lee wagoner ijp 2008

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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 stress–strain 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) 545–582

www.elsevier.com/locate/ijplas

mailto:[email protected]:[email protected]


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and asymmetry of the initial yield stress, the Drucker–Prager’s 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 Drucker– Prager 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 (Young’s 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 of magnesium 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, thestress–strain 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 stress–strain 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 Prager’s 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 Amstrong–Fredrick 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

Prager’s 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 stress–strain curve is piecewise linear because

Fig. 1. Schematic view of typical stress–strain 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 of yield 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 of describing the yielding asymmetry and anisotropy in stress–strain 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 Drucker–Prager 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 stress– strain 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 Hill’s48 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 isotropic–kine-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 Barlat’s96 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 stress–straincurves 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 stress–strain 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 stress–strain 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 stress–strain 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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Here, r is the Cauchy stress and a is the back-stress, which defines the central position of the 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

oðr aÞ dr oU

oðr aÞ da hisoðr aÞ

risodep ¼ 0 ð2Þ

where hiso drisode

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

da ¼ darisoðm Þ m ¼

da

de de

m

risoðm Þ : ð3Þ

where the relation UðdaÞ ¼ Uðdm Þ ¼ dc1risoðm Þ ¼ 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 ¼ C dee ¼ C ðde depÞ ð4Þwhere C is the elastic modulus, while d e and d ee are total and elastic strain increments,respectively. The plastic strain increment is defined by the normality rule as

dep

¼ dk

oU

oðr aÞ ¼ dk

oriso

oðr aÞ ¼ de

oriso

oðr aÞ;

ð5Þ

After some manipulations,

de ¼oriso

oðraÞ C de daoriso

oðraÞ C orisooðraÞ þ hiso: ð6Þ

Therefore, for a given strain increment d e 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

PðR AÞ Riso ¼ 0 ð7Þwhere 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

¼ Riso

riso ð

r

a

Þ ð8

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

dA da ¼ dlðR rÞ or dA ¼ da dlðR rÞ ¼ dA1 dA2 ð9Þ



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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 of Eq. (9),

dA2 ¼ d A2riso

ðR rÞ ð10Þ

where d A2 ¼ risoðdA2Þ and riso ¼ risoðR rÞ 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 ¼

r þ

d or dR

de ¼ dr

de þ dd

de ð11Þ

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,RisoðeÞ, and translation, AðeÞ (or A2ðeÞÞ with proper separation. Then, rðeÞ is obtained toaccount for the transient behavior every time reverse loading occurs, considering the pre-scribed gap function dðeÞ, which is dependent on din, the initial gap distance measured atthe start of reverse loading. The separation of r into the expansion and translation, risoðeÞand aðeÞ, is executed to properly account for the Bauschinger effect. For the hardeningmodel ðdr > 0Þ, the simple decomposition as following,

dr ¼ mldr þ ð1 mlÞdr ¼ driso þ da ð12ÞdR ¼ mbdR þ ð1 mbÞd R ¼ dRiso þ d A ð13Þ

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 ¼ risoðR rÞ ¼ UðR rÞ ð14Þwhich 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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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 ¼ dðdin; cold; cnew; epÞ ð17Þwhere cold and cnew are the vectors of c-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 <

ffiffiffi3

p , the tensile twin that elongates the c-axis during the deformation dominates.

Therefore, the criterion for the ‘‘twinning” mode includes the stress state which causes

elongation of the ‘‘current”

direction of c-axis. Here, ‘‘current”

means the c-axis beforerotation and usually thickness direction of sheet plate. This deformation mode can be alsocalled as ‘‘thickening” mode. In the plane stress state, the ‘‘thickening” mode is determinedby the principal increment of plastic strain rate. During the deformation, the yield stresssurface will translate and expand according to the isotropic–kinematic hardening rule,which is shown in Fig. 4a. By the normality rule, the plastic strain increment is normalto the yield surface at the corresponding stress point. Therefore, the two components of principal plastic strain increment can be produced as shown in Fig. 4(b). This is oftencalled as strain rate potential. The ‘‘thickening” or ‘‘twinning” mode is called when the fol-lowing conditions are satisfied:

Dep1 þ Dep2 P 0 ð18Þwhere Dep1;2 ¼ _ep1;2Dt are principal values of plastic strain increments during the time stepDt. Therefore, under plane stress condition,


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Dep1;2 ¼

Dep xx þ Dep yy 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Dep xx þ Dep yy

2 2

þ Dep xy s

ð19Þ

due to relation Dep1 þ Dep2 ¼ Dep xx þ Dep yy . During the twinning deformation, the c-axes ro-tate onto the loading plane. The amount of texture evolution (or rotation) may be ex-pressed by the functional relation with the critical strain, etwin;cr for the evolution to becompleted. Linear proportional factor is introduced in this study:

r twin ¼1 etwin

etwin;crðetwin 6 etwin;crÞ

0 ðetwin > etwin;crÞ

( ; ð20Þ

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

residual of c-axis remaining in the original direction. Thus, when etwin reaches etwin;cr, all of the 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

ðr twinÞ 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:

r twin 6 r twin and De

p1 þ Dep2 e0untwin;crÞ(

; ð22Þ

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 r untwin 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 stress–strain responses, the 1-D flow curves are generalized with a

