a macroscopic constitutive model for shape-memory alloys: theory and finite-element simulations

Comput. Methods Appl. Mech. Engrg. 198 (2009) 1074–1086

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Comput. Methods Appl. Mech. Engrg.

journal homepage: www.elsevier .com/locate /cma

A macroscopic constitutive model for shape-memory alloys: Theoryand finite-element simulations

P. Thamburaja a,*, N. Nikabdullah b

a Department of Mechanical Engineering, National University of Singapore, Block E1, #05-25, Singapore 117576, Singaporeb Department of Mechanical Engineering, University Kebangsaan Malaysia, Bangi, Malaysia 43600, Malaysia

a r t i c l e i n f o

Article history:Received 24 December 2007Received in revised form 31 July 2008Accepted 18 November 2008Available online 3 December 2008

Keywords:A. Shape-memory alloysB. Constitutive behaviorPlasticityC. Finite elements

0045-7825/$ - see front matter � 2008 Elsevier B.V. Adoi:10.1016/j.cma.2008.11.016

* Corresponding author. Tel./fax: +65 6779 6559.E-mail address: [email protected] (P. Thamburaja

a b s t r a c t

In this work, we develop a non-local and thermo-mechanically-coupled constitutive model for poly-crystalline shape-memory alloys (SMAs) capable of undergoing austenite $ martensite phase transfor-mations. The theory is developed in the isotropic metal-plasticity setting using fundamental thermody-namic laws and the principle of micro-force balance [E. Fried, M. Gurtin, Dynamic solid–solid transitionswith phase characterized by an order parameter, Physica D 72 (1994) 287–308]. The constitutive model isthen implemented in the ABAQUS/Explicit (2007) finite-element program by writing a user-material sub-routine. The results from the constitutive model and numerical procedure are then compared to repre-sentative physical experiments conducted on a polycrystalline rod Ti–Ni undergoing superelasticity.The constitutive model and the numerical simulations are able to reproduce the stress–strain responsesfrom these physical experiments to good accuracy. Experimental strain–temperature–cycling and shape-memory effect responses have also shown to be qualitatively well-reproduced by the developed consti-tutive model.

With the aid of finite-element simulations we also show that during phase transformation, the depen-dence of the position i.e. the thickness of the austenite–martensite interface on the mesh density is heav-ily minimized when a non-local constitutive theory is used.

� 2008 Elsevier B.V. All rights reserved.

1 In this work, we neglect the effect of martensitic reorientation and detwinning inMAs. For recent crystal-plasticity-based constitutive models which describe the

1. Introduction

Polycrystalline shape-memory alloys (SMAs) are one the mostresearched smart materials due to its various technological appli-cations which are primarily in the biomedical, MEMS, aerospaceand civil structures area [3,8,14,22]. Some examples of SMAs in-clude the Ti–Ni, Cu–Al–Ni, Cu–Zn–Al and Au–Cd systems. Depend-ing on temperature and/or stress state, SMAs have the ability toexist in different phases. For practical purposes, the two mostimportant phases are the austenite phase and the martensite phase.The austenite phase (high-symmetry phase) is stable under lowstresses and/or high temperatures whereas the martensite phase(low-symmetry phase) is stable under high stresses and/or lowtemperatures. In SMAs, the transformation between the austeniticphase and the martensitic phase is reversible and diffusionless.

Due to its ability in undergoing reversible austenite$martens-ite phase transformations, SMAs exhibit: (a) superelasticity orpseudoelasticity by transformation, and (b) the shape-memoryeffect.

Superelasticity is the stress-induced transformation from aus-tenite to martensite and back due to cyclic extension between zero

ll rights reserved.

).

and a finite (but small) amount of strain at an ambient tempera-ture above the SMA sample’s austenite finish temperature, haf . Thistype of deformation typically results in a flag-like stress–strain re-sponse. Although the deformation due to phase transformation isrecovered at the end of the loading cycle, there is hysteresis dueto the motion of interfaces between the austenite and martensitephases.

The shape-memory effect occurs when a SMA sample is initiallyaustenitic and tested under isothermal conditions at a temperaturebetween its martensite start temperature, hms, and haf . Forward load-ing will cause a transformation from austenite to martensite. How-ever, upon reverse loading to zero stress, the transformation strainobtained due to the austenite to martensite transformation is notrecovered until the sample’s temperature is raised to above haf .

There has been extensive activity in the development of consti-tutive models to describe the austenite$ martensite phase trans-formations in SMAs1. Examples of one-dimensional work include

artensitic reorientation and detwinning in initially-martensitic single-crystal SMAs,lease refer to the works of Thamburaja [27], Thamburaja et al. [32] and Pan et al.8]. An isotropic-plasticity-based constitutive model which describes the martensiticorientation in initially-martensitic SMAs has also been developed by Pan et al. [19].

Smp[1re

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3 Notation: The terms div,r andr2 denote the divergence, gradient and Laplacianrespectively. All the tensorial variables in this work are second-order tensors unlessstated otherwise. For a tensor B, B> denotes its transpose. We also write traceB for the

P. Thamburaja, N. Nikabdullah / Comput. Methods Appl. Mech. Engrg. 198 (2009) 1074–1086 1075

the models of Liang and Rogers [12], Mueller and Xu [16], Abey-aratne and Knowles [2], and Bekker and Brinson [6] which capturesthe necessary response characteristics of SMAs. Some earlier three-dimensional modeling of SMAs include the works of Sun and Hwang[24,25], Boyd and Lagoudas [7], Patoor et al. [20], Auricchio and Tay-lor [4], Gall and Sehitoglu [10] and others. However, these aforemen-tioned models were verified for only uniaxial-type loadingconditions. The three-dimensional constitutive models of Lim andMcDowell [13], Thamburaja and Anand [28,29] and Moumni andZaki [35,15] have been shown to successfully model austenite $martensite phase transformations under multi-axial loading condi-tions.2 Two common aspects among these aforementioned modelsare that they are not formulated to study the coupled thermo-mechanical response during austenite $ martensite phase transi-tions, and their models were developed using local theories.

Since the overall behavior of SMAs are very sensitive to changesin temperature, it is of paramount importance to develop a set ofconstitutive equations which are able to model the coupled ther-mo-mechanical response of SMAs under multi-axial deformations.The constitutive model must also be capable of tracking the posi-tion of the austenite–martensite interface front(s) during phasetransformation as it is the motion of these interfaces which causesthe observed hysteresis during the deformation of SMAs. Further-more, depending on the polycrystalline crystallographic textureSMAs also exhibit the tension–compression asymmetry or strength-differential effect during austenite$martensite phase transforma-tions (cf. Orgeas and Favier, [17]). Hence, the developed constitutivemodel needs to also recognize this effect to accurately model thephase transformation behavior in tension versus compression.

Therefore, the main focuses of this work are to: (1) develop aconstitutive model capable of describing the austenite$martens-ite phase transitions in SMAs that also reproduces the aforemen-tioned key points; (2) develop a robust numerical algorithmbased on the constitutive model, and implement it in the commer-cially-available finite-element program ABAQUS [1]; and (3) verifythe constitutive model and its numerical implementation with re-spect to physical experimental results.

