| |

| |

| |

Introduction | |

| |

| |

| |

Hooke's Law | |

| |

| |

| |

Linear Solids with Memory | |

| |

| |

| |

Sinusoidal Oscillations in Viscoelastic Material: Models of Viscoelasticity | |

| |

| |

| |

Plasticity | |

| |

| |

| |

Vibrations | |

| |

| |

| |

Prototype of Wave Dynamics | |

| |

| |

| |

Biomechanics | |

| |

| |

| |

Historical Remarks | |

| |

| |

| |

Tensor Analysis | |

| |

| |

| |

Notation and Summation Convention | |

| |

| |

| |

Coordinate Transformation | |

| |

| |

| |

Euclidean Metric Tensor | |

| |

| |

| |

Scalars, Contravariant Vectors, Covariant Vectors | |

| |

| |

| |

Tensor Fields of Higher Rank | |

| |

| |

| |

Some Important Special Tensors | |

| |

| |

| |

The Significance of Tensor Characteristics | |

| |

| |

| |

Rectangular Cartesian Tensors | |

| |

| |

| |

Contraction | |

| |

| |

| |

Quotient Rule | |

| |

| |

| |

Partial Derivatives in Cartesian Coordinates | |

| |

| |

| |

Covariant Differentiation of Vector Fields | |

| |

| |

| |

Tensor Equations | |

| |

| |

| |

Geometric Interpretation of Tensor Components | |

| |

| |

| |

Geometric Interpretation of Covariant Derivatives | |

| |

| |

| |

Physical Components of a Vector | |

| |

| |

| |

Stress Tensor | |

| |

| |

| |

Stresses | |

| |

| |

| |

Laws of Motion | |

| |

| |

| |

Cauchy's Formula | |

| |

| |

| |

Equations of Equilibrium | |

| |

| |

| |

Transformation of Coordinates | |

| |

| |

| |

Plane State of Stress | |

| |

| |

| |

Principal Stresses | |

| |

| |

| |

Shearing Stresses | |

| |

| |

| |

Mohr's Circles | |

| |

| |

| |

Stress Deviations | |

| |

| |

| |

Octahedral Shearing Stress | |

| |

| |

| |

Stress Tensor in General Coordinates | |

| |

| |

| |

Physical Components of a Stress Tensor in General Coordinates | |

| |

| |

| |

Equations of Equilibrium in Curvilinear Coordinates | |

| |

| |

| |

Analysis of Strain | |

| |

| |

| |

Deformation | |

| |

| |

| |

Strain Tensors in Rectangular Cartesian Coordinates | |

| |

| |

| |

Geometric Interpretation of Infinitesimal Strain Components | |

| |

| |

| |

Rotation | |

| |

| |

| |

Finite Strain Components | |

| |

| |

| |

Compatibility of Strain Components | |

| |

| |

| |

Multiply Connected Regions | |

| |

| |

| |

Multivalued Displacements | |

| |

| |

| |

Properties of the Strain Tensor | |

| |

| |

| |

Physical Components | |

| |

| |

| |

Example--Spherical Coordinates | |

| |

| |

| |

Example--Cylindrical Polar Coordinates | |

| |

| |

| |

Conservation Laws | |

| |

| |

| |

Gauss' Theorem | |

| |

| |

| |

Material and Spatial Descriptions of Changing Configurations | |

| |

| |

| |

Material Derivative of Volume Integral | |

| |

| |

| |

The Equation of Continuity | |

| |

| |

| |

Equations of Motion | |

| |

| |

| |

Moment of Momentum | |

| |

| |

| |

Other Field Equations | |

| |

| |

| |

Elastic and Plastic Behavior of Materials | |

| |

| |

| |

Generalized Hooke's Law | |

| |

| |

| |

Stress-Strain Relationship for Isotropic Elastic Materials | |

| |

| |

| |

Ideal Plastic Solids | |

| |

| |

| |

Some Experimental Information | |

| |

| |

| |

A Basic Assumption of the Mathematical Theory of Plasticity | |

| |

| |

| |

Loading and Unloading Criteria | |

| |

| |

| |

Isotropic Stress Theories of Yield Function | |

| |

| |

| |

Further Examples of Yield Functions | |

| |

| |

| |

Work Hardening--Drucker's Hypothesis and Definition | |

| |

| |

| |

Ideal Plasticity | |

| |

| |

