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Simulation of texture evolution and macroscopic properties in Mg alloys using the crystal plasticity finite element method
Materials Science and Engineering A 527 (2010) 1151–1159
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Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Simulation of texture evolution and macroscopic properties in Mg alloys using the crystal plasticity finite element method S.-H. Choi a,∗ , D.H. Kim a , H.W. Lee a , E.J. Shin b a b
Department of Materials Science and Metallurgical Engineering, Sunchon National University, 315 Maegok, Sunchon, Jeonnam 540-742, Republic of Korea Korea Atomic Energy Research Institute, Neutron Physics Department, Daejeon 305-600, Republic of Korea
a r t i c l e
i n f o
Article history: Received 2 July 2009 Received in revised form 21 September 2009 Accepted 26 September 2009
Keywords: Crystal plasticity Finite element Macroscopic properties Mg alloys Texture
a b s t r a c t A crystal plasticity finite element method (CPFEM), considering both crystallographic slip and deformation twinning, was used to simulate texture evolution and macroscopic properties of AZ31 Mg alloys. To capture grain reorientation due to deformation twinning in twin-dominated deformation, a predominant twin reorientation (PTR) model was considered. The validity of the proposed theoretical framework was demonstrated through comparison of simulated results, such as texture evolution and macroscopic properties, with the experimental results and measurements. The simulation of texture evolution and macroscopic properties of AZ31 Mg alloys was shown to be in good agreement with the corresponding experimental results. © 2009 Elsevier B.V. All rights reserved.
1. Introduction The various deformation modes, such as basal a slip, prismatic a slip, pyramidal a, pyramidal c + a slip and tensile twinning, complicate the deformation behavior of hexagonal close packed (HCP) polycrystalline Mg alloys. It is known that critical resolved shear stress (CRSS) of non-basal slip systems at near room temperature (RT) is much higher than that of a basal slip system [1–3]. The limited number of operative slip systems at near RT, is responsible for the poor formability of Mg alloys. Therefore, a high temperature and a low strain rate have been used to improve the formability of Mg alloys. Modification of crystallographic texture is another method to improve the formability of Mg alloys. A sharp basal fibre texture has been characterized as the main texture component in wrought Mg alloys. It was reported that the addition of alloy elements into Mg alloys induces spreading of the basal poles in the rolling direction (RD) [2,4]. The studies reveal that spreading of the basal plane enhances the activation of basal slip during plastic deformation. √ As the c/a ratio of the hexagonal Mg lattice (1.624) is less than 3, a tensile twin is easily activated by caxis tension [5,6]. Deformation twinning can affect the evolution of deformation texture and macroscopic properties [1,7–11]. To understand texture evolution in Mg alloys during plastic defor-
∗ 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 © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2009.09.055
mation, a number of simulation studies have been conducted in various length scales. In particular, many studies have been conducted in an attempt to understand the effect of twin reorientation on the evolution of texture and macroscopic properties during plastic deformation. A predominant twin reorientation (PTR) model [12] has been suggested to explain the reorientation of the crystallographic orientation by deformation twinning. 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 PTR model has been successfully implemented in visco-plastic self-consistent (VPSC) models [13–15] to capture grain reorientation due to deformation twinning [2,15–19]. The VPSC model is reportedly based on a homogenization scheme that can successfully predict macroscopic properties and texture evolution during plastic deformation. However, the VPSC model has limited application to metal-forming simulations due to their complicated geometry. Crystal plasticity finite element methods (CPFEM) based on an inhomogenization scheme have been developed to simulate heterogeneous plastic deformation of hexagonal close packed (HCP) polycrystalline materials [20–23]. A critical issue of the current theoretical framework of CPFEM is the incorporation of a reorientation scheme by deformation twinning into the constitutive equations. A probabilistic approach [21] was used to simulate the texture evolution and stress–strain response of a polycrystalline Mg alloy. In the approach, the orientations of grains were replaced by twin-related orientations only if the twinned volume fraction exceeded a certain random number. A total Lagrangian
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approach [24] was proposed to simulate rolling textures in a polycrystalline Zr alloy. The same theoretical framework was used to simulate the texture evolution and flow stress in high purity ␣-Ti [23,25]. The PTR model has been used to simulate to simulate earing profiles after deep drawing [26] and texture evolution during hot rolling [27]. 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. The validity of the proposed model framework was verified by comparing the predicted texture evolution, twin-volume fraction and macroscopic properties of AZ31 Mg alloys with the measured experimental results. 2. Experimental procedure and theoretical methods
Taking into account the condition of volume constancy, the R-value is defined by [30] R=
εw ln(wf /wi ) εw =− = εt (εl + εw ) −ln(lf /li ) − ln(wf /wi )
(1)
where εl , εw and εt are the strains in the length, width and thickness directions, respectively; l, w and t represent sample length, width, and thickness, respectively, and i and f represent the initial and final values. All R-value measurements were conducted at true strains of 0.04, 0.08 and 0.1. 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.
