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 DOI: 10.1177/1045389X11431745

2012 23: 107Journal of Intelligent Material Systems and StructuresMostafa Baghani, Reza Naghdabadi, Jamal Arghavani and Saeed Sohrabpour

A constitutive model for shape memory polymers with application to torsion of prismatic bars  

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Journal of Intelligent Material Systemsand Structures23(2) 107–116� The Author(s) 2011Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1045389X11431745jim.sagepub.com

A constitutive model for shapememory polymers with application totorsion of prismatic bars

Mostafa Baghani1, Reza Naghdabadi1,2, Jamal Arghavani3,1 and SaeedSohrabpour4,1

AbstractIn this article, satisfying the second law of thermodynamics, we present a 3D constitutive model for shape memory poly-mers. The model is based on an additive decomposition of the strain into four parts. Also, evolution laws for internalvariables during both cooling and heating processes are proposed. Since temperature has considerable effect on theshape memory polymer behavior, for simulation of a shape memory polymer–based structure, it is required to performa heat-transfer analysis. Commonly, an experimentally observed temperature rate–dependent behavior of shape memorypolymers is justified by a rate-dependent glassy temperature, but using the heat-transfer analysis, it is shown that theglassy temperature could be considered as a constant material parameter. To this end, implementing the constitutivemodel within a nonlinear finite element code, we simulate torsion of a shape memory polymer rectangular bar and a cir-cular tube. Moreover, we compare the predicted results with experimental data recently reported in the literature,which shows a good agreement.

KeywordsShape memory polymers, torsion, temperature rate

Introduction

Shape memory materials are stimuli-responsive materi-als that possess the capability to recover their originalshape upon application of an external stimulus. Achange in shape caused by a change in temperature iscalled thermally induced shape memory effect (SME).This behavior has been observed in metals, ceramics,and polymers (Diani et al., 2006; Gall et al., 2009;Lendlein and Behl, 2009; Meng and Hu, 2010). Shapememory materials have been developed and utilized inan extensive range of applications such as advancedtechnologies in the medical and oil exploration indus-tries (Lendlein and Behl, 2009; Small et al., 2010).Constitutive modeling of shape memory materials hasbeen studied by many researchers (see Arghavani etal., 2010, 2011; Nguyen et al., 2010; among others). Inthis work, we limit our study to shape memory poly-mers (SMPs). In contrast to other smart materialssuch as shape memory alloys, SMPs have the advan-tages of ability of large elastic deformation, low cost,low density, low energy consumption for shape pro-gramming, high biocompatibility, biodegradability,and great manufacturability. Because of these charac-teristics, SMPs have become the center of attention

for their potential applications (Beloshenko et al.,2005; Tey et al., 2001).

Constitutive modeling of polymer materials has beenconsiderably investigated by several researchers.However, in special case of SMPs, this field of researchis still growing. In addition to experimental effortsattempted to characterize the behavior of SMPs(Abrahamson et al., 2003; Atli et al., 2009; Baer et al.,2007; Kim et al., 2010; Kolesov et al., 2009; Liang etal., 1997; Liu et al., 2006; Tobushi et al., 1997, 1998;Volk et al., 2010a, 2010b; Yu et al., 2011), an increasing

1Department of Mechanical Engineering, Sharif University of Technology,

Tehran, Iran2Institute for Nano-Science and Technology, Sharif University of

Technology, Tehran, Iran3Department of Mechanical Engineering, Golpayegan University of

Technology, Golpayegan, Iran4Department of Engineering Sciences, The Academy of Sciences of IR

Iran, Tehran, Iran.

Corresponding author:

Reza Naghdabadi, Department of Mechanical Engineering and Institute

for Nano-Science and Technology, Sharif University of Technology, Tehran

11155-9567, Iran.

Email: naghdabd@sharif.edu

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research effort has also been devoted to the predictionof the SMP behavior using either phenomenological(Chen and Lagoudas, 2008a; Diani et al., 2006; Kimet al., 2010; Liu et al., 2006; Qi et al., 2008; Reese et al.,2010; Xu and Li, 2010) or physical constitutive models(Barot et al., 2008; Nguyen et al., 2008, 2010;Srivastava et al., 2010).

