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Double groupoids of composites: applications to uniformity

V. M. Jiménez, M. De León, M. Epstein

TL;DR

The paper develops a double groupoid framework to study uniformity in composite materials by representing each constituent with a transitive material groupoid $\Omega_i(\mathcal{B})$ and forming their commuting-square double groupoid $\boxdot(\Omega_1(\mathcal{B}), \Omega_2(\mathcal{B}))$. Uniformity of a composite is analyzed through transitivity notions on the double groupoid, with precision via surjectivity conditions of anchor maps and the introduction of horizontal/vertical, strong, and weak uniformity. The approach formalizes how common material isomorphisms encode compatibility between constituents and provides concrete examples (e.g., crystalline and triclinic crystals) to illustrate the hierarchy and limitations of these notions. This framework paves the way for generalizing to multi-materials and microstructured media, and it connects uniformity with geometric structures such as double algebroids and G-structures to enable broader physical and mathematical applications.

Abstract

In this paper we present a geometrical framework to study the uniformity of a composite material by means of double groupoid theory. The notions of vertical and horizontal uniformity are introduced, as well as other weaker ones that allows us to study other possible notions of more general uniformity.

Double groupoids of composites: applications to uniformity

TL;DR

The paper develops a double groupoid framework to study uniformity in composite materials by representing each constituent with a transitive material groupoid and forming their commuting-square double groupoid . Uniformity of a composite is analyzed through transitivity notions on the double groupoid, with precision via surjectivity conditions of anchor maps and the introduction of horizontal/vertical, strong, and weak uniformity. The approach formalizes how common material isomorphisms encode compatibility between constituents and provides concrete examples (e.g., crystalline and triclinic crystals) to illustrate the hierarchy and limitations of these notions. This framework paves the way for generalizing to multi-materials and microstructured media, and it connects uniformity with geometric structures such as double algebroids and G-structures to enable broader physical and mathematical applications.

Abstract

In this paper we present a geometrical framework to study the uniformity of a composite material by means of double groupoid theory. The notions of vertical and horizontal uniformity are introduced, as well as other weaker ones that allows us to study other possible notions of more general uniformity.

Paper Structure

This paper contains 5 sections, 10 theorems, 50 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Simple materials
  3. The structure of a double groupoid
  4. Material double groupoid
  5. Conclusions and further work

Key Result

Proposition 1

Let $\mathcal{B}$ be a body. Two body points $X$ and $Y$ are materially isomorphic if, and only if, there exist two (local) configurations $\phi_{1}$ and $\phi_{2}$ such that where $W_{i}$ is the mechanical response associated to $\phi_{i}$ for $i=1,2$.

Figures (2)

  • Figure 1: Artistic rendition of the building block of a double groupoid eps02.
  • Figure 2: Groupoid operations

Theorems & Definitions (37)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Proposition 1: VMMDME
  • Definition 4: Transitivity
  • Definition 5: Total intransitivity
  • Example 1: Groups
  • Example 2: Pair groupoid
  • Example 3: Frame groupoid and 1-jets groupoid
  • ...and 27 more