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Irwin arrived at the definition of \(K\) as a near-crack-tip approximation to Westergaard's complete solution for the stress field surrounding a crack [2]. Recall that Westergaard used complex numbers and Airy stress functions to do so. We will first review Westergaard's solution, and then see how Irwin used it to develop the stress intensity factor.

The complete set of equations for the stress field is

\[ \begin{eqnarray} \sigma_{xx} & = & \text{Re}\,Z - y \, \text{Im} \, Z' \\ \\ \sigma_{yy} & = & \text{Re}\,Z + y \, \text{Im} \, Z' \\ \\ \tau_{xy} & = & - y \, \text{Re} \, Z' \end{eqnarray} \]

where \(Z\) and its derivative, \(Z'\), are

\[ Z(z) = {\sigma_\infty \over \sqrt{1 - \left( a \over z \right)^2}} \quad \qquad \text{and} \qquad \quad Z'(z) = {- \sigma_\infty \, a^2 \over z^3 \left[ 1 - \left( a \over z \right)^2 \right]^{3/2} } \]

and \(a\) is crack length, and \(z\) equals \(x + i y\).

When \(y = 0\), the equation for \(\sigma_{yy}\) reduces to

\[ \sigma_{yy} \; = \; { \sigma_\infty \over \sqrt{1 - \left( a \over x \right)^2} } \]

A plot is shown below. It shows the stress value quickly dropping from infinity at the crack tip, \(x = a\), to the far-field value of \(\sigma_\infty\). Keep in mind that this is along the crack plane, \(y=0\).

The equation and its graph are the key results of Westergaard's solution that are shared by most authors, and with good reason. Computing the stress at any other position near the crack requires a Taylor series expansion of the function in order to partition it into its real and imaginary parts... a great deal of work. This imposes a significant hurdle to the general understanding of the stress state surrounding a crack.

\[ z = a + r e^{i \theta} \]

The key feature of this expression is that \(r = 0\) at the crack tip (\(x=a\)). This is reflected by the red components in the figure. And note that \(r \lt \lt a\) in the region very near the crack tip. This inequality was used by Irwin to find simple expressions (well, compared to the complex functions) for the stress components near the crack tip in terms of polar coordinates, \(r\) and \(\theta\). And we will see how, along the way, the definition of the stress intensity factor naturally falls out.

\[ Z(z) = {\sigma_\infty \over \sqrt{1 - \left( a \over a + r e^{i \theta} \right)^2}} \]

and applying some algebra, specifically obtaining a common denominator, gives

\[ Z(z) = {\sigma_\infty \over \sqrt{ 2 a r e^{i \theta} + r^2 e^{i 2 \theta} \over a^2 + 2 a r e^{i \theta} + r^2 e^{i 2 \theta} }} \]

It's at this point that Irwin imposed some physical insight on the problem in order to simplify the equation. He recognized that the area immediately near the crack tip corresponds to very small values of \(r\), so small in fact that \(a \gt \gt r\). (This is exactly why he expressed \(z\) as \(a + r e^{i \theta}\), after all.) Therefore, \(a^2 \gt \gt a \, r \gt \gt r^2\), and all terms following the first in both the numerator and denominator can be neglected, leaving

\[ Z(z) = {\sigma_\infty \over \sqrt{ 2 \, a \, r \, e^{i \theta} \over a^2 } } \]

which can be further simplified to

\[ Z(z) = \sigma_\infty \sqrt{ a \over 2 \, r } \; e^{-i {\theta \over 2} } \]

Euler's famous identity, \(e^{i \phi} = \cos \phi + i \, \sin \phi\), is next used to get

\[ Z(z) = \sigma_\infty \sqrt{ a \over 2 \, r } \; \left( \cos {\theta \over 2} - i \, \sin {\theta \over 2} \right) \]

