Figure 9: Evolution of (a ) differential stress, \(\sigma\), (b ) porosity,\(\varphi\) and (c ) the number of segmented voids, \(N\), with increasing axial strain, \(\epsilon\) for the untreated sample (blue circles) and the heat-treated sample (orange circles). Dash-dot lines show the strain at which each sample failed, while dashed lines show the onset of damage localization as seen the μCT volumes (Figure 4 and Figure 5). Solid lines in (a ) show the region of data used to calculate the Young’s moduli (11.3 and 9.5 GPa respectively). Solid lines in (b ) and (c ) show the preferred simple power-law models with 95% confidence intervals shown as dashed lines; see text for exponents.

Micro-crack geometry

To establish empirically how the micro-crack geometry evolves with increasing deformation, we present the variation with strain of the mean value of the major, minor and medium ellipsoid radii from the population of voids in each µCT sub-volume (Figure 10a,b). We also show the mean ellipsoid eigenvalue ratios, used to infer the evolution of void aspect ratio (Figure 10c,d). We present two ratios: (i) the smallest to the medium eigenvalue of the covariance matrix (Section 2.5.2 and Text S1b in SI), where flatter objects have smaller values, and (ii) the medium to the largest eigenvalue, where more elongated objects have smaller values.
Corresponding mean void radii, \(\overset{\overline{}}{r}\), were about the same size in both samples. In the untreated sample (Figure 10a)\({\overset{\overline{}}{r}}_{\text{major}}\) and\({\overset{\overline{}}{r}}_{\text{medium}}\) (blue and orange circles respectively) began to increase at the onset of localization (Figure 4F). In the heat-treated sample (Figure 10b) they began to increase as micro-cracks localized along the optimally oriented damage zone (Figure 5K), after the onset of localization. In both samples,\(r_{\text{major}}\) were oriented approximately parallel to the strike of the eventual fault plane (Figure 6c,d), with their mean values,\({\overset{\overline{}}{r}}_{\text{major}}\), increasing more quickly than \({\overset{\overline{}}{r}}_{\text{medium}}\), showing that micro-cracks grew twice as fast along strike (i.e., perpendicular to\(\sigma_{1}\)) than down dip, becoming more elongate as deformation progressed (Figure 10c). Voids in the heat-treated sample were marginally flatter than those in the untreated sample, with voids in both samples becoming flatter as failure approached (Figure 10d). This implies that the scaling of crack growth, while scale-invariant in length, may be self-affine (variable aspect ratio) rather than self-similar (constant aspect ratio). The down-dip extent (\({\overset{\overline{}}{r}}_{\text{medium}}\)) increased from 2.5 to 4 times the crack aperture (\({\overset{\overline{}}{r}}_{\text{minor}}\)– yellow circles). In the heat-treated sample, this was due to continued crack growth down dip relative to a constant crack aperture (Figure 10b). However, in the untreated sample, crack growth down dip stopped altogether close to failure and a small decrease in aperture accounted for the voids becoming flatter (Figure 10a). Growth along strike also stopped (within error), and the continued increase in the number of cracks at this stage (Figure 9c) confirms that nucleation of new cracks accounted for almost all the porosity generation close to failure in this sample.
In summary, the average growth pattern of individual micro-cracks is independent of heterogeneity in the early stages of localization. The behavior changes close to failure, with continued crack growth in the heat-treated sample accounting for the faster acceleration in porosity than void number, while crack growth in the untreated sample was effectively halted in favor of crack nucleation, which accounted for the entire acceleration in porosity and void number as failure approached.