# Brittle Crack Initiation Seen as a Dynamic Crack Propagation Phenomenon

Aurélien Doitrand, Rafael Estevez, François Bonnel

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Aurélien Doitrand, Rafael Estevez, François Bonnel. Brittle Crack Initiation Seen as a Dynamic Crack Propagation Phenomenon. Journal of the Mechanics and Physics of Solids, In press, 10.1016/j.jmps.2026.106550. hal-05505991

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Submitted on 11 Feb 2026

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# Brittle Crack Initiation Seen as a Dynamic Crack Propagation Phenomenon

Aurélien Doitrand<fnref n="1" />, Rafael Estevez<fnref n="2" />, and François Bonnel<fnref n="3" />

While dynamic propagation of long cracks is well established, the early stage of brittle crack initiation remains to be clarified. We address the identification of the crack velocity profile during initiation under quasi-static loading conditions within the coupled criterion framework. Based on stress and energy conditions, the load level and crack length at initiation are obtained. An additional dynamic crack propagation condition is proposed to ensure the crack length and velocity continuity between the end of the initiation stage and the beginning of subsequent crack propagation. This defines a minimum admissible initiation crack length, only attained for quasi-static crack initiation followed by stable crack growth. If inertial effects are not negligible, the fulfillment of the energy equilibrium during initiation constrains the crack to reach a minimum velocity at the end of initiation and a larger initiation crack length. The proposed approach thus makes it possible to estimate the validity of the quasi-static application of the coupled criterion. It enables determining the crack velocity profile during initiation, which is found to be similar to the crack velocity profiles measured experimentally in unnotched and notched specimens.

**Keywords** Brittle crack initiation, Coupled criterion, Dynamic crack propagation

## 1 Introduction

Dynamic crack propagation under quasi-static loading can be addressed using Linear Elastic Fracture Mechanics (LEFM), from the balance of the dynamic energy release rate (ERR) and the material critical ERR. This extends Griffith's approach (Griffith 1921; Griffith 1924) to consider the crack propagation velocity and related inertial effects. The energy balance provides the variations of the crack velocity and length as a function of time (Fineberg et al. 1992; Goldman et al. 2010; Bouchbinder et al. 2014). They were for instance obtained based on an analytical derivation of the ERR for a finite crack in an infinite medium with forces applied to the crack faces (Freund 1990) or a semi-infinite crack propagating in an infinitely long strip (Marder 1991). The experimental characterization of dynamic crack propagation under quasi-static loading may be performed by first loading a specimen up to a given level before introducing a crack in the pre-loaded specimen (Fond and Schirrer 1997; Yeoh 2001; Livne et al. 2005; Corre et al. 2020; Fournier et al. 2025). These kinds of experiments enable the identification of the material critical ERR based on the knowledge of the ERR together with crack velocity measurements (Broberg 1960; Fineberg and Marder 1999; Goldman et al. 2010; Kopp et al. 2014; Henry et al. 2025). Such a configuration could not be attained with a monotonic loading of a pre-cracked specimen, so that the crack can be considered as "overloaded". The crack thus further propagates dynamically to maintain the energy equilibrium, and converts a part of the available energy into kinetic energy.

LEFM is however not applicable to study crack initiation as its main underlying assumption is the existence of an initial crack, whose propagation is studied. Brittle crack initiation can actually be studied using the coupled criterion (CC) (Leguillon 2001; Leguillon 2002), which combines stress and energy criteria and succeeds in solving fracture problems for which LEFM is limited, such as fracture size effects (Cornetti et al. 2006; Weißgraeber et al. 2016; Doitrand et al. 2020; Molnár et al. 2020; Aranda et al. 2023), crack initiation in unnotched specimens without initial defects (Parvizi et al. 1978; Leguillon 2002; Doitrand et al. 2021; Girard et al. 2024), facet nucleation under antiplane shear (Doitrand and Leguillon 2018; Doitrand et al. 2023; Molnár et al.

<footnote n="1">¹ Univ Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CNRS UMR5510, MATEIS, F-69621 Villeurbanne, France, aurelien.doitrand@insa-lyon.fr</footnote>

<footnote n="2">² Univ. Grenoble Alpes, Grenoble INP, CNRS UMR5266, SIMAP, F-38000 Grenoble, France</footnote>

<footnote n="3">³ Univ. Grenoble Alpes, Grenoble INP, CMTC, F-38000 Grenoble, France</footnote>
2024), or at singular points such as V-notches (Yosibash et al. 2006; Leguillon et al. 2007; Cornetti et al. 2016; Leguillon and Yosibash 2017; Doitrand and Molnár 2025). Accounting for dynamic effects within a FFM description has received attention in the recent years. Chao Correas et al. (2023) proposed to introduce a characteristic time related to the kinetics of the failure process in the CC implementation so as to study the failure load dependency on the loading rate under dynamic loading regimes. They successfully applied this formulation to study crack initiation in rock specimens under dynamic loading, though this approach does not explicitly consider the kinetic energy in the energy balance. There are only a few configurations for which the quasi-static energy equilibrium is strictly satisfied at initiation (Dormieux et al. 2025). This may happen when the ERR is a non-monotonic function of the crack length (Weißgraeb er et al. 2016; Doitrand et al. 2024; Dormieux et al. 2025) or in the specific case of strong singularities, for which the ERR is a decreasing function of the crack length (Leguillon et al. 2000; Leguillon and Murer 2012; Aranda and Leguillon 2023). In other configurations, it is thus essential to consider a dynamic description that accounts for the kinetic energy variation due to crack initiation (Doitrand et al. 2022). A key ingredient that is required to establish such a description is the crack velocity profile during initiation which has a first order influence on the initiation remote stress predicted by the CC (Doitrand et al. 2022; Chen et al. 2023; Chen et al. 2024). So far, no method have been proposed to determine the velocity profile during crack initiation.

Even when considering the dynamic approach of the CC, the initiation configuration may correspond to a case where the crack is "overloaded", which means that unstable crack propagation occurs immediately after initiation until a larger crack length is reached (Weißgraeb er et al. 2016; Doitrand et al. 2019). If the crack stops propagating, the arrest crack length can be measured experimentally (Doitrand et al. 2019; Bidaud et al. 2025). The cracking configuration at the end of crack initiation is a running crack whose velocity is commonly measured in dynamic fracture. The propagation stage occurring immediately after crack initiation can therefore be studied using LEFM. The end of the crack initiation stage coincides with the beginning of further crack propagation. As a consequence, there is a crack length and velocity continuity condition between these two stages, which is addressed in the present paper.

