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# Asymptotic approaches for dealing with distorted crack geometries

Véronique Lazarus

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Véronique Lazarus. Asymptotic approaches for dealing with distorted crack geometries. 2025. hal-05083728v2

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# Asymptotic approaches for dealing with distorted crack geometries

© Véronique Lazarus¹

¹ IMSIA, ENSTA, CNRS, EDF, Institut Polytechnique de Paris, Palaiseau, 91120, France

A growing, albeit not predominant, way to contribute to the climate transition in the field of fracture mechanics is to refine predictions from damage tolerance approaches used to assess the durability and reliability of sensitive components such as those found in railway, aeronautics, aerospace or nuclear industries. If it can be certified that the reduced safety margins remain acceptable, the intervals between maintenance operations could be extended, and the replacement of defective parts delayed, thereby reducing the environmental footprint.

The aim of Damage Tolerance Approaches is to ensure that the existence of unavoidable defects does not compromise safety. This involves considering the most unfavorable case of brittle fracture and determining the propagation of a preexisting crack under cyclic loading until the Griffith energy fracture threshold is reached. Currently, this is done using simplified, smoothed-out crack geometries, relying either on tabulated values or Finite Element Methods. Since meshing of the entire structure is required, the latter currently struggles to accurately account for the small-scale tortuosity of the crack geometry.

This paper aims to show that asymptotic approaches are efficient alternatives to address this challenge. Various aspects of these approaches, along with selected applications, will be reviewed. In addition to supporting the reduction of safety margins, these approaches also help to ensure that small geometrical perturbations do not lead to unexpected catastrophic failure. Asymptotic methods aim to provide analytical formulas for the variation of stress intensity factors caused by small-scale crack shape perturbations. Several applications are presented including crack shape evolution, the influence of heterogeneities, and propagation in general mixed-mode I+II+III conditions.

Keywords Linear Elastic Fracture Mechanics, Stress Intensity Factors, small geometric perturbations, crack propagation

## Contents

1 Introduction 2

2 Linear Elastic Fracture Mechanics 3

2.1 Definitions 3

2.2 Crack propagation laws 4

3 In-plane perturbations of planar cracks 5

3.1 First order formulas for small perturbations 5

3.2 Large perturbations 6

3.3 Applications 7

3.3.1 Shape stability problem 7

3.3.2 Equilibrium crack front shapes 7

3.3.3 Apparent toughness for heterogeneous materials 8

3.3.4 Fatigue propagation of distorted closed cracks 8

4 Out-of-plane perturbations 9

4.1 Theory 9

4.1.1 Wavy perturbations 9

4.1.2 Perturbations with angular points 10

4.2 Applications 11

4.2.1 Wavy crack propagation in a 2D mode I setting 11
<table><tr><td>4.2.2</td><td>Kinked crack path</td><td>11</td></tr><tr><td>4.2.3</td><td>Crack propagation in the presence of mode III</td><td>12</td></tr><tr><td>5</td><td>Conclusion</td><td>14</td></tr></table>

# 1 Introduction

Predicting how and when a crack propagates is a key issue in many domains such as earthquake sciences, heritage preservation (e.g. paintings and buildings), civil engineering structures, and transportation systems. These fields involve components subjected to repetitive loads, where failure must be avoided at all costs to ensure human safety, for example, railway tracks, aircraft turbine discs, and nuclear reactor components. Critical components are closely monitored, with particular attention paid to their condition and expected time to failure. These diagnostics are crucial for defining maintenance strategies, including how often to conduct inspections, perform repairs, or replace components.

Damage tolerance approaches are key to estimating component lifetimes. Their primary objective is to ensure that catastrophic component failure never occurs. To achieve this, the worst-case scenario is assumed; any defect is treated as a sharp crack that introduces a singularity in the stress field, and brittle fracture is generally supposed. The number of loading cycles leading to failure is then estimated using the Linear Elastic Fracture Mechanics (LEFM) framework. Finally, a safety margin is added to account for uncertainties, modeling inaccuracies, and to eliminate any residual risk.

In aeronautics, this approach has proven effective, as shown by the low number of component-related accidents in recent years. However, changes are now necessary to meet environmental impact reduction targets. New technologies such as the Rise open fan architecture or advanced materials, such as single crystals or additively manufactured components, are being explored to achieve significant fuel savings. Therefore, it is essential to ensure that these innovations are fracture resistant. In addition, to reduce component and equipment replacements and extend the intervals between maintenance operations, validating a longer service life through improved modeling is also under consideration. The development of materials with lower environmental impact will also play a key role in this transition. In this paper, I aim to demonstrate how asymptotic approaches can contribute to addressing these challenges.

At present, lifetime is mostly based on simplified, idealized crack geometries, using either tabulated values or Finite Element Methods. However, since FEM typically requires meshing the entire structure, it struggles to accurately capture the small-scale tortuosity of crack geometry. The objective of this paper is to demonstrate that asymptotic approaches offer efficient alternatives to address this limitation. By asymptotic approaches, we refer to methods that aim to describe the variation of certain quantities as a small parameter tends to zero (Sanchez-Hubert et al. 1992). In this paper, the focus is on small geometric perturbations of the crack geometry and the corresponding variations of the Stress Intensity Factors. Only continuous perturbations will be considered, although asymptotic methods can also handle discontinuous phenomena such as the nucleation of new disconnected cracks or defects (Leguillon 2002; Doitrand et al. 2023; Doitrand et al. 2024; Jiménez-Alfaro et al. 2025). In this context, we review different aspects of the asymptotic approach and present several illustrative applications.

The outline of this work is as follows. The paper begins by recalling the definitions of Stress Intensity Factors (SIFs) and the Energy Release Rate (ERR), highlighting their central role in Linear Elastic Fracture Mechanics (section 2). It then describes the computation of these quantities for small and large in-plane perturbations of a flat crack (section 3), followed by an analysis of smooth and angular out-of-plane perturbations (section 4). In each section, the theoretical framework is presented, followed by selected applications that illustrate how these approaches improve structural safety while supporting environmental sustainability. In particular, the methods are shown to improve the treatment of shape distortions, material heterogeneities, and fracture anisotropy, as well as address complex mixed-mode loading, including mode III.
## 2 Linear Elastic Fracture Mechanics

The aim of this section is to recall the main concepts of Linear Elastic Fracture Mechanics. A more detailed version, using similar notation, is available in Chapter 5 of the open-access fracture mechanics book by Besson et al. (2023).

