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Simulation of stress concentration in Mg alloys using the crystal plasticity finite element method
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Acta Materialia 58 (2010) 320–329 www.elsevier.com/locate/actamat
Simulation of stress concentration in Mg alloys using the crystal plasticity finite element method S.-H. Choi a,*, D.H. Kim a, S.S. Park b, B.S. You b a
Department of Materials Science and Metallurgical Engineering, Sunchon National University, Sunchon 540-742, Republic of Korea b LightMetals Research Group, Korea Institute of Materials Science, Changwon 641-831, Republic of Korea Received 17 June 2009; received in revised form 4 August 2009; accepted 6 September 2009
Abstract A crystal plasticity finite element method (CPFEM), considering both crystallographic slip and deformation twinning, was developed to simulate the spatial stress concentration in AZ31 Mg alloys during in-plane compression. A predominant twin reorientation (PTR) model was successfully implemented to capture grain reorientation due to deformation twinning in twin-dominated deformation. By using the direct mapping technique for electron backscatter diffraction (EBSD) data, CPFEM can capture the heterogeneity of stress concentration at the grain boundaries in AZ31 Mg alloys during in-plane compression. The model demonstrated that deformation twinning enhances the local stress concentration at the grain boundaries between untwinned and twinned grains. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Crystal plasticity; Finite element; Stress concentration; Texture; Twinning
1. Introduction Mg alloys exhibit excellent strength-to-weight and stiffness-to-weight ratios. Therefore, wrought Mg alloys have been used in electric and lightweight structural parts for decades [1,2]. However, their low ductility and formability at room temperature (RT), and near RT, limit their broader application [3,4]. Critical resolved shear stress (CRSS) of non-basal slip systems at RT, and near RT, is much higher than that of a basal slip system [5–7]. The limited number of operative slip systems at RT, and near RT, is responsible for the poor ductility and formability. It is known that a high temperature and moderate strain rates are required to enhance ductility and formability of Mg alloys. It is also known that grain size and crystallographic texture play important roles in the ductility and formability of Mg alloys. To improve the ductility and formability of Mg alloys, processing technologies using severe plastic
*
Corresponding author. Tel.: +82 61 750 3556; fax: +82 61 750 3550. E-mail address: [email protected] (S.-H. Choi).
deformation have been investigated for reduction of grain size [8–10]. These results indicate that grain refinement enhances the activity of non-basal systems and reduces the volume fraction of deformation twins during plastic deformation. Modification of crystallographic texture is another method by which the ductility and formability of Mg alloys are improved. Casting, rolling and annealing conditions all contribute to the development of crystallographic texture in Mg alloys. The main texture component in wrought Mg alloys can be characterized as a sharp basal texture. Addition of alloy elements into Mg alloys induces spreading of the basal poles in the rolling direction (RD) [6,11]. Spreading of the basal plane enhances the activation of basal slip during plastic deformation. The various deformation modes, such as basal hai slip, prismatic hai slip, pyramidal hai, pyramidal hc + ai slip and tensile twinning, complicate the deformation behavior of Mg alloys. As the c/a pffiffiffi ratio of the hexagonal Mg lattice (1.624) is less than 3, a tensile twin is easily activated by c-axis tension [12,13]. Many theoretical studies have been conducted in an attempt to understand the effect of twin reorientation on the evolution of texture and macroscopic properties
1359-6454/$36.00 Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2009.09.010
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during plastic deformation [5,14,15–19]. Models have been suggested to explain the reorientation of the crystallographic orientation by deformation twinning [20–22]. A predominant twin reorientation (PTR) model tracks the new orientation created by deformation twinning relatively easily [21,23]. In the PTR model, twinning is considered to be a pseudo-slip mechanism and a grain is allowed to reorient rapidly if a specific accumulated value reaches the threshold. The crystal plasticity finite element method (CPFEM) was developed to simulate heterogeneous plastic deformation of hexagonal close packed (hcp) polycrystalline materials [24–27]. Numerical formulation and verification of CPFEM have been well developed for fcc and bcc materials deformed by crystallographic slip [28–32]. A difficult aspect of the current theoretical framework of CPFEM for hcp materials is the incorporation of a reorientation scheme by deformation twinning into the constitutive equations. Staroselsky and Anand [25] used a probabilistic approach to simulate the texture evolution and stress–strain response of a polycrystalline Mg alloy. In this approach, the orientations of grains were replaced by twin-related orientations only if the twinned volume fraction exceeded a certain random number. Kalidindi [22] proposed a total Lagrangian approach to simulate rolling textures in a polycrystalline Zr alloy. The same theoretical framework was used to simulate texture evolution and stress–strain response in high purity a-Ti [27,33]. Walde and Riedel [34] used the PTR model to simulate earing profiles after deep drawing. They tried to predict the influence of the initial texture components on the evolution of the earing profiles. Unfortunately, the authors did not compare experimental and theoretical results. Graff et al. [26] conducted a simulation of channel die compression that revealed a strong anisotropy and asymmetric yield behavior. This model did not take the crystallographic reorientation of the twinned volume into account. To improve the ductility and formability of AZ31 Mg alloys, the effect of deformation twinning on the heterogeneous stress concentration at grain boundaries or twin boundaries must be understood theoretically. Theoretical studies using the CPFEM have not been conducted to explain how the deformation twinning affects the stress concentration at grain or twin boundaries in AZ31 Mg alloys during plastic deformation. In the present work, we present a theoretical framework for including crystallographic slip and deformation twinning in the CPFEM. The PTR model was implemented to capture grain reorientation due to deformation twinning. A topological mapping method based on electron backscatter diffraction (EBSD) data was used to simulate the spatial stress concentration in AZ31 Mg alloys during in-plane compression. The validity of the proposed model framework was verified by comparing the predicted texture evolution and twin volume fraction of AZ31 Mg alloys with the measured experimental results.
