Thermodynamics of materials with memory : theory and applications / Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden.

Format:

Book

Author/Creator:

Amendola, Giovambattista, author.

Fabrizio, Mauro, 1940- author.

Golden, J. M. (John M.), 1945- author.

Language:

English

Subjects (All):

Thermodynamics--Mathematical models.

Thermodynamics.

Physical Description:

1 online resource (756 pages)

Edition:

Second edition.

Place of Publication:

Cham, Switzerland : Springer, [2021]

Summary:

This monograph deals with the mechanics and thermodynamics of materials with memory, including properties of the dynamical equations that describe their evolution in time under varying loads. A work in four parts, the first is an introduction to continuum mechanics, including classical fluid mechanics, linear and non-linear elasticity. The second part considers continuum thermodynamics and its use to derive constitutive equations of materials with memory, including viscoelastic solids, fluids, heat conductors and some examples of non-simple materials. In the third part, free energies for materials with linear memory constitutive relations are discussed. The concept of a minimal state is introduced. Explicit formulae are presented for the minimum and related free energies. The final part deals with existence, uniqueness, and stability results for the integrodifferential equations describing the dynamical evolution of viscoelastic materials, including a new approach based on minimal states rather than histories. 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. The second edition includes a new chapter on thermoelectromagnetism as well as recent findings on minimal states and free energies. It considers the case of minimum free energies for non-simple materials and dielectrics, together with an introduction to fractional derivative models.

Contents:

Intro

Preface to Second Edition

Preface to First Edition

Contents

Introduction

Part I Continuum Mechanics and Classical Materials

1 Introduction to Continuum Mechanics

1.1 Introduction

1.2 Kinematics

1.2.1 Continuous Bodies: Deformations-Strain Tensors

1.2.2 Small Deformations: The Saint-Venant Compatibility Conditions

1.2.3 Transformation of Areas and Volumes: Transport Theorems

1.3 Principles of Continuum Mechanics

1.3.1 Principle of Conservation of Mass

1.3.2 Momentum Balance Principles

1.3.3 Consequences of Momentum Balance Laws

1.3.4 The Piola-Kirchhoff Stresses

1.4 Constitutive Equations

1.4.1 Objectivity

1.4.2 Principle of Material Objectivity

1.4.3 Fading Memory

2 Materials with Constitutive Equations That Are Local in Time

2.1 Introduction

2.2 Fluids: Ideal Fluids

2.2.1 Elastic Fluids

2.2.2 Newtonian Fluids: The Navier-Stokes Equations

2.2.3 Uniqueness of Solutions

2.3 Elastic Solids

2.3.1 Finite Elasticity

2.3.2 Hyperelastic Bodies

2.4 Linear Elasticity

2.4.1 Linear Elastostatics

2.4.2 Saint-Venant's Problem

Part II Continuum Thermodynamics and Constitutive Equations of Mechanics and Electromagnetism

