Modeling the effect of twinning and detwinning during strain-path changes of magnesium alloy AZ31

International Journal of Plasticity 25 (2009) 861–880

Contents lists available at ScienceDirect

International Journal of Plasticity journal homepage: www.elsevier.com/locate/ijplas

Modeling the effect of twinning and detwinning during strain-path changes of magnesium alloy AZ31 Gwénaëlle Proust a,b,*, Carlos N. Tomé a, Ashutosh Jain c, Sean R. Agnew c a

Los Alamos National Laboratory, MST-8, MS G755, Los Alamos, NM 87545, USA University of Sydney, Sydney, NSW 2006, Australia c Department of Materials Science and Engineering, University of Virginia, 116 Engineering Way, Charlottesville, VA 22904, USA b

a r t i c l e

i n f o

Article history: Received 11 December 2007 Received in final revised form 17 May 2008 Available online 14 June 2008 Keywords: Twinning Polycrystal modeling Hardening Hexagonal materials Magnesium

a b s t r a c t Hexagonal materials deform plastically by activating diverse slip and twinning modes. The activation of such modes depends on their relative critical stresses, and the orientation of the crystals with respect to the loading direction. To be reliable, a constitutive description of these materials has to account for texture evolution associated with reorientations due to both dislocation slip and twinning, and for the effect of the twin boundaries as barriers to dislocation propagation. We extend a previously introduced twin model, which accounts explicitly for the composite character of the grain formed by a matrix with embedded twin lamellae, to describe the influence of twinning on the mechanical behavior of the material. The role of the twins as barriers to dislocations is explicitly incorporated into the hardening description of slip deformation via a directional Hall–Petch mechanism. We introduce here an improved hardening law for twinning, which discriminates for specific twin/dislocation interactions, and a detwinning mechanism. We apply this model to the interpretation of compression and tension experiments done in rolled magnesium alloy AZ31B at room temperature. Particularly challenging cases involve strain-path changes that force strong interactions between twinning, detwinning, and slip mechanisms. Ó 2008 Elsevier Ltd. All rights reserved.

* Corresponding author. Address: University of Sydney, School of Civil Engineering, Sydney, NSW 2006, Australia. Tel.: +61 2 9036 5498; fax: +61 2 9351 3343. E-mail address: [email protected] (G. Proust). 0749-6419/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2008.05.005

862

G. Proust et al. / International Journal of Plasticity 25 (2009) 861–880

1. Introduction The purpose of the present work is twofold: develop a crystallographic model for the plastic response of Mg AZ31, applicable to non-monotonic deformation conditions and, in the process, increase our basic understanding of the role that slip and twin modes play in texture and hardening evolution. Wrought magnesium and magnesium alloys show, especially at room temperature, high asymmetry and anisotropy in their mechanical properties as a result of texture, the polar nature of twinning, and the fact that different deformation modes are active depending on the loading direction (Kelley and Hosford, 1968; Avedesian and Baker, 1999; Jain and Agnew, 2007; Lou et al., 2007). Several studies have been realized to understand the occurrence of the various slip and twin modes and their role onto the hardening behavior and texture evolution of Mg and Mg alloys during monotonic deformation (see for example Klimanek and Potzsch, 2002; Agnew et al., 2003; Agnew and Duygulu, 2005; Jiang et al., 2007). From these studies, it is clear that Mg and its alloys possess two ‘‘easy” deformation   modes: hai slip on basal planes and tensile f1012gh 1011i twinning. In particular, tensile twinning has been associated with the increased hardening of the material that is observed when deformation takes place along the main basal component (Kelley and Hosford, 1968; Barnett, 2007a). These two modes alone, however, are insufficient to accommodate arbitrary deformation, and it has been found exper  imentally that other slip modes contribute to strain accommodation: pyramidal hai slip f1011gh11 20i   (e.g., Schmid and Boas, 1968), prism hai slip f1010gh1120i (e.g., Ward-Flynn et al., 1961) and pyrami  (Stohr and Poirier, 1972; Obara et al., 1973; Ando and Tonda, 2000). dal hc + ai slip f1122gh11 23i The relative activity of the various slip and twinning modes described above depends on the specific loading conditions and initial texture; in turn, it determines texture evolution. In addition, hardening depends, in a complex manner, on the texture and interactions between slip and twin modes. Central to this effect is the fact that tensile twins change the texture of the material by reorienting domains of the grain by 86.6°. Moreover, twin boundaries can also act as obstacles to further slip and twinning deformation (Christian and Mahajan, 1995; Serra and Bacon, 1995; Serra et al., 2002). As a consequence, not only the monotonic loading response varies much depending on texture and testing direction and sense, but the mechanical response associated with strain-path changes (such as the ones which take place during forming) cannot be deduced from the knowledge of the monotonic response. This strain-path change behavior has been characterized experimentally for AZ31 Mg by Jain and Agnew (2006) and Lou et al. (2007). These authors observe that in sheet pre-deformed mainly by slip, twinning is not prevented by the presence of dislocations in the material, but the reloading yield strengths are slightly higher than for the annealed material. The influence of twins introduced by pre-straining on the reloading behavior of several magnesium alloys has also been explored (Caceres et al., 2003; Kleiner and Uggowitzer, 2004; Jain and Agnew, 2006; Brown et al., 2007; Lou et al., 2007; Mann et al., 2007; Wang and Huang, 2007). These authors report the phenomenon of detwinning (also referred to as untwinning) upon reversal or strain-path changes: the twins created during preload disappear during reload, and texture evolution is reversed to a large extent. The objective of this paper is to predict the mechanical behavior at room temperature of the Mg alloy AZ31B during strain-path changes. Recently, we published a similar analysis for pure Zr deformed at 76 K, a regime where tensile and compressive twins are active, and where secondary twinning plays an important role in increasing ductility (Proust et al., 2007). In that paper, where we present our new composite grain (CG) twin model, we argue that only a crystallography-based model that accounts for the orientation of slip and twin systems in each grain can describe the mechanical response for arbitrary deformation routes. Mg differs from Zr in that it twins more easily – after only 2% strain, Mg alloys can already exhibit as much as 14% twinned volume fraction (Chino et al., 2008) – and that it has been observe to undergo prolific detwinning. Therefore, it was necessary to extend the previously described CG twin model to properly describe these unique twinning behaviors associated with Mg alloys. In our previous paper (Proust et al., 2007), we provided a comprehensive review of polycrystal models addressing twinning in HCP materials. In short, early models were only concerned with describing texture evolution associated with monotonic loading (Van Houtte, 1978; Tomé et al., 1991; Lebensohn and Tomé, 1993; Philippe et al., 1995). More recently, researchers started developing

