (1 + ~P/e o) ~
(1)
where a ~ is the yield stress, % the intercept on the negative strain axis and n the h a r d e n i n g index. Hill [3] simplified the solution to a quadratic equation in ~v. ( ~ e ) 2 + [%_ (4/11)(2n + 1)]~.e- 4e o/11 = 0
(2)
A solution to ~ e follows for an annealed material, when e o = 0 in Eqs (1) and (2): ~ e = (4/11)(2n + 1)
(3)
There is little difference between Eqs (2) and (3) when instability occurs at large strain. Recent work has extended a similar analysis to othotropic sheet metals [4]. The sub-tangent S for circular bulging of orthotropic sheet metal is: 1/S = (4ro + 7rgo)Ni32/3)(r o + rgo)(ro + rgo + r o rgo) -- 1/2~ p
(4)
where r 0 and rg0 are the width to thickness tensile strain increment ratios for the sheet's rolling and transverse directions. Eq.(4) can provide a closed solution to T/e when combined with H011omon's hardening law [5]: ~= K(g ~ ) ~
(5)
where K is a strength coefficient. Instability gives a gradient to the flow curve: d ~ / d ~ P = a / S = n f f /~i e
Eq.(6) shows t h a t the equivalent instability strain is; ~ e = with Eq.(4) this gives: ~P = [2(2n + 1)/(4r0 + 7r90)1 ~](2/3)(r0 + r90)(r0 + rgo + r 0 r90)
(6) nS. When combined
(7)
Eq.(7) reduces to Eq.(3) when r 0 = rg0 = 1 for an isotropic sheet material. Such estimates to ~-:v lead to the two principal surface strains used in the construction of the forming limit diagram and to the corresponding ultimate pressure, In circular bulging failure will always occur at the pole once the m a x i m u m pressure has been reached [6- 10]. The theory of bulging through non-circular dies is less complete. Experiments using square [11] and rectangular dies [12,13] on mild steel, brass and copper showed that local instability failures occurred under rising pressure close to the die rim. The elliptical bulge test [1.4] revealed transitional behaviour; where failure occurred either at the pole or at the edge depending upon the die size, sheet thickness and material properties. For example, pole failures were observed for a die axis length ratio of 0.75 when testing 0.75 - 0.80 m m sheets of these materials with their rolling directions aligned with either die axis. Edge failures occurred for a die axis length ratio of 0.25 in each
237 material. Where failures occurred under a rising pressure, predictions to the pressure versus height curve and to the burst pressure assumed plane strain. Pressure maxima were reached for elliptical bulging of mild steel sheet [14]. More recently pole failure and pressure maxima were observed in the pressure versus height plots for elliptical bulging of stainless and orthotropic sheet steels [1,15]. The purpose of the present work is to provide a theory for this.
2. T H E O R E T I C A L Let 1' and 2' define the respective minor and major die axes in Fig.1. This shows a height H above the die surface and the biaxial stress state at pole. The radii of curvature, R o and Re enclose the bulge height, H, and the lengths of the semimajor and semi-minor axes, A and B, respectively. o;
2 2'
1'
I Figure 1. Ellipsoidal bulge geometry The major and minor principal stresses at the bulge pole are aligned with the minor and major axes of the ellipse respectively. They are given by [1]: a~'= p R i t and o'z'= Qcr1'
(8a,b)
where R and the stress ratio; Q = ~2'/a 1' < l, are given as: R = R o R , / ( R o + QR~, ) Rq, = (H 2 + B 2)/2H, R o = ( H 2 + A 2 ) / 2 H Q = [R 0 + (1 + l/ro)R~]/[R~,+ Ro(1 + 1/roo)]
The derivative of Eq.(8a) is;
(9a) (9b,c)
(10)
238 (11a)
b'a ~'l a ~' = 8p/p + 8R/ R - 8 t It
In E q . ( l l a ) there appears the incremental thickness strain, 8e 31" = ~ t / t . At the point of instability, where there is no further increase in pressure, 8 p / p = 0 and: (1 lb)
&rt'laI' = 8 R I R - ~ 3 v
F r o m Eqs (9a, b and c) we define: q = R o/Rq, = [ 1 + ( H / A ) 2 ] / [ ( H / A ) 2 + ( B
]A) 2]
(12a) (12b)
8 R / R = (~R/3R~)(6R~, /R) + (OR/ORo)(SR o /R) = F)(~R~, m ~ ) + o(817, o /Re)
in which the coefficients 0 and o depend upon r 0 , rg0 and q as: O = [ 2 + ( l + l/rgo)[2q-(l + l/ro)+q2(l + llr90)]/[ l +q(l + l/rgo)][2+q(l + l / ~ o ) + ( l + liro)/q]
(13a)
o =[(1 + 1/r90) + 2(1 + l/to)(1 + 1/rgo)/q + (l+l/ro)/q2]/[ 1/q + (1+1/r90)][2 + q(l+l/rgo) + (l+l/ro)/q]
(13b)
Dividing Eq.(11b) by ~3 P and and substituting from Eq.(12b)" (14)
(1/a~')(&q'/Se3v) =- 1 - O ( 1 - R J H ) / ( 1 + R J R o ) - o ( 1 - Ro/H)/(1 + R o / R ~
Within Eq.(14)we have from Eq.(10): 1 +Rc/Ro=(llr 0 + Q/rgo)/[(1 + l/to)- Q], 1 + R o / R , = ( l l r 0 +Qtrgo)l[Q(1 + I/rgo)- 11
(15a,b)
Using Eqs (9b,e) the in-plane, plastic strain increments may be integrated: 8e 11e ' = 8 H / R , and 8~22 v ' = 8 H / R 0 R/H=
891 - exp(- ell e ,)]- 1 and R o / H = 891 - exp(- e22e ')]- l
(16a,b) (17a,b)
3.1. Incremental Strain Ratios Eq.(14) refers to the pole of the bulge where el~ P ' and ~22 P ' in Eq.(17a,b) are in-plane strains aligned with the major and minor elliptical bulge axes. Where the latter axes are inclined at /9 to the material's rolling and transverse directions 1 and 2 (see Fig.l) the principal axes of stress and plastic strain will not, in general, coincide. Strain increment ratios follow from orthotropic plasticity theory and strain transformation [16] as; de22 P '/dell e '= w(O,Q), de.22e '/d%V= r(O,Q), d?,12v '/dell e '= g(O,Q)
where w, r and g are the the following functions of 0 and ~.
(18a,b,c)
239 w( o,Q) = de 2.2P . / de11 p ' = (b4cos4o+ b5sin4o+ b6sin220 ) I(blCOS4O+ b2sin40+ b3sin220 )
(19a)
r (o,Q) =de22 P ' / d e 3 P = - (bgCOS40 + b5sin40 + b6sin220 )/(bloCoS2o + b 1 lsin20)
(195)
g(o,Q) = d71.2 P ' / de I 1P ' = (bTCOS2O + basin20 - b9cos20)sin20/(blcos4o + b2sin40 + b3sin220 )
(19c)
b 1 = rg0[r0(l - Q) + 1], b 3 = 88 1) - (r o + rgo)[2r45(Q- 1)- 1]}, b 3 --- rgo[ro(Q- 1) + Q], b 7 --[2r0rg0(1- Q) + r 9 0 - Qro], b 9 = 89 - Q)(1 + 2r 45)(to + r90),
b 2 = ro[r90(l - (2) + I ]
b4 = ro[r90(Q- I) + (2] b 6 = ~{2rorgo(1 - Q) - ( r 0 + r90)[2r45(1 - Q ) - (2]}
b8 -[2rorgo(Q - 1) + Qr90- to] bl0 -- (/90 + Olr0), bll -- (QTg0+ r0)
Eqs (19a-c) become constants when, for a given Q and 0, the plot between the two plastic strains remains linear. When bulging thin sheet t h r o u g h elliptical apertures, it has been shown [1] that Q remains sensibly constant at the pole as the pressure rises to its m a x i m u m at instability. 3.2. E q u i v a l e n t S t r e s s a n d S t r a i n In order to determine the instability strains for elliptical bulging of orthotropic sheet Eq.(14) must be written in terms of equivalent stress and plastic strain. For off-axis bulging stress and strain transformations [16] lead to equivalent stress and plastic strain expressions: !
"ff = x(O, Q)cr 1 , ~=
d~'= x da 1'
(20a,b)
z ( O , Q ) e l , P ,, d ~ = z de~l v,
(21 a,b)
The increments apply to when x = x(O,Q) and z = z(O,Q) a r e constants for a given 0 and Q within the following expressions: [(213)(r0+ r90 + ror90]l/2x(~
(1 + ro ) ( c ~
Q sin2~ 2 " 2rorgo(C~176 Qsin2e)(Q c~
sin2e)
+ to(1 + r90)(Qcos2e + sin20)2 + (l/4)(ro + rgo)(i + 2r45)( 1 . Q)2sin220 } 1/2
(22)
z(~)
(23)
={A1(~)+A2(~)W(~Q)+A3[W(~Q)]2 + A 4 ( ~ ) g ( ~ ) + A 5 ( ~ ) W ( ~ ) g ( ~ Q ) + A6(~)[g(~Q)~2} ~/2
AI(O) = a c o s 4 0 + b s i n 4 o + ( d 4 + d ) s i n 2 2 0 , A2(o) = ( a / 2 + b/2- 2 d ) s i n 2 2 0 + c ( c o s 4 o + s i n 4 0 ) A3(o) = a sin4o + b cos4o + (c/4 + d)sin220, A4(o) = [(a - c/2)cos2o - (b- c/2)sin20 - 2dcos20]sin20 As(O) = [ ( a - c/2)sin20- ( b - d2)cos2o + 2d cos20Jsin20, A6(o) = (1/4)(a + b - c)sin220 + d cos220 a --- (2/3)(ro+rgo+ror90)( l+,bo)/rgo( 1+to+r90), b = (2/3)(ro+r90+rorgo) ( l+ro)/ro(1+to+r90) c = (4/3)(r 0 + r90 + rorgo)/(1 + r0 + rgo) , d = (2/3)(r 0 + rgo + rot90)/(1 + 2r45)(r0 + ~0)
Also, for a given sheet orientation, 8, it follows from Eqs(19a, b,c) t h a t W, r and g are constants. The condition w + r + wr = 0 ensures that the following total plastic strains sum to zero. Combining Eqs (19a,b,c) with Eq.(21a): e.3P = ( w l r z ) g ~,
e l l e , = ( l / z ) g ~ . ezz e ' = ( w I z ) g P
(24a,b,c)
3.3. S u b - t a n g e n t and i n s t a b i l i t y s t r a i n s Combining Eqs (14), (15a,b) and (17a,b) with Eqs (20a,b) and (24a, b,c) eventually leads to an instability condition:
240 (1/#)(d~/dg v) = . (w/rz){ 1 + 0[(1 + l/r0) - Q][ 90 8 9 v + ~ e / 4 z ) l / ( 1 / r o + Q/rgo) + 0[ (2(1 + l/rgo) - 1][ 9 0 Vz(zlw~.. v + w~" e l 4 z ) ] l ( l l r o + Qlrgo ) }
(25)
C o m p a r i n g Eq.(25) with Eq.(6) provides the sub-tangent for ellipsoidal bulging: 1/S = - [w/rz(1/r o + Q/rgo)]{(1/r o + (~rgo) + 0[(1 + 1/rl) - Q][ 90 8 9 + O[Q(1 + 1/rgo)-l][ 90 8 9 w~e/4z)]}
v + Ue/4z)]
(26)
The equivalent instability strain is t h e n found from the condition: e'-,.e = nS. S u b s t i t u t i n g from Eq.(26) and noting t h a t w < 1 and z > 1 we may neglect the two t e r m s in ( ~ e ) 2 to give the following closed solution to ~ v . e~.e = (z/2w){C)w[(1 + l/to) - Q] + 0[(1 + 1/r~9) - 11- 2nr (1/r o + Q/rgo)} (1/ro + Q/rgo) + 90 + l/to) - (2] + 010(1 + 1/rgo) - 1]}
(27)
In calculating ~ v from Eq.(27), the input data required are: r o , rg0 and r45, the Hollomon contants n, A, the initial sheet thickness, t o, and the ratio between the lengths of the minor and major ellipse axes; a . = B / A < 1. For a given orientation, the radii of c u r v a t u r e (9b,c) and the stress ratio (10) are calculated from an a s s u m e d dimensionless height (h = H / A ) . Then, w, r and z follow from Eqs (19a,b,c) and (23). The equivalent strain is also found from Eqs(16a) and (21b) as: H
~v= z f dH/R~ = z 0
In (1 + h 2
~at2 )
(28)
If e~e from Eqs (27) and (28) do not agree then h is incremented and w, r and z are re-calculated until the difference in ~i v becomes negligible. The equivalent stress at instability t h e n follows from Eq.(5). Eqs (24b,c) allow ~ v to be converted to surface strains, e.l~e ' a n d e22v ' aligned with the ellipse axes. There also exists a shear s t r a i n y12P ' w h e n 0 ~ < 0 < 90" which is found from combining Eq.(18c) with Eq.(24b): Y12 v ' = g e ~ v ' = ( g / z ) ~ v. The three strain components: e l f ', %2v ' and ~q2v ', are sufficient to d e t e r m i n e the principal, in-plane pole strains. The t h r o u g h - t h i c k n e s s strain follows from the incompressibility condition: e3v = - (e I f '+ e22v '). 3.4. P r e s s u r e v e r s u s b u l g e h e i g h t The analysis of instability has been associated with m a x i m u m pressure being reached. This leads to a theoretical prediction to the p versus H curve which m a y be compared to a test recording. Using h = H / A and a r = B / A , to normalise Eqs (9a,b,c), the radii of c u r v a t u r e become: r = Rr
(h : + arz)/2h and r o = R o / A = (h 2 + 1)/2h
p = R / A = r 0 % [%+(.1 + 1/r90)r 0 ]/ro[r~+ (1 + llr9o~r 0 ]+re [to+(1 + 1/ro)r~ ]
(29a,b) (3o)
241 Eq.(28) supplies the equivalent pole strain from which ~ is found from Eq.(5). The relationship between p and h requires the current pole thickness, t to be found. Noting that s3P = ln(t/t) and using Eqs(24a) and (28): (31)
t= to exp [(w/rz)g el = to(1 + tiZ/a~.Z)~'~ Combining Eqs (31) and (20a) with Eq.(8a) gives:
(32)
p = t a l A p x = (~tolApx)(1 + h21~2) ~
where w, r and x follow from Eqs (19a, b), and (22) respectively, given Eq.(lO).
Q from
4. E X P E R I M E N T A L Table 1 gives details of the test materials: annealed 18/8 stainless steel, as-rolled 60/40 brass sheet and soft CR1 steel sheets with and without zinc cladding. Bulge tests were restricted to orientations of 0 = 0 ~ and 90 ~ in dies of aspect ratios; a r = 1.0, 0.89, 0.78, 0.63, 0.50 and 0.42. The length of the major axis of all die apertures was 100 mm. The bulge height H was measured by positioning an lvdt transducer vertically above the pole. The lvdt was supported on two legs giving 2A = 50.8 m m i n contact with the major axis. The voltage outputs from the lvdt and a pressure transducer were plotted together during each test to establish a continuous p versus H curve through to failure. We have seen t h a t the calculations require the K and n - values and the three r - values for a material. These were determined from tension testpieces machined at orientations of 0 ~ 45" and 90 ~ to the rolling direction. Dimensional changes to the parallel length and width dimensions were measured following unloading from the plastic range using 10 increments of increasing plastic strain. The r - values and the Hollomon constants as found from true stress and n a t u r a l strain plots are given in Table 1. Table 1 Tensile properties of four sheet materials Material
Stainless Brass C-Steel Zn-plated
toirrm
steel
steel
0.50 0.40 0.78 0.81
r o
1.00 0.89 2.17 1.70
r45
rgo
K
1.00 0.73 1.47 1.86
1.00 0.59 2.20 2.02
1260 650 600 600
n
0.42 0.20 0.21 0.25
Note that for the present predictions average Holloman constants K and n in Table 1 were used to described both bulge flow and tensile flow where e x a c t equivalence was not achieved.
242 The pressure versus height curves for dies of various aspect ratios are given in Figs 2 - 6 . Stainless steel (Fig.2) shows a maximum pressure particularly for dies of lower aspect ratios. 200 -
/
bor
4.9 4-~ § +~+'~
150-
-
,oo-
,r
/2
S~ )I
so-
,,~ ~_.~*-
, ~+ i~ x , X
.fi(. ~ .'~
_/- /_.o~.-
-x-
..ISy~-bo~
. /I~,o
xh
./
l9
X
I~
+
~
o
I---1
8 ,I o.~Io,~ o.~, o.~o o.~i,~~
f
22~" 0
I
I
2
i+
h/ram
I
1
6
8
Figure 2. Pressure versus H for ellipsoidal bulging 18/8 stainless steel. In contrast, brass (Fig.3) failed on a rising pressure curve. A pressure plateau appears for bulging CR steel in all dies (Figs 4 - 6). Theoretical predictions from Eq.(32) for extreme aspect ratios, contain all the observed curves, reproducing their shape but yielding some error in the maximum pressure attained.
/
80
P bar
60 9
.§ .. ,a'~ §
-~
~+/+ 4.,,§
r
+I + ~
_.qyx~
20
15 2.
hlm m
5SO
Figure 3. Pressure versus H for ellipsoidal bulging 60/40 brass.
243 We see from Eqs (20) and (21) that the theory depends upon equivalent pole flow being achieved when bulging sheet through dies of all aspect ratios. The K and expressions for @ = 0 ~ and 90 ~ orientations in the 100 mm die tests did not produce a unique flow curve for each material investigated [1]. ~ 200 -
P
100
I
I
2
~
h/mm
I
I
6
8
Figure 4. Pressure versus H for ellipsoidal bulging CR1 steel. Of these, isotropic stainless steel exhibited the least divergence in its bulge flow behaviour at large strain. It appeared for all materials t h a t pole flow was restricted by the near die wall when bulging through the dies of low a .
200 ~
P bor
- - - - - - - - - -
o
P
100
I
0
2
~
I
h/ram
6
Figure 5. Pressure versus H for bulging zinc-plated CR1 steel r = 0~
244 Also, as the radii of curvature decrease with increasing pressure, ultrasonic thickness measurement taken at the sharply curved pole become unreliable.
200 0
~
P bar ..+."+
§ +
"~§
"-§ ~ ' §
~W
+-
100
0
4
hlmm
6
8
Figure 6. Pressure versus H for bulging zinc-plated CR1 steel (0= 90"). A preliminary investigation of bulging Zn-plated steels through larger 180 mm dies has achieved the desired equivalence and these will be employed for a future appraisal of the theory presented here.
REFERENCES ~
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
D.W.A. Rees, Int. J. Mech. Sci., 37 (1_995)373. H.W. Swift, J. Mech. Phys. Solids 1 (1952) 1. R. Hill, Phil. Mag., 41 (1_950) 1138. D.W.A. Rees, J. Mats Process Tech., 40 (1994) 173. J.H. Hollomon, Trans AIME, 162, (1945) 268. N.A. Weil and N.M. Newmark, J. Appl. Mech., 22 (1955) 533. A.B. Haberfield and M.W. Boyles, Sheet Met. Ind., 50 (1973) 400. W.F. Brown and G. Sachs, Trans ASME, 70 (1948) 241. P.B. Mellor, J. Mech. Phys Solids, 5 (1956) 41. A.N. Gleyzal, Trans ASME 70 (1948) 288. J.L. Duncan and W. Johnson, Int. J. Mech. Sci., 10 (1968) 157. J.L. Duncan and W. Johnson, Int. J. Mech. Sci., 10 (1968) 143. J.L. Duncan and W. Johnson, Int. J. Mech. Sci., 9 (1967) 681. M.I. Yousif, J.L. Duncan and W. Johnson, Int. J. Mech. Sci., 12 (1970) 959. D.W.A. Rees, J. Mats Process Tech., 55 (1995) 146. D.W.A.Rees J. Strain Anal., 30, (1995) 305.
FORMABILITY CHARACTERIZATION
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Advanced Methods in MaterialsProcessingDefects M. Predeleanuand P. Gilormini(Editors) 9 1997 Elsevier Science B.V. All rights reserved.
247
C o m p r e s s i o n o f a b l o c k b e t w e e n cylindrical dies and its a p p l i c a t i o n to the workability diagram S. Alexandrova, N. Chikanovab and D. Viloticc alnstitute for Problems in Mechanics RAS, 101 Prospect Vernadskogo, Moscow 117526, Russia* bDepartment of Applied Mechanics, Bauman Moscow State Technical University, Baumanskaya 5, Moscow 107005, Russia
2 nd
clnstitution for Production Engineering, University of Novi Sad, V.Perica-Valtera 2, Novi Sad 21000, Yugoslavia 1. INTRODUCTION In engineering practice two methods are commonly used for modeling technological processes of metal forming, the slab method and the upper bound method (see, for example, [1, 2]). Despite the recent development of the upper bound method [3, 4], its applications are restricted to the special friction laws and material models for which the necessary variation principle is proven. On the other hand, the slab method often gives reasonably good predictions for average stresses. However, this method does not permit the description of the kinematic variables, which are of great importance for initiation and growth of microdefects during technological processes. Therefore, in the present paper Hill's method [5] is used to determine the stress-strain components in the deforming region during upsetting performed with two identical cylindrical dies. The material is assumed to be rigidplastic, hardening. The friction law proposed by Carter [6] is applied on the die surface. This problem was investigated by Vilotic and Shabaik [7] using the slab method. Brovman [8] found a slip line solution for a rigid perfectly plastic material. The results of calculations and experimental data are used to find a point on the workability diagram introduced by Vujovic and Shabaik [9]. The experimental determination of stress distribution on the die surface was performed by the method described by Vilotic [10] and Plancak et al. [11]. Steel specimens (0.35 percent C) with initial height 18 mm and length of 40 mm are used in this study. The stress - strain curve for the material is determined experimentally using tension and compression tests according to the Rastegaev method, the results are approximated in the form of the Ludwick relation. The upsetting experiments with cylindrical dies were conducted on a Sack and Kieselback hydraulic press with a capacity of 6.3 MN. *Present address: Alcoa Technical Center, 100 Technical Drive, Alcoa Center, PA 15069-0001. The main part of this work was performed at GKSS Research Centre, Geesthacht, Germany.
248 2. VELOCITY FIELD Due to Hill's method it is necessary to choose a kinematically admissible velocity field satisfying the kinematic boundary conditions. Moreover, this velocity field must satisfy the incompressibility condition. For the process under consideration (Fig.l) it is natural to adopt a bipolar coordinate system. The transformation equations for this coordinate system (see,
rigid-plasticd~=~obOUndary]D[ U E rigid zone
vo
vo
v=v0 die surface
Figure 1. Geometric representation for upsetting. for example, Flugge [12]) are:
x = Asinhg,/(coshg, +cos~);
y = Asin(~/(coshg, +cos~)
(1)
and the components of the metric tensor and the Christofel symbols of the second kind are given by
g ~ = gr162= A2/(cosh V/+ cos r ~, F~,~ = F~ = - F # _
_
sinhg, cosh~ +cos~'
(2) ~, ~ = F~ = sin~ F~,~=-F~,~, # cosh~, + c o s ~
(3)
where A is the distance between the die centers, a function of time, t. It follows from the geometrical considerations (Fig. 1) that
A = Rsinh 9'o,
H = R(cosh 9'o - 1)
(4)
249 where R is the die radius and W0 is the W-coordinate of a line coinciding with the die surface. Let us assume that the rigid-plastic boundary coincides with a coordinate line of the ~b-family, ~ = ~0, which passes through a point N (Fig.l). This coordinate line is determined by the equation
cOSr
= Asinh ~o I H
o
(s)
-cosh~ o
where H0 is the height of the undeformed specimen. The outward normal to the tool surface is given by
(cosh ~'o + fi = -
cos
~)~
A
1 + cosh ~o cos ~_ -
A sin
cosh ~o + c o s ~ / + R(cosh ~o + c o s ~) }
(6)
where v0 is the given magnitude of the tool velocity, v0 > 0. The velocity vector of the tool in terms of Cartesian basis is 9 = -Voi from which it follows that v . n = v n = -Von x
(7)
On the other hand, in bipolar coordinates we have v
9
n
=
Vn
:
v~n ~ + V2 n 2
(8)
Here and in what follows, 1 corresponds to the w-direction and 2 to the d~-direction. Combining (7) and (8) and taking into account (6) one can obtain the boundary condition at W = W0
v~ = - voA(1 + cosh ~o cos ~)//(cosh ~o + cos ~)2
(9)
Analogously, on the rigid-plastic boundary at ~ = ~o
u : v~(cos,,,,,§247
(10)
where u is the velocity of rigid material moving along the y-axis (Fig.l). The incompressibility condition in bipolar coordinates has the form A ' , / v ~ + A'2/d~ = 0
(11)
To satisfy boundary condition (9) and the condition on the symmetry axis, Vl = 0 at W = 0, we assume
v~
=
- ~(~, ~o)V0~0 § oo~. ~o oo~ ~)/[~(~o, ~o)(CO~. ~o § oos~) ~]
~1~
250 where F(W,~0) is an arbitrary function of ~ and ~0 satisfying the condition F = 0 at ~ = 0. Substitution of (12) into (11 ) gives
v2=
F'(~,~o)voAsin~/[F(~o,
~o)(cosh ~'o + cos ~) 2] + C(~, ~o)
(13)
where F' = ct=/c~ and C(~,W0) is an arbitrary function which is determined by virtue of (10)
ruo+cos,
v~
c~ COS ~0
~'~
~~
}
(14)
- F(~o
The magnitude of u can be obtained from the incompressibility condition in integral form
~vknkdl =0
(15)
L
where L is a contour bounding the plastic zone, CNBO (Fig. 1), dl is the length element, n k is the outward unit normal to L. From symmetry, vl = 0 at t~ = 0 and v2 = 0 at t~ = 0. Therefore, substitution of (10) and (12) into (15) leads to u = v o sin ~o/sinh ~/o
(16)
Then, C(~,~o) can be rewritten as
C(~',~'o) = Avo sinr
1+ cosh ~cOSr )2 sinh~,o(cosh~, +cOSr
'o)(c~
+ cos #o)
(17)
Thus, expressions (12), (13) and (17) determine a kinematically admissible velocity field involving one arbitrary function F(~,~0). The components of the strain-rate tensor can be found by means of tensor analysis sinh ~,
sin~
~ =v~,~ + cosh~ +cos~ v~ + cosh~ +cos~ v2;
oc'22 -- V2, 2
-
-
sinh~ sin~ cosh ~, + cos ~ v~ - cosh ~, + cos ~ v2;
Vl, 2 -I- V2,1 ~12 =
sin~
sinh ~,
cosh ~ + cos ~ v~ + cosh V/+ cos ~ v2
(18)
251 3. THIN PLATES If the plate is sufficiently thin that ~o 2 (( 1 then analysis is essentially simplified. We put sinh~0 ~ ~g0, sinh~g ~ ~g, cosh~g0.~ 1, cosh~g ~ 1
(19)
Then the geometrical relationships (4) take the form
A = R~' o, H = Rgo 2//2
(20)
As F(~,~g0) must be an odd function of ~g, we take F(~g,~g0)=V Ago
(21)
Using (2), (19) and (21) the expressions for velocities (12) and (13) and for strain-rates (18) in terms of physical components become v~ = - V o g / / ~
=-.,
o,
v~ = v o
sin~/~,o
(22)
cos )cos /(A o)
v~ [3sin~+ sin ~~(c~
e~,~ - 2A go
~~(1+ - c~ cos~~ ~o)?)(1+ cos ~) 2
1
(23)
4. SISTEM OF EQUATIONS Due to Hill's method [5] it is necessary to choose orthogonalizing motion. Since the kinematically admissible velocity field (22) does not involve any arbitrary functions and, on the other hand, this field is a solenoidal one, orthogonalizing motion must contain one arbitrary function corresponding to the hydrostatic stress, or. This stress is an even function of ~g therefore we assume c = cr(~,H)
(24)
to the leading order. Hence, a comparable orthogonalizing motion in the deforming zone may be chosen in the form
v; = o. v; = v(o..)
(25
where v is an arbitrary continuous function of ~ and H satisfying v(0,H) = 0. According to (20) the independent variable H may be replaced by ~o and vice versa. Applying the
252 Hill's method with the orthogonalizing motion (25) and the boundary conditions on the surface ~ = 0 cry,, = 0
at
~ =0
(26)
and on the surface ~ = 0 ~v, = 0
at
~ = 0
(27)
and on the surface ~ = ~0 (friction surface) where the friction law proposed by Carter [6] is adopted i
i
rt =-o" w =/~Is~,I
(28)
we obtain the approximate equilibrium equation in the form
2S~ sin cosh~, + cosr
d~ ~'o + ~L or
=0
'
(29)
_1_~=~, 0
where the upper sign "~" denotes the stresses calculated from the velocity field (22) and from the constitutive equations; S~, and Sr are the deviatoric portions of the normal stresses. To determine the boundary condition for o, we integrate the equilibrium equation over the rigid zone BNED (Fig. 1). Then, ~r
cr=-~
d~
at
r162
(30)
~o The yield criterion under plane strain condition is \/ 2
2
S~ - S~ } + 4cr;,~ = 4k
2
(31)
where k is the shear yield stress. Its dependence of the equivalent strain % has been determined experimentally using tension and compression tests according to the Rastegaev method. The results can be approximated by the equation k = 213.45 + 385.87 eeq 0"38 MPa
(32)
The equivalent strain rate is given by
[
Oeq = 2(6~,G~, + ~6~ + 2 % %
(33)
253 Applying the associated flow rule we obtain from equation (31) that (34) Eliminating ~ from (34) we find (35) Then, taking into account (23) and (19) we obtain from (35)
S~,=-S~=-k,
cos~)2cos~ sinr176176 ~~162162176 o'~,~=k~,(l+
2) + 3sin~
]
(36)
and from (33) eeq -" Vo COS~l
4- COS ~)/(%~-")
(37)
Due to (36) equation (29) takes the form
~,0rolk
2k sin r 7_,
(38
to the leading order. Taking into account (32), equations (37) and (38) constitute the closed form set with respect to o and eeq. Since eeq should be an even function of ~, it does not depend on W to the leading order. In this case integration with respect to ~ in (30) and (38) may be done. Using (36) it leads to
dcr cik 2k~o sin r ~'O-d--~-+~-~o+ l+cos~ -/zk=O
(39)
o = -k
(40)
at
~ = ~b0
5. SOLUTION We begin with equation (37) which does not contain o. This equation can be transformed to ~eq
--~--
(1 q- COS ~)sin ~ ~e,eq 2H
---~- +
(1 -!- COS ~)sin 4r3H
=0
(41)
The exact initial condition to this equation is eeq = 0 at H = H0. However, the adopted approach does not allow one to consider the initial stage of the process when the plastic
254 deformations are localized near the die surface. Therefore, neglecting this stage we assume the initial condition in the form eeq = 0
at
h
(42)
= cz
where cz = const and h = H/H0. The general solution of equation (41) can be obtained by the characteristics method in the following form
# 1 ;)]sin eeq = ----~ln{OIhtan ~exp(~tan2
r
(43)
where 9is an arbitrary function of its argumentand htan[~/2)exp[O.5tan2[~b/2)]=C / _ \
r
/ _ x l
defines the characteristics of equation (41). The characteristic curves are shown in Fig.2 (except the line AB which shows the coordinate of the moving rigid-plastic boundary found from (4) and (5)). The function 9 is determined by the different analytical expressions on the left and on the right sides from the characteristic passing through the point B (Fig. 2). On the left side, the condition (42) must be satisfied, then
~[a tan(#/2)exp(0.5tan2(#/2))]sin # =1
(44)
On the right side, the condition on the rigid-plastic boundary, where
eeq-" O,
leads to
010.5(1+ cos #)tan(#/2)exp(0.5tan2(#/2))]sin #: 1 h
(45) q)
A
10
0.7 0.6
/
4
Be 0.5
I
1
1.5
Figure 2. Characteristic curves and position of rigid-plastic boundary (AB line).
2 0.2
0.4
0.6
0.8
Figure 3. 9 as function of its argument (solid line from eq. (44) and dashed line from eq.(45)).
Functions tg(z), where z is their argument, are shown in Fig. 3. From the structure of equation (41) it follows that it should be treated near point ~ = O. If ~ is sufficiently small
255 quantity then the solution (43) may be rewritten as (46) Then the condition (42) leads to eeq = 2/~/31n(~/h)
(47)
in the vicinity ~ = 0. Thus, the analytical solution of the equation (41) is found. Hence, the magnitude of k can be calculated from (32). After that, equation (39) can be solved by the characteristics method. It is clear that H = const are the characteristics of equation (39) which is the relation along the characteristics. 6. A P P L I C A T I O N TO T H E W O R K A B I L I T Y D I A G R A M We apply our results to the workability diagram proposed in [9]. This workability diagram is defined by effective strain at fracture to be a function of the stress ratio 13, where [3 is given by (48)
[~ = 3~/~eq
For a Mises material
(Yeq ---
~/3 k. Therefore, we may rewrite (48) in the form
13= ~/3~/k
(49)
The upsetting experiments have shown that fracture initiates at the point ~ = 0 at h ~ 0.5. For this value of h equation (39) has been solved numerically at R = 50 mm, ~t = 0.3 and cz = 0.75 (this value of cz has been taken because the experimental observations showed that the plastic zone reaches the symmetry axis, x = 0, at this value of cz). After this the value of [3 has been determined from (49). At ~b= 0 we have obtained 13= -0.46
(50)
As we have said above, at the beginning ot~the process the analysis is not correct. However, the local deformations at ~ = 0 on this stage influence fracture conditions. We may take them into account approximately since the average effective strain at ~ = 0 is still given by (47) even if a = 1. In this case eeq ~ 0.8
at
h = 0.5
(51)
The values of [3 and eeq given by (50) and (51) determine a point on the workability diagram.
256 7. CONCLUSION Using Hill's method a semi-analytical solution for compression of a block between cylindrical dies is found. This solution combined with the experimental data determines a point on the workability diagram. ACKNOLEDGMENT The authors would like to thank Dr. P.A.Hollinjhead for his help with English. REFERENCES
1. W.F. Hosford and R.M. Caddel, Metal Forming: Mechanics and Metallurgy, Prentice-Hall, New York, 1983. 2. B. Avitzur, Metal Forming: The Applications of Limit Analysis, Dekker, New York and Basel, 1980. 3. A. Azarkhin and O. Richmond, Trans. ASME J. Appl. Mech., 58 (1991) 493. 4. B.A. Druyanov, Technological Mechanics of Porous Bodies, Clarendon Press, Oxford, 1993. 5. R. Hill, J. Mech. Plys. Solids, 11 (1963) 305. 6. W.T. Carter, Trans. ASME J. Eng. Mater. Technol., 116 (1994) 8. 7. D. Vilotic and A.H. Shabaik, Trans ASME J. Eng. Mater. Technol., 107 (1985) 261. 8. M.Y. Brovman, Kuznechno Shtampovochnoy Proizvodstvo, 9 (1966) 1. (in Russian) 9. V. Vujovic and A.H. Shabaik, Trans ASME J. Eng. Mater. Technol., 108 (1986) 245. 10. D. Vilotic, Ponasanje Celicnih Materijala u Razlicitim Obradnim Sistemima Hladnog Zapreminskog Deformisanja, Naucno-Obrazovni, Novi Sad, 1987. (in Serbian) 11. M. Plancak, A.N. Bramley and F.H. Osman, J. Mater. Processing Technol., 60 (1996) 339. 12. W. Flugge, Tensor Analysis and Continuum Mechanics, Springer-Verlag, Berlin, 1972.
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
257
Sheet metal formability predicted by using the new (1993) Hill's yield criterion
D. Banabic a,b "Technical University of Cluj-Napoca, 15 C. Daicoviciu St., 3400 Cluj-Napoca, Romania b present address: Institut of Metal Forming, University of Stuttgart, Holzgartenstrasse 17, 70174 Stuttgart, Germany
The paper attempts to settle a mathematical model in order to predict theoretically Forming Limit Diagrams (FLD's) by using the Marciniak-Kuczynski analysis with the new orthotropic yield criterion (transversaly isotropic version) proposed by Hill in 1993. The model is solved numerically. The algorithm is very flexible, so that it allows to study the influence of various parameters upon FLD's: strain hardening coefficient (n), strain-rate sensitivity index (m), normal anisotropy coefficient (r), geometrical non-homogeneity coefficient (f), ratio between uniaxial and biaxial yield stress (o JOb) (a).
1. INTRODUCTION After the introduction of the Forming Limit Diagram concept by Keeler [1] and Goodwin [2], the research in the field of sheet-metal formability has focused mainly on the development of some mathematical models for the theoretical determination of FLD's. The first realistic mathematical model has been proposed by Marciniak and Kuczynski (M-K model) [3]. They have made the assumption that the strain localization, in case of the biaxial straining, appears in the region of a geometric non-homogeneity of the sheet-metal. Marciniak and Kuczynski have considered this non-homogeneity as a variation of the thickness (a notch) directed along the minimum principal stress axis. The M-K model has been used for analysing the influence of some material parameters (such as the strain hardening coefficient n, normal anisotropy coefficient r, strain-rate sensitivity index m etc.) or process parameters (strain rate, strain path, punch curvature, temperature, vibrations etc.) upon the shape and position of the FLD's. A survey of the research performed in this field has been presented in [4]. The limit strains computed on the basis of the M-K model (using the Mises yield criterion) are overestimated in the domain of biaxial straining and underestimated in the domain of plane straining. The shape and position of the FLD's are strongly influenced by the expression of the yield criterion used in the model. As a consequence, the above-mentioned drawback may be attenuated by considering an appropriate yield function. Several authors have tested this possibility by introducing different yield criteria in the M-K model: Parmar and Mellor [5], Marciniak and Ike [6], Lian [7] have used the criterion proposed
258 by Hill in 1979 [8]; Bassani et al. [9], Neale and Chater [ 10] have used the criterion proposed by Bassani [11]; Gotoh [12] has used the criterion proposed by himself [13]; Graf and Hosford [ 14,15], Padwal and Chaturvedi [16] have used the criterion proposed by Hosford [ 17]; Lian et al. [18] have used the Barlat - Richmond criterion [ 19] extended by Barlat and Lian [20]; Ferron and Touati [21] have used the criterion proposed by Budiansky [22]. Further information concerning these yield criteria and their introduction into the M-K model can be found in [23-25]. The above-mentioned yield criteria has been extended recently in order to make them more flexible. Thus, Weixian [26] has extended the Hill's 1979 yield criterion, Tourki et al. [27] have extended the Budianski's yield criterion, Chu [28] has extended the Barlat-Lian yield criterion.
2. HILL'S 1993 YIELD CRITERION Hill has proposed in 1993 a new yield criterion [29] for orthotropic sheets, valid in the first quadrant. This criterion is expressed by a function containing only quadratic and cubic terms. The principal stresses o~, 02 does not appear in the cubic terms. Thus, it is possible to express one of the principal stresses with respect to the other by means of a quadratic relationship. The expression of the Hill's 1993 yield criterion is O12 _ r 002
+ 022 + [(p+q) - (po,+qo2) ] 0,02
00090
0902
Ob
- 1,
(1)
00090
where: p and q are non-dimensional parameters that may be obtained from the following relationships: (___1+ I -- ___])p = 2ro(Ob-090) -- 2rgOOb + ~C, ~ 090 Ob (1 +r0)o~ (1 +rg0)o~0 ~
(1 + 1 _l)q
. 2rgo(Ob-Oo) . . . (1 +r90)o~0
O0 090 Ob
2roOb , C ; (1 +r0)o ~ 090
(2)
(3)
c may be obtained from the relationship c 00090
1 2
00
1
1
090
0b
(4)
01 and 02 are the principal stresses; 0 0 and 09o are the yield stresses determined by uniaxial straining along the rolling and transverse directions, respectively; Ob is the yield stress obtained by biaxial straining; r 0 and rgo are the anisotropy coefficients corresponding to the rolling and transverse directions, respectively. By rewriting En. (1) for the two cases of uniaxial straining (01 = o . o2 = 0 and o 2 = %0, o~ = 0), one may obtain different values for the anisotropy coefficients
259 ro * rgo even ifoo = 09o. The yield criteria mentioned in the previous section cannot describe such a situation. Assuming that oo = 09o = o= is the yield stress in uniaxial straining and ro = rgo = r is the normal anisotropy coefficient, the yield criterion (in a transversaly isotropic version) proposed by Hill may be rewritten as ol 2 -
where
p
2 2 = o~, o----;)oio2+ 022 + [(p+q) - (POl+q02)]OlO o
(2 - ~
and
q
(5)
may be obtained from the relationship 2
P
= q =
2 (1 +-"-~
-~)/(2 - ~ Ob 0b
-
(6)
Relation (5) is valid only in the case o~ > 0, 0 2 > 0. Its extension to other stress states imposes the cubic terms to be rewritten as
(plo~ I + qlo~l)o~o~
(7)
The following notations are useful when introducing the Hill's yield criterion in the M-K model" OI
t-
Ou
; o
a-
.
