Continuum Mechanics of Single-Substance Bodies

List of Contributors

Preface

Contents of Volume I

Part I. Basic Principles


1. Deformation and Motion


1.1 Scope of the Chapter


1.2 Coordinates


1.3 The Motion, Deformation, Strain Measures


1.4 Length and Angle Changes


1.5 Strain Ellipsoids of Cauchy


1.6 Strain Invariants, Principal Directions


1.7 Rotation


1.8 Area and Volume Changes


1.9 Compatibility Conditions


1.10 Kinematics, Time Rates of Tensors


1.11 Deformation Rate, Spin, Vorticity


1.12 Rates of Strains and Rotations


1.13 Material and Spatial Manifolds


1.14 Kinematics of Line, Surface, and Volume Integrals


2. Balance Laws


2.1 Scope of the Chapter


2.2 Global Balance Laws


2.3 Master Law for Local Balance


2.4 Local Balance Laws


2.5 Stress Quadratic, Stress Invariants


2.6 Stress Flux


3. Thermodynamics of Continua


3.1 Scope of the Chapter


3.2 Thermodynamic Processes


3.3 The First and the Second Laws of Thermodynamics


3.4 Thermodynamic Restrictions on Some Simple Materials


3.5 Discontinuous Thermodynamic Processes


3.6 Thermodynamics of Materials with Memory


3.7 Onsager Forces and Fluxes


3.8 Onsager Force Potential, Variational Principle

References

Part II. Constitutive Equations for Simple Materials


1. General Theory


1.1 Scope of the Chapter


1.2 Raison d'Etre


1.3 Axioms of Constitutive Theory


1.4 Thermomechanical Materials


1.5 Thermoelastic Materials


1.6 Thermoviscous Fluids


1.7 Simple Thermomechanical Materials

References


2. Thermoelastic Solids


2.1 Scope of the Chapter


2.2 Resume of the Fundamental Equations


2.3 Constitutive Relations for Thermoelastic Solids


2.4 Isotropic Thermoelastic Solids


2.5 Linear Constitutive Relations


2.6 Linear Theory for Isotropic Thermoelastic Solids


2.7 Temperature-Rate-Dependent Thermoelastic Solids


2.8 Constitutive Relations for Elastic Materials. Hyperelasticity


2.9 Various Forms of Constitutive Relations


2.10 Anisotropic Elastic Solids


2.11 Restrictions on the Strain Energy Function for Isotropic Materials


2.12 Work Relations for Elastic Equilibrium


2.13 Formulation of Boundary-Value Problems. Elasticities


2.14 Formulation of Boundary-Value Problems in Isotropic Materials


2.15 Approximate Theories for Hyperelastic Solids


2.16 Variational Theorems of Elastostatics


2.17 Small Motions Superimposed on Large Static Deformations


2.18 Stability of Elastic Equilibrium

References


3. Thermoviscous Fluids


3.1 Scope of the Chapter


3.2 Equations of Balance


3.3 Entropy Inequality


3.4 Definition and Constitutive Relations of a Temperature-Rate-Independent Thermoviscous Fluid


3.5 Limitations Placed on the Constitutive Functions by the Entropy Inequality


3.6 Connection with the Classical Theory of Linear Thermoviscous Fluids

References


4. Simple Materials with Fading Memory


4.1 Scope of the Chapter


4.2 Linear Viscoelasticity


4.3 Mathematical Prerequisites


4.4 Nonlinear Constitutive Relations


4.5 Material Symmetries


4.6 Fading Memory Space


4.7 Finite Linear Viscoelasticity


4.8 Materials of Integral Type


4.9 Thermodynamics of Kelvin-Voigt Materials


4.10 Thermodynamics of Materials with Fading Memory


4.11 Thermodynamical Restrictions on the Mechanical Constitutive Relations


4.12 Small Deformations


4.13 Material Testing


4.14 Fluids

References

Part III. Method of Solution


1 Exact Solutions in Fluids and Solids


1.1 Scope of the Chapter


1.2 Historical Precis


1.3 Erickson's Theorems in Finite Elasticity for Static Deformations


1.4 Viscometric Flows


1.5 Universal Motions for Isotropic, Homogeneous, Incompressible, Simple Materials


1.6 Sundry Mathematical Representation Theorems


1.7 Simple Fluids


1.8 Simple Shearing in a Reiner-Rivlin Fluid


1.9 Simple Shearing in a Simple Fluid


1.10 Radial Flow in a Simple Fluid


1.11 On the Thermodynamic Impossibility of a Steady Poiseulle Flow in a General Simple Fluid


1.12 Simple Isotropic Solids


1.13 Dynamic Simple Shearing in an Elastic Body


1.14 Motions in Simple Solids; Response Functionals Determined by Homogeneous Motions


1.15 Radial Oscillations in a Simple Solid Hollow Sphere


1.16 Static Deformations

References


2. Singular Surfaces and Waves


2.1 Scope of the Chapter


2.2 Compatibility Conditions on a Moving Singular Surface


2.3 Classification of Singular Surfaces


2.4 Basic Laws of Continuum Mechanics


2.5 Propagation of Acceleration Waves in Definite Conductors


2.6 The Variation of the Amplitudes of Acceleration Waves in Definite Conductors


2.7 Propagation of Acceleration Waves in Nonconductors


2.8 Acceleration Waves in Isotropic Materials


2.9 The Influence of Hydrostatic Pressure on the Propagation of Acceleration Waves


2.10 Second-Order Effects in Wave Propagation


2.11 Relations of Acceleration Waves to Plane Waves of Infinitesimal Amplitude


2.12 Waves in Incompressible Materials


2.13 Simple Waves


2.14 Undirectional Simple Waves in Isotropic Media


2.15 Shock Waves in Elastic Nonconductors


2.16 Shock Waves in Infinitesimal Amplitude


2.17 Shock Waves in Isotropic Media


2.18 Solution of Initial Boundary Value Problems

References


3. Complex Function Technique


3.1 Scope of the Chapter


3.2 Definitions: Dual Series, Dual Integral Equations, Potential, Flux


3.3 Methods of Solution of Mixed Boundary Value Problems


3.4 Direct Application of Complex Potentials


3.5 Nature of the Kernel in Mixed Boundary Value Problems


3.6 Reduction of Dual Series Equations to Singular Integral Equations


3.7 Reduction of Dual Integral Equations to Singular Integral Equations


3.8 Dual Integral Equations Leading to Singular Integral Equations of the Second Kind


3.9 A System of Dual Series-Integral Equations


3.10 Singular Integral Equations with a Generalized Cauchy Kernel


3.11 Numerical Solution of the Singular Integral Equations of the First Kind


3.12 Solution of Singular Integral Equations of the Second Kind


3.13 Solutions by Gauss-Chebyshev and Gauss-Jacoby Integration Formulas

References

Author Index

Subject Index