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Rough Surfaces thomas4

The surface as a random process | The surface as a random process The mean summit curvature Km at any summit on a surface is defined as the mean of the principal curvatures at that point. The curvatures at a summit in the x and y directions are -d2x dx2 and -d2z dy2 respectively and so d2z d2z z - -7 7-2 2-------- 6.47 ay j Using the probability distribution for summits with height and equivalent mean curvature Km derived from 2 . it is possible to derive the expected value of the mean curvature for summits of height . In dimensionless form this is plotted in Fig. 6.5 for further details see Nayak 1971 . The magnitude of the surface gradient is defined by f 1 2 and by a standard method the probability density for is P 6 exp-- m2 zm2 The expected value of the gradient is thus i 00 _ irm2 1 2 Now the expected value of the absolute slope of a profile is 7r 6.48 6.49 6.50 130 o Surface statistics and so Cl 1 5 l l 6.51 Thus the mean surface gradient is greater than the mean profile slope. Discussion The probability density psum is plotted in Fig. 6.3 for a range of a values. Since a 5 1.5 Longuet-Higgins 1957a no values of a less than 1.5 appear in that figure. As a - 1.5 the probability of a high peak increases. Similarly Fig. 6.1 gives plots of ppeak f f r a range of a values for one-dimensional random processes a 1 . The significance of the parameter a is in general related to the breadth of PSDF a broad spectrum large a values has waves with a large number of wavelengths a narrow spectrum small a values has waves of approximately equal wavelength though see sect. 6.4 below . Figures 6-. 4 show a comparison of ppeak and psum - Clearly the profile distorts the surface in such a way as to show far fewer high peaks C and far more low peaks than appear on the surface. The distortion is greatest when a 1.5 and tends to zero as a In fact a profile- measuring instrument will travel over the shoulder of an asperity on the surface rather than the summit. A peak will still appear on the profile but of a .