# Fatigue crack nucleation at a stress concentration point

Dominique Leguillon, Sébastien Murer

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Dominique Leguillon, Sébastien Murer. Fatigue crack nucleation at a stress concentration point. CP2012, 2012, Rome, Italy. hal-04691477

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# Fatigue crack nucleation at a stress concentration point

D. Leguillon¹ and S. Murer²

¹ IJLRA – CNRS UMR 7190, Université Pierre et Marie Curie, 4 place Jussieu, case 162, 75252 PARIS Cedex 05 (France).

² GRESPI/MAN – EA 4301, Université de Reims-Champagne-Ardenne, Campus du Moulin de la Housse - BP 1039 - 51687 REIMS Cedex 2 (France).

**ABSTRACT.** The coupled criterion, using both stress and energy conditions, satisfactorily predicts the crack nucleation starting from a stress concentration point in brittle or quasi-brittle materials under monotonic loading. But it is a priori difficult to generalize to fatigue. A first fatigue model was established based on a Dugdale cohesive zone model but in turn it proved difficult to be extended to complex loadings. The present work is twofold: (i) showing how to generalize the coupled criterion to take into account both shear and tensile strengths as well as mode I and II toughness to predict crack nucleation under monotonic complex loadings; (ii) extending this criterion to the crack nucleation under fatigue cycles by considering a gradual degradation along the presupposed crack path. One parameter is identified so that the rate of advance coincides with that of a Paris law in case of a pre-existing long crack. As derived from the model, the growth is intermittent which provides an explanation for the striations observed in experiments.

A relationship is established between the initial crack velocity and the exponent of the singularity characterizing the stress concentration, showing that the weaker the singularity and the smaller the crack advance rate. From these considerations one can deduce that the short crack range can be characterized by the distance required to reach a steady velocity.

## INTRODUCTION

The coupled criterion, using both stress and energy conditions [1], was developed to predict crack nucleation at stress concentration points in brittle materials under monotonic loading. It was established using asymptotic expansions and theory of singularity and gave satisfactory predictions. However, in its original form, it appeared difficult to generalize to fatigue. It was a priori dedicated to brittle fracture and did not seem able to integrate concepts such as the accumulation of damage or plasticity. That is why we tried in a first step to develop a fatigue criterion [2] based on the use of Dugdale's cohesive zone model (CZM) [3]. It was originally developed as a simplified model of crack tip plasticity: the traction acting ahead of the crack tip cannot exceed a threshold value denoted here $\sigma_c$ (the tensile strength), but corresponding to the plastic flow threshold in the original model. It was extended to V-notches in homogeneous
materials [4] and proved that, under symmetric monotonic loadings, such a model gave similar answers to the coupled criterion. The generalization to fatigue was obtained by exploiting the following idea: there is an accumulation of the successive elementary openings of the cohesive zone until it reaches a critical value [5]. This model is similar to a kind of cumulative plasticity. However, this approach has had to be modified because it leads systematically to a Paris exponent [6] equal to 4 which is a special case poorly adapted to quasi-brittle materials. For this purpose, an alternative interpretation of the accumulation law was made which is more consistent with the concept of cumulative damage. We proposed that only an adjustable part of the opening energy is converted into damage while the other is restored. The partition is achieved by using a parameter which can be identified in the particular case of a pre-existing long crack using a known relationship with the Paris exponent.

Unfortunately these results are difficult to generalize to complex loadings or other geometries that will inevitably require the concept of mode mixity. All these statements will be adapted to develop the model presented below.

## COMPLEX MONOTONIC LOADINGS

Before focusing on fatigue, the first stage is to generalize the coupled criterion [2] to monotonic complex loadings. This was required in many situations examined in different contexts such as the study of bonded structures under complex loading using the Arcan test [7] (Fig. 1) for instance.

<figure><img src="image_2.png" /><figcaption>Figure 1. The Arcan test (left) and a PMMA specimen with an adhesive joint (right). Such a setup allows a complete range of mixity from a pure symmetric mode (loading angle 0°) to a pure antisymmetric one (loading angle 90°).</figcaption></figure>

