Geometric series solution for the plane elastostatic problem in the presence of a cavity

Abstract: This paper presents an analytic series solution method for the elastic inclusion problem in a two-dimensional unbounded isotropic medium with a cavity. Generalizing the work of Mattei and Lim \cite{Mattei:2021:EAS}, this study develops an analytic series solution method for the elastic inclusion problem to encompass a cavity problem. The central mathematical challenge tackled in this research is to deal with the conormal derivative condition. By using the complex-variable formulation for the conormal derivative, we effectively deal with the boundary condition and derive an explicit series solution for the plane elastostatic problem with a cavity of arbitrary shape subject to arbitrary far-field loading. The solution is expressed as a series expansion in terms of the given far-field loading and the exterior conformal mapping associated with the cavity.