a 3d finite deformation constitutive model for amorphous shape memory polymers: a multi-branch...

Mechanics of Materials 43 (2011) 853–869

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Mechanics of Materials

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A 3D finite deformation constitutive model for amorphous shapememory polymers: A multi-branch modeling approach fornonequilibrium relaxation processes

Kristofer K. Westbrook, Philip H. Kao, Francisco Castro, Yifu Ding, H. Jerry Qi ⇑Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309, United States

a r t i c l e i n f o

Article history:Received 26 October 2010Received in revised form 27 June 2011Available online 6 October 2011

Keywords:Shape memory polymersConstitutive modelsThermomechanical behaviorsFinite deformation

0167-6636/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.mechmat.2011.09.004

⇑ Corresponding author.E-mail address: [email protected] (H. Jerry Qi).

a b s t r a c t

Shape memory polymers (SMPs) are materials that can recover a large pre-deformed shapein response to environmental stimuli. For a thermally activated amorphous SMP, the pre-deformation and recovery of the shape require the SMP to traverse its glass transition tem-perature (Tg) to complete the shape memory (SM) cycle. As a result, the recovery behaviorof SMPs shows strong dependency on both the pre-deforming temperature and recoverytemperature. Generally, to capture the multitude of relaxation processes, multi-branchmodels (similar to the 1D generalized viscoelastic model or Prony series) are used to modelthe time-dependent behaviors of polymers. This approach often requires an arbitrary (usu-ally numerous) number of branches to capture the material behavior, which results in asubstantial number of material parameters. In this paper, a multi-branch model is devel-oped to capture the SM effect by considering the complex thermomechanical propertiesof amorphous SMPs as the temperature crosses Tg. The model utilizes two sets of nonequi-librium branches for two fundamentally different modes of relaxation: the glassy modeand Rouse modes. This leads to a significant reduction in the number of material parame-ters. Model simulation comparisons with a range of thermomechanical experiments con-ducted on a tert-butyl acrylate-based SMP show very good agreement. The model isfurther utilized to explore the intrinsic recovery behavior of an SMP and the size effectson the free recovery characteristics of a magneto-sensitive SMP composite.

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1. Introduction

Shape memory polymers (SMPs) are a class of polymerscapable of demonstrating large recoverable shape changesdue to an environmental stimulus, such as temperature(Chung et al., 2008; Gall et al., 2005; Mather et al., 2009;Westbrook et al., in press), light (Lendlein et al., 2005; Scottet al., 2005), magnetic field (Buckley et al., 2006; Schmidt,2006) or humidity (Huang et al., 2005; Jung et al., 2006).The latter two stimuli can be thought of as indirect ther-mally triggered behavior. In literature, the first and mostwidely studied class of SMPs is thermally triggered. Most

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polymers can exhibit some degree of shape memory (SM)behavior but to achieve large recoverable shape changescrosslinked polymers are preferred as the crosslinks can re-tain the original permanent shape. Compared to shapememory alloys and ceramics, SMPs are inexpensive, easyto manufacture and chemically tunable to achieve biocom-patibility and biodegradability. Most significantly, SMPshave the capability of over 400% strain and strain recovery(Wei et al., 1998). Based on these advantageous features ofSMPs, a wide-range of potential biomedical, aerospace andMEMS applications exist, such as arterial stents, deploy-able structures and actuators and sensors, respectively.

A typical thermomechanical history during an SM cycleis shown in Fig. 1 where both constrained and free recov-ery conditions are shown. Two steps occur during an SM

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Fig. 1. A schematic depicting the thermomechanical history for a shape memory cycle showing the two steps of programming (shape fixing) and recovery(deploying) under both a constrained and free recovery scenario.

854 K.K. Westbrook et al. / Mechanics of Materials 43 (2011) 853–869

cycle: a programming (or shape fixing) step and a recov-ery (or deploying) step. In the programming step, it is re-quired that the polymer first be deformed at atemperature above its glass transition temperature (Tg)into the desired deformed state. This is followed by cool-ing the polymer down to a temperature below Tg whilemaintaining the initial deformation constraint. The recov-ery step, depending on the application where the recov-ery strain or stress is important, can be either underfree or constrained conditions respectively. During con-strained recovery, the constraint provided by the initialdeformation is maintained while the polymer is heatedto a temperature above Tg. A stress overshoot responseis typically observed where a peak stress is larger thanthe stress measured during the initial deformation (Castroet al., 2010; Liu et al., 2003; Qi et al., 2008). Alternativelyduring free recovery, the initial deformation constraint isremoved at a temperature below Tg and then the polymeris heated to a temperature above Tg while the change inthe strain is measured.

Since SMPs can have multiple shapes in their service cy-cle, it is necessary to develop constitutive models that canultimately be used together with simulation tools to assistdesigns of SMP applications. The modeling efforts havesplit in two directions. In one direction, the SM effect isattributed to the formation of phases during cooling whichserve to lock the temporary shape and a loss of the phasesdue to heating when shape recovery occurs. Models basedon this concept include the 1D model by Liu et al. (2006),3D models by Barot et al. (2006, 2008) for semicrystallineSMPs, Qi et al. (2008) and Chen and Lagoudas (2008a,b)for general SMPs. These types of models are consideredphenomenological for amorphous SMPs but can find phys-ical meaning in semicrystalline SMPs. Recently, Westbrooket al. (in press) successfully applied the phase-based mod-eling approach to the one-way and two-way SM effect insemicrystalline poly(cyclooctene) (PCO) SMP. In amor-phous SMPs, the SM effect is due to the material behavior

during the glass transition where the shape deformed athigh temperatures can be fixed at low temperatures asthe material enters the nonequilibrium state (glassy state).The second direction for SMP modeling approaches arebased on this concept and include the early model byTobushi et al. (1996) and recent models by Nguyen et al.(2008) and Castro et al. (2010). Although these modelshave improved the understanding of SMPs, more sophisti-cated models are needed to capture the complicated ther-momechancial behaviors of SMPs. Specifically, most of theprevious models focused on capturing the overall SMbehaviors of SMPs during a thermomechanical cycle whilethe most important feature in SM behavior, the recoverycharacteristics, was not the focal point. For example, Castroet al. (2010), showed that to capture the observed stressovershoot during constrained recovery the nonequilibriumvolume changes and temperature dependent stress relaxa-tion times must be considered. In addition, as will beshown in this paper, the free recovery of an SMP demon-strates strong dependence on temperature, which has notbeen studied experimentally or theoretically.

Thermomechanical behaviors of amorphous SMPs arecomplex as the temperature traverses their Tg duringwhich different relaxation processes are activated. Thisleads to very complicated recovery behaviors in SMPs suchas stress overshoot and temperature dependent recovery.Generally, to capture multiple relaxation processes in poly-mers, multi-branch models (resembling the 1D generalizedviscoelastic model or Prony series) should be used to mod-el the time-dependent behaviors of polymers under iso-thermal conditions (Anand and Ames, 2006; Engels et al.,2009). However, two major drawbacks of this type of mod-els are: (1) the number of branches required to capture thematerial relaxation behavior seems to be arbitrary andlarge and (2) each branch has two or more independentmaterial parameters. These drawbacks often lead to a largeamount of material parameters and limit the practicalapplication of the model. In this paper, based on previous

K.K. Westbrook et al. / Mechanics of Materials 43 (2011) 853–869 855

work on thermo-viscoelasticity for SMPs (Castro et al.,2010; Nguyen et al., 2008), a new multi-branch thermo-viscoelastic model for SMPs is developed. This model re-quires two distinct sets of nonequilibrium branches. Thefirst contains one nonequilibrium branch representingthe glassy mode of relaxation and the second containsmultiple nonequilibrium branches representing the Rousemodes in the rubbery state. Although multiple branchesare used in the second set, they are not independent andtherefore only two material parameters are needed. Inaddition, this new model considers the nonlinear volumechange due to structural relaxation of a glassy polymeraround its Tg. The paper is arranged in the following order.Section 2 presents the results from the thermomechanicalfinite deformation material characterization experimentson SMPs, with a focus on free recovery behavior, whichwas not studied comprehensively in previous modelingwork in literature. Section 3 introduces the proposed 3D fi-nite deformation constitutive model. Section 4 presentsthe comparisons between the experimental results andthe model predictions. The model is further utilized to ex-plore the intrinsic recovery behavior of an SMP and the sizeeffects on the free recovery characteristics of a magneto-sensitive SMP composite. The results for these applicationsare also presented in Section 4.

