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International Journal of Plasticity 24 (2008) 1307–1332

www.elsevier.com/locate/ijplas

Micromechanical modelling of the effectof plastic deformation on the mechanical behaviour

in pseudoelastic shape memory alloys

X.M. Wang, B.X. Xu, Z.F. Yue *

School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical

University, Xi’an 710072, PR China

Received 12 May 2007; received in final revised form 10 September 2007Available online 29 September 2007

Abstract

Except for the recoverable strain induced by phase transformation, NiTi alloys are very ductileeven in the martensite phase. The purpose of the present paper is to study the influence of permanentdeformation, which results from plastic deformation of martensite, on the mechanical behaviour ofpseudoelastic NiTi alloys. Based on phenomenological theory of martensitic transformation andcrystal plasticity, a new three dimensional micromechanical model is proposed by coupling boththe slip and twinning deformation mechanisms. The present model is implemented as User MATerialsubroutine (UMAT) into ABAQUS/Standard to study the influences of plastic deformation on thestress and strain fields, and on the evolution of martensite transformation. Results show that withthe increasing of plastic deformation the residual strain increases and the phase transformationstress–strain curves from the martensite to austenite become steeper and less obvious. Both charac-teristics, stabilisation of martensite and impedance of the reverse transformation, due to plasticdeformation are captured.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Pseudoelasticity; NiTi alloys; Permanent deformation; Phase transformation; Finite element

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

doi:10.1016/j.ijplas.2007.09.006

* Corresponding author. Tel./fax: +86 29 88460251.E-mail address: zfyue@nwpu.edu.cn (Z.F. Yue).

Nomenclature

RVE representative volume elementa, a0, b, c, b lattice parameters for a Ti-49.75 at.% Ni alloyB constant in the chemical energy termd(s) unit vector denoting the slip directiond(a) unit vector denoting the a-deformation twinning system direction.EM elastic modulus of martensiteEA elastic modulus of austeniteEtr average transformation strain in RVEEp average plastic strain in RVEf total volume fraction of deformation twinned martensitef(a) volume fraction of a-deformation twinning system_f ðaÞ0 reference deformation twinning rateFC critical driving force for phase transformationg(s) hardness of slip systemg(a) hardness of a-deformation twinning systemgtr magnitude of phase transformationGA shear modulus of austeniteh(s) coefficient of slip hardeninghðsÞ0 , q1 material constants for slip hardeningh(ab) components of the hardening matrix for deformation twinning systemhðaÞ0 , q2 material constants for deformation twinning hardeningHmn components of interaction energy matrixI fourth rank identity tensork1 material rate sensitivity of slip systemk2 material rate sensitivity of deformation twinning systemK latent heat of transformation per unit volumel unit vector denoting the habit plane normalm unit vector denoting the transformation directionP(s) tensorial direction of the shear caused by slipP(a) tensorial direction of the shear caused by a-deformation twinning systemDR volume increment of heat generation per unit times(s) unit vector denoting the normal of a slip planes(a) unit vector denoting the normal of a- deformation twinning planeS average effective elastic complianceSA elastic compliance of austeniteSM elastic compliance of martensiteDS difference of elastic compliance between two phasesT test temperatureT0 phase equilibrium temperaturet Poisson’s ratioee local elastic strainen stress free transformation strain for the nth martensite varianteT internal transformation strain

1308 X.M. Wang et al. / International Journal of Plasticity 24 (2008) 1307–1332

nn volume fraction of the nth martensite variantn total volume fraction of martensite in phase transformationw complementary free energywel elastic energywpo potential energy of loading systemwch chemical energy due to phase transitionwsur surface energyr local stressR average stress in RVEs internal stress related to the incompatibilities in the transformation fields(s) resolved shear stress of slip systems(a) resolved shear stress of a-deformation twinning systemsðsÞ0 value of g(s) at the beginning of the deformationsðaÞ0 value of g(a) at the beginning of the twinning deformationsðsÞs saturation strength for slip systemstw

s saturation strength of deformation twinning system_cðsÞ plastic shear rate of the slip system_cðsÞ0 reference shear rate of the slip system_cðaÞtw plastic shear rate of the a- deformation twinning systemcðaÞT constant shear strain of the a- deformation twinning systemk0, kn Lagrange multiplierssgn ( ) sign of ( )

X.M. Wang et al. / International Journal of Plasticity 24 (2008) 1307–1332 1309

1. Introduction

Shape memory alloys (SMAs) have shape memory effect and pseudoelasticity, whichmake them remarkably different from other materials. With the persistent research ontheir thermo-mechanical properties (e.g. Shaw and Kyriakides, 1995; Sehitoglu et al.,2001; Gastien et al., 2005), their phase transition characteristics (e.g. Sittner et al., 1998;Hane and Shield, 1999; Ohba et al., 2006) and associated shape memory effects (e.g.Cingolani et al., 1995; Lexcellent et al., 2000; Thamburaja et al., 2005), shape memoryalloys are developed rapidly in both engineering and medical applications. NiTi shapememory alloys are the most successful shape memory alloys so far because they combinegood structural and functional properties with good corrosion resistance, biocompatibilityand repeatability of effects (Saburi, 1998). Recently, Otsuka and Ren (2005) have reviewedthe fundamental issues about NiTi based alloys from the viewpoint of physical metallurgy.

An important phenomenon related to the thermomechanical properties of shape mem-ory alloys is plasticity. Strnade et al. (1995), Sehitoglu et al., 2001, and Gall and Maier(2002) found plastic strain accumulation during cyclic loading in stress induced phasetransformation behaviour of NiTi alloys. Brinson et al. (2004) observed localised plasticdeformation after a few loading cycles via in situ optical microscopy. Miller and Lagoudas(2000) studied the influence of plastic strain on the two-way shape memory effect. Besidesthese researches of their functional properties (pseudoelasticity and shape memory effect),the failure characteristics has become a popular topic recently (for instance, refer toMcKelvey and Ritchie, 1999, 2001; Gall et al., 2001; Chen et al., 2005). McKelvey and

