General solution to the Hertz–Mindlin problem via Preisach formalism

Abstract

In this paper we analyze the contact problem for two elastic spheres with friction (the Hertz–Mindlin or HM problem) and present an original solution expressed in terms of the Preisach description. The major obstacle for an efficient use of the HM mechanics in the case of general loading histories lies in the complex memory-dependent and hysteretic behavior of the shear stress (traction) caused by the presence of irreversible slip in the contact zone. On the other hand, the Preisach formalism provides a versatile and powerful tool that is applicable to a wide class of systems with rate-independent hysteresis of any nature. The approach consists in a representation of a hysteretic system as a collection of bi-stable elements which can be switched from one state to another by an external action. An ensemble of the states of the elements determines all memory information contained in the system. Switching of elements is governed by a simple and easy-to-program procedure applicable to any kind of external action (protocol). We show that the two crucially identifying criterions of a generic Preisach system are verified within the HM mechanics and we calculate the characteristic parameters of the Preisach description. As a result, we obtain a compact and efficient analytical solution that allows to compute the hysteretic response in a direct way, i.e. without consideration of cumbersome traction distribution expressions.