This work therefore aims to determine the velocity profile during crack initiation by combining (i) the study of crack initiation based on the CC and (ii) further crack propagation based on LEFM. The methodology to ensure the crack length and velocity continuity between the initiation stage and further propagation, in the CC solution is presented in Section 2 and illustrated by two case studies. In Section 3, we investigate crack initiation in unnotched specimens subjected to uniform tension. Crack propagation in plates with an initial central sharp notch is then evaluated in Section 4.

## 2 Brittle crack initiation and further crack propagation

We aim to determine the crack velocity profile during brittle crack initiation under quasi-static loading by combining the dynamic approach of the coupled criterion i.e., considering inertial effects associated with crack initiation (Doitrand et al. 2022; Chen et al. 2023; Chen et al. 2024), and the analysis of further dynamic crack propagation immediately after initiation. The proposed approach will be illustrated through two examples: i) the initiation of a crack in a 2D infinite plate under tension (Figures 1(a) and 1(b), Section 3) and ii) the propagation of a central crack in a 2D infinite plate under tension (Figure 1(c) and section 4). The proposed approach aims to determine:

- The remote stress at initiation $\sigma_{\infty}$,

- The initiation crack length $l_i$,

- The crack length variation ($l(t)$) and related velocity profile ($v(t) = d\ell/dt$) during initiation. We assume that crack initiation occurs during a time interval $\Delta t_c$ between the times $t = t_0$ and $t = t_c$ and that the crack velocity profile writes

$$v(t) = v_c \left( \frac{t - t_0}{\Delta t_c} \right)^{\alpha}, \qquad (1)$$

with $\alpha$ a positive exponent to be determined, $v_c$ the crack velocity at the end of initiation (i.e., for $t = t_c$) and $t_0$ the time at the onset of crack initiation. A power-law description was adopted
<figure><img src="image_2.png" /><figcaption>Figure 1: Configurations under investigation: (a) infinite plate under remote tensile stress ($\sigma_{\infty}$) in which (b) a crack of length $2\ell_i$ initiates. (c) Initiation of a crack increment ($2\ell_i$) in infinite plate with an initial central sharp notch (length $2\ell_0$) under remote tensile stress ($\sigma_{\infty}$).</figcaption></figure>

but another description for the crack velocity profile could be used, for instance considering a non-zero initial crack velocity. The corresponding initiation crack length profile is given by

$$ \ell(t) = \frac{v_c \Delta t_c}{\alpha + 1} \left( \frac{t - t_0}{\Delta t_c} \right)^{\alpha+1}. \qquad (2) $$

The initiation crack length is $\ell_i = \ell(t_c)$, so that

$$ \ell(t) = \ell_i \left( \frac{t - t_0}{\Delta t_c} \right)^{\alpha+1}, \qquad (3) $$

from which we derive the following relation:

$$ \Delta t_c = \frac{\alpha + 1}{v_c} \ell_i. \qquad (4) $$

The time $t_0$ can be arbitrarily chosen when considering quasi-static loadings. Since we are only considering quasi-static loading in this work, we will choose $t_0 = 0$ for the sake of simplicity. In the sequel, we recall the formulation of the CC (Section 2.1), present the dynamic crack propagation analysis (Section 2.2), and then detail how they are combined to determine the initiation crack velocity profile (Section 2.3).

## 2.1 Crack initiation

Classical LEFM focuses on the infinitesimal propagation of long pre-existing cracks by comparing the ERR with the material critical ERR. For stress concentrators or singularities other than a crack, the ERR of an infinitesimal crack extension may be zero, preventing the study of crack initiation (Leguillon 2002; Weißgraeb er et al. 2016; Doitrand et al. 2024). Finite Fracture Mechanics (FFM) overcomes this limitation by relaxing the assumption of infinitesimal crack growth and instead considering the formation of finite cracks (Hashin 1996; Nairn 2000). Within the FFM framework, the CC states that crack initiation occurs provided the simultaneous fulfillment of stress and energy conditions (Leguillon 2001; Leguillon 2002). The stress condition requires that before initiation, the material strength surface is attained along the entire crack path $\Gamma$. For instance, it corresponds to the segment of length $2\ell_i$ in the example depicted in Figure 1(b). For materials having a Rankine-like strength surface, the normal stress component to the crack path can be compared to the material tensile strength $\sigma_c$ as

$$ \sigma_{yy}(\mathbf{x}, \sigma_{\infty}) \geq \sigma_c \quad \forall \mathbf{x} \in \Gamma. \qquad (5) $$

The energy condition of the CC derives from the energy conservation principle between the states before and after crack initiation

$$ \Delta W_k + \Delta W_{el} + G_c \ell = \Delta W_{ext}, \qquad (6) $$
where $G_c \ell$ is the crack surface creation energy with $\ell$ the crack length corresponding to $\Gamma$ and $G_c$ the material critical ERR, $\Delta W_{ext}$ denotes the variation of external force work, $\Delta W_k$ the variation of the kinetic energy, $\Delta W_{el}$ the variation of the elastic strain energy. These quantities are expressed per unit thickness since we are considering 2D problems. The energy condition of the CC defines the incremental energy release rate (IERR) as

$$G_{\text{inc}}^{\text{DYN}}(\sigma_{\infty}, \ell, v) = \frac{\Delta W_{\text{ext}}(\sigma_{\infty}, \ell, v) - \Delta W_{\text{el}}(\sigma_{\infty}, \ell, v) - \Delta W_{k}(\sigma_{\infty}, \ell, v)}{\ell} \geq G_c. \quad (7)$$

The IERR is related to the ERR so that

$$G_{\text{inc}}^{\text{DYN}}(\sigma_{\infty}, \ell, v) = \frac{1}{\ell} \int_{0}^{\ell} G^{\text{DYN}}(\sigma_{\infty}, \ell, v) d\ell. \quad (8)$$

In the sequel, we assume isotropic linear elastic material behavior and that $G_c$ is a constant material parameter, although a crack velocity-dependent critical ERR could also be considered (Fournier et al. 2025). In the case of a quasi-static analysis of crack initiation, $G_c$ includes both the kinetic energy variation and the crack-velocity dependence, since it reverts to putting the term $(\Delta W_k/\ell)$ to the right-hand side of Equation (7). As a consequence, $G_c$ depends not only on the material but also on the studied geometry (which possibly affects $\Delta W_k$) and may thus not be considered solely as a material parameter, similarly to what is observed when considering crack initiation in presence of plasticity (Doitrand and Leguillon 2025). When considering the kinetic energy variation in the energy balance (Zimmermann et al. 1984; Ivankovic and Williams 1993; Fond and Schirrer 2001; Scheibert et al. 2010), the dependence of $G_c$ to the crack velocity represents microstructural phenomena, which can further be characterized experimentally to explain the fracture toughness dependence on the crack velocity (Heinzmann et al. 2023; Fournier et al. 2025). If the critical ERR variation as a function of the crack velocity is known, there is no methodological limitation to consider it in the present approach. For a given crack velocity profile $v(t)$, solving the CC consists in determining the minimum remote stress $\sigma_{\infty}$ as well as the crack length $\ell_i$ for which both the stress and the energy conditions are fulfilled.

