modeling the relaxation mechanisms of amorphous shape memory polymers

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Modeling the Relaxation Mechanisms of Amorphous Shape Memory Polymers

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By Thao. D. Nguyen , * Christopher M. Yakacki , Parth D. Brahmbhatt , and Matthew L. Chambers

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In this progress report, we review two common approaches to constitutive modeling of thermally activated shape memory polymers, then focus on a recent thermoviscoelastic model that incorporates the time-dependent effects of structural and stress relaxation mechanisms of amorphous networks. An extension of the model is presented that incorporates the effects of mul-tiple discrete structural and stress relaxation processes to more accurately describe the time-dependent behavior. In addition, a procedure is developed to determine the model parameters from standard thermomechanical experi-ments. The thermoviscoelastic model was applied to simulate the uncon-strained recovery response of a family of (meth)acrylate-based networks with different weight fractions of the crosslinking agent. Results showed signifi -cant improvement in predicting the temperature-dependent strain recovery response.

1. Introduction

Shape memory polymers (SMPs) are unique among active materials in that they can be programmed thermomechanically to undergo large shape changes from the programmed shape to the manufactured permanent shape in response to a spe-cifi c temperature stimulus. Most polymers exhibit some degree of shape memory behavior, but SMPs are distinguished from conventional polymers by their ability to recover nearly 100% of the programmed deformation under useful temperature and mechanical loading conditions. Recently, Liu et al. [ 1 ] intro-duced a classifi cation system that divides SMPs into four basic

© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimAdv. Mater. 2010, 22, 3411–3423

[∗] Prof. T. D. Nguyen Department of Mechanical EngineeringThe Johns Hopkins UniversityBaltimore, MD 21218 (USA) E-mail: [email protected] Dr. C. M. Yakacki , P. D. Brahmbhatt , M. L. Chambers Research and DevelopmentMedShape Solutions Inc., Atlanta, GA 30318 (USA) Dr. C. M. Yakacki School of Materials Science and EngineeringThe Georgia Institute of TechnologyAtlanta, GA 30332 (USA) P. D. Brahmbhatt , M. L. Chambers Woodruff School of Mechanical EngineeringThe Georgia Institute of TechnologyAtlanta, GA 30332 (USA)

DOI: 10.1002/adma.200904119

types. The simplest type, class I, encom-passes covalently crosslinked amorphous networks, which can be homopolymers or random copolymers consisting of func-tional monomers and crosslinking agents. The permanent shape is maintained by the equilibrium confi guration of the net-work and shape storage and recovery are driven by the glass transition and chain dynamics. Class II materials are covalently crosslinked semicrystalline networks, which can exhibit two distinct shape changes. Class III and IV materials con-sist of physically crosslinked amorphous and semicrystalline polymers. The perma-nent shape of these materials is set by the network of physical entanglements of the hard domains, which decay with time and temperature. Consequently, the perma-

nent shape of class III and IV polymers can be reprocessed by heating the material above the thermal transition of the hard domains.

The shape memory performance of thermally activated poly-mers depends on the complex interactions of the polymer struc-ture and morphology (e.g., the crosslink density, chain length), programming conditions, and deployment conditions. [ 2 ] Thus, considerable opportunities exist to tailor the polymer structure and morphology and the thermomechanical programming conditions to achieve the desired shape memory performance. SMPs are being investigated for a variety of biomedical devices and smart engineering structures. [ 3 – 5 ] Potential applica-tions of thermally activated SMPs require controlling the shape change response and possibly coordinating multiple shape change events. Because of the inherent complexity of the shape memory response, a conventional build-and-test approach to designing SMP materials and devices can be pro-hibitively ineffi cient. For these complex materials, a numerical simulation tool can provide an inexpensive and effi cient test plat-form to explore different combinations of material and design options. A simulation-based design approach requires the devel-opment of constitutive models for the large deformation, time-dependent and temperature-dependent behavior of SMPs that can be implemented into a fi nite element framework.

Though research on shape memory polymers has grown tremendously in the past fi ve years, most efforts have been dedicated toward developing new materials and devices with increasingly sophisticated properties and functionalities and experimental methods to characterize the shape memory

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Thao D. Nguyen received her S.B. in mechanical engi-neering (1998) from MIT. She obtained an M.S. (2001) and Ph.D. (2004) in mechanical engineering from Stanford University, where she studied the dynamic fracture behavior of glassy polymers with Dr. Huajian Gao and Dr. Sanjay Govindjee. Immediately after, Dr. Nguyen obtained

a position as Senior Member of the Technical Staff at Sandia National Laboratories, Livermore, CA, where she worked on modeling and experimental characteriza-tion of the viscoelastic behavior of soft tissues, shape memory polymers, and polymer composites. Dr. Nguyen joined the Department of Mechanical Engineering at The Johns Hopkins University in Fall of 2007 as an Assistant Professor.

performance. By comparison, efforts to model and predict shape memory performance have lagged. The earliest modeling efforts were one-dimensional, small-strain formulations that had limited applicability and predictive capability. However, recent years have seen a proliferation of three-dimensional, fi nite-deformation models that can be implemented numeri-cally for fi nite element computation. [ 3 ] Most SMP models have been developed for T g activated class I materials and these tend to follow either a phase transition or a thermoviscoelastic approach.

1.1. Phase Transition Modeling Approaches

The phase transition approach considers an SMP as a continuum mixture of a compliant rubbery phase, a stiff glassy phase, and various intermediate phases (e.g., refer-ences [6–8]). The volume fraction of the constituent phases depends on temperature, such that only the rubbery phase exists at temperatures well above T g and the glassy phase at temperatures well below T g . Shape memory behavior is mod-eled as a temperature induced transition between the phases that transfers the deformation of the rubbery phase into the immobile glassy phase during cooling then releases it back into the mobile rubbery phase during reheating. Qi et al. [ 7 ] considered three distinct phases, the rubbery phase, the initial glassy phase present in the undeformed confi guration at the beginning of the analysis, and the frozen glassy phase newly formed during cooling. The phases were assumed to deform in parallel, and the rule of mixtures was used to calculate the stress response of the SMP from those of the individual phases. A phenomenological relation was developed for the rate-independent temperature evolution of the volume fraction of each phase. The relation was based on the Vogel-Tammann-Fulcher (VTF) equation for the temperature dependence of the relaxation time of glassy polymer networks, and the param-eters were fi t to the temperature-dependence of the Young’s modulus measured under isothermal uniaxial compression. The model was able to reproduce the experimentally meas-ured stretch-temperature curves of an unconstrained recovery experiment, though with a slightly higher onset temperature and steeper slope. It was not able to capture the hysteresis in the stress-temperature response observed in constrained recovery experiments.

