Mathematical and Physical Simulation of the Properties of Hot Rolled Products

Vy=Vxtg7 TyS/zcTpSin 7 Tx = T y = 0

<;p=a(T-To)+qJ

-rx«Ty=0

^=aa^(T-To)

f<^«a/v
t<;g>=<^awO"-i"oj|

Tx=Ty=0

<;p=a(T-To)-K|

Figure 5.1 Finite-element mesh and boundary conditions for a general case of flat rolling The discretization of the thermal part of the problem is performed using the same 4-node quadrilateral elements. In some cases, involving very steep temperature gradients, especially near the surfaces of the rolled samples, 12-node quadrilateral elements are used and the meshes for thermal and mechanical solutions are identical in the comer nodes. Using 12-node elements for the thermal part allows for a more accurate simulation of the temperature gradients in the deformation zone, while the memory required to store the conductivity matrix H remains approximately equal to that necessary to store the tangent stiffness matrix K. MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

133 5.4.1

Hot rolling of steel

Hot rolling of a 50 mm thick C-Mn steel slab, reduced by 25% in one pass, is presented below. The slab is heated to a uniform temperature of 1050°C and cooled for 15 seconds in air. The process parameters are: roll radius 340 mm, roll velocity 22 rpm, and the friction coefficient is 0.25. Shida's equations (Shida, 1979), describing the yield stress as a fiinction of the carbon content, temperature, strain rate and strain, are used as the constitutive model. Figures 5.2 and 5.3 show the strain rate, effective strain, average stress and temperature distributions. Strain rate concentrations, characteristic of the rolling process, are observed at the initial contact point between the roll and the plate. The shear strain rate concentrations near the contact cause a non-uniform strain distribution. The largest strains appear near the surface.

effective strain rate, s*''

0

10

20

30

40

50

60

70

80

90

X, mm shear strain rate, S'^

0

10

20

30

40

50

60

70

80

90

X, mm

Figure 5.2 Calculated distributions of effective strain rate, shear strain rate and effective strain for hot rolling of steel THE FINITE ELEMENT METHOD IN METAL FORMING

134

X, mm Figure 5.3 Calculated distributions of average stress and temperature during hot rolling of steel

Analysis of the average stress field in Figure 5.3 shows that compressive stresses appear almost in the whole deformation zone. Some tensile stresses are observed at the entry plane at the center of the plate and at the exit close to the surface. The temperature field, also shown in Figure 5.3, agrees with the numerous experimental data presented by Pietrzyk and Lenard (1991). As expected, the coolest part of the plate is the one near the contact zone and a sharp temperature gradient towards the plate center is observed. Low rate of heat loss of the central portion, before entering to the roll gap, is also noticed. Recall that the plate was heated to a uniform temperature of 1050°C in the furnace and then cooled for 15 seconds in air. The plate also experiences some heating in the roll gap, the result of the work done on the plastically deforming metal. The thermal-mechanical finite-element model described in this chapter has also been validated under industrial conditions. Simulation of hot rolling a C-Mn steel strip, rolled in 12 passes, including five roughing passes and seven continuous finishing passes, has been performed. The rolling parameters and results of calculations are given in Table 5.3. Rows marked "pyro" or "coil" in this table contain temperatures measured by an optical pyrometer (first number) and the calculated surface temperatures at this point (second number). The final column in Table 5.3 contains the resuhs of calculations of the austenite grain size, which have been performed using the microstructure evolution model, described in the next chapter.

^M THEMA TJCAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

135 Table 5.3 The results of simulation of the hot strip rolling process aver, No thickn. red. roll interpass roll torque force time diameter temp velocity mm mm s kN/mm Nm/mm rpm 150.00 1314 95.1 120.00 0.200 1 940.0 9.4 1268 21.00 7.9 730 72.00 0.400 2 18.7 911.0 1256 20.00 1310 9.8 54.00 0.250 3 910.0 14.4 1228 33.00 610 7.3 46.00 0.148 4 14.0 655.7 71.00 1195 200 4.1 35.00 0.239 5 614.8 40.5 71.00 1160 6.1 320 pyro 1097/1087 24.20 0.309 749.0 6 10.8 987 4.9 28.32 610 13.97 0.423 7 2.6 631.8 58.17 12.7 970 650 8 11.99 0.142 2.4 647.5 957 66.13 100 4.4 9.48 0.209 9 1.9 663.5 938 81.62 180 6.9 7.36 0.224 10 1.4 652.0 924 170 106.99 7.3 6.16 0.163 11 632.3 131.81 908 1.2 90 5.2 6.00 0.026 12 679.1 3.3 893 126.00 6 1.0 pyro 881/885 coil 600/610

grain size jLun

50.3/49.6 28.0/27.5 34.7/33.6 33.3/32.4 30.3/29.6 30.6/29.8 30.6/29.8

1400 The grain size was calculated for the ] roughing train •-I + 1 u * *• front (first number) and the tail (second 1300 number) end of the plate. Figures 5.4 and 1200 5.5 show the time-temperature profiles for the rolling schedule of Table 5.3. Two val- o 1100 o ues of the temperature are presented, one 1000 3 at the center of the plate and the other, at 2 900 the surface. The results of measurements E using the optical pyrometer behind stands 800 number 5 and 12 are shown in Figures 5.4 700 and 5.5, as well. The tail end of the strip 600 enters the finishing train at a temperature lower, by about 60**C, than the front end. 500 1 1 1 1 \ 1 This difference disappears at the coiler, 20 40 60 80 100 120 140 160 due to the acceleration of the mill when time, s the front end leaves the last stand. Analysis of the results in Table 5.3 and Figure Figure 5.4 Time-temperature profiles for strip 5.4 shows that the model is capable of rolling in the roughing mill according to the simulating industrial rolling processes with rolling schedule in Table 5.3 good accuracy. Similar studies have also been performed for hot rolling of a thick niobium steel plate in a two-stand reversing mill (Pietrzyk et al., 1996) and the resuhs are presented in the next chapter together with the evolution of the microstructure.

THE FINITE ELEMENT METHOD IN METAL FORMING

136

1200

-+A

[•

measurement

ZA

1100

surface

800

"^

700

A Zl

1100

ID

surface measurement

1000 900

I 900 laminar cooling

"TO

g. E o

center

1200 centEr

[water descaier

800 4 700 600 H

600

finishing train - front end I \ 1 1 \ 1 \ r~ 500 160 170 180 190 200 210 220 230 240 time, s

500

finishing train - tail end —1 1 1 1 1 I f — 190 200 210 220 230 240 250 260 time, s

Figure 5.5 Time-temperature profiles for strip rolling in the finishing mill according to the rolling schedule in Table 5.3

5.4.2

Asymmetrical rolling

Simulation of the asymmetrical rolling process presents particular difficulties connected with thefi-eebending of the plate after exitfi-omthe roll gap. Asymmetric rolling is an important part of industrial rolling technology. Asymmetry can be deliberately introduced to reduce the rolling force (Sinicyn, 1984) or to control the development of a required curvature (Too and Barnes, 1989; Dyja and Pilarczyk, 1993). In many instances, however, the development of asymmetric deformation in a rolled plate is an undesirable phenomenon, and should be avoided. The asymmetry may be caused by variations in roll speed, roll diameter or roll surface conditions or by non-uniform temperature distributions or properties through the plate thickness. Bi-metallic rolling is an extreme example, where differing material properties cause asymmetric deformation. Whatever the cause of this problem, control strategies to minimize the asymmetry would be of significant industrial value. Such strategies can be developed through the use of numerical simulation techniques that examine, in detail, the effect of a very wide range of process and material parameters. Before the data can be gathered, however, the models used to generate the data must be carefully validated against experimental measurements. Published reports on asymmetric rolling (Pietrzyk, 1986; Hamauzu et al., 1987; Cao et al., 1993; Park et al., 1993; Richelsen, 1994; Lenard et al., 1994) have produced little evidence of experimental validation. Therefore, providing a careful experimental validation of the results for asymmetrical rolling is very important. The tests performed by Pietrzyk et al. (1996a) are reported briefly below. Two finite-element models are validated experimentally. These are: (i) elastic-plastic nonsteady state model by Pillinger (1992) which will be referred to as epfep3 and, (ii) a rigid-

MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

m plastic steady state model described in this chapter, referred to as elroll. Both models include thermal-mechanical coupling. The experiments used for the validation include both hot and cold rolling processes. Thick aluminum plates were used in the cold rolling tests. The plates were made in two halves joined along a longitudinal, vertical mid-section. This mating face contained a grid, the deformation of which was measured after rolling in order to determine the strain distribution. The asymmetry was introduced by various roll diameters. Two diameter ratios were used (183/202 and 193/202). The smaller roll in the second set was prepared with an increased surface roughness. The rolls were ground after the cold rolling experiments and the diameter ratios in hot rolling were 182/201 and 192/201. The plates were rolled to four different reductions. The stress-strain curve of aluminum used in the models was determinedfi-omthe compression test to be: or^ =94.5 + 218.7/-^^^

(5.115)

where <jp is the flow stress, and s is the true strain. The hot rolling of thick steel samples with various thicknesses was undertaken to assess the amount of curvature, and whether the plate turned up or down. The steel composition was 0.15%C, 0.55%Mn for 6 mm thick samples and 0.17%C, 0.44%Mn for 4 mm thick samples. The hyperbolic sine equation, Eq. (3.24), with the constants K = 100, A = 2xl0'^^ m = 0.2, n = 0.2 and activation energy in the Zener-HoUomon parameter Q = 350000 J/mol, were used in the calculations for both steels. In all cases, different amounts of reduction were examined. The samples measuring 4 or 6 mm in thickness were rolled to three different reductions at various temperatures. The samples were heated to 890°C or 1 lOO^'C in the furnace, transported to the rolling mill and rolled. The air-cooling during the transport of the sample from the fiimace to the mill was included in the calculations. The time necessary to transport the sample varied between 5 and 15 s, resulting in a temperature drop of 40''C - lOO^'C, depending on the plate thickness. The roll angular velocity was 20 rpm. Cold rolling of aluminum: The resuhs of calculations of the strain distribution in the roll gap during cold rolling of aluminum strips are compared with the measurements in Figures 5.6, 5.7, 5.8 and 5.9. Symbol D in these figures represents the roll diameter. The analytical results, by elroll, were obtained using a friction coefficient of 0.2, determined from the ring compression test. The graphs show that the finite-element models predict the metal flow reasonably well. The results obtained from epfep3 account for the influence of the non-stationary part of the process, when the front end of the plate is rolled and dynamic effects are important. Beyond the stress and strain fields in the roll gap, the bending of the plate after exit is an important parameter, which is to be controlled during rolling. As shown by Pietrzyk et al. (1993) and Pietrzyk et al. (1996b), the proper prediction of the plate curvature presents significant difficulties. Comparison of the measured and predicted (by elroll) plate curvature for the current experiment is shown in Figure 5.10. Positive curvature indicates turning towards the upper (smaller) roll. Figure 5.10a demonstrates that the results are very sensitive to the fiiction coefficient used in the model. In general, the curve obtained for the friction coefficient of 0.2 gives results closest to the measurements. The largest discrepancies are observed for large reductions, probably due to the fact that in the real process, the friction coefficient THE FINITE ELEMENT METHOD IN METAL FORMING

138 crease with increasing pressure/reduction. Figure 5.10b shows the results of calculations for a non-workhardening, ideally plastic, material, a workhardening material with the flow stress described by Eq. (5.115) and a workhardening material with a significantly increased slope of the strain-hardening curve. Figure 5.10b indicates that, as may be expected, the effect of hardening increases with increasing reduction. The most pronounced effect is observed at the point when hardening is introduced in the model, while a further increase of the intensity of hardening has a negligible influence on the bending of the plate.

8 10 12 14 16 18 20 22 24 26 28 30 X, mm

Figure 5.6 Comparison of the measured and calculated strain fields during asymmetric rolling of aluminum strips, using a roll diameter ratio of 184/202 mm and a reduction of 13%

prediction > epfep3 [reduction 0.13

10

12

14

16

18

20

22

24

X, mm

4 E E

prediction - eiroli reduction 0.13

2 0

""-2-1 -4

2

4

6

8 10 12 14 16 18 20 22 X, mm

The results obtained from the epfep3 program are not presented in Figures 5.10 and 5.11. As far as the epfep3 code's predictions of bending are considered, the non-steady state model showed good correlation with the low reduction experiments but did not reproduce the turndown observed at high reductions (see Figures 5.6-5.9). The curvatures predicted by epfep3 for the experimental data in Figure 5.10 were 10, 5,10 and 6 m"^ for the reductions 0.13, 0.24, 0.33 and 0.49, respectively. A friction factor of 0.4 was used in these calculations.

MA THEMA TICAL AND PHYSICAL SIMULA HON OF THE PROPERTIES OF HOT ROLLED PRODUCTS

139

Figure 5.7 Comparison of the measured and calculated strain fields during asymmetric rolling of aluminum strips, using a roll diameter ratio of 184/202 mm and a reduction of 24%

-"*

i? =202 mm

15

20

30

X, mm

15

20

25

30

35

40

45

X, mm

Figure 5.8 Comparison of the measured and calculated strain fields during asymmetric rolling of aluminum plates, using a roll diameter ratio of 184/202 mm and a reduction of 33%

8^ E

E

[prediction-epfep3 reduction 0.33

X, mm ^:^:::30.50-|

^t^^^ 30

35

THE FINITE ELEMENT METHOD IN METAL FORMING

140

.Figure 5.9 Comparison of the measured and calculated strain fields during asymmetric rolling of aluminum plates, using a roll diameter ratio of 184/202 mm and a reduction of 0.49% 4i predictbn • eiroll

V\~~2:--c—P^1M mm

reduction 0.49

0 4

1

1 —

10

'!^iS--^-^''^D^2mm

15

20

25

30

35

40

45

X, mm

Q

measurement

-A—

FEM.f=0.18

-0—

FEM,f=0.20

-B—

FEM.f=0.22

f° A /S V s

nU

measurement ideal plasticity hardening, eq. (5.115) increased hardening

Figure 5.10 Measured and calculated curvature of the aluminum plate exiting the roll gap during cold asymmetric rolling (roll diameter ratio 184/202); (a) various friction coefficients; (b) various constitutive laws in the model

Similar results obtained for the roll diameter ratio of 193/202 are presented in Figure 5.11. In this experiment the smaller roll's surface roughness was increased, and because of this, the asymmetry was due to both the different diameters and the different friction coefficients. The MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

141 results show agreement for the friction coefficient of 0.18 for the larger roll and 0.21 for the smaller roll. The coincidence is good for medium reductions. The discrepancies are larger for both small reductions and heavy ones. This confirms the conclusion that the discrepancies may be due to the friction coefficient's dependence on the roll pressure and the reduction, neither of which is not accounted for in the models. This dependence, however, is somewhat controversial. Results showing both an increase of the friction coefficient with increasing reduction (Karagiozis and Lenard, 1985; Guang-Ying and Huml, 1993), and a decrease of the coefficient with increasing roll pressures (Dobrucki, 1962), or with increasing yield stress (Roberts, 1978), have been published. Beyond this, Schey (1987) shows that the maximum possible coefficient of friction decreases when interface pressures increase above the yield stress of the material. The current measurements and calculations of plate bending support the latter hypothesis. The results of calculations, carried out for the friction coefficient, varying from 0.22 for light reductions to 0.18 for heavy reductions, are shown in Figure 5.12. The improvement of the similarity between the predictions and the measurements is obvious. Hot rolling of steel: The objective of the steel rolling experiment was the as15 sessment of the model's ability to predict D measurement the bending of the plate after exit fi-om FEM,fs0.16/0.19j -2^ the roll gap. Due to a larger number of 10 H FEM,f^ai8A).21 factors affecting hot rolling (temperaFEM, fi=0.22«.25 ture, strain rate), the analysis is even more complex than for the cold rolling experiment. The results obtained for to stress-strain Eq. (3.24) and various but constant values of the friction coefficient /are shown in Figures 5.13-5.16. Analysis of the results shows that the reduction and the entry thickness are the parameters which most affect the turning -ion __l j _ I of the plate. The influence of the tem0.3 0.4 0.5 0.6 0.2 0.1 perature, however, is negligible. The reduction qualitative agreement between the measurements and the model's predictions is Figure 5.11 Measured and calculated curvature good. Beyond the small reduction, the of the aluminum plate exiting the roll gap durmodel predicts the bending quantitatively ing cold asymmetric rolling using a roll diquite well when the friction coefficient of ameter ratio of 193/202; various friction coeffi0.2 is assumed. The discrepancies appear- cients/for the upper and the lower roll ing at the small reductions, require fiirther investigation. The validation of the finite-element codes permit several observations and conclusions regarding the application of these programs to asymmetrical rolling. The asymmetrical process is difficult to model. Indeed, an adaptation of the existing codes does not present numerical problems, but the results appear to be extremely sensitive to boundary conditions, in particular to the friction coefficient. The steady-state model clearly shows a change in the direction of the turning of the plate as the reduction increased. The non-steady state model shows good correlation with the low reduction experiments, but does not reproduce the turn-down observed at high reductions. The THE FINITE ELEMENT METHOD IN METAL FORMING

142 models are expected to work very well for asymmetrical cold rolling with tensions, when the turning of the strip is constrained. Difficulties appear when rolling thick plates with unconstrained motion before entry to, and after exit from, the roll gap is considered.

a)

b) n ^

measurements ]

10-

prediction, FEMj

Q

measurements^

^—•

prediction, FEMj

£

f s

Figure 5.12 Curvature of the plate calculated for thefrictioncoefficient decreasing with increasing reduction compared with the measured values using a roll diameter ratio of (a) 187/202 and (b)193/202

15

thickness 6 mm temperature 850°C

10

5

thickness 4 mm temperature SOOoC

D

measurement]

-A—

FEM,f==0.15

•^—

FEM,fs0.20 I

1—B—

FEM,f=0.30 ]

[D A

measurement

ZA

FEM, f=0.15

•s

FEM. 1^0.20

\/ n I U

FEM,f=0.30

-10

-15

roil diameter ratio 182/201 0.0

n

\

0.1

0.2

1

r~

0.3 0.4 reduction

0.5

0.2

0.3 0.4 reduction

Figure 5.13 Measured and calculated curvature of the steel plate exiting the roll gap during hot asymmetric rolling, heating temperature 890*" using a roll diameter ratio of 182/201; (a) entry thickness 6 mm; (b) entry thickness 4 mm MA THEMA JJCAL AND PHYSICAL SIMULA HON OF THE PROPERTIES OF HOT ROLLED PRODUCTS

143

thickness 6 mm temperature 1000°C roll diameter ratio 182/201 D -^—

0.2

thickness 4 mm temperature 965°C roll diameter ratio 182/201 Q

measurement! FEM,f=0.15

-^—

FEM,f=0.20

- Q -

FEM.f^O.aO

"1 0.1

I I 0.3 0.4 reduction

\ 0.2

measurement]

^s—

FEM.f=0.15

^0—

FEM, f=0.20

g —

FEM, f=0.30

\ r 0.3 0.4 reduction

Figure 5.14 Measured and calculated curvature of the steel plate exiting the roll gap during hot asymmetric rolling, heating temperature 1100°C using a roll diameter ratio of 182/201; (a) entry thickness 6 mm; (b) entry thickness 4 mm

a)

b) thickness 6 mm temperature 800°C

15

Q

10

15

thickness 4 mm temperature 800°C Q

measurement]

10 H

-^s—

FEM,f=0.15

measurement]

7^—

FEM.f=0.15

^^—

FEM, f=0.20

g —

FEM, f=0.30

% ^

-0—

FEM,f^0.20

g—

FEM.f=0.30 J

I 0 O

-10 H

-5H

roll diameter ratio 192/201

0.0

0.1

T" ~1 0.2 0.3 0.4 reduction

~r

0.5

roll diameter ratio 192/201

-10 0.0

I 0.1

"1 0.2

r 0.3 0.4 reduction

I 0.5

Figure 5.15 Measured and calculated curvature of the steel plate exiting the roll gap during hot asymmetric rolling, heating temperature 890**C using a roll diameter ratio of 192/201; (a) entry thickness 6 mm; (b) entry thickness 4 mm Analysis of the results for the cold rolling of aluminum plates shows that the prediction of the turning of the plate after exit from the roll gap is very sensitive to the friction coefficient in the model. It is difficult to obtain good agreement betv^een the measurements and calculaTHE FINITE ELEMENT METHOD IN METAL FORMING

144 tions when a constant value of this coefficient is assumed. It is shown that the introduction of a coefficient, decreasing with increasing roll pressures, improves the accuracy significantly. It is observed further that, when the turning of the plate is predicted correctly, both models predict the strain fields in the roll gap reasonably well. Investigation of the sensitivity of the results to the material's resistance to deformation shows that the slope of the hardening curve has little effect on the turning of the plate. However, neglecting the hardening and assuming ideal plasticity lead to a noticeable change of the results.

a)

b) thickness 6 mm temperature 1000°C

15 10H

ID

measurement

thickness 4 mm temperature 940°C

15

Q

10 H

measurement]

A

FEM.f^0.15

-£s— FEM,f^0.15

/\ \y

FEM, f=0.20

-0—

FEM.M).20

-a-

REM. M).30

g—

FEM,f^0.30

Z^

-5H -10 H

roll diameter ratio 192/201 0.0

0.1

0.2

0.3 0.4 reduction

0.5

roll diameter ratio 192/201 ~-i 1 r~~ 0.1 0.2 0.3 0.4 reduction

0.0

0.5

Figure 5.16 Measured and calculated curvature of the steel plate exiting the roll gap during hot asymmetric rolling, heating temperature 1 lOO^'C, using a roll diameter ratio of 192/201; (a) entry thickness 6 mm; (b) entry thickness 4 mm

Analysis of the hot rolling of steel shows very good agreement between the measurements and calculations, as far as the character of the curvature versus reduction relationship is considered. Quantitatively, over-estimation of the curvature for very small reductions was observed in all cases. Since these reductions are not used in industrial practice, the predictive ability of the model can be considered to be good. In order to demonstrate further the abilities of the rigid-plastic finite-element thermalmechanical model as far as the simulation of asymmetrical rolling is considered, calculations for asymmetrical cold rolling of a 10 mm thick plate were performed. The remaining process parameters were as follows: lower roll radius 250 mm, upper roll radius 238 mm, entry angle 3°, friction coefficient 0.2. Calculated fields of the effective strain and the average stress are presented in Figure 5.17. The results show that due to the lack of symmetry, the plate is turning dovm at the exit. The calculated curvature is 7.8 m\ The resuhs further show that the model is capable of simulating the fields of strain rates, strains, stresses and temperatures for various asymmetrical processes, when different factors causing the lack of symmetry, such as various roll diameters, roll rotational velocities, friction coefficients of upper and lower roll etc., exist separately or together. Accounting for the asymmetry, resulting from the nonMATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

145^ uniform distribution of plate properties or temperature through the thickness, can be accounted for as well.

effective strain

0

5

10

15

20

25

30

35

40

45

50

X, mm average stress, MPa

X. mm Figure 5.17 Calculated fields of the effective strain and the average stress during asymmetrical rolling

5.4.3

Porosity closure

The increasing application of continuous casting introduces additional problems to the simulation of metal forming processes. The elimination of porosity is essential in industrial practices, and because of this, the ability to predict the behavior of voids should be one of the features of the finite-element models. The objectives of the present section are twofold. The first objective is to show an application of the finite-element model to the simulation of the behavior of a single void in a metal matrix subjected to plastic deformation. The influence of such factors as the material's strain and strain rate sensitivities on the behavior of voids is investigated. An attempt to account for the influence of the pressure of the gases in the pore on its closure is also undertaken. The second objective of the section is to analyse the behavior of the voids in industrial processes. An experimental validation of the approach is performed by comparing the results with the data of Wang et al. (1996) and Wang et al. (1997). The behavior of voids is first investigated in general, independently of the process. The assumption is made that a void is located in a uniform stress field, and a distortion of that field, caused by the void, is evaluated. This field is generated by the rigid-plastic thermalmechanical finite-element model, formulated for plane strain compression. A detailed deTHE FINITE ELEMENT METHOD IN METAL FORMING

146 scription of the model was given in section 5.1, and its application to the plane strain compression process is shown by Pietrzyk et al. (1993b). In order to create the uniform stress field, the width of the die is assumed to be larger than the width of the sample. Round voids, with one or two axes of symmetry, are considered. The finite element mesh used in both cases is shown in Figure 5.18. The two-dimensional model is used in the calculations. Thus, an inclusion is represented by a cylinder with a unit length and a diameter equal to that of the void. The behavior of that cylinder in the plane-strain field is assumed to be representative of the behavior of the spherical void. Results of the simulation of the behavior of voids depend on a description of the flow stress of the matrix (Pietrzyk, 1998). The following constitutive equation: A 1^ 023.0214

^38000"^

Gp =4.16£'^"f"-^*^exp\ - RT )

(5.116)

is used in the present calculations. Pietrzyk et al. (1995c) showed that the behavior of voids during the plastic deformation process also depends strongly on the state of stress, which in the current work, is expressed by the ratio of the invariants of the stress tensor /j / / j , represented by: ^^^CTi-KT;

(5.117)

where a^ and a^ are the dominant stresses.

a)

Figure 5.18 Finite element mesh for pores with two axes (a) and one axis (b) of symmetry

During compression, the required state of stress is created by imposing a relevant pressure on thefi-eesurfaces. The history of stress state, necessary for the simulation of the void's cloMATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

147 sure during rolling, has been obtained from the rigid-plastic finite-element model for the flat rolling, described in section 5.L The pressure of gases present in the void slows down the elimination of porosity and its effect should be accounted for in modeling. However, there is a certain lack of information regarding changes of this pressure. The present calculations show that for initial pressures of about 0.1 MPa, the influence of the gas pressure is observed only at the final stage of the closure. Complete closure is found to be impossible without diffiision of the gases in the metal. The relation describing the decrease of the mass of the gas in the void is suggested: M = MQ exp{-cpt)

(5.118)

where MQ is the initial mass of gas in the void, p is the pressure, / is the time, and c is a constant. Eq. (5.118) gives a qualitative description of the phenomenon only, assuming that the decrease of the mass of gases in the void is proportional to the pressure. This assumption is realistic and it should not affect the numerical results in any significant manner. Preliminary results, showing the influence of the material's properties and process parameters on the closure of the pores, are given by Pietrzyk et al. (1995c) and by Pietrzyk (1998). Further results are shown below. The relation between the strain needed to close the pore and the state of stress for various material models is shown in Figure 5.19. Accounting for the strain rate and strain sensitivity is shown to slow the elimination of the pores. The calculated decrease of the relative pore volume, during rolling of an 80 mm thick continuously cast slab in rolls with various diameters, is presented in Figure 5.20. The effect of the gas pressure in the pore, assuming an initial pressure of 0.1 MPa, is shown in this figure as well The remaining parameters are: entry temperature 1180°C, reduction 0.22, average strain rate 3.6 s\ fi-ictioncoefficient 0.25, initial porosity 7%. 0.8-

-G-

ideal plasticity

0.6

2

•

roil radius:

constitutive model (5.116)J

0.4

480 mm

-#-»..^.

350 mm

h&

230 mm

230 mm 480 mm

0.2

0.0 -5

-4 -3 -2 -1 0 stress state coefficient ^

Figure 5.19 Strain necessary to close the pore as a function of stress state and material's model

"1

1

\

r-—\

r

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 strain

Figure 5.20 Changes of the relative volume of a pore during rolling, calculated for various roll radii

THE FINITE ELEMENT METHOD IN METAL FORMING

148

—

Calculations show that larger rolls create larger average stresses in the roll gap and foster closure of the pores. The current results agree qualitatively with the studies of Keife and Stahlberg (1980) and of Tanaka et al. (1986), who concluded that the effective strain and the time integral of the hydrostatic pressure are the main factors governing the closure of voids. Validation of the model was done by a comparTable 5.4 jj^g j^^ predictions with the data of Wang et al. Predictions of pore closure compared (1995^ 1997) j^iQ experiments included the with the observations of Wang et ai. ^QUi^g of steel samples with holes drilled trans(1996, 1997); rolljradius 300 mm ^ ^^^g^jy ^nd longitudinally. Examination of samno r/c V, Sc obser- ples, sectioned across the axis of the hole after m/s mm vation rolling, supplied information regarding the cor8.6 900 0.1 0.26 c+nw relation between the rolling parameters and the 1 8.6 1050 0.1 0.24 2 PC closing and welding of the holes. Eq. (5.116) 8.6 1200 0.1 0.22 c+pw was used as the flow stress model. Some of the 3 8.6 1200 0.4 0.25 c-fpw results for a reduction of 0.3 are given in Table 4 900 0.4 0.29 pc+pw 5.4, where h is the entry thickness, T is the tem5 20 6 20 1050 0.4 0.26 c+pw perature, Ec is the strain necessary to close the 7 20 1200 0.4 0.26 c+pw pore, and v is the roll velocity. The symbols c, w, p and n designate closed, welded, partly, welded and not welded pores, respectively. The experiments of Wang et al. (1996, 1997) involved various locations of pores, temperatures and thickness of the samples. Since the pores were closed during rolling, a comparison of theoretical and experimental results allowed only for the confirmation of the model's ability to predict the closure correctly. The model predicts closure for strains smaller than those used in the experiments. In Tests 2 and 5, partial closure of the void was observed, indicating that neglecting the influence of the gas pressure caused a slight underestimate of the strain necessary for closure. The available data did not allow the validation of the model with respect to changes of porosity during rolling are considered. The influence of the gas pressure on the behavior of the pores is demonstrated in Figure 5.21. The calculations were performed for various initial pressures. The pressure was changing during the process, following the changes of the volume of the pore and of the mass of the gas in the pore, calculated from Eq. (5.118) with c = 2 [MPa s]"^ Analysis of the resuhs in Figure 5.21 shows that the influence of the gas pressure at the primary stage of pore closure is negligible. The effect of this pressure becomes more important at the final stage of the closure, when the volume of the pore becomes small and causes an increase in the pressure. The observation of the results of the calculations shows that for larger pressures, complete closure of the pore is not possible. It should be emphasized, however, that the resuhs shown in Figure 5.21 depend strongly on the constant c in Eq. (5.118) which governs diffusion of gases. There are no reliable data regarding this phenomenon. Therefore, modeling of the closure of pores can be treated as a qualitative assessment of the problem, and thus allows comparative analysis only. Further analysis includes simulation of rolling after continuous casting. The initial state of the slab is described on the basis of the data published by Brimacombe and Samarasekera (1989). The porosity distribution through the thickness at entry to the roll gap has been assumed to vary between 7% and 3%. The current porosity influences on the thermal properties of the material are included in the computations. Figures 5.22 and 5.23 show the calcuhU THEMA nCAL AND PHYSICAL SIMULA HON OF THE PROPERTIES OF HOT ROLLED PRODUCTS

149 lated distributions of the temperature, effective strain, average stress and porosity during rolling of an 80 mm thick continuously cast slab. The remaining process parameters are: entry temperature 1180°C, reduction 0.3, roll radius 350 mm, average strain rate 3.6 s'\ friction coefficient 0.25, and rotational velocity of 22 rpm. Analysis of the results shows that there is no significant difference between temperatures and strains calculated for solid and continuously cast material. Therefore, a possibility of the prediction of pore closure becomes the most important aspect of the simulation.

pressure:

—•—

0.

-#-

0.1 MPa

-•-

0.25 MPa

•

[

A

0.5 MPa

I \ I \ r 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 strain

Figure 5.21 Changes of the relative volume of a pore during rolling, calculated for various pressures in the pore

40 E E 20

effective strain Ij^ J.,4.„iJ,,A..j,:A; 20 40 60 80 100 120 140 160 180

X, mm

Figure 5.22 Calculated distributions of effective strain and average stress during rolling of an 80 mm thick iJj^imJilESr slab obtained from continuous casting (roll radius 300 mm, reduction 0.3)

80 100 120 140 160 180 X, mm

THE FINITE ELEMENT METHOD IN METAL FORMING

150

100 120 140 160 180 X, mm

Figure 5.23 Calculated distributions of temperature and relative density during rolling of 80 mm thick slab obtained from continuous casting (roll radius 300 mm, reduction 0.3)

100 120 140 160 180 X, mm Testing of the method based on the simulation of the behavior of a single void in a matrix, deformed under a local stress state, characteristic of rolling, leads to a few conclusions. The results qualitatively confirmed the predictive ability of the method. Quantitative agreement with the experimental data published by Wang (1996, 1997) was observed for the pore closure predictions only, when the gas pressure in the pore was accounted for. The simulation of changes of the pore volume during rolling was not validated. The method allows an investigation of the influence of rolling process parameters on its ability to eliminate porosity, resuhing from the continuous casting of steel.

MA THEMA TICAL AND PHYSICAL SIMULA HON OF THE PROPERTIES OF HOT ROLLED PRODUCTS

151

Chapter 6 Microstructure Evolution and Mechanical Properties of the Final Product One of the objectives of the hot rolling process is to create steel with small, uniform ferrite grains. The properties required of the product are high strength, good ductility, good weldability and formability. The process of controlled rolling, in which the process parameters are chosen to suit a particular steel and a particular mill in order to attain those attributes, is the preferred technique followed to attain those attributes. The parameters available are the temperature, the strain and the strain rate per pass, the interstand and the pre-coiler cooling rates. The primary objective of the steel mill engineer is to design the draft schedule, referred to as the thermal-mechanical treatment. In doing so, the knowledge and understanding of the three critical temperatures are essential, as these affect the hardening and softening processes. These processes include precipitation hardening, recrystallization and recovery, each of which may be static or dynamic, depending on whether loads are applied. As well, work hardening, resuhing fi-om deformation below the recrystallization stop temperature, which causes pancaking of the grains, is to be considered. The critical temperatures, affecting hardening and softening, are: • • •

the precipitation start and stop temperature; the recrystallization start and stop temperature; and the transformation start and stop temperature.

All three temperatures are functions of the process and material parameters. Their importance in understanding the problems associated with the control of the steel's microstruture and structure are best examined by studying the iron-carbon phase equilibrium diagram, see for example Roberts (1983). The phase diagram shows the phases and their compositions as a fimction of the temperature and the alloy composition. The upper curve on the diagram represents the liquidus temperature, above which the alloy is in the liquid phase. The liquid begins to solidify when the temperature cools to the liquidus temperature. On solidification, the amorphous liquid phase changes into a crystalline solid, and grains nucleate and grow. This transformation does not happen instantly. On reaching the liquidus, nuclei form at several locations in the melt. The temperature remains constant while the nuclei grow. The solid phase just formed is called austenite, designated by y, and is a face-centered-cubic structure (FCC). Onftirthercooling to the ATS temperature, the first ferrite (a) grains appear and the steel reaches the two-phase region. The structure of the ferrite grains is body-centered-cubic (BCC). As the temperature drops again, the transformation stops and the steel becomes fiilly ferritic. This temperature is identified as the ATI. Depending on the carbon content and cooling rates, other phases such as pearlite or bainite may appear, as well. The two temperatures, Ars and Ari, are affected by the

152

chemical composition, pre-strain, cooling rate and initial austenite grain size. These effects are measurable, and for most purposes, the phase diagram gives sufficient information regarding transformation temperatures. Further, thermomechanical processing yields non-equilibrium microstructures, causing the modeling of these processes to be particularly difficuh. The objective of this chapter is a presentation and critical evaluation of models, describing the above mentioned phenomena, while paying particular attention to their compatibility with the thermomechanical approach to hot metal forming processes. The preceding chapters show that modeling of thermal and mechanical phenomena during rolling has improved significantly during recent years. Models of various complexities and various abilities have been developed, and accurate predictions of the metal flow and temperature fields in two- and three-dimensional domains do not present particular difficulties. Simultaneous with the development of thermal-mechanical models, the problem of the modeling of recrystallization and grain growth phenomena has also been of interest, and a number of models, based mainly on closed form equations, have been developed for various steels. Lack of information regarding strain and temperature distribution in the deformation zone, for more complex geometry of tooling, forced the assumption of uniform and isothermal deformation in the processes. Development of the finite-element technique and its applications to the simulation of metal forming processes provided a new perspective for the modeling of microstructure evolution, as shown in Figure 6.1. The information supplied by the finite-element models regarding the evolution of strain rates, strains, stresses and FINITE ELEMENT METHOD temperatures is useful input for the mod[fields of: eling of microstructural phenomena, field of strain rates| which take place in the deformation ttemperatureJ strains zone during hot rolling. The importance stresses of modeling of the evolution of the microstructure is discussed by Sellars (1990) and can be summarized: •

•

for a given composition of alloy, the high temperature flow stress is influenced to a large extent by the microstructure. Proper prediction of the rolling force is possible only if the relevant microstructure is known and the microstructure present at the end of the rolling and cooling operations controls the product properties.

MICROSTRUCTURE EVOLUTION

Figure 6.1 Usefulness of information supplied by finite-element models for a prediction of microstructure evolution during hot forming of steels

Industrial trials are expensive, difficult to control and monitor, and are necessarily constrained by the capabilities of existing plants. Laboratory simulation tests are unable to reproduce all conditions of industrial hot rolling. Taking only the most favorable test conditions of torsion and compression, there are limitations on attainable strain rates, particularly in relation to strip or rod rolling. These limitations are even more pronounced in torsion, which also develops different textures from those in flat rolling. On the other hand, the plane-strain and MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

153^ axisymmetrical compression tests cannot achieve the total strains of complete industrial rolling schedules. The advantages of using the finite-element method in the microstructure evolution modeling are twofold. First, is a possibility of investigation of local situations in the deformation zone, using conventional closed-form microstructure evolution equations. Solving these equations using temperatures, strain rates and strains, calculated by the finite-element method, allows the prediction of the distribution of the grain size, recrystallization kinetics and other microstructural phenomena in the volume of the deformed body. Variations of the temperature in time can be accounted for using the additivity rule. The second advantage is a possibility of using more advanced phenomenological models to examine the evolution of the microstructure, which are usually based on the internal variable method. Both of these possibilities are discussed in this chapter.

6.1

CONVENTIONAL MICROSTRUCTURE EVOLUTION MODELS

The problem of the correlation between the parameters of hot deformation processes and the development of the resulting microstructure has been investigated extensively and a number of papers have been published in the scientific literature. Among them, the works of Sellars (1990), Roberts et al. (1983), Laasraoui and Jonas (1991a and 1991b), Choquet et al., (1990), Hodgson and Gibbs (1992), Yada (1989), Beynon and Sellars (1992), Sakai (1995), Kuzi^ (1997) and Devadas et al., (1991) should be mentioned. In these, various closed form equations are presented, describing the processes of recrystallization and grain growth. Some of these equations, concerning various chemistry of steels, are reported in this section and they are used together with the finite-element models for the predictions of microstructure evolution. The focus is on the description of the models. Readers interested in the experimental techniques may find the necessary information in the original publications. In general, one refers to all softening phenomena, which take place during the deformation, as dynamic processes. The softening phenomena, taking place after deformation is complete, are called static processes. The following nomenclature is used in this chapter: SRX is the static recrystallization, SRV is static recovery, DRX is dynamic recrystallization, DRV is dynamic recovery, MDRX is metadynamic recrystallization, and MDRV stands for metadynamic recovery. The balance between dislocation generation and the removal of dislocations determines the rate of the hardening of the austenite during deformation by dynamic recovery. Austenite does not undergo dynamic recovery to the same degree as do other metals. This ability to have extensive work hardening, without softening by recovery, may lead to dynamic recrystallization at higher strains. This kind of recrystallization is more probable at higher temperatures and at lower strain rates. In industrial hot deformation processes, however, often the strain rates are so high that there is not enough time to trigger dynamic softening of the work hardened material. Therefore, it is much more common for the material to be deformed only in the work hardening regime, that is, the strain per pass is not large enough (<0.4) to initiate dynamic recrystallization. Hence, concurrent static recovery, accompanied by static recrystallization, usually occurs after deformation. There is a high driving force for static softening to take place between rolling passes and during cooling after the final pass prior to transformation. Both static

MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

154 recovery and recrystallization have been observed in austenite, although the extent of the former is rather limited. 6.1.1

Static changes of the microstnicture

Static recovery is defined as a softening process in which the decrease of density and the change in the distribution of the dislocations after hot deformation or during annealing are the operating mechanisms. In a low temperature range, the acting mechanism involves vacancy motion. Those operating in the intermediate temperature or in the high temperature range (>0.5r;„, where T^ is the melting temperature) involve dislocation climb and cross slip. As mentioned above, the extent of static recovery in hot rolling processes is rather limited. There is a general consensus that the maximum amount of softening during holding, attributable to recovery, is approximately 20%. The hot deformation of austenite at strains typically encountered in plate or strip rolling processes leads to significant work hardening, which is usually not removed by either dynamic softening processes or by static recovery. This hardening creates a high driving force for static softening processes. The mechanism of these processes is explained clearly by Hodgson (1993 a). Some of his observations are presented briefly below. Following static recovery, there is partial or complete softening of the microstnicture by static recrystallization, usually described as taking place in two stages: nucleation of new grains and the growth of these grains at the expense of deformed ones. Some features of static recrystallization are: • • • •

a minimum amount of deformation (critical strain) is necessary before static recrystallization can take place; the lower the degree of deformation, the higher the temperature required to initiate static recrystallization; the final grain size depends on the degree of deformation and to a lesser extent, on the annealing temperature; and the larger the original grain size, the slower the rate of recrystallization.

There is evidence that the nucleation of recrystallization occurs at the austenite grain boundaries by a bulge mechanism under deformation conditions, encountered in hot rolling. As with most grain surface nucleation, grain comers, edges and surfaces, in that order of preference, are the preferred sites for recrystallization. The bulge mechanism leads to a formation of critical size nuclei, which then grow by grain boundary migration due to a high driving force ahead of the growing point. This driving force is predominantly related to the dislocation density of the subgrain walls, the degree of misorientation across the subgrain boundaries and the size of the subgrains. At very large initial grain sizes (> 100 |im), there appears to be a change in the recrystallization mechanism, with the initial stage of recrystallization still occurring at the grain boundaries. At an early stage of transformation, however, the original grain boundaries are completely covered by new recrystallized grains, which have impinged upon one another, stopping fiirther growth. Further softening by recrystallization would require intragranular nucleation, which can occur at subgrain boundaries or deformation bands. At high temperatures (> 1200**C), only large grains are present and it takes only a fi-action of a second for the material to recrystallize completely. There is, therefore, little interest in the MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

][5^

kinetics of recrystallization under these conditions. This type of recrystallization may become important, however, at lower preheating temperatures or in hot direct rolling. During conventional preheating at high temperatures, incomplete recrystallization can take place at an early stage of the rolling process when small reductions are applied. The accumulation of strains then leads to full recrystallization in subsequent passes and, in consequence, the effect of the initial conditions on the downstream final microstructure is very small and is usually neglected. An exception is the thick plate rolling process, when the total reduction ratio can be as low as 3:1, and thus, insufficient to remove the reheated grain structure. Numerous experiments show that for grain sizes below 100 |im, recrystallization in carbon-manganese steels is very rapid above lOOO'^C. Slowing down recrystallization to the extent important from the practical point of view usually takes place below 950°C. The situation is different for microalloyed steels, in which recrystallization is retarded by the precipitates. The modeling of static recrystallization is an important part of rolling technology design. The most popular static recrystallization models are presented below. The models describing kinetics of static recrystallization are usually based on the JohnsonMehl-Avrami-Kolmogorov equation. The recrystallized volume fraction X in this equation is expressed as a function of the holding time after deformation:

X = \- exp

^ t

(6.1)

where: / is the holding time, tx is the time for a given volumefractionZ to recrystallize, A = \n{X), and k is the Avrami exponent. The majority of microstructure evolution models has been developed for X = 0.5, indicating that tx in Eq. (6.1) represents the time for 50% recrystallization and constant A = -0.639. Various empirical and semi-empirical equations, describing time for 50% recrystallization {h.sx) have been published. The most commonly used form of this equation is:

^..^Be'DTs'^xJ^^

(6.2)

where e is the strain, D is the grain size prior to deformation, Z is the Zener-HoUomon parameter, € is the strain rate, QRX is the apparent activation energy for recrystallization, R is the gas constant, and 7 is the absolute temperature. The constants B, p, q, r, s and QRX in Eq. (6.2) and Avrami exponent k in Eq. (6.1) in some commonly used models are given in Table 6.1. In the equation, developed by Choquet et al. (1990), the strain sensitivity of the time for 50 % recrystallization depends on the austenite grain size prior to deformation, according to the formulaic = -0.95D^'^^. The Japanese scientists (Suehiro et al., 1988, Yada, 1987) introduce a grain surface to grain volume ratio as an important parameter affecting the microstructural phenomena. This ratio is calculated as: S^ =—[0.491exp(f) + 0.155exp(-f)+0.1433exp(-3f)] MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

(6.3)

156 Table 6.1 Coefficients in equations (6.1) and (6.2) describing kinetics of staticJ recrystallization s r B source QRX, q P J/mole 300000 0 0 -4 2 Sellars (1990) 2.5x10-^^ 330000 0 Roberts et ah, (1983) 0 -4 2 5x10*^^ -0.28 270000 Choquet et al, (1990) 0 -0.14 6.1x10-^^ Pc 230000 Hodgson and Gibbs 0 0 -2.5 2 2.3x10'*^ (1992) Laasraoui and Jonas 0 -3.8 0 -0.41 252000 1.14x10"^^ (1991a) Suehiro et al., (1988) 0.286x10*'^^;;^^ 149650 -2 0 -0.2 0 Yada(1987) Nanbaetal., (1992) 1.12x10-^5',-^^ 0 0 -0.19 172850 -2.1 Kwon (1992) 3.32x10-^^ 285000 -3.14 1.4145 -0.12 0

k 1.7 1.7 1.7 1 1 2 2 2

The rate of recrystallization is only one important aspect when modeling the microstructural evolution, occurring during intervals between deformations and during cooling from the last pass to the transformation temperature. The ability to predict the recrystallized grain size is also necessary. Several authors have investigated the influence of strain rate, strain and temperature on the fully recrystallized grain size. General observations regarding this effect are given by Hodgson (1993 a) and they are repeated below. A power-law relationship between the recrystallized grain size and the applied strain was observed. To some extent, using a constant value of this power for different temperatures, is possible,. Furthermore, there appears to be no effect of the composition over the range of conventional carbon-manganese steels, confirming the lack of the effect of the composition on static recrystallization kinetics, observed earlier. There is also agreement with a study by Suehiro et al., (1988) where no effect of composition on the recrystallized grain size was observed for carbon contents varying from 0.1 to 0.8. Kuziak, (1997) observed no effect of carbon on recrystallized grain size for small initial grains only. However, for larger grains (> 100 ^m) he claims that there is a substantial influence of the carbon content on the size of recrystallized grains. Another feature concerning the prediction of the recrystallized grain size is its sensitivity to temperature, observed by a majority of authors. There are, however, models (Sellars, 1990) which do not incorporate a temperature term. The most commonly used form of the equation, describing the grain size after recrystallization {Dr) is: Z), =Cl+C2f'"5''Z)'exp•

RT

(6.4)

Not all components of Eq. (6.4) appear in all the models. The constants Ci, Ci, m, n and Qd, in Eq. (6.4), are presented, for some models developed for carbon-manganese steels, in Table 6.2. Several models contain equations with a structure different from that in equations (6.2) and (6.4). There are also models dealing with other than carbon-manganese steels. Some of these models, for the chemical compositions listed in Table 6.3, are given in Tables 6.4 and MATHEMAJJCAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

157 6.5. In Kuziak's equation (1997), the strain sensitivity of the time for 50% recrystallization depends on the austenite grain size prior to deformation, according to the formula p - 1.006£)^"^^. The compositions of steels given in Table 6.3 correspond to the materials, that were used in the experiments carried out to develop the models. Their appUcability is expected to extend to steels with chemical composition, similar to those used in the experiments.

Table 6.2 Coefficients in Eg. (6.4) describing the recrystallized grain size for carbon-manganese steels source n Ci C2 m J/mole Sellars (1990) 0 0.67 0 0 0.5 -1 Roberts et al. (1983) 0.5 35000 6.2 55.7 -0.65 0 Choquet et al (1990) -0.1 0.374 25000 0 45 -0.6 Hodgson and Gibbs (1992) 0.4 45000 0 343 -0.6 0

Table 6.3 Chemical steel C-Mnl C-Mn2 C-Mo Ti-V eutectoid VT lowNb high Mb

compositions of steels, models for which are giverI in this Chapter C Mn Si Al N Mo Nb Ti V 0.18 1.33 0.33 0.0003 0.003 0.003 0.16 1.04 0.36 0.001 0.004 0.003 0.06 1.2 0.22 0.18 0.029 0.0027 0.02 0.13 1.4 0.01 0.01 0.04 0.72 1.2 0.28 0.002 0.20 1.36 0.26 0.09 0.001 0.006 0.15 1.23 0.19 0.026 0.0024 0.024 0.002 0.07 1.33 0.33 0.035 0.006 0.084

Cu

Cr 0.012 0.09

0.08 -

0.21 0.012

0.032 0.022

Table 6.4 Equations describing time for 50% recrystallization for various steels equation source steel Roucoules et al, (1994)

C-Mo

Roberts et al. (1983)

Ti-V

Kuziak(1997) Kuziak et al, (1997)

eutectoid

Kuziak et al, (1997)

VT

W

=1-5x10-^2

^-0.28^-2.18^-0.878/232000^

W=5xlO-^^(£:-0.085)"''D' exp

^o.5j^ = 2 . 4 x 1 0

'280000^ .

RT >

-.^^-9^-exp[i^]

ro.3;,=9.3x lO-'^^-^P'exp

^230000^ .

RT J

MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

158 Table 6.5 Equations describing the recrystallized grain size for various steels equation source steel Sellars and Whiteman (1979)

C-Mn

Yada(1987)

C-Mn

Nanba et al, (1992)

C-Mn

D,=34^y

Laasraoui and Jonas (1991)

C-Mn

D,=0.5D^^'e-^''

Roberts etal ,(1983)

Ti-V

Kuziak(1997) Kuziaketal, (1997)

eutectoid

-0.67

D =25 14.925 In-i^i-^

^

Kuziaketal.,(1997)

VT

Sellars (1980)

niobium

I

8.5 J z),=5Kr

\ D,=9.9lD'''s-'''l~''exp D,=4.54D'''e-'''l-'\xp

X

e

RT ]

^-17540^

,

RT J

'-15000 "j

D,=1.1D^^V^-^'

The models, described above, deal with the recrystallization behavior of carbon-manganese steels, including some with additions of molybdenum, vanadium and/or titanium. However, an important class of modem structural HSLA steels containing niobium has been designed to benefit from this element. The addition of niobium in steels causes retardation of recrystallization during the interpass period and during cooling after the final pass. Since it is the most potent element in retarding recrystallization on a weight percentage basis, niobium is added to steels in very small quantities, below 0.1 wt%. Strain induced precipitation is the operative mechanism when recrystallization is retarded to times, observed during holding at temperatures below 900°C. Retardation of recrystallization can be caused by either solute drag or strain induced precipitation (Sellars, 1980). There is still debate regarding the relative contribution of these two processes in industrial rolling. The difficulties with experimentally distinguishing the mechanism of retardation are discussed by Hodgson, (1993a). Further, numerical modeling of these processes is of importance. The models, proposed for static recrystallization of niobium steels in the absence of strain induced precipitation, are of the same character as those for carbon-manganese steels, except that the constants are different. The first model for niobium microalloyed steels, using a modified Avrami equation, was proposed by Sellars, (1980). The change in recrystallization kinetics, caused by strain induced precipitation due to the presence of niobium, was handled by taking the constant B and the activation energy QRX in Eq. (6.2) as functions of the temperature. However, the appearance of precipitation changes the character of recrystallization when compared to the solute drag effect only. Dutta and Sellars (1987) developed a model for strain induced precipitation which is the basis of several later models. In the Dutta and Sellars model, the change in recrystallization behavior is handled in a recursive manner. The amount of recrystallization during a time inMA THEMA TICAL AND PHYSICAL SIMULA HON OF THE PROPERTIES OF HOT ROLLED PRODUCTS

159 crement is modeled using the solute drag equation, followed by a calculation of the time for precipitation to start, using the process conditions for the austenite at that time interval. The relative values of the time for 5% recrystallization r^osx ^ time for 5% precipitation tQ^^p, and time for 95% recrystallization to,qsx' control the material's behavior. It is assumed that for ^o.osx < ^o.osp < ^o.95;rrecrystalHzationis retarded by precipitation and for t^Q^^ > TQ^^^ recrystallization does not take place and there would be no further recrystallization once precipitation is predicted to occur. Figure 6.2 demonstrates the flow of calculations during modeling of the evolution of the microstructure in niobium microalloyed steels. RX in this figure stands for recrystallization and tp is the interpass time.

