Mechanics of Materials 73 (2014) 1–10

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Mechanics of Materials

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A finite deformation constitutive model for shape memorypolymers based on Hencky strain

http://dx.doi.org/10.1016/j.mechmat.2013.11.0110167-6636/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Department of Mechanical Engineering,Sharif University of Technology, Tehran, Iran. Tel.: +98 21 6616 5546; fax:+98 21 6600 0021.

E-mail address: naghdabd@sharif.edu (R. Naghdabadi).

M. Baghani a, J. Arghavani b, R. Naghdabadi b,c,⇑a School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iranb Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iranc Institute for Nano-Science and Technology, Sharif University of Technology, Tehran, Iran

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

Article history:Received 12 March 2013Received in revised form 6 October 2013Available online 13 March 2014

Keywords:Shape memory polymerFinite deformationHencky strain tensorContinuum thermodynamicsFinite element

In many engineering applications, shape memory polymers (SMPs) usually undergo arbi-trary thermomechanical loadings at finite deformation. Thus, development of 3D constitu-tive models for SMPs within the finite deformation regime has attracted a great deal ofinterest. In this paper, based on the classical framework of thermodynamics of irreversibleprocesses, employing the logarithmic (or Hencky) strain as a more physical measure ofstrain, a 3D large-strain macromechanical model is presented. In the constitutive modeldevelopment, we adopt a multiplicative decomposition of the deformation gradient intoelastic and stored parts. In addition, employing the averaging scheme, the logarithmic elas-tic strain tensor is decomposed into the rubbery and glassy parts. The evolution equationsfor internal variables are introduced for both cooling and heating processes. The time-dis-crete form of the proposed model in the implicit form is also presented. Comparing the pre-dicted results with experimental data reported in the literature, the model is validated.Finally, using the finite element method, two boundary value problems e.g., a 3D beamand a medical stent made of SMPs are numerically simulated.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Shape memory materials opened unexplored horizonsin the materials science and engineering, giving access tonumerous unconventional functions in different materialsclass: metals, ceramics and polymers (Schmidt et al.,2011) These materials are able to return from a temporaryshape to their permanent shape. The shape memory effect(SME) is typically induced by external stimuli such as heat,electricity or magnetism (Lendlein and Langer, 2002;Baghani et al., 2012a). This gives shape memory materialsan increasing potential for application in sensors,

actuators, smart devices (e.g., medical stents) and mediarecorders (Reese et al., 2010). The mechanisms controllingthe thermally-induced SME in shape memory polymers arevery different from those operating in shape memory al-loys or ceramics, resulting in differences in the materialbehavior (Diani et al., 2006).

Many attempts have been made to experimentallycharacterize the behavior of SMPs (see e.g., Liu et al.,2004, 2006; Atli et al., 2009; Kim et al., 2010; Volk et al.,2010b,a; Prima et al., 2010a,b,c among others), but a con-siderable part of the research has also been devoted tothe description of the SMPs performance through thedevelopment of constitutive models.

Most SMPs constitutive models have been developedfor thermally-activated ones. They usually tend to followeither a phase transition approach (see Liu et al., 2006; Bar-ot et al., 2008; Chen and Lagoudas, 2008a; Qi et al., 2008;

Fig. 1. Stress–strain-temperature diagram showing the thermomechan-ical behavior of a pre-tensioned SMP under different strain or stressrecovery conditions.

2 M. Baghani et al. / Mechanics of Materials 73 (2014) 1–10

Kim et al., 2010; Reese et al., 2010; Xu and Li, 2010;Baghani et al., 2012b among others) or a thermoviscoelas-tic approach (see Diani et al., 2006; Nguyen et al., 2008,2010; Westbrook et al., 2011; Ghosh and Srinivasa, 2011,2013 among others). The phase transition approach pos-sesses a distinct advantage in that the concepts can beapplied to an extensive variety of SMPs with different tran-sition mechanisms (Nguyen et al., 2010).

Based on the phase transition concept, Liu et al. (2006)proposed a macroscopic constitutive model for SMPs in thesmall strain regime. In this model, the strain is decomposedinto thermal, elastic, and a stored part. The stored strain is de-fined to capture the strain storage/release phenomenon.However, the model does not present the mathematical evo-lution equations upon the heating process (Chen and Lagou-das, 2008a,b). Based on this model, Chen and Lagoudas(2008a,b) proposed a constitutive model to capture the char-acteristics of SMPs behavior in large strain loadings. Ghoshand Srinivasa (2011) proposed a one-dimensional constitu-tive model for SMPs in the range of small strains. Their modelwas able to simulate the response of the material duringheating and cooling cycles and the sensitive dependence ofthe response on thermal expansion. Recently, based on theframework of continuum thermodynamics, Baghani et al.(2012c) have proposed a phenomenological model, in whichevolution laws have been introduced for stored strain notonly in the cooling process but also in the heating process un-der 3D loadings. In another work, Baghani et al. (2012b) haveextended the model to the time-dependent regime. Theyhave also added a hard segment in the constitutive modelto predict the behavior of the SMP-based foams.

