Conductivity tensor of anisotropic composite media from the microstructure

Perturbation expansions and rigorous bounds on the effective conductivity tensor σe of d‐dimensional anisotropic two‐phase composite media of arbitrary topology have recently been shown by the authors to depend upon the set of n‐point probability functions S(i)1,..., S(i)n. S(i)n gives the probability of simultaneously finding n points in phase i (i=1,2). Here we describe a means of representing these statistical quantities for distributions of identical, oriented inclusions of arbitrary shape. Our results are applied by computing second‐order perturbation expansions and bounds for a certain distribution of oriented cylinders with a finite aspect ratio. We examine both cases of conducting cylindrical inclusions in an insulating matrix and of insulating cracks or voids in a conducting matrix.