Theory of Elasticity – Amenzade

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The theory of elasticity is concerned with the mechanics of deformable media which, after the removal of the forces producing deformation, completely recover their original shape and give up all the work expended in the deformation.

The first attempts to develop the theory of elasticity on the basis of the concept of a continuous medium, which enables one to ignore its molecular structure and describe macroscopic phenomena by the methods of mathematical analysis, date back to the first half of the eighteenth century.
The fundamental contribution to the classical theory was made by R. Hooke, C. L. M. H. Navier, A. L. Cauchy, G. Lame, G. Green, B. P. E. Clapeyron. In 1678 Hooke established a law linearly con- necting stresses and strains.

After Navier established the basic equations in 1821 and Cauchy developed the theory of stress and strain, of great importance in the development of elasticity theory were the investigations of B. de Saint Venant. In his classical work on the theory of torsion and bending Saint Venant gave the solution of the problems of torsion and bending of prismatic bars on the basis of the general equations of the theory of elasticity. In these investigations Saint Venant devised a semi-inverse method for the solution of elasticity problems, formulated the famous Saint Venant’s principle, which enables one to obtain the solution of elasticity problems. Since then much effort has been made to develop the theory of elasticity and its applications, a number of general theorems have been proved, the general methods for the integration of differential equations of equilibrium and motion have been proposed, many special problems of fundamental interest have been solved. The development of new fields of engineering demands deeper and more extensive studies of the theory of elasticity. High velocities call for the formulation and solution of complex vibrational problems. Lightweight metallic structures draw particular attention to the question of elastic stability. The concentration of stress entails dangerous consequences, which cannot safely be ignored.

Notation 9
Introduction 13

Chapter I. ELEMENTS OF TENSOR CALCULUS 15

Scalars, Vectors, and Tensors 16
Addition, Multiplication, and Contraction of Tensors. The Quotient Law of Tensors 19
The Metric Tensor 22
Differentiation of Base Vectors. The Christoffel Symbols 28
A Parallel Field of Vectors 30
The Riemann-Christoffel Tensor. Derivative of a Veetor. The Gauss-Ostrogradsky Formula. The 𝜀-tensor 32

Chapter II. THEORY OF STRESS 39

Types of External Forces 39
The Method of Sections. The Stress Vector 41
The Stress Tensor 43
Equations of Motion and Equilibrium in Terms of the Components of the Stress Tensor 44
Surface Conditions 47
Equations of Motion and Equilibrium Referred to a Cartesian Co-ordinate System 48
Equations of Motion and Equilibrium Referred to Cylindrical and Spherical Co-ordinates 49
Determination of the Principal Normal Stresses 52

Chapter III. THEORY OF STRAIN 55

The Finite Strain Tensor 55
The Small Strain Tensor 59
Strai Compatibility Equations 60
The Strain Tensor Referred to a Cartesian Co-ordinate System 61
Components of the Small Strain and Rotation Tensors Referred to Cylindrical and Spherical Co-ordinates 62
Principal Extensions 64
Strain Compatibility Equations in Some Co-ordinate Systems (Saint Venant’s Conditions) 65
Determination of Displacements from the Components of the Small Strain Tensor 66

Chapter IV. STRESS-STRAIN RELATIONS 69

Generalized Hooke’s Law 69
Work Done by External Forces 70
Stress Tensor Potential 71
Potential in the Case of a Linearly Elastic Body 75
Various Cases of Elastic Symmetry of a Body 75
Thermal Stresses 80
A Energy Integral for the Equations of Motion of an Elastic Body 80
Betti’s Identity 82
Clapeyron’s Theorem 82

Chapter V. COMPLETE SYSTEM OF FUNDAMENTAL EQUATIONS IN THE THEORY OF ELASTICITY 84

Equations of Elastic Equilibrium and Motion in Terms of Displacements 84
Equations in Terms of Stress Components 90
Fundamental Boundary Value Problems in Elastostatics. Uniqueness of Solution 93
Fundamental Problems in Elastodynamics 95
Saint Venant’s Principle (Principle of Softening of Boundary Conditions) 96
Direct and Inverse Solutions of Elasticity Problems. Saint Venant’s Semi-inverse Method 98
Simple Problems of the Theory of Elasticity 99

Chapter VI. THE PLANE PROBLEM IN THE THEORY OF ELASTICITY 108

Plane Strain 108
Plane Stress 111
Generalized Plane Stress 113
Airy’s Stress Function 115
Airy’s Function in Polar Co-ordinates. Lamé’s Problem 120
Complex Representation of a Biharmonic Function, of the Components of the Displacement Vector and the Stress Tensor 127
Degree of Determinancy of the Introduced Functions and Restrictions Imposed on Them 132
Fundamental Boundary Value Problems and Their Reduction to Problems of Complex Function Theory 138
Maurice Lévy’s Theorem 141
Conformal Mapping Method 142
Cauchy-type Integral 145
Harnack’s Theorem 151
Riemann Boundary Value Problem 151
Reduction of the Fundamental Boundary Value Problems to Functional Equations 154
Equilibrium of a Hollow Circular Cylinder 155
Infinite Plate with an Elliptic Hole 159
Solution of Boundary Value Problems for a Half-plane 164
Some Information on Fourier Integral Transformation 170
Infinite Plane Deformed Under Body Forces 174
Solution of the Biharmonic Equation for a Weightless Half-plane 177

Chapter VII. TORSION AND BENDING OF PRISMATIC BODIES 182

Torsion of a Prismatic Body of Arbitrary Simply Connected Cross Section 182
Some Properties of Shearing Stresses 187
Torsion at Hollow Prismatic Bodies 188
Shear Circulation Theorem 190
Analogies in Torsion 191
Complex Torsion Function 196
Solution of Special Torsion Problems 198
Bending of a Prismatic Body Fixed at One End 206
The Centre of Flexure 211
Bending of a Prismatic Body of Elliptical Cross Section 216

Chapter VIII. GENERAL THEOREMS OF THE THEORY OF ELASTICITY. VARIATIONAL METHODS 219

Betti’s Reciprocal Theorem 219
Principle of Minimum Potential Energy 220
Principle of Minimum Complementary Work—Castigliano’s Principle 222
Rayleigh-Ritz Method 224
Reissner’s Variational Principle 228
Equilibrium Equations and Boundary Conditions for a Geometrically Non-linear Body 230

Chapter IX. THREE-DIMENSIONAL STATIC PROBLEMS 232