The structure of this paper is as follows: In Section 2 we formu-late our thermo-mechanically-coupled, isotropic-plasticity-basedconstitutive model to describe austenite$ martensite phase tran-sitions in polycrystalline SMAs using basic thermodynamics princi-ples and the concept of micro-force balance [9]. The constitutivemodel is then implemented in the ABAQUS/Explicit (2007) finite-element program by writing a user-material subroutine. Thealgorithmic details of the time-integration procedure given inAppendix A. In Section 3, we perform finite-element simulationsto model the superelastic deformation for selected experimentsconducted on polycrystalline rod. Furthermore, we also performnumerical simulations to show that the constitutive model is ableto qualitatively reproduce the experimental strain–temperaturecycling and shape-memory effect responses exhibited by SMAs.We study the motion of the austenite–martensite interfaces duringphase transformation in Section 4. Here we demonstrate the abilityof our constitutive model in tracking the position and the motionof these interfaces during superelastic deformation. Finally, weconclude in Section 5.

2. Constitutive model

Here we develop a constitutive model for shape-memory alloyscapable of undergoing austenite–martensite phase transformation

2 Some work on modeling the deformation behavior of shape-memory alloys usingthe Young measures technique have also been pursued by Mielke et al. [11] andKruzik et al. [21].

trace of the tensor B. The symmetric portion of tensor B is denoted bysymB � ð1=2ÞðBþ B>Þ. The deviatoric portion of tensor B is denoted byB0 � B� ð1=3ÞðtraceBÞ1. The scalar product of two tensors A and B is denoted byA � B ¼ traceðB>AÞ. The scalar product of two vectors u and v is also denoted by u � vThe magnitude of vector u and tensor B is denoted by juj and jBj, respectively. The

using the theory of isotropic metal plasticity cf. Thamburaja andEkambaram [31] for a recent example.

The list of the governing variables in the constitutive modelare3: (i) The Helmholtz free energy per unit volume, w. (ii) The Cau-chy stress tensor, T. (iii) The total strain tensor, E. (iv) The totalinelastic strain tensor, Ep. It represents the cumulative strain due toaustenite $ martensite phase transformations. (v) The total elasticstrain tensor, Ee. It describes the elastic stretches that gives rise tothe Cauchy stress T. The elastic strain is given by

Ee ¼ E� Ep: ð1Þ

(vi) The total martensite volume fraction, n with 0 6 n 6 1.To construct the constitutive equations we focus on a contin-

uum body occupying a region R with n the outward unit normalon its boundary denoted by oR. We also denote oA and oV as thearea and volume integral, respectively.

2.1. Kinematics and kinetics

Let x and t respectively denote the position vector of a materialpoint and time. The total strain is then given by

E ¼ symðruÞ ! _E ¼ symðr _uÞ; ð2Þ

where u ¼ uðx; tÞ and _u ¼ _uðx; tÞ represents the displacement vec-tor and velocity vector, respectively. Assuming that phase transfor-mations are accompanied by no changes in volume i.e. traceEp ¼ 0[24], we take the inelastic strain-rate tensor to be purely deviatoric,and augment it with a term to describe the tension–compressionasymmetry [17] :

_Ep ¼ kð1þ a/ÞX2

i¼1

_niNi: ð3Þ

The flow direction tensors Ni with i ¼ 1;2 are restricted by Ni ¼ N>iand traceNi ¼ 0. We define N1 and N2 as the forward transformationand reverse transformation flow direction, respectively. Further-more, we denote _n1 P 0 and _n2 6 0 as the transformation rates. For-ward transformation occurs if _n1 > 0 whereas reverse transformationoccurs if _n2 < 0. Following Auricchio et al. [5], the total rate ofchange of martensite volume fraction is then given by

_n ¼X2

i¼1

_ni ¼ _n1 þ _n2: ð4Þ

A transformation from austenite to martensite takes place when_n > 0. Conversely, a transformation from martensite to austenitetakes place when _n < 0. No phase transformation occurs when_n ¼ 0. With k > 0 being a constant of proportionality, we will en-force jNij ¼ �T for each i where �T > 0 denotes the transformationstrain due to austenite–martensite phase transformation (to bedetermined experimentally). The scalar / with �1 6 / 6 1 repre-sents the J3 i.e. the third stress-invariant measure with the dimen-sionless constitutive parameter a (to be calibrated fromexperiments) controlling the extent of the tension–compressionasymmetry. The functional form for / will be described later.

From microscopic considerations, the reverse transformation isthe crystallographic recovery of the deformation which is inducedduring forward transformation i.e. the reverse transformation is re-

second-order identity tensor is denoted by 1.

,

.

1076 P. Thamburaja, N. Nikabdullah / Comput. Methods Appl. Mech. Engrg. 198 (2009) 1074–1086

stricted by the forward transformation history. Furthermore from aphenomenological point of view, the deformation experienced dueto austenite ! martensite (forward) phase transformation can becompletely recovered by the reversal of the forward loadinghistory.

Therefore, with N1 to be defined we take the flow direction N2

to be given by (Sun and Hwang, [24], Boyd and Lagoudas, [7]):

N2 ¼ �TEp

jEpj

� �:

2.2. Micro-force balance

For the martensite volume fraction n, the micro-force systemwhich describe the forces that perform work associated to austen-ite $ martensite phase transformations consist of: (a) the micro-traction vector, c, measured per unit area; (b) the scalar internalmicro-force, pint, measured per unit volume; and (c) the scalarexternal micro-force, pext, measured per unit volume. Followingthe work of Fried and Gurtin [9], we write the corresponding mi-cro-force balance equation associated with these micro-force sys-tems asZ

oR

c � noAþZR

pextoV ¼ZR

pintoV : ð5Þ

Applying the divergence law on Eq. (5) and localizing the resultwithin R results in

divc� pint þ pext ¼ 0: ð6Þ

We take c, pint and pext to be functions of the variables n, _n and rn:

c ¼ cðn; _n;rnÞ; pint ¼ pintðn; _n;rnÞ and pext ¼ pextðn; _n;rnÞ:

2.3. Balance of linear momentum

The balance of linear momentum is given byZoR

TnoAþZR

boV ¼ o; ð7Þ

with b being the macroscopic body force vector per unit volume.Inertial forces are also included in the body force b. Using the diver-gence law on Eq. (7) and localizing the result within R yields

divTþ b ¼ o: ð8Þ

2.4. Balance of angular momentum

From the balance of angular momentum we haveZoR

x� TnoAþZR

x� boV ¼ o: ð9Þ

Applying the divergence law on Eq. (9) and localizing the resultwithin R while using Eq. (8) yields

T ¼ T> ! the Cauchy stress is symmetric: ð10Þ

2.5. Balance of energy

The first law of thermodynamics (the balance of energy) is sta-ted as

ZoR

Tn � _uþ ðc � nÞ _n� q � nh i

oAþZR

b � _uþ pext_nþ r

� �oV

¼ ddt

ZR

�oV ; ð11Þ

where � is the internal energy per unit volume. Here q is the heatflux vector measured per unit area and r is the heat supply per unitvolume. Applying the divergence law on Eq. (11) and localizing theresult within R while using Eqs. (2), (6), (8) and (10) yields

