Lee Wagoner IJP 2008.pdf

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Constitutive modeling for anisotropic/asymmetrichardening behavior of magnesium alloy sheets

Myoung-Gyu Lee a,1, R.H. Wagoner b, J.K. Lee c,K. Chung d, H.Y. Kim e,*

a Eco-Materials Research Center, 66 Sangnam, Korea Institute of Machinery and Materials,

Changwon, Kyungnam 641-010, Republic of Koreab Department of Materials Science and Engineering, 2041 College Road, Ohio State University,

Columbus, OH 43210, USAc Department of Mechanical Engineering, Scott Laboratory, 201 West 19th Avenue,

Ohio State University, Columbus, OH 43210, USAd School of Material Science and Engineering, Seoul National University, 56-1, Shinlim-Dong,

Kwanak-Ku, Seoul 151-742, Republic of Koreae Division of Mechanical Engineering and Mechatronics, Kangwon National University, 192-1 Hyoja 2-Dong,

Chunchon, Gangwon-Do 200-701, Republic of Korea

Received 8 March 2007; received in final revised form 25 May 2007Available online 14 June 2007

Abstract

Magnesium alloy sheets have been extending their field of applications to automotive and elec-tronic industries taking advantage of their excellent light weight property. In addition to well-knownlower formability, magnesium alloys have unique mechanical properties which have not been thor-oughly studied: high in-plane anisotropy/asymmetry of yield stress and hardening response. The rea-

son of the unusual mechanical behavior of magnesium alloys has been understood by the limitedsymmetry crystal structure of HCP metals and thus by deformation twinning. In this paper, thephenomenological continuum plasticity models considering the unusual plastic behavior of magne-sium alloy sheet were developed for a finite element analysis. A hardening law based on two-surfacemodel was further extended to consider the general stressstrain response of metal sheets includingBauschinger effect, transient behavior and the unusual asymmetry. Three deformation modesobserved during the continuous in-plane tension/compression tests were mathematically formulatedwith simplified relations between the state of deformation and their histories. In terms of the anisotropy

0749-6419/$ - see front matter 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijplas.2007.05.004

* Corresponding author. Tel.: +82 33 250 6317; fax: +82 33 242 6013.

E-mail address: [email protected] (H.Y. Kim).1 Department of Materials Science and Engineering, Ohio State University, Columbus, OH 43210, USA.

Available online at www.sciencedirect.com

International Journal of Plasticity 24 (2008) 545582

www.elsevier.com/locate/ijplas
mailto:[email protected]:[email protected]


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and asymmetry of the initial yield stress, the DruckerPragers pressure dependent yield surface wasmodified to include the anisotropy of magnesium alloy. The numerical formulations and character-ization procedures were also presented and finally the correlation of simulation with measurementswas performed to validate the proposed theory.

2007 Elsevier Ltd. All rights reserved.

Keywords: Magnesium alloy sheet; Asymmetry; Two-surface model; Bauschinger effect; Modified DruckerPrager model

1. Introduction

Over the last several decades, increasing attention has been paid to understand thedeformation behavior of magnesium alloys due to the lowest density of all metallic con-

structional materials (Roberts, 1960; Mordike and Ebert, 2001). In addition to their excel-lent light property, magnesium alloys have several other advantages: high specific strength,good welding capability and corrosion resistance. Wrought magnesium alloys have beenreported to have better mechanical properties including tensile and fatigue resistance thanthe casting counterparts (Duygulu and Agnew, 2003; Agnew and Duygulu, 2005; Bettlesand Gibson, 2005; Easton et al., 2006; Lou et al., 2007 ) and therefore more promisingfor the potential applications. Taking advantage of these benefits, magnesium alloys havegreat potential for high performance automotive applications such as steering wheels,seats, gear box housing and so on. Due to better mechanical properties, resistance toaging, electrical and thermal conductivity, magnesium alloys have also been substitutingthe polymeric materials in the electronic devices industries. In sheet metal forming appli-cation with magnesium alloys, the lower formability and high springback due to the lowerelastic property (Youngs modulus = 45 GPa, Cubberly et al., 1979; Lou et al., 2007) atroom temperature are major hurdles by which magnesium alloys have limited applications(Gradinger and Stolfig, 2003). Therefore, the studies on the process optimization based onaccurate characterization of the material properties of magnesium alloys are significantlynecessary to broaden their applications.

In spite of potential applications of magnesium alloys in various areas, limitedresearches have been performed, especially for their unusual mechanical properties.

