On Perturbation Solutions for Nearly Circular Inclusion Problems in Plane Thermoelasticity

An approximate analytical solution to the nearly circular inclusion problems of arbitrary shape in plane thermoelasticity is provided. The shape of the inclusion boundary considered in the present study is assumed to have the form $r=a0[1+Aθ],$ where $a0$ is the radius of the unperturbed circle and $Aθ$ is the radius perturbation magnitude that is represented by a Fourier series expansion. The proposed method in this study is based on the complex variable theory, analytical continuation theorem, and the boundary perturbation technique. Originating from the principle of superposition, the solution of the present problem is composed of the reference and the perturbation terms that the reference term is the known exact solution pertaining to the case with circular inclusion. First-order perturbation solutions of both temperature and stress fields are obtained explicitly for elastic inclusions of arbitrary shape. To demonstrate the derived general solutions, two typical examples including elliptical and smooth polygonal inclusions are discussed in detail. Compared to other existing approaches for elastic inclusion problems, our methodology presented here is remarked by its efficiency and applicability to inclusions of arbitrary shape in a plane under thermal load.