Geometrical families of mechanically stable granular packings

TL;DR

This paper classifies and analyzes the geometrical families of mechanically stable granular packings in small sheared systems, revealing deterministic dynamics for fewer than 16 particles and complex bifurcations for larger systems.

Contribution

It introduces a classification of MS packings into geometrical families and studies their dynamics under shear, highlighting differences between small and large systems.

Findings

MS packings form continuous geometrical families.

Dynamics are deterministic and contracting for N<16.

Bifurcations cause complex behavior in larger systems.

Abstract

We enumerate and classify nearly all of the possible mechanically stable (MS) packings of bidipserse mixtures of frictionless disks in small sheared systems. We find that MS packings form continuous geometrical families, where each family is defined by its particular network of particle contacts. We also monitor the dynamics of MS packings along geometrical families by applying quasistatic simple shear strain at zero pressure. For small numbers of particles (N < 16), we find that the dynamics is deterministic and highly contracting. That is, if the system is initialized in a MS packing at a given shear strain, it will quickly lock into a periodic orbit at subsequent shear strain, and therefore sample only a very small fraction of the possible MS packings in steady state. In studies with N>16, we observe an increase in the period and random splittings of the trajectories caused by bifurcations in configuration space. We argue that the ratio of the splitting and contraction rates in large systems will determine the distribution of MS-packing geometrical families visited in steady-state. This work is part of our long-term research program to develop a master-equation formalism to describe macroscopic slowly driven granular systems in terms of collections of small subsystems.