a finite-deformation-based phenomenological theory for shape-memory alloys

International Journal of Plasticity 26 (2010) 1195–1219

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International Journal of Plasticity

journal homepage: www.elsevier .com/locate / i jp las

A finite-deformation-based phenomenological theory forshape-memory alloys

P. Thamburaja *

Department of Mechanical Engineering, National University of Singapore, Singapore 117576, Singapore

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

Article history:Received 24 September 2009Received in final revised form 10 December2009Available online 4 January 2010

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

0749-6419/$ - see front matter � 2010 Elsevier Ltddoi:10.1016/j.ijplas.2009.12.004

* Corresponding author. Fax: +65 6779 6559.E-mail address: [email protected]

In this work we develop a finite-deformation-based, thermo-mechanically-coupled andnon-local phenomenological theory for polycrystalline shape-memory alloys (SMAs) capa-ble of undergoing austenite$martensite phase transformations. The constitutive model isdeveloped in the isotropic plasticity setting using standard balance laws, thermodynamiclaws and the theory of micro-force balance (Fried and Gurtin, 1994). The constitutivemodel is then implemented in the ABAQUS/Explicit (2009) finite-element program by writ-ing a user-material subroutine. Material parameters in the constitutive model were fittedto a set of superelastic experiments conducted by Thamburaja and Anand (2001) on a poly-crystalline rod Ti–Ni. With the material parameters calibrated, we show that the experi-mental stress-biased strain–temperature-cycling and shape-memory effect responses arequalitatively well-reproduced by the constitutive model and the numerical simulations.We also show the capability of our constitutive mode in studying the response of SMAsundergoing coupled thermo-mechanical loading and also multi-axial loading conditionsby studying the deformation behavior of a stent unit cell.

Finally, with the aid of finite-element simulations we also show that our non-local con-stitutive theory is able to accurately determine the position and motion of austenite–mar-tensite interfaces during phase transformations regardless of mesh density and without theaid of jump conditions.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Shape-memory alloys (SMA) are a class of metallic alloys which have the ability to exist in different phases depending onthe temperature and/or stress state i.e. its current state is very sensitive to stress states and temperature changes. At hightemperatures and/or low stresses, SMAs will exist in the high symmetry austenite phase. However, at low temperaturesand/or high stresses SMAs will exist in the low-symmetry martensite phase. SMAs can also exist in a metastable state termedas the rhombohedral phase or R-phase (Shaw and Kyriakides, 1995; Otsuka and Wayman, 1999). Of the multitude of smart/functional materials being used in the recent years, polycrystalline shape-memory alloys (e.g. Ti–Ni, Cu–Al–Ni, Cu–Zn–Al,Au–Cd systems etc.) remain one of the more popular choices with its variety of applications in the Micro-Electro-MechanicalSystems (MEMS), biomedical, aerospace and civil structures area (Aizawa et al., 1998; Duerig et al., 1999; Machado and Savi,2003; Seelecke and Mueller, 2004).

The phase transformation between the austenite phase and the martensite phase in SMAs is classified as a first-orderphase transformation. Austenite $ martensite phase transformations in SMAs are also diffusionless and reversible. Due toits ability in undergoing reversible first-order phase transformation, SMAs exhibit exotic behaviors such as (a) superelasticity

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1196 P. Thamburaja / International Journal of Plasticity 26 (2010) 1195–1219

or pseudoelasticity by transformation, and (b) the shape-memory effect. For a detailed discussion regarding superelasticityand the shape-memory effect, please refer to the works of Shaw and Kyriakides (1995) and Otsuka and Wayman (1999).Although austenite $ martensite transformations in SMAs are reversible, there is hysteresis that accompanies these typesof phase transformations. The mechanism responsible for the exhibited hysteresis is the motion of interfaces between thetwo phases.

There is a world-wide activity in the development of models to describe the deformation behavior of SMAs. Concentratingon the continuum-mechanics-based constitutive theories, these models can broadly be classified under two categories: (1)phenomenologically-based models, and (2) crystal-plasticity-based constitutive theories. Some examples of phenomenolog-ically-based theories include the works of Liang and Rogers (1990), Brinson (1993), Sun and Hwang (1993), Abeyaratne et al.(1994), Boyd and Lagoudas, 1994, 1996, Auricchio and Taylor (1997), Lim and McDowell (1999), Mueller and Bruhns (2006),Helm (2007), Ziolkowski (2007), Moumni et al. (2008) and Reese and Christ (2008). The constitutive models of Patoor et al.(1996), Lu and Weng (1998), Fang et al. (1999), Gall and Sehitoglu (1999), Thamburaja and Anand (2001), Anand and Gurtin(2003a), Jung et al. (2004), Peng et al. (2008) and Wang et al. (2008) are examples of crystal-mechanics-based SMA consti-tutive models.1 Although crystal-plasticity-based theories are more physical i.e. it takes into account the crystallographic infor-mation and texture effects, phenomenological models are still very useful in determining the deformation behavior of SMAsbecause: (a) phenomenological models are easier to implement numerically, and (b) simulations using phenomenological mod-els run much faster. This makes the usage of phenomenological models very relevant in the design of especially complex com-ponents utilizing the exotic behavior exhibited by SMAs. Some recent work regarding the use of phenomenologically-basedSMA models to study the deformation behavior of biomedical structures such as orthodontic wires and arterial stents can befound in Auricchio (2001) and Zaki and Moumni (2007), respectively. Since these SMA structures have the ability to undergorelatively large deformations during its deployment, it is of paramount importance to develop a theory within a finite-deforma-tion setting (cf. Auricchio and Taylor, 1997; Auricchio, 2001) to accurately model the response of these structures whendeformed.

As also mentioned previously, the behavior of SMAs is very sensitive to changes in temperature. Since the numericalimplementation of the phenomenologically-based constitutive models described above were not performed in a thermo-mechanically-coupled setting, the response of structures and components which have spatially-varying temperature fieldswill not be accurately modeled. Furthermore, the above mentioned phenomenological models are derived based on local the-ories i.e. there is no material length scale in the constitutive equations. Hence, these models are unable to track the propa-gation of austenite–martensite interfaces during phase transformation without the aid of jump conditions as the calculationsare heavily dependent on mesh size or grid size.2 A comprehensive theory for SMAs should also be able to model the coupledthermo-mechanical response of SMAs during deformation and also accurately track the motion of the austenite–martensiteinterfaces.

Therefore, the main tasks of this present work are to: (a) develop a finite-deformation-based phenomenological theory todescribe the coupled thermo-mechanical behavior of SMAs,3 and (b) develop a non-local-based constitutive model which willminimize the effect of mesh density when tracking of the austenite–martensite interface motion during phase transformation.

The plan of this paper is as follows: In Section 2, we will develop a finite-deformation-based phenomenological consti-tutive model based on isotropic plasticity theory. The derivation of this constitutive model is performed with the aid of stan-dard balance laws, thermodynamic laws and the theory of micro-force balance (Fried and Gurtin, 1994). We have alsoimplemented our constitutive model in the ABAQUS/Explicit (2009) finite-element program. Algorithmic details of thetime-integration procedure used to implement the model are given in Appendix A. In Section 3, we fit the material param-eters in the constitutive model to the superelastic experiments of Thamburaja and Anand (2001) conducted on a polycrys-talline rod Ti–Ni. With the material parameters calibrated, we proceed to show that our constitutive model is able toreproduce other exotic shape-memory alloy behaviors such as stress-biased strain–temperature-cycling and the austen-ite!martensite! austenite shape-memory effect. Here we also demonstrate the capability of our constitutive model inperforming coupled thermo-mechanical analysis/simulations and qualitatively analyzing the multi-axial deformationbehavior of a stent unit cell. In Section 4, we showcase the non-local version of our theory by studying the austenite–mar-tensite interface propagation during superelastic deformation. We show that our theory is able to heavily minimize the ef-fect of mesh density in the tracking of the austenite–martensite interface motion during superelastic deformation. Weconclude and provide directions for future research in Section 5.

