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Über dieses Buch

This is a work in four parts, dealing with the mechanics and thermodynamics of materials with memory, including properties of the dynamical equations which describe their evolution in time under varying loads. The first part is an introduction to Continuum Mechanics with sections dealing with classical Fluid Mechanics and Elasticity, linear and non-linear. The second part is devoted to Continuum Thermodynamics, which is used to derive constitutive equations of materials with memory, including viscoelastic solids, fluids, heat conductors and some examples of non-simple materials. In part three, free energies for materials with linear memory constitutive relations are comprehensively explored. The new concept of a minimal state is also introduced. Formulae derived over the last decade for the minimum and related free energies are discussed in depth. Also, a new single integral free energy which is a functional of the minimal state is analyzed in detail. Finally, free energies for examples of non-simple materials are considered. In the final part, existence, uniqueness and stability results are presented for the integrodifferential equations describing the dynamical evolution of viscoelastic materials. A new approach to these topics, based on the use of minimal states rather than histories, is discussed in detail. There are also chapters on the controllability of thermoelastic systems with memory, the Saint-Venant problem for viscoelastic materials and on the theory of inverse problems.

Inhaltsverzeichnis

Frontmatter

Chapter 0. Introduction

Abstract
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.
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Continuum Mechanics and Classical Materials

Frontmatter

Chapter 1. Introduction to Continuum Mechanics

Abstract
In this initial chapter, we introduce various fundamentals: description of deformation, definition and interpretation of the strain and stress tensors, balance laws, and general restrictions on constitutive equations. These provide the foundation for later developments.
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Chapter 2. Materials with Constitutive Equations That Are Local in Time

Abstract
We now consider the constitutive equations relating to fluids and solids for which memory effects are negligible. This contrasts with subsequent chapters, which are devoted almost entirely to materials with memory. In fact, however, one example included in the discussion, namely viscous fluids, can be visualized as possessing very short-term memory, expressed by the presence of time derivatives of certain field quantities. This is consistent with the general correlation that will arise throughout the present work between memory effects and energy dissipation.
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Continuum Thermodynamics and Constitutive Equations

Frontmatter

Chapter 3. Principles of Thermodynamics

Abstract
In this chapter, we discuss various fundamental concepts and results in continuum thermodynamics. Some examples are given in terms of the materials discussed in Part I, generalized to a nonisothermal context.
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Chapter 4. Free Energies and the Dissipation Principle

Abstract
We present in this chapter an axiomatic formulation of thermodynamics in order to introduce free energies in a very general manner and to prove certain fundamental properties of these quantities. For most of the discussion, no underlying model is assumed, in contrast to the previous chapter. However, in the context of an equivalence relation between states, we ascribe a form to the work function consistent with the general nonisothermal theory introduced in Chapter 5.
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Chapter 5. Thermodynamics of Materials with Memory

Abstract
We now apply thermodynamic principles to field theories with memory. For general nonlinear, nonisothermal theories, we assume that a free energy is given, this being the fundamental constitutive assumption. Applying a generalization of the approach of Coleman [36], Coleman and Mizel [38], Gurtin and Pipkin [130], we derive the constitutive equations for the theory in Section 5.1. Also, fundamental properties of free energies are derived. Furthermore, some observations are made on the case of periodic histories and in relation to constraints on the nonuniqueness of free energies. In Section 5.2, an expression for the maximum recoverable work is given for general materials, together with an integral equation for the process yielding this maximum. Finally, in Section 5.3, we discuss how free energies can be constructed from combinations of simpler free energies.
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Free Energies for Materials with Linear Memory

Frontmatter

Chapter 6. A Linear Memory Model

Abstract
We now address the problem of finding explicit forms for the free energy of materials with constitutive relations given by linear memory functionals. Such materials are referred to in this work as linear memory materials. As we will see, the equilibrium (or alternatively, the instantaneous) contribution, which is to say the portion of the constitutive equation without memory effects, need not be linear. If the part of the constitutive equation without memory is also linear, we use the description a completely linear material. A linear viscoelastic material is understood to be completely linear, while a viscoelastic material with linear memory need not have this property.
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Chapter 7. Viscoelastic Solids and Fluids

Abstract
We now consider special cases of the constitutive relations (6.1.15), namely linear viscoelastic solids and fluids with linear memory under isothermal conditions in the present chapter and an approximate version of rigid heat conductors in Chapter 8. Some of the formulas are similar to those derived in the general case, and detailed proofs are omitted or a different version is given. Other formulas are specific to completely linear materials, for example.
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Chapter 8. Heat Conductors

Abstract
To remove the paradox of classical Fourier theory relating to the instantaneous propagation of thermal disturbances, Cattaneo [29] suggested a generalized Fourier law, which he justified by means of statistical considerations. This constitutive equation relates the heat flux, its time derivative, and the temperature gradient. It is referred to as the Cattaneo–Maxwell relation, since Maxwell [170] previously obtained it but immediately eliminated the term involving the time derivative of the heat flux. It leads to a hyperbolic heat equation.
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Chapter 9. Free Energies on Special Classes of Material

Abstract
We present in this chapter functionals that are free energies, provided certain assumptions on the relaxation function are valid. In the first section, the general nonisothermal model introduced in Chapter 6 is considered, while in subsequent sections, these functionals are discussed for materials introduced in Chapters 7, 8.
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Chapter 10. The Minimum Free Energy

Abstract
Breuer and Onat [24] considered the following question: what is the maximum amount of work recoverable from a body that has undergone a specified strain history? They found that the answer for linear viscoelastic memory materials is provided by the solution of an integral equation of Wiener–Hopf type, which is in fact a special case of the result given in Section 5.2. They gave a detailed solution by elementary means for a material with relaxation function in the form of a finite sum of decaying exponentials. The nonuniqueness problem was also explicitly exposed by these authors [25].
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Chapter 11. Representation of the Minimum Free Energy in the Time Domain

