The Determination of the Elastodynamic Fields of an Ellipsoidal Inhomogeneity

Elastic fields of a single ellipsoidal inhomogeneity embedded in an infinite elastic matrix subjected to plane time-harmonic waves are studied by employing the concept of eigenstrain and the extended version of Eshelby’s method of equivalent inclusion. Using the dynamic version of the Betti-Rayleigh reciprocal theorem, an integral representation of the displacement field, due to the presence of inhomogeneity, is given in terms of the eigenstrains. Two types of eigenstrains arise in the elastodynamic case. Expanding the eigenstrains and applied strains in the polynomial form in the position vector r and satisfying the equivalence conditions at every point, the governing simultaneous algebraic equations for the unknown coefficients in the eigenstrain expansion are derived. Elastodynamic field outside an ellipsoidal inhomogeneity in a linear elastic isotropic medium is given as an example. The angular and frequency dependence of the induced displacement field, which is in fact the scattered displacement field, the differential and the total cross sections are formally given in series expansion form for the case of uniformly distributed eigenstrains.