Souriau's Relativistic general covariant formulation of hyperelasticity revisited

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. We focus as much as possible on the Continuum Mechanics point of view and describe the general covariant formulation of Hyperelasticity in (Variational) General Relativity. At this level of generality, different relativistic strain and stress tensors are formulated and discussed. Then, the choice of an observer, through the introduction of a spacetime structure allows us to make deeper insights into the foundations of Continuum Mechanics. We extend Souriau’s calculations, initially performed in the flat Minkowski spacetime, to the Schwarzschild spacetime accounting for gravitation. Finally, we recover the Classical Galilean Hyperelasticity with gravity, as the Newton–Cartan infinite light speed limit of this formulation.