On the Dynamic Response of Disordered Composites

A formulation is obtained that is to be satisfied by the mean (i.e., statistical averaged) field quantities in a statistical sample of heterogeneous, linearly elastic solids. Inertia effects are included in the analysis. A low frequency-long wavelength theory is extracted from the general formulation as an approximation to be used when spatial variations of the mean field quantities are slow relative to spatial variations of the material properties of the inhomogeneous solids. The temporal variations are restricted to slow variations on a time scale defined by the spatial variations of material properties and a characteristic wave speed. The predictions of the low frequency-long wavelength theory can be given a purely deterministic interpretation. Some aspects of the latter formulation are investigated. In particular, it is shown that the infinite wavelength limit reduces to an effective modulus theory. The effective elastic moduli tensor is identical to one that is obtained on ignoring inertia effects from the outset; the mass density to be used is the “averaged” mass density. By retaining correction terms it is then shown that elastic wave propagation will always exhibit both dispersion and decay over large enough propagation distances.