Periodic travelling waves and compactons in granular chains

TL;DR

This paper investigates unique periodic travelling waves in granular chains with Hertzian contact forces, demonstrating their existence, asymptotic behavior, and the emergence of compactons, along with stability analysis.

Contribution

It proves the existence of these waves near binary oscillations, explores their long wave limit, and introduces the concept of compactons in granular chains.

Findings

Existence of periodic travelling waves close to binary oscillations.

Convergence of waves to shock profiles with solitary wave regions.

Identification of compactons as finite support compression waves.

Abstract

We study the propagation of an unusual type of periodic travelling waves in chains of identical beads interacting via Hertz's contact forces. Each bead periodically undergoes a compression phase followed by a free flight, due to special properties of Hertzian interactions (fully nonlinear under compression and vanishing in the absence of contact). We prove the existence of such waves close to binary oscillations, and numerically continue these solutions when their wavelength is increased. In the long wave limit, we observe their convergence towards shock profiles consisting of small compression regions close to solitary waves, alternating with large domains of free flight where bead velocities are small. We give formal arguments to justify this asymptotic behaviour, using a matching technique and previous results concerning solitary wave solutions. The numerical finding of such waves implies the existence of compactons, i.e. compactly supported compression waves propagating at a constant velocity, depending on the amplitude and width of the wave. The beads are stationary and separated by equal gaps outside the wave, and each bead reached by the wave is shifted by a finite distance during a finite time interval. Below a critical wavenumber, we observe fast instabilities of the periodic travelling waves leading to a disordered regime.