2. Constitutive model

In this section we construct a constitutive model for shape-memory alloys capable of undergoing austenite$martensitephase transformation using finite-deformation-based isotropic plasticity theory. To construct the constitutive theory, we

1 There are also detailed micromechanics-based constitutive models developed by Levitas and co-workers to describe various phenomena exhibited byshape-memory alloys. The goal of the present work is to develop the simplest phenomenologically-based finite-deformation-based constitutive model todescribe the deformation behavior of polycrystalline SMAs under thermo-mechanical loading conditions.

2 The present discussions regarding the mesh sensitivity on the prediction of the austenite–martensite interface positions were made under the assumptionof isothermal conditions.

3 We have also developed a small-strain-based phenomenological theory for polycrystalline SMAs in Thamburaja and Nikabdullah (2009).

P. Thamburaja / International Journal of Plasticity 26 (2010) 1195–1219 1197

focus on an arbitrary subregion R of a continuum body in the reference configuration with n denoting the outward unit nor-mal of the body’s boundary specified by @R. All the balance and imbalance equations in this work are formulated in the ref-erence configuration.

The list of the governing variables in the constitutive model are4: (i) The Helmholtz free energy per unit reference volume,w. (ii) The Cauchy stress tensor, T. (iii) The total deformation gradient tensor, F with det F > 0. (iv) The absolute temperature, h.(v) The inelastic distortion tensor, Fp with det Fp > 0. It measures the cumulative deformation due to austenite$martensitephase transformations. (vi) The elastic distortion tensor, Fe with det Fe > 0. It describes the elastic stretches that gives rise tothe Cauchy stress T. From the theory of Kröner (1960) and Lee (1969)), the elastic distortion tensor is given by Fe ¼ FFp�1.(vii) The total martensite volume fraction, n with 0 6 n 6 1.

2.1. Motion, kinematics and flow rule

Let y denote the position vector of a material point in the current configuration. The velocity vector of the material point isthen given by v ¼ _y. Hence, the material time derivative of the deformation gradient tensor F can be written as

4 Notare secotensor BskwB �symB0.magnitu

_F � rv ¼ LF ð1Þ

where L represents the total velocity gradient. Using the definition F ¼ FeFp, we can further write the total velocity gradientas

L ¼ Le þ FeLpFe�1 ð2Þ

where Le ¼ _FeFe�1 and Lp ¼ _FpFp�1 represent the elastic and inelastic velocity gradients, respectively. With Ce ¼ Fe>Fe, we de-fine a measure for the elastic strain, Ee by

Ee ¼ ð1=2ÞðCe � 1Þ ! _Ee ¼ Fe>ðsymLeÞFe: ð3Þ

In formulating the flow rule, we make two key assumptions: (1) from the work of Gurtin and Anand (2005), the inelasticvelocity gradient, Lp is taken to be spinless i.e. skwLp ¼ 0, and (2) small volume changes accompanying austenite$martens-ite phase transformations are neglected (Sun and Hwang, 1993) i.e. the inelastic velocity gradient, Lp is purely deviatoric.

With these assumptions, we generalize the work of Sun and Hwang (1993) to a finite-deformation setting and set theinelastic velocity gradient to be

Lp ¼ kð1þ a/ÞX2

i¼1

_niNi: ð4Þ

Here _n1 P 0 and _n2 6 0 denote the martensitic transformation rates. Forward transformation occurs if _n1 > 0 whereas re-verse transformation occurs if _n2 < 0. The flow direction tensors Ni with i ¼ 1;2 are restricted by Ni ¼ N>i and traceNi ¼ 0.We define N1 and N2 as the forward transformation and reverse transformation flow direction, respectively. Following Auric-chio et al. (1997), the total rate of martensitic volume fraction change is given by

_n ¼X2

i¼1

_ni ¼ _n1 þ _n2: ð5Þ

A transformation from austenite to martensite takes place when _n > 0. Conversely, a transformation from martensite toaustenite takes place when _n < 0. No net phase transformation occurs if _n ¼ 0.

With k > 0 being a constant of proportionality, we will further enforce jNij ¼ �T for each i where �T > 0 denotes the trans-formation strain due to austenite–martensite phase transformation (to be determined experimentally). Furthermore, guidedby the work of Orgeas and Favier (1998) we have the scalar / with �1 6 / 6 1 to represent the J3 i.e. the third stress-invari-ant measure with the dimensionless constitutive parameter a (to be calibrated from experiments) controlling the extent ofthe tension–compression asymmetry exhibited by SMAs during deformation. The functional form for / will be describedlater.

From microscopic considerations, the reverse transformation is the crystallographic recovery of the deformation which isinduced during forward transformation i.e. the reverse transformation is restricted by the forward transformation history.The deformation experienced due to austenite ! martensite (forward) phase transformation can be completely recoveredby the reversal of the forward loading history. Therefore, with N1 to be defined we take the flow direction N2 to be givenby (Sun and Hwang, 1993; Boyd and Lagoudas, 1996):

ation: The terms Div, r and r2 denote the referential divergence, gradient and Laplacian operators, respectively. All the tensorial variables in this worknd-order tensors unless stated otherwise. For a tensor B;B> denotes its transpose. We also write trace B for the trace of the tensor B. The determinant ofis denoted by det B. The symmetric portion of tensor B is denoted by sym B � ð1=2ÞðBþ B>Þ. The skew-symmetric portion of tensor B is denoted byð1=2ÞðB� B>Þ. The deviatoric portion of tensor B is denoted by B0 � B� ð1=3ÞðtraceBÞ1. The symmetric and deviatoric portion of tensor B is denoted byThe scalar product of two tensors A and B is denoted by A � B ¼ trace ðB>AÞ. The scalar product of two vectors u and v is also denoted by u � v. Thede of vector u and tensor B is denoted by juj and jBj, respectively. The second-order identity tensor is denoted by 1.


N2 ¼ �TBjBj

� �with _B ¼ Lp:

Here B is a tensor defined in the configuration determined by Fp i.e. the relaxed configuration.

2.2. Micro-force balance

For the martensite volume fraction n, the micro-force system which describe the forces that perform work associated toaustenite$ martensite phase transformations consist of: (a) the micro-traction vector, c, measured per unit area in the ref-erence configuration; (b) the scalar internal micro-force, pint , measured per unit volume in the reference configuration; and (c)the scalar external micro-force, pext , measured per unit volume in the reference configuration. Following the work of Fried andGurtin (1994), we write the corresponding micro-force balance equation associated with these micro-force systems as
Z
@R

c � ndAþZR

pext dV ¼ZR

pint dV ð6Þ

where dA and dV denotes the area and volume integral in the reference configuration, respectively. Applying the divergencelaw on Eq. (6) and localizing the result within R results in

Divc� pint þ pext ¼ 0: ð7Þ

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 by
Z@R
SndAþZR

bdV ¼ o ð8Þ

where S � ðdet FÞTF�> denotes the First Piola–Kirchoff stress tensor and b the macroscopic body force vector per unit refer-ence volume. Inertial forces are also included in the body force b. Using the divergence law on Eq. (8) and localizing the resultwithin R yields

DivSþ b ¼ o: ð9Þ

2.4. Balance of angular momentum

The balance of angular momentum is written as
Z@R
y � SndAþZR

y � bdV ¼ o: ð10Þ

Applying the divergence law on Eq. (10) and localizing the result within R while using Eq. (9) yields

SF> ¼ FS>: ð11Þ

Substituting S ¼ ðdet FÞTF�> into Eq. (11) results in T ¼ T> i.e. the Cauchy stress is symmetric.

