Force-displacement relationships for spherical inclusions in finite elastic media

Here we present derivations of the force-displacement relationship for a rigid spherical inclusion embedded in homogeneous, isotropic, linear solids. Formulas are given for both the case of perfect interfacial bonding (no slip) and the case of a sliding contact (slip) between the medium and the inclusion. The formulas are applicable to both compressible and incompressible solids and are applicable for elastic media of finite extent. The results allow the determination of the elastic modulus of the medium from force-displacement measurements on the inclusion provided Poisson's ratio is known. We find that, when the size of the medium is much larger than that of the inclusion, the inclusion displacement in response to an applied force is 50% larger when slip is present.