The Green functions of an elastic medium surrounding a rigid spherical inclusion

Abstract

A homogeneous isotropic elastic medium surrounds and adheres to a rigid inclusion of spherical shape and otherwise fills the whole of space. A point force is located at an arbitrary one of its points. We determine the associated elastic field explicitly for four fundamental cases, in order to identify the Green tensor functions which are needed for the construction of the more complex elastic fields of other singularities, or of an internal fracture. The first case has a fixed inclusion, with no permitted translation or rotation, the second has an unconstrained one, not subject to any external force or couple, the third has an untranslated one which is not subject to any external torque about its centre, and the fourth has an unrotated inclusion which is subject only to an external couple. When the elastic medium is incompressible (with its Poisson's ratio equal to one-half), we are able to verify for the first case that, in keeping with the analogy noted by Rayleigh, our Green function is coincident in mathematical terms with the one given long ago by Oseen, for slow viscous flow outside a fixed spherical boundary.