And since Westergaard's solution only needs the real part of \(Z(z)\), then

\[ \text{Re} \, Z = \sigma_\infty \sqrt{ a \over 2 \, r } \; \cos {\theta \over 2} \]

\[ \begin{eqnarray} Z'(z) & = & {- \sigma_\infty \, a^2 \over z^3 \left[ 1 - \left( a \over z \right)^2 \right]^{3/2} } \\ \\ & = & {- \sigma_\infty \, a^2 \over (a + r e^{i \theta})^3 \left[ 1 - \left( a \over (a + r e^{i \theta}) \right)^2 \right]^{3/2} } \\ \\ & = & {- \sigma_\infty \, a^2 \over (a + r e^{i \theta})^3 \left[ 2 a r e^{i \theta} + r^2 e^{i 2 \theta} \over a^2 + 2 a r e^{i \theta} + r^2 e^{i 2 \theta} \right]^{3/2} } \\ \end{eqnarray} \]

And neglect negligible terms again because \(a \gt \gt r\).

\[ Z'(z) = {- \sigma_\infty \, a^2 \over a^3 \left[ 2 a r e^{i \theta} \over a^2 \right]^{3/2} } \]

This simplifies to

\[ Z'(z) = {- \sigma_\infty \over 2 \, r} \sqrt{ a \over 2 \, r} \; e^{-i {3 \theta \over 2}} \]

Using the Euler identity gives

\[ Z'(z) = {- \sigma_\infty \over 2 \, r} \sqrt{ a \over 2 \, r} \; \left( \cos {3 \theta \over 2} - i \sin {3 \theta \over 2} \right) \]

\[ \sigma_{xx} = \text{Re}\,Z - y \, \text{Im} \, Z' \]

Inserting the appropriate real and imaginary components gives Irwin's near crack tip approximation

\[ \sigma_{xx} = \sigma_\infty \sqrt{ a \over 2 \, r } \; \cos {\theta \over 2} - y \; {\sigma_\infty \over 2 \, r} \; \sqrt{ a \over 2 \, r} \; \sin {3 \theta \over 2} \]

which simplifies to

\[ \sigma_{xx} = \sigma_\infty \sqrt{ a \over 2 \, r } \; \left( \cos {\theta \over 2} - {y \over 2 \, r} \; \sin {3 \theta \over 2} \right) \]

And \(y\) can be replaced by \(r \sin \theta\). This gives

\[ \sigma_{xx} = \sigma_\infty \sqrt{ a \over 2 \, r } \; \left( \cos {\theta \over 2} - {1 \over 2} \; \sin \theta \; \sin {3 \theta \over 2} \right) \]

The following trig identity is used next: \({1 \over 2} \sin \theta = \sin {\theta \over 2} \cos {\theta \over 2}\)

\[ \sigma_{xx} = \sigma_\infty \sqrt{ a \over 2 \, r } \; \left( \cos {\theta \over 2} - \sin {\theta \over 2} \cos {\theta \over 2} \sin {3 \theta \over 2} \right) \]

which permits a \(\cos {\theta \over 2}\) to be factored out of both terms.

\[ \sigma_{xx} = {\sigma_\infty \sqrt{a} \over \sqrt{ 2 r }} \cos {\theta \over 2} \left( 1 - \sin {\theta \over 2} \sin {3 \theta \over 2} \right) \]

Irwin's development stopped here, and this is indeed the natural place to stop. Nevertheless, in the years following Irwin's publication, it became popular to include \(\sqrt{(\pi/\pi)}\) in the expressions as follows.

\[ \sigma_{xx} = {\sigma_\infty \sqrt{\pi a} \over \sqrt{ 2 \pi r }} \cos {\theta \over 2} \left( 1 - \sin {\theta \over 2} \sin {3 \theta \over 2} \right) \]