Liu et al. (2006) introduced a phenomenologicalconstitutive model for small-strain deformation ofSMPs. They used a first-order phase transition conceptand considered the material as a mixture of a frozenand an active phases. This model additively decom-poses the strain into thermal, elastic, and a stored term.The stored strain was introduced to capture the strainstorage and release mechanisms. However, evolutionlaw during heating process was not presented (Chenand Lagoudas, 2008a, 2008b). Based on this model,Chen and Lagoudas (2008a, 2008b) developed a modelto capture the characteristics of SMP behavior in thelarge-deformation regime.

In addition, Xu and Li (2010) extended the modelpresented by Liu et al. (2006) into time-dependentregime in 1D modeling of SMP structures. Also, Kimet al. (2010) used the phase transition concept (Liu etal., 2006) and introduced a 1D constitutive model inthe large-strain regime.

Up to now, the characterization of the SMP beha-vior has been carried out with traditional uniaxial ten-sion and compression tests; however, torsionalthermomechanical tests can play an important role inthe characterization of an SMP constitutive model.Besides, torsional test may be useful to verify and eval-uate the validity of a 3D constitutive model as well asits numerical counterpart.

In this study, we improve the model presented byLiu et al. (2006) and present the evolution law for inter-nal variables not only in cooling but also in heating.The main focus of this article is to apply the presentedconstitutive model in simulating the torsion of rectangu-lar bars and circular tubes. While temperature is a cru-cial variable in modeling of thermally induced SMPs,the thermal analysis is also performed to show the effectof temperature distribution on the results. In modelingof SMP structures, it is a common practice to assumeTg as a rate-dependent material parameter (Volk et al.,2010a, 2010b). In this work, we investigate the effect ofsuch an assumption on the behavior of SMP structures.Implementing the proposed constitutive model within auser-defined material subroutine (UMAT) in the non-linear finite element software ABAQUS/Standard, wesimulate the torsion of a rectangular bar and a circulartube and compare the predicted results with the experi-mental data reported in the literature.

This article is organized as follows. We first describethe behavior of an SMP in a full cycle for both fixed-strain stress recovery and stress-free strain recoveryprocesses. Then, a 3D thermodynamically consistent

constitutive model for SMPs is developed. Subsequently,we employ the presented model into a finite element pro-gram and solve the torsion problem of rectangular barsand circular tubes and compare the predicted results withthe experimental data available in the literature. Finally,we present a summary and draw conclusions.

SME in SMPs under a thermomechanicalcycle

From a macroscopic point of view, SME can be char-acterized in a stress–strain–temperature diagram asdepicted in Figure 1. The thermomechanical cycle startsin a strain- and stress-free state at a high temperatureTh (point a, permanent shape). At this point, a mechan-ical loading is applied to SMP and the material exhibitsan active behavior up to point b. At point b, strain iskept fixed (external loadings), and the temperature isdecreased until the active polymer gradually turns intoa frozen polymer at a low temperature Tl (point c, tem-porary shape). In fact, around the glassy temperatureTg, SMP shows a combination of active and frozenbehaviors. In this step, the material is unloaded.According to very high stiffness of the frozen phasecompared to the active phase, after unloading, strainschange slightly (point d). Finally, we increase the tem-perature up to Th. It is seen that the strain will relaxand the original permanent shape can be recovered(point a). This cycle is called a stress-free strain recoveryin SMP applications. In practice, other types of recov-eries may happen. If at point d, the strain is fixed andthe temperature is increased, the fixed-strain stressrecovery happens (point e). Such a recovery has beenshown in Figure 1 with a dashed line.

εσ

TFigure 1. Stress–strain–temperature diagram illustrating thethermomechanical behavior of a pretensioned shape memorypolymer under different strain or stress recovery conditions.

108 Journal of Intelligent Material Systems and Structures 23(2)

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Constitutive model development for SMPs

We use an equivalent representative volume element(RVE) of the material consisted of a frozen and anactive phases (Figure 2). Assuming small strains, weuse the mixture rule in the RVE and decompose addi-tively the total strain as

ε = uaεa + uf ε

F + εT ð1Þ

where εa and εF stand for the elastic strain in the activephase and strain in the frozen phase, respectively, whileεT denotes the thermal strain and is defined byaT (T � Th), where aT is the effective thermal expansioncoefficient. Also, ua and uf are volume fractions of theactive and frozen phases, respectively, with constraintua + uf = 1. It is assumed that ua and uf are only func-tions of temperature.

