Discrete rearranging disordered patterns, part I: Robust statistical tools in two or three dimensions

TL;DR

This paper develops a set of robust statistical tools to analyze and quantify local distortions and rearrangements in disordered patterns across two and three dimensions, linking microscopic structure to macroscopic behavior.

Contribution

It introduces a coherent, self-consistent formalism for six statistical tools based on the texture matrix, unifying local and global pattern analysis in disordered systems.

Findings

Six tools provide a direct link between local distortions and large-scale material behavior.

The tools enable comparison of experiments, simulations, and models within a unified framework.

Application to foam plasticity demonstrates the tools' practical utility.

Abstract

Discrete rearranging patterns include cellular patterns, for instance liquid foams, biological tissues, grains in polycrystals; assemblies of particles such as beads, granular materials, colloids, molecules, atoms; and interconnected networks. Such a pattern can be described as a list of links between neighbouring sites. Performing statistics on the links between neighbouring sites yields average quantities (hereafter "tools") as the result of direct measurements on images. These descriptive tools are flexible and suitable for various problems where quantitative measurements are required, whether in two or in three dimensions. Here, we present a coherent set of robust tools, in three steps. First, we revisit the definitions of three existing tools based on the texture matrix. Second, thanks to their more general definition, we embed these three tools in a self-consistent formalism, which includes three additional ones. Third, we show that the six tools together provide a direct correspondence between a small scale, where they quantify the discrete pattern's local distortion and rearrangements, and a large scale, where they help describe a material as a continuous medium. This enables to formulate elastic, plastic, fluid behaviours in a common, self-consistent modelling using continuous mechanics. Experiments, simulations and models can be expressed in the same language and directly compared. As an example, a companion paper (Marmottant, Raufaste and Graner, joint paper) provides an application to foam plasticity.