The Physics of the Deformation of Densely Packed Granular Materials. M A C Koenders
integrals
Appendix BList of Symbols and Notations
B.1List of symbols
Chapter 1
General Concepts
1.1Introduction
Granular materials play a role in nearly all human activities. Users of, for example, sand, from children in sandpits to sophisticated geotechnical engineers, know that it is a fascinating — and to some extent, unpredictable — material. Many groups are concerned professionally with granular materials: chemical engineers, pharmacists, food technologists, agriculturalists, biologists, geologists, geophysicists, astronomers even, are obliged to study their behaviour under a wide variety of circumstances. In addition to sand, which itself may be of many compositions, the types of materials include gravel, fine-particle aggregates as employed in cosmetics, pharmaceuticals, dust, crushed rock and granules that occur in a domestic environment, such as breakfast cereals, sugar, salt and (instant or ground) coffee granules.
It is important to distinguish between the various states in which these materials may be encountered. The possible range of regimes is extensive. The delineation of regimes is accomplished by specifying first the packing density, second the grain-size, or size distribution and particle shape, then the type of medium in the interstices between the grains (fluid, gas, vacuum) and finally the stress and temperature environment. Depending on any combination of these factors, examples of phenomena that may take place come in a wide variety. A few celebrated ones are landslides, blocked silos and sewers, rubble asteroids breaking up, segregation effects in breakfast cereals, dust-storm propagation after a terrorist attack, the spreading of sun-cream over skin and the formation of dunes. The list is by no means exhaustive; not only do people continually invent new applications for granulates, they also discover new processes where these materials may be deployed. The sheer diversity of effects illustrates the range of professionals that may be engaged with the subject.
The mechanical behaviour of an assembly of grains depends first and foremost on the interaction between the particles. For a low packing density the grains are fairly free to move and interactions may take place in a similar way to the molecules in a gas: the interactions are short-duration ‘events’. When, on the other hand, the material is densely packed the grains are locked in enduring interaction with each other. This does not preclude relative motion between the particles. In a dense slurry, for example, the interstitial fluid is the interactive medium. Particles may move, while the interactive strength varies with motion, but the interaction continues to be relevant for particles in close proximity. When the medium is dense and dry, on the other hand, particles must make contact. Their relative motion may be sliding, or even suffer a very slight indentation when two particles are being pressed together hard, but there is only a non-zero interaction while the contact endures.
In order to describe the motion in various states, distinctly different branches of mechanics are required. For a dilute flow in which collisions are prevalent, for example, concepts of gas dynamics have to be invoked: a temperature field is needed to describe the velocity fluctuations while motion takes place. For dense (but not too dense) slurry flow in which a fluid mediates the interaction of the grains the relative velocity difference of the particles needs to be described. For very small particles Brownian motion will play a role too. For a dense packing, in which the grains are in enduring contact, the physics of the interaction is quite different. As this is the field of interest in the publication to hand a small study of the background of this subject is of use.
It could be argued that the densely packed state is fairly boring, as the displacements tend to be so insignificant. Essentially, one might say, a densely packed granular material behaves like a solid. There are, however, certain features that relate to this régime that are quite unlike traditional solids. In fact, it is one of the most difficult to describe problems in materials science. The reason for this is that the material properties change dramatically under certain specific loading conditions.
Figure 1.1. Picture of an assembly of photo-elastic discs. Experiment by [Konishi, 1978].
The easiest way to see where the problems arise is by considering an experiment of dry dense sand on a slope. When the sand is initially deposited and densified the slope is horizontal. Now imagine an experiment in which the slope angle is gradually increased. There comes a point when the angle is so great that the sand can no longer support a stable configuration and a landslide ensues. The changes in the sand up until this point are almost imperceptible, yet, internally, changes must have taken place in order for the sand to go into a state that cannot support a stable equilibrium. The question is: what physics underlies the internal change of state and how can its mechanics be captured?
This problem is, of course, the province of soil mechanics. Tribute must be paid to the tremendous body of useful work that has been produced by civil engineers, especially in the area of experimentation. One type of test in particular is very common in soil testing and that is the so-called triaxial-cell test. In this test a cylindrical sample of soil is subjected to a stress path in which — after initially building up a compressive pressure — the stress on the cylinder wall is kept constant while the stress on the ends of the cylinder is increased (precise definitions of stress and strain are explored in Chapter 2). The same type of test can be done in two dimensions on a sample of an assembly of discs. The latter case is illustrated in a picture of photo-elastic discs in which the contact forces are made visible by means of polarised light. Figure 1.1 provides an example.
A typical response of the assembly so stressed is depicted below in somewhat stylised form (stylised to remove the inevitable experimental noise). The stress ratio (the ratio of the major and minor principal stress) goes up with increased principal strain until it appears to remain more or less constant. Now look at the tangent modulus (ratio of stress increment to strain increment). While the stress ratio is close to unity the assembly is quite stiff and behaves just like a solid block of material. As the stress ratio increases, however, the tangent modulus rapidly decreases till it reaches zero — a dramatic change in only a few percent of deformation!
Figure 1.2. Stress ratio and volume strain as a function of the major principal strain in a biaxial cell test.
Even more bizarre is the behaviour of the volume strain. Initially, at a stress ratio close to unity, the sample contracts, as one would expect from an ordinary solid that is compressed in one direction. At higher stress ratios a peculiar effect becomes manifest: the sample expands. This is completely counter-intuitive behaviour. The effect is called dilation. The reader may carry out a very simple experiment to verify the phenomenon. Go to a wet beach with well-compacted sand and simply step on it. One can see the sand go dry underneath one’s feet. The soil expands, causing there to be more space in the interstices, and in so doing it sucks the water in from the neighbourhood, making it temporarily drier. The effect was first described by [Reynolds, 1885].
The amount of motion involved in this development is minimal; the strain is in the order of a few percent. The mechanical features that occur here are very important not only for the geotechnical industry, but also for the understanding of, for example, the motion of burrowing animals — see for example, [Dorgan et al., 2006]. While a further discussion is only possible when grain assemblies are considered that contain an interstitial fluid, it is obvious that such creatures are adapted to employ the mechanical properties of granular deposits,