Periodic elastic plane problems have many applications in the practice. The present book gives a systematic treatment of this subject using the method of complex variable (Kolosov-Muskhelishvili’s formulation). The knowledge of linear elasticity and the basic theory of boundary value problems of analytic functions is assumed.
The book consists of six chapters. In chapter I, the authors discuss two kinds of fundamental boundary value problems for analytic functions, namely the periodic Riemann and Riemann-Hilbert problems for closed contours and for open arcs. Here, the authors present some integral formulae for Hilbert kernel in the case of half-plane. A general expression for the complex potential of isotropic infinite elastic plane with periodically distributed holes of arbitrary shapes is discussed in chapter II, together with periodic welding problems and contact problems. Chapter III deals with periodic problems for anisotropic media. The authors reduce periodic contact problems in anisotropic plane elasticity to periodic Riemann-Hilbert boundary value problems in terms of certain integrals with Hilbert kernel. The problems of periodic moving loads on the boundary of an isotropic half-plane are studied in chapter IV. Crack problems for periodic rectilinear and collinear cracks are topics of chapter V, where the shape and number (in a period) of cracks may be arbitrary. Using conformal mappings, the authors study the crack problems by means of singular integral equations. Chapter VI gives a brief sketch of double periodic elastic problems for isotropic two-dimensional media in the general case, deriving expressions for complex potentials in double-periodic fundamental problems.
The book can be highly recommended as an advanced graduate text to students in applied mathematics or mechanics. It is an excellent addition to previous books on the applications of complex variable methods in elasticity.