Thermodynamics of Materials with Memory: Theory and Applications

Introduction. In this work, we consider materials the constitutive equations of which contain a dependence upon the past history of kinetic variables. In particular, we deal with the constraints imposed upon these constitutive equations by the laws of thermodynamics. Such materials are often referred to as materials with memory or with hereditary effects. The study of materials with memory arises from the pioneering articles of Boltzmann [20, 21] and Volterra [211, 212, 213], in which they sought an extension of the concept of an elastic material. The key assumption of the theory was that the stress at a time t depends upon the history of the deformation up to t. The hypothesis that the remote history has less influence than the recent history is already implicit in their work. This assumption, later termed the fading memory principle by Coleman and Noll [40], is imposed by means of a constitutive equation for the stress, of integral type, which in the linear case involves a suitable kernel (relaxation function) that is a positive, monotonic, decreasing function. In the classical approach to materials with memory, the state is identified with the history of variables carrying information about the input processes.We show in this book how Noll’s definition of state [188] is more convenient for application to such materials. Indeed, Noll takes the material response as the basis for the definition of state: if an arbitrary continuation of different given histories leads to the same response of the material, then the given histories are equivalent and the state is represented as the class of all such equivalent histories. We refer to this class as the minimal state. The concept of a minimal state is developed and applied in [116] to the case of linear viscoelasticity with scalar relaxation functions given by a sum of exponentials. A subsequent paper [57] presents a treatment in three dimensions and in the more general context of thermodynamically compatible (tensor-valued) relaxation functions, taking into account weak regularity of histories and processes. A generalization of minimal states to materials under nonisothermal conditions is discussed in Section 6.4 of the present work. A functional It is introduced, given by (6.4.2)with the crucial property expressed by (6.4.3). This quantity characterizes the minimal state. Special cases of it are used in a variety of contexts in later chapters. It is closely related to the response of the material after time t, where the input variable is null for a finite period after this time on the material element (i.e., a “small” neighborhood of a fixed and arbitrary point of the body) under consideration. This characterization of the state is an interesting alternative to the usual one based on knowledge of the deformation history. It seems more appropriate to refer to materials with states characterized in this way as materials with relaxation rather than materials with fading memory. For the usual definition of state, a fading-memory property of the response functional [36] is required, as opposed to the case in which the minimal state is adopted, where indeed the relaxation property of the response functional suffices. Obviously, whenever the stress-response functional is such that knowledge of the minimal state turns out to be equivalent to knowledge of the past history, the property of relaxation of the stress response implies fading memory of the related functional. In this sense, the class of materials with relaxation is larger than the one described by constitutive equations with fading memory. A significant advantage of the response-based definition of state relates to the physical features of the state itself. Indeed, the “future stress” It(τ) can be determined through measurements and does not require knowledge of the past history at all. For materials with memory, there are in general many different functional forms with the required properties for a free energy. Some of these are functions of the minimal state, while others do not have this property (see, e.g., [57]). In Part III, these functional forms are explored for different categories of materials with memory. We note that for materials whose constitutive relation for the response functional has a linear memory term, all free energies associated with this material have memory terms that are quadratic functionals. A new class of single-integral-type free energies, for certain categories of relaxation functions, is introduced in Section 9.1.3 as a quadratic form of the time derivative of the state variable It (see, e.g., [128, 129] for discussion and analysis of single-integral type free energies that are quadratic forms of histories). For exponentially decaying relaxation functions, it can be shown that the dissipation associated with such energies is bounded below by a time-decay coefficientmultiplied by the purely memory-dependent part of the free energy. This property turns out to be crucial in the analysis of PDEs relating to linear viscoelastic materials, which is developed in Part IV. An analogous property holds for a family of multiple-integral free-energy functionals that are the generalization of the previous single-integral-type free energy. We may refer to such a family as the n-family. For n = 1 one recovers the singleintegral case. In Chapters 10–14, explicit forms of the minimum free energy are derived both in the general nonisothermal case and, more specifically, for viscoelastic solids, fluids, and rigid heat conductors. Different forms of relaxation functions are also considered. The minimum free energy is always a function of the minimal state. Indeed, an explicit formula is derived in Section 11.2 for this quantity as a quadratic functional of minimal-state variables related to It . In Chapters 15 and 16, relaxation functions consisting of sums of decaying exponentials multiplying polynomials are considered. A family of free energies, including the minimum, maximum, and intermediate forms, are given explicitly. All of these are functions of state and derivable from an optimization procedure. In Part IV, we observe that the new approach outlined above and the new free energies, in both cases adapted to the theory of viscoelasticity, have interesting applications to the PDEs governing the motion of a suitable class of viscoelastic bodies. In particular, the use of the new free energies given by quadratic forms of the minimal state variables yields results relating to well-posedness and stability for the IBVP. This formulation allows for initial data belonging to broader functional spaces than those usually considered in the literature, which are based on histories. Indeed, the response-based definition of state is useful for both the study of IBVP on the one hand and the evolution of linear viscoelastic systems on the other hand. Furthermore, an application of semigroup theory to this class of materials is presented. Here, besides having the system of equations in a more general form than for the classical approach, results on asymptotic stability are again obtained for initial data belonging to a space broader than the one usually employed when states and histories are identified. The book is divided into four parts, Part I dealing with the general principles of continuum mechanics and with elastic materials and classical fluids, which of course provide the foundation for developments in later chapters. A general treatment of continuum thermodynamics is presented in Part II. In Part III, materials that are described by constitutive equations with linear memory terms are discussed in some detail. The specific cases included are viscoelastic solids and fluids, together with rigid heat conductors. Also, as noted earlier, the derivation of explicit forms of free energies is considered in depth. Part IV deals with the application of results and ideas from Part III to the equations of motion of linear viscoelastic materials. Notation conventions are described at the beginning of Appendix A. Relevant mathematical topics are summarized in Appendices A, B, and C