2.1. Materials and microstructure
3. Theoretical procedure
The present study used strip-cast AZ31 (3 wt.% Al, 1 wt.% Zn, balance Mg) Mg alloy followed by hot rolling. The sheet of 2 mm thick had a grain size of approximately 10.16 m. To examine macroscopic texture of as-rolled AZ31 Mg alloy sheet, pole figure measurements were carried out using a Rigaku D Max 2500 ¯ ¯ ¯ X-ray diffractometer. The five ((0 0 0 2), (1010), (1011), (1012), ¯ and (1120)) incomplete pole figures (tilting angle: 0–70◦ ) were
Texture evolution and macroscopic properties in HCP polycrystalline Mg alloys was simulated using the finite element code, ABAQUS/Standard [31], with the material model programmed based on continuum crystal plasticity theory. A rate-dependent constitutive relation has been implemented into the user material subroutine UMAT in ABAQUS. The model is fundamentally based on a 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.
used to calculate ODF. Based on the five incomplete pole figures, the crystallographic orientation distribution function (ODF) was calculated using the WIMV method [28] and Beartex software. 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 the 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 300 m × 100 m at a step size of 0.5 m. The EBSD data were analyzed using the TSL software to evaluate pole figure, ODF and twin-volume fraction. 2.2. Mechanical properties To measure the macroscopic mechanical properties of hotrolled AZ31 Mg alloy sheets, tensile specimens were cut from the sheet to the RD. Tensile specimens were prepared by laser cutting in accordance with ASTM standard B557M-94 (gauge length: 25 mm; width: 6 mm). Compressive specimens were machined by laser cutting from the 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 in-plane compression was the buckling phenomenon. To avoid buckling, either through-thickness sheet stabilization [29] or a thicker Mg sheet have been used [3]. 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 GLEEBLE® 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 s−1 . 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. The uniaxial loading tests were interrupted by unloading and, with the aid of a micrometer, the plastic strains were measured relative to the width and thickness directions to determine the R-value (i.e., the plastic strain ratio).
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
˙ ˛ s˛ ⊗ m˛
(4)
˛=1
The summation represents all of the deformation modes, N (=Ns + Nt ), consisting of slip, Ns , and twin systems, Nt . The plastic part of the velocity gradient is decomposed further into symmetric and antisymmetric parts (Lp = Dp + p ) to yield the formula Dp =
p =
1 ˛ ˛ ˙ (s ⊗ m˛ + m˛ ⊗ s˛ ) = ˙ ˛ P˛ 2 N
N
˛=1
˛=1
1 ˛ ˛ ˙ (s ⊗ m˛ − m˛ ⊗ s˛ ) = ˙ ˛ W˛ 2 N
N
˛=1
˛=1
(5)
where Dp is the plastic part of the rate of deformation tensor and p is the plastic spin. The evolution of the slip/twin directions and slip/twin plane normals can be expressed in terms of the elastic part of the deformation gradient as s˛ = F e · s˛ o
and
e−1 m˛ = m˛ o ·F
(6)
˛ where s˛ o and mo are the slip/twin vectors in the reference orientation of the crystal lattice. The rate of change of Fe is given by e F˙ = Le · Fe = (L − Lp ) · Fe
(7)
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The resolved shear stress, ˛ , is chosen such that it is work conjugate of the slip rate, ˙ ˛ . Expressing the plastic work rate in the reference volume, : DP =
N
˙ ˛ : P˛ =
˛=1
N
˙ ˛ ˛
(8)
˛=1
Here, is the symmetric Kirchhoff stress, which is related to the Cauchy stress by = J, where J = Det(F). As described in [32], the Jaumann rate of Kirchhoff stress can be expressed as ˆ = K : D −
N
˙ ˛ R ˛
Table 1 Microscopic hardening coefficients used in the CPFEM simulation. Mode
o˛ (MPa)
ho (MPa)
sat (MPa)
a
Basal a Prism a Pyramidal a Pyramidal c + a 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
summation of shear strain in the elements (e) as follows: R.A.˛ =
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 R˛ 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, 5, independent elastic constants of pure Mg are used in the present work [33]: C11 = 58 GPa, C12 = 25 GPa, C13 = 20.8 GPa, C33 = 61.2 GPa,
e=1
˛ 1/m ˙ ˛ = ˙ o˛ ˛ sign( ˛ )
(10)
o
Self- and latent-hardening are readily accounted for by a suitable evolution of the reference o˛ values in the constitutive law in Eq. (10). The present work employs a microscopic hardening law [34,35] for this purpose, as follows:
˙ o˛ =
N ˇ
H ˛ˇ |˙ ˇ |
= q˛ˇ ho 1 −
˛, ˇ = 1, . . . , (Ns + Nt ) o˛ sat
H ˛ˇ
a (11)
where H˛ˇ is a hardening matrix that is introduced to account for interaction between the slip and twin systems. q˛ˇ accounts for the hardening rate of the slip/twin system, ˛, due to slip/twin activity on system, ˇ. Here, it is assumed that the self-hardening term (diagonal term of q˛ˇ ) equals the latent-hardening term (off-diagonal term of q˛ˇ ), i.e. (q˛ˇ = 1). The fitting simulation was carried out by varying the CRSS values and microscopic hardening parameters (ho , sat and a) until agreement was achieved between the predicted and the measured uniaxial loading curves. In this study, four slip systems and one twin system were considered: basal a ({0 0 0 1} ¯ ¯ ¯ ¯ ¯ 1120), prismatic a ({1100}11 20), pyramidal a ({1100}11 20), ¯ 1¯ 123) ¯ ¯ 1011). ¯ pyramidal c + a({1122} and tensile twin ({1012} CRSS values and microscopic hardening parameters were determined by fitting the experimental stress–strain curve. The set of parameters listed in Table 1 was used in the theoretical simulation. The relative contribution of deformation modes as a function of true strain gives information that is useful for analysis of plastic deformation. The relative activity (R.A.˛ ) of each deformation mode, ˛, among slip/twin systems (N = Ns + Nt ), was determined by
ˇ=1
(12)
ˇ,e
t,g t
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
˛,e
where Ne is the total number of elements in the current simulation. To consider the effect of deformation twinning on texture evolution and macroscopic properties, the predominant twin reorientation (PTR) model [12] was implemented in the CPFEM. This requires tracking of the shear strain, t,g , contributed by each twin system t, and of the associated volume fraction Vt,g = t,g /St (St = 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: V acc,mode =
C55 = 16.6 GPa.
