Finite Elements in Analysis and Design 47 (2011) 166–174

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Finite Elements in Analysis and Design

0168-87

doi:10.1

� Corr

E-m1 Cu

Univers

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

An improved, fully symmetric, finite-strain phenomenological constitutivemodel for shape memory alloys

J. Arghavani a,�,1, F. Auricchio b,c,d, R. Naghdabadi a,e, A. Reali b,d

a Department of Mechanical Engineering, Sharif University of Technology, 11155-9567 Tehran, Iranb Dipartimento di Meccanica Strutturale, Universit �a degli Studi di Pavia, Italyc Center for Advanced Numerical Simulation (CeSNA), IUSS, Pavia, Italyd European Centre for Training and Research in Earthquake Engineering (EUCENTRE), Pavia, Italye Institute for Nano-Science and Technology, Sharif University of Technology, 11155-9567 Tehran, Iran

a r t i c l e i n f o

Article history:

Received 31 May 2010

Accepted 6 September 2010Available online 28 September 2010

Keywords:

Shape memory alloys

Finite strain

Solution algorithm

Finite element

Exponential mapping

UMAT

4X/$ - see front matter & 2010 Elsevier B.V.

016/j.finel.2010.09.001

esponding author. Tel.: +98 21 6616 5511; fa

ail address: arghavani@mech.sharif.ir (J. Argh

rrently visiting Ph.D. student, Dipartiment

it�a degli Studi di Pavia, Italy.

a b s t r a c t

The ever increasing number of shape memory alloy applications has motivated the development of

appropriate constitutive models taking into account large rotations and moderate or finite strains. Up to

now proposed finite-strain constitutive models usually contain an asymmetric tensor in the definition

of the limit (yield) function. To this end, in the present work, we propose an improved alternative

constitutive model in which all quantities are symmetric. To conserve the volume during inelastic

deformation, an exponential mapping is used to arrive at the time-discrete form of the evolution

equation. Such a symmetric model simplifies the constitutive relations and as a result of less

nonlinearity in the equations to be solved, numerical efficiency increases. Implementing the proposed

constitutive model within a user-defined subroutine UMAT in the nonlinear finite element software

ABAQUS/Standard, we solve different boundary value problems. Comparing the solution CPU times for

symmetric and asymmetric cases, we show the effectiveness of the proposed constitutive model as well

as of the solution algorithm. The presented procedure can also be used for other finite-strain

constitutive models in plasticity and shape memory alloy constitutive modeling.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

Shape memory alloys (SMAs) are unique materials with theability to undergo large deformations regaining the original shapeeither during unloading (superelastic or pseudo-elastic (PE)effect) or through a thermal cycle (shape-memory effect (SME))[1,2]. Since such effects are in general not present in standardalloys, SMAs are often used in innovative applications. Forexample, nowadays pseudo-elastic Nitinol is a common andwell-known engineering material in the medical industry [3,4].

The origin of SMA features is a reversible thermo-elasticmartensitic phase transformation between a high symmetry,austenitic phase and a low symmetry, martensitic phase.Austenite is a solid phase, usually characterized by a body-centered cubic crystallographic structure, which transforms intomartensite by means of a lattice shearing mechanism. When thetransformation is driven by a temperature decrease, martensite

All rights reserved.

x: +98 21 6600 0021.

avani).

o di Meccanica Strutturale,

variants compensate each other, resulting in no macroscopicdeformation. However, when the transformation is driven by theapplication of a load, specific martensite variants, favorable to theapplied stress, are preferentially formed, exhibiting a macroscopicshape change in the direction of the applied stress. Uponunloading or heating, this shape change disappears through thereversible conversion of the martensite variants into the parentphase [1,5].

For a stress-free SMA material, four characteristic tempera-tures can be identified, defined as the starting and finishingtemperatures during forward transformation (austenite to mar-tensite), Ms and Mf, and as the starting and finishing temperaturesduring reverse transformation, As and Af. Accordingly, in a stress-free condition, at a temperature above Af, only the austeniticphase is stable, while at a temperature below Mf, only themartensitic phase is stable. As a consequence, applying a stress ata temperature above Af, SMAs exhibit a pseudo-elastic behaviorwith a full recovery of inelastic strain upon unloading, while at atemperature below Ms, the material presents the shape-memoryeffect with permanent inelastic strains upon unloading whichmay be recovered by heating.

In most applications, SMAs experience a general thermo-mechanical loading conditions more complicated than uniaxial or

J. Arghavani et al. / Finite Elements in Analysis and Design 47 (2011) 166–174 167

multiaxial proportional loadings, typically undergoing very largerotations and moderate strains (i.e., in the range of 10% forpolycrystals [1]). For example, with reference to biomedicalapplications, stent structures are usually designed to significantlyreduce their diameter during the insertion into a catheter;thereby, large rotations combined with moderate strains occurand the use of a finite deformation scheme is preferred.

