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 curveCFor our disk problem, choose

Cto be the perimeter of the circle , whenceNow, since lies inside the

C, must lie outside it, and soSubtract these expressions and note that on

CSimplify and note on

CFinally, to find the displacement field, take the real part