Paris' law

Paris' law (also known as the Paris–Erdogan equation) is a crack growth equation that gives the rate of growth of a fatigue crack. The stress intensity factor ${\displaystyle K}$ characterises the load around a crack tip and the rate of crack growth is experimentally shown to be a function of the range of stress intensity ${\displaystyle \Delta K}$ seen in a loading cycle. The Paris equation is^[1]

${\displaystyle {\begin{aligned}{da \over dN}&=C\left(\Delta K\right)^{m},\end{aligned}}}$

where ${\displaystyle a}$ is the crack length and ${\displaystyle {\rm {d}}a/{\rm {d}}N}$ is the fatigue crack growth for a load cycle ${\displaystyle N}$. The material coefficients ${\displaystyle C}$ and ${\displaystyle m}$ are obtained experimentally and also depend on environment, frequency, temperature and stress ratio.^[2] The stress intensity factor range has been found to correlate the rate of crack growth from a variety of different conditions and is the difference between the maximum and minimum stress intensity factors in a load cycle and is defined as

${\displaystyle \Delta K=K_{\text{max}}-K_{\text{min}}.}$

Being a power law relationship between the crack growth rate during cyclic loading and the range of the stress intensity factor, the Paris–Erdogan equation can be visualized as a straight line on a log-log plot, where the x-axis is denoted by the range of the stress intensity factor and the y-axis is denoted by the crack growth rate.

The ability of ΔK to correlate crack growth rate data depends to a large extent on the fact that alternating stresses causing crack growth are small compared to the yield strength. Therefore crack tip plastic zones are small compared to crack length even in very ductile materials like stainless steels.^[3]

The equation gives the growth for a single cycle. Single cycles can be readily counted for constant-amplitude loading. Additional cycle identification techniques such as rainflow-counting algorithm need to be used to extract the equivalent constant-amplitude cycles from a variable-amplitude loading sequence.