1310 X.M. Wang et al. / International Journal of Plasticity 24 (2008) 1307–1332

Ritchie (1999, 2001) carried out a series of experimental study on the growth of fatiguecracks in NiTi alloy. They have experimentally found that plastic deformation after for-ward transformation could stabilise martensite and hinder the reverse transformation.Thus, the effect of plastic deformation after the stress induced phase transformation onthe mechanical behaviour of NiTi shape memory alloys is important not only to the func-tional properties but also to the failure mechanism. A schematic stress–strain curve of aninitially austenite material including pseudoelastic and plastic deformation is shown inFig. 1. As done by Liu and Xiang (1998), the deformation behaviour is divided into IVstages. The first stage is the elastic deformation of Austenite. The deformation mechanismof the stage II is stress induced martensitic transformation, which is characterized by amacroscopic stress plateau. The mechanism of deformation for the stage III is rather com-plex. It is a mixed process of elastic deformation, reorientation/detwinning and slip ofmartensite, as well as further stress-induced phase transformation of residual austenite(Miyazaki et al., 1981; Liu and Tan, 2004; Otsuka and Ren, 2005). It is acknowledged thatin stage IV plastic deformation of martensite via dislocation slip or deformation twinningtakes place (Miyazaki et al., 1981; Sehitoglu et al., 2000; Karaman et al., 2005).

In parallel, researches on constitutive relations for shape memory effect and pseudoelas-ticity are promoted. Many models have been proposed in recently years (e.g. Patoor et al.,1996; Boyd and Lagoudas, 1996; Huang and Brinson, 1998; Auricchio and Taylor, 1997;Siredey et al., 1999; Gall and Sehitoglu, 1999; Shaw, 2002; Thamburaja and Anand, 2001;Lim and McDowell, 2002; Muller and Bruhns, 2006). The models proposed by Gall andSehitoglu (1999) and Lim and McDowell (2002) are remarkable, which are based onmicromechanics of a single crystal and can capture the multiaxial behaviour of shapememory alloys. These constitutive models are still being developed in order to describeall details of SMAs thermomechanical behaviour (e.g. Sittner and Novak, 2000; Iadicolaand Shaw, 2004; Thamburaja, 2005; Pan et al., 2007; Auricchio et al., 2007). It should bementioned that the reorientation and detwinning of martensite has been incorporated inthe crystal-based constitutive model of Thamburaja (2005) for NiTi SMAs initially in mar-tensite phase. There are also a few models presented to deal with plasticity. Tanaka et al.(1995) related the plastic strain to residual stress caused by cyclic loading in their

Fig. 1. Schematic illustration showing stress-induced martensitic transformation and plasticity after phasetransformation. (l) habit plane normal, (m) transformation direction.

X.M. Wang et al. / International Journal of Plasticity 24 (2008) 1307–1332 1311

one-dimensional model. Bo and Lagoudas (1999) developed a thermomechanical modelthat can capture thermally induced transformation and its interaction with plastic strainsunder cyclic loading. The stress-induced phase transformation and plastic strain develop-ment were taken into account by Lagoudas and Entchev (2004). In both models, the devel-opment of the plastic strain is connected to the detwinned martensitic volume fraction andis induced by cyclic loading conditions. Savi et al. (2002), Yan et al. (2003) and Paiva et al.(2005) considered the plasticity after the stress induced phase transformation in their mod-els, respectively. In the model of Yan et al. (2003), plasticity is described by the von Misesisotropic hardening model. Plastic strain is included by a combination of the kinematicand isotropic hardening model by Savi et al. (2002) and Paiva et al. (2005). It should bementioned that plastic strain described by all these models is introduced based on a phe-nomenological macroscopic relation.

Most of the models for phase transformation (including all those mentioned above) arerate independent, except for the model proposed by Achenbach (1989) and later developedby Govindjee and Hall (2000). From a metallurgical viewpoint, martensitic transformationis a time independent transformation. The strain rate sensitivity of the stress–strain char-acteristics for phase transformation is induced by the latent heat during transformationthat strongly affects the temperature field inside the material (Entemeyer et al., 2000).Therefore, the rate sensitivity is considered by thermomechanical coupling effect in mostof the models (e.g. Gall and Sehitoglu, 1999; Entemeyer et al., 2000; Lim and McDowell,2002).

In this paper, a physically based micromechanical model is presented which takes intoaccount not only stress induced phase transformation but also plasticity after the stressinduced phase transformation (stage IV in Fig. 1). The modelling of stress induced phasetransformation is based on the models of Gall and Sehitoglu (1999), Lim and McDowell(2002), and Wang and Yue (2006), in which 24 martensite variants are considered. Theplastic strain is introduced by the crystal plasticity with the combination of slip and twin-ning. The deformation of stage III is linearised and treated as elastic deformation of mar-tensite. Since the model is developed at crystal level, the effect of plastic deformation onthe pseudoelastic behaviour of NiTi shape memory alloys should be captured automati-cally. To validate the model and illustrate its characteristics, the model is implementedas UMAT (User MATerial subroutine) into ABAQUS/Standard (Abaqus standard user’smanual, 2000). For nearly all commercial polycrystalline NiTi is textured to some degree,the mechanical behaviour after the stress induced martensite transformation and the effectof plasticity on the reverse phase transformation behaviour are studied on textured as wellas untextured SMAs to demonstrate the capability of the model.

2. Micromechanical model

The martensite in NiTi binary alloy is B19’ monoclinic that belongs to space group P21/m.Lattice parameters for a Ti-49.75 at. %Ni alloy are: for the austenite phase a0 = 0.3015 nm,and for the martensite phase a = 0.2889 nm, b = 0.4120 nm, c = 0.4622 nm, b = 96.8�(Otsuka et al., 1971) and they are slightly composition dependent (Hehemann and Sandrock,1971; Michal and Sinclair, 1981; Kudoh et al., 1985). A comparison of lattice parameters fordifferent NiTi alloys is presented by Prokoshkin et al. (2004).

Wechsler et al. (1953), and Bowles and Mackenzie (1954) developed a theory named‘phenomenological crystallographic theory of martensitic transformation’, respectively,

1312 X.M. Wang et al. / International Journal of Plasticity 24 (2008) 1307–1332

which is generally accepted to account for the martensite phase transformation from thecrystallography point of view. In this theory, the lattice parameters of martensite and aus-tenite, the lattice correspondences between parent and martensite, and the lattice invariantshears are input parameters. The habit plane, orientation relationship between martensiteand austenite, the magnitude and direction of transformation, etc. can all be determined.