2.5. Balance of energy

The first law of thermodynamics (the balance of energy) is stated as
Z@R
½Sn � v þ ðc � nÞ _n� q � n�dAþZR

ðb � v þ pext_nþ rÞdV ¼ d

dt

ZR

�dV ð12Þ

where � is the internal energy per unit reference volume. Here q is the heat flux vector measured per unit area in the referenceconfiguration and r is the heat supply per unit reference volume. Applying the divergence law on Eq. (12) and localizing theresult within R while using Eqs. (1), (7) and (9) yields

SF> � L þ c � r _nþ pint_n� Divqþ r ¼ _�: ð13Þ

Assuming that the external micro-force vanishes i.e. pext ¼ 0, substituting S � ðdet FÞTF�>, Eqs. (2), (3) and (7) into Eq.(13) while using the result of Eq. (11) yields

T� � _Ee þ T � Lp þ c � r _nþ ðDivcÞ _n� Divqþ r ¼ _� ð14Þ


where

5 Frami.e. theconsidetranslatabove,det F!Lp ! Lp

w! w;

T� ¼ ðdet FÞFe�1TFe�> and T ¼ CeT� ð15Þ

denote frame-invariant5 measures of stresses. Note that T� is symmetric whereas T is generally not symmetric.

2.6. Entropy imbalance

The second law of thermodynamics is written as

ddt

ZR

gdV PZ@R

�qh� ndAþ

ZR

rh

dV ð16Þ

with g representing the entropy per unit reference volume. Using the divergence law on Eq. (16) and localizing the resultwithin R yields

_ghþ Divq� qh� rh� r P 0: ð17Þ

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

w ¼ �� gh! _w ¼ _�� _gh� g _h: ð18Þ

Denoting m ¼ rn, we use the functional expression for the free energy density of a shape-memory alloy (Helm and Hau-pt, 2003) and augment it with a gradient energy (Fried and Gurtin, 1994) viz.

w ¼ wðEe;m; n; hÞ ! _w ¼ @w

@Ee � _Ee þ @w@m� _mþ @w

@n_nþ @w

@h_h: ð19Þ

Substituting Eqs. (18) and (19) into Eq. (14) yields

T� � @w

@Ee

� �� _Ee � gþ @w

@h

� �_hþ c� @w

@m

� �� _mþ C ¼ _gh ð20Þ

where

C � T � Lp þ ðDivcÞ _n� @w@n

_n� Divqþ r: ð21Þ

Further substitution of Eq. (20) into inequality (17) results in the dissipation inequality:

T� � @w

@Ee

� �� _Ee � gþ @w

@h

� �_hþ c� @w

@m

� �� _mþP P 0 ð22Þ

where

P � T � Lp þ ðDivcÞ _n� @w@n

_n� qh� rh: ð23Þ

From rational thermodynamic arguments, inequality (22) yields

T� ¼ @w

@Ee ; g ¼ � @w@h

and c ¼ @w@m

: ð24Þ

Eq. (24)1, (24)2 and (24)3 are the constitutive equations for the stress, entropy and the micro-traction vector, respectively.

2.7. Phase transformation criteria and Fourier’s law

Using Eq. (24), we obtain the reduced dissipation inequality from inequality (22):

P � T � Lp þ ðDivcÞ _n� @w@n

_n� qh� rh P 0: ð25Þ

Here P represents the total dissipation and it is always non-negative. Recall that each Ni is defined to be symmetric anddeviatoric. Substituting Eqs. (4) and (5) into Eq. (25), and assuming each dissipative mechanism to be strongly dissipative(Anand and Gurtin, 2003a,b) yields:

e-invariance (Anand and Gurtin, 2003b): Let x and z denote the position of the material point in the reference configuration and relaxed configurationconfiguration determined by Fp , respectively. As mentioned previously, y denotes the position of the material point in the current configuration. Now,r the transformations of the form x! x; z! z and y! Q ðtÞy þ aðtÞ where t denotes time, Q ðtÞ is a proper orthogonal rotation tensor and aðtÞ aional vector. The reference and relaxed configurations are independent of the choice of such changes in frame. Under changes in frame of the form giventhe variables: (a) F! QF, (b) Fp ! Fp , (c) Lp ! Lp , (d) Fe ! QFe , (e) Ce ! Ce since Fe ! QFe , (f) Ee ! Ee since Ce ! Ce , (g) T� ! T� sincedet F; Fe ! QFe and T! QTQ> , (h) T! T since Ce ! Ce and T� ! T� , (i) B! B since B is defined in the relaxed configuration, (j) _B! _B since _B ¼ Lp and

, (k) c! c and q! q since the micro-traction vector and the heat flux vector are referentially-defined, and finally (l)pint ! pint ; pext ! pext ; h! h; n! n and �! � since these variables are scalar fields.

6 In f


f1_n1 > 0 whenever _n1–0 ð26Þ

where f1 � kð1þ a/Þ½symT0 � N1� þ Divc� @w@n

� �denotes the driving force for forward transformation,

f2_n2 > 0 whenever _n2–0 ð27Þ

where f2 � kð1þ a/Þ½symT0 � N2� þ Divc� @w@n

� �denotes the driving force for reverse transformation, and finally

�qh� rh > 0 whenever rh–o: ð28Þ

The inequalities (26)–(28) are assumed to be obeyed at all times so that the reduced dissipation inequality (25) is con-currently satisfied.

To satisfy inequality (26) under the assumption of rate-independence, we choose an expression for f1 as follows:

f1 ¼ fc1ðsignð _n1ÞÞ ! f1 ¼ fc1 whenever _n1 > 0 ð29Þ

where fc1 ¼ f c1ðhÞ > 0 represents the critical resistance to forward transformation.Similarly, to satisfy inequality (27) under the assumption of rate-independence, we choose an expression for f2 as follows:

f2 ¼ fc2ðsignð _n2ÞÞ ! f2 ¼ �fc2 whenever _n2 < 0 ð30Þ

where fc2 ¼ f c2ðhÞ > 0 represents the critical resistance to reverse transformation.Eqs. (29) and (30) are the criteria for forward and reverse phase transformations, respectively. Generally, we can have

fc1–fc2.Finally, assuming the material obeys Fourier’s law of heat conduction we enforce

q ¼ �kthrh ð31Þ

to satisfy inequality (28) where kth ¼ kthðn; hÞ > 0 denotes the coefficient of thermal conductivity. For simplicity, we will as-sume the coefficient of thermal conductivity to be equal and constant at all times regardless of martensite volume fractionand temperature.

2.8. Free energy density and specific constitutive functions

The free-energy density of the material is taken to contain the free energy of a conventional shape-memory alloy (Helmand Haupt, 2003) augmented with a gradient energy (Fried and Gurtin, 1994). We take the free energy per unit reference vol-ume, w to be in the separable form

w ¼ weðEe; hÞ þ wgðmÞ þ wnðn; hÞ þ whðhÞ where ð32Þ

weðEe; hÞ ¼ l Ee0

�� 2 þ jðtraceEeÞ2 � 3jathðh� hoÞðtraceEeÞ; wgðmÞ ¼ 12

snjmj2; ð33Þ

wnðn; hÞ ¼ kT

hTðh� hTÞnþ

12

hn2 and whðhÞ ¼ cðh� hoÞ � ch logðh=hoÞ: ð34Þ

Here we denotes the classical thermo-elastic free energy density with l ¼ lðn; hÞ; j ¼ jðn; hÞ and ath ¼ athðn; hÞ denotingthe shear modulus, bulk modulus and the coefficient of thermal expansion, respectively.

