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\DOI{10.5802/crmeca.296}
\datereceived{2025-02-11}
\daterevised{2025-04-10}
\dateaccepted{2025-04-10}
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\dateposted{2025-05-12}
\begin{document}

\begin{noXML}

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\shortrunauthors

\title{Some remarks concerning the nucleation  and initiation lengths
of fractures in brittle materials in the light of Dominique Leguillon's
energy criterion}

\alttitle{Quelques remarques sur la nucl\'eation et les longueurs
d'initiation des fissures dans les mat\'eriaux fragiles \`a la lumi\`ere
du crit\`ere \'energ\'etique de Dominique Leguillon}

\author{\firstname{Luc} \lastname{Dormieux}}
\address{Laboratoire Navier (UMR 8205), \'Ecole nationale des ponts et 
chauss\'ees - Institut Polytechnique de Paris, CNRS, Universit\'e 
Gustave Eiffel, 6 et 8 avenue Blaise Pascal 77455 cedex-2 Marne-la-Vall\'ee, 
France}
\email[L. Dormieux]{luc.dormieux@enpc.fr}

\author{\firstname{Eric} \lastname{Lemarchand}\CDRorcid{0000-0003-1788-9857}\IsCorresp}
\addressSameAs{1}{Laboratoire Navier (UMR 8205), \'Ecole nationale des ponts et 
chauss\'ees - Institut Polytechnique de Paris, CNRS, Universit\'e 
Gustave Eiffel, 6 et 8 avenue Blaise Pascal 77455 cedex-2 Marne-la-Vall\'ee, 
France}
\email[E. Lemarchand]{eric.lemarchand@enpc.fr}

\author{\firstname{Laurent} \lastname{Jeannin}}
\address{STORENGY, 12 rue Raoul Nordling - Djinn - CS 70001, 92274 Bois Colombes Cedex, France}
\email[L. Jeannin]{laurent.jeannin@storengy.com}

\author{\firstname{Djimedo}\nobreakauthor\lastname{Kondo}}
\address{Sorbonne Universit\'es, Institut Jean Le Rond d'Alembert, UMR 7190 CNRS-SU, F-75005, Paris, France}
\email[D. Kondo]{djimedo.kondo@sorbonne-universite.fr}

\keywords{\kwd{Crack initiation}
\kwd{Crack nucleation}
\kwd{Energy criterion}
\kwd{Dynamic}
\kwd{Bounds}}

\altkeywords{\kwd{Initiation de fissure}
\kwd{Nucl\'eation de fissure}
\kwd{Crit\`ere \'energ\'etique}
\kwd{Dynamique}
\kwd{Bornes}}

\begin{abstract}  
Crack nucleation has been the subject of important contributions in the
two last decades. Starting from the energy criterion of
Leguillon (\textit{Eur. J. Mech.---A/Solids} \textbf{21} (2002)), the
present paper examines some consequences that follow directly from the
perfectly brittle model. This enables to discuss whether a quasi-static
modelling of nucleation is possible. Rigorous bounds for the loading
level that triggers nucleation as well as for the set of observable
crack lengths under dynamic conditions are established.
\end{abstract}

\begin{altabstract}
La nucl\'eation des fissures a fait l'objet de contributions importantes
au cours des deux derni\`eres d\'ecennies. En partant du crit\`ere
\'energ\'etique propos\'e par Leguillon
(\textit{Eur. J. Mech.---A/Solids}
\textbf{21} (2002)), le pr\'esent article examine
certaines cons\'equences qui d\'ecoulent directement du mod\`ele
parfaitement fragile. Cela permet de discuter de la possibilit\'e d'une
mod\'elisation quasi-statique de la nucl\'eation. Des bornes
rigoureuses sont \'etablies pour le niveau de chargement d\'eclenchant
la nucl\'eation ainsi que pour l'ensemble des longueurs de fissures
observables dans des conditions dynamiques.
\end{altabstract}

\maketitle

\end{noXML}

\newcommand{\uv}[1]{\underline{#1}}
\newcommand{\fe}{\Psi}

\def\eqref#1{(\ref{#1})}

\vspace*{-4pt}

\section{Introduction}\label{sec}
The pioneering paper~\cite{dominik} of Leguillon was, without a doubt,
a major step in the direction of understanding and quantifying the
crack nucleation process. Its starting point was a seemingly simple
remark concerning the potential energy release occuring during the
process:
{\begin{equation}\label{eq:NC}
-{\delta \mathscr{E}_{\mathrm{pot}}}\ge 
G_c\delta S
\end{equation}}\unskip
in which $G_c$ denotes the fracture energy per unit surface. As opposed
to the Griffith concept of rate of energy release~\cite{griffith}, it is emphasized
that the above inequality refers to a finite variation $\delta S$ of
surface. The significance of \eqref{eq:NC}, that is generally referred
to as energy criterion, is now widely acknowledged.  
In~\cite{dominik}, it was proposed that \eqref{eq:NC} should be
combined with a stress criterion to be fulfilled in the area where the
crack is nucleated. This is intended to determine the critical loading
level of nucleation as well as its extent. This approach is therefore
often referred to as the double or coupled criterion (CC) method. Ever
since, the latter was applied to deal with a number of different
situations, first in its initial quasi-static framework and more
recently with account for dynamic effects~\cite{doitrand}.

More recently,  in the case of stable nucleation
processes,~\cite{Carlioz} argued that  Leguillon's energy criterion was
sufficient for characterizing the occurrence and the extent of
nucleation. This new methodology was first applied  to study fracture
nucleation that occurs during tunnelling in the context of nuclear
waste storage. It was also illustrated in various problems of rock
mechanics (e.g.~\cite{JDC,JDLC}). Finally, an extension to the case of
saturated porous materials was proposed~\cite{DormieuxLemarchandCJ}.

The modelling of crack nucleation must address the determination of the
geometry of the crack and its evolution in time. For simplicity,
considering the case of a straight line crack, it amounts to determine
the interval $\mathscr{I}$ of observable crack lengths $\ell$ together
with the crack velocity $\dot{\ell}$. The next question is how the
loading level affects the minimum and maximum (if any) in $\mathscr{I}$
as well as the evolution law.

Among these questions, the determination of the minimum of 
$\mathscr{I}$, which can be termed the initiation crack length, is of
paramount importance. More precisely, the question is whether it has a
non-zero value or if the interval $\mathscr{I}$ of observable crack
lengths truly starts at 0. It has been argued~\cite{doitrand} that a
non zero initiation length is an artefact of the quasi-static
description of nucleation.  It was also suggested in~\cite{doitrand}
that an appropriate dynamic description of crack initiation could
account for a progressive extension of crack length starting from 0. It
seems however that the issue is still opened and that some arguments
are in favor of a strictly positive initiation length, irrespective of
whether the description is dynamic or not.  Furthermore, the question
that comes next is whether the model of brittle material is able, by
itself, to describe the transition from the initial state (no crack) to
that of a cracked state. We shall see that the energy criterion in the
form  \eqref{eq:NC} is a good starting point to contribute to this
discussion.

