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Granular Matter and the Marginal Rigidity State

Model experiments are reported on the build-up of granular piles in two dimensions. These show that as the initial density of falling grains is increased, the resulting pile has decreasing final density and its coordination number approaches the low value predicted for the theoretical marginal rigidity state. This provides the first direct experimental evidence for this state of granular matter. We trace the decrease in the coordination number to the dynamics within an advancing yield front between the consolidated pile and the falling grains. We show that the front's size increases with initial density, diverging as the marginal rigidity state is approached.


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The key idea behind this state is that rigid grains falling to form a pile will rearrange and consolidate, until their mean coordination number z reaches a critical value z c at which the net force and torque on each grain can first be balanced by the intergranular forces. At this Marginal Rigidity State the forces, and hence the transmission of stress, are exactly determined by balance alone, without reference to any internal constitutive behaviour of the individual grains. This state separates fluid from conventional solid: so long as z < z c , mechanical balance cannot in general be obtained and the system is fluid. By contrast when z > z c the system can be mechanically balanced, but additional constraints are needed to determine the intergranular forces.
The extra constraints usually come from constitutive stress-strain relations and consistency of the strain field, leading to stress transmission typical of traditional solids.
For convex grains with friction the critical coordination number predicted is notably low, z c = 3 and 4 in two and three dimensions respectively 4 . The corresponding values for sequential packing are 4 and 6, and regular sphere close packing gives even higher values of 6 and 12 respectively. Therefore, the coordination number is a key feature to test in order to identify the Marginal Rigidity State.
Our experiments are illustrated in Figure 1. In an analogue of free fall, grains were conveyed by the moving base until they encountered either the collecting boundary or its accumulating pile. Then the force of sliding friction relative to the moving base played a role analogous to gravitational force on a stationary mass. Figure 2a shows how the final density ρ p of the resulting pile decreases with increasing the initial density ρ i at which grains were fed in. We believe this decrease may be characteristic of a jamming system, and our discussion of the front below offers some insight as to how this comes about. A good empirical fit to the density given by ρ p = 0.446 + 0.072 e -2.7 ρ i (1) and the extrapolated value at which the two densities match is given by ρ c = 0.465 (5). This latter is the natural limiting density of our experimental series. 3 Figure 2b shows how the coordination number z varies with initial density ρ i in our experiments. The striking feature is that as ρ i increases, z decreases towards an extrapolated value z(ρ c ) = 3.1, in qualitative agreement with the simple prediction z c = 3 of marginal rigidity. We know of no other interpretation of how starting at higher density leads to lower coordination number of this value. Due to their shape, our grains are just capable of forming double contacts, a complication that is not addressed by the theory for convex grains. We have counted these as two contacts each, whereas a double contact imposes only three constraints in total rather than four for two separate contacts: one constraint is thus 'wasted' per double contact. Counting coordination number as we have done, this means that the theoretically predicted critical coordination number z c should be shifted (in two dimensions) to z c = 3 + w 2 (2) where w 2 is (without any double counting) the number of double contacts per grain.
From our experiments we measured w 2 ≈ 0.1 with no significant dependence on ρ i , leading to quantitative agreement of z c with the extrapolated value z(ρ c ) .
At lower initial densities the measured coordination number rises with decreasing initial density, approaching z 0 = 3.7 as ρ i tends to zero. This is in fair agreement with the value 4 expected for sequential packing, where each arriving grain would come fully to rest with two contacts before another encounters the pile. Another way to view this would be that since each grain encounters an already rigid pile, one constraint is wasted out of the four from two contacts, in precise analogy to our discussion of double contacts above. In the experiments, we observe that occasionally an incoming grain will dislodge other previously stationary grains before settling down.
Thus the positioning is not strictly sequential and this explains why the coordination number does not reach as high as 4.  Figure 3a shows the front for three different values of ρ i . It can be observed that the higher the initial density the deeper the front. In the sequential packing regime ρ i →0, the front is less than a monolayer and each falling grain collides with an almost completely rigid pile, hence the significant constraint wastage and higher coordination number. At higher ρ i , falling grains encounter a layer that is still rearranging: then the constraints from their contacts are less likely to be wasted, resulting in a lower final coordination number.
We characterised the front by its mass per unit width of the base, m, whose increase with ρ i can be seen in the main plot of Figure 3b. On the basis of this data and a simple theory presented below, we conjecture that the front size in an infinite system diverges as ∆ρ=ρ p -ρ i →0. In our experiments the front size approaches our (limited) system size by ρ i =0.3.
We can estimate theoretically the divergence as ∆ρ→0 of the front depth ξ by considering the time available for grains to rearrange whilst in the front. From 5 conservation of grains it follows that the mean velocity of grains relative to the moving front is approximately v r = v 0 (ρ p +ρ i )/(2∆ρ), where v 0 is the speed of the falling grains relative to the pile. The time a grain resides in the front is then estimated by τ ≈ ξ/v r . In this time the maximum sideways displacement that a grain can achieve is of order ∆r ≈ τv 0 , in terms of which we can now express the front width as ξ ≈ ∆r (ρ p +ρ i )/(2∆ρ) (3).
We would expect ∆r to be of order a grain size for reorganisation of the grain distribution to be achieved from the feed into the pile, and certainly by experimental observation we are confident that if ∆r→0 as ∆ρ→0, it does so only very slowly.
Hence we are led to the front width diverging as with ν=1 if ∆r is set by the grain size.
The very existence of a diverging front size as the marginally rigid state is approached strongly suggests that the latter is a continuous critical point, analogous to At the low density the front is less than one grain deep while at the high density it is more than half the pile and the consolidation is highly cooperative. b, The measured front mass m, in grains per unit of base, as a function of the initial density. This shows a diverging trend, and the point at ρ=0.236 should be regarded a lower bound. The inset shows 1/m plotted against (ρ p -ρ i )/ (ρ p +ρ i ), which a simple analysis (see text) would predict to be a straight line through the origin corresponding to a critical divergence m ∝ ∆ρ -1 .