Williams crack-tip expansion

Statement

Let $\Omega \subset \R^2$ contain the slit $\set{(x_1, 0) : -\delta < x_1 \le 0}$ and let $(r, \theta)$, $-\pi < \theta < \pi$, be polar coordinates centred at its tip. Let the stress $\sigma$ solve the equations of plane linear elasticity for an isotropic material in $B_\delta(0)$ minus the slit, without body forces. Assume that the crack faces $\theta = \pm \pi$ are traction-free and that the elastic energy near the tip is finite. Then there are real numbers $a_n$, $b_n$ such that, for $r < \delta$, $$\sigma_{ij}(r, \theta) = \sum_{n=1}^{\infty} r^{n/2 - 1} \left( a_n f_{ij}^{(n)}(\theta) + b_n g_{ij}^{(n)}(\theta) \right).$$ Here $f^{(n)}$ and $g^{(n)}$ are fixed trigonometric functions, symmetric (mode I) and antisymmetric (mode II) with respect to the crack line. The leading term is the inverse square root singularity with the stress intensity factors; in mode I it reads $$\sigma_{11} = \frac{K_I}{\sqrt{2 \pi r}} \cos\frac{\theta}{2} \Big(1 - \sin\frac{\theta}{2} \sin\frac{3\theta}{2}\Big), \quad \sigma_{22} = \frac{K_I}{\sqrt{2 \pi r}} \cos\frac{\theta}{2} \Big(1 + \sin\frac{\theta}{2} \sin\frac{3\theta}{2}\Big), \quad \sigma_{12} = \frac{K_I}{\sqrt{2 \pi r}} \cos\frac{\theta}{2} \sin\frac{\theta}{2} \cos\frac{3\theta}{2}.$$ The term with $n = 2$ is a uniform stress $\sigma_{11} = T$ parallel to the crack, the T-stress.