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. 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:

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

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. 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plasticity typically occurs at stresses exceeding the material's yield strength and the linear elastic solution is no longer applicable.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.

In 1957, G. Irwin found that the stresses around a crack could be expressed in terms of a scaling factor called the stress intensity factor. He found that a crack subjected to any arbitrary loading could be resolved into three types of linearly independent cracking modes. These load types are categorized as Mode I, II, or III as shown in the figure. Mode I is an opening (tensile) mode where the crack surfaces move directly apart. Mode II is a sliding (in-plane shear) mode where the crack surfaces slide over one another in a direction perpendicular to the leading edge of the crack. Mode III is a tearing (antiplane shear) mode where the crack surfaces move relative to one another and parallel to the leading edge of the crack. Mode I is the most common load type encountered in engineering design.

Different subscripts are used to designate the stress intensity factor for the three different modes. The stress intensity factor for mode I is designated ${\displaystyle K_{\rm {I}}}$ and applied to the crack opening mode. The mode II stress intensity factor, ${\displaystyle K_{\rm {II}}}$, applies to the crack sliding mode and the mode III stress intensity factor, ${\displaystyle K_{\rm {III}}}$, applies to the tearing mode. These factors are formally defined as:

In plane stress conditions, the strain energy release rate (${\displaystyle G}$) for a crack under pure mode I, or pure mode II loading is related to the stress intensity factor by: