A theoretical prediction of twin variants in extruded AZ31 Mg alloys using the microstructure based crystal plasticity finite element method

Materials Science and Engineering A 538 (2012) 190–201

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Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea

A theoretical prediction of twin variants in extruded AZ31 Mg alloys using the microstructure based crystal plasticity finite element method E.J. Shin a , A. Jung b , S.-H. Choi b,∗ , A.D. Rollett c , S.S. Park d a

Neutron Physics Department, Korea Atomic Energy Research Institute, Daejeon 305-600, Republic of Korea Department of Materials Science and Metallurgical Engineering, Sunchon National University, Sunchon, Jeonnam 540-742, Republic of Korea Materials Science and Engineering Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA d School of Mechanical and Advanced Materials Engineering, Ulsan National Institute of Science and Technology, Ulsan 689-798, Republic of Korea b c

a r t i c l e

i n f o

Article history: Received 1 August 2011 Received in revised form 29 November 2011 Accepted 1 January 2012 Available online 18 January 2012 Keywords: Crystal plasticity Finite element Resolved shear stress Twin variants

a b s t r a c t A resolved shear stress (RSS) criterion and the microstructure-based-crystal plasticity finite element method (MB-CPFEM) were used to analyze the activation of twin variants in extruded AZ31 Mg alloys during ex situ uniaxial compression. The RSS criterion, which is simply based on the Schmid factor, failed to predict the activation of twin variants consisting of the second-highest RSS and the third-highest RSS. In contrast to the RSS criterion, the MB-CPFEM based on a quasi-3D finite element mesh successfully predicted the activation of twin variants consisting of the highest RSS and the second-highest RSS. The MB-CPFEM demonstrated that local fluctuation of the stress field induces the activation of twin variants with the second-highest RSS during uniaxial compression. © 2012 Elsevier B.V. All rights reserved.

1. Introduction In Mg and its alloys at temperatures near RT, the critical resolved shear stress (CRSS) of non-basal slip systems, which are needed for ductility, is much higher than that of the basal slip system [1–3]. Therefore, only a limited number of slip systems can be activated to accommodate external deformation during plastic deformation at temperatures near RT. Since the c/a ratio (=1.624) of Mg is less √ than 3 (∼ =1.732), a tensile twin on the {1 0 1¯ 2} plane is more easily activated by c-axis tension [4,5]. A number of researchers have also observed a compression twin activated by c-axis compression [6–9]. In most cases, primary twinning on the {1 0 1¯ 1} plane is followed by secondary twinning (or retwinning) on the {1 0 1¯ 2} plane. Twinning involves pseudo-shear on specific crystallographic planes and in specific crystallographic directions. The main difference between slip and twinning is that twinning shear is directional in nature. Recently, electron back-scattered diffraction (EBSD) studies have been conducted on the activation of twin variants for primary and secondary twinning in HCP materials [7–9,15–20]. The activation of twin variants in parent grains rotates the basal plane into a position where it is much more favorably oriented for glide. Rapid rotation of crystallographic orientation in primary and secondary

∗ Corresponding author. Tel.: +82 61 750 3556; fax: +82 61 750 3550. E-mail address: [email protected] (S.-H. Choi). 0921-5093/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2012.01.028

twinning can affect strain hardening behavior, texture evolution [7,10–18], stress concentration near grain boundaries [16], strain localization [19], and fatigue failure [17]. Accurate prediction of active twin variants is, therefore, very important for optimizing the microstructure under different loading conditions. Schmid factor analysis is the most common approach to understanding the mechanism that causes the activation of twin variants of primary and secondary twinning in HCP materials under different loading conditions [7–9,18,20,21]. In experiments, the twinning planes of twin bands correspond to those with high Schmid factor values. Mg alloys with a sharp initial texture satisfactorily follow the simple criterion for the selection of active twin variants of primary twinning [8,17,18]. However, this type of analysis has been conducted on the few grains in which twin variants with a high Schmid factor have been observed. Statistically meaningful experimental observation has revealed that the AZ31 Mg alloy departs from Schmid-type behavior in both primary and secondary twinning [9]. A statistical analysis of high-purity polycrystalline Mg [20] also revealed that the twin variants with the highest Schmid factor had the highest probability of activation, but the probability of activation of other twin variants was also significant. The non-Schmid behavior of primary twinning and secondary twinning in polycrystalline Mg materials can be attributed to two main sources, as follows: (i) a deviation of the local stress state within the parent grain from the macroscopic stress state, as a result of interaction with neighboring grains or twin bands, and (ii) local stress

