Книга: Applications of Potential Theory in Mechanics

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It is not easy to find something new in mathematics. It is difficult to find something new in something very old, like the Potential Theory, which was studied by the greatest scientists in the past centuries. And it is extremely difficult to make this find on an elementary level, with no mathematical apparatus involved which would be considered new even in the times of, say, Poisson.

This is exactly what the author claims to have done in this book: a new and elementary method is described for solving mixed boundary value problems, and their applications in engineering. The method can solve non-axisymmetric problems as easily as axisymmetric ones, exactly and in closed form. It enables us to treat analytically non-classical domains. The major achievements of the method comprise the derivation of explicit and elementary expressions for the Green’s functions, related to a penny-shaped crack and a circular punch, development of the Saint-Venant type theory of contact and crack problems for general domains, and investigation of various interactions between cracks, punches, and external loadings.

The method also provides, as a bonus, a tool for exact evaluation of various two-dimensional integrals involving distances between two or more points. It is believed that majority of these results are beyond the reach of existing methods.