2. Thermomechanical behavior of shape memorypolymers

2.1. Material and specimen preparations and testingequipment

The SMP material used in this study is an acrylate-basednetwork polymer following Yakacki (2007). The SMP wascast in long rods and machined into various shaped speci-mens depending on the experiments: rectangular speci-mens (20 � 3.6 � 1 mm) for the Dynamic MechanicalAnalysis (DMA) and (22 � 9.72 � 2.81 mm) for the coeffi-cient of thermal expansion (CTE) experiments; cylindricalspecimens (10 mm diameter and 10 mm height) for theisothermal uniaxial compression experiments as well asfree and constrained recovery experiments. For both theDMA and CTE experiments, a DMA (TA Instruments, ModelQ800) was used. For isothermal uniaxial compressionexperiments and SM recovery experiments, an MTS Uni-versal Materials Testing Machine with a load capacity of10 kN (Model Insight 10) equipped with a customizedthermal chamber (Thermcraft, Model LBO) and a tempera-ture controller (Model Euro 2404) were used. A laserextensometer (EIR, Model LE-05) and a mechanical contactextensometer (MTS, Model 632.24F-50) were used to mea-sure the deformation. Following another study (Westbrooket al., 2010), the thermal management is controlled via athermocouple located inside and near the compressionplaten surface and the compression platens have beenredesigned to eliminate large temperature gradients with-in the specimen during testing. Due to the location of thethermocouple near the surface of the compression platens,the thermal management of the specimen is therefore lim-ited by the thermal chamber’s control system. In order to

reduce the effects of friction and hence barreling, Teflon�

sheets were placed between the specimen and compres-sion platens.

2.2. DMA and CTE experiments

DMA and CTE experiments are performed to character-ize the glass transition behavior of the SMP. For the DMAexperiment, the specimen was heated to 100 �C and al-lowed 30 min to reach thermal equilibrium. A preload of5 kPa was applied and then an initial strain with a magni-tude of 0.2% was applied and oscillated at 1 Hz with apeak-to-peak amplitude of 0.1% while the temperaturewas decreased from 100 to 0 �C at a rate of 1 �C/min. Thetemperature in the chamber was held at 0 �C for 30 minand then increased to 100 �C at the same rate. This was re-peated multiple times and the data from the last coolingstep is reported. Fig. 2A shows the DMA experiment resultsof the temperature dependence of storage modulus and tand, which is defined as the ratio between the loss modulusand the storage modulus. The Tg with a value of 42 �C isidentified as the peak temperature of the tan d curve. TheCTE measurement was performed utilizing the same ten-sile setup in conjunction with the DMA hardware. Thetemperature in the DMA chamber was set to 120 �C for30 min to reach thermal equilibrium. To measure thechange in the specimen’s length during thermal expan-sion/contraction, a constant and relatively small tensileforce of 1 mN was applied and maintained by the top gripduring the entire experiment. The temperature was de-creased from 120 to �20 �C at a rate of 1 �C/min. Afterreaching �20 �C, the temperature was then increased to120 �C at the same rate. This thermal cycle was repeatedthree times and only the data from the last cooling stepis reported. Fig. 2B shows the experimental results of theCTE measurement for the temperature range from 0 to100 �C. As shown in the figure, the linear regressions attemperatures above and below Tg are taken to be therespective rubbery and glassy linear CTE values. Here theglassy and rubbery linear CTE values are determined tobe ag = 1.25 � 10�4/�C and ar = 2.52 � 10�4/�C,respectively.

2.3. Isothermal uniaxial compression experiments

In an isothermal test, the specimen was placed at thecenter of the bottom compression platen and the chambertemperature was set to the desired value. After allowingthe specimen 10 min for thermal equilibration, the topcompression platen was brought into contact with thetop surface of the specimen. Uniaxial compression experi-ments were conducted at strain rates of 0.01/s and 0.1/s. Toexplore the rubbery and glassy behavior, the isothermaluniaxial compression tests were performed for tempera-tures between 0 and 100 �C in 10 �C increments. Fig. 3Ashows the stress–strain behavior at a strain rate of 0.01/sfor various temperatures. The isothermal compression re-sults show that the SMP displays typical hyperelasticbehavior in the rubbery region above Tg and typical glassybehavior below Tg. For temperatures below Tg with increas-ing strain, a yield point is exhibited followed by a flow

(A)

0 0.1 0.2 0.3 0.40

10

20

30

40

50

60

70

Strain

Com

pres

sive

Str

ess

(MPa

) 0oC

10oC

20oC

30oC

60oC

(B)

0 0.1 0.2 0.3 0.40

10

20

30

40

50

60

70

Strain

Com

pres

sive

Str

ess

(MPa

)10oC

30oC

60oC

0.1/s0.01/s

Fig. 3. Isothermal compression results for: (A) different temperatures at astrain rate of 0.01/s and (B) the comparison between strain rates of 0.01/sand 0.1/s.

(A)

0 20 40 60 80 100100

101

102

103

104

Tg = 42oC

Stor

age

Mod

ulus

(MPa

)

Temperature (oC)0 20 40 60 80 100

0

0.5

1

1.5

2

tan

δ

Storage Modulustan δ

(B)

0 20 40 60 80 100−0.025

−0.02

−0.015

−0.01

−0.005

0

αr = 2.52x10−4 / oC

Tg = 33 oC

αg = 1.25x10−4 / oC

Temperature (oC)

Ther

mal

Str

ain

Fig. 2. Experimental results from the (A) DMA, (B) CTE (with rubbery andglassy CTE values). For each experiment, the respective method fordetermining the Tg is used and its value is shown on the graph.

Fig. 4. Thermomechanical history schematic showing both constrained and free recovery experiments during a shape memory cycle. The differencebetween the constrained and free recovery experiments occurs at the end of stabilization step at TL where the platen remains at its initial position duringloading or is removed entirely to allow for thermal expansion for TH2 > TH1 experiments, respectively.


(A) 1.5


behavior (softening and then hardening). As is often thecase with glassy polymers, the mechanical response highlydepends on the strain rate as shown in Fig. 3B.

10 20 30 40 50 600

0.5

1

Temperature (oC)

Com

pres

sive

Str

ess

(MPa

)

Cooling

Heating

(B)

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

Time Since Beginning of Heating (min)

Stra

in R

ecov

ery

Rat

io

TH2 = 30oC

TH2 = 35oC

TH2 = 40oC

TH2 = 50oC

Fig. 5. Shape memory cycle recovery experiment results for: (A) con-strained recovery and (B) free recovery at various recovery temperatures.In (B), the times where the temperature reaches TH2 in increasing orderoccurs at 4, 6, 8 and 12 min.

2.4. Shape memory cycle: constrained and free recoveryexperiments

During an SM cycle experiment, both constrained andfree conditions during the recovery step were explored;the thermomechanical history for this test is shown inFig. 4. As shown in the figure, the platen position dictateswhich of these conditions is explored. For these experi-ments, the cylindrical SMP specimen was initially allowedto equilibrate at the programming temperature TH1 andthen deformed to a compressive strain of 20% at a strainrate of 0.01/s. The specimen was then allowed 10 min torelax before the temperature was reduced to the shape fix-ing temperature TL at a rate of 2.5 �C/min. Once the fixingtemperature was reached, the specimen was given60 min to stabilize. It was here at the end of the stabiliza-tion step where the difference between constrained andfree recovery occurs. For a constrained recovery experi-ment, the platen was held in place still using the mechan-ical extensometer control while the stress evolution wasrecorded using the load cell. For a free recovery experi-ment, the upper compression platen providing the initialdeformation was withdrawn and the evolution of theshape (or strain) recovery in the specimen was measuredusing the laser extensometer. After the stabilization stepat TL, the temperature was increased to the recovery tem-perature TH2 at the same rate of cooling and once the tem-perature reached TH2 the material was given time tostabilize so that the total combined time for the heatingand stabilize steps was 60 min.

