Results: Micro-fracture network
evolution
Here we present our 4D observations of the evolving segmented fracture
network in each sample, together with the distributions of micro-crack
orientations. This is followed by results from our quantitative analysis
of the fracture network. We first show the influence of heterogeneity on
the evolution with strain of (i) stress, porosity and the number of
voids, and (ii) void geometry, which reveals how the initial, small
anisotropy of the crack network increased in the lead-up to failure.
Following this, we test our hypothesis regarding the type of phase
transition undergone by each sample by showing the transition to failure
of the correlation length as a function of stress. Finally we present
the evolution with strain of the correlation length and the scaling
exponents of void size and separation. For the purposes of testing our
hypothesis and for clarity of presentation we have analyzed data up to
failure but not beyond.
4D observations of micro-fractures and their
orientations
The spatial evolution of micro-cracks differs significantly between the
two samples (Figure 4 and Figure 5). Although the untreated sample
appeared to fail along a localized shear fault (Figure 4P), pre-failure
damage accumulated in a spalling pattern of radial damage zones
sub-parallel to \(\sigma_{1}\) (Figure 4F-O). Conversely, the
heat-treated sample failed along a localized shear fault, inclined at
30° to \(\sigma_{1}\), along which pre-failure damage had already
accumulated (Figure 5K-P). In both samples, the localized damage zones
consisted mainly of shear and axial micro-fractures oriented between 5°
and 30° to \(\sigma_{1}\) (Figure 6a,b) with bridging ligaments. Local
fracture aperture increased as fractures propagated within the localized
zones of damage.
In the untreated sample, damage localization (established visually from
the segmented CT volumes in Figure 4) first occurred along four narrow
zones simultaneously (orange ellipses in Figure 4F) at 1.37% strain and
64% of the failure stress, \(\sigma_{c}\), (Table 3). Damage was
concentrated along these zones until further radial zones developed on
the other side of the sample (pink ellipses in Figure 4J) at 85%\(\sigma_{c}\). It is not clear exactly how the eventual fault developed
because the sample failed before the next scheduled image time. The
post-failure µCT sub-volume (Figure 4P) indicates that slip occurred
along a plane not previously localized and located slightly above where
the radial damage had accumulated, which is consistent with the sudden,
abrupt nature of the failure inferred from the stress-strain data
(Section 3.2.1). However, close to peak stress (97% \(\sigma_{c}\);
Figure 4O) cracks had begun to localize along a shear zone that was
above and formed from the tip of some of the radial damage zones, and
was conjugate to the eventual fault plane.
In the heat-treated sample, damage localization (established visually
from the segmented CT volumes in Figure 5) first occurred along a shear
zone conjugate to the eventual fault plane (orange ellipse Figure 5H) at
1.24% strain and 72% \(\sigma_{c}\) (Table 4). Damage progressively
concentrated along this plane until localization around the incipient
fault plane became apparent (pink ellipse in Figure 5K) at 90%\(\sigma_{c}\) and 1.38% strain; the same amount of strain as the
initial localization in the untreated sample. Fracture nucleation and
propagation within the initial damage zone then stopped, continuing
instead along this more favorably-oriented zone until failure. This flip
in orientation between two optimally-oriented, conjugate, shear planes
has previously been seen by Lennartz-Sassinek et al. (2014). Here it
coincides with reduced sample stiffness and strain hardening inferred
from the stress-strain data (Section 3.2.1).
Microcrack dips, \(\phi\), became progressively more vertical with
increasing stress in both samples (Figure 6a,b), indicating the
preferential nucleation of tensile cracks. These cracks formed
en-echelon (Tapponnier and Brace, 1976; Kranz, 1979; Olson and Pollard,
1991; Reches and Lockner, 1994) and wing-crack ( Fairhurst and
Cook, 1966; Nemat-Nasser and Horii, 1982; Horii and Nemat-Nasser, 1985;
1986; Ashby and Hallam, 1986; Nemat-Nasser and Obata, 1988; Ashby and
Sammis, 1990) arrays (Figure 7), concentrated in the heat-treated case
at the tip of the propagating fault zone. All radial damage zones in the
untreated sample grew in this manner immediately after their initial
localization (Figure 4F onwards). In contrast, this process occurred
only in the heat-treated sample during localization around the eventual
fault plane (Figure 5L onwards), not during the initial localization
around the unfavorable conjugate. En-echelon and wing-crack arrays
formed at 1.45\(\pm\)0.01% strain in both samples (Figure 4G and Figure
5L). At this point the untreated sample was only at 70% \(\sigma_{c}\),
compared to 90% for the heat-treated sample.