More generally, the choice between quasi-static and dynamic modelling
of nucleation is in itself an important issue that the present paper
would like to consider. Indeed, as pointed out in~\cite{doitrand}, the
very nature of nucleation is the creation of a crack in short amount of
time. Therefore, a quasi-static modelling of such a process seems a
priori questionable, although it is widely used. At least, it deserves
a thorough investigation in order to clarify in which case it can be
implemented. 

The methodology adopted in this paper consists to strictly rely on the
model of perfectly brittle material without additionnal assumption. The
purpose is to draw the necessary conditions following from this basis.
Accordingly, as regards the description of the crack nucleation
process, the relevant material constant is the fracture energy $G_c$
with which both the Griffith criterion as well as \eqref{eq:NC} can be
formulated. In short, the main objective is to determine what can be
derived from this model as well as what is out of reach and would
require a more elaborated framework or further hypotheses.

\section{Dynamic energy release rate}
Let $\Omega$ denote the mechanical system in which possible crack
nucleation is considered. The geometry of the crack is described by a
formal parameter $\ell$. This is for instance the length in 2D or the
area in 3D.  It controls the energy $G_c\ell$ that is dissipated during
nucleation. During the nucleation process, $\ell$ increases from 0 to a
limit value $\ell^\ast$ for which stability is recovered under the
loading that has triggered the nucleation.

The starting point is the Clausius--Duhem inequality~\cite{dorko,rice}: In isothermal
conditions, it states that the external power  ($\mathscr{P}_e$)
provided to the system $\Omega$ is transformed into the rate of change
of free energy ($\dot{\fe}$) and the rate of change of the kinetic
energy ($\dot{K}$) or dissipated in the crack nucleation process
(dissipation rate  $\mathscr{D}$):
{\begin{equation}\label{premier}
\mathscr{D}=\mathscr{P}_e-\dot{\fe}-\dot{K} \geq 0.
\end{equation}}\unskip
As already stated in the introduction of the present paper, this is
also the starting point of the celebrated work of 
Leguillon~\cite{dominik} in which \eqref{premier} is considered in an
integrated form with respect to time:
{\begin{equation}\label{premierDL}
\Delta\fe-\mathscr{W}_e+K+\int\mathscr{D}\,\mathrm{d}t=0.
\end{equation}}\unskip
For simplicity, we assume steady state conditions for the mechanical
loading $\mathscr L$, the latter being defined by body forces $\uv{F}$,
prescribed surface forces $\uv{T}^d$ on  $\partial \Omega_T$,
prescribed displacements  $\uv{\xi}^d$ on  $\partial \Omega_\xi$, the
boundary of the material system being $\partial \Omega=\partial
\Omega_\xi\cup\partial \Omega_T$. The external power provided to the
system thus reads:
{$$
\mathscr{P}_e =
\int_\Omega\uv{F}\cdot\dot{\uv{\xi}}\,\mathrm{d}V+
\int_{\partial\Omega_T}\uv{T}^d\cdot\dot{\uv{\xi}}\,\mathrm{d}S.
$$}\unskip
Since we aim at describing the crack propagation occuring in the
framework of crack nucleation, we focus on a time interval of crack
length increase ($\dot{\ell}&gt;0$).  This makes the relation between time
and crack length a bijective one, so that the velocity $\uv{\dot{\xi}}$
can be put in the form:
{$$
\dot{\uv{\xi}}=
\dfrac{\partial \uv{\xi}}{\partial \ell}_{|\mathscr{L}}\dot{\ell}.
$$}\unskip
For the same reason, the time derivatives of the free energy and the
kinetic energy are replaced by derivatives with respect to the crack
length:
{\begin{equation}\label{second} 
\dot{\fe}=\dfrac{\partial \fe}{\partial\ell}_{|\mathscr L}
\dot{\ell};\quad 
\dot{K}=\dfrac{\partial K}{\partial\ell}_{|\mathscr L}\dot{\ell}.	
\end{equation}}\unskip
Under steady loading conditions, the external power is due to crack
propagation only:
{\begin{equation}\label{eq:phi}
\mathscr{P}_{e|\mathscr{L}}=
\dfrac{\partial \Phi}{\partial \ell}_{|\mathscr{L}}\dot{\ell}
\quad\mbox{with}\quad
\Phi=\int_\Omega\uv{F}\cdot{\uv{\xi}}\,\mathrm{d}V+
\lambda_T(t)
\int_{\partial\Omega_T}\uv{T}^0\cdot{\uv{\xi}}\,\mathrm{d}S
\end{equation}}\unskip
where $\Phi$ is the potential of the given loads.  Eventually, the
combination of \eqref{premier}, \eqref{second} and \eqref{eq:phi}
yields:
{\begin{equation}\label{premier-0}
\mathscr{D}+
\dfrac{\partial K}{\partial\ell}_{|\mathscr{L}}\dot{\ell}=
-\dfrac{\partial \mathscr{E}_{\mathrm{pot}}^{\mathrm{dyn}}}
{\partial\ell}_{|\mathscr{L}}
\dot{\ell}
\end{equation}}\unskip
which introduces the potential energy
$\mathscr{E}_{\mathrm{pot}}^{\mathrm{dyn}}=\fe-\Phi$, where the
upperscript $\underline{\mathrm{dyn}}$
recalls that it is evaluated from the dynamic
evolution of the displacement. If the dissipation process can be
described by the concept of specific fracture energy, 
\eqref{premier-0} takes the form:
{\begin{equation}\label{premier-1}
G_c+
\dfrac{\partial K}{\partial\ell}_{|\mathscr{L}}=
-\dfrac{\partial \mathscr{E}_{\mathrm{pot}}^{\mathrm{dyn}}}
{\partial\ell}_{|\mathscr{L}}.
\end{equation}}\unskip
The derived expression is formally very similar to the classical rate
of energy release (see e.g.~\cite{Leblond}), except for the fact that
it comprises a term of kinetic energy.  Furthermore,  unlike the
quasistatic case, the rate of potential energy release cannot be
evaluated a priori. Indeed, it must be calculated from the strain and
displacement fields which are solution to the dynamic evolution of the
system. It is therefore appropriate to refer to a dynamic  potential
energy release rate. Qualitatively, \eqref{premier-1} states that the
released energy is not entirely dissipated since a part of it is
transformed into kinetic energy.

