Maxwell's far-field methodology applied to the prediction of properties of multi-phase isotropic particulate composites

The far-field methodology developed by Maxwell, when estimating the effective electrical conductivity of isotropic particulate composites, is used to estimate effective thermoelastic properties of multi-phase isotropic composites. In particular, Maxwell's methodology applied to the analogous thermal conduction problem is described, extending the approach to multi-phase spherical particles having different sizes and properties. The methodology is also used to estimate the effective bulk and shear moduli, and the thermal expansion coefficients, of multi-phase isotropic particulate composites. Results correspond with expressions derived in the literature, and coincide with, or lie between, variational bounds for all volume fractions. These characteristics, relating to isotropic effective properties, indicate that results obtained using the methodology, are not necessarily restricted to low volume fractions, as originally suggested by Maxwell. It is concluded that Maxwell's methodology is a unifying optimum technique to estimate the properties of multi-phase isotropic particulate composites, because it provides closed form estimates that are fully consistent with other methods, without imposing restrictions, except that the particles must be spherical ( but can have a range of size and properties) and the resultant effective properties must be isotropic.