Hyperstaticity and loops in frictional granular packings

TL;DR

This paper investigates the hyperstaticity in rigid granular packings, highlighting the role of elementary loops and their differing stress behaviors based on loop parity, with statistical properties linked to friction.

Contribution

It introduces an algebraic and numerical analysis of loops in granular packings, revealing how odd and even loops differ in stress distribution and their statistical Gibbsian nature.

Findings

Odd loops have exterior latent stresses characterized by sum of forces.

Even loops have interior latent stresses characterized by alternating sum.

Loop force statistics follow a Gibbs distribution with a temperature proportional to friction coefficient.

Abstract

The hyperstatic nature of granular packings of perfectly rigid disks is analyzed algebraically and through numerical simulation. The elementary loops of grains emerge as a fundamental element in addressing hyperstaticity. Loops consisting of an odd number of grains behave differently than those with an even number. For odd loops, the latent stresses are exterior and are characterized by the sum of frictional forces around each loop. For even loops, the latent stresses are interior and are characterized by the alternating sum of frictional forces around each loop. The statistics of these two types of loop sums are found to be Gibbsian with a "temperature" that is linear with the friction coefficient mu when mu<1.