FULL RX

PARTIAL RX TOO SHORT INTERPASS TIME

NORX, STOPPED BY SOLUTE DRAG EFFECT

PARTIAL RX STOPPED BY I PRECIPITATION

NO RX. STOPPED BY SOLUTE DRAG EFFECT

NO RX, STOPPED BY PRECIPITATION

Figure 6.2 Flow chart of calculations of microstructure evolution in niobium steels

The relevant equations describing the kinetics of precipitation, as well as recrystallization in niobium steels, are given in Table 6.6. Notice that some of the equations describe the time for 5% recrystallization ^005;^, and some the time for 50% recrystallization /o.5x • ^"^ should remember that when foos;^ ^s used in the Avrami equation, Eq. (6.1), the constant A = 0.0513. In Table 6.6 [Nb] represents the niobium content in solution. In some cases, the models are valid for one steel composition only. The majority of them, however, introduce coefficients in the equations as a function of the niobium content. There are several more models for niobium steels (Choquet et al., 1990, Siwecki et al, 1995), in which not all material constants are given. These are not referred to in Table 6.6. MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

160

Table 6.6 Equations describing recrystallization and precipitation times for Nb microalloyed steels source recrystallization and precipitation times Dutta and Sellars .^^ ,^-20^^2 -4 300000 'I21i^^x^^ [Nb] (1987) W x =6.75x10 '°Z)'f ^ ' ^ P ^ ^ ^ exp 270000

^o.osp = 3 X 10-^[Nb]-'f-^Z-^-^ exp

RT

exp

2.5»10'^

[Nb]f[c]-.1^[N] where the supersaturation ratio is:

k^2.26

10 Hodgson (1993a) Hodgson (1993b)

'„,. = (- 5.24 + 550[Nb])x lO-'P^^-^^^t^" e x p [ ^ ^ ]

O.OSp

Laasraoui and Jonas(1991b)

'

:6xlO~^[Nbr£-'Z""'exp

270000

2.5.10*'

RT

U'On^J'j

-exp

for 0.05%Nb ,,.=1.27xl0-.-r-expf^^^^ \ RT ) 9 QA V in-19 .-3.8 .-o.42_r4360001 for 0.055%Nb, 0.003%B 5J^ = 2.86 X10 s s exp RT -24^-3 55^-0.33_/559000V 05, =1.06xl0-'^f-'-''f-'''exp _" |for 0.058%Nb, 0.003%B

RT J

Kwon(1992)

0.03. = 6 . 7 5 x 1 0 - / ) ^ . - e x p f ^ l e x p l

^275000

I- RT

-185 [Nb]

^1.534x10' _ 206300^[Nb][c] exp

RT

r

]

r

The models developed for microalloyed steels allow the prediction of the rolling temperature at which precipitates stop the recrystallization. Evaluation of the no-recrystallization temperature is presented by Bai et al, (1993). The no-recrystallization temperature, in **C, may be calculated from (Boratto et al., 1988): T^UBX = 887+464[C]+(6445I>fbh644^/[Nb])+ H-(732[V]-23(V[V])-f890[Ti]+363[Al]-357[Si

(6.5)

where the content of elements is in weight %. MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

161^ Situations when partial recrystallization takes place during interpass times are common in the industrial rolling processes. A majority of researchers use a simple weighted average Dp = XDr + (1 - Z)) to calculate the grain size at the entry to the next pass; in the relation D represents the grain size prior to deformation, Dp is the recrystallized grain size and X is the recrystallized volume fraction. Beynon and Sellars, (1992), claim that the following equation gives better results: Dp=D,X^"^UD{\-Xy

(6.6)

All considerations presented above are limited to carbon-manganese, eutectoid and microalloyed steels. Information is harder to find for other metals and alloys, probably because of the complexity of their behavior during thermomechanical processing. Significant limitations in the equations, describing the effects of the principal process variables and prior grain size on recrystallization kinetics during hot rolling of aluminum alloys, are caused by the influence of solutes, second phase particles and precipitates, which is more complex than that for steels. The particles may influence recrystallization by pinning boundaries and by acting as preferential nucleation sites. Sellars et al., (1985), reports that the kinetics of recrystaUization differed significantly in two samples of an Al-lMg alloy, of similar compositions, but slightly different distribution of second phase particles. Chen at al., (1992) discuss the main principles of modeling the microstructure evolution during hot rolling of aluminum. The microstructure evolution equations for some aluminum alloys are presented by Sellars et al., (1985, 1990). 6.1.2

Dynamic softening

All softening processes that take place during plastic deformation are referred to as dynamic ones. These include dynamic recovery and dynamic recrystallization. Dynamic recovery involves the rearrangement of dislocations and consists of two processes. Dislocations of opposite signs annihilate each other or rearrange to form cells of relatively low dislocation density, surrounded by boundaries of high dislocation density. At high temperatures, the mechanisms responsible for dynamic recovery are the cross slip of screw dislocations or the climb of the edge dislocations (Kocks, 1976). Since in the conventional approach dynamic recovery has only an indirect influence on modeling the microstructure evolution by controlling the onset of dynamic recrystallization, it will not be discussed fiirther. More information about modeling dynamic recovery is found in section 6.4, which treats the internal variable method. In metals of high stacking fault energy, dynamic recovery takes place rapidly and a steady state of stress is reached. This is the result of the balance between work hardening and recovery (Figure 3.3). The steady state is characterized by a subgrain size, which in general depends on the Zener-Hollomon parameter, Z. Deformation of materials with medium or low stacking fault energy is characterized by slow dynamic recovery. Thus, usually dislocation density is permitted to increase to an appreciable level, causing the onset of dynamic recrystallization before the steady state is reached, as shown in Figure 3.4. Dynamic recrystallization is a well researched phenomenon. Although the major features have been delineated, there still remains a number of unresolved matters (McQueen, 1993). A survey of research on hot deformation of steels shows that there is agreement that, generally, softening is caused by dynamic recrystallization, and that plastic instabilities become less likely or frequent as the MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

162 temperature increases and/or the strain rate diminishes. Contrary observations for other materials have been reported. Wierzbinski and Korbel (1993) observed that, in the sequence of structural changes during hot deformation of Cu-lONi alloy, the shear bands play the most important role in mechanical performance while dynamic recrystallization is a secondary phenomenon. When modeling thermomechanical processing of steels, assuming that when the dislocation density reaches its critical value, dynamic recrystallization starts and becomes the dominant softening phenomenon, is valid. This critical value of the density corresponds to the critical strain, Sc. For a given stacking fauh energy, the critical strain is a function of the temperature, the strain rate and the austenite grain size. The characteristics of microstructure evolution during and after dynamic recrystallization is connected to the mechanism of recrystallization, as discussed by Roucoules, (1992). Nucleation of dynamic recrystallization is said to begin at a critical strainfib,which corresponds to a critical dislocation density. Once the critical density, which depends on strain rate, temperature and steel chemical composition, is reached, dynamic recrystallization is initiated by the bulging of preexisting grain boundaries at low strain rates. At higher strain rates, dynamic recrystallization is initiated by the growth of the high angle cell boundaries, formed by dislocation accumulation. The driving force for the grov^h of the nuclei is the difference in dislocation density in front of and behind the boundary. The mechanism of nucleation differs for single peak and multiple peak behavior. In the single peak case (grain refinement), nucleation essentially occurs along existing grain boundaries and is referred to as the necklace structure. The growth of each grain is stopped by the concurrent deformation. When all the grain boundary sites are exhausted, further new grains are nucleated within the original grains at the interface of the recrystallized and unrecrystallized grains. In the multiple peak case, the growth of each new grain is terminated by boundary impingement and not by the concurrent deformation. In industrial hot rolling processes, the strain rate is relatively high, such that only single peak dynamic recrystallization is likely to occur, if any. In a given material, the characteristics of dynamic recrystallization may be assumed to depend on only three parameters: the initial grain size A the temperature 7, and the strain rate e. The initial grain size affeas the critical strain ec, the peak strain Sp, and the kinetics of dynamic recrystallization. The finer the initial grain size, the lower the critical and peak strains. This is because dislocations accumulate more rapidly and the higher specific grain boundary area per unit volume, leads to faster recrystallization kinetics. Peak stress is also found to be dependent on the initial grain size, while the steady state stress andfinalgrain size are independent of it. Sakai (1995) presents a different theory regarding dynamic recrystallization processes. One of the contributions of this theory is a criterion distinguishing between the multiple and single peak behavior. Sakai states that if the peak strain Sp is greater than the strain necessary for the completion of flow softening GS, which takes place at low Z, dynamic recrystalization is cyclic. Conversly, at high Z when the peak strain Sp is lower than Ss, dynamic recrystallization is continuous in all areas of the material. Sakai (1995) also claims that multiple peak deformation is observed for finer grains prior to deformation A and a single peak is displayed by the coarse grain material. Since the stable, dynamically recrystallized grain size A , for a given temperature is independent of D, this suggests that tiie multiple peak or a single peak behavior can be controlled by the ratio A ID rather than by Z. When Ds>2D grain refinement and single peak flow take place. When A < 2D, multiple peak flow appears during grain coarsening. When dynamic recrystallization starts during deformation, after deformation other processes, leading to a decrease of dislocation density, take place. These processes include MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

163^ metadynamic recrystallization, static recrystallization, metadynamic recovery, and static recovery. Metadynamic processes, researched by Roucoules, (1992), Roucoules et al, (1994), Roucoules and Hodgson, (1995), Sakai, (1995) and Kuziak et a!., (1997), are presented briefly in Section 6.3, where the main equations for these processes are given. The conventional models of dynamic recrystallization involve equations describing the critical strain, kinetics of dynamic recrystallization and the grain size after dynamic recrystallization. Numerous models have been developed for various materials and some of them are presented below. The equation describing the critical strain for dynamic recrystallization is: s, = AZ^D^

(6.7)

Coefficients^,/? and q and activation energy QDRX in the Zener-Hollomon parameter for various materials are given in Table 6.7.

Table 6.7 Coefficients in Eq. (6.7) describing the critical strain for the dynamic recrystallization source A material QDRX ^ P J/mole 312000 Sellars (1980) 0.5 0.15 C-Mn steel 4.9x10"* 66500 Yada(1987) C-Mn steel 4.76x10"* 312000 Laasraoui and Jonas (1991a) 0.13 C-Mn steel 6.82x10"* 312000 0.18 Kuziak et al. (1997) 0.15 VT 4.4x10"^ 315000 0.3 Kuziak et al. (1997) 0.18 eutectoid 4.3x10-^

Kinetics of dynamic recrystallization is often described using strain as an independent variable instead of time. The equation describing dynamically recrystallized volume fraction is: X^DRX • 1 - expB\ s-s.

(6.8)

K 'P J

where Sp is the strain at the peak stress, usually calculated as Sp = Csc. The coefficients in Eq. (6.8) for various materials are given in Table 6.8. Yada, (1987) and Anan et al., (1992), describe the kinetics of dynamic recrystallization by:

^D/yr = l - e x p • 0.693 V **o.5jr J

MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

(6.9)

164 Table 6.8 Coefficients in Eg. (6.8) describing kinetics of dynamic material source C-Mn steel Hodgson (1993a) VT Kuziaketal., (1997) Kuziak et al, (1997) eutectoid

recrystallization B k 1.4 -0.8 1.5 -0.4 1.3 -1.7

C 1.23 1.12 1.05

The strain for 50% recrystallization is calculated as: s,,^

= 1.144 X lO-'D'^'s'"'

(6.10)

expf ^' ^ ^ ° ^

Experiments show that the fully dynamically recrystallized grain size is insensitive to the strain and grain size prior to deformation, but that it depends on a joint effect of the temperature and the strain rate, expressed by the Zener-HoUomon parameter. The general equation describing the grain size after dynamic recrystallization is: Dn

(6.11)

••BZ'

Coefficients in Eq. (6.11) for various steel compositions are presented in Table 6.9. Table 6.9 Coefficients in Eq. (6.11) describing the dynamically recrystallized grain size r B source material Sellars (1980) Yada (1987) Hodgson and Gibbs (1992) Kuziak et al. (1997) Kuziak et al. (1997)

6.1.3

C-Mn steel C-Mn steel C-Mn steel VT eutectoid

1.8x10^ 2.26x10^ 1.6x10^ 1.4x10^ 1.6x10^

-0.15 -0.27 -0.23 -0.16 -0.2

QDRX

J/mole 312000 267100 312000 312000 315000

Metadynamic Recrystallization

Once the dynamic recrystallization is initiated during the deformation, the dynamically recrystallized nuclei continue to grow even after the deformation is interrupted. This mechanism is identified as metadynamic recrystallization. Three distinct softening processes take place after dynamic recrystallization, described as static recovery, metadynamic recrystallization and static recrystallization (Roucoules, 1992). While the dynamic recrystallization nuclei grow by metadynamic recrystallization, the rest of the material undergoes static recovery and static recrystallization. Unlike static recrystallization, metadynamic recrystallization apparently does not require an incubation time, because it makes use of the nuclei formed by dyMATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

16^ namic recrystallization. As a consequence, dynamically recrystallized microstructures are subject to rapid changes after unloading and this may resuh in a coarser grain size. One of the most common interpretations of restoration after dynamic recrystallization as a distinct process is proposed by Hodgson, (1993a). The fact that static recrystallization laws do not apply once the strain exceeds the critical value was observed earlier by Sellars, (1979). He noticed that for strains above the critical strain, the time for 50% recrystallization becomes independent of the strain, leading to much lower rates of recrystallization than predicted by the static recrystallization models. The kinetics of microstructure restoration processes after the dynamic recrystallization, measured by Hodgson, (1993 a), is strongly dependent on strain rate. In contrast, during static recrystallization, the transformation kinetics depends strongly on strain and temperature and not so strongly on strain rate. At the early stage of static recrystallization the grain size decreases, irrespective of whether thefinalfiiUyrecrystallized grains arefineror coarser that the original ones. Quite a different behavior is observed for post dynamic recrystallization. The fiilly recrystallized grains are always coarser than those observed right after dynamic recrystallization at the point of unloading. There is no tendency for the microstructure to be refined during post dynamic softening. Rather, there is a continual increase in the average grain size with the recrystallizedfi-action.Observations (Hodgson, 1993a) show that increasing strain rate and decreasing temperature refine the metadynamic grain size. Similar effect on the dynamically recrystallized grains is also observed. Sakai (1995) presents another view of the mechanism of metadynamic recrystallization, outlined here briefly. Sakai (1995) observes two stages of softening in the material, deformed below the critical strain. The first involves static recovery and the second, static recrystallization. Softening following dynamic recrystallization consists of three stages: i) almost instantaneous initial softening leading to a plateau, ii) substantial softening with an Avrami slope of 0.36 leading to a second plateau, and iii) gradual softening leading to a third plateau and to X of about 0.7. Such unique softening behavior of the DRX matrices is caused by the heterogeneous nature of the dynamic substructure. When the hot deformation ceases, the DRX nuclei continue to grow, leading to metadynamic recrystallization. Classical nucleation is not possible in these grains, and they soften only by metadynamic recovery in their interiors. Finally, the fully work hardened and dynamically recrystallized grains undergo static recovery and static recrystallization. Sakai (1995) writes that although the four mechanisms (MDRX, MDRV, SRX and SRV) can operate concurrently, their effect in the DRX matrix can be distinguished. His investigation shows that static recovery has the largest contribution to the softening of the DRX matrix. The first stage of softening involving MDRX leads to a softening plateau at Jr= 0.2. Substantial softening takes place in the second step associated with the classical SRX. All observations form the bases for the metadynamic recrystallization models, which are quite different than the static recrystallization ones. The contradictory theories of Hodgson (1993 a) and Sakai (1995), pertaining to the relative contribution of different mechanisms in the post dynamic softening processes, have to be considered in modeling. The softening curve for the metadynamic recrystallization can be adequately described by the Avrami equation with the Avrami constant of approximately 1.5, higher than reported by Hodgson and Gibbs, (1992) and slightly lower than reported by others (Sellars, 1990; Yada, 1987, Choquet et al., 1990; Roberts et al, 1983) for static recrystallization.

MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

166 The general form of the equation describing the time for 50% metadynamic recrystallization is: ^0.5 = A^' expl

(6.12) RT

The constants in Eq. (6.12) for carbon-manganese steels are given in Table 6.10, with Qd being the activation energy of deformation in the Zener-Hollomon parameter. Table 6.10 Coefficients in Eq. (6.12) describing time for 50% metadynamic recrystallization for various steels s source steel Ai Q J/mole J/mole 300000 312000 -0.6 Sellars (1979) C-Mn 1.06x10-^ 230000 -0.8 300000 Hodgson (1993) C-Mn 1.12

Most researchers have not considered models for post dynamic recrystallization and the resulting grain size. Metadynamic grain size depends on the Zener-Hollomon parameter and is larger than that after dynamic recrystallization. The general equation describing the metadynamic grain size is: (6.13)

D^=AZ" The constants in Eq. (6.13) for various steels are given in Table 6.11.

Table 6.11 Coefficients in Eq.(6.13) describing the grain size after metadynamic recrystallization for various steels source u A steel J/mole 312000 Sellars (1979) -0.11 C-Mn 1.06x10^ Hodgson (1993) -0.23 312000 C-Mn 2.6x10^ Kuziaketal. (1997) -0.16 315000 VT 2.3x10^ Kuziaketal. (1997) -0.23 312000 eutectoid 2.5x10^

Changes of the grain size during metadynamic recrystallization are usually calculated as the weighted average of the contributing grains: D{t)=I)DRxH^MD-^DRX%

MD

(6.14)

MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

^

167

In Eq. (6.14), XMD is the volume fraction after metadynamic recrystallization, calculated from the Avrami equation with k = 1.5. More detailed analysis of metadynamic recrystallization shows that it grain growth, different from normal growth, following static recrystallization. Senuma and Yada (1986) propose one of the more advanced models:

^ ( 0 = ^D/?;.+l.l(^A/D-^Di^)^l -exp

-295^«Wr^^^

(6.15)

In all equations above the grain growth model is used to calculate the average grain size D{t) starting from the completion of the dynamic recrystallization until the critical post dynamic grain size DMD is reached. From here, normal grain growth equations should be used. 6.1.4

Grain Growth

Following complete static or metadynamic recrystallization, the equiaxed austenite microstructure coarsens by grain growth. Since there is a lack of models dealing with an abnormal grain growth, the assumption of uniform growth is made. The models for uniform growth are, in general, based on the isothermal law:

^W"=^^+Vexp RT

(6.16)

where DRX is the fully recrystallized grain size, / is the time after complete recrystallization, Qg is the apparent activation energy for grain growth, and n and kg are constants. Table 6.12 contains these, obtained by Nanba et al., (1992) and by Hodgson and Gibbs, (1992) for various materials. A slightly different approach to the grain growth problem is proposed by Roberts et al, (1983), observing that a characteristic feature of grain growth after recrystallization is an abrupt decrease in the growth rate, some time after the completion of recrystallization. A parabolic type equation with two sets of parameters for two stages of growth is used to characterize this behaviour in a qualitative manner:

H)

D{tf =Z)^+/xlo"

(6.17)

The values of constants a and b for various steels are given in Table 6.13. The first stage of grain growth is assumed to last approximately 20 s. Table 6.12 Coefficients in Eq. (6.16) describing the grain growth source steel Nanba et al., (1992) C-Mn Hodgson and Gibbs (1992) C-Mn-V Hodgson and Gibbs (1992) C-Mn-Ti Hodgson and Gibbs (1992) C-Mn-Nb

n 2 7 10 4.5

% 4.27x10^^ 1.45x10^"^ 2.6x10^^ 4.1x10^^

Qz, J/mole 66600 400000 437000 435000

MCROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

168 Table 6.13 Constants a and h in grain growth Eq. (6.17) for various steels source steel stage 1 a b Roberts et al. (1983) C-Mn 6.6 6200 Kuziaketal. (1997) VT 7.1 7180 Kuziak et al. (1997) eutectoid 7.0 5900

stage 2 a 8.1 9.5 8.4

b 9000 10920 8520

Fast development of the finite-element method in the 80s allowed taking advantage of the possibility of accurate evaluation of the distributions of the temperatures, strain rates and strains in the deformation zone. This, in turn, lead to the connection of the finite-element programs with the conventional microstructural equations (Figure 6.1), described by Pietrzyk, (1990). Solutions have been proposed by Zouhar and Hofgen, (1988), Beynon and Sellars (1992) and Buchmayr et al, (1993) who used finite difference techniques and Korhonen et al., (1991), Karhausen and Kopp, (1992), Glowacki et al., (1992), Grossteriinden et al, (1994) and Kuziak et al., (1997), who used the finite-element method. Both approaches allow the evaluation of the distribution of the grain size in the volume of the product after hot forming. The validation of the predictions of the models is, of course, essential While the models perform reasonably well under laboratory conditions, their application to industrial forming processes often presents difficuhies. Substantiation of various conventional microstructural equations connected with the finiteelement model, using laboratory plastometric tests, with axially symmetrical samples, and laboratory and industrial rolling tests, is presented in the next section.

6.2

PROPERTIES AT ROOM TEMPERATURES

The objective of modem processes is to produce high strength products. Modeling the correlation between thermomechanical history and the mechanical properties of the final product, especially after experimental substantiation of the predictions, would be very usefiil during the development process. Computer simulations of the thermomechanical processing of hot rolled steels are reasonably advanced, as presented in Section 6.1. However, progress in the prediction of transformed microstructures, and in the attempts to predict the resuking mechanical properties, taking into account the transformation, is still quite limited. Existing models, describing the influence of thermomechanical history on the mechanical properties and their distribution in the final product, are discussed in this section. 6.2.1

Ferrite grain size

As the steel continues to cool after hot deformation, it transforms fi-om austenite to lower temperature phases. In the predominantly ferritic microstructures (steels with carbon equivalent Ceq = (C + Mn)/6 < 0.45), transformation to ferrite and pearlite takes place. Here the main parameter of interest is the ferrite grain size, as the pearlite and interlamellar spacing do not contribute significantly to the strength of these compositions. Ferrite grains have been shown to nucleate at austenite grain boundaries, at deformation bands, at second phase partiMA THEMA TJCAL AND PHYSICAL SIMULA TION OF THE PROPER TIES OF HOT ROLLED PRODUCTS

169 cles, and at recovered subgrain boundaries, especially if decorated by precipitates. The factors that affect the ferrite grain size are the final austenite grain size and the retained strain, both of which relate to the deformation history, the composition and cooling rate, which are external influences. The final austenite grain size is the last fully recrystallized grain size, increased by growth between the last pass and the beginning of the transformation. The retained strain applies to that strain not removed by recrystallization prior to transformation. The strain may accumulate over a number of passes, in which case the retained strain is the total accumulated strain after the last pass of full recrystallization. Various relationships for the ferrite grain sizes have been suggested in the literature. Some of them are presented in Table 6.14 where Da is the ferrite grain size in jim, G is the cooling rate in K/s, D is the austenite grain size, also in |im, 8r is the accumulated strain, ^ i s the ferrite fraction, and Ti.os/ is the temperature at which transformation begins. 6.2.2

Lower yield stress

The mechanical properties are represented by the yield strength, indicating the beginning of plastic deformation or the end of pure elastic behavior of the loaded material. The lower yield stress and yield strength definitions are taken to be identical. One of the most successful approaches to the prediction of mechanical properties in products made of plain carbon steels was developed by Gladman et al, (1972), relating the yield strength, ultimate tensile strength, and impact transition temperature to the microstructural parameters of the ferrite-pearlite microstructure. When trying to rationalize the results obtained by Kuziak et al., (1997) and Majta et al, (1995, 1996b), the equations of Gladman and co-workers were unsatisfactory for predicting the yield and ultimate tensile strengths for a variety of steels. More detailed studies of this problem, based on published information is presented below. According to the Hall-Petch equation, the lower yield stress Oy for a homogeneous microstructure is expressed as: ory=<7o+KyD^^

(6.18)

where OQ is the lattice fi-iction stress, Ky is the grain boundary unlocking term for high angle grain boundaries, taken as 15.1 -18.1 Nmm"^^^, and Da is the ferrite grain size. In microalloyed steels the yield strength is a combination of the strengthening contributions connected with the thermomechanical history of the deformed material. Employing all possible strengthening contributions, ao in Eq. (6.18) is given by (Majta et al., 1995):
(6.19)

where <Jo is the inherent friction stress, equal to 32 MPa for pure ferrite and 70 MPa for low carbon steel, ass is the solid solution strengthening, C7p is the dispersion strengthening (precipitation hardening), <jd is the dislocation strengthening, due to forest dislocations, Gsg is the unaccounted for strengthening mainly due to subgrain strengthening, and o? is the texture strengthening.

MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

170 Table 6.14 Equations describing the ferrite grain size steel equation source Sellars and C-Mn Z)„=(l-0.45^;'')x Beynon ^.4 + 5C,'^^^-f>22[l-exp(-L5xlO-^£))] (1984) C-Mn Hodgson and Ceq<0.35 Gibbs(1992) C-Mn Hodgson and Ceq > 0.35 Gibbs (1992) ^ C-Mn Suehiro et al., (1988) C-Mn

Nb

V-Ti V-Ti

V-T

Donnay et al. (1996) Sellars and Beynon (1984) Roberts et al. (1983) Sellars and Beynon (1984) Kuziak et al. (1997)

f- 0.4 + 6.37C«,)+ (24.2 - 59C^)C;°' + 22[l ~ exp(- 0.015£))] D = ( l - 0 45f *^^)x . \ 0/ r / M^ |^22.6-57C,J+3C;°'+22[l-exp(-0.015D)]j D = 5.51xlO*°Z)^'^Sxp

-21430 \

To. .05/

;

Z)„ = {l3 - 0.73^000 ([C] + 0.\\Mnf jJD^'C Z)„=(l-0.45^;'')x {2.5+ 3C,"^^^+2o[l-exp (-1.5x10-^/))]} D^=3.75 + 0.18Z) + 1.4C;®^ D^=(l-0.45^;'')x ^ + 1.4C,"^'^4-17[l-exp (-1.5x10-^1))]} D ^ "^ 1 +(0.036 +0.0233 C ° ^ ) D

The superposition law based on the Hall-Petch relationship is applicable when the glide resistance is relatively low over most of the volume of the material, and is quite high over thin sections, such as grain boundaries. When the contribution of the dislocation density is higher it is convenient to separate the strengthening components due to interactions between forest dislocations and mobile dislocations, and collect all other contributions into the second major strengthening component, to which it is necessary to add the grain size stress. When experimental resuhs are fitted by the root of the sum of the squares summation (r.s.s.), the agreement between the measured and calculated yield stress is much better than with a linear summation. This improvement is especially noticeable when the dislocation density is high, when rolling is finished at low temperatures. Using the r.s.s. summation, the yield stress is expressed as a sum of two parts (Kocks et al., 1975): matrix strengthening (oo*, (J„, Op) together with grain boundary strengthening (cTsg,ag) and dislocation strengthening (cTd): tf^l + ^ « +^p +CT^g + a , + 0 - ^ ' +C7/J

(6.20)

MATHEMA7JCAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

171 where:

This model is restricted to steels, in which the volume fraction of pearlite does not contribute significantly to the strength, when pearlite content is less than 30%. When the yield strength of the second phase is known, the total strengthening is obtained by the weighted average rule: c7y={l-Xyay,-^X"cTy^

(6.21)

where X is the volume fraction of the second microstructural constituent (pearlite, bainite, etc.), and ay, cFyi and
(6.22)

where k is solid solution strengthening coefficient for the /th solute, C/ is the concentration of the 7th solute (wt.%), and n is the number of solutes. Numerous semi-empirical equations describing the lower yield stress of carbon-manganese and microalloyed steels have been developed on the basis of the principles discussed above. Some of the commonly used equations for the carbon-manganese steels are presented in Table 6.15 where Da is the ferrite grain size in jam, Xf is the ferrite fraction, G is the cooling rate in K/s, and So is the pearlite interlamellar spacing. The symbols of the elements refer to the content of these elements in the steel. Some of the equations in Table 6.15 account for an influence of pearlite on mechanical properties of steels. This influence is represented by two parameters, volume fraction of ferrite XfdXid interlamellar spacing of pearlite So, calculated from (Kuziak et al, 1997): Xj.=X}-

5.476[l - exp(- 0.0106CJ]- 0.723[l - exp(- 8.8 x 10"^ Z))]

(6.23)

Z ; = 1 ~ [C](0.789 - 0.1671[M«]+ 0A607[Mnf - 0.0448[/l/«f )"'

(6.24)

^0 =0.1307 + 1.027[c]-1.993[cf -0.1108[M«]+0.0305C,"^"

(6.25)

MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

172 Table 6.15 Equations describing lower yield stress for carbon-manganese steels source equation Gladman ^^ = 63[Si] + 425[Nf' + ^^^^^^

X'J' (35 + 58[Mn] -H7(0.001Z)„ ^ )-f (l - X'J' )(l79 + 3 . 9 ^ ^ )

Gladman (1975)

a^ =88 + 37[Mn] + 83[Si]+2900[Nf' + 1 7 ( O . 0 0 1 D J " ' '

Le Bon and Saint Martin (1974)

cr^ = 190 + 15.9(0.001Z)^)"^^

Baker7l993)

^y " WQ + 37[Mn]+83[Si]+1500[N]+18.6(o.001Z)^)'°^y +q-j

Choquet et al. (1990)

a^ = 63 + 23[Mn]+ 53[Si]+ 5000[N]+700[p]+ .

Xf[i5A - 30[c]+^|;5^J(o.ooii),r^+(1 - jr^)(360+260o[cr) Hodgson and Gibbs(1992)

a^ = 62.6 + 26.l[Mnl+60.2[Si]+759[p]+212.9[Cu]+3286[N] r ^-o5 +19.7(0.00 ID^)^^

Majta et al.,

a^ = 75.4[Si]+478[N]+1200[p]+ X^(77.7 + 59.5[Mn] + 9.l(0.001i)J-^^)4-(l-A^^)(l45.5 4 - 3 . 5 V ' )

Kuziaketal, (1997)

^ = j ^ (77 7 ^ 59.5[Mn]+ 9.l(0.001i)„ r ' ) + >' /V \ ^ \ 478 N f + 1200[P]+ (1 - J^/A1^5.5 + 3.5 V ]

In Eq. (6.24) Xf- is the equilibrium content of the ferrite. The mean true interlamellar spacing is the most important parameter affecting the strength properties of the pearlite microstructure. Traditionally, a Hall-Petch type equation is used to characterize the effect of this parameter on the lower yield stress and ultimate tensile strength. However, as shown by Kuziak et al, (1997) for eutectoid steels containing 0.6-0.8%C, the approach suggested by Dollar et al., (1988) correlating the mechanical properties to ^o instead of S^'^ gives better resuhs. In this approach, the mean free path for slip M in the pearlitic ferrite is used instead of the mean true interlamellar spacing to characterize the strength. These conclusions were reached in experiments which included measurements of the properties of rails and bars after hot rolling. Different strengths were produced by varying the chemical composition and the cooling rate of the experimental samples. The best correlation was obtained with two sets of constants, for small and large values of the mean free path for slip in the ferrite. These resuhs can be associated with the different deformation mechanisms in fine and coarse pearlites (Dollar et al., 1988). The equations relating the lower yield stress to the mean free path for slip in the ferrite are: MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

173

J_ a^ =308 + 0.07— M

for

SQ>0.\S\im (6.26)

<j^ =259 + 0.087-^

for

5'o<0.15fjLW

where A/= 2^o(l-0.15[c]). On the basis of published data, Majta et al., (1995, 1996b), developed a complex model which combines thefinite-elementthermal-mechanical approach with the equations describing microstructure evolution and mechanical properties of steels during and after hot forming. This model was validated using carbon-manganese and niobium steels. Some more details regarding the equations in Table 6.15, as well as the basis for the equations used in the present model are given below. When the deformation is completed in the austenite region the dislocation density in the ferrite structure is relatively low. When the deformation is extended to the austenite-ferrite or the ferrite region, the dislocation strengthening becomes much more significant. The dislocation contribution to strengthening is very complex, and usually is an effect of the interactions among forest dislocations, mobile dislocations and the substructure. When finish rolling is in the ferrite region, it is difficult to distinguish between the contributions due to precipitation and dislocation hardening. The dislocation contribution to strengthening at a specific temperature is well described by the equation: a,=aGbp'"'

(6.27)

where G is the shear modulus for pure iron, equal to 8.1x10"^ MPa, b is the Burgers vector for ferrite, 0.248 nm, p is the forest dislocation density, and a is a constant, depending on the interaction between dislocations at the considered temperature, varying from 0.38 to 1.33 (in the current model a = 0.76). Since dislocation density measurements are tedious and often inaccurate, the dislocation contribution to strength is usually assessed on the basis of the difference between observed and calculated values. Existing formulae for calculating the dislocation density, which involves changes in the yield strength, are usually valid below the temperature range of dynamic recovery. In the present model the correlation between dislocation density and deformation temperature is expressed as:

for TFT> >Ar3

P = Pc

(6.28)

and

}=

p^Bexp{£-\i

A_

for TFD
MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

(6.29)

174 where pbisthe dislocation density of the annealed steel, 4x10^ cm•^ f is the amount of deformation below >4/-3, and B, A, and u are material constants. Majta et al., (1995) suggested .4 = 1030,5 =1.8 and M = 8.8. The subgrain contribution to the yield strength is important as well. The extent of strengthening derived from the subgrains depends on their size and the volume fraction of ferrite grains with subgrains, both of which depend on the finish rolling temperature. Usually the mechanism of substructural strengthening is presented in a form similar to the Hall-Petch equation for high angle grain boundaries: ^sg=ksr

(6.30)

where m is an exponent, usually varying from -1 to -0.5, and ks is a constant, associated with subgrain boundary strength: k ^ = l ^ = 2A^ ' 2;r(l-v)

(6.31) ^ ^

In Eq. (6.31), 0 is the subgrain boundary misorientation, / is the subgrain boundary average intercept distance, and vis Poisson's ratio. Computations using Eq. (6.31) indicate that small variations in the value of 0 (v^athin the range 1° - 5^ have a large effect on k^. Furthermore, estimates of the fraction of grains containing subgrains can lead to significant errors because of the experimental problems associated with sampling and the determination of the subgrain boundary misorientation. In the present model, the average value of subgrain contribution to strengthening is used, based on experimental resuhs (Baker, 1979). When the last deformation is below Ar3, the subgrain strengthening component will increase the yield strength by approximately: -sP\^-\\

(6.32)

where s is the strain, defined as in Eq. (6.29), and P is a constant which depends mostly on the ferrite grain size. For a steel containing 0.076% Nb and having grains of d = 5-13 [im, the constants are given as P = 300 MPa, w = 0.3. In microalloyed steels, accounting for the influence of precipitates on an increase of mechanical properties is essential. The precipitation contribution in strengthening can be expressed by the modified Ashby-Orowan equation: 1/2 Gp=

'-

{

In

^

Ih

(6.33)

\ where/is the volume fraction of NbC, VC and VN particles when complete precipitation occurs, and X is the mean planar intercept diameter of the precipitates. MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

175 Eq. (6.33) is difficult to manipulate because of the need to estimate the size of the particles. In most practical metal forming processes involving microalloyed steels, the effect of precipitation can be described by using the dispersion strengthening contribution coefficient K of the alloying elements. In that case the precipitation strengthening is obtained in the form: (J

= Z^/[M,]

(6.34)

where Kt are constants in the range of 1500 and 3000 MPa/wt.%, and [M] designates the Nb, V, Ti contents in vA%. Beyond the general principles outlined above, some particular features of the precipitation hardening phenomenon are pointed out. The contribution of the vanadium carbonitride precipitates to the overall strengthening effect may be assumed to depend linearly on the nitrogen and vanadium contents and logarithmically on the cooling rate. The dependence of the strengthening effect on the cooling rate is known to reach a maximum within a cooling-rate range of 5 to 10°C/s, depending on the steel composition and the processing route. However, since the cooling rate of air-cooled products is typically less than 5°C/s, a logarithmic representation of the cooling-rate effect on the strength properties is justified for all rolling processes considered in this study. The precipitation of niobium nitrides and carbides can be significantly modified, because of the complex Ti-Nb nitrides or carbonitrides formed as a result of small titanium additions to niobium steels. The reduction of the yield strength in these steels is due to the formation of coarse complex Ti-Nb nitrides and carbides which remove the niobium and reduce the dispersion hardening by fine niobium carbide particles, depending on the Ti/N ratio and process route. Accounting for the features of the precipitation hardening equations, describing the contribution of precipitation to the yield stress, have been suggested by researchers. Some of these equations are presented in Table 6.16 where [Nb]* is the niobium content available in the solution at the transformation temperature Ary Assuming that 0.01% Ti additions, on average, reduce the yield strength in C-Mn-Nb steels by at least 12.5 MPa, Majta et al., (1995) suggested the precipitation contribution in strengthening, modifying that by Kejian and Baker (1993) in Table 6.16.

Table 6.16 Equations describing the contribution of precipitation to the lower yield stress steel source equation V Hodgson and Up = 57logC, + 700[V]+ 7 8 0 0 [ N ] + 19 Gibbs(1992) Nb Hodgson and Gp = 2500[Nbr Gibbs (1992) V Kuziak et al. o-^ =19.9[c] + 552.8[cf +590[v]-f8650[N]+19.91n(C,) (1997) Nb-Ti Kejian and (j^ = 1500[Nb]* Baker (1993) Nb-Ti Majta et al. o-^=2500([Nb]*-0.5[Ti]) (1995) MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

176 The model, for precipitation-induced increases in the yield strength and in the ultimate tensile strength, developed by Hodgson and Gibbs, (1992) is adequate for practical observations. Although not found in this model, the contribution of carbon to precipitationstrengthening should be included in the analysis of the effect of vanadium carbonitride on strength properties (Kuziak et al, 1997). The effect of carbon content on the transformation temperature justifies its inclusion. Most vanadium precipitation occurs during ferrite transformation (interphase precipitation) and in the ferrite after transformation. Increasing the carbon content should lead to a decrease of the mean particle size of vanadium carbonitride by depressing the precipitation start temperature. Consequently, precipitation-strengthening components should be increased as the carbon content increases. A correlation analysis, which was performed for the model's development, is based on the results of Kuziak et al., (1997) and on the data of Sawada et al, (1994) and Burnet, (1986). The increase in yield strength and ultimate tensile strength, originating from the vanadium carbonitride precipitates was estimated by subtracting the structure-related values of these parameters (calculated from the equations of Kuziak et al., (1997), listed in Table 6.16) from the measured values of yield strength and ultimate tensile strength. Large, complex deformations usually involve the creation of a pronounced crystallographic texture. This could arise in two ways: inheritance from the austenite phase or deformation of the ferrite. However, texture studies show that above 600'*C there is no development of a preferential texture, which could significantly increase the strength. Thus, an increase of the strength due to texture usually does not exceed 10% of o^ and, therefore, at will not be analyzed separately. In the current model, possible eventual contribution of texture in strengthening is included in the value of cxsg . Using the relationships described above in the finite-element model of the process, it is possible to predict the yield strength distribution in the final product as a result of different thermomechanical histories. 6.2.3

Tensile strength

Selected equations describing the tensile strength are presented in Table 6.17 where Da is the ferrite grain size in |im, A/is the ferrite fraction, Cr is the cooling rate in K/s, and So is the pearlite interlamellar spacing). Kuziak et al., (1997) investigated the uhimate tensile strength for the eutectoid steels, containing about 0.6-0.8%C. They suggest that, similar to the lower yield stress, the tensile strength for these steels should be correlated to SQ^ rather than to S^'^. In consequence, the following equations describing the tensile strength as a fimction of the free path for slip in the ferrite are obtained: C7„ =706 + 0 . 0 7 2 - ^ + 122[Si]

for

^o >0.15pL/w

M

(6.35) a„ =773 + 0 . 0 5 8 - ^ + 122[Si]

for

SQ <0.15pL/w

M

where M= 25'o(l-0.15[c]) MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

177 So = ^29.3 - 54.4[Mn]- 4.4[Cr]- 17.5[S/]- (0.18 - 0.07[Mn]- 0.012[Cr]- 0.027[Si])r^ }''

Table 6.17 Equations describing tensile strength for carbon-manganese steels source equation Gladmanetal. o-„ =Zy'(246 + 1143[Nf' +18.l(0.001Z)J"°')-f 97[Si]H (1972) {^-X^/^\719 + 3.5S^-^) Choquet et al, o-„ = 237 + 29[Mn]+ 79[Si]+ 5369[N]+700[P]4Hodgson and Gibbs(1992)

7.24X^(0.00 LD„ y-^ + 500(l - Xf) o-„ = 164.9+ 634.7[c]+53.6[Mn]+99.7[Si]+651.9[p]+ . i r i / v-0 5 472.6[N?]4- 3339.[N]+1 l(0.001D^r ^

Kuziaketal, (1997)

C7„ =Z^(20 4-2440[Nf' +18.5(0.001Z)J-'')+

6.3

750(l - Xf )+ 3(1 - X^/ )S^' -f 92.5[Si]

SUBSTANTIATION OF THERMAL-MECHANICAL-MICROSTRUCTURAL MODELS

The microstructure evolution models described in the previous section are semi-empirical in their nature. Therefore, the results obtained from various models can often vary significantly, making the choice of the most suitable approach difficult. A number of experiments have been carried out to substantiate the models and selected results of this substantiation are presented in this section. In all considered cases the microstructure evolution equations were implemented into the thermal-mechanical finite element programs. 6.3.1

Carbon-manganese steels

Experimental validation of the thermal-mechanical-microstructural models under industrial conditions is expensive, therefore, researchers usually used torsion (eg. Hodgson, 1993a, 1993b; Samuel et al., 1990) or compression (e.g. Kopp, 1988; Majta et al., 1995) tests as a substitution for rolling. Kedzierski et al., (1996) made an attempt of using two kinds of processes, namely, axisymmetric hot compression and hot strip drawing, as physical models of rolling processes. The objective of the experiment was to investigate the evolution of the microstructure in various hot forming conditions and to validate the microstructure evolution models, described above. The experimental setting for the drawing-rolling test includes the furnace, three roller dies and a quenching device (Figure 6.3). The rollers are placed in a special cage with five sections, enabling changing the distance between them. The roll diameter is 60 mm, the roll width is 20 mm and the maximum roll gap is 5.5 mm. During the experiment, the samples are heated in the furnace to the test temperature and then pulled through the roller dies and quenched. The MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

178 initial thickness of the samples was between 5,5 and 5.8 mm. The thermal conditions and the straining in the drawing process are similar to those in the continuous rolling.

Figure 6.3 Experimental setting; 1. Furnace, 2. Sample 3. Cage, 4. Roller dies, 5. Insulator, 6. Quenching device

Finite element analysis has shown that the distributions of temperatures, strain rates and strains in the rolling and drawing processes are identical. The only difference between these processes is connected with the state of stress. As shown in Figure 6.4, the drawing process involves more tensile stresses compared to rolling. However, since the influence of the stress state on the microstructure evolution is not accounted for in the models, the differences between the stresses in drawing and rolling are neglected here, as well. Thus, drawing through the set of roller dies is considered a good representative of the industrial strip rolling process.

Figure 6.4 Distribution of the average stress in the deformation zone in the processes of hot rolling and drawing of strips MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

m Various combinations of reductions and inteq)ass times were used in the investigation. Two carbon-manganese steels, with the chemical compositions given in Table 6.3 (designated as C-Mnl and C-Mn2), were tested in the experiments with the strains being similar in both compression and drawing processes. The time-temperature profiles were different. Various heating procedures prior to the deformation were employed leading to different initial grain sizes. Finer but slightly nonuniform grains were observed for steel C-Mnl, making the analysis less reliable. More uniform, coarser grains were obtained after preheating steel C-Mn2. The evaluation of the time-temperature profile is essential for the accuracy of the predictions of microstructure evolution. In the present tests, temperatures were determined by the finite-element model described in Chapter 5. Some of the drawing experiments were performed with the thermocouples inserted in the sample and the monitored temperatures compared well with the calculated ones (Kedzierski et al., 1996). The heat transfer coefficient at the roll/workpiece interface used in the calculations was 20 kW/m^K, as suggested by Pietrzyk et al, (1994). A typical air-cooling model, accounting for convection and radiation, (Lenard and Pietrzyk, 1990), was used during the interpass times. The results of the measurements of the grain size in the multistage compression process are compared with the calculations in Tables 6.18 and 6.19, where Q indicates quenching. Strain rates of about 2 s'^ were used in all tests. Five equations describing the grain size after the static recrystallization were used. These equations are presented in Tables 6.2 and 6.5 and the following symbols are used for distinguishing the references: SW indicates Sellars and Whiteman (1979), RR stand Roberts et al. (1983), CH for Choquet et al. (1990), YY for Yada (1987), and HG for Hodgson and Gibbs (1992). M in the tables stands for the measurements.

Table 6.18 Measured and calculated austenite grain size (|Lim) in the compression test (steel C-Mnl in Table 6.3, M stands for measurements) No source M schedule RR CH YY SW 1 20.3 lOOOOC-Q 2 llOOOC-Q 39.2 3 4 5 6 7 8 9 10 11

1000OC-24%-2.3s-Q 1100OC.24%-2.3s-Q 1000OC-24%-l.7s-Q 1100OC-24%-1.7s-Q lOOOOC - 24% - Is -16% - l.7s - Q 1100OC-24%-ls-16%-1.7s- Q lOOQOC - 24% - l.7s -16% - 1.7s -Q llOOOC - 24% -1.7s -16% - 1.7s -Q 1 lOQOC - 24% - Is - 16% - Is -16% -1.7s-Q

17.4

15.7

26.3

28.9

26.9 15.5

41.5 26.1

26.7

41.3

21.0 35.0

18.7 30.5

22.2 34.0 30.5

18.8 30.6 29.6

34.4 51.3 34.5 51.4 54.4

19.9 37.3

25.8 40.0

20.5 30.4

25.6 38.8

20.3 30.2 25.3

33.3 46.4 33.3 46.5 46.3

MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

32.1 25.4 32.3 32.3

180 Table 6.19 Measured and calculated austenite grain size (^lm) in the compression test (steel C-Mn2 in Table 6.3, M stands for measurements) source M No schedule SW RR CH HG YY 1 lOOOOC-Q 24.2 2 llOOOC-Q 32.0 3 1000OC-24%-2.3s-Q 22.7 17.1 28.0 27.4 31.8 18.8 4 1100OC.24%-2.3s-Q 27.0 24.5 38.0 36.1 48.4 26.9 5 1000OC-24%-1.7s-Q 20.6 17.5 27.9 27.3 31.7 18.7 6 1100OC-24%-1.7s-Q 23.8 24.2 37.8 35.9 48.3 26.6 7 1000OC-24%-1.0s-16%-1.7s-Q 23.3 19.5 35.4 34.1 42.1 19.8 8 1100OC-24%-1.0s-16%-1.7s-Q 22.6 29.1 45.9 41.9 67.8 30.2 9 1000OC-24%-1.7s-16%-1.7s-Q 21.2 19.6 35.4 34.1 42.3 18.7 10 IIOOOC.24%-1.7s-16%-1.7s-Q 21.8 29.2 49.6 45.1 67.8 30.3 11 1100OC-19%-0.9s-20%-0.9s-21%-1.9s-Q 190 18.7 36.9 33.7 44.6 19.4 12 1100OC-20%-0.4s-19%-0.4s-20%-1.3s-Q 20.2 29.0 53.7 46.0 75.1 31.7 The grain size equations were incorporated into thefinite-elementmodel for the compression test, accounting for the inhomogeneity of deformation. The local strains in the center of the sample, where the microstructure was measured, exceed the homogeneous strain, observed in Figure 6.5, where the distributions of the effective strain at the cross section of the sample, during a 3-stage compression (Test #12 in Table 6.19), are presented. The strain inhomogeneity in the sample leads to a nonuniform distribution of the grain size, shown in Figure 6.6. The results in this figure are based on the equation developed by Sellars and Whiteman, (1979), see Table 6.5. Strain inhomogeneity leads to the prediction of finer grains in the center of the sample, compared with those of Kedzierski et al., (1995), obtained using the assumption of uniform deformation. In the present calculations coarser grains are observed in the area located below the die in the center of the sample. Small strains (Figure 6.5) have been predicted in this area. A comparison of the measured and calculated grain size, obtained in continuous drawing through the roller dies, is given in Tables 6.20 and 6.21. Measurements and calculations at several locations on the cross section have been performed and are shown in the center of the strip. Slightly finer grains are observed closer to the surface. Figure 6.7a shows a comparison of the measured and calculated austenite grain sizes during 3-pass drawing of steel C-Mnl, the samples of which were heated to an initial temperature of 1100°C (Test #11 in Table 6.20). Similar results, obtained for the steel C-Mn2 samples heated to the initial temperatures of llOO^C and lOOO^C (Tests #14 and #13 in Table 6.21), are presented in Figures 6.7b and 6.8. These figures show some discrepancies among the values of the grain sizes, calculated using different models. The kinetic recrystallization model predicts partial recrystallization in the last pass of tests #13 and #15. In the experiment, elongated grains are observed after these tests. In these situations, the grain size is evaluated using the grain dimensions in two perpendicular directions. MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

181

0.0

0.5

1.0

1.5

2.0

2.5

r, mm

Figure 6.5 Calculated distribution of the effective strain during 3-stage compression (test 12 in Table 6.19)

Table 6.20 Measured and calculated austenite grain size (|im) in the continuous drawing test (steel CMnl in Table 6.3, M stands for measurements) No

schedule

source

M SW

RR

CH

HG

-

-

-

20.5 38.0 19.7 36.6 26.2 26.4 40.8 39.9

16.6 29.8 15.8 28.6 20.0

15.6 30.8 14.9 30.0 20.2 20.7 31.9 32.8

1

lOOQOC-Q

11.5

2

IIGGOC-Q 1000OC-24%-2.3s-Q 1100OC-24%-2.3s-Q 1000OC-24%-1.7s-Q 1100OC-24%-1.7s-Q lOOOOC - 24% - 0.3s -16% - 1.7s •Q lOOOOC - 24% - 0.6s -16% - 1.7s •Q • 1 lOOOC - 24% - 0.6s -16% - 1.7s•Q 1 lOOOC - 24% - 0.7s -16% - 0.6s - 16%-1.7S-Q

41.9

-

12.4 32.6 11.8 35.7 10.9 12.5 23.7 25.5

12.2 22.8 11.2 21.5 13.1 13.4 21.4 20.0

3 4 5 6 7 9 10 11

20.2 29.9 28.3

MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

182

0.0

0.5

10

1.5

2.0

25

r, mm

Figure 6.6 Calculated distribution of the austenite grain size at the cross section of the sample during 3-stage compression (Test 12 in Table 6.19)

The microstructures at the center of the sample for all the experiments in Table 6.21 using the initial temperature of 1 lOOT are shown in Figure 6.9. The numbers in the figure refer to the test numbers in Table 6.21. Beyond test 15 and possibly 16, full recrystallization is observed. The microstructure in test 16 reveals some not folly recrystallized grains. The results presented in Tables 6.18-6.21 and in Figures 6.5-6.9 show significant discrepancies among various microstructure evolution models for carbon-manganese steels. Since the equations describing microstructural phenomena are semi-empirical in their nature, and were obtained for various chemical compositions of steels, conclusions regarding the accuracy of the models should not be drawn directly from these results. However, the resuhs of Glowacki et al., (1992), Pietrzyk et al, (1993c) and by Kedzierski et al., (1995, 1996), as well as the data presented in this chapter, lead to the conclusion that the Sellars and Whiteman, (1979) grain size model gives results, closest to the measurements obtained under different conditions for carbon-manganese steels C-Mnl and C-Mn2. The equation of Choquet et al., (1990) also gives reasonably good resuhs when the predictions are compared to experimental data for a majority of tests.

MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

183 Table 6.21 Measured and calculated austenite grain size (|im) in the continuous drawing test (steel CMn2, M stands for measurements). Italics are used for elongated grains, calculated as a geometrical average from two perpendicular dimensions No

schedule

SW

RR

CH

HG

YY

90.0

-

-

-

-

-

24.6

19.4

28.9

25.0

31.9

23.7

76.9

59.2

17.0

1 2

lOOOOC-Q UOOOC-Q

3 4 5 6 7 8 9 10 11

1000OC-18%-2.3s-Q 1100OC-18%-1.4s-Q 1000OC-19%-2.7s-Q 1100OC-18%-2.5s-Q lOOOOC -18% - 0.85s - 17% - 1.5s - Q 1 lOOOC -17% - 0.8s - 20% - 2.0s - Q lOOOOC -19% - 0.5s -19% - 1.5s - Q 1 lOOOC -18% - 2.3s - 18% - 1.8s - Q lOOO^C -19% - 3.5s - 20% - 1.9s - Q

12 13 14 15 16

1 lOO^C -19% - 0.4s -19% - 1.7s - Q lOOOOC -19% - 0.9s -19% - 0.9s - 18% - 3.4s - Q 1 lOO^C - 19% - 0.9s - 20% - 0.9s - 21% - 1.9s - Q lOOO^C - 19% - 0.5s -19% - 0.5s - 22% - 1.3s - Q 1 lOOOC - 20% - 0.4s -19% - 0.4s - 20% - 1.3s -Q

a)

40.7

39.6

65.7

50.4

24.8 43.0

18.3 42.9 17.7

27.7 69.2 31.7

24.0 53.2 25.3

30.0 81.3 34.8

23.2 61.5 27.4

26.9 15.7 27.3 16.0 29.4 15.2 19.5 12.3 21.1

55.3 29.7 56.4 22.9 52.6 29.3 40.8 27.2 43.4

39.2 23.6 39.2 28.6 37.5 22.7 28.9 20.4 30.4

68.0 32.8 65.1 30.4 64.9 31.9 49.5 29.3 53.7

47.4 24.9 50.1 26.3 45.0 27.2 35.9 23.9 36.6

17.1 33.2 13.2 29.5 20.9 25.8 15.0 19.6 14.0 19.0

b) AR —1

110-n

40-

10090-

35 -

£

source

M

__.

G—

#

//

80-

30-


E

#

•^

3. oi" .ti!

/

(0

c

- - 0 - • measurement

[--•• • •

measurement

j—^nLJ -

Sellars & Whiteman, i y / y

7060-

I

Roberts et al., 1983

A -

Choquetetal., 1990

\J -

Yada, 1987

504030-

10 -1

-

510 00

-^^

20-

101 1020

1 1040

1 1060

temperature. °C

1 1080

11 00

1000

1 1020

1 1040

1 1060

1 1080

11 00

temperature. °C

Figure 6.7 Measured and calculated grain size during 3-pass drawing through the roller dies, steel C-Mnl (a) and C-Mn2; (b) initial temperature 1100°C MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

184 All the remaining formulae yield grain sizes, that exceed those measured for higher starting temperatures (llOO^C) when coarse 25 grains prior to deformation (90 jum) are observed in the experiment. The accuracy of E 20 H those formulae is much higher for the finer initial grains (Figure 6.7a) and for the lower 15H starting temperature (Figure 6.8). The accuasurement racy of the models of the kinetics of recrys10 Sellars & Whiteman, 1979 tallization affects directly the calculations of { -B- Roberts etal., 1983 the recrystallized grain size. Analysis of the Choquet et al., 1990 micrographs shows that for the starting temYada. 1987 perature of 1100°C the recrystallization was 1 I ~n not complete in the surface layer in test 16 920 900 940 960 980 1000 temperature. ^C in Table 6.21 (see Figure 6.9). Partial recrystallization was also observed in the Figure 6.8 Measured and calculated grain size whole volume of all samples, deformed in 3 during 3-pass drawing through the roller dies, passes at a starting temperature of 1000**C. steel C-Mn2; initial temperature 1000°C The equation describing the time for 50% recrystallization given by Roberts et al., (1983), did not predict partial recrystallization in the tests mentioned above. On the other hand, the equation developed by Sellars, (1990), and presented in Table 6.1, yields longer times for 50% recrystallization. In consequence, it properly predicted partial recrystallization in the whole volume of the samples after 3-pass drawing at the starting temperature of 1000°C and in the surface layer of the sample after test 16 in Table 6.21. Note that the current grain size is the major parameter affecting the kinetics of the recrystallization. Thus, the accuracy of the predictions of both the grain size and the recrystallization time is mutually dependent and the models should not be validated separately. Therefore, in order to enable the comparison of the grain size models independently of the predictions of the kinetics of recrystallization, the measured grain size was used in the calculations of the time for 50% recrystallization. 30 H

6.3.2

« pancaked grains

Niobium microalloyed steels

In the following section the full model is used to describe the microstructure evolution during hot rolling of a Nb microalloyed steel (Pietrzyk et al, 1995a). The model has been compared with laboratory data which have the advantage of representing the true rolling conditions. This also allows accurate measurements of the temperature, load and the microstructure, both during rolling and after cooling to room temperature. The complete model involves a combination of the finite-element approach, described in Chapter 5, and a microstructure model, which predicts the evolution of the austenite microstructure to calculate the thermal and mechanical aspects of rolling. The predictions of loads during rolling depend on the choice of the stress-strain relationship used in the calculations. The constitutive equation for the current model is based on the work by Hodgson and Collinson (1990), who suggested the hyperbolic sine equation, Eq. (3.24), with the coefficients K = MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

185 100 , p - 0.2, A = 2x10 and m - 0.2. This function was obtained direcdy from torsion tests. In the original work, the constants given above were for the mean yield strength, rather than for the stress at a given strain, strain rate and temperature. However, Pietrzyk et al., (1995a) has noted that there is no significant difference, for the particular steel studied and the range of deformation conditions used, between the plane strain mean yield strength at a given strain and the equivalent tensile stress.

