Engineering Tribology Episode 2 Part 10 pot

Theintroduction of statistical methods to the analysis of surface topography was probably due toAbbott and Firestone in 1933 [11] when they proposed a bearing area curve as a means ofpro

FIGURE 10.3 Self-similarity of surface profiles.

It has been observed that surface roughness profiles resemble electrical recordings of whitenoise and therefore similar statistical methods have been employed in their analysis Theintroduction of statistical methods to the analysis of surface topography was probably due toAbbott and Firestone in 1933 [11] when they proposed a bearing area curve as a means ofprofile representation [9] This curve representing the real contact area, also known as theAbbott curve, is obtained from the surface profile It is compiled by considering the fraction ofsurface profile intersected by an infinitesimally thin plane positioned above a datum plane.The intersect length with material along the plane is measured, summed together andplotted as a proportion of the total length The procedure is repeated through a number ofslices The proportion of this sum to the total length of bearing line is considered to representthe proportion of the true area to the nominal area [9] Although it can be disputed that thisprocedure gives the bearing length along a profile, it has been shown that for a randomsurface the bearing length and bearing area fractions are identical [12,9] The obtained curve is

in fact an integral of the height probability density function ‘p(z)’ and if the height

distribution is Gaussian then this curve is nothing else than the cumulative probability

function ‘P(z)’ of classical statistics The height distribution is constructed by plotting the

number or proportion of surface heights lying between two specific heights as a function ofthe height [9] It is a means of representing all surface heights The method of obtaining thebearing area curve is illustrated schematically in Figure 10.4 It can be seen from Figure 10.4that the percentage of bearing area lying above a certain height can easily be assessed

Although in general it is assumed that most surfaces exhibit Gaussian height distributionsthis is not always true For example, it has been shown that machining processes such asgrinding, honing and lapping produce negatively skewed height distributions [13] whilesome milling and turning operations can produce positively skewed height distributions [9]

In practice, however, many surfaces exhibit symmetrical Gaussian height distributions.Plotting the deviation of surface height from a mean datum on a Gaussian cumulativedistribution diagram usually gives a linear relationship [14,15] A classic example of Gaussianprofile distribution observed on a bead-blasted surface is shown in Figure 10.5 The scale ofthe diagram is arranged to give a straight line if a Gaussian distribution is present

0

h z

∆z

perpendicular to the plane of the surface, ∆z is the interval between two heights,

h is the mean plane separation, p(z) is the height probability density function, P(z) is the cumulative probability function [9].

Height above arbitrary datum [µm]

from [14])

During mild wear the peaks of the surface asperities are truncated resulting in a surfaceprofile consisting of plateaux and sharp grooves In such profiles the asperity heights aredistributed according to not one but two Gaussian constants, i.e Gaussian surface profileexhibits bi-modal behaviour [40] Truncation of surface asperities is found to be closelyrelated to a ‘running-in’ process where a freshly machined surface is worn at light loads inorder to be able to carry a high load during service

It should also be realized that most real engineering surfaces consist of a blend of randomand non-random features The series of grooves formed by a shaper on a metal surface are aprime example of non-random topographical characteristics On the other hand, bead-blastedsurfaces consist almost entirely of random features because of the random nature of thisprocess The shaped surface also contains a high degree of random surface features whichgives its rough texture In general, non-random features do not significantly affect the contactarea and contact stress provided that random roughness is superimposed on the non-random features

Characterization of Surface Topography

A number of techniques and parameters have been developed to characterize surfacetopography The most widely used surface descriptors are the statistical surface parameters Anew development in this area involves surface characterization by fractals

· Characterization of Surface Topography by Statistical Parameters

Real surfaces are difficult to define In order to describe the surface at least two parameters areneeded, one describing the variation in height (i.e height parameter), the other describinghow height varies in the plane of the surface (i.e spatial parameter) [9] The deviation of asurface from its mean plane is assumed to be a random process which can be described using

a number of statistical parameters

Height characteristics are commonly described by parameters such as the centre-line-average

or roughness average (CLA or ‘R a ’), root mean square roughness (RMS or ‘R q’), mean value

of the maximum peak-to-valley height (‘R tm ’), ten-point height (‘R z’) and many others Inengineering practice, however, the most commonly used parameter is the roughnessaverage Some of the height parameters are defined in Table 10.1