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2. Experimental procedure and theoretical methods 2.1. Materials and microstructure The present study used strip-cast AZ31 (3 wt.% Al, 1 wt.% Zn, balance Mg) Mg alloy followed by hot rolling. The 2 mm thick sheet had a grain size of approximately 10.16 lm. The EBSD technique was used to analyze the microtexture and twin volume fraction of as-rolled and deformed specimens. The specimens, deformed to true strains of 0.05 and 0.1, were cut in a rolling direction (RD)–normal direction (ND) section parallel to the compression axis. The specimens were prepared using colloidal silica as the polishing medium for the intermediate stage. The final specimens were prepared by electro-polishing in AC2 electrolyte for the final stage. Automated EBSD scans were measured in the stage-control mode using TSL data acquisition software. Microtexture and twin volume fraction were examined by scanning an area of 100 100 lm2 at a step size of 0.25 lm. The EBSD data were analyzed using the TSL software to evaluate the pole figure, ODF and twin volume fraction. 2.2. In-plane compression Compressive specimens were machined by laser cutting from the as-rolled sheet. The initial height and width of the compressive specimens were 4 mm and 10 mm, respectively. The main experimental difficulty encountered during evaluation of the in-plane compression was the buckling phenomenon. To avoid buckling, either through-thickness sheet stabilization [35] or a thicker Mg sheet has been used [7]. In the present study, a thicker Mg sheet with a low ratio (=2) of height to thickness was used to avoid the buckling phenomenon. The specimens were installed in a GLEEBLEÒ 3500C thermo-mechanical simulator and heated by means of an inductive heating device at a rate of 5 °C/s. After holding at 200 °C for 1 min, they were deformed at a strain rate of 0.1 s1. The loading direction of specimens was parallel to the RD of the sheet and allowed to shrink or expand in the transverse direction (TD) and ND of the sheet. To capture the effect of strain on the evolution of crystallographic texture and deformation twinning, the tests were stopped at true strains of 0.05 and 0.1. After the plastic deformation, the deformed specimens were cooled in water to maintain their deformed microstructures. 3. Theoretical procedure This study employed the crystal plasticity theory, developed by Peirce et al. [36] and by Asaro and Needleman [37], as the constitutive model. A rate-dependent constitutive relationship was implemented into the user material subroutine UMAT in the commercial finite element code, ABAQUS/Standard [38]. The model was fundamentally based on a multiplicative decomposition of the deforma-
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tion gradient, F, into a plastic part characterized by shearing rates on active slip and twin systems, as well as a part that accounts for the rotation and elastic distortion of the crystal lattice. F ¼ Fe Fp
ð1Þ
This formula leads to additive decomposition of the velocity gradient into elastic and plastic parts, L ¼ Le þ Lp
ð2Þ
with the plastic part determined by slip rates, c_ a , on slip/ twin planes with normals, ma , and slip/twin directions, sa Lp ¼
N X
c_ a sa ma
ð3Þ
a¼1
The summation represents all of the deformation modes, N (=Ns + Nt), consisting of slip, Ns, and twin systems, Nt. As described in [36], the Jaumann rate of Kirchhof stress can be expressed as ^s ¼ K : D
N X
c_ a Ra
ð4Þ
a¼1
where K is a fourth-order tensor based on the anisotropic elastic modulus, C. D is the rate of deformation tensor (symmetric part of the velocity gradient), and Ra is a tensor that depends on the current slip/twin plane normal and direction, the applied stress and the elastic modulus. For materials with hexagonal crystal symmetry, five, independent elastic constants of pure Mg were used in the present work [39]: C 11 ¼ 58 GPa; C 12 ¼ 25 GPa; C 13 ¼ 20:8 GPa; C 33 ¼ 61:2 GPa; C 55 ¼ 16:6 GPa For rate-dependent materials, shear rates are given explicitly in terms of the resolved shear stress on the active slip/twin systems and the resistance of the active slip/twin systems to shear. For these simulations, this dependence is given by a 1=m s ð5Þ c_ a ¼ c_ ao a signðsa Þ so Self- and latent-hardening are readily accounted for by a suitable evolution of the reference sao values in the constitutive law in Eq. (5). The present work employed a microscopic hardening law [40,32] for this purpose, as follows: s_ ao ¼
N X
H ab j_cb j a; b ¼ 1 . . . ðN s þ N t Þ
H ab
equals the latent-hardening term (off-diagonal term of qab), i.e. (qab = 1). The fitting simulation was carried out by varying the CRSS values and microscopic hardening parameters (ho, ssat and a) until agreement was achieved between the predicted and the measured flow curves. A 3-D mesh (20 20 10 = 4000 elements) of a polycrystal model was used to predict the macroscopic flow curve [41]. An ODF measured X-ray diffraction was used to generate a set of 4000-grain orientation for polycrystal modeling with the help of orientation repartition functions [42]. To impose initial orientation of the elements, each orientation selected from the 4000-grain orientation was mapped onto each integration point in the finite element mesh. The set of parameters listed in Table 1 was used in the theoretical simulation. In this study, four slip systems and one twin system were considered: basal hai ({0 0 0 1}h1 1 2 0i), prismatic hai (f1 1 0 0g h1 1 2 0i), pyramidal hai (f1 1 0 0g h1 1 2 0i) pyramidal hc + ai (f1 1 2 2g h1 1 2 3i) and tensile twin (f1 0 1 2g h1 0 1 1i). At RT, the existence of pyramidal hc + a i slip is not established in Mg alloys, but the slip system is an active deformation mechanism at elevated temperature when twinning is suppressed [4]. To consider the effect of deformation twinning on texture evolution and stress concentration in Mg alloys, the predominant twin reorientation (PTR) model [21] was implemented in the CPFEM. This requires tracking of the shear strain, ct,g, contributed by each twin system t, and of the associated volume fraction V t;g ¼ ct;g =S t ðS t ¼ 0:129 is the characteristic twin shear) within each orientation, g. By summation of all systems in each element, the accumulated twin fraction, Vacc,mode, in each orientation can be determined as follows: X V acc;mode ¼ ct;g =S t ð7Þ t
At each incremental step, the fractions accumulated in the individual twinning systems of each orientation are compared against a threshold fraction, Vth,mode, defined as follows: V th;mode ¼ C th1 þ C th2 V acc;mode
ð8Þ
After each deformation increment, the twin system with the highest accumulated volume fraction is identified. If the accumulated volume fraction is greater than the threshold fraction, Vth,mode, the orientation is allowed to reorient. After all, the threshold fraction, Vth,mode, increases gradually, and further reorientation by twinning can be inhibited by large deformations. The algorithm prevents reorienta-
b
a sao ¼ q ho 1 ssat ab
ð6Þ
where Hab is a hardening matrix that is introduced to account for interaction between the slip and twin systems. qab accounts for the hardening rate of the slip/twin system, a, due to slip/twin activity on the system, b. Here, it is assumed that the self-hardening term (diagonal term of qab)
Table 1 Microscopic hardening coefficients used in the CPFEM simulation. Mode
sao (MPa)
ho (MPa)
ssat (MPa)
a
Basal hai Prism hai Pyram hai Pyram hc + ai Twin
25 68 68 68 40
100 130 130 130 50
70 210 210 210 50
1.1 0.8 0.8 0.8 1.1
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tion by twinning until a threshold value, Cth1, is attained. The threshold value, Cth2, determines the evolution of the twin volume fraction during plastic deformation. Reportedly, the volume fraction of twinning is strongly dependent on the initial microstructure and deformation conditions, such as strain, strain rate and temperature [43,44]. Therefore, the optimal combination of constants in Eq. (8) should be determined by consideration of the initial microstructure and deformation conditions. Experimental observation indicated that one or more twin variants accommodate plastic deformation within each orientation [45,23]. However, the PTR model assumes that each orientation is allowed to reorient with respect to the normal direction of a mirror plane in the most active twin system, as determined by the CPFEM. The transformation matrix, T, between the lattice orientation in the matrix and the lattice orientation in the twinned region can be defined as [20] T ij ¼ 2ni nj dij ;
dij ¼ 0
if
i–j;
dij ¼ 1
ð9Þ
where n represents the unit vector of the twin plane normal in orthogonal coordinates.The integration of the stress rate (Eq. (4)) with the unknowns, c_ a , requires small time intervals for stable numerical integration. The implicit time integration algorithm for the slip rates is similar to the rate tangent method of Peirce et al. [36]. Moreover, a material Jacobian (@D^s=@De) for the implicit method can be obtained from Eq. (4). This is equivalent to the modulus given by Peirce et al. [36]. The stress vector in the UMAT subroutine can be updated by multiplying the material Jacobian by the strain increment. It should be noted that the material Jacobian is required in the iterative procedure in minimizing the force residual. If the equilibrium tolerances had not been satisfied, ABAQUS iterates by providing the constitutive routine with updated estimates of the strain increment.
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4. Results and discussion Fig. 1a shows the microtexture of the as-rolled AZ31 Mg alloy. The alloy exhibited a relatively inhomogeneous and coarse distribution of grain size. The microtexture evolution of AZ31 Mg alloy during in-plane compression to a true strain of 0.05 is shown in Fig. 1b. After uniaxial compression to a true strain of 0.05, twin bands were nucleated and propagated through parent grains. The shape of twinned regions can be classified into lenticular and partially propagated types. At a true strain of 0.1, as shown in Fig. 1c, the propagation of twin bands was almost prohibited by collision with high angle grain boundaries and other twin bands at the twin boundaries. The as-rolled AZ31 Mg alloy can be characterized as a sharp basal fiber texture, as shown in Fig. 2a–c shows the measured deformation textures for compressive specimens of the AZ31 Mg alloy deformed to a true strain of 0.05 and 0.1 along the RD, respectively. The microtexture revealed that many crystallographic lattices rotate to the RD by twin-induced reorientation. The main texture components developed in deformed AZ31 Mg alloys under in-plane compression can be represented by pole figures, as shown in Fig. 2d. Analysis showed that the F texture component is a stable orientation in the specimen deformed to a true strain of 0.05. It is also clear that the near F and H texture components are relatively stable orientations in the specimen deformed to a true strain of 0.1. Crystallographic orientations of untwined lattices were analyzed as the E and G texture components. Volume fractions of parent grains and twin bands were identified using twin-analysis technique implemented in the TSL software. In the case of partially propagated twins, it is easy to identify the twin volume fraction using the orientation relationship between parent grains and twin bands. Fig. 1b shows the volume fraction
Fig. 1. Inverse pole figure map (the position of the ND in the colored stereographic triangle) showing the texture evolution of AZ31 Mg alloys: (a) asrolled, (b) true strain = 0.05, (c) true strain = 0.1.
