Comptes Rendus
***
Mécanique

Luc Dormieux, Eric Lemarchand, Laurent Jeannin and Djimedo Kondo

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

Volume 1, issue 0 (0000), p. 000-000

https://doi.org/10.5802/crmeca.296

<figure><img src="image_1.png" /></figure>

This article is licensed under the
CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.
http://creativecommons.org/licenses/by/4.0/
Research article / *Article de recherche*

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

## Quelques remarques sur la nucléation et les longueurs d'initiation des fissures dans les matériaux fragiles à la lumière du critère énergétique de Dominique Leguillon

Luc Dormieux<sup>a</sup>, Eric Lemarchand<sup>●,*,a</sup>, Laurent Jeannin<sup>b</sup> and Djimedo Kondo<sup>c</sup>

**Abstract.** Crack nucleation has been the subject of important contributions in the two last decades. Starting from the energy criterion of Leguillon (Eur. J. Mech.—A/Solids 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.

**Résumé.** La nucléation des fissures a fait l'objet de contributions importantes au cours des deux dernières décennies. En partant du critère énergétique proposé par Leguillon (Eur. J. Mech.—A/Solids 21 (2002)), le présent article examine certaines conséquences qui découlent directement du modèle parfaitement fragile. Cela permet de discuter de la possibilité d'une modélisation quasi-statique de la nucléation. Des bornes rigoureuses sont établies pour le niveau de chargement déclenchant la nucléation ainsi que pour l'ensemble des longueurs de fissures observables dans des conditions dynamiques.

**Keywords.** Crack initiation, Crack nucleation, Energy criterion, Dynamic, Bounds.

**Mots-clés.** Initiation de fissure, Nucléation de fissure, Critère énergétique, Dynamiques, Bornes.
Manuscript received 11 February 2025, revised and accepted 10 April 2025.

<footnote><sup>a</sup> Laboratoire Navier (UMR 8205), ENPC/CNRS/UGE, 6 et 8 avenue Blaise Pascal, 77455, cedex-2 Marne-la-Vallée, France</footnote>

<footnote><sup>b</sup> STORENGY, 12 rue Raoul Nordling - Djinn - CS 70001, 92274 Bois Colombes Cedex, France</footnote>

<footnote><sup>c</sup> Sorbonne Universités, Institut Jean Le Rond d'Alembert, UMR 7190 CNRS-SU, F-75005, Paris, France</footnote>

<footnote>E-mails: luc.dormieux@enpc.fr (L. Dormieux), eric.lemarchand@enpc.fr (E. Lemarchand), laurent.jeannin@storengy.com (L. Jeannin), djimedo.kondo@sorbonne-universite.fr (D. Kondo)</footnote>

<footnote>*Corresponding author</footnote>
# 1. Introduction

The pioneering paper [1] 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:

$$-\delta\mathcal{E}_{\text{pot}} \geq G_c\delta S \quad (1)$$

in which $G_c$ denotes the fracture energy per unit surface. As opposed to the Griffith concept of rate of energy release [2], it is emphasized that the above inequality refers to a finite variation $\delta S$ of surface. The significance of (1), that is generally referred to as energy criterion, is now widely acknowledged. In [1], it was proposed that (1) 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 [3].

More recently, in the case of stable nucleation processes, [4] 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. [5,6]). Finally, an extension to the case of saturated porous materials was proposed [7].

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 $\mathcal{I}$ 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 $\mathcal{I}$ as well as the evolution law.

Among these questions, the determination of the minimum of $\mathcal{I}$, 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 $\mathcal{I}$ of observable crack lengths truly starts at 0. It has been argued [3] that a non zero initiation length is an artefact of the quasi-static description of nucleation. It was also suggested in [3] 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 (1) 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 [3], 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 additional 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 (1) 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.
## 2. 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 $\ell^*$ for which stability is recovered under the loading that has triggered the nucleation.

