In this talk I will introduce a new four-parameters family of constitutive functions for spherically symmetric elastic bodies which extends the two-parameters class of polytropic fluid models widely used in several applications of fluid mechanics. The four parameters in the polytropic elastic model are the adiabatic index, the bulk modulus, the shear parameter and the Poisson ratio. The two-parameters class of polytropic fluid models arises as a special case when the Poisson ratio equals 1/2. In contrast to the standard Lagrangian approach in elasticity theory, the polytropic elastic model is formulated directly in physical space, which is particularly useful for the applications e.g. to astrophysics where the reference state of the bodies of interest (stars, planets, etc.) is not observable. After discussing some general properties of the polytropic elastic model, I will present some numerical and analytical results on steady states and the homologous motion of a polytropic elastic ball interacting with its own self-generated Newtonian gravitational field.