The various (programming, fixing and recovery) tem-peratures used in the recovery experiments are summa-rized in Table 1. TL is chosen to be well below Tg toensure a good fixity can be obtained. The results from theconstrained and free recovery experiments are shown inFig. 5. During constrained recovery (Fig. 5A) a stress over-shoot peak for an initial compression deformation is ob-served on the stress–temperature graph. The stressovershoot response refers to the peak stress occurring dur-ing heating in the recovery step that is larger than thestress measured during the initial deformation. Alterna-tively, during free recovery (Fig. 5B) the amount of strainrecovered is shown. Here, as proposed by Lendlein andLanger (2002), the amount of the initial fixed deformationthat is recovered during a free recovery test is quantifiedby using a normalized strain measurement. A strain recov-ery ratio �eðtÞ ¼ 1� eðtÞ=emax is used where emax is the strain

Table 1Programming, fixing and recovery temperatures for shape memory cyclerecovery experiments.

Recovery description Temperatures (�C)

Programming,TH1

Fixing,TL

Recovery,TH2

Constrained recovery 60 10 60Free recovery 40 20 30, 35, 40, 50

measured at the start of the heating step and e(t) is theinstantaneous strain (both strains are measured using thelaser extensometer). The recovery behavior strongly de-pends on the recovery temperature. For instance, a recov-ery temperature of 30 �C can barely recover 20% of theprogrammed deformation after 60 min. These observationsare in qualitative agreement with what was observed in anepoxy-based SMP along with the recovery behavior beinghighly dependent on the programming, fixing and recoverytemperatures as well as the cooling and heating rates (Cas-tro et al., 2011).

3. Constitutive model

3.1. Overall model description

Here, a 3D SMP body is considered. A material point inthe initial configuration occupies point X and has a tem-perature T0. At time t, this material point moves to a spatial

Eeq

Rubbery Nonequilibrium

Branches

Eg Er Er

τg τ1 τ2

...

Er

τm

Glassy Nonequilibrium

Branch

Equilibrium Branch

Thermal Expansion Component

Fig. 6. 1D rheological representation of the proposed model.


point x and the temperature changes to T. Fig. 6 shows the1D rheological schematic of the proposed model. A thermalexpansion element is arranged in series with mechanicalelements. As shown previously (Holzapfel, 2000), the totaldeformation gradient F can be decomposed into:

F ¼ FMFT ; ð1Þ

where F = dx/dX, FM is the mechanical deformation gradi-ent and gives rise to stresses and FT is the thermal defor-mation gradient. The mechanical elements consist of anequilibrium branch and several nonequilibrium branchesplaced in parallel. Each nonequilibrium branch is a nonlin-ear Maxwell element where an elastic spring and a dashpotare placed in series. In the set of nonequilibrium branches,only one is used to represent the relaxation behavior of theglassy mode, which describes the structural or segmentalrelaxation of polymers. This mode is universal for all glassforming systems, and is the most fundamental relaxationprocess for polymers. In particular, it defines the monomerfriction coefficient which in term enslaves all the longertime scale chain relaxation processes. The remaining non-equilibrium branches are used to represent the relaxationprocesses in the rubbery state (or melt state), which canbe represented by a series of Rouse modes. Detailed discus-sions on these nonequilibrium branches will be presentedlater. The total Cauchy stress r is:

r ¼ req þ rg þXm

i¼1

rir ; ð2Þ

where req, rg and rir are the Cauchy stresses in the equilib-

rium branch, in the glassy nonequilibrium branch, and inthe ith rubbery nonequilibrium branches, respectively.

3.2. Thermal expansion

Thermal expansion is assumed to be isotropic, i.e.,

FT ¼ JT I; ð3Þ

where I is the second order unit tensor and JT is the volumechange due to thermal expansion/contraction, i.e.:

JT ¼VðT; tÞ

V0; ð4Þ

where V(T, t) is the volume at time t and temperature T,V0 is the reference volume at the reference temperatureT0. It has been well documented that an amorphouspolymer is in a nonequilibrium state at temperatures be-low its Tg (Hutchinson, 1995; Kauzmann, 1948; Kovacs,1964; Kovacs et al., 1979; McKenna, 1989). This occursas a result of the reduction in free volume upon coolingthe material from a temperature above to below Tg. Oncethe temperature is reduced below Tg, the characteristictime scale for the adjustment of the material structurebecomes increasingly longer. During this process, theinstantaneous volume V(T, t) begins to depart from theequilibrium volume Veq(T) which is considered to bethe material volume if the experimental timescale wasinfinitely long allowing the material to fully equilibrate.Following Tool (Tool, 1946, 1948; Tool and Eichlin,1931), below Tg, Veq(T) can be considered a linearextrapolation of the temperature dependence of theequilibrium volume above Tg and therefore the nonequi-librium volume is given by Vneq(t) = V(T, t) � Veq(T). Thedegree of the volume departure from equilibrium d isgiven by

d ¼ VneqðT; tÞVeqðTÞ

¼ VðT; tÞVeqðTÞ

� 1; ð5Þ

where the Veq(T) is given by

VeqðTÞ ¼ 1þ 3arðT � T0Þ½ �V0; ð6Þ

where ar is the linear CTE in the rubbery state and is as-sumed to be the linear equilibrium CTE in glassy stategiving:

JT ¼ 1þ 3arðT � T0Þ½ �ð1þ dÞ: ð7Þ

At a constant temperature below Tg, the relaxation ofthe nonequilibrium volume toward equilibrium dependson the free volume. As the material evolves towards theequilibrium volume, the characteristic time scale forstructural relaxation substantially increases due to thereduction in free volume (Hutchinson, 1995). Differenttheories (Moynihan et al., 1976; Robertson et al.,1984) have been developed in the past to representthe evolution of volume departure from equilibrium,i.e. d. Here, the well-known KAHR (Kovacs et al., 1979)33-parameter theory is used, which defines the evolu-tion of d by

d ¼X33

j¼1

dj; ð8aÞ

ddj

dt¼ �3gjDa

dTdt� dj

sjV ðT; dÞ

; ð8bÞ

gj ¼0:1516

for j 6 16 and gj ¼ 0:05 for j P 17;ð8cÞ

where Da = ar � ag, ag is the linear glassy CTE and sjV is

the structural relaxation time of the jth region and it isa function of current T and d. According to the KAHRmodel, the structural relaxation time of the jth region isgiven by


sjV ðT; dÞ ¼ sj

V0bT bd; ð9aÞ

bT ¼ exp �hðT � TV Þ½ �; ð9bÞbd ¼ exp �ð1� xÞhd=Da½ �; ð9cÞ

where sjV0

is the relaxation time in equilibrium at the ref-erence temperature TV, bT is a shift factor that incorporatesthe temperature dependence of sj

V at equilibrium (consid-ered constant d) through the parameter h and bd is a secondshift factor that incorporates the effect of the structure-dependent adjustment on the time scale. To control the de-gree of the contribution from temperature and structure tosj

V , the parameter x in Eq. (9c) is used that ranges in valuefrom 0 and 1. The structural relaxation times were equallydistributed in logarithmic space over four decades with amedian value of sj¼17

V0¼ sV . As a consequence, the fastest

structural relaxation time sj¼1V0¼ 0:01sV and the slowest

sj¼33V0¼ 100sV .

3.3. Hyperelastic component: equilibrium behavior

At the temperatures above Tg, the SMP is in rubberystate. Hyperelastic material models typically for rubbersare used for the equilibrium behaviors. Here, the Cauchystress tensor for the Langevin chain based Arruda-Boyceeight chain model (Arruda and Boyce, 1993) for hyperelas-ticity is defined as:

req ¼nkBT3JM

ffiffiffiffiNp

kchainL�1 kchainffiffiffiffi

Np

� �B0 þ KðJM � 1ÞI; ð10Þ

where n is the crosslinking density, kB is Boltzmann’s con-stant, T is the temperature, N is the number of Kuhn seg-ments between two crosslink sites (and/or strongphysical entanglements). The temperature dependentshear modulus lr(T) of the elastomer (which is an indica-tion of entropic elasticity) is given by nkBT. K is the bulkmodulus and is typically orders of magnitude larger thanlr to ensure material incompressibility (JM = 1). The devia-toric part of the isochoric left Cauchy-Green tensor is givenby B0 ¼ B� 1=3trðBÞI where B ¼ FMFM

T and BM ¼ J�1=3M FM .

