A geometric examination of the development of critical states in soil
BGC Engineering Inc., Victoria, BC
This paper examines the development of critical states in soil from a geometric perspective, using concepts from percolation theory, a mathematical field studying the relationships between random clusters of objects. Simulations of randomly generated soils with different grain size distributions suggest that percolation principles apply to granular soils. At a critical density, or percolation threshold, the soil specimen is spanned by a single large cluster of interconnected soil grains, having a maximum perimeter length with fractal properties, minimum cross sectional area, and minimum shear strength and stiffness. Plastic flow with no change in volume or effective stress can occur coincident with the development of this geometric critical state. It is suggested that reliance on fractal geometric models may be fruitful in further examination of specific problems in geotechnique that are not well explained by traditional mechanical models derived from normal assumptions of Euclidean geometric characteristics.
Percolation theory, critical state soil mechanics, fractal geometry, granular media, effective stress
The use of soil mechanics in geotechnical engineering involves a significant measure of both art and science. Successful engineering practice requires considerable reliance on experience, either personal experience on the part of the practitioner, or more general experience of the profession reported in the literature as case studies, models or experiments. Regardless of the level of care taken in site investigations, laboratory testing or related inquiries, significant uncertainty is typical in geotechnical engineering problems, hence the usual use of large factors of safety, often ranging between 1.5 and 3, sometimes greater. This uncertainty is largely due to real variability in complex natural materials, combined with the uncertainty associated with limited spatial coverage of economic site investigation programs. However, it is suggested that there are certain classes of problems where the core physical assumptions behind geotechnical engineering analyses are incompletely framed by standard analytical techniques, thus contributing to the inherent uncertainty.
This paper proposes that the basic physics, or mechanics, used in standard geotechnique may be inappropriate in certain circumstances. More specifically, it is suggested that the geometric assumptions underlying the physics of soil mechanics can benefit from closer examination for certain classes of problems. First, it is generally assumed that the soil can be modeled as a continuum, with uniform material properties within some predetermined scale, and with behavior depending on the effective stress operating on the continuous soil element. Further, it is often assumed that behaviour of the unit soil element can be scaled up or down with no change in material behavior, hence no scale dependence. In reality, soil is comprised of an interconnected skeleton of soil particles separated by pore spaces that may be filled by some pore fluid. Expectations about stress within this soil skeleton necessarily rely on a set of simplifying assumptions, such as those made by Terzaghi in developing the principle of effective stress. Since material behavior in soil frequently matches the predictions made based on the principle of effective stress within an acceptable and predictable margin of error, such assumptions are usually judged appropriate for practical engineering purposes. There are, however, certain problems in geotechnique where scale dependence is observed (e.g. brittle failure, progressive failure), or where the principle of effective stress is observed to be less suitable (e.g. saturated rocks, concrete).
The physics of soil mechanics are derived from basic geometric precepts consistent with Euclidean geometry. This paper will suggest that fractal geometric principles may be useful in certain applications in geotechnique, and will focus on examination of the geometric basis for existence of critical states in soil, using principles from fractal geometry. The argument in support of fractal behavior will make specific use of the idea of percolation, which is known to generate fractal structures, and is associated with important changes of state in a wide range of physical applications.
The basic thesis of this paper is that the existence of critical states in soil mechanics coincides with, and can be explained by, the development of a fractal network of interlocking soil particles at a critical density, where this critical density is a factor of the grain size distribution of the soil under study, and is also 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 or durability. The development of a fractal structure will be examined by modeling percolation of the soil skeleton with changes of density.
2 FRACTAL GEOMETRY – BASIC CONCEPTS
Development of analytical models for behavior of a physical system relies on mathematical abstraction, including a geometric description of the problem space. Usually the simplest and most familiar geometric description of a physical problem can be developed using notions of Euclidean geometry, dating to ancient Greece. Euclidean geometry relies on familiar concepts of points, lines, curves, plane shapes and solids, which can be described and drawn within a framework an integer number of spatial dimensions. Such simple abstractions are usually suitable for describing real physical systems, even though there is no physical object precisely equivalent to a point, perfectly straight line, or perfect cube or sphere.
