Core-shell spheres under diametrical compression: An analytical solution

The analytical solution for the stress distributions within core-shell spheres or layered spheres under diametrical compression is obtained. The solution reduces analytically to the classical solution for solid spheres by Hiramatsu and Oka (1966) and the analytical solution for hollow spheres by Wei et al. (2015) in the two limiting special cases. The numerical results of the present solution show that a drastic increase of tensile stress is usually observed for a core-shell sphere with a stiff shell or a shell with the same stiffness but a small Poisson's ratio, while a drastic decrease of tensile stress is usually observed for a core-shell sphere with a soft shell or a shell with the same stiffness but a large Poisson's ratio. The maximum tensile stress may be either induced at the interface or near the loading areas, which is more likely induced at the interface of a core-shell sphere with a thin and stiff shell, otherwise it is more likely induced near the loading areas of a core-shell sphere. Moreover, the maximum tensile stress is affected by the size of loading areas, the ratio of the Young's moduli of the core and the outer shell, Poisson's ratio and the thickness of the shell. Since more and more composite materials made up of core-shell spheres are used for some advanced devices to achieve multi-functions or some intelligent abilities, the present solution can be used as a benchmark or a basic solution for analyzing the failure mechanism of composite materials made up of core-shell spheres or layered spheres.