Stress intensity factor

In fracture mechanics, the stress intensity factor (K) is used to predict the stress state ("stress intensity") near the tip of a crack or notch caused by a remote load or residual stresses.^[1] It is a theoretical construct usually applied to a homogeneous, linear elastic material and is useful for providing a failure criterion for brittle materials, and is a critical technique in the discipline of damage tolerance. The concept can also be applied to materials that exhibit small-scale yielding at a crack tip.

The magnitude of K depends on specimen geometry, the size and location of the crack or notch, and the magnitude and the distribution of loads on the material. It can be written as:^[2][3]

${\displaystyle K=\sigma {\sqrt {\pi a}}\,f(a/W)}$

where ${\displaystyle f(a/W)}$ is a specimen geometry dependent function of the crack length, a, and the specimen width, W, and σ is the applied stress.

Linear elastic theory predicts that the stress distribution (${\displaystyle \sigma _{ij}}$) near the crack tip, in polar coordinates (${\displaystyle r,\theta }$) with origin at the crack tip, has the form ^[4]

${\displaystyle \sigma _{ij}(r,\theta )={\frac {K}{\sqrt {2\pi r}}}\,f_{ij}(\theta )+\,\,{\rm {higher\,order\,terms}}}$

where K is the stress intensity factor (with units of stress × length^1/2) and ${\displaystyle f_{ij}}$ is a dimensionless quantity that varies with the load and geometry. Theoretically, as r goes to 0, the stress ${\displaystyle \sigma _{ij}}$ goes to ${\displaystyle \infty }$ resulting in a stress singularity.^[5] Practically however, this relation breaks down very close to the tip (small r) because plasticity typically occurs at stresses exceeding the material's yield strength and the linear elastic solution is no longer applicable. Nonetheless, if the crack-tip plastic zone is small in comparison to the crack length, the asymptotic stress distribution near the crack tip is still applicable.