Jamming graphs: A local approach to global mechanical rigidity

TL;DR

This paper introduces jamming graphs, a local construction method for understanding global mechanical rigidity in sphere packings, revealing new insights into stability, destabilization, and lengthscales at the jamming transition.

Contribution

It presents a novel local move-based construction of jamming graphs that incorporate both global and local stability properties, linking rigidity to correlated percolation.

Findings

Jamming graphs can be constructed via local moves respecting rigidity constraints.

Destabilization by bond deletion reveals a diverging lengthscale.

Adding redundant contacts can destabilize the system, indicating complex stability behavior.

Abstract

We revisit the concept of minimal rigidity as applied to soft repulsive, frictionless sphere packings in two-dimensions with the introduction of the jamming graph. Minimal rigidity is a purely combinatorial property encoded via Laman's theorem in two-dimensions. It constrains the global, average coordination number of the graph, for example. However, minimal rigidity does not address the geometry of local mechanical stability. The jamming graph contains both properties of global mechanical stability at the onset of jamming and local mechanical stability. We demonstrate how jamming graphs can be constructed using local moves via the Henneberg construction such that these graphs fall under the jurisdiction of correlated percolation. We then probe how jamming graphs destabilize, or become unjammed, by deleting a bond and computing the resulting rigid cluster distribution. We also study how the system restabilizes with the addition of new contacts and how a jamming graph with extra/redundant contacts destabilizes. The latter endeavor allows us to probe a disc packing in the rigid phase and uncover a potentially new diverging lengthscale associated with the random deletion of contacts as compared to the study of cut-out (or frozen in) subsystems.