On finite-length cracks and inclusions in poroelasticity

Abstract

A procedure is devised for the asymptotic solution of finite-length crack and inclusion problems under arbitrary time-dependent stress, pore pressure or pressure gradient loadings in porous elastic materials. The stress-intensity factors are identified explicitly as an expansion in real time for model problems involving uniformly loaded cracks. In the most general case this involves matched asymptotic expansions and rescalings that take advantage of the singular perturbation nature of the governing equations. The problems are written in terms of coupled Cauchy singular integral equations containing a small parameter ; these can be combined to form a single subsidiary integral equation. This approach is used to split the problems into pieces driven by elastic, or pore-pressure, dominated solutions. Outer solutions that are driven by elastic solutions and eigensolutions are identified ; these are matched to inner problems. The inner problems require the solution of a system of coupled integral equations ; these are formulated in terms of functional equations. Fortunately the system uncouples and allows the inner solutions to be found explicitly. The pore-pressure driven pieces are deduced, to leading order, by a direct rescaling.