On singular integral equations and fundamental solutions of poroelasticity

This paper presents a unified formulation of the various singular integral equations used in the boundary element methods (BEM) for the solution of linear, quasi-static, anisotropic poroelasticity. In particular, a derivation is provided that connects the "direct method" with the "indirect methods". The presentation begins with an alternative derivative of the time and space dependent reciprocal integral. The Somigliana-type integral equations for the direct BEM are first constructed. By summing integral equations representing an interior and an exterior domain problem, the Somigliana (displacement discontinuity) and Volterra (stress discontinuity) type dislocation equations for indirect BEM are obtained. An extension to the edge dislocation method is discussed. These stress and displacement discontinuity equations are then combined to construct a symmetric Galerkin integral equation system. Through such construction, many intriguing connections among Green's functions of fluid source, dipole, dilatation, fluid body force, total body force, and displacement discontinuity are revealed. Finally, a complete compilation of fundamental solutions for the isotropic case is provided.