EN224: Linear Elasticity
Division of Engineering
5.4 Solving Anti-Plane Shear Problems using Cauchy Integral Formulae
The Cauchy integral formula sometimes allows us to find explicit solutions to Laplaces equation. For example, consider a displacement boundary value problem for a circular cylinder in anti-plane shear.
Recall the Cauchy integral formulae
for any f(z) analytic inside a closed curve C
For our disk problem, choose C to be the perimeter of the circle , whence
Now, since lies inside the C, must lie outside it, and so
Subtract these expressions and note that on C
Simplify and note on C
Finally, to find the displacement field, take the real part