E.J. Shin et al. / Materials Science and Engineering A 538 (2012) 190–201

fluctuations within the parent grain produced by volume defects (precipitates and inclusions). The present study was conducted to determine how much the first source affects the activation of twin variants in Mg alloys. The second source appears to be important in the precipitation-hardened Mg alloys [22,23]. Twin nucleation can be promoted by precipitates in age-hardened Mg alloys [22]. Crystal plasticity finite element methods (CPFEMs) were developed specifically to simulate heterogeneous plastic deformation of HCP polycrystalline materials [16,24–27]. In the microstructurebased (MB)-CPFEM [16,28,29], EBSD data were directly mapped onto a quasi-3D finite element mesh of hexahedral elements. In a previous paper [16], MB-CPFEM was developed to simulate the spatial stress concentration in hot-rolled AZ31 Mg alloy under in-plane compression. A modified predominant twin reorientation (MPTR) scheme was successfully implemented to capture grain reorientation due to deformation twinning in twin-dominated deformation. The MB-CPFEM successfully simulated more than one twin orientation in the parent grains. It also successfully modeled the heterogeneous stress concentration at the grain level during inplane compression. Since the quasi-3D mesh of the model could not be used to consider the interactions with neighboring grains above and below the model grains, a complete 3D mesh was required to capture the interactions. Recently, a microstructure mapping technique that considered both the average grain size and microtexture, as measured by an EBSD technique, was used to create a statistically representative 3D digital microstructure for the initial configuration [30]. However, the previous papers [16,30] did not provide a direct comparison of the simulation results with experimental observations to verify the theoretical framework. In the present study, ex situ compressive tests were conducted to capture the microstructure evolution in the same region of extruded AZ31 Mg alloy during uniaxial compression. The activation of twin variants in deformed grains was analyzed using two theoretical methods. In the first method, the resolved shear stress (RSS) for each variant of the {1 0 1¯ 2} tensile twins was calculated in each grain under uniaxial compression. The RSS criterion, in which interaction with neighboring grains is neglected and the RSS is computed from the far field (average) stress, can capture the activation of twin variants in each grain as a first approximation. In the second method, MB-CPFEM was used to consider the interaction with neighboring grains during plastic deformation such that the RSS was computed with the local stress state at each integration point. The MB-CPFEM was in relatively good agreement with the experimental results with respect to twin variant selection of twin bands in deformed grains of extruded AZ31 Mg alloy, compared to the simple RSS criterion.

2. Experimental The composition of the AZ31 Mg alloy used in the present study was Mg–2.90 wt.% Al–0.69 wt.% Zn–0.32 wt.% Mn. In order to obtain extruded bars with circular cross-section, indirect extrusion experiments were conducted at initial billet temperatures of 250 ◦ C and at ram speeds of 1.3 mm/s with a fixed extrusion ratio of 25. A detailed explanation of the indirect extrusion can be found in an article written by one of the co-authors [31]. Ex situ compressive tests were conducted to measure the evolution of deformation twins in the same region during plastic deformation. Fig. 1(a) shows a schematic diagram of the procedure for the ex situ compressive tests. Compressive specimens of a rectangular parallelepiped shape were machined by laser cutting from the extruded bar (=16 mm) of AZ31 Mg alloy. The initial height and width of the compressive specimens were 10 and 8 mm, respectively. The EBSD technique was used to analyze the evolution of microtexture in extruded AZ31 Mg alloy during uniaxial compression. The

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microtexture of the initial specimen was examined by scanning an area of 100 ␮m × 100 ␮m at a step size of 0.25 ␮m. The specimen was installed in a GLEEBLE® 3500 C thermo-mechanical simulator and was heated by means of an inductive heating device at a rate of 5 ◦ C/s. After holding at 200 ◦ C for 1 min, it was deformed at a strain rate of 0.1 s−1 . The loading direction of the specimen was parallel to the extrusion direction (ED), and it was allowed to expand in the orthogonal directions. To capture the effect of strain on the evolution of deformation twinning, the test was stopped at a true strain of 0.05. After unloading, the deformed specimen was cooled to RT. The deformed specimen was the prepared using a Hitachi’s IM-3000 ion polisher to analyze the evolution of microtexture. Microtexture deformed to a true strain of 0.05 was examined by automated EBSD scans of an area 120 ␮m × 110 ␮m at a step size of 0.25 ␮m using the TSL software. The specimen was the deformed again in a GLEEBLE system under the same experimental conditions. The uniaxial compression test was stopped at a true strain of 0.1. The deformed specimen was then prepared using ion polisher to analyze the evolution of microtexture. Microtexture deformed to a true strain of 0.1 was examined by automated EBSD scans of an area 150 ␮m × 110 ␮m at a step size of 0.25 ␮m. 3. Theoretical procedure Two theoretical methods were used to understand the mechanisms that lead to the selection of specific twin variants. The first method is a criterion that is simply based on the Schmid factor (M). For this purpose, the resolved shear stress (RSS) for twin variants of {1 0 1¯ 2} tensile twin is defined as:  = cos  · cos  ·  = M · 

