Modeling of deformation behavior and texture evolution in magnesium alloy using the intermediate ϕ-model

International Journal of Plasticity 52 (2014) 77–94

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Modeling of deformation behavior and texture evolution in magnesium alloy using the intermediate /-model D.S. Li a, S. Ahzi b,c,⇑, S. M’Guil b, W. Wen b, C. Lavender a, M.A. Khaleel d a

Pacific Northwest National Laboratory, CSMD, Richland, WA 99352, USA ICube Laboratory, University of Strasbourg, CNRS, 2 Rue Boussingault, 67000 Strasbourg, France c Materials Science and Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA d Qatar Energy and Environment Research Institute, Qatar Foundation Research and Development, Qatar b

a r t i c l e

i n f o

Article history: Received 11 October 2012 Received in final revised form 21 June 2013 Available online 2 July 2013 Keywords: /-Model Magnesium alloy Crystal plasticity Texture

a b s t r a c t The viscoplastic intermediate /-model was applied in this work to predict the deformation behavior and texture evolution in a magnesium alloy, an HCP material. We simulated the deformation behavior with different intergranular interaction strengths and compared the predicted results with available experimental results. In this approach, elasticity is neglected and the plastic deformation mechanisms are assumed as a combination of crystallographic slip and twinning systems. Tests are performed for rolling (plane strain compression) of random textured Mg polycrystal as well as for tensile and compressive tests on rolled Mg sheets. Simulated texture evolutions agree well with experimental data. Activities of twinning and slip, predicted by the intermediate /-model, reveal the strong anisotropic behavior during tension and compression of rolled sheets. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction As light weight structural materials, several hexagonal metals have been used in industries mainly for a significant reduction of weight. For example, magnesium and its alloys, with a low density, have been used as structural components in automotive, computer, communication and consumer electronic appliances (Wang and Huang, 2003). Another example is titanium and its alloys, applied in high performance engineering industries such as aerospace industry. Zirconium alloys, also hexagonal materials, are used as cladding materials in nuclear reactor fuels. The main challenges of hexagonal metals as structural materials are the limited ductility and the poor room temperature formability, which are primarily due to the restricted number of slip systems (Agnew and Duygulu, 2005; Parks and Ahzi, 1990; Schoenfeld et al., 1995). At room temperature, hexagonal materials possess fewer easy glide systems than cubic metals, resulting in greater crystal anisotropy. However, at high temperatures and moderate strain rates, hexagonal metals can be ductile and readily formable due to lower flow resistance and the activation of additional slip systems (Jain and Agnew, 2007). The main crystallographic slip families in hexagonal close-packed (HCP) structures are the basal hai, prismatic hai and pyramidal hai slip systems. The first- and the second-order pyramidal hc þ ai slip systems occur mainly at high temperature and were held responsible for the good elevated temperature ductility of HCP metals such as magnesium alloys. The hai slip comprises only four independent slip systems and, thus, cannot accommodate plastic deformation in the crystallographic c direction of the single crystal. The basal hai and the prismatic hai planes are mutually orthogonal. However, because the h1120i (or hai) slip directions are perpendicular to the c-axis and confined to the basal plane, there are only two independent ⇑ Corresponding author at: ICube Laboratory, University of Strasbourg, CNRS, 2 Rue Boussingault, 67000 Strasbourg, France. Tel.: +33 368852952; fax: +33 368852936. E-mail address: [email protected] (S. Ahzi). 0749-6419/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijplas.2013.06.005

