Conduction degradation in anisotropic multi-cracked materials

The electrical and thermal conduction properties of disordered solids and the possible degradation processes induced by the generation of cracks are central issues in the field of the heterogeneous materials. However, most of the existing theories are unable to consider an arbitrary density of cracks. We obtained an exact result for the fields induced within an elliptic anisotropic inhomogeneity embedded in a different anisotropic (two-dimensional) conductor. Then, we applied it to show that the degradation process strongly depends on the statistical orientational distribution of defects: in particular we theoretically prove that parallel cracks lead to the power law decay log σ ∼ − log N while random oriented cracks lead to the exponential law decay log σ ∼ −N (where σ is the effective conductivity of a region with a large number N of defects), as recently predicted by numerical findings.