A crystal plasticity model for hexagonal close packed (HCP) crystals including twinning and de-twinning mechanisms

International Journal of Plasticity 49 (2013) 36–52

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A crystal plasticity model for hexagonal close packed (HCP) crystals including twinning and de-twinning mechanisms H. Wang a,⇑, P.D. Wu a, J. Wang b, C.N. Tomé b a b

Department of Mechanical Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4L7 Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

a r t i c l e

i n f o

Article history: Received 11 September 2012 Received in final revised form 4 February 2013 Available online 7 March 2013 Keywords: De-twinning Twinning Crystal plasticity Cyclic loading

a b s t r a c t Together with slip, deformation twinning and de-twinning are the plastic deformation mechanisms in hexagonal close packed (HCP) crystals, which strongly affect texture evolution and anisotropic response. As a consequence, several twinning models have been proposed and implemented in the existing polycrystalline plasticity models. De-twinning is an inverse process with respect to twinning, which is relevant to cycling, fatigue and complex loads but is rarely incorporated into polycrystalline plastic models. In this paper, we propose a physics-based twinning and de-twinning (TDT) model that has the capability of dealing with both mechanisms during plastic deformation. The TDT model is characterized by four deformation mechanisms corresponding to twin nucleation, twin growth, twin shrinkage and re-twinning. Twin nucleation and twin growth are associated with deformation twinning, and twin shrinkage and re-twinning are associated with de-twinning. The proposed TDT model is implemented in the Elasto-Visco-Plastic Self-Consistent (EVPSC) model. We demonstrate the validity and the capability of the TDT model by simulating cyclic loading of magnesium alloys AZ31B plate and AZ31 bar. Comparison with the measurements indicates that the TDT model is able to capture the key features observed in experiments, implying that the mechanical response in the simulated materials is mainly associated with twinning and de-twinning. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction The deformation mechanisms of HCP materials, such as zircaloy, titanium, zinc, beryllium and magnesium, etc., have been investigated both experimentally and theoretically (Barrett and Haller, 1947; Bell and Cahn, 1957; Kelley and Hosford, 1968; Akhtar, 1973; Wang et al., 2010g,h; Abdolvand et al., 2011; Ghaderi and Barnett, 2011; Brown et al., 2012). It has been found that plastic deformation in materials with HCP structure is accommodated by both slip and twinning. Among the HCP materials, magnesium alloys are of the current interest due to the potential energy saving from their low density. Tremendous experimental works have been carried out for magnesium alloys to understand the plastic deformation under various temperature and strain rate (Chapuis and Drivers, 2011; Khan et al., 2011) and cyclic loading (Guillemer et al., 2011), and texture effects (Chun and Davies, 2011; Stanford et al., 2011), and twinning effects (Xin et al., 2012; Ma et al., 2012). Meanwhile, theoretical and numerical modellings have been focused on describing the stress–strain response and texture evolution

⇑ Corresponding author. Tel.: +1 (905) 525 9140 27329. E-mail address: [email protected] (H. Wang). 0749-6419/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijplas.2013.02.016

H. Wang et al. / International Journal of Plasticity 49 (2013) 36–52

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by developing crystal plasticity models with different focuses (Izadbakhsh et al., 2011; Mayama et al., 2011; Homayonifar and Mosler, 2012; Ostapovets et al., 2012; Wang et al., 2012a), for example, spatial stress distribution effects (Choi et al., 2011), twin variants effects (Jonas et al., 2011), and solid solution alloying effects (Raeisinia et al., 2011). Nevertheless,  0i) and the plastic deformation of magnesium alloys at room temperature is mostly related to basal hai slip (f0 0 0 1gh1 1 2  2gh1 0 1  1i). The other slip modes, such as prismatic hai slip (f1 0 1  0gh1 1 2  0i) and pyramidal hc + ai slip extension twin (f1 0 1  2gh  1 1  2 3i), will be activated to accommodate arbitrary plastic deformation, but are rare due to the low mobility of the (f1 1 2 associated dislocations. Due to the polar nature of twinning, the extension twin can only be activated by a tensile component of stress along the c-axis. Because the extension twinning results in an 86.3° reorientation of the basal pole, de-twinning will occur in a twin if the load is reversed (Roberts, 1960). Under cyclic loading and strain path changes, twinning and detwinning appear alternately (Gharghouri et al., 1999; Lou et al., 2007; Wu et al., 2008a,b; Proust et al., 2009; Wu et al., 2010; Yu et al., 2011; Zhang et al., 2011; Park et al., 2012). Experimental studies have focused on the role of twinning and de-twinning in fatigue testing by analyzing the hysteresis loops (Wu et al., 2008b, 2010; Hong et al., 2010a,b). With respect to strain path change, the effect of pre-compression along extrusion direction of bars on the subsequent tension has also been studied, revealing a lower yield stress upon load reversal (or no apparent yielding) associated with de-twinning. The stress differential was ascribed to the residual stresses caused by pre-compression. If the residual stress was removed before the subsequent tension, the yield stress associated with de-twinning is found to be the same as the one associated with twinning (Kleiner and Uggowitzer, 2003, 2004). Early materials models consider slip as the major plastic deformation mechanisms (Taylor, 1938; Peirce et al., 1982, 1983; Asaro and Needleman, 1985). Material models including twinning have been developed but they were not meant to describe both twinning and de-twinning. For example, the models developed in Refs. (Van Houtte, 1978; Tomé et al., 1991; Lebensohn and Tomé, 1993; Kalidindi, 1998; Staroselsky and Anand, 2003; Clausen et al., 2008; Wu et al., 2008c; Abdolvand et al., 2011; El Kadiri and Oppedal, 2010) mainly describe the reorientation and hardening evolution associated with twinning under monotonic loading conditions (Agnew and Duygulu, 2005; Graff et al., 2007; Neil and Agnew, 2009; Signorelli et al., 2009; Knezevic et al., 2010; Levesque et al., 2010; Wang et al., 2010a; Oppedal et al., 2012). Recently, Li et al. (2010) proposed a phenomenological approach for describing both twinning and de-twinning under cyclic loading of AZ31B sheets. This model employs a von Mises yield surface with initial non-zero backstress and introduces plastic yielding asymmetry according to an isotropic and nonlinear kinematic hardening law. A weighted discrete probability density function of c-axis orientations, which evolves with twinning or de-twinning according to explicit rules, was adopted to represent the texture evolution during deformation. This model has been implemented in ABAQUS/Standard through UMAT (User-defined Material) with constitutive parameters obtained from three in-plane tests: uniaxial tension, uniaxial compression and uniaxial tension after compression. Various tests showed good agreement with experiments. However, the phenomenological model gives an abrupt transition during strain path changes instead of a gradual elastic–plastic transition, and the simplified description of texture is not able to truly capture the texture evolution, either. A sophisticated composite grain (CG) model for describing twinning has been proposed by Tomé and Kaschner (2005) and Proust et al. (2007) based on the lamellar grain model of Lebensohn (1999). The CG model accounts for twin shape and twin-parent interaction, and was implemented in the Polycrystal Plasticity Self-Consistent code (VPSC) (Lebensohn and Tomé, 1993) and applied to address the role of twinning during strain path changes in Zr. The CG model has been extended to include de-twinning when studying magnesium subjected to strain path changes (Proust et al., 2009). In the early version of the CG model (Proust et al., 2007) the continuity of the shear stress parallel to the twin interface was enforced in order to deal with the laminar twin structure in Zr. When grown-up twins are assumed to coalesce and become separate inclusions (Proust et al., 2009), the continuity of the shear stress across twin boundaries was relaxed in studying the mechanical response of Mg. Recently, one major development of the CG model is that the empirical twin nucleation model used in Refs. (Proust et al., 2007, 2009) has been updated with newly developed physics-based nucleation models (Capolungo et al., 2009; Beyerlein and Tomé, 2010) that are inspired by atomistic simulations (Wang et al., 2010f; Tomé et al., 2011) and experiments (Aydiner et al., 2009; Balogh et al., in press). For what follows, it is important to notice that the CG model assumes that twin growth accommodates strains inside the un-twinned domain (matrix) and is driven by the average stress in the matrix, while de-twinning accommodates strains inside the twin domain and is driven by the average stress in the twin. This treatment assigns the contribution of plastic deformation that results from twinning or de-twinning to either the matrix or the twin domain, respectively, rather than assigning the plastic deformation to the composite entity formed by both the matrix and twin domains. To overcome this shortage, the model that we present here will adopt both the average stresses in the matrix and in the twin to calculate the twin growth and the twin shrinkage (de-twinning). In addition, the continuity of stress across the twin interface was relaxed in the present model, and we treat parent and twin as separate inclusions from the start. As a consequence, the model is not strictly a composite grain one, and allows accounting for more than one twin variant per grain. Thus, the purpose of this work is to develop a physics-based crystal plasticity model including twinning and de-twinning, correspondingly named TDT, to study plastic deformation of HCP metals during cyclic loading and strain path changes. The paper is organized as follows. Section 2 formulates the crystal plasticity TDT model and reviews the EVPSC model in which it is implemented. In Section 3 we demonstrate the predictive capability of the TDT model by simulating several tests and comparing the predictions with existing experiments. Finally, we draw conclusions in Section 4.