Chen and Lagoudas [ 8 ] modeled the interaction of an active and frozen phase during the glass transition. At any mate-rial point, the phases experience the same macroscopic stress state but each material point may undergo the glass transition at different temperatures. It was assumed that the deforma-tion remained continuous during cooling and the deformation of the frozen phase was instantaneously released to the active phase during reheating. From these assumptions, Chen and Lagoudas [ 8 ] developed a relation for the time-dependent aver-aged deformation gradient as a function of the stress state. The model required three constitutive functions to be determined from experiments, for the temperature dependence of the volume fraction of the frozen phase and for the stress and tem-perature dependence of the deformation gradient of the frozen and active phases. The authors presented a linearized version

© 2010 WILEY-VCH Verlag G

of the generalized model and applied it to successfully predict the uniaxial deformation experiments of Liu et al. [ 6 ] The model did not consider the time-dependent effects of the glass transi-tion, thus the stress response was independent of the cooling rate and strain rate, both of which may have signifi cant effects on the shape memory performance.

The phase transition approach has a distinct advantage in that the concepts can be applied to a wide variety of SMP mate-rials with different transition mechanisms. Barot et al. [ 9 ] applied the concepts of phase transition to model the shape memory behavior of crystallizable (class II and IV) SMPs. Long et al. [ 10 ] recently developed a model for photo-activated shape memory elastomers that treat the material as a mixture of two evolving networks, the original network present at the beginning of the analysis and the photo-stimulated reformed network. For class I materials, the phase transition approach does not avail of the rich literature for the thermomechanical behavior of amor-phous networks, which has provided many successful models for the glass transition and associated time-dependent mecha-nisms involved in the recovery behavior of class I materials.

1.2. Thermoviscoelastic Modeling Approaches

Early thermoviscoelastic treatments of amorphous SMPs applied one-dimensional, small strain, rheological models with temperature-dependent viscosity and/or modulus parameters. Tobushi et al., [ 11 , 12 ] Morshedian et al., [ 13 ] and Khonakdar et al. [ 14 ] used an Arrhenius equation for glassy materials to describe the increase in the viscosity with decreasing temperature, while Buckley et al. [ 15 ] used the Williams-Landel-Ferry (WLF) equation for rubbery materials to describe the temperature dependence of the retardation time. The underlying mechanism of shape memory behavior is the temperature dependence of the chain mobility, represented either by the viscosity or relaxation time of the dashpot element. The viscosity is low at high temperatures,

mbH & Co. KGaA, Weinheim Adv. Mater. 2010, 22, 3411–3423


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T »T g , such that the polymer network can relax quickly to equi-librium. It increases during cooling, which causes the network to exhibit noticeable viscoelasticity. Considerable time is needed for the deformed network to relax toward equilibrium. As a result, deformation imposed at high temperatures becomes locked in the dashpot element of the rheological model during cooling producing internal stresses. Reheating softens the vis-cosity and allows the internal stresses to drive the material to recover its equilibrium confi guration.

Three-dimensional, fi nite-deformation, thermoviscoelastic models have been developed by Diani et al. [ 16 ] and the author and coworkers [ 17 ] using an internal state variable framework. In addition to viscoelasticity, Nguyen et al. [ 17 ] incorporated the time-dependent effects of structural relaxation and viscoplastic fl ow below the glass transition temperature. The model also integrated mechanisms of thermal expansion and entropic elas-ticity of the rubbery network. Structural relaxation describes the time-dependent process for the polymer chains to rearrange to a new equilibrium confi guration in response to a temperature change. The Aruda and Boyce model [ 18 ] with Langevin chain statistics was used to describe the high temperature stress response of the rubbery material, while a Neo-Hookean model described the stiff elastic response of the glassy material. For structural relaxation, the nonlinear Adams-Gibbs model was adapted to describe the divergence in the temperature depend-ence of the viscosity from the WLF equation above T g to the Arrhenius equation below T g . A modifi ed Eyring fl ow rule [ 19 , 20 ] was developed to describe the high temperature viscoelastic and low temperature viscoplastic behavior. The Eyring model was modifi ed by replacing the Arrhenius temperature dependence of the viscosity with the temperature and structure dependence of the nonlinear Adam-Gibbs model. This allowed the model to represent the temperature and strain-rate dependent transi-tion of the inelastic behavior from viscoplastic for T « T g with a distinct yield point and post-yield fl ow to viscoelastic for T » T g with no distinct yield point and perfect recovery.

© 2010 WILEY-VCH Verlag GmAdv. Mater. 2010, 22, 3411–3423

Figure 1 . Comparing experimental and simulation results from the thermovisof constrained recovery, and (b) the stretch-temperature response of uncons

0 20 40 60 80 1000

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1.5

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By incorporating structural and stress relaxation, the model was able to predict many important features of the recovery response. Figure 1a compares the experimental data and model prediction for the stress-temperature curve of a constrained recovery experiment for a tBA-co-PEGDMA material, a class I copolymer of tert -butyl acrylate and poly(ethylene glycol) dimethacrylate crosslinkers. [ 17 ] The model parameters were determined from a series of standard thermomechanical exper-iments including isothermal compression at different tempera-tures spanning the glass transition and dynamic mechanical analysis measurements of the temperature dependence of the storage modulus. The model was able to predict key features of the stress hysteresis of the constrained recovery response, including the peak stress and associated temperature. Subse-quent parameter studies showed that both strongly depended on the cooling rate and heating rate as a result of viscoelastic and structural relaxation. Figure 1b compares the experimental data and model prediction for the strain-temperature curve of an unconstrained recovery experiment. The onset tempera-ture of strain recovery predicted by the model agreed well with experiments. However, the model signifi cantly underestimated the recovery time, as evidenced by the steeper slope of the stretch-temperature curve. The time-dependent behavior was described in the model by a single characteristic relaxation time, whereas amorphous polymers typically display a broad spectrum of relaxation times. We hypothesized that this sim-plifi ed treatment of the time-dependent behavior of the SMP caused the discrepancy in the recovery time.

To improve the predictive capability of the model, we devel-oped an extension that incorporated a broad relaxation spec-trum through multiple discrete relaxation processes, each with a characteristic temperature- and structure- dependent relaxa-tion time. The new model was applied to predict the large strain unconstrained recovery response of tBA-co-PEGDMA networks with different weight percent PEGDMA crosslinking agents. A procedure developed by Haupt et al. [ 21 ] was extended to

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coelastic model of Nguyen et al. [ 17 ] for (a) the stress-temperature response trained recovery.

