In this post, we will look at the book *Integral Equations In Elasticity* by V. Z. Parton, P. I. Perlin.

**About the book**

This book presents the fundamentals of the theory of regular and singular integral equations in the case of one and two variables. The general principles of the theory of approximate methods are considered as well as their application for the efficient solution of both regular and singular integral equations. The necessary information is given on the three-dimensional and two-dimensional equations of the theory of elasticity including the formulation of boundary value problems. The book contains the derivation and analysis of various integral equations of the plane problem for both fundamental boundary value problems and mixed problems, and also for bodies with cuts. In the three-dimensional case the construction and analysis of integral equations are carried out for the first and second fundamental problems.

Emphasis is placed on efficient methods for solving integral equations for the plane and three-dimensional problems of elasticity. Examples are given illustrating the advantages of a particular approach. The book is appended with an extensive list of references giving comprehensive information of the subject of investigation.

The emphasis on numerical methods for the solution of integral equations for elastostatic problems corresponds to the author’s conviction that this approach has considerable promise, particularly with the advent of the nearest-generation computers.

The scope of the book is limited to elastostatic problems though the extension of the methods described to dynamic problems apparently involves no fundamental difficulties.

The book was translated from the Russian by ???? and was published by Mir in 1982.

Many thanks to *Akbar Azimi* for the scans.

The Internet Archive Link

**Contents **

Preface to the English Edition 7

Preface to the Russian Edition 8

On the Formation of Integral Equation Methods in the Theory of Elasticity by D. I. Sherman 10

Notation 19

**Chapter 1 ELEMENTS OF THE THEORY OF ONE-DIMENSIONAL AND MULTIDIMENSIONAL INTEGRAL EQUATIONS **

1. Analytic Theory of a Resolvent 21

2. Cauchy-type Integral 35

3. Riemann Boundary Value Problem 48

4. Singular Integral Equations 52

5. Riemann Boundary Value Problem in the Case of Discontinuous Coefficients and Unclosed Contours 64

6. Singular Integral Equations in the Case of Discontinuous Coefficients and Unclosed Contours 71

7. Two-dimensional Singular Integrals 75

8. Two-dimensional Singular Integral Equations 89

**Chapter II APPROXIMATE METHODS FOR SOLVING INTEGRAL EQUATIONS **

9. General Principles of the Theory of Approximate Methods 98

10. Method of Successive Approximations 105

11. Mechanical Quadrature Method for Regular Integral Equations 111

12. Approximate Methods for Solving Singular Integral Equations 114

13. Approximate Methods for Solving Singular Integral ^

Equations (Continued) 120

**Chapter III FUNDAMENTAL PRINCIPLES OF THE MATHEMATICAL THEORY OF ELASTICITY **

14. Three-dimensional Problem 137

15. Plane Problem 137

16. Bending of Thin Plates 143

17. On Singular Solutions of Elastic Equations 148

**Chapter IV INTEGRAL EQUATIONS FOR TWO-DIMENSIONAL PROBLEMS OF THE THEORY OF ELASTICITY **

18. Muskhelishvili’s Integral Equations 155

19. Sherman-Lauricella Integral Equations 159

20. Sherman-Lauricella Integral Equations (Continued) 164

21. Multiply (Doubly) Connected Regions 168

22. Problems of the Theory of Elasticity for Piecewise Homogeneous Bod ies 171

**Chapter V SOME SPECIAL TOPICS OF TWO-DIMENSIONAL ELASTICITY **

23. Problems of the Theory of Elasticity for Bodies with Cuts 175

24. Integral Equations for Mixed (Contact) Problems 179

25. Problems of the Theory of Elasticity for Bodies Bounded by Piecewise Smooth Contours 182

26. Method of Linear Relationship 186

27. Method of Linear Relationship (Continued) 189

**Chapter VI INTEGRAL EQUATIONS FOR FUNDAMENTAL THREE-DIMENSIONAL PROBLEMS OF THE THEORY OF ELASTICITY **

28. Generalized Elastic Potentials 199

29. Regular and Singular Integral-Equations for Fundamental Three-dimensional Problems 206

30. Extension of the Fredholm Alternatives to Singular Integral Equations of the Theory of Elasticity 215

31. Spectral Properties of Regular and Singular Integral Equations. Method of Successive Approximations 217

32. Differential Properties of Solutions of Integral Equations and Generalized Elastic Potentials 223

33. Approximate Methods of Solving Integral Equations for Fundamental Three-dimensional Problems 224

34. Problems of the Theory of Elasticity for Bodies Bounded by Several Surfaces 239

35. Three-dimensional Problems of the Theory of Elasticity for Bodies with Gut 244

36. Piecewise Homogeneous Bodies 253

37. Solution of Problems of the Theory of Elasticity for Bodies Bounded by Piecewise Smooth Surfaces 262

38. Mixed (Contact) Problems 269

Conclusion 274

References 277

Author Index 299

Subject Index 302