A thermoviscoelastic model for amorphous shape memory polymers: Incorporating structural and stress relaxation

ARTICLE IN PRESS Journal of the Mechanics and Physics of Solids 56 (2008) 2792– 2814

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A thermoviscoelastic model for amorphous shape memory polymers: Incorporating structural and stress relaxation Thao. D. Nguyen a,, H. Jerry Qi b, Francisco Castro b, Kevin N. Long b a b

Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA Department of Mechanical Engineering, University of Colorado, UCB 427, Boulder, CO 80309, USA

a r t i c l e i n f o

abstract

Article history: Received 15 January 2008 Received in revised form 29 April 2008 Accepted 30 April 2008

A thermoviscoelastic constitutive model is developed for amorphous shape memory polymers (SMP) based on the hypothesis that structural and stress relaxation are the primary molecular mechanisms of the shape memory effect and its time-dependence. This work represents a new and fundamentally different approach to modeling amorphous SMPs. A principal feature of the constitutive model is the incorporation of the nonlinear Adam–Gibbs model of structural relaxation and a modified Eyring model of viscous flow into a continuum finite–deformation thermoviscoelastic framework. Comparisons with experiments show that the model can reproduce the strain–temperature response, the temperature and strain-rate dependent stress–strain response, and important features of the temperature dependence of the shape memory response. Because the model includes structural relaxation, the shape memory response also exhibits a dependence on the cooling and heating rates. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Shape memory polymers Structural relaxation Glass transition Thermoviscoelasticity Viscoelasticity

1. Introduction Thermally activated shape memory polymers (SMP) are an emerging class of active materials that respond to a specific temperature event by generating a shape change (Nakayama, 1991; Otsuka and Wayman, 1998; Monkman, 2000; Lendlein and Kelch, 2002; Lendlein et al., 2005). Compared to shape memory alloys, SMPs are inexpensive to manufacture, malleable and damage tolerant, and can undergo large shape changes in excess of 100% strain. A thermally active SMP device is processed into its permanent shape using conventional techniques. The permanent shape is determined by the network of crosslinks, loosely defined here as junctions in the macromolecular network that persist through the temperature and deformation range of operation. They can be chemical bonds (thermosets), physical entanglements (thermoplastics), or small crystalline domains (phase-segregated block copolymers) (Lendlein and Kelch, 2002; Lendlein et al., 2005). The temporary shape of an SMP device can be programmed by a thermomechanical cycle, in which the undeformed device first is heated to T high 4T trans . The heated device is deformed to the desired shape, cooled to T low oT trans , and mechanically unloaded. This causes a small elastic spring-back and produces the programmed (temporary) shape, which is fixed as long as the device is kept below T trans. Heating to above T trans causes it to deploy and recover its permanent shape. The molecular mechanism underpinning the shape memory phenomena of thermally activated amorphous SMPs is the dramatic change in the temperature dependence of the chain mobility induced by the glass transition (i.e., T trans ¼ T g ). The chain mobility describes the ability of the chain segments to rearrange locally to bring the macromolecular structure and

 Corresponding author.

E-mail address: [email protected] (T.D. Nguyen). 0022-5096/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2008.04.007

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log10 η

Isostructural

Equlibrium

η (T1)

η (T0) 1/T

2

4

6

8

10

t/τR Fig. 1. (a) Illustration of the temperature dependence of the viscosity showing the transition from equilibrium to isostructural behavior for a glass forming material (Scherer, 1986, 1990). (b) Illustration of the time evolution of the viscosity Z in response in to a step decrease in temperature from T 0 to T 1 .

stress response to equilibrium. The mobility is high for temperatures above the T g such that the structure can relax quickly to an equilibrium configuration in response to a temperature change. As a result, the material viscosity, a macroscopic measure of the chain mobility, changes nearly instantaneously with the temperature. The temperature dependence of the viscosity of a glass forming material is illustrated in Fig. 1(a). Cooling reduces the chain mobility and causes the structure to respond more sluggishly in the glass transition region. A finite time is required for the structure, and thus the viscosity, to relax towards equilibrium for a given temperature as shown in Fig. 1(b). Below the transition temperature T g , the structure is prevented by the vanishing mobility and reduced thermal energy from relaxing to equilibrium in an observable time frame in response to a temperature change. This effectively freezes the structure in a nonequilibrium configuration and allows the material to store a deformed shape. Reheating to above T g restores the mobility and allows the structure to relax again to an equilibrium configuration and the material to recover its permanent shape. It is proposed that in addition to heat transfer, the important molecular mechanisms determining the time-dependence of the shape memory response of amorphous SMPs are structural relaxation in the glass transition region, and stress relaxation in the form of viscoelasticity in the high temperature (rubbery) and glass transition regions and viscoplasticity in the low temperature (glassy) region. Structural relaxation describes the time-dependent response to temperature and pressure changes, while stress relaxation describes the time-dependent response to a change in the mechanical, particularly deviatoric, loading. Both occur because the microstructure is unable to rearrange instantaneously to an equilibrium configuration in response to an external stimuli. A decrease in the temperature and increase in the pressure reduces the configurational entropy, and thus the molecular mobility, by reducing the free volume. One of the central ideas of structural relaxation is that the mobility depends not only on the temperature and pressure but also on the evolving structure. As a result, the viscosity and related properties cannot change instantaneously with a change in the temperature or pressure, but instead evolve with time to an equilibrium value, as illustrated in Fig. 1(b). Moreover, the time-dependence of the dilatation response to a temperature and pressure change is inherently nonlinear and nonexponential. Most polymers under normal operating conditions are much less sensitive to pressure changes than temperature changes. Thus, the effects of pressure on the mobility and the time-dependence of the dilatational response are not considered here. Deviatoric stress relaxation is inherently different than structural relaxation in that above the glass transition temperature, the deviatoric deformation does not alter significantly the free volume. The characteristic stress relaxation time is little affected by small to moderate deviatoric strains, and the time-dependent response can be modeled as a structure-independent viscoelastic phenomenon. Large deviatoric deformation can alter the configurational entropy and molecular mobility by straightening the chain segments to their contour lengths. However, the effects of chain straightening on the deviatoric stress relaxation response are not considered here for simplicity. Below the glass transition temperature, the time-dependent deviatoric response is governed by stress-activated molecular processes that result in viscoplastic flow. To examine the relative importance of the structural and stress relaxation mechanisms, a constitutive model has been developed for the finite-deformation, time-dependent thermomechanical behavior of thermally active amorphous SMPs

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that include structural relaxation in the glass transition region, viscoelasticity in the rubbery and transition regions and viscoplasticity in the glassy region. This work represents a fundamentally different and new approach to modeling the behavior of SMPs, one which offers physical explanations for the temperature and time-dependence of the shape memory response. Currently, most constitutive models for thermally active SMPs do not consider the time-dependent effects from structural relaxation. Moreover, many are limited either to one-dimension or small strains (see for example Tobushi et al., 2001; Liu et al., 2006). Three dimensional finite-deformation models have been developed very recently by the authors (Qi et al., 2008) and also by Diani et al. (2006). The work of Diani et al. (2006) assumes a thermoviscoelastic approach that applies a phenomenological temperature dependence of the viscosity. Extending the work of Liu et al. (2006); Qi et al. (2008) applies a phenomenological, first-order, phase transition approach that models the SMP as a continuum mixture of a glassy and rubbery phase. Each is characterized by a volume fraction, and a homogenization scheme is used to formulate the stress response of the SMP from the stress response of the phases. Finally, constitutive relations are proposed for the temperature evolution of the volume fractions. The main disadvantage of this approach is that it is not representative of the physical processes of the glass transition and thus results in nonphysical parameters, such as the volume fractions of the glassy and rubbery phases. The structural relaxation of glass forming materials is an active research field rooted in the pioneering work of Tool and Eichlin (1925, 1931) and Tool (1946). It has produced many different approaches to modeling the glass transition and related structural relaxation phenomena. An extensive review of the field of the glass transition and structural relaxation is provided by Scherer (1990), Hutchinson (1993), and Hodge (1997). The distinguishing thermomechanical characteristics of structural relaxation, namely the nonexponentiality and asymmetry of the time-dependent response to a temperature change, are attributed to the dependence of the characteristic structural relaxation time on the temperature and the nonequilibrium structure of the glass. To describe the nonequilibrium structure, Tool (1946) introduced the concept of the fictive temperature, T f , as the temperature at which the nonequilibrium structure at T is in equilibrium. Structural relaxation to an equilibrium configuration is modeled as the time evolution of T f to T. Many phenomenological approaches have been developed for modeling the dependence of the structural relaxation time on the structure and temperature (see for example, Narayanaswamy, 1971; Moynihan et al., 1976; Kovacs et al., 1979). The physical approach of Adam and Gibbs (1965) has proven successful in modeling the features of the glass transition and annealing of glasses far below T g. The Adam–Gibbs model is based on the Gibbs and DiMarzio (1958) thermodynamic model that states that the glass transition is a time-dependent manifestation of an equilibrium second order phase transition at T 2 . This theoretical equilibrium state can never be reached because the mobility vanishes as T ! T 2 , and is observed instead at a higher temperature T g 4T 2 as the glass transition. Adam and Gibbs (1965) hypothesized that the increase in the relaxation time during cooling to T 2 is caused by the progressive reduction in the number of available configurations. They proposed a formulation for the relaxation time that is a function of the temperature dependent configurational entropy. Scherer (1984) developed a nonlinear generalization of the Adam–Gibbs model by calculating the configurational entropy at T f instead of T. This allowed the configurational entropy to depend on the actual rather than equilibrium structure. Finally, Hodge (1987, 1997) developed an analytical approximation for nonlinear Scherer formulation of the structural relaxation time. The result predicts the transition in the temperature dependence of the relaxation time from an Arrhenius behavior for T5T g, where structural relaxation effectively is arrested, to the Williams–Landel–Ferry (WLF) behavior for TbT g, where the structure is in equilibrium. A principal feature of the constitutive model presented here is the incorporation of the nonlinear Adam–Gibbs model of structural relaxation into a continuum finite-deformation thermoviscoelastic model for amorphous SMPs. To keep matters simple, the developments neglect the effects of heat conduction and of pressure on the structural relaxation and inelastic behavior of the material. The paper begins with a description of the SMP material and experimental methods applied to characterize the thermomechanical properties and shape memory performance. The following section presents the development of the constitutive model. The model parameters are either directly measured from or fitted to thermomechanical characterization experiments. Finally, the model with the fitted parameters is applied to simulate the free and constrained recovery experiments. Comparisons with experiments show that the model can reproduce the stress-free strain–temperature response, the temperature and strain-rate dependent stress–strain response, and important features of the temperature dependence of the shape memory response. 2. Experimental method The material sample preparation, experimental methods, and experimental results have been described in detail in Qi et al. (2008). They are briefly presented here to facilitate later discussions comparing the modeling results and experimental data. 2.1. Material and sample preparations The SMP material was synthesized following the procedure of Yakacki et al. (2007). Tert-butyl acrylate (tBA) monomer and crosslinker poly(ethylene glycol) dimethacrylate in liquid forms, and photo polymerization initiator (2, 2-dimethoxy-2phenylacetophenone) in powder form were mixed according to a pre-calculated ratio and shaken for 10 s to ensure good

