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Method of reduction of dimensionality in contact mechanics

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2007, Geike and Popov have shown [1], that there is a wide class of contacts between tree dimensional bodes which can be mapped either exactly or without loss of essential information to one dimensional systems (one-dimensional elastic or visco-elastic foundations). 2011, Markus Heß proved many of the mapping theorems and have shown that the exact mapping is always possible for any axis-symmetrical body, both without and with adhesion [2]. The equivalence of three dimensional systems to one dimensional ones is valid for relations of relative approach of the surfaces (or indentation depth), the contact area and the contact force. Tangential contact problem with and without creep is also mapped exactly to one-dimensional system. Another class of systems, to which the mapping can be applied, are bodies with randomly rough surfaces. It can further 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 of reduction of dimensionality means a huge reduction of computational time for simulation of contact and friction between rough surfaces with account of complicated rheology and adhesion. Because of independence of single “springs” of equivalent elastic foundations, it is predestinated for parallel calculation on graphic cards. The method allows for the first time to combine microscopic contact mechanics with simulation of macroscopic system dynamics without determining the “law of friction” as an intermediate step. Using the possibility to simulate both the frictional law and the macroscopic dynamics of a system in the framework of the same numerical model we illustrate the method on an example of a nano robot driven by oscillating spherical contacts both with smooth and rough surface. References 1. Geike T. and V.L. Popov, Mapping of three-dimensional contact problems into one dimension. – Phys. Rev. E., 2007, v. 76, 036710 (5 pp.). 2. Hess, M.: Über die exakte Abbildung ausgewählter dreidimensionaler Kontakte auf Systeme mit niedrigerer räumlicher Dimension. (About exact mapping of some contacts to systems of lower spatial dimension), Cuvillier, 172 p., 2011.

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