Method of reduction of dimensionality in contact and friction mechanics: A linkage between micro and macro scales

Computer simulations have been an integral part of the technical development process for a long time now. Industrial tribology is one of the last fields in which computer simulations have, until now, played no significant role. This is primarily due to the fact that investigating tribological phenomena requires considering all spatial scales from the macroscopic shape of the contact system down to the micro-scales. In the present paper, we give an overview of the previous work on the so-called method of reduction of dimensionality (MRD), which in our opinion, gives a key for the linking of the micro-and macro-scales in tribological simulations. MRD in contact mechanics is based on the mapping of some classes of three-dimensional contact problems onto one-dimensional contacts with elastic foundations. The equivalence of three-dimensional systems to those of one-dimension is valid for relations of the indentation depth and the contact force and in some cases for the contact area. For arbitrary bodies of revolution, MRD is exact and provides a sort of "pocket edition" of contact mechanics, giving the possibility of deriving any result of classical contact mechanics with or without adhesion in a very simple way. A tangential contact problem with and without creep can also be mapped exactly to a one-dimensional system. It can be shown that the reduction method is applicable to contacts of linear visco-elastic bodies as well as to thermal effects in contacts. The method was further validated for randomly rough self-affine surfaces through comparison with direct 3D simulations. MRD means a huge reduction of computational time for the simulation of contact and friction between rough surfaces accounting for complicated rheology and adhesion. In MRD, not only is the dimension of the space reduced from three to one, but the resulting degrees of freedom are independent (like normal modes in the theory of oscillations). Because of this independence, the method is predestinated for parallel calculation on graphic cards, which brings further acceleration. The method opens completely new possibilities in combining microscopic contact mechanics with the simulation of macroscopic system dynamics without determining the "law of friction" as an intermediate step.