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

Abstract

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.

Three-point correlation bounds on the effective bulk modulus of isotropic multicomponent materials

“…1b) without distinct inclusion and matrix phases, we have ζ α = v α [6,7]. The upper and lower bounds for those mixtures are projected in Fig.…”

“…1a), the parameters ζ α have been determined exactly and analytically [7,8]. In particular ζ 2 = 1 if phase-2 is the matrix phase, and ζ 2 = 0 if it is the inclusion phase.…”

“…In the case of two-component materials, the three-point correlation parameters A βγ α can be expressed through just one independent parameter [7] A 11 For Hashin-Shtrikman poly-dispersed two-phase coated sphere model where the coated sphere with the same volume proportions of phases fill all the material space (Fig. 1a), the parameters ζ α have been determined exactly and analytically [7,8].…”

A modified strong-contrast expansion for estimating the effective thermal conductivity of multiphase heterogeneous materials

“…Developing a powerful approach for estimating the effective properties of microstructure with high-contrast constituents led to the strongcontrast expansions. 4,16,17 In the literature, closed form relations are available to estimate the second and third order bounds on the effective properties of simple microstructures, e.g., coated-sphere Hashin-Shtrikman structures. 16 However, for more complex microstructure, strong-contrast approximations are usually good alternatives.…”

“…4,16,17 In the literature, closed form relations are available to estimate the second and third order bounds on the effective properties of simple microstructures, e.g., coated-sphere Hashin-Shtrikman structures. 16 However, for more complex microstructure, strong-contrast approximations are usually good alternatives. The d-dimensional strong-contrast formulation was developed to evaluate the effective elastic, electric, thermal, and permeability properties of a microstructure comprising two isotropic phases.…”