In this post, we will see the book *Some Basic Problems of The Mathematical Theory of Elasticity* by N. I. Muskhelishvili.

# About the book

This book reproduces, in a considerably revised and enlarged form, the contents of a course of lectures, delivered by the author in Spring 1931 at the invitation of the Seismological Institute of the Academy of Sciences of the U.S.S.R. before the scientific workers of the Institute, and of lectures delivered in 1932 before post-graduate students of the Physico- Mathematical Institute of Mathematics and Mechanics at the University of Leningrad. The lectures were intended for persons acquainted with the principles of the theory of elasticity and were to be devoted to separate fundamental questions the choice of which was largely left to me; author naturally dwelt on subject matter in which author had been working myself.Thus, this book deals only with a few chapters of the theory of elasticity each of which receives fairly complete treatment.

The book was translated from Russian by JRM Radok and was published in 1948.

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You can get the book here.

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# Contents

# Part I – Fundamental equations of the mechanics of an elastic body. 1

## CHAPTER 1. ANALYSIS OF STRESS. 5

## CHAPTER 2. ANALYSIS OF STRAIN 28

## CHAPTER 3. THE FUNDAMENTAL LAW OF THE THEORY OF ELASTICITY; THE BASIC EQUATIONS. 52

# Part II – General formulae of the plane theory of elasticity 85

## CHAPTER 4. BASIC EQUATIONS OF THE PLANE THEORY OF ELASTICITY. 89

## CHAPTER 5. STRESS FUNCTION. COMPLEX REPRESENTATION OF THE GENERAL SOLUTION OF THE EQUATIONS OF THE PLANE THEORY OF ELASTICITY. 104

## CHAPTER 6. MULTI-VALUED DISPLACEMENTS. THERMAL STRESSES. 157

## CHAPTER 7. TRANSFORMATION OF THE BASIC FORMULAE FOR CON-

FORMAL MAPPING. 166

# PART III – Solution of several problems of the plane theory of Aran by means of power series 187

## CHAPTER 8. ON FOURIER SERIES. 189

## CHAPTER 9. SOLUTION FOR REGIONS, BOUNDED BY A CIRCLE. 194

## CHAPTER 10. THE CIRCULAR RING. 218

## CHAPTER 11. APPLICATION OF CONFORMAL MAPPING. 237

# Part IV – On Cauchy Integrals 251

## CHAPTER 12, FUNDAMENTAL PROPERTIES OF CAUCHY INTEGRALS 253

## CHAPTER 13. BOUNDARY VALUES OF HOLOMORPHIC FUNCTIONS. 283

# Part V – Application of Cauchy integrals to the solution of boundary problems of plane elasticity

## CHAPTER 14, GENERAL SOLUTION OF THE FUNDAMENTAL PROBLEMS FOR REGIONS BOUNDED BY ONE CONTOUR. 303

## CHAPTER 15. SOLUTION OF THE FUNDAMENTAL PROBLEMS FOR

REGIONS MAPPED ON TO A CIRCLE BY RATIONAL FUNCTIONS. EXTENSION TO APPROXIMATE SOLUTION FOR REGIONS OF GENERAL SHAPE

## CHAPTER 16. SOLUTION OF THE FUNDAMENTAL PROBLEMS FOR THE HALF-PLANE AND FOR SEMI-INFINITE REGIONS 373

## CHAPTER 17. SOME GENERAL METHODS OF SOLUTION OF BOUNDARY VALUE PROBLEMS. GENERALIZATIONS. 395

# Part VI – Solution of the boundary of the plane theory of elasticity

by reduction to the problem of linear relationship 425

## CHAPTER 18. THE PROBLEM OF LINEAR RELATIONSHIP 427

## CHAPTER 19. SOLUTION OF THE FUNDAMENTAL PROBLEMS FOR THE HALF-PLANE AND FOR THE PLANE WITH STRAIGHT CUTS. 451

## CHAPTER 20. SOLUTION OF BOUNDARY PROBLEMS FOR REGIONS, BOUNDED BY CIRCLES, AND FOR THE INFINITE PLANE, CUT ALONG CIRCULAR ARCS. 504

## CHAPTER 21. SOLUTION OF THE BOUNDARY PROBLEMS FOR REGIONS, MAPPED ON TO THE CIRCLE BY RATIONAL FUNCTIONS 525

# Part VII –

Extension, torsion and bending of homogeneous and compound bars 557

## CHAPTER 22, TORSION AND BENDING OF HOMOGENEOUS BARS (PROBLEM OF SAINT-VENANT). 561

## CHAPTER 23. TORSION OF BARS CONSISTING OF DIFFERENT

MATERIALS. 597

## CHAPTER 24. EXTENSION AND BENDING OF BARS, CONSISTING OF DIFFERENT MATERIALS WITH UNIFORM POISSON’S RATIO 614

## CHAPTER 25. EXTENSION AND BENDING FOR DIFFERENT POISSON’S RATIOS 624

APPENDIX l. On the concept of a tensor 656

APPENDIX 2. On the determination of functions from their differentials in multiply connected regions 671

APPENDIX 3. Determination of a function of a complex variable from its real part. Indefinite integrals of holomorphic functions 682

Author Index 687

Subject Index 701

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