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\dateposted{2025-01-08}
\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{review}

\title{A review of characteristic lengths in the coupled criterion
framework and advanced fracture models}

\alttitle{Une revue des longueurs caract\'{e}ristiques dans le cadre du
crit\`{e}re coupl\'{e} et dans les mod\`{e}les de fracture avanc\'{e}s}

\author{\firstname{Gergely} \lastname{Moln\'{a}r}\CDRorcid{0000-0002-7274-5373}\IsCorresp}
\address{CNRS, INSA Lyon, LaMCoS, UMR5259, 69621 Villeurbanne, France}
\email[G. Moln\'{a}r]{gergely.molnar@insa-lyon.fr}


\author{\firstname{Aur\'{e}lien} \lastname{Doitrand}\CDRorcid{0000-0003-1000-3507}\IsCorresp}
\address{Universit\'e Lyon, INSA-Lyon, UCBL, CNRS, MATEIS, UMR5510, F-69621 Villeurbanne, France}
\email[A. Doitrand]{aurelien.doitrand@insa-lyon.fr}


\author{\firstname{Rafael} \lastname{Estevez}\CDRorcid{0000-0001-5036-6420}}
\address{Univ. Grenoble Alpes, CNRS, Grenoble INP, SIMAP, F-38000 Grenoble, France}
\email[R. Estevez]{rafael.estevez@grenoble-inp.fr}

\author{\firstname{Anthony} \lastname{~Gravouil}\CDRorcid{0000-0002-5967-8452}}
\address{Univ Lyon, CNRS, INSA Lyon, LaMCoS, UMR5259, 69621 Villeurbanne, France}
\email[A. Gravouil]{anthony.gravouil@insa-lyon.fr}


\begin{abstract} 
The review paper explores the significance of characteristic lengths in
fracture mechanics, focusing on the Coupled Criterion framework. It
addresses limitations in traditional Linear Elastic Fracture Mechanics,
which struggle to predict small-scale crack behaviors, and highlights
the need for models that allow characteristic lengths to emerge from
material properties and geometry rather than being predefined inputs.

The review covers two main characteristic lengths: the initiation crack
length and Irwin's length, examining their interactions with lengths
used in other fracture approaches such as Phase-Field methods, Cohesive
Zone Models, and atomic-scale simulations. The findings show that
Irwin's length consistently appears in models that combine stress and
energy criteria, indicating its fundamental role in fracture
prediction.

The study identifies limitations in current models, especially in cases
involving strong singularities or where the energy condition dominates,
and suggests improvements by incorporating process zone descriptions or
regularization techniques from Phase-Field models. These enhancements
could better capture the complex behaviors at smaller scales.

The paper concludes by advocating for a combined approach that
integrates various fracture models, which could provide a more
comprehensive understanding of crack initiation and propagation across
different scales. This integrative strategy would allow for more
accurate predictions and a deeper insight into the mechanics of
fracture.
\end{abstract}

\begin{altabstract} 
Cet article de revue explore l'importance des longueurs
caract\'{e}ristiques en m\'{e}canique de la rupture, en se concentrant sur le
cadre du Crit\`{e}re Coupl\'{e}. Il met en lumi\`{e}re les limites des approches
traditionnelles de la m\'{e}canique de la rupture \'{e}lastique lin\'{e}aire,
qui peinent \`{a} pr\'{e}dire les comportements des fissures \`{a} petite
\'{e}chelle, et souligne le besoin de mod\`{e}les permettant aux longueurs
caract\'{e}ristiques d'\'{e}merger des propri\'{e}t\'{e}s des mat\'{e}riaux et de la
g\'{e}om\'{e}trie, plut\^{o}t que d'\^{e}tre d\'{e}finies a priori.

La revue couvre deux longueurs caract\'{e}ristiques principales : la
longueur de fissure d'initiation et la longueur d'Irwin, en examinant
leurs interactions avec les longueurs utilis\'{e}es dans d'autres
approches de rupture, telles que les m\'{e}thodes de champ de phase, les
mod\`{e}les de zone coh\'{e}sive et les simulations \`{a} l'\'{e}chelle atomique.
Les r\'{e}sultats montrent que la longueur d'Irwin appara\^{i}t
syst\'{e}matiquement dans les mod\`{e}les combinant des crit\`{e}res de
contrainte et \'{e}nerg\'{e}tique, soulignant son r\^{o}le fondamental dans la
pr\'{e}diction de la rupture.

L'\'{e}tude identifie les limites des mod\`{e}les actuels, en particulier
dans les cas impliquant des singularit\'{e}s fortes ou lorsque la
condition \'{e}nerg\'{e}tique domine, et propose des am\'{e}liorations en
incorporant des descriptions de zone de processus ou des techniques de
r\'{e}gularisation issues des mod\`{e}les de champ de phase. Ces
am\'{e}liorations pourraient mieux capturer les comportements complexes \`{a} 
des plus petites \'{e}chelles.

L'article conclut en pr\^{o}nant une approche combin\'{e}e int\'{e}grant divers
mod\`{e}les de rupture, ce qui pourrait offrir une compr\'{e}hension plus
compl\`{e}te de l'initiation et de la propagation des fissures \`{a} 
diff\'{e}rentes \'{e}chelles. Cette strat\'{e}gie int\'{e}grative permettrait des
pr\'{e}dictions plus pr\'{e}cises et une compr\'{e}hension approfondie des
m\'{e}canismes de la rupture.
\end{altabstract}


\shortrunauthors

\keywords{\kwd{Finite fracture mechanics}
\kwd{Coupled criterion}
\kwd{Characteristic length}
\kwd{Crack initiation}
\kwd{Irwin's length}}

\altkeywords{\kwd{M\'{e}canique de la rupture finie}
\kwd{Crit\`{e}re coupl\'{e}}
\kwd{Longueur caract\'{e}ristique}
\kwd{Initiation de fissure}
\kwd{Longueur d'Irwin}}

\maketitle

\end{noXML}

\newcommand{\eg}{{e.g.}}
\newcommand{\ie}{{i.e.}}
\newcommand{\Ki}{K_{\mathrm{i}}}
\newcommand{\KI}{K_{\mathrm{I}}}
\newcommand{\KIc}{K_{\mathrm{Ic}}}
\newcommand{\Ginc}{\mathcal{G}_{\mathrm{inc}}}
\newcommand{\Gc}{\mathcal{G}_{\mathrm{c}}}
\newcommand{\G}{\mathcal{G}}
\newcommand{\epsilonc}{\varepsilon_{\mathrm{c}}}
\newcommand{\sigmac}{\sigma_{\mathrm{c}}}
\newcommand{\sigmann}{\sigma_{\mathrm{nn}}}
\newcommand{\Wext}{W_{\mathrm{ext}}}
\newcommand{\Wel}{W_{\mathrm{el}}}
\newcommand{\elli}{\ell_{\mathrm{i}}}
\newcommand{\ellmat}{\ell_{\mathrm{mat}}}

\section{Introduction}
Linear Elastic Fracture Mechanics (LEFM), introduced by Griffith
\cite{Griffith1921, Griffith1924}, addresses the limitations of
traditional mechanics in predicting failure in structures containing
sharp defects. Stress-based methods suffice for smooth, flaw-free
materials, but they fail around cracks, where stresses theoretically
peak to infinity. Griffith's energy-based approach relies on the
energy release rate to predict crack growth, thus avoiding the
consideration of stress singularities. However, LEFM assumes a defect
large enough to disregard smaller-scale phenomena near the crack tip,
leading to the concept of a non-linear transition length scale, which
defines the boundary between the linear elastic region and more complex
fracture processes like plasticity or other dissipative mechanisms.