σ

∑

t

δ

0ϑ

1=

10 <

t σ

=

1

10


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mathematical description. Under the current two-surface scheme, two hardening curvesneed to be defined: bounding curve ðRÞ and loading curve ðrÞ. In order to account forasymmetry in tension and compression, two different hardening curves or gap distancesare assigned to satisfy the plastic work rate equivalence. If the thinning mode is chosen

for a reference hardening curve which is usually obtained by the uni-axial tensiontest from the undeformed state, the gap distance between bounding and loading curvesis

d ¼ UðR xx r xxÞ ¼ R xx r xx ¼ R r ¼ R ðrt þ #ðR rt ÞÞ ¼ ð1 #ÞðR rt Þð23Þ

where Rxx and rxx are uni-axial tensile stresses on the loading and bounding curves,respectively, rt is the reference stress as indicated in Fig. 5, and # is a sigmoid functionwith S-shape, which gives a value between 0 and 1.

On the other hand, in terms of the hardening data (effective value of gap distance) forthe thickening mode or twinning mode, the curve is obtained from the uni-axial compres-sive test with the following conditions:

d ¼ r xx;0rC xx;0

dC e p ;C ¼ rC xx;0

r xx;0e

! ð24Þ

where dC is measured gap distance during the twinning (compressive) deformation which issimilar for defined in Eq. (23),

r xx;0

rC xx;0

is the asymmetry ratio which shows difference in initial

tensile and compressive stresses, and superscript ‘C’ denotes the ‘compressive’deformation.

The Eq. (23) can be re-written as a rate form:

dd

de ¼ d#

de ðR rt Þ þ ð1 #Þ drt

de þ # dR

de : ð25Þ

The material parameters for the functions rt and # are characterized by fitting the stress– strain curves of continuous in-plane tension/compression (or compression–tension) testsof magnesium alloys with different pre-strains (Boger et al., 2005). Note that these param-

eters are functions of initial gap distance after load reversal in the conventional two-sur-face model as well as the history of deformation which decides three different modes.Therefore,

rt ¼ rtðdin; cold; cnew; epÞ ð26Þ# ¼ #ðdin; cold; cnew; epÞ ð27Þ

3.3. Yield function of the anisotropic/asymmetric materials

In order to consider yielding asymmetry in tension and compression, classicalapproaches have introduced hydrostatic pressure. For instance, the Coulomb-Mohr yieldcriterion modifies the Tresca’s isotropic yield criterion by adding the effect of mean stresson yielding, while the Drucker–Prager criterion is based on the von Mises yield criterion



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with hydrostatic stress term. The models can reasonably represent the yielding behavior of pressure sensitive materials like porous medium.

More recently, the yield criterion for the HCP metals has been developed by Cazacuet al. (2004) where the linear transformation of the deviatoric Cauchy stress is adopted.

The yield surface is defined as

r ¼ ðUc=bÞ1a; Uc ¼ ðjS 1j kS 1Þa þ ðjS 2j kS 2Þa þ ðjS 3j kS 3Þa ð28Þ

where r is effective stress and S 13 are the principal values of the following matrix underthe plane stress condition:

~s ¼ C0 s; C0 ¼

C 011 C 012 C

013 0

C 012 C 022 C

023 0

C 013 C 023 C

033 0

0 0 0 C 066

0BBB@

1CCCA

ð29Þ

where C0 is a fourth-order tensor containing anisotropy coefficients and s is the deviatoricCauchy stress. Therefore, the principal values of ~s becomes

S 1;2 ¼ ~ s xx þ ~ s yy 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

~ s xx þ ~ s yy 2

2þ ~ s2 xy

s ; S 3 ¼ ~ s zz ð30Þ

The coefficient of anisotropy and other material constants can be obtained by the tensionand compression tests along three different directions. For more details, refer to the articleby Cazacu et al. (2004).

In the present study, to further simplify the criterion by reducing the material param-eters under the plane-stress condition, the Drucker–Prager model is slightly modified byadding the anisotropic coefficients. Note that the criterion in Eq. (28) can be also quadraticform with a = 2. The criterion adopted in the paper is

U ¼ p ðr2 xx b2r xxr yy þ b22r2 yy þ 3b23r2 xy Þ1=2 þ qðr xx þ b4r yy Þ riso ¼ 0; ð31Þ

where riso denotes the size of the yield surface and the five parameters p, q, b2, b3 and b4are the material constants to be determined experimentally. Note that the proposed yieldcriterion is a first-order homogeneous function of the stress tensor for the effective stress.

The yield criterion reproduces the von Mises yield criterion if b2 = b3 = 1, q = 0 and theclassical Drucker–Prager yield criterion if b2 = b3 = b4 = 1 and q 6¼ 0. The above yield cri-terion was applied to predict the bending behavior of polymeric composite material whichhas high anisotropy and asymmetry (Kim et al., 2007b).

Besides the yield criterion’s ability in describing anisotropy and asymmetry (or eccen-tricity), it can also prescribe shear yield stress independently. Therefore, five materialparameters can be determined from the two tensile yield stresses rT xx, r

T yy , two compressive

yield stresses rC xx, rC yy in the x, y directions, and the shear yield stress rxy or tensile yield

stress rT 45 xx in the 45 direction.With tensile and compressive yield stresses and Eq. (31):

p rT xx þ qrT xx ¼ riso p rC xx qrC xx ¼ riso

ð32Þ



16/38

where the reference value is the tensile yield stress in the x-direction, riso ¼ rT xx and ‘T’ and‘C’ denote tension and compression, respectively. For the tensile and compressive tests inthe y direction, Eq. (31) gives

p jb2jrT yy þ qb4rT yy ¼ riso p jb2jrC yy qb4rC yy ¼ riso

ð33Þ

From the experimentally measured yield stresses in tension and compression, the param-eters p, q, b2 and b4 are obtained.