<figure><img src="image_3.png" /><figcaption>Figure 1: A typical three-dimensional Linear Elastic Fracture Mechanics scenario</figcaption></figure>

### 2.1 Definitions

The objective of Linear Elastic Fracture Mechanics is to predict when and how a preexisting crack will propagate in a linear elastic body $\Omega$. In this framework, a crack is defined as a discontinuity surface of the displacement field, bounded by a crack front $\mathcal{F}$.

In this paper, we assume quasi-static crack evolution. Crack surfaces are considered stress-free, and body forces negligible compared to the applied tractions $\vec{T}^d$ on the boundary $\partial\Omega_t$ and prescribed displacements $\vec{u}^d$ on the complementary boundary $\partial\Omega_T$. The material is assumed to be isotropic, characterized by Young's modulus $E$ and Poisson's ratio $\nu$ (Fig. 1). Then for each crack position, the mechanical displacement $\vec{u}$ and stress fields $\sigma$ are the solution of the following equilibrium linear elastic Boundary Value Problem (BVP) defined by:

* partial differential equations at all points of $\Omega$:

$$ \left\{ \begin{array}{l} \mathrm{Div}\sigma = \vec{0} \\ \varepsilon = \frac{1+\nu}{E} \sigma - \frac{\nu}{E}(\mathrm{tr}\sigma)\mathbf{1} \\ \varepsilon(\vec{u}) = \frac{1}{2}(\mathrm{Grad}\vec{u} + \mathrm{Grad}\vec{u}), \end{array} \right. \qquad (1) $$

* boundary conditions:

$$ \left\{ \begin{array}{l} \sigma \cdot \vec{n} = \vec{T}^d \quad \text{on } \partial\Omega_T \\ \vec{u} = \vec{u}^d \quad \text{on } \partial\Omega_d \\ \sigma \cdot \vec{n} = \vec{0} \quad \text{on the remaining part of } \partial\Omega \text{ and on the crack surfaces} \end{array} \right. \qquad (2) $$

Denote $s$ a curvilinear abscissa along the crack front and $(s, \vec{e}_1(s), \vec{e}_2(s), \vec{e}_3(s))$ the corresponding Frenet frame as defined in Figure 1. The polar coordinates $(r, \theta)$ are defined in the local plane $(s, \vec{e}_1(s), \vec{e}_2(s))$ originating at the crack tip and oriented with respect to the tangent to the crack lips. As $r \to 0$, the stress field $\sigma$ exhibits the following asymptotic behavior:

$$ \sigma_{ij}(r, \theta) = \frac{K_I(s)}{\sqrt{2\pi r}} f_{ij}^I(\theta) + \frac{K_{II}(s)}{\sqrt{2\pi r}} f_{ij}^{II}(\theta) + \frac{K_{III}(s)}{\sqrt{2\pi r}} f_{ij}^{III}(\theta) + O(1) \qquad (3) $$
The pre-factors $K_I(s)$, $K_{II}(s)$, $K_{III}(s)$ are the mode I, mode II and mode III Stress Intensity Factors. As solutions to the elasticity boundary value problem, these quantities depend on the elastic constants $E$ and $v$, the geometry of both the body and the crack, and the applied loading conditions ($\vec{T}^d, \vec{u}^d$).

Functions $f_{ij}^p(\theta)$ are universal (in the sense they do not depend on the BVP) functions depending on $\theta$ only, given by:

$$ \left\{ \begin{array}{ll} f_{xx}^I = \cos\left(\frac{\theta}{2}\right) \left(1 - \sin\left(\frac{\theta}{2}\right) \sin\left(\frac{3\theta}{2}\right)\right) & f_{xx}^{II} = -\sin\left(\frac{\theta}{2}\right) \left(2 + \cos\left(\frac{\theta}{2}\right) \cos\left(\frac{3\theta}{2}\right)\right) \\ f_{yy}^I = \cos\left(\frac{\theta}{2}\right) \left(1 + \sin\left(\frac{\theta}{2}\right) \sin\left(\frac{3\theta}{2}\right)\right) & f_{yy}^{II} = \sin\left(\frac{\theta}{2}\right) \cos\left(\frac{\theta}{2}\right) \cos\left(\frac{3\theta}{2}\right) \\ f_{xy}^I = \cos\left(\frac{\theta}{2}\right) \sin\left(\frac{\theta}{2}\right) \cos\left(\frac{3\theta}{2}\right) & f_{xy}^{II} = \cos\left(\frac{\theta}{2}\right) \left(1 - \sin\left(\frac{\theta}{2}\right) \sin\left(\frac{3\theta}{2}\right)\right) \end{array} \right. \quad (4) $$

and

$$ \left\{ \begin{array}{l} f_{xz}^{III} = -\sin\left(\frac{\theta}{2}\right) \\ f_{yz}^{III} = \cos\left(\frac{\theta}{2}\right) \end{array} \right. \qquad (5) $$

the other components being zero. Note that these functions are the same under both the plane stress and the plane strain assumptions, while the asymptotic behavior of the displacement field $\vec{u}$ depends on the chosen hypotheses (Bui 2006). A practical method for identifying the appropriate modeling assumption in a given experiment, based on Digital Image Correlation (DIC), is presented in Triclot et al. (2023).

Define the potential energy of the cracked body $\Omega$ as:

$$ \mathcal{P} = \mathcal{E} - W_{\text{ext}}, \quad \text{where} \quad \mathcal{E} = \frac{1}{2} \int_{\Omega} \sigma \cdot \varepsilon \, dv \quad \text{and} \quad W_{\text{ext}} = \int_{\partial \Omega_T} \vec{T} \cdot \vec{u} \, dS \qquad (6) $$

The Energy Release Rate is defined as the potential energy released per unit area $A$ of crack advance around the crack front location $s$:

$$ \mathcal{G}(s) = - \left. \frac{\partial \mathcal{P}}{\partial A} \right|_{\vec{T}^d \text{ and } \vec{u}^d} \qquad (7) $$

Irwin (1957)'s formula relates the SIF to the ERR through:

$$ \mathcal{G}(s) = \frac{1 - v^2}{E} (K_I(s)^2 + K_{II}(s)^2) + \frac{1 + v}{E} K_{III}(s)^2 \qquad (8) $$

## 2.2 Crack propagation laws

If the size of the Process Zone (PZ) - the region where dissipative processes occur to counteract the stress singularity - is small, the SIF $K_p(s)$, $p = I, II, III$ and the ERR $\mathcal{G}(s)$ are relevant for determining the local driving forces available for crack propagation. Most crack growth laws are based on these quantities.

