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Journal of the Mechanics and Physics of Solids 56 (2008) 1730–1751

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Finite deformation thermo-mechanical behavior of thermallyinduced shape memory polymers

H. Jerry Qia,�, Thao D. Nguyenb, Francisco Castroa, Christopher M. Yakackia,Robin Shandasa

aDepartment of Mechanical Engineering, University of Colorado, 427 UCB, ECME 124, Boulder, CO 80309, USAbDepartment of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA

Received 12 June 2007; received in revised form 10 December 2007; accepted 13 December 2007

Abstract

Shape memory polymers (SMPs) are polymers that can demonstrate programmable shape memory effects. Typically, an

SMP is pre-deformed from an initial shape to a deformed shape by applying a mechanical load at the temperature TH4Tg.

It will maintain this deformed shape after subsequently lowering the temperature to TLoTg and removing the externally

mechanical load. The shape memory effect is activated by increasing the temperature to TD4Tg, where the initial shape is

recovered. In this paper, the finite deformation thermo-mechanical behaviors of amorphous SMPs are experimentally

investigated. Based on the experimental observations and an understanding of the underlying physical mechanism of the

shape memory behavior, a three-dimensional (3D) constitutive model is developed to describe the finite deformation

thermo-mechanical response of SMPs. The model in this paper has been implemented into an ABAQUS user material

subroutine (UMAT) for finite element analysis, and numerical simulations of the thermo-mechanical experiments verify

the efficiency of the model. This model will serve as a modeling tool for the design of more complicated SMP-based

structures and devices.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Shape memory polymers; Constitutive modeling; Finite deformation behavior; Stress–strain behavior; Thermo-mechanical

behavior

1. Introduction

Shape memory polymers (SMPs) have been investigated intensively because of their capability to recover apredetermined shape in response to environmental changes, such as temperatures and light irradiations(Lendlein et al., 2005a, b; Monkman, 2000; Otsuka and Wayman, 1998; Scott et al., 2005). Compared to othershape memory materials, such as NiTi alloy, where the reported maximum deformation due to shape change is�8%, SMPs can exhibit large deformations exceeding 400% (Lendlein et al., 2005b). This advantage permitsapplications such as microsystem actuation components, biomedical devices, aerospace deployable structures,and morphing structures (Tobushi et al., 1996; Liu et al., 2004; Yakacki et al., 2007).

e front matter r 2007 Elsevier Ltd. All rights reserved.

ps.2007.12.002

ing author. Tel.: +1 303 492 1270; fax: +1 303 492 3498.

ess: [email protected] (H.J. Qi).

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dx.doi.org/10.1016/j.jmps.2007.12.002

mailto:[email protected]

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Permanent ShapePredeform (Program)

TL < Tg

TL < Tg

Temporary Shape

Deploy (Recover)

TD >Tg

TH >Tg TH >Tg

Fig. 1. A typical thermo-mechanical loading/unloading cycle in a SMP application.

H.J. Qi et al. / J. Mech. Phys. Solids 56 (2008) 1730–1751 1731

For a device design using thermally induced SMPs, the SMP will undergo a thermo-mechanical loading-unloading cycle illustrated in Fig. 1. The SMP is isothermally predeformed (or programmed) from an initialshape to a deformed shape by applying a mechanical load at the temperature TH. The material will maintainits deformed shape after subsequently lowering the temperature to TL and removing the external mechanicalload. The unloaded deformed shape at TL is commonly referred to as the temporary or programmed shape.The SMP can largely maintain this shape as long as the temperature does not change. The shape memoryeffect is activated by raising the temperature to TD where the initial shape is recovered. In general, TH and TD

are above the glass transition temperature Tg, and TL is below Tg. Recent advances in polymer science hasmade it possible to vary the Tg by controlling the chemistry and/or the structure of SMPs for a variety ofapplications (Yakacki et al., 2007).

In principle, most polymers demonstrate a certain degree of shape memory behavior. However, in order toachieve a highly recoverable and programmable shape change, crosslinking polymers are typically used.1 Thecrosslinking can be chemical crosslinking, physical crosslinking, or macromolecular chain entanglements.Lendlein et al. (2005b) and Liu et al. (2006) gave in-depth discussions of the underlying physical mechanism ofthermally induced shape memory effects. The shape memory effect is caused by the transition of a crosslinkingpolymer from a state dominated by entropic energy (rubbery state) to a state dominated by internal energy(glassy state) as the temperature decreases. At temperatures above the glassy transition temperature Tg,individual macromolecular chains undergo large random conformational changes, which are constrained bythe crosslinking sites formed during material processing. Deforming the material reduces the possibleconfiguration and hence the configurational entropy of the macromolecular chains, leading to the well-knownentropic behavior of elastomers. After the removal of the external load at a temperature above Tg, thetendency of the material to increase its entropy will recover the undeformed (processed) shape defined bythe spatial arrangement of crosslinking sites. However, this shape recovery can be interrupted by lowering thetemperature below Tg. There, the mobility of macromolecular chains is significantly reduced by the reductionin free volume, and the conformational change of individual macromolecules becomes increasingly difficult.Instead, cooperative conformational change of neighboring chains becomes dominant, and deformation thusrequires much higher energy. Therefore, the removal of the mechanical load at temperatures below Tg only

1There are some other polymers that can be termed as ‘‘thermally induced shape memory polymers’’ but not necessarily be cross-linking

polymers. For example, the shape change of some liquid crystal elastomers can be activated by temperatures. However, the shape memory

effects in liquid crystal elastomers are due to a molecular transition between trans to cis state under proper temperature conditions. The

mechanism of shape memory effects is therefore distinct from SMPs studied in this paper. Significant research efforts have been carried on

liquid crystal elastomers, including constitutive modeling and computational implementation. The readers are referred to Wanner and

Terentjev (2003) for liquid crystal elastomers.

ARTICLE IN PRESSH.J. Qi et al. / J. Mech. Phys. Solids 56 (2008) 1730–17511732

induces a small amount of shape recovery and most of the deformation incurred at the temperature above Tg

is retained (stored, or frozen). The shape memory effect is invoked as the temperature increases above Tg,where the individual macromolecular chains become active again and the shape recovery mechanism describedabove is permitted. In this sense, shape memory effect is simply a temperature-delayed recovery.

In the applications of SMPs, because of large and complicated deformation involved, it is highly desirablethat the deformation history of SMPs can be predicted and the recovery properties can be optimized. Thisrequires a finite deformation constitutive model that is based on the fundamental understanding ofstructure–function relationships and can capture the thermo-mechanical response of SMPs. Most of theexisting constitutive models of SMPs have been limited to one-dimensional (1D) small deformations (Tobushiet al., 1996; Liu et al., 2006). For example, considering the SMP as a mixture of two phases (active phase andfrozen phase), Liu et al. (2006) developed a 1D small deformation model. There, a ‘‘stored strain’’ was used tomemorize the predeformation. Note that the intensive research on shape memory alloys (SMAs) in the pasthas resulted in the developments of sophisticated constitutive models for SMAs (such as Thamburaja andAnand, 2002; Lagoudas et al., 2006, etc.). However, these models cannot be applied to SMPs because of thefundamental differences in the underlying mechanism for shape memory effects. In this paper, thermo-mechanical experiments were conducted to identify key features of finite deformation behaviors of SMPs.Based on the experimental observations and the concept of phase transitions, a three-dimensional (3D) finitedeformation constitutive model that describes the thermo-mechanical response of SMPs is developed. Thismodel is implemented into a user material subroutine (UMAT) in the finite element software packageABAQUS. The paper was arranged as the following. Section 2 presents finite deformation thermo-mechanicalexperiments performed to explore the properties and shape memory behavior of SMPs. Based on observationsfrom these experiments, a 3D finite deformation constitutive model is proposed in Section 3. Comparisonsbetween model prediction and experimental results are presented in Section 4, and future work is discussed inthe concluding section.

