An analytical solution for the stress distribution within a spherically isotropic hollow sphere under diametrical compression

Abstract

An analytical solution for the stress distributions within a spherically isotropic hollow sphere under diametrical compression is derived. The deformation of the spherically isotropic hollow sphere is divided into axis-symmetric deformation and non-symmetric deformation. The analytical solution for the axis-symmetric deformation is obtained by following the displacement method, while that for non-symmetric deformation of the spherically isotropic hollow sphere is obtained by employing the displacement function method. When the isotropic limit is considered, the analytical solution for isotropic hollow spheres is recovered identically. Numerical results show that tensile stress concentrations are usually developed both at the inner surface and near r/R = 0.85 within a hollow sphere along the axis of loading. A small value of the anisotropy in Young’s modulus or Poisson’s ratio usually leads to a large value of the maximum tensile stress at the inner surface. While a smaller value of the anisotropy in Poisson’s ratio or a large value of the anisotropy in the shear modulus may lead to a large value of the maximum tensile stress near r/R = 0.85 for a thick and anisotropic hollow sphere. In addition, it is found that the anisotropy in the shear modulus has drastic influence on the tensile stress distributions within a thin hollow sphere, and a particular value of the anisotropy in the shear modulus may reduce greatly the tensile stress concentrations both at inner surface and near r/R = 0.85 of a thin hollow sphere. This result may provide us a very attractive method to optimize the elastic constants of anisotropic hollow spheres by synthesis process technique so as to improve the load capacity of a hollow sphere, and extend the fatigue life of composite material made of thin hollow spheres.