Fracture Mechanics Online Class

Energetic Approach > J-Integral

As seen in the previous pages, the crack closure integral has some limitations: It can only be "easily" computed for linear and elastic materials, and is useful for cracks growing straight ahead only. The following introduces a more general energy-related concept called the J-Integral.

The J-integral concept

Homogeneous bodies

Picture III.25: Definition of the contour.

In 1968, Rice suggested to compute the energy that flows toward the crack tip. To do so he considered a homogeneous un-cracked body $B$ with the following conditions:

Using these notations the J-integral is a vector defined as:

\begin{equation} \mathbf{J} = \int_{\partial D} \left[U\left(\mathbf{\nabla}\mathbf{u}\right) \mathbf{n} - \left(\mathbf{\nabla}\mathbf{u}\right)^T \mathbf{T}\right]dA,\label{eq:Jintegral} \end{equation}

or, in the indicial form:

\begin{equation} \mathbf{J}_i = \int_{\partial D} \left[U\left(\mathbf{\varepsilon}\right) \mathbf{n}_i - \mathbf{\nabla}_i \mathbf{u}_k \mathbf{\sigma}_{km} \mathbf{n}_m\right] dA.\label{eq:JintegralIndices}

As $\mathbf{\sigma}^T=\mathbf{\sigma}$ and $\nabla \cdot \mathbf{\sigma} = 0$, applying the Gauss theorem on this last expression leads to

\begin{eqnarray} \mathbf{J}_i &=& \int_{D} \left[\underbrace{\frac{\partial U}{\partial \mathbf{\varepsilon}_{km}}}_{\mathbf{\sigma}_{km}}\frac{\mathbf{u}_{k,mi}+\mathbf{u}_{m,ki}}{2} - \mathbf{\sigma}_{km}\mathbf{u}_{k,im}-\underbrace{\mathbf{\sigma}_{km,m}}_{=0}\mathbf{u}_{k,i} \right] dD=\mathbf{\sigma}_{km}\mathbf{u}_{k,mi}- \mathbf{\sigma}_{km}\mathbf{u}_{k,im}= 0. \label{eq:Jhomog}\end{eqnarray}

This relation means that the energy flowing through a closed surface in an homogeneous medium is equal to zero. But what happens if the body is heterogeneous or cracked?

Heterogeneous bodies

For heterogeneous bodies the internal material potential $U(\mathbf{\nabla} \mathbf{u},\, \mathbf{X})$ depends on the position $\mathbf{X}$, so when applying the Gauss theorem we should consider that

\begin{equation} \frac{D U\left(\mathbf{\nabla}\mathbf{u},\mathbf{X}\right)}{D\mathbf{X}_i} = \frac{\partial U}{\partial \mathbf{\varepsilon}}:\mathbf{\varepsilon}_{,i} +\frac{\partial U}{\partial \mathbf{X}_i}. \end{equation}

As a result $\mathbf{J} \neq 0$ even for un-cracked bodies as the second term remains in (\ref{eq:Jhomog}).

Cracked homogeneous bodies (2D)

Picture III.26: J-integral for a cracked homogeneous material.

Let us now consider a cracked homogeneous material. We can still use (\ref{eq:Jhomog}) if the contour $\Gamma$ embeds a homogeneous part, meaning if it does not intercept the crack. Considering a crack parallel to the x-axis with the contour $\Gamma=\Gamma_1+\Gamma^++\Gamma^--\Gamma_2$ going around the crack, see Picture III.26, then (\ref{eq:Jhomog}) is satisfied and

\begin{equation} \mathbf{J}_x = \oint_\Gamma \left[U\left(\mathbf{\varepsilon}\right) \mathbf{n}_x - \mathbf{\mathbf{u}}_{,x} \cdot \mathbf{T}\right] dl = 0, \label{eq:JCrackedFull} \end{equation}

with $\oint_\Gamma = \int_{\Gamma_1} + \int_{\Gamma^+}+ \int_{\Gamma^-} - \int_{\Gamma_2}$. We can now analyze the contribution on the different curves. Along $\Gamma^+$ and $\Gamma^-$ we have

