Richard Leach - Fundamental prinsiples of engineering nanometrology (778895), страница 55
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There are many ways of analysingfractal profiles [45]. Fractal parameters utilize information about the heightand the spacing characteristics of the surface, making them hybridparameters. Fractal profiles and surfaces usually have the followingcharacteristics:-they are continuous but nowhere differentiable;-they are not made up of smooth curves, but rather maybe described asjagged or irregular;-they have features that repeat over multiple scales;-they have features that repeat in such a way that they are self-similarwith respect to scale over some range of scales.Many, if not most, measured profiles appear to have the above characteristics over some scale ranges; that is to say that many profiles and surfacesof practical interest may be by their geometric nature more easily describedby fractal geometry rather than by Euclidian geometry.Fractals have some interesting geometric properties.
Most interesting isthat fractal surfaces have geometric properties that change with scale. Peak251252C H A P T ER 8 : Surface topography characterizationand valley radii, inclination of the surface, profile length and surface area, forexample, all change with the scale of observation or calculation. This meansthat a profile does not have a unique length. The length depends on the scaleof observation or calculation. This property in particular can be effectivelyexploited to provide characterization methods that can be used to modelphenomena that depend on roughness and to discriminate surfaces thatbehave differently or that were created differently.
The lack of a unique lengthis the basis for length-scale analysis.Fractals are often characterized by a fractional, or fractal, dimension,which is essentially a measure of the complexity of the surface or profile. Thefractal dimension for a line will be equal to or greater than one and less thantwo. The fractal dimension for a surface will be equal to or greater than twoand less than three. For mathematical fractal constructs, this characterization by fractal dimension can be scale-insensitive [46].
However, mostsurfaces of engineering interest are smooth if viewed at a sufficiently largescale, and the fractal dimension can change with respect to scale.Two approaches have been used to adapt fractal analysis to engineeringprofiles and surfaces. One approach is to treat the profiles as self-affine,meaning that they have a scaling exponent that varies with scale [47]. Theother approach is to divide the scales into regions. For example, most surfacesare rough at fine scales, and smooth at larger scales, and a smooth–roughcrossover (SRC) scale can be used to define the boundary between rough(described by fractal geometry) and smooth (described by Euclidean geometry). In the rough region the fractal dimension can be used to characterizeroughness; however, the relative lengths and relative areas at particularscales, which are used to determine the fractal dimension, may be moreuseful.
The SRC is determined as the scale at which the relative lengths orareas exceed a certain threshold. There may be other crossover scales,separating scale regions where different surface creation mechanisms havecreated geometries with different complexities.8.4.1 Linear fractal methodsThe fractal dimension and the length-scale fractal complexity are determinedfrom the slope of a log-log plot of relative lengths against scale [48].
Therelative lengths are the calculated lengths, determined from a series of virtualtiling exercises, divided by the nominal length (see Figure 8.23). The nominallength is the straight line length, or the length of the profile used in the lengthcalculation projected onto the datum. In a virtual tiling exercise the length ofthe profile at a certain scale is calculated by stepping along the measuredprofile with a line segment whose length is that scale.
The exercise isFractal methodsFIGURE 8.23 Line segment tiling on a profile.repeated in the series by using progressively different lengths and plotting thelogarithm of the relative lengths against the logarithm of the correspondingscale for determining that relative length. Linear interpolation is usedbetween measured heights to maintain consistency in the length of the linesegments.The slope of the graph in Figure 8.23 is determined over some appropriaterange of scales where the plot is approximately linear.
The scale region isindicated with the slope. Slope multiplied by minus 1000 is the linear fractalcomplexity parameter,Rlfc ¼ 1000 ðslopeÞ:(8.38)One minus the slope of the length-scale plot, whose value is a negativenumber, is the fractal dimension,Dls ¼ 1 ðslopeÞ:(8.39)While the slope of the length-scale plot is generally negative or zero whenthere are periodic structures on the surface, aliasing can result in small-scaleregions with positive slopes. In these cases local minima in relative lengthscan be found at integer multiples of the wavelength of the periodic structures[49].
The finest linear scale in this analysis that has meaning is the samplinginterval, and the largest is the length of the measured profile.Length-scale fractal analysis has found fewer applications than area-scalefractal analysis. Some examples of its use include determining anisotropy fordiscriminating different kinds of dental microwear [50], discrimination oftool usage and there is some indication that length-scale fractal analysis maybe useful in understanding the skin effect in high-frequency electrical253254C H A P T ER 8 : Surface topography characterizationtransmissions.
The relative lengths as a function of scale have also been usedto compare instruments [51].The relative length at a particular scale is related to the inclination on thesurface, f, at that scale. Inclinations on a surface vary as a function of scale(see Figure 8.24). The relative length parameter is given byRrel ¼X 1 pi Lcosfi(8.40)where L is the total nominal length of the profile and pi is the nominal, orprojected length of the ith segment.The relative length can give an indication of the amount of the surfacethat is available for interaction. The relative area, calculated from an arealmeasurement, however, gives a better indication, because it contains moretopographic information.When the analysed profile is sufficiently long, an SRC can be observed.
Atthe largest scales the relative lengths will tend towards a minimum weightedaverage of the reciprocal of the cosine of the average inclination of the analysed profile. If the profile is levelled, this will be one, the minimum relativelength.
In any case, the slope at the largest scales will be zero so that thefractal dimension will be one, the minimum for a profile. At larger scales therelative lengths deviate significantly from one and the SRC has been reached.A threshold in relative length can be used to determine the crossover in scale.There may be other crossover scales dividing regions of scale that havedifferent slopes on the relative length-scale plot. This possibility of multipleFIGURE 8.24 Inclination on a profile.Fractal methodsslopes on the length-scale plot is a characteristic of a scale-sensitive fractalanalysis.8.4.2 Areal fractal analysisThe areal fractal methods are in many ways similar to the linear methodsdiscussed in section 8.4.1.
As with the profile analyses there are manymethods that can be used to estimate the fractal dimension of a rough arealsurface. Two areal methods can be found in ISO 25178 part 2 [25], volumescale and area-scale methods.8.4.2.1 Volume-scale analysisVolume-scale analysis, also known as the variation method, estimates thevolume between morphological opening and closing envelopes abouta surface. The volume is estimated using nominally square, structuringelements. The size of the structuring elements is varied and the change ofvolume (Svs) is noted. The logarithm of the volume is plotted against thescale of the elements, i.e. the length of the sides of the square structuringelements. As the scale increases so does the volume.
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