The ‘R a’ represents the average roughness over the sampling length The effect of a singlespurious, non-typical peak or valley (e.g a scratch) is averaged out and has only a small effect

on the final value Therefore, because of the averaging employed, one of the maindisadvantages of this parameter is that it can give identical values for surfaces with totally

different characteristics Since the ‘R a’ value is directly related to the area enclosed by thesurface profile about the mean line any redistribution of material has no effect on its value.The problem is illustrated in Figure 10.6 where the material from the peaks of a ‘bad’ bearing

surface is redistributed to form a ‘good’ bearing surface without any change in the ‘R a’ value[9]

−a

The ‘good’ bearing surface illustrated schematically in Figure 10.6 in fact approximates tomost worn surfaces where lubrication is effective Such surfaces tend to exhibit thefavourable surface profile, i.e quasi-planar plateaux separated by randomly spaced narrowgrooves

The problem associated with the averaging effect can be rectified by the application of theRMS parameter since, because it is weighted by the square of the heights, it is more sensitive

than ‘R a’ to deviations from the mean line

Spatial characteristics of real surfaces can be described by a number of statistical functions.Some of the commonly used functions are shown in Table 10.2 Although two surfaces canhave the same height parameters their spatial arrangement and hence their wear and

TABLE 10.1 Commonly used height parameters.

peak-to-Ten-point height

Average separation of the five highest peaks and the five lowest valleys within the sampling length

R t= 1 5ΣR maxi

i = 1 5

where:

L is the sampling length [m];

z is the height of the profile along ‘x’ [m].

frictional behaviour can be very different To describe spatial arrangement of a surface the

autocovariance function (ACVF) or its normalized form the autocorrelation function (ACF), the structure function (SF) or the power spectral density function (PSDF) are commonly used

[9] The autocovariance function or the autocorrelation function are most popular inrepresenting spatial variation These functions are used to discriminate between the differingspatial surface characteristics by examining their decaying properties Their limitation,however, is that they are not sensitive enough to be used to study changes in surfacetopography during wear Wear usually occurs over almost all wavelengths and thereforechanges in the surface topography are hidden by ensemble averaging and the autocorrelation

functions for worn and unworn surfaces can look very similar as shown in Table 10.2 [9].

This problem can be avoided by the application of a structure function [9,16] Although thisfunction contains the same amount of information as the autocorrelation function it allows

a much more accurate description of surface characteristics The power spectral density

TABLE 10.2 Statistical functions used to describe spatial characteristics of the real surfaces

τ is the spatial distance [m];

β* is the decay constant of the exponential autocorrelation function [m];

ω is the radial frequency [m-1], i.e ω = 2π/λ, where ‘λ’ is the wavelength [m]

function as a spatial representation of surface characteristics seems to be of little value.Although Fourier surface representation is mathematically valid the very complex nature ofthe surfaces means that even a very simple structure needs a very broad spectrum to be wellrepresented [9].

A detailed description of the height and spatial surface parameters can be found in, forexample, [9,41]

Multi-Scale Characterization of Surface Topography

A characteristic feature of the engineering surfaces is that they exhibit topographical detailsover a wide range of scales; from nano- to micro-scales It has been shown that surfacetopography is a nonstationary random process for which the variance of height distribution(RMS2) depends on the sampling length (i.e the length over which the measurement istaken) [17] Therefore the same surface can exhibit different values of the statisticalparameters when a different sampling length or an instrument with a different resolution isused This leads to certain inconsistencies in surface characterization [18] The main problem

is associated with the discrepancy between the large number of length scales that a roughsurface contains and the small number of particular length scales, i.e sampling length andinstrument resolution, that are used to define the surface parameters Therefore traditionalmethods used in 3-D surface topography characterization provide functions or parametersthat strongly depend on the scale at which they are calculated This means that theseparameters are not unique for a particular surface (e.g [42-44]) Since this ‘one-scale’characterization provided by statistical functions and parameters is in conflict with themulti-scale nature of tribological surfaces new ‘multi-scale’ characterization methods stillneed to be developed Recent developments in this area have been concentrated on threedifferent approaches:

