An exact expression of three-body system for the complex shear modulus of frictional granular materials

TL;DR

This paper introduces a simple three-particle model to derive an exact analytical expression for the complex shear modulus of jammed frictional granular materials, accurately capturing the behavior near the jamming transition.

Contribution

The study provides an exact analytical formula for the shear modulus of many-particle systems based on a three-particle model, revealing a scaling law near jamming.

Findings

Exact expression reproduces many-body shear modulus at low strains.

Model captures behavior of disordered systems with a single fitting parameter.

Results align with numerical simulations of granular materials.

Abstract

We propose a simple model comprising three particles to study the nonlinear mechanical response of jammed frictional granular materials under oscillatory shear. Owing to the introduction of the simple model, we obtain an exact analytical expression of the complex shear modulus for a system including many mono-dispersed disks, which satisfies a scaling law in the vicinity of the jamming point. These expressions perfectly reproduce the shear modulus of the many-body system with low strain amplitudes and friction coefficients. Even for disordered many-body systems, the model reproduces results by introducing a single fitting parameter.