Fracture Mechanics Online Class

Cohesive Zone > Cohesive Zone Models

Introduction to the Cohesive Zone Models (CZMs)

Picture VII.5: Validity region of the asymptotic LEFM solution.
Picture VII.6: Cohesive Zone Model (CZM) at the crack tip.

The idea behind the cohesive zone models is to remove the stress singularity of the asymptotic solution, see Picture VII.5, by introducing a plastic zone at the crack tip, see Picture VII.6. The cohesive zone models assume that the nonlinearities are localized at the crack tip, in the so-called process zone which extends on a length $r_p$ ahead of crack tip, see Picture VII.6. The opening of the crack at crack tip is thus non-zero and is referred to as the Crack Tip Opening Displacement (CTOD) $\delta_t$. The two main Cohesive Zone Models are now introduced.

Dugdale's Cohesive Zone/Yield Strip Model

Picture VII.7: Variation of the yield Stress with the Crack Opening Displacement (COD) for the Dugdale's model.
Picture VII.8: Typical stress-strain curves of polymers.
Picture VII.9: Typical stress-strain curve for a specimen exhibiting Lüders' band.

Dugdale's model assumes that the material is elastic-perfectly plastic, that is, the material of the specimen under consideration yields with minimal or zero strain hardening. As a result, the stress $\sigma_y$ in the process zone is uniform for all process zone opening $\delta$ until reaching the crack tip opening displacement $\delta_t$. This stress corresponds to the initial yield stress of the material $\sigma_p^0$ as illustrated in Picture VII.7.

Picture VII.10: Lüders' bands formation (By Errel67 (Image capturing and analysis with DIC system) [Copyrighted free use], via Wikimedia Commons).

Dugdale's model is valid

Barenblatt's Cohesive Model

Picture VII.11: Variation of the yield Stress with the Crack Opening Displacement (COD) for the Barenblatt's model.

Barenblattt's model represents the decrease of the atomic or molecular attraction with the increase of the separation $\delta$, until vanishing at the CTOD $\delta_t$, see Picture VII.11. This model also allows avoiding the stress singularity at the crack tip. Moreover, the fracture response is uniquely governed by the shape of the decreasing curve. However, as the model is directly related to the evolution of the bonding between atoms, this model is valid only for brittle materials.

Description of Dugdale's model

Picture VII.12: Yield Strip model.

Dugdale has developed a model in which the process zone is subjected to a constant surface traction corresponding to the yield stress $\sigma_p^0$, see Picture VII.7 and Picture VII.12. Since the non-linearities are limited in the process zone and are replaced by boundary conditions, the problem can be solved using LEFM. The length of the cohesive zone $r_p$ is then defined in order to remove the stress singularity inherent to LEFM.

Since LEFM is used for solving this problem, the superposition principle holds and the solution of the problem illustrated in Picture VII.12 is the superposition of the three cases shown in Picture VII.13 to Picture VII.15. After having recalled the main ingredients to solve linear elasticity in a cracked plate, the three cases are successively solved.

Picture VII.13: Case 1 of Dugdale's model.
Picture VII.14: Case 2 of Dugdale's model.
Picture VII.15: Case 3 of Dugdale's model.

LEFM resolution using Westergaard functions

Under the assumptions of 2D linear elasticity, the problem can be stated in terms of the Airy function and is thus governed by the bi-harmonic equation:

\begin{equation} \nabla^2 \nabla^2 \Phi = 0 .\label{eq:biharmonic}\end{equation}

One solution of this equation has the form:

\begin{equation} \Phi = \frac{\bar{\zeta}\Omega+\zeta\bar{\Omega} + \omega + \bar{\omega}}{2} ,\label{eq:PhiCrack}\end{equation}

where the functions $\omega(\zeta)$ and $\Omega(\zeta)$ have to be determined so that the boundary conditions are satisfied. The solution fields can be expressed in terms of these functions as follows:

Picture VII.16: Mode I crack lips loading.

