On the use of Somigliana's formulae and series of surface spherical harmonics for elasticity problems with spherical boundaries

Abstract

This paper discusses applications of Somiglina's identities to the solutions of elasticity problems with spherical boundaries. The components of the boundary displacements and tractions involved in the identities are represented as truncated series of surface spherical harmonics, and all of the integrals involved in the formulae are evaluated analytically. The classical problems of a solid sphere, a spherical cavity, and a perfectly bonded spherical inhomogeneity (an inclusion with the elastic properties different from those of the surrounding material) are solved with the use of Somiglina's identities. Extensions of the new solutions to more complicated three-dimensional problems with spherical boundaries are discussed.