# Damage spreading in quasi-brittle disordered solids: II. What the statistics
of precursors teach us about compressive failure

Estelle Berthier  Institut Jean Le Rond d’Alembert (UMR 7190), CNRS and
Sorbonne Université, 75005 Paris, France
Current address: Arnold-Sommerfeld-Center for Theoretical Physics and Center
for NanoScience, Ludwig-Maximilians-Universität München, D-80333 München,
Germany  Ashwij Mayya and Laurent Ponson<EMAIL_ADDRESS>Institut Jean
Le Rond d’Alembert (UMR 7190), CNRS and Sorbonne Université, 75005 Paris,
France

###### Abstract

We investigate numerically and theoretically the precursory intermittent
activity characterizing the preliminary phase of damage accumulation prior to
failure of quasi-brittle solids. We use a minimal but thermodynamically
consistent model of damage growth and localization developed by Berthier et
al. [2017]. The approach accounts for both microstructural disorder and non-
local interactions and permits inferring a complete scaling description of the
spatio-temporal structure of failure precursors. By developing a theoretical
model of damage growth in disordered elasto-damageable specimen, we
demonstrate that these scaling relations emerge from the physics of elastic
manifolds driven in disordered media, while the divergence of these quantities
close to failure is reminiscent of the loss of stability of the specimen at
the localization threshold. Our study sorts out a long-standing debate on the
nature of the compressive failure point and the origin of the universal
statistics of the precursors preceding it. Our analysis rules out a critical-
point scenario in which the divergence of the precursor size close to failure
is signature of a second-order phase transition governed by the
microstructural disorder. Instead, we show that while the jerky evolution of
damage prior to failure results from the presence of material disorder, the
latter does not significantly change the nature of the localization process,
which is an instability well described by standard bifurcation theory of
homogeneous systems. Finally, we harness our detailed understanding of the
precursory statistics to design a methodology to estimate the residual
lifetime of a structure from the statistical analysis of precursors. This
method relevant for structural health monitoring is shown to perform rather
accurately on our data.

###### keywords:

Compressive failure , damage accumulation and localization , quasi-brittle
materials , failure precursors , statistical aspects of failure , disordered
materials , failure prediction , structural health monitoring

Quasi-brittle failure takes place through the accumulation and then the
localization of a large number of microcracks in interaction. This failure
mode is ubiquitous in a large range of materials such as concrete, rocks,
woods, ceramics, especially under compressive loading conditions [Ashby and
Sammis, 1990, Lockner et al., 1991, Kachanov, 2003]. However, the way
microcracks evolve and organize over time to ultimately lead to the failure of
a specimen or a structure remains poorly understood.

Quasi-brittle failure generally proceeds in the following way [Fortin et al.,
2006, Manzato et al., 2014, Tal et al., 2016]: First, damage spreads rather
uniformly within the specimen. Then, microcracks progressively coalesce and
organize into fracture patterns of increasing size. Finally, at the
localization threshold, the damage activity concentrates into a macroscopic
band, ultimately leading to the complete failure of the specimen. Noticeably,
damage accumulation preceding localization is accompanied by an intense
acoustic activity [Lockner, 1993, Petri et al., 1994, Fortin et al., 2009].
These acoustic emissions are reminiscent of bursts of damage activity, also
referred to as precursors, separated by silent periods reminiscent of elastic
reloading phases. As localization approaches, the intermittency in the damage
growth, together with the acoustic activity intensify: The size and duration
of the bursts both increase as a power law with the distance to failure
[Guarino et al., 1998, Girard et al., 2010, Kun et al., 2013, Baró et al.,
2018, Vu et al., 2019]. As a result, close to failure, they largely exceed the
characteristic microstructural size of the material and its associated time
scale.

The statistical features of the mechanical and acoustic bursts have been
extensively investigated both experimentally [Garcimartin et al., 1997,
Guarino et al., 2002, Deschanel et al., 2006, Davidsen et al., 2007, Rosti et
al., 2009, Baró et al., 2013, Vu et al., 2019] and numerically [Herrmann and
Roux, 1990, Zapperi et al., 1997b, Tang, 1997, Amitrano, 2006, Alava et al.,
2006, Pradhan et al., 2010, Girard et al., 2010, Kun et al., 2014]. It turns
out that they follow power-law statistics involving scaling exponents that are
robust and independent, to a large extent, of the type of materials and
loading conditions. These observations have fed a long-standing debate on the
nature of failure by damage accumulation. Several studies interpreted failure
as a first-order transition [Zapperi et al., 1997a, Alava et al., 2006, da
Rocha and Truskinovsky, 2020]. While consistent with the observation of a loss
of material rigidity at the failure point, this approach does not account for
the universal scaling relations characterizing the precursors, like the power
laws relating their size, duration and spatial extent, as well as the
divergence of these quantities close to failure, two features that are
generally associated with critical phenomena. As a result, it has also been
argued that compressive failure can be interpreted as a second-order phase
transition where the critical point corresponds to the failure threshold
[Garcimartin et al., 1997, Moreno et al., 2000, Girard et al., 2010, Weiss et
al., 2014, Vu et al., 2019]. In this scenario, the scaling exponents involved
in the precursors statistics are critical exponents. Their value depends on
the range of the elastic interactions, while the divergence of the bursts size
and duration close to failure is reminiscent of the classical phenomenology of
critical phenomena. This scenario is particularly attractive as it entails
universality of the scaling exponents. Yet, a theoretical proof of the
critical nature of compressive failure in elasto-damageable solids is still
lacking. As a result, the statistical features of the precursors observed
experimentally, such as the value of the scaling exponents, still remain
largely unexplained.

In the present work, we revisit this debate by performing a thorough analysis
of the precursors statistics in a simplified model of failure by damage
accumulation. We do not seek to provide a comprehensive modeling of the
intermittent evolution of damage observed during compression experiments in
all its complexity. Instead, we aim to identify and characterize the
elementary mechanisms that underlie intermittency during compressive failure
and account qualitatively for the statistical features of precursors observed
experimentally. For that purpose, we use a simplified damage model that
qualitatively captures the various aspects of damage accumulation in
disordered solids. We discuss the discrepancies between this simple model and
real materials response and their implications in our interpretation. We
consider the simplified model of damage accumulation and failure proposed in
Berthier et al. [2017], which we review in the first section. In this
approach, quasi-brittle specimens are described at a continuum mesoscopic
scale by a 1D array of interacting elasto-damageable elements loaded in
parallel with randomly distributed damage thresholds. Interactions between
neighboring elements emerging from elasticity are described qualitatively by
introducing a cooperative length scale through a non-local damage variable
[Pijaudier-Cabot and Bazant, 1987, Frémond and Nedjar, 1996, Pijaudier-Cabot
and Grégoire, 2014]. This length scale governs the spatial extent of the
stress redistribution taking place in the specimen after an individual damage
event. As shown by Berthier et al. [2017], this approach is sufficiently
simple to reveal and identify the basic mechanisms underlying quasi-brittle
failure and rich enough to capture the main features of their mechanical
response. In particular, it predicts damage localization and catastrophic
failure, which was showed to result from the unstable evolution of specific
growth modes of the damage field that are selected by the spatial structure of
the non-local interactions. Here, we first show that this model also provides
a realistic description of the precursory activity preceding failure. We
investigate the size, duration and spatial extent of damage bursts as
predicted by this approach and demonstrate that they follow scaling relations.
We also demonstrate that these quantities, on average, diverge as the specimen
is driven closer to failure, thus reproducing qualitatively the main features
observed experimentally [Garcimartin et al., 1997, Guarino et al., 2002, Baró
et al., 2018, Vu et al., 2019].

Taking advantage of the simplicity of the considered model, we then explore
theoretically the elementary mechanisms at the origin of the jerky evolution
of damage during the accumulation phase preceding failure. We first retrieve
that precursors are cascades of elementary failure events, also called
avalanches, triggered by each other through the non-local stress
redistribution following each individual event. We then derive an evolution
equation of the damage field within the specimen and show that it behaves like
an interface, the elasticity of which derives from the non-local nature of the
interactions considered in our damage model. This interface is driven in a
disordered medium reminiscent of the disordered fracture properties of the
specimen with a driving speed that diverges as the specimen approaches
failure. This mapping between quasi-brittle failure and the realm of driven
elastic interfaces [Barabási and Stanley, 1995], already theorized by Weiss et
al. [2014] and Vu et al. [2019], successfully accounts for the statistical
features of the precursors observed at some finite distance to failure. In
particular, it explains the scaling relationships between the size, the
duration and the spatial extent of precursors as observed in our simulations.
However, at odds with Weiss et al. [2014] and Vu et al. [2019], our analysis
shows that the divergence of these quantities close to the localization
threshold is not reminiscent of the depinning transition, a dynamic phase
transition emerging from the competition between disorder and elasticity met
by driven elastic interfaces [Narayan and Fisher, 1993]. Instead, this
behavior derives from the unstable nature of damage localization [Rudnicki and
Rice, 1975, Bigoni, 2012, Dansereau et al., 2019], a standard bifurcation that
does not result from the presence of material disorder.

This interpretation of the divergence of the precursor size and duration close
to failure has several important implications that are thoroughly discussed in
our article. In particular, it implies that final failure does not result per
se from the micro-instabilities observed during the precursory accumulation
phase. Precursors are simply by-products of the collective growth of the
damage field in interaction with the material disorder. As such, shutting down
the disorder would also shut down the intermittency and would result in a
smooth homogeneous growth of damage until the localization threshold is
reached. From a modeling perspective, it means that the approach proposed by
Berthier et al. [2017] and extended more recently to 2D elasto-damageable
solids by Dansereau et al. [2019] using standard bifurcation theory of
homogeneous systems is sufficient to account for damage localization and
failure, as well as to predict the load bearing capacity of quasi-brittle
solids with a reasonable accuracy. The proposed explanation of the divergence
of the burst size and duration close to failure suggests that this mechanism
is ubiquitous in compressive failure and follows the same law in a large range
of materials.

In the last part of our study, following the seminal ideas of Sornette [2002],
we harness this property to design a methodology that predicts the residual
lifetime of structures from the statistical analysis of precursors. We test
the approach on our numerical data set and show that the methodology
successfully predicts failure from the evolution of the precursor sizes
measured over a short loading window far from localization. Hence, this
approach paves the way for quantitative and predictive methods of structural
health monitoring in more complex situations [Mayya et al., 2020].

Our article is organized as follows. In the first section, we review the model
proposed by Berthier et al. [2017] which is used to simulate the intermittent
evolution of damage in quasi-brittle solids. The second part is dedicated to
the thorough statistical characterization of the precursors. Their magnitude,
duration and spatial extent are related to each other by simple scaling laws
and their variations over time display a power-law increase as failure
approaches. The third section deals with the theoretical description of the
intermittency and the interpretation of these properties. We provide an
evolution law of the damage field that accounts for the collective evolution
of the damage within the specimen and that fully captures the statistical
properties observed numerically. This evolution equation sheds light on the
complex connection between quasi-brittle failure and the theoretical framework
of driven elastic interfaces. It also provides explanation for the divergence
of the precursor size close to failure as a result of the unstable nature of
the localization process. The last section is dedicated to the application of
these concepts in structural health monitoring. We bring the numerical proof
of concept that the non-stationary nature of the precursor statistics can be
harnessed to predict the residual lifetime of progressively damaging
specimens.

##  1 Model and numerical implementation

In this section, we provide the main features of the model used numerically to
investigate the intermittent evolution of damage during compressive failure.
We consider a material description where each element damages progressively
owing to the microfracturing processes at lower length-scales. Hence, both
heterogeneities in damage resistance and elastic redistributions after damage
are described at a mesoscopic continuum scale. The reader is invited to refer
to the work of Berthier et al. [2017] for a more detailed review and
discussion on the average failure behavior predicted by this approach.

