Thermodynamics of Materials with Memory: Theory and Applications

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
			17.3.2 A Double Integral Model
			17.3.3 The Work Function
		17.4 Time Domain Representation of Free Energies in Terms of the Kernel K(·,·)
			17.4.1 Some Examples
		17.5 Frequency Domain Representations of Free Energies in Terms of the Kernel K+-(·,·)
			17.5.1 Example: Discrete-Spectrum Materials
			17.5.2 Non-uniqueness of the Kernels
		17.6 General Dissipative Materials for Specified Histories
			17.6.1 Free Energy and Dissipation Functionals for Particular Histories
				17.6.1.1 Step Function Histories
				17.6.1.2 SSE Histories
				17.6.1.3 Purely Sinusoidal Histories
				17.6.1.4 Exponential Histories
		17.7 Product Formulae in the Time and Frequency Domains
			17.7.1 The Time Domain
				17.7.1.1 New Category of Free Energies: Time Domain
			17.7.2 The Frequency Domain
				17.7.2.1 New Category of Free Energies: Frequency Domain
		17.8 Approximation of a Discrete-Spectrum Material bya Day Functional
		17.9 Single-Integral Free Energies in Terms of It Derivatives
	18 Minimal States and Periodic Histories
		18.1 A New Linear Condition for Determining If a Free Energy Is a FMS
			18.1.1 Some Examples of Application of the New Condition
			18.1.2 Corresponding Frequency Domain Results
			18.1.3 Application of Product Formulae in the Time and Frequency Domains
		18.2 Free Energies for Singleton Minimal States
			18.2.1 Approximating Continuous-Spectrum Behavior by Discrete-Spectrum Formulae
			18.2.2 The Minimum Free Energy for Continuous-Spectrum Materials
			18.2.3 Proposed Method for Approximating Free Energies for Materials with Singleton Minimal States
			18.2.4 Free Energy Functionals for Sinusoidal/Exponential Histories Which Vanish for t < 0
			18.2.5 Numerical Results Relevant to the Method for Approximating Continuous-Spectrum Materials
	19 Second-Order Approximation for Heat Conduction: Dissipation Principle and Free Energies
		19.1 Introduction
		19.2 A Fading Memory Constitutive Equation and the Second Law
			Second Law of Thermodynamics
		19.3 Fundamental Relations
		19.4 The Minimum Free Energy for Second-Order Heat Conduction
		19.5 Free Energies Related to the Minimum Free Energy
	20 Free Energies for Nonlinear Materials with Memory
		20.1 Introduction
		20.2 A Generalized Quadratic Model
		20.3 Dissipation
		20.4 General Form of a Free Energy for Nonlinear Materials
			20.4.1 Generalizing Specified Linear Memory Free Energies
		20.5 Constitutive Relations
	21 Free Energies for Nonlocal Materials
		21.1 Second-Gradient Thermoviscoelastic Fluids
			21.1.1 The Nonlocal Graffi–Volterra Free Energy for Thermoviscoelastic Fluids
			21.1.2 A Single-Integral Free Energy in Terms of the Minimal State
			21.1.3 The Nonlocal Minimum Free Energy for Thermoviscoelastic Fluids
		21.2 Heat Flux in a Rigid Conductor with Nonlocal Behavior
			21.2.1 The Graffi–Volterra Free Energy for Nonlocal Rigid Conductors
			21.2.2 A Nonlocal Free Energy in Terms of the Minimal State
		21.3 Free Energies in a General Nonlocal Theory of a Material with Memory
			21.3.1 Derivation of the Field Equations
			21.3.2 A Nonlocal Quadratic Model for Free Energies
	22 The Minimum and Related Free Energies for Dielectric Materials
		22.1 Introduction and General Relations
		22.2 A Linear Memory Model for Dielectric Materials
			22.2.1 The Kernel L(u) for Dielectric Materials
			22.2.2 The Work Function for Dielectric Materials
		22.3 The Minimum Free Energy for Dielectric Materials
		22.4 Free Energies for Non-magnetic Materials
			22.4.1 The Kernel G(u) for Non-magnetic Materials
			22.4.2 Factorization of H(w) for Non-magnetic Dielectrics
			22.4.3 The Free Energy for the Non-magnetic Case Associated with a Particular Factorization
			22.4.4 A Detailed Dielectric Model
	23 Fractional Derivative Models of Materials with Memory
		23.1 Introduction
		23.2 Fractional Derivatives
			23.2.1 The Caputo Fractional Derivative
			23.2.2 Fractional Derivatives Without Singular Kernels
				23.2.2.1 A New Fractional Time Derivative
				23.2.2.2 Some Results for Given Histories
				23.2.2.3 The Laplace Transform of the NFDt
				23.2.2.4 Fractional Gradient Operator
