Conductivity tensor of anisotropic composite media from the microstructure

Perturbation expansions and rigorous bounds on the effective conductivity tensor σ_e 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.