###  1.1 A thermodynamics based damage model

In our description, we consider a uni-dimensional structure ($\Sigma$) made of
elasto-damageable elements in parallel. Such a system is loaded between two
rigid plates: The bottom one is clamped while a quasi-static uni-axial
loading, controlled in displacement $\Delta$, is applied to the top plate that
experiences a reaction force $F$. We consider that each element is
homogeneously deformed and undergoes a displacement equal to the macroscopic
one. Individual elements are characterized by a scalar damage parameter $d$.
This quantity, analogous to a microcracks density, ranges from zero when the
element is intact to one when fully broken. Such a damage level affects the
elastic response of each element as its elastic stiffness $k$ decreases with
$d$. Each element represents a mesoscopic heterogeneity of randomly drawn
damage energy $Y_{c}$ that vary with the damage level in the element. As a
result, the total energy required to fully break an element is given by
$\int_{0}^{1}Y_{c}(\tilde{d})d\tilde{d}$. The total energy of the system
comprising the elasto-damageable specimen and the loading device is thus
written as the sum of three contributions: The elastic energy $E^{el}$ stored
in the materials elements, the energy $E^{d}$ dissipated due to damage growth
and the work $W$ of the external force. It follows

$E=E^{el}+E^{d}-W=\int_{\Sigma}\dfrac{1}{2}\Delta^{2}k(\overline{d}(x))dx+\int_{\Sigma}\int_{0}^{d(x)}Y_{c}(x,\tilde{d})d\tilde{d}dx-\int_{0}^{\Delta}F(\tilde{\Delta})d\tilde{\Delta}.$
(1)

As expressed in this equation, the local stiffness is chosen to depend on a
non-local damage parameter $\overline{d}$ rather than on its local counterpart
$d$. This non-local damage field corresponds to the weighed average

$\overline{d}(x)=\alpha(x)\ast
d(x)=\int_{\Sigma}\alpha(x-x^{\prime})d(x^{\prime})dx^{\prime}.$ (2)

As shown in Berthier et al. [2017], the introduction of such a non-local
damage variable allows for a simple and practical implementation of the
elastic interactions within the 1D structure. Indeed, each individual damage
event triggers a redistribution of stress, the spatial extent of which is set
by the so-called interaction function $\alpha(x)$. Interestingly, with this
approach elastic interactions can be tuned to explore their impact on the
failure response of the specimen, as done in Berthier et al. [2017]. In
addition, this formulation ensures that the total energy is conserved all
along the process of damage accumulation.

A damage criterion is then obtained by differentiating Eq.(1) with respect to
the damage parameters, and invoking energy conservation. In practice, the rate
of mechanical energy composed of work of the external force and the elastic
energy released in the specimen must compensate the energy dissipated by
damage for the damage increment. Denoting $\mathcal{F}(x)=-\dfrac{\delta
E}{\delta d}$ the total driving force for damage, the damage criterion writes
as

$\left\\{\begin{array}[]{lcl}\vspace{5pt}\mathcal{F}(x)&=&\overline{Y}(x)-Y_{c}(x)<0\quad\Rightarrow\quad\mathrm{No\,damage}\\\
\mathcal{F}(x)&=&\overline{Y}(x)-Y_{c}(x)=0\quad\Rightarrow\quad\mathrm{Damage\,growth}\end{array}\right.$
(3)

where $\overline{Y}(x)$ is the non-local elastic energy release rate. It
writes as the convolution of the interaction function $\alpha(x)$ with the
local rate of energy restitution $Y(x)$ (see Berthier et al. [2017]):

$\begin{split}\overline{Y}(x)&=\alpha(x)\ast Y(x)\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{where}\\\
Y(x)&=-\dfrac{1}{2}\Delta^{2}k^{\prime}(d(x)).\end{split}$ (4)

Here, $k^{\prime}$ denotes the derivative of the stiffness with respect to the
damage parameter $d$. Note that we adopt here a thermodynamically-consistent
damage mechanics framework [Frémond and Nedjar, 1996, Pham et al., 2011b], in
a similar manner to fracture mechanics predicting crack propagation from the
balance of mechanical and fracture energy [Lawn, 1993, Rice, 1978]. The
particularity of the proposed model relies in that the driving force involved
in the damage criterion is non-local. This arises from the dependency of the
local stiffness on the non-local damage parameter. It means that the driving
force in one element depends on the damage level within the surrounding
elements, in a manner defined by the interaction function $\alpha$. It also
implies that an increase of damage in one element induces an update of the
damage driving force in an extended region around the damaging element.
Therefore, the tunable function $\alpha$ controls the mechanism of stress
redistribution following damage events.

Energy conservation is illustrated in Fig. 1(a). Below the elastic limit
$\Delta_{\mathrm{el}}$, the elastic energy release rate is not sufficient to
damage the material: therefore the work of the external force is entirely
stored as elastic energy. Above $\Delta_{\mathrm{el}}$, the mechanical energy
injected within the medium via the work of the external force is shared among
stored elastic energy and released dissipated energy due to damage. As we will
see in the following, the dissipation of mechanical energy through damage
occurs through sudden bursts separated by silent elastic phases. This
intermittency during quasi-brittle failure of disordered materials is the main
focus of this study.

###  1.2 Material parameters

The microstructural heterogeneities of the material are accounted for by
spatial variations in the damage energy field $Y_{c}(x,d)$, described by the
quenched noise $y_{\mathrm{c}}(x,d)$. For each element located at $x$ and
having a damage level $d$, $y_{\mathrm{c}}(x,d)$ is drawn from a uniform
distribution of standard deviation $\sigma$ and zero mean value. The field of
resistance to damage hence writes as

$Y_{c}(x,d)=Y_{c0}\left[1+y_{\mathrm{c}}(x,d)+\eta\,d\right]$ (5)

where $\eta>0$ is a hardening parameter and $Y_{c0}$ is the average damage
energy of the intact material. Such a toughening behavior has been observed
experimentally [Berthier, 2015, Renard et al., 2018] and considered in other
damage models [Pham et al., 2011a]. This hardening behavior may be related to
our modeling choice that consists in describing damage growth at a mesoscopic
scale. Indeed, as the mesoscopic element progressively damages, the
distribution of damage thresholds within this material element is explored
from its lowest values to its highest ones. As a result, the value of the
parameter $\eta$ may emerge from the spreading of microcracking thresholds at
the microscopic scale.

In terms of statistical modeling, our description can be seen as intermediary
between original fiber bundle models comprising brittle fibers and continuum
damage mechanics model with homogeneous resistance to failure. It allows
exploring material behaviors with extended post-peak response.111 Small
hardening parameters $\eta\simeq 0$ result in a brittle failure of the
material element and, as a consequence, of an abrupt failure of the specimen
after the first elements break down The population of precursors in this case
is dominated by the material disorder: Catastrophic and sudden failure without
precursors is observed for weakly disordered solids whereas strong disorder
leads to delayed fracture with power-law distributed precursors [Shekhawat et
al., 2013]. The mechanical response remains qualitatively unaffected for a
broad range of parameter values as long as disorder is present and a
transition from a stable to an unstable growth of damage in the constituting
elements takes place as $d(x)$ increases (see Berthier et al. [2017] for the
determination of the ranges of material parameter verifying this condition).

The stiffness decay with damage, representative of the degradation of the
elastic properties as microcracks develop, is introduced via a polynomial
expression. For a single element, such a softening writes as

$k(d)=k_{0}\left[ad^{3}-(a+1)d+1\right]$ (6)

where $k_{0}$ is the stiffness of the intact material and $a$ is a constant
the sign of which, in combination with $\eta$, sets the stability of
individual elements (see Berthier et al. [2017]). Such a polynomial expression
allows for the exploration of a large range of mechanical response at the
element scale. Different softening laws $k(d)$ can arise from a large variety
of microscopic damage processes, and we can for example refer to Kachanov
[1993] for a thorough investigation of the dependency of the elastic moduli on
the considered damage processes. Moreover, in combination with an appropriate
choice of hardening parameter $\eta$, the expression (6) with $a<0$ may result
in an extended regime of stable damage growth even in the limit of small
disorder, unlike in conventional fiber bundle models where such a ductile-like
response is strongly dependent on the disorder level [Shekhawat et al., 2013,
da Rocha and Truskinovsky, 2020]. The linear decrease of the elastic modulus
with damage (recovered here by setting $a=0$) classically used in continuum
damage mechanics [Lemaitre, 1992] leads to local damage models that do not
capture the intermittent evolution of damage in disordered materials.
Nonlinear expressions as used in Girard et al. [2010] and Thilakarathna et al.
[2020] results in non-local interactions during damage growth, giving rise to
more realistic dynamics without losing the generality of the model.

The interaction function $\alpha$ introduced in Eq. (7) to define the non-
local damage parameter is expressed as

$\alpha(x)=\alpha_{0}\exp\left(-\dfrac{|x|}{2\ell}\right)$ (7)

In this expression, $|x|$ represents the distance between elements,
$\alpha_{0}$ is a normalization constant ensuring energy conservation such
that$\int_{\Sigma}\alpha(x)dx=1$ and $\ell_{o}$ is an internal length which
controls the spatial extent of interactions within the specimen. In
particular, it provides the range over which the driving force is
redistributed after a damage event. Here also, as illustrated subsequently,
the main results of our study remain robust to the choice of $\ell_{o}$ as
long as it is much larger than the element size and much smaller than the
specimen size. In practice, we consider internal lengths several times larger
than the heterogeneity size and system sizes several thousand times larger.
While real materials display long-range interactions [Bazant, 1994, Dansereau
et al., 2019], similar forms of interaction functions have been assumed in
non-local models of damage evolution in quasi-brittle materials [Frémond and
Nedjar, 1996, Pham et al., 2011b, Pijaudier-Cabot and Grégoire, 2014]. The
expression in Eq. (7) permits analytical calculations of failure and
localisation threshold and qualitatively captures the main aspects of damage
evolution resulting from the cooperative action of disorder and interactions.

###  1.3 Numerical resolution

We take advantage of the quasi-static loading conditions and adopt an extremal
dynamics inspired by Schmittbuhl et al. [1995] based on the following rules:
The imposed displacement is increased until the failure criterion is reached
for at least one element. The damage level of this element is then increased
by an elementary increment $\delta d_{0}\ll 1$, and the spatial distribution
of driving forces in the material is updated using Eqs. (4), (5) and (7). As
schematically shown in Fig. 1(d), the redistribution of driving force results
in additional damage, leading to additional driving force redistributions
until such a cascade stops. This occurs when the driving force $Y$ is below
its critical value $Y_{c}$, everywhere in the specimen. The external load is
then further increased until the weakest element of the array reaches its
failure threshold, triggering another cascade of damage events.