The starting point is the Clausius-Duhem inequality [8,9]: In isothermal conditions, it states that the external power ($\mathcal{P}_e$) provided to the system $\Omega$ is transformed into the rate of change of free energy ($\Psi$) and the rate of change of the kinetic energy ($\dot{K}$) or dissipated in the crack nucleation process (dissipation rate $\mathcal{D}$):

$$ \mathcal{D} = \mathcal{P}_e - \dot{\Psi} - \dot{K} \geq 0. \qquad (2) $$

As already stated in the introduction of the present paper, this is also the starting point of the celebrated work of Leguillon [1] in which (2) is considered in an integrated form with respect to time:

$$ \Delta\Psi - \mathcal{W}_e + K + \int \mathcal{D} dt = 0. \qquad (3) $$

For simplicity, we assume steady state conditions for the mechanical loading $\mathcal{L}$, the latter being defined by body forces $F$, prescribed surface forces $T^d$ on $\partial\Omega_T$, prescribed displacements $\xi^d$ 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:

$$ \mathcal{P}_e = \int_{\Omega} \underline{F} \cdot \dot{\xi} dV + \int_{\partial\Omega_T} \underline{T}^d \cdot \dot{\xi} dS. $$

Since we aim at describing the crack propagation occurring in the framework of crack nucleation, we focus on a time interval of crack length increase ($\dot{\ell} > 0$). This makes the relation between time and crack length a bijective one, so that the velocity $\dot{\xi}$ can be put in the form:

$$ \dot{\xi} = \frac{\partial \xi}{\partial \ell} \bigg|_{\mathcal{L}} \dot{\ell}. $$

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:

$$ \dot{\Psi} = \frac{\partial \Psi}{\partial \ell} \bigg|_{\mathcal{L}} \dot{\ell}; \quad \dot{K} = \frac{\partial K}{\partial \ell} \bigg|_{\mathcal{L}} \dot{\ell}. \qquad (4) $$

Under steady loading conditions, the external power is due to crack propagation only:

$$ \mathcal{P}_{e|\mathcal{L}} = \frac{\partial \Phi}{\partial \ell} \bigg|_{\mathcal{L}} \dot{\ell} \quad \text{with} \quad \Phi = \int_{\Omega} \underline{F} \cdot \dot{\xi} dV + \lambda_T(t) \int_{\partial\Omega_T} \underline{T}^0 \cdot \dot{\xi} dS \qquad (5) $$

where $\Phi$ is the potential of the given loads. Eventually, the combination of (2), (4) and (5) yields:

$$ \mathcal{D} + \frac{\partial K}{\partial \ell} \bigg|_{\mathcal{L}} \dot{\ell} = - \frac{\partial \mathcal{E}_{\text{pot}}^{\text{dyn}}}{\partial \ell} \bigg|_{\mathcal{L}} \dot{\ell} \qquad (6) $$

which introduces the potential energy $\mathcal{E}_{\text{pot}}^{\text{dyn}} = \Psi - \Phi$, where the upperscript 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, (6) takes the form:

$$ G_c + \frac{\partial K}{\partial \ell} \bigg|_{\mathcal{L}} = - \frac{\partial \mathcal{E}_{\text{pot}}^{\text{dyn}}}{\partial \ell} \bigg|_{\mathcal{L}}. \qquad (7) $$

The derived expression is formally very similar to the classical rate of energy release (see e.g. [10]), 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, (7) states that the released energy is not entirely dissipated since a part of it is transformed into kinetic energy.

Note that (7) can be put in the equivalent form:

$$G_c = - \frac{\partial \mathcal{E}_{\text{mec}}^{\text{dyn}}}{\partial \ell} \Big|_{\mathcal{L}} \qquad (8)$$

where $\mathcal{E}_{\text{mec}}^{\text{dyn}}$ is the mechanical energy $\mathcal{E}_{\text{pot}} + K$. This suggests an alternative definition to the dynamic energy release rate.

### 3. The assumption of quasistatic nucleation process

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<fnref n="1" /> by a single scalar parameter $Q$, and that the corresponding work of external forces reads $\Phi = Q\xi$. The free energy takes the form $\Psi = (1/2)C(\ell)\xi^2$ where $C(\ell)$ represents the elastic stiffness of the cracked structure. In view of further use, let us introduce the functions $C(\ell)$ and $\mathcal{C}(\ell)$:

$$C(\ell) = \frac{1}{\ell}(C(0) - C(\ell)); \quad \mathcal{C}(\ell) = \frac{1}{\ell}\left(\frac{1}{C(\ell)} - \frac{1}{C(0)}\right). \qquad (9)$$

In particular, the function $\mathcal{C}(\ell)$ 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:

$$\mathcal{E}_{\text{pot}}^{\text{stat}} = \min_{\xi} \left( \frac{1}{2} C(\ell) \xi^2 - Q\xi \right) = -\frac{Q^2}{2C(\ell)}.$$