The stretch on each chain in the eight-chain network is gi-ven by kchain ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitrðBÞ=3

q;L is the Langevin function defined

as LðbÞ ¼ cothb� 1=b.

3.4. Viscoelastic branches: viscous flow rules

For the nonequilibrium behaviors in the viscoelasticbranches, it is assumed that all branches follow the sameviscous flow rules but with different relaxation times. Forthe ith branch (i = 1, . . . ,m + 1 where 1 6 i 6m representsthe m nonequilibrium rubbery branches and i = m + 1 rep-resents the single nonequilibrium glassy branch), thedeformation gradient can be further decomposed into anelastic part and a viscous part:

FiM ¼ Fi

eFiv ; ð11Þ

where Fiv is a relaxed configuration obtained by elastically

unloading by Fie. The Cauchy stress can be calculated using

Fie:

ri ¼ 1

Jie

LieðTÞ : Ei

e

h i; ð12aÞ

with

Jie ¼ det Fi

e

� �; Ei

e ¼ ln Vie; Vi

e ¼ FieRiT

e ; ð12bÞ

and LieðtÞ is the fourth order isotropic elasticity tensor

which is taken to be temperature dependent in general, i.e.

LieðTÞ ¼ 2GiðTÞ I� 1

3I� I

� �þ KiðTÞI� I; ð13Þ

where I is the fourth order identity tensor, Gi(T) and Ki(T)are shear and bulk moduli for the ith branch, respectively.In the above equations, the symbol (:) represents a tensordouble contraction and (�) represents the tensor dyadicproduct. For the nonequilibrium rubbery branches(i = 1, . . . ,m), it is assumed that all the rubbery brancheshave the same shear modulus, i.e.

GiðTÞ ¼ nRkBT for 1 6 i 6 m; ð14Þ

where nR is the crosslinking density for the nonequilibriumrubbery branches associated with the Rouse modes(Rubinstein and Colby, 2003). Since the bulk modulus isused to enforce a nearly incompressible condition, Ki(T) ischosen to be independent of temperature and be equiva-lent to K in the equilibrium branch (Eq. (10)). For the non-equilibrium glassy branch (i = m + 1), the shear modulus istaken to be independent of temperature, i.e.

Gmþ1ðTÞ ¼ lg ; ð15aÞ

and Km+1(T) is calculated by using Gm+1(T) with the Poissonratio vm+1 = vg:

Kmþ1ðTÞ ¼ 2ð1þ vgÞ3ð1� 2vgÞ

lg : ð15bÞ

In each nonequilibrium branch (1 6 i 6m + 1), the sec-ond Piola–Kirchoff stress in the intermediate configuration(or elastically unloaded configuration) and the Mandelstress are given by

Si ¼ Jie Fi

e

� ��1ri Fi

e

� ��Tand Mi ¼ Ci

eSi; ð16Þ

where Cie ¼ FiT

e Cie is the right Cauchy-Green deformation

tensor. Typically for inelastic materials (Holzapfel, 2000),the Mandel stress is used to drive the viscous flow _ci

v viathe equivalent shear stress:

_civ ¼

Mi

GiðTÞsiM T;Mi� � ; ð17Þ

where the equivalent shear stress is defined as:

Mi ¼ 12ðMiÞ0 : ðMiÞ0

� �1=2

; ð18Þ

where (Mi)0 = Mi � 1/3tr(Ci)I. In Eq. (17), the temperatureand stress dependent material stress relaxation timesi

MðT;MiÞ will be discussed in the next section.

The viscous stretch rate Div is constitutively prescribed

to be:


Div ¼

_civffiffiffi

2p

MiMi: ð19Þ

As discussed previously in Boyce et al. (1988), Div can be

made equal to the viscous spatial velocity gradientliv ¼ _Fi

v ðFiv Þ�1 by ignoring the spin rate Wi

v and therefore:

_Fiv ¼ Di

vFiv : ð20Þ

3.5. Relaxation time and time–temperature shift factor

As the temperature is varied, the relaxation times inindividual branches also vary. Here, it is assumed thatthe time–temperature shift for each branch follows thesame rule, according to the well-established ‘‘thermorhe-ological simplicity’’ principle (Rubinstein and Colby,2003), i.e.

siMðTÞ ¼ si

0aTðTÞ for i ¼ 1; . . . ;mþ 1; ð21Þ

where aT(T) is the time–temperature superposition shiftfactor and si

0 is the reference relaxation time at the tem-perature when aT(T)=1 (discussion below). FollowingO’Connell and McKenna (1999), the method for calculatingthe temperature influence on the viscoelastic behavior de-pends on whether the material temperature is above or be-low Tg. At temperatures close to or above Tg, the WLFequation (Williams et al., 1955) is used:

log aT ¼C1ðT � TMÞ

C2 þ ðT � TMÞ; ð22aÞ

where log is the base 10 logarithm, C1 and C2 are materialconstants and TM is the WLF reference temperature. On theother hand, when temperatures are below Tg, an Arrhe-nius-type behavior developed by Di Marzio and Yang(1997) is used where it is assumed the configurational en-ergy Fc is constant below Tg giving:

ln aT ¼ �AFc

kB

1T� 1

Tg

� �; ð22bÞ

where ln is the natural logarithm and A is a material con-stant. In order to ensure a continuous aT value for all tem-peratures, a switching temperature TS is introduced andcalculated by equating aT in Eqs. (22a) and (22b). For tem-peratures greater than TS the WLF equation (Eq. (22a)) isused and for temperatures less than TS the Arrhenius-typebehavior (Eq. (22b)) is used.

For the nonequilibrium glassy branch (i = m + 1), at atemperature below Tg, the mobility of the polymer chainsis reduced and the change in chain conformation is hin-dered. Alternatively, cooperative change is possible but re-quires substantially higher energy. As the applied stressincreases, the energy barrier for the cooperative changeis reduced allowing the chain energy to overcome it andas result a yield-type behavior is exhibited. The yield pointin polymeric materials corresponds to complex behaviors,including chain sliding and/or chain kinking, both of whichincrease the chain mobility and therefore decrease therelaxation time. Therefore, the relaxation time, given inEq. (21) (i = m + 1), is taken to be not only a function of

temperature but also stress and can be evaluated by anEyring type of function (Treloar, 1958), i.e.

smþ1M T;Mmþ1� �

¼ smþ10 aTðTÞ exp � DG

kBTMmþ1

S

!; ð23Þ

where smþ10 ¼ sg

0 is the material relaxation time at the tem-perature when aT(T) = 1, DG is the activation energy and sis the athermal shear strength representing the resistanceto the viscoplastic shear deformation in the material. In or-der to adequately account for the experimentally observedsoftening effects, the evolution rule for s is defined as:

_s ¼ h0 1� s=ssð Þ _cv ; ð24aÞs ¼ s0 when cv ¼ 0; ð24bÞ

where s0 is the initial value of the athermal shear strength,ss is the saturation value, h0 is a prefactor and _cv is definedin Eq. (17). For the case when s0 > ss, Eq. (24) represents anevolution rule that characterizes the experimentally ob-served softening behavior of the material.

For the nonequilibrium rubbery branches (i = 1, . . . ,m),each branch is taken to represent a relaxation mode. Therelaxation behaviors of an unentangled polymer melt canbe modeled by the Rouse model where a macromolecularchain is divided into several entropic segments. Underthe mean-field approximation, the dynamic behavior ofthese segments under the action of the frictional force(or viscous force) yields an eigenvalue problem. This leadsto the relaxation of polymer melts be decomposed into aseries of (normal) Rouse modes. Specifically, the relaxationtimes associated with these Rouse modes are given byRubinstein and Colby (2003):

si0 ¼

sR

i2 for i ¼ 1; . . . ;m: ð25Þ

Here, Eq. (25) is adopted in order to determine the relaxa-tion times in the rubbery branches. When i = 1, s1

0 ¼ sR andhence the relaxation time in the first nonequilibrium rub-bery branch is equal to longest relaxation time in the Rousemodes. In Eq. (25), a practical question is how manybranches or how many Rouse modes should be included.In the Rouse model, the number of segments is not clearlydefined. Typically, the Kuhn segment is regarded as theshortest segment. Therefore, the number of nonequilibri-um rubbery branches m can be chosen to be equal to thenumber of Kuhn segments in a macromolecular chain. Inelastomers, m is equal to the number of the Kuhn segmentsbetween two crosslinking sites (or m = N introduced in Eq.(10)). Here, N can be estimated by determining the lockingstretch of the elastomer in a stress-stretch curve (seeAppendix A).