To simulate damage growth, we consider a discretized specimen made of $N$
elasto-damageable elements loaded in parallel. Initially, damage $d(x)=0$ is
zero everywhere and the field $Y_{c}(x,d)$ of damage resistance is randomly
distributed owing to the disorder term $y_{c}(x,d)$. We compute the non-local
damage field described in Eq. (7) using the convolution property of Fourier
transforms,

$\overline{d}(x)=\mathrm{F}^{-1}(\tilde{\alpha}(q)\times\tilde{d}(q))$ (8)

where, $\mathrm{F}^{-1}$ denotes the inverse Fourier transform and
$\tilde{\alpha}$ is the Fourier transform of the function $\alpha$. This is
then used to calculate the damage driving force of Eq. (4)

$\overline{Y}(x)=-\frac{1}{2}\mathrm{F}^{-1}(\tilde{\alpha}(q)\times\tilde{k^{\prime}}(\overline{d}))\Delta^{2}.$
(9)

The displacement $\Delta$ is increased such that exactly one element satisfies
the damage criterion $\overline{Y}(x,d)=Y_{c}(x,d)$. The damage level for this
element is then increased by $\delta d_{0}\ll 1$. After that, the field of
damage driving force is recalculated. The heterogeneous field of damage energy
is also redrawn after each damage event as

$Y_{c}(x,n_{0})=Y_{c0}[1+y_{c}(x,n_{0})+n_{0}\overline{\eta}]$ (10)

where, $n_{0}=d/\delta d_{0}$ is the total number of instances that an element
has damaged, $\overline{\eta}=\eta\delta d_{0}$ and $y_{c}(x,n_{0})$ is a
random number drawn from a centered uniform distribution with standard
deviation $\sigma$. Additional damage events may follow due to the
redistribution. The cascade process stops when $\overline{Y}(x)<Y_{c}(x,d)$
for all elements. Macroscopically, a load drop at constant displacement is
registered due to the increase of the damage level. The displacement is then
again increased so that one element satisfies the damage criterion. This
sequence of damage growth and reloading continues till damage variable is one
in all material elements.

There is no explicit time in the simulation. Yet, we can define a posteriori
the rate $v_{\mathrm{ext}}$ at which the displacement imposed to the specimen
is increased. We consider the limit $v_{\mathrm{ext}}\rightarrow 0$ that
corresponds to quasi-static loading conditions. Thus, the elastic reloading
phases are much longer than the dissipative phases. This justifies our
numerical procedure. In addition, it results in two specific features: Damage
cascades take place under fixed imposed displacement and precursors are well
separated in time so they appear as isolated dissipative events. In the
following, we use $\eta=9$, $a=-0.3$, $\delta d_{0}=0.005$ and $\sigma=0.2$.
The reader is invited to refer to Berthier et al. [2017] for a detailed study
of the effect of the parameter values $\eta$ and $a$ on the damage evolution.
Note however that the statistical properties of the precursors revealed by our
study remain identical for other parameter values as long as failure is
preceded by a preliminary regime of damage accumulation.

##  2 Intermittency and precursors statistics

###  2.1 Global and local scale manifestations of the intermittent growth of
damage

As observed experimentally by Petri et al. [1994], Guarino et al. [1998],
Davidsen et al. [2007], Baró et al. [2013] or Vu et al. [2019], the energy
dissipated through damage during quasi-brittle failure evolves in a step-wise
fashion, even though the applied load amplitude is continuously increased.
This phenomenon is captured by our approach, as illustrated in the inset of
Fig. 1(a). The mechanical response of the specimen under slowly increasing
load consists of slow reloading phases where the material behaves elastically
(identified by plateau regimes in the evolution of the accumulated dissipated
energy) and micro-instabilities taking place at constant displacement
(identified by instantaneous increase of the accumulated dissipated energy).
Owing to the description of damage as a transfer of mechanical energy into
dissipated energy, the evolution of the elastic energy stored in the specimen
also reflects such a jerky dynamics (not shown here). The signature of this
intermittent behavior is also observable from the macroscopic force-
displacement curve, as shown in the inset of Fig. 1(b). Here, the elastic
reloading phases are separated by sudden drops of force resulting from the
decrease in elastic stiffness associated with the degradation of the material.
As further highlighted later, a remarkable feature of these fluctuations is
the broad range of scales that they cover. This is in contrast with standard
Gaussian statistics that describes mild fluctuations, for example observed in
at-equilibrium systems, and that are reminiscent of thermal fluctuations.

Such an intermittent dynamics, also referred to as crackling noise [Sethna et
al., 2001], is reported for a large range of mechanics problems involving
disorder, such as tensile failure of brittle disordered solids through crack
growth [Ponson, 2016], wetting of disordered surfaces [Eggers et al., 2009] or
imbibing of porous media [Planet et al., 2009]. It is also observed beyond
mechanics, for example during the magnetization of ferromagnets, displaying
the so-called Barkhausen noise [Zapperi et al., 1998], or even in social
systems as illustrated by the application of these concepts for describing the
abnormally large fluctuations of the stock markets [Sornette et al., 1996]. As
a common denominator to all these very different physical systems, long-range
interactions in interaction with disorder result in feedback loops that
trigger micro-instabilities with broadly distributed sizes.

The mechanism underlying such avalanches in compressive failure is illustrated
in Fig. 1(c). It consists of a cascade of local failure events triggered by
each other by the redistribution of stress in the specimen that follows each
elementary damage. A remarkable feature of these cascades is the evolution of
their energy (referred to as their size $S$) that increases as the specimen
approaches failure, as shown in the inset of Fig. 1(b). This evolution reveals
the progressive loss of stability of the specimen, a connection further
explored in Section 3.

Beyond the time-series features of the damage evolution, we also investigate
the spatial structure of the damage cascades. To that end, we study damage
spatial organization at the local scale by comparing the damage field before
(blue) and after (red) a certain number (typically ten) of successive
avalanches, as illustrated in Fig. 1(d) for four different load levels. Far
from failure, i.e. at low damage levels $\langle d\rangle\lesssim 0.55$,
damage spreading is dominated by material disorder. Damage grows rather
uniformly within the specimen, and the incremental variation of the damage
field is characterized by small clusters of dimension comparable with the
characteristic size of the material heterogeneity. On the contrary, close to
failure for $\langle d\rangle\simeq 0.9$, we observe compact localized
avalanches, the size of which is much larger than the heterogeneity size. At
this stage, the redistribution process dominates over material disorder,
leading to large and structured precursors. This competition between material
disorder and elastic interactions, as well as its evolution as the specimen is
driven towards failure, will be thoroughly investigated in the following.

Figure 1: Observation of precursors and basic damage mechanism: (a) Evolution
of the elastic energy $E_{\mathrm{el}}$, the dissipated energy
$E_{\mathrm{d}}$ and the work $W$ of the external force with the imposed
loading amplitude $\Delta$, normalized by the elastic limit
$\Delta_{\mathrm{el}}=\sqrt{-2Y_{c0}/k^{\prime}(0)}$. Note that the sum of the
three terms remains constant, as a result of the energy conservation principle
from which derives our model. The inset emphasizes the intermittent nature of
damage evolution that displays bursts of dissipated energy; (b) macroscopic
force-displacement response during damage growth normalized by the elastic
limits $F_{el}$ and $\Delta_{\mathrm{el}}$ of the specimen displaying
catastrophic failure for $\Delta_{\mathrm{c}}\approx
2.1\,\Delta_{\mathrm{el}}$. The size $S$ of damage bursts is shown in the
lower inset as a function of the distance to failure. The effect of the damage
cascades on the force-displacement response of the specimen is evidenced by
the sudden force drops shown in the upper inset; (c) schematic of the feedback
loop at the origin of the precursory micro-instabilities; (d) damage field
before (blue) and after (red) ten successive avalanches at different distances
to failure. The damage field is normalized by the critical damage level
$d_{\mathrm{c}}$ at failure.

###  2.2 Statistical characterization of precursors

To investigate the intermittent evolution of damage, we characterize each
precursor by three quantities: its spatial extent $\ell_{\mathrm{x}}$, its
size $S_{\mathrm{d}}$ and its duration $T$. $\ell_{\mathrm{x}}$ corresponds to
the distance between the two most distant damaged elements belonging to the
same damage cascade. $S_{\mathrm{d}}$ is defined as the total number of
elementary damage increments (the quantity $\delta d_{0}$ in Section 1.3)
constituting the cascade. As shown from its linear variation with the
dissipated energy $S$ during the whole cascade (see Fig. 2(a)), $S_{d}$
measures the energy dissipated through a precursor. In the following, we
consider equivalently $S$ or $S_{\mathrm{d}}$ to measure the precursor size.
Despite the discrete dynamics adopted in our numerical scheme, we can also
define the duration $T$ of a precursor from the total number of damage
redistribution loops involved during the cascade. This amounts to assume that
an individual damage event and the resulting redistribution of driving force
take place over some material specific time scale that is much smaller than
the characteristic time of the applied driving, an assumption that is
supported by the limit $v_{\mathrm{ext}}\rightarrow 0$ considered in this
study.

Remarkably, these three quantities are related through power laws as shown in
Fig. 2. In particular, the spatial extent $\ell_{\mathrm{x}}$ of the
precursors is related to its size $S_{\mathrm{d}}$ via the fractal dimension
$d_{\mathrm{f}}$ as

$\ell_{x}\sim S_{\mathrm{d}}^{1/d_{\mathrm{f}}}$ (11)

where $d_{\mathrm{f}}=2.35\pm 0.15$. As shown by the renormalization of the
ordinates of Fig. 2(c), $\ell_{\mathrm{x}}$ scales linearly with the
interaction length $\ell_{\mathrm{0}}$ introduced in the redistribution
function. Note also that the smallest precursors following such a scaling
behavior are of the order of $10\,\ell_{\mathrm{0}}$, suggesting that such
damage cascades emerge from the cooperative response of a sufficiently large
number of material elements in interaction. Investigating now the duration of
the precursors in Fig. 2(b), we observe that it scales with the avalanche size
as

$T\sim S_{\mathrm{d}}^{z/d_{\mathrm{f}}}$ (12)

where $z=1.40\pm 0.15$ is the so-called dynamic exponent. Therefore, these
three quantities (size, duration and length scale) can be related to each
other and, by virtue of these scaling, we can equally use one or the other to
characterize the magnitude of a damage cascade.

Figure 2: Scaling behavior of precursors as observed during the whole process
of damage evolution: Relationship between (a) the dissipated energy $S$, (b)
the duration $T$ and (c) the spatial extent $\ell_{\mathrm{x}}$ of the
precursors to the total number $S_{\mathrm{d}}$ of elementary damage
increments per cascade. The dashed lines indicate the scaling relationships
$S\sim S_{\mathrm{d}}$, $T\sim S_{\mathrm{d}}^{z/d_{\mathrm{f}}}$ and
$\ell_{\mathrm{x}}\sim S_{\mathrm{d}}^{1/d_{\mathrm{f}}}$, respectively,
leading to the value of the fractal dimension $d_{\mathrm{f}}=2.35\pm 0.15$
and the dynamic exponent $z=1.40\pm 0.15$.

As noticed previously, the intermittency during damage spreading is non-
stationary. We now explore this feature through the detailed analysis of the
avalanche size distribution computed at different distances $\delta$ to
catastrophic failure. We define $\delta$ as

$\delta=\dfrac{\Delta_{\mathrm{c}}-\Delta}{\Delta_{\mathrm{c}}-\Delta_{\mathrm{el}}}$
(13)

where $\Delta_{\mathrm{el}}$ and $\Delta_{\mathrm{c}}$ are the elastic limit
and the failure load, respectively. This quantity ranges from one as the first
damage event takes place to zero at failure.

The precursors statistics is then investigated at different instants during
the progressive failure of the specimen. To do so, we divide the specimen’s
lifetime into bins and, considering several realizations of the disorder, we
compute the distribution of precursor sizes in each bin. Because the focus of
this paper is on the precursors to failure and not the failure event itself,
the final cascade during which catastrophic failure takes place is left aside.
The distributions $P_{\delta}(S)$ of avalanche sizes are shown in Fig. 3(a)
for $\ell_{0}=5$. Irrespective of the value of the interaction length, a power
law distribution with exponent $\beta=1.5\pm 0.1$ is obtained for avalanches
of size $S\ll S^{\star}$. Above the cut-off $S^{\star}$ which increases as
failure is approached, the probability density decays exponentially fast.
Hence, the avalanche size distributions are well described by

$P_{\delta}(S)\sim S^{-\beta}e^{-S/S^{\star}}.$ (14)

The variations of $S^{*}$ with the distance to failure $\delta$ is shown for
the interaction lengths $\ell_{0}=5$ and $\ell_{0}=10$ in the inset of Fig.
3(a). In both cases, a power-law behavior

$S^{*}\sim 1/\delta^{\gamma}$ (15)

is observed with $\gamma=1.0\pm 0.1$.