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mixing. The solution was injected either onto the surface of a specially designed glass slide or a glass tube, then placed under the UV lamp to polymerize. After 10 min, the polymer was removed from the glass slide or tube and cured in an oven for 1 h at 80 1C. The material formed on the glass slides were cut into thin strips measuring 10 mm  2 mm  1 mm for three-point bend tests, while those formed in the glass tubes were cut into plugs 5 mm in diameter and 5 mm in height for uniaxial compression tests. To minimize experimental error from sample variations, the samples from the same batch were used for all of the thermomechanical tests. For each test described below, at least three experiments were conducted to confirm the repeatability. 2.2. Dynamic mechanical analysis The dynamic mechanical analysis (DMA) tests were conducted using a Perkin-Elmer DMA Tester (Model 7 Series) with a Perkin-Elmer Intracooler 2 cooling system to measure the coefficients of thermal expansion (CTE), T g , and the dynamic mechanical properties. For the CTE measurements, a cylindrical SMP plug was placed between two platens to measure the displacement of the sample during cooling at a constant cooling rate. A small compressive force of 1 mN was applied to ensure contact between the sample and the platens. The sample was first heated from room temperature ð25  CÞ to 70  C at 5  C=min and allowed to equilibrate at 70  C for 20 min. Then, the heated sample was cooled to 0  C at qcool ¼ 1  C=min. To measure the temperature dependence of the storage and loss moduli and tan d, the thin strip samples was placed in a three-point bending fixture in the DMA and subjected to a dynamic temperature scan at a constant temperature rate of 1 1C/min. The dynamic temperature scan applied a small dynamic load to the three-point bending specimen at a frequency of 1 Hz. 2.3. Isothermal uniaxial compression tests Isothermal uniaxial compression tests at different temperatures and strain rates were performed to investigate the temperature dependence of the large deformation viscoplastic and viscoelastic behavior of the SMP material. The cylindrical SMP plug was placed between two compression platens in an Instron Universal Testing Machine (Model 5565) equipped with an Instron SFL Temperature Controlled Chamber (Model 3119-405-21) and a temperature controller (Euro 2408). Temperature control was achieved by a thermocouple placed close to the sample. The results presented in later sections report the temperatures measured by the thermocouple and thus should be interpreted as air temperatures. The sample and platens were heated to the test temperature at 1 1C/min and equilibrated at the test temperature for 30 min before being compressed at a constant strain rate. In order to reduce the effects of friction, Teflon sheets were placed between the upper and lower surfaces of the plug and the platens. The heated samples were compressed to 50% engineering strain at a constant strain rate e_ . The experiments examined two different engineering strain rates, e_ ¼ 0:01 s1 ; 0:1 s1 . 2.4. Thermomechanical cycle tests Thermomechanical cycle tests were performed on two groups of samples to characterize the unconstrained strain recovery and constrained stress recovery response. These tests were conducted using the load frame setup of the isothermal compression tests. A cylindrical SMP plug was placed in the Instron at room temperature. The sample and the compression platens were heated to T high ¼ 100  C at 1  C=min and equilibrated for 30 min. The heated sample was subjected to uniaxial compression at an engineering strain rate e_ ¼ 0:01 s1 to the test strain level and allowed to relax for 600 s at constant strain. The compressed sample then was cooled to T low ¼ 0  C at cooling rate qcool ¼ 1  C=min, then reheated to T high ¼ 100  C at heating rate qheat ¼ 1  C=min. For the constrained stress recovery tests, the platens were held in the lowered position during reheating to prevent recovery of the original height. The unconstrained strain recovery experiment followed the same procedure as the constrained recovery experiment, except that the upper platen was raised above the starting position before reheating to allow free recovery. The experiments measured the force exerted by the sample on the platen, and pictures were taken of the sample and used to measure the strain of the sample throughout the thermomechanical cycle. A nominal stress response was calculated by dividing the force by the undeformed area. 3. Constitutive model formulation In the following, a finite-deformation continuum constitutive model is developed for the thermoviscoelastic behavior of amorphous SMPs that incorporates the Adam–Gibbs theory of structural relaxation in the glass transition region. The model extends the features of the linear thermoviscoelastic rheological model shown in Fig. 2(a) to finite deformation. This is accomplished by assuming a series of multiplicative decompositions of the deformation gradient, first into thermal and mechanical components, then into elastic and viscous mechanical component. The sequence of deformation maps produced by the successive decompositions of the deformation gradient is illustrated in Fig. 2(b). The constitutive relations for the mechanical stress response and thermal deformation are additively split into equilibrium and nonequilibrium parts, and time evolution equations are specified for the nonequilibrium thermal and viscous deformation.

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ε εT

Reference Ω0

εM

μneq

εeM

μeq

ηs

εvM

FvM

F

Relaxed ∼ ΩM

FeM Spatial Ω

Heated ∼ ΩT FM

eT, Fig. 2. (a) A linear thermoviscoelastic rheological model. (b) An analogous decomposition scheme for the deformation gradient to define the heated, O e M , intermediate configurations. and the stress-free, O

3.1. Kinematics Let O0 denote the reference configuration of an undeformed and unheated continuum body at time t 0 and temperature T 0 . It is assumed that the configuration O0 is in thermodynamic equilibrium. This is a strong assumption in that it restricts the initial state of the material to be in the rubbery state, where relaxation events occur nearly instantaneously. To be more general and allow events to start in the glassy state (e.g., cold drawing) additional assumptions must be made to describe the initial nonequilibrium state of the glass.1 Otherwise, the model as proposed below can be cooled in a mechanically unconstrained manner to T5T g to achieve a stress-free glassy configuration. The nonlinear mapping of a point X 2 O0 to the spatial configuration O of the deformed and heated body is defined as x ¼ /ðXðtÞ; TðtÞÞ, and the tangent of the deformation map defines the deformation gradient F ¼ q/=qX. To separate the thermal and mechanical deformation we introduce the multiplicative split of the deformation gradient into thermal and mechanical components (Lu and Pister, 1975; Lion, 1997): F ¼ FM FT .

(1)

e T . It is assumed As shown in Fig. 2(b), the component FT is a mapping from O0 to an intermediate heated configuration O that the thermomechanical response is isotropic such that the thermal deformation gradient can be expressed as 1=3

FT ¼ YT 1,

(2)

where YT ¼ detðFT Þ is the thermal volumetric deformation. To model the stress relaxation response, FM is split further multiplicatively into elastic and viscous components (Sidoroff, 1974; Lion, 1997; Reese and Govindjee, 1998): FM ¼ FeM FvM ,

(3)

e T to a stress-free intermediate configuration O e M . The decomposition in Eq. (3) can be where is a mapping from O generalized as FMi ¼ FeMi FvMi for i ¼ 1; . . . ; P to model a discrete spectrum of P nonequilibrium processes (Govindjee and Reese, 1997). However to keep matters simple, the following developments consider only one stress relaxation process. Because most amorphous polymers exhibit vastly different volumetric and deviatoric behavior, it is convenient to formulate the constitutive relations in terms of separate volumetric and deviatoric contributions. The split formulation is enabled by the multiplicative decomposition of FM into volumetric and deviatoric components as described by Flory (1961) and Simo et al. (1985), FvM

1=3

FM ¼ YM

FM ,

(4)

where YM ¼ det½FM  is the mechanical component of the volumetric deformation. The total volumetric deformation is given by Y ¼ YM YT. It is assumed that the stress relaxation response is purely deviatoric such that the volumetric response caused by mechanical deformation is time-independent. Though not strictly true, this is a good approximation for most polymers, which exhibit only small changes in the bulk modulus, usually about a factor of two, over a wide range of time and frequencies Ferry (1980, Chapter 2, Section B). In contrast, the shear modulus can vary by orders of magnitudes over the same time and frequency range. Assuming that the volumetric/deviatoric split also applies to FeM and FvM , the isochoric flow assumption results in the following relations: YvM ¼ 1;

YM ¼ YeM ;

e

FM ¼ FM FvM .

(5)

1 A common source of discrepancy in the thermomechanical characterization of glasses and glassy polymers is the uncertainty in the thermomechanical history of the sample (e.g., curing and aging). The thermomechanical history is an important determinant of the nonequilibrium state. Consequently, this uncertainty can lead to significant differences in the thermomechanical response.