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Fig. 2. Experimentally measured pole figures and ODF sections showing (a) the initial texture of as-rolled specimen and deformation texture of the CA// RD specimen,: (b) true strain = 0.05, (c) true strain = 0.1 and (d) a graphical representation of the main texture components.
of the twinned region in the specimen deformed to a true strain of 0.05. In the case of fully propagated twins, more careful analysis is required to identify the twin volume fraction. The twin-analysis technique implemented in the TSL software does not work for the deformed specimen containing fully propagated twins. Since the as-rolled AZ31 alloy was characterized as a strong fiber texture, a direct comparison of pole figures can distinguish the twin regions undergoing a drastic change of crystallographic orientation from the untwinned region undergoing a gradual change of crystallographic orientation. Fig. 1c shows the volume fraction of the twinned region in the specimen deformed to a true strain of 0.1. To evaluate the spatial stress concentration in the deformed AZ31 Mg alloys, a direct mapping method based on EBSD data was used in the CPFEM. The dashed line in Fig. 1a shows the cropped region from the as-rolled microtexture for the simulation. All scan data was directly mapped onto the quasi 3-D finite element meshes (200 200 1 = 40,000 elements), as shown in Fig. 3a. The initial length of the model region is given by lo = 50 lm, wo = 50 lm, to = 0.25 lm. A true compressive strain of e = 0.1 was simulated using appropriate boundary conditions. The boundary conditions were applied to the four planes comprising the quasi 3-D mesh, as shown in Fig. 3b. Since the cropped region represents the small part of a larger body, boundary conditions considering neighboring grains should be applied. Although several ways are available to impose boundary conditions, none rigorously represent a real condition. In the present study, for the simulation of
uniaxial compression, prescribed displacement in the RD was imposed on the 2-3-6-7 face. The three faces (1-2-3-4, 1-2-5-6 and 1-4-5-8) were constrained so as not to displace. The two moving faces (5-6-7-8 and 3-4-7-8) were ascribed to the remaining plane. However, simple boundary conditions can induce a considerable shear strain at the boundary regions if a significant displacement is imposed on the 2-3-6-7 face. In addition, since the quasi 3-D mesh of the model cannot be used to consider the interactions with neighboring grains above and below the model grains, a fully 3-D mesh was required to capture the interaction. In further work, a 3-D EBSD scan using the serial sectioning technique could be used to construct a fully 3-D mesh. Fig. 4a shows the spatial distribution of the Schmid factor (mSF) for the cropped region from the as-rolled microtexture. For the calculation of m values, the basal hai slip system was only considered under the boundary conditions of uniaxial compression. As a first approximation, grains having high mSF values were expected to be deformed by slip-dominated deformation at a low-strain level. By contrast, grains having low mSF values were expected to be deformed by twin-dominated deformation at a low-strain level. To capture which grains were favorable for the twin-dominated deformation under uniaxial compression, grains having relatively low mSF values ranging between 0.2 mSF,max and mSF,min are highlighted, as shown in Fig. 4b. Fig. 5a shows the simulated microtexture for the CA// RD specimen to a true strain of 0.05. To represent the deformed microstructure using the inverse pole figure
S.-H. Choi et al. / Acta Materialia 58 (2010) 320–329
325
b
a
8 4
(1-2-3-4) : UTD= 0 (1-2-5-6) : UND= 0 (1-4-5-8) : URD= 0 (2-3-6-7) : URD= compression
5 1
7 3
TD
ND
TD
ND
RD
RD
6 2
20
Fig. 3. A direct mapping method of EBSD data for the simulation of stress concentration in deformed AZ31 Mg alloy sheets: (a) cropped region for finite element meshing, (b) boundary condition for uniaxial compression.
Fig. 4. Spatial distribution of the Schmid factor (mSF) of as-rolled AZ31 Mg alloys: (a) mSF,max mSF,min (b) 0.2 mSF,max mSF,min.
map instead of finite element meshes, crystallographic orientations calculated by CPFEM were mapped onto regular grids to satisfy the data format of TSL software. Fig. 5b1 shows the twin volume fraction after uniaxial compression to a true strain of 0.05. Blue and red colors represent portions of untwinned and twinned elements, respectively. The CPFEM successfully predicted the twin volume fraction of the CA//RD specimen to a true strain of 0.05. It was found that the shape of twinned regions simulated by the CPFEM could be classified into lenticular (1–7) and partially propagated (8–25) types. However, comparison of Figs. 4 and 5 b a reveals that the mSF value has a limitation to predict grains to be deformed by twin-dominated deformation at
1 For interpretation of color in Fig. 5b, the reader is referred to the web version of this article.
a low-strain level. Grains having a low mSF value ranging between 0.2 mSF,max and mSF,min did not exactly match with the twinned regions calculated by the CPFEM. It should be noted that the partial twinned regions (1, 2, 5, 9, 15, 16, 17, 24) correspond to the grains having a low mSF value. Even some twinned regions (8, 10) corresponded with the grains having high mSF values. These results indicate that the Schmid factor cannot be used to explain the heterogeneous distribution of twinned regions during plastic deformation at a low-strain level. However, the CPFEM successfully predicted the heterogeneous distribution of twinned regions during plastic deformation at a strain level, as shown in Fig. 5b. Heterogeneous distribution of accumulation twin fraction in Eq. (7) seemed to be related to the heterogeneous nucleation of twin bands during plastic deformation. To understand the heterogeneous nucleation of twin bands, the associated volume
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Fig. 5. Spatial distribution of simulated microstructure for the CA//RD specimen to a true strain of 0.05: (a) crystallographic orientation map and (b) deformation twinning.
fraction accumulated in the individual twinning system, ct;g =S t , in Eq. (7) was analyzed for the deformed microstructure. Fig. 6 shows the spatial distribution of the individual associated volume fractions in the deformed microstructure after uniaxial compression to a true strain of 0.05. It is noteworthy that the accumulation of shear strain in each twin system exhibited a distinct heterogeneity
in the deformed microstructure. Since the twin system having the highest accumulated volume fraction determines the transformation matrix, T, in Eq. (9), the various crystallographic orientations in twinned region were attributed to the heterogeneous distribution of the associated volume fraction. Fig. 7a shows the simulated deformation texture of the CA//RD specimen to a true strain of 0.05. Compar-
Fig. 6. Spatial distribution of the individual associated volume fractions in the deformed microstructure after uniaxial compression to a true strain of 0.05.
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(i) intergranular stress between untwinned grains (b– c, c–e, g–h); (ii) intragranular stress between an untwinned grain and a twinned grain (a-8, b-2, b-9, c-1, d-12, d-13, e-6, e14, f-14, f-18, g-3, g-4, g-15, h-15, h-18, i-18, i-19, j-19, j-24); and, (iii) intragranular stress between twinned grains (2–9, 15– 18, 17–18).