The majority of the currently available 3D macroscopicconstitutive models for SMAs has been developed in the smalldeformation regime (see, e.g., [6–16] among others). Finitedeformation SMA constitutive models proposed in the literaturehave been mainly developed by extending small strain models.The approach in most cases is based on the multiplicativedecomposition of the deformation gradient into an elasticand an inelastic or transformation part [17–25], thoughthere are some models which have utilized an additive decom-position of the strain rate tensor into an elastic and an inelasticpart [26].

In the present work we focus on a finite-strain extension of thesmall-strain constitutive model initially proposed by Souza et al.[8] and extensively studied in Refs. [27–29]. We first develop afinite-strain constitutive model containing an asymmetric tensor,also observed in the constitutive equations of [20–24]. To thisend, we propose an improved alternative constitutive modelwhich is expressed in terms of symmetric tensors only. We thenimplement the proposed model in a user-defined subroutine(UMAT) in the nonlinear finite element software ABAQUS/Standard and compare the solution CPU times for differentboundary value problems. The results show the increasedcomputational efficiency (in terms of solution CPU time), whenthe proposed alternative symmetric form is used. This is mostlydue to the simplification in computing fourth-order tensorsappearing when a tensorial equation is linearized.

The structure of the paper is as follows. In Section 2, based on amultiplicative decomposition of the deformation gradient intoelastic and transformation parts, we present the time-continuousfinite-strain constitutive model. In Section 3, we propose analternative constitutive equation which includes only symmetrictensors. In Section 4, based on an exponential mapping, the time-discrete form and the solution algorithm are discussed. In Section5, implementing the proposed integration algorithm within thecommercial nonlinear finite element software ABAQUS/Standard,we simulate different boundary value problems. We finally drawconclusions in Section 6.

2. A 3D finite-strain SMA constitutive model: time-continuousframe

We use a multiplicative decomposition of the deformationgradient and present a thermodynamically consistent finite strainconstitutive model as done by Reese and Christ [20,21],Evangelista et al. [22] and, more recently, by Arghavani et al.[23,24]. The finite-strain constitutive model takes its origin fromthe small-strain constitutive model proposed by Souza et al. [8]and improved and discussed by Auricchio and Petrini [27–29].

2.1. Constitutive model development

Considering a deformable body, we denote with F thedeformation gradient and with J its determinant, supposed to bepositive. The tensor F can be uniquely decomposed as

F ¼ RU ¼VR ð1Þ

where U and V are the right and left stretch tensors, respectively,both positive definite and symmetric, while R is a proper

orthogonal rotation tensor. The right and left Cauchy–Greendeformation tensors are then, respectively, defined as

C ¼ FT F , b¼ FFTð2Þ

and the Green–Lagrange strain tensor, E, reads as

E¼C�1

2ð3Þ

where 1 is the second-order identity tensor. Moreover, thevelocity gradient tensor l is given as

l¼ _F F�1ð4Þ

The symmetric and anti-symmetric parts of l supply the strainrate tensor d and the vorticity tensor w, i.e.,

d¼ 12 ðlþ lT

Þ, w¼ 12ðl�lT

Þ ð5Þ

Taking the time derivative of Eq. (3) and using (4) and (5), it canbe shown that

_E ¼ FT dF ð6Þ

Following a well-established approach adopted in plasticity[30,31] and already used for SMAs [17–24], we assume a localmultiplicative decomposition of the deformation gradient into anelastic part Fe, defined with respect to an intermediate config-uration, and a transformation one Ft , defined with respect to thereference configuration. Accordingly,

F ¼ FeFtð7Þ

Since experimental evidences indicate that the transformationflow is nearly isochoric, we impose detðFt

Þ ¼ 1, which after takingthe time derivative results in

trðdtÞ ¼ 0 ð8Þ

We define Ce¼ FeT

Fe and Ct¼ FtT

Ft as the elastic and thetransformation right Cauchy–Green deformation tensors, respec-tively, and using definitions (2) and (7), we obtain

Ce¼ Ft�T

CFt�1

ð9Þ

To satisfy the principle of material objectivity, the Helmholtz freeenergy has to depend on Fe only through the elastic right Cauchy–Green deformation tensor; it is moreover assumed to be afunction of the transformation right Cauchy–Green deformationtensor and of the temperature, T, in the following form [22–24]

C¼CðCe,Ct ,TÞ ¼ceðCeÞþct

ðEt ,TÞ ð10Þ

where ceðCeÞ is a hyperelastic strain energy function and

Et¼ ðCt

�1Þ=2 is the transformation strain. We remark that inproposing decomposition (10), we have assumed the samematerial behavior for the austenite and martensite phases. Inaddition, we assume ce

ðCeÞ to be an isotropic function of Ce; it can

be therefore expressed as

ceðCeÞ ¼ce

ðICe ,IICe ,IIICe Þ ð11Þ

where ICe ,IICe ,IIICe are the invariants of Ce. We also define ct in thefollowing form [8,22,27–29]

ctðEt ,TÞ ¼ tMðTÞJEtJþ1

2hJEtJ2þI eLðJEtJÞ ð12Þ

where tMðTÞ ¼ b/T�T0S and b, T0 and h are material parameters;the MacCauley brackets calculate the positive part of theargument, i.e., /xS¼ ðxþjxjÞ=2, and the norm operator is definedas JAJ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiA : AT

p, with A : B¼ AijBij.

