Method of Dimensionality Reduction in Contact Mechanics and Friction
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This book describes for the first time a simulation method for the fast calculation of contact properties and friction between rough surfaces in a complete form. In contrast to existing simulation methods, the method of dimensionality reduction (MDR) is based on the exact mapping of various types of three-dimensional contact problems onto contacts of one-dimensional foundations. Within the confines of MDR, not only are three dimensional systems reduced to one-dimensional, but also the resulting degrees of freedom are independent from another. Therefore, MDR results in an enormous reduction of the development time for the numerical implementation of contact problems as well as the direct computation time and can ultimately assume a similar role in tribology as FEM has in structure mechanics or CFD methods, in hydrodynamics. Furthermore, it substantially simplifies analytical calculation and presents a sort of “pocket book edition” of the entirety contact mechanics. Measurements of the rheology of bodies in contact as well as their surface topography and adhesive properties are the inputs of the calculations. In particular, it is possible to capture the entire dynamics of a system – beginning with the macroscopic, dynamic contact calculation all the way down to the influence of roughness – in a single numerical simulation model. Accordingly, MDR allows for the unification of the methods of solving contact problems on different scales. The goals of this book are on the one hand, to prove the applicability and reliability of the method and on the other hand, to explain its extremely simple application to those interested.
requiring that the displacement at the edge of the contact approaches zero: (3.32) The normal force is the sum of the spring forces (3.33) which provides (3.34) after integration and rearranging with the help of (3.32). The results (3.32) and (3.34) obtained by using the reduction method are exactly those derived by Ejike  for the three-dimensional problem. For the sake of completeness, let us state the relationship between normal force and indentation depth, which after solving (3.32)
Insertion of (4.32) into (4.23) and taking the separation condition (4.21) into account, results in (4.33) and after simple rearrangement, the contact radii for which the system is stable are obtained: (4.34) The marginally stable case characterizes the critical state at which the calculation of the critical values is possible: the minimum normal force and minimum indentation depth. In order to accomplish this, the contact radius in Eq. (4.34) must be take into account in Eqs. (4.30) and
Linearly Viscous Elastomers In this section, the results thus far will be used to demonstrate the application of the reduction method on elastomers. As in the previous section, the procedure will be first shown using a concrete example, the indentation of a rigid indenter into a linearly viscous incompressible half-space. The comparable elastic problem was closely examined in the previous chapter using the reduction method for an elastic half-space. The elastic half-space is mapped to a chain of
characteristic contact time can be approximated as , where is the linear velocity (driving speed). Then, the quasi-static state condition simply means (2.3) For a rough contact with a characteristic wavelength of , the characteristic time is , so that condition for the quasi-static state is much more restrictive: or (2.4) In most tribological systems, we are dealing with the movements of components whose relative velocities (e.g., a train at around 50 m/s) are orders of magnitude smaller than
could be done are discussed in . 16.6 Fracture and Plastic Deformation in the Method of Dimensionality Reduction The simple feature indicating that a physical quantity can be simulated very easily using a one-dimensional mapping, is the proportionality of this quantity to the diameter of the contact. This property can be trivially “one-dimensionalized.” Therefore, parameters such as stiffness and electrical/thermal conductivity are easily mappable to one-dimensional systems. Also the