This property rationalizes the observation made in the inset of Fig. 1(b): As
the specimen approaches failure, the characteristic precursor size set by the
cut-off $S^{\star}$ increases. Owing to the scaling relations shown in Fig. 2,
it implies that the characteristic spatial extent and characteristic duration
of the precursors also increase as failure is approached. This is indeed
verified in Fig. 3(b): The cutoffs $T^{\star}$ and $\ell^{\star}$ as extracted
from the distributions of duration $P_{\delta}(T)$ and spatial extent
$P_{\delta}(\ell_{\mathrm{x}})$ of avalanches also increase with the distance
to failure as

$\left\\{\begin{array}[]{lcl}\vspace{3pt}T^{\ast}&\sim&1/\delta^{\phi}\\\
\ell_{\mathrm{x}}^{\ast}&\sim&1/\delta^{\kappa}\end{array}\right.$ (16)

with $\phi=0.53\pm 0.10$ and $\kappa=0.37\pm 0.10$.

We can now connect these relationships with the ones previously evidenced in
Fig. 2 between the size, the duration and the spatial extent of all the
precursors. Applying them to the cut-off of the statistical distributions at
some given distance $\delta$ to failure, one obtains
$T^{\star}\sim(S^{\star})^{z/d_{\mathrm{f}}}$ and
$\ell_{\mathrm{x}}^{\star}\sim(S^{\star})^{1/d_{\mathrm{f}}}$ that, together
with Eqs. (15) and (16), lead to
$T^{*}(\delta)\sim(S^{*}(\delta))^{z/d_{\mathrm{f}}}\sim\delta^{-\gamma
z/d_{\mathrm{f}}}$ and
$\ell_{x}^{*}(\delta)\sim(S^{*}(\delta))^{1/d_{\mathrm{f}}}\sim\delta^{-\gamma/d_{\mathrm{f}}}$
from which we infer the relationships between exponents

$\left\\{\begin{array}[]{lcl}\vspace{3pt}\phi&=&\gamma\,z/d_{\mathrm{f}}\\\
\kappa&=&\gamma/d_{\mathrm{f}}.\end{array}\right.$ (17)

Interestingly, a third relationship $\phi=\kappa\,z$ relating the exponents
$\phi$ and $\kappa$ characterizing the non-stationary nature of the damage
evolution with the dynamic exponent $z$ can be derived from the previous
equations. We can verify that all these scaling relations are consistent with
the values of the exponents measured in our simulations.

Figure 3: Statistics of avalanches at different distances to failure: (a)
normalized probability density function computed at different distances
$\delta$ to catastrophic failure. The dashed line indicates the scaling
behavior $S^{-1.5}$. The variations of the cutoff avalanche size $S^{*}$ as
defined by Eq. (14) is shown in inset. It varies as $S^{\star}\sim 1/\delta$
as highlighted by the dashed line; (b) evolution of the cutoffs duration
$T^{\star}$ and spatial extent $\ell_{x}^{\star}$ extracted from the
distributions $P_{\delta}(T)$ and $P_{\delta}(\ell_{\mathrm{x}})$ (not shown
here). They also increase as a power-law $T^{\star}\sim 1/\delta^{\phi}$ and
$\ell_{\mathrm{x}}^{\star}\sim 1/\delta^{\kappa}$ with the distance to failure
with the exponents $\phi\simeq 0.53$ and $\kappa\simeq 0.37$.

We now focus on the spatial characterization of the precursors at the local
scale. As the specimen is driven closer to failure, damage cascades extend
over material regions of increasing size. We aim to find the footprint of this
increasing cooperative length scale on the accumulated damage field by
introducing the correlation function

$C(\delta x)=\sqrt{\langle\langle d(x)\rangle^{2}_{\delta
x}\rangle_{x}-\langle\langle d(x)\rangle_{\delta
x}\rangle^{2}_{x}}=\mathrm{std}({\langle d(x)\rangle_{\mathrm{\delta x}}}).$
(18)

This function is then averaged over different disorder realizations, at some
given distance $\delta$ to failure. From the central limit theorem, one
expects the correlation function to decay as $1/\sqrt{\delta x}$ for an
uncorrelated damage field, as awaited from a purely random process of damage
growth. On the contrary, if the damage field is correlated over some length
scale, a deviation from this behavior is expected. Hence, this correlation
function reveals deviation from pure random processes and determines the
scales at which such deviations take place.

Figure 4 shows the correlation function as a function of the box size $\delta
x$ after normalization by $\sqrt{\delta x}$, for $\ell_{0}=5$ and various
distances to failure. As expected, far from failure (see for example
$\delta=0.98$), the normalized correlation function is constant, pointing out
an uncorrelated damage field. As failure is approached, a deviation from this
plateau behavior is observed. The crossover length $\xi$ between a square root
behavior $C(\delta x)\sim 1/\sqrt{\delta x}$ at large length scales ($\delta
x>\xi$), and a non-trivial regime at shorter length scales ($\delta x<\xi$),
defines the correlation length $\xi$ of the damage field. At a given distance
to failure, the cross-over is defined as the length-scale at which deviation
from the plateau behavior is larger than $5\%$. The cut-off was not clear for
early stages of damage and therefore, not considered for studying the
variation of correlation length with distance to failure. The spatial extent
of correlated damage increases as failure is approached, and so does the
amplitude of the deviation from the random field behavior. Hence, the
accumulated damage field shows correlations over an increasing range of length
scales, up to the correlation length $\xi$. The latter increases as failure is
approached and follows a power law

$\xi\sim 1/\delta^{\rho}$ (19)

where $\rho=0.35\pm 0.10$ (see the inset of Fig. 4). Interestingly, this
behavior is similar to the one of the avalanche spatial extent that also
increases as $\ell_{x}^{*}\sim 1/\delta^{\kappa}$ with $\kappa\simeq\rho$,
suggesting that both quantities are signatures of the same process (see Fig.
3(c)). Moreover, the lowest value of $\xi$ that could be identified at the
earliest stage of the damage spreading is of the order of $10\,\ell_{0}$, a
length scale that compares with the precursor spatial extent above which
scaling behaviors between size, duration and spatial extent starts to emerge
(see Figs. 2(c) and 3(b)). Hence, we conclude that the cooperative dynamics of
damage, which accumulates through bursts localized over some material region
of characteristic size $\xi$, shapes the accumulated damage field through the
introduction of non-trivial correlations over a length scale
$\ell_{\mathrm{x}}\simeq\xi$ that increases as failure is approached.

Figure 4: Normalized correlation function of the cumulative damage field at
several distances $\delta$ to failure. The inset shows the evolution the
extracted correlation length $\xi$ that is found to vary as $\xi\sim
1/\delta^{\rho}$ with $\rho\simeq 0.35$.

In summary, the failure precursors can be characterized by the dissipated
energy per avalanche $S$, the accumulated damage $S_{d}$, the duration $T$ or
the spatial extent $\ell_{x}$ that are related all together by scaling
relationships. The scaling exponents involved are independent of the spatial
extent of the redistribution function, in the range investigated in our study.
In addition, the statistical characterization of precursors reveal that the
progressive degradation of the specimen is a non-stationary phenomenon. In
particular, the characteristic size, duration and spatial extent of the
precursors diverges as a power law with the distance to catastrophic failure.
The presence of large structured avalanches is encoded in the accumulated
damage field from which emerges also a diverging correlation length Eq. (19).

Overall, our detailed analysis of the precursors statistics reveals the
existence of a time scale $T^{\star}$ and a length scale
$\ell_{\mathrm{x}}^{*}\simeq\xi$ encoded in the damage field fluctuations.
Both characterize the process of damage spreading and diverge at failure. Such
a behavior is often considered as a hallmark of critical phenomena. In this
scenario proposed by Garcimartin et al. [1997], Moreno et al. [2000], Girard
et al. [2010], Weiss et al. [2014] and Vu et al. [2019], compressive failure
is interpreted as a second-order phase transition between a specimen able to
sustain a mechanical load and a fully broken specimen. This interpretation is
appealing as the observation of universal scaling exponents is a key feature
of second order phase transitions. Nevertheless, and despite the strong
similarities with the phenomenology of critical phenomena, we will see in the
following that this conclusion is too hasty and that the behavior of the
precursors as the specimen is driven closer to failure is actually not
consistent with the response of a system that is driven towards a critical
point.

##  3 Theoretical modeling : Deciphering the statistics of precursors

We now aim at providing a quantitative explanation for the statistical
properties of the precursors observed in our simulations.We first rewrite the
damage model described in Section 1.1 under the form of an evolution equation
of the damage field at a fixed distance to failure. This evolution equation
describes strictly the same damage process and can be numerically resolved
following the procedure described in Sec. 1.3. Yet, this evolution equation,
which includes all the model ingredients described in Section 1, reveals
explicitly the connection between the theoretical framework of driven
disordered elastic interfaces [Narayan and Fisher, 1993, Barabási and Stanley,
1995, Leschhorn et al., 1997] and the process of damage growth in disordered
elasto-damageable solids, a connection already theorized by Weiss et al.
[2014]. First, this mapping is used in Section 3.2 to interpret the
statistical features of the precursors observed in our simulations at a fixed
distance to failure, such as the scaling relations between the size, the
duration and the spatial extent of the damage cascades. In a second step, in
Section 3.3, we explicitly write the dependance of each term of the evolution
equation with the distance to failure and analyse it to determine the origin
of the non-stationary evolution of precursors statistics observed in our
simulations in Section 2.2. We show that some parameters controlling the
damage evolution equation, for example the driving speed of the elastic
interface, diverge at failure. This observation accounts for the divergence of
the duration and spatial extent of the precursors observed in our simulations,
without invoking the role of material disorder. Our analysis thus rules out a
scenario in which quasi-brittle failure is interpreted as a critical
phenomenon emerging from the competition between elasticity and disorder and
provides an alternative explanation for the divergence of the characteristic
time and length scale of the damage cascades close to failure.

###  3.1 Evolution equation of the damage field under fixed loading amplitude

We start our theoretical analysis by deriving an evolution equation of the
damage field as it evolves from a reference damage level $d(x,t_{0})=d_{0}$ at
the reference time $t_{0}$ under a fixed loading amplitude
$\Delta=\Delta_{\mathrm{0}}$. We consider the damage field perturbation
$\delta d(x,t)=d(x,t)-d_{0}$ and assume a so-called over-damped dynamics
$\dot{d}(x,t)=\dot{\delta d}(x,t)\propto\mathcal{F}(d(x,t),\Delta_{0})$ where
$\mathcal{F}$ is the damage driving force introduced in Eq. (3). This
assumption is commonly used for deriving kinetic laws from thermodynamic
driving force, e.g. in brittle fracture or plasticity problems to relate the
crack speed [Gao and Rice, 1989, Ponson and Pindra, 2017] or the plastic flow
[Puglisi and Truskinovsky, 2005] to the local driving. As shown by Chopin et
al. [2018] in the context of crack propagation under tensile loading
conditions, it actually derives from the linearization of the rate-dependency
of the dissipative term. This amounts here to assume that the resistance to
damage $Y_{\mathrm{c}}$ is an increasing function of the damage rate
$\dot{d}$.

