Packing geometry and statistics of force networks in granular media

Arches and contact forces in a granular pile

Assemblies of granular particles mechanically stable under their own weight contain arches. These are structural units identified as sets of mutually stable grains. It is generally assumed that these arches shield the weight above them and should bear most of the stress in the system. We test such hypothesis by studying the stress born by in-arch and out-of-arch grains. We show that, indeed, particles in arches withstand larger stresses. In particular, the isotropic stress tends to be larger for in-arch grains whereas the anisotropic component is marginally distinguishable between the two types of particles. The contact force distributions demonstrate that an exponential tail (compatible with the maximization of entropy under no extra constraints) is followed only by the out-of-arch contacts. In-arch contacts seem to be compatible with a Gaussian distribution consistent with a recently introduced approach that takes into account constraints imposed by the local force balance on grains.

Amplitude dependence of elasticity for the assembly of SiO2 powders under shear oscillation strain

We experimentally report the elastic response of an assembly of SiO2 powders under a shear oscillation strain by a modified inverted torsion pendulum with a Couette-like setup. We find that the shear restoring force displayed by SiO2 assembly is of strong amplitude dependence, which is ascribed to the dynamic behavior of the stress network inside and could be used to classify the states of the SiO2 assembly, i.e., elastic solid, plastic solid, viscoelastic solid, and viscous fluid. The underlying mechanisms for different states of the SiO2 assembly are tentatively proposed.

Tail of the contact force distribution in static granular materials

We numerically study the distribution P(f) of contact forces in frictionless bead packs, by averaging over the ensemble of all possible force network configurations. We resort to umbrella sampling to resolve the asymptotic decay of P(f) for large f , and determine P(f) down to values of order 10{-45} for ordered and disordered systems in two (2D) and three dimensions (3D). Our findings unambiguously show that, in the ensemble approach, the force distributions decay much faster than exponentially: P(f) approximately exp(-cf(alpha)}) , with alpha approximately 2.0 for 2D systems, and alpha approximately 1.7 for 3D systems.

Granular packs under vertical tapping: structure evolution, grain motion, and dynamical heterogeneities

The compaction dynamics of a granular media subject to a sequence of vertical taps made of fluid pulses is investigated via molecular dynamics simulations. Our study focuses on three different levels: macroscopic (volume fraction), mesoscopic (Voronoï volumes, force distributions), and microscopic (grain displacements). We show that the compaction process has many characteristics which are reminiscent of the slow dynamics of glass forming systems, as previously suggested. For instance, the mean volume fraction slowly increases in time and approaches a stationary value following a stretched exponential law, and the associated compaction time diverges as the tapping intensity decreases. The study of microscopic quantities also put in evidence the existence of analogies with the dynamics of glass formers, as the existence of dynamical heterogeneities and spatially correlated motion of grains; however, it also shows that there are important qualitative differences, as, for instance, in the role of the cage effect. Correlations between geometry and dynamics of the system at the grain level are put in evidence by comparing a particle Voronoï volume and its displacement in a single tap.

Scale separation in granular packings: stress plateaus and fluctuations

It is demonstrated, by numerical simulations of a 2D assembly of polydisperse disks, that there exists a range (plateau) of coarse-graining scales for which the stress tensor field in a granular solid is nearly resolution independent, thereby enabling an "objective" definition of this field. Expectedly, it is not the mere size of the system but the (related) magnitudes of the gradients that determine the widths of the plateaus. Ensemble averaging (even over "small" ensembles) extends the widths of the plateaus to subparticle scales. The fluctuations within the ensemble are studied as well. Both the response to homogeneous forcing and to an external compressive localized load (and gravity) are studied. Implications to small solid systems and constitutive relations are briefly discussed.

Elegance of disordered granular packings: a validation of Edward's hypothesis

We have found a way to analyze Edwards' density of states for static granular packings in the special case of round, rigid, frictionless grains assuming a constant coordination number. It obtains the most entropic density of single grain states, which predicts several observables including the distribution of contact forces. We compare these results against empirical data obtained in dynamic simulations of granular packings. The agreement is quite good, helping validate the use of statistical mechanics methods in granular physics. The differences between theory and empirics are mainly related to the coordination number, and when the empirical data are sorted by that number we obtain several insights that suggest an underlying elegance in the density of states.

Evolution of distributions and spatial correlations of single-particle forces and stresses during compression of ductile granular materials

Uniaxial compression of disordered packings of millimeter-sized ductile particles formed from microcrystalline cellulose is investigated experimentally, at compression pressures in the vicinity of the minimum pressure required to form a coherent compact. Distributions of normal forces and stresses exerted by individual particles on a confining wall are determined. Spatial force and stress correlations are investigated. The distribution of normal forces is found to narrow with increasing pressure, but no indication of a crossover to a Gaussian decay at high forces is observed. The distribution of normal stresses is found to be considerably more Gaussian in shape for all pressures investigated. This finding may be interpreted as resulting from a positive correlation between the area corresponding to each particle and the force it experienced during compression. Spatial force and stress correlations are observed for distances smaller than three particle diameters. The spatial stress correlations indicate that the mode of stress transmission changes when the compression pressure exceeds the minimum pressure required to form a coherent compact.

Ensemble theory for force networks in hyperstatic granular matter

An ensemble approach for force networks in static granular packings is developed. The framework is based on the separation of packing and force scales, together with an a priori flat measure in the force phase space under the constraints that the contact forces are repulsive and balance on every particle. In this paper we will give a general formulation of this force network ensemble, and derive the general expression for the force distribution P(f). For small regular packings these probability densities are obtained in closed form, while for larger packings we present a systematic numerical analysis. Since technically the problem can be written as a noninvertible matrix problem (where the matrix is determined by the contact geometry), we study what happens if we perturb the packing matrix or replace it by a random matrix. The resulting P(f) 's differ significantly from those of normal packings, which touches upon the deep question of how network statistics is related to the underlying network structure. Overall, the ensemble formulation opens up a different perspective on force networks that is analytically accessible, and which may find applications beyond granular matter.

Granular contact force density of states and entropy in a modified Edwards ensemble

A method has been found to analyze Edwards' granular contact force probability functional for a special case. As a result, the granular contact force probability density functions are obtained from first principles for this case. The results are in excellent agreement with the experimental and simulation data. The derivation assumes Edwards' flat measure--a density of states (DOS) that is uniform within the metastable regions of phase space. The enabling assumption, supported by physical arguments and empirical evidence, is that correlating information is not significantly recursive through loops in the packing. Maximizing a state-counting entropy results in a transport equation that can be solved numerically. For the present this has been done using the "mean-structure approximation," projecting the DOS across all angular coordinates to more clearly identify its predominant nonuniformities. These features are (1) the grain factor Psi related to grain stability and strong correlation between the contact forces on the same grain and (2) the structure factor Upsilon related to Newton's third law and strong correlation between neighboring grains.