On entropic characterization of granular materials

Blumenfeld, Raphaël

doi:10.1142/9789812771995_0003

Cited by 12 publications

(15 citation statements)

References 15 publications

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“…These constraints lead to a robust conservation law involving the force-moment tensor [4,15,46] of the granular packing,Σ, defined in Eq. 2, which is related to the Cauchy stress tensor [43]σ:…”

Section: Conservation Principlesmentioning

confidence: 99%

The Statistical Physics of Athermal Materials

Henkes

Daniels

et al. 2015

Annu. Rev. Condens. Matter Phys.

139

140

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At the core of equilibrium statistical mechanics lies the notion of statistical ensembles: a collection of microstates, each occurring with a given a priori probability that depends only on a few macroscopic parameters such as temperature, pressure, volume, and energy. In this review article, we discuss recent advances in establishing statistical ensembles for athermal materials. The broad class of granular and particulate materials is immune from the effects of thermal fluctuations because the constituents are macroscopic. In addition, interactions between grains are frictional and dissipative, which invalidates the fundamental postulates of equilibrium statistical mechanics. However, granular materials exhibit distributions of microscopic quantities that are reproducible and often depend on only a few macroscopic parameters. We explore the history of statistical ensemble ideas in the context of granular materials, clarify the nature of such ensembles and their foundational principles, highlight advances in testing key ideas, and discuss applications of ensembles to analyze the collective behavior of granular materials.

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Section: Conservation Principlesmentioning

confidence: 99%

The Statistical Physics of Athermal Materials

Henkes

Daniels

et al. 2015

Annu. Rev. Condens. Matter Phys.

139

140

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“…Concomitant with a given network geometry there is a distribution of contact forces or stresses in the particulate medium. This means that the V-ensemble must be supplemented by the force or stress ensemble (F-ensemble) determined by the contact forces [16,43,44] for a full characterization of jammed matter.…”

Section: Understanding Jammingmentioning

confidence: 99%

“…with the stress σ ij = 1/(2V ) c f c i r c j describing the F-ensemble [16,43,44], where f c i , r c i are the force and position at contact c. For simplicity only the isotropic case is described. Thus, only the pressure σ = σ ii /3 is necessary to describe force fluctuations.…”

Section: Statistical Mechanics Of Jammed Mattermentioning

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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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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“…In three-dimensional systems of mean coordination number z ) 4 (the marginal rigidity value for grains with high friction coefficients), the number of degrees of freedom per grain is much smaller than the number of quadrons. 8,10 This already suggests that simply summing in the volume function over grains is inaccurate. Indeed, a recent analysis has shown that using the quadrons as the basic "quasiparticles" of the statistical mechanical formalism leads to a better insight than using grains into the role of grain characteristics in forming the structure of a granular system.…”

Section: The Distribution Function Exp (F -H/kt) Is Then Extended To mentioning

confidence: 99%

On Granular Stress Statistics: Compactivity, Angoricity, and Some Open Issues

Blumenfeld

Edwards

2009

J. Phys. Chem. B

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We discuss the microstates of compressed granular matter in terms of two independent ensembles: one of volumes and another of boundary force moments. The former has been described in the literature and gives rise to the concept of compactivity: a scalar quantity that is the analogue of temperature in thermal systems. The latter ensemble gives rise to another analogue of the temperature: an angoricity tensor. We discuss averages under either of the ensembles and their relevance to experimental measurements. We also chart the transition from the microcanoncial to a canonical description for granular materials and show that one consequence of the traditional treatment is that the well-known exponential distribution of forces in granular systems subject to external forces is an immediate consequence of the canonical distribution, just as in the microcanonical description E ) H leads to exp (-H/kT). We also put this conclusion in the context of observations of nonexponential forms of decay. We then present a Boltzmann-equation and Fokker-Planck approaches to the problem of diffusion in dense granular systems. Our approach allows us to derive, under simplifying assumptions, an explicit relation between the diffusion constant and the value of the hitherto elusive compactivity. We follow with a discussion of several unresolved issues. One of these issues is that the lack of ergodicity prevents convenient translation between time and ensemble averages, and the problem is illustrated in the context of diffusion. Another issue is that it is unclear how to make use in the statistical formalism the emerging ability to exactly predict stress fields for given structures of granular systems.

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On entropic characterization of granular materials

Cited by 12 publications

References 15 publications

The Statistical Physics of Athermal Materials

The Statistical Physics of Athermal Materials

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

On Granular Stress Statistics: Compactivity, Angoricity, and Some Open Issues

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