Abstract:

We derive closed-form solutions to the three-dimensional Eshelby’s problem of a spherical Eshelby inclusion undergoing uniform deviatoric eigenstrains concentrically embedded in an isotropic elastic finite spherical domain with a traction-free or rigidly clamped boundary. The interface between the inclusion and its surrounding domain is assumed to be of Steigmann-Ogden type. Our solutions indicate that the stresses and strains within the spherical inclusion are generally nonuniform because of the effects of the finite spherical domain and the Steigmann-Ogden imperfect interface. The internal elastic field of stresses and strains is uniform within the spherical inclusion when a condition that relates the single interface parameter to the geometric parameter and Poisson’s ratio of the finite domain is satisfied. When the spherical edge is rigidly clamped, a Gurtin-Murdoch interface is found to be sufficient to achieve this interior uniformity property. In contrast, when the spherical edge is traction-free, a Steigmann-Ogden interface with nonzero and positive bending stiffness parameters must be used to achieve the interior uniformity property.

Key words: spherical inclusion, spherical domain, Steigmann-Ogden interface, Gurtin-Murdoch interface, Neumann problem, Dirichlet problem, uniformity property