Fig. 7. Simulated deformation texture of the CA//RD specimens: (a) true strain = 0.05 and (b) true strain = 0.1.
ing EBSD data, as shown in Fig. 2b, it is clear that CPFEM accurately simulated the rotation of the c-axis from the ND to the RD on the (0 0 0 2) pole figure. It appears that the activity of the tensile twin played an important role in rotation of the c-axis from the ND to the RD during plastic deformation. An effective stress computed by the von Mises’ formulation was used to evaluate the spatial stress concentration in the deformed AZ31 alloy. For a grain-level crack nucleation criterion, a more rigorous effective stress based on slip system stress fields [46] can be used to understand crack nucleation due to stress concentration. It should be noted that the present study concentrated on the grain-level stress field instead of the grain-level crack nucleation criterion. Fig. 8 shows the spatial distribution of effective stress for the CA//RD specimen to a true strain of 0.05. During plastic deformation, heterogeneous stress distribution occurred at regions near the grain boundaries of untwinned grains (a-j). Heterogeneous stress concentration can be classified into three types:
Fig. 8. Spatial distribution of effective stress computed by the von Mises’ formulation for the CA//RD specimen to a true strain of 0.05: untwinned regions (a–l) and twinned regions (1–25).
These results indicate that plastic deformation induces stress concentration at the regions in the vicinity of the grain boundary separating two untwinned grains. Analysis showed that the heterogeneous stress concentration can be attributed to the activation of non-basal slip systems having high CRSS values to satisfy strain compatibility at the grain boundary. These results also indicate that plastic deformation induces stress concentration along grain boundaries of twinned grains as well as within twinned grains. Analysis also showed that the heterogeneous stress concentration in twinned grains can be attributed to the activation of non-basal slip systems to satisfy strain compatibility at the grain boundary after twin-induced reorientation. According to the CPFEM simulation of Mg alloys using the representative volume elements [41], plastic deformation began with the activation of a tensile twin and a basal hai slip as the primary mode during in-plane compression. Subsequently, the tensile twin decreased continuously and the pyramidal hc + ai slip increased gradually during plastic deformation. Apparently, stress gradients can be found in the twinned grains from the grain boundary to the grain interior. Fig. 9a shows the simulated microtexture for the CA//RD specimen to a true strain of 0.1. The deformed grains consist mainly of twinned grains and partly untwinned grains, as shown in Fig. 9b.2 Blue and red regions represent portions of untwinned and twinned elements, respectively. The twin volume fraction predicted by the CPFEM is in very good agreement with the experimental data, as shown in Fig. 1. Twinned regions consisting of fully propagated twin bands and partially propagated regions are rarely found in the microstructure. Fig. 7b shows the simulated deformation texture of a CA// RD specimen to a true strain of 0.1. Comparing EBSD data, as shown in Fig. 1b, it is clear that CPFEM successfully simulated the rotation of the c-axis from the ND to the RD on the (0002) pole figure. These results indicate that a near-F texture component is a relatively stable orientation in the CA//RD specimen. Fig. 10 shows the distribution of effective stress for the CA//RD specimen to a true strain of 0.1. Analysis showed that heterogeneous stress distribution had occurred at the grain boundaries of untwinned grains (a–o0 ). The untwinned grains were surrounded either by twinned grains (10 –130 ) or by untwinned
2 For interpretation of color in Fig. 9b, the reader is referred to the web version of this article.
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Fig. 9. Spatial distribution of simulated microstructure for the CA//RD specimen to a true strain of 0.1: (a) crystallographic orientation map and (b) deformation twinning.
However, it should be noted that the density of type (ii) stress concentration increased as the true strain increased. Similar to the specimen deformed to a true strain of 0.05, stress gradients also could be observed in the twinned grains from the grain boundary to the grain interior. The twinned grains also exhibited a distinct stress concentration compared to the untwinned grains. These results showed that deformation twinning enhanced the local stress concentration at regions near the grain boundaries neighboring untwinned grains. 5. Conclusions
Fig. 10. Spatial distribution of effective stress computed by the von Mises’ formulation for he CA//RD specimen to a true strain of 0.1: untwinned regions (a0 –o0 ) and twinned regions (10 –130 ).
grains. The heterogeneous stress concentration can be also classified into three types: (i) intergranular stress between untwinned grains (b0 –c0 , c0 –g0 , k0 –l0 ); (ii) intragranular stress between an untwinned grain and a twinned grain (a0 -10 , b0 -20 , b0 -30 , c0 -30 , f0 -100 , g0 -70 , g0 80 , h0 -70 , h0 -80 , h0 -90 , i0 -110 , j0 -120 , k0 -70 , m0 -50 , m0 -60 , n0 40 , o0 -50 ); (iii) intragranular stress between twinned grains (90 –130 ). A comparison of Figs. 8 and 10 shows that the stress concentrations in the specimen deformed to a true strain of 0.1 are similar to that in the specimen deformed to a true strain of 0.05, regardless of the type of stress concentration.
A crystal plasticity finite element method (CPFEM) that considers both crystallographic slip and deformation twinning was developed to simulate the spatial stress concentration in AZ31 Mg alloys during in-plane compression. The major feature of the model is implementation of a PTR model to capture grain reorientation due to deformation twinning. The CPFEM was used to simulate spatial stress concentration in deformed AZ31 Mg alloy under uniaxial compression. A direct mapping method of EBSD data onto quasi 3-D finite element meshes was employed to consider a real microstructure as an initial configuration. The CPFEM successfully simulated the shape of twinned regions such as lenticular and partially propagated types that are typical shapes of in-plane compression specimens at a low-strain level. Simulation results indicated that the stress concentration can be classified into three types: (i) intergranular stress between untwinned grains; (ii) intragranular stress between an untwinned grain and a twinned grain; and (iii) intragranular stress between twinned grains. Theoretical analysis showed that type (ii) exhibited the
S.-H. Choi et al. / Acta Materialia 58 (2010) 320–329
highest stress concentration and stress gradients in the twinned grains from the grain boundary to the grain interior, regardless of the true strain at a low-strain level. These results showed that deformation twinning, and twininduced reorientation in particular, should be considered in the CPFEM analysis of heterogeneous stress concentration in deformed Mg alloys. Acknowledgments This study was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korean government (MEST) (R01-2007-000-10015-0) and a grant from the Fundamental R&D Program for Core Technology of Materials funded by the Ministry of Knowledge Economy, Republic of Korea. References [1] Kainer KU, Bohlen J, Letzig D, Mueller S, Mueller K, Reimers W. In: IMA conference proceedings; 2007. p. 23. [2] Luo AA, Sachdev AK, Mishra RK. In: IMA conference proceedings; 2007. p. 67. [3] Yin DL, Zhang KF, Wang GF, Han W-B. Mater Sci Eng 2005;A392:320. [4] Agnew SR, Duygulu O. Int J Plast 2005;21:1161. [5] Kelly EW, Hosford WF. Trans Metall Soc AIME 1968;242:5. [6] Agnew SR, Yoo M-H, Tome´ CN. Acta Mater 2001;49:4277. [7] Jain A, Agnew SR. Mater Sci Eng 2007;A462:29. [8] Koike J, Kobayashi K, Mukai T, Watanabe H, Suzuki M, Maruyama K, et al. Acta Mater 2003;51:2055. [9] Kim W-J, Park J-D, Wang J-Y, Yoon W-S. Scripta Mater 2007;57:755. [10] Somekawa H, Osawa Y, Mukai T. Scripta Mater 2006;55:593. [11] Mackenzie LWF, Pekguleryuz M. Mater Sci Eng 2008;A480:189. [12] Yoo M-H. Metall Trans A 1981;12A:409. [13] Yoo M-H, Lee J-K. Philos Mag A 1991;63:987. [14] Reed-Hill RE. Trans Metall Soc AIME 1960;218:554. [15] Wonsiewicz BC, Backofen WA. Trans Metall Soc AIME 1967;239:1422.