Magnesium alloys have unique plastic behavior compared to other alloys. First, magne-sium alloys have significant difference in initial yield stress for tension and compression.The initial yield stress for tension is much higher than that for compression ( Ball andPrangnell, 1994). For example, the yield stress of AZ31B magnesium sheet in the trans-verse direction is 192 MPa, while 110 MPa for the compression (Lou, 2005; Lou et al.,2007). The difference in tensile and compressive yield stresses is often denoted as (yield)asymmetry. In addition to the yield asymmetry, magnesium alloys show asymmetry inthe plastic flow stress (flow asymmetry). Literature survey on the plastic behavior ofmagnesium alloys with strong basal texture reveals the following common mechanisticbehavior.

During the uniaxial tensile deformation from undeformed state, hardening curve is nor-mal concave-down shape. Basal slip dominates during the deformation with other con-tributions of non-basal slip and twinning to maintain local compatibility. Therefore, the

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conventional constitutive equations can contribute to the slip deformation. That is, thestressstrain curve during the slip deformation can be fitted by exponential or power-law type hardening equations.

Unusual concave-up or S-shape hardening curves are observed during in-plane com-

pression or tension following compression. The unique inflected flow curves are causedby the activation of twinning or exhaustion of twinning. When the loading is reversedfrom twinning, untwining initiates due to the contraction of twinned regions withoutnucleation. For more details, refer to Lou et al. (2007).

Fig. 1 shows the schematic view of typical stressstrain curves of cubic crystal alloysand textured magnesium alloy sheets. The figure shows reduced yield stress during reverseloading, which is associated with the Bauschinger effect (Bauschinger, 1886) for both mate-rials. For the cubic metals such as aluminum alloys, the magnitude of yield stress in ten-sion and compression is assumed to be same and flow stress curve during tension andcompression is also symmetric, while strong asymmetries in yield stress and flow curvesare shown for the textured magnesium alloy sheets. In sheet metal forming applications,the non-monotonous deformation is especially important because reverse loading is com-monly observed when sheet element moves through the tool radii and draw beads. Also,when sheet parts are removed from tools after forming, material elements experience elas-tic unloading and springback.

Two main approaches have been made to describe the reverse loading behavior inthe continuum phenomenological plasticity: one based on kinematic hardening involv-ing shifting of a single-yield surface and the other involving multiple yield surfaces

(Khan and Huang, 1995). The simplest one in the former group is based on linearkinematic hardening proposed by Prager (1956), Ziegler (1959) and Hodge (1957)to describe the Bauschinger effect. Note that these phenomenological models ignorethe microplastic effects and introduce mathematical rule of translation of yield sur-face. To add the transient behavior, the linear model was modified to nonlinear formsby Amstrong and Frederick (1966) and Chaboche (1991) by introducing an additionalterm to Pragers linear kinematic hardening model. Nonlinear and smooth deforma-tion during loading and reverse loading were reproduced by introducing additionalback stress term which makes total back stress decrease gradually with deformation.Several nonlinear kinematic hardening models based on AmstrongFredrick model

have been emerged by introducing multiple back-stress terms (Ohno and Wang,1993a,b) and translating limiting surface (Ohno and Kachi, 1986; Bower, 1989).The Chaboche model was further generalized recently as a combined type model, uti-lizing a non-quadratic anisotropic yield function and the Ziegler kinematic hardeningmodel (1959), based on the plastic work equivalence principle modified for kinematichardening to properly define effective (or equivalent) quantities in stress and plasticstrain rate (Chung et al., 2005; Lee et al., 2005a,b). This modified Chaboche modelonly accounts for the Bauschinger and transient behavior, not the permanent soften-ing (Kim et al., 2005). The nonlinear kinematic hardening model has been furthermodified to incorporate the permanent softening as well as the Bauschinger effect

and transient behavior (Geng and Wagoner, 2002; Geng et al., 2002; Chun et al.,2002a,b).

Another frequently used family of phenomenological hardening models is multi-surface model. The classical Mroz model involves multiple numbers of yield surfaces

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in which piecewise linear variation of hardening is defined (Mroz, 1967) according to

Pragers kinematic hardening model. On the other hand, two-surface models indepen-dently proposed by Krieg (1975) and Dafalias and Popov (1976) define the continuousvariation of hardening between two yield surfaces. In the original multi-surface modelproposed by Mroz (1967), the predicted stressstrain curve is piecewise linear because

Fig. 1. Schematic view of typical stressstrain curves of: (a) cubic crystal alloy sheets which show symmetric yieldstress in tension and compression as well as reduced reverse yield stress (Bauschinger effect) and (b) texturedmagnesium alloys which show asymmetric yield stress in tension and compression, Bauschinger effect andasymmetry in flow curves with three deformation modes.



