Lie groupoids and algebroids applied to the study of uniformity and homogeneity of Cosserat media

TL;DR

This paper introduces a novel mathematical framework using Lie groupoids and algebroids to analyze the homogeneity of Cosserat media without relying on a reference crystal, providing new insights into material uniformity.

Contribution

It develops a new definition of homogeneity for Cosserat media using second-order non-holonomic Lie groupoids and algebroids, advancing the mathematical tools for material analysis.

Findings

New Lie groupoid and algebroid structures for Cosserat media

A reference-crystal-independent definition of homogeneity

Connections with non-holonomic second-order G-structures

Abstract

A Lie groupoid, called \textit{second-order non-holonomic material Lie groupoid}, is associated in a natural way to any Cosserat media. This groupoid is used to give a new definition of homogeneity which does not depend on a reference crystal. The corresponding Lie algebroid, called \textit{second-order non-holonomic material Lie algebroid}, is used to characterize the homogeneity property of the material. We also relate these results with the previously ones in terms of non-holonomic second-order $\overline{G}$-structures.