Measurement of Indeterminacy in Packings of Perfectly Rigid Disks

TL;DR

This paper investigates the indeterminacy of contact forces in static packings of perfectly rigid disks, combining theoretical formulation and numerical simulations to analyze force sets, sliding contacts, and force indeterminacy.

Contribution

It introduces a mathematical framework for contact force sets, linking their boundary to sliding contacts, and compares theoretical predictions with simulation results.

Findings

The set of possible contact forces is convex and its boundary relates to sliding contacts.

The number of sliding contacts in stable packings is limited by 2M-3N in two dimensions.

Contacts with high force indeterminacy are located on force chains.

Abstract

Static packings of perfectly rigid particles are investigated theoretically and numerically. The problem of finding the contact forces in such packings is formulated mathematically. Letting the values of the contact forces define a vector in a high-dimensional space enable us to show that the set of all possible contact forces is convex, facilitating its numerical exploration. It is also found that the boundary of the set is connected with the presence of sliding contacts, suggesting that a stable packing should not have more than 2M-3N sliding contacts in two dimensions, where M is the number of contacts and N is the number of particles. These results were used to analyze packings generated in different ways by either molecular dynamics or contact dynamics simulations. The dimension of the set of possible forces and the number of sliding contacts agrees with the theoretical expectations. The indeterminacy of each component of the contact forces are found, as well as the an estimate for the diameter of the set of possible contact forces. We also show that contacts with high indeterminacy are located on force chains. The question of whether the simulation methods can represent a packing's memory of its formation is addressed.