(1)

where  and  are the angles between the load axis and the twinning plane normal and shear direction, respectively. To simplify the criterion, uniaxial compression of −1 MPa as a macroscopic stress state () was imposed to evaluate RSS in each grain. Considering the polarity of twinning, the twin variants that are positive and have the highest RSS will have the highest probability of activation. Twin variants with negative RSS are not activated during plastic deformation. Six different variants of the tensile twin were consid¯ ¯ 111], 3: (1¯ 1 0 2)[1 1¯ 0 1], 4: ered, i.e., 1: (1 0 1¯ 2)[1¯ 0 1 1], 2: (0112)[0 (1¯ 0 1 2)[1 0 1¯ 1], 5: (0 1¯ 1 2)[0 1 1¯ 1], and 6: (1 1¯ 0 2)[1¯ 1 0 1]. It should be noted that the RSS criterion does not consider the interaction with neighboring grains since a constant stress was assumed. In the second method, MB-CPFEM was used to consider the interaction with neighboring grains in the initial microstructure of extruded AZ31 Mg alloy during uniaxial compression. A ratedependent constitutive relationship was implemented in a user material subroutine UMAT in the commercial finite element code, ABAQUS/Standard [32]. The model was fundamentally based on the multiplicative decomposition of the deformation 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 = Fe · Fp

(2)

This formula leads to additive decomposition of the velocity gradient into elastic and plastic parts, L = Le + Lp

(3)

with the plastic part determined by slip rates, ˙ ˛ , on slip/twin planes with normals, m˛ , and slip/twin directions, s˛ Lp =

N  ˛=1

˙ ˛ s˛ ⊗ m˛

(4)

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Fig. 1. (a) A schematic diagram showing the procedure for the ex situ compressive tests. (b) Inverse pole figure map (the position of the ED in the colored stereographic triangle) showing the evolution of microtexture in extruded AZ31 Mg alloys. (c) (0 0 0 1) pole figures showing the evolution of crystallographic orientation in extruded AZ31 Mg alloys under uniaxial compression.

The summation represents all of the deformation modes, N (=Ns + Ntw ), consisting of the slip, Ns , and twin systems, Ntw . As described in [33], the Jaumann rate of Kirchhof stress can be expressed as: ˆ = K : D −

N 

˛ ˛

˙ R

(5)

˛=1

where K is a fourth-order tensor based on the anisotropic elastic modulus, C. D is the rate of deformation tensor (the symmetric part of the velocity gradient), and R˛ is a tensor that depends on the current slip/twin plane normal, direction, the applied stress and the elastic modulus. 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:

 1/m  ˛   sign ( ˛ ) 0˛ 

˙ ˛ = ˙ 0˛  

(6)

In the present study, as we assume twinning as pseudo-slip, a critical resolved shear stress should be imposed to activate the twinning system on the twinning plane and along the twinning direction. However, it differs from slip in its directionality, which we model by allowing activation only if the resolved shear stress is positive. Self- and latent-hardening are readily accounted for by a suitable evolution of the reference 0˛ values in the constitutive law expressed by Eq. (6). The present work employed a microscopic hardening law [34,35] for this purpose, as follows: ˙ 0˛ =

N  ˇ

  H ˛ˇ ˙ ˇ  ˛, ˇ = 1 · · · (Ns + Ntw )

(7)

