Stress correlations in granular materials: An entropic formulation

Abstract

We study the response of dry granular materials to external stress using experiment, simulation, and theory. We derive a Ginzburg-Landau functional that enforces mechanical stability and positivity of contact forces. In this framework, the elastic moduli depend only on the applied stress. A combination of this feature and the positivity constraint leads to stress correlations whose shape and magnitude are extremely sensitive to the nature of the applied stress. The predictions from the theory describe the stress correlations for both simulations and experiments semiquantitatively. DOI: 10.1103/PhysRevE.80.060303 PACS number͑s͒: 45.70.Ϫn, 64.60.Ϫi, 83.80.Fg A striking feature of dry granular materials and other athermal systems is that they form force chain networks in response to applied stress, with large forces distributed inhomogeneously into linear chainlike structures ͓1,2͔. A number of experimental studies have visualized and quantified these networks in granular systems using carbon paper ͓3͔ and photoelastic techniques ͓4,5͔. These studies demonstrated that geometrical and mechanical properties of force chain networks are acutely sensitive to preparation procedures, especially near the jamming transition ͓6͔. For example, in isotropically compressed systems, force networks are ramified with only short-ranged spatial correlations of the stress. In contrast, in sheared systems, aligned force chains give rise to longer-ranged stress correlations in the compressive direction.Developing theoretical descriptions for the mechanical properties of granular media is challenging for several important reasons ͓7-10͔: ͑1͒ since tensile stresses are absent in dry granular materials they only remain intact via applied stress, making the limiting zero-stress isostatic state-where the number of degrees of freedom matches the number of constraints ͓11-13͔-problematic; ͑2͒ forces at the microscopic level are indeterminate due to friction and disorder; ͑3͒ granular materials are athermal, so that conventional energy-based statistical approaches are not appropriate; and ͑4͒ near isostaticity we expect fluctuations to be important, both within a single realization of a system and from realization to realization. New methods are needed to bridge the gap between force networks at small length scales and continuum elastoplastic theory at large scales, and to capture the highly sensitive, fluctuating behavior of granular systems near jamming.In this Rapid Communication, we construct a model for stress fluctuations based on grain-scale force and torque balance, positivity of contact forces, and entropy maximization. We then calculate stress correlations and predict differences for systems under isotropic compression versus pure shear. We also perform complementary numerical simulations and experimental studies of jammed granular systems in twodimensions ͑2D͒ subject to isotropic compression and pure shear. The stress correlation functions from theory, simulation, and experiment are in qualitative and in some cases quantitative...

The Statistical Physics of Athermal Materials

“…A field theoretical model [47] of the spatial stress fluctuations was developed based on numerical data for frictionless disks, and this has been used to predict the stress fluctuations of numerical and experimental packings under pure shear [57].…”

Local Anisotropy in Globally Isotropic Granular Packings

“…One of the most promising recent approaches, proposed by Henkes and Chakraborty, is based on a statistical ensemble which equally weights all continuum stress fields which satisfy force balance and positivity constraints [18]. Previous work within this framework focused on pressure-pressure correlations where no interesting lengthscale was detected (in the isotropically prepared state) in either experiments or simulations [4].…”

“…figure 1 below), quantitative characterizations of them have only recently been reported. These recent approaches have focused on topological characterizations [3], 2-point correlations in the local hydrostatic pressure [4], and statistics of spatially averaged forces [5]. These works showed that there are some characteristics in the spatial structure of the stress field which are independent of density, but they left open the important question of whether one can define a characteristic lengthscale associated with the general visual impression one gets when observing the contact force network.…”

Numerical test of the Edwards conjecture shows that all packings are equally probable at jamming

“…The relative fluctuations χ P ≡ N σ 2 (P/ P B ), can be non-zero, and a diverging χ P would then imply a diverging χ Λ . Because of the bounded nature of P [23][24][25], however, χ P can only diverge at the unjamming transition where P B → 0 (see Methods). We find that χ P does diverge ( Fig.…”