In recent years, much attention has been devoted to thedevelopment of constitutive models for large deformationsof solids. The logarithmic strain (natural or Hencky strain),introduced by Hencky (1928), is a favored measure ofstrain due to its remarkable properties at finite deforma-tions in solid mechanics. An important feature is thatamong all finite strain measures, only the spherical andthe deviatoric parts of the Hencky strain can additivelyseparate the volumetric and isochoric deformations fromthe total deformation (Xiao et al., 2004).

Furthermore, the Hencky strain has certain intrinsicproperties that shows its important position among allstrain measures. For instance, the Eulerian logarithmicstrain is the unique strain measure that its corotational rate(associated with the so-called logarithmic spin) is the strainrate tensor (Xiao et al., 2006). In other words, the strain ratetensor, d, is the corotational rate of the Hencky strain tensorassociated with the logarithmic spin tensor. This result hasbeen presented by Reinhardt and Dubey (1995) as D-rateand by Xiao et al. (1997) as log-rate. The work-conjugatepair of Eulerian Hencky strain and Cauchy stress are of mostinterest in constitutive modeling in large strain regime (Xiaoet al., 2006). Due to these remarkable properties, Henckystrain has been used in constitutive modeling of solids bymany researchers (see, e.g., Naghdabadi et al., 2005; Linand Schomburg, 2003). There are still more motivations todevelop a Hencky-strain based constitutive model. Forexample, for a Hookean-type constitutive relation, onlythe Hencky-based one is useful at moderately large elasticstretches (Anand, 1979, 1986).

Large strain constitutive models are normally devel-oped based on appropriate small strain ones (Arghavaniet al., 2010, 2011). The recently proposed 3D model byBaghani et al. (2012c), based on the classical frameworkof continuum thermodynamics, is a good candidate to beextended to large strains. This model is interesting for itscapability to capture SME, beside its thermodynamicalconsistency. Starting from this basis, the development ofa Hencky based finite deformation model, which is alsoable to take into account large deformation effects, is themain goal of the present work.

The current paper is presented in the following sections.Section 2 describes the behavior of an SMP in a full cyclefor different (strain or stress) recovery processes. Section3 presents the kinematic formulations. In Section 4, wedevelop a thermodynamically-consistent constitutivemodel for SMPs in the large deformation regime. Time-dis-crete form of the introduced model is presented within thefinite element framework in Section 7. We then, in Section6, apply the proposed constitutive model to some exam-ples and investigate the ability of the model to reproducethe features of SMPs. Using the proposed finite elementmodel, two boundary value problems, i.e., a 3D beam anda medical stent are simulated. Finally, we present asummary and draw conclusions in Section 7.

2. Shape memory effect in a thermomechanical SMPcycle

In this section, we describe SME in a stress–strain-tem-perature diagram as depicted in Fig. 1. The thermome-chanical cycle starts at a strain- and stress-free state ata high temperature Th (point �a , permanent shape). Atthis state, SMP undergoes a mechanical loading andexhibits a rubbery behavior up to point �b . At point �b ,strain is held fixed and the temperature is decreased untilthe rubbery polymer gradually converts to a glassy poly-mer at a low temperature Tl (point�c , temporary shape).In fact, around the glassy temperature Tg , SMP possesses acombination of rubbery and glassy behaviors. In this step,the material is unloaded. According to the very high stiff-ness of the glassy phase compared to the rubbery phase,after unloading, strains vary slightly (point �d). Finally,the temperature is increased up to Th. It is observed that

M. Baghani et al. / Mechanics of Materials 73 (2014) 1–10 3

the strain relaxes and the permanent shape is recovered(point �a ). This cycle is called a stress-free strain recoveryin SMP applications. Commonly, other kinds of recoveriesmay occur. If at point �d, the strain is kept fixed and thetemperature is increased, the fixed-strain stress recovery(point �e ) happens (shown in Fig. 1 with a dotted-line).

