Discussion
The results presented above reveal key aspects of the evolving nature of
compressive failure of brittle rocks through the accumulation of
micro-cracks that spontaneously organize themselves along localized
damage zones. Our synchrotron x-ray micro-tomography (μCT) observations
of in-situ compressive rock deformation reveal the underlying
processes – in particular the nature of the phase transition between
intact and failed states in materials with different degrees of starting
heterogeneity. Both our post-failure samples contained a localized shear
fault, but the preceding accumulation of micro-cracks was very different
between the samples, especially in their spatial distribution and their
growth characteristics close to failure. We confirm our hypothesis that,
in terms of stress and within the time-resolution of our experiments,
the transition to failure is abrupt and unpredictable (first-order) in
the homogeneous sample, but continuous and predictable (second-order) in
the heterogeneous sample.
Microcrack network
evolution
Prior to failure, our initially crack-free, and therefore more
homogeneous, sample accumulated damage in a spalling pattern of
localized zones distributed radially around the sample with no preferred
strike direction. This damage pattern was completely overprinted during
failure, highlighting the drawback of analyzing failed samples
retrospectively to gain insights into pre-failure damage accumulation.
Pre-failure behavior in this sample resembles strain localization
observed from in-situ µCT images of deforming mono-minerallic,
fine-grained and uniformly graded (i.e., structurally homogeneous) sand
specimens (Desrues et al., 1996). The macroscopic fault localized
abruptly at >97% of peak (failure) stress, \(\sigma_{c}\),
as microcracks transitioned from being broadly distributed throughout
the sample (albeit along several radially oriented zones) to being
organized along an emerging shear zone.
In contrast, our pre-cracked, and therefore more heterogeneous, sample
accumulated damage around, and subsequently failed along, a localized
shear zone. This behavior resembles the observations of Lockner et al.
(1991; 1992) who showed progressive localization of AE along a shear
zone in deforming Westerly granite samples from peak stress onwards.
However, in our experiment the shear zone localized earlier, at 90% of\(\sigma_{c}\), with a subsequent period of stable crack nucleation and
growth along the damage zone during strain hardening prior to dynamic
rupture at peak stress. This behavior resembles fault nucleation and
propagation from AE in Berea (Lockner et al., 1992) and Clashach
(Liakopoulou-Morris, et al. 1994; Lennart-Sassinek et al., 2014)
sandstone samples (arguably more heterogeneous than granite samples in
terms of their porosity), where a diffuse damage zone appeared and
gradually localized around an incipient fault plane prior to\(\sigma_{c}\).
Our results show that heterogeneity exerts a strong control on the
evolution of crack network anisotropy, with homogeneity acting to
stabilize the system prior to dynamic failure, generating more complex
patterns of strain localization with more isotropic global
characteristics, as suggested by Desrues et al., (1996). Under
axi-symmetric triaxial loading conditions, sample homogeneity is a
constraint that favors a transversely isotropic spalling pattern until
very close to peak stress, whereas the presence of heterogeneity acts to
amplify the pre-existing anisotropy with the formation of a shear fault.
Radial spalling patterns are rarely observed in studies of AE,
potentially due to limits on their location accuracy, where microcracks
occurring along several radially distributed, but localized, damage
zones might give the impression of being distributed throughout the
sample.
In both of our samples, damage accumulated via the nucleation and
sub-critical growth of micro-cracks along localized damage zones.
En-echelon and wing-crack arrays formed at different stages in the
deformation process in each sample (at initial localization in the
untreated sample but only once the optimally oriented shear zone
localized in the heat-treated sample), and formed at the same degree of
strain (Figure 4F and Figure 5L), implying a degree of strain control.
The main direction of individual micro-crack growth in the localized
zones was along strike rather than down dip (Figure 10a,b,c). Models of
damage accumulation under tri-axial compression are usually based on AE
locations and microstructural observations of post-failure samples, from
which it is difficult to quantify the relative proportion of
progressive, pre-failure axial to radial micro-crack growth.
Along-strike growth is consistent with our conventional tri-axial
compressional stress configuration
(\(\sigma_{1}>\ \sigma_{2}=\sigma_{3}\)), in which it is
energetically more favorable for tensile micro-cracks to open radially
against the axes of minimum principal stress and close against the axis
of maximum principal stress. Down-dip fault propagation occurred instead
by the nucleation, growth and then linkage of an increasing number of
small, tensile en-echelon and wing cracks forming at the fault tip
(Figure 6a,b, Figure 7 and Figure 9c). This is consistent with previous
experimental and modelling work (e.g., Tapponnier and Brace, 1976;
Kranz, 1979; Nemat-Nasser and Horii, 1982; Horii and Nemat-Nasser, 1985;
1986; Sammis and Ashby, 1986; Ashby and Hallam, 1986; Nemat-Nasser and
Obata, 1988; Rundle and Klein, 1989; Ashby and Sammis, 1990; Reches and
Lockner, 1994; Potyondy and Cundall, 2004; Cho et al., 2007) and recentin-situ observations of damage accumulation in strong rocks
(Renard et al., 2017; 2018).