We then decompose the total damage driving force defined in Eq. (3) into two
contributions: (i) A homogeneous term
$\mathcal{F}_{hom}=Y(d_{0},\Delta_{0})-Y_{\mathrm{c}}(d_{0})$ (where $Y$ and
$Y_{c}$ are respectively given by Eqs. (4) and (5) ), that depends only on the
imposed loading amplitude (or equivalently to the distance to failure) and the
initial damage level $d_{0}$, and (ii) an inhomogeneous contribution
$\delta\mathcal{F}$ that depends on the damage field perturbations $\delta
d(x)$. The perturbation is assumed small in comparison to the average damage
level. This hypothesis was found well satisfied even close to localization
(Berthier et al. [2017]).

$\dot{\delta
d}(x,t)\propto\mathcal{F}(d(x,t),\Delta_{0})=\mathcal{F}_{hom}(d_{0},\Delta_{0})+\delta\mathcal{F}(\delta
d(x,t),d_{0},\Delta_{0})$ (20)

that provides the damage growth rate as a function of the distribution
$d(x,t)$ of damage within the specimen at the prescribed displacement
$\Delta_{0}$. Since our damage model does not allow healing of the material,
i.e. $\dot{\delta d}\geq 0$, a more rigorous formulation of the damage
evolution law is $\delta\dot{d}(x,\Delta)\propto\max[0,\mathcal{F}]$. But for
the sake of simplicity, this positiveness condition on $\delta\dot{d}$ is not
explicitly written in the following. The above equation can finally be written
as

$\dot{\delta
d}(x,t)\propto\mathcal{F}_{hom}(d_{0},\Delta_{0})+\psi(x,d_{0},\Delta_{0})\ast~\delta
d(x)-y_{\mathrm{c}}(x,d(x,t))$ (21)

where the disorder term $y_{\mathrm{c}}$ introduced in Eq. (5) describes the
spatial variations of damage energy that depends both on the position in the
specimen and the current damage level. The inhomogeneous contribution to the
driving force consists of a convolution product accounting for the non-local
interactions describing the redistributions of driving force taking place
within the specimen after each individual damage event. As shown by Berthier
et al. [2017], the so-called redistribution kernel $\psi$ is obtained from a
linearization of the driving force around the reference damage level $d_{0}$
and reads as

$\psi(x,d_{0},\Delta_{0})=Y^{\prime}(d_{0},\Delta_{0})\alpha_{2}(x)-Y_{c}^{\prime}(d_{0})$
(22)

where $\alpha_{2}(x)=\exp(-|x|/(2\ell_{0}))(2\ell_{0}+|x|)$ denotes the
convolution of the interaction function $\alpha(x)$ with itself. In the former
expression, the prime represents the derivative with respect to the reference
damage level $d_{0}$. We can verify that an increase $\delta d(x)=\delta
d_{0}\,\delta(x-x_{0})$ of the damage field localized in $x=x_{0}$ where
$\delta(u)$ is the Dirac function results in a variation
$\delta\mathcal{F}(x)=\delta d_{0}\,\psi(x-x_{0})$ of the driving force. In
other words, the spatial structure of the load redistribution following an
elementary damage event is controlled by the kernel $\psi$, and therefore, by
the interaction function $\alpha(x)$ introduced in the definition of the non-
local damage variable $\bar{d}=\alpha\ast d$ in Eq. (7).

In line with the model of Section 1 which comprises three contributions to the
total damage driving force, namely a slowly increasing load amplitude
corresponding to a local contribution, a non-local contribution $\overline{Y}$
emerging from the interactions and finally a disordered term
$Y_{\mathrm{c}}(x,d)$ source of strong non-linearities, we note that the
evolution equation (21) of the damage field consists of three terms too: (i) A
local contribution $\mathcal{F}_{hom}$, which predicts how the average damage
level increases with the imposed loading, (ii) a non-local contribution
$\delta\mathcal{F}=\psi\ast\delta d$ that couples the damage growth rate in
$x$ with the damage level in the other regions of the specimen, and (iii) a
quenched noise $y_{\mathrm{c}}$ that describes the resistance to damage that
depends both on the position $x$ in the specimen and the current damage level
$d(x)$. Owing to the dependence of $y_{\mathrm{c}}$ with the damage level,
this equation is strongly non-linear and cannot be solved exactly (see for
example Doussal et al. [2004] for advanced methods of resolution based on
functional renormalization group). It turns out that such equations are
reminiscent of a broader class of problems referred to as driven disordered
elastic interfaces that deal with elastic manifolds driven in disordered media
[Narayan and Fisher, 1993, Barabási and Stanley, 1995, Leschhorn et al., 1997,
Wiese, 2021]. Following an analogy already used by Vandembroucq and Roux
[2011], Lin et al. [2014] and Weiss et al. [2014], the damage field behaves
then as a uni-dimensional interface of position $\delta d(x)$ driven in a two-
dimensional plane $(x,d)$ of disordered resistance $y_{\mathrm{c}}(x,d)$. The
evolution equation (21) predicts the interface speed $\dot{\delta d}(x)$ as a
function of the current interface position $\delta d(x)$. Note that in this
framework, the redistribution kernel $\psi$ describes the elasticity of the
interface. Indeed, as one point of the interface located in $x=x_{0}$ moves
forward of an increment $\delta d_{0}$, the other regions of the interface
feels an increase $\delta\mathcal{F}(x)=\delta d_{0}\,\psi(x-x_{0})>0$ of the
applied driving force.

###  3.2 Scaling behavior of the precursors at fixed distance to failure

The behavior of driven disordered elastic interfaces has been thoroughly
investigated both theoretically and numerically (see Wiese [2021] for a recent
review). Such systems exhibit an intermittent dynamics characterized by micro-
instabilities localized both in space and time that can be elegantly described
by scaling laws. In particular, the driven elastic interface (and so the
damage field that it represents) grows through avalanches, the size, duration
and spatial extent of which scale with each other as

$\left\\{\begin{array}[]{l c l}S\sim\ell_{\mathrm{x}}^{d_{\mathrm{f}}}\\\
T\sim\ell_{\mathrm{x}}^{z}.\end{array}\right.$ (23)

The exponents $d_{\mathrm{f}}$ and $z$ take values that depend only on two
features of the interface: Its dimensionality and its elasticity. Here, the
interface is a uni-dimensional line while its elasticity is referred to as
short-range, as the redistribution kernel $\psi\sim\alpha_{2}$ decays
exponentially fast with the distance $x$ resulting in a short-range coupling
between the constituting elements of the specimen. Considering 1D interfaces
with short-range elasticity, Rosso et al. [2003] and Duemmer and Krauth [2005]
predicted a fractal dimension $d_{\mathrm{f}}\simeq 9/4$ and a dynamic
exponent $z\simeq 3/2$. Our numerical investigation of the statistics of
precursors carried in Section 2 led to $d_{\mathrm{f}}=2.35\pm 0.15$ and
$z=1.40\pm 0.15$, two values that are in good agreement with these
theoretically predicted exponents. Moreover, because the scalings are
determined at a fixed distance to failure, which can be defined in terms of
applied displacement or force, we do not expect variations in the scaling
relations between size, duration and spatial extent if considering force as
the controlled loading parameter. This successful comparison suggests that the
theoretical framework of driven disordered elastic interface captures
adequately the intermittent dynamics of damage growth, at least at some fixed
distance to failure. This confirms the claim made by Petri et al. [1994],
Zapperi et al. [1997b] that fracture precursors are reminiscent of the
critical dynamics of damage growth, even though, as we will show later, the
catastrophic failure of the specimen does not correspond to a critical point.
As an additional remark, we can notice that the measured exponent $\beta\simeq
1.5$ characterizing the distribution of avalanche sizes in (14) corresponds to
the mean-field exponent, and not the value $\beta\simeq 1.1$ predicted for
disordered elastic interfaces with short-range interactions [Rosso et al.,
2009, Le Priol et al., 2021]. This observation, consistent with the exponent
value reported by Raischel et al. [2006] that considered a local fiber model
with progressively damaging elements, may be connected to the non-stationary
nature of depinning model underlying compressive failure, a point thoroughly
discussed in the next section.

Before investigating the non-stationary dynamics of damage accumulation in
more detail, we would like to emphasize that the nature of interactions
between the constituting elements of the specimen, described by the kernel
$\psi$ and the function $\alpha$ from which it derives, control both the onset
of specimen failure, as thoroughly discussed by Berthier et al. [2017], and
the statistical properties of the damage cascades that precede it, as shown
here. As a result, the precursory damage activity carry information on the
final failure event, an idea that we will use thereafter to design a method of
failure prediction from the statistical analysis of damage cascades.

###  3.3 Non-stationary evolution equation of the damage field under
monotonically increasing loading amplitude

We now go a step further and focus on a more realistic scenario of damage
growth as the loading amplitude increases. We seek to explain the non-
stationary features of damage evolution, observed numerically in Section 2. As
in Section 3.1, we start by decomposing the fields in terms of homogeneous and
perturbation contributions. We then explicitly formulate the time dependency
to evaluate the velocity, and other characteristic features, of the interface
as it is externally driven towards failure.

We first decompose the damage field in two contributions

$d(x,t_{0})=d_{0}(t_{0})+\delta d(x,t_{0})$ (24)

where $d_{0}(t_{0})=\langle d(x,t_{0})\rangle_{x}$ is the mean damage level at
the reference time $t_{0}=0$. In the following, as previously, we assume a
damage field perturbation that is small in comparison to the average damage
level, at all distance to localization. We assume that the damage field
perturbations $\delta d\ll d_{0}$ are small in comparison to the reference
damage level $d_{0}$. This amounts to consider rather small values $\sigma\ll
1$ of disorder amplitude, as damage field fluctuations scales linearly with
$\sigma$, as well as rather short amount of time $(t-t_{0})$ before the
reference time $t_{0}$ must be updated again.

We can first solve the homogeneous problem that consists in predicting how
$d_{0}$ varies with the imposed displacement $\Delta_{0}$. The governing
equation writes as

$\mathcal{F}_{hom}(d_{0},\Delta_{0})=0\quad\Rightarrow\quad d_{0}(\Delta_{0})$
(25)

from which we derives the relation between $d_{0}$ and $\Delta_{0}$. As a
result, we can indifferently use one or the other variable to describe the
reference state of the specimen, and thus the distance to catastrophic
failure.

We are now interested in predicting the evolution of the damage field
perturbations. To do so, we assume that the imposed displacement increases
linearly with time

$\Delta(t)=\Delta_{0}+\delta\Delta(t)=\Delta_{0}+v_{\mathrm{ext}}\,t$ (26)

where the displacement increment $\delta\Delta(t)$ is assumed to be small with
respect to the reference displacement $\Delta_{0}$. This amounts to consider
the evolution of the damage field on a rather short amount of time
$t-t_{0}=t\ll\Delta_{0}/v_{\mathrm{ext}}$, before the reference displacement
$\Delta_{0}$ must be updated again.

