A macroscopic constitutive model of shape memory alloy considering plasticity

Mechanics of Materials 48 (2012) 71–81

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Mechanics of Materials

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A macroscopic constitutive model of shape memory alloyconsidering plasticity

Bo ZhouCollege of Aerospace and Civil Engineering, Harbin Engineering University, Harbin 15001, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 29 May 2011Received in revised form 23 August 2011Available online 16 February 2012

Keywords:Shape memory alloyPlastic strainMacroscopic constitutive modelThermo-mechanical behavior

0167-6636/$ - see front matter � 2012 Elsevier Ltddoi:10.1016/j.mechmat.2012.02.001

E-mail address: [email protected]

This paper presents a macroscopic constitutive model which is able to reproduce thethermo-mechanical behaviors of the super-elastic SMA undergoing plastic strain. Amechanical constitutive equation, which predicts the stress–strain response of the SMAundergoing plastic strain, is developed based on the expression of Gibbs free energy withplastic strain. A linear plastic constraint equation is supposed to describe the effect of plas-ticity on the phase transformation behaviors of SMA. A sine-type phase transformationequation is established to describe the phase transformation behaviors of the SMA under-going plastic strain. The mechanical constitutive equation, plastic constraint equation, andphase transformation equation together compose the presented macroscopic constitutivemodel which reproduces the thermo-mechanical behaviors of the SMA undergoing plasticstrain. Especially all material constants related to the presented macroscopic constitutivemodel can be determined through macroscopic experiments. Therefore it is easy to use thispresented model for the practical applications of SMA. The mechanical behaviors of thesupper-elastic SMA undergoing plastic strain and the effect of plasticity are numericallysimulated by the presented macroscopic constitutive model. Results show that the pre-sented macroscopic constitutive model can effectively reproduce the thermo-mechanicalbehaviors of the super-elastic SMA and express the effect of plasticity.

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1. Introduction

Shape memory alloy (SMA) has two unique thermo-mechanical characteristics, shape memory effect andsuper-elasticity, in comparison to the ordinary metals andalloys (Ford and White, 1996; Auricchio and Taylor, 1997;Gastien et al., 2005). Both shape memory effect andsuper-elasticity are due to the reversible changes of crystal-lographic structure occurring in such material. Thesechanges are called as the phase transformations betweenan austenite of crystallographic more-ordered phase anda martensite of crystallographic less-ordered phase. Theaustenite is stable at high temperature and low stress,while the martensite is stable at low temperature and highstress. SMA has been applied to a wide variety of practical

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applications in various fields due to not only the shapememory effect and super-elasticity but also the goodbiocompatibility, corrosion resistance and mechanicalproperties (Pasquale, 2003; Morgan, 2004; Yeung et al.,2004; Dilibal et al., 2004).

The constitutive model describing the thermo-mechan-ical behaviors of SMA is important for its practical applica-tions in various engineering fields. Many such constitutivemodels have been established, of which the constitutivemodels from practical viewpoints played important rolesin the practical applications of SMA. Tanaka (1986) devel-oped an exponent-type phase transformation equation anda one-dimensional mechanical constitutive equation inrate form. Liang and Rogers (1990) supposed a cosine-typephase transformation equation, and developed a one-dimensional mechanical constitutive equation. Sun andHwang (1993) explained the super-elasticity and shapememory effect induced by non-proportional loading, and

72 B. Zhou / Mechanics of Materials 48 (2012) 71–81

established a three-dimensional micromechanical consti-tutive equation. Boyd and Lagoudas (1994) separated thetotal strain of SMA into three parts: the elastic strain, phasetransformation strain and thermo-expanding strain, anddeveloped a three-dimensional micromechanical constitu-tive equation. Brinson (1993) separated the martensiticvolume fraction into two parts induced by stress andtemperature, and developed another cosine-type phasetransformation equation. This equation overcomes thelimitation that both exponent-type and cosine-type phasetransformation equations fail to describe the processthat SMA translates from twined martensite to det-wined martensite upon loading. She also developed a one-dimensional mechanical constitutive equation. Broccaet al. (2002) established a three-dimensional constitutivemodel based on micro-plane theory and the exponent-typephase transformation equation. Zhou and Yoon (2006a)developed a one-dimensional phase transformation equa-tion which can express the influences of phase transforma-tion peak temperatures on the phase transformationbehaviors of SMA. Zhou et al. (2009) established athree-dimensional macroscopic mechanical constitutiveequation and a three-dimensional phase transformationequations based on the differential relationship ofGibbs free energy, stress and strain, and the differentialrelationship of martensitic volume fraction and phasetransformation free energy, respectively.

