Reducing stress concentrations in unidirectionally tensioned thick-walled spheres through embedding a functionally graded reinforcement

Abstract

This work aims to investigate the effects of a radially graded sphere on the reduction of stress concentrations in homogeneous hollow spheres. Both hydrostatic boundary pressures and a uniaxial outer tension are considered. For the spherically symmetric loading, closed-form solutions were successfully developed by assuming a power-law gradation in shear modulus and a constant Poisson’s ratio. For the case of uniaxial tension, the problem is tackled by discretizing the graded sphere into a number of homogeneous sublayers and by solving a simultaneous system about the series coefficients in Boussinesq displacement potentials. The sensitivity of the semianalytical solutions on the number of discretized sublayers is first assessed. Stress distributions and concentration factors are subsequently evaluated for a representative geometry and a few power-law gradation indices. Optimal indices are also identified, for which the stress concentration factors at the inner surface of the graded reinforcement sphere and the reinforcement/matrix interface become well balanced. Wherever possible, finite element solutions are also presented for the purpose of verification and validation. When the outer radius of the homogeneous hollow sphere is beyond four times that of its inner surface, the solutions well approximate the limiting case of an infinite matrix. Both the semianalytical and finite element solutions suggest the promising means of tailoring stress concentrations in thick-walled spheres through internally coating a thin layer of graded reinforcement.