The Williams expansion of the elastic solution $\underline{U}(r, \theta)$, in polar coordinates emanating from the notch root, is written with two singular terms, and a mixity parameter $m$
$$
\left\{
\begin{array}{l}
\underline{U}(r, \theta) = \underline{U}(O) + k_1 r^{\lambda_1} (\underline{u}_1(\theta) + m \underline{u}_2(\theta)) + \dots \\
\sigma(r, \theta) = k_1 r^{\lambda_1-1} (s_1(\theta) + m s_2(\theta)) + \dots \quad \text{with } m = \frac{k_2}{k_1} r^{\lambda_2 - \lambda_1} \\
\tau(r, \theta) = k_1 r^{\lambda_1-1} (t_1(\theta) + m t_2(\theta)) + \dots
\end{array}
\right.
\qquad (1)
$$

Where $\sigma$ and $\tau$ hold for the tensile ($\sigma_{\theta\theta}$) and shear ($\sigma_{r\theta}$) components of the stress tensor. The coefficients $k_1$ and $k_2$ are the generalized stress intensity factors (GSIF) of the two singular terms characterized by the exponents $\lambda_1$ and $\lambda_2$ and the two modes $\underline{u}_1(\theta)$ and $\underline{u}_2(\theta)$, respectively symmetric and antisymmetric with respect to the bisector. The functions $s_1(\theta)$ and $t_1(\theta)$ (resp. $s_2(\theta)$ and $t_2(\theta)$) are associated with the components $\theta\theta$ and $r\theta$ of the stress field derived from $\underline{u}_1(\theta)$ (resp. $\underline{u}_2(\theta)$) through the elastic constitutive law. Note that the definition of $m$ implies $k_1 \neq 0$, the particular case $k_1 = 0$ refers to the pure mode II that can be treated similarly to the pure mode I. Energy and stress conditions give

$$
\delta W^P(l) \geq \int_0^l G_c(\mu) dr ; \sigma \geq \sigma_f(\mu) ; |\tau| \geq \tau_f(\mu) \text{ for } 0 \leq r \leq l \text{ with } \mu = \frac{|\tau|}{\sigma} = \frac{|t_1 + m t_2|}{s_1 + m s_2} \quad (2)
$$

Where $G_c$ is the toughness, and where $\sigma_f$ and $\tau_f$ are the tensile and shear strength at failure under mix mode loading (the mixity is characterized by $\mu$). According to (1)

$$
\left\{
\begin{array}{l}
k_1^2 l^{2\lambda_1} (A_1(\theta_0) + m(l)A_{12}(\theta_0) + m(l)^2 A_2(\theta_0)) \ge \int_0^l G_c(r) dr \\
k_1 r^{\lambda_1-1} (s_1 + m(r)s_2) \ge \sigma_f(r) \quad \text{for } 0 \le r \le l \\
k_1 r^{\lambda_1-1} |t_1 + m(r)t_2| \ge \tau_f(r) \quad \text{for } 0 \le r \le l
\end{array}
\right.
\qquad (3)
$$

The scaling functions $A_i$ were defined in [8] and $\theta_0$ is the crack direction. With the additional relation ($q=2$ holds for an elliptical criterion)

$$
\begin{align}
& \left( \frac{\sigma_f(\mu)}{\sigma_c} \right)^q + \left( \frac{\tau_f(\mu)}{\tau_c} \right)^q = 1 ; \quad \tau_f(\mu) = \mu \sigma_f(\mu) \tag{7} \\
\Rightarrow \quad & \sigma_f(\mu) = \frac{\sigma_c \tau_c}{(\tau_c^q + \mu^q \sigma_c^q)^{1/q}} ; \quad \tau_f(\mu) = \mu \sigma_f(\mu) \nonumber
\end{align}
$$
With the Hutchinson and Suo [9] condition, setting $\psi = \tan^{-1}(\mu)$ and where $G_{Ic}$ and $G_{IIc}$ respectively denote mode I and II toughness

$$ 
\begin{aligned}
G_c &= G_{Ic} (1 + \tan^2(c\psi)) \\
\text{if } \psi=0 \quad G_c &= G_{Ic} \\
\text{if } \psi=\pm\pi/2 \quad G_c &= G_{IIc} \Rightarrow c = \frac{2}{\pi} \tan^{-1}\left(\sqrt{\frac{G_{IIc}}{G_{Ic}}} - 1\right)
\end{aligned} 
\quad (8) $$

The general form of the criterion can be written

$$ k_1 \ge k_{1f} = \left( \frac{\bar{G}_c}{A_1(\theta_0) + m_0 A_{12}(\theta_0) + m_0^2 A_2(\theta_0)} \right)^{1-\lambda_1} \left( \frac{\sigma_c \tau_c}{( (s_1+m_0 s_2)^q \tau_c^q + |t_1+m_0 t_2|^q \sigma_c^q )^{1/q}} \right)^{2\lambda_1-1} \quad (9) $$