| |

Flow Rule for Work-Hardening Materials | |

| |

| |

| |

Subsequent Loading Surfaces--Isotropic and Kinematic Hardening Rules | |

| |

| |

| |

Mroz's, Dafalias and Popov's, and Valanis' Plasticity Theories | |

| |

| |

| |

Strain Space Formulations | |

| |

| |

| |

Finite Deformation | |

| |

| |

| |

Plastic Deformation of Crystals | |

| |

| |

| |

Linearized Theory of Elasticity | |

| |

| |

| |

Basic Equations of Elasticity for Homogeneous Isotropic Bodies | |

| |

| |

| |

Equilibrium of an Elastic Body Under Zero Body Force | |

| |

| |

| |

Boundary Value Problems | |

| |

| |

| |

Equilibrium and Uniqueness of Solutions | |

| |

| |

| |

Saint Venant's Theory of Torsion | |

| |

| |

| |

Soap Film Analogy | |

| |

| |

| |

Bending of Beams | |

| |

| |

| |

Plane Elastic Waves | |

| |

| |

| |

Rayleigh Surface Wave | |

| |

| |

| |

Love Wave | |

| |

| |

| |

Solutions of Problems in Linearized Theory of Elasticity by Potentials | |

| |

| |

| |

Scalar and Vector Potentials for Displacement Vector Fields | |

| |

| |

| |

Equations of Motion in Terms of Displacement Potentials | |

| |

| |

| |

Strain Potential | |

| |

| |

| |

Galerkin Vector | |

| |

| |

| |

Equivalent Galerkin Vectors | |

| |

| |

| |

Example--Vertical Load on the Horizontal Surface of a Semi-Infinite Solid | |

| |

| |

| |

Love's Strain Function | |

| |

| |

| |

Kelvin's Problem--A Single Force Acting in the Interior of an Infinite Solid | |

| |

| |

| |

Perturbation of Elasticity Solutions by a Change of Poisson's Ratio | |

| |

| |

| |

Boussinesq's Problem | |

| |

| |

| |

On Biharmonic Functions | |

| |

| |

| |

Neuber-Papkovich Representation | |

| |

| |

| |

Other Methods of Solution of Elastostatic Problems | |

| |

| |

| |

Reflection and Refraction of Plane P and S Waves | |

| |

| |

| |

Lamb's Problem--Line Load Suddenly Applied on Elastic Half-Space | |

| |

| |

| |

Two-Dimensional Problems in Linearized Theory of Elasticity | |

| |

| |

| |

Plane State of Stress or Strain | |

| |

| |

| |

Airy Stress Functions for Two-Dimensional Problems | |

| |

| |

| |

Airy Stress Function in Polar Coordinates | |

| |

| |

| |

General Case | |

| |

| |

| |

Representation of Two-Dimensional Biharmonic Functions by Analytic Functions of a Complex Variable | |

| |

| |

| |

Kolosoff-Muskhelishvili Method | |

| |

| |

| |

Variational Calculus, Energy Theorems, Saint-Venant's Principle | |

| |

| |

| |

Minimization of Functionals | |

| |

| |

| |

Functional Involving Higher Derivatives of the Dependent Variable | |

| |

| |

| |

Several Unknown Functions | |

| |

| |

| |

Several Independent Variables | |

| |

| |

| |

Subsidiary Conditions--Lagrangian Multipliers | |

| |

| |

| |

Natural Boundary Conditions | |

| |

| |

| |

Theorem of Minimum Potential Energy Under Small Variations of Displacements | |

| |

| |

| |

Example of Application: Static Loading on a Beam--Natural and Rigid End Conditions | |

| |

| |

| |

The Complementary Energy Theorem Under Small Variations of Stresses | |

| |

| |

| |

Variational Functionals Frequently Used in Computational Mechanics | |

| |

| |

| |

Saint-Venant's Principle | |

| |

| |

| |

Saint-Venant's Principle-Boussinesq-Von Mises-Sternberg Formulation | |

| |

| |

| |

Practical Applications of Saint-Venant's Principle | |

| |

| |

| |

Extremum Principles for Plasticity | |

| |

| |

| |

Limit Analysis | |

| |

| |

| |

Hamilton's Principle, Wave Propagation, Applications of Generalized Coordinates | |