The resolution of a loaded crack can be found using the Westergaard functions as detailed in the SIF lecture. For a mode I loading, see Picture VII.16, the choice $\omega''=-\zeta\Omega''$ leads to cancel the shear stress $\mathbf{\sigma}_{xy}$ along the $x$-axis as shown in the SIF lecture. Indeed, the stress (\ref{eq:stressAiry}) and displacement fields (\ref{eq:uAiry}) become, see details in the SIF lecture, respectively

\begin{equation} \left\{ \begin{array}{r c l} \mathbf{\sigma}_{xx} &=& 2\mathcal{R}\left(\Omega'\right)-2 y\mathcal{I}\left(\Omega''\right),\\ \mathbf{\sigma}_{yy} &=& 2\mathcal{R}\left(\Omega'\right)+2 y\mathcal{I}\left(\Omega''\right),\\ \mathbf{\sigma}_{xy} &=& -y2\mathcal{R}\left(\Omega''\right),\text{ and}\\ \mathbf{u}_x &=& \mathcal{R}\left(\mathbf{u}\right) = \frac{1+\nu}{E}\left[\left(\kappa-1\right)\mathcal{R}\left(\Omega\right)-2y\mathcal{I}\left(\Omega'\right)\right],\\ \mathbf{u}_y &=& \mathcal{I}\left(\mathbf{u}\right) = \frac{1+\nu}{E}\left[\left(\kappa+1\right)\mathcal{I}\left(\Omega\right)-2y\mathcal{R}\left(\Omega'\right)\right]. \end{array} \right.\label{eq:WeestMode1}\end{equation}

At this level only $\Omega$ has to be defined so that the stress and displacement fields satisfy the boundary conditions shown in Picture VII.16.

Resolution of Case 1

Picture VII.17: Case 1.

The solution of the first case, see Picture VII.17, is directly obtained using elasticity theory following the derivation in the SIF lecture:

\begin{equation} \left. \begin{array}{l c l} \mathbf{\sigma}_{yy}&=&\mathbf{\sigma}_\infty\\ \mathbf{\sigma}_{xx}&=&\mathbf{\sigma}_{xy}=0\end{array}\right\}\Longrightarrow\left\{ \begin{array}{ r c l} \epsilon_{xx}& = &\frac{\left(1+\nu\right)\left(3-\kappa\right)}{4E}\mathbf{\sigma}_\infty\,,\\ \epsilon_{yy}& =& \frac{\left(1+\nu\right)\left(1+\kappa\right)}{4E}\mathbf{\sigma}_\infty\,,\\ \epsilon_{xy}& =& 0\,,\\ u_{x} &=&\frac{\left(1+\nu\right)\left(3-\kappa\right)}{4E}\mathbf{\sigma}_\infty x\,,\\ u_{y} &=&\frac{\left(1+\nu\right)\left(1+\kappa\right)}{4E}\mathbf{\sigma}_\infty y\,.\\ \end{array}\right. \label{eq:SIFcase1} \end{equation}

Resolution of Case 2

Picture VII.18: Case 2.

A similar problem as the second case, Picture VII.18, has been solved in the SIF lecture, but for a different crack length, and the solution is directly obtained by substituting $a$ by $a+r_p$

\begin{equation} \left\{\begin{array}{l c l} \Omega&=&\frac{\mathbf{\sigma}_\mathbf{\infty}}{2}\bigg(\sqrt{\mathbf{\zeta}^2-{\left(a+r_p\right)}^2}-\mathbf{\zeta}\bigg)\,,\\ \Omega'&=&\frac{\mathbf{\sigma}_\mathbf{\infty}}{2}\bigg(\frac{\mathbf{\zeta}}{\sqrt{\mathbf{\zeta}^2-{\left(a+r_p\right)}^2}}-1\bigg)\,,\\ \Omega''&=&-\frac{\mathbf{\sigma}_\mathbf{\infty}}{2}\bigg(\frac{(a+r_p)^2}{\sqrt{\mathbf{\zeta}^2-{\left(a+r_p\right)}^2}^3}\bigg)\,.\\ \end{array}\right. \label{eq:OmegaCase2}\end{equation}

Picture VII.19: Polar coordinates linked to the cohesive zone tip.

