Constitutive Relations from Particle Simulations

  • Stefan LudingEmail author
Conference paper
Part of the Springer Series in Geomechanics and Geoengineering book series (SSGG)


Particulate systems like powders, soil or granular matter are discrete, disordered systems displaying dynamic and static, fluid - and solid-like states. The transients between fluid - and solid-like behavior can be intermittent and sometimes both states coexist in steady-state. Bridging the gap between the particulate, microscopic picture (velocities, forces) on the particle scale and their continuum description (strain and stress) via a so-called micro-macro transition is the goal of this paper. The generalized local constitutive relation for the stress in critical state granular flows involves not only density and strain rate but also the jamming-density and the granular temperature.


Particle models Micro-macro transition Continuum rheology 


  1. 1.
    Luding, S.: The effect of friction on wide shear bands. Part. Sci. Tech. 26, 33–42 (2008)CrossRefGoogle Scholar
  2. 2.
    Luding, S., Alonso-Marroquin, F.: The critical-state yield stress (termination locus) of adhesive powders from a single experiment. Granular Matter 13, 109–119 (2011)CrossRefGoogle Scholar
  3. 3.
    Weinhart, T., Hartkamp, R., Thornton, A.R., Luding, S.: Coarse-grained local and objective continuum description of 3D granular flows down an inclined surface. Phys. Fluids 25, 070605 (2013)CrossRefGoogle Scholar
  4. 4.
    Hartkamp, R., Todd, B., Luding, S.: A constitutive framework for the non-Newtonian pressure tensor of a simple fluid under planar flows. J. Chem. Phys. 138, 244508 (2013)CrossRefGoogle Scholar
  5. 5.
    Singh, A., Saitoh, K., Magnanimo, V., Luding, S.: Role of gravity or confining pressure and contact stiffness in granular rheology. New J. Phys. 17, 043028 (2015)CrossRefGoogle Scholar
  6. 6.
    Singh, A., Magnanimo, V., Saitoh, K., Luding, S.: Effect of cohesion on shear banding in quasi-static granular material. Phys. Rev. E 90(2), 022202 (2014)CrossRefGoogle Scholar
  7. 7.
    Vescovi, D., Luding, S.: Merging fluid & solid granular behavior. Soft Matter 12, 8616 (2016)CrossRefGoogle Scholar
  8. 8.
    Roy, S., Singh, A., Luding, S., Weinhart, T.: Micro-macro transition and simplified contact models for wet granular materials. Comp. Part. Mech. 3(4), 449–462 (2016)CrossRefGoogle Scholar
  9. 9.
    Roy, S., Luding, S., Weinhart, T.: A general(ized) local rheology for wet granular materials. New J. Phys. 19, 043014 (2017)CrossRefGoogle Scholar
  10. 10.
    Göncü, F., Luding, S.: Effect of particle friction and polydispersity on the macroscopic stress-strain relations of granular materials. Acta Geotech. 8, 629–643 (2013)CrossRefGoogle Scholar
  11. 11.
    Kumar, N., Luding, S., Magnanimo, V.: Macroscopic model with anisotropy based on micro-macro informations. Acta Mech. 225(8), 2319–2343 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Goldhirsch, I.: Stress, stress asymmetry and couple stress: from discrete particles to continuous fields. Granular Matter 12, 239–252 (2010)CrossRefGoogle Scholar
  13. 13.
    Weinhart, T., Thornton, A.R., Luding, S., Bokhove, O.: From discrete particles to continuum fields near a boundary. Granular Matter 14, 289–294 (2012)CrossRefGoogle Scholar
  14. 14.
    Midi, G.D.R.: On dense granular flows. Eur. Phys. J. E 14, 367–371 (2004)CrossRefGoogle Scholar
  15. 15.
    Ries, A., Wolf, D.E., Unger, T.: Shear zones in granular media: three-dimensional contact dynamics simulations. Phys. Rev. E 76, 051301 (2007)CrossRefGoogle Scholar
  16. 16.
    Nguyen, V.B., Darnige, T., Bruand, A., Clément, E.: Creep and fluidity of a real granular packing near jamming. Phys. Rev. Lett. 107, 138303 (2011)CrossRefGoogle Scholar
  17. 17.
    Kamrin, K., Koval, G.: Nonlocal constitutive relation for steady granular flow. Phys. Rev. Lett. 108, 178301 (2012)CrossRefGoogle Scholar
  18. 18.
    Hennan, D.L., Kamrin, K.: A predictive, size-dependent continuum model for dense granular flows. Proc. Natl. Acad. Sci. USA 110, 6730–6735 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Shi, H., Luding, S., Magnanimo, V.: Limestone powders yielding and steady state resistance under shearing with different testers. In: Mallick, S.S. (ed.) Proceedings of PGBSIA 2016, 1–3 December 2016Google Scholar
  20. 20.
    Shi, H., et al.: Effect of particle size and cohesion on powder yielding and flow. KONA 2018014 (2017)Google Scholar
  21. 21.
    Göncü, F., Duran, O., Luding, S.: Constitutive relations for the isotropic deformation of frictionless packings of polydisperse spheres. C. R. Mecanique 338(10–11), 570–586 (2010)CrossRefGoogle Scholar
  22. 22.
    Luding, S.: Granular matter: so much for the jamming point. Nat. Phys. 12, 531 (2016)CrossRefGoogle Scholar
  23. 23.
    Kumar, N., Luding, S.: Memory of jamming – multiscale models for soft and granular matter. Granular Matter 18, 58 (2016)CrossRefGoogle Scholar
  24. 24.
    Pons, A., Darnige, T., Crassous, J., Clement, E., Amon, A.: Spatial repartition of local plastic processes in different creep regimes in a granular material. EPL 113(2), 28001 (2016)CrossRefGoogle Scholar
  25. 25.
    Koval, G., Roux, J.N., Corfdir, A., Chevoir, F.: Annular shear of cohesionless granular materials: from the inertial to quasistatic regime. Phys. Rev. E 79, 021306 (2009)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Multi-Scale Mechanics (MSM), Faculty of Engineering Technology, MESA+University of TwenteEnschedeThe Netherlands

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