Conductivity of a composite containing uniformly oriented penny-shaped cracks or perfectly conducting discs

We consider the problem of determining the effective conductivity of a composite material consisting of parallel discs embedded in an isotropic matrix. A method of multipole expansion is used for computing the conductivity for the cases of perfectly conducting discs and non-conducting discs or penny-shaped cracks. Both the cases of periodic and random arrangements of discs or cracks are considered. It is shown that, in general, the conductivity is quite sensitive to the details of the arrangement of the discs. The results of computations are compared with those obtained using an effective-medium approximation and with a bound on conductivity.