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, such as dilatancy, in their survival. Another example is the motion of sheared layers of granular materials in geological settings — see [Petford and Koenders, 2003] — in which hot magma is sucked up under volcanoes.
Further scrutiny of Fig. 1.1, the photo-elastic assembly of discs, shows another interesting feature: the force distribution is very heterogeneous. Some regions are entirely force-free, while other regions experience high inter-particle forces that frequently — but not exclusively — line up to form ‘force bridges’. The variability in contact forces points to an accompanying variability in local deformations. Here is something that will prove very important in the study of the mechanics of granular media that are not packed in a regular lattice (which is only possible if there is only one grain size or for a very particular combination of sizes), which is the norm in any naturally occurring sample: granular media are intrinsically heterogeneous. The consequences of this for the mechanics of a granular assembly will be explored in forthcoming chapters.
When the material reaches the plateau of the stress ratio in Fig. 1.2 another feature may become apparent. As the tangent modulus becomes poorly defined the material may find, depending on the precise boundary conditions, a mode of motion that is localised. Such ‘rupture layers’ and ‘failure’ are very important for the engineering community, as illustrated in the example of a landslide occurring as described earlier in this section.
Literature on soil mechanics is plentiful: [Lambe and Whitman, 1969] is a classic text, as is [Terzaghi, Peck and Mesri, 1996]; [Powrie, 2004] is a more modern textbook.
1.2The isostatic state and jamming
A static packed assembly of grains in contact confined by a compressive stress is equivalent to a network of forces. As it is static, force and moment equilibrium will hold. The question being addressed in this section is: how many forces in the network can be specified in such a way that force and moment equilibrium alone are sufficient to determine them, given the detailed geometry of the conformation?
A few conditions need to be laid down to come to a non-trivial answer. The first is that a regular packing is excluded from the analysis; an assembly in a regular packing satisfies certain symmetry rules which need to be imposed in addition to the equilibrium equations. Thus, a medium that consists of identical spherical particles is not accounted for at this stage. Rather, a polydisperse grain-size distribution is envisaged, making for a random packing. Alternatively, rough particles may make up the assembly. No isotropic condition imposition is necessary, though this is often (sometimes tacitly) assumed in the literature. Furthermore, it is assumed that the assembly is very large, so that the number of forces on the perimeter of the sample is small compared to the number of forces in the network. Basically, any condition that somehow constrains the forces in the network is excluded for the moment, implying that the equilibrium equations alone are sufficient to do the analysis. Specific constraining assumptions are discussed below.
In a random packing with N interacting particles in d dimensions there are Nd force equilibrium equations, as each particle that participates in the network is in equilibrium. The force moment equilibrium for each particle provides d(d – 1)/2 equations, so for N particles there are Nd + Nd(d – 1)/2 = Nd(d + 1)/2 equations. Each contact force will have d components and is shared by two particles. Equating the two gives the result that it is possible to calculate N(d + 1) forces, or an assembly coordinate number, that is the number of contacts per particle, of Nc,iso = d + 1 forces on average (the subscript ‘iso’ refers to the isostatic case). Note that this average pertains to particles that participate in the force network. It is well possible that a fair percentage of particles have no contact and these obviously do not contribute to the evaluation of the isostatic coordinate number.
When there are more force-carrying contacts, the equilibrium equations alone will not be able to permit the calculation of the forces. The system is then statically indeterminate. When there are fewer than d + 1 contact forces per particle there are more equations than unknowns and the system cannot be in static equilibrium. The isostatic state is therefore a very precarious, marginally stable state. The slightest disruption that results in the loss of even one contact will make the structure change until the number of force-carrying contacts is at least equal to the required number.
The number d + 1, which equals 3 in two dimensions and 4 in three dimensions, compared to any experimental result for a densely packed material shows that for practical purposes the statically indeterminate state is much more relevant. However, the analysis changes somewhat when constraints are imposed. So, the assumption of randomness is still maintained, but a constraint may follow from the fact that certain contacts slip. In that case the nature of friction must be considered.
Particles interact via their surfaces and these need not be smooth. As long as the surfaces are ‘infinitely sticky’ the force component that is tangential to the surface is free to take any value. In cases where slip is relevant, a Coulomb-type constraint reigns in which the magnitude of the tangential force remains equal to μs times the normal force. Contact forces must then be classified according to those that stick and those that slip. Let the ratio of slipping contacts in the assembly be given as a fraction fμ of all the contact forces, then the number of sticking forces populates a fraction 1 – fμ.
