Traveling waves and their tails in locally resonant granular systems

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

doi:10.1088/1751-8113/48/19/195204

Title: 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

DOI:https://doi.org/10.1088/1751-8113/48/19/195204· OSTI ID:1233282

Xu, H. ^[1]; Kevrekidis, P. G. ^[2]; Stefanov, A. ^[3]

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

In the present study, we revisit the theme of wave propagation in locally resonant granular crystal systems, also referred to as mass-in-mass systems. We use three distinct approaches to identify relevant traveling waves. In addition, the first consists of a direct solution of the traveling wave problem. The second one consists of the solution of the Fourier tranformed variant of the problem, or, more precisely, of its convolution reformulation (upon an inverse Fourier transform) in real space. Finally, our third approach will restrict considerations to a finite domain, utilizing the notion of Fourier series for important technical reasons, namely the avoidance of resonances, which will be discussed in detail. All three approaches can be utilized in either the displacement or the strain formulation. Typical resulting computations in finite domains result in the solitary waves bearing symmetric non-vanishing tails at both ends of the computational domain. Importantly, however, a countably infinite set of anti-resonance conditions is identified for which solutions with genuinely rapidly decaying tails arise.

View Accepted Manuscript (DOE)

Cite

Export

Save

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

Sponsoring Organization:: USDOE

Grant/Contract Number:: 1313107; DMS1312856; 2010239; 605096; FA9550-12-10332; AC52-06NA25396

OSTI ID:: 1233282

Alternate ID(s):: OSTI ID: 1239032

Report Number(s):: LA-UR-14-29563; TRN: US1600510

Journal Information:: Journal of Physics. A, Mathematical and Theoretical, Vol. 48, Issue 19; ISSN 1751-8113

Country of Publication:: United States

Language:: English

Citation Metrics:

Cited by: 29 works

Citation information provided by
Web of Science

References (57)

