# Three-Dimensional Problems of Elasticity and Thermoelasticity, Volume 25

## 1st Edition

**Editor:**V.D. Kupradze

**eBook ISBN:**9780080984636

**Imprint:**North Holland

**Published Date:**1st January 1979

**View all volumes in this series:**North-Holland Series in Applied Mathematics and Mechanics

## Table of Contents

Preface

Chapter I Basic Concepts and Axiomatization

§1. Stresses

1. Internal and External Forces

2. Mass and Surface Forces

3. Force- and Couple-Stresses

§2. Components of stress

1. Components of Force- and Couple-Stress Tensors

2. Expression of the Force-Stress Vector in Terms of Components of the Forcestress Tensor

3. Expression of the Couple-Stress Vector in Terms of Components of the Couplestress Tensor

§3. Displacements and Rotations

1. Displacement Vector

2. Rotation Vector

§4. Basic Equations in Terms of Stress Components

1. Equations of Motion in Classical Elasticity

2. Equations of Motion in the Couple-Stress Theory

§5. Hooke’s Law in Classical Elasticity

1. Components of the Strain Tensor

2. Formulation of Hooke’s Law

3. Isotropic Medium

4. Transversally Isotropic Medium

§6. Strain Energy in Classical Elasticity

1. Law of Energy Conservation

2. Specific Energy of Strain

§7. Strain Energy and Hooke’S Law in the Couple-Stress Theory

1. Law of Energy Conservation

2. Specific Energy of Strain

3. Hooke’s Law

4. Isotropic Medium (with the Centre of Symmetry)

§8. Thermoelasticity. Duhamel-Neumann’s Law

1. Deformation Produced by Temperature Variation

2. Law of Energy Conservation

3. Duhamel-Neumann’s Law

4. Isotropic Medium

§9. Heat Conduction Equation

§10. Stationary Elastic Oscillations

1. Classical Theory of Elasticity

2. Couple-Stress Theory

3. Theory of Thermoelasticity

§11. Axiomatization of the Theory

1. Classical Elasticity

2. Couple-Stress Elasticity

3. Thermo Elasticity

§12. Matrix Representation of the Basic Equations

1. Classical Elasticity

2. Couple-Stress Elasticity

3. Thermo Elasticity

§13. Stress Operator

1. Classical Theory

2. Couple-Stress Elasticity

3. Thermo Elasticity

§ 14. Formulation of the Basic Problems

1. Classical Theory

2. Couple-Stress Elasticity

3. Thermo Elasticity

4. Problems for Inhomogeneous Media

§15. Some Additional Remarks and Bibliographic References

1. Specific Energy of Strain and Basic Equations in Displacement Components for an Anisotropic Inhomogeneous Medium

2. On Differential Operators of the Theory of Elasticity

3. Ellipticity of the Basic Boundary-Value Problems of Statics and Oscillation. Lopatinski’S Condition

4. On Some Functional Spaces of Class πκ(α)

5. Bibliographic References

Chapter II Basic Singular Solutions

§1. Fundamental Solutions of Classical Elasticity

1. Statics. Kelvin’s Matrix

2. Harmonic Oscillations. Kupradze’s Matrix

3. Basic Properties of Kupradze’s Matrix

§2. Fundamental Solutions of the Couple-Stress Theory

1. Harmonic Oscillations

2. Statics

§3. Fundamental Solutions of Thermoelasticity

1. Harmonic Oscillations

2. Associated Equation

3. Statics

§4. Singular Solutions of Classical Elasticity

1. Statics

2. Harmonic Oscillations

§5. Singular Solutions of the Couple-Stress Theory

1. Harmonic Oscillations

2. Statics

§6. Singular Solutions of Thermo Elasticity

1. Harmonic Oscillations

2. Statics

§7. Various Remarks and Bibliographic References

Problems

Chapter III Uniqueness Theorems

§1. Static Problems in Classical Elasticity

1. Green’s Formulas

2. Solution of an Auxiliary Equation

3. Basic Lemma

4. Uniqueness Theorems

5. Equation A(∂x)u-ρτ2u=0

6. Inhomogeneous Medium

§2. Problems of Steady Elastic Oscillations

1. General Representation of Regular Solutions in D+

2. Expansion of Regular Solutions

3. Radiation Condition in Elasticity

4. Representation of Solutions in D

5. Uniqueness Theorems for External Problems

6. Uniqueness Theorems for Inhomogeneous Media

