Figure 11: (a,b ) Cumulative complementary probability distributions of void volumes, \(V_{i}\), computed with a bin width of 4000 μm3, and (c,d ) cumulative probability distributions of inter-void lengths, \(L_{i}\), computed with 256 bins, at each increment of stress and strain in the (a,c ) untreated and (b,d ) heat-treated samples. As described in Section 2.5.3, we obtained maximum likelihood estimates for \(\beta\) from the cumulative complementary \(V_{i}\) data (see Figure S1 in SI) using the models of Kagan (2002), and fitted the incremental \(L_{i}\) data using linear regression in log-log space (see Figure S3 in SI) to find \(D\), after Turcotte (1997). Blue distributions overlaying the rest are post-failure.

An inverse power-law acceleration to failure?

Parameters for an inverse power-law acceleration to failure for the normalized correlation length, \(\xi/l\), were obtained for both samples (Figure 12), using data observed in segmented μCT volumes between Figure 4F-O and Figure 5H-O and the method described in Section 2.5.3. While an inverse power-law acceleration is commonly only distinguishable from an exponential acceleration within 10% of the singularity (Bell et al., 2013b), restricting the data to this region (stages L-O in the untreated sample and K-O in the heat-treated sample) would have left very few data points for the analysis. In the untreated sample the exponential and inverse power-law models are indistinguishable over the data range, and the predicted failure stress, \(\sigma_{p}\), is far from the observed failure stress, \(\sigma_{c}\). The likelihood that the inverse power-law model fits the data as well as the exponential model is just 3% (Table S5 in SI). Thus, it is impossible to define an accurate failure point in this sample. The sample failed abruptly, long before the predicted singularity, after an exponential acceleration in the correlation length. Conversely, in the heat-treated sample,\(\sigma_{p}\) is accurate to within 3% of \(\sigma_{c}\), while the asymptote of the exponential model is further from \(\sigma_{c}\).The likelihood of the exponential model fitting the data as well as the inverse power-law is 72% (Table S5 in SI), which although relatively high, is not significant (>95%). These differences are diagnostic of a first (abrupt) and second (continuous) order phase transition respectively (Figure 1), validating our hypothesis that, within the temporal resolution of our experiments, the transition to failure is first-order in the untreated sample and second-order in the heat-treated sample.