This chapter is devoted to the properties of granular media immersed in a liquid. Mixtures of grains and fluids are used in many industrial processes, for example in civil engineering projects with concrete. In environmental problems, the coupling between a granular soil and water controls the soil stability and is important in the understanding of many natural disasters such as landslides and mud flows. The physics of two-phase flows involving grains and liquid is a vast area of research. In this book dedicated to granular media, we will restrict ourselves to the high-concentration regime, for which the grains are in contact and interact primarily through contact interactions. Our goal is to illustrate through examples how the concepts developed in the previous chapters on dry granular materials are modified in the presence of an interstitial fluid. In the first part (Section 7.1) we introduce two-phase-flow equations, the relevant theoretical framework for studying immersed granular media. In this approach, the granular medium and the liquid are described as two interpenetrating continuum media, which interact. In the second part, the use of two-phase-flow equations is illustrated in simple examples for which the granular skeleton is static (Section 7.2) and for which it is slightly deformed (Section 7.3). Finally, the influence of the interstitial fluid on the rheology of sheared granular media is discussed in Section 7.4.

A granular medium without external perturbation can be considered primarily as a solid. A pile of sand, the soil on which a house is built and a silo filled with grains are examples of situations in which the grains do not move. The material supports external forces without flowing, just like a solid. This chapter is dedicated to the statics of granular media: how are grains organized in a packing? Howare the forces distributed among the particles to ensure the mechanical balance of the pile? Is it possible to describe the granular medium as a continuum and to define stresses? The chapter starts with the description of the geometrical properties of packings by introducing the concepts of volume fraction and compaction of a granular medium (Section 3.1). Then the problem of the mechanical equilibrium of a sand pile is addressed, and the statistical properties of the inter-particle force distribution are presented (Section 3.2). Following the analysis at the microscopic level, the possibility of a continuum description is discussed in Section 3.3. The concept of stresses in granular media and the relation between inter-particle forces and macroscopic stresses are presented. Simple cases for which the stress distribution can be calculated are studied (Section 3.4). Finally, the issue of elasticity and sound propagation in a granular packing is discussed in Section 3.5.

Granular packings

Packings of grains have been studied since antiquity. Mathematicians, physicists and engineers, in a quest to optimize the storage of granular matter, have been interested in these issues. This section is an introduction to the concepts which are useful for characterizing packings of grains. For more details, the reader is referred to more specialized works, such as the book by Cumberland and Crawford (1987).

Sand dunes, screes, river deltas … Most geological structures at the Earth's surface involve granular materials. This final chapter proposes an application of the various properties of granular media developed throughout this book to geomorphology, i.e. to the study of the nature and origin of landforms, particularly regarding the formative processes of weathering and erosion that occur in the atmosphere and hydrosphere. These processes continually shape the Earth's surface, and generate the sediments that circulate in the rock cycle. Landforms are the result of the interactions among the geosphere, atmosphere and hydrosphere.

On the one hand, we will propose a description of sedimentary landscape elements that is based on natural history. On the other hand, we will analyse various sedimentary structures in terms of physical mechanisms. When possible, we will detail the associated scaling laws, which allow one to reproduce geophysical phenomena in the lab, at small scale, in a controlled way. In particular, we have established in the previous chapter a description of erosion and sediment transport in a homogeneous flow. We will now use these results by considering the coupling between relief and transport. We will pay special attention to linear instabilities, which can explain the emergence and organization of geological objects. We will study successively gravity-driven flows (Section 9.1), ripples and dunes (Section 9.2), coastal instability (Section 9.3) and, finally, rivers (Section 9.4).

The behaviour of a granular material is closely related to the nature of the interactions between grains. In this chapter, we focus on these forces at the grain level. We first discuss solid contact, which is dominant in the case of dry granular media made of macroscopic particles (Section 2.1). The basics of Hertz elastic contact, solid friction and the rules of inelastic collisions between solid particles are given. We then discuss other kinds of interaction between grains such as electrostatic and adhesive forces, capillary cohesion and solid bridges (Section 2.2). The last part of the chapter gives a brief overview of the hydrodynamic forces produced on a particle immersed in a fluid (Section 2.3). Our aim in this chapter is to provide some background in contact physics and hydrodynamics that will be useful for our study of granular media. More detailed treatments can be found in the classical books given in the text.

