Construction and justification of Paris-like fatigue laws from Dugdale-type cohesive models

Abstract

We propose a construction of fatigue laws from cohesive forces models in the case of a crack submitted to a mode I cyclic loading. Taking the cumulated opening as the memory variable and the surface energy density associated with Dugdale’s model, we explicitly construct the fatigue law which gives the crack growth rate by cycle dℓ/dN in terms of the stress intensity factor K _I . In particular, we recover a Paris law with an exponent 4, i.e., dℓ/dN = C K ⁴ _I , when K _I is small, the coefficient C being explicitly expressed in terms of the material parameters. Furthermore, the law can be applied in the full range of values of K _I and can be extended to non simple cycles.

Appendix 1: The generic problem in the neighborhood of the crack tip

The plane is equipped with the cartesian coordinate system (x, y), the associated canonical basis is (i, j) and $z=x+{\tt i} y$ denotes the affixe of the complex number associated with the point (x, y). Let ${\tt K}$, ${\tt L}$ and ${\tt D}$ be three given real numbers with ${\tt K}>0$ and ${\tt D}>0$. Let us consider the following plane-strain elastic problem whose unknowns are the displacement and stress field U and $\varvec{\Upsigma}$:

$$ \begin{aligned} &{\bf div}\,\varvec{\Upsigma}=0, \quad \varvec{\Upsigma}=\lambda \hbox{div}{\bf U I}+2\mu \varvec{\varepsilon} ({\bf U})\quad\hbox{in}\quad {\mathbb{R}}^2\setminus(-\infty,{\tt L}+{\tt D})\times\{0\},\\ &\Upsigma_{12}=\Upsigma_{22}=0\quad\hbox{ on }\quad (-\infty,{\tt L})\times \{0\}, \quad \quad \Upsigma_{12}=0,\Upsigma_{22}=\sigma_c \quad \hbox{ on } \quad ({\tt L, L}+{\tt D})\times \{0\}, \end{aligned} $$

with the condition at infinity

$$ \hbox{lim}_{r\to\infty}\left({\bf U}(x,y)-\frac{\tt K}{2\mu} \sqrt{\frac{r}{2\pi}}{\bf u}^S(\theta)\right)=0 $$

where $x=r\cos\theta, y=r\sin\theta$ and ${\bf u}^S(\theta)=(3-4\nu-\cos\theta)\left(\cos \frac{\theta}{2} {\bf i}+ \sin \frac{\theta}{2} {\bf j}\right)$. This problem admits a unique solution which can be obtained in a closed form by using the theory of complex potentials, cf [26]. We simply recall here the main results. The fields U et $\varvec{\Upsigma}$ are related to the function φ(z) of the complex variable z by

$$ \begin{aligned} \Upsigma_{22}(x,y)-{\tt i}\Upsigma_{12}(x,y)&=\varphi^\prime(z)+\varphi^\prime\left(\overline{z}\right)+ \left(z-\overline{z}\right)\overline{\varphi^{\prime\prime}(z)},\\ 2\mu\left(U_1(x,y)+{\tt i} U_2(x,y)\right)&=(3-4\nu)\varphi(z)-\varphi\left(\overline{z}\right)- \left(z-\overline{z}\right) \overline{\varphi^\prime(z)}, \end{aligned} $$

φ being holomorphic in the plane without the half-line $(-\infty,\,{\tt L}+{\tt D})\times\{0\}$, the bar denoting the complex conjugate. By a standard procedure, we get

$$ \varphi^\prime(z)=\frac{\sigma_c}{2\pi\sqrt{z-{\tt L}-{\tt D}}}\int\limits_{{\tt L}}^{{\tt L}+{\tt D}}\frac{\sqrt{{\tt L}+{\tt D}-x}}{x-z} dx+\frac{\tt K}{2\sqrt{2\pi( z-{\tt L}-{\tt D})}}. $$

(51)