3. Kinematics

In accordance with the polar decomposition theorem,the deformation gradient F with J ¼ detðFÞ > 0 may bedecomposed as:

F ¼ RU ¼ VR ð1Þ

where U and V are symmetric positive definite right andleft stretch tensors, respectively. Also, R is a proper orthog-onal tensor. Moreover, the right and left Cauchy-Greendeformation tensors are, respectively, defined as:

C ¼ FT F ¼ U2; b ¼ FFT ¼ V2 ð2Þ

The material (Lagrangian) and spatial (Eulerian) Henckystrain tensors H and h are defined, respectively, as Lubarda(2001):

H ¼ log U ¼ 12

log C; h ¼ log V ¼ 12

log b ð3Þ

The velocity gradient tensor l is also defined in the follow-ing form:

l ¼ _FF�1 ð4Þ

The strain rate tensor d and the vorticity tensor w are sym-metric and skew symmetric parts of l, respectively, asfollow1:

d ¼ symðlÞ; w ¼ skewðlÞ ð5Þ

Taking the time derivative of relation C ¼ FT F yields:

_C ¼ 2FT dF ð6Þ

Corotational rate of an Eulerian tensor A associated withthe rotating frame having spin X is defined as Naghdabadiet al. (2012):

A�¼ _A�XAþAX ð7Þ

where _A is the material time derivative of A in the fixedframe and A

�is the corotational rate of A. Double contract-

ing (7) with an arbitrary second-order symmetric tensor B

(we also assume A to be symmetric) and after some math-ematical manipulations, we obtain2:

A�

: B ¼ _A : Bþ 2X : ðABÞ ð8Þ

If A and B are coaxial3 (and symmetric), AB is also sym-metric. Thus, we arrive at the following identity for any arbi-trary spin tensor X:

1 symðAÞ ¼ ð1=2ÞðAþ AT Þ and skewðAÞ ¼ ð1=2ÞðA� AT Þ compute thesymmetric and skew symmetric parts of an arbitrary tensor A, respectively.

2 Double contraction is represented by :, defined as A : B ¼ AijBij .3 Tensors A and B are coaxial if AB ¼ BA.

_A : B ¼ A�

: B ð9Þ

The corotational rate of the Eulerian Hencky strain h asso-ciated with the so-called logarithmic spin Xlog is the sameas the strain rate tensor d, and h is the only strain with thisproperty. In view of (7), we obtain (Naghdabadi et al.,2012; Teeriaho, 2013):

h�

log ¼ _h�Xloghþ hXlog ¼ d ð10Þ

Finally, the second Piola–Kirchhoff stress tensor S isobtained from the Cauchy stress as:

S ¼ JF�1rF�T ð11Þ

4. Constitutive model development

Employing a multiplicative decomposition, we first sep-arate the thermal expansion effects. The total thermome-chanical deformation gradient, F, is defined as F ¼ FmFth

where Fm is the mechanical part and Fth represents thethermal deformation gradient and is defined as:

Fth ¼ Fth1 ¼ 1þZ T

Th

aeff dT

!1 ð12Þ

where aeff denotes the effective thermal expansion coeffi-cient. The second-order identity tensor is also representedby 1.

Considering Eq. (12), total velocity gradient tensor isobtained by d ¼ dm þ dth.

In this study, we separate the material domain intoglassy and rubbery phases as previously performed in sim-ilar works (e.g., Liu et al., 2006; Baghani et al., 2012c,b).Following a well-established approach adopted in plastic-ity (Lubarda, 2001; Haupt, 2002), we assume a local multi-plicative decomposition of the mechanical deformationgradient in the rubbery phase, Fr as well as the glassyphase, Fg , into elastic parts Fer and Feg , defined with respectto intermediate configurations, and inelastic stored part Fs,defined with respect to the reference configuration.Accordingly:

Fm ¼ FeFs; Fr ¼ FerFs; Fg ¼ FegFs ð13Þ

Assuming finite deformations, we employ the averagingscheme (on the current volume) and decompose the elasticHencky strain tensor, he, into two parts: elastic Henckystrain tensor in the rubbery phase, her , and elastic Henckystrain tensor in the glassy phase, heg .

he ¼ urher þugheg;

ur þug ¼ 10 6 ur ;ug

6 1

�ð14Þ

where superscripts r and g stand for the rubbery and glassyphases, respectively. Volume fractions of the rubbery andglassy phases are represented by ur and ug , respectively.Such an averaging scheme has previously been adoptedfor constitutive modeling of multi-phase materials for dif-ferent quantities (for example, for F by Chen and Lagoudas,2008a,b; Sengupta and Papadopoulos, 2011, for d by Ahziet al., 2003; Makradi et al., 2005; Wu et al., 2005, for E

4 M. Baghani et al. / Mechanics of Materials 73 (2014) 1–10

(Green strain) by Baghani et al., 2013 or for stretch, k, byKim et al., 2010).

