2009 - proust et al - cg constitutive law for mg az31 - ijp

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Author's personal copy

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

Gwnalle Proust a,b,*, Carlos N. Tom a, Ashutosh Jain c, Sean R. Agnew c

a

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

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:

TwinningPolycrystal 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 todislocation 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 defor-

mation via a directional HallPetch mechanism. We introduce here

an improved hardening law for twinning, which discriminates for

specific twin/dislocation interactions, and a detwinning mecha-

nism. We apply this model to the interpretation of compression

and tension experiments done in rolled magnesium alloy AZ31B

at room temperature. Particularly challenging cases involvestrain-path changes that force strong interactions between twin-

ning, detwinning, and slip mechanisms.

2008 Elsevier Ltd. All rights reserved.

0749-6419/$ - see front matter 2008 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijplas.2008.05.005

* 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).

International Journal of Plasticity 25 (2009) 861880

Contents lists available atScienceDirect

International Journal of Plasticity

j o u rn a l h o me p a g e : w w w . e l s e v i e r. c o m/ l o c a t e / i j p l a s


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1. Introduction

The purpose of the present work is twofold: develop a crystallographic model for the plastic re-

sponse 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 asymme-try 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 deforma-

tion (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:haislip on basal planes and tensile f1012gh1011itwinning. 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: pyramidalhaislip f1011gh1120i

(e.g.,Schmid and Boas, 1968), prism haislipf1010gh1120i(e.g.,Ward-Flynn et al., 1961) and pyrami-

dalhc+ ai slipf1122gh1123i (Stohr and Poirier, 1972; Obara et al., 1973; Ando and Tonda, 2000).

The relative activity of the various slip and twinning modes described above depends on the spe-

cific loading conditions and initial texture; in turn, it determines texture evolution. In addition, hard-

ening 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 mono-tonic response. This strain-path change behavior has been characterized experimentally for AZ31

Mg byJain and Agnew (2006)andLou 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 ex-

plored (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 de-formed at 76 K, a regime where tensile and compressive twins are active, and where secondary twin-

ning 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; Tom et al.,1991; Lebensohn and Tom, 1993; Philippe et al., 1995). More recently, researchers started developing

862 G. Proust et al. / International Journal of Plasticity 25 (2009) 861880


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constitutive models that, in addition to texture, also addressed the hardening response due to twin-

ning associated with monotonic loading (Agnew et al., 2001; Kalidindi, 2001; Kaschner et al., 2001;

Tom 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 inter-

esting to notice that models for martensitic transformation in TRIP steels share many commonalitieswith 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 stressstrain 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 Tom, 1993) combined with the predom-

inant twin reorientation (PTR) scheme (Tom 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 evo-

lution 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 re-

ceived 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 X-ray 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 dif-

fraction (EBSD). Detailed experimental procedures were published previously (Jain and Agnew, 2006).

2.2. Monotonic deformation

Fig. 1b shows the stressstrain response of the alloy deformed monotonically by in-plane tension

(IPT), in-plane compression (IPC) and through-thickness compression (TTC). The respective final tex-

tures are shown inFig. 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 par-

allel 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

basalhaiand pyramidalhc+ aislip; the latter mechanism was first observed in Mg duringc-axis com-

pression of single crystals (Obara et al., 1973). However, recent studies (Koike, 2005; Jiang et al., 2006;

Barnett, 2007b) have shown thatf1011gcompressive twinning,f1011gf1012gdouble twining and

f1013gf1012gdouble twinning, can also accommodate compressive strains along the c-axis at room

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

G. Proust et al. / International Journal of Plasticity 25 (2009) 861880 863


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tensile twinning to shear accommodation. As noted byJain et al. (2008)these twins cause reorienta-

tions 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+ aislip (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

thec-axis distribution has been reduced, as can be seen on the texture profile shown inFig. 3. To ob-tain 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 haiand 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 de-

formed 10% in IPT (seeFig. 2), the presence of (0002) 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 dras-

tically, as can be seen inFig. 4a, showing a micrograph of a sample deformed 7% in IPC. Most of thegrains present twin lamellae and some of them are heavily twinned. The texture of the deformed

material is also very different from the initial one (seeFig. 2). The basal component along the ND

