Three-dimensional analysis of surface cracks in an elastic half-space

A numerical method is presented for analyzing arbitrary planar cracks in a halfspace. The method is based on the fundamental solution for a dislocation loop in a half-space, which is derived from the Mindlin solution (Mindlin, Physics, Vol. 7, 1936) for a point force in a half-space. By appropriate replacement of the Burgers vectors of the dislocation by the differential crack-opening displacement, a singular integral equation is obtained in terms of the gradient of the crack opening. A numerical method is developed by covering the crack with triangular elements and by minimizing the total potential energy. The singularity of the kernel, when the integral equation is expressed in terms of the gradient of the crack opening, is sufficiently weak that all integrals exist in the regular sense and no special numerical procedures are required to evaluate the contributions to the stiffness matrix. The integrals over the source elements are converted into line integrals along the perimeter of the element and evaluated analytically. Numerical results are presented and compared with known results for both surface breaking cracks and near surface cracks.