# On the homogeneity of non-uniform material bodies

## TL;DR

This paper introduces the concept of a material groupoid and distribution to analyze the homogeneity of non-uniform bodies, enabling a rigorous subdivision into uniform parts and new measures of uniformity.

## Contribution

It develops a novel framework using material groupoids and distributions to study non-uniform bodies and define homogeneity.

## Key findings

- Subdivision of bodies into uniform sub-bodies, laminates, and points.
- A measure of uniformity for simple bodies.
- Rigorous definitions of homogeneity for non-uniform bodies.

## Abstract

A groupoid $\Omega \left( \mathcal{B} \right)$ called material groupoid is naturally associated to any simple body $\mathcal{B}$. The material distribution is introduced due to the (possible) lack of differentiability of the material groupoid. Thus, the inclusion of these new objects in the theory of material bodies opens the possibility of studying non-uniform bodies. As an example, the material distribution and its associated singular foliation result in a rigorous and unique subdivision of the material body into strictly smoothly uniform sub-bodies, laminates, filaments and isolated points. Furthermore, the material distribution permits us to present a "measure" of uniformity of a simple body as well as more general definitions of homogeneity for non-uniform bodies.