Despite its effectiveness for large cracks, LEFM cannot fully capture
behaviors at this smaller scale. Additionally, most structures are not
designed with pre-existing macroscopic cracks, making Griffith's
theory impractical in such cases, where it would predict an unrealistic
infinite load-bearing capacity. As a result, engineering standards
continue to predominantly rely on stress-based criteria for materials
where significant defects are not anticipated.


The first empirical observations of the size effect date back to
Leonardo da Vinci \cite{daVinci1504, Williams1957}, who noticed that
shorter cable segments were stronger than longer ones, though he did
not provide a practical explanation for this phenomenon. It was Galileo
Galilei who later formulated the correct scaling laws for materials
under tension and bending \cite{Galilei1638}, emphasizing how size
effects limit the structural integrity of large natural and man-made
structures. Centuries later, as iron and steel became more widely used,
concerns about brittle fracture grew, prompting early material failure
testing \cite{Kirkaldy1864}. Around the same time, Mariotte
\cite{Mariotte1886}, through extensive experimentation, suggested that
the size effect observed by da Vinci was likely due to internal faults,
concluding that larger structural elements have a higher probability of
containing weak spots, thus reducing their overall strength.



Alongside the development of fracture mechanics, researchers began
exploring statistical theories to explain the power-law scaling
observed in experimental data. Peirce \cite{Peirce1926} introduced the
weakest-link model for chains, building on extreme value statistics by
Tippett \cite{Tippett1925}. This line of work reached a milestone with
Weibull \cite{Weibull1939a, Weibull1939b}, who developed the Weibull
distribution to model failure probability based on low-strength
extremes, establishing a power-law relationship between material
strength and failure probability, especially in materials with
microscopic flaws or microcracks \cite{Freudenthal1946}. This
statistical approach has since been applied across various materials
and fracture phenomena \cite{Kittl1990, Evans1978, Beremin1983,
Lei1998}. While widely accepted, in this paper, we focus on a
physically-based deterministic approach, recognizing that a combined
statistical-deterministic framework can provide a more comprehensive
view of material failure across scales.



The non-linear scaling law, first documented by Irwin in the 1950s
\cite{Irwin1958}, was initially overlooked or considered a statistical
anomaly. Motivated by the observation that large concrete structures
(such as dams and bridges) behave differently from small laboratory
specimens, Ba{\v{z}}ant conducted a series of experiments
\cite{Bazant1984}. He eventually published his theoretical explanation
in 1986, describing a non-linear scaling law in fracture mechanics
\cite{Bazant1986}. Ba{\v{z}}ant emphasized the need for non-linear
analysis to account for the significant size effects observed in
various engineering structures. This phenomenon has since become
critical for the design of large-scale composite structures such as
ship hulls or structural fuselages, as well as in fields like
geotechnical and arctic engineering. For example, evaluating fault slip
stability in the Earth's crust involves scale transitions that span
multiple orders of magnitude.

Ba{\v{z}}ant \cite{Bazant1999} further showed that fracture resistance
in many materials deviates from the power-law predictions of linear
elastic fracture mechanics, especially when the initial flaw size is
smaller than a critical value. In such cases, stress-based criteria
should be applied. The existence of this critical length scale has
since been demonstrated in various materials, including ceramics
\cite{Kimoto1985, Usami1986, Leguillon2018, Martin2018}, polymers
\cite{Doitrand2020IJNLM, Sapora2018}, silica glass \cite{Luo2016},
silicon carbide \cite{Bazant1990}, fiber composite laminates
\cite{Bazant1996}, wood \cite{Aicher2010}, concrete, rock
\cite{Shah1987}, spider silk~\cite{Keten2010} and even sea ice
\cite{Dempsey1999}. However, experimentally demonstrate this
non-linear scale transition is challenging, as it requires testing
specimens across multiple size ranges.



The transition length scale is often compared to the size of the
fracture process zone (FPZ), a region around a crack tip where complex,
nonlinear deformation occurs. The FPZ, characterized by a transition
from elastic to inelastic behavior, plays a critical role in fracture
mechanics. In the 1950s, Irwin \cite{Irwin1948} and Orowan
\cite{Orowan1949} used X-ray measurements to demonstrate that even in
brittle materials, there is evidence of regularization along crack
surfaces. They independently concluded that the true critical energy
release rate should be several orders of magnitude larger than
Griffith's original proposal. Later, Barenblatt \cite{Barenblatt1959}
and Dugdale \cite{Dugdale1960} theorized that material near the crack
yields, and this local cohesive traction limits the otherwise infinite
stress\break peak.

Since then, numerous experimental techniques \cite{Labuz1983,
Chengyong1990, Denarie2001, Du1990, Guo1993, Yu1994, Zang2000,
Otsuka2000, Zietlow1998, Labuz1987} have been developed to measure the
size and shape \cite{Neimitz1987} of the FPZ in brittle materials.
These studies commonly assume that the FPZ is a damaged region around
the crack tip linked to irreversible microstructural changes. The FPZ
has been observed in materials such as concrete \cite{Cedolin1983},
granite \cite{Labuz1983}, natural faults \cite{Vermilye1998}, wood
\cite{Yu2019}, model materials \cite{Haidar2005}, and silica glass
\cite{Rountree2010}. A comprehensive review of the FPZ can be found in
the thesis of Brooks \cite{Brooks2013}. Today, digital image
correlation \cite{Rethore2013} is the primary technique used to
quantify the FPZ, although other methods exist for transparent
materials like polycarbonate \cite{Cortet2005} or for X-ray measurement
in concrete \cite{Otsuka2000}. D{\"o}ll and K{\"o}ncz{\"o}l
\cite{Doll1983, Doll1990} used optical interferometry to measure crack
tip opening displacement in polymethyl methacrylate, polystyrene, and
polycarbonate. Their findings indicate that, for these materials, the
experimentally measured opening profile aligns well with predictions
from an appropriate Dugdale model.

While the non-linear scaling law is widely accepted, its underlying
cause remains a topic of active debate. This is particularly important
given the rise of advanced manufacturing techniques that allow the
creation of architected materials with structural elements smaller than
the critical length scale of bulk materials, resulting in exceptionally
strong overall responses \cite{Schaedler2011}.

In this review paper, we will explore recent methods and theories that
effectively account for this non-linear length scale. This length scale
can either be explicitly incorporated as a material parameter or arise
through the coupling of different criteria.

