Crystal Plasticity Finite Element Analysis of Texture Evolution during Rolling of fcc Polycrystalline Metal

Crystal Plasticity Finite Element Analysis of Texture Evolution

during Rolling of fcc Polycrystalline Metal

Kyung-Hwan Jung

, Dong-Kyu Kim

, Yong-Taek Im

and Yong-Shin Lee

1_{National Research Laboratory for Computer Aided Materials Processing, Department of Mechanical Engineering,}

KAIST 373-1 Guseong-dong, Yuseong-gu, Daejeon 305-701, South Korea

2_{School of Mechanical Engineering, Kookmin University, 861-1 Jungneung-dong, Sungbuk-gu, Seoul 136-702, South Korea}

Localized strain and crystallographic orientation distribution during rolling process have a significant effect on anisotropicflow behavior in sheet forming of aluminum alloy, resulting in local thinning. In this study, crystal plasticityfinite element method (CPFEM), which incorporates a crystal plasticity constitutive law into the three-dimensionalfinite element method, was used to investigate strain localization and textural evolution during theflat rolling process of the face-centered-cubic material. A rate-dependent polycrystalline theory based on the Taylor model was fully implemented into an in-house program, CAMProll3D. The through-thickness texture evolution depending on the degree of draught was predicted by using the developed CPFEM program and compared well with the experimental data available in the literature. The orientation distributions not only in the thickness direction but also in the width direction of theflat rolled sheet were investigated depending on the amount of reduction during the multi-passflat rolling in terms of polefigure, orientation distribution function andflow potential surface in the³-plane. Finally, the effect of friction condition between the rolls and the material on rotation about the transverse direction was found to be important to determine the texture evolution at the surface of the rolled sheet. [doi:10.2320/matertrans.M2012346]

(Received October 17, 2012; Accepted February 20, 2013; Published April 25, 2013)

Keywords: texture evolution, draught,flat rolling, crystal plasticity,finite element method

1. Introduction

Texture or preferred orientation is fundamental phenom-enon resulting from the microstructure evolution that takes place during various processes such as thin film fabrication and thermo-mechanical processing of materials. A strong texture developed in the material results in anisotropy in properties of the material. The crystal lattices of the individual grains for a polycrystalline metal produced in an initially isotropic state by appropriate heat treatment tend to reorient in preferred directions resulting in the strong crystallographic texture in the material as deformation continues. This might cause deformation behavior of the polycrystalline material to be highly anisotropic as well.1) That is, drawing formability of the sheet materials highly relies on Lankford value which is strongly related to the texture developed in the materials.2)

Especially, crystallographic texture develops during theflat rolling process. It is well known that textures are inhomoge-neous through the thickness of rolled plates. Many researches have been conducted to experimentally investigate the rolling deformation behavior of the face-centered-cubic aluminum.

Lee et al.3)_{reported that the formation of deformation band}

in the cold rolling depended on several parameters including orientation, grain-size and strain. Liu et al.4) _{described an}

analysis such that shear strain in the plane perpendicular to the transverse direction (TD) during rolling resulted in rotation of the crystal around the TD direction. Sekineet al.5) reported that the characteristic features of rolling texture development in FCC alloys are dependent on the level of stacking fault energy.

According to the studies of Truszkowski et al.6) _and

Mishinet al.,7)_{it was shown that the geometry of the roll gap}

influenced through-thickness homogeneity of rolling texture.

The geometry of the roll gap is frequently represented by “draught”. Here, draught means the linear reduction, which is generated by the compressive force, in the thickness direction and can be expressed byld/hm. The projected contact length

(ld) and mean thickness (hm= {h1+h2}/2) shown in Fig. 1,

have been used for characterizing the roll-gap geometry. Here,h1is the entrance thickness, andh2is the exit thickness

of the material. Especially,ffiffiffiffiffiffiffiffiffiffi ld can be simplified as ld¼

Rh p

, only when ¦h=h1¹h2 is smaller than 0.08R,

whereRis the radius of the roll.

