Conductivity of inhomogeneous materials represented by discrete resistor networks

We examine the effective resistivity of inhomogeneous materials, in particular two-phase materials, which are statistically homogeneous and isotropic. A review of methods used to determine the effective resistivity is given. Comparisons are made between the traditional methods for continuous materials and resistor network calculations. The purpose is to clarify to what extent it is possible to describe a continuous, inhomogeneous material, with significantly differing constituent resitivities by discrete resistor models. Four discretization procedures, the aim of which are to determine the effective resistivity of such materials, are proposed and discussed. The applicability of discretization procedures is tested against a specific example, the checkerboard. It is shown that none of the discretization models may be expected to give a good description of the material. This fact is intimately related to the difference in duality properties of continuous and discrete systems.