Modeling friction

The calculation of the lifetime is thus reduced to the problem of calculating (F(t)F(O)). This is a problem that has had a fairly long association with studies of solvation dynamics, where it usually appears in the context of efforts to model friction coefficients. A great deal of activity in this field has been directed at using the methods of density functional theory (83) to derive expressions for the correlation function that involve the thermodynamic parameters of the system (72,84), which themselves are often amenable to further analytical treatment or else may be determined experimentally or through simulations. In the treatment of vibrational relaxation... 

Kulczycki, A. (1985) The correlation between results of different model friction tests in terms of an energy analysis of friction and lubrication. Wear 103 67-75. 

R. L. Goetz and K. V. Jata, Modeling Friction Stir Welding of Titanium and Aluminum Alloys , in Friction Stir Welding and Processing, eds. K. V. Jata, M. W. Mahoney, R. S. Mishra, S. L. Semiatin, and D. P. Field, TMS, November 2001. 

W.J. Arbegast, Modeling Friction Stir Joining as a Metal Working Process, Hot Deformation of Aluminum Alloys, Z. Jin, Ed., TMS, 2003... 

Chapter 14 deals with plucking, striking, bowing, and rubbing excitations for physical models. Friction is discussed at length. MIDI and other protocols for controlling physical models are discussed. Finally, some custom-built physical modeling synthesis controllers are shown. 

McCarthy CT, McCarthy MA, Stanley WF, Lawlor VP. Experiences with modelling friction in composite bolted joints. J Compos Mater 2005 39(21) 1881—908. 

Figure 8.16 shows a biaxial yield contour for principal stresses a and 02 (with (73 = 0) of PVC, PC, PS, and PMMA at room temperature, normalized with the uniaxial yield stresses of these four polymers (Raghava et al. 1973). Thus, the center of the typical elliptical von Mises yield contour is displaced toward the biaxial compression quadrant characteristic of the pressure-dependent yield formalisms of Coulomb (1773) and Mohr (1900). For PC, discussed above, a model friction coefficient /if = 0.297 is obtained from eq. (8.40). In contrast, the displaced elliptical yield contour of experiments shown in Fig. 8.16 was obtained with a best fit of /if = 0.23, which gives a somewhat larger discrepancy than that for PC alone, given through eqs. (8.36) above. This could be attributed to the fact that the Oy used for normalization is itself dependent on (Tm. 

The microchannel geometric characteristics are length (2L) and hydraulic diameter (Dh, equal to four times the area divided by the perimeter of a section), shown in Fig. la. The model relates the efficiency of the compression process to the velocity, pressure, and temperature of the gas at the entrance of the channel (station 1 Mj, pi, Ti) the pressure ratio across the shock 11s the friction coefficient fi and channel dimensicms. fri Fig. la, a shockwave is shown that moves in the opposite direction to the flow and is positioned in the middle of the channel. It can be shown that a snapshot evaluatimi at the mid position is a good representation of the overall results and does not affect the accuracy of the model. Friction is considered along the lengths L before and after the shock. The frictional effect is modeled as shear stress at the wall acting on a fluid with uniform properties over the cross section. 

An additional amount of pressure must be applied to overcome frictional force. The Washburn equation [190] can be used to model frictional forces ... 

The two main approaches used to model frictional pressure gradients in macro- and microscale two-phase gas-liquid flow are the homogeneous model and the separated flow model. The homogeneous model is based on the assumption that the two-phase fluid behaves as a pseudo single-phase fluid with pseudo-properties that are weighted relative to the vapor and Uquid flow fraction. Different ways to define the pseudo-properties, usually called mixture properties, have been proposed that are well detailed by Collier and Thome [54] and Thome [55]. 

Dimensionless constant in turbulent viscosity model (—) Friction factor (—)... 

The radioautographic work suggests another model illustrated in Fig. XII-11. The load is supported over area A, with metal contacts of shear strength s over a portion of the area ctA and film-film contacts of shear strength Sf over the rest of the area. In analogy to Eq. XII-9, one can write the total frictional force, F as... 

This ensures the correct connection between the one-dimensional Kramers model in the regime of large friction and multidimensional imimolecular rate theory in that of low friction, where Kramers model is known to be incorrect as it is restricted to the energy diflfiision limit. For low damping, equation (A3.6.29) reduces to the Lindemann-Flinshelwood expression, while in the case of very large damping, it attains the Smoluchowski limit... 

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. 

Multidimensionality may also manifest itself in the rate coefficient as a consequence of anisotropy of the friction coefficient [M]- Weak friction transverse to the minimum energy reaction path causes a significant reduction of the effective friction and leads to a much weaker dependence of the rate constant on solvent viscosity. These conclusions based on two-dimensional models also have been shown to hold for the general multidimensional case [M, 59, and 61]. 

The relation between the microscopic friction acting on a molecule during its motion in a solvent enviromnent and macroscopic bulk solvent viscosity is a key problem affecting the rates of many reactions in condensed phase. The sequence of steps leading from friction to diflfiision coefficient to viscosity is based on the general validity of the Stokes-Einstein relation and the concept of describing friction by hydrodynamic as opposed to microscopic models involving local solvent structure. In the hydrodynamic limit the effect of solvent friction on, for example, rotational relaxation times of a solute molecule is [ ]... 

Zhu S-B, Lee J, Robinson G W and Lin S H 1988 A microscopic form of the extended Kramers equation. A simple friction model for cis-trans isomerization reactions Chem. Phys. Lett. 148 164-8... 

Carpick et al [M] used AFM, with a Pt-coated tip on a mica substrate in ultraliigh vacuum, to show that if the defonnation of the substrate and the tip-substrate adhesion are taken into account (the so-called JKR model [175] of elastic adliesive contact), then the frictional force is indeed proportional to the contact area between tip and sample. Flowever, under these smgle-asperity conditions, Amontons law does not hold, since the statistical effect of more asperities coming into play no longer occurs, and the contact area is not simply proportional to the applied load. 

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