G. Proust et al. / International Journal of Plasticity 25 (2009) 861–880

863

constitutive models that, in addition to texture, also addressed the hardening response due to twinning associated with monotonic loading (Agnew et al., 2001; Kalidindi, 2001; Kaschner et al., 2001; Tomé et al., 2001; Salem et al., 2003, 2005; Staroselsky and Anand, 2003; Barnett et al., 2006; Clausen et al., 2008; Wu et al., 2007). Most of these approaches accounted for twin reorientation and used the concept of latent hardening to capture the role that twin interfaces play in hardening. It is also interesting to notice that models for martensitic transformation in TRIP steels share many commonalities with the crystallography-based twin models described above (Cherkaoui et al., 1998; Cherkaoui, 2003; Kubler et al., 2003) and have been adapted (Cherkaoui, 2003) for modeling twinning in fcc materials. The models described above reasonably predict the stress–strain response and texture evolution of magnesium alloys during monotonic deformation but predicting the mechanical behavior of these materials is more challenging once one considers changes in the loading path. To the best of our knowledge, only one attempt has been made to model the strain-path change behavior of Mg alloys. Jain and Agnew (2006) used the VPSC model (Lebensohn and Tomé, 1993) combined with the predominant twin reorientation (PTR) scheme (Tomé et al., 1991) and fitted the hardening parameters to monotonic deformation data. Although this particular model allows for latent hardening between the various deformation modes, it ignores the hardening directionality due to the microstructure evolution during twinning. The predictions for the strain-path changes did not match the experimental results, which demonstrated the need for a new model. This paper describes the use and extension of the CG twin model to predict the hardening, texture and twin volume fraction evolution of rolled Mg alloy AZ31B during monotonic and strain-path change deformations at room temperature. Although the focus is on modeling issues, we also interpret the experimental data to understand how the various deformation modes interact during strain-path change. We have also made a first attempt at modeling detwinning and our simple initial approach captures the main features associated with that process. 2. Experimental results 2.1. Material Commercial magnesium alloy AZ31B (3 wt% Al, 1 wt% Zn and balance Mg) sheet material was received in the stress relieved H24-temper. The material was annealed for an hour at 345 °C to reduce the presence of mechanical twins. After the heat treatment the microstructure of the material was an equiaxed grain structure with an average grain size of 13 lm. The initial texture was measured by Xray diffraction (XRD) and is shown in Fig. 1a. Compression and tension tests were performed at room temperature with an initial strain rate of 5  103 s1 using a computer controlled MTS screw-driven machine. The final texture of each deformed sample was then measured by electron backscattered diffraction (EBSD). Detailed experimental procedures were published previously (Jain and Agnew, 2006). 2.2. Monotonic deformation Fig. 1b shows the stress–strain response of the alloy deformed monotonically by in-plane tension (IPT), in-plane compression (IPC) and through-thickness compression (TTC). The respective final textures are shown in Fig. 2. As the initial texture is not axisymmetric about the sheet normal direction, some anisotropy was observed for tests along different in-plane directions (Jain and Agnew, 2007); however, the in-plane results reported in the present paper are solely obtained for a load applied parallel to the rolling direction (RD) of the plate. The IPT and TTC samples exhibit the typical hardening behavior associated with slip dominated deformation. During TTC, the material deforms mainly via basal hai and pyramidal hc + ai slip; the latter mechanism was first observed in Mg during c-axis compression of single crystals (Obara et al., 1973). However, recent studies (Koike, 2005; Jiang et al., 2006;  compressive twinning, f1011g—f10   double twining and Barnett, 2007b) have shown that f1011g 12g   double twinning, can also accommodate compressive strains along the c-axis at room f1013g—f10 12g temperature. However, those twinning systems never grow to reach the size or volume fraction of the  f1012g tensile twins (Jiang et al., 2007) and, therefore, do not contribute to the same extent as the

864

G. Proust et al. / International Journal of Plasticity 25 (2009) 861–880

400

RD

RD

TD

TD

1.0 2.0 4.0 8.0

Stress (MPa)

10 1 0

0002

300 200

IPT TTC IPC

100 0 0.00

0.05

0.10

0.15

0.20

Strain 400

300

Stress (MPa)

Stress (MPa)

400

IPC Reloads

200 100 0 0.00

0.05

0.10

Strain

0.15

0.20

300 200

TTC Reloads

100 0 0.00

Strain-path change Monotonic 0.05

0.10

0.15

0.20

Strain

Fig. 1. (a) Basal and prismatic pole figure showing the texture of the as-annealed AZ31B Mg; (b) stress–strain curves for monotonic IPT, IPC and TTC; (c) strain-path change stress–strain curves for the samples deformed first in TTC to 5% and 10% strain and then deformed in IPC (the monotonic IPC stress–strain curve is represented by the dotted line for comparison); and (d) strain-path change stress–strain curves for the samples deformed first in IPC to 5% and 10% strain and then deformed in TTC (the monotonic TTC stress–strain curve is represented by the dotted line for comparison).

tensile twinning to shear accommodation. As noted by Jain et al. (2008) these twins cause reorientations within the main texture components and do not result in marked texture evolution. Hence, those systems will not be considered in the present simulations, rather it will be assumed that the required shear accommodated by these twins is reasonably approximated by hc + ai slip (Agnew et al., 2006). Before deformation, the material shows a strong basal texture with most of the grains having their c-axes within 40° from the TT direction. After 5% TTC, the basal component of the texture has been reinforced and now most grains have their c-axes within 30° from the TT direction. The spread in the c-axis distribution has been reduced, as can be seen on the texture profile shown in Fig. 3. To obtain the profiles from the experimental the XRD and EBDS texture data, we enforce axisymmetry on the pole figure by averaging the intensity along the azimuthal direction between 0 and 360°. During IPT, basal hai and prismatic hai slip accommodate most of the deformation (Barnett et al., 2006; Jain et al., 2008). But due to the c-axis spread of the initial texture, some grains are favorably oriented for tensile twinning (Jain et al., 2008). In the basal pole figure obtained for the sample deformed 10% in IPT (see Fig. 2), the presence of (0 0 0 2) intensity in the direction perpendicular to both the rolling and TT directions confirms the existence of twins in the material. Integrating the area under the basal pole intensity curve of Fig. 3, gives an estimated 7% volume fraction of twins after 10% deformation. The hardening displayed by the sample deformed in IPC shows the characteristic increase in the hardening rate associated with twinning. The microstructure of the material has been changed drastically, as can be seen in Fig. 4a, showing a micrograph of a sample deformed 7% in IPC. Most of the grains present twin lamellae and some of them are heavily twinned. The texture of the deformed material is also very different from the initial one (see Fig. 2). The basal component along the ND

865

G. Proust et al. / International Journal of Plasticity 25 (2009) 861–880

Measured 0001

Predicted 10 1 0

0001

10 1 0

5% TTC

10% IPC

10% IPT 0.4 1.0 2.0 5% TTC

4.0

+ 5% IPC

8.0 RD

5%IPC+ 10%TTC TD Fig. 2. Comparison of measured and predicted basal and prismatic pole figures for monotonic and strain-path change deformations. The measured pole figures were obtained by EBSD. IPC and IPT were realized along the RD. The 0.4 intensity line is included in the 10% IPT case to reveal the ‘anomalous’ twinning effect.

has disappeared and now the c-axes of the crystals are aligned with the loading direction due to the 86.6° reorientation caused by tensile twinning. A comparison of the texture profiles obtained for the as-annealed material and the sample deformed in IPC, allows us to identify that there is a separation between the c-axis orientations belonging to the matrix or to the twinned portion of the material at a tilt angle of about 50°. Integrating the intensity profiles between 50° and 90°, and subtracting the initial volume fraction in the same interval, allows us to evaluate the twin volume fractions in deformed samples. Results are reported in Table 1, where it can be seen that after 10% IPC, 90% of the aggregate has twinned. 2.3. Effect of prior slip on subsequent twinning In order to study the effect of dislocation substructure on subsequent deformation dominated by twinning, samples were pre-strained in TTC up to strains of 5% and 10%, and then reloaded in IPC. The stress–strain curves corresponding to these experiments are shown in Fig. 1c. The final texture corresponding to the sample deformed 5% in TTC and then 5% in IPC is shown in Fig. 2. During TTC pre-straining, the material deforms primarily by basal hai and pyramidal hc + ai slip, though there is  compressive twinning as noted by Jiang et al. (2006). likely some f1011g

866

G. Proust et al. / International Journal of Plasticity 25 (2009) 861–880

Measured

Predicted 0.3

Normalized intensity

Normailized Intensity

0.3

0.2

0.1

0.0

0.2

0.1

0.0 0

10

20

30 40 50

60 70

80 90

0

10

angle Initial

20

30 40 50

60 70

80 90

angle 5% TTC

5% TTC+5% IPC

10% IPT

10% IPC

5% IPC +10% TTC

Fig. 3. Comparison of the measured and predicted texture profiles used to estimate the twin volume fractions in the samples deformed by IPC (monotonic or strain-path change) and by IPT. The angle represents the orientation of the c-axis of the various crystals in reference to the normal direction of the plate. The solid black line represents the texture profile of the as-annealed material and is used as the base line in the twin volume fraction calculations.

Fig. 4. Micrographs showing the microstructure of (a) a sample deformed by in-plane compression to a strain of 7% and (b) a sample first deformed by in-plane compression to a strain of 7% and then by through-thickness compression to a strain of 6%.

