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Simulation of stored energy and orientation gradients in cold-rolled interstitial free steels
Acta Materialia 51 (2003) 1775–1788 www.actamat-journals.com
Simulation of stored energy and orientation gradients in cold-rolled interstitial free steels S.-H. Choi ∗ Technical Research Laboratories, POSCO 699 Gumho-dong, Gwangyang-si, Jeonnam, 545-090, South Korea Received 11 July 2002; received in revised form 21 November 2002; accepted 30 November 2002
Abstract The plane strain compression of polycrystalline interstitial free steels is simulated using the crystal plasticity finite element method. In order to capture the interaction between neighboring grains, a simplified quasi 3-D geometric mesh is used in the simulation. The analysis shows that the orientation gradients depend not only on the initial orientation of the grain, but also on the neighboring grains. The simulation shows that the distribution of orientation in the deformed grains can be classified into three types based on the main texture components. The simulations provide a method to quantitatively evaluate the orientation-dependent stored energy in cold-rolled IF steels. The accumulation of stored energy at the specific grain boundaries is also a function of initial orientations as well as neighboring grains. 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Plane strain; IF steels; Crystal plasticity; Stored energy
1. Introduction Macroscopic anisotropy, such as r-value (plastic strain ratio) and yield stress, of polycrystalline sheet materials affects the formability in metal forming processes. The metallurgical parameters which affect the macroscopic anisotropy of polycrystalline materials are the crystallographic texture [1,2] and the morphological texture, i.e. grain shape [3–5], precipitate distribution [6,7] and dislocation structure [8]. In the case of the fully annealed interstitial free (IF) steel sheets, crystallographic texture is the dominant metallurgical para-
∗
Tel.: +82-61-790-8705; fax: +82-61-790-8799. E-mail address: [email protected] (S.-H. Choi).
meter for determining the macroscopic anisotropy of the materials. Much research has been conducted to optimize the macroscopic r-value and its anisotropy of IF steels [9–11]. It was found that the annealing texture of the IF steels depends on the chemical composition and thermo-mechanical processing condition. In order to understand the evolution of the texture in IF steels during hot rolling, cold rolling and recrystallization processes, many experimental and simulation works have been conducted in several length scales [12–16]. In recent years, there have been some results indicating that the stored energy of subgrains in a deformed specimen is an important metallurgical factor in static recrystallization [17,18]. Many experimental and theoretical works have been carried out to evaluate the orientations-dependent
1359-6454/03/$30.00 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. doi:10.1016/S1359-6454(02)00576-1
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stored energy of cold deformed polycrystalline materials [15,19–24]. Rajmohan et al. [19,20] derived the stored energy distribution function in the Euler angle space for cold-rolled IF steels by measuring the neutron line broadening. However, since the method provides average stored energy for each orientation, it cannot be used to study the heterogeneous distribution of stored energy inside the deformed grains. The pattern quality (or image quality) determined by electron back scattered diffraction (EBSD) analysis was used to estimate the orientation-dependent stored energy in cold-rolled materials [23,24]. However, since the pattern quality can be affected by experimental procedures such as specimen preparation and contamination, it can only be used as a semi-quantitative value for the stored energy term. In theoretical works, the Taylor model was used to calculate the Taylor factor as an orientation-dependent stored energy term in cold-rolled IF steels [16,19,20]. The Taylor factor is a first order approximation for the stored energy induced by plastic deformation of polycrystalline materials. Bacroix et al. [25] and Diligent et al. [26] calculated the orientation-dependent stored energy in cold-rolled IF steels with the visco-plastic self-consistent and elasto-plastic self-consistent polycrystal models, respectively. The polycrystal models assume that the homogenization scheme (stress and strain rate are uniform in each grain) cannot properly simulate the heterogeneous distribution of stored energy in the deformed grains. In order to overcome the shortcoming of the models, Sarma et al. [27] simulated the plane strain compression of FCC polycrystalline materials to determine the orientation-dependent stored energy using a 3-D crystal plasticity finite element model with a real configuration for microstructure. The results of their simulation provided a means to obtain quantitative information on the inhomogeneous distribution of stored energy and orientations among the different grains comprising FCC polycrystalline materials. Recently, Raabe et al. [28] investigated the influences of intrinsic (initial grain orientation) and extrinsic (neighboring grains) origins on the formation of orientation gradients using a 3-D crystal plasticity finite element model with BCC and FCC bi-crystal structures. However, no study has been
carried out to obtain the orientation-dependent stored energy in BCC polycrystalline materials such as IF steels using crystal plasticity finite element model. In the present work, plane strain compression of polycrystalline IF steel was simulated using a quasi 3-D crystal plasticity finite element model in a similar approach to Becker [29,30]. Moreover, the inhomogeneous stored energy in cold-rolled IF steel was analyzed quantitatively as a function of orientation. In order to estimate the magnitude of the orientation gradient in deformed grains, misorientation angle was calculated for all elements and compared for all grains.
2. The simulation procedure 2.1. Constitutive laws for rate dependent crystals The plane strain compression was simulated using the finite element code, ABAQUS [31] with the material model programmed by a continuum crystal plasticity theory. The constitutive model accounts for deformation by slip on active slip systems and for the rotation of the crystal lattice at finite deformation. The detail kinematics of the deformation and lattice rotation are described in the paper of Asaro and Rice [32]. The numerical implementation as user material subroutine UMAT in ABAQUS using a rate-dependent slip system constitutive relation follows the work of Peirce et al. [33]. The model is fundamentally based on a multiplicative decomposition of the deformation gradient into a plastic part characterized by slip on active slip systems and an elastic part that accounts for the rotation and elastic distortion of the crystal lattice. This formulation leads to an addictive decomposition of the velocity gradient into elastic and plastic parts, Table 1 Microscopic hardening coefficients determined from the fitting procedure ho(MPa)
a
tao (MPa)
tsat(MPa)
2100
4.5
117
340
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Fig. 2. (a) Deformed finite element meshes (b) distribution of stored energy after reduction in thickness of ⑀ = 0.37.
