A singular perturbation approach to integral equations occurring in poroelasticity

Singular perturbation theory is used to solve the integral equationswhich occur when treating finite-length crack problems in porouselastic materials. The method provides the stress intensityfactors which characterize the near crack tip stress and displacementfields for small times. The method also gives the stress andpore pressure fields on the fracture plane for small times relativeto the diffusive time scale. In this paper, the authors treatcrack problems which are unmixed in the pore pressure boundarycondition on the fracture plane. The Abelian result that smalltimes correspond, in Laplace transform space, to large valuesof the transform variable is used to formulate the problemsin terms of a small parameter. Rescaling on this small parameterleads to inner problems which are eigensolutions of the semi-infiniteproblems treated earlier by the authors. The outer solutionsare given by elastic eigensolutions together with appropriatefluid dipole responses. These outer solutions give the completestress and pore pressure fields except in the neighbourhoodof the crack tips; in this region the outer solutions are asymptoticallymatched with inner solutions. The full outer solutions are givenhere as an asymptotic expansion for small times and enable thedevelopment of the outer fields to be followed in real time.A reciprocal theorem in Laplace transform space is used to checkthe small-time solutions. The inner problem is rescaled to asemi-infinite crack problem, so eigensolutions of this semi-infiniteproblem are used together with the known asymptotic behaviourof the real solution to identify the stress intensity factor.The stress intensity factor is then related to an integral involvingthe inner limit of the outer solution together with the eigensolutionof the semi-infinite problem. Using this integral, we recoverthe result for the stress intensity factor found using singularperturbation theory. A ‘nearly’ invariant integralanalogous to the invariant M integral used in elastostaticsis derived. Unfortunately, the poroelastic analogue is not invariant,although it is used to verify the small-time results.