2. Thermo-mechanical behavior of SMPs

2.1. Material and sample preparations

Sheets of SMPs were synthesized following the procedure in Yakacki et al. (2007). Briefly, tert-butyl acrylate(tBA) monomer and crosslinker poly(ethylene glycol) dimethacrylate (PEGDMA) in liquid forms, and photopolymerization initiator (2, 2-dimethoxy-2-phenylacetophenone) in powder form were mixed in a beakeraccording to a pre-calculated ratio. The beaker was shaken for about 10 s to ensure a good mixture. Thesolution was injected onto the surface of a specially designed glass slide or a glass tube, which then was placedunder the UV lamp for polymerization. After 10min, the SMP material was removed from the glass slide ortube and was put into an oven for 1 h at 80 1C. The SMP materials were machined into the proper shapes fordifferent experiments. In order to eliminate variations in the material properties caused by processing, thesamples from the same prepared batch were used for a series of thermo-mechanical experiments, includingdynamic mechanical analysis (DMA), isothermal uniaxial compression, and cooling/heating experiments.Each experiment is described below, together with the corresponding result. For each type of experiment, atleast three tests were conducted to confirm the repeatability, but only result from one test was shown for thesake of clarity.

2.2. DMA tests

DMA tests were conducted using a Perkin-Elmer DMA Tester (Model 7 Series) with Perkin-ElmerIntracooler 2 cooling system to determine the glass transition temperature Tg of the SMP specimens. The testprocedure follows the one applied in Liu et al. (2006). A rectangular bar with dimensions of 10� 2� 1mm3

was placed into a DMA three point bending device. A small dynamic load at 1Hz was applied to the platenand the temperature was lowered at a rate of 1 1C/min. Fig. 2 shows the storage modulus and tan d vstemperature curves from one DMA test. It was determined that the glass transition temperature Tg of the SMPsamples was �49 1C.

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0

40 60 80 100

Sto

rag

e M

od

ulu

s (

MP

a)

103

102

101

100

20

Temperature (°C)

tan

�

0.4

0.8

1.2

1.6

2

tan �Storage Modulus

Fig. 2. DMA test of the SMP dynamic mechanical properties.

0

1.48×10−4

1

2.53×10−4

(L/L

0-1)

(%

)

0.2

0

-0.2

-0.4

-0.6

-0.8

-1

-1.2

-1.4

Temperature (°C)

10 20 30 40 50 60 70

1

Fig. 3. CTE measurement of the SMP during cooling.


2.3. CTE measurement

The coefficients of thermal expansion (CTE) were measured using the Perkin-Elmer DMA Tester.Cylindrical samples of 10mm in diameter and 10mm in height were cut from the synthesized SMPs. Thesample was placed between two plates inside the DMA tester. A small compressive force of 1mN was appliedand maintained to ensure the contact between the sample and the plates during the CTE measurement. As thetemperature was varied, the height of the sample changed due to thermal expansion. The displacement of thetop plate was recorded by the DMA tester. To measure the CTE, a sample was first heated from roomtemperature (25 1C) to 70 1C at 5 1C/min and allowed to equilibrate at 70 1C for 20min. The sample then wascooled from 70 to 0 1C at 1 1C/min. The sample was kept at 0 1C for 10min. It was then reheated to 70 1C at1 1C/min. The height change of the sample during cooling and reheating was recorded.

Fig. 3 shows the measurement of CTEs during the cooling from 70 to 0 1C. The curve for heating from 0 to70 1C shows similar result and is not presented here for the sake of clarity. In Fig. 3, the thermal expansionstrain (L/L0�1) is plotted versus the temperature. The reference height (L0) is the sample height at 70 1C.


The portions below and above Tg are used to obtain the slopes, which are the CTEs. In this paper, we usedlinear regressions of the curve between 10 and 20 1C to obtain the CTE below Tg, and the curve between 50and 60 1C to obtain the CTE above Tg. From Fig. 3, the CTEs were measured as a1 ¼ 2:53� 10�4 fortemperatures above Tg and a2 ¼ 1:48� 10�4 for temperatures below Tg.

2.4. Isothermal uniaxial compression tests

In order to investigate the large deformation behavior of SMPs under different temperature conditions,isothermal uniaxial compression tests were conducted by using a universal materials testing machine (InstronModel 5565 with load capacity of 5 kN). The machine is equipped with a temperature-controlled chamber(Instron Model 3119-405-21 with a Euro 2408 controller). A thermocouple was placed close to the sample

0 10

10

20

30

40

50

60

70

00

10

20

30

40

50

60

70

-Tru

e S

tress (

MP

a)

-True Strain

0.750.50.25

-Tru

e S

tress (

MP

a)

-True Strain

0.25 0.5 0.75 1

T = 60°C

T = 20°C

T = 0°C

rate = 0.1s-1rate = 0.01s-1

T = 0°C

T = 10°C

T = 20°C

T = 30°C

T = 60°C

Fig. 4. (A) Stress–strain behaviors of the SMP from isothermal uniaxial compression experiments at different temperatures. The strain

rate is 0.01/s. (B) Stress–strain behaviors of the SMP at different strain rates and at different temperatures.

ARTICLE IN PRESSH.J. Qi et al. / J. Mech. Phys. Solids 56 (2008) 1730–1751 1735

surface to maintain the chamber temperature. Cylindrical samples of the same dimensions as those in the CTEmeasurements were used. Previous studies showed that the effects of barreling and buckling could be reducedas long as the ratio of sample height and diameter was within the range of 0.5–2.0 (Qi and Boyce, 2005;Bergstrom and Boyce, 1998). In order to further reduce the effects of friction, Teflon sheets were placedbetween the sample and the platens. In an isothermal test, the sample was first placed on the bottom platen;the chamber temperature then was changed to the desired temperature; finally, the sample was given �20minto reach a thermal equilibrium before the uniaxial compression started. Uniaxial compression experimentswere conducted at two different strain rates of 0.01 and 0.1/s. Whilst most of the samples were tested to break,a few samples were unloaded before their broken to observe the unloading behaviors. Only the results inloading were presented in this paper.

Fig. 4A shows the stress–strain plots from uniaxial compression tests at T ¼ 0, 10, 20, 30 and 60 1C at thestrain rate of 0.01/s. Fig. 4B shows the stress–strain behaviors at T ¼ 0, 20, and 60 1C with two strain rates of0.01 and 0.1/s at each temperature. The SMP demonstrates distinct behaviors at temperatures below andabove its glassy transition temperature Tg. Above Tg, it exhibits a typical hyperelastic behavior, with littleviscous effects for tested strain rates and almost no permanent deformation after unloading (unloading curvenot shown); at temperatures below Tg, it exhibits a typical glassy polymer behavior with a yield point, followedby a softening, and then a slight hardening with significant permanent deformation observed after unloading(unloading curve not shown). In addition, strong dependency on the mechanical loading rates can be observedat temperatures below Tg (Fig. 4B).

