Skip to main content
SpringerLink
Log in

Universal design law of equivalent systems for Nesterenko solitary waves transmission

  • Original Paper
  • Published:
Granular Matter Aims and scope Submit manuscript

Abstract

Wave propagation behavior within 1D granular chain has been a much-studied topic in the past decades due to its novel physical wave phenomena. The initial motivation of the present study was prompted by how to construct a granular system with desired wave properties and particle motions. Here, we systematically report the methodology to construct equivalent systems (systems with identical output) using theoretical analysis, experiments and numerical simulation. We investigate the properties of Nesterenko solitary wave supported by one-dimensional granular chains and achieve an equivalent wave transmission among various materials and dimensions. The results of typical experiments and finite element analysis fully support the established theoretical predictions based on Hertz contact and long wavelength approximation theory, which yields a broad class of equivalent systems. Finally, an instructive mechanism map, containing various equivalent systems, is proposed to clarify the relationship between geometric parameters and material properties, providing insights into designing a desired nonlinear dynamic system and guidance for potential engineering applications.

Graphic abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Nesterenko, V.: Dynamics of Heterogeneous Materials. Springer, Berlin (2013)

    Google Scholar 

  2. Porter, M.A., Kevrekidis, P.G., Daraio, C.: Granular crystals: nonlinear dynamics meets materials engineering. Phys. Today 68(11), 44 (2015). https://doi.org/10.1063/PT.3.2981

    Article  Google Scholar 

  3. Man, Y., Boechler, N., Theocharis, G., Kevrekidis, P., Daraio, C.: Defect modes in one-dimensional granular crystals. Phys. Rev. E 85(3), 037601 (2012)

    Article  ADS  Google Scholar 

  4. Coste, C., Falcon, E., Fauve, S.: Solitary waves in a chain of beads under Hertz contact. Phys. Rev. E 56(5), 6104 (1997)

    Article  ADS  Google Scholar 

  5. Daraio, C., Nesterenko, V., Herbold, E., Jin, S.: Tunability of solitary wave properties in one-dimensional strongly nonlinear phononic crystals. Phys. Rev. E 73(2), 026610 (2006)

    Article  ADS  Google Scholar 

  6. Doney, R., Sen, S.: Decorated, tapered, and highly nonlinear granular chain. Phys. Rev. Lett. 97(15), 155502 (2006)

    Article  ADS  Google Scholar 

  7. Achilleos, V., Theocharis, G., Skokos, C.: Energy transport in one-dimensional disordered granular solids. Phys. Rev. E 93(2), 022903 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  8. Manciu, M., Sen, S., Hurd, A.J.: Crossing of identical solitary waves in a chain of elastic beads. Phys. Rev. E 63(1), 016614 (2000)

    Article  ADS  Google Scholar 

  9. Hong, J., Xu, A.: Effects of gravity and nonlinearity on the waves in the granular chain. Phys. Rev. E 63(6), 061310 (2001)

    Article  ADS  Google Scholar 

  10. Jayaprakash, K., Starosvetsky, Y., Vakakis, A.F., Peeters, M., Kerschen, G.: Nonlinear normal modes and band zones in granular chains with no pre-compression. Nonlinear Dyn. 63(3), 359–385 (2011)

    Article  MathSciNet  Google Scholar 

  11. Detroux, T., Starosvetsky, Y., Kerschen, G., Vakakis, A.F.: Classification of periodic orbits of two-dimensional homogeneous granular crystals with no pre-compression. Nonlinear Dyn. 76(1), 673–696 (2014)

    Article  Google Scholar 

  12. Hasan, M.A., Cho, S., Remick, K., Vakakis, A.F., McFarland, D.M., Kriven, W.M.: Experimental study of nonlinear acoustic bands and propagating breathers in ordered granular media embedded in matrix. Granul. Matter 17(1), 49–72 (2015)

    Article  Google Scholar 

  13. Zhang, Y., McFarland, D.M., Vakakis, A.F.: Propagating discrete breathers in forced one-dimensional granular networks: theory and experiment. Granul. Matter 19(3), 59 (2017)

    Article  Google Scholar 

  14. Zhang, Y., Pozharskiy, D., McFarland, D.M., Kevrekidis, P.G., Kevrekidis, I.G., Vakakis, A.F.: Experimental study of nonlinear resonances and anti-resonances in a forced, ordered granular chain. Exp. Mech. 57(4), 505–520 (2017)