When the shear stress rxy is known, Eq. (31) gives ffiffiffi3

p p jb3jr xy ¼ riso ð34Þ

Since the pure shear stress is hardly measured with accuracy, an alternate 45 tension test

can be utilized to determine b3 by solving the following equation:

p rT45 xx

2

2 b2

rT45 xx2

2þ b22

rT45 xx2

2þ 3b23

rT45 xx2

2 !1=2þ q r

T45 xx

2 þ b4

rT45 xx2

¼ riso

ð35ÞThe solutions of Eqs. (32)–(35) are given as follows:

p ¼ 12

1 þ rT xx

rC xx ð36Þ

q ¼ 12

1 rT xx

rC xx

ð37Þ

b2 ¼ðrT yy þ rC yy Þ:rT xx:rC xxðrT xx þ rC xxÞ:rT yy :rC yy

ð38Þ

b4 ¼ðrC yy rT yy Þ:rT xx:rC xxðrC xx rT xxÞ:rT yy :rC yy

ð39Þ

b3 ¼ 2

ffiffiffi3p rT xx:r

C xx

ðrT xx:r

C xx

Þr xy

or ð40Þ

b3 ¼ 1 ffiffiffi

3p 1 p 2

2 riso

rT 45 xx qð1 þ b4Þ

2 ð1 b2 þ b22Þ

" #1=2

3.4. Numerical implementation

For the numerical formulation, the incremental deformation theory (Chung andRichmond, 1993) was applied to the elasto-plastic formulation based on the materiallyembedded coordinate system. For a given total strain increment De and the other statevariables from the previous time step, the numerical formulation provides increments of elastic and plastic strain, Cauchy stress and back stress. The stored state variables at



17/38

the end of the previous time step are stress, back stress, plastic strain, and the informa-tion on the orientation of c-axes along with their residues. During the incremental step,the magnitudes of all of these variables are functions of the incremental effective strainDe only. Therefore, the following nonlinear equation for De is valid for the loading

surface:

Uðr0 a0 þ DrðDeÞ DaðDeÞÞ ¼ risoðe0 þ DeÞ ð41Þalong with two hardening curves, risoðeÞ and aðeÞ, which are prescribed in advance oncereloading criterion is satisfied and the deformation mode is determined as discussed inthe previous section. After De is obtained as a solution of Eq. (41), updated stress andback-stress are obtained from risoðeÞ and aðeÞ, respectively, while the new configurationof the bounding surface is also updated at the end of each step considering RisoðeÞ, A2ðeÞ and aðeÞ for De. The two hardening curves of the loading surface, risoðeÞ and aðeÞ,are newly updated every time reloading occurs, considering the new initial gap distance

din, corresponding gap function and deformation history which decides one of three defor-mation modes.

The predictor–corrector scheme was used to solve De in Eq. (41) for the loading surface;i.e.,

F ¼ Uðrnþ1 anþ1Þ risoðDeÞ ¼ 0; ð42Þwhere

rnþ1 ¼ rTnþ1 DeC oriso

oðrnþb anþbÞ ð43Þ

and

anþ1 ¼ an þ DaðDeÞ ðrnþb anþbÞriso

; ð44Þ

where 0 6 b 6 1.In Eq. (43), the superscript ‘T’ stands for a trial state and the subscript denotes the pro-

cess time step. Therefore,

rTnþ1 ¼ rn þ C De: ð45Þ

Eq. (42) is a nonlinear equation to solve for De, when De is given. Then, linearization of

Eq. (42) leads to

dðDeÞk þ1 ¼ F k o F oDe

k ð46Þ

for the k th iteration and

o F

oDe ¼ o F

ornþ1

ornþ1oDe

þ o F oanþ1

oanþ1oDe

þ o F oriso;nþ1

oriso;nþ1oDe

; ð47Þ

where

ornþ1

oDe ¼ C or

isooðrnþb anþbÞ ð48Þ

oanþ1oDe

¼ oDaoDe

rnþb anþbriso

ð49Þ



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o F

ornþ1¼ o F

oanþ1¼ oriso

oðrnþ1 anþ1Þ ð50Þo F

oriso¼ 1: ð51Þ

After enþ1 (therefore, along with rn+1 and an+1) is obtained for the loading surface, the cur-rent stress on the bounding surface Rn+1 and its center An+1 (or DA2) are obtained fromthe following two conditions:

Anþ1 ¼ An þ Dan D A2ðDenþ1Þ ðRnþ1 rnþ1ÞrisoðRnþ1 rnþ1Þ ð52Þ

Rnþ1 Anþ1 ¼ RisoðDenþ1ÞrisoðDenþ1Þ ðrnþ1 anþ1Þ ð53Þ

which are two simultaneous equations for the stress on the bounding surface and the cen-ter of the surface: Rn+1 and An+1. Therefore, adding Eqs. (52) and (53), the following non-linear equation is obtained for the unknown quantity Rn+1:

U ¼ Rnþ1 RisoðDenþ1ÞrisoðDenþ1Þ ðrnþ1 anþ1Þ An Dan þ D A2ðD

enþ1Þ ðRnþ1 rnþ1ÞrisoðRnþ1 rnþ1Þ ¼ 0

ð54ÞLinearizing Eq. (51) for the Newton–Raphson method provides, for the k th iteration,

dRk þ1nþ1 ¼ Uk

oUoRnþ1

k ; ð55Þwhere

oU

oRnþ1¼ I þ D A2ðDenþ1Þ

risoðRnþ1 rnþ1Þ I D A2ðDenþ1Þ

r2isoðRnþ1 rnþ1Þ oriso

oRnþ1: ð56Þ

Here, I is the second-order identity tensor. After solving Rn+1 from Eq. (56), An+1 is ob-tained from Eq. (52).

4. Model calibration

4.1. Characterization of material parameters of AZ31B magnesium alloy sheets

One-dimensional continuous in-plane tension–compression and compression–tensiontests are performed in order to verify the implementation of formulations developed in

Table 1Chemical compositions of Mg alloys

Al Cu Mn Zn Si Mg

AZ31B-O tempered 3.0 – 0.2 1.0 – BalanceAZ31B 3.0 0.05 0.2 1.0 0.1 Balance


http://-/?-http://-/?-


19/38

the previous sections. Two model materials are adopted to investigate the effect of temper-

ing on the cyclic behavior of magnesium alloys: O-tempered AZ31B magnesium alloy with3.2 mm thickness and AZ31B magnesium alloy with 2 mm thickness. The experimentaldata for the O-tempered AZ31B were reproduced from the previous work by Lou et al.(2007) and as-received AZ31B alloy was newly adopted in the present work. The chemicalcompositions of the two materials are listed in Table 1. The stress–strain responses of O-tempered AZ31B in tension–compression–tension (T–C–T) and compression–tension(C–T) are shown in Fig. 6a and b, respectively, while in Fig. 7a and b for AZ31B withouttempering. For the continuous tension/compression tests, the specially designed devicewhich prevents the sheet from being buckled during the compression was utilized. Notethat the stress–strain curves shown in Figs. 6 and 7 are corrected curves after considering

friction between sheets and clamping device and bi-axial effect from clamping force. Formore details on the experimental procedure, refer to Boger et al. (2005). From these fig-ures, three important unique features are observed, which should be properly consideredwith the current constitutive models.

Strain

-0.10 -0.05 0.00 0.05 0.10 0.15 0.20

S t r e s s ( M P a )

-300

-200

-100

0

100

200

300

400T-C-T(Lou et al., 2007)

strain

-0.12 -0.08 -0.04 0.00 0.04 0.08 0.12


-300

-200

-100

0

100

200

300

400

C-T(Lou et al., 2007)

b

a

Fig. 6. The stress–strain responses of O-tempered AZ31B magnesium alloy sheets in (a) tension–compression– tension (T–C–T) and (b) compression–tension (C–T) (Lou et al., 2007).



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strain

-0.08 -0.04 0.00 0.04


-300

-200

-100

0

100

200

300

400

AZ31B, 2mm thickC-T

Strain

-0.10 -0.05 0.00 0.05 0.10 0.15


-300

-200

-100

0

100

200

300

400

AZ31B, 2mm thickT-C-T

a

b

Fig. 7. The stress–strain responses of AZ31B Mg alloy sheets (no tempering, 2 mm thickness) in (a) tension– compression–tension (T–C–T) and (b) compression–tension (C–T).

Table 2

Initial yield stresses in uni-axial tension and compression of Mg alloy (unit: MPa)

rT xx rC xx r

T yy r

C yy r

45T xx

AZ31B-O tempereda 192 110 164 104 180AZ31B 220 120 250 140 210

a Lou et al. (2007).

Table 3Anisotropic parameters for the modified Drucker–Prager yield surface

p q b2 b3 b4

AZ31B-O tempered 1.373 0.373 1.099 1.03 0.906AZ31B 1.318 0.318 1.138 0.994 0.943



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(1) Strong ‘‘asymmetry” in tensile and compressive yield stresses- the ratios of initial

yield stresses in tension and compression are 192 MPa: 110 MPa for O-temperedAZ31B and 220 MPa:120 MPa for AZ31B.

(2) Unusual ‘‘concave-up” shape during compression and ‘‘S-shape” during tension fol-lowing compression, which is explained by deformation twinning.

Normalized Yield Stress at TD

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

N o r m a l i z e d Y i e l d S t r e s s a t R D

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0Calculated (no offset)

fs

Measured (no offset)

Calculated (0.2% offset)

Calculated (0.2% offset)

AZ31B-O tempered

Normalized Yield Stress at RD

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5

N o r m a l i z e d Y i e l d S t r e s s a t T D

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

C

AZ31Bt=2mm

a

bCalculated (0.2% ffset)

Measured (0.2% ffset)

Fig. 8. Initial yield locus of (a) AZ31B-O tempered (3.2 mm thick): Closed squares are experimentally measuredyield stress by no-offset and 0.2% offset method. The uni-axial tests data were reproduced from Lou et al. (2007)and balanced bi-axial test data was reproduced from Jain and Agnew (2005). (b) AZ31B (no tempering, 2 mmthick) with modified Drucker–Prager model. Zero shear stress is assumed.



