Elasticity/Wedge with boundary tractions

Wedge with Boundary Tractions

Elastic wedge with normal and shear surface tractions

Suppose

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

Then

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

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 $n\neq 0$ and $n\neq -2$,

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

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.

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