Local Waiting Time Fluctuations along a Randomly Pinned Crack Front

The propagation of an interfacial crack along a heterogeneous weak plane of a transparent Plexiglas block is followed using a high resolution fast camera. We show that the fracture front dynamics is governed by local and irregular avalanches with very large size and velocity fluctuations. We characterize the intermittent dynamics observed, i.e., the local pinnings and depinnings of the crack front by measuring the local waiting time fluctuations along the crack front during its propagation. The deduced local front line velocity distribution exhibits a power law behavior, $P(v)\propto v^{-\eta }$ with $\eta =2.55\pm 0.15$, for velocities $v$ larger than the average front speed $⟨v⟩$. The burst size distribution is also a power law, $P(S)\propto S^{-\gamma }$ with $\gamma =1.7\pm 0.1$. Above a characteristic length scale of disorder $L_d\sim 15\mu \mathrm{m}$, the avalanche clusters become anisotropic providing an estimate of the roughness exponent of the crack front line, $H=0.66$.

Figure 1

(a) Typical example of a picture recorded by the high speed camera (Photron Ultima) during an experiment with an average crack front speed $⟨v⟩=28.1\mu \mathrm{m}{\mathrm{s}}^{-1}$, and a pixel size $a=3.5\mu \mathrm{m}$. The solid line represents the interface separating the uncracked (in black) and cracked parts extracted after image analysis. (b) Gray scale map of the waiting time matrix deduced from 10 000 front positions recorded at a rate of 1000 fps. The darker parts show the longer waiting times. (c) Spatial distribution of clusters (in white) corresponding to velocities 10 times larger than the average crack front speed.

Figure 2

The velocity distribution $P(v/⟨v⟩)$ as a function of the scaled velocity $v/⟨v⟩$ for different experimental conditions (various average crack front speeds $⟨v⟩$ and pixel sizes $a$). A fit (solid line) to all data for $v>⟨v⟩$ has a slope $-2.55$. Inset shows the corresponding waiting time distribution $P(w/⟨w⟩)$ as a function of the scaled waiting time $w/⟨w⟩$. The solid line represents a fit to all the data for $w<⟨w⟩$ with a slope 0.55.

Figure 3

Burst size $S$ distribution $P(S/⟨S⟩)$, normalized by the average burst size $⟨S⟩$, for different experimental conditions (the various symbols correspond to those on Fig. 2). The bursts detected for each experiment correspond to clusters of velocities 3 times larger than the average crack front speed. A fit on all the data (dashed line) gives a slope equal to 1.71. Inset: Normalized bursts size distribution $P(S/⟨S⟩)$ averaged over all the different experimental conditions, for a wide range of different threshold levels $C$. A fit to all the data, cutting the largest clusters at which a cutoff appears due to the lack of statistics (solid line), gives a slope equal to 1.67.

Figure 4

Average length scale $⟨L_y⟩$ as function of the length scale $⟨L_x⟩$, for different threshold levels $C$, averaged over all the different experiments performed. The solid line is a fit to the data points for $L_x>15\mu \mathrm{m}$ and has a slope $H=0.66$, consistent with the roughness exponent of the fracture front line. The dotted line represents the curve $⟨L_y⟩=⟨L_x⟩$ and serves as a guide for the eye.