Elasticity/Wedge with boundary tractions

Wedge with Boundary Tractions

File:Wedge with surface traction.png Elastic wedge with normal and shear surface tractions

Suppose

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

Then

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

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

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

If $n\ne 0$ and $n\ne -2$,

$\text{(3)}\phi =r^{n+2}[a_1\cos \{(n+2)\theta \}+a_2\cos (n\theta )+a_3\sin \{(n+2)\theta \}+a_4\sin (n\theta )]$

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.