A SINGULAR PERTURBATION APPROACH TO INTEGRAL-EQUATIONS OCCURRING IN POROELASTICITY

Singular perturbation theory is used to solve the integral equations w hich occur when treating finite-length crack problems in porous clasti c materials. The method provides the stress intensity factors which ch aracterize the near crack tip stress and displacement fields for small times. The method also gives the stress and pore pressure fields on t he fracture plane for small times relative to the diffusive time scale . In this paper, the authors treat crack problems which are unmixed in the pore pressure boundary condition on the fracture plane. The Abeli an result that small times correspond, in Laplace transform space, to large values of the transform variable is used to formulate the proble ms in terms of a small parameter. Rescaling on this small parameter le ads to inner problems which are eigensolutions of the semi-infinite pr oblems treated earlier by the authors. The outer solutions are given b y elastic eigensolutions together with appropriate fluid dipole respon ses. These outer solutions give the complete stress and pore pressure fields except in the neighbourhood of the crack tips; in this region t he outer solutions are asymptotically matched with inner solutions. Th e full outer solutions are given here as an asymptotic expansion for s mall times and enable the development of the outer fields to be follow ed in real time. A reciprocal theorem in Laplace transform space is us ed to check the small-time solutions. The inner problem is rescaled to a semi-infinite crack problem, so eigensolutions of this semi-infinit e problem are used together with the known asymptotic behaviour of the real solution to identify the stress intensity factor. The stress int ensity factor is then related to an integral involving the inner limit of the outer solution together with the eigensolution of the semi-inf inite problem. Using this integral, we recover the result for the stre ss intensity factor found using singular perturbation theory. A 'nearl y' invariant integral analogous to the invariant M integral used in el astostatics is derived. Unfortunately, the poroelastic analogue is not invariant, although it is used to verify the small-time results.