Boundary Integral Equations In Elasticity Theory

by the author to the English edition The book aims to present a powerful new tool of computational mechanics, complex variable boundary integral equations (CV-BIE). The book is conceived as a continuation of the classical monograph by N. I. Muskhelishvili into the computer era. Two years have passed since the Russian edition of the present book. We have seen growing interest in numerical simulation of media with internal structure, and have evidence of the potential of the new methods. The evidence was especially clear in problems relating to multiple grains, blocks, cracks, inclusions and voids. This prompted me, when preparing the English edition, to place more emphasis on such topics. The other change was inspired by Professor Graham Gladwell. It was he who urged me to abridge the chain of formulae and to increase the number of examples. Now the reader will find more examples showing the potential and advantages of the analysis. The first chapter of the book contains a simple exposition of the theory of real variable potentials, including the hypersingular potential and the hypersingular equations. This makes up for the absence of such exposition in current textbooks, and reveals important links between the real variable BIE and the complex variable counterparts. The chapter may also help readers who are learning or lecturing on the boundary element method.

DIVHigh-level treatment of one-dimensional singular integral equations covers Holder Condition, Hilbert and Riemann-Hilbert problems, Dirichlet problem, more. 1953 edition. /div

Many problems in mathematical physics rely heavily on the use of elliptical partial differential equations, and boundary integral methods play a significant role in solving these equations. Stationary Oscillations of Elastic Plates studies the latter in the context of stationary vibrations of thin elastic plates. The techniques presented here reduce the complexity of classical elasticity to a system of two independent variables, modeling problems of flexural-vibrational elastic body deformation with the aid of eigenfrequencies and simplifying them to manageable, uniquely solvable integral equations. The book is intended for an audience with a knowledge of advanced calculus and some familiarity with functional analysis. It is a valuable resource for professionals in pure and applied mathematics, and for theoretical physicists and mechanical engineers whose work involves elastic plates. Graduate students in these fields can also benefit from the monograph as a supplementary text for courses relating to theories of elasticity or flexural vibrations.

This 2000 book provided the first detailed exposition of the mathematical theory of boundary integral equations of the first kind on non-smooth domains.

In preparing this translation for publication certain minor modifications and additions have been introduced into the original Russian text, in order to increase its readibility and usefulness. Thus, instead of the first person, the third person has been used throughout; wherever possible footnotes have been included with the main text. The chapters and their subsections of the Russian edition have been renamed parts and chapters respectively and the last have been numbered consecutively. An authors and subject index has been added. In particular, the former has been combined with the list of references of the original text, in order to enable the reader to find quickly all information on anyone reference in which he may be especially interested. This has been considered most important with a view to the difficulties experienced outside Russia in obtaining references, published in that country. Russian names have been printed in Russian letters in the authors index, in order to overcome any possible confusion arising from transliteration.

The present book deals with the finite-part singular integral equations, the multidimensional singular integral equations and the non-linear singular integral equations, which are currently used in many fields of engineering mechanics with applied character, like elasticity, plasticity, thermoelastoplasticity, viscoelasticity, viscoplasticity, fracture mechanics, structural analysis, fluid mechanics, aerodynamics and elastodynamics. These types of singular integral equations form the latest high technology on the solution of very important problems of solid and fluid mechanics and therefore special attention should be given by the reader of the present book, who is interested for the new technology of the twentieth-one century. Chapter 1 is devoted with a historical report and an extended outline of References, for the finite-part singular integral equations, the multidimensional singular integral equations and the non-linear singular integral equations. Chapter 2 provides a finite-part singular integral representation analysis in Lp spaces and in general Hilbert spaces. In the same Chapter are investigated all possible approximation methods for the numerical evaluation of the finite-part singular integral equations, as closed form solutions for the above type of integral equations are available only in simple cases. Also, Chapter 2 provides further a generalization of the well known Sokhotski-Plemelj formulae and the Nother theorems, for the case of a finite-part singular integral equation.

This book is devoted to the mathematical foundation of boundary integral equations. The combination of ?nite element analysis on the boundary with these equations has led to very e?cient computational tools, the boundary element methods (see e.g., the authors [139] and Schanz and Steinbach (eds.) [267]). Although we do not deal with the boundary element discretizations in this book, the material presented here gives the mathematical foundation of these methods. In order to avoid over generalization we have con?ned ourselves to the treatment of elliptic boundary value problems. The central idea of eliminating the ?eld equations in the domain and - ducing boundary value problems to equivalent equations only on the bou- ary requires the knowledge of corresponding fundamental solutions, and this idea has a long history dating back to the work of Green [107] and Gauss [95, 96]. Today the resulting boundary integral equations still serve as a major tool for the analysis and construction of solutions to boundary value problems.