Truszkowski et al.6) and Mishin et al.7) experimentally studied the textural changes in aluminum during cold flat rolling according to the draught levels. According to their studies, shear texture has been frequently observed at the surface of the sheet after rolling for large draughts (ld/

hm>5). Rolling with small draughts (ld/hm<0.5) resulted

in shear texture in the quarter-thickness layer of the sheet. Strong texture is developed as it undergoes multi-passflat rolling as well. Moreover, the rolling sequence might be a

R

h₁ h₂

α

d

l

2 2 1 2

2

2

− − −

=R R h h

ld

2 1 h h h

where Δ = −

h R ld≈ Δ

Fig. 1 Parameters characterizing roll-gap geometry of draught in theflat rolling.

+_{Corresponding author, E-mail: yslee@kookmin.ac.kr}

crucial factor for local texture formation in the material, affecting the proportions of the shear and rolling texture in different layers. When the material deforms during rolling, the stress or strain of each grain is, in general, different from the macroscopic stress or strain. The operating slip systems vary with the orientation of each grain. The compatibility conditions and force equilibrium among grains should be satisfied simultaneously. To simulate this kind of phenomenon, crystal plasticity finite element method (CPFEM) introduced by Peirce et al.8) is known to be effective by Wenk and Van Houtte.9)

For prediction of texture in the rolled plates, Chenet al.10)

reported that the rolling texture evolution of pure aluminum with initial texture could be reproduced by means of the Taylor-type model. For better texture prediction, there are a few attempts to link the crystal plasticity model with thefinite element (FE) analysis. Mathur and Dawson11) _{showed that}

crystal plasticity could allow a detailed analysis of through-thickness texture gradients in rolled aluminum alloys. Si

et al.12) reported that the crystal predominantly rotated

around the TD after rolling. However, the evolution of texture gradient along the width direction was rarely investigated since most researches carried out so far in rolling studies have been limited to the two-dimensional simulation in which the variation along the width direction of the sheet was not taken into account based on plane strain assumption in the plane perpendicular to the transverse direction of the sheet.

Recently, Jung et al.13) _{implemented a Taylor-type}

poly-crystalline constitutive model14) _{to the rigid-viscoplastic}

three-dimensional finite element program CAMProll3D developed by Kim and Im15) _{to simulate the effects of}

texture evolution and deformation heterogeneity during the equal channel angular processing and sequential upsetting of aluminum alloy with the decoupled and coupled approach to increase the computational efficiency.

In this study, the developed CPFEM model has been applied to investigate texture evolution during the cold rolling process of aluminum alloy AA1050-O. Firstly, fully-coupled simulations were carried out in order to investigate the effect of draught on the through-thickness materialflow and texture evolution in the rolled sheet. The simulation results were compared with the reported experimental works available in the literature.6,7)_{Secondly, the multi-pass rolling}

was simulated to investigate the effect of reductions on the lattice rotation not only in the thickness direction but also in the width direction of theflat rolled sheet. The polefigure and orientation distribution function are obtained and compared to the available experimental results reported by Sekine

et al.5) and Hansen and Mecking.16) Variations of flow

potential surface in the ³-plane were also determined and compared depending on the local deformation level during the process. Lastly, the effect of the friction between the workpiece and rolls on the texture evolution was inves-tigated.

2. CPFE Simulations

In the current study, three-dimensional CPFE simulations were carried out using the program fully-coupled with

CAMProll3D, an in-house Lagrangian FE program devel-oped based on the thermo-rigid-viscoplastic formulation with a constant shear friction model. A rate-dependent crystal plasticity theory8,11) _{was fully implemented into the}

FE program by Junget al.,13)_{as shown in Fig. 2. The Taylor}

model, in which all the crystals in the aggregate are assumed to undergo identical deformation was adopted in the present development. Thus, the deformation rate of each crystal in the polycrystalline aggregate was assumed to be the same as the macroscopic deformation rate. Since the detailed derivation is available in Jung et al.,13) it will be omitted here.

