Inclusion and Inhomogeneity Problems in the Hexagonal Electroelastic Medium

Abstract

The elastic Eshelby tensor plays a fundamental role for elastic inclusion problems [1]. For electroelastic media a generalization exists [2, 3], namely the electroelastic analogue of Eshelby tensor (EAET), a linear operator consisting of four tensors of fourth, third and second ranks. In order to derive explicit closed-form expressions for the EAET we assume a linear material law connecting stress and dielectric displacements with strain and electric field. The governing field equations for stress and dielectric displacements are determined by the elastic equilibrium and the conservation of free electric charges. In the modelling of the properties of such linear piezoelectric material Systems the absence of explicit Green's functions for electroelastic media has hindered progress for a long time. Using Michelitsch's recently derived explicit electroelastic 4 4 Green's function1† of the hexagonal (transversely isotropic) infinite medium [4] we calculated the EAET in explicit compact form for a spheroidal inclusion with the same ten hexagonal electroelastic moduli and orientation as the sourrounding medium [5]. The inclusion is assumed to be rotational symmetric with respect to the hexagonal c-axis with semi-axes (a1 ˆ a2 ˆ a, a3), a3 being parallel to the c-axis (x3-axis). The EAET generally is an operator with the structure [2, 3, 5]