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of the constant plastic moduli (Khan and Huang, 1995). Therefore, an infinite number ofyield surfaces are needed to predict a smooth nonlinear curve as proposed by Mroz andNiemunis (1987), while the two-surface model can represent realistic smooth hardeningwith the continuous plastic modulus. However, the two-surface model may lead to the

discontinuous change of elasto-plastic stiffness after partial unloading and also may pro-duce unrealistic behavior such as too strong ratcheting as addressed by Hashiguchi(1997) and Chaboche (1986). Several multi/two-surface models have been proposed later(Mroz et al., 1979; Hashiguchi, 1981; McDowell, 1985; Hashiguchi, 1988; Khoei andJamali, 2005; Lee et al., 2007) to analyze the one-dimensional cyclic behavior of solidstructures at small strains.

In the continuum plasticity, in order to describe initial yield anisotropy, mathematicalrepresentations of the yield surfaces are combined with hardening behavior for numericalmodeling. For the FCC and BCC materials which do not have strong asymmetry in yield-ing, several yield criteria have been proposed from isotropic (von Mises, 1928; Tresca,1864; Hosford, 1972) to anisotropic yield surfaces (Hill, 1948; Hill, 1979; Barlat et al.,1991; Barlat et al., 2003).

Although considerable efforts have been made for the constitutive equations which canreproduce more realistic mechanical behavior of metallic materials, few phenomenologicalmodels have been reported for the asymmetric materials including magnesium alloys (Li,2006; Kim et al., 2007a). Modeling constitutive equations for the magnesium alloysrequires understanding of plastic behavior of more complex deformation paths includingreverse loading condition. The challenge is met by their unusual hardening curves withlarge asymmetry and considerable Bauschinger effect. Other attempts to explain yielding

asymmetry and anisotropy of HCP metals have also been made by crystal plasticity.For example, self-consistent (SC) methods were adopted to simulate tension and compres-sion behavior of textured HCP alloys (Lebenson and Tome, 1993; Brown et al., 2005) andfinite element analysis with crystal plasticity model was performed to incorporate defor-mation twinning (Kalidindi, 1998). More recently, a crystal-mechanics-based constitutivemodel to account for the twinning was developed for polycrystalline magnesium alloyAZ31B (Staroselsky and Anand, 2003). Although the texture or crystal plasticity modelswere partially successful in predicting deformation behavior of magnesium alloys, it is nec-essary for these models to contain large number of orientations at each integration point.Thus, these approaches are computationally inefficient in forming application with finite

element method.In the present study, constitutive models for the sheet magnesium alloys are devel-

oped based upon the phenomenological continuum plasticity, which is capable ofdescribing the yielding asymmetry and anisotropy in stressstrain response. For thehardening model, two-surface model is modified to include ability to model the threedeformation modes: slip, twinning and untwining. In terms of asymmetry in the initialyield stress and anisotropy, the modified DruckerPrager model is adopted with aniso-tropic coefficients under plane stress conditions. Based on the developed theory, charac-terization procedures for the material parameters from continuous in-plane uni-axialtension/compression tests are presented. The models have been implemented into a

finite element program, ABAQUS/Standard with user material subroutine that success-fully reproduces the main features of the experimental results for the unusual stressstrain responses of O-tempered AZ31B and AZ31B without tempering magnesium alloysheets.



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2. Mechanical behavior of magnesium alloys

2.1. Anisotropy/asymmetry

In the phenomenological continuum plasticity, the yield surface is one of the main con-stitutive equations which complete the response of materials. Magnesium (Mg) alloy sheetand HCP metals show high anisotropy and strong eccentricity in yielding. In other words,the yield stress shows directional difference (anisotropy) and difference in tension and com-pression (asymmetry or eccentricity). It has been well known that the yielding asymmetryin in-plane tension and compression is associated with the strong basal texture and theactivation of twinning during compression. For example, the tensile yield stress in texturedmagnesium alloy is much larger than that of compressive stress (Hosford, 1993; Lou et al.,2007).

