Static friction model

This is just the empirical eqn. (25.1) we started with, with = 1/2, but this time it is not empirical - we derived it from a model of the sliding process. The value = 1/2 is close to the value of coefficients of static friction between unlubricated metal, ceramic and glass surfaces - a considerable success. 

P. Bordarier, B. Rousseau, A. H. Fuchs. A model for the static friction behavior of nanolubricated contacts. Thin Solid Films 330 21-26, 1998. 

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. 

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. 

In reality, static friction is always observed regardless of whether the surfaces in contact are commensurate or not. This raises a new question as to why the model illustrated in Fig. 29 fails to provide a satisfactory explanation for the origin of static friction. 

It has been proposed recently [28] that static friction may result from the molecules of a third medium, such as adsorbed monolayers or liquid lubricant confined between the surfaces. The confined molecules can easily adjust or rearrange themselves to form localized structures that are conformal to both adjacent surfaces, so that they stay at the energy minimum. A finite lateral force is required to initiate motion because the energy barrier created by the substrate-medium system has to be overcome, which gives rise to a static friction depending on the interfacial substances. The model is consistent with the results of computer simulations [29], meanwhile it successfully explains the sensitivity of friction to surface film or contamination. 

Finally, it deserves to be mentioned that considerable numbers of models of static friction based on continuum mechanics and asperity contact were proposed in the literature. For instance, the friction at individual asperity was calculated, and the total force of friction was then obtained through a statistical sum-up [35]. In the majority of such models, however, the friction on individual asperity was estimated in terms of a phenomenal shear stress without involving the origin of friction. 

A number of analogous compounds to BA have been reported, including 5,5 -dibenzo-[a]-pyrenyl (BBPY) [116]. These compounds exhibit emission spectra similar to BA. It would be interesting to explore the ultrafast dynamics of BBPY in order to test the generality of the GLE model. It would also be interesting to study the femtosecond dynamics of BA as a function of applied pressure. Static experiments on the emission of BA, reported by Hara et al. [123], demonstrate that in low viscosity solvents an increase of pressure affects the emission similarly to an increase of solvent polarity. As the pressure is increased, however, the LE/CT interconversion is slowed down. It would be interesting to measure C(r) in these environments and compare the solvation dynamics with LE/CT dynamics, in order to test the generality of the GLE dielectric friction model. 

When this force exceeds the static friction force 0(0), the block slips. We assume in this model that Fij becomes zero after a local slip of the block... 

Figure 10. Variation of steady state velocity v with force per atom F in dimensionless natural units. Results for a commensurate case (squares) follow static friction law, v is zero, until a direshold force is exceeded. The force is then relatively insensitive to velocity. Circles show results for a model of Kr on Au in an incommensurate crystalline state at 7=77 K (open) and in a fluid state at 160 K(filled). In both states the friction follows a viscous law, F v. The frictional force on the crystal is less than on the fluid [14].

One central issue in tiibology is why static friction is so universally observed between solid objects. How does any pair of macroscopic objects, placed in contact at any position and orientation, manage to lock together in a local free energy minimum A second issue is why experimental values of Fj and tend to be closely correlated. The two reflect fundamentally different processes and their behavior is qualitatively different in many of the simple models described below. 

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.

As argued by Fisher, pinned and sliding solutions can only coexist in some range of the externally applied force if the inertial term exceeds a certain threshold value [29]. This can lead to stick-slip motion as described in Section VI.A. For sufficiently small inertial terms, Middleton [85] has shown for a wide class of models, which includes the PT model as a special case, that the transition between pinned and sliding states is nonhysteretic and that there is a unique average value of F which does depend on vq but not on the initial microstate. The instantaneous value of Fk can nevertheless fluctuate, and the maximum of Fk can be used as a lower bound for the static friction force Fg. The measured values of Fj can also fluctuate, because unlike Fk they may depend on the initial microstate of the system [85]. 

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. 

The third threshold amplitude, Vq, is important for the precise meaning of the static friction Fj. Below this threshold the ground state of the undriven FKT model is the only mechanically stable state. For Vo > Vq additional metastable states appear. The first metastable state is not very different from the ground state. It can be described as a ground state plus a defect, which separates two equally sized domains. The motion of this defect allows sliding to occur. 

Miiser [25] examined yield of much larger tips modeled as incommensurate Lennard-Jones solids. The tips deformed elastically until the normal stress became comparable to the ideal yield stress and then deformed plastically. No static friction was observed between elastically deformed surfaces, while plastic deformation always led to pinning. Sliding led to mixing of the two materials like that found in larger two-dimensional simulations of copper discussed in Section IV.E. 

Figures 17b and 17c show the response in the lateral and normal directions to a lateral constant velocity drive for the stick slip regime that occurs at low driving velocities. This behavior is similar for the presently discussed model. The separation between the plates, which is initially Zq at equilibrium, starts growing before slippage occurs and stabilizes at a larger interplate distance as long as the motion continues. Since the static friction is determined by the amplitude of the potential corrugation exp(l — Z/A), it is obvious that the dilatancy leads to a decrease of the static friction compared to the case of a constant distance between plates.

Over a wide range of system parameters the dilatancy is smaller than the characteristic length A. Under this condition the generalized Prandtl Tomlinson model predicts a linear increase of the static friction with the normal load, which is in agreement with Amontons s law. It should be noted that, in contrast to the multi-asperity surfaces discussed in Section VII, here the contact area is independent of the load. The fulfillment of Amontons s law in the present model results from the enhancement of the potential corrugation, a2C7oexp(l — Z/K), experienced by the driven plate with an increase of the normal load. 

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