Where $\bar{G}_c = \frac{1}{l} \int_0^l G_c(r)dr$ corresponds to the mean toughness along the presupposed crack path and $m_0 = m(l_0)$, $l_0$ being the solution to the implicit equation

$$ l = \frac{\bar{G}_c \left( (s_1+m(l)s_2)^q \tau_c^q + |t_1+m(l)t_2|^q \sigma_c^q \right)^{2/q}}{\sigma_c^2 \tau_c^2 (A_1(\theta_0) + m(l)A_{12}(\theta_0) + m^2(l)A_2(\theta_0))} \quad (10) $$

A more general form of the criterion (yet slightly heavier to handle) was established by Garcia and Leguillon [10], it avoids the asymmetry between mode I and mode II that appears through the mixity parameter $m$ defined in (4).

## EXTENSION TO FATIGUE LOADINGS

<figure><img src="image_3.png" /><figcaption>Figure 2. Schematic view of a fatigue loading vs. time $t$. The load intensity is characterized by the GSIF $k_1$ of the singularity at the root of the V-notch, $k_{Ic}$ is its critical value under monotonic loading and $k_{Im}$ the maximum reached during a cycle.</figcaption></figure>
We assume that $G_c$ remains approximately constant along the crack path (it is not a strong assumption since the crack extension lengths are very small especially in fatigue, which is easy to check a posteriori). For simplicity we note $G_f = G_c = \bar{G}_c$ and $G_m$ is the toughness of a material having the same elastic moduli but a failure corresponding to $k_{1m}$ (the GSIF at the peak of a fatigue cycle) and we set $\alpha = k_{1m} / k_{1f}$.

To be consistent with the fatigue model derived from the Dugdale CZM, it is assumed that the toughness is degraded at each cycle of a quantity $\Delta G$ proportional to $G_m$ ($\beta$ is an adjustable parameter), then for $n$ cycles during the nucleation phase

$$G_m = G_f - n\Delta G, \quad \text{if } \Delta G = \alpha^\beta G_m \quad \text{then} \quad G_m(1+n\alpha^\beta) = G_f \quad (11)$$

According to (9), it comes

$$ \left\{ \begin{aligned} k_{1f} &= \left( \frac{G_f}{A_f^*} \right)^{1-\lambda_1} \left( \frac{\sigma_c \tau_c}{S_f^*} \right)^{2\lambda_1 - 1} & \text{with} \\ k_{1m} &= \left( \frac{G_m}{A_m^*} \right)^{1-\lambda_1} \left( \frac{\sigma_c \tau_c}{S_m^*} \right)^{2\lambda_1 - 1} \end{aligned} \right. \quad (12) \\ \begin{cases} A_a^* = A_1 + m_a A_{12} + m_a^2 A_2; & S_a^* = ( (s_1+m_a s_2)^q \tau_c^q + |t_1+m_a t_2|^q \sigma_c^q )^{1/q} \\ m_a = \frac{k_2}{k_1} l_a^{\lambda_2-\lambda_1} = \frac{k_{2a}}{k_{1a}} l_a^{\lambda_2-\lambda_1} & a=f,m \end{cases} $$

Where $l_f$ and $l_m$ are the extension lengths derived from the coupled criterion, they are solution to (10) where $G_f$ and $G_m$ successively replace $\bar{G}_c$

$$l_a = \frac{G_a S_a^{*2}}{\sigma_c^2 \tau_c^2 A_a^*} \quad a = f, m \quad (13)$$

It is important to note that the intensity of the cyclic loading occurs only through $G_m$. From (12), we deduce

$$\alpha = \left( \frac{G_m}{G_f} \right)^{1-\lambda_1} \left( \frac{A_f^*}{A_m^*} \right)^{1-\lambda_1} \left( \frac{S_f^*}{S_m^*} \right)^{2\lambda_1 - 1} \quad (14)$$

Inserting in (11) leads to
$$n = \frac{1}{\alpha^{\beta}} \left( \frac{1}{\alpha^{\frac{1-\lambda_1}{1-\lambda_4}} F^*} - 1 \right) \approx \frac{1}{\alpha^{\frac{\beta+\frac{1}{1-\lambda_4}}{1-\lambda_4}} F^*} \quad \text{where} \quad F^* = \frac{A_m^*}{A_f^*} \left( \frac{S_m^*}{S_f^*} \right)^{\frac{2\lambda_1-1}{1-\lambda_4}} \quad (15)$$

The case of a symmetric loading (pure mode I) corresponds to $m_m = m_f = 0$ then $\sigma_f = \sigma_c$, $\tau_f = 0$, $G_f = G_{lc}$, $A_m^* = A_f^* = A_1$, $S_m^* = S_f^* = \tau_c$ and $F^* = 1$, through proper normalization of the singular modes.

