A geometric formulation of Schaefer’s theory of Cosserat solids

Abstract

The Cosserat solid is a theoretical model of a continuum whose elementary constituents are notional rigid bodies, having both positional and orientational degrees of freedom. In this article, we present a differential geometric formulation of the mechanics of a Cosserat solid, motivated by Schaefer’s “motor field” theory. The solid is modeled as a special principal fiber bundle (a Cartan space) and its configurations are related by bundle maps. We show that the classical Lagrangian strain measure of a Cosserat solid is the difference of two Cartan connections on the bundle. The infinitesimal strain is derived by a rigorous linearization and is revealed to be the Lie derivative of a Cartan connection along the vector field representing the infinitesimal deformation. Incompatibilities in a Cosserat solid are characterised by a non-flat Cartan connection whose curvature is recognized as the density of topological defects. Stresses are defined as vector bundle-valued differential forms that are work-dual to strains and balance laws in the limit of vanishing inertia are obtained via a d’Alembert principle. Constitutive equations, with an emphasis on recent applications to active oriented solids, are briefly discussed.