· Fourier transform methods;

· wavelet transformation methods, and

· fractal methods

For the characterization of surfaces by wavelet and fractal methods the 3-D surfacetopography data is presented in the form of range images [45,46] In these images the surfaceelevation data is encoded into a pixel brightness value, i.e the brightest pixel, depicted by thegrey level of ‘255’, represents the highest elevation point on the surface, while the darkestpixel, depicted by the grey level of ‘0’, represents the lowest elevation point on the surface[46]

· Characterization of Surface Topography by Fourier Transform

Fourier transform methods allow to decompose the surface data into complex exponentialfunctions of different frequencies The Fourier methods were used to calculate the powerspectrum and the autocorrelation function in order to obtain the surface topographyparameters [e.g 41,47-49] However, the problem with the application of these methods tosurfaces is that they provide results which strongly depend on the scale at which they arecalculated, and hence they are not unique for a particular surface This is because the Fouriertransformation provides only the information whether a certain frequency component exists

or not As the result, the surface parameters calculated do not provide information about thescale at which the particular frequency component appears

· Characterization of Surface Topography by Wavelets

Wavelet methods allow to decompose the surface data into different frequency componentsand then to characterize it at each individual scale The wavelet methods were used todecompose the topography of a grinding wheel surface into long and small wavelengths [50],

to analyse 3-D surface topography of orthopaedic joint prostheses [51] and others Whenapplying wavelets the surfaces are usually first decomposed into roughness, waviness andform, and then the changes in surface peaks, pits and scratches together with their locationsare obtained at different scales However, there are still major difficulties in extracting theappropriate surface texture parameters from wavelets [44]

An example of the application of a wavelet transform to decompose a titanium alloy surfaceimage at two different levels is shown in Figure 10.7 Each decomposition level contains alow resolution image and three images containing the vertical, horizontal and diagonaldetails of the original image at a particular scale The low resolution images and the detailimages were obtained by applying a combination of low-pass and/or high-pass filters alongthe rows and columns of the original image and a downsampling operator The originalimage can be reconstructed back from these images obtained by using mirror filters and anupsampling operator [52]

Original image

Level 1 wavelet decomposition

Level 2 wavelet decomposition

Figure 10.7 Example of application of wavelet transform to titanium alloy surface image

· Characterization of Surface Topography by Fractals

Fractal methods allow to characterize surface data in a scale-invariant manner Usuallyfractal dimensions, since they are both ‘scale-invariant’ and closely related to self-similarity,are employed to characterize rough surfaces [19] The basic difference between thecharacterization of real surfaces by statistical methods and fractals is that the statisticalmethods are used to characterize the disorder of the surface roughness while the fractals areused to characterize the order behind this apparent disorder [10]

The variation in height ‘z’ above a mean position with respect to the distance along the axis

‘x’ of the surface profile obtained by stylus or optical measurements, Figure 10.8, can becharacterized by the Weierstrass-Mandelbrot function which has the fractal dimension ‘D’and is given in the following form [19]:

z ( x ) is the function describing the variation of surface heights along ‘x’;

G is the characteristic length scale of a surface [m] It depends on the degree of

surface finish For example, ‘G’ for lapped surfaces was found to be in the range

of 1 × 10 -9 to about 12.5 × 10 -9 [m], for ground surfaces about 0.1 × 10 -9 - 10 × 10 -9 [m]

while for shape turned surfaces ‘G’ is about 7.6 × 10 -9 [m] [20];

n 1 is the lowest frequency of the profile, i.e the cut-off frequency, which depends

on the sampling length ‘L’, i.e γ n1 = 1/L [m-1] [19,17];

γ is the parameter which determines the density of the spectrum and the relative

phase difference between the spectral modes Usually γ = 1.5 [20];

γn are the frequency modes corresponding to the reciprocal of roughness

wavelength, i.e γn = 1/λn [m-1] [19];

D is the fractal dimension which is between 1 and 2 It depends on the degree of

surface finish For example, ‘D’ for lapped surfaces was found to be in the range

of 1.7 to about 1.9, for ground surfaces about 1.6 while for shape turned surface

‘D’ is about 1.8 [20].

x z

The Weierstrass-Mandelbrot function has the properties of generating a profile that does notappear to change regardless of the magnification at which it is viewed As the magnification

is increased, more fine details become visible and so the profile generated by this functionclosely resembles the real surfaces In analytical terms, the Weierstrass-Mandelbrot function

is non-differentiable because it is impossible to obtain a true tangent to any value of the

function The fractal dimension and other parameters included in the Mandelbrot function provide more consistent indicators of surface roughness thanconventional parameters such as the standard deviation about a mean plane This is becausethe fractal dimension is independent of the sampling length and the resolution of theinstrument which otherwise directly affect the measured roughness [17].

