Fluctuations and the Effective Moduli of an Isotropic, Random Aggregate of Identical, Frictionless Spheres

TL;DR

This paper investigates how fluctuations affect the effective elastic moduli of a random aggregate of frictionless spheres under isotropic compression and shear, improving agreement with numerical simulations.

Contribution

It introduces a model incorporating fluctuations in particle displacements, leading to more accurate predictions of the aggregate's shear modulus.

Findings

Effective shear modulus decreases when fluctuations are included.

Model predictions align better with numerical simulation results.

Fluctuations significantly influence the aggregate's elastic response.

Abstract

We consider a random aggregate of identical frictionless elastic spheres that has first been subjected to an isotropic compression and then sheared. We assume that the average strain provides a good description of how stress is built up in the initial isotropic compression. However, when calculating the increment in the displacement between a typical pair of contaction particles due to the shearing, we employ force equilibrium for the particles of the pair, assuming that the average strain provides a good approximation for their interactions with their neighbors. The incorporation of these additional degrees of freedom in the displacement of a typical pair relaxes the system, leading to a decrease in the effective moduli of the aggregate. The introduction of simple models for the statistics of the ordinary and conditional averages contributes an additional decrease in moduli. The resulting value of the shear modulus is in far better agreement with that measured in numerical simulations.