On the solution of the dynamic Eshelby problem for inclusions of various shapes

In many dynamic applications of theoretical physics, for instance in electrodynamics, elastodynamics, and materials sciences (dynamic variant of Eshelby s inclusion and inhomogeneity problems) the solution of the inhomogeneous Helmholtz equation ( dynamic or Helmholtz potential) plays a crucial role. In materials sciences from such a solution the dynamical fields due to harmonically transforming eigenfields can be constructed. In contrast to the static Eshelby s inclusion problem (Eshelby, 1957), due to its mathematical complexity, the dynamic variant of the problem is comparably little touched. Only for a restricted set of cases, namely for ellipsoidal, spheroidal and continuous fiber-inclusions, analytical approaches exist. For ellipsoidal shells we derive a 1D integral representation of the Helmholtz potential which is useful to be extended to inhomogeneous ellipsoidal source regions. We determine the dynamic potential and dynamic variant of the Eshelby tensor for arbitrary source densities and distributions by employing a numerical technique based on Gauss quadrature. We study a series of examples of Eshelby problems which are of interest for applications in materials sciences, such as for instance cubic and prismatic inclusions. The method is especially useful to be applied in self-consistent methods (e.g. the effective field method) if one looks for the effective dynamic characteristics of the material containing a random set of inclusions. 2004 Elsevier Ltd. All rights reserved.