A Complex Variable Solution for a Deforming Circular Tunnel in an Elastic Half-Plane

An analytical solution is presented of problems for an elastic half-plane with a circular tunnel, which undergoes a certain given deformation. The solution uses complex variables, with a conformal mapping onto a circular ring. The coefficients in the Laurent series expansion of the stress functions are determined by a combination of analytical and numerical computations. As an example the case of a uniform radial displacement of the tunnel boundary is considered in some detail. It appears that a uniform radial displacement is accompanied by a downward displacement of the tunnel as a whole. This phenomenon also means that the distribution of the apparent spring constant is strongly non-uniform.