Long-wavelength propagation in composite elastic media II. Ellipsoidal inclusions

A self-consistent method of estimating effective macroscopic elastic constants for inhomogeneous materials with ellipsoidal inclusions is formulated using elastic-wave scattering theory. The method is a generalization of the method for spherical inclusions presented in the first part of this series. The results are compared to the Kuster–Toksöz estimates for the elastic moduli and to the rigorous Hashin–Shtrikman bounds and Miller bounds. For general ellipsoidal inclusions, our self-consistent estimates satisfy both the Hashin–Shtrikman bounds and the more stringent Miller bounds, whereas the Kuster–Toksöz estimates for nonspherical inclusions do not satisfy even the Hashin–Shtrikman bounds in general. Our self-consistency conditions for the cases of needle and disk inclusions differ from those of Wu, Walpole, and Boucher. Since our results are in better agreement with the rigorous bounds, it is concluded that our results are to be preferred. The method is also briefly compared to other self-consistent scattering theory approaches. The theory is used to calculate velocity and attenuation of elastic waves in fluid-saturated media with oblate and prolate spheroidal inclusions.