In view of (13)1, the strain rate tensor could be rewrit-ten as:

dm ¼ sym _FmFm�1� �

¼ de þ sym FelsFe�1� �

ð15Þ

where ls ¼ _FsFs�1. Taking derivative of (14), we obtain:

_he ¼ ur _her þug _heg þ _ug heg � her� �ð16Þ

To satisfy the principle of material objectivity, the Helm-holtz free energies, in the both phases, should depend onFer and Feg only via the elastic Hencky strain tensors her

and heg , respectively:

W Fm; Tð Þ ¼ W her;heg

; T� �

ð17Þ

Using the averaging scheme, the Helmholtz free energyfunction is introduced as:

W her;heg

;ug ; T� �

¼ urWr her� �þugWg heg� �

þWT Tð Þ ð18Þ

where Wr and Wg are hyperelastic strain energy functionsof the rubbery and glassy phases, respectively, and WT isthe thermal energy. It is also assumed that Wr and Wg areisotropic functions of her and heg , respectively.

s : d� _Wþ g _T� �

� 0 ð19Þ

in which g represents the entropy. In this section it is as-sumed that the Kirchhoff stress, s, and elastic Henckystrain, he, are coaxial. Thus, we obtain:

s : de ¼ s : _he ð20Þ

Substituting (15), (20) and (16) in (19) gives:

s : dth þurdother þug _heg þ _ug heg � her� �þ symðFelsFe�1 Þ

h i� @W

@her : _her þ @W@heg : _heg þ @W

@T_T þ g _T

� �P 0

ð21Þ

Using relation (18), we have:

ur s� @Wr

@her

� �: _her þug s� @Wg

@heg

� �: _hegþ

s : dth þ _ug heg � her� �þ symðFelsFe�1 Þ

h i� @W

@T_T þ g _T

� �P 0

ð22Þ

Following standard arguments, we finally may write:

s ¼ @Wr

@her ¼@Wg

@heg ð23Þ

From (23), it is observed that Kirchhoff stress and elasticstrains her and heg are coaxial. Thus, as the storage/releasemechanism is considered fully thermal-driven, entropy, g,is easily obtained.

Motivated from the small strain model (Baghani et al.,2012b,c), it is assumed that the Hencky stored strain fol-lows the evolution Eq. (24), in the intermediate configura-tion, during the cooling and heating processes:

hsD

¼ ds ¼_ugher; _T < 0_ug

ug hs; _T > 0

0; _T ¼ 0

8>><>>: ð24Þ

where hs ¼ log Vsð Þ and her ¼ log Uerð Þ. The appropriateobjective corotational rate is denoted by D, alsoˆ meansthat the quantity is defined with respect to the intermedi-ate configuration. In fact, in the cooling process, evolutionEq. (24)1 allows the stored strain to take the value of theelastic strain in the rubbery phase, her . On the other hand,during the heating process, relation (24)2 makes it possiblefor the stored strain to be recovered in a trend similar toug . It is noted that we employ a prescribed evolution equa-tion for ug . This equation is derived using the uncon-strained strain recovery of the material as a function oftemperature. The evolution Eq. (24) can be transferred tothe initial configuration in the following form:

_Cs ¼ 2FsT dsFs ¼_ug log CrCs�1

� �Cs; _T < 0

_ug

ug Cs log Cs� �; _T > 0

8<: ð25Þ

To develop a fully-symmetric constitutive model with rea-sonable computational cost, we should avoid computationof tensor log CrCs�1

� �which is generally unsymmetric. For

this purpose, we can use the following identity (Naghdaba-di et al., 2012):

log CrCs�1� �

Cs ¼ Us log Us�1CrUs�1� �

Us ð26Þ

which contains only symmetric tensors. As observed from(24), the strain storage/release is a fully thermal-drivenphenomenon.

Up to now, the relations have been derived in a com-pletely general manner without specifying the form ofthe Helmholtz free energies Wg and Wr , apart from the factthat they are isotropic functions of heg and her , respec-tively. The hyperelastic strain energy functions Wg andWr can take any well-known form in finite elasticity. Forthe numerical examples to be discussed in the next sec-tions we employ the commonly-used Saint–Venant Kirch-hoff strain energy function:

Wr her� �¼ 1

2her

: Kr : her; Wg heg� �

¼ 12

heg: Kg : heg

ð27Þ

where Kr and Kg denote the fourth order elasticity tensorsin the rubbery and glassy phases, respectively. With the aidof Eqs. (23) and (27), the stress is calculated as:

s ¼ Kr : her ¼ Kg : heg ð28Þ

Recasting relation (28) in a matrix form leads to:

Kr½ � her� �¼ Kg½ � heg� �

) heg� �¼ Kg½ ��1

Kr½ � her� �A½ � ¼ Kg½ ��1

Kr½ �; heg ¼ A : herð29Þ

Substitution of (29) in (14), we may write:

he ¼ urher þugheg ¼ urher þugA : her ¼ B : her;

B ¼ urIþugA ð30Þ

Table 1Material parameters adopted from experiments reported by Liu et al.

