Isotropic elasticity of dilatational conical inclusion. An analytical approach

Abstract

In this article, the analytical solution for the isotropic elasticity problem for the cone-shaped inclusion with dilatational eigenstrain in an infinite medium is given. The conical inclusion is modeled by dilatational infinitesimally thin circular disks distributed continuously along the cone z-axis with the radii of the disks being proportional to z-coordinate. The displacements, strains, and stresses of the conical inclusion are given in spherical coordinates with the origin in the cone-apex in the form of the series with Legendre polynomials. The maps of the displacements and the stresses are presented. A comparison of the displacements of the conical inclusion with the displacements of the finite cylindrical and hemispherical inclusions is provided. It is also shown, as expected, that the energy of the dilatational inclusion does not depend on its shape. In Discussion section, the specific features of an “hourglass” dipole inclusion consisting of two conical inclusions with different sign eigenstrains are demonstrated.