Hertz–Mindlin problem for arbitrary oblique 2D loading: General solution by memory diagrams

Abstract

In this paper we present a new general solution to the fundamental problem of frictional contact of two elastic spheres, also known as the Hertz–Mindlin (HM) problem. The description of spheres in contact is a central topic in contact mechanics. It became a foundation of many applications, such as the friction of rough surfaces and the mechanics of granular materials and rocks, etc. Moreover, it serves as a theoretical background in modern nonlinear acoustics and elasticity, e.g. seismology and nondestructive testing. However, despite many efforts, a rigorous analytical solution for the general case when arbitrary normal and tangential forces are present is still missing, mainly because the traction distribution within the contact zone is convoluted and hardly tractable, even under relatively simple external action. Here, accepting a number of traditional limitations such as 2D loading and the existence of a functional dependence between normal and tangential forces, we propose an original way of replacing the complex traction distributions by simple graphical counterparts called memory diagrams, and we formulate a procedure that enables initiating and maintaining these memory diagrams following an arbitrary loading history. For each memory diagram, the solution can be expressed by closed-form analytical formulas that we have derived using known techniques suggested by Mindlin, Deresiewicz, and others. So far, to the best of our knowledge, arbitrary loading histories have been treated only numerically. Implementation of the proposed memory diagram method provides an easy-to-use computer-assisted analytical solution with a high level of generality. Examples and results illustrate the variety and richness of effects that can be encountered in a geometrically simple system of two contacting spheres.