Constitutive relations for the isotropic deformation of frictionless packings of polydisperse spheres

Fatih Göncü; Orencio Durán; Stefan Luding

doi:10.1016/j.crme.2010.10.004

Constitutive relations for the isotropic deformation of frictionless packings of polydisperse spheres
[Lois de comportement pour déformations isotrope d'assemblage de sphères polydisperses sans frottement]

Fatih Göncü ^1,² ; Orencio Durán ^1,³ ; Stefan Luding ^1,²

¹ Multi Scale Mechanics (MSM), Faculty of Engineering Technology, University of Twente, P.O. Box 217, NL-7500 AE Enschede, The Netherlands
² NanoStructured Materials (NSM), ChemTech, Delft University of Technology, Delft, The Netherlands
³ PMMH, UMR7636 (CNRS), ESPCI Univ. P6–P7, 10, rue Vauquelin, 75005 Paris, France

Comptes Rendus. Mécanique, Volume 338 (2010) no. 10-11, pp. 570-586.

Résumé
Abstract

La compression isotrope d'assemblages polydisperses de sphères sans frottement est modélisée par une méthode aux éléments discrets (DEM). L'évolution du nombre de coordination, de la fraction de « rattlers » (les particules instables, sans contactes), de la texture isotrope et de la pression (contrainte isotrope) est étudiée en fonction de la fraction volumique pour différentes valeurs des paramètres du système. Une relation en loi puissance, avec un exposé proche de 0,5, entre le nombre de coordination et la fraction volumique est confirmée en régime de blocage pour une large gamme de fractions volumiques et pour différentes polydispersités. La polydispersité de l'assemblage induit un décalage de la fraction volumique critique, c'est-à-dire que les assemblages plus hétérogènes se bloquent à des fractions volumiques plus élevées. Au voisinage du jamming, le nombre de coordination et la fraction volumique de blocage dépendent à la fois de l'histoire et de la vitesse de chargement. A des densités plus élevées, ni l'histoire des déformations et ni la vitesse de chargement ont un effet significatif sur l'évolution du nombre de coordination.

En ce qui concerne le tenseur de texture, la comparaison de nos résultats DEM avec les prédictions théoriques est satisfaisante pour différentes polydispersités. Une expression analytique de la pression en fonction des déformations volumiques est proposée pour différents assemblages polydisperses, fondée sur une hypothèse de déformation uniforme. On notera que, outre la proportionnalité implicite vis-à-vis de la densité de nombre de contacts, aucune loi puissance ne peut être mise en évidence dans la relation donnant la pression. Cependant, partant d'une pression nulle au point de blocage (jamming), un terme linéaire peut décrire, avec une correction quadratique, l'évolution de la contrainte de manière satisfaisante, pour une large gamme de densités et pour diverses polydispersités. Finalement, une équation d'évolution incrémentale est proposée à la fois pour la texture et la contrainte, en fonction de la déformation volumique, et impliquant le nombre de coordination et la fraction de rattlers. Elle constitue un point de départ pour de futurs travaux en relation avec les déformations anisotropes.

The isotropic compression of polydisperse packings of frictionless spheres is modeled with the Discrete Element Method (DEM). The evolution of coordination number, fraction of rattlers, isotropic fabric, and pressure (isotropic stress) is reported as function of volume fraction for different system parameters. The power law relationship, with power $\approx 1 / 2$ , between coordination number and volume fraction is confirmed in the jammed state for a broad range of volume fractions and for different (moderate) polydispersities. The polydispersity in the packing causes a shift of the critical volume fraction, i.e., more heterogeneous packings jam at higher volume fractions. Close to jamming, the coordination number and the jamming volume fraction itself depend on both history and rate. At larger densities, neither the deformation history nor the loading rate have a significant effect on the evolution of the coordination number.

Concerning the fabric tensor, comparing our DEM results to theoretical predictions, good agreement for different polydispersities is observed. An analytical expression for the pressure as function of isotropic (volumetric) strain is proposed for polydisperse packings, based on the assumption of uniform deformation. We note that, besides the implicit proportionality to contact number density (or fabric), no single power-law is evidenced in the relation between pressure and isotropic strain. However, starting from zero pressure at the jamming point, a linear term with a quadratic correction describes the stress evolution rather well for a broad range of densities and for various polydispersities. Finally, an incremental evolution equation is proposed for both fabric and stress, as function of isotropic strain, and involving the coordination number and the fraction of rattlers, as starting point for further studies involving anisotropic deformations.

Publié le : 2010-10-01

DOI : 10.1016/j.crme.2010.10.004

Keywords: Granular media, Polydisperse, Isotropic compression, Constitutive models, Rattlers
Mot clés : Milieux granulaires, Matériaux granulaires polydisperses sans frottement, Compression isotrope, Lois de comportement, Particules instables (rattlers)

Affiliations des auteurs :

Fatih Göncü ^{1, 2} ; Orencio Durán ^{1, 3} ; Stefan Luding ^{1, 2}

¹ Multi Scale Mechanics (MSM), Faculty of Engineering Technology, University of Twente, P.O. Box 217, NL-7500 AE Enschede, The Netherlands
² NanoStructured Materials (NSM), ChemTech, Delft University of Technology, Delft, The Netherlands
³ PMMH, UMR7636 (CNRS), ESPCI Univ. P6–P7, 10, rue Vauquelin, 75005 Paris, France

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     author = {Fatih G\"onc\"u and Orencio Dur\'an and Stefan Luding},
     title = {Constitutive relations for the isotropic deformation of frictionless packings of polydisperse spheres},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {570--586},
     publisher = {Elsevier},
     volume = {338},
     number = {10-11},
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     doi = {10.1016/j.crme.2010.10.004},
     language = {en},
}

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Fatih Göncü; Orencio Durán; Stefan Luding. Constitutive relations for the isotropic deformation of frictionless packings of polydisperse spheres. Comptes Rendus. Mécanique, Volume 338 (2010) no. 10-11, pp. 570-586. doi : 10.1016/j.crme.2010.10.004. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2010.10.004/

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