ANALYTICAL SOLUTIONS FOR STRESSES AND DISPLACEMENTS IN ROTATING SOLID AND HOLLOW SPHERES UNDER INTERNAL PRESSURE

Abstract

This study determines the exact solution through which stresses and displacements in both solid and hollow sphere rotates under internal pressure. We use an indirect method, which assumes a deformation pattern containing some parameters. We substitute this deformation form to the standard elasticity equation describing spherical elasticity for homogeneous isotropic deformation for compressible structural material. We considered typically, a Blatz-ko material and analysed both the solid and hollow spherical form of this material when the force that cause deformation is internal pressure such that the external environment is stress free. Consequently we determine the stress components as we allow the material to deform. By using Cauchy elasticity we noted that the only non-trivial component of stress is in the radial direction. Substitution in the non-zero component of continuity equation resulted into a non-linear second order partial differential equation for explicit determination of stresses and displacement the model gave rise to a boundary value problem where both the equation and boundary conditions were non-linear. A transformation was introduced which linearized the boundary conditions, this allowed the method of asymptotes. The minimization of the error in sobolove norm helped us to control both the function and its gradient. Consequently, exact solutions of the stresses and displacements at every section of the compressible spherical solid and hollow sphere deforming under internal pressure were determined.