Ne
N Ne e=1
(9)
˛=1
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(13)
St
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 = C th1 + C th2 · V acc,mode
(14)
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 reorientation 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 [10,36,37]. Therefore, the optimal combination of constants in Eq. (14) 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 [38,15]. However, it should be noted that 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 CPFEM. The transformation matrix, T, between the lattice orientation in the matrix and the lattice orientation in the twinned region can be defined as [39]: Tij = 2ni nj − ıij ;
ıij = 0
if i = / j;
ıij = 1
(15)
where n represents the unit vector of the twin plane normal in orthogonal coordinates. The implicit time integration algorithm for the slip rates is similar to the rate tangent method of Peirce et al. [32]. Moreover, a material Jacobian (∂ˆ /∂ε) for the implicit method can be obtained from Eq. (9). 4. Results and discussion The recalculated pole figures and ODF sections of the as-rolled sheet are shown in Fig. 1. The main texture components developed
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Fig. 1. Macrotexture of the as-rolled AZ31 alloy sheet measured by X-ray diffractometer: (a) pole figures (stereographic projection) and (b) ODF sections.
Fig. 2. Graphical representation of the main texture components that are typical in as-rolled and deformed AZ31 Mg alloys.
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Fig. 3. Inverse pole figure map (the position of the ND in the colored stereographic triangle) showing the texture evolution of AZ31 Mg alloys: (a) as-rolled, (b) true strain = 0.05 and (c) true strain = 0.1.
in as-rolled and deformed AZ31 Mg alloys can be represented by pole figures and ODF sections, as shown in Fig. 2. Fig. 1(a) clearly shows that the as-rolled AZ31 alloy sheet exhibits asymmetrical distribution of basal texture components (0001//ND) with respect
to ND on the (0 0 0 2) pole figure. A comparison of Fig. 1(b) and Fig. 2(b) shows that the as-rolled AZ31 Mg alloy sheet had a relatively even distribution of basal texture components (A, B, C and D orientations) along the ϕ1 direction of the ϕ2 = 0◦ section.
Fig. 4. Experimentally measured pole figures and ODF sections showing the deformation texture of the CA//RD specimen: (a) true strain = 0.05 and (b) true strain = 0.1.
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Fig. 5. Finite element mesh and boundary conditions used in the simulation of uniaxial loading of as-rolled AZ31 Mg alloy sheet.
Fig. 3(a) shows the microtexture of the as-rolled AZ31 Mg alloy. The alloy exhibited a relatively inhomogeneous and coarse distribution of grain size. Since uniaxial tension did not significantly affect the deformation texture of AZ31 Mg alloy, only the microtexture for the compression specimen is shown in the present study. Microtexture evolution of AZ31 Mg alloy during uniaxial compression to a true strain of 0.05 is shown in Fig. 3(b). After uniaxial compression to a true strain of 0.05, twin bands were nucleated and propagated through their parent grains. It was observed that 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. 3(c), the propagation of twin bands was almost prohibited by collision with high angle grain boundaries of their parent
Fig. 6. Simulated stress–strain curves for AZ31 alloy sheets together with experimental measurements.
grains and other twin bands at twin boundaries. Fig. 4 shows the measured deformation textures for compressive specimens of the AZ31 Mg alloy deformed to a true strain of 0.05, 0.1 along the RD, respectively. The microtexture revealed that many crystallographic lattices rotate to the RD by twin-induced reorientation. A comparison of Figs. 2 and 4 shows that the intensity of F texture component was enhanced in the specimen deformed to a true strain of 0.05. It was also clear that the near F and H texture components were relatively stable orientations in the specimen to a true strain of 0.1. Crystallographic orientations of untwinned lattices were characterized as the E and G texture components. The result indicates that untwinned lattices accommodate the macroscopic deformation by a gradual rotation of crystallographic orientation. As an example of CPFEM verification, simulation studies have been conducted to predict the texture evolution and macroscopic properties of the AZ31 Mg alloy sheet measured at 200 ◦ C, as mentioned in Section 2.2. The initial mesh (20 × 20 × 10 = 4000 elements) of the polycrystal model is shown in Fig. 5. The initial length of the model region is given by lo = 2 mm, wo = 2 mm, to = 1 mm. A 3-D solid element C3D8R (ABAQUS) with eight nodes and one integration point was used. 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 [40]. To impose initial orientations of the elements, each orientation selected from the 4000-grain orientations was mapped onto each integration point in the finite element mesh. Other sophisticated mapping methods can be classified into a probability assignment method [41] and mathematical mapping method using a spherical Gaussian texture components [42]. Based on the PTR model explained in Section 3, it seems that the current mapping method is more favorable for simulation of texture evolution and macroscopic properties in Mg alloys. The total true strain for tension and compression was ε = 0.2. The boundary condition was applied to the four planes comprising the 3-D mesh, as shown in Fig. 5. For the uniaxial loading simulation, prescribed displacements in the RD were imposed on the 2-3-6-7 face. Because the tensile axis (TA) and compressive axis (CA) were
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Fig. 7. Relative activities of the five deformation modes of the uniaxial loading specimens measured at 200 ◦ C: (a) TA//RD and (b) CA//RD.