Although slip and twinning are both possible candidates as invariant shear modes,twinning is generally used in NiTi alloy, since many deformation twinning modes areobserved experimentally in NiTi martensite. In Otsuka and Ren’ paper (2005), the twin-ning modes of NiTi martensite in the literature available are summarised. Whether thetwinning mode has a solution for the phenomenological crystallographic theory is alsoprovided, as listed in Table 1. The crystallographic quantities are in the cubic crystal basis.From Table 1, it can been seen that h011i Type II and {011}, f�1�11g Type I twinning areall possible invariant shear modes. However, there is a consensus among researchers thath01 1i Type II twinning mode is dominant, in which there are 24 habit plane variants. Thehabit plane normals and transformation directions for the 24 habit plane variants havebeen calculated by Matsumoto et al. (1987) with the Lattice parameters determined byOtsuka et al. (1971). Then they are employed in the model of Lu and Weng (1998), Galland Sehitoglu (1999), Gall et al., 2000, and Lim and McDowell (2002). The magnitudes forthe habit plane normal l and transformation direction m are listed in Table 2. The stressfree transformation strain en for each of the 24 martensite variants is defined:

en ¼ 1

2gtrðl�mþm� lÞ ð1 6 n 6 24Þ ð1Þ

where gtr is the magnitude of transformation.A representative volume element (RVE) is chosen to define all the variables, which con-

tains martensite variants nucleated from a single crystal austenite.The volume average of transformation strain is defined as the sum of transformation

strain contributed by all martensite variant,

Etr ¼X24

n¼1

ennn ð2Þ

Table 1Twinning modes of B19’ martensite (Otsuka and Ren, 2005)

Number Twinning mode Twin plane (s(a)) Twinning direction (d(a)) Twinning shear (cT) Solution

1 f�111g ð�1�10Þ ½0:540430:540431:45957� 0.30961 Yes2 Type I ð�101Þ ½0:540431:459570:54043� 0.30961 Yes3 {111} (101) ½1:511720:488281:51172� 0.14222 No4 Type I ð1�10Þ ½1:511721:511720:48828� 0.14222 No5 {011} (001) [1.5727120] 0.28040 Yes6 Type I (010) ½1:5727102� 0.28040 Yes7 h011i (0.7250310) [001] 0.28040 Yes8 Type II ð0:7205301Þ [010] 0.28040 Yes9 {001} ð0�11Þ [100] 0.23848 No10 Compound (100) ½01�1� 0.23848 No11 f20�1g ð41�1Þ ½�12�2� 0.4250 No

The column indicated by ‘solution’ represent whether a solution for the phenomenological crystallographictheory exists or not.

Table 2Crystallographic data for the 24 martensite variants in NiTi

n Habit plane normal, l Transformation direction, m

1 �0.8889 �0.4044 0.2152 0.4114 �0.4981 0.76332 �0.4044 �0.8889 �0.2152 �0.4981 0.4114 �0.76333 0.8889 0.4044 0.2152 �0.4114 0.4981 0.76334 0.4044 0.8889 �0.2152 0.4981 �0.4114 �0.76335 �0.8889 0.4044 �0.2152 0.4114 0.4981 �0.76336 0.4044 �0.8889 0.2152 0.4981 0.4114 0.76337 0.8889 �0.4044 �0.2152 �0.4114 �0.4981 �0.76338 �0.4044 0.8889 0.2152 �0.4981 �0.4114 0.76339 0.2152 0.8889 0.4044 0.7633 �0.4114 0.498110 0.2152 �0.8889 �0.4044 0.7633 0.4114 �0.498111 �0.2152 �0.4044 �0.8889 �0.7633 �0.4981 0.411412 �0.2152 0.4044 0.8889 �0.7633 0.4981 �0.411413 �0.2152 0.8889 �0.4044 �0.7633 �0.4114 �0.498114 �0.2152 �0.8889 0.4044 �0.7633 0.4114 0.498115 0.2152 0.4044 �0.8889 0.7633 0.4981 0.411416 0.2152 �0.4044 0.8889 0.7633 �0.4981 �0.411417 0.8889 �0.2152 0.4044 �0.4114 �0.7633 0.498118 �0.8889 �0.2152 �0.4044 0.4114 �0.7633 �0.498119 0.4044 0.2152 0.8889 0.4981 0.7633 �0.411420 �0.4044 0.2152 �0.8889 �0.4981 0.7633 0.411421 0.8889 0.2152 �0.4044 �0.4114 0.7633 �0.498122 �0.8889 0.2152 0.4044 0.4114 0.7633 0.498123 �0.4044 �0.2152 0.8889 �0.4981 �0.7633 �0.411424 0.4044 �0.2152 �0.8889 0.4981 �0.7633 0.4114

X.M. Wang et al. / International Journal of Plasticity 24 (2008) 1307–1332 1313

where nn is the martensite volume fraction of the nth martensite variant. The total mar-tensite volume fraction, n, is the sum of volume fraction of each martensite variant. Thefollowing natural physical constraints must be satisfied:

n ¼X24

n¼1

nn6 1 and 0 6 nn

6 1 ð3Þ

Complementary free energy w is composed of elastic energy wel, potential energy ofloading system wpo, chemical energy due to the phase transition wch and surface energywsur.

wðR; T ; nn;EpÞ ¼ �ðwel þ wpo þ wch þ wsurÞ ð4ÞElastic energy is expressed as follows:

wel ¼ 1

2

Zv

r : eedv ¼ 1

2R : S : R� R : Ep � 1

2

Zv

s : eTdv ð5Þ

where r is the local stress; ee is the local elastic strain; R is the average stress tensor of localstress field; S is the average effective elastic compliance; Ep is the average value of localplastic strain; s is the internal stress related to the incompatibilities in the transformationfield; eT is the internal transformation strain. The last term in Eq. (5) is interaction energywint related to transformation strain field. According to Gall and Sehitoglu (1999), theinteraction energy wint is defined by:

1314 X.M. Wang et al. / International Journal of Plasticity 24 (2008) 1307–1332

wint ¼ � 1

2

X24

m;n¼1

Hmnnmnn ð6Þ

where Hmn is the interaction energy matrix, as given in Table 3.Potential energy caused by the loading system is defined as:

wpo ¼ �R : ðS : Rþ EtrÞ ¼ �R : S : R� R :X24

n¼1

ennn ð7Þ

Chemical energy is linearly proportional to temperature change:

wch ¼ BðT � T 0Þn ¼ BðT � T 0ÞX24

n¼1

nn ð8Þ

where B is a constant, T is the test temperature and T0 is the phase equilibriumtemperature.