So (\ref{eq:JCrackedFull}) simplifies into the so-called J-integral, which is the energy that flows toward the crack tip:

\begin{equation} J = \int_{\Gamma_1} \left[U\left(\mathbf{\varepsilon}\right) \mathbf{n}_x - \mathbf{\mathbf{u}}_{,x} \cdot \mathbf{T}\right] dl = \int_ {\Gamma_2} \left[U\left(\mathbf{\varepsilon}\right) \mathbf{n}_x - \mathbf{\mathbf{u}}_{,x} \cdot \mathbf{T}\right] dl.\label{eq:J} \end{equation}

As it can be deduced from (\ref{eq:J}), this integral

This concept is thus particularly general and is widely used in analytical and numerical analyzes, as it will be shown later.

Particular cases and the J-integral

Although the concept is general as long as an internal potential exists, the J-Integral can be specialized and in particular can be related to the concepts of energy release rate and of SIFs studied before.

For cracks growing straight ahead

Picture III.27: Recall of the crack closure integral.

Heading back to the crack closure integral, we have found the following potential energy variation for a growing cavity, see Picture III.27:

\begin{eqnarray} \Delta \Pi_T &=& \int_{B-\Delta B} U\left(\mathbf{\nabla}\mathbf{u}+\mathbf{\nabla}\Delta\mathbf{u}\right)-U\left(\mathbf{\nabla}\mathbf{u}\right) dB-\nonumber\\&&\int_{\Delta B} U\left(\mathbf{\nabla}\mathbf{u}\right) dB - \int_{\partial_N B} \bar{\mathbf{T}} \cdot\Delta\mathbf{u}d\partial B.\label{eq:DpiCCI1} \end{eqnarray}

The energy release rate for a crack (no change of volume) can thus be evaluated as

\begin{eqnarray} G = \lim_{\Delta a \rightarrow 0} - \frac{\Delta \Pi_T}{ t \Delta a} = \lim_{\Delta a \rightarrow 0} \frac{1}{\Delta a}\left\{ \int_{B}
U\left(\mathbf{\nabla}\mathbf{u}\right) -U\left(\mathbf{\nabla}\mathbf{u}+\mathbf{\nabla}\Delta\mathbf{u}\right) d B + \int_{\partial_N B} \bar{\mathbf{T}} \cdot \Delta\mathbf{u} d \partial B\right\}.\label{eq:JToG1} \end{eqnarray}

As the crack is stress-free, as $\Delta \mathbf{u}=0$ on $\partial_DB$, and as $\Delta\mathbf{\sigma}=0$ on $\partial_N B$, the last term of this equation can be simplified into

\begin{equation} \int_{\partial_N B}\bar{\mathbf{T}} \cdot \Delta \mathbf{u} d \partial B = \int_{\partial_N B+\partial_D B + S+\Delta S} \Delta \mathbf{u}\cdot \left(\mathbf{\sigma}+\Delta\mathbf{\sigma}\right)\cdot \mathbf{n} d \partial B = \int_{B} \mathbf{\nabla}\cdot\left( \Delta \mathbf{u}\cdot \left(\mathbf{\sigma}+\Delta\mathbf{\sigma}\right)\right) d B.\label{eq:JToG2}, \end{equation}

or, as $\mathbf{\nabla}\cdot\mathbf{\sigma}=\mathbf{\nabla}\cdot\left(\mathbf{\sigma}+\Delta\mathbf{\sigma}\right)=0$, into

\begin{equation} \int_{\partial_N B}\bar{\mathbf{T}} \cdot \Delta \mathbf{u} d \partial B = \int_{B} \left(\mathbf{\sigma}+\Delta\mathbf{\sigma}\right):\left(\mathbf{\nabla} \Delta \mathbf{u}\right) d B\label{eq:JToG3}. \end{equation}

Using this last result in (\ref{eq:JToG1}) yields

G = \lim_{\Delta a \rightarrow 0} \frac{1}{\Delta a}\left\{ \int_{B} U\left(\mathbf{\varepsilon}\right) - U\left(\mathbf{\varepsilon}+\Delta\mathbf{\varepsilon}\right) +
\left(\mathbf{\sigma}+\Delta\mathbf{\sigma}\right):\Delta\mathbf{\varepsilon} d B\right\}.\label{eq:JToG4} \end{equation}

Picture III.28: Crack growing straight ahead.