 H ˛ˇ = q˛ˇ h0 1 −

0˛

a (8)

sat

where H˛ˇ is a hardening matrix that was introduced to account for the interaction between the slip and twin systems. q˛ˇ accounts for the hardening rate of the slip/twin system, ˛, due to the slip or twin activity of the system. Here, it was assumed that the self-hardening term (diagonal term of q˛ˇ ) equals the latent hardening term i.e. (q˛ˇ = 1). ˇ. In the present study, four slip systems and one twin system were considered: basal a ({0 0 0 1}1 1 2¯ 0), prismatic a ({1 1¯ 0 0}1 1 2¯ 0), pyramidal a ({1 1¯ 0 0}1 1 2¯ 0) pyramidal c + a ({1 1 2¯ 2}1¯ 1¯ 2 3), and tensile twin ({1 0 1¯ 2}1¯ 0 1 1). The relative contribution of the deformation modes as a function of true strain yields useful information for the analysis of plastic N deformation. The relative activity (˙ ˛,e / ˇ=1 ˙ ˇ,e ) of each defor-

mation mode, ˛, among the slip/twin systems (N = Ns + Ntw ), was determined by summation of the shear strain in each element (e). To consider the effect of deformation twinning on the spatial distribution of the stress concentration, the original (PTR) scheme [36] was modified and implemented in the MB-CPFEM [16]. This required tracking of both the shear strain,  tw,e , contributed by each twin system, tw, and the associated volume fraction,  tw,e /Stw (Stw = 0.129 is the characteristic twin shear) within each element, e. By summation of all twin systems in each element, the accumulated twin fraction, Vacc , in each element was determined as follows:

 V acc = 0

⎛ t

⎜ ⎜ ⎝

|| ˙ tw,e

tw

S tw

⎞ ⎟ ⎟ dt ⎠

(9)

E.J. Shin et al. / Materials Science and Engineering A 538 (2012) 190–201 Table 1 Microscopic hardening coefficients used in the CPFEM simulation. Mode

0˛ (MPa)

h0 (MPa)

 sat (MPa)

a

Basal a Prism a Pyram a Pyram c + a Twin

25 65 65 65 40

100 130 130 130 50

70 200 200 200 50

1.1 0.8 0.8 0.8 1.1

At each incremental step, the fractions accumulated in the individual twinning systems of each orientation were compared against a threshold fraction, Vth , defined as follows: V th = C th1 + C th2 · V acc

(10)

After each deformation increment, the twin system with the highest accumulated volume fraction was identified. If the total accumulated volume fraction was greater than the threshold fraction, Vth , the orientation was allowed to reorient. The threshold fraction, Vth , increased gradually and further reorientation by twinning was inhibited by large deformations. The threshold value, Cth2 , determines the evolution of the twin volume fraction during plastic deformation. The optimal combination of constants in Eq. (10) should be determined by consideration of both the initial microstructure and the deformation conditions. The MPTR model assumes that each orientation is allowed to reorient by 180◦ about the normal direction of the twin plane in the most active twin system, as determined by MB-CPFEM. The transformation matrix, T, between the lattice orientation in the matrix and the lattice orientation in the twinned region can be defined as follows [37]: Tij = 2ni nj − ıij ; ıij = 0 if i = / j; ıij = 1

(11)

where n represents the unit vector of the twin plane normal in orthogonal coordinates. If the stress state in a deformed grain is strongly affected by interaction with neighboring grains, it is expected that more than one twin variant can be generated during plastic deformation. Here, we assumed that the element reoriented by twinning does not undergo a second reorientation by retwinning. The fitting simulation was carried out by varying the CRSS values, microscopic hardening parameters (h0 ,  sat and a), and twinning parameters (Cth1 and Cth2 ) until agreement was achieved between the predicted and the measured flow curves. A simple 3D mesh (20 × 20 × 10 = 4000 elements) of a polycrystal model was used to predict the microscopic and macroscopic responses [27]. An ODF measured by X-ray diffraction was used to generate a set of 4000-grain orientations for polycrystal modeling with the help of orientation repartition functions. The set of parameters used in the theoretical simulation are shown in Table 1. 4. Results and discussion Fig. 1(b) shows the evolution of microtexture in the extruded AZ31 Mg alloy during ex situ compressive tests. During uniaxial compression to a true strain of 0.05, twin bands were nucleated and propagated through their parent grains. The shape of twinned regions can be classified into lenticular and partially propagated types. During uniaxial compression to a true strain of 0.1, twin bands were also nucleated and propagated through their parent grains. Some twin bands impinged against other twin bands at the twin boundaries. The (0 0 0 1) pole figures of initial and deformed specimens of the AZ31 alloy after uniaxial compression are shown in Fig. 1(c). The as-extruded AZ31 Mg alloy can be characterized as a fiber texture in which c-axis is preferentially aligned perpendicular to the ED, which is typical of extruded Mg alloys. Fig. 1(c) also shows the crystallographic orientations of the AZ31 Mg alloy

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Table 2 A comparison of twin variants observed by experiment and predicted by RSS criterion and CPFEM simulation. Grain

Exp.

