Introduction to English Edition
This book, which was first published in Russian in 1955, treats the
solution of certain classical problems of the mathematical theory of
elasticity. Beside the revised derivation of previously known solutions,
it presents material on three-dirnensional problems of the theory of
elasticity published by the author over the past years.
Assuming that the reader is familiar with the general foundations of
the mathematical theory of elasticity and the solution of the simplest
problems, the author has concentrated attention on more specialized
questions; they have been enumerated in the Contents, and make repeti-
tion here superfluous. The author has striven to give an independent
exposition of the separate chapters. However, a uniformity of method
has been maintained, namely the representation of the solutions in terms
of the Papkovich-Neuber harmonic functions. In many cases, this per-
mits a systematic derivation of the solutions and the avoidance of "guess-
ing" and trial substitujions, from which success can be expected only if
one has already sufficient experience in the solution of complicated
boundary-value probÍems. The method of Papkovich-Neuber is not used
in Chapters 3 and 4, because the application of the symbolic method
adopted in these chapters to the displacement equations of equilibrium
in the theory of elasticity leads more quickly to the representation of the
solutions of problems concerning the elastic layer and plate.
Of course, the book does not give a complete coverage of the topic;
there is no treatment of the questions of existence of solutions and the
application of integral equations, no exposition of the problems of Saint
Venant, or of Michell and Almansi on the deformation of prismatic beams.
Together with other omissions, the reader will also find no account of the
excellent achievements of H. Neuber on the problem of stress concentra-
tion in notches and cavities.
The German translation of the book appeared in 1963 (Akademie-
Verlag, Berlín).
Leningrad, 1963
A. I. LUR'E
Author(s): Anatoli Isakievich Lur´e
Publisher: Interscience Publishers (Wiley)
Year: 1964
Language: English
Commentary: Original edition: Moscow, 1955
Pages: 506
City: New York, London, Sydney
Preface
Introduction to English Edition
Chapter 1. THE FUNDAMENTAL EQUATIONS OF THE MATHEMATICAL
THEORY OF ELASTICITY
1.1 The stress tensor. The static equations of a continuous medium
1.2 The deformation of a continuous medium . . . . . . . .
1.3 Certain tensor operations . . . . , . . . . . . . . . . . .
1.4 The determination of the displacements from the strain tensor
1.5 Stress functions . . . . . . . . . . . . . . . . . . . .
1.6 Orthogonal curvilinear coordinates . . . . . . . . . . . .
1. 7 The basic equations for the mechanics of continuous media in
curvilinear coordinates . . . . . . . . . . . . . . . . .
1.8 The relationship between the stress and strain tensors in an iso-
tropic elastic ,body (generalized Hooke's law) . . . . . . .
1.9 The differential equations of the theory of elasticity in terms
of displacements . . . . . . . . . . . . . . . . . . . .
1.10 The Papkovich-Neuber form of the solutions of the equilib-
rium equations of the theory of elasticity in displacements
1.11 The differential equations of the theory of elasticity in terms
of stresses . . . . . . . . . . . . . . . . . . . . . . .
1.12 The relationship between the general solution and the tensor
stress functions . . . . . . . . . . . . . .
1.13 Body forces with a potential. Thermal stresses
Notes and Bibliography to Chapter 1. . . . . .
Chapter 2. THE INFINITE ELASTIC MEDIUM AND THE ELASTIC
HALF-SPACE
2.1 Effect of a concentrated force in an infinite elastic medium
68
2.2
Solutions of the equations of the theory of elasticity corre-
sponding to singular points .
74
2.3
The effect of a system of forces distributed over a small volume
78
2.4
The effect of distributed singularities .
83
2;5
Effect of a concentrated force and distributed loading normal
to the boundary plane of an elastic half-space 87
2.6 Continuous load distributions
97
2.7 Non-uniform loading over an elliptic region.
110
2.8 The state of stress in an elastic half-space with given surface
tractions. . . . . . . . . . . . . . . . . 119
2.9 A concentrated force in an elastic half-space . 132
2.10 Problems of the equilibrium of an elastic cone 137
Notes and Bibliography to Chapter 2. . . . . . 142
Chapter 3. THE EQUILIBRIUM OF AN ELASTIC LAYER
3.1 Formulation of the problem. Extension and bending of a layer 145
3.2 A symbolic method for the construction of the solutions 148
3.3 The introduction of stress functions .
3.4 Determination of the stress functions
3.5 Compression of an elastic layer. . .
3.6 Flexure of an elastic layer . . . . .
3.7 The action of body forces. Thermal stresses in a layer
Notes and Bibliography to Chapter 3. . . . . .
