Material groupoids and algebroids

TL;DR

This paper explores how Lie groupoids and algebroids model the properties and defects of continuous media in mechanics and geometry, providing an intuitive understanding of their role in material analysis.

Contribution

It introduces a framework linking material properties to Lie groupoids and algebroids, highlighting their significance in studying material defects.

Findings

Material properties can be associated with Lie groupoids.

Lie algebroids help determine the existence of material defects.

The paper offers an intuitive treatment of these mathematical structures.

Abstract

Lie groupoids and their associated algebroids arise naturally in the study of the constitutive properties of continuous media. Thus, Continuum Mechanics and Differential Geometry illuminate each other in a mutual entanglement of theory and applications. Given any material property, such as the elastic energy or an index of refraction, affected by the state of deformation of the material body, one can automatically associate to it a groupoid. Under conditions of differentiability, this material groupoid is a Lie groupoid. Its associated Lie algebroid plays an important role in the determination of the existence of material defects, such as dislocations. This paper presents a rather intuitive treatment of these ideas.