Nonsmoothness, indeterminacy, and friction in two-dimensional arrays of rigid particles

Continuum simulation of the discharge of the granular silo: a validation test for the μ(I) visco-plastic flow law

Using a continuum Navier-Stokes solver with the μ(I) flow law implemented to model the viscous behavior, and the discrete Contact Dynamics algorithm, the discharge of granular silos is simulated in two dimensions from the early stages of the discharge until complete release of the material. In both cases, the Beverloo scaling is recovered. We first do not attempt a quantitative comparison, but focus on the qualitative behavior of velocity and pressure at different locations in the flow. A good agreement for the velocity is obtained in the regions of rapid flows, while areas of slow creep are not entirely captured by the continuum model. The pressure field shows a general good agreement, while bulk deformations are found to be similar in both approaches. The influence of the parameters of the μ(I) flow law is systematically investigated, showing the importance of the dependence on the inertial number I to achieve quantitative agreement between continuum and discrete discharge. However, potential problems involving the systems size, the configuration and "non-local" effects, are suggested. Yet the general ability of the continuum model to reproduce qualitatively the granular behavior is found to be very encouraging.

Friction and the oscillatory motion of granular flows

This contribution reports on numerical simulations of two-dimensional granular flows on erodible beds. The broad aim is to investigate whether simple flows of model granular matter exhibit spontaneous oscillatory motion in generic flow conditions, and in this case, whether the frictional properties of the contacts between grains may affect the existence or the characteristics of this oscillatory motion. The analysis of different series of simulations shows that the flow develops an oscillatory motion with a well-defined frequency which increases like the inverse of the velocity's square root. We show that the oscillation is essentially a surface phenomenon. The amplitude of the oscillation is higher for lower volume fractions and can thus be related to the flow velocity and grains' friction properties. The study of the influence of the periodic geometry of the simulation cell shows no significant effect. These results are discussed in relation to sonic sands.

The influence of grain shape, friction and cohesion on granular compaction dynamics

This article is a review of our recent and new experimental works on granular compaction. The effects of various microscopic parameters on the compaction dynamics are addressed, in particular the influence of the grain shape, the friction and the cohesion between the grains. Two dimensional and three dimensional systems are analysed. And the role of dimensionality will be emphasized. Theoretical and numerical investigations provide additional informations about that phenomenon. Indeed numerical models permit us to study the influence of some parameters not easily accessible experimentally. Our results show that the above mentioned parameters have a deep impact on the compaction dynamics. Anisotropic grains lead to two different compaction regimes separated by a "burst" of the packing fraction. Friction is observed to modify how the grains are arranged in the pile. This is confirmed by numerical simulations. Cohesive forces between particles inhibit compaction and lead to extremely low values of the packing fraction.

Stress transmission in wet granular materials

We analyze stress transmission in wet granular media in the pendular state by means of three-dimensional molecular-dynamics simulations. We show that the tensile action of capillary bonds induces a self-stressed particle network organized in two percolating "phases" of positive and negative particle pressures. Various statistical descriptors of the microstructure and bond force network are used to characterize this partition. Two basic properties emerge: 1) the highest particle pressure is located in the bulk of each phase; 2) the lowest pressure level occurs at the interface between the two phases, involving also the largest connectivity of the particles via tensile and compressive bonds. When a confining pressure is applied, the number of tensile bonds falls off and the negative phase breaks into aggregates and isolated sites.

Quasirigidity: some uniqueness issues

Quasirigidity means that one builds a theory for assemblies of grains under a slowly changing external load by using the deformation of those grains as a small parameter. Is quasirigidity a complete theory for these granular assemblies? Does it provide unique predictions of the assembly's behavior, or must some other process be invoked to decide between several possibilities? We provide evidence that quasirigidity is a complete theory by showing that two possible sources of indeterminacy do not exist for the case of disk-shaped grains. One possible source of indeterminacy arises from zero-frequency modes present in the packing. This problem can be solved by considering the conditions required to obtain force equilibrium. A second possible source of indeterminacy is the necessity to choose the status (sliding or nonsliding) at each contact. We show that only one choice is permitted, if contacts slide only when required by Coulomb friction.

Indeterminacy and the onset of motion in a simple granular packing

We examine the relatively simple problem of a disk placed in a symmetric V-shaped channel, and subjected to gravity and a torque. We obtain analytic predictions of the contact forces and disk motion using two different models. In the first model, the disk is assumed to be perfectly rigid, leading to force indeterminacy. In the second model, the disk is assumed to interact with the walls of the channel via linear springs, leading to a unique solution for the contact forces. The results of these two models are compared. It is shown that there are two possible ways motion can occur--through the appearance of a null eigenvector of the stiffness matrix, or through an instability. When motion occurs through an instability, the first model cannot predict when the disk will rotate; it is necessary to know the undetermined forces in order to predict the motion of the disk. It is also shown how indeterminacy in the first model is linked to memory in the second. The analytical results are also compared with numerical simulations using two different methods, each related to one of the models.

