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Boundary integral equations in elasticity theory. (English) Zbl 1046.74001
Solid Mechanics and Its Applications 99. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-0574-1/hbk). xiii, 268 p. (2002).
The application of complex variables to the biharmonic partial differential equation goes back to Goursat and Filon. Kolosov derived and applied complex variables to the solutions of two-dimensional problems of mathematical theory of elasticity. N. I. Muskhelishvili continued and developed this work. Complex variable methods in the theory of elasticity became internationally known with the publication of I. S. Sokolnikoff’s “Mathematical theory of elasticity” from 1946 [2nd ed. Rep. of the 1956 orig. publ. by McGraw-Hill, New York (Robert E. Krieger Publishing Company, Malabar) (1983; Zbl 0499.73006)] and the English edition of N. I. Muskhelihvili’s “Some basic problems of the mathematical theory of elasticity” [(P. Noordhoff Ltd., Groningen) (1953; Zbl 0052.41402)]. At that time numerical computation was extremely cumbersome. But when powerful personal computers are freely available, it is possible to get a deeper insight into the problems and to obtain numerical solutions together with knowledge of their accuracy.
The present book aims to present a powerful new tool of computational mechanics, complex variable boundary integral equations. The book is conceived as a continuation of the classical monograph by N. I. Muskhelishvili into the computer era, with many examples showing the potential and advantage of the analysis. The author belongs to a small group of mathematicians who are able to undertake the difficult task of continuing and extending Muskhelishvili’s monograph.
The first chapter contains a simple exposition of the theory of real variable potentials, including the hypersingular potential and hypersingular equations. This makes up for the absence of such exposition in current textbooks, and reveals important links between the real variable boundary integral equations and the complex counterparts. The chapter may also help readers who are learning or lecturing on the boundary element method.
The book emphasizes the importance of hypersingular equations and the computational advantage of using complex variables. These two concepts are combined in the complex variable hypersingular boundary integral equations. They are given for piecewise homogeneous media with cracks, inclusions and voids, for finite and infinite regions, for periodic and doubly periodic problems and for bonded half-planes. Also treated are a simple theory of complex variable hypersingular integrals of arbitrary order. The complex variable boundary element method is used to solve problems involving discontinuities on multiple surfaces, and the method is presented in detail. The method of mechanical quadratures, traditionally employed for isolated cracks, is also presented.
The monograph consists of an introduction and four parts.
Introduction (6 pp.): Puts the subject into a historical perspective by giving many references. Explains the scope of the book.
Part I: Method of potentials. (4 chapters, 64 pp): Reproduces the real boundary integral equations in a complex variable form.
Part II: Methods based on the theory by Kolosov-Muskhelishvili. (5 chapters, 95 pp): Follows the path of Kolosov-Muskhelishvili beginning from their classical formulae and employing the analytical nature of their functions. Contains sections concerned with systems of blocks with cracks, holes and inclusions and with integral representations of Kolosov-Muskhelishvili functions, periodic and doubly periodic problems, and problems for bonded half-planes and for circular inclusions. Complex hypersingular equations are introduced and widely used.
Part III: Theory of complex integral equations. (2 chapters, 33 pp): Contains the theory of complex variable hypersingular integral equations.
Part IV: Numerical solution of complex variable boundary integral equations. (3 chapters, 60 pp): Concentrates on numerical implementations of complex variable integral equations. Examples illustrate the wide applicability and high efficiency of the complex variable method of boundary elements.
Index (2 pp): Is not extensive. References (8 pp): The list is short: it includes 163 items. Consequently, it contains only a limited number of representative publications, many of review nature.
The subject matter is presented carefully, step by step with many helpful introductory pieces of text. And it is therefore a pleasure to read the book; the pleasant printing and binding also contribute to the pleasure.
The book is addressed to a wide range of readers: graduate students, academics, researchers and engineers. It may be of use to those who wants to calculate stresses, strains, stress intensity factors and effective properties of a medium with internal structure when dealing with problems of material science, fracture mechanics, micromechanics, soil and rock mechanics, geomechanics, civil and mechanical engineering. It may also serve as a textbook on the theory of real potentials, on the complex variable boundary integral equations and on the complex variable boundary element method. The theoretical results on the relation between real and complex variable boundary integral equations, on the equations for periodic and doubly periodic systems, for bonded half-planes, for circular inclusions, and on the theory of hypersingular equations are relatively new. They may be of interest to specialists in the theory of elasticity.

74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
74S15 Boundary element methods applied to problems in solid mechanics
65R20 Numerical methods for integral equations
74B05 Classical linear elasticity