Mathematical Modelling of Continuum Physics

Preface
Contents
Part I Basic Principles and Balance Equations
1 Kinematics
1.1 Frames of Reference and Configurations
1.2 Deformation
1.2.1 Effects on Lengths, Areas, and Volumes
1.2.2 Some Identities Involving the Deformation Gradient
1.3 Motion
1.3.1 Stretching and Spin Tensors
1.4 Rotations and Angular Velocities
1.4.1 Active Rotations
1.5 Transport Relations for Convecting Sets
1.5.1 Identities About the Total Time Derivative of Spatial Gradients
1.5.2 Transport Relations for Vorticity and Velocity Gradient
1.6 Transport Relations for Non-Convecting Sets
1.6.1 Transport Theorems for Discontinuous Fields
1.7 Kinematics of Singular Surfaces
1.7.1 Geometrical Conditions of Compatibility
1.7.2 Kinematical Conditions of Compatibility
1.7.3 Jumps at Acoustic Waves
1.8 Transport Theorems for Surface Integrals
1.8.1 Transport Theorems for Discontinuous Fields
1.9 Objectivity
1.9.1 Transformation Rules for Kinematic Fields
1.9.2 Time Derivatives Relative to Generic Frames of Reference
1.10 Objective Time Derivatives
1.10.1 General Form of Objective Derivatives
1.10.2 Rivlin–Eriksen Tensors
1.10.3 Spins and Angular Velocities Under Euclidean Transformations
2 Balance Equations
2.1 Balance of Mass
2.2 Balance of Linear Momentum
2.3 Balance of Angular Momentum
2.4 Balance of Energy
2.4.1 First Law of Thermodynamics
2.4.2 Balance of Virtual Power
2.5 Material Forms of the Balance Equations
2.5.1 Equation of Motion in Pre-Stressed Materials
2.5.2 Lie Derivative of the Cauchy Stress
2.5.3 Eshelby Stress and Linear-Angular Momentum
2.6 Balance of Entropy
2.6.1 Material Formulation of the Second Law
2.6.2 Third Law of Thermodynamics
2.6.3 Exergy
2.6.4 Entropy Production in Stochastic Kinetics
2.7 Second Law and Phase-Field Models
2.8 Bernoulli's Law and Balance Equations for Fluids
2.8.1 Bernoulli's Law
2.8.2 Variational Derivation of the Equation of Motion
2.9 Balance in a Control Volume
2.10 Basic Principles in Electromagnetism
2.10.1 Balance Equations in Free Space
2.10.2 Balance Laws in Matter
2.11 The Invariant Form of Maxwell's Equations
2.12 Maxwell's Equations for the Fields at the Rest Frame
2.13 Material Description of Electromagnetic Fields
2.14 Formulations of Electromagnetism in Matter
2.15 Poynting's Theorem
2.16 Balance Equations in Electromagnetism
2.16.1 Forces, Torques, and Energy Supply
2.16.2 Angular Momentum and Magnetic Moment
2.16.3 Maxwell Stress Tensor
2.16.4 Balance of Entropy
2.17 Conservation Laws Across a Singular Surface
2.17.1 Imbalance Laws Across a Singular Surface
2.17.2 Jump Conditions
2.17.3 Rankine–Hugoniot Condition
2.17.4 Interaction Between Discontinuities of Different Order
2.17.5 Eigenvector Form of Weak Discontinuities
2.17.6 Jump Conditions of Integrals
2.17.7 Jump Conditions in the Reference Configuration
2.18 Balance Laws for Discontinuous Electromagnetic Fields
2.18.1 Boundary Conditions
Part II Constitutive Models of Simple Materials
3 Generalities on Constitutive Models
3.1 Constitutive Equations
3.2 Objectivity
3.2.1 Dependence on the Velocity Gradient
3.2.2 Dependence on the Deformation Gradient
3.2.3 Thermoelastic Variables
3.2.4 Thermo-Viscous Variables
3.2.5 Dependence on Histories
3.2.6 Variables in Rate-Type Equations
3.2.7 Objectivity and Invariants
3.3 Objectivity and Euclidean Invariants in Electromagnetism
3.4 Consistency with the Second Law
3.4.1 Exploitation of the Second Law
3.4.2 Other Formulations of the Second Law
3.4.3 Remarks on Onsager's Reciprocal Relations
3.5 Entropy Equation and Gibbs Equations