T � _Eþ c � r _nþ pint_n� divqþ r ¼ _�: ð12Þ

Assuming that the external micro-force vanishes i.e. pext ¼ 0,substituting _E ¼ _Ee þ _Ep and Eq. (6) into Eq. (12) results in

T � _Ee þ T � _Ep þ c � r _nþ ðdivcÞ _n� divqþ r ¼ _�: ð13Þ

2.6. Dissipation inequality

The second law of thermodynamics is written as

ddt

ZR

goV PZ

oR

�qh� noAþ

ZR

rhoV ; ð14Þ

with g representing the entropy per unit volume. Using the diver-gence law on Eq. (14) and localizing the result within R yields

_ghþ divq� qh� rh� r P 0: ð15Þ

The Helmholtz free energy per unit volume, w, is defined as

w ¼ �� gh! _w ¼ _�� _gh� g _h: ð16Þ

Denoting m ¼ rn, we use the functional expression for the free en-ergy density of a shape-memory alloy [27] and augment it with agradient energy [9] viz

w ¼ w Ee;m; n; hð Þ ! _w ¼ ow

oEe � _Ee þ owom� _mþ ow

on_nþ ow

oh_h: ð17Þ

Substituting Eqs. (16) and (17) into Eq. (13) yields

T� ow

oEe

� �� _Ee � gþ ow

oh

� �_hþ c� ow

om

� �� _mþ C ¼ _gh; ð18Þ

where

C � T � _Ep þ ðdivcÞ _n� owon

_n� divqþ r: ð19Þ

Further substitution of Eq. (18) into inequality (15) results in thedissipation inequality:

T� ow

oEe

� �� _Ee � gþ ow

oh

� �_hþ c� ow

om

� �� _mþP P 0; ð20Þ

where

P � T � _Ep þ ðdivcÞ _n� owon

_n� qh� rh: ð21Þ

From the principle of equipresence, inequality (20) yields

T ¼ ow

oEe ; g ¼ � owoh

and h c ¼ owom

: ð22Þ

Eqs. (22)1, (22)2 and (22)3 are the constitutive equations for theCauchy stress, the entropy and the micro-traction vector,respectively.

2.7. Phase transformation criteria and Fourier’s law

Using Eq. (22), we obtain the reduced dissipation inequality frominequality (20):

P � T � _Ep þ ðdivcÞ _n� owon

_n� qh� rh P 0: ð23Þ

Here P represents the total dissipation and it is always non-nega-tive. Recall that each Ni is defined to be deviatoric. Substituting


Eqs. (3) and (4) into Eq. (23), and assuming each dissipative mech-anism to be strongly dissipative [36] yields:

f1_n1 > 0 whenever _n1–0; ð24Þ

where f1 � ðkð1þ a/Þ½T0 �N1� þ divc� owonÞ denotes the driving force

for forward transformation

f2_n2 > 0 whenever _n2–0; ð25Þ

where f2 � ðkð1þ a/Þ½T0 �N2� þ divc� owonÞ denotes the driving force

for reverse transformation, and finally

�qh� rh > 0 whenever rh–o: ð26Þ

The inequalities (24)–(26) are assumed to be obeyed at all times sothat the reduced dissipation inequality (23) is concurrentlysatisfied.

To satisfy inequality (24) under the assumption of rate-indepen-dence, we choose an expression for f1 as follows:

f1 ¼ fc1ðsignð _n1ÞÞ ! f1 ¼ fc1 whenever _n1 > 0; ð27Þ

where fc1 > 0 represents the critical resistance to forwardtransformation.

Similarly, to satisfy inequality (25) under the assumption ofrate-independence, we choose an expression for f2 as follows:

f2 ¼ fc2ðsignð _n2ÞÞ ! f2 ¼ �fc2 whenever _n2 < 0; ð28Þ

where fc2 > 0 represents the critical resistance to reversetransformation.

Eqs. (27) and (28) are the criteria for forward and reverse phasetransformations, respectively. In general, we can have fc1–fc2.

Finally, assuming the material obeys Fourier’s law of heat con-duction we enforce

q ¼ �kthrh ð29Þto satisfy inequality (26) where kth > 0 denotes the constant coeffi-cient of thermal conductivity.

2.8. Free energy density and specific constitutive functions

The free-energy density of the material is taken to contain thefree energy of a conventional shape-memory alloy [27] augmentedwith a gradient energy [9]. We take the free energy per unit vol-ume, w to be in the separable form

w ¼ weðEe; hÞ þ wgðmÞ þ wnðn; hÞ þ whðhÞ where ð30ÞweðEe; hÞ ¼ ljEe

0j2 þ jðtraceEeÞ2 � 3jathðh� hoÞðtraceEeÞ;

wgðmÞ ¼ 12

snjmj2; ð31Þ

wnðn; hÞ ¼ kT

hTðh� hTÞnþ

12

hn2 and

whðhÞ ¼ cðh� hoÞ � ch logðh=hoÞ: ð32Þ

Here we denotes the classical thermo-elastic free energy densitywith l, j and ath denoting the shear modulus, bulk modulus andthe coefficient of thermal expansion, respectively. Following thework of Fried and Gurtin [9], we introduce a gradient energy wg

where sn P 0 denotes a material parameter with units of energyper unit length. The gradient energy acts to penalize to presenceof austenite–martensite interfaces. The austenite $ martensitephase transformation energy is denoted by wn where kT and hT rep-resents the latent heat released/absorbed (units of energy per unitvolume) during the austenite $ martensite phase transformationand the phase equilibrium temperature, respectively.4 The energetic

4 In formulating the phase transformation free energy, we are guided by the one-dimensional model of Abeyaratne and Knowles [2].

interaction coefficient, h has units of energy per unit volume. Final-ly, wh represents the purely thermal portion of the free energy withc > 0 being the specific heat per unit volume.

For simplicity, we will treat the material parameters fl;j;ath;

sn; kT ; hT ;h; cg to be constants. Furthermore, we will also assumethat there are no mismatches between the austenite phase andthe martensite phase material parameters.

2.9 Constitutive equation for the stress, entropy and micro-tractionvector

Substituting Eq. (30) into Eq. (22)1 yields the constitutive equa-tion for the Cauchy stress:

T ¼ 2lEe0 þ j½traceEe � 3athðh� hoÞ�1: ð33Þ

The constitutive equation for the entropy density and micro-trac-tion vector are respectively given substituting Eq. (30) into Eq.(22)2 and (22)3:

g ¼ c logðh=hoÞ þ 3jathðtraceEeÞ � ðkT=hTÞn andc ¼ snðrnÞ: ð34Þ

2.10. Flow direction N1 and the J3 parameter

Substituting Eqs. (30) and (34)2 into the expressions for thedriving force for phase transformation yield:

f1 ¼ kð1þ a/ÞðT0 � N1Þ þ snðr2nÞ � kT

hTðh� hTÞ � hn; ð35Þ

f2 ¼ kð1þ a/ÞðT0 � N2Þ þ snðr2nÞ � kT

hTðh� hTÞ � hn: ð36Þ

From the expression for the driving force shown in Eq. (35) and (36),we can see that the transformation between the austenite and mar-tensite phase is affected by local terms (due to the stress and phasetransformation energy) and non-local terms (due to the gradient en-ergy). Hence, these exists an intrinsic material length scale in theconstitutive model.

