A discrete time evolution model for fracture networks

TL;DR

This paper models the evolution of geophysical crack patterns using mean field theory, analyzing how local fracture and rearrangement steps influence the overall network structure and cell density over time.

Contribution

It introduces a discrete time evolution model for fracture networks based on convex mosaics and proves the existence of limit points and monotonic behavior of cell density.

Findings

Limit points exist for various trajectory types.

Cell density increases monotonically under all admissible trajectories.

The model provides a mathematical framework for fracture pattern evolution.

Abstract

We examine geophysical crack patterns using the mean field theory of convex mosaics. We assign the pair $(\bar n^*,\bar v^*)$ of average corner degrees to each crack pattern and we define two local, random evolutionary steps $R_0$ and $R_1$, corresponding to secondary fracture and rearrangement of cracks, respectively. Random sequences of these steps result in trajectories on the $(\bar n^*,\bar v^*)$ plane. We prove the existence of limit points for several types of trajectories. Also, we prove that cell density $\rho = \bar v^*/\bar n^*$ increases monotonically under any admissible trajectory.