Following the work of Fried and Gurtin (1994), we introduce an isotropic gradient energy wg where sn ¼ snðn; hÞP 0 de-notes a material parameter with units of energy per unit length. The gradient energy acts to penalize to presence of austen-ite–martensite interfaces. Hence, due to the gradient energy there exists an intrinsic material length scale in the constitutivemodel. As a first-cut assumption, we will treat sn as a constant.

The austenite$ martensite phase transformation energy is denoted by wn where kT and hT represents the latent heat re-leased/absorbed (units of energy per unit volume) during the austenite $ martensite phase transformation and the phaseequilibrium temperature, respectively.6 The energetic interaction coefficient, h has units of energy per unit volume.

Finally, wh represents the purely thermal portion of the free energy with c ¼ cðn; hÞ being the specific heat per unit refer-ence volume. Following the modeling assumptions adopted by Abeyaratne and Knowles (1993), we will suppress the depen-dence of l;j;ath and c on the martensite volume fraction and temperature, and treat them as constants.

2.9. Constitutive equation for the stress, entropy and micro-traction vector

Substituting Eq. (32) into Eq. (24)1 yields the constitutive equation for the stress:

T� ¼ 2lEe0 þ j½traceEe � 3athðh� hoÞ�1: ð35Þ

ormulating the phase transformation free energy, we are guided by the one-dimensional model of Abeyaratne and Knowles (1993).


The constitutive equation for the entropy density and micro-traction vector are respectively given substituting Eq. (32)into Eq. (24)2 and (24)3:

g ¼ c logðh=hoÞ þ 3jathðtraceEeÞ � ðkT=hTÞn and c ¼ snðrnÞ: ð36Þ

2.10. Flow direction N1 and the J3 parameter

Since the material is assumed to be elastically-isotropic, we can conclude from Eq. (35) that Ee and T� are co-axial. Thus,the stress tensor T � CeT� is also symmetric. Using this result and substituting Eqs. (32) and (36)2 into the expressions for thedriving force for phase transformation yield:

f1 ¼ kð1þ a/ÞðT0 � N1Þ þ snðr2nÞ � kT

hTðh� hTÞ � hn; ð37Þ

f2 ¼ kð1þ a/ÞðT0 � N2Þ þ snðr2nÞ � kT

hTðh� hTÞ � hn: ð38Þ

From the expression for the driving force shown in Eqs. (37) and (38), we can see that the transformation between theaustenite and martensite phase is affected by local terms (due to the stress and phase transformation energy) and non-localterms (due to the gradient energy).

During forward transformation i.e. _n1–0, substituting Eq. (37) into Eq. (29) results in

kð1þ a/ÞðT0 � N1Þ ¼kT

hTðh� hTÞ þ hn� snðr2nÞ þ fc1ðsignð _n1ÞÞ: ð39Þ

To satisfy Eq. (38), we take

ð�TÞ2½kð1þ a/Þ�T0 ¼kT

hTðh� hTÞ þ hn� snðr2nÞ þ fc1ðsignð _n1ÞÞ

N1 ð40Þ

since jN1j ¼ �T . Taking the magnitude on both sides of Eq. (40) yields

N1 ¼ �TT0

jT0j

( )! f1 ¼ �r�Tð1þ a/Þ þ snðr2nÞ � kT

hTðh� hTÞ � hn ð41Þ

where �r ¼ kjT0j represents an equivalent stress. From the work of Orgeas and Favier (1998), the J3 parameter is then given by

/ ¼ffiffiffi6p½N1 � N2

1�ð�TÞ�3:

2.11. Flow rule: revisited

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

�Tð1þ a/Þ _n1 �ffiffiffiffiffiffiffiffi2=3

pjsymLpj ! k ¼

ffiffiffiffiffiffiffiffi3=2

pand �r ¼

ffiffiffiffiffiffiffiffi3=2

pjT0j:

Therefore �r denotes the equivalent tensile stress or Mises stress. The final form for the inelastic strain-rate i.e the flow rule isthen given by

Lp ¼ffiffiffiffiffiffiffiffi3=2

pð1þ a/Þ

X2

i¼1

_niNi: ð42Þ

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 for all values of f1. For n ¼ 0, the driving force for reversetransformation is defined for all values of f2.

In a rate-independent theory, the variables ff1; _n1g and ff2; _n2g must 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: ð43Þ

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

Fig. 1.Also sh


_n2_ðf2 þ fc2Þ ¼ 0: ð44Þ

� End conditions: If n ¼ 1 and f1 ¼ fc1, then _n1 ¼ 0. If n ¼ 0 and f2 ¼ �fc2, then _n2 ¼ 0.

Eqs. (43) and (44) are the consistency conditions for forward and reverse phase transformation, respectively. The consis-tency conditions are used to determine the transformation rates _n1 and _n2.

2.13. Balance of energy: revisited

Substituting Eqs. (31), (32), (35), (36)1 and (36)2 into Eq. (20) yields

T0 � Lp þ snðr2nÞ _n� kT

hTðh� hTÞ _nþ kthðr2hÞ þ r ¼ _gh: ð45Þ

Taking the time-derivative of Eq. (36)1 results in

_g ¼ ch

� �_hþ 3jathðtrace _EeÞ � kT

hT

_n: ð46Þ

The evolution equation for the temperature is given by substituting Eqs. (5), (42) and (46) into Eq. (45):

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

i¼1

fi_ni þ r: ð47Þ

On the right-hand side of Eq. (47), the first term describes the heat conduction contribution, the second and third termdescribes the heat source due to entropic contributions, and the fourth term describes the dissipation due to the motion ofaustenite–martensite interfaces.

The list of constitutive parameters/functions that needed 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 hasbeen developed and implemented in the ABAQUS/Explicit (Abaqus reference manuals, 2009) finite-element program bywriting a user-material subroutine. Algorithmic details for the time-integration procedure used to implement the modelin the finite-element code are given in Appendix A.

3. Experiments and finite-element simulations

To calibrate the material parameters in the constitutive model, we fit the constitutive theory to the stress–strain curvesobtained from simple tension and simple compression superelastic experiments conducted on a polycrystalline rod Ti–Ni ata temperature of 298 K (Thamburaja and Anand, 2001). Isothermal conditions were maintained by conducting the experi-ments under very low strain-rates. With hmf and has denoting the martensite finish temperature and the austenite start tem-perature, respectively, the transformation temperatures for the polycrystalline Ti–Ni material were determined to be

Stress Stress

Strain StrainActual behavior Idealized behavior

(a) (b)

austenite to martensite

martensite to austenite

The schematic stress–strain response of a polycrystalline shape-memory alloys undergoing superelastic deformation under uniaxial stress states.own is the idealized stress–strain response used to calibrate the material parameters in the constitutive model.


hms ¼ 251:3 K;hmf ¼ 213:0 K;has ¼ 260:3 K and haf ¼ 268:5 K. Hence the material is initially in the fully-austenitic phase attemperature 298 K.

Fig. 1a shows a schematic stress–strain response of an actual polycrystalline shape-memory alloy undergoing superelasticdeformation under uniaxial loading. The fitting of the constitutive parameters were performed on the idealized version ofthe actual experimental stress–strain response cf. Fig. 1b. The idealized stress–strain response contains the key featuresof superelastic deformation, namely: (1) Initial loading which causes the elastic deformation of the austenitic material;

STRAIN

STR

ESS

[MPa

]

EXPERIMENTSIMULATION

STRAIN

STR

ESS

[MPa

]

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

50

100

150

200

250

300

350

400

450

500

SHEAR STRAIN

SHEA

R S

TRES

S [M

Pa] EXPERIMENT

SIMULATION

STRAIN

STR

ESS

[MPa

]

TENSIONCOMPRESSION

(b) (c)

(d) (e)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

100

200

300

400

500

600

700

800

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

(a)

Fig. 2. (a) An initially-undeformed single ABAQUS C3D8R continuum-three-dimensional brick element. Numerical and experimental superelastic stress–strain curves in (b) simple tension, (c) simple compression and (d) simple shear. The data from the tension and compression experiments were used to fitthe material parameters. The simulated shear stress-shear strain response corresponds to an independent prediction. (e) Comparison of the stress–strainresponse from the tension and compression simulations to demonstrate the tension–compression asymmetry.


