Predicting Defects in Soft Sphere Packings near Jamming Using the Force Network Ensemble

Local stability of spheres via the convex hull and the radical Voronoi diagram

Jamming is an emergent phenomenon wherein the local stability of individual particles percolates to form a globally rigid structure. However, the onset of rigidity does not imply that every particle becomes rigid, and indeed some remain locally unstable. These particles, if they become unmoored from their neighbors, are called rattlers, and their identification is critical to understanding the rigid backbone of a packing, as these particles cannot bear stress. The accurate identification of rattlers, however, can be a time-consuming process, and the currently accepted method lacks a simple geometric interpretation. In this manuscript, we propose two simpler classifications of rattlers in hard sphere systems based on the convex hull of contacting neighbors and the maximum inscribed sphere of the radical Voronoi cell, each of which provides geometric insight into the source of their instability. Furthermore, the convex hull formulation can be generalized to explore stability in hyperstatic soft sphere packings, spring networks, nonspherical packings, and mean-field non-central-force potentials.

Training precise stress patterns

We introduce a training rule that enables a network composed of springs and dashpots to learn precise stress patterns. Our goal is to control the tensions on a fraction of "target" bonds, which are chosen randomly. The system is trained by applying stresses to the target bonds, causing the remaining bonds, which act as the learning degrees of freedom, to evolve. Different criteria for selecting the target bonds affects whether frustration is present. When there is at most a single target bond per node the error converges to computer precision. Additional targets on a single node may lead to slow convergence and failure. Nonetheless, training is successful even when approaching the limit predicted by the Maxwell Calladine theorem. We demonstrate the generality of these ideas by considering dashpots with yield stresses. We show that training converges, albeit with a slower, power-law decay of the error. Furthermore, dashpots with yielding stresses prevent the system from relaxing after training, enabling to encode permanent memories.

First-contact-breaking distributions in strained disordered crystals

We derive exact probability distributions for the strain (ε) at which the first stress drop event occurs in uniformly strained disordered crystals, with quenched disorder introduced through polydispersity in particle sizes. We characterize these first stress drop events numerically as well as theoretically and identify them with the first-contact-breaking event in the system. Our theoretical results are corroborated with numerical simulations of quasistatic volumetric strain applied to disordered near-crystalline configurations of athermal soft particles. We develop a general technique to determine the distribution of strains at which the first stress drop events occur, through an exact mapping between the cumulative distribution of first-contact-breaking events and the volume of a convex polytope whose dimension is determined by the number of defects N_{d} in the system. An exact numerical computation of this polytope volume for systems with small numbers of defects displays a remarkable match with the distribution of strains generated through direct numerical simulations. Finally, we derive the distribution of strains at which the first stress drop occurs, assuming that individual contact-breaking events are uncorrelated, which accurately reproduces distributions obtained from direct numerical simulations.