Assuming the crack grows straight ahead we can consider the following change of variables, see Picture III.28,

\begin{equation} \begin{cases} x' = x- a\\ y' = y \end{cases} \rightarrow \begin{cases}f(x',y',a) = f(x-a,y,a) \\ \frac{d f}{d a} = - \partial_{x'} f +\partial_a f\end{cases}. \label{eq:JToGCV}\end{equation}

The energy release rate (\ref{eq:JToG4}) involves the whole body $B$. However as $\Delta a\rightarrow 0$, the non-vanishing contributions are around the crack tip. So we can limit the integral to a fixed region $D$ of boundary $\Gamma$, see Picture III.28, and this equation becomes

\begin{eqnarray} G = \lim_{\Delta a \rightarrow 0} \frac{1}{\Delta a}&&\left\{ \int_{D} U\left(\mathbf{\varepsilon}\right) - U\left(\mathbf{\varepsilon}+\Delta\mathbf{\varepsilon}\right) +\right.\nonumber\\&&\left.\left(\mathbf{\sigma}+\Delta\mathbf{\sigma}\right):\Delta\mathbf{\varepsilon} d D\right\}.\label{eq:JToG5} \end{eqnarray}

We can integrate by parts and apply the Gauss theorem on the last term of (\ref{eq:JToG5}), and as $\mathbf{\nabla}\cdot\left(\mathbf{\sigma}+\Delta\mathbf{\sigma}\right)=0$, it yields

\begin{equation} \int_D \left(\mathbf{\sigma}+\Delta \mathbf{\sigma}\right): \Delta \mathbf{\varepsilon} dD = \int_\Gamma \Delta \mathbf{u}\cdot\left(\mathbf{\sigma}+\Delta \mathbf{\sigma}\right)\cdot\mathbf{n} dl ,\label{eq:JToG6}\end{equation}


\begin{equation}\lim_{\Delta a\rightarrow 0}\frac{1}{\Delta a} \int_D \left(\mathbf{\sigma}+\Delta \mathbf{\sigma}\right): \Delta \mathbf{\varepsilon} dD =\lim_{\Delta a\rightarrow 0}\frac{1}{\Delta a} \int_\Gamma \Delta \mathbf{u}\cdot\left(\mathbf{\sigma}+\Delta \mathbf{\sigma}\right)\cdot\mathbf{n} dl .\label{eq:JToG8}\end{equation}

Considering the change of variables (\ref{eq:JToGCV}), this last equation reads

\begin{equation}\lim_{\Delta a\rightarrow 0}\frac{1}{\Delta a} \int_D \left(\mathbf{\sigma}+\Delta \mathbf{\sigma}\right): \Delta \mathbf{\varepsilon} dD =\int_\Gamma \frac{d
\mathbf{u}}{d a} \cdot \mathbf{T} dl = \int_\Gamma \left( - \partial_{x'} \mathbf{u} +\partial_a \mathbf{u} \right) \cdot \mathbf{T} dl. \label{eq:JToG9}\end{equation}

One more time using $\mathbf{\nabla}\cdot\mathbf{\sigma}=\mathbf{\nabla}\cdot\partial_{\mathbf{\varepsilon}}U=0$ we have

\begin{equation} \int_\Gamma \partial_a \mathbf{u} \cdot \mathbf{T} dl = \int_\Gamma \partial_a\mathbf{u} \cdot \partial_{\mathbf{\varepsilon}} U \cdot \mathbf{n} dl = \int_D
\partial_a\mathbf{\varepsilon} : \partial_{\mathbf{\varepsilon}} U d D = \int_D \partial_a U d D,\label{eq:JToG10} \end{equation}

and (\ref{eq:JToG9}) reads

\begin{equation}\lim_{\Delta a\rightarrow 0}\frac{1}{\Delta a} \int_D \left(\mathbf{\sigma}+\Delta \mathbf{\sigma}\right): \Delta \mathbf{\varepsilon} dD = - \int_\Gamma \partial_{x'}
\mathbf{u} \cdot \mathbf{T} dl + \int_D \partial_a U d D. \label{eq:JToG11}\end{equation}

Picture III.29: Crack growing straight ahead: moving domain.

