Tensor Electromagnetism and Emergent Elasticity in Jammed Solids

TL;DR

This paper develops a novel stress-only theoretical framework linking emergent tensor electromagnetism to elasticity in jammed solids, supported by numerical simulations, providing new insights into their mechanical response.

Contribution

It introduces an emergent $U(1)$ tensor electromagnetism model for amorphous jammed solids, connecting force balance constraints to elastic behavior and electromagnetic analogies.

Findings

Exact mapping of mechanical response to dielectric response in tensor electromagnetism

Numerical simulations confirm stress correlations and responses in 2D and 3D

External forces act as electric charges in the emergent electromagnetic framework

Abstract

The theory of mechanical response and stress transmission in disordered, jammed solids poses several open questions of how non-periodic networks -- apparently indistinguishable from a snapshot of a fluid -- sustain shear. We present a stress-only theory of emergent elasticity for a non-thermal amorphous assembly of grains in a jammed solid, where each grain is subjected to mechanical constraints of force and torque balance. These grain-level constraints lead to the Gauss's law of an emergent $U(1)$ tensor electromagnetism, which then accounts for the mechanical response of such solids. This formulation of amorphous elasticity has several immediate consequences. The mechanical response maps exactly to the static, dielectric response of this tensorial electromagnetism with the polarizability of the medium mapping to emergent elastic moduli. External forces act as vector electric charges whereas the tensorial magnetic fields are sourced by momentum density. The dynamics in the electric and magnetic sectors, naturally translate into the dynamics of the rigid jammed network and ballistic particle motion respectively. The theoretical predictions for both stress-stress correlations and responses are borne out by the results of numerical simulations of frictionless granular packings in the static limit of the theory in both 2D and 3D.