Granular Matter

, 10:235 | Cite as

Cohesive, frictional powders: contact models for tension

  • Stefan Luding
Open Access


The contacts between cohesive, frictional particles with sizes in the range 0.1–10 μm are the subject of this study. Discrete element model (DEM) simulations rely on realistic contact force models—however, too much details make both implementation and interpretation prohibitively difficult. A rather simple, objective contact model is presented, involving the physical properties of elastic–plastic repulsion, dissipation, adhesion, friction as well as rolling- and torsion-resistance. This contact model allows to model bulk properties like friction, cohesion and yield-surfaces. Very loose packings and even fractal agglomerates have been reported in earlier work. The same model also allows for pressure-sintering and tensile strength tests as presented in this study.


Granular materials Molecular dynamics (MD) and discrete elementmodel (DEM) force-laws Friction Rolling- and torsion-resistance Adhesion Plastic deformation 


Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


  1. 1.
    Allen M.P. and Tildesley D.J. (1987). Computer Simulation of Liquids. Oxford University Press, Oxford zbMATHGoogle Scholar
  2. 2.
    Bartels G., Unger T., Kadau D., Wolf D.E. and Kertesz J. (2005). The effect of contact torques on porosity of cohesive powders. Granular Matter 7: 139 CrossRefzbMATHGoogle Scholar
  3. 3.
    Bashir Y.M. and Goddard J.D. (1991). A novel simulation method for the quasi-static mechanics of granular assemblages. J Rheol 35(5): 849–885 CrossRefADSGoogle Scholar
  4. 4.
    Berger, F.: Das Gesetz des Kraftverlaufes beim Stoß. Friedr. Vieweg & Sohn AG (1924)Google Scholar
  5. 5.
    Brendel L. (2006). Modeling of caked contacts in DEMs. Chem Eng Technol 29(11): 1355–1359 CrossRefGoogle Scholar
  6. 6.
    Brendel, L., Dippel, S.: Lasting contacts in molecular dynamics simulations. In: Herrmann, H.J., Hovi, J.P., Luding, S. (eds) Physics of Dry Granular Media. Kluwer, Dordrecht, p 313 (1998)Google Scholar
  7. 7.
    Brilliantov N.V., Spahn F., Hertzsch J.M. and Pöschel T. (1996). Model for collisions in granular gases. Phys Rev E 53(5): 5382 CrossRefADSGoogle Scholar
  8. 8.
    Butt H.J., Cappella B. and Kappl M. (2005). Force measurements with the atomic force microscope: Technique, interpretation and applications. Surf. Sci. Rep. 59(1–6): 1–152 CrossRefADSGoogle Scholar
  9. 9.
    Castellanos A. (2005). The relationship between attractive interparticle forces and bulk behavior in dry and uncharged fine powders. Adv. Phys. 54(4): 263–376 CrossRefADSGoogle Scholar
  10. 10.
    Coste, C., Falcon, E., Fauve, S.: Propagations d’ondes non-linéaires dans une chaîne de bille s en contact de Hertz. In: Petit C., Pijaudier-Cabot G., Reynouard J.M. (eds) Des géomatériaux aux ouvrages: expérimentations et modélis ations, Hermes, Paris, pp 33–52 (in french, 1995)Google Scholar
  11. 11.
    Coste C., Falcon E. and Fauve S. (1997). Solitary waves in a chain of beads under Hertz contact. Phys. Rev. E 56(5): 6104–6117 CrossRefADSGoogle Scholar
  12. 12.
    Cundall P.A. and Strack O.D.L. (1979). A discrete numerical model for granular assemblies. Géotechnique 29(1): 47–65 Google Scholar
  13. 13.
