On the effective conductivity of a class of random dispersions

A new class of random dispersions is considered in which not only the location of the spheres is random, but their conductivity is random as well. The classical variational principles are employed in which classes of trial fields in the form of suitably truncated functional series are introduced. In this way three-point variational bounds on the effective conductivity of the dispersion are derived and discussed in more detail for some particular statistical distributions of sphere conductivity. A rigorous formula for the effective conductivity, correct to the order square of sphere fraction, is finally obtained which contains only absolutely convergent integrals.