Figure 12: Evidence for an inverse power-law acceleration (solid lines) with respect to stress in the correlation length (normalized by the length of the analyzed sub-volume): \(\xi/l=k{(\sigma_{p}-\sigma)}^{-p}\) in the heat-treated sample (orange), to a predicted failure stress,\(\sigma_{p}\), within 3% of the observed failure stress,\(\sigma_{c}\). This compares with the same model yielding a much poorer prediction in the untreated sample (blue), where\(\sigma_{p}=2.4\ \sigma_{c}\). Dotted lines correspond to the exponential model, \(\xi/l=h\ \exp(q\sigma)\), which in the heat-treated sample is less likely and in the untreated sample is more likely than the inverse power-law model (the relative likelihood of each model is expressed in the main text and also in Table S5 in SI, along with the AICc values). The dash-dot lines show \(\sigma_{c}\)for both samples and the dashed line shows \(\sigma_{p}\) for the heat-treated sample.

Evolution of crack population metrics with respect to strain

Both samples show a systematic increase in the correlation length\(\xi\) towards failure as a function of strain, with failure occurring when \(\xi\) increased beyond 200 µm (Figure 13a). This limit marks the longest crack supported by the sample volume without a runaway instability developing, and falls just short of the mean grain size (i.e., the length between grain boundaries) of the groundmass (250 μm – see Section 2.1). This implies that the sample breaks when whole grains break. The nature of the increase in \(\xi\left(\epsilon\right)\) in the untreated sample (blue) was exponential (i.e., had the lowestAICc – see Table S4 in SI) up to a finite \(\xi\) that fluctuated around 200 µm before failure. Conversely, in the heat-treated sample (orange) \(\xi\left(\epsilon\right)\) preferred a simple power-law acceleration (Figure S4 and Table S4 in SI) to failure, failing when \(\xi>\) 200 μm. The exponent (6.9) is the same (within error) as the exponents for the evolution of \(\varphi\) and \(N\) with strain (Section 3.2.1; Tables S2-S4 in the SI), independently confirming that in this sample crack growth played an increasingly important role closer to failure. The power-law acceleration emerges only once fractures began to localize along the optimally oriented damage zone (at 90% \(\sigma_{c}\) and \(\xi>\ \xi_{0}\)), representing a strong self-organization in the crack network over all distances to concentrate on the damage zone that controls the eventual fault plane.
The two samples had different initial exponents \(\beta\) for their volume distributions (Figure 13b). In the untreated sample (blue)\(\beta\) rose sharply to a peak at the transition from the TRP to the GR model. This shows that, at this point, the largest cracks in the taper were growing or opening, while simultaneously many more small cracks were becoming active above the segmentation detection threshold. The number of voids and the porosity were constant in this phase, implying other voids were simultaneously closing in compaction (Figure 9b,c). This trade-off is consistent with independent observations from acoustic emissions (Graham et al., 2010) and models (e.g. Brantut et al., 2012; 2014) of the competition between compaction and dilatancy during the quasi-elastic region of the stress-strain curve (Figure 9a). Beyond this peak, \(\beta\) decreased smoothly to the first of two local minima once the additional radial zones had localized (Figure 4K). This indicates instability in the sample-related crack nucleation (Figure 9c and Figure 10a) and might be considered a precursor to failure, albeit without evidence of quasi-static damage zone propagation within the temporal resolution of the method. Conversely, in the heat-treated sample (orange) \(\beta\) decreased throughout, reflecting an increase in the relative proportion of larger micro-cracks. This change occurred gradually at first and then more sharply once cracks localized along the optimally oriented shear zone, similar to that observed in numerical simulations (Kun et al., 2013) and as inferred from AE magnitude distributions (Sammonds et al., 1992) in dry porous media. The sharp drop in \(\beta\) is a clear precursor to failure, corresponding to propagation of the shear zone through the sample (Figure 5M-O). This supports our hypothesis that the heat-treated sample exhibits the clear precursors associated with a second-order phase transition.
The evolution of the two-point correlation (fractal) dimension \(D\) was very different between the two samples (Figure 13c). Initially there was a greater degree of clustering in the untreated sample (blue) than the heat-treated one (orange). In both samples \(D\) shows a minimum in the two time windows before the onset of localization, demonstrating the sensitivity of \(D\) to localization (see also Bonnet et al., 2001). The degree of clustering at this point, reflected in the value of \(D\), was very similar between the two samples. From this point on, \(D\) in the untreated sample increased significantly as micro-cracks became more distributed (less clustered) along the radial zones. Conversely, in the heat-treated sample \(D\) shows increased clustering that was sustained throughout localization. It recovered (implying decreased clustering) to a relatively constant value as the optimally oriented shear zone propagated stably through the sample before accelerating at the point of failure as the shear zone spanned the sample. Thus, \(D\) highlights clear differences in the spatial distribution of the micro-crack network between the increasingly distributed damage zones in the untreated sample and localization of an asymmetric shear fault in the heat-treated one. The increasingly distributed nature of crack damage in the untreated sample gives no indication of potential failure, while increased clustering due to localization in the heat-treated sample is a clear and early precursor to failure associated with the development of a damage zone optimally oriented to encourage system-sized shear failure. While both samples show precursory changes, only the heat-treated sample has precursors capable of accurately forecasting the point of system sized catastrophic failure.