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
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
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
Tunability of solitary wave properties in one-dimensional strongly nonlinear phononic crystals Daraio, C.; Nesterenko, V. F.; Herbold, E. B. Physical Review E, Vol. 73, Issue 2 https://doi.org/10.1103/PhysRevE.73.026610	journal	February 2006
Universal Power-Law Decay of the Impulse Energy in Granular Protectors Hong, Jongbae Physical Review Letters, Vol. 94, Issue 10 https://doi.org/10.1103/PhysRevLett.94.108001	journal	March 2005
Optimal Design of Composite Granular Protectors Fraternali, Fernando; Porter, Mason A.; Daraio, Chiara Mechanics of Advanced Materials and Structures, Vol. 17, Issue 1 https://doi.org/10.1080/15376490802710779	journal	December 2009
Decorated, Tapered, and Highly Nonlinear Granular Chain Doney, Robert; Sen, Surajit Physical Review Letters, Vol. 97, Issue 15 https://doi.org/10.1103/PhysRevLett.97.155502	journal	October 2006
Sound propagation in a constrained lattice of beads: High-frequency behavior and dispersion relation Coste, Christophe; Gilles, Bruno Physical Review E, Vol. 77, Issue 2 https://doi.org/10.1103/PhysRevE.77.021302	journal	February 2008
Strongly Nonlinear Waves in 3D Phononic Crystals Daraio, C. SHOCK COMPRESSION OF CONDENSED MATTER - 2003: Proceedings of the Conference of the American Physical Society Topical Group on Shock Compression of Condensed Matter, AIP Conference Proceedings https://doi.org/10.1063/1.1780215	conference	January 2004
Experimental study of sound propagation in a chain of spherical beads de Billy, M. The Journal of the Acoustical Society of America, Vol. 108, Issue 4 https://doi.org/10.1121/1.1289365	journal	October 2000
On the low vibration modes observed in a sphere submitted to a tangential contact force De Billy, M. Ultrasonics, Vol. 45, Issue 1-4 https://doi.org/10.1016/j.ultras.2006.07.019	journal	December 2006
Low-Frequency Behavior of Beads Constrained on a Lattice Gilles, Bruno; Coste, Christophe Physical Review Letters, Vol. 90, Issue 17 https://doi.org/10.1103/PhysRevLett.90.174302	journal	May 2003
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
Highly nonlinear solitary waves in periodic dimer granular chains Porter, Mason A.; Daraio, Chiara; Herbold, Eric B. Physical Review E, Vol. 77, Issue 1 https://doi.org/10.1103/PhysRevE.77.015601	journal	January 2008
Highly nonlinear solitary waves in heterogeneous periodic granular media Porter, Mason A.; Daraio, Chiara; Szelengowicz, Ivan Physica D: Nonlinear Phenomena, Vol. 238, Issue 6 https://doi.org/10.1016/j.physd.2008.12.010	journal	April 2009
Pulse velocity in a granular chain Rosas, Alexandre; Lindenberg, Katja Physical Review E, Vol. 69, Issue 3 https://doi.org/10.1103/PhysRevE.69.037601	journal	March 2004
Dynamic photoelastic studies of wave propagation in granular media Shukla, Arun Optics and Lasers in Engineering, Vol. 14, Issue 3 https://doi.org/10.1016/0143-8166(91)90047-W	journal	January 1991
Stationary shocks in periodic highly nonlinear granular chains Molinari, Alain; Daraio, Chiara Physical Review E, Vol. 80, Issue 5 https://doi.org/10.1103/PhysRevE.80.056602	journal	November 2009
Energy Trapping and Shock Disintegration in a Composite Granular Medium Daraio, C.; Nesterenko, V. F.; Herbold, E. B. Physical Review Letters, Vol. 96, Issue 5 https://doi.org/10.1103/PhysRevLett.96.058002	journal	February 2006
Solitary and shock waves in discrete strongly nonlinear double power-law materials Herbold, E. B.; Nesterenko, V. F. Applied Physics Letters, Vol. 90, Issue 26 https://doi.org/10.1063/1.2751592	journal	June 2007
Nondestructive identification of impurities in granular medium Hong, Jongbae; Xu, Aiguo Applied Physics Letters, Vol. 81, Issue 25 https://doi.org/10.1063/1.1522829	journal	December 2002
How Hertzian Solitary Waves Interact with Boundaries in a 1D Granular Medium Job, Stéphane; Melo, Francisco; Sokolow, Adam Physical Review Letters, Vol. 94, Issue 17 https://doi.org/10.1103/PhysRevLett.94.178002	journal	May 2005
Solitary wave trains in granular chains: experiments, theory and simulations Job, Stéphane; Melo, Francisco; Sokolow, Adam Granular Matter, Vol. 10, Issue 1 https://doi.org/10.1007/s10035-007-0054-2	journal	September 2007
Experimental evidence of shock mitigation in a Hertzian tapered chain Melo, Francisco; Job, Stéphane; Santibanez, Francisco Physical Review E, Vol. 73, Issue 4 https://doi.org/10.1103/PhysRevE.73.041305	journal	April 2006
Discrete Breathers in One-Dimensional Diatomic Granular Crystals Boechler, N.; Theocharis, G.; Job, S. Physical Review Letters, Vol. 104, Issue 24 https://doi.org/10.1103/PhysRevLett.104.244302	journal	June 2010
