Mechanics of Systems of Affine Bodies. Geometric Foundations and Applications in Dynamics of Structured Media

TL;DR

This paper develops a geometric framework for the mechanics of systems of affinely-rigid bodies, introducing invariant deformation measures and exploring their applications in modeling complex physical media.

Contribution

It introduces the concept of mutual deformation tensors and affinely-invariant scalars for systems of affine bodies, expanding the modeling capabilities beyond single-body mechanics.

Findings

Existence of nontrivial invariant potentials for systems of affine bodies.

Construction of scalar invariants from mutual deformation tensors.

Hierarchy of interaction models based on invariance groups.

Abstract

In the present paper we investigate the mechanics of systems of affinely-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 are dynamical models invariant under the group underlying geometry of degrees of freedom. In contrary to the single body case there exist nontrivial potentials invariant under this group (left and right acting). The concept of relative (mutual) deformation tensors of pairs of affine bodies is discussed. Scalar invariants built of such tensors are constructed. There is an essential novelty in comparison to deformation scalars of single affine bodies, i.e., there exist affinely-invariant scalars of mutual deformations. Hence, the hierarchy of interaction models according to their invariance group, from Euclidean to affine ones, can be considered.