Therefore, considering the polar coordinates linked to the end of the process zone, Picture VII.19, the stress field ahead of the cohesive zone tip follows from the same argumentation as in the SIF lecture, leading to:

\begin{equation} \left\{\begin{array}{l c l} \mathbf{\sigma}_{xy}\left(\mathbf{\theta}=0\right)&=&0\,,\\ \mathbf{\sigma}_{xx}\left(\mathbf{\theta}=0\right)&=&2\mathcal{R}\left(\mathbf{\Omega}'\right)=\mathbf{\sigma}_\mathbf{\infty}\bigg(\frac{x}{\sqrt{x^2-{\left(a+r_p\right)}^2}}-1\bigg)\,,\\ \mathbf{\sigma}_{yy}\left(\mathbf{\theta}=0\right)&=&2\mathcal{R}\left(\mathbf{\Omega}'\right)=\mathbf{\sigma}_\mathbf{\infty}\bigg(\frac{x}{\sqrt{x^2-{\left(a+r_p\right)}^2}}-1\bigg)\,,\\\end{array}\right. \label{eq:SIFCase2}\end{equation}

and the displacement field of the crack lips and process zone also follows from the same argumentation as in the SIF lecture, leading to:

\begin{equation} \lim_{\theta\rightarrow\pm\pi} u_y =\pm \frac{\left(1+\nu\right)\left(\kappa+1\right)\sigma_\infty} {2E} \sqrt{\left(a+r_p\right)^2-x^2} \,.\label{eq:uCase2}\end{equation}

Resolution of Case 3

Picture VII.20: Case 3.

The resolution of the third case is stated as finding $\Omega$ so that the boundary conditions illustrated on see Picture VII.20 are satisfied, i.e. such that

Considering the right part of the sample ($x\geq0$), and using the complex referential represented in Picture VII.19, the solution $\Omega$ which satisfies these conditions is

\begin{equation} \left\{\begin{array}{rcl} \Omega &=&-\frac{\sigma_p^0}{\pi}\sqrt{\mathbf{\zeta}^2-{\left(a+r_p\right)}^2}\text{arcotan}\frac{a}{\sqrt{r_p^2+2ar_p}}-\frac{\sigma_p^0a}{\pi}\text{arcotan}\sqrt{\frac{\mathbf{\zeta}^2-{\left(a+r_p\right)}^2}{r_p^2+2ar_p}}+\\&&\frac{\sigma_p^0\mathbf{\zeta}}{\pi}\text{arcotan}\bigg(\frac{a}{\mathbf{\zeta}}\sqrt{\frac{\mathbf{\zeta}^2-{\left(a+r_p\right)}^2}{r_p^2+2ar_p}}\bigg)\,, \\ \Omega'&=&-\frac{\sigma_p^0}{\pi}\frac{\mathbf{\zeta}}{\sqrt{\mathbf{\zeta}^2-{\left(a+r_p\right)}^2}}\text{arcotan}\frac{a}{\sqrt{r_p^2+2ar_p}}+\frac{\sigma_p^0}{\pi}\text{arcotan}\bigg({\frac{a}{\mathbf{\zeta}}\sqrt{\frac{\mathbf{\zeta}^2-{\left(a+r_p\right)}^2}{r_p^2+2ar_p}}}\bigg)\,,\end{array}\right.\label{eq:OmegaPrimCase3}\end{equation}

as it will be shown here below. The derivative $\Omega'$ is obtained from $\Omega$ using the identity $d_{\mathbf{\zeta}} \text{arcotan} \mathbf{\zeta} = -\frac{1}{1+\mathbf{\zeta}^2}$.

To demonstrate that Eq. (\ref{eq:OmegaPrimCase3}) is the solution of the third problem, it is first necessary to show that this function induces two discontinuities, one between the crack lips and the cohesive zone, and one between the cohesive zone and the sound material. Then the verification of the boundary conditions can be assessed.

We can now evaluate the stress field ahead of the cohesive zone. For $\mathbf{\zeta} = x \pm|\varepsilon| i$, with $\varepsilon\rightarrow 0$ and $a+r_p < x$, Eq. (\ref{eq:OmegaPrimCase3}) is rewritten

\begin{equation} \Omega'=-\frac{\sigma_p^0}{\pi}\frac{x}{\sqrt{x^2-{\left(a+r_p\right)}^2}}\text{arcotan}\frac{a}{\sqrt{r_p^2+2ar_p}}+\frac{\sigma_p^0}{\pi}\text{arcotan}\left(\frac{a}{x}\sqrt{\frac{x^2-{\left(a+r_p\right)}^2}{r_p^2+2ar_p}}\right) \,, \label{eq:OmegaPrimCase3theta0}\end{equation}

and the stress field ahead of the cohesive zone is given from Eq. (\ref{eq:WeestMode1}) as,

\begin{equation}\begin{array}{rcl} \mathbf{\sigma}_{yy}\left(\theta=0\right)=2\mathcal{R}\Omega'\left(\theta=0\right)&=&-\frac{2\sigma_p^0}{\mathbf{\pi}}\frac{x}{\sqrt{x^2-{\left(a+r_p\right)}^2}}\text{arcotan}\frac{a}{\sqrt{r_p^2+2ar_p}}+\\&&\frac{2\sigma_p^0}{\pi}\text{arcotan}\left(\frac{a}{x}\sqrt{\frac{x^2-{\left(a+r_p\right)}^2}{r_p^2+2ar_p}}\right)\,.\end{array}\label{eq:SIFCase3} \end{equation}

