Traveling waves for the mass in mass model of granular chains

Kevrekidis, Panayotis G.; Stefanov, Atanas G.; Xu, Haitao

doi:10.1007/s11005-016-0854-6

Title: Traveling waves for the mass in mass model of granular chains

Journal Article · Fri Jun 03 00:00:00 EDT 2016 · Letters in Mathematical Physics

DOI:https://doi.org/10.1007/s11005-016-0854-6· OSTI ID:1259964

Kevrekidis, Panayotis G. ^[1]; Stefanov, Atanas G. ^[2]; Xu, Haitao ^[3]

Univ. of Massachusetts, Amherst, MA (United States); Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Univ. of Kansas, Lawrence, KS (United States)
Univ. of Massachusetts, Amherst, MA (United States)

In this work, we consider the mass in mass (or mass with mass) system of granular chains, namely, a granular chain involving additionally an internal (or, respectively, external) resonator. For these chains, we rigorously establish that under suitable “anti-resonance” conditions connecting the mass of the resonator and the speed of the wave, bell-shaped traveling-wave solutions continue to exist in the system, in a way reminiscent of the results proven for the standard granular chain of elastic Hertzian contacts. Finally, we also numerically touch upon settings, where the conditions do not hold, illustrating, in line also with recent experimental work, that non-monotonic waves bearing non-vanishing tails may exist in the latter case.

View Accepted Manuscript (DOE)

Cite

Export

Save

Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE; National Science Foundation (NSF)

Grant/Contract Number:: AC52-06NA25396

OSTI ID:: 1259964

Report Number(s):: LA-UR-15-21320

Journal Information:: Letters in Mathematical Physics, Journal Name: Letters in Mathematical Physics; ISSN 0377-9017

Publisher:: SpringerCopyright Statement

Country of Publication:: United States

Language:: English

Citation Metrics:

Cited by: 19 works

Citation information provided by
Web of Science

References (37)