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H. Wang et al. / International Journal of Plasticity 49 (2013) 36–52

2. Constitutive model Plastic deformation of a crystal with the HCP structure is in general ascribed to crystallographic slip, twinning and detwinning. The slip or twinning system a is represented by the slip or twinning direction (sa ) and the direction normal to the slip or twinning plane (na ). The plastic strain rate tensor for the crystal can be expressed as: p

d ¼

X

c_ a P a

ð1Þ

a

in terms of the shear rate c_ a and the Schmid tensor P a ¼ ðsa na þ na sa Þ=2 for system a. Regardless of slip or twinning, the driving force for shear rate c_ a is the resolved shear stress sa ¼ sa  r  na ¼ r : P a , where r is the Cauchy stress tensor. For slip, as suggested by Asaro and Needleman (1985), the shear rate can be described by the rate-sensitive constitutive law

c_ a ¼ c_ 0 jsa =sacr j1=m sgnðsa Þ

ð2Þ

where c_ 0 is a reference shear rate, sacr is the threshold shear stress (or critical resolved shear stress (CRSS)), and m is the strain rate sensitivity. In later applications, Eq. (2) has been used (with a high enough 1=m values) as an efficient numerically continuous law for describing system activation when the resolved shear is close to the threshold, while rate sensitivity effects are included in the dependence with strain rate of sacr (Brown et al., 2012). Such approach is well suited for twinning, which usually exhibits little rate-sensitivity. 2.1. TDT model In this section, we present a detailed description of the TDT model for describing the plastic deformation that results from twinning and de-twinning (preliminary results have been reported in Ref. (Wang et al., 2012b)). The most relevant difference from the CG model is that we adopt both the stresses in the matrix and in the twin to drive twinning and de-twinning. In addition, we treat twins as separate inclusions from the start (rather than transitioning from a ‘true’ CG to uncoupled matrix and twin inclusions, as done in Proust et al. (2009)), and allow for any active twin variants to become a twin (as opposed to consider only the predominant twin system in the grain). In the TDT model, the deformation processes of twinning and de-twinning can take place by four operations as illustrated in Fig. 11: twin nucleation, twin growth, twin shrinkage and re-twinning. It should be pointed out that all these four processes are associated with the conventional shear-shuffle or glide-shuffle mechanism entailing the glide of twinning disconnections (Wang et al., 2009a,b, 2011b, 2012d, 2013c). Starting with a twin-free grain (referred to as ‘matrix’) in Fig. 1(a), the grain starts twinning by twin nucleation (TN)2 associated with the twin system ðsa ; na Þ (Fig. 1(b)), when the resolved shear stress (RSS) in the matrix, saM ¼ sa  r  na , equals the critical resolved shear stress (CRSS) for twin nucleation (sTN cr ). The grain is then split into an un-twinned domain (matrix) and a twinned domain (twin). The corresponding crystallographic lattices are mirrored across the twin boundary (TB). Due to the polar nature of twinning, the twinning dislocations (TDs) can only glide in the twinning direction sa . It is clear that the matrix Cauchy stress rM must be used to calculate the RSS of twin nucleation because the twin is not born yet. The RSS acting on the twinning system in the matrix ðsaM ¼ sa ; naM ¼ na Þ is obtained through saM ¼ saM  rM  naM (Fig. 1(b)), and must be positive for twin nucleation to take place. After a twin nucleates, the twinning system a in the twin becomes saT ¼ Q  saM ¼ sa and naT ¼ Q  naM ¼ na , where Q ¼ 2na  na  1. Twin growth (TG) starts after twin nucleation when the RSS on the twinning system exceeds the CRSS for twin growth sTG cr (Fig. 1(c)), which is usually less than the CRSS for twin nucleation sTN cr (Proust et al., 2009). Twin growth is accomplished through gliding of TDs in the matrix on the TB, and the driving force is the stress acting on TDs at the TB interface. Because the TDT model is implemented in an effective medium approximation, it does not provide the local stresses at the TB, only the average stress in the ellipsoids that represent matrix and twin. Experimental measurements of average stresses in twin and matrix done by Aydiner et al. (2009) and of local stresses in the vicinity of twins done by Balogh et al. (in press), plus local Finite Element (FE) and Fast Fourier Transform (FFT) simulations of twins performed by Zhang et al. (2008) and Kanjarla et al. (in preparation), respectively, indicate that important stress gradients are present at the TB. For thin twin lamellae it is likely that the stress at the interface will be closer to the average stress in the twin domain than to the average stress in the parent. As a consequence, in order to incorporate the effect of the stress difference in the matrix and twin and the repartition of twin shear among the matrix and the twin, twin growth is treated as two parts: matrix reduction (MR) and twin propagation (TP), which are respectively activated by the average stresses in the matrix and in the twin. MR is accomplished through the migration of twin boundary induced by the resolved shear stress saM ¼ saM  raM  naM in the matrix and reduces the current volume of the matrix; TP is accomplished through the migration of twin boundary towards the matrix induced by the resolved shear stress saT ¼ saT  raT  naT in the twin and increases the current volume of the twin. The twinning 1 The layered matrix-twin configuration shown in Fig. 1 is meant for discussing our twin criteria in terms of the interface, and is not a representation of the system that we solve. In the model the matrix and the twin are treated as separate individual inclusions. We verified that the treatment of the twin as an inclusion uncoupled from the parent and interacting with the Effective Medium, predicts twin stresses that differ by less than 2% (5% for flat twin inclusions) from the stress that results from assuming that the twin only interacts with the parent. 2 In this work, nucleation refers to the introduction of an initial ellipsoid representing a small fraction of the grain volume.

