Elasticity/Wedge with boundary tractions

Wedge with Boundary Tractions
Suppose Then
 * The tractions on the boundary vary as $$r^n\,$$.
 * No body forces.
 * $$\text{(1)} \qquad

\varphi = r^{n+2} f(\theta) $$ To find $$f(\theta)$$ plug into $$\nabla^4{\varphi} = 0$$.
 * $$\text{(2)} \qquad

\left(\frac{d^2}{d\theta^2} + (n+2)^2\right) \left(\frac{d^2}{d\theta^2} + n^2\right) f(\theta) = 0 $$ If $$n \ne 0$$ and $$n \ne -2$$,
 * $$\text{(3)} \qquad

\varphi = r^{n+2} \left[ a_1 \cos\{(n+2)\theta\} + a_2 \cos(n\theta) + a_3 \sin\{(n+2)\theta\} + a_4 \sin(n\theta)\right] $$ The corresponding stresses and displacements can be found from the tables associated with Michell's solution. We have to take special care for the case where $$n = 0$$, i.e., the traction on the surface is constant.

Sample Homework Problem
Find the stresses and displacements for a wedge loaded in constant shear $$ S $$ on its surface.