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Journal Articles Nonlinearity Year : 2016

Strongly nonlinear waves in locally resonant granular chains

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Abstract

We explore a recently proposed locally resonant granular system bearing harmonic internal resonators in a chain of beads interacting via Hertzian elastic contacts. In this system, we propose the existence of two types of configurations: (a) small-amplitude periodic traveling waves and (b) darkbreather solutions, i.e. exponentially localized, time-periodic states mounted on top of a non-vanishing background. A remarkable feature distinguishing our results from other settings where dark breathers are observed is the complete absence of precompression in the system, i.e. the absence of a linear spectral band. We also identify conditions under which the system admits long-lived bright breather solutions. Our results are obtained by means of an asymptotic reduction to a suitably modified version of the so-called discrete p-Schrödinger (DpS) equation, which is established as controllably approximating the solutions of the original system for large but finite times (under suitable assumptions on the solution amplitude and the resonator mass). The findings are also corroborated by detailed numerical computations. Long-lived bright breathers are proved to exist over long but finite times, after which numerical simulations indicate that the breathers disintegrate. In line with these results, we prove that the only exact time-periodic bright breathers consist of trivial linear oscillations, without contact interactions between discrete elements.

Dates and versions

hal-01417957 , version 1 (16-12-2016)

Identifiers

Cite

Lifeng Liu, Guillaume James, Panayotis Kevrekidis, Anna Vainchtein. Strongly nonlinear waves in locally resonant granular chains. Nonlinearity, 2016, 29 (11), pp.3496-3527. ⟨10.1088/0951-7715/29/11/3496⟩. ⟨hal-01417957⟩
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