Stick-slip model

We discuss the various dynamical models of earthquake-like failures in Chapter 4. Specifically, the properties of the Burridge-Knopoff stick-slip model (Burridge and Knopoff 1967) and of the self-organised criticality models, the Guttenberg-Richter type power laws, for the frequency distribution of earthquakes in these models are discussed here. 

Burridge-Knopoff stick-slip model of earthquakes 4.2.1 Laboratory simulation model... 

Velocity-Dependent Friction Research and development work on the numerical specification of contact and friction conditions may include mathematical formulation and implementation of friction models as well as adaptation of the numerical solution methods (Heisel et al. 2009 Neugebauer et al. 2011). Standard implementations may be illustrated using Coulomb s Law and the Friction Factor Law. These two basic models were modified using a stick-slip model. Using these models enables consideration of the relative sliding velocity between the tool and the workpiece. 

In the studies that attribute the boundary friction to confined liquid, on the other hand, the interests are mostly in understanding the role of the spatial arrangement of lubricant molecules, e.g., the molecular ordering and transitions among solid, liquid, and amorphous states. It has been proposed in the models of confined liquid, for example, that a periodic phase transition of lubricant between frozen and melting states, which can be detected in the process of sliding, is responsible for the occurrence of the stick-slip motions, but this model is unable to explain how the chemical natures of lubricant molecules would change the performance of boundary lubrication. 

The example demonstrates that the instability and consequent energy dissipation, similar to those in the Tomlinson model, do exist in a real molecule system. Keep in mind, however, that it is observed only in a commensurate system in which the lattice constants of two monolayers are in a ratio of rational value. For incommensurate sliding, the situation is totally different. Results shown in Fig. 21(b) were obtained under the same conditions as those in Fig. 21 (a), but from an incommensurate system. The lateral force and tilt angle in Fig. 21(b) fluctuate randomly and no stick-slip motion is observed. In addition, the average lateral force is found much smaller, about one-fifth of the commensurate one. 

Macroscopic stick-slip motion described above applies to the center of mass movement of the bodies. However, even in situations where the movement of the overall mass is smooth and steady, there may occur local, microscopic stick-slip. This involves the movement of single atoms, molecular groups, or asperities. In fact, such stick-slip events form the basis of microscopic models of friction and are the explanation why the friction force is largely independent of speed (see Section 11.1.9). 

Figure 11.9 Friction force microscope pictures (a, b) of a graphite(OOOl) surface as obtained experimentally with FFM and results of simulations (c, d) of the stick-slip friction using a two-dimensional equivalent of the Tomlinson model. The friction force parallel to the scan direction (a, c) and the lateral force perpendicular to the scan direction (b, d) are shown. The scan size is 20 Ax 20 A. Pictures taken from Ref. [481] with kind permission from R. Wiesendanger.

The validity of Coulomb s law has been verified also on the nanoscale Zworner et al. [484] showed that, for different carbon compound surfaces, friction does not depend on sliding velocity in the range between 0.1 /xm/s and up to 24 /xm/s. At low speeds, a weak (logarithmic) dependence of friction on speed was observed by Gnecco et al. [485] on a NaCl(lOO) surface and by Bennewitz et al. [486] on a Cu (111) surface. This can be modeled when taking into account thermal activation of the irreversible jumps in atomic stick-slip [487],... 

Figure 7 Examples of nanotribology on dry carbon surfaces for atomic force microscopy (AFM) (a) schematic description of the out-of-plane graphene deformation with the sliding AFM (Lee et al., 2010), (b) nanotube without tip (left) and tip-nanotube interaction under 2.5 nN normal force (right) (Lucas et al., 2009), (c) stick-slip rolling model with a step rotation of a C60 molecule (Miura et al., 2003), and (d) ballistic sliding of gold nanocluster on graphite (Schirmeisen, 2010).

F. Heslot, T. Baumberger, B. Caroli, and C Caroli, Creep, Stick-slip and Dry Friction Dynamics Experiments and a Heuristic Model, Phys. Rev., E49, 4973 (1994). 

The EVSS-G method introduced by Brown et al. uses the velocity gradient as an additional unknown ([7]). In order to come back to primitive variable, Guenette and Fortin ([20]) have introduced a (U, p, o, D) method where no explicit change of variable is performed in the constitutive equation. Hence this method is easier to implement. The elements used by these authors are continuous for velocity, discontinuous Pj for and pressure and continuous Qi for G and D. This method was tested on the 4 1 contraction and the stick-slip problem. This method seems robust and no limiting Weissenberg number was reached when using the PTT model for the stick-slip problem. 

Leonov A. I., Rheol. Acta, "A linear model of the stick slip phenomena in... 

It may be mentioned here that a recent study (Vasconcelos 1996) of a simple noncooperative (one-block) model of stick-slip motion (described by eqn (4.2) with / o = 0 or eqn (4.4) with k = 0) shows discontinuous velocity-dependent transition in the block displacement, for generic velocity-dependent friction forces. Naive generalisation of this observation for the coupled Burridge-Knopoff model would indicate a possible absence of criticality in the model. 

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]. 

As discussed in Section I.D, the dependence of friction on past history is often modeled by the evolution of a state variable (Eq. 6) in a rate-state model [50,51]. Heslot et al. [53] have compared one such model, where the state variable changes the height of the potential in a finite-temperature PT model, to their detailed experimental studies of stick-slip motion. They slid two pieces of a special type of paper called Bristol board and varied the slider mass M, pulling... 

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.

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