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.