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[16] Hartt WH, Reed-Hill RE. Trans Metall Soc AIME 1968;242:1127. [17] Nave MD, Barnett MR. Scripta Mater 2004;51:881. [18] Jiang L, Jonas JJ, Luo AA, Sachdev AK, Godet S. Mater Sci Eng 2007;A445:302. [19] Wang YN, Huang JC. Acta Mater 2007;55:897. [20] Houtte PV. Acta Metall 1978;26:591. [21] Tome´ CN, Lebensohn RA, Kocks UF. Acta Metall Mater 1991;39: 2667. [22] Kalidindi SR. J Mech Phys Solids 1998;46:267. [23] Choi S-H, Shin E-J, Seong B-S. Acta Mater 2007;55:4181. [24] Balasubramanian S, Anand L. Acta Mater 2002;50:133. [25] Staroselsky A, Anand L. Int J Plast 2003;19:1843. [26] Graff S, Brocks W, Steglich D. Int J Plast 2007;23:1957. [27] Wu X, Kalidindi SR, Necker C, Salem AA. Acta Mater 2007;55:423. [28] Becker R. Acta Metal Mater 1991;39:1211. [29] Becker R, Panchanadeeswaran S. Acta Metal Mater 1995;43:2701. [30] Sarma GB, Radhakrishnan B, Zacharia T. Compos Mater Sci 1998;12:105. [31] Raabe D, Zhao Z, Park SJ, Roters F. Acta Mater 2002;50:421. [32] Choi S-H. Acta Mater 2003;51:1775. [33] Salem AA, Kalidindi SR, Semiatin SL. Acta Mater 2005;53: 3495. [34] Walde T, Riedel H. Acta Mater 2007;55:867. [35] Lou XY, Li M, Boger RK, Agnew SR, Wagoner RH. Int J Plast 2007;237:44. [36] Peirce D, Asaro RJ, Needleman A. Acta Metall 1983;31:1951. [37] Asaro RJ, Needleman A. Acta Metall 1985;33:923. [38] ABAQUS user0 s manual, version 6.5. Providence, Hibbit, Karlsson & Sorenson, RI; 2004. [39] Simmons G, Wang H. Single crystal elastic constants and calculated aggregate properties: a handbook. Cambridge, (MA): MIT Press; 1971. [40] Kalidindi SR, Bronkhorst CA, Anand L. J Mech Phys Solids 1992;40:537. [41] Choi S-H, Kim DH, Lee HW, Shin EJ, in press. [42] Francois M, Sprauel JM, Lebrun JL. Texture Microstruct 1991;169: 14. [43] Jiang L, Jonas JJ, Luo AA, Sachdev AK, Godet S. Scripta Mater 2006;54:771. [44] Choi S-H, Kim JK, Kim BJ, Park YB. Mater Sci Eng 2008;A488: 458. [45] Yang P, Yu Y, Chen L, Mao W. Scripta Mater 2004;50:1163. [46] Kirane K, Ghosh S. Int J Fatigue 2008;30:2127.
Acta Materialia 58 (2010) 320–329 www.elsevier.com/locate/actamat
Simulation of stress concentration in Mg alloys using the crystal plasticity finite element method S.-H. Choi a,*, D.H. Kim a, S.S. Park b, B.S. You b a
Department of Materials Science and Metallurgical Engineering, Sunchon National University, Sunchon 540-742, Republic of Korea b LightMetals Research Group, Korea Institute of Materials Science, Changwon 641-831, Republic of Korea Received 17 June 2009; received in revised form 4 August 2009; accepted 6 September 2009
Abstract A crystal plasticity finite element method (CPFEM), considering both crystallographic slip and deformation twinning, was developed to simulate the spatial stress concentration in AZ31 Mg alloys during in-plane compression. A predominant twin reorientation (PTR) model was successfully implemented to capture grain reorientation due to deformation twinning in twin-dominated deformation. By using the direct mapping technique for electron backscatter diffraction (EBSD) data, CPFEM can capture the heterogeneity of stress concentration at the grain boundaries in AZ31 Mg alloys during in-plane compression. The model demonstrated that deformation twinning enhances the local stress concentration at the grain boundaries between untwinned and twinned grains. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Crystal plasticity; Finite element; Stress concentration; Texture; Twinning
1. Introduction Mg alloys exhibit excellent strength-to-weight and stiffness-to-weight ratios. Therefore, wrought Mg alloys have been used in electric and lightweight structural parts for decades [1,2]. However, their low ductility and formability at room temperature (RT), and near RT, limit their broader application [3,4]. Critical resolved shear stress (CRSS) of non-basal slip systems at RT, and near RT, is much higher than that of a basal slip system [5–7]. The limited number of operative slip systems at RT, and near RT, is responsible for the poor ductility and formability. It is known that a high temperature and moderate strain rates are required to enhance ductility and formability of Mg alloys. It is also known that grain size and crystallographic texture play important roles in the ductility and formability of Mg alloys. To improve the ductility and formability of Mg alloys, processing technologies using severe plastic
*
Corresponding author. Tel.: +82 61 750 3556; fax: +82 61 750 3550. E-mail address: [email protected] (S.-H. Choi).
deformation have been investigated for reduction of grain size [8–10]. These results indicate that grain refinement enhances the activity of non-basal systems and reduces the volume fraction of deformation twins during plastic deformation. Modification of crystallographic texture is another method by which the ductility and formability of Mg alloys are improved. Casting, rolling and annealing conditions all contribute to the development of crystallographic texture in Mg alloys. The main texture component in wrought Mg alloys can be characterized as a sharp basal texture. Addition of alloy elements into Mg alloys induces spreading of the basal poles in the rolling direction (RD) [6,11]. Spreading of the basal plane enhances the activation of basal slip during plastic deformation. The various deformation modes, such as basal hai slip, prismatic hai slip, pyramidal hai, pyramidal hc + ai slip and tensile twinning, complicate the deformation behavior of Mg alloys. As the c/a pffiffiffi ratio of the hexagonal Mg lattice (1.624) is less than 3, a tensile twin is easily activated by c-axis tension [12,13]. Many theoretical studies have been conducted in an attempt to understand the effect of twin reorientation on the evolution of texture and macroscopic properties
1359-6454/$36.00 Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2009.09.010
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during plastic deformation [5,14,15–19]. Models have been suggested to explain the reorientation of the crystallographic orientation by deformation twinning [20–22]. A predominant twin reorientation (PTR) model tracks the new orientation created by deformation twinning relatively easily [21,23]. In the PTR model, twinning is considered to be a pseudo-slip mechanism and a grain is allowed to reorient rapidly if a specific accumulated value reaches the threshold. The crystal plasticity finite element method (CPFEM) was developed to simulate heterogeneous plastic deformation of hexagonal close packed (hcp) polycrystalline materials [24–27]. Numerical formulation and verification of CPFEM have been well developed for fcc and bcc materials deformed by crystallographic slip [28–32]. A difficult aspect of the current theoretical framework of CPFEM for hcp materials is the incorporation of a reorientation scheme by deformation twinning into the constitutive equations. Staroselsky and Anand [25] used a probabilistic approach to simulate the texture evolution and stress–strain response of a polycrystalline Mg alloy. In this approach, the orientations of grains were replaced by twin-related orientations only if the twinned volume fraction exceeded a certain random number. Kalidindi [22] proposed a total Lagrangian approach to simulate rolling textures in a polycrystalline Zr alloy. The same theoretical framework was used to simulate texture evolution and stress–strain response in high purity a-Ti [27,33]. Walde and Riedel [34] used the PTR model to simulate earing profiles after deep drawing. They tried to predict the influence of the initial texture components on the evolution of the earing profiles. Unfortunately, the authors did not compare experimental and theoretical results. Graff et al. [26] conducted a simulation of channel die compression that revealed a strong anisotropy and asymmetric yield behavior. This model did not take the crystallographic reorientation of the twinned volume into account. To improve the ductility and formability of AZ31 Mg alloys, the effect of deformation twinning on the heterogeneous stress concentration at grain boundaries or twin boundaries must be understood theoretically. Theoretical studies using the CPFEM have not been conducted to explain how the deformation twinning affects the stress concentration at grain or twin boundaries in AZ31 Mg alloys during plastic deformation. In the present work, we present a theoretical framework for including crystallographic slip and deformation twinning in the CPFEM. The PTR model was implemented to capture grain reorientation due to deformation twinning. A topological mapping method based on electron backscatter diffraction (EBSD) data was used to simulate the spatial stress concentration in AZ31 Mg alloys during in-plane compression. The validity of the proposed model framework was verified by comparing the predicted texture evolution and twin volume fraction of AZ31 Mg alloys with the measured experimental results.