RSS

CPFEM

g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 g12 g13 g14 g15 g16

1(#) 1, 4(#) 3(#,*) 2(#,*) 2, 3(#,*) 6 1(#,*) , 2,5(#) 5(#) 1(#) , 4 1(#,*) , 2 2(#,*) 2 1, 4(#) 2(#,*) , 5, 6(#) 2(#,*) 5(#)

4 1 2 5 2 6 2 6 4 2 5 2 1 5 5 6

1, 4 1, 4 2 5 2 6 2, 5 5, 6 1, 4 2 5 2 1, 4 5, 6 5 5, 6

deformed to a true strain of 0.05 and 0.1 along the ED. The (0 0 0 1) pole figures revealed that the c-axis of many grains rotated to the ED by twin-induced reorientation. The orientation spread in each grain increased as the true strain increased, as is commonly observed. In order to investigate the orientation relationship between twin bands and their parent grains, sixteen grains (g1–g16) were examined, as shown in Fig. 1(b). The boundaries of twin bands were identified using a twin-analysis technique implemented in the TSL software. Fig. 2 represents the crystallographic orientations of sixteen parent grains (g1–g16) in a specimen deformed to a true strain of 0.1 and main twin bands existing in each parent grain. Fig. 2 also includes the type of twin variants observed in sixteen parent grains (g1–g16) in a specimen deformed to a true strain of 0.1. In order to identify the type of the observed twin variants, theoretical orientations of twin variants were calculated using Eq. (11). Six independent orientations (tv: 1–6) calculated from twin variants in the parent grains (g1–g16) are illustrated on a pole figure, (0 0 0 1), as shown in Fig. 3. It should be noted that the final orientations fell into three pairs (1–4, 2–5, 3–6). The members of each pair corresponded to 86◦ rotation about the same 1 2¯ 1 0 axis, but with opposite signs. The theoretical analysis in a previous paper [15] revealed that the misorientation relationship between the members of each pair was 7.4◦ about the 2 1¯ 1¯ 0 axis. The misorientation relationship between the members of different pairs was analyzed as mostly 60◦ or 60.4◦ about the rotation axes near the 1 0 1¯ 0 axis. A comparison of Figs. 2 and 3 reveals that the number and type of twin variants generated in each grain were significantly dependent on the parent grain. The observed twin variants are listed in Table 2. Half of the parent grains had twin bands exhibiting one type of twin variant. Parent grains denoted as g2, g5, g9, g10 and g13 contained twin bands categorized by two types of twin variants. Parent grains, denoted as g7 and g14, contained many twin bands categorized by three variants. In order to confirm the orientation relationship between the parent grain and the twin bands, or between the twin bands, the twin bands in deformed parent grain, g14 were examined in detail. Fig. 4(a) shows a crystallographic orientation map for deformed parent grain, g14, which was strained to a true strain of 0.1. A line profile through the grain, as shown in Fig. 4(b), indicates that the boundary types are dependent on the type of twin variants. It is clear that the misorientation between the members of a different pair (6–2) is a high angle grain boundary (HAGB), while the misorientation between the members of a same pair (5–2) is a low angle grain boundary (LAGB). The RSS, as explained in Section 3, was used to theoretically evaluate the activity of the twin variants in each grain. Fig. 5 shows the spatial distribution of the RSS for the extruded AZ31 Mg alloy under uniaxial compression to the ED. For the calculation of RSS, the six

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Fig. 2. Crystallographic orientations of a parent grain (g1–g16) of a specimen deformed to a true strain of 0.1 and main twin bands existing in each deformed grain enveloped by high angle grain boundaries (>boundary level: 10◦ ).

independent twin variants were considered under the boundary conditions of uniaxial compression. As a first approximation, the positive twin variant having the highest RSS was expected to be favorable for twin-induced reorientation. Fig. 6(a) shows the spatial distribution of the highest RSS in each grain. The white region represents grains with a negative RSS. Spatial distribution of the type of twin variant with the highest RSS is illustrated in Fig. 6(b). The calculated twin variants with the highest RSS in each grain are listed in Table 2. Comparison of these results with the EBSD data, as shown in Fig. 1(b), indicates that the twin variants with the highest RSS occupy the largest fraction (56.25%) and the twin variants with the second- (25.0%) and third-highest (18.75%) RSSs also occupy significant fractions. These fractions had a trend similar to the results of a statistical analysis of active twin variants in high-purity polycrystalline Zr [20]. Twin variants that were not predicted by the RSS criterion are indicated with a (#) symbol in Table 2. It should be noted that grains activating the twin variant with the secondhighest RSS had a relatively high ratio of the second-highest RSS to the highest RSS (=RSS21); 0.982 for g1, 0.915 for g4, 0.992 for g9 and 0.9791 for g15. The two variants (1–4 for g1, 2–5 for g4, 1–4 for g9 and 2–5 for g15) were analyzed as members of the same pair. The results indicate that a small deviation of the local stress state from the macroscopic stress state can activate the twin variants with the second-highest RSS. On the other hand, the grains activating twin variants with the third-highest RSS had relatively low ratios

