Bimaterial Eshelby's inclusion problem for polyhedra

Abstract

This paper presents the closed-form Eshelby's tensor for an arbitrarily oriented polyhedral inclusion in a bimaterial domain under general uniform eigenstrain. Existing bimaterial solutions are mainly restricted to special inclusion shapes or dilatational eigenstrains, because the bimaterial Green's function contains two Boussinesq's displacement potentials in addition to the harmonic and biharmonic potentials in Kelvin's solution. This paper derives the missing domain integrals of the two Boussinesq's potentials by reducing the volume integrals to surface and elementary line integrals. The formulae provide the complete elastic and thermoelastic bimaterial Eshelby's tensors, which are verified against analytical solutions for spherical and cuboidal inclusions parallel to the bimaterial interface, and finite element results of an inclined cuboid. Singularity analysis demonstrates that the interface-related contribution remains regular when the inclusion is separated from the bimaterial interface, while additional logarithmic singularities arise when an edge or vertex touches the interface without increasing the dominant singularity order.