Stress in planar cellular solids and isostatic granular assemblies: Coarse-graining the constitutive equation

TL;DR

This paper develops a coarse-graining method for stress equations in planar cellular and granular systems, enabling analysis of macroscopic structures with microstructural complexity.

Contribution

It introduces a renormalization approach to handle frustrated staggered order, allowing the application of a constitutive equation to realistic systems.

Findings

Successfully couples stress to a reduced geometric tensor.

Enables stress analysis in systems with frustrated staggered order.

Demonstrates the method by calculating stress around a defect in a foam.

Abstract

A recent theory for stress transmission in isostatic granular and cellular systems predicts a constitutive equation that couples the stress field to the local microstructure. The theory could not be applied to macroscopic systems because the constitutive equation becomes trivial upon straightforward coarse-graining. This problem is resolved here for arbitrary planar structures. The solution is based on the observation that staggered order makes it possible to couple the stress to a reduced geometric tensor that can be coarse-grained. The method proposed here makes it possible to apply this idea to realistic systems whose staggered order is generally 'frustrated'. This is achieved by a renormalization procedure which removes the frustration and enables the use of the upscalable reduced tensor. As an example we calculate the stress due to a defect in a periodic solid foam.