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The right and left Cauchy–Green deformation tensors are defined for FM and its components as T

CM ¼ FM FTM ;

CeM ¼ FeM FeM ;

bM ¼ FTM FM ;

bM ¼ FeM FeM ;

T

e

T

CvM ¼ FvM FvM , T

v

bM ¼ FvM FvM . 2=3 YM CM v

(6) e CM

2=3

CeM ,

2=3 YM bM

e bM

2=3

e

and ¼ YeM and likewise for bM ¼ and ¼ YeM bM . Substituting Eq. (4) into Eq. (6) gives, CM ¼ Finally, the rate of the viscous deformation tensor C_ M in O0 can be expressed as an objective rate in the spatial configuration O as _1 e dv bM ¼ FM ðCvM ÞFTM ,

(7)

where the operator dv ðÞ is the Lie time derivative. 3.2. Constitutive relations 3.2.1. Thermal strains and structural relaxation A variety of approaches have been developed to model the glass transition phenomena. Here, an internal variable approach is adopted to describe the nonequilibrium structure of the polymer in the formulation of the free energy density. We choose for the internal variable the fictive temperature T f . First introduced by Tool (1946) to model the annealing of glass, the fictive temperature is the temperature at which the nonequilibrium structure at T is in equilibrium. The significance of T f is illustrated in Fig. 3, which also shows that the limiting value of T f is T g . To describe the observed timedependent response to a temperature jump, Tool (1946) proposed the following evolution equation for T f to its equilibrium value T: dT f 1 ¼  ðT f  TÞ. tR dt

(8)

For a material initially in equilibrium at T 0 , T f ðT 0 ; t 0 Þ ¼ T 0 . The parameter tR, commonly referred to as the structural relaxation time, is the characteristic retardation time of the volume creep. By applying the definition of T f illustrated in Fig. 3, the isobaric volumetric deformation in response to a temperature change from T 0 to T can be evaluated as YT ðT; T f Þ ¼ 1 þ ar ðT  T 0 Þ þ ðDaðT  T f ÞÞ , |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} deq

(9)

dneq

where ar is the CTE of the rubbery material ag is the CTE of the glassy material and Da ¼ ar  ag . The expression for YT is split into a time-independent equilibrium part deq that represents the long-time thermal dilatation, and a time-dependent nonequilibrium part that represents the departure from equilibrium. An evolution equation for dneq can be obtained by substituting the expression for dneq of Eq. (9) into Eq. (8) for T f : ddneq 1 dT . ¼  dneq  Da tR dt dt

(10)

Θ (T0)

αr

αg

Θ (T1) αg

T1

Tf (T1) T0 Tg

Fig. 3. Illustration of the volumetric deformation as a function of temperature for cooling from T 0 to T 1 . The glass transition temperature T g is found by the intersection of the high temperature line, with slope ar , and low temperature line, with slope ag . The fictive temperature at T 1 , T f ðT 1 Þ, is found by extrapolating a line from YðT 1 Þ with slope ag to the high temperature line. The limiting value of T f is T g (Scherer, 1986, 1990).

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The result is similar to the evolution equation for the departure from equilibrium ðv  veq Þ=veq , defined by the KAHR model for structural relaxation (Kovacs et al., 1979). The differences between the two are negligible for small dneq 51. The evolution Eq. (10) can be written alternatively to remove the explicit dependence on T_ as neq

dd dt

¼

1 neq ðd  DaðT  T 0 ÞÞ, tR

(11)

neq

¼ dneq þ DaðT  T 0 Þ is the departure from the instantaneous response ag ðT  T 0 Þ. Following Kovacs et al. (1979), where d Eq. (11) can be generalized to include multiple relaxation processes, neq

ddj

dt

¼

1 neq ðd  Daj ðT  T 0 ÞÞ; tR j j

neq

d

¼

Q X

neq

dj

j

;

Da ¼

Q X

Daj ,

(12)

j

where j ¼ 1; . . . ; Q is the number of structural relaxation processes. Only one process is assumed here for simplicity.

3.2.2. The structural relaxation time The structural relaxation time tR is a macroscopic measure of the molecular mobility of the polymer, and is related to the viscosity through a modulus-like quantity. It depends on the temperature and polymer structure, which is represented by T f (or equivalently by dneq ) in the model. As illustrated in Fig. 1(a), the isostructural log viscosity curve below the glass transition region is linearly related to 1=T, which indicates an Arrhenius relationship. Above the glass transition temperature, the equilibrium log viscosity curve has a significantly steeper slope and a slightly curved shape, which suggests a Vogel–Fulchur–Tamman (VFT) relationship. The steeper slope of the equilibrium curve signifies that above the glass transition, the relaxation time decays significantly faster with temperature. In the context of the description of Lendlein and Kelch (2002) and Lendlein et al. (2005) of the molecular shape memory mechanism, this change in the temperature dependence of the relaxation time upon the glass transition is the switching mechanism of shape storage and recovery of amorphous SMPs. Adam and Gibbs (1965) developed a model for the temperature dependence of tR based on the thermodynamic theory of Gibbs and DiMarzio (1958) for the glass transition. The model proposes that viscous flow during cooling involves the cooperative motion of progressively larger number of molecules. The number of cooperatively rearranging molecular groups depends inversely on the configurational entropy and become infinite at T ¼ T 2 where the configurational entropy becomes zero. The equilibrium state at T 2 defines a second order phase transition where the temperature derivatives of the volume and enthalpy, i.e., the CTE and heat capacity, are discontinuous. Adam and Gibbs hypothesized that the increase in the tR as T ! T 2 is caused by the progressive reduction of the number of available configurations with cooling and proposed the following formulation for the retardation time as a function of the configurational entropy Sc, tR ¼ B1 exp



 B2 ; TSc ðTÞ

Sc ðTÞ ¼

Z

T T2

DC p ðT 0 Þ dT 0 . T0

(13)

The parameter B1 is a scaling constant, B2 is an activation parameter and C p ðTÞ is the isobaric heat capacity. Scherer (1984) proposed a nonlinear generalization of Eq. (13) that evaluates Sc at the fictive temperature T f instead of at T. This allows the mobility to depend on the nonequilibrium rather than the equilibrium configuration. To obtain an analytical expression for tR , Hodge (1987, 1997) approximated the temperature dependence of the heat capacity as DC p ¼ a=T, where a is a constant. Substituting this into the nonlinear generalization of Eq. (13) gives   QR , (14) tR ðT; T f Þ ¼ tR0 exp Tð1  T 2 =T f Þ where tR0 is a reference value of the structural relaxation time and Q R is an activation parameter. Eq. (14) can reproduce the transition from equilibrium to isostructural behavior shown in Fig. 1(a). At equilibrium, T f ¼ T and Eq. (14) reduces to the VFT relation,   QR . (15) tR ðT; T f Þ ¼ tR0 exp T  T2 Well below the glass transition temperature, where structural relaxation effectively is curtailed and T f ! T g , Eq. (14) reduces to the Arrhenius relation,   A , (16) tR ðT; T f Þ ¼ tR0 exp T where A ¼ Q R T g =ðT g  T 2 Þ. A more convenient form of Eq. (14) can be obtained by considering that the VFT equation is equivalent to the WLF equation for the time/temperature shift factor (Ferry, 1980, Chapter 11). Using a reference glass transition temperature T ref g ,

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the WLF equation can be written as aT ¼ log

tR ðT ref g Þ

! ¼

tR ðTÞ

C 1 ðT  T ref g Þ C 2 þ T  T ref g

,

(17)

where C 1 and C 2 are the two WLF constants. Eq. (17) is obtained from Eq. (15) simply by applying the following relations: T 2 ¼ T g  C2;

QR ¼

C1C2 ; log e

tR0 ¼ tref R exp

C 1 , log e

(18)

where tRg is the retardation time measured at T ref g . These relations can be used to rewrite the Hodge–Scherer equation in terms of the WLF constants and T ref as g "

tR ðT; T f Þ ¼

tref R

C 2 ðT  T f Þ þ TðT f  T ref C1 g Þ exp  log e TðC 2 þ T f  T ref Þ g

!# .

(19)

Eq. (19) is more convenient than Eq. (14), for the purpose of parameter determination because C 1 , C 2 and T ref can be g determined from standard thermomechanical tests. 3.2.3. Stress relations To model the time-dependent mechanical stress response, it is assumed that the Helmholtz free energy density can be expressed as the sum of equilibrium and nonequilibrium components, e

e

CðCM ; CM ; YM ; TÞ ¼ W neq ðCM Þ þ W eq ðCM Þ þ UðYM Þ. eq

(20)

neq

are the time-independent equilibrium and time-dependent nonequilibrium parts of the deviatoric The terms W and W component of the free energy density, while U is the volumetric component. The equilibrium component W eq describes the rubbery stress response of the SMP at high temperatures. For T4T g, the mobility is sufficiently high for macroscopic deformation of the polymer to occur by the entropic configurational rearrangement of the chain segments. The process involves the straightening and reorientation of the chain segments towards the loading direction. To capture the effects of chain straightening and alignment, the eight-chain network model of Arruda and Boyce (1993b) with Langevin chain statistics is applied for W eq,     x  l l  w0 ; x ¼ L1 eff , (21) W eq ðIM1 Þ ¼ mN l2L eff x þ ln sinh x lL lL where w0 is a constant related to the energy of the undeformed chain. The function L1 is the inverse Langevin function where LðuÞ ¼ coth u  u1 , leff is the effective chain stretch given by qffiffiffiffiffiffiffiffiffi leff ¼ 13IM1 (22) and lL is the locking stretch. The parameter mN represents the characteristic stiffness of the polymer network. At low temperatures ToT g , the polymer chains are effectively ‘‘frozen’’ by the vanishing mobility. Consequently, macroscopic deformation occurs primarily by the enthalpic stretching of intermolecular and intramolecular bonds. The resulting stiff stress response of the glassy material is modeled by applying a neo-Hookean strain energy density potential for W neq, e