The precepts of Euclidean geometry are sometimes inadequate to fully describe complex physical systems. A famous example of this is Einstein’s use of curved space-time in development of his theory of relativity. This geometric abstraction, due to famous 19th century mathematicians Carl Friedrich Gauss and Bernhard Riemann, diverged in subtle yet profound ways from the normal assumptions of Euclidean geometry, and represented a necessary refinement to explain fine relativistic effects that could not be explained otherwise by normal mathematical abstraction. More recently, theoretical physicists are using further refinements of higher dimensional geometry with string theory in an attempt to bridge the currently incompatible yet apparently equally valid theories of relativity and quantum physics. While these fields of study bear little resemblance to that of soil mechanics, the common thread is the notion of complexity that may not be adequately idealized by Euclidean abstraction.
The study of fractal geometry has its theoretical roots in the 19th century, was popularized in the 1980s by Mandelbrot (1983), and has gained momentum in recent decades. The word “fractal” comes from the idea that fractal objects occupy fractional dimensions, rather than integer dimensional spaces (e.g. 1-D, 2-D and 3-D spaces). This idea may seem particularly abstract, but is a relatively simple modification of basic Euclidean geometric concepts.
Fractal shapes are those that have self-similarity, or self-affinity, at all scales. With a self-similar fractal, one can magnify the problem space to any scale and find the same shapes represented, although possibly rotated, reflected or translated. With self-affine fractals, the same precise shapes are not necessarily present at all scales, but shapes that are statistically indistinguishable will be present.
Perhaps the most important first idea to absorb is the notion that fractal shapes have a complex texture, rather than perfectly straight lines or smooth surfaces, and that this texture exists, in similar fashion, at all observable scales. A second important idea is that the behavior of fractal systems tends to be described by power laws, and these power laws often have a simple arithmetic relationship to the system’s fractal dimension. The fractal dimension, which is usually a non-integer, higher than the embedding dimension (or “normal” dimensional space in the traditional Euclidean sense), can be defined in many different ways, often as the Hausdorff dimension. Interested readers can consult any of a number of standard references (e.g. Peitgen et al. 2004, Sornette, 2006) for formal definition of the Hausdorff dimension and mathematical treatment of other notions of fractal dimension.
The basic idea of fractal dimension, in an intuitive sense, is that a geometric object occupies some fractional component of a higher dimension. For example, consider a curved line, or 1D shape, with fractal properties, having ever finer and self-similar textural detail when viewed at progressively higher magnification. Such a curve may be confined within a finite area, but can have infinite length, and will “fill” some measurable fraction of the 2D area within which it is confined. Thus it will have a fractal dimension greater than 1, but less than or potentially equal to 2 (i.e. equal to 2 in the case of space filling curves that fill the entire 2D area). Such a fractal curve may be closed, bounding a 2D shape. This shape will have finite area, but an infinitely long perimeter. Thus, the normal, Euclidean, integer power law relationship between length, area and volume (i.e. area and volume vary with the square and cube of perimeter, respectively), do not apply for fractal shapes. Thus, some care would be necessary in, for example, calculating stresses within a fractal body due to point, line or surface loads applied as boundary conditions.
Fractal objects lack a characteristic scale, since they have the same general appearance regardless of the scale of observation. This absence of a characteristic scale is, perhaps counter-intuitively, associated with observation of scale effects, or different behavior with changing scale. Therefore, when scale effects are apparent in nature, the existence of some fractal geometric character might be suspected. This absence of a characteristic scale leads to some unexpected statistical properties; for example, with a true fractal object, the mean, mode or median values of some physical property may have no mathematical meaning, being entirely dependent on the scale of observation.
