Figure
10: (a,b ) Evolution with strain of the mean void ellipsoid
radii, \({\overset{\overline{}}{r}}_{\text{major}}\) (blue circles),\({\overset{\overline{}}{r}}_{\text{medium}}\) (orange circles) and\({\overset{\overline{}}{r}}_{\text{minor}}\) (yellow circles), for the
(a ) untreated and (b ) heat-treated samples.
(c,d ) Evolution of mean void eigenvalue ratios (c )\({\overset{\overline{}}{r}}_{\text{med}}/{\overset{\overline{}}{r}}_{\text{maj}}\)and (d )\({\overset{\overline{}}{r}}_{\min}/{\overset{\overline{}}{r}}_{\text{med}}\)with strain in the untreated (blue circles) and heat-treated (orange
circles) samples. Voids become flatter or more elongate as the
respective ratio \(\rightarrow 0\). Error bars show the standard error
of the mean in each μCT sub-volume. Dash-dot lines show the failure
strain for each sample while dashed lines show the onset of localization
as seen in the μCT volumes.
Evidence for phase transition style
To establish the type of phase transition undergone by each sample, we
present the evolution to failure of the correlation length and the
scaling relations as a function of both differential stress \(\sigma\)and axial sample strain \(\epsilon\). Renard et al. (2018) argue that
stress is a stronger control variable than strain, but strain is usually
the only directly-observable control parameter in real Earth
applications. We first present the scaling relationships for void volume
and inter-void length, and then show how the correlation length, \(\xi\)(linear dimension of the largest void) evolves as a function of stress.
We then analyze the evolution of \(\xi\), \(\beta\) and \(D\) (the void
volume and inter-void length exponents respectively, defined in Section
2.5.3) as a function of strain.
Microcrack volume and inter-crack length
distributions
Both samples show an approximately power-law complementary probability
distribution in void volume, \(V_{i}\), (Figure 11a,b), with the
proportion of larger voids increasing systematically with respect to
strain and stress. Both samples also show an approximately power-law
distribution in their inter-void lengths, \(L_{i}\) (Figure 11c,d),
within a finite range, identified as \(30<L_{i}<1350\) μm (close to
half the sample diameter), with little apparent difference in the shape
of the distributions as stress and strain increase. We can therefore
define power-law scaling exponents \(\beta\) from the frequency-volume
distributions and the correlation dimension \(D\) from the inter-void
length distributions.
Values of completeness volume, \(V_{t}\) (defined in Section 2.5.3),
ranged from 3000 to 4000 μm3, roughly equivalent to a
void aperture of 14-16 μm. This is much larger than the theoretical
detection threshold of half the pixel size (1.3 μm) consistent with
under-sampling of very narrow cracks during segmentation. Void volumes
in the untreated sample are best described (i.e., have the lowestBIC) by the truncated Pareto distribution (TRP) at the three
earliest steps of deformation and then by the characteristic Pareto
distribution (GR), with the transition between the two models occurring
at 43% \(\sigma_{c}\), two stages before the onset of localization
(Figure S2 in the SI). In contrast, void volumes in the heat-treated
sample are best described by the GR distribution throughout the
experiment.