Effective conductivities of heterogeneous media containing multiple inclusions with various spatial distributions

Abstract

A scheme is proposed for predicting the effective conductivities of heterogeneous media containing ellipsoidal inclusions of diverse shapes, spatial distributions, and orientations. This scheme yields explicit expressions for the effective conductivity tensor in terms of three microstructural parameters that characterize the shape, distribution, and orientation of the inclusions. By expanding the effective conductivity tensor in terms of the volume fraction of the inclusions, it is found that the effect of the shape of the distribution ellipsoid on the effective conductivity tensor is of a higher order in the volume fraction than the effect of the shape of the inclusions. The scheme proposed here generalizes the Maxwell formula to heterogeneous media containing multiple inclusions while also taking into account the orientation of the inclusions. Thus, the existing formulas in the literature are special cases of the general formulas given by the present scheme. The predicted effective conductivities of heterogeneous media containing aligned ellipsoidal inclusions, randomly oriented ellipsoidal inclusions, spheroidal inclusions with orientational distributions, and mixtures of cavities and cracks are found to agree well with the experimental results and the results of other schemes.

Frequency-Dependent Effective Material Parameters of Composites as a Function of Inclusion Shape

“…(25) and (26), the MG mixing rule retains the shape of effective susceptibility loss peak characteristic for the intrinsic susceptibility of inclusions. From the standpoint of the BM spectral theory, the reason is that additional loss due to mixing may arise over the entire range of effective form factors, where the spectral function has non-zero values.…”

“…The reason is agglomeration or deagglomeration of inclusions, which depends on the properties of the interface between inclusions and the host matrix. The importance of spatial distribution of inclusions in a composite for validity of mixing rules is discussed in [25].…”

Singular harmonic problems at a wedge vertex: mathematical analogies between elasticity, diffusion, electromagnetism, and fluid dynamics

“…Singular stress states occur in many boundary value problems of linear elasticity where different materials are present (see [England 1971;Paggi and Carpinteri 2008;Sinclair 2004a;2004b] for a broad overview). In this context, problems involving multimaterial wedges or junctions have received a great attention from the scientific community, since they are commonly observed in composite materials.…”

“…Mathematical analogies among elasticity, electromagnetism, and conductivity have been known and exploited for a long time (see [Hashin and Shtrikman 1962;Duan et al 2006], for instance). Sinclair [1980] pointed out the mathematical analogy between the singular steady-state heat transfer and the Keywords: singularities, multimaterial wedges, elasticity, diffusion, electromagnetism, fluid dynamics.…”

The Thermal Conductivity of Periodic partIculate Composites as Obtained from a Crystallographic Mode of Particle Packing

“…On the other hand, Maxwell [4] performed a general basis for estimating the effective thermal conductivity of particulate composites, whereas in Refs. [5][6][7] some remarkable theoretical and empirical approaches were presented to analyze the thermal conductivity of composites. Further, in Ref.…”