of the third-highest RSS to the highest RSS (=RSS31); 0.826 for g3, 0.711 for g8, 0.892 for g16. The non-Schmid behavior of primary twinning in extruded AZ31 Mg alloy seems to be mainly due to the deviation of the local stress state within the parent grain from the macroscopic stress state, as a result of interaction with neighboring grains. MB-CPFEM was used to consider the effect of neighboring grains on the activity of twin variants in the extruded AZ31 Mg alloy. Fig. 7 shows the pre-processing procedure for microstructure mapping into finite element (FE) mesh. The technique of direct mapping of EBSD data onto a regular FE mesh induced right-angle segments along the grain boundaries, as shown in Fig. 7(a). The stress and strain fields along the irregular grain boundaries may be inaccurate. Moreover, the crystallographic orientation collected by EBSD was not uniform within a single grain. Several factors, including measurement error and imperfections in the material can contribute to the heterogeneity of crystallographic orientation. First, an average orientation scheme was used to make a uniform orientation within each grain. Smooth segments along the grain boundaries were then constructed using the commercial AutoCAD® 2D software. Fig. 7(b) shows the 2D microstructure having smooth segments along the grain boundaries. Finally, the 2D microstructure was used to construct a quasi-3D FE mesh, including one element layer through the thickness direction (Y). Fig. 7(c) shows the constructed quasi3D FE mesh and boundary conditions for uniaxial compression.

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Fig. 3. Six independent orientations (tv: 1–6) calculated from twin variants in the parent grains (g1–g16).

Fig. 4. (a) Crystallographic orientation map for identification of the orientation relationships of twin boundaries generated by impingement of twin variants in the parent grain, g14. (b) Line profiles of misorientation along a line scan.

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Fig. 5. Spatial distribution of the RSS for the extruded AZ31 Mg alloy under uniaxial compression to the ED.

It should be noted that the quasi-3D FE mesh can only consider the effect of neighboring grains existing on the X–Z plane within one element layer to the Y direction. True compressive strains of ε = 0.05 and 0.1 were simulated using appropriate boundary conditions. The boundary conditions were applied to the four planes comprising the quasi-3D FE mesh, as shown in Fig. 7(c). To simulate uniaxial compression, prescribed displacement in the ED was imposed on the 3–4–7–8 face. The three faces (1–2–3–4, 1–2–5–6 and 1–4–5–8) were constrained to zero displacement. The moving face (5–6–7–8) was 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 3–4–7–8 face. In addition, since the quasi-3D FE mesh of the model cannot be used to consider the interactions with neighboring grains below the model grains, a full-3D mesh was required to capture the interaction. Fig. 8(a) shows the crystallographic orientation map and (0 0 0 1) pole figure of simulated microstructure for the compression specimen to a true strain of 0.05. Crystallographic orientations calculated by MB-CPFEM were directly mapped onto regular grids to satisfy

Fig. 6. (a) Spatial distribution of the highest RSS in each grain. (b) Spatial distribution of the type of twin variant with the highest RSS.

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Fig. 7. A procedure for pre-processing for microstructure mapping into finite element mesh. (a) Technique for direct mapping of EBSD data onto a regular FE mesh. (b) Construction of 2D microstructure having smooth segments along the grain boundaries. (c) Construction of a quasi-3D FE mesh and boundary condition for uniaxial compression.

Fig. 8. (a) Crystallographic orientation map and (0 0 0 1) pole figure of the simulated microstructure of the specimen compressed to a true strain of 0.05. (b) Spatial distribution of twinned region in the simulated microstructure of the specimen compressed to a true strain of 0.05. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

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Fig. 9. Spatial distribution of relative activity of each deformation mode in the simulated microstructure of the specimen compressed to a true strain of 0.05.

Fig. 10. Spatial distribution of individual-associated volume fractions,  tw,e /Stw in the simulated microstructure of the specimen compressed to a true strain of 0.05.