Even if well-described by various fracture mechanics models
through numerical experiments, it is not always straightforward to
explain this transition between strength-based and energy-based
descriptions of fracture. This transition is clearly understood using
the Coupled Criterion (CC) concept \cite{Leguillon2002}, that was
proposed to rationalize transverse cracking experiments in laminated
composites by Parvizi, Garrett, and Bailey \cite{Parvizi1978}. Their
results showed two distinct regimes: for sufficiently thick plies, the
strain at crack initiation remained constant, whereas it increased with
decreasing ply thickness for thin plies. A criterion based solely on
critical stress or strain---first introduced by Lam\'{e} and Clapeyron
\cite{Lame1831}---could only account for the failure of thick plies.
Extending this approach to a non-local stress evaluation, either
averaged over a finite volume \cite{Neuber1936} or at a specific
distance \cite{Peterson1938,Whitney1974,Tanaka1983}---known as the
Theory of Critical Distances \cite{Taylor2007}---did not explain the
increase in strain at failure for thin plies. Since Griffith's energy
approach cannot predict crack initiation, an energy-based criterion
applied to a finite crack surface increment was introduced,
representing crack initiation across the full ply thickness. This idea
was already suggested in Aveston and Kelly's model \cite{Aveston1973}
and was later formalized as Finite Fracture Mechanics (FFM) by Hashin
\cite{Hashin1996} and then by Nairn \cite{Nairn2000}, successfully
explaining the size effect in thin ply fracture. However, it
underestimated the load level for crack initiation in thick plies.
Dominique Leguillon \cite{Leguillon2002} eventually proposed combining
a non-local stress criterion with an energy criterion applied to a
finite crack surface increment, asserting that both conditions must be
met simultaneously for crack initiation. This approach effectively
explains the transition from tensile strength-driven failure in thick
plies to energy release rate-driven failure in thin plies.

The CC highlights that the nonlinear fracture resistance scaling
is driven by the ratio between the initial flaw size and the Irwin's
length. It results from the initiation length emerging from the
coupling between the strength-based and energy-based criteria. Our
primary focus will thus be on the CC \cite{Leguillon2002}, one of the
earliest approaches to offer a mechanics-based explanation for the
emergence of a process zone. We will then relate macroscopic theories,
such as the theory of critical distances, the Phase-Field method, and
the Cohesive Zone Model, to the fundamental concept of the Coupled
Criterion. We also demonstrate how approaches that model atomic-scale
behavior can bridge the gap between the continuum scale and the actual
material properties.




The paper is structured as follows. First, in
Section~\ref{sec:CClength} the basics of the Coupled Criterion is
presented with particular attention to the emerging length scale. Then
in Section~\ref{sec:Corr}, we summarize the literature comparing the
results of the Coupled Criterion with other theories and numerical
methods with allow us to have a length scale explicitly or by
emergence.  In Section~\ref{sec:discussion}, we compare and contrast
the results obtained with the methods, highlighting their similarities
and differences. Finally, in Section~\ref{sec:conclus}, we draw
conclusions based on our findings.


\section{The characteristic lengths in the Coupled Criterion}
\label{sec:CClength}

The Coupled Criterion (CC) \cite{Leguillon2002} is an approach in
fracture mechanics that enables us to study fracture analysis in a wide
range of applications \cite{Weissgraeber2016review,Doitrand2024JTCAM}.
The underlying concept of the CC is that crack initiation occurs when
two conditions are simultaneously met. The first condition arises from
an energy equilibrium between the states before and after crack
initiation over a specified surface $S$. This allows for the definition
of the incremental energy release rate (IERR), given linear elastic
material behavior and negligible inertial effects, as $\Ginc=(
\delta \Wext - \delta \Wel ) / S$, where $\Wext$ represents the
work done by external forces, $\Wel$ the elastic strain energy, and $S$
the crack surface area. The IERR approaches the energy release rate
(ERR) $\G$ as the crack surface area approaches zero. The CC model is
thus consistent with Linear Elastic Fracture Mechanics (LEFM) for
sufficiently long cracks (under LEFM assumptions), while also enabling
the study of crack initiation, which LEFM cannot handle as $\G$ tends
toward zero when the crack surface tends toward zero. The energy
condition for the CC is expressed as:
{\begin{eqnarray}
\Ginc \geqslant \Gc,
\label{eq:Energy_condition}
\end{eqnarray}}\unskip
where $\Gc$ is the material's critical ERR.

The second condition of the CC states that the stress along the
prospective crack path must be sufficiently large. This introduces a
non-local stress criterion, which can be expressed as a function of the
stress tensor components and the material strength surface. For a
material adhering to a Rankine's strength surface \cite{Rankine1857},
it is given by:
{\begin{eqnarray}
\sigmann (\mathbf{x}) \geqslant \sigmac\forall \mathbf{x}
\quad \mbox{in }\Gamma, 
\label{eq:Stress_condition}
\end{eqnarray}}\unskip
where $\sigmann$ is the stress normal to the crack path $\Gamma$ before
initiation, and $\sigmac$ is the material's tensile strength.

Applying the CC involves combining both the stress and energy
conditions to determine the minimum load magnitude at which both
criteria are simultaneously met for at least one given crack surface.

An emblematic illustration of the CC is the initiation of a crack at
the tip of a V-notch, where the free surfaces form an angle $\beta$, as
seen in a notched specimen under bending, illustrated in
Figure~\ref{fig:CHAPCC_Vnotch}.

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig:CHAPCC_Vnotch}(a) Three-point bending of a specimen with a V-notch. (b)
Characteristic exponent of the singularity $\lambda$ corresponding to
the opening mode as a function of $\beta$, the angle of the
V-notch.}
\vspace*{-3pt}
\end{figure}

The case $\beta = 0\text{\textdegree}$ corresponds to an initial crack and reverts
to LEFM (\ie, crack propagation based solely on energy) provided the
crack is sufficiently long \cite{Molnar2020}. The case $\beta =
180\text{\textdegree}$ corresponds to a straight edge with no notch; here, the
stress is homogeneous, allowing a criterion based on material tensile
strength if the specimen is large enough \cite{Leguillon2002,
Doitrand2020JACS}. An asymptotic approach yields the stress normal to
the crack path before initiation $\sigmann$ and the IERR for a crack of
length $\ell$ at the notch tip \cite{Leguillon1987}:
{\begin{eqnarray}
\left\lbrace
\begin{array}{l} 
\sigmann(r,\theta=0) = K \ell^{\lambda-1}, \\
\displaystyle
\Ginc(\ell) = \frac{K^2}{E} \ell^{2\lambda-1} A_\beta,
\end{array}
\right.
\label{eq:CHAPCC_sigma_G_asy}
\end{eqnarray}}\unskip
where $r$ and $\theta$ are the polar coordinates, $E$ is Young's
modulus, and $\lambda$ is an exponent characterizing the singularity,
varying between $0.5$ for a crack ($\beta=0\text{\textdegree}$) and $1$ for a
straight edge ($\beta=180\text{\textdegree}$). In-between cases are presented in
Figure~\ref{fig:CHAPCC_Vnotch}b. The dimensionless function $A_{\beta}$ depends on the
problem's geometry (notably the notch angle $\beta$) and the local
loading at the singular point, represented by the generalized stress
intensity factor (GSIF) $K$. The GSIF drives the magnitude of the
local stress field variation near the V-notch tip. It is a
characteristic parameter that can be used~to study the nucleation of a
crack, \ie,~crack initiation occurs when a critical GSIF is reached. It
is calculated based on a path-independent contour integral
\cite{Leguillon1987,Doitrand2020FINEL}. Under a linear elastic
framework and assuming small deformations, the GSIF is proportional to
the applied load $F$, shown in Figure~\ref{fig:CHAPCC_Vnotch}a.