3 Principles of Thermodynamics

3.1 Heat Equation

3.2 Definition of a Material as a Dynamical System

3.3 First Principle of Thermodynamics

3.4 Second Principle of Thermodynamics

3.4.1 The Absolute Temperature Scale

3.4.2 Entropy Action

3.5 Applications to Elastic Bodies

3.6 Thermodynamic Restrictions for Viscous Fluids

3.7 Principles of Thermodynamics for Nonsimple Materials

3.7.1 First Law of Thermodynamics

3.7.2 Second Law of Thermodynamics

4 Free Energies and the Dissipation Principle

4.1 Axiomatic Formulation of Thermodynamics

4.2 Minimum and Maximum Free Energies.

5 Thermodynamics of Materials with Memory

5.1 Derivation of the Constitutive Equations

5.1.1 Required Properties of a Free Energy

5.1.2 Periodic Histories for General Materials

5.1.3 Constraints on the Nonuniqueness of the Free Energy

5.2 The Maximum Recoverable Work for General Materials

5.3 Generation of New Free Energies

6 Thermoelectromagnetism of Continuous Media

6.1 Electromagnetism of Continuous Media

6.1.1 Balance Laws in Electromagnetic Media

6.1.2 Constitutive Equations

6.1.3 Boundary Conditions

6.1.4 Balance of Energy and the First Law of Thermodynamics

6.1.5 Second Law of Thermodynamics and the Clausius-Duhem Inequality

6.1.6 Thermodynamics of Nonlocal Materials

6.1.7 Two Potentials Related to the Electromagnetic Fields

6.2 Electromagnetic Systems with Memory

6.2.1 Memory Effects Justified by Waves in Water

6.2.2 Some Simple Models to Study Material Behavior

6.2.2.1 Dielectrics

6.2.2.2 Magnetic Materials

6.2.2.3 Metals

6.2.2.4 The Ionosphere

6.2.3 The Clausius-Duhem Inequality and Its Consequences

6.3 Thermodynamics of Simple Electromagnetic Materials

6.3.1 Electromagnetic Materials

6.3.2 Materials with Fading Memory

6.3.2.1 Dielectrics with Memory

6.3.2.2 Conductors with Memory

6.3.3 Thermodynamic Laws in Terms of Cycles

Part III Free Energies for Materials with Linear Memory

7 A Linear Memory Model

7.1 A Quadratic Model for Free Energies

7.1.1 Constitutive Relations

7.1.2 Dissipation Rate

7.1.3 Complete Material Characterization

7.1.4 Linear Equilibrium Response

7.1.5 Time-Independent Eigenspaces

7.1.6 Short-Term Memory

7.2 Constitutive Equations in the Frequency Domain

7.2.1 Sinusoidal Histories for the General Theory

7.2.2 Properties of L'

7.2.3 Frequency-Domain Representation of the History.

7.2.4 Constitutive Equations in Terms of Frequency-Domain Quantities

7.3 The Form of the Generalized Relaxation Function

7.3.1 Isolated Singularities

7.3.2 Branch Cuts

7.3.3 Essential Singularities

7.4 Minimal States in the Nonisothermal Case

7.5 Forms of the Work Function

8 Viscoelastic Solids and Fluids

8.1 Linear Viscoelastic Solids

8.1.1 Thermodynamic Restrictions for Viscoelastic Solids

8.2 Decomposition of Stress

8.3 Equivalence and Minimal States

8.4 State and History for Exponential-Type Relaxation Functions

8.5 Inversion of Constitutive Relations

8.6 Linear Viscoelastic Free Energies as Quadratic Functionals

8.6.1 General Forms of a Free Energy in Terms of Stress

8.6.2 The Work Function as a Free Energy

8.7 The Relaxation Property and a Work Function Norm

8.8 Viscoelastic Fluids

8.9 Compressible Viscoelastic Fluids

8.9.1 A Particular Class of Compressible Fluids

8.9.2 Representation of Free Energies for Compressible Fluids

8.9.3 Thermodynamic Restrictions for Compressible Fluids

8.10 Incompressible Viscoelastic Fluids

8.10.1 Thermodynamic Restrictions for Incompressible Viscoelastic Fluids

8.10.2 The Mechanical Work

8.10.3 Maximum Free Energy for Incompressible Fluids

9 Heat Conductors

9.1 Constitutive Equations for Rigid Heat Conductors

9.1.1 States in Terms of t(s) and gt

9.1.2 Constitutive Equations in Terms of States and Processes

9.1.3 Equivalent Histories and Minimal States

9.2 Thermodynamic Constraints for Rigid Heat Conductors

9.3 Thermal Work

9.3.1 Integrated Histories for Isotropic Heat Conductors

