REDUCTION OF THE GENERAL FRACTURE COMPLIANCE MATRIX-Z TO ONLY 5 INDEPENDENT ELEMENTS

In a system of plane parallel fractures, it is often assumed that resistance to slip on the fractures is independent of the direction of that slip. The fracture system then is transversely isotropic in its elastic properties and can be characterized by just two numbers: the compliances normal and tangential to the fractures, for example. But any relief on the fracture surfaces will destroy the symmetry and demand additional elastic constants. In general, a system of plane parallel fractures has a 3 x 3 matrix Z of fracture compliances which contributes to the overall 6 x 6 compliance matrix S of the fractured solid. For transversely isotropic fractures Z has only two independent elements, but for general fractures all that is known is that Z must be symmetric. This implies that six parameters are needed to describe a fracture system with triclinic behaviour. We find, however, that there is always a coordinate rotation which sets a symmetric pair of the off-diagonal terms of Z to zero, so Z has, in fact, only five independent elements. The zeros of the rotated Z show that particle displacements tangential to the fractures and parallel to the new coordinate axes are decoupled from each other. Despite this decoupling, the medium is still fully triclinic because displacement normal to the fractures still couples with all the other displacements.