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Asymmetric scattering effect of solitary wave in a two-section composite granular chain

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Abstract

Granular chain serves as a fundamental component for controlling the nonlinear solitary wave propagation in acoustic metamaterials. The understanding of the nonlinear scattering at the mismatched interface plays a crucial role in designing novel acoustic devices. In this study, numerical simulations are conducted to investigate the asymmetric scattering effect of solitary waves in a two-section composite granular chain. Building upon Nesterenko’s work on solitary waves in monodisperse granular chains and using continuous acoustic wave theory in linear medium, we argue that the mismatches of acoustic velocity and acoustic impedance, because of the differences of mass density and elastic coefficient, dominate the asymmetric scattering effect at the mismatched interface. The simulation results confirm the occurrence of a multipulse structure for the transmitted solitary waves when the solitary wave passes through the mismatched interface with a small–large wave velocity. The overshooting effect occurs for the reflected solitary waves because of the mismatched interface with high–low acoustic impedance. The phase diagram in the space of the mass density and the elastic coefficient ratio of the right-section granular chain to the left-section granular chain further validates the predictions.

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The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

References

  1. Guo, Y., Curtis, J.S.: Discrete element method simulations for complex granular flows. Annu. Rev. Fluid Mech. 47, 21 (2015)

    MathSciNet  Google Scholar 

  2. Hou, M., Chen, W., Zhang, T., Lu, K., Chan, C.K.: Global nature of dilute-to-dense transition of granular flows in a 2D channel. Phys. Rev. Lett. 91, 204301 (2003)

    Google Scholar 

  3. Zhou, X.Y., Liu, S.K., Zhao, Z.H., Li, X., Li, C.H., Sun, M., Huang, D.C.: Dilute-to-dense flow transition and flow-rate behavior of lateral bifurcated granular flow. Powder Technol. 383, 536 (2021)

    Google Scholar 

  4. Umbanhowar, P.B., Lueptow, R.M., Ottino, J.M.: Modeling density segregation in granular flow. Annu. Rev. Chem. Biomol. Eng. 10, 5.1 (2019)

    Google Scholar 

  5. Breu, A.P.J., Ensner, H.M., Kruelle, C.A., Rehberg, I.: Reversing the Brazil\(-\)Nut effect: competition between percolation and condensation. Phys. Rev. Lett. Eng. 90, 014302 (2003)

    Google Scholar 

  6. Shao, Y.F., Li, A.H., Wang, Z.Z., Sun, M., Huang, D.C.: Resonance\(-\)induced acceleration of the RBNE\(-\)BNE segregation inversion. AIChE J. e18101 (2022)

  7. Fronk, M.D., Fang, L., Packo, P., Leamy, M.J.: Elastic wave propagation in weakly nonlinear media and metamaterials: a review of recent developments. Nonlinear Dyn. 111, 10709 (2023)

    Google Scholar 

  8. Lydon, J., Jayaprakash, K.R., Ngo, D., Starosvetsky, Y.L., Vakakis, A.F., Daraio, C.: Frequency bands of strongly nonlinear homogeneous granular systems. Phys. Rev. 88, 012206 (2013)

    Google Scholar 

  9. Jiao, T.F., Zhang, S.T., Sun, M., Huang, D.C.: Occurrence of gradual resonance in a finite-length granular chain driven by harmonic vibration. Nonlinear Dyn. 111, 9049 (2023)

    Google Scholar 

  10. Zhou, J.Z., McFarland, D.M., Cheng, X.L., Lu, H.C., Vakakis, A.F.: One-dimensional granular chains as transmitted force attenuators. Nonlinear Dyn. 111, 14713 (2023)

    Google Scholar 

  11. Burgoyne, H.A., Newman, J.A., Jackson, W.C., Daraio, C.: Guided impact mitigation in 2D and 3D granular crystals. Procedia Eng. 103, 52 (2015)

    Google Scholar 

  12. Breindel, A., Sun, D., Sen, S.: Impulse absorption using small, hard panels of embedded cylinders with granular alignments. Appl. Phys. Lett. 99, 063510 (2011)

    Google Scholar 

  13. Boechler, N., Theocharis, G., Daraio, C.: Bifurcation-based acoustic switching and rectification. Nat. Mater. 10, 665 (2011)

    Google Scholar 

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

    Google Scholar 

  15. Liu, Y.F., Qin, Z.Y., Chu, F.L.: Nonlinear forced vibrations of FGM sandwich cylindrical shells with porosities on an elastic substrate. Nonlinear Dyn. 104, 1007 (2021)

    Google Scholar 

  16. Liu, Y.F., Qin, Z.Y., Chu, F.L.: Nonlinear forced vibrations of rotating cylindrical shells under multi-harmonic excitations in thermal environment. Nonlinear Dyn. 108, 2977 (2022)

    Google Scholar 

  17. Nesterenko, V.F.: Propagation of nonlinear compression pulses in granular media. J. Appl. Mech. Tech. Phys. 24, 733 (1983)