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slip systems of each type. Thus, basal hai and prismatic hai slips together possess only four independent slip systems. Therefore, HCP metals are often nearly inextensible along their c-axis. Pyramidal hc þ ai systems, if activated, will provide the additional fifth independent slip system, necessary for the accommodation of an arbitrary plastic deformation. However, the slip resistance of pyramidal hc þ ai slip systems are usually much higher than those of basal and prismatic systems. Even when five independent slip systems are available, the difference of the critical resolved shear stresses between different deformation mechanisms (basal, prismatic and hc þ ai pyramidal) can be large enough to introduce an important plastic anisotropy at the single crystal level. Therefore, the mechanical behavior of HCP metals is controlled by the relative strengths and substantially different hardening responses of the various slip modes (Francillette et al., 1998; Fundenberger et al., 1997). For many HCP crystals, the resistance to plastic shearing can vary by an order of magnitude (or even higher) from one mechanism to another (Francillette et al., 1998; Fundenberger et al., 1997). A random aggregate of such crystals will deform only by their soft modes (e.g. hai slip). As the deformation proceeds, the aggregate becomes highly textured and the activation of the hard modes is possible only after a high degree of alignment that leads to materials’ locking, resulting into microcraking and materials failure. It is well known that in addition to crystallographic slip from dislocation movement, HCP materials exhibit a greater tendency to mechanically twin than cubic materials. In the absence of pyramidal hc þ ai slip, twinning may supplement the hai slip for full kinematic freedom. Twinning provides additional deformation which relaxes the requirements for five independents slip modes and may help a material to satisfy the Taylor criterion (Agnew et al., 2001; Brown et al., 2005; Kocks and Westlake, 1967). However, as a polar mechanism (Agnew and Duygulu, 2005), twinning depends strongly on temperature, alloying content, stacking fault energy and crystal lattice structure, making the modeling of the interaction between slip and twinning complicated (Prantil et al., 1995). At low temperature, twinning will compete with slip to accommodate the crystal motion. In warmer processing regimes, however, twinning may become a less favorable deformation mechanism than slip. Many crystal plasticity models have been applied to simulate the deformation behavior and texture evolution of polycrystalline metals. These models are based mainly on the physical deformation mechanisms of slip and twinning and account for hardening and reorientation of single-crystal grains. The main crystal plasticity models are Taylor type (Asaro and Needleman, 1985; Kalidindi et al., 1992; Kocks, 1970; Parks and Ahzi, 1990; Schoenfeld et al., 1995; Taylor, 1938), Sachs type (Ahzi et al., 1993, 2002; Leffers, 1968; Leffers and Pedersen, 2002; Leffers and Ray, 2009; Sachs, 1928) and self-consistent type (Abdul-Latif and Radi, 2010; Berveiller and Zaoui, 1978; Dingli et al., 2000; Hill, 1965; Hutchinson, 1976; Molinari et al., 1987; Molinari et al., 1997; Mercier and Molinari, 2009; Nemat-Nasser and Obata, 1986; Lebensohn and Tomé, 1993, 1994; Oppedal et al., 2012; Rousselier et al., 2010; Segurado et al., 2012). Polycrystal plasticity and texture evolution of hexagonal materials are characterized by the diversity of possible deformation mechanisms such as basal slip, prismatic slip, pyramidal slip and several twining modes. The low symmetry nature of these polycrystals may presents high anisotropy, kinematic deficiencies, and associated locking nature at high strains. The difficulty of modeling of the plastic behavior of HCP metals is mainly due to the fact that less than five independent soft modes are available owing to the hexagonal symmetry of the lattice cell. In other words, hard glide systems should be activated to accommodate any arbitrary deformation (Kocks, 1970). The mechanical deformation behavior and texture evolution of HCP metals have been simulated using the self-consistent theory (Agnew and Duygulu, 2005; Agnew et al., 2003; Beausir et al., 2008; Castelnau et al., 2001; Hutchinson, 1976; Lebensohn and Canova, 1997; Lebensohn and Tomé, 1993, 1994; Philippe et al., 1998; Suwas et al. 2011; Wang et al., 2010), the Taylor model (Fundenberger et al., 1997; Hutchinson, 1976; Gehrmann et al., 2005; Inal and Mishra, 2012; Jung et al., 2013; Philippe et al., 1995, 1998; Staroselsky and Anand, 2003) and the Static model (Fundenberger et al., 1997). We note that self-consistent approaches, such as the VPSC approach (Molinari et al., 1987; Tomé et al., 1987), are well-suited to HCP crystals since the hard systems may not be activated as long as the polycrystal is not highly textured. Based on this model, a fully anisotropic VPSC code was developed by Lebensohn and Tomé (1993) which has been applied by several authors to a wide variety of hexagonal metals and alloys, such as zirconium (Lebensohn and Tomé, 1993; Plunkett et al., 2006), titanium (Lebensohn and Canova, 1997; Suwas et al., 2011), and magnesium (El Kadiri et al., 2013; Neil and Agnew, 2009; Steglich et al., 2012; Yi et al., 2006). More recently, the developed large strain elastic visco-plastic self-consistent (EVPSC) model was also used for Magnesium (Wang et al., 2012a,b; Wang et al., 2011). Obviously, other types of modeling were applied for HCP metals such as the finite element based models (CPFEM) (see for instance the work of Inal and Mishra (2012), Abdolvand and Daymond (2013), Jung et al. (2013)). In the work of M’Guil and Ahzi (2006), the Sachs, the self-consistent and the Constrained-Hybrid models were utilized to predict texture evolution in a class of low symmetry crystals comprising less than five independent slip systems. In our previous work, we have used the /-model developed by Ahzi and M’Guil (2008) and the VPSC model to simulate the deformation behavior of HCP metals deforming by both soft and hard slip modes (M’Guil et al., 2009). The /-model is a nonlinear intermediate homogenization approach accounting for the interactions between the grains and the surrounding polycrystals. A new interaction law was formulated by minimizing an error function of the normalized deviations of the single-crystal fields (stress and strain rate) from their corresponding macroscopic fields (Ahzi and M’Guil, 2008). The obtained interaction law is self-consistent like. However, the interaction tensor does not depend on the Eshelby tensor but it includes a parameter that allows for weighting the strength of the intergranular interaction from a stiff to a more compliant one (for details, see Ahzi and M’Guil, 2008). The advantage of the /-model against the most usually used crystal plasticity model is that it can predict a very large range in textures results by simply varying the interaction strength parameter. Note that in the case of

