Subsampling of avalanches in the fiber bundle models of fracture

TL;DR

The paper addresses how partial detection of avalanche events distorts fracture statistics. It introduces a 1D fiber bundle model with tunable load sharing ($\gamma$-model) and a block-based sub-sampling scheme; they quantify similarity between full and partial data with Normalized Mutual Information. Key findings: there is a crossover at $\gamma_c \approx \frac{4}{3}$ between mean-field and local-load regimes; avalanche size distributions shift from $S^{-5/2}$ to exponential; NMI increases with observation block length and peaks near the crossover, especially for larger systems. Near the elastic failure regime the distortion is minimized, implying limited observation can still reflect failure dynamics; the results inform acoustic emission detection strategies and advocate for connected observation patches.

Abstract

We study the subsampling of the avalanches in the fiber bundle model of fracture. In cases where only a part of the system is observed for the micro-failure events, the recorded avalanche statistics gets distorted compared to the actual fracture events. We show that, particularly in the cases where the load redistribution is localized, this distortion is significant. Surprisingly, however, near an elastic failure regime, the distortion is minimized, suggesting a much reduced observational capacity could still represent the actual failure dynamics in the case of fracture of elastic solids.

Figures (5)

• Figure 1: The schematic diagram for the partially observed fiber bundle model is shown. The open circles represent the observed fibers, whereas the crossed circles represent unobserved fibers. The three cases are for $M=1$, $M=2$ and $M=3$ from the top to the bottom. In each case, the observed part is $50\%$, but the effects of having consecutive observation points are seen in the recorded avalanche statistics.
• Figure 2: The size distributions of avalanches are shown for various values of $\gamma$, when the parameter controlling the range of load distribution, $\gamma$, is varied. The observation patch length $M=1000$. The size distribution starts deviating from power law for $\gamma>\gamma_c$.
• Figure 3: The normalized mutual information (Eq. (\ref{['nmi_def']})) between the fully observed and partially observed avalanche series is plotted aginst $\gamma$ for different lengths of observation patches $M$ for a given system size $L=50000$. The mutual information show a monotonic dependence with $M$ and a non-monotonic dependence with $\gamma$. The peak appears near $-\gamma_c$, where the model is known to crossover from mean-field to local load sharing behavior. We argue that in general having a connected observation patch is more useful than not scattered observation points.
• Figure 4: The finite size analysis of the variation of normalized mutual information with $\gamma$, keeping the $M/L$ ratio fixed. The peak gradually shifts towards $\gamma_c=4/3$, although it starts from a higher $\gamma$ value for smaller system sizes.
• Figure 5: The variations of normalized mutual information are shown with different values of random observation fraction $p$ of the model. In all cases, there is significant drop in the NMI values beyond $\gamma=1$.