9.3.2 Finite Work Processes and w-Equivalence for States

9.3.3 Free Energies as Quadratic Functionals for Rigid Heat Conductors

9.3.4 The Work Function

10 Free Energies on Special Classes of Material.

10.1 The General Nonisothermal Case

10.1.1 The Graffi-Volterra Free Energy

10.1.2 Dill/Staverman-Schwarzl Free Energy

10.1.3 Single-Integral Quadratic Functionals of It

10.2 Free Energies for Restricted Classes of Solids

10.3 Free Energies for Restricted Classes of Fluids

10.4 Free Pseudoenergies for Restricted Classes of RigidHeat Conductors

11 The Minimum Free Energy

11.1 Factorization of Positive Definite Tensors

11.1.1 The Scalar Case

11.2 Derivation of the Form of the Minimum Free Energy

11.2.1 A Variational Approach

11.2.2 The Wiener-Hopf Method

11.2.3 Histories Rather Than Relative Histories

11.2.4 Confirmation That ψm Is a Free Energy

11.2.5 Double Frequency Integral Form

11.3 Characterization of the Minimal State in the Frequency Domain

11.4 The Space of States and Processes

11.5 Limiting Properties of the Optimal Future Continuation

11.6 Time-Independent Eigenspaces

11.7 The Minimum Free Energy for Sinusoidal Histories

11.8 Example: Viscoelastic Materials

11.9 Explicit Forms of the Minimum Free Energy for Discrete-Spectrum Materials

12 Representation of the Minimum Free Energy in the Time Domain

12.1 The Minimum Free Energy in Terms of Time-Domain Relative Histories

12.2 The Minimum Free Energy Expressed in Terms of It

13 Minimum Free Energy for Viscoelastic Solids, Fluids, and Heat Conductors

13.1 Maximum Recoverable Work for Solids

13.1.1 Minimum Free Energy for Solids

13.1.2 Minimum Free Energies in Terms of Stress History

13.2 Maximum Recoverable Work for Fluids

13.2.1 The Minimum Free Energy for Fluids

13.3 The Minimum Free Energy for Incompressible Fluids

13.3.1 The Minimum Free Energy in Terms of It

13.4 The Maximum Recoverable Work for Heat Conductors

13.4.1 The Minimum Free Energy for Heat Conductors.

13.4.2 The Discrete-Spectrum Model for Heat Conductors

14 The Minimum Free Energy for a Continuous-Spectrum Material

14.1 Introduction

14.2 Continuous-Spectrum Materials

14.3 Factorization of H for a Continuous-Spectrum Material

14.3.1 Properties of the Factorization Formulas

14.4 The Minimum Free Energy

14.5 An Alternative Approach

14.6 Minimal States

15 The Minimum Free Energy for a Finite-Memory Material

15.1 Introduction

15.2 Finite Memory

15.3 The History Dependence of the Minimum Free Energy

15.4 Factorization of H(ω)

15.5 Explicit Forms of the Minimum Free Energy

16 Free Energies for the Case of Isolated Singularities

16.1 Constitutive Relations, Histories, and Free Energy Properties for the Scalar Case

16.1.1 Frequency-Domain Quantities for the Scalar Case

16.1.2 Defining Properties of Free Energies

16.2 Materials with Only Isolated Singularities

16.3 Free Energies as Discrete Quadratic Forms

16.3.1 Discrete-Spectrum Materials

16.4 The Minimum and Related Free Energies

16.5 Equivalent States and the Maximum Free Energy

16.5.1 Minimal States

16.5.1.1 Explicit Examples of Minimal States

16.5.1.2 The Maximum Free Energy

16.6 Scalar Product Notation for ψf and Related Quantities as Quadratic Functionals

16.6.1 Confirmation That ψf Is a Free Energy

16.7 Asymptotic Behavior and Discontinuities

16.8 Partial Orderings of the ψf

16.9 Explicit Forms for ψf

16.9.1 Explicit Forms of the Minimum and Related Free Energies for Discrete-Spectrum Materials

16.10 The Central Free Energy and Related Dissipation

16.11 Plots of Free Energies

17 Constructing Free Energies for Materials with Memory

17.1 Two Equivalent Interpretations of the Set of Free Energies

17.2 Unique Characterization of Materials with Memory

17.3 Quadratic Models for Free Energies.

17.3.1 A Single-Integral Model.

Notes:

Description based on print version record.

ISBN:

3-030-80534-4

OCLC:

1281681251