A thermodynamically consistent, nonlinear viscoelastic approach for modeling glassy polymers

Abstract

A thermodynamically consistent nonlinear viscoelastic constitutive theory is derived to capture the wide range of behavior observed in glassy polymers, including such phenomena as yield, stress/volume/enthalpy relaxation, nonlinear stress–strain behavior in complex loading histories, and physical aging. The Helmholtz free energy for an isotropic, thermorheologically simple, viscoelastic material is constructed, and quantities such as the stress and entropy are determined from the Helmholtz potential using Rational Mechanics. The constitutive theory employs a generalized strain measure and a material clock, where the rate of relaxation is controlled by the internal energy that is likewise determined consistently from the viscoelastic Helmholtz potential. This is perhaps the simplest model consistent with the basic requirements of continuum physics, where the rate of relaxation depends upon the thermodynamic state of the polymer. The predictions of the model are compared with extensive experimental data in the following companion paper.

Introduction

Polymers in the glassy state exhibit complex, nonlinear, time-dependent relaxation of volume, enthalpy, and stress. A constitutive equation should be able to describe all of this relaxation behavior for arbitrary temperature and deformation histories, yet existing theories (see Refs. [1], [2] and references cited therein) are only able to capture subsets of the material's response. Our goal is quite simple but quite broad. We wish to develop a constitutive theory consistent with the fundamental requirements of continuum physics that can describe the full range of relaxation behavior of glassy polymers under arbitrary time/temperature/deformation histories. A small subset of the types of behavior to be addressed includes stress relaxation, yield under a constant strain rate loading, volume relaxation after quenching into the glass, and the various manifestations of physical aging. We are not, however, interested in simply developing a framework that is theoretically capable of describing these various phenomena; neither are we content to be able to just fit one or two tests. Rather, the constitutive formalism must be quantitatively predictive for all relaxation experiments to engineering accuracy using one set of material parameters. It is this goal that distinguishes our work from previous efforts and obliges us to validate it under an equally wide range of experimental conditions to assess its accuracy. The purpose of this paper is to provide a complete discussion of the theory leading to a working, quantitative, constitutive equation in a form that is accessible to a polymer scientist rather than a continuum mechanics specialist. In the following paper [3], we will present an extensive set of data from four different polymers that critically examines the ability of the constitutive equation to predict volume, enthalpy and mechanical relaxation and demonstrates its universality.

Two frameworks have been proposed for describing glassy polymers, plasticity and nonlinear viscoelasticity. While these two approaches may appear similar under certain situations, they have significant differences. Amorphous polymers are linear viscoelastic for infinitesimal strains, and no distinct change in the linear viscoelastic response is observed as rubbers are gradually cooled into the glass except that the relaxation times grow longer [4]. Nonlinear viscoelastic formalisms preserve this continuous transition between the rubbery and glassy states, and view mechanical yield as a natural consequence of the nonlinear relaxation behavior induced by loading. In contrast, plasticity theories by their very construction predict yield [5], [6], [7], but need to propose a second mechanism distinct from yield to describe the glass transition. Aside from the inherent problem of predicting a glass transition, plasticity theories also encounter difficulties explaining the concept of plastic flow in cross-linked materials that yield yet return to their original state when heated above the glass transition temperature, T_g. Consequently, we have chosen to develop a viscoelastic framework, believing that the use of a single underlying mechanism to describe both uncross-linked and cross-linked polymers provides a more physically based constitutive description of glassy polymers.