Two different ways have been proposed to represent the eccentricity. The first method isto introduce the pressure dependent or invariant of stress into the existing anisotropicyield functions. For example, Hosford (1966, 1972) added linear stress terms into Hills48 yield surface (Hill, 1948), while Cazacu and Barlat (2001, 2004) added third invariantof stress deviator into the orthotropic yield surface which was modified from the isotropicDrucker yield criterion (Drucker, 1949). The other method is to introduce initial transla-tion of yield surface by assuming non-zero back-stress in the combined isotropickine-matic hardening model. Yoon et al. (1998) represented different in-plane tension andcompression yield stresses of aluminum alloy 2008-T4 by using asymmetry model incorpo-rated into Barlats96 anisotropic yield function (Barlat et al., 1997) and the earring predic-

tion was in good accordance with the measurement. More recently, Li (2006) also usednon-zero initial back stresses along with the von Mises isotropic yield function and propertranslation rule consistent with texture evolution of magnesium alloys to simplify theformulation.

2.2. Hardening behavior

Besides the yielding asymmetry and high in-plane anisotropy, magnesium alloy sheetshave unique hardening behavior during the plastic deformation. Inflected stressstraincurves during in-plane compression and subsequent tension tests are reported for the mag-

nesium alloys. These unusual behaviors are due to the twinning and untwining deforma-tion which induce abrupt grain reorientation, creation and disappearance of twinboundaries. As schematically shown in Fig. 1b, the typical stressstrain curves of magne-sium alloy sheets have different behavior for the following three deformation modes: (a)in-plane tension, (b) in-plane compression and (c) compression followed by tension. Thetensile response is similar to the behavior of FCC or BCC metals which have strong sym-metry in slip activities and at least five independent slip systems exist. The dislocation slipmechanism is dominant and the shape of stressstrain curve is normal concave-down inthis deformation mode. For the in-plane compression mode, twinning mechanism is pre-dominant with abrupt texture change and exhausted with continuous compression. When

the twinning is exhausted, the slip initiates again and the flow stress rises rapidly. There-fore, the stressstrain curve during in-plane compression shows unusual concave-up shape.For the tension following compression, the deformation is similar to the in-plane compres-sion. During the early tensile deformation, untwining mode dominates so that the stress



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strain curve in this region shows concave-up shape. When the untwining exhausts with fur-ther tensile deformation and no further texture change exists, the dislocation slip modetakes over again. Therefore, the shape of stressstrain curve during in-plane tension fol-lowing compression is unusual S-shape. The summary of the stressstrain responses and

evolution of microstructure for the magnesium alloy sheet is well presented in the previouswork by Lou et al. (2007).

3. Theory

3.1. Summary of the two-surface theory

In the two-surface model, two yield stress surfaces are employed as shown in Fig. 2: (1)the loading surface on which the current stress at point a in Fig. 2 is defined and (2)bounding surface on which a corresponding stress at point A in Fig. 2 is defined. In gen-eral, these two surfaces have the same shape and translate and/or expand at the same time.The correspondence between points a and A is defined by the common yield surface nor-mal directions. The gap between the current and corresponding stresses determines thehardening rate, which is initially steep but becomes smaller as the gap decreases. Sincethe two surfaces can make contact only at points sharing the same normal direction, con-tact is imposed to occur at the current stress and corresponding stress points in the model.For the detailed description of the two-surface model and its problematic issues arereported in the previous article by Lee et al. (2007).

The general yield function for the loading surface is

Ur a riso 0: 1

Fig. 2. A schematic view of the two-surface model and two gap distances aA and bB.



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Here, r is the Cauchy stress and a is the back-stress, which defines the central position ofthe current yield stress surface (with a = 0 initially). Also, riso is the effective stress, a mea-sure of the size of the yield surface. Note that the function used in the present paper is first-order homogeneous and will be described in the later section.

From Eq. (1) and the plastic work equivalence principle,

oU

or a dr oU

or a da hisor ariso

dep 0 2

where hiso drisode

is the slope of the isotropic hardening curve, riso, as a function of theeffective plastic strain e Rde , while

da darisom m

da

dede

m

risom : 3

where the relation Uda Udm dc1risom da is utilized for the first-order homoge-neous function and the translational direction of the loading surface m$ (r a) isadopted.

Considering linear elasticity and the additive decomposition of the strain increment,

dr C dee C de dep 4where C is the elastic modulus, while de and dee are total and elastic strain increments,respectively. The plastic strain increment is defined by the normality rule as

dep

dk

oU

or a dk

oriso

or a de

oriso

or a;

5

After some manipulations,

de oriso

ora C de daoriso

ora C orisoora hiso: 6

Therefore, for a given strain increment de prescribed at every time increment, Eqs. (5) and(6) determine the plastic strain increment, while the stresses are updated by Eqs. (4) and (5)on the loading surface as Jaumann increments.