    Article  Google Scholar 

  15. Zhang, Y., Moore, K.J., McFarland, D.M., Vakakis, A.F.: Targeted energy transfers and passive acoustic wave redirection in a two-dimensional granular network under periodic excitation. J. Appl. Phys. 118(23), 234901 (2015)

    Article  ADS  Google Scholar 

  16. Job, S., Melo, F., Sokolow, A., Sen, S.: Solitary wave trains in granular chains: experiments, theory and simulations. Granul. Matter 10(1), 13–20 (2007)

    Article  MATH  Google Scholar 

  17. Wu, D.T.: Conservation principles in solitary impulse propagation through granular chains. Phys. A 315(1–2), 194–202 (2002)

    Article  ADS  MATH  Google Scholar 

  18. Anco, S.C., Przedborski, M.: Long-wavelength solitary waves in Hertzian chains and their properties in different nonlinearity regimes. Phys. Rev. E 98(4), 042208 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  19. Coste, C., Gilles, B.: On the validity of Hertz contact law for granular material acoustics. Eur. Phys. J. B Condens. Matter Complex Syst. 7(1), 155–168 (1999)

    Article  Google Scholar 

  20. Leonard, A., Chong, C., Kevrekidis, P.G., Daraio, C.: Traveling waves in 2D hexagonal granular crystal lattices. Granul. Matter 16(4), 531–542 (2014)

    Article  Google Scholar 

  21. Ngo, D., Khatri, D., Daraio, C.: Highly nonlinear solitary waves in chains of ellipsoidal particles. Phys. Rev. E 84(2), 026610 (2011)

    Article  ADS  Google Scholar 

  22. Khatri, D., Ngo, D., Daraio, C.: Highly nonlinear solitary waves in chains of cylindrical particles. Granul. Matter 14(1), 63–69 (2012)

    Article  Google Scholar 

  23. Kim, E., Yang, J.: Wave propagation in single column woodpile phononic crystals: formation of tunable band gaps. J. Mech. Phys. Solids 71, 33–45 (2014)

    Article  ADS  MATH  Google Scholar 

  24. Leonard, A., Daraio, C.: Stress wave anisotropy in centered square highly nonlinear granular systems. Phys. Rev. Lett. 108(21), 214301 (2012)

    Article  ADS  Google Scholar 

  25. Job, S., Santibanez, F., Tapia, F., Melo, F.: Wave localization in strongly nonlinear Hertzian chains with mass defect. Phys. Rev. E 80(2), 025602 (2009)

    Article  ADS  Google Scholar 

  26. Takato, Y., Sen, S.: Long-lived solitary wave in a precompressed granular chain. EPL (Europhys. Lett.) 100(2), 24003 (2012)

    Article  ADS  Google Scholar 

  27. Vergara, L.: Delayed scattering of solitary waves from interfaces in a granular container. Phys. Rev. E 73(6), 066623 (2006)

    Article  ADS  Google Scholar 

  28. Nesterenko, V., Daraio, C., Herbold, E., Jin, S.: Anomalous wave reflection at the interface of two strongly nonlinear granular media. Phys. Rev. Lett. 95(15), 158702 (2005)

    Article  ADS  Google Scholar 

  29. Li, F., Anzel, P., Yang, J., Kevrekidis, P.G., Daraio, C.: Granular acoustic switches and logic elements. Nat. Commun. 5, 5311 (2014)

    Article  ADS  Google Scholar 

  30. Boechler, N., Theocharis, G., Job, S., Kevrekidis, P., Porter, M.A., Daraio, C.: Discrete breathers in one-dimensional diatomic granular crystals. Phys. Rev. Lett. 104(24), 244302 (2010)

    Article  ADS  Google Scholar 

  31. Serra-Garcia, M., Lydon, J., Daraio, C.: Extreme stiffness tunability through the excitation of nonlinear defect modes. Phys. Rev. E 93(1), 010901 (2016)

    Article  ADS  Google Scholar 

  32. de Billy, M.: Frequency analysis of the acoustic signal transmitted through a one-dimensional chain of metallic spheres. J. Acoust. Soc. Am. 110(2), 710–716 (2001)

