Preisach analysis of the Hertz–Mindlin system

In this paper we apply the Preisach hysteretic formalism to the Hertz–Mindlin system dealing with frictional and elastic interactions of two spheres in contact subjected to a varying oblique force. The classical solution for this fundamental mechanical problem, obtained by Mindlin and Deresiewicz over 50 years ago, has become the foundation for many applications, including friction of rough surfaces, physics of granular materials, rock mechanics, nonlinear acoustics, seismology, geological exploration, monitoring of oil reservoirs and mining processes, etc. However, due to the complex nature of the general case of a two-sphere interaction subjected to an arbitrary force protocol, researchers have commonly utilized only the basic results of the Hertz–Mindlin theory, i.e. the formulations of the normal and tangential stiffnesses and of the hysteretic loss of energy. The full description with arbitrarily varying forces has been possible only on a case-to-case basis. To overcome this hurdle, we propose to link the Hertz–Mindlin system to the fundamental theory introduced by Preisach over 70 years ago, which provides the tools to mathematically describe arbitrary hysteretic functions, i.e. non-unique and history-dependent functional relations with irreversible behavior. We show how the Preisach formalism simplifies the mathematical complexity of the Hertz–Mindlin problem and makes it possible to consider complicated force protocols in a universal manner. We prove that the Hertz–Mindlin interaction with dN/dT=const is a particular case of a Preisach system and we derive the essential parameters of the corresponding Preisach formalism. Finally, we demonstrate the analogy between force-driven systems and displacement-driven systems, and we propose a new solution for the displacement-driven Hertz–Mindlin friction.