Wave localization in strongly nonlinear Hertzian chains with mass defect Job, Stéphane; Santibanez, Francisco; Tapia, Franco Physical Review E, Vol. 80, Issue 2 https://doi.org/10.1103/PhysRevE.80.025602	journal	August 2009
Localized breathing modes in granular crystals with defects Theocharis, G.; Kavousanakis, M.; Kevrekidis, P. G. Physical Review E, Vol. 80, Issue 6 https://doi.org/10.1103/PhysRevE.80.066601	journal	December 2009
Damped-driven granular chains: An ideal playground for dark breathers and multibreathers Chong, C.; Li, F.; Yang, J. Physical Review E, Vol. 89, Issue 3 https://doi.org/10.1103/PhysRevE.89.032924	journal	March 2014
Dynamics of metastable breathers in nonlinear chains in acoustic vacuum Sen, Surajit; Mohan, T. R. Krishna Physical Review E, Vol. 79, Issue 3 https://doi.org/10.1103/physreve.79.036603	journal	March 2009
How solitary waves collide in discrete granular alignments Ávalos, Edgar; Sen, Surajit Physical Review E, Vol. 79, Issue 4 https://doi.org/10.1103/PhysRevE.79.046607	journal	April 2009
Traveling waves and localized modes in one-dimensional homogeneous granular chains with no precompression Starosvetsky, Yuli; Vakakis, Alexander F. Physical Review E, Vol. 82, Issue 2 https://doi.org/10.1103/PhysRevE.82.026603	journal	August 2010
New family of solitary waves in granular dimer chains with no precompression Jayaprakash, K. R.; Starosvetsky, Yuli; Vakakis, Alexander F. Physical Review E, Vol. 83, Issue 3 https://doi.org/10.1103/PhysRevE.83.036606	journal	March 2011
Energy equipartition in two-dimensional granular systems with spherical intruders Szelengowicz, I.; Hasan, M. A.; Starosvetsky, Y. Physical Review E, Vol. 87, Issue 3 https://doi.org/10.1103/PhysRevE.87.032204	journal	March 2013
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
Justification of leading order quasicontinuum approximations of strongly nonlinear lattices Schneider, Guido; Kevrekidis, P. G.; Chong, Christopher Discrete and Continuous Dynamical Systems, Vol. 34, Issue 9 https://doi.org/10.3934/dcds.2014.34.3403	journal	March 2014
Traveling waves in 2D hexagonal granular crystal lattices Leonard, A.; Chong, C.; Kevrekidis, P. G. Granular Matter, Vol. 16, Issue 4 https://doi.org/10.1007/s10035-014-0487-3	journal	April 2014
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
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
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
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
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
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
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
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
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
Site-Specific Quantification of Bone Quality Using Highly Nonlinear Solitary Waves Yang, Jinkyu; Sangiorgio, Sophia N.; Borkowski, Sean L. Journal of Biomechanical Engineering, Vol. 134, Issue 10 https://doi.org/10.1115/1.4007364	journal	October 2012
Monitoring the hydration of cement using highly nonlinear solitary waves Ni, Xianglei; Rizzo, Piervincenzo; Yang, Jinkyu NDT & E International, Vol. 52 https://doi.org/10.1016/j.ndteint.2012.05.003	journal	November 2012
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
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
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
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
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 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
Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics: Generalized Solitons and Hyperasymptotic Perturbation Theory Boyd, John P. Mathematics and Its Applications https://doi.org/10.1007/978-1-4615-5825-5	book	January 1998
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

Cited By (3)

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
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
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

Similar Records

Wave-number shocks for the tail of Korteweg-de Vries solitary waves in slowly varying media. Doctoral thesis

Technical Report · Mon Apr 07 00:00:00 EST 1986 · OSTI ID:1233282

Allgaier, D E

Formation of rarefaction waves in origami-based metamaterials

Journal Article · Fri Apr 15 00:00:00 EDT 2016 · Physical Review E · OSTI ID:1233282

Yasuda, H.; Chong, C.; Charalampidis, E. G.; +2 more

The numerical calculation of traveling wave solutions of nonlinear parabolic equations

Journal Article · Tue Jul 01 00:00:00 EDT 1986 · SIAM J. Sci. Stat. Comput.; (United States) · OSTI ID:1233282

Hagstrom, T; Keller, H B

Related Subjects

36 MATERIALS SCIENCE
97 MATHEMATICS AND COMPUTING

Title: Traveling waves and their tails in locally resonant granular systems

Citation Formats

References (57)

Cited By (3)

Similar Records

Related Subjects