Many physical systems or objects can be well represented by fractals over some meaningful range of scales. An early example of a real physical system with fractal characteristics is the coastline of Britain. Mandelbrot (1983) demonstrated the fractal nature of this and other coastlines by plotting its measured length against the scale of measurement. When the nation is viewed in whole, many fine features like small points or bays are not evident. Thus when measuring the length of the coast with a long measuring stick, say hundreds of kilometers in length, one obtains a shorter length than if one were to measure it by walking the whole length with a surveyor’s wheel. By plotting total length versus length of measuring stick, one obtains a power law relationship, whereby log10(total length) varies with roughly -0.36log10(length of measuring stick), implying a fractal dimension of about 1.36 for the British coast. By contrast, the state of Utah, with straight line borders, has the same total perimeter regardless of scale of measurement, and a fractal dimension equal to 1, the same as the Euclidean dimension.
As with any mathematical abstraction, there are limits to the applicability of fractal geometry for modeling real physical systems. A basic notion in fractal geometry is that self-similarity or self-affinity occurs at all scales, implying infinite application of associated power law relationships. In reality, there are upper and lower practical limits to self-similarity; for example, when considering self-similar patterns of fracture in rock materials on Earth, no single fracture could be larger than the planet nor smaller than individual atoms, implying a possible range of up to 17 orders of magnitude, from about 10-10 to 107 m. Similarly, there must be some upper limit on the theoretically measurable length of the British coast, suggesting its fractal nature extends over some limited (but large) range in scale. There is therefore expected to be an upper limit of potential range of fractal behavior for any natural terrestrial phenomenon, and most physical systems with apparent fractal characteristics likely have a smaller range than 17 orders of magnitude; however, it is proposed that fractal geometry is a useful mathematical abstraction for some complex physical systems over some practical range of working scales. In particular, fractal geometry is expected to be of use in addressing certain challenging problems in geotechnique.
Several phenomena in geotechnique have apparently fractal characteristics. The geometry of fractures in rock demonstrate self-similarity across a wide range of scales, from micro-fractures in small test specimens to large faults in the earth’s crust. The strength and deformability of rock discontinuities is known to be affected by scale (see for example Bandis, 1990), which is consistent with the idea of a fractal geometry over some scale range of engineering significance. Similarly, the grain size distribution of block size in certain geological deposits known as block-in-matrix rocks, or “bimrocks,” have been described as being fractal. Medley (2002) shows how the most common (i.e. mode) block size observed in a bimrock exposure in Franciscan melange is a function of the scale of observation, being equal to roughly 5 % of the square root of surface area of exposure examined, with the largest block being approximately 75 % of the square root of surface area. For example, if one were to map block size by scan line on a randomly selected bimrock exposure 10 m wide by 10 m high, the most common block would be roughly 50 cm, and the largest on the order of 7.5 m. However, mapping a 1 m by 1 m exposure would likely reveal largest and most common block sizes of about 0.75 m and 0.05 m, respectively. Hence, the observed statistical properties of the block size distribution are scale dependent.
The author has observed similar fractal characteristics in a volcanic debris avalanche deposit in the high Andes of Chile. The deposit, roughly 1 km by 2 km in plan, and perhaps 50 m thick on average, has a similar apparent grain size distribution across a wide range of scales, as illustrated in the photographs in Figure 1. Grain size analyses on grab samples of the matrix show a broadly graded material, with all grain sizes present including gravel, sand, silt and clay size particles. The upper particle size limit, or the median grain size, both depend on the scale of observation, as clearly shown in the exposures in Figure 1, which show frequent cobbles and boulders in small exposures, and much larger boulders, up to perhaps 75 m across, exposed at the surface of the deposit. Thus the grain size distribution of the whole deposit is a scaled up version of that of a small grab sample, indicating self-affinity in the deposit from micron scale to tens of metres.
