General Covariant Relativistic Gradient Hyperelasticity

Assuming in the Variational Relativity framework that the contribution of the matter Lagrangian density depends only on the Universe metric and on the first jet of the perfect matter field, Souriau's 1958 theorem states that General Covariance implies that this density depends only on the 0-jet of the matter field and on a secondary variable, the conformation, an invariant of the diffeomorphism group, which generalizes the inverse of the right Cauchy-Green tensor of Classical Continuum Mechanics. We extend Souriau's result to a second order gradient theory in General Relativity, considering the case of a matter Lagrangian density which depends on the second jets of the Universe metric and the perfect matter field. We introduce accordingly higher order diffeomorphisms invariants, which play the same role as Souriau's conformation. Among them are exhibited three novel gravitation/matter field coupling invariants. Their classical limits are calculated, showing that the 3D Continuum Mechanics second gradient theory can be derived from such a relativistic theory. Some of these invariants converge to objective quantities (modeling solids) in the Galilean limit, others to non-objective quantities (modeling fluids). The present work aims thus at precising the theoretical foundation of higher gradient Continuum Mechanics theory.