In this chapter, we seek to explain the nature of experimentally observed avalanche statistics from a more event-based point of view than in the earlier chapter. In some ways, the difference between the two approaches is akin to that between Monte Carlo and molecular dynamics approaches. In the last chapter (as in Monte Carlo simulations), the dynamics is ‘simulated’ – real grains do not, after all, topple as a result of height thresholds – with a view to matching only the end results of, in this case, avalanche statistics. In this chapter, we try to model an (albeit simplified) version of the real dynamics that occurs when grains avalanche. Interestingly, though both approaches are totally different, the results are robustly similar – we find via both approaches the prediction of a special scale for large avalanches, and, in this chapter, propose a dynamical mechanism which leads to their being unleashed.

Avalanches in a rotating cylinder

Here we describe a model of an experimental situation which forms the basis of many traditional as well as modern experiments; a sandpile in a rotating cylinder. Consider the dynamics of sand in a half-cylinder that is rotating slowly around its axis. Supposing that the sand is uniformly distributed in the direction of the axis, we are dealing with an essentially one-dimensional situation. The driving force arising from rotation continually affects the stability of the sand at all positions in the pile and is therefore distinct from random deposition.

Introduction

The term ‘rapid granular flows’ is short for ‘rapidly sheared granular flows’. Indeed, the paradigm for a fluidised granular system had been for a long time a strongly sheared granular system, as in the classic experiments of Bagnold. In recent years it seems that the main method of fluidisation in research laboratories is vertical vibration, see, e.g., and, at times, horizontal shaking of collections of grains, see, e.g., or even electromagnetic fluidisation. The fluidised state of a granular assembly is recently referred to as a ‘granular gas’, probably following the terminology introduced in. Although most granular gases on earth are ‘man made’, there are naturally occurring granular gases, as part of snow and rock slides are fluidised. In outer space one finds interstellar dust and planetary rings (the latter being composed of ice particles).

In many cases, the grains comprising a granular gas are embedded in a fluid, hence technically they are part of a suspension. However, as noted by Bagnold, when the stress due to the grains sufficiently exceeds the fluid stress (the ratio of the two is known as the Bagnold number) one can ignore the effect of the ambient fluid (clearly, when the air is pumped out of a granular system, as in, e.g, or when one considers celestial granular gases, one need not worry about the ambient fluid). Suspensions will not be considered in this chapter.

Introduction

Many granular and particulate systems have been studied in the literature and there is a wide range of parameters and physical states that they support. Here we confine ourselves to jammed ensembles of perfectly hard particles. There are extensive studies in the literature of suspensions of particles in liquids or gases using various methods, including Stokes or Einstein fluid mechanics and Boltzmann or Enskog gas mechanics. These, however, are not jammed and we therefore discuss them no further. This chapter is not intended as a comprehensive review but rather as an interim report on the work that has been done by us to date.

The simplest material for a general jammed system is that of hard and rough particles, ideally perfectly hard and infinitely rough. To a lesser extent it is also useful to study perfectly hard but smooth particles. The former is easily available in nature, for example sand, salt, etc., and we prefer to focus on this case. Nevertheless, the discussion can be readily extended to systems of particles of finite rigidity, as has been shown recently. In jammed systems particles touch their neighbours at points, which have to be either predicted or observed. At these contact points the particles exert on one another forces that must obey Newton's laws. In general, determination of the structure and the forces requires prior knowledge about the history of formation of the jammed system.

In this chapter we extend the model of the previous chapter in two different directions; the first and most important aim is to introduce long-range interactions with a view to obtaining properly glassy behaviour, and the second is to explore the role of grain shapes in granular compaction.

The model is based on the following picture. Consider a box of sand in the presence of gravity in the jamming limit. Adopting, as in the previous chapter, a lattice-based viewpoint, we visualise this box as being constituted of rows and columns of grains. When this box is shaken along the direction of gravity, the predominant dynamical response of the sandbox is known to be in the vertical direction – recall the on- and off-lattice computer simulation results presented in earlier chapters which show that correlations in the transverse plane (i.e. along rows of grains) are negligible compared to these. Another important aspect of the jamming limit is that grain-sized voids are typically absent. The dominant dynamical mechanism in this regime is therefore grain reorientation within each column to minimise the size of the partial voids that persist. We thus focus on a column model of grains in the jamming limit.

We now extend the concept of disorder to include the effect of grain shapes. Each ordered grain occupies one unit of space, while each disordered grain occupies 1 + ∈ units of space, with ∈ a measure of the partial void trapped by misorientation.

This book was commissioned seven years ago, in Oxford, where I was an EPSRC Visiting Fellow at my alma mater, by Cambridge University Press. Its completion in Cambridge, Massachusetts, where I am a Radcliffe Fellow at Harvard University, owes a lot to the tranquillity of my initial and final conditions of work, where I am away from the regular pressures of my permanent position in India.

In the seven years since its conception, many things took priority over its writing, including, to a large extent, the research that has been presented in it. I feel this delay has been largely beneficial. In 1999, many of the developments that now seem obvious, that have now allowed granular media to be the focus of many conferences or multiple sessions at large meetings, were yet to happen. In particular, they changed the conception of the book itself, in my mind.

My initial idea, when I was approached to write a monograph on granular media, was to focus only on those areas where I had some understanding, or where I had myself been active. At that time, it was the so-called statistical mechanics of granular media, pioneered by Edwards, that held centre stage; people like myself were trying to make inroads into the dynamics of these fascinating systems. We focused in particular on what is now known as the jamming limit, which I thought even at the time had fascinating analogies to glasses.