## 2.2 Rapid crack propagation

Dynamic crack propagation at a given velocity converts part of the available energy into kinetic energy, so that the larger the crack velocity, the smaller the dynamic ERR $G_c^{DYN}$. For instance, the dynamic ERR of a crack of length $\ell$ propagating at a velocity $v$ in an infinite medium subjected to remote tension $\sigma_{\infty}$ can be expressed as a function of the quasi-static ERR $G_c^{QS}$ and a function $g(v)$ as (Broberg 1960; Freund 1990; Fineberg and Marder 1999; Goldman et al. 2010; Kopp and Girardot 2020)

$$G_c^{\text{DYN}}(\sigma_{\infty}, \ell, v) = g(v)G_c^{\text{QS}}(\sigma_{\infty}, \ell) \approx (1 - v/v_{\text{lim}})G_c^{\text{QS}}(\sigma_{\infty}, \ell), \quad (9)$$

where $v_{lim}$ is the maximum crack velocity which may be equal to the Rayleigh wave velocity $c_R$ (Broberg 1960; Freund 1990), which is observed experimentally, for instance, in polyacrylamide gels (Livne et al. 2005; Goldman et al. 2010). Experimentally, this measured maximum crack velocity can be smaller than $c_R$ (around $0.6c_R$ in PMMA (Fineberg and Marder 1999) or $0.9c_R$ in tungsten (Hull and Beardmore 1966) and MgO (Field 1971)) or even exceed it in presence of geometric nonlinearities (Svetlizky et al. 2020; Pundir et al. 2024; Wang et al. 2025; Tian and Gong 2025). Note that the function relating $G_c^{DYN}$ and $G_c^{QS}$ may differ from a linear decrease as a function of the normalized crack velocity for other geometries (Obrezanova et al. 2002; Slepyan 2010; Kopp et al. 2014) and can be identified based on numerical simulations (Croquelois et al. 2021; Doitrand et al. 2022; Henry et al. 2025; Fournier et al. 2025). This function may also depend on the crack acceleration $\dot{v} = dv/dt$, for instance in the case of strip band specimens (Marder 1998; Bouchbinder et al. 2014).

The dynamic crack propagation condition

$$G_c^{\text{DYN}}(\sigma_{\infty}, \ell, v) = G_c \quad (10)$$

provides a differential equation to be solved to determine the propagation velocity profile of a crack. For instance, the ERR for a crack of length $2\ell$ in infinite medium under quasi-static remote
tension under plane strain conditions is given by $G^{QS}(\sigma_{\infty}, 2\ell) = \pi \frac{1-\nu^2}{E} \sigma_{\infty}^2 \ell$, which yields from Equations (9) and (10):

$$ \pi \frac{1 - \nu^2}{E} \sigma_{\infty}^2 \ell(t) (1 - v(t)/v_{\text{lim}}) = G_c. \qquad (11) $$

Under a given remote stress $\sigma_{\infty}$, the crack velocity increases monotonically up to the maximum velocity $v_{\text{lim}}$ as the crack length increases in order to maintain the energy balance condition given in Equation (11). The dynamic crack propagation condition thus yields a relation between the applied remote stress, the crack length and the crack velocity.

## 2.3 Continuity between crack initiation and further propagation

The central aspect of the proposed approach consists in ensuring the continuity of the crack length and velocity between the end of initiation and the beginning of further propagation. This condition ensures the energy balance at all times during initiation and subsequent crack propagation. In other words, crack initiation requires three conditions to be fulfilled. In addition to the stress (Equation (5)) and energy (Equation (7)) conditions of the CC, the third condition is related to the dynamic crack propagation after initiation (Equation (10)). This condition is thus applied to the crack configuration at the end of initiation: remote stress $\sigma_{\infty}$, crack length $\ell_i$ and crack velocity $v_c$. We thus recall the set of equations that must be solved in the proposed approach:

$$ \left\{ \begin{array}{l} \sigma_{yy}(\mathbf{x}, \sigma_{\infty}) \geq \sigma_c \quad \forall \mathbf{x} \in \Gamma. \\ G_{\text{inc}}^{\text{DYN}}(\sigma_{\infty}, \ell, v) \geq G_c. \\ G_{\text{inc}}^{\text{DYN}}(\sigma_{\infty}, \ell, v) = G_c \end{array} \right. \qquad (12) $$

With the objective of determining the initiation crack velocity profile and the remote stress, the following approach can be adopted:

- the stress condition (Equation (5) and Equation (12)-first line) provides, for a given initiation crack length $\ell_i$, the minimum imposed stress at initiation $\sigma_{\infty}$;

- then, the propagation condition (Equation (10) and Equation (12)-third line) provides, for given $\ell_i$ and $\sigma_{\infty}$, the crack velocity at the end of initiation $v_c$;

- finally, the energy condition (Equation (7) and Equation (12)-second line) applied for given $\ell_i$, $\sigma_{\infty}$ and $v_c$, enables to determine the existence of an admissible initiation crack velocity profiles $v(t)$ and the corresponding exponent $\alpha$.

Among the admissible velocity profiles, the initiation crack velocity profile is determined as the one that corresponds to a minimum remote stress.