∂x˙ ⫽ L ⫽ Le ⫹ Lp ∂x Fig. 1. Initial finite element mesh (24 × 60 × 1 elements) and distribution of grain orientations in the model geometry.
(1)
with the plastic part determined by slip rates, g˙ a, on slip planes with normals ma, and slip directions sa
冘 n
Lp ⫽
Table 2 Euler angles of the initial orientations
G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16 G17
g˙ asa丢ma
(2)
a⫽1
f1
f
f2
25 35 45 45 30 20 70 10 25 25 25 85 70 20 45 55 85
80 25 90 90 50 35 70 35 15 45 45 80 80 75 75 70 60
65 30 60 80 30 20 50 50 20 50 60 30 85 35 65 70 55
The summation is over all of the slip systems, n. The plastic part of the velocity gradient is decomposed further into symmetric and antisymmetric parts (Lp ⫽ Dp ⫹ wp)
冘 冘 n
Dp ⫽
n
wp ⫽
冘 冘
1 g˙ a(sa丢ma ⫹ ma丢sa) ⫽ 2a ⫽ 1 1 g˙a(sa丢ma⫺ma丢sa) ⫽ 2a ⫽ 1
n
g˙ aPa
a⫽1 n
g˙ aWa
a⫽1
(3) p
where D is the plastic part of the rate of deformation tensor and wp is the plastic spin. The evolution of the slip directions and slip plane normals can be expressed in terms of the elastic part of the deformation gradient as
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(a) Deformed finite element meshes (b) distribution of stored energy after reduction in thickness of ⑀ = 1.05.
Fig. 3.
s˙a ⫽ F˙ e·sa and m ˙ a ⫽ ma·F˙ e⫺1
(4)
where the rate of change of Fe is given by F˙ e ⫽ Le·Fe ⫽ (L⫺Lp)·Fe
(5)
As described in Pierce et al. [33], the Jaumann rate of Kirchhof stress can be expressed as
冘
present work, the approach of Kalidindi et al. [34] has been employed for this purpose, which used the following hardening law
冘
冉 冊
n
t˙ ao ⫽
Qab|g˙ b| a,b ⫽ 1...n Qab ⫽ qijho 1⫺
b
g˙ aRa
(6)
a⫽1
where K is a fourth order tensor based on the anisotropic elastic modulus, D is the rate of deformation tensor (symmetric part of the velocity gradient), and Ra is a tensor depending on the current slip plane normal and direction, the applied stress and the elastic modulus. For a rate-dependent material, this slip rate is given explicitly in terms of the resolved shear stress on the active slip systems and the resistance of the active slip systems to shear. For these simulations, this dependence is given by g˙ a ⫽ g˙ ao
ta tao
1/m
||
a
(8)
n
tˆ ⫽ K:D⫺
tao tsat
sign(ta)
(7)
Self and latent hardening can be readily accounted for by a suitable evolution of the reference tao values in the constitutive law by Eq. (7). In the
ab
where Q is a (n×n) hardening matrix, which is introduced to account for the interaction between slip systems and qab accounts for the hardening rate of slip system a due to slip activity on system b. The diagonal parts of qab are the self-hardening term, and the off-diagonal parts of qab are the latent hardening term. The effect of the latent hardening term on the stored energy or macroscopic hardening behavior has been investigated by several researchers [25,35,36]. The different value of latent hardening term does not affect much the stored energy of low carbon steels [25]. It is assumed here that the self-hardening term equals the latent hardening term, i.e. (qab=1). The fitting simulation was carried out by varying the hardening parameters (a, ho and tsat) until an agreement was obtained between the predicted and the measured uniaxial tension curve. Table 1 shows the determined hardening coefficients for the crystal plasticity modeling.
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Fig. 4. The (111) pole figures of ideal single orientations commonly found in IF steels.
The rate equations given by Eq. (6) using Eq. (7) for the slip rate are numerically stiff and require very small time steps for stable time integration. The rate tangent method of Peirce et al [33] is used to increase the stable time step size. During plane strain deformation, the stored energy values were calculated based on the resolved shear stresses in each element. The aver-
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Fig. 5. Comparison between the texture evolution predicted from the Taylor model (left) and the texture evolution predicted from the crystal plasticity finite element model (right) for type I grain.
age resolved shear stress can be expressed to be proportional to the square root of the dislocation density, ρ [18,27]
冘 n
(
a
1 tao ) / n ⫽ to ⫽ Gbr1/2 2
(9)
where G is the shear modulus of the material and b is the magnitude of the Burgers vector. The sum-
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3. Results
Fig. 6. Comparison between the texture evolution predicted from the Taylor model (left) and the texture evolution predicted from the crystal plasticity finite element model (right) for type II grain.
mation of the resolved shear stress is over all of the slip systems, n. The stored energy (per unit volume) can be expressed as a relationship between dislocation density and dislocation energy Edisl 1 S ⫽ rEdisl⬇ rGb2 2
(10)
This relationship highly approximates dislocation structure in real materials. Eqs. (9) and (10) can be combined to write the stored energy in terms of the resolved shear stress and the shear modulus as S⫽
2(to)2 G
(11)
The crystal plasticity FE model discussed in the previous section has been used to simulate cold deformation corresponding to different strains under plane strain compression. The initial mesh ( 24 × 60 × 1 elements) of the polycrystal model is shown in Fig. 1. The initial length of the model region is given by Lo / Ho = 1.35 (Lo = 162 µm, Ho = 120 µm, Wo = 3.5 µm). The simple hexagonal prisms are introduced to represent idealized grains with 17 different initial orientations (from G1 to G17). In order to impose initial orientations of the elements, single orientations of hot-rolled IF steel samples were used. Pole figure measurement was carried out by Rigaku D-Max 2500 X-ray diffractometer at mid-plane thickness of the hot-rolled steel sample. From the {110}, {200}, {211} incomplete pole figures, the ODF was calculated using the WIMV method [37]. The ODF was used to generate a set of grain orientations. The initial grain orientations are shown in Table 2. The spatial distribution of the initial orientations for polycrystalline material is selected arbitrarily. Reductions in thickness of 31% and 65%, corresponding to compressive true strains of ⑀ = 0.37 and ⑀ = 1.05, respectively, were simulated using an appropriate boundary condition and discretization. The boundary conditions were applied to the six planes consisting of the 3-D mesh. The interior nodes on the X1–X2 plane (X3 = Ho at initial) were just free to move to the X3 direction. This boundary condition gives rise to out of plane shear tractions. It is possible that the boundary conditions imposed on this plane can be applied in such a way that compatibility and equilibrium with neighboring material are maintained [30]. This periodic boundary condition can be done by enforcing uniform normal displacement and maintaining zero shear tractions at interior nodes on the plane. In this study, the periodic boundary condition was not applied in the plane. The interior nodes on the X2–X3 plane (X1 = Lo at initial) were constrained to exist on plane normal to the direction X1. The interior nodes on the planes with normal to the X2 direction were just free to move in the X1–X3 planes (X2 = 0 and X2 = Wo), but not normal to the X2 direction.
