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):