    Article  ADS  Google Scholar 

  33. Ni, X., Rizzo, P., Daraio, C.: Laser-based excitation of nonlinear solitary waves in a chain of particles. Phys. Rev. E 84(2), 026601 (2011)

    Article  ADS  Google Scholar 

  34. Carretero-González, R., Khatri, D., Porter, M.A., Kevrekidis, P., Daraio, C.: Dissipative solitary waves in granular crystals. Phys. Rev. Lett. 102(2), 024102 (2009)

    Article  ADS  Google Scholar 

  35. Khatri, D., Daraio, C., Rizzo, P.: Coupling of highly nonlinear waves with linear elastic media. In: Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems 2009, p. 72920P. International Society for Optics and Photonics (2009)

  36. Lazaridi, A., Nesterenko, V.: Observation of a new type of solitary waves in a one-dimensional granular medium. J. Appl. Mech. Tech. Phys. 26(3), 405–408 (1985)

    Article  ADS  Google Scholar 

  37. Nesterenko, V.F.: Waves in strongly nonlinear discrete systems. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 376(2127), 20170130 (2018)

    Article  ADS  Google Scholar 

  38. Starosvetsky, Y., Jayaprakash, K., Hasan, M.A., Vakakis, A.F.: Topics on the Nonlinear Dynamics and Acoustics of Ordered Granular Media. World Scientific, Singapore (2017)

    Book  MATH  Google Scholar 

  39. Job, S., Melo, F., Sokolow, A., Sen, S.: How Hertzian solitary waves interact with boundaries in a 1D granular medium. Phys. Rev. Lett. 94(17), 178002 (2005)

    Article  ADS  Google Scholar 

  40. Melo, F., Job, S., Santibanez, F., Tapia, F.: Experimental evidence of shock mitigation in a Hertzian tapered chain. Phys. Rev. E 73(4), 041305 (2006)

    Article  ADS  Google Scholar 

  41. Santibanez, F., Munoz, R., Caussarieu, A., Job, S., Melo, F.: Experimental evidence of solitary wave interaction in Hertzian chains. Phys. Rev. E 84(2), 026604 (2011)

    Article  ADS  Google Scholar 

  42. Job, S., Santibanez, F., Tapia, F., Melo, F.: Nonlinear waves in dry and wet Hertzian granular chains. Ultrasonics 48(6–7), 506–514 (2008)

    Article  Google Scholar 

  43. Sen, S., Hong, J., Bang, J., Avalos, E., Doney, R.: Solitary waves in the granular chain. Phys. Rep. 462(2), 21–66 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  44. Sun, D., Sen, S.: Nonlinear grain–grain forces and the width of the solitary wave in granular chains: a numerical study. Granul. Matter 15(2), 157–161 (2013)

    Article  Google Scholar 

  45. Przedborski, M.A., Harroun, T.A., Sen, S.: Localizing energy in granular materials. Appl. Phys. Lett. 107(24), 244105 (2015)

    Article  ADS  Google Scholar 

  46. Nesterenko, V.: Propagation of nonlinear compression pulses in granular media. J. Appl. Mech. Tech. Phys. 24(5), 733–743 (1983)

    Article  ADS  Google Scholar 

  47. Przedborski, M., Anco, S.C.: Traveling waves and conservation laws for highly nonlinear wave equations modeling Hertz chains. J. Math. Phys. 58(9), 091502 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. Lin, W.-H., Daraio, C.: Wave propagation in one-dimensional microscopic granular chains. Phys. Rev. E 94(5), 052907 (2016)

    Article  ADS  Google Scholar 

  49. Xu, J., Zheng, B., Liu, Y.: Solitary wave in one-dimensional buckyball system at nanoscale. Sci. Rep. 6, 21052 (2016)

    Article  ADS  Google Scholar 

  50. Xu, J., Zheng, B.: Highly effective energy dissipation system based on one-dimensionally arrayed short single-walled carbon nanotubes. Extreme Mech. Lett. 9, 336–341 (2016)

    Article  Google Scholar 

  51. Xu, J., Zheng, B.: Stress wave propagation in two-dimensional buckyball lattice. Sci. Rep. 6, 37692 (2016)

    Article  ADS  Google Scholar 

  52. Xu, J., Zheng, B.: Quantitative tuning nanoscale solitary waves. Carbon 111, 62–66 (2017)

    Article  Google Scholar 

  53. Zheng, B., Xu, J.: Enhanced stress wave attenuation of single-walled carbon nanotube lattice via mass mismatch-induced resonance. Carbon 116, 391–397 (2017)

