Effective conductivities of some multi-color, isotropic, regular tessellations

Abstract: Algebraic expressions are found for the effective conductivities of some infinite tessellations composed of conducting square, triangular, or hexagonal tiles. A tessellation is further characterized by the number N of different colors (different tile conductivities) represented. Tiles of a color are distributed randomly, and constitute an areal fraction 1/N of the tessellation. The expressions take account of the percolation threshold associated with the tile type. This generates a series of expressions for the three-color case, that suggests this approach gives lower bounds for the true effective conductivities.