On Mechanical and Motion Behavior of the Normal Impact Interface between a Rigid Sphere and Elastic Half-Space

Abstract

This paper presents exact solutions for the mechanical behavior of the interface during the normal collision between a rigid sphere and an elastic half-space based on kinematics and particle dynamics theory. The interfacial contact stress is significantly different from the static solution obtained from the Hertz contact theory. Firstly, according to the kinematics theory, the elastic half-space interface deformation of the sphere and the half-space interface in the collision process is divided into the deformation of the contact area and the non-contact area. The curve of the non-contract area is strictly antisymmetric to the deformation curve of the contact area, and the lateral deformation for each particle at the interface can be neglected when the collision depth is relatively small. Then, the vertical deformation equation of the contact area of the half-space interface, the dynamic equation of the rigid sphere, and the dynamic equation of the interface particle in the contact region are established. The proportional relationship between the stress or strain of any particle in the contact area, the stress or strain of the collision center point, and the method for determining the maximum collision depth are obtained. The equal deformation depth in the contact and non-contact regions, and the proportional relationship between the stress or strain in the contact area and the center point of the collision at any moment are consistent with the Hertz contact theory, which verifies the reliability of the current study. Taking the sphere of no initial velocity collides with the half-space under pure gravity as an example, when the elastic half-space Poisson’s ratio is taken as 0.2–0.4, the ratio of the maximum contact stress determined by the Hertz theory and the current solution when the sphere reaches the maximum collision depth is 0.58–0.54. Based on this study, the contact stress and its distribution in the interfacial contact region can be obtained when the motion state of the sphere and the interface are determined.