Although these constitutive models mentioned aboveare able to express the shape memory effect and super-elasticity of SMA, they fail to take into account the effectof plasticity on the characteristics of the SMA undergoingplastic strain. However the plasticity is one of the impor-tant phenomena related to the thermo-mechanicalfeatures of SMA. McKelvey and Ritchie (1999, 2001) exper-imentally found that the plastic strain in SMA can stabilizestress-induced martensite and hinder the inverse phasetransformation from martensite to austenite during theunloading process. Miller and Lagoudas (2000) investi-gated the influence of plastic strain on the two-way shapememory effect of SMA. The plastic strain in SMA can beattributed to the results from the dislocation or twinningoccurring in martensite upon loading (Miyazaki et al.,1981; Sehitoglu et al., 2001a,b; Otsuka and Ren, 2005). Arelatively small amount of literature has been reportedon the constitutive model of SMA related to the plasticdeformation. Bo and Lagoudas (1999) developed a ther-mo-mechanical constitutive model, which can describethe accumulation of plastic strain in the SMA under cyclicloading, by introducing the internal state variables of backand drag stresses. Savi et al. (2002) proposed a one-dimensional constitutive model to consider the effect ofplasticity based on the assumption that there are fourphases, which include three variants of martensite andan austenitic phase, in SMA. Yan et al. (2003) developeda constitutive model to take into account the effect ofplasticity on the reverse phase transformation in thesuperelastic SMA by using the Drucker–Prager type phasetransformation functions and the Von Mises isotropichardening model. Lagoudas and Entchev (2004) developeda three-dimensional rate-independent thermo-mechanicalconstitutive model which can account for the plasticity and

cyclic effect of SMA. Paiva et al. (2005) proposed aconstitutive model to consider both tensile-compressiveasymmetry and plastic strains occurring in the thermo-mechanical behaviors of SMA. Wang et al. (2008a) devel-oped a micro-mechanical constitutive model to expressthe influences of plastic deformation on the mechanicalbehavior of super-elastic SMA by coupling both the slipand twinning deformation mechanisms.

In this paper we make effort to establish a macroscopicconstitutive model which can express the influences ofplasticity on the thermo-mechanical behaviors of SMA.The influences of plasticity on the thermo-mechanicalbehaviors of SMA are attributed to the effects of plasticityon the phase transformation from austenite to martensite(i.e. martensite inverse phase transformation). The macro-scopic constitutive is based on the phenomenon that theplastic strain occurring in SMA can stabilize the martensiteand hinder the martensite inverse phase transformation.

A three-dimensional mechanical constitutive equationis developed to predict the stress–strain response of theSMA undergoing plastic strain based on the expression ofGibbs free energy with plastic strain. A plastic constraintequation, which formulates a linear relationship betweenunrecoverable stress-induced martensitic volume fractionand equivalent plastic strain, is supposed to take accountof the effect of plasticity on the phase transformationbehaviors of SMA undergoing plastic strain. A phase trans-formation equation of sine-type is established to describethe phase transformation behaviors of the SMA undergoingplastic strain based on the differential relationship betweenthe heat energy increment and martensitic volume fraction.The mechanical constitutive equation, plastic constraintequation, and phase transformation equation togethercompose the constitutive model which can express theinfluences of plasticity on the thermo-mechanical behav-iors of SMA. The thermo-mechanical behaviors of the sup-per-elastic SMA undergoing plastic strain and the effect ofplasticity are numerically simulated by the presentedmacroscopic constitutive model. Results show that the pre-sented macroscopic constitutive model can effectivelyreproduce the thermo-mechanical behaviors of the super-elastic SMA undergoing plastic strain and express the effectof plasticity.

2. Mechanical constitutive equation

Fig. 1 shows a diagram of the stress–strain response ofthe super-elastic SMA undergoing plastic strain accordingto the experimental results (McKelvey and Ritchie, 2001;Wang et al., 2008b). Focusing on the phenomenal descrip-tion of the mechanical behaviors without detailing themechanism of phase transformation and plasticity ofSMA, we divide the stress–strain response of the super-elastic SMA undergoing plastic strain into seven variousstages. The four stages in the loading process are respec-tively indicated by the curve segments of ‘oa’, ‘ab’, ‘bc’,and ‘cd’, and called as the stages of initial elastic deforma-tion, forward phase transformation, subsequent elasticdeformation, and the plastic deformation.

The three stages in the unloading process are respec-tively indicated by the curve segments of ‘de’, ‘ef’, and

Fig. 1. Diagram of stress–strain response of super-elastic SMA undergo-ing plastic strain.

B. Zhou / Mechanics of Materials 48 (2012) 71–81 73

‘fg’, and called as the stages of initial elastic recovery ofdeformation, inverse phase transformation, and subse-quent elastic recovery of deformation.