(8)
o
With this notations, Eqn. (5) becomes
012 + [p(1-t)+q-(2-a2)]olo2
02
+ (1-qt)o2 2 = o2,
1"I 0.6
0.4
0.2
o'o" o:.:/;' oo.4. 4 016" 11 0.2 O.6 ola' o.a
Eqn. (9) will be used for obtaining the relations between stress and strain increments in the flow theory (the so-called Levy - Mises equations) and the expression of the equivalent stress increment. Figure 1 shows a graphic representation of the yield locus (in a transversaly isotropic version) corresponding to the first quadrant (normal anisotropy coefficient is assumed r=l). One may notice that the yield curve moves towards the origin when the value of a is raised. For a = 1, the yield curve becomes the locus described by the von ~1.2 Mises yield criterion.
0.90
0.8
oI
(9)
Figure 1. The influence of the a parameter on the yield locus
260 3. THE MARCINIAK-KUCZYNSKI MODEL The M-K model assumes that the strain localization appears in the region of a geometrical or structural non-homogeneity. The model presented in this paper assumes the existence of a geometric non-homogeneity in the form of a notch (zone b) perpendicular to the direction of the maximum principal stress at. The initial thickness of the sheet-metal t'0 is greater than the initial thickness in the region of the notch tbo (see Figure 2). The sheet-metal is stretched by the principal stresses o I and o~. The current value of the nonhomogeneity coefficient is expressed by the relationship f-
Figure 2. Geometrical model of the Marciniak-Kuczynski approach
tt, to,
(10)
where t. and t b are the current values of the thickness in the regions a and b, respectively~
The basic equations of the model The following equations are valid for each of the two regions of the sheet: a) The yield criterion:
012 + [p(l_t)+q_(2_a2)]ol02 + (l_qt)022
2
(11)
b) The Levy- Mises equations: de I 2 0 l+[p(1
-t)+q-(2-a2)]o2
de 2
de e
2(1-qt)o 2+[p(1-t)+q-(2-a2)]Ol
2o~
(12)
The increment of the equivalent strain may be expressed from the equality of the incremental work done by the principal stresses and the incremental work done by the equivalent stress. c) The strain - hardening law:
o e = k (e0+e,)" ~m, where: K is a material parameter; e0 is a pre-strain; [~is the equivalent strain-rate.
(13)
261
d) Volume constancy condition: dl~ 1 + d ~ 2 + d ~ 3 = 0.
(14)
The model is completed with two equations expressing the link between regions a and b: - equation expressing the equilibrium of the forces acting along the interface of the two regions: o1~ to = Olb tb;
(15)
- equation expressing the fact that the strains parallel to the notch are equal in both regions: de2a = de2b-
(16)
In addition, the model assumes that the strain ratio in zone a is constant during the whole process:
de2a = O de1,,.
(17)
The model has been solved by using the numerical procedure presented in [30-32]. The determination of the strain and stress state in zone b has been made by using a double predictorcorrector algorithm (a detailed presentation of this alghoritm is in the paper [33 ]). 4. RESULTS AND DISCUSSIONS The mathematical model presented in this paper has a broad generality, which allows the analysis of the influence of several material parameters upon FLD's. Figure 3 shows the influence of the strain-hardening coefficient upon the FLD's. One may notice an increase of the limit strains when the this coefficient is raised. This fact is in good agreement with the experimental and theoretical results presented in [ 14, 34] etc. Figure 4 shows the influence of the normal anisotropy coefficient upon FLD's. According to this figure, the influence of the parameter r is very strong for the case ofbiaxial straining. One may see on Figure 4 that an increase ofr causes a decrease of the limit strains. This fact contradicts the experimental and theoretical results presented in [ 14-16] (Hosford's yield criterion is used in these papers). The curves shown in Figure 4 agree with the theoretical results presented in [34-36] etc. (Hill's 1948 yield criterion [37] is used in these papers). Figure 5 shows the influence of the strain-rate sensitivity index m upon FLD's. One may notice an increase of the limit strains when m is raised. A suggestive explanation of this phenomenon is presented by Marciniak and Duncan in an excellent monograph published recently [38]. The raise of the non-homogeneity factor f (which means the reduction of the thickness variations)causes an increase of the limit strains, as may be seen on Figure 6. This conclusion is in agreement with the conclusions of other papers dealing with the problem of FLD calculation [3, 14, 34, 35] etc. The parameter a defined by Eqn. (8) influences the FLD's shape and position in the manner presented in Figure 7. One may notice that the FLD moves downwards when a is raised. The phenomenon is more significant for the case of equi-biaxial tension. Figure 8 shows the influence
262
0.5 eps 1
e~sl
0.4 ............................................................................................................................ ~ i
!
0.3 ...........................................i..................................... ,',~
n-
0.6 .............................................................~............................................-.................~..........................................................................................
........................
i
i
......................................................................................................
0.4
............................................................. i ........................................................; ............................................................... i...............................
0.2 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.1
~ 1 , 5
0
0
02
0.l
01
0.3 eps
0.4
0.5
.
.
.
0
.
02
.
.
.
.
.
.
0.4
0.6
et~
2
Figure 3. Influence of the strain-hardening coefficient, n, on the FLD (r=l, m=0.01,f=-0.95, a=l)
Figure 4. Influence of the anisotropy coefficient, r, on the FLD (n=0.2, m=0.01,f=-0.95,a=l) 0.6. epsl
0"4i~1 0 . 3 t ............
: ................................... !
o.4t. ..................................~....................................!................................
0.2
"
.
"
0 . 3 ......................................-.................................................................................................. :..................................~...................................
...............................i . . . .
i....................................i ...................................i....................................i ..................................
o.1 ............................................
i....................................i ...................................i....................................i....................................
o.z 0.1
O
0.2
0.I
0.3
0
0.4
eps2 Figure 5. Influence of the strain-rate sensitivity index, m, on the FLD (n=0.2, r=l, f=0.95, a=l)
0.30e~l
0
0.I
02
0.3
0.4
0.5
0.6
eps2 Figure 6. Influence of the non-homoge-neity coefficient, f, on the FLD (n=0.2, r=l, m=0.01, a=l) 0.30
epsl
0.25
i
i
ia--
i
i
io. iio
0.20
0.10 0.05 0
.................... .........................................................................................................
0.05
0.10
~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.15
0.20
0.25
0.10
0.05
0.30
epsa
Figure 7. Influence of the parameter a on the FLD (n=0.2, ~ 1 , m=O.O1, f=0.95)
i]
84 .....................................................................................................................
...........................! .......................................................................................................................................................................................
0
0.80
'
'
'
'
0.85
0.90
0.95
1.00
1.05
1.10
1.15
a
Figure 8. Dependence of computed limit strains on the parameter a (n=0.2, r=l, m=O.O1, f=0.95)
263 of the parameter a upon the limit strain epsl for the case of equi-biaxial tension. One may notice that the variation is approximately linear. 5. CONCLUSION The paper presents the use of a new yield criterion (in a transversaly isotropic version) introduced by Hill in 1993 in the M-K model. The influence of different material parameters upon FLD's is studied. The influence of the yield curve shape upon FLD's is analyzed by using the parameter a (expressed as the ratio of the uniaxial yield stress and biaxial yield stress). The increase of a causes an approximately linear increase of the limit strain for the case of equi-biaxial tension. ACKNOWLEDGEMENTS This research was supported by the Humboldt Foundation at the Institut of Metal Forming, University of Stuttgart. The author are grateful to Professor Klaus Siegert (Director of this Institut) and Mr. Eckart Dannenmann for their help.
REFERENCES S. P. Keeler and W.A. Backofen, Trans. of the A.S.M., 56 (1963) 25. G. M. Goodwin, La Metallurgia Italiana, (1968) 767. Z. Marciniak and K. Kuczynski, Int. J. Mech. Sci., 9 (1967) 609. R. M. Wagoner, K. S. Chan and S. P. Keeler, Forming Limit Diagrams: Concepts, Methods and Applications, TMS, Warendale, 1989. A. Parmar and P. B. MeUor, Int. J. Mech. Sci., 20 (1978) 385. 6. Z. Marciniak and H. Ike, Proceedings of the IDDRG Meeting, Helsinki, 1983. 7. Lian, J., Instabilite plastique, surface de plasticite et endommagement au cours de la defomation plastique et superplastique (These de doctorat), Inst. Nat. Politehnique Grenoble. 1987. R. Hill, Math. Proc. Cambr. Phil. Sot., 85 (1979) 179. 9. J. L. Bassani, J. W. Hutchinson and K. W. Neale, In: Metal Forming Plasticity, Edited by H. Lippmann, Springer Verlag, Berlin, 1989, 1. 10. K. W. Neale and E. Chater, Int. J. Mech. Sci., 22 (1980) 563. 11. J. L. Bassani, Int. J. Mech. Sci., 19 (1977) 651. 12. M. Gotoh, Int. J. Solids and Struct., 21 (1985) 1131. 13. M. Gotoh, Int. J. Mech. Sci., 19 (1977) 505. 14. A. Graf and W. F. Hosford, In: Forming Limit Diagrams: Concepts, Methods and Applications, Edited by R.M. Wagoner, K.S. Chan, S. P. Keeler, TMS, Warendale, 1989, 153. 15. A. Graf and W. E. Hosford, Met. Trans., 21A (1990) 87. 16. S. B. Padwal and R. C. Chaturvedi, Int. J. Mech. Sci., 34 (1992) 541. 17. W. F. Hosford, Proc. 7th NAMRC, Dearborn, 1979, 191. 18. J. Lian, F. Barlat and B. Baudelet, Int. J. of Plasticity, 5 (1989) 131. 19 F. Barlat and O. Richmond, Mat. Sci. Eng., 95 (1987) 15. 20. F. Barlat and J. Lian., Int. J. of Plasticity, 5 (1989) 51. 21. G. Ferron and M. Mliha Touati, Int. J. Mech. Sci., 27 (1985) 121. 22. B. Budiansky, In: Mechanics of Material Behaviour, Edited by: G. J. Dvorak and R. T. Shield, .
.
264 Elsevier, Amsterdam, 1984, 15. 23. F. Barlat, Mat. Sci. Eng., 91 (1987) 55. 24. D. Banabic and I. R. D6rr, Sheet Metal Formability, O.I.D.I.C.M. Publishing House, Bucharest, 1992, (in Romanian). 25. D. Banabic and I. R. Dorr, Mathematical Modelling of the Sheet Metals Processes, Transilvania Press Publishing House, ClujoNapoca, 1995, (in Romanian). 26. Z. Weixian, Int. J. Mech. Sci., 32 (1990) 513. 27. Z. Tourki, R. Makkouk, A. Zeghloul andG. Ferron, J. Mater. Process. Technol., 45 (1994) 453. 28. E. Chu, J. Mater. Process. Technol., 50 (1995) 207. 29. R. Hill, Int. J. Mech. Sci.,35 (1993) 19. 30. D. Banabic and S. Valasutean, J. Mater. Process. Technol., 34 (1992) 431. 31. D. Banabic, Research on the Thin Sheet Metal Formability, Ph.D. Thesis, Technical University of Cluj-Napoca, 1993, (in Romanian). 32. D. Banabic and I. R. Dorr, J. Mater. Process. Technol., 45 (1994) 551. 33. D. Banabic, Proc. Int. Conf. NUMISHEET'96, Dearborn, 1996, 240. 34. Q. Q. Nie and D. Lee, J. Mater. Shaping Technol., 9 (1991) 233. 35. Z. Marciniak, K. Kuczynski and T. Pokora, Int.J.Mech.Sci., 15 (1973) 789. 36. S. N. Rasmussen, Limit Strains in Sheet Metal Forming, (Master Thesis), Technical University of Denmark, Lyngby, 1981. 37. R. Hill, Proc. Royal Society of London, 193A (1948) 281. 38. Z. Marciniak and J. Duncan, Mechanics of Sheet Metal Forming, E.Arnold, LondonMelbourne-Auckland, 1992.
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
265
Characterization of the formability for aluminum alloy and steel sheets F. Barlat a, J.C. Brem a, D.J. Lege a and K. Chung b aAlcoa Technical Center, 100 Technical Drive, Alcoa Center, PA 15060-0001, USA bDepartment of Fiber and Polymer Science, College of Engineering, Seoul National University, 56-1 Shinlim-Dong, Kwanak-Ku, Seoul, 151-742, Korea
In this work, the formabilities of an aluminum and steel sheet samples were assessed both experimentally and theoretically. The forming limit diagram (FLD) and the limiting dome height (LDH) tests were simulated using mathematical models incorporating suitable constitutive equations for the steel and aluminum alloys. Experimental and predicted results were compared and discussed. It was suggested that this type of mathematical modeling can be used to design and optimize forming processes, especially for aluminum alloy sheets.
1. INTRODUCTION The automotive industry, which has relied primarily on steel as a material of choice for structure and body applications, is beginning to use alternate materials for lightweighting purposes. This trend is driven by environmental concerns and energy saving needs. Because aluminum alloys exhibit a lower density with a relatively high strength, they can be used as good substitutes for steel in autobody sheet applications. However, steel has some economical and technological advantages over aluminum alloys. In particular, drawing quality steels have better formability. Moreover, sheet forming practitioners have used drawing quality steels for over a century in stamping operations and they have empirically optimized processing conditions for such materials. Comparatively, little experience has been gained in optimizing forming processes for aluminum alloys. However, numerical simulations using finite element methods (FEM) can be used to reduce the number of experimental trials. As an important input, these simulations require information regarding the material properties, i.e. constitutive equations. Because steel and aluminum sheets exhibit different formabilities, the constitutive equations must be specific for each material. The purpose of this work is to use constitutive equations which differentiate between steel and aluminum sheets, to characterize the formabilities of these materials with numerical models, and to validate the results with experiments.
266 2. EXPERIMENTAL
An aluminum AA2008-T4 sheet sample, a drawing quality aluminum killed (DQAK) steel and a drawing quality interstitial free (DQIF) steel were investigated in this study. The chemical compositions of these materials are listed in Table 1. For each material, the stress strain curves were measured in uniaxial tension at 0 ~ 45 ~ and 90 ~ to the rolling direction (RD). The r values, the width to thickness plastic strain ratio in uniaxial tension, were obtained in these three directions. The stress-strain curves were also measured in balanced biaxial tension with the bulge test at two different constant strain rates. This material information is necessary to calculate the different coefficients of the constitutive equations. The initial portion of the shear stress-strain curve (up to about 5% shear strain) was measured with the Iosipescu shear test [ 1] in the 0 ~ and 45 ~ orientations.
Table 1 Chemical composition in weight% of AA2008-T4, DQIF and DQAK steels Si Cu Mg A1 Fe Mn Cr AA2008-T4 0.68 0.96 0.41 Bal 0.17 0.06 0.01 A1 Fe Mn C DQAK steel 0.051 Bal 0.21 0.06 DQIF steel 0.042 Bal 0.17 0.01
(remelt analysis) Zn Ti 0.01 0.02 P S 0.005 0.02 0.008 0.02
Many processes and material parameters affect formability and, consequently, no one single parameter can be used to characterize it. However, specific formability tests are conducted in material processing or stamping plants to assess formability. The forming limit diagram (FLD), the limiting dome height (LDH), the stretch-bend and guided-bend tests are performed rather routinely to evaluate sheet formability. For autobody applications these tests correlate generally well with field performance. The FLD represents the limit strains above which localized necking occurs. Rectangular specimens are stretched using a hemispherical punch and different strain paths are achieved by varying the aspect ratio of the specimen and the lubrication conditions. Grids are applied onto the sheet surface prior to deformation and are measured after deformation, near necked or fractured regions. These measurements provide strains which are reported on a diagram whose axes correspond to the in-plane strains. This information is used by metallurgists to characterize formability, by sheet forming engineers to design processes, and by stamping engineers to assess safety margins during product manufacturing. The FLD gives information about the strains that a material element (about the size of the grid spacing which is on the order of the sheet thickness) can sustain, but it does not give information about how the material distributes strain in a larger area. For this purpose, the LDH is used. Again, a rectangular specimen is stretched using a hemispherical punch and the height at which the specimen fails is recorded. Several specimen widths are tested and the minimum height corresponds to the LDH. This minimum is associated to limit strains in a
267 state of plane strain tension, where one of the in-plane strains is zero. It is well known that plane strain is the critical state in practical stamping operations. Other tests, such as the stretch-bend and guided bend tests are conducted because they are used to assess the material performance in drawing or hemming operations. These tests characterize the ability of the material to resist fracture under bending in various conditions. However, these tests are not discussed further in this paper
1.5
'0
800
I
700
i
I
I
I
I
I
I
0.1 0.2 0.3
I
-
).5
ID
51111 ~9 ~ -
a~.
~
.400
2
f
300
T ~
f
100
x5r
0 0.0
_
D
.
.
'
\~ !
.
.
.
!
!
,o-=
)Jj
-
- ..... 0 ~ - - 45 ~ t e n s i o n
D Q A K steel --o- Bulge D Q I F steel --U-- Bulge
- - - 90 ~ t e n s i o n ~ 0~ shear --o-- 45 ~ s h e a r -
,I
I
I
I
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
iiiiii
-
2008-T~1 (S.N. 7 1 4 8 6 4 ) 1.5 I . J -1.5 -1.0 -0.5 0.0
1.0
Effective strain
i I 0.5
0
sh~r
I 1.0
1.5
x
Figure 1. Stress- strain cuves for AA2008T4, DQIF and DQAK steels.
3. CONSTITUTIVE
0.4 0.5
i [ '
__~._ Bu!g e
J ~ " r/f.."
200
4
._, I I I Gla i xy i i contours ! ! i 0.0 ............. i ............. . . . . . . . . . . . . . . . . . . . .~ "
Figure 2. Yield surface for AA2008-T4 predicted with Yld91 and Yld94.
MODELING
For the modeling aspect, the material behavior was assumed to be adequately described with a yield surface and its associated flow rule, to be rate sensitive, and to harden isotropically. Yield surface, as well as strain and strain rate hardening have been identified as the macroscopic parameters that influence the FLD the most [2-3]. Under these conditions, it is necessary to define a yield function and a stress-strain curve giving the rate of isotropic expansion of the yield surface with the strain and strain rate. The yield surface was described with an anisotropic non-quadratic yield function which contains a parameter that characterizes the crystal structure of the materials (BCC for steel and FCC for aluminum) [4,
5]. , = c~llsz -s3l a + ~zls3 -Sl[ a + ~31s1 -s2l a
= 2~ a
(1)
In Equation (1), the components S 1 , S 2 and S 3 a r e the principal values of a traceless tensor, s, obtained by linear transformation of the Euler stress components. This tensor reduces to the stress deviator in the isotropic case. The exponent a (6 for the steels and 8 for AA2008-T4)
268 characterize the crystal structure of the materials. The terms ~ k are functions of the orientation of the principal directions of s with respect to the anisotropy axes of the materials, rolling (RD), transverse (TD) and normal (ND) directions. ~ was selected to represent the flow stress in balanced biaxial tension (bulge test) because the hardening curve is evaluated over a larger strain range in this deformation mode (see Figure 1). The yield function defined by Equation (1) and subsequently referred to as Yld94, generalizes a previous version (referred to as Yld91, [6]) where the terms 0~kare constant (0t 1 = ~ 2 = 0 t 3 = l ) The coefficients of the yield function were derived from the flow curves obtained in balanced biaxial tension (bulge test) and uniaxial tension in the 0 ~ 45 ~ and 90 ~ orientations (Figure 1). The shear flow curves were just used to assess the validity of the yield function. The yield surface for AA2008-T4 is shown in Figure 2. The balanced biaxial stress-strain curves were represented by the Swift and the Voce equations for steel and aluminum alloy sheets, respectively. In general, these equations lead to the best fit with experimental data for these respective materials. Finally, the strain rate effect was assumed to be represented by a reference strain rate, ~o and a strain rate sensitivity coefficient, m. This coefficient is usually different for steel and aluminum alloys. The mathematical representation for strain and strain rate hardening is given by the following equations and the corresponding material coefficients are listed in Table 2. = [ A - B exp(-CE)](~/to}m : K(Eo + ~n){~/~o }m
(AA2008-T4)
(2)
(DQIF and DQAK steels)
(3)
Table 2 Strain and strain rate hardening parameters A (MPa) B (MPa) C 2008-T4 396.0 230.6 6.024 K (MPa) eo n DQIF steel 686.5 0.0037 0.274 DQAK steel 694.1 0.0032 0.261 R
..1%
Eo
m
0.005 eo 0.005 0.005
0.00 m 0.(12 0.02
4. FORMABILITY MODELING Using the constitutive equations, the forming limit diagram (FLD) and the limiting dome height (LDH) were predicted for the aluminum alloy and the two steel sheet samples. The FLD was calculated using a model in which a local defect is growing as plastic deformation increases until all the deformation localizes in this imperfection [7]. The FLD was computed for each of the three sheet materials using the Yld94 yield function and the respective hardening function given above. This model had been successfully used in the past to predict the FLD for many aluminum alloy sheets based on crystallographic texture measurements [8].
269 The LDH tests were simulated using the finite element method (FEM) code ABAQUS/EXPLICIT 5.3. In this model, the geometry of the blank and tools were given as input. The friction was accounted for by a single coefficient, ~, typical for the kind of material/tool interfaces employed in the experiments (It = 0.1). For numerical efficiency, the draw bead was not modeled. Rather, a large friction coefficient was used to characterize the interface between the blank holder and the blank. As with the physical experiments, dome heights were computed for several specimen widths. For a given specimen width, the dome height was defined as the height at which the largest strain in the sample reached the predicted forming limit. The width leading to plane strain deformation in the critical area of the specimen was reported and the corresponding dome height was used to define the LDH.
0.40 IC
t\
r~
]
~
/ ~~176
i
~ :
~
o.8o,,
t
oo-'oF ~7~-~,W,/ ~ 1 7 6 ~lJ / :: o.oo7~i ~:~:~ -0.10
0.00
0.10 0.20 TD strain
,
,
,
,
.,
~ r oo,~.,~176
\'~
~I~ ~176_ _ J
-0.20
0.00
to I :: o.oo ~176 ~lJ / [~_~:,o~ , }~~..~,o,.R
,
-0.20
,
~\o~
, o,o F
0.30
Figure 3. Experimental and predicted FLDs for AA2008-T4.
0.4
-0.40
0.20 0.40 TD strain
0.60
0.8
Figure 4. Experimental and predicted FLDs for DQIF steel.
The dependence of strain and strain rate hardening on the strain distribution in formed parts is well-known. The influence of the yield surface shape on the strain distribution has been shown using FEM codes [9]. The LDH test has been modeled with FEM codes using Yld94 [ 10] or a more sophisticated material model using a polycrystal approach [ 11]. All of these models indicated that the crystal structure and the crystallographic texture that affect the shape of the yield surface have a strong influence on the strain distribution during forming processes. In this work, because the yield function Yld94 was not implemented into ABAQUS when these computations were performed, Yld91 was used for the LDH calculations. This yield function is not as accurate as Yld94 over the entire stress range. However, the stresses and strains involved in the LDH test are close to the plane strain mode and the coefficients of Yld91 were determined so that Yld91 and Yld94 lead to almost identical yield surface shapes near plane strain tension (indistinguishable in Figure 2).
270 As a summary of this section, the FLD and LDH tests can be assessed theoretically if a minimal amount of information is known about materials. A coefficient of friction between the punch and blank, and three uniaxial tension and two balanced biaxial stress-strain curves are needed to adequately describe the friction and plasticity aspects of the forming process. Everything else is accounted for by mathematical modeling and numerical simulations.
5. DISCUSSION
The predicted FLDs for AA2008-T4 alloy, DQAK and DQIF steels are generally in good agreement with experimental FLDs reported in the literature [8, 12]. Figures 3 and 4 show the predicted FLD curves and experimental data obtained for AA2008-T4 and DQIF steel, respectively. The experimental FLDs cannot be drawn accurately as a curve because of the scatter in the data. The experimental FLD which lies somewhere below the experimental data points labeled "neck affected," agrees reasonably well with the predictions. Figure 5 shows the predicted FLDs for AA2008-T4, DQAK and DQIF steels. This plot indicates that the FLD is lower for the aluminum alloy and that the DQIF steel performs slightly better than the DQAK steel, which is in agreement with current knowledge.
0.80
i
I
i
i
/
"7:
0.60
0.40
~176 f
=
e-
m
0.20
FLo
:t
0.35
o
0.30
9 108.0 m m , 6.0 o 108.0 m m , 6.5
.,
120.7 m m , 6.0 s
0.25 0.20
0.00
0.10 211t18-1"4
-0.20 -
-0.40 -0.40
_
DQAK steel DOIF steel I -0.20
I 0.00
0,05 N 714864 ).
I
i
I
0.20
0.40
0.60
TD strain
Figure 5. Predicted FLDs for AA2008-T4, DQIF and DQAK steels.
0.80
0.00 -0.15 -0.10 -0.05 0.00 0.05
0.10
0.15
0.20
0.25
TD strain
Figure 6. Predicted strain distributions during LDH test for AA2008-T4.
Figure 6 shows how the minimum (plane strain) punch height in the LDH test was theoretically determined. The test was simulated with a constant punch displacement rate (250 mm/min) for specimens that were 177.8 mm long, while different widths were used to determine the plane strain conditions, as in experimental trials. In this figure, the predicted FLD is plotted with the strain profiles taken along the specimen length for three combinations of specimen width and displacement duration. For the sample with a width of 120.7 mm, the
271 maximum strain does not occur in plane strain condition, where one principal surface strain is zero (y-intercept). Figure 6 shows that for this width, the material is subjected to biaxial expansion. However, the maximum strain is clearly in the plane strain condition for a width of 108.0 mm. The LDH is obtained for this width when the maximum strain reaches the forming limit, i.e. after a punch displacement of 6.5 sec which produces a height of 27.8 mm. 0.50
I
I
1.60
I
DQIF (S.N. 714875)
.
0.40
i
L
I 1.50 I'-
"
I
"
I
"
I
"
I
"
I
o 2oo8-T4 [] DQIF steel
"
I
"
I
/
"/3 /
/ j
J
]
1.40 1.30
0.30
1.20
=
0.20
1.10 I
0.10
0.00 -0.10
iI ~ / ii H ~ i ~ ~/ ,
~
DQIF steel / o 120.7mm, 8.Os-1 " 120.7mrn, 9.Os| o 120.Tmm, 9.5s] I
0.00
0.10 0.20 TD strain
I
0.30
0.40
Figure 7. Predicted strain distributions during LDH test for DQIF steel.
1.00 0.90 0.80 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 Predicted LDH
Figure 8. Experimental vs. predicted LDH for AA2008-T4, DQIF and DQAK steels.
Figure 7 shows the same kind of information for the DQIF steel. For this material, the predicted height was found to be 39.6 mm with a specimen width of 120.7 mm. The predicted LDH for DQAK steel was determined with the same method. Figure 8 shows the predicted versus experimental LDH values. This figure indicates that the absolute LDH is not perfectly predicted with the model. This is not surprising because many assumptions have been made concerning the material properties, friction and boundary conditions. However, the trends given by the modeling work are excellent. The predicted LDH is higher for steel than for aluminum and slightly higher for DQIF than for DQAK steel, in total agreement with the experiments. The specimen width corresponding to the LDH (plane strain) is smaller for aluminum than for steel, again in full agreement with experiments (Table 3). These results concerning the FLD and the LDH clearly validate the theoretical approach. The modeling work is able to accurately describe the formability of the steels and the aluminum alloy with minimal mechanical information needed as an input (2 bulge tests, 3 uniaxial tension tests).
Table 3 Experimental and predicted widths (mm) for LDH specimens 2008-T4 DQIF steel DQAK steel Experimental 114.3 133.4 133.4 Predicted 108.0 120.7 120.7
272 The experimental and predicted FLD and LDH results were found to be in good agreement for the three materials studied. In particular, it was found that the model was able to rank the relative value of the LDH very well. This result is particularly interesting because it shows that this kind of relatively simple constitutive modeling can be used in FEM simulations of forming processes to differentiate material forming performance according to plastic behavior. Thus, it is likely that this type of modeling can be successfully used in FEM codes to design and optimize forming processes for specific materials, particularly for aluminum alloy sheets.
ACKNOWLEDGMENTS
The authors would like to thank Shawn Murtha for his many helpful discussions, Greg Fata, Harry Zonker and Jerry Morson for their help in obtaining materials and conducting/analyzing experiments, and Rich Becker for implementing the constitutive equations into ABAQUS.
REFERENCES
1. 2. 3.
N. Iosipescu, J. Materials 2 (1967) 537. F. Barlat, Mat. Sci. Eng. 91 (1987) 55. F. Barlat, Forming Limit Diagrams - Concepts, Methods, and Applications', ed. R. H. Wagoner et al., The Metallurgical Society, 1989, p. 275. 4. F. Barlat, Y. Maeda, M. Yanagawa, K. Chung, J.C. Brem, Y. Hayashida, D.J. Lege, K. Matsui, S.J. Murtha, S. Hattori, Proc. Fourth International Conference on Computational Plasticity, (Complas IV), Barcelona, Spain, April 1995, ed. D.R.J. Owen, E. Ofiate and E. Hinton, Pineridge Press, Swansea (UK), p. 879. 5. F. Barlat, R.C. Becker, Y. Hayashida, Y. Maeda, M. Yanagawa, K. Chung, J.C. Brem, D.J. Lege, K. Matsui, S.J. Murtha, S. and Hattori, Submitted to Int. J. Plasticity (1997). 6. F. Barlat, D.J. Lege and J.C. Brem, Int. J. Plasticity, 7 (1991) 693. 7. Z. Marciniak and K. Kuczynski, Int. J. Mech. Sci. 9 (1967) 609. 8. H.R. Zonker, J.C. Brem and J.L. Morson SAE Technical Paper 950701 (1995). 9. K.N. Shah and R.E. Dick, SAE Technical Paper 950925 (1995). 10. Y. Hayashida, Y. Maeda, K. Matsui, N. Hashimoto, S. Hattori, M. Yanagawa, K. Chung, F. Barlat, J.C. Brem, D.J. Lege, S.J. Murtha, Simulation of Materials Processing: Theory, Methods and Applications, ed. Shen and Dawson, Balkema, Rotterdam, 1995, p. 722. 11. J. Bryant, A.J. Beaudoin and Van Dyke, SAE Technical Paper 940161 (1994). 12. A.K. Ghosh, S.S. Hecker and S.P. Keeler, Workability Testing Techniques, Ed. G.E. Dieter, ASM, 1984.
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
273
Material Plastic Properties Defects And The Formability O f Sheet Metal J. D. Bressan Departamento de Engenharia Mec~.nica- Faculdade de Engenharia de Joinville Campus Universithrio - 89.223-100 - Joinville - SC - Brasil. Abstract
A main thecnological concern of sheet metal forming is the analysis of the influence of material imperfections on its formability. The imperfections can be defined as local variations in thickness and in plastic properties which can affect the plastic flow and, therefore, influence the neck formation phenomena. From the micro structural point of view this could be due to the variations in grain size and orientation, inclusions and second phase contents, porosity, as well as residual cold work, originating from the melting, solidification and rolling processes. As consequence of these imperfections within the material, the thickness, strain hardening ,strain rate sensitivity and strength coefficients and initial prestrain may exhibit local variations and thereby initiation of the neck and limit strains be influenced. Using the concept of strain gradient at the neck, the limiting strains are investigated with respect to the material parameters mentioned above. It is discussed the role of these defects on the development of the local neck and the forming limit diagram FLD. The predicted results are compared with Marciniak Theory and wtih the experimental results obtained by Azfin.
Keywords :forming limit diagram FLD, defects, sheet metal. I. INTRODUCTION The forming of sheet metal is a widespread technological process which requires a deep understanding of the mechanics of materials deformation and failure in order to optimise this process and to develop new high formability materials. The Forming Limit Diagram FLD is a well established aid in the development of sheet metal. It displays the combination of major and minor strains, e 1 and g2 respectively, which represent the limit of performance for a sheet material in principal strain space; or it thus display, in principal strain space, the combinations of useful strains that can be achieved before the onset of local necking. FLDs obtained from laboratory sheet stretching and punch forming agree reasonably well with those from pressshop experience. Although the FLD was originally empirical [ 1] a number of theoretical treatments have been developed aimed at predicting the forming limit curve in terms of readily measured material parameters and the conditions of the forming operation itself. For a material with a strain hardening eponent n, within the biaxial stretching region the limiting strain is approximately n when theory predicts the formation of a dif~se neck [2]. Within the drawing region of the FLD the Hill's theory [3] predicts an increase in the limiting strain el from n to 2n in uniaxial tension. In general, for constant strain path, the agreement between Hill's theory
274 and experimental points is reasonably good whereas in the positive quadrant the diffuse necking prediction is a lower bound limit for the limiting strains which may increase to 1.Sn in the balanced biaxial stretching case. Therefore, most theoretical effort has been concentrated on the biaxial stretching region of the FLD. Within the biaxial stretching region experimental observations [1] on sheet metal deformation clearly show that failure is generally preceded by local necking. The first theorectical approach which has overcome the problem of no direction of zero extension in the plane of the sheet under biaxial stretching was that presented by Marciniak and Kuczynski [4]. They argued that under the actual conditions of press forming local necking or fracture can occur either before or after the classical instability conditions or the applied load has attained its maximum. The authors postulate that local necking originates at initial inhomogeneities of thickness and/or material properties within the sheet which lead to conditions at the failure site moving towards those of plane strain and thus local necking could occur. These initial inhomogeneities or imperfections are characterised by a conceptual parameter f which might include thickness variations, non-uniform distribution of impurities, varying texture, different size and orientation of grains, porosity and others. Although this concept provides a powerful and fruitful model there are discrepancies not yet fully explained. A different approach to the onset of local necking in sheet metal forming by the development of strain gradients was presented by Bressan and Williams [5]. The approach describes neck initiation and growth as a continous process of flow or strain localisation owing to initial inhomogeneities in thickness or porosity. These initial inhomogeneities are characterised by the parameter ~t which is the initial gradient in area. The limit strain or local necking occurs when the strain gradient ~ attains a critical value or when the ratio M~t is approximately 100. Later the analysis is extended to variations in the plastic properties in a similar approach. 2. THEORETICAL ANALYSIS In this work an attempt is made to analyse the influence of the defects in thickness and material on the limit strains and, thus, affecting the FLD. The approach analysis consider strain hardening and strain rate hardening materials having a constitutive equation of the form,
~=
k ( 6 o + ~ ) " ~ "M
(1)
where k is the strength coefficient, s o is the prestrain, n is strain hardening coefficient, M is the strain rate sensitivity coefficient, ~ is the equivalent strain, ~ is the strain rate and ~- is the equivalent or flow stress. Assuming that the phenomena of local necking is a flow localisation process it is straightforward to conclude from eqn.(1) that necking is likely to be highly influenced by both the n-value and M-value as the flow stress ~" has strong dependence on these plastic flow parameters [5]. However, other parameters of eqn.(1) can also influence the limit strain. The imperfections that may influence the necking formation phenomena can be defined as local variations in thickness and in the flow parameters above. From the micro structural point of view these could be due to variations in grain size and orientation, porosity, inclusions
275 and second phase content as well as residual cold work, originating from the production processes. As a consequence of these imperfections within the material the plastic flow parameters n, M, k, R and e o may exhibit local variations and thereby neck initiation be influenced. A straightforward way to investigate the influence of these inhomogeneities on the neck growth behaviour is to consider the plastic flow parameters as a function of the coordinate x perpendicular to the neck, i.e., n(x), M(x), k(x), and eo(X). From eqn.(1) the equivalent stress gradient is given by,
.
.
-~ dx
.
.
k dx
~
6o+-~ dx
,,xl
+
+ -.-~
~" dx
/
~" /
/
--v-g '
/
+ en(Co + a ) ' - ~
n dx
~/
~
,
+ en ~
(2)
M dx
A
U / . ~ ~l (/
~SF1
/ / / / / / s2
F2
I s,~_A ~s,+.s, "A+dA b----'-~-- ~~176- - ~ 2Ax
I I
Fig. 1 - Sheet element with a local neck From the analysis presented by the author [5], the governing equation for the necking formation in sheet metal forming is,
d-~- M
~
(l+a)Z
.}
(60+?) 2
(3)
where 2 = c~/6x is the equivalent strain gradient at the neck, a = 661 is the strain path and
&2
Z is the critical value of the subtangent for necking. Equation (3) was derived from the analysis of an element of the sheet exhibiting a current neck. See Fig.1. This analysis lead to the following equation,
276 ld-~ dx
a =
c7~
(4)
+12
(1+ a)Z dx
Equating equation (2) to equation (4) we arrive at,
-
mE o
n]
d2 =__p + ~ ~ _ LY~ M M (1+ a)Z
2
(5)
6o+~
where the equivalent defect parameter ~ is,
12 =12 + 12k + (6 '0%+ -~)12 6 o +
1 OA o
,u
Ao Ox
12 , I l k , 12n,
12n +
1 Ok
Pk
12e~
_n
e,,( ~o + e)
,12M
k Ox
gn(~)M
12M
(6)
1 8~ o
tUeo
eo d x
1 On
'Un
ndx
1
t'tM
dM
MOx
are the material imperfections or defects. Equation (5) is similar to
equation (3). Therefore, it is quite clear that the K-type of imperfection, 12k , give a similar effect as thickness imperfections bt and both play the most decisive role on the development of the strain gradients and, thus, on the FLD. For a drawing quality steel, we can assume n = 0.25, M - 0.012, 60= 0.01, ~ - 0.001, and at the onset of the diffuse necking strain ~ is approximately 0.5. Thus, equation (6) can be reduced to,
the
(7)
= 12 + 12k +0"005126o-0"16812n- 0"08212M
The effect of the initial defects can be better explained by considering the point of instability or at the onset of diffuse necking, that is at ~= zdn -- Co . Thus, from equation (5), at this point, the strain gradient development is,
c~
=
(8)
M
If the initial imperfections 12k =126~ = ,Un = 12M =0
then, ~ = p
, and, for thickness
imperfection only, equation (8) yields, 32
=
12 M
(9)
277 But, if this imperfections are not zero and have the same size, then the strain gradient development intensity at the begining of diffuse necking is, for #k =#60
=,Un = t i M = t t ,
d2 _ 1.755~ M
(10)
Therefore, the velocity of localization of the strain gradient is almost double the value for thickness imperfection only, That is, the velocity of the necking process considering all types of imperfections is 1.755 times greater than that for a thickness defect only. However, if we have k-type and thickness type of imperfections only, the velocity of the necking process at instability point is, for #k = P equation (8) becomes, dA
_
2.0/t
(11)
M
Thus, the k-type and thickness-type of defects are the most severe in sheet metal forming. 3. RESULTS
The influence of the defect type on the limit strain for a = 3.6 is presented in Fig.2 bellow. 0.00 0.60 I
0.04 '
Defect size f 0.08
I
'
I
n =0.24" .X..~ 0~
0.12
'
!
0.16
I
'
M =0.017"
I
'
R =1.5
~o = 0.01
.= 0.40
~ 0.20 -
gM
~
gn
_
kt or ktk
1 0.00
i
0.00
/
5.00
i'
I
i
I
10.00 15.00 Defect size
I
j
I
I
20.00
25.00
Fig.2 - Influence of Defect type on the limit strain.
278 The present theory predicts that n-type and M-type of defects are quite similar. The worst defect are the thickness-type and k-type as mentioned above. In order to validate the present theory, comparisons were made with experimental results and with the Marciniak and Kuczinski theory (M-K model). The experimental results obtained by Azrin [6] using sheets of aluminium-killed steel with pre-machined grooves are compared with both theories. The co-relation between the defect parameters f and IX is,
where h is the thickness and 2 A x is the width of the machined groove and f is the M-K defect parameter. For the strain path 62/61 = 2.8 , Azrin used constant groove width of 0.015in. Therefore IX = 133.3 f/in. In Fig.3 bellow comparisons of the present theory are made with the experimental results of aluminium-killed steel from Azrin [6]. It is assumed a critical value for ~, at the limit strains, that is, 2=, = 20 in the present analysis. The predicted curve for the limit strains decrease with the defect size and is greatly affected by the strain rate sensitivity parameter M. For M = 0.017 the predicted curve agree closely with the experimental points. It is clear that M is beneficial to increase the limit strain in the presence of defects up to approximately f-- 5%.. For defect size greater than this the benefit vanishes.