For instance, to determine the mode I crack advance in fatigue under cyclic loading, Paris' law is the most used one:

$$ \frac{da(s)}{dN} = C(\Delta K_I(s))^m \qquad (9) $$

where $C, m$ are material constants.

To determine the propagation threshold, Griffith (1920)'s criterion can be used:

$$ \mathcal{G}(s) = \mathcal{G}_c \qquad (10) $$

where $\mathcal{G}_c$ is the surface fracture energy.

In mode I, it is equivalent to Irwin (1957)'s criterion:

$$ K_I(s) = K_{Ic} \qquad (11) $$
<figure><img src="image_4.png" /><figcaption>Figure 2: Planar crack in an infinite linear isotropic medium under remote tensile $\sigma$ and shear $\tau$ loading.</figcaption></figure>

where $K_{Ic}$ is the fracture toughness.

The crack propagation direction can be determined using the Principle of Local Symmetry (PLS) stating that $K_{II} = 0$ along the crack propagation path (Goldstein et al. 1974) or the maximum energy release rate (MERR) criterion stating that the propagation direction maximizes the ERR $G(\varphi)$ among the possible propagation direction $\varphi$ (Erdogan et al. 1963).

Further details on crack path determination can be found in Amestoy et al. (1992).

## 3 In-plane perturbations of planar cracks

This section focuses on in-plane propagation of flat cracks, with particular attention given to perturbations of the crack front relative to a reference configuration.

First, formulas describing the variation of the SIFs under small perturbations is revisited. Second, their usefulness is illustrated through several applications. A comprehensive review is provided in Lazarus (2011), which is extended here to incorporate more recent developments.

### 3.1 First order formulas for small perturbations

Consider a planar crack in an infinite body, subjected to some remote tensile $\sigma$ and shear $\tau$ loading (Fig.2). Disturb the crack front $\mathcal{F}$ by a small normal advance $\delta(s)$. Bueckner (1987)-Rice (1985) weight function theory provides the first order variation in $\delta(s)$, of the SIF $\delta K_p(s_0)$, $p = I, II, III$, at each point $s_0$ of $\mathcal{F}$ by:

$$ \delta K_p(s_0) = N_{pq}(\nu) \cdot K_q(s_0)\delta'(s_0) + \frac{1}{2\pi} \mathrm{PV} \int_{\mathcal{F}} \frac{W_{pq}(s_0, s)}{D^2(s_0, s)} \cdot K_q(s) [\delta(s) - \delta(s_0)\vec{e}_2(s_0) \cdot \vec{e}_2(s)] \mathrm{d}s \quad (12) $$

This formula has been derived by Rice (1989) for $p = I$ and later generalized by Leblond et al. (1999) for $p = II, III$. It involves several quantities defined along the initial crack front $\mathcal{F}$: $D(s, s_0)$ is the distance between two points $(s, s_0)$; $W_{pq}(s, s_0)$, the kernel of the integrals. The functions $W_{pq}$, $p, q = I, II, III$ are linked to the Bueckner weight functions, that is to the $p$-th SIF at $s$ for unit point forces $\vec{e}_q$ near the crack front at $s_0$. The matrix $N$ of components $N_{pq}(\nu)$, $p, q \in I, II, III$ is:

$$ N = \frac{2}{2 - \nu} \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 - \nu & 0 \end{pmatrix}. \quad (13) $$

Formula (12) enables an updated evaluation of $K_p(s_0)$ for all $s_0$, provided that $K_q(s)$ and $W_{pq}(s, s_0)$ are known along the initial crack front $\mathcal{F}$. A review of such configurations can be found in
Lazarus (2011), which mainly covers straight or circular crack fronts. As an example, for a circular crack of radius a:

$$ 
\left\{
\begin{aligned}
K_I &= \sigma \sqrt{\pi a} \\
K_{II}(\theta) &= -\frac{4}{2 - \nu} \tau \sqrt{\frac{a}{\pi}} \cos \theta \\
K_{III}(\theta) &= \frac{4(1 - \nu)}{2 - \nu} \tau \sqrt{\frac{a}{\pi}} \sin \theta;
\end{aligned}
\right.
\qquad (14) $$

and

$$ 
\left\{
\begin{aligned}
W_{II}(\theta_0, \theta_1) &= 1 \\
W_{II,II}(\theta_0, \theta_1) &= \frac{2 \cos(\theta_0 - \theta_1) - 3\nu}{2 - \nu} \\
W_{III,III}(\theta_0, \theta_1) &= \frac{2(1 - \nu) \cos(\theta_0 - \theta_1) + 3\nu}{2 - \nu} \\
W_{II,III}(\theta_0, \theta_1) &= \frac{1}{1 - \nu} \\
W_{III,II}(\theta_1, \theta_0) &= \frac{2 \sin(\theta_0 - \theta_1)}{2 - \nu}.
\end{aligned}
\right.
\qquad (15) $$