We can now consider the first term of (\ref{eq:JToG5}). The domain $D$ is fixed to the initial crack tip, but we can define a domain $D^\star$ moving with the crack tip such that $D=D^\star+\Delta D_L-\Delta D_R$, see Picture III.29. The first term of (\ref{eq:JToG5}) reads

\begin{eqnarray} \lim_{\Delta a \rightarrow 0} \frac{1}{\Delta a}\int_{D} U\left(\mathbf{\varepsilon}\right) - U\left(\mathbf{\varepsilon}+\Delta\mathbf{\varepsilon}\right) d D =\quad\quad\quad\quad\nonumber\\
\lim_{\Delta a \rightarrow 0} \frac{1}{\Delta a}\left\{ \int_{D}
U\left(\mathbf{\varepsilon}\left(a\right)\right) dD -\right.\quad\quad\nonumber\\
\left. \int_{D*+\Delta D_L -
\Delta D_R} U\left(\mathbf{\varepsilon}\left(a+\Delta a\right)\right) d D\right\}.\label{eq:JToG12} \end{eqnarray}

As $D^\star \rightarrow D$ for infinitesimal crack growth, this relation becomes (formally, one should use derivatives & limits of integrals with non-constant intervals)

\begin{eqnarray} \lim_{\Delta a \rightarrow 0} \frac{1}{\Delta a}\int_{D} U\left(\mathbf{\varepsilon}\right) - U\left(\mathbf{\varepsilon}+\Delta\mathbf{\varepsilon}\right) d D &=&
\lim_{\Delta a \rightarrow 0} \left\{ \int_{D} \frac{U\left(\mathbf{\varepsilon}\left(a\right)\right)-U\left(\mathbf{\varepsilon}\left(a+\Delta a\right)\right) }{\Delta a} dD -\right.\nonumber\\
&&\left. \frac{1}{\Delta a} \int_{\Delta D_L - \Delta D_R} U\left(\mathbf{\varepsilon}\left(a+\Delta a\right)\right) d D\right\} .\label{eq:JToG13}\end{eqnarray}

Picture III.30: Crack growing straight ahead: moving left domain.

These integrals can be successively computed:

Picture III.31: Crack growing straight ahead: moving right domain.

Combining (\ref{eq:JtoGtmp1}-\ref{eq:JtoGtmp3}), and as $\Gamma_L + \Gamma_R^\star\rightarrow $\Gamma$, the relation (\ref{eq:JToG13}) becomes

\begin{equation} \lim_{\Delta a \rightarrow 0} \frac{1}{\Delta a}\int_{D} U\left(\mathbf{\varepsilon}\right) - U\left(\mathbf{\varepsilon}+\Delta\mathbf{\varepsilon}\right) d D=-\int_{D} \partial_a U dD +\int_{ \Gamma} U\left(\mathbf{\varepsilon}\left(a\right)\right)\mathbf{n}_x d l .\label{eq:JToG14} \end{equation}

If the crack grows straight ahead, for $\Delta a \rightarrow 0$, we have found (\ref{eq:JToG11}) and (\ref{eq:JToG14}) which are summarized as

\begin{equation} \begin{cases} \lim_{\Delta a \rightarrow 0} \frac{1}{\Delta a}\int_{D} U\left(\mathbf{\varepsilon}\right) - \left(\mathbf{\varepsilon}+\Delta\mathbf{\varepsilon}\right) d D = -\int_{D} \partial_a U dD +\int_{ \Gamma} U\left(\mathbf{\varepsilon}\left(a\right)\right)\mathbf{n}_x dl \\ \lim_{\Delta a\rightarrow 0}\frac{1}{\Delta a} \int_D
\left(\mathbf{\sigma}+\Delta \mathbf{\sigma}\right): \Delta \mathbf{\varepsilon} dD = - \int_\Gamma \partial_{x'}
\mathbf{u} \cdot \mathbf{T} dl + \int_D \partial_a U d D \end{cases}, \end{equation}

so that (\ref{eq:JToG5}) finally reads

\begin{equation} G = \int_{ \Gamma} U\left(\mathbf{\varepsilon}\left(a\right)\right)\mathbf{n}_x d l- \int_\Gamma \partial_{x}
\mathbf{u} \cdot \mathbf{T} dl = J. \end{equation}

So, if the material is defined by an internal potential and if the crack grows straight ahead, $G=J$.

