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.