    Article  Google Scholar 

  54. Zheng, B., Xu, J.: Mechanical wave propagation within nanogold granular crystals. Extreme Mech. Lett. 15, 17–25 (2017)

    Article  Google Scholar 

  55. Takato, Y., Benson, M.E., Sen, S.: Small nanoparticles, surface geometry and contact forces. Proc. R. Soc. A Math. Phys. Eng. Sci. 474(2211), 20170723 (2018)

    Article  ADS  Google Scholar 

  56. Sun, D., Daraio, C., Sen, S.: Nonlinear repulsive force between two solids with axial symmetry. Phys. Rev. E 83(6), 066605 (2011)

    Article  ADS  Google Scholar 

  57. Hertz, H.: Ueber die Beruhrung fester elastischer Korper. J. Reine Angew Math. 92, 156–171 (1882)

    MathSciNet  MATH  Google Scholar 

  58. Chong, C., Porter, M.A., Kevrekidis, P.G., Daraio, C.: Nonlinear coherent structures in granular crystals. J. Phys. Condens. Matter 29(41), 413003 (2017)

    Article  Google Scholar 

  59. Porter, M.A., Daraio, C., Herbold, E.B., Szelengowicz, I., Kevrekidis, P.: Highly nonlinear solitary waves in periodic dimer granular chains. Phys. Rev. E 77(1), 015601 (2008)

    Article  ADS  MATH  Google Scholar 

  60. Rosas, A., Lindenberg, K.: Pulse dynamics in a chain of granules with friction. Phys. Rev. E 68(4), 041304 (2003)

    Article  ADS  Google Scholar 

  61. Herbold, E., Nesterenko, V.: Shock wave structure in a strongly nonlinear lattice with viscous dissipation. Phys. Rev. E 75(2), 021304 (2007)

    Article  ADS  Google Scholar 

  62. Daraio, C., Nesterenko, V., Herbold, E., Jin, S.: Strongly nonlinear waves in a chain of Teflon beads. Phys. Rev. E 72(1), 016603 (2005)

    Article  ADS  Google Scholar 

  63. Porter, M.A., Daraio, C., Szelengowicz, I., Herbold, E.B., Kevrekidis, P.: Highly nonlinear solitary waves in heterogeneous periodic granular media. Phys. D 238(6), 666–676 (2009)

    Article  MATH  Google Scholar 

  64. Starosvetsky, Y., Vakakis, A.F.: Traveling waves and localized modes in one-dimensional homogeneous granular chains with no precompression. Phys. Rev. E 82(2), 026603 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  65. English, J., Pego, R.: On the solitary wave pulse in a chain of beads. Proc. Am. Math. Soc. 133(6), 1763–1768 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  66. Yang, J., Silvestro, C., Khatri, D., De Nardo, L., Daraio, C.: Interaction of highly nonlinear solitary waves with linear elastic media. Phys. Rev. E 83(4), 046606 (2011)

    Article  ADS  Google Scholar 

  67. Ashby, M.F., Cebon, D.: Materials selection in mechanical design. Le J. Phys. IV 3(C7), C7-1–C7-9 (1993)

    Google Scholar 

  68. Yin, S., Chen, D., Xu, J.: Novel propagation behavior of impact stress wave in one-dimensional hollow spherical structures. Int. J. Impact Eng. 134, 103368 (2019)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering and Engineering Science, The University of North Carolina at Charlotte, Charlotte, NC, 28223, USA

    Wen Zhang & Jun Xu

  2. Vehicle Energy and Safety Laboratory (VESL), North Carolina Motorsports and Automotive Research Center, The University of North Carolina at Charlotte, Charlotte, NC, 28223, USA

    Wen Zhang & Jun Xu

Authors
  1. Wen Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jun Xu.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary file1 (DOCX 207 kb)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, W., Xu, J. Universal design law of equivalent systems for Nesterenko solitary waves transmission. Granular Matter 22, 46 (2020). https://doi.org/10.1007/s10035-020-1011-6

Download citation

  • Received:

  • Published:

  • DOI: https://doi.org/10.1007/s10035-020-1011-6

Keywords

Search

Navigation