Frenkel Kontorova friction model

The FK model accounts for the effects that have been ignored in the Tomlinson model, resulting from the interactions of neighboring atoms. For a more realistic friction model of solid bodies in relative sliding, the particles in the harmonic chain have to be connected to a substrate. This motivates the idea of combining the two models into a new system, as schematically shown in Fig. 24, which is known as the Frenkel-Kontorova-Tomlinson model. Static and dynamic behavior of the combined system can be studied through a similar approach presented in this section. 

Weiss, M. and Elmer, F. J., "A Simple Model for Wearless Friction The Frenkel-Kontorova-Tomlinson Model, Physics of Friction, B. N. J. Persson and E. Tosatti, Eds., Kluwer Academic Publishers, 1996,pp. 163-178. 

Another simple model that includes elasticity is the Frenkel Kontorova (FK) model [98], which plays an important role in understanding various aspects of solid friction. The simplest form of the FK model [98] consists of a onedimensional chain of N harmonically coupled atoms that interact with a periodic substrate potential (see Fig. 7). The potential energy is... 

The comparison reveals that in addition to the activity of individual atoms, the next-neighbor interaction and collective atomic motion must play an important role in creating friction. This mechanism can be investigated more efficiently via the Frenkel-Kontorova model. 

Studies based on the Frenkel-Kontorova model reveal that static friction depends on the strength of interactions and structural commensurability between the surfaces in contact. For surfaces in incommensurate contact, there is a critical strength, b, below which the depinning force becomes zero and static friction disappears, i.e., the chain starts to slide if an infinitely small force F is applied (cf. Section 3). This is understandable from the energetic point of view that the interfacial atoms in an incommensurate system can hardly settle in any potential minimum, or the energy barrier, which prevents the object from moving, can be almost zero. 

There have been relatively few experimental tests of the Frenkel Kontorova model because of the difficulty in making sufficiently flat surfaces and of removing chemical contamination from surfaces. In one of the earliest experiments, Hirano et al. [126] examined the orientational dependence of the friction between atomically flat mica surfaces. They found as much as an order of magnitude decrease in fi iction when the mica was rotated to become incommensurate. Although this experiment was done in vacuum, the residual friction in the incommensurate case may have been due to surface contamination. When the surfaces were contaminated by exposure to air, there was no significant variation in friction with the orientation of the surfaces. 

Applications The HMM has been applied to the study of friction between two-dimensional atomically flat crystal surfaces, dislocation dynamics in the Frenkel-Kontorova model (i.e., considering a one-dimensional chain of atoms in a periodic potential, coupled by linear springs ), and crack propagation in an inhomogeneous medium. ... 

Frenkel and Kontorova were not the hrst ones to use the model that is now associated with their names. However, unlike Dehlinger, who suggested the model [99], they succeeded in solving some aspects of the continuum approximation. Like the PT model, the FK model was first used to describe dislocations in crystals. Many of the recent applications are concerned with the motion of an elastic object over (or in) ordered [100] or disordered [101] structures. The FK model and generalizations thereof are also increasingly used to understand the friction between two solid bodies. 

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