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Fig. 7. Comparison between the texture evolution predicted from the Taylor model (left) and the texture evolution predicted from the crystal plasticity finite element model (right) for type III grain.
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Fig. 8. Histograms showing the distribution of (a) stored energy (left) and (b) misorientation (right) calculated from the crystal plasticity finite element model for type I grain.
In order to capture the inhomogeneous deformation of the grains, each grain was discretized with 52–104 elements per grain (52 elements for four grains, 84 elements for six grains, 104 elements for seven grains). The 3-D solid linear element type C3D8 [31] with eight nodes and eight integration points was used. Reduced integration element can be used to reduce computation time.
However, the reduced integration element can lead to hourglassing. The artificial stiffness parameter can be used to control the hourglass modes in the reduced integration element. In this study, the reduced integration scheme was not used since the hourglass control may introduce artificial response for the true deformation. In addition, volumetric locking can occur in fully integrated elements
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Fig. 9. Histograms showing the distribution of (a) stored energy (left) and (b) misorientation (right) calculated from the crystal plasticity finite element model for type II grain.
when the material behavior is incompressible or almost incompressible. With the brick element C3D8, this problem is overcome by assumed uniform volume strain combined with full integration for deviatoric strain, which is called a selectively reduced integration scheme [31]. The grain boundaries are assumed as perfectly bonded structures, and compatibility and equilibrium are satisfied across the grain boundaries by the finite element method. The deformed finite element mesh after a reduction in thickness of ⑀ = 0.37 is shown in Fig. 2(a). At the deformation, the elements distorted non-uniformly and the magnitude of the distortion strongly depended on the initial orientation of the grain as well as the neighboring grains. The stored energy for all elements was evaluated using the
procedure described in Section 2. The stored energy distribution on the X1–X3 section of the deformed element mesh is shown in Fig. 2(b). It can be observed that the stored energy exhibits inhomogeneous distribution in each grain and between neighboring grains. It should be noted that some grain boundaries exhibit much higher stored energy than other grain boundaries. The accumulation of the stored energy at the grain boundaries compared to the grain interior is closely related to the orientation relationship with neighboring grains. For a given external strain, each grain subdivides into several subgrains. A grain more favorably oriented to accommodate the external strain can be constrained by an unfavorably oriented neighbor grain. After all, the constrained grain boundaries will exhibit high stored energy com-
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Fig. 10. Histograms showing the distribution of (a) stored energy (left) and (b) misorientation (right) calculated from the crystal plasticity finite element model for type III grain.
pared to the unconstrained grain boundaries. As shown in Fig. 2(b), the grain boundaries between two neighboring grains (e.g. G1 and G2 or G4 and G5 or G8 and G9 or G13 and G14) exhibit relatively high stored energy accumulation. Fig. 3 shows the deformed finite element meshes and stored energy distribution after a reduction in thickness of ⑀ = 1.05. At this deformation, some
elements are highly distorted and the stored energy distribution strongly depends on the initial orientation of the grain as well as the neighboring grains in the case of ⑀ = 0.37. The accumulation of stored energy at the specific grain boundary is similar to that observed in the case of ⑀ = 0.37. It should be noted that not all grain boundaries have high stored energy.
S..-H. Choi / Acta Materialia 51 (2003) 1775–1788
Fig. 10.
3.1. Texture evolution Fig. 4 shows the (111) pole figure of the typical texture components, the γ-fibre (normal direction parallel to ⬍111⬎) and the α-fibre (rolling direction parallel to ⬍110⬎) orientations, which can be found in cold-rolled sheets of BCC polycrystalline materials [11,12].
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continued
The evolution of deformation texture after a reduction in thickness of ⑀ = 1.05 is shown in Figs. 5, 6 and 7. The (111) pole figures on the left side in Figs. 5, 6 and 7 show the orientations from trace of orientation predicted from the Taylor model [38]. The square symbol denotes the initial orientation of grain. It is clear that the homogenization scheme such as the Taylor model is not properly
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Fig. 10.
able to simulate the subdivision of the initial grain by plane strain compression. It also means that the homogenization scheme has a limitation in estimating orientation-dependent stored energy in coldrolled IF steels. The (111) pole figures on the right side in Figs. 5, 6 and 7 show the orientations simulated for all elements (actual integration points) in the crystal plasticity FE model. Based on the careful comparison between orientations simulated from the crystal plasticity FE model, it is found that the initial grains can be classified into three types (I, II, and III) of grains. It can be seen that grains of type I (Fig. 5) subdivide into the α-fibre and γ-fibre orientations as the main texture component by plane strain compression. For type II, grains (Fig. 6) subdivide into several γ-fibre orientations as the main texture component by plane strain compression. Grains of type III (Fig. 7) subdivide only into the single γfibre orientation (partially near γ-fibre orientation) as the main texture component by plane strain compression. In the Taylor model, the external strain imposed on the polycrystal is exactly the same for each grain. The predicted orientations from the Taylor model exist close to or far from
continued
the orientations simulated from the crystal plasticity FE model. It can be observed that the difference between predictions of the Taylor model and the crystal plasticity FE model varies from grain to grain. The deformation orientations of the type I grain simulated from the crystal plasticity finite element model exhibit spread by a rotation about the X1 axis. Comparing Figs. 4 and 5, it is clear that the deformation orientations of the type I grain consist of several α-fibre orientations and the single γ-fibre orientation as the main texture component. The orientation spread of the type II grain is different to that of the type I grain. The deformation orientations of the type II grain simulated from the crystal plasticity finite element model exhibit spread by a rotation about the X3 axis as shown in Fig. 6. It can be seen that the type II grain consists of several γ-fibre orientations as the main texture component. In contrast, most of the type III grains consisting of single γ-fibre orientation as the main texture component exhibit no directionality about a specific rotation axis. Particularly, it should be noted that the type III grain (e.g. G1, G2 and G12) exhibits a weak directionality about a specific rotation axis.
