Interior and exterior solutions for boundary value problems in composite elastic and viscous media

We present the solutions for the boundary value problems of elasticity when a homogeneous and isotropic solid of an arbitrary shape is embedded in an infinite homogeneous isotropic medium of different properties. The solutions are obtained inside both the guest and host media by an integral equation technique. The boundaries considered are an oblong, a triaxial ellipsoid and an elliptic cyclinder of a finite height and their limiting configurations in two and three dimensions. The exact interior and exterior solutions for an ellipsoidal inclusion and its limiting configurations are presented when the infinite host medium is subjected to a uniform strain. In the case of an oblong or an elliptic cylinder of finite height the solutions are approximate. Next, we present the formula for the energy stored in the infinite host medium due to the presence of an arbitrary symmetrical void in it. This formula is evaluated for the special case of a spherical void. Finally, we analyse the change of shape of a viscous incompressible ellipsoidal region embedded in a slowly deforming fluid of a different viscosity. Two interesting limiting cases are discussed in detail