Linear Elasticity

Picture III.32: J-Integral with a circular contour.

Let us now consider the case of linear elastic materials. The general expression of $J$ reads

\begin{equation} J = \int_{\Gamma} \left[U\left(\mathbf{\varepsilon}\right) \mathbf{n}_x - \mathbf{u}_{,x} \cdot \mathbf{T}\right] dl, \label{eq:Jelast1}\end{equation}

and can be particularized for linear elasticity ($U=\frac{\mathbf{\sigma}\mathbf{\varepsilon}}{2}$) as

\begin{equation} J = \int_{\Gamma} \left[\frac{\mathbf{\sigma}_{ij}\mathbf{u}_{i,j}}{2}\delta_{xk} - \mathbf{u}_{i,x} \mathbf{\sigma}_{ik}\right]\mathbf{n}_k dl .\label{eq:Jelast2}\end{equation}

Let the contour $\Gamma$ be a circle as shown in Figure III.32. Thus as

\begin{equation} \begin{pmatrix} \partial_x \\ \partial_y \end{pmatrix} = \begin{pmatrix} \cos \theta & -\frac{\sin \theta}{r} \\ \sin \theta & \frac{\cos \theta}{r} \end{pmatrix} \begin{pmatrix} \partial_r \\ \partial_\theta \end{pmatrix},\label{eq:Jelast3} \end{equation}

the J-Integral (\ref{eq:Jelast2}) becomes

\begin{eqnarray} J &=& \int_{-\pi}^{\pi} \frac{r\mathbf{\sigma}_{ix}\cos{\theta}u_{i,r}-\mathbf{\sigma}_{ix}\sin{\theta}u_{i,\theta}+ r \mathbf{\sigma}_{iy} \sin{\theta}u_{i,r} + \mathbf{\sigma}_{iy}\cos{\theta}u_{i,\theta}}{2}\cos{\theta} d\theta-\nonumber\\
&& \int_{-\pi}^{\pi} \left[r\mathbf{\sigma}_{ix}\cos{\theta}u_{i,r}-\mathbf{\sigma}_{ix}\sin{\theta}u_{i,\theta}\right]\cos{\theta}
d\theta-\nonumber\\&& \int_{-\pi}^{\pi} \left[r\mathbf{\sigma}_{iy}\cos{\theta}u_{i,r}-\mathbf{\sigma}_{iy}\sin{\theta}u_{i,\theta}\right]\sin{\theta} d\theta \label{eq:Jelast4}.\end{eqnarray}

As J is path-independent we can take $r$ tending to 0 and can thus use the asymptotic solution

\begin{equation} \begin{cases} \mathbf{\sigma}^\text{mode i} = \frac{K_i}{\sqrt{2 \pi r}} \mathbf{f}^\text{mode i}(\theta) \\ \mathbf{u}^\text{mode i} = K_i \sqrt{\frac{r}{2 \pi}} \mathbf{g}^\text{mode i}(\theta) \end{cases}. \end{equation}

Superposing the fracture modes yields

\begin{equation} \begin{cases} \mathbf{u} = \sqrt{\frac{r}{2\pi}}\sum_{i=I}^{III} K_i \mathbf{g}^{\text{mode i}}\left(\theta\right) \\ \mathbf{\sigma} = \frac{1}{\sqrt{2\pi r}}\sum_{i=I}^{III} K_i \mathbf{f}^{\text{mode i}}\left(\theta\right) \end{cases}, \end{equation}

and after some manipulations the J-integral (\ref{eq:Jelast4}) becomes

\begin{equation} J = \frac{K^2_I}{E'}+\frac{K^2_{II}}{E'}+\frac{K^2_{III}}{2\mu}.\label{eq:Jelast5} \end{equation}

So, if the material is linear and elastic for any direction of the crack growth $J=\frac{K^2_I}{E'}+\frac{K^2_{II}}{E'}+\frac{K^2_{III}}{2\mu}$.