 Linear elastic fracture mechanics

Linear elastic fracture mechanics predicts the loading of a crack using linear elasticity outside a small region near its tip. The near-tip stress field has a characteristic $r^{-1/2}$ dependence; its amplitude is the stress intensity factor. For an opening crack under nominal tensile stress $\sigma$,where $a$ is a specified crack-length parameter and $Y$ a dimensionless geometry factor. A central crack in an effectively infinite plate has total length $2a$ and $Y=1$. Crack advance is assessed by comparing $K_I$ with the appropriate measured fracture toughness.^[ref]

For $a=1\mathrm{mm}$ and $\sigma =100\mathrm{MPa}$ in that ideal geometry, $K_I\simeq 5.60\mathrm{MPa}\sqrt{\mathrm{m}}$. Increasing the crack's half-length fourfold doubles $K_I$ without changing the applied stress. This explains why a structure can fail at a stress below the strength of an uncracked specimen.

A toughness measurement loads a precracked specimen of known geometry and records the force associated with crack extension. Thickness, remaining ligament and crack length must be large relative to the crack-tip plastic region before the plane-strain value $K_{Ic}$ is transferable in this model. Large-scale plastic deformation requires a different description; the elastic singularity is a useful outer approximation, not an assertion of physically infinite stress inside the material.^[ref]