In this thesis we consider a thick shell, i.e. the walls of the shell are of arbitrary finiteness, which is composed of a hyperelastic material and we undertake investigations of its behaviour in its own static gravitational field.

First, we deduce the field equations in both the spatial and the material description together with their boundary conditions. The problem is studied within the framework of the Einsteinian theory of gravity and afterwards we take the Newtonian limit of the derived system. Using the material description and by means of the implicit function theorem, we prove an existence theorem for the elastic equilibria of the static self-gravitating shell in the Newtonian case for small gravitational constants and for materials other than auxetic materials.

In addition, we investigate numerically the behaviour of the solution to the system of linearised Newtonian field equations for various realistic elastic materials.