Considering any stage in the propagation process associated with some crack length $\ell$, the energy balance reads:

$$\frac{Q^2}{2} \left( \frac{1}{C(\ell)} - \frac{1}{C(0)} \right) = \int \mathcal{D} \, dt.$$

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:

$$\frac{Q^2}{2} \mathcal{C}(\ell) = \frac{\mathcal{E}_{\text{pot}}^{\text{stat}}(0) - \mathcal{E}_{\text{pot}}^{\text{stat}}(\ell)}{\ell} = G_c. \qquad (10)$$

Note that the above equation is also obviously retrieved from (3) in which the kinetic energy term is omitted. (10) 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 $\mathcal{C}(\ell)$. The determination

<footnote n="1">¹The boundary conditions could also be defined on the displacement. This would affect the definition of the potential energy.</footnote>
of the latter only requires the resolution of standard elastostatic problems. It is worth noticing that the first equality in (10), that is:

$$ \frac{Q^2}{2} \mathcal{C}(\ell) = \frac{\mathcal{E}_{\text{pot}}^{\text{stat}}(0) - \mathcal{E}_{\text{pot}}^{\text{stat}}(\ell)}{\ell} $$

can be regarded as a more general definition of function $\mathcal{C}(\ell)$. 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

$$ G_{\text{inc}}(\ell) = \frac{\mathcal{E}_{\text{pot}}^{\text{stat}}(0) - \mathcal{E}_{\text{pot}}^{\text{stat}}(\ell)}{\ell} $$

introduced by Leguillon overcomes this shortcoming. Besides, in many cases, an analytical form of $\mathcal{C}(\ell)$ can be derived from the stress intensity factors (see e.g. [11]) by means of Irwin equation (see for example Section 4.2). Indeed, according to the latter, the incremental energy release rate of Leguillon is immediately derived by integration of the square of the stress intensity factor.

Nevertheless, it may be useful to observe that it is equivalent to give oneself the stiffness function $C(\ell)$ or the function $\mathcal{C}(\ell)$ appearing in the static incremental energy release rate. Indeed, the latter can obviously be derived from the former by (9). 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 $C(\ell) = 1/(a + \ell\mathcal{C}(\ell))$. The choice of the positive constant $a$ is arbitrary provided that $\lim_{\ell \to 0} \ell\mathcal{C}(\ell) = 0$. This constant vanishes in the expressions of energy criteria.

Similarly, it is equivalent to give oneself the compliance function $S(\ell) = 1/C(\ell)$ or the function $\mathcal{C}(\ell)$ appearing in the static incremental energy release rate. Again, the latter can be derived from the former by (9). 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 $S(\ell) = a + \ell\mathcal{C}(\ell)$. As before, the choice of the positive constant $a$ is arbitrary.

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

### 3.1. The case of a monotonic $\mathcal{C}(\ell)$ function

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 $\mathcal{C}(\ell)$ shares this property. We first consider the case where the $\mathcal{C}(\ell)$ function is indeed a strictly increasing function (see Figure 1 case (a)).

From a mathematical point of view, this is in particular the case if the stiffness function is a decreasing strictly concave function:

$$ (\forall \ell \ge 0) \quad C'(\ell) \le 0 \quad \text{and} \quad C''(\ell) < 0. $$

From a second order Taylor-Lagrange expansion:

$$ (\exists \theta \in [0, 1]) \quad C(0) = C(\ell) - \ell C'(\ell) + \frac{1}{2} C''(\theta \ell) \ell^2 $$

it is found that $C(\ell) + C'(\ell) < 0$. Observing that

$$ \mathcal{C}'(\ell) = \frac{1}{C(0)C(\ell)\ell} \left( -\frac{C(0)}{C(\ell)} C'(\ell) - \mathcal{C}(\ell) \right) $$

and recalling that $C(0) > C(\ell)$, we conclude that $\mathcal{C}'(\ell) > 0$. It can be also readily shown that $\mathcal{C}(\ell)$ 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
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Various patterns of the rate of change of structural compliance $\mathcal{C}(\ell)$: (a) example of the monotonic increasing case; (b) example of the existence of a maximum.</figcaption></figure>