W neq ðIM1 Þ ¼

mneq e ðI  3Þ. 2 M1

(23)

Finally, the following strain energy potential is assumed for the volumetric stress response, U neq ðYM Þ ¼ kðYM  ln YM  1Þ,

(24)

where k is the characteristic bulk modulus. The second Piola Kirchhoff stress relation is defined from the free energy density as, S ¼ 2qC=qC. Evaluating S from C in Eq. (20) and applying the Piola transformation to the result gives the following expressions for the deviatoric and volumetric components of the Cauchy stress response r ¼ s þ p1,       e 1 1 1 1 e lL l s ¼ meq bM  IM1 1 þ mneq bM  IM1 1 ; meq ¼ mN L1 eff , J 3 J 3 lL leff |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} seq

1 p ¼ kðYM  1Þ. J

sneq

(25)

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Eq. (25) shows that mneq þ meq is the shear modulus of the short-time deviatoric stress response. To complete the e constitutive formulation, the following evolution equation is applied for bM (Reese and Govindjee, 1998), 1 1 neq e e1 s ,  dv bM bM ¼ 2 2ZS

(26)

where ZS is the shear viscosity and dv is the Lie time derivative defined in Eq. (7). 3.2.4. The viscous flow rule It is assumed that the shear viscosity ZS is related to the structural relaxation time tR , and thus can depend on the temperature T and polymer structure through T f . In addition, it can depend on the flow stress sneq . Experiments have shown that amorphous SMPs in the glassy state exhibit large viscoplastic deformation. The molecular processes of viscoplastic flow for moderate strain rates and temperatures below T g of glassy polymers have been attributed to the local intermolecular resistance to segmental rotation and the network resistance to molecular alignment (Haward and Thackray, 1968; Argon, 1973; Ward, 1984). Yielding occurs when the ‘‘frozen’’ polymer chains overcome the intermolecular resistance to segmental rotation for local rearrangement. The initial rearrangements of the chain segments alters the local structure in a manner that decreases the shear resistance of the material, possibly by increasing the local free volume, and leads to a macroscopic post-yield strain-softening behavior. As the material flows, the chain segments begin to align along the flow direction, and strain hardening occurs when the chain alignment reaches the limit of chain extensibility. A number of models have been developed for the finite-deformation viscoplastic response of glassy polymers at moderate strain rates and temperatures that incorporate the molecular processes of yielding described above (e.g., Boyce et al., 1989a; Arruda and Boyce, 1993a; Wu and van der Giessen, 1993; Anand and Ames, 2006). Here, the Eyring (1936) model is applied to describe the temperature-dependent, stress-activated nature of viscoplastic flow process. The Eyring equation can be written for the effective viscous shear stretch rate as     sy T E0 QS s g_ v ¼ pffiffiffi ref sinh exp , (27) kB T T sy 2ZS Q S g

pffiffiffi where is the characteristic shear viscosity, s ¼ ð12 sneq  sneq Þ1=2 ¼ ksneq k= 2, is the flow stress, sy is the yield (activation) stress, and E0 and Q S are activation parameters. In the limit of small stresses, a Taylor’s expansion of Eq. (27) about s ¼ 0 gives the following linear flow rule:   1 E0 ksneq k. (28) g_ v ¼ ref exp kB T 2ZS Zref Sg

g

Eq. (28) exhibits an Arrhenius temperature dependence which is exhibited only by the glassy material. To extend the viscoplastic flow rule to the glass transition and rubbery temperature regions, the Scherer–Hodge Eq. (19) is applied to modify the temperature dependence of Eq. (27) to give " !#   C 2 ðT  T f Þ þ TðT f  T ref sy T C1 QS s g Þ v . (29) exp g_ ¼ pffiffiffi ref sinh log e T sy 2ZSg Q S TðC 2 þ T f  T ref g Þ It is assumed that the effective viscous stretch rate is related to the viscosity through g_ v ¼

1 ksneq k, 2ZS

(30)

such that the viscosity can be formulated as " !#  1 C 2 ðT  T f Þ þ TðT f  T ref C1 QS s g Þ ref Q S s ZS ¼ ZSg exp  . sinh T sy log e T sy TðC 2 þ T f  T ref g Þ

(31)

In the rubbery region, the structure is near equilibrium such that T f ! T and ksneq ! 0k. As a result the viscosity ZS ðTÞ in Eq. (31) reduces to the WLF equation for the time and temperature shift factor in Eq. (17). Substituting Eq. (31) into Eq. (26) e gives an evolution equation for the internal deformations bM , 1 e e1  dv bM bM ¼ g_ v nv ; 2

nv ¼

sneq . ksneq k

(32)

The evolution Eq. (32) states that the magnitude of the rate of inelastic deformation tensor scales with the effective viscous stretch rate while its direction is determined by the direction nv of the flow stress. Similar unified treatments of structural and stress relaxation have been suggested previously for small-strain thermoviscoelastic models (Rekhson, 1985; Aklonis, 1981) and finite-deformation convolution integral models (Caruthers et al., 2004) for glassy polymers. To model the postyield softening response, the phenomenological evolution equation used by the Boyce et al. (1989b) model is proposed for

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Table 1 Thermo-viscoelastic model Kinematics 1=3

1=3

FM ¼ YT YM YM ¼ J=YT

F

Thermal deformation response neq

YT ðT; dneq Þ ¼ 1 þ ag ðT  T 0 Þ þ d neq 1 neq neq d_ ¼  ðd  DaðT  T 0 ÞÞ; d ðt ¼ 0Þ ¼ 0 tR " !# ref C 1 C 2 ðT  T f Þ þ TðT f  T g Þ neq ; tR ðT; T f ðd ÞÞ ¼ tRg exp  ref log e TðC 2 þ T f  T g Þ

Tf ¼

1 neq d þ T0 Da

Stress response 1 p ¼ kðYM  1Þ J      e 1 lL 1 leff 1 1 1e s ¼ mN L bM  IM1 1 þ mneq bM  IM1 1 J leff 3 J 3 lL |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} seq

sneq

1 sneq e e1 e  dv bM bM ¼ g_ v neq ; bM ðt ¼ 0Þ ¼ 1 2 ks k " !# ! ref sy T C 1 C 2 ðT  T f Þ þ TðT f  T g Þ Q S ksneq k pffiffiffi sinh exp g_ v ¼ pffiffiffi ref ref Q log e T 2ZS S 2sy TðC 2 þ T f  T g Þ  g  sy s_ y ¼ h 1  g_ v ; sy ðt ¼ 0Þ ¼ sy0 syss

the yield strength,   sy g_ v ; s_ y ¼ h 1  syss

sy ðt ¼ 0Þ ¼ sy0 ,

(33)

where sy0 at t ¼ 0 and syss osy0 is the steady-state yield strength. The temperature dependence of the stress relaxation response of the thermoviscoelastic model can be understood from the point of view of the rheological model in Fig. 2(a). For TbT g, ZS is small, and the effects of viscoelasticity are negligible. Consequently, the rheological model reduces to the equilibrium spring in series with the temperature element. This situation corresponds to the mobile rubbery state, where stress relaxation can occur quickly to prevent the development of large nonequilibrium stresses. As the temperature is lowered and the material begins to enter the glass transition region, the dashpot in the Maxwell element stiffens and the nonequilibrium spring becomes progressively engaged in the mechanical response. Viscoelastic stress relaxation becomes important. Decreasing the temperature to below the T g causes the dashpot to stiffen further and the material to build up more nonequilibrium stress. The ability to build up large nonequilibrium stress allows the glassy material to exhibit a dramatically stiffer stress response than the rubbery material. This process continues until the nonequilibrium stress exceeds the yield strength ðs4sy0 Þ of the nonequilibrium spring. The resulting viscous flow causes the dashpot to soften according to Eq. (33). 3.3. General remarks The constitutive relations for the thermoviscoelastic behavior of amorphous SMPs are summarized in Table 1. It is important to note that the thermoviscoelastic model presented here is relatively simple compared to the real behavior of SMPs. Currently, the model does not consider the effects of heat transfer, deformation-induced changes in the configurational entropy, and of pressure on the structural and stress relaxation response. The model also assumes a single nonequilibrium process for the structural relaxation response. Consequently, the model cannot reproduce the ‘‘memory effects’’ of the thermal deformation, YT , in response to successive temperature changes as described by Kovacs et al. (1979). A single nonequilibrium process is also assumed for the stress relaxation response, yet multiple relaxation mechanism likely are needed to describe the long range viscoelastic flow processes of the rubbery material and the short-range viscoplastic flow processes of the glassy material. Finally, the model assumes that the same mechanism governs the material response to chain alignment at high and low temperatures. Consequently, W eq is applied to model both the entropy-driven stress response of the rubbery material and the flow-induced strain-hardening response of the glassy material. A few authors have commented on the conceptual difficulties of applying an entropic model to describe the strain-hardening behavior of the glassy material (Hoy and Robbins, 2006; Anand and Ames, 2006). Below the T g , the chains lack sufficient mobility to sample the configurational space. However, the eight-chain model has been successful at describing experimental data, though the values for mN and lL may not agree with theoretical predictions. For this reason mN is treated as a temperature-independent phenomenological parameter and not the physical quantity mN ¼ nkT derived from

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Table 2 Parameters of the preliminary thermoviscoelastic constitutive model for amorphous SMPs Parameter

Values

Physical significance Glass transition temperature for qcool ¼ 1  C=min

25

 T ref g ð C)

