Friction law

It seems that indeed the answers to many fundamental questions are obtained, at least in qualitative form. Perhaps, the most important exception are thixotropic phenomena. There are many of them and the necessary systematization and mathematical generalization are absent here. Thus, it is not clear how to describe the effect of an amplitude on nonlinear dynamic properties. It is not clear what is the depth and kinetics of the processes of fracture-reduction of structure, formed by a filler during deformation. Further, there is no strict description of wall effects and a friction law for a wall slip is unknown in particular. 

The model proposed by Bowden and Tabor has been regarded as the most successful one for presenting a simple and logical theory capable of explaining the Amontons friction law. However, suspicions concerning the two fundamental assumptions in the model were gradually aroused over past years. Friction has been attributed, in Bowden and Tabor s model, to the adhesion between asperities in contact and torn-off of the adhesive junctions when the shear stress exceeds a critical value. This implies that plastic flow and surface destruction may occur at the moment of slip, and that friction is dominated by the shear strength of the adhesive conjunctions, which is material dependent. 

For the tangential component of the contact force, a Coulomb-type friction law is used ... 

The discussion begins below with an overview of proposed energy dissipation mechanisms that lead to friction. This is followed by brief discussions of phenomenological friction laws that describe the dependence of friction upon normal load and sliding velocity. The dependence of friction on the symmetry of the surfaces that are in contact is discussed later. 

Apart from the elastic stress transfer at the perfectly bonded interface, another important phenomenon that must be taken into account is the stress transfer by friction, which is governed by the Coulomb friction law after the interface bond fails. Furthermore, matrix yielding often takes place at the interface region in preference to interfacial debonding if the matrix shear yield strength, Xm is significantly smaller than the apparent interface bond strength, tb. It follows thus... 

In the debonded regions L z - (L - ) and (L - )<,z L), frictional slip occurs between the fiber and matrix and the stress transfer is governed by the Coulomb friction law for a constant coefficient of friction, p... 

Based on the Coulomb friction law, which governs the frictional stress transfer in the debonded interface, and combining Eqs. (4.12) and (4.18) yield the MAS at the interface (r = a)... 

However, some theoretical treatment considers only the special case of friction sliding of a single fiber along a mechanically bonded interface, particularly for some ceramic matrix composites, where the Coulomb friction law applies. See for example Zhou and Mai (1995) and Shetty (1988). Assuming a constant friction at the fiber-matrix interface and neglecting the Poisson effects, Shetty (1988) reported a simple force balance equation for the frictional shear strength, Tfr... 

This discrepancy can be due to a breakdown in the hydrodynamics friction law, Eq. (11.45), i.e., Stokes law, and/or a breakdown of the basic assumptions of Kramers theory. As we will see in the following section, a problem with Kramers theory is that the Langevin equation does not provide a sufficiently accurate description of the dynamics associated with the reaction coordinate. 

Under partial slip conditions, the estimate of the stress conditions at the edge of the contacts is complicated due to the unknown frictional behaviour within the partial slip annulus. If Coulomb s friction law is assumed to apply locally within this area, some contact mechanics calculation can, however, be... 

For elastic materials, the contact problem is usually solved as a unilateral contact problem obeying Coulomb s friction law. The algorithms used here are based on those pioneered by Kalker [66]. The contact area, the stick and slip regions, the pressure and traction distributions are numerically determined first and then the stress and displacement distributions within the elastic bodies can be established at the various stages of the tangential cyclic loading. On the basis of these calculations, the occurrence of crack initiation processes can subsequently be analysed in the meridian plane of the contact, y = 0 (Fig. 12), where the cracks first initiate. As a first approach, parameters based on the amplitude of the shear stress, rm, along a particular direction and the amplitude of the tensile stress, [Pg.174]

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].

MOLECULAR-LEVEL EXPLANATION OF MACROSCOPIC FRICTION LAWS ... 

For the turbulent boundary layer on a flat plate, von Karman s friction law with modified numerical coefficients [212,289],... 

For weakly nonlinearly viscous fluids that obey the viscous friction law t = r(7), the local friction coefficient can be found by using the approximate formulas... 

The slow variable models are ultimately based on purely macroscopic measurements. For example Onsager s diffusional model of Eq. (A.53) derives from Eq. (A. 15), which in turn is based on macroscopic phenomenology. Similarly, Onsager s Langevin model of Eq. (A.52) derives from Eq. (A.28), which in turn is suggested by the empirical friction law —for a macroscopic particle. 

In particular, slipping material becomes weaker during rupture and recovers to the static level at the end of the rupture. This behavior of the strength is known as the static-kinetic friction law. 

Such a dependence is characteristic of the dry friction, i.e. it agrees with Coulomb s friction law, Ffr = /frFN. Consequently, the model of plastic behavior of a material or disperse system may be represented by two surfaces (two plates) with a mutual friction coefficient, fx, pressed against each other with normal force, FN, causing the tangential force, Ffr, to be equal to the critical shear stress of material (Fig. IX-7). 

The above skin friction relationship was deduced intuitively many years earlier by von K5rman [10], who assumed local wall properties would control the skin friction law when property variations occur. Thus, Eq. 6.54 is equivalent to... 

There may also be a motion in x-direction, as indicated in Fig. 20.1. The first term describes the acceleration of the particle in y-direction, as the inert force. The second term contains the velocity of the particle and a friction term, i.e., the friction force rises with the velocity, counteracting the acceleration. The third term describes the electric force due to the voltage across the plates of the condenser. The friction law is similar to the law of Stokes, that is, the friction force is proportional to the velocity of the particle. / is a friction factor, in terms of the law of Stokes... 

Merchant has proposed an analytical approach elegantly appUcable to determine the force components in orthogonal cutting (Merchant 1944). He assumes the shear plane model and Coulomb s friction law. In Fig. 12, the force... 

However, as seen in [392] and [393], the Amonton law and the Coulomb criterion hold when the forces of intermolecular interaction between the shifted bodies are not taken into account. This interaction was taken into consideration by Derjagin [392, 393] in a binomial friction law. Analyzing the applicability of a binomial friction law, Deijagin notes that it should be used when equilibrium conditions are considered between two adjacent bodies with large area of actual contact, which are subjected not only to the forces perpendicular to the contact surface but also to forces parallel to it. An adhesive joint is just the contact of at least two surfaces by means of adhesive that fills the microrelief and ensm-es a large area of the true contact. 

It should be noted that the binomial friction law of Derjagin states that the adhesion bonds of two bodies and the friction forces determine a tangential load. The aggregate of these values is represented by the concept of static friction. Experimental validation [392, 393] of the binomial friction law allows the determination of this sum, but not of each item separately. As has been noted [392], this method serves only as an indirect estimation of adhesive joint strength. 

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