Brittle fracture is a scale-dependent process affecting some geological materials. In many geotechnical problems, shear failure is examined from a perspective of limiting equilibrium, which relies on the assumption that shear failure occurs instantaneously along a continuous failure surface. While this assumption is believed to be reasonable with ductile materials, it is known to be inappropriate for brittle failure, where failure propagates over time. In ductile materials, failure occurs when the shear stress acting on the body equals or exceeds the available shear strength. In brittle materials, failure depends to some degree on the shear stress and strength, in that shear stress must equal or exceed the residual shear strength of the material. However, brittle failure also depends on the degree of existing damage to the system, or length of existing developing failure surface in relation to the scale of the system, and on the brittleness of the material (Anderson, 1995). Thus brittle failure is affected by scale. Certain geological materials experience brittle failure under certain stress states when perturbed via specific stress paths or strain paths. There may therefore be merit in exploring fractal geometric concepts in relation to brittle failure of geologic materials.
A number of other geologic or geomorphic phenomena have apparent fractal characteristics across some scale range of engineering interest, and thus might benefit from examination via fractal geometry. Examples include: landslide distributions in time and space; spatial variability of soil properties, including index properties and strength-deformation properties; strain rate effects in stress-deformation behavior of geologic materials; variability in ductility versus brittleness over different time scales; groundwater flow in fractured and granular media; grain size distribution of various materials; advance and retreat of glaciers; and earthquake occurrence. This paper examines the example of critical states in soil, using the concept of percolation as an entry point, hypothesizing that the occurrence of the critical state coincides with development of a fractal network of interconnected particles at a critical density.
3 PERCOLATION – A UNIVERSAL PHENOMENON ASSOCIATED WITH CRITICAL CHANGES OF STATE
The following section is synthesized from discussions of percolation phenomena in Schroeder (1991) and Peitgen et al. (2004). Percolation is a term used to describe a wide variety of physical phenomena associated with distinct changes of state, including, for example: formation of galaxies and clusters of galaxies; propagation of epidemics; and, fragmentation of atomic nuclei. Percolation theory addresses the spatial relationships of connected and disconnected clusters in a random space. Given a specific geometric arrangement (e.g. regular lattice) of randomly occurring possible features, there exists a critical percolation density, or percolation threshold, pc. Below this density, there will be a disconnected set of clusters of interconnected particles, with the largest cluster having a maximum dimension smaller than the problem space. Above the critical density, or percolation threshold, there will be a single large cluster that spans the problem space, surrounded by smaller clusters within the spaces not occupied by the largest cluster. As density (or probability of occupation of a given lattice point) increases, the largest cluster will grow toward eventually filling the whole space. Thus, below the percolation density, system behavior is governed by the interaction of a number of disconnected clusters, and above that density behavior is governed by the behavior of a single large connected cluster. At the critical density, behavior is influenced by the existence of a single space-spanning cluster with unusual geometric properties.
The transition that occurs at the percolation density is often marked by a profound change in behavior, sometimes observed as a change in state. This marked change in behavior coincides with the initial development of a space-spanning cluster. The largest cluster at the percolation threshold has a fractal structure, with self-affinity at all scales from the size of the individual particle or lattice point to the size of the complete problem space. The percolation threshold is not a function of the size of the problem space, but is rather only a function of the spatial arrangement and number of degrees of freedom. The site percolation threshold for a square lattice is known from extensive numerical simulation trials to be about pc ~ 0.59, and an exact mathematical solution of pc = ½ is available for the triangular lattice.
One can develop an intuitive sense for the concept of percolation by considering the following hypothetical example of forest fire affecting a planned forest, where trees have been planted on a fixed grid, and a tree will be present at any given grid point, having either died or survived, with a fixed probability, p. The forest occupies several hectares arranged in a square, and is affected by fire along one boundary. Imagine that if a burning tree has a neighbor in any immediately adjacent grid point, the fire will move on to all such neighbouring trees. If a tree has no un-burned immediate neighbours, it will burn out and the fire will not propagate from it. The potential for a forest fire to burn across the complete property, from one boundary to the opposite, clearly depends on the existence of at least one completely connected cluster of trees spanning across the property. Percolation theory suggests this will occur at site occupation probabilities equal to or greater than the critical density, and this is explored in Figures 2, 3, 4 and 5.
