In-plane perturbation of a system of two coplanar slit-cracks – II: Case of close inner crack fronts or distant outer ones

The study of the in-plane perturbation of a system of two coplanar slit-cracks carried out in Part I is specialized to the case where the distance between the inner crack fronts is small, or equivalently that between the outer fronts large. The limit process involved is complex because of appearance of a “boundary layer” in the limiting case considered; this boundary layer occurs near the origin in the Fourier space used to determine the unknown components of the fundamental kernel looked for. A technique of matched asymptotic expansions is used to tackle this difficulty.The problem is thus reduced to determining two unknown functions only, which characterize the “interactions” between the two inner fronts. These functions obey a system of nonlinear differential equations in Fourier’s space, which are solved analytically near the origin and numerically in general. The results evidence a very slow decrease of long-range interactions between distinct points on the same front or distinct ones. This represents a striking difference with respect to the cases considered earlier of a single semi-infinite crack and a single slit-crack.