0.00
0~0
0.02
I
Defect Size f 0 . 0 4 0.06 0 . 0 8 0 . 1 0 0 . 1 2 0.14
I
I
I
0.40
I
I
l
PresentTheory n=0.24"R= 1.5"~=0
r.x)
01
~
o.oo I 0.00
i I0.00
,' 20.00
Defect size ~ or Ix Fig.3 - Influence of the thickness defect size ix and strain rate sensitivity M on the limit strain 6~* compared with the experimental results [6] for aluminium-killed steel.
279 Defect Size f 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
I
0.50
0.40
I
I
I
\
9
-
I
Experimental
_ _
Present Theory
....
- -
\
0.20
I
n = 0 . 2 4 , R = 1.5 ; ~ = 0
--
0.30
I
M-K model
\
-
0.10 -
0.00
i
I
5.00
0.00
i 15.00
10.00
20.00
Defect size Fig. 4 - Comparisons between theories and the experimental results for aluminium-killed steel [7].
In Fig.4 the present theory is compared with the M-k theory for 62/6~ 2.8 .The correlation is good and shows the same trend with the defect size. The predicted curve for the M-K model was obtained from the paper by Ghosh [7]. In the paper by Azrin the parameter M was not measured but it is reasonable to suppose M = 0.017 for aluminium-killed steel. The comparisons between both theories and the experimental results for aluminium-killed steel are made assuming M = 0.017 and the critical =
strain gradient value I=, = 20 for the Present Theory. For the M-K model it is adopted M = 0 [7]. The correlations are good except that for the M-K predicted curve M = 0. The discrepancies between the theories should increase with the parameter M.
4. Conclusions
The present theory defines an equivalent defect size ~ that takes into account the imperfections in thickness, strength coefficient k , workhardening coefficient n , strain rate sensitivity parameter M and pre-strain. The limit strains decrease with the equivalent deflect size. The worst defects are imperfections in thickness and in the strength coefficient k and the least important are small variations in M and pre-strain. Thickness type and k-type have equivalent contribution to decrease the limit strains in sheet metal forming. For an equivalent defect size of 7.5/in or 5%/in the limit strains can decreased by approximately 100%. On the
280 other hand, positive variations in n are beneficial to balance the imperfections in thickness and coefficient k. The present approach takes into account the width of the local defect besides its thickness size f. It is clear that both have large influence on the limit strains of sheet metals. For aluminium-killed steel a realistic critical strain gradient is 2or, = 20. The random distribution of defects in the directions of the sheet plane should produce scatter in limit strains as seen from the experimental FLD curves.
5. Acknowledgements The author would like to thanks the University of CNPq for the financial support.
Santa Catarina State - UDESC and
6. References [1] S. P. Keeler, Sheet Metal Industries, Sept., 633 (1968). [2] H. W. Swirl, J. Mech. Phys. Solids, 1 (1952) 1. [3] R. Hill, J. Mech. Phys. Solids, 1 (1952) 19. [4] Z. Marciniak and K. Kuczynski, Int. J. Mech. Sci., 9 (1967) 609. [5] J. D. Bressan and J. A. Williams, J. Mech. Working Tech., 11 (1985) 291. [6] M. Azrin and W. A. Backofen, Met. Trans., 1 (1970) 2857. [7] A. K. Ghosh, The Mechanics of Sheet Metal Forming, Ed. Koistinen and Wang, Plenum Press, New York, (1978).
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
281
Formability analysis based on the anisotropically extended Gurson model E. Doege, A. Bagaviev and H. Dohrmann Institute for Metal Forming and Metal Forming Machine Tools, University of Hannover, Welfengarten 1A, 30167 Hannover, Germany A formability analysis of anisotropic sheet metal is carried out based on the anisotropically extended rate independent constutive equations suggested by Gurson and subsequently modified by Tvergaard and Needleman. The advanced model takes into account the influence of the porosity as well as and anisotropic properties of the matrix material. The integration scheme was implemented in the ABAQUS general purpose finite element program. The validity of the anisotropic extended model is proved by simulating special cases of the implemented laws as e.g. yon Mises, Hill and original Gurson plasticity. These constitutive models can be obtained from the extended model as particular cases. 1. I N T R O D U C T I O N The numerical simulation of complex sheet metal forming processes by means of the finite element method (FEM) is of great relevance for the industry. Redundant trial and error steps can be avoided during the die design period and thus manufacturing costs can be lowered. For an accurate computational prediction of the sheet metal behaviour, it is indispensable to choose an adequate constitutive model. Indeed it might be necessary to take into account the complex nature of the problem including the onset of instability phenomena, because its formation is an important precursor to failure. Workability of a sheet metal can be defined as the ability of the material to be deformed without failure. There is no consensus among researchers on the best criteria to be used. The approach advocated in the present paper utilises a porosity (damage) as such a parameter. The various ways of the occurrence of the instability phenomena are the reason to take into account the combination of anisotropy and damage as the factors affecting a sheet metal forming process.
2. C O S T I T U T I V E B E H A V I O U R OF P O R O U S D U C T I L E M A T E R I A L The main instability phenomena in a deep drawing process are wrinkling caused by the influence of circumferential compressive stresses and necking as a localised plastic deformation due to internal material inhomogenities. The computational prediction of instability phenomena in plastic solids under deformation depends substantially on the
282 assumed material response. In the following the Gurson's plasticity model and its possible anisotropic extension are briefly outlined. 2.1. G u r s o n ' s plasticity m o d e l The Gurson-type yield function for a plastic material containing randomly distributed voids gets popular for considering the ductile damage development during sheet metal forming. Based on an upper-bound solution for deformations around a single spherical void under the assumption of rigid-plastic isotropic material, Gurson proposed yield function [1], which has the following form after modification [5]:
42 --
q
-~1
+ 2qlfcosh
(
2k] ] - (1 + q3f2) ,
(1)
where k / i s the equivalent tensile flow stress of the matrix material and f is the current void volume fraction, which is small for real sheet metals. The parameters ql, q2, and q3 were introduced by Tvergaard [5] to agree the predictions of the Gurson model with his numerical studies of materials containing periodically distributed circular cylindrical and spherical voids, q is the root of the second stress deviator, p is the hydrostatic pressure. This model takes into account the progressive damage due to void nucleation and growth of an initially dense material [7,8]. The limit case with zero porosity for the isotropic Gurson model is the Levy-Mises plasticity. 2.2. F o r m a l anisotropic extension In this work it is adopted [4] that the deviatoric part of the model can be extended to an anisotropic plasticity model with the fourth rank anisotropy tensor Ai/kz, i.e.
q= i3a.TAer.
(2)
The constants in the Hill's stress function [2,3], representing planar anisotropy, are calculated based on the r-values obtained by tests of the material under different orientations c~ to the rolling direction [6]:
r~
S 9
In so
(3)
"
Therefore in the limit case with zero porosity the modified model should turn into the Hill quadratic yield condition. Additionally, when Aijkl = Iijkl, i.e., an isotropic fourth rank tensor, the conventional isotropic Gurson model is obtained. 3. C O M P U T A T I O N A L
PROCEDURE
The constitutive library provided in the ABAQUS general purpose finite element program enables the description of the behaviour of porous (mildly voided) metals [7,8]. An implicit and unconditionally stable method for the numerical integration of a class of isotropic pressure dependent plasticity models has been developed by Aravas. The procedure
283 has been applied to the material model for void-containing ductile solids with isotropic hardening developed by Gurson and modified by Tvergaard and Needleman. Following Aravas, the same Euler-backward integration method was applied. As it is characteristical for metals, undergoing small elastic and large plastic deformations, the additive decomposition of the strain increment into an elastic and an inelastic part is assumed. de = de ~; + ds pt.
(4)
For the case of linear isotropic elasticity it is assumed (r = C d ~ d
,
(5)
where C ~t is the fourth order elasticity tensor C~]kt -
K - 5G
(6)
5ijSkt + 2GSikSit.
G and K are the elastic shear and bulk moduli respectively, and 5ij is the Kronecker delta. The equation (5) gives cr~+~ = cr d -
C ~ I A e p.
(7)
The use of the Euler-backward integration scheme yields the expression for the plastic strain increment
0+] t+~t"
(8)
As p t - A,~ ~
The combination of the equations (5), (7) and (8) results in the following expression for the stress increment
The numerical algorithm proposed by Aravas is based on an assumption that is only valid for isotropic plasticity. Thus, for the integration of (1), using (2), the continuum tangent modulus C EP is used. The continuum tangent modulus is derived from the continuum rate equations by the enforcement of the consistency condition that upon yielding the stress point must remain on the yield surface if no unloading occurs. The stress increment can be calculated implicitly by using the continuum tangent moduli for any given strain increment. The procedure is straightforward and well documented thus only the expression for C EP is presented:
cel oo (cel O(~~T cEP = C~l_
'~
(10)
oo'J
Ggur wherein
(OI~I T Ga,.,,. --
~
2 ~
- ( 1 - f)~-~-~-~ + (1 - f ) k ) \
-~]
"
(11)
284 To solve the non-linear system of constitutive equations for the new approach (12)
O't+At --- ~ t -~- c E P t A ~ ,
an integration procedure based on the method of successive approximation was used. The implemented routine is verified on an one-element model for various load cases. Very good consistency with the particular cases such as isotropic Gurson's plasticity, anisotropic Hill's plasicity and isotropic Mises plasticity models, was achieved, comparising the results directly with those gained with ABAQUS. 4. S H E E T T H I C K N E S S
I N C A S E OF " P L A N E S T R E S S "
For the simulations of complex sheet metal forming processes shell elements were employed. These elements use the "plane stress" assumption 0.33 = 0.
(13)
The out-of-plane strain component is not defined kinematically. In ABAQUS the component A:33 is treated as one of the unknowns to be calculated. In the present paper another approach was employed. On the one side, for the pore-free metals the following physically based equation is adopted:
/~g33- --(/~Cll "JI-Z~C22).
(14)
On the other side, under the assumption (8) one gets
t A : o = AAa~j, where
(15)
0.~j is the deviator of the stress tensor. Consequently 1
/~Cll + /~C22- 5i)~(0.11 -1t- 0"22)
(16)
and 1
A:33 -- -~AA(0":: -t- 0"22).
(17)
Therefore, it is possible to calculate the out-of plane strain component for the porous metal formally as 0r A:33 - A ~ - -
(18)
00"33
and then to take 0"33= 0 in the above equation. 5. P R A C T I C A L
APPLICATION
The simulations were carried out with ABAQUS/Explicit. The flange of a trapezoidal cup after 40 mm punch-travel can be seen in Figure 1. This is in very good agreement with practical results. Figure 2 shows the distribution of the sheet thickness of the deformed cup. The considered material is a St1503 with an initial sheet thickness of 1 mm. The initial porosity
285 is assumed as f = 0.001. As the sheet thickness is connected with the porosity, Figures 2 and 3 have to be discussed together. The sharper corner of the cup shows the more critical value as the thinning of the sheet reaches approximately 27 % and the porosity is increasing upon 0.117 what allows the assumption that here a fracture will occure with great possibility. Thus, the porosity (damage variable as a material parameter) distribution allows to indicate localisation of the failure onset. The second example is the simulation of the S-rail deep drawing process according to the NUMISHEET'96 Benchmark geometry. In the Figures 4 and 5 are shown the sheet thickness and porosity distributions. One can see that the porosity defines precisely localisation of the damage occurrence and possible failure, while the sheet thickness might lead to ambiguously indicated place of necking. 6. C O N C L U S I O N S The numerical simulation of the deep drawing process of the anisotropic sheet metal was carried out taking into account the influence of anisotropy on the damage during the forming process. From the results it may be concluded that as a precursor of the area endangered by ductile failure, porosity (damage variable), predicted from the anisotropic extended Gurson's model, is a material-dependent refined criteria for the formability estimation of a metal sheet and the process reliability. The results emphasise that the anisotropic extension ~of the Gurson model is a good method for failure assessment in the design of metal forming processes of metal sheets with anisotropic properties. REFERENCES
1. GURSON, A.L.: Plastic Flow and Fracture Behaviour of Ductile Materials Incorporating Void Nucleation, Growth, and Interaction. Ph.D.-thesis, Division of Engineering, Brown University, 1975 2. HILL, R.: A theory of the yielding and plastic flow of anisotropic metals. Proc. of the Royal Society, London, 1948 3. HILL, R.: Theoretical plasticity of textured aggregates. Math. Proc. Camb. Phil. Soc. 85 (1979), S. 179-191 4. SEIBERT,D.: Untersuchung des duktilen Versagens von Feinblechen beim Tiefziehen. Dissertation, Universit/it Hannover, VDI-Verlag, Dfisseldorf, 1994 5. TVERGAARD,V.: MechanicM Modelling of Ductile Fracture. Mechanica 26 (1991), S. 11-16 6. IDDRG: Ermittlung der senkrechten Anisotropie (r-Wert) yon Feinblech im Zugversuch. Stahl-Eisen-Prfifblatt 1126-84, Nov. 1984 7. ABAQUS/Explicit. User manual. Hibbitt, Karlsson'& Sorensen, Inc., 1994 8. ABAQUS. Theory manuM. Hibbitt, Karlsson & Sorensen, Inc., 1994
286
Figure l. Flange of a trapeziodal cup after 40 mm punch-travel.
Figure 2. Sheet thickness distribution after 40 mm punch-travel.
287
Figure 3. Porosity distribution after 40 mm punch-travel.
Figure 4. Deep drawing of the S-rail. Sheet thickness distribution at the end of deformation.
288
Figure 5. l)eep drawing of the S-rail. Porosity distribution at the end of deformation.
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
289
R u p t u r e c r i t e r i a d u r i n g d e e p d r a w i n g of a l u m i n u m a l l o y s J. Proubet a and B. Baudelet b apechiney Centre de Recherches de Voreppe, BP 27, 38340 Voreppe, France bGPM2 Department, INPG, BP 46, 38402 St-Martin d'H~res, France
1. INTRODUCTION The drawing operation is a key process for the packaging industry as well as for the automotive i n d u s t r y : every day, tens of millions of food and beverage cans are m a n u f a c t u r e d by Pechiney which also promotes the use of aluminium for drawing car body panels. Hence, this industry often faces the problem of designing new series of tools for making cans or car body panels either newly designed, or with decreased thickness, or out of a different alloy. Another recurrent problem is choosing the best alloy for a given drawing operation. For all these reasons, research is going on at Pechiney to try and find out failure criteria during deep drawing of aluminium alloys. Aims of this research are diverse : have a better understanding of why failures occur, reduce the lengthy and costly procedures necessary to characterize the formability of aluminium alloys, and reduce the trial and error experiments necessary to design new series of tools each time a different type of can or car body panel has to be manufactured. Two approaches came out of this research : numerical simulations and a simplified failure criterion for deep-drawing. These two approaches are presented here. Theoretical results obtained are compared with experimental results. 2. NUMERICAL SIMULATIONS OF R U F r U R E IN D E E P DRAWING 2.1. Application of existing theories on rupture to a l u m i n i u m alloys A lot of work has been published in literature on forming limits of metal during drawing. Most of the work available, however, is concerned with local failure criteria : the aim is to understand why a flat sheet of metal will break under given stress and strain histories. Well-known theories on the subject include those of Hill [1] and Marciniak-Kuczinski [2]. Hill theory predicts that failure can only occur for states of strains varying from plane strain to pure shear.
290 This theory was extended to all states of strains by Marciniak et al. who introduced an imperfection in the sheet and simply applied to it equations from the mechanics of continuous media. Questions were raised on why such a defect is necessary and what magnitude should be taken for it. Another difficulty with the Marciniak approach is that results depend also on the strain rate sensitivity of the material. Many authors tried to link the defect with metallurgical damage [3]. In our opinion, the introduction of a defect is indeed necessary for explaining the rupture of a fiat sheet of metal submitted to stress and strain histories but m a y not be necessary any more when dealing with ruptures during deep drawing : it that case, ruptures occur in areas which do show defects but these defects may be created by complex stress and strain histories leading to thickness heterogeneities and not necessarily by metallurgical damage. This means that for alloys with little metallurgical defects, mechanics of continuous media should be able to predict rupture. Indeed, most of the aluminium alloys used for deep drawing have a very small initial damage compared to imperfections in thickness created during deep drawing. Typical damage in an AA 3004 H19 aluminium alloy was measured to about 5.10 -4 [4] while thinnings of 10% are frequently observed in deep drawing. Thus, it should not be necessary to introduce artificial defects in a finite element simulation of deep drawing for predicting rupture of most aluminium alloys : thickness heterogeneities will be naturally generated by the complex stress and strain histories. Since finite elements simulations use the same equations of the mechanics of continuous media as Marciniak et al., running such simulations is indeed equivalent to solving the complex equations of the M-K theory. Also, aluminium alloys are not much sensitive to deformation rate, which simplifies further their simulation. On the contrary, rupture of alloys which do have some large imperfections will not be correctly simulated without introducing into finite element codes an appropriate failure criterion. Such theories, not based on the mechanics of continuous media, exist and try to link metallurgical damage with failure [3,5].
2.2. Experimental results A common way of investigating the drawing ability of a given alloy is to determine the failure curve giving the maximum blank holder force (BHF) allowable for drawing without breaking a blank with a given diameter. Pechiney has determined such curves for many of its alloys. Problem with these curves is that they not only depend on the material and geometry used but also on the lubrication. Fig.1 below shows two curves obtained for an AA 5754-0 alloy used at Pechiney. One curve ('high lub') was obtained by lubricating carefully the blanks, while for the other ('low lub') blanks were wiped. These curves themselves are approximate due to high dispersion in the experimental results.
291 Geometry used []
H i g h lub lub
punch diameter" Dp = 69.9 mm
3
punch radius 9 rp=3mm 2 ~J
die diameter 9 Dm = 72.5 mm 1
die radius 9 rm = 5 mm 0
. 138
I
"
140
I
142
'
I
'
144
I
146
'
148
blank thickness 9 t = I mm
B l a n k d i a m e t e r (mm) F i g u r e 1. E x p e r i m e n t a l failure curves for AA 5 7 5 4 - 0 P r e s s u r e s P s h o w n on t h e c u r v e s a r e h y d r a u l i c p r e s s u r e s a p p l i e d in press. A p r e s s u r e P of 1 b a r c o r r e s p o n d s to a force F of 15.7 k N a p p l i e d on blank holder. T h e slopes of t h e two curves are respectively a b o u t - 0 . 7 b a r / m m for 'high a n d - 0 . 3 b a r / m m for 'low lub'. Rheology of t h e AA 5754 O u s e d is t h e following : - y i e l d s t r e n g t h " Ro,2 = 103 M P a - L a n k f o r d coefficient : r = 0.61 - h a r d e n i n g d e s c r i b e d by Voce law : (~ = A - B exp (-C e) w h e r e (~ is
the the lub'
the
e q u i v a l e n t s t r e s s in MPa, e t h e e q u i v a l e n t d e f o r m a t i o n a n d A = 285.4 M P a , B = 194.3 MPa, C = 12.9
2.3. Description of numerical simulations F i n i t e e l e m e n t s i m u l a t i o n s w e r e c a r r i e d o u t to t r y a n d r e p r o d u c e e x p e r i m e n t a l f a i l u r e curves. T h e explicit code O p t r i s TM w a s used. R h e o l o g y d e s c r i b e d above w a s m o d e l l e d u s i n g t h e available Hill model. Since i n t e r e s t w a s not in m o d e l l i n g t h e effect of p l a n a r a n i s o t r o p y , a n d to m i n i m i z e C P U t i m e , only 4 ~ of t h e a x i s y m m e t r i c g e o m e t r y w e r e m e s h e d w i t h a p p r o p r i a t e b o u n d a r y conditions. P u n c h a n d die w e r e a s s u m e d u n d e f o r m a b l e ; t h e i r r a d i i m e s h e d w i t h a b o u t 10 e l e m e n t s . V a r y i n g forces w e r e a p p l i e d on a d e f o r m a b l e , elastic b l a n k holder. P r e l i m i n a r y r u n s s h o w e d t h a t b e s t r e s u l t s a r e o b t a i n e d w h e n t h e b l a n k is m e s h e d as t h i n as possible u s i n g available shell e l e m e n t s : t h e m i n i m a l l e n g t h of a n e l e m e n t m u s t be l a r g e r t h a n its t h i c k n e s s to p r e v e n t self-contact a n d to k e e p t h e a s s u m p t i o n of 'shell' e l e m e n t s (whose t h i c k n e s s is s u p p o s e d to be m u c h s m a l l e r t h a n t h e i r span). Only one e l e m e n t w a s u s e d in t h e o r t h o r a d i a l d i r e c t i o n to describe t h e 4 ~ s p a n (4 ~ w e r e chosen so t h a t t h e o r t h o r a d i a l l e n g t h
292 be always larger t h a n the blank thickness). The part of the blank u n d e r the punch was meshed with a single 3-node shell element. P u n c h speed was accelerated to 10 m/s (this speed was found optimal to minimize CPU time without introducing perturbing inertial forces). Average CPU time for such a drawing simulation was about 10 min on a Silicon Graphic R8000 processor. 2.4. Influence of friction coefficients
E x p e r i m e n t a l results (see Fig.l) clearly showed t h a t friction has a large influence on failure curves. Three different friction coefficients (Coulomb's law) are defined in the simulations : IXfor the part of the blank under the blank holder, Ixm on the die radius and ~tp on the punch radius. P r e l i m i n a r y trials showed that in order to have a correct t h i n n i n g of the blank under the punch (0.9 mm in the experiments), ~tp should be higher t h a n 12 %. Otherwise, simulations predict too much thinning. Other simulations also showed t h a t in order to prevent too much t h i n n i n g u n d e r the punch radius, a 'good' value for ~tp is 20 %. This value may seem quite high, especially for lubricated blanks, but may be explained by the very high local pressures exerted by the blank on the punch radius. Then, simulations were run to determine the influence of Ix and Ixm on failure curves. To compute such a curve, simulations with v a r y i n g b l a n k holder pressures are run for two or three blank diameters D (typically close to d r a w i n g ratios of 2) until pressures at which r u p t u r e occurs are reached. Ruptures do occur when pressures get too large and a r e r e v e a l e d in numerical simulation by the localisation of deformation on a given set of finite elements which get thinner and thinner. All ruptures simulated occur on the part of the b l a n k located between the punch and die radii. Failure curves are a s s u m e d s t r a i g h t (this a s s u m p t i o n is justified for drawing ratios close to the limit drawing ratio usually situated between 1.9 and 2.1 for aluminium alloys). The following table shows the slopes and elevation of the failure curves obtained for varying friction coefficients : Table 1 : slope and elevation of failure curves (alloy AA 5754-0) ~m Ixp pressure (bar) slope for D= 142 m m (bar/mm) 0.02 0.02 0.2 11.5 -1.25 0.04 0.04 0.2 4.2 -0.6 varying 0.06 0.06 0.2 2.2 -0.5 0.08 0.08 0.2 0.5 -0.25 0.06 0.04 0.2 3.5 -0.5 l.tm varying 0.06 0.08 0.2 0.9 -0.5 0.04 0.06 0.2 3.2 -0.75 ~t varying 0.08 0.06 0.2 1.2 -0.25 .
_
_
It is seen t h a t " - the slope of the failure curve depends on IXbut not (much) on lXm,
293 - the height of the failure curve depends on both }~ and ~tm. By c o m p a r i n g all these curves with e x p e r i m e n t a l curves (Fig.l), it is seen t h a t the 'low lub' curve is best fitted using (~, ~tm)=(8%, 6%) and t h a t the 'high lub' curve is best fitted using (~, pro)=(4%, 6%). These curves are s h o w n in Fig.2 below.
9
4
Low lub (NS) (~,~m)=(8%,6%)
%
~-
High lub (NS) 6%)
~2
~2 r~
Low lub (exp) 0
'
138
I
140
'
I
142
'
I
144
~''
I
146
0
'
148
'
138
B l a n k d i a m e t e r (mm)
High lub (exp) I
140
'
I
142
'
I
"~ '
144
I
146
'
148
B l a n k d i a m e t e r (mm)
Figure 2. F a i l u r e curves for AA 5754-0 9experiments vs. simulations 2.5. C o n c l u s i o n
It can t h u s be concluded t h a t n u m e r i c a l s i m u l a t i o n s are able to reproduce failure curves as long a s " - the alloy s i m u l a t e d does not have large metallurgical defects, - the finite e l e m e n t m e s h is fine enough, - friction coefficients are adapted. The modelling of the rheology of the m a t e r i a l should be accurate b u t is less critical. 3. S I M P L I F I E D A N A L Y T I C A L C R I T E R I O N 3.1. P r i n c i p l e
Most r u p t u r e s d u r i n g deep drawing occur in the cup wall (between the die a n d p u n c h radii) : only this p a r t is u n c o n s t r a i n e d . A r u p t u r e criterion can t h e n be found if computed stresses in this p a r t of the cup are h i g h e r t h a n limit s t r e s s e s allowable. This criterion will only be valid if, as for n u m e r i c a l s i m u l a t i o n s , no large m e t a l l u r g i c a l defects are p r e s e n t in the initial flange. It will not be able to predict failures occurring u n d e r the blank holder but it can be t h o u g h t t h a t if a failure occurs there, t h e n the criterion should detect a failure in the cup wall for n e a r b y b l a n k holder pressures.
294 3.2. C o m p u t a t i o n o f s t r e s s i n t h e w a l l
The r a d i a l s t r e s s gr due to s h e a r in the flange above the wall can be computed as [6] 9 c
r
Rm = malog~ Ri
(1)
where ~ is the equivalent flow stress, 'm' is a coefficient enabling adjusting the Tresca criterion (used for establishing the formula) to the von Mises criterion (m = 1.1), Rm is the flange radius (at the given draw depth) and Ri the die i n t e r n a l radius. To this stress m u s t be added the stress asfdue to the blank holder pressure 9 ~t-F Csf - 2. ft. Ri. t m
(2)
where F is the blank holder force and t the flange thickness (it can be t a k e n to the initial flange thickness as an approximation). W h e n the flange slides over the die radius, these stresses are t r a n s f o r m e d by the so-called rope formula, so t h a t the stress ~ in the can wall becomes 9 ~ = (Gr + Osf)" el'tm"0
(3)
w h e r e 0 is the angle over which the flange is sliding on the die radius. This angle can be c o m p u t e d as a function of the d r a w i n g d e p t h (0 at d r a w beginning, n/2 at the end). An a d d i t i o n a l s t r e s s gbub m u s t be added to this s t r e s s due to the b e n d i n g / u n b e n d i n g of the flange :
(~LULou= ~" 2" rm --
(4)
where rm is the die radius and E is the Young's modulus. The stress (~t in the flange wall is thus [7] 9 crt = (~r + ~ s f ) "el'tm'0 +~bub
(5)
S t r a i n h a r d e n i n g can be t a k e n into account by varying the equivalent flow stress (~ as a function of an equivalent strain. This equivalent s t r a i n can, for example, be calculated in the middle of the flange (at radius (Rm+Ri)/2).
295 3.3. F a i l u r e
criterion
The s t a t e of stress at failure is necessarily located on t h e yield curve. The m a x i m u m positive s t r e s s is w h e r e the t a n g e n t to the yield curve is vertical. W i t h an anisotropic Hill criterion, this stress is computed as [8] 9
max
=o
-
z
l+r
=~ ~ ~/2r + 1
(6)
w h e r e r is the L a n k f o r d coefficient. At failure, the criterion is 9 ot = (~max
(7)
And the corresponding orthoradial stress o0 is" r o 0 = ~ o
r+l
(8)
z
It is not null, as frequently a s s u m e d but positive (tensile). Therefore, close to r u p t u r e , compressive o r t h o r a d i a l stresses in the flange become tensile after the flange h a s gone over the die radius. This fact w a s confirmed by n u m e r i c a l simulations. Also, the stress Crmax is an increasing function of r, w h i c h p r o v i d e s a n e x p l a n a t i o n to the fact t h a t alloys with a high L a n k f o r d coefficient are best for drawing. 3.4. F a i l u r e
curves
A p p l y i n g the above criterion, the following failure curves are found (blank holder force F at r u p t u r e for varying b l a n k radii Rm) 9
F - 2 gRi t(( (~max - Gbub )e -]'tm0 - m~ log -~1 Rm )
(9)
Since failure curves are u s u a l l y d r a w n for d r a w i n g ratios b e t w e e n 1.5 a n d 2.5, the l o g a r i t h m can be a p p r o x i m a t e d by" mlog Rm -- 0.5 R m - 0.35 Ri
(with m=l.1)
(10)
Ri
so t h a t failure curves are about s t r a i g h t with slopes K a p p r o x i m a t e d by 9 g
2.rt.tx.t.~ K -- -0.5 g -- - ~ g g Results found experimentally explained by Eqs.9 and 11 9
(11) a n d in n u m e r i c a l
simulation
are
thus
296 - the lower the friction under the blank holder, the more vertical the failure curve get. In a perfect case with no friction, the curve would be vertical. - the height of the curve depends on both friction coefficients ~t and ~trn. The higher the friction ~tm on the die radius, the lower the failure curve.
3.5. Comparison between analytical criterion and numerical simulations First comparison is made on the slopes of the failure curves. Table 2 below compares slopes found in numerical simulation for alloy AA 5754-0 (see Table 1) and slopes K' found from Eqs.9 and 10 (t = 1.mm, ~ = A = 0.285 GPa). Slopes K' are expressed in bars per mm of blank diameter so that E q . l l becomes : P (bar) = K'. Dm (mm) with
K ' = K / 2 / 15.74
(12)
Table 2 9slope of failure curves (alloy AA 5754-0) slope K' (bar/mm) simulation analytical 0.02 -1.25 -1.4 0.04 -0.6/-0.75 -0.7 0.06 -0.5 -0.5 0.08 -0.25 -0.35 It is concluded t h a t given uncertainties on slopes computed n u m e r i c a l l y (due to small n u m b e r of cases computed), a g r e e m e n t b e t w e e n a n a l y t i c a l criterion and numerical simulations is very good. A F o r t r a n program was written to compute failure curves according to Eq.9. F a i l u r e condition Eq.7 is computed for each blank diameter, p r e s s u r e and drawing depth. It was found t h a t in order to match finite element curves, the friction coefficient ~tm t a k e n for the simulations has to be increased by about 0.08 when applied in Eq.3. Some authors [8] pointed out that the so-called rope formula is not well adapted to deep drawing due to bending stiffness of the sheet. Fig.3 below shows a comparison between failure curves o b t a i n e d in simulation and analytically in the case of alloy AA 5754-0 with low and high lubrication 9
297
-
Low lub (Anal)
(p,pm)=(4%,6%)
Low lub (NS) 3 ~2 2 00
1 ~
Hi High lub (NS)
(~t,pm)=(8%,6%) 0
'
138
I
140
'
!
142
'
I
144
'
I
146
Blank diameter (mm)
0
'
148
'
138
I
140
'
I
142
'
I
144
"
(iPI
146
'
148
Blank diameter (mm)
Figure 3. Alloy AA 5754-0. Numerical simulation vs. analytical criterion Here again, the agreement is quite good. It should be noted that slopes found with the Fortran program do not match exactly slopes computed according to Eq.12 since this equation is a simplified version of Eq.9. 4. CONCLUSION This research has shown that deep-drawing failure curves can be predicted both by numerical simulation and an analytical criterion as long as the metal studied does not have large metallurgical defects - as in the case of commercial aluminium alloys used for drawing cans or car body panels. Results of the simulations are influenced by mesh density, which should be as fine as possible, by rheology but mainly by the friction coefficients defined. The analytical criterion confirms and explains the importance of the friction coefficients. In particular, the slope of failure curves strongly depends on the friction under the blank holder. As a first approach, the analytical criterion provides accurate enough failure curves but problem remains to enter adequate friction coefficients to use this model in a predictive way. REFELtENCES 1. R. Hill, J. Mech. Phys. Solids, Vol.1 (1952), 19-30. 2. Z. Marciniak & K. Kuczinski, Int. J. Mech. Science, Vol.15 (1973), 789. 3. I.L. Dillamore, J.G. Roberts, A.C. Bush, Metal Science, 2 (1979), 73-77. 4. B. Baudelet & B. Grange, Scripta Met. et Mat., Vol.26 (1992), 375-379. 5. L. Felg~res, Mem. Sci. Rev. Met., 77 (1980), 327-342. 6. W. Johnson & P.B. Mellor, Engineering Plasticity, Ellis Horwood Ltd (eds), England, 1983. 7. K. Manabe & H. Nishimura, Proc. of the 2nd Inter. Conf. on Techn. of Plasticity, Ed. K.Lange, Stuttgart, 24-28 Aug., 1987, Vol.II, pp. 1297-1304. 8. J.A.H. Ramaekers et al., Proc. IDDRG'94, Lisbon, 403.
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PREDICTION OF SHAPE INACCURACIES
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Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
301
Prediction of flange w r i n k l e s in deep d r a w i n g Jian Cao a, Apostolos Karafillis b and Michael Ostrowski b aDepartment of Mechanical Engineering, Northwestern University, Evanston, IL 60208-3111, U.S.A. bResearch and Development Center, General Electric Company, Schenectady, NY 12301, U.S.A.
ABSTRACT A numerical method for predicting the onset of flange wrinkles of small wavelength during deep drawing process is presented. The method is based on the approach developed by Cao and Boyce (1997) for predicting the buckling behavior of sheet metal under lateral constraint using a combination of energy conservation and finite element method. Continuum elements are used in a simple Finite Element Analysis model to study wrinkles with a maximum wavelength of ten times the sheet thickness. The analysis provides the critical buckling stress and the resulting buckling wavelength as functions of normal pressure. Such relationships are then implemented in a Finite Element Method (FEM) package that uses membrane elements to simulate the workpiece deformation during a forming process. The use of membrane elements significantly reduces the amount of computation time required in comparison to using structural shell elements with multiple integration points through the thickness. The stress histories calculated from the FEM membrane analysis are used to predict the onset of buckling during the forming process. The application of the forming of a rectangular pan is examined. The comparison between numerical simulation and the experimental results is presented. Our approach predicts the onset of buckling in excellent agreement with the experimental observations.
1. INTRODUCTION Wrinkles during a sheet metal forming process is a major consideration due to the fact that it alters the ability to impose stretching during processing and also adversely affects final part appearance, assembling and function. Computational prediction of the onset and growth of the wrinkles has significant ramification for optimizing the design of parts and tooling, selecting materials and improving part formability. The problem that we are particularly interested in is the buckling in the flange (Fig. 1) which is held between a blankholder and a die. This buckling can be combined with subsequent draw-in of the already wrinkled part of the flange, thereby causing the appearance of wrinkles at the sidewalls of the part by the end of the process. The methodologies for predicting the
302 onset of buckling can be mainly divided into two categories. One is the bifurcation analysis initiated from Hill's (1958) general theory of bifurcation and uniqueness, and later detailed by Hutchinson (1974) in the plastic buckling range. Triantafyllidis and Needleman (1980) studied the problem by assuming sheet metal resting on an elastic foundation whose stiffness relates to the binder pressure. Although their results were found to compare favorably with some previous empirical models for the cases where binder stiffness K=0 (no binder constraint), no comparison between the numerical results and experiments was given when K:~0. However, they calculated the effect of binder stiffness on the critical buckling stress and the wave number. A similar approach for elastic rectangular plates on a non-linear uni-laterial elastic foundation can be found in recent literature such as Elisakoff et. al (1994) and Shahwan et. al (1994) whose work was directed towards understanding the buckling of films bonded to a substrate. Overall, the bifurcation analysis is essentially an eigenvalue approach. The major obstacle of this method is that numerically it becomes extremely complicated if large deformation theory and anisotropic material constitutive law are involved. Nevertheless, the analytical method using shell theory is limited to shallow buckling, i.e., the buckling wavelength is large compared to the sheet thickness.
Fig. 1 A rectangular cup (left) having buckling and tearing failure (right shows a corner). The other methodology is Finite Element Method (FEM) with either implicit or explicit integration method, which becomes a prime tool to predict buckling behavior for complicated geometry and boundary conditions. Implicit method is essentially an eigenvalue approach and the post-buckling behavior has to be traced by initial imperfections built in the original mesh which usually is a specific modal shape. Unlike the implicit method, the nature of the explicit integration as a dynamic code and the numerical error accumulated in the analysis, which acts as imperfections, can automatically generate wrinkled deformed shape. However, the predicted onset and post-buckling behavior is affected by variations in the FEM model, such as, material density, simulation speed and mesh density, etc. Therefore, the robustness of the FEM simulations is not quite reliable. On top of these, an accurate prediction of buckling in the order of sheet thickness requires a very fine mesh which is considered impossible or unrealistic to have for a complicated three-dimensional forming simulation. Recently, a different approach which promises the ability to predict sheet buckling under lateral constraint was developed by Cao and Boyce (1997) using a combination of energy conservation and the implicit finite element method. They simplified the flange buckling
303 behavior in 3-D sheet metal forming process into a rectangular plate under lateral constraint. The approach was tested for shallow flange buckling using shell elements and the predicted results, which are critical buckling stress and the resulting buckling wavelength, match the experimental conical cup forming results extremely well. In this paper, the same analytical method is adopted to study short-wavelength flange buckling. The results are then compared to the forming of a rectangular pan conducted at GE.
2. N U M E R I C A L S O L U T I O N
2.1 Review of the Buckling Criterion As known, for plates of length L under in-plane compression (Ux) with no lateral constraint, the strain energy in a perfect flat plate (Go) is greater than that in a buckled plate (ew). Such difference can be interpreted as the work (gZn) executed by external lateral force (F) to suppress the buckling. Specifically, (1)
Wn-- G o - Ew n = f n ( U x ) -- I0 2~n~ w F d u z
where n represents the buckling mode (mode 1 is a half sinusoidal wave), Ux is the in-plane edge displacement, w is the width of the plate, Uz is the out-of-plane displacement, and ~max is the maximum buckled height at a displacement of Ux. By assuming F = a (Uz - ~max) 2 -F b , where a and b are fitting parameters to a parabola, the maximum pressure required to suppress the buckling of the n th mode at a given ux/L can be calculated as Pmax n =
3 (Co -
(2)
Ew n ) / 4 ( ~ m a x n L W )
m
t~ O
Mode 1 ....
/
Mode 2
f
E b E LU x~ E 9
.'/
.,
**./ .,/1
p**
4
0~
~
~. 2
O
.
/
z
/ ' ~ " d " I" " "" "
,-
.
.
.
.
.
.
.
.
.
.
.
.
,
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
p*
,
0.0000 0.0010 0.0020 0.0030 Normalized edge displacement Ux/L Fig. 2 Maximum binder pressures as functions of the edge displacement for the first and second mode buckling to illustrate the concept of the buckling criterion.
304 Figure 2 shows the calculated Pmax] and pmax2 as a function of ux/L. The two curves are found to cross over each other at a specific point, the transition pressure p]2, which is the binder pressure where the favored mode of buckling transits from mode 1 to mode 2. For example, consider the case of a binder pressure p* applied on the plate: the plate would buckle in mode 1 and mode 2 at displacements of u]* and u2*, respectively. For the case where p* < P12, we have Ul* < u2* (Fig.2), i.e., the critical buckling stress for mode 1 is lower than that for mode 2, and therefore, the plate would favor mode 1. On the other hand, for the case of p** > p12, we have u2** < Ul**, and therefore, the plate would buckle in mode 2. This criterion thus quantitatively defines the critical buckling stress (the associated stress at the critical or transit u/L) and also the mode of buckling which will occur in the presence of a given binder pressure. The criterion was tested for the drawing of aluminum conical cup and excellent predictions were obtained as shown in Fig. 3.
40
.-=
~o
7
m
O
t.-
30
Conical Cup Forming Data
o = ..Q
Numerical Results
i n
6 TL-O
8~5
, m
"E O
"o ~ 2 0 --~
~
O~ t_
O
.-- ~ ~ a
~_ ~ 10 E o z 0
Conical Cup Forming Data Numerical Results
0
z
0
20
40
60
80
2
0
100
Normalized binder pressure P/Oyo (10 4)
20
40
60
80
100
Normalized binder pressure P/~yo (10 4)
Fig. 3 Calculated critical buckling parameters compared to experimental results.