Compactons and chaos in strongly nonlinear lattices Ahnert, Karsten; Pikovsky, Arkady Physical Review E, Vol. 79, Issue 2 https://doi.org/10.1103/PhysRevE.79.026209	journal	February 2009
Bifurcation-based acoustic switching and rectification Boechler, N.; Theocharis, G.; Daraio, C. Nature Materials, Vol. 10, Issue 9 https://doi.org/10.1038/nmat3072	journal	July 2011
Wave propagation in granular chains with local resonances Bonanomi, Luca; Theocharis, Georgios; Daraio, Chiara Physical Review E, Vol. 91, Issue 3 https://doi.org/10.1103/PhysRevE.91.033208	journal	March 2015
A general rearrangement inequality for multiple integrals Brascamp, H. J.; Lieb, Elliott H.; Luttinger, J. M. Journal of Functional Analysis, Vol. 17, Issue 2 https://doi.org/10.1016/0022-1236(74)90013-5	journal	October 1974
Introduction: The Fermi–Pasta–Ulam problem—The first fifty years Campbell, David K.; Rosenau, Phillip; Zaslavsky, George M. Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 15, Issue 1 https://doi.org/10.1063/1.1889345	journal	March 2005
Asymptotic solution for solitary waves in a chain of elastic spheres Chatterjee, Anindya Physical Review E, Vol. 59, Issue 5 https://doi.org/10.1103/PhysRevE.59.5912	journal	May 1999
Strongly nonlinear waves in a chain of Teflon beads Daraio, C.; Nesterenko, V. F.; Herbold, E. B. Physical Review E, Vol. 72, Issue 1 https://doi.org/10.1103/PhysRevE.72.016603	journal	July 2005
On the solitary wave pulse in a chain of beads English, J. M.; Pego, R. L. Proceedings of the American Mathematical Society, Vol. 133, Issue 6, p. 1763-1768 https://doi.org/10.1090/S0002-9939-05-07851-2	journal	January 2005
Existence theorem for solitary waves on lattices Friesecke, Gero; Wattis, Jonathan A. D. Communications in Mathematical Physics, Vol. 161, Issue 2 https://doi.org/10.1007/BF02099784	journal	March 1994
Granular metamaterials for vibration mitigation Gantzounis, G.; Serra-Garcia, M.; Homma, K. Journal of Applied Physics, Vol. 114, Issue 9 https://doi.org/10.1063/1.4820521	journal	September 2013
An iterative method for the calculation of narrow solitary excitations on atomic chains Hochstrasser, D.; Mertens, F. G.; Büttner, H. Physica D: Nonlinear Phenomena, Vol. 35, Issue 1-2 https://doi.org/10.1016/0167-2789(89)90107-3	journal	April 1989
NONLINEAR WAVES IN NEWTON'S CRADLE AND THE DISCRETE p-SCHRÖDINGER EQUATION James, Guillaume Mathematical Models and Methods in Applied Sciences, Vol. 21, Issue 11 https://doi.org/10.1142/S0218202511005763	journal	November 2011
Breathers in oscillator chains with Hertzian interactions James, Guillaume; Kevrekidis, Panayotis G.; Cuevas, Jesús Physica D: Nonlinear Phenomena, Vol. 251 https://doi.org/10.1016/j.physd.2013.01.017	journal	May 2013
Non-linear waves in lattices: past, present, future Kevrekidis, P. G. IMA Journal of Applied Mathematics, Vol. 76, Issue 3 https://doi.org/10.1093/imamat/hxr015	journal	April 2011
Interaction of traveling waves with mass-with-mass defects within a Hertzian chain Kevrekidis, P. G.; Vainchtein, A.; Garcia, M. Serra Physical Review E, Vol. 87, Issue 4 https://doi.org/10.1103/PhysRevE.87.042911	journal	April 2013
Highly nonlinear waves' sensor technology for highway infrastructures Khatri, Devvrath; Daraio, Chiara; Rizzo, Piervincenzo The 15th International Symposium on: Smart Structures and Materials & Nondestructive Evaluation and Health Monitoring, SPIE Proceedings https://doi.org/10.1117/12.775848	conference	April 2008
Highly Nonlinear Wave Propagation in Elastic Woodpile Periodic Structures Kim, E.; Li, F.; Chong, C. Physical Review Letters, Vol. 114, Issue 11 https://doi.org/10.1103/PhysRevLett.114.118002	journal	March 2015
Wave propagation in single column woodpile phononic crystals: Formation of tunable band gaps Kim, Eunho; Yang, Jinkyu Journal of the Mechanics and Physics of Solids, Vol. 71 https://doi.org/10.1016/j.jmps.2014.06.012	journal	November 2014
Granular acoustic switches and logic elements Li, Feng; Anzel, Paul; Yang, Jinkyu Nature Communications, Vol. 5, Issue 1 https://doi.org/10.1038/ncomms6311	journal	October 2014
Tunable Unidirectional Sound Propagation through a Sonic-Crystal-Based Acoustic Diode Li, Xue-Feng; Ni, Xu; Feng, Liang Physical Review Letters, Vol. 106, Issue 8 https://doi.org/10.1103/PhysRevLett.106.084301	journal	February 2011
Acoustic Diode: Rectification of Acoustic Energy Flux in One-Dimensional Systems Liang, Bin; Yuan, Bo; Cheng, Jian-chun Physical Review Letters, Vol. 103, Issue 10 https://doi.org/10.1103/PhysRevLett.103.104301	journal	September 2009
The concentration-compactness principle in the Calculus of Variations. The Locally compact case, part 2 Lions, P. L. Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Vol. 1, Issue 4 https://doi.org/10.1016/S0294-1449(16)30422-X	journal	July 1984
Solitary waves in a chain of beads under Hertz contact MacKay, R. S. Physics Letters A, Vol. 251, Issue 3 https://doi.org/10.1016/S0375-9601(98)00867-6	journal	January 1999
Dynamics of Heterogeneous Materials Nesterenko, Vitali F. Shock Wave and High Pressure Phenomena https://doi.org/10.1007/978-1-4757-3524-6	book	January 2001
Anomalous Wave Reflection at the Interface of Two Strongly Nonlinear Granular Media Nesterenko, V. F.; Daraio, C.; Herbold, E. B. Physical Review Letters, Vol. 95, Issue 15 https://doi.org/10.1103/PhysRevLett.95.158702	journal	October 2005
Propagation of nonlinear compression pulses in granular media Nesterenko, V. F. Journal of Applied Mechanics and Technical Physics, Vol. 24, Issue 5 https://doi.org/10.1007/BF00905892	journal	January 1984
Solitary waves in the granular chain Sen, S.; Hong, J.; Bang, J. Physics Reports, Vol. 462, Issue 2 https://doi.org/10.1016/j.physrep.2007.10.007	journal	June 2008
Generation and control of sound bullets with a nonlinear acoustic lens Spadoni, A.; Daraio, C. Proceedings of the National Academy of Sciences, Vol. 107, Issue 16 https://doi.org/10.1073/pnas.1001514107	journal	April 2010
On the Existence of Solitary Traveling Waves for Generalized Hertzian Chains Stefanov, Atanas; Kevrekidis, Panayotis Journal of Nonlinear Science, Vol. 22, Issue 3 https://doi.org/10.1007/s00332-011-9119-9	journal	January 2012
Traveling waves for monomer chains with precompression Stefanov, Atanas; Kevrekidis, Panayotis Nonlinearity, Vol. 26, Issue 2 https://doi.org/10.1088/0951-7715/26/2/539	journal	January 2013
Traveling waves and their tails in locally resonant granular systems Xu, H.; Kevrekidis, P. G.; Stefanov, A. Journal of Physics A: Mathematical and Theoretical, Vol. 48, Issue 19 https://doi.org/10.1088/1751-8113/48/19/195204	journal	April 2015
Embedded Solitons in Second-Harmonic-Generating Systems Yang, J.; Malomed, B. A.; Kaup, D. J. Physical Review Letters, Vol. 83, Issue 10 https://doi.org/10.1103/PhysRevLett.83.1958	journal	September 1999
Discrete embedded solitons Yagasaki, Kazuyuki; Champneys, Alan R.; Malomed, Boris A. Nonlinearity, Vol. 18, Issue 6 https://doi.org/10.1088/0951-7715/18/6/010	journal	August 2005
Compactons and Chaos in Strongly Nonlinear Lattices Ahnert, Karsten; Pikovsky, Arkady arXiv https://doi.org/10.48550/arxiv.0808.3059	text	January 2008
Nonlinear waves in Newton's cradle and the discrete p-Schroedinger equation James, Guillaume arXiv https://doi.org/10.48550/arxiv.1008.1153	preprint	January 2010
Breathers in oscillator chains with Hertzian interactions James, Guillaume; Kevrekidis, Panayotis; Cuevas, Jesus arXiv https://doi.org/10.48550/arxiv.1111.1857	preprint	January 2011
Strongly nonlinear waves in a chain of Teflon beads Daraio, C.; Nesterenko, V. F.; Herbold, E. arXiv https://doi.org/10.48550/arxiv.cond-mat/0503299	text	January 2005

