Optimal path cracks in correlated and uncorrelated lattices

TL;DR

This paper investigates the optimal path crack model on both uncorrelated and correlated lattices, computing critical exponents, exploring effects of correlations, and extending analysis to three dimensions, revealing new fractal properties.

Contribution

It provides the first detailed computation of percolation exponents for the model and explores the impact of spatial correlations and three-dimensional extensions.

Findings

Computed main percolation exponents for uncorrelated surfaces.

Found non-universal fractal dimensions with spatial correlations.

Identified the fractal dimension of the main crack in 3D as approximately 2.46.

Abstract

The optimal path crack model on uncorrelated surfaces, recently introduced by Andrade et al. (Phys. Rev. Lett. 103, 225503, 2009), is studied in detail and its main percolation exponents computed. In addition to beta/nu = 0.46 \pm 0.03 we report, for the first time, gamma/nu = 1.3 \pm 0.2 and tau = 2.3 \pm 0.2. The analysis is extended to surfaces with spatial long-range power-law correlations, where non-universal fractal dimensions are obtained when the degree of correlation is varied. The model is also considered on a three-dimensional lattice, where the main crack is found to be a surface with a fractal dimension of 2.46 \pm 0.05.