intensity factor is of the form $\sigma_{\infty}\sqrt{\pi\ell}F$ where $F$ is a scalar coefficient ($F \approx 1.12$, [11]). By integration of Irwin equation, this implies, in the present example, that the $\mathcal{C}(\ell)$ function is proportional to the crack length:

$$ \mathcal{C}(\ell) = F^2 \pi \frac{1 - \nu^2}{E} \ell. \qquad (11) $$

Let us go back to the general case of a strictly increasing function $\mathcal{C}(\ell)$. It follows that (10) with unknown $\ell$ has a unique solution $\ell^*$ (provided that $Q^2 < 2G_c/\mathcal{C}(0)$, if $\mathcal{C}(0) > 0$). This obviously implies that the nucleation process is not the propagation of a crack from 0 to the final length $\ell^*$. In other words, within the quasistatic framework, it is impossible to observe any crack having a length $\ell < \ell^*$.

Let us assume that the crack with length $\ell^*$ 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:

$$ G_{\text{diff}}(\ell^*) = -\frac{\partial \mathcal{E}_{\text{pot}}^{\text{stat}}}{\partial \ell}(\ell^*) = \frac{Q^2}{2} \frac{d}{d\ell}(\ell\mathcal{C}(\ell))(\ell^*) \leq G_c. \qquad (12) $$

Since $\mathcal{C}'(\ell^*) > 0$, we have:

$$ \frac{d}{d\ell}(\ell\mathcal{C}(\ell))(\ell^*) = \mathcal{C}(\ell^*) + \ell^*\mathcal{C}'(\ell^*) > \mathcal{C}(\ell^*) $$

from which it follows that (10) and (12) are not compatible. In other words, in the case of a strictly concav stiffness function, or more generally when $\mathcal{C}(\ell)$ is a monotonic increasing function, the quasistatic assumption fails to describe the nucleation process. As stated above (see (11)), 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.

## 3.2. The case of a non monotonic $\mathcal{C}(\ell)$ function

We now assume that $\mathcal{C}(\ell)$ is increasing on some interval $[0, \ell_{cr}]$ and decreases for $\ell > \ell_{cr}$ (see Figure 1 case (b)). This implies that $\mathcal{C}'(\ell_{cr}) = 0$. For given elastic properties, it is important to note that this property and in particular the critical length $\ell_{cr}$ are, by nature, independent of the loading level. Indeed, the practical determination of $\mathcal{C}(\ell)$ incorporates informations concerning the type of boundary conditions and the geometry of the body and of the crack. This is all the
<figure><img src="image_3.png" /><figcaption>**Figure 2.** Characterization of the nucleated crack length $ℓ^*$: differential $G_{\text{diff}}(ℓ)$ and incremental $G_{\text{inc}}(ℓ)$ energy release rates for the critical loading level $Q^*$ of (14).</figcaption></figure>

more clear when $\mathcal{C}(ℓ)$ 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 4.2). 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 $ℓ_{cr}$ 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 $ℓ^*$ complies with (10) and (12) which are repeated here for convenience:

$$ \frac{\mathcal{E}_{\text{pot}}^{\text{stat}}(0) - \mathcal{E}_{\text{pot}}^{\text{stat}}(\ell^*)}{\ell^*} = \frac{Q^2}{2} \mathcal{C}(\ell^*) = G_c; \quad -\frac{\partial \mathcal{E}_{\text{pot}}^{\text{stat}}}{\partial \ell}(\ell^*) = \frac{Q^2}{2} (\mathcal{C}(\ell^*) + \ell^* \mathcal{C}'(\ell^*)) \le G_c. \quad (13) $$

The compatibility of these conditions appears to be $\mathcal{C}'(\ell^*) = 0$, which means that the nucleation crack length $ℓ^*$ is none but the critical length $ℓ_{cr}$. 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 (13):

$$ Q^* = \sqrt{\frac{2G_c}{\mathcal{C}(\ell_{cr})}}. \quad (14) $$

We thus note that both the nucleation crack length and the loading level are determined in a unique way [12]. If $Q < Q^*$, the energy criterion is not fulfilled, in the sense that the energy release is not sufficient for the creation of the crack. If $Q > Q^*$, 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 $Q^*$ is such that the plot of the incremental energy release rate $G_{\text{inc}}(ℓ)$ is tangent to the line $G = G_c$ at the point with abscissa $ℓ_{cr}$. The relative position of the plots of $G_{\text{inc}}(ℓ)$ and $G_{\text{diff}}(ℓ)$ in Figure 2 is obtained from the expressions of these quantities recalled in (13).

