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.