    David, C.T., Rojo, R.G., Herrmann, H.J., Luding, S.: Hysteresis and creep in powders and grains. In: Garcia-Rojo, R., Herrmann, H.J., McNamara, S. (eds) Powders and Grains 2005, Balkema, Leiden, Netherlands, pp. 291–294 (2005)Google Scholar
  14. 14.
    David C.T., Garcia-Rojo R., Herrmann H.J. and Luding S. (2007). Powder flow testing with 2d and 3d biaxial and triaxial simulations. Particle Particle Syst. Charact. 24(1): 29–33 CrossRefGoogle Scholar
  15. 15.
    Derjaguin B.V., Muller V.M. and Toporov Y.P. (1975). Effect of contact deformation on adhesion of particles. J. Colloid Interf. Sci. 53: 314–326 CrossRefGoogle Scholar
  16. 16.
    Dintwa E., van Zeebroeck M., Tijskens E. and Ramon H. (2005). Torsion of viscoelastic spheres in contact. Granular Matter 7(2–3): 169–179 CrossRefzbMATHGoogle Scholar
  17. 17.
    Els, D.: Definition of roll velocity for spherical particles. Granular Matter (2006, submitted)Google Scholar
  18. 18.
    Foerster S.F., Louge M.Y., Chang H. and Allia K. (1994). Measurements of the collision properties of small spheres. Phys. Fluids 6(3): 1108–1115 CrossRefADSGoogle Scholar
  19. 19.
    Grof Z., Lawrence C.J. and Stepanek F. (2008). Computer simulation of evolving capillary bridges in granular media. Granular Matter 10(2): 93–103 CrossRefGoogle Scholar
  20. 20.
    Heim L.O., Butt H.J., Blum J. and Schrapler R. (2008). A new method for the analysis of compaction processes in high-porosity agglomerates. Granular Matter 10(2): 89–91 CrossRefGoogle Scholar
  21. 21.
    Herrmann, H.J., Hovi, J.P., Luding, S.: (eds) Physics of dry granular media—NATO ASI Series E 350. Kluwer, Dordrecht (1998)Google Scholar
  22. 22.
    Hertz H. (1882). Über die Berührung fester elastischer K örper. J für die reine u angew Math 92: 136 Google Scholar
  23. 23.
    Janssen H.A. (1895). Versuche über Getreidedruck in Silozellen. Zeitschr d Vereines deutscher Ingenieure 39(35): 1045–1049 Google Scholar
  24. 24.
    Jenkins J.T. and Koenders M.A. (2005). Hydrodynamic interaction of rough spheres. Granular Matter 7(1): 13–18 CrossRefzbMATHGoogle Scholar
  25. 25.
    Johnson K.L. (1989). Contact Mechanics. Cambridge University Press, Cambridge Google Scholar
  26. 26.
    Johnson K.L., Kendall K. and Roberts A.D. (1971). Surface energy and contact of elastic solids. Proc R Soc Lond Ser A 324(1558): 301 ADSGoogle Scholar
  27. 27.
    Johnson P.C. and Jackson R. (1987). Frictional-collisional constitutive relations for granular materials, with application to plane shearing. J Fluid Mech 176: 67 CrossRefADSGoogle Scholar
  28. 28.
    Kadau D., Schwesig D., Theuerkauf J. and Wolf D.E. (2006). Influence of particle elasticity in shear testers. Granular Matter 8: 34–40 CrossRefGoogle Scholar
  29. 29.
    Kafui K.D. and Thornton C. (2000). Numerical simulations of impact breakage of spherical crystalline agglomerate. Powder Technol 109: 113–132 CrossRefGoogle Scholar
  30. 30.
    Kappl, M., Heim, L., Butt, H.J., Luding, S., Tykhoniuk, R., Tomas, J.: From grains to powders: from single particle contact mechanics measurements to bulk powder properties. In: Garcia-Rojo, R., Herrmann, H.J., McNamara, S. (eds) Powders and Grains 2005, Balkema, Leiden, Netherlands, pp. 493–497 (2005)Google Scholar
  31. 31.
