Some Basic Problems of the Mathematical Theory of Elasticity

TO THE FIRST ENGLISH EDITION. In preparing this translation, I have taken the liberty of including footnotes in the main text or inserting them in small type at the appropriate places. I have also corrected minor misprints without special mention .. The Chapters and Sections of the original text have been called Parts and Chapters respectively, where the latter have been numbered consecutively. The subject index was not contained in the Russian original and the authors' index represents an extension of the original list of references. In this way the reader should be able to find quickly the pages on which anyone reference is discussed. The transliteration problem has been overcome by printing the names of Russian authors and journals also in Russian type. While preparing this translation in the first place for my own informa­ tion, the knowledge that it would also become accessible to a large circle of readers has made the effort doubly worthwhile. I feel sure that the reader will share with me in my admiration for the simplicity and lucidity of presentation.

Fundamental Equations of the Mechanics of an Elastic Body.- Analysis of Stress.- Analysis of Strain.- The Fundamental Law of the Theory of Elasticity. The Basic Equations.- General Formulae of the Plane Theory of Elasticity.- Basic Equations of the Plane Theory of Elasticity.- Stress Function. Complex Representation of the General Solution of the Equations of the Plane Theory of Elasticity.- Multi-Valued Displacements. Thermal Stresses.- Transformation of the Basic Formulae for Conformal Mapping.- Solution of Several Problems of the Plane Theory of Elasticity by Means of Power Series.- On Fourier Series.- Solution for Regions, Bounded by a Circle.- The Circular Ring.- Application of Conformal Mapping.- On Cauchy Integrals.- Fundamental Properties of Cauchy Integrals.- Boundary Values of Holomorphic Functions.- Application of Cauchy Integrals to the Solution of Boundary Problems of Plane Elasticity.- General Solution of the Fundamental Problems for Regions Bounded by One Contour.- Solution of the Fundamental Problems for Regions Mapped on to a Circle by Rational Functions. Extension to Approximate Solution for Regions of General Shape.- Solution of the Fundamental Problems for the Half-Plane and for Semi-Infinite Regions.- Some General Methods of Solution of Boundary Value Problems. Generalizations.- Solution of the Boundary Problems of the Plane Theory of Elasticity by Reduction to the Problem of Linear Relationship.- The Problem of Linear Relationship.- Solution of the Fundamental Problems for the Half-Plane and for the Plane with Straight Cuts.- Solution of Boundary Problems for Regions, Bounded by Circles, and for the Infinite Plane, Cut Along Circular Arcs.- Solution of the Boundary Problems for Regions, Mapped on to the Circle by Rational Functions.- Extension, Torsion and Bending of Homogeneous and Compound Bars.- Torsion and Bending of Homogeneous Bars (Problem of Saint-Venant).- Torsion of Bars Consisting of Different Materials.- Extension and Bending of Bars, Consisting of Different Materials with Uniform Poisson’s Ratio.- Extension and Bending for Different Poisson’s Ratios.