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2. Experimental procedure and theoretical methods 2.1. Materials and microstructure The present study used strip-cast AZ31 (3 wt.% Al, 1 wt.% Zn, balance Mg) Mg alloy followed by hot rolling. The 2 mm thick sheet had a grain size of approximately 10.16 lm. The EBSD technique was used to analyze the microtexture and twin volume fraction of as-rolled and deformed specimens. The specimens, deformed to true strains of 0.05 and 0.1, were cut in a rolling direction (RD)–normal direction (ND) section parallel to the compression axis. The specimens were prepared using colloidal silica as the polishing medium for the intermediate stage. The final specimens were prepared by electro-polishing in AC2 electrolyte for the final stage. Automated EBSD scans were measured in the stage-control mode using TSL data acquisition software. Microtexture and twin volume fraction were examined by scanning an area of 100 100 lm2 at a step size of 0.25 lm. The EBSD data were analyzed using the TSL software to evaluate the pole figure, ODF and twin volume fraction. 2.2. In-plane compression Compressive specimens were machined by laser cutting from the as-rolled sheet. The initial height and width of the compressive specimens were 4 mm and 10 mm, respectively. The main experimental difficulty encountered during evaluation of the in-plane compression was the buckling phenomenon. To avoid buckling, either through-thickness sheet stabilization [35] or a thicker Mg sheet has been used [7]. In the present study, a thicker Mg sheet with a low ratio (=2) of height to thickness was used to avoid the buckling phenomenon. The specimens were installed in a GLEEBLEÒ 3500C thermo-mechanical simulator and heated by means of an inductive heating device at a rate of 5 °C/s. After holding at 200 °C for 1 min, they were deformed at a strain rate of 0.1 s1. The loading direction of specimens was parallel to the RD of the sheet and allowed to shrink or expand in the transverse direction (TD) and ND of the sheet. To capture the effect of strain on the evolution of crystallographic texture and deformation twinning, the tests were stopped at true strains of 0.05 and 0.1. After the plastic deformation, the deformed specimens were cooled in water to maintain their deformed microstructures. 3. Theoretical procedure This study employed the crystal plasticity theory, developed by Peirce et al. [36] and by Asaro and Needleman [37], as the constitutive model. A rate-dependent constitutive relationship was implemented into the user material subroutine UMAT in the commercial finite element code, ABAQUS/Standard [38]. The model was fundamentally based on a multiplicative decomposition of the deforma-
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tion gradient, F, into a plastic part characterized by shearing rates on active slip and twin systems, as well as a part that accounts for the rotation and elastic distortion of the crystal lattice. F ¼ Fe Fp
ð1Þ
This formula leads to additive decomposition of the velocity gradient into elastic and plastic parts, L ¼ Le þ Lp
ð2Þ
with the plastic part determined by slip rates, c_ a , on slip/ twin planes with normals, ma , and slip/twin directions, sa Lp ¼
N X
c_ a sa ma
ð3Þ
a¼1
The summation represents all of the deformation modes, N (=Ns + Nt), consisting of slip, Ns, and twin systems, Nt. As described in [36], the Jaumann rate of Kirchhof stress can be expressed as ^s ¼ K : D
N X
c_ a Ra
ð4Þ
a¼1
where K is a fourth-order tensor based on the anisotropic elastic modulus, C. D is the rate of deformation tensor (symmetric part of the velocity gradient), and Ra is a tensor that depends on the current slip/twin plane normal and direction, the applied stress and the elastic modulus. For materials with hexagonal crystal symmetry, five, independent elastic constants of pure Mg were used in the present work [39]: C 11 ¼ 58 GPa; C 12 ¼ 25 GPa; C 13 ¼ 20:8 GPa; C 33 ¼ 61:2 GPa; C 55 ¼ 16:6 GPa For rate-dependent materials, shear rates are given explicitly in terms of the resolved shear stress on the active slip/twin systems and the resistance of the active slip/twin systems to shear. For these simulations, this dependence is given by a 1=m s ð5Þ c_ a ¼ c_ ao a signðsa Þ so Self- and latent-hardening are readily accounted for by a suitable evolution of the reference sao values in the constitutive law in Eq. (5). The present work employed a microscopic hardening law [40,32] for this purpose, as follows: s_ ao ¼
N X
H ab j_cb j a; b ¼ 1 . . . ðN s þ N t Þ
H ab
equals the latent-hardening term (off-diagonal term of qab), i.e. (qab = 1). The fitting simulation was carried out by varying the CRSS values and microscopic hardening parameters (ho, ssat and a) until agreement was achieved between the predicted and the measured flow curves. A 3-D mesh (20 20 10 = 4000 elements) of a polycrystal model was used to predict the macroscopic flow curve [41]. An ODF measured X-ray diffraction was used to generate a set of 4000-grain orientation for polycrystal modeling with the help of orientation repartition functions [42]. To impose initial orientation of the elements, each orientation selected from the 4000-grain orientation was mapped onto each integration point in the finite element mesh. The set of parameters listed in Table 1 was used in the theoretical simulation. In this study, four slip systems and one twin system were considered: basal hai ({0 0 0 1}h1 1 2 0i), prismatic hai (f1 1 0 0g h1 1 2 0i), pyramidal hai (f1 1 0 0g h1 1 2 0i) pyramidal hc + ai (f1 1 2 2g h1 1 2 3i) and tensile twin (f1 0 1 2g h1 0 1 1i). At RT, the existence of pyramidal hc + a i slip is not established in Mg alloys, but the slip system is an active deformation mechanism at elevated temperature when twinning is suppressed [4]. To consider the effect of deformation twinning on texture evolution and stress concentration in Mg alloys, the predominant twin reorientation (PTR) model [21] was implemented in the CPFEM. This requires tracking of the shear strain, ct,g, contributed by each twin system t, and of the associated volume fraction V t;g ¼ ct;g =S t ðS t ¼ 0:129 is the characteristic twin shear) within each orientation, g. By summation of all systems in each element, the accumulated twin fraction, Vacc,mode, in each orientation can be determined as follows: X V acc;mode ¼ ct;g =S t ð7Þ t
At each incremental step, the fractions accumulated in the individual twinning systems of each orientation are compared against a threshold fraction, Vth,mode, defined as follows: V th;mode ¼ C th1 þ C th2 V acc;mode
ð8Þ
After each deformation increment, the twin system with the highest accumulated volume fraction is identified. If the accumulated volume fraction is greater than the threshold fraction, Vth,mode, the orientation is allowed to reorient. After all, the threshold fraction, Vth,mode, increases gradually, and further reorientation by twinning can be inhibited by large deformations. The algorithm prevents reorienta-
b
a sao ¼ q ho 1 ssat ab
ð6Þ
where Hab is a hardening matrix that is introduced to account for interaction between the slip and twin systems. qab accounts for the hardening rate of the slip/twin system, a, due to slip/twin activity on the system, b. Here, it is assumed that the self-hardening term (diagonal term of qab)
Table 1 Microscopic hardening coefficients used in the CPFEM simulation. Mode
sao (MPa)
ho (MPa)
ssat (MPa)
a
Basal hai Prism hai Pyram hai Pyram hc + ai Twin
25 68 68 68 40
100 130 130 130 50
70 210 210 210 50
1.1 0.8 0.8 0.8 1.1
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tion by twinning until a threshold value, Cth1, is attained. The threshold value, Cth2, determines the evolution of the twin volume fraction during plastic deformation. Reportedly, the volume fraction of twinning is strongly dependent on the initial microstructure and deformation conditions, such as strain, strain rate and temperature [43,44]. Therefore, the optimal combination of constants in Eq. (8) should be determined by consideration of the initial microstructure and deformation conditions. Experimental observation indicated that one or more twin variants accommodate plastic deformation within each orientation [45,23]. However, the PTR model assumes that each orientation is allowed to reorient with respect to the normal direction of a mirror plane in the most active twin system, as determined by the CPFEM. The transformation matrix, T, between the lattice orientation in the matrix and the lattice orientation in the twinned region can be defined as [20] T ij ¼ 2ni nj dij ;
dij ¼ 0
if
i–j;
dij ¼ 1
ð9Þ
where n represents the unit vector of the twin plane normal in orthogonal coordinates.The integration of the stress rate (Eq. (4)) with the unknowns, c_ a , requires small time intervals for stable numerical integration. The implicit time integration algorithm for the slip rates is similar to the rate tangent method of Peirce et al. [36]. Moreover, a material Jacobian (@D^s=@De) for the implicit method can be obtained from Eq. (4). This is equivalent to the modulus given by Peirce et al. [36]. The stress vector in the UMAT subroutine can be updated by multiplying the material Jacobian by the strain increment. It should be noted that the material Jacobian is required in the iterative procedure in minimizing the force residual. If the equilibrium tolerances had not been satisfied, ABAQUS iterates by providing the constitutive routine with updated estimates of the strain increment.