We observed significant anisotropy of void strike in the pre-existing
porosity in both samples (Figure 6c,d), despite visual inspection of
thin sections and compressional wave velocities of the same rock type
showing only 1% anisotropy in bench-top tests on the original material
(Meredith et al., 2005; Meredith, pers comm.). This indicates that a
small velocity anisotropy represents substantial void anisotropy. The
pre-existing void anisotropy is more pronounced in the heat-treated
sample than in the untreated sample, possibly due to thermal expansion
during the heat-treatment acting to close the isolated, mainly round
voids in the feldspar micro-phenocrysts (Meredith et al., 2012). This
may also account for the otherwise counter-intuitive smaller overall
porosity in the heat-treated sample compared with the untreated one. The
application of confining pressure may also have contributed to the
porosity difference by acting to close the thermally-induced cracks in
the heat-treated sample more effectively than the stiffer pores in the
untreated sample. In the heterogeneous sample, the preferred strike of
the pre-existing porosity corresponds almost exactly to the strike of
the emerging fault plane (Figure 6d). There was also significant
amplification of the pre-existing anisotropy of the rock fabric (from
33% to 96% just before failure; Table S1 in SI). This was not the case
in the homogeneous sample, where the degree of anisotropy remained
approximately constant throughout the lead-up to failure (Table S1 in
SI), consistent with the lack of an overall preferred strike in the
pre-failure localized zones in this sample.
The results in the previous paragraph prove that the initial
microstructure, specifically the orientation and anisotropy of
pre-existing porosity dictated the geometry and location of the future
(post-failure) fault, particularly in the heat-treated sample. We
speculate that this happens via a modification of the local stress field
with respect to the principal stress axes. In true tri-axial
configurations (\(\sigma_{1}>\ \sigma_{2}>\sigma_{3}\)), shear wave
velocity anisotropy measurements have shown that micro-cracks in general
propagate parallel to \(\sigma_{2}\) as they open parallel to\(\sigma_{3}\) (Crawford et al., 1995), while polymodal faulting is also
often seen (Healy et al., 2015). Thus, although the global stress
configuration is axi-symmetric in our case, both heterogeneity and void
anisotropy in the microstructure appear to cause the local development
of truly tri-axial stresses such that a particular strike is preferred.
One possible mechanism for this may be stress rotation around
microstructural discontinuities (Faulkner et al., 2006), possibly
reflected in our experiments in the rotation of the void ellipsoids with
respect to the principal stress axes (Figure 6c,d). In this case, the
pre-existing network of anisotropic micro-cracks with a preferred
orientation would have generated an emergent, locally dominant
true-triaxial stress field within the body of our heterogeneous sample,
even though the confining pressure was isotropic around the vertical
(\(\sigma_{1}\)) axis. Conversely, in our homogeneous sample, some
complex interplay between local true tri-axial stresses and global
axi-symmetry would be required to generate several radially distributed
damage zones simultaneously. We speculate that the global axi-symmetry
initially counteracts the rotation of internal stresses in this sample,
acting to prevent an increase in crack anisotropy and thereby increasing
the uniformity of the strike distribution as the experiment progresses.
Thus, the relationship between the evolving anisotropy of the
micro-cracks and their preferred orientation is likely to be a
controlling factor on the geometry and location of an asymmetric shear
fault, on the timing of the formation of this fault and on whether
pre-failure damage is localized along this fault or not.