Moreover, in Eq. (12) we also use the indicator function I eL

defined as

I eLðJEtJÞ ¼

0 if JEtJreL

þ1 otherwise

(ð13Þ

J. Arghavani et al. / Finite Elements in Analysis and Design 47 (2011) 166–174168

to satisfy the constraint on the transformation strain norm(i.e., JEtJreL ). The material parameter eL is the maximumtransformation strain norm in a uniaxial test.

Taking the time derivative of Eq. (9) and using (4), the materialtime derivative of the elastic right Cauchy–Green deformationtensor is obtained as

_Ce¼�ltT

CeþF t�T _CF t�1

�Celtð14Þ

We now use the Clausius–Duhem inequality form of the secondlaw of thermodynamics

S : 12_C�ð _CþZ _T ÞZ0 ð15Þ

where S is the second Piola–Kirchhoff stress tensor related to theCauchy stress r through

S ¼ JF�1rF�Tð16Þ

and Z is the entropy. Substituting (10) into (15), we obtain

S :1

2_C�

@ce

@Ce :_C

eþ@ct

@Et :_E

tþ@ct

@T_T

!�Z _T Z0 ð17Þ

Now, substituting (14) into (17), after some mathematicalmanipulations, we obtain

S�2Ft�1 @ce

@Ce Ft�T

� �:

1

2_Cþ2Ce @c

e

@Ce : lt�X : _E

t� Zþ @c

t

@T

!_T Z0

ð18Þ

where

X ¼ hEtþðtMðTÞþgÞN ð19Þ

and

N ¼Et

JEtJð20Þ

The positive variable g results from the indicator functionsubdifferential @I eL

ðJEtJÞ and it is such that

gZ0 if JEtJ¼ eL

g¼ 0 otherwise

(ð21Þ

In deriving (18) we have used the isotropic property of ceðCeÞ,

implying that Ce and @ce=@Ce are coaxial. Following standardarguments, we finally conclude that

S ¼ 2Ft�1@ce

@Ce Ft�T

Z¼�@ct

@T

8>>><>>>:

ð22Þ

Using a property of second-order tensor double contraction, i.e.,A : ðBCÞ ¼ B : ðACT

Þ ¼ C : ðBT AÞ, we obtain

X : _Et¼ ðF tXFtT

Þ : dtð23Þ

In deriving (23) we have used the following relation:

_Et¼ FtT

dtFtð24Þ

Substituting (22) and (23) into (18) and taking advantage of thesymmetry of Ce

ð@ce=@CeÞ, the dissipation inequality can be

written as

ðP�KÞ : dtZ0 ð25Þ

where

P ¼ 2Ce@ce

@Ce

K ¼ FtXF tT

8>>><>>>:

ð26Þ

To satisfy the second law of thermodynamics, (25), we define thefollowing evolution equation:

dt¼ _zðP�KÞD

JðP�KÞDJð27Þ

where we denote with a superscript D the deviatoric part of atensor. Moreover, we highlight that (27) satisfies the inelasticdeformation incompressibility condition (8).

To describe phase transformation, we choose the followinglimit function:

f ¼ JðP�KÞDJ�R ð28Þ

where the material parameter R is the elastic region radius.Similarly to plasticity, we also introduce the consistencyparameter and the limit function satisfying the Kuhn–Tuckerconditions

f r0, _zZ0, _zf ¼ 0 ð29Þ

2.2. Representation with respect to the reference configuration

In the previous section, using a multiplicative decomposition ofthe deformation gradient into elastic and transformation parts, wederived a finite-strain constitutive model. However, while F and Ft

are defined with respect to the reference configuration, Fe is definedwith respect to an intermediate configuration. It is then necessary torecast all equations in terms of C (control variable) and Ct (internalvariable) which are described with respect to the reference config-uration, allowing to express all equations in a Lagrangian form.

Since a hyperelastic strain energy function ce has beenintroduced in terms of Ce (i.e., with respect to the intermediateconfiguration), we first investigate it.

