TRANSPORT IN MACROSCOPICALLY INHOMOGENEOUS MATERIALS.

We consider electrical and thermal transport in macroscopically inhomogeneous materials, when the two components forming a mixture have very different conductance properties. Because of their complexity, such systems are sometimes modeled by resistor networks. It is shown that the most natural models violate the Hashin-Shtrikman bounds to the effective conductivity of continuous composite materials. The distribution of the Joule heat between the phases, in the case of electrical conductance, is also largely erroneous. Thus, better estimates of conductance properties are obtained by disregarding detailed information about the phase geometry and instead using general methods for continuous materials, valid for a wide class of geometries.