Free Energies for Nonlinear Materials with Memory

An exploration of representations of free energies and associated rates of dissipation for a broad class of nonlinear viscoelastic materials is presented in this work. Also included are expressions for the stress functions and work functions derivable from such free energies. For simplicity, only the scalar case is considered. Certain standard formulae are generalized to include higher power terms. It is shown that the correct initial procedure in this context is to specify the rate of dissipation as a positive semi-definite functional and then to determine the free energy from this, rather than the other way around, which would be the traditional approach. Particularly detailed versions of these formulae are given for the model with two memory contributions in the free energy, the first being the well-known quadratic functional leading to constitutive relations with linear history terms, while the second is a quartic functional yielding a cubic term for the stress function memory dependence. Also, the discrete spectrum model, for which each memory kernel is a sum of exponentials, is generalized from the quadratic functional representation for the free energy to that with the quartic functional included. Finally, a model is considered, allowing functional power series with an infinite number of terms for the free energy, rate of dissipation and stress function

Viscoelastic and Electromagnetic Materials with Nonlinear Memory

A method is presented for generating free energies relating to nonlinear constitutive equations with memory from known free energies associated with hereditary linear theories. Some applications to viscoelastic solids and hereditary electrical conductors are presented. These new free energies are then used to obtain estimates for nonlinear integro-differential evolution problems describing the behavior of nonlinear plasmas with memory

A Mathematical Model for Visco-Ferromagnetic Materials

Visco-ferromagnetic materials represented by non-local constitutive equation are considered in the paper. We use fractional derivatives in order to describe memory and spatial effects. Also, thermodynamic principles are formulated and studied

Second Gradient Viscoelastic Fluids: Dissipation Principle and Free Energies

We consider a generalization of the constitutive equation for an incompressible second order fluid, by including thermal and viscoelastic effects in the expression for the stress tensor. The presence of the histories of the strain rate tensor and its gradient yields a non-simple material, for which the laws of thermodynamics assume a modified form. These laws are expressed in terms of the internal mechanical power which is evaluated, using the dynamical equation for the fluid. Generalized thermodynamic constraints on the constitutive equation are presented. The required properties of free energy functionals are discussed. In particular, it is shown that they differ from the standard Graffi conditions. Various free energy functionals, which are well-known in relation to simple materials, are generalized so that they apply to this fluid. In particular, expressions for the minimum free energy and a more recently introduced explicit functional of the minimal state are proposed. Derivations of various formulae are abbreviated if closely analogous proofs already exist in the literature

General dissipative materials for simple histories

A material with memory typically has a set of many free energy functionals associated with it, all members of which yield the same constitutive relations. An alternative interpretation of this set is explored in the present work. Explicit formulae are derived for the free energy and total dissipation of an arbitrary material in the cases of step function and sinusoidal/exponential histories. Expressions for the fraction of stored and dissipated energy are deduced. Also, various formulae are given for discrete spectrum materials. For materials with relaxation function containing one decaying exponential, the associated Day functional is the physical free energy. For more general materials, we seek a best fit of the relaxation function with one decaying expo- nential to that chosen for the general case. The free energy, total dissipation and fractions of stored and dissipated energies relating to the Day material are derived for the various histories. Similar data, in the case of the general mate- rial, are explored for the minimum and maximum free energies and also for a centrally located free energy given in the literature. Various plots of aspects of this data, including comparisons between the behaviour for general and Day materials, are presented and discussed

Viscoelastic and Electromagnetic Materials with Nonlinear Memory

A method is presented for generating free energies relating to nonlinear constitutive equations with memory from known free energies associated with hereditary linear theories. Some applications to viscoelastic solids and hereditary electrical conductors are presented. These new free energies are then used to obtain estimates for nonlinear integro-differential evolution problems describing the behavior of nonlinear plasmas with memory

Free Energies and Minimal States for Scalar Linear Viscoelasticity

The concept of a minimal state was introduced in recent decades, based on earlier work by Noll. The property that a given quantity is a functional of the minimal state is of central interest in the present work. Using a standard representation of a free energy associated with a linear memory constitutive relation, a new condition, involving linear functionals, is derived which, if satisfied, ensures that the free energy is a functional of the minimal state. Using this result and recent work on constructing free energy functionals, it is shown that if the kernel of the rate of dissipation functional is given by sums of products, the associated free energy functional is a functional of the minimal state

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