Note that \eqref{premier-1} can be put in the equivalent form:
{\begin{equation}\label{premier-10}
G_c=
-\dfrac{\partial \mathscr{E}_{\mathrm{mec}}^{\mathrm{dyn}}}
{\partial\ell}_{|\mathscr{L}}
\end{equation}}\unskip
where $\mathscr{E}_{\mathrm{mec}}^{\mathrm{dyn}}$ is the mechanical
energy $\mathscr{E}_{\mathrm{pot}}+K$. This suggests an alternative
definition to the dynamic energy release rate.

\section{The assumption of quasistatic nucleation process}\label{sec3}
To begin with, let us however leave the dynamic aspect of the process
aside which amounts to assuming that the kinetic energy term is
negligible. In this section, we shall therefore examine the assumption
of a quasistatic nucleation process under a given constant loading
level. The quasistatic assumption means that each state in the
nucleation process is an equilibrium state.  From a practical point of
view, the great advantage of this assumption is clearly that, by
nature, it avoids the determination of a dynamic problem. The potential
energy therefore becomes a function of the crack length.

For instance, let us assume that the loading is defined\footnote{The
boundary conditions could also be defined on the displacemnt. This
would affect the definition of the potential energy.} by a single
scalar parameter $Q$, and that the corresponding work of external
forces reads  $\Phi=Q\xi$. The free energy takes the form
$\Psi=\sofrac{1}{2}C(\ell)\xi^2$ where $C(\ell)$ represents the elastic
stiffness of the cracked structure. In view of further use, let us
introduce the functions $\mathbb{C}(\ell)$ and  $\mathscr{C}(\ell)$:
{\begin{equation}\label{defC}
\mathbb{C}(\ell)=
\dfrac{1}{\ell}
({C(0)}-{C(\ell)});\quad
\mathscr{C}(\ell)=
\dfrac{1}{\ell}
\left(\dfrac{1}{C(\ell)}-
\dfrac{1}{C(0)}\right).
\end{equation}}\unskip
In particular, the function $\mathscr{C}(\ell)$ is going to play a very
important role in the following discussion. It involves informations
concerning the geometry of the structure and the boundary conditions
used for defining the loading. It represents the rate of change of the
structural compliance. The static potential energy of the solution
reads:
{$$
\mathscr{E}_{\mathrm{pot}}^{\mathrm{stat}}=
\min_{\xi}
\left(\dfrac{1}{2}C(\ell)\xi^2- Q\xi\right)=
-\dfrac{Q^2}{2C(\ell)}.
$$}\unskip
Considering any stage in the propagation process associated with some
crack length $\ell$, the energy balance  reads:
{$$
\dfrac{Q^2}{2}
\left(\dfrac{1}{C(\ell)}-\dfrac{1}{C(0)}\right)=
\int\mathscr{D}\,\mathrm{d}t.
$$}\unskip
We now assume that the dissipation process can be described by the
concept of a constant fracture energy $G_c$ per unit surface. 
Accordingly, the above equation takes the form:
{\begin{equation}\label{carlioz}
\dfrac{Q^2}{2}\mathscr{C}(\ell)=
\dfrac{\mathscr{E}_{\mathrm{pot}}^{\mathrm{stat}}(0)-
\mathscr{E}_{\mathrm{pot}}^{\mathrm{stat}}(\ell)}{\ell}=
G_c.
 \end{equation}}\unskip
Note that the above equation is also obviously retrieved from
\eqref{premierDL} in which the kinetic energy term is omitted. 
\eqref{carlioz} indeed very much recalls the incremental form of the
energy criterion of  Leguillon. However, attention is drawn to the fact
that we herein consider an equality which is made possible from the
fact that we handle with a quasistatic process. Once again, such a 
framework is necessary for a practical determination of the energy
release associated with the crack nucleation.  This information is
namely provided by the function $\mathscr{C}(\ell)$. The determination
of the latter only requires the resolution of standard elastostatic
problems. It is worth noticing that the first equality in
\eqref{carlioz}, that is:
{$$ 
\dfrac{Q^2}{2}\mathscr{C}(\ell)=
\dfrac{\mathscr{E}_{\mathrm{pot}}^{\mathrm{stat}}(0)-
\mathscr{E}_{\mathrm{pot}}^{\mathrm{stat}}(\ell)}{\ell}
$$}\unskip
can be regarded as a more general definition of function 
$\mathscr{C}(\ell)$.  Its interest is to avoid  to rely upon the
stiffness function $C(\ell)$ which cannot be defined in the case of an
infinite body. In contrast, the incremental energy release rate
{$$
G_{\mathrm{inc}}(\ell)=
\dfrac{\mathscr{E}_{\mathrm{pot}}^{\mathrm{stat}}(0)-
\mathscr{E}_{\mathrm{pot}}^{\mathrm{stat}}(\ell)}{\ell}
$$}\unskip
introduced by Leguillon overcomes this shortcoming.  Besides, in many
cases, an analytical form of $\mathscr{C}(\ell)$ can be derived from
the stress intensity factors (see e.g.~\cite{sih}) by means of Irwin
equation (see for example Section~\ref{sec:hole}). Indeed, according to
the latter, the incremental energy release rate of  Leguillon is
immediately derived by integration of the square of the stress
intensity\break factor.

Nevertheless, it may be useful to observe that it is equivalent to give
oneself the stiffness function $C(\ell)$ or the function
$\mathscr{C}(\ell)$ appearing in the static incremental  energy release
rate. Indeed, the latter can obviously be derived from the former by
\eqref{defC}.  And in return, even if the stiffness function  does not
exist as such (typically the case of an infinite body), it is always
possible to introduce $C(\ell)=1/(a+\ell\mathscr{C}(\ell))$. The choice
of the positive constant $a$ is arbitrary provided that $\lim_{\ell\to
0}\ell\mathscr{C}(\ell)=0$. This constant vanishes in the expressions
of energy\break criteria.

Similarly, it is equivalent to give oneself the compliance function
$S(\ell)=1/C(\ell)$ or the function $\mathscr{C}(\ell)$ appearing in
the static incremental energy release rate. Again, the latter can be
derived from the former by \eqref{defC}.  And in return, even if the
compliance function  cannot be defined (typically the case of an
infinite body where the compliance tends to 0), it is possible to
introduce $S(\ell)=a+\ell\mathscr{C}(\ell)$. As before, the choice of
the positive constant $a$ is arbitrary.

We shall see now that the quasistatic assumption has drastic
consequences.

