Percolation and Fractals in Granular Media

This is a bit of follow up to an initial post from a few days ago, on 6 February, here:
(sorry, I still haven’t figured out how to insert hyperlinks)

So the idea I am trying to explore is that as the density of a granular medium increases to some critical value (i.e. the “critical state” in soil mechanics), there will be some dramatic change in geometric properties, similar to what is encountered in percolation phenomena, where phenomenological behaviour takes on a dramatically different character near the critical density, where the geometry of the setting takes on fractal characteristics within some narrow range of density near the critical value.

I’ve started examining this problem by first generating a set of 30,000 points randomly located within a 1000 x 1000 block, with each point then becoming a circle with radius varying randomly between 0.05 and 5 (dimensionless – call them cm if you like, so a 10 m x 10 m block with particles from 0.5 mm up to 5 cm). The resulting random granular medium looks like this, with a zoomed view in the inset:

At this particular density, you have clusters of connected particles separated from each other by space, which in reality could be some kind of pore fluid, like water. This image shows individual clusters as different colours, with a closeup in the inset, and the largest cluster shown in black:

Now we can increase the density by making each particle slightly larger, or bringing the slightly closer together. The following image shows the largest cluster in red, with the rest of the granular medium in black, after a slight increase in density:

With continued slight increases in density, we get the following series of snapshots, with the largest cluster continuing to grow until it reaches the point where it spans the whole model, after which its growth begins to slow:

Now we can examine the development of several properties as we vary the density of the matrix, both above and below the initial condition shown in the first image at the top of the post. The following graph shows the relationship between the total number of discrete clusters with density:

There is a near linear relationship for low density, below about 0.5. Now let’s look at the size of the largest cluster as a function of density. The following two graphs show first the area, then the perimeter length, of the largest cluster as a function of density:

There is evidently a dramatic change occurring at somewhere near 0.65, with the size of the largest cluster changing very rapidly with small changes in density, as compared to very small changes in largest cluster size with changes in density at lower densities. And we see a peak perimeter length for the largest cluster at a density of about 0.75, suggesting this may be a critical density of some sort.

We can also look at the area of the largest cluster as a function of the number of clusters:

… and the relationship between perimeter length and area of the largest cluster:

The specific details of the relationships shown in these preceding images are a function of the specific nature of the model setup (number of points, size of model, size range of particles etc), but I believe we are seeing some basic core behaviour of discontinuous granular media, and the obvious potential for dramatic changes in geometric character with small changes in density near some critical densities. I believe this is the starting point for an explanation as to why we see some chaotic behaviour near “critical states” especially when approached from below (i.e. from lower densities).

More to follow… πŸ™‚


About petequinn

I'm a Canadian geotechnical engineer specializing in the study of landslides. I started this page to discuss some mathematical topics that interest me, initially this involved mostly prime numbers, but more recently I've diverted focus back to a number of topics of interest in geotechnique, geographic information systems and risk. I completed undergraduate training in engineering physics at Royal Military College (Kingston, Ontario), did a masters degree in civil (geotechnical) engineering at University of British Columbia (Vancouver), and doctorate in geological engineering at Queen's University (Kingston). I was a military engineer for several years at the beginning of my career, and did design and construction work across Canada and abroad. I've worked a few years for the federal government managing large environmental clean up projects in Canada's arctic, and I've worked across Canada, on both coasts and in the middle, as a consulting geotechnical engineer. My work has taken me everywhere in Canada's north, to most major Canadian cities and many small Canadian towns, and to Alaska, Chile, Bermuda, the Caribbean, Germany, Norway, Sweden, Bosnia, and Croatia. My main "hobby" is competitive distance running, which I may write about in future.
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