The gradual removal of Hertz pressure from the surface of elastic half-space

Abstract

Contact stress determination in non-stationary dynamic loading of elastic bodies is crucial for modelling structures at high speeds, but it presents mathematical challenges due to the time-dependent and often unknown contact area size and shape. The study aims to obtain an energy remainder estimation that forms waves during the contact interaction of elastic bodies, based on the exact solutions of non-stationary problems for an elastic half-space. For this purpose, the problem of the instantaneous loading half-space as an additional research problem was reconstructed using the Hankel transform concerning a radial coordinate and the Laplace transform concerning a time variable. The method of derivation of the displacements at an elastic half-space loaded (unloaded) gradually by Hertz's contact pressure has been proposed. Its availability made it possible to pass to the solution of the main problem – the problem of gradual loading of the half-space surface by Hertz pressure. The possibility of changing of the order of differentiation and integration operations in the obtained representation is substantiated based on the integrand properties. The cases when the speed of the indenter was constant when its motion was uniformly accelerated and when the motion corresponded to the law of the first quarter of the cosine period in the time were considered. It was concluded that the distribution of dynamic contact stresses is similar to the Hertz distribution. An estimation of the part of the energy spent on the formation of elastic waves was made for various laws of unloading. The practical significance of this study lies in its development of an effective method for calculating normal displacements on a loading area in dynamic contact interactions of elastic bodies, which can be valuable for modelling structures at high speeds