As introduced in (11), ce depends on Ce only through itsinvariants, which are equal to those of CCt�1

. In fact, for the firstinvariant the following identity holds

ICe ¼ trðCeÞ ¼ trðFt�T

CFt�1

Þ ¼ tr ðCFt�1

Ft�T

Þ

¼ trðCCt�1

Þ ¼ ICCt�1 ð30Þ

and similar relations hold for the second and the third invariants.Accordingly, we may write

@ce

@Ce ¼ a11þa2Ceþa3Ce2

ð31Þ

where ai ¼ aiðICCtr�1 ,IICCt�1 ,III

CCt�1 Þ. Substituting (31) into (22)1 andusing (9), we conclude that

S ¼ 2ða1Ct�1

þa2Ct�1

CCt�1

þa3Ct�1

ðCCt�1

Þ2Þ ð32Þ

which expresses the second Piola–Kirchhoff stress tensor interms of quantities computed with respect to the referenceconfiguration.

In order to find the Lagrangian form of the evolution equation,we substitute (27) into (24) and obtain

_Et¼ _zFtT ðP�KÞD

JðP�KÞDJFt

ð33Þ

or equivalently,

_Ct¼ 2 _zFtT ðP�KÞD

JðP�KÞDJFt

ð34Þ

Focusing on the first term of the right hand side of Eq. (34), wemay observe that

FtT

PFt¼ ðFtCeFt

Þ 2Ft�1 @ce

@Ce F t�T

� �Ct¼ CSCt

ð35Þ

J. Arghavani et al. / Finite Elements in Analysis and Design 47 (2011) 166–174 169

while, focusing on the second term of the right hand side ofEq. (34), we may observe that

FtT

KFt¼ FtT

FtXFtT

Ft¼ CtXCt

ð36Þ

We may then conclude that

FtT

ðP�KÞFt¼ YCt

ð37Þ

and obtain from (35)–(37)

Y ¼ CS�CtX ð38Þ

Moreover, we have

FtT

ðP�KÞDFt¼ FtT

ðP�K�13trðP�KÞ1ÞFt

¼ ðCSCt�CtXCt

�13 trðCSÞCt

þ13trðCtXÞCt

Þ ¼ YDCtð39Þ

yielding

ðP�KÞD ¼ Ft�T

YDFtT

ð40Þ

and

JðP�KÞDJ¼ JYDJ ð41Þ

Now, we substitute (40) and (41) into (28) and (33) to obtain

f ¼ JYDJ�R ð42Þ

and

_Ct¼ 2 _z

YD

JYDJCt

ð43Þ

Table 1 presents the time-continuous finite-strain constitutivemodel expressed only in terms of Lagrangian quantities, whichhas already been presented in [22,24].

3. An alternative form of the constitutive model in terms ofsymmetric quantities

We observe that due to the asymmetry of CS, the quantity Y isnot symmetric. The asymmetric tensor Y also appears in theconstitutive equations proposed in [20–24]. To this end, we

Table 1Finite-strain constitutive model in the time-continuous frame: original form

[22,24].

External variables: C ,T

Internal variable: Ct

Stress quantities:

S ¼ 2a1Ct�1þ2a2Ct�1

CCt�1

þ2a3Ct�1ðCCt�1

Þ2

Y ¼ CS�CtX

X ¼ hEtþðtMþgÞN

with

gZ0 ifJEtJ¼ eL

g¼ 0 otherwise

(

and

N ¼ Et

JEtJ

where Et¼ ðCt

�1Þ=2

Evolution equation:

_Ct¼ 2 _z YD

JYDJCt

Limit function:

f ¼ JYDJ�R

Kuhn–Tucker conditions:

f r0, _zZ0, _zf ¼ 0

Norm operator:

JYDJ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiYD : YDT

p

present an alternative formulation which is in terms of symmetrictensors only. According to (32), we may write

CS ¼ 2ða1CCt�1

þa2ðCCt�1

Þ2þa3ðCCt�1

Þ3Þ ð44Þ

Eq. (44) shows that the asymmetry in CS is due to the asymmetricterm CCt�1. We now present the following identity:

CCt�1

¼UtðUt�1

CUt�1

ÞUt�1

¼UtCUt�1

ð45Þ

where

C ¼Ut�1

CUt�1

ð46Þ

Substituting (45) into (44), we obtain

CS ¼UtC SUt�1

ð47Þ

where

S ¼ 2ða11þa2Cþa3C2Þ ð48Þ

Moreover, due to the coaxiality of Ct and X, we may write

CtX ¼UtCtXUt�1

ð49Þ

Substituting (47) and (49) into (38), we obtain

Y ¼UtQUt�1

ð50Þ

where Q is defined as

Q ¼ C S�CtX ð51Þ

Using the property trðUtQUt�1

Þ ¼ trðQ Þ, we also obtain

YD¼UtQ DUt�1

ð52Þ

We now substitute (52) into (42) and (43) and obtain (53) and(54), respectively, as follows:

YD : YDT

¼Q D : Q D or f ¼ JYDJ�R¼ JQ DJ�R ð53Þ

and

_Ct¼ 2 _zUt Q D

JQ DJUt

ð54Þ

We remark that in the proposed formulation we can express S as

S ¼Ut�1

SUt�1

ð55Þ

Finally, we summarize the proposed improved time-continuousfinite-strain constitutive models in Table 2.