Solid contact forces

The contact force between two dry grains is usually split into a normal force and a tangential force. The physical origin of these forces at the microscopic level involves many phenomena, such as surface roughness, local mechanical properties (elasticity, plasticity, viscoelasticity) and physical and chemical properties (the presence of electrical charge, oxidation, temperature, the presence of lubricant film). In the following, we will not consider these microscopic features in detail but rather focus on the macroscopic laws of solid contact. At the macroscopic level, these laws are dominated by elastic repulsion (Hertz contact) and solid friction (Coulomb’s law).

Sand, gravel, rice, sugar … Granular matter is familiar and abounds around us. However, the physics of granular media is still poorly understood and continues to fascinate scientists and other people, more than three centuries after the work of Coulomb on slope stability. A pile of grains actually exhibits a great variety of behaviours with unique properties. Strong enough to support the weight of a building, grains can also easily flow like water in an hourglass or be transported by wind to sculpt dunes and deserts. For a long time, the study of granular materials has remained the preserve of engineers and geologists. Therefore, important concepts arose from the need to build structures on solid ground, store grains in a silo or predict the history of a sediment. More recently, the study of granular media has entered the field of physics, at the crossroads of statistical physics, mechanics and soft-matter physics. The combination of results from laboratory experiments on model materials, discrete numerical simulations and theoretical approaches from other fields has enriched and renewed our understanding of granular materials.

This book has been written in this context. Our goal is to provide an introduction to the physics of granular media that takes into account recent advances in this field, while describing the basic concepts and tools useful in many industrial and geophysical applications. This book is intended primarily for students, researchers and engineers willing to become familiar with the fundamental properties of granular matter.

The previous chapters show that a granular medium can behave as a solid. In the opposite limit, when grains are strongly shaken in a box, particles are agitated and interact mainly by binary collisions. The medium is then more similar to a gas. This chapter is devoted to this ‘gaseous’ regime of granular matter. The analogy between agitated grains and molecules in a gas was the basis for the development of kinetic theories of granular media that provide constitutive equations for rapid and diluted granular flows. In this chapter, we first introduce the notion of granular temperature and briefly discuss the analogies and differences between a granular gas and a classical molecular gas (Section 5.1). We then present a first approach to the kinetic theory, which gives insight into the physical origin of the transport coefficients (Section 5.2). A more formal presentation of the kinetic theory that is based on the Boltzmann equation for inelastic gases is given in the next section (Section 5.3). We then apply the hydrodynamic equations of the kinetic theory to various situations that highlight the role of inelastic collisions in the behaviour of granular gases (Section 5.4). Finally, some limits of the kinetic theory are discussed, in particular concerning dense media (Section 5.5).

Analogies and differences with a molecular gas

Figure 5.1 gives two examples of granular materials in a ‘gaseous’ state. The first one is obtained by shaking vertically a box containing beads. The second example shows steel beads flowing down a steeply inclined plane under the action of gravity. In both cases, the medium looks like a gas. Particles are strongly agitated and move independently, except when collisions occur.

Most granular flows encountered in nature and industry lie between the quasistatic and gaseous regimes seen in the previous chapters. In this intermediate ‘liquid’ regime, particles remain closely packed and interact both by collision and through long-lived contacts. Understanding and modelling the flow of dense granular media is challenging and many questions remain to be answered, despite important advances having been made during the last decade. In this chapter, we first present the basic features of dense granular flows (Section 6.1), before focusing on the rheology of this peculiar liquid (Section 6.2). A phenomenological constitutive law that is based on dimensional analysis is presented, in which the medium is described as a viscoplastic fluid with a frictional behaviour. The success and limitations of this approach are then discussed, in particular close to the solid–liquid transition where complex collective behaviours are observed. The second part of the chapter presents a hydrodynamic description of dense flows that is valid for a shallow layer flowing under gravity (the Saint-Venant equations) (Section 6.3). This depth-averaged approach enables one to gather the complex rheology into a single basal friction term and is commonly used in geophysics to describe rock avalanches and landslides. We close the chapter with a presentation of the phenomenon of size segregation, which occurs when the medium is composed of particles of different sizes. The consequences of segregation for polydisperse granular flows in various configurations are presented (Section 6.4).

