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Densest arrangement of frictionless polydisperse sphere packings with a power-law grain size distribution

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Abstract

In this work we use DEM simulations to find the densest packing of frictionless spheres subject to mechanical compression by changing systematically the grain size distribution (GSD). This is done by modeling the GSD as a power law and varying the GSD describing parameters: the size span and the distribution shape, while relating these parameters to the density and other micro-structural parameters. For large size spans, the optimal GSD resembles the century old Fuller-Thomson distribution that is commonly used in the field of cements and pavements. This optimal power-law GSD produces not only the densest packing but also presents good connectivity, measured from a small proportion of floating particles, and reduces the packing internal spatial correlations, measured by the pair correlation function. Thus, tuning the GSD appropriately allows for designing mechanically stable packings that have the highest density, a small number of floating particles, and less local correlations.

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Notes

  1. It must be noted that the jamming density can be protocol-dependent, as shown in [33] and several of the references therein.

  2. The model proposed in [27, 39] suggests that the reference solid fraction \(\phi _{\text {ref}}\) should be close to that of the well known Random Close Packing of monodisperse systems (i.e., 0.64). Instead, we used a slightly higher value, taking into account some degree of local ordering.

References

  1. Fuller, W.B., Thompson, S.E.: The laws of proportioning concrete. Trans. Am. Soc. Civ. Eng. 59(2), 67 (1907)

    Google Scholar 

  2. Taylor, F.W., Thompson, S.E.: A Treatise on Concrete, Plain and Reinforced. Wiley, Hoboken (1905)

    Google Scholar 

  3. A.N. Talbot, F.E. Richart: The strength of concrete-its relation to the cement, aggregates and water. Bulletin 137, University Of Illinois, University of Illinois, Urbana (1923)

  4. C. Furnas: The relations between specific volume, voids, and size composition in systems of broken solids of mixed sizes. Technical Report Serial No. 2894, Department of Commerce, Bureau of Mines, Report of Investigation Serial No. 2894 (1929)

  5. Furnas, C.: Grading aggregates-I.-Mathematical relations for beds of broken solids of maximum density. Ind. Eng. Chem. 23(9), 1052 (1931). https://doi.org/10.1021/ie50261a017

    Article  Google Scholar 

  6. Agnolin, I., Roux, J.N.: Internal states of model isotropic granular packings. II. Compression and pressure cycles. Phys. Rev. E (Stat. Nonlinear Soft Matter Phys.) 76(6), 061303 (2007). https://doi.org/10.1103/PhysRevE.76.061303

    Article  ADS  MathSciNet  Google Scholar 

  7. Agnolin, I., Roux, J.N.: Internal states of model isotropic granular packings. III. Elastic properties. Phys. Rev. E (Stat. Nonlinear Soft Matter Phys.) 76(6), 061304 (2007). https://doi.org/10.1103/PhysRevE.76.061304

    Article  ADS  MathSciNet  Google Scholar 

  8. Khalili, M.H., Roux, J.N., Pereira, J.M., Brisard, S., Bornert, M.: Numerical study of one-dimensional compression of granular materials. II. Elastic moduli, stresses, and microstructure. Phys. Rev. E 95(3), 032908 (2017). https://doi.org/10.1103/PhysRevE.95.032908

    Article  ADS  Google Scholar 

  9. Troadec, H., Radjai, F., Roux, S., Charmet, J.: Model for granular texture with steric exclusion. Phys. Rev. E 66(4), 041305 (2002). https://doi.org/10.1103/physreve.66.041305

    Article  ADS  Google Scholar 

  10. Radjai, F., Delenne, J.Y., Azéma, E., Roux, S.: Fabric evolution and accessible geometrical states in granular materials. Granul. Matter 14(2), 259 (2012). https://doi.org/10.1007/s10035-012-0321-8