M. Baghani et al. / Mechanics of Materials 73 (2014) 1–10 5

where I denotes the fourth order symmetric identitytensor.

(2006).

Materialparameters

Values Units

Er ; Eg 8.8, 813 [MPa]mr ; mg 0.48, 0.35 [–]Tl; Tg ; Th 273, 343, 358 [K]

Fth expð7:1� 10�7T2 � 3:16� 10�4T þ 2:213� 10�2Þ [–]

ug1= 1þ 2:76� 10�5 Th � Tð Þ4� �

[–]

5. Time-discrete form of the SMP constitutive model

In this section, we investigate the numerical solution forthe constitutive model presented in Section 4, with the fi-nal goal of applying it in a finite element program. Themain task is to apply an appropriate numerical time-inte-gration scheme to the evolution equations. In general, im-plicit schemes are preferred due to their stability at largertime step sizes. We subdivide the time interval of interest½0; t� into sub-increments and solve the evolution problemover the generic interval ½tn; tnþ1� with tnþ1 > tn. We denotewith the subscript n a quantity evaluated at time tn, andwith no subscript a quantity evaluated at time tnþ1. Wealso show the increment of time by Dt. Having the solutionas well as Cn at time tn and C at time tnþ1, the stress and theinternal variables are updated from the deformationhistory.

Discretization of Eqs. (25)1 and (30), during the coolingprocess yields:

Cs ¼ Csn þ Dug log CrCs�1

� �Cs

log FmCs�1FmT� �

¼ B : log FerFer T� �

8><>: ð31Þ

where Dug ¼ ug �ugn. Also, it is assumed that the rotation in

each phase is equal to the total rotation, Rr ¼ Rg ¼ R. It couldbe shown that log RUmCs�1UmRT

� �¼ R log UmCs�1Um

� �RT

and also log RUrCs�1UrRT� �

¼ R log UrCs�1Ur� �

RT . Further-more, for any fourth order isotropic tensor such as B we haveBijkl ¼ RpiRqjRrkRslBpqrs, in which R is an orthogonal arbitrarytensor (de Souza Neto et al., 2011). As a result, Eq. (31) canbe recast in the following form:

Cs ¼ Csn þ DugUs log Us�1CrUs�1

� �Us

log UmCs�1Um� �

¼ B : log UrCs�1Ur� �

8><>: ð32Þ

On the other hand, discretization of Eqs. (24)2 and (32)2

during the heating process yields:

0.8 0.85 0.9 0.95 1−1

0

1

2

3

4

T* [ K/K ]

Stre

ss [

MPa

]

ε0 = 9.1%

ε0 = −9.1%

ε0 = 0%

−

−Stre

ss [

MPa

]

Fig. 2. Left: Stress–temperature response in the cooling process in different pre-denote large strain model, small strain model and experimental data, respectdifferent pre-strains e0’s of 9:1%; 0% and �9:1% in fixed-strain stress recovery. A

Cs ¼ Csn þ

Dug

ug Cs log Cs� �log UmCs�1Um

� �¼ B : log UrCs�1Ur

� �8<: ð33Þ

Both Eqs. (32) and (33) are nonlinear in terms of Us and Ur

and should be numerically solved. To this end, theNewton–Raphson method is employed. The numericalsolution of the proposed model is presented in detail inAppendix A.

6. Numerical results and model verification

The present section deals with some loading paths aswell as boundary value problems. In particular, Example1 presents the results for uniaxial tests to show the modelcapability in reproducing the shape memory effect, com-paring them with experimental data available in the liter-ature. In Example 2, to show the difference between smalland large strain theory, the simple shear test is simulated.In Examples 3 and 4, a double-clamped 3D SMP beam andan SMP medical stent which undergo large strains are sim-ulated to show the validity of the numerical solution pro-cedure adopted in this work for the proposed constitutivemodel. For the simulation, we use the nonlinear finite ele-ment software ABAQUS/Standard, implementing thedescribed algorithm within a user-defined subroutineUMAT.

The aim for simulating these 3D boundary value prob-lems (Examples 3 and 4 is to show the capability of the pro-

0.8 0.85 0.9 0.95 1−2

1.5

−1

0.5

0

0.5

1

1.5

2

T* [ K/K ]

ε0 = 9.1%

ε0 = 0%

ε0 = −9.1%

strains e0’s of �9:1%; 0% and 9:1%. Solid lines, dashed lines and symbolsively. Right: Stress–temperature diagram during the heating process atll experiments are reported from Liu et al. (2006).