Cited By (4)

Nonlinear mechanisms of two-dimensional wave-wave transformations in the initially coupled acoustic structure Vorotnikov, K.; Starosvetsky, Y. Journal of Applied Physics, Vol. 123, Issue 2 https://doi.org/10.1063/1.4986282	journal	January 2018
Waves in strongly nonlinear discrete systems Nesterenko, Vitali F. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 376, Issue 2127 https://doi.org/10.1098/rsta.2017.0130	journal	July 2018
Travelling breathers and solitary waves in strongly nonlinear lattices James, Guillaume Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 376, Issue 2127 https://doi.org/10.1098/rsta.2017.0138	journal	July 2018
Nonlinear Coherent Structures in Granular Crystals Chong, C.; Porter, Mason A.; Kevrekidis, P. G. arXiv https://doi.org/10.48550/arxiv.1612.03977	text	January 2016

Similar Records

Strongly nonlinear waves in locally resonant granular chains

Journal Article · Fri Sep 23 00:00:00 EDT 2016 · Nonlinearity · OSTI ID:1259964

Liu, Lifeng; James, Guillaume; Kevrekidis, Panayotis; +1 more

Traveling waves and their tails in locally resonant granular systems

Journal Article · Wed Apr 22 00:00:00 EDT 2015 · Journal of Physics. A, Mathematical and Theoretical · OSTI ID:1259964

Xu, H.; Kevrekidis, P. G.; Stefanov, A.

Breathers in a locally resonant granular chain with precompression

Journal Article · Thu Sep 01 00:00:00 EDT 2016 · Physica. D, Nonlinear Phenomena · OSTI ID:1259964

Liu, Lifeng; James, Guillaume; Kevrekidis, Panayotis; +1 more

Related Subjects

36 MATERIALS SCIENCE
97 MATHEMATICS AND COMPUTING
Mathematics
Material Science

Title: Traveling waves for the mass in mass model of granular chains

Citation Formats

References (37)

Cited By (4)

Similar Records

Related Subjects