## 3 Crack initiation in an infinite plate under remote tension

The proposed approach is illustrated to study crack initiation in an infinite plate under quasi-static remote tensile stress $\sigma_{\infty}$ (Figure 1(a)). We thus study the initiation of a crack of length $2\ell_i$ in a 2D linear elastic infinite medium (Young's modulus $E$, Poisson's ratio $\nu$), subjected to a remote tensile stress $\sigma_{\infty}$ (Figure 1(b)). The objective is to determine the remote tensile stress at initiation as well as the crack velocity profile during initiation. Under quasi-static loading conditions, we recall that the energy release rate (ERR) of a crack of length $2\ell$ is given by

$$ G^{\text{QS}}(\sigma_{\infty}, 2\ell) = \pi \frac{1 - \nu^2}{E} \sigma_{\infty}^2 \ell. \qquad (13) $$

Following dynamic crack propagation in an infinite medium (Broberg 1960; Freund 1990), the dynamic ERR is given as a function of the crack velocity and its maximum value $v_{\text{lim}}$ as

$$ G^{\text{DYN}}(\sigma_{\infty}, 2\ell, v) \approx \left(1 - \frac{v(t)}{v_{\text{lim}}}\right) G^{\text{QS}}(\sigma_{\infty}, 2\ell) = \pi \frac{1 - \nu^2}{E} \sigma_{\infty}^2 \ell \left(1 - \frac{v(t)}{v_{\text{lim}}}\right). \qquad (14) $$
In the following, we focus on the initiation of a crack of length $2\ell_i$ to determine the crack velocity profile $v(t)$ during initiation, i.e., for $t_0 < t < t_c$ where $\ell(t_c) = \ell_i$. It thus requires to ensure the fulfillment of (i) the stress criterion, (ii) the energy criterion and (iii) the continuity in the velocity profile between the end of crack initiation and the beginning of further crack propagation.

## 3.1 Stress criterion

Before initiation, the tensile stress in the plate is constant and equal to the remote stress $\sigma_\infty$. Whatever the initiation crack length, the remote stress at crack initiation $\sigma_\infty$ must be at least equal to $\sigma_c$ to fulfill the stress criterion, which provides the relation between the initiation crack length and the material tensile strength

$$ \sigma_\infty(\ell_i) \geq \sigma_c. \qquad (15) $$

This condition defines, for a given initiation crack length, the minimum remote stress for crack initiation.

## 3.2 Crack propagation condition

The next step consists in determining the crack velocity at the end of initiation, i.e., $v(t = t_c) = v_c$. At $t = t_c$, crack propagation subsequent to initiation thus starts from a length $2\ell_i$, with a crack tip velocity $v_c = (\alpha + 1)\ell_i/\Delta t_c$, assuming the continuity of the crack velocity between the end of crack initiation and the beginning of further crack propagation. We can thus write the propagation condition for a crack of length $2\ell_i$ and velocity $v_c$ based on LEFM, $G^{\text{DYN}}(\sigma_\infty, 2\ell_i, v_c) = G_c$, which yields from Equation (14)

$$ \pi \frac{1 - v^2}{E} \sigma_{\infty}^2 \ell_i \left( 1 - \frac{v_c}{v_{\text{lim}}} \right) = G_c. \qquad (16) $$

Since the stress criterion is fulfilled, $\sigma_\infty \geq \sigma_c$, the minimum crack velocity at the end of initiation can therefore be expressed as a function of the Irwin's length $\ell_{\text{mat}} = \frac{E G_c}{(1 - v^2) \sigma_c^2}$ as

$$ \frac{v_c}{v_{\text{lim}}} \geq 1 - \frac{\ell_{\text{mat}}}{\pi \ell_i}. \qquad (17) $$

Since the crack velocity at the end of initiation must be positive, this condition also provides a lower bound for the initiation crack length, given by $\ell_i \geq \ell_{\text{mat}}/\pi$, which corresponds to quasi-static crack initiation with no kinetic energy variation during the initiation phase.

## 3.3 Energy criterion

We now use the energy condition of the CC to determine, among admissible initiation crack lengths given by the crack propagation condition, those for which the energy balance corresponding to crack initiation is fulfilled. We follow the crack velocity profile and corresponding crack length profile given, respectively, in Equation (1) and Equation (2). The energy condition of the CC derives from an energy conservation principle between the states before and after initiation, it writes $G_{\text{inc}}^{\text{DYN}}(\sigma_\infty, 2\ell_i, v) = G_c$ where

$$ G_{\text{inc}}^{\text{DYN}}(\sigma_{\infty}, 2\ell_i, v) = \frac{1}{\ell_i} \int_{0}^{\ell_i} G^{\text{DYN}}(\sigma_{\infty}, 2\ell, v) d\ell = \frac{1}{\ell_i} \int_{0}^{\ell_i} \pi \frac{1 - v^2}{E} \sigma_{\infty}^2 \ell \left( 1 - \frac{v}{v_{\text{lim}}} \right) d\ell, \qquad (18) $$

which rewrites (with the change in variable $d\ell = vdt$)

$$ G_{\text{inc}}^{\text{DYN}}(\sigma_{\infty}, 2\ell_i, v) = \pi \sigma_{\infty}^2 \frac{1 - v^2}{E \ell_i} \int_{\ell_0}^{t_c} \ell(t)v(t) \left(1 - \frac{v(t)}{v_{\text{lim}}}\right) dt. \qquad (19) $$

By substituting $\ell$ and $v = d\ell/dt$ by their expressions given in Equations (1) and (2), and with the stress condition $\sigma_\infty = \sigma_c$, we obtain

$$ G_{\text{inc}}^{\text{DYN}}(\sigma_c, 2\ell_i, v) = \frac{\pi(1 - v^2)\sigma_c^2 \Delta t_c v_c (v_{\text{lim}}(3\alpha + 2) - 2v_c(\alpha + 1))}{E 2v_{\text{lim}}(\alpha + 1)(3\alpha + 2)}. \qquad (20) $$
The condition $G_{inc}^{DYN} = G_c$ provides the expression for the velocity profile exponent $\alpha$ as a function of the initiation crack length $\ell_i$, using Equation (4) and Equation (17)

$$ \alpha = \frac{2(1 - v_c/v_{lim})}{4v_c/v_{lim} - 3} = \frac{2}{\pi \ell_i / \ell_{mat} - 4}. \qquad (21) $$

It results in a lower bound for the initiation crack length $\ell_i \ge \frac{4}{\pi}\ell_{mat}$ that is larger than the quasi-static initiation crack length ($\frac{1}{\pi}\ell_{mat}$) determined by the propagation condition in Section 3.2. The variation of the exponent $\alpha$ in the velocity profile with the normalized initiation crack length is presented in Figure 2(a). It decreases with increasing initiation crack length and tends towards infinity when the initiation crack length tends towards its lower bound defined by Equation (21). Figure 2(b) shows the variation of the normalized crack velocity at the end of initiation, which has