The finite element mesh used in simulations consisted of 900 brick elements, and each integration point in an element was considered to be a polycrystalline aggregate consisting of 200 grains to impose the computation efficiency similar to the work by Jung et al.13) _{The distribution of the initial}

hardness of all the slip systems for each grain before the rolling was assumed to be uniform. The initial orientation distributions of the grains were chosen to be random. The predicted grain orientations and the slip system hardness were stored and transferred over to the simulations of subsequent passes.

In this study, the commercially pure aluminum alloy (AA1050-O) was used for the numerical simulations. Figure 3 shows both experimentally measured and numeri-cally predicted stress­strain curves. The experimental flow stress curve in this figure was obtained by the simple compression test at a constant ram speed of 0.6 mm/min at room temperature. The numerical flow stress curve was predicted by the crystal plasticity constitutive law in the following equations:

2 (TD) 3 (ND)

1 (RD)

EDGE CENTER

SURFACE

C

C

Fig. 2 Schematic of theflat rolling simulation.

Strain,

ε

Flo

w stress, / MP

a

σ

_ £ð¡Þ_¼

_ £ð¡Þ

0

¸ð¡Þ

¸ð¡Þ R

¸ð¡Þ

¸ð¡Þ R 1 m1

ð1Þ

where£_ð₀¡Þis the reference shear strain rate for the plastic slip,

¸ð¡Þ_{the resolved shear stress,¸}ð¡Þ

R the reference shear stress on the¡-th slip system, and mthe rate sensitivity parameter.

The overall strain hardening characteristic of the material is reflected in the evolution of the crystal hardness¸R. In the

present formulation, an evolution law firstly proposed by Voce17)and modified by Kocks18)was used as follows:

_ ¸R¼H0

¸sð£_Þ ¸avg

¸sð£_Þ ¸0

_

£_{; £_ ¼}X

¡ j£_ð¡Þj;

¸sð£_Þ ¼¸s0 _ £

_ £s

A=® ð2Þ

where £_is the net shearing on all the slip systems, ¸s the

saturation value of the current strain hardened state of the slip system,®the elastic shear modulus, andH0,¸0,¸s0andAare

material parameters.

Material parameters used for the formulation of the evolution of critical resolved shear strength in eq. (2) were determined by comparing theflow stress curves between the compression experiment and the numerical data. Material parameters determined by trial and error are summarized in Table 1.

At the very incipient stage of low strain level in the stress­ strain curve, the obvious discrepancy between experimental and fitted data was observed due to the characteristics of the rate dependent constitutive law in eq. (1) in which the shearing rate on a slip system is related to its resolved shear stress based on the power law. However, such a perturbation within a narrow range of strain is less influential on the texture evolution because the strain level in the rolling is normally high enough to neglect the elastic behavior.

According to the experimental results available in Refs. 6, 7), it was found that the through-thickness homogeneity of the texture depended on the draught in the flat rolling. In order to examine validity of the computational accuracy of the developed rate-dependent polycrystalline finite element program, numerical simulations were carried out for the cases with two different levels of draught similar to the rolling conditions reported in the same references. Simulation results were then compared with the experimental data obtained by Truszkowski et al.6)_{and Mishinet al.}7)_for

large and small draughts, respectively. These two particular

draughts are important as they produce two distinctly different through-thickness textures depending on local deformation fields.

Then, three rolling passes with a nominal thickness reduction of 30, 28.6 and 30%, respectively, were simulated to achieve an overall reduction of 65%during the multi-pass flat rolling. The initial thickness and width of the entering sheet of AA1050-O was assumed to be 20 and 200 mm, respectively. The rolls were assumed to be rigid and its diameter and speed were 250 mm and 16 rpm, respectively. Due to the symmetry, one quarter of the problem domain was modeled in the present investigation. Constant shear friction factor of 0.8 obtained from the Ref. 15) was used in the simulation.

After finishing the CPFE analysis, a particular cross-section of the sheet was selected by checking the steady state of deformation. At such cross-sections, three elements were chosen whose Gauss integration points were close to the sheet center, quarter-thickness (intermediate) layer and sheet surface.