    Article  Google Scholar 

  11. Estrada, N.: Effects of grain size distribution on the packing fraction and shear strength of frictionless disk packings. Phys. Rev. E 94(6), 062903 (2016). https://doi.org/10.1103/PhysRevE.94.062903

    Article  ADS  Google Scholar 

  12. Estrada, N., Oquendo, W.F.: Microstructure as a function of the grain size distribution for packings of frictionless disks: effects of the size span and the shape of the distribution. Phys. Rev. E 96(4), 042907 (2017). https://doi.org/10.1103/physreve.96.042907

    Article  ADS  Google Scholar 

  13. Delaney, G.W., Hutzler, S., Aste, T.: Relation between grain shape and fractal properties in random Apollonian packing with grain rotation. Phys. Rev. Lett. 101(12), 120602 (2008). https://doi.org/10.1103/PhysRevLett.101.120602

    Article  ADS  Google Scholar 

  14. Herrmann, H., Mahmoodi Baram, R., Wackenhut, M.: Searching for the perfect packing. Physica A Stat. Mech. Appl. 330(1–2), 77 (2003). https://doi.org/10.1016/j.physa.2003.08.023

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Manna, S.: Space filling tiling by random packing of discs. Physica A Stat. Mech. Appl. 187(3–4), 373 (1992). https://doi.org/10.1016/0378-4371(92)90001-7

    Article  ADS  MathSciNet  Google Scholar 

  16. Manna, S.S., Herrmann, H.J.: Precise determination of the fractal dimensions of apollonian packing and space-filling bearings. J. Phys. A Math. Gen. 24(9), L481 (1991). https://doi.org/10.1088/0305-4470/24/9/006

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Varrato, F., Foffi, G.: Apollonian packings as physical fractals. Mol. Phys. 109(22), 2663 (2011). https://doi.org/10.1080/00268976.2011.640039

    Article  ADS  Google Scholar 

  18. Bardenhagen, S.G., Brackbill, J.U., Sulsky, D.: Numerical study of stress distribution in sheared granular material in two dimensions. Phys. Rev. E 62(3), 3882 (2000). https://doi.org/10.1103/PhysRevE.62.3882

    Article  ADS  Google Scholar 

  19. Shaebani, M.R., Madadi, M., Luding, S., Wolf, D.E.: Influence of polydispersity on micromechanics of granular materials. Phys. Rev. E 85(1), 011301 (2012). https://doi.org/10.1103/PhysRevE.85.011301

    Article  ADS  Google Scholar 

  20. Muthuswamy, M., Tordesillas, A.: How do interparticle contact friction, packing density and degree of polydispersity affect force propagation in particulate assemblies? J. Stat. Mech. Theory Exp. 2006(09), P09003 (2006). https://doi.org/10.1088/1742-5468/2006/09/P09003

    Article  MATH  Google Scholar 

  21. Voivret, C., Radjaï, F., Delenne, J.Y., El Youssoufi, M.S.: Space-filling properties of polydisperse granular media. Phys. Rev. E 76(2), 021301 (2007). https://doi.org/10.1103/PhysRevE.76.021301

    Article  ADS  MATH  Google Scholar 

  22. Voivret, C., Radjaï, F., Delenne, J.Y., El Youssoufi, M.S.: Multiscale force networks in highly polydisperse granular media. Phys. Rev. Lett. 102(17), 178001 (2009). https://doi.org/10.1103/PhysRevLett.102.178001

    Article  ADS  MATH  Google Scholar 

  23. Nguyen, D.H., Azéma, É., Radjai, F., Sornay, P.: Effect of size polydispersity versus particle shape in dense granular media. Phys. Rev. E 90(1), 012202 (2014). https://doi.org/10.1103/PhysRevE.90.012202

    Article  ADS  Google Scholar 

  24. Nguyen, D.H., Azéma, E., Sornay, P., Radjai, F.: Effects of shape and size polydispersity on strength properties of granular materials. Phys. Rev. E 91(3), 032203 (2015). https://doi.org/10.1103/PhysRevE.91.032203

