A geometric formulation of Schaefer's theory of Cosserat solids

TL;DR

This paper develops a modern differential geometric framework for Cosserat solids, linking their deformation and defect structures to principal fibre bundles, Lie derivatives, and curvature, enhancing understanding and modeling of these complex materials.

Contribution

It introduces a coordinate-independent geometric formulation of Cosserat solid mechanics using principal fibre bundles and Lie derivatives, extending Schaefer's motor field theory.

Findings

Finite strain is obtained by integrating infinitesimal strain along a path.

Zero curvature corresponds to path-independent strain, indicating absence of topological defects.

Non-zero curvature represents the density of topological defects and influences material behavior.

Abstract

The Cosserat solid is a theoretical model of a continuum whose elementary constituents are notional rigid bodies. Here we present a formulation of the mechanics of a Cosserat solid in the language of modern differential geometry and exterior calculus, motivated by Schaefer's "motor field" theory. The solid is modelled as a principal fibre bundle and configurations are related by translations and rotations of each constituent. This kinematic property is described in a coordinate-independent manner by a bundle map. Configurations are equivalent if this bundle map is a global Euclidean isometry. Inequivalent configurations, representing deformations of the solid, are characterised by the local structure of the bundle map. Using Cartan's magic formula we show that the strain associated with infinitesimal deformations is the Lie derivative of a connection one-form on the bundle, revealing it to be a Lie algebra-valued one-form. Extending Schaefer's theory, we derive the finite strain by integrating the infinitesimal strain along a prescribed path. This is path independent when the curvature of the connection one-form is zero. Path dependence signals the presence of topological defects and the non-zero curvature is then recognised as the density of topological defects. Mechanical stresses are defined by a virtual work principle in which the Lie algebra-valued strain one-form is paired with a dual Lie algebra-valued stress two-form to yield a scalar work volume form. The d'Alembert principle for the work form provides the balance laws, which is shown to be integrable for a hyperelastic Cosserat solid. The breakdown of integrability, relevant to active oriented solids, is briefly examined. Our work elucidates the geometric structure of Cosserat solids, aids in constitutive modelling of active oriented materials, and suggests structure-preserving integration schemes.