0 20 40 60 80 1000.82

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determine the model parameters from standard thermome-chanical experiments. The method involved fi rst linearizing the discrete-spectrum model for the case of structural equilibrium (i.e., high temperature) then constructing a corresponding linear fractional damping viscoelastic model. The latter had a continuous relaxation spectrum and signifi cantly fewer param-eters. The parameters of the fractional damping model were fi t to data from dynamic mechanical analysis experiments, then used to defi ne the parameters of the large strain discrete-spectrum model. The results showed signifi cantly improved agreement with experimental data for the recovery time. The following sections present the development of the improved thermoviscoelastic model. The materials and experimental methods are described briefl y in the next section, followed by the development of the constitutive model, and a discussion of the results.

2. Methods

2.1. Materials and Sample Preparations

The materials were synthesized by purchasing tert-Butyl acr-ylate (tBA), poly(ethylene glycol) n dimethacrylate (PEGDMA) with typical molecular weight of Mn = 550, di(ethylene glycol) dimethacrylate (DEGDMA), and photoinitiator 2,2-dimethoxy-2-phenylacetophenone (DMPA) from Aldrich and used in their as-received conditions. A crosslinker solution (XLS) was prepared by mixing 30 wt% DEGDMA with 70 wt% PEGDMA, and tBA-co-XLS networks with 2 wt%, 10 wt%, 20 wt%, and 40 wt% crosslinks were the synthesized by free-radical polymerization using 0.1 wt% of DMPA photoinitiator. The polymer solutions of various wt% crosslinks were injected between two glass slides separated by a spacer and polymerized in a UV crosslinker oven (UVP, CL1000) for 15 minutes. Following UV curing, all polymer samples were thermally cured in an oven at 90 ° C for one hour to ensure com-plete polymerization. The samples were cut into 1 mm × 5 mm × 25 mm specimens using a laser cutter for mechanical testing.

2.2. Experimental Methods

All thermomechanical experiments were performed in a TA Q800 Dynamic Mechanical Analyzer (DMA) under uniaxial tension using the 1 mm × 5 mm × 25 mm specimens. The gauge length of the samples between the tensile grips was set approximately to 10 mm for all tests.

2.2.1. Dynamic Thermal Scan

A dynamic thermal scan at 1Hz was used to characterize the thermomechanical properties of the networks of different % crosslinks. Samples were thermally equilibrated at 0 ° C and heated to 120 ° C at a rate of 3 ° C min − 1 while subjected to a 0.2% dynamic strain at 1 Hz. Samples were run in triplicate. For the purpose of the unconstrained recovery experiments described below, the midpoint glass transition temperature T g was defi ned at the peak of the tan delta curve. The onset of glass transition T onset was calculated using an intersecting line


method with the starting point of the left line being 50 ° C less than T g and the starting point of the right line being T g . The rubbery modulus was measured at the lowest point of the rub-bery plateau, and the glassy modulus was measured at the onset of the glass transition.

2.2.2. Coeffi cients of Thermal Expansion

Coeffi cient of thermal expansion (CTE) tests were performed in the DMA to measure the thermal strain of each polymer net-work as a function of temperature. With the machine under a controlled zero-force mode, samples were equilibrated at 120 ° C before being cooled to 0 ° C at a rate of 1 ° C min − 1 . The samples were then reheated back to 120 ° C at the same rate. The rubbery CTE ( α r ) was calculated from the slope of the linear portion of the displacement-temperature curve measured during cooling, from 80–100 ° C, while the glassy CTE was calculated from the slope of the curve between 0–20 ° C. The reference glass transi-tion temperature T g ref was defi ned from the intersection of two lines fi tted to the rubbery and glassy regions.

2.2.3. Time-Temperature Superposition

A time-temperature superposition (TTS) test was performed to construct the master curve of the storage modulus for the dif-ferent networks. The DMA was set to run in multi-frequency strain mode, applying a 0.2% dynamic strain at various frequen-cies across a temperature range. The samples were dynamically strained at 0.32 Hz, 1.00 Hz, 3.20 Hz, 10.00 Hz, and 31.60 Hz to cover two decades of frequency. Samples were equilibrated at each testing temperature, ranging from 0 to 120 ° C in 5 ° C increments, for fi ve minutes before running a frequency sweep at each testing temperature. The frequency dependence of the storage modulus and tan delta curves were recorded to com-pare the shift in the glass transition temperature as a function of frequency. The master curve of the storage modulus ( G ′ ) was constructed by shifting the storage modulus curves at dif-ferent temperatures using the procedure described by Ferry. [ 22 ] For all the networks, a reference temperature of T 0 = 80 ° C was chosen, which corresponds to the beginning of the rubbery pla-teau. The storage modulus was plotted for each temperature as a function of the logarithmic frequency, and the horizontal distance was calculated between curves of adjacent temperature and recorded as the incremental time-temperature shift factor, Δ log a T . The incremental shift factors were summed progres-sively at each temperature from T 0 to obtain the temperature dependence of the shift factor log a T . For thermorheolgically simple materials, the temperature dependence of log a T for temperatures above the glass transition temperature follows WLF empirical relation,

log aT = −C0

1 (T − T0)

C02 + T − T0

,

(1)

where C 0 1 and C 0 2 are the WLF constants for the reference tem-perature T 0 . To determine C 0 1 and C 0 2 , ( T – T 0 )/log a T was plotted against T – T 0 , and a linear fi t was performed to determine the slope m and intercept a of the best fi tting line through the points. The WLF constants at T were calculated from the results as



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Figure 2 . A linear rheological model, where E eq and E neq I are the equilib-rium and nonequilibrium Young’s moduli and ε represent the strain. The rheological model has a thermal (T) element with N structural relaxa-tion mechanisms and a mechanical element (M) with N stress relaxa-tion mechanisms. The present nonlinear thermoviscoelastic model is an extension of the rheological model to fi nite deformation.

αg

Eeq

E1

neq

EN

neq

τSN

(T, δ)

τS1

(T, δ)

τR1

(T, δ) τRN

(T, δ)

Δα1

ΔαN

δ εeM

εvM

εε

Tε

M

εoT

C0

1 = − 1

m, C0

2 = a

m.

(2)

Finally, the WLF constants for T ref g as the reference temperature were calculated for the thermoviscoelastic model in Section 2.3 using the transformation,

Cg

2 = C02 + T ref

g − T0, Cg1 = C0

1 C02

Cg2

.