Table 1 Estimation of the twin volume fraction using the measured and predicted texture profiles shown in Fig. 3 Twin volume fraction

10% IPT 10% IPC 5 IPC + 10% TTC 5% TTC + 5% IPC

Experimental

Prediction

0.07 0.90 0 0.65

0.08 0.86 0 0.72

The IPC reload curve following 5% and 10% TTC pre-load is similar to the monotonic IPC curve: the deformation is still largely accommodated by twinning, as the hardening and texture evolution show, but the onset of twinning happens at a higher stress. The value of the reload yield strength has in-

G. Proust et al. / International Journal of Plasticity 25 (2009) 861–880

867

creased by 40 and 60 MPa, respectively, and there is a Bauschinger-like transition within the first 2% deformation. In addition, both reload curves show a more extended transition from easy to hard deformation, probably as a result of a more balanced twin and slip activity, which also leads to a slower texture evolution. Lou et al. (2007) proposed that dislocation multiplication affects the twin nucleation stress. In this case, during the preloading, hai and hc + ai dislocations are introduced. These dislocations may act as barriers to twin nucleation or/and twin propagation. This would explain both the increase of the stress corresponding to the onset of twinning with the increase in pre-strain, as well as the more extended hardening plateau associated with twinning. 2.4. Detwinning In order to study the effect of twinning on subsequent deformation, samples were pre-strained in IPC up to strains of 5% and 10%, and then reloaded in TTC. The stress–strain curves corresponding to these experiments are shown in Fig. 1d, and the texture corresponding to 5% IPC followed by 10% TTC is shown in Fig. 2. During pre-straining, the material deforms primarily by tensile twinning, but reverses the twins (detwins) during the reload stage. The integration of the texture profiles is consistent with this understanding; Table 1 indicates that the material is twin free after 10% TTC reload. During monotonic TTC, primarily basal hai and pyramidal hc + ai slip are active; however, once twins have been introduced by previous IPC, the hardening behavior of the material reloaded in TTC changes drastically. The reload flow curves display the sigmoidal shape associated to deformation twinning. The reason is that the grains that have twinned during IPC are now properly oriented to twin again (or detwin) during subsequent TTC since the basal poles are roughly at 90° from the TT direction. The reversal of the texture (see Fig. 2) associated with TTC reloads is a strong indication of detwinning. To prove that detwinning is actually happening during this strain-path change, micrographs of the microstructure were taken after 7% IPC (Fig. 4a) and after 7% IPC followed by 6% TTC (Fig. 4b). By comparing Fig. 4a and b, we see that the amount of twins has decreased during TTC reload and though some twin lamellae are still visible in Fig. 4b, there are many grains that are twin free and very few grains are heavily twinned. Detwinning may not be complete because the test was stopped before full strain reversal (Wu et al., in press). Both reloading curves in Fig. 1d are similar to the monotonic IPC stress–strain curve except for the value of the reload yield strength. After 5% IPC pre-strain, the yield stress upon reloading is actually lower than the initial yield for IPC indicating that detwinning is easier to activate than twinning, as noted previously (Lou et al., 2007). As the amount of pre-straining increases, the yield strength increases. This phenomenon could be explained by the fact that slip is activated inside and around the newly created twins and, as the density of dislocations increases, detwinning becomes harder to activate. 3. Polycrystal model The Visco-Plastic Self Consistent (VPSC) polycrystal model is used as a platform for implementing a mesoscopic Composite Grain (CG) model that accounts for twinning evolution inside the grain. The reader can find a detailed description of VPSC and of the recently proposed CG model in our recent paper (Proust et al., 2007), where the model was applied to describe the response of Zr subjected to strain-path changes. Within the VPSC approach, each grain is regarded as a visco-plastic inclusion embedded in and interacting with the visco-plastic effective medium that represents the aggregate. When the medium is subjected to externally imposed loading conditions, the relative stiffness of grain and medium determine the deformation of the former. The strain rate is assumed to be uniform inside the grain, and is accommodated by the shear rates provided by slip and twin systems. The strain rate of the grain, e_ , is related to the shear rates c_ s contributed by slip and twinning systems through a rate sensitive law

e_ ij ¼

X s

msij c_ s ¼ c_ 0

X s

msij



n ms : r

ss

¼ M sec ijkl rkl :

ð1Þ

868

G. Proust et al. / International Journal of Plasticity 25 (2009) 861–880

where r is the stress tensor, and ss is the threshold resolved shear stress associated with the system s. Msec is a linearized visco-plastic compliance tensor (secant approximation) that is accurate for a discrete range of rates and stresses. Its evolution with deformation is a complex function of the slip and twinning activity in the grain and is discussed below. 3.1. The composite grain twin model The CG twin model was introduced to describe the strain-path change behavior of Zr at 76 K (Tomé and Kaschner, 2005; Proust et al., 2007). Differences in how twinning operates in Mg by comparison to Zr, forced us to extend the model and revise some of the assumptions used in Proust et al. (2007). We refer the interested reader to the two papers mentioned above, while here we focus on what is new about the model. Fig. 5 illustrates the characteristics of the CG model. During a deformation simulation, the Predominant Twin System (PTS) is identified in each grain, and a layered structure of twins parallel to the twin plane of the PTS is assumed to form and to evolve with twin activity. The interaction between this composite grain and the surrounding effective medium is characterized by the CG effective mechanical properties. The layers are assumed to be equidistant and two parameters are introduced: the separation dc of the center planes of the lamellae, and the maximum volume fraction of the grain that may PTS . Because twins are assumed (as a first approximation) to pose impenbe reoriented by twinning fmax etrable barriers to dislocations or to other twins, the separation of the twin interfaces is relevant to the hardening response, as we will see below. The first adaptation we have introduced in the CG model to reproduce the behavior of Mg concerns the spacing between the twin lamellae. For the Zr we assumed that the parameter dc was constant throughout the entire deformation, which was in agreement with our experimental data showing that by 30% deformation less than 50% of the material had twinned (Proust et al., 2007). In the case of Mg, the experimental data shows that after 10% deformation almost 90% of the material has twinned, and that twins ‘coalesce’ inside the grain. Therefore, it does not seem appropriate to have twin boundaries acting as barriers when almost the entire grain is transformed by twinning. For this reason, the value of dc is made to increase with the PTS volume fraction until reaching a value of one (which corresponds to having only one twin domain), when 70% of the grain has twinned. By creating this layered twin–matrix structure inside grains, we are able to account for the directional barrier effect created by the twin boundaries. If slip is occurring, inside the matrix or inside the twin, on a plane non-parallel to the twin/matrix interface, the dislocation mean free path is reduced

Fig. 5. Schematics of the CG showing the characteristic lengths used in the model and the evolution of these lengths with the PTS volume fraction when the material twins and detwins. (a) The material has not started to twin and the grain is only constituted of a matrix region, (b) when the material starts to twin, several thin lamellae are created, (c) as the PTS volume fraction increases, some lamellae merge together increasing the mean free path (dmfp) for the dislocation motion inside the twinned domains, and (d) the grain has almost completed twinned and we have now a single twinned region.

G. Proust et al. / International Journal of Plasticity 25 (2009) 861–880

869

due to the barrier effect and that particular slip system will be harder to activate. This idea is implemented in the CG model through a Hall–Petch term that depends on the calculated mean free path for each slip system in the matrix and in the twins. The second change we have introduced in the extended CG twin model concerns the coupling between twin and matrix. In the previous paper presenting the CG model, we introduced two different methods to calculate the deformation of matrix and twin: the coupled and uncoupled twin lamellae. In the former case, continuity of stress and strain is enforced across the twin interface that separates twin and matrix, once the PTS is identified in the grain and the twins are created. In the latter approach, twin and matrix representative ellipsoids are assumed to interact independently with the effective medium, although the volume transfer between matrix and twin and the shape update (due to the evolution of the twin volume fraction, the thickness of the matrix and twin lamellae is changing) are still enforced. To predict the behavior of this Mg alloy we use the coupled deformation scheme at the beginning of deformation, when twins can be regarded as thin lamellae (as in Fig. 4a). When the twin volume fraction inside the grain is higher than 70% and twins coalesce, we transition to the uncoupled scheme. 3.2. Hardening of slip systems Within the CG model, three mechanisms may contribute to the hardening of slip systems inside the matrix and the twin: the evolution of the statistical dislocations with strain, the evolution of geometrically necessary dislocations (GND) and a directional Hall–Petch effect:

ss ¼ ssSTAT þ ssGND þ ssHP :

ð2Þ

Each of these terms is updated incrementally at each straining step. The first term is a classical saturation Voce law associated with statistical dislocations, to which a latent hardening effect has been added:

^s X ss0 s0 ds h Dc ; dC s0    Ch ^s ¼ s0 þ s1 1  exp  0 : where s

DssSTAT ¼

s1

ð3aÞ ð3bÞ

0

Here C is accumulated shear in the grain, Dcs the shear increment in the slip or twinning system s0 0 and hss the latent hardening coefficient, coupling hardening of s due to activity of s0 . While only the barrier effect of the PTS is explicitly accounted for, the other twinning systems can contribute to the hardening of slip systems through the latent hardening parameter. s The second term of Eq. (2) depends on the directional mean free path dmfp defined by the assumed lamellar spacing of the PTS for dislocations on system s. It represents the influence of the GNDs over the threshold stress (Karaman et al., 2000):

DsSGND ¼

HsGND s s dmfp ð STAT þ

s

ssGND Þ

Dcs :

ð4Þ

The last term of Eq. (2) describes the directional Hall–Petch effect:

Hs

HP ssHP ¼ qffiffiffiffiffiffiffiffiffi : s

ð5Þ

dmfp

While the GND contribution is not so relevant at the strains considered here, the Hall–Petch term plays an important role on hardening, especially during strain-path changes. In addition, the Hall– Petch effect introduces length scaling into the model. It has been proposed that originally mobile dislocations will become sessile once the lattice in which they reside has been twinned (Basinski et al., 1997). Basinski et al. further proposed that these ‘‘inherited dislocations” will harden slip inside the twins. However, because the relative effects of these hardening mechanisms have yet to be experimentally differentiated, only the Hall–Petch effect will be used in our model.

