Eshelby problem of an arbitrary polygonal inclusion in anisotropic piezoelectric media with quadratic eigenstrains

This paper presents an exact closed-form solution for the Eshelby problem of a polygonal inclusion with quadratic eigenstrains in an anisotropic piezoelectric full plane. Based on the equivalent body-force concept of eigenstrains and the line-source Green's functions, the solution is first expressed in terms of line integrals along the inclusion's boundary. This involved boundary integration is then carried out analytically, with the final expression containing only elementary functions. The accuracy of the solution is verified using two different methods, and then, the solution is applied to analyze the induced fields in semiconductor quantum wires of elliptical and square shapes. Numerical results clearly illustrate the effect of different types of eigenstrains on the induced fields and on the singularity at the vertex. The exact closed-form solution should be useful to the analysis of nanoscale structures where non-uniform eigenstrains are involved due to the lattice mismatch between the substrate and inclusion.