Near the tip $z={\tt L}+{\tt D}, \varphi^\prime(z)$ behaves like

$$ \varphi^\prime(z)\approx \frac{\sqrt{2\pi}{\tt K}-4\sigma_c\sqrt{{\tt D}}}{4\pi\sqrt{z-{\tt L}-{\tt D}}} $$

and hence the stresses are singular with the usual singularity in $1/\sqrt{r}$ except if the factor $\sqrt{2\pi}{\tt K}-4\sigma_c\sqrt{{\tt D}}$ vanishes. Specifically, the jump of the normal displacement just behind the tip $x={\tt L}+{\tt D}$ and the normal stress just ahead the tip read as

$$ \lbrack\!\lbrack U_2 \rbrack\!\rbrack (r)=\frac{2\left(1-\nu^2\right)}{\pi E}\left(\sqrt{2\pi}{\tt K}-4\sigma_c\sqrt{{\tt D}}\right)\sqrt{r}+\cdots,\quad \Upsigma_{22}(r)=\frac{\sqrt{2\pi}{\tt K}-4\sigma_c\sqrt{{\tt D}}}{4\pi\sqrt{r}}+\cdots. $$

Therefore, if it is required that $\Upsigma_{22}\le\sigma_c$ on the half-line $({\tt L}+{\tt D},+\infty) \times \{0\},$ then ${\tt K}$ and ${\tt D}$ must be such that $\sqrt{2\pi}{\tt K}\le4\sigma_c\sqrt{{\tt D}}$. On the other hand, if it is required that $\lbrack\!\lbrack U_2 \rbrack\!\rbrack \ge 0$ holds everywhere (by a non interpenetration condition, for instance), then ${\tt K}$ and ${\tt D}$ must satisfy the converse inequality $\sqrt{2\pi}{\tt K}\ge4\sigma_c\sqrt{{\tt D}}$. Accordingly, in order that both conditions are satisfied, the solution must be non singular at the tip ${\tt L}+{\tt D}$. In such a case ${\tt D}$ and ${\tt K}$ are related by

$$ {\tt K}=4\sigma_c\sqrt{\frac{\tt D}{2\pi}}. $$

(52)

Assuming from now on that (52) holds, (51) becomes

$$ {\varphi}^\prime(z)=\frac{\sigma_c}{2} + \frac{{\tt i} \sigma_c}{2\pi} \left(\hbox{Log}\left(\sqrt{{\tt D}}+{\tt i}\sqrt{z-{\tt L}-{\tt D}}\right)-\hbox{Log}\left(\sqrt{{\tt D}}-{\tt i}\sqrt{z-{\tt L}-{\tt D}}\right)\right). $$

(53)

In (53), Log denotes the principal determination of the complex logarithm. After some calculations, one obtains that the normal jump of the displacement along the x-axis reads as

$$ \lbrack\!\lbrack U_2 \rbrack\!\rbrack (x)= V \left(\frac{x-{\tt L}} {\tt D}\right)\frac{8\left(1-\nu^2\right)}{\pi}\frac{\sigma_c}{E}{\tt D},$$

(54)

where V denotes the dimensionless real-valued function defined by

$$ V(\zeta)= \left\{ \begin{array}{ll} \sqrt{1-\zeta}-\zeta\ln\left(1+\sqrt{1-\zeta}\right)+\zeta\ln \sqrt{|\zeta|}&\hbox{if }\; \zeta\le 1, \zeta\ne0\\ 0&\hbox{if }\;\zeta\ge 1 \end{array} \right. $$

(55)

and V(0) = 1. Let us note that V is continuously differentiable everywhere (even at ζ = 0 and ζ = 1), is concave for ζ ≤ 0 and is strictly decreasing from ∞ to 0 when ζ goes from −∞ to 1. When ζ→ −∞, $V(\zeta)=2\sqrt{|\zeta|}+o(1)$. The non interpenetration condition $\lbrack\!\lbrack U_2 \rbrack\!\rbrack \ge 0$ is satisfied everywhere. The normal stress $\Upsigma_{22}$ along the half-line $({\tt L}+{\tt D},+\infty) \times \{0\}$ is given by

$$ \Upsigma_{22}(x,0)=\left(1-\frac{2}{\pi} \arcsin\sqrt{1-\frac{\tt D}{x-{\tt L}}}\right)\sigma_c. $$

(56)

It decreases from σ _c to 0 and, therefore, the condition $\Upsigma_{22}\le\sigma_c$ is satisfied.