    Kun F. and Herrmann H.J. (2000). Damage development under gradual loading of composites. J. Mater. Sci. 35(18): 4685–4693 CrossRefGoogle Scholar
  32. 32.
    Kuwabara G. and Kono K. (1987). Restitution coefficient in a collision between two spheres. Jpn. J. Appl. Phys. 26(8): 1230–1233 CrossRefADSGoogle Scholar
  33. 33.
    Labous L., Rosato A.D. and Dave R. (1997). Measurements of collision properties of spheres using high-speed video analysis. Phys. Rev. E 56: 5715 CrossRefADSGoogle Scholar
  34. 34.
    Lätzel M., Luding S., Herrmann H.J., Howell D.W. and Behringer R.P. (2003). Comparing simulation and experiment of a 2d granular couette shear device. Eur. Phys. J. Eng. 11(4): 325–333 CrossRefGoogle Scholar
  35. 35.
    Leroy B. (1985). Collision between two balls accompanied by deformation: A qualitative approach to Hertz’s theory. Am. J. Phys. 53(4): 346–349 CrossRefADSGoogle Scholar
  36. 36.
    Lian G., Adams M.J. and Thornton C. (1996). Elastohydrodynamic collisions of solid spheres. J Fluid. Mech. 311: 141 CrossRefADSzbMATHGoogle Scholar
  37. 37.
    Lorenz A., Tuozzolo C. and Louge M.Y. (1997). Measurements of impact properties of small, nearly spherical particles. Exp. Mech. 37(3): 292–297 CrossRefGoogle Scholar
  38. 38.
    Lubachevsky B.D. (1991). How to simulate billards and similar systems. J. Comp. Phys. 94(2): 255 CrossRefADSMathSciNetzbMATHGoogle Scholar
  39. 39.
    Luding S. (1998). Collisions and contacts between two particles. In: Herrmann, H.J., Hovi, J.P. and Luding, S. (eds) Physics of dry granular media—NATO ASI Series E350, pp 285. Kluwer, Dordrecht Google Scholar
  40. 40.
    Luding, S.: Micro-macro models for anisotropic granular media. In: Vermeer, P.A., Ehlers, W., Herrmann, H.J., Ramm, E. (eds) Modelling of cohesive-frictional Materials, Balkema, pp 195–206 (ISBN 04 1536 023 4) (2004a)Google Scholar
  41. 41.
    Luding S. (2004b). Molecular dynamics simulations of granular materials. In: Hinrichsen, H. and Wolf, D.E. (eds) The Physics of Granular Media, pp 299–324. Weinheim, Wiley VCH Google Scholar
  42. 42.
    Luding S. (2005). Anisotropy in cohesive, frictional granular media. J. Phys. Condens. Matter 17: S2623–S2640 CrossRefADSGoogle Scholar
  43. 43.
    Luding, S.: About contact force-laws for cohesive frictional materials in 2d and 3d. In: Walzel, P., Linz, S., Krülle, C., Grochowski, R. (eds) Behavior of Granular Media, Shaker Verlag, pp 137–147, band 9, Schriftenreihe Mechanische Verfahrenstechnik, ISBN 3-8322-5524-9 (2006)Google Scholar
  44. 44.
    Luding, S.: Contact models for very loose granular materials. In: Eberhard P. (ed) Symposium on Multiscale Problems in Multibody System Contacts, Springer, Heidelberg, pp. 135–150. ISBN 978-1-4020-5980-3 (2007)Google Scholar
  45. 45.
    Luding, S., Herrmann, H.J.: Micro-macro transition for cohesive granular media. In: Diebels S. (Ed.) Bericht Nr. II-7, Inst. für Mechanik, Universität Stuttgart (2001)Google Scholar
  46. 46.
    Luding, S., Suiker, A.: Self-healing of damaged particulate materials through sintering. Philos. Mag. (2008, submitted)Google Scholar
  47. 47.
    Luding S., Clément E., Blumen A., Rajchenbach J. and Duran J. (1994). Anomalous energy dissipation in molecular dynamics simulations of grains: The “detachment effect”. Phys. Rev. E 50: 4113 CrossRefADSGoogle Scholar
  48. 48.
