Granular Matter

, Volume 9, Issue 6, pp 465–469 | Cite as

Coefficient of restitution and linear–dashpot model revisited

  • Thomas SchwagerEmail author
  • Thorsten Pöschel
Brief Communication


With the assumption of a linear–dashpot interaction force, the coefficient of restitution, \(\varepsilon_d^0(k, \gamma)\) , can be computed as a function of the elastic and dissipative material constants, k and γ by integrating Newton’s equation of motion for an isolated pair of colliding particles. If we require further that the particles interact exclusively repulsive, which is a common assumption in granular systems, we obtain an expression \(\varepsilon_d(k, \gamma)\) which differs even qualitatively from the known result \(\varepsilon_d^0(k, \gamma)\) . The expression \(\varepsilon_d(k, \gamma)\) allows to relate Molecular Dynamics simulations to event-driven Molecular Dynamics for a widely used collision model.


Particle collisions Coefficient of restitution 


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  1. 1.
    Brilliantov N.V. and Pöschel T. (2004). Kinetic Theory of Granular Gases. Clarendon, Oxford zbMATHGoogle Scholar
  2. 2.
    Brilliantov N.V., Spahn F., Hertzsch J.M. and Pöschel T. (1996). A model for collisions in granular gases. Phys. Rev. E 53: 5382 CrossRefADSGoogle Scholar
  3. 3.
    Campbell C.S. (2002). Granular shear flows at the elastic limit. J. Fluid Mech. 465: 261 zbMATHCrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Engel P.A. (1978). Impact Wear of Materials. Elsevier, Amsterdam Google Scholar
  5. 5.
    Goldenberg C. and Goldhirsch I. (2004). Small and large scale granular statics. Granul. Matter 6: 87 zbMATHCrossRefGoogle Scholar
  6. 6.
    Goldhirsch I. and Goldenberg C. (2004). Stress in dense granular materials. In: Hinrichsen, H. and Wolf, D.E. (eds) The Physics of Granular Media., pp 3. Wiley, Weinheim Google Scholar
  7. 7.
    Hertz H. (1882). Über die Berührung fester elastischer Körper. J. f reine u angewandte Math. 92: 156 CrossRefGoogle Scholar
  8. 8.
    Kadau D., Schwesig D., Theuerkauf J. and Wolf D.E. (2006). Influence of particle elasticity in shear testers. Granul. Matter 8: 35 CrossRefGoogle Scholar
  9. 9.
    Kruggel-Emden H., Simsek E., Rickelt S., Wirtz S. and Scherer V. (2007). Review and extension of normal force models for the Discrete Element Method. Powder Technol. 171: 157 CrossRefGoogle Scholar
  10. 10.
    Luding S. (1997). Stress distribution in static two dimensional granular model media in the absence of friction. Phys. Rev. E 55: 4720 CrossRefADSGoogle Scholar
  11. 11.
    Luding S. (1998). Collisions & contacts between two particles. In: Herrmann, H.J., J.-P. Hovi, J.P. and Luding, S. (eds) Physics of dry granular Media., pp 285. Kluwer, Dordrecht Google Scholar
  12. 12.
    Luding S. (1998b). Die Physik kohäsionsloser granularer Medien (Habilitation thesis; in german). Logos Verlag, Berlin Google Scholar
  13. 13.
    Luding S., Clément E., Blumen A., Rajchenbach J. and Duran J. (1994a). Anomalous energy dissipation in molecular dynamics simulations of grains: The “detachment” effect. Phys. Rev. E 50: 4113 CrossRefADSGoogle Scholar
  14. 14.
    Luding S., Clément E., Blumen A., Rajchenbach J. and Duran J. (1994b). Onset of convection in molecular dynamics simulations of grains. Phys. Rev. E 50: R1762 CrossRefADSGoogle Scholar
  15. 15.
