Modeling the elastic impact of a body with a special point at its surface

Abstract

We have considered the elastic straight impact along a flat border of the stationary half-space of the body bounded in a zone of contact interaction by the surface of rotation, whose order is smaller than two. The feature of the problem is that for the selected case an infinite curvature of the boundary surface at a point of initial contact, from which the process of dynamic compression of bodies in time starts. In addition to basic assumptions from the quasi-static theory of elastic impact between solid bodies, we have used a known solution to the static axisymmetric contact problem from the theory of elasticity. The process of an impact at a small initial velocity is divided into two stages: the dynamic compression and the dynamic decompression. For each of them, we have built an analytic solution to the nonlinear differential equation of relative convergence of the centers of bodies' masses in time. A solution to the non-linear problem with initial conditions for the differential equation of second order at the first stage was expressed through the Ateb-sinus, and at the second stage ‒ through the Ateb-cosine. To simplify calculations, we have compiled separate tables for the specified special functions, as well as proposed their compact approximations using basic functions. It was established that an error of analytical approximations of both special functions is less than one percent. We have also derived closed expressions for computing the maximum values: compression of a body, impact strength, radius of the circular contact area, and pressure, which is limited in the center of this area. We have considered a numerical example related to the impact of a rigid elastic body against a rubber half-space. Problems of this type arise when modeling the dynamic action of pieces of a solid mineral on rubber, when they fall on the rolls of a vibratory classifier lined with rubber. Based on the results from comparing the calculated parameters of an impact, we have received good agreement between numerical results, obtained from the constructed analytical solutions, and the integration of a nonlinear equation at a computer. This confirms the reliability of the built analytical solutions to the problem on impact, which provide for the convolution of a brief process over time