During forward transformation i.e. _n1–0, substituting Eq. (35)into Eq. (27) results in

kð1þ a/ÞðT0 � N1Þ ¼kT

hTðh� hTÞ þ hn� snðr2nÞ þ fc1ðsignð _n1ÞÞ:

ð37ÞTo satisfy Eq. (37), we take

ð�TÞ2½kð1þ a/Þ�T0 ¼kT

hTh� hTð Þ þ hn� snðr2nÞ þ fc1ðsignð _n1ÞÞ

� N1;

ð38Þ

since jN1j ¼ �T . Taking the magnitude on both sides of Eq. (38)yields

N1 ¼ �TT0

jT0j

� ! f1

¼ �r�Tð1þ a/Þ þ snðr2nÞ � kT

hTðh� hTÞ � hn; ð39Þ

where �r ¼ kjT0j represents an equivalent stress. From the work of[17], the J3 parameter is given by

/ ¼ffiffiffi6p½N1 � N2

1�ð�TÞ�3:

2.11. Flow rule

During forward transformation i.e. _n1–0 and _n2 ¼ 0, we define

�Tð1þ a/Þ _n1 �ffiffiffiffiffiffiffiffi2=3

pj _Epj ! k ¼

ffiffiffiffiffiffiffiffi3=2

pand �r ¼

ffiffiffiffiffiffiffiffi3=2

pjT0j:


Therefore, �r denotes the equivalent tensile stress or Mises stress. Thefinal form for the inelastic strain-rate i.e. the flow rule is then givenby

_Ep ¼ffiffiffiffiffiffiffiffi3=2

pð1þ a/Þ

X2

i¼1

_niNi: ð40Þ

2.12. Conditions on the driving forces and phase transformation rates

The driving forces f1 and f2 are defined to be within the ranges:

f1 6 fc1 for 0 6 n < 1; f 2 P �fc2 for 0 < n 6 1:

For n ¼ 1, the driving force for forward transformation is defined forall values of f1. For n ¼ 0, the driving force for reverse transforma-tion is defined for all values of f2.

In a rate-independent theory, the variables ff1; _n1g and ff2; _n2gmust satisfy the following conditions:

� Elastic range conditions: If f1–fc1, then _n1 ¼ 0. If f2–� fc2, then_n2 ¼ 0.

� Forward transformation: If 0 6 n < 1 and f1 ¼ fc1, then

_n1_ðf1 � fc1Þ ¼ 0: ð41Þ

� Reverse transformation: If 0 < n 6 1 and f2 ¼ �fc2, then

_n2_ðf2 þ fc2Þ ¼ 0: ð42Þ

� End conditions: If n ¼ 1 and f1 ¼ fc1, then _n1 ¼ 0. If n ¼ 0 andf2 ¼ �fc2, then _n2 ¼ 0.

Eqs. (41) and (42) are the consistency conditions for forward andreverse phase transformation, respectively. The consistency condi-tions are used to determine the transformation rates _n1 and _n2.

2.13. Balance of energy: revisited

Substituting Eqs. (29), (30), (33), (34)1, and (34)2 into Eq. (18)yields

T0 � _Ep þ snðr2nÞ _n� kT

hTðh� hTÞ _nþ kthðr2hÞ þ r ¼ _gh: ð43Þ

Taking the time-derivative of Eq. (34)1 results in

_g ¼ ch

� �_hþ 3jathðtrace _EeÞ � kT

hT

_n: ð44Þ

The evolution equation for temperature is given by substituting Eqs.(4), (40) and (44) into Eq. (43):

Stress

StrainActual behavior

a

Fig. 1. Schematic stress–strain response of an actual polycrystalline shape-memoryexperimental stress–strain response of the aforementioned polycrystalline shape-memo

c _h ¼ kthðr2hÞ þ ðkT=hTÞh _n� 3jathðtrace _EeÞhþX2

i¼1

fi_ni þ r: ð45Þ

To summarize, the list of constitutive parameters/functions thatneeded to calibrated/specified are

fl;j;ath; sn; kT ; hT ;h; c; a; �T ; fc1; fc2; kth; rg:

A time-integration procedure based on the isotropic-plasticity-based constitutive model for shape-memory alloys has been devel-oped and implemented in the ABAQUS/Explicit [1] finite-elementprogram by writing a user-material subroutine. Algorithmic detailsfor the time-integration procedure used to implement the model inthe finite-element code are given in Appendix A.

3. Physical experiments and FEM simulations

The material parameters for the constitutive model developedin Section 2 were fitted to the stress–strain responses obtainedfrom superelastic simple tension, simple compression and tubulartorsion experiments conducted on a polycrystalline rod Ti–Ni at anambient temperature of 298 K [28]. These experiments were con-ducted under very-low strain-rates to ensure that isothermal con-ditions prevail during the testing period. With hmf and has denotingthe martensite finish temperature and the austenite start tempera-ture, respectively, the transformation temperatures for the poly-crystalline Ti–Ni material were determined to be hms ¼ 251:3 K,hmf ¼ 213:0 K, has ¼ 260:3 K and haf ¼ 268:5 K. Thus, the materialis initially in the fully-austenitic phase.

Fig. 1a shows a schematic stress–strain response of an actualpolycrystalline shape-memory alloy undergoing superelasticdeformation under uniaxial loading. The fitting of the constitutiveparameters were performed on an idealized version of the actualexperimental stress–strain response shown in Fig. 1a. Fig. 1bshows the idealized experimental superelastic stress–strain re-sponse which we have assumed. The idealized stress–strain re-sponse contains the key features of superelastic deformation,namely: (1) Initial loading which causes the elastic deformationof the austenitic material; (2) continued loading which results ina stress plateau due to austenite ! martensite phase transforma-tion; (3) further loading which will cause the elastic deformationof the martensitic material; (4) reverse loading which causes theelastic unloading of the martensitic material; (5) continued reverseloading which results in another stress plateau due to reverse mar-tensite ! austenite phase transformation; and (6) further reverseloading which results in the elastic unloading of the austeniticmaterial.

Stress

StrainIdealized behavior

b

alloy exhibiting uniaxial superelastic deformation. Also shown is the idealizedry alloy undergoing superelastic uniaxial deformation.


As a first-cut assumption, we will take the critical resistances toforward and reverse transformation to be equal i.e. fc1 ¼ fc2 ¼ fc

where fc > 0 has units of energy per unit volume. Since the sizeof the test specimens are several orders of magnitude larger thanthe thickness of the austenite–martensite interface, we will ignorethe effect of the gradient energy in our calculations as a first-cutassumption i.e. we set sn ¼ 0 J/m. The influence of the gradient en-ergy on the deformation behavior of shape-memory alloys under-going austenite $ martensite phase transformations will beinvestigated in Section 4. In the spirit of modeling elastic per-fectly-plastic materials, we set the energetic interaction coefficient,h ¼ 0 J/m3. Finally, the heat supply per unit volume, r, is ignored inall of our calculations by setting it to be zero.

