Exact elastic solution of the axisymmetric and deviatoric loaded hollow sphere

This paper provides the analytical solution of the elastic hollow sphere subjected to axisymmetric and pure deviatoric surface tractions within the framework of the infinitesimal strains. The expressions of the stress and displacement fields are derived in closed-form in terms of spherical harmonics by using Boussinesq-Neuber-Papkovitch potentials. The obtained solution is valid for thin and thick hollow sphere. It is shown that, for the J2-plasticity, the hollow spherical shell undergoes incipient first plastic strains at the pole θ=π/2 located on the internal surface boundary. In the perspective of shakedown analysis of ductile porous materials, the macroscopic stress and strain fields of the hollow sphere model are obtained from their local counterparts by the volume average operator.