3.6 Second Law and Representation of Constitutive Functions
4 Solids
4.1 Thermoelastic Solids
4.1.1 Linear Theory
4.1.2 Free Energy and Invariants
4.2 Elastic Solids
4.2.1 Material Symmetries
4.2.2 Elastic Waves
4.2.3 Linear Theory
4.2.4 Elastostatics and Compatibility of Strains
4.2.5 Vibrating Strings, Chains, and Phonons
4.3 Internal Constraints
4.4 Hyperelastic Solids and Rubber-Like Materials
4.4.1 Compressible Mooney-Rivlin Solid
4.5 Modelling of Dissipative Solids
4.5.1 A Model for Viscosity and Heat Conduction
4.6 Modelling via Dissipation Potentials
4.6.1 Convex Dissipation Potential
4.6.2 Model with a Scalar Internal Variable
4.6.3 Model with a Vector Internal Variable
5 Fluids
5.1 Elastic Fluids
5.1.1 Water Wave Theories
5.2 Thermoelastic Fluids
5.2.1 Gibbs Relations and Entropy Equation
5.2.2 Conjugate Variables and Maxwell's Relations
5.2.3 Specific Heats
5.3 Ideal Gas
5.3.1 Specific Heats and Entropy Functions
5.4 Models of Real Gases
5.4.1 Van der Waals Model
5.4.2 Peng-Robinson Model
5.5 Heat-Conducting, Viscous Fluids
5.5.1 Stokesian Fluids
5.5.2 Boundary Conditions
5.5.3 Incompressible Fluids
5.5.4 Oberbeck–Boussinesq Approximation
5.6 Newtonian Fluids
5.6.1 Models with Thermal Expansion and Pressure-Dependent Viscosities
5.6.2 Fluids with Pressure-Dependent Viscosities
5.6.3 Vorticity and Enstrophy Transport Equation
5.6.4 Viscosity and Energy Decay
5.6.5 Incompressible Newtonian Fluids
5.7 Generalized Newtonian Fluids
5.8 Viscoplastic and Viscoelastic Fluids
5.8.1 Viscoplastic Fluids
5.9 Models of Turbulence
5.9.1 Incompressible Fluids
5.9.2 Compressible Fluids
Part III Non-simple Materials
6 Rate-Type Models
6.1 Rheological Models
6.1.1 Rigid-Perfectly Plastic Model with Kinematic Hardening
6.1.2 Elastic-Perfectly Plastic Solid
6.1.3 Elastic-Plastic Model with Kinematical Hardening
6.1.4 Maxwell-Wiechert Fluid
6.1.5 Bingham Model
6.1.6 Bingham-Maxwell Model
6.1.7 Kelvin-Voigt Solid
6.1.8 Jeffreys' and Burgers' Fluids
6.1.9 Standard Linear Solid
6.1.10 Generalized Models
6.1.11 Relaxation Modulus and Creep Compliance
6.2 Rate-Type Models of Fluids
6.2.1 Rate Equation for Thermo-Viscous Fluids in the Eulerian Description
6.2.2 Invariant Fields and Objective Rates
6.2.3 Rate Equations in the Material Description
6.2.4 Nonlinear Rate-Type Models of Fluids
6.3 Rate-Type Models of Solids
6.3.1 The Kelvin-Voigt-Fourier Solid
6.3.2 Thermo-Viscoelastic Materials
6.3.3 Thermo-Viscoplastic Models
6.4 Higher-Order Rate Models
6.4.1 Burgers' Fluid
6.4.2 Oldroyd-B Fluid
6.4.3 White-Metzner Fluid
6.4.4 Wave Features in Higher-Order Rate Models
6.5 Wave Equation in Hereditary Fluids
6.6 Further Thermoelastic Models
6.6.1 Maxwell-Cattaneo Equation; a Generalized Form
6.6.2 Temperature-Rate Dependent Thermoelastic Materials
6.6.3 Thermoelasticity Based on an Integral Variable
7 Materials with Memory
7.1 Materials with Fading Memory
7.1.1 Fading Memory Space of Histories
7.1.2 Difference and Summed Histories
7.1.3 Properties About Histories and Norms
7.2 Thermoelastic Materials with Memory
7.3 Rigid Heat Conductors
7.3.1 Models Dependent on Thermal Histories
7.3.2 Models Dependent on Summed Thermal Histories
7.3.3 Moore–Gibson–Thompson Temperature Equation
7.4 Linear Viscoelasticity
7.4.1 Linear Viscoelastic Solids
7.4.2 Thermodynamic Restrictions for the Linear Viscoelastic Solid
7.4.3 Free Energies and Minimal States
7.4.4 Viscoelastic Solids with Unbounded Relaxation Functions
7.4.5 Nonlinear Viscoelastic Models
7.4.6 Some Examples
7.5 Viscous Fluids with Memory
7.5.1 Incompressible Viscoelastic Fluids