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simple models such as Taylor model, only one single result can be obtained for a given materials and for fixed hardening parameters. The /-model (for relevant value of /) is able to reproduce similar results as the VPSC model (for relevant value of the effective interaction parameter used in VPSC), particularly the plastic locking of HCP metals at high strains (M’Guil et al., 2009). The viscoplastic /-model of Ahzi and M’Guil. (2008) (see also M’Guil et al., 2009; M’Guil et al. 2011; Wen et al., 2012) is used here for the analysis of the plastic deformation and texture evolution in HCP polycrystals with specific application to AZ31 magnesium alloy. In this alloy, hai slip (basal and prismatic) as well as hc þ ai pyramidal slip and tensile twinning are considered as the mechanisms of plastic deformation (Jain and Agnew, 2007; Proust et al., 2009; Styczynski et al., 2004; Wang et al., 2010). We have applied the /-model to simulate rolling texture development using an initially isotropic polycrystal. Tensile and compressive deformation behaviors of a rolled sample are also simulated. Comparing with experimental results in the literature (Jain and Agnew, 2007; Styczynski et al., 2004; Ulacia et al., 2010a,b; Wang et al., 2010), simulation results using the /model are discussed in terms of the effect of interaction strength, via the / parameter, on texture evolution and activation of different deformation mechanisms. 2. Modeling aspects 2.1. Single crystal behavior The present work focuses on modeling texture evolution in hexagonal polycrystals deforming by crystallographic slip and twinning. It is assumed that these mechanisms are rate sensitive.The plastic-shear rate, c_ a , of a given slip system family a, depends on the corresponding resolved shear stress, sa, according to a viscoplastic power law (Hutchinson, 1976): c_ a ¼ c_ 0 ðsa =g a Þjsa =g a jn1 . Here, c_ 0 is a reference shear rate assumed to be the same for a given slip systems family (basal, prismatic and pyramidal for HCP structures), but can differ from one family to another. The parameter n is the inverse rate sensitivity coefficient and ga is the critical resolved shear stress of the system a. This threshold stress controls the activation of slip systems and its evolution for each system is given by a microscopic hardening law (see Eq. (3)). In the viscoplastic power law, sa is resolved shear stress of the system a and is obtained by the following relation: sa = PaS. Here, the symbol ‘’ denotes the scalar product, Pa represents the symmetric part of the Schmid tensor and S deviatoric Cauchy stress of the single crystal. During plastic forming the contribution to deformation from elasticity is negligibly small. Therefore, elasticity is disregarded and only the plastic contribution to deformation is described. The inclusion of elasticity extends the use of the models but does not affect the predicted textures and the trend of the stress–strain behavior (Schoenfeld et al., 1995). The viscoplastic constitutive law of a single crystal is thus described by a rate-sensitive constitutive power law given by the following nonlinear relationship between the plastic strain rate tensor D and the deviatoric Cauchy stress tensor S:

D¼

Xc_ 0  Pa  Sn1  a   P  Pa  S  MðSÞ  S ga ga a

ð1Þ a

In Eq. (1), M is the fourth order viscoplastic compliance tensor of the crystal. For the slip system a, we denote by b and sa the slip plane normal and the slip direction respectively. These vectors are orthogonal and are assumed to rotate with the elastic spin of the lattice. Note that the geometry of a slip system a is represented by the symmetric and antisymmetric parts     a a a a a of Schmid tensor sa  b : Paij ¼ 1=2 sai bj þ saj bi and Aaij ¼ 1=2 sai bj  saj bi . This decomposing of the Schmid tensor allows us to decompose the velocity gradient L into a strain rate and a spin:

L ¼DþX

ð2Þ T

P

T



P

P

P

where D ¼ 1=2ðL þ L Þ  D and X ¼ 1=2ðL  L Þ ¼ X þ X . D and X are the plastic deformation rate and plastic spin, P respectively. These tensors are associated with the microscopic shear velocity by the following relations: DPij ¼ a c_ a Paij P a a P  and Xij ¼ a c_ Aij . The tensor X is the lattice spin. 2.2. Polycrystal averaging The macroscopic behavior can be described by a similar relationship to Eq. (1): D ¼ M  S, where D, S and M are the macroscopic plastic strain rate, the macroscopic deviatoric Cauchy stress and the macroscopic fourth order viscoplastic compliance tensor, respectively. The global conditions are given by the following averaging conditions: D ¼ hDi and S ¼ hSi, where hi indicates the volume average. 2.3. Voce law for the single-crystal hardening In the case of large deformation, the flow stress of polycrystals may be described by Voce hardening law which allows the description of high hardening rate observed at the onset of plasticity and its decrease toward a constant rate at large strain.

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For this, the instantaneous critical resolved shear stress, g a , evolves as a function of the total accumulated shear strain (C) in the grain, according to the following empirical Voce hardening rule:

g a ðCÞ ¼ sa0 þ









sa1 þ ha1 C 1  exp C

ha0

sa



with Ca ¼

1

Z

t

X

c_ a dt

ð3Þ

o

In Eq. (3), the parameters sa0 , sa1 , ha0 and ha1 could be determined from experimental curves. Parameters sa0 and ha0 describe the initial flow stress and the initial hardening rate, respectively in the crystal. The parameters ha1 and sa1 describe the asymptotic characteristic of the hardening. The conditions: ha0 P ha1 and sa1 P 0 correspond to increasing yield stress and decreasing hardening rate tending to linear saturation. A linear hardening is a limit case of this law and takes place with sa1 approaching zero. 2.4. Modeling of twinning mechanism with the Predominant Twin Reorientation scheme Even if deformation twinning does not contribute substantially to the macroscopic strain, it plays an important role in the development of deformation texture (anisotropy) and in the microstructural (such as grain refinement) evolution of polycrystalline materials. In order to accommodate the applied deformation, twinning is usually activated in combination with crystallographic slip. In general, this mechanism can be activated in the case of large grain size, of high strain rate, of low temperature, and low-symmetry crystallographic structures. The Predominant Twin Reorientation (PTR) scheme, proposed by Tomé et al. (1991), is used in our work to account for the crystallographic reorientation by twinning. In each twinning system of each grain, the associated volume fraction is defined as: V t;g ¼ ct;g =St , where ct;g represents the shear strain contributed by the twin and St is the characteristic twin shear. The twinning system with the highest associated volume fraction is identified as the Predominant Twin System (PTS). The sum of the associated volume fraction over all twinning systems in a given twinning mode and over all grains represents PP the accumulated twin fraction: V acc ¼ t g ct;g =St . At each deformation incremental step, a grain is picked randomly, and the grain will be fully reoriented if the accumulated twin fraction exceeds a threshold value which is defined as (Tomé et al., 1991):

