Introduction and Motivation of Models

  • Christopher ChongEmail author
  • Panayotis G. Kevrekidis
Part of the SpringerBriefs in Physics book series (SpringerBriefs in Physics)


In the present short volume our aim will be to explore a variety of nonlinear wave structures that are found in a particular physical system, i.e., granular crystals. Many of the readers have almost certainly encountered a variant of the granular crystal.


Granular Crystals Nonlinear Wave Structures Dispersive Shock Waves Solitary Traveling Wave Discrete Breathers 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    V.F. Nesterenko, Dynamics of Heterogeneous Materials (Springer, New York, 2001)CrossRefGoogle Scholar
  2. 2.
    H. Hertz, Über die Berührung fester elastischer Körper. J. Reine. Angew. Math. 92, 156 (1881)zbMATHGoogle Scholar
  3. 3.
    K.L. Johnson, Contact Mechanics (Cambridge University Press, Cambridge, 1985)CrossRefzbMATHGoogle Scholar
  4. 4.
    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
  5. 5.
    G. Friesecke, J.A.D. Wattis, Existence theorem for solitary waves on lattices. Commun. Math. Phys. 161, 391 (1994)Google Scholar
  6. 6.
    R.S. MacKay, Solitary waves in a chain of beads under Hertz contact. Phys. Lett. A 251, 191 (1999)Google Scholar
  7. 7.
    A. Merkel, V. Tournat, V. Gusev, Experimental evidence of rotational elastic waves in granular phononic crystals. Phys. Rev. Lett. 107, 225502 (2011)ADSCrossRefGoogle Scholar
  8. 8.
    J. Cabaret, P. Béquin, G. Theocharis, V. Andreev, V.E. Gusev, V. Tournat, Nonlinear hysteretic torsional waves. Phys. Rev. Lett. 115, 054301 (2015)ADSCrossRefGoogle Scholar
  9. 9.
    F. Allein, V. Tournat, V. Gusev, G. Theocharis, Tunable magneto-granular phononic crystals. Appl. Phys. Lett. 108, 161903 (2016)ADSCrossRefGoogle Scholar
  10. 10.
    W. Lin, C. Daraio, Wave propagation in one-dimensional microscopic granular chains. Phys. Rev. E 94, 052907 (2016)ADSCrossRefGoogle Scholar
  11. 11.
    J. Yang, S. Dunatunga, C. Daraio, Amplitude-dependent attenuation of compressive waves in curved granular crystals constrained by elastic guides. Acta Mech. 223, 549 (2012)CrossRefzbMATHGoogle Scholar
  12. 12.
    J. Yang, M. Sutton, Nonlinear wave propagation in a hexagonally packed granular channel under rotational dynamics. Int. J. Solids Struct. 77, 65 (2015)CrossRefGoogle Scholar
  13. 13.
    P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies. Géotechnique 29, 47 (1979)CrossRefGoogle Scholar
  14. 14.
    D. Zabulionis, R. Kačianauskas, D. Markauskas, J. Rojek, An investigation of nonlinear tangential contact behaviour of a spherical particle under varying loading. Bull. Pol. Acad. Sci. Tech. Sci. 60, 265 (2012)Google Scholar
  15. 15.
    N. Boechler, G. Theocharis, C. Daraio, Bifurcation based acoustic switching and rectification. Nat. Mater. 10, 665 (2011)ADSCrossRefGoogle Scholar
  16. 16.
    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. Euro. Phys. Lett. 101, 44003 (2013)ADSCrossRefGoogle Scholar
  17. 17.
    A. Rosas, A.H. Romero, V.F. Nesterenko, K. Lindenberg, Observation of two-wave structure in strongly nonlinear dissipative granular chains. Phys. Rev. Lett. 98, 164301 (2007)ADSCrossRefGoogle Scholar
  18. 18.
    A. Rosas, A.H. Romero, V.F. Nesterenko, K. Lindenberg, Short-pulse dynamics in strongly nonlinear dissipative granular chains. Phys. Rev. E 78, 051303 (2008)ADSCrossRefGoogle Scholar
  19. 19.
    M. Peyrard, I. Daumont, Statistical properties of one-dimensional "turbulence". Europhys. Lett. 59, 834 (2002)ADSCrossRefGoogle Scholar
  20. 20.
    N.V. Brilliantov, A.V. Pimenova, D.S. Goldobin, A dissipative force between colliding viscoelastic bodies: rigorous approach. EPL (Europhys. Lett.) 109, 14005 (2015)Google Scholar