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Twin free

sαM

nαM

Twin nucleation

sαM

(a)

nαM

(b)

Twin growth

nαT

Twin shrinkage

Re-twinning

(d)

(e)

sαT

(c)

Fig. 1. Schematic representation of twinning and de-twinning in a grain. (a) The twin-free grain (matrix). (b) Twin Nucleation: introduction of twin nucleation. Solid green lines represent twinning boundaries. Lattices in the matrix and twin are represented by dotted blue lines and dotted red lines, respectively. (c) Twin Growth: growth of a twin through the nucleation and glide of twinning dislocations on the twin boundaries in the matrix. (d) Twin Shrinkage: Shrinkage of a twin through the nucleation and glide of twinning dislocations on the twin boundaries. (e) Re-Twinning: introduction of a twin variant in a twinned grain.

system activated by both mechanisms is the one in the matrix, that is (saM ; naM ). Therefore saM and saT have to be respectively positive and negative in order to activate the unidirectional TD associated with twin growth. Twin shrinkage (TS), accomplished by gliding of TDs in the twin along the TB, is the opposite operation to twin growth (Fig. 1(d)). Twin shrinkage operates when the RSS corresponding to the de-twinning is greater than the CRSS for twin shrinkTG age sTS cr . The CRSS for twin shrinkage could be the same as the CRSS for twin growth scr . Twin shrinkage is the reverse operation of twin growth, and thus consists of two parts: matrix propagation (MP) and twin reduction (TR). MP is accomplished through the migration of twin boundary towards the interior of the twin induced by the resolved shear stress saM in the matrix and increases the current volume of the matrix. TR, on the other hand, is induced by the resolved shear stress saT in the twin and shrinks the current volume of the twin. The twinning system activated (saT ; naT ) is the one of the twin and saM and saT have to be respectively negative and positive in order to activate the unidirectional TD associated with twin shrinkage. Re-twinning (RT) represents the activation of a twin within a twin (Fig. 1(e)). The twin variant could be same as or different from the pre-existing twin variant. If the variant is the same as the pre-existing twin, it corresponds to de-twinning through the nucleation of a twin. If the variant is different from the pre-existing twin, it corresponds to secondary twinning through the nucleation of a new twin variant in the twin. Because re-twinning corresponds to the introduction of a new twin, TN aT correspondthe CRSS for re-twinning sRT cr could be the same as the CRSS for twin nucleation scr . The resolved shear stress s ing to this operation is calculated using the stress state inside the twin. Although a new twin is treated as a new grain in the CG and TDT models, a difference is that all twin variants are allowed in the TDT model, while only the predominant twinning variant is considered in the CG model. The latter limitation follows from enforcing shear stress continuity across the TB in the CG model, a condition that cannot be fulfilled in multiple twin planes unless the stress is assumed the same in twins and matrix. While continuity conditions have to be met locally at the TB, Aydiner et al. (2009) show that they are not fulfilled by the average stress in the matrix and twin domains. As a consequence of relaxing such condition, newly ‘born’ twins are treated in the TDT model as new inclusions embedded in the effective medium and uncoupled from the matrix. When a twin variant a is nucleated into a grain denoted as i, the new a grain, denoted as i , is assigned an initial orientation related to the twin orientation by the rotation matrix a Q ¼ 2na na  1. The shape of the new grain i is assumed to be ellipsoidal. The weight of the new grain wga is the product a of the twin volume fraction f and the weight of the twin-free grain wg , i.e. wga ¼ f a wg . The numerical treatment of twinning in the TDT model is illustrated in the flowchart in Fig. 2. The grain i in a polycrystal containing n grains is initially a twin free grain with volume fraction w0i . The twin is first introduced by twin nucleation (TN) in the twin free grain with volume fraction of wai corresponding to the twin variant a. As discussed above, the resolved shear stress of twin nucleation acting on the twinning system is calculated from the matrix Cauchy stress rM as saM ¼ saM  rM  naM , which must be positive for twin nucleation to take place. The shear rate of twinning system a in the matrix grain associated with twin nucleation is

(

c_ aTN ¼

c_ 0 jsa =sacr j1=m sa > 0 0 sa 6 0

The resolved shear stress is sa ¼ saM , the CRSS for twin nucleation is calculated with respect to the matrix grain as

ð3Þ

sacr ¼ sTN cr and the change of twin volume fraction is

f_ aTN ¼ jc_ aTN j=ctw where ctw is the characteristic twinning shear strain and is 0.129 for extension twinning systems of magnesium alloys.

ð4Þ

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H. Wang et al. / International Journal of Plasticity 49 (2013) 36–52

Fig. 2. Flowchart of the numerical treatment of twinning in CG and TDT models (Note: de-twinning is the reverse process of twinning).

Regarding twin growth, both the CG and TDT models are based on the same dislocation mechanism, i.e., the glide of twinning dislocations on twin boundaries. A difference between the CG model and the TDT model is illustrated in Fig. 2. In the CG model, the plastic deformation associated with twin growth is assigned to a shear taking place in the matrix domain, only the matrix stress induces twin growth, and the localized twin shear is homogenized across the matrix grain. This component of twin growth is called matrix reduction (MR) in the TDT model and the corresponding shear rate is

(

c_ aMR ¼

c_ 0 jsa =sacr j1=m sa > 0 0 sa 6 0

The resolved shear stress is sa ¼ saM , the CRSS for MR is that for twin growth fraction is calculated with respect to the matrix grain as

f_ aMR ¼ jc_ aMR j=ctw

ð5Þ

sacr ¼ sTG cr , and the change of twin volume

ð6Þ

In the TDT model, in addition to the contribution to strain from the matrix reduction mechanism described above, we also introduce a contribution from twin propagation (TP) to twin growth. The stress state inside the twin propagates the twin and the corresponding localized twin shear is homogenized across the twin domain grain. It must be pointed out that for TP the a stress in the twin grain i is used to activate the twinning system in the matrix i. Thus, the corresponding shear rate in the twinned grain is

(

c_ aTP ¼

c_ 0 jsa =sacr j1=m 0

sa < 0 sa P 0

ð7Þ

The resolved shear stress is sa ¼ saT , the CRSS for TP is that for twin growth sacr ¼ sTG cr , and the change of twin volume fraca tion is represented with respect to the twinned grain i by

f_ aTP ¼ jc_ aTP j=ctw

ð8Þ

In addition to twin growth, if the resolved shear stress in the matrix becomes greater than the CRSS of twin nucleation sTN cr , twin nucleation can also be activated in the matrix again. However for a single twin variant, twin nucleation and matrix shrinkage are unlikely to be activated simultaneously unless the motion of the TB is impeded (hardened) by the presence of immobile point/line defects due to the dislocation-TB reactions, resulting in the increase of the CRSS of twin growth sTG cr (Wang et al., 2012d, 2013c). The competition between twin nucleation and matrix reduction makes use of either Eq.

