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David Roylance developed an engineering treatment of fracture mechanics connecting cracks, loading and a material's resistance to failure. His notes derive an energy-balance description and introduce the stress intensity factor as a measure of crack-tip loading.^[ref]

In a simple linear-elastic mode-I geometry, $K_I=Y\sigma \sqrt{\pi a}$, with crack scale $a$ and dimensionless geometry coefficient $Y$. Comparing $K_I$ with the relevant fracture toughness shows why a nominally moderate bulk stress can be critical when a sufficiently large flaw is present. The geometry and elastic regime are essential assumptions.

Fracture toughness measures resistance to crack advance under specified loading and environmental conditions. In linear elasticity, $K_c$ is the critical stress intensity factor. For opening-mode loading, a valid plane strain measurement is denoted $K_{Ic}$, with units $\mathrm{Pa}\sqrt{\mathrm{m}}$, commonly $\mathrm{MPa}\sqrt{\mathrm{m}}$. It differs from a tensile yield strength: a crack changes the near-tip stress concentration.

For a fixed geometry with $K_I=Y\sigma \sqrt{\pi a}$, the threshold $K_I=K_{Ic}$ givesFor example, $K_{Ic}=10\mathrm{MPa}\sqrt{\mathrm{m}}$, $\sigma =100\mathrm{MPa}$ and $Y=1$ give $a_c\simeq 3.18\mathrm{mm}$ in this idealized geometry. The crack-tip plastic zone must remain small for the elastic intensity description to apply; extensive plasticity in continuum mechanics requires a different characterization.^[ref]

A thin elastic plate model may instead prescribe a critical front curvature ${\kappa }_f$. Such a model parameter cannot be substituted numerically for $K_{Ic}$ without a relation specific to that fracture model.

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]