Abstract
The method of dimensionality reduction (MDR) is based on the observation that certain types of three-dimensional contacts can be exactly mapped to one-dimensional linearly elastic foundations. The MDR consists essentially of two simple steps: (a) substitution of the three-dimensional continuum by a uniquely defined one-dimensional linearly elastic or viscoelastic foundation (Winkler foundation) and (b) transformation of the three-dimensional profile of the contacting bodies by means of the MDR-transformation. As soon as these two steps are done, the contact problem can be considered to be solved. For axially-symmetric contacts, only a small calculation by hand is required which not exceed elementary calculus and will not be a barrier for any practically-oriented engineer. In spite of its simplicity, all results are exact. The present chapter describes the basic ideas of MDR in its application to normal contacts.
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Notes
- 1.
This result can be found in any book dealing with contact mechanics (see, for example [1]).
- 2.
Strictly speaking, a parabolic profile with the radius of curvature R is considered.
- 3.
German–Russian Workshop “Numerical simulation methods in tribology: possibilities and limitations”, Berlin University of Technology, March 14–17, 2005. Published in [2].
- 4.
Let it be pointed out here that, as in the introductory examples, a one-dimensional profile is generally denoted with \( g\left( x \right) \) and a three-dimensional profile with \( f\left( r \right) \). Both are defined as being positive from the tip of the indenter upwards, which is additionally introduced as the coordinate \( \tilde{z} \) (see Fig. 3.4).
- 5.
For self-affinity, the following property is understood: If the profile (3.14) is stretched in the horizontal direction by the factor \( C \) and simultaneously in the vertical direction by a factor \( C^{n} \), then one obtains the original profile. The exponent \( n \) is known as the Hurst exponent.
- 6.
Frequently, the one-dimensional profile is referred to in the following; this is to be understood, of course, as the profile in the one-dimensional model.
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Popov, V.L., Heß, M. (2015). Normal Contact Problems with Axially-Symmetric Bodies Without Adhesion. In: Method of Dimensionality Reduction in Contact Mechanics and Friction. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-53876-6_3
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