An automaton for fractal patterns of fragmentation

Abstract

FRACTURES in the Earth's crust have a fractal structure over a wide range of length scales. A micromechanical model has been proposed¹ for the formation of fractal patterns of fragmentation in fault zones, based on the preferential fracture, at all length scales, of neighbours of a particle that have the same size as the particle itself. Here we explore this model in two and three dimensions using computer automata which implement these nearest-neighbour fracture rules. The automata produce random fractals which have capacity dimensions between 1.1 and 1.7 in two dimensions, and between 2.0 and 2.8 in three dimensions, the precise value depending on the packing geometry and the presence of long-range interactions imposed by uniform strain conditions. The fractal fragmentation patterns observed in natural systems tend to have dimensions between 2.5 and 2.7; we suggest that our model may permit an interpretation of these values in terms of the packing configuration (number of nearest neighbours) of the constituent particles.