Stress analysis of a hollow sphere compressed between two flat platens

Abstract

The theoretical basis for determination of tensile strength and elastic constants by compression of hollow spheres between two flat platens is provided in this paper. The method of solution follows the displacement function approach, and three displacement functions are introduced so that the fifteen governing equations can be uncoupled analytically. The Hertz contact between the out surface of the hollow sphere and the flat loading platens is considered, and Fourier-Legendre expansion technique is applied on the Hertz contact stress in order to determine the unknown constants in the general analytical expression for stress components. An analytical solution for the stress and displacement components within a hollow sphere compressed between two flat platens is obtained. Numerical results of the analytical solution show that the stress distributions within hollow spheres are not uniform, two tensile stress concentration zones are usually developed at the inner surface and near the loading area along the loading axis of the hollow spheres within both thick and thin hollow spheres, but the maximum tensile stress is usually developed at the inner surface, except few cases for relatively thick hollow sphere with a very small Poisson ratio. The maximum tensile stress developed along the axis of loading can be used to estimate the tensile strength of hollow spheres. The curves obtained by the analytical solution for the ratio between the vertical and horizontal strains at the equator of the out surface of the hollow sphere, together with those for nominal stress versus the normalized displacement between the two flat loading platens, may provide optional way to determine Poisson's ratio and Young's modulus of the hollow sphere, respectively.