Double groupoids in the theory of material uniformity

TL;DR

This paper introduces double groupoids and related structures as tools to analyze and quantify the uniformity and misalignment in continuous media with dual geometric or constitutive properties.

Contribution

It proposes a novel framework using double groupoids and double Lie algebroids to describe and measure uniformity and misalignment in complex materials.

Findings

Measures of misalignment and non-uniformity are defined.

Concept of coupled symmetries is introduced and illustrated.

Framework links geometric structures with material uniformity analysis.

Abstract

The use of double groupoids and their associated double Lie algebroids and characteristic distributions is proposed for the description and analysis of continuous media that carry two different constitutive or geometric structures. Various measures of misalignment and lack of uniformity that may arise when two individually perfectly uniform structures are combined are suggested, and the concept of coupled symmetries is proposed and illustrated.