| |

| |

| |

Hamilton's Principle | |

| |

| |

| |

Example of Application--Equation of Vibration of a Beam | |

| |

| |

| |

Group Velocity | |

| |

| |

| |

Hopkinson's Experiment | |

| |

| |

| |

Generalized Coordinates | |

| |

| |

| |

Approximate Representation of Functions | |

| |

| |

| |

Approximate Solution of Differential Equations | |

| |

| |

| |

Direct Methods of Variational Calculus | |

| |

| |

| |

Elasticity and Thermodynamics | |

| |

| |

| |

The Laws of Thermodynamics | |

| |

| |

| |

The Energy Equation | |

| |

| |

| |

The Strain Energy Function | |

| |

| |

| |

The Conditions of Thermodynamic Equilibrium | |

| |

| |

| |

The Positive Definiteness of the Strain Energy Function | |

| |

| |

| |

Thermodynamic Restrictions on the Stress-Strain Law of an Isotropic Elastic Material | |

| |

| |

| |

Generalized Hooke's Law, Including the Effect of Thermal Expansion | |

| |

| |

| |

Thermodynamic Functions for Isotropic Hookean Materials | |

| |

| |

| |

Equations Connecting Thermal and Mechanical Properties of a Solid | |

| |

| |

| |

Irreversible Thermodynamics and Viscoelasticity | |

| |

| |

| |

Basic Assumptions | |

| |

| |

| |

One-Dimensional Heat Conduction | |

| |

| |

| |

Phenomenological Relations-Onsager Principle | |

| |

| |

| |

Basic Equations of Thermomechanics | |

| |

| |

| |

Equations of Evolution for a Linear Hereditary Material | |

| |

| |

| |

Relaxation Modes | |

| |

| |

| |

Normal Coordinates | |

| |

| |

| |

Hidden Variables and the Force-Displacement Relationship | |

| |

| |

| |

Anisotropic Linear Viscoelastic Materials | |

| |

| |

| |

Thermoelasticity | |

| |

| |

| |

Basic Equations | |

| |

| |

| |

Thermal Effects Due to a Change of Strain; Kelvin's Formula | |

| |

| |

| |

Ratio of Adiabatic to Isothermal Elastic Moduli | |

| |

| |

| |

Uncoupled, Quasi-Static Thermoelastic Theory | |

| |

| |

| |

Temperature Distribution | |

| |

| |

| |

Thermal Stresses | |

| |

| |

| |

Particular Integral: Goodier's Method | |

| |

| |

| |

Plane Strain | |

| |

| |

| |

An Example--Stresses in a Turbine Disk | |

| |

| |

| |

Variational Principle for Uncoupled Thermoelasticity | |

| |

| |

| |

Variational Principle for Heat Conduction | |

| |

| |

| |

Coupled Thermoelasticity | |

| |

| |

| |

Lagrangian Equations for Heat Conduction and Thermoelasticity | |

| |

| |

| |

Viscoelasticity | |

| |

| |

| |

Viscoelastic Material | |

| |

| |

| |

Stress-Strain Relations in Differential Equation Form | |

| |

| |

| |

Boundary-Value Problems and Integral Transformations | |

| |

| |

| |

Waves in an Infinite Medium | |

| |

| |

| |

Quasi-Static Problems | |

| |

| |

| |

Reciprocity Relations | |

| |

| |

| |

Large Deformation | |

| |

| |

| |

Coordinate Systems and Tensor Notation | |

| |

| |

| |

Deformation Gradient | |

| |

| |

| |

Strains | |

| |

| |

| |

Right and Left Stretch Strain and Rotation Tensors | |

| |

| |

| |

Strain Rates | |

| |

| |

| |

Material Derivatives of Line, Area, and Volume Elements | |

| |

| |

| |

Stresses | |

| |

| |

| |

Example: Combined Tension and Torsion Loads | |

| |

| |

| |

Objectivity | |

| |

| |

| |

Equations of Motion | |

| |

| |

| |

Constitutive Equations of Thermoelastic Bodies | |

| |

| |

| |

More Examples | |

| |

| |

| |

Variational Principles for Finite Elasticity: Compressible Materials | |

| |

| |

| |

Variational Principles for Finite Elasticity: Nearly Incompressible or Incompressible Materials | |