  21. 21.
    R. Carretero-González, D. Khatri, M.A. Porter, P.G. Kevrekidis, C. Daraio, Dissipative solitary waves in granular crystals. Phys. Rev. Lett. 102, 024102 (2009)ADSCrossRefGoogle Scholar
  22. 22.
    L. Vergara, Model for dissipative highly nonlinear waves in dry granular systems. Phys. Rev. Lett. 104, 118001 (2010)ADSCrossRefGoogle Scholar
  23. 23.
    M. Gonzalez, J. Yang, C. Daraio, M. Ortiz, Mesoscopic approach to granular crystal dynamics. Phys. Rev. E 85, 016604 (2012)ADSCrossRefGoogle Scholar
  24. 24.
    R.K. Pal, J. Morton, E. Wang, J. Lambros, P.H. Geubelle, Impact response of elasto-plastic granular chains containing an intruder particle. J. Appl. Mech. 82, 38 (2015)Google Scholar
  25. 25.
    H.A. Burgoyne, C. Daraio, Elastic-plastic wave propagation in uniform and periodic granular chains. J. Appl. Mech. 82, 081002 (2015)ADSCrossRefGoogle Scholar
  26. 26.
    H. Burgoyne, C. Daraio, Strain-rate-dependent model for the dynamic compression of elastoplastic spheres. Phys. Rev. E 89, 032203 (2014)ADSCrossRefGoogle Scholar
  27. 27.
    H.A. Burgoyne, Dynamics of granular crystals with elastic–plastic contacts, 2016. Ph.D. Dissertation, California Institute of Technology (2016).
  28. 28.
    T. On, P.A. LaVigne, J. Lambros, Development of plastic nonlinear waves in one-dimensional ductile granular chains under impact loading. Mech. Mater. 68, 29 (2014)CrossRefGoogle Scholar
  29. 29.
    T. On, E. Wang, J. Lambros, Plasticwaves in one-dimensional heterogeneous granular chains under impact loading: single intruders and dimer chains. Int. J. Solids Struct. 62, 81 (2015)CrossRefGoogle Scholar
  30. 30.
    J. Yang, M. Gonzalez, E. Kim, C. Agbasi, M. Sutton, Attenuation of solitary waves and localization of breathers in 1D granular crystals visualized via high speed photography. Exp. Mech. 54, 1043 (2014)CrossRefGoogle Scholar
  31. 31.
    S. Sen, J. Hong, J. Bang, E. Avalos, R. Doney, Solitary waves in the granular chain. Phys. Rep. 462, 21 (2008)Google Scholar
  32. 32.
    G. Theocharis, N. Boechler, C. Daraio, Nonlinear Phononic Periodic Structures and Granular Crystals, Acoustic Metamaterials, Phononic Crystals (Springer, Berlin, 2013), pp. 217–251CrossRefGoogle Scholar
  33. 33.
    A.F. Vakakis, Analytical methodologies for nonlinear periodic media, Wave Propagation in Linear and Nonlinear Periodic Media, (International Center for Mechanical Sciences (CISM) Courses and Lectures) (Springer, Berlin, 2012), p. 257CrossRefGoogle Scholar
  34. 34.
    M.A. Porter, P.G. Kevrekidis, C. Daraio, Granular crystals: nonlinear dynamics meets materials engineering. Phys. Today 68, 44 (2015)CrossRefGoogle Scholar
  35. 35.
    Y. Starosvetsky, K. Jayaprakash, M.A. Hasan, A. Vakakis, Dynamics and Acoustics of Ordered Granular Media (World Scientific, Singapore, 2017)Google Scholar
  36. 36.
    M.J. Ablowitz, M. Hoefer, Dispersive shock waves. Scholarpedia 4, 5562 (2009)Google Scholar
  37. 37.
    G.A. El, M.A. Hoefer, M. Shearer, Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws. SIAM Rev. 59, 3–61 (2017)Google Scholar
  38. 38.
    J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer, New York, 1983)Google Scholar
  39. 39.
    S. Flach, A. Gorbach, Discrete breathers: advances in theory and applications. Phys. Rep. 467, 1 (2008)Google Scholar
  40. 40.
    S. Aubry, Discrete breathers: localization and transfer of energy in discrete Hamiltonian nonlinear systems. Phys. D 216, 1 (2006)Google Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Bowdoin CollegeBrunswickUSA
  2. 2.University of MassachusettsAmherstUSA

Personalised recommendations