All the finite-element simulations in this work were conductedusing a single ABAQUS C3D8R continuum-three-dimensional brick

STRAIN

STR

ESS

[MPa

]

EXPERIMENTSIMULATION

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

50100

150

200

250

300

350

400

450

500

SHEAR STRAIN

SHEA

R S

TRES

S [M

Pa] EXPERIMENT

SIMULATION

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

100

200

300

400

500

600

700

800

1

3

2

γ

1

4

5

8

d

cb

a

Fig. 2. (a) An initially-undeformed single ABAQUS C3D8R continuum-three-dimensicorresponding corner nodes numbering system. Numerical and experimental superelastshear. The data from the tension and compression experiments were used to fit the mateindependent prediction. (e) Comparison of the stress–strain response from the tension a

element shown in Fig. 2a under unless stated otherwise. This ele-ment is meshed using eight corner nodes with the node numberingscheme shown in Fig. 2a, and is integrated numerically using a re-duced integration scheme. Letting a; b and c to represent the iso-parametric element coordinates (see Fig. 2a) with a; b and cspanning between �1 and 1 in an element, the interpolation func-tion for the eight noded brick element can be written as

u ¼X8

j¼1

Njða;b; cÞuj;

where u represents the displacement vector, Nj the isoparametricshape function for the jth node, uj the displacement vector of thejth node, and j ¼ 1; . . . ;8 the elemental node number. The isopara-metric shape functions are given as

STRAIN

STR

ESS

[MPa

]

STRAIN

STR

ESS

[MPa

] TENSIONCOMPRESSION

0 0.01 0.02 0.03 0.04 0.05 0.060

100

200

300

400

500

600

700

800

900

1000

EXPERIMENTSIMULATION

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

100

200

300

400

500

600

700

800

900

1000

α

2

3

6

7

β

e

onal brick element shown with the isoparametric coordinate system and theic stress–strain curves in (b) simple tension, (c) simple compression and (d) simplerial parameters. The simulated shear stress-shear strain response corresponds to annd compression simulations to demonstrate the tension–compression asymmetry.

Table 1Material parameters for the polycrystalline rod Ti–Ni.

l ¼ 23:31 GPa j ¼ 60:78 GPa ath ¼ 10� 10�6 K�1 �T ¼ 0:046a ¼ 0:13 h ¼ 0 J=m3 c ¼ 2:08 MJ=Km3 kth ¼ 18 W=mKhT ¼ 255:8 K kT ¼ 97 MJ=m3 fc ¼ 7:8 MJ=m3 sn ¼ 0 J=mr ¼ 0 W=m3

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

100

200

300

400

500

600

700

800

900

STRAIN

STR

ESS

[MPa

]

TENSION : 308 K TENSION : 298 K TENSION : 288 K

0 0.01 0.02 0.03 0.04 0.05 0.060

200

400

600

800

1000

1200

STRAIN

STR

ESS

[MPa

]

COMPRESSION : 308 K COMPRESSION : 298 K COMPRESSION : 288 K

a

b

Fig. 3. Simulated superelastic stress–strain responses in (a) simple tension and (b)simple compression at ambient temperatures of 288 K, 298 K and 308 K.


N1 ¼ ð1=8Þð1� aÞð1� bÞð1� cÞ; N2 ¼ ð1=8Þð1þ aÞð1� bÞð1� cÞ;N3 ¼ ð1=8Þð1þ aÞð1þ bÞð1� cÞ; N4 ¼ ð1=8Þð1� aÞð1þ bÞð1� cÞ;N5 ¼ ð1=8Þð1� aÞð1� bÞð1þ cÞ; N6 ¼ ð1=8Þð1þ aÞð1� bÞð1þ cÞ;N7 ¼ ð1=8Þð1þ aÞð1þ bÞð1þ cÞ; N8 ¼ ð1=8Þð1� aÞð1þ bÞð1þ cÞ:

The material is taken to be fully-austenitic at the beginning of eachsimulation. The constitutive parameters were determined by fittingthe model to the simple tension and compression experimentsusing a similar methodology outlined in [28]. Using the materialparameters listed out in Table 15 the isothermal stress–strain curvesat a temperature of 298 K obtained from the simple tension and sim-ple compression finite-element simulations are plotted in Fig. 2b andc, respectively. The fit from the numerical simulations are in good ac-cord with the experimental stress–strain responses.

With the constitutive parameters calibrated, a finite-elementsimulation in simple shear was performed and the resulting iso-thermal shear stress-shear strain response at a temperature of298 K is plotted in Fig. 2d. The experimental shear stress-shearstrain curve is well-predicted by the constitutive model.

The numerical stress–strain curves obtained from the simpletension and simple compression simulations conducted aboveare repeatedly plotted in Fig. 2e for comparison. As shown by thesestress–strain responses, the present constitutive theory is able tomodel the tension–compression asymmetry exhibited by polycrys-talline rod Ti–Ni namely: (i) the stress level required to nucleatethe martensitic phase from the parent austenitic phase is consider-ably higher in compression than in tension; (ii) the transformationstrain measured in compression is smaller than that in tension;and (iii) the hysteresis loop generated in compression is wider(measured along the stress axis) than the hysteresis loop generatedin tension. These features in the stress–strain responses are exhib-ited due to the influence of the J3 parameter.

With the model calibrated, we perform a set of superelasticsimulations in simple tension and simple compression under iso-thermal conditions at two other ambient temperatures: 288 Kand 308 K. The stress–strain curves from these finite-element sim-ulations are plotted in Fig. 3a and b together with the simple ten-sion and simple compression stress–strain curves obtained fromthe simulations conducted at an ambient temperature of 298 K(as also shown in Fig. 2b and c). The stress–strain responses plottedin Fig. 3a and b show that the stress required to induce austenite tomartensite transformation or martensite to austenite transforma-tion increases with increasing ambient/test temperature. This con-curs very well with experimental findings (e.g. [30]).

Next we conduct a series of strain–temperature cycling simula-tions which can be described as follows: At a temperature of 320 K,the material is first pre-stressed to a desired stress level. With thepre-stress maintained, the temperature of the material is reducedto 220 K and then increased back again to 320 K. Depending onthe pre-stress level, a transformation from the austenite to mar-tensite phase will occur at a particular temperature as result of asufficient reduction in temperature. At this point, a sufficient in-crease in temperature will then cause a transformation from themartensite to austenite phase to take place at another criticaltemperature.

