Hysteretic Friction

An expression for the hysteretic friction coefficient in the small viscoelasticity approximation is derived in the next section. 

We now give expressions for the coefficient of hysteretic friction, introduced in Sect. 2.11, for the problem of a moving, cylindrical punch discussed in the previous two sections. Let us consider the frictionless case first. We will confine the discussion to steady-state problems. Equation (2.11.12), adapted to plane strain conditions, gives that the coefficient of hysteretic friction has the general form 

Both of these are of interest. Consider (3.8.10) to begin with. If the material is incompressible, then v =, and we have 

We remarked at the end of Sect. 3.5 that for incompressible materials, the equations of the problem with friction reduced to those of the frictionless case, so that p(x) is given by the form for a lubricated surface. The only effect of the frictional surface is to attach the factor (1 +/ ) to /h- Therefore, if / is small, we can neglect its effects completely since it occurs in second order. 

The form given by (3.8.11) is also convenient in that it indicates immediately that /h must vanish in the instantaneous elastic limit. The point is that y x) becomes a delta function in this limit so that /h vanishes, because of the antisymmetry of the Hilbert kernel. We remark that in fact it vanishes in both limits discussed after (3.7.1). Let 

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This formula was given by Hunter (1961) for the special case of plane strain conditions and a cylindrical punch. For an elastic medium, this force is zero. However, for a viscoelastic medium, we shall see that this is not the case. The deformation caused by the moving load results in mechanical energy loss, which is manifested by the presence of a resisting force. This is the well-known force of hysteretic friction, first demonstrated experimentally by Tabor (1952). 

The expression for the hysteretic friction coefficient also simplifies greatly in this limit, as we shall see in Sect. 3.8. 

Equation (3.8.15) is a convenient general expression for the coefficient of hysteretic friction of a cylindrical indentor on a lubricated, slightly viscoelastic half-plane. 

IV. Hysteretic Friction. The hysteretic friction coefficient due to a moving inden-tor in lubricated contact with the half-plane is given by (3.8.1) and in the frictional case, by (3.8.9) or alternatively by (3.8.10) or (3.8.11). On an incompressible half-plane, it reduces to (3.8.12) which has the same form as for lubricated contact, apart from a factor (1 + / ). These expressions cannot be evaluated until the implicit equations governing the problem are solved, since the pressure is required. 

We consider the moving load problem in this section to the extent of deriving an expression for the coefficient of hysteretic friction in the small viscoelasticity and small velocity approximations, respectively. 

Let us now consider the coefficient of hysteretic friction, given by (5.4.1). This becomes... 

Three-dimensional contact problem solutions are obtained and analyses of impact and hysteretic friction are made. 

Sect. 5.4 contains approximate formulae for the coefficient of hysteretic friction. The approximations apply in the cases of small viscoelasticity (5.4.4,13) and small velocity (5.4.22, 23, 26). 

Golden, J.M. (1975) A molecular theory of adhesive rubber friction. J. Phys. A. 8, 966-979 Golden, J.M. (1977) Hysteretic friction of a plane punch on a half-plane with arbitrary viscoelastic behaviour. Q. J. Mech. Appl. Math. 30, 23-49... 

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