(3)

2.2.4. Unconstrained Strain Recovery Experiments

The unconstrained strain recovery experiments involved a two-step process that included programmed deformation and recovery. The samples were equilibrated fi rst at T g + 5 ° C for 4 min then deformed at a constant engineering stress rate of 0.0220 MPa min − 1 , 0.0588 MPa min − 1 , and 0.1020 MPa min − 1 to an engineering strain of 20% for the 2 wt%, 10 wt%, and 20 wt% specimens, and a rate of 0.3630 MPa min − 1 to a strain of 10% for the more brittle 40 wt% specimen. The loading rates cor-responded to an average strain rate of 0.06% s − 1 for all of the specimens. After deformation, the samples were cooled to 0 ° C at a rate of 10 ° C min − 1 then unloaded to a nominally zero load to allow for unconstrained recovery. Recovery was monitored as the temperature was ramped to 100 ° C at a rate of 3 ° C min − 1 .

2.3. Constitutive Model Formulation

A thermoviscoelastic model is developed below that includes the effects of thermal expansion, temperature-dependent viscoelas-ticity, and structural relaxation. To simplify matters, the model does not consider the effects of yielding and viscoplastic fl ow of the glassy material as did the early model of Nguyen et al. [ 17 ] Preliminary studies showed that the viscoplastic mechanism of the glassy material did not signifi cantly affect the unconstrained recovery response of SMP specimens programmed by high tem-perature deformation. The model formulation is based on the nonlinear viscoelastic theory of Reese and Govindjee, [ 23 ] and is an extension of the rheological model shown in Figure 2 to fi nite deformation. The rheological model consists of a thermal element in series with a mechanical element. The thermal ele-ment is described by a generalized Kelvin element with a spring, representing the instantaneous (low temperature, short time) thermal strain response in series with N Voigt elements for the time-dependent structural relaxation of the thermal strain to equilibrium. The mechanical part is described by a spring rep-resenting the equilibrium (high temperature, long-time) stress response in parallel with N Maxwell elements for the time-dependent stress relaxation toward equilibrium. For simplicity, we have assumed that both structural and stress relaxation can be described the same discrete numbers of relaxation processes.

2.4. Kinematics

The deformation gradient tensor is defi ned as F = ∂ x∂ X , where

x ( X ) is a continuous one-to-one mapping of the reference posi-tion X to the deformed position. In fi nite deformation, the


serial arrangement of the thermal and mechanical elements of the rheological model (Figure 2 ) corresponds to a multiplicative decomposition of the deformation gradient into thermal and mechanical parts,

F = FMFT. (4)

It is assumed that the thermal deformation is isotropic such that F = �

13T 1 , where Θ T is the thermal volumetric deforma-

tion. Moreover, the thermal deformation is small for the consid-ered temperature range, which allows the Θ T to be decomposed additively as,

�T (t, T ) = 1 + αg (T − Tinit)︸︷︷︸

δ 0

+N

∑

k

δk,

(5)

where δ 0 is the instantaneous thermal strain response charac-terized by the glassy CTE, and α g and δ k ( t , T ) are N time- and temperature-dependent internal variables describing the state of structural relaxation. The temperature T init denotes the tem-perature at the beginning of the analysis.

To model the potentially large deformation resulting from stress relaxation over a broad relaxation spectrum, the mechanical deformation gradient is further decomposed multi-plicatively into elastic and viscous components for each relaxa-tion process,

FM = FeMk

FvTk

, for k = 1 . . . N. (6)

Because the polymer can exhibit different volumetric and isochoric behavior, we also decompose F M and F e M k into volu-metric and isochoric parts,

FM = �− 1

3M FM, F

eMk

= �e− 13

MkFe

Mk,

(7)

where Θ M = det[ F M and �eMk

= det[

FeMk

]

signify the mechan-ical and elastic components of the volumetric deformation. The product Θ = det[ F ] = Θ M Θ T gives the total volumetric deformation.


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The left and right Cauchy-Green deformation tensors are defi ned for the mechanical deformation and its components as,

bM = FMFTM, be

Mk= Fe

MkFeT

Mk, bv

Mk= Fv

MkFvT

Mk

CM = FTMFM, Ce

Mk= FeT

MkFe

Mk, Cv

Mk= FvT

MkFv

M (8)

Combining Equation ( 6 ) into Equation ( 7 ) gives the iso-choric deformation tensors, CM = �

− 23

M CM , CeMk

= �e− 23

MkCe

Mk ,

bM = �− 2

3M bM , and b

e

Mk= �e− 2

3Mk

beMk

. Finally, the rate of the vis-cous deformation tensor of each stress relaxation process, C. v

Mk m can be expressed as an objective rate in the deformed

confi guration by the transformation,

£vb

eMk

= FM

(

Cv−1

Mk

)

FTM,

.

(9)

where the operator £v (.) is the Lie time derivative.

2.5. Constitutive Relations

2.5.1. Thermal Strains and Structural Relaxation

The internal variables δ k describe the nonequilibrium state of the polymer structure. We assume that their evolution toward equilibrium can be described by the following nonlinear rate equation, inspired by the KAHR model for structural relaxation, [ 24 ]

∂δk

∂t= − 1

τ Rk(T, δ)

(δk − �αk (T − T0)) , δk (0, T ) = 0,

(10)

where τ R k are the temperature-dependent and structure-dependent characteristic structural relaxation times and Δ α k are parameters characterizing structural relaxation spectrum. The parameters Δ α k sum to α r – α g , while

∑Nk δk = δ gives the total

departure from the instantaneous response. The latter is related to the fi ctive temperature T f introduced by [ 25 ] as,

Tf = 1

αr − αgδ + T0.

(11)

The fi ctive temperature can be understood physically as the temperature at which a quenched structure at T would be in equilibrium. The Hodge-Scherer nonlinear extension of the Adam-Gibbs model [ 17 , 26– 28 ] is adapted to describe the tempera-ture and structure dependence of the characteristic structural relaxation time in terms of the WLF constants,

τ Rk(T, Tf )

= τgRk

exp

⎡

⎣− Cg1

log e

⎛

⎝

Cg2 (T − Tf ) + T

(

Tf − T refg

)

T(

Cg2 + Tf − T ref

g

)

⎞

⎠

⎤

⎦ ,

(12)

where T g ref is the glass transition temperature measured from CTE experiments (Section 2.2.2) for a reference cooling rate, and C g 1 and C g 2 are the WLF parameters for T ref g measured by the TTS experiments (Section 2.2.3). The characteristic relaxation


time of each nonequilibrium process is dependent on the internal state variable of all N processes. Thus, Equation ( 10 ) is a coupled system of nonlinear evolution equations for δ k .