22/38

(3) Significant ‘‘reduction” in the size of elastic range during compression and compres-sion following tension.

The yield parameters for the modified Drucker–Prager yield surface are calculated

from the measured data and Eqs. (36)–(40). Five initial yield stresses for the tensionand compression of different loading directions (Table 2) are utilized to calculate aniso-tropic coefficients (Table 3). Considering uni-axial tension tests as reference state, theyield surfaces normalized by initial tensile yield stress are shown in Fig. 8. Zero shearstress is assumed for both model materials. In case of O-tempered AZ31B sheet, bothcriteria with zero and 0.2% offset methods in determining yield stress are shown in thefigure. For the yield surface of O-tempered AZ31B Mg sheet shown in Fig. 8a, the uni-axial test data were reproduced from Lou et al. (2007), which are used for the calcu-lation of yield surface, and balanced bi-axial yield stress is used from Jain and Agnew(2005) for the verification. The figures show that the strong anisotropy and asymmetryin the initial yield stress are well represented by the current modified yield surface forboth materials.

The hardening parameters defined in Section 3.2 are characterized from the measureddata by applying specific phenomenological forms for the Eqs. (26) and (27). Since thebounding surface is arbitrarily defined, constant bounding surface with large enough ini-tial size is assumed to simplify the characterization procedure. That is, dR ¼ 0 in Eq. (25).Therefore, the bounding surface does not expand or move during the deformation and thegap distance is measured from the static bounding curve (with constant value) and loadingcurve. Therefore,

R ¼ R: ð57ÞHere, for the model material AZ31B alloy sheets, constant value R ¼ 520 MPa is chosen.By considering the common shape of bounding and loading surface and the asymmetry of the loading surface, the constant bounding values in tension and compression are 520 MPaand 300 MPa for O-tempered AZ31B and 520 MPa and 284 for AZ31B.

Acummulated strain0.00 0.02 0.04 0.06 0.08 0.10

δ ( M P a )

150

200

250

300

350

AZ31B O-tempered, measured

AZ31B O-tempered, fitted

AZ31B, measured

AZ31B, fitted

Fig. 9. Fitting curve of gap distance from the measured uni-axial tensile curves (closed circle: AZ31B-Otempered, open square: AZ31B without tempering).



23/38

As for the loading surface, combined isotropic–kinematic hardening is assumed withthe gap distance for different deformation modes. The gap distance during the slip defor-mation from undeformed state is calculated from the measured stress–strain curve andbounding surface defined in Eq. (57). The gap function for slip mode is well defined with

the following function as used for the conventional alloy sheets such as aluminum alloy(Lee et al., in press):

d ¼ asðdinÞ þ bsðdinÞ expðcsðdinÞelÞ; ð58Þwhere as, bs, cs are material parameters which depend on initial gap distance and e

l is theplastic strain whose value is re-initialized for each reverse loading. From the experimentaldata, the material parameters in Eq. (58) are obtained by considering gap distances be-tween bounding and loading stresses. Due to the limited availability of the experimentaldata, constant parameters are assumed and obtained by curve fitting hardening curvesduring the initial tension. The fitted curve is shown in Fig. 9 for two materials and the

δ in (MPa)300 350 400 450 500 550

a U T

( M P a )

0

1000

2000

3000

4000

5000

6000

AZ31B

AZ31B-O tempered

δin (MPa)300 350 400 450 500

b U T

( M P a )

0

20

40

60

80

100

120

140

AZ31B

AZ31B-O tempered

a

b

Fig. 10. Fitting parameters for rt: (a) aUT, (b) bUT, (c) aT, (d) bT with varying din (closed circle: AZ31B-Otempered, open square: AZ31B without tempered).



24/38

constant parameters are as = 216.2 MPa, bs = 115.4 MPa, and cs = 16.2, for O-temperedAZ31B and as = 186.0 MPa, bs = 114.0 MPa, and cs = 15.4 for AZ31B.

In terms of hardening curve during the compression or compression following tension,the ‘‘sigmoid” or S-shaped function is introduced to represent particular stress–straincurve of magnesium alloys during deformation twinning. The two functions, rt and #are characterized by fitting the stress–strain curves of continuous T–C–T and C–T withrespect to several different initial gap distances. For the function rt , the following linearfunction with accumulated plastic strain is used.

rt ¼ aT or UTðdinÞ þ bT or UTðdinÞel ð59Þwhere aT or UT and bT or UT are material parameters which depend on the initial gap dis-tance with load reversal. The subscripts ‘T’ and ‘UT’ denote the twinning and untwiningmodes, respectively. Fig. 10 shows the linear dependence of the two parameters on the ini-tial gap for the untwinning mode (Fig. 10a,b) and twining mode (Fig. 10c,d).

δin (MPa)120 140 160 180 200 220

a T

( M P a )

0

100

200

300

400

500

600

AZ31B

AZ31B-O tempered

δin (MPa)100 120 140 160 180 200 220

b T ( M P a )

0

50

100

150

200

AZ31B

AZ31B-O tempered

d

c

Fig. 10 (continued )



25/38

For the function #, the following sigmoid type curve fitting is utilized:

# ¼ y 0;T or UTðdinÞ þ cT or UTð

dinÞ1 þ exp el x0;TorUT ðdinÞ

d T or UTðdinÞ

ð60Þwhere y0,T or UT, cT or UT, x0,T or UT, and d T or UT are material parameters and obtained byconsidering initial gap distance. The dependence of these four parameters on the initialgap stress is assumed piecewise linear as shown in Fig. 11a for the untwining andFig. 11b for the twinning mode, respectively. Note that Eqs. (57)–(60) are empirically ob-tained forms which conform the cyclic behavior of the present two materials in roomtemperature.