The surface energy is small in comparison with other terms and is negligible. From Eqs.(5)–(8), complementary free energy in a RVE is expressed as:

wðR; T ; nn;EpÞ ¼ 1

2R : S : Rþ R : Ep þ R

:X24

n¼1

ennn � 1

2

X24

m;n¼1

H mnnmnn � BðT � T 0ÞX24

n¼1

nn ð9Þ

Table 3Interaction matrix for NiTi

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

1 C C C C C I C I I I I C I I C I C I I I C I I I

2 C C C C I C I C C I I I C I I I I I C I I I I C

3 C C C C C I C I I I C I I I I C I C I I I C I I

4 C C C C I C I C I C I I I C I I I I I C I I C I

5 C I C I C C C C I I C I I I I C C I I I C I I I

6 I C I C C C C C C I I I C I I I I I I C I I C I

7 C I C I C C C C I I I C I I C I I C I I I C I I

8 I C I C C C C C I C I I I C I I I I C I I I I C

9 I C I I I C I I C C C C C C I I I C I I C I I I

10 I I I C I I I C C C C C C C I I C I I I I C I I

11 I I C I C I I I C C C C I I C C I I C I I I C I

12 C I I I I I C I C C C C I I C C I I I C I I I C

13 I C I I I C I I C C I I C C C C C I I I I C I I

14 I I I C I I I C C C I I C C C C I C I I C I I I

15 C I I I I I C I I I C C C C C C I I C I I I C I

16 I I C I C I I I I I C C C C C C I I I C I I I C

17 C I I I C I I I I C I I C I I I C C C C C C I I

18 I I C I I I C I C I I I I C I I C C C C C C I I

19 I C I I I I I C I I C I I I C I C C C C I I C C

20 I I I C I C I I I I I C I I I C C C C C I I C C

21 C I I I C I I I C I I I I C I I C C I I C C C C

22 I I C I I I C I I C I I C I I I C C I I C C C C

23 I I I C I C I I I I C I I I C I I I C C C C C C

24 I C I I I I I C I I I C I I I C I I C C C C C C

Here C = GA/3000, I = GA/750, where GA is the shear modulus of austenite.

X.M. Wang et al. / International Journal of Plasticity 24 (2008) 1307–1332 1315

It is experimentally observed that Young’s modulus of the martensite is about one-thirdto one-half of Young’s modulus of the austenite for binary NiTi shape memory alloys(Hodgson et al., 1991; Thamburaja and Anand, 2001). Therefore, the differences betweenthese two phases must be considered. Here, we use the identical method to Bo and Lagou-das (1999).

SðnÞ ¼ ð1� nÞSA þ nSM ð10ÞSA, SM are elastic compliances of austenite and martensite, respectively.

In order to satisfy the second-law of thermodynamics, a Lagrangian function is intro-duced based on Eq. (3) to derive the driving forces for phase transformation (Patoor et al.,1996; Huang and Brinson, 1998.).

LðR; T ; nn;EpÞ ¼ wðR; T ; nn;EpÞ � k0

X24

n¼1

nn � 1

!�X24

n¼1

knð�nnÞ ð11Þ

where k0 and kn are Lagrangian multipliers and have positive values.The driving force for the transformation from austenite to a particular martensite var-

iant is assumed to satisfy the following condition:

F n ¼oLonn ¼

1

2R : DS : Rþ R : en �

X24

m¼1

Hmnnm � BðT � T 0Þ � k0 þ kn ¼ F C ð12Þ

where DS = SM � SA, FC is the constant critical driving force for phase transformation.For this particular martensite variant to transform back to austenite, the following con-

dition must be met:

F n ¼oLonn ¼

1

2R : DS : Rþ R : en �

X24

m¼1

Hmnnm � BðT � T 0Þ � k0 þ kn ¼ �F C ð13Þ

To satisfy Eq. (3), the Lagrange multipliers are enforced to satisfy (Patoor et al., 1996):

k0 ¼1

2R : DS : Rþ R : en �

X24

m¼1

Hmnnm � BðT � T 0Þ þ kn � F C P 0 ð14aÞ

kn ¼ �1

2R : DS : R� R : en þ

X24

m¼1

Hmnnm þ BðT � T 0Þ þ k0 þ F C P 0 ð14bÞ

for the transformation from austenite to a particular martensite variant, and

k0 ¼1

2R : DS : Rþ R : en �

X24

m¼1

Hmnnm � BðT � T 0Þ þ kn þ F C P 0 ð15aÞ

kn ¼ �1

2R : DS : R� R : en þ

X24

m¼1

Hmnnm þ BðT � T 0Þ þ k0 � F C P 0 ð15bÞ

for this particular martensite variant to transform back to austenite.During transformation, the consistency conditions are Fn � FC = 0 for austenite to a

martensite variant transformation and Fn + FC = 0 for this variant to transform back toaustenite. Since FC = constant, the following equation can be derived:

1316 X.M. Wang et al. / International Journal of Plasticity 24 (2008) 1307–1332

oF n

oR: _Rþ

X24

m¼1

oF n

onm_nm þ oF n

oT_T ¼ 0 ð16Þ

which leads to:

DS : R : _Rþ _R : en �X

m

H mn _nm � B _T ¼ 0 ð17Þ

At any given moment, there may be arbitrary number of martensite variants that trans-form simultaneously. As a result, there are q simultaneous equations of (17), where q isthe sum of all actively transforming variants determined by Eqs. (12) and (13).

The total strain is defined by:

E ¼ owoR¼ S : Rþ EP þ Etr ð18Þ

The total strain rate can be expressed by:

_E ¼ oE

oR_Rþ oE

on_n ¼ R : _Rþ _EP þ oS

on: R _nþ _Etr ¼ S : _Rþ _EP þ DS : R _nþ _Etr ð19Þ

The third-term on the right-hand side of Eq. (19) represents the effects of the change of theelastic compliance between austenite and martensite.