2.5. Cooling/heating experiments

Cooling/heating experiments were conducted on two groups of samples for constrained recovery and freerecovery, respectively. The inset in Fig. 5 shows the mechanical loading and temperature histories for theconstrained recovery. The sample was first compressed by 20% at T ¼ 100 1C, then allowed to relax for10min. After that, the sample was cooled down to T ¼ 0 1C at the temperature rate of _T ¼ �1 oC=min, thenreheated back to T ¼ 100 1C at _T ¼ 1 oC=min. During the cooling and heating processes, the position of theplaten was maintained in its lowered position by using an extensometer, such that the sample was constrainedfrom recovering to its original shape during reheating. The free recovery experiment followed the same

0 20 40 60 80 1000

0.4

0.8

1.2

1.6

2

No

rmal

ized

Tru

e S

tres

s

Temperature (°C)

Cooling

Heating

Time

Strain

Temperature

Fig. 5. Stress–temperature plot during cooling-heating for the constrained recovery case. The stress is normalized by the maximum

compressive stress immediately after loading. The inset shows the mechanical and thermal loading history for the constrained recovery.

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Fig. 6. Snapshot images from the free recovery case: (A) before compression, T ¼ 100 1C; (B) immediately after compression, T ¼ 100 1C;

(C) after cooling, T ¼ 0 1C; (D) after the constraint was removed, T ¼ 0 1C; and (E) after reheating, T ¼ 100 1C.

H.J. Qi et al. / J. Mech. Phys. Solids 56 (2008) 1730–17511736

procedure as the constrained recovery, except that the top platen was raised above its starting position aftercooling to permit a full recovery.

Fig. 5 shows the stress vs temperature plot for the case of constrained recovery. Fig. 6 shows the images atdifferent points of thermo-mechanical history for the free recovery case. During cooling, the constrained stressgradually decreases to zero. This is due to two reasons: (1) the deformation at high temperatures being frozenas the temperature passes the glassy transition temperature and (2) thermal contraction during cooling. Whilstthese two reasons cause the stress to decrease, heat transfer may delay the stress decrease. The heat transferproblem is currently investigated by the authors. When reheating, the recovery stress does not follow the samepath as the one during cooling: It first overshoots to a much higher stress than the stress during cooling at thesame temperature, then decreases to the stress value close to cooling, follows the same path as cooling, andfinally reaches the stress value before the thermal cycle. The distinct force paths during cooling and reheatingare conjectured to be due to the structural relaxation and viscoelastic stress relaxation of the polymerundergoing a glass transition, which depends on the temperature rate and the idling time between cooling andreheating (McKenna, 1989). The effects of structural relaxation on the shape memory effects are currentlyunder investigation by the authors and are not in the scope of the model proposed in this paper.

3. Constitutive model

3.1. General considerations

As discussed previously, it is proposed that the shape memory effect of an SMP is due to two concurrentprocesses: (1) the transition from rubbery behavior dominated by entropic energy at high temperatures toglassy behavior dominated by internal energy at low temperatures and (2) the storage of the deformationincurred at high temperatures during cooling. In the first process, the transition of energy can be described asthe change in Helmholtz energy:

H total ¼ f rðTÞHr þ f gðTÞHg, (1)

where Htotal is the total Helmholtz energy at a given temperature T; Hr is the Helmholtz energy of the rubbermaterial at TbTg; Hg is the Helmholtz energy of the glassy material at T5Tg; and fr and fg are functions ofthe temperature and fr+fg ¼ 1. At TbTg, frE1 and fgE0 so that HtotalEHr; at T5Tg, frE0 and fgE1 so thatHtotalEHg. Therefore, evolving fr and fg will capture the transition of Helmholtz energy.

Noting that this energy consideration resembles a description of a first order phase transition process, forthe sake of model description, we phenomenologically assume there exist a rubbery phase (RP) and a glassyphase in the material and the phase transition between these two phases is realized through the change ofvolume fraction of each phase. Here, we point out that such a phase transition description between RP andglassy phase is phenomenological and in a real polymer systems, there may or may not exist distinct phases.We also note that the phase transition concept has been used previously for the development of a 1D model ofSMPs (Liu et al., 2006), where an SMP was considered as a mixture of two phases.


The second process, or the deformation storage process, implies that the glassy phase formed during coolingmay have different deformation history. We therefore further divide the glassy phase into two phases: theinitial glassy phase (IGP) and frozen glassy phase (FGP). Therefore, there are three phases in the model2:

1.

2

rub

one

SM

Rubbery phase (RP): The RP dominates at temperatures well above the glass transition temperature Tg.The volume fraction of the RP is fr. As the temperature decreases, the volume fraction of this phasedecreases, and vice versa. In the limit, TbTg, fr ¼ 1.0, and when T5Tg, fr ¼ 0.0.

2.
Frozen glassy phase (FGP): FGP refers to the newly formed glassy phase caused by a decrease in thetemperature. The volume fraction of the FGP is fT. As the temperature decreases, the volume fraction ofthis phase increases, and vice versa.
3.
Initial glassy phase (IGP): IGP refers to the glassy phase in the initial configuration of the material. Thisphase of the material deforms when an external load is applied at the beginning of an analysis. The volumefraction of the IGP is fg0. Since decreasing the temperature will induce new glassy phases, which is the FGP,the volume fraction of the IGP will not change. On the other hand, as the temperature increases, the volumefraction of the IGP decreases. Note that in most of SMP applications, the predeformation step is performedat TbTg. In such cases, fg0E0. However, it is also possible that one start with loading the sample at atemperature in the vicinity of Tg. For the completeness of the model, we consider a general case where ananalysis starts from a temperature in the vicinity of Tg and fg0 6¼0.
The volume fractions of the phases should satisfy f g0 þ f T ¼ f g, and f r þ f g0 þ f T ¼ 1:0. Here, although fg,fg0, and fT are the volume fractions of the glassy phases, they refer to glassy phases with different thermo-mechanical history: fg is the total volume fraction of the glassy phase in the material at a given temperature; fg0is the glassy phase existing since the start of an analysis; fT is the glassy phase formed during cooling. Asdiscussed below, due to the different thermo-mechanical history, the FGP carries the deformation only afterits formation whilst the IGP carries the deformation since an analysis starts.

With three phases defined above, the total Helmholtz energy can be written as

H total ¼ f rHr þ f g0Hg0 þ f T HT . (2)

The total stress therefore is

T ¼ f rTr þ f g0Tg0 þ f TTT . (3)

The evolution rule for volume fractions and the determination of deformations and stresses for individualphases are described below. It is noted that Eqs. (1)–(3) follow the simple Voigt mixing rule for composite.Although more sophisticated composite theories can be used to capture the transition of the energy, this paperfocuses on developing a theoretical framework that captures the shape memory effects and we therefore usethis simplest model.

3.2. Evolution rules for volume fractions

As the temperature changes, the volume fractions of individual phases will change. In this model, we onlyconsider the equilibrium phase separation condition, i.e., the volume fractions of RP and glassy phase are solefunctions of the temperature. The volume fraction of each phase as a function of the temperature is defined as

f r ¼1

1þ exp½�ðT � T rÞ=A�, (4a)

f g ¼ 1�1

1þ exp½�ðT � T rÞ=A�, (4b)

Following the argument of dividing the glassy phase into an initial glassy phase and a frozen glassy phase, one can also distinguish the

bery phase into an initial rubbery phase and a softening rubbery phase. Such a distinction for the rubbery phase is only necessary when

increases the temperature after the initial mechanical loading. However, such a case is not a typical thermo-mechanical loading in

P applications. We therefore do not distinguish these two rubbery phases.


where A is a parameter that characterizes the width of the phase transition zone, Tr is a reference temperatureand is close to Tg. Note that these equations are similar to the VTF functions (Fulcher, 1925; Vogel, 1921;Tammann and Hesse, 1926), which are typically used to characterize the viscosity of a liquid as a function ofthe temperature at temperatures above Tg. As in VTF equations, Tr is typically a temperature some distancebelow the Tg.