If $0<\beta<180\text{\textdegree}$, the stress tends to infinity as $r$ approaches
$0$, making a stress-based criterion alone ineffective, as it predicts
crack initiation under an infinitesimal load. Conversely, the IERR
approaches $0$ as the crack length approaches $0$, meaning an
energy-based criterion alone does not predict crack initiation either.
This indicates a missing element in crack initiation studies: a
\textit{length scale}. One approach is to introduce this length as an
additional input parameter to the stress criterion (yielding the Theory
of Critical Distances \cite{Taylor2007}) or in conjunction with the
energy criterion, resulting in an incremental energy approach within
Finite Fracture Mechanics \cite{Hashin1996,Nairn2000}.

Another possibility is to use the CC, \ie,~to combine the stress
criterion and the energy criterion, which are already valid for the two
extreme cases ($\beta=0\text{\textdegree}$ and $\beta=180\text{\textdegree}$). We then look for
the minimum loading level and the corresponding crack length for which
both criteria are met. The combination of the two criteria introduces a
characteristic length into the problem, which is not an input parameter
but rather the result of this coupling. We then obtain the initiation
crack length $\elli$ and the GSIF $\Ki$ at initiation
\cite{Leguillon2002,Leguillon2007}:
{\begin{eqnarray}
\left\lbrace
\begin{array}{l} 
\displaystyle
\elli = \frac{E\Gc}{A_\beta \sigmac^2}=\frac{\ellmat}{A_\beta}, \\
\displaystyle
\Ki = \left( \frac{E\Gc}{A_\beta}\right)^{1-\lambda} (\sigmac )^{2\lambda-1}.
\end{array}
\right.
\label{eq:CHAPCC_li_Ki}
\end{eqnarray}}\unskip

We note that the GSIF is homogeneous to the product of a stress and a
length to the power ($1-\lambda$). For a straight edge, $\lambda=1$ and
the GSIF at initiation becomes $\Ki=\sigmac$ ($\Ki$ is then homogeneous
to a stress), thus reducing to a stress criterion. In the case of a
crack, $\lambda=1/2$ and we return to Irwin's criterion
$\Ki=\KIc=\sqrt{E\Gc/A_{\beta}}$ ($\Ki$ is then homogeneous to a
critical stress intensity factor). LEFM is therefore included within
the CC formulation, allowing for the study of both crack initiation and
the propagation of an existing crack. The proposed formulation remains
general, as it can address crack initiation with or without a
singularity.

In addition to the initiation length, the combination of the two
criteria introduces another characteristic length, Irwin's length
$\ellmat = E\Gc / \sigmac^2$ \cite{Irwin1958}, which is intrinsic to
the material since it depends only on its elastic and fracture
properties. The asymptotic approach reveals that the initiation length
is actually related to Irwin's length through the dimensionless
coefficient $A_\beta$, which depends on (i) the geometry (here the
V-notch angle) and (ii) the loading (here the GSIF)
\cite{Martin2018,Doitrand2020FINEL}. In the following, we examine how
these two lengths in the CC formulation (\ie, the initiation length and
Irwin's length) correspond to other characteristic lengths encountered
in various fracture mechanics approaches.


\section{Correlation with lengths involved in other fracture approaches}
\label{sec:Corr}

In this section, we focus on different approaches able to describe the
nonlinear fracture resistance scaling, similarly to the CC. Even if
based on different fracture description and involving different input
parameters, we thus provide an insight on how the characteristic and
intrinsic length scales involved in these approaches are related to the
one obtained in the CC: the initiation length and Irwin's length.

Figure~\ref{fig:summary} illustrates schematically how various major
techniques facilitate the emergence of a characteristic length scale,
which subsequently aids in describing the nonlinear scaling observed in
experiments.

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig:summary}Summary of the key methods discussed, highlighting how each
technique integrates material length scales into its framework and how
these are reflected in simulation outcomes. In the figure $d$
represents damage, $l_{\mathrm{c}}$ is the phase-field length scale, $a$ is the
crack length and $\delta_{\mathrm{c}}$ is the critical separation when the crack
is fully open.}
\end{figure}

For the Coupled Criterion (CC), a characteristic length emerges as
the distance where both energy and stress criteria are simultaneously
satisfied. This length can consistently be correlated to Irwin's
fundamental metric. Notably, studies have shown that if the initial
crack is sufficiently sharp and large compared to the material's
intrinsic scale, the resulting length remains constant
\cite{Molnar2020}. This observation provides a possible
explanation for the effectiveness of the Theory of Critical Distances
(TCD): evaluating the stress at a fixed distance produces results
comparable to more advanced criteria.

In smeared damage approaches, such as the Phase-Field (PF) and
Thick Level Set (TLS) methods, a regularization length is inherently
introduced. This regularization modifies the stress profile, resulting
in a slight alteration not only near the crack tip but also in the tail
of the theoretically elastic stress response for the same macroscopic
equilibrium. This phenomenon raises an important question: at what
distance does the calculation of experimental toughness, based on a fit
to the singular solution, remain valid?

The Cohesive Zone Model (CZM) operates in a manner similar to
smeared damage methods, defining the tension--separation law as an input
parameter. However, CZM offers greater flexibility by allowing the
critical separation to be specified, which results in a certain damage
diffusion length representing the process zone ahead of the crack tip.
The latter also depends on the geometry, boundary conditions and local
stress state. Despite this advantage, the technique suffers from a
significant limitation: it requires prior knowledge of the crack path.

Finally, particle-based methods, such as molecular dynamics (MD)
and peridynamics, treat the material as a discrete system of particles.
In peridynamics, the horizon explicitly defines the characteristic
length scale. In atomic-scale simulations, the interaction between
interatomic potentials and the realistic atomic structure leads to the
emergence of a localized, inelastic zone around the crack tip
\cite{Rountree2010}.

This section discusses these various methods and evaluates their
potential for comparison with the Coupled Criterion. We will highlight
results from the literature that address the nonlinear scaling
transition and examine the underlying mechanisms involved.


\subsection{Theory of critical distances}\label{ssec:TCD}

The approaches based on the Theory of Critical Distances (TCD) are
commonly used for engineering failure prediction
\cite{Taylor2005,Taylor2007}. Since a local maximum stress criterion is
unsuitable for predicting the experimentally observed size effect, even
for non-singular stress fields \cite{Chao2021}, TCD compares the stress
at a specified distance from a stress concentration or singular point
to the material tensile strength.

