## Theory of Elasticity – Amenzade

In this post, we will see the book Theory Of Elasticity by Yu. A. Amenzade. 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.

The book was translated from Russian by M. Konyaeva and was published in 1979 by Mir.

You can get the book here.

Follow us on The Internet Archive: https://archive.org/details/@mirtitles
Write to us: mirtitles@gmail.com
Fork us at GitLab: https://gitlab.com/mirtitles/
Add new entries to the detailed book catalog here.

## Contents

Notation 9
Introduction 13

Chapter I. ELEMENTS OF TENSOR CALCULUS 15

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

Chapter II. THEORY OF STRESS 39

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

Chapter III. THEORY OF STRAIN 55

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

Chapter IV. STRESS-STRAIN RELATIONS 69

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

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

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

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

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

Chapter VII. TORSION AND BENDING OF PRISMATIC BODIES 182

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

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

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

Chapter IX. THREE-DIMENSIONAL STATIC PROBLEMS 232

75. Kelvin’s and the Boussinesq-Papkovich Solutions 232
76. Doursinesa’s Elementary Solutions of the First and Second Kind 236
77. Pressure on the Surface of a Semi-infinite Body 238
78. Hertz’s Problem of the Pressure Between Two Bodies in Contact 240
79. Symmetrical Deformation of a Bedy of Revolution 246
80. Thermal Stresses 256

Chapter X. THEORY OF PROPAGATION OF ELASTIC WAVES 258

81. Two Types of Waves 258
82. Rayleigh Surface Waves 262
83. Love Waves 265

Chapter XI. THEORY OF THIN PLATES 268

84. Differential Equation for Bending of Thin Plates 268
85. Boundary Conditions 271
86. Bending Equation for a Plate Referred to Polar Co-ordinates 274
87. Symmetrical Bending of a Circular Plate 276
Literature 278
Subject Index 279 