    Luding S., Clément E., Blumen A., Rajchenbach J. and Duran J. (1994). The onset of convection in molecular dynamics simulations of grains. Phys. Rev. E 50: R1762 CrossRefADSGoogle Scholar
  49. 49.
    Luding S., Clément E., Blumen A., Rajchenbach J. and Duran J. (1994). Studies of columns of beads under external vibrations. Phys. Rev. E 49(2): 1634 CrossRefADSGoogle Scholar
  50. 50.
    Luding S., Manetsberger K. and Muellers J. (2005). A discrete model for long time sintering. J. Mech. Phys. Solids 53(2): 455–491 CrossRefADSzbMATHGoogle Scholar
  51. 51.
    Luding, S., Suiker, A., Kadashevich, I.: Discrete element modeling of self-healing processes in damaged particulate materials. In: Schmets, A.J.M., van der Zwaag, S. (eds) Proceedings of the 1st International Conference on Self Healing Materials, Springer series in Material Science, Berlin, Germany (ISBN 978-1-4020-6249-0 (2007)Google Scholar
  52. 52.
    Matuttis H.G., Luding S. and Herrmann H.J. (2000). Discrete element methods for the simulation of dense packings and heaps made of spherical and non-spherical particles. Powder Technol. 109: 278–292 CrossRefGoogle Scholar
  53. 53.
    Mindlin R.D. (1949). Compliance of elastic bodies in contact. J. Appl. Mech. 16: 259 MathSciNetzbMATHGoogle Scholar
  54. 54.
    Mindlin R.D. and Deresiewicz H. (1953). Elastic spheres in contact under varying oblique forces. J. Appl. Mech. 20: 327 MathSciNetzbMATHGoogle Scholar
  55. 55.
    Moreau, J.J.: New computation methods in granular dynamics. In: Powders and Grains, vol. 93. Balkema, Rotterdam, p 227 (1993)Google Scholar
  56. 56.
    Moreau J.J. (1994). Some numerical methods in multibody dynamics: application to granular materials. Eur J Mech A 13: 93 MathSciNetzbMATHGoogle Scholar
  57. 57.
    Oda M. and Iwashita K. (2000). Study on couple stress and shear band development in granular media based on numerical simulation analyses. Int. J. Eng. Sci. 38: 1713–1740 CrossRefGoogle Scholar
  58. 58.
    Oda M. and Kazama H. (1998). Microstructure of shear bands and its relation to the mechanism of dilatancy and failure of dense granular soils. Géotechnique 48(4): 465–481 Google Scholar
  59. 59.
    Oda, M., Iwashita, K., Kazama, H.: Micro-structure developed in shear bands of dense granular soils and its computer simulation—mechanism of dilatancy and failure. In: Fleck, N.A., Cocks, A.C.E. (eds) IUTAM Symposium on Mechanics of Granular and Porous Materials. Kluwer, Dordrecht, pp 353–364 (1997)Google Scholar
  60. 60.
    Pao Y.H. (1955). Extension of the Hertz theory of impact to the viscoelastic case. J. Appl. Phys. 26: 1083 CrossRefADSzbMATHGoogle Scholar
  61. 61.
    Pöschel T. and Schwager T. (2005). Computational Granular Dynamics. Springer, Berlin Google Scholar
  62. 62.
    Pöschel T., Schwager T. and Brilliantov N.V. (1999). Rolling friction of a hard cylinder on a viscous plane. Eur. J. Phys. 10: 169–175 ADSGoogle Scholar
  63. 63.
    Radjai F., Jean M., Moreau J.J. and Roux S. (1996). Force distribution in dense two-dimensional granular systems. Phys. Rev. Lett. 77(2): 274 CrossRefADSGoogle Scholar
  64. 64.
    Radjai F., Schäfer J., Dippel S. and Wolf D. (1997). Collective friction of an array of particles: A crucial test for numerical algorithms. J. Phys. I France 7: 1053 CrossRefGoogle Scholar
