Granular Matter

, Volume 16, Issue 2, pp 209–216 | Cite as

Vibrations of jammed disk packings with Hertzian interactions

  • Carl F. Schreck
  • Corey S. O’Hern
  • Mark D. Shattuck
Original Paper

Abstract

Contact breaking and Hertzian interactions between grains can both give rise to nonlinear vibrational response of static granular packings. We perform molecular dynamics simulations at constant energy in 2D of frictionless bidisperse disks that interact via Hertzian spring potentials as a function of energy and measure directly the vibrational response from the Fourier transform of the velocity autocorrelation function. We compare the measured vibrational response of static packings near jamming onset to that obtained from the eigenvalues of the dynamical matrix to determine the temperature above which the harmonic approximation breaks down. We compare packings that interact via single-sided (purely repulsive) and double-sided Hertzian spring interactions to disentangle the effects of the shape of the potential from contact breaking. Our studies show that while Hertzian interactions lead to weak nonlinearities in the vibrational behavior (e.g. the generation of harmonics of the eigenfrequencies of the dynamical matrix), the vibrational response of static packings with Hertzian contact interactions is dominated near jamming by contact breaking as found for systems with repulsive linear spring interactions.

Keywords

Granular materials Acoustics Jamming Vibrations 

References

  1. 1.
    Makse, H.A., Gland, N., Johnson, D.L., Schwartz, L.: Granular packings: nonlinear elasticity, sound propagation, and collective relaxation dynamics. Phys. Rev. E 70, 061302 (2004)ADSCrossRefGoogle Scholar
  2. 2.
    van den Wildenberg, S., van Hecke, M., Jia, X.: Evolution of granular packings by nonlinear acoustic waves. Eur. Phys. Lett. 101, 14004 (2013)ADSCrossRefGoogle Scholar
  3. 3.
    Boechler, N., Theocharis, G., Daraio, C.: Bifurcation-based acoustic switching and rectification. Nat. Mater. 10, 665–668 (2011)ADSCrossRefGoogle Scholar
  4. 4.
    Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1985)CrossRefMATHGoogle Scholar
  5. 5.
    Tournat, V., Zaitsev, V., Gusev, V., Nazarov, V., Béquin, P., Castagnéde, B.: Probing weak forces in granular media through nonlinear dynamic dilatancy: clapping contacts and polarization anisotropy. Phys. Rev. Lett. 92, 085502 (2004)ADSCrossRefGoogle Scholar
  6. 6.
    Schreck, C.F., Bertrand, T., O’Hern, C.S., Shattuck, M.D.: Repulsive contact interactions make jammed particulate systems inherently nonharmonic. Phys. Rev. Lett. 107, 078301 (2011)ADSCrossRefGoogle Scholar
  7. 7.
    Bertrand, T., Schreck, C.F., O’Hern, C.S., Shattuck, M.D.: Vibrations in jammed solids: beyond linear response. unpublished (2013); xxx.lanl.gov/abs/1307.0440.Google Scholar
  8. 8.
    Capozza, R., Vanossi, A., Zapperi, S.: Triggering frictional slip by mechanical vibrations. Tribol. Lett. 48, 95–102 (2012)CrossRefGoogle Scholar
  9. 9.
    Henkes, S., van Hecke, M., van Saarloos, W.: Critical jamming of frictional grains in the generalized isostaticity picture. Europhys. Lett. 90, 14003 (2010)ADSCrossRefGoogle Scholar
  10. 10.
    D’Anna, G., Sellerio, A.L., Mari, D., Gremaud, G.: Friction and Hertzian contact in granular glass. J. Stat. Mech. 05, P05009 (2013)Google Scholar
  11. 11.
    D’Anna, G., Mayor, P., Gremaud, G., Barrat, A., Loreto, V.: Observing brownian motion in vibration-fluidized granular matter. Europhys. Lett. 61, 60–66 (2003)ADSCrossRefGoogle Scholar
  12. 12.
    Gao, G.-J., Blawzdziewicz, J., O’Hern, C.S.: Frequency distribution of mechanically stable disk packings. Phys. Rev. E 74, 061304 (2006)ADSCrossRefGoogle Scholar
  13. 13.
    Schreck, C.F., O’Hern, C.S., Silbert, L.E.: Tuning jammed frictionless disk packings from isostatic to hyperstatic. Phys. Rev. E 84, 011305 (2011)ADSCrossRefGoogle Scholar
  14. 14.
    Tanguy, A., Wittmer, J.P., Leonforte, F., Barrat, J.-L.: Continuum limit of amorphous elastic bodies: a finite-size study of low-frequency harmonic vibrations. Phys. Rev. B 66, 174205 (2002)ADSCrossRefGoogle Scholar
  15. 15.
    Henkes, S., Brito, C., Dauchot, O.: Extracting vibrational modes from fluctuations: a pedagogical discussion. Soft Matter 8, 6092–6109 (2012)ADSCrossRefGoogle Scholar
  16. 16.
    O’Hern, C.S., Silbert, L.E., Liu, A.J., Nagel, S.R.: Jamming at zero temperature and zero applied stress: the epitome of disorder. Phys. Rev. E 68, 011306 (2003)ADSCrossRefGoogle Scholar
  17. 17.
    Jayaprakash, K.R., 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, 359–385 (2011)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Schreck, C.F., Bertrand, T., O’Hern, C.S., Shattuck, M.D.: Response to comment on repulsive contact interactions make jammed particulate systems inherently nonharmonic’. (unpublished) 2013; http://xxx.lanl.gov/abs/1306.1961v1
  19. 19.
    Silbert, L., Liu, A.J., Nagel, S.R.: Normal modes in model jammed systems in three dimensions. Phys. Rev. E 79, 021308 (2009) Google Scholar
  20. 20.
    Vakakis, A.F.: Non-linear normal modes (NNMs) and their applications in vibration theory: an overview. Mech. Syst. Signal Process. 11, 3–22 (1997)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Carl F. Schreck
    • 1
    • 2
  • Corey S. O’Hern
    • 2
    • 3
    • 1
  • Mark D. Shattuck
    • 4
    • 1
  1. 1.Department of Mechanical Engineering and Materials ScienceYale UniversityNew HavenUSA
  2. 2.Department of PhysicsYale UniversityNew HavenUSA
  3. 3.Department of Applied PhysicsYale UniversityNew HavenUSA
  4. 4.Physics Department, Benjamin Levich InstituteThe City College of the City University of New YorkNew YorkUSA

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