\subsection{The case of a monotonic $\mathscr{C}(\ell)$
function}\label{sec:monoton}
The very concept of energy release suggests that the stiffness function
is monotonically decreasing, or that the elastic compliance function is
monotonically increasing. This obviously does not imply that
$\mathscr{C}(\ell)$ shares this property.  We first consider the case
where the $\mathscr{C}(\ell)$ function is indeed a strictly increasing
function (see  Figure~\ref{Lafigure} case (a)).

\begin{figure}
{\vspace*{3pt}}
\includegraphics{fig01}
{\vspace*{3pt}}
\caption{\label{Lafigure}Various patterns of the rate of change of
structural compliance $\mathscr{C}(\ell)$: (a) example of the monotonic
increasing case; (b) example of the existence of a maximum.}
{\vspace*{3pt}}
\end{figure}

From a mathematical point of view, this is in particular the case if
the stiffness function is a decreasing strictly concave function:
{$$
(\forall \ell\geq 0)\quad 
C'(\ell)\leq 0
\quad\mbox{and}\quad
C''(\ell)&lt;0.
$$}\unskip
From a second order Taylor--Lagrange expansion:
{$$
(\exists\theta\in[0,1])\quad 
C(0)=C(\ell)-\ell C'(\ell)+
\tfrac{1}{2}C''(\theta\ell)\ell^2
$$}\unskip
it is found that $\mathbb{C}(\ell)+C'(\ell)&lt;0$. Observing that 
{$$
\mathscr{C}'(\ell)=
\dfrac{1}{C(0)C(\ell)\ell}
\left(-\dfrac{C(0)}{C(\ell)}C'(\ell)-\mathbb{C}(\ell) \right)
$$}\unskip
and recalling that $C(0)&gt;C(\ell)$, we conclude that
$\mathscr{C}'(\ell)&gt;0$. It can be also readily shown that
$\mathscr{C}(\ell)$ is monotonically increasing if the compliance
function $S(\ell)$ is convex.

More concretely speaking, let us consider the example of a
semi-infinite body subjected to a remote traction stress
$\sigma_\infty$ with an edge-crack normal to the boundary. The (mode I)
stress intensity factor is of the form $\sigma_\infty\sqrt{\pi\ell}F$
where $F$ is a scalar coefficient ($F\approx 1.12$,~\cite{sih}). By
integration of Irwin equation, this implies, in the present example,
that the $\mathscr{C}(\ell)$ function is proportional to the crack
length:
{\begin{equation}\label{eq:ell}
\mathscr{C}(\ell)=
F^2\pi\dfrac{1-\nu^2}{E}\ell.
\end{equation}}\unskip
Let us go back to the general case of  a strictly increasing function
$\mathscr{C}(\ell)$. It follows that \eqref{carlioz} with unknown
$\ell$ has a unique solution $\ell^\ast$ (provided that
$Q^2&lt;2G_c/\mathscr{C}(0)$, if $\mathscr{C}(0)&gt;0$). This obviously
implies that the nucleation process \textbf{is not} the propagation of
a crack from 0 to the final length $\ell^\ast$. In other words, within
the quasistatic framework, it is impossible to observe any crack having
a length $\ell&lt;\ell^*$. 

Let us assume that the crack with length $\ell^\ast$ has been indeed
created. Following the quasistatic assumption, the current state is a
(possibly not stable) equilibrium state. In any case, the rate of
energy release in its differential form must comply with the standard
Griffith criterion:
{\begin{equation}\label{griffithform}
G_{\mathrm{diff}}(\ell^\ast)=
-\dfrac{\partial \mathscr{E}_{\mathrm{pot}}^{\mathrm{stat}}}
{\partial\ell}(\ell^\ast)=
\dfrac{Q^2}{2}
\dfrac{d}{d\ell}
(\ell\mathscr{C}(\ell))(\ell^\ast)\leq G_c.
\end{equation}}\unskip
Since $\mathscr{C}'(\ell^\ast)&gt;0$, we have:
{$$
\dfrac{\mathrm{d}}{\mathrm{d}\ell}
(\ell\mathscr{C}(\ell))
(\ell^\ast)=\mathscr{C}(\ell^\ast)+
\ell^\ast\mathscr{C}'(\ell^\ast)&gt;\mathscr{C}(\ell^\ast)
$$}\unskip
from which it follows that \eqref{carlioz} and \eqref{griffithform}
\textbf{are not} compatible. In other words,  in the case of a strictly
concav stiffness function, or more generally when $\mathscr{C}(\ell)$
is a monotonic increasing function, the quasistatic assumption fails to
describe the nucleation process. As stated above (see \eqref{eq:ell}),
this approach is therefore disqualified for a wide class of structures.
This suggests that any method that aims at predicting the crack
formation should take into account inertia effects, that is, should be
developed in a dynamic framework.

\subsection{The case of a non monotonic $\mathscr{C}(\ell)$
function}\label{sec:nonmonoton}
We now assume that $\mathscr{C}(\ell)$ is increasing on some interval
$[0,\ell_{\mathrm{cr}}]$ and decreases for $\ell&gt;\ell_{\mathrm{cr}}$ 
(see Figure~\ref{Lafigure} case (b)). This implies that
$\mathscr{C}'(\ell_{\mathrm{cr}})=0$. For given elastic properties, it
is important to note that this property and in particular the critical
length $\ell_{\mathrm{cr}}$ are, by nature, independent of the loading
level. Indeed, the practical determination of $\mathscr{C}(\ell)$
incorporates informations concerning the type of boundary conditions
and the geometry of the body and of the crack. This is all the more
clear when $\mathscr{C}(\ell)$ is determined by integration of the
(square of the) stress intensity factor in Irwin equation.

An example of this situation is the (mode I) crack that appears in a
plate with circular hole under compression (see 
Section~\ref{sec:hole}). The crack is expected to develop from the
cavity boundary along the direction of compression. The analysis of the
physical dimensions at stake indicates that the critical length
$\ell_{\mathrm{cr}}$ is expected to be proportional to the hole
diameter $R$.