As a specific case of interest, consider a semi-infinite planar crack with a straight front, embedded in an infinite medium and subjected to a uniform loading such that the SIFs are constant and equal to $K_I^0, K_{II}^0, K_{III}^0$ along the entire front (Fig. 3, left). Suppose that the crack front is perturbed to a new position by a normal advance $\delta(z) = \epsilon\phi_x(z)$. Then as demonstrated by Rice (1985) and Gao et al. (1986), the variations of the SIF to the first order in $\epsilon$ is given by

$$ 
\begin{aligned}
\delta_x K_I(z) &= \frac{K_I^0}{2\pi} \mathrm{PV} \int_{-\infty}^{+\infty} \frac{\epsilon\phi_x(z') - \epsilon\phi_x(z)}{(z'-z)^2} dz' \\
\delta_x K_{II}(z) &= -\frac{2}{2-\nu} K_{III}^0 \epsilon \frac{\partial\phi_x}{\partial z}(z) + \frac{2-3\nu}{2-\nu} \frac{K_{II}^0}{2\pi} \mathrm{PV} \int_{-\infty}^{+\infty} \frac{\epsilon\phi_x(z') - \epsilon\phi_x(z)}{(z'-z)^2} dz' \\
\delta_x K_{III}(z) &= \frac{2(1-\nu)}{2-\nu} K_{II}^0 \epsilon \frac{\partial\phi_x}{\partial z}(z) + \frac{2+\nu}{2-\nu} \frac{K_{III}^0}{2\pi} \mathrm{PV} \int_{-\infty}^{+\infty} \frac{\epsilon\phi_x(z') - \epsilon\phi_x(z)}{(z'-z)^2} dz'
\end{aligned}
\qquad (16) $$

The aforementioned formulas (12) and (16) suppose that the body is infinite, which is pertinent for small cracks located far away from any boundary. Finite thickness plates are considered by Legrand et al. (2011) and free edge cracks by Gao et al. (1991). However, the generalization of this approach to any finite size body is not straightforward, the reason being that the translatory motion $\delta(s_0)\vec{e}_2(s_0)$. $\vec{e}_2(s)$ introduced in formula (12) gives rise to a none zero variation of the SIF when the crack interacts with the boundaries.

## 3.2 Large perturbations

Besides the first order formulas (Eq. 12) on $\delta K_p$, formulas for $\delta W_{pq}$ are available within the same hypotheses of the infinite body. Written on the matrix $\delta W$ of their components, we have:

$$ 
\begin{aligned}
\delta \mathbf{W}(s_0, s_1) &= \mathbf{N} \cdot \mathbf{W}(s_0, s_1) [\delta'(s_0) - \delta_{**}(s_0)] \\
&\quad - \mathbf{W}(s_0, s_1) \cdot \mathbf{N} [\delta'(s_1) - \delta_{**}(s_1)] \\
&\quad + \frac{D^2(s_0, s_1)}{2\pi} \mathrm{PV} \int_F \frac{\mathbf{W}(s_0, s) \cdot \mathbf{W}(s, s_1)}{D^2(s_0, s)D^2(s_1, s)} [\delta(s) - \delta_{**}(s)]ds,
\end{aligned}
\qquad (17) $$

In this formula, $\delta_{**}(s)$ is a combination of a translatory motion, a rotation and a homothetical transformation that maps the points $s_0$ and $s_1$ from their initial to their final positions. This motion is specifically chosen to leave the crack geometry unchanged, resulting in an invariant dimensionless quantity $\mathbf{W}$, i.e., $\delta_{**}\mathbf{W} = 0$. The formulas for the case $p = I, q = I$ were derived by Rice (1989), while those involving shear modes ($p = II, III$ or $q = II, III$) were established by Favier et al. (2006).

When combined, formulas Eq. 12 and Eq. 17 enable the incremental analysis of large in-plane disturbances (Rice 1989), which can be used in two ways. First, starting from a shape for which
$K_p$ and $W_{pq}$ are known (for instance, circular, Eq. 14-15), they can be updated incrementally until reaching a targeted front shape $\mathcal{F}$ by construction of successive small perturbations. Second, once $K_p$ and $W_{pq}$ are known for $\mathcal{F}$, the propagation path following Paris' law (Eq. 9), Griffith's (Eq. 10) or Irwin's (Eq. 11) criterions, as well as any law based on the SIF can be obtained.

In the particular mode I case, only $K_I = K$ and $W_{I,I} = W$ have to be updated, as the other components are zero. Formulas Eq. 12 and Eq. 17 then simplify as follows:

$$ \delta K(s_0) = \frac{1}{2\pi} \text{PV} \int_{\mathcal{F}} \frac{W(s_0, s)}{D^2(s_0, s)} \cdot K(s) [\delta(s) - \delta(s_0)\vec{e}_2(s_0) \cdot \vec{e}_2(s)] \, ds \\ \delta W(s_0, s_1) = \frac{D^2(s_0, s_1)}{2\pi} \text{PV} \int_{\mathcal{F}} \frac{W(s, s_0) W(s, s_1)}{D^2(s, s_0) D^2(s, s_1)} [\delta(s) - \delta_{**}(s)] \, ds \quad (18) $$

This incremental method was first implemented numerically by Bower et al. (1990), and then used in the case of propagation through an array of tough micro-particles (Bower et al. 1991), tough particles and micro-cracks (Bower et al. 1993b), as well as non-uniform remote stresses and residual stresses (Bower et al. 1993a). We adapted the method to study the propagation of closed crack fronts of any shape in a homogeneous material in both mode I (Lazarus 2003; David et al. 2022) and mode II+III (Favier et al. 2006) loading conditions, but also for a tensile crack propagating in a highly heterogeneous solid (Vasoya et al. 2016) and for the coalescence of two circular cracks (Legrand et al. 2015).

## 3.3 Applications

### 3.3.1 Shape stability problem

<figure><img src="image_5.png" /><figcaption>**Figure 3:** Examples of simplified crack configurations under remote loading: half-plane crack, circular (penny-shape) crack, and tunnel crack.</figcaption></figure>

The problem as stated by Rice (1985) is as follows. Instead of using the true crack geometry, circular or straight approximations are often used. Is this safe? If the crack front is perturbed, will the perturbation increase or decrease over time? If it grows, the geometry becomes unstable, and any prediction that assumes a fixed crack shape may no longer be valid.

Many configurations have been studied, ranging from half-plane cracks (Rice 1985; Gao et al. 1986) to circular cracks (Gao et al. 1987; Gao 1988) and tunnel cracks (Leblond et al. 1996; Lazarus et al. 2002). An extensive review is available in Lazarus (2011), which shows that under linear elastic and homogeneous conditions, a straight or circular crack is always stable when isolated, that is, located far from any boundary or other crack (Fig. 3). However, this stability is not guaranteed, at least when the crack interacts with another crack (Pindra et al. 2010; Legrand et al. 2010), approaches a free boundary (Gao et al. 1991) or is embedded in a thin film (Adda-Bedia et al. 2006).

