Solitary waves in a chain of beads under Hertz contact

Abstract

International audienceWe study experimentally the propagation of high-amplitude compressional waves in a chain of beads in contact, submitted or not to a small static force. In such a system, solitary waves have been theoretically predicted by Nesterenko J. Appl. Mech. Tech. Phys. USSR 5, 733 1984. We have built an impact generator in order to create high-amplitude waves in the chain. We observe the propagation of isolated nonlinear pulses, measure their velocity as a function of their maximum amplitude, for different applied static forces, and record their shape. In all experiments, we find good agreement between our observations and the theoretical predictions of the above reference, without using any adjustable parameter in the data analysis. We also show that the velocity measurements taken at three different nonzero applied static forces all lie on a single curve, when expressed in rescaled variables. The size of the pulses is typically one-tenth the total length of the chain. All the measurements support the identification of these isolated nonlinear pulses with the solitary waves predicted by Nesterenko. S1063-651X9704211-

On the solitary wave pulse in a chain of beads

“…Motivation for studying equation (1) comes from the work of Coste et al [3]. These authors studied experimentally the propagation of solitary wave pulses through a long chain of identical steel spheres.…”

“…In such a chain, the laws of motion for the system state that the acceleration of each ball is proportional to the difference between the compression force of the two neighboring balls. For two adjacent elastic steel spheres with radius a, the force between them according to Hertz's law is F comp ∝ r 3 2 , where r is the distance of approach of their centers. Thus, the equation of motion for the n th bead in the chain has the form…”

Cohesive, frictional powders: contact models for tension

“…For spheres, typically contact models in the spirit of Hertz [4,22,35,60,66,70,79] seem appropriate-but only when the forces are small enough so that the yield stress is reached nowhere close to the contact area. For rather large metal spheres, the details of contact models are even measurable, when waves propagate along chains of particles [10,11,49,75], and a Hertz based contact law is recommmended. However, Hertz models will not be discussed in this study, since finer powders only have a negligible range of elastic Hertz-like behavior [87] and, furthermore, are never perfectly spherical at the contact anyway.…”

Mechanical wave propagation within nanogold granular crystals

“…The simplest form of granular crystals, one-dimensional (1D) chains of identical elastic spheres, were analytically, numerically and experimentally shown to possess the concept of "sonic vacuum" where classical phonons are not supported to highly nonlinear solitary waves (Nesterenko soliton or compacton) [5][6][7][8] . The nonlinear response of 1D granular crystals can be tuned in a wide range from linearity, weakly nonlinearity to strongly nonlinearity by the application of a variable precompression [7][8][9] .…”

“…The nonlinear response of 1D granular crystals can be tuned in a wide range from linearity, weakly nonlinearity to strongly nonlinearity by the application of a variable precompression [7][8][9] . Furthermore, by altering the geometry 10-13 , material properties 14,15 and spatial distribution [16][17][18] of component granules, one is rendered more freedom to manipulate the propagation of mechanical signals, which makes granular crystals building blocks for a broad range of novel applications such as impact-protection devices 13,19,20 , acoustic diodes 21 and switches 22 , and tunable vibration filters 23,24 , just to name a few.…”