Near the tip of a sharp crack in a linear-elastic body, the stress field has a universal leading form

$$\sigma_{ij}(r,\theta)\approx \frac{K}{\sqrt{2\pi r}}f_{ij}(\theta),$$

where $r$ is distance from the crack tip, $\theta$ locates direction around the tip, and the dimensionless functions $f_{ij}(\theta)$ describe the angular pattern for a chosen loading mode. The amplitude $K$ is the stress-intensity factor.

For opening-mode loading (Mode I), many geometries can be written

$$K_I=Y\sigma\sqrt{\pi a},$$

where $a$ is a characteristic crack size and $Y$ is a dimensionless geometry factor.

The crack becomes unstable in linear-elastic fracture mechanics when

$$K_I\ge K_{Ic},$$

where $K_{Ic}$ is fracture toughness measured under sufficiently high crack-tip constraint.

Example

For $Y=1$, $a=2,\mathrm{mm}$ and $K_{Ic}=40,\mathrm{MPa}\sqrt{\mathrm m}$,

$$\sigma_c=\frac{K_{Ic}}{\sqrt{\pi a}} \approx \frac{40}{\sqrt{\pi(0.002)}} \approx 505,\mathrm{MPa}.$$

The same material therefore tolerates a lower nominal stress when the crack is longer.

Linear-elastic fracture mechanics requires the nonlinear crack-tip region to be small compared with crack and component dimensions. If large-scale plasticity develops, elastic $K$ no longer describes the entire fracture process and elastic-plastic measures are needed.