We extend the previous approach of one of the authors on exact strong-contrast expansions for the effective conductivity e 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 material͒, 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 e 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 assemblage͒ are given. It is shown that the effective conductivity of the multicoated spheres model can be determined exactly.

No abstract

No abstract

We derive exact expressions for so-called "void" bounds on the trapping constant gamma and fluid permeability k for coated-sphere and coated-cylinder models of porous media. We find that in some cases the bounds are optimal. In these instances, exact expressions are obtained for the relevant length scale that arises in the void bounds, which depends on a two-point correlation function that characterizes the porous medium. This is the first time that model microstructures have been found that exactly realize bounds on either the trapping constant or fluid permeability.

Summary.Simple estimations for the overall conductivity and elastic properties of some isotropic locally-ordered composites are deduced from two different variational approaches. The estimations based on limited information about microgeometry of the composites lie inside the Hashin-Shtrikman bounds over the whole range of parameters.