Introduction to 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 $n \ne 0$ 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.