Figure 2 shows the distribution of clusters of connected “trees” in a 200 x 200 square lattice (i.e. managed forest) with an occupation probability at lattice points of p = 0.5. The largest single cluster is much smaller than the complete grid; in this case with a length perhaps 10 % the length across the grid. Experiments with a larger lattice of 1000 x 1000 yields similar sized “typical” clusters, with the largest cluster still being much smaller than the complete grid. Figure 3 shows the same lattice with an occupation probability of 0.58, approaching the expected percolation threshold. The largest cluster is much larger than in Figure 2, but remains a fraction of the size of the whole grid. It can also be seen that the “typical” clusters have grown, as many other clusters are much larger than the largest cluster in Figure 2.
Figure 4 shows the largest cluster when p = 0.59. This cluster fully spans the model from top to bottom (but does not quite span the model from side to side). Therefore, if a fire were lit along the bottom boundary, and the fire behaved according to the simple rules previously described, it would burn across to the upper boundary before burning out. With slight increases in density, the largest cluster continues to grow and fill the model space, as illustrated in Figure 5, therefore at any density above the percolation threshold, the forest fire is certain to cross the whole forest.
At densities lower than critical, the system behaves as a set of disconnected clusters. At densities higher than critical, the system behaves like a solid continuum, with behavior dominated by the largest single space-spanning cluster. At or near the critical density, or percolation threshold, the largest cluster has fractal properties, with self-affine shapes present at all scales from that of the individual tree to the whole model. When the largest cluster has this fractal shape, system behavior can be dramatically different. In the case of the forest fire, the duration of burning is a function of tree density, being relatively low above and below the percolation threshold, and being substantially higher at the threshold. This concept is illustrated in Figure 6, and can be explained by the fractal nature of the largest cluster, with nearly infinite detail, containing numerous smaller clusters along its perimeter, and thus nearly infinite travel path lengths for the fire to burn along. At low densities, the burning terminates as soon as the longest cluster in contact with the initial fire has burned out. At high densities, fire will progress across the forest fairly quickly due to the presence of numerous broad paths across the forest. At the percolation threshold, the largest cluster contains a theoretically infinite amount of detail, and a small number of relatively long paths for the fire to travel across. Hence the duration of burning is markedly higher at close to the percolation threshold.
Percolation theory is well developed for regular geometric arrangements of objects (Stauffer and Aharony, 1994), like the regular square grid considered in the preceding discussion of the hypothetical managed forest. This paper will examine the idea of percolation in an irregular geometric arrangement: that of a simulated random soil matrix, where particle locations are randomly located within a problem space (i.e. not located on fixed lattice points), and assigned random radii, to yield a pre-defined grain size distribution. This exploration is intended to test the idea that percolation would occur in a randomly located set of randomly sized particles, similar to a real granular soil material. The results of this exploration are then examined in light of the core concepts of effective stress and critical states in soil mechanics. It is shown that the existence of a critical state for a granular media like soil can be anticipated from percolation theory.
4 SPATIAL MODELLING OF PERCOLATION IN A SOIL MATRIX
This section discusses simulations of percolation in a synthetic soil, comprised of a spatially random distribution of variable diameter circular soil particles within a rectangular problem space. Three different grain size materials were simulated, as illustrated in Figure 7. Construction of the initial models involved the following steps: generation of a large number of particle locations, typically 64,000 to 512,000, at random locations within the rectangular problem space; random selection of diameter for each particle within the desired range, with selection designed to yield the appropriate grain size distribution on the basis of percent passing by weight; and, spatial comparison of particle locations to generate clusters of connected or overlapping particles. The initial models were created with densities low enough that maximum cluster size was much smaller than the problem space. Percolation simulations involved several incremental increases in density, and examination of the geometric and statistical properties of the resulting soil particle clusters.