The volume fraction of RP is defined by Eq. (4a) during cooling and heating. For the glassy phase, however,the IGP and the FGP evolve differently during cooling and reheating. Prior to a thermal loading, the initialvalues of the IGP and the FGP are

f g0jt¼0 ¼ f gjt¼0, (5a)

f T jt¼0 ¼ 0:0. (5b)

During cooling, Dfg volume fraction of the RP transforms into the glassy phase. Here, we define that the entireDfg becomes the FGP, i.e.,

f g0 ¼ f gjt¼0, (6a)

f T jt¼t2 ¼ f T jt¼t1 þ Df g, (6b)

where t24t1, t2 is the time immediately after t1.During reheating, Dfg volume fraction of glassy phase transforms into the RP. Here, both the IGP and the

FGP transform into RP. We assume the IGP and the FGP transform into the RP in a similar way. Therefore,the partition of these two phases depends on the relative volume fraction ratio, i.e.,

Df g0 ¼f g0

f g0 þ f T

Df g, (7a)

Df T ¼f T

f g0 þ f T

Df g, (7b)

where Dfg0 is the volume fraction from the IGP, DfT is the volume fraction from the FGP. Therefore,

f g0jt¼t4 ¼ f g0jt¼t3 � Df g0, (8a)

f T jt¼t4 ¼ f T jt¼t3 � Df T , (8b)

where t44t3, t4 is the time immediately after t3.

3.3. Deformations and stresses

3.3.1. Rubbery phase

As demonstrated in Section 2, the material response at the temperature above Tg shows rubber-likehyperelastic behavior. It is therefore possible to capture this behavior using a hyperelastic model, such as theMooney–Rivlin model, Arruda–Boyce eight chain model, and the Ogden model, etc. For comparisons of thesemodels, one is referred to Treloar (1958) and a review paper by Boyce and Arruda (2000). For an isotropichomogeneous elastomer, the Langevin chain based Arruda–Boyce eight-chain model (Boyce and Arruda,1993) captures the hyperelastic behavior of the material up to large stretches and is used here to represent thematerial behavior of the RP. The Cauchy stress tensor is defined as

Tr ¼mr3J

ffiffiffiffiffiffiNr

p

lchainL�1

lchainffiffiffiffiffiffiNr

p

� �B0þ kr½J � 1� 3a1ðT � T0Þ�I, (9)

where mr is the initial shear modulus, and Nr is the number of ‘‘rigid links’’ between the two crosslink sites(and/or strong physical entanglements), k is bulk modulus, a1 is the thermal expansion coefficient attemperature above Tg, and T0 is the initial temperature in an analysis. Among these parameters, mr and Nr arethe fitting parameters that can be determined by the stress–strain behavior at high temperatures; a1 isdetermined from experimental measurements; the bulk modulus can be chosen to be two or three order of

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componentViscoplasticspring

Hyperelastic

Fig. 7. One-dimensional rheologic representation of viscoplastic model of a glassy polymer.


magnitude higher than mr to represent the incompressibility of rubbery behavior. In this model, F is the overalldeformation gradient and contains a small volumetric strain due to both material compressibility and thermal

expansion. The volumetric strain is taken out through F ¼ ð1=J1=3ÞF, where J ¼ det[F]. B is the isochoric left

Cauchy–Green tensor, B ¼ FFT, and B

0¼ B� 1

3trðBÞI is the deviatoric part of B. lchain ¼

ffiffiffiffiffiffiffiffiffiffiI1=3

qis the stretch

on each chain in the eight-chain network, and I1 ¼ trðBÞ is the first invariant of B. L is the Langevin functiondefined as

LðbÞ ¼ coth b�1

b. (10)

3.3.2. Initial glassy phase

The deformation of the IGP is represented by the total deformation gradient, F. The viscoplastic behaviorof SMPs at low temperatures can be captured by many models (Simo, 1987; Govindjee and Simo, 1991;Govindjee and Reese, 1997; Reese and Govindjee, 1998; Miehe and Keck, 2000; Lion, 2000; Boyce et al.,1988a–c, 1989, 2001; Bergstrom and Boyce, 1998; Qi and Boyce 2005; Weber and Anand, 1990). The methodby Boyce and co-workers is used here, though the constitutive framework can accommodate other internalvariable based viscoplastic models. The viscoplastic behaviors of a polymer can be modeled by decomposingthe stress response into an equilibrium time-independent behavior and a non-equilibrium time-dependentbehavior. Fig. 7 shows a 1D rheological representation. The total stress is

Tg0 ¼ Trg0 þ Tve

g0. (11)

The hyperelastic spring (equilibrium response) can be modeled using the Arruda–Boyce eight-chain model, asoutlined in the previous section,3 but with different material parameters, i.e.,

Trg0 ¼

mg3J

ffiffiffiffiffiffiNg

plchain

L�1lchainffiffiffiffiffiffi

Ng

p !

B0þ kg½J � 1� 3a2ðT � T0Þ�I. (12)

Note here, since the Eq. (12) is for the material behaviors below Tg, the thermal expansion coefficient attemperatures below Tg a2 is used here. For the viscoplastic deformation, the elastic deformation gradient isdetermined from the multiplicative decomposition of F into elastic and viscoplastic contributions,

Fe ¼ FFv�1 , (13)

3Since hyperelastic models (with different material parameters) are used for both RP and IGP, one can combine these two into one

model, which will reduce the total number of material parameters. However, as discussed in Appendix B, the hyperelastic model used in

the glassy polymer model and the hyperelastic model for rubbery behavior captures different features of stress-strain behaviors of the

material at different temperatures, using two separate models offers the flexibility to capture the more general behavior of SMPs. In

addition, the short range deformation mechanisms of the glass and long range deformation mechanisms of the rubber are fundamentally

different and warrant different descriptions.


where Fv is a relaxed configuration obtained by elastically unloading Fe. The stress due to viscoplasticdeformation can be calculated using F

e, i.e.,

Tveg0 ¼

1

Je ½Le : Ee � a2ð3lg þ 2GgÞðT � T0ÞI�, (14a)

where Je ¼ detðFeÞ, Ee ¼ ln Ve, Ve ¼ FeRe, and Le is the fourth order isotropic elasticity tensor and

Le ¼ 2GgIþ lgI� I, (14b)

where Gg and lg are Lame constants, I is the fourth order identity tensor and I is the second order identitytensor.