The CC and the TCD have been compared in various configurations, \eg,
to study the fatigue limit of V-notch specimens \cite{Liu2020}.
Regardless of the notch radius, the critical distance remains constant
in the TCD, whereas it decreases with increasing notch radius in the
CC. By basing the TCD critical distance on Irwin's length, a
relationship between the critical crack advance for both
approaches---independent of material parameters---was derived. This
result aligns with the fact that the initiation crack length in the CC
is proportional to Irwin's length \cite{Martin2018,Doitrand2020FINEL}.

Chao Correas et~al.~\cite{Chao2021} demonstrated that both the CC and
TCD describe the gradual transition between two stress-driven solutions
for crack initiation at a spherical void (for small and large void
radii, respectively). The transition between these regimes falls within
the same range of void radii relative to the material characteristic
length.

Campagnolo et~al.~\cite{Campagnolo2016} compared the CC to the Strain
Energy Density (SED) approach for crack initiation at a V-notch under
in-plane shear loading. The SED model considers the strain energy
density over a control volume around the crack initiation site as the
critical parameter. Both methods showed that the apparent SIF at crack
initiation is proportional to powers of $\KIc$ and $\sigmac$, differing
only in the proportionality factor, which depends on the notch angle in
the CC and Poisson's ratio in the SED approach. Both methods
predicted similar apparent SIFs at crack initiation for this
configuration, with the control volume radius based on Irwin's length
\cite{Yosibash2004,Campagnolo2016}.

The primary distinction between the TCD and CC is that in TCD, the
characteristic length is an input parameter, while in CC it is an
output derived from combining the stress and energy conditions.
Several ways of nonlocal stress evaluation exist, such as
integration over a volume or the use of pure nonlocal functions
\cite{Gillham2023,Raghavendra2024}. One of the first method
proposed was to evaluate the stress at a given distance from a singular
point. For example, applying the TCD as a point stress criterion at a
distance equal to Irwin's length ($\sigmann(\ellmat)=\sigmac$) yields:
{\begin{eqnarray}
\Ki = (E\Gc )^{1-\lambda} \sigmac^{2\lambda-1}.
\label{eq:CHAPCC_sigma_TCD}
\end{eqnarray}}\unskip

This expression only differs from the one obtained using the CC by the
coefficient $A_\beta$ (see Equation~(\ref{eq:CHAPCC_li_Ki})), reflecting
that the TCD disregards geometry, whereas the CC accounts for it
through $A_\beta$. Figure~\ref{fig:CHAPCC_TCD} illustrates the
differences between the TCD and CC in predicting crack initiation at a
sharp V-notch.

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig:CHAPCC_TCD}(a) Initiation length normalized by Irwin's length as a
function of the V-notch angle predicted using the CC for a V-notch in
an infinite medium. (b) Ratio between the GSIF obtained using either
the CC ($\Ki^{\mathrm{CC}}$) or the TCD ($\Ki^{\mathrm{TCD}}$) as a
function of the V-notch angle.}
\end{figure}

The normalized initiation length obtained using the CC for different
V-notch angles is shown in Figure~\ref{fig:CHAPCC_TCD}a. It indicates
that the initiation length using the CC is generally smaller than
Irwin's length and depends on the geometry. In addition,
Figure~\ref{fig:CHAPCC_TCD}b shows the ratio between the GSIF obtained
using the CC or the TCD as a function of the V-notch angle. Based on
Equations~(\ref{eq:CHAPCC_li_Ki}) and~(\ref{eq:CHAPCC_sigma_TCD}), 
this ratio is given by
$\KI^{\mathrm{CC}}/\KI^{\mathrm{TCD}}=A_\beta^{\lambda-1}$. Using LEFM
normalization of displacement fields, a ratio of $1$ is achieved for
$\beta=0\text{\textdegree}$ and $\beta=180\text{\textdegree}$, with a
non-monotonic trend as $\beta$ increases. Specifically, $A_\beta=1$ for
$\beta=0$, decreasing with increasing $\beta$, while $\lambda$
increases from $1/2$ ($\beta=0\text{\textdegree}$) to $1$
($\beta=180\text{\textdegree}$). Overall, differences of less than 20\%
can be expected between the CC and TCD in predicting crack initiation
at a V-notch in an infinite medium.

Since the TCD disregards the overall specimen geometry, questions arise
about its application for small-scale specimens, where size may be
smaller than Irwin's length. This method may also be unsuitable for
initiation configurations driven by the energy criterion, such as
transverse cracking in thin composite laminates
\cite{Leguillon2002,Doitrand2022EJMAS,Hamam2022}.


\subsection{Phase-field}\label{ssec:Phase-Field}

Fracture Phase-Field (PF) models \cite{Bourdin2000, Bourdin2008}
approximate crack discontinuities through a smeared damage field
controlled by a length scale parameter ($l_{\mathrm{c}}$), which defines the
extent of damage. These models balance elastic energy with diffused
fracture energy to identify the energetically favorable crack front,
using fracture toughness ($\Gc$) and the regularization length ($l_{\mathrm{c}}$)
as primary inputs.

Both the CC and PF approaches use the critical energy release rate
($\Gc$) as an input parameter, but they differ in how they handle
tensile strength: CC explicitly incorporates it, while PF replaces it
with the regularization length ($l_{\mathrm{c}}$). In simulations of notched thin
ply laminate fractures, Reinoso et~al.~\cite{Reinoso2017} showed that
the CC method effectively captures the size effect and accurately
predicts failure stress, whereas the PF approach slightly
underestimates failure stress, with $l_{\mathrm{c}}$ chosen to match the
experimentally measured displacement around stress concentrators.

Early studies comparing CC and PF results \cite{Bleyer2018} found good
correspondence between the two methods, although the choice of $l_{\mathrm{c}}$
was often based on matching maximum tensile stress under uniaxial
loading to the material's tensile strength. Strobl
et~al.~\cite{Strobl2017, Strobl2020} simulated Hertzian
indentation-induced fractures using both CC and PF, noting consistent
trends in crack location and critical displacement, with $l_{\mathrm{c}}$
determined using the homogeneous Phase-Field solution under uniaxial
tension.

Kumar et~al.~\cite{Kumar2020a} addressed PF's tendency to overestimate
critical loads at the onset of damage by explicitly incorporating a
stress criterion, effectively creating a CC-inspired PF approach.
Similarly, Abaza et~al.~\cite{Abaza2022} calibrated $l_{\mathrm{c}}$ in PF models
for notched ceramic specimens to match apparent stress intensity
factors at crack nucleation with those obtained using CC. Jimenez
et~al.~\cite{Jimenez2022ECF} demonstrated that for small-scale
specimens, critical displacements or forces largely depend on $\Gc$,
using CC to guide the load range selection in PF models and suggesting
PF as a preliminary step for CC when the crack path is not known
{a priori}.