§3. Problems of Steady Thermoelastic Oscillations

1. Expansion of a Regular Solution

2. Green’s Formulas

3. Condition of Thermoelastic Radiation

4. Uniqueness Theorems for External Problems

5. Uniqueness Theorems for Problems of Thermoelastic Pseudo-Oscillations

§4. Static Problems in the Couple-Stress Theory

1. Green’s Formulas

2. Solution of an Auxiliary Equation

3. Uniqueness Theorems in Static Problems

§5. Problems of Steady Couple-Stress Oscillations

1. Expansion of the Regular Solution of M(∂x, σ)U=0

2. Radiation Condition

3. Auxiliary Estimates

4. Uniqueness Theorems

§6. Uniqueness Theorems in Dynamic Problems

1. Energy Identities

2. Uniqueness Theorems

§7. Some Remarks and Bibliographic References

Problems

Chapter IV Singular Integrals and Integral Equations

§1. Introductory Notes. Special Classes of Functions and their Properties

1. On Singular Integrals and Integral Equations

2. Functions of Class G and Z

3. Singular Kernel and Singular Integral

§2. Integral with the Kernel having a Weak Singularity

1. Elementary Properties

2. On Derivatives of Integrals with the Kernel having a Weak Singularity

§3. Singular Integrals

1. Singular Integrals in Classes C0,β. Giraud’s Theorem

2. Singular Integrals in Classes C5,α

3. Integrals with the Kernel of a Special Construction

4. Singular Integrals on Manifolds

5. Singular Operators in Spaces Lp. Calderon and Zygmund Theorem

§4. Formula of Inversion of Integration Order in Iterated Singular Integrals. Composition of Singular Kernels

1. General Inversion Formula

2. Example

§5. Regularization of Singular Operators

1. Giraud’s Method

2. Mikhlin’s Method

3. Regularization of Singular Operators Over Closed Surfaces

§6. Basic Theorems

1. Noether Theorems

2. Differential Properties of Solutions of Singular Integral Equations. Embedding Theorems

§7. Singular Resolvent. Properties and Applications

1. Transformation of the Kernel

2. Mapping on the Circle

3. Mapping on the Infinite Plane

4. Local Regularizat Ion

5. Operator of Global Regularization

6. Functional Equations of the Resolvent. Fredholm’S First Theorem

7. Fredholm’s Second Theorem

8. Biorthonormalization of the Fundamental Solutions of Associated Systems

9. Fredholm’s Third Theorem

§8. Concluding Remarks

Problems

Chapter V The Potential Theory

§1. Some Auxiliary Operators, Formulas and Theorems

1. Definition of III+(y, δ) and III-(y, δ)

2. Definition of the operators Dk, Mkj and ∂/∂Sk

3. Stokes’ Theorem and its Applications

4. Representation Formula for x ∈ S

§2. Boundary Properties of Some Potential-Type Integrals

§3. Single- and Double-Layer Potentials. Angular Boundary Values

§4. Double-Layer Potential with Density of Class C0,β(S)

§5. Boundary Properties of the First Derivatives of the Single-Layer Potential

§6. Derivatives of Single- and Double-Layer Potentials with Differentiable Density

§7. On Differential Properties of Elastic Potentials

§8. Liapunov-Tauber Theorems in Elasticity

1. Liapunov-Tauber Theorems for a Harmonic Double-Layer Potential

2. Liapunov-Tauber Theorems in Elasticity

3. An Auxiliary Theorem

§9. Boundary Properties of Potentials of the Third and the Fourth Problems

1. Boundary Conditions for the Third and the Fourth Problems. Reduction to the Equivalent Conditions

2. Potentials of the Third and the Fourth Problems

3. Liapunov-Tauber Theorems for the Potentials of the Third and the Fourth

4. Somigliana’s Formulas for the Third and the Fourth Problems

§10. Volume Potentials

1. Definitions. Elementary Properties

2. Evaluation of the Second Order Derivatives

3. Theorem on Extension of Functions

4. Volume Potentials with Differentiable Densities

5. Behaviour of the Integral of the Volume Potential Type at Infinity

§11. Bibliographic References

Problems

Chapter VI Boundary Value Problems of Elastic Equilibrium (Statics)