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3.2. Stored energy and misorientation angle
Figs. 8, 9 and 10 show the distribution of stored energy and misorientation angle simulated for all elements in the form of a histogram. As explained in Section 2, the distribution of stored energy for all elements in the deformed grain is not homogeneous. From Figs. 8, 9 and 10, it is found that the distribution of stored energy depends on the initial orientation as well as the neighboring grains. The type II grain exhibits a relatively weak gradient of stored energy compared to the type I and type III grains. From Fig. 3, it can be seen that the stored energy accumulates at the specific grain boundaries in grains having a high stored energy. Comparing Figs. 1 and 3, the grain boundaries between two specific grains (e.g. G1 and G2 or G4 and G5 or G8 and G9 or G11 and G12) exhibit accumulation of stored energy. From Figs. 8, 9 and 10, it is found that the neighboring two grains having common grain boundaries do not exhibit a high stored energy. Only two grains (e.g. G4 and G5) exhibit a similar high stored energy at the common grain boundary. Particularly, the two grains (e.g. G10 and G11) having almost the same crystallographic orientation after deformation exhibit fairly different distribution of stored energy in each grain. This result indicates the fact that the homogeneous scheme is not suitable to evaluate the orientation-dependent stored energy in the deformed grains. The misorientation angle calculated for all the elements in the deformed grains is also a function of the initial orientation as well as of the neighboring grains. Most of the type III grains exhibit many low angle grain boundaries under 15° as a misorientation angle compared to type I and type II grains. The results seem to be due to the weak orientation gradients in most of the type III grains after plane strain compression. It is known that the subgrains having a high stored energy and high angle grain boundaries are the most likely nucleation sites for static recrystallization after plane strain compression [18]. Therefore, it is likely that the subgrains exhibiting high stored energy and high misorientation angle will recover rapidly and subgrain growth results in the merging
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of the deformed microstructure by the movement of high angle grain boundaries.
4. Discussion and conclusions The plane strain compression of IF steels has been simulated by using the crystal plasticity finite element model. A simplified geometry of a polycrystal material is used in order to consider the grain interaction between neighboring grains during deformation. The finite element discretization in each grain can capture the subdivision of orientation and accumulation of stored energy at grain boundaries during plane strain compression of IF steels. In order to satisfy the compatibility and stress equilibrium, each grain undergoes inhomogeneous deformation which is observed in experimental results [13]. The Taylor model, which neglects stress equilibrium between neighboring grains, cannot provide subdivision of orientation during plane strain compression. This indicates that the homogenization scheme such as the Taylor model is not proper to estimate the orientationdependent stored energy in cold-rolled IF steels. It is clear that the orientation gradients calculated from the crystal plasticity finite element model strongly depend on the initial orientation of the grain as well as the neighboring grains. The distribution in the spreading of orientations simulated by the crystal plasticity finite element model can be classified into three types of grains. The type I grain including α-fibre and γ-fibre orientations as a main texture component exhibits relatively strong gradients of the stored energy. The type II grain including several γ-fibre orientations as a main texture component exhibits strong gradients of the stored energy and high density of the high angle grain boundaries. The type III grain including mostly single γ-fibre orientation as a main texture component exhibits weak gradients of the stored energy and a high density of the low angle grain boundaries. It was also found that the accumulation of the stored energy at the grain boundaries depends significantly on the local orientations of the neighboring grains. Grain boundaries between more favorably orientated grains and more unfavorably orientated grains
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exhibit high stored energy accumulation compared to the other grain boundaries. It is expected that such regions are more likely to be nucleation sites than the other grain boundaries during static recrystallization of cold-rolled IF steels. On the other hand, because of the quasi 3-D mesh of the model, the interactions with grains above and below the model grains cannot be considered in this study. In order to capture the interaction, a fully 3-D mesh is required in the model. However, the present work can be used in the coupling of the deformation model with recrystallization model such as the 2-D Monte Carlo technique [16,22] to simulate static recrystallization in cold-rolled IF steels.
