Anomalous force diffusion in nearly-ordered packings of frictionless discs

TL;DR

This paper analytically investigates force propagation in nearly-ordered, frictionless disc packings, revealing superdiffusive and subdiffusive regimes, and characterizing load response fluctuations with depth.

Contribution

It provides the first analytical expressions for force distribution and response fluctuations in nearly-ordered frictionless disc packings, extending understanding beyond ordered solutions.

Findings

Force distribution P(f) broadens with depth

Response fluctuations depend on height, not lateral position

Force propagation exhibits superdiffusive and subdiffusive regimes

Abstract

We derive analytic expressions for force propagation in packings of frictionless discs with a narrow distribution of disc sizes, by expanding to first order about the known ordered solution. The distribution of contact forces P(f) is found to be narrow at the upper surface, and broaden at a rate that varies with depth, being superdiffusive near the surface until crossing over to a subdiffusive regime near the fixed base. Furthermore, the response to an isolated load propagates along the edge of a `cone,' as in the ordered case, but fluctuates under ensemble averaging by an amount that depends purely on height, not on the lateral position. Finally, we comment on ways in which the analytical framework presented here could be extended to a wider range of granular packings.