§1. Boundary Value Problems For Inhomogenfeous Equations

§2. Integral Equations of the Boundary Value Problems

§3. Fredholm’s Theorems and Embedding Theorems

§4. Theorems on Eigenvalues

§5. Existence of Solutions of Boundary Value Problems

1. Problems (I)+ and (II)-

2. Problems (II)+ and (I)-

3. Alternative Method of Proving Existence Theorems for Problems (I)- and (II)+

4. Problems (III)+ and (IV)+

5. Problems (III)- and (IV)+

6. Problems (VI)+ and (VI)-

7. Problem (V)+

§6. Problems of Correctness

1. Formulation of the Problem

2. First Internal Problem of Statics

3. Second Internal Problem of Static

§7. Bibliographic References

Problems

Chapter VII Boundary Value Problems of Steady Elastic Oscillations

§1. Internal Problems

1. Reduction to Integral Equations

2. Green’s Tensors

3. Representation Formulas

4. Homogeneous Internal Problems. The Eigenfrequency Spectrum

§2. Basic Theorems of the Oscillation Theory

1. First Liapunov-Tauber Theorem in Elasticity

2. Properties of Eigenfrequencies and Eigenfunctions

3. Theorems on Simplicity of Resolvent Poles

4. Investigation of the Internal Problems. The Resonance Case

§3. External Problems

1. Solvability for Arbitrary Frequencies. Problems (I)ω- and (II)ω-

2. Other Problems

§4. Bibliographic References

Problems

Chapter VIII Mixed Dynamic Problems

§1. First Basic Problem

1. Conditions for the Data. The Basic Theorem

2. Reduction to the Special Case

3. Averaging Kernel. Properties of the Mean-Valued Function

4. Proof of the existence of σ(x, t)

5. Laplace Transform. Reduction to an Elliptic Problem

6. Uniqueness, Existence, Representation and Differential Properties of ũ0(x, τ)

7. Smoothness of ũ0(x, τ) relative to τ ∈ ∏σ0

8. Asymptotic Estimates of ũ0(x, τ) with Respect to τ

9. Some Elementary Inequalities

10. Asymptotic Estimates with Respect to τ for the First Derivatives ∂u0/∂xi

11. Asymptotic Estimates with Respect to r for the Second Derivatives ∂2ũ(x, τ)/∂xi∂xi

12. Some Properties of the Laplace Transform

13. Proof of the Existence of u0(x, t) and u(x, t)

14. Calculation of u(x, t) and Completion of the Proof of the Basic Theorem

§2. Second Basic Problem

1. Basic Theorem

2. Laplace Transform. Solution of an Elliptic Problem

3. Smoothness of ũ0(x, τ) with Respect to τ ∈ ∏σ0

4. Asymptotic Estimates of ũ0(x, τ) and its Derivatives with Respect to τ

5. Proof of the Existence and Calculation of the Solution of the Second Problem

§3. External Problems

1. Formulation of the Problems

2. Basic Lemma

3. Investigation of the First External Problem

§4. Concluding Remarks. Bibliographic References

Problems

Chapter IX Couple-Stress Elasticity

§1. Introduction

1. Basic Equations

2. Stress Operator

3. Basic Problems

3. Somigliana Formulas

5. Potentials

6. Liapunov-Tauber Theorem

§2. Investigation of Static Problems

1. Reduction to Integral Equations

2. Investigation of Integral Equations

3. Existence Theorems for Problems (I)+ and (II)-

4. Existence Theorems for Problems (II)+ and (I)-

5. Existence Theorems for Problems (III)+ and (IV)-

6. Existence Theorems for Problems (III)- and (IV )+

§3. Oscillation Problems

1. Reduction to Integral Equations

2. Investigation of Integral Equations

3. Green’s Tensors

4. Internal Problems

5. External Problems

§4. Dynamic Problems

1. Formulation and Reduction to a Special Form

2. Laplace Transform. Solution of an Elliptic Problem. Analyticity of the Solution