In order to represent texture evolution after the CPFE simulation, orientation distribution function (ODF) and pole figure were plotted since ODF plots can separate overlapped components in the pole figures, allowing for better comparison of the individual components at the selected locations. An ODF was obtained by fitting a Gaussian distribution with a half scatter width 5° using the EDAX TSL OIM Analysis v.5.31.

3. Results and Discussion

Figure 4 shows the comparisons of (111) stereographic polefigures between the CPFE predictions and experiments depending on the levels of draught as described in Refs. 6, 7). In Figs. 4(a) and 4(b), the textures for the large draught rolling are given compared to the textures for the small draught rolling in Figs. 4(c) and 4(d). To avoid visual confusion, plotting types of the polefigures for the prediction and experiment were matched each other in thisfigure. Every texture at the sheet center in this figure is the well-known plane strain compression texture composed of mainly Brass

f110gh112i and Copperf112gh111i ideal orientations. Even though the pole figures numerically predicted at the other positions did not perfectly coincide with the experimental results, the overall tendency was fairly good. Especially, shear texture was observed at the surface of the sheet after rolling with large draughts, as seen in Fig. 4(a), while rolling with small draughts resulted in shear texture in the quarter-thickness layer, as seen in Fig. 4(d). Although the Taylor model was quite successful in predicting the rotations of the individual grains, there are some discrepancies such as a stronger Copper orientation and a weaker S orientation in the prediction than the experimental results in the mid-thickness in Fig. 4(a). It seems to be due to the common problem of the Taylor-type averaging method, which tends to overpredict peak texture intensities and to shift the position of texture components, especially at large strains.19)

Figure 5 shows stereographic pole figures for initial, 30, 50 and 65% reduction in the center of the flat rolled sheet during three stages of theflat rolling. It is shown in thisfigure

Table 1 Material parameters of the Voce-Kocks model17,18) _{used in} simulations for AA1050-O.

Model parameter Value Unit

m 0.05

_

£0 1.0 s¹1

¸0 23.50 MPa

¸s0 55.80 MPa

H0 30.41 MPa

_

£s 5.0©1010 s¹1

A 0.129 GPa

(b)

(c)

(d)

(a)

max=3.057

max=4.342

max=6.064

max=9.347

Fig. 5 Stereographic polefigures for (a) the initial, (b) 30%reduction, (c) 50%reduction and (d) 65%reduction in the center of theflat rolled sheet. Contours: 0.69/1.0/1.45/2.11/3.07/4.47/6.5.

(a)

(b)

(c)

(d)

that texture evolution becomes very strong near the center of the workpiece as deformation progresses.

The ODF predicted from the CPFE analysis is plotted in Fig. 6 and compared with the main orientations which were generally developed after multi-passflat rolling. The textures at the center of the rolled sheet are shown in thisfigure using the¤2=constant (0­90° in steps of 5°) ODF sections. It was

found that the predicted texture at the center after the 65% reduction was characterized by the plane strain compression mode in Fig. 6(b).5)

The texture variations predicted after three passes through the thickness of the workpiece are shown in Figs. 7(a) and 7(b). Since considerable shear deformation occurred near the surface in addition to the compression required to decrease the thickness, rotation of the texture about the TD was observed as the material point moved to the surface. In result,

it was found that thef111gh112i,f111gh110iandf001gh110i

orientations increased in these figures. However, since there was no shear effect at the center, the texture near the center was very similar to the ideal case for the plane strain condition known as mainly Brass f011gh211i and Copper

f112gh111i orientation as seen in Fig. 7(c), similar to the work presented by Hansen and Mecking.16)

Variations of shear strain rate components along the thickness direction are illustrated in Fig. 8(a). In thisfigure, the magnitude of the shear strain rate component in the plane perpendicular to the TD, «D31«increased along the thickness

direction compared to other shear strain rate components, «D12«and«D23«in the plane perpendicular to the ND and the

