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.