THE SCATTERING OF A PLANE ELASTIC WAVE BY SPHERICAL ELASTIC INCLUSIONS

A method has been developed by Gaunaurd, Uberall and others to describe the behaviour of elastic composites. Results from the theory Of (low-frequency) scattering by a single sphere are used to obtain three (uncoupled) equations for the effective bulk modulus K*, shear.modulus mu* and density rho* of an elastic matrix containing spherical inclusions. In the Gaunaurd and Uberall work, K* is frequency dependent whereas mu* and rho* are constant. Dynamic estimates are obtained for the effective speed and attenuation of longitudinal waves in the composite and static estimates for the effective speed of shear waves. However the results do not compare very favourably with experimental data for other than very low frequency. The Gaunaurd and Uberall theory is examined here in some detail for the particular case of identical elastic spheres embedded in an elastic matrix. It is shown that an analytical error in the equation for K* leads to incorrect estimates for longitudinal speed and attenuation. For specific choices of material parameters, the numerical estimates are computed using both the corrected and the original versions of the effective equations. The results are also compared to the multiple scattering method of Waterman and Truell which has been shown to yield accurate predictions for the materials studied here. Finally, the GU estimates are shown to be highly dependent upon the choice of radius R of an effective sphere but it is argued that the GU choice of setting R = 0 in the appropriate equations is generally no worse than that of setting R equal to some positive constant.