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 per...

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

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.

Optimal Bounds on the Trapping Constant and Permeability of Porous Media

Upper and lower bounds on the effective conductivity of statistically isotropic multicomponent materials in d dimensions (d=2 or 3) are constructed from the minimum energy principles and appropriate trial fields. The trial fields, involving harmonic potentials and free parameters to be optimized, lead to the bounds containing up to three-point correlation information about the microgeometry of a composite. The bounds are applied to give estimates for the symmetric cell materials, which are optimal over some ranges of parameters, and asymmetric multicoated spheres, which yield the exact effective conductivity in certain cases. The results also agree with many known ones. New bounds for random cell polycrystals are obtained and illustrated on a number of polycrystalline aggregates.

Bounds on the effective conductivity of statistically isotropic multicomponent materials and random cell polycrystals

Bounds on the elastic moduli of statistically isotropic multicomponent materials and random cell polycrystals

Variational results on the macroscopic conductivity (thermal, electrical, etc.) of the multi-coated sphere assemblage have been used to derive the explicit expression of the respective field (thermal, electrical, etc.) within the spheres in d dimensions (\(d=2,3\)). A differential substitution approach has been developed to construct various explicit expressions or determining equations for the effective spherically symmetric inclusion problems, which include those with radially variable conductivity, different radially variable transverse and normal conductivities, and those involving imperfect interfaces, in d dimensions. When the volume proportion of the outermost spherical shell increases toward 1, one obtains the respective exact results for the most important specific cases: the dilute solutions for the compound inhomogeneities suspended in a major matrix phase. Those dilute solution results are also needed for other effective medium approximation schemes.

Solutions for the conductivity of multi-coated spheres and spherically symmetric inclusion problems

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.

D. C. Pham

Papers

Estimations for the overall properties of some locally-ordered composites