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Heterogeneous Media

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Coherent Structures in Granular Crystals

Part of the book series: SpringerBriefs in Physics ((SpringerBriefs in Physics))

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Abstract

Of the three primary structures of interest in this book, solitary waves and breathers have been studied in heterogeneous granular chains, while dispersive shock waves have not. The latter represents an area of open and active research and the former is the focus of this chapter. While we will focus on traveling solitary waves and breathers (each with its own section) in this chapter, we briefly mention other work that has been done in heterogeneous chains. For example, traveling waves in granular chains with impurities can lead to transmitted and reflected waves (Sen, Hong, Bang, Avalos, Doney in Phys Rep 462:21, 2008). In chains with multiple impurities and precompression, resonant interactions can result in perfect transmission of waves with small amplitude through the impurities (Martínez, Yasuda, Kim, Kevrekidis, Porter, Yang in Phys Rev E 93:052224, 2016). This is similar to the Ramsauer–Townsend resonance in quantum physics. Chains with special heterogeneities can be used as impact mitigators (Fraternali, Porter in Mech Adv Mat Struct 17(1):1, 2010). Tapered chains have also been thoroughly studied (Doney, Sen in Phys Rev Lett 97:155502, 2006, Doney, Agui, Sen in J Appl Phys 106:064905, 2009, Harbola, Rosas, Romero, Esposito, Lindenberg in Phys Rev E 80:051302, 2009, Harbola, Rosas, Esposito, Lindenberg, Phys Rev E 80:031303, 2009), which are chains where some feature, such as the mass of the particles, changes gradually along the chain. One possible analytical approach to study such heterogeneous chains is the binary collision approximation (Lindenberg, Harbola, Romero, Rosas in Pulse propagation in granular chains. American Institute of Physics, 2011). Here, one studies the dynamics between two adjacent nodes for certain intervals of time. Disorder in granular chains has also been studied, where the tuning of the balance of disorder to nonlinearity strength can lead to subdiffusive or superdiffusive transport of energy through the lattice (Martínez, Kevrekidis, Porter in Phys Rev E 93:022902, 2016, Achilleos, Theocharis, Skokos, Phys Rev E 93:022903, 2016) (see also, e.g., (Sokolow, Sen, Ann Phys 322:2104, 2007, Ponson, Boechler, Lai, Porter, Kevrekidis, Daraio, Phys Rev E 82:021301, 2010)).

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References

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

    Google Scholar 

  2. A.J. Martínez, 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)

    Article  ADS  Google Scholar 

  3. F. Fraternali, M.A. Porter, C. Daraio, Optimal design of composite granular protectors. Mech. Adv. Mat. Struct. 17(1), 1 (2010)

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  5. R.L. Doney, J.H. Agui, S. Sen, Energy partitioning and impulse dispersion in the decorated, tapered, strongly nonlinear granular alignment: a system with many potential applications. J. Appl. Phys. 106, 064905 (2009)

    Article  ADS  Google Scholar 

  6. 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)

    Article  ADS  Google Scholar 

  7. U. Harbola, A. Rosas, M. Esposito, K. Lindenberg, Pulse propagation in tapered granular chains: an analytic study. Phys. Rev. E 80, 031303 (2009)

    Article  ADS  Google Scholar 

  8. K. Lindenberg, U. Harbola, A.H. Romero, A. Rosas, Pulse propagation in granular chains, in International Conference on Applications in Nonlinear Dynamics — ICAND 2010 (AIP Conference Proceedings 1339) (American Institute of Physics, 2011), p. 97

    Google Scholar 

  9. A.J. Martínez, P.G. Kevrekidis, M.A. Porter, Superdiffusive transport and energy localization in disordered granular crystals. Phys. Rev. E 93, 022902 (2016)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  11. A. Sokolow, S. Sen, Exact solution to the problem of nonlinear pulse propagation through random layered media and its connection with number triangles. Ann. Phys. 322, 2104 (2007)

    Article  ADS  MATH  Google Scholar 

  12. L. Ponson, N. Boechler, Y.M. Lai, M.A. Porter, P.G. Kevrekidis, C. Daraio, Nonlinear waves in disordered diatomic granular chains. Phys. Rev. E 82, 021301 (2010)

    Article  ADS  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  14. M.A. Porter, C. Daraio, I. Szelengowicz, E.B. Herbold, P.G. Kevrekidis, Highly nonlinear solitary waves in heterogeneous periodic granular media. Physica D 238, 666 (2009)

    Article  ADS  MATH  Google Scholar 

  15. K.R. Jayaprakash, Y. Starosvetsky, A.F. Vakakis, New family of solitary waves in granular dimer chains with no precompression. Phys. Rev. E 83, 036606 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  16. K.R. Jayaprakash, A.F. Vakakis, Y. Starosvetsky, Solitary waves in a general class of granular dimer chains. J. App. Phys. 112, 034903 (2012)