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199

Fig. 11. (a) Spatial distribution of the accumulated twin fraction, Vacc , in each element. (b) Spatial distribution of the type of twin variant with the highest Vacc .

the data format of TSL software. A comparison of Figs. 1(b) and 8(a) shows that MB-CPFEM accurately simulated the rotation of the c-axis toward the near ED, as shown in the (0 0 0 1) pole figure. Fig. 8(b) shows the spatial distribution of the twinned region in the simulated microstructure for the specimen compressed to a true strain of 0.05. Blue and red colors represent portions of untwinned and twinned elements, respectively. As shown, twin bands were nucleated mainly on grain boundaries and partly on twin boundaries or within parent grains. Fig. 9 shows the spatial distribution of the relative activity of each deformation mode in the simulated microstructure for the specimen compressed to a true strain of 0.05. A comparison of Figs. 8(b) and 9 reveals that the basal a slip and tensile twin were the dominant deformation modes in the untwinned region. However, the twinned regions exhibited significant activation of the pyramidal c + a slip. It seems that reorientation induced by twinning contributed to high pyramidal c + a slip activity in the twinned region. The deformed grains that included twin bands exhibited high activation of the tensile twin under uniaxial compression. Fig. 10 shows the spatial distribution of individual-associated volume fractions,  tw,e /Stw , in the simulated microstructure for the specimen compressed to a true strain of 0.05. The accumulation of shear strain in each twin variant exhibited a distinct heterogeneity in the twinned regions. Because the twin system with the highest accumulated volume fraction determines the transformation matrix, T, described by Eq. (11), the various crystallographic orientations in the twinned region were attributed to the heterogeneous distribution of the associated volume fraction. Fig. 11(a) shows the spatial distribution of the accumulated twin fraction, Vacc , in each element described by Eq. (9). The element with the highest Vacc is expected to be a favorable site for nucleation and propagation of twin bands. A comparison of Figs. 8(b) and 11(a) shows that elements with a relatively high Vacc corresponded to the twinned region. However, a comparison of Figs. 1(b) and 11(a) also shows that even grains (g6, g10 and g14) with relatively low Vacc included twin bands in the microstructure deformed by uniaxial compression to a true strain of 0.05. This result indicates that a quasi-3D FE mesh has a limited ability to capture twinning activity accurately in AZ31 Mg alloy, even at a low strain level. The type of twin variant with the highest Vacc is illustrated in Fig. 11(b). A comparison of Figs. 6(b) and 11(b) reveals that MB-CPFEM provided a heterogeneous distribution of the types of twin variants in deformed grains, in particular, near grain

boundaries, compared with the RSS criterion. This result seems due to interaction with neighboring grains existing on the X–Z plane during plastic deformation. Fig. 12(a) shows the crystallographic orientation map and (0 0 0 1) pole figure of the simulated microstructure for the specimen compressed to a true strain of 0.1. Comparison of these results with the EBSD data, which is shown in Fig. 1(b), indicates that MBCPFEM successfully simulated rotation of the c-axis toward the near ED on the (0 0 0 1) pole figure. Fig. 12(b) shows the spatial distribution of the twinned region in the simulated microstructure for the specimen compressed to a true strain of 0.1. The twin volume fraction predicted by the MB-CPFEM was higher than that obtained from the experimental data, as shown in Fig. 1(b). It appears that material parameters determined using the non-topological FE mesh, as explained in Section 2, are not fully optimized for the topological FE mesh. MB-CPFEM provided one or two twin variants activated in parent grains, contrary to the RSS criterion. The twin variants in each grain calculated using the MB-CPFEM are listed in Table 2. It should be noted that the twin variants in each grain predicted by the MB-CPFEM for the most part included the same twin variants as predicted by the RSS criterion. However, comparison of these results with the EBSD data revealed that the MB-CPFEM predicted twin variants that were not predicted by the RSS criterion: 1 for g1, 4 for g2, 5 for g8, 1 for g9, 4 for g13, 6 for g14, and 5 for g16. However, MB-CPFEM failed to predict a few twin variants in deformed grains, as indicated by the (*) symbol in Table 2. The number for (*) was lower than the number for (#). It is likely that the failure in prediction of the twin variants can be overcome by consideration of a full-3D FE mesh. Moreover, MB-CPFEM overestimated the number of active twin variants in parent grains g1, g8 and g16. Using 5◦ as the identification angle of the grain boundary, LAGBs were found in the region between twin bands in deformed grains (g1, g2, g7, g9 and g13). It should be noted that the LAGBs disappeared when 10◦ was used as the identification angle for grain boundaries, as shown in Fig. 12(b). The two neighboring variants in the deformed grains were analyzed as members of the same pair: 1–4 for g1, 1–4 for g2, 2–5 for g7, 1–4 for g9, and 1–4 for g13. As explained in Section 2, the misorientation relationship between the members of a same pair for twin variants was 7.4◦ about the 2 1¯ 1¯ 0 axis. The members were identified as twin variants exhibiting the highest RSS and the second-highest RSS, respectively. Moreover, the ratio of RSS values (second-highest

