Periodic travelling waves and compactons in granular chains

Guillaume James 1
1 CASYS
LJK - Laboratoire Jean Kuntzmann
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.
Document type :
Journal articles
Journal of Nonlinear Science, Springer Verlag (Germany), 2012, 22 (5), pp.813-848. <10.1007/s00332-012-9128-3>


https://hal.archives-ouvertes.fr/hal-00676076
Contributor : Guillaume James <>
Submitted on : Friday, March 2, 2012 - 9:52:58 PM
Last modification on : Tuesday, October 28, 2014 - 6:33:54 PM

Files

James_jnls_granular.pdf
fileSource_public_author

Identifiers

Collections

Citation

Guillaume James. Periodic travelling waves and compactons in granular chains. Journal of Nonlinear Science, Springer Verlag (Germany), 2012, 22 (5), pp.813-848. <10.1007/s00332-012-9128-3>. <hal-00676076>

Export

Share

Metrics

Consultation de
la notice

202

Téléchargement du document

68