2.5.2. Stress Relations

To model the viscoelastic stress response, it is assumed that the Helmholtz free energy density can be expressed as the sum of a time-independent volumetric component, an equilib-rium deviatoric component, and N nonequilibrium deviatoric components,

�(

CM, CeMk

,�M, T)

=N

∑

k

Wneqk

(

CeMk

)

+W eq(

CM

) + U (�M) . (13)

The equilibrium component W eq describes the rubbery stress response of the SMP at high temperatures, and is represented by the Arruda and Boyce [ 18 ] network model with Langevin chain statistics,

Weq(

IM1

) = μNλ2L

[

λeff

λLx + ln

( x

sinh x

)

]

− w0,

x = L−1

(

λeff

λL

)

, λeff =√

1

3IM1

(14)

Uneq (�M) = κ (�M − ln �M − 1) , (15)

where κ is the characteristic bulk modulus. Lastly, the stiff glassy response at low temperatures is described by a Neo-Hookean strain energy density potential for W k neq ,

Wneq

k

(

IeMk 1

)

= μneqk

2

(

IeMk 1

− 3)

.

(16)

where μ neq k are the characteristic shear moduli, and I

eMk 1

= trace[

CeMk

]

are the fi rst invariant of the isochoric elastic deformation tensors. The second Piola-Kirchhoff stress response is defi ned from the free energy density as, S = 2 ∂ �

∂ C ,

and the Cauchy stress, which signifi es the true stress, is calcu-lated from S using the Piola transformation, r = 1

�FSFT . The

Cauchy stress evaluated for the free energy density defi ned above can be written in terms of deviatoric and hydrostatic pressure components as σ = s + pp 1 , where,

s = 1

�μeq

(

bM − 1

3IM1 1

)

︸︷︷︸

seq

+N

∑

k

1

�μ

neqk

(

be

Mk− 1

3I

eMk 1

1)

︸︷︷︸

sneqk

,

μeq = μNλL

λeff

L−1

(

λeff

λL

)

p = 1

�κ (�M − 1) .

(17)



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The elastic deformation components, b e M k are internal vari-ables that evolve to equilibrium according to the following rateequation, [ 23 ]

−1

2£ vb

eMk

be−1

Mk= 1

2ηSk

sneqk ,

(18)

where η S k ( T , δ ) are the temperature and structure-dependentshear viscosities and £ v is the Lie time derivative defi ned inEquation ( 9 ). The characteristic stress relaxation times aredefi ned as τ Sk = ηSk

/μneqk . We assume that τ S k exhibit the same

temperature and structure dependence as the structural relaxa-tion time τ R k in Equation ( 12 ),

τ Sk(T, Tf )

= τgSk

exp

⎡

⎣− Cg1

log e

⎛

⎝

Cg2 (T − Tf ) + T

(

Tf − T refg

)

T(

Cg2 + Tf − T ref

g

)

⎞

⎠

⎤

⎦ ,

(19)

where τ gSk

are the characteristic relaxation times at T refg . Unlike

in the single process model of Nguyen et al. [ 17 ] the character-istic stress relaxation time does not include a stress activationterm. Thus, the stress response for T « T g does not experienceyielding, though unloading at low temperatures effectivelywill produce permanent deformation since τ S k grow to infi nityduring cooling.

2.5.3. Determining the Parameters of the Discrete Spectrum for Viscoelastic Relaxation

The time-dependent behavior of the fi nite-deformation thermov-iscoelastic model is described by two discrete relaxation spectra.To determine the parameters μ neq k and τ S k of the viscoelasticrelaxation spectrum from experiments, we fi rst linearize thefi nite-deformation model for the case of structural equilibrium(i.e., high temperatures) to obtain a small-strain model, thenconstruct a corresponding small-strain viscoelastic model witha continuous relaxation spectrum. This approach was proposedby Haupt et al. [ 21 ] to model the dynamic behavior of polymersunder fi nite deformation. The procedure is summarized briefl yhere, though the reader is directed toward Haupt et al. [ 21 ] for adetailed presentation.

Linearizing to small strain causes the large deforma-tion tensors to reduce to their small strain counterparts, �T → trace [εT ] , bM → 1 + 2εM, and be

Mk→ 1 + 2εe

Mk . The

thermal and mechanical strain tensors sum to give the totalstrain ε = εM + εT . Moreover, εM = εe

Mk+ εv

Mk , for each

k th nonequilibrium process. The isochoric deformation ten-sors similarly reduce to bM → 1 + 2eM and b

e

Mk→ 1 + 2ee

Mk ,

where eM = εM − trace [εM] 1 and eeMk

= εeMk

− trace[

εeMk

]

1 .These reduce the deviatoric stress relations in Equation ( 17 ) to,

s = 2μeqeM +

N∑

k

2μneqk ee

Mk

(20)

Similarly, the nonlinear evolution equation in Equation ( 18 )becomes,

e. v

Mk= 1

τ Sk

(

eM − evMk

)

, evMk

(0, T ) = 0,

(21)


where evMk

= eM − eeMk

is the deviatoric viscous strain. The storage and loss moduli of the linearized discrete model can be calculated for a harmonic deformation as,

G′disc (ω) = μeq +

N∑

k

μneqk ω2τ 2

Sk

1 + ω2τ 2Sk

,

G′′disc (ω) =

N∑

k

μneqk ωτ Sk

1 + ω2τ 2Sk

,

(22)

where G disc ∗ ( i ω ) = G ′ disc ( ω ) + iG ″ disc ( ω ) is the complex modulus of the frequency response.