We now decompose the strain in the frozen phase, εF

into two parts as

uf εF = uf ε

f + εis ð2Þ

where εf is the elastic strain in the frozen phase and εis

denotes the inelastic stored strain that will be definedmore specifically in the following. A schematic rheolo-gical illustration is shown in Figure 3 to follow the deri-vation of the equations in this section. We remark thatthe total strain is the weighted summation of the strainin each phase (weights are shown under each elementin Figure 3).

In this model, the internal variables are uf and εis.Thus, evolution equations should be defined for theinternal variables within the framework of continuumthermodynamics. Moreover, a prescribed evolutionequation is employed for uf . This equation is derivedusing the unconstrained strain recovery of the materialas a function of temperature.

We now consider the phase transformation (activeto frozen and vice versa) in the RVE. Assuming thattemperature decreases, the strain in the newly generatedfrozen phase, which was already in the active phase,experienced εa in the previous time step. Thus, uf ε

F isdefined as

uf εF = uf εf + �εf

� �= uf εf + 1

Vf

ÐVf

εa dv

!= uf ε

f + 1V

ÐVf

εa dv

ð3Þ

where Vf and V are volumes of the frozen phase andthe whole RVE, respectively. In equation (3), strain inthe frozen phase is decomposed into two parts: strainin the old frozen phase, εf , and strain in the newly gen-erated frozen phase, �εf . We call the term ‘‘ uf �ε

f ’’ as thestored strain and denote it by εis. We now recast equa-tion (3) as

uf εF = uf ε

f +

ðεa duf = uf ε

f + εis ð4Þ

Consequently, in the cooling process εis is defined as

εis =

ðεa duf ð5Þ

Such a strain storage in the cooling process has pre-viously been introduced by Liu et al. (2006). In contrastto the cooling process, in the heating process, the storedstrain in the frozen phase should be relaxed. This ismathematically shown as

Figure 2. Equivalent representative volume element. (a) At T = Tl , dominant frozen phase. (b) At T = Tg , combination of bothphases. (c) At T = Th, dominant active phase.

and release

ε ε εε

Figure 3. Schematic rheological illustration of the proposedconstitutive model.

Baghani et al. 109

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uf εF = uf εf + �εf

� �= uf εf +

1

Vf

ðVf

εis

uf

dv

0B@

1CA= uf ε

f +1

V

ðVf

εis

uf

dv

ð6Þ

We may write equation (6) in a more compact form as

uf εF = uf ε

f +

ðεis

uf

duf = uf εf + εis ð7Þ

As a result, during the heating process, εis is defined as

εis =

ðεis

uf

duf ð8Þ

We remark that the strain storage/release occursonly in the frozen phase. However, by definition, εis isassigned to the whole RVE. As a result, a division byuf appears in the integrals of equations (6) to (8)1.From equations (5) and (8), it is concluded that εis is afully thermal-driven variable.

We now express the convex free energy density func-tion C for an SMP. Based on the mixture rule, we intro-duce the following form for the energy function

C ε, T , uf , εa, εf , εis� �

= uaCa εað Þ+ uf Cf εf� �

+ Cl ε, T , ua, uf , εa, εf , εis� �

+ CT Tð Þ ð9Þ

where Ca and Cf denote Helmholtz free energy densityfunctions of the active and frozen phases, respectively. CT

represents the thermal energy and Cl enforces the kine-matic constraint of equation (2) in the following form

Cl ε, T , ua, uf , εa, εf , εis� �

= l : ε� uaεa + uf ε

f + εis� �

� εT� �

ð10Þ

where l is the (tensorial) Lagrange multiplier.We now apply the second law of thermodynamics in

the sense of Clausius–Duhem inequality to derive con-straints on the evolution equation (Haupt, 2002) as