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[16] Choi S-H, Barlat F, Chung JH. Scripta Mater 2001;45:1155. [17] Doherty RD, Hughes DA, Humphreys FJ, Jonas JJ, Juul Jensen D, Kassneer ME, King WE, McNelley TR, McQueen HJ, Rollet AD. Mater Sci Eng 1997;A238:219. [18] Humphreys FJ, Hatherly M. Recrystallization and Related Annealing Phenomena. Oxford: Pergamon, 1995. [19] Rajmohan N, Hayakawa Y, Szpunar JA, Root JH. Acta Mater 1997;45:2485. [20] Rajmohan N, Hayakawa Y, Szpunar JA, Root JH. Physica B 1998;1225:241-243. [21] Radhakrishnan B, Sarma GB, Zacharia T. Acta Mater 1998;46:4415. [22] Radhakrishnan B, Sarma GB, Zacharia T. Scripta Mater 1998;39:1639. [23] Engler O. In: Carstensen JV et al., editors. Proceedings of the 19th Riso International Symposium on Materials Science: Modeling of Structure and Mechanics of Materials from Microscale to Product. Riso National Laboratory, 1998:253. [24] Choi S-H. Material Science Forum 2002;469:408-412. [25] Bacroix B, Miroux A, Castelnau O. Modelling Simul Mater Sci Eng 1999;7:851. [26] Diligent S, Gautier E, Lemoine X, Berveiller M. Acta Mater 2001;49:4079. [27] Sarma GB, Radhakrishnan B, Zacharia T. Computational Mater Sci 1998;12:105. [28] Raabe D, Zhao Z, Park SJ, Roters F. Acta Mater 2002;50:421. [29] Becker R. Acta Metal Mater 1991;39:1211. [30] Becker R, Panchanadeeswaran S. Acta Metal Mater 1995;43:2701. [31] ABAQUS User’s Manual, version 6.1. Providence, RI: Hibbit, Karlsson and Sorenson, 2000. [32] Asaro RJ, Needleman A. Acta Metall 1985;33:923. [33] Peirce D, Asaro RJ, Needleman A. Acta Metall 1983;31:1951. [34] Kalidindi SR, Bronkhorst CA, Anand L. J Mech Phys Solids 1992;40:537. [35] Zhou Y, Neal KW, Toth LS. Int J Plasticity 1993;9:961. [36] Toth LS. In: Liang Z et al., editors. Proceedings of the ICOTOM11. Inter Academic Pub, 1996: 347. [37] Matthies S, Muller J, Vinel GW. Textures and Microstructures 1988;10:77. [38] Choi S-H, Cho J-H, Oh KH, Chung K, Barlat F. Int J Mech Sci 2000;42:1571.
Simulation of stored energy and orientation gradients in cold-rolled interstitial free steels S.-H. Choi ∗ Technical Research Laboratories, POSCO 699 Gumho-dong, Gwangyang-si, Jeonnam, 545-090, South Korea Received 11 July 2002; received in revised form 21 November 2002; accepted 30 November 2002
Abstract The plane strain compression of polycrystalline interstitial free steels is simulated using the crystal plasticity finite element method. In order to capture the interaction between neighboring grains, a simplified quasi 3-D geometric mesh is used in the simulation. The analysis shows that the orientation gradients depend not only on the initial orientation of the grain, but also on the neighboring grains. The simulation shows that the distribution of orientation in the deformed grains can be classified into three types based on the main texture components. The simulations provide a method to quantitatively evaluate the orientation-dependent stored energy in cold-rolled IF steels. The accumulation of stored energy at the specific grain boundaries is also a function of initial orientations as well as neighboring grains. 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Plane strain; IF steels; Crystal plasticity; Stored energy
1. Introduction Macroscopic anisotropy, such as r-value (plastic strain ratio) and yield stress, of polycrystalline sheet materials affects the formability in metal forming processes. The metallurgical parameters which affect the macroscopic anisotropy of polycrystalline materials are the crystallographic texture [1,2] and the morphological texture, i.e. grain shape [3–5], precipitate distribution [6,7] and dislocation structure [8]. In the case of the fully annealed interstitial free (IF) steel sheets, crystallographic texture is the dominant metallurgical para-
∗
Tel.: +82-61-790-8705; fax: +82-61-790-8799. E-mail address: [email protected] (S.-H. Choi).
meter for determining the macroscopic anisotropy of the materials. Much research has been conducted to optimize the macroscopic r-value and its anisotropy of IF steels [9–11]. It was found that the annealing texture of the IF steels depends on the chemical composition and thermo-mechanical processing condition. In order to understand the evolution of the texture in IF steels during hot rolling, cold rolling and recrystallization processes, many experimental and simulation works have been conducted in several length scales [12–16]. In recent years, there have been some results indicating that the stored energy of subgrains in a deformed specimen is an important metallurgical factor in static recrystallization [17,18]. Many experimental and theoretical works have been carried out to evaluate the orientations-dependent
1359-6454/03/$30.00 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. doi:10.1016/S1359-6454(02)00576-1
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stored energy of cold deformed polycrystalline materials [15,19–24]. Rajmohan et al. [19,20] derived the stored energy distribution function in the Euler angle space for cold-rolled IF steels by measuring the neutron line broadening. However, since the method provides average stored energy for each orientation, it cannot be used to study the heterogeneous distribution of stored energy inside the deformed grains. The pattern quality (or image quality) determined by electron back scattered diffraction (EBSD) analysis was used to estimate the orientation-dependent stored energy in cold-rolled materials [23,24]. However, since the pattern quality can be affected by experimental procedures such as specimen preparation and contamination, it can only be used as a semi-quantitative value for the stored energy term. In theoretical works, the Taylor model was used to calculate the Taylor factor as an orientation-dependent stored energy term in cold-rolled IF steels [16,19,20]. The Taylor factor is a first order approximation for the stored energy induced by plastic deformation of polycrystalline materials. Bacroix et al. [25] and Diligent et al. [26] calculated the orientation-dependent stored energy in cold-rolled IF steels with the visco-plastic self-consistent and elasto-plastic self-consistent polycrystal models, respectively. The polycrystal models assume that the homogenization scheme (stress and strain rate are uniform in each grain) cannot properly simulate the heterogeneous distribution of stored energy in the deformed grains. In order to overcome the shortcoming of the models, Sarma et al. [27] simulated the plane strain compression of FCC polycrystalline materials to determine the orientation-dependent stored energy using a 3-D crystal plasticity finite element model with a real configuration for microstructure. The results of their simulation provided a means to obtain quantitative information on the inhomogeneous distribution of stored energy and orientations among the different grains comprising FCC polycrystalline materials. Recently, Raabe et al. [28] investigated the influences of intrinsic (initial grain orientation) and extrinsic (neighboring grains) origins on the formation of orientation gradients using a 3-D crystal plasticity finite element model with BCC and FCC bi-crystal structures. However, no study has been
carried out to obtain the orientation-dependent stored energy in BCC polycrystalline materials such as IF steels using crystal plasticity finite element model. In the present work, plane strain compression of polycrystalline IF steel was simulated using a quasi 3-D crystal plasticity finite element model in a similar approach to Becker [29,30]. Moreover, the inhomogeneous stored energy in cold-rolled IF steel was analyzed quantitatively as a function of orientation. In order to estimate the magnitude of the orientation gradient in deformed grains, misorientation angle was calculated for all elements and compared for all grains.