Weierstrass-Although the Weierstrass-Mandelbrot function appears to be very similar to a Fourier series,there is a basic difference The frequencies in a Fourier series increase in an arithmeticprogression as multiples of a basic frequency, while in a Weierstrass-Mandelbrot functionthey increase in a geometric progression [19] In Fourier series the phases of some frequenciescoincide at certain nodes which make the function appear non-random With theapplication of a Weierstrass-Mandelbrot function this problem is avoided by choosing a non-integer ‘γ’, and taking its powers to form a geometric series It was found that γ = 1.5 providesboth phase randomization and high spectral density [20]

The parameters ‘G’ and ‘D’ can be found from the power spectrum of the

Weierstrass-Mandelbrot function (10.1) which is in the form [19,21]:

S( ω) is the power spectrum [m3];

ω is the frequency, i.e the reciprocal of the wavelength of roughness, [m-1], i.e the

low frequency limit corresponds to the sampling length while the highfrequency limit corresponds to the Nyquist frequency which is related to theresolution of the instrument [19]

The fractal dimension ‘D’ is obtained from the slope ‘m’ of the log-log plot of ‘S(ω)’ versus ‘ω’,

The parameter ‘G’ which determines the location of the spectrum along the power axis and is

a characteristic length scale of a surface is obtained by equating the experimental variance ofthe profile to that of the Weierstrass-Mandelbrot function [19,20]

The constants ‘D’, ‘G’ and ‘n 1’ of the Weierstrass-Mandelbrot function form a complete set ofscale independent parameters which characterize an isotropic rough surface [20] When theyare known then the surface roughness at any length scale can be determined from theWeierstrass-Mandelbrot function [20]

It should also be mentioned that in the fractal model of roughness, as developed so far, thescale of roughness is imagined to be unlimited For example, if a sufficient sampling distance

is selected, then macroscopic surface features, i.e ridges and craters would be observed Inpractice, engineering surfaces contain a limit to roughness, i.e the surfaces are machined

‘smooth’ and in this respect, the fractal model diverges from reality

There have been various techniques developed to evaluate the fractal dimension from aprofile, e.g horizontal structuring element method (HSEM) [53], correlation integral [54,55]and fast Fourier transform (FFT) [e.g 56], modified 1D Richardson method [57] and others.However, it was found that fractal dimensions calculated from surface profiles exhibit somefundamental limitations especially when applied to the characterization of worn surfaces[56,58,59] For example, it was shown that the fractal dimensions calculated fail to distinguishbetween the two worn surfaces [60] Also tests conducted on artificially generated profilesdemonstrated that the problem of choosing any particular algorithm for the calculation offractal dimension from a profile is not a simple one since there is no way of knowing the

‘true’ or even ‘nominal’ fractal dimension of the surface profile under consideration [59].Attempts have also been made to apply fractal methods to characterization of 3-D surfacetopographies For example, it has been shown that surface fractal dimensions could be used

to characterize surfaces exhibiting fractal nature [56], surface profiles produced by turning,electrical discharge and grinding [61], isotropic sandblasted surfaces and anisotropic groundsurfaces [62], engineering surfaces measured with different resolutions [63], etc The mostpopular methods used to calculate surface fractal dimension are: the ε-blanket [64], box-counting [65], two-dimensional Hurst analysis [56], triangular prism area surface [66] andvariation method [67], generalized fractal analysis based on a Ganti-Bhushan model [63] andthe patchwork method [68] The basic limitation of these methods is that they work well onlywith isotropic surfaces, i.e with surfaces which exhibit the same statistical characteristics inall directions [45] Majority of surfaces, however, are anisotropic, i.e they exhibit differentsurface patterns along different directions