There are often two approaches to develop the constitu-tive relationship of stress, strain and temperature for SMAbased on thermodynamics (Zhou et al., 2009). One is to for-mulate the stress as a function of strain and temperatureusing the expression of Helmholtz free energy. The otheris to formulate the strain as a function of stress and tem-perature using the expression of Gibbs free energy. Gener-ally the Gibbs free energy of SMA is composed of elasticenergy, loading potential energy, phase transformationchemical energy, and surface energy (Wang et al., 2008a).The surface energy has a very small value in comparisonto the other three terms and is negligible, so

U ¼ Uel þUpo þUch ð1Þ

where U, Uel, Upo and Uch denote the Gibbs free energy,elastic energy, loading potential energy, and phase trans-formation chemical energy, respectively.

Considering the plasticity and thermal expansionsimultaneously, we can express the elastic energy of SMAas

Uel ¼ 12

Seijklrijrkl � rij½ep

ij þKijðT � T0Þ� �Uint ð2Þ

where Seijkl and Kij respectively denote elastic compliance

and thermal expansion tensors; rij and epij respectively de-

note macroscopic stress and plastic strain tensors; T and T0

respectively denote current and reference temperatures;Uint is the interaction energy of phase transformation.

The loading potential energy of SMA is a function ofstress and phase transformation strain, expressed as

Upo ¼ �rijðSeijklrkl þ etr

ij Þ ð3Þ

where etrij is the macroscopic phase transformation strain

tensor. The phase transformation chemical energy of SMAis a function of temperature and martensitic volume frac-tion, expressed as

Uch ¼ BðT � TeqÞn ð4Þ

where B is a constant; n and Teq denote martensitic volumefraction and phase equilibrium temperature, respectively.Substituting Eqs. (2)–(4) into Eq. (1), we have

U ¼ � 12

Seijklrijrkl � rijep

ij � rijKijðT � T0Þ � rijetrij

þ BðT � TeqÞn�Uint ð5Þ

There are 24 kinds of martensite variant occurring inSMA during its phase transformations between martensiteand austenite (Gall and Sehitoglu, 1999; Lim and McDo-well, 2002; Wang and Yue 2006). The martensitic volumefraction of SMA should be expressed as

n ¼X24

n¼1

nn ð6Þ

where nn stands for the volume fraction of the nth mar-tensite variant. The macroscopic phase transformationstrain of SMA reads as

etrij ¼

X24

n¼1

enijn

n ð7Þ

where enij stands for the phase transformation strain of the

nth martensite variant. The interaction energy of phasetransformation is defined as (Gall and Sehitoglu, 1999)

Uint ¼ �X24

m;n¼1

Hmnnmnn ð8Þ

where Hmn is the interaction energy matrix (Wang et al.,2008a). Substituting Eq. (8) into Eq. (5), we have

U ¼ � 12

Sijklrijrkl � rij½epij þKijðT � T0Þ� � rijetr

ij

þ BðT � TeqÞX24

n¼1

nn þX24

m;n¼1

Hmnnmnn ð9Þ

According to thermodynamics, the relationship be-tween macroscopic strain and Gibbs free energy in SMAreads as

eij ¼ �@U@rij

ð10Þ

Substituting Eq. (9) into Eq. (10), we have

eij ¼ Seijklrkl þ etr

ij þ epij þKijðT � T0Þ ð11Þ

where the terms Seijklrkl and Kij(T � T0) denote the elastic

strain and thermal expansion strain, respectively.Therefore the total macroscopic strain of SMA includesfour parts, which are elastic strain, phase transformationstrain, plastic strain, and thermal expansion strain,respectively.

Based on the linear hardening model of plastic mechan-ics (Zhou et al., 2010), the plastic strain rate of SMA is for-mulated as

_epij ¼ Hðre � rsÞðSp

ijkl � SeijklÞ _rkl ð12Þ

74 B. Zhou / Mechanics of Materials 48 (2012) 71–81

where ‘.’ denote the derivative with respect to time; Spijkl is

called as the plastic compliance tensor; rs is the plasticyield stress;

re ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32

rij �13rkk

� �rij �

13rkk

� �sð13Þ

is the equivalent stress;

Hðre � rsÞ ¼1 ðre > rsÞ0 ðre � rsÞ

�ð14Þ

is the Heaviside function. The integral operation on Eq. (12)leads to

epij ¼ ðS

pijkl � Se

ijklÞZ t

0Hðre � rsÞ _rkldt ð15Þ

where t stands for time.Brinson (1993) macroscopically resolved the martensite

occurring in SMA into two parts, which are respectively in-duced by temperature and stress. The martensitic volumefraction of SMA is also resolved into two parts:

n ¼ nt þ ns ð16Þ

where nt and ns respectively denote temperature-inducedmartensitic volume fraction and stress-induced martesniticvolume fraction. The macroscopic phase transformationstrain of SMA is a linear function of stress-inducedmartesnitic volume fraction, expressed as

etrij ¼ eL

ijns ð17Þ

where eLij is called as extreme phase transformation strain.

