Turbulent Fracture Surfaces: A Footprint of Damage Percolation?

We show that a length scale $\xi$ can be extracted from the spatial correlations of the “steep cliffs” that appear on a fracture surface. Above $\xi$, the slope amplitudes are uncorrelated and the fracture surface is monoaffine. Below $\xi$, long-range spatial correlations lead to a multifractal behavior of the surface, reminiscent of turbulent flows. Our results support a unifying conjecture for the geometry of fracture surfaces: for scales larger than $\xi$, the surface is the trace left by an elastic line propagating in a random medium, while for scales smaller than $\xi$, the highly correlated patterns on the surface result from the merging of interacting damage cavities.

Figure 1

Maps of $h$ and ${\omega }_{\epsilon }$ for the three materials studied. Top: the height $h$ of the measured fracture surface. Bottom: transformation providing the field ${\omega }_{\epsilon }$ of local slopes computed at a scale $\epsilon$. In both cases, the quantiles of the distribution are represented by a gray hue, the largest values being represented by the lightest hue. ${\omega }_{\epsilon }$ is computed at the scales $\epsilon =3$, 50, and $8\mu \mathrm{m}$ for the aluminum, mortar, and ceramics fracture surfaces, respectively.

Figure 2

Distribution of height fluctuations $p(\delta h|\delta r)$ at various scales $\delta r$ for the three samples considered.

Figure 3

Spatial correlations of ${\omega }_{\epsilon }$ for the three materials considered. The correlations are represented for ${\omega }_{\epsilon }$ computed at different scales $\epsilon$. The cutoff length $\xi$ is represented for each case.

Figure 4

Multifractal spectrum of the fracture surfaces. The spectrum is computed both below (left) and above $\xi$ (right). The spectrum predicted for $\delta r<\xi$ by the multifractal model of Eq. (2) is represented by a straight line.

Figure 5

Cumulative distribution function $P_{>}(S)$ of cluster size for different thresholds $p_{\mathrm{th}}$. The normalization factor $S_0$ is the average size of a cluster with a gyration radius equal to $\xi$.