Classical and Computational Solid Mechanics

The following sections are included:

• DEDICATION

• PREFACE

• CONTENTS

The following sections are included:

• HOOKE'S LAW

• LINEAR SOLIDS WITH MEMORY

• SINUSOIDAL OSCILLATIONS IN A VISCOELASTIC MATERIAL: MODELS OF VISCOELASTICITY

• PLASTICITY

• VIBRATIONS

• PROTOTYPE OF WAVE DYNAMICS

• BIOMECHANICS

• HISTORICAL REMARKS

The following sections are included:

• 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

The following sections are included:

• 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

The following sections are included:

• 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

The following sections are included:

• GAUSS' THEOREM

• MATERIAL AND SPATIAL DESCRIPTIONS OF CHANGING CONFIGURATIONS

• MATERIAL DERIVATIVE OF VOLUME INTEGRAL

• THE EQUATION OF CONTINUITY

• THE EQUATIONS OF MOTION

• MOMENT OF MOMENTUM

• OTHER FIELD EQUATIONS

The following sections are included:

• 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: THE EXISTENCE OF A YIELD FUNCTION

• 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

The following sections are included:

• 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

The following sections are included:

• 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

The following sections are included:

• 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

The following sections are included:

• 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

The following sections are included:

• 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

The following sections are included:

• THE LAWS OF THERMODYNAMICS

• ENERGY EQUATION

• THE STRAIN ENERGY FUNCTION

• 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

The following sections are included:

• 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

The following sections are included:

• 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

The following sections are included:

• 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

The following sections are included:

• 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

The following sections are included:

• UPDATED LAGRANGIAN DESCRIPTION

• LINEARIZED RATES OF DEFORMATION

• LINEARIZED RATES OF STRESS MEASURES

• INCREMENTAL EQUATIONS OF MOTION

• CONSTITUTIVE LAWS

• INCREMENTAL VARIATIONAL PRINCIPLES IN TERMS OF

• INCREMENTAL VARIATIONAL PRINCIPLES IN TERMS OF

• INCOMPRESSIBLE AND NEARLY INCOMPRESSIBLE MATERIALS

• UPDATED SOLUTION

• INCREMENTAL LOADS

• INFINITESIMAL STRAIN THEORY

The following sections are included:

• 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 (MacNeal 1994)

• PERSPECTIVE

The following sections are included:

• MIXED FORMULATIONS

• HYBRID FORMULATIONS

• HYBRID SINGULAR ELEMENTS (SUPER-ELEMENTS)

• ELEMENTS FOR HETEROGENEOUS MATERIALS

• ELEMENTS FOR INFINITE DOMAIN

• INCOMPRESSIBLE OR NEARLY INCOMPRESSIBLE ELASTICITY

The following sections are included:

• 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 STABLIZATION IN SHELL APPLICATIONS

The following sections are included:

• UPDATED LAGRANGIAN SOLUTION FOR LARGE DEFORMATION

• INCREMENTAL SOLUTION

• DYNAMIC SOLUTION

• NEWTON-RAPHSON ITERATION METHOD

• VISCOELASTICITY

• PLASTICITY

• VISCOPLASTICITY

• CREEP

The following sections are included:

• BIBLIOGRAPHY

• AUTHOR INDEX

• SUBJECT INDEX