3. Asymptotic Estimates with Respect to τ of U0(x, τ) and its Derivatives. Solution of a Dynamic Problem

§5. Concluding Remarks. Bibliographic References

Problems

Chapter X Theory of Thermoelasticity

§1. Introduction

§2. Steady Thermoelastic Oscillations

1. Associated System. Properties of the Fundamental Solutions. Green’S Identities

2. General Representation of Regular Solutions of the Homogeneous Equation

3. Basic Properties of Thermoelastopotentials

4. Basic Boundary Value Problems. Reduction to Integral Equation

5. Fredholm’s Theorems

6. Internal Problems. The Eigenfrequency Spectrum. Uniqueness Theorems

7. Investigation of Integral Equations of External Problems

8. Applications in the Theory of External Problems. Proof of the Existence Theorems

§3. Static and Pseudo-Oscillation Problems

1. Static Problems

2. Pseudo-Oscillations

§4. Dynamic Problems of Thermoelasticity

1. First Problem. Formulation and Reduction to a Special Form

2. Laplace Transform. Solution of an Elliptic Problem

3. Smoothness of Ũ0(x, τ) with respect to τ ∈ ∏σ0

4. Asymptotic Estimates for Ũ0(1)(x, τ) and its Derivatives with Respect to τ

5. Asymptotic Estimates for Ũ0(2)(x, τ) with Respect to τ

6. Estimates for the τ-Derivatives of Ũ0(2)(x, τ) with Respect to xi

7. Completion of the Solution of the Dynamic Problem

§5. Additional Remarks. Bibliographic References

Problems

Chapter XI Boundary Value Problems for Media Bounded by Several Surfaces

§1. Boundary Value Problems of Elastic Equilibrium

1. Formulation of the Problems and the Uniqueness Theorems

2. Solution of Problem (I)±

3. Solution of Problems (I)±, (III)±

4. Green’s Tensors for Domains Bounded by Several Closed Surfaces

§2. Mixed Static Problems

1. Existence Theorems for Mixed Static Problems (IV)±

2. Solution of Mixed Problem (V)+

3. Existence Theorems for Static Mixed Problems (VI)+, (VII)+, (V)-

§3. Oscillation Problems

1. Homogeneous Internal Problems. Eigenfrequency Spectrum

2. Oscillation Problems (I)ω-, (II)ω-, (III)ω-. Reduction to the Integral Equations. Basic Theorems

3. Existence Theorems for the External Oscillation Problems (I)ω-, (II)ω-, (III)ω-

4. Mixed Boundary Oscillation Problems (IV)ω-, (V)ω-

§4. Concluding Remarks

Problems

Chapter XII Boundary-Contact Problems for Inhomogeneous Media

§1. Basic Boundary-Contact Problems

§2. Integral Equations of the Basic Contact Problem

§3. Solution of Static Boundary-Contact Problems

§4. Solution of Boundary-Contact Problems for Oscillation Equations

§5. Functional Equations of Boundary-Contact Problems

1. First Static Problem (I)+

2. Second Static Problem (II)+

3. Mixed Boundary-Contact Problem of Statics

4. Boundary-Contact Oscillation Problems

5. Equivalence Theorems

6. Cauchy’s Hypothesis. Investigation of Static Problems. Generalized Solutions

7. Dynamic Problems. The Eigenfrequency Spectrum. Generalized Solutions

8. Proof of the Existence Theorems for the General Case

9. Problems for an Unbounded Domain

§6. Concluding Remarks

Chapter XIII Method of Generalized Fourier Series

§1. First and Second Problems of Elasticity (Statics)

1. Theorem on Completeness for Problem (I)+

2. First Version (Problem I)

3. Second Version (Problem I)

4. Third Version (Problem I)

5. Theorem on Completeness for Problem (II)+

6. First Version (Problems (II)+ and (II)-)

7. Second Version (Problem II)

8. Third Version (Problem II)

§2. Other Problems (Statics)