A comprehensive comparison of CC and PF approaches was provided in Ref.
\cite{Molnar2020}, which studied tensile opening and in-plane shear
fractures, proposing a correlation between tensile strength and $l_{\mathrm{c}}$
that depends on the stress state:
{\begin{eqnarray}\label{eq:Homo}
    {\sigma _{\mathrm{c}}} \simeq {\sigma _{\max }} = \eta \left( {\nu ,\frac{{{\sigma _2}}}{{{\sigma _1}}},\frac{{{\sigma _3}}}{{{\sigma _1}}}} \right)\sqrt {\frac{{E\Gc}}{{{l_{\mathrm{c}}}}}},
\end{eqnarray}}\unskip
where $\eta$ accounts for the stress state. This work was later
extended to antiplane shear \cite{Molnar2024}. In these studies, 
different aspects were compared, such as critical initiation load in
tension, branching angle in simple shear, and facet spacing in
antiplane shear.

The results are summarized in Figure~\ref{fig:CC_PF}. The correlations
for tensile opening, in-plane, and antiplane shear fracture modes show
a general trend: as the regularization length ($l_{\mathrm{c}}$) decreases, the
material strength ($\sigma_{\mathrm{c}}$) increases. However, this relationship
cannot be captured by a single master curve; instead, it forms a
failure surface varying within the range defined by
Equation~(\ref{eq:Homo}), shown by the shaded area in Figure~\ref{fig:CC_PF}a.
Although the initiation length $\Delta a_{\mathrm{c}}$ does not explicitly appear
in the PF, it is crucial for determining where the stress and energy
criteria are satisfied simultaneously, with its correlation to $l_{\mathrm{c}}$
being linear across different fracture modes.

\begin{figure}
\includegraphics{fig04}
\caption{\label{fig:CC_PF}(a) Summarized correlations between the tensile strength
($\sigma_{\mathrm{c}}$) and the internal length scale ($l_{\mathrm{c}}$). The blue shade
represents the accessible space based on the homogeneous solution. (b)
Correlation between $l_{\mathrm{c}}$ and the initiation lengths and characteristic
distances.}
\end{figure}

The results suggest that $l_{\mathrm{c}}$ serves as an intermediate parameter
between Irwin's intrinsic length and the actual process zone size,
though it cannot fully account for the effects of macroscopic geometry.
This raises the question of whether materials have a single tensile
strength (as per Rankine's theory \cite{Rankine1857}) or if the maximum
tensile strength is influenced by the stress state (corresponding to
another strength surface in the principal stress space).

Furthermore, studies \cite{Molnar2020,Kristensen2021} showed that,
similar to the CC, the PF method satisfies both energy and stress
criteria due to an indirect correlation between the maximum tensile
stress and $l_{\mathrm{c}}$. As $l_{\mathrm{c}}$ increases, the process zone enlarges,
resulting in lower maximum stress and an earlier satisfaction of
Griffith's criterion. Thus, higher $l_{\mathrm{c}}$ values correlate with lower
maximum tensile stresses needed for fracture.

Additionally, in the PF technique, the stress field is non-singular,
requiring the stress tail to be slightly higher than the singular
solution used in the CC to maintain equilibrium. The study of antiplane
shear highlighted the need for an enhanced CC model incorporating a
regularized stress field inspired by PF regularization, which would
allow the CC to better address three-dimensional antiplane cracking by
introducing a third parameter to account for the crack geometry, as
suggested in Ref. \cite{Doitrand2024IJF}.



\subsection{Cohesive zone}\label{ssec:CZM}
When used in fracture mechanics, a Cohesive Zone Model (CZM)
\cite{Barenblatt1959,Dugdale1960} is designed to describe the formation
and evolution of both the crack (corresponding to the traction-free
region) and the process zone ahead of the crack tip. Given, as input
parameters, the material tensile and shear strengths and critical
energy release rate, the CZM defines the traction--separation behavior
between two surfaces.

The cohesive response is triggered once a local critical traction is
reached, leading to separation and a discontinuity in the displacement
field across the surfaces, governed by a distribution of traction. The
traction--separation profile mechanically replicates the underlying
failure mechanisms and remains active until a critical separation
$\delta_{\mathrm{c}}$ is reached, at which point a crack nucleates locally. This
critical separation $\delta_{\mathrm{c}}$ provides an intrinsic length scale that
characterizes the failure process. As the separation between the two
surfaces increases, the traction--separation behavior progresses to this
critical point, defining a characteristic length associated with the
displacement jumps necessary for crack formation.

As an example, considering only the opening mode and a bilinear
traction--separation profile in a linear elastic isotropic material, the
critical displacement jump is given by:
{\begin{eqnarray}
\delta_{\mathrm{c}} = \frac{2\Gc}{\sigmac} = \frac{2\sigmac}{E} \ellmat = 2\epsilonc\ellmat, 
\label{eq:CHAPCC_deltac}
\end{eqnarray}}\unskip
where $\epsilonc$ is the material tensile strain at failure. The
critical displacement jump is thus related to the material's Irwin's
length and critical strain, making it an intrinsic material property.

In addition, to this characteristic length, another length is involved
in CZM which is the extent of the process zone ahead of the crack.
Indeed, the CZM and the CC mainly differ concerning the description of
the cracking process. While the CC relies on a binary description of
fracture considering two possible states, namely undamaged or cracked
material, the CZM defines an intermediate state: the process zone
through the description of a traction--separation profile. This process
zone induces another main difference between both approaches since the
stress can locally be larger than the material tensile strength in the
CC whereas in CZM, the stress is always bounded by the material
strength within the process zone. The process zone length depends on
the Irwin's length \cite{Alfano2001,Alfano2006}, but contrary to the
critical displacement jump, it is not an intrinsic material property as
it also depends on the specimen geometry and boundary conditions.
Cornetti et~al.~\cite{Cornetti2016short} observed that for an initial
crack in infinite medium or at a V-notch, the CZM process zone length
was significantly different from the initiation length obtained using
the CC, even if both length variations followed almost identical trends
with respect to the normalized initial crack length. This analysis was
then refined \cite{Cornetti2019} by introducing a weight function in
the stress condition of the CC in order to match the CZM. The CZM with
cohesive laws exhibiting earlier softening showed satisfactory
correspondence with the CC stress conditions, modified by weight
functions that were elevated near the crack tip and tapered off with
distance. Actually, it was shown that the difference between the
failure load predicted by the CC and the CZM differs as the critical
separation (and equivalently the Irwin's length) increases
\cite{Martin2016}.

Nevertheless, an equivalence may possibly be determined between the CC
and a given traction--separation CZM profile. Summarizing previous works
aiming at the comparison between the CC and CZM
\cite{Henninger2007,Murer2010,Garcia2014,Cornetti2015,Stein2015,Tavara2016,Dimitri2017,Rosendahl2017,Gentieu2018,Doitrand2019TAFMEC,MunozReja2020Interfaces,Chao2021},
it appears that there is not a unique CZM traction--separation profile
that enables retrieving the failure loading and crack length predicted
using the CC. The CZM traction--separation profile corresponding to the
CC actually depends on the geometry, the type of loading, the cracking
mechanism and thus has to be identified for a given configuration. A
further comparison between the CC and the CZM was established based on
another extrinsic length, \ie,~the length of the crack after the
unstable propagation following initiation \cite{Doitrand2019TAFMEC}. It
was shown that the range of crack lengths after unstable propagation
obtained with various traction--separation profiles respectively
comprised the crack length lower bound obtained using the CC, and that
crack lengths similar to those obtained using the CC were obtained
using a bilinear traction--separation profile in that case. 


