DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Strongly nonlinear waves in locally resonant granular chains

Journal Article · · Nonlinearity
 [1];  [2];  [3];  [1]
  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 ScienceCopyright Statement
Country of Publication:
United States
Language:
English
Citation Metrics:
Cited by: 20 works
Citation information provided by
Web of Science

References (30)

Wave localization in strongly nonlinear Hertzian chains with mass defect journal August 2009
Periodic Travelling Waves and Compactons in Granular Chains journal April 2012
Discrete Breathers in One-Dimensional Diatomic Granular Crystals journal June 2010
Breather Solutions of the Discrete p-Schrödinger Equation book October 2013
Experimental study of nonlinear acoustic bands and propagating breathers in ordered granular media embedded in matrix journal December 2014
Solitary waves in a chain of repelling magnets journal May 2014
sine-Gordon Equation: From Discrete to Continuum book January 2014
Rocking Newton’s cradle journal December 2004
Shock wave structure in a strongly nonlinear lattice with viscous dissipation journal February 2007
From Newton's Cradle to the Discrete $p$-Schrödinger Equation journal January 2013
Dark breathers in granular crystals journal April 2013
Wave propagation in single column woodpile phononic crystals: Formation of tunable band gaps journal November 2014
Damped-driven granular chains: An ideal playground for dark breathers and multibreathers journal March 2014
Breathers in nonlinear lattices: Existence, linear stability and quantization journal April 1997
Localized breathing modes in granular crystals with defects journal December 2009
NONLINEAR WAVES IN NEWTON'S CRADLE AND THE DISCRETE p-SCHRÖDINGER EQUATION journal November 2011
Interaction of traveling waves with mass-with-mass defects within a Hertzian chain journal April 2013
Highly Nonlinear Wave Propagation in Elastic Woodpile Periodic Structures journal March 2015
Solitary waves in a chain of beads under Hertz contact journal November 1997
Granular metamaterials for vibration mitigation journal September 2013
Discrete breathers — Advances in theory and applications journal October 2008
Dynamics of Heterogeneous Materials book January 2001
Stationary shocks in periodic highly nonlinear granular chains journal November 2009
Formation of rarefaction waves in origami-based metamaterials journal April 2016
Intrinsic energy localization through discrete gap breathers in one-dimensional diatomic granular crystals journal November 2010
Spatially inhomogeneous time-periodic propagating waves in anharmonic systems journal May 1997
Breathers in oscillator chains with Hertzian interactions journal May 2013
Solitary waves in the granular chain journal June 2008
Wave propagation in granular chains with local resonances journal March 2015
Observation of Two-Wave Structure in Strongly Nonlinear Dissipative Granular Chains journal April 2007

Cited By (8)

Nonlinear Excitations in Magnetic Lattices with Long-Range Interactions text January 2018
Waves in strongly nonlinear discrete systems journal July 2018
Nonlinear Coherent Structures in Granular Crystals text January 2016
Nonlinear mechanisms of two-dimensional wave-wave transformations in the initially coupled acoustic structure journal January 2018
Nonlinear excitations in magnetic lattices with long-range interactions journal June 2019
Quasiperiodic granular chains and Hofstadter butterflies
  • Martínez, Alejandro J.; Porter, Mason A.; Kevrekidis, P. G.
  • Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 376, Issue 2127 https://doi.org/10.1098/rsta.2017.0139
journal July 2018
Nonlinear excitations in magnetic lattices with long-range interactions text January 2019
Travelling breathers and solitary waves in strongly nonlinear lattices journal July 2018