The equation governing the evolution of $\delta d(x,t)$ derives from a
linearization of the evolution law

$\dot{d}(x,t)\propto\mathcal{F}(d(x,t),\Delta(t))\quad\Rightarrow\quad\delta\dot{d}(x,t)\propto\mathcal{F}(d_{0}+\delta
d(x,t),\Delta_{0}+\delta\Delta(t))$ (27)

where the total damage driving force decomposes into two contributions
$\mathcal{F}_{\mathrm{hom}}(d_{0},\Delta_{0})+\delta\mathcal{F}(\Delta_{0},\delta
d(x,t),\delta\Delta(t))$. At the first order in the damage field perturbation
$\delta d(x,t)$, the evolution equation of the damage field perturbation
writes as

$\dot{\delta
d}(x,t)\propto\frac{\partial\mathcal{F}_{\mathrm{hom}}}{\partial\Delta}(d_{0},\Delta_{0})\,\delta\Delta(t)+\psi(d_{0},\Delta_{0})\ast\delta
d(x,t)-y_{\mathrm{c}}(x,d(x,t)).$ (28)

The last two terms represent the perturbation in the damage driving force
under fixed loading amplitude $\Delta(t)=\Delta_{0}$, which directly derives
from Eq. (21). Using the expression of the homogeneous damage driving force
$\mathcal{F}_{\mathrm{hom}}(d_{0}\Delta_{0})=Y(d_{0},\Delta_{0})-Y_{\mathrm{c}}(d_{0})$
and the expression (22) of the interaction kernel $\psi(d_{0},\Delta_{0})$,
one obtains

$\dot{\delta
d}(x,t)\propto\frac{\partial\mathcal{F}_{\mathrm{hom}}}{\partial\Delta}(d_{0},\Delta_{0})\,v_{\mathrm{ext}}t-Y_{c}^{\prime}(d_{0})\delta
d(x)+Y^{\prime}(d_{0},\Delta_{0})(\alpha_{2}\ast\delta
d)(x)-y_{\mathrm{c}}(x,d).$ (29)

Introducing the parameters $\mathcal{K}$, $v_{\mathrm{m}}$ and $\mathcal{H}$,
the evolution equation takes its final form

$\dot{\delta
d}(x,t)\propto\mathcal{K}(\delta)\left[v_{\mathrm{m}}(\delta)t-\delta
d(x,t)\right]+\mathcal{H}(\delta)\displaystyle\int_{\Sigma}\alpha_{2}(|x-x^{\prime}|)\left[\delta
d(x^{\prime},t)-\delta d(x,t)\right]dx^{\prime}-y_{\mathrm{c}}(x,d(x,t))$ (30)
$\mathrm{where}\quad\left\\{\begin{array}[]{ll}\displaystyle\mathcal{K}(\delta)=\mathcal{K}(d_{0},\Delta_{0})=Y_{\mathrm{c}}^{\prime}(d_{0})-Y^{\prime}(d_{0},\Delta_{0})=-\frac{\partial\mathcal{F}_{\mathrm{hom}}}{\partial
d}(d_{0},\Delta_{0})\\\ \\\ \displaystyle
v_{\mathrm{m}}(\delta)=v_{\mathrm{m}}(d_{0},\Delta_{0})=\frac{\partial\mathcal{F}_{\mathrm{hom}}}{\partial\Delta}(d_{0},\Delta_{0})\,\frac{v_{\mathrm{ext}}}{\mathcal{K}(d_{0},\Delta_{0})}\\\
\\\
\mathcal{H}(\delta)=\mathcal{H}(d_{0},\Delta_{0})=-Y^{\prime}(d_{0},\Delta_{0}).\end{array}\right.$
(31)

Here, the dependence of the newly introduced parameters $\mathcal{K}$,
$v_{\mathrm{m}}$ and $\mathcal{H}$ with the sole distance to failure
$\displaystyle\delta=\frac{\Delta_{\mathrm{c}}-\Delta_{0}}{\Delta_{\mathrm{c}}-\Delta_{\mathrm{el}}}$
derives from the homogeneous solution (25) that is used to replace the
reference damage level by its expression $d_{0}(\Delta_{0})$ as a function of
$\Delta_{0}$, and then $\Delta_{0}$ by $\delta$ in the previous expressions.
The integral term describes the elastic interactions between different regions
of the specimen as the interface moves forward, ultimately controlling the
scaling properties of the avalanches, like the relations between their size,
their duration and their spatial extent. Separately from this mechanism, the
mean temporal variation of these three quantities is controlled by the first
term of the equation that describes the external loading conditions. The
details of the interaction function thus affect the spatial structure of the
avalanches but not the divergence of their size on approaching failure that is
robust to the particular choice of interaction function.

Under this form, the evolution equation also sheds light on the analogy
between the damage field evolution and the problem of an elastic interface
driven in a disordered medium, analogy that is schematically illustrated in
Fig. 5. First of all, the elasticity of the interface is described by the
integral term of Eq. (30). It provides the driving force distribution along
the interface as a function of its geometry $\delta d(x)$. For a flat
interface, this contribution is zero and the driving force is constant along
the interface. On the contrary, if a point located in $x$ is in advance (resp.
behind) with respect to the rest of the interface ($\delta d(x)>\delta
d(x^{\prime})$ for all $x^{\prime}$), the driving force in $x$ is then smaller
(resp. larger) than in the other regions of the interface.222As discussed in
Section 3.1, this term also provides the spatial structure of the load
redistribution taking place after each individual damage event. The spatial
structure is described by the kernel $\mathcal{H}(\delta)\alpha_{2}(x)$ that
is equal to the redistribution kernel $\psi$ of Eq. (22) up to a constant
$-Y_{\mathrm{c}}^{\prime}(d_{0})$. This local contribution to the damage
driving force is accounted for in the first term
$\mathcal{K}(\delta)\left[v_{\mathrm{m}}(\delta)t-\delta d(x,t)\right]$ of the
evolution equation (30), together with the other local contributions. Second
of all, the first term in the evolution equation (30) describes the driving
conditions imposed to the interface. The interface (and thus the damage field
it represents) is driven by a rigid bar moving at the speed $v_{\mathrm{m}}$,
as schematically represented in Fig. 5. The load transfer from the bar to the
interface is ensured by Hookean springs of stiffness $\mathcal{K}$. This
particular driving can also be pictured by considering that the interface is
trapped by a quadratic potential of width $1/\mathcal{K}$ that moves with a
speed $v_{\mathrm{m}}$.

Figure 5: Representation of the damage field evolution by the problem of an
elastic interface driven in a random medium. A rigid bar pulls through Hookean
springs on an elastic interface of position $d(x,t)$ which describes the
damage accumulated in the specimen. The springs stiffness varies with local
interface position - advanced (resp. nascent) positions experience small
(resp. large) stiffness. The interface elasticity, that represents the elastic
interactions within the specimen, tries to maintain a homogeneous damage
level, while the disorder traps the elastic line in some rough configurations.
The competition between elasticity and disorder results in the crackling
dynamics with broadly distributed precursors described in Fig. 1. The non-
stationary behavior of the precursory activity evidenced in Fig 3 can then be
understood as a result of variations in the driving conditions imposed to the
interface, both in terms of driving speed and spring stiffness.

Contrary to the analysis carried in Section 3.1, the parameters involved in
Eqs. (30), like the spring stiffness $\mathcal{K}$ and the speed
$v_{\mathrm{m}}$ at which the interface is driven, now vary with distance to
failure, see Eq. (31). We then examine these variations and their effect on
the precursors statistics. Catastrophic failure taking place at the critical
damage level $d_{0}=d_{\mathrm{c}}$ results from the unstable growth of damage
under fixed loading conditions. The failure condition thus writes as
$\displaystyle\frac{\partial\mathcal{F}_{\mathrm{hom}}}{\partial
d}(d_{\mathrm{c}},\Delta_{\mathrm{c}})=0$ that translates, from Eq. (31), into
a vanishing spring stiffness $\mathcal{K}=0$ at the failure point $\delta=0$.
This also implies that on approaching failure, the driving speed
$v_{\mathrm{m}}\sim 1/\mathcal{K}$ blows up.

What is the effect of the variations of $v_{\mathrm{m}}$ and $\mathcal{K}$ on
the statistics of precursors ? In standard models of elastic interfaces, the
driving controls the distance to the so-called depinning transition. In
particular, considering vanishingly small driving speed and spring stiffness
drives the interface in a critical state where the characteristic time
$T^{\star}$ and length scale $\xi$ characterizing the intermittent dynamics
diverge. Such a behavior is strikingly similar to the one observed in our
simulations and in experiments like the ones of Garcimartin et al. [1997],
Baró et al. [2018], Vu et al. [2019], leading Weiss et al. [2014] and Vu et
al. [2019] to interpret compressive failure as a critical depinning
transition. However, the essential condition of vanishingly small driving
speed $v_{\mathrm{m}}\rightarrow 0$ is not met, as instead, $v_{\mathrm{m}}$
diverges close to failure (interpreted as a critical point). As a result, such
scenario cannot account neither for the increase of the precursor size
observed in our simulations, nor the one reported in experiments, as we will
argue later. The role of varying $\mathcal{K}$, which arises from the non-
stationary nature of damage and was not considered in Weiss et al. [2014], is
in contrast to standard depinning models. In the following, we present an
alternative explanation based on the interpretation of compressive failure as
a standard bifurcation.

###  3.4 Acceleration of the precursory activity as signature of the unstable
nature of compressive failure

We first come back to the evolution equation (30) of the damage field and
derive from it the evolution of the precursor size with distance to failure.
On the one hand, we invoke an important property of driven disordered
interfaces, namely that the rate $\dot{N}$ of precursors is constant and
independent of the interface speed Wiese [2021], a property that is verified
in our simulations, as shown in Fig. 6(c). This implies that the interface
speed is set by the average avalanche size, i.e.
$\langle\delta\dot{d}\rangle=\dot{N}\,\langle S\rangle\propto\langle
S\rangle$. On the other hand, following the evolution equation (30), the
interface velocity is imposed by the speed $v_{\mathrm{m}}$ at which the rigid
bar moves, so that $\langle S\rangle\propto v_{\mathrm{m}}$.333This property
can be deduced by noticing that the first term in the evolution equation (30)
of the interface must remain finite, so that $\delta d\propto
v_{\mathrm{m}}\,t$ The expression (31) of the speed $v_{\mathrm{m}}$ then
leads to

$\langle S\rangle\sim\dfrac{1}{\mathcal{K}}.$ (32)

We now derive the variations of the stiffness $\mathcal{K}$ with the distance
to failure $\delta$. As noticed earlier,
$\displaystyle\mathcal{K}=-\frac{\partial\mathcal{F}_{\mathrm{hom}}}{\partial
d}(d_{0},\Delta_{0})$ provides the stability of the process of damage growth
for a homogeneous specimen. As a result, $\mathcal{K}$ vanishes when
$\delta=0$. It turns out that close to failure,
$\mathcal{K}\propto\sqrt{\delta}$, a property that is verified in Fig. 6(b)
and demonstrated in Appendix A. Therefore, the average avalanche size diverges
as

$\langle S\rangle\sim\dfrac{1}{\sqrt{\delta}}.$ (33)

As shown in Fig. 6(a), this behavior accounts for the variations of $\langle
S\rangle$ measured in our simulations that shows divergence with an exponent
$\alpha_{\mathrm{s}}\simeq 0.48\pm 0.05$. Note that while we have considered
here displacement-imposed conditions, we can show, following the same logic
detailed in Eqs.(25)-(30), that a similar scaling behavior holds under force-
imposed conditions, after defining the distance to failure from the distance
$F_{\mathrm{c}}-F$ to force at failure instead of $\Delta_{\mathrm{c}}-\Delta$
as in Eq. (13)

Figure 6: (a) Variations of the precursor size measured in the simulations
with the distance to failure. The behavior $\langle
S_{\mathrm{d}}\rangle\propto\langle
S\rangle\propto\delta^{-\alpha_{\mathrm{s}}}$ with an exponent
$\alpha_{\mathrm{s}}=0.48\pm 0.05$ (dashed line) is compatible with the
square-root singularity predicted in Eq. (33) from the governing equation
(30); (b) The spring stiffness $\mathcal{K}$ involved in the governing
equation (30) and computed from Eq. (31) is represented as a function of the
distance to failure $\delta$. The dashed line
$\mathcal{K}\propto\sqrt{\delta}$ confirms that it vanishes as the specimen
approaches failure. (c) The rate $\dot{N}$ of precursors is shown as a
function of the distance to failure. This behavior is compatible with a
standard property of driven elastic interfaces, namely that the rate of
avalanches is constant and independent of the interface speed.