A corresponding continuous-spectrum model is constructed using the concepts of fractional calculus following Haupt et al. [ 21 ] (see also Caputo and Mainardi [ 29 ] and Bagley and Torvik). [ 30 ] The constitutive formulation is developed from a sim-pler rheological model consisting of an equilibrium spring ele-ment in parallel with a Maxwell fl uid with a fractional damping element. The fractional damping element is characterized by the parameters 0 < α < 1 and ξ , which denote the breadth of the relaxation spectrum and the characteristic relaxation time. For the fractional damping model, the deviatoric stress response can be written as,

s = 2μeqeM + 2μneq(

eM − evM

)

, (23)

where μ neq is the nonequilibrium modulus of the continuous-spectrum model and corresponds to

∑Nk μ

neqk of the discrete-

spectrum model. The evolution equation for the viscous strain of the fractional damping element is given by,

d α evM

d tα= 1

ξα

(

eM − eeM

)

, evM (0, T ) = 0,

(24)

where d α ( ⋅ )/d t α is the Riemann-Liouville fractional derivative operator of order 0 < α < 1 defi ned as,

d ds dsα f(s)

d tα� 1

� (1 − α)

∫ t

0

f (s)

(t − s )−(α−1)

(25)

The storage and loss moduli of the frequency response for the fractional model can be calculated for a harmonic deforma-tion as,

G′frac (ω) = μeq + μ

neqk

(

(ωξ)2α + (ωξ)α cos(

απ2

))

1 + (ωξ)2α + (ωξ)α cos(

απ2

) ,

G′′frac (ω) = μ

neqk

(

(ωξ)α sin(

απ2

))

1 + (ωξ)2α + (ωξ)α cos(

απ2

) ,

(26)

A viscoelastic relaxation spectrum is calculated for both the discrete and fractional models from their respective complex moduli using the Stieltjes transform (see references [31,32]),


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G

wtd

H

ls

H

tDro

H

w

dfa Tlgor

λ

wtfTve

τ

d

∗ (iω)

iω=

∫ ∞

0

h (λ)

λ + iωd λ,

here the relaxation frequency λ ision time. This allows a cumulative efi ned as follows,

(λ) =∫ λ

0h (z) d z,

For the fractional model, Equatioowing analytical expression for tpectrum,

frac (λ) = μneq[

arctan(

(λξ)α + co

(27)

the inverse of the relaxa-relaxation spectrum to be

(28)

n ( 26 )– (27) yield the fol-he cumulative relaxation

απ

s (απ)

sin (απ)

)

−π

(

1

2− α

)]

.

(29)

The relaxation spectrum of the discrete model calculated for he storage and loss moduli in Equation ( 22 ) is a series of Delta irac functions evaluated at λ k = 1/ τ S k . Thus, the cumulative

elaxation spectrum of the discrete model evaluates to the sum f step functions,

disc (λ) =N

∑

k

μneqk 〈λ − λk〉 ,

(30)

here ⟨ λ – λ k ⟩ = 0 for λ < λ k . The procedure for determining the parameters of the

iscrete-spectrum model fi rst fi ts the storage modulus of the ractional model in Equation ( 26 ) to the master curve gener-ted by the TTS tests (Section 2.2.3) for a reference temperature 0 to determine α and ξ . The parameters μ neq and μ eq are calcu-

ated from the glassy and rubbery Young’s moduli assuming a lassy Poisson’s ratio of ν g = 0.35 and a rubbery Poisson’s ratio f ν r = 0.5. Next, we assume a power-law distribution of the elaxation frequencies for T 0 ,

0k = λ0

min

(

λ0max

λ0min

) k−1N−1

,

(31)

here λ 0 k = λ k ( T 0 ) = 1/ τ S k ( T 0 ). The upper and lower bounds of he frequency range, λ 0 max and λ 0 min are chosen based on the requency range of the master curve of the storage modulus. he characteristic viscoelastic relaxation times of the thermo-iscoelastic model can be computed from λ k 0 using the WLF quation as,

gSk

= 1

λ0k

exp

[

Cg1

log e

(

T0 − T refg

Cg2 + T0 − T ref

g

)]

,

(32)

Finally, the parameters μ neq k of the discrete model are etermined for the selected λ 0 k such that cumulative relaxation

© 2010 WILEY-VCH Verlag Gm

spectrum of the discrete model, H disc ( λ ), forms a stair-case approximation of the cumulative relaxation spectrum of the continuous fractional-damping model, H disc ( λ ), as follows,

λ01 < λ < λ0

2 :

Hdisc (λ) = μneq1 := 1

2

(

Hfrac

(

λ01

) + Hfrac

(

λ02

))

,

λ02 < λ < λ0

3 :

Hdisc (λ) = μneq1 + μ

neq2 := 1

2

(

Hfrac

(

λ02

) + Hfrac

(

λ03

))

,

λ0k < λ < λ0

k+1 :

Hdisc (λ) =k

∑

l=1

μneql := 1

2

(

Hfrac

(

λ0k

) + Hfrac

(

λ0k+1

))

.

(33)

Evaluating (33) for each k interval and solving for μ neq k gives,

μneq1 = 1

2

(

Hfrac

(

λ01

) + Hfrac

(

λ02

))

,

μneqk = 1

2

(

Hfrac

(

λ0k+1

) − Hfrac

(

λ0k−1

))

, 1 < k < N − 1,

μneqN = μneq −

N−1∑

k

μneqk ,

(34)

for the discrete viscoelastic relaxation spectrum.

2.5.4. Determining the Discrete Structural Relaxation Spectrum

To determine the parameters Δ α k and τ R k , it is assumed that the structural relaxation spectrum has the same shape as the vis-coelastic relaxation spectrum. This assumption has not been demonstrated in general for amorphous polymers, though both viscoelastic and structural relaxation arise from the motion of molecular chains. Experiments for polyvinylacetate obtained dis-tinctly different viscoelastic and structural relaxation spectra. [ 33 ] However, we assume for simplicity that the two have the same shape at equilibrium, and assume an expression analogous to Equation ( 29 ) for the cumulative structural relaxation function at equilibrium,

L frac (λ) = αr − αg

απ

[

arctan(

(λP )α + cos (απ)

sin (απ)

)

−π

(

1

2− α

)]

.

(35)

The parameter α defi nes the breadth of the relaxation spectrum and is the same as in Equation ( 29 ) for the cumu-lative viscoelastic relaxation spectrum. The parameter χ is the characteristic structural relaxation time analogous to ξ in

bH & Co. KGaA, Weinheim Adv. Mater. 2010, 22, 3411–3423


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Equation ( 29 ). Then for the same power-law distribution of relaxation frequencies in Equation ( 31 ), the parameters Δ α k of the discrete structural relaxation spectrum are defi ned analo-gous to Equation ( 34 ) as,

�α1 = 1

2(L frac (λ1) + L frac (λ2)) ,

�αk = 1

2(L frac (λk+1) − L frac (λk−1)) , 1 < k < N − 1

�αN = (

αr − αg

) −N−1∑

k

�αk .