In soil, volume change does not occur by adding or subtracting soil particles, as in the previous example of the forest fire. Density of soil varies with volume changes in the sample, and such volume changes occur either through isotropic volumetric expansion or contraction with applied isotropic loads, or through shear-induced dilation or contraction. Simulation of volume change in this work was done in several ways, as illustrated in Figure 8. Case 1 shows isotropic compression, resulting in higher density. Case 2 shows one-dimensional compression, or constrained compression, similar to compression in an oedometer. Case 3 shows direct shear with no evident volume change, and case 4 shows direct shear with coincident contraction, as might occur with a loose material. Case 1, 2 and 4 all yield qualitatively similar results. Case 3 yields no volume change, as expected, and no change in the statistical properties of soil particle clusters. Since case 1, 2 and 4 were all effectively identical in behavior, and case 1 was the easiest to model, this approach was used in the rest of the simulations.
Figure 9 shows the initial problem state for a uniformly graded simulated soil. The largest twenty-one clusters are highlighted in white with black outline, and the single largest cluster is shown in black. It can be seen that all soil particle clusters are disconnected from each other, and the largest cluster is some small fraction of the total model size. The simulated soil has a 2D void ratio of 0.54 at this stage, where this void ratio is the total surface area of voids divided by the total surface area of soil particle clusters.
Figure 10 shows the same simulated soil after volumetric contraction to increase density. The single largest cluster, shown in black, spans the whole model space, and void ratio has decreased to 0.24. This largest cluster contains numerous interior voids, each enclosing smaller clusters. The outer perimeter of the largest cluster has significant textural detail, with numerous small “points” and “bays,” with each of the bays containing smaller, separate clusters. Experiments with various size specimens show that the critical density at which the first specimen-spanning cluster occurs is independent of the size of the specimen, and seems rather to depend only on the grain size distribution, being different for the other two simulated soils. Therefore if the specimen were infinitely large, or sufficiently large as to be virtually infinite in relation to particle size, the largest specimen at the critical density would have a fractal shape, with infinite perimeter length.
Figure 11 shows the broadly graded material at a density higher than the percolation threshold, with a corresponding 2D void ratio of 0.07. The largest cluster nearly fills the problem space, and there are relatively few small voids within and around the largest cluster, containing a small number of very small disconnected clusters. It may be noted that the simulation trials involving the two other simulated soils yielded qualitatively similar results, with percolation thresholds occurring at different densities or void rations.
These trials demonstrate that a percolation phenomenon does indeed occur when changes in density are made to a simulated soil, with random diameter particles conforming to a given grain size distribution distributed randomly within a problem space. Figure 12 shows the variation of perimeter length around the outside of the largest cluster with changes in 2D void ratio. As density increases from some initial low value, and correspondingly the void ratio decreases (i.e. moving from right to left on the graph in Figure 12), the perimeter length of the largest cluster gradually increases, then suddenly begins to increase dramatically at void ratios below 0.3, before reaching a peak and falling off at void rations below 0.2. The dramatic increase in perimeter length occurs as the largest cluster begins to grow to span the specimen, and thereafter decreases as the largest cluster begins to fill the specimen space. In the eventual limit of zero void ratio and a single cluster filling the specimen, the perimeter length of the largest cluster is equal to that of the specimen.
The dramatic increase in perimeter length near the critical density, or percolation threshold, is indicative of very high surface complexity around the largest cluster’s perimeter. The largest cluster takes an apparently fractal shape for this irregular geometric arrangement of random soil particles, consistent with percolation theory for regular geometric arrangements. The relationships between perimeter length and cluster area and volume near the critical density are therefore complex, diverging significantly from normal Euclidean relationships. The distribution of stresses through the system must therefore be expected to be significantly more complex at this critical density than normally assumed.
5 CRITICAL STATES, EFFECTIVE STRESS AND CRITICAL DENSITY IN THE SOIL MATRIX
In soil mechanics, the critical state is a stress state at which plastic shearing will continue indefinitely with no change in volume or effective stress (Wood, 1990). This stress state is expected to be uniquely associated with a given density, or void ratio, and can be approached via any number of different stress or strain paths.