  65. 65.
    Radjai F., Wolf D.E., Jean M. and Moreau J.J. (1998). Bimodal character of stress transmission in granular packings. Phys. Rev. Lett. 80(1): 61–64 CrossRefADSGoogle Scholar
  66. 66.
    Raman C.V. (1918). The photographic study of impact at minimal velocities. Phys. Rev. 12: 442–447 CrossRefADSGoogle Scholar
  67. 67.
    Rapaport D.C. (1995). The Art of Molecular Dynamics Simulation. Cambridge University Press, Cambridge Google Scholar
  68. 68.
    Richefeu V., Radjai F. and Youssoufi M.S.E. (2006). Stress transmission in wet granular materials. Eur. Phys. J. Eng. 21(4): 359–369 CrossRefGoogle Scholar
  69. 69.
    Röck M., Morgeneyer M., Schwedes J., Brendel L., Wolf D.E. and Kadau D. (2008). Visualization of shear motions of cohesive powders in the true biaxial shear tester. Partic. Sci. Technol. 26: 43–54 CrossRefGoogle Scholar
  70. 70.
    Roux, S.: Quasi-static contacts. In: Herrmann, H.J., Hovi, J.P., Luding, S. (eds) Physics of dry granular media—NATO ASI Series E 350, Kluwer, Dordrecht, p. 267 (1998)Google Scholar
  71. 71.
    Sadd M.H., Tai Q.M. and Shukla A. (1993). Contact law effects on wave propagation in particulate materials using distinct element modeling. Int. J. Non-Lin. Mech. 28(2): 251 CrossRefzbMATHGoogle Scholar
  72. 72.
    Savkoor A.R. and Briggs G.A.D. (1977). The effect of tangential force on the contact of elastic solids in adhesion. Proc. R. Soc. Lond. A 356: 103 ADSzbMATHCrossRefGoogle Scholar
  73. 73.
    Schäfer J., Dippel S. and Wolf D.E. (1996). Force schemes in simulations of granular materials. J. Phys. I France 6: 5–20 CrossRefGoogle Scholar
  74. 74.
    Severens I.E.M., de Ven A.A.F.V., Wolf D.E. and Mattheij R.M.M. (2006). Discrete element method simulations of toner behavior in the development nip of the oce direct imaging print process. Granular Matter 8(3–4): 137–150 CrossRefGoogle Scholar
  75. 75.
    Sinkovits R.S. and Sen S. (1995). Nonlinear dynamics in granular columns. Phys. Rev. Lett. 74(14): 2686 CrossRefADSGoogle Scholar
  76. 76.
    Spahn F., Hertzsch J.M. and Brilliantov N.V. (1995). The role of particle collisions for the dynamics in planetary rings. Chaos Solitons Fractals 5: 1945 CrossRefADSGoogle Scholar
  77. 77.
    Sperl M. (2006). Experiments on corn pressure in silo cells. Translation and comment of Janssen’s paper from 1895. Granular Matter 8(2): 59–65 CrossRefzbMATHGoogle Scholar
  78. 78.
    Suiker A.S.J. and Fleck N.A. (2004). Frictional collapse of granular assemblies. J. Appl. Mech. 71: 350–358 CrossRefzbMATHGoogle Scholar
  79. 79.
    Tanakov M.Y., Trusov L.I., Belyi M.V., Bulgakov V.E. and Gryaznov V.G. (1993). Elastically stressed state in small particles under conditions of Hertzian contacts. J. Phys. D 26: 997 CrossRefADSGoogle Scholar
  80. 80.
    Thornton C. (1997). Force transmission in granular media. KONA Powder Particle 15: 81–90 Google Scholar
  81. 81.
    Thornton C. (2000). Numerical simulations of deviatoric shear deformation of granular media. Géotechnique 50(1): 43–53 CrossRefGoogle Scholar
  82. 82.
    Thornton C. and Antony S.J. (2000). Quasi-static deformation of a soft particle system. Powder Technol. 109(1–3): 179–191 CrossRefGoogle Scholar
  83. 83.