To evolve Fe in Eq. (14a), we apply the decomposition of the spatial velocity gradient

l ¼ _FF�1¼ _F

eFe�1 þ FelvFe�1, (15)

where _F is the material velocity gradient, lv ¼ _FvFv�1 is the spatial velocity gradient. The lv can be further

decomposed into:

lv ¼ _FvFv�1 ¼ Dv þWv, (16)

where Dv and Wv are the rate of stretching and the spin, respectively. We take Wv¼ 0 without losing

generality in the isotropic case as shown in Boyce et al. (1988b). The viscoplastic stretch rate Dv isconstitutively prescribed to be

Dv ¼_gvffiffiffi2p

tT0

g0, (17)

where Tg0 is the stress acting on the viscoplastic component convected to its relaxed configurationTg0 ¼ ReTg0R

e, the prime denotes the deviator; t is the equivalent shear stress defined as

t ¼ 12T0

g0 � T0

g0

h i1=2. (18)

_gv Denotes the viscoplastic shear strain rate and is constitutively prescribed to take the form

_gv ¼ _g0 exp �DG

kT1�

ts

� �� , (19)

where _g0 is the pre-exponential factor, DG is the zero stress level activation energy, and s is the athermal shearstrength, which represents the resistance to the viscoplastic shear deformation in the material. To furtherconsider the softening effects observed in the experiments, the evolution rule for s as defined below can beused,

_s ¼ h0ð1� s=ssÞ_gv, (20a)

and the initial condition

s ¼ s0 when gv ¼ 0, (20b)

where s0 is the initial value of athermal shear strength, and ss is the saturation value. h0 is a parameter, and _gv

is defined in Eq. (19). When s04ss, Eqs. (20) represent an evolution rule that characterizes a softeningbehavior of a material.

3.3.3. Frozen glassy phase

During cooling, new glassy phase will be formed (‘‘frozen’’ from the RP). As discussed earlier, thedeformation in the RP is also ‘‘frozen’’, implying that the newly formed glassy phase does not inherit thedeformation of the rubber phase and will behave as an undeformed material, i.e., the newly formed glassyphase takes the current configuration of the RP as its initial configuration. The initial deformation of the FGPupon formation is zero. However, due to the 3D nature of deformation, the vanishing deformation of thetransforming the RP will cause a redistribution of deformation in the material, which in turn will cause a newdeformation in the FGP. Here, we emphasize that this new deformation is very different from the overall

ARTIC

LEIN

PRES

S

Table 1

Summary of the model

Total stress T ¼ frTr+fg0Tg0+fTTT

Phases Rubbery phase fr Initial glassy phase: fg0 Frozen glassy phase: fT

Volume fraction evolution rulef r ¼

1

1þ exp½�ðT � T rÞ=A�f g ¼ 1�

1

1þ exp½�ðT � T rÞ=A�

Df g0 ¼

0 cooling

f g0

f g

Df g heating

8><>: Df T ¼

Df g cooling

f T

f g

Df g heating

8><>:

Deformation gradient F FDFnþ1

T ¼Fnþ1ðFnÞ

�1 if DTa0

I if DT ¼ 0

(and Fnþ1

T ¼ DFnþ1T Fn

T

StressTr ¼

mr3J

ffiffiffiffiffiNr

p

lchainL�1

lchainffiffiffiffiffiffiNr

p

� �B0þ kr½J � 1� 3a1ðT � T0Þ�I Tr

g0 ¼mg3J

ffiffiffiffiffiNg

p

lchainL�1

lchainffiffiffiffiffiffiffiNg

p !

B0þ kg½J � 1� 3a2ðT � T0Þ�I

Same as IGP but using FT

Tveg0 ¼

1

Je½Le : Ee � a2ð3lg þ 2GgÞðT � T0ÞI�

_gv ¼ _g0 exp �DG

kT1�

ts

� ��

H.J

.Q

iet

al.

/J

.M

ech.

Ph

ys.

So

lids

56

(2

00

8)

17

30

–1

75

11741


deformation of the material. Since this new deformation is due to the redistribution of overall deformation,the incremental deformation gradient for the FGP DFnþ1

T is

DFnþ1T ¼

Fnþ1ðFnÞ�1 if DTa0;

I if DT ¼ 0;

((21)

where Fn and Fn+1 are the overall deformation gradient at the increment n and n+1, respectively. Here, weonly consider a monotonic thermal loading, i.e., the cooling or reheating processes are not interrupted byisothermal mechanical loading. This implies that during the cooling or reheating processes the onlydeformation is due to the temperature change and mechanical constraints. Note that such a monotonicthermal loading is a typical loading condition in most SMP applications.

Strictly speaking, the FGPs formed at different time during the cooling process have a different deformationhistory. Computationally, tracking the phases formed at different time will be prohibitively expensive.Therefore, we simplify this process by assuming a temporal and spatial average, i.e., we assume that all theFGPs have the same deformation gradient and therefore will not distinguish the FGPs formed at differenttime. The total deformation gradient acting on the FGP is

Fnþ1T ¼ DFnþ1

T FnT , (22)

where FnT and Fnþ1

T are the deformation gradient for the FGP at the increment n and n+1, respectively. It isnoted that in a typical SMP applications, the programming is achieved by holding the sample in a deformedshape then lowering the temperature. In this process, the new deformation in the FGP should be small ascompared to the deformation imposed to the material at high temperatures. Therefore, Eq. (22) provides areasonable approximation of the deformation gradient in the FGP.

The stress in the FGP can then be calculated using Eqs. (9)–(20), with FT instead of F in Eqs. (9) and (13).It is also noted that the glassy phases formed at different temperature and time history may haveslightly different mechanical properties. In this model, for the sake of simplicity, we assume all the glassyphases have the same mechanical properties, and therefore, the material parameters for all the glassy phasesare the same.

The definition of the FGP plays a key role in capturing the shape memory effects. The advantage of thisdefinition is that it does not require the introduction of a 3D finite deformation equivalent ‘‘stored strain’’,which was used in a 1D small deformation constitutive model (Liu et al., 2006).

3.4. Summary of the model

As demonstrated in the previous section, SMPs demonstrate very complicated thermo-mechanicalbehaviors. To capture these behaviors, a comprehensive model is necessary. Table 1 summarizes the model.One important feature of this model is that it considers separately material behaviors at temperatures aboveand below of Tg. This allows the use of other type of models for rubbery behavior and glassy behavior andcapture the shape memory effects. This feature also simplifies the process of material parameter identification.For example, as shown in Appendix A, using the experimental stress–strain curve at the temperature wellabove Tg, the material parameters associated with the RP can be identified. Also, using the experimentalstress–strain curve at the temperature well below Tg, the material parameters associated with the glassy phasecan be identified.

4. Results

The constitutive model was implemented in ABAQUS as a UMAT. In this section, finite elementsimulations using this UMAT are compared with the experimental results presented in Section 2. Followingthe procedure described in Appendix A, the material parameters in the model were identified and listed inTable 2.

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Table 2

Material parameters used in the simulations

Model components Material parameters Values

Thermal expansion coefficients a1(K�1) 2.53� 10�4

a2(K�1) 1.48� 10�4

Volume fraction evolution A 7.0

Tr (K) 297

Rubbery phase mr (MPa) 0.8

Nr 17

kr (MPa) 1� 103

Glassy phase mg (MPa) 0.1

Hyperelastic spring Ng 17

kg (MPa) 1� 103

Viscoplastic component Gg (MPa) 0.47� 103

lg (MPa) 1.89� 103

DG (� 10�19) 0.92

_g0 (s�1) 52

s0 (MPa) 56

ss (MPa) 28

h0 (MPa) 400


4.1. Isothermal uniaxial compression simulation

Fig. 8 shows the comparison of the results from numerical simulations and experiments of isothermaluniaxial compressions at different temperatures and strain rates. In these simulations, four asymmetric fournode elements were used. The uniaxial deformation was imposed by using a rigid surface to compress the topsurface of the SMP. It is noted that since the deformation in this case is homogenous, the choice of number ofelements does not affect the simulation results. It can be seen that the model captures the isothermalstress–strain behavior of the material at different temperatures and different strain rates.

