Gaussian Statistics of Fracture Surfaces

TL;DR

This study investigates the statistical properties of fracture surface heights across various materials, revealing Gaussian behavior at large scales and multi-affine deviations at smaller scales, linked to material discreteness.

Contribution

It introduces a new analysis method based on structure functions to characterize the Gaussian and multi-affine regimes of fracture surface height fluctuations.

Findings

Gaussian distribution for height increments at large scales

Scaling of standard deviation with a unique roughness exponent

Deviation from Gaussian at small scales due to material discreteness

Abstract

We analyse the statistical distribution function for the height fluctuations of brittle fracture surfaces using extensive experimental data sampled on widely different materials and geometries. We compare a direct measurement of the distribution to a new analysis based on the structure functions. For length scales $\delta$ larger than a characteristic scale $\delta^*$, we find that the distribution of the height increments $\Delta h = h(x+ \delta) -h(x)$ is Gaussian. Self-affinity enters through the scaling of the standard deviation $\sigma$, which is proportional to $\delta^\zeta$ with a unique roughness exponent. Below the scale $\delta^*$ we observe an effective multi-affine behavior of the height fluctuations and a deviation from a Gaussian distribution which is related to the discreteness of the measurement or of the material.