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Applications of potential theory in mechanics. A selection of new results. (English) Zbl 0744.73016
Mathematics and its Applications (Dordrecht). 51. Dordrecht etc.: Kluwer Academic Publishers. xiii, 467 p. (1989).

This book carries on the tradition of looking for integral representations of solutions to boundary value problems for the Laplacian. Just as the solution of the Poisson equation in a regular domain ${\Omega }$: ${\Delta }u=f$ in ${\Omega }$, $u=0$ on $\partial {\Omega }$, is a mixture of Green functions solutions of ${\Delta }u={\delta }_{x}$ in ${\Omega }$, $u=0$ on $\partial {\Omega }$, and the solution of the Dirichlet problem ${\Delta }u=0$ in ${\Omega }$, $u=g$ on $\partial {\Omega }$ is a mixture of solutions of ${\Delta }u=0$ in ${\Omega }$, $u={\delta }_{x}$ on $\partial {\Omega }$, such fundamental solutions having explicit analytic forms for some geometry of the domain ${\Omega }$, similarly, the solutions to mixed problems with boundary conditions partially of Dirichlet type partially of Neumann type possess integral representations by means of fundamental solutions which are explicit for some special geometrical cases. This is the subject of the book of V. I. Fabrikant as well as applications in fracture mechanics and to punch and contact problems.

In chapter 1 a closed form exact solution is given to the non- axisymmetric mixed homogeneous problem for the Laplacian in the half- space, with Dirichlet conditions prescribed inside a circle and Neumann conditions on the outside, or vice-versa. In chapter 2 these formulas are used to express the solution of several mixed boundary problems for the elastic half-space. Methods of chapters 1 and 2 are extended in chapter 3 to more general situations allowing to obtain the field of stresses and deplacements and yielding therefore an answer to crack problems studied in chapter 4 and to punch problems and contact problems studied in chapter 5.

The style is difficult: There is sometimes an accumulation of formulas without any references to mathematical concepts which could clarify the setting. Furthermore, the exposition is based entirely on the author’s works presented as ‘the new method’ so that the interest of the reader, which could be a little bit wider, is somewhat forgotten.

Nevertheless this book is certainly an important contribution which brings efficient tools for engineers especially with the development of symbolic calculus on computers.

##### MSC:
 74B05 Classical linear elasticity 74-01 Textbooks (mechanics of deformable solids) 74B99 Elastic materials 74H99 Dynamical problems in solid mechanics 31-01 Textbooks (potential theory) 31A25 Boundary value and inverse problems (two-dimensional potential theory) 74R99 Fracture and damage 74A55 Theories of friction (tribology) 74M15 Contact (solid mechanics) 31A10 Integral representations of harmonic functions (two-dimensional)