Repeating the process for \(\sigma_{yy}\) and \(\tau_{xy}\) gives

\[ \sigma_{yy} = {\sigma_\infty \sqrt{\pi a} \over \sqrt{ 2 \pi r }} \cos {\theta \over 2} \left( 1 + \sin {\theta \over 2} \sin {3 \theta \over 2} \right) \]

and

\[ \tau_{xy} = {\sigma_\infty \sqrt{\pi a} \over \sqrt{ 2 \pi r }} \cos {\theta \over 2} \sin {\theta \over 2} \cos {3 \theta \over 2} \]

\[ \sigma_{yy} = { \sigma_\infty \over \sqrt{1 - \left( a \over x \right)^2} } \]

and Irwin's approximation (with \(\theta = 0\)) is

\[ \sigma_{yy} = { \sigma_\infty \sqrt{ \pi a} \over \sqrt{2 \pi r} } \]

The two equations are shown in the graph below. It is clear that both are very close at the crack tip and diverge as the distance from the tip increases. The region of close agreement is approximately \(r \le a/10\). Beyond this, the approximate expression continues to decrease toward zero because \(1/\sqrt{r}\) always decreases as \(r\) increases. In contrast, the exact solution levels out at \(\sigma_\infty\).

One might ask, why even bother with Irwin's approximate solution when Westergaard's exact solution is available? There are several reasons. First, the approximate solution is indeed accurate at the crack tip, and this is all that really matters because only the conditions at the crack tip dictate (i) how fast the crack grows, (ii) in which direction it grows, and (iii) whether or not it fails catastrophically. Second, the approximate solution clearly reveals the dependence of the stress components on \(\theta\), something that is present, but masked in complexity in the exact solution. Finally, the approximate solution leads to the definition of the stress intensity factor, one of the most important parameters in all of fracture mechanics. It is explained in the next section.

\[ K = \sigma_\infty \sqrt{\pi a} \]

Note also that all three expressions have a \(\sqrt{r}\) in the denominator, which dictates the dependence of stress on distance from the crack tip and reflects the singularity at \(r=0\). And finally, all three expressions contain functions of \(\theta\) alone, not intermingled with \(r\), \(\sigma_\infty\), or \(a\). Therefore, all three expressions can be written compactly as

\[ \sigma_{xx} = {K \over \sqrt{ 2 \pi r }} f(\theta) \qquad \qquad \sigma_{yy} = {K \over \sqrt{ 2 \pi r }} g(\theta) \qquad \qquad \tau_{xy} = {K \over \sqrt{ 2 \pi r }} h(\theta) \]

\[ G_c \; = \; { \sigma_f^2 \, \pi \, a \over E} \]

The numerator is the square of the critical Stress Intensity Factor, \(K_c\), defined as

\[ K_c = \sigma_f \sqrt{\pi \, a} \]

and this means that \(G_c\) and \(K_c\) are related according to

\[ G_c = { K_c^2 \over E} \]

More generally, noncritical values of \(G\) and \(K\) are related in the same way

\[ G = { K^2 \over E} \]

when \(\sigma\) is less than \(\sigma_f\).

The above result relating Griffith's energy release rate, \(G\), to Irwin's stress intensity factor, \(K\), In polar coordinates, the biharmonic equation is

\[ \left( {1 \over r} {\partial \over \partial r} \left( r \, {\partial \over \partial r} \right) + {1 \over r^2} {\partial^2 \over \partial \theta^2} \right)^2 \phi = 0 \]

and the relationships for the stress components are

\[ \sigma_{rr} = {1 \over r} {\partial \phi \over \partial r} + {1 \over r^2} {\partial^2 \phi \over \partial \theta^2} \qquad \qquad \sigma_{\theta \theta} = {\partial^2 \phi \over \partial r^2} \qquad \qquad \tau_{r \theta} = - {\partial \over \partial r} \left( {1 \over r} {\partial \phi \over \partial \theta} \right) \]

- Irwin, G.R., "Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate,"
*Journal of Applied Mechanics,*Vol. 24, pp. 361-364, 1957. - Westergaard, H.M., "Bearing Pressures and Cracks,"
*Journal of Applied Mechanics,*Vol. 6, pp. A49-53, 1939. - Griffith, A.A., "The Phenomena of Rupture and Flow in Solids,"
*Philosophical Transactions, Series A,*Vol. 221, pp. 163-198, 1920.