  A new four-parameters family of constitutive functions for spherically
symmetric elastic bodies is introduced 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
polytropic exponent $\gamma$, the bulk modulus $\kappa$, the shear exponent
$\beta$ and the Poisson ratio $\nu\in (-1,1/2]$. The two-parameters class of
polytropic fluid models arises as a special case when $\nu=1/2$ and
$\beta=\gamma$. In contrast to the standard Lagrangian approach in elasticity
theory, the polytropic elastic model in this paper is formulated directly in
physical space, i.e., in terms of Eulerian state variables, 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, the steady states and the homologous motion of Newtonian
self-gravitating polytropic elastic balls are investigated. It is shown
numerically that static balls exist when the parameters $\gamma,\beta$ are
contained in a particular region $\mathcal{O}$ of the plane, depending on
$\nu$, and proved analytically for $(\gamma,\beta)\in\mathcal{V}$, where
$\mathcal{V}\subset\mathcal{O}$ is a disconnected set which also depends on the
Poisson ratio $\nu$. Homologous solutions describing continuously collapsing
balls are constructed numerically when $\gamma=4/3$. The radius of these
solutions shrinks to zero in finite time, causing the formation of a center
singularity with infinite density and pressure. Expanding self-gravitating
homologous elastic balls are also constructed analytically for some special
values of the shear parameter $\beta$.