870

G. Proust et al. / International Journal of Plasticity 25 (2009) 861–880

3.3. Hardening of twinning systems When the CG twin model was applied to Zr, the hardening law given by Eq. (2) was used to describe the hardening behavior of both slip and twinning systems. While this hardening law is appropriate to describe the hardening associated with dislocation creation and annihilation, in connection to twinning it should only be regarded as an empirical law that provides a dependence of twin hardening with accumulated shear. We verified that, in the case of Mg, it is not possible to adjust the parameters of such law to the experimental hardening behavior observed during strain-path changes. As a consequence, we present here a ‘deformation mode specific’ law to describe the hardening of twinning systems, of the form: TWS

s

¼s

TWS 0

þ

X M

(

s

M

1  exp 

h M CM

sM

!)

;

ð6Þ

where the sum is made over all the slip modes M (basal hai, prism hai, pyramidal hc + ai) that can be activated during deformation, and CM represents the accumulated shear for all the slip systems belonging to the slip mode M. While this new hardening law is empirical, it provides for a selective influence of different deformation modes upon twinning, which is more consistent with the sparse experimental and theoretical information available. For example, it has been proposed by Mendelson (1970) that non-planar dislocation dissociations may be the mechanism of twin nucleation for HCP materials. As a consequence: (a) twin nucleation is delayed until specific dislocation structures develop and (b) such dissociation may be favorable for one type of dislocation but not for another. In what concerns twin growth, atomic scale simulations done for Mg and a-Ti (Serra and Bacon, 1996; Pond et al., 1999) suggest that twin interfaces may propagate (grow) via mixed basal dislocations that, upon arrival at the twin interface, dissociate into dipolar twin dislocations which propagate and advance the interface. As a consequence, twin propagation should be coupled to dislocation activation (Serra and Bacon, 1996). Moreover, during strain-path changes, it was observed that the onset of twinning happens at a higher stress once the material has been deformed by a slip dominated process. Lou and coworkers (Lou et al., 2007) proposed that pyramidal hc + ai dislocations increase the twin nucleation stress. The empirical hardening law that we propose here for tensile twins reflects all these observations. The twinning threshold stress value increases or decreases with the amount of shear accommodated by the various slip modes. The nucleation and growth phases of twinning are simulated by lowering the value of the twinning threshold stress when strain is accommodated by specific slip systems. For this purpose, negative values are given in Eq. (6) to the parameters sM and hM of basal and prism slip. In the case studied here, it is assumed that hai dislocations on basal planes are the most likely to induce twin nucleation and growth, based on atomistic simulations studies (Serra and Bacon, 1996) and the observations that basal slip is easy to activate in most orientations. Tensile twinning and hc + ai dislocations on pyramidal planes are competing mechanisms. However, during strain-path changes, it is possible to have previously induced hc + ai dislocations interact with twins and harden them. In the new hardening law it is possible to describe this phenomenon by giving positive values to the parameters sM and hM. Fig. 6b shows the evolution of the twinning threshold stress with the amount of shear accommodated by each slip mode. The determination of the hardening parameters associated with these curves is explained in Section 4 of this paper. The same hardening laws are applied in the twinned regions to predict the evolution of the slip and twinning threshold stresses as in the matrix. For slip, the same hardening parameters are used in the matrix and in the twin. However, the experimental evidence suggests that detwinning is easier than twinning (Lou et al., 2007); therefore, the hardening parameters, which describe the threshold stress evolution for twinning are different inside the matrix and inside the twins (detwinning), as discussed below. 3.4. Detwinning It has been experimentally observed for Mg that when the loading direction is changed, grains that had previously twinned can detwin easily (Caceres et al., 2003; Kleiner and Uggowitzer, 2004; Lou

871

250

250

200

200

τTWS (MPa)

τS (MPa)

G. Proust et al. / International Journal of Plasticity 25 (2009) 861–880

150

100

Basal Prismatic Pyramidal

50

0 0.00

0.05

0.10

0.15

Basal Prismatic Pyramidal

150

100

50

0.20

0 0.00

0.05

Γ

0.10

0.15

0.20

ΓM

Fig. 6. (a) Evolution of the threshold stress ss with the total shear strain C for the three slip modes, (b) influence of the three slip modes on the twinning threshold stress sTWS. CM represents the shear strain associated with the slip mode M.

et al., 2007; Wang and Huang, 2007). The detwinning mechanism may require less stress to be activated because, since the twin already exists, no nucleation is necessary. In addition, back-stresses engendered by the twin growth, may aid the detwinning process (Wu et al., in press). To model this mechanism we favor the activation of the PTS inside the twin by setting a high value, upon reloading, of the CRSS’s of the other twin systems inside the twinned region. Such procedure prevents them from being activated upon reloading. Once the PTS has been activated inside the twin, instead of creating a secondary twin, the volume of the twin that should be occupied by this secondary twin is transferred from the original twin to the matrix. This process can continue until the entire twin volume has been transferred back to the initial grain, at which point the grain is twin free. 4. Application of the CG twin model to AZ31B Mg 4.1. Hardening parameters We determine one set of hardening parameters, given in Table 2, to reproduce all our experimental data. The single crystal hardening parameters described in the previous section are obtained for each deformation mode by fitting the experimental stress–strain curves obtained for monotonic TTC, IPC and IPT and one of the IPC and TTC reloading experiments (cf. Fig. 7). The validity of these parameters is confirmed by verifying that the predicted deformed textures correspond to the measured ones and, when experimental data is available, that the observed deformation modes are predicted. Moreover, we used the second TTC and IPC reloading curves as well as experimental data for monotonic compression and tension realized parallel to the transverse direction of the initial plate (these curves

Table 2a Single crystal hardening parameters for AZ31B Mg deformed at room temperature, basal hai, prismatic hai and pyramidal hc + ai slip parameters

Basal Prism Pyra

s0 (MPa)

s1 (MPa)

h0 (MPa)

HGND (MPa lm)

HHP (MPa lm1/2)

Latent hardening Basal

Prism

Pyra

2 60 50

52 60 70

3000 600 2200

105 0 0

100 100 380

1 1 1

2 1 1

35 1 1

872

G. Proust et al. / International Journal of Plasticity 25 (2009) 861–880

Table 2b Single crystal hardening parameters for AZ31B Mg deformed at room temperature, tensile twinning parameters in the matrix and in the twins

In matrix In twins

s0 (MPa)

sbasal (MPa)

hbasal (MPa)

sprism (MPa)

hprism (MPa)

spyra (MPa)

hpyra (MPa)

