Conclusions
Our in-situ time-resolved x-ray µCT images of very small samples
of deforming granite show that the heterogeneity of the starting
material exerts a strong control on the evolution of the statistical
properties of crack size and spatial distribution during fracture
network evolution. The accumulating micro-cracks have power-law
frequency-volume and inter-crack length distributions over a finite
scaling range, irrespective of the degree of starting heterogeneity,
with well-determined scaling exponents \(\beta\) (the size exponent) and\(D\) (the correlation dimension). The inferred correlation length\(\xi\) increases exponentially with respect to stress in the
homogeneous case, with sudden-onset, unpredictable failure, analogous to
the behavior expected for a first-order (discontinuous) phase
transition. In contrast, the heterogeneous sample shows an inverse
power-law acceleration to a predictable failure point at the asymptote,
diagnostic of a second-order (continuous) phase transition (Equation 1
and Figure 1). The second-order transition is linked to the distinct
physical process of quasi-static, asymmetric accumulation of damage
within an optimally-oriented zone increasingly localized around the
eventual fault plane, with associated reliable precursors to failure in
the evolution of \(\beta\) and \(D\). The correlation dimension is a key
early indicator of localization on such a shear zone for the
heterogeneous sample. This is not observed within the time resolution of
our observations for the homogeneous sample, where the precursory damage
takes the form of more radially isotropic zones of spalling, and shear
localization must occur very close to the point of dynamic failure
itself to explain the post-failure observation of a shear fault.
Crack nucleation dominates the frequency-size statistics in the
homogeneous case and crack growth in the heterogeneous case. In both
cases, the transition to localized damage occurs by a combination of
nucleation and growth. The timing of the onset of crack coalescence
defines the order of the phase transition, and hence the predictability
of the failure time. Nevertheless, catastrophic failure occurs in both
cases as the correlation length approaches the grain size, which in turn
controls the failure of local bridges between aligned en-echelon and
wing-cracks in the shear damage zone in the heterogeneous sample. The
initial rock microstructure, specifically the anisotropy of pre-existing
porosity, dictates the geometry and orientation of the emergent fault
plane, independent of starting heterogeneity. This reflects the strong
control of starting microstructure on the rock’s internal stress state,
despite the axi-symmetric external loading conditions and the very low
anisotropy (1%) inferred from acoustic velocity measurement.