4.2. DMA simulations to determine Tg

The temperature-dependent mechanical properties of the proposed model were evaluated by simulating aDMA experiment dynamic under uniaxial compression and tension. In the simulations, a cylindrical sample(modeled by four axisymmetric four node elements) was subjected to an isothermal uniaxial cyclic loadingfollowing a sinusoidal waveform at 1Hz with a maximum strain of 4%. The imposed strain was achieved byimposing a uniform vertical displacement on the top surface of the model. Fig. 9 shows tan d vs temperaturebehavior from the experiment and the numerical simulations. tan d was obtained from the phase differencebetween the imposed strain and the resulting stress. Due to the difference between the model (uniaxial loading)setup and experimental setup (three point bending), tan d values were normalized by the maximum values inthe experiment and numerical simulation. From Fig. 9, the model shows the general features of a tan d curvewhich starts from a small value and increases to a peak then decreases. The numerical simulations predict a Tg

of �53 1C, which is slightly higher than the experimentally obtained Tg of �49 1C. Note that DMA test resultwas not used to obtain the material parameters in the model and hence provides a verification of the modelprediction.

4.3. Free recovery

The numerical simulation of a free recovery experiment as described above was conducted to demonstratethe shape memory effect of the model. Here, the same model for uniaxial compression simulations was used.Fig. 10 shows the comparison of images from experiment and numerical simulations at different points in the

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00

10

20

30

40

50

60

70

00

10

20

30

40

50

60

70

-Tru

e S

tress (

MP

a)

-True Strain

0.25 0.5 0.75 1

-Tru

e S

tress (

MP

a)

-True Strain

0.25 0.5 0.75 1

� = 0.01s-1.T = 20°C � = 0.1s-1.

T = 20°C

� = 0.01s-1.T = 0°C

� = 0.1s-1.T = 0°C

Simulations

Experiments

T = 60°C

T = 30°C

T = 20°C

T = 10°C

T = 0°C

Simulations

Experiments

Fig. 8. Numerical simulations of stress–strain behavior of the SMP from isothermal uniaxial compressions: (A) at different temperatures

and (B) at different strain rates. The figures also show the comparisons with experiments.


thermo-mechanical history. Fig. 11 shows the comparison of the deformation recovery measured from theimages taken during the experiment and from numerical simulations. It can be seen that the constitutive modelcaptures the shape memory effects after cooling and the shape recovery after reheating.

4.4. Constrained recovery

The numerical simulation of a constrained recovery experiment as described above was also conducted tocharacterize the force recovery feature of the constitutive model. Fig. 12 shows the comparison between thenumerical simulation and the experiment. The stress was normalized by the maximum compressive stressimmediately after loading at the high temperature. Numerical simulation predicts the decrease of the forceduring cooling and the increase of the force to the level before cooling during heating. It also captures that thestress recovery after the cooling-heating cycle. However, the simulation predicts the zero reaction force at a

ARTICLE IN PRESS

00

tan�

/ (t

an�)

ma

x

1.2

1

0.8

0.6

0.4

0.2

Temperature (°C)

20 40 60 80 100

Simulation

Experiment

Fig. 9. tan d vs temperature behavior from the experiment and the numerical simulations.

Rigid Plate

SMP

Fig. 10. Snapshot images of SMP deformations during the free recovery experiment and simulation. A and A0: before compression,

T ¼ 100 1C; B and B0: immediately after compression, T ¼ 100 1C; C and C0: after cooling, T ¼ 0 1C; D and D0: after the platen was

removed, T ¼ 0 1C; E and E0: after reheating, T ¼ 100 1C. The prime denote images from the simulation.


higher temperature (�33 1C) than the temperature observed in the experiment (�7 1C). In addition, duringheating, a large overshoot in the stress was observed in the experiments but is not captured by the model. Asdiscussed in Section 2.5, the stress decrease during cooling may be delayed by the heat transfer and the stressovershoot observed in the experiments is conjectured to be caused by structural relaxation during glasstransition, which are not considered in current model. Heat transfer and structural relaxation are currentlyunder investigations by the authors and will be included into the model in the future.

5. Conclusions

The finite deformation thermo-mechanical behaviors of thermally induced SMPs were investigated in thispaper. It was shown experimentally that SMPs undergo dramatic stress–strain behavior change from arubbery hyperelastic behavior at temperatures above Tg to a glassy viscoplastic behavior at temperaturesbelow Tg. These results were applied to guide the development of a 3D finite deformation constitutive model.The model employed the concept of the first order phase transition to describe the change in the deformationmechanism from entropic elasticity of the rubbery state to viscoplasticity of the glassy state as the temperature

ARTICLE IN PRESS

0 20 40 60 80 1000.8

0.84

0.88

0.92

0.96

1

Str

etc

h R

ati

o

Temperature (°C)

Heating

Simulation

Experiment

Fig. 11. Deformation recovered during reheating in the free recovery case.

00

0.4

0.8

1.2

1.6

2

Norm

aliz

ed T

rue S

tress

Temperature (°°C)

20 40 60 80 100

Simulation

Experiment

Heating

Heating

CoolingCooling

Fig. 12. Stress response in the constrained recovery case. The stress is normalized by the maximum compressive stress immediately after

loading.


traverses Tg. It was assumed further that the newly formed FGP does not inherit the deformation of thetransforming rubber phase; the new deformation incurred by the FGP was modeled as a redistribution of totaldeformation. The advantage of this assumption is that it eliminates the requirement of defining a ‘‘storedstrain’’ used in a previous 1D small deformation constitutive model. The model was shown to capture theshape memory effects and shape recovery of SMPs. A significant feature of this constitutive framework is thatone is not limited to the models for elastomers and glassy polymers used in this paper; in fact, one can useother models that are suitable for the specific material systems. This feature also allows modeling many othermaterial systems that demonstrate the shape memory effect.

The model in the paper was implemented into a UMAT in ABAQUS. Numerical simulations of uniaxialcompression experiments under isothermal conditions using this UMAT agreed with the experimentallyobtained stress–strain behavior. For the case with thermal cooling/heating, the model captured the shapememory effects and shape recovery. However, the model did not fully capture the stress recovery, implying acomplicated transition behavior that was not captured by the simple volume fraction evolution rule proposed


in the current model. In addition, the model does not account the temperature rate effects to the thermo-mechanical behaviors of SMPs. The effects of thermal load are currently under investigation and will bereported in the future.

This model enables the investigations of more complicated 3D deformation behaviors of SMPs, which arecommon in most of material applications. We are currently investigating a few representative 3D deformationcases, such as indentation of SMPs; the results from these studies will be presented in the future.

Acknowledgments

The authors gratefully acknowledge the support from a NIH Grant (EB 004481), a NSF career award toHJQ (CMMI-0645219), NSF-Sandia initiative (Sandia National Laboratories, 618780), Laboratory DirectedResearch and Development program at Sandia National Laboratories (105951), and US Army ResearchOffice (W31PQ-06-C-0406). Discussions with Professors Martin Dunn, Kenneth Gall, and Drs. Richard Vaia,Jeffery Baur, Richard Hall, and Mr. Jason Hermiller are gratefully appreciated.

Appendix A. Guideline for material parameter identification

In the model developed in this paper, there are 17 material parameters, among which 15 are fittingparameters. Here, we provide a procedure to estimate these parameters, which serve as initial values. The finalvalues of these parameters should be subject to fitting experimental curves, but should be close to the initialestimated values.