    Google Scholar 

  18. Nesterenko, V.F.: Dynamics of Heterogeneous Materials. Springer-Verlag, New York (2001)

    Google Scholar 

  19. Nesterenko, V.F.: Waves in strongly nonlinear discrete systems. Phil. Trans. R. Soc. A 376, 20170130 (2018)

    Google Scholar 

  20. Sen, S., Hong, J.B., Bang, J.H., Avalos, E., Doney, R.: Solitary waves in the granular chain. Phys. Rep. 462, 21 (2008)

    MathSciNet  Google Scholar 

  21. Rosas, A., Lindenberg, K.: Pulse propagation in granular chains. Phys. Rep. 735, 1 (2018)

    MathSciNet  Google Scholar 

  22. Ngo, D., Fraternali, F., Daraio, C.: Highly nonlinear solitary wave propagation in Y-shaped granular crystals with variable branch angles. Phys. Rev. E 85, 036602 (2012)

    Google Scholar 

  23. Leonard, A., Ponson, L., Daraio, C.: Wave mitigation in ordered networks of granular chains. J. Mech. Phys. Solids 73, 103 (2014)

    Google Scholar 

  24. Liu, S.S., Yang, Y.Y., Duan, W.S., Yang, L.: Pulse reflection and transmission due to impurities in a granular chain. Phys. Rev. E 92, 013202 (2015)

    Google Scholar 

  25. Tzirtzilakis, E., Xenos, M., Marinakis, V., Bountis, T.C.: Interactions and stability of solitary waves in shallow water. Chaos Solitions Fractals 14, 87 (2002)

    MathSciNet  Google Scholar 

  26. Chen, Y.Y., Kharif, C., Yang, J.H., Hsu, H.C., Touboul, J., Chambarel, J.: An experimental study of steep solitary wave reflection at a vertical wall. Eur. J. Mech. B-Fluid. 49, 20–28 (2015)

    Google Scholar 

  27. Su, C.H., Mirie, R.M.: On head\(-\)on collisions between two solitary waves. J. Fluid Mech. 98, 509–525 (1980)

    MathSciNet  Google Scholar 

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

    Google Scholar 

  29. Manciu, M., Sen, S., Hurd, A.J.: Impulse propagation in dissipative and disordered chains with power-law repulsive potentials. Phys. D 157, 226 (2001)

    Google Scholar 

  30. Takato, Y., Sen, S.: Long-lived solitary wave in a precompressed granular chain. EPL 100, 24003 (2015)

    Google Scholar 

  31. Jiao, T.F., Chen, W.Z., Takato, Y., Sen, S., Huang, D.C.: Revisiting Nesterenko’s solitary wave in the precompressed granular alignment held between fixed ends. Granul. Matter. 25, 17 (2023)

    Google Scholar 

  32. Chen, Y., Yeh, H.: Laboratory experiments on counter-propagating collisions of solitary waves. Part 1. Wave interactions. J. Fluid Mech. 749, 577 (2014)

    Google Scholar 

  33. Liu, X.Y., Wen, W., Mao, X.Z., Wang, X.L.: Optimal and efficient generation of sine-Gordon breathers. Phys. Rev. E 104, 014209 (2021)

    MathSciNet  Google Scholar 

  34. Jin, T., Zhao, H., Hu, B.: Spatial shift of lattice soliton scattering in the Fermi-Pasta-Ulam model. Phys. Rev. E 81, 037601 (2010)

    Google Scholar 

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

    Google Scholar 

  36. Manciu, F.S., Sen, S.: Secondary solitary wave formation in systems with generalized Hertz interactions. Phys. Rev. E 66, 016616 (2002)

    MathSciNet  Google Scholar 

  37. Ávalos, E., Sen, S.: How solitary waves collide in discrete granular alignments. Phys. Rev. E 79, 046607 (2009)

    Google Scholar 

  38. Deng, G., Biondini, G., Sen, S.: Interactions of solitary waves in integrable and nonintegrable lattices. Chaos 30, 043101 (2020)

    MathSciNet  Google Scholar 

  39. Tichler, A.M., Gómez, L.R., Upadhyaya, N., Campman, X., Nesterenko, V.F., Vitelli, V.: Transmission and reflection of strongly nonlinear solitary waves at granular interfaces. Phys. Rev. Lett. 111, 048001 (2013)

    Google Scholar 

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

    Google Scholar 

  41. Shen, Y., Kevrekidis, P.G., Sen, S., Hoffman, A.: Characterizing traveling-wave collisions in granular chains starting from integrable limits: the case of the Korteweg-de Vries equation and the Toda lattice. Phys. Rev. E 90, 022905 (2014)

    Google Scholar 

  42. Ma, L., Huang, D.C., Chen, W.Z., Jiao, T.F., Sun, M., Hu, F.L., Su, Y.J.: Oscillating collision of the granular chain on static wall. Phys. Lett. A 381, 542 (2017)

    Google Scholar 

  43. Wu, Q.Q., Liu, X.Y., Jiao, T.F., Sen, S., Huang, D.C.: Head-on collision of solitary waves described by the toda lattice model in granular chain. Chin. Phys. Lett. 37, 074501 (2020)

