Hypersingular Integral Equations Arising in the Boundary Value Problems of the Elasticity Theory

Abstract

The exact solutions of a class of hypersingular integral equations with kernels $\left( {s-x} \right)^{-2}$ , $\left( {\sin \frac{s-x}{2}} \right)^{-2}$ , $\left( {\sinh \frac{s-x}{2}} \right)^{-2},\cos \frac{s-x}{2}\left( {\sin \frac{s-x}{2}} \right)^{-2}$ , $\cosh \frac{s-x}{2}\left( {\sinh \frac{s-x}{2}} \right)^{-2}$ are obtained where the integrals must be interpreted as Hadamard finite-part integrals. Problems of cracks in elastic bodies of various canonical forms under antiplane and plane deformations, where the crack edges are loaded symmetrically, lead to such equations. These problems, in turn, lead to mixed boundary value problems of the mathematical theory of elasticity for a half-plane, a circle, a strip and a wedge.