Consider three different samples of the same soil, with very different densities, as illustrated in Figure 13. For simplicity assume the soils are saturated, so the voids are filled with water. The first sample on the left has a density lower than the percolation threshold, and is comprised of a matrix of disconnected small clusters of interconnected soil particles. This may represent a completely unconsolidated material during deposition in water, or possibly a fully liquefied soil. The second sample, in the middle, is at close to the percolation threshold, possessing a single, large cluster with fractal character spanning the specimen. This cluster has a very large perimeter area, but one can also see that its cross section parallel to the shear forces varies, ranging from very small – possibly as small as a single soil particle – to some moderate fraction of the width of the specimen. The third sample, at a higher density, is effectively filled by the single largest cluster.
Figure 13. Shear force applied at upper and lower boundaries of specimen at three different densities: lower than critical (only large clusters shown for emphasis); close to critical (only largest single cluster is shown); and, higher than critical (largest cluster shown).
Imagine an applied shear force, T, at the upper and lower boundaries of the three samples. Since water (i.e. the pore fluid) cannot transmit shear forces, it is clear that the first system, with density lower than the percolation threshold, has no shear strength or stiffness, and will deform freely in shear. The third, densest sample, will have the highest strength and stiffness, as shear forces will be transmitted across a well connected cluster with multiple broad paths. One might expect the shear strength and stiffness to be governed by the minimum cross section parallel to the applied shear force, which in this case is very nearly equal to the width of the specimen. The middle sample, at close to the percolation threshold, is fully connected across the specimen from top to bottom, but only just barely. This specimen will be spanned by a cluster with the smallest geometrically possible minimum horizontal cross section. Thus this sample will have the lowest possible non-zero strength and stiffness. Hence, the critical density obtained at the percolation threshold coincides with the onset of zero strength, and therefore plastic flow can occur at this state with no further changes in density.
One can also examine this scenario through basic concepts of the principle of effective stress, which 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 neutral stress, which is the pressure acting in any pore fluid. In elaborating Terzaghi’s original derivation of the principle of effective stress, Skempton (1960) wrote the principle of effective stress as:
 sig’ = sig – (1-a)u
sig’ = effective stress
sig = total stress
u = pore pressure
a = Ac/A = contact area (Ac) between soil particles per unit gross area (A) of the soil.
The physical meaning of ‘a’ can be understood by examining Figure 14. The original assumption in Terzaghi’s work was that inter-particle contact areas approach point areas, so ‘a’ approaches zero, and equation  can be written in the regular form, sig’ = sig – u. Since soil particles are not incompressible, clearly ‘a’ cannot be precisely zero, and the modeling presented previously shows that ‘a’ must increase substantially (although perhaps still remain relatively small) with increases in density beyond the percolation threshold, or critical density. In fact ‘a’ will decrease to zero very rapidly as the critical density is approached, and thus the value of effective stress will suddenly drop to a minimum at the critical density. Since strength is a function of effective stress in a frictional material, shear strength will take a minimum value at the critical density. This interpretation, based on considerations of effective stress, is consistent with that drawn from purely geometric considerations earlier in the section. Therefore, using the concept of percolation, one can anticipate that a granular soil material will tend toward zero strength at the percolation threshold or critical density. Thus the critical state in soil mechanics, or possibility of plastic flow with no further change in volume of effective stress, coincides with the development of a fractal complex of interconnected soil particles at a critical density, or percolation threshold.
The preceding discussion uses a relatively simplistic argument, relying on simple, idealized models, to defend a fractal geometric basis for the existence of critical states in granular soil matrices. The argument has relied on a number of simplifying assumptions, and is not intended to claim a precise representation of real soil behavior, but rather is intended to present a concept, and stimulate discussion, potentially leading to further development of fractal-based analytical methods for complex problems in geotechnique.
