Introduction

Catastrophic failure of rocks in the brittle Earth is a critically-important driving mechanism for phenomena such as landslides, volcanic eruptions and earthquakes, including induced seismicity. Such failure often happens suddenly and with devastating consequences, occurring when structural damage, in the form of smaller faults and fractures, concentrates within localized zones. Damage localization leads to weakening and stress redistribution, eventually resulting in system-sized brittle failure along a distinct and emergent fault plane. Localized damage is pervasive at all scales throughout the brittle crust (Mitchell and Faulkner, 2012) and is therefore a fundamental control on catastrophic failure. Crack nucleation and growth, and crack coalescence within already localized zones, are relatively well-understood from microstructural and field observations of damaged rocks, and from monitoring and locating earthquakes and acoustic emissions (elastic wave packets released during laboratory-scale micro-fracturing events). However, the process of localization remains elusive. Smaller cracks spontaneously self-organize along the incipient fault plane, often immediately before failure, but the precise mechanisms involved in this self-organization have yet to be determined. Open questions include: (i) how do cracks, pores and grain boundaries interact locally with the applied stress field to cause catastrophic failure to occur at a specific place, orientation and time? (ii) why can we detect precursors to catastrophic failure only in some cases?
Fractures and faults have a self-similar structure; they are scale-invariant in their length and spatial distributions (Main et al., 1990; Main, 1996; Bonnet et al., 2001), and in the way their size relates to the energy released during rupture (Abercrombie, 1995; Goodfellow and Young, 2014). Remarkably, earthquakes and acoustic emissions (AE) are indistinguishable apart from the absolute source size, with scaling characteristics that are invariant over 15 orders of magnitude (Goodfellow and Young, 2014). This behavior is controlled by the local stress state and rock mass properties. Classically, brittle rock deformation and failure have been characterized by AE, with progressive cracking in heterogeneous materials under stress leading to systematic changes in the AE event rate and its frequency-magnitude distribution. Experiments have shown that pervasive microcracking accumulates in the sample sub-critically, i.e., without causing system-sized failure (Lockner et al., 1991; Lei et al., 2000), until the accumulating cracks self-organize along an asymmetric, localized damage zone. System-scale failure then occurs when nucleating micro-cracks have localized sufficiently for a runaway positive feedback of self-sustaining crack propagation and coalescence to take over (Main et al., 1993). In some cases, this self-organization becomes evident in the emergence of an inverse power-law acceleration of event rate with a well-defined failure time. In others, system-sized failure of rock samples is commonly associated with the transition from an exponential increase to a sudden, rapid acceleration in the AE event rate close to peak stress (Sammonds et al., 1992; Moura et al., 2005; Vasseur et al., 2015). This transition occurs exactly when cracks begin to localize along the incipient fault plane (Lockner et al., 1991). At this crucial point, nucleated cracks grow by jumping geometrical and rheological barriers, so regions of stress concentration must already be correlated at the scale of the incipient fault network (Sammis and Sornette, 2002). The organized fracture network then propagates dynamically, with macroscopic failure occurring at a well-defined, finite time as the power-law reaches its asymptote.
This behavior indicates a transition from pervasive but stable crack growth, controlled by the sample’s microstructure, to an unstable regime of dynamic rupture along an organized fracture network, controlled by stress and fracture mechanics (Guéguen and Schubnel, 2003; Alava et al., 2008). The inverse power law transition can be described as a critical or second-order phase transition; a continuous transition from one state to another, during which the system becomes extremely susceptible to external factors. It is second-order if the first derivative of the free energy of the system (an entropy term) changes continuously as a function of the control parameter, e.g., temperature (Stanley, 1971, Fig. 2.6) or, in the case of a constant strain (or stress) rate rock deformation experiment, strain (or stress). This is associated with an inverse power law acceleration of the correlation length towards the critical point (Bruce and Wallace, 1989). At this point, strong correlations exist between all parts of the system (including at long-range) and many length scales become relevant (resulting in a self-similar structure and power-law scaling), with events occurring at all relevant length scales (associated with broadband self-similarity of correlations). The transition to an inverse power-law in the AE event rate, with its ‘finite-time singularity’ at failure, is also characteristic of a second-order or critical phase transition (Sammis and Sornette, 2002). If this occurs in the lead up to macroscopic failure, then the failure time can be forecast accurately (Vasseur et al., 2015; 2017). Inverse power-law acceleration to a well-defined failure time has also been seen in the evolving microstructure (micro-crack porosity and the volume of the largest micro-fracture) of crystalline rocks undergoing brittle deformation (Renard et al., 2017; 2018).
However, the evolution of damage does not always allow a fit to a model with a well-defined failure time. In structurally homogeneous materials, there is no emergent, smooth power-law acceleration to failure, as shown experimentally by Vasseur et al. (2015; 2017) for a range of rock types and material analogues. In the extreme case of a single flaw in an otherwise uniform starting material, there is no precursor, and catastrophic system-sized failure occurs suddenly when the flaw propagates dynamically at a maximum in the system free energy. In turn, this depends on the applied stress, the length of the starting flaw, and the specific surface energy of the material (Griffith, 1921; 1924). This results in a discontinuous or first-order transition between intact and failed states. In real materials that possess only a small amount of microstructural disorder, progressive subcritical cracking, i.e., cracking which does not fulfil the conditions for sustained propagation, shows only an exponential increase in the event rate time-to-failure behavior (Vasseur et al., 2015; 2017). Failure occurs suddenly and early; often much earlier than expected from an exponential asymptote (Vasseur et al., 2015; 2017). This behavior is also characteristic of an abrupt first-order transition, with the exponential behavior reflecting local correlations (Stanley, 1971; Sethna, 2006).
Phase transitions are often characterized by the evolution of the correlation length and scaling exponents of the system in question (Stanley, 1971). The correlation length, \(\xi\), is the distance over which the effects of a local perturbation of the system will spread (Thouless, 1989). Close to a critical point, the system can be viewed as made up of regions of size \(\xi\) in the critical state. In this case,\(\xi\) can be interpreted as the size of the regions of the nucleating phase, or the typical linear dimension of the largest piece of correlated spatial structure (Bruce and Wallace, 1989), which in our case is approximately the length of the largest fracture. As the two phases (intact and failed) compete to select the final equilibrium state, regions closer than \(\xi\) are correlated and those further apart are uncorrelated. Approaching the critical point, the correlated (nucleating) regions become comparable to the system size. Thus, the maximum correlation length, and associated parameters such as the maximum AE magnitude, are restricted by the system size.
During a first-order transition, the correlation length \(\xi\) becomes macroscopically large but remains finite until the discontinuity at the sudden change of state (Stanley, 1971). In the case of a single Griffith crack, the correlation length is simply the length of the starting flaw, which suddenly becomes system-sized at failure as the flaw propagates instantaneously through the material (Figure 1, blue line). When real, structurally homogeneous materials, with a dilute population of material flaws, undergo progressive subcritical cracking (e.g., Vasseur et al., 2015; 2017), we expect the correlation length to increase exponentially but remain finite until it becomes system-sized at a sudden-onset discontinuity (Figure 1, orange line). Conversely, during a continuous phase transition, the correlation length \(\xi\) becomes effectively infinite (Figure 1, green line), growing as an inverse power-law function of the control parameter (e.g., temperature), \(T\), approaching the critical point, \(T_{c}\) (Bruce and Wallace, 1989):