Þ

7.67

Rubbery coefficient of volumetric thermal expansion

ag ð104  C1 Þ

3.85

Glassy coefficient of volumetric thermal expansion

ar ð10

4 

C

1

1100

tref Rg (s) C1 C 2 (  C) mN (MPa) lL k (MPa) mneq (MPa) ref neq (s) tref Sg ¼ ZSg =m

Q S =sy0 ð K=MPaÞ syss =sy0 h0 (MPa)

Structural relaxation time at T ¼ T ref g

17.44 90 0.28 4.12 611 406 34.9

First WLF constant Second WLF constant Shear modulus of equilibrium polymer network Limiting chain stretch of equilibrium network Bulk modulus Nonequilibrium shear modulus of glassy material

101 0.43 250

Activation parameter for viscous flow Ratio of initial to steady-state yield strength Flow softening modulus

Deviatoric stress relaxation time at T ¼ T ref g

-0.4 Experiments

Thermal Strain (1-L/L0) %

Simulations Low Temp Fit

-0.6

High Temp Fit

-0.8 Tg = 25°C -1

-1.2

5

15

25 Temperature °C

35

45

Fig. 4. Thermal strain response for a stress-free, constant cooling rate (qcool ¼ 1  C=min) test comparing experiments and simulations. The glass transition temperature, obtained from the extrapolated high and low temperature lines, is T g ¼ 25  C.

entropic elasticity, where n is the density of network chains. The temperature changes examined here are small. Thus, neglecting the temperature dependence of mN will not have a significant effect. The entropic stiffening and viscoplastic strain-hardening behaviors can be distinguished by prescribing separate relaxation processes (i.e., separate Maxwell elements in Fig. 2(a)) for viscous flow in the glassy and rubbery regions. 4. Thermomechanical response To demonstrate the ability of the thermoviscoelastic model to reproduce the shape memory behavior, the constitutive relations in Table 1 were implemented in a finite element (FE) program using the algorithm briefly described in Appendix A and applied to simulate the stress-free thermal deformation response and the isothermal uniaxial compression stress response. These simulations were applied to fit the parameters of the model to the corresponding experimental data (see Section 2). The results of the parameter fitting is described in below, while details of the parameter fitting procedure and a parameter study are described in Appendix B. Table 2 lists the model parameters, their physical significance, and fitted values. 4.1. Thermal strain response The CTEs, ar and ag , and reference glass transition temperature T ref g were determined for the SMP material from the thermal strain response to the stress-free, constant cooling rate experiments described in Section 2.2. The linear rubbery and glassy CTEs were computed from the slopes of the thermal strain vs. temperature plots at high and low temperatures.

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The volumetric CTEs were approximated as three times the linear CTEs. This resulted in ar ¼ 7:67  104  C1 and ag ¼ 3:85  104  C1 . The T g was determined from the intersection of the extrapolated high and low temperature lines as  shown in Fig. 4. The result was used for the reference glass transition temperature, T ref g ¼ 25 C. The WLF constants can be determined by applying the procedure described by Ferry (1980, Chapter 11, Section B) to calculate the temperature dependence of the shift factor aT ðTÞ from DMA tests conducted over a wide range of temperatures and frequencies. However, lacking the data, FE simulations of the constant cooling rate tests, described in Appendix B.1, were applied to fit C 1 , C 2 , and the characteristic structural relaxation time tref R to the measured thermal strain vs. temperature curves. The parameters associated with the mechanical deformation response do not affect the stress-free thermal strain response and thus were not considered. It has been shown that C 1 does not vary significantly for amorphous polymers. Thus, the simulations used the universal value for C 1 as given in Ferry (1980, Chapter11, Section C) and only varied the parameters C 2  ref and tref Rg to fit the experimental data. This resulted in C 2 ¼ 90 C and tRg ¼ 1100 s. Fig. 4 shows the thermal strain response comparing the experimental data and simulation results. Plotted is 1  L=L0 , where L and L0 are the deformed and original height of the FE geometry. The model was able to reproduce well the T g and the gradual transition from the rubbery to glassy response. 4.2. Stress response 4.2.1. Glassy and rubbery moduli The equilibrium and nonequilibrium moduli, meq mneq and k, were determined from the initial Young’s modulus measured in the uniaxial compression tests described in Section 2. It was assumed that the SMP material exhibited a Poisson’s ratio of 0.35 in the glassy state which is typical of amorphous polymers and 0.5 in the rubbery state. The former assumption gave k ¼ 611 MPa and mneq ¼ 406 MPa from the Young’s modulus Eg ¼ 1:0 GPa measured at T ¼ 20  CoT g , while the latter gave meq ¼ 0:88 MPa for the Young’s modulus Er ¼ 2:6 MPa measured at T ¼ 60  CbT g . The limiting chain stretch lL was fitted to the stress–strain curve at a high temperature T4T g . This gave lL ¼ 4:12 (Qi et al., 2008). Finally, mN ¼ 0:28 MPa was computed from meq using Eq. (25), lL ¼ 4:12 and leff ¼ 1. 4.2.2. Characteristic stress relaxation time ref neq The characteristic stress relaxation time defined as tref was determined from the temperature dependence of Sg ¼ ZSg =m the storage modulus measured by the DMA experiments described in Section 2.2 using FE simulations of small strain cyclic uniaxial compression. The details of the simulations are described in Appendix B.2. The simulations used the tref Rg , C 1 and C 2 fitted to the thermal strain response and the meq mneq and k fitted to the uniaxial compression tests as described above. The parameters Q S , sy0 , syss and h0 governing the yielding and post yielding behavior did not affect the small strain response and were not considered. The results yielded a value of tref Sg ¼ 34:9 s. Fig. 5 plots the storage modulus and tan d comparing the experimental data and simulation results of the fitted model. The glassy storage modulus measured by the DMA experiments was a factor of two smaller than the Young’s modulus measured in the constant strain-rate uniaxial compression experiments. Since the simulations used the Young’s modulus measured by the uniaxial compression experiments, the computed glassy storage modulus was a factor of two higher than measured by the DMA experiments. The sources of the discrepancy between the two experimental measurements were unclear. It was likely that the stiffness of the sample below T g was comparable to the stiffness of the DMA testing machine. This would have resulted in an artificially low stiffness measurement of the glassy storage modulus. Regardless, the simulations were able to reproduce well the temperature range and starting temperature of the glass transition region. The same was observed for the tan d. Though the maximum tan d calculated by the simulation was significantly larger than the measured value, the temperature of the peak value and the temperature range of the tan d variation agreed well with

8

Experiment Simulations

Experiment Simulations

7 6

103

5 Tan δ

Storage Modulus (MPa)

104

102

4 3 2

101

1

100

-1

0 0

20

40 60 80 Temperature (°C)

100

0

20

40 60 80 Temperature (°C)

100

Fig. 5. Temperature dependence of the (a) storage modulus and (b) tan d comparing experimental data and simulation results.

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70 50

Experiment

True Stress (MPa)

True Stress (MPa)

40 10°C

30

20°C

20

30°C

10 50°C

0

0.2 0.4 0.6 0.8 True Strain (mm/mm)

Simulation

50 10°C

40

20°C

30

30°C

20

40°C

10

40°C

0

Experiment

60

Simulation

1

50°C

0 0

0.2 0.4 0.6 0.8 True Strain (mm/mm)

1

Fig. 6. Isothermal uniaxial compression stress response at different temperatures spanning the glass transition region for applied engineering strain rates (a) e_ ¼ 0:01 s1 and (b) e_ ¼ 0:1 s1 comparing simulations and experiments.

the experimental data. The larger tan d peak indicated that the model was more dissipative in the glass transition region than the SMP material. This may be the result of using only one stress relaxation and one structural relaxation process to model the complex thermomechanical behavior. A more extensive parameter study is needed to understand the source of this discrepancy and its effect on the ability of the model to reproduce the shape memory performance. 4.2.3. Viscoplastic parameters The remaining parameters Q S , sy0 , syss and h0 governed the yield and post-yield behavior of the glassy material, and were fitted to the isothermal constant strain-rate uniaxial compression experiments described in Section 2.3. Details of the FE simulations of the uniaxial compression experiments and the fitting procedure are described in Appendix B.3. The results of the parameter fitting were: Q S =sy0 ¼ 101  K=MPa, syss =sy0 ¼ 0:43 and h0 ¼ 250 MPa. Fig. 6 compares the isothermal uniaxial stress response of the model and experiments for two different strain rates. The model was able to reproduce qualitatively the temperature dependence and strain-rate dependence of the yield strength. In particular, the model captured the temperature transition of the inelastic behavior from the viscoplastic behavior of the glass ‘with a distinct yield point’ to the viscoelastic behavior of the rubber. In addition, the model was able to capture the strain-rate dependence of the transition temperature. The viscoplastic/viscoelastic transition in the inelastic behavior was obtained at T ¼ 40  C for e_ ¼ 0:01 s1 and T ¼ 50  C for e_ ¼ 0:1 s1. However, the model generally computed a larger increase in the yield strength for increasing strain rates, and a larger drop in the yield strength with increasing temperature than measured by experiments. It also calculated a stiffer glassy response, which resulted in smaller yield strains. The difference in the yield strains between the experiments and simulations increased with the temperature. The overly stiff response suggested that additional short-time relaxation processes may be needed to describe the glassy stress response. 5. Shape memory behavior To demonstrate the shape memory performance of the thermoviscoelastic model, the axisymmetric FE model of a quarter section of a cylindrical SMP plug and compression platens shown in Fig. 7 was applied to simulate the thermomechanical cycle experiments described in Section 2.4. The plug measured H ¼ 1 mm in height and R ¼ 0:5 mm in radius, while the platen was Hp ¼ 0:5 mm in height and Rp ¼ 1:5 mm in radius. The plug and platen were discretized using bilinear C 0 square elements of size h ¼ 0:05H. The interface between the plug and platen was modeled by frictionless penalty contact elements to allow for unrestricted lateral deformation during compression. Displacements were fixed on the bottom surface, uz ðz ¼ 0Þ ¼ 0, and central axis of the plug, ur ðr ¼ 0Þ ¼ 0, to prevent rigid body motions. Otherwise, the plug was free to deform in response to an applied load and temperature change. The thermoviscoelastic constitutive model with the parameters in Table 2 was applied for the plug while a hyperelastic Kirchhoff–St. Venant model was used for the platen. The Young’s modulus of the platen was 1000 times the glassy modulus of the SMP plug. No additional fitting of the parameters were performed to obtain the following results. 5.1. Unconstrained strain recovery To model thermomechanical cycle experiments with unconstrained strain recovery, the plug initially in equilibrium at T H ¼ 100  C was compressed by the platen to an engineering strain of uZ =H ¼ 17% at a strain rate e_ ¼ u_ Z =H ¼ 0:01 s1 and relaxed for 10 min. Little stress relaxation was observed during this time. The deformed plug was cooled at a rate of qcool ¼ 1  C=min to a low temperature T L ¼ 0  C. Next, the platen was raised to above its starting position, and the plug was reheated at a rate qheat ¼ 1  C=min to T H ¼ 100  C. Fig. 8 plots the stretch ratio l ¼ 1  uZ =H of the plug during the reheating

