= 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 $\sigma$,
$$
K_I=Y\sigma\sqrt{\pi a},
$$
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>.<Introduction to Fracture Mechanics by David Roylance>[printed pp.6–9, near-tip field, geometry factors and equation (5)]{ref}
For $a=1\,\mathrm{mm}$ and $\sigma=100\,\mathrm{MPa}$ in that ideal geometry, $K_I\simeq5.60\,\mathrm{MPa}\sqrt{\mathrm m}$. 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.<Introduction to Fracture Mechanics by David Roylance>[printed pp.9–11, specimen geometry, plastic zone and plane strain]{ref}
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