Figure 8: Stereonet projection of void ellipsoid orientations (poles to planes) projected down the loading axis (i.e., in the \(x,y\) plane) for both samples. Letters refer to the stress-strain steps for the untreated (Table 3 and Figure 4) and the heat-treated (Table 4 and Figure 5) samples. Clustering contours were calculated from uniform kernel density estimation to show significant departures from a uniform distribution (Kamb, 1959). Kernel radius, \(r=3/\sqrt{\pi(9+n)}\), where \(n\) is the number of data points. Contour intervals are given as a multiple of the standard deviation to emphasize the statistical significance of the number of points falling into each kernel (Haneberg, 2004). In the untreated sample, strikes were dominated by the pre-existing porosity until scan F, when micro-crack localization initiated along the steeply dipping, radially distributed zones (orange ellipses in Figure 4). The radial pattern became increasingly symmetrical around the sample throughout the rest of the experiment as these zones propagated and as micro-cracks localized along new zones (pink ellipses in Figure 4). In the heat-treated sample, strikes were dominated by the pre-existing porosity throughout deformation, although the initial localized damage zone observed at scan H (orange ellipse in Figure 5) was conjugate to the eventual fault zone seen at scan J (pink ellipse in Figure 5).

Evolution of specimen and microcrack characteristics with strain

Stress, porosity and the number of voids

Both samples had a similar stiffness early on (Figure 9a) but the heat-treated sample became stiffer at 0.84% strain when the number of voids decreased slightly (Figure 9c). This is consistent with compaction of the compliant thermal microcracks. The onset of localization, as determined visually from the CT volumes, is evident in both samples as a yield point in the stress-strain curve; at 1.37% strain and 1.24% in the untreated (Figure 4F) and heat-treated (Figure 5H) samples respectively. Further yielding occurred once the damage zone propagated at 95% \(\sigma_{c}\) in the heat-treated sample (Figure 5M), but only from 97% \(\sigma_{c}\) in the untreated sample (Figure 4O).
The heat-treated sample had lower pre-existing porosity than the untreated sample (\(\varphi_{0\ HT}=0.62\varphi_{0\ UT}\)) and fewer but slightly larger voids (\(N_{0\ HT}=0.54N_{0\ UT}\)), with half the number of voids accounting for two-thirds of the porosity seen in the untreated sample (Figure 9b,c). However, this observation only accounts for voids visible above the detection threshold of the segmentation algorithm (a void volume of 3000-4000 μm3 – see Section 3.3.1), and does not include unresolved nano-scale thermally-induced cracks. The observed differences may be accounted for by natural sample variation within these very small samples and/or some void closure from thermal expansion during the heat-treatment.
Both samples showed a ten-fold increase in porosity, \(\varphi,\) over the duration of their respective deformation experiment (Figure 9b), but only a two-fold increase in the total number of voids, \(N\), in the heat-treated sample, compared with a nearly three-fold increase in the untreated sample (Figure 9c). This indicates that crack nucleation was more dominant in the untreated sample, compared with crack growth in the heat-treated sample. The untreated sample showed no evidence of compaction in the early stages of deformation and the onset of localization (Figure 4E-F) is evident as a large jump in \(N\) of 600 voids at 1.37% strain, and a corresponding three-fold increase in\(\varphi\) (Figure 9). Conversely, in the heat-treated sample a small decrease in \(N\) of approximately 50 voids provides evidence for some early compaction due to void closure, although this equates to only a tiny proportion (0.005%) of \(\varphi_{0}\). This was associated with the closure of some optimally oriented (shallow dipping) voids prior to localization (Figure 8 – orange stereonets). The onset of localization is evident as a minimum in both \(\varphi\) and \(N\) at 1.24% strain (Figure 5H) and both variables exceeded their initial values when the optimally oriented damage zone localized (Figure 5K). Once localization initiated, both samples showed an overall acceleration towards failure in both \(\varphi\) and \(N\). However, in the untreated sample there were two occasions where the acceleration was temporarily arrested. The first of these corresponded to the propagation of new localized zones (Figure 4J), while the second corresponded to the change in orientation of the bridging zone (described in Section 3.1). The heat-treated sample showed a slight slow-down in acceleration that corresponded to the nucleation of new micro-cracks between the two ends of the eventual fault (described in Section 3.1), followed by a final acceleration immediately before dynamic rupture.
In both samples the evolution of both \(\varphi\) and \(N\) with strain is best described with simple power-law models (Figure 9b,c); i.e., they have the lowest AICc, (Tables S2 and S3 in SI). The exponent is 3.1 for both variables in the untreated sample, compared with 8.8 and 7.7 for \(\varphi\) and \(N\) respectively in the heat-treated sample, showing an acceleration towards failure that was almost three times faster in the heat-treated sample than the untreated one. These exponents also show that the acceleration in \(N\) accounted for all of the acceleration in \(\varphi\) in the untreated sample, confirming that crack nucleation was the dominant damage mechanism throughout deformation, whereas in the heat-treated sample, the acceleration in\(N\) did not completely account for all of the acceleration in\(\varphi\), confirming that crack growth played an increasingly important role closer to failure.