				23.2.2.5 Fourier Transform of the Fractional Gradient and Divergence
				23.2.2.6 Fractional Laplacian
				23.2.2.7 Memory Operators
		23.3 The Fractional Derivative Memory Model
			23.3.1 Power Laws and Fractional Derivatives
		23.4 Thermodynamical Constraints and Free Energies
			23.4.1 The Graffi–Volterra Free Energy
		23.5 Frequency-Domain Quantities for Scalar Fractional Derivative Materials
			23.5.1 Complex Modulus for the Fractional Derivative Model
			23.5.2 The Work Function for Fractional Derivative Materials
		23.6 The Minimum Free Energy for Fractional Derivative Models
			23.6.1 General Form of the Minimum Free Energy
			23.6.2 The Minimum Free Energy for Simple Histories
				23.6.2.1 Sinusoidal Histories
				23.6.2.2 Exponential History
				23.6.2.3 The Physical Free Energy
		23.7 Application to Viscoelastic Systems
			23.7.1 Viscoelastic Solids
			23.7.2 Viscoelastic Fluids
		23.8 Application to Rigid Heat Conductors
			23.8.1 UFDt Fractional Cattaneo Equation
			23.8.2 The NFDt Model
		23.9 Application to Electromagnetic Systems
			23.9.1 Visco-Ferromagnetic Materials
			23.9.2 Nonlocal Visco-Ferromagnetic Materials
Part IV The Dynamical Equations for Materials with Memory
	24 Existence and Uniqueness
		24.1 Introduction to Existence and Uniqueness
		24.2 Dynamics of Viscoelastic Solids
			24.2.1 Existence and Uniqueness of Solutions
			24.2.2 Quasistatic Problem in Linear Viscoelasticity: Fichera's Problem
			24.2.3 Dynamical Problem in Linear Viscoelasticity
				24.2.3.1 Weak Solution in a Variational Sense
				24.2.3.2 Virtual Power Solution
				24.2.3.3 Existence and Uniqueness
				24.2.3.4 Transformed Problem
				24.2.3.5 Domain-of-Dependence Inequality
				24.2.3.6 Hyperbolicity
	25 Controllability of Thermoelastic Systems with Memory
		25.1 The Controllability Problem: Generalities and Types
		25.2 Exact Controllability Under an Assumption on the Smallness of k
		25.3 Exact Controllability with No Restriction on the Size of k
	26 The Saint-Venant Problem for Viscoelastic Materials
		26.1 Problem Description
		26.2 A Generalized Plane Strain State
		26.3 Analysis of the Saint-Venant Problem by Plane Cross-Section Solutions
		26.4 Primary Solution Class
		26.5 Secondary Solution Class
		26.6 Solution of the Relaxed Saint-Venant Problem
		26.7 The Saint-Venant Problem for an Isotropic and Homogeneous Cylinder
		26.8 The Saint-Venant Principle
	27 Exponential Decay
		27.1 Differential Problem with Nonconvex Kernels
			27.1.1 Transformed Problem and Some Useful Preliminaries
			27.1.2 The Resolvent of the Kernel
			27.1.3 Stability Results
	28 Semigroup Theory for Abstract Equations with Memory
		28.1 Introduction
			28.1.0.1 Notation
		28.2 The History Formulation
		28.3 The State Formulation
		28.4 The Semigroup in the Extended State Space
		28.5 The Original Equation Revisited
		28.6 Proper States
		28.7 State Versus History
	29 Identification Problems for Integrodifferential Equations
		29.1 Problem Specification
		29.2 Solving the First Identification Problem
		29.3 Solving the Second Identification Problem
		29.4 Solving the Third Identification Problem
	30 Dynamics of Viscoelastic Fluids
		30.1 Introduction
		30.2 An Initial Boundary Value Problem for an Incompressible Viscoelastic Fluid
			30.2.1 Transformed Problem
			30.2.2 Counterexamples to Asymptotic Stability
A Conventions and Some Properties of Vector Spaces
	A.1 Notation
	A.2 Finite-Dimensional Vector Spaces
		A.2.1 Positive Definite Tensors
		A.2.2 Differentiation with Respect to Vector Fields
		A.2.3 The Vector Space Sym
B Some Properties of Functions on the Complex Plane
	B.1 Introduction
		B.1.1 Cauchy's Theorem and Integral Formula
		B.1.2 Analytic Continuation
		B.1.3 Liouville's Theorem
		B.1.4 Singularities
		B.1.5 Branch Points
		B.1.6 Evaluation of Contour Integrals
	B.2 Cauchy Integrals
		B.2.1 Cauchy Integrals on the Real Line
C Fourier Transforms
	C.1 Definitions
	C.2 Fourier Transforms on the Complex Plane
		C.2.1 Laplace Transforms
		C.2.2 The Fourier Transform of Functions with Compact Support
		C.2.3 Functions that Do Not Belong to L1 L2
		C.2.4 The Form of f at Large Frequencies
		C.2.5 Expressions for f in Terms of fF
	C.3 Parseval's Formula and the Convolution Theorem
References
Index