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Fractal Fragmentation and Frictional Stability in Granular Materials

Charles G. Sammis

Published July 15, 1996, SCEC Contribution #422

The wall rocks of most natural faults are separated by a layer of finely crushed rock called “fault gouge”. Particle-size studies of such unaltered gouge have found a fractal (power-law) distribution with a fractal dimension of 1.6±0.1 in 2D planar section (or 2.6±0.1 for the isotropic distribution in 3D). A simple fragmentation mechanism which leads to a fractal distribution with this dimension can be formulated. It assumes that a particle’s fracture probability is based solely on the relative size of its nearest neighbors — a particle loaded at opposing poles by neighbors of the same size develops the largest internal tension, and is most likely to fragment. The ultimate result is a particle distribution in which no particle has a same-sized neighbor at any scale. Such a distribution is fractal; a perfect, geometrical, discrete fractal having this property is the Sierpinski gasket which has a dimension of 1.58. We have coined the term “constrained comminution” for the neighbor-dominated process which leads to a fractal distribution. Constrained comminution has been simulated in a double-direct shear friction apparatus and has been shown to offer a physical explanation for the observed evolution of the frictional behavior of a granular layer from stable velocity strengthening behavior to potentially unstable velocity weakening as shear deformation proceeds. The initial velocity strengthening occurs when strain is mostly accommodated by the crushing of grains which has no velocity dependent weakening mechanism. As the fractal distribution emerges, an increasing proportion of the strain is accommodated by slip between the grains. Such slip is enhanced by the geometrical fact that a fractal distribution minimizes the dilatancy associated with slip. Slip between grains produces the observed velocity weakening and consequent stick-slip instability.

Sammis, C. G. (1996). Fractal Fragmentation and Frictional Stability in Granular Materials. Dordrecht, Netherlands: Springer Netherlands. doi: 10.1007/978-94-011-5520-5_3.