Local volume fraction fluctuations in heterogeneous media

The volume fractions of multiphase heterogeneous media fluctuate on a spatially local level even for statistically homogeneous materials. A general formulation is given to represent the standard deviation associated with the local volume fraction of statistically homogeneous but anisotropic D-dimensional two-phase media for arbitrary-shaped observation regions. The standard deviation divided by the macroscopic volume fraction, termed the coarseness, is computed for D-dimensional distributions of penetrable as well as impenetrable spheres, for a wide range of densities and observation-region sizes. The effect of impenetrability of the particles, for fixed observation-region size, is to reduce the coarseness relative to that of the penetrable-sphere model, especially at high densities. For either sphere model, increasing the dimensionality D decreases the coarseness.