We may compare the original and the proposed alternativetime-continuous forms summarized in Tables 1 and 2, respec-tively. It is clearly observed that in the proposed alternativeconstitutive model, all quantities are symmetric and the normoperator argument is also a symmetric tensor.

Remark 1. We note that according to (20), the variable N is notdefined for the case of vanishing transformation strain. To thisend, we use a regularization scheme proposed by Helm and Haupt[9] and also used in [7,20,21] to overcome this problem:

JEtJ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiJEtJ2

þdq

ð56Þ

where d is a user-defined parameter (typical value: 10�7).

4. A 3D finite-strain SMA constitutive model: time-discreteframe

In this section we investigate the numerical solution of theconstitutive model derived in Section 3 and summarized inTable 2, with the final goal of using it within a finite elementprogram. The main task is to apply an appropriate numericaltime-integration scheme to the evolution equation of the internalvariable, considering that, in general, implicit schemes are

Table 2Finite-strain constitutive model in the time-continuous frame: proposed alter-

native form.

External variables: C,T

Internal variable: Ct

Stress quantities:

S ¼Ut�1

SUt�1

S ¼ 2ða11þa2C þa3C2Þ

Q ¼ C S�CtX

X ¼ hEtþðtMþgÞN

with

gZ0 if JEtJ¼ eL

g¼ 0 otherwise

(

and

N ¼ Et

JEtJ,C ¼Ut�1

CUt�1

where Et¼ ðCt

�1Þ=2

Evolution equation:

_Ct¼ 2 _zUt Q D

JQ DJUt

Limit function:

f ¼ JQ DJ�R

Kuhn–Tucker conditions:

f r0, _zZ0, _zf ¼ 0

Norm operator:

JQ DJ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ D : Q D

q

J. Arghavani et al. / Finite Elements in Analysis and Design 47 (2011) 166–174170

preferred because of their stability at larger time step sizes.Moreover, the present section provides some details about thestress update and the computation of the consistent tangentmatrix, which are the two points where the material model isdirectly connected to the finite element solution procedure.

We treat the nonlinear problem described in Section 3 as animplicit time-discrete deformation-driven problem. Accordingly,we subdivide the time interval of interest [0, t] in sub-incrementsand we solve the evolution problem over the generic interval[tn,tn +1] with tnþ14tn. To simplify the notation, we indicate withthe subscript n a quantity evaluated at time tn, and with nosubscript a quantity evaluated at time tn +1.Assuming to know thesolution and the deformation gradient Fn at time tn as well as thedeformation gradient F at time tn +1, the stress and the internalvariable have to be updated from the deformation history.

4.1. Time integration

Exponential-based integration schemes are frequently appliedto problems in plasticity and isotropic inelasticity [32,33], sincethe use of the exponential mapping allows to exactly conserve thevolume during an inelastic deformation. Thus, larger time stepsizes than any other first-order accurate integration scheme maybe used.

We now consider an evolution equation in the following form:

_Ct¼ _zA¼ _zA1Ct

ð57Þ

where A1 ¼ACt�1

with A being a symmetric tensor, while A1 is notsymmetric in general.

Applying the exponential mapping scheme to the evolutionequation (57), we obtain

Ct¼ expðDzA1ÞC

tn ð58Þ

In the equation above, we need to compute the exponential of anasymmetric tensor, which is a problematic computation due tothe impossibility of using a spectral decomposition method.2

2 The exponent of an asymmetric tensor is computed with a series expansion.

However, the exponent of a symmetric tensor can also be computed in closed form

by means of a spectral decomposition.

Accordingly, using the strategy initially proposed in [20–22,34]and also exploited in [35,24], we can find an alternativeexpression where the argument of the exponential operator is asymmetric tensor which can be computed through spectraldecomposition. Following [20], we can write

expðDzA1Þ ¼ 1þDzA1þDz2

2!A2

1þ � � �

¼ 1þDzACt�1

þDz2

2!ðACt�1

Þ2þ � � �

¼Ut 1þDzUt�1

AUt�1

þDz2

2!ðUt�1

AUt�1

Þ2þ � � �

!Ut�1

¼UtexpðDzUt�1

AUt�1

ÞUt�1

ð59Þ

We now right- and left-multiply (58) by Ct�1

n and Ct�1

, respec-tively, to obtain

Ct�1

n ¼ Ct�1

expðDzA1Þ ð60Þ

Then, substituting (59) into (60), we can write the integrationformula as

�Ct�1

n þUt�1

expðDzUt�1AUt�1

ÞUt�1

¼ 0 ð61Þ

which is the time-discrete form associated to (57) presented so farin the literature [20–22,34,35,24].

