For the past 2-3 years, I’ve been losing sleep trying to get my head around notions of fractal geometry in relation to geotechnique, looking for a meaningful entry point. I think a productive line of investigation has come to me. This came to me as I was getting close to the end of Manfred Schroeder’s book, “Fractals, Chaos, Power Laws,” where he has a chapter on percolation.

A couple of important things I drew from his discussion of percolation:

– percolation phenomena, occurring near some “critical density,” are governed by power laws in space and time, and the power law exponents are independent of the physical process at work, depending only on the geometry of the problem space, the embedding spatial dimension and the number of degrees of freedom

– the behaviour of the system near the critical density is markedly different than “typical behaviour” further from the critical density

– aspects of the system become distinctly fractal at near the critical density, resulting in absence of a characteristic scale, and thus behaviour that depends on scale. In the example he uses of a 2D lattice, at close to the critical density there exists connectivity across the lattice, with connected clusters that are self-similar at all scales (up to the size of the lattice). Below this critical density, there exist smaller, disconnected clusters, and above the critical density, the majority of the lattice becomes connected. In either case, the fractal aspects disappear as density diverges away from the critical density in either direction.

Schroeder mentions a number of physical phenomena that can be governed by percolation phenomena, including spread of forest fires, formation of galaxies, spread of disease, nuclear fragmentation as examples. These phenomena take on a markedly different character when close to the critical density, and this character has chaotic aspects due to the fractal nature of the physical system.

Fractal systems can have a scale dependence, and can also lead to deterministic chaos, where the general nature of the physical behaviour may be understood, but the precise details cannot be predicted from initial conditions.

There are several phenomena in geotechnique that are scale dependent and chaotic

In soil mechanics there are several phenomena that defy precise mathematical treatment, suggesting deterministic chaos, and that may have scale dependence. Brittle fracture is a scale dependent phenomenon. Many geological materials have brittle behaviour under certain circumstances, either involving a specific stress state or time scale (i.e. rate of strain or rate of load application).

In soil mechanics we also speak of critical states, being states of density and stress that can lead to collapse, bifurcation, liquefaction etc. At the critical state, the soil matrix will neither dilate nor contract upon shearing. Wet of the critical state (meaning lower density of the soil skeleton, and higher proportion of water, assuming a saturated soil medium), the material will tend to contract on shearing, with collapse occurring as density moves to the critical state. Dry of the critical state (or higher density), the material will tend to dilate on shearing, with density decreasing toward the critical state. Catastrophic collapse is far less likely to occur from an initially denser state; however, the term “dense” is relative, and under a higher state of stress, the critical density will be higher.

Progressive failure is a brittle failure phenomenon, whereby the development of a continuous failure surface occurs progressively along the eventual full length over some finite time (rather than instantaneously along the full length, as is the case with ductile failure). Brittle failure is also a scale-dependent phenomenon, whereby the potential for failure to propagate to ultimate collapse depends not only on the working stresses and available strengths (both peak and residual), but also the size and geometry of the problem setting (e.g. slope that may fail, triaxial test sample that may fail), AND the current length of a developing failure surface. Once the failure surface has grown to a certain length (and this critical length depends on the model size/geometry and the acting stress state), it will then propagate to failure even if external loads are removed, with no need for additional perturbation.

The scale dependence and temporal aspects suggest some fractal aspect at work. Here’s what I think is going on, as a first attempt at an explanation, in very simplistic terms:

– at a density greater than some “critical density” or critical state, there is intimate connectivity between all soil particles within the soil matrix, or at least there is continuous contact, through multiple different paths, through the model. Therefore shearing can occur instantaneously along some continuous failure surface through the whole model, hence assumptions of limiting equilibrium, and instantaneous failure are valid

– at a density lower than the critical density, the soil matrix is disconnected, or at least not fully connected across the model. As the model deforms, shearing induces contraction. On the one hand, this increases the pressure in the pore fluid, with resulting decrease in effective stress. However, on another hand, we can imagine that as the density increases, a greater number of soil particles come into contact. We know from computer simulations related to percolation phenomena that as this density increases, the interconnectivity between particles in the matrix will increase, and at some point the interconnected clusters of particles will develop a fractal shape, with self-affinity at all scales (from single particle to scale of the model)

– at the density where the interconnected soil matrix has taken on a fractal geometry, the behaviour will become chaotic. Transmission of stresses and deformations through this fractal soil matrix will not be instantaneous, but will rather occur over time, in a fashion that cannot be determined precisely from initial conditions

OK, that’s all I got for now.ðŸ™‚