Strong-contrast expansions and approximations for the effective conductivity of isotropic multiphase composites

We extend the previous approach of one of the authors on exact strong-contrast expansions for the effective conductivity σe of d-dimensional two-phase composites to case of macroscopically isotropic composites consisting of N phases. The series consists of a principal reference part and a fluctuation part (a perturbation about a homogeneous reference or comparison material), which contains multipoint correlation functions that characterize the microstructure of the composite. The fluctuation term may be estimated exactly or approximately in particular cases. We show that appropriate choices of the reference phase conductivity, such that the fluctuation term vanishes, results in simple expressions for σe that coincide with the well-known effective-medium and Maxwell approximations for two-phase composites. We propose a simple three-point approximation for the fluctuation part, which agrees well with a number of analytical and numerical results, even when the contrast between the phases is infinite near percolation thresholds. Analytical expressions for the relevant three-point microstructural parameters for certain mixed coated and multicoated spheres assemblages (extensions of the Hashin–Shtrikman coated-spheres assemblage) are given. It is shown that the effective conductivity of the multicoated spheres model can be determined exactly.