Figure 6.9 Micrographs at the center of the sample after hot drawing of carbon-manganese steel sample according to the schedules in Table 6.21, initial temperature 1100°C

MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

186 Two microstructure evolution models, suggested by Hodgson, (1993a, 1993b) and by Dutta and Sellars, (1987), both presented in Table 6.4, were used in the finite-element program for the analysis of the recrystallization and precipitation kinetics. When full recrystallization occurs, the recrystallized grain size is calculated. When partial recrystallization occurs, a simplified approach assuming a simple weighted average is used to determine the grain size and the retained strain is calculated as: €,=A£{\~X)

(6.36)

where X is the recrystallized volume fraction, £ is the strain in a pass, and A is a, coefficient between 0.5 and 1 (Hodgson and Gibbs, 1992). The possibility of grain growth after complete recrystallization is included in this analysis, using Eq. (6.16). Since the steel used in the experiment contained some niobium, the coefficients for C-Mn-Nb steel in Table 6.12 are employed. The presence of Ti was expected to require some alteration to the grain growth equation. However, the multiple reheats used during the initial rolling of the slab to plate and then rolling as described in this work, appear to have eliminated this need. The microstructural equations by Dutta and Sellars, (1987) and by Hodgson, (1993a), given in Table 6.6, were incorporated into the finite-element program, and calculations of the recrystallization kinetics and grain sizes were performed at different locations through the strip thickness. Local values of the temperatures, strain rates and strains were used in the microstructural equations. The additivity rule was introduced to account for the temperature variations during the time intervals. Both precipitation and recrystallization processes were simulated according to the following principle: ^ = ± ^

(6.37)

where n is the number of time increments, and /, is the time, calculated from the relevant equation for the current temperature and the time increment At.. A low Nb HSLA steel, with the composition given in Table 6.3, was used. The samples were reheated for 30 minutes at 1250°C in a small furnace in stainless steel envelopes to reduce scale formation. The samples were then rolled at a rolling speed of 0.36m/s according to four rolling schedules. The temperature was measured using two N-type thermocouples inserted in the mid-thickness and mid-length of the samples. Roll forces were recorded during the tests. Tests Tl, T2 and T3 were based on an approximation of a plate rolling schedule, involving continuous cooling with a constant interpass time of 10s and a constant strain for each pass. Such an approach has been termed a no-recrystallization temperature (TNRX) test. The remaining tests, T4 and T5, were based on an approximation of a hot strip mill schedule, with separate roughing and finishing phases. The strains and interpass times were not those encountered in actual strip rolling, but were designed to test the conditions of a large number of passes with strain accumulation. The details of each test were: •

Tl consisted of 10 passes, each of 0.15 true strain, separated by a 10s interpass time; MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

187 • • •

•

T2 consisted of 8 passes, each of 0.25 true strain, separated by a 10s interpass time; T3 consisted of 6 passes, each of 0.35 true strain, separated by a 10s interpass time; T4 consisted of 3 roughing passes, each of 0.25 true strain and given at fixed temperature; then 5 finishing passes, each of 0.25 true strain and separated by a 6s interpass time, started at two entry finishing temperatures; and T5 consisted of the same schedule as in T3, however the amount of true strain given during the finishing was decreased to 0.15.

The temperatures at which the tests were started, the entry finishing temperature (tests T4 and T5 only) and the finishing temperatures as well as the number of passes for each test are given in Table 6.22. Typical time-temperature history for one case of each test is presented in Figure 6.10. Samples were either quenched after a certain number of passes or air cooled to determine the prior austenite and ferrite structure, respectively. The longitudinal section corresponding to the position of the thermocouples was examined. The average grain size was estimated by the linear intercept method at the center of the strip and at the surface.

Table 6.22 Start and end temperatures for the tests Tl, T2, T3, T4 and T5 test Tl T2 T3 start 1. 1187 (10) 1. 1189(6) 1. 1190 (6) temperature, °C 2. 1190(6) 2. 1188(8) 2. 1042 (6) 3. 1168 (10) 3. 1185(3) (number 4. 1208 (10) 4. 1193(5) of passes) 5. 1163(8) 6. 1209 (8) 7. 1169(8) entry finishing temperature, *^C end finishing 1. 825 temperature, **C 2. 1002 3. 819 4. 845

1. 2. 3. 4. 5. 6. 7.

823 804 1097 1015 793 835 803

1. 845 2. 796

T4 T5 1. 1169(3/5) 1. 1169(3/5) 2. 1171(3/5) 3. 1177(3/5)

1. 2. 3. 1. 2. 3.

937 920 921 803 773 785

1. 922

1. 836

The temperature drop measured during the five schedules was compared to the FEM calculation. The heat transfer coefficient at the roU-workpiece interface was chosen as 20 kW/m^K on the basis of the information, presented in Chapter 2. Since the samples were protected fi'om scaling during the heating, the maximum value of the heat transfer coefficient reported by Pietrzyk et al, (1994) has been chosen. The calculated history of the temperature agreed closely with the measurements (Figure 6.10), confirming again the predictive ability of the model. With the exception of test 3-1, in MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

188

^

which the large strains caused breaking of the thermocouple after the 5 pass, the measurements and calculations agree quite well during the whole process.

1300 n 0

1200

2

1100 H

test 2-1 (strains 0.25) +

measurement prediction

1 1000 test 1-4 (strains 0.15) I 900 H -4- measurement S

800 prediction

700 20

40

—i r100 60 80 time, s

700 120

nuenched after 6th pass 1 1

20

40

* \

60 80 time, s

\

100

\—

120

test 3-1 (strains 0.35) -f-

measurement prediction

Figure 6.10 Selected measured and calculated time-temperature profiles for each test time, s

Due to air cooling of the surface and the chilling effect of the contact between the plate and the rolls, the calculated temperature at the surface was 80 to 140°C less than the center temperature. The temperature of the strip, averagedfi-omthe surface to the center to represent the gradient through the plate, was slightly less than the calculated central temperature. At this stage of the calculations, the resuhs obtained from the finite-element method were compared with those fi'om the one-dimensional model (slab method) described in Chapter 4. In both calculations, it was assumed that full recrystallization was taking place after each pass. An example of the comparison between these two methods and the measured rolling loads, presented by Pietrzyk et al., (1995a), shows that while the FE calculation gave slightly higher values than the slab method, the differences are negligible.

MA THEMA TICAL AND PHYSICAL SIMULA HON OF THE PROPERTIES OF HOT ROLLED PRODUCTS

189 Typical results of measurements and calculations of the roll force for the tests 800 n, T2-6 in Table 6.22 are presented in Figure measurements test 2-6 6.11. Both models give a good prediction calculations, 1D model of the rolling loads from the 1^ pass to the II-A" calculations, FEM 6005**" pass. The spread of rolling data during a given pass is quite large, in particular for the 4* and subsequent passes, and this is due to the impact between the rolled plate £ 400 -Jf and the rolls, which becomes more profull recry$te(lizat]on nounced as the steel hardens. The compariassumed son between measurements and predictions 200 was, therefore, based on the loads measured after the transient caused by the impact. When full recrystallization was assumed, the experimental roll forces in the last three 0 10 20 30 40 50 60 70 80 90 passes increased at a much higher rate than time, s the model's predictions, presumably due to the strain accumulation from one pass to Figure 6.11 Roll forces measured and calcuanother. Good agreement was obtained lated by the FEM and the ID model for T2-6 when the kinetics of recrystallization was accounted for. A more convenient way of representing this data, and supplying more information regarding the contribution of the microstruaure, is to consider the ratio of the measured loads to those calculated by the FEM, assuming complete recrystallization. This ratio is presented for each schedule as a function of the temperature (Figures 6.12a-6.16a). At high temperatures, the ratio remained within 5% of the measurements for each schedule. For lower temperatures, a deviation from the prediction was observed. The experimental rolling loads were up to 30% higher than the predictions. The temperatures at which the deviations were observed, were 1010°C, 960°C and 940°C for the tests Tl, T2 and T3, respectively. Application of heavy roughing reductions and long interpass times led to the decrease of this temperature to about 940OC for the tests T4 and T5.

nHfi -t

For the tests T2 and T4, above 1000°C, the predicted mean hot strength was in closer agreement with the experiments than for Tl, which involved a lower reduction per pass. As the temperature decreased, the experimental mean hot strength deviated to higher values than the model's predictions. Below 1000°C, the solute drag effect and the precipitation of NbC take place and inhibit or completely stop the recrystallization process. As a result, strain accumulation took place and the hot strength increased. Therefore, all the calculations were repeated with microstructure evolution model introduced into the finite-element code. Two recrystallization and precipitation kinetics models, developed by Dutta and Sellars, (1987) and by Hodgson, (1993a), presented in Table 6.6, were tested, as shown by Pietrzyk et al, (1995a). Only the results for the latter model are presented in Figures 6.11 and 6.12b-6.16b. This approach accounts for the influence of the strain accumulation on the rolling loads. An example of the comparison of loads predicted by the one-dimensional model and the fiMICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

190 nite-element method for test T2-6 is shown in Figure 6.11. Loads calculated, accounting for the strain accumulation, are plotted using thick lines. Some influence of the strain accumulation on the predictions of rolling forces is observed in Figures 6.12b-6.16b, where the measured to calculated load ratio is plotted as a function of the temperature for all the tests. Accounting for the evolution of the microstructure caused significant improvement in the agreement between measurements and calculations at low temperatures. a)

b)

1.3

[ •

^^ 1.2

•

^

1.1

1.25-1

1 11870C

•

2

1190«C

• I•

3

11680C

4

1208«C

rw

test1 1.20- strains 0.15

1.15-

m

3-1168«»C 4 -1208OC

1.10-

• •

1.05-

•

•

•

1.00-J • 1^-5 •

testi Istrains 0.15 0.9 1 800 900

2-1190OC

•

u.

1.0-+

1 -11870C

A

v-i::--

0.95-

. • •

0.900.851 \— 1000 1100 temperature, oc

1200

A

• *

•

• 1

1

800

900

1

1

1000 1100 temperature, ^C

1 1 20

Figure 6.12 Measured (Fm) to calculated (Fc) rolling load ratio for the test Tl; (a) foil recrystallization assumed, (b) kinetics of recrystallization accounted for using Hodgson's model These results lead to some conclusions regarding microstructure evolution models for niobium steels. Dutta and Sellars' (1987) model tends to predict faster recrystallization kinetics than does Hodgson's (1993a) model. The former model predicts lower loads as the temperature decreases, especially for large reductions. It also predicts the appearance of retained strains at lower temperatures than observed in the experiments (Figure 6.17), while failing to predict partial recrystallization observed metallographically after the 5* pass in T2-4. Using the Hodgson (1993 a) model, the rolling loads over the entire temperature range (Figures lib-15b) are predicted within a ±5% range for all tests, except for some individual passes in Tl-2, Tl-4, T2-1 and T2-7. The accuracy of the recrystallization kinetics model was found to have a greater effect on the prediction of the rolling loads than the actual expression of the strain accumulation. The coefficient A in the strain accumulation Eq. (6.36) is reported to be between 0.5 and 1 (Hodgson and Gibbs, 1992). Simulations with each factor gave only minor differences, except for a lower applied strain. Overall, however, using a coefficient of unity in Eq. (6.36) and the Hodgson recrystallization model, (1993 a, 1993b), gave the most accurate predictions over the entire range of conditions studied. MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

191 a)

b)

1.3-

[ •

test 2 strains 0.25

1.2 H

^ 1.H

1.25-1

1 11890C

+ •

2 1188°C

•

4 11930c

•

5 •11630C

1.10

*

6 12090c

1.05 H

[m

7 -11690C

test 2 strains 0.25

1.20

3 -11850C

1.15H

800

11890C

+

118800

• A

3 .11850C 4

11930c

• •

5

11630C

1.00 4 - - - - - - - * - - - V

1 1— 1000 1100 temperature, oc

6

12090c

7

11690C^

•

*"

/%>i

0.90 H

^' ^ « * i 1 900

1 2

[ •

0.95

1.0-^

0.9

[ •

0.85 800

1200

900

1000 1100 temperature, oc

1200

Figure 6.13 Measured (Fn,) to calculated (Fc) rolling load ratio for the test T2; (a) full recrystallization assumed, (b) kinetics of recrystallization accounted for, using Hodgson's model

a)

b)

1.41-1190OC

2 - 1042ocJ

1.3 H

1.25-

pF

1.20-

1-1190OC]

•

2 - 1042ocJ

•

•

1.151.10-

12 H

1.05-

:^ 1.1

.%-

1.00-

-

• •

•

•

0.95-

1.04 0.9

• 0.90-

test 3 strain 0.35 —I

800

900

\

1

\—

1000 1100 temperature, °C

test 3 strains 0.35

1200

8()0

900

1

1

1100 1000 temperatureJ, oc

12

Figure 6.14 Measured (Fm) to calculated (Fc) rolling load ratio for the test T3; (a) full recrystallization assumed, (b) kinetics of recrystallization accounted for, using Hodgson's model In Figure 6.12, (a) and (b) show the results obtained assuming full recrystallization and with Hodgson's model, respectively. The strains were increased and the effect of a strain of 0.25 is shown in Figure 6.13, (a) and (b) and of 0.35, in Figures 6.14, (a) and (b). Similar results are given in Figures 6.15 and 6.16, for tests 4 and 5. MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

192 a)

b) 1.5-|

1.4-1

f •

1-1169<<:

• i#

2-1171«K:

1.25 n

r •

2-1171
[ •

3-11770C

1.15

1.3

1-11690C

•

1.20

3-117700^

1.10 ^E 105-1

1.2

1.00 H

#--fr

1.1 -I

••

0.95 1.0 Htest 4 finishing strains 0.15 0.9 1 \ \ 700 800 900 1000 1100 temperature, ^C

0.90 0.85

test 4 finishing strains 0.15 1 \ 900 1000 1100 1200 temperature, oc

n 800

700

1200

•% ^

Figure 6.15 Measured (Fm) to calculated (Fc) rolling load ratio for the test T4; (a) full recrystallization assumed, (b) kinetics of recrystallization accounted for, using Hodgson's model

a)

b) 1.3n

( •

i-ii6yc)

( •

1-1169
1.201.15-

1.2 H

1.10-

^

1.H

E

1.05-

LU

1.000.95-

1.0-h test 5 finishing strains 0.15 0.9 n \— 800 900 1000 1100 temperature, oC

^*

1200

V"" •

0.90- • 0.85800

V

• •

test 5 finishing strains 0.15 1

900

i

1

1000 1100 temperature, OQ

1

1200

Figure 6.16 Measured (Fm) to calculated (Fc) rolling load ratio for the test T5; (a) fiill recrystallization assumed, (b) kinetics of recrystallization accounted for, using Hodgson's model

The time for 5% strain induced precipitation was estimated by both models and compared to the time for 5% recrystallization. Typical results obtained for the test T2-2 are presented in Figure 6.18. Precipitation was predicted to stop recrystallization at about 900°C using the Hodgson model (1993a) and at about 840''C, using Dutta and Sellars' model (1987). ThereMA THEMA TICAL AND PHYSICAL SIMULA HON OF THE PROPERTIES OF HOT ROLLED PRODUCTS

193^ fore, the precipitation does not account for the deviation in load observed earlier in Figures 6.12a-6.16a, when assuming full recrystallization during all interpass times. It can be concluded that for the low niobium steel, the solute drag effect is responsible for the retardation of recrystallization in the temperature range between 950 and lOOO^C, where the strain accumulation is predicted.

Dutta and Sellars, 1987 Hodgson, 1993a

5% precipitation 5% recrystallization 95% recrystallization] n 900

1 — — t 1000 1100 temperature, oc

Figure 6.17 Comparison of the predicted fi"actional softening by 2 models for the test T2-2

900

1000 1100 temperature, oC

1200

Figure 6.18 Evolution of recrystallization and precipitation times predicted by two models for the test T2-2

The evolution of the microstructure during rolling of the niobium steel was also investigated (Pietrzyk et al., 1995a). Samples quenched after pass number 3 (1040°C) and 5 (940''C) for the tests T2 showed fully and partially recrystallized microstructures in the center of the strip, respectively. However, when quenched after pass 6 at 870°C, the microstructure was pancaked (i.e. non-recrystallized). This change in microstructure corresponds to the deviation of the experimental rolling loads from those predicted assuming complete recrystallization between passes. The average austenite grain size for the tests T2, calculated using the FEM and the equation given in Table 6.5 for niobium steels, was in good agreement with the measurements (Figure 6.19). The starting austenite grain size was measured to be 100 jLon. As the number of deformations increased and the temperature decreased, the austenite grain size was refined to lOjjm. These measurements reflect the combination of both recrystallization and grain growth. The FE model predicted that grain growth, calculated using Eq. (6.16) with the coefficients relevant for a C-Mn-Nb steel, had an important contribution during the interpass time at high temperatures. Without the incorporation of this component, the prediction of the microstructure and changes in recrystallization behavior were inaccurate. The investigation presented above was performed for a low niobium steel. It is expected that higher niobium steels will behave in a different manner. Additional experiments for these MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

194 steels have been performed by Majta et al, (1996). Since Hodgson's model (1993a) is not valid for the niobium contents above 0.035%, only Dutta and Sellars' model (1987) has been validated. The experiment included hot rolling of a high niobium steel with the chemical composition given in Table 6.3, obtained as 12 mm thick strips. The hyperbolic sine equation, Eq. (3.24), with K = 100, p = 0.2, A = 2xl0" and m = 0.2, was used to calculate the flow stress. The experiment was conducted on a two-high laboratory mill with work rolls of 150 mm diameter. The mill was instrumented to measure the rolling loads. Since previous experiments confirmed good accuracy of the thermal model, the temperatures were not measured in the present tests and predictions of the model were used in the analysis. Four passes were performed with strains of about 0.35. The process was interrupted at various stages, the sample was quenched, and the microstructure at the cross section was measured. The objective of the analysis was an assessment of the importance of accounting for the microstructure evolution in the modeling of rolling loads. Similarly, as for the low niobium steel, thefirstset of calculations was performed assuming full recrystallization during interpass times. The results are presented in the form of measured to calculated roll force ratios, as shown in Figure 6.20. This ratio is close to unity for the first two passes, and it increases in the remaining two passes. It is concluded that recrystallization is not completed after passes 2 and 3. 100

I

Q—

80-11|- -^- 0

60

calculations, T2-41

1.4

calculations, T2-5i measurements

1-2-1

A

.1.1-1

+

A

full recrystallization assumed

-}-

retained strain accounted for^

40 H 1.0 20

0.9

0 700

800

Q pancacked grains 1 1 1 1 900 1000 1100 1200 temperature, °C

Figure 6.19 Evaluation of the measured and predicted austenite grain size

0.8 900

1 950

1 1 1000 1050 temperature, °C

1 1100

1150

Figure 6.20 Measured (Fm) to calculated (Fc) roll force ratio during rolling of a high Nb steel

The calculations were repeated using the complete model, accounting for the recrystallization kinetics and the retained strain. Some improvement in the rolling force calculations was observed (Figure 6.19), but at low temperatures the measured force still exceeded the calculated one by 18%, suggesting that in the real process, recrystallization was slower than predicted by the model. Measurements and calculations of the grain size supplied additional information regarding the accuracy of the model. Calculated temperatures, austenite grain size, recrystallized volume fraction and retained strain, as well as the measured austenite grain size in four passes, are presented in Table 6.23. The detailed applicability of the strain induced precipitation model is not fiilly validated. For low niobium steels the recrystallization was initially halted by solute drag. Running the MA THEMA TJCAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

195 model for some scenarios without the strain induced precipitation included some cases where recrystallization due to the strain accumulation would have been expected. Hence, at least qualitatively, it appears that strain induced precipitation, which then eliminates any subsequent recrystallization, is occurring during the rolling experiments. From a modeling perspective this only requires the strain induced precipitation to be predicted at any time prior to the onset of recrystallization through strain accumulation.

Table 6.23 Austenite grain size, recrystallized volume fraction and retained strains during 4-pass rolling of high Nb steel pass temperature grain size recrystallized retained strain grain size measured, [xm calculated, jam volume fraction 1 1128 0.00 1.0 82 100±15 2 1045 0.8 0.21 74 66±8 3 0.43 956 0.75 66 56±10 4 0.69 907 0.1 61 56+15

All tests described above involve reasonably long interpass times, characteristic of the reverse plate rolling processes. Some more information regarding the performance of the microstructure evolution models is supplied by the experiments carried out by Kedzierski et al. (1996) and by Pietrzyk et al. (1997). The objective of these experiments was to investigate the behavior of low niobium steels during hot deformation with short interpass times. The set of three roller dies, described in section 6.3.1 (Figure 6.3), was used in these tests. The drawing schedules were similar to those performed for the carbon-manganese steel. Strains and interpass times used in all tests are given in Table 6.24. As before, the models by Dutta and Sellars (1987) and by Hodgson (1993a, 1993b) were tested. The results agreed with expectations, and because of this, the observations and conclusions were not very revealing. Metallography has shown that recrystallization has not started in any of the tests, even during the longest interpass times, confirmed by Hodgson's model (1993a). Typical austenite microstructure after the 3-pass test is shown in Figure 6.21a. The results of calculations and measurements of the process parameters are presented in Table 6.25, where DS is the Dutta and Sellars model (1987), and H indicates Hodgson's model (1993a, 1993b). Dutta and Sellars' model (1987), predicted the start of recrystallization in some of the tests correaly. The results concerning precipitation during rolling with short interpass times are not very revealing, either. Replicas did not show precipitates for all variations, in which the calculated time for 5% precipitation /o.o5/> exceeded the interpass time. Precipitates were observed when to.osp was much shorter than the interpass time, see for example Figure 6.21b, where the replica for the test 9 in Tables 6.24 and 6.25 is shown. Lack of consistency in the comparison between the predicted /o.osp and metallography was observed when to.osp was comparable with the interpass time. It should be pointed out, however, that precipitation kinetics depends strongly on the relative contents of such elements as carbon, nitrogen, niobium and aluminum in steel. Small fluctuations of these contents can have a noticeable influence on the precipitaMICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

196 tion start time and as a result, precipitation processes are very complex and difficult to model. Since the tested models are semi-empirical, they cannot be generalized. Table 6.24 Parameters of the experimental hot strip drawing tests for low niobium steels interpass times strains test number of thickness s passes mm 1.6 0.248 1 1 6.48/4.87 0.8/1.4 2 0.248/0.255 2 6.45/4.85/3.76 1.7 3 0.253 1 6.48/4.84 4 2 1.45/2.25 0.245/0.237 6.42/4.85/3.7 5 0.25/0.24/0.207 1.8/1.5/1.6 3 6.48/4.85/3.69/2.93 6 2 0.254/0.241 0.7/8.3 6.5/4.85/3.69 7 1.6/1.7 2 0.246/0.239 6.44/4.86/2.69 8 1.6 1 0.411 6.44/3.79 9 0.254/0.231/0.192 2.1/1.6/3.6 3 6.51/4.856/3.73/3.015 10 3 0,25/0.235/0.192 0.9/0.8/1.2 6.48/4.86/3.72/3.01 11 2 0.25/0.0293 0.9/2.0 6.48/4.86/3.434 12 2 1.7/1.2 0.25/0.386 6.48/4.86/2.98

Figure 6.21 (a) Elongated austenite grains after the test no.lO and (b) precipitates observed after the test no. 9, in Tables 6.19 and 6.20

Recapitulating the validation of the microstructure evolution models for the niobium steels, the conclusion can be drawn that the kinetics of recrystallization for these steels is affected by both solute drag and precipitation. The data presented are too limited to specify precisely under what conditions the solute drag effect prevails. It was observed by Pietrzyk et al. (1995a) that during multipass deformation of a low niobium steel, the recrystallization was retarded at the temperatures above the precipitation start temperature. This conclusion was confirmed by MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

197 metallography as none of the tested samples revealed precipitates. Precipitation was predicted by the model and observed by metallography in a few cases of strip drawing, in Tables 6.24 and 6.25 (Kedzierski et al., 1996). Precisely, these were the tests with longer interpass times (tests 9 and 10 in Table 6.20), when the temperature dropped while the straining was rather low.

Table 6.25 Results of calculations and measurements for hot drawing of low niobium steel T k:95x k. 05x ^.05/7 X 0 C test s S H DS DS H H meas. calc. DS H DS 1 0.32 0.02 16.7 32.9 0.48 8.53 6.27 105.9 984 2 0.28 0.02 42.26 81.6 0.28 5.19 3.67 66.9 1003 1.00 0.02 2.56 5.56 0.14 7.44 1.40 92.4 962 3 0.32 0.02 17.87 35.3 0.42 1026 7.21 130.5 965 975 4 0.28 0.02 56.76 53.0 0.43 8.62 7.09 109.4 975 984 0.48 0.01 2.41 3.85 0.34 22.20 6.30 301.9 900 917 5 0.28 0.02 17.03 33.6 0.54 11.20 8.87 143.4 968 972 0.28 0.01 1.96 3.13 0.53 23.30 7.38 329.4 908 913 0.04 0.00 0.72 0.88 2.49 64.40 20.7 972.3 860 855 6 0.21 0.01 22.68 47.7 0.35 5.90 4.37 79.3 980 994 1.00 0.03 2.39 3.74 0.09 12.90 1.25 241.9 865 875 7 0.34 0.02 25.06 50.7 0.4 8.70 6.36 18.0 980 983 0.36 0.01 2.42 3.70 0.42 18.80 6.37 259.7 915 921 8 0.78 0.01 3.52 6.90 0.08 17.70 2.34 243.6 920 927 9 0.03 0.00 8.04 16.2 18.3 92.10 125.9 1371 888 889 0.02 0.00 0.72 1.44 5.4 187.9 43.4 2835 842 841 0.01 0.00 0.36 0.72 7.7 497.2 60.1 7487 760 776 10 0.06 0.00 4.82 9.50 0.9 18.60 20.0 267.0 952 948 0.22 0.00 0.93 1.76 0.4 23.40 4.80 340.0 905 913 0.06 0.00 0.42 0.66 0.88 43.00 7.70 649.0 868 871 11 0.06 0.00 4.71 9.31 0.9 19.05 20.6 274.0 950 948 0.33 0.00 0.80 1.60 0.2 28.90 3.10 445.0 873 894 12 0.009 0.01 4.52 8.88 1.7 24.20 26.2 340.0 910 940 0.41 0.00 0.66 1.31 0.12 31.20 1.98 452.0 870 891

strips D fxm DS 100.2 103.4 38.5 99.9 103.7 82.1 103.0 99.3 123.0 107.9 36.0 99.3 92.8 54.2 120.8 123.0 123.0 118.5 103.9 123.0 118.6 93.0 116.4 84.0

H 121.6 121.9 121.2 121.7 121.7 122.0 121.7 122.4 123.0 122.2 123.0 121.5 122.2 122.1 122.8 123.0 123.0 122.7 122.7 123.0 122.7 123.0 122.5 123.0

Also beyond the scope of the limited number of hot rolling and drawing experiments described above is a discussion of the potential causes for deviation between the recrystallization models and the measurements, except to suggest that in the case of low niobium steels Hodgson's model performs reasonably well. In the case of data used for this model and the data generated by Pietrzyk et al, (1995a), there was significant thermomechanical processing of the austenite prior to the pass where the recrystallization was halted. Dutta et al., (1992) suggested that roughing can have a major effect on the kinetics of strain induced precipitation. However, this would not be expected to be a major cause here, since the difference appeared MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

198

.

in the early passes where solute drag seems to be the main cause for the retardation of recrystallization. The potential segregation of solutes and other impurities in reheated and nondeformed steels compared with extensively deformed steels may account for some of the differences. Also, the recrystallization model developed by Hodgson (1993a) was based on steels with low niobium while the data used by Dutta and Sellars (1987) was for higher niobium steels. Experiments carried out for 0.084%Nb steel (Figure 6.19) showed that the latter model improves, to some extent, the results of calculations of the rolling loads. It also predicts the grain size, which agrees well with the measurements. 6.3.3

Room temperature properties

Some typical experiments carried out to validate the models describing the metal's microstructure and mechanical properties at room temperature are presented in this section. The main focus is placed on niobium microalloyed steels. The rolling tests carried out by Pietrzyk et al., (1995a), and described above, involved measurements of the ferrite microstructure after cooling the samples in air. Analysis of micrographs for all tests shows that the microstructures were more nonhomogeneous for T2 and T4 than for Tl and T3 tests. A finer microstructure with a smaller volume fi-action of pearlite was observed at the surface, than at the center. The ferrite grain sizes for the tests Tl, T2, T3 and T4 for different start temperatures were measured and compared to the prediction and good agreement was observed as shown in Table 6.26. A lower start temperature, which resulted in a lower finishing temperature, yielded a larger ferrite grain size for the test Tl in the center. The surface displayed a smaller grain size than the center, especially for the tests T3 andT4.

Table 6.26 Measured and calculated ferrite grain size (^im) test center (start temperature) measured calculated Tl-l(118rC) 7.3 8 T1-3(1168°C) 8.4 8.2 T2-5(1163°C) 8.4 7.4 T2-6 (1209^0 8.4 7.4 7.4 T4.1(1169°C) 6.1 4.8 T4-1 (1187^0 6.1 T5-1 (1169^0 6.3 7.7

surface measured 5.5 6.4 7.0 7.0 6.3 3.3 3.6

calculated 6.8 6.7 6.5 6.5 5.0 4.6 6.0

Other experiments, aiming at an evaluation of mechanical properties at room temperatures, were carried out by Majta et al., (1995). The objective of the experiments, which included three-stage compression tests, was to obtain data concerning the effect of the thermomechanical history on room temperature properties. The isothermal tests were performed under constant true strain rate conditions with different length of interruption times between the compression cycles. The development of the microstructure, as affected by the delay times, was observed on the ferrite grain sizes and, in consequence, on the yield stress distribution in the MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

199 final product. The tested material was a high niobium steel containing 0.067%C, 1.3%Mn, 0.076%Nb, 0.024%Ti, 0.34%Si, 0.037%A1., 0.005%V, 0.035%Cr, 0.013%Cu and 0.0054%N. All samples were annealed and solution treated before testing. Each was fitted with a type K thermocouple, embedded at midpoint. A glass powder - alcohol emulsion was used as the lubricant.

200 n strain rate 2 s^*" temperature 975° C 160H

120-

r.

I STAGE

t

II STAGE

80

'

III STAGE

f

td^Os^

—

td= 0.3 s

-—-

td=20s^

40-

0.0

td=3s

0.2

0.4

0.6 strain

0.8

1.0

I 1.2

Figure 6.22. Continuous and interrupted stress-strain curves obtained for various delay times td.

600

700

780

850

950

finish deformation temperature, °C

^Jg^^e 6.23. Contribution of various components to strengthening as a function of the finishing temperature.

In the multi-stage tests, a strain of 0.37 was applied in every stage of the compression, chosen to avoid dynamic recrystallization during prestraining. The resulting stress-strain curves are shown in Figure 6.22, indicating, in the uninterrupted test, the presence of dynamic recovery only. The sample was deformed to a true strain of 0.37, unloaded and reloaded to a total strain of 0.74, then unloaded and reloaded again to the total final strain of 1.1. Each of the deformations was performed at a constant strain rate of 2 s"V The interruption times were 0.3, 3 and 20 s, respectively. The same delay times were used after full compression and before quenching in order to allow the estimation of the recrystallized austenite grain diameters. During interrupted compression tests, the delay time was constant for each stage. After deformation, the specimens were cooled in air at a cooling rate of 4 K/s. The amounts of fractional softening after each interruption are shown in Figure 6.22 and it is observed that, as expected, the softening increases with increasing interruption times. Of course, the fractional softening changes if the deformation per stage changes. The contribution of the various components to the strengthening of a microalloyed niobium steel is presented in Figure 6.23, as a function of the temperature at the last stage of deformation. As indicated, the increase in yield stress with decreasing finishing temperature Tm results primarily from the increase of the dislocation density. Beginning at the Ars temperature, which for the present steel is expected to be about 785°C, subgrain formation causes further increases in strength. The effect of using Eq. (6.22), the root of the sum of the squares sumMICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

200

mation, is also observable, showing that the effects of the other strengthening components decrease below Ars. The microstructures after various delay times are shown in Fig. 6.24. The lack of refinement of the ferrite structure is observed when the delay time is increased to 20 seconds, see Figures 6.24e and f The amount of statically recrystallized austenite also increased and the ferrite grain diameters are over 10 jim at some of the locations. Away from the center, the grains are observed to be still larger, probably indicating the decreasing axial pressures with increasing radial distances.

Center

2 mm from the center

J v^i'V't*"""!

"' ^i^.4-ii?f 0.3 s de/ay

-^V'" 20 urn

3 s delay

^ i.^

20unfi

e) :-;->

"{^' >;

20 urn

^ ^ y^

r-•%.. 20 s delay r y ^

20 urn

< ">'

20 ^m

Figure 6.24 Microstructure after various delay times during interrupted compression tests

A comparison of the yield stress calculated by Eq. (6.22) and those measured elsewhere by various researchers is shown in Figure 6.25. A large number of different steels of various chemicar compositions, as well as different reheating, deformation and cooling conditions have been incorporated in the figure. The comparison indicates good agreement between calculated and measured yield strengths, showing that Eq. (6.22), using the root of the sum of the MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPER TIES OF HOT ROLLED PRODUCTS

201^ squares summation, is a reliable way of calculating the room temperature properties of steel subjected to thermal-mechanical treatment. Further proof of the predictive ability provided by the root of the sum of the squares summation is given in Figure 6.26. There, a comparison of the calculated yield strength by linear and r.s.s. method is illustrated, showing that as the finish deformation temperature decreases, the yield strength calculated by the linear summation predicts significantly higher values. This could lead to difficulties during subsequent planning using material strength. These differences increase when the TFD decreases, especially below ^rs.

800 CL

Figure 6.25 Computed yield stress compared with the measurements carried out by various authors

700 600

500 3

400

300 300

f A°

MO

D •

IR2

A

BA

+ T

GR

lo 400 500 600 700 measured yield strength, MPa

IR1

HE

C01 C02

MO - Morrison et al. (1993) IRl - Irvine and Baker (1976) IR2 - Irvine etal. (1967) HE - Keijan and Baker (1993) BA - Baker and McPherson (1979) GR - Greday and Lambergist (1976) COl-Coldrenetal. (1981) C02 - Coleman et al. (1973)

800

Figure 6.27 shows the comparison between yield strength calculated using Eq. (6.22) developed by Majta et al, (1995) and those proposed by several researchers. When the last deformation is at a temperature in the recrystallization region (finishing temperature 7}© = 975°C), the agreement among the calculated results is reasonably good. However, the predictions are more difficult for the lower TFDFigures 6.26, 6.27 and 6.28 show results obtainedfi-omthe finite-element model described in Chapter 5, coupled with the empirical microstructural relations given in this chapter. The results include the strain distribution after the second and the third stage, as well as final austenite and ferrite grain sizes. In thefigures,all grain sizes are given in |j,m. The measured austenite and ferrite grain diameters are also shown in the figures, written using bold font at the appropriate positions. Since the difference between strains for interpass times of 3 and 20 s was small (Figures 6.27a and b and 6.28), the grain sizes for the latter case are not shown. In order to demonstrate the abilities of the model. Figures 6.28, 6.29 and 6.30 show the development of the retained strain distributions for the three tested delay times. The true strains, calculated by the usual definition of Ludwik, are 0.37, 0.74 and 1.1, after each of the three stages of compression. However, the values of strain given in Figures 6.28, 6.29 and 6.30 include in this measure the effect of static recrystallization, occurring during the unloading periods. If the delay times during the interruptions are long enough (3 and 20 secMICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

202

^

onds, as in Figures 6.29 and 6.30), the distributions of strains after stages n and III are typical for a fully annealed material. When, due to a short delay time (0.3 seconds in Figure 6.28), the static recrystallization does not occur, the accumulated strains are expected to achieve larger values and their distributions are more complex. It is important to state here that these distributions have been calculated as a result of the complete history, including the deformation and the microstructural development. 900 800 (0 Q-

S

700

k

•

\

tjl- ———

\\

k

500

Irvine and BaKer (1984)

800

linear summation QL

2 700-1

\\

^

Gladman et al. (1976) Hodgson and Gibbs (1992) Le Bon and Saint-Martin (1976) Majta et al. (1995)

t 600 2

900

1 1 1.8. summation 1 1

teooH \ "^

(A ' * * •

•*-. 2 500

400

400 Hfinishing temperature 975^0

300 600 650 700 750 800 850 900 9501000 finishing temperature, °C

Figure 6.26 Yield stress as a function of thefinishingtemperature calculated using linear summation and r.s.s. method

300

-1 \ \ r 2 4 6 8 ferrite grain size, ^im

10

Figure 6.27 Yield stress as a function of the ferrite grain size calculated using various formulae

The distributions of austenite grain sizes after each of the loading cycles are presented in Figures 6.28b and 6.29b. When the deformation history is as presented in Figure 6.28, the final austenite grain size is generally smaller than in the case with longer unloading times. The deformation is more homogeneous in the latter case. There is, in general, an increase in the austenite grain diameters as the cylindrical surface or the tool/workpiece interface is approached. This increase is quite small for 0.3 and 3 seconds, in both the radial and axial directions, but reaches 25 - 30% when the delay time is 20 seconds, (Majta et al, 1995). Several competing mechanisms are operating here. As the cylindrical surface is approached, the cooling rate increases, leading to smaller grains. The strains are decreasing, however, with increasing radial distances, leading to less pancaking. The second mechanism appears to be dominant in the present instance in the radial direction. In the axial direction the cooling rates are smaller, since the compression rams are, at least initially, at the same temperature as the sample. As a result both mechanisms are co-operating in creating progressively larger grains. These figures also show that there is good agreement between the predictions and the experimental data. Because of the direct correlation between austenite grain size and ferrite grain size, similar differences in ferrite distributions are demonstrated in Figures 6.28c and 6.29c. The ferrite grains are approximately 20 - 25% smaller than the austenite grains. The variations in ferrite grain size in the axial and radial directions are considerably smaller than MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

203 those observed for the austenite grains. Again, the agreement between the calculated and measured grain size distributions is quite good. The differences are a few percent.

a)

b) rlO.68 £

tp = 0.3 s P

austenite grain size, ^(,m

E A

Figure 6.28 Distribution of the effective strain after the second and third stages (a), austenite grain size (b) and ferrite grain size (c) after the third stage for the interpass times of 0.3 s; Isolines represent the calculated, and bold face numbers, the measured grain size

a)

b) tp = 3 s

effective strain

tp = 3 s

austenite grain size, /xm

Figure 6.29 Distribution of the effective strain after the second and third stages (a), austenite grain size (b) and ferrite grain size (c) after the third stage for the interpass time of 0.3 s; Isolines represent the calculated, and bold face numbers, the measured grain size MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

204

The yield strength distributions for the tp = 20 s effective strain investigated tests, as predicted by Eq. (6.20), are presented by Majta et al. (1995). The effect of increasing ferrite refinement is seen there, with the finest ferrite grains corresponding to the highest values of yield strength. As expected, the largest yield strength, 460 MPa at the center, corresponds to the shortest delay time when essentially no softening occurred after the first loading. As the delay was increased by a factor of ten, the yield strength at the r, mm center decreased to 450 MPa, a drop of about 2%. With the 20-second delay the Figure 6.30 Distribution of the effective yield strength dropped by another 2% to strain after the second and third stages 442 MPa. While these changes are not exof compression cessive, the possibility of changing the metal's strength by thermo-mechanical treatment was demonstrated and correctly predicted. A number of experiments was performed to illustrate the predictive capability of the equations, describing the mechanical properties of steels at room temperatures, developed by Kuziak et al., (1997). The selected data for 13 specimens, containing the type of product and steel chemical composition, are listed in Table 6.27. Table 6.28 contains results of calculations of microstructural parameters, as well as a comparison of the measured and predicted yield stress for these specimens. The best-fit equations developed for the average yield stress and ultimate tensile strengths, on the basis of data obtained from all tested specimens, are given in Table 6.15 (lower yield stress). Table 6,16 (contribution of precipitates) and Table 6.17 (tensile strength). Major differences distinguishing these equations from those developed by Gladman, (1972) consist of i) a linear law of mixtures for the yield-strength calculation; and ii) an inclusion of the nonlinear dependence of the ultimate tensile strength on the pearlite interlamellar spacing. The latter followed earlier observations that the pearlite interlamellar spacing plays a significant role in shaping the uhimate tensile strength when the ferrite loses its continuity in the microstructure. The predictive ability of the equations, developed by Kuziak et al., (1997), is confirmed by the experimental results given in Table 6.28. Notation in this table is as follows: Xf is the volumefractionof pearlite, Da is the ferrite grain size. So is the interlamellar spacing in pearlite, ay is the lower yield stress, and cju is the ultimate tensile strength. Zurek et al., (1998) investigated the effect of strain rate and two-phase deformation on the mechanical properties after thermomechanical processing of a niobium steel, containing 0.067%C, 1.3%Mn, 0.34%Si 0.037%A1. 0.076%Nb and 0.024%Ti. The uniaxial compression tests were conducted in a range of strain rates from 10 s'* to 50 s"^ using a tensioncompression testing machine. A split Hopkinson pressure bar was used for the strain rate of 2500 s" Each of the experiments was performed at a range of temperatures corresponding to austenite, two-phase and ferrite regions. Typical stress-strain curves obtained during these MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

205^

experiments are presented in Figure 6.31. As expected, only a small influence of the temperature on the flow stress is observed in the fastest test.

Table 6.27 Selected chemical compositions and forms of the steel products used for the development of the microstructure versus mechanical properties relationships composition

steel product form

No

C

Mn

Si

P

S

N

1.

0.04

0.23

0.05

0.010

0.015

0.008

2.

0.05

0.31

0.05

0.010

0.016

0.007

10-mm-diam. rod

3.

0.18

0.57

0.26

0.028

0.026

0.009

12-mm-diam. rod

4.

0.65

0.52

0.21

0.014

0.012

0.006

10-mm-diam. rod

5.

0.19

1.20

0.33

0.019

0.021

0.007

12-mm-diam. rod

6.

0.22

1.23

0.34

0.059

0.038

0.009

14-mm-diam. rod

7.

0.33

1.20

0.52

0.015

0.028

0.008

8-mm-diam. rod

8.

0.20

1.15

0.35

0.044

0.029

0.005

8-mm-diam. rod

9.

0.19

1.31

0.36

0.024

0.030

0.007

I-beam

10.

0.18

1.32

0.35

0.023

0.028

0.008

I-beam

11.

0.20

1.32

0.39

0.033

0.022

0.009

I-beam

12.

0.48

0.86

0.22

0.026

0.024

13.

0.42

1.01

0.61

0.029

0.042

0.009 0.007

20-mm-thick plate

8-mm-diam. rod

I-beam

Metallography carried out by Majta et al, (1997) and by Zurek et al, (1998) shows that the refinement of the ferrite grains is much stronger for the specimens strained at 750°C and 800°C than those strained at 850°C. Moreover, a low finishing temperature results in more nonhomogeneous ferrite microstructure. This is probably due to the different amounts of strain concentrated in austenite and ferrite phases at these temperatures. The influence of the strain rate on the ferrite microstructure is shown in Figure 6.32. The ferrite grains in the specimens deformed at low strain rates are finer than expected. The coarser grains obtained after faster deformation (50 s"^) are probably due to the faster static recrystallization and grain growth after recrystallization. Again, the nonhomogeneity of the microstructure is very pronounced at 700°C, probably due to nonuniform conditions of transformation in the volume of the sample. More uniform microstructure is observed in the experiments, conducted at 850°C. Simulation of microstructural phenomena, which take place in the two-phase region, is very difficuh. There is still a lack of reliable models able to describe this process adequately.

MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

206 Table 6.28 Measured microstructural parameters and strength properties of the products specified in Table 6.27 compared with the predictions based on the equations by Kuziak et al, (1997) given in Tables 6.15, 6.16 and 6.17 No

strength properties

measured microstructural parameters

CJy,

cr„, MPa

MPa predicted

measured

predicted

Xf

Da, |im

&, \im

measured

1 2

0.90

23.6

0.220

278

232

390

408

0.91

18.8

0.201

272

240

417

405

3

0.78

18.4

0.168

331

328

530

509

4

0.15

8.20

0.245

450

437

630

793

5

0.71

13.9

0.155

380

395

582

542

6

0.65

10.6

0.182

421

447

635

600

7

0.51

9.73

0.158

482

461

690

654

8

0.57

9.10

0.158

430

452

636

613

9

0.61

17.2

0.182

402

398

550

575

10

0.67

15.1

0.185

400

402

567

565

11

0.64

19.2

0.221

390

393

582

576

12

0.20

9.43

0.230

460

453

770

780

13

0.34

10.5

0.213

448

452

750

740

b) 400-| strain rate 2500 s -"'

800 *C .700°C VSOoC

strain rate: ^2500 s-^

350300-

^

S. 250-

50 s -1 1.0 s-""

CO 2 0 0 co % 150-

"0.001 s-""

10050-|r^

0.00

0.05

0.10

"1 1 0.15 0.20 Strain

1 0.25

1 0.30

temperature SOQoC

fi f

U 1

0.00

0.05

0.10

1

1

0.15

0.20

strain

1 0.25

1 0.30

Figure 6.31 Effect of the strain rate and temperature on the stress-strain curves for a Nb steel MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

207

700X 0.001 s"'

700°C 50 s''

850°C 0.001 s'^

850°C 50 s-'

Figure 6.32 Ferrite microstructure of specimens deformed at strain rates 0.001 s^' (a,c) and 50 s'^ (b,d) at temperatures 700°C (a,b) and 850°C (c,d).

6.4

THE INTERNAL VARIABLE METHOD

In the previous chapters, the modeling of material properties and the evolution of the microstructure during hot metal forming processes have been based on semi-empirical equations, which describe the yield stress controlled by the hardening and softening phenomena. The external variables, including the temperature, strain rate and strain, are the input parameters for the models. The shortcomings of this approach are obvious and they derive from the inability of the strain to represent the state of a hot-worked metal under non-steady conditions. Current finite-element models are able to quantify the complete thermomechanical history of the workpiece. They show that all elements undergo variations in strain rate and temperature, and these variations depend on the locations of the elements (see the typical results in Chapter 5). Furthermore, some of the elements have complex strain path histories. Even in geometrically simple processes like flat rolling, elements near the surface are subjected to a reverse shear strain. Figure 6.33 shows the calculated distributions of the longitudinal strain, the shear strain and the effective strain, during rolling of a 20 mm thick plate with a reduction of 0.2. Reversing shear strains are clearly seen in this figure. The shear strains have been calculated by an integration of the shear strain rates along the flow lines. The integration leads to almost zero shear strains at the exit plane. Distribution of the longitudinal strain through the thickness at the exit is unifonn. Effective strains, which account for the contribution of the reversing shear strains, are distributed non-uniformly, with the lowest values in the center of the plate. The results in Figure 6.33 show that accounting for the history of deformation is necessary in the modeling of hot deformation processes.

MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

208 longitudinal strain Ey 10 E E

V^^YY1=?

5

if? o o i> o ai^ ^

^

%'%'<^%'^;^\% S^ :r:.:

10 15 20 25 30 36 40 45 50 55 60 65 X, mm shear strain ^^v J

AL

10 15 20 25 30 35 40 45 50 55 60 65 X, mm effective strain Sj 10

E E >i

5

V^^m..

0 10

20

30

40

X, mm Figure 6.33 Calculated distributions of the longitudinal strain, shear strain and effective strain during rolling of 20 mm thick plate, reduction 0.2

The conventional microstructure evolution models are not capable of taking advantage of the ability of thefinite-elementtechniques to predict variations of the process parameters and complexity of the strain path in various forming processes. Since the microstructure evolves through thermally activated processes, which continue in the absence of deformation, the strain is not a state variable at elevated temperatures. Other variables are, therefore, necessary to represent the true state of the material. In recent years, modeling hot deformation behavior has increasingly been based on the physical phenomena within the material. These models are based on the so-called internal state variables, which, may be scalars, vectors or tensors. The most commonly used internal state variable models involve features such as internal dislocation density, subgrain size and subgrain boundary misorientation (see for example Brown and Wlassich, 1992, Sellars, 1997). Both internal and external state variables provide sufficient information to describe a particular phenomenon accurately. Sandstrom and Lagneborg (1975a and 1975b) suggested a model for hot working and recrystallization, based on the dislocation density as an internal state variable. More recently, Mecking and Kocks, (1981), and Estrin and Mecking, (1984), have proposed similar approaches. These publications give a description of the stress-strain relationship, which considers the dependence of the process on strain hardening and fractional softening. Possibilities of the application of numerical techniques to the solution of complex problems in materials MA THEMA TJCAL AND PHYSICAL SIMULA IJON OF THE PROPERTIES OF HOT ROLLED PRODUCTS

209 science were limited in the 1970s and early 1980s. Therefore, the solutions of the dislocation density equations were performed for isothermal conditions, and the advantages of this model could not be exposed. Rapid development of computers and numerical techniques during late the 1980s and 1990s allowed for an extensive qualitative change in this approach. The objective of further work was the implementation of the internal variable model into finite-element programs, which simulate metal flow and heat transfer in industrial forming processes (Pietrzyk 1993, Pietrzyk, 1994). In what follows, a description of the internal variable approach, based on the dislocation density is given, as presented in numerous publications. Notable among these are the works of Sandstrom and Lagneborg, (1975a and 1975b); Mecking and Kocks (1981); Estrin and Mecking, (1984); Alden, (1987); Brown and Wlassich, (1992); Pietrzyk, (1994); Davies, (1994); Estrin (1996); and Sellars, (1997). Further examples of the application of this model to the simulation of hot forming processes are presented. 6.4.1

General frame of the model

A general fi-ame for the model is given on the basis of its application as a constitutive law, as described in detail by Estrin, (1996). He shows how the problem of constitutive modeling can be reduced to operating with scalar instead of tensorial quantities. The components of the tensor of total strain rate e, are given by the sum of the elastic and plastic components, f *and e^, respectively: e = e^+e^

(6.38)

The elastic component obeys Hooke's law, while the plastic component of the strain rate tensor is given by the Levy-Mises Eq. (5.2). In this equation, the Huber-Mises equivalent quantities are used: sf = j—sfjsfj V3 -^ -^ and

-if

(effective plastic strain rate)

(JijCJij (effective stress)

(6.39)

(6.40)

This implies plastic isotropy of the deformed material. The specifics of the model is due to a particular form of the equation, relating the effective plastic strain rate and the effective stress referred to as a kinetic equation (Mecking and Kocks, 1981; Estrin and Mecking, 1984). Thus, the problem of constitutive modeling is reduced to operating with scalar, rather than tensorial quantities. The character of the experimental data obtained in plastometric tests offers a suitable form of the kinetic equation. A power-law is the most commonly used form: (6.41)

MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

210 where according to Estrin (1996), G^ is the internal variable representing the state of the material, while ^0 ai^d f^ are material parameters. Since the temperature dependence of the plastic strain rate is included in the stress exponent w, Eq. (6.41) obeys the Arrhenius law for thermally activated plastic flow by dislocation glide. The parameter e^ is proportional to the density of mobile dislocations. Another variation of the kinetic equation, in which the Arrhenius law is preserved, is (Kocks et al., 1975):

- ^ - ^ o e x p ^ - - ^ 1-

(6.42)

where AGo is the Gibbs free energy of activation at zero stress, <j is the Boltzman constant, p and q are material parameters determined from fitting the sf versus o} curve, and ^o ^s a constant. At the limit of 0 Eq. (6.42) yields a finite plastic strain rate. Thus, a definite nonzero plastic strain rate corresponds to any value of stress, however small that may be. The model is thus characterized by no yielding or loading/unloading conditions. A number of constitutive models (Mecking and Kocks, 1981; Estrin and Mecking, 1984) share this property. The kinetic equation refers to a stable microstructure, that is, to a constant value of the internal variable a^. However, since the microstructure varies in hot metal forming processes, a separate equation is needed to describe the evolution of G^ :

^

= f{p,,erj)

(6.43)

According to this equation the rate of change of the internal variable G^ is determined by its current value and no memory or path dependent effects are included. Once the concrete form of the function/is specified, the constitutive formulation is complete. The analysis presented above implies that the quantity G^ is the sole internal state variable that represents the microstructural state of a material (Estrin and Mecking, 1984). However, it can generally be expected that several internal structural variables are needed to describe the mechanical response of a material. These variables are characterized by different rates of relaxation towards their steady state values. The steady state value of a particular internal variable can be considered to be dynamic one, since it is governed by current values of slower internal variables still evolving towards their steady state (Estrin and Mecking, 1984). As the deformation proceeds, more and more internal variables will reach steady state. Eventually, at large strains, the slowest-evolving internal variable will govern the rest. In the above formulation, the total dislocation density is considered to be that slowest variable. The examples of modeling plastic deformation given below demonstrate that for continuous monotonic straining, a one^intemal-variable model is sufficient. More sophisticated models with two and more internal variables can be invoked if rapid changes in deformation conditions or cyclic loading are to be accounted for (Estrin, 1996). MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

2n^ 6.4.2

Hardening and recovery

The general framework of the one-internal variable model is presented on the basis of the research of Mecking and Kocks, (1981). Their model, known in the scientific literature as the Kocks-Mecking model (K-M model) is the fundamental approach and has pointed the direction for later developments and more recent theories. The model is based on the kinetic relationship in the form of Eq. (6.42), where the internal variable <J^ is related to the total dislocation density p: a^=Mj,aGby[p

(6.44)

where G is the shear modulus, b is the Burgers vector, MT is the average Taylor factor and a is a constant. The model implies that the strength of the material is determined by dislocation-dislocation interactions. All other sources of resistance to dislocation glide are disregarded at this stage. Arguments justifying this choice of the internal variable and proving that Eq. (6,44) can be assumed to have general validity are given by Kocks et al., (1975). The Taylor factor Mr accounts for the texture evolution and will not be considered here. The evolution equation for the dislocation density p is derived accounting for the concurrent effects of storage and recovery. The storage term is written by expressing a shear strain increment, in terms of the dislocation density increment associated with immobilization of mobile dislocations at impenetrable obstacles after they have traveled a distance /:

In Eq. (6.45), s represents the plastic strain. Since elastic deformation will not be discussed further, the index p is omitted. To include dynamic recovery, one can consider the annihilation of stored dislocations, which involves their leaving the glide planes on which they are stored. This is frirnished by the cross-slip of screw dislocations or the climb of the edge dislocations. The first process prevails at low and the second at high temperatures. Inclusion of a dynamic recovery term in Eq. (6.45) yields:

where the recovery coefficient is strain rate and temperature dependent:

^2 ~ ""20

H-i

(6.47)

In Eq. (6.47) feo is a constant. The temperature dependence of fe at high temperatures is contained in the Arrhenius term in Eq. (6.47), while /? is a constant, typically about 4. In the low temperature range, the temperature dependence is contained in «, for which case n is inMICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

212

versely proportional to T, while the Arrhenius term is omitted. In Eq. (6.47) Qs represents the activation energy for dislocation climb, equal to that for self-diffusion (Estrin and Mecking, 1984). Eq. (6.46) can be expressed in the form of the time derivative of the dislocation density: ^ = ±.-k^pe at bl

(6.48)

The recovery processes included in the above equations are of dynamic origin. Static recovery, in which a decrease of the dislocation density is proportional to the corresponding time increment, can be included in Eq. (6.48): ^-^ = fj-k.ps-K

(6.49)

Assuming that static recovery is driven by the stress determined by the square root of the current dislocation density, a reasonable phenomenological model for the static recovery coefficient, 7?v, is (Estrin, 1996): i^=/?,oexp|-^|sinh ' ^ 1

(6-50)

where i?vo, Co and C\ are constants. The form of the evolution Eq. (6.50) for the dislocation density provides a possibility of incorporating metallurgical characteristics and microstructural features of a material into the model. In a material which is coarse-grained and single-phase the only kinds of obstacles to moving dislocations will be those related to the dislocation structure itself and those provided by the grain boundaries. Regardless of how the dislocations are arranged, whether completely at random or in a cell or subgrain boundary structure, the mean free path of dislocations / is proportional to p~^-^. Since the free path / is usually much smaller than the grain size of the material, Eq. (6.48) becomes:

-^^Ke4p~k^pe

(6.51)

where k\ and k2 are constants, with k2 being given by Eq. (6.47). The constitutive Eq. (6.51) can be integrated analytically, at least for the case of uniaxial deformation with constant plastic strain rate (Mecking and Kocks, 1981) and for constant stress creep (Estrin and Mecking, 1984). The hardening term in Eq. (6.51) has to be changed when the density of geometrical obstacles becomes larger than that of the obstacles caused by other dislocations in the population. The mean free path / is then identified with the spacing between these geometrical obstacles, d. Assuming that the distance between the obstacles does not change during the deformation and, further, that the obstacles do not affect the recovery coefficient ^2, their only influence on MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

213 the flow stress will be through the effect on the rate of dislocation density. The kinetic equation is then still given by Eq. (6.51), whereas the evolution equation for dislocation density is written as:

f = ±-.,p.

