A component can survive many cycles while a small crack grows incrementally even when the maximum applied stress is below the monotonic fracture stress.

For Mode-I cycling, define the stress-intensity range

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

In a broad intermediate growth regime, many materials approximately follow the Paris law

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

where $a$ is crack length, $N$ is cycle count, and $C,m$ are experimentally determined material/environment parameters.

Because $K\propto \sigma\sqrt{a}$, crack growth can accelerate as the crack becomes longer even if the external load range remains constant.

Remaining-life calculation

If geometry and loading give a known function $\Delta K(a)$, the cycles to grow from $a_i$ to $a_f$ are estimated by

$$N=\int_{a_i}^{a_f}\frac{da}{C[\Delta K(a)]^m}.$$

The Paris relation is not valid over the entire fatigue-growth curve. At low $\Delta K$, growth approaches a threshold below which propagation can become extremely slow. At high $K_{\max}$, growth accelerates toward unstable fracture as the material's fracture resistance is approached.

This crack-growth viewpoint complements S-N fatigue: S-N methods treat total life empirically, while fracture-mechanics methods track an existing flaw and support damage-tolerant inspection planning.