Dmech = s : _ε� _C + h _T� �

� 0 ð11Þ

where h represents the entropy and _() = ∂=∂t representsthe derivative with respect to time. Substituting equa-tion (9) into equation (11), we obtain the followinginequality

s : _ε� ua

∂Ca

∂εa: _εa � uf

∂Cf

∂εf: _εf � ∂C

∂uf

u9f_T

�CT 9 _T � ε� uaεa + uf ε

f + εis� �

� εT� �

: _l

� _ε� ua _εa + uf _εf + _εis� �� �

: l� h _T � 0

ð12Þ

where ()9 = ∂=∂T denotes the derivative with respect totemperature. Moreover, combination of equations (5) and(8) leads to one evolution equation in the rate form as

_εis = u9f_T ks1

εa + ks2

εis

uf

!;

ks1= 1, ks2

= 0; _T\0

ks1= 0, ks2

= 1; _T.0

ks1= 0, ks2

= 0; _T = 0

8<:

ð13Þ

Inequality equation (12) must be fulfilled for arbi-trary thermodynamic processes, that is, for arbitrary _ε,_εa, _εf , _l, and _T . For arbitrary choices of the variables _ε,_εa, _εf , _l, and _T , we may arrive at (Coleman and Gurtin,1967)

s = l =∂Ca

∂εa=∂Cf

∂εf

ε = uaεa + uf ε

f + εis + εT

h = �CT 9 + s : εT 9 + u9f ks1εa + ks2

εis

uf

! !+

∂C

∂uf

u9f

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

ð14Þ

Equations (14)1 and (14)2 are consequences of thebasic assumption of simultaneous existence of theactive and frozen phases, together with satisfying thesecond law of thermodynamics. In fact, Liu et al.(2006) assumed that the stresses in both phases are thesame, while in this work, employing the rule of mixture,we define Helmholtz free energy density function thatnecessitates the stresses in both phases to be the same.

In the following, we use quadratic forms for the freeenergy density functions as

Ca εað Þ=1

2εa : Ka : εa, Cf εf

� �=

1

2εf : Kf : εf ð15Þ

where Ka and Kf are fourth-order positive definite elas-ticity tensors of active and frozen phases, respectively.In the case of isotropic materials, we only need to knowelastic modulus E, and Poisson’s ratio, n, of each phase.

We use a numerical integration scheme in an implicitform to solve the nonlinear system of equations (13)and (14). We should remark that small-strain constitu-tive models can be successfully applied to the solutionof large displacement problems, where strains are smallthough rotations can be arbitrarily large. The numeri-cal manipulation of this procedure follows a standardapproach in the literature (Hartl and Lagoudas, 2009),thanks to the well-known Hughes–Winget algorithm(Hughes and Winget, 1980). Thus, in order to capturesuch effects in the presented model, we employ theHughes–Winget algorithm.

Finite element simulation results

The constitutive equations presented in the previoussection are implemented in the finite element analysissoftware (ABAQUS/Standard) using a UMAT.Numerical modeling of torsion of rectangular bars andcircular tubes are done, and the results are comparedwith the experiments performed by Diani et al. (2011).

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Initial configuration of the rectangular bars and circu-lar tubes has been shown in Figure 4. The rectangularbar has a length of 100 mm, width of 10 mm, andthickness of 1 mm, while the circular bar has a lengthof 200 mm, inner radius of Ri = 8 mm, and outerradius of Ro = 10 mm.

While experimental results for the rectangular barare in hand, as the first step, simulation of the rectangu-lar bar is performed, and the results of numerical simu-lations are compared with the experimental data. In thesecond step, numerical simulation of the circular tubewith the same material parameters is presented. In thefollowing examples, we simulate the stress-free strainrecovery process through the path a-b-c-d-a shown inFigure 1.

Heat-transfer analysis by finite element method

In the structural analysis of SMPs, the ambient tem-perature is commonly used instead of the exact tem-perature in any part of the structure. In order to makean improvement in modeling of structures made ofSMPs, we calculate the temperature, T , through a time-dependent heat-transfer finite element analysis. In fact,in some studies, glassy temperature has been assumedas a temperature rate (TR)–dependent parameter, whilewe believe that if the heat-transfer analysis is per-formed, there is no need for such an assumption.

Material parameters applied in this simulationare adopted for experiments recently performed byDiani et al. (2011). It should be emphasized that

parameters used in heat-transfer analysis of the struc-tures are not reported by Diani et al. (2011). Thus,these parameters (density r, specific heat capacity Cp,heat conductivity k, and heat convection coefficient ofair h) are approximated using similar materials avail-able in the literature.2 Moreover, we use the reportedtemperature in the experiments as the ambient tempera-ture T‘. It is noteworthy to mention that characteristictemperatures Tl, Tg, and Th as well as the elastic moduliof the active and frozen phases are measured usingdynamic mechanical analysis (DMA) tests (Diani et al.,2011).