0.8 0.85 0.9 0.95 1 1.05−2

0

2

4

6

8

10

12

Cau

chy

stre

ss [

MP

a ]

T*

Large strainSmall strain

0.80.9

1

0

0.5

1−2

0

2

4

6

8

10

12

T*Displacement

Cau

chy

stre

ss [

MPa

]

Small strainLarge strain

Fig. 3. Stress–temperature-displacement diagrams in stress-free strain recovery with 100% pre-deformed displacement.

1

κ

0 0.1 0.5 0.6 1

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

t / tf

Stor

ed s

trai

nsHs

11

Hs22

Hs33

Hs12

εs12

Fig. 4. Loading of the element in the simple shear test. The changes of (Lagrangian) stored strain components, Hs . es12 denotes the small shear strain.

3

4

σ11

σ

6 M. Baghani et al. / Mechanics of Materials 73 (2014) 1–10

posed model as well as its time-integration scheme in anal-ysis of SMP structures. In all examples, we use the materialparameters reported in Table 1. It is noted that in Examples2 to 4 the thermal expansion has been neglected.

0 0.1 0.5 0.6 1 −2

−1

0

1

2

t / tf

Stre

sses

[ M

Pa ]

22

σ33

σ12

, Large strain

σ12

, Small strain

Fig. 5. Cauchy stress components in different steps of the stress-free strainrecovery cycle in the simple shear test with j ¼ 1.

6.1. Example 1: Reproduction of ‘‘stress-free strain recovery’’

To show the ability of the model to produce SME instress-free strain recovery, we use the experiments doneby Liu et al. (2006) and adopt the material parametersreported in Table 1. As shown in Fig. 2, the model is capa-ble of reproducing the SMP stress response in the coolingprocess. Results of the present work are compared to thoseof the small strain model and also the experimental data.As observed in Fig. 2, results of the present work are ingood correspondence to the experiments.

Furthermore, according to Fig. 3, the proposed modeland small strain model results in a fixed-strain stress recov-ery are compared. As observed, the small strain model pre-dicts larger values for stress in different parts of the cycle.

6.1.1. Example 2: Simple shear test in ‘‘stress-free strainrecovery’’ process

In this example, to show the difference between thelarge and small strain models, we simulate the simpleshear test (Fig. 4). In this problem, deformation gradient

tensor in the loading step (to reach the maximum shearvalue of j) obeys the following relation:

F ¼1 j 00 1 00 0 1

264

375 ð34Þ

It is noted that the thermal expansion effects are ignored inthis example to only investigate the effect of phase trans-

0 0.3226 0.5 0.6452 1 −0.2

0

0.2

0.4

0.6

0.8

1

1.2

t / tf

|P| /

|Pm

ax|,

|U| /

|Um

ax|,

(T

− T

l) / (

Th −

Tl)

(T − Tl) / (T

h − T

l)

|U| / |Umax

|

|P| / |Pmax

|

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(T − Tl) / (T

h − T

l)

|U| /

|Um

ax|

Fig. 6. Temperature and displacement in node A and pressure in element B versus time. jUj; jUmax j; jPj and jPmax j denote the displacement magnitude,maximum displacement magnitude, pressure and maximum pressure, respectively (left). Displacement in node A versus temperature (right).

Fig. 7. Initial and temporary configurations of the SMP beam.

Fig. 8. Initial and temporary shape of the stent beside the logarithmic maximum principal strain distribution in the temporary state.

M. Baghani et al. / Mechanics of Materials 73 (2014) 1–10 7

formation and storage/release phenomenon. In 0:1 of thetotal time, j at Th is linearly increased up to j ¼ 1. Then,in time between 0:1 and 0:5, the material is cooled downto Tl. It is noteworthy that during the cooling process,the deformation gradient is kept fixed. Now, the systemis unloaded and the temporary shape of the material isobtained. Finally, in time between 0:6 and 1:0, the materialis freely heated up to Th. It is observed that the materialrecovers its initial shape upon heating. The changes of(Lagrangian) stored strain components, Hs, is illustratedin Fig. 4. Results are also compared to those of the smallstrain model. In Fig. 5, Cauchy stress components aredepicted in different steps of the cycle. Results are also

compared to those of the small strain model. As shownin Fig. 5, in the cooling process, due to the existence ofthe external constraints (to maintain F fixed during thephase transformation), stress values increase considerably.In addition, the small strain model predicts larger valuesfor the shear stress in the SMP cycle.