Substituting Eqs. (17) and (15) into Eq. (11), we have

eij ¼ Seijklrkl þ eL

ijns þ ðSpijkl � Se

ijklÞZ t

0Hðre � rsÞ _rkldt

þKijðT � T0Þ ð18Þ

This is the integral-form mechanical constitutive equationexpressing the relationship of stress, strain and tempera-ture of the SMA undergoing plastic strain. Generally arate-form constitutive equation is more convenient thanan integral-form constitutive equation for the numericalcalculation in practical applications. Taking the derivativewith respect to time on both sides of Eq. (18) and assumingthe material parameters of SMA are rate-independent, wehave the rate-form mechanical constitutive equation ofthe SMA undergoing plastic strain, expressed as

_eij ¼ Seijkl

_rkl þ eLij

_ns þ ðSpijkl � Se

ijklÞHðre � rsÞ _rkl þKij_T ð19Þ

In this paper, SMA is assumed to be an isotropic mate-rial. According to continuum mechanics, the elastic com-pliance tensor of the isotropic SMA is expressed as

SeijklðnÞ ¼

1E

12ð1þ mÞðdikdjl þ djkdilÞ þ mdijdkl

� �ð20Þ

where E is the elastic modulus, v is the Poisson’s ratio, and

dij ¼1 i ¼ j

0 i – j

�ð21Þ

is the Kronecker delta. Similarly the plastic compliancetensor of the isotropic SMA is expressed as

SpijklðnÞ ¼

1P

12ð1þ mÞðdikdjl þ djkdilÞ þ mdijdkl

� �ð22Þ

where P is called as the plastic hardening modulus (Zhouet al., 2010). The extreme phase transformation strain ofthe isotropic SMA is expressed as

eLij ¼ TbiTbjeL ð23Þ

where the material constant eL is called as maximum resid-ual strain, Tbi/Tbj is the coordinate transformation tensor,the subscript of b stands for the principle coordinate, andthe subscript of i/j stands for the arbitrary coordinate.The thermal expansion tensor of the isotropic SMA is ex-pressed as

Kij ¼ adij ð24Þ

where a is the thermal expansion coefficient of the isotro-pic SMA.

Substituting Eqs. (24), (23), (22), and (20) into Eq. (18),we express the integral-form mechanical constitutiveequation of the isotropic SMA as

eij ¼ mijklrkl

Eþ 1

P� 1

E

� �Z t

0Hðre � rsÞ _rkldt

� �þ TbiTbjeLns þ dijaðT � T0Þ ð25Þ

where

mijkl ¼12ð1þ mÞðdikdjl þ djkdilÞ þ mdijdkl ð26Þ

Similarly substituting Eqs. (24), (23), (22), and (20) into Eq.(19), we express the rate-form mechanical constitutiveequation of the isotropic SMA as

_eij ¼ mijkl1Eþ 1

P� 1

E

� �Hðre � rsÞ

� �_rkl þ TbiTbjeL

_ns þ dija _T

ð27Þ

The elastic modulus of the isotropic SMA is a function ofmartensitic volume fraction (Brinson, 1993), expressed as

E ¼ Ea þ ðEm � EaÞn ð28Þ

where Ea and Em denote the moduli of austenite and mar-tensite, respectively.

Eqs. (25) and (27) are the new integral-form and rate-form three-dimensional mechanical constitutive equationswhich predict the relationship of stress, strain and temper-ature of the SMA undergoing plastic strain. It should beemphasized that all the material constants related to thisnew mechanical equation can be determined through thetensile tests at different temperatures (Zhou and Yoon,2005; Wang et al., 2008b).

3. Plastic constraint equation

The influences of plasticity on the thermo-mechanicalbehaviors are the results that the effect of plasticity onthe martensite inverse phase transformation occurring inSMA. The plastic strain in the super-elastic SMA is able tostabilize the stress-induced martensite appearing duringthe loading process, which leads to an incomplete inversephase transformation from martensite to austenite during

B. Zhou / Mechanics of Materials 48 (2012) 71–81 75

the unloading process (McKelvey and Ritchie, 2001).Therefore the effect of plasticity on the martensite inversephase transformation of SMA can be expressed by the rela-tionship between unrecoverable stress-induced martens-itic volume fraction and plastic strain occurring in SMA.