The displacement field on the crack lips and on the cohesive zone can be obtained by rewriting Eq. (\ref{eq:OmegaPrimCase3}) for $\mathbf{\zeta} = x \pm|\varepsilon| i$, with $\varepsilon\rightarrow 0$ and $0 \leq x < a+r_p$, as

\begin{equation} \begin{array}{rcl}\Omega\left(\theta\rightarrow\pm\pi\right) &=& \mp \frac{\sigma_p^0 i}{\pi}\sqrt{\left(a+r_p\right)^2-x^2} \text{arcotan} \frac{a}{\sqrt{r_p^2+2ar_p}} - \frac{\sigma_p^0 a }{\pi} \text{arcotan} \left( \pm i \sqrt{\frac{\left(a+r_p\right)^2-x^2}{r_p^2+2ar_p}}\right) +\\&& \frac{ \sigma _p^0{x}}{\pi} \text{arcotan} \left(\pm \frac{ai}{x} \sqrt{\frac{\left(a+r_p\right)^2-x^2}{r_p^2+2ar_p}}\right)\,, \end{array}\label{eq:OmegaPrimCase3thetaPi}\end{equation}

or again using the identity $\text{arcotan}\,\mathbf{\zeta}=\frac{i}{2}\ln\frac{\mathbf{\zeta}-i}{\mathbf{\zeta}+i}$, as

\begin{equation} \begin{array}{rcl} \Omega\left(\theta\rightarrow \pm \pi\right) &=& \mp \frac{\sigma_p^0 i }{\pi}\sqrt{\left(a+r_p\right)^2-x^2} \text{arcotan} \frac{a}{\sqrt{r_p^2+2ar_p}} \mp \frac{ \sigma _p^0 ai }{2\pi} \ln \left(\frac{ \sqrt{ \frac{\left(a+r_p\right)^2-x^2}{r_p^2+2ar_p}}-1} {\sqrt{\frac{\left(a+r_p\right)^2-x^2}{r_p^2+2ar_p}}+1}\right) \pm \\ && \frac{\sigma_p^0{xi}}{2\pi} \ln \frac{\frac{a}{x} \sqrt{ \frac{\left(a+r_p\right)^2-x^2}{r_p^2+2ar_p}}-1} {\frac{a}{x}\sqrt{\frac{\left(a+r_p\right)^2-x^2}{r_p^2+2ar_p}}+1} \,.\end{array}\label{eq:OmegaPrimCase3thetaLn}\end{equation}

We now need to consider separately the displacement fields on the crack lips and on the process zone:

By combining Eq. (\ref{eq:uCase3CrackLips}) and Eq. (\ref{eq:uCase3CohesiveZone}), the displacement field can be written as

\begin{equation}\begin{array}{rcl} u_y\left(\theta\rightarrow \pm \pi\right) &=& \pm \frac{\left(1+\nu\right)\left(1+\kappa\right) \sigma_p^0}{E\pi} \bigg[-\sqrt{\left(a+r_p\right)^2-x^2} \text{arcotan} \frac{a}{\sqrt{r_p^2+2ar_p}} + a \text{arcoth} \sqrt{ \frac{\left(a+r_p\right)^2 - x^2}{r_p^2 + 2ar_p}}\\ &&- x f\left(\frac{a}{x}\sqrt{ \frac{\left(a+r_p\right)^2 - x^2}{r_p^2 + 2ar_p}}\right) \bigg]\,,\end{array} \label{eq:uCase3}\end{equation}


\begin{equation}f(x) = \left\{\begin{array}{ll} \text{arcoth}(x) &\text{ if } x<a\, ;\\ \text{arctanh}(x) &\text{ if } a<x<a+r_p\,.\end{array}\right. \label{eq:fUCase3}\end{equation}