4. Results

The constitutive model was implemented into a usermaterial subroutine (UMAT) in the finite element softwarepackage ABAQUS (Simulia, Providence, RI). Using theimplemented UMAT, results from finite element simula-tions are compared to the experimental results introducedin Section 2. Taking advantage of symmetry in the com-pression specimens and with the frictionless boundary

Table 2Model parameters.

Description Parameter Value

Thermal component parametersGlass transition temperature Tg (�C) 40Linear rubbery coefficient of thermal

expansionar (1/�C) 2.35 � 10�4

Linear glassy coefficient of thermalexpansion

ag (1/�C) 1.15 � 10�4

Structural glass transition temperature Tv (�C) 40Material temperature parameter h (1/�C) 0.7Material partition parameter x (�) 0.7Structural relaxation time sv (s) 1 � 10�3

Equilibrium branch parametersCrosslinking density n (m�3) 1.7 � 1026

Number of kuhn segments between twocrosslinks

N (�) 2.49

Bulk modulus K (Pa) 1 � 109

Nonequilibrium rubbery branch parametersCrosslinking density nR (m�3) 1 � 1026

Rouse mode longest relaxation time sR (s) 1 � 105

Nonequilibrium glassy branch parametersShear modulus lg (Pa) 370 � 106

Poisson ratio vg (�) 0.4Initial material relaxation time sg

0ðsÞ 60

Zero stress level activation energy DG (J) 5 � 10�20

Initial shear strength s0 (Pa) 55 � 106

Saturation shear strength ss (Pa) 24 � 106

Prefactor parameter h0 (Pa) 400 � 106

Time–temperature shifting parametersWLF reference temperature TM (�C) 30WLF constant C1 (–) 17.44WLF constant C2 (�C) 51.6Pre-exponential arrhenius factor AFck�1

B (K) �15500

0 0.1 0.2 0.3 0.40

10

20

30

40

50

60

70

Strain

Com

pres

sive

Str

ess

(MPa

) 0oC

10oC

20oC

30oC

60oC

ExperimentSimulation

(B)

(A)

0 0.1 0.2 0.3 0.40

10

20

30

40

50

60

70

Strain

Com

pres

sive

Str

ess

(MPa

)


Fig. 7. Numerical simulation stress–strain results compared to theisothermal uniaxial compression experimental results at: (A) differenttemperatures at a strain rate of 0.01/s and (B) different strain rates of0.01/s and 0.1/s.


conditions assumption (due to the Teflon� sheets placedbetween the sample and the compression platens), a single8-node biquadratic, hybrid with linear pressure axis-sym-metric element (CAX8H) was used. Also, all of the bottomcentral node’s degrees of freedom are fixed while theremaining edges were modeled with rollers. For all simula-tions, an analytically rigid surface was used to deform thematerial. For all non-isothermal simulations, the thermalexpansion/contraction of the portion of the compressionplatens located between the mechanical extensometermounting positions was included during the cooling andheating steps.

Using the parameter identification protocol (describedin detail in Appendix B) as an initial guide to determiningthe model parameters, the simulation results were simul-taneously fit to the experimental results and the final mod-el parameter set are listed in Table 2. Importantly, thecurve fitting emphasis was placed on several key featuresfrom the array of characterization experiments introducedabove: (1) the overall behavior of the isothermal uniaxialcompression results at a strain rate of 0.01/s, (2) the loca-tion and magnitude of the stress overshoot peak in theconstrained recovery experiments and (3) the overallbehavior for all recovery temperatures for the free recoveryexperiments. Once a good fit is achieved for these key fea-tures, the set of material parameters is used to predict: (1)the effects of strain rate on the isothermal uniaxial com-pression experiments and (2) the DMA measurement.

The number of nonequilibrium rubbery branches neededto capture the various experimental behaviors in this paperwas two. It should be also noted that increasing the num-ber of branches would increase the simulation time as itwill need more memory and calculations. This will havea significant impact for large scale simulations.

4.1. Isothermal uniaxial compression simulations

Using the model parameters in Table 2, the results fromisothermal uniaxial compression simulations are com-pared to the experimental results. The curve fits at a strainrate of 0.01/s at different temperatures, represented inFig. 7A, show good agreement with the experimental re-sults. As shown in Fig. 7B, the model is able to predictthe strain rate dependence of the material behavior. Over-all, the model is able to capture the temperature depen-dent stress–strain trend both above and below Tg andpredict the behavior at different strain rates.

10 20 30 40 50 600

0.5

1

1.5

Temperature (oC)

Com

pres

sive

Str

ess

(MPa

)

Cooling

Heating


Fig. 8. Comparison between numerical simulation and experimentalresults for the stress response during a constrained recovery thermome-chanical shape memory cycle.

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1


Stra

in R

ecov

ery

Rat

io

TH1 = 40oC

TH2 = 30oC

TH2 = 35oC

TH2 = 40oC

TH2 = 50oC


Fig. 9. Comparison between numerical simulation and experimentalresults for the strain recovery during a free recovery thermomechanicalshape memory cycle at different recovery temperatures for a program-ming temperature of TH1 = 40 �C.

0 20 40 60 80 100100

101

102

103

104

Stor

age

Mod

ulus

(MPa

)

Temperature (oC)

StorageModulus

NormalizedTan Delta

0 20 40 60 80 1000

0.25

0.5

0.75

1

Nor

mal

ized

Tan

Del

ta


Fig. 10. Storage modulus and normalized tan d curves as a function oftemperature for the DMA numerical simulation predictions compared tothe experimental results.


4.2. Constrained recovery simulation

In an SMP application for which constrained recoveryboundary conditions exist, the knowledge of an SMP’sforce producing response during recovery is required.Numerical simulations were performed during the curvefitting process under these boundary conditions and thesimulation results are compared to the experimental re-sults as shown in Fig. 8. For the constrained recovery con-ditions, the model is able to capture the magnitude andtemperature location of the stress overshoot peak duringheating. However, the overall behavior above �35 �C dur-ing cooling or heating and the location of the temperatureat which the stress overshoot begins needs furtherrefinement.

4.3. Free recovery simulations

Numerical simulations were performed to describe thefree recovery conditions. The curve fitting results of thenumerical simulations compared to the experimental re-sults are shown in Fig. 9 for a programming temperatureof TH1 = 40 �C. As a result of putting the emphasis on simul-taneously curve fitting the multitude of experimental re-sults, the overall fit for the free recovery conditions afteran SM cycle captures the experimental results well. Forrecovery temperatures greater than or equal to the mate-rial’s Tg, the free recovery behavior (both the rate of recov-ery and the time to begin recovery after heating begins) isequivalent excluding the thermal expansion for recoverytemperatures greater than programming temperatures.

4.4. DMA simulation

The temperature and time dependent mechanical prop-erties of the material were simulated under dynamic uni-axial tension and compression boundary conditionscharacteristic of a DMA experiment. Specifically, in the

simulations the specimen was subjected to an oscillatoryvertical displacement applied on the top surface at a fre-quency of 1 Hz with a peak-to-peak amplitude of 1% strainto negate the effects of yielding at temperatures below Tg.Fig. 10 shows the comparison between the experiment andsimulations results for the temperature dependent storagemodulus and tan d. The tan d response, for both the exper-iment and simulation results, has been normalized to themaximum tan d value. From Fig. 10, it is seen that the mod-el under predicts the storage modulus at low temperaturesbut it provides a good prediction for the overall behavior atall temperatures. Additionally, from the normalized tan dcomparison, the simulation is able to predict the experi-mentally measured Tg value (temperature corresponding


to maximum tan d value) of 41.5 �C. The tan d responsefrom the simulation shows a narrower glass transition re-gion compared to the experiment. Such discrepancy islikely caused by the distribution and cooperativity of struc-tural relaxations that underlie the glass transition pro-cesses. It is well known that the distributed structuralrelaxations become increasingly cooperative as tempera-ture approaches Tg. Most often, an average relaxation timeis used to represent all the structure relaxation processes,as used in the model presented here. Such a simplificationwill undoubtedly cause the narrowing of the glass transi-tion process as shown in Fig. 10. This discrepancy can bedirectly addressed, if a distribution of structural relaxationprocess fsg;ig for glassy modes is used. However, this willinevitably increase the number of parameters used in themodel. This will be explored in the near future. Neverthe-less, the numerical simulations show that the model is ableto predict the complex temperature and time dependentmechanical response of an SMP.

