POLYCRYSTALLINE CONFIGURATIONS THAT MAXIMIZE ELECTRICAL-RESISTIVITY

A LOWER bound on the effective conductivity tensor of polycrystalline aggregates formed from a single basic crystal of conductivity sigma was recently established by Avellaneda, Cherkaev, Lurie and Milton. The bound holds for any basic crystal, but for isotropic aggregates of a uniaxial crystal, the bound is achieved by a sphere assemblage model of Schulgasser. This left open the question of attainability of the bound when the crystal is not uniaxial. The present work establishes that the bound is always attained by a rather large class of polycrystalline materials. These polycrystalline materials, with maximal electrical resistivity, are constructed by sequential lamination of the basic crystal and rotations of itself on widely separated length scales. The analysis is facilitated by introducing a tensor s = theta(theta-I+sigma)-1 where theta > 0 is chosen so that Tr s = 1. This tensor s is related to the electric field in the optimal polycrystalline configurations.