After the phase transformation, the deformation of martensite occurs with increasingloading. Since the phase transformation is discussed from the crystallography point ofview, it is rational to analyse the plastic deformation of martensite based on crystal plas-ticity theory. Although the slip system of martensite has not been confirmed experimen-tally, it can be reasonably assumed from the crystal structure of martensite because theslip system depends only on structure. The most possible slip system in NiTi martensiteis [100](0 01) (Kudoh et al., 1985). According to Taylor’s criterion, normally 5 indepen-dent slip systems have to be activated to accommodate a certain strain increment in poly-crystal. Apparently, the number of independent slip systems in monoclinic martensite isnot enough. Nishida et al. (1998) found not only slip but also twinning deformation with(100), (001) and ð20�1Þ compound twins of martensite under heavy deformation, which issufficient for large deformation. Therefore, formation of the twinning deformation modesmay provide martensite with sufficient independent slip systems.

Liu et al. (2002) investigated the rate dependence of martensitic NiTi shape memoryalloys and found that the mechanical behaviour of the material exhibited strain ratedependence during plastic deformation caused by dislocation. Although, the proposedmodel for phase transformation is rate independent, the strain rate sensitivity can be cap-tured by coupling thermal effect (cf. Entemeyer et al., 2000; Lim and McDowell, 2002).Based on these considerations, the plastic deformation of martensite is taken into accountusing a rate dependent relationship. Asaro and Needleman (1985) have established a rig-orous constitutive framework for rate dependent crystalline materials undergoing plasticdeformation by considering crystallographic slip alone. In order to predict the evolutionof both anisotropic stress strain response and the crystallographic texture for the materialwith low stacking energies, Schlogl and Fischer (1996), Kalidindi (1998a,b), Staroselskyand Anand (1998) incorporated deformation twinning as a deformation mode into crystalplasticity models. The main issues and challenges involved in modelling anisotropic strainhardening and deformation textures in low staking fault energy fcc metals were reviewedand summarized by Kalidindi (2001). These efforts extended crystal plasticity models to a

X.M. Wang et al. / International Journal of Plasticity 24 (2008) 1307–1332 1317

wider range of polycrystalline metals and also made it possible to model the plastic defor-mation behaviour of martensite from crystallography point of view.

In the present work, only the [100] (001) slip system is included in the model. All theexperimentally observed deformation twinning modes (listed in Table 2) are considered.The average plastic strain rate is expressed by

_EP ¼ ð1� f ÞPðsÞ _cðsÞ þX11

a¼1

PðaÞ _cðaÞtw ð20Þ

where f is the total volume fraction of deformation twinned martensite in the RVE; P(s)

and P(a) represent the tensorial directions of the shear caused by slip and an twinning sys-tem, respectively; _cðsÞ is the plastic shear rate of the slip system; _cðaÞtw is the plastic shear rateof the a-deformation twinning system.

f ¼X11

a¼1

f ðaÞ; 0 6 f 6 1; 0 6 f ðaÞ 6 1 ð21Þ

f(a) is the volume fraction of the a-deformation twinning system.

PðsÞ ¼ 1

2ðsðsÞ � dðsÞ þ dðsÞ � sðsÞÞ; PðaÞ ¼ 1

2ðsðaÞ � dðaÞ þ dðaÞ � sðaÞÞ ð22Þ

s(s) and d(s) represent the normals to the slip plane and vector of the slip direction, respec-tively; s(a) and d(a) are the normals to the a-deformation twinning plane and the directionof the a-deformation twinning system.

For simplicity, it is assumed that further slip or twinning does not occur inside twinnedregions. The slip rate is expressed with the rate dependent law (Asaro and Needleman,1985):

_cðsÞ ¼ _cðsÞ0 sgnsðsÞsðsÞ

gðsÞ

��������

1k1

ð23Þ

where _cðsÞ0 is the reference shear rate of slip system; k1 is the material rate sensitivity of slipsystem; s(s) is the resolved shear stress of slip system; g(s) is the slip system hardness; sgn ( )denotes the sign of ( ).

sðsÞ ¼ R : PðsÞ ð24ÞThe evolution of the hardness is defined by the following relations:

_gðsÞ ¼ hðsÞj _cðsÞj ð25Þ

hðsÞ ¼ hðsÞ0 1� gðsÞ

sðsÞs

� �q1

ð26Þ

h(s) is the coefficient of slip hardening; hðsÞ0 and q1 are material constants; sðsÞs is the satura-tion strength of slip system; the value of g(s) at the beginning of the deformation must bespecified and the symbol sðsÞ0 is used.

The plastic shear rate _cðaÞtw is expressed as:

_cðaÞtw ¼ cðaÞT_f ðaÞ ð27Þ

cðaÞT is the constant twinning shear of the a- twinning system.

1318 X.M. Wang et al. / International Journal of Plasticity 24 (2008) 1307–1332

The rate of the twinned martensite volume fraction _f ðaÞ is expressed using the same for-mula of that of slip rate. However, it is important to note that deformation twinning isunidirectional as opposed to dislocation slip which is bi-directional.

_f ðaÞ ¼_f ðaÞ0

sðaÞ

gðaÞ

� � 1k2 ; sðaÞ > 0

0; sðaÞ 6 0

8<: ð28Þ

sðaÞ ¼ R : PðaÞ ð29Þ

_gðaÞ ¼X11

b¼1

hðabÞ _cðaÞ; hðabÞ ¼ hðaÞ0 1� gðaÞ

stws

� �q2

ð30Þ

where _f ðaÞ0 is the reference twining rate; k2 is the material rate sensitivity of twinningsystem; stw

s is the saturation strength of twinning system; g(a) is the twinning hardness.h(ab) are components of the hardening matrix. hðaÞ0 and q2 are material constants. Thevalue of g(a) at the beginning of the deformation must be specified and the symbol sðaÞ0

is adopted.

3. Time-discrete model and implementation into ABAQUS

The presented model is implemented as User MATerial subroutine (UMAT) into ABA-QUS/Standard (ABAQUS, 2000). At the start of each increment, ABAQUS passes strain,stress, strain increment, temperature and user defined state variables (SDV) into UMAT.At the end of each increment, the updated stress, volumetric heat generation per unit time(RPL) and SDV must be fed back to ABAQUS. The Jacobian matrix of the constitutivemodel, variation of the stress increments with respect to the temperature, variation of RPLwith respect to the strain increments, and variation of RPL with respect to the temperaturemust be defined. In order to do this, the model is discretized.