2.2 Study of short-wavelength buckling under constraint
The buckling criterion reviewed in the previous section provides a general principal for characterizing sheet buckling under lateral constraint. The methodology itself has no limitation on the applications of using various material constitutive laws or considering small or large deformation. However, the results presented in Fig. 3 are valid only for shallow buckling, since shell elements were used in a simple FEM analysis to obtain eo and ewn. Here, we will present an approach to study short-wavelength buckling under lateral constraint for a forming quality steel sheet. 2.2.1 Finite Element Model Using the commercial finite element code ABAQUS, a plate of length L is modeled with 8-node reduced integration plane stress continuum elements (ABAQUS CPS8R) when considering a uniaxial in-plane stress state, and 20-node reduced integration brick elements (ABAQUS type C3D20R) when considering a biaxial in-plane stress state. The reduced
305 integration element CPS8R has four integration points whereas C3D20R has 8 integration points. The finite element code uses an implicit solver where equilibrium is converged upon in each increment using a Newton-Raphson method. For the uniaxial stress state case, which corresponds to the stress condition at the outer edge of the flange in a sheet metal forming operation, the plate, having unit width, is modeled with 12 CPS8R elements along the length (x direction) and 3 elements through the thickness (y direction). The loading condition consists of a monotonically increasing compressive displacement Ux of edges at x=0 and x=2L. Two cases are simulated for each 2L: a perfect plate and a plate buckled to mode 2. Notice that mode 2 buckling of a length 2L is equivalent to mode 1 buckling of a length L. In Cao and Boyce (1997), mode 1 was used in the simulations to perform the analysis. Here, mode 2 is used due to the simplicity of setting the boundary conditions associated with continuum elements. As known, buckling initiates from some form of imperfection in the structure which can be a geometric imperfection (such as lack of flatness), a material imperfection (e.g., non-uniform thickness), or loading imperfection (e.g., off-center loading). Cao and Boyce (1997) have shown that the effect of geometric imperfection on the initiation of buckling is more significant than that of material imperfection. Here to obtain the buckled plates, the form of imperfection is chosen to be a geometric imperfection which positions nodes to a sinusoidal mode shape (Zo = Ato (1-cos (2rex/L))) of a very small amplitude A=0.005, where to is the plate thickness. Incorporation of the imperfection in this manner acts to predefine the buckling mode obtained where A=0 for obtaining a perfect flat plate under in-plane compression. The deformed mesh of a buckled plate with the above imperfection form (A~:0) is shown in Fig. 4. The strain energies of a flat plate (eo)and a buckled plate (ew), and the buckled amplitude are recorded as functions of edge compression ux/L.
z u z[mx Fig.4 Deformed mesh in uniaxial stress state.
Fig.5 Deformed mesh in biaxial stress state.
For studying the buckling behavior under biaxial loading (compression along the length direction and tension in the width direction), which represents the stress condition near the binder radius in sheet metal forming, the plate is modeled with twelve C3D20R elements along the length direction (x direction), one element in the width (y) direction and three elements through the thickness (z direction). The width is defined as one-twelfth of the total length in x direction. The loading condition consists of a constant tensile stress in y direction and a
306 monotonically increasing compressive displacement Ux of nodes at x=0 and x=2L. A transverse tensile stress (-- 0.5 (~yo) is used to examine the effect of tensile stress on the onset of sheet buckling. Again, two simulations of a perfect and a wrinkled plate are calculated for each length 2L where the same imperfection form as described before is adopted. The deformed shape of a wrinkled plate using three dimensional brick element is shown in Fig. 5.
2.2.2 Buckling stress prediction Following the calculation procedure reviewed in Section 2.2.1, the critical buckling wavelength normalized to the sheet thickness as a function of applied normal pressure normalized to the initial yield stress is plotted in Fig. 6a for both uniaxial and biaxial loading cases. Fig. 6b shows the critical buckling stress normalized to the initial yield stress versus normalized pressure. Two curves are plotted in this diagram. The solid lines correspond to a uniaxial compression stress state, whereas the dashed lone corresponds to the biaxial stress state described in Section 2.2.1. By comparing the two curves in the diagram of Fig. 6, we see that a transverse tension in the plane of the sheet metal will initiate buckling earlier and will result in a longer buckling wavelength. A buckling severity index is then defined a s (]applied/(~crit as a function of binder pressure. A value higher than 1.0 means that buckling will occur. We call the diagram of Fig. 6b the Buckling Limit Diagram (BLD). The BLD of a material can be used for the prediction of the initiation of small wavelength buckling, once the stress state in the material is known. 12
._=
"| .
.
10
~l
0:3 t'-
8
~.
-o ~9 N
6
~
o Z
~
.
.
.
.
.
uniaxial ......
~
biaxial
N~
m"
. B m
--.
4
~f
"-----uniaxial ......
z.~
2
i
0
J
0
~
0.1
Normalized binder pressure p/O'y o
0.2
0
biaxial
I
0.1
i
0.2
Normalized binder pressure P/~yo
Fig. 6 Calculated buckling parameters as functions of binder pressure
3. A P P L I C A T I O N AND C O M P A R I S O N W I T H E X P E R I M E N T A L R E S U L T S The proposed buckling prediction method was implemented in a finite element analysis code used for sheet metal forming process simulation. The code is based on a dynamic explicit analysis of the forming process and uses three node membrane elements to model the sheet metal workpiece. The contact calculations area effected by using a cubic polynomial based description of the forming tool surfaces in combination with a predictor-corrector algorithm for
307 contact boundary conditions enforcement. During the dynamic explicit calculations, strains and stresses are calculated in every element using the combination of an explicit integration scheme, a plasticity flow theory, and an elastic-plastic material model.
3.1 Defect prediction The forming process simulation is used in order to predict the deformation behavior of the workpiece during a sheet metal forming operation. One of the major uses of our computer simulations of sheet metal forming processes is the prediction of two types of forming defects: tearing, due to excessive stretching of the workpiece, and buckling, due to in-plane compressive stresses. The methods used for the prediction of the occurrence of these defects are explained below. 9 Tearing prediction: To predict tearing, we use the concept of the Forming Limit Diagram (FLD), already introduced by Keeler (1964). The FLD of a material provides the magnitude of the major strain allowed prior to failure by localized thinning for a given value of the minor strain. A schematic representation of an FLD is shown in Fig. 7. Using the FLD of a material, and the calculated strains at an integration point of our finite element analysis process simulation, we can define the Forming Severity (FS) as shown in Fig. 7, i.e. (FS) = Emax/E* where Emax is the calculated maximum strain at a material point, and e* is the maximum allowed strain prior to failure by localized thinning. Therefore, a value of the forming severity higher than 1 indicates failure by tearing. During our finite element analysis simulation of the process, the highest value of the forming severity obtained in every material point is recorded as a state variable. Contours of forming severity can then be displayed upon completion of the simulation.
~max
o,
t
J
S~
~max
0 Minimumstrain, Fig. 7: A schematic representation of the Forming Limit Diagram.
308
Buckling severity: The measure of the buckling severity for the prediction of buckling initiation has been already introduced in Section 2.2.2. During the finite element analysis simulation of a sheet metal forming process, we calculate the principal stresses at every material integration point. Then, based on the Buckling Limit Diagram of Fig. 6b, we calculate the forming severity as ~applied/~crit, where ~applied is the absolute value of the minimum principal stress, and CYcdt is the critical stress for the initiation of small wavelength for a given normal blankholder pressure. The buckling severity is updated at every time increment of the finite element analysis only in the elements that are in the blankholder area and in which 0< ]~trans/(Yappliedl<0.5, and Crtrans>0, and ~applied<0 and p>O.Ol~yo, where 6t~ans is the major principal stress. This range of stresses corresponds to a typical stress state in the blankholder area of a sheet metal forming operation. The maximum value of the buckling severity during the forming process simulation is recorded. Contours of the value of the buckling severity distribution in the workpiece can be plotted upon completion of the forming simulation. Note here that an element that is initially in the blankholder area can register high values of the buckling severity during the initial steps of the process and then draw in the forming area. In this case, the highest value Of the buckling severity will be retained throughout the process. This feature can be used to predict side wall wrinkles that initiate in the flange area and then draw in the forming area. However, secondary effects such as contact pressure changes and localized stress and strain gradients due to post-buckling behavior are not modeled.
3.2 Process simulation results
The finite element analysis simulation and the process defect prediction techniques introduced and discussed in this paper were used in the modeling of the forming of the square pan of Fig. 1. The contour of the buckling severity, as obtained from our finite element analysis, is shown in Fig. 8, where only a quarter of the actual part is presented. Only the values of buckling severity which are higher than 1.0 are displayed in this case, in order to identify the areas where buckling of small wavelength had initiated. Therefore all areas with a buckling severity contour level higher than 1.0 will incur small wavelength buckling during the forming process. By comparing Figs. 1 and 8, it can be seen that our prediction of buckling initiation is in excellent agreement with the experimental observation. We were able to successfully predict that all the flange will undergo the development of small wavelength wrinkles, as the buckling severity exceeds the value of 1.0 in the whole flange. Although not very clear in Fig. 1, wrinkles from the flange area were drawn in the forming area during the deep drawing process. This phenomenon was also predicted (compare Fig. 1 and Fig. 8). In general our model predicted the areas at which buckling of small wavelength occurred in very good agreement with the experimental observations on the test part of Fig. 1. Additional forming simulations and experiments need to be performed in order to further validate our buckling initiation modeling and process simulation capability used to predict the development of small wavelength wrinkles for different blankholder pressures and different forming geometry.
309
Fig. 8: A contour of the buckling severity of the part of Fig. 1 It is worth noting here that substantial tearing was also developed in the part of Fig. 1. In Fig. 9 we plot the forming severity of the part of Fig. 1 upon completion of forming. The maximum predicted value of severity was equal to 0.92, developed in the area of the vertical comer of the box, see Fig. 9. As it can be also seen in Fig. 1 this is an area where cracking has occurred. The maximum predicted forming severity was slightly lower than 1.0 (=0.92) indicating a high likelihood of failure by localized thinning. The area of high forming severity was also an area of buckling severity higher than 1.0 implying that both buckling and tearing were developed in the same area, compare Figs. 8 and 9. This prediction was verified by our experimental observations, see Fig. 1.
Fig. 9: A contour of the forming severity of the part of Fig. 1.
310 4. CONCLUSIONS A numerical method for predicting the onset of wrinkles of small wavelength was developed. This method uses an energy based approach to predict the contact pressure required to suppress small wavelength buckling in the flange area of a sheet metal formed part. Based on this approach, a stress based Buckling Limit Diagram (BLD) for a drawing quality steel was created. The Buckling Limit Diagram was then used in conjunction with a Finite Element Analysis code to predict the initiation of small wavelength buckling during the forming of a square box test part. Based on the BLD used and the stress calculations of the Finite Element Analysis code, we were able to successfully predict the initiation of small wavelength buckling. Our method was also capable of predicting side wall wrinkles that initiated in the flange area demonstrating the effectiveness of our approach.
ACKNOWLEDGEMENTS The authors would like to acknowledge the efforts of William Carter and Michael Graham of GE Corporate Research and Development in the development of the Finite Element Analysis code used in this work. Philip Stine of GE Appliances provided the square box test part and all the data for its forming.
REFERENCES Cao, J. and Boyce, M. (1997) Wrinkling behavior of rectangular plates under lateral constraint, Int. J. Solids Structures, 34 (2), 153-176. Elishakoff, I. and Cai, G.Q. (1994) Non-linear buckling of a column with initial imperfectio via stochastic and non-stochastic convex models, Int. J. Non-linear Mech., 29(1), 71-82. Keeler, S. and Backofen W.A., (1964), Plastic instability and fracture in sheet stretched over rigid punches, Trans. ASM, 56, 25-48. Hill, R. (1958) A general theory of uniqueness and stability in elastic/plastic solids, J. Mech. Phys. Solids, 6, 236-249. Hutchinson, J. W. (1974) Plastic Buckling, Adv. in Appl. Mech., 14, 67-144. Shahwan K. W., and Waas A. M. (1994) A mechanical model for the buckling of unilaterally constrained rectangular plates, Int. J. Solids Structures, 31(1), 75-87. Triantafyllidis, N. and Needleman A. (1980) An analysis of wrinkling in the Swift cup test, J. Eng. Mater. and Tech., 102, 241-248.
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
311
F i l l i n g d e f e c t s in a c e r a m i c s f o r m i n g p r o c e s s F. Chinesta, R. Torres, I. Mont6n ~ A. Poitou b and F. Olmos c Universidad Polit~cnica de Valencia. Camino de Vera s/n. 46071 Valencia. Spain. bLMT, ENS de Cachan, 61 Avenue du President Wilson, 94235 Cachan Cedex. France. Universitat de Valencia. Campus de los Naranjos. 46071 Valencia. Spain. This paper whishes to present a simple semi-heuristic method to predict mold filling defects in a porcelain forming process. It involves both a mechanical modelling of this process as a squeezing flow, and a postprocessing calculus which unables a defects prediction as soon as the shape and the size of artistic reliefs are known. This shape analysis is carried out with a multiresolution wavelet analysis. 1. I N T R O D U C T I O N Traditionnal porcelain processing is mainly achieved in two steps. During the first one, a paste is formed in a mold with a process similar to "turning". The difficulty of this stage is to fill every superimposed drawing correctly (Figure 1). A second step consists in drying the molded part so that the schrinkage allows for its demolding. The aim of this paper is to model the filling stage and to propose a criterium to avoid any filling defect which could waste the quality of the drawings. In the following, section 2 deals with a simplified modelling of the process during which the "turning" stage is approximated by the deformation under pressure of a Norton Hoff ceramics paste squeezed between the mandrel and the averaged surface of the mold. The problem is solved with the Hele Shaw equations generalized to the case of a Norton Hoff's materials on an axisymetrical shell with appropriate boundary conditions. Section 3 proposes a criterium for a possible mold filling defect which can be applied as a postprocessing procedure as soon as the average velocity of the flow front has been computed and as the shape and location of the artistic reliefs are quantified. The technic of localisation and analysis of the reliefs involves a wavelet decomposition of the exact shape of the mold which is assumed here to have been previously numerized for its machining. 2. A V E R A G E M O L D F I L L I N G The mold filling operation is threedimensionnal. An excentered mandrel rotates both around its axis and around the axis of symetry of the bell. Its spacing from the midsurface of the mold is reduced progressively in order to control the width of the part (Figure 2). This stage has been modelled as follows.
312
.j i i ! |
I Z+ Z--
i |
|
I !
Figure 1" Porcelain bell
Figure 2: The forming process
2.1. G e o m e t r y E denotes the average mid-surface of the mold. It is a surface of revolution which is generated by rotation around the z axis of a curve parametrized by its curvilinear coordinate s. A point _x~ of this surface is parametrized by two orthogonal coordinates s and ~o 9 (1) We note 9 g-* =
g-v =
dx_r. ds =
dr(s) ds
cos~pi +
dr(s) ds
sin ~ j + d g - l ( r 2 ( s ) ) k -
= - r ( s ) sin~p i + r ( s ) cos~ j
d~
(2)
ds
(3)
Thus g_,.g_, = 1, g_ .g_~ = r 2 et g_,.g_~ = 0. The domain occupied by the material is = E x [-h, hi, so that a point in 1~ is denoted by _~ = ~
+ o~
(4)
In equation 4 _zr. is the projection of _z on E, 0 lies in [-h, h] and g-0 is the unit vector normal to E, defined by"
x~
(5)
= I1~ x ~ l l The equations are classically written in the orthogonal coordonate system (s, ~, 0) associated to the basis ( ~ , g_~,go) with use of the metric tensor g" 10
O)
g_ =
0
r2
0
-
0
0
1
(6)
313
2.2. T h e t h r e e d i m e n s i o n n a l m o d e l l i n g The material is assumed to be incompressible and to exhibit a Norton Hoff's behavior. Moreover, the contact between the mandrel and the paste is assumed to be perfect (i.e. a normal velocity as well as a zero shear stress are assumed to be prescribed), the contact between the paste and the mold is described by a Norton's law F:
ak"'lk = 0
# = k
with v.n_=0
(r
diJdij
)n-1
(flow front)
(n_.=a.n__)= F(Vg)
on E_
a,-k = _pg,.k + 2 #g mr g ksd m8
where
on E_
z . n = 0 on Eut a.n-
vkl k = 0
and
1
(7)
and
dij = ~ (viii + viii)
(8)
v.n=U
on E+
(9)
(n.=a.n) = 0 on E+
(10)
a.n-
where V_g is the slipping velocity on E_
(11)
2.3. H e l e S h a w e q u a t i o n s The threedimensionnal equations are solved with approximations similar as those described in [4]. These approximations are an extension of Hele Shaw equations and consists in searching, with use of the virtual work theorem, a solution for which the 0 dependance would be that of a simple shearing motion. Thus we search a velocity field v whose projection w on the tangent plane writes:
w__ =
v,p
= f(O) u + V_g = f(O)
u~(s, qp)
+
(12)
Va~(s,~)
with
f(O) = l + 2n { 1 _ ( O - h ) ~--~""} 1+ n
(2h) ~-t~-. "
(13)
w and Vg can then be written as functions of the pressure p which is assumed to depend on s and T only:
1
_w-
(2h) Itpl~ll~pl~ -~
with
2hplk =-F(V__g)
and
a=
h
21o1 o /14)
p(s, r is then calculated in solving the generalised Reynolds equation: (2h giJuj + 2h giJVgj) I` = U(s,~)
(15)
Equation 15 is solved numerically with a finite element method. At each time step, the computation provides the averaged flow front velocity _u_u,which is used simultaneously to actualise the flow front location and to predict with the following results an eventual mold filling defect.
314
Figure 4: Local variance pattern
Figure 3: Criterium for a mold filling defect 3. M O L D F I L L I N G D E F E C T S
3.1. C r i t e r i u m for a d e f e c t The above computation is achieved in assuming that the mold wall do not exhibit any artistic drawing. Thus it is impossible through this calculation to predict directly if the reliefs of the mold are correctly filled in. To answer this question, we propose here a simplified semi-empirical criterium. Let's assume that the averaged front velocity has been calculated in solving the Reynolds equation (equation 15). Let's assume moreover t h a t the material arrives at the edge of a relief (Figure 3). If to denotes the time required for the material, without accounting for the relief, to go from the entry to the exit of the relief in the flow direction (to = a/u). We state that a mold filling defect can possibly occur if the time to is not sufficient to fill in the area ab = S of the relief's section in the flow direction: ~otO
U s(t) dt =
~otO
t~ U u t dt = U u -2-
=
U a2 2 u
<
a b
(16)
From equation 16 it is deduced that in order to avoid any defect, the velocity field must not exceed a critical value 9 U a vcr=2 b
=
Ua 2 2S
This semi-empirical criterium can be applied as soon as the average velocity u and the shape of the relief are known. 3.2. L o c a t i o n a n a l y s i s a n d d e t e c t i o n of t h e d r a w i n g s The shape detection of the relief is in itself a noneasy problem. This is achieved in two steps. We first adjust a smooth polynomial curve which represents the shape of the mold in the flow direction, without taking into account the local details. Artistic reliefs are not accounted for at this level and appear on a smaller scale. In order to determine this polynome, we have used a minimum square technique, by means of Tchebyshev's polynomes (Figure 5). We define a coordinate system (1,s) on the reference curve in the
315 to
io
7o in Ia
io
Pl
4o
3o
30
1o
o
x
.lO -m
v
Figure 5: Reference curve.
Figure 6" {1,s} coordinates fmdmm
as, a4
as I! al
o .al
uooo
11900
laDO0
12gO0
13000
13500
14000
Figure 7: Scaling function
1,4m0
-,O.II 11000
11n00
lJO00
1~i0
1~000
13i500
14000
|4100
Figure 8: Wavelet functions
following way: We determine a point Q(xq, yq) on the polynomial curve so that the line defined between the point of the part P(xr,,yp) and this point Q is orthogonal to the reference curve described by the polynome. The distance between these two points will be considered as coordinate "s"; and the arc length from a point of the reference curve as coordinate "1". In this way, a reference system (1,s) is defined, which is associated with the reference curve (Figure 6). The detection of the reliefs is then achieved in using a wavelet analysis (Figures 7, 8). The algorithm is the Fast Wavelet Transform (FWT). This decomposition provides informations about location and size of the reliefs. The choice of the kind of wavelets to be chosen is a half-way option between a spatial location and a good frequency discrimination. Due to this, we choose the Battle-Lemarie wavelet. The location of the reliefs boundaries, can be achieved in appliing the local maximum criterium to the decomposition wavelet coefficients. This procedure localizes the edges of the reliefs correctly, but does not allow to establish any connexion between them. It is thus impossible by this way to recognize a drawing which would be inserted in an another one. To overcome this difficulty, we propose to calculate the local variance of the wavelet coefficients of each analysis level, and to use this information in order connect the edges, that is to determine a sort of.shape of the reliefs. Firstly, at each level j, the coefficients are normalized, so that max Idol = 1,
316 (~ are the wavelet coefficients of the analysis). Then, the local variance viJ is calculated according to the equation: ViJ
= v r{4
Jk=i-N
(18)
N is the number of coefficients next to ~ used in order to calculate the variance. It is often chosen as N = 1, because the location capability disappears with high N values. The results of this procedure appear on figure 11, where we can see that the behaviour of the variance v iJ calculated, is the same for all the reliefs: Two edge points of a relief are connected if it gives rise to the pattern represented on figure 4. The level at which this pattern appears, provides an approximation of the width of the relief. 4. E X A M P L E S 4.1. P r o f i l e o f a c e r a m i c b e l l
In this test (figure 9-11), a typical profile of the porcelain bell is studied. This profile is symmetric and has 7 reliefs; 6 of them have the same size, and one of them is between 4 and 6 times wider. Thus, these reliefs must appear represented with a difference of 2 or 3 levels in the multiresolution analysis. The decomposition coefficients in the wavelet bases have been represented on figure 9, the decomposition by a multiresolution analysis is shown in figure 10 and the local variance of the wavelet coefficients in figure 11. 4.2. A r e a s c a l c u l a t i o n
In order to calculate the relief area, we use the values of the scale coefficients included within the interval L, where the relief is located. Considering Lj as the set of indices i of the level J included within the local maximums of the variance, we calculate the area in taking into account the following relationships. From the condition
f ~o,o dx = 1
(19)
and using the relation 1 ~,,0(x) = ~ ~0,0(x/2)
(20)
we can write the area of the function f in the interval L as
A = f . f dx
(21)
which approached in a level J, is calculated according to
f = ~ c~j,,
(22)
iELj
so that the area expression becomes
A = Z c{ (v/2) a
(23)
iELj
The level J chosen for the approach of the relief area must be one in which the relief is correctly represented. In the proposed examples, it has been proved that if the relief is located at the level JR, a level J = JR + 5 is sufficient to do the approach and the area approach is obtained with an error smaller than 1%
317
9
e~
lmo
,eoo
~ee
~
eom
oeeo ,e
r-
_
~
7 9
9
to
is
so
oe
e 9
ao
s
9
9
9
,e
Is
,6
.o
,,..~,.o6
m
7 9
1~
~
9
._. ~
~
a~ ,,o.,.....o,,s ~
~o
,
,~
9
.'.
.~,e.me ....~..,.oe
D
w.,.w.ls
i! [ .e
9
am
,eo
,eo
am
~o ooo
Figure 9: Decomposition coefficients.
u~...ls
.
318 1
41 9
i i
I
I
I
iooo
GOOD
~
1 I
a~o
I
~oo
m
I
1~o
9
I I
I
I
~
I
I
~
9
I I
m
I
I
l
u 1,4 u u
:9
~2
.l
I .....
J
I
I
[
~
I
.o.o
-t
:
u
I
?
m
L4
,,o~
9
sow
~
~oo
dmO0
mOO
e~O
TOW
e~O
Figure 10" Multiresolution Analysis.
m
~
m
319
o.~ o.s o.~
o.ls
":"
I
I
,,
I 0.7
O.~S
o.o
o., o.~ o.a o~ oJ o.ts o.s o.os
,:o
J.~
I.
,~
I
,_
I
.
.
.
,..
.
.
.
.
.
-
vw.os --
o.4
oO:o !
j,..,j
J_
o.o ..........
0.4
O.a
0"0:
O.m
O.SS
O.a
o ~
.
.
.
.
.
.
o
1
;
~
1
~
.
, vw.ss
~ "2 = 0: "-
o o
-
-
:i = 22 2:1 ,/~
so
kAA)
1oo
lao
M,
xoo
.
....
o o
~s
I
1.s
9
F i g u r e 11: Local v a r i a n c e of t h e wavelet coefficients. Levels 1 - 11
x~
=
320 REFERENCES
1. L. Chevalier and A. Poitou. Quatre m$thodes pour presenter l'$quation de Reynolds. LMT. Internal report 149, (1993). 2. F. Chinesta, F. Olmos, A. Poitou and R. Torres. Mecs de los Medios Continuos. Universidad Polit6cnica de Valencia, (1996). 3. I. Daubechies. Ten Lectures on Wavelets. Number 61 in CBMS-NSF Series in Applied Mathematics. SIAM, (1992) 4. C.C. Lee et C.L. Tucker Journal of Engineering Industry, vol. 106, 114 (1984).. 5. Y. Meyer, editor. Wavelets and Applications. Number 20 in Research Notes in Applied Mathematics. Springer Verlag, (1991).
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
321
L o c a l i s a t i o n of d e b i n d i n g zone for fluid-particle flows in m e t a l injection m o l d i n g M.Dutilly and J.C.Gelin Laboratoire de M6canique Appliqu6e R.Chal6at Universit6 de Franche-Comt6 24, Chemin de l'Epitaphe, 25030 Besanqon Cedex, France
Abstract One of the most difficult problem in Metal Injection Molding is phase separation effect. The mixture theory is used to solve this problem. To localize the debinding zone, momentum and mass conservation equations are solved by the finite elements method. To take into account the solid-fluid interaction, phase exchange terms are introduced in the conservation equations. A free surface tracking method is developped from volume of fluid method.
1. INTRODUCTION In many processes, a two-phase mixture of solid particles and viscous fluid is used. In Metal Injection Molding (MIM), a metallic powder is mixed with a thermoplastic binder and injected to produce metallic parts like plastics ones. The MIM process is composed of four steps. In the first one, powder and binder are mixed together to form the injection molding feedstock for the second step. Next, the organic part is removed in the debinding step. The porous media is shrinked by sintering in the final step. The injection step is very important. Indeed if a part is molded with heterogeneity, distorsion and defects could appear in subsequently steps. So, the localisation of debinding zone is very important for the success of the process. Several different approaches have been developped to simulate Fluid-Particle flows. The granular media mechanics [ 1-2] calculates the motion of each particle in the fluid. Powder characteristics, as grain size and distribution, can be taken into account. But, such an approach limits the number of particle that can be treated, therefore industrial cases are difficult to be simulated. The continuum approach try to describe the macroscopic behavior of the mixture. For dilute suspensions, several authors [3-5] use a single fluid approach in defining an effective viscosity of the mixture. For highly concentrated media, only empirical viscosities were used [6-7]. Nevertheless, these methods neglect phase separation effect upon flowability of the mixture. To solve the problem, we adopt in this paper a multiphase flow approach [8]. Each phase has its own density, velocity field and volume fraction. In order to account for interaction between two phases, a term of momentum exchange is introduced in equations of motion of each phase [9]. Debinding zones are localized by solving the mass conservation equation of
322 powder. Equations are solved by a finite elements method. The method used for the modelling of mould fillng is based upon volume of fluid method [ 10]. At each time step, the filling rate of each element is updated by calculating the flow through each side of element. The free surface is localized in any partial filled element. This method allow to solve problems with any geometry of elements (triangle, quadrilateral, cubic, tetrahedron .... ). 2. M U L T I P H A S E F L O W M O D E L
2.1. Conservation laws An Eulerian scheme is adopted in this paper. Mixture theory considers two interpenetrating continuum media and at each material point, the solid and the liquid phase are both present. Therefore the solid ~s and the fluid ~f tvolume fraction are defined as new field variables related to partial densities as Ps = ~sPs0 and pf = ~fPf0" Since for high concentration, powder flows are dominated by particle collisions, the solid phase behaves like a non Newtonian viscous fluid [11-12]. Within domain f~, conservation laws enable to write: 9mass conservation
9momentum conservation
/)--i + vs. V ps + Psdivvs = 0
(1)
OPf ~-i + Vf" V pf + pfdivvf = 0
(2)
~)vs Ps ( ~ + Vs" V Vs) = m s + Psbs + diver s ~vf pf (~-~ + vf. V Vf) -- mf + pfbf + divtJf
(3) (4)
where m i are interaction forces which result from momentum exchange between both phases. In this paper, we consider the following form for the interaction equation: mf = - m s = k (v s - vf)
(5)
where k = k (q~s' Vs' Vf) is an interaction parameter [13]. When Vsvanishes, powder is gf considered like an opened porous media and k tends to m , a well-known result of flow K: through porous media [14]. We denote ~l,f as unloaded fluid viscosity and ~: as the porous media permeability. When t~s vanishes, it results that the interaction forces are similar to the classical drag forces. As materials are considered incompressible, so equations (1) and (2) are respectively reduced to:
Or + v s
9V ~s + ~sdivvs = 0
(6)
323
~t~f ~--~ + Vf" V ~)f + q~fdivvf = 0
(7)
(8)
The mixture is always saturated resulting in ~s + ~f --" 1
By summing equantion (6) and equation (7) and using (8), one can derive the following relation" (9)
div (t~sVs + CfVf) -" 0 that can be interpreted as the global incompressibility of the mixture.
2.2. Material behavior One have to distinguish both phases in the media in order to describe the material behavior of the mixture. The behavior of the fluid phase can be represented by a linear viscous constitutive equation: af = 2gfgf where gf is the strain rate of fluid.
(10)
On the other hand, the solid phase is characterized by a non linear viscous constitutive equations"
~s = 2gs (~s' ~s) es
(11)
where ~sf is the strain rate of solid phase. The resulting mixture's behavior is non linear due to the presence of the solid phase [ 15]. Taking account to this fact, the volume fraction and strain rate dependencies are introduced directly in the powder viscosity gs 9 2.3. Boundary conditions Concerning the boundary conditions, several cases can be distinguished. One can write the following relations on surface 1-' of domain s 9 9distinctly imposed velocity on each phase: v s = Vs0 o n F u
(12)
vf = vf0
(13)
For example on a wall, a slip condition can be chosen for the solid phase and an adherence condition for the fluid one. 9distinctly hydrodynamic friction on a wall: tJts = l~s" t =--fsVs
c~ = Gf. t = -ffvf
on F t
(14)
(15)
where t is the unit tangential vector to the surface and fi friction parameters of each phases.
324 9imposed mixture flux: Q = f (r
F
+ t~fVf) dS
(1 6)
where n is the unit normal vector to the surface. In fact, this condition implies a linear relationship between degrees of freedom. 9pressure P of the mixture on a surface. P is defined further like a Lagrange multiplier for the continuity equation (9). 3. FINITE ELEMENT FORMULATION FOR MULTIPHASE FLOWS The conservation equations are discretized using a Galerkin weighted residual technique. A mixed velocity-pressure formulation or reduced integration penalty method is used to solve the incompressibility constraint (9). The following equations set is obtained (in the case of the penalty method):
I Mf
0I I Vf ] FKff + k + ""ff Kfs | lr div - k ] i Vf ] IIF Fft + div div = Ms Vs LKfs - k Kss+k+Kss J Y s
(17)
where M i is the mass matrix of phase i, Kii is the stiffness matrix of phase, k is the interaction matrix and K div is the penalty matrix for incompressibility constraint (9).
Mob +
(Kadvee + Kdiv) r = 0
(18)
where M is the mass matrix for volume fraction equations, Kadvec is the stiffness matrix for advection phenomena and Kdi v is the stiffness matrix for diffusion phenomena. An implicit backward scheme is used to discretize both systems with respect to the time. The stages to determine velocity fields and volume fraction are: 1. The filling state of domain is updated. 2. Navier-Stokes equations of the coupled system (17) are solved with current solid volume fraction. 3. The new solid velocity field is transferred to volume fraction system. 4. Advection equation of the volume fraction system (18) are solved with current solid velocity field. 5. The new solid volume fraction field is transferred to Navier-Stokes system. 6. Repeat (2) to (5) until velocities and volumic fraction stabilization. 7. Go to (1) if mold is not completely filled.
325 4. A L G O R I T H M F O R F R E E SURFACE T R A C K I N G The simulation of mould filling involves the tracking of mixture free surface. The Volume of Fluid Method is used to solve this well-known problem [ 16]. At each material point of the domain, a pseudo-concentration F is defined. F is governed by the following advection equation: /)F --+v-VF=0 Ot where v = ~sVs + (~fVf is the volume mixture velocity.
(19)
F = 1 means that free surface has already reached this point, F = 0 for other points. The pseudo-concentration field is calculated as following: 1 F i = V. [ FdV
(20)
in each element i.
IV i
F i is updated by solving the following equation: At.-. k A F i = ~ii k ~ ( Z k q i
(21)
where q~ the flow across side k into element i and etk the fractionnal area of side k over which fluid can cross side k.This parameter depends on the filling state of element i and his neighbors connected to sides k (see figure 1).
~ 1 ~ oq = 1
'
E empty element PF partially filled element
53=0 0r = r
direction of flux
F
= a
filled element
Fig 1" fractionnal area calculation
5. RESULTS AND DISCUSSIONS The first example concerns an extrusion problem where the influence of model parameters like viscosities or interaction term is studied. Numerical simulations are performed upon a profile extrusion die (figure 2a) for different values of the parameters. The results are illustrated in figure 2b and 2c.
326
debinding calculated point
L=3mm
reduction ratio = 1 / 2 gf = 1 Pa.s (10 -3) Inflow = lmm3/s Constant model for k and ILtp
Inflow
L a) Test conditions 0.8
1.1(-.
o
0.8
i t = 100 It= 50
r
tl:i
~-~ 0.9-: ._o E 0.8-
~0.7
~ .o
lap= 10
o 0.7-
-o
~
"~ 0.60.5
'''1''''1
0
5
''''
I''''1''''1'
k=lO
0.7 0.6
0.6
k=50
0.5
0
'''1''''1
10 15 20 25 30 35
. 0
time(s)
b) debinding for k = 10 Pa.s/mm 2
5
~ 10 15 20 25 30 35 time(s)
c) debinding for lap = 10 Pa.s (10 -3)
Fig 2 9Illustration of model parameters influence. From these results, one can conclude that interaction term and viscosites ratio have opposite influence upon debinding. Now these results are in agreement with experiments showing that when solid volume fraction increases the mixture viscosity increases [6] and permeability of opened porous media decreases [ 14]. Therefore, when solid volume fraction tends to critical volume of solid loading, one can predict upon debinding, a competition between these two phenomena. To illustrate the method, a casting mold is filled with a fluid-particle flow. The geometry and the mesh are represented in figure 3. inlet
Exterior radius" 5 mm Inlet flux" 1 mm3/s gl = 2 Pa.s (10 3) gp = 20 Pa.s (10 3) k = 1e 5 Pa.s/mm 2 outlet Fig 3 9Mold geometry and filling test conditions.
327
Fig 4 9filling state and solid volume fraction.
328
Fig 4 : filling state and solid volume fraction. Figure 4 shows phase separation effect during the mold filling. At beginning, particles migrate to the outer radius of the part whereas at the center, mixture has a weak loading of particles.When front reaches the outlet, the problem becomes non symetric and mixture flow is accelerated on the right side of the part. Therefore fluid flow carried more quickly solid particles accumulated on the outer right wall. 6. CONCLUSION It has been shown that the multiphase flow model is able to simulate the debinding effect during mold filling. The main difficulty arises from evaluation of powder viscosity and interaction terms. Up to now, only empirical laws can be used to simulate solid volume dependency.
329 All the results presented in this paper were obtained with the assumption of an isothermal model. A possible extension consists to write a thermal coupled model. Thermal gradient in the mold creates heterogeneities in the viscosity which can increase the problem of phase separation. Three dimensionnal extensions of the algorithms used in this paper are easy to be achieved with object oriented capabilities of EF++ code [ 17]. The main problem is the high requirement in computational memory for complex geometries problems. REFERENCES [ 1] T. Iwai, T. Aizawa and J. Kihara, Proc. of Powder Metallurgy (1994). [2] H.H. Hu, Int. J. Multiphase Flow, Vol 22, 15 (1996) 335-352. [3] A. Einstein, Ann. der Phys., 19 (1906) 289-306. [4] J. Happel, J. of Applied Phys., Vol 28, 11 (1957) 1288-1292. [5] G.K. Batchelor, J. Fluid. Mech., 52 (1972) 245-268. [6] H. Eilers, Kollo~'d Zim., 97 (1941) 313-321. [7] J.S. Chong and E.B. Christianensen, J.App. Polymer Sci., 15 (1971) 2007-2021. [8] R.M. Bowen, R.A. Grot and G.A. Maugin, C.Eringen Ed., Academic Press (1976). [9]A. Poitou, PhD Thesis in Materials Engineering, ENSMP, France (1988). [ 10]C.W. Hirt and B.D. Nichols, J. Computational Phys., 39 (1981) 201-225. [11] P.K. Haff, J. Fluid Mech., 134 (1983) 401-430. [12] A. Khelil and J.C. Roth, European J. Mech. B/Fluid, 13 (1994) 57-72. [13] J.W. Nunziato, in theory of dispersed multiphase flow R.E. Meyer Ed Academic Press (1983) 191-203. [ 14] A.E. Scheidegger, Encyclopedia of Physics vol 9 (Springer, Berlin, 1960). [15] D. Bialo and Z.Ludynski, in Proc of Powder Metallurgy World Congress, Paris 6-9 Juin, (1994) 1121-1124. [16] J.M. Floryan and H.Rasmussen, Appl. Mech. Rev, vo142 n~ (1989) 323-341. [ 17] L. Walterthum, PhD Thesis, Franche-Comte University (France), 1996.
This Page Intentionally Left Blank
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
331
Simplified approaches for the prediction of deep-drawing ears P. Gilormini a and B. Bacroix b aLaboratoire de M6canique et Technologie, ENS de Cachan/CNRS/Universit6 Paris 6, 61 av. Pr6sident Wilson, 94230 Cachan, France bLaboratoire des Propri6t6s M6caniques et Thermodynamiques des Mat6riaux, CNRS/Universit6 Paris-Nord, av. J.-B. C16ment, 93430 Villetaneuse, France 1. I N T R O D U C T I O N
The deep-drawing of cups is a very common forming process, where a circular flange is pushed by a punch through a die to give a cylindrical cup. As an example, this is the first step in the manufacturing of the billions of beverage cans produced annually from aluminum alloys or steel. One defect t h a t may appear in this process is a wavy profile of the rim of the cup, with troughs alternating with peaks (or 'ears'), as illustrated in Figure 1.
Figure 1. An example of deep-drawing ears obtained on a steel cup. It has been established clearly that this phenomenon is mainly due to the in-plane anisotropy of the flange, which is closely connected with the crystallo-graphic texture of the material, as illustrated in Figure 2 (see also the papers in [1], for example). Consequently, the simulation of earing in order to optimize the texture and reduce this defect usually combines two approaches: (i) a more or less simplified stress and strain analysis of the process, and (ii) a model for relating the sheet mechanical behaviour to its texture. The present paper describes some of these analyses.