In order to overcome this limitation and characterize the surface in all directions a modifiedHurst Orientation Transform (HOT) method was developed [69] The HOT method allowscalculation of Hurst coefficients (H), which are directly related to surface fractal dimensions,i.e D = 3-H, in all possible directions These coefficients, when plotted as a function oforientation, reveal surface anisotropy [45,69]

The problem is that none of the methods mentioned provide a full description of surfacetopography since they were designed to characterize only particular morphological surfacefeatures such as surface roughness and surface directionality Even though a modified HOTmethod allows for a characterization of surface anisotropy it still does not provide a fulldescription of the surface topography It seems that fractal methods currently used only workwell with surfaces that conform to a fractional Brownian motion (FBM) model and are self-similar with uniform scaling

Recently a new approach, called a Partitioned Iterated Function System (PIFS), has been tried.This approach is based on the idea that, since most of the complex structures observed innature can be described and modelled by a combination of simple mathematical rules [e.g.70,71], it is reasonable to assume that, in principle, it should be possible to describe a surface

by a set of such rules

It can be observed that any surface image, containing 3-D surface topography data, exhibits acertain degree of ‘self-transformability’, i.e one part of the image can be transformed intoanother part of the image reproducing itself almost exactly [72] In other words, a surfaceimage is composed of image parts which can be converted to fit approximately other partslocated elsewhere in the image [45] This is illustrated in Figure 10.9 which shows a mild steelsurface with the ‘self-transformable’ parts marked by the squares

PIFS method is based on these affine transformations and allows to encapsulate the wholeinformation about the surface in a set of mathematical formulae [44,45] These formulaewhen iteratively applied into any initial image result in a sequence of images whichconverge to the original surface image This is illustrated in Figure 10.10 where the sets of

rules found for a surface image shown in Figure 10.9 were applied to some starting image, i.e.black square.

Figure 10.9 Range image of a mild steel surface with marked ‘self-transformable’ parts

Figure 10.10 The application of the PIFS data obtained for the mild steel surface image

shown in Figure 10.9; a) initial image, b) one iteration, c) four iterations, d)twelve iterations

It can be seen from Figure 10.10 that an almost exact replica of the original image has beenobtained from the ‘black’ image only after 12 iterations of PIFS data Since a relativelyaccurate description of the whole surface is obtained the PISF method may also be used toclassify the surfaces into specific groups

Optimum Surface Roughness

In practical engineering applications the surface roughness of components is critical as itdetermines the ability of surfaces to support load [22] It has been found that at high or very

low values of ‘R q ’ only light loads can be supported while the intermediate ‘R q’ values allowfor much higher loads This is illustrated schematically in Figure 10.11 where the optimumoperating region under conditions of boundary lubrication is determined in terms of theheight and spatial surface characteristics If surfaces are too rough then excessive wear andeventual seizure might occur On the other hand, if surfaces are too smooth, i.e when β* < 2[µm], then immediate surface failure occurs even at very light loads [22]

It is found that most worn surfaces, where lubrication is effective, tend to exhibit afavourable surface profile, i.e quasi-planar plateaux separated by randomly spaced narrowgrooves In this case, the profile is still random which allows the same analysis of contactbetween rough surfaces as described below, but a skewed Gaussian profile results

0.01 0.02 0.05 0.1 0.2 0.5

Safe Unsafe

and spatial surface characteristics [22]

10.3 CONTACT BETWEEN SOLIDS

Surface roughness limits the contact between solid bodies to a very small portion of theapparent contact area The true contact over most of the apparent contact area is only found

at extremely high contact stresses which occur between rocks at considerable depths below thesurface of the earth and between a metal-forming tool and its workpiece Contact betweensolid bodies at normal operating loads is limited to small areas of true contact between thehigh spots of either surface The random nature of roughness prevents any interlocking ormeshing of surfaces True contact area is therefore distributed between a number of micro-contact areas If the load is raised, the number of contact areas rather than the ‘average’individual size of contact area is increased, i.e an increase in load is balanced by newlyformed small contact areas A representation of contact between solids is shownschematically in Figure 10.12