Kinetic friction model

Tomlinson model that results in a finite kinetic friction. 

At finite velocity kinetic friction behaves quite differently in the sense that the commensurability plays a less significant role. Besides, the system shows rich dynamic properties since Eq (16) may lead to periodic, quasi-periodic, or chaotic solutions, depending on damping coefficient y and interaction strength h. Based on numerical results of an incommensurate case [18,19], we outline a force curve of F in Fig. 23 asafunction ofv, in hopes of gaining a better understanding of dynamic behavior in the F-K model. 

A paper by Prandtl [18] on the kinetic theory of solid bodies, which was published in 1928, one year prior to Tomlinson s paper [17], never achieved the recognition in the tribology community that it deserves. PrandtI s model is similar to the Tomlinson model and likewise focused on elastic hysteresis effects within the bulk. Nevertheless, Prandtl did emphasize the relevance of his work to dry friction between solid bodies. In particular, he formulated the condition that can be considered the Holy Grail of dry, elastic friction If the elastic coupling of the mass points is chosen such that at every instance of time a fraction of the mass points possesses several stable equilibrium positions, then the system shows hysteresis. In the context of friction, hysteresis translates to finite static friction or to a finite kinetic friction that does not vanish in the limit of small sliding velocities. Note that the dissipative term that is introduced ad hoc in Eq. (19) does vanish linearly with small velocities. 

Figure 9. Average kinetic friction F (independent of a) in the athermal Prandtl Tomlinson model at low velocities v for two different spring strengths k and various damping coefficients 7. The symbols at r o = 0 indicate the static friction force for k = 0.1k. All units are reduced units.

For Vo below the second threshold denoted by Vq, the kinetic friction is zero in the limit of quasi-static sliding that is, for sliding velocity v Q. That is, for Vo < Vq" the kinetic friction behaves like a viscous drag. For Vo > the dynamics is determined by the Prandtl Tomlinson-like mechanism of elastic instability, which leads to a finite kinetic friction. The threshold amplitude Vq increases with k and is always larger than zero. Therefore, in the commensurate case, vanishing kinetic friction does not imply vanishing static friction just like in the PT model. The FKT model for Vj, < Vo < is an example of a dry-friction system that behaves dynamically like a viscous fluid under shear even though the static friction is not zero. 

Physisorbed molecules also provide a natural explanation for the logarithmic increase in kinetic friction with sliding velocity that is observed for many materials and represented by the coefficient A in the rate-state model of Eq. (5). Figure 16 shows calculated values of tq and a as a function of log for a sub monolayer of chain molecules between incommensurate surfaces [195]. The value of To becomes independent of v at low velocities. The value of a, which... 

For reactive flows the governing equations used by Lindborg et al [92] resemble those in sect 3.4.3, but the solid phase momentum equation contains several additional terms derived from kinetic theory and a frictional stress closure for slow quasi-static flow conditions based on concepts developed in soil mechanics. Moreover, to close the kinetic theory model the granular temperature is calculated from a separate transport equation. To avoid misconception the model equations are given below (in which the averaging symbols are disregarded for convenience) ... 

The introduction to the concept of static and kinetic friction in Chapter 7, Section 2 is admittedly simplistic. Familiarity with the experimental details of measuring friction leads to a more realistic view in behavioristic terms and also to some theoretical questions. In particular, the theory of stick-slip friction requires that be greater than and distinct from and indeed Fig. 7-5a shows a discontinuity between static and kinetic friction. But the model for the fundamental adhesive mechanism of friction does not predict such a discontinuity. 

The factor that is least controllable in the deduction of static and kinetic friction values from stick-slip experiments is the condition of the contacting surfaces. This explains why Brockley and Davis [20] in their study of the influence of the time of contact on were unable to obtain repeatable results, as shown in Fig. 8-15a. For each series of determinations made with a single placement of the rider on the track, the plot of against the time of quiescent contact shows satisfactory self-consistency. But when the experiment is repeated with a fresh placement of the rider on the track, a different curve is obtained, self-consistent but not a duplicate of the previous experiment. This was traced to the variability of the surface with location on the rubbing track. Furthermore, it was demonstrated that a model of time-dependent junction growth could be made to yield the following expression ... 

The load-deflection response from both friction models (i.e. the continuous and stick-slip friction models) are shown in Figure 11.11 together with the experimental curve. Both friction models represent the behaviour in the experiment reasonably well. The sharp change from static fiiction Slope 1) to kinetic fiiction transition region) is slightly better represented by the stick-slip friction model. This is because... 

The close relationship between the dissipated energy and the kinetic friction force can be imderstood in terms of a model due to Israelachvili et al. that was recently proposed and shown to be valid for several systems. - According to this model the kinetic fiiction force can be written as... 

Anisotropic friction and molecular stick-slip friction were observed on a number of polymer systems by LFM using non-functionalized and functionalized tips. The observations described in this paper can be understood within the frame of the cobblestone (interlocking asperity) model. For a quantitative description, however, important parameters such as the energy fraction dissipated during the kinetic friction process (e) are still lacking. 

Predicting the solvent or density dependence of rate constants by equation (A3.6.29) or equation (A3.6.31) requires the same ingredients as the calculation of TST rate constants plus an estimate of and a suitable model for the friction coefficient y and its density dependence. While in the framework of molecular dynamics simulations it may be worthwhile to numerically calculate friction coefficients from the average of the relevant time correlation fiinctions, for practical purposes in the analysis of kinetic data it is much more convenient and instructive to use experimentally detemiined macroscopic solvent parameters. 

Eigure 20 compares the predictions of the k-Q, RSM, and ASM models and experimental data for the growth of the layer width 5 and the variation of the maximum turbulent kinetic energy k and turbulent shear stress normalized with respect to the friction velocity jp for a curved mixing layer... 

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