H. Wang et al. / International Journal of Plasticity 49 (2013) 36–52

41

(3) or Eq. (5) to compute the shear rate of the corresponding twin variant. The volume fraction associated with twin nuclea ¼ 0:005). Therefore, the matrix reduction is the main mechanism to ation of a grain is set to be small in the TDT model (fTN grow the twins associated with the matrix grain stress. When loading in opposite direction, the twin will be reduced by twin shrinkage and re-twinning (RT). Re-twinning and twin shrinkage are the inverse operations of twin nucleation and twin growth, respectively. Twin shrinkage is correspondingly described in two parts: matrix propagation (MP) into the twin and twin reduction (TR). Therefore the governing equations associated with de-twinning are similar with those in twinning. Similar to twin nucleation, the volume fraction a ). Currently, we do not allow for secondary twinning associated with re-twinning is also set to be small in the TDT model (fRT in the TDT model, i.e., the twinning system b cannot be activated inside the twin a if b – a. Correspondingly, the shear rate associated with the de-twinning mechanisms of MP is:

( _a

cMP ¼

c_ 0 jsa =sacr j1=m 0

sa < 0 sa P 0

ð9Þ

The resolved shear stress is sa ¼ saM , the CRSS for MP is that for twin shrinkage sacr ¼ sTS cr . De-twinning decreases the twin volume fraction, therefore the change of twin volume fraction is calculated with respect to the matrix grain by

f_ aMP ¼ jc_ aMP j=ctw

ð10Þ

The shear rate associated with the de-twinning mechanisms of TR is:

( _a

cTR ¼

c_ 0 jsa =sacr j1=m sa > 0 0 sa 6 0

ð11Þ

The resolved shear stress is sa ¼ saT , the CRSS for TR is that for twin shrinkage a fraction is calculated with respect to the twinned grain i by

sacr ¼ sTS cr , and the change of twin volume

f_ aTR ¼ jc_ aTR j=ctw

ð12Þ

The shear rate associated with the re-twinning mechanisms of RT is:

(

c_ aRT ¼

c_ 0 jsa =sacr j1=m sa > 0 0 sa 6 0

ð13Þ

The resolved shear stress is sa ¼ saT , the CRSS for RT is that for re-twinning sacr ¼ sRT cr , and the change of twin volume fraca tion is calculated with respect to the twinned grain i by

f_ aRT ¼ jc_ aRT j=ctw

ð14Þ

Because the twin volume fraction and plastic strain associated with twin nucleation and re-twinning are small, the net evolution of twin volume fraction f a (with respect to composite grain including the corresponding matrix grain and all twin grains) associated with twinning system a is mainly due to twin growth (MR and TP) and twin shrinkage (MP and TR):

    f_ a ¼ f M f_ aMR þ f_ aMP þ f a f_ aTP þ f_ aTR

ð15Þ

P where f M is the volume fraction of the matrix, i.e. f M ¼ 1  f tw ¼ 1  a f a . It is worth mentioning that the available volume th fraction of a composite grain for a twinning system is the entire volume fraction of the matrix and the ath twin. Once a twin forms, the available volume fraction for other twinning systems is accordingly reduced and, as a consequence, their ability to accommodate strain. For a single twinning system, twinning and de-twinning are impossible to operate simultaneously. If twinning operates (c_ aMR and c_ aTP are non-zero), de-twinning cannot operate (c_ aMP and c_ aTR must be zero) and vice versa. Therefore, the plastic strain rates induced by twinning in the matrix grain and the ath twin grain are: M

dtw ¼

X







c_ aMR þ c_ aMP P aM ; datw ¼ c_ aTP þ c_ aTR P aT

ð16Þ

a

Then the plastic strain rate of the composite grain induced by twinning or de-twinning is: M

dtw ¼ f M dtw þ

X a f a dtw

ð17Þ

a

A threshold twin volume fraction is defined in the model to terminate twinning because it is rare that a grain can be fully twinned. Correspondingly, we introduce two statistical variables: accumulated twin fraction V acc and effective twinned fraction V eff . More specifically, V acc and V eff are the weighted volume fraction of the twinned region and volume fraction of twin terminated grains, respectively. The threshold volume fraction V th is defined as

V th ¼ minð1:0; A1 þ A2 V eff =V acc Þ

ð18Þ

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where A1 and A2 are two material constants. For both slip and twinning, the evolution of the critical resolved shear stress (CRSS) sacr is given by:

s_ acr ¼

^ a X ab b ds h jc_ j dC b

ð19Þ

P R ab where C ¼ a jc_ a jdt is the accumulated shear strain in the grain, and h are the latent hardening coupling coefficients ^a is the threshold stress and is defined which empirically account for the obstacles on system a associated with system b. s by an extended Voce law:

s^a ¼ sa0 þ ðsa1 þ ha1 CÞð1  expðha0 C=sa1 ÞÞ

ð20Þ

Here, s0 , h0 , h1 and s0 þ s1 are the initial CRSS, the initial hardening rate, the asymptotic hardening rate, and the backextrapolated CRSS, respectively. 2.2. EVPSC model Here, the TDT model is implemented in the EVPSC polycrystal model (Wang et al., 2010d), although the concept of the TDT model is general and can be implemented into any polycrystal code. Within the EVPSC model, each grain is regarded as an elastic-viscoplastic inclusion embedded and interacting with the elastic-viscoplastic effective medium that represents the aggregate. The strain rate is assumed to be uniform inside the grain, and is accommodated by both the elastic strain rate and plastic strain rate. The elastic strain rate is the distortion of the crystallographic lattice and is related to the stress rate of the grain through the single crystal constitutive law: r