RD, respectively. Contrary to the through-thickness texture gradient,«D12«increased along the width direction, as shown

in Fig. 8(b). As a result, the crystal predominantly rotated around the normal direction (ND), as shown in Figs. 7(d) and 7(e). Since the surface at the end of the workpiece in the width direction was not constrained, material could deform

(a)

(b)

90° 0°

Φ

90° 1

ϕ

60° 30°

60°

30° (4411)[11118]

(112)[111]

(110)[112]

(110)[001]

Dillamore Copper

ϕ2 =45°

Goss Brass

90° 0°

Φ

90° 1 ϕ

max=27.189

Fig. 6 (a)¤2=constant ODF sections of textures predicted after the 65% reduction and (b) the position of important FCC orientations at¤2=45°.5) Contours: 0.57/1.0/1.76/3.11/5.47/9.65/17.0.

CENTER

SURFACE EDGE

THICKNESS DIRECTION WIDTH DIRECTION

(b)

(c)

(d)

(a)

(e)

Fig. 7 Stereographic polefigures for the 65%reduction at (a) the surface and (b) quarter-thickness in the thickness direction, (c) the center, (d) quarter-width and (e) edge position in the width direction of theflat rolled sheet. Contours: 1.0/1.26/1.59/2.0/2.5/3.18/4.0.

-0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05

23

D

31

D

12

D

(a)

Shear strain rate,

D

/ s

-1

0 2 4 6 8 10 12

0 50 100 150 200 250 300 -0.14

-0.12 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 (b)

23

D

31

D

12

D

Width from the center, w/ mm

Shear strain rate,

D

/ s

-1

Thickness from the center, t/ mm

freely in the TD. Therefore, it was found that the elliptic shape of the (111) polefigure was transformed into the circle as we proceeded towards the edge from the center.

The local strain rate makes us easier understand the reason why the local textures develop differently. At all the locations such as the center, surface and edge, as shown in Fig. 2, the workpiece underwent the tension in the rolling direction, and the compression in the normal direction, resulting in the plane strain compression mode. Additional shear strain rate exists at the surface and edge positions, while there is no more shear strain rate component owing to the friction in the center of the workpiece. Due to these shear strain rate components, crystals rotate around the specific direction. Especially, the pole figure shape changes into the circle close to the well-known uni-axial compression texture since material deforms freely in the TD.

In the flat rolling experiment, in which the width of the workpiece is not wide enough to satisfy the plane strain condition, the lattice rotation could be influenced by local deformation near the free edge in the width direction. In this study, it was found that theflat rolling simulation using the CPFEM could simulate locally different pole figures in the width direction as shown in Figs. 7(d) and 7(e).

In order to emphasize the macroscopic effect on the deformation, textured aggregates calculated from the rolling simulation were reloaded to calculate yield surfaces. Figure 9(a) shows the comparison of flow potential surface evolution in the ³-plane at the center of the flat rolled sheet at 30, 50 and 65% reductions. Since the material initially had a random orientation, intermediate forms of Tresca and von Mises yield surfaces were shown in this figure. As deformation proceeds, it is observed that the surface keeps evolving and losing the initially symmetric shape. In addition, the hardening rate decreased drastically as pass increased, while the initial hardening rate in the 30% reduction was relatively high.

In addition,flow potential surface evolution in the³-plane at the center and the surface of theflat rolled sheet at the 65% reduction is also compared in Fig. 9(b). It was found in this figure that the flow potential surfaces were formulated differently according to the location, resulting in more distorted one at the surface compared to the center of the sheet.

In order to investigate the effect of the friction condition between the rolls and the material on texture evolution in the workpiece, an additional simulation was also tried with the

friction factor of 0.3. While no significant difference was observed at the center of the sheet, the smaller amount of shear strains was accumulated at the surface of the sheet since the nodes at the contact with the rolls were subject to the lower friction. Hence, the texture developed in the case of mf=0.3 showed a less rotation about the TD compared to

the case ofmf=0.8 as shown in Fig. 10.

