Laws of Granular Solids. Geometry and Topology

TL;DR

This paper develops an exact discrete calculus to connect microscopic force balance in granular solids to macroscopic rigidity, revealing how topology and geometry constraints govern stress distribution.

Contribution

It introduces a novel discrete calculus framework that links local force balance to global stress fields using topology and geometry in granular materials.

Findings

Derived Airy's stress function for disordered media

Established exact relations between topology, geometry, and force balance

Provided intrinsic and extrinsic formulations of force equilibrium

Abstract

In a granular solid, mechanical equilibrium requires a delicate balance of forces at the disordered grain scale. To understand how macroscopic rigidity can emerge in this amorphous solid, it is crucial that we understand how Newton's laws pass from the disordered grain scale to the laboratory scale. In this work, we introduce an exact discrete calculus, in which Newton's laws appear as differential relations at the scale of a single grain. Using this calculus, we introduce gauge variables which describe identically force- and torque-balanced configurations. In a first, intrinsic formulation, we use the topology of the contact network, but not its geometry. In a second, extrinsic formulation, we introduce geometry with the Delaunay triangulation. These formulations show, with exact methods, how topology and geometry in a disordered medium are related by constraints. In particular, we derive Airy's expression for a divergence-free, symmetric stress tensor in two and three dimensions.