Oscillatory Instabilities in 3-Dimensional Frictional Granular Matter

TL;DR

This paper demonstrates the existence of oscillatory instabilities in three-dimensional frictional granular materials, revealing complex eigenvalues and dynamic growth leading to catastrophic failure, extending previous 2D findings to 3D systems.

Contribution

The study analytically derives and confirms oscillatory instabilities in 3D granular packings, a phenomenon previously observed only in 1D and 2D systems.

Findings

Complex eigenvalues emerge at instability onset.

Oscillatory exponential growth in mean-square displacement.

Instabilities lead to catastrophic system failure.

Abstract

The dynamics of amorphous granular matter with frictional interactions cannot be derived in general from a Hamiltonian and therefore displays oscillatory instabilities stemming from the onset of complex eigenvalues in the stability matrix. These instabilities were discovered in the context of one and two dimensional systems, while the three dimensional case was never studied in detail. Here we fill this gap by deriving and demonstrating the presence of oscillatory instabilities in a three dimensional granular packing. We study binary assemblies of spheres of two sizes interacting via classical Hertz and Mindlin force laws for the longitudinal and tangent interactions, respectively. We formulate analytically the stability matrix in 3D and observe that a couple of complex eigenvalues emerges at the onset of the instability as in the case of frictional disks in two-dimensions. The dynamics then shows oscillatory exponential growth in the Mean-Square-Displacement, followed by a catastrophic event. The generality of these results for any choice of forces that break the symplectic Hamiltonian symmetry is discussed.