LAUSR

Abstract This paper presents an analytical solution for a thick-walled hollow sphere interiorly reinforced by a soft graded layer subjected to an arbitrary uniform axial symmetric outer traction. The elastic modulus of the graded reinforcement is assumed to vary through its thickness as a power-law function. The resultant inhomogeneous Navier’s equations of equilibrium are in the form of a system of second-order differential equations of the Euler type. The differential system is subsequently solved as an eigenvalue problem by the use of variable replacement and differential theory. Stress distributions and stress concentrations are analyzed for five inhomogeneity indices due to pure-shear, unidirectional, all-around and centrosymmetric outer tractions. For each loading configuration, the effects of property gradation of the reinforcement layer on stress reductions of hollow spheres are analyzed. The optimal inhomogeneity index that leads to the best stress distributions and therefore the least stress concentrations is also determined. For the purpose of verification and validation, finite element modelings are also conducted for some numerical examples. The fundamental mechanism of reducing stress concentrations is to drive the high-stress zone typically occurring along the inner surface of hollow spheres toward their outer boundary such that stresses may distribute more uniformly through the thickness dimension.