V th ¼ Ath1 þ Ath2

V eff V acc

ð4Þ

where V eff , called effective twinned fraction, represents the volume fraction of the grains that are already reoriented and the values of Ath1 and Ath2 can be determined either by single-crystal experiments or by fitting to a known polycrystal response. Once the condition is fulfilled, the grain will be completely reoriented and only the PTS is considered in the reorientation. Then, both V eff and V acc will be updated. This process will be repeated until either all grains are randomly picked or the V eff exceeds the accumulated twin volume. 2.5. Micro–Macro modeling using the /-model In this section, we briefly describe the viscoplastic homogenization model that is used to simulate plastic deformation in the considered materials (the details can be found elsewhere (Ahzi and M’Guil, 2008). To account for intergranular heterogeneities during plastic deformations of polycrystals, the viscoplastic /-model can be used. The /-model belongs to the long range interactions type models. It is a viscoplastic self-consistent like model (one site). However, unlike the VPSC, the interaction between the mean fields and the grains are not based on Eshelby tensor. Therefore, grain shape is not explicitly accounted for in the /-model. In this model, the interaction strength is tuned via the parameter / which comes naturally into the interaction law. This tuning of the interaction strength was also addressed by several authors for the one-site VPSC interaction by introducing a parameter that tunes the interaction strength (see for instance the work of Tóth and Molinari (1994) and Lebensohn and Tomé (1993). This idea of tuning the interaction strength goes back to the work of Berveiller and Zaoui (1978). In the /-model, each grain deforms differently, depending on its orientation and its anisotropy relative to that of the matrix. This model accounts for deviations in the grains behavior from the average behavior of the polycrystal. Such deviations can be significant for materials with marked plastic anisotropy (such as HCP metals). The /-model is formulated by the minimization of a specific function combining the local fields’ deviations from the macroscopic ones. This function depends also on a parameter, / between 0 (Taylor) and 1 (static). The parameter 0 < / < 1 allows spanning the entire solution domain between the upper and lower bounds. The obtained dual interaction laws are given by Ahzi and M’Guil (2008):

D ¼ A < A>1  D or S ¼ B < B>1  S

ð5Þ

with

   1    /1 /1 f M 1  bn M M A ¼ M1  bn / /

ð6aÞ

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D.S. Li et al. / International Journal of Plasticity 52 (2014) 77–94 Table 1 Model parameters describing critical resolved shear stress and hardening parameters. Mode

s0 (MPa)

s1 (MPa)

h0 (MPa)

h1 (MPa)

Basal Prismatic Pyramidal Twin

16 70 110 30

4 11 23 0

2300 450 7350 0

110 77 54 0

Table 2 CPU time calculation for the /-model, tangent VPSC and Taylor model. CPU time (s)

Tangent VPSC model

/-model

Without twinning With twinning

Taylor

/ ¼ 0:2

/ ¼ 0:8

Average grain shape

Individual grain shape

10.4 12.1

11.2 12.0

19.0 35.0

90.0 170.0

6.3 8.0

and

   1    /1 /1 ~ MM MM B ¼ I  bn I  bn / /

ð6bÞ

2

In Eqs. (6a), (6b), b ¼ ðs0 =c_ 0 Þ is a normalizing coefficient (s0 is a critical resolved shear stress), I is the fourth-order identity tensor, A and B are fourth-order strain-rate and stress interaction tensors, respectively and the inverse of the average com~ 1 ¼< B>1  < M1  A >. To complete these interaction laws, the local spin is simply equated pliance tensor is equated to: M to the macroscopic one: X ¼ X. We note that the /-model respects the averaging conditions (global equilibrium and global compatibility). The details of this model can be found in the work of Ahzi and M’Guil (2008). In addition, we note that the non-use of Eshelby tensor makes the /-model more attractive from CPU time point of view. We reported below, a comparison of the CPU time calculations for the tangent VPSC model, the /-model and the Taylor model. In this way, we have analyzed one case study (plane strain compression until 30% strain) with the same inputs for all models. The obtained CPU time results are summarized in Table 2. We can see that the CPU time of the /-model is at least half of the CPU time for the tangent VPSC model which constitutes a real advantage for the simulation of large deformation of polycrystalline materials. 3. Results and discussion The material studied here is magnesium alloy AZ31, a hexagonal material with parameter ratio c/a = 1.624. The intermediate /-model was applied to simulate the deformation behavior of AZ31, validated with experimental results from the literature (Jain and Agnew, 2007; Styczynski et al., 2004; Ulacia et al., 2010a,b; Wang et al., 2010). Two series of tests were carried out in this study. First we simulated cold rolling of heat treated AZ31 sheet with initial random texture. We note that, in this paper, the rolling test is approximated by the plane strain compression test which neglects the effects of shear involved in the real rolling process. Second, we conducted simulations of compression and tension tests, of rolled AZ31, in different directions. For each test, we imposed the macroscopic velocity gradient, L corresponding respectively to a plane strain compression, compression and tension. The plastic deformation mechanisms in AZ31 is assumed to be composed of three kinds of slip systems: (0 0 0 1)<1 1 2 0> basal slip, (1 0 0 0)<1 1 2 0> prismatic slip, (1 1 2 2)<1 1 2 3> pyramidal slip and tensile twinning (1 0 1 2)<1 0 1 1>. In all the numerical simulation, the reference shear rate and rate sensitivity parameters (see Eq. (1)) are assumed to be the same for all the considered systems: c_ 0 = 0.001 s1 and n = 19. The other hardening law parameters used in Eq. (3) are listed in Table 1. Here, the hc þ ai pyramidal slip system has a much higher critical resolved shear stress than that of basal slip and prismatic slip systems. The hardening parameters used in Table 1 are obtained by backfitting the experimental mechanical behavior for the case of high interaction with the parameter / ¼ 0:1. Fig. 4 shows the stress strain curve of rolled magnesium sheet under uniaxial tension along rolling direction (RD). The initial texture for these simulations is that of rolled sheet. Experimental data are obtained from Jain and Agnew’s work (2007). Using parameters in Table 1, simulation results from Taylor model and intermediate model with / ¼ 0:1 agrees well with the experimental results. With the increase of interaction parameter /, stress strain curves drop down and deviate from experimental data. It is possible to fit the curves for all cases of interaction parameters by changing the hardening law parameters, just like it is possible to simulate the stress–strain response by different models by changing the parameters. However, we will not fit the experimental data under different interaction parameters. Instead, a parametric study will be performed to investigate the influence of the interaction parameters on deformation systems activities and texture evolution. The goal of this paper is to validate the intermediate model in simulating mechanical behavior and texture evolution of hexagonal magnesium alloys.

