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 Laplace’s 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