Statistical Mechanics of Stress Transmission in Disordered Granular Arrays

Edwards, S. F.; Grinev, D. V.

doi:10.1103/physrevlett.82.5397

Cited by 144 publications

(124 citation statements)

References 10 publications

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“…Numerical simulations and theoretical work suggest that at the jamming transition the system becomes exactly isostatic [29,31,27,34,61,33,60]. But no rigorous proof of this statement exists.…”

Section: Definition Of Jamming: Isostatic Conjecturementioning

confidence: 99%

“…In practice, it is widely believed that the isostatic condition is necessary for a jammed disordered packing following the Alexander conjecture [59,60,61] which was tested in several works [27,29,30,33] It is well known that mechanical equilibrium imposes an average coordination number larger or equal than a minimum coordination where the number of force variables equals the number of force and torque balance equations [59,61,60]. The so-called isostatic condition.…”

Section: Definition Of Jamming: Isostatic Conjecturementioning

confidence: 99%

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Jamming II: Edwards’ statistical mechanics of random packings of hard spheres

Wang

Jin

et al. 2011

Physica A: Statistical Mechanics and its Applications

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The problem of finding the most efficient way to pack spheres has an illustrious history, dating back to the crystalline arrays conjectured by Kepler and the random geometries explored by Bernal in the 60's. This problem finds applications spanning from the mathematician's pencil, the processing of granular materials, the jamming and glass transitions, all the way to fruit packing in every grocery. There are presently numerous experiments showing that the loosest way to pack spheres gives a density of ∼ 55% (named random loose packing, RLP) while filling all the loose voids results in a maximum density of ∼ 63 − 64% (named random close packing, RCP). While those values seem robustly true, to this date there is no well-accepted physical explanation or theoretical prediction for them. Here we develop a common framework for understanding the random packings of monodisperse hard spheres whose limits can be interpreted as the experimentally observed RLP and RCP. The reason for these limits arises from a statistical picture of jammed states in which the RCP can be interpreted as the ground state of the ensemble of jammed matter with zero compactivity, while the RLP arises in the infinite compactivity limit. We combine an extended statistical mechanics approach 'a la Edwards' (where the role traditionally played by the energy and temperature in thermal systems is substituted by the volume and compactivity) with a constraint on mechanical stability imposed by the isostatic condition. We show how such approaches can bring results that can be compared to experiments and allow for an exploitation of the statistical mechanics framework. The key result is the use of a relation between the local Voronoi volumes of the constituent grains (denoted the volume function) and the number of neighbors in contact that permits to simple combine the two approaches to develop a theory of volume fluctuations in jammed * Corresponding author

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Section: Definition Of Jamming: Isostatic Conjecturementioning

confidence: 99%

Section: Definition Of Jamming: Isostatic Conjecturementioning

confidence: 99%

Jamming II: Edwards’ statistical mechanics of random packings of hard spheres

Wang

Jin

et al. 2011

Physica A: Statistical Mechanics and its Applications

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“…There might remain the possibility that the system selects degenerate case configurations for which at least one of the constitutive equations becomes differential in form, but to achieve this appears to place conditions on the sample history or, leading to contradiction, the present stress tensor. Work on disordered arrays of rigid grains is in progress [18].…”

Section: Grains With Frictionmentioning

confidence: 99%

The stress transmission universality classes of periodic granular arrays

Ball

Grinev

2001

Physica A: Statistical Mechanics and its Applications

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The transmission of stress is analysed for static periodic arrays of rigid grains, with perfect and zero friction. For minimal coordination number (which is sensitive to friction, sphericity and dimensionality), the stress distribution is soluble without reference to the corresponding displacement fields. In non-degenerate cases, the constitutive equations are found to be simple linear in the stress components. The corresponding coefficients depend crucially upon geometrical disorder of the grain contacts.

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“…Isostatic states are characterized by low mean coordination numbers per particle which depend on the dimensionality of the system and on the particles roughness. For rough and infinitely rigid particles in d-dimensional systems (d = 2, 3) this number is z c = d + 1, for smooth infinitely rigid particles of arbitrary shape z c = d(d + 1) [16][17][18][19], and for smooth infinitely rigid spheres z c = 2d. Isostatic packings of particles are marginally rigid and such states have been shown to be easy to approach experimentally [19], making them interesting more than only theoretically.…”

mentioning

confidence: 99%

“…Isostatic packings of particles are marginally rigid and such states have been shown to be easy to approach experimentally [19], making them interesting more than only theoretically. Several empirical [20][21][22][23] and statistical [12,18] models have been proposed for the macroscopic stress field equations in these systems, suggesting a linear coupling between the components of the stress tensor. This has been recently established from first principles in the twodimensional case for systems of infinitely rigid particles [24].…”

mentioning

confidence: 99%

Stress Transmission and Isostatic States of Non-Rigid Particulate Systems

Blumenfeld

Modeling of Soft Matter

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Abstract. The isostaticity theory for stress transmission in macroscopic planar particulate assemblies is extended here to non-rigid particles. It is shown that, provided that the mean coordination number in d dimensions is d + 1, macroscopic systems can be mapped onto equivalent assemblies of perfectly rigid particles that support the same stress field. The error in the stress field that the compliance introduces for finite systems is shown to decay with size as a power law. This leads to the conclusion that the isostatic state is not limited to infinitely rigid particles both in two and in three dimensions, and paves the way to an application of isostaticity theory to more general systems.

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Statistical Mechanics of Stress Transmission in Disordered Granular Arrays

Cited by 144 publications

References 10 publications

Jamming II: Edwards’ statistical mechanics of random packings of hard spheres

Jamming II: Edwards’ statistical mechanics of random packings of hard spheres

The stress transmission universality classes of periodic granular arrays

Stress Transmission and Isostatic States of Non-Rigid Particulate Systems

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