2. The simulation procedure 2.1. Constitutive laws for rate dependent crystals The plane strain compression was simulated using the finite element code, ABAQUS [31] with the material model programmed by a continuum crystal plasticity theory. The constitutive model accounts for deformation by slip on active slip systems and for the rotation of the crystal lattice at finite deformation. The detail kinematics of the deformation and lattice rotation are described in the paper of Asaro and Rice [32]. The numerical implementation as user material subroutine UMAT in ABAQUS using a rate-dependent slip system constitutive relation follows the work of Peirce et al. [33]. The model is fundamentally based on a multiplicative decomposition of the deformation gradient into a plastic part characterized by slip on active slip systems and an elastic part that accounts for the rotation and elastic distortion of the crystal lattice. This formulation leads to an addictive decomposition of the velocity gradient into elastic and plastic parts, Table 1 Microscopic hardening coefficients determined from the fitting procedure ho(MPa)
a
tao (MPa)
tsat(MPa)
2100
4.5
117
340
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Fig. 2. (a) Deformed finite element meshes (b) distribution of stored energy after reduction in thickness of ⑀ = 0.37.
∂x˙ ⫽ L ⫽ Le ⫹ Lp ∂x Fig. 1. Initial finite element mesh (24 × 60 × 1 elements) and distribution of grain orientations in the model geometry.
(1)
with the plastic part determined by slip rates, g˙ a, on slip planes with normals ma, and slip directions sa
冘 n
Lp ⫽
Table 2 Euler angles of the initial orientations
G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16 G17
g˙ asa丢ma
(2)
a⫽1
f1
f
f2
25 35 45 45 30 20 70 10 25 25 25 85 70 20 45 55 85
80 25 90 90 50 35 70 35 15 45 45 80 80 75 75 70 60
65 30 60 80 30 20 50 50 20 50 60 30 85 35 65 70 55
The summation is over all of the slip systems, n. The plastic part of the velocity gradient is decomposed further into symmetric and antisymmetric parts (Lp ⫽ Dp ⫹ wp)
冘 冘 n
Dp ⫽
n
wp ⫽
冘 冘
1 g˙ a(sa丢ma ⫹ ma丢sa) ⫽ 2a ⫽ 1 1 g˙a(sa丢ma⫺ma丢sa) ⫽ 2a ⫽ 1
n
g˙ aPa
a⫽1 n
g˙ aWa
a⫽1
(3) p
where D is the plastic part of the rate of deformation tensor and wp is the plastic spin. The evolution of the slip directions and slip plane normals can be expressed in terms of the elastic part of the deformation gradient as
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(a) Deformed finite element meshes (b) distribution of stored energy after reduction in thickness of ⑀ = 1.05.
Fig. 3.
s˙a ⫽ F˙ e·sa and m ˙ a ⫽ ma·F˙ e⫺1
(4)
where the rate of change of Fe is given by F˙ e ⫽ Le·Fe ⫽ (L⫺Lp)·Fe
(5)
As described in Pierce et al. [33], the Jaumann rate of Kirchhof stress can be expressed as
冘
present work, the approach of Kalidindi et al. [34] has been employed for this purpose, which used the following hardening law
冘
冉 冊
n
t˙ ao ⫽
Qab|g˙ b| a,b ⫽ 1...n Qab ⫽ qijho 1⫺
b
g˙ aRa
(6)
a⫽1
where K is a fourth order tensor based on the anisotropic elastic modulus, D is the rate of deformation tensor (symmetric part of the velocity gradient), and Ra is a tensor depending on the current slip plane normal and direction, the applied stress and the elastic modulus. For a rate-dependent material, this slip rate is given explicitly in terms of the resolved shear stress on the active slip systems and the resistance of the active slip systems to shear. For these simulations, this dependence is given by g˙ a ⫽ g˙ ao
ta tao
1/m
||
a
(8)
n
tˆ ⫽ K:D⫺
tao tsat
sign(ta)
(7)
Self and latent hardening can be readily accounted for by a suitable evolution of the reference tao values in the constitutive law by Eq. (7). In the
ab
where Q is a (n×n) hardening matrix, which is introduced to account for the interaction between slip systems and qab accounts for the hardening rate of slip system a due to slip activity on system b. The diagonal parts of qab are the self-hardening term, and the off-diagonal parts of qab are the latent hardening term. The effect of the latent hardening term on the stored energy or macroscopic hardening behavior has been investigated by several researchers [25,35,36]. The different value of latent hardening term does not affect much the stored energy of low carbon steels [25]. It is assumed here that the self-hardening term equals the latent hardening term, i.e. (qab=1). The fitting simulation was carried out by varying the hardening parameters (a, ho and tsat) until an agreement was obtained between the predicted and the measured uniaxial tension curve. Table 1 shows the determined hardening coefficients for the crystal plasticity modeling.
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Fig. 4. The (111) pole figures of ideal single orientations commonly found in IF steels.
The rate equations given by Eq. (6) using Eq. (7) for the slip rate are numerically stiff and require very small time steps for stable time integration. The rate tangent method of Peirce et al [33] is used to increase the stable time step size. During plane strain deformation, the stored energy values were calculated based on the resolved shear stresses in each element. The aver-
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Fig. 5. Comparison between the texture evolution predicted from the Taylor model (left) and the texture evolution predicted from the crystal plasticity finite element model (right) for type I grain.
age resolved shear stress can be expressed to be proportional to the square root of the dislocation density, ρ [18,27]
冘 n
(
a
1 tao ) / n ⫽ to ⫽ Gbr1/2 2
(9)
where G is the shear modulus of the material and b is the magnitude of the Burgers vector. The sum-
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3. Results
Fig. 6. Comparison between the texture evolution predicted from the Taylor model (left) and the texture evolution predicted from the crystal plasticity finite element model (right) for type II grain.