332 'copper' texture
'cube' texture
RD
RD
RD
(a)
(b)
(c)
Figure 2. Cups deep-drawn from copper flanges with crystallographic textures (sketched <111> pole figures above) induced by cold rolling (a) and subsequent partial (b) or hill (c) recrystallization. RD: rolling direction. 2. ANALYTICAL AND SEMI-ANALYTICAL A P P R O A C H E S The first attempt to predict the positions of deep-drawing ears is due to Hill [2] in 1948. The quadratic yield criterion was used in reference [2], but further developments of the initial approach were proposed in Hill's book [3] using a nonquadratic yield function. If it is assumed that the final positions of the ears are already determined at the beginning of the process, when the blank starts being drawn, they can be deduced from an analysis of the velocity field in a circular blank. Because of the initial orthotropy of the material (cut from a rolled sheet), ears or hollows, if any, will appear at least along the initial rolling and transverse directions, for s y m m e t r y reasons. Near these points of the blank edge, the (planar) velocity field is symmetric with respect to the radial direction, i.e. the radial velocity passes an extremum and the tangent velocity component is zero. More generally, the basic assumptions of the analysis are that (i) extremas of the radial velocity and zeros of the tangent component occur simultaneously around the edge, and (ii)this still applies at neighbouring points along the radius. Then, ~vr/~0 = 0 and ve = 0, in addition to ~v o/~r = 0, give e r0 = 0. Under plane stress conditions, i.e. if the stress applied by the blank-holder can be neglected, the stress state along the edge writes as ~ = ~ i | i , where i is a unit vector tangent to the edge. Neglecting the elastic part of the strain rate and applying the
333 normality rule to the yield criterion O((~) = ~o, where ~ is a homogeneous function of degree one, the following condition is obtained: (1) which states t h a t the component of e.t along the radial direction (i.e. ~r0 ) is zero. Scalar product with [ and use of the yield condition (~:(OO/O(~)=(~((~)= = (~O({ |
(~o give TI/~. = (~o/(~, and (1) can then be written as
~] .? = a o [ ~ I~| ~" G
(2)
In other words, the tangent vector t where an ear or a valley forms is an eigenvector of 0O/O(~li | ~ ; t h e corresponding eigenvalue equals (~o/(~ and gives the stress state at this point. As noted by Hill from a different derivation, this problem is equivalent to finding the tangent directions where (~ (or, equivalently, (~o/(~ = O(t | {)) is extremal, because the partial derivatives of |
+
(3)
where T can be a non-unit vector and A is a Lagrange multiplier, are simultaneously zero for a unit T and for (2) with A = Co/(~. The above analysis applies to quite general yield criteria, but the quadratic Hill function leads to simple results. It writes as ~((~) = ~/(~:H" (~ or, more explicitly,
r
~F ((J'22--(~33 )2 "~-"-2" G )2 H )2 ((J'33--O"11 "~"-2 ((~11 -- (J'22 "~"L(~2a2 + M(~3~2 + N(~2 2
where axes 1 and 2 are parallel to the rolling and transverse respectively, Then, equation (2) takes the following form
i
(4)
directions,
(5)
which is reminiscent of the classical Christoffel equation for finding the directions along which plane longitudinal waves can propagate in elastic solids. More specifically, the wave direction would be t and the constants of the orthotropic elastic medium would be Cll = H + G , c22 = F + H , c33 = G + F , c23 = - F , c31 = - G , c~2 = - H , c44 = L / 2 , css = 114/2, c6s = N / 2 . As shown by Brugger [4], at most one solution between 0 and rd2 can be obtained in the (1,2) plane, inclined at an angle (z given by
334
a = t a n - l l c 1 1 - 2 c 6 6 - c 1 9 = tan-~ ~ G + 2 H - N c~2 - 2c66 -c19 F + 2H- N
(6)
In these conditions, ears or troughs inclined at rd2-a to the rolling direction form in addition to those at 0 and rd2, i.e. 4 ears (and 4 hollows) appear. If N is between G + 2 H and F + 2 H , only 2 ears form, a result obtained by Hill in a different way. As a consequence, the 6- or 8-ear profiles which have been observed sometimes cannot be obtained, and this has stimulated the development of non-quadratic yield functions. Moreover, the above analysis, even applied to a general yield criterion related in some way to the crystallographic texture, cannot predict the amplitude of earing after complete drawing and cannot decide between an ear and a hollow without additional assumptions. Part of these shortcomings are avoided by the approach of Panchanadeeswaran et al. [5], which is not limited to incipient flow. Neglecting the tangent velocity around the edge gives ~rO
coo
_
_
1 OVr 2v r ~)0
r~,
-
-
,1
lOln[a[O,t)] 2 ~0
(7)
Then, assuming the above ratio remains equal to the ratio F of the shear strain rate to the longitudinal strain rate measured when a uniaxial tensile test is performed on a specimen cut at an angle 0+ n/2 to the rolling direction, the following expression is obtained for the flange radius and incomplete drawing: R ( O , t ) = R o - [ R o - R ( 0 , t ) ] expI2y: F(0 + n/2) d0 1
(8)
This approach predicts only the difference between the radius at an angle O and R(0), the radius along the rolling direction, but the latter is related to the drawing conditions (tools geometry, etc.) in a complex manner beyond such a simple analysis. Ears and troughs are predicted for ~re -- 0 (like with Hill's approach), i.e. at angles a such that F is zero at rd2-a (note that, for symmetry reasons, at least F(0) and F(rd2) are 0 for an orthotropic rolled sheet). The F(0) function, used as a basic data, can be deduced from a yield criterion, from a polycrystal model, or from a series of tensile tests, similarly to the (~(0) curve required by Hill's analysis. A more quantitative prediction of earing requires taking into account the drawing conditions, but this is hardly compatible with analytical calculations. As an example, Kanetake and co-workers ([6], for instance) have proposed such a model based on a simplified analysis of the process and using the initial crystallographic texture of the blank, which can be summarized as follows: (i) First, the average height of the cup is deduced by assuming that its volume and thickness are equal to the volume and thickness of the initial blank. (ii) Then, the angular variation of the cup height is deduced from an evaluation of the average strain of each initially radial line. This uses the texture data for weighting the combination of slips induced in each crystal by the same applied stress (with (~rr and (Sos as only non-zero, uniform and opposite, components). (iii) Finally, the applied stress can be eliminated by equating the average height of the cup calculated above to the value deduced from the strain of all the radial
335 lines, and only two non-dimensional parameters remain for characterising the hardening law of the slip systems. Kanetake and co-workers recommend a set of values which, apparently, can be used without change for various metals (aluminum, copper, steels) and leads to the correct number, positions, height and width of ears when the drawing conditions (punch and die radii) are varied. This approach involves non analytical calculations, although it makes very crude approximations including no texture change and a uniform-stress polycrystal model. 3. S I M P L I F I E D F I N I T E E L E M E N T A N A L Y S E S
The process of deep-drawing can also be simulated in a very complete manner by using the finite element method and the rapidly increasing possibilities of modern computers. The whole set of tools can be considered (punch, die, and blank-holder, see Figure 3 where a flange with quadratic symmetry is considered and Hill's criterion was used), friction can be included (with different values on the various interfaces), as well as unilateral contact conditions. The present limit is in the description of the mechanical behaviour: taking into account the crystallographic texture explicitly by running a polycrystal model with hundreds of grains at each integration point leads to very large computation time. For this reason, phenomenological yield criteria are used in most studies, and they are usually fitted to the initial sheet texture, assuming there is no significant texture change in the process.
punch ,
blank-holder
-- -~ /
die , ~
eighth oftheblank
:
Figure 3. A finite element simulation with the complete set of tools and a blank with a quadratic symmetry.
336 Such complete finite element calculations were carried out by Chung and Shah [7], Becker et al. [8], Yoon et al. [9], Zhou et al. [10], among others. A complete drawing of the flange could not be obtained by Zhou et al. [10], and this may be due to the increasingly high normal stress applied by the blank-holder on the outer elements when most of the blank has been drawn. To avoid this problem, Chung and Shah [7] apparently allowed the blank-holder to move upwards only, and Becker et al. [8] fixed the blank-holder position when a single element was leR to draw. These simulations suggested a small influence of friction on the cup profile, and this may be due to an efficient lubrication and to the small magnitude of the blank-holder force (just sufficient to avoid wrinkling). Moreover, the deformation of the sheet below the punch was found very small. Finally, the observed ear profile could not be reproduced very precisely in these calculations, and this was usually ascribed to an incomplete description of the mechanical behaviour of the sheet. Considering the amount of computations involved in such simulations, it seems reasonable to suggest intermediate approaches that would keep both the essential features of the drawing process and the power and flexibility of the finite element method. Two types of such simplified finite element approaches will be reported here, proposed by Gotoh and Ishis6 [11] and by Bacroix and Gilormini [12].
Figure 4. Simulation of the flow below the blank-holder. Initial mesh (a), and deformed mesh after 2 rows of elements have been pulled and removed (b). The long computation times are partly due to the contact between the blank and the tools: the upper surface of the blank leaves the blank-holder and then m a y contact the punch; its lower surface looses contact with the die. A second reason is the three-dimensional character (loss of axial symmetry) of the process when earing occurs. A possible simplification is then to limit the analysis to the flow in the initial plane of the sheet as shown in Figure 4. The initial mesh has a central hole corresponding to the punch trajectory, and a radial displacement towards the center is prescribed to the inner row of nodes. When a row of elements has been pulled over the central hole, it is removed from the mesh and the procedure is applied to the next row, as illustrated in Figure 4 obtained with the ABAQUS [13] finite element code and Hill's quadratic yield criterion. This method, proposed
337 initially by Gotoh and Ishis6 [11] who used a specific code, has been employed also by Yang and Kim [14] and by Becker et al. [8]. If the latter authors employed three-dimensional elements in some of their simulations, this simplified approach is usually applied with two-dimensional (plane stress) elements, leading to shorter computation times. The validity of the method is confirmed by Becker et al. [8], who compared this simplified approach with a complete finite element simulation of the process and obtained a good agreement for the flange profile under the blank-holder. It was also also reported that the positions of the ears were slightly changing during the process in the strongly anisotropic case of a single crystal sheet, which suggests limiting the validity of Hill's approach to moderate anisotropies. This simplified approach has been applied as a first test of the possibilities and numerical implementation of various yield criteria (2nd, 4th, and 8th order for Yang and Kim [14], Gotoh and Ishis6 [11], and Becker et al. [8], respectively) and of single- or polycrystal models (Becker et al. [8]). Unfortunately, it cannot predict the punch force and the cup profile after complete drawing, and this is why Bacroix and Gilormini [12] proposed another simplified approach recently. Like above, the initial mesh has a central hole with a radius equal to the punch radius, as shown in Figure 5, but it does not remain plane. More precisely, a uniform vertical displacement is applied to the nodes of the inner rim, and the other nodes are constrained to keep contact with the die. The latter condition allows for neglecting the blank-holder, and the punch (under which deformation is neglected) is not explicitly considered as well. By contrast, the die profile radius is taken into account and this three-dimensional simplified finite element approach, where friction is neglected, authorises complete drawing and punch force calculation.
die
Figure 5. Simplified simulation of deep-drawing. Initial mesh (lei~), and deformed mesh (right) after incomplete drawing. Various types of elements were considered in this approach, using the ABAQUS [13] finite element code, and Table 1 compares some of the results. As in Kim and Yang [14], the aluminum-killed steel obeys a Hill quadratic criterion,
338
the flange has a radius of 49 mm and a thickness of 0.8 mm, and the die radius is 20 mm. Moreover, a profile radius of 5 mm was taken. For a single layer of threedimensional elements (lines 1 to 4), it can be observed that low-order elements with reduced integration (line 2) should be avoided because very small ears are obtained (probably because of the hourglass control). Moreover, using sophisticated elements with "incompatible modes" (line 3) or a higher order (line 4) essentially leads to longer computations. It was also found that using a hybrid formulation did not bring any significant change, and that the high-order elements with full integration lead to an excessively elongated cup. The two-dimensional elements (line 5 to 8) lead to a smaller punch force because of bending effects (and this is, of course more evident with membrane elements). The fastest simulation is naturally run with low-order membrane elements (line 5), and reducing integration (line 6) or increasing the order (line 7) increases computation time and gives smaller cups with more earing (but combining both modifications of the m e m b r a n e elements leads to divergence). A similar trend was obtained with shell elements (line 8), for which increasing from 3 to 5 integration points through the thickness did not bring any significant change. Table 1 Comparison of various elements in the simplified approach of [12] Element type (ABAQUS)
CPU time (normalized)
Total force (N) (30 mm draw)
Hmin (mm)
AH]Hminxl00
C3D8
2.5
11,440
50.8
24.6
C3D8R
1.5
11,670
53.7
2.4
C3D8I
6.1
10,390
48.2
25.1
C3D27R
58.3
10,310
47.9
26.6
M3D4
1
7,280
40.0
26.6
M3D4R
1.3
6,900
38.4
29.6
M3D9
9.4
7,280
38.8
28.8
S4RF
2,0
8,700
42.1
25.0
This simplified approach has been applied by Bacroix and Gflormini [12] to study the influence of using a 4th-order strain-rate potential taking the crystallographic texture into account explicitly (and which may lead to 6 ears, as shown in [12]), rather than the quadratic Hill criterion. The method will not be detailed here, but differs from the one proposed by Yoon et al. [9]. Figure 6 ,shows the flange profiles that were obtained with the simplified approach (membrane elements) for the incomplete drawing of an aluminum alloy blank, and experimental measures are indicated. The flange radius and thickness were 165.1 m m and 1.58 mm, respectively, the die radius was 76.2 mm with a profile radius of 9 mm, and drawing was interrupted after a punch travel equal to the die radius. Because the blank was not perfectly centered on the die initially (a similar problem
339 was reported by Chung and Shah [7]), the induced profile did not exhibit the expected s y m m e t r y and the values in Figure 6 are averages taken over sets of equivalent points. Note also that a part of the cup had been cut for texture analyses before the profile could be measured, which explains the incomplete set of values in Figure 6. Despite these limitations, Figure 6 indicates that the simplified approach leads to a reasonable profile. Incidentally, it also suggests that the texture of the material considered does not lead to a very significant difference between the two types of descriptions of the plastic behaviour. This illustrates a typical use of simplified approaches for the prediction of deep-drawing ears: rapidly testing models which relate the crystallographic texture to the mechanical behaviour.
. . . . . . . . ~.2... ~'~ ,,
/
- 4th order
I
-
Hill
..
\
9~
o 118
/
\"
"
- -t
"/ \
114
110
'
0
'
'
'
'
'
'
20 40 60 angle from rolling direction (deg.)
'
80
Figure 6. Flange profiles obtained for an incomplete drawing using the simplified approach of [12] with the quadratic Hill criterion and with a 4th-order strain-rate potential. Comparison with experimental measures.
R E F E R E N C E S
1. H.D. Merchant and J.G. Morris (Editors), Textures in Non-Ferrous Metals and Alloys, The Metallurgical Society (1985). 2. R. Hill, Proc. Roy. Lond., A193 (1948) 281. 3. R. Hill, The Mathematical Theory of Plasticity, Oxford University Press (1950). 4. K. Brugger, J. Appl. Phys., 36 (1965) 759. 5. S. Panchanadeeswaran, O. Richmond, W.G. Fricke and L.A. Lalli, in 8th International Conference on Textures of Materials, J.S. Kallend and G. Gottstein (Editors), The Metallurgical Society (1988) 1103. 6. N. Kanetake and Y. Tosawa, Textures and Microstructures, 7 (1987) 131. 7. K. Chung and K. Shah, Int. J. Plasticity, 8 (1992) 453. 8. R. Becker, R.E. Smelser and S. Panchanadeeswaran, Modelling Simul.
340 Mater. Sci. Eng., 1 (1993) 203. 9. J.W. Yoon, I.S. Song, D.Y. Yang, K. Chung and F. Barlat, Int. J. Mech. Sci., 37 (1995) 733. 10. Y. Zhou, J.J. Jonas, L. Szab6, A. Makinde, M. Jain and S. MacEwen, Int. J. Plasticity (1997) in press. 11. M. Gotoh and F. Ishis~, Int. J. Mech. Sci., 20 (1978) 423. 12. B. Bacroix and P. Gilormini, Modelling Simul. Mater. Sci. Engng, 3 (1995) 1. 13. ABAQUS, Reference Manuals, Hibbitt, Karlsson and Sorensen Inc., Pawtucket RI (1995). 14. D.Y. Yang and Y.J. Kim, Int. J. Mech. Sci., 28 (1986) 825.
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
341
Spring back and 'rebound' phenomenon analysis with the software PLIA GE Fabrice MORESTIN a, Maurice BOIVIN", Franck BUBLEXa, Xiaoling DENGb, Mostafa EL MOUATASSIMb aLaboratoire de M6canique des Solides, b~.t. 304 - INSA de Lyon, 69621 Villeurbanne, France bRENAULT, service 60303 Boulogne Billancourt, France 1. INTRODUCTION At the International Conference Numisheet 93 in Tokyo [ 1], an experimental and numerical benchmark on the shape after spring back of a deep drawn 'U' has been presented in order to check the validity of several F.E.M. softwares. The results were very sparse, so numerical prediction of spring back is still relevant as high yield strength steel sheet and aluminium alloys are in growing use in automotive industry. Since 1989, a very efficient analytic software called PLIAGE, is used in RENAULT Development Department for the spring back modelling for bidimensionnal drawing (plane strain) [2]. This validated software has given really good results for the Numisheet 93's benchmark and its analytical formulation leads to very short calculation times. Nevertheless, for die radius versus thickness ratio lower than 3, PLIAGE no longer gives predictions in good agreement with experimental results. This kind of die radius will be called in the sequel: small die radius. The present study points out the pay lines of the existing modelling with a 'U' drawing tool developed for experimental measurements during the process and with an accurate numerical analysis with the ABAQUS F.E.M. code with its explicit and standard versions. Blank holder ~point (A) ml~
p~
(B)'~ R d ~
Punch. Die
Figure 1. Definition of Rdefect and other parameters For a small die radius, this mechanical assumption of tight thread does not satisfy the true plastic strain path of the material particles of the sheet. Actually, a phenomenon of 'rebound',
342 has been described by HAYASHI [3] for small die radius utilisation and was not formulated in the PLIAGE pre-processor. The corresponding defect radius is denoted Rdefect (See figure 1). This geometrical phenomenon seems to be the main source of discrepancy between predicted and experimental spring back values, so the aim of the experimental investigations presented in this paper is the measurements of the defect radius induced by the rebound phenomenon for different gap-thickness ratio and for two punch shapes, flat headed punch and half-cylindrical punch. The experimental value of this radius is introduced in PLIAGE and the numerical predictions obtained for the spring back are in good agreement with the actual values. This experimental study has been completed by finite element modelling with ABAQUS F.E.M code which shows clearly the rebound phenomenon but its spring back predictions are far from the experimental values and PLIAGE prediction. Erroneous results predicted for a small die radius with PLIAGE may come from two assumptions. The first one is the plane stress state assumed in the thickness of the sheet that is no longer verified for important bending with large plastic strain, see Marciniak [4]. The second one concerns the geometrical deep drawing simulation made by the pre-processor of PLIAGE that considers the metal sheet as a tight thread between the die and the punch. It is false when small die radius are used. (See figure 1). The pre-processor of PLIAGE has to calculate the identical strain history areas during the process (See figure 2).
o
Figure 2. Example of identical strain history areas
2. EXPERIMENTS 2.1. Experimental setup The drawing tool is installed on a tensile compressive test machine SCHENCK RSA 250 with a maximum capacity of 250 kN. This machine is speed controlled from 0.1 to 250 mm per minute by a desktop computer which also saves all the data coming from force and displacement transducers. The drawing tool is composed of two half die with a variable spacing between them in order to allow different gap - thickness ratio (See figure 3). The blank holder are geometrically clamped along vertical axis (corridor effect). The punch supports a fiat head, R = 2 mm, and half-cylindrical head, R=25mm. The drawing force is measured with a precision of 10 N. The displacement of the punch is measured with a precision of 0.1 mm. The shifting of the sheet metal under the blank holder is given by a resistive transducer with a precision of 0.5 mm.
343
Figure 3. Experimental setup 2.2. Description of one whole test The blank is cleaned with alcohol and marked with a number. This number will be the generic name with different extensions for the storage of several data. After that, the gap - thickness ratio is adjusted : the punch is bring down between the two half die which are movable. They are pushed against the punch. The two half die are separated from the punch with two defined thickness blocks. These blocks allow to get accurately the desired gap - thickness ratio. Tree draw parts are made for each configuration in order to measure the residual radius of the side wall. The subsequent specifications are made chronologically for each draw blank : 1 - the blank is cleaned again on its two faces and it is placed between the blank holder and the die plate. 2 the blank holders are laid on the metal sheet and the superior nuts are screw down without locking. There is no pressure on the blank holder. Then; the inferior nuts are screw up. With this method, there is a little gap between the blank holder and the blank which can freely move. 3 - the swallowing sensor is hold on. During the approach stage, the punch goes down at 240 mm/mn. At 5 mm from the blank, its speed is decreased until 10 mm/mn and follows until it touch the blank. At this moment the punch is stopped and displacement is set to zero. After deep drawing, the draw part is extract manually carefully in order to preserve spring back. 2.3. Profile d e t e r m i n a t i o n and residual radius m e a s u r e m e n t
The main goal of the measurements is to determine the residual radius of the side wall, a identical strain history area which is bent to Rd and then to Rw (See figure 2). It is the only intrinsic parameter which can quantify spring back [2]. First the profile is digitalized with a GT 600 Epson scanner in order to obtain numeric pictures. Second, the residual radius is measured with the help of the so,ware GANDALF. The symmetry of the draw is verified.
Resid radius~
Figure 4. Definition of the main residual radius
344
2.4. Estimation of Rdefect (parasitic radius) In order to estimate the defect radius the walls of the die and the punch are painted with a indelible pencil marker. The thin layer is pull away during the friction with the blank and indicates the areas where the blank have touch the punch and the die. The different friction areas are described on the figure 5.
Figure 5. Friction areas after deep drawing. For each test, the dimensions A and B are measured. A mean is done for tree draw part and they are used with a CAD software to determine the defect radius by interpolation. (figure 1).
2.5. Experimental results All results are collect in tables 1 and 2. The defect radius is only estimated for the halfcylindrical punch. Every results are given in millimetres. Table 1 Results fo r the half-cylindrical punch
Gap thickness ratio
point (A)
point (B)
Estimated Defect Radius
1.2 1.4 1.6
104.0 + 0.5 103.8 + 0.5 103.4 + 0.5
92.0 + 1.0 85.7 + 1.0 80.0 • 1.0
230.0 + 30'.0 - 280.0 + 20.0 - 330.0 • 30.0
Residual radius 2636.0 + 5 . 0 934.0 + 5.0 571.0 + 5.0
-
Table 2 Results for the flat headed punch
Gap thickness ratio
point (A)
point (B)
Residual radius
1.2 103.5 + 0.5 90.3 + 1.0 ' 1.4 103.0+0.5 82.7 + 1.0 1.6 103.3 + 0.5 (*) (*) There is no friction marks on the die for a gap - thickness ration equal to
....... ,, ,, ,,
,,
1378.0 + 5.0 473.0 • 5.0 370.0 + 5.0 1.6.
345
3. NUMERICAL SIMULATIONS
3.1. Materials data True stress (MPa) The used steel is a XES steel provided by SOLLAC. The main mechanical parameters are : - Yield stress = 153 MPa - Poisson ratio = 0.3 - Young modulus = 202000 MPa - C = 1195 - ~ =6.8 C and ~ are the parameters which define the hardening.
kinematic
material's
work
400 350 300 250 200 150 100 50 0 0
20
10
30
40
Natural strain (%) Figure 6. True stress - natural strain curve for XES steel.
3.2. Results given by PLIAGE The entrance data for PLIAGE is done with the knowledge of the strain path for each part of the blank. The first case of figure 7 presents the strain path of the blank's area which occurs for the wall area without parasitic radius. This strain path is automatically determined with the help of the pre-processor of PLIAGE. So as to take into account the experimental rebound, a over bending radius, Rdefect, is introduced after the bending on the radius die. First case, the blank fold a straight path like a tight thread between die and punch.
Second case, the blank follows a over bending which modifies its strain path
oo
I oo Rd.
Rd <==>
Rdefect
oo
oo
Figure 7. Different strain path introduced in PLIAGE. The tensile stress in the blank is given by : cy = Effp/2.S, with 'Effp' the punch load measured during the drawing. Table 3 Results given by PLIAGE Gap Residual radius thickness without Rdefect ratio 1.2 200.0 1.4 200.0 1.6 200.0
Deviation from experiments
Residual radius with Rdefect
Deviation from experiments
1218 % 367 % 185 %
1871.0 1146.0 772.0
41% 18 % 26 %
346 The differences between PLIAGE and experimental results can be minimised by considering adapted defect radius while being in the range of uncertainty. Table 4 Results given by PLIAGE with adapted
Gap thickness ratio
Rdefect
Adapted defect radius for PLIAGE
Experimental defect radius
Deviation from experiments
-220.0 -290.0 -350.0
- 230.0 + 30.0 - 280.0 + 20.0 - 330.0 + 30.0
4.3 % 3.6 % 6.0 %
1.2 1.4 1.6
3.3. Results obtained with ABAQUS standard The chosen elements are CPE4R : four nodes bilinear elements used in plane strain. The mesh is made with 400 elements along the blank's length and 2 elements along the thickness. The tools are modelized with rigid surfaces. Gap elements are disposed on the blank where friction will occur. The calculation are made with two separated stages : 1 - The punch is going down for 105 mm. Drawing stage. 2 -.Spring back calculation. Every calculus use a coefficient of friction equal to zero but a tensile load is applied at the extremity of the blank. Table 5 Results given by ABAQUS standard
Gap thickness ratio 1.2 1.4 1.6
ABQ/standard 400 x 2 elements
Deviation from experiments
ABQ/standard 600 x 4 elements
Deviation from experiments
310 300 285
750 % 211% 100 %
606 341 290
335 % 174 % 97 %
The residual radius is the more important that the mesh is coarse. It is depend on the element size. Theses results are not in good agreement with those presented in the reference [5] about the tests performed for the international Conference NUMISHEET'93.
D
.... A
Before springback A : El. 1. = 0.5 mm B : El. 1. = 1.0 mm C : El. 1. = 2.0 mm D : El. 1. = 3.0 mm
Figure 8. Figure issued from [5].
3.4. Results obtained with ABAQUS explicit Simulations are performed with the explicit version of ABAQUS with the same data as the previous simulation with standard version. A mesh with 600 x 4 elements is used. The coefficient of friction used is equal to 0.2.
347
Table 6 Results ~iven by ABAQUSexplicit
Gap thickness ratio
Residual radius ABQ/Explicit ~=0.2
Deviation from experiments
1.2 1.4 1.6
8126.0 4500.0 2742.0
208 % 381 % 380 %
4. COMPARISON BETWEEN EXPERIMENTAL AND NUMERICAL RESULTS The comparison of the residual radius between the two ABAQUS versions shows two spring back shapes which are opposed 9 one is a spring back, the other is a spring go. And unfortunately theses two results are very far from the experimental one. Table 7 Comparison between ABAQUS versions
Residual radius
Residual radius
ABQ/lmplieit p=0 400 x 2 elements
ABQ/lmplieit ltt=0 ABQ/Explieit p=0 600 x 4 elements 400 x 2 elements
Residual radius
Experimental residual radius
G/t ratio = 1.2
310.0
606.0
- 456.0
2636.0
The results given by ABAQUS are not agreed with the experiment but the calculated residual radius decreases when the gap - thickness ratio increases as the experimental results. The factors which can influence the residual radius are : - the size of elements if the blank. - the work hardening law (isotropic or kinematic). - the dynamic effects for explicit codes (drawing speed). The figure 9 summarise all the results obtained in this study. 8000:
Residual radius (mm)
--*-Experimental residual radius
7000
--*-Residual radius given by PLIAGE without Rdeffect
6000
-,,-Residual radius given by PLIAGE with Rdeffect
5000
-a-Residual radius given by ABAQUS/standard with a coefficient of friction equal to 0 --*--Residual radius given by ABAQUS/explicit with a coefficient of friction equal to 0.2
4000 3000 2000~ 1000t__
,1
i.4
0 t.2 Figure 9. Summary of all results
i.6 Gap - thickness ratio
348
5. CONCLUSION This study confirms that PLIAGE gives bad spring back results for die radius on thickness ratio lower than 3, bad results which are due to the rebound phenomenon produced by the parasitic radius Rdefect. Errors from experiment is not due to the plane stress state assumption. However, PLIAGE can be used for die radius on thickness ratio lower than 3 by adding the right value of Rdefect. This is not easy because this radius depends on the material, the gap thickness ratio, the geometry of the tools, .... A solution consists in using ABAQUS like a preprocessor for the analytical sottware PLIAGE in order to obtain the fight identical strain area. This procedure has been used to calculate the residual radius of a blank introduced in a drawbead simulator and it has give very good results.
REFERENCES [ 1] MORESTIN F., BRUNET M. 1993 - "2D Simulations, B-Sim6" Proceedings of the second international conference NUMISHEET93. Tokyo (Japon), p 623-659. [2] MORESTIN F. 1993 - "Contribution ~ l'6tude du retour 61astique lors de la mise en forme.des produits plats" M6moire de Th6se de M6canique. INSA de Lyon, 180 p. [3] HAYASHI, Yutaka. Control of Side Wall Curl in Draw-Bending of High Strength Steel Sheets. Advanced Technology of Plasticity, 1984, Vol. 1,p. 735-740 [4] MARCINIAK Z. and DUNCAN J. 1992 - "Mechanics of sheet metal forming" Edward Arnold ISBN 0-340-56405-9 - London [5] MATTIASSON, Kjell; THILDERKVIST, Per; STRANGE, Anders; SAMUELSSON,
Alf. Simulation of springback in sheet metal forming. Simulation of materials : theory, methods and applications NUMIFORM95. Rotterdam. Pays-Bas. 1995. p. 115-124
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 91997Elsevier Science B.V. All rights reserved.
349
Prediction of elastic springback deflects in sheet stamping processes using finite element m e t h o d s E. Ofiate I and J. Rojek 2 1 International Center for Numerical Methods in Engineering (CIMNE) Edificio C1, Gran Capits s/n, 08034 Barcelona, Spain 2 Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, Poland Summary
Undesiderable spring-back induced deformations are one of the causes for rejection of many stamped sheet parts. The paper presents the alternatives for predicting springback deflects in the context of the most popular finite element (FE) formulations based on rigid-plastic and elasto-plastic material models, quasistatic (implicit) and explicit dynamic algorithms. Examples showing the possibilities and efficiency of the different FE procedures are also given. 1. I N T R O D U C T I O N Considerable effort has been invested in the past ten years in the development of finite element (FE) methods for analysis of the deformation of metal sheets during stamping operations. A survey of recent achievements can be found in the proceedings of latest NUMISHEET [1,2], NUMIFORM [3] and COMPLAS [4] conferences. From a conceptual point of view the different FE methods available for sheet stamping analysis can be grouped into the three following classes accordingly to the basic solution methods and constitutive models chosen: 1. Quasistatic rigid-plastic/viscoplastic flow FE models (QFM) 2. Quasistatic (implicit) elasto-plastic/viscoplastic solid FE models (QSM) 3. Explicit dynamic elasto-plastic/viscoplastic solid FE models (EDM) Codes derived from models 1 and 2 are termed "implicit" as the solution for the non linear deformation process in the sheet invariably requires the solution of a series of linear system of equations for each time step. Model 3 is termed "explicit" as the time integration of the dynamic system of equations is performed via an explicit central difference scheme with a diagonal mass matrix and ,therefore, no solution of a system of equations is required.
350 Commercial FE codes based on the explicit dynamic approach have become very popular in recent years due to the low memory requirements which allows the solution of sheet stamping problems of industrial size in standard workstation and high-end PC's. A description of the basic features of QFM, QSM and EDM procedures can be found in [4-10]. The objective of this paper is to discuss the possibilities of the above three classes of FE models for predicting elastic-spring back deflects in sheet stamping operations. The main ingredients of the FE formulations are briefly described next and the possibilities for an accurate and efficient prediction of spring-back are discussed for each case. Some examples of application are also given. 2. A B O U T E L A S T I C S P R I N G - B A C K D E F F E C T S The term "spring-back" refers here to the deformation induced in a stamped metal sheet once the pressing tools have been removed. Obviously, spring-back is an undesiderable effect as it can lead to distorted final shapes of the stamped part. The control of the spring-back deformation is nowadays one important priority in the design of a stamping process. The need for efficient and accurate numerical prediction of this deflect is therefore obvious in order to minimize the cost of design and production of stamping dies and sheet metal parts. The spring-back deformation is mainly produced by the "bending" energy accumulated in the sheet during the stamping process. Therefore, the effect of spring back is reduced in sheets where membrane (strectching) strains are dominant. Obviously, spring back deflects are more pronounced in open sheet parts with U-type sections subjected to deep drawing. 3.
QUASISTATIC FLOW MODEL (QFM)
This approach is typical of fluid mechanics where a fixed Eulerian frame is defined through which the material flows. The QFM has been extensively used to the analysis of bulk and sheet forming processes [2,3,4-10]. The main variables of the QFM are the velocities/1 of the sheet points. The kinematic and constitutive relationships are given by the standard expressions for an incompressible fluid = L/I
,
a = D(#)k
(I)
where k is the rate of deformation vector, L is a linear gradient operator, a is the Cauchy stress vector and D is the constitutive matrix depending on the non linear flow viscosity # only. The value of # can be readily obtained for rigid-plastic/viscoplastic materials. For the simplest von-Mises material it can be found # = ~ [5,8,11], where Oy is the von-Mises yield stress and ~ - (2/3~ij~ij)l/2. The set of equations is completed by the integral expression defining the rate of virtual work [5-9]. After standard finite element discretization a non linear system of equations can be found in the form r(a, x,
t)
= p(a, t) - f(t, x) = 0
(2)
351 where r, p and f stand for the vectors of residual forces, internal forces and external forces, respectively, a is the nodal velocity vector, x is the cartesian coordinate vector and t is the time. Vector p can be written in the flow approach as p-Ka
with
(3)
K-fvBTDBdv
where K is the stiffness matrix and B is the standard strain rate matrix [5-9]. It is interesting to note that the form of K is analogous to that of infinitessimal elasticity problems [5]. The transient solution of the non linear system (2) is attempted via standard explicit and implicit time integration schemes and Newton-Rapshon techniques [9]. Note also that the application of the flow approach is particularly straight forward if shell elements are used ass the incompressibility condition can be simply enforced in this case by consistently updating the sheet thickness at each time step making use of the plane stress condition. Obviously, the effect of frictional-contact conditions at the tool-sheet interfaces must be properly taken into account in the non linear solution process [8]. The computation of spring-back deformations in the QFM seems, in principle, an impossible task as the necessary effect of elasticity is excluded in the formulation. In a recent paper [10] the authors have suggested a simple procedure for evaluating springback effects in the QFM based in the following sequence of operations. (i) During the loading process the effect of elasticity is neglected in all elements and the flow formulation as previously described is used. (ii) Once the tools are removed all elements are assumed to behave elastically. This simply implies replacing the original constitutive matrix D(#) by that of standard elasticity D(E, v) where E is the Young's modulus and v the Poisson's ratio. (iii) Equilibrium under the initial stress field is obtained by using an iterative updated Lagrangian approach accounting for geometrically non linear effects. The simplest iteration process can be written in the form P
Aa k = _ K-1/,T[Bk]Takdv ,Iv
ak+l _ a k + Aa k ak+l __ak + D B k A a k
(4)
where a ~ are the initial stresses in the last increment during the stamping process using the flow approach, K is the elastic stiffness matrix kept constant during the iterations and B k is the standard strain matrix from small displacement theory which is updated for each iteration. The iterative process stops when the residual forces equal to - f B T a d V satisfy a prescribed norm. This process can be enhanced by scaling the initial stresses which are then applied in an incremental manner. The efficiency of this procedure is proved with an example in a next section.
352 4. Q U A S I S T A T I C S O L I D M O D E L ( Q S M ) The basic variables of the QSM are the displacements u of the sheet points and these are related to the strains e by standard non linear kinematic expressions in solid mechanics of the form e = (L + L(u))u
(5)
where L is the linear operator of (1) and L(u) is a non linear strain operator accounting for large displacement effects. The constitutive equations are based on elasto-plastic/viscoplastic laws relating the appropiate stress and strain measures in an objective manner. A description of the hyperelastic constitutive model used by the authors in recent applications of the QSM can be found in [10]. The finite element formulation of the QSM follows the standard pattern and a number of continuum and shell elements are available for this purpose. The different merits of each option are discussed in [10,12]. The resulting system of equations analogous to eq.(2) is again highly non linear due to the kinematic, constitutive and frictional contact conditions. The non linear solution is attempted via enhanced implicit time integration Newton-Raphson algorithms. Details can be found elsewhere [10]. The QSM is a natural option for predicting spring-back effects as the effect of elasticity is accounted for during the whole solution process. Thus, the overall non linear solution continues in the standard manner after removal of pressing tools, simply taking the existing stresses in the sheet as initial stress values for the solution of the first spring-back increment. The main draw-back of the QSM is the huge demands in computer memory and CPU time required for solving stamping problems of industrial size. A way out to these problems is the use of domain decomposition and parallel computing techniques [15]. Alternatively the explicit dynamic approach described in next section can be used. 5. E X P L I C I T D Y N A M I C M O D E L ( E D M ) Explicit dynamic methods have recently become very popular in the context of the solid approach, as they do not require the solution of a system of equations. The basic idea is the solution of the dynamic equilibrium equations at time t using an explicit integration scheme with a diagonal matrix. The explicit-dynamic algorithm is shown in Box 1. The advantage of this procedure is that the stiffness matrix does not need to be formed and that contact conditions are accurately modelled in a simple manner because of the requirements of small time steps. Moreover they the EDM can be easily parallelized [15]. Both continuum and shell elements have been implemented in the context of the explicit-dynamic model. The constitutive model used is based on hyperelasticity [12]. Continuum elements require much smaller time steps than shell type elements, as the time increment is inversely proportional to the thickness stiffness. Greater time steps can be used in the shell case if shell elements involving only translational degrees of freedom are used [13-15]. Once the final deformation of the sheet is obtained, the dynamic analysis can be
353
EXPLICIT DYNAMIC SOLUTION Discretized dynamic equilibrium equation Mii + p(u) = f Solution at time tn 1) Compute kinematic variables Accelerations: Velocities: Displacements:
~in = MD 1(fn _ pn) , MD -- diag M h n+l/2 = h n-1/2 + anAtn+l/2 an+l = a n _[_~n+1/2itn+l
At n+1/2 = ~(Atln + Atn+l) 2) 3) 4) 5) 6)
Compute strains and stresses e n+l, a n+l Compute internal force vector pn+l Check frictional contact conditions Compute external force vector fn+l Go back to 1) and repeat the process for the next time step Box 1. Flow chart of explicit dynamic solution for the solid approach.
continued with removed contact conditions to obtain the deformed shape after springback. The spring-back analysis is treated in this case as a problem of d a m p e d free vibrations. Introduction of an adequate structural damping an static equilibrium state is found giving the final deformed shape. Let us briefly explain the damping process. The dynamic equation to be solved can be written in the simplest 1D case as
+ 2 ~ + w2a -- 0
(6)
where a denotes a displacement value, w is the natural frequency (w 2 = k with k and m being the stiffness and mass, respectively) and ~ = 2-~c is the d a m p i n g ratio. The solution of eq.(6) can be found to be
a = Ae-~t sin(wrt + B)
(7)
where Wr = w(1 - u2)1/2 , u = ~/w and A, B are appropiate constants. Let us consider now two displacement amplitudes AO and A1 separated by a time #,
tp (>~-2r). The ratio between A0 and A1 can be found using e q . ( 7 ) a s f : ~
= e-~tP
and
~ :-t~Inf
(8)
Both the values of tp and f can be defined by the user. The value of tp can be interpreted as the time required to diminish the energy of the d a m p e d vibrations to a fraction f of the non d a m p e d value. Typically tp = 3~r and f = 0.01 was taken in our analyses.
354 This concept is simply extended to the general multidimensional case by adding a damping matrix C - 2~M to the original undamped system where ~ is given by eq.(8). Experience proves that the time period required for the spring-back analysis with the explicit dynamic code is extremely long and in many occasions exceeds the time of the stamping analysis. The reason is that the critical time step is very small as compared to the period of natural vibrations. A solution to this problem is to switch the explicit dynamic analysis of the spring-back process to either an "implicit" dynamic scheme (thus allowing for larger time steps) or to a quasistatic solver of the QSM type as explained in previous section. This alternative is currently being implemented in most commercial explicit dynamic codes. 6. E X A M P L E S
6.1 D e e p drawing of an a l u m i n i u m profile The first example is the deep drawing of an aluminium U profile (Figure 1). Experimental results for this problem are available from [1]. The material properties used are E = 7 1 G P a , u = 0,33, p = 2 7 0 0 K g / m 3 and a = 579.8(0.01658 + ~ ) 0 " 3 5 9 3 M p a . The Coulomb's friction coefficient chosen was 0.162. [Front view]
unc
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I! ,
'L I ,=
;..~.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
~"
-----,~ X
Die
I
52 ~
Blank size: 350 350<
Figure 1. Definitionof 2D drawing of an aluminium U-profile. The explicity dynamic solution was performed first for a blank holding force of 2.45KN using the code STAMPACK developed at CIMNE. Due to the prismatic geometry of the problem a 2D plane strain solution was attempted. Due to symmetry a half of the blank was modeled with 358 four node quadrilateral elements using 2 element layers through the thickness. The problem was analyzed with the actual mass density and the punch velocity changing armonically with Vmax = l O m / s . The explicit dynamic solution with a constant time increment of 5 x l O - 8 s e c , took 29 CPU min. on a single processor Silicon-Graphics R10.000 whereas the spring-back simulation took 90 min. The numerical and experimental results agree very well in this case as shown in
355
i el: Angle between 0 - X end A - B e2: Angle between A-B end E - F B
--- -~. ,~ A/__[ O~
el-
=~
A_~
"-
'
[Curvature] p: Defined by the radius of o circle through A. B and C ~X
Measuring method numerical (STAMPACK):
STAMPACK ~ , ~
81 = 112-4~
/ ~
p = 106 mm /
~ L__.______fl[
~
experiment(Numisheet '93): ~= = 74.30 p = 100.5 mm
Figure 2. Spring-back results for 2D drawing of an aluminium U-profile. Evolution of the displacements in time during the dynamic relaxation process.