For a very small time step Dt, the stress rate _R, the rate of the volume fraction of nthmartensite _nn, the temperature rate _T , the total strain rate _E, the transformation strain rate_Etr, the slip rate _cðsÞ, the rate of the twinned martensite volume fraction _f ðaÞ and the plasticstrain rate _EP can be approximated by _R ¼ DR=Dt, _nn ¼ Dnn=Dt, _T ¼ DT=Dt, _E ¼ DE=Dt,_Etr ¼ DEtr=Dt, _cðsÞ ¼ DcðsÞ=Dt, _f ðaÞ ¼ Df ðaÞ=Dt and _EP ¼ DEP=Dt, respectively. Eqs. (14)and (16) can then be approximated by

DS : R : DRþ DR : en �X

m

H mnDnm � BDT ¼ 0 ð31Þ

DE ¼ S : DRþ DEP þ DS : RDnþ DEtr ð32Þ

where Dn ¼P

nDnn, DEtr ¼P

nenDnn.

For a given time increment Dt,

R ¼ Rt þ DR; n ¼ nt þ Dn ð33Þ

where the subscript t represents the time that has been passed after the previous incre-ment.

X.M. Wang et al. / International Journal of Plasticity 24 (2008) 1307–1332 1319

Substituting Eq. (33) into Eq. (32), the change of stress DR is deduced:

DR ¼ ½SA þ DSnt þ 2DSX

n

Dnn��1

: DE� DS : Rt

Xn

Dnn �X

n

enDnn � DEP

!ð34Þ

3.1. Evolution of variables updated in UMAT during phase transformation

When the resolved shear stress of the slip system and twinning systems reach their crit-ical values, the plastic deformation can occur. Since the plastic phase transformation ofmartensite are considered, these two critical values are relatively high so that the plasticdeformation does not take place during the phase transformation. Thus, Eq. (34) becomes,

DR ¼ SA þ DSnt þ 2DSX

n

Dnn

" #�1

: DE� DS : Rt

Xn

Dnn �X

n

enDnn

!ð35Þ

From Eq. (31), we have:

Dnm ¼X

n

ðH mnÞ�1ðDS : R : DRþ DR : en � BDT Þ ð36Þ

Differentiating Eq. (35) with respect to DE,

oDRoDE

¼ SA þ DSnt þ 2DSX

n

Dnn

" #�1

: I� DS :X

n

oðRtDnnÞoDE

�X

n

oðenDnnÞoDE

" #

� SA þ DSnt þ 2DSX

n

Dnn

" #�1

: 2DS

8<:

9=;DR�

Xn

oDnn

oDEð37Þ

where I is the fourth rank identity tensor; oDR/oDE is the Jacobian matrix.Differentiating Eq. (35) with respect to DT, variation of the stress increments with

respect to the temperature, oDR/oDT, is obtained,

oDRoDT

¼ SA þ DSnt þ 2DSX

n

Dnn

" #�1

: �DS :X

n

oðRtDnnÞoDT

�X

n

oðenDnnÞoDT

" #

� SA þ DSnt þ 2DSX

n

Dnn

" #�1

: 2DS

8<:

9=;DR�

Xn

oDnn

oDTð38Þ

The rate of heat generation is assumed linearly proportional to the transformation rate.The volume heat generation increment per unit time (RPL), DR, can be written as,

1320 X.M. Wang et al. / International Journal of Plasticity 24 (2008) 1307–1332

DR ¼X24

n¼1

F nDnn þ KT 0

TX24

n¼1

Dnn ð39Þ

where K is the latent heat of transformation per unit volume. Differentiating Eq. (39) withrespect to the strain and the temperature increments respectively, we obtain:

oDRoDE

¼X24

n¼1

oF nDnn

oDEþ K

T 0

TX24

n¼1

oDnn

oDEð40Þ

oDRoDT

¼X24

n¼1

oF nDnn

oDTþ K

T 0

TX24

n¼1

oDnn

oDTð41Þ

It should be noted that the rates are imported to the solution of the coupled heat transferproblem, and are obtained by _R ¼ DR=Dt, _T ¼ DT=Dt, and so on.

3.2. Evolution of variables updated in UMAT during plastic deformation

In order to obtain rigorous stability with high calculation efficiency, the explicit back-ward Euler integration procedure is utilised.

DcðsÞ ¼ _cðsÞDt ð42ÞDf ðaÞ ¼ _f ðaÞDt ð43Þ

To approximate the slip rate, _cðsÞ, and the rate of the deformation twinned martensite vol-ume fraction, _f ðaÞ, at the present increment, we employ a Taylor expansion:

_cðsÞ ¼ _cðsÞt þo_cðsÞ

osðsÞ

����t

DsðsÞ þ o_cðsÞ

ogðsÞ

����t

DgðsÞ

¼ sgnðsðsÞÞ _cðsÞ0

sðsÞ

gðsÞ

� �1=k1

1þ 1

k1

DsðsÞ

sðsÞt

� DgðsÞ

gðsÞt

!" #ð44Þ

_f ðaÞ ¼ _f ðaÞt þo _f ðaÞ

osðaÞ

����t

DsðaÞ þ o _f ðaÞ

ogðaÞ

����t

DgðaÞ

¼ _f ðaÞ0

sðaÞ

gðaÞ

� �1=k2

1þ 1

k2

DsðaÞ

sðaÞt

� DgðaÞ

gðaÞt

!" #ð45Þ

The increment of the resolved stress is expressed:

DsðsÞ ¼ PðsÞ : DR ð46ÞDsðaÞ ¼ PðaÞ : DR ð47Þ

The increment of hardness evolution is

DgðsÞ ¼ hðsÞDcðsÞsgnðDcÞ ð48Þ

DgðaÞ ¼X11

b¼1

hðabÞDcðaÞ ð49Þ

X.M. Wang et al. / International Journal of Plasticity 24 (2008) 1307–1332 1321

At this stage, the phase transformation is finished, i.e. n = 1 and Dnn = 0, Eq. (34)becomes,

DR ¼ ½SA þ DS��1: ðDE� DEPÞ ¼ S�1

M : ðDE� DEPÞ ¼ S�1M

: ðDE� 1� ft �X11

a¼1

Df ðaÞ !