    Thornton, C., Randall, C.W.: Applications of theoretical contact mechanics to solid particle system simulation. In: Micromechanics of granular media. Elsevier, Amsterdam (1988)Google Scholar
  84. 84.
    Thornton C. and Yin K.K. (1991). Impact of elastic spheres with and without adhesion. Powder Technol. 65: 153 CrossRefGoogle Scholar
  85. 85.
    Thornton, C., Zhang, L.: A DEM comparison of different shear testing devices. In: Kishino, Y. (ed) Powders and Grains 2001. Balkema, Rotterdam, pp. 183–190 (2001)Google Scholar
  86. 86.
    Tighe B.P. and Sperl M. (2007). Pressure and motion of dry sand: translation of Hagen’s paper from 1852. Granular Matter 9(3/4): 141–144 CrossRefGoogle Scholar
  87. 87.
    Tomas J. (2000). Particle adhesion fundamentals and bulk powder consolidation. KONA 18: 157–169 Google Scholar
  88. 88.
    Tomas J. (2004). Fundamentals of cohesive powder consolidation and flow. Granular Matter 6(2/3): 75–86 CrossRefzbMATHGoogle Scholar
  89. 89.
    Tykhoniuk R., Tomas J. and Luding S. (2006). A microstructure-based simulation environment on the basis of an interface enhanced particle model. Granular Matter 8(3/4): 159–174 Google Scholar
  90. 90.
    Valverde J.M. and Castellanos A. (2007). Compaction of fine powders: from fluidized agglomerates to primary particles. Granular Matter 9(1–2): 19–24 Google Scholar
  91. 91.
    Vermeer, P.A., Diebels, S., Ehlers, W., Herrmann, H.J., Luding, S., Ramm, E. (eds) Continuous and Discontinuous Modelling of Cohesive Frictional Materials. Lecture Notes in Physics, vol. 568. Springer, Berlin (2001)Google Scholar
  92. 92.
    Vermeer, P.A., Ehlers, W., Herrmann, H.J., Ramm, E.: (eds) Modelling of Cohesive-frictional materials, Balkema, Leiden, Netherlands (ISBN 04 1536 023 4) (2004)Google Scholar
  93. 93.
    Walton K. (1978). The oblique compression of two elastic spheres. J. Mech. Phys. Solids 26: 139 CrossRefADSMathSciNetzbMATHGoogle Scholar
  94. 94.
    Walton O.R. (1989). Force models for particle-dynamics simulations of granular materials. NATO ASI Ser. E Appl. Sci. 287: 367–379 Google Scholar
  95. 95.
    Walton, O.R.: Effects of interparticle friction and particle shape on dynamic angles of repose via particle-dynamics simulation. In: Workshop: Mechanics and Statistical Physics of Particulate Materials (1994)Google Scholar
  96. 96.
    Walton, O.R.: Elastic frictional contact models based on analysis of Mindlin (1949), private communication (1995a)Google Scholar
  97. 97.
    Walton O.R. (1995). Force models for particle-dynamics simulations of granular materials. In: Guazzelli, E. and Oger, L. (eds) Mobile particulate systems, pp 367. Kluwer, Dordrecht Google Scholar
  98. 98.
    Walton O.R. and Braun R.L. (1986). Viscosity, granular-temperature and stress calculations for shearing assemblies of inelastic, frictional disks. J. Rheol. 30(5): 949–980 CrossRefADSGoogle Scholar
  99. 99.
    Zhu C.Y., Shukla A. and Sadd M.H. (1991). Prediction of dynamic contact loads in granular assemblies. J. Appl. Mech. 58: 341 CrossRefGoogle Scholar

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© The Author(s) 2008

Authors and Affiliations

  1. 1.Multi Scale Mechanics, TS, CTW, UTwenteEnschedeNetherlands
  2. 2.Particle Technology, Nanostructured MaterialsDelftChemTech, TNW, TUDelftDelftNetherlands

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