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We present and modernize Souriau's 1958 geometric framework for Relativisticcontinuous media, and enlighten the necessary and the ad hoc modeling choicesmade since, focusing as much as possible on the Continuum Mechanics point ofview. We describe the general covariant formulation of Hyperelasticity inGeneral Relativity, and then in the particular case of a static spacetime.Finally, we apply this formalism for the Schwarzschild's metric, and recoverthe Classical Galilean Hyperelasticity with gravity, as the Newton-Cartaninfinite light speed limit of this formulation.

Souriau's Relativistic general covariant formulation of hyperelasticity revisited