Chapter 4. THE EQUILIBRIUM OF A THICK PLATE
4.1 The homogeneous solution .
4.2 Non-homogeneous solutions .
4.3 Equilibrium of a thick circular disc
4.4 Thermal stresses in plates . . . .
Notes and Bibliography to Chapter 4
Chapter 5. THREE-DIMENSIONAL CONTACT PROBLEMS
5.1 Formulation of the problem of the action of a rigid punch on an
elastic half-space . . . . . . . . . . . . . . . . . . . . 251
5.2 Method of solution of the problem of the rigid punch . . . . 255
5.3 The special case of elliptic coordinates. Determination of the
harmonic functions w and w1 . . . . . . . . . 259
5.4 Flat punch with a circular base . . . . . . . . . 271
5.5 Punch with curved base and circular cross-section 274
5.6 The blunt punch . . . . . . . . . 279
5. 7 The conical punch . . . . . . . . . 284
5.8 The general case of elliptic coordinates 286
5.9 The flat elliptic punch . . . . . . . 298
5.10 The elliptic punch with curved base . . 303
5.11 Contact of elastic bodies (Hertz problem) 314
Notes and Bibliography to Chapter 5. . . . 323
Chapter 6. THE DEFORMATION OF A SYMMETRICALLY LOADED
ELASTIC SPHERE
6.1 The general form of the solution of the problem of the equilib-
rium of a symmetrically loaded body of revolution . . . . . 325
6.2 Expressions for the displacements and stresses in terms of solid
spherical harmonics . . . . . . . . . . . . . . 327
6.3 The particular solution corresponding to a body force 339
6.4 The study of the simplest particular problems 341
6.5 The revolving sphere . . . . . . . . . . . . . . 347
6.6 The internal problem for the sphere . . . . . . . 350
6.7 Compression of an elastic sphere by concentrated forces 361
6.8 The equilibrium of a heavy sphere . . . . . . . . . . 367
6.9 The state of stress in the vicinity of an ellipsoidal cavity 370
Notes and Bibliography to Chapter 6. . . . . . . . . . . 377
Chapter 7. THE DEFORMATION OF A SYMMETRICALLY LOADED ELAS-
TIC CIRCULAR CYLINDER
7.1 Survey of the contents of the chapter. Basic relations 380
7 .2 Elementary solutions . . . . . . . . . . . . . . . 386
7.3 Polynomial solutions of the problem of the equilibrium of a
cylinder . . . . . . . . . . . . . . . . . . . . . . . . 388
7.4 Sinusoidal loading on the side surface of a cylinder. The homo-
geneous solutions . . . . . . . . . . . . . . . . . . . 393
7.5 The deformation of a cylinder of finite length, loaded over
the side surface. The method of trigonometric series . . . . 399
7.6 Deformation of an infinite cylinder, loaded over a strip of its
side surface. Application of Fourier integrals . . . . . . . . 404
7.7 The "banded cylinder". Normal loading of arbitrary sign on
the side surface . . . . . . . . . . . . . 416
7.8 Shear loading over a strip of the side surface . 418
7.9 Boundary conditions at the ends of a cylinder 428
Notes and Bibliography to Chapter 7. . . . . . 437
Chapter 8. THE GENERAL PROBLEM OF THE EQUILIBRIUM OF AN
ELASTIC SPHERE
8.1 Formulation of the problem. Certain properties of spherical
solid harmonics 440
8.2 Boundary value problems for the solid sphere. The case when
displacements are prescribed on the surface of the sphere . . 446
8.3 The case of tractions prescribed on the surface of the sphere . 449
· 8.4 The solution of boundary value problems for a spherical cavity
in an unbounded elastic medium. . . . . . . . . . . . . 459
8.5 The equilibrium of a hollow elastic sphere for displacements
prescribed on the boundaries . . . . . . . . . . . . . . 462
8.6 The equilibrium of an elastic hollow sphere for forces prescribed
on the boundaries . . . . . . . 472
Notes and Bibliography to Chapter 8. 483
Supplementary Bibliography 485
Author Index . 489
Subject Index .491