    Google Scholar 

  44. Zhang, S.T., Liu, S.K., Jiao, T.F., Sun, M., Hu, F.L., Huang, D.C.: Nonlinear interaction of head\(-\)on solitary waves in integrable and nonintegrable systems. arXiv:2310.19356 (2023)

  45. Wen, Z.Y., Wang, S.J., Zhang, X.M., Li, L.: Solitary Wave Interactions in Granular Media. Chin. Phys. Lett. 24, 2887 (2007)

    Google Scholar 

  46. Ávalos, E., Sun, D.K., Doney, D.L., Sen, S.: Sustained strong fluctuations in a nonlinear chain at acoustic vacuum: Beyond equilibrium. Phys. Rev. E 84, 046610 (2011)

    Google Scholar 

  47. Vergara, L.: Scattering of solitary waves from interfaces in granular media. Phys. Rev. Lett. 95, 108002 (2005)

    Google Scholar 

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

    Google Scholar 

  49. Daraio, C., Ngo, D., Nesterenko, V.F., Fraernali, F.: Highly nonlinear pulse splitting and recombination in a two-dimensional granular network. Phys. Rev. E 82, 036603 (2010)

    Google Scholar 

  50. Herbold, E.B., Nesterenko, V.F.: Propagation of rarefaction pulses in discrete materials with strain-softening behavior. Phys. Rev. Lett. 110, 144101 (2013)

    Google Scholar 

  51. Wang, P.J., Xia, J.H., Li, Y.D., Liu, C.S.: Crossover in the power-law behavior of confined energy in a composite granular chain. Phys. Rev. E 76, 041305 (2007)

    Google Scholar 

  52. Wallen, S.P., Lee, J., Mei, D., Chong, C., Kevrekidis, P.G., Boechler, N.: Discrete breathers in a mass-in-mass chain with Hertzian local resonators. Phys. Rev. E 95, 022904 (2017)

    Google Scholar 

  53. Yang, Y.Y., Liu, S.W., Zhang, Z.B., Duan, W.S., Yang, L.: Solitary waves propagation described by Korteweg-de Vries equation in the granular chain with initial prestress. AIP Adv. 6, 075317 (2016)

    Google Scholar 

  54. Przedborski, M., Harroun, T.A., Sen, S.: Granular chains with soft boundaries: slowing the transition to quasiequilibrium. Phys. Rev. E 91, 042207 (2015)

    Google Scholar 

  55. Kinsler, L.E., Frey, A.R., Coppens, A.B., Sanders, J.V.: Fundamentals of Acoustics. John Wiley (1999)

  56. Du, G.H., Zhu, Z.M., Gong, F.X.: Fundamentals of Acoustics. Nanjing University Press (2012)

  57. Cundall, P.A., Strack, O.D.L.: A discrete numerical model for granular assemblies. Geotechnique 29, 47 (1979)

    Google Scholar 

  58. Kuwabara, G., Kono, K.: Restitution coefficient in a collision between two spheres. Japn. J. Appl. Phys. 26, 1230 (1987)

    Google Scholar 

  59. Scha̋fer, J., Dippel, S., Wolf, D.E.: Force schemes in simulations of granular materials. J. Phys. I, 6, 5 (1996)

  60. Xu, Y., Nesterenko, V.F.: Propagation of short stress pulses in discrete strongly nonlinear tunable metamaterials. Philow. Trans. R. Soc. A 372, 20130186 (2014)

    MathSciNet  Google Scholar 

  61. Daraio, C., Herbold, E.B., Nesterenko, V.F., Jin, S.: Energy trapping and shock disintegration in a composite granular medium. Phys. Rev. Lett. 96, 058002 (2006)

    Google Scholar 

  62. Sen, S., Manciu, M., Wright, J.D.: Solitonlike pulses in perturbed and driven Hertzian chains and their possible applications in detecting buried impurities. Phys. Rev. E 57, 2386 (1998)

    Google Scholar 

  63. Sokolow, A., Bittle, E.G., Sen, S.: Solitary wave train formation in Hertzian chains. EPL 77, 24002 (2007)

    Google Scholar 

  64. Liu, X.Y., Jiao, T.F., Ma, L., Su, J.Y., Chen, W.Z., Sun, Q.C., Huang, D.C.: Rectification effect on solitary waves in the symmetric Y-shaped granular chain. Granul. Matt. 19, 55 (2017)

    Google Scholar 

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Funding

This work was supported financially by the National Natural Science Foundation of China (Grant No. 11574153) and the foundation of the Ministry of Industry and Information Technology of China (Grant No.TSXK2022D007).

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Authors and Affiliations

  1. Department of Applied Physics, Nanjing University of Science and Technology, Nanjing, 210094, China

    Xingyi Liu, Tengfei Jiao, Shutian Zhang, Min Sun & Decai Huang

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  1. Xingyi Liu
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Correspondence to Decai Huang.

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Liu, X., Jiao, T., Zhang, S. et al. Asymmetric scattering effect of solitary wave in a two-section composite granular chain. Nonlinear Dyn 112, 6561–6575 (2024). https://doi.org/10.1007/s11071-024-09383-4

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