dc

fmax

55 20

30 0

15,000 0

5 0

20,000 0

50 20

1200 100

0.25 0.25

0.9 0.9

are not shown in this paper). The last verification was made by comparing the amount of twinning measured and predicted by our model. The hardening parameters associated with the three slip modes are determined using the monotonic TTC and IPC flow stress curves. The Hall–Petch coefficients describing the interactions between these slip modes and twins are selectively given by the stress–strain curves obtained during in-plane deformation: during IPT, the small volume fraction of twins which form are oriented favorably to activate prismatic hai, and during IPC, the twins are oriented favorably to activate pyramidal hc + ai slip and significant basal hai slip also occurs. The IPC stress–strain curve is also used to determine most of the hardening parameters associated with tensile twinning. However, we also need to use one of the IPC reload experiment to determine the parameters describing the influence of pyramidal slip on tensile twinning. The last set of parameters to be determined corresponds to the detwinning mechanism and these parameters are fitted using the TTC reload data. The evolution of the threshold stresses with the shear strain accumulated by each slip mode can be seen in Fig. 6a. The initial CRSS values of all the slip modes are given by adding s0 and the Hall–Petch factor corresponding to our initial grain size of 13 lm. For our simulations, the starting CRSS ratios between prismatic and basal slip and between pyramidal and basal slip are 2.9 and 5.2, respectively. These values respect the general trend reported in the literature that shows that the easiest slip mode is basal hai and the hardest one is pyramidal hc + ai (Barnett et al., 2006; Clausen et al., 2008; Jain and Agnew, 2007). The evolution of the twinning threshold stress with the shear accommodated by each slip mode is shown in Fig. 6b. The new hardening law that we have implemented for twinning has two advantages: (1) we can simulate a nucleation phase by assigning a high initial threshold stress for twinning and have this value decrease once some dislocations have been introduced in the material, (2) we can describe the hardening effect of specific dislocations onto twin propagation. The initial value of CRSS for tensile twinning is 55 MPa and that value decreases rapidly when basal and/or prismatic hai dislocations are introduced in the material due to the negative values of sM and hM for both basal and prismatic slip (basal slip affects the twinning threshold stress more than prismatic slip). The values of sM and hM for pyramidal hc + ai are positive, which leads to the hardening of twinning due to the presence of these dislocations. A similar approach is used to reproduce the detwinning mechanism. In the matrix, the twinning hardening parameters associated to basal and prismatic slip were used to simulate a twinning nucleation phase. As this nucleation phase is non-existent in the case of detwinning (Lou et al., 2007), the twinning (or more appropriately detwinning) hardening parameters inside the twins due to basal and prismatic slip are taken as 0. Thus, it is assumed that these two slip modes do not influence detwinning. However, the presence of pyramidal dislocations inside the twins is likely to inhibit detwinning so we have used relatively large values to describe the influence of pyramidal slip onto detwinning. Moreover, it has been shown experimentally that the CRSS for detwinning is equal (Wang and Huang, 2007) or slightly smaller (Lou et al., 2007) than for twinning. In our model, the initial threshold , corresponds to the saturation threshold stress for twinning (Fig. 6b) once value for detwinning, sDETWS 0 basal and prismatic slip effects have been completely accounted for, i.e.

sDETWS ¼ sTWS þ sbasal þ sprism ; 0 0 basal

prism

ð7Þ

where s s and s represent the hardening parameters described in Eq. (6) and associated with twinning inside the matrix. The reader is reminded that the latter two terms are negative, thus the detwinning stress is, in general, lower than the twinning. TWS , 0

873

G. Proust et al. / International Journal of Plasticity 25 (2009) 861–880

300 200

Experiment Simulation

100

0.05

0.10

0.15

MODE ACTIVITY IN MATRIX

STRESS (MPa)

0.6 0.4 0.2 0.0 0.00

0.05

0.10

300 200 100

0.05

0.10

0.15

0.6 0.4 0.2 0.05

0.10

200 100 0.05

0.10

0.15

0.2 0.05

0.10

100 0.05

0.10

0.15

0.6 0.4 0.2 0.05

0.10

MODE ACTIVITY IN MATRIX

200 100

0.05 0.10 STRAIN

0.15

0.05

0.10

0.15

0.05

0.10

0.15

0.05

0.10

0.15

0.05 0.10 STRAIN

0.15

0.4 0.2

0.8 0.6 0.4 0.2

0.8 0.6 0.4 0.2

1.0

0.8 0.6 0.4 0.2 0.0 0.00

0.15

0.6

0.0 0.00

0.15

1.0

300

0.10

1.0

0.8

0.0 0.00

0.05

0.8

0.0 0.00

0.15

MODE ACTIVITY IN TWINS

MODE ACTIVITY IN MATRIX

STRESS (MPa)

200

400

STRESS (MPa)

0.4

1.0

300

0 0.00

0.6

0.0 0.00

400

0 0.00

0.8

MODE ACTIVITY IN TWINS

0 0.00

0.2

1.0

MODE ACTIVITY IN TWINS

MODE ACTIVITY IN MATRIX

STRESS (MPa)

300

0.4

0.0 0.00

0.15

1.0 400

0.6

Twin Vol Frac Prismatic Basal Pyramidal Tensile Twinning

1.0

0.8

0.0 0.00

0.8

0.0 0.00

0.15

1.0

400

0 0.00

0.8

MODE ACTIVITY IN TWINS

0 0.00

1.0

MODE ACTIVITY IN TWINS

1.0

MODE ACTIVITY IN MATRIX

STRESS (MPa)

400

0.05 0.10 STRAIN

0.15

0.8 0.6 0.4 0.2 0.0 0.00

Fig. 7. Comparison between the experimental (symbols) and predicted (solid lines) stress–strain curves, predicted deformation mode activities in matrix and in twins (the prismatic activity is represented by open squares, the basal activity by open triangles, the pyramidal activity by crosses and the tensile twinning activity by solid circles. The solid line in the activity plot in the twins represents the evolution of the twin volume fraction with strain) for (a) monotonic TTC, (b) monotonic IPT, (c) monotonic IPC, (d) TTC followed by IPC, and (e) IPC followed by TTC.

4.2. Monotonic deformation Predicted stress–strain curves and textures for monotonic TTC, IPT and IPC are shown in Figs. 7 and 2, respectively, where they are compared with the experimental data. The predicted behavior of the Mg alloy AZ31B was obtained using the set of hardening parameters given in Table 2. In Fig. 7 we also report the predicted relative deformation mode activities in the matrix and in the twinned regions, and the volume fraction of the twinned material. The relative activity is defined as the ratio of the mode activity and the total activity. The former is the sum over all grains of the shear rates contributed

874

G. Proust et al. / International Journal of Plasticity 25 (2009) 861–880

by a given deformation mode, weighted by the grain volume fraction. The latter is the sum of all mode activities. During TTC, the strain is accommodated by two slip modes, basal hai and pyramidal hc + ai, and twinning is not activated (as explained earlier, compression twinning and double twinning were not considered in these simulations), as can be seen in Fig. 7a. This activity ‘sharpens’ the main basal component and does not change the texture of the material substantially after 5% deformation. Fig. 2 shows that the measured and predicted textures are very similar to the initial texture. IPT is mostly accommodated by prismatic and basal hai slip in the matrix (see Fig. 7b) along with a small contribution of tensile twinning (Jain et al., 2008). The effects of twinning can be observed in both the measured and predicted basal pole figures (Fig. 2) where some of the grains have reoriented with their c-axis parallel to the transverse direction (TD). Notably, the model predicts that 8% of the material is reoriented by twins, which is in agreement with the experimental value of 7% obtained from integrating the texture profile. There is, however, a larger spread of the twin orientations in the measured texture which explains the difference in the pole figure intensities between experiment and simulation. At the beginning of IPC, two deformation modes are active in the matrix: basal slip and tensile twinning (Fig. 7c), with twinning increasing its contribution until about 3% strain, while the basal activity decreases. After 5% strain, the model predicts that about 70% of the aggregate has reoriented by twinning, and that strain is then mostly accommodated by ‘hard’ hc + ai pyramidal slip inside the twins. The initial threshold stress associated with the slip modes inside the twins are high due to the Hall–Petch effect. But once the twin lamellae grow and merge together to create thicker regions inside the grains, the CRSS associated with pyramidal slip decreases (due to the reduction of the Hall–Petch term in the threshold stress expression) and the twin can plastically deform. At 10% strain, the model predicts that the twins represent 86% of the material (the experimental value was estimated at 90%) so most of the deformation is now accommodated by hard modes within the twins, which explains the drastic increase in the flow stress produced by a change from easy deformation of the matrix to hard deformation of the twins. In addition to this basic texture hardening effect, the CG model is able to connect some of the rapid hardening with the directional Hall–Petch effect, as discussed above. Fig. 2 shows that the predicted texture after 10% IPC is in good agreement with the measured texture. Most of the grains have been reoriented such that their c-axis is now parallel to the compression direction. 4.3. Effect of prior slip on subsequent twinning Using the same set of hardening parameters, we were able to predict the TTC followed by IPC strain-path change response (see Fig. 7d for flow stress curves and predicted mode activities in matrix and twins). During the pre-loading stage of the deformation, basal hai and pyramidal hc + ai dislocations are introduced in the material. The hc + ai dislocations experience a low resolved shear stress (are less mobile) once we change the direction of loading and, in addition, they may act as barriers to twin nucleation and propagation, as suggested by Lou et al. (2007) and the hardening parameters in Table 2b reflect that the more hc + ai dislocations in the material, the harder tensile twinning becomes. In Fig. 8 we have plotted the experimental and predicted stress–strain curves for two different amounts of pre-straining. The CG model captures the increase in the reloading yield strength with pre-straining: for monotonic IPC the predicted yield strength is equal to 100 MPa, after 5% TTC the predicted IPC reload yield strength is 135 MPa and after 10% TTC, 160 MPa. These values match the experimental findings. However, as this model does not include back-stresses (or kinematic hardening), we are not able to capture the initial reversal behavior of this material, which may be termed a generalized Bauschinger effect. The comparison of the experimental monotonic and reload IPC stress–strain curves reveals that not only the yield strength changes but also the length of the initial plateau and the slope of the curve before reaching the saturation stress (Fig. 8). These differences seem to be due to a change in the deformation modes activated during the IPC reload. The transition from easy to hard deformation is less abrupt once the material has been pre-strained. As we have shown earlier, this transition from easy to hard deformation is due to the transition from twinning accommodating strain in the matrix to slip