Based on the description supposed by Yan et al. (2003),the relationship of unrecoverable stress-induced martens-itic volume fraction and equivalent plastic strain of thesuper-elastic SMA undergoing plastic strain is formulatedas

nus ¼

eep

ecpþ 1�

eep

ecp

!H ee

p � ecp

� ð29Þ

and called as plastic constraint equation, where nus denote

unrecoverable stress-induced martensitic volume fraction;H is the Heaviside function expressed by Eq. (14); ec

p is thecritical plastic above which the inverse phase transforma-tion from martensite to austenite upon loading will not oc-cur at all, and it can be determined from tensile test (Wanget al., 2008b);

eep ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi23ep

ijepij

rð30Þ

is the equivalent plastic strain according to plastictheorem.

4. Phase transformation equation

The phase transformation behaviors of the SMA in thestate of stress free can be investigated by the differentialscanning calorimetry (DSC) tests. Based on the results ofDSC test, Zhou and Yoon (2006b) described the curve ofheat flow versus temperature from DSC test as

hðTÞ ¼ dDQdT¼ b1 cosðmT �m0Þ ðMs � T � Mf Þ ð31aÞ

during the forward phase transformation from austenite tomartensite upon cooling, and

hðTÞ ¼ dDQdT¼ b2 cosðaT � a0Þ ðAs � T � Af Þ ð31bÞ

during the inverse phase transformation from martensiteto austenite upon heating, respectively. In Eq. (31), DQ de-notes heat energy increment; b1 and b2 are the constants tobe determined;

m ¼ pMf�Ms

m0 ¼ p2

MsþMf

Mf�Ms

8<: ð32aÞ

where Ms and Mf denote the critical temperatures in theforward phase transformation upon cooling, which arecalled as martensitic starting temperature and martensiticfinishing temperature, respectively;

a ¼ pAf�As

a0 ¼ p2

AsþAf

Af�As

8<: ð32bÞ

where As and Af denote the critical temperatures in the in-verse phase transformation upon heating, which are calledas austenitic starting temperature and austenitic finishing

temperature, respectively. In Eq. (32), p denotes circumfer-ence ratio with the value of 1.3415926.

There is a following differential relationship betweenthe heat energy increment and temperature-induced mar-tensitic volume fraction (Zhou et al., 2009).

dnt

dT¼ k

dDQdTþ d ð33Þ

where k and d are the constants to be determined. Substi-tuting Eq. (31a) into Eq. (33) and assuming both values ofdnt/dt at T = Ms and T = Mf are zero, we have

dnt

dT¼ B1 cosðmT �m0Þ ðMs � T � Mf Þ ð34aÞ

where B1 = kb1 is a constant to be determined. SubstitutingEq. (31b) into Eq. (33) and assuming both values of dnt/dt atT = As and T = Af are zero, we have

dnt

dT¼ B2 cosðaT � a0Þ ðAs � T � Af Þ ð34bÞ

where B2 = kb2 is a constant to be determined.Based on the relationship of phase transformation crit-

ical stresses and temperature of SMA supposed by Brinson(1993), the phase transformation critical stresses of mar-tensitic starting stress rms and martensitic finishing stressrmf during the forward phase transformation are respec-tively formulated as

rms ¼rcrs ðT � MsÞrcrs þ CMðT �MsÞ ðT > MsÞ

�ð35aÞ

and

rmf ¼rcrf ðT � MsÞrcrf þ CMðT �MsÞ ðT > MsÞ

�ð35bÞ

where CM, rcrs and rcrf are the material constants express-ing the relationship between the forward phase transfor-mation critical stresses and temperature in SMA. Thephase transformation critical stresses of austenitic startingstress ras and austenitic finishing stress raf during the in-verse phase transformation are respectively formulated as

ras ¼ CAðT � AsÞ ðT > AsÞ ð36aÞ

and

raf ¼ CAðT � Af Þ ðT > Af Þ ð36bÞ

where CA is a material constant expressing the relationshipbetween the inverse phase transformation critical stressesand temperature in SMA.

According to the linear relationship of phase transfor-mation critical stresses and temperature in Eqs. (35) and(36), we can assume that the differential relationship ofstress-induced martensitic volume fraction and stress besame as that of temperature-induced martensitic volumefraction and temperature. Bases on the expression for thederivative of temperature-induced martensitic volumefraction with respect to temperature in Eqs. (34a) and(34b), we formulate the derivative of stress-induced mar-tensitic volume fraction with respect to stress as

dns

dre¼ B1 cosðsmre � sm0Þ ðrms � re � rmf Þ ð37aÞ

Table 1Material constants of SMA for numerical simulations.

rms/MPa rmf/MPa ras/MPa raf/MPa rs/MPa

410 430 230 200 800

Ea/GPa Em/GPa P/GPa ecp/% eL/%

67 55 10 0.1 4.0

76 B. Zhou / Mechanics of Materials 48 (2012) 71–81

during the forward phase transformation from austenite tomartensite upon loading, and

dns

dre¼ B2 cosðsare � sa0Þ ðras � re � raf Þ ð37bÞ

during the inverse phase transformation from martensiteto austenite upon unloading, respectively, where

sm ¼ prmf�rms

sm0 ¼ p2

rmsþrmf

rmf�rms

8<: ð38aÞ

and

sa ¼ praf�ras

sa0 ¼ p2

rasþraf

raf�ras

8<: ð38bÞ

In Eq. (38), p also denotes circumference ratio with the va-lue of 1.3415926.

