Dynamic Normal Stress Determination at the Rigid Sphere-Elastic Half-Space Interface: A Particle Dynamics Approach

The Hertz theory provides a foundation for understanding the contact behavior between elastic bodies. However, the theory's reliance on static equilibrium assumptions limits its applicability to dynamic scenarios, particularly where the contact interface experiences rapid changes in stress due to impact or vibration. This paper addresses the gap in the literature regarding the dynamic normal stress at the contact interface between a rigid sphere and an elastic half-space based on the theory of particle dynamics. Our approach begins by establishing a system of particle dynamics equations for the interface contact zone. By decoupling these equations based on the lateral and tangential motion characteristics of the particles, we derive a unified normal dynamics equation that describes the relative relationship between the normal strain of each particle and the center point of the interface. Utilizing the dynamic equation of the interface collision center and the motion equation of the small ball, we determine the absolute values of the normal strain of each particle in the interface contact zone. The research results demonstrate that the lateral and tangential strains (forces) prior to collision do not directly affect the sphere motion in the normal direction, while the incremental strains (forces) post-collision significantly do. By analyzing the normal dynamic equations of the contact zone, we obtain the lateral stress increment, tangential stress increment, and the distribution law of normal stress for each particle at the interface. In conclusion, the normal strain values obtained using our proposed method, when compared with those from Hertz theory at the maximum collision depth, are significantly greater. This indicates that our method is more effective in explaining contact interface damage phenomena, which are not adequately addressed by Hertz theory. The findings have implications for understanding and predicting contact interface failure in engineering applications, particularly under dynamic loading conditions.