A microscopic description which focuses on the essential discreteness of grains is a fundamental part of understanding many fascinating features of a granular medium. Its response to vibration can be understood in terms of the competition between individual-particle and cooperative dynamical mechanisms intrinsic to the grains in a disordered assembly. We present below computer simulation results for some features of granular structure which depend only weakly on details of grain size or material, e.g. the finite range for the packing fraction or shaking-induced size segregation.

Details of simulation algorithm

In this section, we describe the details of the simulations first reported in. Simulations are performed for a bed of monodisperse hard spheres above a hard base at z = 0. The granular bed is periodic, with a repeat distance of L sphere diameters in two perpendicular directions x and y in the plane. Each primary simulation cell contains N spheres. A unidirectional gravitational field acts downwards, i.e. along the negative z-direction. Initially, spheres are placed in the cell using a sequential random close-packing procedure; the packing is then subjected to a series of nonsequential N-particle reorganisations. Each reorganisation is performed in three parts: first, a vertical expansion or dilation, second, a Monte Carlo consolidation, and finally a nonsequential close-packing procedure. We call each full reorganisation a shake cycle or, simply, a shake. The duration of model shaking processes as well as the lengths of other time intervals are measured in units of the shake cycle.

Introduction

Vibrating sand results in very varied dynamics, ranging from glassy to fluidised. In much of this book, we will focus on the former, while a subsequent chapter will contain a review of the latter regime. Of course, it is important to have a theoretical understanding of how one regime gives way to the other; it is for this reason that the model discussed in this chapter is a simple model of a vibrated sandbox, which interpolates between the glassy and fluidised regimes through an extremely interesting intermediate regime, whose properties are not fully understood to date, and are the subject of current investigations.

The model is based on the generalisation of an earlier cellular automaton (CA) model of an avalanching sandpile. This version of the model contains only near-neighbour interactions, with grains being in one of two orientational states. It shows both fast and slow dynamics in the appropriate regimes and, in its simplest form, reduces to an exactly solvable model in the frozen or jammed regime. Of course, in order to replicate the truly glassy behaviour of the jammed regime, one needs to introduce true long-range interactions. This will be done in the next chapter, where in particular the effect of grain shapes will be probed using an extension of this model.

When grains are deposited on a sandpile, avalanches result. These have much in common with many other varieties of avalanche; for example, snow or rocks, or even the stress releases that result in earthquakes. The unifying phenomenon in all these cases is that of a threshold instability: an overburden builds up, typically that due to surface roughening, to the point where this threshold is crossed, and grains are released in an avalanche. Avalanches can be classed in two main categories; those that do not have intrinsic time or length scales, and those that do. Avalanches relevant to granular media belong to the second category, and we shall discuss their characteristics in depth. We will, however summarise some of the known characteristics of the first category, referring readers who are interested in more details to ref. on the subject.

Avalanches type I – SOC

Bak, Tang and Wiesenfeld, in their now famous theory of self-organised criticality (SOC), suggested that extended systems were marginally stable, such that the slightest overburdening would cause avalanching; a sandpile at its so-called ‘critical’ angle of repose was held to be paradigmatic of this. Although this turned out to be, in retrospect, the wrong paradigm, the explorations that surrounded it in fact greatly enriched the physics of granular avalanches. We touch briefly on the important features of SOC here.

Sand has many avatars – it can behave as a solid, liquid or gas, depending on external circumstances. This multiple identity is one of several reasons why the computer simulation of dry granular materials is difficult. Sand in the solid-like state responds to external stimuli on a very different timescale to sand in its liquidlike avatar – in contrast to most efficient computer simulation methods, which are typically tuned to one particular timescale such as a collision or relaxation time. Other features of sand which are difficult to simulate efficiently include complex, dissipative interparticle and particle–wall interactions, typically irregular grain shapes and strong hysteretic effects. Furthermore, the athermal nature of sand means that grains do not randomly sample all possible states ergodically – as a result, appropriate statistical averages can only be obtained by repeated (computationally demanding) simulations of a granular system.

For normal dry powders, interstitial fluid plays only a minor role – apart from exceptional cases when, say, there are small liquid pools at particle contacts which could seriously alter the pairwise nature of grain interactions. This is a clear distinction between granular systems and dense suspensions – for the former, interparticle interactions are restricted to short-ranged contact forces. In practice, the methods developed for granular simulations are quite similar to classical methods used to simulate simple liquids.

Introduction

Some general remarks

The two previous chapters have dealt, in different guises, with the post-avalanche smoothing of a sandpile which is expected to happen in nature. It is clear what happens physically: an avalanche provides a means of shaving off roughness from the surface of a sandpile by transferring grains from bumps to available voids, and thus leaves in its wake a smoother surface. However, surprisingly little research has been done on this phenomenon so far, despite its ubiquity in nature, ranging from snow to rock avalanches.

In particular, what has not attracted enough attention in the literature is the qualitative difference between the situations which obtain when sandpiles exhibit intermittent and continuous avalanches. In this chapter we examine both the latter situations, via coupled continuum equations of sandpile surfaces. These were originally envisaged as the local version of coupled equations that had been written down using global variables in; subsequently, many versions were introduced in the literature to model different situations. The use of these equations has also since been diversified into many areas, including ripple formation and the propagation of sand dunes, about which we will have something to say at the end of this chapter.

In order to discuss this, we introduce first the notion that granular dynamics is well described by the competition between the dynamics of grains moving independently of each other and that of their collective motion within clusters.