Journal of Nonlinear Science

, Volume 22, Issue 5, pp 813–848 | Cite as

Periodic Travelling Waves and Compactons in Granular Chains

Article
First Online:
Received:
Accepted:

Abstract

We study the propagation of an unusual type of periodic travelling waves in chains of identical beads interacting via Hertz’s contact forces. Each bead periodically undergoes a compression phase followed by free flight, due to special properties of Hertzian interactions (fully nonlinear under compression and vanishing in the absence of contact). We prove the existence of such waves close to binary oscillations, and numerically continue these solutions when their wavelength is increased. In the long wave limit, we observe their convergence towards shock profiles consisting of small compression regions close to solitary waves, alternating with large domains of free flight where bead velocities are small. We give formal arguments to justify this asymptotic behavior, using a matching technique and previous results concerning solitary wave solutions. The numerical finding of such waves implies the existence of compactons, i.e. compactly supported compression waves propagating at a constant velocity, depending on the amplitude and width of the wave. The beads are stationary and separated by equal gaps outside the wave, and each bead reached by the wave is shifted by a finite distance during a finite time interval. Below a critical wave number, we observe fast instabilities of the periodic travelling waves, leading to a disordered regime.

Keywords

Granular chain Hertzian contact Hamiltonian lattice Periodic travelling wave Compacton Fully nonlinear dispersion 

Mathematics Subject Classification

37K60 70F45 70K50 70K75 74J30 
This is a preview of subscription content, log in to check access

Notes

Acknowledgements

The author is grateful to anonymous referees for several suggestions which essentially improved the paper, in particular for pointing out the question of the existence of compactons. Helpful discussions with J. Malick, B. Brogliato, A.R. Champneys and P.G. Kevrekidis are also acknowledged.