PðsÞDcðsÞ �X11

a¼1

cðaÞT PðaÞDf ðaÞÞ ¼ S�1M

: DE� S�1M : ð1� ftÞPðsÞDcðsÞ þ S�1

M :X11

a¼1

Df ðaÞPðsÞDcðsÞ � S�1M

:X11

a¼1

cðaÞT PðaÞDf ðaÞ ð50Þ

Differentiating Eq. (50) with respect to DE gives,

oDRoDE

¼ S�1M : I� ð1� ftÞS�1

M : PðsÞoDcðsÞ

oDEþ S�1

M :X11

a¼1

oDf ðaÞ

oDEPðsÞDcðsÞ þ S�1

M

:X11

a¼1

Df ðaÞPðsÞoDcðsÞ

oDE� S�1

M :X11

a¼1

oDf ðaÞ

oDEPðaÞcðaÞT ð51Þ

3.3. Solving algorithm adopted in UMAT

The integration algorithm for the discrete model is given out in the following steps:

(1) Read in DE, Rt, nt, nnt , ft and f ðaÞt from ABAQUS.

(2) Assign oDR/oDE = S�1 (n). Calculate the trial stress Rtrial = Rt + oDR/ + oDE:DE,which is used to evaluate the driving force for all variants Fn (1 6 n 6 24) usingEqs. (12) and (13), as well as the resolved shear stresses for plastic deformation util-ising Eqs. (24) and (29).

(3) Evaluate: if plastic deformation occurs according to Fn > Fc, n > 1 � e andsðaÞ > 0:9� sðaÞ0 (or sðsÞ > 0:9� sðsÞ0 ). If plastic deformation occurs, calculate the fol-lowing variables: Df(a), f ðaÞ ¼ f ðaÞt þ Df ðaÞ, f ¼

P11a¼1f ðaÞ, oDR/oDE according to

Eq. (51), and DR = oDR/oDE:DE. Then, update R = Rt + DR and write f(a), f, R,oDR/oDE back to ABAQUS and goto step 9. If plastic deformation does not happen,then go to step 4.

(4) Evaluate: if the currently active variants will transform to inactive variants accordingto the rules:(a) A phase to M phase transformation: nn > 1 � e or n > 1 � e.(b) M phase to A phase transformation: nn < e or n < e.

(5) Evaluate: if the currently inactive variants will transform to active variants accordingto the rules:(a) A phase to M phase transformation: FnPFC, with nn < 1 � e and n < 1 � e.(b) M phase to A phase transformation: Fn 6 �FC, with nn > e and n > e.

where e denotes numerical tolerance in numerical simulations, e � 0.001.

1322 X.M. Wang et al. / International Journal of Plasticity 24 (2008) 1307–1332

(6) If there are no transforming variants, DR = S�1(n):DE, calculate R = Rt + DR andwrite it back to ABAQUS, then goto step 9.

(7) If there are transforming variants, calculate Dnn and evaluate if there is any non-active transforming martensite variant according to(a) A phase to M phase transformation: Dnn < 0.(b) M phase to A phase transformation: Dnn > 0.If there are any non-active variants, remove the non-active variants and goto step 7.

(8) Calculate the following variables: nn ¼ nnt þ Dnn, n ¼

P24n¼1n

n, oDR/oDE accordingto Eq. (37), DR, oDR/oDE, oDR/oDTDR = oDR/oDE:DE + oDR/oDT � DT. UpdateR = Rt + DR. Write nn, n, R, oDR/oDE, DR, oDR/oDE, oDR/oDT back toABAQUS.

(9) End the program.

4. Finite element model

The quasi-static behaviours of NiTi alloy are simulated here. All the terms involvingheat generation and heat transfer are omitted in the following calculations. Since polycrys-tal is important from practical point of view, the following simulations will focus on poly-crystalline NiTi alloys.

The parameters for phase transformation are chosen as same as those chosen by Limand McDowell (2002). We consider the difference between elastic modulus of martensiteand austenite. The elastic modulus of martensite EM is assumed one-third of that of aus-tenite EA. Since the Poisson’s ratios t for both phases are approximately the same (Tham-buraja and Anand, 2001), a typical value of 0.33 is chosen. The rate sensitivity parametersfor slip and deformation twinning systems are assumed of taking the same low value of0.02. The reference shear rate and reference twinning rate are chosen as 0.001 s�1 (Asaroand Needleman, 1985; Kalidindi, 2001 and Wu et al., 2007). The input parameters in thefinite element analysis are listed in Table 4.

Due to the manufacture process of NiTi alloys, polycrystalline alloys are usually strongtextured. Texture has great influences on the overall properties of NiTi alloys. Liu et al.(1998) found that different mechanisms of martensite twins occurred in different texturedNiTi alloys. Investigations by Gall and Sehitoglu (1999), Thamburaja and Anand (2001)showed that tension-compression stress–strain behaviour of textured NiTi alloys is asym-metry. Therefore, two models are simulated with one textured and the other without tex-ture. Since both Gall and Sehitoglu (1999), Thamburaja and Anand (2001), and Yawnyet al. (2005) found [111] texture in their polycrystalline NiTi alloys, this kind of textureis used as the example.

In the FE model, each element represents one crystal. There are totally 250 8-node con-tinuous solid brick (C3D8) type elements as shown in Fig. 2a. In the model without tex-ture, the orientations of elements are randomly assigned. In the textured model, eachelement is assigned an orientation, such that a [111] direction was scattered within 0�–5� wobble to the loading direction and the [10 0] directions are randomly distributedamong the elements.

Uniaxial loading condition is simulated, in which the displacement loading is appliedparallel to 2-axial direction.

Table 4Input parameters for finite element calculation

EA EM t gtr FC B K _cðsÞ0 , _f ðaÞ0 k1, k2 hðsÞ0 , hðaÞ0 q1, q2 sðsÞ0 , sðaÞ0 sðsÞs , stws

120 GPa 40 GPa 0.33 0.1308 9.9 MJ m�3 0.6 MJ m�3 �C�1 157 MJ m�3 0.001 0.02 360 MPa 1.3 300 MPa 900 MPa

X.M

.W

an

get

al./In

terna

tiona

lJ

ou

rna

lo

fP

lasticity

24

(2

00

8)

13

07

–1

33

21323

Fig. 2. Finite element model of 250 ABAQUES C3D8 elements (a) polycrystal macroscopic stress–strainresponse of textured and untextured alloys with maximum strain amplitude (b) 4%; and (c) 12%.