Again, it is noteworthy that no crack with length $ℓ < ℓ_{cr}$ can be observed. This suggests that the nucleation process that yields the critical crack length $ℓ^* = ℓ_{cr}$ 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 [13] can be helpful for describing this maturation phase that precedes nucleation.

Anyway, under the loading $Q^*$, the structure is left in a state of stable equilibrium with a created crack of length $\ell_{cr}$. 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.

# 4. The non quasistatic nucleation process

## 4.1. Rigorous bounds for the set of observable crack lengths

At Section 3.1, 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 3.2 if the loading level is greater than the threshold given by (14). 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 $\mathcal{J}^{dyn}$.

The inequality (1) 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 (1) 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 *dynamic* potential energy $\mathcal{E}_{pot}^{dyn}$ by its *static* counterpart $\mathcal{E}_{pot}^{stat}$. This can be highlighted in re-stating (1) in the explicit form:

$$ -\delta \mathcal{E}_{pot}^{dyn} \geq G_c \delta S. \qquad (15) $$

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 $\xi^{dyn}$ to the “real” dynamic evolution and the solution $\xi^{stat}$ to the elastostatic problem. The displacement field $\xi^{dyn}$ is, by nature, kinematically admissible, so that the theorem ensures that $\mathcal{E}_{pot}^{dyn} > \mathcal{E}_{pot}^{stat}$. An immediate consequence is that $-\delta \mathcal{E}_{pot}^{dyn} < -\delta \mathcal{E}_{pot}^{stat}$ and further that:

$$ -\delta \mathcal{E}_{pot}^{stat} > -\delta \mathcal{E}_{pot}^{dyn} \geq G_c \delta S. \qquad (16) $$

This provides a simple justification of Leguillon's seminal intuition to the use of a static estimate of the potential energy in (1). 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 *dynamic* nucleation process must comply with (16):

$$ (\forall \ell \in \mathcal{J}^{dyn}) \quad \frac{Q^2}{2} \mathcal{C}(\ell) > G_c. \qquad (17) $$

Let us first consider the case of the monotonic increasing function $\mathcal{C}(\ell)$.

The equation $Q^2\mathcal{C}(\ell) = 2G_c$ with unknown $\ell$ has therefore a unique solution which, owing to (17), proves to be a **lower bound** for any observable crack length during the dynamic evolution process, that is, of the interval $\mathcal{J}^{dyn}$. It is therefore denoted by $\ell^{lb}(Q)$ and defined by:

$$ \frac{Q^2}{2} \mathcal{C}(\ell^{lb}) = G_c; \quad (\forall \ell \in \mathcal{J}^{dyn}) \quad \ell > \ell^{lb}(Q). \qquad (18) $$
The very fact that $\mathcal{S}^{dyn}$ has a non-zero lower bound leads to introduce the concept of initiation length, denoted by $\ell_{init}^{dyn}$, 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 $\ell_{init}$ that can be determined from the associated static problem:

$$\ell_{init}^{dyn} > \ell^{lb}(Q).$$

Using again that $\mathcal{C}(\ell)$ is an increasing function, it follows from (18) that $\ell^{lb}(Q)$ is a bijective decreasing function of $Q$. As stated at the end of Section 3.1, 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.

Let us now consider the case of a non monotonic function $\mathcal{C}(\ell)$.

Again, the starting point is that observable crack lengths must comply with (17). 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:

$$Q > \sqrt{\frac{2Gc}{\mathcal{C}(\ell)}}.$$

A necessary condition for the existence of a solution $\ell$ in the above inequality is:

$$Q > \sqrt{\frac{2Gc}{\max\mathcal{C}(\ell)}} = \sqrt{\frac{2Gc}{\mathcal{C}(\ell_{cr})}} = Q^* \quad (19)$$

in which we recognize the threshold $Q^*$ introduced at (14) as the unique loading level being compatible with a quasistatic nucleation process. We now gain a new interpretation of the latter: $Q^*$ 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 $Q < Q^*$.

