Crack growth

The Paris equation is given by

$\frac{da}{dN} = A \, (\Delta K)^m$

where $a(N)$ is the crack length at load $N$ . Plug in $\Delta K = Y\,\Delta\sigma \sqrt{a}$ , we have

$da = A \, Y^m \Delta\sigma^m \sqrt{a^m} \, dN$

integrate this general form, we have:

$\begin{split} N_f = & \int dN = \int^{a_f}_{a_0} (A \, Y^m \Delta\sigma^m \sqrt{a^m})^{-1} \, da \\ = & \frac{1}{A \, Y^m \Delta\sigma^m}\left[\bigg(\frac{1}{a_0}\bigg)^\frac{m-2}{2} - \bigg(\frac{1}{a_f}\bigg)^\frac{m-2}{2}\right] \end{split}$

References

Paris’ law - Wikipedia