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The Prandtl-Tomlinson-Model


One of the most popular models in nanotribology widely used as the basis for many investigations of frictional mechanisms on the atomic scale is the so-called "Tomlinson model", schematically shown in the Figure.
The name "Tomlinson model" is, however, historically incorrect: The paper by Tomlinson that is often cited in this context [Phil. Mag, 1929, v.7, p.905] did not contain the model known as the "Tomlinson model" and suggests an adhesive contribution to friction. In reality it was Ludwig Prandtl who suggested in 1928 this model to describe the plastic deformations in crystals
[L. Prandtl, Ein Gedankenmodell zur kinetischen Theorie der festen Körper. ZAMM, 1928, Vol. 8, p. 85-106.].
Following some other researchers we therefore call this model the "Prandtl-Tomlinson-Model". The original paper by Ludwig Prandtl was written in German and was not accessible for a long time for the largest part of international tribological community. Valentin Popov and Joshua Gray have translated this classical paper into English. This translation appeared 2012 in the same Journal as the original paper by Prandtl:

Popov V.L.  and Gray J.A.T., Prandtl-Tomlinson Model: History and applications in friction, plasticity, and nanotechnologies.- ZAMM, Z. Angew. Math. Mech., 2012, v. 92, No. 9, pp. 683-708 / DOI 10.1002/zamm.201200097.


In this paper, Ludwig Prandtl considered not only the simplest deterministic form of the model shown in the Figure, but also the influence of thermal fluctuations. He was the first "tribologist" who came to the conclusion that thermal fluctuations should lead to a logarithmic dependency of the frictional force on velocity. The success of the model as well as its variations and generalizations is due to the fact that it is the simplest usable model of a tribological system. Results concerning the basic properties of the Prandtl-Tomlinson-Model are scattered over many publications. In his book
Contact Mechanics and Friction (2nd Ed., Springer 2017),
Valentin Popov tries to summarize the basic properties of the model.

Chapter 11 "The Prandtl-Tomlinson-Model for Dry Friction"
from: Valentin L. Popov, "Contact Mechanics and Friction", Springer, 2010.

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