On the Wave Transmission in a Gently Perturbed Weakly Inhomogeneous Non-Linear Force Chain
Keywords:wave transport, Hertz chain, resonance modes, multiscale analysis, disorder
We have obtained rigorous analytic and numerical solutions of the equations which govern the transport of mechanical perturbations in a gently precompressed 1D Hertz chain. Both finite-length and infinite-length systems have been studied. We examine both discrete and continuous
formulations of the mentioned problem. A few families of analytic solutions of the problem given in the form of quasinormal waves and specific resonance modes have been obtained in the linear approximation for weakly perturbed inhomogeneous systems. Resonance modes are proposed to be interpreted as the Ramsauer–Townsend effect which happens due to the inhomogeneity. The obtained analytic results have been compared with numerical solutions of the discrete equations. We observe a multiscaled scenario of the impulse transport in an inhomogeneous force chain which could happens either asymptotically or at the intermittency between discrete- and continuous limits of the formulated problem. The role of a disorder has been also analyzed with the help of the Dyson concept.
E. Fermi, J. Pasta, S. Ulam, M. Tsingou. Studies of non-linear problems. Los Alamos Sci. Lab. Rep. LA-1940, 978 (1955). https://doi.org/10.2172/4376203
P.L. Bhatnagar. Nonlinear Waves in One-Dimensional Dispersive Systems (Clarendon, 1979) [ISBN: 978-0198535317].
Y. Xu, V.F. Nesterenko. Propagation of short stress pulses in discrete strongly nonlinear tunable metamaterials. Phil. Trans. R. Soc. A 372, 20130186 (2014). https://doi.org/10.1098/rsta.2013.0186
V. Nesterenko. Dynamics of Heterogeneous Materials (Springer, 2001) [ISBN: 978-1-4419-2926-6]. https://doi.org/10.1007/978-1-4757-3524-6
C. Coste, E. Falcon, S. Fauve. Solitary waves in a chain of beads under Hertz contact. Phys. Rev. E 56, 6104 (1997). https://doi.org/10.1103/PhysRevE.56.6104
S. Sen, J. Hong, J. Bang, E. Avalos. R. Doney. Solitary waves in the granular chain. Phys. Rep. 462, 21 (2008). https://doi.org/10.1016/j.physrep.2007.10.007
E. Hasco¨et, H.J. Herrmann, V. Loreto. Shock propagation in a granular chain. Phys. Rev. E 59, 3202 (1999). https://doi.org/10.1103/PhysRevE.59.3202
U. Harbola, A. Rosas, A.H. Romero, M. Esposito, K. Lindenberg. Pulse propagation in decorated granular chains: an analytical approach. Phys. Rev. E 80, 051302 (2009). https://doi.org/10.1103/PhysRevE.80.051302
E. Somfai, J.-N. Roux, J.H. Snoeijer, M. van Hecke,W. van Saarloos. Elastic wave propagation in confined granular systems. Phys. Rev. E 72, 021301 (2005). https://doi.org/10.1103/PhysRevE.72.021301
O.I. Gerasymov, N. Vandewalle, A.Ya. Spivak, N.N. Khudyntsev, G. Lumay, S. Dorbolo, O.A. Klymenkov. Stationary states in a 1D system of inelastic particles. Ukr. J. Phys. 53, 1128 (2008).
O.I. Gerasymov, N. Vandewalle. On the exact solutions of the problem of impulsive propagation in an inhomogeneous granular chain. Dopov. Nac. akad. nauk Ukr. 8, 67 (2012).
G. Lumay, S. Dorbolo, O. Gerasymov, N. Vandewalle. Experimental study of a vertical column of grains submitted to a series of impulses. Eur. Phys. J. E 36, 16 (2013). https://doi.org/10.1140/epje/i2013-13016-1
L.D. Landau, L.P. Pitaevskii, A.M. Kosevich, E.M. Lifshitz. Theory of elasticity, 3rd ed. (Butterworth-Heinemann, 1986) [ISBN: 978-0750626330].
H. Bateman. Some simple differential difference equations and the related functions. Bull. Amer. Math. Soc. 49, 494 (1943). https://doi.org/10.1090/S0002-9904-1943-07927-X
M. Abramowitz, I.A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964) [ISBN: 0-486-61272-4].
E. Pinney. Ordinary difference-differential equations (University of California Press, 1958)
F.J. Dyson. The dynamics of a disordered linear chain. Phys. Rev. 92, 1331 (1953). https://doi.org/10.1103/PhysRev.92.1331
N.H. March, M. Parrinello. Collective Effects in Solids and Liquids (Adam Hilger, 1982) [ISBN: 978-0852745281].