(6.52)

It should be mentioned here that Sandstrom and Lagneborg, (1975a, 1975b), and Roberts and Ahlblom, (1978), adopt a recovery term derived from Friedel's treatment (1964) of the climb controlled dislocation network. This approach is related to the probability of dislocations meeting and annihilating one another, which is proportional to ^\

The constantfeis defined as: ^3 = I'M

(6.54)

where M is the dislocation mobility which, in a pure metal, is directly related to the selfdiffusion coefficient, and ris the dislocation line energy. Kocks and Mecking (1981) have argued that Friedel's development is based on a model for static rather than dynamic recovery and some caution must be exercised in its application. A particular case in which the model represented by Eq. (6.52) or (6.53) will be applied is that of grain boundary hardening. Estrin, (1996) points out, however, that the limiting case of the grain size being the smallest characteristic length in the struaure applies only for the submicron grain sizes, when J<10p"°^. The modified constitutive model now given by Eq. (6.52) allows simple integration. The above considerations apply if geometrical obstacles outnumber the dislocationstructure related ones. In a more general case, when a superposition of the immobilizing effects of both types of obstacles is considered, the inverse obstacle spacing Ml in Eq. (6.50) can be expressed as a linear combination of the inverse spacings of the two types of obstacles taken separately. The resulting evolution equation for dislocation density (Estrin and Mecking, 1984) is: ^^^s[^^k,fp\^-k^p8

(6.55)

Estrin (1996) presents an integration of the constitutive model Eq. (6.55). He also shows other applications of the model based on the kinetic Eq. (6.42), for example, to describe plastic deformation of other systems where the dislocation mean free path is constrained by microstructural elements, such as precipitates or second-phase particles. The methodology described above can be used to account for the particle effects on the deformation behavior. The constitutive model based on the kinetic Eq. (6.42) can also be formulated for materials conMICROSTRUCTURE EVOLUTION AND AdECHANICAL PROPERTIES OF THE FINAL PRODUCT

214 taining a dispersion of non-shareable second-phase particles, such as non-coherent precipitates, oxide or carbide dispersions. 6.4.3

Recrystallization

The theory described in the previous section is suitable for modeling situations in which dislocation interactions resuh in an immediate response of the system, which is dynamic recovery. In industrial processes, however, this is not always the case. It is well-known, and can be confirmed by experiments, that an excess of stored energy leads to the occurrence of recrystallization (see Section 6.1). This, in turn, implies a delay in the system's response, while the required excess of energy accumulates and can be mathematically simulated by the modification of Eq. (6.52). This modification involves a parameter, that accounts for the development of the dislocation population (stored energy), to the point at which widespread elimination of dislocations is observed. Thus, recrystallization is a discrete process, which requires that some threshold of stored energy is exceeded and which needs some time to be completed. Therefore, there can always be a situation in which neighboring areas in the material have different values of the internal variable. It seems that a proper approach to this problem requires an analysis of the distribution of the internal variable within some small volume of the material. This approach will be discussed in the following sections, while a simplified model, assuming an average dislocation density is described first. Davies (1994) claims that dislocations can be treated en masse without losing any of the detail that determines the behavior of a material. While hardening and recovery in this approach are described by Eq. (6.52), the modeling of recrystallization is not as evident. Davies (1994) proposes the use of a so-called development strain, which accounts for the stored energy and controls the onset of dynamic recrystallization. The most simplified application of this concept assumes that the influence of hardening and recovery on the flow stress is well described by one of the stress-strain functions, discussed in Chapter 3. Kusiak et al. (1996) considered the hyperbolic sine equation, Eq. (3.24) for this purpose. Thus, since the dislocation density is proportional to the square of the flow stress, the following relationship is obtained:

^(0= > w ( O f -\A[a{t-t;j^dt

(6.56)

where a{t) is the flow stress accounting for dynamic recrystallization, (JHRv(t) is the flow stress calculated from Eq. (3.24), but using time t as an independent variable, and tc is the time correlating to the critical value of development strain. Constant A in Eq. (6.56) depends on the temperature, strain rate and grain size. Application of the inverse technique yields the following relationship (Kusiak et al., 1996):

where D is the austenite grain size, s is the strain rate, R is the gas constant and T is the absolute temperature. MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

215 Eq. (6.56) was validated for the compression of cylindrical samples measuring 4.5 mm in diameter and 9 mm in height. A detailed description of the experiment is given by Kusiak et al., (1996). Single-stage and two-stage compression tests were performed at various temperatures and strain rates. The chemical composition of the steel was: 0.49%C, 0.72%Mn, 0.55%Si, 0.01%P, 0.008%S, 0.14%Cu, 3.26%Cr, 0.48%Ni, 0.44%Mo, 0.11%V, 0.009%N and 0.04%A1. The interpass times for the two-stage tests varied between 1 and 10 s. Typical results of the comparison of the measured and calculated compression stress are presented in Figure 6.34 There are two theoretical curves in each plot, both calculated by the finiteelement model. One curve was obtained using constitutive Eq. (3.24), and the second using the current model (6.56) as a constitutive law. Analysis of the plots in Figure 6.34 and all the results presented by Kusiak et al. (1996) shows that significant improvements of the accuracy of simulation were obtained when Eq. (6.56), which accounts for the dynamic recrystallization, was used. The interpass time in the two-stage compression (Figure 6.34b) was 1.4 s. Since the steel contained elements which retard the recrystallization, the material has not completely softened. As seen in Figure 6.35, the recrystallized volume fraction varied between zero below the die to about 0.45 in the center of the sample. In consequence, the stored energy accumulated in the second stage of compression and dynamic recrystallization was triggered.

a)

b)

400-

400-1 I temperature 845°C strain rate 0.5/0.8 s-"" interpass time 1.4 s

temperature 843oC strain rate 0.65 s-*"

300-

soon (0 Q.

Q.

IE

200

200

100-

measurement

100 H I

0-t 0.0

1 0.2

[—O— calculation, eq. (6.56)J \ \ \ 1 0.4 0.6 0.8 1.0 Strain

measurement -0—

calculation, eq. (3.24)

0-f 0.0

calculation, eq. (3.24)

i—O— calculation, eq. (6.56)J T \ \ 1 0.2 0.4 0.6 0.8 1.0 strain

Figure 6.34 Measured and calculated stress for single-stage (a) and two-stage (b) compression

6.4.4

Average dislocation density model

A more advanced solution of the dynamic recrystallization problem, still based on the average dislocation density, is presented by Davies, (1994). He introduced the development strain as a modifying parameter in Eq. (6.55) and assumed that hardening is controlled by two mechanisms. At low strains, the dislocation density is proportional to the imposed plastic MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

216 strain, while at higher strains the generation of dislocations is dependent on the current dislocation density. The annihilation of dislocation in Davies' model (1994) is related to the probability of dislocations meeting. In consequence, he obtained:

de

(6.58)

where c is the development strain and ps is the dislocation density for the current strain e. The constant A\ is determined by the first term in Eq. (6.52), and A2 and A-^ are evaluated from experimental data. The constant A^ is temperature dependent according to the Arrhenius law. Eq. (6.58) shows that for larger values of the development strain c, the accumulation of stored energy is sufficient to trigger recrystallization. If the development strain is low or zero, recrystallization is not possible, and recovery is a dominant softening mechanism. Eq. (6.58) is solved numerically. To obtain the solution, Davies (1994) replaces the strain e by ih, where / represents integer numbers and h is an iteration step. Assuming that low strain behavior is negligible when compared with the overall trend yields a finite difference solution for the instantaneous dislocation density as:

Figure 6.35 Distribution of the recrystallized volume fraction after first stage of compression, strain 0.25

p[(/ + \)h] = p[ih]+ AM^hl^ - 4 P [ ( ' - k)h]

(6.59)

r, mm

with k representing the development strain (kh = c). The critical factors in Eq. (6.59) are A2, A3 and c. The parameters A2 and A3 determine the steady state solution. The equilibrium population of dislocations, representing a balance between generation and annihilation of dislocations, is equal to A^^. Changes of A2 and c affect the stability of this equilibrium. Thus, for ^42^ < e'^ stable overdamping is observed, while for e'* < A2C < 0.57C, a stable overdamped solution, oscillatory in nature, is found. For A2C > O.Sn, unstable, periodic solutions are observed. Thus, within the limit 0 < A2C < 0.5n, a range of dynamic microstructural behavior may be described by the single Eq. (6.59). In consequence, this model has the potential to predict stress-strain curves under a wide range of test conditions. 6.4.5

Model based on the volume distribution of dislocations

To simulate complex microstructural phenomena, the dislocation density cannot be treated as an average value. Rather, the entire spectrum of the dislocation densities is to be considered. To allow this, the distribution fimction G{p,t), suggested by Sandstrom and Lagneborg MA THEMA TICAL AND PHYSICAL SIMULA HON OF THE PROPERTIES OF HOT ROLLED PRODUCTS

217 (1975a, 1975b), and defined as a volume fraction with a dislocation density between p and p•^•dp, is introduced. As a consequence, the equation, which describes the evolution of the dislocation populations and accounts for recrystallization, is:

dt

^

(6.60)

is)-g{s)-^mTpG{pj)

In Eq. (6.60), ^(A^) represents athermal storage (hardening), Ae is a strain increment, g{s) is the thermally activated softening (recovery) and r=//Z>^/2. This equation is solved, for each interval of dislocation density, together with equations describing the kinetics of recrystallization and grain growth (see Table 6.29). The fraction of migrating grain boundary, y in Eq. (6.60), is controlled by the nucleation rate at the beginning of recrystallization and by grain impingement in the final stages. This leads to the assumption that y is qualitatively controlled by the term X{l-X). Pietrzyk et al. (1995b) and Pietrzyk and Kuziak, (1995), give possible equations that describe the mobile fraction of the grain boundary y.

Table 6.29 Cycles of the simulation performed at each time step of the solution No process variables direction condition ref Sandstrom and 1 hardenAAP ^r-^Lagneborg, ^>0 ing 1975a, 1975b 2

recovery

Ap, p

3

recrystallization

4

grain refinement

D

grain growth

D

5

G, X, p

always

< P>

P^ Per

Sandstrom and Lagneborg, 1975a, 1975b Estrin and Mecking, 1984 Sandstrom and Lagneborg, 1975a, 1975b Sandstrom and

P> Prr

^

^^'^

always

T

U

Lagneborg, 1975a, 1975b Sandstrom and Lagneborg, 1975a, 1975b

equation dp _ s dt ~ hi

dt dp

'IMtp^

••-k^p

d£

^ - r - = -^rmG,iPi dt D ,r^ "^

dt

dt — dP

rncjg

dt'

D

All formulae used in the present volume distribution model are given in Tables 6.29 and 6.30. The following notation is used in these tables: b is the Burgers vector, D is the austenite grain size, d^ is the dislocation cell size, G is the volume distribution of dislocation MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

218 density, / is the dislocation mean free path, Q^ is the activation energy for grain boundary mobility, Q^ is the activation energy for recovery, Q^ is the activation energy for selfdiffusion, t is the time, X is the recrystallized volume fraction, Z is the Zener-HoUomon parameter, £• is the strain, e is the strain rate, y is the fraction of subgrain boundary that is migrating, /i is the shear modulus, p is the dislocation density, p^r is the critical dislocation density for nucleation, a^ is the grain boundary energy, and x is the energy per unit length of dislocation.

Table 6.30 Equations in the volume distribution model variable ref cell size (freepath)

Roberts and Ahlblom, 1978

equation ^^^ ^

^ ^ 2^

mobile fraction of boundary

Pietrzyk and Kuziak, 1995

_N / y>^f. yi P r =[l-exp(-^)](l-X)^^^^^

number of new grains per one old grain

Sandstrom and Lagneborg, 1975a, 1975b

^

critical dislocation density

Roberts and Ahlblom, 1978

80-^ Pcr-—r

{D^ ^sj

The differential equations describing the evolution of dislocation populations, kinetics of recrystallization and grain growth are solved in parallel with the simulation of the forming process. Due to the complexity of the equations, only numerical solutions are possible. A detailed description of the numerical solution of partial differential equations in Tables 6.29 and 6.30 is given by Pietrzyk, (1994). The approach is based on the division of the whole spectrum of dislocation densities into a number of intervals measuring Apo • Schematic illustration of the discretization of the distribution function is presented in Figure 6.36. Since all intervals Apo are equal, the volume fraction with the dislocation density between p, and Py + Apo, which is originally given as G(py,/)Apo, will be further referred to as G,. As a n

consequence, the condition ^ G , =1, where n is the number of intervals of dislocation densities, is always met during the simulation. Internal variables such as the grain size and the recrystallized volume fraction are given separately for each interval. Moreover, the average dislocation density in the interval is prescribed. The arrays !>(«), Ap(«) and G ( « ) , where n is the number of the intervals, are introduced into the program. The process of deformation is divided into time steps A/ and five cycles of simulation are performed for each step. The process represented by the cycle, the variable which is involved, the direction of the simulation MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

2^ (towards increasing or decreasing dislocation densities), the condition for the cycle to be performed and the equation describing the process are given in Table 6.29. At the beginning of the deformation of the fully recrystallized material, the dislocation density in the whole volume is within the first interval (with the lowest dislocation density). The distribution function is unity for this interval (Gi = 1) and zero for all remaining intervals. Hardening increases the level of dislocation densities in the interval considered by Ap^ and, eventually, moves the distributionftinctiontowards the right. Further, recovery decreases the level of dislocation densities by Ap^ and, eventually, moves the distribution function towards the lefl. Both of these cycles may eventually shift the volume fraction G, to the next (hardening) or previous (recovery) interval. If such a situation takes place during time step Ar in an arbitrary i-th interval, the new values of variables for hardening are calculated from: M+i - — T ; — — ^

^i+i = G,+i + G,

Ap,,, = Ap,.,G,,,+(AA+A/>,-Apo)G, Ap,=OG,^i=Q^i+G,

^^^^^

G,=0

In Eq. (6.61), Ap, represents the level of dislocation density in the /'^ interval. In order to avoid repeating the hardening cycle for the same fraction of the material in one time step, simulation of this cycle is performed towards the decreasing dislocation densities. By an analogy for the recovery cycle, the equations, concerning the (/ -1)*^ interval, when the decrease of dislocation density due to recovery in the time step A/ in the i^^ interval exceeds the level Ap,, are written:

G,_i+G, Ap,., = M-,G,_,^[Apo-(Ap,^Ap,)]G, G,_i +G, Ap,=OQ,i=G,,j+G,

^^^2^

G,=0

In order to avoid repeating the recovery cycle for the same fraction of the material in one time step, simulation of this cycle is performed towards the increasing dislocation densities. The recrystallization cycle is performed on all intervals above the critical dislocation density for nucleation. The recrystallized volume fraction AX/ in the time interval A^ is calculated from the equation in row 3 in Table 6.29, written in afinitedifference form:

MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

220 bX.^^mTpM

(6.63)

This volume fraction is subtracted from G, and appears as fresh material with the lowest dislocation density in the first interval. The following substitutions follow:

Gi + AA , Changes of grain size in each time step are simulated, as well. Grain refinement and grain growth calculations are performed for all intervals of dislocation density using equations in rows 4 and 5 in Table 6.29, accounting for the local distribution of grain size. The main difficulty in the application of the dislocation density model to the simulation of industrial forming processes is connected to the evaluation of the material parameters. It is assumed that these parameters can be determined by fitting the results of the finite-element calculations to the measurements of stress-strain curves and kinetics of recrystallization in typical plastometric tests. Since all of these constants have physical meanings and their dependence on the temperature is known, the values can easily be extrapolated to other deformation conditions. The inverse technique described in Chapter 8 is used here, with the objective function during optimization being defined as the sum of the squares of the differences between the measured and calculated values of the parameters: T — I T

\

"•

I

X

—

X

»

(6.65) V /=1 V ^mi

J

where ac and am are the calculated and measured flow stress, Xc and Xn, are the calculated and measured fractional softening, / is the number of sampling points in the flow stress measurements, and k is the number of sampling points in the fractional softening measurements. Different materials behave differently and separate tests should be performed for each new chemical composition. Typical results of the validation of the model for selected steel grades are presented below. 6.4.6

Application to carbon-manganese steels

The tests available for the inverse analysis for carbon-manganese steels included the measurements of the fractional softening by Hodgson et al. (1990) and the stress-strain curves determined by Laasraoui and Jonas (1991). A simplified model based on the average dislocation density (Davies, 1994) was validated first, using stress-strain data only. The inverse analysis, based on the flow stress measurements, was applied to evaluate the constants in this model. The objective function was limited to the load error term, as shown in Chapter 8. Comparison of the stress-strain curves, measured by Laasraoui and Jonas (1991) and calculated using the simplified average dislocation density model, is presented in Figure 6.37. Reasonably good agreement was obtained. MA THEMA TICAL AND PHYSICAL SIMULA HON OF THE PROPERTIES OF HOT ROLLED PRODUCTS

221

Pi-2

Pi-1 Pi Pi+1 Pl+2 Pi+3 Pi+4 dislocation density

Figure 6.36 Schematic illustration of the volume distribution function, showing nomenclature used in the text

1 0.0 0.1 0.2

\ 0.3

\ 1 0.4 0.5 Strain

r 0.6 0.7

0.8

Figure 6.37 Flow curves measured (filled points) and calculated by the average dislocation model (open points)

The complete model, based on the dislocation distribution function, is to be used when microstructural parameters are to be determined. Validation of this model is based on both softening data and the stress-strain curves. In all calculations, the austenite grain size prior to deformation was assumed to be 40 jam. Several material constants in the dislocation distribution model are to be determined. They include the dislocation cell size coefficients K^ and q, the mobility of recovery term MQ or ^2» the grain boundary mobility coefficient m^, the number of recrystaUized grains per old grain N, the activation energies for recrystallization Q^ and self-diffusion Qs, the coefficients describing the mobilefi-actionof the subgrain boundary qi and ^2> the shear modulus //, the critical dislocation density for nucleation of recrystallization p^, the grain boundary energy a^, the coefficient a in Eq. (6.44) and the Burgers vector b. The initial values of these constants were taken from the literature and they are given by Pietrzyk, (1994). Frost and Ashby, (1982) give the temperature dependence of the shear modulus. Some of the material's parameters are known with good accuracy, and are not considered to be unknowns. Eight constants (Kd, q, /wo, Mo, Qm, QM or Qs, q\ and a) are introduced in the optimization and become the varying parameters in the inverse technique. The objective function, defined by Eq. (6.67), is very complex within the wider range of unknowns. Because of this, a search was performed in the vicinity of the values of the constants, published in the literature. Two versions of the recovery term, one dependent on p (the equation by Estrin and Mecking, 1984, in Table 6.29) and the second, dependent on f? (the equation by Sandstrom and Lagneborg, 1975a, in Table 6.29) were considered. It was observed that the approach based MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

222 on Sandstrom and Lagneborg's equation (1975a) with QM = 400 kJ/mole and Mo = 5.7 x \d^ m2(Ns)-l (Roberts and Ahlblom, 1978) underestimates the recovery for low temperatures. Therefore, only the recovery term described by Estrin and Mecking's equation (1984) is used in further analysis. For the sake of clarity, the following nomenclature is used in the text below: SRX is static recrystallization, SRV is static recovery, DRV is dynamic recovery, and RX is dynamic recrystallization. The inverse technique, applied to determine the material constants using the stress-strain curves and softening curves, yielded the values shown in Table 6.31. The minimum in the error function was not clearly defined and the search procedure was slow. In order to explain the reason for these problems, optimization was also tried using the stress-strain curves and the softening curves, separately. The latter gave the apparent activation energy Qs equal to 24 kJ/mole for SRX and 14 kJ/mole for the stress-strain curves. Although the error was now below 5%, recrystallization was underestimated during the deformation (see Figure 2 in Pietrzyk et al., 1995b). Recovery became the dominant factor competing with hardening. Thus, this approach failed to model SRX qualitatively during deformation.

Table 6.31 variable units value

K, m/s 0.26 10-3

k. 34

Qs kJ/mole 20

/WQ

m'(Ns)-' 1.4 105

Q. kJ/mole 340

^1

q

-

-

1.75

0.1

a 1.23

The first approach, which consisted of performing the optimization on all the available experimental results, led to an error norm of 13% and to the material constants, shown in Table 6.31. When these constants were used in the calculations, the sensitivity of static recrystallization to strain agreed reasonably well with experimental data, shown in Figure 6.38a for a strain rate of 0.2 s'\ Sensitivity of SRX to the temperature, presented in Figure 6.38b, shows that at higher temperatures, recrystallization is predicted to be faster than measured. This is believed to be due to the high value of activation energy of grain boundary mobility Q^, which is reported to be 360 kJ/mole (Roberts and Ahlblom, 1978). The stress-strain curves are qualitatively in agreement with the experiments (Figure 6.39). Recovery was found to be an important component of hardening, as shown in Figure 6.40. The plots presented were obtained by disconnecting various parts of the model during calculations. Thus, the curve obtained with the recrystallization turned off shows that deformation leads to equilibrium between hardening and recovery, and that a plateau is observed for larger strains. Removing the recovery term from the model involves an increase in the rate of hardening and, in consequence, triggers dynamic recrystallization at a smaller strain. Characteristic oscillations are observed on the calculated stress-strain curve when a lack of recovery is assumed (Figure 6.40). A comparison of the measured and calculated stress-strain curves for the strain rate of 0.2 s'^ is presented in Figure 6.39a. Analysis of these resuhs shows that good agreement between measurements and calculations is obtained for higher temperatures of deformation. Discrepancies are large at 800°C, probably due to the change of activation energy for selfMA THEMA TfCAL AND PHYSICAL SIMULA HON OF THE PROPERTIES OF HOT ROLLED PRODUCTS

223^

diffusion with decreasing temperatures. Constant value of this energy was assumed in the calculations. The recrystallization during slow deformatipn is calculated reasonably well.

0.0

1.0 lg(time)

2.0

-1.5 -1.0

-0.5

1 0.0 0.5 lg(time)

1 1.0

\ 1.5

1 2.0

Figure 6.38 Measured (filled points) and calculated (open points) fractional softening for (a) various strains and (b) various temperatures

strain n \ \ i \ \ r 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Strain

0.8

1 1 0.0 0.1 0.2

\ 0.3

r 0.4 0.5 Strain

0.6

0.7

0.8

Figure 6.39 Measured (filled points) and calculated (open points) stress-strain curves for the (a) strain rate of 0.2 s (b) and for 2.0 s and for various temperatures MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

,

224

The agreement between the measurements and the calculations, however, is not as good for the faster tests, seen in Figure 6.39b where the resuhs for the strain rate of 2 s are presented. Again, the discrepancies are larger at lower temperatures. Evidently, further work must be done to improve the accuracy of the model. On the other hand, however, the model's ability to simulate complex conditions of deformation, involving varying strain rate or/and temperature, is an important advantage. The dislocation distribution model explains numerous microstructural phenomena, observed experimentally. Calculations performed under isothermal conditions show good correlation with published experimental data, demonstrated in Figure 6.41, which shows the influence of the strain and strain rate on the kinetics of recrystallization. The plots agree well with the observation that the sensitivity of the kinetics of recrystallization on strain diminishes for large strains. Increasing the strain rate causes loss of strain sensitivity at larger strains. 1.0

J0OOO-O-O-

O - O - O - <S

5^i

0.9

1 W

I °^ temperature 1000°C strain rate 0.2 s~^

V)

^

calculattons (full model)|

-A— 0.0

I

0.1

0.2

\

\

i

T ^ ^* am:n -%£ -

U

0.4

\

0.3 0.4 0.5 0.6 0.7 strain

I

0.8

Figure 6.40 Measured stress-strain curves compared with calculations for various parts of the model being active

^ c;

A

Za

0.3 H

Strain rate:

0.2 -1.0

-0.5

—1 r~ 0.0 0.5 In (time)

0.4 n7

n

calculations, no DRX \

* ^

Iu 0.5

calculations, no DRV I

/

:fi 0.6 H

measurements

-^—

r/

o

'-^ J - 0 . 2 s-1

-2.0 s-1 —1 1.0 1.5

Figure 6.41 Kinetics of recrystallization calculated for various strains and strain rates, temperature lOOO^C

This phenomenon is demonstrated even better in Figure 6.42, where a relation between the time for 50% recrystallization and the strain for various strain rates and temperatures is given. The plots coincide qualitatively with the well-known relation of /Q 5 versus the strain, which, in the logarithmic scale, is presented in the form of two straight lines (see for example Sellars, 1990). One line for lower strains is inclined and the inclination angle determines the strain sensitivity of ^0.5 The second line, for larger strains, is parallel to the horizontal axis and expresses the insensitivity of /0.5 to the strain. The transition point between these two lines is reported to move towards larger strains with increasing strain rates. Plots in Figure 6.42 give a similar description of this phenomenon, except that the relations in Figure 6.42 are continuous, as they are in a real material. These results, as well as additional numerical experiments MA THEMA TICAL AND PHYSICAL SIMULA HON OF THE PROPERTIES OF HOT ROLLED PRODUCTS

225 performed by Pietrzyk (1994), Pietrzyk et al. (1995b), Pietrzyk (1996), confirmed the predictive ability of the internal variable model based, on the dislocation density distribution function, as far as the simulation of the flow stress and various complex microstructural phenomena is considered.

a)

b) strain rate:

-2.5

-2.0

n -1.5

I I -1.0 -0.5 In(straln)

1 0.0

r 0.5

1.0

-2.5

-2.0

n -1.5

1 1 -1.0 -0.5 In(strain)

1 0.0

1 0.5

I 1.0

Figure 6.42 Time for 50% recrystallization calculated by the internal variable model for various strain rates (a) and temperatures (b)

6.5

CASE STUDIES - INDUSTRIAL APPLICATIONS

An application of the model, coupling the finite-element method of Chapter 5 with the conventional microstructure evolution equations of Section 6.1, to the simulation of metal flow, heat transfer and microstructure evolution during hot rolling of steel plates and strips is described in this section. 6.5.1

Plate rolling

Pietrzyk and Lenard (1991a) presented an application of the finite-element method to the modeling of thermal and mechanical phenomena in the plate rolling processes. In this section the finite-element program described in Chapter 5, combined with the microstructure evolution model of Hodgson (1993a), is applied to the simulation of metal flow, heat transfer and microstructural evolution during the rolling of niobium steel plates in a two-stand reverse mill. The parameters of the mill are as follows: the work roll diameter is 900 mm, the back-up roll diameter is 1900 mm, and the roll length is 3800 mm. The steel contains 0.16%C, 1.37%Mn, 0.28%Si, 0.014%P, 0.025%S, 0.08%Cr, 0.07%Ni, 0.15%Cu, 0.025%Nb, 0.026%A1 and 0.0052%N. The flow stress is described by the hyperbolic sine equation, Eq. (3.24), with the coefficients suggested for low niobium steels, in Chapter 6.3. Typical rolling MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

226 parameters, together with the results of calculations and measurements, are presented in Table 6.32, where R stands for the roughing passes and F for the finishing passes. Analysis of the results shows that discrepancies between the measurements and calculations appear in some of the passes. The roll force was measured by a standard load gauge and the temperatures were measured by pyrometers used in the control system during production. The reliability of these measurements is not known.

Table.6.32 Rolling parameters (schedule I), plate width 2100 mm pass thickness reduc- tempera- tempera- roll force torque force no. tion ture, X nire, X measured predicted predicted mm measured predicted kN/mm kN/mm Nm/mm 0 225.0 1230 Rl 189.0 807 7.5 0.16 1192 1199 6.1 R2 143.0 12.4 1200 0.24 1183 1197 7.7 R3 115.0 653 0.20 1173 9.3 1192 7.9 R4 112.0 130 0.03 1163 1.9 2.0 1187 R5 75.7 15.7 1052 0.32 1146 1178 13.5 R6 43.5 20.4 1170 0.43 1130 1166 20.8 R7 21.6 0.50 22.4 998 1105 1142 21.8 Fl 14.2 0.34 1041 354 1074 13.5 12.8 F2 12.2 0.14 73 5.9 1003 1003 7.2 F3 9.6 0.21 80 962 7.2 940 9.2 F4 7.7 907 0.20 12.4 165 877 9.4 F5 6.5 0.16 852 110 11.8 814 9.7

grain size jim

interpass time s

300 293 138 127 142 85 49 30 17 18 19 19 19

5 9 5 9 7 8 18 12 13 14 15

-

The rolling schedule presented in Table 6.32 is a typical one, taken from the log books of the control system. Analysis of the technological parameters in Table 6.32 shows, however, several inconsistencies. The reductions in passes R4 and F2 are smaller than in the remaining passes. As a consequence, the microstructure development was disturbed at some stages of the draft schedule. The calculated austenite grain size during rolling is plotted as a function of the temperature in Figure 6.43. The unusual increase of grain size is observed in pass R4. Figure 6.44 shows the predicted times for 5% recrystallization, 95% recrystallization and 5% precipitation compared with the interpass time. Since fiill recrystallization was predicted in all roughing passes, only five finishing passes are presented in Figure 6.44. Note that the pass number increases (from Fl to F5) with decreasing temperatures. Due to large differences between the times, a logarithmic time scale is used. Analysis of the plots in Figure 6.44 shows partial recrystallization after pass F2, which is stopped by the precipitation in pass F3. An analysis is performed investigating the possibility of increasing the solute drag effect on the retardation of recrystallization. This should lead to precipitation appearing in the last pass only. Since the effect of rough rolling on the microstructure during the finishing stages is negligible, only the latter were investigated. A distribution of reductions giving reasonably uniform distribution of forces was assumed first (Table 6.33). The plots of the recrystallization and precipitation times for these parameters are shown in Figure 6.45. An increase of the MA THEMA TICAL AND PHYSICAL SIMULA HON OF THE PROPERTIES OF HOT ROLLED PRODUCTS

^

227

precipitation time is observed in this case and the solute drag effect is responsible for the retardation in pass F3. The recrystallization is stopped by precipitates in pass F4. 200-|

6n

schedule I

180 J 160-] 140

120 H 100

Ji

80-1

60-1 20 0 700

800

900 1000 temperature, °C

1100

1200

4 2

oH 5% recrystallization

-jAj—

95% recrystallization I

-(3""

5% precipitation

-4-4

•§•

Interpass tlnr^e

1 800

850

\

n

900 950 1000 temperature, o c

schedule I

"B"

6

—0—

0

A £1

Figure 6.43 Changes of austenite grain size during rolling of niobium steel plate in a two-stand reverse mill

-2

r

-2H ( ^

r^

40

I

r 1050 110

Figure 6.45 Calculated recrystallization and precipitation times during rolling of niobium steel plate in the finishing stand (schedule II)

95% recrystallization 5% precipitation

+I 800

5% recrystallization

850

interpasstime

900 950 1000 temperature.oC

"~1 1050

1 1100

Figure 6.44 Calculated recrystallization and precipitation times during rolling of niobium steel plate in the finishing stand (schedule I)

Further analysis assumed an increase of the entry temperature to the finishing stand and an increase of the reductions in passes Fl and F2. The results presented in Table 6.34 and in Figure 6.46 show a further increase of precipitation time at the primary stages of rolling. However, since the plate is thinner in passes F3 and F4, the temperature drops there are larger. Lower temperatures accelerate the precipitation. The next rolling schedule is designed to prevent the temperature drop in the finishing passes. The rolling velocity is increased by 20%, leading to shorter interpass times. The resulting parameters are given in Table 6.35 and the recrystallization and precipitation times are plotted in Figure 6.47. The precipitation time in pass F4 is longer than in the previous schedules and the recrystallization is stopped by precipitates in the last pass.

MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

228 Table.6.33 Rolling parameters, schedule II pass no. thickness reduction temperature, roll force kN/mm mm Fl 10.4 1074 0.27 16.0 F2 9.9 1010 0.24 12.2 F3 9.6 944 10.1 0.21 F4 7.7 874 12.1 0.20 6.5 F5 805 11.9 0.16

roll torque interpass time s Nm/mm 12 245 13 182 14 137 15 140 101 -

Table.6.34 Rolling parameters, schedule III pass no. thickness reduction temperature. roll force mm kN/mm Fl 13.8 0.36 1088 13.4 F2 9.7 0.30 1021 11.5 F3 8.3 0.14 945 6.0 F4 7.3 0.12 871 6.7 F5 6.5 0.11 803 7.8

roll torque interpass time s Nm/mm 12 366 13 221 14 57 15 50 48 -

2-\

schedule IV

-2H

5% recrystallizati 95% recrystallization

95% recrystallization|

-4H

h-Q-

800

h-B

5% precipitation

interpass time 1 1 850 900 950 1000 temperature, o c

—I 1050 110

Figure 6.46 Calculated recrystallization and precipitation times during rolling of niobium steel plate in the finishing stand (schedule EI)

800

5% precipitation

interpass time n I I — —\ 850 900 950 1000 1050 temperature oc

1 1100

Figure 6.47 Calculated recrystallization and precipitation times during rolling of niobium steel plate in the finishing stand (schedule IV)

MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

229 The results presented in Figures 6.44-6.47 and in Tables 6.32-6.35 show the ability of the finite-element program to simulate the microstructure evolution in the industrial type plate mills. The program allows for the consideration of a number of various rolling schedules and the investigation of the influence of various parameters on microstructure evolution.

Table 6.35 Rolling parameters, schedule IV pass no. thickness reduction temperature, roll force mm kN/mm Fl 0.36 13.8 13.4 1088 F2 0.30 9.7 11.2 1034 F3 8.3 0.14 5.7 970 F4 7.3 0.12 905 5.9 F5 6.5 0.11 6.9 842

6.5.2

roll torque interpass time Nm/mm s 9 366 10 215 11 54 12 43 42

Hot strip rolling

The models have been applied to the simulation of hot strip rolling processes. A hot strip mill, consisting of five roughing stands and seven finishing stands, is considered. Preliminary results obtained fi-om the thermal-mechanical model have been presented in Chapter 5, in Figures 5.4, 5.5 and Table 5.3. The present section contains further results, obtained afl;er the incorporation of the microstructure evolution equations into the finite-element program. The emphasis is put on the possibility of the application of the fiill model to the evaluation of relations between process parameters and product properties. The production of pre-determined properties of the final product is usually complicated and requu-es the selection of numerous technological parameters. The relationships between the product's properties and process parameters are complex, and it is often difficult to predict the changes to be introduced in the rolling schedule to achieve the required goal. Application of the simulation programs, used off'-line, eases this procedure. The simplest way of using the models in computer aided technology is to perform simulations and sensitivity studies for various process parameters and compare the results. Numerous examples of these simulations for hot strip rolling are presented by Pietrzyk and Lenard, (1993). Typical resuhs, showing the predictions of austenite and ferrite grain sizes for various velocities in the last stand are demonstrated in Figure 6.48. The results were obtained during rolling of 2 mm thick carbon-manganese steel strips firom a slab, measuring 170x830x5500 mm. The models of Sellars and Whiteman (1979) and Sellars (1990) for the austenite microstructure evolution, and Sellars and Beynon (1993) for the ferrite grain size were used in the calculations, presented in Figure 6.48, The rolling velocity in the last stand, representing the velocity of the whole finishing train, is an independent variable in the plot. Increasing the velocity yields coarser austenite and ferrite grains, in contradiction with the usual opinions. Two opposite tendencies cause the apparent contradiction. Shorter time intervals between the passes leave less time for grain growth. This phenomenon is overtaken by the temperature increase caused by faster rolling, leading to larger grains after recrystallization and to faster MCROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

230

grain growth. A decrease of the grain size with increasing rolling velocity is obtained, when the entry temperature is changed corresponding to the rolling velocity, such that a constant finishing temperature is maintained. An example, shown in Figure 6.48, as well as the data presented by Pietrzyk and cooling rate 4° C/s Lenard (1993), show that the influence of 25 the process parameters on the final microstructure and mechanical properties cannot be analyzed separately for each parameter. 20 H Instead, the complex influence of all parameters should be investigated. The model •^ 15 H therefore, has been applied to determine the sensitivities of the product's microstructure a> 10 and mechanical properties to such process austenite (surface) parameters as stock temperature, rolling SH velocity, rolling schedule and interpass austenite (center) time between the roughing and finishing ferrite (center) passes (Pietrzyk et al., 1998). I \ I Development of the applications of 5 6 7 rolling velocity, m/s computer techniques to the optimization of technological processes involves new nuFigure 6.48 Austenite and ferrite grain size merical methods. Beyond the commonly vs. rolling velocity for the hot strip mill used finite difference and finite-element methods, sensitivity analyses will have to be included in the simulations involving optimization of processes (see Kleiber and Kowalczyk, 1995). Constitutive parameters or thermal properties are the usual independent variables in the sensitivity analysis. Pietrzyk et al., (1998) considered using the technological parameters in this role. Analytical calculations of derivatives with respect to these variables are not possible and the sensitivities were determined by numerical differentiation. Application of the sensitivity studies can be twofold. They can be used in the on-line control systems for the prediction of changes of the process parameters, introduced to obtain the necessary changes of output parameters. In hot strip rolling processes only the finishing temperature is a controllable output parameter, which is measured during the process. Since they cannot be measured on-line, microstructure or properties cannot be controlled directly, on-line. Therefore, only thefinishingtemperature is considered in the first part of the analysis. The second application of the sensitivity study is in off-line control, when the changes in the rolling technology are introduced on the basis of measurements in previous runs and tests of the final products. The output parameters considered in this part of the analysis include the austenite grain size, the ferrite grain size and the tensile strength. The following partial derivatives are determined in the first part of analysis: 30

ST^ oTf oTj- dP

cD

dP

MA THEMA TJCAL ANP PHYSICAL SIMULA HON OF THE PROPERTIES OF HOTROLLEP PROPUCTS

(6.66)

231 where To is the stock temperature, 7/is the finishing temperature, tp is the interpass time between roughing and finishing, D is the final austenite grain size and v? is the exit velocity. The following partial derivatives are determined in the second part of sensitivity analysis:

where cjy is the lower yield stress Gp is the tensile strength and Da is the ferrite grain size. The calculations were performed for rolling a 2 mm thick C-Mn steel strip from a slab, measuring 175x1250x5500 mm. The chemical composition of the steel is given in Table 6.36. Table 6.36 The chemical composition of the steel, weight% C Mn Si P 0.09 0.66 0.02 0.013

Cr 0.03

Cu 0.04

Al 0.039

N 0.005

The rolling process was performed in 12 passes, consisting of five roughing passes with the entry thickness in each pass given as 175 -> 134 -> 90 ^ 60 -> 48 -> 37 mm, followed by seven finishing passes, with the thickness in each pass: 37.0 -^ 21.0 -^ 10.9 -> 6.58 -> 4.15 ^ 2.91-^ 2.34-> 2.02 mm. The following microstructure evolution models were used: • • • •

Sellars and Whiteman, (1979) and Sellars, (1990) for the austenite microstructure evolution; Sellars and Beynon, (1993) for the ferrite grain size; and Hodgson and Gibbs, (1992) for the mechanical properties at room temperature.

The character of the relationships between output parameters (finishing temperature, microstructural parameters, mechanical properties in room temperature) and process technological parameters (entry temperature, rolling velocity, interpass time between roughing and finishing) can be easily demonstrated in the form of three-dimensional graphs, shown in Figures 6.49 and 6.50. Figure 6.49 shows the dependence of the finishing temperature on the entry temperature, and the rolling velocity, represented by the roll angular velocity in the last stand, in rpm, and the interpass time. As expected, the model predicts that the finishing temperature decreases with decreasing entry temperature, decreasing velocity and increasing interpass time. The relationships obtained for the austenite grain size are shown in Figure 6.50. Finer microstructure is predicted for lower entry temperatures, lower rolling velocities and longer interpass times. Figures 6.49 and 6.50 show that the relationships are monotonous and the optimum conditions of the process are determined by the constraints of the optimization, such as limiting angles of bite, rolling loads, etc.

MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

232

intfirpass time

entry temperature 1240^0

Figure 6.49 Three-dimensional graphs showing temperature dependence on the process technological parameters

interpass time 0

entry temperature 1240^0

Figure 6.50 Three-dimensional graphs showing austenite grain size dependence on the process technological parameters

Three-dimensional plots showing the dependence of the ferrite grain size and the lower yield stress on process parameters are shown in Figures 6.51 and 6.52. These relationships are monotonous as well, and the optimum conditions are determined by other constraints imposed on the optimization. Note that the model of the yield stress neglects the fact that greater austenite grains lead to an increase of pearlite content in the steel and to the formation of acicular ferrite, which may further increase the yield stress. This phenomenon is not accounted for in the present model. Figures 6.53 and 6.54 show the sensitivities of the finishing temperature and the austenite grain size with respect to the rolling velocity and entry temperatures, respectively. These relationships can be approximated and used for the calculations of partial derivatives of selected output parameters with respect to technological parameters. Beyond the entry temperature, the interpass time and the rolling velocity, the rolling schedule is an additional parameter, which can be used to control the microstructure and mechanical properties of the final product. The preceding results were obtained for the rolling schedule given at the previous page.

MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

233 interpass time 0 ^

^

interpass time 0

entry temperature 1240^0 entry temperature 1240, ^C

> ^ Figure 6.51 Three-dimensional graphs showing ferrite grain size dependence on the process technological parameters

Figure 6.52 Three-dimensional graphs showing yield stress dependence on the process technological parameters

0.35 20.5 4-0.30

I 0.05 austenite grain size -| 6.5

1 \ 1 7.0 7.5 8.0 rolling velocity, m^

r 8.5

9.0

Figure 6.53 Finishing temperature and austenite grain size sensitivity to the entry temperature as a function of the velocity

1200

finishing temperalurej 1 1 r 1230 1260 1290 1320 tennperatura wsadu, °C

Figure 6.54 Finishing temperature and aus tenite grain size sensitivity to the velocity as a function of the entry temperature

MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

234

^

The possibility of using the rolling schedule as an optimization parameter is described briefly below. Since the final properties of the product are determined in the finishing train, only this phase is investigated. The representation of the rolling schedule for seven stands by one variable is possible only if the reductions in subsequent passes are not treated individually. Instead, a distribution function is introduced. For linear distribution of reductions in subsequent passes, the functions plotted in Figure 6.55 are obtained. Assuming further that reductions in all passes are equal, corresponding to the horizontal line in Figure 6.55, one obtains: £,=1-7^-

for

/ = 1,

,7

(6.68)

In Eq. (6.68), ho is the thickness at the entry to the finishing train, hj is the final thickness at the exit from the 7* stand and f, represents the reduction in the /* pass. The assumption of linear distribution of reductions allows the use of one coefficient to describe the particular rolling schedule. This coefficient is called the reduction coefficient y. The reduction in the i^ pass is calculated as f, = ^i - iy. In practical applications of this approach, each rolling schedule is represented by the reduction in the first pass ei, while the reduction coefficient ;^is calculated by solving the equation: 7

^Ul^-^^Hi-l)y]-h,=^0

(6.69)

Eq. (6.69) is solved numerically. When the reduction in the first pass eu is assumed to equal the average reduction sj, calculatedfi-omEq. (6.68), the relationship, Eq. (6.69), yields a reduction coefficient y= 0, showing that the reductions in all passes are equal. When si < Si, the relationship (6.69) yields negative values for the coefficient y, indicating that the reductions increase from pass to pass. Further, when £i > et, Eq. (6.69) yields positive values of the coefficient y, leading to reductions that decrease from pass to pass. The former situation is not possible in industrial practice, because of the limitations of the available motor power in the last finishing stands, where rolling velocities are high and the temperatures are low. Therefore, only those situations with positive values of the reduction coefficient y are considered here. Distributions of reductions in thefinishingpasses for various values of the coefficient /are shown in Figure 6.55a and the corresponding thickness of the strip in subsequent stands are presented in Figure 6.55b. The thick line in Figure 6.55b represents the rolling schedule, currently used in the strip mill. Notice that for all cases shown in Figure 6.55, the total reduction in the finishing train is the same, equal to 94.5%, corresponding to the entry thickness of 37 mm and the final thickness of 2.02 mm. Similar analysis can be performed assuming second order distribution of reductions.

MA THEMA TICAL AND PHYSICAL SMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

235

reduction coefficient 7

—+-

—+— 00.020'

reduction coefficient

A £\ A

0.076

--^-&--•---A--

0.096

[- -A- - 0.096

I m.

1

'

0.038

V

1

0

-B-•-

J%

0.020

0.056

1

1

1

3 4 5 pass number

6

7

1 2

1

1

3 4 5 pass number

0.038 0.056 0.076

r 6

Figure 6.55 Distribution of reductions and strip thickness in finishing passes for various values of the coefficient y

Figure 6.56 shows the finishing temperature and the final austenite grain size as a 856 function of the reduction coefficient y. The austenite grain size decreases with decreasB ing 7, while the minimum finishing tem2 852 perature is obtained for y = 0.076. It can be E concluded that the reduction coefficient y 848-J c can be efficiently used as a single optimizaz tion parameter. Sensitivities of the output vt c 844 H parameters with respect to this coefficient can be calculated and used in on-line control 840 of the process. Application of the coefficient 1 1 1 ^-TT 0.02 0.04 0.06 0.08 y is limited in the case of rolling mills, reduction coefficient 7 which have finishing trains equipped in various types of stands, in particular when Figure 6.56 Influence of distribution of rethey differ significantly in the nominal ductions represented by the coefficient ;^on power of motors. Such situation, however, the finishing temperature and on the may not allow monotonous distribution of austenite grain size, for the remaining reductions, which is a basic assumption of parameters constant the current approach. Some of the stands with more powerful motors can accommodate reductions much larger than the remaining stands. Imposing also a load coefficient for each stand as a constraint on MICROSTRUCTURE EVOLUTION AND MECHANICAL PROPERTIES OF THE FINAL PRODUCT

236 the solution allows accounting for differences in nominal power of the motors. The load coefficient represents the rolling power to the motor nominal power ratio. As shown in the chapter, it is possible to predict the evolution of the microstructure of the hot rolled steel, quite accurately, using the models now available. The possibility of determining the properties of the rolled product, after it has cooled to room temperatures, is also possible. Further, using the models, it is possible to optimize the draft schedule with the aim of minimizing the grain sizes and maximizing the strength. Direct substantiation of the predictions of the models in industrial situations is difficult. Grain size measurements at individual stands are simply unavailable. In most hot strip mills, no temperature readings at each stands are taken. At most, the pyrometers are located at the entry to thefinishingtrain and at the coilers.

MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

237

Chapter 71 Shape Rolling I A large group of rolling processes, involving three-dimensional metal forming operations is referred to as shape rolling. These processes can be divided into two groups. One is connected with the rolling of products with cross sections that are close to round or square. These processes are usually performed in series of passes, involving grooves such as round, oval, square, diamond, and octagonal. The second group includes rolling of various structural shapes: I-beams, channels, angles, rails, etc. These processes require carefiil design of each of the passes. Numerical modeling of metal flow and heat transfer in these processes is to be based on three-dimensional analysis. Numerous examples of the application of these models to the simulation of shape rolling processes have been published, see for example Huisman and Huetink, 1985; Pillinger et al., 1988; Mori and Osakada, 1989; and Bertrand-Corsini et al, 1989; it is understood that the list is far from complete. However, these solutions need very long computing times even on modern work stations. Simulation of the complete process, involving a large number of passes, would require such long computing times that efficient analysis of several variables and variations of the roll pass design becomes unfeasible. A new approach, that combines reasonable computing times with acceptable accuracy, is necessary.

7.1

GENERALIZED PLANE STRAIN APPROACH

Approximate methods to analyze the shape rolling process have been developed and applied in several instances (Kastner et al., 1981; Pietrzyk and Glowacki, 1981; Okon et al., 1987). These have been useful in predicting the rolling loads, the overall geometrical changes of deformed pieces and the qualitative modes of metal flow. They are also capable of determining, approximately, the optimum conditions for the shape rolling process. However, most of them have been based on the transformation of the required, arbitrary complex cross section into an equivalent rectangular shape, often using the constant cross-sectional area method. In consequence, one-dimensional models developed for the flat rolling processes have been directly applied to analyze shape rolling. These solutions gave realistic but very approximate results for load calculations. They were not capable of predicting of the temperature fields. Much better and more accurate solutions have been obtained when simplified finiteelement methods have been used. One of the most efficient approaches to the simulation of shape rolling processes is called the generalized plane strain method. This method can be applied to forming processes in which the metals experience a reasonably uniform deformation in one direction. The rolling and drawing of shapes, as well as stretch forging or rotary swaging, exhibit this feature. Due to its efficiency, combined with reasonable accuracy, the generalized plane strain approach has become a very popular method of simulating three-dimensional rolling processes.

238

_^

Among the first applications of this method, the work by Glowacki (1990) should be mentioned and is described below. Similar models have been published for rolling in three-roll grooves, by Komori, (1997) for rolling of tubes, by Yamada et al., (1990), and for rolling in oval and diamond grooves was examined by Xin et al, (1990). 7.1.1

Basic principles of the method

The main assumption of the method is that the two-dimensional finite-element solution is to be performed in the plane, perpendicular to the rolling direction. The process is divided into several subsequent steps, connected with subsequent locations of this plane in the deformation zone. It is assumed that the strains in the rolling direction are distributed uniformly across the sample. This assumption results in constant components of the strain and strain rate tensors in that direction, and in zero (or constant) components of the shear strain and shear strain rate, connected with that direction. In consequence, the plane of the cross section of the sample remains plane during the whole process. This cross section is analyzed using a nonsteady state, two-dimensional model, while keeping in mind that the whole process is threedimensional. The longitudinal strain in each time step is uniform at the cross section. This strain is often obtained from experiments, but such an approach does not allow the prediction of filling of the grooves. Therefore, in more advanced generalized plane strain solutions, the longitudinal strain is introduced as a variable in the optimization procedure. Some phenomena, like forward slip, coupled with the rolling direction, are calculated using existing models and are introduced in the finite-element model. The solution is performed using a mesh of elements in the plane perpendicular to the sample axis. In subsequent steps of the calculations the plane moves between the tools and is deformed. Schematic illustration of the method is presented in Figure 7.1. The tp part of the figure shows the mesh in a typical two-dimensional solution for the flat rolling process, while the lower part of the figure presents the plane with the mesh, moving through the roll gap, as is done in the generalized plane strain approach.

Figure 7.1 Schematic illustration of the finite-element mesh in the conventional twodimensional solution (top) and in the generalized plane strain solution (bottom) for rolling MA THEMA TICAL AND PHYSICAL SIMULA HON OF THE PROPERTIES OF HOT ROLLED PRODUCTS

232 Simulation of the metal flow in the two-dimensional domain is performed using a typical rigid-plastic fmite-element approach, as described in Chapter 5. The deforming body is assumed to obey the Levy-Mises flow rule, given by Eq. (5.2). The effective stress in Eq. (5.2) is taken to equal the yield stress, calculated from the strain hardening curve. Solutions for various material models are then possible, including those accounting for the influence of the strain rate, strain and temperature on the yield stress. The friction stress at the contact surface is usually modeled employing the friction factor model:

^ -

^

(7.1)

where m is the friction factor and a^ is the yield stress. Other friction models can also be used. The basic part of the solution is the optimization of the power functional given by Eq. (5.1). The major problems and difficulties in the application of the generalized plane strain method to the simulation of three-dimensional rolling processes are: • • • • •

prediction of the location of the neutral zone; accounting for the longitudinal component of the friction forces; prediction of the longitudinal strain; calculation of the rolling force; and imposing the incompressibility condition on the solution.

