On deformations of a spherically isotropic hollow sphere of a functionally graded material

By introducing three displacement functions and two stress functions, two independent state equations with variable coefficients are derived from the three-dimensional equilibrium equations of a radially inhomogeneous spherically isotropic elastic medium. By virtue of the laminate approximation theory, the state equations are then transformed to the ones with constant variables, of which the solutions can be easily obtained. The continuity conditions at each fictitious interface then lead to linear algebraic equations about the state variables at the inner and outer spherical surfaces only. Numerical example of a functionally graded hollow sphere Subjected to external loading is considered and the effect of the material gradient index (or the inhomogeneity parameter) is discussed.