In order to specify the evolution equation for thevolume fraction of the frozen phase uf as a function oftemperature, we use the fact that equation (13) duringheating in a 1D stress-free strain recovery (while thestored strain after unloading is ε0, corresponding topoint d in Figure 1) reduces to

_εis �_uf

uf

εis = 0, εis��uf = 1

= ε0 ) εis = ε0uf ð16Þ

Equation (16) shows that the expression for uf

should follow the same trend as the stored strain εis.Thus, employing a curve fitting method, we define afunction that fits the experimental data using the leastsquares method. A combination of exponential and/orpower terms normally leads to a successful curve fit-ting. Such a curve fitting has also been utilized for theexperimental data reported by Diani et al. (2011) whenthe applied twist angle u0 is 360o. In this case, we useda relation for uf in the following form

uf =tanh Th � Tg

� �b

� �� tanh T � Tg

� �b

� �tanh Th � Tg

� �b

� �� tanh Tl � Tg

� �b

� � ð17Þ

where b is calculated by applying a curve fittingmethod. The obtained value is b = 4:8178C. In addition,due to lack of the experimental data, the Poisson’s ratioof both phases are assumed to be 0:4. All materialparameters used in numerical simulations are reportedin Table 1.

Table 1. Material parameters adopted for experiments reported by Diani et al. (2011).

Material parameters Values Units

Ea, Ef 15.2, 2600 MPana, nf 0.4, 0.4 —Tl, Tg, Th 25, 46, 62 �CaT = uaaa + uf af aa = 2310�4, af = 1310�4 1/�C

uf

tanh Th � Tg

� �b

� �� tanh T � Tg

� �b

� �tanh Th � Tg

� �b

� �� tanh Tl � Tg

� �b

� � , b = 4.817 �C —

r 1100 kg/m3

Cp 1400 J/kg �Ck 0.05 W/m �C

Figure 4. Initial configuration of rectangular bar and circulartube used in numerical simulations.

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Torsion of an SMP rectangular bar: comparison withthe experimental data

In Figure 5, temperature–time response of a node atupper central free end of the rectangular bar withapplied twist angle u0 = 3608 is shown at different TRsof 0.9�C/min and 2.5�C/min. It is observed that at therate of 0.9�C/min, the structure is almost in thermalequilibrium. Consequently, the difference between T‘

and T is approximately negligible, while at the rate of2.5�C/min, the bar does not have enough time to reachT‘, thus, there is a gap between T‘ and T .

In Figure 6, Von-Mises stress and maximum loga-rithmic principal strain in temporary shape of the rec-tangular bar (corresponding to point d in Figure 1) areillustrated, while the applied twist angle at Th isu0 = 3608 and the TR is 0.9�C/min. As one may see, themaximum logarithmic principal strain in this case isunder 6%, while the structure undergoes very largerotations about 3608 in some cases. Therefore, it con-firms our assumption of small or moderate strains withlarge rotations in constitutive modeling of torsion ofprismatic bars.

In Figure 7, the angle–temperature curve for differ-ent applied twist angles u0 are illustrated, and theresults are compared with the experimental data

0 25 50 75 100 125 150 17525

30

35

40

46

50

55

62

Time [ min ]

Tem

pera

ture

[ o C

]

T∞

T, h = 15 W/m 2 .oC

a)

0 10 20 30 40 50 60 6525

30

35

40

46

50

55

62

Time [ min ]

Tem

pera

ture

[ o C

]

T∞

T, h = 10 W/m2 .oC

T, h = 15 W/m 2.oC

b)

Figure 5. Temperature history at upper central node of free end of rectangular bar with u0 = 3608 at (a) temperature rate (TR) =0.9�C/min and (b) TR = 2.5�C/min.

Figure 6. (a) Von-Mises stress in pascal and (b) maximum logarithmic principal strain in temporary shape of the rectangular bar atu0 = 3608 and temperature rate (TR) = 0.9�C/min.

