Paris' law

Higher mean stress is known to increase the rate of crack growth and is known as the mean stress effect. The mean stress of a cycle is expressed in terms of the stress ratio ${\displaystyle R}$ which is defined as

${\displaystyle R={K_{\text{min}} \over K_{\text{max}}},}$

or ratio of minimum to maximum stress intensity factors. In the linear elastic fracture regime, ${\displaystyle R}$ is also equivalent to the load ratio

${\displaystyle R\equiv {P_{\text{min}} \over P_{\text{max}}}.}$

The Paris–Erdogan equation does not explicitly include the effect of stress ratio, although equation coefficients can be chosen for a specific stress ratio. Other crack growth equations such as the Forman equation do explicitly include the effect of stress ratio, as does the Elber equation by modelling the effect of crack closure.

The Paris–Erdogan equation holds over the mid-range of growth rate regime, but does not apply for very low values of ${\displaystyle \Delta K}$approaching the threshold value ${\displaystyle \Delta K_{\text{th}}}$, or for very high values approaching the material's fracture toughness, ${\displaystyle K_{\text{Ic}}}$. The alternating stress intensity at the critical limit is given by ${\displaystyle {\begin{aligned}\Delta K_{\text{cr}}&=(1-R)K_{\text{Ic}}\end{aligned}}}$.[6]

The slope of the crack growth rate curve on log-log scale denotes the value of the exponent ${\displaystyle m}$ and is typically found to lie between ${\displaystyle 2}$ and ${\displaystyle 4}$, although for materials with low static fracture toughness such as high-strength steels, the value of ${\displaystyle m}$ can be as high as ${\displaystyle 10}$.

Because the size of the plastic zone ${\displaystyle (r_{\text{p}}\approx K_{I}^{2}/\sigma _{y}^{2})}$ is small in comparison to the crack length, ${\displaystyle a}$ (here, ${\displaystyle \sigma _{y}}$ is yield stress), the approximation of small-scale yielding applies, enabling the use of linear elastic fracture mechanics and the stress intensity factor. Thus, the Paris–Erdogan equation is only valid in the linear elastic fracture regime, under tensile loading and for long cracks.[7]