One advantage of the 3D sampling enabled by µCT imaging is that we can
test the null hypothesis that the initial sample porosity is isotropic.
The optimal strikes of the segmented voids, \(\theta\), show a
predominant orientation in the initial porosity in both samples (Figure
6c,d). This starting anisotropy was more pronounced in the heat-treated
sample than in the untreated sample (33.0±15.1% compared with
14.3±11.8% - see Table S1 in SI). Overall, anisotropy in the void
strike increased steadily throughout deformation in the heat-treated
sample but remained approximately constant in the untreated sample
(Table S1 in SI). The strike of the eventual fault closely followed this
pre-existing anisotropy in both samples, but to a much greater degree in
the heat-treated sample. In the untreated sample, although the
distribution peaks and troughs broaden as the radial zones localized,
the strike of the post-failure fault was oriented within 30° of the
initial preferred strike orientation (Figure 6c). In the heat-treated
sample, the strike of the emerging fault plane tracked the orientation
of the initial crack porosity anisotropy almost exactly (Figure 6d),
while the distribution of peaks and troughs remained stationary, and
became more defined, as the damage zone localized.
Stereonet depictions (Figure 8) of the void orientations (poles to
planes) projected along the axial direction (\(\sigma_{1}\)) confirm
these observations, showing a predominant strike parallel to the
pre-existing porosity in both samples, followed by the development of
mainly vertical cracks at localization, in line with our visual
examination. These cracks initially cluster along the pre-existing
strike in both samples but become increasingly distributed in the
untreated sample (blue stereonets) during yield and approaching failure,
with failure occurring along a fault offset by around 30° to the
pre-existing strike. Conversely, in the heat-treated sample (orange
stereonets), these vertical cracks cluster increasingly along the
pre-existing strike throughout deformation. Closure of some
shallow-dipping voids is seen in both samples.
In the time-lapse video of the untreated sample in the \(x,y\)projection (Video S1 in the SI), the first axial fracture initiated at a
spot on the sample edge close to the bottom of the sample, below and on
the outside of a region of concentrated porosity. Further localization
occurred simultaneously along vertical zones distributed radially around
the sample, which appear to have grown preferentially into the sample
before propagating vertically as crack segments linked up. Approaching
peak stress, an array of micro-cracks with varying orientations formed
around the region of concentrated initial porosity, bridging three
radial zones in a curved damage zone. The same process occurred again at
97% \(\sigma_{c}\), bridging four radial zones adjacent to the previous
three, but in a conjugate orientation. This bridging fault propagated up
the sample (Video S2 in the SI – \(y,z\) projection), at a different
strike to the post-failure fault.
Time-lapse video of deformation in the heat-treated sample in the\(x,y\) projection (Video S3 in the SI) shows that localization
initiated within the sample, not at the boundary, on the site of a
pre-existing void, precisely as anticipated by Griffith (1921, 1924).
Subsequent micro-cracks that localized along the damage zone nucleated
between the initial site and the sample edge. The emerging fault plane
initiated at the sample boundary and grew horizontally into the sample,
as subsequent micro-cracks localized along it, before propagating down
the sample parallel to the \(z,x\) plane (Video S4 in the SI – \(y,z\)projection). Simultaneously, micro-cracks localized on the opposite side
of the sample along the same strike as the initial, arrested damage
zone. As deformation continued and the sample reached peak stress,
micro-cracks nucleated in the center of the sample. These cracks joined
the optimally-oriented damage zone to the conjugate damage zone on the
other side of the sample, resulting in a curved shear zone, consisting
of arrays of micro-crack segments linked by bridges of intact rock,
along which the sample failed.
Table
3: Differential stress, \(\sigma\), and axial strain, \(\epsilon\), for
each μCT sub-volume in the untreated sample [ACfresh02]. Letters A-P
refer to the image volumes shown in Figure 4. Localization first
appeared in scan F along several vertical zones distributed radially
around the sample. Additional zones localized in scan J but scan P shows
that failure occurred along an unrelated shear fault.