(36)

Finally, the parameters τ gRk

are evaluated from Equation ( 12 ) for the power-law frequency distribution as,

τ

gRk

= 1

λkexp

[

Cg1

log e

(

T0 − T refg

Cg2 + T0 − T ref

g

)]

,

(37)

An iterative procedure is employed for determining the discrete spectrum Δ α k from the thermal expansion data of the CTE experiments (Section 2.2.2). The procedure begins by assuming a distribution of relaxation frequencies λ 0 k in Equation ( 31 ) and calculating τ g

Rk from Equation ( 37 ). An ini-

tial estimate is assumed for the characteristic relaxation time χ for the continuous cumulative spectrum in Equation (35), and Equation ( 36 ) is used to calculate Δ α k . The parameters Δ α k and τ g

Rk are used by the fi nite deformation thermoviscoe-

lastic model to simulate the thermal expansion during cooling from 100 ° C to 0 ° C at a constant rate of 1 ° C min − 1 . The glass transition temperature is calculated for the simulation results and compared to the experimentally determined T g ref (Section 2.2.2). The results are used to update the estimate for Δ α k , and the procedure is repeated until the two agree within a toler-ance of 0.1 ° C.

2.5.5. General Comments

The advantage of using the parameter identifi cation pro-cedure developed by Haupt et al. [ 21 ] is that the fractional damping model provides analytic expressions for the storage and loss moduli and for the cumulative relaxation spectra. However, other phenomenological relaxation models such as the stretched exponential model can be used to construct a corresponding, continuous, viscoelastic spectrum model. A numerical method may be needed to solve for the cumulative relaxation spectrum from the complex modulus for alternate relaxation models. Alternatively, the parameters μ neq k can be fi t directly to the master curve assuming a distribution of relaxa-tion times.

For the structural relaxation spectrum, the thermal expansion measurements for a constant cooling rate provide a less sensi-tive means for determining the distribution of Δ α k compared to isothermal recovery measurements for a step change in tem-perature (e.g., Kovacs et al. [ 24 ] ). We used the former because the constant cooling rate data was conveniently available and the results produced reasonable agreement with experiments for


the recovery response as shown below in Section 3.2. However, we are working currently to develop a more accurate procedure to characterize the structural relaxation spectrum from iso-thermal recovery experiments.

3. Results and Discussions

3.1. Parameter Determination

The fi nite-deformation thermoviscoelastic model and the cor-responding small-strain fractional damping model, developed for the case of structural equilibrium, were implemented in Mathematica and applied to determine the model parame-ters for the tBA networks with 2 wt%, 10 wt%, 20 wt%, and 40 wt% crosslinking agents using the procedure described in Sections 2.2, 2.5.3, and 2.5.4. The results are listed in Table 1 . Figure 3a shows the master curve of the storage modulus for the different networks comparing the data from the TTS tests (Section 2.2.3) and results for Equation ( 26 ) of the frac-tional damping model. The fractional damping model, with a continuous relaxation spectrum, corresponds to the linear-ized thermoviscoelastic model, with a discrete spectrum, for the case of structural equilibrium. Thus, the parameters α for the frequency span and ξ for the characteristic stress relaxa-tion time of Equation ( 26 ) were fi t to data measured at high temperatures, above 45 ° C. The simulation results agreed well with the experimental data up until the onset of the glassy pla-teau, where the WLF time-temperature superposition relation begins to break down. The model achieved a glassy plateau at lower frequencies than observed in the data. The best fi t was obtained for the 40 wt% material, while the largest discrepan-cies were observed at higher frequencies for the 10 wt% and 20 wt% materials.

Figure 3b plots the thermal strain as a function of tem-perature comparing data from the CTE experiments (Section 2.2.2) and results from simulations using the nonlinear ther-moviscoelastic model. The data and simulation results for the 10 wt% material were nearly the same as for the 20 wt% mate-rial, and were omitted from Figure 3b for clarity. The glassy and rubbery CTE, α g and α r , were calculated from the slope of the strain-temperature data for the temperature range of 0–20 ° C and 80–100 ° C. The simulations were used to fi t the charac-teristic structural relaxation time χ in Equation ( 35 ) using the iterative procedure describe in Section 2.5.4. A good fi t was obtained for the 2 wt% specimen over the entire temperature range. For the other specimens, good agreement was achieved at high temperatures but noticeable discrepancies developed in the glass transition region. The data generally showed a broader glass transition region than predicted by the model, and the largest disagreement was observed for the 10 wt% and 20 wt% specimens. This discrepancy likely stemmed from the assumption that both viscoelastic and structural relaxation can be described by the same cumulative relaxation spectrum given in Equation ( 29 ) and ( 35 ). The same parameter α was used for both, and the two differed only in the characteristic relaxation time. We are working improve the model and characterization of structural relaxation using isothermal recovery tests.


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Table 1. Model parameters for tBA-co-PEGDMA networks with different wt% crosslinks.

Parameter 2 wt% 10 wt% 20 wt% 40 wt% Physical signifi cance

T ref g ( ° C) 31 36 38 40 T g for q cool = 1 ° C min − 1 .

α r (10 − 4 ° C − 1 ) 7.52 7.48 7.47 6.90 rubbery CTE.

α g (10 − 4 ° C − 1 ) 2.58 2.67 2.93 2.98 glassy CTE.

χ (10 − 6 s) 8.0 200 700 1200 structural relaxation time of continuous model.

ξ (10 − 6 s) 0.02 1.0 2.0 3.0 stress relaxation time of continuous model.

α 0.62 0.70 0.70 0.62 fractional order/breadth of spectrum

C g 1 17.25 13.76 13.63 16.21 fi rst WLF constant.

C g 2 ( ° C) 51.59 32.46 33.75 49.50 second WLF constant.

μ eq (MPa) 0.167 0.533 1.166 3.43 equilibrium shear modulus.

κ (MPa) 1666.67 1666.67 1666.67 1666.67 bulk modulus of continuous model.

μ neq (MPa) 555.56 555.56 555.56 555.56 non-equilibrium shear modulus of continuous

model.