Comparing (54) and (57), we conclude that

A¼ 2Ut Q D

JQ DJUt

ð62Þ

We now substitute (62) into (61) and after right- and left-multiplying by Ut , we obtain the time-discrete form of theproposed evolution Eq. (54) as

�UtCt�1

n Utþexpð2DzWÞ ¼ 0 ð63Þ

where

W ¼Q D

JQ DJð64Þ

4.2. Solution algorithm

As usual in computational inelasticity problems, to solve thetime-discrete constitutive model we use an elastic predictor-inelastic corrector procedure. The algorithm consists of evaluatingan elastic trial state, in which the internal variable remainsconstant, and of verifying the admissibility of the trial function. Ifthe trial state is admissible, the step is elastic; otherwise, the stepis inelastic and the transformation internal variable has to beupdated through integration of the evolution equation.

In order to solve the inelastic step, we use another predictor-corrector scheme, that is, we assume g¼ 0 (i.e., we predict anunsaturated transformation strain case with JEtJreL) and wesolve the following system of nonlinear equations (we refer to thissystem as PT 1 system):

Rt¼�UtCt�1

n Utþexpð2DzWÞ ¼ 0

Rz ¼ JQ DJ�R¼ 0

(ð65Þ

If the solution is not admissible (i.e., JEtJPT14eL), we assume gZ0(i.e., we consider a saturated transformation case with JEtJ¼ eL)and we solve the following system of nonlinear equations (werefer to this system as PT2 system):

Rt¼�UtCt�1

n Utþexpð2DzWÞ ¼ 0

Rz ¼ JQ DJ�R¼ 0

Rg ¼ JEtJ�eL ¼ 0

8>><>>: ð66Þ

Table 3Solution algorithm for the proposed improved finite-strain constitutive model.

1. compute C ¼ FT F2. set UtTR

¼Utn to compute fTR and trial solution

3. if ðf TR o0Þ then

Accept trial solution, i.e., Ut¼UtTR

elseSolve PT1 system.

if ðJEtJPT1 oeLÞ then

Accept PT1 solution, i.e., Ut¼ ðUt

ÞPT1

else

Solve PT2 system and set Ut¼ ðUt

ÞPT2

end ifend if

J. Arghavani et al. / Finite Elements in Analysis and Design 47 (2011) 166–174 171

The solution of Eqs. (65) and (66) is, in general, approachedthrough a straightforward Newton-Raphson method.

Remark 2. The corresponding PT 1 and PT 2 systems for theoriginal form of the constitutive model are, respectively, asfollows [22,24]:

Rt¼�Ct�1

n þUt�1expðDzUt�1

AUt�1

ÞUt�1

¼ 0

Rz ¼ JYDJ�R¼ 0

(ð67Þ

Rt¼�Ct�1

n þUt�1

expðDzUt�1

AUt�1

ÞUt�1

¼ 0

Rz ¼ JYDJ�R¼ 0

Rg ¼ JEtJ�eL ¼ 0

8>><>>: ð68Þ

where

A¼ 2YD

JYDJCt

4.3. Consistent tangent matrix

We now linearize the nonlinear equations as it is required forthe iterative Newton-Raphson method. For brevity, we report theconstruction of the tangent matrix only for the case of thesaturated phase transformation, corresponding to (66). Lineariz-ing (66), we obtain

RtþRt

,Ut : dUtþRt

,Dz dDzþRt,g dg¼ 0

RzþRz,Ut : dUt

þRz,Dz dDzþRz

,g dg¼ 0

RgþRg,Ut : dUt

þRg,Dz dDzþRg

,g dg¼ 0

8>>><>>>:

ð69Þ

where subscripts following a comma indicate differentiation withrespect to that quantity. The derivatives appearing in the aboveequations are detailed in [36].

Utilizing the linearized form (69), after converting it to matrixform, a system of eight nonlinear scalar equations is obtained thatis solved for dUt , dDz and dg.

We now address the construction of the tangent tensorconsistent with the time-discrete constitutive model. The use ofa consistent tensor preserves the quadratic convergence of theNewton–Raphson method for the incremental solution of theglobal time-discrete problem, as it is done in the framework of afinite element scheme.

The consistent tangent is computed by linearizing the secondPiola–Kirchhof tensor, i.e.,

dS ¼D : dE¼D : 12dC ð70Þ

Recalling that S is a function of C and Ct , we can write

dS ¼@S

@C: dCþ

@S

@Ut : dUtð71Þ

Recasting (71) into a matrix form, we obtain

½dS� ¼@S

@C

� �½dC�þ

@S

@Ut

� �½dUt� ð72Þ

where we have used ½�� to denote the matrix form of the tensorialargument.