Similarly to the loading surface, the bounding surface is described as

PR A Riso 0 7where R and A are the stress and back-stress of the bounding surface, respectively. Also,Riso represents the size of the bounding surface and it is pre-determined by the properdecomposition. Since the bounding surface P shares the same shape with the loading sur-face U and the corresponding stress on the bounding surface R shares the same normaldirection with the current stress r at the loading surface,

R

A

Riso

riso

r

a

8

As for the back-stress evolution, the following condition is imposed in addition to Eq. (3):

dA da dlR r or dA da dlR r dA1 dA2 9



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The condition in Eq. (9) specifies that two surfaces relatively translate along the line be-tween the current and corresponding stresses, which ensures that contact between two sur-faces occur at current and corresponding stresses. For the second term of the last term ofEq. (9),

dA2 dA2riso

R r 10

where dA2 risodA2 and riso risoR r as similarly done in Eq. (3).In the two-surface model, the expansion and translation of the two surfaces are param-

eterized such that the Bauschinger, transient and permanent softening behaviors are prop-erly represented and are matched to the 1-D reference state. For the reference state, thestress relationship becomes

R

r

d ordR

de dr

de dd

de 11

where d is the gap between R and r, the stress states at the bounding and loading surfaces,respectively. Initially, the hardening of the bounding surface is prescribed for expansion,Risoe, and translation, Ae (or A2e with proper separation. Then, re is obtained toaccount for the transient behavior every time reverse loading occurs, considering the pre-scribed gap function de, which is dependent on din, the initial gap distance measured atthe start of reverse loading. The separation ofr into the expansion and translation, risoeand ae, is executed to properly account for the Bauschinger effect. For the hardeningmodel dr > 0, the simple decomposition as following,

dr mldr 1 mldr driso da 12dR mbdR 1 mbdR dRiso dA 13

where ml and mb are the ratios of the isotropic hardening for the loading and boundingsurfaces, respectively, which are the functions of the accumulative plastic strain in general,however, constant values are assumed here for simplicity. When the loading path is notproportional, caution should be made for the separation of the isotropic hardening andthe kinematic hardening in the loading surface. The gap d is the distance between the cur-rent stress on the loading surface and the corresponding stress on the bounding surface(marked a and A in Fig. 2), while the gap n is the distance between the corresponding stres-

ses (marked b and B in Fig. 2) aligned with the line connecting two centers of the loadingand bounding surfaces. Note that premature contact at b and B should be avoided in thetwo-surface model with the proper separation of the isotropic and kinematic hardening inorder not to penetrate the bounding surface.

The scalar parameter d to measure the gap between the current stress at the loading sur-face and the corresponding stress at the bounding surface is defined here as

d risoR r UR r 14which is the effective stress value obtained by replacing r

a with R

r.

In order for the current two-surface model to be used in a practical way for theplane stress problem, it is efficient to properly define the reverse loading criterion suchthat the new initial gap distance din is updated only when the reverse loading criterion issatisfied. Fig. 3 shows the reverse loading criterion introduced here, in which hd is the



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angle between two subsequent stresses on the loading surface, while hr is a prescribedreference angle for reverse loading: the reverse loading condition that (06) hr 6 hd(6p)where

hd cos1 r ajr aj

new

r ajr aj

old

cos1dnew dold: 15

The hardening update which involves the new gap function and reverse loading condi-

tion is performed considering the following linear combination associated with new andprevious initial gaps and the parameters ca and cb, which are the functions of the anglehd:

d 1 ca ddoldin ;ei 1 cbeold ca ddnewin ;ei 1 cbeold; 0 6 ca;b 6 1

d ddnewin ;ei 0; ca;b > 116

Here, doldin anddnewin are initial gap distances for the previous and the current loading curves,

respectively, while ei and eold are the initial plastic strain for the reversed hardening curve

and the strain at the load reversal from the previous loading curve, respectively. Theparameters are supposed to be experimentally determined, however, ca = cb = jhd/hrj is as-sumed here for simplicity.

3.2. Hardening behavior of Mg alloys

Although the conventional two-surface models can effectively represent realistic hard-ening behavior such as Bauschinger effect, transient behavior and permanent softening,it is still necessary for new or modified constitutive models developed to represent unusualasymmetric stressstrain response in in-plane tension, compression and continuous ten-

sioncompression (or compressiontension). As discussed in Section 2.2, the magnesiumalloys (or HCP metals) have three different deformation modes: slip mode, twinning mode,and untwining mode. Therefore, the gap distance d should be updated for different defor-mation modes in addition to the initial gap distance din when the loading is reversed. The

r

d

oldd

newd

reverse loading criterion

Fig. 3. Reverse loading criterion for the two-surface model.