<figure><img src="image_3.png" /><figcaption>Figure 2: (a) Velocity profile exponent as a function of the normalized initiation crack length and b-c) Velocity at the end of crack initiation $v_c$ as a function of (b) the normalized initiation crack length and (c) the velocity profile exponent. The hatched zones in (a) and (b) corresponds to crack lengths for which crack initiation cannot occur because the energy balance during initiation is not fulfilled.</figcaption></figure>

an increasing trend as a function of the normalized initiation crack length (see Equation (17)). The hatched zone in Figure 2(b) corresponds to initiation crack lengths smaller than $\frac{4}{\pi}\ell_{mat}$ for which crack initiation cannot occur because the energy criterion is not fulfilled. The energy equilibrium fulfillment during initiation thus implies the definition of a minimum velocity $v_c \ge \frac{3}{4}v_{lim}$ that the crack has to reach at the end of initiation. The variation of the normalized crack velocity at the end of initiation as a function of the crack velocity profile exponent is shown in Figure 2(c). For a given initiation crack length $\ell_i$, the initiation time interval $\Delta t_c$ is obtained knowing the parameters $\alpha$ and $v_c$ (see Equation (4)). This fully defines the crack velocity profile. Among all the admissible crack velocity profiles, the initiation crack velocity profile is determined as the one that minimizes the remote stress at initiation. In the present example of crack initiation in an infinite plate under remote tension, all the admissible profiles correspond to a remote stress $\sigma_{\infty} = \sigma_c$ and may therefore be observed. Figure 3 shows an example of different initiation crack velocity (Figure 3(a)) and length (Figures 3(b) and 3(c)) profiles. The normalized crack velocity

<figure><img src="image_4.png" /><figcaption>Figure 3: (a) Normalized crack velocity variation during initiation as a function of normalized time and (b-c) crack length as a function of time obtained for several velocity profile exponents (the circular markers correspond to the initiation crack lengths), (b) absolute and (c) shifted time values (the time shift is made so that the curves coincide in the propagation part for $\ell = 1$ mm).</figcaption></figure>
as a function of normalized time during initiation is shown in Figure 3(a) for various profile exponents, which correspond to different initiation lengths, crack velocities and time intervals. The corresponding crack length variations as a function of time are presented in Figure 3(b) for material properties corresponding to extruded PMMA ($E = 2700$ MPa, $v = 0.39$, $\rho = 1190$ kg/m$^3$, $G_c = 92$ J/m$^2$ and $\sigma_c = 54$ MPa). The configuration at the end of crack initiation is shown by the circular marker, the initiation crack length profile up to this point corresponds to the integrated normalized velocity profile in Figure 3(a) and further crack length variation as a function of time is computed based on linear elastic fracture mechanics applied to dynamic crack propagation (see Section 2.2). After initiation, the crack is expected to accelerate up to its limiting velocity, following Equation (11). It can be observed that the exponent $\alpha$ has a limited influence on the predicted crack length profile, increasing $\alpha$ mainly shifts the velocity profile towards larger initiation times and slightly decreases the initiation crack length. Figure 3(c) shows the crack length profiles that were shifted so that the propagation parts at the maximum crack velocity coincide (i.e., for $l = 1$ mm). This visualization confirms that the velocity profile exponent only has limited influence on the initiation crack length profile.

## 3.4 Crack initiation in unnotched specimens

### 3.4.1 PMMA specimens

Tensile tests on three unnotched extruded polymethyl methacrylate (PMMA) specimens are performed using a 20 kN Zwick machine. The specimen dimensions are 100 ± 0.1 mm (length) × 15.4 ± 0.1 mm (width) × 8 ± 0.1 mm (thickness). The material elastic and fracture properties are given in Section 3.3. The prescribed displacement rate is 1 mm/min, which ensures a quasi-static loading state. The specimens are loaded monotonically up to brittle fracture, which is optically recorded by a Phantom-T4040 camera with 256 pixels × 64 pixels resolution at 444444 frames per second. The image recording is manually triggered and allows recording images over a 11-second interval spanning both before and after the trigger time. Then, the crack length is optically measured from the recorded images as shown in Figure 4. The crack length variation as

<figure><img src="image_5.png" /><figcaption>**Figure 4:** Crack length variation as a function of time measured experimentally on three unnotched PMMA specimens (error bars corresponding to minimum and maximum measured crack lengths for the three specimens) from optical observations (shown as inset for different crack lengths) and numerically (dashed line).</figcaption></figure>

a function of time obtained with the proposed approach is also shown for a maximum crack velocity $v_{lim} = 0.6c_R$, which is the maximum crack velocity measured in PMMA by several authors (Fineberg et al. 1992; Fineberg and Marder 1999; Fond and Schirrer 2001; Molnár et al. 2020). The proposed approach enables a correct overall description of the crack length variation as a function of time. Nevertheless, the model predicts that a 0.2 mm crack length initiates in around 1 µs (see Figure 3(b)). The pixel size of the images recorded by the camera is 0.13 mm, each image being recorded every 2.25 µs. The initiation configuration is therefore at the limit of the time and space measurement resolution. Such measurements could be improved by using a high-speed and high-resolution camera, which nevertheless requires a complex setup for the
camera triggering (Eid et al. 2023). Consequently, it is not straightforward to fully validate the predicted initiation crack velocity profile, contrary to further unstable crack propagation for which the model prediction aligns correctly with experiments. The beginning of the propagation stage after initiation (up to 5 mm / 10 µs) is well-represented by the proposed model. From crack lengths larger than 5 mm, there is a slight deviation between the model prediction and the measured profile. This difference could be explained by a finite size geometry, a non-perfectly planar crack or microstructural events such as micro-branching that are not considered in the model.