0 0.1 0.2 0.3 0.40

10

20

30

40

50

60

70

Strain

Com

pres

sive

Str

ess

(MPa

)

10oC

30oC

60oC

m = 0m = 1m = 2m = 5

(B)

(A)

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1


Stra

in R

ecov

ery

Rat

io

TH1 = 40oC

TH2 = 30oC

TH2 = 35oC

TH2 = 40oC

m = 0m = 1m = 2m = 5

Fig. 11. Simulation results showing the effects of the number of branches(m = 0,1,2,5) on: (A) isothermal uniaxial compression behavior and (B)the free recovery behavior after an SM cycle with a programmingtemperature of TH1 = 40 �C and recovery temperatures of TH2 = 30, 35 and40 �C.

4.5. Effects of number of nonequilibrium rubbery branches

It is important to understand the effects of including thenonequilibrium rubbery branches and how the number ofbranches affects the SM behavior in particular. Recall thenumber of nonequilibrium rubbery branches determinedfrom the curve fitting process above was two (m = 2). Todo so, simulations for isothermal uniaxial compressionand free recovery for an SM cycle are performed. Here,the effect of neglecting all the rubbery branches (m = 0)is also considered. Fig. 11A shows the isothermal uniaxialcompression simulation results for temperatures of 10,30 and 60 �C. For these results, it is seen that in generalincluding rubbery branches does not affect isothermalbehaviors significantly. But rather, it only affects the lowtemperature, post-yielding behavior, i.e., the morebranches that are included the larger the effect of post-yielding strain stiffening. However, for free recovery the ef-fect is significant. Fig. 11B shows the free recovery simula-tion results for an SM cycle for a programmingtemperature of TH1 = 40 �C and recovery temperatures ofTH2 = 30, 35 and 40 �C. By including the nonequilibriumrubbery branches, a significant increase in the time toreach an equilibrated recovered state at a constant recov-ery temperature is observed when compared to the caseof m = 0. As the number of branches is increased, the timeto reach an equilibrated recovered state increases and therate of recovery is decreased. The most significant effectof including even just one nonequilibrium rubbery branch,which corresponds to the longest Rouse relaxation mode,is seen for recovery temperatures just below Tg (for exam-ple TH2 = 35 �C). Therefore, it is concluded that in order tocapture the complex thermomechanical behavior of anSMP, it is necessary to include the additional relaxationmechanisms attributed to the Rouse modes of the materialthrough the nonequilibrium rubbery branches.

4.6. Applications of model

4.6.1. Intrinsic recovery of SMPRecovery of an SMP depends on how fast the material

can be heated (thermal conduction pathways) and howfast the material can recover if it could be heated instanta-neously. Here, the latter is termed intrinsic recovery sinceit is effectively a material property. Experimentally decou-pling intrinsic recovery from heat transfer is difficult inregular laboratories as instantaneous heating is almostimpossible. Here, numerical simulations to examine theintrinsic recovery of the SMP studied in this paper are con-ducted. In the simulations, the programming and fixingtemperatures are held constant at 40 and 20 �C while therecovery temperature is varied between Tg � 20 �C andTg + 20 �C. To consider the intrinsic recovery, we assumethe sample is heated to the targeted recovery temperatureswithin 1 ms leading to an instantaneous heating rate of2.4 � 106 �C/min for the largest temperature increase(recovery temperature of Tg + 20 �C).

Fig. 12 shows the simulated dependence of recoverytime on recovery temperatures. Here, the recovery timeis defined as the time when the 95% of strain is recovered.The recovery time strongly depends on the recovery

20 30 40 50 6010−2

100

102

104

106

108

Recovery Temperature (oC)

Tim

e fo

r 95%

Rec

over

y (s

)

Fig. 12. Simulated intrinsic recovery results showing the time to reach95% recovery for programming and fixing temperatures of 40 and 20 �C,respectively, and various recovery temperatures.

Fig. 13. Representative volume element (RVE) for a magnetosensitiveSMP composite. (A) 2D cross-section schematic showing the RVE for theperiodic problem. (B) Finite element model mesh for the RVE for a fillervolume fraction of 10% and a filler diameter of 10 lm.


temperature. Such a dependence also shows two distinc-tive regions: at the temperature above Ts � 28 �C, theintrinsic recovery time scales with temperature as�106 � 10�0.21(T�27.5); at the temperature below Ts � 28 �C,it scales as �106 � 10�0.082(T�27.5). The presence of a transi-tion between these two regions coincides with the transi-tion of shifting factor aT defined by Eqs. (22a) and (22b),indicating that the temperature dependence of recoveryis closely related to the time–temperature shift character-istics. A detailed correlation will be studied in the future.

4.6.2. Magneto-sensitive SMP compositeRecently, efforts have been made to improve the recov-

ery rate of SMPs by using novel heating methods. Forexample, the SM effect is realized through the inductiveheating of magnetic particles dispersed in an SMP (Mad-bouly and Lendlein, 2010; Weigel et al., 2009). This ap-proach offers two potential advantages for shaperecovery. First, heating the material can be achieved remo-tely by alternating a magnetic field. Second, the heatingrate of the SMP structure can be significantly increased. Be-sides these advantages, the inclusions of particles in SMPs(Buckley et al., 2006; Yakacki et al., 2009) allows for med-ical imaging techniques, such as fluoroscopy or computedtomography scans, to detect the implanted device withoutadditional surgeries for proper device placement andfunction.

As an application of the model, the effect of particle sizeon the free recovery behavior of a magnetosensitive SMPcomposite is demonstrated. In particular, in order to focuson the behavior of SMPs, it is assumed that the particlescan be heated to the targeted temperature immediatelyand be maintained at that temperature. The heat is thentransferred into the programmed SMP matrix to triggerthe recovery. The effect of particle size on the shape recov-ery time is investigated. It is noted that it is possible toconsider a more complex system where the heating of

the particle due to magnetic field is also modeled but thisis out of the scope of this paper. Assuming uniform particledispersion, a representative volume element (RVE) is mod-eled in a 2D setting as shown in Fig. 13. Although the 2Dmodel may represent long fiber filler, the general insightfrom this model can be extended to particles. The filler vol-ume fraction can be related to the RVE geometry by

/2D ¼Vfiller

VRVE¼ pd2

16a2 ; ð26Þ

where d is the diameter of the filler and 2a is the RVE edgelength as shown in the schematic in Fig. 13A. In the para-metric studies for a given filler volume fraction, the fillerdiameter is varied over many decades and the correspond-ing RVE edge lengths are calculated.

Using the geometry of the RVE, finite element modelsthat couple heat transfer with finite deformation solidmechanics were created. For the SMP matrix, 4-node bilin-ear displacement and temperature, hybrid with constantpressure elements (CPE4HT) in Abaqus element librarywere used; for the filler, 3-node linear displacement andtemperature elements (CPE3T) were used. A representativefinite element mesh is shown in Fig. 13B for a 10% volumefraction and a filler diameter of 10 lm. Periodic boundaryconditions (PBCs) (Bertoldi et al., 2008; van der Sluiset al., 2000) were applied by using equation constraintsfor the edge nodes. Because of the PBCs, the initial com-pressive displacement is applied on the RVE’s top left node.Additionally, to account for the thermal contraction duringcooling, an analytic rigid surface was included so the topleft node vertical displacement boundary condition could

10−2 10−1 100 101 102 103100

101

102

103

104

Particle Diameter (μm)

Tim

e fo

r Ful

l Rec

over

y (s

)

Recovery at 40oC

Recovery at 50oC

0.1%1%10%

Fig. 14. Model application to a ferromagnetic particle filled SMPcomposite showing the effects of the particle size on the free recoverybehavior after an SM cycle.


be removed after the initial compression. The thermome-chanical history follows the schematic in Fig. 4. Here theprogramming and fixing temperatures were 40 and 20 �C,respectively for all the cases. The recovery temperaturewas 40 and 50 �C, respectively. Following Westbrooket al. (2010) it is assumed the SMP has a heat capacity, den-sity and conductivity of 640 J/(kg �C), 1050 kg/m3 and0.15 W/(m �C), respectively. The particle is assumed tohave a conductivity of 500 J/(kg �C), a Young’s modulus of200 GPa and a value of 0.2 for the Poisson ratio. Duringthe cooling step, the nodal temperature (SMP and filler)are prescribed; whereas, to account for instantaneouslyheating, the temperature of the particle nodes are rampedto the recovery temperature in 0.1 ms.

