Statistical properties of fractures in damaged materials

TL;DR

This paper presents a simple analytical model for mud crack formation in thin layers, revealing how fracture growth time scales with system size and how weakening evolves over time.

Contribution

It introduces a novel minimal model for mud cracking that allows analytical derivation of fracture dynamics and weakening behavior.

Findings

Breakdown time scales as L^2 with system size.

Derived a formula for mean weakening over time.

Fracture process exhibits non-trivial fractal properties.

Abstract

We introduce a model for the dynamics of mud cracking in the limit of of extremely thin layers. In this model the growth of fracture proceeds by selecting the part of the material with the smallest (quenched) breaking threshold. In addition, weakening affects the area of the sample neighbour to the crack. Due to the simplicity of the model, it is possible to derive some analytical results. In particular, we find that the total time to break down the sample grows with the dimension L of the lattice as L^2 even though the percolating cluster has a non trivial fractal dimension. Furthermore, we obtain a formula for the mean weakening with time of the whole sample.