    Article  ADS  Google Scholar 

  25. Azéma, E., Linero, S., Estrada, N., Lizcano, A.: Does modifying the particle size distribution of a granular material (i.e., material scalping) alters its shear strength? EPJ Web of Conferences 140, 06001 (2017). https://doi.org/10.1051/epjconf/201714006001

    Article  Google Scholar 

  26. Ogarko, V., Luding, S.: Equation of state and jamming density for equivalent bi- and polydisperse, smooth, hard sphere systems. J. Chem. Phys. 136(12), 124508 (2012). https://doi.org/10.1063/1.3694030

    Article  ADS  Google Scholar 

  27. Santos, A., Yuste, S.B., López de Haro, M., Odriozola, G., Ogarko, V.: Simple effective rule to estimate the jamming packing fraction of polydisperse hard spheres. Phys. Rev. E 89(4), 040302 (2014). https://doi.org/10.1103/PhysRevE.89.040302

    Article  ADS  Google Scholar 

  28. Cantor, D., Azéma, E., Sornay, P., Radjai, F.: Rheology and structure of polydisperse three-dimensional packings of spheres. Phys. Rev. E 98(5), 052910 (2018). https://doi.org/10.1103/PhysRevE.98.052910

    Article  ADS  Google Scholar 

  29. Kloss, C., Goniva, C., Hager, A., Amberger, S., Pirker, S.: Models, algorithms and validation for opensource DEM and CFD-DEM. Prog. Comput. Fluid Dyn. Int. J. 12(2/3), 140 (2012). https://doi.org/10.1504/PCFD.2012.047457

    Article  MathSciNet  Google Scholar 

  30. Luding, S.: Cohesive, frictional powders: contact models for tension. Granul. Matter 10(4), 235 (2008). https://doi.org/10.1007/s10035-008-0099-x

    Article  MathSciNet  MATH  Google Scholar 

  31. Hertz, H.: Über Die Berührung Fester Elastiches Körper. J. Reine Angew. Math. 92, 156 (1882)

    MathSciNet  MATH  Google Scholar 

  32. Brilliantov, N.V., Spahn, F., Hertzsch, J.M., Pöschel, T.: Model for collisions in granular gases. Phys. Rev. E 53(5), 5382 (1996). https://doi.org/10.1103/PhysRevE.53.5382

    Article  ADS  Google Scholar 

  33. Kumar, N., Luding, S.: Memory of jamming-multiscale models for soft and granular matter. Granul. Matter 18, 3 (2016). https://doi.org/10.1007/s10035-016-0624-2

    Article  Google Scholar 

  34. Lozano, E., Roehl, D., Celes, W., Gattass, M.: An efficient algorithm to generate random sphere packs in arbitrary domains. Comput. Math. Appl. 71(8), 1586 (2016). https://doi.org/10.1016/j.camwa.2016.02.032

    Article  MathSciNet  MATH  Google Scholar 

  35. Agnolin, I., Roux, J.N.: Internal states of model isotropic granular packings. I. Assembling process, geometry, and contact networks. Phys. Rev. E (Stat. Nonlinear Soft Matter Phys.) 76(6), 061302 (2007). https://doi.org/10.1103/PhysRevE.76.061302

    Article  ADS  MathSciNet  Google Scholar 

  36. MiDi, G.D.R.: On dense granular flows. Eur. Phys. J. E 14(4), 341 (2004). https://doi.org/10.1140/epje/i2003-10153-0

    Article  Google Scholar 

  37. Rognon, P.G., Roux, J.N., Naaïm, M., Chevoir, F.: Dense flows of bidisperse assemblies of disks down an inclined plane. Phys. Fluids 19(5), 058101 (2007). https://doi.org/10.1063/1.2722242

