On Self-Gravitating Polytropic Elastic Balls

A new four-parameters family of constitutive functions for spherically symmetric elastic bodies is introduced which extends the two-parameters class polytropic fluid models widely used in several applications mechanics. The four parameters model are exponent \(\gamma \), bulk modulus \(\kappa shear \(\beta \) and Poisson ratio \(\nu \in (-1,1/2]\). arises as a special case when =1/2\) =\gamma \). In contrast to standard Lagrangian approach elasticity theory, this paper formulated directly physical space, i.e., terms Eulerian state variables, particularly useful applications, e.g., astrophysics where reference interest (stars, planets, etc.) not observable. After discussing some general properties model, steady states homologous motion Newtonian self-gravitating balls investigated. It shown numerically that static exist ,\beta contained particular region \({\mathcal {O}}\) plane, depending on proved analytically \((\gamma )\in {\mathcal {V}}\), {V}}\subset disconnected set also depends Homologous solutions describing continuously collapsing constructed =4/3\). radius these shrinks zero finite time, causing formation center singularity with infinite density pressure. Expanding values parameter