On the thermodynamics of non-linear materials with quasi-elastic response (original) (raw)

Free energies in a general non-local theory of a material with memory

Published 23 Communicated by S. Müller 24 A general theory of non-local materials, with linear constitutive equations and memory 25 effects, is developed within a thermodynamic framework. Several free energy and dis-26 sipation functionals are constructed and explored. These include an expression for the 27 minimum free energy and a functional that is a free energy for important categories of 28 memory kernels and is explicitly a functional of the minimal state. The functionals dis-29 cussed have a similar general form to the corresponding expressions for simple materials.

Constructing Free Energies for Materials with Memory

Springer eBooks, 2021

The free energy for most materials with memory is not unique. There is a convex set of free energy functionals with a minimum and a maximum element. Various functionals have been shown to have the properties of a free energy for materials with particular types of relaxation behaviour. Also, over the last decade or more, forms have been given for the minimum and related free energies. These are all quadratic functionals which yield linear memory terms in the constitutive equations for the stress. A difficulty in constructing free energy functionals arises in making choices that ensure a non-negative quadratic form both for the free energy and for the rate of dissipation. We propose a technique which renders this task more straightforward. Instead of constructing the free energy and determining from this the rate of dissipation, which may not have the required non-negativity, the procedure is reversed, which guarantees a satisfactory free energy functional. Certain results for quadratic functionals in the time and frequency domains are derived, providing a platform for this alternative approach, which produces new free energies, including a family of functionals that are generalizations of the minimum and related free energies. 2010 Mathematics Subject Classification. 80A17, 74A15, 74D05. Key words and phrases. Thermodynamics, memory effects, new free energy functionals, rate of dissipation, frequency domain representations, causality. 1 We consider for definiteness here isothermal mechanical problems, indeed those for solid viscoelastic materials. Also, only the scalar case is considered, which simplifies the algebra and allows us to focus on the essential structure of the arguments. It must be emphasized however that similar results can be given with little extra difficulty, for viscoelastic fluids, non-isothermal problems, electromagnetism, non-simple materials etc. as presented in the references noted above, and indeed for the general tensor cases relating to all these materials.

On thermodynamics and intrinsically equilibrated materials

Annali di Matematica Pura ed Applicata, Series 4, 1976

The existence and uniqueness o] ]ree energy ]unctions is demonstrated ]or a class of materials broad enoqtgh to contain as special cases those o] the theory o] ]inite elasticity, the theory o] hypo-elasticity, and the theory o] internal state variables ]or which the path o] evolution is invariant under resealings o] tinge. 1.-Introduction. (*) Entrata in Redazione il 24 giugno 1975.

Some Remarks on Materials with Memory: Heat Conduction and Viscoelasticity

Journal of Nonlinear Mathematical Physics, 2005

Materials with memory are here considered. The introduction of the dependence on time not only via the present, but also, via the past time represents a way, alternative to the introduction of possible non linearities, when the physical problem under investigation cannot be suitably described by any linear model. Specifically, the two different models of a rigid heat conductor, on one side, and of a viscoelastic body, on the other one, are analyzed. In them both, to evaluate the quantities of physical interest a key role is played by the past history of the material and, accordingly, the behaviour of such materials is characterized by suitable constitutive equations where Volterra type kernels appear. Specifically, in the heat conduction problem, the heat flux is related to the history of the temperature-gradient while, in isothermal viscoelasticity, the stress tensor is related to the strain history. Then, the notion of equivalence is considered to single out and associate together all those different thermal histories, or, in turn, strain histories, which produce the same work. The corresponding explicit expressions of the minimum free energy are compared. The introduction of a Volterra type integral to model a rigid heat conductor in such a way to avoid the infinte speed heat propagation goes back to Cattaneo [3], who suggested a new generalized Fourier's law which linearly relates the heat flux, its time derivative and the temperature-gradient. Subsequently, Coleman's results [4] concerning materials with memory, induced Gurtin and Pipkin [20], to proposed a non-linear model, as well as a linearized version of it, which generalizes the previous ones. Among the many results, since then, in the study of heat transfer phenomena, those more directly of interest in connection to the present approach have been obtained by Gurtin [19] and by Coleman and Dill [5]. They studied properties of free energy functionals in the case of materials with memory and, subsequently, Giorgi and Gentili [16] investigated the heat conduction problem in a fading memory material. The thermodynamical model here adopted is the same studied by Fabrizio, Gentili and Reynolds [11], who considered the thermodynamics of a rigid homogeneous linear heat conductor with