4. Conclusions

In this study, the fully-coupled three-dimensional CPFE program was adopted to investigate textural inhomogeneity in both width and thickness directions of the cold rolled sheet for AA1050-O. Firstly, simulations considering the effect of draught on the texture evolution were carried out success-fully. It was reassured that draught played a major role in determining material flow in the roll gap and through-thickness texture evolution in the rolled sheet. The predicted deformation texture showed a good correlation with the reported experimental results available in references. Sec-ondly, the multi-passflat rolling was simulated to investigate texture distributions not only in the thickness direction but also in the width direction of the flat rolled sheet in terms of pole figure, orientation distribution function and flow potential surface in the³-plane. It was found that the crystals rotated around the TD in the thickness direction and rotated around the ND in the width direction due to shear strain rates ofD31andD12, respectively. In addition, the effect of friction

condition between the rolls and the material was found to be important to determine the texture evolution at the surface of the rolled sheet. Since the local texture evolution determined by the CPFEM could have a profound effect on subsequent material deformation in a secondary process, texture predictions made by the current approach could provide important information for modeling subsequent processes where the deformation induced anisotropy might be strong.

Initial random 30% CENTER 50% CENTER 65% CENTER

Initial random 65% CENTER 65% SURFACE

(a) (b)

Fig. 9 Comparisons offlow potential surface evolutions (a) at the center of theflat rolled sheet at the 30, 50 and 65%reductions and (b) at the center and the surface of theflat rolled sheet at the 65%reduction.

C

SURFACE

(b)

(a)

max=8.107

max=6.086

Acknowledgements

The authors wish to acknowledge the grant of POSCO without which this work would not have been possible. Prof. Yong-Shin Lee is grateful for the support from Kookmin University.

REFERENCES

1) T. Abe and Y. Ono:Met. Mater. Int.4(1998) 376­379.

2) H. Tanaka, Y. Nagai, Y. Oguri and H. Yoshida:Mater. Trans.48(2007) 2008­2013.

3) C. S. Lee, B. J. Duggan and R. E. Smallman:Acta Metall. Mater.41

(1993) 2265­2270.

4) Q. Liu, J. Wert and N. Hansen:Acta Mater.48(2000) 4267­4279.

5) K. Sekine and J. Wang: Mater. Trans.40(1999) 1­6.

6) W. Truszkowski, J. Krol and B. Major: Metall. Mater. Trans. A13

(1982) 665­669.

7) O. Mishin, B. Bay and D. Juul Jensen: Metall. Mater. Trans. A31

(2000) 1653­1662.

8) D. Peirce, R. J. Asaro and A. Needleman:Acta Metall.30(1982) 1087­ 1119.

9) H. R. Wenk and P. Van Houtte:Rep. Prog. Phys.67 (2004) 1367­ 1428.

10) Z. Chen, X. Zhang, C. Liu, H. Cai and F. Wang:Mater. Trans. 45

(2004) 2845­2850.

11) K. K. Mathur and P. R. Dawson:Int. J. Plasticity5(1989) 67­94.

12) L. Y. Si, C. Lu, N. N. Huynh, A. K. Tieu and X. H. Liu:J. Mater. Process. Technol.201(2008) 79­84.

13) K. H. Jung, D. K. Kim, Y. T. Im and Y. S. Lee:Int. J. Plasticity42

(2013) 120­140.

14) J. W. Hutchinson:Proc. R. Soc. Lond. A348(1976) 101­127.

15) S. Y. Kim and Y. T. Im:J. Mater. Process. Technol.127(2002) 57­63.

16) J. Hansen and H. Mecking:Proc. 4th ICOTOM meeting, ed. by G. J. Davies, I. L. Dillamore, R. C. Hudd and J. S. Kallend, (Metals Society, Cambridge, London, 1975) pp. 34­47.

17) E. Voce: Acta Metall.51(1955) 219­226.

18) U. F. Kocks: Constitutive Relations for Slip, Constitutive Equations in Plasticity, ed. by A. S. Argon, (The M.I.T. Press, Cambridge, Massachusetts, 1975) pp. 84­115.

19) Z. Zhao, S. Kuchnicki, R. Radovitzky and A. Cuitino:Acta Mater.55