In the present paper we investigate the mechanics of systems ofaffinely-rigid bodies, i.e., bodies rigid in the sense of affine geometry.Certain physical applications are possible in modelling of molecular crystals,granular media, and other physical objects. Particularly interesting aredynamical models invariant under the group underlying geometry of degrees offreedom. In contrary to the single body case there exist nontrivial potentialsinvariant under this group (left and right acting). The concept of relative(mutual) deformation tensors of pairs of affine bodies is discussed. Scalarinvariants built of such tensors are constructed. There is an essential noveltyin comparison to deformation scalars of single affine bodies, i.e., there existaffinely-invariant scalars of mutual deformations. Hence, the hierarchy ofinteraction models according to their invariance group, from Euclidean toaffine ones, can be considered.
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