Let us examine if a quasistatic nucleation is compatible with  this new
framework. Clearly enough, this implies (necessary condition) that 
the sought crack length $\ell^\ast$ complies with \eqref{carlioz} and
\eqref{griffithform} which are repeated here for convenience:
{\begin{equation}\label{eq:compa}
\dfrac{\mathscr{E}_{\mathrm{pot}}^{\mathrm{stat}}(0)-
\mathscr{E}_{\mathrm{pot}}^{\mathrm{stat}}(\ell^\ast)}{\ell^\ast}=
\dfrac{Q^2}{2}
\mathscr{C}(\ell^\ast)=G_c;\quad 
-\dfrac{\partial \mathscr{E}_{\mathrm{pot}}^{\mathrm{stat}}}{\partial\ell}
(\ell^\ast)=
\dfrac{Q^2}{2}
(\mathscr{C}(\ell^\ast)+\ell^\ast\mathscr{C}'(\ell^\ast))
\leq G_c.
\end{equation}}\unskip
The compatibility of these conditions appears to be 
$\mathscr{C}'(\ell^\ast)=0$, which means that the nucleation crack
length $\ell^\ast$ is none but the critical length
$\ell_{\mathrm{cr}}$. Following the previous remarks concerning the
latter, it is emphasized that the nucleation crack length predicted in
this framework is determined a priori, without reference to the loading
level that is responsible for nucleation.

The next question is to determine this loading level. The answer
follows directly from \eqref{eq:compa}:
{\begin{equation}\label{eq:Qstar}
Q^\ast=
\sqrt{\dfrac{2G_c}{\mathscr{C}(\ell_{\mathrm{cr}})}}.
\end{equation}}\unskip
We thus note that both the nucleation crack length and the loading
level are determined in a unique way~\cite{cdl}. If $Q&lt;Q^\ast$, the energy
criterion is not fulfilled, in the sense that the energy release is not
sufficient for the creation of the crack. If $Q&gt;Q^\ast$, a dynamic
process is intuitively expected, in which inertia effects have to be
accounted for. This conclusion will be justified more thoroughly at the
next section.

The loading level $Q^\ast$ is such that the plot of the incremental
energy release rate $G_{\mathrm{inc}}(\ell)$ is tangent to the line
$G=G_c$ at the point with abscissa $\ell_{\mathrm{cr}}$. The relative
position of the plots of $G_{\mathrm{inc}}(\ell)$ and
$G_{\mathrm{diff}}(\ell)$ in  Figure~\ref{fig:G} is obtained from the
expressions of these quantities recalled in~\eqref{eq:compa}.

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig:G}Characterization of the nucleated crack length
$\ell^\ast$: differential  $G_{\mathrm{diff}}(\ell)$ and incremental
$G_{\mathrm{inc}}(\ell)$  energy release rates for the critical loading
level $Q^\ast$ of \eqref{eq:Qstar}.}
\end{figure}

Again, it is noteworthy that no crack with length
$\ell&lt;\ell_{\mathrm{cr}}$ can be observed. This suggests that the
nucleation process that yields the critical crack length
$\ell^\ast=\ell_{\mathrm{cr}}$ cannot be described with the
conventional concept of fracture energy $G_c$ and the corresponding
so-called Griffith criterion. The opinion of the authors is that more
elaborated modellings such as the idea of gradient
damage~\cite{marigo1} can be helpful for describing this maturation
phase that precedes nucleation.

Anyway, under the loading $Q^\ast$, the structure is left in a state of
stable equilibrium with a created crack of length $\ell_{\mathrm{cr}}$.
Here, the physical interpretation of quasistatic nucleation  is that
the energy release  exactly reaches the amount required for the
creation of the crack and, so to speak, nothing is left for kinetic
energy. 

\section{The non quasistatic nucleation process}\label{sec4}

\subsection{Rigorous bounds for the set of observable crack
lengths}\label{sec:boundze}
At Section~\ref{sec:monoton}, we have seen that the quasistatic
assumption fails to describe the nucleation process. This means that
the latter cannot be viewed as a sequence of equilibrium states. It
also fails in the framework of  Section~\ref{sec:nonmonoton} if the
loading level is greater than the threshold given by~\eqref{eq:Qstar}.
In both cases, the reason is obviously related to non negligible
inertia effects. This is also most consistent with the widespread
physical intuition of nucleation as a sudden phenomenon taking place
during a short amount of time. From now on, the set of observable crack
lengths is denoted by $\mathscr{I}^{\mathrm{dyn}}$.

The inequality \eqref{eq:NC} derived by Leguillon, already stated in
the introduction of this paper, appears as an appealing way to approach
the modelling of nucleation in a more realistic manner. The latter
relies upon the simple but very effective remark that the starting
point being an equilibrium state, the variation of kinetic energy is
necessarily positive. The profound interest of~\eqref{eq:NC} is
clearly that it is most general. Nevertheless, the question that arises
is: how can we evaluate the potential energy in the lefthand side,
with due account for the fact that the kinetic energy is not negligible? 
This very fact prohibits an approximation of the \textit{dynamic}
potential energy $\mathscr{E}_{\mathrm{pot}}^{\mathrm{dyn}}$ by its 
\textit{static} counterpart $\mathscr{E}_{\mathrm{pot}}^{\mathrm{stat}}$. This
can be highlighted in re-stating \eqref{eq:NC} in the explicit form:
{\begin{equation}\label{eq:NCdyn}
-\delta \mathscr{E}_{\mathrm{pot}}^{\mathrm{dyn}} 
\ge G_c\delta S.
\end{equation}}\unskip
Some insight in this matter can be gained from an elementary argument,
namely the so-called principle of minimum potential energy in linear
elasticity. More precisely, for a given crack length $\ell$ and for the
associated geometry $\Omega_\ell$, we are faced with two displacement
fields: the solution $\uv{\xi}^{\mathrm{dyn}}$ to the ``real'' dynamic
evolution and the solution $\uv{\xi}^{\mathrm{stat}}$ to the elastostatic
problem. The displacement field  $\uv{\xi}^{\mathrm{dyn}}$ is, by
nature, kinematically admissible, so that the theorem ensures that 
$\mathscr{E}_{\mathrm{pot}}^{\mathrm{dyn}}&gt;
\mathscr{E}_{\mathrm{pot}}^{\mathrm{stat}}$. An immediate consequence
is that $-{\delta \mathscr{E}_{\mathrm{pot}}^{\mathrm{dyn}}}&lt;-{\delta
\mathscr{E}_{\mathrm{pot}}^{\mathrm{stat}}}$ and further that:
{\begin{equation}\label{eq:NCdynstat}
-{\delta \mathscr{E}_{\mathrm{pot}}^{\mathrm{stat}} }&gt;
-{\delta \mathscr{E}_{\mathrm{pot}}^{\mathrm{dyn}} }\ge 
G_c\delta S.
\end{equation}}\unskip
This provides a simple justification of Leguillon's seminal intuition
to the use of a static estimate of the potential energy in 
\eqref{eq:NC}. Let us now interpret the consequence of this result as
far as the modelling of nucleation is concerned. The obvious but also
considerable interest of the static potential energy is that it is a
function of the crack length and that it does not require the solution
to an evolution problem. Accordingly, any crack length in the 
\textit{dynamic} nucleation process must comply with
\eqref{eq:NCdynstat}:
{\begin{equation}\label{eq:init0}(
\forall\ell\in\mathscr{I}^{\mathrm{dyn}})\quad 
\dfrac{Q^2}{2}\mathscr{C}(\ell)&gt; G_c.
\end{equation}}\unskip
Let us first consider the case of the monotonic increasing function
$\mathscr{C}(\ell)$.

