3.8 Fractal surfaces [3.11]

From what has been said, it is clear that the autocorrelation and power spectrum can be used in surface analysis. One fundamental type of spectrum is given below for, say, a ground surface.

Fractal surfaces have a similar spectrum

v, the index for fractal surfaces, can take values between 0 and 2 allowing non-integer values. From this index, one of the two fractal parameters can be obtained. Thus + v = 5-2D; the other parameter called the topothesy ? can be obtained from the structure function [3.11].

There is a big debate at present about the relevance of fractals in surface metrology. Why bother? The essential property of a fractal surface is that the parameters D and A work out to be the same whatever the scale - a bit like 'fleas on fleas' as seen in Figure 3.52. All conventional parameters like curvature, slopes etc. are very scale-sensitive. Using conventional geometry, there is no such thing as a number or numbers representing the surface.

Figure 3.52: Scale invariant fractal surface

So, the idea is that with fractal geometry under certain conditions, a unique number can be assigned to the surface. At first sight, this appears to be a breakthrough in surface characterization but there is a snag. This is that functional uses of surfaces are very scale-dependent as will be seen later. Consequently, surface parameters hoping to predict performance should be equally scale-dependent. In other words, fractal geometry is precisely what...