References

  1. Abraham, R., Marsden, J.E.: Foundations of Mechanics, 2nd edn. Addison-Wesley, Reading (1987) Google Scholar
  2. Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions (1964). National Bureau of Standards (10th corrected printing, 1970). www.nr.com MATHGoogle Scholar
  3. Acary, V., Brogliato, B.: Concurrent multiple impacts modelling: case study of a 3-ball chain. In: Bathe, K.J. (ed.) Proc. of the MIT Conference on Computational Fluid and Solid Mechanics, pp. 1836–1841. Elsevier, Amsterdam (2003) Google Scholar
  4. Ahnert, K., Pikovsky, A.: Compactons and chaos in strongly nonlinear lattices. Phys. Rev. E 79, 026209 (2009) MathSciNetCrossRefGoogle Scholar
  5. Aubry, S., Cretegny, T.: Mobility and reactivity of discrete breathers. Physica D 119, 34–46 (1998) MathSciNetMATHCrossRefGoogle Scholar
  6. Campbell, D.K., et al. (eds.): The Fermi–Pasta–Ulam Problem: The First 50 Years. Chaos, vol. 15 (2005) Google Scholar
  7. Chatterjee, A.: Asymptotic solutions for solitary waves in a chain of elastic spheres. Phys. Rev. E 59, 5912–5918 (1999) CrossRefGoogle Scholar
  8. Chicone, C.: Ordinary Differential Equations with Applications. Texts in Applied Mathematics, vol. 34. Springer, Berlin (1999) MATHGoogle Scholar
  9. Cretegny, T., Aubry, S.: Spatially inhomogeneous time-periodic propagating waves in anharmonic systems. Phys. Rev. B 55, R11929–R11932 (1997) CrossRefGoogle Scholar
  10. Dennis, J.E. Jr., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM Classics in Applied Mathematics, vol. 16. SIAM, Philadelphia (1996) MATHCrossRefGoogle Scholar
  11. Dreyer, W., Herrmann, M., Mielke, A.: Micro-macro transition in the atomic chain via Whitham’s modulation equation. Nonlinearity 19, 471–500 (2006) MathSciNetMATHCrossRefGoogle Scholar
  12. Dreyer, W., Herrmann, M.: Numerical experiments on the modulation theory for the nonlinear atomic chain. Physica D 237, 255–282 (2008) MathSciNetMATHCrossRefGoogle Scholar
  13. English, J.M., Pego, R.L.: On the solitary wave pulse in a chain of beads. Proc. Am. Math. Soc. 133(6), 1763–1768 (2005) MathSciNetMATHCrossRefGoogle Scholar
  14. Falcon, E.: Comportements dynamiques associés au contact de Hertz: processus collectifs de collision et propagation d’ondes solitaires dans les milieux granulaires. PhD thesis, Université Claude Bernard Lyon 1 (1997) Google Scholar
  15. Filip, A.M., Venakides, S.: Existence and modulation of traveling waves in particle chains. Commun. Pure Appl. Math. 52, 693–735 (1999) MathSciNetCrossRefGoogle Scholar
  16. Fraternali, F., Porter, M.A., Daraio, C.: Optimal design of composite granular protectors. Mech. Adv. Mat. Struct. 17, 1–19 (2010) CrossRefGoogle Scholar
  17. Friesecke, G., Pego, R.L.: Solitary waves on FPU lattices: I. Qualitative properties, renormalization and continuum limit. Nonlinearity 12, 1601–1627 (1999) MathSciNetMATHCrossRefGoogle Scholar
  18. Friesecke, G., Pego, R.L.: Solitary waves on FPU lattices: IV. Proof of stability at low energy. Nonlinearity 17, 229–251 (2004) MathSciNetMATHCrossRefGoogle Scholar
  19. Friesecke, G., Wattis, J.A.: Existence theorem for solitary waves on lattices. Commun. Math. Phys. 161, 391–418 (1994) MathSciNetMATHCrossRefGoogle Scholar
  20. Fu, G.: An extension of Hertz’s theory in contact mechanics. J. Appl. Mech. 74, 373–375 (2007) MATHCrossRefGoogle Scholar
  21. Gallavotti, G. (ed.): The Fermi–Pasta–Ulam Problem. A Status Report. Lecture Notes in Physics, vol. 728. Springer, Berlin (2008) MATHGoogle Scholar
  22. Herrmann, M.: Unimodal wave trains and solitons in convex FPU chains. Proc. R. Soc. Edinb. A 140, 753–785 (2010) MATHCrossRefGoogle Scholar
  23. Hinch, E.J., Saint-Jean, S.: The fragmentation of a line of ball by an impact. Proc. R. Soc. Lond. Ser. A 455, 3201–3220 (1999) MathSciNetMATHCrossRefGoogle Scholar
  24. Hoffman, A., Wayne, C.E.: A Simple Proof of the Stability of Solitary Waves in the Fermi–Pasta–Ulam Model Near the KdV Limit (2008). arXiv:0811.2406v1 [nlin.PS]
  25. Iooss, G.: Travelling waves in the Fermi–Pasta–Ulam lattice. Nonlinearity 13, 849–866 (2000) MathSciNetMATHCrossRefGoogle Scholar
  26. James, G.: Nonlinear waves in Newton’s cradle and the discrete p-Schrödinger equation. Math. Models Methods Appl. Sci. 21, 2335–2377 (2011) MathSciNetMATHCrossRefGoogle Scholar
  27. James, G., Kevrekidis, P.G., Cuevas, J.: Breathers in oscillator chains with Hertzian interactions. Physica D (2011, in press). arXiv:1111.1857v1 [nlin.PS]
  28. Ji, J.-Y., Hong, J.: Existence criterion of solitary waves in a chain of grains. Phys. Lett. A 260, 60–61 (1999) MathSciNetMATHCrossRefGoogle Scholar
  29. Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1985) MATHGoogle Scholar
  30. Johnson, P.A., Jia, X.: Nonlinear dynamics, granular media and dynamic earthquake triggering. Nature 437, 871–874 (2005) CrossRefGoogle Scholar
  31. Liu, C., Zhao, Z., Brogliato, B.: Frictionless multiple impacts in multibody systems. I. Theoretical framework. Proc. R. Soc. A, Math. Phys. Eng. Sci. 464, 3193–3211 (2008) MathSciNetMATHCrossRefGoogle Scholar
  32. Liu, C., Zhao, Z., Brogliato, B.: Frictionless multiple impacts in multibody systems. II. Numerical algorithm and simulation results. Proc. R. Soc. A, Math. Phys. Eng. Sci. 465, 1–23 (2009) MathSciNetMATHCrossRefGoogle Scholar
  33. Ma, W., Liu, C., Chen, B., Huang, L.: Theoretical model for the pulse dynamics in a long granular chain. Phys. Rev. E 74, 046602 (2006) CrossRefGoogle Scholar
  34. MacKay, R.S.: Solitary waves in a chain of beads under Hertz contact. Phys. Lett. A 251, 191–192 (1999) CrossRefGoogle Scholar
  35. Nesterenko, V.F.: Propagation of nonlinear compression pulses in granular media. J. Appl. Mech. Tech. Phys. 24, 733–743 (1983) CrossRefGoogle Scholar
  36. Nesterenko, V.F.: Dynamics of Heterogeneous Materials. Springer, Berlin (2001) Google Scholar
  37. Pankov, A.: Travelling Waves and Periodic Oscillations in Fermi–Pasta–Ulam Lattices. Imperial College Press, London (2005) MATHCrossRefGoogle Scholar
  38. Porter, M., Daraio, C., Szelengowicz, I., Herbold, E.B., Kevrekidis, P.G.: Highly nonlinear solitary waves in heterogeneous periodic granular media. Physica D 238, 666–676 (2009) MATHCrossRefGoogle Scholar
  39. Rosenau, P., Hyman, J.M.: Compactons: solitons with finite wavelength. Phys. Rev. Lett. 70, 564 (1993) MATHCrossRefGoogle Scholar
  40. Rosenau, P., Schochet, S.: Compact and almost compact breathers: a bridge between an anharmonic lattice and its continuum limit. Chaos 15, 015111 (2005) MathSciNetCrossRefGoogle Scholar
  41. Schmittbuhl, J., Vilotte, J.-P., Roux, S.: Propagative macrodislocation modes in an earthquake fault model. Europhys. Lett. 21, 375–380 (1993) CrossRefGoogle Scholar
  42. Sen, S., Hong, J., Bang, J., Avalos, E., Doney, R.: Solitary waves in the granular chain. Phys. Rep. 462, 21–66 (2008) MathSciNetCrossRefGoogle Scholar
  43. Sen, S., Manciu, M., Wright, J.D.: Soliton-like pulses in perturbed and driven hertzian chains and their possible applications in detecting buried impurities. Phys. Rev. E 57, 2386–2397 (1998) CrossRefGoogle Scholar
  44. Sepulchre, J.-A., MacKay, R.S.: Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators. Nonlinearity 10, 679–713 (1997) MathSciNetMATHCrossRefGoogle Scholar
  45. Starosvetsky, Y., Vakakis, A.F.: Traveling waves and localized modes in one-dimensional homogeneous granular chains with no precompression. Phys. Rev. E 82, 026603 (2010) MathSciNetCrossRefGoogle Scholar
  46. Stefanov, A., Kevrekidis, P.G.: On the existence of solitary traveling waves for generalized hertzian chains. J. Nonlinear Sci. (2012). doi: 10.1007/s00332-011-9119-9 Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Laboratoire Jean KuntzmannUniversité de Grenoble and CNRSGrenoble Cedex 9France

Personalised recommendations