### 3.4.2 Paper specimens

Fracture analysis in paper focuses on fracture toughness characterization (Seth 1974; Uesaka et al. 1979) and process zone characterization (Yamauchi and Hirano 2000; H. and K 2000; Isaksson et al. 2012) in relation to microstructural features (Villette et al. 2023). From the modeling viewpoint, nonlocal damage models are relevant to capture the observed mechanisms during fracture tests (Isaksson et al. 2004; Isaksson et al. 2012), other methods such as phase-field (Marengo et al. 2024) and statistical/stochastic approaches (Villette et al. 2024; Fallah et al. 2024) are also employed. To the best of our knowledge, the CC was not yet applied for crack initiation prediction in paper. Tensile tests on seven 80 g/m² tecnopro laser paper specimens are performed using a 500 N Tilnius tensile test machine. The specimen dimensions are 80 ± 1 mm (length) × 30 ± 1 mm (width) × 109 ± 3 µm (thickness). All the specimens were tested in the machine direction, so that the material anisotropy is not considered in this work. The material elastic and fracture properties in the tested direction are *E* = 2000 MPa, *ν* = 0.3, *ρ* = 800 kg/m³, *G*<sub>c</sub> = 4.8 kJ/m² and *σ*<sub>c</sub> = 42.5 MPa). The prescribed displacement rate is 1 mm/min for all the tested specimens, avoiding fracture properties dependence to the loading rate (Baumann et al. 2024) and ensuring a quasi-static loading state. The specimens are loaded monotonically up to fracture, which is optically recorded by a Phantom-T4040 camera with 512 pixels × 320 pixels resolution at 92538 frames per second. The image recording is manually triggered and allows recording images over a time interval spanning both before and after the trigger time. Then, the crack length is optically measured from the recorded images as shown in Figure 5(a). The crack path is rather tortuous than straight,

<figure><img src="image_6.png" /><figcaption>**Figure 5:** (a) Observations of crack initiation and propagation in a paper unnotched specimen under tensile loading and (b) crack length variation as a function of time measured experimentally (error bars corresponding to minimum and maximum measured crack lengths for the seven specimens) and obtained with the proposed approach for $v_{\text{lim}} = c_R$ (solid line) or $v_{\text{lim}} = 0.6c_R$ (dashed line).</figcaption></figure>

which is due to local heterogeneity of the material and related fluctuations of the mechanical properties (Villette et al. 2023; Villette et al. 2024). Due to these local crack deviations, the actual total crack length is around 1.6 times the specimen width. In a first analysis of the experimental data, we assume that the crack extension corresponds to the maximum position along the horizontal axis. This neglects the crack meandering and provide an estimation of the extend of a straight crack propagating horizontally. This measure can then be compared to the predictions which are derived from a model with a straight crack. Thus, its extension varies between 0 and the specimen width.. If the tortuous crack can reach a maximum velocity, such as for instance
$c_R$, the maximum velocity of an equivalent straight crack is thus $c_R/1.6 \approx 0.6c_R$. Figure 5(b) shows the crack length variation measured experimentally and obtained with the proposed approach, using either $v_{lim} = c_R$ or $v_{lim} = 0.6c_R$. The predicted initiation crack length attains around 8 mm in 39 µs, which is slightly larger than the crack length measured on image #2 in Figure 5(a). The initiation crack velocity profiles predicted using either $v_{lim} = c_R$ or $v_{lim} = 0.6c_R$ seem to be compatible with experimental observations. However, for larger crack lengths, a $v_{lim} = 0.6c_R$ maximum crack velocity better describes the crack profile measured experimentally during the unstable crack propagation following initiation. Compared to previous dynamic CC implementations, for which the crack velocity profile was either disregarded (Chao Correas et al. 2023) or postulated (Doitrand et al. 2022; Chen et al. 2023; Chen et al. 2024), a main advantage of the proposed approach is the determination of the crack velocity profile during the initiation step.

# 4 Propagation of a central crack in infinite medium

We now illustrate the proposed approach to study the propagation of a central sharp notch of length $2\ell_0$ subjected to a remote tensile stress along the crack plane normal direction in infinite medium (Figure 1(c)). The objective is to determine the remote tensile stress $\sigma_\infty$ at which the initial notch propagates over a crack increment $2\ell_i$ as well as the crack velocity profile during this initiation step. In the sequel, we present the conditions for crack initiation over a crack increment $2\ell_i$ and subsequent crack propagation.

## 4.1 Stress criterion

Before initiation, the tensile stress in the plate is given by (Sun and Jin 2012; Westergaard 1939).

$$ \sigma_{yy}(x) = \frac{\sigma_{\infty}(x + \ell_0)}{\sqrt{x(x + 2\ell_0)}}, \qquad (22) $$

where the axis system origin is located at the right tip of the crack and the x-axis corresponds to the crack direction. The minimum remote stress $\sigma_\infty$ required to fulfill the stress criterion for a given crack increment $\ell_i$ thus writes

$$ \sigma_{\infty}(\ell_i) \geq \sigma_c \frac{\sqrt{\ell_i(\ell_i + 2\ell_0)}}{\ell_i + \ell_0}. \qquad (23) $$

## 4.2 Crack propagation condition

After the crack increment initiation, i.e., at $t = t_c$, crack propagation starts from a length $2(\ell_0 + \ell_i)$ with a crack tip velocity $v(t = t_c) = v_c = (\alpha + 1)\ell_i/\Delta t_c$ (see Equation (4)). The LEFM crack propagation condition for a crack of length $2(\ell_0 + \ell_i)$ is $G^{\text{DYN}}(\sigma_\infty, 2(\ell_0 + \ell_i), v_c) = G_c$, which yields, from Equation (14)

$$ \pi \frac{1 - v^2}{E} \sigma_{\infty}^2 (\ell_0 + \ell_i) \left( 1 - \frac{v_c}{v_{\text{lim}}} \right) = G_c. \qquad (24) $$

This expression provides the crack velocity at the end of initiation $v_c$ as a function of the crack increment length $\ell_i$ using Equations (23) and (24)

$$ \frac{v_c}{v_{\text{lim}}} = \frac{\ell_i^2 + \ell_i(2\ell_0 - \ell_{\text{mat}}/\pi) - \ell_0\ell_{\text{mat}}/\pi}{\ell_i(\ell_i + 2\ell_0)}. \qquad (25) $$

This condition also provides a lower bound for the initiation crack length corresponding to a quasi-static crack initiation (i.e., $v_c/v_{\text{lim}} = 0$) with no kinetic energy variation during the initiation

$$ \text{phase } \ell_i \geq \frac{\ell_{\text{mat}}}{2\pi} - \ell_0 + \sqrt{\ell_0^2 + \left(\frac{\ell_{\text{mat}}}{2\pi}\right)^2}. $$

