# Stress Response of Granular Systems

## TL;DR

This paper introduces a framework for understanding stress response in two-dimensional granular media, highlighting force localization and the influence of network structure and constraints on stability.

## Contribution

It develops a graph Laplacian-based approach to analyze force response and localization in granular systems, incorporating microscopic force balance and gauge degrees of freedom.

## Key findings

- Force localization is demonstrated through spectral analysis.
- Numerical evidence from soft disk packings supports the theory.
- Constraints like torque balance affect packing stability.

## Abstract

We develop a framework for stress response in two dimensional granular media, with and without friction, that respects vector force balance at the microscopic level. We introduce local gauge degrees of freedom that determine the response of contact forces between constituent grains on a given, disordered, contact network, to external perturbations. By mapping this response to the spectral properties of the graph Laplacian corresponding to the underlying contact network, we show that this naturally leads to spatial localization of forces. We present numerical evidence for localization using exact diagonalization studies of network Laplacians of soft disk packings. Finally, we discuss the role of other constraints, such as torque balance, in determining the stability of a granular packing to external perturbations.