Figure 2, where the evolution of the displacements at a point of the sheet during the dynamic relaxation process are also shown. The problem was analyzed next with the quasistatic flow model using 90 plane strain shell-type elements. The CPU time required in this case was 80s. whereas the springback computation took only 8s. Similar results where obtained with the quasistatic solid model. The comparison of the deformed shapes after spring-back for both the explicit dynamic and quasistatic models is shown in Figure 3. A clear conclusion of this study is the ineficiency of the explicit dynamic solution for spring-back analysis as expected. The spring-back deformation for this case is highly sensitive to the values of the blank-holding force. Clearly as this value increases membrane (stretching) effects became more dominant and the overall spring-back is reduced. This effect is clearly seen in Figure 3 where the deformed shape after spring-back for a blank-holding force of 20 KN is also shown.
6.2 Sectional analysis of two stamped parts Despite the great developments of the hardware and finite element software capabilities, it is still difficult to perform 3-D analysis for sheet parts of complex geometry and simplified 2-D models of 3-D parts are usually considered. A section of an industrial fastener (Figure 4) was analyzed assuming the plane strain state. The explicit dynamic approach was chosen in this case. The material of the sheet is a stainless steel with E = 2.1 • 105 MPa, v = 0.3, elasto-plastic model with initial yield stress a(Y) - 850MPa and isotropic linear hardening modulus H = 700MPa were assumed. The thickness of the sheet is 0.3 mm. A friction coefficient # - 0.1 was
356 taken. 200 four node quadrilateral shell elements with the boundary conditions imposing plane strain conditions were used to model the cross-section. Both the forming process and subsequent springback have been simulated. Different stages of deformation are presented in Figure 5. The deformed shape at the end of forming and the deformed shape of the profile after springback are compared in Figure 5 .
G)
Explicit dynamic
Quasistatic
b)
Figure 3. 2D draw bending - deformed shapes after spring-back. Comparison of explicity dynamic and quasistatic analysis. (a) blankholding force 2.45 KN; (b) blankholding force 20 KN. "-- 7 "~3.
I 33.5~o.,
I
~,o.~o., nJ l
j
2.5+0-5
5
....
/
I
~.2,o.,
I
,
lj
7.3_O0.Z .-
Figure 4. Forming of the industrial fastener - geometry of the part.
B
357
Figure 5. Forming of the industrial fastener - different stages of the simulation and spring-back deformation. 6.3 A n a l y s i s o f a n S - R a i l The last example is the spring-back analysis of a curved U profile (S-rail). Details of the mechanical and geometrical parameters can be found in [2] where experimental results are also available. Figure 6 shows a perspective of the finite element discretization of the sheet and tools chosen for the analysis which was performed with the explicit dynamic code STAMPACK simple three node. Shell triangles involving only translational degrees fo freedom were used for discretizing the sheet geometry [13,14,15]. Figures 7 and 8 show a perspective of the deformation of the sheet after spring-back where a slight buckling of the upper surface is detected. This deflect coincides well with experimental observations as clearly seen in Figure 8.
Figure 6. Stamping of a S-rail. Finite element discretization of sheet and tools.
358
Figure 7. S-rail. Deformed shape after spring-back.
Figure 8. S-rail. Experimental results of final shape. 7. C O N C L U D I N G
REMARKS
The paper has briefly described the possibilites of state of the art finite element procedures for capturing deflects originated by spring-back deformations during sheet stamping operations. It is clear that althoug explicit dynamic methods offer distinct advantages versus quasistatic ones in memory reduction for solving large scale industrial problems, their practical use for predicting spring-back deflects is nowadays prohibitibe. The short term future seems to lay in the coupling of explicit-dynamic and quasistatic solvers in the context of a parallel computing enviroment and surely much research in this direction will be reported in next coming years. ACKN OWLED G EMENT S The authors are grateful to Dr. P. Cendoya for providing results for the analysis of the S-rail problem. REFERENCES
1. A. Makinouchi, Nakamachi, E. Ofiate and R. Wagoner (Eds.), NUMISHEET 2d. Int. Conference on Numerical Simulation of 3-D Sheet Metal Forming Processes,
359
o
~
.
,
~
~
10.
11.
12.
13.
14.
15.
16.
Verification of Simulation with Experiment, 31 Aug.-2 Sept. 1993, Isekava, Japan. J.K. Lee, G.L. Kinzel and R.H. Wagoner (Eds.), Numerical Simulations of 319 Sheet Metal Forming Processes, NUMISHEET 96, Ohio State Univ., 1996. S.F. Shen and P. Dawson (Eds.), "Simulation of Materials Processinff', Proceedings of NUMIFORM'95, Balkema, 1995. D.R.J. Owen, E. Ofiate and E. Hinton (Eds.), "Computational Plasticity. Fundamental and Applications", Proceedings of COMPLAS V Conference, CIMNE, Barcelona, 1997. O.C Zienkiewicz, P.C. Jain and E. Ofiate, "Flow of solids during forming and extrusion. Some aspects of numerical solutions", Int. J. Solids Struct., 14, 1428, 1987. E. Ofiate and O.C. Zienkiewicz, "A viscous shell formulation for the analysis of thin sheet metal forming", Int. J. Mech. Sci., 25, pp. 305-335, 1983. E. Ofiate and C. Agelet de Saracibar, "Analysis of sheet metal forming problems using a selective bending-membrane formulation", Int. J. Num. Engng., 30, 15771593, 1990. E. Ofiate and C. Agelet de Saracibar, "Numerical modelling of sheet metal forming problems", Num. Modelling of Material Deformation Processes: Research, Developments and Applications, P.Hartley et al. (eds.), 318-354, Springer-Verlag, 1992. E. Ofiate and C. Agelet de Saracibar, "Alternatives for finite element analysis of sheet metal forming problems", Proceedings of NUMIFORM'92, pp 79-88, Valbonne, France, 14-18 September, 1992. E. Ofiate, J. Rojek and C. Garcia Garino, "NUMISTAMP: a research project for assessment of finite-dement models for stamping processes", J. of Materials Proc. Tech., 50, 17-38, 1995. C. Agelet de Saracibar, and E. Ofiate, "Plasticity for porous metals", Proceedings of the 2nd International Conference on Computational Plasticity, D.R.J Owen, E. Hinton and E. Ofiate (eds.), Pineridge Press, pp. 159-?, 1989. C. Garcia Garino and J. Oliver, "Simulation of a sheet metal forming processes using a frictional finite strain elastoplastic model", in Ch. Hirsch et al. (Eds), Numerical Methods in Engineering 92, Elsevier, Amsterdam, 1992. E. Ofiate, P. Cendoya, J. Rojek and J. Miquel, "A simple thin shell triangle with translational degrees of freedom for sheet stamping analysis", in 3rd Int. Conference on Numerical Simulation of 3D Sheet Forming Processes, NUMISHEET'96, Dearbon, Michigan, USA, 29 Sept.-3 October, 1996. E. Ofiate, P. Cendoya, J. Rojek and J. Miquel, "Non linear explicit dynamic analysis of shell structures using a simple triangle with translational degrees of freedom only", in Int. Conference on Computational Engineering Science (ICES'97), San Josd, Costa Rica, May 4-9, 1997. E. Ofiate, "Possibilities of parallel computing in the finite element analysis of industrial forming processes", 2nd Int. Meeting on Vector and Parallel Processing, VECPAR'96, Porto, Portugal, September 1996. G. Duffet, L. Neamtu, E. Ofiate, J. Rojek and F. Zarate, "Parallel explicit dynamic analysis of sheet stamping processes", in Computational Plasticity, R. Owen, E. Ofiate and E. Hinton (Eds.), CIMNE, Barcelona, 1997.
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Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
361
Creep Deformation in Heat Treated Components* T. C. T szeng and W. T. Wu Scientific Forming Technologies Corporation Columbus, Ohio 43202
Unacceptable microstructure, properties, residual stresses or dimensional accuracy are the common defects in heat treated components. The ability to predict the occurrence of these defects relies upon the accurate calculation of the complex thermal, mechanical and metallurgical changes in response to the volatile heating and quenching cycles in the processes. In this paper, we report our implementation of the mathematical model of creep in finite element code using the concept of effective time. A new formulation is derived in the framework of return mapping technique for the general thermo-elastoplastic with creep deformation. The consistent tangent stiffness is derived and implemented as well. We apply the computer program to the solution treatment (heating, soaking and quenching) of an Inconel 718 component. The calculated results are analyzed with reference to the prediction of components distortion after heat treating.
I. INTRODUCTION Heat treatment is an important stage of the manufacturing processes of today's structural components. In this stage, the components are usually given a solution treatment followed by rapid quench, and, for age-hardenable materials, a subsequent aging treatment to improve the physical and mechanical properties. A good process design is to achieve the desirable combination of microstructure, properties, residual stresses and dimensional accuracy in the final product through cycles of heat treating and machining. In many cases, distortion is the most troublesome issue the process developers are facing. The ability to predict the distortion in a heat treated component relies upon the accurate calculation of the complex thermal, mechanical and metallurgical changes in response to the volatile heating and quenching cycles in the processes. Indeed, heat treatment processes involve relatively complex interactions between metallurgical and thermal-mechanical phenomena. Some sophisticated researchers in the industry's bigger research and development centers are using the FEM programs for predicting the thermal-mechanical response of the workpiece in heat treatment processes [ 1-5]. These efforts have proved to be very helpful in improving the process design. However, the success of the process model seems to be a matter of chance; very often, the * This study is partially funded by a US Air Force/Navy SBIR Award (Contract # F33615-95C-5238).
362 predictions do not agree in a reasonable range with the experimental data. Studies have been taken to identify the critical factor(s) which dictate the accuracy of the predictions. The material constitutive behavior is certainly one of the factors one has to incorporate in the study. Because of the sustaining high temperatures in any heat treating processes, creep is always one of the important deformation mechanisms to be included in the constitutive behavior. However, creep is also the deformation ot~en being neglected in the modeling of heat treated components. The reason may have been related to the belief that the time-dependent creep deformation may be neglected altogether in the short period of quenching. In this paper, we report our implementation of the mathematical model of creep in finite dement code DEFORM. This work is the extension of our earlier work for modeling the heat treating processes for superalloys [3,5,6]. Among the various hardening rules for describing the creep curves at changing stress, we use the concept of effective time discussed in Whirley and Henshall [7]. A new formulation is derived in the framework of return mapping technique for the general thermo-elastoplastic with creep deformation. The consistent tangent stiffness is derived and implemented as well. To some extend, our approach is similar to that of Duxbury et al. [8]. However, due to the different hardening rule we use, the resulting formulations are very different. The calculated results of a sample problem involving uniaxial loading are compared with test results from the literature. We also apply the computer program to the solution treatment (heating, soaking and quenching) of an Inconel 718 component. The calculated results are analyzed with reference to the prediction of components distortion and stress state a~er heat treating.
2. THERMO-ELASTOPLASTIC WITH CREEP DEFORMATION MODEL 2.1. Radial return technique The deformation kinetics is adopted from the theory of large deformation [9], details of the theory will not be elaborated here. The incremental strain is decomposed in the following manner,
ar~-ar~ ~ +ar~ ~ + a ~ r + d ~ c
(1)
where the superscripts E, P, T and C represent the elastic, plastic, thermal and creep strains, respectively. By using the relation: ~ - ~ - ~ r _ ~c _ ~ - 1 ~ , the stress increment d~ is written as d~ - (~(arg - arg e - dg r - d'g c ) + g d T
(2)
where C" is the elastic tensor at temperature T, and ~ = dCc~-. By using the flow rule, the
dT
plastic strain increment and creep strain increment are: a r g p - Y d 2 and arg c - Y@, respectively. In these relations, Y is the stress deviator. It is further shown that the proportionality parameters are
363
d2-~
3arg c
3d~ p
and d y - ~ 2~
2~
(3)
In the context of radial return technique, the stress state used in (5) is the final solution state for the current time increment (state F). A trial stress, st, has the following form, gr - s p + 2 p ( d ~ - d e r ) - sF + 2 # ( d~e + d ' g C ) - [ 1 + 2p(d;L + dT')]~'R
(4)
where la is the shear modulus. By definition, ~z ~'
m2
2 .~ _
crr [1 + 2kt(dA + dr)] 2
3 sF s~
(5)
Therefore, m
O" T m
(6)
dA, + d 7 - cry
2p Again, by definition,
~2
die
1 + 2p(d2 + d ? ' )
_2
-3 ~r'~r
-ard2
3 1 + 2p(d2 + d ? ' )
(7)
Combining equations (6) and (7), we have
~,, _ 1 ( ~
(8)
_ ~)_ ~
3p For strain hardened materials, a R - a8 arge _ ~ r - ~ B - 3 P d-gc 3,u+H
+ Hd-~P
9Replacing
this relation in equation (8) gives (9)
Note that equation (9) can be restored to the original elastoplastic counterpart when argc =0. 2.2. Effective time
This part of the formulation is largely adopted from Whirley and Henshall [7]. The specific form of the creep strain increment, argc , is derived in this section. The creep curves obtained from constant stress tests are represented by the following functional form: -gc _ A - 6 . t m
(10)
364 Whirley and Henshall [7] used the "effective time" to rewrite equation (10) in the following form: -8 cn
-AK" -"
(11)
n+l In
in which the subscript n+ 1 represents the state at time t.+~. The effective time, i., is defined by --C
in _
(A6.),/m
(12)
The incremental form of equation (11) using the backward difference gives d~ c - AKnn+,a7"
(13)
where d[m - ( i n + dr) m - t-nm
(14)
Combine equations (12) and (14), we have --C
--C
d [ = = ( 6. + d r ) = _ ~ 6 . AK"n+l A K,~+1
(15)
Since the stress K.+~ can be expressed in terms of the plastic strain increment (argP), equations (9) and (13) are two equations for the two unknowns: argc and dg P . 2.3. Consistent tangent stiffness matrix Equation (2) can be rewritten as (neglect the thermal strain for now): O "-"- n + l
-- C(~n+l
~ P 1 - - "-~C __ "O~n+ O~ 1 )
(16)
Use the relations arg p - Yd2 and arg c - Ydy in equation (16), and take the first variation, we obtain 6a - c(6~ - (dz + dy)ax - (a(dz) + a(dr))~)
(17)
The work of Duxbury et al. [8] uses the similar procedure as above, but based on different hardening rules. By using equation (3) and some mathematical manipulation, we can show that
6(d2)- 9 4T(H
1 ar~p)~~ -
a
(18)
365 where H is the strain hardening coefficient, and P is the conversion matrix in the relation - P ~ . Similarly, we can also show that
9 (a_d-gC)~ffb ~ 6(d),') = 4_-----T -~
(19)
where ot is in the relation 6(d~ c) - a6-d
(20)
It is easily shown that a
-
(21)
Z]'l-~ n-I dt m
rl+l
Replacing equations (18) and (19) in equation (17), the resulting equation is symbolically written as: 6~ - C'6~
(22)
where the consistent tangent stiffness matrix, C, is given by
_
+
9
- - =
1
4~' " n
+
-
d-~P nt-d~C)p~e--[- 3 (d~ p + d g c ) P
(23)
2~
3. APPLICATIONS 3.1. Example 1 The first example is the same as that used in Whirley and Henshall [7]. The uniform axial stress applied on a cylinder of Ti-6%AI-4%V changes abruptly several times during the loading history, i.e., 10 ksi at time zero; 25 ksi at 100 seconds; 28.8 ksi at 225 seconds; and 40 ksi at 330 seconds. The creep parameters are: A = 1.42• 6, n = 1.58 and m = 0.55. Note that the stress is in ksi, strain in in~in, and time in second. A single axisymmetry element of unity aspect ratio (square) is used. The elastic properties are: E = 11.0xl03 ksi and v = 0.33. The deformation is assumed to be elastic-creep, no plastic deformation is involved. The calculated creep strain and the data of Russell and Kobayashi as cited by Whirley and Henshall [7] are shown in Figure 1. In fact, the results are almost identical to that of Whirley and Henshall [7] which also indicated that the concept of "effective time" offers the better predictions of transient creep than other integration methods (i.e., strain hardening and time hardening). Figure 1 actually also shows the results of using an elastoplastic with creep deformation model. In the second material, the initial yield stress is 20 ksi and the strain hardening coefficient H = 500 ksi. The calculated creep strain is almost exactly the same as that of elastic-creep model. The effective plastic strain is shown in Figure 2.
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Figure 1. The effective creep strain in a cylinder of Ti-6%A1-4%V as a function of time during uniaxial tensile loading which increases instantaneously at times 100, 225 and 330 seconds. Symbols are the data of Russell and Kobayashi cited by Whirley and Henshall [7]. The results of elastic-creep and elastoplastic-creep are almost identical.
Figure 2. The effective plastic strain in the same specimen of Figure 1 when the elastoplastic-creep material model is used.
3.2. Example 2 In the more realistic study, we use a specimen of cylinder which has a geometry of 36mm-IDxl08-mm-ODxl00-mm-height. The model material is nickel-iron based superalloy Inconel 718. We will examine two heat treating processes. A simple mesh system in Figure 3 is used for all of our calculations. The material properties and other data are the same as that used in our previous work [6]. The parameters, A, m and n in describing the creep characteristics have been derived from the measured creep curves of Jain, etal., [10]. The directly calculated values of these parameters are strongly dependent upon temperature and stress. For this study, we use n = 3.7, m = 0.98, and A is a function of temperature: Temperature (C) 20 500 760 871 982 1000
A 5.0E-10 5.0E- 10 5.0E-8 1.0E-4 1.4E-2 1.4E-2
When using these parameters, the stress is in ksi, time in hour, and strain in %. The value of A at any temperature is then obtained by interpolation in log scale. The specimen is first heated and hold at a target temperature of 1100~ for 10 minutes, followed by oil quench to the room temperature (Process 1). The heating stage is simulated by using a uniform HTC of 1 kW/m2K. The environment temperature is 1150~ for the first 136 seconds, and changed to 1100 ~ for the next 500 seconds. At the end of the heating period,
367 the specimen is quenched in oil at a temperature of 25 ~ Although the HTC at each face of the surface are usually different during quenching [4,11,12], we assume the HTCs at different faces are the same. The temperature dependency of the HTC is shown n Figure 4. At the beginning of heating, no residual stress or plastic strain are left in the specimen. No consideration is taken to account for the effects of recrystallization in the soaking stage. The same thermal and tensile properties are used in the heating and cooling stages [6]. The calculated history of temperature at four locations close to the surface (Figure 3) is shown in Figure 5. The temperature drops to the oil temperature in about 30 minutes. The growth of creep strain at the same locations is shown in Figure 6. According to the figure, the creep strain escalates in the first half of the heating period. Certainly, the growth rate of the creep strain strongly depends on the heating power from the surface. In the studied case, the surface HTC of 1 kW/m2K is considered to be very high for furnace heating. In the quenching stage, the creep strain grows immediately after the temperature starts dropping, but the plastic strain (Figure 7) remains unchanged for a short period, followed by a significant growth. For comparison, we conduct another set of simulation (Process 2) whose conditions/parameters are exactly the same as Process 1, except no creep is considered in the entire heating and quenching periods. The history of plastic strain for Process 2 at the same locations as that in Figure 5 are shown in Figure 8. A comparison between Figures 7 and 8 shows that the presence of creep deformation (Process 1 in Figure 7) reduces the development
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x/R Figure 3. A specimen of cylinder which has a geometry of 36-mm-IDx 108-mm-ODx 100-mm-height. The model material is nickel-iron based superalloy Inconel 718. Surface points 1 through 4 are used to track the history of temperature and strains in the heat treating process.
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Figure 6. The history of effective creep strain at the surface points in Figure 3.
Figure 5. The history of temperature at the surface points in Figure 3. The environment temperature is 1150~ for the first 136 seconds, and changed to 1100 ~ for the next 500 seconds. At the end of the heating period, the specimen is quenched in oil at a temperature of 25 ~
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Figure 7. The history of effective plastic strain at the surface points in Figure 3.
0
500
1000
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Figure 8. The history of effective plastic strain at the-surface points in Figure 3 for Process 2 in which no creep deformation is considered.
369 of plastic strain. Literally, the creep deformation in the early stage of quenching prevents the occurrence of plastic deformation. However, the overall inelastic strain (plastic plus creep) is higher in Process 1. This also shows up in the distortion (magnified by 30 times) of the quenched specimens (figure 9). In this figure, the distortion is much more severe in Process 1 which involves creep deformation. As a final comparison, the calculated distribution of hoop residual stress after quenching are shown in Figures 10 for both Process 1 and Process 2. The residual hoop stress after quenching is mainly compressive at the OD and near free at the ID. The overall distributions of residual stresses are very close in the two processes. For the studied cases, the presence of creep does not change greatly the distribution of residual stresses.
4. CONCLUSIONS In this paper, we report our implementation of the mathematical model for creep in finite element code. A new formulation is derived in the framework of return mapping technique for the general thermo-elastoplastic with creep deformation. The consistent tangent stiffness is derived and implemented as well. We apply the computer program to the solution treatment (heating, soaking and quenching) of an Inconel 718 component. In the solution treatment process, we found that the creep deformation plays a very significant role in determining the evolution of strain and distortion in the component.
ACKNOWLEDGMENTS This study is financially supported in part by a US Air Force/Navy SBIR award (No. F33615-95-C-5238) with Dr. L. Semiatin at WPAFB as the Project Engineer. We would also like to acknowledge the help of Mr. Percy Gros, Jr. at the Edison Materials Technology Center, Kettering, OH.
REFERENCES
1. Totten, G. E. (Ed.), 1992, Quenching and Distortion Control, Proceedings of the First International Conference on Quenching and Control of Distortion, ASM International. 2. Inoue, T. and Wang, Z., 1985, "Coupling Between Stress, Temperature, and Metallic Structures During Processes Involving Phase Transformations," Materials Science and Technology, Vol. 1, pp. 845-850. 3. Tszeng, T. C., Wu, W. T. and Tang, J. P., 1995, "An Integrated System for Modeling Heat Treating and Machining Processes," NUMIFORM'95 Proceeding, June 18-21, 1995, Cornell University, Ithaca, New York, pp. 875-881. 4. Majorek, A., et al, 1992, "The Influence of Heat Transfer on the Development of Stresses, Residual Stresses and Distortions in Martensitically Hardened SAE 1045 and SAE 4140," Quenching and Distortion Control (ed. G. E. Totten), ASM International, pp. 171-179. 5. Tszeng, T. C., Wu, W. T. and Tang, J. P., 1996a, "Prediction of Distortion During Heat Treating and Machining Processes," Proceedings of the 16th Heat Treating Society
370
Conference & Exposition, 19-21 March 1996, Cincinnati, Ohio (eds. J. L. Dossett and R. E. Luetje), pp. 9-15. 6. Tszeng, T. C. and Wu. W. T., 1996b, "A Sensitivity Study of the Process Model for Predicting the Distortion During Heat Treating," Second Int. Conference on Quench and Distortion Control, Cleveland, November 4-7, 1996, ASM International. 7. Whirley, R. G. and Henshall, G. A., 1992, "Creep Deformation Structural Analysis Using an Efficient Numerical Algorithm," Int. J. Numerical Methods m Engineering, Vol. 35, pp. 1427-1442. 8. Duxbury, P., Crook, T. and Lyons, P., 1994, "A Consistent Formulation for the Integration of Combined Plasticity and Creep," lnt. J. Numerical Methods m Engineering, Vol. 37, pp. 1277-1295. 9. Hallquist, J. O., 1986, "NIKE2D-A Vectorized, Implicit, Finite Deformation, Finite Element Code for Analyzing the Static and Dynamic Response of 2-D Solids," University of California, UCID-19677. 10. Jain, V. K., Srinivasan, R., Weiss, I. and Srivatsa, S. K., 1991, "High Temperature Materials Forging Data Base," EMTEC Report No. CT-13/TR-91-13, Edison Materials Technology Center, Kettering, Ohio, USA. 11. Segerberg, S. and Bodin, J., 1992, "Variation in the Heat Transfer Coefficient Around Components of Different Shapes During Quenching," Quenching and Distortion Control (ed. G. E. Totten), ASM International, pp. 165-170. 12. Tensi, H. M., Lanier, K., Totten, G. E. and Webster, G. M., 1996, "Quenching Uniformity and Surface Cooling Mechanisms," Proceedings of the 16th Heat Treating Society Conference & Exposition, 19-21 March 1996, Cincinnati, Ohio (eds. J. L. Dossett and R. E. Luetje), pp. 3-8.
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Figure 9. Distortion(magnified by 30 times) of the specimenafter quenched to the room temperature. The geometry of the original, before heating, specimen is shown as borken lines in the figure. The displacement at the lower left comer is choosen to be zero in the axial direction. (a) With creep deformation; (b) without creep deformation.
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This Page Intentionally Left Blank
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
A
373
study of sharkskin defects of linear low density polyethylene
C. V e n e t , J.F. Agassant, B. Vergnes CEMEF, Ecole des Mines de Paris, URA CNRS 1374, BP 207, 06904 Sophia-Antipolis, France
The sharkskin defect appearing during the capillary extrusion of a linear low d e n s i t y p o l y e t h y l e n e resin has been c h a r a c t e r i z e d u s i n g t h r e e complementary techniques (profilometry, optical microscopy and observation of cross sections). The influence of flow parameters on the amplitude and the period of the defects has been accurately quantified. The stress field in the capillary and near the die exit has been computed using a viscoelastic flow simulation software. High elongational stresses are observed at the periphery of the extrudate, which allows to validate the assumption of surface rupture to explain the origin of the sharkskin defect. 1. INTRODUCTION One of the main limitation in the extrusion process of linear polyethylenes with narrow molecular weight distribution is the appearance of surface irregularities, usually refered to as "sharkskin" [1, 2]. Above a certain flow rate, the extrudate gradually looses its transparency, becomes mat and finally surface ridges appear. The amplitude of these irregularities increases with increasing flow rate and the defects become more and more organized and periodic. S h a r k s k i n defects are often described as a high frequency/small amplitude surface defect, although more severe distortions may be observed for very high molecular weight materials. The mechanism leading to the sharkskin defect is still a subject of debate. It was suggested that a loss of adhesion between polymer and die wall could be at the origin of the defect [3-5]. However, it is now well known that improving wall slip, for example by using low enegy surfaces, allows to reduce and even totally suppress the appearance of sharkskin defects [6-8]. Another explaining mechanism was initially proposed by Cogswell [2]. When the polymer extrudate is leaving the die, its surface is submitted to a sudden acceleration and corresponding elongational stresses. This could promote the surface r u p t u r e , leading to the sharkskin defect. Alternative explanations have been proposed. Tremblay [9] suggested that negative pressures close to the die exit could induce the formation of voids by cavitation, the coalescence of which will then produce sharkskin. Other
374 a u t h o r s [10-12] proposed a local m e c h a n i s m of stick-slip or r e l a x a t i o n oscillations restricted to a thin peripheral layer, near the die exit. One of the main dilemma in studying sharkskin defects is the following : if one w a n t s to elucidate its origin, it would be necessary to observe the extrudate j u s t at the die exit, before any possible relaxation of the surface [13]. This however is not compatible with a precise quantification of the defect, which can only be achieved on solid extrudate samples. In the present study, we have focused our efforts on the quantification of s h a r k s k i n defect, but keeping in mind that the observed morphology could be slightly different from the real one at the die exit. 2. MATERIALS AND METHODS
2.1. Presentation of the resin The polyethylene resin was provided by Dow Benelux N.V.. It is a metalocene catalysed polyethylene, characterized by a very narrow molecular weight distribution (Mw/Mn = 1.7) and very low density (p = 870 kg/m3). Its s t r u c t u r e is linear, with 15 % of short chain branches regularly distributed along the chains (comb structure). The rheological behavior of this resin was characterized both in shear and elongational situations. Dynamic oscillatory m e a s u r e m e n t s were carried out on a Rheometric RMS 800 and t r a n s i e n t uniaxial elongational behavior was m e a s u r e d at ETH-Zurich. Results presented the classical behavior of a LLDPE resin, but without any strain-hardening behavior in elongational situation.
2.2. Capillary experiments Capillary experiments were carried out at two t e m p e r a t u r e s (150 ~ and 190 ~ to characterize the development of the sharkskin defect. Capillaries of d i a m e t e r D -" 1.39 mm in t u n g s t e n carbide were used, with L/D r a n g i n g between 16 and zero (orifice die, L/D -- 0.1). This orifice die was chosen to check the influence of u p s t r e a m flow conditions. For each experiment, the a p p a r e n t s h e a r rate was varied between 5 and 5000 s -1, covering the entire range of e x t r u d a t e distortions, from smooth and bright to chaotic. In the smooth and s h a r k s k i n regions, samples were collected at the die exit and rapidly cooled in water, for further observations and quantifications.
2.3. Methods of quantification Most of the published literature concerning the s h a r k s k i n defects suffers the same drawback : m a n y authors use a qualitative description of the aspect of the extrudate, instead of trying to quantify the defect. In practice, the problem consists in m e a s u r i n g accurately surface defects in the range 1-300 ~m, w i t h periods in the r a n g e 30-800 ~m, a n d s o m e t i m e s complex a s y m m e t r i c a l shapes. A single technique is t h u s often insufficient. To overcome this difficulty, we decided to use three different c o m p l e m e n t a r y techniques.
375 2.3.1. P r o f i l o m e t r y : We employed a 3D profilometer developed in the laboratory and also used in preceding studies [7, 14]. A sensor with a diameter of 5 ~m and a top angle of 90 ~ follows the surface of the sample and measures the local variations of altitude. This method is very precise and efficient as soon as the dimensions of the sensor are negligible compared to the dimensions of the defect. In practice, profilometry is therefore interesting for measuring defects of low amplitude (A < 10 ~m) and large period. 2.3.2. O p t i c a l m i c r o s c o p y " Using an optical microscope and focusing alternatively at the top and at the bottom of the ridges, it is possible, through the position of the lens, measured by a micrometric system, to estimate very rapidly the amplitude of the defect. The period is also easily quantified. In t h a t case, the limitations arise from a too small amplitude, or from deep and high frequency ridges, for which brightness intensity may not be sufficient to focus at the bottom of the ridges. This technique is well suited for developed defects. 2.3.3. O b s e r v a t i o n o f cross s e c t i o n s : This technique is very powerfull but much time consuming. It consists in encapsulating the sample in an acrylic resin, and cutting longitudinal cross sections by polishing the surface. The cross sections are then investigated by optical microscopy and quantified. The main advantage of this technique is that it provides the exact shape of the extrudate profile (but after an eventual relaxation, as explained before). As for optical microscopy, it is more adapted for characterizing developed sharkskin. From the m e a s u r e m e n t of the period P, it is easy to compute the frequency F of the defect from the following relationship :
~a
F = R4---~ where
~a is
(1) 4Q the apparent wall shear rate (~a = ~ )
and R the capillary radius.
3. EXPERIMENTAL RESULTS 3.1. G e n e r a l t r e n d s
Figure 1 presents an example of the transformation of the surface of the extrudate when increasing the flow rate (or, similarly, the a p p a r e n t shear rate). At low shear rate, the extrudate is smooth. Then, some ridges appear, which begin to develop and organize themselves, and finally become very regular and periodic. First of all, it is important to check the complementarity of the different quantification techniques. It may be observed in Figure 2 t h a t the results provided by profilometry, optical microscopy and observation of cross sections are in very good agreement and allow to follow accurately the development of the s h a r k s k i n over the whole range of shear rates. In the present case, the total roughness is already developed at low shear rate (RT -- 5 ~m at 43 s -1) and increases rapidly and regularly with the shear rate, until the onset of gross melt fracture above 3000 s -1. At this stage, the defect amplitude is around 270
376
Figure 1. Examples of surface defects observations by optical microscopy (x 120, 190 ~ L]D = 0). a) 24 s -1, b) 43 s -1, c) 76 s -1, d) 137 s -1, e) 244 s -1 rtm. Its frequency starts around 150 ridges/s at low shear rate, and increases up to 1300 ridges/s at high s h e a r rate. Similar increases in a m p l i t u d e and frequency with the shear rate, in agreement with the literature [10, 14], were observed for the different temperatures and die geometries. l0 4 r~ r,D 9
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APPARENT SHEAR RATE (s-l) Figure 2. Evolution of the amplitude and the frequency of the surface defects with increasing apparent shear rate (190 ~ L/D = 0) o 9profilometry, e" optical microscopy, o" observation of cross sections 3.2. Influence of die geometry Four different capillaries, with L/D of, respectively, 16, 8, 4 and 0, were used for testing the influence of the die geometry. The quantification presented in F i g u r e 3 shows t h a t two groups of data may be identified. For a finite capillary length (L/D = 4, 8 and 16), amplitude and period are very close and are hardly influenced by the die length. On the contrary, the orifice die leads to a smaller amplitude (approximately two times smaller, at a fixed flow rate) and a much shorter period. The frequency varies linearly with the amplitude
377 of the defect. For L/D = 16, the frequency is quite i n d e p e n d a n t of the a m p l i t u d e and its value is a r o u n d 100 ridges/s. For the orifice die, the frequency is highly d e p e n d e n t on the a m p l i t u d e and m a y reach 1000 ridges/s for high a m p l i t u d e defects [15]. The values for L/D = 16 are of the s a m e order t h a n those reported by K u r t z [10], r a n g i n g b e t w e e n 10 and 200 ridges/s for a LLDPE. It will be shown in the second p a r t of this study t h a t the u p s t r e a m flow conditions at the vicinity of the die exit are very different between an orifice die and a classical die (L/D ~ 0), which could explain these differences in defect morphology. 10 3
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Figure 3. Influence of capillary geometry on the evolution of the a m p l i t u d e (a) and the period (b) of the surface defects with a p p a r e n t shear rate (190 ~ 0 " IJD = 0, o " L/D = 8, ~" L / D = 16
3.3. I n f l u e n c e
of temperature
As p r e v i o u s l y o b s e r v e d by m a n y a u t h o r s [2, 5, 14], a n i n c r e a s e in t e m p e r a t u r e is shown to shift the onset and the development of the s h a r k s k i n defect to h i g h e r s h e a r rates. At a fixed s h e a r rate, both a m p l i t u d e and period are lower w h e n the t e m p e r a t u r e is higher. As a l r e a d y m e n t i o n e d by K u r t z [10], the frequency of the defect increases with the t e m p e r a t u r e and the slope o f its l i n e a r v a r i a t i o n w i t h the defect a m p l i t u d e is higher [15]. It is possible to s u p e r i m p o s e t h e curves at different t e m p e r a t u r e s to obtain a m a s t e r c u r v e , e i t h e r by a p p l y i n g to t h e s h e a r r a t e the shift factor deduced from l i n e a r viscoelastic m e a s u r e m e n t s , or by plotting a m p l i t u d e and period as a function of the wall s h e a r stress (Figure 4). We can t h u s deduce a critical s h e a r stress of 0.07 M P a , i n d e p e n d e n t of t e m p e r a t u r e . This critical v a l u e is in close a g r e e m e n t w i t h the values usually found for LLDPE in the l i t e r a t u r e [3, 10, 13, 14]. The m a s t e r c u r v e s , for both the a m p l i t u d e and the period of the s h a r k s k i n defect, seem to indicate t h a t the onset and the development of the i n s t a b i l i t y isdirectly controled by the s t r e s s level in the flow a n d by t h e viscoelastic properties of the polymer.
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Figure 4. Evolution of the amplitude (a) and the period (b) of the surface defects as a function of the wall shear stress, for two t e m p e r a t u r e s (L/D = 16). 9 9T = 1 5 0 ~
o" T = 190 ~
5. F L O W C O M P U T A T I O N AND D I S C U S S I O N 5.1. S i m u l a t i o n
software presentation
In order to explain the preceding experimental data, we used a s i m u l a t i o n software to investigate the flow conditions in the vicinity of the die exit, w h e r e the defect is a s s u m e d to be created [2, 5, 13, 14]. This software is based on the finite e l e m e n t m e t h o d and allows to compute the flow of a viscoelastic fluid in the g e o m e t r y described in Figure 5, including the capillary (reservoir a n d die land) a n d t h e free surface [16]. We used a m u l t i m o d e P h a n T i e n - T a n n e r c o n s t i t u t i v e e q u a t i o n w i t h five r e l a x a t i o n times, whose p a r a m e t e r s w e r e deduced from rheological m e a s u r e m e n t s in s h e a r and elongation [17].
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F i g u r e 5. A x i s y m m e t r i c geometry reference s t r e a m l i n e s
.............
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used in the s i m u l a t i o n and location of the
S t r e s s s i n g u l a r i t i e s are encountered along the peripheric s t r e a m l i n e , both at the r e e n t r a n t a n d at the exit corners. Around these points, the stress level is h i g h l y d e p e n d e n t on the finite e l e m e n t m e s h size. To obtain realistic s t r e s s v a l u e s , we r e s t r i c t e d our a n a l y s i s to s t r e a m l i n e s close to the free surface. Some of t h e s e reference s t r e a m l i n e s are indicated in Figure 5. It was checked
379 t h a t m e s h r e f i n e m e n t did not affect t h e v a l u e s c o m p u t e d a l o n g t h e s e s t r e a m l i n e s . In w h a t follows, we will focus on the tangential stresses along the s t r e a m l i n e s a n d the corresponding deformations and deformation rates. 5.2. R e s u l t s
For the long die (L/D = 16) at low flow r a t e (190 ~ ~a = 14 s-l), F i g u r e 6 p r e s e n t s the evolution of the t a n g e n t i a l stress fbr the different s t r e a m l i n e s : 1 is close to the surface (= 26 ~m) and corresponds to 95 % of the total flow rate, 2 is deeper (= 103 ~m) and 3 is the s y m m e t r y axis (= 824 ~m from the surface, due to die swell). Along the centerline, we observe t h a t the s t r e s s is n e g a t i v e (compressive zone) in the capillary and t e n d s progressively to zero in t h e e x t r u d a t e . For the s t r e a m l i n e s close to the free surface, the s t r e s s becomes positive (tensile zone) at the die exit, reaches a m a x i m u m around 0.2 m m from the exit a n d t h e n decreases to zero. Thus, we point out the existence, at the p e r i p h e r y of the e x t r u d a t e , of a small a r e a (in this case, a p p r o x i m a t e l y 50 pm in d e p t h and 360 ~m in length) in which the polymer is s u b m i t t e d to traction stresses. ,..ff 0,1
Die land
l~Freesurface
[-., <
0,0
J2
~
Z Z < [--,
-0,1 -l
0 1 POSITION (mm)
2
Figure 6. Evolution of the t a n g e n t i a l stress along the reference s t r e a m l i n e s W h e n the flow r a t e is increased, the tensile zone increases in dimensions a n d also in i n t e n s i t y , as shown in Figure 7. We m a y t h e n a s s u m e t h a t , at a c e r t a i n flow rate, the m a x i m u m stress exceeds a critical value a n d therefore leads to the r u p t u r e of the e x t r u d a t e skin. The crack m a y t h e n propagate in the whole t r a c t i o n zone. W h a t e v e r the flow rate, the d e p t h of this zone r e m a i n s l i m i t e d (50-150 ~m) and of the order of m a g n i t u d e of the a m p l i t u d e of the s h a r k s k i n defect [17]. If we consider the orifice die, we observe t h a t the computed stress field is similar to the one for L/D = 16 at the same flow r a t e (Figure 8a), with a traction zone of a p p r o x i m a t e l y the s a m e dimensions and the s a m e values. This shows
380
Die land r~ r~
~~ee
surface
0,5
r~
<
-14 s-1 --
0,0
z
o z<
~ 1 3 7
-0,5
S-1
,, |
-l
I
o
' ......
!
2
POSITION (mm) F i g u r e 7. Influence of a p p a r e n t wall s h e a r r a t e on the evolution of the tangential stress along streamline 1 (190 ~ that, contrarily to what is generally admited in the literature, the stress is not the u n i q u e p a r a m e t e r able to explain the morphology of the defect. W h e n considering the elongational deformations along the streamline 1 (Figure 8b), it appears a great difference between the two geometries : for the long die, the elongation starts just before the die exit and reaches a final value of 0.8. For the orifice die, t h e e l o n g a t i o n a l flow begins at the c o n v e r g e n t i n l e t a n d consequently the final deformation is much higher, around 6.4 in the present case. For a long die, the flow along the die land allows a relaxation of the elongation sustained in the convergent, which is not possible in the orifice die. Thus, we m a y assume t h a t the deformation experienced by the polymer before the die exit could also be an important p a r a m e t e r controling the morphology of the surface defect.