\subsection{Atomic scale simulations}\label{ssec:Atomistic}

At the smallest scales currently accessible through computational
resources to study fracture, molecular dynamics (MD) simulations
\cite{Rountree2010} offer valuable insights into crack propagation by
using interatomic potentials and detailed atomic structures, thus
avoiding the need for additional numerical parameters. Recent studies
\cite{Aghababaei2016,Aghababaei2017} have advanced our understanding of
finite-size effects, revealing a critical transition in wear mechanisms
at the atomic scale. Specifically, atomistic simulations show that when
asperity contact junctions surpass a critical size, fracture-induced
debris formation occurs, while smaller junctions result in a gradual
smoothing through plastic deformation. This non-linear behavior
highlights the crucial role of length scale in determining whether
fracture or plastic smoothing dominates.

Later, Brochard et~al.~\cite{Brochard2018} specifically examined brittle
failure in two materials: a simplified 2D toy model and graphene. Using
molecular dynamics (MD) simulations, the authors investigate how the
stress and toughness criteria contribute to the emergence of a length
scale during failure. The toy model, consisting of a regular triangular
lattice with harmonic interatomic interactions, allowed for a
straightforward theoretical analysis of failure processes. In contrast,
graphene, with its more complex atomic structure and realistic
mechanical behavior, served as a case study for a real material.

For both materials, the authors use MD simulations to observe failure
under different conditions, such as varying temperature, system size,
and loading rate. Through these simulations, the study highlighted the
emergence of a nonlinear transition length scale at the process zone
near the crack tip. This process zone grew larger as the temperature
increased, and the material transitions from a stress-based failure
mode to one governed by energy dissipation at the crack tip. The
results showed that, in graphene, this length scale and the
corresponding process zone were much larger compared to the toy model,
demonstrating how atomic interactions and material properties influence
the scaling of strength and toughness. The simulations provided a
deeper understanding of how microscopic bond-breaking processes connect
to macroscopic failure behaviors, particularly in materials with
varying defect types and sizes.

Chao Correas et~al.~\cite{Chao2021} validated their findings by
comparing the FFM approach with both experimental data and atomistic
simulations. In terms of atomistic simulations, the study by Ippolito
et~al.~\cite{Ippolito2007} on $\beta$-silicon carbide is highlighted. In
this work, atomistic simulations were used to create a model free of
intrinsic defects by removing atoms from a crystalline lattice to
simulate spherical voids. The simulations provided critical material
properties, such as fracture toughness and strength, which allowed for
the calculation of the Irwin's length (characteristic length scale).

Chao Correas et~al.~\cite{Chao2021} found that the results of the
atomistic simulations agreed with the predictions made by FFM,
particularly when using the averaged stress variant. This variant
provided the most accurate results when compared to the atomistic data,
which eliminated scattering from experimental flaws and imperfections.
This supported the use of FFM as a robust predictive tool in materials
without inherent defects. The study concluded that atomistic
simulations are crucial for refining predictions of failure in brittle
materials, offering a close match with coupled criteria approaches like
FFM in defect-free scenarios.

\subsection{Thick level set}\label{ssec:TLS}

The underlying idea of the The Thick Level Set (TLS) model is to
constrain the norm of the damage gradient to control the evolution of a
damage field, introducing a characteristic length which represents the
smallest possible distance between a fully damage point and a point
where there is no damage. The characteristic length thus represents the
extent of the regularization zone around a sharp crack. Zghal
et~al.~\cite{Zghal2018} compared the (TLS) approach to the matched
asymptotic approach of the CC considering sharp or blunted notches and
cavities. TLS and CC resulted in close apparent strengths for all cases
provided the assumptions of the matched asymptotic approach were
satisfied. However, no comparison between the TLS characteristic length
and the CC initiation length was provided.

\subsection{Peridynamics}\label{ssec:Peridynamics}

Zhang et~al.~\cite{Zhang2018} implemented the CC to study crack
initiation at circular holes within a peridynamic framework. They
showed that similar strain and stress values were obtained using either
peridynamics or FE modeling, except locally near the hole edge due to a
skin effect arising from the incomplete non-local horizon for nodes
around the hole edge. They also demonstrated that peridynamics yielded
failure stresses similar to those obtained using Finite Fracture
Mechanics \cite{Zhang2018}. Ultimately, in peridynamics, the
characteristic length is defined by the horizon size.

In a subsequent study, Zhang et~al.~\cite{Zhang2019} analyzed the stress
and IERR for various horizon sizes. The main conclusion was that the
skin effect influences both stress and IERR, especially for small crack
lengths near a free edge. The stress and IERR values in this region are
unreliable, as the local material response near a free edge differs
from that in the bulk due to the incomplete horizon. They also observed
that the region where stress and IERR deviate from the finite element
solution increases with a larger horizon size. Ultimately, the horizon
size must be set sufficiently smaller than the initiation length in the
CC. Under this condition, similar stress and IERR values are obtained,
indicating that peridynamics can predict similar initiation loads as
the CC.


\subsection{Gradient elasticity}\label{ssec:gradient_elasticity}

The Gradient Elasticity (GE) model and the CC were compared to predict
borehole crack initiation under combined pressure and biaxial loading
\cite{Sapora2021}. It was highlighted that the CC is local in its
constitutive law but non-local in the failure criterion, as both stress
and energy conditions must be met simultaneously at a specific distance
from the singular point or stress concentration. Conversely, the GE
model is non-local in its constitutive law but local in its failure
criterion, treating the governing failure parameter as the local stress
concentration factor. The GE model introduces a characteristic length
that defines the distance over which non-local effects act, smoothing
high variations in the elastic stress field. Similar to TCD, the GE
model is inapplicable below a threshold size where the internal length
becomes comparable to the specimen's characteristic size. Sapora
et~al.~\cite{Sapora2021} demonstrated that nearly identical failure
stress predictions can be achieved if the internal length in the GE
model is calibrated based on Irwin's length.

\subsection{Continuum damage model}\label{ssec:CDM}

Continuum Damage Mechanics (CDM) \cite{Lemaitre1984} provides a
framework for understanding how micro-damage, like micro-cracks or
voids, impacts material properties at a larger scale. Kachanov
\cite{Kachanov1958} introduced the damage variable concept to quantify
degradation from micro-defects. CDM uses constitutive models that
describe stress--strain relationships and damage evolution equations
based on thermodynamics to predict the transition from micro-defects to
failure. Although classical CDM models uniform damage well, it faces
challenges with discontinuities, prompting the development of gradient
damage models
\cite{Pijaudier1987,Benallal1991,deBorst1996,Peerlings1996,Peerlings1998},
which include regularization terms to simulate phenomena like brittle
fracture \cite{deBorst2002} and localized damage \cite{Geers1998}.
Methods for measuring the characteristic length of nonlocal continua
have also been proposed \cite{Bazant1989}.