| |

| |

| |

Small Deflection of Thin Plates | |

| |

| |

| |

Large Deflection of Plates | |

| |

| |

| |

Incremental Approach to Solving Some Nonlinear Problems | |

| |

| |

| |

Updated Lagrangian Description | |

| |

| |

| |

Linearized Rates of Deformation | |

| |

| |

| |

Linearized Rates of Stress Measures | |

| |

| |

| |

Incremental Equations of Motion | |

| |

| |

| |

Constitutive Laws | |

| |

| |

| |

Incremental Variational Principles in Terms of T | |

| |

| |

| |

Incremental Variational Principles in Terms of r | |

| |

| |

| |

Incompressible and Nearly Incompressible Materials | |

| |

| |

| |

Updated Solution | |

| |

| |

| |

Incremental Loads | |

| |

| |

| |

Infinitesimal Strain Theory | |

| |

| |

| |

Finite Element Methods | |

| |

| |

| |

Basic Approach | |

| |

| |

| |

One Dimensional Problems Governed by a Second Order Differential Equation | |

| |

| |

| |

Shape Functions and Element Matrices for Higher Order Ordinary Differential Equations | |

| |

| |

| |

Assembling and Constraining Global Matrices | |

| |

| |

| |

Equation Solving | |

| |

| |

| |

Two Dimensional Problems by One-Dimensional Elements | |

| |

| |

| |

General Finite Element Formulation | |

| |

| |

| |

Convergence | |

| |

| |

| |

Two-Dimensional Shape Functions | |

| |

| |

| |

Element Matrices for a Second-Order Elliptical Equation | |

| |

| |

| |

Coordinate Transformation | |

| |

| |

| |

Triangular Elements with Curved Sides | |

| |

| |

| |

Quadrilateral Elements | |

| |

| |

| |

Plane Elasticity | |

| |

| |

| |

Three-Dimensional Shape Functions | |

| |

| |

| |

Three Dimensional Elasticity | |

| |

| |

| |

Dynamic Problems of Elastic Solids | |

| |

| |

| |

Numerical Integration | |

| |

| |

| |

Patch Tests | |

| |

| |

| |

Locking-Free Elements | |

| |

| |

| |

Spurious Modes in Reduced Integration | |

| |

| |

| |

Perspective | |

| |

| |

| |

Mixed and Hybrid Formulations | |

| |

| |

| |

Mixed Formulations | |

| |

| |

| |

Hybrid Formulations | |

| |

| |

| |

Hybrid Singular Elements (Super-Elements) | |

| |

| |

| |

Elements for Heterogeneous Materials | |

| |

| |

| |

Elements for Infinite Domain | |

| |

| |

| |

Incompressible or Nearly Incompressible Elasticity | |

| |

| |

| |

Finite Element Methods for Plates and Shells | |

| |

| |

| |

Linearized Bending Theory of Thin Plates | |

| |

| |

| |

Reissner-Mindlin Plates | |

| |

| |

| |

Mixed Functionals for Reissner Plate Theory | |

| |

| |

| |

Hybrid Formulations for Plates | |

| |

| |

| |

Shell as an Assembly of Plate Elements | |

| |

| |

| |

General Shell Elements | |

| |

| |

| |

Locking and Stabilization in Shell Applications | |

| |

| |

| |

Finite Element Modeling of Nonlinear Elasticity, Viscoelasticity, Plasticity, Viscoplasticity and Creep | |

| |

| |

| |

Updated Lagrangian Solution for Large Deformation | |

| |

| |

| |

Incremental Solution | |

| |

| |

| |

Dynamic Solution | |

| |

| |

| |

Newton-Raphson Iteration Method | |

| |

| |

| |

Viscoelasticity | |

| |

| |

| |

Plasticity | |

| |

| |

| |

Viscoplasticity | |

| |

| |

| |

Creep | |

| |

| |

Bibliography | |

| |

| |

Author Index | |

| |

| |

Subject Index | |