### 3.3.2 Equilibrium crack front shapes

Another interesting question is what are the shapes of the equilibrium crack configurations, such as $G(s) = G_c = cst$, along the entire crack front? While it is straightforward that half-plane, tunnel, and penny-shaped cracks satisfy this condition under uniform tensile loading, this solution is not always unique. For instance, a bifurcated wavy solution exists when a finite size enters into play. This is the case for tunnel cracks (Leblond et al. 1996; Lazarus et al. 2002; Pindra et al. 2010; Legrand et al. 2010) and in thin films (Adda-Bedia et al. 2006).
<figure><img src="image_6.png" /><figcaption>Figure 4: Wedge-splitting experiments on two glass plates glued along a strip-patterned interface.</figcaption></figure>

In the case of uniform remote shear loading, the equilibrium shape can be estimated using the perturbation approach incrementally, as outlined in Section 3.2. In this way, Favier et al. (2006) showed that the shape is close to an ellipse with:

$$ \frac{a}{b} \approx 1 - 2A, \quad A \equiv \frac{\nu(2 - \nu)}{2(2 - 2\nu - \nu^2)}, \qquad (19) $$

a, b being the small and large axis of the ellipse.

The analysis can be extended to heterogeneous fracture toughness, seeking configurations of the form $G(s) = G_c(s)$, $\forall s$ with $G_c(s) \neq \text{cst}$. Consider the case of a small toughness contrast and a mode I half-plane crack with uniform line tractions applied at a distance $a$ from the crack front. The problem is then to find the perturbation $\delta(z)$ from straightness such as $K_I(x, z) = K_c(x, z)$ with $K_c(x, z) = K_c^m(1 + \Delta K_c(x, z))$, $|\Delta K_c| \ll 1$. Using Eq. 18, we obtain:

$$ \frac{\delta(z)}{2a} + \frac{1}{2\pi} PV \int_{-\infty}^{\infty} W(z, z') \frac{\delta(z') - \delta(z)}{(z' - z)^2} dz' = \Delta K_c(a, z) \qquad (20) $$

and considering $W = 1$ corresponding to an infinite medium and using Fourier transformation (.):

$$ \frac{\hat{\delta}(a, k)}{a} = -2 \frac{\widehat{\Delta K_c}(a, k)}{1 + ka} \qquad (21) $$

Application of this formula provides a qualitative agreement with wedge-splitting experiments (Dalmas et al. 2009). Using the kernel $W \neq 1$ for a finite thickness of Legrand et al. (2011), a quantitative agreement has been demonstrated. This allowed us to propose a way to measure the adhesion energy of an impenetrable interface from the measure of the front shape (Patinet et al. 2013) using a strip-patterned interface (Fig. 4).

### 3.3.3 Apparent toughness for heterogeneous materials

Continuing with the case of small toughness perturbations, one question arises: what is the apparent toughness resulting from the heterogeneities? As demonstrated by Roux et al. (2003); Demery et al. (2014), the answer is given by:

* $K_c^{app} = K_c^m$ to the first order if $K(x, z) = K_c^m(1 + \Delta K_c(x, z))$, $\forall (x, z) \in \mathcal{F}$

* $K_c^{app} > K_c^m$ if abrupt depinning transitions occur between partial equilibrium configurations, characterized by the existence of certain points $(x, z)$ such that $K(x, z) < K_c^m(1 + \Delta K_c(x, z))$ (Roux et al. 2003). This increase is caused by additional dynamic dissipation associated with snapback instabilities occurring between these states.

If second order terms enter into play, $K_c^{app} \neq K_c^m$ with subtle size effects, that result in an R-curve due to interactions between the crack and the heterogeneity scale (Lebihain et al. 2023).

For large heterogeneity contrasts, the incremental application of the perturbation approach, as described in Section 3.2, has revealed the existence of a fingering instability, which leads to a drastic decrease of the apparent toughness (Vasoya et al. 2016). A similar reduction in toughness was also observed using the same method during the coalescence phase of two coplanar circular cracks (Legrand et al. 2015).

### 3.3.4 Fatigue propagation of distorted closed cracks

The method originally implemented by Lazarus (2003) was improved during L. David's PhD thesis (2024), with the addition of an efficient remeshing procedure and a general approach to
<figure><img src="image_7.png" /><figcaption>Figure 5: Patchwork giving the wide variety of simulated crack front shapes</figcaption></figure>

evolving the crack from its initial circular shape to the final form (David 2024). This advancement enabled the simulation of fatigue crack propagation across a wide variety of shapes, as illustrated in Figure 5. The method’s high efficiency, requiring only line meshing and the evaluation of one-dimensional integrals, enabled us to perform 545 simulations corresponding to $n = 1, 2, 3, 4, 5$ in Paris’ law (Eq. 9) with minimal set-up and computation time (David et al. 2022).

Beyond the method’s efficiency compared to 3D finite element approaches, the key takeaway from these simulations is that, regardless of the initial shape, the crack evolves into a circular form and behaves like a circular crack of the same area. More precisely, supposing the remote tensile load amplitude $\Delta\sigma$ is constant, setting $r = \sqrt{S/\pi}$ where S is the area of the crack surface and $\overline{N} = NC(\Delta\sigma)^n r_0^{n/2-1}$, the evolution of the size with the number of cycles can accurately be obtained using following formulas which correspond to the closed-form solution for the propagation of a circular crack:

$$ \begin{aligned} \frac{r}{r_0} &\sim \exp\left(\frac{4}{\pi}\overline{N}\right), & n = 2 \\ \frac{r}{r_0} &\sim \left(1 - 2^{n-1}\frac{n-2}{\pi^{n/2}}\overline{N}\right)^{\frac{-2}{n-2}}, & n \neq 2 \end{aligned} \qquad (22) $$

# 4 Out-of-plane perturbations

## 4.1 Theory

As in the previous section, we aim to provide analytical solutions for the variations of the SIFs induced by small geometrical perturbations of the crack, using asymptotic approaches. However, this portion of the paper moves beyond the planar crack assumption by considering situations involving out-of-plane perturbations. In Section 4.1.1, we consider the case of wavy perturbations of the crack surface. Then, in Section 4.1.2, we address two perturbation examples featuring a selection of angular points, namely crack kinking and segmentation, that commonly occur during mixed-mode I+II and I+III nominal loading conditions.