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Fig. 12. (a) Crystallographic orientation map and (0 0 0 1) pole figure of the simulated microstructure of the specimen compressed to a true strain of 0.1. (b) Spatial distribution of twinned region in the simulated microstructure of the specimen compressed to a true strain of 0.1.

over highest), RSS21 was relatively high, compared with the other grains: 0.982 for g1, 0.987 for g2, 0.993 for g7, 0.992 for g9, and 0.995 for g13. This result indicates that a small deviation in the stress state due to interaction with neighboring grains can activate members of twin variants exhibiting relatively high RSS. Comparison of these results with the EBSD data revealed that four grains (g2, g7, g9 and g13) have twin boundaries similar to those predicted by MB-CPFEM. MB-CPFEM also predicted twin variants impinging on each other with a HAGB. The twin variants (5–6 for g8, 5–6 for g14, and 5–6 for g16) in the deformed grains were analyzed as members of a different pair. The misorientation relationship between the twin variants was analyzed as mostly 60◦ about the rotation axes near the 1 0 1¯ 0 axis. Comparison of these results with the EBSD data reveals that only one grain (g14) had the same twin boundaries as those predicted by MB-CPFEM. The two twin variants in g8 and g16 were the highest RSS (tv = 6) and the third-highest RSS (tv = 5), but the two twin variants in g14 were the highest RSS (tv = 5) and the second-highest RSS (tv = 6). It should be noted that RSS31 (0.711 for g8 and 0.8924 for g16) was relatively low, compared with RSS21 (0.9857 for g14). The two active twin variants in g14 were not members of the same pair. This result indicates that MB-CPFEM based on a quasi-3D FE mesh was sufficient to predict the twin variants consisting of the highest RSS and the secondhighest RSS, but failed to predict the twin variants consisting of the highest RSS and the third-highest RSS. Fig. 13 shows the type of twin variant with the highest Vacc in the simulated microstructure for the specimen compressed to a true strain of 0.1. The interaction with neighboring grains during uniaxial compression induced a slight change in the type of twin variant with the highest Vacc in the region of HAGBs. In order to improve the predictive ability of MBCPFEM, the interactions with neighboring grains below the model grains should be considered by the construction of a full-3D FE mesh.

Fig. 13. The type of twin variant with the highest Vacc in the simulated microstructure of the specimen compressed to a true strain of 0.1.

5. Conclusions A resolved shear stress (RSS) criterion and the microstructure based-crystal plasticity finite element method (MB-CPFEM) were used to analyze the activation of twin variants in extruded AZ31 Mg alloys during ex situ uniaxial compression. The RSS criterion, which is simply based on the Schmid factor, failed to predict the activation of twin variants consisting of the second-highest RSS and the third-highest RSS. EBSD analysis indicated that the twin variants with the

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highest RSS occupied the largest fraction (56.25%) and the twin variants with the second- (25.0%) and third-highest (18.75%) RSSs also occupied a significant fraction. MB-CPFEM was used to consider the interaction with neighboring grains in extruded AZ31 Mg alloy under uniaxial compression. A method of direct mapping of EBSD data onto quasi-3D finite element (FE) mesh with smooth grain boundaries was employed to consider the initial microstructure. In contrast to the RSS criterion, the MB-CPFEM successfully predicted the activation of twin variants consisting of the highest RSS and the second-highest RSS. The MB-CPFEM demonstrated that local fluctuation of the stress field induces the activation of twin variants with the second-highest RSS during uniaxial compression. MB-CPFEM also predicted twin variants impinging on LAGBs or HAGBs, which are observed in EBSD data. However, MB-CPFEM based on a quasi-3D FE mesh failed to predict the twin variants consisting of the highest RSS and the third-highest RSS. Acknowledgments This study was supported by Nuclear Research & Development Program of the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korean government (MEST). References [1] [2] [3] [4] [5] [6]

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