    Article  ADS  MATH  Google Scholar 

  38. Rycroft, C.H.: VORO++: a three-dimensional voronoi cell library in C++. Chaos Interdiscip. J. Nonlinear Sci. 19(4), 041111 (2009). https://doi.org/10.1063/1.3215722

    Article  Google Scholar 

  39. Santos, A., Yuste, S.B., López de Haro, M., Ogarko, V.: Equation of state of polydisperse hard-disk mixtures in the high-density regime. Phys. Rev. E 96(6), 062603 (2017). https://doi.org/10.1103/PhysRevE.96.062603

    Article  ADS  Google Scholar 

  40. Kumar, N., Magnanimo, V., Ramaioli, M., Luding, S.: Tuning the bulk properties of bidisperse granular mixtures by small amount of fines. Powder Technol. 293, 94 (2016). https://doi.org/10.1016/j.powtec.2015.11.042

    Article  Google Scholar 

  41. Radjaï, F., Dubois, F. (eds.): Discrete Numerical Modeling of Granular Materials. Wiley-ISTE, New York (2011)

    Google Scholar 

  42. Göncü, F., Durán, O., Luding, S.: Constitutive relations for the isotropic deformation of frictionless packings of polydisperse spheres. Comptes Rendus Mécanique 338(10–11), 570 (2010). https://doi.org/10.1016/j.crme.2010.10.004

    Article  ADS  MATH  Google Scholar 

  43. Rothenburg, L., Bathurst, R.: Analytical study of induced anisotropy in idealized granular materials. Geotechnique 39(4), 601 (1989)

    Article  Google Scholar 

  44. Ouadfel, H., Rothenburg, L.: Stress–force–fabric’relationship for assemblies of ellipsoids. Mech. Mater. 33(4), 201 (2001)

    Article  Google Scholar 

  45. Donev, A., Connelly, R., Stillinger, F.H., Torquato, S.: Underconstrained jammed packings of nonspherical hard particles: ellipses and ellipsoids. Phys. Rev. E 75(5), 051304 (2007). https://doi.org/10.1103/PhysRevE.75.051304

    Article  ADS  MathSciNet  Google Scholar 

  46. van Hecke, M.: Jamming of soft particles: geometry, mechanics, scaling and isostaticity. J. Phys. Condens. Matter 22(3), 033101 (2010). https://doi.org/10.1088/0953-8984/22/3/033101

    Article  ADS  Google Scholar 

  47. Goodrich, C.P., Liu, A.J., Nagel, S.R.: Finite-size scaling at the jamming transition. Phys. Rev. Lett. 109(9), 095704 (2012). https://doi.org/10.1103/PhysRevLett.109.095704

    Article  ADS  Google Scholar 

  48. Iliev, P.S., Wittel, F.K., Herrmann, H.J.: Evolution of fragment size distributions from the crushing of granular materials. Phys. Rev. E 99(1), 012904 (2019). https://doi.org/10.1103/PhysRevE.99.012904

    Article  ADS  Google Scholar 

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Acknowledgements

N.E. thanks Catalina María Gutiérrez for her helpful advice and stimulating discussions. W.O. wants to thank the Universidad de la Sabana, Colombia, internal Project No. ING-182-2016 for funding, the National University of Colombia for support to run part of the simulations, and Nohora Cortés for her immense support and patience. This work was supported by the project Ecos Nord No. C19P01-63672.

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Authors and Affiliations

  1. Faculty of Engineering, Universidad de la Sabana, Chía, Colombia

    William Fernando Oquendo-Patiño

  2. Department of Physics, National University of Colombia, Bogotá, Colombia

    William Fernando Oquendo-Patiño

  3. Department of Civil and Environmental Engineering, Universidad de los Andes, Bogotá, Colombia

    Nicolas Estrada

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Oquendo-Patiño, W.F., Estrada, N. Densest arrangement of frictionless polydisperse sphere packings with a power-law grain size distribution. Granular Matter 22, 75 (2020). https://doi.org/10.1007/s10035-020-01043-9

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