The models described in this paper considered uniform volume change across specimens comprised of a random matrix of random diameter soil particles. No such idealized soil exists in nature, and it is not suggested that such uniform volume change occurs in nature, nor is it suggested that boundary forces would be distributed uniformly across the entirety of a real physical soil model. For example, shear failure is often observed as a localized phenomenon, with the critical state developing only within some finite width shear zone, or along distinct shear bands, and significant dilation or contraction limited to that zone. With percolation theory, the percolation threshold exists for the given geometric arrangement of the matrix, regardless of the size or shape of the problem space; therefore, the occurrence of localization does not detract from the proposed development of a fractal network of interconnected soil clusters within a shear zone, rather than within the complete specimen.
The analysis has focused on the kinematics of a simulated random soil matrix, without specific consideration of forces, stresses, displacements, work or other conditions. The analysis therefore excluded neglected local distribution of forces and stresses through the interconnected soil matrix, as well as effects associated with movement of particles, including local dilation or contraction, and associated pore fluid flow, gradients and seepage pressures.
Notwithstanding the limitations just described, the observed geometric behavior follows directly from accepted mathematical abstraction. These additional factors, if adequately considered, will surely alter important details of model behaviour, perhaps leading to stress concentrations and localization; however, they cannot affect the basic finding that a fractal cluster will develop at the critical density, and this fractal cluster will permit plastic flow with no further change in volume or stress. This key finding derives directly from a geometric analysis of the kinematics of the granular soil matrix.
This paper has examined the geometric characteristics of a granular soil medium with changing density. The results suggest that percolation theory can be used to explain the development of critical states in soil. With subtle changes in density of a given random granular matrix with given grain size distribution, there are profound changes in the geometric properties of interconnected clusters of soil grains in contact with each other. At the critical density, or percolation threshold, a soil specimen is at its weakest point, due to the existence of a single largest cluster with arbitrarily large perimeter length and arbitrarily small minimum cross sectional area. The soil sample’s ability to resist shear stress drops toward zero at this critical stress, and this conclusion can be derived either from purely geometric deductions via percolation theory, or by re-examining the basic derivation of Terzaghi’s principle of effective stress. It is hoped that the findings of this paper may spark some discussion about the use of fractal geometry in the mechanics of soil behavior, particularly for complex phenomena where normal analytical methods in soil mechanics break down, including, for example, brittle fracture and progressive failure.
Anderson, T.L. 1995. Fracture mechanics: fundamentals and applications. CRC Press LLC, Boca Raton, FL. ISBN 0-8493-4260-0.
Bandis, S.C. 1990. Scale effects in the strength and deformability of rocks and rock joints. In: Scale Effects in Rock Masses, Pinto da Cunha (ed.). Balkema, Rotterdam. 59-76. ISBN 90-6191-126-5.
Holtz, R.D., and Kovacs, W.D. 1981. An Introduction to Geotechnical Engineering. Prentice-Hall. Englewood Cliffs, N.J. ISBN 0-13-484394-0.
Mandelbrot, B.B. 1983. The fractal geometry of nature. W.H. Freeman, New York.
Medley, E. 2002. Estimating block size distributions of melanges and similar block-in-matrix rocks (bimrocks). Proceedings, 5th North American Rock Mechanics Symposium, University of Toronto, Toronto, Canada. 599–606.
Peitgen, H.-O., Jürgens, H., and Saupe, D. 2004. Chaos and Fractals, New Frontiers of Science. 2nd Edition. Spriner, New York. ISBN 978-0-387-20229-7.
Schroeder, M. 1991. Fractals, Chaos, Power Laws – Minutes from an Infinite Paradise. Dover Publications Inc., Mineola, New York, ISBN 978-0-486-47204-1.
Skempton, A.W. 1960. Terzaghi’s discovery of effective stress. In: From Theory to Practice in Soil Mechanics, John Wiley and Sons, New York. 42-53.
Sornette, D. 2006. Critical Phenomena in Natural Sciences. Springer-Verlag, Berin. ISBN 3-540-40754-5.
Stauffer, D. and Aharony, A. 1994. Introduction to Percolation Theory (2nd ed.), CRC Press, ISBN 978-0-7484-0253-3.
Wood, D.M. 1990. Soil Behaviour and Critical State Soil Mechanics. Cambridge University Press. ISBN 0-521-33782-8.