parallel to the RD, these specimens were denoted as follows: TA//RD or CA//RD. Fig. 6 shows a comparison of the simulated and measured stress–strain curves for the TA//RD and CA//RD specimens. The TA//RD specimen exhibited higher flow stress compared with the CA//RD specimen at the low strain level. The simulation results were consistent with the experimental results. It is clear that CPFEM successfully simulated tension–compression asymmetric yielding behavior, which has been already reported in several studies [29,30]. The macroscopic flow curve of the CA//RD specimen exhibited a sigmoidal hardening behavior, which is typical in twindominated deformation. A sigmoidal hardening behavior in the flow curve of AZ31 Mg alloys is partially responsible for the twininduced formation of barriers to dislocation motion, as shown in Fig. 3(b). It is known that primary and secondary twin boundaries contribute to these barriers [10,37]. A previous paper [15] also suggested that the activation of pyramidal c + aslip in the twinned region can contribute to the high hardening rate of the CA//RD specimen after a true strain of about 0.05. Since specimens will have low
density twin boundaries, the barriers to dislocation motion will not increase significantly during plastic deformation. To consider the effect of weak twin-induced barriers, a relatively low hardening parameter, ho , was imposed on the deformation modes by dislocation slip, as shown in Table 1. Moreover, the threshold values, Cth1 (=0.25) and Cth2 (=0.5), in Eq. (14) were also optimized to fit the compressive flow curve of the AZ31 Mg alloy sheets. Fig. 7(a) shows the variation in the relative activity of each deformation mode for the TA//RD specimen. Plastic deformation began with the activation of the basal a slip as a primary mode. The basal a slip decreased rapidly and prismatic a and pyramidal c + aslips increased rapidly during deformation. Fig. 7(b) shows the variation in the relative activity of each deformation mode for the CA//RD specimen. Plastic deformation began with the activation of a tensile twin and a basal a slip as the primary mode. Subsequently, the tensile twin decreased continuously and the pyramidal c + aslip increased gradually during plastic deformation. Fig. 8 shows the simulated textures for the CA//RD specimens deformed to true strains of 0.05 and 0.1. During uniaxial compression, the c-
Fig. 8. Simulated pole figures and ODF sections showing the deformation texture of the CA//RD specimen deformed at 200 ◦ C: (a) true strain = 0.05 and (b) true strain = 0.1.
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Fig. 9. Comparison of experimentally measured and theoretically simulated Rvalues of AZ31 Mg alloy sheets.
Fig. 11. Comparison of experimentally measured and theoretically simulated twinvolume fraction in the CA//RD specimens.
axis of many orientations were aligned with the RD for the CA//RD specimens on the (0 0 0 2) pole figure. The activity of the tensile twin played an important role in rotation of the c-axis from the ND to the RD on the (0 0 0 2) pole figure during plastic deformation. It is clear that the CA//RD specimen deformed to a true strain of 0.05, exhibits an enhanced intensity at the F texture component, as shown in Fig. 8(a). However, in the CA//RD specimen deformed to a true strain of 0.1, stable orientations moved to the F and H texture components, as shown in Fig. 8(b). Similar results were observed for the deformation texture measured using an EBSD technique, as shown in Fig. 4. A comparison of the measured and simulated R-values of AZ31 Mg alloy sheets is shown in Fig. 9. Regardless of loading sign, it should be noted that R-values increased gradually as true strain increased. The directionality of the simulated R-value for uniaxial loading was similar to the experimental data. It should be noted that AZ31 Mg alloy sheets exhibited asymmetric R-values during plastic deformation. These results indicate that if Mg alloy sheets exhibit twin-dominated plastic deformation, asymmetric R-values, as well as asymmetric yielding, should be considered in simulations of sheet metal forming. Based on the principle of normality, the shape of the yield surface was closely related to the plastic strain
ratio [43]. Therefore, this result also indicates that the yield surface of the textured Mg alloy was distorted and significantly altered by in-plane plastic deformation [44]. Evolution of the twin-volume fraction was also simulated to evaluate the validity of the PTR model implemented in the current CPFEM. Fig. 10 shows the evolution of the twin-volume fraction during uniaxial compression. Blue and red elements represent portions of untwinned and twinned regions, respectively. Recently, microtextural characterization using EBSD verified that twin bands nucleate and propagate through the parent orientation during plastic deformation [9,10,15,37]. It seems that the current mapping method is not appropriate for the simulation of twin band nucleation and propagation in a deformed-parent orientation. A CPFEM based on a real microstructure as an initial configuration should be employed to simulate the heterogeneous deformation behavior in AZ31 Mg alloys [45]. As mentioned in Section 2, volume fractions of deformed specimens can be identified using the twinanalysis technique implemented with TSL software. Fig. 11 shows the evolution of the twin-volume fraction in the CA//RD specimens during uniaxial compression. EBSD results indicate that 200 ◦ C is a favorable temperature for twin-dominated deformation at a low strain level. It was found that the twin-volume fraction of com-
Fig. 10. Evolution of the deformation twinning during uniaxial compression: (a) true strain = 0.05, (b) true strain = 0.1 and (c) true strain = 0.15.
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pression specimens did not change until a critical strain of about 0.03. CPFEM successfully predicted the twin-volume fraction in the CA//RD specimens at a low strain level, when the experimental error was considered.