\begin{quote}
\fontsize{8.65}{10.2}\selectfont
The equation $Q^2\mathscr{C}(\ell)=2G_c$ with unknown $\ell$ has
therefore a unique solution  which, owing to~\eqref{eq:init0}, proves
to be a \textbf{lower bound} for any observable crack length during the
dynamic evolution process, that is, of the interval
$\mathscr{I}^{\mathrm{dyn}}$. It is therefore denoted by $\ell^{lb}(Q)$
and defined by:
{\begin{equation}\label{eq:lb}
\dfrac{Q^2}{2}\mathscr{C}(\ell^{lb})= G_c; \quad 
(\forall\ell\in\mathscr{I}^{\mathrm{dyn}})\quad
\ell&gt;\ell^{lb}(Q).
\end{equation}}\unskip
The very fact that  $\mathscr{I}^{\mathrm{dyn}}$ has a non-zero lower
bound leads to introduce the concept of initiation length, denoted by
$\ell_{\mathrm{init}}^{\mathrm{dyn}}$, and being the smallest crack
length that can be observed in the {dynamic} nucleation process.
Clearly enough,  $\ell^{lb}(Q)$ is also a rigorous lower bound for
$\ell_{\mathrm{init}}$ that can be determined from the associated
static problem:
{$$
\ell_{\mathrm{init}}^{\mathrm{dyn}}&gt;\ell^{lb}(Q).
$$}\unskip
Using again that $\mathscr{C}(\ell)$ is an increasing function, it
follows from \eqref{eq:lb} that $\ell^{lb}(Q)$ is a bijective
decreasing function of $Q$.  As stated at the end of 
Section~\ref{sec:monoton}, and despite the static nature of
$\ell^{lb}(Q)$, it should be remembered that any attempt to determine
the exact value of the initiation length should account for dynamic
effects.
\end{quote}
Let us now consider the case of a non monotonic function
$\mathscr{C}(\ell)$.

\begin{quote}
\fontsize{8.65}{10.2}\selectfont
Again, the starting point is that observable crack lengths must comply
with \eqref{eq:init0}. This is a necessary condition for the very
existence of a crack of length $\ell$ in a dynamic process of
nucleation. For a given value of $Q$, this implies that $\ell$ meets
the following requirement:
{$$
Q&gt;\sqrt{\dfrac{2Gc}{\mathscr{C}(\ell)}}.
$$}\unskip
A necessary condition for the existence of a solution $\ell$ in the
above inequality is:
{\begin{equation}\label{carliozQlb}
Q&gt;\sqrt{\dfrac{2Gc}{\max\mathscr{C}(\ell)}}=
\sqrt{\dfrac{2Gc}{\mathscr{C}(\ell_{\mathrm{cr}})}}
=Q^\ast
\end{equation}}\unskip
in which we recognize the threshold $Q^\ast$ introduced at
\eqref{eq:Qstar} as the unique loading level being compatible with a
quasistatic nucleation process. We now gain a new interpretation of the
latter: $Q^\ast$ is in fact a rigorous lower bound for any loading
level able to initiate a crack. In other words, we draw the
conservative conclusion that no crack nucleation should be observed,
neither quasistatic nor dynamic, is $Q&lt;Q^\ast$.

We now assume that $Q&gt;Q^\ast$. Let $\ell^{lb}(Q)$ and $\ell^{ub}(Q)$ be
the solutions of the equation ${Q}^2\mathscr{C}(\ell)=2G_c$ with
unknown $\ell$ as shown on Figure~\ref{fig:G2}. The condition
\eqref{eq:init0} implies that these lengths indeed provide   bounds for
$\mathscr{I}^{\mathrm{dyn}}$. The latter is therefore a bounded
interval of the form
$[\ell_{\mathrm{init}}^{\mathrm{dyn}},
\ell_{\mathrm{nuc}}^{\mathrm{dyn}}]$ and we have:
{\begin{equation}\label{eq:init2}
\ell^{lb}(Q)\le\ell_{\mathrm{init}}^{\mathrm{dyn}}
\le\ell_{\mathrm{nuc}}^{\mathrm{dyn}}
\le\ell^{ub}(Q).
\end{equation}}\unskip
In particular, for values of $Q&gt;Q^\ast$ in the neighborhood of
$Q^\ast$, it may be concluded that the initiation and nucleation
lengths should be very close and that $\ell_{\mathrm{cr}}$ is a good
estimate of them (see Figure~\ref{fig:G2}).

It is emphasized that the bounds $\ell^{lb}(Q)$ and $\ell^{ub}(Q)$ are
very simply determined within an \textbf{elastostatic framework} and
are therefore elastostatic by nature. Furthermore, for any loading
level $Q&gt;Q^\ast$, we have seen that the nucleation process is not a
quasistatic one. It is all the more remarkable that $\ell^{lb}(Q)$ and
$\ell^{ub}(Q)$ nevertheless provide rigorous bounds for the interval
$\mathscr{I}^{\mathrm{dyn}}$ of observable crack lengths in a dynamic
process. 
\end{quote}

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig:G2}Lower and upper bounds of
$\mathscr{I}^{\mathrm{dyn}}$ in the case of non monotonic
$\mathscr{C}(\ell)$.}
\end{figure}

\subsection{The example of an infinite medium with circular
hole}\label{sec:hole}
Let us apply the developments of Section~\ref{sec:boundze} to the
classical problem of an infinite medium with a circular hole subjected
to remote uniaxial compression in plane strains. In this case, it is
wellknown that the nucleation of a stable crack is expected
(e.g.~\cite{Sammis,JDLC}).