G. Proust et al. / International Journal of Plasticity 25 (2009) 861–880

875

400

Stress (MPa)

300

200

100

0 0.00

Experiments Simulations

0.05

0.10

0.15

0.20

Strain Fig. 8. Effect of the amount of pre-strain on twinning during IPC reloads. The grey curves represent the results for 5% TTC prestrain and the black curves the results for 10% TTC pre-strain.

accommodating strain in the twinned regions. The fact that this transition is less abrupt during reloading suggests that while twinning is occurring in the matrix, another deformation mode is also accommodating strain either in the matrix or in the twins (Morozumi et al., 1976), explaining why twinning is occurring over a larger strain. However, our model does not predict such behavior. The matrix and twin activity plots for this strain-path change show that the matrix deforms mostly by tensile twinning and after 4% reload, pyramidal hc + ai slip in the twins dominate the deformation. This activity pattern is similar as the one obtained during monotonic IPC. The measured and predicted textures are analogous, and the measured and predicted twin volume fractions at the end of deformation are 65% and 72%, respectively. While the final values compare well with the experiment, the stress evolution indicates that twins grow at a faster rate in the model. This proves that our model does not yet capture all the consequences of pre-straining onto the mechanical behavior of the material. Twinning should be slowed down while another deformation mode would complement it to accommodate strain. However, to adapt the model, one will need more experimental information concerning which deformation modes are active during reloading. 4.4. Detwinning This paper presents a first attempt to predict the detwinning mechanism during strain-path changes. In Fig. 7e, the stress–strain curve and predicted activities in the matrix and in the twins are displayed for a sample that has been first subjected to 5% strain in IPC and then 10% strain in TTC. After the pre-loading phase of the test, the texture of the sample has changed drastically and our model predicts that 72% of the material has twinned. As a consequence the TTC reloading behavior of the material is totally different from the monotonic behavior. The reloading yield strength drops from 180 to 90 MPa and the hardening evolution is characteristic of a material deforming by twinning instead of slip, as it is the case for the monotonic TTC deformation. Moreover, microscopy shows that the twins created during the pre-loading phase disappear during re-loading (see Fig. 4b). This was also observed in the study by (Lou et al., 2007). Detwinning occurs when the twin system that is active inside a twin shares the same twin plane with the PTS that originally created the twin. As shear is accommodated by that twin system inside the twin, the twin transforms ‘back’ into the matrix orientation and ‘shrinks’, while the volume fraction of the matrix increases. Our model allows for the twin to disappear completely, as can be seen in the twin fraction evolution shown in Fig. 7e. Once we change the direction of loading, there is twinning activity in the twins and the twin volume fraction decreases until it becomes equal to zero after roughly 4% strain, at which point all the deformation is accommodated by basal hai and pyramidal hc + ai slip in the matrix. for tensile twinning in the twins obtained from Eq. (7), the CG model can Using the value of sDETWS 0 capture the TTC reloading yield strength. We can also reproduce the rapid increase in the hardening

876

G. Proust et al. / International Journal of Plasticity 25 (2009) 861–880

rate that corresponds to the matrix starting to deform. But, again, the model does not reproduce the long initial plateau present in the experimental stress–strain curve after reload. In the case of monotonic IPC, the CG model predicts the plateau by activating easy basal slip in the matrix, in addition to twinning, while slip activity in the twins starts after 4.5% strain and leads to the observed rapid change in the flow stress. During TTC reloading, on the other hand, the model predicts that the matrix starts deforming by basal slip as soon as detwinning has started (see activity plot in Fig. 7e). But the CRSS for basal slip in the matrix is higher than the CRSS for tensile twinning in the twins because of the Hall– Petch effect. As a consequence, the macroscopic stress rapidly reaches the saturation value because basal slip has become now a hard deformation mode. After 4% reload strain, the model predicts that the material is totally detwinned, and that the deformation is now accommodated by basal hai and pyramidal hc + ai in the matrix since, after detwinning, the texture is similar to the initial texture, and the material deforms by activating the same deformation modes as during monotonic TTC. Fig. 2 shows the measured and predicted textures for the material deformed 5% by IPC followed by 10% TTC. The predicted texture reproduces the general experimental features; specifically, it shows the reorientation of the grains such that their c-axis are back to be almost parallel to the normal direction of the plate. We attribute the predicted ‘depletion’ at the center of the basal pole figure, to excessive pyramidal hc + ai slip activity in the twins during pre-loading. This rotates the twinned domains in such a way that, upon detwinning, the c-axis is reoriented away from its original position.

5. Discussion Jain and Agnew (2006) made a first attempt to predict the strain-path change behavior of the AZ31B magnesium alloy using the VPSC model. In their simulations, they treated twinning using the PTR scheme that accounts for twin reorientation but does not incorporate the microstructure features inherent to the presence of twins in the grains. Moreover, twinning was not assumed to harden due to the presence of dislocations. Although they obtained good predictions for the monotonic deformation of the material, the strain-path change predictions were far from the experimental data. The model they used showed the role played by texture evolution on the response of the material but did not capture the different hardening evolution due to the presence of dislocation networks and twins in the material. By incorporating the CG twin model in the VPSC polycrystal model and by defining a new hardening law for twinning, we have dramatically improved the predictions of the hardening and texture evolutions during strain-path change. Although the Hall–Petch parameters that we are characterizing in our model (see Table 2) correspond to the barrier effect associated with twin boundaries, we used our model to predict the monotonic IPT response of the same alloy with different initial grain sizes. During IPT, most of the strain needs to be accommodated by prismatic slip so the Hall–Petch effect observed experimentally (cf. Fig. 9) is mainly caused by the hardening of prismatic slip due to a difference in grain size. Experimentally, the overall Hall–Petch coefficient for Mg alloy, AZ31B, was determined to be equal to 200 MPa lm1/2 (Jain et al., 2008) while our fitting process gave a value of 100 MPa lm1/2 for prism hai slip. (Given the initial texture and loading direction during IPT, the Schmid factor for prismatic slip is near maximum, m  2, so this is viewed as very good agreement.) With this value of the Hall–Petch parameter for prismatic slip, we were able to closely reproduce the stress–strain response of the same Mg alloy having initial grain sizes of 42 and 89 lm as shown in Fig. 9. The new hardening model for twinning that we introduce in this paper has been guided by experimental observations. It has been observed that twinning in polycrystalline magnesium does not start from the onset of deformation but some dislocations are introduced in the material before twins are observed (Agnew et al., 2003; Brown et al., 2007; Clausen et al., 2008). It is likely that twin nucleation requires stress concentration associated with dislocation structures, such as pile-ups. Our model captures this phenomenon by having a high initial value for the tensile twinning CRSS and then decreasing the CRSS once basal (or prismatic) slip has accommodated the initial strain. On the matrix activity plot of Fig. 7c, one can see that at the start of the deformation the activity of basal slip is higher than the activity of tensile twinning but rapidly the situation is reversed and twinning accommodates most of the deformation after 1% strain. Work is in progress to understand the mechanism of twin

G. Proust et al. / International Journal of Plasticity 25 (2009) 861–880

877

Stress (MPa)

300

200

100

0 0.00

Experiments 13 microns 42 microns 89 microns 0.02

0.04

Simulations 13 microns 42 microns 89 microns 0.06

0.08

0.10

strain Fig. 9. Hall–Petch effect due to the initial grain size during IPT. Comparison of the experimental and predicted stress–strain curves.