We now assume that $Q > Q^*$. Let $\ell^{lb}(Q)$ and $\ell^{ub}(Q)$ be the solutions of the equation $Q^2\mathcal{C}(\ell) = 2G_c$ with unknown $\ell$ as shown on Figure 3. The condition (17) implies that these lengths indeed provide bounds for $\mathcal{S}^{dyn}$. The latter is therefore a bounded interval of the form $[\ell_{init}^{dyn}, \ell_{nuc}^{dyn}]$ and we have:

$$\ell^{lb}(Q) \le \ell_{init}^{dyn} \le \ell_{nuc}^{dyn} \le \ell^{ub}(Q). \quad (20)$$

In particular, for values of $Q > Q^*$ in the neighborhood of $Q^*$, it may be concluded that the initiation and nucleation lengths should be very close and that $\ell_{cr}$ is a good estimate of them (see Figure 3).

It is emphasized that the bounds $\ell^{lb}(Q)$ and $\ell^{ub}(Q)$ are very simply determined within an **elastostatic framework** and are therefore elastostatic by nature. Furthermore, for any loading level $Q > Q^*$, 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 $\mathcal{S}^{dyn}$ of observable crack lengths in a dynamic process.

## 4.2. The example of an infinite medium with circular hole

Let us apply the developments of Section 4.1 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 well-known that the nucleation of a stable crack is expected (e.g. [6,14]).

Figure 4 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 [14]:

$$K_I(\ell) \approx -\frac{1.1\sqrt{\ell/R}}{(1+\ell/R)^{3.3}} \sigma_{\infty} \sqrt{\pi R}. \quad (21)$$

Irwin's formula in plane strains relates the stress intensity factor to the rate of energy release $G(\ell)$:

$$G(\ell) = -\frac{\partial \mathcal{E}_{\text{pot}}^{\text{stat}}}{\partial \ell} = \frac{1 - \nu^2}{E} K_I^2. \quad (22)$$
<figure><img src="image_4.png" /><figcaption>Figure 3. Lower and upper bounds of $\mathcal{I}^{dyn}$ in the case of non monotonic $\mathcal{C}(\ell)$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 4. Circular hole in an infinite medium. Incremental energy release rate for two different stress levels: the critical stress level $\sigma_{\infty}^*$ (solid line); $\sigma_{\infty} = 1.5\sigma_{\infty}^*$ (dashed line).</figcaption></figure>

$G_{inc}(\ell)$ is readily obtained by integration:

$$G_{inc}(\ell) = \frac{1 - \nu^2}{E\ell} \int_{0}^{\ell} K_{I}^{2}(u) du \qquad (23)$$

which yields, with $\lambda = \ell/R$:

$$G_{inc}(\ell) = \Gamma \left( \frac{1}{21.29\lambda} \left( 1 - \frac{1}{(1+\lambda)^{4.6}} \right) - \frac{1}{4.63(1+\lambda)^{5.6}} \right) \quad \text{with} \quad \Gamma = \frac{\pi R \sigma_{\infty}^2 (1 - \nu^2)}{E} \qquad (24)$$

Figure 5 presents the variation of the lower and upper bounds of $\mathcal{I}^{dyn}$ 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 $\ell^* = \ell_{cr}$.
<figure><img src="image_6.png" /><figcaption>**Figure 5.** Circular hole in an infinite medium. Normalized lower and upper bounds of $\mathcal{J}_{dyn}$ as functions of the normalized loading level $\sigma^\infty/\sigma_*^\infty$.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 6.** Double cantilever beam. The crack tip is located at $x = -\ell(t)$ (left extremity of the beams).</figcaption></figure>

### 4.3. 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 $\pm Qe_y$ act on two identical masses m located at the symmetric right extremities A and A' of the beams (see Figure 6). 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 $\partial\xi_y/\partial x(-\ell(t), t) = 0$.

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:

$$ \lambda = \sqrt{\frac{EI}{G_c b}}, \quad \ell_0 = \frac{\sqrt{EIG_c b}}{Q}, \quad \kappa = \frac{K}{Q}. $$
Let denote $V(x, t)$ and $M(x, t)$ the shear force and the bending moment. Moment balance yields:

$$ -V + Q = m\ddot{v} \quad \text{and} \quad \frac{dM}{dx} + V = 0 \Rightarrow M(x, t) = (m\ddot{v} - Q)x. $$