The integration operation on Eq. (37a) with the bound-ary condition of ns = 0 at re = rms and ns = 1 at re = rmf leadsto

ns ¼12

sinðsmre � sm0Þ þ12ðrms � re � rmf Þ ð39aÞ

during the forward phase transformation from austenite tomartensite upon loading. Integration on Eq. (37b) with theboundary condition of ns = 1 at re = ras and ns ¼ nu

s at re =raf leads to

ns ¼ �1� nu

s

2sinðsare � sa0Þ þ

1þ nus

2ðras � re � raf Þ

ð39bÞ

during the inverse phase transformation from martensiteto austenite upon unloading.

Eq. (39) is the phase transformation equation describingthe phase transformation behaviors of SMA undergoingplastic strain. Taking the derivative with respect to timeon both sides of Eq. (39), we have the rate-form phasetransformation equation of SMA undergoing plastic strain,which is expressed as

_ns ¼12

sm _re cosðsmre � sm0Þ ðrms � re � rmf Þ ð40aÞ

during the process of forward phase transformation fromaustenite to martensite upon loading, and

_ns ¼ �12ð1� nu

s Þsa _re cosðsare � sa0Þ ðras � re � raf Þ

ð40bÞ

during the process of inverse phase transformation frommartensite to austenite upon unloading, respectively. Eqs.(39) and (40) are the new phase transformation equationand its rate-form, which can describe the phase transfor-mation behaviors of the SMA undergoing plastic strain.All material constants related to the new phase transfor-mation equation can be determined through the tensileand DSC tests (Brinson, 1993; Zhou and Yoon, 2006a).

The integral-form mechanical constitutive equation,Eq. (25), the plastic constraint equation, Eq. (29), and thephase transformation equation, Eq. (39), compose theintegral-form macroscopic constitutive model of SMA

considering plasticity; and the rate-form mechanical con-stitutive equation, Eq. (27), the plastic constraint equation,Eq. (29), and the rate-form phase transformation equation,Eq. (40), compose the rate-form macroscopic constitutivemodel of SMA considering plasticity. All the material con-stants in this macroscopic constitutive model can be deter-mined through macroscopic experiments, so it is easy touse this model to simulate the thermo-mechanical behav-iors of the SMA undergoing plastic strain for the practicalapplications.

5. Numerical simulations

The supposed macroscopic constitutive model of rate-form, Eqs. (27), (29) and (40), is used to numerically simu-late the mechanical behaviors of the super-elastic SMAundergoing different plastic strains. The effect of plasticityon the mechanical properties of SMA is also simulated bysupposed constitutive model of rate-form. The materialconstants of SMA used for the numerical simulations arelisted in Table 1. The curves of stress versus strain, stress-induced martensitic volume fraction versus stress, elasticmodulus versus stress, residual strain after unloadingversus plastic strain, and elastic modulus after unloadingversus plastic strain are plotted based on the results ofnumerical simulation.

Fig. 2 shows the stress-strain curves of the super-elasticSMA with different values of plastic strain. Fig. 2(a) plotsthe curves simulated by the new macroscopic constitutivemodel at the cases that SMA undergoes the plastic strainsof 0%, 0.5%, 1.0%, and 1.5%, respectively. At the case thatthe super-elastic SMA does not undergo plastic strain, aclose hysteretic loop occurs in the stress-strain curve. Dur-ing the loading process, the material initiates elastic strain,and then appears large nonlinear strain due to the forwardphase transformation from austenite to martensite. Thematerial expresses as elastic behavior again after the for-ward phase transformation finishes. During the unloadingprocess, the strain first elastically recovers, and then non-linearly recovers due to the inverse phase transformationfrom martensite to austenite. The strain elastically recov-ers again after the inverse phase transformation finishes.There is no residual strain occurring in the material afterunloading, which illustrates the classical super-elasticityof SMA. At the case that the super-elastic SMA undergoesthe plastic strain of 0.5%, an open hysteretic loop occursin the stress–strain curve. During the loading process, thematerial serially undergoes the four stages of initial elasticdeformation, forward phase transformation, subsequentelastic deformation, and plastic deformation. During theunloading process, the material serially undergoes thethree stages of initial elastic recovery of deformation,