Varying the crosslinker weight percent did not affect the CTE and moduli κ and μ neq . As expected, the rubbery modulus, μ eq , and the characteristic structural and stress relaxation times, χ and ξ , increased signifi cantly with crosslinker content. Neither the WLF constants nor α varied monotonically with the crosslink content. Both the 10 wt% and 20 wt% networks had the largest α and the smallest C g 1 and C g 2 . For all four materials, the values of the WLF constants were near the universal values C 1 g = 17.44 and C g

2 = 51.6 derived from free-volume considerations. [ 22 ]

3.2. Model Validation

The fi nite-deformation thermoviscoelastic model was applied using the parameters in Table 1 to simulate the dynamic


Figure 3 . Experimental data and model fi t for (a) the master curve of the stothermal strain for cooling rate of q = 1 ° C min − 1 for networks of different we

100

105

10

100

101

102

103

Frequency (Hz)

Sto

rage

Mod

ulus

(M

Pa)

Model2wt% Experiment

10wt% Experiment20wt% Experiment40wt% Experiment

2 wt%

10 wt%

20 wt%

40 wt%

(a)

thermal scan experiments and unconstrained recovery experi-ments described in Section 2.2.1. The dynamic thermal scan experiments subjected the sample to a dynamic uniaxial tensile loading at 1.0 Hz while the sample was heated from 0 to 120 ° C at a rate of 3 ° C min − 1 . Figure 4 compares the experimental data and model predictions for the temperature-dependence of the storage modulus for the four different materials. The results showed good quantitative agreement for the 10wt%, 20wt%, and 40wt% specimens. The model was able to reproduce the breadth of the glass transition region and T onset , though the parameters of the relaxation spectrum were fi t only to data at temperatures greater than T onset . For the 2 wt% material, the model predicted a lower onset temperature and a broader glass transition region.

Figure 5 compares the experimental data and model pre-diction for the unconstrained strain recovery response under


rage modulus for a reference temperature of T 0 = 80 ° C, and (b) the ight percent crosslinking agents.

0 20 40 60 80Temperature (oC)

Str

ain

Model

20wt% Experiment2wt% Experiment

40wt% Experiment

(b)


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Figure 4 . Temperature dependence of the uniaxial tension storage modulus for the (a) 2 wt%, (b) 10 wt%, (c) 20 wt%, and (d) 40 wt% networks comparing experimental data and model prediction. The material was subjected to a dynamic strain at 1Hz while heated at a rate of 3 ° C min − 1 .

0 20 40 60 80 100

100

101

102

103

Temperature (oC)

Sto

rage

Mod

ulus

(M

Pa)

ModelExperiment

(a)

0 20 40 60 80 100

100

101

102

103

Temperature (oC)

Sto

rage

Mod

ulus

(M

Pa)

ModelExperiment

(b)

0 20 40 60 80 100

100

101

102

103

Temperature (oC)

Sto

rage

Mod

ulus

(M

Pa)

ModelExperiment

(c)

0 20 40 60 80 100

100

101

102

103

Temperature (oC)

Sto

rage

Mod

ulus

(M

Pa)

ModelExperiment

(d)

uniaxial tension. The simulations used the exact loading history prescribed in the experiments and described in Section 2.2.4. In general, the model prediction for the onset temperature of strain recovery and the recovery time, given by the slope of the strain-temperature curve and the temperature span of the recovery event, agreed well with experiments. Compared to the results in Figure 1b , the addition of multiple relaxation proc-esses signifi cantly improved the prediction of the recovery time, though the model continued to predict a steeper slope and a narrower recovery temperature range. The best prediction was obtained for the 40 wt% material. The model accurately captured the recovery temperature range for the 2 wt% mate-rial but calculated a T onset that was 5 ° C lower. For the 10 wt% and 20 wt% materials, the model predicted a slightly lower onset recovery temperature but a 10 ° C narrower recovery tem-perature range. Recall that the best parameter fi t to the storage


modulus and thermal expansion data was obtained for the 40 wt% and 2 wt% material, as shown in Figure 3 , while the worst was obtained for the 10 wt% and 20 wt%. For these mate-rials, the model produced a sharper glass transition, particularly compared to the thermal strain data, because of simplifying assumptions regarding the structural relaxation spectrum.

4. Conclusions

A fi nite-deformation thermoviscoelastic model was developed that incorporated the time-dependent effects of structural and stress relaxation of multiple discrete processes. The model was based on the previous thermoviscoelastic model developed by Nguyen et al., [ 17 ] which used a single process to describe both relaxation mechanisms. The formulation was based on a


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Figure 5 . The unconstrained recovery response for the (a) 2 wt%, (b) 10 wt%, (c) 20 wt%, and (d) 40 wt% networks comparing data and model predictions.

0 20 40 60 80 1000

5

10

15

20

Temperature (oC)

Str

ain

(%)

ModelExperiment

(a)

0 20 40 60 80 1000

5

10

15

20

Temperature (oC)

Str

ain

(%)

ModelExperiment

(b)

0 20 40 60 80 1000

5

10

15

20

Temperature (oC)

Str

ain

(%)

ModelExperiment

(c)

0 20 40 60 80 1000

2

4

6

8

10

Temperature (oC)

Str

ain

(%)

ModelExperiment

(d)

multiplicative split of the deformation gradient into thermal and mechanical parts. The latter was decomposed further in N parallel multiplicative elastic and viscous components, one for each viscoelastic relaxation process. The thermal part was decomposed additively into an instantaneous parts and N internal variables for nonequilibrium structure of the mate-rial. The innovations of the present work can be summarized as follows.

The thermoviscoelastic model combines established theories • of temperature-dependent nonlinear viscoelasticity and struc-tural relaxation for amorphous polymers near T g to describe shape memory behavior. The model employs the Adam-Gibbs description of tem-• perature and structure dependence of the relaxation times to model the temperature transition from Arrhenius to WLF


behavior, whereas previous thermoviscoelastic models for shape memory polymers have used one or the other. In ad-dition, previous models for structural relaxations have as-sumed a separable dependence on temperature and structure (e.g., Kovacs et al. [ 24 ] ). A method was developed based on the work of Haupt • et al. [ 21 ] to obtain the relaxation spectra from a standard time-temperature-superposition experiment and a constant cool-ing rate, thermal expansion experiment. The main advantage of the method is that it greatly reduces the number of param-eters to be determined from experiments.

Currently, we are working to develop a more accurate charac-terization of the structural relaxation spectrum from isothermal recovery experiments. This will improve the structural relaxa-tion model and the predictions of the recovery response. In



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addition, we plan to characterize the isothermal stress-strain response of the materials to their elongation limit to extend the applicability of the model to larger strains. The model currently cannot describe the viscoplastic yielding and fl ow behavior of the glassy material at low temperatures. Thus, it cannot be used to simulate the recovery response of specimens programmed by cold drawing. We will incorporate the effects of yielding and post-yield softening, a feature of the original model, into the present thermoviscoelastic model with multiple relaxation processes.

AcknowledgementsThis work was funded by the National Science Foundation (CMMI-0758390) and the Laboratory Directed Research and Development program at Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-ACO4-94AL85000.

Received: January 12, 2010 Revised: January 25, 2010

Published online: July 28, 2010

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modeling the relaxation mechanisms of amorphous shape memory polymers

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