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3.1. Regression study to demonstrate the appropriateness to use 1000 grains for this homogenization method First, the regression of the simulation results against the number of grains used in the simulation is studied. Three sets of initial random texture with 100, 500 and 1000 grains were used in this regression study. Using interaction parameter / = 0.1 and all the other parameters fixed, the simulated 0002 pole figures at a rolling strain of 0.2 are presented in Fig. 1. There is a little change in the spread of the texture component. With the increase of number of grain used in the calculation, the texture component intensity regressed to 3 times random. Fig. 2 demonstrated the simulated true stress vs plastic true strain curves using these three sets of polycrystalline aggregate. There is no much difference in the simulated results with the curves cover each other closely. Fig. 3 demonstrated the predicted activities of deformation mechanisms from these three aggregates with different grain numbers. Again, the trends of deformation mechanism activity evolution are same for all three aggregates. All the following simulation uses the aggregate with grain number of 1 0 0 0.

100 crystals (a)

500 crystals (b)

1000 crystal (c)

Fig. 1. Simulated (0 0 0 2) pole figure at a rolling strain of 0.2 using a polycrystalline aggregate with (a) 1 0 0, (b) 5 0 0 and (c) 1 0 0 0 crystals.

Fig. 2. Simulated true stress vs plastic true strain curves using polycrystalline aggregates with different grain numbers.

100 crystals (a)

500 crystals (b)

1000 crystal (c)

Fig. 3. Predicted evolution of activities of deformation mechanisms during rolling for polycrystalline aggregate with (a) 1 0 0, (b) 5 0 0 and (c) 1 0 0 0 crystals.

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Fig. 4. Simulated and experimental stress strain curves of rolled AZ31 sheet under uniaxial tension along rolling direction (experimental data after Jain and Agnew, 2007).

3.2. Simulation of cold rolling of heat treated AZ31 In the first test set, rolling of initially random AZ31 polycrystal was simulated using a plane strain compression approximation. Texture evolution results are listed in Fig. 5. The experimental (0 0 0 1) and (1 1 2 0) pole figures of rolled sheet are shown in Fig. 5a (Styczynski et al., 2004). The simulated polycrystalline aggregate is composed of 1 0 0 0 single crystals with initial random texture as shown by pole figures in Fig. 5b. After rolling, experimental texture (Styczynski et al., 2004) has a strong component with c-axis aligned around the normal direction (ND) (Fig. 5c). Texture evolution and mechanical behavior

RD

ND

(a)

(b)

(c)

(d) φ = 0.1

(e) φ =

TD

0.9

Fig. 5. (0 0 0 1) pole figures of squeeze cast AZ31 sheet with random texture (a), simulated AZ31 with random texture (b), cold rolled AZ31 sheet (c), simulated rolled texture using the /-model with / ¼ 0:1 (d), / ¼ 0:9 (e) (experimental data after Styczynski et al., 2004).

