Hybrid Green's functions for multiple cracks: Numerical extension of single crack analytical Green's functions to multiple cracks

Abstract

The Green's functions for the line force and dislocation that satisfy the traction free boundary condition on the surfaces of arbitrary multiple straight cracks in the isotropic solids are obtained. We develop the hybrid Green's functions combining the analytical and numerical Green's function methods. The Green's function is split into the singular and the image terms. The crack opening displacement, represented as the continuous distribution of dislocation dipoles over each crack, serves as the source of the image term. It is proposed that the image term from each crack is further split into two parts: analytical image term for the individual single crack and the additional numerical image term caused by the presence of other cracks. Although, for the single crack problem, the analytical image term is the only image term needed to satisfy the traction free boundary condition, additional image term is needed to satisfy the boundary condition for multiple cracks. The advantage of the analytical image term is its ability to absorb high stress gradient when the singularity is located near one of the cracks. The additional image term, given in terms of the additional dislocation dipole distribution, needs only to provide small and smooth perturbation, due to the presence of other cracks, to satisfy the traction free boundary condition.