Fracturing the optimal paths

TL;DR

This paper investigates the sequential failure of optimal paths in disordered systems, revealing a self-similar crack structure in strong disorder and a disorder-independent backbone, with implications for various physical phenomena.

Contribution

It introduces a novel fracture problem by analyzing the failure sequence of optimal paths and characterizes the crack structure and backbone behavior across disorder regimes.

Findings

Cracks form a self-similar line with fractal dimension ~1.22 in strong disorder.

In weak disorder, cracks are spread throughout the network before disconnection.

The disconnecting backbone is independent of disorder strength.

Abstract

Optimal paths play a fundamental role in numerous physical applications ranging from random polymers to brittle fracture, from the flow through porous media to information propagation. Here for the first time we explore the path that is activated once this optimal path fails and what happens when this new path also fails and so on, until the system is completely disconnected. In fact numerous applications can be found for this novel fracture problem. In the limit of strong disorder, our results show that all the cracks are located on a single self-similar connected line of fractal dimension $D_{b} \approx 1.22$. For weak disorder, the number of cracks spreads all over the entire network before global connectivity is lost. Strikingly, the disconnecting path (backbone) is, however, completely independent on the disorder.