Two non‐circular Eshelby inclusions in an infinite isotropic elastic plane

We use analytic continuation and conformal mapping techniques to derive closed‐form solutions to the two‐dimensional Eshelby's problem of two interacting equal symmetric non‐circular inclusions undergoing uniform anti‐plane and in‐plane eigenstrains embedded in an infinite elastic matrix. The boundaries of the two inclusions are characterized by a conformal mapping function that maps the doubly connected quadrature domain occupied by the matrix onto an annulus in the image plane. The availability of the closed‐form solutions is due to the property of quadrature domains. The boundary value problem is studied in the physical plane rather than in the image plane. Elementary expressions for the internal anti‐plane and in‐plane stresses within each inclusion are obtained and illustrated.