Abstract
A minimum principle is developed with the aim of determining the contact forces in a random packing of hard frictionless particles. The principle is an extension of the minimum complementary energy principle to this particular non linear mechanical system. Under the assumptions that a convex complementary energy exists and that relative displacements between particles are small, it allows us to synthesize the generation mechanism of the force network from an energetic point of view and provides, besides the existing methods (e.g. Discrete Element Method), a new method to analyze the contact force distribution in discrete particle systems. Here the contact force network in a packing of identical elastic frictionless spheres with Hertz contacts is determined and it is shown that it is unique and independent both by the value of the applied load and by the value of elastic constants. Numerical examples confirm that the contact forces so determined are consistent with previous experimental results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anikeenko, A.V., Medvedev, N.N., Aste, T.: Structural and entropic insights into the nature of the random-close-packing limit. Phys. Rev. E 77, 031101 (2008)
Ammi, M., Bideau, D., Troadec, J.P.: Geometrical structure of disordered packings of regular polygons; comparison with disc packings structures. J. Phys. D 20, 424–428 (1987)
Antony, S.J.: Evolution of force distribution in three-dimensional granular media. Phys. Rev. E 63, 011302 (2000)
Bagi, K.: Stress and strain in granular assemblies. Mech. Mater. 22, 165–177 (1996)
Bernal, J.D., Mason, J.: Packing of spheres: co-ordination of randomly packed spheres. Nature 188, 910–911 (1960)
Blair, D.L., Mueggenburg, N.W., Marshall, A.H., Jaeger, H.M., Nagel, S.R.: Force distributions in three-dimensional granular assemblies: effects of packing order and interparticle friction. Phys. Rev. E 63, 041304 (2001)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2009)
Coppersmith, S.N., Liu, C., Majumdar, S., Narayan, O., Witten, T.A.: Model for force fluctuations in bead packs. Phys. Rev. E 53, 4673–4685 (1996)
Desmond, K.W., Weeks, E.R.: Random close packing of disks and spheres in confined geometries. Phys. Rev. E 80, 051305 (2009)
Gurtin, M.E.: The linear theory of elasticity. In: Truesdell, C. (ed) Linear Theories of Elasticity and Thermoelasticity, pp. 1–295. Springer, Berlin (1973)
Hicher, P.-Y., Chang, C.S.: A microstructural elastoplastic model for unsaturated granular materials. Int. J. Solids Struct. 44, 2304–2323 (2007)
Howell, D., Behringer, R.P.: Stress fluctuations in a 2D granular couette experiment: a continuous transition. Phys. Rev. Lett. 82, 5241–5244 (1999)
Jodrey, W.S., Tory, E.M.: Computer simulation of close random packing of equal spheres. Phys. Rev. A 32, 2347–2351 (1985)
Kramar, M., Goullet, A., Kondic, L., Mischaikow, K.: Persistence of force networks in compressed granular meida. Phys. Rev. E 87, 042207 (2013)
Liu, C., Nagel, S.R., Schecter, D.A., Coppersmith, S.N., Majumdar, S., Narayan, O., Witten, T.A.: Force fluctuations in bead packs. Science 269, 513–515 (1995)
Luding, S.: Stress distribution in static two-dimensional granular model media in the absence of friction. Phys. Rev. E 55, 4720–4729 (1997)
Majmudar, T.S., Behringer, R.P.: Contact force measurements and stress-induced anisotropy in granular materials. Nature 435, 1079–1082 (2005)
Makse, H.A., Johnson, D.L., Schwartz, L.M.: Packing of compressible granular materials. Phys. Rev. Lett. 84, 4160–4163 (2000)
Miller, B., O’Hern, C., Behringer, R.P.: Stress fluctuations for continuously sheared granular materials. Phys. Rev. Lett. 77, 3110–3113 (1996)
Mueth, D.M., Jaeger, H.M., Nagel, S.R.: Force distribution in a granular medium. Phys. Rev. E 57, 3164–3169 (1998)
Nemat-Nasser, S.: A micromechanically-based constitutive model for frictional deformation of granular materials. J. Mech. Phys. Solids 48, 1541–1563 (2000)
Nemat-Nasser, S., Zhang, J.H.: Constitutive relations for cohesionless frictional granular materials. Int. J. Plast. 18, 531–547 (2002)
Radjai, F., Jean, M., Moreau, J.J., Roux, S.: Force distributions in dense two-dimensional granular systems. Phys. Rev. Lett. 77, 274–277 (1996)
Saadatfar, M., Sheppard, A.P., Senden, T.J., Kabla, A.J.: Mapping forces in a 3D elastic assembly of grains. J. Mech. Phys. Solids 60, 55–66 (2012)
Timoshenko, S., Goodier, J.N.: Theory of Elasticity. Mac Graw Hill, New York (1951)
Trentadue, F.: A micromechanical model for a non-linear elastic granular material based on local equilibrium conditions. Int. J. Solids Struct. 38, 7319–7342 (2001)
Trentadue, F.: An equilibrium based approach for the micromechanical modelling of a non linear elastic granular material. Mech. Mater. 36, 323–324 (2004)
Trentadue, F.: A rigid-plastic micromechanical modeling of a random packing of frictional particles. Int. J. Solids Struct. 48, 2529–2535 (2011)
Walsh, S.D.T., Tordesillas, A., Peters, J.F.: Development of micromechanical models for granular media—the projection problem. Granul. Matter 9, 337–352 (2007)
Yang, Z.X., Yang, J., Wang, L.Z.: Micro-scale modeling of anisotropy effects on undrained behavior of granular soils. Granul. Matter 15, 557–572 (2013)
Zhou, J., Long, S., Wang, Q., Dinsmore, A.A.: Measurement of forces inside a three-dimensional pile of frictionless droplets. Science 312, 1631–1633 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liu, H., Zhang, SH., Cheng, M. et al. A minimum principle for contact forces in random packings of elastic frictionless particles. Granular Matter 17, 475–482 (2015). https://doi.org/10.1007/s10035-015-0567-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10035-015-0567-z