Simulation results – percolation of granular material leading to critical states

I started trying to compile my thoughts and recent simulation work on this topic yesterday, but ran out of steam. Here’s what I got written yesterday: percolation part 1, 13 March, and percolation part 2, 13 March.

I previously showed the interconnected soil matrix for a broadly graded “soil” at initial (fully disconnected) conditions (void ratio ~ 0.54), and somewhere close to the “critical density” (void ratio ~ 0.24). This next image will show a much denser material, with void ratio ~ 0.07:

In the first case, we have a completely disconnected soil matrix, which might correspond to a completely unconsolidated soil, perhaps a slurry.

In the second case, we have a material at close to the critical state. The full specimen is spanned by one large cluster of interconnected soil particles, with a large number of much smaller clusters in the remaining void spaces. At the critical density, this cluster would be the minimum size cluster that just barely spans the whole specimen. This cluster has fractal properties, in that it has statistical self similarity at all scales from the full size of the model to something approaching the size of the largest individual particles, and also in that the size of this maximum particle depends only on the size of the specimen. In other words, the largest cluster will span any size “box” at the critical density, and so it has no reference scale.

In the third (densest) case, the problem geometry (and expected behaviour) is dominated by one single cluster that nearly fills the complete specimen.

Anyway, what I would like to show is the relative surface area of the largest cluster as a function of density, which I will show two ways. First, this shows relative surface area (being total perimeter length of the largest cluster divided by total area of the specimen) as a function of the “density” (total area of all clusters divided by total area of the specimen):

Next, I flip this around and show the same relative surface area as a function of “void ratio” (which is the surface area of voids, or spaces, divided by the surface area of all clusters, or solid particles):

This latter graph shows that as we approach a threshold somewhere in the neighbourhood of 0.22-24 (for void ratio), the perimeter length of the largest interconnected cluster drops off quickly toward zero. Therefore, at this point, the relative area of interparticle contact, ‘a’ (see previous posts) drops toward zero, and effective stress will drop suddenly due to an increase in the effect of pore pressure.


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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