r ¼ L : de  rtrðde Þ

ð21Þ e

r

where L is the fourth order elastic stiffness tensor, d is the elastic strain rate tensor and r is the Jaumann rate of the Cauchy stress r based on the lattice spin tensor we . The single crystal elastic anisotropy is included in L through the crystal elastic constants C ij (Wang and Mora, 2008). If the elasticity of a material is assumed to be isotropic, L can be expressed as a function of Young’s modulus, E, and Poisson’s ratio, m. The plastic strain rate is accommodated by shear rates provided by both slip and twinning (Eq. (1)). From Eqs. (1), (17), one can linearize the behavior of the single crystal as follow (Wang et al., 2010d): e

p

d ¼ d þ d ¼ M e : r_ þ M p : r þ d0 e

ð22Þ

p

where M , M and d0 are the elastic compliance, the visco-plastic compliance, and the back-extrapolated term for the single crystal, respectively. The self-consistent approach assumes the behaviour of the effective medium has a linear relation analogical to (22):

 e : R_ þ M  p : R þ D0 D¼M

ð23Þ

 e, M  p , R and D0 are the strain rate, the elastic compliance, the visco-plastic compliance, the Cauchy stress and the where D, M back-extrapolated term for the effective medium, respectively. The strain rates and stresses of single crystals are related selfconsistently to the corresponding values of the effective medium as follows:

~ e : ðr_  RÞ ~ p : ðr  RÞ _ M d  D ¼ M

ð24Þ

~ e and M ~ p are given by: where the interaction tensors M

 e; M ~ p ¼ ðI  S p Þ1 : S p : M p ~ e ¼ ðI  S e Þ1 : S e : M M e

ð25Þ

p

Here, S and S are the elastic and visco-plastic Eshelby tensor for a given grain, respectively (Eshelby, 1957). I is the identity tensor. Different linearization of the single crystal behaviour leads to different self-consistent schemes. Wang et al. (2010a,c,e) have evaluated several self-consistent schemes by studying the large strain behaviour of magnesium alloy AZ31B sheet under tension and compression along different directions. It has been demonstrated that, of the schemes examined, the Affine self-consistent scheme gives the best overall performance. Therefore, the Affine self-consistent scheme is employed in the present study. The applications of the self-consistent models associated with Affine scheme are discussed in Wang et al. (2010b,d, 2011a, 2012c, 2013a,b) and Wu et al. (2012). 3. Results and discussions To demonstrate the validity of the proposed TDT model, we implemented it into the EVPSC model and apply it to two typical magnesium alloys under different loading conditions. The first examination is AZ31B plate under a uniaxial tension–compression cycle. The second examination is AZ31 extruded bar under pre-compression followed by subsequent ten 0i), prismatic hai (f1 0 1  0gh1 1 2  0i) and pyramidal hc + ai sion. In these examinations, we consider the basal hai (f0 0 0 1gh1 1 2       (f1 1 2 2gh1 1 2 3i) slip systems, and f1 0 1 2gh1 0 1 1i extension twin system. The room temperature elastic constants of the

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H. Wang et al. / International Journal of Plasticity 49 (2013) 36–52

magnesium alloy are taken from (Simmons and Wang, 1971): C 11 ¼ 58:0, C 12 ¼ 25:0, C 13 ¼ 20:8, C 33 ¼ 61:2 and C 44 ¼ 16:6 (GPa). The reference slip/twinning rate c_ 0 and the rate sensitivity m are prescribed to be the same for all slip/twinning systems: c_ 0 ¼ 0:001 s1 and m ¼ 0:05, respectively. 3.1. AZ31B plate Cyclic behavior of magnesium alloy AZ31B plate with H24 temper is first simulated and compared with the experiments conducted by Wu et al. (2010). The initial texture reported by Wu et al. (2010) was represented using 2160 orientations, a number which may increase up to sevenfold as new twin grains are created during deformation. Values of the hardening parameters are estimated by fitting numerical simulations of monotonic tension and compression along the rolling direction (RD) to the corresponding experimental flow curves (Fig. 3(a)). It is found that the EVPSC model with the proposed TDT model can well reproduce experimental curves of both the AZ31B plate by using the hardening parameters listed in Table 1 (without losing the generality, the CRSSs associated with twinning/de-twinning mechanisms (TN, TG, TS and RT) are prescribed to be the same). For comparison purposes, the average Cauchy stresses associated with uniaxial compression in the matrix grains and in the twin grains are also included. At small strains, the former exceeds the latter by as much as 50 MPa, indicating that the mechanism of matrix reduction is more likely to be active than the mechanism of twin growth. Such relation is reversed after 10% compression. Fig. 3(b) shows the evolution of twin volume fraction (f tw ) and the accumulated twin/de-twin volume fractions associated with twin growth (matrix reduction and twin propagation) and twin shrinkage (matrix propagation and twin reduction) (fMR , fTP , fMP , fTR ) as a function of strain. The predicted twin volume fraction is in agreement with the experimental observations (Proust et al., 2009; Oppedal et al., 2012). The twin volume fraction increases monotonically and saturates after about 7% strain, mainly due to MR and TP, i.e. initially dominated by MR (fMR ) and later 400

Twin volume fraction

σ

(a) Tension

300

200

Compression

0

0.05

0.1

0.15

0.6

fMR 0.4

ε

1

(c) 0.8

0

0.2

Relative activity

Relative activity

f tw

0.8

0.2

Experiment Simulation Matrix Twin

Extension twin

0.6

MR 0.4

fMP , fTR 0

0.05

0.1

0.15

0.2

1

(d) 0.8

Extension twin

Basal

0.6

0.4

0.2

0

(b)

fTP

100

0

1

MP, TR 0

Pyramidal

0.2

TP 0.05

0.1

0.15

0.2

0

Prismatic 0

0.05

0.1

0.15

0.2

Fig. 3. Uniaxial tension and compression of magnesium alloy AZ31B plates along the RD. (a) Measured (Wu et al., 2010) and fitted stress and strain curves; (b) Evolution of twin volume fraction under uniaxial compression; (c) Relative shear activities of various mechanisms associated with extension twin under uniaxial compression; (d) Relative shear activities of slip deformation mechanisms under uniaxial compression.

