The application of invariant integrals in diffusive elastic solids

Abstract

Three techniques for deducing near crack tip singular fields from far field stress and pore pressure information are developed for the diffusive elastic theories of Biot: ( a ) methods based on a 'pseudo’ energy-momentum tensor in the Laplace transformed domain; as a generalization of the energy-momentum tensor of Eshelby; methods based on a reciprocal theorem in the Laplace transform domain; ( c ) methods based on a reciprocal theorem in real time. All of the methods relate near crack tip singular fields to far field information. In the most difficult cases, method ( a ) gives coefficients of singular stress fields and singular pore pressure gradients combined rather than separately. Nevertheless, this method is used to show that, remarkably, the complicated shear crack tip results derived by Craster & Atkinson can be checked in special circumstances. Methods ( b ) and ( c ) require appropriate dual functions. Versions of these dual functions are determined. Combinations of all three methods can, of course, be used in conjunction with numerical methods. All three methods are illustrated first by using the diffusion equation and then by using the full poroelastic equations.