## What is π doing in the Stress Intensity expression?

Today it is π-day (3/14), a perfect day to wonder what π is doing in the stress intensity factor (SIF) equation. Let us have a look to this SIF equation first:

$K=\beta&space;\cdot&space;S\cdot&space;\sqrt{\pi&space;\cdot&space;a}$

In this equation, K is the stress intensity factor, β is a geometry factor, S is the gross stress and a is the (half) crack length. This SIF describes the severity of the stress field around a crack tip. With increasing SIF, the fatigue crack growth rate will increase as well:

$\frac{da}{dn}=C\cdot&space;\left&space;(\Delta&space;K&space;\right&space;)^{n}$

It seems that the SIF equation could do quite well without the π. To understand why the π was introduced we have to go back in history.

#### Westergaard

Westergaard [1] introduced an Airy stress function of complex numbers to describe the stress field in an infinite plate with a crack. The equations are:

$\sigma&space;_{xx}=Re\left&space;(Z&space;\right&space;)-y\cdot&space;Im\left&space;(&space;Z'&space;\right&space;)$

$\sigma&space;_{yy}=Re\left&space;(Z&space;\right&space;)+y\cdot&space;Im\left&space;(&space;Z'&space;\right&space;)$

$\tau&space;_{xy}=-y\cdot&space;Re\left&space;(Z'&space;\right&space;)$

Z and Z’ are:

$Z\left&space;(&space;z&space;\right&space;)=\frac{S}{\sqrt{1-\left&space;(&space;\frac{a}{z}&space;\right&space;)^{2}}}$

$Z'\left&space;(&space;z&space;\right&space;)=\frac{-S\cdot&space;a^{2}}{z^{3}&space;\left&space;[(1-\left&space;(&space;\frac{a}{z}&space;\right&space;)^{2}&space;\right&space;]^{\frac{3}{2}}}$

a is the half crack length and z equals x+iy. For y=0, so in the plane of the crack:

$\sigma&space;_{yy}=\frac{S}{\sqrt{1-\left&space;(&space;\frac{a}{x}&space;\right&space;)^{2}}}$

At a=x, there is a singularity; σyy is infinite.

#### Irwin

Later Irwin [2] simplified Westergaard’s solution. He proposed the following expression for z:

$z=a+r\cdot&space;e^{^{i\theta&space;}}$

By doing so, Irwin found expressions for the stress field in the close vicinity of the crack tip using the polar coordinates r and θ. After applying a lot of algebra, the following equation for y=0 is found:

$\sigma&space;_{yy}=\frac{S\sqrt{a}}{\sqrt{2r}}\cdot&space;cos\frac{\theta&space;}{2}\left&space;(1+sin\frac{\theta&space;}{2}sin\frac{3\theta&space;}{2}&space;\right&space;)$

Still without a π. In the years after Irwin published his paper, the term √(π/π) was introduced in the equation:

$\sigma&space;_{yy}=\frac{S\sqrt{\pi&space;a}}{\sqrt{2\pi&space;r}}\cdot&space;cos\frac{\theta&space;}{2}\left&space;(1+sin\frac{\theta&space;}{2}sin\frac{3\theta&space;}{2}&space;\right&space;)$

With $\inline&space;K=S\cdot&space;\sqrt{\pi&space;\cdot&space;a}$  the stress field can be expressed as:

$\sigma&space;_{xx}=\frac{K}{\sqrt{2\pi&space;r}}\cdot&space;f\left&space;(&space;\theta&space;\right&space;)\;\;\;\;\;&space;\sigma&space;_{yy}=\frac{K}{\sqrt{2\pi&space;r}}\cdot&space;g\left&space;(&space;\theta&space;\right&space;)\;\;\;\;\;&space;\tau&space;_{xy}=\frac{K}{\sqrt{2\pi&space;r}}\cdot&space;h\left&space;(&space;\theta&space;\right&space;)$

This is only an approximation of the exact solution of Westergaard. Close to the crack tip the approximation is quite good. The advantage of Irwin’s approximation is that it uses the stress intensity factor K. This SIF is the key parameter in linear elastic fracture mechanics.

But we still haven’t seen why we need π in the equation for K.

#### Griffith

Early last century, Griffith [3] worked on an energy-based criterion to predict the failure behaviour of a plate with a crack. in short, the criterion implies that failure (unstable crack growth) occurs when the release of strain energy is larger than the required energy to break the atomic bonds in a material. Griffith critical energy release rate is written as:

$G_{c}=\frac{S_{c}^{2}\pi&space;a}{E}$

$G=\frac{K^{2}}{E}$

And here we have the point where the π was introduced.

### References

1. Westergaard, H.M., Bearing Pressures and Cracks, Journal of Applied Mechanics, Vol. 6, pp. A49-53, 1939.
2. 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.
3. Griffith, A.A., The Phenomena of Rupture and Flow in Solids, Philosophical Transactions, Series A, Vol. 221, pp. 163-198, 1920.

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