Effective conductivity of anisotropic two-phase composite media

We derive a new perturbation expansion for the effective conductivity tensor ${\sigma }_e$ of a macroscopically anisotropic d-dimensional two-phase composite of arbitrary microstructure. The nth-order tensor coefficients ${\mathrm{ssA}}_n^{\left(i\right)}$ of the expansion (termed n-point microstructural parameters) are given explicitly in terms of integrals over the set of n-point probability functions (associated with the ith phase) which statistically characterize the microstructure. Macroscopic anisotropy arises out of some asymmetry in the microstructure, i.e., due to statistical anisotropy (e.g., a distribution of oriented, nonspherical inclusions in a matrix, layered media, such as sandstones and laminates, etc.). General and useful properties of the n-point microstructural parameters are established, and contact is made with the formal results of Milton. We then derive rigorous nth-order bounds on ${\sigma }_e$ (from our perturbation expansion) that depend upon the n-point parameters ${\mathrm{ssA}}_n^{\left(i\right)}$ for n=1, 2, 3, and 4. This is the first time that such bounds (for n>1) have been explicitly given in terms of the ${\mathrm{ssA}}_n^{\left(i\right)}$.