0.00 0.05 0.10 0.15 0.20

0

100

200

300

400

IPT

TTC

IPC

Stress(M

Pa)

Strain

0.00 0.05 0.10 0.15 0.20

0

100

200

300

400

Strain-path change

Monotonic

Stress(MPa)

Strain

0.00 0.05 0.10 0.15 0.20

0

100

200

300

400

Stress(MPa)

Strain

IPC

Reloads TTC

Reloads

0002 0110

1.0

2.0

4.0

8.0

TD

RDRD

TD

Fig. 1. (a) Basal and prismatic pole figure showing the texture of the as-annealed AZ31B Mg; (b) stressstrain curves for

monotonic IPT, IPC and TTC; (c) strain-path change stressstrain curves for the samples deformed first in TTC to 5% and 10%

strain and then deformed in IPC (the monotonic IPC stressstrain curve is represented by the dotted line for comparison); and

(d) strain-path change stressstrain curves for the samples deformed first in IPC to 5% and 10% strain and then deformed in TTC

(the monotonic TTC stressstrain curve is represented by the dotted line for comparison).



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has disappeared and now the c-axes of the crystals are aligned with the loading direction due to the

86.6reorientation 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 separationbetween thec-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 50and 90, and subtracting the ini-

tial volume fraction in the same interval, allows us to evaluate the twin volume fractions in deformed

samples. Results are reported inTable 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 stressstrain curves corresponding to these experiments are shown inFig. 1c. The final texture

corresponding to the sample deformed 5% in TTC and then 5% in IPC is shown in Fig. 2. During TTCpre-straining, the material deforms primarily by basal hai and pyramidal hc+ ai slip, though there is

likely somef1011gcompressive twinning as noted by Jiang et al. (2006).

Measured Predicted

0001 0110

0.4

1.0

2.0

4.0

8.0

0001 0110

RD

TD

5%

TTC

10%

IPT

10%

IPC

5% TTC

+ 5% IPC

5%IPC+

10%TTC

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.



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The IPC reload curve following 5% and 10% TTC pre-load is similar to the monotonic IPC curve: thedeformation 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-

0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 900.0

0.1

0.2

0.3

0.0

0.1

0.2

0.3

Predicted

NormailizedIntensity

Measured

Normalizedinten

sity

Initial 5% TTC 10% IPT 10% IPC

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

angleangle

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 thec-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 inFig. 3

Twin volume fraction

Experimental Prediction

10% IPT 0.07 0.08

10% IPC 0.90 0.86

5 IPC + 10% TTC 0 0

5% TTC + 5% IPC 0.65 0.72



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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 defor-

mation, 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 nucle-

ation stress. In this case, during the preloading,haiandhc+ aidislocations are introduced. These dis-

locations may act as barriers to twin nucleation or/and twin propagation. This would explain both theincrease 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 stressstrain curves corresponding to

these experiments are shown inFig. 1d, and the texture corresponding to 5% IPC followed by 10% TTC

is shown inFig. 2. During pre-straining, the material deforms primarily by tensile twinning, but re-

verses the twins (detwins) during the reload stage. The integration of the texture profiles is consistent

with this understanding;Table 1indicates 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 com-

paringFig. 4a and b, we see that the amount of twins has decreased during TTC reload and though

some twin lamellae are still visible inFig. 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 inFig. 1d are similar to the monotonic IPC stressstrain 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 in-

creases. 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 amesoscopic 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 _cs contributed by slip and twinning systems through a ratesensitive law

_eij Xs

msij_cs _c0

Xs

msij ms

:rss

nMsecijklrkl: 1



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where ris 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 dis-

crete 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 (Tom

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 Predom-

inant 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 mechan-

ical properties. The layers are assumed to be equidistant and two parameters are introduced: the sep-

aration dc of the center planes of the lamellae, and the maximum volume fraction of the grain that may

be reoriented by twinningfPTSmax. Because twins are assumed (as a first approximation) to pose impen-

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

ofdc is made to increase with the PTS volume fraction until reaching a value of one (which corre-sponds to having only one twin domain), when 70% of the grain has twinned.

By creating this layered twinmatrix structure inside grains, we are able to account for the direc-

tional 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 onlyconstituted 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.