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have been simulated up to 20% strain and using / ! 0, 0.1, 0.2, 0.5, 0.7, 0.9. The results for / ! 0 is the same as that from Taylor model (not reported here). We observed a large difference in both simulated stress strain curves and textures for different values of /. Large values of / predict sharper textures (see Fig. 5e). The large difference in predicted stress strain curve using different /, not reported here, is attributed to the predicted activities of different deformation mechanisms. As shown in Fig. 6, contribution from twinning activity is smaller in all cases. Texture evolution is therefore dominated by slip. In the case of low /, shown in Fig. 6a and b, activities of the three slip families are similar and there are no much changes during rolling. The pyramidal slip is strongly activated for low / values and remains at a constant level throughout the deformation process. With the increase of /, the activity of pyramidal slip are smaller at the beginning of deformation and increases with increasing deformation, as seen in Fig. 6c and d. For / ¼ 0:9, the basal slip is very active whereas the activity of prismatic slip is weaker at low strain. During the deformation, the basal slip activity is reduced and the prismatic activity is increased. 3.3. Simulations of compression and tension tests, of rolled AZ31, in different directions In the second set of test, tension and compression of rolled magnesium sheets along different directions were investigated. For the simulation of tensile deformation test up to 20% strain, we used the rolling texture in Fig. 5d as initial texture. Tensile tests were performed along RD. The predicted textures for different values of / are plotted in Fig. 7 in comparison with the experimental results (Ulacia et al., 2010a,b). Fig. 7a shows the experimental (0 0 0 1) and (1 0 1 0) pole figures of rolled Mg sheet uniaxially stretched along RD at room temperature (Ulacia et al., 2010a,b). The deformation texture has a strong texture component with c-axis aligned along ND. (1 0 1 0) axis distributed along the plane of RD (also tension direction here) and transverse direction (TD). Pole figures simulated using intermediate model with / ¼ 0:01, 0.1 and 0.9 are presented in Fig. 7b–d, respectively. Again, simulated pole figures from intermediate model agree well with experimental results, no matter what interaction parameters used here. For low interaction (high / value), the crystal are less

(a) φ = 0.01

(b) φ = 0 . 1

(c) φ = 0.5

(d) φ = 0.9

Fig. 6. Predicted activities of different deformation mechanisms in Mg sheet during rolling using the intermediate /-model, / ¼ 0:01 (a), / ¼ 0:1 (b), / ¼ 0:5 (c) and / ¼ 0:9 (d).

D.S. Li et al. / International Journal of Plasticity 52 (2014) 77–94

85

Fig. 7. Experimental (a) and simulated (0 0 0 1) and (1 0 1 0) pole figures of Mg sheet uniaxially stretched along RD using the /-model with different interaction parameter / (b-c-d) (experimental data after Ulacia et al., 2010a; Ulacia et al., 2010).

constrained and thus the activity of hard modes (pyramidal) is not required for the accommodation of plastic deformation. However, as deformation proceeds, the polycrystal becomes highly textured with the soft modes becoming less favorable for activation and therefore, hard mode start to be more activated. As shown in Fig. 7d, the best correlation with the experimental texture evolution is obtained for high value of / (low interaction strength). We note that our predicted (1 0 1 0) pole figures have some discrepancies with the experimental results of Ulacia et al. (2010a,b). However, there are also discrepancies between different results in the literature for Mg AZ31 (Agnew and Duygulu, 2005; Koike and Ohyama; 2005; Ulacia et al.,

(a) φ = 0.01

(c) φ = 0.5

(b) φ = 0.1

(d) φ = 0.9

Fig. 8. Predicted activities of different deformation mechanisms in Mg sheet during uniaxial tension along RD using the intermediate /-model, / ¼ 0:01 (a), / ¼ 0:1 (b), / ¼ 0:5 (c) and / ¼ 0:9 (d).

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φ = 0.01

experimental

φ = 0.1

φ = 0.9

(0001)

RD

(10 1 0)

TD

Fig. 9. Experimental and simulated (0 0 0 1) and (1 0 1 0) pole figures of Mg sheet uniaxially compressed along TD using the intermediate /-model with different / (experimental data after Jain and Agnew, 2007).

(a) φ = 0.01

(c) φ = 0.5

(b) φ = 0.1

(d) φ = 0.9

Fig. 10. Predicted activities of different deformation mechanisms in Mg sheet during uniaxial compression along TD using the intermediate /-model, / ¼ 0:01 (a), / ¼ 0:1 (b), / ¼ 0:5 (c) and / ¼ 0:9 (d).

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2010; Yi et al., 2010). From these experimental results, it is concluded that the (0 0 0 2) pole figures have a common feature showing c-axis aligned along the ND direction with some cases showing a split of the c alignment. In the ideal case (no splitting), the (1 0 1 0) poles should be aligned in the plane RD–TD. However, because of the splitting, the (1 0 1 0) poles may have preferential orientations in the RD–TD planes (with several components as shown in the experimental results of Agnew and Duygulu (2005), Koike and Ohyama (2005), Ulacia et al. (2010), Yi et al. (2010). As can be observed in the reported experimental (1 0 1 0) pole figures, the positions of preferential orientations of these poles depend on the experimental conditions Therefore, our predicted pole (1 0 1 0) figures show a misorientation about 30° from the experiments of Ulacia et al. (2010a,b) but this disagreement may be much smaller if we have compared to other experimental results for Mg AZ31.

φ = 0 .01

experimental

φ = 0 .1

φ = 0 .9

(0001)

(10 10)

Fig. 11. Experimental and simulated (0 0 0 1) and (1 0 1 0) pole figures of Mg sheet uniaxially compressed along RD using the intermediate /-model with different / (experimental data after Jain and Agnew, 2007).

(a) φ = 0.01

(b) φ = 0 . 1

(c) φ = 0.5

(d) φ = 0.9

Fig. 12. Effect of parameter / on the predicted slip/twinning activities under uniaxial compression along ND.