The displacement $\xi_y(x, t)$ is therefore obtained by two integrations from the curvature $\chi = \partial^2\xi_y/\partial x^2$:

$$ EI \frac{\partial^2 \xi_y}{\partial x^2} (x, t) = (m\ddot{v} - Q)x. $$

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:

$$ EI\xi_y(x, t) = (m\ddot{v} - Q) \left( \frac{x^3}{6} - \frac{\ell(t)^2 x}{2} - \frac{\ell^3}{3} \right) $$

and eventually at $x = 0$:

$$ EI\nu(t) = (Q - m\ddot{v}) \frac{\ell(t)^3}{3}. $$

The dynamic equation of the beam written for $u(t) = 2\nu(t)$ therefore takes a familiar shape:

$$ \mu\ddot{u} + C(\ell)u = Q $$

with $\mu = m/2$ and $C(\ell) = (3/2)(EI/\ell^3)$. The total kinetic energy is $K = (1/2)\mu\dot{u}^2$ and, after integration of the elastic energy density $(1/2)EI\chi^2$, the total elastic energy is $(1/2)C(\ell)u^2$. Eventually, the dynamic potential energy reads:

$$ \mathcal{E}_{\text{pot}}^{\text{dyn}} = \frac{1}{2} C(\ell) u^2 - Qu $$

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 $u^{\text{stat}} = (2Q\ell^3)/(3EI)$ which yields

$$ \mathcal{E}_{\text{pot}}^{\text{stat}} = -\frac{Q^2 \ell^3}{3EI}. \qquad (25) $$

According to definition (9), it is found that

$$ \mathcal{C}(\ell) = \frac{2\ell^2}{3EI} $$

which is monotonic increasing. Following the conclusions of Sections 3.1 and 4.1, a lower bound of the initiation crack length can be derived from (18):

$$ \ell_{\text{init}}^{\text{dyn}} > \sqrt{3\ell_0} \qquad (26) $$

and we know that the damage process that leads to the crack of length $\ell_{\text{init}}^{\text{dyn}}$ cannot be a quasistatic one.

It is recalled that the previous bound is derived from the inequality $\mathcal{E}_{\text{pot}}^{\text{dyn}} > \mathcal{E}_{\text{pot}}^{\text{stat}}$ 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 (26). With the expressions of $\mathcal{E}_{\text{pot}}^{\text{dyn}}$ and $K$, the energy balance (3) reads:

$$ \frac{d}{dt} \left( \frac{1}{2} C(\ell) u^2 - Qu + \frac{1}{2} \mu \dot{u}^2 + G_c b \ell \right) = 0 $$

which can be put in the form:

$$ \dot{\ell}(\frac{1}{2} C'(\ell) u^2 + G_c b) + \dot{u}(\mu\ddot{u} - Q + C(\ell)u) = 0. $$

The combination with the momentum balance equation yields $\dot{\ell}((1/2)C'(\ell)u^2 + G_c b) = 0$. Therefore, once the crack has been created ($\ell \ge \ell_{\text{init}}^{\text{dyn}}$) and propagates ($\ell > 0$), we have to write together

$$ \frac{1}{2} C'(\ell) u^2 + G_c b = 0 $$
and the energy balance in the integrated form:

$$ \frac{1}{2} C(\ell) u^2 - Qu + G_c b \ell + K = 0 $$

where K is the current value of the kinetic energy. Note that these equations are the particular form in the present example of (7) and (3) respectively. This yields

$$ K + \frac{4}{3} G_c b \ell - \frac{2}{3} \ell^2 Q \sqrt{\frac{G_c b}{EI}} = 0 $$

or

$$ \ell(t)^2 - 2\ell_0\ell(t) - \frac{3}{2}\kappa(t)\lambda = 0 \Rightarrow \ell(t) = \ell_0 \left( 1 + \sqrt{1 + \frac{3\lambda\kappa(t)}{2\ell_0^2}} \right). $$

Since $\kappa(t)$ is a positive number, the above expression of $\ell(t)$ provides a new lower bound of the initiation length:

$$ \ell_{\text{init}}^{\text{dyn}} = \ell_0 \left( 1 + \sqrt{1 + \frac{3\lambda\kappa_{\text{init}}}{2\ell_0^2}} \right) \geq 2\ell_0. $$

The improvement of the lower bound as compared to (26) underlines that the initiation state is not an equilibrium state, even when we take the limit $\kappa_{\text{init}} \to 0$. It is noteworthy that the present approach cannot per se determine the value of the constant $\kappa_{\text{init}}$ which is necessary for deriving the exact value of the initiation length.

