Fractal dimension and scaling behavior of cracks in a random medium: ‘‘Frequency-rank’’ distribution described by generalized random walks

Abstract

A dynamical model of cracks in a random medium, specified by the fractal dimension and conformation of the cracks, is proposed. The random medium is composed of many different types of segments. Each segment is sectioned into subsegments in which many microcracks, not directly observable, are randomly arranged. Under an applied stress, the directions of the microcracks partly line up along an easy axis of the stress, and a set of the microcracks forms connected microcracks in the subsegment, and finally the subsegment causes a fracture over the segment. The fractures are counted as cracks in the frequency-rank (FR) distribution. The FR distribution of cracks is characterized by two parameters, corresponding to the fractal dimension D and the ``reciprocal temperature'' \ensuremath{\beta}, respectively.