Dynamic Eshelby tensor and potentials for ellipsoidal inclusions

Abstract

The dynamic Eshelby inclusion problem for an ellipsoidal inclusion in a threedimensional infinite elastic isotropic medium is considered. The dynamic Eshelby tensor is expressed in terms of solutions of the Helmholtz equation (Helmholtz potentials). A compact formulation for the components of the dynamic Eshelby tensor is derived for the inside region of the inclusion. For spheroidal inclusions, one-dimensional integrals similar to the elliptic integrals are obtained. The approach leads to closed-form expressions in cases such as spheres and continuous fibres coinciding with those given in 1990 by Mikata & Nemat-Nasser and in 2002 by Michelitsch et al . by employing other techniques. For the inside region of the inclusion, the static limit is performed in closed form and coincides with Eshelby’s classical 1957 result.