Fluctuations of Global Energy Release and Crackling in Nominally Brittle Heterogeneous Fracture

The temporal evolution of mechanical energy and spatially averaged crack speed are both monitored in slowly fracturing artificial rocks. Both signals display an irregular burstlike dynamics, with power-law distributed fluctuations spanning a broad range of scales. Yet, the elastic power released at each time step is proportional to the global velocity all along the process, which enables defining a material-constant fracture energy. We characterize the intermittent dynamics by computing the burst statistics. This latter displays the scale-free features signature of crackling dynamics, in qualitative but not quantitative agreement with the depinning interface models derived for fracture problems. The possible sources of discrepancies are pointed out and discussed.

Figure 1

(a) Sketch of the Wedge-Splitting test. (b) Zoomed view of the crack tip speed $v(t)$ (black) and stored mechanical energy $E(t)$ (gray) as a function of time in a typical fracture experiment (Here, $V_{\text{wedge}}=16\mathrm{nm}/\mathrm{s}$). (c) Instantaneous power release $\mathcal{P}=-dE/dt$ as a function of $v(t)$ for all $t$. The axes are linear in the main panel, and logarithmic in the inset. In both panels, straight lines indicate proportionality. The proportionality constant $\mathrm{\Gamma }=100\pm 10{\mathrm{J}/\mathrm{m}}^2$ gives the material’s fracture energy. In both panels (b) and (c), the coarse-graining time is $\delta t=0.2\mathrm{s}$

Figure 2

Distribution of instantaneous power release for different coarse-graining time $\delta t$ plotted in logarithmic [panel (a)] and linear scales [panel (b)]. Here, $V_{\text{wedge}}=16\mathrm{nm}/\mathrm{s}$. Plain curves are Gaussian distributions with zero average and a variance $\sigma (\delta t)$ prescribed so as the Gaussian curve fits the experimental data on the small scale plateau. (c) Main panel: Same graphs as in (a) after having withdrawn the noise-dominated Gaussian part of each distribution. Empty and filled symbols correspond to $V_{\text{wedge}}=16$ and $V_{\text{wedge}}=160\mathrm{nm}/\mathrm{s}$, respectively. The curves associated to $V_{\text{wedge}}=160\mathrm{nm}/\mathrm{s}$ have been shifted vertically for the sake of clarity. Distributions involve 3094 events for $V_{\text{wedge}}=16\mathrm{nm}/\mathrm{s}$ and 5299 events for $V_{\text{wedge}}=160\mathrm{nm}/\mathrm{s}$. Leftward (red) and Rightward (blue) straight lines are power-law fits with exponents $a_{\text{small}}=1.4\pm 0.15$ (small scale regime) and $a_{\text{large}}=2.5\pm 0.1$ (large scale regime). Vertical dashed line locates the crossover ${\mathcal{P}}_C\approx 0.34\mathrm{mW}$. (c) Inset: maximum value ${\mathcal{P}}_{\max }$ observed for $\mathcal{P}(t)$ as a function of $\delta t$. Red line shows the $1/\sqrt{\delta t}$ dependency.

Figure 3

(a) Distribution of the avalanche size $S$, defined either as the energy released (black, bottom) or as the area swept (red, top) during the event. In the latter case, the area is normalized by $d^2$ where $d=500\mu \mathrm{m}$ is the grain diameter (b) Scaling between normalized size $S/⟨S⟩$ and duration $D/⟨D⟩$. In both panels, the various symbols correspond to various values for $\{\delta t,C\}$ [specified in the inset of (a)] and $V_{\text{wedge}}$ (empty symbol for $V_{\text{wedge}}=16\mathrm{nm}/\mathrm{s}$, filled ones for $V_{\text{wedge}}=160\mathrm{nm}/\mathrm{s}$; the latter have been shifted vertically for sake of clarity). Analyses involve 899 avalanches for $V_{\text{wedge}}=16\mathrm{nm}/\mathrm{s}$ and 473 avalanches for $V_{\text{wedge}}=160\mathrm{nm}/\mathrm{s}$. Straight lines are power-law fits $P(S)\propto S^{-\tau }$ and $S\propto D^{\gamma }$, with $\tau =1.4\pm 0.1$ ($\tau =1.1\pm 0.15$) and $\gamma =1.38\pm 0.05$ ($\gamma =1.17\pm 0.05$) for $V_{\text{wedge}}=16\mathrm{nm}/\mathrm{s}$ ($V_{\text{wedge}}=160\mathrm{nm}/\mathrm{s}$).

Figure 4

Temporal avalanche shape for different avalanche durations $D$ ($V_{\text{wedge}}=16\mathrm{nm}/\mathrm{s}$). Note the shape flattening and leftward asymmetry that develop as $D$ increases.