Souriau's Relativistic general covariant formulation of hyperelasticity revisited

TL;DR

This paper revisits Souriau's 1958 geometric framework for relativistic hyperelasticity, modernizing it and exploring its applications in general relativity, including static spacetimes and the Newton-Cartan limit.

Contribution

It provides a modernized, covariant formulation of relativistic hyperelasticity and demonstrates its connection to classical mechanics in the Newton-Cartan limit.

Findings

Unified covariant formulation of hyperelasticity in GR

Application to Schwarzschild metric

Recovery of classical hyperelasticity in Newtonian limit

Abstract

We present and modernize Souriau's 1958 geometric framework for Relativistic continuous media, and enlighten the necessary and the ad hoc modeling choices made since, focusing as much as possible on the Continuum Mechanics point of view. We describe the general covariant formulation of Hyperelasticity in General Relativity, and then in the particular case of a static spacetime. Finally, we apply this formalism for the Schwarzschild's metric, and recover the Classical Galilean Hyperelasticity with gravity, as the Newton-Cartan infinite light speed limit of this formulation.