Page 107 - Tribology in Machine Design

Elements of contact mechanics 93

where

</>*(s) being the probability density standardized by scaling it to give a unit
standard deviation. Using these equations one may evaluate the total real
area, load and number of contact spots for any given height distribution.
An interesting case arises where such a distribution is exponential, that is,

In this case

so that

These equations give

where Ci and C 2 are constants of the system. Therefore, even though the
asperities are deforming elastically, there is exact linearity between the load
and the real area of contact. For other distributions of asperity heights, such
a simple relationship will not apply, but for distributions approaching an
exponential shape it will be substantially true. For many practical surfaces
the distribution of asperity peak heights is near to a Gaussian shape.
Where the asperities obey a plastic deformation law, eqns (3.53) and
(3.54) are modified to become

It is immedately seen that the load is linearly related to the real area of
contact by N' = HA' and this result is totally independent of the height
distribution </>(z), see eqn (3.51).
The analysis presented has so far been based on a theoretical model of the
rough surface. An alternative approach to the problem is to apply the
concept of profilometry using the surface bearing-area curve discussed in
Section 3.8.1. In the absence of the asperity interaction, the bearing-area
curve provides a direct method for determining the area of contact at any
given normal approach. Thus, if the bearing-area curve or the all-ordinate
distribution curve is denoted by \j/(z) and the current separation between
the smooth surface and the reference plane is d, then for a unit nominal