In both our samples, the majority of cracks dip steeply within ±15° of
the loading direction, although a few dip less steeply between 15 and
30° (Figure 6a,b). This is consistent with the results of post-failure
sample analysis in early experimental work (Brace et al. 1966; Hallbauer
et al. 1973; Lajtai 1974). The macroscopic fault in our homogeneous
sample dips at a similar angle to the pre-failure micro-cracks, whereas
in our heterogeneous sample it dips less steeply post-failure than it
does at peak stress. Although the effective pressure was relatively low
(15 MPa), which may promote axial failure over shear, it was consistent
across the two experiments. This implies that the differences in fault
dip result from an intrinsic microstructural response, whereby the
emergent internal friction coefficient decreases during failure in the
heterogeneous sample but remains constant in the homogeneous sample,
consistent with DEM models (Kun et al., 2018) that show a decreasing
coefficient of internal friction with increasing heterogeneity. In both
samples, the dip angle increases during quasi-static damage
accumulation, increasing earlier in the homogeneous case (during initial
localization) than the heterogeneous case (only during localization
around the optimally oriented shear zone). In the homogeneous case, the
steep dip of the nucleating cracks (Figure 6a and Figure 7a) and the
eventual fault plane (10°; Figure 6a) indicates that the internal
friction coefficient in this sample is sufficiently high to inhibit
micro-crack damage by shear mechanisms until immediately before dynamic
failure. In the heterogeneous case, the dip, and therefore the internal
friction coefficient, increases only during propagation of the shear
zone and is particularly pronounced immediately before failure (Figure
6b), while the eventual fault plane dips less steeply (30°; Figure 6b).
This indicates that early crack nucleation and failure itself both
involve shear mechanisms, but shear zone propagation is governed
primarily by tensile mechanisms, i.e., the accumulation of en-echelon
tensile cracks (Figure 7b), with a corresponding increase in the
internal friction coefficient. For this reason we have referred to a
‘damage zone’ prior to failure and a ‘fault plane’ afterwards.
In our homogeneous sample, increased clustering (Figure 13c; blue
circles) occurred at 43% \(\sigma_{c}\) with the onset of localization
at 64%. This agrees with observations and models of cracks initiating
earlier than the theoretical shear-sliding threshold for more
homogeneous low porosity, crystalline rocks (70% \(\sigma_{c}\);
Hallbauer et al., 1973; Nicksiar and Martin, 2013; 2014). The
implication is that our more homogeneous sample is weakest in tension
and, once a sufficient number of tensile cracks form, a macroscopic
shear fracture will naturally develop. We therefore conclude that damage
in this sample most likely initiated via the nucleation of
pore-emanating (Sammis and Ashby, 1986; Ashby and Sammis, 1990) or
force-chain controlled (Potyondy and Cundall, 2004; Cho et al., 2007)
tensile micro-cracks due to the re-distribution of stress around equant
compressing pores and grains. Conversely, increased clustering in our
heterogeneous sample (Figure 13c; orange circles) occurred at 62%\(\sigma_{c}\) with the onset of localization at 72%. This is later
than the theoretical shear-sliding threshold for heterogeneous rocks
(60% \(\sigma_{c}\); Hallbauer et al., 1973; Nicksiar and Martin, 2013;
2014). The implication here is that our more heterogeneous sample is
weaker in shear than in tension since shear sliding along preferentially
oriented, pre-existing cracks occurred before tensile cracking. We
therefore conclude that damage in this sample most likely initiated via
the development of tensile ‘wing-cracks’ (Nemat-Nasser and Hori, 1982;
Horii and Nemat-Nasser, 1985; 1986; Ashby and Hallam, 1986; Nemat-Nasser
and Obata, 1988; Ashby and Sammis, 1990) at the tips of pre-existing
defects due to shear-sliding along those defects. Unfortunately, such
shear sliding would not be visible in our images without significant
dilatancy during slip.
In summary, our experimental data confirm that the initial heterogeneity
within a rock sample is a key control over how cracks, pores and grain
boundaries interact locally with the applied stress field, and imply
that the microstructure transitions from being weakest in tension to
being weakest in shear as heterogeneity increases.
Scaling, phase transition style and predictability of
failure
time
Micro-crack volume and inter-crack length distributions follow
power-laws throughout the cycle of deformation and failure in both
samples, characteristic of the scale-invariant (fractal) nature of
natural fault networks (Main et al., 1990; Bonnet et al., 2001) and
consistent with the power-law microcrack volume distributions observed
by Renard et al. (2017; 2018). The transition from the TRP to the GR
model for the micro-crack volume distributions (Figure S2 in SI) in the
homogeneous sample emulates changes in the organization of earthquake
size distributions following the occurrence of extreme or very large
earthquakes (Bell et al., 2013a). Close to failure the void volume
distribution shows a bump at large volumes, indicative of a
supercritical state with an elevated probability of occurrence of large
events (Main, 1996), sometimes known as ‘dragon kings’ (Sornette, 2009).
We have demonstrated that the parameters of these distributions are more
sensitive to heterogeneity than porosity alone, consistent with the
findings of Vasseur et al. (2017) and Kun et al. (2018). In combination
with µCT observations of fault formation, the evolution of these
parameters provides a microstructural explanation for the variation in
the systematic prediction error for the failure time based on acoustic
emissions (Vasseur et al. 2015).
