The transmission of force through granular matter such as sand is a crucial consideration in certain applications. The behaviour observed depends on the particle interactions as well as on the length scale involved.
How is information propagated through granular matter? This is an essential question for researchers investigating the stability of buildings, silos and slopes — particularly for predicting failure and avalanches. Does propagation occur through specific (‘easy’) paths or along a wide front? Is the mechanism similar to that of sound- or light-wave propagation? Is it elastic, or entirely different?
Goldenberg and Goldhirsch1 provide answers to these questions on page 188 of this issue. Their numerical simulations show that, for short distances, forces in granular systems propagate much like waves, but that at longer distances an applied force causes an elastic-like deformation. Furthermore, the regime of wave-like behaviour becomes smaller with increasing disorder and friction — thus, friction and disorder enhance elasticity.
In the general case of a point force applied to an elastic material in static equilibrium, the strength of the force decreases with distance. But even far away from the source, all material points are ‘aware’ of that force. This contrasts with wave propagation, for which practically no information is propagated outside characteristic ‘wave-fronts’ (rays), and also with diffusion, for which information is only ‘present’ behind the diffusive front.
The mathematical equations for elastic behaviour, waves and diffusion have different forms and thus different solutions. They must obey the basic laws of mass-, momentum- and energy-balance, like the Navier–Stokes equations of motion, which contain as special cases all of the above three types of behaviour. Thus, elastic solids and potential flows (for example, in water) behave elastically; sound is a propagating, wave-like mode; and heat is diffusive (Fig. 1).
What is the form of an equation that describes the propagation of information and, more specifically, the force, or stress, distribution in granular media? For granular materials such as sand, soil or snow, material disorder, anisotropy (direction dependence of properties) and friction have to be considered. In a fluid, the stress increases linearly with depth. But in a silo, the stress in the granular material saturates at a certain level that is independent of the depth, because static friction (not present in a fluid) causes the weight of the sand to be partially transferred to, and carried by, the silo's walls2. But still the question remains: how is the weight transported/propagated inside granular material?
When there are no side walls, as in the case of a conical sandpile, the situation is even less clear. One would expect the stress to be maximal under the apex (the highest point). Surprisingly, some experiments3 reveal that the stress actually shows a local minimum — a ‘dip’ — right under the apex. This observation is one of the reasons behind the revival of interest in the problem of force (or stress) propagation in granular materials. Although the dip cannot be explained using the simplest elasticity theory4, wave-like models readily predict it5,6. But the dip can also be reproduced by including anisotropic elasticity, for instance4. Invoking special boundary conditions, such as a deflection of the supporting ground4 or a rough floor7, allows elastic models also to mimic the dip. Furthermore, it turns out that the existence of the dip depends on the structure of the sandpile and the way it was constructed8,9,10. Thus, the problem also concerns the interplay between extrinsic boundary effects and intrinsic granular-material behaviour.
Recent laboratory experiments8,9,10,11,12,13 have probed the more fundamental response of a system to a point-like source of perturbation. But they have not led to a common conclusion. Some find wave-like behaviour whereas others do not, or observe mixed behaviour8,11; in some very small systems, even diffusive-like behaviour seems to exist12. These discrepancies can be due to the effects of the finite size of the laboratory systems13, and the idealized particles (discs or spheres) or the two-dimensional nature of some of the experiments. Another possibility is that all types of behaviour (elastic, wave-like and diffusive) are present and that other details, as mentioned above, determine which one is observed14. Although most experimental observations can be explained with numerical and theoretical models1,4,5,6,15,16,17,18,19,20,21, they all depend on certain basic assumptions and again do not provide conclusive answers.
Goldenberg and Goldhirsch1 investigate a numerical model of two-dimensional slabs composed of disc-shaped grains subjected to a local, downward, vertical force. This approach has a notable advantage over physical experiments — one can systematically vary the material properties, control the system parameters and gather the desired information. In principle, the approach allows the individual effects of different particle sizes and shapes, packing structure, anisotropy and friction to be studied. The authors observe elastic behaviour in granular media by including the effects of sufficiently strong disorder and friction in their simulations1,20,21. In contrast, wave-like propagation is observed in assemblies of frictionless particles17 and/or in regular, lattice-like structures15, or only on rather small scales of 10–20 particle diameters1,14. Thus, even though wave-like behaviour is observed for short distances in laboratory experiments, classical elastic-like behaviour is obtained for the larger length scales of structures such as dams, building foundations and silos. Furthermore, Goldenberg and Goldhirsch find that both disorder and friction — relevant for most realistic, large-scale systems — reduce the short-range wave-like regime and enhance elasticity. But as their conclusions are based on two-dimensional numerical simulations of discs, they will need to be checked in three dimensions, on larger systems and with less-idealized, non-spherical particles.
The puzzling question about the form of information propagation in quasi-static granular media seems to be solved, in that elastic-type behaviour occurs. Nevertheless, it may be that the local or large-scale ‘fragile’ (nearly unstable) states of granular matter (connected to wave-like information propagation) are also relevant. On rare occasions, perhaps such factors are responsible for silo collapse or foundation failure.
Many other properties of granular matter are still not understood. Examples are dynamic sound propagation, jamming (exceedingly slow dynamics), ratcheting (a system's response under cyclic loading, such as trucks on roads or trains on rails), creep and phenomena such as quicksand. Granular matter offers plenty of other puzzles and challenges.
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