Crack problems in a poroelastic medium: an asymptotic approach

Abstract

We consider problems involving semi-infinite cracks in a porous elastic material. The cracks are loaded with a time dependent internal stress, or pore pressure. Either mixed or unmixed pore pressure boundary conditions on the fracture plane are considered. An asymptotic procedure that partly uncouples the elastic and fluid responses is used, allowing an asymptotic expression for the stress intensity factors as time progresses to be obtained. The method allows the physical processes involved at the crack tip and their interactions to be studied. This is an advance on previous methods where results were obtained in Laplace transform space and inverted numerically to obtain real-time solutions. The crack problems are formulated using distributions of dislocations (and pore pressure gradient discontinuities when necessary) to generate integral equations of the Wiener—Hopf type. The resulting functional equations are, of course, identical to those considered by C. Atkinson and R. V. Craster, but with the alternative formulation we develop an asymptotic procedure which should be applicable to other problems (e.g. finite length cracks). This asymptotic procedure can be used to derive asymptotic expansions for more complicated loadings when the numerical effort involved in evaluating results would be excessive. A large-time asymptotic method is also briefly described which complements the small-time method. The operators for poroelastic crack problems are inverted for a particular loading; the reciprocal theorem for poroelasticity is used together with eigensolutions of the fundamental problems to deduce the stress (or where necessary the pore pressure gradient) intensity factors for any loading. These formulae extend previous results allowing a wide range of different loadings to be considered. As an example, the stress intensity factor for a point loaded crack is derived and the asymptotic method is applied to this problem to derive a simple asymptotic formula. Finally, an invariant integral, which is a generalization of the Eshelby energy-momentum tensor, is used to derive integral identities which serve as a check on the intensity factors in some situations.