mation of the resolved shear stress is over all of the slip systems, n. The stored energy (per unit volume) can be expressed as a relationship between dislocation density and dislocation energy Edisl 1 S ⫽ rEdisl⬇ rGb2 2
(10)
This relationship highly approximates dislocation structure in real materials. Eqs. (9) and (10) can be combined to write the stored energy in terms of the resolved shear stress and the shear modulus as S⫽
2(to)2 G
(11)
The crystal plasticity FE model discussed in the previous section has been used to simulate cold deformation corresponding to different strains under plane strain compression. The initial mesh ( 24 × 60 × 1 elements) of the polycrystal model is shown in Fig. 1. The initial length of the model region is given by Lo / Ho = 1.35 (Lo = 162 µm, Ho = 120 µm, Wo = 3.5 µm). The simple hexagonal prisms are introduced to represent idealized grains with 17 different initial orientations (from G1 to G17). In order to impose initial orientations of the elements, single orientations of hot-rolled IF steel samples were used. Pole figure measurement was carried out by Rigaku D-Max 2500 X-ray diffractometer at mid-plane thickness of the hot-rolled steel sample. From the {110}, {200}, {211} incomplete pole figures, the ODF was calculated using the WIMV method [37]. The ODF was used to generate a set of grain orientations. The initial grain orientations are shown in Table 2. The spatial distribution of the initial orientations for polycrystalline material is selected arbitrarily. Reductions in thickness of 31% and 65%, corresponding to compressive true strains of ⑀ = 0.37 and ⑀ = 1.05, respectively, were simulated using an appropriate boundary condition and discretization. The boundary conditions were applied to the six planes consisting of the 3-D mesh. The interior nodes on the X1–X2 plane (X3 = Ho at initial) were just free to move to the X3 direction. This boundary condition gives rise to out of plane shear tractions. It is possible that the boundary conditions imposed on this plane can be applied in such a way that compatibility and equilibrium with neighboring material are maintained [30]. This periodic boundary condition can be done by enforcing uniform normal displacement and maintaining zero shear tractions at interior nodes on the plane. In this study, the periodic boundary condition was not applied in the plane. The interior nodes on the X2–X3 plane (X1 = Lo at initial) were constrained to exist on plane normal to the direction X1. The interior nodes on the planes with normal to the X2 direction were just free to move in the X1–X3 planes (X2 = 0 and X2 = Wo), but not normal to the X2 direction.
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Fig. 7. Comparison between the texture evolution predicted from the Taylor model (left) and the texture evolution predicted from the crystal plasticity finite element model (right) for type III grain.
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Fig. 8. Histograms showing the distribution of (a) stored energy (left) and (b) misorientation (right) calculated from the crystal plasticity finite element model for type I grain.
In order to capture the inhomogeneous deformation of the grains, each grain was discretized with 52–104 elements per grain (52 elements for four grains, 84 elements for six grains, 104 elements for seven grains). The 3-D solid linear element type C3D8 [31] with eight nodes and eight integration points was used. Reduced integration element can be used to reduce computation time.
However, the reduced integration element can lead to hourglassing. The artificial stiffness parameter can be used to control the hourglass modes in the reduced integration element. In this study, the reduced integration scheme was not used since the hourglass control may introduce artificial response for the true deformation. In addition, volumetric locking can occur in fully integrated elements
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Fig. 9. Histograms showing the distribution of (a) stored energy (left) and (b) misorientation (right) calculated from the crystal plasticity finite element model for type II grain.
when the material behavior is incompressible or almost incompressible. With the brick element C3D8, this problem is overcome by assumed uniform volume strain combined with full integration for deviatoric strain, which is called a selectively reduced integration scheme [31]. The grain boundaries are assumed as perfectly bonded structures, and compatibility and equilibrium are satisfied across the grain boundaries by the finite element method. The deformed finite element mesh after a reduction in thickness of ⑀ = 0.37 is shown in Fig. 2(a). At the deformation, the elements distorted non-uniformly and the magnitude of the distortion strongly depended on the initial orientation of the grain as well as the neighboring grains. The stored energy for all elements was evaluated using the
procedure described in Section 2. The stored energy distribution on the X1–X3 section of the deformed element mesh is shown in Fig. 2(b). It can be observed that the stored energy exhibits inhomogeneous distribution in each grain and between neighboring grains. It should be noted that some grain boundaries exhibit much higher stored energy than other grain boundaries. The accumulation of the stored energy at the grain boundaries compared to the grain interior is closely related to the orientation relationship with neighboring grains. For a given external strain, each grain subdivides into several subgrains. A grain more favorably oriented to accommodate the external strain can be constrained by an unfavorably oriented neighbor grain. After all, the constrained grain boundaries will exhibit high stored energy com-
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Fig. 10. Histograms showing the distribution of (a) stored energy (left) and (b) misorientation (right) calculated from the crystal plasticity finite element model for type III grain.
pared to the unconstrained grain boundaries. As shown in Fig. 2(b), the grain boundaries between two neighboring grains (e.g. G1 and G2 or G4 and G5 or G8 and G9 or G13 and G14) exhibit relatively high stored energy accumulation. Fig. 3 shows the deformed finite element meshes and stored energy distribution after a reduction in thickness of ⑀ = 1.05. At this deformation, some
elements are highly distorted and the stored energy distribution strongly depends on the initial orientation of the grain as well as the neighboring grains in the case of ⑀ = 0.37. The accumulation of stored energy at the specific grain boundary is similar to that observed in the case of ⑀ = 0.37. It should be noted that not all grain boundaries have high stored energy.
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Fig. 10.
3.1. Texture evolution Fig. 4 shows the (111) pole figure of the typical texture components, the γ-fibre (normal direction parallel to ⬍111⬎) and the α-fibre (rolling direction parallel to ⬍110⬎) orientations, which can be found in cold-rolled sheets of BCC polycrystalline materials [11,12].
