Self-gravitating static balls of power-law elastic matter

TL;DR

This paper investigates static, self-gravitating elastic balls with power-law energy functions, identifying a minimal parameter subclass and providing numerical evidence on the necessity of certain hypotheses for their existence.

Contribution

It introduces a Lamé-type subclass of power-law elastic materials with minimal parameters and offers numerical validation of theoretical existence conditions for elastic balls.

Findings

Numerical evidence supports some hypotheses of the existence theorem.

Lamé-type models depend only on bulk modulus and Poisson ratio.

Certain hypotheses in the theorem are shown to be necessary or unnecessary.

Abstract

We study a class of power-law stored energy functions for spherically symmetric elastic bodies that includes well-known material models, such as the Saint Venant-Kirchhoff, Hadamard, Signorini and John models. We identify a finite subclass of these stored energy functions, which we call Lam\'e type, that depend on no more material parameters than the bulk modulus $\kappa>0$ and the Poisson ratio $-1<\nu\leq1/2$. A general theorem proving the existence of static self-gravitating elastic balls for some power-law materials has been given elsewhere. In this paper numerical evidence is provided that some hypotheses in this theorem are necessary, while others are not.