44

H. Wang et al. / International Journal of Plasticity 49 (2013) 36–52

Table 1 TDT model parameters determined by fitting the monotonic uniaxial tension and compression stress strain curves of AZ31B plate. The experimental data are taken from Wu et al. (2010). Without losing the generality, the CRSSs associated with twinning/de-twinning mechanisms (TN, TG, TS and RT) are prescribed to be the same. Mode

s1 (MPa)

s2 (MPa)

h0 (MPa)

h1 (MPa)

hab

A1

A2

Basal Prismatic Pyramidal Extension twin

18 75 90 33

3 50 115 0

500 1500 2000 0

10 10 0 0

1 1 1 1

0.85

0.6

σ

400

(a)

Twin volume fraction

dominated by TP (fTP ) after strain of 3%, as shown in Fig. 3(c). The activities of various deformation mechanisms under uniaxial compression are compared in Fig. 3(d). The results are consistent with previous studies (Agnew and Duygulu, 2005; Wang et al., 2010a), i.e., basal and pyramidal slips will dominate the rest deformation after twinning is saturated. Fig. 4(a) compares our simulations with the experiment (Wu et al., 2010) of the uniaxial tension–compression cyclic behavior of AZ31B plate along RD with the strain amplitude of 3%. Results show the capability of the TDT model for well reproducing the experimental stress–strain hysteresis loops. The tension–compression asymmetry observed in the experiments has been ascribed to the fundamental difference of plastic deformation modes, but a quantitative study could not be done in previous models (Proust et al., 2007, 2009). With the proposed TDT model, we quantitatively characterize the

P3 P7

200

P2 P6 P0

0

P0

1

P1

P2

P3

P4

P5

P6

(b)

fMR

f tw

fTP

0.5

-200

fTR -0.5

-0.02

P1

P2

0

P3

ε

0.02

P4

P5

P6

P7

fMP

P8

(c) Extension twin

0.8

0.6

MR

-1

0.04

Relative activity

Relative activity

P0

1

P4 Experiment (1st cycle) Experiment (2nd cycle) Simulation (1st cycle) Simulation (2nd cycle)

-400 -0.04

P8

0

P8 P1 P5

P7

P0

1

ε

-0.03

0.0

0.03

0.0

-0.03

0.0

0.03

0.0

P1

P2

P3

P4

P5

P6

P7

P8

(d) Extension twin

0.8

0.6

MP

0.4

0.4

Basal 0.2

0.2

TP 0

-0.03

0.0

0.03

0.0

Pyramidal

TR -0.03

0.0

0.03

ε

0.0

0

-0.03

0.0

Prismatic 0.03

0.0

-0.03

0.0

0.03

ε

0.0

Fig. 4. (a) Measured and predicted stress and strain curves of the AZ31B plate under cyclic loading along the RD with strain amplitude of 3.0%. The experimental data are taken from Wu et al. (2010); (b) Predicted twin volume fractions as a function of strain; (c) Relative shear activities of various mechanisms associated with extension twin; (d) Relative shear activities of slip deformation mechanisms.

45

H. Wang et al. / International Journal of Plasticity 49 (2013) 36–52

contribution of different deformation mechanisms to the plastic deformation. We find that the deformation modes for processes P0–P1, P1–P2, P2–P3, P3–P5, P5–P7 and P7–P8 correspond to twinning, de-twinning, slip, twinning, de-twinning and twinning, respectively. The twin volume fraction evolution as a function of the strain along RD is shown in Fig. 4(b). The twinning (MR and TP) and de-twinning (MP and TR) are activated alternately in the compression and tension branches of the cyclic loading. The twin volume fraction increases linearly to 35.9% during the initial compressive straining of 3% (state P1), and then decreases linearly to nearly zero (2.1%) at the strain of 0.3%, at which point de-twinning is exhausted. Between e ¼ 0:3% and 2.1% the twin volume fraction remains almost constant (passes the state P3) because the dominant plastic deformation mechanisms are slip modes. Past e ¼ 2:1%, the twin volume fraction increases linearly and reaches 63.6% at 3% strain (state P5). The twin volume fraction decreases linearly to the 4.8% at a strain of 2% (between state P6 and P7), where the de-twinning is exhausted, then remains nearly unchanged up to e ¼ 2:2%, and then increases linearly to 30% at e ¼ 0. The strain required to de-twin completely is approximately equal to the strain accumulated during twinning because de-twinning is exactly the reverse action of twinning. The latter explains the striking difference in stress–strain response between the first and the second tensile cycles. At state P2 de-twinning is exhausted and between P2 and P3 deformation is accommodated by prismatic slip, which leads to an increase in flow stress. In the second cycle, on the other hand, about twice as much twinning accumulates by P5, and de-twinning is still active between P6 and P7, which keeps the plateau in the flow stress. The activity of different deformation mechanisms is compared in Fig. 4(c) and (d). It is clear that: (i) Alternate twinning and de-twinning are observed during the cyclic loading (Fig. 4(c)); (ii) Extension twinning dominates most of the deformations, and prismatic slip is alternatively active when extension twinning is terminated; (iii) Basal slip is active through the whole deformation processes. In addition, the twinning and de-twinning associated with matrix grains (MR and MP) contribute more plastic deformation than the other twinning/de-twinning mechanisms (TP and TR) (Fig. 4(b) and (c)), which can be

P0

P1

P2

P4

P3

P5

P6

P7

P8

TD

{1010}

RD 0.78 1.24 1.971 3.133 4.98 7.92 12.58 20.0

{0001}

{1011}

Fig. 5. Predicted crystallographic textures of the AZ31B plate under cyclic loading along the RD at states of P0 to P8.

P0 P1

Twin volume fraction

1

P2

P3

P4 P5

P6

P7

P8

fMR 0.5

f tw fTP

0

fTR -0.5

fMP -131 -1

-137 0 115

328

0 -152

0 1 11

30 8

σ

0 -120

Fig. 6. Predicted twin volume fraction as a function of applied stress under cyclic loading along the RD.

46

H. Wang et al. / International Journal of Plasticity 49 (2013) 36–52

Twin volume fraction

P0

(a)

200

P7 P3

P6 P0

0

P2 P8 P4

-200

Relative activity

-400 -0.04

P0

P1 P5 -0.02

P1

P2

0

P3

P4

(c)

ε

0.02

P5

P6

P7

P8

0.8

0.6

MP

0.4

P2

P3

P4

P6

P7

P8

fMR f

0.5

P5

tw

fTP 0

fTR fMP

P0

1

ε

0.03

0.0

-0.03

0.0

0.03

0.0

-0.03

0.0

P1

P2

P3

P4

P5

P6

P7

P8

(d)

Extension twin

0.8

0.6

Pyramidal

0.4

TR

TP

0.2

P1

(b)

-1

0.04

Extension twin

MR

1

-0.5

Experiment (1st cycle) Experiment (2nd cycle) Simulation (1st cycle) Simulation (2nd cycle)

Relative activity

σ

400

Basal

0.2

Prismatic 0

0.03

0.0

-0.03

0.0

0.03

0.0

-0.03

ε

0.0

0

0.03

0.0

-0.03

0.0

0.03

0.0

-0.03

ε

0.0

Fig. 7. (a) Measured and predicted stress and strain curves of the AZ31B plate under cyclic loading along the ND with strain amplitude of 3.0%. The experimental data are taken from Wu et al. (2010); (b) Predicted twin volume fractions as a function of strain; (c) Relative activities of various mechanisms associated with extension twin; (d) Relative activities of various deformation mechanisms.

ascribed to the fact that each twinning operation’s contribution is instantaneously proportional to its associated grain volume fraction. The texture evolution during the cyclic loading of AZ31B plate along RD is shown in Fig. 5 and reflects the strong effect of twin reorientation upon texture. For states P0, P2, P3, P7, the pole figures show strong basal textures, which are either untwined or completely de-twinned. As for states P1, P4, P5, P6 and P8, the pole figures show typical partially twinned textures. The pole figure P5 shows the strongest twinned texture because of the largest twin volume fraction at state P5. The predicted texture evolution is in good agreement with the experiments (Fig. 9 in Wu et al. (2010)3). The stress–strain curve, twin volume fraction evolution and texture evolution from our simulations indicate that alternating deformation mechanisms of twinning and de-twinning drives the cyclic straining process. In order to investigate the twinning and de-twinning behavior during elastic–plastic transits (near plastic yielding), the twin volume fraction is plotted as a function of the applied stress under cyclic loading along RD (Fig. 6). The twin volume fraction f tw remains zero until a compressive yield stress of 77 MPa, and then increases rapidly to 35.9% at a stress of 137 MPa. Then f tw remains constant until the material is completely unloaded to a stress of 0 MPa. The AZ31B plate starts 3

Please note that the experimental data indices are in contracted Miller-Bravais notation.