20 25 30 35 40 46 50 55 62

0

45

90

135

180

225

270

315

360

400

Temperature [ oC ]

Ang

le [

°]

Present work at θ0 = 180o

Present work at θ0 = 270o

Present work at θ0 = 360o

Experiment at θ0 = 180o

Experiment at θ0 = 270o

Experiment at θ0 = 360o

Figure 7. Angle recovery of the rectangular bar at different u0

at temperature rate (TR) = 0.9�C/min.Experimental results are reported from Diani et al. (2011).

112 Journal of Intelligent Material Systems and Structures 23(2)

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reported by Diani et al. (2011). There is a good qualita-tive agreement between the numerical results and theexperimental data. However, in the experimentalresults, after unloading at low temperature Tl (corre-sponding to point d in Figure 1), the stresses arereleased and the instantaneous elastic angular recoveryis measured to be about 188, while in our simulations,we found a 28 instantaneous elastic angular recoveryfor u0 = 3608. In fact, as reported in other works (Dianiet al., 2011), the elastic strain resulting from the stressrelease is proportional to the applied elastic strain athigh temperature, and the active-to-frozen modulusratio in our model is (15:2=2600)33608 = 2:18. It seemsthat this difference is the result of time dependency ofthe SMP behavior in the frozen phase, which has notbeen considered in this work. It also could be due tosome plastic residual strains at low temperature, whichhas not been taken into account in the presentedmodel.

In Figure 8, the effect of TR on a stress-free strainrecovery of the rectangular bar is investigated.According to Figure 8, the experimental results depicta 4�C shift in temperature-dependent response of anglerecovery. Furthermore, the effect of heat convectioncoefficient of air h, on angle recovery is shown inFigure 8. Similar effect is also observed in traditional1D compression–tension experiments reported by Volket al. (2010a, 2010b). They justified this effect by select-ing a linear TR-dependent glassy temperature in theform of Tg = a _T + b and did not perform the heat-transfer analysis of the structure. As illustrated inFigure 8, although we took glassy temperature as aconstant material parameter, but numerical results arecapable of capturing the mentioned shift in twist angle–temperature response of the bar. There is a delicatefundamental difference between the approach assuminga TR-dependent Tg and the approach proposed here(assuming Tg to be rate independent and performing

the heat-transfer analysis). In fact, the first approachassumes the observed shift in the response of a struc-ture is a consequence of a phenomena related to amaterial behavior (constitutive model level), while thesecond one assumes the material behavior being inde-pendent of TR, and the observation is related to thestructural behavior through performing a heat-transferanalysis.

In Figure 9, angular velocity of the free end of thebar as a function of temperature during stress-freeheating (corresponding to the path d-a in Figure 1) atTR = 0.9�C/min is depicted. Numerical results arecompared with the experimental data reported byDiani et al. (2011) for two shape memory epoxy sam-ples. It is noted that there is a good agreement betweenthe numerical results and the experimental data.

Torsion of an SMP circular tube

After validating the results of the proposed constitutivemodel in the case of rectangular bar, we solve the tor-sion problem for an SMP circular tube under thermo-mechanical loading. In the following, the stress-freestrain recovery of an SMP circular tube (through thepath a-b-c-d-a as shown schematically in Figure 1) willbe simulated. The circular tube shown in Figure 4 isfixed at one end, while the other end is twisted.

In Figure 10, the applied temperature history, T‘, ofthe circular tube is plotted at different TRs. As onemay expect, at lower rate I (Figure 10(a)), the tube isnearly in thermal equilibrium. Therefore, the differencebetween T‘ and T is almost negligible, while at higherrate II (Figure 10(b)), the tube does not have enoughtime to reach T‘, thus there is a gap between T‘ and T.

The state of shear stress in cross section of the bar(at Tl) in temporary shape is shown in Figure 11, while

20 25 30 35 40 46 50 55 6265 700.0

0.1

0.2

0.3

0.4

0.5

0.6

Temperature [ oC ]

Ang

ular

vel

ocity

[ °/

s]

Present workExperiment−sample 1Experiment−sample 2

Figure 9. Angular velocity–temperature response of the freeend of the rectangular bar during stress-free heating attemperature rate (TR) = 0.9�C/min and comparison ofnumerical simulation results with experimental data for twoshape memory epoxy samples reported by Diani et al. (2011).