We are now in position to explain the divergence of the characteristic time
and length scale of the precursors observed close to failure in our
simulations. We focus first on the size of the largest precursors $S^{\star}$
that can be related to $\langle S\rangle$ from the expression (14) of the
precursor size distribution,

$\langle S\rangle\propto(S^{\star})^{2-\beta}\propto\sqrt{S^{\star}}$ (34)

where the exponent $\beta\simeq 1.5$ is inferred from the simulations.
Combined with the square-root divergence (33) of the average precursor size,
one obtains

$S^{\star}\sim 1/\delta$ (35)

that accounts for the behavior reported in Eq. (15) for the largest
precursors. Using finally the scaling relations
$T^{\star}\sim(S^{\star})^{z/d_{\mathrm{f}}}$ and
$\ell_{\mathrm{x}}^{\star}\sim(S^{\star})^{1/d_{\mathrm{f}}}$ between the
duration, the spatial extent and the size of the largest precursors, we
predict

$\left\\{\begin{array}[]{lcl}\vspace{5pt}T^{\star}&\sim&1/\delta^{\phi}\quad\mathrm{with}\quad\phi=z/d_{\mathrm{f}}\simeq
2/3\\\
\ell_{\mathrm{x}}^{\star}&\sim&1/\delta^{\kappa}\quad\mathrm{with}\quad\kappa=1/d_{\mathrm{f}}\simeq
4/9,\end{array}\right.$ (36)

two equations that account for the numerical observations of Eqs. (16) and
(17).

The ability of our model to describe quantitatively the non-stationary
dynamics of failure precursors observed in our simulations calls for a few
comments. First, the evolution equation (30) derived from the model described
in Section 1.1 explicitly connects the non-stationnary dynamics of precursors
and the loss of stability of the damage growth process, a feature that was not
evident from the simulation results. Consequently, the divergence of the
duration, size and spatial extent of the damage cascades close to failure does
not derive from the depinning transition of the elastic manifold describing
the damage field.444Note that in such a scenario, the exponents $\phi$,
$\kappa$ and $\gamma$ characterizing the divergences $T^{\star}\sim
1/\delta^{\phi}$, $\ell_{\mathrm{x}}^{\star}\sim 1/\delta^{\kappa}$ and
$S^{\star}\sim 1/\delta^{\gamma}$ are set by the so-called correlation length
exponent $\nu\simeq 4/3$ leading to $\phi=\nu\simeq 4/3$, $\kappa=\nu\,z\simeq
2$ and $\gamma=\nu\,d_{\mathrm{f}}\simeq 3$ incompatible with the exponents
measured in the simulations.. Instead the elastic interface that describes the
damage field is driven away from the critical point. This phenomenon takes
place through the divergence of the driving speed and thus the damage rate, a
behavior that is reminiscent of the presence of a bifurcation point in
$\delta=0$ leading to the full failure of the specimen, irrespective of the
presence of material disorder. This conclusion is in line with the ones drawn
by Zapperi et al. [1997a] who investigated failure in disordered materials by
micro-cracking using a discrete approach referred to as random fuse model.
Such a description may also lead to a divergence of the precursor size close
to failure, without the system reaching a critical state. Instead, the failure
process can be described as a standard bifurcation for which the disorder is
irrelevant. Such mechanism, referred to as sweeping of an instability,
encompasses many physical systems including ferromagnetic and granular
materials [Sornette, 2002].

In fracture, the absence of critical behavior at the failure point has
practical consequences. Indeed, the exponents characterizing the precursor
evolution close to failure do not derive from the theory of critical
phenomena, but instead, can be predicted from standard bifurcation theory. As
a result, the exponent that describes the divergence of the avalanche size
close to failure is expected to keep a constant value
$\alpha_{\mathrm{s}}=1/2$, irrespective of the range of the interactions as
well as the system dimension. This means that in principle we expect this
exponent to hold in models of more realistic systems, including 3D elasto-
damageable materials for which elasticity leads to long-range interactions.
Yet, we note that the divergence of the precursors size characterized by
$\alpha_{\mathrm{S}}=0.5$ observed in our simulations and explained in our
model differs from the value $\alpha_{\mathrm{S}}=1.0$ measured in simulations
considering Eshelby-like interaction functions [Girard et al., 2010] and in
molecular dynamics simulations [Karimi et al., 2019]. We now harness this
property to design a methodology that predicts the residual lifetime of
materials and structures from the statistical analysis of the failure
precursors.

##  4 Predicting failure from the statistics of precursors: A numerical proof
of concept

As an application of our theoretical findings, we now would like to discuss
how the acceleration of precursory activity as a signature of an impending
instability can be used in structural health monitoring. Damage accumulation
results in the progressive loss of stiffness of structures, ultimately
threatening their mechanical integrity. Tracking damage and its evolution
inside structures is a very challenging task.555X-ray tomography, although
used to characterize damage inside small specimens, see e.g. Kandula et al.
[2019] and Cartwright-Taylor et al. [2020], is not adapted to study large
structures. As a result, acoustic emissions, that accompany damage growth and
can be recorded by piezoelectric sensors, have been used as a ready-made
source of information for monitoring their mechanical health. Laboratory
experiments show that the amplitude and frequency of the acoustic bursts
increase on approaching failure [Garcimartin et al., 1997, Baró et al., 2013,
Vu et al., 2019]. Taking inspiration from seismology, such an increase has
been described using the so-called time-reversed Omori-law666The standard
Omori-law describes the rate of the after-shocks following large earthquakes
[Hirata, 1987, Hirata et al., 1987, Baró et al., 2013, 2018, Salje et al.,
2017]. It follows Eq. (37) with the noticeable difference that
$t-t_{\mathrm{c}}$ is used instead of $t_{\mathrm{c}}-t$.

$\centering
A(t)=\frac{A_{\mathrm{o}}}{(t_{\mathrm{c}}-t)^{p}}.\@add@centering$ (37)

In this expression, $A(t)$ is the rate of acoustic bursts recorded at time $t$
before catastrophic failure that takes place at time $t_{\mathrm{c}}$.
$A_{\mathrm{o}}$ is a proportionality constant while the exponent $p$ is found
to be close to one [Baró et al., 2013], even though it has been shown to vary
with strain rate and temperature [Ojala et al., 2004]. Inspired by procedures
used for predicting earthquakes, landslides and volcanic eruptions [Voight,
1988, Kilburn and Voight, 1998, Sornette, 2002, Bell et al., 2013, Bell, 2018]
where similar power-law divergence of precursors have been reported, it has
recently been proposed that such a scaling behavior could be used for
monitoring the mechanical health of structures [Mayya et al., 2020], following
the idea of Anifrani et al. [1995].

As a numerical proof of concept, we now use the precursors signals obtained by
numerical implementation of our model described in Section 1.3 to perform a
retrospective failure prediction. Our model does not provide the acoustic
signal accompanying damage growth. Hence, we assume that the size $S$ (in
energy) of damage precursors can be inferred from the acoustic signal recorded
from the real structure. As a result, we consider the time-series $S(t)$ that
follows a scaling behavior similar to Eq. (37) as input data for failure
prediction. We consider that damage starts to grow at $t=0$ (corresponding to
$\Delta=\Delta_{\mathrm{el}}$, i.e. $\delta=1$) and specimen failure occurs at
$t=t_{\mathrm{c}}$ (corresponding to $\Delta=\Delta_{\mathrm{c}}$, i.e.
$\delta=0$). Assuming that the displacement is increased linearly with time,
the distance to failure writes then as
$\delta=(t_{\mathrm{c}}-t)/t_{\mathrm{c}}$. The scaling law (33) then rewrites
as

$\centering\langle
S\rangle=\frac{S_{\mathrm{o}}}{(t_{\mathrm{c}}-t)^{\alpha_{s}}},\@add@centering$
(38)

where $\alpha_{\mathrm{s}}=1/2$. This equation is developed in the form

$\centering\langle S\rangle^{1/\alpha_{\mathrm{s}}}\,t=t_{\mathrm{c}}\ \langle
S\rangle^{1/\alpha_{\mathrm{s}}}-S_{\mathrm{o}}\@add@centering$ (39)

that follows $Y=t_{\mathrm{s}}\,X-S_{\mathrm{o}}$ where both $Y=\langle
S\rangle^{1/\alpha_{\mathrm{s}}}\,t$ and $X=\langle
S\rangle^{1/\alpha_{\mathrm{s}}}$ can be computed at different time steps
during the monitoring so that the failure time $t_{\mathrm{c}}$ can be
subsequently predicted from a linear fit.

Figure 7: (a) Evolution of avalanche energy, $S$ as a function of the time. An
example case of the prediction $t_{\mathrm{c}}^{\mathrm{pred}}$ for a value of
$t_{\mathrm{current}}$ is also presented.(b) Predicted values of time to
failure and (c) the error in prediction
($(t_{\mathrm{c}}-t_{\mathrm{c}}^{\mathrm{pred}})/t_{\mathrm{c}}$) at
different instances of $\delta=(t_{\mathrm{c}}-t)/t$.

In practice, the implementation of the method is as follows. We assume that
the process of damage spreading takes $t_{\mathrm{c}}=1000\leavevmode\nobreak\
\mathrm{s}$ until the specimen fails, as shown in Fig. 7(a). This amounts to
assume a loading rate
$\dot{\Delta}=(\Delta_{\mathrm{c}}-\Delta_{\mathrm{el}})/1000$. The mean
precursors size $\langle S\rangle$ is obtained from an average over non-
overlapping time windows of $20\leavevmode\nobreak\ \mathrm{s}$, a duration
sufficiently small to accurately capture the acceleration of precursory
activity. The monitoring of the damage activity is performed at time
$t_{\mathrm{current}}$. We then record damage events in the immediate history
of $t_{\mathrm{current}}$, i.e. during $100\leavevmode\nobreak\ \mathrm{s}$
before $t_{\mathrm{current}}$. Equation (39) is then used to fit the data and
predicts the failure time. The prediction $t_{\mathrm{c}}^{\mathrm{pred}}$ is
shown in Fig. 7(b) as a function of the residual lifetime
$\delta=(t_{\mathrm{c}}-t_{\mathrm{current}})/t_{\mathrm{c}}$ when the
monitoring is performed. It is also compared with the actual failure time
$t_{\mathrm{c}}=1000\leavevmode\nobreak\ \mathrm{s}$. The relative error
$(t_{\mathrm{c}}-t_{\mathrm{c}}^{\mathrm{pred}})/t_{\mathrm{c}}$ on the
prediction is shown in Fig. 7(c). It rapidly decreases as the prediction is
made closer to final failure. The predictions are within 10% of the actual
failure time when the remaining lifetime is less than 25%.

Interestingly, the proposed methodology is conservative, predicting
systematically failure time smaller than $t_{\mathrm{c}}$. Note also that it
does not require monitoring from the beginning of the damage accumulation
phase, unlike cumulative record based predictions [Carpinteri et al., 2007,
Abdelrahman et al., 2014]. Accurate residual life-assessment can be inferred
at any time from data recorded on a rather short amount of time whereas
empirical law based approaches require detailed experimental protocols for
measuring various material parameters in the lab before providing reliable
predictions for the structure [Godin et al., 2019]. In practice, working with
the raw acoustic signal emitted by the structure and not the mechanical
precursors $S(t)$ renders the post-processing more complex. However, several
strategies have been proposed by Mayya et al. [2020] to circumvent these
difficulties and they are now tested in laboratory experiments where both the
acoustic precursors and the mechanical precursors can be recorded.

