Multi-body Spherically Symmetric Steady States of Newtonian Self-Gravitating Elastic Matter

Abstract

We study the problem of static, spherically symmetric, self-gravitating elastic matter distributions in Newtonian gravity. To this purpose we first introduce a new definition of homogeneous, spherically symmetric (hyper)elastic body in Euler coordinates, i.e., in terms of matter fields defined on the current physical state of the body. We show that our definition is equivalent to the classical one existing in the literature and which is given in Lagrangian coordinates, i.e., in terms of the deformation of the body from a given reference state. After a number of well-known examples of constitutive functions of elastic bodies are re-defined in our new formulation, a detailed study of the Seth model is presented. For this type of material the existence of single and multi-body solutions is established.

Self-gravitating static balls of power-law elastic matter

“…where is a (dimensionless) constant. Except for some particular values of , this stored energy function is of type (1,3,2). From now on we restrict to the case = 0, which we called the quasi-linear John model.…”

“…The existence of static Newtonian self-gravitating elastic bodies, without any symmetry assumption, has been studied in [4,5] using the Lagrangian formulation of elasticity theory. The first theorem proving the existence of static self-gravitating multibody elastic matter distributions with regular boundaries and arbitrarily large strain has been given in [1] for the Seth model in spherical symmetry and it was later extended to more general elastic models for static self-gravitating balls in [2]. One purpose of this paper is to present numerical evidence showing that some of the assumptions made in [2] are necessary, while other are not, see Section 3.…”

“…In this paper we study Equation (1.1) for single balls of elastic matter with stored energy function w : (0, ∞) 2 → R. We use the formulation of elasticity theory for spherically symmetric bodies with natural reference state introduced in [1], see also [6], in which the Euler state variables of elastic balls satisfy the equations of state ρ(r) = Kδ(r), p rad (r) = p rad (δ(r), η(r)), p tan (r) = p tan (δ(r), η(r)),…”

On self-gravitating polytropic elastic balls

“…The present paper is part of a series which aims to extend the theory of self-gravitating fluids to self-gravitating elastic bodies. The articles [1,2,3] focus on self-gravitating spherically symmetric elastic bodies in static equilibrium. In the present paper some first results on the problem of self-gravitating elastic balls in motion will be established.…”

“…In [1] it was shown that these two different approaches of astrophysics and elasticity can be reconciled, namely that the equation of state of spherically symmetric elastic bodies can be defined directly in physical space as a constraint between Eulerian state variables. This formulation has been used to prove the existence of static self-gravitating elastic matter distributions with arbitrarily large strain [1,2], albeit only in the spherically symmetric case. This is the formulation of the theory of self-gravitating elastic balls used in the present paper.…”