Constant volume constraint in the typical two-dimensional solutions is imposed using a Lagrange multiplier, A. This multiplier can also be used to predict the average stress (Pietrzyk and Lenard, 1991). This method does not work correctly for the generalized approach; therefore, in the further analysis in this chapter, X. in Eq. (5,1) is taken as a penalty coefficient. In order to prevent volume losses in the simulation, the constant penalty coefficient X is replaced by its varying value (Malinowski and Lenard, 1993): ^ =

EAt

(7.2) ^ ^

6(1-2v)

where E is the Young's modulus, v is the Poison's ratio and A/ is the time interval. Poison's ratio is close to 0.5, and in a practical solution it is assumed to be 0.499 for steel and 0.4999 for copper, a softer material. The thermal component of the model is based on thefinite-elementsolution of general heat conduction, Eq. (5.68), in the plane, perpendicular to the rolling direction. Pietrzyk and Lenard (1988) give details of this approach. The model also neglects the longitudinal heat conduction. Since during the very short times in the roll gap, the temperature gradients in this direction are small, this assumption does not introduce a significant error to the solution. One finite-element Lagrangian type mesh is used in both mechanical and thermal solutions. The model has the ability to simulate realistically the rolling processes, in particular rolling using 4-roll grooves where the differences in the elongation of various parts of the shape are negligible. However, for processes with less uniform elongation, like rolling in 2-roll grooves

SHAPE ROLLING

240 or stretch forging, the results of the calculations are still reasonably close to the practical observations. Glowacki et al. (1992) incorporated the equations describing the evolution of the microstructure of Chapter 6 into the generalized plane strain model. These equations are solved in each element at the cross section of the product, using the current values of the temperatures, strain rates and strains, as calculated by the model. In consequence, the distributions of all microstructural parameters at the cross section of rolled products are determined.

7.2

CASE STUDIES

Glowacki's model (1990, 1993), has been successfully applied to the simulation of the rolling of rails (Glowacki et al., 1995), of the rolling of eutectoid steel rods (Kuziak et al., 1996), of the rolling of vanadium steel rods (Glowacki et al., 1996) and of the rolling of steel rods covered with copper (Glowacki, 1997). A detailed description of the application to study rolling of vanadium steel rods, is presented in the next section. Because of the high rolling velocities which result in high strain rates, this process is of interest to researchers. The high velocities make the simulation particularly difficult. 7.2.1

Rolling of rods

The rod rolling process has stimulated many experiments and theoretical considerations related to the mechanisms of the restoration processes in deformed austenite. Many researchers advocate the possibility of the initiation of dynamic recrystallization during the final passes, taking place at fairly low temperatures (Hodgson, 1993a; Yue et al., 1995). The hypothesis is based on the accumulation of strains because of the short time intervals between consecutive passes. Despite the tremendous amount of experimental efforts directed at detailed understanding of dynamic and post-dynamic recrystallization mechanisms (see Sections 6.1.2 and 6.1.3), understanding of the role of these processes in microstructure development is still far from satisfactory. This is because knowledge of the material's response in the rod rolling process is mostly based on the results of laboratory simulations, conducted using hotdeformation simulators, in which the range of deformation parameters is very limited. This especially pertains to the very high strain rate during the finishing passes, up to 1000 s"\ a level which is not attained by laboratory testing equipment. Moreover, severe changes of deformation parameters at the cross-section of a rod in the roll bite are expected to occur. To accommodate these facts, a theoretical analysis of the rod rolling process is based on an extrapolation of material behavior during laboratory experiments to the range of drastically changing processing conditions in industry. Mathematical modeling of rod rolling processes is not well developed, since it requires a great deal of specialist knowledge and is an interdisciplinary activity. Thermomechanical models of rod rolling are complicated. They require three-dimensional solutions, which, because of long computing times, do not allow for the efficient simulation of rolling in large number of roughing and finishing passes. Application of the model is presented below. Special attention is paid to the application of the relationships relating the mechanical properties of the product at room temperature to the microstructural parameters and thermal conditions.

MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

241 7,2. LI

Validation

Experiments: The steel contained 0.18%C, 1.3%Mn, 0.33%Si, 0.024%P, 0.021%S, 0.076%V, 0.002%A1. and 0.009%N. Among the microalloying additions, vanadium plays a special role. Although this element is thought to have a moderate influence on the austenite microstructure evolution during thermomechanical processing, a strong effect on ferrite morphology and strength properties was reported by Zajac et al., (1995). Despite this, a quantitative description of the vanadium effect on the microstructural evolution and mechanical properties is still incomplete. Vanadium in the steel is claimed to increase the steel's yield strength to a value within the range of 520-550 MPa. The material was obtained as a 70-ton industrial heat, cast on the continuous caster into billets, having a cross section of 140x140 mm. The ascast billets were rolled into rods of 8 mm in diameter. The rolling process consists of several technological stages, including: preheating, roughing passes, intermediate and finish rolling, and cooling on the cooling bed. Finish rolling in six passes and cooling on the cooling bed are the most important stages for shaping the final microstructure of the rods. Therefore, the research was focussed on this stage. In the course of rolling, specimens were cut and quenched to allow for the austenite grain size measurements at the entry to the last stand and approximately 15 s after the last pass. The temperature of the rods was measured using an infrared pyrometer. Specimens for the measurement of mechanical properties and quantitative metallography were taken after cooling on the cooling bed. The mean linear intercept method was used to determine the austenite and ferrite grain sizes, and for the pearlite interlamellar spacing measurements. Models: The metal flow and the temperature distribution during muhipass rolling were calculated using the generalized plane-strain finite-element analysis, coupled with the conventional approach to microstructure evolution modeling. The equations, developed by Kuziak et al., (1997), pertaining to dynamic, metadynamic and static recrystallization, static recovery, and grain growth after recrystallization, presented in Tables 6.4, 6.5, 6.7, 6.8, 6.9, 6.11, and 6.13, were adopted in the study. As a result, the deformation history of a particular element can be interrelated with the local microstructural changes taking place during and after rolling. Thus, the whole picture of the rolling process is created in the calculations through the superposition of events taking place within individual elements. As a consequence, the influence of the inhomogeneity of the deformation on the microstructure evolution can receive better substantiation than by treating deformation and microstructure evolution independently. During numerical calculations, the condition for dynamic recrystallization initiation is checked. When it is fulfilled, the dynamically recrystallized volume Abaction within the material is calculated. This fraction is assumed to recrystallize metadynamically after deformation. Further, static recrystallization and static recovery are assumed to operate in the regions of the deformed material where dynamic recrystallization is not initiated, or is not completed during the deformation. The investigation shows that the grain growth of dynamically and statically recrystallized grains can be calculated using the same equation (Kuziak et al, 1997; see Table 6.13). The temperature model is used for the simulation of the cooling of the rods on the cooling bed. The heat generated during phase transformation is also accounted for in this model. The most important microstructural parameters of the ferritepearlite microstructure include the ferrite grain size, ferrite volume fraction, and pearlite interlamellar spacing. The set of equations relating the ferrite grmn size, ferrite volume fraction, and pearlite interlamellar spacing, to the chemical composition of a steel, austenite grain size SHAPE ROLLING

242

prior to the transformation, and cooling rate over the transformation temperatures range are given in Table 6.14. The equations describing the mechanical properties of the vanadium steel at room temperature are given in Tables 6.15 and 6.16.

05 10 15 20 25 30

0 0.5 1.0 1.5 2.0 2.5 3.0

X, mm

X, mm

Figure 7.2 Results of calculations of strain (a), average strain rate (b) and temperature (c) distributions at the cross-section of the work piece at the end of the last pass

0.5

1.0

1.5

20

25

3.0

X, mm

Results - metalflowand temperature: Thefinishrolling of the rods consists of six passes. The temperature of the rods entering the first finishing pass is stable, in the range of 900925°C. A very short interval between consecutive passes, not exceeding 0.2 s, is the most important feature of the finish rolling. The results of thefinite-elementcalculations of strain, strain rate and temperature distributions at the cross-section of the work piece at the end of the last pass are shown in Figure 7.2. Due to the two axes of symmetry, only one quarter of the AdA THEMA TJCAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

243.

cross-section is presented in these figures. A strong inhomogeneity of the plastic flow during the rolling is clearly seen. However, the most striking feature of the process is the temperature increase, of approximately 100°C at the exit, which is connected to the plastic work during deformations. Results - microstructure: The influence of the processing conditions on the austenite microstructure evolution is demonstrated in Figure 7.3a. The microstructure evolution model predicts that dynamic recrystallization will be initiated in the material under deformation conditions existing in the industrial rod rolling process. Consequently, metadynamic recrystallization operates during the intervals between passes. However, short interpass times preclude full restoration of the microstructure. This leads to strain accumulation and further triggers the dynamic recrystallization process. Figure 7.3b shows the distribution of the recrystallized volume fraction at the cross-section of the rod, at the exitfi-omthe last pass. Recrystallization kinetics is observed to be nonuniform. Full recrystallization is observed at the upper part of the cross section, while only 20% recrystallized volume is predicted in the center of the rod. Significant grain growth occurred after the last pass, diminishing the beneficial effect of the dynamic recrystallization on the austenite grain refinement. This is associated with a high finish rolling temperature.

recrystallized volume fraction

^•Ro 0.5 1.0 1.5 2.0 2.5 3.0 X, mm Figure 7.3 Results of calculations of (a) distributions of austenite grain size and (b) recrystallized volume fraction, at the exit from the last pass Results - room temperature properties: The cooling process takes place on the cooling bed. The most important feature of this operation pertains to the heat transfer conditions. The rods are cut and collected at the exit of the roll stands in bundles of 10 - 15. The bundles are transferred to the cooling bed. The cooling history of a particular rod depends on its location in the bundle. Approximate cooling curves were measured in the laboratory experiment, in which the rods' arrangement and the heat transfer conditions were close to those prevailing in the rolling mill. The results (Kuziak et al., 1996) for the eutectoid steel rods are presented in SHAPE ROLLING

244 Figure 7.4a. Estimates of the rods' cooling rates were used for the evaluation of the heat transfer coefficient in the thermal model. Measurements and calculations of temperatures during cooling were used for the prediction of microstructural parameters and mechanical properties at room temperature. The results of the strength properties are listed in Table 7.1.

b)

a) 1000

1000

950 900 H

900

850-1 ^^800 3

lie centre

-;^—

measurement, surface]

-^—

measurement, center

-^—

prediction, surface

- 0 —

prediction, centre

800

750 H

to

w 700H

I 650

700 external area

600-|

600

550-] 500 450

1 100

1 200

1 300 tfme,s

1 400

1 500

500 100

200 300 time, s

400

I 500

Figure 7.4 (a) Temperatures measured during cooling of rods in bundles and (b) comparison of measured and calculated temperatures during cooling of a single rod Table 7.1 Measured and predicted strength properties of the rods located in the center and external area of the bundle consisting of 12 rods ultimate tensile strength, MPa yield strength, MPa location measured predicted measured predicted center 675 662 (502+160) 540 530(377+153) external area 693 688 (522+166) 556 550 (388+162)

7.2.2

I-beams and rails

The model has been successfully applied to the simulation of thermal, mechanical and microstructural phenomena during rolling of various other shapes (Figures 7.5 and 7.6). Figure 7.5 shows the calculated distribution of the temperature at the exit from the last pass during rolling of an I-beam. Two plots are presented. One is obtained with an assumption of symmetrical cooling of the I-beam from the top and from the bottom (Figure 7.5a). The second plot represents the results obtained for forced cooling applied to the top surface of the shape. Figure 7.6 shows the results of simulation of rail cooling after the rolling process. Temperature distributions at the cross section of the rail, at 400, 800 and 1400 seconds after the exit from the last pass, are presented in this figure. MATHEMAHCAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

245 a) mmmmmm 860 880 900 920 940 960 980

0

50

0

100 150 200 250 300 350 40

50

100 150 200 250 300 350 40

X, m m

X, m m

Figure 7.5 Distributions of temperature at the cross section of I-beam at the exit from last pass; (a) uniform cooling from the bottom and the top, (b) accelerated cooling from the top

a)

b) 60

60

600 630 660 690 720 750 780 i

40

40

20

20

E

£

0

-20 -40

49C 520 550 580 610 640

fr^

w

H

iiiiDiiiiir)

^

^--^-. ,-, /

^^^m

-60 -80 -60 -40 -20

0

20

40

60

80

-80 -60 -40 -20

6

20

40

60

80

y, mm

X, mm

C)

320 335 350 365 380 395 Figure 7.6 Distributions of temperature at the cross section of rail during cooling after rolling; 400 s (a), 800 s (b) and 1400 s (c) after exit from the last pass (courtesy of Dr. M. Glowacki)

-80 -60 -40 -20

0

20

40

60

80

y, mm

SHAPE ROLLING

246

.

The validation of the method was performed by comparing the measured and predicted microstructural parameters. Beyond this, simulation of the temperature field during the cooling of shapes agreed with the experimental data, as well. Analysis of all the results presented by Glowacki et al., (1995), Kuziak et al., (1996), Glowacki et al., (1996) and Glowacki, (1997), as well as the results shown in Figures 7.2-7.6 confirms the predictive ability of the models, based on the generalized plane strain approach. The savings of computing time and required computer memory are the main advantages of the generalized plane strain model. The tests performed by Glowacki (not published) have shown that the computing time and required memory in this model are about 100 - 150 times smaller than in the full three-dimensional solutions. These advantages make the generalized plane strain model a useful and efficient tool in the simulation of shape rolling processes and in computer aided roll pass design. 7. 2.3

The temperature, roll force and torque during hot bar rolling - simple models

Low carbon steel bars, 16 mm square, were rolled in four passes, consisting of a squarediamond-square sequence, to a 10 mm square cross-section. The roll separating forces and the roll torques were determined as a function of the area reduction at entry temperatures of 900, 950, 1000, 1050 and 1100°C. The predictions of the roll forces and torques by several models were compared to the experimental data, with mixed success. Simplified one dimensional models, derived for the calculation of the roll force and torque, and mostly based on empirical modifications of the flat rolling theories, have also been published (Arnold and Whitton, 1975; Orowan and Pascoe, 1948; Shinokura and Takai, 1986; and Khaikin et al, 1971). The present study is aimed at comparing the predictions of roll forces and torques from the various simplified rolling formulae with experimentally measured values in bar rolling. An attempt is made to control carefully the experimental conditions. Neither front nor back tension is applied. The effects of temperature change and reduction on the predictive abilities of some of the traditional models, developed for bar rolling, are examined. Material and sample preparation: AISI 1018 carbon steel bars, for which the chemical composition is given in Table 7.2, were used in the experiments. The material was obtained in the form of hot rolled squares with 16 mm long sides. All bars were annealed at 900°C for two hours and air cooled to room temperature in order to remove prior texture variations and anisotropy caused by previous operations. Table 7.2 The chemical composition of the material (wt%) C Mn Si S P Fe 0.19 0.75 0.05 0.007 0.009 Rest The samples to be rolled were cut from the annealed bars to 80 mm length. The samples were fitted with three type K (chromel - alumel) thermocouples, with Inconel sheaths of 1.6 mm outside diameter, embedded in 25 mm deep holes drilled in the tail ends. Equipment: All experiments were carried out on a two high, experimental Stanat mill, fitted with grooved rolls of 135 mm maximum diameter x 200 mm face width, made of H13 tool steel hardened to Rockwell C55. The surface finish of the rolls was obtained by sand blasting. The mill is driven by a 15 kW constant torque, DC motor. The two drive spindles MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

247 contain the torque transducers to measure roll torque while the roll separating forces are monitored by two force transducers, located over the bearing blocks of the top work roll. A tube type furnace is used to heat the specimens to the desired deformation temperature. Roll Pass Design: The square - diamond - square sequence is used as a break-down pass to obtain the rapid reduction of sectional area. In this sequence, the roll gap can be controlled and hence the profile geometry can be regulated, and there is little danger of metal twisting in the pass. The design of the above pass profiles followed the recommendations of Wuastowski, (1969). In designing the square or diamond passes, Holzweiler's formulae, presented by Wusatowski are used, as given below: Square-in-diamond pass: /,_!= 1.372a,-0.758^^

(7.3)

w,_^= 1.758

(7.4)

a,-2.024^0^

Diamond-in-square pass: a,_i =0.505r, +0.346w,_,

(7.5)

/,_, =H',_, =1.414a,_,

(7.6)

In the above sequence, square passes alternate with diamond ones. Table 7.3 gives the dimensions of the roll grooves. The total number of passes is four with a total and mean coefficients of elongation Xt = 2.48 and ?ijn = (X,J Table 7.3 Roll groove dimensions pass no. 1 entry section square a 15.97 X 15.97


exit section a (t>i ta

X Wa

area red. %

= 1.26, respectively.

2 square 14.58 X 14.58

3 diamond 13.77

90^ 22.58 X 22.58 square 14.58 X 14.58

90^ 20.62 X 20.62 diamond 13.77

50.99° 11.85x24.85

90^ 20.62 X 20.62 16.65

130^ 11.85x24.85 30.74

90° 14.73 X 14.73

square 10.42 X 10.42

26.26

4 square 10.42 X 10.42 90° 14.73 X 14.73 square 10.15x10.15 90° 14.36 X 14.36 5.12

Wusatowski (1969) recommends a range of elongation coefficients for square-in-diamond and diamond-in-square passes. His values are close to the numbers indicated above. Note that the process of designing the pass sequence, involving the determination of the number of passes, their distribution along the roll and their profile shape is completely empirical.

SHAPE ROLLING

248

,

Experimental procedure: The bars, fully instrumented with the thermocouples, are heated for 15 minutes in the furnace, pre-heated slightly above the testing temperature. The speed of the rolling mill is set at 10 rpm, giving a roll surface velocity of 70.69 mm/s. It was found necessary to use guides for the bars, both at entry and exit, in order to minimize sideways bending. The parameters monitored during the experiments include the temperature, roll force and torque. Time-history of the temperature distribution: A typical temperature history during three passes and subsequent quenching is shown in Figure 7.7.

n—r 15 20 25 30 35 time (s)

45 60

Figure 7.7 The time-temperature history of three passes during rod rolling

The roll speed was 10 rpm, leading to a roll surface velocity of 70.69 mm/s. Three thermocouples were embedded in the bar, at locations shown in the figure. One of the thermocouples is at the center of the bar, marked (A); another (B), is at the mid-height, and the third (C) is placed as close to the surface as possible, without causing fracture due to stress concentration. The temperature of the bar is uniform while in the closed furnace. As soon as it is removed, the surface begins to cool at a rate faster than the center. The properties of the product will depend on the temperature, in addition to the cooling rate, and since the latter varies across the cross-section, a homogeneous product will not be obtained. The cooling rates in air, at all three locations, are quite similar. At entry to the roll gap the thermocouple (C), closest to the surface, indicates a fairly large temperature drop, the magnitude of which depends on the true contact area, the pressure distribution, the temperature of the roll surface, the relative velocity, the interfacial shear stress, the development of scales and finally, on the interfacial heat transfer coefficient. The temperature drop in the first pass is nearly 100°C, accompanying a 16.65% area reduction. In the second pass a 30.74% area reduction is caused and in spite of this larger reduction imparted to the bar, the temperature loss is lower, near 50°C. In the third pass of 26.26% reduction, a fairly large drop of about 180°C is observed.

MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

249 The thermocouple (A), placed in the center of the bar, indicates a small increase in the temperature, reacting to the heat generated by the plastic work done in the pass. The thermocouple at mid-height largely follows the one at the center. Of the three heat transfer mechanisms, the predominant one is due to conduction to the cold roll surface. The heat flux depends on the nature of the contact at the surface, including the temperature difference of the surfaces, the normal pressure, the interfacial shear stress, the relative velocity and most importantly, the random nature of the development of the surface scales. The large normal pressures in pass #2 break up the scale formed prior to entry and when the bar arrives at pass # 3, the thickness of the layers of scale are less than they were at the entry to the second pass. Less scaling has been shown to increase the heat flux (Murata et al, 1984), resulting in the increased drop of the surface temperature. After exit, the hot center acts as a heat source and the surface of the bar regains most of the temperature losses. Roll force and torque: Figures 7.8 and 7.9 show the effect of the average bar temperature on the roll force and torque in the first two rolling passes, respectively. In each figure the roll force in IcN is plotted on the left ordinate, the roll torque in kNm on the right ordinate and the temperature is given on the abscissa. Figure 7.8 is for pass 1, the square-square pass, with an area reduction of 16.65, and Figure 7.9 shows pass 2, with area reductions of 30.74%, and 26.26%, respectively. As expected, the decreasing temperature causes the mill loads to increase. As well, cross-plotting allows one to conclude the expected result, that increasing reduction leads to increasing forces and torques. 70

0.50

1.6

120

60

1.4

100 H

1.2

50

E

f

|80

1.0

0.30 E

"" 4 0 H

§60-1

hO.8

0)

130. & 20 10

0.20 Roil speed =10rpm Area reduction - 16.65 % Pass #1 (Square - Square) a Roll force o Roll torque — 1

0

800

\

900

1 0.10

1 —

1000

2-

^

0.6

1 40

Roll speed =10rpm Area reduction = 30.74% Pass #2 (Square - Diamond) 20H o Roll force o Roll torque 0 750 800 850 average temperature

average temperature fC)

Figure 7.8 The roll force and the roll torque in the first pass

1

0.4 1-0.2 0.0

('C)

Figure 7.9 The roll force and the roll torque in the second pass

The mathematical models. Accurate computation of the roll force and torque when rolling with grooved rolls is considerably more complicated than in flat rolling. There are essentially three problems, present during flat rolling as well, but somewhat easier to handle. These are the material's resistance to deformation, as a fiinction of the strain, strain rate and temperature, the ability to calculate the distributions of the strains, strain rates, stress and

SHAPE ROLLING

250 temperature in the deformation zone and the conditions at the roll/metal interface, that is, the coefficients of heat transfer and friction. There are several empirical formulae, published in the technical literature, that allow the prediction of the roll force and roll torque during the rolling of bars. Most of these are based on models of flat rolling with modifications that introduce expressions for draft, roll radius, contact area and coefficient of friction for various pass shapes. Some of the models, to be used in the comparison that follows, are repeated below for the sake of completeness. Arnold and Whitton's formula (1975) for the roll separating force is derived from Sims's hot flat rolling theory (Sims, 1954). Modifications to account for the projected area of contact and empirical factors result in the relation: P = \.\S{2k)A^Q^S,

(7.7)

where Ik is the mean flow strength in the pass, calculated using Shida's relations (Shida, 1974) in the present work. Sj- is an empirical factor, equal to 1.3 for diamond-square, squarediamond, and all square passes, and Qp is a multiplier, dependent on the reduction and the ratio of the roll radius and the exit thickness. Values for Qp have been given by Larke (1957) as a function of the reduction and the geometry. The projected contact area is given by: ^.=0.85^>.,L,

(7.8)

where the projected length of contact and the final width are given, respectively, by: ^. = V ^ ( ^ 7 ^

and

w^=0.96w

p-

(7.9)

Orowan and Pascoe (1948) modified their simplified theory of flat rolling to be convenient for bar rolling. Assuming sticking friction, their formulae are: P = \.\55(2k)w„L,

(7.10)

where: ^m=

'3

^'

(7.11)

Shinokura and Takai (1986) introduced a method for calculating the effective roll radius, the projected contact area, the non-dimensional roll force and the torque arm coefficient which were expressed as simple functions of the geometry of the deformation zone. These variables are given for square to oval, round to oval, square to diamond, diamond to diamond and oval to oval passes. The authors used a simple formula for computing the roll force: P = Q,A^(2k)

(7.12)

MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

251

where the projected contact area is: A^ =-{0.9w,)L,

=Q.573w,L,

(7.13)

TT

The projected length of contact in the deformation zone is:

^-=JI^-^|('^-'J

(7.14)

the multiplier, Qs, is given by: a=-0.731 + 0 . 7 7 1 M + ^ M

(7.15)

where M is dependent on the projected contact area and the initial and final cross sections:

Khaikin et al., (1971) developed a relation for the projected area of contact for squarediamond-square sequences, given below:

^,=J^«4

(7.17)

In the present study Eq. (7.17) is combined with the equations of Shinokura and Takai to calculate the roll force. Khaikin et al. (1971) refer to over 300 experiments in which the validity of their model was checked. Box passes, diamond - diamond, diamond - square diamond, round - oval and oval - square passes were included in the tests. The probability that the ratio Pcau/Pexp is between 0.8 - 1.2 was determined to be 0.77. The range 0.7 - 1.3 is reached with a probability of 0.93. Unfortunately, neither the details of the experiments nor the actual data are given in their publication. Comparison of the experiments and the predictions: The range of validity of the formulae are quite broad and most of the experimental data can be accommodated. The temperature range is between 700 and 1200°C, eliminating the possibility of using all of the data of the third and fourth passes. The strain rate is calculated using the formula of Arnold andWhitton (1975):

s^2^^^^\-:L^

(7.18)

SHAPE ROLLING

252

The results of the comparisons are shown in Figure 7.10, for the first pass. Only the roll force data are used , since the roll torques are closely dependent on that, obtained as the force, multiplied by an assumed moment arm, given by Tselikov et al., (1980) as 0.375 - 0.5 for diamond-square combinations. As observed, none of the models is capable of accurate prediction of the roll force. Consistency in the predictive capabilities of a model is more important than accuracy, however, and in that regard the models of Shinokura and Takai (1986) and that of Khaikin et al., (1971) are superior to that of Arnold and Whitton (1975) and Orowan and Pascoe (1948). While underpredicting the forces, both models may be multiplied by a constant, approximately equal to two, to raise them to equal the measurements. The differences in the models concern deformation geometry exclusively, since the temperatures used are the measurements, and the resistance to deformation is determined in an identical manner, shown to be accurate and consistent for all. A three-dimensional model of metal flow would be necessary to define the exact contact surfaces in the roll gap and to calculate the vertical separating force.

2.20 2.00 1.80

+ O D A

Arnold and Whitton Orowan and Pascoe Shinokura and Takai Khaikin etal.

1.60 1.40 1.20

800

900

1000

1100

1200

average temperature ('C)

Figure 7.10 The predictive abilities of the empirical models

MA THEMA 77CAL AND PHYSICAL SIMULA TJON OF THE PROPERTIES OF HOT ROLLED PRODUCTS

253^

Chapter 8 Parameter Evaluation in Metal Forming In general, inverse analysis is designed to evaluate either the material constants in various models or the coefficients in the boundary conditions. The application of this technique to the determination of thermal properties of materials and the heat transfer coefficient has already been presented (Malinowski et al., 1994). However, the field in which inverse analysis in metal forming is applied most often is concerned with the identification of rheological parameters in the constitutive laws for the materials, subjected to plastic deformation. Tension, torsion and compression tests are routinely carried out when the rheological parameters of a material in metal forming processes are to be determined. The relative advantages and disadvantages of these three testing techniques are well understood. They are discussed by Pietrzyk et al., (1993b) as well as in Chapter 3. They refer to the ease of conducting a tension test and the difficulty of coping with low levels of possible straining before the triaxiality of the stresses develops. Interfacial friction and the attendant barreling of samples are the factors that affect axisymmetric compression. The need for some interpretation of the resuhs of torsion tests, demanded by the variations of stresses and strains along the radius of the sample, is its main drawback. Deformation heating and heat transfer to the tool are additional phenomena affecting the results in all tests. The major difficulty involved with the interpretation of the results of all tests is connected with the inhomogeneity of deformation, caused by various factors, depending on the type of the test. Finite-element simulation of plastometric tests allows the prediction of the real state of deformation, accounting for all factors causing the inhomogeneity. However, for numerical analysis of metal flow and heat transfer in metal forming processes these models require the description of the actual material properties, which are introduced in the program in the form of a constitutive law. The approach, which assumes that the constitutive parameters of the material are known, is often used and is called the direct method. Since the main objective of performing a mechanical test is to determine the material parameters, the application of the direct method is limited. This fact led to the development of a new parameter estimation method, which is known as the inverse method. The general idea of this method is based on a combination of the finite-element simulation of the test with the measurements of overall parameters, such as the force or torque (Gelin and Ghouati, 1994; Kusiak et al., 1995; Malinowski et al., 1995; Khoddam et al., 1996; Gavrus et al., 1996). Measurements of process parameters are compared with the predictions by the finite-element method. The error norm is defined as the vector of distances between these measured and calculated values. The minimization of error norm is used to determine the unknown parameters in the constitutive law. The objective of the present chapter is to describe the principles of the inverse technique. The definition of the error norm and methods of minimization of this error are discussed. Also, the applications of the parameter estimation method concerning mechanical tests and the uniqueness of the constitutive parameters thus obtained are investigated. The general presentation of inverse analysis is followed by several case studies, in which the description of its implementation to the evaluation of important parameters of metal forming processes is examined.

254 8.1

PARAMETER EVALUATION - THE INVERSE ANALYSIS

8.1.1

General principles of the inverse method

When constitutive laws are described analytically by one of the equations discussed in Chapter 3, the rheologicai analysis is reduced to the computation of the unknown material parameters in these equations. These parameters are grouped in the vector of unknown variables X = (xi, JC2, , Xr}, where r represents the number of variables in the constitutive law. Additional information, obtained from experimental observations, is also required. Starting from the rheologicai measurements one must obtain a systematic set of experimental data d"" =[d^,ci^,d^, ,^3» »^« ^^^ given values of the rheologicai parameters, x. The inverse analysis is formulated in order to determine the constitutive coefficients for which the difference between experimental measurements and computed data is minimal. Quantitatively, the analysis is reduced to the minimization of an objective function with respect to the unknown vector of parameters, x. A classical identification problem is obtained and the objective (cost) function is written in a least squares sense: •iix)=±w,[if{x)-dr]

(8.1)

/=1

where w^ are the weighting factors necessary to scale the cost function value (see section 8.2.4) and n is the number of measurements of the parameter d. Inverse analysis can be based on more than one process parameter. The cost function O is then written in the form:

^*)=ti-.k w-^;r =ii:^,bwf=r^r

(8.2)

where n is the number of measurements performed for k process parameters. The residual vector r is defined as: r = d^-d"

(8.3)

where d"* is the vector of the measured parameters and d'' is the vector of the calculated ones, both multiplied by the vector weighting factors, w,. MA THEMA TICAL AND PHYSICAL SIMULA HON OF THE PROPERTIES OF HOT ROLLED PRODUCTS

____^^^^^_^^^

255

The cost function, equations (8.1) or (8.2), is subject to the minimization procedure leading to the evaluation of the optimal values of variables, which are assumed to be the actual Theological parameters. The flow chart showing schematically the general principles of the inverse technique is shown in Figure 8.1.

INVERSE METHOD

FEM morf6l of a jpfOCeis& calculation of process p^rameter^

I

Measurement of process parameters: dr j=1,...,k i=1,...,n k - number of parameters n - number of sampling points

Comj^utation «l ^ iiBrhf&ihfes with respect fofTujclei

—_

gradient methods

non-gradient methods

minimization of the error norm

NO

RESULT optimal model variables: x

Figure 8.1 Flow chart of the inverse method

8.1.2

Definition of the cost function for practical solutions

The cost flinction depends on the variables to be evaluated by inverse analysis. The most common application of the inverse method in metal forming is the parameter estimation in the rheological models. The experimental data for this analysis are supplied by one of the plastometric tests. These data may include compression loads in a plane strain or in axisymmetric PARAMETER EVALUA T70N IN METAL FORMING

256 compression tests or torques from the torsion test. Estimating the Theological parameters is usually achieved by minimizing the length of the distance vector, which is defined as the sum of the squares of the differences between the measurements and model values. For example, when estimating the constitutive parameters using the measured loads FJ" in the axisymmetric or plane strain compression tests, the cost function has the form:

(^.^2,^3,--,^J=J-i—^-(/^^-^

(8-4)

where X\,X2, ...,x are rheological parameters, n is the total number of sampling points of the measurements, the subscript / designates the number of the measurement and Fl" and Fl" are the calculated and measured loads at a given /, respectively. For an appropriate constitutive equation, the true constitutive parameters minimize the above sum.

8.2

OPTIMIZATION TECHNIQUES

The minimization of an objective function is a major component of the parameter estimation method. In the general case, when the number of parameters is large, the objective function (cost function) may have several minima, and it is necessary to make sure that the minima obtained from the optimization identify the global one. This condition is met when the initial estimates of the optimization variables are close enough to the global minimum. If a good initial guess is provided, then the local minimum obtained from the optimization procedure and the global minimum coincide. The choice of the search method adequate to the given cost function is another important task, influencing the technique's efficiency. Providing a good initial guess for the optimization and selecting the proper search method are also discussed in this chapter. 8.2.1

Minimization of a function

The minimization problem in general form can be stated as follows: Minimize cost function 4>(x) subject to: gj (x) < 0, j = 1, /,

inequality constraints

/?^(x)=0, A: = 1,4

equality constraints

x\ < X, < xl,

side constraints

7 = 1, /,

(8.5)

where x = {x,,X2, ..,x^}^ is the vector of optimization variables, xl,x" are lower and upper limits of the variable x, and t , //, and t , are the numbers of equality, inequality and side constraints, respectively. It is common in optimization to combine the cost function and its conMA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

____^^^^^^

257

straints, leading to a new, unconstrained cost function and the problem can be treated as unconstrained optimization. Whatever strategy has been chosen for the optimization of a function of several variables, the problem can always be broken into two tasks. The first is an evaluation of the search direction, and the second is a determination of the length of the step performed along the search directions. An initial set of optimization variables, x^, is required for a majority of the optimization algorithms. The design is updated iteratively, beginning from this starting point. The most common form of this iterative procedure is given by: x'=x'-U>^s

(8.6)

where / is the iteration number and the scalar quantity p defines the distance to move in the search direction vector s. Different methods of optimization offer distinct schemes for determining s and p. Since any optimization problem involves many steps in which the vector s is maintained constant, while the minimum oi<^(p) is sought, methods for the minimization of a fimction of one variable and the methods for bracketing the minimum become important. Most algorithms, such as the polynomial approximation or golden section method, which are commonly used to find the minimum of a fimction, are based on the knowledge of the bounds of the minimum. In situations without a physical insight to identify the bounds on the solution, it is important that a logical approach be introduced for finding them. Generally, these bounds are not known a priori. If they are overestimated, the number of computations involved increases significantly. In many situations, it is not possible to evaluate the derivative of the cost function with respect to p for a proposed vector s. Instead, an algorithm must be implemented for finding the bounds on the minimum of O using only the function values. In general, three function values are needed to ensure that the correct upper and lower bounds on the minimum are identified. Once these bounds are identified, one can use several methods for finding the minimum for a function of one variable. The golden section method or the Fibonacci method are often chosen for a variety of reasons. First, while the fiinction is assumed to have only one minimum, it need not have continuous derivatives. In special cases, this method can handle functions having more than one minimum or that are discontinuous. Secondly, contrary to polynomial or other curve fitting techniques, the rate of convergence for the Fibbonaci numbers method is known. The method is found to be reliable for poorly conditioned problems. Details for the bracketing of the minimum of the cost function as well as the description of the golden section method can be found in any introductory text on optimization, see for example Schwefel (1981). 8.2.2

Search methods

One way of categorizing optimization algorithms is according to the type of information that must be provided for finding the minimum of the function. The simplest approach to minimize 0(x) is a random selection of large numbers of x vectors and an evaluation of the cost function for each of them. The x corresponding to the minimum of 0(x) obtained from this set is called the optimum, x*. These methods, which require only function evaluation in searching for the optimum, are known as the non-gradient methods. PARAMETER EVALUATION IN METAL FORMING

258 The non-gradient methods are usually reliable and are easy to program. They can often deal with non-convex and discontinuous functions, and in many cases can work with discrete values of design variables. Among non-gradient methods, the stochastic methods based on random search are considered to be less efficient, but they are effective in global optimization problems. A more difficuh, but more efficient approach to the minimization problem is to use gradient information in seeking the optimum. Since the cost function in the inverse analysis often contains an experimental component, which may contain experimental noise, the derivatives of the cost function may give misleading search directions. Moreover, because the gradients are often calculated numerically, the methods need more function evaluations and, in consequence, require longer computation times than non-gradient methods. Selected gradient and non-gradient search methods are discussed briefly. Monte Carlo method. This technique belongs to the group of stochastic methods, which are described in detail by Szmelter, (1980). The points in this method are chosen randomly, and the minimum value is remembered after each choice. After several cycles of choice and the selection of a minimum value, the overall minimum of Q>{i) is localized. This method is efficient in the preliminary approximation of the minimum, in the case when there is a lack of information about possible localization of this minimum. It is also efficient when the objective function is irregular. The Monte Carlo method provides the opportunity to select the points, which can be omitted during a systematic search performed by other methods, to be described below. In consequence, this method increases the probability of finding the global minimum rather than the local minima. Long computing times are the main drawback of the Monte Carlo method. Therefore, it is usually used for the primary search, which is followed by a final search, using one of the following methods. PowelVs method (Powell, 1964) is one of the most efficient and reliable of the zero order methods. Let {s\ sV.., s"} be a set of linearly independent vectors where n is the total number of design variables, and x° is a vector of initial estimate of x. The following steps must be followed to apply the Powell's algorithm: 1. For / = 1, 2,

, A?find0 that minimizes 0(x'"' + 0s') with respect to 0 and define:

x' = x'-'+y5V 2. For 7 = 1, 2 , . . . . , w . 1, replace s' by s'^^ 3. Replaces"by x " - x ^ 4. Find 0 " that minimizes o(x" + y^" [x" - x° ]) with respect to 0 and replace x^ by:

x«+y5"[x"-x«]. 5. Go to step 1 unless the cost function is smaller than the desired value. In order to start the algorithm, the unit vectors in an ^-dimensional system of coordinates can be used in place of {s\ s^ ..., s"}.

MA THEMA JJCAL AND PHYSICAL SIMULA HON OF THE PROPERTIES OF HOT ROLLED PRODUCTS

259 The simplex method. This is one of the most efficient non-gradient search methods. Simplex is a family of closely related methods in optimization. Among several versions of the simplex method, the one proposed by Nedler and Mead, (1965), is discussed below. The name "simplex" is derived from the simplest solid in w - dimensional space. For example, a three dimensional simplex is a tetrahedron. In general, an w-dimensional simplex has w + 1 vertices. In order to start the simplex search, the objective function is evaluated at each of the n + 1 vertices. This is called the initial simplex. It is assumed then that the optimum lies in a direction pointing av^ay from the vertex with the highest cost function and towards the other vertices. If a new point can be found with a lower function value, this point replaces the original vertex with the highest value. The shape of the simplex changes and gradually its size becomes smaller after each iteration. As the search progresses, the simplex adapts itself to the local minimum of the cost function. The search is terminated once the simplex has contracted to a size that is less than the required accuracy. After evaluating the cost function at the w + 1 vertices of the simplex, x'', x' and x^, corresponding to the highest, second highest and lowest values of the cost function can be identified. A new point, x^, is determined as the centroid of the vertices, excluding x^ from the following relationship:

x'=^Z«'

(8-7)

The highest points TL^ and x^ may be used to produce a new point x'^: x"=(l + a y - - a x ' '

(8.8)

where a is a positive constant. Observe that x'^ lies on an extension of the line from x'' to x"" and if a value of a =1.0 is chosen, the distance between x* and T^ equals that between x"" and x'^. If O(x') <
(8.9)

where ;^ is a positive constant, greater than unity and is usually chosen to be 2.0. The point with the maximum value of O is then replaced by whichever of x^ and x'^ has the lowest cost function O. The process is then started again with this new simplex. If O(X^)>a)(x^) but o(x'^)
260

x*=^'-+(l-^y

(8.10)

where ^ is a constant between 0 and I, and usually is chosen as 0.5. The highest point x is replaced by whichever of x^ and x'^ has the lowest cost function Q> and the process is repeated. Finally, if o(x'^)><|)(x^), then no improvement has been made by moving in the current direction. This is probably because the optimum is within the simplex. A contraction of the whole simplex towards the lowest point is performed, by applying the following operation: x' =7jX' -^{l-Tjy

(8.11)

where ;; is a constant, between 0 and 1. The operation, Eq. (8.11), is applied to each of the vertices x'. Small values of TJ may improve the rate of the convergence. A severe contraction, by choosing an exceedingly small value of TJ , however, is likely to cause the optimum to be missed. A typical value for T] is 0.5. The optimization process is then restarted with the contracted simplex. The non-gradient methods described above are used in all examples discussed later in this chapter. Since gradient methods are efficiently applied in several works dealing with the inverse analysis, they are presented briefly below. The gradient methods. Gavrus et al., (1996), use a gradient-based procedure, known as the Gauss-Newton method, combined with a line search algorithm. This requires the computations of the derivatives of the cost function (8.1), with respect to the optimization variables x:

^ dxj

= Z2.,kW-''rl^ ^

"^

(8.12)

'dxj

where dd^'iidxj represents the Sy term of the sensitivity matrix S and n is the number of measurements. Gelin and Ghouati, (1994, 1995) incorporated the parameters directly in the objective function, as follows: 0*(x)=S(x)+iC;(x)

(8.13)

where q is the number of inequality constraints. The weighted penalty functions, Q, are the inverse barrier functions, proposed by Carroll, (1961): 0),

(8.14) where Wj are the non-negative weights and ^X^) ^^^ ^^^ inequality constraints. MA THEMA JICAL AND PHYSICAL SIMULA TJON OF THE PROPERTIES OF HOT ROLLED PRODUCTS

_^^____^^_^^^^

261^

Gelin and Ghouati, (1994, 1995), solve the non-linear least squares problem using a modified Levenberg'Marquardt method, which accounts for the weighted penalty functions in the objective function O (Marquardt, 1963). Starting from an initial parameter guess x^, which satisfies the constraints, the modified Levenberg-Marquardt method determines a sequence of corrections to the parameters until convergence is reached according to specified criteria. The parameter correction, dJ", at iteration k is calculated from the following equations set (Schnur andZabaras, 1992): [(j^f J^ +A^I + H ^ ] ^ ^ = - ( j ^ f r^ +g^

(8.15)

where ^ is the Levenberg-Marquardt parameter, a non-negative scalar, J is the Jacobian matrix of S, and r is the residual vector defined by Eq. (8.3). In Eq. (8.15), g and H contain the first and second order derivatives of the weighted penalty functions with respect to the parameters X, respectively. Gelin and Ghouati, (1994) proposed an algorithm for the solution using the Levenberg-Marquardt method together with the finite-element approach for metal forming processes. This algorithm is presented in Table 8.1. Table 8.1 Algorithm for the solution using the Levenberg-Marquardt method coupled with the finiteelement code (Gelin and Ghouati, 1994) (l).Set the initial values of the parameters, x^"^ and the Levenberg-Marquardt parameter 1^^^ = 0.001. (2). Solve the direct finite element problem at x^®^ for the displacement u*^°^ and evaluate the cost function 0^°l (3).Set the initial penalty function weights coj^^ = O.OOOIC/®^ Q>^^\j = 1, q. Evaluate the penalty function, ^}^\j =l,q and the objective function 0*^®\ (4).For iteration k = 0, 1,2,.. . ,K (a) Calculate the Jacobian matrix, J^^\ using a finite difference approximation by solving six direct finite-element problems. (b) Calculate the penalty function derivatives to from tf *^ and g^^\ (c) Solve equation (8.15) for the parameter increment diP^\ and update the parameters: x<'"^> = x^^^ + ^^^l (d) Solve the finite-element problem at x^^^^\ Evaluate ^f'^^^j =\,qmd O*^^""^^ (e) Check if
262 and Gelin and Ghouati, (1995) discuss various approaches to this problem in detail. One possibility is to use the Gauss-Newton search method, which uses derivatives of the calculated process parameters d"" with respect to the variables x. A general relation, which defines the process parameters, is: d"=G(..,q,x,X)

(8.16)

where q is the vector containing all state variables and X is the vector of coordinates. Vector d"" can be a function of other parameters, such as time, depending on the finite element model. Startingfi*omthe relation, Eq. (8.16), differentiation with respect to x yields (Gavrus et al., 1996):

^

^ M ^ / ax ax ax

where r represents the number of nodal state variables. Identification studies by Tortorelli and Michaleris, (1994) show that the sensitivity analysis can be performed with finite difference computation, adjoint problem formulation, semianalytical method or direct differentiation of the fundamental equations. To define the complete sensitivity matrix, one must compute the nodal parametric sensitivity of all state variables with respect to the parameter vector x. A detailed description of this solution, based on the analytical calculation of derivatives ,is given by Gavrus et al., (1996). The Levenberg-Marquardt method requires the determination of the Jacobian matrix of sensitivities. Gelin and Ghouati, (1995) present this Jacobian for the incremental step-by-step finite-element calculations, using the displacements u as measured process parameters. The following relationship is obtained:

''^ ^i

^i

h\

dxi

for / = 1, . . . , « , ;

/ = 1 , ...,A?

(8.18)

where n is the number of measurements in one time step and rit is the number of time steps. The increment of displacement Au*^'"^^Ms obtained when the equilibrium is satisfied. Accounting for the general finite-element equation, Ku = f, the equilibrium condition is written as:

where m is a number of the time step, and K is the stiffness matrix. Kusiak and Thompson, (1989) give a detailed description of calculations of the derivatives of displacements u with respect to the unknown variables x. In their method these derivatives are calculated indirectly by analytical differentiation of Eq. (8.19), which leads to:

MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

263

aR'"

aK'^

ax

ax

_^y*(»» + l) +K'""*'^

^(^„»(m^i)y

ax

af" = 0 ax

(8.20)

Assuming, however, that this matrix^does not depend on x, solving Eq. (8.20) for the unknown derivatives of displacements u* with respect to variables x, yields:

[K^'^'Y

ax

d('''

(8.21)

ax

The stiffness matrix K""^^ is available from the direct problem. The derivative of the load vector is determined next, calculated accounting for the following relationships: K^=JB^EB^

o = EBu

(8.22)

where E is the stress-strain matrix containing the flow stress, described by the constitutive equation, B is the strain-displacement matrix and o is the stress tensor.

ax

J*

(8.23)

ax

All quantities on the right hand side of Eq. (8.23) are calculated for the m^ time step. The term da/dx must be evaluated in a manner consistent with the algorithm, calculating the stress and state variables. This part of the sensitivity analysis depends on the constitutive behavior under consideration. The details are discussed by Gelin and Ghouati, (1995). 8.2.3

Scaling of the variables

The benefits and possible disadvantages of the scaling approach depend strongly on the search method, used in the optimization procedure. It may be useful to scale the variables in order to enhance the efficiency and reliability of the numerical process. Scaling is necessary when values of the variables x or the derivatives of the cost function O with respect to these variables differ significantly. If this happens for a two-variable function, the lines of constant value of the cost function form a series of ellipses with high aspect ratios. The application of a gradient method causes a complex iteration history with the direction vector changing at each iteration. Due to the shape of the design contours, a true minimum may not be reached during a reasonable number of iterations. The non-gradient techniques lead to the minimum, but also require a large number of iterations. Scaling of the cost function (8.1) is done by introducing a weighting factor wr. 1

for i = 1,. . . . , « (8.24) PARAMETER EVALUATION IN METAL FORMING

264 for / = 1 , . . . ., w In Eq. (8.24), n is the number of measurements of process parameters. In the case of a two-variable problem the cost function, after scaling, forms a set of concentric circles. The minimum is found in a relatively low number of iterations, using gradient methods. The nongradient methods will also be very efficient in this case. 8.2.4

Convergence criterion

An important part of the overall process is determining when to stop the iterations when finding a minimum. The choice of the termination criteria has a major effect on the efficiency and reliability of the optimization process. Two termination criteria are discussed below. The first termination criterion, which should always be used in combination with other convergence criteria, is based on terminating the search when the maximum number of iterations is exceeded. This ensures that if progress is extremely slow due to numerical or algorithm difficulties, the program will not continue to iterate indefinitely. The second criterion is based on the absolute or relative change in the cost function. This criterion checks the progress of optimization. Two variations should be used in this case to ensure that the progress has slowed sufficiently to indicate convergence. The first is to compare the absolute value of 0(x) in successive iterations and convergence is indicated if: \o{x')-Q>{x'-')\<e,

(8.25)

where q is an iteration number and e^ is a specified tolerance, which may be a constant such as s^ = 0.0001, or may be a fraction of the cost function value at x°, say s^ = O.000l|o(x° J|. The other criterion, which should be used, is a check on the relative change in 0(x) between successive iterations. Here, convergence is indicated if:

'

I /A

'^g.

(8.26)

where Sj^ is a specifiedfi-actionalchange such as 0.001. The use of both an absolute and a relative change convergence criterion ensures that convergence is identified if the magnitude of
MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

__^_^ 8.2.5

26£ Constrained functions of n variables

The general problem of optimization is presented in Section 8.2.1. The common approach is to minimize the cost function as an unconstrained function, while providing some penahy to limit constraint violations, because the way in which the penahy function is imposed often leads to a numerically ill-conditioned problem. To avoid this, the problem is minimized several times and the penalty is increased as the optimization progresses. This requires the solution of several unconstrained minimization problems in obtaining the optimum constrained design. The classical approach is to create a pseudo-cost function of the form: r(x,rJ=
(8.27)

where 0(x) is the original cost function and P(x) is an imposed penalty function. The form of the penalty function depends on the chosen constraint method. A scalar, r^, is used as a multiplier for the penalty term to adjust the magnitude of the applied penalty. It is held constant during each unconstrained minimization stage. The subscript p is the unconstrained minimization number. The most important constraints for the parameter estimation method are the side constraints, defined in section 8.2.1. Each side constraint may be broken into two inequality constraints. Depending on the method chosen, the definition of the penalty term is different. One of the methods introduces the penalty function in Eq. (8.27) in the following form: m

PW = S ^ ; W

(8.28)

1=1

where.

8A^) (8.29) -1 -TLA

^*W = 2
fo"" ^ . W ^ ?

**• ^'W>^ (''> for ^.W>?

The parameter ^ is a small negative number, which marks a transition from the interior penalty equation, Eq.(8.29a), to the extended penalty equation, Eq. (8.29b). It is advisable to choose ^ in a way that provides a positive slope for the pseudo-cost function r(x,rp) at the constraint boundary. Further, it is recommended that ^be defined by: g = Cr;

(8.30)

PARAMETER EVALUATION IN METAL FORMING

266 where C is a constant and a varies between 0.33 and 0.66. To begin the optimization, a value of g in the range -0.3 < ^< -0.1 is chosen. Moreover, the initial Vp is determined, so that the two terms in Eq. (8.27) are equal [0(x) = r^ P(x)]. This defines an appropriate value for C in Eq. (8.30). At the beginning of each unconstrained optimization, g is calculated from Eq. (8.30) and remains constant throughout the unconstrained minimization. 8.3

CASE STUDIES

The problem of evaluation of rheological parameters in the cold forming processes is reasonably simple. The yield stress at room temperatures can be assumed to be insensitive to the strain rate. In consequence, constitutive equations contain two or at most, three coefficients which must be evaluated. The situation is more complex in hot forming processes. Evaluation of rheological parameters for hot deformation of metals requires accounting for a large number of tests, which are performed at various temperatures and strain rates. As well, as shown in Chapter 3, the constitutive equations for hot deformation of steels may contain a low of four to a high of seven coefficients in the vector x of Eq. (8.4). The initial estimate for this vector is difficult to establish and it may happen that the starting point is located far from the minimum. These cause the optimization procedure to involve a large number of simulations of the tests. Applying the finite-element technique from the beginning of the optimization would lead to very long computing times and would make the procedure quite inefficient. Therefore, a two-stage approach to this problem is proposed. First, optimization should be performed without the finite-element model, assuming that the stress measured in the test is equal to the flow stress of the material. The results of this solution are then used as a starting point for the final optimization, which uses the finite-element technique and which accounts for all inhomogeneities in the test. Two examples of the application of the inverse analysis to the evaluation of the rheological parameters at hot forming temperatures are described below. One involves plane strain compression tests for aluminum alloys and the second concerns axisymmetric tests for carbonmanganese and niobium microalloyed steels. 8.3.1

Evaluation of rheological parameters for hot forming of aluminum

The objective of the inverse analysis is the estimation of the rheological parameters for an aluminum alloy, using the results of plane strain compression tests, carried out at various temperatures and strain rates. A detailed description of the experimental procedure and the results of the analysis are given by Pietrzyk and Tibbals, (1995). Two sets of the experiments have been used. Both were carried out at temperatures of 350''C, 420°C and 500**C and at strain rates of 0.1 s'\ 1 s'^ and 10 s'^ The initial thickness of the aluminum alloy samples was 10 mm. The widths of the platens and of the sample were different in each of the two sets of tests, 15 mm and 40 mm in the first set of tests, and 5 mm and 17.5 mm in the second set, for the platen and the sample, respectively. MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

^

267

Typical registered true stress-true strain curves, obtained at a strain rate of 1 s'^ are presented in Figure 8.2, indicated by the solid, black symbols. The calculated parameters are shown by the empty circles. Analysis of the results shows that different load-displacement relationships were obtained in the two tests. Some softening was observed for the narrow platen and some strain hardening was registered for the wide platen. Moreover, the differences between stresses measured in the two sets of tests for low strains exceeded 30%. a)

b) 350°C

5

100

^^«o*ooOO

ooooooe sco'c )Ooooooooooooeo o o o o o

• experiment o calculaHons 0.0

0.2

0.4

width of the piaten S mm thicl^ness of the sample 10 mm strain rate 1 s' 0.6 Strain

^4»«^«P6TJ c ia
• experiment 0 calculations

wiatM oi the ploler. 15 rprr* thickness cf the -jomple 10 mm strain rote 1 s ' '

0.4 0.6 stroin

Figure 8.2 Measured and predicted stress-strain curves for plane strain compression of aluminum alloy; strain rate 1 s\ width of the platen (a) 5 mm and (b)15 mm

The explanation for these discrepancies can be found in Figure 8.3, where the distributions of strains in the deformation zone are presented. Due to the two axes of symmetry, only a quarter of the deformation zone is shown. The wide platen involves much more uniform distribution of strains than does the narrow platen, caused by the different values of the shape factor of the deformation zone A, defined as the height-to-width ratio of the deforming sample. This ratio equals 0.66 for the wide platen and two for the narrow platen. It is well known (Backofen, 1972) that the plane strain compression stress depends strongly on the value of A. For large shape coefficients, this stress can significantly exceed the yield stress of the material. On the other hand, the influence of friction on the compression stress is larger for low values of A. Both tests were expected to yield the same stress-strain relationship for the material. In order to obtain that relationship, the inverse analysis was applied. The following constitutive equation was assumed:

a,=4l + B£''>'exp|^-|;j

(8.31)

where e is the strain, s is the strain rate and Tis the absolute temperature. The cost function (8.4) with x = {A, B,p, q, Qf was used in the optimization. The vector of measured loads included several pointsfi"omeach experimental curve. Monte Carlo and simplex non-gradient methods were used in the optimization. The analysis, based on the two PARAMETER EVALUATION IN METAL FORMING

268 sets of tests yielded the following values of the variables: .4 = 1.181 MPa, J5 = 0.174, /? = 0.21, ^ = 0.12, g = 22080 J/mole. The resuhs of the finite-element simulation, using the constitutive equation (8.31) with the optimum coefficients, are shown in Figure 8.2 using the blank points. The constitutive equation thus determined shows very good agreement between the measured and calculated compression loads. Pietrzyk and Tibbals, (1995) present the full set of results, as well as an application of other constitutive models, based on the dislocation density.

