On the possibility to represent effective properties of a material with inhomogeneities in terms of concentration parameters

It is argued that representation of the effective properties of heterogeneous materials in terms of a single parameter of concentration of inhomogeneities (such as volume fraction or crack density) is generally impossible for mixtures of inhomogeneities of diverse shapes; the exceptions – when this is possible – are identified. Application of such parameters to microstructures of “irregular” microgeometries may lead to various inconsistencies. Instead, the effective constants in such cases can be represented in terms of a “tracking” parameter for a specific process of evolution of a microstructural pattern. These issues are discussed in the context of the elastic and conductive properties.