However, the systematic change in the mean void aspect ratios during
crack growth may indicate that the scaling of crack growth is
self-affine (i.e., exhibits scale-invariance in length with different
exponents for individual growth axes, leading to a variable aspect
ratio) rather than self-similar (the same scaling exponent for all
growth axes, with a constant aspect ratio). This is consistent with
observations of fracture surface geometries in rocks (Schmittbuhl et
al., 1995) and other materials (Mandlebrot et al., 1984; Bouchaud et
al., 1990; Russ, 1994; Schmittbuhl and Maloy, 1997; see also Bouchaud,
1997 for a review), which are well-described by self-affine fractals.
These studies have shown that scaling along the aperture axis is
systematically smaller than along the mean crack plane, with the
systematic (Hurst) exponent defining the fracture roughness (Bouchaud,
1997; Weiss, 2001). Our observation that almost no growth at all occurs
along the aperture axis supports the conjecture that the aperture
direction is not physically equivalent to the mean crack plane
(Schmittbuhl et al., 1995). Our results indicate that scaling along the
strike and dip axes may also systematically differ from each other. This
contradicts the notion of strict self-similarity in the mean crack plane
(Schmittbuhl et al., 1995), and implies that the strike and dip
directions are not physically equivalent either. Further work is
required to quantify the scaling anisotropy for crack growth in our
experiments and to test these hypotheses. Since crack surfaces in
crystalline materials require heterogeneities, such as grain boundaries
and dislocations that pin the propagating crack front, in order to
develop self-affine roughness (Schmittbuhl and Maloy, 1997; Bouchaud,
1997; Weiss, 2001), we expect that scaling exponents for the
heterogeneous sample may be more anisotropic than for the homogeneous
sample.
In the heterogeneous (heat-treated) case, we find evidence for a
continuous (second-order) phase transition in the inverse power-law
acceleration to failure of \(\xi\) with respect to stress (Figure 12;
solid orange line), with failure occurring near the asymptote, together
with clear precursors in \(\beta\) and \(D\). The rapid decrease in\(\beta\) corresponds to the formation of a localized damage zone
optimally oriented for macroscopic shear failure, occurring when the
microcrack network self-organizes. This provides a clear precursor to
sample failure related to a distinct physical process, i.e. the emergent
inverse power-law acceleration in \(\xi\). The asymptote defines a
predictable failure time defined by a smooth transition to an infinite\(\xi\) at the sample-scale (Figure 1; orange line). The early and
sustained decrease in \(D\) in 3D is a key precursory indicator of
localization, while its recovery is associated with shear zone
propagation in 2D, as anticipated by the model of Main (1992). This
provides another clear precursor to failure. Such behavior agrees with
statistical physics models of rupture as a critical, second-order
phenomenon (Girard et al., 2010; Kun et al., 2013). Thus, taken
together, these variables show that damage localization along a zone
optimally orientated for macroscopic shear failure is the physical
process that defines whether the phase transition from an intact to a
failed state is second-order, and therefore predictable, with reliable
precursors to failure.
In the homogeneous (untreated) case, we find evidence for an abrupt or
discontinuous (first-order) phase transition, with an unsuccessful
forecast of the failure stress, and a preference for an exponential
model for the evolution of the correlation length, \(\xi\), with respect
to stress. Furthermore, there is very little evidence for reliable
precursors in either the micro-crack volume exponent, \(\beta\), or the
two-point fractal dimension, \(D\), and the bump in the void size
distribution at large volumes is reminiscent of a first-order phase
transition (Lominitz-Adler et al., 1992; Ceva and Perazzo, 1993).
Approaching failure we see small fluctuations in \(\beta\), \(\xi\) and\(D\) that may indicate impending failure as they are associated with
formation of the additional damage zones and subsequent microstructural
instability due to crack nucleation close to failure. However, using
these parameters as precursors may lead to false alarms since they are
not associated with the eventual fault plane. The exponential increase
in \(\xi\) (implying that local correlations dominate) is unusual and
generally associated with the critical regime during phase transitions
across surfaces (Kosterlitz and Thouless, 1973; Kosterlitz, 1974), such
as during large-scale faceting at the surfaces of growing crystals
(Nozières, 1992). Its stabilization to a finite value shortly followed
by abrupt failure is characteristic of a first order phase transition
(Figure 1; green line). In numerical models of fault growth, an
exponential distribution of fault lengths is associated with crack
nucleation, whereas a power-law distribution emerges with nucleation
plus crack growth and coalescence (Cowie et al., 1995). Hence, the
origin of this response in our rock volume may be explained by our
observation that crack nucleation is the dominant damage process in the
homogeneous sample while crack growth becomes increasingly important
closer to failure in the heterogeneous sample (Section 3.2.1). This
behavior corresponds to the existence of a metastable state of crack
nucleation at a system-sized \(\xi\) during a first-order transition,
when the system is vulnerable to the influence of sufficiently large
perturbations (subcritical bifurcation) (Sornette, 2006). This
vulnerability and the resulting discontinuity may be the reason for an
unpredictable failure time (Vasseur et al., 2015).
