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.