| A constitutive law of fault gouge is essential in describing the stability/instability of the
motion of a fault. Extensive experiments and simulations have been conducted to reveal that
the frictional properties of granular matter, such as the fluctuation of frictional force, are
significantly affected by the particle-size distribution (PSD). For example, see Marone and
Scholz 1989, Morgan and Boettcher 1999, Mair et al. 2002, and Abe and Mair 2005. Because
the PSD depends on the nature of comminution in a fault, comminution is one of the key
processes that dominate the frictional properties of fault gouge. However, we still cannot
answer the following fundamental questions: Why is gouge fractal? What sets the fractal
dimension?
Here we show a simple (perhaps the simplest) model that can give answers to the above
questions. Our model is somewhat similar to a classical model by Sammis et al. (1989), but
quite different in modeling successive fragmentation processes. Our model involves the time
evolution equation of PSD so that we can discuss any transient states of PSD during the
comminution process.
One of the important results is that the PSD for a steady state is always fractal irrespective
of the fracture criterion of each particle. This makes a quite contrast to the model of
Sammis et al, which requires a certain condition to the fracture criterion in order
to reproduce the fractal PSD; otherwise the lognormal PSD is obtained in their
model.
Another important prediction of our model is the fractal dimension. It is found that the
fractal dimension depends on the fracture criterion. To reproduce the universal value 2.6, the
fracture probability of a single particle should be proportional to d-0.4, where d denotes a
dimension of a particle. We can explain this fracture probability by taking two
ingredients into account: the strength of single particle and the statistics of force
chains.
Furthermore, by means of discrete element simulation, we investigate the rate dependence
of friction coefficient of granular matter that has fractal PSD. It is found that the
(time-averaged) friction coefficient of fractal systems is lower than that of a non-fractal
system, but is insensitive to the fractal dimension. |