[5] [6] [7] [8] [9] [10]
5. Conclusions
[11] [12]
A crystal plasticity framework that considers both crystallographic slip and deformation twinning was developed to simulate the texture evolution and macroscopic properties of AZ31 Mg alloys. The major feature of the model is the implementation of a PTR model to capture grain reorientation due to deformation twinning during plastic deformation. For comparison with theoretically simulated results, experiments were performed using as-rolled AZ31 Mg alloy sheets. To demonstrate the validity of the proposed theoretical framework, uniaxial tension and compression tests were conducted at deformation temperatures of 200 ◦ C to simulate texture evolution, twin-volume fraction and macroscopic properties of the as-rolled AZ31 Mg alloy sheets. Results of the simulation revealed that texture evolution, twin-volume fraction and macroscopic properties in uniaxial loading were strongly dependent on the loading sign. The relative activities of the deformation modes can be used to theoretically explain the asymmetric behaviors of tension–compression yielding, and the R-value evident in the AZ31 Mg alloy sheets. Acknowledgements This study was supported by Nuclear Research & Development Program of the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korean government (MEST) 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] [2] [3] [4]
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Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Simulation of texture evolution and macroscopic properties in Mg alloys using the crystal plasticity finite element method S.-H. Choi a,∗ , D.H. Kim a , H.W. Lee a , E.J. Shin b a b
Department of Materials Science and Metallurgical Engineering, Sunchon National University, 315 Maegok, Sunchon, Jeonnam 540-742, Republic of Korea Korea Atomic Energy Research Institute, Neutron Physics Department, Daejeon 305-600, Republic of Korea
a r t i c l e
i n f o
Article history: Received 2 July 2009 Received in revised form 21 September 2009 Accepted 26 September 2009
Keywords: Crystal plasticity Finite element Macroscopic properties Mg alloys Texture
a b s t r a c t A crystal plasticity finite element method (CPFEM), considering both crystallographic slip and deformation twinning, was used to simulate texture evolution and macroscopic properties of AZ31 Mg alloys. To capture grain reorientation due to deformation twinning in twin-dominated deformation, a predominant twin reorientation (PTR) model was considered. The validity of the proposed theoretical framework was demonstrated through comparison of simulated results, such as texture evolution and macroscopic properties, with the experimental results and measurements. The simulation of texture evolution and macroscopic properties of AZ31 Mg alloys was shown to be in good agreement with the corresponding experimental results. © 2009 Elsevier B.V. All rights reserved.
1. Introduction The various deformation modes, such as basal a slip, prismatic a slip, pyramidal a, pyramidal c + a slip and tensile twinning, complicate the deformation behavior of hexagonal close packed (HCP) polycrystalline Mg alloys. It is known that critical resolved shear stress (CRSS) of non-basal slip systems at near room temperature (RT) is much higher than that of a basal slip system [1–3]. The limited number of operative slip systems at near RT, is responsible for the poor formability of Mg alloys. Therefore, a high temperature and a low strain rate have been used to improve the formability of Mg alloys. Modification of crystallographic texture is another method to improve the formability of Mg alloys. A sharp basal fibre texture has been characterized as the main texture component in wrought Mg alloys. It was reported that the addition of alloy elements into Mg alloys induces spreading of the basal poles in the rolling direction (RD) [2,4]. The studies reveal that spreading of the basal plane enhances the activation of basal slip during plastic deformation. √ As the c/a ratio of the hexagonal Mg lattice (1.624) is less than 3, a tensile twin is easily activated by caxis tension [5,6]. Deformation twinning can affect the evolution of deformation texture and macroscopic properties [1,7–11]. To understand texture evolution in Mg alloys during plastic defor-
∗ 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 © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2009.09.055
mation, a number of simulation studies have been conducted in various length scales. In particular, many studies have been conducted in an attempt to understand the effect of twin reorientation on the evolution of texture and macroscopic properties during plastic deformation. A predominant twin reorientation (PTR) model [12] has been suggested to explain the reorientation of the crystallographic orientation by deformation twinning. 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 PTR model has been successfully implemented in visco-plastic self-consistent (VPSC) models [13–15] to capture grain reorientation due to deformation twinning [2,15–19]. The VPSC model is reportedly based on a homogenization scheme that can successfully predict macroscopic properties and texture evolution during plastic deformation. However, the VPSC model has limited application to metal-forming simulations due to their complicated geometry. Crystal plasticity finite element methods (CPFEM) based on an inhomogenization scheme have been developed to simulate heterogeneous plastic deformation of hexagonal close packed (HCP) polycrystalline materials [20–23]. A critical issue of the current theoretical framework of CPFEM is the incorporation of a reorientation scheme by deformation twinning into the constitutive equations. A probabilistic approach [21] was used to simulate the texture evolution and stress–strain response of a polycrystalline Mg alloy. In the approach, the orientations of grains were replaced by twin-related orientations only if the twinned volume fraction exceeded a certain random number. A total Lagrangian
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approach [24] was proposed to simulate rolling textures in a polycrystalline Zr alloy. The same theoretical framework was used to simulate the texture evolution and flow stress in high purity ␣-Ti [23,25]. The PTR model has been used to simulate to simulate earing profiles after deep drawing [26] and texture evolution during hot rolling [27]. 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. The validity of the proposed model framework was verified by comparing the predicted texture evolution, twin-volume fraction and macroscopic properties of AZ31 Mg alloys with the measured experimental results. 2. Experimental procedure and theoretical methods
Taking into account the condition of volume constancy, the R-value is defined by [30] R=
εw ln(wf /wi ) εw =− = εt (εl + εw ) −ln(lf /li ) − ln(wf /wi )
(1)
where εl , εw and εt are the strains in the length, width and thickness directions, respectively; l, w and t represent sample length, width, and thickness, respectively, and i and f represent the initial and final values. All R-value measurements were conducted at true strains of 0.04, 0.08 and 0.1. 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.
2.1. Materials and microstructure
3. Theoretical procedure
The present study used strip-cast AZ31 (3 wt.% Al, 1 wt.% Zn, balance Mg) Mg alloy followed by hot rolling. The sheet of 2 mm thick had a grain size of approximately 10.16 m. To examine macroscopic texture of as-rolled AZ31 Mg alloy sheet, pole figure measurements were carried out using a Rigaku D Max 2500 ¯ ¯ ¯ X-ray diffractometer. The five ((0 0 0 2), (1010), (1011), (1012), ¯ and (1120)) incomplete pole figures (tilting angle: 0–70◦ ) were
Texture evolution and macroscopic properties in HCP polycrystalline Mg alloys was simulated using the finite element code, ABAQUS/Standard [31], with the material model programmed based on continuum crystal plasticity theory. A rate-dependent constitutive relation has been implemented into the user material subroutine UMAT in ABAQUS. The model is fundamentally based on a 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.
