Page 103 - Tribology in Machine Design

Elements of contact mechanics 89

So far only variations in the height of the surface have been discussed.
However, spatial variations must also be taken into account. There are
several ways in which the spatial variation can be represented. One of them
uses the r.m.s. slope o m and r.m.s. curvature a k. For example, if the sample
length L of the surface is traversed by a stylus profilometer and the height z
is sampled at discrete intervals of length h, and if z,--i and z i+l are three
consecutive heights, the slope is then defined as

The r.m.s. slope and r.m.s. curvature are then found from

where n = L/h is the total number of heights sampled.
It would be convenient to think of the parameters a, a m and a k as
properties of the surface which they describe. Unfortunately their values in
practice depend upon both the sample length L and the sampling interval h
used in their measurements. If a random surface is thought of as having a
continuous spectrum of wavelengths, neither wavelengths which are longer
than the sample length nor wavelengths which are shorter than the
sampling interval will be recorded faithfully by a profilometer. A practical
upper limit for the sample length is imposed by the size of the specimen and
a lower limit to the meaningful sampling interval by the radius of the
profilometer stylus. The mean square roughness, a, is virtually independent
of the sampling interval h, provided that h is small compared with the
sample length L. The parameters <r m and a k, however, are very sensitive to
sampling interval; their values tend to increase without limit as h is made
smaller and shorter, and shorter wavelengths are included. This fact has led
to the concept of function filtering. When rough surfaces are pressed into
contact they touch at the high spots of the two surfaces, which deform to
bring more spots into contact. To quantify this behaviour it is necessary to
know the standard deviation of the asperity heights, <r s, the mean curvature
of their peaks, k s, and the asperity density, T/ S, i.e. the number of asperities
per unit area of the surface. These quantities have to be deduced from the
information contained in a profilometer trace. It must be kept in mind that
a maximum in the profilometer trace, referred to as a peak does not
necessarily correspond to a true maximum in the surface, referred to as a
summit since the trace is only a one-dimensional section of a two-
dimensional surface.
The discussion presented above can be summarized briefly as follows:
(i) for an isotropic surface having a Gaussian height distribution with