Cosserat media in dynamics

Our aim is to develop a general approach for the dynamics of material bodies of dimension $d$ represented by a matter manifold $N$ of dimension $(d+1)$ embedded into the space–time $M$. It can be specialized for $d=0$ (pointwise object), $d=1$ (arch if it is a solid, flow in a pipe or jet if it is a fluid), $d=2$ (plate or shell if it is a solid, sheet of fluid), $d=3$ (bulky bodies). We call torsor a skew-symmetric bilinear map on the vector space of affine real functions on the affine tangent space to the space–time. We use the affine connections as originally developed by Élie Cartan, that is the connections associated to the affine group. We introduce a general principle of covariant divergence free torsor from which we deduce 10 balance equations. We show the relevance of this general principle by applying it for $d$ from 1 to 4 in the context of the Galilean relativity.