### 4.1.1 Wavy perturbations

Consider the same semi-infinite configuration as before (Fig. 3, left) but now with the crack surface perturbed to a new position $y = \varepsilon\phi_y(x, z)$ while keeping the crack front straight. Then, as demonstrated in Leblond et al. (2011), and based on the previous work of Movchan et al. (1998),
the variation of the SIF to the first order in $\epsilon$ is given by:

$$ 
\begin{aligned}
\delta_y K_I(z) &= -\frac{3}{2} K_{II}^0 \epsilon \frac{\partial \phi_y}{\partial x}(0, z) - 2 K_{III}^0 \epsilon \frac{\partial \phi_y}{\partial z}(0, z) + \delta_y K_I^{\text{skew}}(z) \\
\delta_y K_{II}(z) &= \frac{K_I^0}{2} \epsilon \frac{\partial \phi_y}{\partial x}(0, z) - \frac{2-3\nu}{2(2-\nu)} \frac{K_I^0}{\pi} \text{PV} \int_{-\infty}^{+\infty} \frac{\epsilon \phi_y(0, z') - \epsilon \phi_y(0, z)}{(z' - z)^2} dz' \quad (23) \\
\delta_y K_{III}(z) &= \frac{2(1-\nu)^2}{2-\nu} K_I^0 \epsilon \frac{\partial \phi_y}{\partial z}(0, z)
\end{aligned}
$$

where the $K_I^{\text{skew}}$ is a non-local term depending on $\epsilon\phi_y(x, z)$ along the whole surface:

$$ \delta_y K_I^{\text{skew}}(z) = \frac{\sqrt{2}}{4\pi} \frac{1-2\nu}{1-\nu} \text{Re} \left[ \int_{-\infty}^{0} dx' \int_{-\infty}^{+\infty} \frac{K_{III}^0 - i(1-\nu)K_{II}^0}{(-x' + i(z-z'))^{3/2}} \frac{\epsilon \partial \phi_y / \partial z(x', z')}{\sqrt{-x'}} dz' \right] \quad (24) $$

### 4.1.2 Perturbations with angular points

<figure><img src="image_8.png" /><figcaption>Figure 6: Perturbed geometries with angular points: kinked crack path, segmentation of the crack in echelon/facets</figcaption></figure>

When the perturbed geometry contains some angular points (Fig. 6), the previous formula (Eq. 23) can no longer be used as $\epsilon \frac{\partial \phi_y}{\partial z}(x, z)$ is not defined at these sharp points.

**Kinked cracks** For a kinked crack extension (Fig. 6, left), the formula from Amestoy et al. (1992) can be used to obtain the SIF $K_p(\varphi)$ for a vanishing kinked extensions, assuming that the loading $(\vec{T}^d, \vec{u}^d)$ of the BVP remains constant:

$$ K_p(\varphi) = F_{pq}(\varphi) K_q^0 \quad (25) $$

In this equation, $K_q^0$ are the initial SIF without any kink and $F_{pq}(\varphi)$ are functions of the sole parameter $\varphi$. These functions have been shown to be universal, that is, independent of the boundary value problem defined in Section 2.1), due to dimensional considerations and the universality of the asymptotic stress field near the crack tip (Eq. 3) (Leblond 1989). Their exact values can be found in Amestoy et al. (1992). Note that these results are accurate and applicable to any LEFM BVP problem (Eq. 1 and 2). They should therefore be preferred for determining the crack propagation direction, rather than relying on approximations as those proposed for instance by Nuismer (1975) or Chen et al. (2022).

**Segmented faceted cracks** For segmented crack extensions as depicted in Fig. 6 (right), a two-scale Cohesive Zone (CZ) model can be used to calculate the energy release rate at the tips of the facets. The model yields this value as a function of facet geometry (tilt angle $\alpha$, length $a$, width $2\ell$, period length $2d$), and the initial SIF $K_I^0, K_{III}^0$ along the unperturbed semi-infinite crack configuration (Leblond et al. 2015). It is assumed that the facet slenderness $\eta \equiv \ell/a$ is small $\eta \ll 1$. Then by using matched asymptotic expansions, the 3D problem can be split into two related 2D problems (Fig. 7): an internal problem, where the facets appear as an array of echelon
<figure><img src="image_9.png" /><figcaption>Figure 7: Inner problem at the meso-scale, the facets appear as 2D echelon cracks (Left). Outer problem at the macro-scale, facets are embedded in a cohesive zone (Right).</figcaption></figure>

cracks and an external problem, in which the facets are embedded in a straight cohesive zone whose tip correspond to the tips of the facets. The effective SIF at their tips is given by:

$$ \left\{ \begin{aligned} K_I^{\text{eff}} &= \sqrt{\frac{2}{\pi}} \int_0^a p(X) \frac{dX}{\sqrt{a-X}} \\ K_{III}^{\text{eff}} &= \sqrt{\frac{2}{\pi}} \int_0^a q(X) \frac{dX}{\sqrt{a-X}}. \end{aligned} \right. \qquad (26) $$

where $\pm[p(X)e_Y + q(X)e_Z]$ are pinning forces exerted on the faces of the CZ, resulting from the ligaments between the facets. The functions $p(X)$ and $q(X)$ are solutions of the following integro-differential equations:

$$ \begin{aligned} & (\mathcal{A}_{22} + 2\nu\mathcal{A}_{12})\frac{dp}{dX} - \mathcal{A}_{23}\frac{dq}{dX} + \frac{1}{\pi\eta a}PV\int_0^a p(X')\sqrt{\frac{a-X'}{a-X}\frac{dX'}{X'-X}} \\ & = \frac{-(\mathcal{A}_{22}+2\nu\mathcal{A}_{12})K_I^0+\mathcal{A}_{23}K_{III}^0}{2\sqrt{2\pi}X^{3/2}} \\ & -(\mathcal{A}_{23}+2\nu\mathcal{A}_{13})\frac{dp}{dX} + \mathcal{A}_{33}\frac{dq}{dX} + \frac{1}{\pi(1-\nu)\eta a}PV\int_0^a q(X')\sqrt{\frac{a-X'}{a-X}\frac{dX'}{X'-X}} \\ & = \frac{(\mathcal{A}_{23}+2\nu\mathcal{A}_{13})K_I^0-\mathcal{A}_{33}K_{III}^0}{2\sqrt{2\pi}X^{3/2}}. \end{aligned} \qquad (27) $$

with the boundary conditions $p(a) = \frac{K_I^0}{\sqrt{2\pi a}}$ and $q(a) = \frac{K_{III}^0}{\sqrt{2\pi a}}$.

