Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity

Abstract

A modification of the form of the singular integral equation for the problem of a plane crack of arbitrary shape in a three-dimensional isotropic elastic medium is proposed. This modification consists in the incorporation of the Laplace operator δ into the integrand. The integral must now be interpreted as a finite-part integral. The new singular integral equation is equivalent to the original one, but simpler in form. Moreovet, its form suggests a new approach for its numerical solution, based on quadrature rules for one-dimensional finite part integrals with a singularity of order two. A very simple application to the problem of a penny-shaped crack under constant pressure is also made. Moreover, the case of straight crack problems in plane isotropic elasticity is also considered in detail and the corresponding results for this special case are also derived.