An estimate for the correlation length exponent (1.15) for Carrara
marble (Kandula et al., 2019) falls almost exactly halfway between the
exponents for Ailsa Craig microgranite found here (0.65 for the
heterogeneous sample and 1.75 for the homogeneous sample). However, the
Carrara marble exponent was estimated by assuming the failure stressa priori , so it is not directly comparable with our results. It
is therefore not possible to confirm whether an inverse power-law would
successfully forecast the failure stress in real time and/or whether a
different model would be more likely. Nevertheless, the nature of
Carrara marble may place it halfway between our two end members. It is
chemically pure, composed of 99% annealed calcite crystals (Alber and
Hauptfleisch, 1999), with a homogeneous microstructure (Oesterling,
2004), a very low permeability (10-19m2) and only 0.2-0.5% connected porosity (Zhang,
1994; Alber and Hauptfleisch, 1999; Bandini et al., 2012;
Cartwright-Taylor et al., 2015). However, studies have shown the
presence of micro-discontinuities within grains, including twin lamellae
(Ramez and Murrell, 1964; Bandini et al., 2012; Cartwright-Taylor et
al., 2015) and a high density of dislocations (Fredrich et al., 1989),
while its isotropic texture consists of both well-locked (xenoblastic)
and more mobile (granoblastic) grain boundaries (Bandini et al., 2012;
Cartwright-Taylor et al., 2015). These factors indicate a complex
history of both static and dynamic recrystallization (Molli and
Heilbronner, 1999; Oesterling, 2004) and introduce a degree of
heterogeneity that may be intermediate between our two samples.
In both samples, the critical value of \(\xi\) is 200 μm, marking the
longest crack supported by the sample volume without a runaway
instability developing. Significantly, this falls just short of the mean
grain size of the groundmass (250 μm). That is, catastrophic failure
occurs when whole grains break. This confirms the findings of Vasseur et
al. (2017) from acoustic emissions (AE) data that the grain size
(inter-particle distance) is a better metric for the characteristic void
dimension at failure than the distance between pores (inter-void
distance).
Our observations highlight the strong dependence of the degree of
predictability on material properties that may be unknown in a field
application, as well as the importance of analyzing several independent
parameters for identifying the type of phase transition and predicting
the point of failure (Lei and Satoh, 2007). They may also explain why,
when looking at long time-series of field-scale seismicity or
deformation, clear and reliable precursors to failure are detected only
in some cases, and preferentially in application to forecasting of
landslides and volcanic eruptions. In other cases, notably in
forecasting of individual large earthquakes, fluctuations related to
instability may be present but may not be statistically significant
enough to be detectable as precursors. In both samples, \(D\) shows
increased clustering earlier than localization is visually apparent in
the µCT images, and therefore may provide useful information about the
impending onset of damage localization for a variety of applications and
settings. Finally, the relatively high strain rates analyzed here may
not be representative of the evolution of precursors at lower strain
rates. For example Ojala et al. (2004) showed that the acceleration to
failure in AE rate asymptotically approaches the behaviour expected of a
single Griffith crack (Figure 1) as strain rate is decreased in
laboratory compression tests on porous sandstones. Nevertheless, we have
confirmed that heterogeneity plays a significant role in determining the
style of evolution of the population of micro-cracks, and hence the
predictability of the system-scale failure time.
Suggestions for future
work
This discussion has highlighted some outstanding research questions to
be addressed in future work. The most notable of these are as follows:
(i) Why do previously obtained degrees of anisotropy inferred from
acoustic measurement differ markedly from our newly obtained structural
ones? (ii) How does crack growth scale (in terms of the ellipsoid
radii), and is it self-affine? (iii) Does the predominant local failure
mechanism change from tensile to shear as system-sized failure
approaches, as seen in the AE data of Graham et al. (2010)? (iv) Does
this transition occur earlier in more heterogeneous materials? Given
tensile fractures are easier to see in imaging void space, the latter
two questions would benefit from digital volume correlation techniques
that are the subject of ongoing work, and can detect local changes in
shear and volumetric strain.