1324 X.M. Wang et al. / International Journal of Plasticity 24 (2008) 1307–1332

X.M. Wang et al. / International Journal of Plasticity 24 (2008) 1307–1332 1325

5. Results and discussions

The comparisons between the polycrystal macroscopic stress–strain response of tex-tured and untextured alloys are shown in Fig. 2. The macroscopic stress was calculatedby dividing the sum of the nodal forces from relevant nodes by the original cross-sectionalarea of the element perpendicular to the loading direction. The perfect pseudoelastic prop-erties are demonstrated in Fig. 2b with maximum strain of 4%. Both the phase transfor-mation stresses and the slope of the stress–strain curve of untextured NiTi alloy duringphase transformation are much bigger than those of textured alloys are, which is in accor-

Fig. 3. Polycrystal macroscopic stress–strain response with different maximum strain amplitude of (a) untexturedalloy and (b) textured alloy.

1326 X.M. Wang et al. / International Journal of Plasticity 24 (2008) 1307–1332

dance with that of Gall and Sehitoglu (1999), and Nae et al. (2003). The textured alloy alsohas a flatter plastic stress–strain curve, as shown in Fig. 2c. Fig. 3a and b show the effectsof plastic deformation on the phase transformation behaviour of the untextured and tex-tured alloys, respectively. With the increasing of plastic deformation, the residual strainincreases and the phase transformation stress–strain curves from the martensite to austen-ite becomes steeper and less obvious. From the evolutions of polycrystal average martens-ite volume fraction during loading-unloading cycle with different maximum strain

Fig. 4. Evolution of polycrystal average martensite volume fraction during loading-unloading cycle with differentmaximum strain amplitude of (a) untextured alloy and (b) textured alloy. The residual strain and stabilisedvolume fraction of martensite (value at zero polycrystal average stress) are indicated by symbols.

X.M. Wang et al. / International Journal of Plasticity 24 (2008) 1307–1332 1327

amplitudes in Fig. 4, it can be seen that the martensite is stabilised with increasing plasticdeformation. The average values refer to values averaged over all elements. These simula-tion results are qualitatively consistent with the experimental results obtained by McKel-vey and Ritchie (1999, 2001).

To understand the effect of plastic deformation on martensite transformation, firstly,the constitutive equations are analysed. From Eq. (33), it can be seen that the incrementof martensite volume fraction is related not only to increment of stress but also to thestate of stress itself, which means that the evolution of martensite volume fraction isinfluenced by the loading history. FEM analysis of a one-crystal model (one C3D8 typeelement model) shows that the presence of plastic deformation changes the stress field aswell as the strain field, consequently influences the evolution of reverse martensite trans-formation. We focus the following analysis on the results of FE simulation with 12%total strain. Fig. 5 gives the comparison of evolution of average total deformationtwinned martensite volume fraction for untextured and textured polycrystal. The evolu-tions of the average value of twin volume fraction for different type of deformation twin-ning modes are shown in Fig. 6a and b for untextured and textured polycrystal,respectively. The untextured polycrystal has larger average total deformation twinnedmartensite volume fraction than that textured polycrystal does. The untextured polycrys-tal inspires all the deformation twinning modes during plastic deformation. Amongthem, the volume fraction of {111} type I twinning modes are much larger than thoseof the others are. However, this type of twinning mode has a smaller twinning shearcompared with other types. In the textured polycrystal, only five twinning systems areactivated. The main deformation twinning modes are {011} type I, f�1�11g type I,{00 1} compound which all have large twinning shear to provide the polycrystal withlarge plastic strains.

Fig. 5. Evolution of polycrystal average total deformation twinned martensite volume fraction for untexturedand textured alloys with maximum strain amplitude of 12%.

Fig. 6. Evolution of polycrystal average twin volume fraction for different deformation twinning modes of(a) untextured alloy and (b) textured alloy. The representations of type number are shown in Table 2.

1328 X.M. Wang et al. / International Journal of Plasticity 24 (2008) 1307–1332

6. Conclusions

A micromechanical constitutive model for pseudoelastic NiTi SMAS is expanded toinclude the permanent plastic deformation. To describe the phase transformation behav-iour, the 24 variants usually used in modelling of NiTi SMAs are considered. The formulasto characterise the pseudoelastic behaviour are built. It has been demonstrated that themultiaxial behaviour and tension-compression asymmetry are well predicted. In the pro-posed model, differences of the elastic properties between austenite and martensite aretaken into account. The plastic deformation is modelled based on crystal plasticity theory.

X.M. Wang et al. / International Journal of Plasticity 24 (2008) 1307–1332 1329

The deformation twinning contribution as well as slip is introduced into the modelling ofplasticity.

FE simulations are performed on two kinds of polycrystalline NiTi SMAs. One is ran-domly oriented and the other has [11 1] texture. Results shows that the phase transforma-tion stresses and the slope of the transformation stress–strain curve of untextured alloy aremuch higher than those of textured alloy are. The presence of plastic deformation changesthe stress and strain field, and thus influences the evolution of martensite transformation.The experimentally observed stabilisation of martensite and hindering of the reverse trans-formation due to plastic deformation after forward transformation are captured qualita-tively. In the untextured alloy, all deformation twinning modes are active during plastictransformation. While for the textured alloy, only twinning modes with large twinningshear are activated. The FEM simulations here are intended to illustrate the characteristicsof the model only. A more precise calibration of the model with experimental results is thesubject of our future work.

Acknowledgements

The authors would like to thank Prof. K. Otsuka (National Institute for Materials Sci-ence, Tsukuba, Japan) for the instructive discussion on the deformation modes of NiTialloys and Dr. J. Wang (School of Mechanical and Aerospace Engineering, Queen’s Uni-versity, Belfast, UK) for the help of correcting the English grammar. The work is sup-ported by National Natural Science Fund of China (10472094), the Research Fund forthe Doctoral Program of Higher Education (N6CJ0001) and the Doctorate Foundationof Northwestern Polytechnical University.

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