Strain ( % )0 1 2 3 4 5 6 7 8

Stre

ss (

Mpa

)

0

200

400

600

800

1000

0 0.5 % 1.0 % 1.5 %

(a) Curves by the new model with the plastic strains of 0, 0.5%, 1.0% and 1.5%

Strain ( % )0 1 2 3 4 5 6 7

Stre

ss (

Mpa

)

0

200

400

600

800

NewYan

(b) Comparison between the new and Yan’s models with plastic strain of 0.5%

40M

pa

40M

pa

Fig. 2. Stress–strain curves of super-elastic SMA with different plastic strains.

B. Zhou / Mechanics of Materials 48 (2012) 71–81 77

inverse phase transformation, and subsequent elasticrecovery of deformation. There is a residual strain occur-ring in the material after unloading due to that the plasticstrain stabilizes the martensite induced by loading, andleads to an incomplete inverse phase transformation frommartensite to austenite during the unloading process. Atthe cases that the super-elastic SMA undergoes the plasticstrain of 1% and 1.5%, there is no hysteretic loop occurringin the stress–strain curve because a larger plastic strain,whose value is above the critical plastic strain, will re-straint the inverse phase transformation from martensiteto austenite during the unloading process. In the sameway, the material serially undergoes the four stages of ini-

tial elastic deformation, forward phase transformation,subsequent elastic deformation, and plastic deformationduring the loading process. But the material only under-goes the stage of elastic recovery of deformation duringthe unloading process. Both plastic strain and phase trans-formation strain appearing in the loading process becomethe residual strain after unloading. Fig. 2(b) plots thestress–strain curves at the case that SMA undergoes theplastic strain of 0.5%, which are simulated by the new mac-roscopic constitutive model and the model developed byYan et al. (2003), respectively. The major differences be-tween these two models are in the forward phase transfor-mation from austenite to martensite during the loading

Stress ( Mpa )0 100 200 300 400 500 600

Stre

ss-in

duce

d m

arte

nsiti

c vo

lum

e fra

ctio

n

0.0

.2

.4

.6

.8

1.0

1.2

loading

loading

unloading

unloading

0

0.1%

0.3%

0.5%

0.7%

0.9%

1.0%

Fig. 3. Curves of stress-induced martensitic volume fraction versus stress of the super-elastic SMA with different plastic strain.

78 B. Zhou / Mechanics of Materials 48 (2012) 71–81

process, and the inverse phase transformation from mar-tensite to austenite during the unloading process. This isbecause the new model and the model developed by Yanet al. (2003) respectively formulate the nonlinear andlinear relationships between stress-induced martensiticvolume fraction and the equivalent stress for the super-elastic SMA.

Fig. 3 shows the curves of stress-induced martensiticvolume fraction versus stress of the super-elastic SMA atthe cases that the material undergoes the plastic strainsof 0%, 0.1%, 0.3%, 0.5%, 0.7%, 0.9% and 1.0%, respectively.At the case that the material does not undergo plastic

Stress0 100 200 3

Elas

tic m

odul

us (

Gpa

)

54

56

58

60

62

64

66

680

0.1%

0.3%

0.5%

0.7%

0.9%

1.0%

loa

un

unloadin

Fig. 4. Curves of elastic modulus versus stress of the

strain, a close hysteretic loop occurs in the curve ofstress-induced martensitic volume fraction versus stress.During the loading process, the stress-induced martensiticvolume fraction increases from zero to one upon that thestress increases from martensitic starting stress to mar-tensitic finishing stress. During the unloading process,the stress-induced martensitic volume fraction decreasesfrom one to zero upon that the stress decreases fromaustenitic starting stress to austenitic finishing stress. Atthe cases that SMA undergoes the plastic strains of 0.1%,0.3%, 0.5%, 0.7% and 0.9%, an open hysteretic loop occursin the curve of stress-induced martensitic volume fraction

( Mpa )00 400 500 600

ding

loading

loading

g

super-elastic SMA with different plastic strain.

B. Zhou / Mechanics of Materials 48 (2012) 71–81 79

versus stress due to the unrecoverable stress-induced mar-tensitic volume fraction occurring after unloading. This isbecause that the plastic strain can stabilize the stress-induced martensite during the loading process, whichbrings about an incomplete inverse phase transformationfrom martensite to austenite during the unloading process.The value of unrecoverable stress-induced martensitic vol-ume fraction increases with the increase of that of plasticstrain. At the case that SMA undergoes the plastic strainsof 1.0%, there is no hysteretic loop occurring in the curveof stress-induced martensitic volume fraction versus stressfor the larger plastic strain restraints the inverse phase

Plastic st0.0 .2 .4 .6 .8

Res

idua

l stra

in (

% )

0

1

2

3

4

5

6

Fig. 5. Curve of residual strain after unloading v

Plastic s0.0 .2 .4 .6 .8

Elas

tic m

odul

us (

Mpa

)

54

56

58

60

62

64

66

68

Fig. 6. Curve of elastic modulus after unloading

transformation from martensite to austenite during theunloading process.