20 25 30 35 40 46 50 55 620

60

120

180

240

300

360

Temperature [ oC ]

Ang

le [

°]

TR = 2.5 oC/min, h = 10 W/m2.oC

TR = 2.5 oC/min, h = 15 W/m2.oC

TR = 0.9 oC/min, h = 15 W/m2.oC

Experiment at TR = 0.9 oC/min

Experiment at TR = 2.5 oC/min

Figure 8. Angle recovery of the rectangular bar at differenttemperature rates (TRs).Experimental results are reported from Diani et al. (2011).

Baghani et al. 113

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the applied twist angle at Th is u0 = 1808 with the tem-perature history I. Von-Mises stress and maximumlogarithmic principal strain in temporary shape of thecircular tube prior to unloading are also depicted inFigure 12, while the applied twist angle is u0 = 1808, andthe temperature history I is used. It is seen that themaximum strain is about 8%, thus the assumption ofsmall or moderate strains is realistic in modeling of thecircular tube.

As shown in the previous section, the rate of theapplied temperature affects the twist angle recoveryand produces a shift in angle–temperature response ofthe SMP tube. A similar phenomenon is observed inthe numerical modeling of circular tubes as plotted inFigure 13. Moreover, twist angle–temperature responseof the SMP tube in different applied angles (u0) of 90�,135�, and 180� at temperature histories I and II aredepicted in Figure 13.

Summary and conclusion

In this article, we presented a constitutive model forSMPs, which reasonably captures the main features ofthe SMP behavior. Considering a first-order mixturerule, we utilized an additive decomposition of the straininto four parts. In fact, the material was considered as amixture of the active and frozen phases with theassumption that the phases are able to be transformedto each other through external stimuli of heat. The evo-lution laws for internal variables were derived in anarbitrary thermomechanical loading. For the sake ofcompleteness, free energy function of the proposedmodel was introduced. Satisfying the second law ofthermodynamics implies that the stress in both phasesshould be the same. Implementing the proposed modelwithin a UMAT in the nonlinear finite element software

0 20 50 80 100 125 175 200

25

35

45

55

62

70T

empe

ratu

re [

o C ]

Time [ min ]

T∞

a)

cooling

heatingloading

History I

0 25 50 62.5 87.5 100

25

35

45

55

62

70

Tem

pera

ture

[ o C

]

Time [ min ]

T∞

loading

cooling

heating

b)

History II

Figure 10. History of temperature applied to the circular tube in (a) temperature history I and (b) temperature history II.

Figure 12. State of Von-Mises stress in pascal and maximum logarithmic principal strain in temporary shape of the circular tubeprior to unloading at u0 = 1808 and temperature history I.

Figure 11. Shear stress in cross section of the circular tube intemporary shape at T = Tl.

114 Journal of Intelligent Material Systems and Structures 23(2)

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ABAQUS, we performed a 3D analysis of the rectangu-lar bars and circular tubes made of SMPs. The modelwas validated by comparing the predicted results withthe experimental data available in the literature. It wasshown that the model is capable of capturing the mainfeatures reported in the experimental observations.

In addition to the structural analysis, the thermalanalysis of SMP bars and tubes were also performedvia finite element method. We showed that there is adelicate fundamental difference between the approachassuming a TR-dependent Tg and the approachemployed in this work (assuming Tg to be rate indepen-dent and performing the heat-transfer analysis). Thefirst one assumes the observed shift in the response of astructure is a consequence of a phenomena related tothe material behavior, while the second one assumesthe material behavior being independent of TR, andthe observation is related to the structural behavior byperforming a heat-transfer analysis. Therefore, the rateof the applied temperature in cooling or heating onlyaffects the heat transfer in the structure and has noth-ing to do with the value of the glassy temperature,which is a crucial parameter in investigating the SMPbehavior.

Notes

1. The analytical solution of equation (8) shows that whenuf ! 0, the stored strain εis approaches 0. Therefore, inour solution algorithm, we detect this limit case and setεis = 0.

2. For example, see http://www.huntsman.com. TechnicalBulletin-JEFFAMINE� EDR-148 Polyetheramine.

Funding

Financial support from the Academy of Sciences of IslamicRepublic of Iran to this work is gratefully acknowledged.

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