A geometric formulation of Schaefer's theory of Cosserat solids

TL;DR

The work develops a coordinate-free, differential-geometric formulation of Cosserat solids by modeling the material body as a principal bundle and configurations as $H$-equivariant maps into the Euclidean group, with strain given by the difference between a material Cartan connection and the pullback of the Maurer–Cartan form. Strain is captured as an $W$-valued 1-form via $E=oldsymbol{ extpsi}^{*}oldsymbol{ extomega}-oldsymbol{ extomega}$, and infinitesimal deformations arise from Lie derivatives of the connection, $e=doldsymbol{oldsymbol{\xi}}+ ext{ad}_{oldsymbol{ extomega}}oldsymbol{oldsymbol{\xi}}$, enabling a path-independent finite strain when curvature vanishes. Compatibility and defects are tied to the curvature of the Cartan connection, while stress is defined as a dual 2-form and balance laws follow from a virtual-work principle through the operator $D^{*}$. The framework yields a natural setting for constitutive modelling, preserving geometric structure and suggesting structure-preserving numerical schemes, with extensions to hyperelasticity, chirality, and active oriented materials. Overall, the paper unifies Cosserat elasticity with Cartan geometry, offering a powerful, coordinate-free toolkit for analysis and computation in complex structured media.

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.

Figures (7)

• Figure 1: Configuration of a simple body. In general, $\mathcal{B}$ is assumed to be just a smooth manifold that labels the material particles, but in many applications, $\mathcal{B}$ is identified with a submanifold of $\mathbb{E}^{3}$ with the aid of a reference configuration.
• Figure 2: Configuration of a body with microstructure in the "traditional" formalism Capriz1989.
• Figure 3: Configuration of a Cosserat continuum using principal fibre bundles. $p=(X,S)\in\mathcal{B}\times H=\mathcal{P}$ denotes an arbitrary element of the bundle, which gets mapped to $\psi(p)=(\kappa(X),Q(X)S)\in\psi(\mathcal{P})\subset G$.
• Figure 4: Commutative diagram corresponding to the bundle map $\psi:\mathcal{P}\to G$.
• Figure 5: Rigid transformation of a Cosserat continuum. Observe that the material particles rotate together with spatial points, therefore spatial and microstructural rotations are not independent.