Predictability and granular materials

Abstract

Granular materials present a number of challenges to predictability. The classical description of a dense granular material is based on Coulomb friction. For a static array of grains, the Coulomb friction forces are typically underdetermined. If we are to make useful statements about such arrays, we must develop new approaches, including the development of statistical descriptions. Granular materials also show large fluctuations in the local forces. These fluctuations are quite sensitive to small perturbations in the packing geometry of the grains. In the past, they have typically been ignored. However, recent experiments and models are beginning to shed new light on their characteristics. This article briefly reviews some of this new work, and in particular presents experimental results characterizing fluctuations and the role of friction in granular materials.

Introduction

Granular materials surround us everywhere in daily life, and present a variety of fascinating dynamic and static phenomena [1], [2], [3], [4], [5], [6], [7]. Common examples of these materials include sand, soils, coal, grains, cereal, ores, pharmaceutical powders, pills, even dog food. Granular materials can behave like solids – a heap is stable up to some angle of repose – but they can also flow, at least heuristically, like fluids. For instance, once a heap is raised past the maximum stability angle, avalanches occur. And, shaken granular materials can convect, exhibit traveling waves, and standing wave patterns that bear a remarkable resemblance to patterns seen in Rayleigh–Bénard convection.

However, our ability to predict the state of a granular material lags far behind our ability to predict the dynamics of ordinary fluids. There are a number of reasons for this, including unpredictability associated with static friction and strong fluctuational effects. Here, we will try to give some insight into the difficulties, and we will discuss recent work that is providing new insights that may resolve some of these difficulties.

At the most fundamental level, the interaction forces between grains are relatively simple ones. These include hard body interactions (to a first approximation), friction, and inelasticity, as characterized by a coefficient of restitution, e<1. One might imagine that a granular material resembles a hard sphere gas, with the additional complication of dissipation.

Although these forces seem simple, they lead to indeterminancy and to complex dynamics that cannot be treated in the framework of ordinary statistical mechanics [8], [9], [10]. Due to the dissipative nature of the interactions, even an initially energetic collection of grains quickly coalesces into a dense compact state, so that the concept of ordinary temperature no longer applies, although it is still possible to consider a “granular temperature” defined in terms of the fluctuation components of the kinetic energy [11]. Such a compact state is typically highly inhomogeneous with regard to the forces experienced by individual grains. These forces are carried preferentially on a complex spatial network known as stress chains. Fig. 1 shows an experimental realization of these chains for a 2D granular system that is discussed in more detail below. In this figure, the “grains” (in this case disks) that are carrying the largest force appear the brightest. It is visually clear that this system is not homogeneous as far as force is concerned. A key point is that if grains are displaced by a very small amount, the stress chains will change dramatically [12]. Thus, even moderate uncertainty in the grain configurations is sufficient to lead to large uncertainty in the force configuration. If the systems is driven, for instance by slow shearing, the stress chains evolve in a complex random way [13].

Classical Coulomb friction is an important source of intrinsic unpredictability for granular systems [14], [15]. Some simple examples give insight into the origin of this indeterminancy associated with Coulomb friction. For example, suppose that a block rests on an inclined surface at an angle θ to the horizontal, with an additional unknown force applied along the incline. Coulomb friction is not able to predict the value of the unknown force, since there is a range of forces which would allow the block to remain at rest on the surface. An example that is particularly germane to granular systems consists of asking when a frictional system on an inclined surface with slope $\text{tan}\text{θ}$ will just begin to slip. For one particle, a block, the answer is easy: Coulomb friction is just able to sustain the block at rest for $\text{tan}\text{θ=μ}{}_{\mathrm{<mtext>s</mtext>}}$, where μ_s is the coefficient of static friction. In this case, there are two unknown forces, one normal and one tangential to the contact surfaces, and two constraints if the block does not slide. (The process is more complex if we consider that the block might also rotate.) Now consider two disks in mutual contact resting on an inclined plane. We imagine that these disks were placed on the plane and that θ was gradually increased just to the point of slipping. As in the first example, the forces are determined because there are now six unknown forces and torques that are matched with the six force and torque balance conditions on the two disks. However, if we augment the system to N=3 particles, there are now 3N=9 static balance conditions, but 10 unknown forces at the 5 contacts. Thus, there is a continuous range of forces that satisfies the static balance conditions. If more particles are added, the situation is typically worse. Each new grain implies two new contacts (in 2D) hence four unknown forces. Force and torque balance applied to this disk provides only three additional constraints.

In the past, it has often been assumed that fluctuations in static or slowly evolving systems were small on the scale of the system of interest. This may be true on very large scales such as a large soil embankment. However, it is less clear that this must be so in other commercially important systems. For instance, if large chunks of coal, with a typical size of 10 cm are in a silo with a diameter of 20 m, there are then only 200 particles across the diameter, which is not much larger than chain lengths seen in some experiments.

In particular, some older experiments [16], [17], [18] as well as several experiments in the past few years indicated that fluctuations can be significant on scales of tens to hundreds of grains. In one of these, Liu and Nagel [12] measured the propagation of sound through an array of grains. Sound was injected through a loudspeaker and detected by a microphone in the material. The frequency dependence of the detected sound was complicated but reproducible as long as the arrangement of the particles was undisturbed. However, a minute perturbation of the particle arrangement was sufficient to significantly change the frequency response.

In another set of experiments, Baxter et al. [19], [20] measured the acoustic transmission at the side of a small hopper during outflow of material. Here, the goal was to test for a characteristic time scale, τ, predicted on the basis of continuum models [21]. However, the detected signal was very noisy, and a power spectrum from this signal was roughly a power law in the frequency, P(ω)∝ω^−α. For various conditions, the exponent α lay in the range 1.3≤α≤2.3. By contrast, the expected response would have involved a broadened peak in the vicinity of a frequency ∼τ⁻¹. More recently, experiments on shearing [22], discussed here, on compression [23], and on forces in bead packs near boundaries [24] have also demonstrated strong fluctuations.

Until recently, models for slow flows in granular materials have neglected fluctuations. However, there have been several new models [25], [26], [27], [28], [29], [30] that consider fluctuations. These models fall into two classes. The first is based on lattice models, where grains are placed (with the possible exception of vacancies [28]) on a regular lattice. The unknown/random nature of the contacts is introduced by “propagating” forces through the lattice so that the forces and torques on a particle are provided from neighbors in a random way, but such that some or all of the forces and torques are in balance. Roughly speaking, these models predict exponential distributions of the force on a single grain, at least for large forces, unless there are sufficiently many defects in the lattice. For instance, in the q-model of Coppersmith et al. the distribution of forces is given by ρ_q∝F^N−1exp(−F/F_o), where N is the dimension of the system, and F_o is determined by the applied force. The other class of model is contact dynamics [30] in which the frictional uncertainty is resolved by allowing the system to relax dynamically. Again, this model predicts essentially exponential distributions of forces at large F. Such distributions have been observed in recent experiments [24], [25].