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.