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The predicted activities of different deformation mechanism in the case of tension along RD are presented in Fig. 8. Similar to the rolling case, contribution from twinning is small. This is an accord with the results of Wang et al. (2010), who showed that the activity of tensile twinning is relatively low and decreases with increasing strain. In addition, there is not much change in the activities during deformation for high interaction strength (low /, see Fig. 8a and b). The activities of basal and prismatic modes are strain independent for / ¼ 0:01 and / ¼ 0:1. In the case of intermediate and low interaction strength (/ ¼ 0:5in Fig. 8c and / ¼ 0:9 in Fig. 8d), the activity of basal slip decreases while that of prismatic slip increases during tension. Higher / value leads to lower activity of pyramidal slip. For the uniaxial compression of the rolled sheet, different loading directions were investigated. First simulated texture evolution during uniaxial compression along TD was compared with Jain and Agnew’s experimental results (Jain and Agnew, 2007), as shown in Fig. 9. The experimental results demonstrate a strong texture component with c-axis aligned close to the uniaxial compression direction (parallel to TD). The simulated results reveal a strong dependence on the used / value. The best correlation with the experimental texture is obtained for low interaction strength (/ ¼ 0:9) where the most grains reorient their c-axes along the loading direction (TD). For the very high interaction strength (/ ¼ 0:01), we obtained results which agree less the experimental results. We note that this case with / ! 0 is a case equivalent to the upper-bound Taylor model. Activities of different deformation mechanisms in Mg sheet during compression along TD explains the large difference in texture simulation using different /. As shown in Fig. 10, the difference between different systems activities during compression predicted by low / values is smaller than those predicted from the other simulations for larger /. The activity of twinning increases with increasing /, particularly for low strains. However, this activity decreases drastically for larger strains and for intermediate and large /. Compression of rolled Mg sheet along RD is also simulated. Fig. 11 shows the predicted texture at 20% strain for various / values. These predictions are compared to the experimental results of Jain and Agnew (2007). Experimental data show that

φ = 0.01

φ = 0 .1

φ = 0 .5

φ = 0 .9

(0001)

(10 1 0)

Fig. 13. Effect of parameter / on the predicted textures under uniaxial compression along ND.

Fig. 14. Experimental ð0002Þ and ð1010Þ textures for pure Mg, Mg-1Y and Mg-3Li under plane strain compression test at

etrue ¼ 30% (Agnew et al., 2001).

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the c-axis is aligned with the RD, which is also the loading direction in this case. Using / ¼ 0:01, simulated results presents a basal texture component with c-axis aligned between RD and ND; with about 30° tilt away from RD. With the increase of /, the basal texture component moves closer to the RD (loading direction). However, the best correlation between experimental and predicted textures, in terms of both intensity and alignment, is obtained for low interaction strength (/ ¼ 0:9). Predicted systems activities for compression along RD have a similar tendency to those obtained for compression along TD, and thus not reported here. The compression test along the ND of the Mg sheet is also studied in this work. Fig. 12 shows the predicted slip/twinning activities. For low / values (/ ! 0 and / ¼ 0:1), the plastic deformation is mostly accommodated by pyramidal slip, and then the basal and prismatic slip. For intermediate and high / values (/ ¼ 0:5 and / ¼ 0:9), the deformation is dominated by basal

Fig. 15. Predicted rolling texture by Agnew et al. (2001). ð0002Þ and ð1010Þ pole figures for X = 12, 6 and 3 (sbasal : s : stw ¼ 1 : X : 2),

X=12

X=6

eeq ¼ 34%.

X=3

φ = 0 .1

φ = 0 .3

φ = 0.5

φ = 0 .7

Fig. 16. Predicted rolling texture for X = 12, 6 and 3 (sbasal : s : stw ¼ 1 : X : 2). ð0002Þ and ð1010Þ pole figures using the /-model,

eeq ¼ 34%.

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X=12

X=6

X=3

φ = 0 .1

φ = 0 .3

φ = 0.5

φ = 0 .7

Fig. 17. Effect of / value on the slip and twinning activities for X = 12, 6 and 3 (sbasal : s : stw ¼ 1 : X : 2).

Fig. 18. Predicted rolling texture by Agnew et al. (2001). ð0002Þ and ð1010Þ pole figures for X = 12 and 4 (sbasal : sprism : s : stw ¼ 1 : 3 : X : 2), eeq = 34%.

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91

slip. For strains e > 0:1, the basal slip activity decreases whereas the pyramidal slip activity increases with strain. The activity of the tensile twinning is relatively weak at lower strain and nearly disappeared for e > 0:1. The corresponding predicted textures are presented in Fig. 13. Higher / value leads to sharper textures where c-axes align almost parallel to ND with a slight tendency to tilt around ND. We note that as our initial goal in this paper is to simply report the applicability to the /-model to Mg, we did not attempt to find the best fit of the stress strain response, particularly for compression. Moreover, this is a quite difficult task since it requires a tremendous work to find the appropriate CRSS parameters and their evolution. Moreover, to the best of our knowledge, the best fitting was found with an elastic viscoplastic self-consistent approach (VPSC plus elasticity) (Wang et al., 2012a,b, 2011). Since elasticity is neglected in our work, this makes the fitting of the stress strain curves even more difficult. This fitting was also obtained using the experimental initial texture which is not the case in our study. 3.4. Comparison between experimental and predicted results for rolled Mg alloys Agnew et al. (2001) have proposed an experimental study on rolled Mg alloys containing lithium (Li) or yttrium (Y). As shown in Fig. 14, a typical ð0002Þ basal texture can be obtained from the rolled pure Mg. However, for the textures of Mg-1Y (1 wt% Y) and Mg-3Li (3 wt% Li), the basal poles have been spread and rotated towards the RD. Agnew et al. (2001) suggested that this texture transition can be predicted by varying the relative activity of pyramidal hc þ ai slip. 3.4.1. Simulations without prismatic slip Agnew et al. (2001) carried out a simulation with the VPSC model. They considered the basal slip, pyramidal hc þ ai slip and tensile twinning. In the simulation, the relative CRSS of these systems are set to be sbasal : s : stw ¼ 1 : X : 2. The relative activity of pyramidal hc þ ai slip is therefore controlled by the X value. In the work of Agnew et al. (2001), the X values of 12, 6 and 3 were chosen and the interaction parameter of the VPSC model was set to be 10 (tangent formulation). The predicted results are shown in Fig. 15. Here, we carry out a similar simulation using the /-model. A linear hardening is considering in the simulations and the inverse strain rate sensitivity coefficient is n = 11. The predicted results for various / values are presented in Fig. 16. It is evident that lower / value leads to a stronger spread of basal poles towards RD in the (0 0 0 1) pole figures. This spread of basal poles can be also reduced by increasing the X value. The results for high / values are close to the one predicted by Agnew et al. (2001). The corresponding relative activities of slip and twinning systems are shown in Fig. 17. As expected, the lower X value increase the relative activities of pyramidal hc þ ai slip, and therefore decrease the ones of basal slip. A higher / value will also leads to a stronger basal slip. The activities of twinning are relative weak and decreased with the increasing