In our finite-element simulations, we choose three differentpre-stress levels under simple tension and simple compressionconditions: 50 MPa, 100 MPa and 150 MPa. Fig. 4a shows the strainversus temperature responses for the strain–temperature cyclingsimulations conducted under the aforementioned tensile stress

5 The austenitic phase values are chosen for the material parametersfl;j;ath; c; kthg. As mentioned previously, we have assumed no mismatches betweenthe austenite and martensite phase material parameters for simplicity.

levels. The strain versus temperature curves for the strain–temper-ature cycling finite-element simulations performed under theaforementioned compressive stress levels are plotted in Fig. 4b.From the simulation results shown in Fig. 4a and b, we can con-clude the following trends: with increasing pre-stress level, aphase transformation from austenite to martensite or martensiteto austenite occurs at a higher temperature. The results shown inFig. 4a and b also qualitatively reproduce the strain–temperaturecycling experimental data shown in [30]. During the strain–tem-perature cycling, note that the transformation (actuation) strainobtained due to the tensile pre-stress is higher to that obtainedusing a compressive pre-stress. This is again due to the introduc-tion of the J3 parameter which takes different values under tensileor compressive stress states. Another point to note is that despitethe amount of pre-stress and the sign of the pre-stress i.e. tensile

220 240 260 280 300 320

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

TEMPERATURE [K]

ST

RA

IN

TENSION : 50 MPa

TENSION : 100 MPa

TENSION : 150 MPa

220 240 260 280 300 320

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

TEMPERATURE [K]

ST

RA

IN

COMPRESSION : 50 MPa



a

b

Fig. 4. Strain vs. temperature response obtained from the strain–temperaturecycling simulations conducted under constant (a) tensile and (b) compressivestresses of 50 MPa, 100 MPa and 150 MPa. The thermal cycling is conductedbetween temperatures of 220 K and 320 K.

COMPRESSION TENSION

00.01

0.020.03

0.040.05

0.06

280

275

270

2650

100

200

300

400

500

STRAINTEMPERATURE [K]

STR

ESS

[MPa

]

Fig. 5. Stress–strain–temperature response obtained from the shape-memoryeffect simulations conducted in simple tension and simple compression. Anisothermal stress–strain response from straining occurs at a temperature of265 K. Following this, an increase in temperature to 280 K takes place.


or compressive, the temperature at which the martensite to aus-tenite transformation occurs is always approximately 42 K higherthan the temperature at which the austenite to martensite trans-formation takes place.

To simulate the one-way austenite ! martensite ! austeniteshape-memory effect, we perform the following finite-elementcalculations: With the temperature of the material initially ath1=265 K where hms < 265 K < haf , we perform an isothermalsimple tension and simple compression simulation to cause atransformation from austenite to martensite. When the materialis fully-martensitic after the forward loading process, a reverseloading process to an unstressed state occurs through an elasticunloading of the material. With the applied stress in the materialmaintained at zero, the temperature of the material is then raisedto 280 K. The stress–strain–temperature response from these fi-nite-element simulations are plotted in Fig. 5. Note that once thereverse loading process to an unstressed state has taken place, aresidual strain will exist in the material as the temperature is notsufficiently high enough for reverse transformation from martens-ite to austenite to take place. However, as shown in Fig. 5, theresidual strain obtained from both these simulations during thedeformation process at temperature h1 will be fully recovered once

the temperature is raised above 276.6 K> haf . Therefore, our consti-tutive model is able to qualitatively reproduce the one-way aus-tenite ! martensite ! austenite shape-memory effect.

To investigate the non-isothermal behavior of shape-memoryalloys during tensile superelastic deformation, we perform the fol-lowing fully-coupled thermo-mechanical simulations: At an ambi-ent temperature of 298 K, an initially-undeformed shape-memoryalloy sheet with dimensions of 5 mm by 20 mm by 1 mm ismeshed using 200 ABAQUS C3D8RT elements as shown inFig. 6a. Each C3D8RT element has displacement and temperaturedegrees of freedom. The nodes on both ends of the specimen inthe 1–3 plane act as grip sections, and their temperature is keptfixed at 298 K throughout the duration of the simulations i.e. thegrips serve as a constant temperature bath. One grip section is con-strained against motion along direction-2 and the other grip sec-tion is deformed along direction-2 under strain-rates of1�10�4 s�1 (Simulation A) and 5�10�4 s�1 (Simulation B). Further-more, heat convection from the outer surfaces of the sheet to theambient environment (still air) is taken into account by settingthe surface film heat transfer coefficient to be 12 W/m2K.

The stress–strain response from these two simulations usingthe initially-undeformed finite-element mesh shown in Fig. 6aare plotted in Fig. 6b. The contours of the martensite volume frac-tion in the sheet specimen obtained from Simulation B keyed todifferent points on its corresponding stress–strain curve is shownin Fig. 7. Due to the boundary conditions imposed on the specimenas explained above, the austenite–martensite phase boundariespropagate from the grip sections to the specimen’s center duringthe forward loading and reverse loading process. The contour plotspresented in Fig. 7 show the possibility of multiple austenite–mar-tensite phase transformation fronts propagating in the specimenduring superelastic deformation (cf. [23]).

Referring back to the stress–strain curves plotted in Fig. 6b,Simulation B exhibits these following trends compared to Simula-tion A: (a) a wider hysteresis loop (measured along the stress axis),(b) a significantly larger hardening in the stress–strain responseduring the forward loading process; and (c) a significantly largersoftening in the stress–strain response during the reverse loadingprocess.

The causes for these observed trends are as follows: SimulationA was conducted at a deformation rate which results in a nearlyisothermal response i.e. the austenite$martensite phase transfor-mations occur at nearly constant stress plateaus. However, Fig. 8shows the contours of the temperature in the sheet specimen

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

100

200

300

400

500

600

700

800

900

STRAIN

STR

ESS

[MPa

]

STRAIN RATE = 5 x 10 _4 s (Simulation B)

STRAIN RATE = 1 x 10 _4 s (Simulation A)_1

1

23

a/g

bc

d

ef

1 mm

40 mm

5 mm

_1

b

a

Fig. 6. (a) Initially-undeformed finite-element mesh of an SMA sheet withdimensions of 5 mm by 40 mm by 1 mm meshed using 200 ABAQUS C3D8RTelements. (b) Simulated tensile superelastic stress–strain response of the sheetshown in Fig. 6a conducted under a strain-rate of 1�10�4 s�1 (Simulation A) and5�10�4 s�1 (Simulation B). Both these simulations were performed using a fully-coupled thermo-mechanical analysis.


obtained from Simulation B keyed to different points on its corre-sponding stress–strain response shown in Fig. 6b i.e. Simulation Bwas conducted at a strain-rate which results in a non-isothermaltemperature field in the specimen during phase transformations.As shown in Fig. 8, the temperature in the mid-section of the sheet

3 2

1

1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00

Martensite volume fraction

a

c

e

Fig. 7. Contours of the martensite volume fraction keyed to various points of the stress–conditions i.e. both ends of the specimen are fully-clamped, the forward and reverse pspecimen.

increases by as much as 14 K above the ambient temperature(298 K) during the forward loading process. At a strain-rate of5�10�4 s�1 the heat generated due to the release of the latent heatand mechanical dissipation is not conducted and convected out ofthe specimen quickly enough, and this results in the increase of thespecimen temperature with respect to the ambient temperature.Thus, it is the increase in temperature which causes the hardeningin the stress–strain response during the forward loading process asshown in Fig. 6b.

