Optimal Design of Functionally Graded Incompressible Linear Elastic Cylinders and Spheres

We find closed-form solutions for axisymmetric plane strain radial deformations of a functionally graded circular hollow cylinder and radial expansion/contraction of a hollow sphere loaded on inner and outer surfaces by uniform hydrostatic pressures. The cylinder and the sphere are presumed to be made of isotropic and incompressible linear elasticmaterials with the shearmodulus a general function of the radius. It is found that the optimal value of the hoop or the circumferential stress in a cylinder and a sphere is a constant and occurs for the linear variation in the radial direction of the shear modulus. The analytical results presented here should serve as benchmarks for verifying numerical solutions of problems.