nucleation in hcp metals (Capolungo and Beyerlein, in press) and to incorporate this understanding into VPSC. The second experimental observation that we have used concerns the IPC reloading yield strength increase after the material has been pre-strained in TTC and this observation suggests that pyramidal hc + ai dislocations affect the twin nucleation stress (Lou et al., 2007). In our model, we tested this hypothesis by hardening tensile twinning once pyramidal hc + ai slip has started to accommodate the deformation. This particular feature does allow us to predict the higher IPC yield strength at reloading. Again, it is admitted that there may be a role of {1 0 1 1} compression twinning, which has been observed in Mg alloys (Koike, 2005; Jiang et al., 2006; Barnett, 2007b) but was ignored in the present modeling. Although our model gives reasonable predictions for the strain-path change stress–strain curves, it does not yet capture all the observed phenomena. For example, in the case of TTC followed by IPC (see Fig. 8), the model fails to capture the initial rapid hardening rate (this is especially noticeable after the material has been pre-strained 10% in TTC), the longer initial plateau and the softer increase in the hardening rate observed during the reloads in comparison with the monotonic response. The initial yield of the reload response is believed to be due to a Bauschinger effect due to basal dislocation reversal, which is not accounted for by the model. A version of the VPSC model incorporating cut-through of planar dislocation walls and dislocation-based reversal mechanism has been developed for copper (Beyerlein and Tomé, 2007). It is suggested that future research should seek to extend such dislocation density-based models developed for fcc materials to hcp deformation. In the present work, we neglected the initial 2% strain of the reloading curves while simulating the stress–strain curves. Finally, the experimentally observed longer plateau and softer hardening during IPC reloads seem to be tied to the activity of another deformation mode, either in the matrix or in the twins, complementing the twinning activity. However, there is no experimental data currently available to test this hypothesis and no combination of model parameters was capable of reproducing this aspect of the flow curves. The validity of the model was put to a final test by comparing the experimental and predicted evolutions of the hardening rate as a function of stress (Fig. 10). During monotonic TTC and IPT, the hardening rate decreases with stress, which is typical of slip deformation. Our model quantitatively predicts the same behavior. During the three other experiments, the hardening rate starts by increasing drastically, reaches a maximum at an equivalent stress of about 125 MPa, and then decreases rapidly. Such behavior is typical of twinning dominated strain accommodation, and the monotonic IPC shows the greatest hardening rate. While our model fails to reproduce the magnitude of the maximum hardening rate associated with monotonic IPC, the predicted IPC stress–strain curve is very close to the experimental data and the principal features associated with twinning are captured by this model. Concerning the reload experiments, the initial slope of the hardening rate is lower and the maximum smaller than for monotonic IPC. Our model does not entirely capture this behavior, which we attribute to the influence that the pre-straining state of internal stress has on the reloading hardening rate. For example: the predicted evolution of the hardening rate during IPC reload is almost identical

878

G. Proust et al. / International Journal of Plasticity 25 (2009) 861–880

Simulations 20000

15000

15000

TTC IPC reload IPC

IPT reload TTC

Theta

Theta

Experimental 20000

10000

10000

5000

5000

0

0 0

50

100

150

200

250

Equivalent Stess (MPa)

300

0

50

100

150

200

250

300

Equivalent Stess (MPa)

Fig. 10. Comparison of the experimental and predicted hardening rates for the five stress–strain curves shown on Fig. 7. For the strain-path change experiments only the hardening rate during the reloading phase of the experiments are plotted in this graph and are labeled reload IPC and reload TTC. The y-axis represents the hardening rate (dr/de) and the x-axis represents the equivalent stress (r  ry).

to the monotonic prediction instead of being ‘softer’. The maximum value is obtained for the same equivalent stress. This means that most of the material is predicted to have twinned at the same stress as during monotonic IPC. Experimentally, it is observed that the maximum is reached at a higher value of stress, showing that twinning is delayed or happens over a longer period during the reload. We believe that this is additional proof that another mechanism has to be active at the beginning of the reloading experiment to counterbalance the effect of twinning on the hardening rate. Similarly, we see that the initial increase in the predicted hardening rate of TTC reload is steeper than for monotonic IPC and that the maximum value is obtained for a lower equivalent stress. These two observations contradict the experimental evidence. The effect of detwinning seems to be counterbalanced by the activation of an additional deformation mechanism, which our model does not predict. In this work we present a simple model to predict the detwinning behavior of the material. The twinned region of the grain that has been created during the loading phase of the deformation is allowed to shrink until it disappears totally and the grain is then constituted entirely of matrix. We are able to capture the macroscopic yield strength during reloading but the extended plateau corresponding to detwinning is not well-predicted. The present CG twin model, only one twin system, the PTS, creates a twinned region in the grain. The other twin systems are allowed to accommodate deformation by shear but not to crystallographically reorient the matrix. When the loading direction is changed, this model only allows the domain associated with the PTS to detwin and, furthermore, suppresses other twin activity (secondary twinning) inside the PTS. Since only one twin system is not sufficient to accommodate axial compression, other systems need to be active either in the matrix or in the twins. Our model predicts that from the onset of reloading there is twinning activity in the twins and basal slip activity in the matrix. Due to the Hall–Petch and latent hardening parameters associated with basal slip, the CRSS of the basal slip mode in the matrix is very high, which explains the predicted rapid increase in the flow stress after only 1.5% reload strain. This paper points out the necessity of incorporating in polycrystalline models detwinning as a deformation mechanism to predict complex loadings of Mg alloys. To better understand this phenomenon more experimental evidence is necessary. For example it could be interesting to realize in situ TEM experiments on pre-strained samples to see how twins react when the loading conditions are reversed. Acknowledgements This work was supported by the Office of Basic Energy Sciences, Project FWP 06SCPE401. This material is based in part upon work supported by the National Science Foundation under Grant No. DMI-0322917. The authors wish to thank Rupalee Mulay for the optical micrographs.

