Mathematical Methods of the Theory of Elasticity

Description

This is the second volume of the monograph ‘‘Mathematical Methods of the Theory of Elasticity”. The first volume, consisting of three chapters, included basic mathematical concepts used in the theory of elasticity, the main ideas of the theory of elasticity, and formulation of basic boundary value problems. Besides, some general problems of the theory of elasticity were discussed, including the reduction of the boundary value problems of the theory of elasticity to classical boundary value problems of mathematical physics, an investigation of the peculiarities of the solution in the vicinity of singular points of the boundary, Saint-Venant's principle and its applications in the formulation of two-dimensional problems of the theory of elasticity, etc. The second volume contains the main part of the monograph. The structure of this book, in accordance with its title, basically differs from the prevalent literature on the theory of elasticity which is classified according to the problems (bending and torsion of rods, the plane problem, the three-dimensional problem, etc.) and not according to the mathematical methods employed for solving them. The reverse approach, which was one of the compelling motives behind the writing of this book, helps in drawing the-reader’s attention to the very methods of solving problems. This is in keeping with the modern views which regard the theory of elasticity as a special applied branch of mathematical physics. It should be noted that while systematizing this course on the theory of elasticity in accordance with the mathematical methods, the authors did not strive to attain any kind of uniformity while describing the material of different chapters. In case a full-fledged theory exists, it has been described together with some illustrative examples (the chapters describing the theory of analytical functions and the potential theory belong to this category). On the contrary, the main stress in other cases has been laid on the solution of specific problems. The reason for this approach (adopted, for example, in the chapter on separation of variables) is that this comparatively easy method is fully clarified in the beginning of this treatise (in Ch. 1), and hence only such specific problems of the theory of elasticity are of any interest where important and constructive results can be achieved. A similar situation arises in Ch. 6 (Integral Representation and Transformations), though for quite different reasons. Since no universal methods are available for solving this type of problems, the mathematical apparatus can be developed only for specific problems. While selecting these problems, the authors were guided not only by the above-mentioned general criteria, but also by the novelty and originality of mathematical results, the importance of the problem for some field related to the theory of elasticity (for example, for fracture mechanics), and the possibility of obtaining general qualitative conclusions from the solutions of these problems. The chapter dealing with variational and difference methods (Ch. 8) has also been written in an illustrative manner, and is based on the solution of specific problems. This is so because the variational methods and, in particular, the difference methods, form one of the most exhaustively investigated areas of computational mathematics (even from the point of view of their application to the problems of the theory of elasticity). Hence a detailed description of these methods is not possible here in view of the limitations of space. However, the examples which have been chosen in this chapter for solving specific problems of the theory of elasticity amply demonstrate the advantages and drawbacks of these methods. The appendices contain a brief account of certain problems of continuum mechanics (whose constructive analysis is possible on the basis of linear theory of elasticity), as well as some problems of the linear theory of elasticity for an anisotropic medium. V. Z. Parton, P. I. Perlin Moscow, January, 1980