A Semi-Analytic Solution of Eshelby’s Problem of a Smooth Inclusion with Polynomial Eigenstrain in an Anisotropic Elastic Full Plane

Unlike the uniform eigenstrain case, polynomial eigenstrain distributions capture spatially varying transformation fields, offering higher physical fidelity and direct relevance to the equivalent inclusion method (EIM) for non-elliptical inhomogeneities. This paper is focused on the anisotropic Eshelby problem for a smooth inclusion in an infinite elastic plane, where the inclusion is described by Laurent polynomials and subjected to a polynomial eigenstrain. Within the Stroh formalism, the elastic perturbation is characterized by analytic eigenfunctions formulated as Cauchy-type contour integrals over the inclusion boundary in transformed complex planes. By employing partial fraction decomposition and the residue theorem, these integrals are evaluated explicitly, yielding a finite-term semi-analytic solution involving only algebraic operations and residue evaluations. The resulting displacement and traction fields satisfy the interface continuity condition exactly. Furthermore, compared with the reported series solution, the presented semi-analytic solution offers improved computational efficiency, numerical stability, and applicability to materials with extreme principal modulus contrast. These results provide a rigorous theoretical benchmark for the analysis and design of composites with complex inclusion geometries and extreme principal modulus contrast.