(2) continued loading which results in a phase transformation from austenite! martensite; (3) further loading which willcause the elastic deformation of the martensitic material; (4) reverse loading which causes the elastic unloading of the

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 0:046

a ¼ 0:13 h ¼ 0 J=m3 c ¼ 2:08 MJ=Km3 kth ¼ 18 W=m K

hT ¼ 255:8 K kT ¼ 97 MJ=m3 fc ¼ 7:8 MJ=m3 sn ¼ 0 J=m

r 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

(a)

0 0.01 0.02 0.03 0.04 0.05 0.060

200

400

600

800

1000

1200

STRAIN

STR

ESS

[MPa

]

(b)

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

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 Kunder isothermal conditions.


martensitic material; (5) continued reverse loading which results in the phase transformation from martensite! austenite;and (6) further reverse loading which results in the elastic unloading of the austenitic material.

As a first-cut assumption, we will take the critical resistances to forward and reverse transformation to be equal, constantand suppress its dependence on temperature 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 than the thickness of the austenite–martensite interface, we willignore the effect of the gradient energy in our calculations as a first-cut assumption i.e. we set sn ¼ 0 J=m. The influence ofthe gradient energy on the deformation behavior of shape-memory alloys undergoing austenite $ martensite phase trans-formations will be investigated in Section 4. In the spirit of modeling elastic-perfectly inelastic materials, we set the ener-getic interaction coefficient, h ¼ 0 J=m3. Finally the heat supply per unit volume, r, is ignored in all of our calculations bysetting it to be zero.

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]

STR

AIN

TENSION : 50 MPa

TENSION : 100 MPa

TENSION : 150 MPa

(a)

220 240 260 280 300 320

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

TEMPERATURE [K]

STR

AIN

COMPRESSION : 50 MPaCOMPRESSION : 100 MPaCOMPRESSION : 150 MPa

(b)

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

COMPRESSION

TENSION

STRAIN

TEMPERATURE [K]

ST

RE

SS

[MP

a]

00.01

0.020.03

0.040.05

0.06

280

275

270

2650

100

200

300

400

500

Fig. 5. Stress–strain–temperature response obtained from the shape-memory effect simulations conducted in simple tension and simple compression. Anisothermal stress–strain response from straining occurs at a temperature of 265 K. Following this, an increase in temperature to 280 K takes place.


All the finite-element simulations in this work were conducted using a single ABAQUS C3D8R continuum-three-dimen-sional brick element shown in Fig. 2a under unless stated otherwise. This element is meshed using eight corner nodes and isintegrated numerically using a reduced integration scheme.

The material is taken to be fully-austenitic at the beginning of each simulation. The constitutive parameters were deter-mined by fitting the model to the simple tension and compression experiments using a similar methodology outlined inThamburaja and Anand (2001). Using the material parameters listed out in Table 17 the isothermal stress–strain curves ata temperature of 298 K obtained from the simple tension and simple compression finite-element simulations are plotted inFig. 2b and c, respectively. The fit from the numerical simulations are in good accord with the experimental stress–strainresponses.

With the constitutive parameters calibrated, a finite-element simulation in simple shear was performed and the resultingisothermal shear stress-shear strain response at a temperature of 298 K is plotted in Fig. 2d. The experimental shear stress-shear strain curve is well-predicted by the constitutive model.

The numerical stress–strain curves obtained from the simple tension and simple compression simulations conductedabove are repeatedly plotted in Fig. 2e for comparison. As shown by these stress–strain responses, the present constitutivetheory is able to model the tension–compression asymmetry exhibited by polycrystalline rod Ti–Ni namely: (i) the stresslevel required to nucleate the martensitic phase from the parent austenitic phase is considerably higher in compression thanin tension; (ii) the transformation strain measured in compression is smaller than that in tension; and (iii) the hysteresisloop generated in compression is wider (measured along the stress axis) than the hysteresis loop generated in tension. Thesefeatures in the stress–strain responses are exhibited due to the influence of the J3 parameter.

With the model calibrated to have values as shown in Table 1, we perform a set of superelastic simulations in simple ten-sion and simple compression under isothermal conditions at two other ambient temperatures: 288 K and 308 K. The stress–strain curves from these finite-element simulations are plotted in Fig. 3a and b together with the simple tension and simplecompression stress–strain curves obtained from the simulations conducted at an ambient temperature of 298 K (as alsoshown in Fig. 2b and c). The stress–strain responses plotted in Fig. 3a and b show that the stress required to induce austeniteto martensite transformation or martensite to austenite transformation increases with increasing ambient/test temperature.This concurs very well with experimental findings (e.g. Thamburaja and Anand, 2003).

Next we conduct a series of strain–temperature-cycling simulations which can be described as follows: At a temperatureof 320 K, the material is first pre-stressed to a desired stress level. With the pre-stress maintained, the temperature of thematerial is reduced to 220 K and then increased back again to 320 K. Depending on the pre-stress level, a transformationfrom the austenite to martensite phase will occur at a particular temperature as result of a sufficient reduction in temper-ature. At this point, a sufficient increase in temperature will then cause a transformation from the martensite to austenitephase to take place at another critical temperature.

In our finite-element simulations, we choose three different pre-stress levels under simple tension and simple compres-sion conditions: 50 MPa, 100 MPa and 150 MPa. Fig. 4a shows the strain versus temperature responses for the strain–tem-perature-cycling simulations conducted under the aforementioned tensile stress levels. The strain versus temperature curves

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


for the strain–temperature-cycling finite-element simulations performed under the aforementioned compressive stress lev-els are plotted in Fig. 4b. From the simulation results shown in Fig. 4a and b, we can conclude the following trends: withincreasing pre-stress level, a phase transformation from austenite to martensite or martensite to austenite occurs at a highertemperature. The results shown in Fig. 4a and b also qualitatively reproduce the strain–temperature-cycling experimentaldata shown in Thamburaja and Anand (2003). During the strain–temperature-cycling, note that the transformation (actua-tion) strain obtained due to the tensile pre-stress is higher to that obtained using a compressive pre-stress. This is again dueto the introduction of the J3 parameter which takes different values under tensile or compressive stress states. Another pointto note is that despite the amount of pre-stress and the sign of the pre-stress i.e. tensile or compressive, the temperature atwhich the martensite to austenite transformation occurs is always approximately 42 K higher than the temperature at whichthe austenite to martensite transformation takes place.

To simulate the one-way austenite!martensite! austenite shape-memory effect, we perform the following finite-ele-ment calculations: With the temperature of the material initially at ho ¼ 265 K where hms < 265 K < haf , we perform anisothermal simple tension and simple compression simulation to cause a transformation from austenite to martensite. Whenthe material is fully-martensitic at the completion of the forward loading process, a reverse loading process to an unstressedstate occurs through an elastic unloading of the material. With the applied stress in the material maintained at zero, thetemperature of the material is then raised to 280 K. The stress–strain–temperature response from these finite-elementsimulations are plotted in Fig. 5. Note that once the reverse loading process to an unstressed state has taken place, a residualstrain will exist in the material as the temperature is not sufficiently high enough for reverse transformation from martensite

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) −1STRAIN−RATE = 1 x 10−4 s (Simulation A)−1

1

23

a/i

bc

e

g

1 mm

40 mm

5 mm(a)

(b)

d

fh

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


to austenite to take place. However, as shown in Fig. 5, the residual strain obtained from both these simulations during thedeformation process at temperature ho will be fully recovered once the temperature is raised above 276:6 K > haf . Therefore,our constitutive model is able to qualitatively reproduce the one-way austenite ! martensite ! austenite shape-memoryeffect to good accord.