6.1.2. Example 3: double-clamped 3D SMP beam underdistributed pressure: SME test

To show the capability of the model to predict the SME,a double-clamped 3D SMP beam under distributed pres-sure on the upper surface is simulated. To this end, a beamwith 100 mm length, 10 mm width and 10 mm height is

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

t / tf

|U| /

|Um

ax|,

(T

− T

l) / (

Th −

Tl)

(T − Tl) / (T

h − T

l)

|U| / |Umax

|

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(T − Tl) / (T

h − T

l)

|U| /

|Um

ax|

Fig. 9. Temperature and displacement in node A versus time (left). Displacement variations in node A versus temperature (right).

8 M. Baghani et al. / Mechanics of Materials 73 (2014) 1–10

modeled. The applied pressure on the upper surface at Th iscontrolled linearly to obtain its maximum value att� ¼ 0:3226 (shown in Fig. 6). Keeping the pressure fixedon the upper surface, the beam is cooled down to Tl untilt� ¼ 0:6452. Regarding the glassy behavior of the materialat lower temperatures, after unloading (removing the pres-sure), the structure keeps its temporary shape(t� ¼ 0:6774). During the ‘‘stress-free strain recovery’’ pro-cess, after increasing the temperature up to Th, the beamremember its initial configuration (Fig. 7).

6.1.3. Example 4: medical SMP stentIn this example, an SMP stent is simulated. Initial config-

uration of the stent has been shown in Fig. 8. Due to thesymmetry of the problem we considered only one-fourthof the stent geometry in our simulation. The stent has alength of 20 mm, an inner radius of Ri ¼ 4:8 mm and anouter radius of Ro ¼ 5 mm. The diameter of the holes isd ¼ 0:5 mm. At T ¼ Th, a displacement uy ¼ �5 mm alongline AB is applied to the stent while the displacementuz ¼ 5 mm is applied on the face C simultaneously. Then,the stent is cooled down to T ¼ Tl. Consequently, the struc-ture is abruptly unloaded. The temporary shape of thestent is shown in Fig. 8. As illustrated, in spite of removingthe external loads, the stent memorizes its temporaryshape. This step is followed by heating the stent up toT ¼ Th. Moreover, the temperature history and displace-ment of node A is depicted in Fig. 9.

7. Summary & conclusions

The increasing number of shape memory polymer(SMP) applications motivates the development of constitu-tive models to better predict these material complexbehaviors. Shape memory effect is the most importantfeature in SMPs, responsible for the specific response ofSMPs under thermomechanical loadings. Furthermore,undergoing large rotations and strains by SMP devices isanother important issue in this material behavior model-ing. To this end, in this study, a 3D finite strain constitutivemodel was developed by extending a recently proposedsmall strain model. The material was considered as a mix-ture of rubbery and glassy phases. It was also assumed thatthe rubbery and glassy phases were able to be transformed

to each other through external stimuli of heat. A multipli-cative decomposition of the deformation gradient (in eachphase) into elastic and inelastic stored parts, withemploying a mixture rule relating the Hencky strain tensorto the rubbery and glassy parts, is used within the frame-work of continuum thermodynamics with internalvariables.

Implementing the proposed model within a user-de-fined subroutine (UMAT) in the nonlinear finite elementsoftware ABAQUS/Standard, we solved uniaxial loadingsand simulated a 3D beam made of SMPs and showedthe capability of the proposed constitutive model. In fact,the model is a useful and appropriate computational toolfor design, analysis and optimization of structures madeof SMPs under complicated loading conditions.

Acknowledgment

The first author is grateful for the research support ofthe Iran National Science Foundation (INSF).

Appendix A. Time-discrete form of the proposed model

Time integration scheme for cooling process: Relations(32) can be recast in the indicial form as:

Rij ¼ L1ij � BijklL

2kl ¼ 0;

L1 ¼ log Yð Þ; Yrs ¼ Urt Us�2� �

tmUsm

L2 ¼ log Zð Þ; Zrs ¼ Urrx Us�2� �

xmUr

ms

Ur ¼ X12; Xij ¼ Us

iwMwrUsrj; M ¼ exp Nð Þ

Nxy ¼ 1Du dxy � Us�1

xw Csn;wmUs�1

my

� �

8>>>>>>><>>>>>>>:

ðA:1Þ

where dij is the Kronecker delta function. Eliminating Ur inEq. (A.1), we arrive at a nonlinear equation in terms of Us

which can be solved employing the Newton–Raphsonmethod as:

R þ R : dUs ¼ 0 ðA:2Þ

in which:

Rijpq ¼@Rij

@Uspq

¼@L1

ij

@Uspq

� Bijkl@L2

kl

@Uspq

¼ L1ijpq � BijklL

2klpq ðA:3Þ

M. Baghani et al. / Mechanics of Materials 73 (2014) 1–10 9

L1ijpq ¼

@L1ij

@Uspq

¼@L1

ij

@Yrs

@Yrs

@Uspq

¼ L3ijrsL

4rspq; L3

ijrs ¼@L1

ij

@YrsðA:4Þ

L4rspq ¼

@Yrs

@Uspq

¼ UrtUsm

@ Us�2� �

tm

@Uspq

0@

1A

¼ UrtUsmL6tmpq; L6

tmpq ¼@ Us�2� �

tm

@Uspq

ðA:5Þ

L2klpq ¼

@L2kl

@Uspq

¼ @L2kl

@Zvw

@Zvw

@Uspq

¼ L11klvwL12

vwpq; L11klvw ¼

@L2kl

@ZvwðA:6Þ

L12rspq ¼ L13

rxpq Us�2� �

xmUr

ms þ UrrxL

6xmpqUr

ms þ Urrx Us�2� �

xmL13

mspq

h iðA:7Þ

L13txpq ¼

@Urtx

@Uspq

¼ @Urtx

@Xij

@Xij

@Uspq

¼ L14txij IiwpqMwrU

srj þ Us

iwMwrIrjpq þ UsiwL15

wrpqUsrj

� �ðA:8Þ

L14txij ¼

@Urtx

@Xij; L15

wrpq ¼@Mwr

@Uspq

¼ @Mwr

@Nxy

@Nxy

@Uspq

¼ L16wrxyL

17xypq;L

16wrxy ¼

@Mwr

@NxyðA:9Þ

L17xypq ¼

@Nxy

@Uspq

¼�Cs

n;wm

DuL6

xwpqUs�1

my þ Us�1

xw L6mypq

� �ðA:10Þ

where I is the fourth order symmetric identity tensor.Now, in view of (11), calculating Cs, the second Piola–Kir-chhoff stress is identified as:

S ¼ F�1sF�T ¼ F�1Kr : G : 1

2 log Veð Þ� �

F�T

¼ U�1R�1Kr : G : 1

2 log RUCs�1URT� �� �

RU�1

¼ 12 U�1

Kr : G : log UCs�1U� �� �

U�1

ðA:11Þ

In order to construct the consistent tangent matrix in thecooling process, we may write:

D ¼ 2dSdC¼ 2

@S@Cþ 2

@S@Us :

@Us

@C¼ 2Xþ 2T : Y ðA:12Þ

Taking derivative from (A.1), we obtain:

dR Us;Cð Þ ¼ @R@C

: dCþ @R@Us : dUs ¼ 0 ðA:13Þ

Recasting (A.13) into a matrix form, we obtain:

dUs½ � ¼ � R½ ��1Q½ � dC½ � ðA:14Þ

We also have Q ¼ @R@C. Now, using (A.14), we can compute

the matrix [Y] (the matrix form of the fourth-order tensorY) such that:

dUs½ � ¼ ½Y� dC½ � dUs ¼ Y : dC ðA:15Þ

The indicial form of fourth order tensors X and T are calcu-lated as follows:

Xijxy ¼@Sij

@Cxy¼

L1rsFmkrs Iimpq U�1

� �kjþ U�1� �

imIkjpq

þ U�1� �

imFmkrs

@L1rs

@UpqU�1� �

kj

8>><>>:

9>>=>>;@Upq

@Cxy

ðA:16Þ

Tijkl ¼@Sij

@Uskl

¼ U�1� �

imFmprs U�1

� �pj

@L1rs

@Yuv

@Yuv

@Uskl

¼ U�1� �

imFmprs U�1

� �pjL3

rsuvL4uvkl ðA:17Þ

in which:

@L1rs

@Upq¼ @L1

rs

@Yuv

@Yuv

@Upq

¼ L3rsuv Cs�1� �

rtIurpqUtv þ UurItvpq� �

ðA:18Þ

We now construct the spatial tangent matrix using the fol-lowing relation (Wriggers, 2008):

Cmnkl ¼1J

FmMFnNFkK FlLDMNKL ðA:19Þ

where C is the fourth-order spatial consistent tangenttensor.

Time integration scheme for heating process: In a similarway, Eq. (33) can be solved to find Cs. Substituting Cs from(33)1, in light of (A.11), the stress is also obtained.

In order to construct the consistent tangent matrix inthe heating process, we may write:

D ¼ 2dSdC¼ 2X ðA:20Þ

where X is previously defined in (A.16).

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