Figure~\ref{fig:Ctilde} presents the quasistatic incremental  energy
release rate for two different remote stresses. In the present case, an
analytical expression can be provided. The starting point is  the
stress intensity factor for a given remote stress state~\cite{Sammis}:
{\begin{equation}\label{eq:sammis}
K_I(\ell)\approx 
-\dfrac{1.1\sqrt{{\ell}/{R}}}
{(1+\ell/R)^{3.3}}
\sigma_\infty\sqrt{\pi R}.
\end{equation}}\unskip
Irwin's formula in plane strains relates the stress intensity factor to
the rate of energy release $G(\ell)$:
{\begin{equation}\label{eq:irwin-0}
G(\ell)=
-\dfrac{\partial {\mathcal E}_{\mathrm{pot}}^{\mathrm{stat}}}{\partial \ell}=
\dfrac{1-\nu^2}{E}K_I^2.
\end{equation}}\unskip
$G_{\mathrm{inc}}(\ell)$ is readily obtained by integration:
{\begin{equation}\label{eq:irwin}
G_{\mathrm{inc}}(\ell)=\dfrac{1-\nu^2}{E\ell}\int_0^\ell K_I^2(u)
\mathrm{d}u
\end{equation}}\unskip
which yields, with $\lambda=\ell/R$:
{\begin{equation}\label{eq:irwi-intn}
G_{\mathrm{inc}}(\ell)=\Gamma
\left(\dfrac{1}{21.29\lambda}
\left(1-\dfrac{1}{(1+\lambda)^{4.6}}\right) 
-\dfrac{1}{4.63(1+\lambda)^{5.6}}\right)
\quad \mbox{with}\;
\Gamma=\dfrac{\pi R{\sigma_\infty}^2(1-\nu^2)}{E}.
\end{equation}}\unskip

\begin{figure}
\includegraphics{fig04}
\caption{\label{fig:Ctilde}Circular hole in an infinite medium. 
Incremental energy release rate for two different stress levels: the
critical stress level $\sigma_\infty^{\ast}$  (solid line);
$\sigma_\infty=1.5\sigma_\infty^{\ast}$  (dashed line).}
\end{figure}

Figure~\ref{fig:encad} presents the variation of the lower and upper
bounds of $\mathscr{I}^{\mathrm{dyn}}$ as functions of the remote
compression stress. Interestingly, when the remote stress is slightly
greater than the critical stress level $\sigma_*^\infty$, the dynamic
lower and upper bounds are very close to the static critical length
$\ell^*=\ell_{\mathrm{cr}}$.

\begin{figure}
\includegraphics{fig05}
\vspace*{4pt}
\caption{\label{fig:encad}Circular hole in an infinite medium.
Normalized lower and upper bounds of $\mathscr{I}^{\mathrm{dyn}}$ as
functions of the normalized loading level
$\sigma^\infty/\sigma^\infty_\ast$.}
\vspace*{6pt}
\end{figure}

\subsection{A simplified dynamic example: the double cantilever beam}
Let us investigate the concept of crack initiation length into more
details in the framework of a one-dimensional continuous medium. The
example is that of a double cantilever elastic beam (flexural rigidity
$EI$, width $b$). Two opposite forces ${\pm} Q\uv{e}_y$ act on two
identical masses $m$ located at the symmetric right extremities  $A$
and $A'$ of the beams (see Figure~\ref{fig:demicyl2}). 
The vertical displacement of the upper beam is
denoted by $\xi_y(x,t)$. Let denote $u(t)=2\xi_y(0,t)=AA'$ and
$v(t)=\xi_y(0,t)$. The inertia effects are accounted for through the
kinetic energy of the two point masses while preserving the possibility
to derive an analytical solution. At point $(-\ell,0)$, the boundary
conditions are $\xi_y(-\ell(t),t)=0$ and $\partial\xi_y/\partial
x(-\ell(t),t)=0$.

\begin{figure}
\includegraphics{fig06}
\vspace*{4pt}
\caption{\label{fig:demicyl2}Double cantilever beam. The crack tip is
located at $x=-\ell(t)$ (left extremity of the beams).}
\end{figure}

For symmetry reasons, we focus on the upper beam. We shall introduce
the three following quantities, which have the dimension of a length,
among which the last one $\kappa$ is time dependent:
{$$
\lambda= 
\sqrt{\dfrac{EI}{G_cb}};\quad 
\ell_0=\dfrac{\sqrt{EIG_cb}}{Q};\quad 
\kappa=\dfrac{K}{Q}.
$$}\unskip
Let denote $V(x,t)$ and $M(x,t)$ the shear force and the bending
moment. Moment balance yields:
{$$
-V+Q=m\ddot{v}
\quad\mbox{and}\quad 
\dfrac{\mathrm{d}M}{\mathrm{d}x}+V=
0\quad
\Rightarrow \;M(x,t)=(m\ddot{v}-Q)x.
$$}\unskip

The displacement $\xi_y(x,t)$ is therefore obtained by two integrations
from the curvature $\chi=\sfrac{\partial^2\xi_y}{\partial x^2}$:
{$$
EI\dfrac{\partial^2\xi_y}{\partial x^2}(x,t)=
(m\ddot{v}-Q)x.
$$}\unskip
The term $m\ddot{v}$ incorporates the inertia effect and is responsible
for the discrepancy between the dynamic solution and the static one.
Owing for the boundary conditions at $x=-\ell(t)$, it is found that:
{$$
EI\xi_y(x,t)=(m\ddot{v}-Q)
\left(\dfrac{x^3}{6}-
\dfrac{\ell(t)^2x}{2}-
\dfrac{\ell^3}{3}\right)
$$}\unskip
and eventually at $x=0$:
{$$
EIv(t)=
(Q-m\ddot{v})\dfrac{\ell(t)^3}{3}.
$$}\unskip
The dynamic equation of the beam written for $u(t)=2v(t)$ therefore
takes a familiar shape:
{$$
\mu\ddot{u}+C(\ell)u=Q
$$}\unskip
with $\mu=m/2$ and $C(\ell)=\sofrac{3}{2}\sofrac{EI}{\ell^3}$.  The
total kinetic energy is $K=\sofrac{1}{2}\mu \dot{u}^2$ and, after
integration of the elastic energy density $\sofrac{1}{2}EI\chi^2$, the
total elastic energy is $\sofrac{1}{2}C(\ell)u^2$. Eventually, the
dynamic potential energy reads:
{$$
\mathscr{E}_{\mathrm{pot}}^{\mathrm{dyn}}=
\tfrac{1}{2}C(\ell)u^2-Qu
$$}\unskip
which is formally identical to the expression of the static  potential
energy, except of course for the fact that $u(t)$ is not equal to the
static solution $u^{\mathrm{stat}}= \stbfrac{2Q\ell^3}{3EI}$ which
yields 
{\begin{equation}\label{eq:Estat}
\mathscr{E}_{\mathrm{pot}}^{\mathrm{stat}}=
-\dfrac{Q^2\ell^3}{3EI}.	
\end{equation}}\unskip
According to definition \eqref{defC}, it is found that 
{$$
\mathscr{C}(\ell)=\dfrac{2\ell^2}{3EI}
$$}\unskip
which is monotonic increasing. Following the conclusions of
Sections~\ref{sec:monoton} and~\ref{sec:boundze}, a lower bound of the
initiation crack length can be derived from \eqref{eq:lb}:
{\begin{equation}\label{eq:firstlb}
\ell_{\mathrm{init}}^{\mathrm{dyn}}&gt;
\sqrt{3}\ell_0
\end{equation}}\unskip
and we know that the damage process that leads to the crack of length
$\ell_{\mathrm{init}}^{\mathrm{dyn}}$ cannot be a quasistatic one.

