Explicit analytic solution for the plane elastostatic problem with a rigid inclusion of arbitrary shape subject to arbitrary far-field loadings

TL;DR

This paper presents an explicit analytical solution for the elastostatic problem involving a rigid inclusion of arbitrary shape in a 2D isotropic elastic body, using boundary integral methods and conformal mapping.

Contribution

The main novelty is the use of Faber polynomials for arbitrary-shaped inclusions, enabling explicit series solutions under arbitrary far-field loadings.

Findings

Derived explicit series solutions for arbitrary-shaped inclusions.

Validated the approach with examples of different inclusion geometries.

Provided a new analytical tool for elastostatic problems with complex boundaries.

Abstract

We provide an analytical solution for the elastic fields in a two-dimensional unbounded isotropic body with a rigid inclusion. Our analysis is based on the boundary integral formulation of the elastostatic problem and geometric function theory. Specifically, we use the coordinate system provided by the exterior conformal mapping of the inclusion to define a density basis functions on the boundary of the inclusion, and we use the Faber polynomials associated with the inclusion for a basis inside the inclusion. The latter, which constitutes the main novelty of our approach, allows us to obtain an explicit series solution for the plane elastostatic problem for an inclusion of arbitrary shape in terms of the given arbitrary far-field loading.