Stress anisotropy in shear-jammed packings of frictionless disks

TL;DR

This study investigates how stress anisotropy develops in mechanically stable packings of frictionless disks under different protocols, revealing protocol-dependent distributions and relationships with dilatancy, and highlighting the impact of packing-generation methods on macroscopic properties.

Contribution

It introduces a detailed analysis of stress anisotropy in frictionless disk packings, deriving relationships with dilatancy and comparing effects of different packing protocols.

Findings

Stress anisotropy distribution is Gaussian centered at zero for isotropic compression.

Shear jammed packings exhibit Weibull-based stress anisotropy distributions with nonzero mean.

Different packing protocols lead to distinct probabilities and macroscopic properties of packings.

Abstract

We perform computational studies of repulsive, frictionless disks to investigate the development of stress anisotropy in mechanically stable (MS) packings. We focus on two protocols for generating MS packings: 1) isotropic compression and 2) applied simple or pure shear strain $\gamma$ at fixed packing fraction $\phi$. MS packings of frictionless disks occur as geometric families (i.e. parabolic segments with positive curvature) in the $\phi$-$\gamma$ plane. MS packings from protocol 1 populate parabolic segments with both signs of the slope, $d\phi/d\gamma >0$ and $d\phi/d\gamma <0$. In contrast, MS packings from protocol 2 populate segments with $d\phi/d\gamma <0$ only. For both simple and pure shear, we derive a relationship between the stress anisotropy and dilatancy $d\phi/d\gamma$ obeyed by MS packings along geometrical families. We show that for MS packings prepared using isotropic compression, the stress anisotropy distribution is Gaussian centered at zero with a standard deviation that decreases with increasing system size. For shear jammed MS packings, the stress anisotropy distribution is a convolution of Weibull distributions that depend on strain, which has a nonzero average and standard deviation in the large-system limit. We also develop a framework to calculate the stress anisotropy distribution for packings generated via protocol 2 in terms of the stress anisotropy distribution for packings generated via protocol 1. These results emphasize that for repulsive frictionless disks, different packing-generation protocols give rise to different MS packing probabilities, which lead to differences in macroscopic properties of MS packings.