Bounds on the conductivity of statistically isotropic polycrystals

The problem of the prediction of the effective electrical conductivity of a polycrystal from the conductivity of a single crystal is considered. If the only information known about phase geometry is that the aggregate is statistically homogeneous and isotropic, it is shown that the average of the principal conductivities of the single crystal is the best upper bound on effective conductivity that can possibly be found. A new rigorous lower bound is found for the case of axially symmetric crystals. An exact solution is found for the case of a two-dimensional polycrystal.