47

H. Wang et al. / International Journal of Plasticity 49 (2013) 36–52

P0

P1

P2

P4

P3

P5

P6

P8

P7

TD

{1010}

0.78 1.24 1.971 3.133 4.98 7.92 12.58 20.0

{0001}

{1011}

Fig. 8. Predicted crystallographic textures of the AZ31B plate under cyclic loading along the ND at states of P0 to P8.

1

σ

(a) Relative activity

(b)

Tension

Compression

0.8

Extension twin

0.6

Basal

0.4

Pyramidal 0.2

ε

Prismatic 0

0.05

0.1

0.15

0.2

0.15

0.2

1

(c)

(d) Relative activity

Twin volume fraction

1

0

f tw

0.8

0.6

fMR 0.4

0.8

Extension twin 0.6

MR 0.4

fTP 0.2

0

0.2

fMP , fTR 0

0.05

0.1

0.15

0.2

0

TP MP, TR 0

0.05

0.1

Fig. 9. Uniaxial tension and compression of magnesium alloy AZ31 extruded bar along the ED. (a) Measured (Agnew et al., 2006) and fitted stress and strain curves; (b) Relative activities of various deformation mechanisms under uniaxial compression; (c) Evolution of twin volume fraction under uniaxial compression, experimental data (symbols) are taken from Clausen et al. (2008); (d) Relative activities of various mechanisms associated with extension twin under uniaxial compression.

48

H. Wang et al. / International Journal of Plasticity 49 (2013) 36–52

de-twinning when the stress becomes tensile of 15 MPa, which is the yield stress of the tensile reload. The magnitude of the yield stress for de-twinning (15 MPa) is much lower than that for twinning (77 MPa). A similar result is observed for the second cycle, too. The evolution of twin volume fraction, and the stress strain curve as well (Fig. 4(a)), is more gradual during de-twinning than during twinning. We also examined the influence of the loading direction on texture evolution and stress–strain response by simulating cyclic loading along the normal direction (ND). The cyclic stress–strain curves with strain amplitude of 3% are shown in Fig. 7(a), and reproduce well the experimental stress–strain hysteresis loop. The evolution of twin volume fraction with respect to applied strain is plotted in Fig. 7(b) and the activities of deformation mechanisms are plotted in Fig. 7(c) and (d). The strain associated with de-twinning is equal to the strain associated with the corresponding twinning. The twinning operation MR and de-twinning operation MP produce more change in the twinning and de-twinning volume fraction than operation TP and TR because the twin volume fraction generated by the operations of the TDT model is instantaneously proportional to the corresponding volume fraction of grains. The corresponding texture evolution is shown in Fig. 8. The initial compression stroke along ND is accommodated by pyramidal and basal slip (Fig. 7(d)), which explains the initial rapid hardening. Twinning is activated when the stress reverses to tensile along ND and reorients about 65% of the volume fraction by P3. A comparison between the experimental (Wu et al., 2010) and predicted basal pole figures at P3 shows higher intensities in the latter and suggests that the TDT model over-predicts twinning during the tensile stroke. As a consequence, during the next compression stroke, de-twinning is overly active and leads to an unexpected softening effect between states P4 and P5, which is not observed in the experiment. The reason for this response is that the TDT model allows six twin variants per grain to be active for uniaxial tension along ND, while only two or four were active in uniaxial compression along RD (Fig 4). From the calculation, it is found that the number of twin variants activated under cyclic compression-tension along RD is 2.09 at P1, while that along ND is 3.96 at P2. This result suggests that more sophisticated slip-twin and twin–twin interactions may be required in the model to control activation of multiple twin variants in each grain, and to accurately capture the whole stress and strain behaviour.  0} and {1 0 1  1} pole figures. For states P0, P1, P5, the pole Fig. 8 shows the texture evolution in terms of {0 0 0 1}, {1 0 1 figures show strong basal textures, which are either untwined or completely de-twinned. As for states P2, P3, P4, P6, P7 and P8, the pole figures show typical partially twinned textures. The pole figures P3 and P7 show the strongest twinned texture because of the largest twin volume fraction at these states. The predicted texture evolution should be compared to Fig. 11 in Wu et al. (2010). While they are in good qualitative agreement with the experimental ones, the tendency is to predict sharper textures. Therefore the results shown in Figs. 7 and 8 imply that slip, twinning, de-twinning, twinning and detwinning are the dominant deformation mechanisms associated with the processes P0–P1, P1–P3, P3–P5, P5–P7 and P7–P8, respectively.

3.2. AZ31 extrusion bar Application of our model to Mg AZ31 extruded bar starts with the initial texture reported by Agnew et al. (2006), which is represented here using 2160 orientations. Values of the hardening parameters are estimated by fitting numerical simulations of monotonic tension and compression along the extrusion direction to the corresponding experimental flow curves

σ

300

σ

(a) 250

0%

200

250

2%

50

50

0.01

0.02

0.03

0.04

ε

0.05

0

2%

1%

150

100

0

0%

3%

100

0

(b)

200

1%

150

300

0

0.01

0.02

3%

0.03

0.04

ε

0.05

Fig. 10. Predicted stress and strain curves of AZ31 extrusion bar under uniaxial tension along the ED. (a) Pre-strained (1–3%) in compression; (b) Prestrained (1–3%) in compression and stress relieved. The experimental results for AZ61 (Kleiner and Uggowitzer, 2003, 2004) are included as insets in the figures.

49

H. Wang et al. / International Journal of Plasticity 49 (2013) 36–52

1

1

(b) Basal Relative activity

Relative activity

(a) Extension twin Stress relieved

0.8

Stress not relieved

0.6

0.8

0.6

0.4

0.4

0.2

0.2

0

Without prestrain 0

50

100

150

200

1

σ

0

250

2% 3% 1%

0

Relative activity

Relative activity

0.6

150

200

250

150

200

250

0.6

0.4

0.2

0.2

100

100

0.8

0.4

50

50

(d) Pyramidal

0.8

0

1%

1

(c) Prismatic

0

2% 3%

150

200

250

0

0

50

100

Fig. 11. Relative activity of various deformation mechanisms under uniaxial tension along the ED following (1–3%) pre-compression with and without stress relieved. (a) Extension twin; (b) Basal slip; (c) Prismatic slip; (d) Pyramidal slip. The same line format convention is used in all four figures.