Fig. 14 shows the effects of changing the diameter forthree volume fractions, 0.1%, 1% and 10%. Here, as before,the time for full recovery is taken to be when the recoveryratio reaches 95%. As seen in Fig. 14, for each volume frac-tion, there exists a critical diameter, below which the timefor full recovery is independent of the filler diameter. Thecritical diameters for the 0.1%, 1% and 10% volume frac-tions are approximately 2, 10 and 95 lm for recoveries at50 �C, and 20, 80 and 900 lm for recoveries at 40 �C,respectively. The existence of a critical filler size representsa transition of recovery from one dominated by heat trans-fer to one dominated by material intrinsic recovery. At thesame filler volume fraction, decreasing filler diameter re-duces the RVE size and thus effectively reduces the lengthof the pathway for heat transfer. Below a critical filler size,the size of the RVE becomes unimportant as heating canoccur almost instantaneously and therefore from heatingefficiency and material recovery points of view, it is unnec-essary to further reduce the filler size. Certainly, theremight be consideration due to manufacturing so that smal-ler size particles might be preferred. For example, smallerparticle can be easily suspended in solution that large par-ticles. The current model was a 2D model. Future work willinclude expanding this analysis to a 3D setting in order to

more realistically capture the heat transfer problem from aspherically shaped particle.

5. Conclusions

A 3D finite deformation constitutive model for the ther-momechanical behavior of SMPs based on the changes inmaterial viscoelasticity as the temperature traverses theTg was developed and implemented in ABAQUS. Variousexperiments were performed to fully characterize the com-plex thermomechanical behavior of SMPs and guide thedevelopment of the constitutive model. In order to capturethe multiple relaxation mechanisms occurring in the mate-rial, additional nonequilibrium rubbery Maxwell brancheswere utilized in parallel with a traditional viscoelasticstandard linear solid model. A protocol for determiningthe complete set of model parameters was introducedand deemed effective in predicting the SMP thermome-chanical behavior. The effects of the number of additionalnonequilibrium rubbery branches on the model behaviorunder various thermomechanical histories was investi-gated and discussed. Overall, the model is able to capturethe wide range of experimental results using the modelparameters found from the curve fitting procedure forthe complex behaviors in the SMP material. To explorethe utility of the model, the intrinsic recovery of an SMPis explored and a magnetosensitive SMP composite wassimulated to determine the effects of changing the particlesize on the free recovery behavior over a range of particlevolume fractions and particle sizes.

Acknowledgements

We gratefully acknowledge the support of an NSF ca-reer award (CMMI-0645219), an AFOSR Grant (FA9550-09-1-0195; Dr. Les Lee, program manager) and an NSF-Sandia initiative (Sandia National Laboratories, 618780).

Appendix A. Method for determining the number ofnonequilibrium rubbery branches

As discussed above, the number of nonequilibriumrubbery branches m can be chosen to be equal to thenumber of Kuhn segments in a macromolecular chain.From non-Gaussian chain statistics, the chain limitingstretch ratio is connected to the number of Kuhn seg-ments by

klimc ¼

ffiffiffiffiNp

: ðA1Þ

Since macromolecules in a polymer network can accom-modate large deformation by rotation, the principalstretches ki (i = 1,2,3) in the three directions are differentfrom the chain stretch kc. Following the eight-chain modelargument, the chain stretch is related to the principalstretches by

kc ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

1 þ k22 þ k2

3

3

s: ðA2Þ

Under uniaxial testing conditions, following the incom-pressible argument:


k1 ¼ k and k2 ¼ k31ffiffiffikp ; ðA3Þ

where k is the stretch ratio along the uniaxial testing direc-tion (k < 1 for compression and k > 1 for tension).Therefore:

kc ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13

k2 þ 2k

� �s: ðA4Þ

For an elastomer, as the stretch ratio approaches the limit-ing stretch, the stress increases dramatically. This can beused to estimate the number of Kuhn segments, N. If thelimiting material stretch is klim, then the limiting chainstretch becomes:

klimc ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13ðklimÞ2 þ 2

klim

� �s¼

ffiffiffiffiNp

: ðA5Þ

The limiting material stretch is identified from isothermaluniaxial compression/tension experiments at a tempera-ture above the material’s Tg to a stretch where strain stiff-ening occurs. This maximum stretch where strainstiffening dramatically increases the stress becomes thelimiting material stretch. From Eq. (A5), the number ofKuhn segments is therefore determined to be:

N ¼ 13ðklimÞ2 þ 2

klim

� �; ðA6Þ

and the number of nonequilibrium rubbery branches m isrelated to N by rounding to the nearest integer (showngraphically in Fig. A1). Once the limiting stretch is experi-mentally determined, the number of Kuhn segments N canbe determined from Eq. (A6) and then the number of non-equilibrium rubbery branches m can be found by roundingN to the nearest integer. Fig. A1 shows how this method ofrounding affects the number of nonequilibrium rubberybranches related to the number of Kuhn segments for a

0 1 2 3 4 50

1

2

3

4

5

6

7

8

9

10

Limiting Stretch, λlim

Num

ber

Kuhn Length Segments, NNonequilibrium Rubbery Branches, m

Fig. A1. Graphical representation for determining the number of non-equilibrium rubbery branches from the number of Kuhn segments as afunction of the experimentally determined limiting stretch.

wide range of limiting stretches for both compressionand tension boundary conditions.

Appendix B. Protocol for model parameteridentification

To adequately identify the material parameters, fourdifferent types of experiments should be performed.Briefly, as the details are described in more detail below,a DMA test is run to determine the Tg, a CTE experimentis performed to find the thermal expansion parametersincluding structural relaxation, isothermal uniaxial com-pression experiments at temperatures above and belowTg are run to find equilibrium hyperelastic and the non-equilibrium glassy material behaviors including yieldingand flow behavior and a stress relaxation test is performedat Tg to find the initial material relaxation for the nonequi-librium glassy branch. These four types of experiments willallow for the thermomechanical behavior of SMPs to besufficiently characterized and provide data useful in prop-erly identifying the material parameters used in the pre-sented model. This parameter identification protocolintroduced here is used to determine the starting pointfor the values of the model parameters which may needto be altered to capture the complex thermomechanicalbehavior of the SMP material.

B.1. Thermal component parameters: Tg, ar, ag, Tv, h, x and sV

The material’s glass transition temperature (Tg = 41.6 �Cas shown in Fig. 2A) is determined from a DMA experi-ment. The rubbery and glassy CTE values are determinedfrom the slopes of a CTE experiment above and below Tg,respectively. As shown in Fig. 2B, the linear glassy and rub-bery CTEs are found to be ag = 1.15 � 10�4/�C andar = 2.35 � 10�4/�C, respectively. The structural relaxationparameters are found by fitting the CTE measurementcurve through a simulation where a small force is appliedto the top surface of the specimen and the temperature isdecreased. The structural glass transition temperature Tv

is chosen to equal Tg and the remaining values are obtainedto be h = 0.43/�C, x = 0.7 and sV = 1 � 10�3 s.