Shearing behavior of polydisperse media

We study the shearing of polydisperse and bidisperse media with a size ratio of 10. Simulations are performed with a two-dimensional shear cell using contact dynamics. With a truncated power law for the polydisperse media we find that they show a stronger dilatancy and greater resistance to shearing than bidisperse mixtures. To model additives used to control viscosity we introduce so-called "point-like particles". Even changing the kinematic behavior very little, the point-like particles reduce the force necessary to maintain a fixed shearing velocity.

Measurement of indeterminacy in packings of perfectly rigid disks

Static packings of perfectly rigid particles with Coulomb friction are investigated theoretically and numerically. The problem of finding the contact forces in such packings is formulated mathematically. Letting the values of the contact forces define a vector in a high-dimensional space enables us to consider the set of all possible contact forces as a region embedded in this same space. It is found that the boundary of the set is connected with the presence of sliding contacts, suggesting that a stable packing should not have more than 2M-3N sliding contacts in two dimensions, where M is the number of contacts and N is the number of particles. These results are used to analyze packings generated in different ways by either molecular dynamics or contact dynamics simulations. The dimension of the set of possible forces and the number of sliding contacts agree with the theoretical expectations. The indeterminacy of each component of the contact forces is found, as well as an estimate for the diameter of the set of possible contact forces. We also show that contacts with high indeterminacy are located on force chains. The question of whether the simulation methods can represent a packing's memory of its formation is addressed.

Shearing of loose granular materials: a statistical mesoscopic model

A two-dimensional lattice model for the formation and evolution of shear bands in granular media is proposed. Each lattice site is assigned a random variable which reflects the local density. At every time step, the strain is localized along a single shear band which is a spanning path on the lattice chosen through an extremum condition. The dynamics consists of randomly changing the "density" of the sites only along the shear band, and then repeating the procedure of locating the extremal path and changing it. Starting from an initially uncorrelated density field, it is found that this dynamics leads to a slow compaction along with a nontrivial patterning of the system, with high-density regions forming which shelter long-lived low-density valleys. Further, as a result of these large density fluctuations, the shear band, which was initially equally likely to be found anywhere on the lattice, gets progressively trapped for longer and longer periods of time. This state is, however, metastable, and the system continues to evolve slowly in a manner reminiscent of glassy dynamics. Several quantities have been studied numerically which support this picture and elucidate the unusual system-size effects involved.

Elastic behavior in contact dynamics of rigid particles

The systematic errors due to the practical implementation of the contact dynamics method for simulation of dense granular media are examined. It is shown that, using the usual iterative solver to simulate a chain of rigid particles, effective elasticity and sound propagation with a finite velocity occur. The characteristics of these phenomena are investigated analytically and numerically in order to assess the limits of applicability of this simulation method and to compare it with soft particle molecular dynamics.

Collective response of an array of rotating particles to fluctuating confining forces

We study the influence of fluctuations of confining forces on the rotation patterns in a dense array of cylinders. Our theoretical studies are motivated by new results from detailed time-resolved experimental measurements. In order to calculate the system's evolution in time at each moment, a molecular-dynamics code adapted to the system is developed. The numerical procedure is tested by a comparison with rigorous predictions derived analytically. The chain's reaction on oscillating confining forces is analyzed numerically for different typical cases. Our theoretical results reproduce the striking features of the experimental data. A quantitative analysis of the experimental data is performed by a computation of their power spectrum and of spatial and temporal correlation functions. From our comparison of the theoretical and experimental results we conclude that the experimental rotation patterns result of random superpositions of different steady-state patterns ("collective random walk").

Contact forces in a granular packing

We present the results of a systematic numerical investigation of force distributions in granular packings. We find that all the main features of force transmission previously established for two-dimensional systems of hard particles hold in three-dimensional systems and for soft particles, too. In particular, the probability distribution of normal forces falls off exponentially for forces above the mean force. For forces below the mean, this distribution is either a decreasing power law when the system is far from static equilibrium, or nearly uniform at static equilibrium, in agreement with recent experiments. Moreover, we show that the forces below the mean do not contribute to the shear stress. The subnetwork of the contacts carrying a force below the mean thus plays a role similar to a fluid surrounding the solid backbone composed of the contacts carrying a force above the mean. We address the issue of the computation of contact forces in a packing at static equilibrium. We introduce a model with no local simplifying force rules, that allows for an exact computation of contact forces for given granular texture and boundary conditions. (c) 1999 American Institute of Physics.

Dynamics of spherical particles on a surface: collision-induced sliding and other effects

We present a model for the motion of hard spherical particles on a two-dimensional surface. The model includes both the interaction between the particles via collisions and the interaction of the particles with the substrate. We analyze in detail the effects of sliding and rolling friction, which are usually overlooked. It is found that the properties of this particulate system are influenced significantly by the substrate-particle interactions. In particular, sliding of the particles relative to the substrate after a collision leads to considerable energy loss for common experimental conditions. The presented results provide a basis that can be used to realistically model the dynamical properties of the system, and provide further insight into density fluctuations and related phenomena of clustering and structure formation.