used to calculate ODF. Based on the five incomplete pole figures, the crystallographic orientation distribution function (ODF) was calculated using the WIMV method [28] and Beartex software. 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 the 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 300 m × 100 m at a step size of 0.5 m. The EBSD data were analyzed using the TSL software to evaluate pole figure, ODF and twin-volume fraction. 2.2. Mechanical properties To measure the macroscopic mechanical properties of hotrolled AZ31 Mg alloy sheets, tensile specimens were cut from the sheet to the RD. Tensile specimens were prepared by laser cutting in accordance with ASTM standard B557M-94 (gauge length: 25 mm; width: 6 mm). Compressive specimens were machined by laser cutting from the 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 in-plane compression was the buckling phenomenon. To avoid buckling, either through-thickness sheet stabilization [29] or a thicker Mg sheet have been used [3]. 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 GLEEBLE® 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 s−1 . 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. The uniaxial loading tests were interrupted by unloading and, with the aid of a micrometer, the plastic strains were measured relative to the width and thickness directions to determine the R-value (i.e., the plastic strain ratio).
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
˙ ˛ s˛ ⊗ m˛
(4)
˛=1
The summation represents all of the deformation modes, N (=Ns + Nt ), consisting of slip, Ns , and twin systems, Nt . The plastic part of the velocity gradient is decomposed further into symmetric and antisymmetric parts (Lp = Dp + p ) to yield the formula Dp =
p =
1 ˛ ˛ ˙ (s ⊗ m˛ + m˛ ⊗ s˛ ) = ˙ ˛ P˛ 2 N
N
˛=1
˛=1
1 ˛ ˛ ˙ (s ⊗ m˛ − m˛ ⊗ s˛ ) = ˙ ˛ W˛ 2 N
N
˛=1
˛=1
(5)
where Dp is the plastic part of the rate of deformation tensor and p is the plastic spin. The evolution of the slip/twin directions and slip/twin plane normals can be expressed in terms of the elastic part of the deformation gradient as s˛ = F e · s˛ o
and
e−1 m˛ = m˛ o ·F
(6)
˛ where s˛ o and mo are the slip/twin vectors in the reference orientation of the crystal lattice. The rate of change of Fe is given by e F˙ = Le · Fe = (L − Lp ) · Fe
(7)
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The resolved shear stress, ˛ , is chosen such that it is work conjugate of the slip rate, ˙ ˛ . Expressing the plastic work rate in the reference volume, : DP =
N
˙ ˛ : P˛ =
˛=1
N
˙ ˛ ˛
(8)
˛=1
Here, is the symmetric Kirchhoff stress, which is related to the Cauchy stress by = J, where J = Det(F). As described in [32], the Jaumann rate of Kirchhoff stress can be expressed as ˆ = K : D −
N
˙ ˛ R ˛
Table 1 Microscopic hardening coefficients used in the CPFEM simulation. Mode
o˛ (MPa)
ho (MPa)
sat (MPa)
a
Basal a Prism a Pyramidal a Pyramidal c + a 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
summation of shear strain in the elements (e) as follows: R.A.˛ =
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 R˛ 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, 5, independent elastic constants of pure Mg are used in the present work [33]: C11 = 58 GPa, C12 = 25 GPa, C13 = 20.8 GPa, C33 = 61.2 GPa,
e=1
˛ 1/m ˙ ˛ = ˙ o˛ ˛ sign( ˛ )
(10)
o
Self- and latent-hardening are readily accounted for by a suitable evolution of the reference o˛ values in the constitutive law in Eq. (10). The present work employs a microscopic hardening law [34,35] for this purpose, as follows:
˙ o˛ =
N ˇ
H ˛ˇ |˙ ˇ |
= q˛ˇ ho 1 −
˛, ˇ = 1, . . . , (Ns + Nt ) o˛ sat
H ˛ˇ
a (11)
where H˛ˇ is a hardening matrix that is introduced to account for interaction between the slip and twin systems. q˛ˇ accounts for the hardening rate of the slip/twin system, ˛, due to slip/twin activity on system, ˇ. Here, it is assumed that the self-hardening term (diagonal term of q˛ˇ ) equals the latent-hardening term (off-diagonal term of q˛ˇ ), i.e. (q˛ˇ = 1). The fitting simulation was carried out by varying the CRSS values and microscopic hardening parameters (ho , sat and a) until agreement was achieved between the predicted and the measured uniaxial loading curves. In this study, four slip systems and one twin system were considered: basal a ({0 0 0 1} ¯ ¯ ¯ ¯ ¯ 1120), prismatic a ({1100}11 20), pyramidal a ({1100}11 20), ¯ 1¯ 123) ¯ ¯ 1011). ¯ pyramidal c + a({1122} and tensile twin ({1012} CRSS values and microscopic hardening parameters were determined by fitting the experimental stress–strain curve. The set of parameters listed in Table 1 was used in the theoretical simulation. The relative contribution of deformation modes as a function of true strain gives information that is useful for analysis of plastic deformation. The relative activity (R.A.˛ ) of each deformation mode, ˛, among slip/twin systems (N = Ns + Nt ), was determined by
ˇ=1
(12)
ˇ,e
t,g t
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
˛,e
where Ne is the total number of elements in the current simulation. To consider the effect of deformation twinning on texture evolution and macroscopic properties, the predominant twin reorientation (PTR) model [12] was implemented in the CPFEM. This requires tracking of the shear strain, t,g , contributed by each twin system t, and of the associated volume fraction Vt,g = t,g /St (St = 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: V acc,mode =
C55 = 16.6 GPa.
