Impact model of sphere on the coated plate

Abstract

Impacting the coated plate induces flexural waves on the plate, and such dynamic effect can be considered in the Zener model, which couples the vibration of plate with Hertzian contact relationship. However, Zener model is not applicable for deep impact δm>δcr on a coated plate, because the Hertzian contact law requires an assumption that the contact depth cannot be larger than few percent of the film thickness. This assumption renders the Zener model invalid for the deep impact δm>δcr on plate coated with thin film, where high impact velocity may induce deep contact. In this study, we built up a control equation for the deep impact δm>δcr problem of sphere on the plate coated with soft film, where a nonlinear contact relationship different from Hertzian should be used. Then, we obtained an analytical solution by solving the control equation with homotopy analysis method (HAM). Using the analytical solution, we obtained an explicit expression for the contact history of sphere, as well as the vibration history of plate, which enables us to obtain the minimum distance between the impact location and the boundary applicable to the present model and Zener model. What is more, we proposed a model for a critical impact velocity, below which Zener’s model applies, and above which the present model does. By comparing with experiments with coated plate in the literature, we found that most experiments should use the present model rather than Zener model.