Abstract
Consider formula (10.2.17) for the minimum free energy.We wish to derive an expression for the integral term involving time-domain quantities. The method applies also to the family of free energies derived in Chapter 15.
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Chapter 12. Minimum Free Energy for Viscoelastic Solids, Fluids, and Heat Conductors

Abstract
We now develop formulas for the minimum free energy and related quantities applicable to three categories of linear materials (completely linear for solids and heat conductors). The methods differ in detail from those in Chapter 10, though they are equivalent. As in Chapters 7 and 8, we make more use of the abstract terminology and notation introduced in Chapters 3, 4.
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Chapter 13. The Minimum Free Energy for a Continuous-Spectrum Material

Abstract
We now examine how the formulas emerging from the methodology developed in Chapter 10 apply to materials other than those exhibiting a discrete-spectrum response, in particular for materials with a branch-cut-type singularity. We confine our considerations to the case that the cut is on the imaginary axis. Such a material is said to have a continuous-spectrum response, i.e., thosematerials for which the relaxation function is given by an integral of a density function multiplying a strictly decaying exponential. The results reported in this chapter were first presented in [61].
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Chapter 14. The Minimum Free Energy for a Finite-Memory Material

Abstract
In this chapter, based on work reported in [74], we derive an expression for the minimum free energy corresponding to a relaxation function with the special property that its derivative is nonzero over only a finite interval of time. It will be seen that there are special features associated with the analytic behavior of the frequencyspace representation of such relaxation functions that render this a nontrivial extension, with unique features, of the general treatments presented in Chapters 6, 13. This property of finite memory is of interest in the first instance because finite and infinite memories are not necessarily experimentally distinguishable; also, the assumption of infinite memory can lead to paradoxical results for certain problems.
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Chapter 15. A Family of Free Energies

Abstract
We now focus mainly on the case ofmaterials characterized by memory kernels with only isolated singularities and derive explicit expressions for a family of free energies, including the minimum free energy discussed in Chapter 10 and the maximum free energy, which, in this context, will turn out to be less than the work function. All of these will be shown to be functionals of the minimal state.
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Chapter 16. Properties and Explicit Forms of Free Energies for the Case of Isolated Singularities

Abstract
A formalism was developed [73] for the scalar case that allows expressions for a family of free energies related to a particularminimal state to be derived for discretespectrum models, including minimum and maximum free energies. Generalization of this work to the full tensor, nonisothermal case was presented in [66]. A generalization of the formalism in [73] has been used more recently [107, 108] to propose a closed formula for the physical free energy and rate of dissipation.
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Chapter 17. Free Energies for Nonlocal Materials

Abstract
When nonlocal materials are considered, the classical laws of thermodynamics must be modified by expressing these laws in terms of internal powers or by introducing directly into them suitable extra fluxes, characteristic of the material under consideration [80], as discussed in some detail in Section 3.7. The first formulation, in terms of internal powers, is more general then the second one, since it is defined a priori by means of the constitutive equations, by taking into account the power balance laws; in the second method there is the problem of introducing a posteriori the vector fluxes in order that compatibility with the laws of thermodynamics be satisfied.
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

The Dynamical Equations for Materials with Memory

Frontmatter

Chapter 18. Existence and Uniqueness

Abstract
The study of differential problems related to materials with fading memory began with the work of Graffi [111, 112]. Later on, these studies were considered by many authors, and in particular, a new important description of such phenomena was given by Dafermos in [47, 46], using semigroup theory, where besides existence and uniqueness of the solution, the interesting problem of asymptotic stability was also examined.
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Chapter 19. Controllability of Thermoelastic Systems with Memory

Abstract
The evolution of any material system is described by means of partial differential equations.With a suitable choice of controls, which may be source terms or boundary conditions, we can act on a given state of the material.
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Chapter 20. The Saint-Venant Problem for Viscoelastic Materials

Abstract
The Saint-Venant problem was analyzed in Section 2.4.2 for linear elastic materials. The same problem has been studied also for viscoelastic materials (see, for example, [33, 60]). The theory is developed here in a similar general and systematic manner.
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Chapter 21. Exponential Decay

Abstract
The differential problem of the dynamics of viscoelastic solids, defined for example by the system (18.2.33), can be represented in an abstract form by means of the following integrodifferential equation.
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Chapter 22. Semigroup Theory for Abstract Equations with Memory

Abstract
We consider in this chapter, in a mathematically abstract way, the evolution equations of the kind discussed in Chapters 18 and 21, comparing the new state formulation with the traditional history approach. Relevant background to this discussion is the concept of a minimal state discussed in Part III from Section 6.4 onward, in particular, certain conclusions of Section 15.2.
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Chapter 23. Identification Problems for Integrodifferential Equations

Abstract
This chapter is devoted to outlining some ideas involving that part of the theory of inverse problems that is usually referred to as the identification of parameters (numbers, vectors, matrices, functions) appearing in integrodifferential equations describing the evolution of fading memory materials. We recall the celebrated definition by Hadamard of a well-posed problem: it requires the existence and uniqueness of the solution to the problem and its continuous dependence on data.
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Chapter 24. Dynamics of Viscoelastic Fluids

Abstract
An evolution problem in a bounded domain for viscoelastic fluids of the kind considered in Chapters 7, 9, and 12 is now presented. Our attention is confined to infinitesimal viscoelasticity for isotropic, homogeneous, and incompressible fluids.
Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Backmatter

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