B.2. Equilibrium branch parameters: n, N and K

The equilibrium branch parameters are used to charac-terize the hyperelastic behavior of the material at temper-atures above Tg. Assuming a 1D problem with a materialstretch of k = L/L0, Eq. (10) reduces to the 1D form:

r1Deq ¼

nkBT3

ffiffiffiffiNp

kchainL�1 kchainffiffiffiffi

Np

� �k2 � 1

k

� �; ðB1Þ

where kchain ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðk2 þ 2=kÞ=3

q. Also assuming an incom-

pressible material (K = 1 GPa), the isothermal uniaxialcompression results for T = 60 �C are used to obtainn = 2.35 � 1026 m�3 (providing a shear modulus value of1.05 MPa at T = 60 �C) and N = 2.49 (providing 2 nonequi-librium rubbery branches as described in Appendix A).


B.3. Nonequilibrium rubbery branch parameters: nR and sR

As introduced above, the nonequilibrium rubberybranches are included in order to capture the complexrelaxation mechanisms in the polymer. Here, the crosslink-ing density nR is chosen to be on the same order as theequilibrium branch, i.e. nR = 1.00 � 1026 m�3. Since therelaxation mechanisms are on a longer characteristic timescale, we choose sR = 1 � 105 s as a starting point.

B.4. Nonequilibrium glassy branch parameters: lg, vg, sg0, DG,

s0, ss and h0

The nonequilibrium glassy shear modulus is found fromthe initial slope of the isothermal uniaxial compressionstress–strain response at T < Tg, here 0 �C is chosen, and be-cause for polymers, the Poisson ratio vg is typically 0.4, theglassy shear modulus is found to be lg = 480 MPa. Thestress-independent, temperature shifted material relaxa-tion time is given in Eq. (22) (i = m + 1) where the materialrelaxation time sg

0 is determined from a stress relaxationexperiment when the shift factor aT = 1 (when T = TM as dis-cussed in the next section). A 2% strain is applied under uni-axial compression loading conditions and the stress in thematerial is allowed to relax until an equilibrium stress isreached. Using the stress relaxation solution to the Maxwellelement which is given by r ¼ ðr0 � r1Þ expðt=sg

0Þ þ r1,the glassy initial material relaxation time is found to besg

0 ¼ 56:1 s (curve fit not shown).The material parameters DG, s0, ss and h0 in the non-

equilibrium glassy branch needed to capture the flowbehavior in the glassy state of the polymer can be esti-mated from the isothermal uniaxial compression experi-mental results at different strain rates. A similar methodis shown in Qi and Boyce (2005) and briefly described herefor the viscous flow relationship given by Eq. (18). Thecurves at 0 �C are used to demonstrate this process. In auniaxial compression test, the equivalent shear stress s

0 0.1 0.2 0.3 0.4 0.50

10

20

30

40

50

60

70

True Strain

Com

pres

ssiv

e Tr

ue S

tres

s (M

Pa)

0.1/s0.01/s

Fig. B1. Isothermal uniaxial compression true stress–strain results atdifferent strain rates for 0 �C where the circles are data points used in thediscussion of Appendix B Section 4 to determine certain flow parameters.

and shear strain c are related to the uniaxial stress andstrain by

s ¼ r=ffiffiffi3p

and c ¼ffiffiffiffiffiffi3ep

: ðB2Þ

In order to use the experimental results at 0 �C, the engi-neering stress–strain values from the isothermal uniaxialcompression experiments, shown in Fig. 3, are convertedto true values as shown in Fig. B1.

From Fig. B1, the yield stresses are found to be�r1 ¼ 62:4 MPa for _e1 ¼ 0:1=s, and �r2 ¼ 57:6 MPa for_e2 ¼ 0:01=s. Therefore, using the relationships in Eq. (B2),s1 ¼ M1 ¼ 36:0 MPa for _c1 ¼ 0:173=s, and s1 ¼ M2 ¼33:3 MPa for _c2 ¼ 0:0173=s. Since the yield point in thestress–strain curve corresponds to the onset of viscoplasticflow in the material, and thus the onset for the evolutionof athermal shear strength s, it is reasonable that at theyielding point, s = s0. Using this logic and Eqs. (18) and(24), the viscoplastic shear strain rates become:

_c1ð0�CÞ ¼ AM1 expDGkBT

M1

s0

!; ðB3aÞ

_c2ð0�CÞ ¼ AM2 expDGkBT

M2

s0

!; ðB3bÞ

where A ¼ ðlgsg0aTÞ�1. By combining Eq. (B3) the following

expression is obtained:

DG ¼ kBT

M2 �M1ln

_c2M1

_c1M2

!s0 ¼ 310� 10�29s0: ðB4Þ

From Fig. B1, at e = 0.5, the stress–strain curve is approach-ing a plateau, implying the athermal shear strength s is nearthe saturation value ss, i.e. s � ss. At e = 0.5, �r3 ¼ 35:7 MPafor _e ¼ 0:1=s, and �r4 ¼ 33:3 MPa for _e2 ¼ 0:01=s and simi-larly, the corresponding equivalent shear yield stressesand equivalent shear strain rates are s3 ¼ M3 ¼ 20:6 MPafor _c1 ¼ 0:173=s, and s4 ¼ M4 ¼ 19:2 MPa for _c2 ¼0:0173=s. Following the same procedure to obtain Eq. (B4),the following expression is obtained:

DG ¼ kBTM4 �M3

ln_c2M3

_c1M4

!ss ¼ 601� 10�29ss: ðB5Þ

From Eqs. (B4) and (B5), the ratio of the initial and satura-tion shear strengths becomes:

s0=ss � 2: ðB6Þ

In order to estimate the parameter h0, the evolution rulefor the athermal shear strength given in Eq. (25) is rear-ranged in the following form:

dss=ss � 1

¼ h0dc; ðB7Þ

and by integrating Eq. (B7) to get:

Ss lnsss� 1

� �� ln

s0

ss� 1

� �� ¼ �h0 c� c1ð Þ: ðB8Þ

It is assumed that s decreases to (s0 + ss)/2 as the stress in thestress–strain curve decreases to ð �r1 þ �r3Þ=2. At ð �r1þ�r3Þ=2 ¼ 49:1 MPa; e ¼ 0:13, or c = 0.23, and at �r ¼

62:4 MPa; e1 ¼ 0:07, or c1 = 0.12. From Eqs. (B6) and (B8):


h0 ¼ �ss

ln sss� 1

� �� ln s0

ss� 1

� �h ic� c1

� 6:3ss: ðB9Þ

Eqs. (B4), (B5), (B6), and (B9) provide the guideline to esti-mate s0, DG, and h0 once ss is known. It can be assumed thatss is slightly higher than s3 ¼ M3 resulting in ss = � 25 MPaand therefore s0 = � 50 MPa, DG � 1.6 � 10�19 J andh0 � 150 MPa.

B.5. Time–temperature shifting parameters: TM, C1, C2 andAFckB

-1

The material parameters TM, C1 and C2 are from the WLFequation given in Eq. (23b). We begin by choosing the ref-erence temperature TM to equal Tg. The material constantsC1 and C2 are taken to be the values determined for TM = Tg

as given in Williams et al. (1955) giving C1 = 17.44 and

(A)

0 20 40 60 80 10010−10

10−5

100

105

Temperature [oC]

Mat

eria

l Rel

axat

ion

Shift

ing

Fact

or, a

T

AFckB−1 (K)

−20000−15000−10000

(B)

0 20 40 60 80 10010−10

10−5

100

105

Temperature [oC]

Mat

eria

l Rel

axat

ion

Shift

ing

Fact

or, a

T

TM (oC)

Tg + 10oCTg

Tg − 10oC

Fig. B2. Material relaxation shift factor behavior showing: (A) the effectsof changing the pre-exponential Arrhenius factor AFckb

�1 with TM = Tg and(B) the effects of changing the WLF reference temperature TM withAFckB

�1 = �1.5 � 104 K.

C2 = 51.6 �C. There is currently no apparent way to predictthe value of the remaining parameter, the pre-exponentialArrhenius factor remaining parameter AFckB

�1 found in Eq.(23b), and therefore the behavior of changing this param-eter on the time–temperature shift factor is shown.Fig. B2A shows how changing AFckB

�1 effects the behaviorof the shift factor aT with TM = Tg. Fig. B2B shows the effectof changing the WLF reference temperature while usingAFckB

�1 = �1.5 � 104 K.

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