Fig. 4 shows the curves of elastic modulus versus stressof the super-elastic SMA at the cases that the materialundergoes the plastic strains of 0%, 0.1%, 0.3%, 0.5%, 0.7%,0.9% and 1.0%, respectively. At the case that the materialdoes not undergo plastic strain, a close hysteretic loop oc-curs in the curve of elastic modulus versus stress. Duringthe loading process, the value of elastic modulus graduallydecreases in the region that the stress increases frommartensitic starting stress to martensitic finishing stress.During the unloading process, the value of elastic modulus

rain ( % )1.0 1.2 1.4 1.6

ersus plastic strain of super-elastic SMA.

train ( % )1.0 1.2 1.4 1.6

versus plastic strain of super-elastic SMA.

80 B. Zhou / Mechanics of Materials 48 (2012) 71–81

gradually increases in the region that the stress decreasesfrom austenitic starting stress to austenitic finishing stress.At the cases that the material undergoes the plastic strainsof 0.1%, 0.3%, 0.5%, 0.7% and 0.9%, an open hysteretic loopoccurs in the curve of elastic modulus versus stress, whichresults from the incomplete inverse phase transformationdue to the plastic strain. At the case that the materialundergoes the plastic strains of 1.0%, there is no hystereticloop occurring in the curve of elastic modulus versusstress, which results from that the larger plastic strain re-straints the inverse phase transformation from martensiteto austenite during the unloading process.

Fig. 5 shows the curve of residual strain after unload-ing versus plastic strain of the super-elastic SMA. Whenthe value of plastic strain is below critical plastic strain,the residual strain linearly increases with the increase ofplastic strain. This is because the plastic is unrecoverableand can reduce the inverse phase transformation frommartensite to austenite and the recoverable phase trans-formation strain during the unloading process. Whenthe value of plastic strain is above critical plastic strain,the residual strain also linearly increases with the in-crease of plastic strain. However the slope at the regionthat the plastic strain is above critical plastic strain ismuch smaller than that at the region of plastic strain be-low critical plastic strain, which is because that a plasticstrain above critical plastic strain will restraint the in-verse phase transformation during the unloading process.

Fig. 6 shows the curve of elastic modulus after unload-ing versus plastic strain of the super-elastic SMA. The valueof elastic modulus after unloading decreases with the in-crease of plastic strain in the region that the plastic strainis below critical plastic strain. This is because the plasticcan reduce the inverse phase transformation from mar-tensite to austenite, and the value of elastic modulus ofaustenite is larger than that of martensite. The value ofelastic modulus after unloading does not change in theregion that the plastic strain is above critical plastic strain,which also because a plastic strain above critical plasticstrain will restraint the inverse phase transformation dur-ing the unloading process.

6. Conclusions

The three-dimensional mechanical constitutive equa-tion is developed to predict the stress–strain response ofthe SMA undergoing plastic strain. The effect of plasticityon the phase transformation behaviors of SMA are ex-pressed by the linear plastic constraint equation. Thesine-type phase transformation equation is established todescribe the phase transformation behaviors of the SMAundergoing plastic strain. All the material constants inthe presented macroscopic constitutive model, which iscomposed of the mechanical constitutive equation, plasticconstraint equation, and phase transformation equation,can be determined through macroscopic experiments.The thermo-mechanical behavior of the supper-elasticSMA undergoing plastic strain and the effect of plasticityare numerically simulated by the new macroscopic consti-tutive model. The results of numerical simulation show

that the presented macroscopic constitutive model caneffectively reproduce the thermo-mechanical behaviors ofthe super-elastic SMA undergoing plastic strain and ex-press the effect of plasticity.

Acknowledgements

This work was supported by the Major Project Cultiva-tion Plan of Fundamental Research Funds for Central Uni-versities (Grant No. HEUCFZ1004), the Harbin TalentFoundation of Scientific and Technical Innovation (GrantNo. RC2009QN0170046), the Foundation for ReturnedOverseas Scholars from National Ministry of Education(Grant No. 37) and the National Postdoctoral Science Foun-dation (Grant No. 20080430933) of China. The authorwould like to thank them and the Harbin Engineering Uni-versity and Harbin Institute of Technology of China.

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