Other common features in the cyclic stress–strain curves of sheet alloys are early re-

yielding (or Bauschinger effect) and rapid work hardening rate (or transient behavior)when the material undergoes change of loading path. The Bauschinger effect and transientbehavior could be modeled with combined isotropic–kinematic hardening laws (Chunget al., 2005). Proper measurement of the size of yield surface for the isotropic hardening

δin (MPa)

350 400 450 500

y 0 , U

T , c U T , d U T , x 0 , U

T

0.00

0.05

0.10

0.20

0.40

0.60

0.80

cUT

x0,UT

y0,UT

dUT

δin (MPa)

120 140 160 180 200 220

y 0 , T , c T , d T , x 0 , T

0.0

0.1

0.2

1.0

2.0

3.0

cT

x0,T

y0,T

dT

a

b

Fig. 11. Material parameters of # (Eq. (60)) for: (a) Untwining, (b) Twinning with din (closed circle: AZ31B-Otempered, open square: AZ31B without tempering).



26/38

∆ σ (MPa)

∆ σ (MPa)

∆ σ (MPa)

0 20 40 60 80 100 120

∆ | σ f - σ r | i s o , e x p

( M P

a )

∆

| σ f - σ r | i s o , e x p

( M P a )

∆ | σ f - σ r | i s o , e x p

( M

P a )

0

50

100

150

200

R a t i o , ∆

| σ f - σ r | i s o , e x p /

∆ | σ f - σ r | p u r e i s o

R a t i o , ∆

| σ f - σ r | i s o , e x p /

∆ |

σ f - σ r | p u r e i s o

R a t i o , ∆

| σ f - σ r | i s o , e x p /

∆ | σ f - σ r | p u r e i s o

0.0

0.2

0.4

0.6

0.8

1.0

Ratio

Pure isotropic

Measured

avg: 0.87

avg: 0.88

0 20 40 60 80 100 120 140-300

-200

-100

0

100

200

300

400

-20

-15

-10

-5

0

5

10

Ratio

Pure isotropic

Measured

avg: -6.98avg: -5.96

0 20 40 60 80 100 120 1400

100

200

300

400

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Ratio

Pure isotropic

Measured

avg: 0.84

avg: 0.81

a

b

c

Fig. 12. Measured size change and the ratio of isotropic change to the assumed pure isotropic case for variouspre-strains or different stress increments from initial yield stress: (a) during the slip mode, (b) during the twinningdeformation without assumption of instant shrinkage, and (c) during the twinning deformation with instantshrinkage (closed circle: AZ31B-O tempered, open square: AZ31B without tempering).



27/38

ratio, which is often called as Bauschinger ratio (Boger, 2006) is needed. The size change of yield surface during deformation is measured from the experimental data in Fig. 12. Forthe assumed (pure) isotropic hardening case, the stress increment during deformation com-pletely contributes to the size change of the yield surface. Therefore, the size change for the

pure isotropic Djrf rrjpure iso becomesDjrf rrj pureiso ¼ Drð1 þ f aÞ ð61Þ

where Dr is the stress increment and f a is the factor of asymmetry of the initial yieldsurface. f a is assumed as

rC xrT x

for tension and rT x

rC xfor compression. For the symmetric

material with f a = 1, size change becomes 2Dr. Note that there is no size change forassumed (pure) kinematic hardening. The size change during tensile deformation fromundeformed state is measured and compared with that of the assumed pure isotropichardening in Fig. 12a. For several different pre-strains or different stress incrementsfrom initial yield stress, the following ratio of size change to the assumed (pure) iso-tropic increment is calculated:

ml ¼ Djrf rrjiso;measured=Djrf rrjpure iso ð62ÞThe variations of the ratio during the slip deformation are very small for both mate-rials and their average value are 0.87 and 0.88, respectively. Therefore, constant ratiosof isotropic hardening in Eq. (12) are adopted using the average values for the slipmode.

In terms of the size change of yield surface during the twinning or compression frominitial material state, similar procedure is applied to calculate the assumed pure isotropicchange and size change in measured stress–strain curves of tension following compression(C–T curves in Figs. 6b and 7b) with several pre-strains. In case of O-tempered AZ31B, forthe pre-strain around 2%, the size of yield surface (or size of linear region) is approxi-mately 100 MPa, while initial size of yield surface before compression (or initial size of

Fig. 13. Back stress of loading surface after instant shrinkage.



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True strain

-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08


-300

-200

-100

0

100

200

300

400

AZ31B-O temperedCompression-TensionPre-strain: -0.023

True strain

-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08


-300

-200

-100

0

100

200

300

400


True strain

-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08

S t r e s s ( M P a

)

-300

-200

-100

0

100

200

300

400


a

b

c

Fig. 14. Comparisons of calculated uni-axial compression–tension (C–T) curves of AZ31B Mg alloy sheet withmeasurements for various initial compressive strains: pre-strains are: (a) 0.023, (b) 0.045, and (c) 0.075; Lines andsquares are calculated results and measured data reproduced from Lou et al. (2007), respectively.



