Crack propagation in heterogeneous media

We assume that an in®nite material contains a point inhomogeneity which has slightly di€erent elastic constants from the rest of the otherwise isotropic and homogeneous elastic material. We evaluate the path of a three- dimensional semi-in®nite quasi static crack propagating near this point inhomogeneity. The real space crack path is evaluated as well as the power spectrum of the crack surface. The latter enables us to calculate the roughness exponent for quasi-static fracture. It is also shown how the real space crack surface and its power spectrum can be obtained for an arbitrary distribution of inhomogeneities