Effective conductivity of aggregates of anisotropic grains

General expressions for the effective conductivity σeff of overall isotropic aggregates of grains are derived within the average field approximation. The grains are modeled by ellipsoids with possibly anisotropic conductivity tensors. Explicit expressions for σeff are stated for a polycrystal, which corrects an earlier result of Bolotin and Moskalenko [V. V. Bolotin and V. N. Moskalenko, J. Appl. Mech. Tech. Phys. 8, 3 (1967)], and for systems resembling the structure of nodular and grey cast iron. Percolation of randomly distributed two-dimensional disks in a three-dimensional continuous medium is studied. The percolation threshold for the number density of disks of radius unity is numerically determined to be 0.233±0.002 per unit volume.