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

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Abstract

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.

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 Thigh>Ttrans. The heated device is deformed to the desired shape, cooled to Tlow<Ttrans, 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 Ttrans. Heating to above Ttrans 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., Ttrans=Tg). The chain mobility describes the ability of the chain segments to rearrange locally to bring the macromolecular structure and stress response to equilibrium. The mobility is high for temperatures above the Tg 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 Tg, 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 Tg 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 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, Tool and Eichlin, 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, Tf, 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 Tf 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 Tg. 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 T2. This theoretical equilibrium state can never be reached because the mobility vanishes as TT2, and is observed instead at a higher temperature Tg>T2 as the glass transition. Adam and Gibbs (1965) hypothesized that the increase in the relaxation time during cooling to T2 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 Tf instead of T. This allowed the configurational entropy to depend on the actual rather than equilibrium structure. Finally, Hodge, 1987, Hodge, 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 TTg, where structural relaxation effectively is arrested, to the Williams–Landel–Ferry (WLF) behavior for TTg, 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.

Section snippets

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.

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

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

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=1mm in height and R=0.5mm in radius, while the platen was Hp=0.5mm in height and Rp=1.5mm in radius. The plug and platen were discretized using bilinear C0 square elements of size h=0.05H. The

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

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.

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