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twinning deformation is known to be closely related with c/a ratio (Roberts, 1960; Yoo,1981) in hexagonal close packed (HCP) alloys. The most activating twin system in magne-sium alloys is f1 01 2gh1 011i where c/a (=1.624) is less than

ffiffiffi3

p. Therefore, this twin

mode is a tensile twin which induces the extension of the c-axis in magnesium alloy. Note

that the extension of the c-axis in rolled sheet alloy is equivalent to the in-plane compres-sion. Twinned magnesium alloys may undergo disappearance of twin density by subse-quent deformation, which is called as untwining. In order to simplify the model and forthe practical application in the sheet metal forming analysis, the following assumptionsare made in the present study:

(1) The stress through the thickness is negligible so that plane stress state can be applied.(2) The initial texture of magnesium alloy is perfect basal texture where all crystal c-axes

align parallel to the sheet normal (thickness direction) and all a-axes are randomlydistributed in the sheet plane. This assumption is verified for the annealed AZ31Bmagnesium alloy by measuring pole figures in the previous research (Lou et al.,2007).

(3) There is no texture evolution during the slip mode. On the other hand, the textureevolution for twinning and untwining is same but reverse direction each other.The critical strain for the complete rotation of c-axis during twinning and untwiningis assumed to be known and the angle of rotation is 90.

With the above assumptions, the modified hardening model considering the three defor-mation modes is proposed. The gap distance as a general form is

d ddin; cold; cnew; ep 17where coldand cnew are the vectors ofc-axes before and after rotation, respectively. Eq. (17)denotes that the gap distance is updated by considering the initial gap distance, the historyof texture evolution and the magnitude of plastic deformation. The details on the criteriato determine the three hardening modes are as following.

As well reported in the previous article (Roberts, 1960), deformation twinning occursunder both compressive and tensile deformations. However, for magnesium alloy whichhas c=a etwin;cr

(; 20

where etwin is accumulated plastic strain during the twinning deformation and rtwin denotes

residual ofc-axis remaining in the original direction. Thus, when etwin reaches etwin;cr, all ofthe c-axes completely rotate on the sheet plane.

The criterion for the untwining mode is similar to the twinning mode. However,the criterion should consider the history of deformation before initiating untwining

Fig. 4. (a) Yield stress surface considering translation and expansion by the isotropickinematic hardening ruleand corresponding and (b) locus of principal plastic strain increment and criterion for the thickening and thinning modes.



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process because the untwining can only occur for the material which experienced the twin-ning process in advance. Therefore, in order for the untwining mode to occur, the follow-ing conditions should be satisfied. First, the loading history before untwining shouldsatisfy the twinning criterion. For the simplified first-order approach, the threshold

rtwin of residual of c-axis is introduced. Thus, only when the texture residue is less thanthis threshold during the previous twinning process, the untwining can be initiated. Sincethe untwining mode occurs by the reverse process of twinning mode, the deformationaccompanies thinning mode:

rtwin 6 rtwin and De

p1 Dep2 < 0: 21

In terms of the texture evolution during the untwining process, the similar method as donefor the twinning mode is adopted:

runtwin 1

euntwin

euntwin;cr1rtwin

1

euntwine0untwin;cr

euntwin 6 e0untwin;cr

0 euntwin > e0untwin;cr(

; 22

where euntwin;cr and euntwin are critical strain for the complete untwining from the perfecttwinned deformation state (rtwin = 0) and accumulated plastic strain during the untwin-ing deformation, respectively. Similarly, the threshold runtwin is used to determine theinitiation of twinning mode when the loading is revered from the untwiningdeformation.

The criterion for the slip mode occurs when the deformation state does not satisfyeither the twinning mode or untwining mode. For example, when tensile loading (or

thinning mode) is applied from the initial undeformed state or when the texture evolu-tion exhausts during the twinning or untwining modes or when the residue of c-axis dur-ing the previous twinning (or untwining) is larger than the prescribed threshold value,the slip modes are activated. Note that there is no rotation of c-axis during the slipmode.

To characterize the hardening behavior of three different modes from the experimen-tally obtained stressstrain responses, the 1-D flow curves are generalized with a

t

0

1=

10

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