To investigate the non-isothermal behavior of shape-memory alloys during tensile superelastic deformation, we performthe following fully-coupled thermo-mechanical simulations: At an initial temperature of 298 K, an initially-undeformedshape-memory alloy sheet with dimensions of 5 mm by 20 mm by 1 mm is meshed using 200 ABAQUS C3D8RT elementsas shown in Fig. 6a. Each C3D8RT element has displacement and temperature degrees of freedom. The nodes on both endsof the specimen in the 1-3 plane act as grip sections, and their temperature is kept fixed at 298 K throughout the duration ofthe simulations i.e. the grips serve as a constant temperature bath. One grip section is constrained against motion alongdirection-2 and the other grip section is deformed along direction-2 under strain-rates of 1� 10�4 s�1 (Simulation A) and5� 10�4 s�1 (Simulation B). Furthermore, heat convection from the outer surfaces of the sheet to the ambient environment(still air) is taken into account by setting the surface film heat transfer coefficient to be 12 W=m2 K.

The stress–strain response from these two simulations using the initially-undeformed finite-element mesh shown inFig. 6a are plotted in Fig. 6b. The contours of the martensite volume fraction in the sheet specimen obtained from SimulationB keyed to different points on its corresponding stress–strain curve is shown in Fig. 7. Due to the boundary conditions

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) (b)

(c) (d)

(e) (f)

(g) (h)

(i)

Fig. 7. Contours of the martensite volume fraction keyed to various points of the stress–strain curve obtained from Simulation B as shown in Fig. 6b. Due tothe boundary conditions, both the forward and reverse phase transformations initiate from the ends and move towards the center of the specimen.


imposed on the specimen as explained above, the austenite–martensite phase boundaries propagate from the grip sectionsto the specimen’s center during the forward loading and reverse loading process. The contour plots presented in Fig. 7 showthe possibility of multiple austenite–martensite phase transformation fronts propagating in the specimen during superelas-tic deformation (cf. Shaw and Kyriakides, 1997).

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

The causes for these observed trends are as follows: Simulation A was conducted at a deformation rate which results in anearly isothermal response i.e. the austenite $ martensite phase transformations occur at nearly constant stress plateaus.However, Fig. 8 shows the contours of the temperature in the sheet specimen obtained from Simulation B keyed to differentpoints on its corresponding stress–strain response shown in Fig. 6b i.e. Simulation B was conducted at a strain-rate whichresults in a non-isothermal temperature field in the specimen during phase transformations. As shown in Fig. 8, the temper-ature in the mid-section of the sheet increases by as much as 16 K above the ambient temperature (298 K) during the for-ward loading process. At a strain-rate of 5� 10�4 s�1 the heat generated due to the release of the latent heat and mechanical

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) (b)

(c) (d)

(e) (f)

(i)

(g) (h)

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 forwardloading, the temperature in the specimen increases by approximately 16 K above the ambient temperature of 298 K due to the release of latent heat as aresult of the austenite to martensite phase transformation. During reverse loading, the temperature in the specimen decreases by approximately 14 K belowthe ambient temperature of 298 K due to the absorption of latent heat as a result of the martensite to austenite phase transformation.


dissipation is not conducted and convected out of the specimen quickly enough, and this results in the increase of the spec-imen temperature with respect to the ambient temperature. Thus, it is the increase in temperature which causes the hard-ening in the stress–strain response during the forward loading process as shown in Fig. 6b.

Conversely, as also shown in Fig. 8 the temperature in the mid-section of the sheet decreases to below the ambient tem-perature during the reverse loading process. At this deformation-rate i.e. 5� 10�4 s�1 the heat loss in the specimen due tothe absorption of the latent heat outweighs the amount of heat conducted and convected into the specimen, and hencecauses a reduction in the specimen’s mid-section temperature by as much as 14 K below the ambient temperature(298 K). Therefore, it is this decrease in temperature which causes the softening in the stress–strain response during the re-verse loading process as shown in Fig. 6b.

As mentioned previously, finite-deformation-based constitutive models have great utility in studying the deformationbehavior of flexible structures experiencing large deformations (Auricchio and Taylor, 1997; Auricchio, 2001). To exhibitthe capability of our constitutive model, we model the deformation behavior of a stent unit cell with its initial geometryshown in Fig. 9a (Pan et al., 2007). Fig. 9b shows the initially-undeformed finite-element mesh of the stent unit cell shownin Fig. 9a using 1304 ABAQUS C3D8R elements. The boundary conditions for the stent unit cell are as follows: the nodes onthe bottom face are prevented from motion whereas a displacement profile along direction-2 is prescribed for all the nodeson the top face. All the nodes which make up the stent section have an initial temperature of 298 K and the finite-elementsimulation was conducted under isothermal conditions using the material parameters listed in Table 1.

Fig. 10 shows the force vs. displacement curve obtained from the loading–unloading simulation performed on the stentunit cell. The total force is obtained from the summation of the reaction forces along direction-2 for the nodes making up thetop surface. The force vs. displacement response resembles the typical uniaxial superelastic stress–strain curves exhibited byshape-memory alloys. From the conducted numerical simulations, we can determine the contours of the martensite volumefraction within the stent unit cell at any given point of the deformation process. Fig. 10a and b respectively show the mar-tensite volume fraction contours keyed to points a and b on the force vs. displacement curve plotted in Fig. 10. As expected,the stent unit cell will revert back to the fully-austenitic state upon the full reversal of the deformation back to the initialstate (cf. Fig. 10a). The numerical simulation is also able to predict the regions with the highest concentration of martensitevolume fraction within the stent unit cell in the deformed state (cf. Fig. 10b).

(b)

1.50

8.00

0.50

R 2.0011.00

R 0.50

R 2.00

R 0.90

10.00

unit=mm

(a)

1

3 2

1

2

Top face

Bottom face

Grip section

Grip section

Stent section

Fig. 9. (a) Specimen geometry for a stent unit cell taken from the work of Pan et al. (2007). The stent has a thickness of 0.38 mm. (b) Undeformed mesh ofthe tested section of the stent unit cell using 1304 ABAQUS C3D8R elements. Direction-2 denotes the loading axis.

0 1 1.5 2 2.5 30

2

4

6

8

10

12

14

16

18

20

DISPLACEMENT [mm]

FOR

CE

[N]

FEM SIMULATION

0.5

b

a

(a) (b)

Martensite volume fraction

3 1

2

1.000.900.800.700.600.500.400.300.200.100.00

Fig. 10. The numerically-determined force vs. displacement response of the stent unit cell shown in Fig. 9a under superelastic deformation. Contours ofmartensite volume fraction within the stent unit cell (a) in the initial state and upon full reversal of the deformation to the initial position i.e. point a on theforce vs. displacement curve shown above, and (b) keyed to point b on the force vs. displacement curve shown above.


4. Phase boundary propagation during superelastic deformation

To study the austenite–martensite phase boundary propagation in a shape-memory alloy undergoing superelasticdeformation, we perform the following simulations: Consider an initially-austenitic cuboidal section as shown inFig. 11a with an initially-undeformed dimensions of 2 mm by 10 lm by 10 lm measured along direction-1, direction-2and direction-3, respectively. This cuboidal section is meshed using 100 and 600 ABAQUS C3D8R elements along direc-tion-1. The nodes on the two ends of the cuboidal specimen in the 2–3 plane serve as the grip sections (grip section Aand B). Sites of geometrical imperfection have also been introduced in both the grip sections. Grip section A is preventedfrom motion along direction-1 whereas a deformation profile is imparted on grip section B along direction-1. The finite-element simulations were performed under isothermal conditions with the specimen temperature maintained at 298 Kthroughout the duration of the simulations. Finally, for these set of simulations we use the values for the material param-eters listed in Table 1 except for: (1) a very small energetic interaction coefficient h of �0:24 MJ=m3 introduced to accel-erate the localization process, and (2) the value of sn to be set at 1:25� 10�3 J=m which will correspond to anexperimentally-determined austenite–martensite interface thickness of approximately 100 l (Sun and Li, 2002).8 Forthe two different mesh densities described above, we have also performed simulations with sn ¼ 0 J=m which collapsesour constitutive model to a local theory.