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uz

Hp = 0.15mm

Frictionless contact element

ur (r = 0) = 0 H = 1.0mm

z

uz (z = 0) = 0

r

R = 0.5mm Rp = 1.5mm Fig. 7. Finite element axisymmetric model of uniaxial compression of cylindrical SMP plug.

1

Stretch Ratio (mm/mm)

0.98 0.96 0.94 0.92 0.9 0.88 0.86

Experiments Simulations

0.84 0.82 0

20

40 60 Temperature °C

80

100

Fig. 8. Stretch ratio as a function of temperature during reheating in the unconstrained recovery test, comparing experiments and simulations for qheat ¼ qcool ¼ 1  C=min.

process comparing the simulation results and experimental data. Both showed that the onset of strain recovery occurred  near the glass transition temperature T ¼ T ref g ¼ 25 C. However, strain recovery occurred much more slowly in the experiments than the simulations. Full recovery was achieved in the simulation at T ¼ 43  C, and further increases in the stretch ratio at higher temperatures was caused solely by thermal expansion. Full recovery was not achieved in the experiments until T ¼ 71  C. In comparison, the previous model by the authors (Qi et al., 2008), which applied a phase transformation approach, captured the end temperature for full recovery, but calculated a start temperature that was 25  C higher than observed in experiments. Strain recovery was completed by the thermoviscoelastic and phase transition models roughly within the same temperature span. To investigate the importance of structural and stress relaxation on the unconstrained strain recovery response, the ref simulation was repeated for different values of tref Sg and tRg . The results of the parameter study are plotted in Fig. 9. They showed that the larger structural relaxation time caused the onset of strain recovery to occur at a lower temperature. It also led to a more gradual temperature dependence at the start of strain recovery. The temperature dependence at higher ref temperatures was unaffected by tref Rg . In contrast, the larger stress relaxation time tSg shifted the onset of strain recovery to a slightly higher temperature but produced a noticeably more gradual temperature dependence of the recovered strain and thus a longer recovery time. Structural relaxation strongly influenced the unconstrained strain recovery response only at the start, while stress relaxation played a more prominent role throughout the process. The result of the parameter study suggested that the presence of additional long-time stress relaxation processes of the rubbery material not captured in the model might be a significant contributor to the long strain recovery time observed in the experiments. The characteristic stress relaxation time of the model at T g ¼ 25  C was tref Sg ¼ 34:9 s. It decreased to less than one second at 60  C. By comparison, the cooling process lasted for t ¼ 6000 s. The discrepancies between experiments

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1 0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.82

τref Rg = 1100 s τ ref Rg =

0

20

40

60

11000 s

80

100

Stretch Ratio (mm/mm)

T.D. Nguyen et al. / J. Mech. Phys. Solids 56 (2008) 2792–2814

Stretch Ratio (mm/mm)

2806

1 0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.82

τ Sg = 34.9 s τ ref Sg = 349 s ref

0

20

Temperature °C

40

60

80

100

Temperature °C

Fig. 9. Stretch ratio during reheating for the unconstrained recovery simulations examining the effects of (a) the characteristic structural relaxation time ref tref Rg and (b) the characteristic stress relaxation time tSg .

2 Experiments Simulations

1 Cool Deform

Heat

Normalized Stress

1.5

0.5

0 0

20

40 60 Temperature °C

80

100

Fig. 10. Stress as a function of temperature throughout the thermomechanical cycle of the constrained recovery test, comparing experiments and simulations for qheat ¼ qcool ¼ 1  C=min. The stress is normalized by the stress response of the deformed plug at T ¼ 100  C.

and simulations could have been caused also by the effects of heat transfer, which was not included in the model. Nonuniform cooling and heating from heat transfer would have resulted in a longer recovery time. The model also neglected the effects of pressure and configurational changes from chain straightening on the relaxation times.

5.2. Constrained stress recovery To model the thermomechanical cycle experiments with constrained stress recovery, the plug initially in equilibrium at T H ¼ 100  C was compressed by the platens to an engineering strain of 20% at a e_ ¼ 0:01=s and relaxed for 20 min. The results showed negligible stress relaxation during this time. The deformed plug was cooled at a rate of qcool ¼ 1  C=min to a low temperature T L ¼ 0  C. Next, the plug was reheated at a rate of qheat ¼ 1  C=min to T H ¼ 100  C with the platen held in the lowered position to prevent the plug from recovering the original height. Fig. 10 plots the engineering stress response as a function of temperature throughout the thermomechanical cycle. Also plotted are the experimental results for the same cooling and heating rates. The stress was computed for the FE simulations by summing the nodal reaction forces of the plug on the face z ¼ 0 and dividing by the undeformed area. Both the measured and computed stress response were normalized by the stress of the deformed plug obtained at the end of the 20 min relaxation period. At the start of cooling, thermal contraction caused the stress in the deformed plug to decrease gradually. Near T ¼ 36  C, the stress quickly dropped to zero. The stress–temperature curve during reheating did not follow the cooling curve. Instead, the stress became nonzero at a lower temperature and a large stress overshoot developed before the reheating curve rejoined the cooling curve. The simulation was able to predict many features of the experimental curve including: (1) the temperature T ¼ 8  C when the stress of the reheating curve regained a nonzero value, (2) the

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2.5

τ ref Rg = 1100 s τ ref Rg =

11000 s

Normalized Stress

Normalized Stress

2 1.5 1 0.5

2807

τref Sg = 34.9 s τ ref = 349 s Sg

2 1.5 1 0.5

0

0 0

20

40 60 Temperature °C

80

100

0

20

40 60 80 Temperature °C

100

4

-0.04 Elastic Strain (λM-1)

-0.06

e

Thermal Volumetric Strain (ΘT-1)

Fig. 11. Stress as a function of temperature throughout the thermomechanical cycle for the constrained recovery simulations examining the effects of (a) ref the characteristic structural relaxation time tref Rg and (b) the characteristic stress relaxation time tSg .

-0.08 -0.1 -0.12

x 10-4

2 Cooling

control τ ref Rg= 11000 s τ ref Sg = 349 s

0 -2 -4 -6 Heating -8

0

20

40

Temperature °C

60

0

20

40 60 80 Temperature °C

100

e ref Fig. 12. Effects of different tref Rg and tSg on the (a) thermal strains YT  1 and (b) elastic strain component lM  1. The control case correspond to ref tref ¼ 1100 s and t ¼ 34:9 s. Rg Sg

temperature T ¼ 36  C where the reheating curve joined the cooling curve and (3) the peak stress of the reheating curve. The simulation was not able to capture the extent of stress relaxation during cooling. The larger stress relaxation observed in the experiments further suggested the presence of additional long-time stress relaxation processes of the rubbery material neglected by the model. However, the results in Fig. 12 represented an improvement over the SMP model of Qi et al. (2008), which was unable to reproduce many of these features. To investigate the importance of structural and stress relaxation on the constrained stress recovery response, the ref simulation was repeated for different values of tref Sg and tRg . The results of the parameter study are plotted in Fig. 11. The thermal strain YT  1 and elastic strain component leM  1 in the direction of loading for the cases of the parameter study are plotted as a function of temperature in Fig. 12. The shape of the leM -temperature plots in Fig. 12(b) were the mirror image of the stress–temperature plots in Figs. 10 and 11, which showed that viscoelasticity in the glass transition region ref was responsible for the observed hysteresis in the stress–temperature curves. Both tref Sg and tRg had a significant though ref opposing effect on the stress–temperature response. As shown in Fig. 12(a), the larger tRg inhibited structural relaxation which shifted the T g to a higher temperature and produced smaller thermal strains. The smaller thermal strains led to smaller changes in the mechanical strain, which caused a more gradual stress drop-off during cooling and a smaller stress overshoot during reheating. In contrast, the larger tref Sg inhibited stress relaxation, which resulted in a stiffer response leading to a steeper stress drop-off during cooling and a larger stress overshoot during reheating.