Carr\`{e}re et~al.~\cite{Carrere2015} compared the CC with CDM to
investigate failure in adhesively bonded joints. Despite differing
definitions of final failure, both models produced similar failure
loads under the assumption of small displacements, as crack initiation
occurs just before the specimen reaches final failure. A characteristic
length emerged in the CDM, corresponding to the extent of the process
zone, \ie, the region where the damage variable ranges from $0$
(pristine material) to $1$ (fully damaged material). At crack
initiation, the process zone extent was found to be larger than the
initiation length predicted by the CC, yet it followed a similar trend
with respect to the material's critical energy release rate.


\section{Discussion}\label{sec:discussion}


The Irwin's length appears in all models that couple a stress and an
energy criterion, either directly or indirectly, such as the CC, CZM,
and PF models, or as a direct input parameter in models like TCD. This
parameter is intrinsic to the material, obtained from a combination of
other intrinsic material properties. While linking Irwin's length to
the material microstructure is not always straightforward, connections
can sometimes be established. For example, in polycrystalline ceramics,
the intrinsic tensile strength---determined for specimens with
extrinsic defects that are sufficiently small compared to the Irwin's
length \cite{Leguillon2018}---is related to grain size
\cite{Hall1951,Petch1953} as well as the critical energy release rate
\cite{Usami1986}. This implies that the Irwin's length depends on grain
size and other intrinsic defects present within the microstructure.

Beyond its role as an intrinsic material property, Irwin's length is
crucial for the numerical implementation of the aforementioned models.
It influences computational setups, such as the choice of mesh size in
CZM \cite{Alfano2001,Alfano2006} and CC \cite{Doitrand2020FINEL}, or
the selection of the regularization length in PF
\cite{Molnar2020,Molnar2022AT1}.



If no length scale appears in a fracture model, it is unable to assess
configurations related to crack initiation or propagation outside the
assumptions of Griffith's model. A first example is Linear Elastic
Fracture Mechanics, which predicts infinitely large remote stresses for
a crack with vanishing size. This aligns with the assumption of a
semi-infinite crack in an infinite medium, which does not hold for
finite or diminishing crack lengths. A second example involves applying
the TCD to specimens smaller than Irwin's length. Since TCD is based on
stress evaluation at a specific distance or over a volume defined by
Irwin's length, applying it to small-scale specimens is problematic, as
the characteristic length becomes meaningless. A third example concerns
TCD's application to energy-driven configurations without stress
gradients, such as transverse cracking in laminates with thin plies. In
this case, a homogeneous stress field exists within the plies
transverse to the loading direction, so the TCD would predict a failure
load based solely on the stress within the ply, regardless of ply
thickness or the evaluation length or volume. As a result, TCD would
miss the observed failure load increase with decreasing ply thickness,
which is primarily controlled by energy.



Even for models coupling stress and energy conditions, there exist
pathological configurations where the failure description remains
incomplete. Certain cases highlight where the CC could be enhanced,
specifically when the length effect is effectively ``disabled'' because
the energy criterion predominates, causing the CC to revert to a purely
energy-based criterion. Two primary configurations exhibit this
behavior. 

Firstly, in the presence of strong singularities
\cite{Leguillon2000,Leguillon2012,Aranda2023}, the IERR scales as
$K\ell^{2\lambda-1}$ with $\lambda<1/2$. Here, the IERR approaches
infinity as the crack length tends toward zero, which means the stress
criterion is always satisfied, causing the CC to revert to an
energy-only criterion. 

The second configuration involves a semi-infinite crack under remote
anti-plane shear loading. An asymptotic approach shows that the stress
criterion no longer influences the initiation generalized stress
intensity factor (GSIF), which is predicted solely based on the energy
criterion, $K=(\Gc/A)^{1-\lambda}\sigmac^{2\lambda-1}$, with
$\lambda=1/2$. The IERR reaches its maximum for rectilinear propagation
\cite{Mittelman2014,Mittelman2015,Doitrand2023IJF}, whereas
experimental observations indicate that facets initiate at an angle
inclined with respect to the primary propagation direction of the
initial crack front.


In configurations where the CC fails to predict crack initiation due to
the absence of a length effect, one potentially missing component could
be the description of the process zone prior to initiation. Indeed, the
Phase-Field approach can account for the occurrence of facets under
antiplane shear \cite{Molnar2024}, as observed experimentally, with the
primary distinction from the CC being the presence of a process zone
that develops before crack initiation. Previous studies have shown that
describing this process zone is essential in PF models to accurately
predict other configurations, such as along the two lips of an initial
crack \cite{Klinsmann2015,Molnar2020,Molnar2022AT1}. 

Incorporating this feature into the CC could improve its predictive
capability. For example, Dominique Leguillon \cite{Leguillon2017}
proposed a model to describe a damage zone ahead of a V-notch prior to
initiation. Alternatively, a combination of the crack regularization
provided by the PF model with the CC could be implemented, as
demonstrated in Ref. \cite{Doitrand2024IJF}.


\section{Conclusion}\label{sec:conclus}

In this review, we examined characteristic lengths in fracture
mechanics, with a particular focus on the Coupled Criterion framework
and its interactions with other advanced fracture models. Our analysis
highlighted that the CC, through coupling stress and energy criteria,
uniquely clarifies the conditions under which cracks initiate across
various configurations, even when dealing with theoretical scenarios
that may not manifest in reality. By exploring these boundaries, we
gained insights into transition behaviors that occur between
configurations and the convergence of stress and energy requirements
for crack initiation.



We observed that while the CC provides a comprehensive understanding of
crack initiation, it lacks certain advantages of other models, such as
autonomous crack path determination or the straightforward handling of
multiple cracks initiating and propagating simultaneously. However, it
offers valuable insights into other fracture models through the
established inter-model dialogue. Our position is that fracture models
can mutually benefit by extending this dialogue beyond simple
comparisons of predicted failure loads obtained through different
approaches.

A promising direction for the CC is to incorporate the regularization
provided by the PF model or the process zone (PZ) description available
in CZM. Such integration would allow for a detailed process zone
description before initiation, addressing limitations in configurations
where the length effect vanishes.

The development of such a dialogue between the CC and other fracture
models remains an open avenue, as exemplified by the Discrete Elements
Method \cite{Andre2019,Voisin2024}, which could offer a way to describe
crack initiation in a manner comparable to continuum mechanics and the
CC.

\section*{Declaration of interests}
The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

\section*{Dedication}
The manuscript was written through contributions of all authors. All
authors have given approval to the final version of the manuscript.

\section*{Acknowledgments}
The authors express their deep gratitude to Dominique Leguillon for his
invaluable advice and inspiration. His unwavering willingness to
encourage us will always serve as a guiding example for the community.

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