It is recalled that the previous bound is derived from the inequality
$\mathscr{E}_{\mathrm{pot}}^{\mathrm{dyn}}&gt;
\mathscr{E}_{\mathrm{pot}}^{\mathrm{stat}}$ which follows from the
theorem of minimal potential energy. As previously stated, it holds
true for any dynamic evolution. But the very fact that it does not
require the determination of the dynamic evolution suggests that it
could be improved if the latter were considered.  Following this idea,
we now aim at improving the lower bound \eqref{eq:firstlb}. With the
expressions of $\mathscr{E}_{\mathrm{pot}}^{\mathrm{dyn}}$ and $K$, the
energy balance \eqref{premierDL} reads:
{$$
\dfrac{\mathrm{d}}{\mathrm{d}t}
\left(\dfrac{1}{2}C(\ell)u^2-Qu+
\dfrac{1}{2}\mu \dot{u}^2+G_cb\ell \right)=0
$$}\unskip
which can be put in the form:
{$$
\dot{\ell}(\tfrac{1}{2}C'(\ell)u^2+G_cb)+
\dot{u}(\mu \ddot{u}-Q+C(\ell)u)=0.	
$$}\unskip
The combination with the momentum balance equation yields
$\dot{\ell}(\sofrac{1}{2}C'(\ell)u^2+G_cb)=0$. Therefore, once the
crack has been created ($\ell\ge \ell_{\mathrm{init}}^{\mathrm{dyn}}$)
and propagates ($\dot{\ell}&gt; 0$), we have to write together
{$$
\tfrac{1}{2}C'(\ell)u^2+G_cb=0
$$}\unskip
and the energy balance in the integrated form:
{$$
\tfrac{1}{2}C(\ell)u^2-Qu+G_cb\ell+K=0
$$}\unskip
where $K$ is the current value of the kinetic energy. Note that these
equations are the particular form in the present example of
\eqref{premier-1} and \eqref{premierDL} respectively. This yields
{$$
K+\dfrac{4}{3}G_cb\ell-
\dfrac{2}{3}\ell^2Q
\sqrt{\dfrac{G_cb}{EI}}=0
$$}\unskip
or
{$$
\ell(t)^2-2\ell_0\ell(t)-
\dfrac{3}{2}\kappa(t)\lambda=0
\quad \Rightarrow\quad 
\ell(t)=\ell_0
\left(1+\sqrt{1+\dfrac{3\lambda\kappa(t)}{2\ell_0^2}}\right).
$$}\unskip
Since $\kappa(t)$ is a positive number, the above expression of
$\ell(t)$ provides a new lower bound of the initiation length:
{$$
\ell_{\mathrm{init}}^{\mathrm{dyn}}=
\ell_0\left(1+\sqrt{1+\dfrac{3\lambda\kappa_{\mathrm{init}}}{2\ell_0^2}}
\right)\ge 2\ell_0.
$$}\unskip
The improvement of the lower bound as compared to \eqref{eq:firstlb}
underlines that the initiation state is not an equilibrium state, even
when we take the limit $\kappa_{\mathrm{init}}\to 0$. It is noteworthy
that the present approach cannot per se determine the value of the
constant $\kappa_{\mathrm{init}}$ which is necessary for deriving the
exact value of the initiation length.

\section{Conclusion}\label{sec5}
The purpose of this paper was to highlight what can be said about the
nucleation process with the standard model of perfectly brittle
material as a modelling basis. It is important to underline that the
developments presented herein  are based on the assumption that the
crack is indeed nucleated. For instance, the lower bound of
$\mathcal{I}^{\mathrm{dyn}}$ provided in \eqref{eq:lb} explicitely
assumes this prerequisite.  Similarly, \eqref{carliozQlb} is a
necessary condition for crack nucleation and the model cannot say
whether it is sufficient. To do so, further material parameters, such
as the tensile strength occurring in the CC (coupled criterion), would
be required.  With this in mind, let us briefly review the main
conclusions drawn in this framework.

A very important characteristic involving the geometry of the structure
together with the way the loading is applied is the rate of change of
the structural compliance. In the simple case of a straightline crack,
it becomes a function $\mathscr{C}(\ell)$ of the crack length. The
latter is easily determined analytically from the knowledge of stress
intensity factors or numerically from the solution to elastostatic
problems. It is appropriate to distinguish two cases whether this
function is monotonically increasing or not.

When the function $\mathscr{C}(\ell)$ is a monotonically increasing
one, a quasistatic description of nucleation is not possible
irrespective of the loading level. In contrast, when  the function
$\mathscr{C}(\ell)$ reaches a maximum at $\ell=\ell_{\mathrm{cr}}$,
there exists a unique loading level $Q^\ast$ that is compatible with a
quasistatic process of nucleation. The critical crack length
$\ell_{\mathrm{cr}}$ is only a function of the geometry at stake. Any
loading level $Q&gt;Q^\ast$ will induce dynamic effects.

An appropriate use of the classical theorem of minimum potential energy
reveals that the dynamic potential energy release is upper bounded by
its static counterpart, that is, for the same crack length change. This
enables to use various mathematical descriptors issued from a
quasistatic framework fro the design of dynamic bounds. For instance, a
rigorous strictly positive lower bound of the (dynamic) initiation
length $\ell_{\mathrm{init}}^{\mathrm{dyn}}$ can be determined as a
function of the loading level. In other words, the theoretical
framework of perfectly brittle material in itself contradicts the
existence of a length crack 
$\ell&lt;\ell_{\mathrm{init}}^{\mathrm{dyn}}$. The description of the
damage process yielding the initiation length should resort to more
elaborated material modellings (see e.g.~\cite{BFM,2borst,marigo2}).
\looseness=-1

When the  function $\mathscr{C}(\ell)$ is non monotonic, the set
$\mathcal{I}^{\mathrm{dyn}}$ of observable crack lengths is lower and
upper bounded. The closer the loading level to the critical one
\eqref{eq:Qstar}, the narrower the set  $\mathcal{I}^{\mathrm{dyn}}$ in
the neighborhood of the critical length $\ell_{\mathrm{cr}}$. 

\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.

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