Ne
N Ne e=1
(9)
˛=1
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(13)
St
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 = C th1 + C th2 · V acc,mode
(14)
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 reorientation 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 [10,36,37]. Therefore, the optimal combination of constants in Eq. (14) 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 [38,15]. However, it should be noted that 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 CPFEM. The transformation matrix, T, between the lattice orientation in the matrix and the lattice orientation in the twinned region can be defined as [39]: Tij = 2ni nj − ıij ;
ıij = 0
if i = / j;
ıij = 1
(15)
where n represents the unit vector of the twin plane normal in orthogonal coordinates. The implicit time integration algorithm for the slip rates is similar to the rate tangent method of Peirce et al. [32]. Moreover, a material Jacobian (∂ˆ /∂ε) for the implicit method can be obtained from Eq. (9). 4. Results and discussion The recalculated pole figures and ODF sections of the as-rolled sheet are shown in Fig. 1. The main texture components developed
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Fig. 1. Macrotexture of the as-rolled AZ31 alloy sheet measured by X-ray diffractometer: (a) pole figures (stereographic projection) and (b) ODF sections.
Fig. 2. Graphical representation of the main texture components that are typical in as-rolled and deformed AZ31 Mg alloys.
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Fig. 3. Inverse pole figure map (the position of the ND in the colored stereographic triangle) showing the texture evolution of AZ31 Mg alloys: (a) as-rolled, (b) true strain = 0.05 and (c) true strain = 0.1.
in as-rolled and deformed AZ31 Mg alloys can be represented by pole figures and ODF sections, as shown in Fig. 2. Fig. 1(a) clearly shows that the as-rolled AZ31 alloy sheet exhibits asymmetrical distribution of basal texture components (0001//ND) with respect
to ND on the (0 0 0 2) pole figure. A comparison of Fig. 1(b) and Fig. 2(b) shows that the as-rolled AZ31 Mg alloy sheet had a relatively even distribution of basal texture components (A, B, C and D orientations) along the ϕ1 direction of the ϕ2 = 0◦ section.
Fig. 4. Experimentally measured pole figures and ODF sections showing the deformation texture of the CA//RD specimen: (a) true strain = 0.05 and (b) true strain = 0.1.
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Fig. 5. Finite element mesh and boundary conditions used in the simulation of uniaxial loading of as-rolled AZ31 Mg alloy sheet.
Fig. 3(a) shows the microtexture of the as-rolled AZ31 Mg alloy. The alloy exhibited a relatively inhomogeneous and coarse distribution of grain size. Since uniaxial tension did not significantly affect the deformation texture of AZ31 Mg alloy, only the microtexture for the compression specimen is shown in the present study. Microtexture evolution of AZ31 Mg alloy during uniaxial compression to a true strain of 0.05 is shown in Fig. 3(b). After uniaxial compression to a true strain of 0.05, twin bands were nucleated and propagated through their parent grains. It was observed that 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. 3(c), the propagation of twin bands was almost prohibited by collision with high angle grain boundaries of their parent
Fig. 6. Simulated stress–strain curves for AZ31 alloy sheets together with experimental measurements.
grains and other twin bands at twin boundaries. Fig. 4 shows the measured deformation textures for compressive specimens of the AZ31 Mg alloy deformed to a true strain of 0.05, 0.1 along the RD, respectively. The microtexture revealed that many crystallographic lattices rotate to the RD by twin-induced reorientation. A comparison of Figs. 2 and 4 shows that the intensity of F texture component was enhanced in the specimen deformed to a true strain of 0.05. It was also clear that the near F and H texture components were relatively stable orientations in the specimen to a true strain of 0.1. Crystallographic orientations of untwinned lattices were characterized as the E and G texture components. The result indicates that untwinned lattices accommodate the macroscopic deformation by a gradual rotation of crystallographic orientation. As an example of CPFEM verification, simulation studies have been conducted to predict the texture evolution and macroscopic properties of the AZ31 Mg alloy sheet measured at 200 ◦ C, as mentioned in Section 2.2. The initial mesh (20 × 20 × 10 = 4000 elements) of the polycrystal model is shown in Fig. 5. The initial length of the model region is given by lo = 2 mm, wo = 2 mm, to = 1 mm. A 3-D solid element C3D8R (ABAQUS) with eight nodes and one integration point was used. 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 [40]. To impose initial orientations of the elements, each orientation selected from the 4000-grain orientations was mapped onto each integration point in the finite element mesh. Other sophisticated mapping methods can be classified into a probability assignment method [41] and mathematical mapping method using a spherical Gaussian texture components [42]. Based on the PTR model explained in Section 3, it seems that the current mapping method is more favorable for simulation of texture evolution and macroscopic properties in Mg alloys. The total true strain for tension and compression was ε = 0.2. The boundary condition was applied to the four planes comprising the 3-D mesh, as shown in Fig. 5. For the uniaxial loading simulation, prescribed displacements in the RD were imposed on the 2-3-6-7 face. Because the tensile axis (TA) and compressive axis (CA) were
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Fig. 7. Relative activities of the five deformation modes of the uniaxial loading specimens measured at 200 ◦ C: (a) TA//RD and (b) CA//RD.