    Matuttis G., Luding S. and Herrmann H.J. (2000). Discrete element simulations of dense packings and heaps made of spherical and non-spherical particles. Powder Technol. 109: 278 CrossRefGoogle Scholar
  16. 16.
    Meerson B., Pöschel T. and Bromberg Y. (2003). Close-packed floating clusters: Granular hydrodynamics beyond the freezing point?. Phys. Rev. Lett. 91: 24,301 CrossRefGoogle Scholar
  17. 17.
    Mouraille O., Mulder W.A. and Luding S. (2006). Sound wave acceleration in granular materials. J. Stat. Mech. 2006: P07,023 CrossRefGoogle Scholar
  18. 18.
    Oger L., Savage S.B., Corriveau D. and Sayed M. (1998). Yield and deformation of an assmebly of disks subjected to a deviatoric stress loading. Mech. Mater. 27: 189 CrossRefGoogle Scholar
  19. 19.
    Pöschel T. and Schwager T. (2005). Computational Granular Dynamics. Springer, Berlin Google Scholar
  20. 20.
    Ramírez R., Pöschel T., Brilliantov N.V. and Schwager T. (1999). Coefficient of restitution of colliding viscoelastic spheres. Phys. Rev. E 60: 4465 CrossRefADSGoogle Scholar
  21. 21.
    Schäfer J. and Wolf D.E. (1995). Bistability in simulated granular flow along corrugated walls. Phys. Rev. E 51: 6154 CrossRefADSGoogle Scholar
  22. 22.
    Schäfer J., Dippel S. and Wolf D.E. (1996). Force schemes in simulations of granular materials. J. Phys. I 6: 5 CrossRefGoogle Scholar
  23. 23.
    Schwager T. (2007). Coefficient of restitution for viscoelastic disks. Phys. Rev. E 75: 051,305 CrossRefGoogle Scholar
  24. 24.
    Schwager T. and Pöschel T. (1998). Coefficient of restitution of viscous particles and cooling rate of granular gases. Phys. Rev. E 57: 650 CrossRefADSGoogle Scholar
  25. 25.
    Schwager, T., Pöschel, T.: Coefficient of restitution for viscoelastic spheres: the effect of delayed recovery. arXiv:07081434 (2007)Google Scholar
  26. 26.
    Taguchi Y. (1992a). Powder turbulence: Direct onset of turbulent flow. J. Phys. I 2: 2103 CrossRefMathSciNetGoogle Scholar
  27. 27.
    Taguchi Yh. (1992b). New origin of a convective motion: elastically induced convection in granular materials. Phys. Rev. Lett. 69: 1367 CrossRefADSGoogle Scholar
  28. 28.
    Tanaka T., Ishida T. and Tsuji Y. (1991). Direct numerical simulation of granular plug flow in a horizontal pipe. the case of cohesionless particles (in japanese). Kiron B 57–534: 456 Google Scholar
  29. 29.
    Thompson P.A. and Grest G.A. (1991). Granular flow: friction and the dilatancy transition. Phys. Rev. Lett. 67: 1751 CrossRefADSGoogle Scholar
  30. 30.
    Tsuji Y., Tanaka T. and Ishida T. (1991). Lagrangian numerical simulation of plug flow of cohesionless particles in a horizontal pipe. Powder Technol. 71: 239 CrossRefGoogle Scholar
  31. 31.
    Weir G. and Tallon S. (2005). The coefficient of restitution for normal incident, low velocity particle impacts. Chem. Eng. Sci. 60: 3637 CrossRefGoogle Scholar
  32. 32.
    Wolf, D.E.: How to simulate granular matter. In: Blügel, S., Gompper, G., Koch, E., Müller-Krumbhaar, H., Spatschek, R., Winkler, R.G. (eds.) Computational Condensed Matter Physics—Lecture Manuscripts of the 37th Spring School of the Institute of Solid State Research, IFF-Ferienkurs, vol 32. Forschungszentrum, Jülich, p. B13 (2006)Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.CharitéBerlinGermany
  2. 2.Physikalisches InstitutUniversität BayreuthBayreuthGermany

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