29/38

yield surface) is 302 MPa. Therefore, the conventional isotropic hardening model withphenomenological yield surface cannot explain the size reduction or softening behavior.As shown in Fig. 12b, the average ratios of size change to the assumed pure isotropicincrement are negative and the absolute value are decreasing for two materials. This means

that the yield surface shrinks abruptly just after compression and expands again as thecompressive deformation proceeds.

In order to consider the effect of softening during twinning mode, the initial shrinkageof the yield surface is introduced in the current model. Here, for the AZ31B magnesiumalloy sheets, the shrinkage ratios are assumed as 0.325 (99/302) for O-tempered sheetand 0.54 (185/340), respectively. With the new initial sizes of yield surfaces, the sizechange and corresponding ratio are re-calculated in Fig. 12c. The figure shows that theyield surface expands almost isotropically up to 5% of plastic strain and then isotropic– kinematically. The average ratios 0.84 and 0.81 which are very similar to that of slip modesare adopted as ratios of isotropic hardening in Eq. (12). As for the size change of yield

True strain

-0.10 -0.05 0.00 0.05 0.10 0.15 0.20


-300

-200

-100

0

100

200

300

400

AZ31B-O temperedTension-Compression-TensionPre-strain: 0.017

True strain

-0.10 -0.05 0.00 0.05 0.10 0.15 0.20


-300

-200

-100

0

100

200

300

400


a

b

Fig. 15. Comparisons of calculated uni-axial tension–compression–tension (T–C–T) curves of AZ31B Mg alloysheet with measurements for various initial tensile strains: pre-strains are (a) 0.017, (b) 0.045, (c) 0.075, and (d)0.165; Lines and squares are calculated results and measured data reproduced from Lou et al. (2007), respectively.



30/38

surface during the untwining mode or tension following compression in Figs. 6 and 7, thesame shrinkage ratios and the ratios of isotropic hardening are assumed. The center of yield surface or back stress needs to be determined once the instant softening (or shrink-

age) occurs. Since the current stress shares the same position on the loading surfacesbefore and after instant softening (marked a in Fig. 13), the back stress of loading surfacecan be calculated considering the stress on the bounding surface (marked A in Fig. 13) andrelation in Eq. (8):

a0 ¼ r r

0iso

RisoðR AÞ ð63Þ

where a0 and r0iso are back-stress and size of the yield surface after instant softening,respectively.

4.2. Correlation of uni-axial cyclic tests

The developed constitutive models based on two-surface hardening law and asym-metric yield surface are implemented into a commercial finite element program ABA-

True strain

-0.10 -0.05 0.00 0.05 0.10 0.15 0.20


-300

-200

-100

0

100

200

300

400


True strain

-0.10 -0.05 0.00 0.05 0.10 0.15 0.20


-300

-200

-100

0

100

200

300

400


c

d




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QUS/Standard which allows the new constitutive models to be implemented by UMATsubroutine. In terms of finite element simulations for the uni-axial tension or compres-sion tests, a four-node shell element with reduced integration, S4R is utilized. To sim-ulate a uniform stress–strain response which is consistent with experimental procedure,proper boundary conditions were applied to the one shell element. Note that the cur-rent analysis can be carried out without use of finite element method. However, inorder to validate the developed constitutive laws and their numerical implementation,finite element procedure was adopted in this work. Fig. 14 shows the comparison of the measured continuous uni-axial compression–tension (C–T) tests shown in Fig. 6b

to the results calculated from the finite element simulations with developed modelsfor O-tempered AZ31B sheet. Three different pre-strains are applied before unloading;0.023, 0.045, and 0.075. In general, the developed constitutive models are able toreproduce the main features of the tension following compression curves. Especially,

True Strain

-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06


-300

-200

-100

0

100

200

300

400

AZ31BCompression-TensionPre-strain: -0.02

True Strain

-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06


-300

-200

-100

0

100

200

300

400


a

b

Fig. 16. Comparisons of calculated uni-axial compression–tension (C–T) curves of AZ31B Mg alloy sheet (notempering, 2 mm thick) with measurements for various initial compressive strains: pre-strains are (a) 2%, (b) 4%,(c) 6%, and (d) 8%; Lines are calculated.



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the unusual concave-up shape of flow curves during the initial compression (or twin-ning mode) and sigmoid shape (S-shape) during the tension following compression

are well reproduced for each stress–strain curve with various pre-strains. Also, the con-stant parameter used for the instant shrinkage (or softening) of the initial yield surfaceand constant ratio of isotropic hardening during the subsequent plastic deformationcan be well verified by the good agreement between the results of model and measure-ments. The largest discrepancy is shown in the transient region of the reloading curveas illustrated in Fig. 14c. The measurement curve shows smooth transient from elasticto plastic, while almost linear in the simulated curve.