In order to identify the material parameters, three types of experiments should be conducted. The firstexperiment is a DMA test to determine the glass transition temperature Tg of the material. Althoughdetermining Tg does not directly identify the material parameters in the model, it provides a guide to decidethe temperature ranges used in the following two types of experiments. The second experiment is to determinethe thermal expansion coefficient. The third type of experiment is isothermal uniaxial compression experi-ments at different strain rates and at different temperatures ranging from well below Tg to well above Tg.

A.1. Coefficients of thermal expansion (CTE)

As discussed above, CTE experiences a change as the temperature traverses through the Tg. The CTE attemperatures above Tg gives a1, and the CTE at temperatures below Tg gives a2.

A.2. mr and Nr

mr And Nr are the parameters that characterize the hyperelastic behavior of the material at hightemperatures, therefore can be determined by fitting the isothermal uniaxial compression stress–strain curve attemperatures above Tg. For example, in this paper, we used the stress–strain curve at T ¼ 80 1C to obtainmr ¼ 0.8MPa and Nr ¼ 17.

A.3. Gg, lg, and mg

Lame constants Gg and lg are related to Young’s modulus Eg and Poisson’s ratio vg by Gg ¼ Eg/(2+2vg)and lg ¼ Egvg/b(1+vg)(1�2vg)c. For polymers, Poisson’s ratio is typically 0.4, we therefore take vg ¼ 0.4 sothat Gg ¼ 0.36Eg and lg ¼ 1.43Eg. Since Eg and mg characterize the initial modulus of glassy behavior, theslope of the isothermal uniaxial compression curve at the temperature well below Tg can be used. In this paper,the curve at T ¼ 0 1C was used to obtain Eg+3mgE1300Mpa.

A.4. A and T r

The parameters A and Tr in the volume fraction evolution rule Eq. (4) can be determined from the initialmodulus measurements. From Eqs. (3), (9), and (14a), the initial Young’s modulus under the isothermal

ARTICLE IN PRESS

25 50 75

Init

ial M

od

ulu

s (

MP

a)

103

102

101

100

Temperature (°C)

0 100

Simulation

Experiment

Fig. A1. Dependence of initial modulus on temperatures.


condition is

E ¼ 3f rmr þ f gðEg þ 3mgÞ. (A.1)

Combining with Eq. (4), we have

E ¼1

1þ exp½�ðT � T rÞ=A�3mr þ 1�

1

1þ exp½�ðT � T rÞ=A�

� �ðEg þ 3mgÞ. (A.2)

Fig. A1 shows a plot of the initial Young modulus (small-strain) as a function of the temperature for thematerial studied in this paper. Tr and A can then be identified by fitting the curve using Eq. (A.2), which yieldTr ¼ 24 1C, and A ¼ 7 (see Fig. A1). It is also noticed that Eq. (A.2) cannot provide a best fit for the initialmodulus measured from experiments, implying Eq. (4) might not provide a satisfactory approximation of thetransition. Other models may be used to replace Eq. (4) to further improve the predictability of the model.

A.5. DG, _g0, s0, ss, h0

The material parameters DG, _g0, s0, ss, h0 in the viscoplastic component for the glassy polymer behavior canbe estimated from the isothermal uniaxial compression curves at different strain rates. Here, we use the curvesat T ¼ 0 1C to illustrate this process. In a uniaxial compression test, the equivalent shear stress t and shearstrain g are related to the uniaxial stress and strain by

t ¼ s=ffiffiffi3p

and g ¼ffiffiffi3p

�. (A.3)

From Fig. 4b, the yielding stresses are s1 ¼ 62:4MPa for _�1 ¼ 0:1=s, and s2 ¼ 57:6MPa for _�2 ¼ 0:01=s.Therefore, the corresponding equivalent shear yield stresses and equivalent shear strain rates are: t1 ¼36:0MPa for _g1 ¼ 0:173=s, and t2 ¼ 33:3MPa for _g2 ¼ 0:0173=s. Since the yielding point in the stress–straincurve corresponds to the onset of significant viscoplastic flow, and thus the onset for the evolution of athermalshear strength s, it is reasonable that at the yielding point, s ¼ s0. From Eq. (19), we have

_g1 ¼ _g0 exp �DG

kT1�

t1s0

� �� , (A.4a)

_g2 ¼ _g0 exp �DG

kT1�

t2s0

� �� . (A.4b)


From Eq. (A.4), we obtain

DG ¼ kTln _g1 � ln _g2t1 � t2

s0 ¼ 310� 10�29s0. (A.5)

From Fig. 4b, at e ¼ 0.5, the stress–strain curve reached a plateau, implying the athermal shear strength s

approached the saturation value ss, or s�ss. At e ¼ 0.5, s3 ¼ 35:7MPa for _�1 ¼ 0:1=s, and s4 ¼ 33:3MPa for_�2 ¼ 0:01=s. The corresponding equivalent shear yield stresses and equivalent shear strain rates are: t3 ¼20:6MPa for _g1 ¼ 0:173=s, and t4 ¼ 19:2MPa for _g2 ¼ 0:0173=s. Following the procedure similar to the oneto obtain Eq. (A.5), we have

DG ¼ kTln _g1 � ln _g2t4 � t3

ss ¼ 621� 10�29ss. (A.6)

From Eqs. (A.5) and (A.6), we have

s0 � 2ss. (A.7)

In order to estimate h0, we obtain from Eq. (20a)

ds

s=ss � 1¼ h0 dg. (A.8)

Integrating Eq. (A.8),

ss lns

ss� 1

� �� ln

s0

ss� 1

� �� ¼ �h0ðg� g1Þ. (A.9)

We assume that s decreases to (s0+ss)/2 as the stress in the stress–strain curve decreases to ðs1 þ s3Þ=2.At s ¼ ðs1 þ s3Þ=2 ¼ 49:1MPa, e ¼ 0.13, or g ¼ 0.23, and at s1 ¼ 62:4MPa, e1 ¼ 0.07 or g1 ¼ 0.12. FromEqs. (A.9) and (A.7), we have

h0 ¼ �ssln s

ss� 1

� ln s0

ss� 1

g� g1

� 6:3ss. (A.10)

Eqs. (A.7), (A.6), and (A.10) provide the guideline to estimate s0, DG, and h0, once ss is known. To startwith, we can assume ss is slightly higher than t3, for example, ss�25MPa. From Eqs. (A.7), (A.6), (A.10), wehave s0�50MPa DG�1.5� 10�19 J, h0�150MPa.

Appendix B. Contributions to stress–strain behaviors from the hyperelastic models

In the proposed model for SMPs, the hyperelastic branch in the glassy polymer model contributes mainly tothe post-yielding behavior at low temperatures; the hyperelastic model for RP affects the high temperaturebehavior. Fig. B1 shows the comparison between the model with one hyperelastic model (with mg ¼ mr0.8MPa, and Ng ¼ Nr ¼ 17) for all phases and the model with hyperelastic models with different materialparameters for glassy phase and RP (with mg ¼ 0.1MPa, mr ¼ 0.8MPa, and Ng ¼ Nr ¼ 17). From Fig. B1, itcan be seen that when the same material parameters are used for the hyperelastic models, the model predicts aslightly higher stress at post-yielding at T ¼ 0 1C. At T ¼ 100 1C, there is no difference between the twobecause the hyperelastic model in the glassy polymer model does not function at high temperatures (due tofgE0). Fig. B2 shows the effects of Ng in the glassy polymer model to the stress–strain behaviors at T ¼ 0 1C(with mg ¼ mr 0.8MPa, and Nr ¼ 17). As Ng becomes smaller, post-yield hardening can be observed. For thematerial studied in this paper, the material does not demonstrate the hardening behavior, which results in thesimilar material parameters for the two hyperelastic models. However, it is noted that some other SMPs maydemonstrate the post-yield hardening behavior at low temperatures but do not show dramatic stress increaseat large stretch at high temperatures, the two hyperelastic models offer the capability to capture more generalbehaviors of SMPs.