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continued
The evolution of deformation texture after a reduction in thickness of ⑀ = 1.05 is shown in Figs. 5, 6 and 7. The (111) pole figures on the left side in Figs. 5, 6 and 7 show the orientations from trace of orientation predicted from the Taylor model [38]. The square symbol denotes the initial orientation of grain. It is clear that the homogenization scheme such as the Taylor model is not properly
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Fig. 10.
able to simulate the subdivision of the initial grain by plane strain compression. It also means that the homogenization scheme has a limitation in estimating orientation-dependent stored energy in coldrolled IF steels. The (111) pole figures on the right side in Figs. 5, 6 and 7 show the orientations simulated for all elements (actual integration points) in the crystal plasticity FE model. Based on the careful comparison between orientations simulated from the crystal plasticity FE model, it is found that the initial grains can be classified into three types (I, II, and III) of grains. It can be seen that grains of type I (Fig. 5) subdivide into the α-fibre and γ-fibre orientations as the main texture component by plane strain compression. For type II, grains (Fig. 6) subdivide into several γ-fibre orientations as the main texture component by plane strain compression. Grains of type III (Fig. 7) subdivide only into the single γfibre orientation (partially near γ-fibre orientation) as the main texture component by plane strain compression. In the Taylor model, the external strain imposed on the polycrystal is exactly the same for each grain. The predicted orientations from the Taylor model exist close to or far from
continued
the orientations simulated from the crystal plasticity FE model. It can be observed that the difference between predictions of the Taylor model and the crystal plasticity FE model varies from grain to grain. The deformation orientations of the type I grain simulated from the crystal plasticity finite element model exhibit spread by a rotation about the X1 axis. Comparing Figs. 4 and 5, it is clear that the deformation orientations of the type I grain consist of several α-fibre orientations and the single γ-fibre orientation as the main texture component. The orientation spread of the type II grain is different to that of the type I grain. The deformation orientations of the type II grain simulated from the crystal plasticity finite element model exhibit spread by a rotation about the X3 axis as shown in Fig. 6. It can be seen that the type II grain consists of several γ-fibre orientations as the main texture component. In contrast, most of the type III grains consisting of single γ-fibre orientation as the main texture component exhibit no directionality about a specific rotation axis. Particularly, it should be noted that the type III grain (e.g. G1, G2 and G12) exhibits a weak directionality about a specific rotation axis.
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3.2. Stored energy and misorientation angle
Figs. 8, 9 and 10 show the distribution of stored energy and misorientation angle simulated for all elements in the form of a histogram. As explained in Section 2, the distribution of stored energy for all elements in the deformed grain is not homogeneous. From Figs. 8, 9 and 10, it is found that the distribution of stored energy depends on the initial orientation as well as the neighboring grains. The type II grain exhibits a relatively weak gradient of stored energy compared to the type I and type III grains. From Fig. 3, it can be seen that the stored energy accumulates at the specific grain boundaries in grains having a high stored energy. Comparing Figs. 1 and 3, the grain boundaries between two specific grains (e.g. G1 and G2 or G4 and G5 or G8 and G9 or G11 and G12) exhibit accumulation of stored energy. From Figs. 8, 9 and 10, it is found that the neighboring two grains having common grain boundaries do not exhibit a high stored energy. Only two grains (e.g. G4 and G5) exhibit a similar high stored energy at the common grain boundary. Particularly, the two grains (e.g. G10 and G11) having almost the same crystallographic orientation after deformation exhibit fairly different distribution of stored energy in each grain. This result indicates the fact that the homogeneous scheme is not suitable to evaluate the orientation-dependent stored energy in the deformed grains. The misorientation angle calculated for all the elements in the deformed grains is also a function of the initial orientation as well as of the neighboring grains. Most of the type III grains exhibit many low angle grain boundaries under 15° as a misorientation angle compared to type I and type II grains. The results seem to be due to the weak orientation gradients in most of the type III grains after plane strain compression. It is known that the subgrains having a high stored energy and high angle grain boundaries are the most likely nucleation sites for static recrystallization after plane strain compression [18]. Therefore, it is likely that the subgrains exhibiting high stored energy and high misorientation angle will recover rapidly and subgrain growth results in the merging
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of the deformed microstructure by the movement of high angle grain boundaries.
4. Discussion and conclusions The plane strain compression of IF steels has been simulated by using the crystal plasticity finite element model. A simplified geometry of a polycrystal material is used in order to consider the grain interaction between neighboring grains during deformation. The finite element discretization in each grain can capture the subdivision of orientation and accumulation of stored energy at grain boundaries during plane strain compression of IF steels. In order to satisfy the compatibility and stress equilibrium, each grain undergoes inhomogeneous deformation which is observed in experimental results [13]. The Taylor model, which neglects stress equilibrium between neighboring grains, cannot provide subdivision of orientation during plane strain compression. This indicates that the homogenization scheme such as the Taylor model is not proper to estimate the orientationdependent stored energy in cold-rolled IF steels. It is clear that the orientation gradients calculated from the crystal plasticity finite element model strongly depend on the initial orientation of the grain as well as the neighboring grains. The distribution in the spreading of orientations simulated by the crystal plasticity finite element model can be classified into three types of grains. The type I grain including α-fibre and γ-fibre orientations as a main texture component exhibits relatively strong gradients of the stored energy. The type II grain including several γ-fibre orientations as a main texture component exhibits strong gradients of the stored energy and high density of the high angle grain boundaries. The type III grain including mostly single γ-fibre orientation as a main texture component exhibits weak gradients of the stored energy and a high density of the low angle grain boundaries. It was also found that the accumulation of the stored energy at the grain boundaries depends significantly on the local orientations of the neighboring grains. Grain boundaries between more favorably orientated grains and more unfavorably orientated grains
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exhibit high stored energy accumulation compared to the other grain boundaries. It is expected that such regions are more likely to be nucleation sites than the other grain boundaries during static recrystallization of cold-rolled IF steels. On the other hand, because of the quasi 3-D mesh of the model, the interactions with grains above and below the model grains cannot be considered in this study. In order to capture the interaction, a fully 3-D mesh is required in the model. However, the present work can be used in the coupling of the deformation model with recrystallization model such as the 2-D Monte Carlo technique [16,22] to simulate static recrystallization in cold-rolled IF steels.
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