Similar comparisons are made for the continuous uni-axial tension–compression–ten-sion (T–C–T) tests in Fig. 15 for O-tempered AZ31B sheet. The test involves all threedifferent deformation modes explained in the previous sections: slip mode during the

initial tension, twinning mode during the compression following tension, and untwiningmode during the tension following compression. Four different pre-strains beforeunloading from the initial tension are chosen: 0.017, 0.045, 0.075, and 0.165. For allfour stress–strain curves, excellent agreements are shown with the developed constitu-

True Strain

-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06


-300

-200

-100

0

100

200

300

400


True Strain

-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06


-300

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-100

0

100

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c

d




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tive models. It is shown that the constant averaged ratio of isotropic hardening wellrepresents the isotropic expansion of the yield surface during the slip mode. Also,the rapid decrease and subsequent expansion of the loading surface during the twinning(or untwining) deformation by introducing instant shrinkage ratio and the ratio of iso-tropic hardening are well verified from good agreements in the compression curves andsizes of the linear regions during the reloading. Note that these two parameters arecharacterized from the compression–tension curves shown in Fig. 6b. The same proce-dures have been applied to AZ31B sheet for the compression–tension and tension–com-pression–tension tests. Similarly, the current material models are able to reproduce theexperimentally observed behavior except for the transient regions as shown in Figs. 16

and 17. Note that the current model has limited validity only to the measured stress– strain curves with limited strain range. However, the present constitutive equations maybe effectively utilized to predict the forming and springback behavior in a typicalstamping process where the strain ranges are moderate.

True Strain

-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12


-300

-200

-100

0

100

200

300

400

500

AZ31BTension-Compression-TensionPre-strain: 0.04

True Strain

-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12


-300

-200

-100

0

100

200

300

400

500


a

b

Fig. 17. Comparisons of calculated uni-axial tension–compression–tension (T–C–T) curves of AZ31B Mg alloysheet (no tempering, 2 mm thick) with measurements for various initial tensile strains: pre-strains are: (a) 4%,(b) 6%, (c) 8%, and (d) 10%; Lines are calculated results.



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5. Conclusions

The constitutive modeling for the magnesium alloy sheets was developed to practicallyrepresent their unusual mechanical properties. The developed constitutive equationsinclude the modified anisotropic yield function and advanced hardening model. First of these is the modified yield criterion with pressure dependent term and can represent highdirectional differences in the initial yield stress (anisotropy) and also high asymmetry intension and compression. The unusual hardening behavior during the non-monotonousdeformation was well represented by the practical two-surface model. The summary of the works done in the present paper is as follows:

1. A practical two-surface plasticity model developed for symmetric materials has been

further extended to represent hardening behavior of magnesium alloy sheets. Thetwo-surface model is based on classical Dafilias/Popov and Krieg concepts and is ableto represent complex hardening effects for non-monotonous loading such as Bauschin-ger effect and transient.

True Strain

-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12


-300

-200

-100

0

100

200

300

400

500


True Strain

-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12


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2. To represent anisotropy and asymmetry between the yield stresses in tension and com-pression, the classical yield surface, Drucker–Prager criterion, was modified by intro-ducing the coefficients of anisotropy in the plane stress condition. The yield surfaceis a conical surface with additional hydrostatic stress term in the three-dimensional

space, thus can represent difference in tension and compression in the plane stress state.3. The developed theory was numerically formulated for the finite element analysis and

implemented into the commercial finite element program ABAQUS/Standard with usermaterial subroutine UMAT.

4. In order to characterize material parameters for the developed constitutive equations,two AZ31B magnesium alloy sheets were utilized: one with O-tempered conditionand the other without tempering. Continuous in-plane uni-axial cyclic tests wereadopted for the characterization purpose. As reported previously, three deformationmodes are observed during the tests. Uni-axial tensile deformation is dominated bythe slip mode and the flow curve is normal concave-down shape. The deformations dur-ing the in-plane compression and tension following compression are dominated by thetwinning and untwining modes which show unusual concave-up or S-shape stresscurves. Besides the different type of hardening behavior, the size of elastic region is alsounusual for each deformation mode. The size of linear region increases from the initialstate during the slip mode, while abrupt reduction of the linear region and subsequentincrease are observed for the twinning and untwining modes.

5. Five experimentally measured initial yield stresses were utilized for the anisotropic/asymmetric yield surface: tensile yield stresses in the rolling and transverse directions,compressive yield stresses in the rolling and transverse directions and tensile yield stress

in the 45

.6. Since the hardening behavior is updated every time when the reversal of loading direc-tion occurs in the two-surface model, different hardening curves (or gap distance) areused for each deformation mode. For the slip mode, normal exponential type gap func-tion was used. On the other hand, for the twinning and untwining modes, sigmoid typefunction which represents S-shape hardening was adopted.

7. To effectively consider the abrupt reduction of linear elastic region during the twinningor untwining deformation, constant shrinkage ratio of the yield surface was introducedand then the yield surface increases with constant ratio.

8. Calculated stress–strain curves for the uni-axial compression–tension (C–T) tests and

tension–compression–tension (T–C–T) tests with various pre-strains were comparedwith measurements. In general, the model could reproduce the experimental behaviorwith great accuracy. Both the unusual hardening curve during the twinning modeand size change of yield surface were well predicted although small discrepancy inthe transient behavior was observed.

9. Finally, based on the promising results of the present constitutive modeling work, fur-ther research on the prediction of real forming and springback behavior needs to beexplored as an application part of the current paper.

Acknowledgements

This work was supported by the sabbatical program of KNU, the Int. Joint R&D Pro-gram by MCIE (10028109), by the SRC/ERC Program of MOST/KOSEF (R11–2005–



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065) in Korea and by the National Science Foundation (DMI–0355429). MGL also appre-ciates the partial support from the center for Advanced Materials Processing of the 21st

Century Frontier R&D Program by MOST.

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