On motions which preserve ellipsoidal holes

Let ℰ be an ellipsoid in ℝ³ contained in a region Ω. Suppose one body occupies the region Ω-ℰ in a certain stress-free reference configuration while a second body, the inclusion, occupies the region ℰ in a stress-free reference configuration. Assume the inclusion is free to slip at ∂ℰ. Now suppose that by changing some variable such as the temperature, pressure, humidity, etc., we cause the trivial deformation y(x)=x of the inclusion to become unstable relative to some other deformation. For example, the inclusion may be made out of such a material that if it were removed from the body, it would suddenly change shape to another stress-free configuration specified by a deformation y=Fx, F^ΥF=C, C being a fixed tensor characteristic of the material, at a certain temperature. However, with an appropriate material model for the surrounding body, we expect it will resist the transformation, and both body and inclusion will end up stressed. In a recent paper, Mura and Furuhashi [1] find the following unexpected result within the context of infinitesimal deformations: certain homogeneous deformations of the ellipsoid which take it to a stress-free configuration also leave the surrounding body stress-free. These are essentially homogeneous, infinitesimal deformations which preserve ellipsoidal holes. In this paper, we find all finite homogeneous deformations and motions which preserve ellipsoidal holes.