The Griffith formula and the Rice–Cherepanov integral for crack problems with unilateral conditions in nonsmooth domains

Abstract

As a paradigm for non-interpenetrating crack models, the Poisson equation in a nonsmooth domain in R² is considered. The geometrical domain has a cut (a crack) of variable length. At the crack faces, inequality type boundary conditions are prescribed. The behaviour of the energy functional is analysed with respect to the crack length changes. In particular, the derivative of the energy functional with respect to the crack length is obtained. The associated Griffith formula is derived, and properties of the solution are investigated. It is shown that the Rice–Cherepanov integral defined for the solutions of the unilateral problem defined in the nonsmooth domain is path-independent. Finally, a non-negative measure characterising interaction forces between the crack faces is constructed.