A conservation law formulation of nonlinear elasticity in general relativity

TL;DR

This paper develops a new Eulerian conservation law framework for nonlinear hyperelasticity in general relativity, extending existing formalisms and validating it through numerical simulations of Riemann problems.

Contribution

It reformulates Carter and Quintana's hyperelasticity formalism as Eulerian conservation laws suitable for numerical relativity, ensuring symmetric hyperbolicity and connecting to Newtonian elasticity.

Findings

Equations can be made symmetric hyperbolic near the unsheared state.

The formalism accurately reproduces Newtonian and relativistic elasticity results.

Numerical tests validate the framework's effectiveness in Minkowski spacetime.

Abstract

We present a practical framework for ideal hyperelasticity in numerical relativity. For this purpose, we recast the formalism of Carter and Quintana as a set of Eulerian conservation laws in an arbitrary 3+1 split of spacetime. The resulting equations are presented as an extension of the standard Valencia formalism for a perfect fluid, with additional terms in the stress-energy tensor, plus a set of kinematic conservation laws that evolve a configuration gradient. We prove that the equations can be made symmetric hyperbolic by suitable constraint additions, at least in a neighbourhood of the unsheared state. We discuss the Newtonian limit of our formalism and its relation to a second formalism also used in Newtonian elasticity. We validate our framework by numerically solving a set of Riemann problems in Minkowski spacetime, as well as Newtonian ones from the literature.