Bounds to the conductivity of some two-component composites

Calculation of third-order bounds to the conductivity of isotropic two-component composites is discussed. Coincidence of the Beran bounds and bounds derived using trial fields based on the solution of a single-body electrostatic boundary-value problem is demonstrated for a random distribution of impenetrable ellipsoids. This extends a proof of Beasley and Torquato [J. Appl. Phys. 60, 3576 (1986)]. A structural parameter related to third-order bounds is calculated for a face-centered cubic array of cubes in a matrix. For an array of rectangular blocks an upper bound in one direction is derived. This bound, and its two-dimensional analogs, become very sharp in the limit of strong inhomogeneity. Improved third- and fourth-order bounds for the three-dimensional checkerboard are presented.