5.3. Dependence on heating and cooling rate The shape memory response of the model exhibited a dependence on the cooling and heating rate because of structural relaxation. To demonstrate this feature, the unconstrained and constrained recovery simulations were performed for a faster heating rate qheat ¼ 5  C=min and a faster cooling rate qcool ¼ 5  C=min. The results are shown in Fig. 13 for the

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1 0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.82

qcool = 1°C/min qcool = 5°C/min

0

20

40 60 80 Temperature °C

Stretch Ratio (mm/mm)

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Stretch Ratio (mm/mm)

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1 0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.82

100

qheat = 1°C/min qheat = 5°C/min

0

20

40 60 80 Temperature °C

100

Fig. 13. Stretch ratio as a function of temperature during reheating in the unconstrained recovery simulations for (a) qheat ¼ 1  C=min and different cooling rates qcool and (b) qcool ¼ 1  C=min and different heating rates qheat .

2

3 qheat = 1°C/min

qcool = 5°C/min

1.5

1

0.5

Normalized Stress

Normalized Stress

qcool = 1°C/min

qheat = 5°C/min

2.5 2 1.5 1 0.5

0

0 0

20

40 60 Temperature °C

80

100

0

20

40 60 Temperature °C

80

100

Fig. 14. Stress as a function of temperature throughout the thermomechanical cycle of the constrained recovery simulations for (a) qheat ¼ 1  C=min and different cooling rates qcool and (b) qcool ¼ 1  C=min and different heating rates qheat .

unconstrained recovery simulations and in Fig. 14 for the constrained recovery simulation. For the unconstrained recovery simulations, the cooling rate had little effect on the recovery response. In contrast, a higher heating rate shifted the onset of strain recovery to a higher temperature, but had little effect on the slope of the stretch–temperature curve. Both the cooling and heating rates had a significant effect on the peak stress of the reheating curve in the constrained recovery simulations. The peak stress increased for a higher heating rate and a lower cooling rate. We are currently designing experiments to measure the unconstrained and constrained recovery response for different heating and cooling rates. The challenge of these experiments is to minimize the effects of heat transfer, which likely play a more significant role with higher heating and cooling rates, to validate the results shown in Figs. 13 and 14. 6. Conclusions A constitutive model was developed for the thermomechanical behavior of amorphous SMPs. The model incorporated the effects of structural relaxation, and stress relaxation in the form of viscoelasticity in the glass transition and rubbery regions and viscoplasticity in the glassy region, on the hypothesis that these mechanisms (in addition to heat transfer) underpin the time-dependence of the shape memory response. The important innovations of the model included:

 The incorporation of a time-dependent model of the glass transition and associated structural relaxation phenomena in



a finite-deformation thermoviscoelastic framework for the molecular ‘‘switching’’ mechanism of the shape memory phenomena. The structural relaxation model used the fictive temperature concept of Tool (1946), and the Scherer–Hodge nonlinear formulation of the Adam–Gibbs model for the temperature and structure dependence of the structural relaxation time. The development of an WLF form of the Scherer–Hodge equation for the structural relaxation time to facilitate parameter determination from thermomechanical experiments.

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 Development of a modified Eyring temperature-dependent flow rule for stress relaxation that can span the glass transition region. The model was implemented in a FE program and applied to simulate the constrained and unconstrained recovery response. A systematic procedure was developed to fit the model parameters to a limited set of data for the strain–temperature and stress–strain response (see Appendix B). The model produced good fits to the stress-free strain–temperature response and temperature and strain-rate dependent, isothermal, uniaxial stress-strain response. For the latter it was able to reproduce the temperature transition of the inelastic behavior from the viscoplastic response of the glassy material, with a distinct yield point and post-yield softening, to the viscoelastic behavior of the rubbery material. Moreover, it captured the dependence of the transition temperature on the strain rate. The model with the fitted parameters was applied to simulate the constrained stress recovery and unconstrained strain recovery experiments. A parameter study was conducted varying the characteristic stress and structural relaxation times to examine the influences of structural and stress relaxation on the shape memory performance. The main results of the simulation and parameter study were:

 For the unconstrained strain recovery simulations, the model achieved full strain recovery. The model was able to    

reproduce the starting temperature of strain recovery, but not the recovery time, which was significantly faster than observed in experiments. Structural relaxation influenced the unconstrained strain recovery behavior only at the start of strain recovery. It had little effect on the recovery time. Stress relaxation strongly influenced the strain recovery time of the unconstrained recovery. For the constrained recovery experiments, the model predicted many features of the hysteresis in the stress–temperature curve, including the peak stress during reheating. Both structural and stress relaxation played important though opposing roles in the constrained recovery stress response. A longer structural relaxation time and shorter stress relaxation time produced a more gradual stress decay and smaller stress overshoot.

Though the model can reproduce many important features of the stress and strain recovery response, some quantitative differences remain, particularly in the strain recovery time of the unconstrained recovery response. Currently, we are investigating the causes of the discrepancies by conducting additional thermomechanical experiments to determine the WLF constants from the time–temperature shift factor; to characterize the hysteresis in the strain–temperature cyclic response; and to characterize the high temperature creep and stress relaxation response. The latter will elucidate whether additional long-time stress relaxation processes of the rubbery material are needed to improve the prediction for the recovery time. Results of the parameter study suggest that long-time stress relaxation processes of the rubbery material strongly influence the strain recovery time. Furthermore, we plan to investigate the effects of pressure and chain straightening from deviatoric deformation on the structural and stress relaxation response, and of heat transfer on the shape recovery response.

Acknowledgments T.D. Nguyen and H.J. Qi gratefully acknowledge Laboratory Directed Research and Development program at Sandia National Laboratories (105951). H.J. Qi also gratefully acknowledges the support from NIH (EB 004481), US ARO (W31PQ-06-C-0406), NSF-Sandia initiative (Sandia National Laboratories, 618780), a NSF career award (CMMI-0645219), as well as discussions with Prof. Martin Dunn, and Drs. Richard Vaia, Jeffery Baur, and Mr. Jason Hermiller. Appendix A. Numerical implementation The coupled evolution equations (11), (26) and (33) for the internal state variables can be solved in a FE framework at the integration point. Given a deformation gradient Fnþ1 at the current step and the internal variables at the previous step, neq syn , CvMn and dn , the updated values of the internal variables can be computed by applying a backwards Euler scheme to Eqs. (11) and (33) and the predictor/corrector integration scheme of Reese and Govindjee (1998) to Eq. (26). This yields the following nonlinear system of equations: neq

dnþ1 þ eeMA

Dt neq neq ðdnþ1  DaðT nþ1  T 0 ÞÞ  dn ¼ 0, tref Rgnþ1

Dt e sneq Anþ1  eMAtr ¼ 0, 2Zref Sgnþ1   sy  Dth 1  nþ1 g_ vnþ1  syn ¼ 0, syss þ

nþ1

synþ1

(A.1)

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where eeMA

nþ1

e

e

¼ 12 ln bMA

nþ1

and eeMA

nþ1

e

e

¼ 12 ln bMA are the log strain of the eigenvalues of the current bnþ1 and the trial solution tr

1

bMtr ¼ FMnþ1 CvMn FTMnþ1 . The nonlinear system of equations can be solved using an iterative method such as Newton–Raphson. In a FE setting, the consistent tangent, defined as DSnþ1 ¼ C : DCnþ1 , can be computed by applying the method of Reese and Govindjee (1998). The key feature of the method is the application of the local update Eqs. (A.1) to relate the increment DCeM to DC. Appendix B. Parameter determination B.1. Thermal strain response: C 1 , C 2 and tref Rg FE simulations of the constant rate cooling tests were applied to fit the WLF parameters C 1 and C 2 , and the characteristic structural relaxation time tref R to the thermal strain–temperature curves of the constant cooling rate experiments. The FE simulations used a cube geometry of dimension L0 ¼ 1 mm to model the test sample. The FE cube was discretized by trilinear C 0 hexahedrons of size h ¼ 0:5L0 . Displacements uz ðz ¼ L0 =2Þ, uy ðy ¼ L0 =2Þ, ux ðx ¼ L0 =2Þ were applied to prevent rigid body motions. Otherwise, the cube was free to deform in response to a temperature change. The boundary conditions produced a homogeneous deformation and temperature fields. Thus, the FE simulations were equivalent to a material point simulation. The initial temperature of the cube was set to 60  C, which was well in the rubbery region. The cube was cooled at qcool ¼ 1  C/s (same as in the experiments) from the initial temperature T ¼ 60  C to T ¼ 0  C, then immediately reheated at qheat ¼ 1  C/s to the initial temperature. An initial guess of the parameters used the universal WLF constants C 1 ¼ 17:44 and C 2 ¼ 51:6  C (Ferry, 1980, Chapter 11, Section C), and tref Rg ¼ 100 s. It has been shown that the parameter C 1 does not vary significantly for amorphous polymers. Thus, the simulations only varied the parameters C 2 and  ref tref Rg to fit the experimental data. This resulted in C 2 ¼ 90 C and tRg ¼ 1100 s. To investigate the importance of the parameters on the thermal deformation response, the simulations considered different values of C 1 , C 2 and tref Rg . The resulting plots of the thermal strain, 1  L=L0 , as a function of temperature are shown in Figs. B1(a)–(c). Each parameter demonstrated a distinct effect on the thermal strain hysteresis and glass transition temperature. The characteristic structural relaxation time did not alter significantly the hysteresis. However, it had the largest effect on the T g , with higher tref Rg corresponding to higher T g. The WLF parameter C 1 had a slight but appreciable effect on the T g , and a dramatic effect on the hysteresis. Increasing values of C 1 decreased the hysteresis. Increasing C 2

Fig. B1. Thermal strain response for a stress-free, constant cooling rate thermocycle comparing different (a) WLF parameter C 1, (b) WLF parameter C 2, (c) characteristic structural relaxation time tref Rg and (d) cooling and heat rates qcool ¼ qheat .

