A solvable model of fracture with power-law distribution of fragment sizes

TL;DR

This paper introduces a stochastic cascade fracture model that analytically predicts a power-law fragment size distribution, accounting for probabilistic stopping at each fracture stage, with exponents influenced by fracture probability and size dependence.

Contribution

It presents a novel analytical model of fracture that incorporates size-dependent stopping probabilities, extending previous cascade fracture models.

Findings

Power-law fragment size distribution with exponents between -1 and 0 for constant stopping probability.

Exponent less than -1 when stopping probability depends on fragment size.

Analytical expressions for the distribution based on model parameters.

Abstract

The present paper describes a stochastic model of fracture, whose fragment size distribution can be calculated analytically as a power-law-like distribution. The model is basically cascade fracture, but incorporates the effect that each fragment in each stage of cascade ceases fracture with a certain probability. When the probability is constant, the exponent of the power-law cumulative distribution lies between -1 and 0, depending not only on the probability but the distribution of fracture points. Whereas, when the probability depends on the size of a fragment, the exponent is less than -1, irrespective of the distribution of fracture points.