G. Proust et al. / International Journal of Plasticity 25 (2009) 861–880

879

References Agnew, S.R., Duygulu, O., 2005. Plastic anisotropy and the role of non-basal slip in magnesium alloy AZ31B. International Journal of Plasticity 21 (6), 1161. Agnew, S.R., Brown, D.W., Tomé, C.N., 2006. Validating a polycrystal model for the elasto-plastic response of magnesium alloy AZ31 Using in-situ neutron diffraction. Acta Matererialia 54 (18), 4841–4852. Agnew, S.R., Tomé, C.N., Brown, D.W., Holden, T.M., Vogel, S.C., 2003. Study of slip mechanisms in a magnesium alloy by neutron diffraction and modeling. Scripta Materialia 48 (8), 1003. Agnew, S.R., Yoo, M.H., Tomé, C.N., 2001. Application of texture simulation to understanding mechanical behavior of Mg and solid solution alloys containing Li or Y. Acta Materialia 49 (20), 4277. Ando, S., Tonda, H., 2000. Non-basal slips in magnesium and magnesium–lithium alloy single crystals. Materials Science Forum 350-351, 43–48. Avedesian, M.M., Baker, H. (Eds.), 1999. Magnesium and Magnesium Alloys. ASM Speciality Handbook. Barnett, M.R., 2007a. Twinning and the ductility of magnesium alloys. Part I: Tension twins. Materials Science and Engineering A, Structural Materials 464 (1–2), 1–7. Barnett, M.R., 2007b. Twinning and the ductility of magnesium alloys. Part II. Contraction twins. Materials Science and Engineering A, Structural Materials 464 (1–2), 8. Barnett, M.R., Keshavarz, Z., Ma, X., 2006. A semianalytical Sachs model for the flow stress of a magnesium alloy. Metallurgical and Materials Transactions A, Physical Metallurgy and Materials Science 37A (7), 2283. Basinski, Z.S., Szczerba, M.S., Niewczas, M., Embury, J.D., Basinski, S.J., 1997. Transformation of slip dislocations during twinning of copper–aluminum alloy crystals. Revue de Metallurgie Cahiers D’Informations Techniques 94 (9), 1037–1044. Beyerlein, I.J., Tomé, C.N., 2007. Modeling transients in the mechanical response of copper due to strain path changes. International Journal of Plasticity 23 (4), 640–664. Brown, D.W., Jain, A., Agnew, S.R., Clausen, B., 2007. Twinning and detwinning during cyclic deformation of Mg alloy AZ31B. Materials Science Forum 539–543 (4), 3407. Caceres, C.H., Sumitomo, T., Veidt, M., 2003. Pseudoelastic behaviour of cast magnesium AZ91 alloy under cyclic loading– unloading. Acta Materialia 51, 6211–6218. Capolungo, L., Beyerlein, I.J., in press. Nucleation and stability of twins in hcp metals. Physical Review B. Cherkaoui, M., 2003. Constitutive equations for twinning and slip in low-stacking-fault-energy metals: a crystal plasticity-type model for moderate strains. Philosophical Magazine 83 (31-34), 3945–3958. Cherkaoui, M., Berveiller, M., Sabar, H., 1998. Micromechanical modeling of martensitic transformation induced plasticity (TRIP) in austenitic single crystals. International Journal of Plasticity 14 (7), 597–626. Chino, Y., Kimura, K., Hakamada, M., Mabuchi, M., 2008. Mechanical anisotropy due to twinning in an extruded AZ31 Mg alloy. Materials Science and Engineering A, Structural Materials 485, 311–317. Christian, J.W., Mahajan, S., 1995. Deformation twinning. Progress in Materials Science 39 (1–2), 1–157. Clausen, B., Tomé, C.N., Brown, D.W., Agnew, S.R., 2008. Reorientation and stress relaxation due to twinning: modeling and experimental characterization for Mg. Acta Materialia 56 (11), 2456–2468. Jain, A., Agnew, S.R., 2006. Effect of twinning on the mechanical behavior of a magnesium alloy sheet during strain path changes. Magnesium Technology 2006, 219. Jain, A., Agnew, S.R., 2007. Modeling the temperature dependent effect of twinning on the behavior of magnesium alloy AZ31B sheet. Materials Science and Engineering A, Structural Materials 462 (1–2), 29. Jain, A., Duygulu, O., Brown, D.W., Tomé, C.N., Agnew, S.R., 2008. Grain size effects on the tensile properties and deformation mechanisms of a magnesium alloy, AZ31B, sheet. Materials Science and Engineering A 486 (1–2), 311–317. Jiang, L., Jonas, J.J., Luo, A.A., Sachdev, A.K., Godet, S., 2006. Twinning-induced softening in polycrystalline AM30 Mg alloy at moderate temperatures. Scripta Materialia 54, 771–775. Jiang, L., Jonas, J.J., Mishra, R.K., Luo, A.A., Sachdev, A.K., Godet, S., 2007. Twinning and texture development in two Mg alloys subjected to loading along three different strain paths. Acta Materialia 55 (11), 3899. Kalidindi, S.R., 2001. Modeling anisotropic strain hardening and deformation textures in low stacking fault energy fcc metals. International Journal of Plasticity 17 (6), 837–860. Karaman, I., Sehitoglu, H., Beaudoin, A.J., Chumlyakov, Y.I., Maier, H.J., Tomé, C.N., 2000. Modeling the deformation behavior of Hadfield steel single and polycrystals due to twinning and slip. Acta Materialia 48, 2031–2047. Kaschner, G.C., Bingert, J.F., Liu, C., Lovato, M.L., Maudlin, P.J., Stout, M.G., Tomé, C.N., 2001. Mechanical response of zirconium – II. Experimental and finite element analysis of bent beams. Acta Materialia 49, 3097–3108. Kelley, E.W., Hosford, W.F.J., 1968. The deformation characteristics of textured magnesium. Transactions of the Metallurgical Society of AIME 242, 654–660. Kleiner, S., Uggowitzer, P.J., 2004. Mechanical anisotropy of extruded Mg–6% Al–1% Zn alloy. Materials Science and Engineering A, Structural Materials 379 (1–2), 258. Klimanek, P., Potzsch, A., 2002. Microstructure evolution under compressive plastic deformation of magnesium at different temperatures and strain rates. Materials Science and Engineering A, Structural Materials A324 (1–2), 145. Koike, J., 2005. Enhanced deformation mechanisms by anisotropic plasticity in polycrystalline Mg alloys at room temperature. Metallurgical and Materials Transactions A 36A, 1689–1696. Kubler, R., Berveiller, M., Cherkaoui, M., Inal, K., 2003. Transformation textures in unstable austenitic steel. Journal of Engineering Materials and Technology 125, 12–17. Lebensohn, R.A., Tomé, C.N., 1993. Self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: application to zirconium alloys. Acta Metallurgica et Materialia 41 (9), 2611–2624. Lou, X.Y., Li, M., Boger, R.K., Agnew, S.R., Wagoner, R.H., 2007. Hardening evolution of AZ31B Mg sheet. International Journal of Plasticity 23 (1), 44. Mann, G.E., Sumitomo, T., Caceres, C.H., Griffiths, J.R., 2007. Reversible plastic strain during cyclic loading-unloading of Mg and Mg–Zn alloys. Materials Science and Engineering A, Structural Materials 456 (1–2), 138. Mendelson, S., 1970. Dislocation dissociations in hcp metals. Journal of Applied Physics 41 (2), 1893–1910.

880

G. Proust et al. / International Journal of Plasticity 25 (2009) 861–880

Morozumi, S., Kikuchi, M., Yoshinaga, H., 1976. Electron microscope observation in and around (1 1 0 2) twins in magnesium. Transactions of the Japan Institute of Metals 17 (3), 158–164. Obara, T., Yoshinga, H., Morozumi, S., 1973. {1 1 2 2}h1 1 2 3i slip system in magnesium. Acta Metallurgica 21, 845–853. Philippe, M.J., Serghat, M., Van Houtte, P., Esling, C., 1995. Modelling of texture evolution for materials of hexagonal symmetry – II. Application to zirconium and titanium a or near a alloys. Acta Metallurgica et Materialia 43 (4), 1619–1630. Pond, R.C., Serra, A., Bacon, D.J., 1999. Dislocations in interfaces in the hcp metals – II. Mechanisms of defect mobility under stress. Acta Materialia 47 (5), 1441–1453. Proust, G., Tomé, C.N., Kaschner, G.C., 2007. Modeling texture, twinning and hardening evolution during deformation of hexagonal materials. Acta Materialia 55, 2137–2148. Salem, A.A., Kalidindi, S.R., Doherty, R.D., 2003. Microstructure evolution and strain hardening mechanisms in titanium. Acta Materialia. Salem, A.A., Kalidindi, S.R., Semiatin, S.L., 2005. Strain hardening due to deformation twinning in a-titanium: constitutive relations and crystal-plasticity modeling. Acta Materialia 53, 3495–3502. Schmid, E., Boas, W., 1968. Plasticity of Crystals. Chapman & Hall Ltd., London. pp. 55–76. Serra, A., Bacon, D.J., 1995. Computer simulation of screw dislocation interactions with twin boundaries in hcp metals. Acta Metallurgica et Materialia 43 (12), 4465–4481. Serra, A., Bacon, D.J., 1996. A new model for {1 0 1 2} twin growth in hcp metals. Philosophical Magazine A 73 (2), 333–343. Serra, A., Bacon, D.J., Pond, R.C., 2002. Twins as barriers to basal slip in hexagonal-close-packed metals. Metallurgical and Materials Transactions A 33, 809–812. Staroselsky, A., Anand, L., 2003. A constitutive model for hcp materials deforming by slip and twinning: application to magnesium alloy AZ31B. International Journal of Plasticity 19 (10), 1843. Stohr, J.-F., Poirier, J.-P., 1972. Etude en microscopie electronique du glissement pyramidal {1 1 2 2}h1 1 2 3i dans le magnesium. Philosophical Magazine 25 (6), 1313–1329. Tomé, C.N., Kaschner, G.C., 2005. Modeling texture, twinning and hardening evolution during deformation of hexagonal materials. Materials Science Forum, 1001–1497. Tomé, C.N., Lebensohn, R.A., Kocks, U.F., 1991. A model for texture development dominated by deformation twinning: application to zirconium alloys. Acta Metallurgica et Materialia 39 (11), 2667–2680. Tomé, C.N., Maudlin, P.J., Lebensohn, R.A., Kaschner, G.C., 2001. Mechanical response of zirconium – I. Derivation of a polycrystal constitutive law and finite element analysis. Acta Materialia 49 (15), 3085–3096. Van Houtte, P., 1978. Simulation of the rolling and shear texture of brass by the Taylor theory adapted for mechanical twinning. Acta Metallurgica et Materialia 26 (4), 591–604. Wang, Y.N., Huang, J.C., 2007. The role of twinning and untwinning in yielding behavior in hot-extruded Mg–Al–Zn alloy. Acta Materialia 55 (3), 897. Ward-Flynn, P., Mote, J., Dorn, J.E., 1961. On the thermally activated mechanism of prismatic slip in magnesium single crystals, Trans. TMS-AIME 221, 1148–1154. Wu, L., Agnew, S.R., Brown, D.W., Stoica, G.M., Clausen, B., Jain, A., Fielden, D.E., Liaw, P.K., in press. Internal-stress relaxation and load redistribution during the twinning–detwinning dominated cyclic deformation of a wrought magnesium alloy, ZK60A. Acta Materialia. Wu, X., Kalidindi, S.R., Necker, C., Salem, A.A., 2007. Prediction of crystallographic texture evolution and anisotropic stress– strain curves during large plastic strains in high purity a-titanium using a Taylor-type crystal plasticity model. Acta Materialia 55, 423–432.