On motions which preserve ellipsoidal holes (Q1108847)

Let \({\mathcal E}\) be an ellipsoid in \({\mathbb{R}}^ 3\) contained in a region \(\Omega\). Suppose one body occupies the region \(\Omega\)-\({\mathcal E}\) in a certain stress-free reference configuration while a second body, the inclusion, occupies the region \({\mathcal E}\) in a stress-free reference configuration. Assume the inclusion is free to slip at \(\partial {\mathcal E}\). 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 \(y=Fx\), \(F^ TF=C\), 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, \textit{T. Mura} and \textit{R. Furuhashi} [J. Appl. Mech. 51, 308-310 (1984; Zbl 0544.73018)] found 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.