Figure 8.3 Distributions of strains after plane strain compression; (a) width of the platen 15 mm and (b) 5 mm

8.3.2

Evaluation of rheological parameters for hot forming of steel

Kusiak et al., (1995) used the inverse technique for the evaluation of rheological coefficients using carbon-manganese and niobium steels. The mathematical model, which they applied in their work, is composed of two parts. The first is the thermal-mechanical finite-element program, described in Chapter 5. This program is used to simulate metal flow and heat transfer during hot compression of axisymmetrical samples. The program also calculates compression loads. The second component of the model contains the optimization procedures, which were used in the inverse analysis of the problem. The objective function was defined as the difference between the measured and calculated loads during the tests, according to Eq. (8.4). As discussed in Chapter 3, the behavior of steels during hot plastic deformation is determined by the combined influence of athermal strain hardening, thermally activated recovery and thermally activated recrystallization (Mecking and Kocks, 1981). These three phenomena lead to the characteristic shape of the stress-strain curves, with a plateau (Kuziak, 1997). Depending on the temperature and the strain rate, the curves may exhibit a state of saturation, one peak or oscillations. The type of curve depends on the history of deformation conditions and it is very difificuh to find a reasonably simple function, which describes all possible types of behavior. In what follows, the hyperbolic sine rate equation, Eq. (3.24), is employed. Two sets of experimental data are considered for testing the inverse analysis. The first, published by Laasraoui and Jonas, (1991), includes hot compression of axisymmetrical low carbon steel (0.03%C, 1.54%Mn) samples. The second was perJFormed in the University of Waterloo, Waterloo, Canada. It involved hot compression of axially symmetrical samples, made of a miMATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

269^ croalloyed steel, containing 0.043%C, 1.43%Mn, 0.312%Si and 0.075%Nb. The 8 mm diameter specimens of 12 mm height were compressed to a true strain of 1.1, at a temperature range of 860 - lOOOT, and at constant rates of strain, varying from 0.0001 to 0.1 s\ Results of both experiments show that, at some test conditions, the dynamic recrystallization leads to the characteristic shape of the stress-strain curve, with the stress reaching a peak at some strain and decreasing after that to a steady state value. Table 8.2 Coefficients in stress-strain equations (3.24), (8.32), and (8.33) obtained for various conditions figure steel eq. K AxlO'^^ A3 A4 m « Q MPa s kJ/mole 8.4 C-Mn (3.24) 355.3 171 1.87 0.393 0.086 8.5 Nb 306.8 (3.24) 156 2.62 0.138 0.108 8.6 C-Mn (8.32) 374.3 167 2.0 1.858 0.573 0.218 8.7 C-Mn (8.33) 0.171 372.7 114 0.212 3.46 1.25 0.26 8.8 275.8 Nb 233 3.13 0.729 0.089 0.081 0.159 (8-33) Application of the inverse technique to the experimental data of Laasraoui and Jonas, (1991) leads to the activation energy in the Zener-Hollomon parameter Q = 329600 J/mole and the constants, given in Table 8.2. The resuhs presented by Kusiak et al, (1995) show clearly that equation (3.24) cannot properly describe the stress-strain relationship when dynamic recrystallization is involved. In order to find the equation which accurately describes the strain hardening itself, compression to the strain below the peak value was considered. An inverse technique resulted in the values of the material constants given in thefirstrow of Table 8.2. Comparison of the measured and calculated stress-strain curves below the strain of 0.25 is shown in Figure 8.4.

a)

b)

200

• • • measurements o • A calculations

200-1 • • • measurements o D A calculations

160

160

I 120

Q. 120

80

80

40

40 strain rate 0.2 s-i 1 —r 0.2 0.1 strain

0.0

0.3

strain rate 2 s-i —1 0.1

0.0

r^ 0.2 strain

0.3

Figure 8.4 Stress-strain curves for C-Mn steel obtained from the inverse analysis for function (3.24) below the strain 0.25 compared with the measurements by Laasraoui and Jonas, (1991); strain rate (a) 0.2 s'^ and (b) 2 s'^ PARAMETER EVALUATION IN METAL FORMING

_ —

270

Similar analysis was performed using the niobium steel and the optimum parameters are given in the secx)nd row of Table 8.2. Figure 8.5 shows the resulting comparison of the measured and calculated compression loads. It can be concluded at this stage that Eq. (3.24) properly describes the strain hardening of the steel when the influence of dynamic recrystallization is negligible. The equation is adequate for the simulation of a single pass in hot rolling. Multipass rolling, however, often leads to the accumulation of strains and to dynamic recrystallization. Moreover, other processes such as hot forging may involve large local deformation, far beyond the strain at the peak stress. In these cases, simulation of metal flow based on Eq. (3.24) may lead to erroneous results. Therefore, an attempt was made to introduce a stress-strain function, which properly describes the yield stress over a wide range of strains. This aim is accomplished by introducing the strain dependence of g and n (Wang and Lenard, 1994, and Rao et al., 1993), requiring two additional parameters in the optimization procedure. A similar problem is connected with the Voce approach, described by Hodgson and Collinson, (1990). The latter equation has the form of the sum of two terms, which may be inconvenient. A simpler approach introduces an additional exponential term with one constant. The following equation was suggested:

(8.32)

crp=Ke'' sinh "^ \AZf ]exp(- A^s) where e is the strain and Z is the Zener-Hollomon parameter.

b)

160

• • measurements o D calculations

140 120-] 0. 100-j

«

80 H ° strain rate:

60 H 40

temperature SSQoC 0.0

0.1

0.2 strain

0.3

Figure 8.5 Stress-strain curves obtainedfromthe inverse analysis for Eq. (3.24) below the strain of 0.25 compared with the measurements for niobium steel (Kusiak et al., 1995); temperature (a)910^Cand(b)860^C

MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

271 The coefficients in Eq. (8.32), determined from the inverse analysis, are given in the third row of Table 8.2. One exponential term in the stress-strain equation leads to the situation presented in Figure 8.6, which is qualitatively incorrect. For larger strains, the theoretical curve tends to zero instead of a saturation steady value. Proper description of the dynamic recrystallization behavior can be obtained by introducing two exponential terms, as suggested by Beynon et al, (1993). The influence of these terms starts above the peak strain:

ap = Ks''sinh-i{AZyllii +

exp[A3R{e'ep)]-exp[AAe'ep)]

140 n

where:

R{e'ep)=0 R\S'8p)=s-ep

(8.33)

fore<ep

120 H

for €>6p Q. 100

(0 In Eq. (8.33) Sp is the strain corresponding strain rate 0.2 s'** 80 to the peak stress. Calculations based on temperature 900<»C Eq. (8.33) lead to the coefficients given in the last two rows of Table 8,2. Comparison 60 measurements] of the measured and calculated stress-strain curves for these coefficients is presented in r-O- calculations 40 Figure 8.7. It can be concluded that the I I 0.0 0.4 0.6 0.2 conventional hyperbolic sine equation, strain multiplied by an expression with two exponential terms, allows the simulation of the Figure 8.6 Stress-strain curves obtained from the behavior of the materials beyond the start inverse analysis for Eq. (8.32) compared with of dynamic recrystallization. It is pointed out, however, that closed form relationships the measurements by Laasraoui and Jonas (1991); temperature 900°C, strain rate 0.2 s"* of the type of Eq. (8.33) cannot describe stress-strain conditions in general. Recapitulating the work described in this section, it is concluded that combining the inverse technique with the optimization procedures allows the determination of the coefficients in the stress-strain equation, which give the best correlation between measured and calculated loads during compression. The method accounts for the influence of the deformation heating and friction on the test with no explicit need to determine the coefficient of friction or thefrictionfactor. The importance of these phenomena and the errors resulting if they are not accounted for are well-known. The choice of the optimization technique has also been shown to be important. Since they do not require the calculation of derivatives of the objective function, the non-gradient methods are preferable. In the present work, the Monte Carlo method was used at the beginning of the search to localize the minimum. The purpose was to avoidfindingthe local minima, which may appear,

PARAMETER EVALUATION IN METAL FORMING

_^

272

in particular when functions with larger numbers of coefficients are used. The simplex method was used at thefinalstage of the search tofindthe precise solution.

a)

b) 160

• • measurements o D calculations

160

• • • measurements o D A calculations

140 H

140 H

120

120 Q.

100 100

80 H

^/^ strain rate:

60 40 H

-

0.1 s-1

-

0.01 s-1

n nri'i e-i temperature 91 O^C U.UUl S ' J m n i 1 — 0.4 0.6 0.8 0.2 Strain A

20

[ 0.0

Figure 8.7 Stress-strain curves obtainedfi-omthe inverse analysis for function (8.33) compared with the measurements: a) C-Mn steel, strain rate 0.2 s'^; b) niobium steel, temperature 910''C.

8, J. 2.1 Concluding remarks The case studies presented above show an application of thefinite-elementtechnique to the interpretation of compression tests. Thefinite-elementanalysis predicts the distributions of strain rates, strains and temperatures in the deformation zone, and accounts for the influence of deformation heating and friction on the stress-strain relationship. The optimization technique with the cost function (8.4) leads to the local strain-hardening relationship. The accuracy of the method depends on the selection of the function describing the stressstrain curve. It is shown that the commonly used sine hyperbolic function (3.24) or the Arrhenius type function (8.31) properly describe the strain hardening only below the peak strain. These functions fail, however, when dynamic recrystallization becomes a dominant factor affecting the flow stress. Introduction of the additional exponential terms allows for the accounting of the dynamic recrystallization. The efficiency of the inverse technique can be understood in two ways. Since the method requires the running of thefinite-elementcode for each experiment available at each iteration during the optimization, it is time consuming. On the other hand, the starting point for optimization is takenfi-omthe direct approximation of the experimental curves and is usually close to the solution. The aim of the inverse technique is only to introduce changes in the coefficients, which account for the influence of friction and heat generation. These changes are small and often, only a few iterations are required tofindthe solution. MATHEMAHCAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

8.3.3

Estimation of the parameters in microstructural equations

The general idea of this approach is based on the fact that an inverse analysis may use the measurements of various process parameters together. Thus, it can be applied to the evaluation of material constants in the models of different but mutually dependent phenomena, such as flow stress and microstructure evolution. The general objective of the present section is to apply the inverse technique to the evaluation of the material constants in both the constitutive equation and the microstructure evolution model. Experiment: The idea of the test design is to create as large an inhomogeneity of deformation, as possible. The cylindrical samples measuring 12 mm in height and 8 mm in diameter were compressed in two ways. First was a typical upsetting of axisymmetrical samples. In the second set of tests, the samples were put on their sides and compressed along the diameter, as shown schematically in Figure 8.8. The latter method yielded large inhomogeneity of deformation, with strains varying from zero close to the side of the sample to a maximum at the center. The compression force was monitored during each test. The samples were quenched directly following the deformation and the grain size was measured at several locations on the cross-section. The recrystallized volume fraction was evaluated based on the measurement of recrystallized and non-recrystallized grains. The tested material was carbon-manganese steel containing 0.16%C and 1.3%Mn. The average austenite grain size after preheating was 44 inm but the microstructure was inhomogeneous. Some large grains were observed in the samples quenched after preheating. The parameters of all the tests are given in Table 8.3 where Tp is the preheat temperature, Tt is the test temperature, r is the reduction, and e is the strain rate. The model is based on coupling of the advanced finite-element approach (see Chapter 5) simulating metal flow and heat transfer during hot compression with the conventional equations describing microstructural phenomena for steels. The hyperbolic sine yield stress equation, Eq. (3.24), was used in the constitutive model. The Voce approach (Hodgson and CoUinson, 1990) was added to account for dynamic recrystallization. Microstructure evolution models used in this section include the semi-empirical closed form equations given in Tables 6.4 and 6.5. The kinetics of recrystallization is calculated from Eq. (6.2) with the coefficients suggested by Sellars, (1990). This is combined with the recrystallized grain size equations by Sellars and Whiteman, (1979), given in Table 6.5, Roberts et al., (1983), from Table 6.2, Choquet et al., (1990) and Hodgson and Gibbs, (1992), from Table 6.2. The possibility of dynamic recrystallization is considered, as well. The equations describing the critical strain and the grain size after dynamic and metadynamic recrystallization, which were used in the present work, are given in Tables 6.7 and 6.9. The coefficients in these equations are determined from experiments using inverse analysis. Parameter evaluation: The resuhs of the experiments are used to determine the coefficients in the flow stress and microstructure evolution equations, achieved by minimizing the error norm, defined as the sum of the differences between the measured and calculated compression force and the recrystallized volume fraction in all experiments:

,\ 2

Fr

/

\2

1^

+-Z

x:

PARAMETER EVALUATION IN METAL FORMING

(8.34)

274 where i^^F^'" are the calculated and measured force, respectively, n is the number of force sampling points, X^.X"^ are the calculated and measured recrystallized volume fractions, respectively, and k is the number of points at which the recrystallization volume fraction was measured. The optimization procedure yields the values of the coefficients p, q and A in the flow stress equation as well as coefficients B in the dynamic recrystallization equations. Since the sensitivities of the cost fiinction O with respect to the parameters A could not be calculated analytically, the non-gradient optimization techniques were applied. There is no direct noticeable correlation between the flow stress and the recrystallized grain size. The parameters in the equations describing the grain size were therefore determined separately using the cost function:

(8.35)

^ 2 = .

where D^^DJ" are the calculated and measured recrystallized grain sizes, respectively, and / is the number of points at which the grain size was measured. Table 8.3 Parameters of experiments 8

no. X

Figure 8.8 Schematic illustration of the sideways compression test

mm

s'^

upsetting of cylindrical samples 1 1095 0.1 850 4.2 2.0 4.7 2 1100 850 3 1068 885 0.1 5.2 2.0 4.9 4 1089 910 0.1 4.8 5 1130 959 2.0 4.8 6 1100 955 5.0 7 1087 1047 0.1 compression of samples lying on sides 4.2 0.1 860 8 1093 2.5 2.0 9 1130 870 3.2 0.1 10 1090 909 2.7 2.0 11 1080 912 3.3 0.1 953 12 1063 3.6 2.0 13 1043 960 3.3 0.1 14 1102 1040

MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

275

The finite-element method supplies information regarding the distributions of strain rates, strains, stresses and temperatures in the deformation zone. This is further used in the inverse analysis, according to the flow chart, presented in Figure 8.1. Typical results obtained from the finite-element simulation for the sideways compressed samples are shown in Figure 8.9. The mesh, used in these calculations, is presented in Figure 8.10, showing a quarter of the deformation zone. Significant inhomogeneity of deformation is observed in Figure 8.9. The inverse technique is designed to handle this inhomogeneity. The recrystallized volume fi-action and grain size measurements were performed at several locations on the cross-section of the sample. These were combined with local values of the process parameters, calculated by the finite-element model. In consequence, one sample supplied information, which in a conventional experimental technique, would necessitate the running of a number of tests.

a)

b) strain rate 0.1 s'^

strain rate 0.1"

Figure 8.9 Effective strain distribution (a) and temperature distribution (b) after sideways compression of cylinder; test 8 in Table 8.3

Typical results of measurements for the upsetting of cylindrical samples are shown in Figure 8.11. These results, together with similar data for the sideways compression, were used in the inverse analysis. Optimization of fiinction Oi yielded the following values of coefficients in Eq. (3.24): iT = 98, ^ = 2x10'^^ m = 0.25, w = 0.2. The dynamic recrystallization coefficients in Table 6.7 are: A = 6x10"^ q = Q3p = 0.17 and in Table 6.9, A = 1.8xlO^ p = 0.15, &i?Ar= 310000 J/mole. The results obtained from the inverse analysis were tested against the experimental data, not used in the optimization. The results for the upsetting of a cylinder are shown in Figure 8,12 and for the sideways compression in Figure 8.13. The second part of the analysis included the identification of the recrystallized grain size models. Detailed description of this part is given by Pietrzyk et al., (1997). Briefly, the inverse analysis yielded the values of the coefficients in the microstructure evolution equations, which give the minimum of the cost function (8.35). These coefficients are presented in Table 8.4 where they are compared with those given in original publications (numbers in parenthesis). Sellars and Whiteman's, (1979), equation is considered here in the following general form:

Dr=C,

Cjln

7 Y no.5 10-^9 Z e m

PARAMETER EVALUA TION IN METAL FORMING

(8.36)

276 where D is the grain size prior to deformation, e is the strain and Z is the Zener-HoUomon parameter. The activation energy in this is identified by Qd. The symbols in the remaining three investigated equations are as in Table 6.2. Table 8.4 shows that the differences between the coefficients in the microstructure evolution models, determined from the inverse analysis, and those given in the original publications, are small. Pietrzyk et al., (1997) also demonstrated that using the four equations with the coefficients determined from the inverse analysis yields results that are more consistent than when using the original coefficients. In the latter situation, the discrepancies between the results obtained from various models are larger. Therefore, the inverse technique can be recommended as an efficient and accurate method for the interpretation of tests in the evaluation of the microstructure evolution models.

120o

90(0 Q.

^,^.-"^--—....,^,^^,,^^,^ 0)

30-

850OC

4"

8850C

b a

0 ]

r

0.0

Figure 8.10 Finite element mesh for sideways compression (a quarter of the cross section is considered)

O

1

0.1

•

9590c

A

10470c 1

1

1

1

0.2

0.3 strain

0.4

0.5

Figure 8.11 Measured stress-strain curves in upsetting

Table 8.4 Coefficients in the equations in Tables 6.2 and 6.5 obtained from the inverse analysis. Hodgson and source Sellars and Choquet et al, Roberts et al., Whiteman (1990) Gibbs (1992) (1983) (1979) Ci 25.4(25) 6.9(6.2) C2 15.2(14.9) 375(343) 36(45) 56.9(55.7) m 8.7(8.5) -0.55(-0.6) -0.57(-0.6) -0.72(-0.65) n -0.65(-0.67) -0.07(-0.1) / 0.36(0.4) 0.68(0.374) 0.56(0.5) &,kJ/mole 312(312) 49(45) 38(38) 46(35) MA THEMA 77CAL AND PHYSICAL SIMULA HON OF THE PROPERTIES OF HOT ROLLED PRODUCTS

0.6

^

277

The equations with coefficients in Table 8.4 have been used in the simulation of the tests, not used in the optimization procedure. Figure 8.14 shows a typical calculated distribution of the austenite grain size and recrystallized volume fraction, compared with the results of measurements for the sideways compression. Metallography shows that dynamic recrystallization began in the center of the sample in all slow tests, an observation that was confirmed by theoretical analysis. Due to the large inhomogeneity of deformation in this process, the distribution of the grain size in the volume of the sample is complex (Figure 8.14a). Figure 8.14b indicates that in the areas where recrystallization is completed or there is no recrystallization, the measured and predicted grain sizes coincide perfectly. In the areas of partial recrystallization, the predicted grain size is a weighted average of non-recrystallized and recrystallized grains and it agrees well with the measured grain size.

temperature 850°c

Sideways compression 2 0 - temperature 860oc

7 ^

f

V °

15-

OJ/JI

ojgjip 10measurement, 2 s-^ —

measurement, 0.1 s-^

—

prediction, 2 s-""

• R — prediction, 0.1 s 1 T 2 3 4 reduction, mm

5-

1^^—

measurement, 860°C

—H~

measurement, 909°C

O 6

Figure 8.12 Measured and calculated compression forces for upsetting of cylinders, tests 1 and 2 in Table 8.3

0 -1F

1

prediction, 860°c

n prediction, 909°C T i l l 2 3 reduction, mm

Figure 8.13 Measured and calculated compression forces for sideways compression, tests 8 and 10 in Table 8.3

Similar results have been obtained for other tests in Table 8.3 and the agreement between the measurements and the calculations was very good. The analysis of all results showed that the inverse technique is capable of evaluating the coefficients A, describing recrystallized grain size in Table 6.4 and Table 6.5, and the coefficients B, in the equations describing dynamic recrystallization in Table 6.6. Coefficients B are very close to those suggested by Sellars, (1990). Application of the inverse analysis to the interpretation of plastometric tests presents numerous advantages. For example, no effort is to be spent on avoiding inhomogeneity of deformation. On the other hand, the nonuniform tests supply much more information than the tests with uniform deformation. One set of compression tests allows the determination of the coefficients in both equations describing the flow stress and the kinetics of recrystallization. PARAMETER EVALUATION IN METAL FORMING

278 using the cost function 0)1. The same set of tests allows the determination of the coefficients in the recrystallized grain size models using the cost function
a)

b) strain rate 2 s""*

strain rate 2 s'^

Figure 8.14 Measured (bold face numbers) and calculated (isolines) austenite grain size (a) and recrystallized volume fraction (b) for test 11 in Table 8.3, at a strain rate of 2 s'*

MA THEMA TJCAL AND PHYSICAL SIMULA HON OF THE PROPERTIES OF HOT ROLLED PRODUCTS

279

Chapter 9 Knowledge Based Modeling Chapters 1-8 show that mathematical and physical models are able to give a reasonably accurate description of the domain in the rolling process. The distributions of the stresses, strains, strain rates, temperatures, etc. in the billet in the roll gap, using models of various complexities, can be determined well, provided care is taken in the mathematical description of the physical phenomena. The models' accuracy and consistency also depend, of course, on the rigor of the description of the boundary and initial conditions. The models describing the entire rolling process must contain certain assumptions, necessary because of the extreme complexity of the process, which unavoidably, introduce errors. The use of less restrictive and less arbitrary assumptions would, of course, lessen but not eliminate the discrepancies and would increase the quality of the predictions. Further attempts to overcome or minimize these errors further are the objectives o^ knowledge based modeling. The major concern in this process is the closed logical loop in the technological design, since every technological operation can and will influence the quality of the product. Technology affects the material properties that further influence the geometry that determines the design of the technology. Disciplines of mechanics and kinetics cannot easily handle the problems of friction, wear, fracture, metallurgical phenomena, and heat loss and gain in multistage processes. Modeling such sophisticated phenomena needs new approaches. For process planning, measured and predicted data are also necessary. The on-line data acquisition systems, used in modem mills, can follow only the local changes, and can control the process according to a limited closed-loop system. The control unit cannot predict the occurrence of and the reasons for some defects, because the reasons are outside of the scope of the data acquisition systems. Knowledge processing, using methods of artificial intelligence (AI), opens novel ways in the "efficient use of deficient knowledge" (Hatvany and Lettner, 1983). A survey of knowledge-based methods in metal forming in 1992 (Cser, 1991, 1992) contained less than 200 publications. The growing interest in these methods is shown by Korhonen et al., (1998), which contains 50 reference items about only one AI method, artificial neural networks in rolling.

9.1

KNOWLEDGE IN HOT ROLLING

The factors influencing the microstructure and thus the properties, of the hot rolled strip involve the entire forming system, including the rolling mills, frames, work rolls, back-up rolls, transfer tables, cooling devices, descalers, etc., even if the exact effect of a particular component on the parameters and variables of the process is not immediately known or cannot yet, be quantified. It is convenient to divide the components and the phenomena into two subsections: those concerning the rolling mills, referred to as the "in-stand" events and those that are between stands, called "inter-stand" events. This approach, separating the issues at

280 and between the stands enables the description of different rolling trains, containing dififerent numbers of roughing and finishing stands. These phenomena are not independent of each other. In fact, they possess a very complex confluence. For example, when having changed the temperature, not only the roll force changes, but the torque, the surface phenomena, as well as the boundary conditions, among others, are all affected. The same is valid for the geometrical parameters, such as the thickness, profile, flatness, width and the surface roughness. In order to predict the expected quality of the strip or plate, detailed analyses of all these factors are necessary. As the first step the task is to: collect all necessary data from the industry, research and literature; store the data and their interconnections - the knowledge; and find similar cases, where the results can be transferred from the known case to the new, unknown one.

9.2

KNOWLEDGE ACQUISITION

The most difficult task in developing a knowledge-based application is knowledge acquisition. The definition of knowledge acquisition, according to the Concise Oxford Dictionary (1995) is as follows: "The process of acquiring knowledge from a human expert for an expert system, which must he carefully organized into IF-THEN rules or some other form of knowledge representation." Techniques of knowledge acquisition include: •

•

knowledge collection from the human experts; and on-line data collection with additional learning and knowledge extraction.

As the name suggests, the first method of acquiring knowledge attempts to collect and code the experience of the processes' users. The main disadvantage of the technique is that a third person, the so-called knowledge engineer is included. The knowledge engineer is familiar with the methods of knowledge processing and with the formal syntax and semantics of knowledge, but not with the specific topics or processes; in this instance, with the technology of hot and subsequent cold rolling. The advantage of this method is the ability of the human expert to give an explanation of the decisions which can also be included in a rule-based expert system. However, the expectations of collecting and storing the knowledge, drawn fi-om the lifelong experience of industrial experts and included in a Knowledge Based Expert System have not achieved the necessary results. The task of sharing between the "experts" and the "knowledge engineer" led to results of low practical value, see Figure 9.1. The maintenance of knowledge, that is, adding new rules or actualizing the old ones, causes diflScuIties in the expert systems. The second approach is very effective in the case of on-line measurement systems, but additional knowledge extraction from the collected data is necessary. The knowledge base, MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

28j_ built using this approach, embodies the knowledge in an implicit form, and so does not possess the ability to explain the decisions to the user.

Machine

Knowledge engineer

Knowledge acquisition & maintenance

MacliJf»«

11 Knowledge 3ase

11

00

Knowledge Base

DB Teaching

Using

Figure 9.1 The knowledge engineer and the user

9.2.1

Collecting knowledge from experts

The "classical" method of knowledge processing, used in different theorem proving systems, is the rule-based cq)proach with ifA^B^.

thenC

type rules, mentioned in the previous definition. The "left part" of the rule (here A ^ B '^ ...) describes the premise, the "right-hand part" the actton (here C). A complete rule, describing for example, the reduction necessary to initiate dynamic recrystallization of a microalloyed steel, containing 0.03% Nb, in a roughing mill within a 3 s inter-pass time may be • • •

if the Nb content is 0.03; and the grain size is 80 ^im; then the necessary reduction is 25%;

• • •

if the Nb content is 0.03; and the grain size is 250 nm; then the necessary reduction is 75%.

Using the rules an Expert System can give a good answer to the question, matching fiiUy the statement in the premise of the rule. It can also explain the reason of a decision answering the question: KNOWLEDGE BASED MODELING

282 Why has 25% reduction been choosen? However, no answer can be found, if the grain size is between 80 and 250 |im, say 150 ^m. The knowledge used above is a typical example of declarative knowledge representation. Declarative knowledge is stored as a set of statements about the phenomenon to be described. These statements are static but can be added to, deleted or modified. Note that there are some other types of knowledge representation, such as procedural, symbolic, subsymbolic, etc. Rule-based systems separate the declarative knowledge from the code that controls the inference and search procedures. The declarative knowledge is stored in a knowledge base (KB) while the control knowledge is kept in a separate area called an inference engine. An inference engine is an algorithm that dynamically directs or controls the system when it searches its knowledge base (Harmon and Hall, 1993). The inference engine matches the domains of the premise part in the rules, stored in the knowledge base, with the action parts building up logical chains forward or backward {forward chaining, or backward chaining), depending on the task (Harmon and King, 1985, Buchanan et al., 1985). Forward chaining is a method that finds every conclusion possible based on a given set of premises. A typical question to be answered using forward chaining is: What happens, if..? Forward chaining is used primarily for the prediction of events, where all possible outcomes based on a given input are under consideration. For all other purposes forward chaining is only feasible when the number of possible outcomes is small. When there exists the probability of too many conclusions, forward chaining becomes too inefficient and time consuming to be of any practical use. When the number of possible outcomes is large, one may choose backward chaining. In backward chaining, one starts with the desired goal, and then attempts to find evidence to justify that goal. A typical question is: What was the reason for... 7 Backward chaining is useful in situations where the quantity of data is very large and where some specific characteristics of the system under consideration are of interest. Typical situations are various problems of diagnosis, such as fault finding in the rolling process based on product quality, when asking the question: what was the reason for the lack of consistent geometry of the product? The rule-based expert systems do not possess true learning capability. They are unable to adjust their expertise to take into account their earlier errors or new situations. Once a system has been installed, it is important to continue the development and extension of the knowledge base. New knowledge gained during the implementation or test period may necessitate modification with the deletion of certain rules or the addition of new features. These operations can be error-prone because of the "spaghetti rule", indicating that when one rule implies another, the modification can destroy the entire structure of the rules. Thus, checking the influence of the modification on every situation becomes necessary. The MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

283 modifications can result in closed logical loops in the inference, as demonstrated formally by the logical if-statements: if A then B,

if B then C, ifCthen A. 9.2.2

Storing the knowledge

Knowledge for an Expert System is stored in a knowledge base (KB), incorporating the knowledge acquired from human experts. The rules can be verbal, making storage and supervision extremely difficult. One solution to overcome this difficulty is the structuring of the rules. Using structured rules the entire knowledge base becomes more compact. In the example above, separating the rules corresponding to different types of steel provides a clearer, better organized structure. Much better results have been achieved in decision-making by using the object-oriented knowledge representation. In such systems, efficient knowledge retrieval can be combined with easy knowledge maintenance (e.g. adding new knowledge to the system), where the syntax of the knowledge corresponds to the structure of the real world. In the case of metal forming tasks, factors that influence the process - the entire forming system - should be described as sets of objects connected with each other and having their own proper attributes. The so-called ^owe^ give the formal description of the knowledge containing the properties of the object in their slots. Slots can be identified by their names and in complex cases, they can ht frames themselves. Characteristic of object-oriented knowledge representation is that it separates the concrete attributes from their relations. The relations - the inside connections of different objects - are described by the generic frame. This empty frame describes the possible connections. The stored cases are the instances in the generic frame. In many cases the value to be estimated (e.g. grain size in hot rolling, or tool life in cold forging) is a result of the confluence of many factors. These factors, practically the entire forming system, are stored in the generic frame. All data extracted from the literature, collected from the industry, as well as simulated or measured in laboratory experiments can form parts of the knowledge base. However, sets of data are mostly incomplete, and some slots remain empty. In these cases, the empty slots can befilledwith default values from similar cases. An advantage of the object-oriented knowledge representation is its ability to define classes with partially common attributes, as well as to inherit the attributes. All members of the class of HSLA steels inherit the properties of the micro-alloyed steels, and it is enough to define a steel as belonging to this class. It is not necessary to define common attributes for them. The greatest advantage of the object-oriented approach is that it supports analogy building. This approach corresponds to human thinking. The basis of the expert's decision is mostly a "similar" case from his former experience and the analysis of differences. The difference between the thinking of the human expert and the Expert System is that the human expert can decide, from his or her own previous experience, whether a difference in one of the compared values is KNOWLEDGE BASED MODELING

284

essential or not. (e.g., difference of 0.01 % in carbon content is not as important as 0.01 % of boron, or niobium). The most important and troublesome part of the software work is connected not with the kernel part of the system incorporating the technical intelligence or with the development of the knowledge base, but with the formalization of the knowledge. However, the software market offers different tools, making the development of expert systems easier. A shell is a program that facilitates the development of expert systems. A good shell is a complete development environment for building and maintaining knowledge-based applications. It provides a systematic methodology for knowledge engineering that allows the domain experts themselves to be directly involved in structuring and encoding the knowledge. (The direct involvement of the domain expert improves the quality, completeness and accuracy of acquired knowledge, lowers the development and maintenance costs and increases their control over the form of the software application.) Features include: a structured approach to knowledge acquisition; a model of knowledge acquisition based on pattern recognition; knowledge represented as objects, production rules and decision tables; handling uncertainty by qualitative, non-numerical procedures; extremely thorough knowledge bases; sophisticated report writing facilities; and self documenting knowledge bases in a hypertext environment (for example, ACQUIRE).

9.3

NUMERICAL DATA FOR DECISION-MAKING

The easiest way to represent knowledge is when the knowledge domains are expressed in explicit form. These are the rules shown above. A very high grade of abstraction is necessary to formulate a rule. During the knowledge acquisition the expert and the knowledge engineer try to formulate the rules. These general rules are mostly "rules of thumb". Sometimes it is extremely difficult to formulate a universally valid rule, because of the large amount of premises and the difficulties in the general abstraction. Some subjective or emotional issues can also appear, and it is likely that another expert may formulate a totally different set of rules for the same case. The bases of all abstractions formulated in the rules are the many years of experience, and the professional background of the human expert. The experience is gained from the cases met by the expert during the entire professional career. Another possible approach is based on trying to collect as many case studies as possible. Having collected a large number of cases, the professional background of the human expert^ as well as the general rules of the professional area, appear as a smaller number of rules, bridging the gap between known cases and new situations. Such a process of knowledge acquisition is called case based learning. A special type of program can also be used to allow the deduction of the rules. The human expert's task is then only to supervise the learning process and correct the conclusions, a form of supervised learning. The main problem in the process of case based learning, following the formal logic of human thinking, is in finding analogies and differences. A human expert, possessing personal experience and knowledge of the process, can evaluate the situation, and decide whether it is similar to one from the past, or whether it is generally new. To find two cases with the same MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

2^ set of numerical data about the whole forming system is practically impossible. Once an input/output map has been constmcted, linking certain inputs to certain outputs, even very small differences may introduce difficulties. 9.3.1

Fuzzy sets and fuzzy logic

The methodology that permits more flexible mapping, and thus the finding of analogies in the cases is fuzzy logic. Fuzzy logic is a superset of conventional (Boolean) logic that has been extended to handle the concept of partial truth - truth values between "completely true" and "completely false" (Meech, 1995). Fuzzy Systems are Expert Systems that use fuzzy sets and fuzzy logic to perform qualitative modeling and/or to control processes when no model has yet been identified or developedfi-omfundamental principles. Fuzzy logic is concerned with the reasoning about "Fuzzy" events or concepts. Examples of fiizzy concepts are "temperature is high" and "C-content is low". When is a rolling temperature high? If the threshold of the high temperature is defined as lOOO'^C, the implication is that a rolling temperature of 998°C is not high. When humans reason with terms such as "high" they do not normally have a fixed threshold in mind, but a smooth, fuzzy definition. Humans can reason very effectively with such fuzzy definitions; therefore, in order to capture human fuzzy reasoning fuzzy logic is needed. An example of a fiizzy rule that involves a fiizzy premise and a fuzzy conclusion is: IF 77 content is high TBEN precipitation hardening is very high. The fuzzy value high designates the range of numerical values from 0.20% to 0.25% in the content of Ti. The term very high indicates that precipitation hardening is present and it is contributing 17 MPa to the material's strength for each 0.01% of Ti. Fuzzy reasoning involves three steps: • • •

fuzzification of the terms that appear in the premises of rules; inference from fiizzy rules; and de-flizzification of the fuzzy terms that appear in the conclusions of rules.

Transformation of an objective term into a fuzzy concept is called fuzzification. Fuzzification is a method of modeling human imprecise reasoning using fuzzy sets. Using this technique, the concept high is related to the underlying objective term, which it is attempting to describe; namely the actual Ti content in %. The transformation of an objective term into a fuzzy concept is called fuzzification. The degree to which the statement: 0.20% Ti is high (x is in F) is true is determined by finding the ordered pair whose first element is x. The degree of truth of the statement is the second element of the ordered pair. But 0.18 % Ti also is high with a degree of truth lower than the 0.20%. Just as there is a strong relationship between Boolean logic and the concept of a subset, there is a similarly strong relationship between fuzzy logic and fiizzy subset theory. KNOWLEDGE BASED MODELING

286^ In classical set theory, a subset U of a set S can be defined as a mapping from the elements of S to the elements of the set {0, 1}: U:S-^{OA}

(9.1)

This mapping may be represented as a set of ordered pairs, with exactly one ordered pair present for each element of S. The first element of the ordered pair is an element of the set S, and the second element is an element of the set {0, 1}. The value zero is used to represent non-membership, and the value one is used to represent membership. The truth or falsity of the statement: X is in U

(9.2)

is determined by finding the ordered pair whose first element is x. The statement is true if the second element of the ordered pair is 1, and the statement is false if it is 0. Similarly, a fuzzy subset F of a set S can be defined as a set of ordered pairs, each with the first element from S, and the second element from the interval [0,1] with exactly one ordered pair present for each element of S. This defines a mapping between elements of the set S and values in the interval [0,1]. The value zero is used to represent complete non-membership, the value one is used to represent complete membership, and the values in between are used to represent intermediate degrees of membership. The set S is referred to as the universe of discourse for the fuzzy subset F, and the mapping is described as the membership function of F (In practice, the terms membership function &nd fuzzy subset are used interchangeably).

0

0.2

0.25

0.3

TJ(%)

Figure 9.2 Membership function of high in context with Ti in HSLA steels

The real power of fuzzy logic systems, compared to crisp logic systems, lies in their ability to represent a concept using a small number of fuzzy values. This therefore reduces the number of rules required to capture the knowledge relating to that concept. To achieve the same accuracy with crisp logic, a large number of logical values would be required, resulting in a large rule base. Inference from a set of fuzzy rules involves fuzzification of the premises of the rules, and then propagating the membership values (confidence factors) of the premises to the conclusions (outcomes) of the rules. MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

^

287 Consider the following rule:

IF initial grain size is medium AND pass reduction is large THEN the re crystallized austenite grain size is small. Inference from this rule involves (using fiizzification) looking up the membership value of the premise '"initial grain size is medimf\ given the initial grain size before the rolling operation, and the membership value of "the pass reduction is large", given the degree of deformation. Fuzzy inference involves having a weight for each rule between 0 and 1, and then multiplying the membership value assigned to the outcome of the rule. By default, each rule weight is set to 1.0. For example, for the initial austenite grain size three categories may be defined: • • •

large. medium small

when the ASTM grain number is < 2; 2-6; >6.

For each category the center of the interval corresponds to the membership value =1, and the middle of the neighboring interval to 0. In a fuzzy rule base a number of rules with the outcome recrystallized austenite grain size is small will be fired. The inference engine will assign the outcome recrystallized austenite grain size is small the maximum membership value from all fired rules. So, the fijzzy inference involves: • • •

de-fuzzification of the premises of each rule and assigning the outcome of each rule the minimum membership value of its premises multiplied by the rule weight; assigning each outcome the maximum membership value from its fired rules; and f^Tzy inference resulting in confidence factors (membership values) assigned to each outcome in the rule base.

If the conclusion of the fiazzy rule set involves fiizzy concepts, these concepts will have to be translated back into objective terms before they can be used in practice. For a rule set including the recrystallized austenite grain size rule described in the previous section, fiizzy inference will result in the terms "'recrystallized austenite grain size is smalT\ "'recrystallized austenite grain size is medium'' and "recrystallized austenite grain size is large'' etc. being assigned membership values. In practice, however, to use the conclusions from a rule base it is necessary to de-fijzzify the conclusions into a crisp recrystallized austenite grain size figure. To do this it is necessary to define the membership fijnctions for the recrystallized austenite grain size outcomes, as shown in Figure 9.3. One method of de-fiizzification is to place the membership fijnctions (confidence factors) generated by inference for each fiizzy outcome at the point where the membership fiinction has its highest value. The required de-fijzzified value can then be calculated as the center of gravity of the membership vectors. This is illustrated in the example below, assuming that fiizzy inference results in membership values of 0.1 for medium, and 0.8 for large. The de-fuzzified value of the recrystallized austenite grain size number is calculated as the center of gravity of the area framed by the bold line. Its ordinate is near 5.8 and this KNOWLEDGE BASED MODELING

288 corresponds to measured values (Ginzburg, 1989). While the main principles of fuzzy logic are broadly accepted, there are a number of various methods of fiizzy inference and defuzzification.

grain size #

Figure 9.3

9.4

Membership functions for the recrystallized austenite grain size number

DATA AND DATA MINING IN HOT ROLLING

In a modem rolling mill, each device, including the furnace, the mill stands, coilers etc., has its own computer control with sensors. Computers controlling the devices are supervised and are controlled on the next hierarchical level by other computers, co-ordinating the devices and uniting them into a technological unit (e.g. finishing mill). These units have a control system of their own, connected with sensors measuring the process parameters before entering the current unit. The work of the technological units is co-ordinated by the computers on the next hierarchical level, which store all activities of the controlled lower level. The production data and the events during the rolling process are collected and stored by the next higher level of computers. In an average modem hot strip mill, a full temperature map is measured behind the fiimace, before the roughing stand, before the first finishing stand, behind the first finishing stand, before the last finishing stand, and before the coiler, etc. After exit from the last finishing stand, the thickness, profile, width, flatness, and wedge are also measured. At each stand the roll separating force, roll bending force, mill motor power, tension, strip bending, etc. values are measured, and the torque is calculated and stored. Data concerning the surface quality, the grain size or mechanical properties are computed or collected later. 9.4.1

What to do with the sampled data?

The measurements listed above are taken continuously during the passage of each strip or plate. The data-sampling interval is between 0.2-2.0 sec. The temperature is measured, h4A THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

289 preferably over the whole width, before the finishing mill on the top and bottom side of the strip. Millions of data points are collected for each strip, and - depending on the performance thousands of strips are rolled in a particular campaign. Following each pass, averages are recorded and the standard deviations are determined. The rest of the data are often discarded. An interesting situation appears with a flood of data on the workshop level, while the research personnel, trying to predict the product quality, suffers because of the lack of detailed data. The basis of the quality prediction is further dependent on human experience collected in practice. There is a gap between the data and the knowledge. Using the definition of the Concise Oxford Dictionary: "KNOWLEDGE: • knowing, familiarity gained by experience, (of person, thing, fact) • person's range of information • theoretical or practical understanding (of subject, language etc.)". Another classical definition of knowledge states that it is the: "Capacity to solve problems, innovate, or otherwise create value on the basis of previous experiences, skills, or learning." These definitions show the necessity offindingthe hidden dependencies in the measurement information collected on-line, to understand them and to use them properly. In this way the gained knowledge allows the prediction of the quality and leads to explanations of the deviations from the required quality parameters. The term "t/orto mining" is just one of several terms, including knowledge extraction, data archaeology, information harvesting, sift-ware and even data dredging, that actually describe the concept of knowledge discovery in databases. The idea behind data mining, then, is the "non-trivial process of identifying valid, novel, potentially useful, and ultimately understandable patterns in data" (Srikant and Agrawal, 1996). Therefore, data mining is the automated discovery of hidden patterns, cross correlation and trends in large amounts of data, and the knowledge discovery in databases. The task of data mining is the clustering/segmentation of the information: •

•

•

identifying homogeneous and separable groups (clusters), i.e.: maximum similarity within a group, maximum difference between groups, identifying rules of co-occurrence of items, to each rule is associated: confidence level (i.e. probability of occurrence): "if a record contains A and B, then it also contains C with probability p = 0.7" support level (i.e. number of records in the database), identifying recurrent patterns over time: discovery of frequent episodes (i.e. collections of events), association rules through time, induction of prediction models, statistical / neural network models. KNOWLEDGE BASED MODELING

290

Data mining is closely connected with the machine learning, pattern recognition, databases, statistics, artificial intelligence (AI), and knowledge acquisition for expert systems and data visualization. Many of the techniques and algorithms are derived from these fields. The underlying basis for these fields is the extraction of knowledge or patterns of information from data in the context of large databases. Traditional database queries contrast with data mining since these are typified by simple questions such as What is the width deviation within the strips? Multidimensional analysis enables one to do much more complex queries, such as comparing actual and planned width for different thickness and nominal width categories. Again, the emphasis in both these cases is that the derived results are values, which are an extraction or aggregation of existing data. Data mining, on the other hand, through the use of specific algorithms or "search engines", attempts to source out discernible patterns and trends in the data and infers rulesfi"omthese patterns. With these rules or fiinctions, the user is then able to support, review, and examine decisions. The main steps and methods of the data mining process are as follows: • •

• •

•

•

• •

object definition; data integration: integrating the existing sources of data, merge data fi'om multiple files and databases (usually fi-om different operational environments), resolve semantic ambiguities; data selection: identify relevant variables and ranges among the available data; pre-processing the data: preparing the data, data cleaning, finding the errors or inconsistencies, handle the noise, data scrubbing, mappings and data conversions, deriving new attributes; visual inspection, distribution and significant outliers, statistical analysis (correlation between attribute pairs, higher order correlation between groups of attributes), feature analysis, clustering, discretization of attributes; knowledge discovery: task classes, clustering/segmentation, association, classification, pattern detection/prediction in time-series, evaluating the error rates, model validation, statistical cross-validation, variance analysis of the results; interpreting the results; and integrating the solutions.

Technology of data mining covers the visualization (data visualization, discovery by visualization as well as visualization of discoveries), the discovery algorithms (e.g. decision trees, clustering, neural networks, association rules, sequential patterns), and data warehousing. 9.4.2 Self-organizing maps (SOM) The Self-Organizing Map (SOM) method is a new, powerfiil software tool for the visualization of multi-dimensional data. It converts complex, non-linear statistical relationships MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

291_ among high-dimensional data into simple geometric relationships on a low-dimensional display (Kohonen and Oja, 1996). Similar to other data mining methods, it also applies and incorporates statistical procedures for modeling and handling of noisy data. As it thereby compresses information while preserving the most important topological and metric relationships of the primary data elements on the display, it may also be considered, as a qualitative abstraction. The SOM is a neural network algorithm which is based on unsupervised learning in a datadriven way (Kohonen, 1995). Unlike supervised learning methods, the SOM can be used for clustering data without knowing the class memberships of the input data. Therefore it can be used to detect features inherent to the problem. The SOM has been successfully applied in various engineering applications, as written by Kohonen, et al., (1996) covering, for instance, areas like pattern recognition, image analysis, process monitoring and control, and fault diagnosis (Tryba, et al., 1991; Simula and Kangas, 1995; Simula, et al., 1998). The SOM has also proved to be a valuable tool in data mining and knowledge discovery with applications in full-text and financial data analysis (Kaski, 1997). The essential point in the applicability of SOM is the topological character of the mapping: similar patterns are mapped in the nearby locations on the map. Therefore, it is a method for the visualization of multi-dimensional patterns. The Self-Organizing Map (SOM) is a neural network model. It consists of neurons organized in array. The number of the neurons may vary fi*om a few dozen up to several thousand. Consider a rolling mill from which several measurements are taken, as mentioned above. Denote the measurement vector by:

X={^J„..4/

(9.3)

wherej indicates a measured parameter (e.g. temperature, thickness, flatness, etc.) The SOM consists of a regular, usually two-dimensional, grid of neurons. Each neuron / of the SOM is represented by an n-dimensional weight, or model vector:

where n is equal to the dimension of the input vectors. The scale of each measurement is normalized so that either the maximum and minimum of each §, respectively, are equal, or the variance of every § is the same. If enough measurement values are collected (say, 10 values for every cell), the weight vectors /w,, are determined. Any component plane J of the SOM, that is, the array of scalar values ///, representing they* components of the weight vectors W/ and having the same format as the SOM array, can be displayed separately, and the values of the //y can be represented on it using colour scale or shades of grey. The weight vectors of the SOM form a code-book. The SOM algorithm performs a topology preserving mapping from the multi-dimensional input space onto map units so that relative distances between data points are preserved. Data points lying near each other in the mput space will be mapped onto nearby map units. The SOM can thus serve as a clustering tool of high-dimensional data. It also has capability to generalize; that is, the network can interpolate between previously encountered inputs. KNOWLEDGE BASED MODELING

292

^

The neurons of the map are connected to adjacent neurons by a neighborhood relation, which dictates the topology of the map. Usually rectangular or hexagonal topology is used. Immediate neighbors (adjacent neurons) belong to the neighborhood N, of the neuron i. In the basic SOM algorithm, the topological relations and the number of neurons are fixed fi-om the beginning. The number of neurons determines the granularity of the mapping, which affects accuracy and the generalization capability of the SOM. During the iterative training procedure, the SOM forms an "elastic" net that folds onto the "cloud" formed by the input data. The net tends to approximate the probability densities of the data (Kohonen, 1995); code-book vectors tend to drift to where the data are dense, while there are only a few code-book vectors where data are sparse. At each training step, one sample vector X is randomly drawnfi-omthe input data set. Distances (i.e., similarities) between the x and all the code-book vectors are computed. The Best-Matching Unit (BMU) is the map unit whose weight vector is closest to x. After finding the BMU; the weight vectors of the SOM are updated. The BMU and its topological neighbors are moved closer to the input vector in the input space. The SOM can be used eflBciently in data visualization due to its ability to approximate the probability density of input data and to represent it in two dimensions. Several different ways to visualize the network are developed (Simula and Kangas, 1995), two of which are introduced in the following: •

•

9.4.3

The unified distance matrix (U-matrix) method by Ultsch, et al., (1990) visualizes the structure of the SOM. First, the matrix of distances (U-matrix) between the weight vector of each neuron and its neighbors is calculated. Then some suitable presentation, such as a grey-level image (livarinen, et al., 1994), is selected to visualize the matrix. Component plane representation visualizes relative component values of the weight vectors of the SOM. It can be considered as a "sliced" version of the SOM, where each plane shows the distribution of one weight vector component. Using the component plane representation, dependencies between different process parameters can be investigated. A proper choice of the colors used in visualization will make this task easier. Discovering hidden dependencies

The analysis is designed to discover how some of the measured values depend on each other, how they change together, how they can be simplified by slicing the n-dimensional statespace of the measured values on component planes, and how to seek for similar patterns at the same places of the planes. If a similar pattern has been found, it shows that the weight values are similarly high or low, showing the co-incidence and the potential co-dependence of the two components. This does not necessarily indicate that these parameters or factors have a causal connection. They can appear together, depending on a third factor or several other factors, indicating that their inter-dependence. The components of the input vector can be continuous or they can be discrete values.

MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

293 9.5

NUMEMCAL DATA COVERING THE GAPS IN KNOWLEDGE

The area of computational intelligence offering true learning capability is that of Artificial Neural Networks. Artificial Neural Networks are capable of modeling extremely large and complex problems, involving hundreds of variables. The models they generate can identify patterns and relationships in data that were previously unknown. Neural networks are nonparametric and are capable of taking on any form that the data requires. This ability makes neural networks extremely accurate predictors of non-linear real events. The paradigms in this field are based on direct modeling of the neural system in living organisms. The I/O space map is modeled as a series of interacting nodes in a network. When a signal appears at a node, it is amplified by a "weight" assigned to each connection between this node and others. At the receiving node, all incoming signals are summed up to give an overall force acting on the neuron. 9.5.1 Artificial neural networks The advantage of neural networks is their ability to learn or adapt to changing conditions. Learning begins by selecting a set of random weights and presenting the network with a set of known I/0-valuesfi-oma large set of training data. The network predicts the output. These are compared with the desired output. The error is calculated in order to modify the weights. The new set of I/O data is examined, weights are modified using the regression or gradient-method and so on. This process continues until the overall error is within a pre-defined tolerance. This is a kind of supervised learning using the back-propagation technique. The number of publications dealing with neural networks in rolling applications has increased during the past few years and the trend in Figure 9.4 suggests that the number may be doubled by 1997 or 1998. However, the number is very small when compared to papers dealing with neural networks in other industrial applications.

90

91

92

93 Year

94

95

Figure 9.4 Number of papers dealing with NEURAL NETWORKS in rolling applications

KNOWLEDGE BASED MODELING

294 9.5.2

Training the artificial neural networks

Multi-layer perceptrons are feed forward type neural networks. The calculation units, called perceptrons, shown in Figure 9.5, perform the following ftinction: (9.5)

>' = / ( S H ' , X , )

where/is usually a sigmoid function of the form: /(i/) = ( l 4 - . - ) '

(9.6)

Figure 9.5 Illustration of a perceptron unit In neural networks the perceptron units are arranged in layers (Figure 9.6), where the outputs;^i are calculated from the input vector X=(xi, ...,XM), using the weights w stored in the network. Most common neural networks architecture consists of two layers of perceptrons.

Bias

Input layer

Bias

Hidden layer

Output layer

Figure 9.6 One hidden layer neural network architecture with fully connected nodes MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

295 Finding an optimal set of weights that perform the desired mapping from inputs to outputs requires training. The back propagation learning rule has been widely accepted as the standard training method for neural networks. It incorporates the least mean squares method to minimize the following equation: Ep = y2i:Sp

(9.7)

where p refers to a pair of input and output vectors, and ^ is the difference between actual and desired outputs. By determining the gradient of Ep with respect to the weights w, the weights can be adjusted in order to minimize the total error of produced outputs. The update rule for the weights can be written as:

Wy{t + l)=Wyit)-^

(9.8)

where / refers to the weights from input layer to the hidden layer, y refers to the weights from hidden layer to the output layer and rj is the learning rate parameter, which is usually a positive number between 0 and 1. The neural network learns by examples, that is, data, presented to it during training. By definition, a neural network with one hidden layer can learn any function with infinite accuracy. However, in applications involving variables of finite accuracy, it is not advisable to train the neural network to the lowest possible prediction error. Instead of memorizing the available data, the training should aim at a good generalization (Rumelhart, et al., 1994). A measure of generalization is the performance of the neural network on previously unseen data. A common way of testing the generaUzation capabilities is to divide the available data into two sets; one for training and one for testing. At the beginning of the training, the average prediction error usually decreases for both data sets. At some point the error on the test data starts to increase while the training error continues to decrease. The sets of weights that produce the lowest test error are generally considered to give the best generalization. The input and output layer sizes depend on the application (the numbers of independent and dependent variables). The hiddenlayer size is set up between the input layer size and output layer size. The input layer size is generally much larger than the output layer size (which is possibly quite small). The backpropagation network has the ability to learn any arbitrary complex non-linear mapping due to the introduction of the hidden layer. The greatest benefit of the neural networks is their interpolation ability. When one of the data sources provides less than expected, fiirther data generation is required, and properly trained neural networks are capable of supplying the needed information. In order to estimate the flow stress at elevated temperatures a large number of measurements are necessary. The use of neural networks can help in covering the knowledge gaps. The results of the interpolation should be handled with special care. An example concerns the chemical content of steels. Some predictions may be made by the neural network, concerning the material attributes, but if the training was not performed carefiilly, potential metallurgical changes may be ignored. Precipitation or new chemical connections may appear, and singularities in the mechanical properties may make the predictions completely unreliable. KNOWLEDGE BASED MODELING

296 9.5.3

Hybrid models in hot rolling

The neural network models are used frequently for estimating correction factors in the physically based rolling models that take into account the features responsible for the variations in the process (Bresson, 1993; Larkiola, et al., 1995; Cho, 1996; Kappen, and Gielen, 1995; Too, et al., 1996; Postlethwaite, et al., 1996). These hybrid models either solve the difficulties of mapping between inputs and outputs, by leaving only unknown dependencies for the neural network to solve, or they are used as fast models from numerically simulated data in order to avoid the long computations in the real-time environment (Sartori, and Antsaklis, 1991; Wiklund, 1996). By using simulated data it is possible to overcome the common problem of uneven distribution, associated with logged process data. In the automation and control of the strip flatness and the determination of mill settings needed for an incoming strip before the strip actually enters the mill, neural networks, based on both measured and simulated data are used. An efficient process model should be able to predict rolling forces, torque, material properties etc. with sufficient accuracy and reliability. It should be possible to update the network reasonably fast. Additional real time corrections, based on continuous measurements, compensate for the pre-calculation error by adapting the model to the process. Neural networks are able to do both modeling and adaptation equally efficiently (Portman, 1995). For the control task a reasonable task sharing between the mathematical modeling and neural network based correction is necessary. On-line computations should be performed by the neural networks, but the possibilities of mechanically correct mathematical modeUng should also be utilized. A typical task sharing is as follows: • • •

prediction of the width changes in the roughing mill; corrections, based on the measurements on exit from the finishing train; and correction of the strip or plate temperature, calculated by analytical models.

