Non-Gaussian Nature of Fracture and the Survival of Fat-Tail Exponents

We study the fluctuations of the global velocity $V_l(t)$, computed at various length scales $l$, during the intermittent mode-I propagation of a crack front. The statistics converge to a non-Gaussian distribution, with an asymmetric shape and a fat tail. This breakdown of the central limit theorem (CLT) is due to the diverging variance of the underlying local crack front velocity distribution, displaying a power law tail. Indeed, by the application of a generalized CLT, the full shape of our experimental velocity distribution at large scale is shown to follow the stable Levy distribution, which preserves the power law tail exponent under upscaling. This study aims to demonstrate in general for crackling noise systems how one can infer the complete scale dependence of the activity—and extreme event distributions—by measuring only at a global scale.

Figure 1

The spatiotemporal velocity field $v(x,t)$ during a fracture experiment (right panel). The corresponding spatially averaged signal $V_L(t)=⟨v(x,t){⟩}_L$, over the full system size $L$ (left panel). The solid red line indicates the full space-time average velocity $⟨V_L{⟩}_T=\overline{v}$.

Figure 2

Average maximum velocity (upper panel) $⟨v_{\max }^l{⟩}_T/\overline{v}$ and standard deviation (lower panel) $\sigma /\overline{v}$ of $V_l(t)$ as functions of the measuring window rescaled by the correlation length $l/x^{*}$. The upper solid line corresponds to the slope $\gamma =1/1.7\approx 0.58$. The lower solid line is a fit of Eq. (4) for $l/x^{*}>3$, giving $\xi =0.42\pm 0.03$ and thus $\alpha =1.7\pm 0.1$. Circular black markers show, for one experiment, $\sigma (l)$, obtained from $U_l(t)$ (smoothed over 15 time steps) as defined in the text. The inset shows the cumulative PDF of the local velocity $R(v/\overline{v})$. The dashed line is a fit to the asymptotic tail, using $\alpha =1.7$, to obtain ${\lambda }^{+}$ as defined in the text.

Figure 3

PDF $\Psi (V^{\prime })$ of the global velocity $V_l(t)$ (upper panel) for three different experiments, shown as dark gray, light gray, and black (red, green, and blue), at four different averaging scales $l/x^{*}\in \{40,20,13,10\}$, in semilog representation. For one of these experiments (blue), we obtain $\Psi (V^{\prime })$ from $U_l(t)$, smoothed over 15 time steps, at the same averaging scales (lower panel). The solid lines show the PDF $\Psi (V^{\prime };1.7,1)$, obtained from Eq. (5), whereas the dashed lines correspond to $\Psi (V^{\prime };1.7\pm 0.1,1)$, displaying the effect of the variations in $\alpha$.

Figure 4

PDF $\Psi (V^{\prime })$ for one experiment in log-log representation, at the scales $l/x^{*}\approx \{40,20,13,10\}$ (upper collapsed curves, empty markers), and in filled markers from top to bottom, at $l/x^{*}\approx \{3,1,0.5,0.2,0.05\}$, respectively. The latter distributions with the theoretical curves, $\Psi (V^{\prime };1.7,1)$ shown as solid lines, have been shifted for visual clarity.