We now consider Eq. (66) as a function of C,Ut ,Dz and g, andthen the corresponding linearization gives

Rt,C : dCþRt

,Ut : dUtþRt

,Dz dDzþRt,g dg¼ 0

Rz,C : dCþRz

,Ut : dUtþRz

,Dz dDzþRz,g dg¼ 0

Rg,C : dCþRg

,Ut : dUtþRg

,Dz dDzþRg,g dg¼ 0

8>>><>>>:

ð73Þ

Recasting (73) into a matrix form, we obtain

dUt

dzdg

264

375¼�

Rt,Ut Rt

,Dz Rt,g

Rz,Ut Rz

,Dz Rz,g

Rg,Ut Rg

,Dz Rg,g

26664

37775�1

Rt,C

Rz,C

Rg,C

26664

37775½dC� ð74Þ

Now, using (74), we can compute the matrix ½B� such that

½dUt� ¼ ½B�½dC� ð75Þ

We then substitute (75) into (72) to obtain the consistent tangentmatrix as

½D� ¼ 2@S

@C

� �þ2

@S

@Ut

� �½B� ð76Þ

Finally, Table 3 presents the solution algorithm for the proposedimproved, fully symmetric, finite-strain constitutive model.

Remark 3. We highlight that in [22,24] and similarly in [20], Ct ,Dz and g are considered as unknowns in (68), while in this workwe assume Ut , Dz and g as unknowns in (66) as it has alreadybeen done in a similar works [21,34].

Remark 4. Up to now, all relations have been derived in acompletely general manner without specifying the form of theHelmholtz free energy ce, apart from the fact that it is an isotropicfunction of Ce. Despite the hyperelastic strain energy function ce

can take any well-known form in finite elasticity, for thenumerical examples discussed in Section 5 we use the commonlyused Saint–Venant Kirchhoff strain energy function:

ce¼

l2ðtr Ee

Þ2þm tr Ee2

ð77Þ

which yields

a1 ¼l4ðC : 1�3Þ�

1

2m, a2 ¼

1

2m, a3 ¼ 0 ð78Þ

where l and m are the Lam�e constants.

5. Boundary value problems

In this section, we solve four boundary value problems tovalidate the adopted fully symmetric model as well as theproposed integration algorithm and the solution procedure. Ahelical spring and a medical stent are simulated at two differenttemperatures to show the model capability of capturing bothpseudo-elasticity and shape-memory effect. We have recentlyused these boundary value problems [24] as benchmarks toinvestigate the efficiency and robustness of different integrationschemes. Although the results presented here are the samepresented in [24], in this work we are interested in comparing theCPU times for both the original asymmetric and the proposedsymmetric constitutive models. In all boundary value examples

J. Arghavani et al. / Finite Elements in Analysis and Design 47 (2011) 166–174172

we use the following material properties:

E¼ 51 700 MPa, n¼ 0:3, h¼ 750 MPa,

eL ¼ 7:5%, b¼ 5:6 MPa 3C�1

T0 ¼�25 3C, R¼ 140 MPa, Af ¼ 0 3C, Mf ¼�25 3C

The temperature in the pseudo-elastic simulations is set to 37 1C,while in the shape-memory effect simulations a temperature of�25 1C is adopted.

For all simulations, we use the commercial nonlinear finiteelement software ABAQUS/Standard, implementing the describedalgorithm within a user-defined subroutine UMAT. In order todemonstrate the 1D SMA behavior, we first simulate a cube withan edge length of 1 mm under an applied displacement on oneface while the opposite face is fixed. The applied displacement isincreased from zero to a maximum value of 0.1 mm andsubsequently decreased to zero and increased in the oppositedirection to a value of 0.1 mm and finally decreased back to zero.Fig. 1 shows the SMA behavior at two different temperatures.

0

200

400

600

800

1000

1200

1400

1600

Forc

e [N

]T = 37 °C

5.1. Helical spring: pseudo-elastic test

A helical spring (with a wire diameter of 4 mm, a springexternal diameter of 24 mm, a pitch size of 12 mm and with twocoils and an initial length of 28 mm) is simulated using 9453quadratic tetrahedron (C3D10) elements and 15 764 nodes. Anaxial force of 1525 N is applied to one end while the other end isfully clamped. The force is increased from zero to its maximumvalue and unloaded back to zero. Fig. 2 shows the spring initialgeometry, the adopted mesh and the deformed shape under themaximum force. After unloading, the spring recovers the originalshape as it is expected in the pseudo-elastic regime. Fig. 3 (top)shows the force-displacement diagram. It is observed that thespring shape is fully recovered after load removal.

−11−10 −8 −6 −4 −2 0 2 4 6 8 10−2000

−1000

0

1000

2000

3000

True (Logarithmic) Strain [%]

Cau

chy

Stre

ss [

MPa

] T = 37° CT = −25° C

Fig. 1. 1D SMA behavior: pseudo-elasticity at T¼37 1C and shape-memory effect

at T¼�25 1C.

Fig. 2. Pseudo-elastic spring: comparison of ini

5.2. Helical spring: shape-memory effect test

We also simulate the same spring of Section 5.1 in the case ofshape-memory effect. An axial force of 427 N is applied atT¼�25 1C (Fig. 4, top) and, after unloading, the spring does notrecover its initial shape (Fig. 4, bottom). After heating, up to atemperature of 10 1 C, the spring recovers its original shape. Fig. 3(right) shows the force-displacement-temperature behavior.According to Fig. 3 (bottom), heating the spring leads to fullrecovery of the original shape at T¼Af (0 1C) and subsequentheating does not change its shape anymore.