8 Although the interface energy used in this work is about two orders of magnitude larger than that calculated by Levitas et al. (2003), the main purpose ofthis Section is to show that the non-local version of our phenomenological theory is able to eliminate the dependence of the austenite–martensite interfacethickness on mesh density.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

100

200

300

400

500

600

700

STRAIN

ST

RE

SS

[MP

a]

100 elements : Non-local600 elements : Non-local

3

1 2

(a)

Grip section A

Grip section B

(b)

a

b

100 elements : Local600 elements : Local

Fig. 11. (a) A cuboidal specimen with initially-undeformed dimensions of 2 mm by 10 lm by 10 lm measured along direction-1, direction-2 and direction-3, respectively. (b) The simulated tensile stress–strain curves using the local ðsn ¼ 0 J=mÞ vs. the non-local ðsn ¼ 1:25� 10�3 J=mÞ version of theory. Thesimulations were conducted on the cuboidal specimen shown in Fig. 11a meshed uniformly along direction-1 using 100 and 600 ABAQUS C3D8R elements.All the calculations were conducted at a temperature of 298 K under isothermal conditions.


Fig. 11b shows the calculated superelastic tensile stress–strain curves for the above described simulations. The stress–strain responses for the finite-element simulations conducted on the cuboidal section shown in Fig. 11a meshed usingthe two different mesh densities for both the local vs. non-local version of the constitutive theory are equivalent.

Fig. 12a and b show the contours of the martensite volume fraction for the simple tension simulation using the local the-ory i.e. sn ¼ 0 J=m conducted on the cuboidal specimen section shown in Fig. 11a meshed using 100 and 600 ABAQUS C3D8Relements keyed to points a and b, respectively on the corresponding stress–strain curves plotted in Fig. 11b. These contourplots are drawn in the reference i.e. undeformed configuration. During the forward loading process, the austenite–martensiteinterface9 propagates from grip section A towards grip section B where the austenitic phase is gradually transforming into themartensitic phase. However, during the reverse loading process the austenite–martensite interface propagates from grip sectionB towards grip section A where the martensitic phase is now gradually transforming back into the austenitic phase. This prop-agation pattern is enforced numerically through the sites of geometrical imperfections introduced into the specimen at the gripends.

During forward transformation, Fig. 12a shows that the austenite–martensite interface thickness predicted using the twodifferent mesh densities to be significantly different. Similarly, during reverse transformation, Fig. 12b shows that the aus-tenite–martensite interface thickness predicted using the two different mesh densities to also be significantly different.Hence the local theory ðsn ¼ 0 J=mÞ is predicting different austenite–martensite interface thickness with varying mesh den-sity i.e. its calculated position is heavily dependent on the mesh density.

From the simulations conducted using the non-local theory i.e. sn ¼ 1:25 � 10�3 J=m, Fig. 13a and b show the contoursof the martensite volume fraction obtained from the simple tension simulation conducted on the cuboidal specimen sec-tion shown in Fig. 11a meshed using 100 and 600 ABAQUS C3D8R elements keyed to points a and b, respectively, on the

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

100 elements

Martensite volume fraction1.0000.9170.8330.7500.6670.5830.5000.4170.3330.2500.1670.0830.000

600 elements

100 elements

600 elements

(a)

(b)

3 1

2

Fig. 12. The contours of the martensite volume fraction keyed to points a and b on the stress–strain curve shown in Fig. 11b during (a) forwardtransformation, and (b) reverse transformation obtained from the simulations conducted using the local version of the theory with sn ¼ 0 J=m. The contoursare plotted on the undeformed mesh.


corresponding stress–strain curves plotted in Fig. 11b. During forward transformation, Fig. 13a shows that the austenite–martensite interface thickness predicted from the simulations conducted using the two different mesh densities to beapproximately equal. Hence the non-local version of the constitutive theory is predicting equal austenite–martensite inter-face thickness during forward transformation regardless of mesh density. Similarly, during reverse transformation, Fig. 13bshows that the austenite–martensite interface thickness predicted from the simulations conducted using the two differentmesh densities to also be approximately equal. Thus, the non-local version of the constitutive theory is also predictingequal austenite–martensite interface thickness during reverse transformation independent of mesh density. From the re-sults shown in Fig. 13 we can see that at a given deformation level during superelasticity, the non-local version of thetheory predicts the same position for the austenite–martensite interface regardless of mesh density. A more detailed com-parison of the martensite volume fraction contours in the vicinity of the austenite–martensite interface is shown inFig. 14.

Also note that from the non-local version of the theory, the simulated austenite–martensite interface thicknessesshown in Fig. 14 during reverse loading is larger than the calculated austenite–martensite interface thicknesses duringforward loading. This trend can be explained as follows: From Fig. 11b, we can see that the stress to cause forward

100 elements

Martensite volume fraction1.0000.9170.8330.7500.6670.5830.5000.4170.3330.2500.1670.0830.000

600 elements

100 elements

600 elements

(a)

(b)

3 1

2

Fig. 13. The contours of the martensite volume fraction keyed to points a and b on the stress–strain curve shown in Fig. 11b during (a) forwardtransformation, and (b) reverse transformation obtained from the simulations conducted using the non-local version of the theory withsn ¼ 1:25� 10�3 J=m. The contours are plotted on the undeformed mesh.


transformation is larger than the stress to cause reverse transformation. Hence the resistance to forward transformation,sþ is larger than the resistance to reverse transformation, s�. Since the thickness of the austenite–martensite interfacescales with s�1=2

þ and s�1=2� during forward and reverse transformation, respectively, it follows that the thickness of

the austenite–martensite interface is larger during the reverse transformation process as sn and h are treated as con-stants at all times.

It is a well-known fact that local theories are not able to eliminate mesh sensitivity in regards to the prediction of interfacethicknesses during deformation.10 From the contour plots shown in Fig. 14 we can conclude that for the mesh densities usedfor the calculations in this section, a non-zero gradient energy i.e. a non-local theory with sn–0 heavily minimizes the effect ofmesh density on the prediction of the austenite–martensite interface thicknesses. Hence the position of the austenite–mar-tensite interface region can now be accurately tracked during phase transformations without the aid of jump conditions.

5. Conclusion

A thermo-mechanically-coupled, non-local and isotropic-plasticity-based constitutive model for shape-memory alloyscapable of undergoing austenite $ martensite phase transformations has been developed with the aid of standard balancelaws, thermodynamic laws and the principle of micro-force balance (Fried and Gurtin, 1994). The constitutive model hasbeen implemented in the ABAQUS/Explicit (2009) finite-element program by writing a user-material subroutine.

10 For instance, see the recent works of Borg (2007) and Lele and Anand (2009) on shear banding in metallic alloys.

Martensite vol. frac.1.0000.8330.6670.5000.3330.1670.000

(a)

100 elements (forward loading)


vs.

100 elements (reverse loading)


vs.



vs.

(b)



vs.

Fig. 14. The contours of the martensite volume fraction in the vicinity of the austenite–martensite interface obtained from the simulations conductingusing the (a) local version of the constitutive theory i.e. sn ¼ 0 J=m, and (b) non-local version of the constitutive theory i.e. sn ¼ 1:25� 10�3 J=m.