7.5.2 Compressible Viscoelastic Fluids
7.5.3 Maxwell-Like Viscoelastic Fluids
7.5.4 Acceleration Waves in Viscous Fluids with Memory
7.6 Electromagnetic Materials with Memory
7.6.1 Dielectrics with Memory
7.6.2 Conductors with Memory
7.6.3 Polarizable Conductors with Memory
7.6.4 Magneto-Viscoelasticity
7.6.5 Rate Equations in the Eulerian Description
7.6.6 Solids
7.7 Causality and Kramers–Kronig Relations
7.8 Memory Models via Fractional Derivatives
7.8.1 Preliminaries on Fractional Derivatives
7.8.2 Fractional Derivatives
7.8.3 Heat Conduction via Fractional Derivatives
7.8.4 Wave Propagation Properties
7.9 Viscoelastic Models of Fractional Order
8 Aging and Higher-Order Grade Materials
8.1 Aging
8.2 Aging of Thermoelastic Solids
8.3 Aging of Rate-Type Materials
8.3.1 Aging Properties
8.4 Aging of Thermo-Viscoelastic Materials
8.5 An Aging Model of Viscous Fluid with Memory
8.6 Damage
8.7 Fluids of Higher-Order Grade
8.7.1 Second Grade Fluid
8.7.2 Third Grade Fluid
8.8 Interaction Effects via Materials of Higher Order
8.9 Modelling via the Interstitial Working
8.9.1 Modelling via the Extra-Entropy Flux
8.10 Materials of Korteweg Type
8.11 Hyperstress and Materials of Higher-Order Grade
8.11.1 Solids of Higher-Order Grade
8.12 A New Scheme Associated with the Hyperstress
9 Mixtures
9.1 Kinematics
9.2 Balance Equations
9.2.1 Balance Equations in the Eulerian Description
9.2.2 Mass Density in the Reference Configuration and Incompressibility
9.3 Second Law of Thermodynamics
9.3.1 Principle of Phase Separation
9.3.2 Remarks on the Thermodynamic Restrictions
9.4 Balance Equations for the Whole Mixture
9.4.1 Stoichiometry Requirements on the Mass Growths
9.4.2 Balance Equation for the Diffusion Flux
9.5 Constitutive Models for Fluid Mixtures
9.5.1 Extent of Reaction and Law of Mass Action
9.5.2 Non-reacting Mixtures with Several Temperatures
9.5.3 Reacting Mixtures with a Single Temperature
9.5.4 Gibbs Equations
9.5.5 Second-Law Inequalities
9.6 Mixtures of Ideal Gases
9.6.1 Chemical Potentials
9.6.2 Mixing Entropy and Mixing Free Energy
9.7 Diffusion
9.7.1 Dynamic Diffusion Equation
9.7.2 Fourier-Like and Rate-Type Diffusion Equations
9.7.3 Maxwell–Stefan Diffusion Model
9.7.4 Coupled Effects Among Diffusion and Heat Conduction
9.7.5 Evolution Problems
9.8 Solid Mixtures
9.8.1 Constitutive Assumptions and Thermodynamic Restrictions
9.9 Immiscible Mixtures
9.10 Entropy Inequality and Models for the Whole Mixture
9.10.1 Gibbs Equations and Restrictions on the Diffusion Flux
9.10.2 Gibbs–Duhem Equation
10 Micropolar Media
10.1 Kinematics of Micropolar Media
10.1.1 Orientational Momentum
10.2 Balance Laws
10.2.1 Balance Laws in the Spatial Description
10.2.2 Balance Laws in the Material Description
10.2.3 Thermoviscous Micropolar Media
10.3 Liquid Crystals
10.3.1 Modelling of Thermotropic Liquid Crystals
10.3.2 Director Field, Energy, and Objectivity
10.4 Nematics
10.4.1 Relation to Other Models Involving the Director Field
10.5 Smectics and Cholesterics
10.6 Mixtures of Micropolar Constituents
10.6.1 Second Law Inequality
10.7 Nanofluids
10.7.1 Brownian Diffusion and Thermophoresis
10.7.2 Model Approximations and Nanofluid Properties
11 Porous Materials
11.1 Porous Materials as Mixtures
11.2 Constitutive Models of Porous Materials
11.2.1 Darcy's Law
11.2.2 Darcy-Like Equations
11.3 Special Models of Porous Materials
11.4 Materials with Voids
11.5 Porous Media with Double Porosity
11.5.1 Porous Material with Undeformable Solid
11.5.2 Porous Material with a Thermoelastic Solid