5.3. Crimping of a medical stent: pseudo-elastic test

In this example, the crimping of a pseudo-elastic medical stentis simulated. To this end a medical stent with 0.216 mm thicknessand an initial outer diameter of 6.3 mm (Fig. 5, top-left) is crimpedto an outer diameter of 1.5 mm. Utilizing the ABAQUS/Standardcontact module, the contact between the catheter and the stent isconsidered in the simulation. A radial displacement is applied to

tial geometry and deformed configuration.

0 10 20 30 40 50 60 70 80

Displacement [mm]

020

4060

80−25−20

−15−10−50510

0

100

200

300

400

500

Temperature [°C]

Displacement [mm]

Forc

e [N

]

Fig. 3. Force-displacement diagram for the SMA spring: pseudo-elasticity (top)

and shape-memory effect (bottom).

Fig. 4. Shape-memory effect in the simulated spring: deformed shape under maximum load (top) and after unloading (bottom).

Fig. 5. Stent crimping: initial geometry (top-left), crimped shape in PE case (top-right), crimped shape in SME case (bottom-left) and deformed shape in SME case after

uncrimping (bottom-right).

J. Arghavani et al. / Finite Elements in Analysis and Design 47 (2011) 166–174 173

the catheter and is then released to reach the initial diameter. Inthis process, the stent recovers its original shape after unloading.Fig. 5 (top-right) shows the stent deformed shape when crimped.

5.4. Crimping of a medical stent: shape-memory effect test

In this example, the same stent is crimped at a temperature of�25 1C as it is shown in Fig. 5 (bottom-left). Due to the lowtemperature, while the catheter is expanded, the stent remains ina deformed state as shown in Fig. 5 (bottom-right). The initialshape is, however, recovered after heating.

5.5. Comparison of CPU times

We now compare the CPU times for three cases, i.e., originalunsymmetric model in which Ct is considered as unknown

[22,24], the original unsymmetric model in which Ut is con-sidered as unknown and the proposed symmetric model in whichUt is considered as unknown. We denote them as original-C,original-U and improved, respectively, in the following.

We remark that we normalize all CPU times with respect to thosefor the case original-C [22,24]. Moreover, the models have beenprogrammed in a consistent manner, so that reliable comparisonsconcerning the speed of computation could be obtained.

The simulation CPU times are reported in Table 4. From suchresults we may conclude that using Ut instead of Ct , decreases theCPU time of approximately one third compared with that adoptedin [22,24].

Moreover, comparing original-U and improved CPU times, weobserve a slightly increased efficiency for the proposed symmetricmodel.

We finally, conclude that using the proposed symmetricconstitutive model and solution algorithm can improve the

Table 4Normalized CPU time comparison.

Spring Spring Stent Stent

(PE) (SME) (PE) (SME)

original-C 1.00 1.00 1.00 1.00

original-U 0.68 0.72 0.65 0.72

improved 0.64 0.69 0.63 0.73

J. Arghavani et al. / Finite Elements in Analysis and Design 47 (2011) 166–174174

numerical efficiency in the order of one third when comparedwith the previously adopted ones in [22,24].

6. Conclusions

In this paper, we investigate a 3D finite-strain constitutivemodel which is fully expressed in terms of symmetric tensors. Thederivation is based on a multiplicative decomposition of thedeformation gradient into elastic and transformation parts andsatisfying the second law of thermodynamics in Clausius–Duheminequality form. Based on an exponential mapping, we propose atime-discrete form of the evolution equation which conserves thevolume during inelastic deformations. Implementing the pro-posed model within a user-defined subroutine (UMAT) in thecommercial nonlinear finite element software ABAQUS, we solvedifferent boundary value problems. Comparing the CPU times, weshow the efficiency of the proposed constitutive model as well asthe solution algorithm, implying a solution time reduction ofaround one third (33%) with respect to the original asymmetricmodel. Moreover, we show that the improved efficiency is mostlydue to the simplification in computing fourth-order tensorsappearing in the linearization procedure.

Acknowledgements

J. Arghavani has been partially supported by Iranian Ministryof Science, Research and Technology, and partially by Diparti-mento di Meccanica Strutturale, Universit�a degli Studi di Pavia.

F. Auricchio and A. Reali have been partially supported bythe Cariplo Foundation through the Project no. 2009.2822; by theMinistero dell’Istruzione, dell’Universit�a e della Ricerca throughthe Project no. 2008MRKXLX; and by the European Research Councilthrough the Starting Independent Research Grant ‘‘BioSMA: Mathe-matics for Shape Memory Technologies in Biomechanics’’.

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