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.

本文提出了一个由软梯度层在内部加固的厚壁空心球体的解析解，该球体受到任意均匀的轴对称外牵引力。假定分级钢筋的弹性模量作为幂律函数随其厚度变化。由此产生的非齐次 Navier 平衡方程采用欧拉类型的二阶微分方程系统的形式。微分系统随后通过使用变量替换和微分理论作为特征值问题解决。由于纯剪切、单向、全方位和中心对称的外牵引力，分析了五个不均匀指数的应力分布和应力集中。对于每个加载配置，分析了增强层的性能级配对空心球应力降低的影响。还确定了导致最佳应力分布并因此最小应力集中的最佳不均匀性指数。为了验证和验证的目的，还对一些数值例子进行了有限元建模。减少应力集中的基本机制是将通常沿空心球内表面出现的高应力区推向其外边界，从而使应力在整个厚度维度上分布更均匀。为了验证和验证的目的，还对一些数值例子进行了有限元建模。减少应力集中的基本机制是将通常沿空心球内表面出现的高应力区推向其外边界，从而使应力在整个厚度维度上分布更均匀。为了验证和验证的目的，还对一些数值例子进行了有限元建模。减少应力集中的基本机制是将通常沿空心球内表面出现的高应力区推向其外边界，从而使应力在整个厚度维度上分布更均匀。