Paris' law

Remarks

The law is an empirical fit to measured crack growth curves, not a consequence of the equations of mechanics; for metals the exponent $m$ is typically between $2$ and $4$. Example: for a through crack in a wide plate, $\Delta K = \Delta\sigma \sqrt{\pi a}$, where $\Delta \sigma$ is the stress range. For $m \ne 2$, separating variables and integrating from an initial length $a_0$ to a final length $a_f$ gives the number of cycles $$N = \frac{a_0^{1 - m/2} - a_f^{1 - m/2}}{(m/2 - 1) \, C \, (\Delta\sigma \sqrt{\pi})^m}.$$ For $m > 2$ this depends mainly on $a_0$, which is why the detection of small initial cracks matters so much. Outside the intermediate range the law fails: near the threshold the growth rate drops faster, and near the fracture toughness it increases faster.

Going deeper

For a student

Paris' law is the basis of damage-tolerant design of fatigue-loaded structures such as aircraft. One estimates the largest crack that inspections might miss. Integrating the law gives the number of cycles until the crack reaches the critical length set by the fracture toughness. The inspection interval is chosen below that number. It links fatigue to linear elastic fracture mechanics through $\Delta K$. Refinements, such as the Forman equation, add the effect of the load ratio and of the approach to fracture.