The functions $\mathcal{A}_{pq} = \mathcal{A}_{pq}(\alpha, \ell/d)$ are related to CZ behavior. They are linked to the geometry of the facets in the inner problem. They have been obtained in closed-form for $l/d \ll 1$ and $\alpha \ll 1$ (Leblond et al. 2015) and by XFEM in a more general case (Lazarus et al. 2020).

## 4.2 Applications

### 4.2.1 Wavy crack propagation in a 2D mode I setting

In the experiments of Yuse et al. (1993); Ronsin et al. (1995), a hot pre-cracked plate is pulled into a colder bath. Although the load is purely mode I, a transition from straight to wavy propagation can still be observed. A perturbation approach for small-wave amplitudes can be used to explain this transition (Adda-Bedia et al. 1995). This result was confirmed by FEM (Yang et al. 2001). Phase-field simulations highlighted the supercritical aspect of this instability (Corson et al. 2009).

### 4.2.2 Kinked crack path

Kinked crack paths occur under mode I+II loading (Erdogan et al. 1963). Amestoy and Leblond's formula can then be used to determine the crack path and the critical load. We are currently revisiting the pioneering experiments of Erdogan et al. (1963) using state-of-the art modeling approaches, including DIC, crack path prediction via LEFM and the MERR criterion, as well as phase-field simulations (Roucou et al. Draft).

Kinking also occurs in anisotropic materials even in mode I when the fracture energy is anisotropic as in wood (Ostapska et al. 2021), single crystals (Cormier et al. 2021; Nagaraja et al. 2023), composites (Mesgarnejad et al. 2020), architected materials (Nguyen et al. 2017) or 3D
printed materials (Kiendl et al. 2020; Corre et al. 2021; Roucou et al. 2023). We are currently taking advantage of the versatility of fused filament fabrication to design experiments that challenge advanced experimental and numerical approaches to fracture mechanics for quasi-static propagation (Zhai et al. 2025) and fatigue advances (Zhai et al. 2024). Experimentally, advanced measurement techniques from Digital Image Correlation to state-of-the-art Digital Volume Correlation approaches are challenged (Corre et al. 2025). Numerically, concepts of linear elastic fracture mechanics, from the classical Griffith variational energy approach (Hakim et al. 2005; Takei et al. 2013) to phase-field simulations developed for anisotropic materials (Li et al. 2019), are tested (Zhai 2023; Zhai et al. Draft).

### 4.2.3 Crack propagation in the presence of mode III

**Experimental observations** In the presence of mode III loading, it has been well established since the pioneer experiments of Sommer (1969) and Knauss (1970) that, even when initially smooth, the crack front tends to segment into echelons or facets during propagation.

<figure><img src="image_10.png" /><figcaption>**Figure 8:** Four-point bending experiments on beams with an inclined slit: experimental setup (top) and variation of SIFs along the crack front for different inclination angles $\Gamma_0$ (bottom).</figcaption></figure>

We experimentally investigated the influence of mode III using four-point bending (4PB) tests on PMMA beams containing an inclined slit (Lazarus et al. 2008). In Figure 8, the setup is shown on the left and the evolution of the SIF along the crack front, as obtained by FEM is presented in the two subsequent figures. Due to symmetry, mode II changes sign on either side of the crack front, as expected. It explains that at the macro-scale (Lazarus et al. 2008), the front twists to align with the mode I plane (Fig. 9, d). A significant amount of mode III is present at the center. This mode III component drives the segmentation into facets observed at the mesoscale (Fig. 9a, b, c) with each facet rotated toward the shear-free direction (Fig. 9b, top row). Progressive coalescence (Fig. 9b, second row) between them results in river lines on the postmortem fracture surface (Fig. 9c). Similar observations have been reported in other materials and experimental setups (Lin et al. 2010; Goldstein et al. 2012; Ronsin et al. 2014), demonstrating that segmentation and coalescence are structural effects.

We perform a systematic study increasing the load until the beam breaks (Hattali et al. 2021). The critical force $F_c$ at failure as a function of $\Gamma_0$ for different initial crack lengths are shown in
<figure><img src="image_11.png" /><figcaption>**Figure 9:** abc: Mesoscale segmentation observed in images of three samples halted at various stages of fatigue crack propagation (the black line scale corresponds to 1 mm): a) global view, b) first row-bottom view, c) second raw-front view, d) Macroscopic twisting</figcaption></figure>

Fig. 10, left. We also performed a preliminary study under cyclic load (Ben Hadj Khaled 2018). The crack length as a function of the number of cycles for different values of $\Gamma_0$ is given in Fig. 10, right (internal preliminary unpublished work). In both cases, toughening is observed as $\Gamma_0$, and thus mode III, increases.

<figure><img src="image_12.png" /><figcaption>**Figure 10:** Increase of the fracture resistance with mode III: increase of the critical force to failure (left), decrease of the advance rate (right)-unpublished, raw data are lost explaining the poor quality of the figure and the notation $\alpha$ instead of $\Gamma_0$.</figcaption></figure>

**Why do facets appear?** Consider the following thought experiment: a semi-infinite crack, like the one in Figure 3, loaded uniformly in mode I+III, where $K_I^0 = \text{cst}$, $K_{II}^0 = 0$, $K_{III}^0 = \text{cst}$ along the entire front. A trivial solution of both Griffith's criterion (Eq. 10) and the PLS, is a coplanar translatory motion of the straight crack front since this motion preserves the geometry of the problem. However, this solution is neither unique nor stable as first shown by Pons et al. (2010) using Phase-Field (PF) simulations. A linear stability analysis based on Eq. 16 and 23, combining in- and out-plane wavy perturbations, resulting in a helical crack front shape, has confirmed this result by showing that above a threshold on $K_{III}^0/K_I^0$ (v) depending on v, the straight propagation becomes unstable (Leblond et al. 2011). But this threshold has turned out to be too large in comparison to the segmentation usually observed in the presence of mode III (Pham et al. 2014). Further linear stability analysis has shown that mode II contribution (Vasudevan et al. 2020), T-stresses (Lebihain et al. 2022), or the dependence of $G_c$ with $K_{III}^0/K_I^0$ (Leblond et al. 2019) can reduce this threshold.

