Stress intensity factor

Definition

Let a linear elastic isotropic body contain a crack with traction-free faces. Let $P$ be a point of the crack front where the front is smooth. Use the frame at $P$ of the modes of fracture: $x_1$ ahead of the crack, $x_2$ normal to the crack plane, $x_3$ along the front. Let $(r, \theta)$ be polar coordinates in the $x_1 x_2$-plane, with $\theta = 0$ ahead of the crack. Let $\sigma_{ij}$ be the components of the Cauchy stress. The stress intensity factors at $P$ are $$K_I = \lim_{r \to 0} \sqrt{2 \pi r} \, \sigma_{22}(r, 0), \quad K_{II} = \lim_{r \to 0} \sqrt{2 \pi r} \, \sigma_{12}(r, 0), \quad K_{III} = \lim_{r \to 0} \sqrt{2 \pi r} \, \sigma_{23}(r, 0),$$ whenever these limits exist.