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due to the barrier effect and that particular slip system will be harder to activate. This idea is imple-

mented in the CG model through a HallPetch 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 be-

tween 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. Inthe 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 ap-

proach, 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 inFig. 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 geomet-

rically necessary dislocations (GND) and a directional HallPetch effect:

ss ssSTATssGNDs

sHP: 2

Each of these terms is updated incrementally at each straining step. The first term is a classical satu-

ration Voce law associated with statistical dislocations, to which a latent hardening effect has been

added:

DssSTATdss

dCXs0

hss0Dcs

0

; 3a

where ss s0s1 1 exp Ch0s1

: 3b

Here C is accumulated shear in the grain, Dcs0

the shear increment in the slip or twinning system s0

andhss0

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.

The second term of Eq.(2)depends on the directional mean free path ds

mfpdefined 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 Hs

GND

dsmfps

sSTATs

sGND

Dcs: 4

The last term of Eq.(2)describes the directional HallPetch effect:

ssHP HsHPffiffiffiffiffiffiffiffiffidsmfp

q : 5While the GND contribution is not so relevant at the strains considered here, the HallPetch 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 dis-

locations 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 thetwins. However, because the relative effects of these hardening mechanisms have yet to be experi-

mentally differentiated, only the HallPetch effect will be used in our model.



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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 twin-

ning it should only be regarded as an empirical law that provides a dependence of twin hardeningwith 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 conse-

quence, we present here a deformation mode specific law to describe the hardening oftwinningsys-

tems, of the form:

sTWS sTWS0 XM

sM 1 exp hMCM

sM

!( ); 6

where the sum is made over all the slip modes M (basal hai, prismhai, pyramidalhc+ 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 defor-

mation modes upon twinning, which is more consistent with the sparse experimental and theoretical

information available. For example, it has been proposed byMendelson (1970)that non-planar dislo-

cation 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 anda-Ti (Serra and Bacon, 1996; Pond et al., 1999) suggest that twininterfaces 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, dur-

ing 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) pro-

posed 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 andhM of basal and prism slip. In thecase studied here, it is assumed that haidislocations on basal planes are the most likely to induce twin

nucleation and growth, based on atomistic simulations studies (Serra and Bacon, 1996) and the obser-

vations that basal slip is easy to activate in most orientations. Tensile twinning and hc+ aidislocations

on pyramidal planes are competing mechanisms. However, during strain-path changes, it is possible

to have previously inducedhc+ aidislocations interact with twins and harden them. In the new hard-

ening law it is possible to describe this phenomenon by giving positive values to the parameterss

M

andhM.Fig. 6b shows the evolution of the twinning threshold stress with the amount of shear accom-

modated by each slip mode. The determination of the hardening parameters associated with these

curves is explained in Section4 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



11/20


et al., 2007; Wang and Huang, 2007). The detwinning mechanism may require less stress to be acti-

vated 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 CRSSs 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 beentransferred 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 stressstrain 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 com-

pression and tension realized parallel to the transverse direction of the initial plate (these curves

0.00 0.05 0.10 0.15 0.200

50

100

150

200

250

Basal

Prismatic

Pyramidal

0.00 0.05 0.10 0.15 0.200

50

100

150

200

250Basal

Prismatic

Pyramidal

M

S(MPa)

TWS(MPa)

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

Table 2a

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

slip parameters

s0 (MPa) s1 (MPa) h0 (MPa) HGND (MPalm) HHP (MPa lm1/2) Latent hardening

Basal Prism Pyra

Basal 2 52 3000 105 100 1 2 35Prism 60 60 600 0 100 1 1 1

Pyra 50 70 2200 0 380 1 1 1



12/20


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 mono-

tonic TTC and IPC flow stress curves. The HallPetch coefficients describing the interactions between

these slip modes and twins are selectively given by the stressstrain curves obtained during in-plane

deformation: during IPT, the small volume fraction of twins which form are oriented favorably to acti-

vate prismatichai, and during IPC, the twins are oriented favorably to activate pyramidal hc+ ai slip

and significant basal hai slip also occurs. The IPC stressstrain 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 mech-