An alternative view is to consider a non-linear stability analysis, which can be achieved using PF simulations. While the PF simulations of Pons et al. (2010) were conducted above the linear stability threshold, we performed additional simulations below this threshold (Chen et al. 2015). In this case, starting from a perfectly straight crack, no instability appears. Both observations are in agreement with the linear stability analysis. However, we observed that the instability can be initiated by decreasing $K_{III}^0/K_I^0$ from above to below the threshold. This shows that geometric imperfections can trigger instability even below the linear stability threshold, revealing its subcritical nature. Further studies confirmed this result (Henry 2016; Pham et al. 2017; Molnár et al. 2024). This sensitivity to imperfections is in agreement with experiments where facets may or may not appear (Ronsin et al. 2014; Eberlein et al. 2017; Pham et al. 2014).
How imperfections trigger the instability and how facets emerge from the initially straight front remain open questions. A recent study (Doitrand et al. 2023), based on the coupled criterion of Leguillon (2002), showed that the nucleation of new facets may be favored in some cases in comparison to straight propagation. Another recent study (Molnár et al. 2024) revealed how toughness heterogeneities can initiate facet propagation in pure mode III loading. It also highlights the relationship between the inter-facet distance at initiation to the PF regularization length, as well as its correlation to Irwin length $ℓ = K_c^2/σ_c$ through the coupled criterion.

**Impact of the facets on the apparent toughness** Regardless of the cause of segmentation, once it occurs, crack propagation is no longer governed by the advance of a straight front, but rather by the propagation of the facet tips. Because of the remaining ligaments between the facets, their openings are smaller than those of the initial crack. This qualitatively explains the toughening observed in 4PB experiments both in term of critical force $F_c$ required for failure under monotonic loading, and in terms of the crack growth rate $da/dN$ under cyclic fatigue loading (Fig. 10).

By accounting in Griffith’s energy balance for the effectively broken surface length $ℓ$ over one period $d$, we obtain:

$$ \frac{F_c}{F_0} = \sqrt{\frac{\ell}{d} \frac{G_0}{G_{eff}}} \qquad (28) $$

where $G_0$ is the initial ERR, $F_c$ is the critical force in the presence of facets, and $F_0$ without.

Using the two-scale cohesive zone model (Section 4.1.2), the apparent ERR $G_{eff}/G_0$, can be obtained as a function of $(α, η, ℓ/d, K_{III}^0/K_I^0)$ based on Eq. 26-27 and 8. By measuring the facet geometry with a profilometer (Cambonie et al. 2014), we achieve quantitative agreement with the experimentally observed critical force $F_c$ (Hattali et al. 2021). Further work is needed to experimentally investigate quasi-static crack propagation, since propagation in the 4PB experiments becomes unstable above the Griffith threshold, but also to predict the facet geometry. Additionally, applying the CZ model to fatigue experiments would allow for direct comparison with the data shown in Figure 10, right.

# 5 Conclusion

This paper provides a review of asymptotic approaches aiming to determine the variation of the Stress Intensity Factors and the Energy Release Rate induced by small perturbations of the crack. Several configurations are considered: in-plane perturbations of planar cracks, wavy out-of-plane variations of the crack surface, crack kinking and segmentation into facets. Unlike standard finite element method calculations, these approaches allow for accurate and efficient treatment of geometric distortions. This enables a detailed analysis of crack geometry evolution during propagation and a quantitative assessment of its influence on fracture resistance.

Some key selected results from these approaches are as follows. Regarding in-plane perturbations of flat cracks, perturbation approaches based on Bueckner-Rice weight functions have allowed us to demonstrate that:

* Straight or circular crack fronts are stable in homogeneous materials and for isolated cracks, but this stability does not necessarily hold in heterogeneous materials or interacting crack systems.

* The apparent toughness can significantly deviate from the mean material toughness in the presence of large geometrical distortions.

* The surface area is the relevant geometric parameter for analyzing isolated cracks with distorted shapes; using this measure leads to more accurate lifetime predictions.

Regarding crack propagation in the presence of mode III loading, thanks to asymptotic methods, we observe that:

* The complex 3D faceted crack pattern observed during propagation is triggered by a subcritical instability, in which material defects play an essential role.

* Mode III loading has a reinforcing effect due to the formation of facets at the mesoscale.

* The propagation of facets demands a higher load than that needed for smooth crack growth, owing to the presence of ligaments between them.
* The toughening effect can be quantitatively evaluated using a two-scale cohesive model based on perturbation methods.

Although these approaches show promising results, further development is needed before they can be widely and reliably applied in engineering practice for finer-grained lifetime predictions in damage tolerance approaches. Significant challenges remain, including capturing the full complexity of crack morphologies, accounting for general mixed-mode loading evolution, addressing small-scale heterogeneities, and incorporating material anisotropy. The design of dedicated experiments to validate these approaches across multiple scales is another key challenge. Nonetheless, these methods hold significant potential, not only for improving predictive capabilities, but also for guiding the optimization of apparent fracture toughness through the tailored distribution of heterogeneities.

## Backmatter information

**Funding** The work was supported by the Agence de l'Innovation de Défense – AID – via Centre Interdisciplinaire d'Etudes pour la Défense et la Sécurité – CIEDS – (projects 2022 - FracAddi), and by Agence Nationale de la Recherche (ANR-23-CE51-0054 3FAM).

**Authors' contributions and acknowledgements** The author has dedicated over 25 years to research on this topic. She would like to sincerely thank all the collaborators and colleagues who have contributed to this journey. A full appreciation of these valued partnerships can be found by carefully exploring the references. She would also like to warmly thank Bethany Cagnol (IP Paris ELEVATE Center) for her thorough proofreading of the English language.

**Competing interests** The authors declare that they have no competing interests.

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a full copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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