(Fig. 9(a)). The EVPSC model with the proposed TDT model and the hardening parameters listed in Table 2 reproduces well the experimental curves of the AZ31 extrusion bar under tension and compression. The yield stresses for uniaxial compression is 55 MPa, and is caused by basal slip and extension twin (Fig. 9(b)). This yield stress is difficult to identify in macroscopic stress strain curves of experiments, but it becomes apparent (Agnew et al., 2006) when the lattice strains are considered. In order to quantitatively compare with the experimental results, the more visible yield stress of 125 MPa associated with basal slip, prismatic slip and extension twin (Fig. 9(a) and (b)) is adopted in our simulations. Fig. 9(c) represents the predicted twin volume fraction evolution of the AZ31 extruded bar under uniaxial compression. The twin volume fraction monotonically increases and eventually saturates to f tw ¼ 82% after strain of 10%. The predicted result is compared with the experimental data (symbols) taken from Agnew et al. (2006) for magnesium alloy AZ31 extrusion bar. The predicted twin volume fraction is in reasonable agreement with the experimental observations. Similar to the AZ31B plate, the twin volume fraction of the AZ31 extrusion bar is accumulated via the twinning operations MR (fMR ) and TP (fTP ). The twin evolution is initially dominated by matrix reduction because there are few twins. Twin propagation plays a more important role after strain of 3%. Fig. 9(b) and (d) show the activities of various deformation mechanisms under uniaxial compression, which is similar with that in the AZ31B plates (Fig. 3(c) and (d)). Fig. 10(a) shows the predicted stress–strain curves during tension for an AZ31 extruded bar is first pre-compressed to 1%, 2%, and 3% without relieving the residual stress. Monotonic tension (labeled as ‘0%’) and monotonic compression (labeled as ‘compression’) are plotted together as a reference. It can be seen that the tensile response following the pre-compression (1– 3%) does not exhibit an elastic region. Fig. 10(b) gives similar plots for the case when the residual stress is removed before the subsequent tension. In the simulation, the residual stress is removed by unloading from compression and resetting the

50

H. Wang et al. / International Journal of Plasticity 49 (2013) 36–52

Table 2 TDT model parameters determined by fitting the monotonic uniaxial tension and compression stress strain curves of AZ31 extrusion bar. The experimental data are taken from Agnew et al. (2006). Without losing the generality, the CRSSs associated with twinning/de-twinning mechanisms (TN, TG, TS and RT) are prescribed to be the same. Mode

s1 (MPa)

s2 (MPa)

h0 (MPa)

h1 (MPa)

hab

A1

A2

Basal Prismatic Pyramidal Extension twin

15 85 95 20

5 25 95 0

200 750 700 0

10 20 0 0

1 1 1 1

0.4

0.65

stresses in all the grains to zero while retaining all other information of the grains (such as CRSSs, twin volume fractions, etc.). It is observed that, upon tensile reloading, the yield stresses are equal to the corresponding ones for uniaxial compression. However, the critical resolved shear stress for both twinning and de-twinning are prescribed to be the same. This indicates that the lower yield stress for de-twinning is due to the residual stress that remains from the previous deformation, and not because of a lower resistance for de-twinning. This result is consistent with the experimental observation in magnesium alloy AZ61 extruded bar by Kleiner and Uggowitzer (2003, 2004). The corresponding experimental results are shown as insets in Fig. 10(a) and (b). Because of the different alloying composition, the magnitude of the flow stresses differs between AZ31 and AZ61. Fig. 11 plots the activities of various deformation mechanisms ((a) extension twin, (b) basal slip, (c) prismatic slip and (d) pyramidal slip) as a function of the applied stress under different deformations processes (1–3% pre-strain with and without the residual stress relieved). Before the yield stress of 125 MPa, prismatic slip and pyramidal slip are not active at all, while basal slip and extension twin are active. In addition, basal slip is more active in the deformation processes without residual stress relief. Thus, the initial slope in tension after pre-compression is actually an elasto-plastic regime for AZ31 extruded bar if the residual stress is not relieved. Our results indicate that the residual stress from the previous deformation can activate the basal slip and de-twinning, leading to the disappearance of the elastic region (Wu et al., 2008a; Hong et al., 2010a). Because basal slip and extension twin (de-twin) are the most active deformation mechanisms and have relatively low critical resolved shear stresses, they can be activated by the residual stress upon reload. 4. Conclusions We propose a physics-based crystal plasticity model including deformation mechanisms of both twinning and de-twinning (TDT). To examine the predictive capability of the TDT model, the cyclic deformation of AZ31B plate and tension after pre-compressions of AZ31 extruded bar are numerically studied. Our results are in good agreement with the available experimental results. The contribution associated with twinning and de-twinning to the plastic deformation is displayed clearly through the twin volume fraction evolution, relative activities of different deformation mechanisms and texture evolution. The reasonable numerical results imply that the introduced deformation mechanisms of twin nucleation, twin growth, twin shrinkage and re-twinning can capture the key feature of plastic deformation in HCP metals. The TDT model introduces several improvements by comparison with the CG model: it does not enforce average stress continuity across twin boundaries, in agreement with recent experimental evidence; it allows for several twin variants to form in each grain, as opposed to only the predominant twin variant; it introduces two sources of driving forces for twinning dislocations, the stress in the matrix and the stress in the twin, which has the effect of capturing de-twinning without having to assume a different threshold stress for twinning and de-twinning. Finally, we would like to point out that the TDT model in the current form can be considered, to some extent, as a purely geometrical twinning model because it does not directly account for effects of twinning on the hardening of the slip modes, and vice versa, although the TDT model rigorously calculates the shear rates, twin volume fraction and reorientation due to twinning and de-twinning. More specifically, the same hardening Eq. (20) with the same values of material parameters is used to describe hardening of materials in twinned and untwined regions. Such simplification may be justified for modeling deformation processes where twinning and de-twinning play a relevant role, such as during cyclic loading of Mg alloys (Yu et al., 2011; Zhang et al., 2011). Future efforts will be devoted to include the interaction between twinning, de-twinning and slip in the TDT model. Acknowledgements This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and by the Ontario Ministry of Research and Innovation. CNT and JW were supported by the US Department of Energy, Office of Basic Energy Sciences (Project No: FWP-06SCPE401). The authors thank Dr. Sean R. Agnew and Dr. Liang Wu for sharing their texture data. References Abdolvand, H., Daymond, M.R., Mareau, C., 2011. Incorporation of twinning into a crystal plasticity finite element model: evolution of lattice strains and texture in Zircoloy-2. International Journal of Plasticity 27, 1721–1738.

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