Section: New Results

Nonlinear waves in granular chains

Participants : Guillaume James, Bernard Brogliato.

Granular chains made of aligned beads interacting by contact (e.g. Newton's cradle) are widely studied in the context of impact dynamics and acoustic metamaterials. While much effort has been devoted to the theoretical and experimental analysis of solitary waves in granular chains, there is now an increasing interest in the study of breathers (spatially localized oscillations) in granular systems. Due to their oscillatory nature and associated resonance phenomena, static or traveling breathers exhibit much more complex dynamical properties compared to solitary waves. Such properties have strong potential applications for the design of acoustic metamaterials allowing to efficiently damp or deviate shocks and vibrations. In the work [29], the existence of static breathers is analyzed in granular metamaterials consisting of hollow beads with internal masses. Using multiple scale analysis and exploiting the unilateral character of Hertzian interactions, we show that long-lived breather solutions exist but time-periodic breathers do not (breather solutions actually disperse on long time scales). In [28], we consider the effect of adding precompression to the above system and establish that the envelope of small ampliude oscillations is governed by a nonlinear Schrödinger equation. This allows us to show that, depending on the applied precompression, normal modes can become modulationally unstable and evolve towards traveling breathers. Moreover, in a collaboration with Y. Starosvetsky and D. Meimukhin (Technion), we numerically study the persistence of traveling breathers in granular chains with local potentials under the effect of contact damping. Using a viscoelastic damping model (Hertz-Kuwabara-Kono model), we show that breathers can be generated by simple impacts in granular chains made from various materials (breathers propagate over a significant number of sites before being damped). The design of an experimental setup to test these theoretical predictions is underway. Another work in progress concerns more specifically the modeling and numerical analysis of dissipative impacts (James, Brogliato). The methodology is based on the introduction of appropriate variables and simplifications for different models of contact damping. A postdoctoral fellow will work on this topic in the team, starting January 2017.