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0 0.5 10

10

20

30

40

50

60

70

-Tru

e S

tress (

MP

a)

-True Strain

0.25 0.75

�g = 0.8MPa, T = 0°C

�g = 0.1MPa, T = 0°C

�g = 0.8MPa, T = 100°C

�g = 0.1MPa, T = 100°C

Fig. B1. Comparison between the model with one hyperelastic model (with mg ¼ mr 0.8MPa, and Ng ¼ Nr ¼ 17) for all phases and the

model with hyperelastic models with different material parameters for the glassy phase and the rubbery phase.

0 10

10

20

30

40

50

60

70

-Tru

e S

tress (

MP

a)

0.750.50.25

-True Strain

Ng = 17.0

Ng = 2.0

Ng = 1.6

T = 0°C

Fig. B2. Effects of Ng in the glassy polymer model to the stress–strain behaviors at T ¼ 0 1C (with mg ¼ mr 0.8MPa, and Nr ¼ 17).


References

Bergstrom, J.S., Boyce, M.C., 1998. Constitutive modeling of the large strain time-dependent behavior of elastomers. J. Mech. Phys. Solids

46 (5), 931–954.

Boyce, M.C., Arruda, E.M., 1993. Three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J. Mech.

Phys. Solids 41 (2), 389–412.

Boyce, M.C., Arruda, E.M., 2000. Constitutive models of rubber elasticity: a review. Rubber Chem. Technol. 73 (3), 504–523.

Boyce, M.C., Parks, D.M., Argon, A.S., 1988a. Computational modeling of large strain plastic-deformation in glassy-polymers. Abstr.

Pap. Am. Chem. Soc. 196 156-POLY.

Boyce, M.C., Parks, D.M., Argon, A.S., 1988b. Large inelastic deformation of glassy-polymers. 1: Rate dependent constitutive model.

Mech. Mater. 7 (1), 15–33.


Boyce, M.C., Parks, D.M., Argon, A.S., 1988c. Large inelastic deformation of glassy-polymers. 2: Numerical-simulation of hydrostatic

extrusion. Mech. Mater. 7 (1), 35–47.

Boyce, M.C., Weber, G.G., Parks, D.M., 1989. On the kinematics of finite strain plasticity. J. Mech. Phys. Solids 37 (5), 647–665.

Boyce, M.C., Kear, K., Socrate, S., Shaw, K., 2001. Deformation of thermoplastic vulcanizates. J. Mech. Phys. Solids 49 (5), 1073–1098.

Fulcher, G.S., 1925. Analysis of recent measurements of the viscosity of glasses. J. Am. Ceram. Soc. 8, 789–794.

Govindjee, S., Reese, S., 1997. A presentation and comparison of two large deformation viscoelasticity models. J. Eng. Mater. Technol.

Trans. ASME 119 (3), 251–255.

Govindjee, S., Simo, J., 1991. A micro-mechanically based continuum damage model for carbon black-filled rubbers incorporating

Mullins effect. J. Mech. Phys. Solids 39 (1), 87–112.

Lagoudas, D.C., Entchev, P.B., Popov, P., Patoor, E., Brinson, L.C., Gao, X.J., 2006. Shape memory alloys, part II: modeling of

polycrystals. Mech. Mater. 38 (5–6), 430–462.

Lendlein, A., Jiang, H.Y., Junger, O., Langer, R., 2005a. Light-induced shape-memory polymers. Nature 434 (7035), 879–882.

Lendlein, A., Kelch, S., Kratz, K., Schulte, J., 2005b. Shape memory polymers. In: Encyclopedia of Materials: Science and Technology.

Lion, A., 2000. Constitutive modelling in finite thermoviscoplasticity: a physical approach based on nonlinear rheological models. Int. J.

Plasticity 16 (5), 469–494.

Liu, Y.P., Gall, K., Dunn, M.L., McCluskey, P., 2004. Thermomechanics of shape memory polymer nanocomposites. Mech. Mater. 36

(10), 929–940.

Liu, Y.P., Gall, K., Dunn, M.L., Greenberg, A.R., Diani, J., 2006. Thermomechanics of shape memory polymers: uniaxial experiments

and constitutive modeling. Int. J. Plasticity 22 (2), 279–313.

McKenna, G.B., 1989. Glass formation and glassy behavior. In: Booth, C.P.C. (Ed.), Comprehensive Polymer Science. Pergamon,

Oxford, pp. 311–362.

Miehe, C., Keck, J., 2000. Superimposed finite elastic–viscoelastic–plastoelastic stress response with damage in filled rubbery polymers.

Experiments, modelling and algorithmic implementation. J. Mech. Phys. Solids 48 (2), 323–365.

Monkman, G.J., 2000. Advances in shape memory polymer actuation. Mechatronics 10 (4–5), 489–498.

Otsuka, K., Wayman, C.M., 1998. Shape Memory Materials. Cambridge University Press, Cambridge, New York.

Qi, H.J., Boyce, M.C., 2005. Stress–strain behavior of thermoplastic polyurethanes. Mech. Mater. 37 (8), 817–839.

Reese, S., Govindjee, S., 1998. A theory of finite viscoelasticity and numerical aspects. Int. J. Solids Struct. 35 (26–27), 3455–3482.

Scott, T.F., Schneider, A.D., Cook, W.D., Bowman, C.N., 2005. Photo-induced plasticity in cross-linked polymers. Science 308 (5728),

1615–1617.

Simo, J.C., 1987. On a fully 3-dimensional finite-strain viscoelastic damage model—formulation and computational aspects. Comput.

Methods Appl. Mech. Eng. 60 (2), 153–173.

Tammann, G., Hesse, W., 1926. Dependence of viscosity on temperature in supercooled liquids. Z. Anorg. Allgem. Chem. 156, 245–257.

Thamburaja, P., Anand, L., 2002. Superelastic behavior in tension–torsion of an initially textured Ti–Ni shape-memory alloy. Int. J.

Plasticity 18 (11), 1607–1617.

Tobushi, H., Hara, H., Yamada, E., Hayashi, S., 1996. Thermomechanical properties in a thin film of shape memory polymer of

polyurethane series. Smart Mater. Struct. 5 (4), 483–491.

Treloar, L.R.G., 1958. The Physics of Rubber Elasticity. Clarendon Press, Oxford.

Vogel, H., 1921. The law of viscosity change with temperature. Physik Z 22, 645–646.

Warner, M., Terentjev, E.M., 2003. Liquid Crystal Elastomers. Oxford University Press, Oxford, New York.

Weber, G., Anand, L., 1990. Finite deformation constitutive-equations and a time integration procedure for isotropic, hyperelastic

viscoplastic solids. Comput. Methods Appl. Mech. Eng. 79 (2), 173–202.

Yakacki, C.M., Shandas, R., Lanning, C., Rech, B., Eckstein, A., Gall, K., 2007. Unconstrained recovery characterization of shape-

memory polymer networks for cardiovascular applications. Biomaterials 28 (14), 2255–2263.

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