In this chapter, we study erosion and sediment transport from the point of view of the physics of granular media. Situations involving erosion, transport and deposition of particles subjected to fluid flow cover a wide range of applications, from the transport of grains in a pipe to the evolution of landscape on geological scales. We focus in this chapter on the study of erosion and transport of natural sediments under the influence of a water flow (streaming, fluvial erosion, tides, waves and glaciers) or of the wind (dunes, sand invasion, desertification). Besides, we wish to describe sediment transport in the perspective of understanding the geological phenomena that will be discussed in the next chapter. The goal is to propose a description of these phenomena, to model them through basic equations and to explain the dynamical mechanisms at the scale of grains. To do this, we will use the concepts introduced throughout this book.

We begin by briefly outlining the characteristics of the different modes of transport and the most important concepts which allow one to characterize erosion and sediment transport (Section 8.1). We then discuss the nature of the threshold above which a flow may entrain grains into motion (Section 8.2), before presenting the formalism used to describe erosion and transport starting from conservation laws (Section 8.3). Once we have introduced the concepts of saturated transport and saturation transient, we apply them to the different modes of transport: bed load (Section 8.4), aeolian transport (saltation and reptation) (Section 8.5) and turbulent suspension (Section 8.6).

In the previous chapter, we discussed the statics and the elasticity of granular media, when deformations are small and reversible. In this chapter, we address the plasticity of granular media, i.e. irreversible deformations occurring beyond the elastic regime. The two issues associated with plasticity are the following: what is the maximum stress level a granular medium can sustain before being irreversibly deformed and how does the deformation take place beyond the threshold? These questions are covered by soil mechanics, which aims to predict and understand soil stability in nature or during construction in civil engineering. The approaches are mainly based on macroscopic and phenomenological models derived from continuum mechanics. More recently, physicists have been interested in the plasticity of disordered materials, focusing on the microscopic features and trying to understand how rearrangements occur at the grain scale. The link with the continuum models proposed in soil mechanics is still a challenge. In this chapter we will focus on simple macroscopic continuum models, and will only briefly discuss the microscopic properties in a box. The first section (Section 4.1) is dedicated to the phenomenology of plasticity. Several configurations that are used for studying the deformation of a granular medium are described. Section 4.2 is dedicated to the plane shear configuration, for which all the properties of the plasticity of granular media can be introduced using scalar quantities. Tensors, which are necessary to model plasticity, are introduced in Section 4.3. The Mohr–Coulomb model is described and Mohr’s circle used to represent the stress tensor is introduced. In Sections 4.4 and 4.5, we discuss briefly more complex models and unresolved questions. Finally, the plasticity of cohesive materials is presented in Section 4.6.

Definition and examples of granular media

From sand to cereals, from rock avalanches to interplanetary aggregates like Saturn's rings and the asteroid belt (Fig. 1.1), granular media form an extremely vast family, composed of grains with very different shapes and materials, which can span several orders of magnitude in size. However, beyond this great diversity, all these particulate media share fundamental features. They are disordered at the grain level but behave like a solid or a fluid at the macroscopic level, exhibiting phenomena such as arching, avalanches and segregation.

In this book, we shall broadly define a granular medium as a collection of rigid1 macroscopic particles, whose particle size is typically larger than 100 μm (Brown & Richards, 1970; Nedderman, 1992; Guyon & Troadec, 1994; Duran, 1997; Rao & Nott, 2008). As we shall see in Chapter 2, this limitation in size corresponds to a limitation in the type of interaction between the particles (Fig. 1.2). In this book, we will focus on non-Brownian particles that interact mainly by friction and collision. For smaller particles, of diameter between 1 μm and 100 μm, other interactions such as van der Waals forces, humidity and air drag start to play an important role as well. This is the domain of powders. Finally, for even smaller particles, those of diameter below 1 μm, thermal agitation is no longer negligible. The world of colloids then begins (Russel et al., 1989).