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caused a more gradual transition from the rubbery to glassy response and delayed the return to equilibrium during reheating. In general, the cooling curve is relatively insensitive to C 1 and C 2 compared to the reheating curve. Thus, experimental data for both the cooling and reheating curves are needed to confidently fit C 1 and C 2 . Finally, the model was applied to examine the effects of cooling and heating rates on the thermal response. The results, plotted in Fig. B1(d), showed that increasing the cooling and heating rates had the same effect as increasing tref Rg in Fig. B1(c). It increased the T g , but did not affect the hysteresis.

B.2. Characteristic stress relaxation time: tref Sg ref neq The characteristic stress relaxation time, defined as tref , was fitted to the temperature dependence of the Sg ¼ ZSg =m storage modulus as measured by the DMA experiments described in Section 2.2 using FE simulations of cyclic uniaxial compression. For the simulations, displacements uz ðz ¼ L0 =2Þ, uy ðy ¼ L0 =2Þ, ux ðx ¼ L0 =2Þ were fixed to prevent rigid body motions. The initial temperature of the cube was set to a high temperature T high ¼ 90  C. The cube was cooled to T low ¼ 0  C and annealed at 0  C for an hour. To approximate the temperature and displacement history of the DMA experiments, the cube was heated from the annealing temperature to the test temperature T test at qheat ¼ 1  C=min. A sinusoidal displacement uy ðy ¼ L0 =2Þ ¼ 0:005L0 sinð2ptÞ was applied to the top surface of the cube while the radial direction was left free to deform in response. This corresponded to an applied strain of 0:5% at f ¼ 1 Hz. The simulation was conducted for test temperatures in the range of T test ¼ 10280  C. The resulting stress response achieved steady-state within one to three cycles depending on the temperature. The storage and loss moduli were computed by dividing the inphase and out-of-phase components of the stress response by the strain magnitude. The tan d was computed from the ratio of the loss modulus to the storage modulus. The cyclic simulations used the tref Rg , C 1 and C 2 fitted to the thermal strain response in Section B.1 and the equilibrium and nonequilibrium moduli which were fitted to the uniaxial compression tests in Section 4.2.1. The Q S , sy0 , syss and h0 governing the yielding and post yielding behavior did not affect the small strain response and were not considered. The simulations attempted to fit tref Sg to the start temperature and temperature range of the glass transition region as observed in the temperature dependence of the storage modulus. This produced a value of tref Sg ¼ 34:9 s.

Fig. B2. Temperature dependence of the storage modulus comparing the effects of different (a) WLF parameter C 1, (b) WLF parameter C 2, (c) characteristic ref viscosity Zref Sg and (d) characteristic structural relaxation time tRg .

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Fig. B3. Temperature dependence of the storage modulus for two heating rates qheat ¼ ð1; 5Þ  C=min. ref Fig. B2(a) plots the temperature dependence of the storage modulus for various tref Sg . Higher tSg shifted the start of the

glass transition region to higher temperatures and slightly increased the temperature range of the transition region. Figs. B2(b)–(d) show the effects of C 1 , C 2 and tref Sg on the mechanical response. A smaller C 1 resulted in a broader transition region and higher start temperature of the glass transition region. The parameter C 2 had a smaller but opposite effect. The characteristic structural relaxation time tref Rg slightly shifted the start of the glass transition region to a higher temperature but had little effect beyond the midpoint T445  C of the glass transition region. The model also was applied to examine the effects of heating rate on the mechanical response. Fig. B3 plots the temperature dependence of the storage modulus for two heating rates, qheat ¼ ð1; 5Þ  C=min. As for the strain–temperature response, increasing the heating rate had the same effect on the temperature dependence of the storage modulus as increasing the tref Rg . It changed the starting temperature and thus the temperature range of the glass transition region, but had little effect beyond the midpoint of the glass transition region. B.3. Viscoplastic parameters: Q S =sy0 , syss =sy0 , h0 The remaining parameters determined the yield and post-yield behavior of the glassy material. They were fitted to the uniaxial compression tests for different temperatures and strain rates. For the FE simulations, the displacement constraints uz ðz ¼ L0 =2Þ ¼ uy ðy ¼ L0 =2Þ ¼ ux ðx ¼ L0 =2Þ ¼ 0 were applied to prevent rigid body motions, and the initial temperature was set to T ¼ 90  C. The cube was cooled to and annealed at the test temperature T test for an hour. Then, the top surface of the cube was displaced at a constant strain rate to an engineering strain of 50%. The simulations applied the two strain rates, e_ ¼ 0:01 and 0:1 s1 , and the temperature range, T test ¼ 10250  C, of the experiments. Given Zref Sg , the ratio Q S =sy0 determines the temperature and strain-rate dependence of the yield strength. The parameters Q S and sy0 do not affect the model independently and thus were fitted as a ratio. An initial guess was calculated from the yield point of the stress–strain curves for the two strain rates measured at T ¼ 10  C. At the yield point, the equivalent viscous stretch rate and yield strength were estimated as g_ v  g_ and sy  sy0 . Since T ¼ 10  C was below the glass transition p temperature, the limiting value T f  T g was used for the fictive temperature. In addition, it was assumed that ffiffiffi Q =Tjsneq j= 2sy0 41 at the yield point such that the sinh function in Eq. (29) can be approximated by an exponential function. With these assumptions, Eq. (29) was approximated at the yield point as !!   T ref sy T C1 QS s g 1 g_ ¼ pffiffiffi ref exp , (B.1) exp log e T T sy 2ZS Q S g

The flow pffiffiffi stress s and pffiffiffi equivalent strain rate g_ were computed from the stress and strain rate in the loading direction as s ¼ s= 3 and g_ ¼ e_ 3. Then from Eq. (B.1), an initial guess for Q S =sy0 was computed from the yield points s1 ¼ 50:21 and s2 ¼ 28:28 for the two strain rates e_1 ¼ 0:01=s and e_2 ¼ 0:1=s for T ¼ 10  C as QS ln g_ 1  ln g_ 1 T ;  97:9  K=MPa. sy0 s1  s2

(B.2)

The parameter syss =sy0 is the ratio of the saturation yield strength to the initial yield strength. An inspection of the stress–strain curve for e_ ¼ 0:01 s1 and T ¼ 0  C yielded an initial guess of syss =sy0  0:5 . The parameter h0 characterizes the

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Fig. B4. Isothermal uniaxial stress response in unconfined compression at different temperatures spanning the glass transition region for different parameters (a) tref Sg , (b) Q S =sy0 , (c) syss =sy0 and (d) h0 .

rate of strain-softening following yield. An initial guess of h0  100 MPa was used. The FE simulations applied these initial guesses along with the previously fitted parameters to iterate for Q S =sy0 , syss =sy0 and h0 . This yielded Q S =sy0 ¼ 101 K=MPa, syss =sy0 ¼ 0:43 and h0 ¼ 250 MPa. Fig. B4 plots the stress–strain response at T ¼ 10 and T ¼ 50  C for different values of Q S =sy0 , syss =sy0 , h0 and Zref Sg . An increase in Zref produced higher flow stresses in both the glassy and rubbery responses. However, changes in the remaining Sg parameters affected only the glassy response. References Adam, G., Gibbs, J.H., 1965. On the temperature dependence of cooperative relaxation properties in the glass-forming liquids. J. Comput. Phys. 43, 139. Aklonis, J.J., 1981. Kinetic treatments of glass transition phenomena and viscoelastic properties of glasses. Polym. Eng. Sci. 23, 896–902. Anand, L., Ames, N.M., 2006. On modeling the micro-indentation response of amorphous polymer. Int. J. Plasticity 22, 1123–1170. Argon, A.S., 1973. A theory for the low-temperature plastic deformation of glassy polymers. Philos. Mag. 28, 839. Arruda, E.M., Boyce, M.C., 1993a. Evolution of plastic anisotropy in amorphous polymers during finite straining. Int. J. Plasticity 9, 697–720. Arruda, E.M., Boyce, M.C., 1993b. A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J. Mech. Phys. Solids 41, 389–412. Boyce, M.C., Parks, D.M., Argon, A.S., 1989a. Plastic flow in oriented glassy polymers. Int. J. Plasticity 5, 593–615. Boyce, M.C., Weber, G.G., Parks, D.M., 1989b. On the kinematics of finite strain plasticity. J. Mech. Phys. Solids 37, 647–665. Caruthers, J.M., Adolf, D.B., Chambers, R.S., Shrikhande, P., 2004. A thermodynamically consistent, nonlinear viscoelastic approach for modeling glassy polymer. Polymer 45, 4577–4597. Diani, J., Liu, Y., Gall, K., 2006. Finite strain 3d thermoviscoelastic constitutive model for shape memory polymers. Polym. Eng. Sci. 46, 486–492. Eyring, H., 1936. Viscosity, plasticity, and diffusion as examples of absolute reaction rates. J. Comput. Phys. 4, 283–291. Ferry, J.D., 1980. Viscoelastic Properties of Polymers. Wiley, New York, NY. Flory, P.J., 1961. Thermodynamic relations for highly elastic materials. Trans. Faraday Soc. 57, 829–838. Gibbs, J.H., DiMarzio, E.A., 1958. Nature of the glass transition and the glassy state. J. Comput. Phys. 28, 373. Govindjee, S., Reese, S., 1997. A presentation and comparison of two large deformation viscoelasticity models. Trans. ASME J. Eng. Mater. Technol. 119, 251–255. Haward, R.N., Thackray, G., 1968. The use of mathematical models to describe isothermal stress strain curves in glassy thermoplastics. Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 302, 453. Hodge, I., 1987. Effects of annealing and prior history on enthalpy relaxation in glassy polymers: Adam–Gibbs formulation of nonlinearity. Macromolecules 20, 2897. Hodge, I., 1997. Adam–Gibbs formulation of enthalpy relaxation near the glass transition. J. Res. Natl. Inst. Stand. Technol. 102, 195–205.

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