Characteristic distribution: An application to material bodies

TL;DR

This paper explores the geometric structure of material bodies via groupoids, linking uniformity and homogeneity to algebraic and Lie groupoid properties, enabling new ways to analyze material properties.

Contribution

It introduces a framework connecting material body uniformity to algebraic and Lie groupoids, allowing the study of homogeneity through geometric methods.

Findings

Uniformity corresponds to transitive groupoids.

Material bodies can be covered by transitive foliations.

Even algebraic subgroupoids can generate transitive Lie groupoids.

Abstract

Associated to each material body $\mathcal{B}$ there exists a groupoid $\Omega \left( \mathcal{B} \right)$ consisting of all the material isomorphisms connecting the points of $\mathcal{B}$. The uniformity character of $\mathcal{B}$ is reflected in the properties of $\Omega \left( \mathcal{B} \right)$: $\mathcal{B}$ is uniform if, and only if, $\Omega \left( \mathcal{B} \right)$ is transitive. Smooth uniformity corresponds to a Lie groupoid and, specifically, to a Lie subgroupoid of the groupoid $\Pi^1\left({\mathcal B}, {\mathcal B} \right)$ of 1-jets of $\mathcal B$. We consider a general situation when $\Omega \left( \mathcal{B} \right)$ is only an algebraic subgroupoid. Even in this case, we can cover $\mathcal{B}$ by a material foliation whose leaves are transitive. The same happens with $\Omega \left( \mathcal{B} \right)$ and the corresponding leaves generate transitive Lie groupoids (roughly speaking, the leaves covering $\mathcal B$). This result opens the possibility to study the homogeneity of general material bodies using geometric instruments.