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.