A number of nonlinear viscoelastic constitutive equations that have been developed for glassy polymers use the concept of a ‘material clock’ [8], [9] to describe how relaxation rates depend on the state of the glass. Such clock models affect nonlinear behavior through free volume [10], [11], [12], entropy [13], stress [14] or strain [15] just as the more familiar ‘WLF shift factor’ defines the dependence of the relaxation rates on temperature. Although existing clock models can predict some features of the nonlinear viscoelastic response well, they are deficient in other areas, having been designed to capture only a portion of the overall response. For example, volumetric [16], [17] and enthalphic [18], [19], [20] behaviors are predicted accurately in some models, but no consideration is given to the mechanical response (e.g., free volume clocks cannot predict compressive yield). In other models, the mechanical behavior [21], [22], [23] has been targeted, while the thermodynamic response has been neglected (e.g., stress clocks are unable to describe the complex, isobaric volume relaxation exhibited by polymers [24]). Of course, one could construct clocks that are ad hoc combinations of the various models (e.g., a stress plus free volume clock, etc.). However, to-date no such hybrid clock has been developed that can fit the full range of relaxation data for glassy polymers. The material clock model appears to have promise, but significant improvements are needed to describe the full range of viscoelastic behavior exhibited by glassy polymers.

Existing clock models are based on two key assumptions. First, the instantaneous rate of relaxation is controlled by the current state of the material, and second, the material is thermorheologically simple (i.e., the shape of the relaxation spectrum does not change with temperature, specific volume or whatever feature of the current state controls the rate of relaxation). Thermorheological simplicity seems to be a reasonable assumption, although there are data indicating that polymers in the glass transition region may exhibit slight deviations [25]. Thus, the main challenge resides in identifying the variable that controls the rate of relaxation. If a thermodynamic variable is to be employed in the clock, then the overall description of the volume, enthalpy, and mechanical relaxation must be carefully defined to avoid creating energy. Noll and Coleman [26], [27] developed the Rational Mechanics framework to ensure thermodynamic consistency for nonlinear viscoelastic materials. The key result of Rational Mechanics is that all time-dependent thermodynamic quantities can be derived from a single time-dependent free energy. However, Coleman and Noll's initial constitutive equations were not defined in terms of a material clock. Lustig et al. [2] extended this approach to include a history-dependent material clock. Specifically, they showed that if the clock depends upon a viscoelastic thermodynamic quantity, then the clock itself must be a function of the thermal and deformation histories to be consistent. Following upon the ideas of Adam and Gibbs [28], Lustig et al. [2] proposed using configurational entropy as the variable that controls the clock. While they were able to show that important features of relaxation in the glass transition region could be predicted qualitatively, the predictions were not quantitative over the breadth of glassy phenomena. Our subsequent investigations of this constitutive equation demonstrated that some predictions were, in fact, quite poor. In this paper, we will describe the important theoretical developments needed to take this basic approach and produce a working, quantitative model for glassy polymers.

Let us first examine several significant assumptions imbedded in the constitutive model of Lustig et al. [2]. First, the independent variables were assumed to be the temperature history and the deformation history, where the deformation history was specified by the right Cauchy Green deformation tensor. The right Cauchy Green deformation tensor, however, is only one of the infinite family of deformation tensors that can be employed. Second, the rate of relaxation was assumed to be a function of the configurational entropy (approximated in that paper by the total entropy); however, the rate of relaxation may depend upon other thermodynamic state variables. Third, even though the free energy was expanded through double integral terms, the stress and entropy were only evaluated through the single integral terms, thus limiting some of the temperature and strain dependencies.

In this paper, we identify and develop three critical advances that are necessary to obtain a theory capable of quantitative predictions: (i) the use of the Hencky strain measure to ensure reasonable volume changes during deformation, (ii) a new potential energy clock based on the ideas of Adam and Gibbs [28], and (iii) the incorporation of temperature and deformation dependencies for specific properties. By incorporating these improvements, accurate predictions are enabled, making the theory a useful tool for polymer engineers. Moreover, the physically based relationship between polymer relaxation rates and the thermodynamic state implies that all model inputs are uniquely defined by equilibrium and linear viscoelastic properties. Therefore, this viscoelastic formalism requires, in principle, no free fitting parameters to adjust the ‘strength’ of the nonlinearities in distinct contrast to most existing investigations [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. Thus, if the proposed constitutive equation is able to describe the full range of nonlinear viscoelastic behavior observed experimentally using only linear viscoelastic properties as input, it provides a strong indication that the underlying postulates of the constitutive equations are physically reasonable.