By calibrating the material parameters in the constitutive model to the superelastic stress–strain responses of represen-tative physical experiments i.e. tension, compression and simple shear, exotic behavior exhibited by SMAs such as the stress-biased strain–temperature-cycling and the shape-memory effect are also qualitatively well-reproduced by the theory. Wehave also shown that our present theory is able to model the coupled thermo-mechanical response during superelasticdeformation and also qualitatively model the reversible behavior of a stent unit cell undergoing superelastic response.

Finally, we also show that the non-local version of our theory i.e. with sn–0 allows the accurate tracking of the austenite–martensite interface motion during superelastic deformation independent of mesh density. Hence the exact position(s) ofthe austenite–martensite interface can be determined without the aid of jump-conditions.

Some directions for future work include: (a) the prediction of complicated multi-axial and coupled thermo-mechanicalexperiments (e.g. Tokuda et al. (1999, 2002)) using our developed constitutive model, and (b) extending the theory to modelmartensitic reorientation/detwinning which may occur during superelasticity under non-proportional loading conditions.

Acknowledgements

The financial support for this work was provided by the Ministry of Science, Technology and Innovation, Malaysia underGrant 03-01-02-SF0257. The ABAQUS finite-element software was made available under an academic license from HKS, Inc.Pawtucket, R.I.

Appendix A. Time-integration procedure

In this appendix we summarize the explicit time-integration procedure that we have developed for our constitutive mod-el presented in Section 2. With t denoting the current time, Dt is an infinitesimal time increment, and s ¼ t þ Dt. The algo-rithm is as follows:


Given: (1) fFðtÞ; FðsÞ; hðtÞ; hðsÞg11; (2) fTðtÞ;FpðtÞg; (3) fBðtÞ;N1ðtÞ;N2ðtÞ;/ðtÞg; (4) the martensite volume fraction nðtÞ. Attime t ¼ 0, we initialize BðtÞ ¼ 0.Calculate: (a) fTðsÞ; FpðsÞg, (b) fBðsÞ;N1ðsÞ;N2ðsÞ;/ðsÞg, (c) the martensite volume fraction nðsÞ, and march forward intime.

The steps used in the calculation procedure are:

Step 1. Calculate the trial elastic strain EeðsÞtrial:

11 The12 The

conditio

FeðsÞtrial ¼ FðsÞFpðtÞ�1;

CeðsÞtrial ¼ ðFeðsÞtrialÞ>FeðsÞtrial;

EeðsÞtrial ¼ ð1=2ÞðCeðsÞtrial � 1Þ:

Step 2. Calculate the trial stress T�ðsÞtrial:

T�ðsÞtrial ¼ 2lEe0ðsÞ

trial þ j½traceEeðsÞtrial � 3athðhðsÞ � hoÞ�1: ð48Þ

Step 3. Calculate the trial driving forces fiðsÞtrial. In our explicit numerical algorithm presented here, we approximate12

NiðsÞ NiðtÞ; /ðsÞ /ðtÞ and r2nðsÞ r2nðtÞ: ð49Þ

For infinitesimal elastic stretches, we can also use the approximation TðsÞ T�ðsÞ. Hence, the trial driving forces for phasetransformation are then given by

fiðsÞtrial ¼ffiffiffi32

rð1þ a/ðtÞÞ½T�0ðsÞ

trial � NiðtÞ� þ sn½r2nðtÞ� � kT

hTðhðsÞ � hTÞ � hnðtÞ: ð50Þ

Step 4. Determine the set PA of potentially active transformation 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. (49), we calculate

FpðsÞ ¼ 1þffiffiffi32

rð1þ a/ðtÞÞ

Xj2PA

DnjNjðtÞ( )

FpðtÞ where j ¼ 1; . . . ;Q : ð51Þ

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 A with elements M 6 Q , may actually be active (non-zero increments). This set is determined in an iterativefashion described below.

During phase transformation, the active transformation systems must satisfy the consistency conditions

f1ðsÞ � fc ¼ 0 and=or f 2ðsÞ þ fc ¼ 0 ð52Þ

for forward transformation and/or reverse transformation, respectively. Using Eqs. (48)–(51), it is straightforward to showthat

fiðsÞ ¼ fiðsÞtrial �X

j2PA

Dnj½3lð1þ a/ðtÞÞ2symðCeðsÞtrialNjðtÞÞ �NiðtÞ þ hdij� ð53Þ

where dij is the Kronecker delta. Substituting Eq. (53) into the consistency conditions (52) give

quantities FðtÞ; FðsÞ; hðtÞ and hðsÞ are inputs provided by the ABAQUS (2009) finite-element program.Laplacian of the martensite volume fraction is calculated via a finite-difference scheme. At the free boundaries, we assume a Neumann-type boundaryn for the martensite volume fraction i.e rn � n ¼ 0.


Xj2PA

Aijxj ¼ bi; i 2 PA; ð54Þ

with

Aij ¼ 3lð1þ a/ðtÞÞ2NiðtÞ � symðCeðsÞtrialNjðtÞÞ 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. (54) is a system of linear equations for the transformation increments xj � Dnj (for j 2 PA). Assuming the matrix A to

be invertible i.e. non-singular, the transformation rates are determined by

x ¼ A�1b ð55Þ

where A�1 is the inverse matrix of the matrix A. If x1 6 0 when b1 > 0 (forward transformation), then this system is inactiveand it is removed from the set of potentially active systems PA and a new matrix A is calculated. Similarly, if x2 P 0 whenb2 < 0 (reverse transformation), then this system is also inactive and it is not included in the set PA used to determine thenew matrix A. The final size 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 inelastic deformation gradient FpðsÞ:

FpðsÞ ¼ 1þffiffiffi32

rð1þ a/ðtÞÞ

Xj2A

DnjNjðtÞ( )

FpðtÞ:

Step 8. Compute the elastic strain EeðsÞ and the stress T�ðsÞ:

FeðsÞ ¼ FðsÞFpðsÞ�1;

CeðsÞ ¼ FeðsÞ>FeðsÞ;EeðsÞ ¼ ð1=2ÞðCeðsÞ � 1Þ;T�ðsÞ ¼ 2lEe

0ðsÞ þ j½traceEeðsÞ � 3athðhðsÞ � hoÞ�1:

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

N1ðsÞ ¼ �TT�0ðsÞjT�0ðsÞj

� �and /ðsÞ ¼

ffiffiffi6p½N1ðsÞ � ðN1ðsÞÞ2�ð�TÞ�3

:

Step 10. Update the tensors BðsÞ and N2ðsÞ:

BðsÞ ¼ BðtÞ þffiffiffi32

rð1þ a/ðtÞÞ

Xj2A

DnjNjðtÞ and N2ðsÞ ¼ �TBðsÞjBðsÞj

:

Step 11. Calculate the driving forces fiðsÞ:

fiðsÞ ¼ffiffiffi32

rð1þ a/ðsÞÞ T�0ðsÞ �NiðsÞ

� þ snðr2nðsÞÞ � kT

hThðsÞ � hTð Þ � hnðsÞ:

Step 12. Calculate the inelastic work increment Dxp:


Dxp ¼ hT

hT

� �hðsÞ

Xi2A

Dni � 3jathhðsÞðtraceðDEeÞÞ þXi2A

fiðsÞDni

where DEe ¼ EeðsÞ � EeðtÞ;EeðtÞ ¼ ð1=2ÞðCeðtÞ � 1Þ and CeðtÞ ¼ FeðtÞ>FeðtÞ. The inelastic work increment is treated as the heatsource which causes heating/cooling at a material point during deformation.

Step 13. Calculate the Cauchy stress TðsÞ:

TðsÞ ¼ ½det FðsÞ��1FeðsÞT�ðsÞFeðsÞ>:

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a finite-deformation-based phenomenological theory for shape-memory alloys

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