General dissipative materials for simple histories

Quarterly of Applied Mathematics

We consider for definiteness here isothermal mechanical problems, indeed those for solid viscoelastic materials. Also, only the scalar case is considered, which simplifies the algebra and allows us to focus on the essential structure of the arguments. It must be emphasized, however, that similar results can be given with little extra difficulty, for viscoelastic fluids, certain non-isothermal problems, electromagnetism, non-simple materials, etc., as presented in the references noted above, and also for the general

Internal dissipation, relaxation property, and free energy in materials with fading memory

Journal of Elasticity, 1995

This paper examines some features of the standard theory of materials with fading memory and emphasizes that the commonly-accepted notion of dissipation yields unexpected consequences. First, application of the Clausius-Duhem inequality to linear viscoelasticity shows that there is a free energy functional such that the so-called internal dissipation vanishes in spite of the dissipative character of the model. Second, upon the choice of a suitable function norm, the relaxation property is proved not to hold for viscoelastic solids. Finally, the particular case is considered when the relaxation function is a superposition of exponentials. Different descriptions of state are then possible which prove to be inequivalent as far as the free energy is concerned.

Free energies for materials with memory in terms of state functionals

Meccanica

The aim of this work is to determine what free 8 energy functionals are expressible as quadratic forms of 9 the state functional I t which is discussed in earlier 10 papers. The single integral form is shown to include 11 the functional w F proposed a few years ago, and also a 12 further category of functionals which are easily 13 described but more complicated to construct. These 14 latter examples exist only for certain types of materials. 15 The double integral case is examined in detail, against 16 the background of a new systematic approach developed 17 recently for double integral quadratic forms in terms of 18 strain history, which was used to uncover new free 19 energy functionals. However, while, in principle, the 20 same method should apply to free energies which can be 21 given by quadratic forms in terms of I t , it emerges that 22 this requirement is very restrictive; indeed, only the 23 minimum free energy can be expressed in such a manner. 24 Keywords Thermodynamics Á Memory effects 25 Á Free energy functional Á Minimal state 26 functional Á Rate of dissipation 27 28 29 1 Introduction 30 Free energy functionals that are expressible as 31 quadratic forms of the state functional I t are explored in the present work. The quantity I t is discussed in [1, 6, 7] and elsewhere. Such free energies have applications in proving results concerning the integro-partial differential equations describing materials with memory. They may also be useful for physical modeling of such materials. However, these applications generally require that the free energy functionals involved have compact, explicit analytic representation. The single integral form is shown to include the functional w F , proposed some years ago [1, 6]. There is also however a further category of functionals of this kind for materials with non-singleton minimal states. These functionals are easily described but more difficult to construct, since basic inequalities relating to thermodynamics must be explicitly imposed; they are therefore not so useful for practical applications. The double integral quadratic form is examined in detail. In this context, a recent paper [10] deals with determining new free energies that are quadratic functionals of the history of strain, using a novel approach. This new method is based on a result showing that if a suitable kernel for the rate of dissipation is known, the associated free energy kernel can be determined by a straightforward formula, yielding a non-negative quadratic form. It allows us to determine previously unknown free energy functionals by hypothesizing rates of dissipation that are non-negative, and applying the formula. In particular, new free energy functionals related to the minimum free energy are constructed. In principle, the methods developed in [10] apply to quadratic forms in terms of I t , and should lead to new

Consequences of non-uniqueness in the free energy of materials with memory

International Journal of Engineering Science, 2001

It is well known that for many materials with memory, the free energy and entropy are not uniquely de®ned. Such arbitrariness cannot extend to directly measurable quantities. The consequences of this requirement are explored in the work reported here both in a general framework and within the context of a quadratic model of free energy. In the latter case, it is shown that a quantity, which reduces in the linear isothermal case to the time-dependent part of the relaxation function, must be proportional to the absolute temperature.