The stress conditions (6) and (7) lead to

$$\left\{ \begin{array}{l} k_{1m} l_m^{\lambda_1 - 1} (s_1 + m_m s_2) \ge \sigma_f(l_m) = \frac{\sigma_c \tau_c}{(\tau_c^q + \mu_m^q \sigma_c^q)^{1/q}} \\ k_{1f} l_f^{\lambda_1 - 1} (s_1 + m_f s_2) \ge \sigma_f(l_f) = \frac{\sigma_c \tau_c}{(\tau_c^q + \mu_f^q \sigma_c^q)^{1/q}} \end{array} \right. \quad \text{with } \mu_a = \frac{|t_1 + m_a t_2|}{s_1 + m_a s_2} \quad a = f, m \quad (16)$$

These inequalities refer only to tensions but by virtue of (7) inequalities in tension and shear are equivalent. It comes finally

$$\alpha = \left( \frac{l_m}{l_f} \right)^{1-\lambda_1} \frac{S_f^*}{S_m^*} \Rightarrow l_m = l_f \alpha^{\frac{1}{1-\lambda_1}} \left( \frac{S_m^*}{S_f^*} \right)^{\frac{1}{1-\lambda_1}} \quad (17)$$

$$\dot{a} = \frac{l_m}{n} = l_f \alpha^{\frac{\beta+\frac{2}{1-\lambda_1}}{1-\lambda_1}} \frac{A_m^*}{A_f^*} \left( \frac{S_m^*}{S_f^*} \right)^{\frac{2\lambda_1}{1-\lambda_1}} = l_f \alpha^{\frac{\beta+\frac{2}{1-\lambda_1}}{1-\lambda_1}} H^* \quad (18)$$

In the symmetric case $H^* = 1$. Parameter $\beta$ can be identified so that (18) coincides with the Paris law of the material in case of a pre-existing long crack: $\lambda_1 = 1/2$ then $\beta = p - 4$ where $p$ is the Paris exponent ($p \ge 4$).

**Remark 1:** If $k_{1m} = k_{1f}$ then $G_m = G_f$, $\alpha = 1$, $F^* = 1$ and $n = 1$, failures occurs at the first cycle. In other terms, the fatigue criterion remains valid for a monotonic loading.

**Remark 2:** In this model, the crack growth under fatigue loading is intermittent, every $n$ cycles, the crack length increases by $l_m$. We put forward the idea that this length could be identified to the striations spacing observed in fatigue in quasi-brittle materials [12].
## AN EXAMPLE

Numerical simulations have been carried out for a V-notch in a PMMA specimen under symmetric loading (loading angle 0° in Fig. 1). Parameters for PMMA are: $E=3250$ MPa (Young's modulus), $\nu=0.3$ (Poisson's ratio), $\sigma_c=70$ MPa (tensile strength), $G_{lc}=0.35$ MPa.mm (toughness), $p=6$ (Paris' exponent [2,12]).

Table 1 shows the invariant parameters that do not depend on the intensity of loading and table 2 those specifically related to fatigue. The remarkable features of these results are, first that the parameter $l_m$ seems to remain approximately constant and depends only on the intensity of the loading, except for wide openings; and second that the crack velocity varies more with respect to the opening angle for low intensity loadings than for higher ones.

<table><caption>Table 1. The singular exponent $\lambda$, the dimensionless scaling coefficient $A'_1 = E A_1$ and the characteristic length at failure $l_c$ under monotonic loading for different openings $\omega$ of the V-notch.</caption><thead><tr><th>ω (°)</th><th>λ</th><th>A'<sub>1</sub></th><th>l<sub>c</sub> (mm)</th></tr></thead><tbody><tr><td>0</td><td>0.500</td><td>6.28</td><td>0.033</td></tr><tr><td>30</td><td>0.502</td><td>6.16</td><td>0.033</td></tr><tr><td>60</td><td>0.512</td><td>5.85</td><td>0.035</td></tr><tr><td>90</td><td>0.545</td><td>5.26</td><td>0.038</td></tr><tr><td>120</td><td>0.616</td><td>4.26</td><td>0.047</td></tr><tr><td>160</td><td>0.819</td><td>2.65</td><td>0.076</td></tr></tbody></table>