Conversely, as also shown in Fig. 8 the temperature in the mid-section of the sheet decreases to below the ambient temperatureduring the reverse loading process. At this deformation-rate i.e.5�10�4 s�1 the heat loss in the specimen due to the absorption ofthe latent heat outweighs the amount of heat conducted and con-vected into the specimen, and hence causes a reduction in thespecimen’s mid-section temperature by as much as 13 K belowthe ambient temperature (298 K). Therefore, it is this decrease intemperature which causes the softening in the stress–strain re-sponse during the reverse loading process as shown in Fig. 6b.

4. Austenite–martensite phase boundary propagation duringsuperelastic deformation

As mentioned previously, the tracking of the position of the aus-tenite–martensite interfaces during phase transformation is notpossible using a locally-based constitutive model i.e. the thicknessof the austenite–martensite phase boundary is highly meshdependent.

To study the austenite–martensite phase boundary propagationin a shape-memory alloy undergoing superelastic deformation, weperform the following simulations: Consider an initially-austeniticcuboidal section as shown in Fig. 9a with an initially-undeformeddimensions of 1 mm by 5 lm by 5 lm measured along direction-1, direction-2 and direction-3, respectively. This cuboidal sectionis meshed using 50, 100 and 200 ABAQUS C3D8R elements alongdirection-1. The nodes on the two ends of the cuboidal specimenin the 2–3 plane serve as the grip sections (grip section A and B)where a very slight taper is provided to introduce sites of geomet-rical imperfection. Grip section A is prevented from motion alongdirection-1 whereas a deformation profile is imparted on gripsection B along direction-1. The finite-element simulations wereperformed under isothermal conditions with the specimen tem-perature maintained at 298 K throughout the duration of the sim-

b

d

f

gstrain curve obtained from Simulation B as shown in Fig. 6b. Due to hard boundaryhase transformations initiate from the ends and move towards the center of the

3 2

1

319.0 315.3311.6 307.9 304.2 300.5 296.8 293.1289.4 285.7 282.0

Temperature [K]

a

c

e f

g

d

b

Fig. 8. Contours of the temperature keyed to various points of the stress–strain curve obtained from Simulation B as shown in Fig. 6b. During forward loading, thetemperature in the specimen increases by approximately 14 K above the ambient temperature of 298 K due to the release of latent heat as a result of the austenite tomartensite phase transformation. During reverse loading, the temperature in the specimen decreases by approximately 13 K below the ambient temperature of 298 K due tothe absorption of latent heat as a result of the martensite to austenite phase transformation.

100 elements

700

600

500

400

300

200

100

00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

200 elements

STR

ESS

[MPa

]

STRAIN

a

b

3

1 2

Grip section A

Grip section B

50 elements

b

a

Fig. 9. (a) A cuboidal specimen with initially-undeformed dimensions of 1 mm by5 lm by 5 lm measured along direction-1, direction-2 and direction-3, respec-tively. (b) The tensile stress–strain curves obtained from the simulations conductedon the cuboidal specimen shown in Fig. 8a meshed uniformly along direction-1using 50, 100 and 200 ABAQUS C3D8R elements. The simulations were conducted atan ambient temperature of 298 K under isothermal conditions.

6 The austenite–martensite interface is defined as a region where a mixture of theaustenite and martensite phases are present.


ulations. Finally, for these set of simulations we use the values forthe material parameters listed in Table 1 except for: (1) a verysmall energetic interaction coefficient h of �0.24 MJ/m3 introducedto accelerate the localization process, and (2) the value of sn to be

set at 2.5 � 10�3 J/m which will correspond to an experimentally-determined austenite–martensite interface thickness of approxi-mately 100 lm [26].

Fig. 9b shows the calculated superelastic tensile stress–straincurves for the above described simulations. The stress–strain re-sponses for the finite-element simulations conducted on the cuboi-dal section shown in Fig. 9a meshed using two different meshdensities are equivalent.

Fig. 10a and b shows the contours of the martensite volumefraction for the simple tension simulation conducted on the cuboi-dal specimen section shown in Fig. 9a meshed using 50, 100 and200 ABAQUS C3D8R elements keyed to points a and b, respectivelyon the corresponding stress–strain curves plotted in Fig. 9b. Duringthe forward loading process, the austenite–martensite interface6

propagates from grip section A towards grip section B where theaustenitic phase is gradually transforming into the martensiticphase. However, during the reverse loading process the austenite–martensite interface propagates from grip section B towards gripsection A where the martensitic phase is now gradually transformingback into the austenitic phase.

During forward transformation, Fig. 10a shows that the austen-ite–martensite interface thickness predicted from the simulationsconducted on the bar shown in Fig. 9a meshed using 50, 100 and200 ABAQUS C3D8R elements to be approximately equal(�75 lm). Similarly, during reverse transformation, Fig. 10b showsthat the austenite–martensite interface thickness predicted fromthe simulations conducted on the bar shown in Fig. 9a meshedusing 50, 100 and 200 ABAQUS C3D8R elements to also be approx-imately equal (�120 lm). Also note that the austenite–martensiteinterface thickness predicted during forward loading is smallerthan the calculated austenite–martensite interface thickness dur-ing reverse loading. This trend is predicted since (1) our calcula-tions have utilized a constant value of sn, and (2) the magnitudeof stress required to cause forward transformation is larger than

1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00

Martensite volume fraction

3 1

2

50 elements

100 elements

200 elements

50 elements

100 elements

200 elements

Grip section B Grip section A

a

b

Fig. 10. Simulated contours of the martensite volume fraction keyed to points a and b on the stress–strain curve plotted in Fig. 9b during (a) forward transformation and (b)reverse transformation, respectively.

7 The quantities EðsÞ and hðsÞ are inputs provided by the ABAQUS (2007) finite-element program.


the magnitude of stress at which the reverse transformation occurs(see Eqs. (27) and (28)).

From the contour plots shown in Fig. 10 we can conclude thatfor the mesh densities used for the calculations in this section, anon-zero gradient energy i.e. a non-local theory with sn–0 heavilyminimizes the effect of mesh density on the prediction of the aus-tenite–martensite interface thickness. Hence the position of theaustenite–martensite interface region can now be accuratelytracked during phase transformation.

5. Conclusion

A thermo-mechanically-coupled and isotropic-plasticity-basedconstitutive model for shape-memory alloys capable of undergoingaustenite $ martensite phase transformation has been developedusing fundamental thermodynamic laws and the principle of mi-cro-force balance ([9]). The constitutive model has been imple-mented in the ABAQUS/Explicit (2007) finite-element program bywriting a user-material subroutine.

The stress–strain curves obtained from several uniaxial andmulti-axial superelastic experiments conducted on a polycrystal-line rod Ti–Ni were well predicted by the constitutive model andthe finite-element simulations. The constitutive model is also ableto qualitatively reproduce the experimental strain–temperaturecycling and one-way shape-memory effect responses exhibitedby shape-memory alloys.