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4. Results and discussion Fig. 1a shows the microtexture of the as-rolled AZ31 Mg alloy. The alloy exhibited a relatively inhomogeneous and coarse distribution of grain size. The microtexture evolution of AZ31 Mg alloy during in-plane compression to a true strain of 0.05 is shown in Fig. 1b. After uniaxial compression to a true strain of 0.05, twin bands were nucleated and propagated through parent grains. The shape of twinned regions can be classified into lenticular and partially propagated types. At a true strain of 0.1, as shown in Fig. 1c, the propagation of twin bands was almost prohibited by collision with high angle grain boundaries and other twin bands at the twin boundaries. The as-rolled AZ31 Mg alloy can be characterized as a sharp basal fiber texture, as shown in Fig. 2a–c shows the measured deformation textures for compressive specimens of the AZ31 Mg alloy deformed to a true strain of 0.05 and 0.1 along the RD, respectively. The microtexture revealed that many crystallographic lattices rotate to the RD by twin-induced reorientation. The main texture components developed in deformed AZ31 Mg alloys under in-plane compression can be represented by pole figures, as shown in Fig. 2d. Analysis showed that the F texture component is a stable orientation in the specimen deformed to a true strain of 0.05. It is also clear that the near F and H texture components are relatively stable orientations in the specimen deformed to a true strain of 0.1. Crystallographic orientations of untwined lattices were analyzed as the E and G texture components. Volume fractions of parent grains and twin bands were identified using twin-analysis technique implemented in the TSL software. In the case of partially propagated twins, it is easy to identify the twin volume fraction using the orientation relationship between parent grains and twin bands. Fig. 1b shows the volume fraction
Fig. 1. Inverse pole figure map (the position of the ND in the colored stereographic triangle) showing the texture evolution of AZ31 Mg alloys: (a) asrolled, (b) true strain = 0.05, (c) true strain = 0.1.
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Fig. 2. Experimentally measured pole figures and ODF sections showing (a) the initial texture of as-rolled specimen and deformation texture of the CA// RD specimen,: (b) true strain = 0.05, (c) true strain = 0.1 and (d) a graphical representation of the main texture components.
of the twinned region in the specimen deformed to a true strain of 0.05. In the case of fully propagated twins, more careful analysis is required to identify the twin volume fraction. The twin-analysis technique implemented in the TSL software does not work for the deformed specimen containing fully propagated twins. Since the as-rolled AZ31 alloy was characterized as a strong fiber texture, a direct comparison of pole figures can distinguish the twin regions undergoing a drastic change of crystallographic orientation from the untwinned region undergoing a gradual change of crystallographic orientation. Fig. 1c shows the volume fraction of the twinned region in the specimen deformed to a true strain of 0.1. To evaluate the spatial stress concentration in the deformed AZ31 Mg alloys, a direct mapping method based on EBSD data was used in the CPFEM. The dashed line in Fig. 1a shows the cropped region from the as-rolled microtexture for the simulation. All scan data was directly mapped onto the quasi 3-D finite element meshes (200 200 1 = 40,000 elements), as shown in Fig. 3a. The initial length of the model region is given by lo = 50 lm, wo = 50 lm, to = 0.25 lm. A true compressive strain of e = 0.1 was simulated using appropriate boundary conditions. The boundary conditions were applied to the four planes comprising the quasi 3-D mesh, as shown in Fig. 3b. Since the cropped region represents the small part of a larger body, boundary conditions considering neighboring grains should be applied. Although several ways are available to impose boundary conditions, none rigorously represent a real condition. In the present study, for the simulation of
uniaxial compression, prescribed displacement in the RD was imposed on the 2-3-6-7 face. The three faces (1-2-3-4, 1-2-5-6 and 1-4-5-8) were constrained so as not to displace. The two moving faces (5-6-7-8 and 3-4-7-8) were ascribed to the remaining plane. However, simple boundary conditions can induce a considerable shear strain at the boundary regions if a significant displacement is imposed on the 2-3-6-7 face. In addition, since the quasi 3-D mesh of the model cannot be used to consider the interactions with neighboring grains above and below the model grains, a fully 3-D mesh was required to capture the interaction. In further work, a 3-D EBSD scan using the serial sectioning technique could be used to construct a fully 3-D mesh. Fig. 4a shows the spatial distribution of the Schmid factor (mSF) for the cropped region from the as-rolled microtexture. For the calculation of m values, the basal hai slip system was only considered under the boundary conditions of uniaxial compression. As a first approximation, grains having high mSF values were expected to be deformed by slip-dominated deformation at a low-strain level. By contrast, grains having low mSF values were expected to be deformed by twin-dominated deformation at a low-strain level. To capture which grains were favorable for the twin-dominated deformation under uniaxial compression, grains having relatively low mSF values ranging between 0.2 mSF,max and mSF,min are highlighted, as shown in Fig. 4b. Fig. 5a shows the simulated microtexture for the CA// RD specimen to a true strain of 0.05. To represent the deformed microstructure using the inverse pole figure
S.-H. Choi et al. / Acta Materialia 58 (2010) 320–329
325
b
a
8 4
(1-2-3-4) : UTD= 0 (1-2-5-6) : UND= 0 (1-4-5-8) : URD= 0 (2-3-6-7) : URD= compression
5 1
7 3
TD
ND
TD
ND
RD
RD
6 2
20
Fig. 3. A direct mapping method of EBSD data for the simulation of stress concentration in deformed AZ31 Mg alloy sheets: (a) cropped region for finite element meshing, (b) boundary condition for uniaxial compression.