## 4.3 Energy criterion

We now use the energy condition of the CC to determine, among admissible initiation crack lengths given by the crack propagation condition, the ones for which the energy balance corresponding to
crack initiation is fulfilled. The energy condition of the CC writes $G_{\text{inc}}^{\text{DYN}}(\sigma_{\infty}, 2\ell_i, v) = G_c$, where

$$G_{\text{inc}}^{\text{DYN}}(\sigma_{\infty}, 2\ell_i, v) = \frac{1}{\ell_i} \int_{0}^{\ell_i} G^{\text{DYN}} d\ell = \frac{1}{\ell_i} \int_{0}^{\ell_i} \pi \frac{1-v^2}{E} \sigma_{\infty}^2 (\ell_0 + \ell) \left(1 - \frac{v}{v_{\text{lim}}}\right) d\ell, \quad (26)$$

which rewrites

$$G_{\text{inc}}^{\text{DYN}}(\sigma_{\infty}, 2\ell_i, v) = \pi \frac{1-v^2}{E\ell_i} \sigma_{\infty}^2 \int_{\ell_0}^{t_c} (\ell_0 + \ell)v \left(1 - \frac{v}{v_{\text{lim}}}\right) dt. \quad (27)$$

Replacing $\ell$ and $v = d\ell/dt$ by their expressions given in Equations (1) and (2), we obtain

$$G_{\text{inc}}^{\text{DYN}}(\sigma_{\infty}, 2\ell_i, v) = \pi \frac{1-v^2}{E\ell_i} \sigma_{\infty}^2 \left( \frac{\ell_0 v_c t_c^{\alpha}}{\alpha+1} - \frac{\ell_0 v_c^2 t_c^{2\alpha}}{2\alpha+1} + \frac{\ell_i v_c t_c^{2\alpha+1}}{2\alpha+2} - \frac{\ell_i v_c^2 t_c^{3\alpha+1}}{3\alpha+2} \right). \quad (28)$$

equivalent to

$$G_{\text{inc}}^{\text{DYN}}(\sigma_{\infty}, 2\ell_i, v) = \pi \frac{1-v^2}{E} \sigma_{\infty}^2 \left( \ell_0 \left( 1 - \frac{v_c(\alpha+1)}{v_{\text{lim}}(2\alpha+1)} \right) + \frac{\ell_i}{2} \left( 1 - \frac{2v_c(\alpha+1)}{v_{\text{lim}}(3\alpha+2)} \right) \right). \quad (29)$$

There is no analytical expression for $\alpha$ that can be obtained from the energy condition of the CC $G_{\text{inc}}^{\text{DYN}}(\sigma_{\infty}, 2\ell_i, v) = G_c$, which has to be solved numerically, for instance using Newton's method.

An example of solution is provided for an initial notch length $2\ell_0 = 1.0$ mm in Figure 6. Figure 6(a) shows the variation of the velocity profile exponent as a function of the normalized

<figure><img src="image_7.png" /><figcaption>Figure 6: (a) Crack velocity profile exponent and (b) normalized crack velocity at the end of initiation variations as a function of the normalized initiation crack length. The hatched zones correspond to crack length for which initiation cannot occur since the energy condition is not fulfilled. (c) Normalized crack velocity at the end of initiation and (d) normalized initiation remote stress variations as a function of the velocity profile exponent. Results are obtained for a $2\ell_0 = 1.0$ mm initial notch length.</figcaption></figure>

initiation crack length. It is a decreasing function and tends towards infinity when the initiation crack length tends towards a lower bound around $0.21\ell_{\text{mat}}$ (corresponding to $2\ell_0 = 1.0$ mm initial
notch length, the influence of the initial notch length is further investigated in Figure 7). Since the crack velocity at the end of initiation increases with increasing initiation crack length (see Figure 6(b)), it defines a lower bound for the crack velocity at the end of initiation, which has to be attained so as to enable crack initiation. Figure 6(c) shows that the larger the velocity profile exponent, the smaller the crack velocity at the end of initiation. As a consequence, less available energy is converted into kinetic energy and the corresponding initiation remote stress slightly decreases with increasing $\alpha$ (see Figure 6(d)). The most favorable velocity profile is thus obtained when $\alpha \to \infty$. Since the crack length profile only weakly depends on $\alpha$ (see Figure 3(c)), in practice, a finite value of the velocity exponent can be chosen. For instance, $\alpha = 10$ yields differences in the initiation crack velocity and remote stress smaller than 1 % compared to larger $\alpha$ values.

Similarly to the case shown in Section 3.3, this example also evidences that crack initiation must occur over a minimum finite length and by reaching a minimum crack velocity. We now investigate the fracture size effect, i.e., how the initial notch length $2\ell_0$ influences these quantities as well as the magnitude of the initiation remote stress and compare the results obtained with the present approach to the quasi-static CC application (Weißgraeber et al. 2016; Molnár et al. 2020). Figure 7(a) shows the normalized initiation crack length ($\ell_i/\ell_{mat}$) as a function of the normalized initial notch length ($\ell_0/\ell_{mat}$) obtained using either the proposed approach (referred to as "Dynamic" in Figure 7(a)) or the quasi-static approach of the CC (disregarding the dynamic crack propagation during initiation). For both analyses, the larger the initial notch length, the

<figure><img src="image_8.png" /><figcaption>**Figure 7:** (a) Normalized initiation crack length obtained using the proposed approach (Dynamic) and the quasi-static CC application (Quasi-static), (b) ratio of the normalized initiation crack lengths obtained with both approaches, (c) normalized crack velocity at the end of initiation and (d) normalized initiation remote stress variations as a function of the normalized initial notch length.</figcaption></figure>

smaller the crack increment. However, the initiation crack increment obtained using the present approach is larger than the one obtained using the quasi-static CC solution. This result was also observed in a previous implementation of the dynamic CC solution with a fixed crack velocity profile (Doitrand et al. 2022; Chen et al. 2023). The variation of the ratio between the initiation crack lengths obtained with both approaches (shown in Figure 7(b)) tends towards 4 for small
initial notch lengths and decreases up to 1 for increasing initial notch lengths. The normalized
crack velocity at the end of initiation also decreases as a function of the normalized notch crack
length, as shown in Figure 7(c). It means that for sufficiently large initial notch length, the
proposed approach reverts to the quasi-static application of the CC, since the initiation crack
velocity is small and that inertial effects can be neglected. However, this is not the case for
smaller initial notch lengths for which a larger initiation crack velocity is obtained, evidencing a
more pronounced influence of inertial effects. A main outcome of the present approach is thus
the ability to estimate whether inertial effects have a noticeable influence or not and evaluate the
validity of the quasi-static CC application. Figure 7(d) shows the normalized initiation remote
stress variation as a function of the initial notch length, which confirms the results previously
discussed for sufficiently large crack lengths. It is also notable that the proposed approach
coincides with LEFM when the initial notch length is sufficiently large compared to the Irwin's
length. For sufficiently small initial notches, fracture occurs for a remote stress equal to the
material tensile strength, which is obtained for both the present approach and the quasi-static CC
application. Finally, the difference in the initiation remote stress between both approaches is the
largest in the transition region $0.1l_{\text{mat}} \le l_0 \le l_{\text{mat}}$.