With the aid of finite-element simulations, we also show thatthe non-local version of our theory (which takes into account a gra-dient energy i.e. sn–0) is able to track the position(s) of the austen-ite–martensite interface during phase transformation without theuse of jump-conditions.

Some future work includes: (a) the comparison of our presentconstitutive model to the response obtained from other modelse.g. [35,15]; (b) the prediction of complicated three-dimensionaland coupled thermo-mechanical experiments (e.g. [34,33]) usingour developed constitutive model.

Acknowledgements

The financial support for this work was provided by the Minis-try of Science, Technology and Innovation, Malaysia under Grant03-01-02-SF0257. The ABAQUS finite-element software was madeavailable under an academic license from HKS, Inc. Pawtucket, R.I.PT gratefully acknowledges the help of R. Ekambaram (NUS) withsome of the finite-element simulations.

Appendix A. Time-integration procedure

In this appendix, we summarize the explicit time-integrationprocedure that we have used for our constitutive model presentedin Section 2. With t denoting the current time, Dt is an infinitesimaltime increment, and s ¼ t þ Dt. The algorithm is as follows:

Given: (1) fEðtÞ;EðsÞ; hðtÞ; hðsÞg7; (2) fTðtÞ;EpðtÞg; (3)fN1ðtÞ;N2ðtÞ;/ðtÞg; (4) the martensite volume fraction nðtÞ.

Calculate: (a) fTðsÞ;EpðsÞg, (b) fN1ðsÞ;N2ðsÞ;/ðsÞg, (c) the mar-tensite volume fraction nðsÞ, and march forward in time.

The steps used in the calculation procedure are

Step 1. Calculate the trial elastic strain EeðsÞtrial:

EeðsÞtrial ¼ EðsÞ � EpðtÞ:

Step 2. Calculate the trial Cauchy stress TðsÞtrial:

TðsÞtrial ¼ 2lEe0ðsÞ

trial þ j½traceEeðsÞtrial � 3athðhðsÞ � hoÞ�1: ð46Þ

Step 3. Calculate the trial driving forces fiðsÞtrial. For our explicitnumerical algorithm presented here, we approximate

NiðsÞ ¼ NiðtÞ; /ðsÞ ¼ /ðtÞ and r2nðsÞ ¼ r2nðtÞ: ð47Þ


The trial driving forces for phase transformation are then given by

fiðsÞtrial ¼ffiffiffi32

rð1þ a/ðtÞÞ T0ðsÞtrial �NiðtÞ

h iþ snðr2nðtÞÞ

� kT

hTðhðsÞ � hTÞ � hnðtÞ: ð48Þ

Step 4. Determine the set PA of potentially active transforma-tion systems which satisfy

f1ðsÞtrial � fc > 0 and 0 6 nðtÞ < 1

for forward transformation, and

f2ðsÞtrial þ fc < 0 and 0 < nðtÞ 6 1

for reverse transformation.Step 5. Using the approximations given in Eq. (47), we calculate

EpðsÞ ¼ EpðtÞ þffiffiffi32

rð1þ a/ðtÞÞ

Xj2PA

DnjNjðtÞ( )

where j ¼ 1; . . . ;Q : ð49Þ

Here Q 6 2 is the total number of potential transformation systems.Of the Q potentially active systems in the set PA, only a subset Awith elements M 6 Q , may actually be active (non-zero incre-ments). This set is determined in an iterative fashion describedbelow.During phase transformation, the active transformation sys-tems must satisfy the consistency conditions

f1ðsÞ � fc ¼ 0 and=or f 2ðsÞ þ fc ¼ 0 ð50Þ

for forward transformation and/or reverse transformation, respec-tively. Using Eqs. (46)–(49), it is straightforward to show that

fiðsÞ ¼ fiðsÞtrial �X

j2PA

3lDnjð1þ a/ðtÞÞ½NiðtÞ �NjðtÞ�( )

�X

j2PA

Dnjhdij

( ); ð51Þ

where dij is the Kronecker delta. Substituting Eq. (51) into the con-sistency conditions (50) gives

Xj2PA

Aijxj ¼ bi; i 2 PA; ð52Þ

with

Aij ¼ 3lð1þ a/ðtÞÞ NiðtÞ � NjðtÞ� �

þ hdij:

We also have

b1 ¼ f1ðsÞtrial � fc > 0 and x1 � Dn1 > 0

for forward transformation, and

b2 ¼ f2ðsÞtrial þ fc < 0 and x2 � Dn2 < 0

for reverse transformation.Eq. (52) is a system of linear equationsfor the transformation increments xj � Dnj (for j 2 PA). Assumingthe matrix A to be invertible i.e. non-singular, the transformationrates are determined by

x ¼ A�1b; ð53Þ

where A�1 is the inverse matrix of the matrix A. If x1 6 0 whenb1 > 0 (forward transformation), then this system is inactive andit is removed from the set of potentially active systems PA and anew matrix A is calculated. Similarly, if x2 P 0 when b2 < 0 (reversetransformation), then this system is also inactive and it is not in-cluded in the set PA used to determine the new matrix A. The finalsize of the matrix A is M �M.

Step 6. Update the martensite volume fraction:

nðsÞ ¼ nðtÞ þX

i

Dni; i 2A:

If nðsÞ > 1, then set nðsÞ ¼ 1. If nðsÞ < 0, then set nðsÞ ¼ 0.Step 7. Update the plastic strain EpðsÞ and the flow direction

N2ðsÞ:

EpðsÞ ¼ EpðtÞ þffiffiffi32

rð1þ a/ðtÞÞ

Xi2A

DniNiðtÞ( )

and

N2ðsÞ ¼ �TEpðsÞjEpðsÞj

� �:

Step 8. Compute the elastic strain EeðsÞ and update the Cauchystress TðsÞ:

EeðsÞ ¼ EðsÞ � EpðsÞ;TðsÞ ¼ 2lEe

0ðsÞ þ j traceEeðsÞ � 3athðhðsÞ � hoÞ½ �1:

Step 9. Update the flow direction N1ðsÞ and the J-3 parameter/ðsÞ:

N1ðsÞ ¼ �TT0ðsÞjT0ðsÞj

� �and /ðsÞ ¼

ffiffiffi6p

N1ðsÞ � ðN1ðsÞÞ2h i

ð�TÞ�3:

Step 10. Calculate the driving forces fiðsÞ:

fiðsÞ ¼ffiffiffi32

rð1þ a/ðsÞÞ T0ðsÞ �NiðsÞ½ � þ snðr2nðsÞÞ

� kT

hTðhðsÞ � hTÞ � hnðsÞ:

Step 11. Calculate the inelastic work increment Dxp:

Dxp ¼ hT

hT

� �hðsÞ

Xi2A

Dni � 3jathhðsÞðtraceðDEeÞÞ þXi2A

fiðsÞDni;

where DEe ¼ EeðsÞ � EeðtÞ and EeðtÞ ¼ EðtÞ � EpðtÞ. The inelasticwork increment is treated as the heat source which causes heat-ing/cooling at a material point during deformation.

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a macroscopic constitutive model for shape-memory alloys: theory and finite-element simulations

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