Networks used in roUing mill control can be divided into two groups: networks trained offline in development laboratories and built into the control system, and networks that are "trained on the job".

9.6

CASE STUDIES

In what follows, seven studies are reviewed. These include a framework for the prediction of the quality of the product using the methods of artificial intelligence and the use of selforganizing maps in a hot strip rolling mill. The use of neural networks is next and these are illustrated by a method to predict the austenite grain size in hot rolling of steels, to predict the constitutive behavior of steels, to predict the roll force in hot strip rolling and to predict the parameters in hot rolling of aluminum. In each of these the emphasis is on considering the quality of the predictions and how they relate experimental or industrial data. MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

297 9.6.1

Prediction of the product quality based on analogies

An example for the expert system usingfijzzylogic and fuzzy inference to find analogies for the prediction of the grain size in hot rolling is described in (Cser, Lenard and Farkas, 1994). The governing idea of the expert system for predicting the grain size and other quality parameters of hot rolled products has been that a similar set up of a similar forming system produces similar quality. Experience of the rolling industry shows that the grain size is influenced not only by the forming system inside the rolling mill, but by the environment between the rolling operations. It is the main idea in the description of the multi-operation rolling processes. The multi-pass rolling process is a sequence of rolling and "between stands" stages and events. The data collected, for example the object Material final properties after the first pass becomes the input data for the second pass, and so on. Such an approach enables the build up of a knowledge base, which is conforms to the physical system. material properties lubrication, speed gap control

inference + contro)

J

quality

I operation-specific rules

T

/"——^

STANDI

STAMD2

Interstandl

interstand2 InterstandS

STAND3 L

J

V

)

description of the stand-related forming system domains

Figure 9.7 Structure of the Expert System predicting the grain size in hot rolling However, the attributes stored in this frame are the results of the influences of pass data, described as objects Stand, Material, Workpiece, Temperature (in the deformation zone). Technology, and "between stands" data, such as Layout, Physical/metallui^ical data, and Cooling. All these are slots in aframecalled the Forming system, and they are frames with slots at the same time. The slots are also frames and so on. This hierarchical structure contains all the material and technological parameters as well as the parameters of the equipment. Structure of the program is given, schematically, in Figure 9.7. A distributed knowledge base has been used in a segmented structure: • • •

forming system is described separately for each stand and for each inter-stand state; pass-specific knowledge domains are defined; and general rules, known in the AI literature as Deep knowledge, are established. KNOWLEDGE BASED MODELING

298 In accordance with this structure, data describing the elements of the forming systems in each pass and inter-pass state are stored in the object oriented database, serving as the knowledge base of the forming system. For data acquisition, the facilities of this data base management system are used. Teaching the system proceeds by manual feeding using case studies, collected in the given rolling mill. The subsequent analysis of the answers given by the expert system is enabled by the identification numbers (ED) or the dates of the knowledge acquisition. The fuzzification proceeds only before the inference, during the search for the analogies, as shown in Figure 9.8. Having found them, the de-fuzzification gives numerical results with confidence factors. The inference is built up on the results collected from the literature and industrial practice, as well as from the results of modeling.

Knowledge |-»{Fuzzification) F u z 2 y | < Defuzzification>-»- Results & [acquisition jnferencej evaluation

KNOWLDEGE BASE

Figure 9.8 Fuzzification and de-fuzzification in the Expert System The final goal of the expert system is to predict the expected material properties of the plate or strip, rolled with a given chemical content as well as with a given set of technological parameters. The cornerstone of the truth in such predictions is the number of case studies stored in the data/knowledge base. If this number is small, the system gives predictions with very low confidence factors and they do not have any practical value. 9.6.2

SOMin a hot strip mill

In order to meet the challenge of improving the product quality, steel manufacturers use highly advanced rolling systems with high grade of automation and on-line data sampling. However, the large amount of collected data can be used not only for registrations required by ISO 9000, but is a rich source of knowledge about the extremely complex processes. Figure 9.9 shows a scheme of a hot rolling mill with continuous casting machines, furnaces, a roughing mill and a multi-stand finishing train. The methods of data mining can be used for improving the geometric accuracy of the strips. When searching for the reasons of the scatter in the geometric characteristics (quality parameters) of the strip, self organizing maps represent an efficient tool when they are applied to the control of information and to the quality parameters, such as the thickness, profile (crown, defined as the difference between the center gauge and the specified edge thickness), width deviation, flatness, measured in I units (I=AL/L X 10 , where AL/L is the strip waviness, AL is the difference between the longest and shortest ribbon across the width of the strip, and L is the wavelength), and the wedge. MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

299 statistically preprocessed data

2

8

48 -40Mbyte/strip

Sampled data (1/sec)

1

Scanned data .

1

casting machines

^m^

111

chemical composition (alloying e

4

4

4

6

furnaces

EEEEl

l\

F. F,. F,. S

F,

roughing mill

F„

If t

T T,

F,

Tn. H. W. We. R. Pr Tc T^

Tail Body Head

Tail Body Head {1/sec) max. deviation min. deviation / \ avg. deviation min max sL deviation 1/sec

Figure 9.9 On-line measurements (data sources) in hot rolling (Ci - casting machines, Fi - furnaces, Tbar, Wbar - bar temperature and bar width, T temperatures of the strip, upside, downside, head, body and tail, Fi (F, Fb, Ft) - force vector in stand I (roll separating force, roll bending force, strip tension force), !„ - temperature, measured before the stand, H, W, We, FI, Pr - geometric quality parameters: thickness, width, wedge, flatness, profile, Tc - coiling temperature) The first task is to analyze the influence of the continuous casting machines on the average value and the standard deviation as an example for using one discrete parameter in the analysis. The question is whether the control of the continuous casting machines influences the flatness value. As is well known, poor flatness or shape, of both hot and cold rolled strips is caused by differential elongation across their width, (Ginzburg, 1989). These differences produce corresponding internal stresses within the strip. Since the HSLA steels are much more sensitive to the overall history of processing, the collected data of more than 16000 strips were separated into two groups: low-carbon steels and HSLA steels. The analysis can be done separately for the head, body, and tail of the strip in order to estimate the necessary corrections in the set-up values. The relative values (e.g. AWAV for the width, or AH/H for the thickness) are used in order to enable a comparison of the dimensionless variables. Comparing the patterns on the parameter maps, no similarity can be established with any of the patterns on the plans, corresponding to the casting machines. The conclusion is clear: the control of casters works equally well in all three units. Similarly, no connection is seen between the geometric quality of the strip and the type of the re-heat fiimace, even in the case of the HSLA steels, though it is well known that they are very sensitive to the temperature and the duration of re-heating.

KNOWLEDGE BASED MODELING

300 9.6.3 Neural networks for grain size prediction Applications of neural networks in rolling cover a wide field of applications ranging from flatness control to the prediction of mechanical properties of the rolled material. In addition, in rolling the most effective way of applying neural networks may not be the global neural network model but a combination of the neural network and the classical mathematical model (Larkiola, et al., 1996). Industrial applications of the neural networks can be divided into the groups: • • •

predictive type applications, for example, product quality parameters; estimation type applications, for example, roll separating force, torque etc.; and control and optimization.

A typical case of the prediction/estimation type applications is the prediction of microstructure and final mechanical properties of hot rolled steels. The problem is well known by the difficulties it presents during modeling. A knowledge based approach for this task is represented by the first case study, in Section 9.6.1. The extremely large number of parameters as inputs and many characteristic parameters as outputs make this problem highly complex. Furthermore, some of these parameters do not directly affect the microstructure but indirectly through others that have their own effect, as well. The neural network used for modeling the grain size (Farkas and Arvai, 1995) of carbon steels after processing has 11 inputs and 7 outputs. The inputs are: roll radius, entry thickness, exit thickness, rolling speed, entry temperature, exit temperature, times and cooling rates, the heat transfer coefficient, coefficient of friction, initial grain size and carbon content. The outputs are: final grain size at 5 different locations of the cross section of the sample, roll force and roll torque. One hidden layer is used, depending on the configuration of the network program. The learning process uses a back propagation scheme. 9.6.4

A study of neural networks for the prediction of constitutive behavior of HSLA and carbon steels

Back propagation neural networks are utilized to store and predict the flow stresses of several steels. A convergence algorithm using a varying learning factor is developed, which has been shown to save one sixth of the learning time when compared with an algorithm in which a constant leammg factor is utilized. A performance test shows that the well-trained neural network can interpolate flow stresses very well if the information for interpolation is sufficient in the training pairs. The ability of the network to extrapolate is found not to be impressive. The neural network can handle several groups of data during adaptive learning simultaneously without losing accuracy. The time needed for adaptive learning to reach a reasonable level of accuracy is short. Comparing the predicted resuhs to other models, the output of the neural network is shown to have the highest accuracy. In both cold and hot flat rolling processes the thickness control of a plate or a strip is governed by the mill stretch formula;

MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

hexit=So+j

(9.9)

where hexn is the exit thickness, So is the roll gap distance, K is the mill stiffness, known for the particular mill and F is roll force. The latter is a function of a number of process parameters, such as the temperature, entry thickness, reduction, strain rate, coefficient of friction, front and back tensions, the viscosity of the lubricant and the flow stress of the rolled material. During hot rolling, the roll force is also affected by the evolution of the microstructure In the above equation K and So are known values while F and hexn are unknown. Once the roll force is predicted accurately, the stretch of the mill is calculated and the correct roll gap can be set to make the exit thickness satisfy the target requirement. Therefore, the roll force model and the prediction of flow stress play important roles in gauge control. The prediction of the flow stress is the subject of the present study. Experimental results: Six steels, one IF steel (Steel A), one extra-low carbon steel (Steel B), three low carbon steels (Steel C, D and E) and one Nb steel (F), are used in this work. Their chemical compositions are listed in Table 9.1. The flow curves of Steel C were obtained from isothermal ring compression tests (Hwu et al., 1993). The flow curves of the remaining steels were obtained by conducting isothermal compression tests with cylindrical samples. The experimental parameters are shown in Table 9.2, in which X indicates that the flow stress data for that particular condition are used for training the neural network. The symbol O indicates the data used for testing the performance of the trained neural network. The terms in the brackets (A, Bl, B2, etc.) v^U be referred to later. Table 9.1 Chemical composition (wt%) steel A B C D E F

C 0.0026 0.003 0.045 0.19 0.343 0.043

Mn 0.16 0.27 0.28 0.72 0.70 1.43

Si 0.006 0.35 0.01 0.05 0.09 0.312

P 0.002 0.056

S 0.011 0.005

0.009 0.008 0.001

0.007 0.009 0.006

Ni

Cr

Mo

Cu

— —

— —

— —

.-_ —

—

—

— 0.006 0.014

0.023 0.003 0.139 0.252

— 0.021 0.037

Ti 0.013

Nb 0.029

«_

-__

— — ...

— — 0.075

Fe balance balance balance balance balance balance

Effect of input variables and number of nodes in the hidden layer: In a previous study (Hwu et al, 1994) three process variables - temperature, strain rate and strain - were selected as input for training the neural networks, but the chemical composition of the steel was not considered since only one steel was used. In the present investigation, however, six different steels are involved and this makes it necessary to include their chemical compositions among the input variables. For carbon steels, carbon, manganese and silicon are dominant elements which affect the flow stress significantly. For microalloyed steels, niobium forms precipitates which retard the restoration mechanisms and increase the flow stress. Hence, the contents of carbon, manganese, silicon and niobium of each steel in Table 9.1 are selected as the extra four input variables. Adding to the three process variables, there are now seven inputs and one output, stress, in the training pairs. Traditionally the carbon equivalent Ceq, is used to indicate the level of strength of steels and is expressed, following Laasraoui and Jonas, (1991c):

KNOWLEDGE BASED MODELING

302

^

•-C +

Mn Ni+Cu + 6 15

+

Cr + Mo+V 5

(9.10)

Table 9.2 Experimental conditions steel

A

B

C

D

E

F

conditions temperature 1100°C 1000°C 950°C

1.0 s' X

temperature llOO'^C 1050°C 1000°C 950°C

1.0 s' X X X X

temperature 1050°C 1000°C 900°C

1.0 s-' X 0(C) X

temperature 1100°C 1000°C 900°C

1.0 S-' X X X

temperature 1100°C 1000°C 900°C

1.0 S-' X X X

temperature 1000°C 970°C 910°C 860°C

0.1 s'* X 0(F1) X X

strain rate 0.1 s'^ X 0(A) X strain rate 0.5 s'^ X 0(B1) X X strain rate 0.1s-' X X X strain rate 0.1 S-' X 0(D) X strain rate 0.1 S-' X X 0(E) strain rate 0.01 S-' X X X 0(F2)

O.Ols-' X

0.05 S-' X X X 0(B2) O.Ols-' X X X 0.01 S-' X X 0.01 S-' X X 0.001 s' X X X X

In order to reduce the number of variables, the contents of carbon and manganese are replaced by the carbon equivalent and the number of input variables is reduced to six. The flow stresses, adopted for training, are selected from the flow curves of each steel at various experimental conditions, indicated by X in Table 9.2. They are taken at several values of strain, starting at 0.05, then 0.1, followed by increments of 0.1 to the steady state strain. The total number of training pairs for the six steels is thus 523, referred to as Pattern I. In the first attempt, three neural networks with seven inputs are trained to examine the effect of the number of nodes in the hidden layer. The results show that 25 is an optimum number of nodes, MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

303^

as given in Table 9.3. Keeping this number of nodes in the hidden layer, a neural network with six inputs (6x25x1) is then trained to examine the effect of the input variables. After 5000 iterations, the sum of square of the errors, the E value, of the network reaches 0.12896, higher than the results obtained with seven inputs, indicating that the training results become worse when the carbon equivalent is adopted in the training pairs. In Table 9.3 ;/ is a constant learning factor while T/O is the initial value of the learning factor that is automatically adjusted during the training. Table 9.3 Training results type of learning factor network Tl 7x15x1 77 = 0.3, constant 7 X 25 X 1 7 = 0.3, constant 7x35x1 77 = 0.3, constant 6 X 25 X 1 77 = 0.3, constant 7 X 25 X 1 770=0.3, Algorithm II 7 X 25 X 1 770=0.3, Algorithm I 7 X 25 X 1 770=0.3, Algorithm I 7 X 25 X 1 77 = 0.3, constant

iteration 5000 5000 5000 5000 20000 20000 60000 60000

sum of squares of error (£) 0.12085 0.11861 0.12319 0.12896 0.07837 0.05622 0.02547 0.02731

The effect of a convergence algorithm: Previous experience (Hwu et al, 1994) shows that if the learning factor 77 is kept constant, the convergence rate becomes very slow when E is small, and that convergence can be improved by reducing the value of learning factor. Because 77 is used to control the updatmg step of the weighting factors, w,y, during searching for the minimum o^E, it is helpful for the network to find a steeper path to reach the minimum when the updating step is reduced. Hence, two convergence algorithms are designed to make the neural network adjust the learning factor automatically during training. The first one (Algorithm I) is based on the comparison of the convergence rate to a limiting criterion. When the limit, such as 5x10'^, is reached the learning factor is reduced by 20%. In the second algorithm (Algorithm II) the learning factor is reduced by 20% after each 2000 iterations. The iterations are stopped when either 77 reaches a specific value (for example, 0.005) or when E reaches the convergence criterion. Using the second algorithm with an initial value of 77 as 0.3, the learning factor reaches 0.005 after 20000 iterations and the sum of the squares of the error is 0.07837. With the first algorithm and the same initial value of 77, E becomes 0.05622 after 20000 iterations. The final E value fiirther reduces to 0.02547 after 60000 iterations, as shown in Table 9.3. Compared with the algorithm with a constant learning factor during training. Algorithm I can save at least one sixth of the learning time without losing accuracy. Performance Test: The drawback of the back propagation learning algorithm is that the global minimum o^E cannot be formally shown to have been reached during learning (White, 1989) . To overcome this problem, the performance test after training is essential. Two groups of data are then prepared to test the performance of the trained neural network. The flow KNOWLEDGE BASED MODELING

304 stresses in the first group are picked from six of the training flow curves, the conditions being shown in Table 9.4, at internal strains between the sampling strains which were used for training. The total number of the data is 223. The average errors of the testing data and the standard deviations of each curve are listed in Table 9.4. Except for one point with a high relative error over 10% which belongs to the strain of 0.06 and steel C, the relative errors of the rest of the data are within acceptable range.

Table 9.4 Testing data and results steel temperature strain rate (°c) A 950 0.1 B 1000 0.5 1050 C 1.0 D 900 0.01 E 1100 1.0 F 900 0.001

average of error (%) -0.0166 0.2116 0.4831 0.1627 4.5766 2.5746

standard deviation of error (%) 0.5060 2.4735 2.6450 3.3529 3.5771 1.5435

Table 9.5 Testing results of the second group of data condition average of error (%) STD* of error (%) A -14.657 2.9651 Bl 6.610 1.5086 B2 -20.487 7.2799 C 1.412 6.5351 D 10.352 2.7652 E 8.611 4.7036 Fl -0.534 1.4757 F2 -6.935 1.8835 STD : Standard Deviation

The second group of testing data included the flow stresses of the conditions marked with the symbol O in Table 9.2. These experimental conditions, which were not included in the training pairs, were adopted to test the performance of interpolation and extrapolation of the well-trained neural network. The average values of errors and standard deviations of each condition are shown in Table 9.5. Observe that the neural network can interpolate for the flow stresses successfully if the mformation in the training pairs is sufficient, such as Bl, C, Fl and F2, as shown in Table 9.2. For steel B, 10 conditions were used for training and condition Bl is in the center of the matrix. For steel C, eight conditions were used for training. Although MATHEMAHCAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

305 condition C is on the edge of the matrix, the neural network is able to interpolate for the stresses reasonably well with the exception of one point. The situations of conditions Fl and F2 are similar to the condition C. For steel D, seven conditions were used in the training and condition D is in the center of the matrix. The predicted results are not as good as for condition Bl, however, because of the deficit of mformation at the neighboring comer. For steel E, again seven conditions were used for training. Note that condition E is on the bottom edge, at the lowest temperature of 900°C. The situation of condition E is similar to that of F2, but the predicted results are worse because of the same reason as condition D: insufficient amount of information exists in the neighborhood. For steel A, only four conditions are involved in the training and the predicted resuhs are quite poor with relative errors between -10% and -20%. The condition B2 is used to test the performance when extrapolation is required and the predicted results are not impressive. A well-trained neural network can interpolate flow stresses quite successfully regardless where the condition is located in the center or on the edge of the condition matrix, provided the information is sufficient. The ability of extrapolation of the neural network, however, is not as good as for interpolation. Adaptability: Adaptability is one of the major considerations for a model to be used for online control. In this section, the adaptability of the neural network will be examined in two ways. In the first try, only the data of condition B2 are used. The testing data of condition B2 are added into the training pairs and the adaptive training starts by using the same weighting matrix Wij, obtainedfi-omthe training and the convergence Algorithm I with an initial value of 77 of 0.3. The resuhs are shown in Table 9.6. After 1000 iterations in adaptive learning, the average value of errors is reduced fi-om -20.48% to -6.08% and the standard deviation is decreased by about 1%. The relative errors, with two exceptions, are within ±10%. The average of errors and standard deviation of the original training data change only slightly. Thus, it is concluded that adaptive learning does not affect the predictive ability of the network when new data are added into the training pairs. Further, when the adaptive learning proceeds, the errors of prediction are reduced even more. Table 9.6 Adaptive training results of condition B2 after training condiafter adapting tion iteration =1000 average STD STD average training data B2

%

%

%

%

0.05703 -20.486

3.0754 7.2799

0.2676 -6.0800

3.5449 6.2396

after iteration average

% 0.1983 -4.4796

adapting =3000 STD

after iteration average

adapting =5000 STD

%

%

%

3.3780 6.4721

0.1693 -3.736

3.3227 6.7284

In the second trial the five conditions. A, B2, D, E, and Fl, are added and the network is adapted simuhaneously. The initial conditions of adaptive training are the same as in the first try. The adaptive training resuhs are shown in Table 9.7. A notable improvement in accuracy is obtainedfi-omthe first 1000 iterations for conditions A and B2. For condition Fl, the accuracy improvement is limited, indicating that for the cases that are already highly accurate, the accuracy will not be lost when new data are added. Although the number of adapted new data KNOWLEDGE BASED MODEUNG

306 has been increased from one to five conditions in the second try, the average of errors and standard deviation of the original training data change little. The adaptive training results of conditions B2 in these two tries are practically identical, see Tables 9.6 and 9.7. The neural network can handle several groups of data during adaptive training simultaneously without losing accuracy. Table 9.7 Testing and adaptive training results after condiafter training iteration tion average STD* average % % % training 0.05703 3.0754 0.09321 -14.657 2.9651 -3.236 A 6.610 1.5086 Bl -20.487 7.2799 -5.990 B2 1.412 6.5351 C 10.352 2.7652 D 7.012 8.611 4.7036 2.308 E -0.534 1.4757 -0.600 Fl -6.935 1.8835 F2

adapting =1000 STD % 3.9078 2.5456

after iteration average % 0.05435 -1.199

adapting =3000 STD % 3.5302 2.1252

after iteration average % 0.05030 -0.202

adapting =20000 STD % 3.3219 2.3017

5.9861

-4.102

5.0261

-3.044

3.9384

3.2641 3.3371 1.8248

4.157 1.190 -0.648

2.7395 2.6948 1.7485

3.478 1.064 -0.479

2.1865 2.4776 1.6204

The conditions Bl, C and F2 are used for testing the performance of the adapted neural network after 3000 iterations. Comparison with the testing results after training, show no improvement in accuracy for condition C is found because there is no extra condition added into training pairs during adaptive learning. The relative errors of condition F2 decrease by about 2% after adaptive training, because the condition Fl has been included in the training pairs to make the information more complete. When the comer condition B2 is added into the training pairs, the errors of predictions for condition Bl are reduced by about 4%, showing that the comer conditions in the matrix are necessary in contributing to the accuracy of the network. It is concluded that the information included in the training pairs is very important and will affect the predictive capability of the neural network. Effect of input data: In order to emphasize the importance of input data, new training pairs are selected. Flow stresses are randomly selected from the flow curves of the six steels and used as new training pairs - a total of 489 groups - and the remaining data -158 groups are used for testing. The new training pairs are referred to as Pattern 11 while the previous training pairs are called Pattem I. The training procedure and conditions are the same with both pattems. The training results after 60000 iterations show that the average of errors and standard deviation are slightly higher with Pattem 11 than with Pattem I. Compared with the testing results of Pattem I, the accuracy of predictions has been significantly improved. It is observed that although the number of the training data is less, the selection of input data for training the neural network not only influences the accuracy of prediction, but also avoids the possibility of extrapolation. Comparison of the accuracy of the neural network with that of statistical methods: In order to compare the predictive abilities of some of the models reviewed above with those of MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPER TIES OF HOT ROLLED PRODUCTS

307 the neural network approach, the flow stresses of steel B are also modeled by Wang's model (Wang and Lenard, 1991), the Voce Equation (Beynon and Sellars, 1992) and Jonas' model (Laasraoui and Jonas, 1991c). The outputs from the neural network have the highest accuracy. 9.6.5 Application of neural networks in the prediction of roll force in hot rolling Three neural networks are utilized to predict the roll force during hot rolling. The first network is used to model the average flow stress of an extra-low carbon steel in the austenite, ferrite and the two-phase regions, obtained from experimental results. A finite-element program is used to calculate the average temperature in the roll bite and a database of the temperatures is thus generated. The second neural network is employed to handle this database. The third network is used for adaptive learning while predicting the roll separating forces. Combining these networks with Alexander's roll force model, the roll separating forces are predicted with good accuracy. In what follows, the hot compression and the hot rolling experiments are described first. The construction of the three neural networks is then given, followed by a discussion of the capabilities of the combined model. Equipment: The rolling tests were conducted on a two-high, Stanat experimental rolling mill. The roll diameters were 150 mm. The rolls were hardened to Re = 60. The surface roughness was 0.8 jxm. The temperatures were monitored by type K, shielded thermocouples, embedded in the slabs. The compression tests were conducted on a computer-controlled servohydraulic testing system. Material: The chemical composition of the steel is given in Table 9.8. Table 9.8 Chemical composition of the steel, by weight% C Si Mn P S 0.003 0.3 0.27 0.056 0.005

Fe balance

Procedure: Slabs with different initial thickness were rolled at different temperatures in the austenite and ferrite regions in a single pass. Slabs were also rolled in a single pass in the twophase region with different amounts of ferrite, obtained by controlling the delay times afl:er the start of the phase transformation and determined, assuming that an Avrami-type relation can be used to model the austenite-to-ferrite transformation. Two slabs were rolled in three passes, in the austenite, two-phase and ferrite regions, respectively. Compression tests: 8 mm diameter, 12 mm high cylindrical samples were compressed at various rates of strain to a total strain of unity. True stress-true strain curves were obtained. The tests were conducted on a computer-controlled servohydraulic testing system, in a radiant fiimace. Friction was reduced by applying a glass powder-alcohol emulsion to the recessed ends of the compression samples. Hot rolling tests: 270 mm slabs were hot rolled into 25 mm thick plates on an industrial rolling mill and then annealed at 1200°C for 4 hours. Specimens, 40 mm in width, 70 mm in length and different thickness were machined from the annealed plate. Two K-type (chromelalumel) shielded thermocouples, with an outside diameter of 1.54 mm and 0.26 mm KNOWLEDGE BASED MODELING

308

^

thermocouple wires, were embedded in the centre and 2 mm beneath the surface, respectively, of specimens with thickness 12 mm and 15 mm. One thermocouple was embedded centrally in the specimens with thickness under 12 mm. Specimens were reheated to 1100°C, held for 10 minutes, cooled in air and rolled at specific temperatures in a single pass. The flow stress network: The average flow stress of the extra-low carbon steel, in the austenite, two-phase and ferrite regions, at different temperatures, strains and strain rates were calculated from the true stress-strain curves (Hwu et al., 1994). A two-hidden-layer neural network, with 12 nodes in each layer, has been developed to model the average flow stress. In addition to the three process variables of the temperature, strain and strain rate, a phase index has also been introduced as the fourth input variable in order to distinguish among the flow curves in the various regimes. The only output variable is the average flow stress. 348 groups of data, randomly selected from a total of 482 groups of data, are utilized to train the network. After 60000 iterations, the sum of the sQuare of the errors between the outputs from neural network and targets reached 0.0105 and the training process is ended. The relative errors of most training pairs are within ± 5%. The rest of the experimental data is used for testing the performance of this network. The results show that the average error reaches -0.066%, the standard deviation is 2.551%, and even the maximum error is less than ±10%. This welltrained flow stress network is then used to predict the average flow stress in subsequent calculations of the roll separating forces. The temperature of the strip in the roll bite is one of the important process parameters and determines the magnitude of the flow stress of the rolled strip during the pass. As mentioned above, in order to accurately predict the flow stress by the flow stress network, the exact temperature in the roll bite, rather than the initial temperature of strip, is necessary. The average temperature in the roll bite is determined by several process variables, such as the heat transfer coefficient in the contact area, the initial temperature of the strip, the flow stress, roll speed, initial thickness and reduction. The finite-element program, Elroll, developed by Pietrzyk and Lenard, (1991), is used to calculate the average temperature in the roll bite. The parameters used in the computations, covering the experimental rolling conditions, are listed in Table II. During the calculations, the roll temperature and roll diameter are kept constant, at 25°C and 150 mm, respectively. A database, having a total of 1944 conditions, is thus built. Table 9.9 Conditions for the calculation of the average temperature variables values heat transfer coefficient ( W W K ) 4000, 6000, 8000 750, 800, 850 initial temperature (°C) 900, 950, 1050 average flow stress (MPa) 30, 80, 130, 180 roll speed (rpm) 5, 15, 50 initial thickness (mm) 6, 10, 15 reduction (%) 10, 30, 50 A single hidden layer neural network with 25 nodes in the hidden layer is used to handle this database. The six variables, listed in Table 9.9, are selected for inputs and the only output is the MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

309 average temperature. Three quarters of the data are arbitrarily selected to train this network and the rest are used for testing. The training procedure is finished after 60000 iterations and satisfactory results have been obtained. 9,6.5.1

Rolling Tests

The slabs were rolled in both the austenite and the ferrite regions in a single pass. The rolling conditions are listed in Table 9.10. The temperature changes in the roll bite are simulated by the finite-element model, described in Chapter 5. When the heat transfer coefficient is 7000 W/m^K, the simulation results show that the measured temperature is close to the calculated values. Table 9.10 Rolling Conditions No h red. (mm) (%) 1 15.0 44 2

12.0

14

3

12.0

31

4

12.0

46

5

9.72

16

6

8.10

30

7

6.2

16

8 9

9.20 12.0

15 24

temperature (°C) 950, 928, 907, 865, 850, 798 1043,950,928,911,875, 852, 798 1026, 962, 925, 910, 907, 869, 859, 802 1021, 946, 930, 915, 878, 860, 851, 795 1018, 947, 926, 868, 848, 800 1031,954,930,870,835, 782 1035, 940, 921, 860, 845, 785 947, 928, 809, 871 960, 956, 952, 950

Roll Force Predictions: Alexander's force model (Ford and Alexander, 1964) is one of the simplest and has been widely used for calculating the roll separating forces in hot strip mills. According to the model the roll force is expressed as: (9.11)

F = k„Qpld^

where F is roll separating force, k^ is the mean constrained flow stress in the roll bite for plane strain conditions. Id is the contact length, w is the width and Qp is a geometric factor which can be expressed as: Qp

(9.12)

=0.25(;r+Q) KNOWLEDGE BASED MODELING

310 in which Q, the average aspect ratio, is the ratio of the contact length and the average of the entry and exit thickness of the rolled piece. The product k„,Qp is the deformation resistance, which is equivalent to the total energy per unit volume, needed to roll the piece. Because k^ is the plastic energy per unit volume of the rolled material, the term Qp is considered to be the multiplier, expressing the redundant work. Hence Qp is always greater than or equal to unity and it is strongly affected by the geometry of the roll bite and the interface conditions between the rolls and rolled piece. When the mill conditions, such as the lubricant, roughness of work roll and scale change, Qp will also change. This term, in fact, reflects the dynamic conditions of the rolling mill. The relative errors of the predicted roll forces by Alexander's model for some of the conditions in Table 9.10 are not satisfactory. In order to improve the accuracy of the predictions, changing or adapting the geometric multiplier, Qp, to the current conditions is necessary. The new values of Qp are calculated in an inverse fashion by equating the roll separating forces, computed by Eq. (9.11), and the measured forces for all of the above conditions and thus, a database is built. The average values of the adapted Qp, for conditions from one to seven, deviate significantly from the values predicted by Alexander's model. The linear relationship between Qp and the aspect ratio of contact area still exits for constant reduction or constant initial thickness. Ideally, if the interface condition is kept constant, the adapted Qp should vary within a very narrow range for a specific geometric condition, even though the temperature changes. But the adapted Qp varies randomly with temperature for conditions seven and four, and varies within a narrow range for conditions one and two. A neural network, similar to the flow stress network, is selected to model the adapted Qp. Not only does the adapted Qp have a strong relationship with the aspect ratio, entry thickness and reduction, it also depends on the initial temperature. Hence, these four variables are chosen as the inputs and the only output is adapted Qp. A total of 52 groups of data, from conditions one to seven, are utilized to train this network. After 80000 iterations, the training procedure is stopped. The results show that the errors of most of the data are within ±2% and the maximum error is within ±6%. This welltrained network, called the adaptive network, is then used for adaptive learning of Q^,. Two groups of data are used to validate the performance of this adaptive network. The first group of data is condition eight, having four experimental points, with the entry thickness equal to 9.2 mm, and the reduction equal to 15%. The flow chart for predicting roll force is shown in Fig. 9.10. First, the initial flow stress of rolled material before rolling is predicted by the flow stress network. This is followed by calculating the average temperature in the roll bite using the temperature network. The exact average flow stress in the roll bite is then re-calculated by the flow stress network and the value of Q^ is predicted by the adaptive network. Finally, the roll force is calculated by Eq. (9.11). The relative errors of these four testing data are shown in Table 9.11. Compared to the resuhs predicted directly by Alexander's model, the accuracy of the predicted roll force by this adaptive network is significantly improved. Four experiments, having the same entry thickness, reduction and temperature, are included in the second group of testing data, as shown in Table 9.12. The predicted resuh of the first data by the adaptive network is worse than the predicted result by the Alexander model, as shown in the table. Thus the new adapted Qp of this testing data is calculated from the MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

m^ measured force and is then put into the training pairs to train the adaptive network to acquire this new information. When the adaptive learning is finished, the adaptive network is used for predicting roll force of the second testing data. Satisfactory results cannot be obtained for the second set of data until 40000 iterations during adaptive learning are finished, as shown in Table 9.12. The same procedure is repeated for predicting the roll force in the third and fourth set of experiments. It is observed that during the adaptive learning, not only is the required number of iterations in adaptive learning reduced, but the accuracy is also improved. The prediction for the first slab is not impressive because the information in the training pairs is not sufficient. But the neural network can improve the accuracy very quickly after an appropriate adaptive learning process. From this example, the adaptive network reveals its powerful ability in adaptation.

Inputs: strain, strain rate, tennperature, chemical composition

(Flowstress network

)

[Temperature network]

[Adaptive network

[Alexander's model ] ( Roll force}

Figure 9.10 The flow chart of the adaptive roll force model

Table 9.11 Relative errors of the first group of testing data No temp. relative error relative error (%) (%) (adaptive network) (Alexander's model) 11.63 1 947 0.99 2 956 17.60 4.33 9.58 3 871 5.86 4.60 4 809 4.97

KNOWLEDGE BASED MODELING

312 Table 9.12 Relative error of the second group of testing data relative No temp. iterations of relative error (%) error (%) adaptive (adaptive learning (Alexander's network) model) 0 1 952 10.84 15.12 2 950 1000 6.94 10.70 2 950 5000 6.94 10.10 2 950 4.00 40000 6.94 3 960 1000 2.00 8.86 500 4 956 0.51 7.23

9.6.6 Using neural networks to predict parameters in hot working of aluminum alloys The ability of an artificial neural network model, using a back propagation learning algorithm, to predict the flow stress, roll force and roll torque obtained during hot compression and rolling of aluminum alloys, is studied. The well-trained neural network models are shown to provide fast, accurate and consistent results, making them superior to other predictive techniques. Back propagation neural netivorks: The multi-layered feedforward back-propagation algorithm is central to much work on modeling and classification by neural networks. This technique is currently one of the most often used supervised learning algorithms. Supervised learning implies that a good set of data or pattern associations is needed to train the network. Input-output pairs are presented to the network, and weights are adjusted to minimize the error between network output and actual value. The knowledge that a neural network possesses is stored in these weights. The back propagation model, presented in Figures 9.5 and 9.6, has three layers of neurons: an input layer, a hidden layer and an output layer. The backpropagation training algorithm is an iterative gradient algorithm, designed to minimize the mean square error between the predicted output and the desired output. It requires continuously differentiable non-linearities. The algorithm of training a back-propagation network is summarized as follows. initialize weights and threshold value: Set all weights and threshold to small random values. Present input and desired output; compute the output of each node in the hidden layer; compute the output of each node in the output layer; compute the output layer error between the target and the observed output; compute the hidden layer error, adjust the weights and thresholds in the output layer; adjust the weights and thresholds in the hidden layer; and repeat the above steps on all pattern pairs until the output layer error is within the specified tolerance for each pattern and for each neuron. MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

313 Deformation resistance of the material: Cylindrical samples of the Al 1100-H14 alloy of 20 mm diameter and 30 mm height have been used to determine the metal's resistance to deformation. The specimens have been machined from plates with the longitudinal direction parallel to the rolling direction. The flat ends of each specimen were machined to a depth of 0.1-0.2 mm to retain the lubricant, boron nitride. A type K thermocouple in an INCONEL shield, with outside diameter of 1.54 mm and 0.26 mm thermocouple wires, was embedded centrally in each specimen. The chemical composition of the material is given in Table 9.13. The compression tests were carried out on a servohydraulic testing system, at different true, constant strain rates and temperatures, as presented in Table 9.14. Temperatures were set to 400, 450, and 500°C and the strain rate ranged from 0.97 to 11.53 s'\

Table 9.13 The chemical composition of the material (weight %) Mn Si Zn Cu 0.05 1.00 0.1 0.05

Table 9.14 Experimental matrix used in flow stress evaluation tests temperature-O-- strain rate "=> 5.04 0.97 2.97 400 °C A3 Al A2 450 °C B3 Bl B2 500 T C3 CI C2

Al remainder

7.58 A4 B4 C4

11.53 A5 B5 C5

Hot rolling of an aluminum alloy: A commercially available 3000 aluminum was used in this phase of the study. Strips were prepared, in 6.12 - 6.16mm thickness, 50 - 52 mm width and 310 mm in length, cut along the direction of rolling of the plates. The chemical composition of the strips is given in Table 9.15. The surface roughness of the strips, as delivered, was in the order of 0.3 fjm. Each strip has a type K (chromel-alumel) thermocouple embedded to a depth of 15 mm in its tail end. Table 9.15 The chemical composition of the material (weight %) Mn Fe Zn Si Mg 1.00 0.63 0.016 .20 0.005

Cu 0.097

Al remainder

Lubricants with different emulsion concentration are used in the hot rolling tests. The base oil is one of two kinds of natural and semi-synthetic oil. The lubricants are made of four base oils, referred to as natural A, natural B, semi-synthetic A and semi-synthetic B. In the case of natural oil, the lubricants were prepared with water and 1% and 3% of the oils by volume, and for semi-synthetic oil they were prepared of 1% and 10% of the oils by volume. Lubricant A KNOWLEDGE BASED MODELING

314 was designed for low friction application and B for higher friction. Both lubricants are based on synthetic esters. The viscosities of the oils at 40 °C and 100 °C are given in the Table 9.16.

Table 9.16 lubricant Natural A Natural B Semi-synthetic A Semi-synthetic B

density (g/ml) 0.904 0.914 0.886 0.883

viscosity at 40°C (mm^/s) 37.9 76.3 28.5 29.6

viscosity at 100°C (mmVs) 5.5 8.6 5.5 5.7

A two-high mill having rolls of 250 mm diameter and 100 mm length, driven by a DC motor of 42 kW power is used. The rolls are hardened to Rc=52 and are ground circumferentially to a surface finish of Ra=0.18 jiim. Two industrial hot-air guns are used to heat the rolls to approximately 90°C. The roll force is measured by two load cells located under the bearing blocks of the lower roll. Two torque transducers in drive spindles measure the roll torque of the upper and lower rolls. The roll speed is measured by a digital shaft encoder, installed in the top drive spindle. In order to measure the forward slip, two photosensitive diodes are installed at exit a known distance apart, the signals of which allow the determination of the exit strip velocity, leading to the original definition of the forward slip. Two reductions of nominally 15 and 35% magnitude are used. The rolling speed is varied from a low of 20 rpm to 160 rpm, giving surface velocities of 0.26 and 2.1 m/s, the higher value of which is near commercial operating speeds. Flow stress prediction: The purpose of modeling is to develop an effective representation of material behavior at high temperatures. From each set of compression test data, 15 data points were picked in equal strain intervals of 0.05. The total number of training data sets for 13 conditions is 195. Conditions B2 and B4 were set aside as network generalization test data and the rest of the patterns were used in training the network. All the input and output values were normalized into the range [-0.9 to 0.9], to avoid premature saturation of the sigmoid function. The learning rate and momentum rate were both initially set to 0.5, then incremented to 0.9. The learning rate of 0.9, with a momentum rate of 0.7, resulted in the fastest convergence. The logistic sigmoid function with a constant steepness factor of 0.5 was chosen as the activation function. Both one hidden layer and two hidden layer networks were examined to investigate the effect of extra hidden layers. The error measure for network performance evaluation was the mean relative error of all the training data points. It was found that the two-hidden layer topology has no advantage over one hidden layer for equal numbers of total processing nodes. Increasing the number of hidden nodes to five increased the accuracy, however, further increase in the number of hidden nodes had no considerable benefit. Therefore, the five hidden nodes network was concluded to be the most efficient design. Training results after 5,000 iterations are shown in Table 9.17. MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

315 Table 9.17 Mean relative error in the prediction of the flow stress - training data temperature (°C) mean relative error (%) 400 2.584 450 2.218 500 1.799 The plot in Figure 9.11 shows the comparison of the experimental values of flow stress with those predicted by the use of the fully trained neural network, at the strain rate of 5.04 s'\ It is found that the predicted flow curves follow the experimental flow curves very closely. The mean relative error was calculated to be less than 1.9%. Similar accuracy was also found at other strain rates. This clearly indicates that the network was able to learn accurately the training data set. The main quality indicator of a neural network is its generalization ability, that is, its ability to predict accurately the output of unseen test data. The network was trained using data at 400°C and 500°C; and then tested and compared using the data at 450°C, shown in Figure 9.12. Good predictive ability was observed. The mean relative error was found to be 2.83%. This error is less than the errors that usually arise in flow stress measurements due to unavoidable variations in temperature, strain rate, and interfacial frictional resistance. 50[8=5.04 s-*]

(£=7.58 s-^3

Temperature

Temperature

ir^y-*-^^^^^^ 400 "C ^4> • » t - ^ 400 °C

450'C »-.^-» w w n-K 500 *C

Neural network Experiment 1

0.00

0.20

\

\

0.40 0.60 true strain

1

0.80

Figure 9.11 The predictions of the neural network

neural network experiment

10 H

1.00

0-f 0.00

-]

0.20

\

1

0.40 0.60 true strain

\

0.80

1.00

Figure 9.12 The predictions of the neural network on unseen data

Roll force prediction: A data base of the roll separating forces and roll torque during hot rolling of the 3000 aluminum alloy strips was developed, as described above. In the first step of neural network modeling, the effect of the lubricant type was excluded, and a network was trained to predict roll forces for a given lubricant. The model inputs were: reduction, roll KNOWLEDGE BASED MODELING

316

_ « _ «

speed, strip temperature, and emulsion concentration. The network was trained using 45 data points, and learned the force variation very closely. The predictive ability of the network was then tested and the results of these computations are shown using the solid diamonds. The network roll force prediction for Natural-A lubricant is shown in Figure 9.13. The maximum relative error during testing was approximately 10% with a mean relative error of 4.1%. Again, this level of error is satisfactory and smaller than errors that normally arise due to experimental variations and the accuracy of instrumentation. After successful development of the model for a given lubricant, the lubricant type was incorporated into the force prediction model. A configuration of one hidden layer with 8 nodes, five input and one output node with a learning rate of 0.7 and momentum rate of 0.7, was found to perform best. After 10000 iterations, the network converged to a solution and fiirther iterations had an insignificant effect on error reduction. The results are given in Figure 9.14 for two sets of reductions, 15% and 35%, at various rolling velocities. The emulsion contained 3%, by volume, of the Natural-A lubricant. The trained network predicted the roll force for 66% of conditions within 5% relative error band, 95% within 10% error band, and only three of the conditions were predicted with errors up to a maximum of 13%. As observed, the predictive ability of the network is satisfactory. The average error, obtained while testing the network, is 4.46% and 3.77%, respectively, for the two reductions, lower than is usually obtained usingthe more classical modeling methods.

1.60-

1.60 Nominal reduction = 32% Natural A lubricant 3% emulsion concentration! [ 4.1% average error ;

1.40 1.20-

1 1.00 10 .0

feVAo^V

I 1.00

r ^ o V %^** •V*^*

t .

0.80

£

0.60-

40 60 80 experiment number

0.80-1

i 0.40 H E 0.20

0.20 20

i*xwwSy

O Training data -15% reduction • Testing data -15% reduction

o Training data • Testing data

0.40

Natural A lubricant 3% emulsion concentration 4.46% average enror at 15% 3.77% average error at 35%

fi 1.40 "o i 1.20

100

Figure 9.13 Testing and training data for roll force

D Training data - 35% reduction • Testing data - 35% reduction -1 10

\ \ T" 20 30 40 experiment number

50

Figure 9.14 Testing and training data for 15% and 35% reductions

Roll torque prediction: A network with one hidden layer is applied in the prediction of roll torque, with five input parameters and one output parameter. The experimental matrix included 66 data points, of which 45 were used for network training and 21 were randomly selected to test the performance of the trained neural network. The learning rate and momentum rate were both initially set to 0.3 and then increased to 0.9. One hidden layer with 8 MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

317 nodes was found to perform best with a learning rate 0.7 and momentum rate of 0.7 for the optimal condition.. The training and testing results, after 15000 iterations, produced relative errors within a 10% error band, shown in Figure 9.15. The network also predicted well the test data, with errors less than 10%. The points with high relative error are the training data.

1.60 B

1.40 H

•o

1.20

o Training data • Testing data

1.00

"^^^VnT

0.80 H

0.60-1 0.40 0.20

Nominal reduction = 32% Natural A lubricant 3% emulsion concentration [ 4.18% average error, training —I 20

I I ^ 40 60 80 experiment number

100

Figure 9.15 Testing and training data for roll torque

Traditional models: Empirical models, one-dimensional models and finite-element based models have been used in a large number of publications to predict roll forces and torques, in addition to other variables of importance in the flat rolling process. The predictions of these have been shown to be reasonably accurate and consistent, providing the metal* s resistance to deformation, as well as the boundary and initial conditions have been described in an adequate manner. The material's flow strength may be measured with good accuracy and may be modeled using well established constitutive models. The problems encountered concern the boundary conditions; specifically the coefficients of friction and heat transfer at the roll/workpiece contact surface, which are notoriously difficult to measure. Using inappropriate magnitudes inevitably results in poor predictions. Thus, this is one of the major advantages of the neural network approach: neither of these parameters must be known to produce accurate modeling. The small price to pay for the accuracy is the lack of an empirical relation which may be used in thefixture,however, as long as a computer and the software are available, that relation is no longer necessary. 9.6.7

Comparison to statistical methods

A limited amount of comparison of the predictions a neural networks and statistical methods has been performed and the results are shown in Figure 9.16. The work was done to KNOWLEDGE BASED MODELING

318

model the constitutive behavior of a microalloyed steel, subjected to compression at high temperatures (875°C) and at constant, true rates of strain (0.01, 0.1 and 1 s'\ respectively) . The flow curves are shown in the figure. A large number of tests were conducted and a back propagation neural network was trained to predict the stress-strain curves. As well, non-linear regression analysis was performed to model the metals' behavior. The comparison shows that the predictions of the neural network are somewhat better than that of the statistical analysis. The mean difference of the predictions of the neural network was 2.17%, while the statistical approach gave a mean difference of 2.99%.

250 200 e

150 100

0

0.2 0.4 0.6 0.8 true strain

1

Figure 9.16 A comparison of the predictions of the neural network, non-linear regression analysis and the experiments.

MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPER TIES OF HOT ROLLED PRODUCTS

319

Chapter 10\ Conclusions The objectives of this publication were to discuss, in a fairly comprehensive manner, the problems associated with the production oi designer steels, defined as those demanded by the customer to fit exacting specifications. These may include the predetermined chemical composition, the yield strength, the ductility or formability, the texture, the ferrite, pearlite or bainite structure with a certain grain size and the precipitates. Geometrical accuracy and consistency and superior surface quality are also required of the products. The possibilities of satisfying the customer completely at the present time were discussed in terms of the technological difficulties the receipt of such an order would pose. The first part of the study, leading up to this book, involved the collection of opinions of engineers of several steel companies, regarding the treatment the demands of a customer would receive. The majority of responses indicated that the order would be matched to the company's product list and the steel whose attributes are closest to the one requested would be offered. There was one exception, in which the respondent commented: "of course we could satisfy the customer, exactly. However, the costs of the development of the new steel, the design of the processing schedule, the testing sequence to prove that the request has been matched, etc. would have to covered." The conclusion was that at the present time, engineers are capable of developing new alloys and processes but the associated costs are prohibitive. The information presented in the book has been designed to indicate the reasons for the expenses and to aid in the process to overcome the difficulties and to reduce the expenses. The importance of the steel industry to a. nation's economic well-being is discussed in Chapter 1. The steel industry has been around for quite some time and it is well known. Its contribution to the gross national product and to the gross domestic product has been well documented and acknowledged. The basic ideas of the rolling process have not changed since its inception, several hundred years ago. The level of technology and the demands it is to satisfy have changed in very significant terms, though. High technology has been introduced in hot strip mills and cold mills. Use of the most advanced data acquisition systems, predictiveadaptive computer control software and material testing techniques are now available and, at least in modem mills, are in constant use. The results have been spectacular in several ways. Productivity has increased and the number of person-hours to produce and roll a ton of steel steadily decreased. The associated human cost is also notable as the total number of steel workers has shrunk. In light of continuously increasing competition further improvements are still needed, however, and these indicate that even more innovations and changes are going to be introduced. These should include improved process control, based on better understanding of the phenomena involved in the rolling process. The phenomena contributing to the quality of the product are discussed next. These include the conditions at the interface between the work roll and the rolled metal, the location where the transfer of mechanical and thermal energy takes place, in addition to the material's

320 resistance to deformation. The first of these is treated in Chapter 2 where tribology is defined to include fiiction, lubrication, heat transfer and their effect on the process and the product: the wear of work rolls and the defects of the surface of the strips. Friction is discussed first. The methods used to determine the magnitude of the coeflBcient, either its average value or its variation through the roll gap, are critically discussed and the lack of applicability of bench tests to the rolling process is mentioned. The parameters and variables, involved with the process and with the material, that define and affect it are listed. The effects of some of these on the coefficient of fiiction are given, including the effect of the load, relative velocity, temperature and the roughness. In general, the coefficient of fiiction increases when the surface roughness increases. It is reduced when the relative velocity is increased. The effects of the temperature and the load are not as straightforward as the interaction of several variables are to be considered. When steel strips are rolled, increasing the reduction and hence, the load, results in decreasing coefficients of fiiction. When soft aluminum is rolled, increasing loads cause increasing fiictional resistance. The effect of the temperature also depends on more than one variable as the thickness of the scale and its chemical composition contribute to the magnitude of fiictional forces. Considering carbon and alloy steels, increasing temperatures appear to reduce fiiction. When aluminum is hot rolled with no lubricants, fiiction increases with temperature. Introducing lubricants brings in additional variables, including the viscosity, the viscositypressure and the viscosity-temperature parameters, and the oil film thickness. The following may then be concluded: M. decreases when the relative velocity increases because: • more oil is drawn in the contact zone; and • less time is available to form adhesive bonds. [i is affected by the interfacial pressure because: • higher pressures increase the viscosity; • higher viscosity decreases ji; • higher pressures decrease (i in the boundary and mixed regimes but increase it under hydrodynamic conditions; and • the effect of increasing number of bonds is counteracted by the effect of pressure. Softer metals create more bonds so the opposite effect may be observed; the interaction of the rate of change of the two mechanisms - rate of increase of viscosity and the rate of increase of the bonds - will influence whether \i increases or decreases with pressure. |Li is affected bv the direction as well as the magnitude of the roughness: • roughness direction around the roll: oil is not distributed and no effect is observed; • roughness direction along the roll: oil is distributed better and the changes depend on the nature of the lubrication regime; and • best direction: random roughness direction, obtained possibly by surface EDM. The magnitude of the coefficient of friction, during dry rolling of soft aluminum is in the range of 0.15 - 0.3. During dry rolling of steel somewhat higher magnitudes are observed. When lubricants, either in solid form or as emulsions, are introduced, the magnitudes decrease THE MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

321 and the coefficient varies from a low of 0.03 to a high of 0.2. At hot rolling temperatures, the coefficient of friction varies between 0.15 and 0.45, lower than suggested by earlier studies. The heat transfer coefficient was examined next. Published values of the coefficients were reviewed and results obtained in a laboratory setting were presented. Since the heat transfer coefficient affects surface phenomena as well, it was expected to be dependent on the same process and material parameters as was friction, including the reduction, rubbing velocity, temperature, real contact area and the presence of scales and/or lubricants. The heat transfer coefficient at the roll/strip interface was much more dependent, than was friction, on the conditions of the surface, however. Some specific recommendations for the magnitude of the heat transfer coefficient, to be used in modeling the rolling process, may be made at this point. When cold rolling aluminum in a laboratory, using a small experimental mill of 150 - 250 mm diameter, a magnitude of 20000 to 30000 W W K appears to be appropriate. During cold rolling of steel similar numbers may be used. When hot rolling of carbon steel is to be modeled, using rolls of 150 - 175 mm diameter, the best magnitude of the heat transfer coefficient is 7000 - 8000 W W K . In hot strip mills, the corresponding numbers vary from 50000 to 90000 W W K . AS for friction, the presence or absence of a layer of scale and its mechanical and thermal attributes will have a significant effect on the numbers. Roll cooling, whether using water or oil-in-water emulsions, will also contribute to surface heat transfer. As with friction, the coefficient of heat transfer may be determined using the inverse method or it may be inferred from temperature measurements. When the inverse method is used, measured temperature data are to be matched with calculations and the coefficient that gives the best match is chosen. When measurements of the temperature changes of the strip and the roll are available, the coefficient may be calculated directly. Both approaches require care and both are error-prone. The error bands associated with the results of either technique should be studied and should be taken into account. In an ideal situation, empirical relations for the two coefficients, in terms of the significant parameters, would be helpful. The relation, due to Wankhede and Samarasekera (1997), is one of those available, in which the coefficient of heat transfer is related to the roll pressure. New equations should be developed that include the effect of the other significant parameters on the coefficients as well. The most important parameters are the load, the temperature and the relative velocity of the contacting surfaces. At high temperatures, the layer of the scales should be included in the formulation. The equations may be obtained using statistical methods, or a neural network may be trained for prediction. Roll wear is caused by contact at the roll/strip interface and is inevitable. Understanding the causes would help in reducing the rate of wear of the rolls, and would lead to cost savings. An empirical relation, given by Roberts (1983), appears to estimate roll wear quite well. The surfaces of worn rolls indicate that wear is not uniform. Its dependence on the pressure, temperature, friction and transfer of heat is cleariy observable. The rate of wear and the need for roll changes could be predicted if the relations, referred to above, were available. The case studies in Chapter 2 concentrate on the effect of lubricants on the rolling process. Aluminum and steel strips, at low and high temperatures are considered. Dry and lubricated passes are examined. As expected, lubricants reduce the loads on the mill, quite considerably, by 20-30%. The use of lauryl and stearyl alcohols and lauric and stearic acids in rolling of aluminum strips is examined. Alcohols appear to be more useful than acids as boundary CONCLUSIONS

322 additives. The chapter closes with a description of the philosophy used in the formulation of the mechanical and thermal boundary conditions of the flat rolling process. The other parameter whose accurate determination and mathematical representation will affect the predictive abilities of the models of the rolling process is the metal's resistance to deformation, discussed in Chapter 3. There are two objectives in requiring this information. The first is to study the constitutive behavior of the metal. The shape of the true stress-true strain curve can be used to infer the metallurgical phenomena occurring during the process. The relative contributions of the hardening and restoration mechanisms are observable on the curves and, of course, can be substantiated by metallography. The second, and more relevant in the present context, is to use the information, gathered in a systematic set of tests, to model the rolling process. The methods for the determination of the metals' flow strength and the difiicuhies encountered are presented first. The available experiments include the tension, compression and torsion tests. The relative advantages and disadvantages are presented and the conclusion drawn suggests that the test chosen should try to reproduce the state of stress, strain and strain rate, existing in the actual process. For rolling, this points to the use of plane strain compression, in which the stress and strain distributions are close to that in rolling, provided the geometry of the sample and the compression platens are selected with care: the shape factors of the compression and the rolling processes should be similar. When single passes are to be studied, straining is not excessive and the compression process can simulate rolling adequately. When the complete strip rolling process, including the roughing and the finishing stages, is to be simulated the large strains obtainable in the torsion test make it the preferred route to follow, in spite of the problems associated with the dependence of the distribution of the field variables on the radius of the sample. Most of the difficulties associated with the tests concern the measurement of temperature. Thermocouples, embedded in the sample are useful but they disturb the heat flow and introduce stress concentration. Optical pyrometers measure surface temperature only. Both instruments respond fast to changes but the response is not instantaneous. It is inevitable that one will have to accept temperature data that is close to the actual values but which carries some experimental scatter, the magnitude of which will determine the observed temperature sensitivity of the material's resistance to deformation. The rate of strain is to be kept constant during the testing process. This is fairly simple when torsion is the technique. When compression of plane or axially symmetrical samples is used, computer control of the process is necessary. After the stress-strain curves have been determined, their behavior is traditionally represented by a mathematical model. At low temperatures, the flow strength is usually treated as dependent on the strain only, and equations that represent the strain hardening curve are not too difficult to find. At high temperatures, there are a large number of available relations. The parameters that affect the strength now include the strain, the rate of strain and the temperature as well and in several instances, the chemical composition and metallurgical parameters are also involved. The most successful relation appears to be the hyperbolic sine form, attributed to Arrhenius, a Swedish chemist. The Arrhenius equation includes the rate of strain, the peak strength, the temperature and the activation energy for deformation as the parameters, in addition to several THE MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

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material constants. These constants have often been expressed in terms of the strain, thus bringing in another variable. A simple empirical relation that describes the resistance to deformation of carbon steels is due to Shida. This equation includes the strain, the rate of strain, temperature and the carbon content and has been used successfully in modeling the hot rolling process. When the carbon content is replaced by the carbon equivalent, the equation is also usable for alloy steels. The case studies of Chapter 3 include a presentation of the true stress-true strain curves of a low niobium steel, indicating the metallurgical phenomena. A description of a method to determine the activation energy of deformation from experimental data follows. A few specific set of constants, collected from the technical literature for use in the hyperbolic sine equation, is also given. The information concerning the boundary conditions of the rolling process, given in Chapter 2, and the initial conditions, given in Chapter 3, are put to use by the mathematical models, presented in Chapters 4 and 5. One-dimensional modeling, applicable when the ratio of the roll diameter to the thickness of the rolled strip is large, is the topic in Chapter 4. In this chapter, the possible objectives of modeling the rolling process are given. These include the design of the rolling mills and/or the rolling schedule. An empirical model, giving the roll separating force as a function of the geometry of the roll gap, friction and the metal's average flow strength in the pass, is described first. The traditional models are presented next, beginning with the Orowan model, dating back to 1943. That model was and continues to be quite successful when roll separating forces and roll torques are to be determined. Its success in describing the rolling process, of course, is dependent on the proper use of the informationfromthe previous two chapters. Using Orowan's model without computers was time consuming and several variations have since been developed by simplifying Orowan's equations. The simplifications carry a price tag in some loss of accuracy and consistency. One of the models in widespread use is Sims' relations for hot rolling, based on the assumption of the existence of sticking friction in the roll gap. The second, developed for cold rolling, is due to Bland and Ford, also based on the Orowan model and also obtained by using some simplifications. All three models have drawbacks. The most significant of these is that all are based on the fiiction hill idea, and this makes their predictions highly unrealistic when large rolls are used to roll thin, hard metals. The models simply do not produce usable results. Still using the friction hill, but introducing somefiirtherrefinements is the model, developed by Rouchoudhury and Lenard. The deformation of the roll and the elastic entry and exit regions are analyzed by using the theory of elasticity in the method. A further refinement, that avoids the use of the friction hill, is also shown in Chapter 4. The presentation of the models is followed by an examination of the sensitivity of the predictions to several parameters. As friction and the reduction increase so do the roll force and the torque. Increasing roll radius and entry thickness also cause larger mill loads. This information may be taken into consideration when the mills are designed. The predictive ability of some of these models is given in the last section. Data, obtained from cold and hot rolling tests are used. In each of the calculations, the coefficient of friction is chosen such that the calculated and measured roll force should agree closely. By choosing the coefficient carefully, all models are able to predict the roll force well. The technique that avoids CONCLUSIONS

324 the friction hill and employs the "shooting problem" approach is the only one that successfully calculates the roll force, the torque and the forward slip. Chapter 4 closes with the argument that the ability of any model to predict significant parameters should not be tested in a one-to-one situation, by measuring and calculating a few parameters. Instead, one should use statistical methods and calculate the average and the st2mdard deviation of the differences of the predictions and the measurements to verify the accuracy and the consistency of the predictive abilities. As well, the mathematical rigor of all components, including the field equations, boundary conditions and the metals' resistance to deformation, should be at the same level. Thefinite-elementmethod is discussed in Chapter 5. After a few historical notes, the solid state incremental approach and the flow formulation are mentioned as the two approaches usually followed. The flow formulation is used in the present book. The principles of the rigidplastic and the elastic-plastic formulations are detailed next and the advantages and disadvantages are given. Also, the instances when one or the other should be applied are presented. The distributions of the velocities, strains, strain rates and temperatures are predicted to be similar by the two techniques. The surface stress distributions are not comparable and when those are required, use of the elastic-plastic model and an elastic-plastic stress-strain relation for the material model are recommended. Several constitutive models, for use in the FE method are mentioned, including the NortonHoff law, the Sellars-Tegart law and the Prandtl-Reuss relations. The equations that describe the transfer of heat at the contact surfaces and the boundary conditions are also shown, including the thermal properties of several materials. The importance of including in the computations the temperature dependence of the thermal properties is emphasized. Three applications are described in the case studies. The first concerns hot, flat rolling of steel. The ability of the method is demonstrated by showing the computed distributions of the effective strain, effective strain rate, effective shear strain rate, the average stress and the temperature. The passage of a steel strip through a hot strip mill, accountmg for the events at the roughing mill, the descaler, thefinishingmill, and laminar water cooling, follows. The next topic is asymmetrical rolling. Two models are validated. The first is the elasticplastic non-steady state model of Pillinger and the second is therigid-plasticmodel, described in Chapter 5. Cold rolling of aluminum plates and hot rolling of steel are studied. The dimensional changes at the exit of plates, rolled using various work roll diameter ratios are calculated. The effects of different coefficients of friction at the contact with the top and with the bottom roll and different roll diameter ratios are given. The last of the case studies deals with the closure of porosities during rolling of continuously cast slabs. The effect of the entrapped gases on closure is studied. The calculations are in good quantitative agreement with published data. The major part of the effort to create steels to match exacting mechanical and metallurgical specifications is dependent on the ability to plan the draft schedule for a particular mill, such that the aims are realized. This may be accomplished by off'-line modeling of the process, making use of relations, proposed to track the metallurgical phenomena as the rolling process continues. Chapter 6 deals with these events in presenting the state-of-the-art, describing the mathematical models for the evolution of the microstructure and the development of the attributes of the cooled product. These descriptions are possible by carefiil analysis of the critical temperatures: the precipitation start and stop temperature, the recrystallization start and THE MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

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stop temperature and the transformation start and stop temperature and the effects process and material parameters exert on them. The microstructure of the steel may change statically, dynamically or, when the changes that are initiated during loading are followed by the removal of the loads, metadynamically. The equations available to track these changes are detailed in the chapter. The parameters and variables are identified, including the history of thermal events, the initial austenite grain size, the potential for precipitation and recrystallization. These equations are empirical in nature. They contain coefficients and exponents and their forms are quite arbitrary. Most of them make use of the exponential nature of phenomena that are activated by thermal energy. These coefficients and exponents are given in Chapter 6, in a large number of tables, for a variety of steels and conditions. A flow chart, designed to examine and track the evolution of the microstructure of a steel, alloyed with niobium, is offered. Thermal-mechanical treatment of the steels during hot rolling is discussed in terms of that flow chart. The model calculates whether full or partial recrystallization occurred, and whether precipitation affected the softening processes. After hot rolling, cooling, at some rate, follows. Transformation and precipitation during the cooling period are the two metallurgical mechanisms that determine the attributes of the product after it reaches room temperatures. The relations, provided to evaluate the sizes and the distributions of the ferrite, pearlite or bainite structures, and the yield and tensile strengths, are given. The model to trace the evolution of the structure is applied in several examples. Hot flat rolling of plates and strips are examined. The possibilities of optimizing the draft schedule are explored. Plate rolling is studied in the first of the case studies. The roughing and finishing stages are considered, and the calculated and measured roll forces and temperatures are compared. The sizes of the austenite grains, during the 12-pass process, are calculated. The predictions and the measurements compare well. The next example concerns hot strip rolling. The effect of the interpass time on the finish temperature, the size of the austenite and ferrite grains and on the yield strength after cooling is demonstrated. The application of mathematical modeling to three-dimensional problems is demonstrated in Chapter 7, dealing with shape rolling. Planning the details of this process, in spite of the large amount of scrutiny and study they received, is still considered an art, not science. The problems and the unknowns are daunting. They involve the ability of the hot metal to fill the grooves of the rolls and the attendant effects of the process and material parameters. Finiteelement analysis can reduce these difficulties. The problems that remain are connected with the boundary conditions, that is, the coefficients of friction and heat transfer at the contact surfaces, which in the case of shape rolling, are the basic unknowns. The question has been raised before: how exactly are these coefficients affected by the process parameters? The application of three-dimensional analysis to shape rolling is extremely time-consuming. The method, presented in Chapter 7, is designed to reduce the time needed while retaining the ability to provide the distributions of thefieldvariables in the deformation zone. The application of a fairly new approach, the evaluation of parameters by a form of inverse analysis, to the flat rolling process is described in Chapter 8. The method is used to determine material constants in constitutive behavior or in boundary conditions. It is also applicable to compute thermal properties. The focus of the chapter is the former topic: the evaluation of rheological parameters in modeling material behavior. After giving the general principles of the CONCLUSIONS

326 technique, some of the components are discussed. These include the minimization processes, search methods, scaling, convergence and constraints. The applications deal with hot forming of aluminum and steel and the determination of parameters in equations, that deal with the evolution of the microstructure. The method is shown to be powerful in describing the processes. Knowledge based modeling, using artificial intelligence, expert systems and neural networks, as applied to flat rolling of steels, is presented in Chapter 9. There is no unique method for knowledge based modeling of rolling processes, especially on the industry level, however, the knowledge based modeling is a promising approach. Expert systems and neural networks can be used very efficiently to cover the gaps and the unknowns in physically based mathematical modeling. The acquisition of knowledge is discussed. Data mining, as applied to the flat rolling process is presented. Self organizing maps (SOM) are given and their output is examined. The possibilities in finding the influence of one parameter on another, using the SOM, are also described. Steps in improving the product quality in a rolling mill, using as minimum a three-level computer control with data acquisition about each strip are given in Chapter 9. These are: • finding the relevant data influencing the required quality parameters using methods of data mining, for example, SOM; • setting up the conventional neural network models for model experiments; and • validation of the models and optimizing the model using fiizzy logic. Neural networks provide a powerful mechanism which is able to handle databases in an efficient manner. In the case studies, the back propagation learning algorithm is utilized to train feedforward neural networks to store and predict the flow stresses of six steels at elevated temperatures. The chemical compositions (C, Mn, Si and Nb) and three process variables (temperature, strain rate and strain) are selected as input variables and the only output is the strength. The accuracy of predictions and the adaptability of the neural networks are examined and illustrated by using different training pairs. Several conclusions are drawn, as follows: • • •

•

For fixed training pairs an optimum number of nodes in the hidden layer exists; When the carbon equivalent, Ccq, is selected as input variable to replace other two chemical compositions, C and Mn, the quality of training is lower; A convergence algorithm is designed to make the neural network adjust the learning factor automatically. In this algorithm, the learning factor is reduced by multiplying it by a constant ratio when the convergence rate is below a specific value. Compared with the algorithm using a constant learning factor, this approach can save one sixth of the learning time without losing accuracy; After 60000 iterations, this algorithm shows very good training results; the average of errors is 0.05703% and the standard deviation of errors is only 3.0754%;

The well-trained neural network can predict flow stresses at internal strains, which are in between the sampling strains used for training, for all conditions very successfully. The flow stresses ofconditions not included in the training pairs, are also used for testing. The neural network can interpolate the flow stresses very well for the conditions no matter where they are THE MA THEMA TICAL AND PHYSICAL SIMULA TION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

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327 located in the matrix, provided the information for interpolation is sufficient in the training pairs. But the capability of extrapolation of the well-trained neural network is not very good. A second neural network is trained by new training pairs which are selected randomly from the experimental data. The accuracy of prediction is significantly improved when it is tested by the same groups of testing data. It is concluded that the selection of input data for the training pairs is important in improving the predictive ability and is critical in avoiding the need for extrapolation. The second case study deals with a new method, involving three neural networks combined with Alexander's force model to predict the roll force during hot rolling. The first neural network is used to model the average flow stresses in the austenite and ferrite regions, a traditionally difficuU problem. The model is shown to have an accuracy better than ±10%. A second neural network is utilized to handle a database of the average temperatures in the roll bite, which is generated by the finite-element program, elroW, described in Chapter 5. The relative errors of the second network, for 99.5% of the data are within ±1.3%. The errors of predictions by Alexander's model are improved by adaptive calculations of the geometric multiplier (Qp) which is shown to depend on the process variables. A database of Qp is then built up from experimental data and third neural network, called adaptive network, is used to handle this database. In the proposed technique, the average temperature which determines the flow stress in roll bite is calculated by the temperature network, the flow stress is predicted by the flow stress network and the Qp term is predicted by the adaptive network. Finally, the roll force is calculated by Alexander's model. The testing results from the first group of testing data show that compared with the resuhs predicted directly by Alexander's model, the accuracy of predicted roll force by the adaptive network is significantly improved. The testing results from the second group of testing data show that although the prediction for the first slab is not impressive, the adaptive network can improve the accuracy for the second slab efficiently after appropriate adaptive learning. For the third and fourth slabs, not only are the required iterations reduced, but the accuracy is also improved. In the third study, the back propagation learning algorithm is used to predict the flow stress, roll force and the roll torque during the hot compression and rolling of two aluminum alloys. First, the ability of the network to reproduce the experimental results is tested and the ability of the model is found to be impressive. The network is then presented with data, not used in the training process and again, its predictive ability is shown to be good. The network is then trained to predict the roll separating forces during hot rolling of aluminum strips, lubricated with various emulsions. The predictions, when tested against unseen data, are within ± 10%. The same conclusions are reached when the predicted and measured roll torques are compared. The major competitor of the neural network approach to predictions is the use of non-linear regression analysis. The predictions of the network are compared to those of a statistical approach and the neural network method appears to be superior. The neural network based model clearly indicates that it is able to learn the training data set and accurately predict the output of unseen test data. Well-trained neural network models provide fast, accurate and consistent results, making them superior to all other techniques. A general conclusion, when either mathematical or physical modeling of the rolling process is considered and the aim is to satisfy the demands of customers, may be given here. It is

CONCLUSIONS

328 possible to produce what the customer wants, exactly. The methods, while there is room for improvement, exist now. The process would be expensive but definitely possible. There is need for further studies, of course. A data base, involving a large number of materials should be compiled. Using careful testing, the constants and coefficients in the relations for constitutive and metallurgical behavior should be developed. Systematic testing to determine the exact dependence of the boundary conditions on process and material parameters should be conducted and another data base should be prepared. These data bases could then be used in the models now available to study hot or cold deformation processing. These steps are necessary to ascertain that the designer steels are available.

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346 Roberts, W.L., Sandberg, A., Siwecki, T. and Welefors, T., 1983, Prediction of Microstructure Development during Recrystallization Hot Rolling of Ti-V Steels, Proc. Conf HSLA Steels, Technology and Applications, Philadelphia, ASM, 67-84. Roberts, W. and Ahlblom, B., 1978, A Nucleation Criterion for Dynamic Recrystallization during Hot Working, Acat Metal, 26, 801-813. van Rooyen, G.T. and Backofen, W.A., 1960, A Study of Interface Friction in Plastic Compression, Int. J. Mech. Sci., 1-27. Roucoules, C, 1992, Dynamic and Metadynamic Recrystallization in HSLA Steels, PhD. Thesis, McGill University, Montreal. Roucoules, C, Hodgson, P.D., Yue, S. and Jonas, J.J., 1994, Softening and Microstructural Change Following the Dynamic Recrystallization of Austenite, Metall. Mater. Trans. A, 25A., 389-400. Roucoules, C. and Hodgson, P.D., 1995, Post-Dynamic Recrystallization after Multiple Peak Dynamic Recrystallization in C-Mn Steels, Mat. Sci. Techn., 11, 548-556. Rowe, G.W., 1977, Principles of Industrial Metalworking Processes, Edward Arnold, London. Roychoudhury, R. and Lenard, J.G., 1984, A Mathematical Model for Cold Rolling Experimental Substantiation, Proc. 1"* Int. Conf Techn. Plast., Tokyo, 1138-1145. Rumelhart, D.E., Widrow, B. and Lehr, M.A., 1994, Communications of the ACM, 87-92, 37. Sa, C-Y. and Wilson, W.R.D., 1994, Full Film Lubrication of Strip Rolling, ASME, J. Trib., 116,569-576. Sadok, L., Pietrzyk M,. and Packo, M., 1991, Strains in the Tube Sinking Process Evaluated by the Finite Element Method and Experimental Technique, steel research, 62, 256-260. Sakai, T., 1995, Dynamic Recrystallization Microstructures under Hot Working Conditions, J. Mat. Proc. Techn., 53, 349-361. Samarasekera, I., Jin, D.Q. and Brimacombe, J.K., 1997, Critical Challenges in the Prediction of Mechanical Properties of Hot-Rolled Steel Strips, Proc. THERMEC'97, (eds), Chandra, T. and Sakai, T., Wollongong, 57-66. Samuel, F.H., Yue, S., Jonas, J.J., Barnes, K.R., 1990, Effect of Dynamic Recrystallization on Microstructural Evolving during Strip Rolling, ISIJ Int., 30, 216-225. Sandstrom, R. and Lagneborg, R., 1975a, A Model for Hot Working Occurring by Recrystallization, Acta Metall., 23, 387-398. Sandstrom, R. and Lagneborg, R., 1975b, A Model for Static Recrystallization after Hot Deformation, Acta Metall., 23,481-488. Sartori, M.A. and Antsaklis, P.J., 1991, IEEE Transactions on Neural Networks, 2, 467-471. Sawada, Y., Foley, R.P., Thompson, S.W. and Krauss, G., 1994, Microstructure-Property Relationships in Plane-Carbon and V and V+Nb Microalloyed Medium Carbon Steels, Proc. 35"*^ MWSP Conf, Hamilton, 263-286. Schey, J.A., 1983, Tribology in Metalworking, ASM, Metals Park, Ohio. MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

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353

Author Index Ackert, R., 6 Adebanjo, R.O., 79 Alden, T.H., 207 Alexander, J.M., 90, 92, 94, 304, 306, 307, 308,309,318 Al-Salehi, F.A.K., 15 Alden, T.H., 209 Altan, T., 66, 88, 89 Anan, C , 163 Arvai, L., 294, 297 Ashby, M. F., 174 Atack, P.A., 57 Avitzur, B., 16, 63 Azushima, A., 26 Backofen, W.A., 15,267 Bai, D.Q., 160 Baker, T.N., 174, 199 Banerji, A., 15 Baragar, D.L., 63, 78 Barber, JR., 44 Batchelor, A.W., 44-46 Bertrand-Corsini, C. 235 Beynon, J.H., 78, 92,153, 161, 170,227, 229, 268, 304 Birks, N., 42 Bland, D.R., 85, 90, 92 Blazevic, D.T., 43 Booser, E.R., 23, 24, 26, 42, 54 Boratto, F., 160 Bowden, F.P., 12,13 Bresson, P., 293 Brimacombe, J.K., 148 Brooks, A.N., 126 Brown, S., 206, 207 Bryant, G.F., 29 Buchanan, B.G., 282 Buchmayr, B. 168 Bugini, A., 57 Burnet, M.E., 176

Cao, G.R, 136 Carroll, C.W., 258 Chakrabarty, J., 112 Chen, B.K., 28, 30, 32, 105, 108, 109 Chen, C.C, 161 Chenot,J.-L., 105,110 Cho, S., 295 Choquet,P., 153, 155, 156, 157, 160, 165, 177, 180, 271,274 Codina, R., 126, 127, 128,129 Coldren, A.P., 199 Coleman, T., 199 CoUinson, D.C., 183,268,271 Cser,L.,277,294,318 Cuong, N.D., 71 Czichos, H., 44 Dadras, P., 28 Davies, C.H.J., 207,212, 213, 214, 218 DePierre, V., 16, 63 Devadas, C. 78, 153 Devenpeck, M.L., 40 Dobrucki, W., 141 Dollar, M., 172 Donnay, B., 170 Dutta,B., 158, 160, 184, 187, 188, 190, 192, 193, 195, 196 Dyja, H., 136 Ekelund, S., 20, 21, 56 El-Kalay, A.K.E.H.A., 43 Estrin, Y., 206, 207, 108, 210, 211, 215, 219, 220 Evans, W., 15 Farkas, K., 294, 297 Fitzpatrick, J.J., 46 Ford, H., 85, 90, 92, 305 Frost, H.J., 219 Gavrus, A., 251, 258, 259, 260 Gelin, J.C, 257, 58, 259, 260, 261

354 Gibbs,R.K., 13, 156, 157, 164, 165, 167, 170, 172, 175, 176, 177, 184, 188, 229, 271, 274 Ginzburg, V.B., 285, 296 Gladman, T., 169, 171, 172, 202 Glowacki, M., 1, 111, 168, 235, 238, 243, 244 Greday, T., 199 Grosman, F., 70, 71 Grossterlinden, R., 168 Guang-Ying, L., 141 Guminski, R.D., 25 Gunasekera, J.S, 16 Hamauzu, S., 136 Harding, R.A., 29 Harmon, P., 282 Hatamura, Y., 15, 17, 18 Hatta, N , 78, 83 Hatvany, J., 277 Heinrich, J.C, 118, 128 Hishikawa, S.,8 Hlady, CO., 32 Hodgson, P.D., 72, 153, 154, 156, 157, 158, 160, 163, 164, 165, 166, 167, 170, 172, 175, 176, 177, 182, 184, 187, 188, 189, 190, 192, 193, 195, 196, 218, 223, 229, 238, 268, 271, 274 Hoff,NJ, 110,111 Hoffman, O., 18 Huetink, J., 235 Hughes, T.J.R, 126 Huisman, HJ., 235 Hum,B., 15, 16, 17,57,96, 141 Hwu, Y.J., 7, 79, 298, 300, 305 livarinen, J., 290 Irvine, I , 199 Jackson, J.E., Jr., 106 Jarl, M., 21 Johnson, C , 126 Jonas, J.J., 78, 79, 153, 156, 158, 160, 163, 218, 266, 267, 269, 299, 304

Kaftanoglu, B., 79 Kappen, B., 293 Karagiozis, A.N., 15, 28, 29, 30, 38, 49, 54, 141 Karhausen, K., 168 von Karman, T., 85, 90 Karhausen, K., 168 Kaski, S., 288 Kastner, P., 235 Kedzierski, Z., 177, 178, 180, 193, 195 Keife, H., 148 Kejian,H., 172, 175 Khoddam, S.,251 Kihara, J., 26 Kikuchi, F., 128 Kim,Y.J., 113 King, D., 280 King, LP., 105 Klafs, U., 28 Kleiber, M. 228, 259 Kliber, J., 72 Kobasa, D., 86 Kobayashi, S., 105, 106,109, 111, 130 Kocks, U.F., 161, 170, 206, 207, 208, 209, 210,211,266 Kohonen, T., 288, 289 Komori, K., 236 Kondo, S., 26 Kopp, R., 168 Korhonen, A.S., 168, 277, 291, 318 Kudo, H., 63 Kusiak, J., 30, 73, 212, 213, 251, 259, 260, 266, 267, 268 Kuziak, R, 67, 153, 156, 157, 158, 164, 166, 168, 169, 170, 171, 172, 175, 176, 202, 204, 215, 216, 238, 239, 241, 244, 266 Kwon, O., 156, 160 Laasraoui, A., 78, 79, 153, 156, 158, 160, 163,218,266,267,299,304 Larkiola, J., 291, 293, 297 Lang, G., 57 LeBon,A.B., 172 Lee,C.H., 105,106, 111 Lehnert, W., 71

MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

355 Lenard, J.G., 15, 16,18, 21, 28, 29, 30, 36, 38,40,47,48,49,50,51,53,54, 55, 56, 57, 70, 74, 78, 79, 82, 91, 93,94,95,98,102, 106, 110,112, 114,117,118,120,125,132,134, 136, 141, 223, 227, 228, 237, 268, 294, 304, 305 Li, a , 105 Lim,L.S., 15 Mahrenholtz, O., 105 Majta, J., 164, 172, 173, 174, 175, 192, 196, 194, 200, 202, 203 Male, A.T., 16, 39, 63 Malinowski, Z., 15, 29, 57, 112, 114, 117, 118,119,237,251 Malvem,L.E., 113, 117 Marcal, P.V., 105 Marquardt, D.W., 258, 259, 260 Matsui, K., 26 Matsuno, F., 42 McG. Tegart, W.J., 78, 105, 116 McKay, J.C., 2 McQueen, HJ., 161 Mecking, H., 206, 207, 208, 209, 210, 211, 215,219,220,266 Mead, R., 256 Meech, J.A., 283 Michell, J.H., 95 Misaka, Y., 78 Mori, K., 105, 235 Morrison, W.B., 199 Mortier, R.M., 22 Mrowec, S., 42 Munther, P., 55, 56, 102 Murata, K., 29, 249 Nagamatsu, A., 15 Nakano, T., 44 Nanba, S., 256, 158, 167 Nautiyal, P.C, 26 Ndumu, A.N., 293 Nedler, J.A, 256 von Neumann, J., 126 Norton, F.H., 110 Okon, R., 235 Orowan, E., 51, 52, 85, 90, 92-95,101,

174, 246, 250, 252 Osakada, K., 235 Park, B.H., 136 Pawelski, O., 15,40 Pietrzyk, M., 28, 29, 30, 31, 55, 70, 73, 74, 91,105,106,110, 111, 120, 125, 132, 134, 135, 136, 137, 146, 147, 168, 177, 180 182, 183, 185, 186, 187, 191, 193, 194, 195, 196, 207, 215, 216, 219, 220, 223, 227, 228, 235,251,264,265,273,305 Pillinger, L, 12, 136, 235 Portman, P., 293 Postlethwaite, I., 293 Powell, M.J.D., 256, 262 Rabinowicz, E., 12, 15, 36, 37, 39, 40, 42, 44 Rao, K.P., 77, 268 Rastegaev, M.V., 63 Rebelo, N., 130 Reid, J.v., 16 Richards, P., Richelsen, AB., 136 Richtmyer, R.D., 128 Rice,W.B., 15 Roache, PJ., 128 Roberts, CD., 11 Roberts, W.L., 16,18,20, 21, 43,49, 58, 141, 153, 156, 157, 158, 165, 167, 170, 177, 182, 211, 220, 271, 274 van Rooyen, G.T., Roucoules, C, 83, 157, 162, 163, 164 Roverso, D., 287 Rowe, G.W., 19, 20 Roychoudhury, R, 51, 93-95 Rumelhart, D.E., 292 Sadok,L., I l l Sakai, T., 153,162, 163, 165 Samarasekera, L, 30, 32, 148 Samuel, F.H., 177 Sandstrom, R, 206, 207, 211, 214, 215, 216, 219, 220 Sartori, M. A , 295

AUTHOR INDEX

356 Sawada, Y., 176 Schey, J.A., 1, 12, 15, 16, 26, 40, 88, 102, 103, 141 Schnur, D.S., 259 Schroeder,W., 151 Schunke, J,N., 43 Schwefel, H.P., 255 Sellars, CM., 28, 43, 92, 105, 110, 111, 152, 153, 156, 157, 158, 160, 161, 163, 164, 165, 166, 168, 170, 177, 178, 180, 182, 184, 187, 188, 190, 192, 193, 196, 206, 207, 222, 227, 229, 271, 273, 274, 275, 304 Semiatin, S.L., 28 Senuma, T., 79, 167 Shaesby, IS., 42 Shaw, L., 43 Shida, S., 55, 74, 132, 248 Siebel,E., 15, 16 Silvonen, A., 29 Sims, R.B., 43, 85, 90, 92, 93, 102, 103, 250 Simula, O., 288, 289, 318 Sims, R.B., 43, 85, 90, 92, 93, 102, 103, 248 Sinicyn, W.G., 136 Siwecki, T., 160 Sluzalec, A., 124, 125 Srikant, R., 287 Stachowiak, G.W., 44 Stevens, P.G., 29 Suehiro,M., 155, 156, 170 Suh, N.P., 46 Szmelter, J., 256

Wang, A, 15, 16, 63, 64, 78, 145, 148, 150,268,304 Wankhede, U., 30, 32 Washizu,K., 112, 117 Wierzbinski, S., 162 Wiklund, O., 293 White, H., 301 Wusatowski, Z., 20, 40, 41, 247 Xin, H., 236 Xu, W.L., 79 Yada, H., 79, 153,155, 156, 158, 163, 164, 165, 167, 177 Yamada, M., 236 Yang, G., 80 Yoneyama,Y., 15,17, 18 Zhang, S., 16, 18, 38, 47, 50, 53, 54 Zienkiewicz, O.C, 105, 110, 111, 118, 120, 121, 124, 127, 128 Zouhar, G., 108, 168 Zurek, A, 202, 203 Zyuzin, W.L, 73, 74

Tabor, 12 Tajima, M., 82 Tanaka, M., 148 Thompson, E.G., 112, 259, 260 Too, J.J.M., 136, 293 Touloukian, Y.S., 130 Tryba, V., 288 Tsoi, A.C., 79 Ultsch, A., 290

MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

357

Subject Index Activation energy For deformation, 156,166 For grain boundary mobility, 168 For recovery, 168 For recrystallization, 130, 155, 158, 159, 163, 221,275 For self diffusion, 162, 221, 22, 223 Additivity rule, 153, 186 Arrhenius law, 76, 210-212, 216, 273 Asymmetrical rolling, 9, 136-145 Austenite, 6, 76, 80, 151, 153, 154, 173, 187, 195, 196, 200, 204, 205, 241, 243 Austenite grain size, 76, 70, 85, 154, 156158, 161, 164-169, 179-184, 187, 193-197, 199, 210, 205, 208, 212, 214, 220, 226-227, 230-233, 235, 241, 243, 274-275, 277-278, Average Error, 303, 307, 315 Flow strength, 18, 27, 85, 88 Strain rate, 73, 84, 88, 147, 149, 240 Thickness (strip), 109 Stress, 133, 134, 144, 145, 148, 149, 239 Temperature, 11, 65, 306-309, 320 Avrami equation, (see Johnson-MehlAvrami-Kolmogorov equation) Back tension, 86, 87, 106, 246, 300 Backup roll, 8 Bainite, 151, 171 Bearing block, 16, 247, 313 Biharmonic equation, 52, 94 Bi-metallic rolling, 136 Body-centered cubic, 151 Boltzman constant, 210

Boundary additives, 26 Boundary condition, 9, 11, 27, 28, 29, 52, 58, 59, 61, 87, 91, 92, 96, 97, 98, 106, 117-121, 124-146, 130-132, 141,253,280,316 Boundary surface, 120 Burgers vector, 173, 211, 221 Cantilever, 15 Carbon content, 55, 73-76, 78, 132,151, 156, 178, 299 Carbon equivalent, 75, 168, 301-302, 319 Carbon steel, 3, 5, 19-20, 26, 28-30, 40-43, 49, 55-56, 58, 73-74, 78-80, 98, 101-102, 169, 246, 299-300 Cast iron, 11 Chemical reactivity, 12 Closed form Solution (formula), 85, 93 Equations, 152-153, 273 Integration, 92, 94 Relationship, 271 Coefficient of heat transfer, 29-30, 35 Cold rolling Mills, 6 Of aluminum Of flat products, 6 Comparison of Calculations and measurements Mathematical models Measurements and predictions Compression platens, 63 Compression test, 15, 16, 39, 40, 61-64, 73, 79-80, 82, 137, 153, 177-178, 196-198, 202, 212, 251-254, 264, 270, 272, 276, 298, 304, 310-311 Constant strain rate, 61, 197, 310 Compressive stress, 134 Computer control, 62, 80,286, 315, 318 Computer memory, 106, 244

358 Computer program, 71, 90, 92 Conductivity, 29, 120, 127, 130, 131, 132 Consistency of model, 47, 102, 193, 250, 279,317 Constitutive parameters, (see rheological parameters) Constitutive relation (equation), 70, 78, 88, 98, 105, 112, 146, 184, 256, 266267, 274 Constraints, 256, 265 Contact spot, 13 Contact surface, 16, 19, 24, 28-30, 33, 38, 46,239,252,316 Contact zone, 14, 15, 19, 22, 25, 28, 30, 38-39,61,99, 134,320 Content of Carbon, 55, 73-76, 78, 132, 151, 156, 178,299 Continuous casting, 3, 7, 145, 148-150, 296 Continuous drawing test, 180, 181, 183 Continuous rolling, 6, 7, 130, 147, 149 Control of temperature, 63-64 Controlled rolling, 151 Conventional microstructure evolution model (equation), 153, 168, 208, 225, 241, 273 Conventional models of dynamic recrystallization, 173 Convergence algorithm, 300, 302, 304, 319 Convergence criterion, 109, 264, 302 Convergence of model, 99, 109, 260 Convergence rate, 260, 302, 313, 319 Cooling rate, 151, 168, 169, 171-172, 175, 177-178, 199, 202, 242, 244, 248, 299 Cost function, (see objective function) Coulomb friction, 63 Crank-Nicholson scheme, 125 Critical strain for dynamic recrystallization, 162, 163, 165 Criterion of plastic flow, 91, 93, 112 Cyclic dynamic recrystallization, 162 Data acquisition system, 33, 279, 317

Deformation heating, 66, 253, 272 Dislocation density, 67, 69, 79, 130, 154, 162, 170, 173, 199,208, 212, 213,215-219,221225 Dislocation strengthening, 169, 170 Draft schedule, 67, 151, 226-229, 230 Elastic energy, 12 Elastic entry (compression) 52, 92-93 Elastic exit (recovery), 52, 92-93 Elastic-plastic formulation, 9, 105, 111119,136 Elastic-viscoplastic formulation, 105, 111119,136 Elastic region (zone) 48, 87, 95, 115 Energy per unit volume, 309, 218 Entry plane, 134 Equation of equilibrium, 48, 52, 90, 92-97 Error norm, 222, 253, 255, 273 Eulerianformulation. 111, 114 Eulerian mesh, 105, 117,118 Euler's formula (equation), 105, 112, 114, 118,121,125-126 Exit surface (plane), 205 Exit (final) thickness, 8, 84, 89, 92, 233234, 250, 299, 300, 309 Exit velocity, 27, 97-98, 230 Explicit method, 125 Extreme pressure additives, 26, 38 Extremum principle, 106 Face-centered-cubic, 151 Fatty acid, 26 Fatty alcohol, 26 Ferrite, 76, 80, 151, 168-170, 172, 173, 176, 187, 198, 200, 204, 205, 207, 241 Ferrite grain size, 87, 169, 170, 171, 174, 176, 198-206, 229-233, 241 Ferrous metal, 66, 89 Fibonacci method, 257 Field of Strain, 137-140, 144,146 Strain rate Stress, 113,117-119, 135, 145-146 Temperature, 27,121, 125,130,

MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

359 134, 152,235,244 Velocity, 109, 111, 113,117,118 Finishing train, 32, 42, 56, 62, 85, 135, 229, 233-234 Finite difference method, 128 Finite element mesh, 126, 146, 276 Flow curve, 65, 79-80, 221, 300, 302-307, 314 Flow strength, 9, 18-19, 27, 44, 63-64, 6667, 69-70, 75, 84, 88, 91, 250, 316 Force transducer, 16,247 Forward slip (measurement of), 11, 16-17, 21-22, 43, 47-48, 51-53, 55, 57, 9698, 101-102,313 Four-high mill, 13 Fourier series, 93-95 Four-node quadrilateral element, 107 Fractional softening, 193, 199, 208, 220, 223 Free surface, 146 Friction factor, 15, 40, 43, 57, 63-64, 110, 138, 239 Friction hill, 52, 85, 92, 95, 99,101 Friction losses, 101, 122 Friction Stress (force), 15, 26, 86-87, 106107, 110, 119-120, 171,239 Galerkin method, 123, 125-128 Gauss integration point, 109, 118 Gauss-Newton method, 260-261 Generalized plane strain method, 237-241, 246 Gibbs free energy, 210 Golden section method, 257 Gradient of temperature, 132,134,237 Grain (subgrain) boundary strengthening, 169, 170 Grain coarsening, 162 Grain refinement, 6, 162, 215, 218, 241 Grain size, 8, 66, 68, 70, 85, 134-135, 151, 153-157, 163-173, 176, 178-184, 186-187, 193-195, 198-199, 201204, 208, 212-213, 217-218, 220221, 226-227, 229-233, 235, 241, 243, 273-278, 281-283, 286-288, 296-297,299,317

Grain growth, 152, 154, 166, 167, 168, 186, 193, 205, 217, 220, 242 Gradient methods, 258, 260, 263, 264 Grooves (rolling) Box passes, 251 Diamond, 237, 238, 247, 249, 250, 251 Octagonal, 237 Oval, 237, 238, 250, 251 Round, 237, 250, 251 Square, 237, 247, 249, 250, 251 Ground steel roll, 20 Hall-Petch equation, 169, 170, 172, 174 Hardening, 63, 67-70, 79, 110, 112, 117118,137-138, 144,151,153-154, 161, 169, 173, 175, 207-208, 211214, 216-219, 222, 239, 266, 268269, 272, 285 Hardness, 12, 32, 36 Heat conduction, 12, 119, 239 Longitudinal, 239 Heat flux, 11, 27-28, 30, 34-35, 120, 132, 249 Heat gain, 59, 279 Heat generation Due to plastic work (deformation), 119,122 Heatloss, 30, 59, 119, 134,279 Heat treatment, 5 High temperature flow, 152 Homogeneous compression, 88, 93 Hooke's law, 95, 209 Hot strip mill, 6, 7, 11, 20-21, 32, 46, 56, 85, 130, 186, 229-230, 288, 298, 308,317 HSLA steel, 78, 158, 186, 283, 286, 299 Huber-Mises yield criterion, 91, 106, 112 Hydrodynamic lubrication, 22, 38, 44, 98 Impact transition temperature, 169 Implicit method, 125 Incompressibility constraint, 108 Infrared pyrometer, 241 Initial Conditions, 9, 28, 61, 85, 155, 279, 305,316

SUBJECTINDEX

360 Temperature, 33, 180, 182-185, 254, 307, 309 Instron testing machine, 197 Interfacial pressure (force), 25, 29-30, 3334,318 Interfacial shear stress, 11, 16, 87, 90, 9295, 248-249 Interlamellar spacing, 168, 171, 172, 204, 241 Interrupted (compression) test, 197 Inverse analysis (method, approach), 218, 212, 253-255, 269-272, 275, 276 Jaumann rate of stress, 112, 114 Johnson-Mehl-Avrami-Kolmogorow Equation, 79, 154, 159, 167 Lagrange multiplier, 106, 108, 239 Lagrangianformulation, 111, 114 Length of contact, 17, 27, 44, 84, 86-89, 92,250-251 Levenberg-Marquardt method, 260-262 Levy-Mises flow rule, 105, 106, 110, 239 Liquid phase, 151 Liquidus temperature, 151 Low carbon steel, 3, 19-20, 26, 30, 42-43, 49, 55-56, 58, 79, 98, 101, 169, 246, 268, 300, 306-307 Material parameters, 11-12, 22, 29, 32, 36, 39,89,136, 151,210,220,253254, 38 Material properties, 14, 40, 136, 207, 253, 279, 295, 297 Mean flow strength, 88,250 Mean free path of dislocations, 172, 212, 213,218 Mean yield strength, 185 Mechanical Properties, 12, 102, 151, 168-169, 171-174, 198, 204-205, 230-231, 233, 240-242, 244, 288, 295, 299 Medium carbon steel, 73 Metallurgical properties, 67 Microalloying elements, 6, 73, 75 Microstructure evolution, 134, 151-153,

155, 159, 161-162, 173, 179, 182, 184, 186, 189-190, 194-196,208, 225, 228-229, 231, 241, 243, 272273, 275 Mill frame, 85 MilHoad, 7, 9, 50, 54, 98-99, 247 Mill stretch, 300 Mineral oil, 24, 26 Model One-dimensional, 9, 18, 28, 48, 52, 56, 85, 90, 98, 100, 126-127, 188189,237,246,316 Monte-Carlo method, 258, 267, 271 Multidimensional analysis, 289 Multiple peak deformation, 70, 162, 269 Multistage test, 61, 199, 279 Natural lubricant (oil), 312 Neutral point (plane), 16, 48, 52, 87, 90, 92-93,96-99,101,109-110 Newton-Raphsontechnique, 108-109, 111 Niobium (bearing) steel, 135, 158-159, 173, 175, 190, 193-199, 204, 225, 227-228, 268-271 Nitrogen content, 175,195 Nodal temperatures, 121-124, 130 Nodal velocities, 105, 107-109, 111 Non-ferrous metals, 66, 89 Non-gradient methods, 257, 258, 260, 263, 264, 274 Non-Newtonian fluid, 105 Non-stationary heat transfer, 122, 137 Non-steady state model, 125, 136, 138, 141 Non-symmetric rolling, (see asymmetric rolling) Non-uniform strain distribution, 133, 207 Non-uniform temperature, 136, 145 No-recrystallization temperature, (see Recrystallization stop temperature) Normal pressure, 11, 13, 19, 33, 62-63, 249 Norton-Hoff law, 105, 110, 111 Numerical methods, 126, 230

MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

361 Objective cost function, 218, 254-261, 263-265, 267, 268, 272, 274, 275, 279 Oleic acid, 26 Oxide layer, 28, 43 One-dimensional model (problem, analysis), 9, 18, 28, 48, 52, 56, 85, 90, 98, 100, 126-127, 188-189, 237, 246, 316 Oscillatory solution, 216 Peak strain, 162, 270, 272 Peak stress, 65, 68, 72, 78-83, 162-163, 270-271 Pearlite, 151, 168, 169, 171, 172, 204, 241 Peclet number, 127-129 Penalty factor (coefficient method), 106, 111, 119,239 Penalty function, 260, 261, 265 Penetration hardness, 12 Petrov-Galerkin approach, 126 Phase diagram, 152 Phase transformation, 79, 241,306 Physical properties, 7, 12, 66 Pin transducer, 15-17,36, 57, 96 Plane strain, 25, 64, 88, 107, 119, 146, 185, 237-240, 246, 254-256, 266268, 309 Plane strain compression test, 25, 64, 146, 256, 266-267 Plastic flow, 49, 61, 69, 87, 91, 93, 110, 112-113,210,243 Plastic potential, 110-111 Plastic region (zone), 115 Plastic strain, 105, 112, 209-212, 215 Plastic work, 119-120, 122, 130, 243, 249 Poisson's ratio, 91, 98, 112, 117-118, 174 Porosity closure, 145-149 PowelPs method, 258, 264 Prandtl-Reuss law, 112 Precipitates 85, 151, 154, 158-160,170, 173-175, 186, 189, 193-197 Precipitation start temperature, 151 Precipitation strengthening, 169, 173-176 Predictive capability of model (equation), 90, 97, 100, 144, 150, 187, 201, 204, 225, 246, 304-305, 314-315

Projected contact length, 27, 84, 88-89, 92 Pure (commercially) aluminum, 26, 56-57, 96 Pure shear, 91 Radius (of curvature) of deformed roll, 18, 27, 44, 84, 86, 90-93 Rate of cooling, 151, 169, 171-172, 175176, 199, 202, 244, 248, 299 Heat generation, 122, 130 Real area of contact, 36 Recovery, 67, 68, 153, 211-213, 216, 217, 219, 221-224, 268 Static, 68, 153, 154, 164, 212, 222, 241, 242 Dynamic, 68, 69, 79, 80, 153, 161, 211,214,222 Recrystallization, 67, 85, 151-153, 155157, 159-167, 180, 182, 184, 186, 188-195, 198, 201-203, 212, 217, 220, 223, 224, 227, 230, 232, 233, 235, 240, 268, 279 Dynamic, 6, 68-70, 78-80, 83, 153, 161-165, 167, 199. 214, 215, 222, 240, 241, 243, 269, 271-273, 275, 279 Metadynamic, 68, 163, 163-166, 240, 241, 242, 273 Partial (incomplete), 154,159,161, 180, 184, 186 Static, 6, 68, 70, 153, 154, 156, 163-166, 200, 205, 222, 241, 242 Recrystallization between passes, 193 Recrystallization stop (no-recrystallization) temperature, 151,161,186 Recrystallized volume fraction, 154, 166, 195, 216, 218, 219, 242, 243, 273, 274, 275, 277 Refined model, 49 Regression analysis, 20, 35, 66, 78, 81-82 Relative velocity, 14, 19, 21-22, 25, 29, 36, 38-39,54,56,248-249,318 Resistance to deformation, 12, 14, 22, 30, 55, 61,66, 69-70, 73-75, 79-80, 8485, 88-89, 98, 106, 110, 144, 247, 252,312,316-317 Retained (accumulated) strain, 168, 169,

SUBJECTINDEX

362 190, 193, 195, 202, 240, 243 Reverse rolling, 125 Rheological parameters, 253-258, 266 Rigid-plastic approach, 9, 106-110, 113,144,145,147,239 Rigid-plastic material, 111 Ring upsetting (compression), 16, 39-40, 63, 79, 137, 146, 300 Roll angular velocity, 137, 232 Roll bending, 8, 288, 300 Roll deformation (flattening), 13, 48, 95 Roll diameter, 12, 29, 31, 44, 58, 85, 90, 132, 136-144, 225, 306-307 Rolling of Aluminum, 38, 57, 137, 161, 309 Aluminum slab, 14 Aluminum strip, 14, 26, 28, 36, 53, 137-140 Angles, 235 Billets, 3, 7, 125, 241 Channels, 235 Copper, 3 I-beams, 237, 244, 245 Ingots, 125 Low carbon steel, 43, 55 Rails, 237, 240, 244, 245 Rods (bars), 240-244, 246, 248 Slab (rectangular), 125 Steel, 20, 40, 47, 56, 70, 132-134, 141, 144, 148, 225, 240, 296, 318 Vanadium steel, 240 Rollmaterial, 31, 87,91,98 Roll pressure, 11, 16-17, 28-30, 32, 36, 48, 52, 53, 58, 87, 90-93, 95-97, 119, 141, 144 Roll radius, 17, 44, 56, 84, 86, 89-90, 93, 98-100, 132, 144, 148-150,250, 299 Roll temperature, 32, 55, 248, 307 Roll torque, 11, 13, 16-17, 21-22, 47-48, 52-55, 57, 85, 87, 91-93, 95-103, 228-229, 246-247, 249-250, 252, 299,311,313-316,321 Roll velocity (speed), 19, 30,47-48, 55-56, 89, 132, 136, 148, 250, 307, 313-

314 Roll wear, 2, 6, 11, 13, 44, 46-47, 57-58 Roll-workpiece interface, 131, 185 Rolling schedule (see draft schedule) Roll pass design, 237, 246 Roughing mill, 62, 86, 135, 281, 295, 298 Rubbing velocity (speed) 318 Runge-Kutta technique, 118 Scale, 7-8, 12, 20, 29, 32, 40-43, 55, 186, 248-249,291,309,318 Search direction, 257 Sellars-Tegartlaw, 105, 110, 111 Sensitivity, 230, 233, 235, 261, 262, 263 Servohydraulic testing system, 33, 62, 80, 306,312 Shape coefficient, 109, 267 Shape function, 107, 121, 123 Shear band, 162 Shear modulus, 12, 112, 173, 209, 216, 221 Shear strain, 133, 207, 208, 211, 238 Shear strain rate, 133, 207, 238 Simplex method, 259,267 Single peak deformation, 69, 70, 162, 269 Slab method (analysis), 93, 188 Sliding (slip) velocity, 15, 110, 120 Softening, 67, 78-79,151,153-154, 161162, 64-165, 193, 199, 207-208, 216-217, 220-223, 266 Solid phase, 151 Solid solution strengthening, 170 Solute drag effect, 158, 193, 194, 196 Specific heat, 12, 30, 65, 120, 130-131 Spindle, 15,85,99,246, 313 Split Hopkinson bar, 204 Stacking fault energy, 68, 161, 162 Stainless steel, 3, 32-33, 40, 186 Stainless steel envelope, 186 Standard deviation, 288, 298, 303-305, 307,319 Statistical analysis, 290 Steady-state flow, 79,118 Steady-state model (solution), 120, 125126, 136, 138, 141, 216 Steel roll, 2, 6, 11, 20-21, 30, 36, 47, 57,

MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS

363 70, 92, 141 Sticking friction, 43, 55, 57, 92, 102, 250 Stiffness matrix, 132, 262 Strain hardening (curve) 63, 79, 208, 239, 266, 268, 269, 272 Strain homogeneity, 66, 180 Strain rate, 6, 9, 30, 61-64, 66, 68-88, 105, 111,130,132-133, 141,144-145, 147, 149, 151-153, 156, 162 164165, 168, 179, 186, 198-199, 204214, 217, 222-225, 238-242, 249, 251, 266-269, 272-274, 279, 300, 301,303,307,312,314,319 Strain rate concentration, 133 Strain rate sensitivity, 110 Strengthening effect, 175 Stress-strain curve (relationship), 6, 9, 64, 67-73, 80-81, 84, 137, 199, 204, 206, 216, 220-224, 267-272, 281, 307 Stress tensor, 105, 112, 114, 118-119, 146, 263 Strip temperature, 21, 54, 314 Stripvelocity, 22, 311 Subgrain boundary misorientation, 174, 208 Subgrain size, 161, 218 Substantiation of model, 15, 168, 240 Surface Energy, 12 Hardness, 36 Roughness, 12, 22, 27-28, 33, 36, 42-43, 47, 137, 140, 280, 306 Temperature, 24, 28, 30, 32, 57, 120, 132, 134, 249 Synthetic ester, 26, 313 Synthetic lubricant, 47, 50 Temperature calculations Temperature difference, 28, 35, 249 Temperature measurement, 30, 65, 66 Temperature variation, 186 Tensile strength, 169, 176, 177, 204, 206, 231,244 Tension test, 61-62, 98, 117, 253 Texture strengthening, 169, 176 Time for 5% recrystallization, 159, 193, 197,

Thermal properties, 14, 32 130-131, 148, 230, 253 Thermocouple, 28-33, 39, 65-66, 179, 186188, 199, 246, 248-249, 306-307, 312 Three-dimensional analysis, 237 Time for 5% precipitation, 159, 192, 193, 195, 197, 226-228 Timeofcontact, 30, 33 Time-temperature profile, 28, 30-31, 135136, 179, 188 Tool pressure. 111 Tool steel, 2, 6, 11, 28, 30-31, 47, 73, 246 Torsion test, 62, 185, 253-255 Total reduction, 155,234 Transfertable, 7, 279 Two-dimensional analysis, 48, 85, 102, 106, 146, 238-239 Uniaxial compression test, 15-16, 29, 117, 204 Uniform deformation, 180, 237, 277 Uniform (non) strain distribution, 133 Uniform (non) temperature distribution, 132, 134, 136 Upsetting, 272, 274-277 Upwinding, 126, 128 Vanadium (bearing steel), 82, 158,175176, 240-242 Variational principle, 120 Velocity dependent friction, 109 Voce equation, 270 Water spray, 7 Wavelength, 298 Weighted residual method, 123 Weighting function, 129 Workhardening, 137-138, 151, 153-154, 161 Workhardening material, 137-138 Yield stress, 169-172, 175, 199-202, 204, 206,231,233,244,266 Young's modulus, 12, 239

SUBJECTINDEX

364 226, 227, 228 Time for 50% recrystallization, 155, 157, 165, 166, 224, 225 Time for 95% recrystallization, 159, 193, 197, 226-228

MATHEMATICAL AND PHYSICAL SIMULATION OF THE PROPERTIES OF HOT ROLLED PRODUCTS