Forces static friction

For a monolayer film, the stress-strain curve from Eqs. (103) and (106) is plotted in Fig. 15. For small shear strains (or stress) the stress-strain curve is linear (Hookean limit). At larger strains the stress-strain curve is increasingly nonlinear, eventually reaching a maximum stress at the yield point defined by = dT Id oLx x) = 0 or equivalently by c (q x4) = 0- The stress = where is the (experimentally accessible) static friction force [138]. By plotting T /Tlx versus o-x/o x shear-stress curves for various loads T x can be mapped onto a universal master curve irrespective of the number of strata [148]. Thus, for stresses (or strains) lower than those at the yield point the substrate sticks to the confined film while it can slip across the surface of the film otherwise so that the yield point separates the sticking from the slipping regime. By comparison with Eq. (106) it is also clear that at the yield point oo. 

The maximum value of the static friction force is proportional to the normal force as... 

The conhned liquid is found to exhibit both viscous and elastic response, which demonstrates that a transition from the liquid to solid state may occur in thin hlms. The solidihed liquid in the him deforms under shear, and hnally yields when the shear stress exceeds a critical value, which results in the static friction force required to initiate the motion. 

Figure 1 Force against time for a sliding system. Sliding is initiated once the threshold corresponding to the static friction force is surpassed when F/Fs = 1.

Kinetic friction force Static friction force Phase... 

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... 

There are two important issues addressed in the literature. One is the size of the prefactors Vn- These prefactors determine the maximum possible lateral force between the two objects, which provides a meaningful upper bound for the static friction force. The other issue is how the friction is altered through thermal fluctuations/activation that stem from the not explicitly treated internal degrees of freedom. These two questions will be discussed separately in Sections ILA and II.B before some applications are presented in Section II.C. 

Here x is the position of the slider relative to the substrate, h /Zn is the substrate s period, and/o is the (zero-temperature) static friction force, whose scaling with the area of contact and normal load we just discussed. In order to incorporate the effects of thermal fluctuations on the motion of the slider, one can exploit the isomorphism to the motion of a Brownian particle moving on a substrate. A nice description of that problem is given by Risken in Chapter 11 of Ref. 64. Here we will discuss some of the aspects that we believe to be important for friction. 

Figure 6. Static friction force versus normal load L for different tip geometries. In all cases, the radius of curvature was Rc = 70 A and contacts were nonadhesive. Straight lines are fits according to oc with the results P = 0.91 0.005 (commensurate) and p = 0.63 0.01 (amorphous). [With permission from Ref. 74. Europhys. Lett. 54, 663 (2001) additional data provided by L. Wenning (open diamonds).]...

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]. 

Experimentally, the static friction force is determined a.s a maximal force needed to initiate sliding motion. The question arises whether the static friction obtained in this way is a unique (inherent) property of the system, or whether it depends also on the conditions of the measurement. Section 11.C.2 only discussed the maximum possible shear force between a tip and a substrate, but the effect of thermal fluctuations and the effect of the driving device for instance, the stiffness of the driving device or the tip itself was not included. 

Nitta Y, Haga H, Kawabata K (2002) Time dependent static friction force of agar gel-on-glass plate immersed in water. J Phys IV France 12 319-320... 

Figure 4 Static frictional force, in arbitrary units, against...

All tests initially show fairly linear behaviour up to about 2 kN appUed load. The slope (marked Slope V in Figure 11.10(b) and (c)) is similar for both clearances. This is most likely because this region is dominated by static friction forces which would be independent of bolt-hole clearance and therefore the same for all joints. 

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