This is the type of behavior associated with progressively more
heterogeneous materials undergoing brittle deformation (Vasseur et al.,
2015; 2017). In summary, the degree of microstructural disorder of a
material exerts a strong control on the type of phase transition from
subcritical crack growth to dynamic rupture, and consequently the
predictability of the transition. In particular, experiments (Vasseur et
al., 2017) and models (Kun et al., 2018) have shown that heterogeneity
strongly influences the spatial distribution of micro-cracks at failure.
Here we investigate the impact of material heterogeneity on the nature
of the phase transition between intact and failed states, and the
associated predictability of failure, at the micron-scale. We show how
the micro-crack network evolves within a deforming crystalline rock with
different amounts of disorder. Since pre-existing cracks are the most
dominant factor of all heterogeneities that govern the fault nucleation
process in laboratory rock samples (Lei et al., 2000), we deformed two
samples of Ailsa Craig micro-granite: one being an as-received control
(nominally crack-free), and the other containing a pre-existing
nano-scale crack network, induced by thermal stress, as a proxy for
increased heterogeneity. Ailsa Craig samples, as received from the
quarry, have no detectable cracks on thin sections under both optical
and scanning electron microscopes (Meredith et al., 2005; 2012). They
are an extreme end member of lowest crack density in natural rocks.
Through the analysis of 4D, in-situ synchrotron x-ray
micro-tomography (μCT) images of the two samples undergoing tri-axial
deformation (see Cartwright-Taylor et al. (2020) for access to the
dataset), we test the hypothesis that the transition to failure is
abrupt and unpredictable (first-order) in the as-received sample (our
initially crack-free end member), but is continuous and predictable
(second-order) in the pre-cracked sample. In-situ observation of
the deforming microstructure allows us to measure directly the relevant
parameters such as the correlation length and the scaling exponents.
We find that increasing the microstructural disorder affects the
geometry, size and spatial distribution of the evolving micro-fractures.
Using a combination of visual inspection of the μCT images, geometrical
analysis of the evolving crack network, and techniques used in
statistical seismology, we show that the micro-crack network evolution
varies significantly between the two samples. The degree of starting
heterogeneity controls (i) the evolving spatial clustering and
anisotropy of the micro-cracks, and (ii) the order of the phase
transition. The initially crack-free sample exhibits an exponential
increase in damage that reflects local correlations, a finite
correlation length, and no obvious precursors to failure. In contrast,
the pre-cracked sample exhibits emergent power-law behavior, an inverse
power-law acceleration to infinite correlation length and clear
precursors to failure. However, though the parameters may be different,
the power-law scaling of the micro-crack volume and inter-crack length
distributions, and some crack growth characteristics, appear independent
of heterogeneity. Allowing for the fact that such microscopic failure
characteristics may not be detectable above ambient noise in a field
experiment, this may explain why measureable geophysical precursors to
catastrophic failure events are detected only in some cases.