A texture tensor to quantify deformations: the example of two-dimensional flowing foams

TL;DR

This paper introduces a texture tensor to measure deformations in materials, demonstrating its application to flowing foams and showing that such foams behave as elastic, continuous media.

Contribution

It extends classical deformation theories by introducing a statistical strain descriptor applicable to complex materials like flowing foams.

Findings

Foam exhibits elastic properties of a linear, isotropic medium.

The texture tensor effectively quantifies deformations in complex flowing materials.

The approach bridges elastic and fluid descriptions for materials with mixed properties.

Abstract

In a continuum description of materials, the stress tensor field $\bar{% \bar{\sigma}}$ quantifies the internal forces the neighbouring regions exert on a region of the material. The classical theory of elastic solids assumes that $\bar{\bar{\sigma}}$ determines the strain, while hydrodynamics assumes that $\bar{\bar{\sigma}}$ determines the strain rate. To extend both successful theories to more general materials, which display both elastic and fluid properties, we recently introduced a descriptor generalizing the classical strain to include plastic deformations: the ``statistical strain'', based on averages on microscopic details (``A texture tensor to quantify deformations'' M.Au., Y.J., J.A.G., F.G, companion paper, {\em Granular Matter}, same issue). Here, we apply such a statistical analysis to a two-dimensional foam steadily flowing through a constriction, a problem beyond reach of both theories, and prove that the foam has the elastic properties of a (linear and isotropic) continuous medium.