Mean Field Theory of Localization in the Fuse Model

TL;DR

This paper develops a mean field theory to analyze damage localization in a quasistatic fuse model, revealing different regimes of damage spread depending on disorder, with analytical phase diagrams and numerical validation.

Contribution

It introduces an analytical mean field framework for damage localization in fuse models, classifying regimes based on disorder and deriving phase diagrams.

Findings

Identifies regimes of percolation, localized, and diffuse damage.

Derives phase diagrams for damage localization regimes.

Validates theoretical predictions with numerical results.

Abstract

We propose a mean field theory for the localization of damage in a quasistatic fuse model on a cylinder. Depending on the quenched disorder distribution of the fuse thresholds, we show analytically that the system can either stay in a percolation regime up to breakdown, or start at some current level to localize starting from the smallest scale (lattice spacing), or instead go to a diffuse localization regime where damage starts to concentrate in bands of width scaling as the width of the system, but remains diffuse at smaller scales. Depending on the nature of the quenched disorder on the fuse thresholds, we derive analytically the phase diagram of the system separating these regimes and the current levels for the onset of these possible localizations. We compare these predictions to numerical results.