Figure 1

(a) An example of the avalanche map $A\left(x,y\right)$, for $k=0.05$ and ${\Gamma }_0=1$, with the area swept over by different avalanches denoted by different shades of gray (time is running from black to white). (b) The same data as in (a) represented by the velocity matrix $V\left(x,y\right)=1/W\left(x,y\right)$ with a zero threshold. The crack front is moving to the positive $y$ direction. Black corresponds to zero velocity, while regions of finite velocity are shown in white. (c) An example of a structure of a single avalanche as identified by the AM method. (d) The same avalanche as in (c) extracted by using the WTM method.Reuse & Permissions

Figure 2

(Color online) The main figure shows the size distributions of the avalanches for ${\Gamma }_0=1$ and various values of $k$ (open symbols), as well as the size distributions of the avalanches for a fixed $k=0.0125$ and various values of ${\Gamma }_0$ ranging from ${\Gamma }_0=0.4$ to ${\Gamma }_0=1.0$ (filled symbols, with the latter set of distributions displaced vertically for clarity). The solid line is a guide to the eye and corresponds to $\tau =1.25$. The inset shows a data collapse of the distributions with ${\Gamma }_0=1$ and various values of $k$, with $\tau =1.25$ and $1/{\sigma }_k=0.725$.Reuse & Permissions

Figure 3

(Color online) Top: the size distributions of the AM clusters for ${\Gamma }_0=1$ and various values of $k$. The solid line is a guide to the eye and corresponds to ${\tau }_a=1.52$. The inset shows a data collapse with ${\tau }_a=1.52$ and $1/{\sigma }_k=0.7$. Bottom: the size distributions of the WTM clusters for ${\Gamma }_0=1$ and various values of $k$. The solid line is a guide to the eye and corresponds to ${\tau }_a=1.53$. The inset shows a data collapse with ${\tau }_a=1.53$ and $1/{\sigma }_k=0.8$.Reuse & Permissions

Figure 4

(Color online) The scaled size distributions of the WTM clusters for various threshold values $v_{th}$, with ${\tau }_a=1.5$ and $1/{\sigma }_v=1.8$. Filled symbols correspond to the simulations, while open symbols are from an experiment on planar crack propagation in Plexiglas plates [6], with $⟨v⟩=11\mu \text{m}/\text{s}$ and a pixel size of $1.7\mu {\text{m}}^2$. The two data collapses (simulations and experiments) have been shifted on top of each other to facilitate comparison.Reuse & Permissions

Figure 5

(Color online) The scaling of the average number of clusters within a global avalanche of size $s$, for various values of $k$, and ${\Gamma }_0=1$. The solid line is a power-law fit of the form $⟨N\left(s\right)⟩\sim s^{\alpha }$, with $\alpha =0.47$. The inset shows a data collapse according to Eq. (7), with $\alpha =0.47$ and $1/{\sigma }_k=0.76$.Reuse & Permissions

Figure 6

(Color online) Top: the scaling of the aspect ratio of the local clusters (open symbols) and the root-mean-square height fluctuations ${⟨\Delta h{\left(\delta \right)}^2⟩}^{1/2}$ of the line (filled symbols), for $k=0.0125$ and different values of ${\Gamma }_0$. For large clusters their aspect ratio scales as $l_y\sim l_x^{\zeta }$, with $\zeta =0.39\pm 0.03$ (solid line), while smaller clusters are characterized by ${\zeta }_L\approx 0.55$ (dashed line). Also the roughness of the line, scaling like ${⟨\Delta h{\left(\delta \right)}^2⟩}^{1/2}\sim {\delta }^{\zeta }$, exhibits two distinct scaling regimes, with ${\zeta }_L\approx 0.48$ for small scales and $\zeta =0.37\pm 0.03$ for large scales. Both sets of data have been collapsed by rescaling $l_x$ and $\delta$ by the Larkin length $L_c\sim {\Gamma }_0^2$. Bottom: the power spectra of the line profiles for different values of ${\Gamma }_0$. The main figure shows a collapse of the spectra, according to $S\left(q\right)={\Gamma }_0^2\tilde{S}\left({\Gamma }_0^{2}q\right)$, while the inset displays the unscaled power spectra. The power spectra scale as $S\left(q\right)\sim q^{-\left(2\zeta +1\right)}$, with a crossover separating regimes with $2{\zeta }_L+1\approx 1.96$ for short length scales and $2\zeta +1\approx 1.76$ for long length scales.Reuse & Permissions