1. Problems III and IV

2. Problem VI

3. Mixed Problems

§3. Static and Oscillation Boundary-Contact Problems

1. Static Boundary-Contact Problems

2. Oscillation Problems

§4. Boundary Value Problems of Thermoelasticity

§5. Numerical Examples

§6. Method of Successive Approximations

1. Problems for Homogeneous Media

2. Boundary-Contact Problems

§7. Concluding Remarks. Bibliographic References

Problems

Chapter XIV Representation of Solutions by Series and Quadratures

§1. Effective Solution of Boundary Value Problems of Elasticity for a Sphere and a Spherical Cavity in an Infinite Medium

1. Problems (I)±

2. Problems (II)±

3. Problems (III)±

4. Problems (IV)±

§2. Boundary Value and Some Other Problems for a Transversally-Isotropic Elastic Half-Space and an Infinite Layer

1. Auxiliary Formulas

2. Boundary Value Problems and Uniqueness Theorems for a Half-Space

3. Solution of Problem (I)+ for a Half-Space

4. Solution of Problem (II)+ for a Half-Space

5. Solution of Problem (III)+ for a Half-Space

6. Problem of the Action of a Rigid Punch on the Elastic Half-Space and Related Problems

7. Effective Solution of the Problem of a Rigid Punch for Some Specific Cases

8. Effective Solution of the Problem of a Crack for Some Specific Cases

9. Solution of Problem (II)+ for an Infinite Layer

§3. Application of Some New Representations of Harmonic Functions and of the Symmetry Principle for the Effective Solution of Elasticity Problems

1. Some Specific Functions Related to Elastic Displacements

2. Continuation of the Solutions

3. Effective Solution of Some Three-Dimensional Boundary Value Problems

§4. Problems of Thermoelasticity in Infinite Domains Bounded by a System of Planes

1. Formulation of Problems for a Half-Space

2. Fundamental Solutions and Representation Formulas for System (4.3), (4.4)

3. Solution of Problems A and B for Systems (4.11). Uniqueness Theorems

4. Solution of Problems V and VI for a Half-Space

5. Theorems on the Symmetry Principle for System (4.11)

6. Solution of Some Boundary Value Problems for System (4.11) in a Quarter of Space

7. Solutions of Dirichlet, Neumann and Mixed Problems for the Metaharmonic Equation in a Quarter of Space

8. Solution of Dirichlet, Neumann and Mixed Problems for the Inhomogeneous Metaharmonic Equation in a Quarter of Space

9. Solution of Problems V, VI and the Mixed Problem in a Quarter of Space for Thermoelastic Equations

10. Solution of Boundary Value Problems for System (4.11) in a Rectangular Trihedron (One Eighth of Space)

11. Solution of Problems V, VI and Mixed Problems for Equations of Thermoelasticity in D+

§5. (Continuation). Application of Fourier’s Integral

1. Representation of Solutions of Thermoelasticity Equations

2. Solution of Problem I for a Half-Space

3. Solution of Problem II for a Half-Space

4. Other Problems

5. Theorems on the Symmetry Principle for Equations of Thermoelasticit

6. Boundary Value Problems for a Quarter of Space

7. Boundary Value Problems for an Infinite Rectangular Trihedron

Problems

Bibliography

List of Institutions

Subject Index

Author Index

List of Principal Notations

## Description

North-Holland Series in Applied Mathematics and Mechanics, Volume 25: Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity focuses on the theory of three-dimensional problems, including oscillation theory, boundary value problems, and integral equations.

The publication first tackles basic concepts and axiomatization and basic singular solutions. Discussions focus on fundamental solutions of thermoelasticity, fundamental solutions of the couple-stress theory, strain energy and Hooke’s law in the couple-stress theory, and basic equations in terms of stress components. The manuscript then examines uniqueness theorems and singular integrals and integral equations.

The book ponders on the potential theory and boundary value problems of elastic equilibrium and steady elastic oscillations. Topics include basic theorems of the oscillation theory, existence of solutions of boundary value problems, integral equations of the boundary value problems, and boundary properties of potential-type integrals. The publication also reviews mixed dynamic problems, couple-stress elasticity, and boundary value problems for media bounded by several surfaces.

The text is a dependable source of data for mathematicians and readers interested in three-dimensional problems of the mathematical theory of elasticity and thermoelasticity.

## Details

- Language:
- English

- Copyright:
- © North Holland 1979

- Published:
- 1st January 1979

- Imprint:
- North Holland

- eBook ISBN:
- 9780080984636