X=12

X=4

φ = 0 .1

φ = 0 .3

φ = 0.5

φ = 0 .7

Fig. 19. Predicted rolling texture for X = 12 and 4 (sbasal : sprism : s : stw ¼ 1 : 3 : X : 2). ð0002Þ and ð1010Þ pole figures using the /-model,

eeq ¼ 34%.

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X=12

X=6

φ = 0 .1

φ = 0 .3

φ = 0.5

φ = 0 .7

Fig. 20. Effect of / value on the slip and twinning activities for X = 12 and 4 (sbasal : sprism : s : stw ¼ 1 : 3 : X : 2).

strain. We can see that the strong basal slip can reduce the spread of basal poles and respond to the formation of typical basal textures. 3.4.2. Simulations with prismatic slip Agnew et al. (2001) have also proposed a similar simulation but take into account the prismatic slip. The relative CRSS of these systems are set to be sbasal : sprism : s : stw ¼ 1 : 3 : X : 2. All other parameters are set to be the same than the previous simulation. The predicted results of Agnew et al. (2001) are presented in Fig. 18. By reducing X from 12 to 4, the basal poles can still be spread but not as evident as the case without prismatic slip. Similar simulations are carried out by using the /-model and the predicted textures are shown in Fig. 19. A strong spread of basal poles can still be found for low / values or low X values. Results of high / values fit well the textures predicted by Agnew et al. (2001). The relative activities of the slip and twinning systems are presented in Fig. 20. The effects of X and / on the activities of basal slip, pyramidal slip and tensile twinning are similar to the previous simulations. The activity of prismatic slip is weaker than the other two slip systems, and it can be reduced by higher X values or by higher / values. 4. Conclusion and future works The viscoplastic intermediate /-model was applied in this study to simulate the deformation behavior and texture evolution in magnesium alloys AZ31. Different cases, including rolling of initially random textured magnesium, tension as well as compression of rolled magnesium sheets along different directions were investigated. The texture evolution and mechanical deformation behavior were simulated using the /-model. Comparing with the experimental results, the simulated results are close to the observation. Further parameter study will be carried out in the future to increase the prediction accuracy. The influence of different intergranular interaction strengths on the predicted results (texture evolution and deformation systems activity) was studied.

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A universal set of crystal plasticity hardening parameters was used in this study along with various values for the interaction strength parameter. However, further study of the physical meaning of the interaction parameter / is necessary to maximize the capability of this intermediate model. In addition, the choice of the hardening law parameters should be optimized to quantitatively improve the model predictions. For this, a detailed parameter studies is necessary to minimize the simulation errors and increase prediction accuracy. Another future research direction is on implementation of hardening law obtained from the lower length scale models, like dislocation dynamics, to enhance our knowledge of the mechanical deformation behavior. Computational efficiency comparing with other homogenization models (Taylor, Sachs and VPSC) will be investigated in detail when imbedded as a materials subroutine in finite element method to solve larger length scale engineering problems In simulating rolling of initially random textured Mg sheet, the effect of / texture development up to 20% strain is not significant. However, for both uniaxial tension and compression of rolled sheet, the simulated texture results are highly dependent on the value of /. Predicted deformation mechanism activities provide good explanation for the texture development. For rolling, the twinning activity is very fairly low for all / values, which affects slightly the texture development as observed by the tilt of the basal texture. For tension and compression of rolled sheet, mechanism activity is highly depending on the value of /. As expected, tensile twinning contributes significantly to plasticity under compression and negligibly under tension. For the uniaxial compression, high values of / induce high activity of the hard modes (twinning or pyramidal). In this case, twinning is highly activated at the lower strains and decreases at large strains where it becomes compensated by a higher activity of the pyramidal slip. Based on our predicted textures, particularly for uniaxial tension and compression, it is concluded that the best correlation with the experimental results was found for low interaction strength (high value of /). With the advantage of easy implementation and fast regression, the intermediate /-model will play an important role in integrated computational materials engineering (ICME) to provide guidance on processing optimization Acknowledgements Support for this work was provided by the DOE Office of Energy Efficiency and Renewable Energy. The Pacific Northwest National Laboratory is operated by Battelle Memorial Institute for the United States Department of Energy (U.S. DOE) under Contract DE-AC06-76RLO 1830. 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