12 Electromagnetism of Continuous Media
12.1 A Thermodynamic Setting for Electromagnetic Solids
12.2 Electroelasticity
12.3 Electroelastic Materials
12.4 Dielectrics with Polarization Gradient
12.5 Fluids in Electromagnetic Fields
12.6 Magnetoelasticity
12.7 Micromagnetics and Magnetism in Rigid Bodies
12.7.1 Rate Equations and Thermodynamic Restrictions
12.7.2 Evolution Equations of Magnetization
12.8 Ferrofluids
12.8.1 Magneto-Rheological Fluids
12.9 Magneto-, Electro-, and Mechanical-Optical Effects
12.9.1 Magneto-Optical Effects
12.9.2 Frequency Dependence, Closed Processes, Memory Functionals
12.9.3 Electro-Optical Effects
12.9.4 Mechanical-Optical Effects
12.9.5 Magnetocaloric and Electro-Caloric Effects
12.10 Plasmas
12.10.1 One-Fluid Plasma Theory and Magnetohydrodynamics
12.11 Chiral Media and Optical Activity
12.11.1 Wave Solutions in Chiral Media
12.12 Anisotropic Non-Dissipative Magnetic Materials (Ferrites)
12.12.1 Magnetization Model with Damping Effects
Part IV Hysteresis and Phase Transitions
13 Plasticity
13.1 Qualitative Aspects of the Stress–Strain Curve
13.1.1 Yield Criteria
13.2 Rate-Independent Scheme of Plastic Flow
13.2.1 Consistency of Mises–Hill Model with a Defect Energy
13.3 A Temperature-Dependent Model
13.4 Gradient Theories of Plasticity
13.5 Model of Plasticity via the Kröner Decomposition
13.6 A Decomposition-Free Thermodynamic Scheme
13.6.1 Duhem-Like Solids
13.6.2 Hyperelastic and Hypoelastic Solids
13.6.3 Dissipative Duhem-Like Solids
13.6.4 Elastic–Plastic Models
13.6.5 One-Dimensional Models
13.6.6 The Helmholtz Free Energy
13.6.7 Some Hysteretic Models
13.6.8 Models with Additive Decomposition of the Strain Rate
13.7 Constitutive Models of Polymeric Foams
14 Superconductivity and Superfluidity
14.1 Superconductors
14.1.1 Discoveries About Superconductivity
14.1.2 The London Equations
14.1.3 An Alternative to London Equations
14.1.4 A Nonlocal Model
14.1.5 Phase Transition Curve
14.1.6 Ginzburg–Landau Theory
14.2 Superfluids
14.2.1 History of Superfluidity and Properties of Superfluids
14.2.2 First and Second Sound
14.2.3 Discontinuity Waves
14.2.4 Mass Fractions and Chemical Potentials
14.2.5 Rotation of Superfluids
14.2.6 The Ginzburg–Landau Theory
15 Ferroics
15.1 Evolution Function and Hysteretic Effects in Ferroelectrics
15.1.1 One-Dimensional Models of Ferroelectric Hysteresis
15.2 Ginzburg–Landau–Devonshire Theory
15.2.1 Landau Free Energy
15.2.2 Landau–Devonshire Free Energy
15.3 Ferromagnetism
15.3.1 Thermodynamic Approach to Magnetic Hysteresis
15.3.2 One-Dimensional Models of Ferromagnetic Hysteresis
15.3.3 Curie–Weiss Law
16 Phase Transitions
16.1 Jump Conditions at Interfaces
16.2 Phase Transitions at Sharp Interfaces with T= - p 1
16.2.1 The Stefan Model
16.3 Liquid–Vapour Transition
16.3.1 Continuity of the Gibbs Free Energy
16.3.2 Viscous Phases and Surface Tension
16.3.3 The Phase Rule
16.4 Solid–Fluid Transition
16.4.1 Transition in a Finite Layer—A Mixture Model
16.4.2 Evolution of the Mass Fraction
16.5 Solidification-Melting of Binary Alloys
16.6 Brine Channels Formation in Sea Ice
16.6.1 Transition Models via Minimization of Functionals
16.7 Phase-Field Model of Liquid-Solid Transitions
16.8 Phase Transitions in SMA
Appendix A Notes on Vectors and Tensors
A.1 Vector and Tensor Algebra
A.1.1 Eigenvalues and Eigenvectors
A.1.2 Length and Induced Norm
A.1.3 Representation Formulae for Vectors and Tensors
A.2 Isotropic Tensors
A.3 Differentiation
A.4 Integration
A.5 Harmonic Waves and Complex-Valued Functions
A.6 Fourier Transform
Appendix References
Index