anism 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 inFig. 6a. The initial CRSS values of all the slip modes are given by adding s0and the HallPetchfactor corresponding to our initial grain size of 13 lm. For our simulations, the starting CRSS ratiosbetween 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 basalhaiand 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 isshown 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 de-

scribe 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 prismatichaidisloca-

tions are introduced in the material due to the negative values ofsM andhM for both basal and pris-matic slip (basal slip affects the twinning threshold stress more than prismatic slip). The values ofsM

andhM 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 nucle-

ation 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 detwin-

ning. 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

value for detwinning,sDETWS0 , corresponds to the saturation threshold stress for twinning (Fig. 6b) oncebasal and prismatic slip effects have been completely accounted for, i.e.

sDETWS0 sTWS0 s

basal sprism; 7

wheresTWS0 ,sbasal andsprism represent the hardening parameters described in Eq. (6) and associatedwith 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.

Table 2b

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

in the twins

s0(MPa) sbasal (MPa) hbasal (MPa) sprism (MPa) hprism (MPa) spyra (MPa) hpyra (MPa) dc fmax

In matrix 55 30 15,000 5 20,000 50 1200 0.25 0.9

In twins 20 0 0 0 0 20 100 0.25 0.9



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4.2. Monotonic deformation

Predicted stressstrain 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 inTable 2. InFig. 7we 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

0.00 0.05 0.10 0.150

100

200

300

400

Experiment

SimulationSTR

ESS(MPa)

0.00 0.05 0.10 0.150.0

0.2

0.4

0.6

0.8

1.0

MODEACTIVITY

INMATRIX

MODEACTIVITY

INMATRIX

0.00 0.05 0.10 0.150.0

0.2

0.4

0.6

0.8

1.0

MO

DEACTIVITY

INTWINS

Twin Vol Frac

Prismatic

Basal

Pyramidal

Tensile Twinning

0.00 0.05 0.10 0.150

100

200

300

400

STRESS(MPa)

0.00 0.05 0.10 0.150.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.10 0.150.0

0.2

0.4

0.6

0.8

1.0

MODEACTIVITY

INTWINS

0.00 0.05 0.10 0.150

100

200

300

400

STRESS(MPa)

0.00 0.05 0.10 0.150.0

0.2

0.4

0.6

0.8

1.0

MODEACTIVITY

INMATRIX

0.00 0.05 0.10 0.150.0

0.2

0.4

0.6

0.8

1.0

MODEACTIVIT

Y

INTWINS

0.00 0.05 0.10 0.150

100

200

300

400

STRESS(MPa)

0.00 0.05 0.10 0.150.0

0.2

0.4

0.6

0.8

1.0

MODEACTIVITY

INMATRIX

0.00 0.05 0.10 0.150.0

0.2

0.4

0.6

0.8

1.0

MODEACTIVITY

INTWINS

0.00 0.05 0.10 0.150

100

200

300

400

STRESS(MPa)

STRAIN0.00 0.05 0.10 0.15

0.0

0.2

0.4

0.6

0.8

1.0

MODEACTIVITY

INMATRIX

STRAIN0.00 0.05 0.10 0.15

0.0

0.2

0.4

0.6

0.8

1.0

MODEACTIVITY

INTWINS

STRAIN

Fig. 7. Comparison between the experimental (symbols) and predicted (solid lines) stressstrain curves, predicted deformationmode 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.



14/20


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. 2shows that the measured and predicted textures are very similar to the initial texture.

IPT is mostly accommodated by prismatic and basalhaislip in the matrix (seeFig. 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 theirc-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+ aipyramidal slip inside the

twins. The initial threshold stress associated with the slip modes inside the twins are high due to the

HallPetch 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 HallPetch

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 HallPetch effect, as discussed above.