parallel to the RD, these specimens were denoted as follows: TA//RD or CA//RD. Fig. 6 shows a comparison of the simulated and measured stress–strain curves for the TA//RD and CA//RD specimens. The TA//RD specimen exhibited higher flow stress compared with the CA//RD specimen at the low strain level. The simulation results were consistent with the experimental results. It is clear that CPFEM successfully simulated tension–compression asymmetric yielding behavior, which has been already reported in several studies [29,30]. The macroscopic flow curve of the CA//RD specimen exhibited a sigmoidal hardening behavior, which is typical in twindominated deformation. A sigmoidal hardening behavior in the flow curve of AZ31 Mg alloys is partially responsible for the twininduced formation of barriers to dislocation motion, as shown in Fig. 3(b). It is known that primary and secondary twin boundaries contribute to these barriers [10,37]. A previous paper [15] also suggested that the activation of pyramidal c + aslip in the twinned region can contribute to the high hardening rate of the CA//RD specimen after a true strain of about 0.05. Since specimens will have low
density twin boundaries, the barriers to dislocation motion will not increase significantly during plastic deformation. To consider the effect of weak twin-induced barriers, a relatively low hardening parameter, ho , was imposed on the deformation modes by dislocation slip, as shown in Table 1. Moreover, the threshold values, Cth1 (=0.25) and Cth2 (=0.5), in Eq. (14) were also optimized to fit the compressive flow curve of the AZ31 Mg alloy sheets. Fig. 7(a) shows the variation in the relative activity of each deformation mode for the TA//RD specimen. Plastic deformation began with the activation of the basal a slip as a primary mode. The basal a slip decreased rapidly and prismatic a and pyramidal c + aslips increased rapidly during deformation. Fig. 7(b) shows the variation in the relative activity of each deformation mode for the CA//RD specimen. Plastic deformation began with the activation of a tensile twin and a basal a slip as the primary mode. Subsequently, the tensile twin decreased continuously and the pyramidal c + aslip increased gradually during plastic deformation. Fig. 8 shows the simulated textures for the CA//RD specimens deformed to true strains of 0.05 and 0.1. During uniaxial compression, the c-
Fig. 8. Simulated pole figures and ODF sections showing the deformation texture of the CA//RD specimen deformed at 200 ◦ C: (a) true strain = 0.05 and (b) true strain = 0.1.
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Fig. 9. Comparison of experimentally measured and theoretically simulated Rvalues of AZ31 Mg alloy sheets.
Fig. 11. Comparison of experimentally measured and theoretically simulated twinvolume fraction in the CA//RD specimens.
axis of many orientations were aligned with the RD for the CA//RD specimens on the (0 0 0 2) pole figure. The activity of the tensile twin played an important role in rotation of the c-axis from the ND to the RD on the (0 0 0 2) pole figure during plastic deformation. It is clear that the CA//RD specimen deformed to a true strain of 0.05, exhibits an enhanced intensity at the F texture component, as shown in Fig. 8(a). However, in the CA//RD specimen deformed to a true strain of 0.1, stable orientations moved to the F and H texture components, as shown in Fig. 8(b). Similar results were observed for the deformation texture measured using an EBSD technique, as shown in Fig. 4. A comparison of the measured and simulated R-values of AZ31 Mg alloy sheets is shown in Fig. 9. Regardless of loading sign, it should be noted that R-values increased gradually as true strain increased. The directionality of the simulated R-value for uniaxial loading was similar to the experimental data. It should be noted that AZ31 Mg alloy sheets exhibited asymmetric R-values during plastic deformation. These results indicate that if Mg alloy sheets exhibit twin-dominated plastic deformation, asymmetric R-values, as well as asymmetric yielding, should be considered in simulations of sheet metal forming. Based on the principle of normality, the shape of the yield surface was closely related to the plastic strain
ratio [43]. Therefore, this result also indicates that the yield surface of the textured Mg alloy was distorted and significantly altered by in-plane plastic deformation [44]. Evolution of the twin-volume fraction was also simulated to evaluate the validity of the PTR model implemented in the current CPFEM. Fig. 10 shows the evolution of the twin-volume fraction during uniaxial compression. Blue and red elements represent portions of untwinned and twinned regions, respectively. Recently, microtextural characterization using EBSD verified that twin bands nucleate and propagate through the parent orientation during plastic deformation [9,10,15,37]. It seems that the current mapping method is not appropriate for the simulation of twin band nucleation and propagation in a deformed-parent orientation. A CPFEM based on a real microstructure as an initial configuration should be employed to simulate the heterogeneous deformation behavior in AZ31 Mg alloys [45]. As mentioned in Section 2, volume fractions of deformed specimens can be identified using the twinanalysis technique implemented with TSL software. Fig. 11 shows the evolution of the twin-volume fraction in the CA//RD specimens during uniaxial compression. EBSD results indicate that 200 ◦ C is a favorable temperature for twin-dominated deformation at a low strain level. It was found that the twin-volume fraction of com-
Fig. 10. Evolution of the deformation twinning during uniaxial compression: (a) true strain = 0.05, (b) true strain = 0.1 and (c) true strain = 0.15.
S.-H. Choi et al. / Materials Science and Engineering A 527 (2010) 1151–1159
pression specimens did not change until a critical strain of about 0.03. CPFEM successfully predicted the twin-volume fraction in the CA//RD specimens at a low strain level, when the experimental error was considered.
[5] [6] [7] [8] [9] [10]
5. Conclusions
[11] [12]
A crystal plasticity framework that considers both crystallographic slip and deformation twinning was developed to simulate the texture evolution and macroscopic properties of AZ31 Mg alloys. The major feature of the model is the implementation of a PTR model to capture grain reorientation due to deformation twinning during plastic deformation. For comparison with theoretically simulated results, experiments were performed using as-rolled AZ31 Mg alloy sheets. To demonstrate the validity of the proposed theoretical framework, uniaxial tension and compression tests were conducted at deformation temperatures of 200 ◦ C to simulate texture evolution, twin-volume fraction and macroscopic properties of the as-rolled AZ31 Mg alloy sheets. Results of the simulation revealed that texture evolution, twin-volume fraction and macroscopic properties in uniaxial loading were strongly dependent on the loading sign. The relative activities of the deformation modes can be used to theoretically explain the asymmetric behaviors of tension–compression yielding, and the R-value evident in the AZ31 Mg alloy sheets. Acknowledgements This study was supported by Nuclear Research & Development Program of the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korean government (MEST) 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] [2] [3] [4]
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