A geometric basis for critical states in soil


Following up briefly from the last post on this topic here: percolation, I’ve completed some additional geometric modelling work, and also managed to do some thinking on mechanics, and I think I’ve come up with a fruitful line of reasoning.

The thesis I am trying to advance is the following:

THE EXISTENCE OF CRITICAL STATES IN SOIL MECHANICS COINCIDES WITH, AND CAN BE EXPLAINED BY, THE DEVELOPMENT OF A FRACTAL SYSTEM OF INTERLOCKING SOIL PARTICLES AT A CRITICAL DENSITY, WHERE THIS CRITICAL DENSITY IS LARGELY A FACTOR OF THE GRAIN SIZE DISTRIBUTION OF THE SOIL UNDER STUDY, BUT WOULD ALSO BE INFLUENCED BY THE STRESS STATE IN THE SOIL (and may also be influenced by other important, but subordinate factors, including grain angularity, grain stiffness, and grain crushing strength/durability).

To make this argument, we can go straight back to Terzaghi’s derivation of his principle of effective stress, which is widely believed (based on a solid body of physical evidence) to govern the behaviour of soil. By way of brief reminder, this principle states that mechanical behaviour of soil depends on the EFFECTIVE STRESS, rather than the TOTAL STRESS, where effective stress is equal to the total stress minus the pressure acting in any pore fluid that exists in the pore spaces between soil grains. Such a fluid is usually water, but may be water with air bubbles, or may be some other fluid.

In elaborating Terzahi’s original derivation of the principle of effective stress, Skempton (1960) wrote the principle of effective stress as:

sig’ = sig – (1 – a)u — [equation 1]

where sig’ = effective stress
sig = total stress
u = pore pressure
a = contact area between soil particles per unit gross area of the soil, where ‘a’ can be illustrated as follows (redrawn from Holtz and Kovacs, 1981):


This representation of the principle of effective stress can be reduced to the more readily accepted version:

sig’ = sig – u — [equation 2]

by assuming that ‘a’ approaches zero, which is generally assumed to be true in granular materials.

The argument I will try to develop will rely on ‘a’ being a non-trivial finite value when the soil is denser than some critical density, rapidly approaching zero as the critical density is approached. By examining equation 1, we can readily observe that a rapid increase in ‘a’ results in a corresponding and equal rapid decrease in effective stress, sig’. Since the shear strength of a granular material is a function of effective stress, increasing with increasing effective stress, this rapid decrease in ‘a’ to zero results in a rapid decrease in shear strength.

The success of this argument will depend on my ability to show that ‘a’ increases to some non-trivial finite value when density increases past the critical state. This suggestion may be difficult for geotechnical engineers to accept, but I believe the percolation simulations I’ve done shows just that.

I will follow with a brief snapshot of selected pieces from the simulations.

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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