Strongly nonlinear waves in locally resonant granular chains

Journal Article · Fri Sep 23 00:00:00 EDT 2016 · Nonlinearity

DOI:https://doi.org/10.1088/0951-7715/29/11/3496· OSTI ID:1360694

Liu, Lifeng ^[1]; James, Guillaume ^[2]; Kevrekidis, Panayotis ^[3]; Vainchtein, Anna ^[1]

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1. Univ. of Pittsburgh, PA (United States). Dept. of Mathematics
2. INRIA Grenoble-Rhone-Alpes, Saint Ismier (France)
3. Univ. of Massachusetts, Amherst, MA (United States). Dept. of Mathematics and Statistics; Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

In this paper, 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) dark-breather 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. Finally, 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.

• Research Organization: Univ. of Pittsburgh, PA (United States); Univ. of Massachusetts, Amherst, MA (United States); Los Alamos National Laboratory (LANL), Los Alamos, NM (United States); INRIA Grenoble-Rhone-Alpes, Saint Ismier (France)
• Sponsoring Organization: USDOE; National Science Foundation (NSF); US Air Force Office of Scientific Research (AFOSR); US Army Research Office (ARO); Rhone-Alpes Complex Systems Inst. (IXXI) (France)
• Grant/Contract Number: AC52-06NA25396; DMS-1007908; DMS-1506904; FA9550-12-1-0332; W911NF-15-1-0604
• OSTI ID: 1360694
• Alternate ID(s): OSTI ID: 1326243
• Report Number(s): LA-UR-15-25011
• Journal Information: Nonlinearity, Vol. 29, Issue 11; ISSN 0951-7715
• Publisher: IOP Science
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
locally resonant granular chain
periodic traveling wave
discrete breather