    Article  ADS  Google Scholar 

  17. K.R. Jayaprakash, A. Shiffer, Y. Starosvetsky, Traveling waves in trimer granular lattice I: Bifurcation structure of traveling waves in the unit-cell model. Commun. Nonlinear Sci. Numer. Simul. 38, 8–22 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  18. K.R. Jayaprakash, Y. Starosvetsky, A.F. Vakakis, O.V. Gendelman, Nonlinear resonances leading to strong pulse attenuation in granular dimer chains. J. Nonlinear Sci. 23, 363 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. S.Y. Wang, V.F. Nesterenko, Attenuation of short strongly nonlinear stress pulses in dissipative granular chains. Phys. Rev. E 91, 062211 (2015)

    Article  ADS  Google Scholar 

  20. E. Kim, R. Chaunsali, H. Xu, J. Castillo, J. Yang, P.G. Kevrekidis, A.F. Vakakis, Nonlinear low-to-high frequency energy cascades in diatomic granular crystals. Phys. Rev. E 92, 062201 (2015)

    Article  ADS  Google Scholar 

  21. A. Vainchtein, Y. Starosvetsky, J.D. Wright, R. Perline, Solitary waves in diatomic chains. Phys. Rev. E 93, 042210 (2016)

    Article  ADS  Google Scholar 

  22. T.E. Faver, J.D. Wright, Exact diatomic Fermi–Pasta–Ulam–Tsingou solitary waves with optical band ripples at infinity (2015), arXiv:1511.00942

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

    Article  ADS  Google Scholar 

  24. R.F. Wallis, Effect of free ends on the vibration frequencies of one-dimensional lattices. Phys. Rev. 105, 540 (1957)

    Article  ADS  MATH  Google Scholar 

  25. M. Makwana, R.V. Craster, Localised point defect states in asymptotic models of discrete lattices. Q. J. Mech. Appl. Math. 66, 289 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. G. Theocharis, M. Kavousanakis, P.G. Kevrekidis, C. Daraio, M.A. Porter, I.G. Kevrekidis, Localized breathing modes in granular crystals with defects. Phys. Rev. E 80, 066601 (2009)

    Article  ADS  Google Scholar 

  27. T. Zibold, E. Nicklas, C. Gross, M.K. Oberthaler, Classical bifurcation at the transition from Rabi to Josephson dynamics. Phys. Rev. Lett. 105, 204101 (2010)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  29. C. Kittel, Introduction to Solid State Physics (Wiley, Hoboken, 2005)

    MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  31. G. Theocharis, N. Boechler, P.G. Kevrekidis, S. Job, M.A. Porter, C. Daraio, Intrinsic energy localization through discrete gap breathers in one-dimensional diatomic granular crystals. Phys. Rev. E 82, 056604 (2010)

    Article  ADS  Google Scholar 

  32. G. Huang, B. Hu, Asymmetric gap soliton modes in diatomic lattices with cubic and quartic nonlinearity. Phys. Rev. B 57, 5746 (1998)

    Article  ADS  Google Scholar 

  33. Y.S. Kivshar, G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003)

    Google Scholar 

  34. C. Hoogeboom, P. Kevrekidis, Breathers in periodic granular chains with multiple band gaps. Phys. Rev. E 86, 061305 (2012)

    Article  ADS  Google Scholar 

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

    Google Scholar 

  36. C. Hoogeboom, Y. Man, N. Boechler, G. Theocharis, P.G. Kevrekidis, I.G. Kevrekidis, C. Daraio, Hysteresis loops and multi-stability: from periodic orbits to chaotic dynamics (and back) in diatomic granular crystals. Eur. Phys. Lett. 101, 44003 (2013)

    Google Scholar 

  37. E.G. Charalampidis, F. Li, C. Chong, J. Yang, P.G. Kevrekidis, Time-periodic solutions of driven-damped trimer granular crystals. Math. Probl. Eng. 2015, 830978 (2015)

    Article  MathSciNet  Google Scholar 

  38. J. Lydon, G. Theocharis, C. Daraio, Nonlinear resonances and energy transfer in finite granular chains. Phys. Rev. E 91, 023208 (2015)

    Article  ADS  Google Scholar 

  39. C. Chong, F. Li, J. Yang, M.O. Williams, I.G. Kevrekidis, P.G. Kevrekidis, C. Daraio, Damped-driven granular chains: an ideal playground for dark breathers and multibreathers. Phys. Rev. E 89, 032924 (2014)

    Article  ADS  Google Scholar 

  40. D. Pozharskiy, Y. Zhang, M.O. Williams, D.M. McFarland, P.G. Kevrekidis, A.F. Vakakis, I.G. Kevrekidis, Nonlinear resonances and antiresonances of a forced sonic vacuum. Phys. Rev. E 92, 063203 (2015)

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

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Authors and Affiliations

  1. Bowdoin College, Brunswick, ME, USA

    Christopher Chong

  2. University of Massachusetts, Amherst, MA, USA

    Panayotis G. Kevrekidis

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  1. Christopher Chong
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Correspondence to Christopher Chong .

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Chong, C., Kevrekidis, P.G. (2018). Heterogeneous Media. In: Coherent Structures in Granular Crystals. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-77884-6_5

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