~_~
a)
Z
b)
5
L/D = 0
o
< 9
Die land
Free surface
2 L/D = 16
[..,
0 -2
-1 0 1 POSITION (mm)
2
-2
-1
' 0 1 POSITION (mm)
2
Figure 8. Influence of die length (L/D) on the evolution of the tangential stress (a) and the deformation (b) along the streamline 1 (190 ~ 137 s -1)
381 5. CONCLUSION In the present study, we have quantified the sharkskin defect of a linear polyethylene resin. Using profilometry, optical microscopy and observations of cross sections of extrudates, we have been able to measure the evolution of amplitude and period (or frequency) of sharkskin defects over a wide range of shear rates and in different flow conditions. We have shown the specific behavior of the orifice die, in comparison to other flow situations. The numerical study of the flow patterns near the die exit points out the existence of a small traction zone at the periphery of the extrudate, where could be initiated the rupture of the skin, subjected to excessive elongational stresses. However, the stress is not sufficient to explain all the e x p e r i m e n t a l observations and other parameters, such as elongational deformations, are also probably concerned in the mechanism of sharkskin formation. ACKNOWI,EDGEMENTS This study was carried out through the framework of a BRITE-EURAM project. The founding of grants by the European Community is gratefully acknowledged. The resin was provided by Dow Benelux N.V.. We are very gratefull to Rudy J. Koopmans from Dow Benelux, to Professor J. Meissner, from ETH-Zurich, for the elongational characterization of the resin and to C. Carrot from Universit~ de St. Etienne for providing the data for the PTT model.
RE~'EgENCES ~
2. 3. 4. 5. ~
7. o
,
10. 11. 12. 13. 14. 15. 16. 17.
N. Bergem, VIIth Congress on Rheology, GSteborg, Sweden (1976) 50. F.N. Cogswell,J. Non-Newt. Fluid Mech., 2 (1977) 37. A. V. Ramamurthy, J. Rheol., 30 (1986) 337. D.S. Kalika and M.M. Denn, J. Rheol. 31, (1987) 815. S.J. Kurtz, in : Advances in Rheology, B. Mena, A. Garcia-Rejon, C. Rangel Nafaile eds., UNAM press, Mexico (1984) 399. S. Nam, Int. Polym. Proc., 1 (1987) 98. G. Sornberger, J.C. Quantin, R. Fajolle, B. Vergnes and J.F. Agassant, J. Non-Newt. Fluid Mech., 23 (1987) 123. N. E1 Kissi, N., L. L~ger, J.M. Piau and A. Mezghani, J. Non-Newt. Fluid Mech., 52 (1994) 249. B. Tremblay, J. Rheol., 35 (1991) 985. S.J. Kurtz, in: Theoretical and Applied Rheology, P. Moldenaers, R. Keunings eds., Elsevier (1992) 377. J. Molenaar and R.J. Koopmans, J. Rheol., 38 (1994) 99. S.Q. Wang, P. A. Drda and Y.W. Inn, J. Rheol., 40 (1996) 875. N. E1 Kissi and J.M. Piau, J. Rheol., 38 (1994) 1447. P. Beaufils, B. Vergnes and J.F. Agassant, Int. Polym. Proc., 4 (1989) 78. C. Venet and B. Vergnes, submitted to J. Rheol. C. B~raudo, A. Fortin, T. Coupez, Y. Demay, B. Vergnes and J.F. Agassant, submitted to J. Non-Newt. Fluid Mech. C. Venet, Ph D dissertation, Ecole des Mines de Paris (1996).
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INFLUENCE OF DF_~'~',CTS ON STRUCTURAL INTEGRITY
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Advanced Methods in MaterialsProcessingDefects M. Predeleanuand P. Gilormini(Editors) 9 1997 ElsevierScienceB.V. All rights reserved.
385
Influence of initial imperfections on the collapse of thin walled structures
A Combescure Laboratoire de Mrcanique et Technologie, ENS de Cachan/CNRS/Universit6 Paris 6, 61 Av. President Wilson 94235 CACHAN CEDEX (FRANCE)
ABSTRACT The collapse prediction of thin walled structure is a rather difficult topic, especially if we consider the buckling case. It has nethertheless been studied since a long time. One can refer here to names like Euler or Von Karman and more recently Koiter [ 1] who first introduced a systematic method to study the effects of intial imperfections on buckling. The aim of this paper is not to give new results on buckling but we would like to show how initial imperfections, introduced by fabrication processes, can induce a collapse of the structure for much lower loads than those estimated using a "perfect" structure modelisation. In this paper we shall give examples and some simple ideas on how initial shape imperfections, thickness imperfections, and boundary condition imperfections can lead to a drastic decrease of the load carrying capacity of a structure under compression. 1 INTRODUCTION It is well known to people working in the prediction of buckling that initial imperfections can dractically reduce the maximum load supported by a structure. The most famous example is the case of a cylinder submitted to an axial compression. But the same type of drastic decrease has been observed for hemispheres under external pressure. The following Figure 1 plots the experimental buckling load compared with the theoretical one as a function of the slenderness of the cylinder. One sees clearly that the maximum loading capacity can be reduced by more than a factor five.
386
O'/O'cl 1.0 ....
~ cl = E e/R,,/3 ( 1 - v z)
0.8
i"
0.6
ii
9|;.
G/O'cl = 0.635
"
-44
== o . . . . , = , =
,=,~....
Moyenne
9 9 : o,s i-~
%.
l
des essais d e b o n n e q u a l i t e
9
_o
0.4
':N i::: '|';~ 'i
"
I
i
,~_~o , . ~ .
;"
o ! o o
s I
9
0.2 "o
o
o
~ I
o
o
!
,
500
1500
'~
(1}
.... ~- ..... :---c~ ,
2500
~--
R/e
Figure 1 Summary of experimental buckling loads compared to theoretical one The first questions that arises is the following: why such an variability? Is the theory not valid? If the theory is valid is the modelisation insufficient? In this case what would be the ingredients to improve the modelisation? What are the main parameters to take into account to have a closer prediction of the experimental results? We shall try to give some tracks to answer to these questions in the following part 2. 2 EFECTS OF DIFFERENT TYPE OF IMPERFECTIONS One first reason of this discreapency is that the models used in many buckling predictions omit the plasticity and hence the prediction of instability is largely overestimated in the case of thick structures. We do not need to consider this case in our paper because we shall always take into account the effects of non-linearities of the material in the buckling predictions. We shall focalise our attention on the fabrication process induced effects; these effects are hence present in the initial state of the structure. The main idea followed in this part is that the "real" structure is different of the modelised one. The differences can be classified in three main categories. First the shape of the "real" structure is different from that of the modelised one; we shall call this difference "shape imperfections". Second, the "real" thicknesses are often different from those of the model; we shall call them "shape imperfections". Third the "real" boundary condition is often very different from the "modelised" one; we shall call this mismatch, the "boundary condition imperfection".
387 21 Shape For the case of cylinder under axial compression the first explanation was given by Koiter [1] who introduced a Taylor expansion of the critical load around the elastic buckling load )~e
Y~ - 2~ 1-
4 3 ( 1 - v~) 2 +~-3 43(1 _ v~)
+ -~-43 (1 - ~7)
(1)
m
In this formula/~e is the buckling load of the imperfect cylinder, v is the Poisson's ratio, and is the normalised amplitude of the initial imperfection (related to the thickness of the cylinder). This formula was computed analytically for a cylinder having an initial axisymmetric or non axisymmetric imperfection. If we look for the influence of such an imperfection on the elastic buckling load of a cylinder having a poisson's ratio of 0.3 the reduction is of 13% when ~ is taken as 0.1 and of 64% when ~ equals 1. This effect is really very important. It is very common, for the thin structures, that the amplitude of the initial imperfections, due to the fabrication process, is close to 1. For a cylinder or a spherical structure one defines the slenderness by the R/t ratio where R is the main radius and t the thickness; a thin cylinder is a cylinder for which the R/t ratio is greater than 100. The previous results were obtained with a cylinder having a linear material behavior and a linear prebuckling deformation. We have later studied the influence of non-linear prebuckling deformations on the buckling and of elastoplastic behavior of the material. The initial imperfections introduce generally bending stresses in the structure and these bending stresses are very harmfull for the buckling because they precipitate the plastification and hence the collapse of the structure for a load which is still lower than the one predicted by equation(I) for the case of a cylinder under axial compression. The effect of initial imperfection is really drastic when the collapse is in the intermediate range between purely plastic regime and purely elastic regime. If we call PE the elastic buckling load, and PY the load for which the plastification appears the ratio q=PE/PL is the key number. If q is greater then 5 the buckling is of the elastic type and the imperfection have in some cases a crucial effects: this is the case of the cylinder under axial compression studied just above. If we now take the same cylinder but submit it to an uniform external pressure the effect of initial imperfections is much smaller; for ususal cases the decrease of the buckling pressure is not more than 20% with an initial imperfection amplitude up to one time the thickness. If q is less than 0.2, we are in the plastic buckling regime and shape initial imperfections have a smaller influence. If q is between the two values the effect of initial imperfection is drastic because the collapse is induced by an hyperbolic growth of the initial imperfection leading to a plastification due to bending, which produces the collapse of the shell. Equation (2) underneath shows typical evolution of the initial imperfection ~0 with the applied load P:
388
P
~-~0
PEp l~~ PE
(2)
In practice the main problem is that we generally know the typical initial imperfection amplitude given by the fabrication process, but we do not now the shape of this imprefection. To overcome this difficulty we suggest to take an imperfect shell obtained by adding to the perfect shell an imperfection of the shape of the elastic buckling mode with an amplitude given by the fabrication process (eg the tolerance specification). A series of experiments were conducted to check this empirical method for the prediction of realistic buckling loads. A typical example for a cylinder under external pressure is given now. This cylinder has a radius of lm a thickness of 0.002m a height of lm. It is clamped at its base and simply supported at its top. The material is a typical 304 stainless steel (Young's modulus 200000Mpa, Poisson's ratio 0.3, yield stress sy 300Mpa). It is loaded by an uniform external pressure P. The cylinder was tested for buckling and the experimental buckling pressure was found to be 0.030Mpa. The initial imperfection was also measured and its maximum amplitude was found to be 0.002m(~=l.). The maximum amplitude relative to the buckling modeshape was found to be 0.0004m(~=0.2). The prediction of the buckling loads, by non linear finite element analysis is given by the following table 1.
O.
P/Pexp 1.29
0.2 1.
1.04 0.67
Imperfection amplitude
Table 1 predicted buckling load compared with experimental one Figure 2 gives the predicted buckling shape which was found very close to the experimental one.
r cylinder under external pressure
389 One observes on this case that the initial imperfection reduce the buckling load of about 30%. This effect is due to a combination of imperfections with plastic hinging. The second example is a cylinder under axial compression. The radius of the cylinder is .075m the thickness is .2mm the heigth is .15m. It is clamped at both ends and and axial compression is imposed at the top. The Young's modulus, E, was 125170Mpa, the Poisson's ratio, v, was 0.3, the yield stress was 77.3Mpa. This cylinder was fabricated and tested without and with on purpose initial imperfection; the initial imperfection was chosen parallel to the most critical buckling mode. The computations have also been performed taking into account the initial imperfections and plasticity. Table 2 contains the comparison of the test and theoretical values. All axial compression stress values are normalised by the theoretical critical stress given by:
cr -E i 3(1-1
,3,
In our case the value is 202Mpa. We can observe that this cylinder is in the dangerous interaction zone between plasticity and initial imperfections. If we apply equation (1) to estimate the effect of initial imperfection on the buckling load we find:
2~ = 1.-~/2.48~'+ 1.54r + 1.24~' 2E
(4)
Table 2 underneath indicated the different predicted values: Type of prediction elastic buckling experimental prediction on perfect strcuture (plasticity) fom equation (4) computed on imperfect strcuture (~ = 2)
Pcr/PE % 100 27 49 16 25
Table 2 Cylinder under axial compression: critical axial stress compared to theory
If we look at results given by table 2, we observe that the critical stress of the ideal structure is nearly four times higher than the experimental critical stress. The initial imperfection being two times the thickness this is not very surprising. The equation (4) gives a pessimistic estimate of the critical stress even if it does not take into account plasticity. If the plasticity only is taken into account the stability prediction is optimistic when compared to the experiment: one sees only a drop of 50% of the buckling load. The influence of intial imperfections leads to a drop to 25%. Figure 3 contains the computed buckling shape which is very close to the observed experimental buckling shape.
390
Figure 3 computed buckled structure From these examples we can derive that in a certain number of cases the initial imperfections induced by the fabrication process have a drastic effect on the integrity of the shells. 22 Thickness A second typical initial imperfection is a variation in the thickness of the shell. The initial fabricated shell has not the specified thickness. We shall consider the case where the fabricated thickness is smaller than the specified one we can make the following remarks: a shell is generally mainly sollicitated by membrane stresses, but the buckling is determined by the bending stiffness. The membrane stresses can be considered as proportionnal to the inverse of the thickness. Let us suppose 10% thickness variation this will lead to 10% increase in membrane stresses. The bending stiffness is proportionnal to the third power of the thickness. Hence a 10% thickness variation shall lead to a 30% bending stiffness variation. If the shell is submitted to loads which can produce buckling, we can deduce form these simple considerations that the load carrying capacity can be decreased by 30%, for 10% thickness decrease. This can be easily shown for a ring (radius R, thickness t) under external pressure which has a uniform reduction of thickness; the membrane circumferential stress is given by:
cr~ - p _R t
(5)
391 The buckling pressure is given by:
PE
-
4 ( 1 - 0 2)
(6)
The situation is more drastic when the shell can plastify because plasticity can be obtained for a smaller pressure and hence precipitate the buckling. A systematic study of the influence of thickness reduction on the buckling of cylinders under external pressure is now being done. For instance the follwing cylinder has been studied [3]: radius R=0.075m heigth H=.15m, Young's modulus E=105000Mpa, nominal thickness t0=165 microns, thickness reduction 20% on a circumferential angle of 80 degrees. The reference pressure buckling PE is 0.0178Mpa. The buckling pressure obtained with the imperfection of thickness is PI=0.77 PE. The reduction is rather important taking into account the fact that only one quarter of the cylinder has lost its origninal thickness. The buckling mode of the imperfect shell is plotted on the following figure 4.
Figure 4 Computed buckled cylinder with 20% thickness decrease 23 boundary condition An other crucial fabrication effect on buckling is the quality of the boundary condition. A shell, when it is short, is very sensitive to the tangential stiffness of the boundary condition. One knows that it is very difficult to produce a real clamping: in practical cases, the boundary condition produces a stiffness which is not infinite. In some cases this effect can be very important. This happened when we tried to design an "improved" cylindrical shell [4], which would have a better resistance to external pressure, than the classical cylindrical shell: for that purpose we invented to replace two third of the heigth by two conical shells as shown on figure 5.
392
L1 I
h
L2 i
i
R
rl
Figure 5 Improved cylinder design. Then we fabricated this improved cylinder and measured the initial imperfections which were very usual for this type of shell (less then the thickness). The Young's modulus was 172300Mpa, the poisson's ratio 0.3. After that, we tested for buckling under external pressure. Let us call PE the buckling pressure of the original cylinder. The "improved" cylinder was predicted to buckle for 2.3 PE. The experiment produced a buckling load of 0.6 PE. When we looked for reasons of this surprise we found that the initial imperfections could not explain the result. We then looked carefully at the "clamping" at the base of the model and found that the shell was attached to the support via a system which was no very stiff. Then we modelised the buckling with two different boundary conditions. For the first case we made the hypothesis that the cylinder was clamped at the base and found a very big overestimation of the buckling load (2.3PE). For the second case we made the hypothesis that the tangential displacements were free and get a pessimistic, but closer, value to the experiment (0.36PE). This indicates that the imperfections due to the boundary conditions fabrication can have a very large influence on the load carrying capacity. Figure 6 underneath show the two buckling modes associated with the two boundary conditions.
6a cumputed clamped
6b computed free
6c experimental
Figure 6 Effect of boundary conditions on buckling mode
393 3 PROSPECTIVE
The results given in the previous paragraphs show the crucial influence of initial fabrication state on the buckling resistance of shells. If the fabrication process produces damage this influence should also be important but this has not been investigated yet. For instance in the case of explosion metal forming there is substantial plastic deformations and damage associated with the fabrication process. If the design of these fabricated shells ignores the initial imperfections, stress and damage state, associated with the fabrication, the integrity assesement is be very optimistic, in some cases. The influence of initial damage on stability could be rather easily modelised by a technique close to that used for plastic buckling evaluation. The tests performed up to now show little influence of welding residual stresses on buckling loads. Nethertheless this effect has not been studied sufficiently systematically to ensure that this result is general: we think that this is due to the fact that the residual stresses associated with welding are mainly bending stresses and hence have little influence on buckling loads. If the fabrication process induces large membrane initial deformations like those obtained in metal forming this process should have an important effect on buckling loads, because these initial deformations are associated with substantial variation of the yield stresses because of the hardening of the material. This point has not yet been studied with great attention. 4 CONCLUSION In this paper we have presented the effect of fabrication process on the buckling capacity of shell structures. The paper shows that these effects can be very important, in case of loads leading to buckling. The effects on buckling loads of initial states such as initial damage, or initial strains have not been studied sufficiently to have a general tendancy. The initial state can also have a very important effect of other types of collapse modes like plastic failure under tensile loads: one knows for instance that a small thickness imperfection leads to a drastic change in the tensile collapse mode. The effect of residual stresses on the crack propagation, and on fatigue, is also very important. The paper nethertheless indicates that the quality of design can be highly improved if one take into account the initial state of the fabricated structure. It is of course difficult to describe in detail he "exact" initial state because the simulation of the fabrication process shall remain approximate for a long time; but there is a real interest to take into account the main indications coming from the fabrication process into the design phase, and the integrity assessement. The industries working with dangerous products for the environnement are interested by this improvement leading to a higher safety: the same trend is observed among the industries concerned by the safety of people. REFERENCES 1 W T. Koiter Over de stabilitiet van het elastisch evenwicht. PhD thesis university of Delf, Amsterdam, Holland, 1945
394 2 A Combescure Static and dynamic buckling of large thin shells Nuclear Engineering and Design 92 (1986) pp 339-354 3 B Schauder Coque cylindrique isotrope sous flexion et pression interne Phd Thesis Insa Lyon 13/3/1997 4 N Debbaneh flambage de coque de revolution a m6ridienne bris6e sous pression lat6rale exteme Phd Thesis Insa Lyon 21/7/1988
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
On Modeling of Laminated terlaminae Imperfections
395
Composite Structures Featuring
In-
M. Di Sciuva* - U. Icardi* - L. Librescu + * Department of Aerospace Engineering-Politecnico di Torino Corso Duca degli Abruzzi 24- 10129 Torino (ITALIA) + Virginia Polytechnic Institute and State University-Engng Science and Mechanics Department Blacksburg, VA 24061-0219, USA Abstract A theory of multilayered anisotropic plates featuring interlaminae imperfections in the form of interlayer slips is presented. The theory rests upon a representation of the displacement field which: i) fulfills a priori the transverse shear stress continuity conditions at each interface of the laminate and the free traction on the top and bottom external planes of the plate, ii) satisfies the requirement of continuous displacements at the perfectly bonded interfaces, and iii) allows for jumps in the in-plane displacements when slip-type interlayer imperfections are present. Numerical results illustrating their implications upon the linear and nonlinear static response of laminated plates are displayed and pertinent conclusions are outlined. 1
INTRODUCTION
It is a well known fact that in contrast to their homogeneous isotropic counterparts, the anisotropic laminated composite structures can exhibit phenomena that may occur at totally different geometric scales, i.e., at the global level, at the ply level or at the reinforcement-matrix level. Since the static and dynamic response characteristics of such structures are significantly affected by the properties of the interlaminae interfaces, the perfect inteTface assumptions, implying continuous displacements and tractions across the interfaces, as used traditionally in the modeling of laminated composite structures, can be inadequate in many important instances. For example, one of the possible ways yielding stiffness degradation in laminated composite structures is typically associated with the interlayer slip. The interlayer slip causes separation of bonded layers and, as a result, stiffness degradation. This constitutes in fact a fracture mode for the laminated structures with detrimental repercussions upon the overall behavior of the structure encompassing its static, dynamic and stability responses. It has to be remarked that, in spite of its importance, the theory of laminated structures
396 with nonrigidly bonded interfaces was developed, mainly, for laminated beams. Few investigations devoted to the modeling and response behavior of laminated plates featuring interlayer slips can be signaled in the specialized literature (see e.g. [1]). One of the goals of the present paper is to fill the existing gap by supplying pertinent information on this topic. To this end, a third-order zig-zag theory for laminated composite plates featuring arbitrary lamination configurations and nonrigidly bonded interfaces is presented. Making use of the principle of the virtual work in conjunction with the postulated displacement field, the pertinent field equations and variationally consistent boundary conditions are obtained. These are completed with the constitutive equations of arbitrarily laminated configuration plates. In addition to the above mentioned features, the obtained equations incorporate geometrical nonlinearities considered in the von Ks163 sense and the existence of an arbitrary temperature field. As a special case, this theory reduces to the generalized zig-zag theory for multilayered anisotropic plates developed by Di Sciuva and Icardi [2,3]. It is also shown that the first-order counterpart of the present plate model is similar to that recently advanced by Schmidt and Librescu [1] and that the present third-order plate model in the absence of temperature effects is similar to that developed by Cheng et al. [4,5,6]. It is hoped that the presented developments and numerical results will contribute to a better understanding and reliable prediction of the response, load carrying capacity and failure of composite structures weakened by interlayer slips.
2
PRELIMINARIES
Consider a composite laminated plate consisting of a finite number N of linearly elastic anisotropic layers, each of them exhibiting different physico-mechanical properties. The existence of the imperfect bonding between the surfaces of two arbitrary contiguous layers is postulated. We denote by 13 the volume of the plate in the undeformed (reference) configuration, f/+ and f/- denote the upper and bottom external planes of the plate, while S is the lateral boundary surface of ]2 generated by the normal to ~ along its boundary curve F (with arc length s). Moreover, ,S'p, ,S',, and Fp, Fu denote the two parts of ,5" and F, where tractions and displacements, respectively, are prescribed. The thickness of the kth lamina and of the entire plate are denoted as (k)h (k = 1,2,...,N) and h, respectively, and are assumed to be constant. For the sake of convenience, the undeformed bottom surface of the plate is selected as the reference plane f/(see fig.l). The points of the 3-D medium are referred to an orthogonal Cartesian coordinate system xj (j = 1,2, 3) with associated unit vectors ij, where x~ (a = 1,2) is the set of in-plane coordinates on f / a n d x3 is the coordinate normal to f/. The distances (along x3) between the reference plane and the undeformed upper and bottom faces of the kth layer are denoted by (k)Z+ and ( k ) z - , respectively (see, figure 1). Associated with the boundary curve F of ~ we define the unit tangent and outward unit normal vectors t and n, respectively, by t - t~i~,
n-
n~i~ - t x i3.
(1)
397 t~ and n~ are the direction cosines of the unit tangent and the unit outward normal with respect to the a-axis, n, t, and i3 are oriented in such a way that (n, t, x3) form a righthanded co-ordinate system. Unless otherwise specified, throughout the paper the usual Cartesian indicial notation is employed with Latin indices ranging from 1 to 3 and Greek indices ranging from 1 to 2, respectively. Repeated indices imply the Einsteinian summation convention, and (..),~ is used to denote partial differentiation with respect to xi. Superscript (k) placed in brackets on the left of any quantity identifies its affiliation to the kth layer, while superscript (r) placed in brackets on the right of any quantity identifies the element in the series expansion of the displacement components. Moreover, Sij denotes the second Piola-Kirchhoff stress tensor, and Eij the Lagrange strain tensor. Under the assumptions that each layer possesses a plane of elastic symmetry parallel to the reference plane (xl,x2) and that 5'33 = 0, the following stress-strain relations hold for each layer
(2) where Q~-y6 are the reduced components of the stiffness tensor; they are symmetric in the indices a and/3, -7 and 6 and the pairs aft and 76. ~t~ are the reduced thermal expansion coefficients and (9 = 0(xi) denotes the stationary temperature rise (see, Librescu [7]). It is postulated that the elastic properties are temperature independent. 3
KINEMATICS
In the spirit of the von Kgrmgn partially nonlinear theory, the following expressions for the Lagrangian strain-displacement relationships are used 1
E~j = ~ (Y~,j + Vj,~ + 89189
(3)
where Vi(xj) is the displacement in the xi-direction. In the case of a plate composed of N layers perfectly bonded together, the following geometric and static conditions have to be fulfilled at the interface between the two consecutive kth and (k + 1)th layers = (k+~)y~l~__(~,,)z_ s~l~=(~)z+ = s~l~=(~+,)z-
(k)v~l~=(~)z,
(4) (5)
However, when at the interface between the l and (l + 1)th layers there is an interlaminar slip, Eq.(4) should be replaced by ( l ) ~ =(t+x) fi~l=3=c,+~)z- _(0 u~l=3=('~z+
(6)
where U~ expresses the interlayer jump of tangential displacement components. In developing a displacement field that fulfills transverse shear stress continuity conditions (5), the following representation for V~(xj) in the kth layer is postulated
~f~(X/3, X3)
--
Uc~(X/3, X3) -~- Uc~(xl3 , x3) -~- ~)r~(x/3 , x3)
(7) y~(~,x~)
-
~(2)(x~)
398 where (7)2 implies that the transversal deflection is uniform throughout the entire laminate thickness. In Eq.(7)l 3
u.(xZ, x3) = E L(~)(xa)u(r)(xz)
(8)
r--0
represents the contribution to the inplane displacement which is continuous with respect to the thickness coordinate x3. This is in fact the classical expansion used in the thirdorder single-layer or smeared laminate models; N-1
U~(x~,x3) = ~
(k)r
-(k) Z+)Hk
(9)
k-1
gives the contribution which is continuous with respect to x3, but with jumps in the first derivative at the interfaces between adjacent layers. This is the expansion used to model multilayered plates by enforcing the continuity of the transverse shear stresses at each layer interfaces in the zig-zag models, Di Sciuva [8,9,10]. Finally, N-1
U~(x~,x3)- E
(k)(].(xz) Hk
(10)
k-1
represents the inplane displacement jumps across each interface enabling interracial imperfection of the slip type to be incorporated (see Schmidt and Librescu [1]). Here, Hk = H(x3 - (k)z+) is the Heaviside unit function and (k)r are yet unknown functions to be determined by satisfying the contact conditions (5) on the transverse shearing stresses at the interfaces. The functions L(r)(x3) can be represented by any set of linearly independent functions, at least continuous with their first derivatives with respect to x3. For a general treatment of this topic, see Di Sciuva [11]. In this paper, we choose the power expansion, i.e.,
L(r)(x3) =x;
r = 0,3.
(11)
The functions (k)r in Eq.(9) are determined such that the continuity conditions for the transverse shearing stresses are satisfied at the layer interfaces. The result is 3
(k)Ca = (k) as u 3,# (~ + E
r--O
(12)
(k)aaBu (~) B (~)
(k).(*) Herein (k)a.~, . ~ are known constants, hereafter called continuity constants, depending only on the transverse shear mechanical properties of the constituent layers: k-1
(k)aa~
__
(k)A. ~ +(k) A.8 ~
(13)
(q)a6f3
q--1 k-1
(k).(r) =
(k)A .~L(r)
+ (k)
E
(q)a~
(141
q=l
with =
(15)
Q.3~3 and S.3~3 being the components of the transverse shear stiffness and their compliance counterpart, respectively. For the interlayer displacement jump we postulate a linear
399 shear slip law (see, Cheng et al. [4] and Schmidt and Librescu [1] for a broad discussion on this topic)
(~)~.(~.,~ =(~)z+) = (~)R.,(~.,~ =(~)z+)(~)s,~(~.,~ =(~)z+)
(~6)
where (k)R,z >_ 0 denote the sliding constants (spring-layer interface) between the kth and (k + 1)th layers. In addition to the extreme situations corresponding to the rigidly bonding interfaces ((k)R,/3 - 0 yielding (k)~, = 0) and completely debonding interfaces ((k)R,z cx~ yielding (k)S~3 = 0), eqn (16) covers also the intermediate cases of imperfectly bonded interfaces ((k)Ro,~ 7s 0, oo). By enforcing the zero shear traction conditions on the top and bottom planes of the laminate, Eq.(7)l can be rewritten in the following form --
(17)
-- w3 3,o~ -}- ]"c~/3Ufl
where N-1
(3)
(18) k=l
and N-1 "Pc~(3)
5
A (2)A(3)'~
h(3)
(
(k)
(19)
k=l (k)l.(a) ~'c~/? =
(k)a(3) ~(2)A(3) (k)a~2 4/3 -- ~'v "Y/3
(20)
a(2) and In Eqs.(19) and (20) zi(2) .t .LOt/3 is the inverse of the square matrix (2x2) ,.,/3 N-1
n(r) ~/3 __ Fhr_l~c~t3-t- E
(k)a(:~);
(21)
r = 2, 3
k=l
4
EQUATIONS
OF
MOTION
The equations of motion are formulated in a weak form using the dynamic version of the principle of virtual displacements. The result is
~(o))
~(o))
N~
,
-<~176176176
<~176
r mr ~ (o)
-t
--
-t- (1) m(~176g" (~ (~lt(3))
/~(3) ol __ T ( 3 ) _ ,
-),/3
(*/3
--
<'m<~176(~
-It- (1)m(~ 3)g(3)'*/3,~ _ (2)m(~176176
~3,'1,
-l- (~176176 ~
(O)m(flo;3)/~(o) + (0)m(3,3)/~(3) (1) (0,3)..(0) /3-y /3 -77"t/3.1, tt3,/3
(22)
(23) (24)
Here, ~ and p~ are the transverse loads applied on the top, f~+, and bottom, f~-, surfaces of the plate, and (k)pi are the in-plane loads acting on the lateral cylindrical surface of the kth layer, (k)Sp; p is the material mass density; the overdot indicates differentiation with respect to the time, and the overbar the prescribed value of a quantity. Moreover,
(25)
400 are the force and moment stress resultants measured per unit lenght;
((')m(~176 (i)m(O,3). (i) (3,3)~ j =
(~c~/3~(a3);~(3) f,(3)~>
(26)
are the inertia resultants, and (-N2n; -m- 2 n ; n--~3) R~In) -" ;
V3--{P3>
(27)
denote the resultants of the applied tractions. Moreover, on F
(.).. = ne(.).e;
(.). = n.(.)o.;
(.).~ = n~(.):. - n:(.)~.
(28)
In Eq.(28), as well as in the forthcoming ones, the repetition of the indices n and t does not imply summations over these indices. Moreover, N [(k)Z+ <'") = E j(k, Z_ ( ' " ) dx3 k=l
The above equations are represented in terms of stress resultants and stress couples. In order to express them in terms of the generalized displacements, we have to use the plate constitutive equations. These are N.~
-
A.~ u(0) + n ( 3 ) u(a) ~ P2. 2,. ~c~e')'. 2,. -- t-)~
.(o)
1A u(O)u(O) o "Jl- ~ ores'. 3,~" 3,. 7!- N2f ~
M. e
=
~(3) u(3) ~ B otp-r. " u(O) %. + a':~ote'r. %. -- ("~
. (o) 1 "rlr + _~B,~e..u(O). (o) + M~e o 3,1r~3,.
R(3) aft
=
r)(3) o (0) F(3,3) u(3) JJ.rea~r,. nt- aerr r , . -
,~(3) (o) i r~(3) (0) (0) | 15aer.tt3,r. n(- ~YotBa'.Ua,~rU3,. nt- So~e
(29)
(30) (31)
(32)
---- *e3r3 ~r
Herein A~oru, B~or. , etc. are the 2-D stiffness quantities defined (A~e~.; B . a . . ; C . e r . ) n(3) p(3) ~-'-o-.; ~ - 0 - . ) a 3,3)
= =
Fe(3,3) 3",-3 =
(O.2~u6~aa..(1; ( Q.2.. ge s A s s
by
X3; *32)} X3) >
f) p(3) /,(3) ( ~323~f~,a*"2r,3 )
and
denote the thermal stress resultants and stress couples. When expressed in terms of generalized displacement components, the equations of motion result in
u(2))
A.-
u (~
+n(3)
u(3)
D
.(0)
(0)m(w0a0)/~(w0) +(0)m(f(ow3)/~(w3) _(1)~,(0,0)/~(0) "'we 3,w --
B a2r# u r,>2a (~ + E ( ~ . u r,#2a (3) - C ~ u
(~ 3,r~r2a
1 O N;e,.
~U3,'xotU3,. + lZ3,rrU3,.oe) -(33)
401
1B
+~ ~
/ (o)
(o)
(0,0)g.(0) _~_(1)
+(1)rnt~-y ~t~,'v
D(3)u~-y.u(~ ~-~(3,3) U(r3)
---~,3"r3 PLATE
(o) (o) ~
~U3,r~olU3,~ .3ff ~3.~.~~
(0,3)-(3)
(2)
rn.y~ uo. v --
(o) + (N~,~3..)., (0,0)..(0)
rno.v u3,~.y
+ F(3'3) ~.u. - E(3).'v~uu(~ (0)
--
(0,3) ..(0)
rnt~~ ut~ +
MODELS
(0)
WITH
:
-t
__-+
--P3 - P3 +
_~_(0) m(O,O)fi(o)
+21D(3)'~'Y~u I, (0)
(3,3) ..(3) __(I)
m&v ut~
PERFECTLY
(0,3) ..(0)
m~
o
--M~.y,.y~
(34)
(0) + u3.~u3.u (0) O
u3,~ -R~.y,~
BONDED
(35) INTER-
FACES If in the displacement field (7) and in the subsequent derivation the effect due to interlayer slips, (/,~(xj), is neglected, the governing equations of multilayered plates with general layup and perfectly bonded interfaces are obtained. These equations correspond to the plate model developed and discussed in Di Sciuva [10] and Icardi and Di Sciuva [3].
6
NUMERICAL
RESULTS
AND
DISCUSSION
In order to shed light on the implications of interfacial defects and their distribution, the global structural response of solid cross-ply and sandwich beams of length L in the xl-direction and of unit width in the x2-direction is investigated. The beams are symmetrically laminated, simply-supported at both ends and are either loaded by a sinusoidally distributed transverse loading on the top surface, p~ = P3 = _pO sin ~L ' or compressed by inplane loading applied to the ends (for buckling analyses). In the solid cross-ply beams considered in the numerical illustrations, all the layers have equal thickness. In the sandwich beams, the faces are assumed to be made by single unidirectional layers; the core is assumed to be orthotropic. The material properties are displayed in the figures, where EL and ET are the elastic moduli of the individual layer in the directions parallel and normal to the fibres, respectively; GLTand GTT are the shear moduli, and lILT is the Poisson's ratio measuring transverse strain under uniaxial stress parallel to the fibres. Moreover, the indices f and c refer to the face and core of sandwich beam. The plotted results are normalized as follows
W* -- lOOu(~
O)ETsh3/(p~ 4) (36)
N~lc," - N11c,.L2/(ETjh2). Following Cheng et al. [4,5,6], we introduce the dimensionless sliding constant R defined by (k)R~ = 5 ~ (k)Rh/Et. The values attached to R stand for the values taken by R at all interfaces. Figures 2 and 3 refer to a three-layered symmetric cross-ply solid beams. Figure 2 gives the central deflection of the beam in bending, as predicted by small deflection theory, for different values of the length-to-thickness ratio, L / h and of the sliping constant, R. For the same beam, figure 3 gives the compressive buckling load parameter. As expected, the detrimental effect of sliping is, in percentage, more pronounced at lower values of the
402 slenderness ratio, L/h, where the transverse shear effects are more important. The results of figure 2 appear to be quantitatively similar to the ones given in Refs. [4]. The results plotted in figures 4 and 5 pertain the behavior of sandwich beams with stiff faces and soft core. It appears that the previuos conclusions apply also for this case, at least qualitatively. The results plotted in figures 3 to 5 appear to be novel in the open literature. Numerical work is in progress to quantitatively assess the effect of interfacial defects on the behavior of unsymmetric cross-ply beams under large deflections and on the natural frequencies.
7
CONCLUDING
REMARKS
A third-order zig-zag discrete-layer theory of laminated composite plates featuring interlayer slips was presented. The theory incorporates the dynamic and thermal effects as well as the geometric non-linearities. The pertinent equations of motion and consistent boundary conditions are derived by means of the dynamic version of the principle of virtual work. The theory represents a generalization of the first-order theory proposed by Schmidt and Librescu [1] and the third-order theory proposed by Cheng, et alii [4,5,6] for laminated plate featuring interlayer slips. The detrimental implications of interlayer slips on the flexural behavior and buckling loads of plates in cylindrical bending have been numerically emphasized and specific conclusions have been outlined. It is hoped that the present paper, together with the referenced ones by Schmidt and Librescu [1] and Cheng et alii [4,5,6] will stimulate further studies on the modeling of laminated composite structures featuring interfacial defects and will contribute to a better understanding of their implications upon the static and dynamic behavior of structures. A c k n o w l e d g m e n t . M. Di Sciuva and U. Icardi would like to thank the Consiglio Nazionale delle Ricerche for the partial support of this research by Grant CNR 96.01765.CTll. Liviu Librescu acknowledges partial support of this research by NATO Grant, CRG 960118.
References [1] Schmidt, R.; Librescu, L. "Geometric Nonlinear Theory of Laminated Anisotropic Composite Plates Featuring Interlayer Slips," Nova Journal of Mathematics, Game Theory and Algebra, 5(2), 131-147, (1996). [2] Di Sciuva, M. "A Generalization of the Zig-Zag Plate Models to Account for General Lamination Configurations.", Atti Accademia delle Scienze di Torino-Classe di Scienze Fisiche, Matematiche e Naturali, 128(3-4), 81-103 (1994). [3] Icardi, U.; Di Sciuva, M. "Large-deflection and Stress Analysis of Multilayered Plates with Induced-Strain Actuators.", Smart Materials and Structures, 5, 140-164 (1996).
403
[4] Cheng, Z.Q.; Jemah, A.K.; Williams, F.W. "Theory for Multilayered Anisotropic Plates with Weakened Interfaces," paper to appear in the J. Appl. Mechanics, Trans. ASME (1996).
[5] Cheng, Z.Q.; Kennedy, D.; Williams, F.W. "Effect of Interfacial Imperfection on Buckling and Bending Behavior of Composite Laminates," AIAA J., 33(12), 2590-
2595,
996).
[6] Cheng, Z.Q.; Howson, W.P. Williams, F.W. "Effect of Interfacial Imperfection on Buckling and Bending Behavior of CompositeLaminates," AIAA J., 33(12), 25902595, (1996).
[7] Librescu, L. Elastostatics and Kinetics of Anisotropic and Heterogeneous Shell-type Structures, Noordhoff Internt. Publishing, Leyden (1975).
Is] Di Sciuva M. "Bending, vibration and buckling of simply-supported thick multilayered orthotropic plates. An evaluation of a new displacement model," Journal of Sound and Vib, 105(3), 425-442, (1986).
[9] Di Sciuva M. "An Improved Shear-Deformation Theory for Moderately Thick Multilayered Anisotropic Shells and Plates," J. Appl. Mechanics, Trans. ASME, 54, 589596, (1987).
[10] Di Sciuva M. "Multilayered Anisotropic Plate Models with Continuous Interlaminar Stresses," Composite Structures, 22(3), 149-167, (1992).
[11] Di Sciuva M. "A Geometrically Nonlinear Theory of Multilayered Plate with Interlayer Slips," Submitted.
404
layer N
h
(k)h
layer k (k) z+
li
,a,er I
B
Figure1: Plategeometryand numberingof layersand interfaces.
5
18
4.5
Laminaeofequalthickness;Lay.up:0/90/0 MechanicalpropeYdesof unidirectionallamina EUEt=25;GLT/ET=O.5;GTT/ET=0.2;vLT=0.25
.it
4 .,.1... o 3.5
Data as in figure 2.
,~16 Z
R=0.2
,._-14 (3) ,.6.-, (1)
R=0
l::: 12
---~
3
R=0,6
"1= 2.5 ~
2
~
1.5
03 10
R=0.4
R=0.4
03 8 _oo
R=0.6
o) 6 .c::
R=0.2
4
1
::3 CX2
21
0.5 4
6
8
10
12
14
16
18
4
20
6
Length-to-thickness ratio, L/h
8
10
12
14
16
18
20
Length-to-thickness ratio, IJh
Rgure2: Simply-supportedbeamundertransversesinueoidalloading.
R~u'e3: Simply.supportedbeamundercompression..