Fig. 4. Spatial distribution of the Schmid factor (mSF) of as-rolled AZ31 Mg alloys: (a) mSF,max mSF,min (b) 0.2 mSF,max mSF,min.
map instead of finite element meshes, crystallographic orientations calculated by CPFEM were mapped onto regular grids to satisfy the data format of TSL software. Fig. 5b1 shows the twin volume fraction after uniaxial compression to a true strain of 0.05. Blue and red colors represent portions of untwinned and twinned elements, respectively. The CPFEM successfully predicted the twin volume fraction of the CA//RD specimen to a true strain of 0.05. It was found that the shape of twinned regions simulated by the CPFEM could be classified into lenticular (1–7) and partially propagated (8–25) types. However, comparison of Figs. 4 and 5 b a reveals that the mSF value has a limitation to predict grains to be deformed by twin-dominated deformation at
1 For interpretation of color in Fig. 5b, the reader is referred to the web version of this article.
a low-strain level. Grains having a low mSF value ranging between 0.2 mSF,max and mSF,min did not exactly match with the twinned regions calculated by the CPFEM. It should be noted that the partial twinned regions (1, 2, 5, 9, 15, 16, 17, 24) correspond to the grains having a low mSF value. Even some twinned regions (8, 10) corresponded with the grains having high mSF values. These results indicate that the Schmid factor cannot be used to explain the heterogeneous distribution of twinned regions during plastic deformation at a low-strain level. However, the CPFEM successfully predicted the heterogeneous distribution of twinned regions during plastic deformation at a strain level, as shown in Fig. 5b. Heterogeneous distribution of accumulation twin fraction in Eq. (7) seemed to be related to the heterogeneous nucleation of twin bands during plastic deformation. To understand the heterogeneous nucleation of twin bands, the associated volume
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Fig. 5. Spatial distribution of simulated microstructure for the CA//RD specimen to a true strain of 0.05: (a) crystallographic orientation map and (b) deformation twinning.
fraction accumulated in the individual twinning system, ct;g =S t , in Eq. (7) was analyzed for the deformed microstructure. Fig. 6 shows the spatial distribution of the individual associated volume fractions in the deformed microstructure after uniaxial compression to a true strain of 0.05. It is noteworthy that the accumulation of shear strain in each twin system exhibited a distinct heterogeneity
in the deformed microstructure. Since the twin system having the highest accumulated volume fraction determines the transformation matrix, T, in Eq. (9), the various crystallographic orientations in twinned region were attributed to the heterogeneous distribution of the associated volume fraction. Fig. 7a shows the simulated deformation texture of the CA//RD specimen to a true strain of 0.05. Compar-
Fig. 6. Spatial distribution of the individual associated volume fractions in the deformed microstructure after uniaxial compression to a true strain of 0.05.
S.-H. Choi et al. / Acta Materialia 58 (2010) 320–329
327
(i) intergranular stress between untwinned grains (b– c, c–e, g–h); (ii) intragranular stress between an untwinned grain and a twinned grain (a-8, b-2, b-9, c-1, d-12, d-13, e-6, e14, f-14, f-18, g-3, g-4, g-15, h-15, h-18, i-18, i-19, j-19, j-24); and, (iii) intragranular stress between twinned grains (2–9, 15– 18, 17–18).
Fig. 7. Simulated deformation texture of the CA//RD specimens: (a) true strain = 0.05 and (b) true strain = 0.1.
ing EBSD data, as shown in Fig. 2b, it is clear that CPFEM accurately simulated the rotation of the c-axis from the ND to the RD on the (0 0 0 2) pole figure. It appears that the activity of the tensile twin played an important role in rotation of the c-axis from the ND to the RD during plastic deformation. An effective stress computed by the von Mises’ formulation was used to evaluate the spatial stress concentration in the deformed AZ31 alloy. For a grain-level crack nucleation criterion, a more rigorous effective stress based on slip system stress fields [46] can be used to understand crack nucleation due to stress concentration. It should be noted that the present study concentrated on the grain-level stress field instead of the grain-level crack nucleation criterion. Fig. 8 shows the spatial distribution of effective stress for the CA//RD specimen to a true strain of 0.05. During plastic deformation, heterogeneous stress distribution occurred at regions near the grain boundaries of untwinned grains (a-j). Heterogeneous stress concentration can be classified into three types:
Fig. 8. Spatial distribution of effective stress computed by the von Mises’ formulation for the CA//RD specimen to a true strain of 0.05: untwinned regions (a–l) and twinned regions (1–25).
These results indicate that plastic deformation induces stress concentration at the regions in the vicinity of the grain boundary separating two untwinned grains. Analysis showed that the heterogeneous stress concentration can be attributed to the activation of non-basal slip systems having high CRSS values to satisfy strain compatibility at the grain boundary. These results also indicate that plastic deformation induces stress concentration along grain boundaries of twinned grains as well as within twinned grains. Analysis also showed that the heterogeneous stress concentration in twinned grains can be attributed to the activation of non-basal slip systems to satisfy strain compatibility at the grain boundary after twin-induced reorientation. According to the CPFEM simulation of Mg alloys using the representative volume elements [41], plastic deformation began with the activation of a tensile twin and a basal hai slip as the primary mode during in-plane compression. Subsequently, the tensile twin decreased continuously and the pyramidal hc + ai slip increased gradually during plastic deformation. Apparently, stress gradients can be found in the twinned grains from the grain boundary to the grain interior. Fig. 9a shows the simulated microtexture for the CA//RD specimen to a true strain of 0.1. The deformed grains consist mainly of twinned grains and partly untwinned grains, as shown in Fig. 9b.2 Blue and red regions represent portions of untwinned and twinned elements, respectively. The twin volume fraction predicted by the CPFEM is in very good agreement with the experimental data, as shown in Fig. 1. Twinned regions consisting of fully propagated twin bands and partially propagated regions are rarely found in the microstructure. Fig. 7b shows the simulated deformation texture of a CA// RD specimen to a true strain of 0.1. Comparing EBSD data, as shown in Fig. 1b, it is clear that CPFEM successfully simulated the rotation of the c-axis from the ND to the RD on the (0002) pole figure. These results indicate that a near-F texture component is a relatively stable orientation in the CA//RD specimen. Fig. 10 shows the distribution of effective stress for the CA//RD specimen to a true strain of 0.1. Analysis showed that heterogeneous stress distribution had occurred at the grain boundaries of untwinned grains (a–o0 ). The untwinned grains were surrounded either by twinned grains (10 –130 ) or by untwinned
2 For interpretation of color in Fig. 9b, the reader is referred to the web version of this article.
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Fig. 9. Spatial distribution of simulated microstructure for the CA//RD specimen to a true strain of 0.1: (a) crystallographic orientation map and (b) deformation twinning.
However, it should be noted that the density of type (ii) stress concentration increased as the true strain increased. Similar to the specimen deformed to a true strain of 0.05, stress gradients also could be observed in the twinned grains from the grain boundary to the grain interior. The twinned grains also exhibited a distinct stress concentration compared to the untwinned grains. These results showed that deformation twinning enhanced the local stress concentration at regions near the grain boundaries neighboring untwinned grains. 5. Conclusions
Fig. 10. Spatial distribution of effective stress computed by the von Mises’ formulation for he CA//RD specimen to a true strain of 0.1: untwinned regions (a0 –o0 ) and twinned regions (10 –130 ).
grains. The heterogeneous stress concentration can be also classified into three types: (i) intergranular stress between untwinned grains (b0 –c0 , c0 –g0 , k0 –l0 ); (ii) intragranular stress between an untwinned grain and a twinned grain (a0 -10 , b0 -20 , b0 -30 , c0 -30 , f0 -100 , g0 -70 , g0 80 , h0 -70 , h0 -80 , h0 -90 , i0 -110 , j0 -120 , k0 -70 , m0 -50 , m0 -60 , n0 40 , o0 -50 ); (iii) intragranular stress between twinned grains (90 –130 ). A comparison of Figs. 8 and 10 shows that the stress concentrations in the specimen deformed to a true strain of 0.1 are similar to that in the specimen deformed to a true strain of 0.05, regardless of the type of stress concentration.
A crystal plasticity finite element method (CPFEM) that considers both crystallographic slip and deformation twinning was developed to simulate the spatial stress concentration in AZ31 Mg alloys during in-plane compression. The major feature of the model is implementation of a PTR model to capture grain reorientation due to deformation twinning. The CPFEM was used to simulate spatial stress concentration in deformed AZ31 Mg alloy under uniaxial compression. A direct mapping method of EBSD data onto quasi 3-D finite element meshes was employed to consider a real microstructure as an initial configuration. The CPFEM successfully simulated the shape of twinned regions such as lenticular and partially propagated types that are typical shapes of in-plane compression specimens at a low-strain level. Simulation results indicated that the stress concentration can be classified into three types: (i) intergranular stress between untwinned grains; (ii) intragranular stress between an untwinned grain and a twinned grain; and (iii) intragranular stress between twinned grains. Theoretical analysis showed that type (ii) exhibited the
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