<table><caption>Table 2. The characteristic length at failure under fatigue loading $l_m$ (the striations spacing) and the crack advance rate $\dot{a}$, function of the loading intensity $\alpha$ and the opening $\omega$ of the V-notch.</caption><thead><tr><th>α</th><th colspan="2">0.3</th><th colspan="2">0.6</th></tr><tr><th>ω (°)</th><th>l<sub>m</sub> (mm)</th><th>\dot{a} (mm/cycle)</th><th>l<sub>m</sub> (mm)</th><th>\dot{a} (mm/cycle)</th></tr></thead><tbody><tr><td>0</td><td>0.0029</td><td>2.4 10<sup>-5</sup></td><td>0.0117</td><td>0.0015</td></tr><tr><td>30</td><td>0.0029</td><td>2.3 10<sup>-5</sup></td><td>0.0118</td><td>0.0015</td></tr><tr><td>60</td><td>0.0029</td><td>2.2 10<sup>-5</sup></td><td>0.0121</td><td>0.0015</td></tr><tr><td>90</td><td>0.0027</td><td>1.7 10<sup>-5</sup></td><td>0.0125</td><td>0.0015</td></tr><tr><td>120</td><td>0.0021</td><td>0.8 10<sup>-5</sup></td><td>0.0126</td><td>0.0012</td></tr><tr><td>160</td><td>0.0001</td><td>1.0 10<sup>-8</sup></td><td>0.0045</td><td>0.0001</td></tr></tbody></table>
Obviously for wide openings, the initial crack velocity is low and the crack tip accelerates to reach a steady state growth rate corresponding to long cracks. Starting from asymptotic considerations, we can show that this regime is reached for $\omega=160^\circ$ (resp. $\omega=120^\circ$) when the crack length exceeds $5l_m^0$ (resp. $2l_m^0$), where $l_m^0$ holds for $l_m$ at $\omega=0^\circ$. For smaller openings, the steady state seems to be reached very quickly.

## CONCLUSION

The theoretical results established here still await experimental confirmation for both complex monotonic loadings and fatigue. Investigations are underway to perform tests such as those illustrated in Figure 1 with specimens made of PMMA.

Only mechanical loadings were included in the above presentation, but the generalization to thermal loads should not introduce any insuperable difficulties except for some minor technical problems almost solved [11]. The special case of complex exponents (including interface cracks) has not yet been considered.

## REFERENCES

1. Leguillon D. (2002). Strength or toughness? A criterion for crack onset at a notch. Eur. J. of Mechanics – A/Solids, 21, 61-72.

2. Murer S., Leguillon D. (2010). Static and fatigue failure of quasi brittle materials at a V-notch using a Dugdale model. Eur. J. Mech. - A/Solids, 29, 109-118.

3. Dugdale D.S. (1960). Yielding of steel sheets containing slits. J. Mech. Phys. Solids, 8, 100-104.

4. Henninger C., Leguillon D., Martin E. (2007). Crack initiation at a v-notch – Comparison between a brittle fracture criterion and the Dugdale cohesive model, C.R. Mécanique, 335(7), 388-393.

5. Abdelmoula R., Marigo J.J., Weller T. (2009). Construction des lois de fatigue à partir de modèles de forces cohésives : cas d'une fissure en mode I. C.R. Mécanique, 337, 166-172.

6. Paris P.C. (1964). Fatigue, an interdisciplinary approach. 10th Sagamore Army Materials research Conference, Burke, Reed and Weiss Eds., 107-127.

7. Richard H., Benitz K. (1983). A loading device for the creation of mixed mode in fracture mechanics. Int. J. Fract. 22(3), 3591-3618.

8. Yosibash, Z., Priel E., Leguillon D. (2006). A failure criterion for brittle elastic materials under mixed mode loading. Int. J. Fract. 141(1), 289-310.

9. Hutchinson, J., Suo Z. (1992). Mixed mode cracking in layered materials. Advances in Applied Mechanics 29, 63-191.

10. Garcia I.G., Leguillon D. (2011). Mixed-mode crack initiation at a v-notch in presence of an adhesive joint. Submitted to Int. J. Solids Structures.

11. Henninger C., Leguillon D. (2008). Adhesive fracture of an epoxy joint under thermal and mechanical loadings. J. Thermal Stresses, 31(1), 59-76.

12. Suresh S. (2004). Fatigue of materials. Second edition, Cambridge University Press, Cambridge, UK.