A.D. Polyanin, V.F. Zaitsev. Handbook of Exact Solutions for Ordinary Differential Equations, 2nd ed. (Chapman and Hall/CRC, 2003) [ISBN: 978-1584882978].
A. Rosas, K. Lindenberg. Pulse dynamics in a chain of granules with friction. Phys. Rev. E 68, 041304 (2003). https://doi.org/10.1103/PhysRevE.68.041304
R. Carretero-Gonzalez, D. Khatri, M.A. Porter, P.G. Kevrekidis, C. Daraio. Dissipative solitary waves in granular crystals. Phys. Rev. Lett. 102, 024102 (2009). https://doi.org/10.1103/PhysRevLett.102.024102
U. Fano. Effects of configuration interaction on intensities and phase shifts. Phys. Rev. 124, 1866 1961. https://doi.org/10.1103/PhysRev.124.1866
H. Yasuda, C. Chong, J. Yang, P.G. Kevrekidis. Emergence of dispersive shocks and rarefaction waves in power-law contact models. Phys. Rev. E 95, 062216 (2017). https://doi.org/10.1103/PhysRevE.95.062216
O.I. Gerasymov. Physics of Granular Materials (TES, 2015) [ISBN: 978-617-7054-82-4].
A.J. Martinez, H. Yasuda, E. Kim, P.G. Kevrekidis, M.A. Porter, J. Yang. Scattering of waves by impurities in precompressed granular chains. Phys. Rev. E 93, 052224 (2016). https://doi.org/10.1103/PhysRevE.93.052224
How to Cite
License to Publish the Paper
The corresponding author and the co-authors (hereon referred to as the Author(s)) of the paper being submitted to the Ukrainian Journal of Physics (hereon referred to as the Paper) from one side and the Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, represented by its Director (hereon referred to as the Publisher) from the other side have come to the following Agreement:
1. Subject of the Agreement.
The Author(s) grant(s) the Publisher the free non-exclusive right to use the Paper (of scientific, technical, or any other content) according to the terms and conditions defined by this Agreement.
2. The ways of using the Paper.
2.1. The Author(s) grant(s) the Publisher the right to use the Paper as follows.
2.1.1. To publish the Paper in the Ukrainian Journal of Physics (hereon referred to as the Journal) in original language and translated into English (the copy of the Paper approved by the Author(s) and the Publisher and accepted for publication is a constitutive part of this License Agreement).
2.1.2. To edit, adapt, and correct the Paper by approval of the Author(s).
2.1.3. To translate the Paper in the case when the Paper is written in a language different from that adopted in the Journal.
2.2. If the Author(s) has(ve) an intent to use the Paper in any other way, e.g., to publish the translated version of the Paper (except for the case defined by Section 2.1.3 of this Agreement), to post the full Paper or any its part on the web, to publish the Paper in any other editions, to include the Paper or any its part in other collections, anthologies, encyclopaedias, etc., the Author(s) should get a written permission from the Publisher.
3. License territory.
The Author(s) grant(s) the Publisher the right to use the Paper as regulated by sections 2.1.1–2.1.3 of this Agreement on the territory of Ukraine and to distribute the Paper as indispensable part of the Journal on the territory of Ukraine and other countries by means of subscription, sales, and free transfer to a third party.
4.1. This Agreement is valid starting from the date of signature and acts for the entire period of the existence of the Journal.
5.1. The Author(s) warrant(s) the Publisher that:
– he/she is the true author (co-author) of the Paper;
– copyright on the Paper was not transferred to any other party;
– the Paper has never been published before and will not be published in any other media before it is published by the Publisher (see also section 2.2);
– the Author(s) do(es) not violate any intellectual property right of other parties. If the Paper includes some materials of other parties, except for citations whose length is regulated by the scientific, informational, or critical character of the Paper, the use of such materials is in compliance with the regulations of the international law and the law of Ukraine.
6. Requisites and signatures of the Parties.
Publisher: Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine.
Address: Ukraine, Kyiv, Metrolohichna Str. 14-b.
Author: Electronic signature on behalf and with endorsement of all co-authors.