Long-wavelength propagation in composite elastic media I. Spherical inclusions

A self-consistent method of estimating effective macroscopic elastic constants for inhomogeneous materials with spherical inclusions is formulated based on elastic-wave scattering theory. The method for general ellipsoidal inclusions will be presented in the second part of this series. The case of spherical inclusions is particularly simple and therefore provides an elementary introduction to the general method. The self-consistent effective medium is determined by requiring the scattered, long-wavelength displacement field to vanish on the average. The resulting formulas are simpler to apply than previous self-consistent scattering theories due to the reduction from tensor to vector equations. In the limit of long wavelengths, our results for spherical inclusions agree with statically derived self-consistent moduli of Hill and Budiansky. Our self-consistent formulas are also compared both to the estimates of Kuster and Toksöz and to the rigorous Hashin–Shtrikman bounds. (For spherical inclusions and long wavelengths, the Kuster–Toksöz effective moduli are known to be identical to the Haskin–Shtrikman bounds.) A result of Hill for two-phase composites is generalized by proving that the self-consistent effective moduli always lie between the Haskin–Shtrikman bounds for n-phase composites. Numerical examples for a two-phase medium with viscous fluid and solid constituents show that the real part of our self-consistent moduli always lie between the rigorous bounds, in agreement with the analytical results. Some of the practical details in the numerical solution of the coupled, nonlinear self-consistency equations are discussed. Examples of velocities and attenuation coefficients estimated when the solid constituent possesses intrinsic absorption are also presented.