Introduction

The problem of wave propagation in composite materials has many important applications. The solution of this problem allows us to predict the response of composite materials to various types of dynamic loadings; this problem forms the theoretical background for non-destructive ultrasonic evaluation of microstructures of composites. The main objectives of the theory in this problem are the dependencies of the phase velocity and attenuation coefficient of the mean (coherent) wave field propagating in the composite on the frequency of the incident field (dispersion curves) and on the details of the composite microstructure. For composite materials with random microstructures, this problem cannot be solved exactly, and only approximate solutions are available. Self-consistent methods are widely used for the construction of such approximate solutions.

In self-consistent methods, complex actual wave fields propagating in heterogeneous media are approximated by simple ones using physically reasonable hypotheses. All the self-consistent methods are based on two types of such hypotheses. The first one reduces the problem of interactions between many inclusions in the composite to a problem for one inclusion (the one particle problem). The second hypothesis is the condition of self-consistency. For application of these methods, the heterogenous medium should have specific features: a typical element (particle) should exist in the medium. Such a particle may be an inclusion in the matrix-inclusion composites, a grain in random polycrystalline materials, a crack in materials with defects, etc.