## 4.4 Central crack in paper specimens

A similar experimental setup as described in 3.4.2 is used to characterize crack initiation and propagation in 80 g/m² Tecnopro laser paper notched specimens under tensile loading (1 mm/min displacement rate). The specimen dimensions are 80 ± 1 mm (length) × 60 ± 1 mm (width) × 109 ± 3 µm (thickness). Crack initiation and propagation are optically recorded by a Phantom-T4040 camera with 512 pixels × 64 pixels resolution at 440000 frames per second. Then, the crack length is optically measured from the recorded images as illustrated for a specimen with a 1 mm initial notch in Figure 8(a). The crack path is slightly tortuous due to local heterogeneities of the

<figure><img src="image_9.png" /><figcaption>**Figure 8:** (a) Observations of crack initiation and propagation in a paper specimen with a 1 mm initial notch under tensile loading and (b) crack length variation as a function of time measured experimentally (error bars corresponding to measurements from three different specimens) and obtained with the proposed approach for $v_{\text{lim}} = c_R$ (solid line) or $v_{\text{lim}} = 0.9c_R$ (dashed line).</figcaption></figure>

material. Due to these local crack deviations, the actual total crack length is approximately
1.15 times the specimen width. Assuming that the tortuous crack can at most reach a velocity
equal to $c_R$, the maximum velocity of the equivalent straight crack is thus $c_R/1.15 \approx 0.9c_R$.
Figure 8(b) shows the crack length variation measured experimentally and obtained with the
proposed approach, using either $v_{\text{lim}} = c_R$ or $v_{\text{lim}} = 0.9c_R$. The predicted initiation crack length
attains 6.5 mm in around 30 µs, which can be compared to crack length measurements on several
images due to the high frame rate recording. The initiation crack velocity profiles predicted using
either $v_{\text{lim}} = c_R$ or $v_{\text{lim}} = 0.9c_R$ are consistent with the crack lengths measured experimentally,
and a $v_{\text{lim}} = 0.9c_R$ maximum crack velocity slightly better describes the crack profile measured
experimentally for larger crack lengths.

We now compare the results obtained experimentally and numerically in relation to the
influence of the initial notch length in Figure 9. Figure 9(a) shows the variation of the normalized
<figure><img src="image_10.png" /><figcaption>**Figure 9:** (a) Normalized remote stress at initiation and (b) initiation crack velocity as a function of the normalized initial notch length obtained using the proposed approach (Dynamic CC, plain line) and experimentally (symbols).</figcaption></figure>

remote failure stress as a function of the normalized initial notch length. The transition between a regime driven by LEFM for large notch cracks and a strength plateau for small initial notches (Weißgraeber et al. 2016; Molnár et al. 2020) is well retrieved experimentally. Interestingly, for one of the tested specimens having a 0.5 mm initial notch length, cracking did not occur from the initial notch, but at another location within the bulk. This is a good experimental illustration of the fact that the material becomes insensitive to extrinsic initial defects smaller than $\ell_{mat}/10$, which was also observed in (Leguillon et al. 2007; Martin et al. 2018). The proposed approach slightly overestimates the remote stress at failure measured experimentally in the transition zone, which could be attributed to the fact that the process zone prior to cracking is not considered in the present approach (Molnár et al. 2020; Doitrand and Molnár 2025). Actually, including the process zone description is expected to decrease the remote stress at failure in the transition zone, as already observed when studying the quasi-static fracture size effect (Molnár et al. 2020). The variation of the initiation crack velocity as a function of the normalized initial notch length is shown in Figure 9(b). Similarly to the model predictions, the experimental measurements show a decreasing crack velocity when increasing the size of the initial notch. The model thus qualitatively captures the crack velocity variation as a function of the initial notch length, but for a range of initial notch lengths that is shifted with respect to the ones measured experimentally. This difference could partly be explained by the absence of a process zone in the proposed approach. A perspective of the present work would be to study the influence of the process zone on dynamic crack initiation (Doitrand and Molnár 2025). Paper finally appears as a good model material for fracture mechanics considerations as it follows LEFM for long initial cracks and the transition from energy-driven to strength-driven crack initiation (see Figure 9(a)). Otherwise, it shows a quasi-brittle fracture behavior, a main interest for crack velocity measurements being that the characteristic length is sufficiently large so that the spatial resolution allows estimating the velocity profile during initiation.

# 5 Conclusion

Ensuring crack length and velocity continuity conditions between the end of crack initiation and the beginning of subsequent crack propagation defines that the larger the initiation crack length, the larger the crack velocity at the end of initiation. In particular, it yields a minimum admissible initiation crack length that actually corresponds to the one obtained under quasi-static conditions (i.e., for a sufficiently small crack velocity).

This is possible for a crack to initiate over this minimum admissible length only if further stable crack propagation occurs. This is achieved, e.g., for a sufficiently long crack, thus retrieving Griffith's configuration. It could also be the case for configurations where the IERR is non-monotonic (Weißgraeber et al. 2016; Doitrand et al. 2024; Dormieux et al. 2025) or a decreasing function of the crack length, such as for strong singularities (Leguillon et al. 2000; Leguillon and
Murer 2012; Aranda and Leguillon 2023).

Otherwise, the fulfillment of the energy equilibrium during initiation constrains the crack to reach a minimum velocity at the end of initiation. This minimum crack velocity depends on the geometry. It is $0.75c_R$ for an infinite plate without initial crack, and decreases to 0 (stable crack propagation) as the initial notch length increases to retrieve Griffith's configuration. Depending on the initiation crack velocity, the initiation crack length may be larger than the minimum length defined by stable crack initiation, up to four times for an infinite plate without initial crack.

The proposed approach enables determining a range of admissible exponents for the power law describing the initiation crack velocity profile rather than a single value. However, the exponent value has a limited influence on the predicted crack length variation as a function of time. A main outcome of the proposed approach is the ability to estimate whether inertial effects have to be accounted for or not in the analysis of the crack initiation stage. In other words, it allows to evaluate the validity of the quasi-static CC application. It also provides the initiation crack velocity profile, which is found to be compatible with experimental observations of crack initiation in PMMA and paper unnotched and notched specimens.

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