Elasticity/Wedge with boundary tractions

Suppose

• The tractions on the boundary vary as ${\displaystyle r^{n}\,}$.
• No body forces.

Then

${\displaystyle {\text{(1)}}\qquad \varphi =r^{n+2}f(\theta )}$

To find ${\displaystyle f(\theta )}$ plug into ${\displaystyle \nabla ^{4}{\varphi }=0}$.

${\displaystyle {\text{(2)}}\qquad \left({\frac {d^{2}}{d\theta ^{2}}}+(n+2)^{2}\right)\left({\frac {d^{2}}{d\theta ^{2}}}+n^{2}\right)f(\theta )=0}$

If ${\displaystyle n\neq 0}$ and ${\displaystyle n\neq -2}$,

${\displaystyle {\text{(3)}}\qquad \varphi =r^{n+2}\left[a_{1}\cos\{(n+2)\theta \}+a_{2}\cos(n\theta )+a_{3}\sin\{(n+2)\theta \}+a_{4}\sin(n\theta )\right]}$

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 ${\displaystyle n=0}$, i.e., the traction on the surface is constant.

Find the stresses and displacements for a wedge loaded in constant shear ${\displaystyle S}$ on its surface.