Stability and roughness of tensile cracks in disordered materials

TL;DR

This paper investigates the stability and roughness of tensile cracks in disordered materials, deriving a new equation of motion and analyzing crack path statistics to clarify their self-affinity properties.

Contribution

It extends stability analysis to disordered materials and critically examines methods for measuring crack path roughness, revealing biases in real-space fractal analysis.

Findings

Crack paths in disordered materials are stochastic and not self-affine.

Real-space fractal analysis can misinterpret crack path roughness.

The derived equation of motion enhances understanding of crack dynamics in heterogeneous media.

Abstract

We study the stability and roughness of propagating cracks in heterogeneous brittle two-dimensional elastic materials. We begin by deriving an equation of motion describing the dynamics of such a crack in the framework of Linear Elastic Fracture Mechanics, based on the Griffith criterion and the Principle of Local Symmetry. This result allows us to extend the stability analysis of Cotterell and Rice to disordered materials. In the stable regime we find stochastic crack paths. Using tools of statistical physics we obtain the power spectrum of these paths and their probability distribution function, and conclude they do not exhibit self-affinity. We show that a real-space fractal analysis of these paths can lead to the wrong conclusion that the paths are self-affine. To complete the picture, we unravel the systematic bias in such real-space methods, and thus contribute to the general discussion of reliability of self-affine measurements.