##  5 Conclusions

Compressive failure of quasi-brittle solids results from the complex evolution
of a large number of dissipative events, such as micro-cracking, in
interaction with each other. Here, we examine such a collective dynamics and
its connection with final failure through the statistical analysis of the
damage precursors taking place during the phase of damage accumulation. In
practice, we analyze the intermittent mechanical response of a disordered
elasto-damageable specimen under compression as predicted from a minimal
thermodynamically-consistent damage model developed by Berthier et al. [2017].
This model captures the co-action of material disorder and elastic
interactions conveyed by the non-local stress distribution taking place in the
specimen after each individual damage event. The major results of our study
are the following:

  1. 1.

The proposed description predicts not only localization and catastrophic
failure, as shown by Berthier et al. [2017], but also captures also
qualitatively the main features of the jerky evolution of damage observed
experimentally in quasi-brittle solids under compression [Petri et al., 1994,
Garcimartin et al., 1997, Guarino et al., 2002, Rosti et al., 2009, Baró et
al., 2013, Vu et al., 2019]. Damage grows through bursts the size, duration
and spatial extent of which are related to each other by scaling laws. As the
specimen is driven towards failure, these quantities increase following power
law relationships with the distance to final failure.

  2. 2.

To rationalize these numerical observations, we then investigate theoretically
the intermittent growth of damage. We derive an evolution equation of the
damage field within the specimen that captures quantitatively all the
statistical properties of the precursors evidenced in our simulations. It also
elicits the connection between compressive failure and the theoretical
framework of elastic interfaces driven in disordered media used in former
studies [Weiss et al., 2014] to represent the damage field. This mapping
accounts for the scaling relations between the size, the duration and the
spatial extent of the failure precursors. The exponents involved in these
scaling relations are critical exponents emerging from the so-called depinning
transition, a second order phase transition taking place when the driving
speed of the elastic interface goes to zero. However, and contrary to the
discussion by Weiss et al. [2014] and Vu et al. [2019], the increase of these
quantities close to failure does not emerge from the depinning transition.
Instead, it results from the divergence of the damage rate (that translates
into the divergence of the interface driving speed), a feature that arises
from the presence of a standard bifurcation at the compressive failure
threshold Berthier et al. [2017], Dansereau et al. [2019]. This new
explanation for the non-stationary features of the damage evolution accounts
for our numerical observations and in particular for the square-root
divergence of the precursor size with distance to failure.

  3. 3.

Last but not least, we finally harness the universal statistical properties of
failure precursors to design an innovative technique of structural health
monitoring inspired from the method proposed by Anifrani et al. [1995]. Using
our numerical data, we show that short time-series of precursors size
available ahead of final failure can be processed to predict the residual
lifetime with accuracy. As the proposed methodology is based on the physics of
damage spreading in quasi-brittle solids, we expect its predictions to be more
robust than the methods used currently in structural health monitoring.

Finally, we would like to discuss how the results derived in this study for a
model elasto-damageable solid may apply to more realistic situations. The
nature of the interactions in real materials being different, i.e. long-ranged
while we here considered short-range interactions, the exponents
characterizing the relations between the size, the duration and the spatial
extent are expected to differ. Nonetheless, the scaling relations between
these quantities are expected to hold and the exponents can be predicted from
the theory of driven disordered interfaces using the appropriate interaction
kernel [Dansereau et al., 2019] and specimen dimensionality – 1D vs. 2/3D. The
scaling laws characterizing the divergence of these quantities with the
distance to failure are shown to be independent of the dimensionality of the
specimen and the nature of the elastic interactions. Hence, the evolution of
precursors towards failure is expected to be characterized by the simple and
robust scaling laws deduced from the stability analysis, reinforcing further
the robustness of the proposed methodology of structural health monitoring
[Mayya et al., 2020]. Interestingly, the depinning scenario discussed in Weiss
et al. [2014] was used to explain the experimental observations of statistical
size effects of strength in quasi-brittle materials, in particular the non-
vanishing strength at large size. Building on the new understanding of the
precursors derived from the present study, we are now revisiting the
experimentally observed size effect. In addition, we aim to apply these ideas
to explain quantitatively the scaling properties of precursors observed in
real materials through experiments on model elasto-damageable solids.

In summary, the minimal model of damage growth used in this study paves the
way for a deep understanding of the physics of compressive failure. Indeed, it
could describe, at least qualitatively, all its salient features, namely
localization, intermittency and non-stationary dynamics. It turned out that
these different phenomena emerged from different mechanisms, albeit connected
to each other through the long-range elastic interactions within the specimen
that control both the statistics of damage cascades during the accumulation
phase and the final failure. The scenario derived from our work suggests that
damage precursors are simple by-products of the damage growth, that could
potentially be shut down if the material disorder could be shut down too. As a
result, they can’t explain the final failure, but instead, constitute ready-
made signals informing about the ongoing damage accumulation. This idea has
major implications in structural health monitoring, in particular when it
comes to decipher the acoustic emission data to predict the residual lifetime
of structures. The next challenges lie certainly in understanding in depth the
complex relationship between the acoustic precursors and the mechanical ones
investigated in this study as well as investigating experimentally at the
microstructural scale the complex spatio-temporal evolution of damage in 2D
and 3D elasto-damageable materials in order to challenge and refine the
proposed scenario.

## Acknowledgements

The authors gratefully acknowledge financial support from Sorbonne Université
through the Emergence grant for the research project From damage spreading to
failure in quasi-brittle materials as well as CNRS and Satt-Lutech through the
tech transfer project - Development of a technology of predictive maintenance
for materials and structures under compression.

##  Appendix A Evolution of spring stiffness $\mathcal{K}$ close to failure

In this appendix, we describe the effect of the non-stationarity of loading on
the stability of damage evolution given by stiffness
$\mathcal{K}(d_{0},\Delta_{0})$ of the springs connecting the rigid bar to the
interface. In the mean-field limit, we have

$\mathcal{K}(d_{0},\Delta_{0})=-\frac{\partial{\mathcal{F}_{\mathrm{hom}}}}{\partial{d}}(d_{0},\Delta_{0})=Y_{c0}\eta+\frac{1}{2}\Delta^{2}_{0}k^{\prime\prime}(d_{0})$
(40)

For instances when $d_{0}\rightarrow d_{c}$ corresponding to
$\Delta_{0}\rightarrow\Delta_{c}$, the stiffness $\mathcal{K}$ can be
expressed by a series expansion around $(d_{c},\Delta_{c})$ as follows.

$\mathcal{K}(d_{0},\Delta_{0})=-\frac{\partial{\mathcal{F_{\mathrm{hom}}}}}{\partial{d}}(d_{c},\Delta_{c})-\frac{\partial^{2}{\mathcal{F}_{\mathrm{hom}}}}{\partial{d^{2}}}(d_{c}-d_{0})-\frac{\partial^{2}{\mathcal{F}_{\mathrm{hom}}}}{\partial{d}\partial{\Delta}}(\Delta_{c}-\Delta_{0})$
(41)

where, $d_{c}$ and $\Delta_{c}$ are the localization threshold and critical
loading amplitude, respectively. From Berthier et al. [2017], the damage
evolution was understood to become unstable at localization threshold implying
$\frac{\partial{\mathcal{F_{\mathrm{hom}}}}}{\partial{d}}(d_{c},\Delta_{c})=0$
leaving the second and third terms of Eq. (41). Further, quantities
$(d_{c}-d_{0})$ and $(\Delta_{c}-\Delta_{0})$ in these terms are measures of
the distance to failure in terms of damage levels and loading amplitude,
respectively. We now proceed to obtain a relation between these two quantities
to decipher the effect on $\mathcal{K}$. The damage driving force in the mean-
field limit $\mathcal{F_{\mathrm{hom}}}$ for damage levels $d_{0}\rightarrow
d_{c}$ follows as

$\centering\begin{split}\mathcal{F_{\mathrm{hom}}}(d_{0},\Delta)=&\leavevmode\nobreak\
\mathcal{F_{\mathrm{hom}}}(d_{c},\Delta_{c})+\frac{\partial{\mathcal{F_{\mathrm{hom}}}}}{\partial{d}}(d_{c}-d_{0})+\frac{\partial{\mathcal{F_{\mathrm{hom}}}}}{\partial{\Delta}}(\Delta_{c}-\Delta_{0})\\\
&+\frac{1}{2}\frac{\partial^{2}{\mathcal{F_{\mathrm{hom}}}}}{\partial{d^{2}}}(d_{c}-d_{0})^{2}+\frac{1}{2}\frac{\partial^{2}{\mathcal{F_{\mathrm{hom}}}}}{\partial{\Delta^{2}}}(\Delta_{c}-\Delta_{0})^{2}\end{split}\@add@centering$
(42)

From governing equation, we obtain both
$\mathcal{F_{\mathrm{hom}}}(d_{0},\Delta)$ and
$\mathcal{F_{\mathrm{hom}}}(d_{loc},\Delta_{c})=0$. Re-invoking the failure
criterion,
$\frac{\partial{\mathcal{F_{\mathrm{hom}}}}}{\partial{d}}(d_{c},\Delta_{c})=0$,
we obtain,

$\centering(d_{c}-d_{0})^{2}\approx
A_{0}(\Delta_{c}-\Delta_{0})\longrightarrow(d_{c}-d_{0})\sim(\Delta_{c}-\Delta_{0})^{0.5}\@add@centering$
(43)

where, $A_{0}$ is a constant and the term containing
$(\Delta_{c}-\Delta_{0})^{2}$ is neglected for being small when
$\Delta_{0}\rightarrow\Delta_{c}$. Comparing Eqs. (41) and (43), we obtain

$\mathcal{K}(d_{0},\Delta_{0})\sim\sqrt{\delta}$ (44)

where, $\delta\sim(\Delta_{c}-\Delta_{0})$ and the third term in Eq. (41) is
neglected for being small when $\Delta\rightarrow\Delta_{c}$.

##  Appendix B Summary of exponents and scaling laws.

We here summarize the scaling laws relating exponents and compare measured
values to scaling-inferred and theory prediction values. In the table, the
avalanche size $S_{\mathrm{d}}$ can be replaced by $S$, owing to their linear
relationship.

Equation | Relation | Exponent | Measure | Scaling
---|---|---|---|---
(8) | $\ell_{x}\sim S_{d}^{1/d_{f}}$ | $d_{f}$ | $2.35\pm 0.15$ | (20) Theory: $9/4=2.25$
(9) | $T\sim S_{d}^{z/d_{f}}$ | $z$ | $1.40\pm 0.15$ | (20) Theory: $3/2=1.5$
(12) | $S^{\ast}_{d}\sim 1/\delta^{\gamma}$ | $\gamma$ | $1.00\pm 0.10$ | (30) & (31)$\Rightarrow$(32)
|  |  |  | $\simeq 1.0$
(13) | $T^{\ast}\sim 1/\delta^{\phi}$ | $\phi$ | $0.52\pm 0.10$ | (9), (12) & (13)$\Rightarrow$(14), (9) & (32)$\Rightarrow$(33)
|  |  |  | $\simeq 0.6$
(14) | $\ell_{x}^{\ast}\sim 1/\delta^{\kappa}$ | $\kappa$ | $0.37\pm 0.10$ | (8), (12) & (13)$\Rightarrow$(14), (8) & (32)$\Rightarrow$(33)
|  |  |  | $\simeq 0.42$
(13) | $\xi\sim 1/\delta^{\rho}$ | $\rho$ | $0.35\pm 0.10$ |
(30) | $\langle S\rangle\sim 1/\delta^{\alpha_{s}}$ | $\alpha_{s}$ | $0.48\pm 0.05$ | Theory: $1/2=0.50$

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