## 5. Conclusion

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 $\mathcal{I}^{\text{dyn}}$ provided in (18) explicitly assumes this prerequisite. Similarly, (19) 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 $\mathcal{C}(\ell)$ 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 $\mathcal{C}(\ell)$ is a monotonically increasing one, a quasistatic description of nucleation is not possible irrespective of the loading level. In contrast, when the function $\mathcal{C}(\ell)$ reaches a maximum at $\ell = \ell_{cr}$, there exists a unique loading level $Q^*$ that is compatible with a quasistatic process of nucleation. The critical crack length $\ell_{cr}$ is only a function of the geometry at stake. Any loading level $Q > Q^*$ 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 $\ell_{\text{init}}^{\text{dyn}}$ 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 $\ell < \ell_{\text{init}}^{\text{dyn}}$. The description of the damage process yielding the initiation length should resort to more elaborated material modellings (see e.g. [15–17]).
When the function $\mathcal{C}(\ell)$ is non monotonic, the set $\mathcal{J}^{\text{dyn}}$ of observable crack lengths is lower and upper bounded. The closer the loading level to the critical one (14), the narrower the set $\mathcal{J}^{\text{dyn}}$ in the neighborhood of the critical length $\ell_{\text{cr}}$.

**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.

**References**

[1] D. Leguillon, "Strength or toughness: a criterion for crack onset at a notch", *Eur. J. Mech.—A/Solids* **21** (2002), pp. 61–72.

[2] A. A. Griffith, "The phenomenon of rupture and flow in solids", *Philos. Trans. R. Soc. Lond.* **221** (1921), pp. 163–198.

[3] A. Doitrand, G. Molnar, D. Leguillon, E. Martin and E. Carrère, "Dynamic crack initiation assessment with the coupled crterion", *Eur. J. Mech. A, Solids* **93** (2022), article no. 104483.

[4] T. Carlioz, *Nucléation et propagation de fissures en conditions anisotropes*, Thèse de doctorat, Université Paris-Est, 2017.

[5] L. Jeannin, L. Dormieux and T. Carlioz, "Nucleation of thermal cracks at the wall of a rock mass", *Mech. Res. Commun.* **97** (2019), pp. 57–62.

[6] L. Jeannin, L. Dormieux, E. Lemarchand and T. Carlioz, "Analysis of fracture nucleation around wellbores", *Int. J. Fracture* **230** (2021), pp. 157–178.

[7] L. Dormieux, E. Lemarchand, T. Carlioz and L. Jeannin, "Crack nucleation in saturated porous media", *Int. J. Numer. Anal. Meth. Geomech.* **5** (2020), pp. 553–574.

[8] L. Dormieux and D. Kondo, *Micromechanics of Fracture and Damage*, ISTE-Wiley: London, 2016.

[9] J. R. Rice, "Thermodynamics of the quasi-static growth of Griffith cracks", *J. Mech. Phys. Solids* **26** (1978), pp. 61–78.

[10] J. B. Leblond, *Mécanique de la rupture fragile et ductile*, Hermes Science Publications: Lavoisier, Paris, 2003.

[11] G. C. Sih, *Handbook of Stress Intensity Factors for Researchers and Engineers*, Lehigh University: Bethlehem, 1973.

[12] T. Carlioz, L. Dormieux and E. Lemarchand, "Thermodynamics of crack nucleation", *Contin. Mech. Thermodyn.* **32** (2020), pp. 1515–1531.

[13] K. Pham and J.-J. Marigo, "Gradient damage models and their use to approximate brittle fracture", *Int. J. Damage Mech.* **4** (2011), pp. 618–652.

[14] C. G. Sammis and M. F. Ashby, "The failure of brittle porous solids under compressive stress states", *Acta Metall.* **34** (1986), no. 3, pp. 511–526.

[15] B. Bourdin, G. A. Francfort and J.-J. Marigo, "The variational approach to fracture", *J. Elast.* **91** (2008), pp. 5–148.

[16] R. De Borst and C. V. Verhoosel, "Gradient damage vs phase-field approaches for fracture: similarities and differences", *Comput. Meth. Appl. Mech. Engrg.* **312** (2016), pp. 78–94.

[17] K. Pham and J.-J. Marigo, "From the onset of damage to rupture: construction of responses with damage localization for a general class of gradient damage models", *Contin. Mech. Thermodyn.* **25** (2013), no. 2, pp. 147–171.