Fig. 2shows 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 (seeFig. 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+ aidisloca-

tions 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 byLou et al. (2007)and the hardening parametersinTable 2breflect that the morehc+ ai dislocationsin the material, the harder tensile twinning be-

comes. In Fig. 8we have plotted the experimental and predicted stressstrain 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 harden-

ing), 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 stressstrain curves reveals that not

only the yield strength changes but also the length of the initial plateau and the slope of the curve be-

fore reaching the saturation stress (Fig. 8). These differences seem to be due to a change in the defor-

mation modes activated during the IPC reload. The transition from easy to hard deformation is lessabrupt 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



15/20


accommodating strain in the twinned regions. The fact that this transition is less abrupt during reload-

ing suggests that while twinning is occurring in the matrix, another deformation mode is also accom-

modating 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 twin-

ning 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 wouldcomplement 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. InFig. 7e, the stressstrain 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 dropsfrom 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 inFig. 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.Using the value ofsDETWS0 for tensile twinning in the twins obtained from Eq.(7), the CG model can

capture the TTC reloading yield strength. We can also reproduce the rapid increase in the hardening

0.00 0.05 0.10 0.15 0.200

100

200

300

400

Experiments

Simulations

Stress(MPa)

Strain

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

strain and the black curves the results for 10% TTC pre-strain.



16/20


rate that corresponds to the matrix starting to deform. But, again, the model does not reproduce the

long initial plateau present in the experimental stressstrain curve after reload. In the case of mono-

tonic 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 forbasal 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. 2shows 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 direc-

tion of the plate. We attribute the predicted depletion at the center of the basal pole figure, to exces-

sive 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 fea-

tures 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 defor-

mation 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 butdid 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 defin-

ing a new hardening law for twinning, we have dramatically improved the predictions of the harden-

ing and texture evolutions during strain-path change.

Although the HallPetch parameters that we are characterizing in our model (seeTable 2) corre-

spond to the barrier effect associated with twin boundaries, we used our model to predict the mono-

tonic 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 HallPetch effect observed experimentally (cf.

Fig. 9) is mainly caused by the hardening of prismatic slip due to a difference in grain size. Experimen-

tally, the overall HallPetch coefficient for Mg alloy, AZ31B, was determined to be equal to

200 MPa lm

1/2

(Jain et al., 2008) while our fitting process gave a value of 100 MPa lm

1/2

for prismhaislip. (Given the initial texture and loading direction during IPT, the Schmid factor for prismatic slip

is near maximum,m2, so this is viewed as very good agreement.) With this value of the HallPetch

parameter for prismatic slip, we were able to closely reproduce the stressstrain response of the same

Mg alloy having initial grain sizes of 42 and 89 lm as shown inFig. 9.The new hardening model for twinning that we introduce in this paper has been guided by exper-

imental 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 cap-

tures this phenomenon by having a high initial value for the tensile twinning CRSS and then decreas-

ing the CRSS once basal (or prismatic) slip has accommodated the initial strain. On the matrix activity

plot ofFig. 7c, one can see that at the start of the deformation the activity of basal slip is higher thanthe 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



17/20


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 {10 11} 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 stressstrain 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 thematerial 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 rever-

sal, 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 Tom, 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 ne-

glected the initial 2% strain of the reloading curves while simulating the stressstrain 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 hypothesisand 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 evo-

lutions of the hardening rate as a function of stress (Fig. 10). During monotonic TTC and IPT, the hard-

ening 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 increas-

ing drastically, reaches a maximum at an equivalent stress of about 125 MPa, and then decreases rap-

idly. 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 stressstrain 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 maxi-

mum smaller than for monotonic IPC. Our model does not entirely capture this behavior, which weattribute 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

0.00 0.02 0.04 0.06 0.08 0.100

100

200

300

Experiments Simulations

13 microns 13 microns



Stress(

MPa)

strain

Fig. 9. HallPetch effect due to the initial grain size during IPT. Comparison of the experimental and predicted stressstrain

curves.



18/20


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 be-

lieve 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 theactivation 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 al-

lowed 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 correspond-

ing 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 deforma-

tion by shear but not to crystallographically reorient the matrix. When the loading direction is chan-

ged, 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 HallPetch 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 defor-

mation 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. Thismaterial 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.

0 50 100 150 200 250 3000

5000

10000

15000

20000

Theta

TTC IPC IPT

reload IPC reload TTC

0

5000

10000

15000

20000

Theta

Equivalent Stess (MPa)

0 50 100 150 200 250 300

Equivalent Stess (MPa)

Experimental Simulations

Fig. 10. Comparison of the experimental and predicted hardening rates for the five stressstrain 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 theequivalent stress (rry).



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