16 9'
R=0.6
14,
R=0.4
.--12 .~
,
o10 c,
R=0.2
8
Sandwichbeam (h f=0.1h; hc=0.8h) Face lay-up: 0190R Materialproperties: Faces: EUET=25; GLTIET=0.5; GTTIET=0.2; v LT=0.25 Core: EUET=I" GLT/ET=0.4;
R=O
Data as in figure 4.
Z
R=0.2
(1) (1)
E 03 03
GTTIET=1.5;vLT=0.25
R=0.6
ETfacelETcore = 25
03
o
~4 m
R=O
!---2
_
r ::3
0 4
6
8
10
12
14
16
18
20
Length-to-thickness ratio, L/h Figure4: Simply-supportedbeamundertransversesinusoidalloading.
01 4
6
8
10
12
14
16
Length.to-thickness ratio, L/h RgureS:Simply.=pportedbeamunder~ n . .
18
20
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
405
Delamination, instability and failure of multilayered composites E. Stein and J. Tegmer Institute of Structural and Computational Mechanics, University of Hannover, Appelstr. 9A, D-30167 Hannover, Fed. Rep. of Germany A numerical method for the nonlinear analysis of thin-walled composite structures is presented within the finite element method. In addition to the static analysis, onset and propagation of delaminations are considered. The variational formulation is described by a multi-director approach with piecewise polynomial functions of the displacements. Initiation of delamination is controlled by a stress based fracture criterion. For its propagation the energy release rate is calculated as a further criterion. An isoparametric quadrilateral multilayer shell element is used for the numerical calculation. It leads to a comparable number of degrees of freedom as by using brick elements but additionally has some main advantages, as 2D-mesh generation is possible, coupling with conventional 2D-shell elements is easier, bending behavior is better, description of interlaminar interfaces is simple. For the kinematic expansion of delaminated zones, additional degrees of freedom are introduced. They are not adjoined to a distinct interface in advance of the delamination analysis. The theory holds for a complete 3D-stress state. Applications to glulam beams are given. 1. I N T R O D U C T I O N Composites are typically used for light weight structures. They are employed in many engeneering fields. Due to anisotropy and stacking sequences of laminates, i.e. heterogenities, the material shows rather complicated states of strains and stresses. Several models for thin composites have been developed over the past, and the generalisation of the elastic shear deformation theory for thin composites leads to the classic laminate theory which takes into consideration the effects of anisotropy and stacking sequence, e.g. coupling of bending and stretching, [1]. In many cases FE methods were applied, [2,4-7,10-15]. The problems of delamination have been studied in [2,4-7,11,13,15] because they have a major impact on the failure bahavior of the considered structures. Especially at free edges or cutouts, 3D-states of stresses arise, and especially the interlaminar stress components can induce delamination growth between adjacent layers. Typically for thin-walled structures, external loads don't lead to large or critical stresses in thickness direction of the structure. Such the existence of a delamination doesn't directly initiate failure of the system. Yet, under compression stresses a delamination can strongly diminish the critical buckling load of the structure, [2,4,7]. Other problems of delamination result from the brittle behavior of composite structures. Stress singularities, due to the ideal elastic material model or due to incompatibilities between adjacent layers with different fiber orientations, can lead
406 to zipper-like crack propagation. These effects have to be analysed if a further use of a structure is intended under admission of delaminations within certain bounds. Goodminded properties of classic materials like plastification of steel have led to many simplifying calculation rules in engeneering disciplines. These rules have to be reconsidered when composite materials are used. In this paper a multi-director shell element with a deformation mode in thickness direction is used for the nonlinear analysis of composite structures, [4,14]. The interpolation in thickness direction with piecewise polynomial functions over each layer is independent from the bilinear tangential interpolation within the middle surface. The complete 3D-stress tensor is computed by this model. The extended kinematics of delaminated elements are realised by introducing additional degrees of freedom at each node within the delamination zone. The onset of delamination is described by a stress based fracture criterion. Propagation is controlled by energy relase rates. Typically, in most parts of thin composite structures, normal stresses S 33 in thickness direction are much smaller than normal stresses in tangential direction. Such, S 3a is neglected in the classical laminate theory. Furthermore, the Kirchhoff-Love hypothesis holds for undisturbed subdomains which are usually predominant. In those regular parts of a structure a conventional one-director kinematic for the shell is sufficient. Therefore we couple standard 2D-shell elements with multi-director elements by using transition elements, [7]. In the sequal only the multi-director theory for shells is outlined and used. 2. V A R I A T I O N A L F O R M U L A T I O N For the analysis of composite materials we restrict the calculations to thin-walled beams and shells with a layerwise build up. The considered shells consist of n physical layers j with thickness h j and N n u m e r i c a l layers i. Therefore, a subdivision or collection of layers for the numerical calculation is possible. The position vector X0 of the reference surface So is labeled with convected coordinates | An orthonormal basis system tk(| ~) is attached to this surface where ta is a normal vector and | the coordinate in thickness direction. The transformation between the different base systems is given by tk(O~) = Ro(O~) ek
(1)
where R0 is a proper orthogonal tensor. Then the position vectors of the reference and the current configuration of the body are given by x(o
o
=
Xo(O
x(O
o
-
x(o
+ o t (o ,o
h , < 0 a < ho
(2)
+ u(O ,
The kinematics of the shell are based on the assumption of a multiplicative decomposition of the displacement field in shell space with independent shape functions in thickness direction and shape functions in tangential direction, defined on the reference surface of the shell. For the numerical layer i the displacement vector is interpolated through the
407 1ilickIless, [14]
u'(O ~, o ~)
-
~~(o
~) u~ -~(o~) _ ~ ( o ~ ) a ~ ( o
~)
(3)
l=1 f l i ( o a)
__
[ f l ~ , - u2, i
" T ,fi~]
...
(2<m<
4) .
The shape functions r are arranged in a matrix with hierarchical shape functions
Oi_ r162 r
i)
-
[r162 1
9..
, r
~(1 - c ~)
r
~) -
1 - C'~
1 ( 1 + C i)
r
~) -
(1 - r162
r
-
2 03/h ~
(4)
Thus polynomial functions are used up to third order to describe the deformed cross sections. The covariant base vectors of the reference configuration are introduced as Gi = OX/O| i. The contravariant base vectors G i are defined by G i . G j - 5~. In the same manner one obtains the convected base vectors gi and gJ of the current configuration. Top and bottom surface of the shell are loaded with loads 15 - ~ ek, whereas body forces are neglected. Then the principle of virtual work in a material description reads G(u, S u ) -
f [ f S k' 'Ekt JdO 3]dSo - f ~ 'uk dS = O, (So) (03)
(5)
(r,)
with J - (X,1 x X , 2 ) . X,3. Note that integration of (5) has to be performed considering the different material parameters or fibre angles through the thickness. The covariant components of the Green-Lagrangian strain tensor E = Ekz G k O G l and the associated variation are obtained from Ekl
-
~1( x , k ' x , l
--X, k "X,l )
~Ekz
-
l(~x,k'x,l
+~X,l'X,k).
(6)
The work conjugate stresses S kl are the contravariant components of the Second PiolaKirchhoff stress tensor S - SktGk | Gt. Each physical layer j is considered as a homogeneous transversally isotropic continuum, where the axes of orthotropy coincide with the material principal axes. Hence the stresses of the actual elastic layer j are given for small strains by the St. Venant-Kirchhoff model S kl
- C k l ' ~ Em~.
(7)
The constitutive parameters C klmn are functions of the fibre angle ~J and the material parameters El, E2, u, G12, G23 of layer j. In our calculations the fibre angle is defined as an angle between fibre direction and base vector tl. We restrict ourselves to small strains, thus direction transformations of the stress components can be neglected, and material parameters of the linear theory can be used, (7). The components expressed in matrix notation refer to an orthonormal coordinate system which is a local fibre oriented basis. The material matrix has to be transformed to a global fibre independent basis system, for more details see [4]. In this way all stress- and strain-components of the 3D-theory are taken into consideration.
408 3. F E - F O R M U L A T I O N
OF THE MULTI-DIRECTOR
THEORY
The four node quadrilateral Ql-element with isoparametric bilinear shape functions within the reference surface is used. Independent interpolation of different order is made in thickness direction. The approximation of the geometry follows as 4
NK(~, ,7)Xg
Xh "- E
with
XK -- X0K + 0 3 t3K,
(8)
K=I
and for the displacment in layer i we get with (3) 4
=
(9) K=I
Here the nodal displacement vector VK consists of n = 3(N + 1) + 3(m - 2) d.o.f.. To avoid shear locking the transverse shear strains ")' - E ~ 3 G 1 H- E,3G2 are independently interpolated as assumed shear strains,. [3] 2Er
--
1 ~[(1 - T])TB -3t-(1 -3t- T])•D ]
2E,3
=
~[(1--~)TA+
1
(1
+~)7C]
.
(10)
The shear strains ~'M are evaluated at the midside nodes M - A, B, C, D of a four-node element. Inserting these interpolations into the virtual work expression yields the discretized weak form of equillibrium numel
a(v,
nel
{Z
e----1
K=I n
(So.) = (o,)
} - 0, 2
(11)
= (
where Jz denotes the surface element of the top and bottom surface Ft~ with applied loads 151. The stress vector S is given by (7), and matrix BK is specified in detail in [4]. The nonlinear algebraic equation (11) is iteratively solved by Newton's method. The residual vectors and tangent matrices are integrated using Gauss quadrature with 2 x 2 integration points within the shell surface. The order of integration in thickness direction is l - m, see (3), and depends on the chosen polynomial degree n p = m - 1. 4. A F E - M O D E L
FOR ONSET AND GROWTH
OF DELAMINATION
For the condition of onset and growth of delaminations we presume: 9 The strains are small. 9 Only one in-plane crack is admitted along the thickness. 9 Onset of delamination is initiated by the criterion of Hashin.
409 9 Growth of delamination is controlled by energy release rates. 9 Interpolations in thickness direction are restricted to linear shape functions
The fracture criterion of Hashin [9] is quadratic in terms of the interlaminar stresses
F(o') - ($13)2 -31-($23)2 ($33)2 R2 4 Zo2
1,
(12)
Z0 is the laminate strength in thickness direction for tension, and R0 is the shear strength. The crack criterion considers the general case of mixed mode stresses. Pure mode-I, mode-II or mode-III stress states are included as special cases. For this criterion the interlaminar stresses er = IS 13, S 23, $33] T at the layer interfaces are needed. By standard calculation, stresses are evaluated at the integration points; then they are projected to the layer boundaries under the assumption that er is C~ along the thickness direction. This is reached by application of the least square method,
[6]. Once the fracture criterion is fullfilled the geometric boundary conditions change. Thus the nodal displacement vector is extended by three additional parameters which allow to describe crack opening displacements. The vector of crack opening displacement g is introduced as the difference of displacements of adjacent nodes k and k + 1 for an assumed crack at layer boundary i, i + 1 g(X)
:= u k+l -
u k
.
(g(X)
without
~ 0
crack)
(13)
."'"'"'""-.
.'""'"'...
_
-
e/ 9
K 9
9
......"-..
9
,."
.
Figure 1. Vector of crack opening displacement
@/
.
i
........"
IgK
:0
...... ..
...
K
..
....'"..
:. 9
..
:
..........
410 The interpolation for the displacement vector 4
u},
E
follows as
4
NK({, r/),I~' ~K
;
gh - E
K=I
~g
Uh
NK({, ~)gg
K=I
[ v~''
with
" " " ' vk'
" " " ' v/~'' vk~l
(14)
] 3(N+2) xl
v~: +1" -v~- ,ifgK--0
If the stress criterion is fullfilled also an energy based criterion has to be checked in order to predict the growing of a crack. There are some criterions in fracture mechanical approaches, such as energy release rates, stress intensity factors or J-integrals [8]. We choose the energy release rate in the sense of a crack closure technique because it is generally valid for anisotropic material, and it is energy-based. For the growth of a crack a critical energy release rate ~'c has to be reached. Gc is a material parameter and has to be determined independently by experiments. The actual rate G - --g~ d~ is numerically calculated as finite ratio {~ =
Art AA
with
ATr-
1 -~Pu'g,
(15)
where AA is the area of the propagating crack, and Air is the difference of the potential before and after the crack progress. Pu are the nodal forces in order to close the crack area, see figure 1. The algorithm is outlined in figure 2. After computation of A~r three different cases have to be distinguished. For all cases the sets 7- and 7"* are defined as
7" := {All nodal points K within a trial crack area}, defined by (12). r*
.-
{K
[
g E T
A
~AAK >- -
~C}
,
7-* C 7- .
(16)
1. G(K C T) < Go: The energy criterion is neither fullfilled for the whole trial crack area nor for each single node of the trial crack area. ==. T* = 0.
(17)
Thus, the trial crack is completely deleted, and the structure remains unchanged for the next load step. 2. G(K C T) > Go: The energy criterion is fullfilled for the whole trial crack area. 7"*= 7".
(18)
Thus, the trial crack area is confirmed, and the structure is changed. Fi
' Ft , the crack produces a new surface.
(19)
With the changed geometric boundary conditions the stress criterion has to be tested again in an iterativ process.
411
g k, k+l "-- 0
"
initial system
without
crack
]
"1~'-I P = P + AP 9next load increment
[
G(v, Av) = 0" nonlin, static analysis ~__
] no r,
I I
g k,k+l:/: 0 : system with trial crack
'
'/liy
Compute Pu = Po G(v, Av) = 0 : nonlin, static analysis ~- t 1
*,-~-,,
al crack
Compute g in nodes K An ~= 0.5 P u,:g
ri
Figure 2. Flow chart of the crack propagation algorithm
3. G(K E T*) >_ Gc: The energy criterion is fullfilled for some nodes of the trial crack area.
----~T*CT. If
~
A TrK AAK > Gc
set
T* = T
~
proceed with Step 2,
(20)
K
('Condition A' in figures 5 and 6), else close nodes K C { T \ T*} and compute the energy release rate for the remaining trail crack area.
5. E X A M P L E We consider a curved glulam beam under prescribed deflection w of the apex as shown in figure 3. For the numerical analysis one half of the system is discretized due to symmetry. We apply the deflection in 20 load steps. Results are presented in figure 4 to 6. The load-deflection curve F(w) is given in figure 4 and shows the influence of growing cracks which ends at point E.
412
~W
I3,0m
E s= 11000 M N / m**2 Et = 300 M N / m**2 G = 500 MN / m**2 v=0,3 Zo= 0,2 MN / m**2 Ro= 0,9 MN / m**2 Gc= 0,001 MN / m
crack s u ~ ~ ~ ,
3
t,Om
....
,Om
O,lm
lO,Om
Figure 3. System of a curved Glulam Beam with material data, displacement controlled loading at apex. The crack begins in point P and grows in both directions until the points Q and R.
-0.3
GlulamBeam M e s h 6~ w i t h Q u t c r a c k ~ M e s h 6, w l t h crack Mesh 6, energy release rate-m--
0.25
In figure 5 and 6 different finite element meshes are compared. They have one element in y-direction, 10 layers in thickness direction and a variable number of elements in s-direction. Mesh Mesh Mesh Mesh Mesh Mesh
1: 2: 3: 4: 5: 6:
10 elements in s-direction 20 elements in s-direction 40 elements in s-direction 80 elements in s-direction 160 elements in s-direction 320 elements in s-direction
-0.2
0.15
0.05
0
0
i
-0[05 -0'.1 w(s-20.115tm I -o'.2
i
-0 25
-0.3
Figure 4. Load-Deflection curve for Glulam Beam without crack, with prescribed crack and with calculated crack-propagation until the points Q and R in figure 3 which corresponds to point E.
413
Glulam
Beam
-0.2 t
~
., . Mesh Mesh
~
I
$
'
{
~'"
.~
~g
~''~&
-e--~-. -B--
Mesh
-.~.....
Mesh
~--
Mesh
~ ....
-0.15
o" u
§
-0.I
l .....
o
~- ~ -~-..-4-.~
........
.............=.= ......................
o
o
§ ,~
-0.05
.~i o o i
0 0
i
1
i
2
3
crack
lenght
1
i
i
4 [m]
5
6
Figure 5. Energy-Release-Rate-algorithm for Glulam Beam with 'Condition A' in (20)
GlulamBeam
-0.2 ti $
,---, ,..., 4~
'
~
~
~
~
,
.
:
+
~
~
'
?,-0
15
-t
~o
+
T
"~~o -o.1
~" ~....
~ .....
_ ......
iMesh + Mesh Mesh ~ Mesh ," M e s h
~-....
3-s-~ ......... ~--
+
a--
_~ .....
;~i..................+;'.....................~""; ~/ :::::::ii :~...........................:::::::::::: ~,.---
.~
.............• .............x"
..$
'13
-0.05 o o l
0 0
1
i
2 crack
i
i
3 lenght
1
4 [m]
i
5
Figure 6. Energy-Release-Rate-algorithm for Glulam Beam w i t h o u t 'Condition A' in
(20) Acknowledgment The financial support of the Deutsche Forschungsgemeinschaft (DFG) for project Ste 238/25-TP.2 is gratefully acknowledged.
414 REFERENCES
1. ALTENBACH, H., ALTENBACH, J. and RIKARDS, R. Einf~hring in die Mechanik der Laminat- und Sandwichtragwerke, Deutscher Verlag f~r Grundstomndustrie, Stuttgart, 1996. 2. COCHELIN, B. and POTIER-FERRY, M. A numerical model for buckling and growth of delaminations in composite laminates. Comput. Methods Appl. Mech. Engrg. 89, 361-380, 1991. 3. DVORKIN, E. N. and BATHE, K.-J. A continuum mechanics based four-node shell element for general nonlinear analysis. Engineering Computations, 1, 77-88, 1984 4. GRUTTMANN, F. and WAGNER, W. On the numerical analysis of local effects in composite structures. Composite Structures, 29, 1-12, 1994. 5. GRUTTMANN, F., STEIN, E. and WAGNER, W. A Generalized FE-Method for Non-Linear Composite Shells with 2D- and 3D-Modeling, in S.N. Atluri, G. Yagawa, T.A. Cruse (Eds.), Proceedings of the Int. Conf. on Comput. Eng. Science, 2533-2538, Hawaii, 1995. 6. GRUTTMANN, F. Theorie und Numerik dfinnwandiger Faserverbundstrukturen. Habilitationsschrift am Fachbereich Bauingenieur- und Vermessungswesen der Universit/it Hannover, 1996. 7. GRUTTMANN, F. and WAGNER, W. Coupling of 2d- and 3d-composite shell elements in linear and nonlinear applications, Comput. Methods Appl. Mech. Engrg. 129, 271-278, 1996. 8. HAHN, H.G. Bruchmechanik, Teubner Verlag, 1976. 9. HASHIN, Z. Failure criteria for unidirectional composites, Journal for Applied Mechanics,47, 329-334, 1980. 10. KLARMANN, R. Nichtlineare FE-Berechnungen yon Schalentragwerken mit geschichtetem anisotropen Querschnitt, Heft 12 d. Schriftenreihe d. Inst. f. Baustatik, UniversitS~t Karlsruhe, 1991. 11. KRUGER, R. Delaminationswachstum in Faserverbundlaminaten, Bericht 13-96 d. Inst. f. Statik u. Dynamik d. Luft- und Raumfahrtkonstruktionen, Universitgt Stuttgart, 1996. 12. LOGEMANN, M. Abschs der Tragfiihigkeit von Holzbauten m. Ausklinkungen und Durchbrfichen, Fortschritt-Berichte Reihe 4, Nr. 102, VDI, Dfisseldorf, 1991. 13. RIKARDS, R., BUCHHOLZ, F.-G. and WANG, H. Finite element analysis of delamination cracks in bending of cross-ply laminates, Mechanics of composite materials and structures 2, 281-294, 1995. 14. ROBBINS, D. H. and REDDY, J. N. Modeling of thick composites using a layerwise laminate theory. International Journal for Numerical Methods in Engineering, 36, 655-677, 1993. 15. WAGNER, W. and GRUTTMANN, F. A Computational Model for the Delamination Analysis of Composite Shells, in D.R.J. Owen, E. Ofiate (eds), Proceedings of the Fourth Int. Conf. on COMPUTATIONAL PLASTICITY: Fundamentals and Applications, 1191-1202, Barcelona, 1995.
Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
Statistical Damage Tolerance for Cast Iron Under Fatigue Loadings H. Yaacoub Agha a, A.-S. B~ranger b, R. Billardon a and F. Hild a aLaboratoire de M~canique et Technologie E.N.S de Cachan / C.N.R.S / Universit~ Paris 6 61, Avenue du President Wilson, F-94235 Cachan Cedex, France. bRenault - Direction de la Recherche- Service 60152 860 quai Stalingrad, F-92109 Boulogne Billancourt, France.
In this paper, a statistical model accounting for the presence of initial flaws is introduced to study the fatigue failure of Spheroidal Graphite cast iron. An expression of the cumulative failure probability of a structure is proposed for cyclic loading conditions. The proposed model uses a modified Paris' law and is valid when no new flaw nucleates during cycling. An identification procedure is developed to determine the flaw distribution as well as the crack growth law from a series of standard fatigue tests. A post-processing approach is developed to study the failure probability of any complex structure, and applied to analyze a suspension arm.
1. I N T R O D U C T I O N
For competitive reasons the automotive industry tries to reduce the cost of its products without repercussions on the final quality of the cars. The cost may be reduced by using cheap manufacturing processes such as casting, where complex shapes can be obtained with little difficulties and the costs are lower than for any other manufacturing procedure such as machining. Due to its good properties, the Spheroidal Graphite cast iron is widely used now in automotive industry for safety components. For example, it is utilized in ground link elements such as steering knuckle holder, suspension arms. Despite the developments of the production technology, flaws (e.g. pin-holes, shrinkage, cavities) are unavoidable. Flaws occur in solidifying cast due to negative pressures generated during solidification contraction, and pressure developed by gases dissolved in the molten metal. These flaws are usually undesirable and are randomly distributed within the material. The cast components are frequently subjected to high cycle fatigue conditions, and their fatigue strength may be reduced by the presence of these initial casting flaws whereas, conventional design procedures to assess the structural integrity of a component use deterministic crack initiation criteria and ignore micro-inhomogeneities within the material. It is important when studying the fatigue properties of cast material to consider the scatter of experimental
416
results which is observed and develop an evaluation method which accurately calculates the effect of casting flaws. In this paper, it is proposed to model the presence of these inhomogeneities, their possible evolution with the number of cycles, and if needed their statistical distribution. An expression of the cumulative failure probability of a component is proposed on the basis of a modified Paris' law. This failure probability is related to the statistical distribution of flaws. An identification procedure is developed to determine the flaw size distribution as well as the micro crack propagation law. This method is applied to an SG cast iron supplied by Renault car company. The identified results obtained from fatigue tension-tension tests are then compared with independent experimental results. A post-processing approach is developed to study the failure probability of structures, such as a suspension arm.
2. R E L I A B I L I T Y OF S T R U C T U R E S
CONTAINING
FLAWS
It is assumed that initial flaws are randomly distributed within a structure and that the flaw distribution is characterized by a probability density function f. The function f depends upon the flaw size a. Other parameters such as orientation are not considered, since only flaws in pure mode I are considered (i.e. with their orientation perpendicular to the maximum principal stress). In the case of cyclic loading conditions, the evolution of the flaw size leads to the evolution of the flaw size distribution. After N cycles, it is assumed that the flaw distribution is described by a function fN 9 At this stage, it is useful to introduce a function ~ that relates the initial flaw size a0 to the flaw size after N cycles aN
ao = ~(aN)
(1)
If the flaw size evolution is deterministic, the probability of finding a flaw of size aN after N cycles is equal to the probability of finding an initial flaw of size q~(aN). Therefore the cumulative failure probability PRO can be written as [i] PRO =
f~
(ac) fo(a)da
(2)
where 9 (ac) denotes the initial flaw size that, after N cycles of loading, reaches the critical flaw size ac. The flaws are supposed to be described bycracks of size a, whose geometry is taken into account by a dimensionless factor Y such that the energy release rate G is given by y2a2a
G=
E
(3)
where a stands for an equivalent uniaxial stress (for instance the maximum principal stress). It is worth noting that the values of the parameter Y depend upon the geometry of the initial defect and the fact that this flaw intersects or not a free surface. Under monotonic and cyclic loading conditions, local failure can be described by a criterion referring to a critical value of the energy release rate
G _> G
(4)
417 In the case of ductile materials subjected to cyclic loading conditions, stable crack propagation can be described by a generalized Paris' law [2]
da ~_ C (V/~maxg!~)" = -V/-'~thl n .V/ P~cT _ ~g(R) /
(5)
dN
where Gmax (resp. gm~) stands for the maximum (resp. minimum) energy release rate over one cycle, Gth denotes the threshold energy release rate under which ~maxg(R) < Gth) no propagation occurs, and N the number of cycles. The parameters C and n are material dependent, and the function g models the influence of the load ratio R = ~~m/n " An expression for the function g has been proposed by Pellas et al [2] 1-R g(R) = 1 m R
(6)
-
where m is a material parameter. From Eq. (5) the following closed-form solution can be derived [1]
)(
- ~ \ V aM
)n(
V~c -- ~ g(R)
Sth ] NF
(7)
where aM denotes the maximum flaw size in the structure, the dimensionless constant C* is equal to ~--~u' c and Sth the cyclic threshold stress, defined as the lowest value of the stress level below which no failure occurs (i.e. the failure probability is equal to zero). The cyclic threshold stress, Sth, is related to the threshold energy release rate Gth. Its expression, when g(R) = 1, is given by
1 ~/EGth Sth = Y V a----M
(8)
The value of the function ~ depends upon the power n. When n -~ 1 and n -~ 2, the function ~ is given by
~(x) -- 2 ( x - Xth)l-n(Xth --(TL- 1)X)
(n- l)(n- 2)
(9)
where Xth is the normalized threshold defect size given by
Xth =
~
a~h
aM
=
S~h amo~g(R)
(10)
If the interaction between flaws is negligible, an independent events assumption can be made. The expression of the cumulative initiation probability P1 of a structure ft of volume V can be derived in the framework of the weakest link theory. The expression of P1 can be related to the cumulative failure probability PRO of a single link by
418
In the case of global unstable propagation, the structural failure corresponds to the initiation and the expression of the cumulative failure probability PF is given by
PF = PI
(12)
It is worth noting that in the case of high cycle fatigue, the propagation stage tends to become negligible when compared, in terms of number of cycles, with the initiation stage (i.e., local failure). Since the propagation stage is neglected when Eq. (12) is used, this equation corresponds to a lower bound to the cumulative failure probability of the structure. Hence, in the following, 'failure' refers to local failure i.e. macroscopic initiation. 3. A N A L Y S I S
OF A FATIGUE
TESTS
In this section, a series of experiments performed on specimens made of ferritic SG cast iron are analyzed in details. These experiments have been carried out at different stress levels. The ratio between the threshold energy rate and the critical energy rate is on the order of 1/9. The specimens contain 'controlled' initial flaws. These specimens are tested under cyclic tension with two different load ratios (R : -I,R = 0.i). Each curve of a standard S-N plot can be associated with a constant failure probability. When the fatigue limits are known, the identification can be performed in two different stages. The first stage consists in the identification of the flaw size distribution. A minimization scheme is used to determine the minimum error between all the available experimental data on fatigue limits [3]. If we assume that the maximum flaw size is bounded by aM, the flaw size distribution f0 can, for instance, be fitted by a beta distribution
fo(a)=
aa-l(aM_a) ~-1 B , ~ a ~+z-1 , w h e n 0 < a < a M ,
a > 0, /3 > 0
(13)
where a and ~ are the parameters of the beta function, and B~Z is equal to B(a,/3), and where B(., .) is the Euler function of the first kind. The parameters to identify are the powers c~ and fl of the beta distribution, the volume ratio ~, v and the threshold stress Sth. The first step of the identification is applied to the experimental results obtained for the load ratio R = 0.I. It leads to the following values: a : 4.7, /3 = 25, ~V = I, and Sth -- 105 MPa. The second stage of the identification concerns the crack growth law (parameters C, n and m). In tension, this identification is performed by studying one constant cumulative failure probability (e.g. 50~ A constant value of PF is described by a constant cumulative failure probability PRO. The cumulative failure probability 5070 is used to minimize an error function [3]. The following values are obtained: n : 2.0, and C* = : 5.9 x 10 -5. In Figure I predictions of the number of cycles to failure are compared with the experimental observations. Three points were used for the identification and the remaining points are predictions. This figure shows that the identified laws are in good agreement with the experimental results.
419
~" 400 ----350
'~ ~w.
,.
.......
,
9 PF=90 % (Experiments)
........
-'QQ.~.~ " "~$~
...__
~300 "o, " T : ~
9mma
9
PF = 50% (Experiments)
9
PF = 1 0 %
(Experiments)
250 r~
200 o~,,~
o
9 9o 9
- -o-
- PF = 90
%
(Identification)
150
" " 0 " ~ PF = 5 0 %
(Identification)
:~ loo 105 106 107 10 4 Number of cycles to failure, N F
.... o " - PF = 10 %
(Identification)
Figure 1. Predicted failure probabilities compared with experiments (R - 0.1).
4. V A L I D A T I O N O F T H E I D E N T I F I E D
MODEL
The previous results are validated by comparing them with other experimental results obtained independently. The two stages of the numerical identification are compared separately. The flaw distribution is compared with an experimentally identified one and the propagation law is also compared with a measured propagation law. Finally these results are used to predict other experimental results. 4.1. Flaw distribution Experimental investigations are performed to quantify the identified flaw distribution, and also to get more information about the distribution such as the size of the largest defect aM. Systematic microscopic observations of 50 fractured surfaces were performed using a Scanning Electron Microscope. The initial defects on the fractured surfaces can be distinguished with no difficulty since the stable propagation area has different morphological characteristics as compared to those of the initial defects. Pictures of the fractured surfaces were stored in a SUN workstation and an image analysis program was used to determine the defect distribution. Flaws with a diameter less than 80 pm were not considered in order to avoid confusion with graphite nodules (with maximum size on the order of 60 pm in diameter [4]). In Figure 2 the experimental flaw distribution is drawn and compared with the identified distribution. The identified distribution is in good agreement with the experimental one. The experimentally measured value of aM is 400 #m. This result shows that we are dealing with short cracks, and that the threshold energy release rate can be considered as a constant for flaws of this size as shown in [5].
420
Identification
o
7
'
0
t
'
'
'
t
'
'
'
I
Image analysis '
'
'
J
'
'
'
9
6 =
=
9
5 4
m
3
2
o Z
inmmhmmmmmimmmmm-,.m~
0 0
0.2 0.4 0.6 0.8 Normalized flaw size, a/a M
Figure 2. Predicted failure probabilities compared with experiments (R = 0.1). Identification
[]
a o = 6. mm
ao = 0 . 2 4 m m
o
ao = l . m m
10 -7
10_8
-~
10_ 9
~a
Q
10-10
~
) (3
10-1] 10-12
[]
t' o ,.
, A,
A
, ,A
I
2 4 6 8 10 20 Stress intensity range, AK (MPa m 1/2)
Figure 3. Crack growth rate as a function of stress intensity range.
421 4.2. P r o p a g a t i o n
law
Figure 3 shows the crack growth rate as a function of the stress intensity range. The solid curve is the identified one. The other curves represent experimental results obtained on specimens made of SG cast iron. The solid squares concern an artificial short crack of initial length a M - - 240 #m [6]. The open circles correspond to an artificial crack of initial size a0 = 1 mm. The open squares concern an artificial long crack of initial length a0 = 6 mm. [6]. The identified curve is in good agreement with the experimental one for short cracks especially near the threshold regime. The distinction with the curve for long cracks is mainly described by threshold differences. 4.3. P r e d i c t i o n
of experimental
results
The comparison between the threshold values calculated by Eq. (11) for the load ratio R = - 1 and R = 0.1 allows to identify the value of parameter m (m = 0.59). Figure 4 shows the comparison between the experimental results for R = - 1 and the predicted results using the parameters identified previously (R = 0.1) and m. The predictions are
400 350 E
'
'
'
9
....
I
.
.
.
.
.
.
.
.
I
.
.
.
.
.
.
.
9
PF = 90 %
9
PF = 50 %
(Experiments)
.
(Experiments)
o * 9149
300
9
250
--o--PF=
~g
PF= 10% (Experiments) 90 %
Prediction
200 . . . . . . . . . "0. . . . . . . . -0 o~-~
150
0 O001D
----c)--- PF = 50 %
Prediction
9 O00O~ .... 0.--- PF = 10 %
100
' ' ....... 104 105 106 10 N u m b e r of cycles to failure, N F . . . . . . . .
. . . . . . . .
Prediction
Figure 4. Predicted failure probabilities compared with experiments (R = - 1 ) .
in reasonable agreement with the experimental data. This result shows that function g accounts for the influence of load ratio for different cumulative failure probabilities. 5. A P P L I C A T I O N
TO A STRUCTURE
The identification, validation of the present model allows to make an extension of the model to real structures. A post-processing program is developed to compute the failure
422 probability of a cast structure. Figure 5 shows the flow-chart of the program called ASTAR.
Loading
ALTAR I (Stress fieldand volumeof eachelemen9
~
uivalent
stressfield at each integrationpoint)
~umerical computationof ~(ac)) i
~~"u~eprobaUilityofeachelement9 ~ ( [
ii
Flaw ) distribution
(Failureprobabilityof thestructurePF)
Figure 5. Flow-chart of the program ASTAR.
From the results of an elastic computation of a structure through a Finite Element analysis, ASTAR evaluates the equivalent stress at each integration point. A critical flaw size is associated to this stress level and to a given number of cycles, N, which corresponds to the flaw size which becomes critical after N cycles. The failure probability at the integration point is then calculated by numerical integration of Eq. (2). The integration over the total volume of a finite element gives the failure probability of the element. Then the failure probability of the structure can be calculated according to Eq. (II). This procedure has been applied to a suspension arm designed by Renault car company. The mesh consists of 3712 triangular shell elements. The industrial FE package
423
Figure 6. contours of the failure probabilities PROfor a maximum stress level on the order of 300 M P a and a number of cycles N = 107 cycles.
424 ABAQUS [7] is used to perform the elastic analysis of the structure. ASTAR is run by using the results of the FE computation. Figure 6 shows the contours of the failure probabilities PFO for a maximum stress level on the order of 300 MPa and a number of cycles N - 107 cycles. It must be noted that this procedure could take account of different flaw size distribution in different parts of the structure.
6. C O N C L U S I O N S A reliability analysis taking account of flaw size distributions has been developed for components subjected to cyclic loading conditions. Emphasis is put on the initiation stage, which is directly related to the evolution of initial flaws. An expression of the cumulative initiation probability is derived in the framework of the weakest link theory and by assuming that the flaws do not interact. Experimental data on SG cast iron in tension are analyzed within this framework. The predictions of the whole set of data is in good agreement with the experimental number of cycles to failure. This last result shows that the expression of cumulative failure probability proposed herein is able to model fatigue data obtained on SG cast iron and that the model can describe the influence of the load ratio. Typical applications of this approach concern the reliability analysis of cast components. More and more tools are available to predict the different flaw size distributions in the different parts of a cast, whereas the fatigue behavior of the material is in general mainly derived from experiments performed on so-called flawless specimens. These two tools should eventually be coupled. The post-processing approach can predict the reliability of the whole component under cyclic loading conditions, in other words, it enables to predict the number of cycles to macrocrack initiation (local failure), or the probability of reaching a certain number of cycles without failure at any point of the component.
ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support of Renault through contract CNRS/109 (H5-24-12) with the Laboratoire de M~canique et Technologie, Cachan. REFERENCES
1. F. Hild and S. Roux, Mech. Res. Comm. 18(6) (1991) 409-414. 2. J. Pellas, G. Baudin and M. Robert, Recherche A~rospatiale 3 191-201 (1977). 3. A.-S. B~ranger, R. Billardon, F. Hild and H. Yaacoub Agha, FATIGUE '96, Berlin (Germany) (1996)1269-1274. 4. P. Clement, J. P. Angeli and A. Pineau, Fatigue Eng. Mat. Struct. 7 (4) (1984) 251265. 5. P. CMment and A. Pineau, Journ~es Internationales de printemps de la SFM, Paris 22-23 May (1984) 203-218. 6. P. Clement, CNAM Report (1984) Paris. 7. H.D. Hibbitt, B. I. Karlsson and P. Sorensen, Abaqus, version 5.5. (1995).
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AUTHOR INDEX
A g a s s a n t , J . E ..................... 373 A h m e d , M ......................... 197 A l e x a n d r o v , S .................... 247 A l m e i d a C a m a r g o , N . H ...... 83 B a c r o i x , B ......................... 331 B a d e a , L ................................ 3 Bagaviev, L ........................ 281 B a m m a n n , D.J ..................... 99 B a n a b i c , D ......................... 257 Barlat, E ............................ 265 Barri~re, T .......................... 165 B a u d e l e t , B ........................ 289 B e g u m , S ........................... 143 B e n n a n i , B ........................ 165 B6ranger, A.S ................... 415 B i l l a r d o n , B ....................... 415 B i t t e n c o u r t , E ...................... 83 B o g a t o v , A . A ....................... 71 Boivin, M .......................... 341 B o u d e , S ............................ 125 B o u d e a u , N ........................ 215 B o u r g a i n , E ......................... 23 Boyer, J.C ............................ 13 B r e m , J.C ........................... 265 B r e s s a n , J.D ....................... 273 B r e t h e n o u x , G ..................... 23 Brunet, M .......................... 205 B u b l e x , E .......................... 341 C a o , J ................................. 301 C e s c o t t o , S ........................... 33 C h a n d r a , A .......................... 89 C h a r l e s , J . E ......................... 33 C h i k a n o v a , N ..................... 247 C h i n e s t a , E ...................... 311 C h i o u , J . M ......................... 135 C h u n g , K ........................... 265 C o m b e s c u r e , A .................. 385 D a w s o n , P.R ........................ 99
D e n g , X ............................. 341 Di Sciuva, M ..................... 395 D o e g e , E ............................ 281 D o h r m a n n , H ..................... 281 D o l t s i n i s , I.St ..................... 111 D o m i l o v s k a y a , T.V. ............. 71 Drazetic, E ........................ 165 Dutilly, M .......................... 321 E1 M o u a t a s s i m , M ............. 341 F e d o t o v , V.E ............. ........... 51 Gelin, J.C .................. 215, 321 G i l o r m i n i , P. ...................... 331 Giusti, J ............................... 23 H a b r a k e n , A . M .................... 33 Hall, E R ............................ 135 H a m b l i , R .......................... 125 Hartley, E .......................... 135 H a s h m i , M . S . J ............ 143,197 Hild, E ............................... 415 H u a n g , Y. ............................. 89 Icardi, U ............................. 395 Karafillis, A ....................... 301 K a r i m , A . N . M ................... 143 Karr, D . G ........................... 225 K i m , Y.S ............................ 155 Kiselev, A . B ......................... 43 K o l m o g o r o v , V.L ........... 51, 61 L a u r o , E ............................ 165 L e g e , D.J ........................... 265 Li, Y. .................................. 185 L i b r e s c u , L ........................ 395 M a z a t a u d , E ........................ 23 M g u i l - T o u c h a l , S .............. 205 M o n t 6 n , I ........................... 311 M o r e s t i n , E ............... 205, 341 M o s h e r , D . A ........................ 99 M u r a t , M ............................. 83 M u z z i , M ............................. 23
O l m o s , E ........................... 311 Ofiate, E ............................. 349 O s t r o w s k i , M ..................... 301 O u d i n , J ..................... 165, 175 Park, J.Y. ............................ 155 Picart, E ............................. 175 Piechel, G .......................... 175 Pillinger, I .......................... 135 Poitou, A ............................ 3 1 1 Potiron, A .......................... 125 P r e d e l e a n u , M ....................... 3 P r o u b e t , J ........................... 289 R a b e e h , B .......................... 185 R e e s , D . W . A ...................... 235 R e s z k a , M .......................... 125 R o j e k , J ......... ~.................... 349 R o k h l i n , S.I ....................... 185 Smirnov, S.V. ................ 61, 71 S o b o y e j o , A . B . O ............... 185 S o b o y e j o , W . O .................. 185 Spevak, L . E ......................... 51 Staub, C ............................... 13 Stein, E .............................. 405 S u b h a s h , G .......................... 89 TefSmer, J ........................... 4 0 4 Torres, R ............................ 311 Traversin, M ........................ 33 T s z e n g , T.C ....................... 361 Venet, C ............................. 373 Vergnes, B ......................... 373 Vilotic, D ........................... 247 W a n g , K . E ........................... 89 W i m m e r , S . A ..................... 225 Wu, W.T. ........................... 361 Y a a c o u b A g h a , H .............. 415 Z h u , Y.Y. .............................. 33
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