Engineering Tribology Episode 2 Part 10 pot

450 ENGINEERING TRIBOLOGY FIGURE 10.3 Self-similarity of surface profiles. It has been observed that surface roughness profiles resemble electrical recordings of white noise and therefore similar statistical methods have been employed in their analysis. The introduction of statistical methods to the analysis of surface topography was probably due to Abbott and Firestone in 1933 [11] when they proposed a bearing area curve as a means of profile representation [9]. This curve representing the real contact area, also known as the Abbott curve, is obtained from the surface profile. It is compiled by considering the fraction of surface profile intersected by an infinitesimally thin plane positioned above a datum plane. The intersect length with material along the plane is measured, summed together and plotted as a proportion of the total length. The procedure is repeated through a number of slices. The proportion of this sum to the total length of bearing line is considered to represent the proportion of the true area to the nominal area [9]. Although it can be disputed that this procedure gives the bearing length along a profile, it has been shown that for a random surface 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 of the height [9]. It is a means of representing all surface heights. The method of obtaining the bearing area curve is illustrated schematically in Figure 10.4. It can be seen from Figure 10.4 that 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 distributions this is not always true. For example, it has been shown that machining processes such as grinding, honing and lapping produce negatively skewed height distributions [13] while some 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 cumulative distribution diagram usually gives a linear relationship [14,15]. A classic example of Gaussian profile distribution observed on a bead-blasted surface is shown in Figure 10.5. The scale of the diagram is arranged to give a straight line if a Gaussian distribution is present. TEAM LRN FUNDAMENTALS OF CONTACT BETWEEN SOLIDS 451 P(z) 100% p(z) zz 50%0 hz ∆z FIGURE 10.4 Determination of a bearing area curve of a rough surface; z is the distance 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]. 0.01 99.99 1 5 0.1 20 50 80 95 99 99.9 Cumulative height distribution [%] 1µm 1mm 5µm Hei g ht above arbitrar y datum [ µ m] FIGURE 10.5 Experimental example of a Gaussian surface profile on a rough surface (adapted from [14]). During mild wear the peaks of the surface asperities are truncated resulting in a surface profile consisting of plateaux and sharp grooves. In such profiles the asperity heights are distributed according to not one but two Gaussian constants, i.e. Gaussian surface profile exhibits bi-modal behaviour [40]. Truncation of surface asperities is found to be closely related to a ‘running-in’ process where a freshly machined surface is worn at light loads in order 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 random and non-random features. The series of grooves formed by a shaper on a metal surface are a prime example of non-random topographical characteristics. On the other hand, bead-blasted surfaces consist almost entirely of random features because of the random nature of this process. The shaped surface also contains a high degree of random surface features which gives its rough texture. In general, non-random features do not significantly affect the contact area and contact stress provided that random roughness is superimposed on the non- random features. TEAM LRN 452 ENGINEERING TRIBOLOGY Characterization of Surface Topography A number of techniques and parameters have been developed to characterize surface topography. The most widely used surface descriptors are the statistical surface parameters. A new 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 are needed, one describing the variation in height (i.e. height parameter), the other describing how height varies in the plane of the surface (i.e. spatial parameter) [9]. The deviation of a surface 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. In engineering practice, however, the most commonly used parameter is the roughness average. 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 single spurious, 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 main disadvantages 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 the surface 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]. Bad bearing surface Good bearing surface R q R a = 0.58a (RMS) = 0.25a (CLA) R q R a = 0.37a = 0.25a +a −a FIGURE 10.6 Effect of averaging on ‘R a ’ value [9]. The ‘good’ bearing surface illustrated schematically in Figure 10.6 in fact approximates to most worn surfaces where lubrication is effective. Such surfaces tend to exhibit the favourable surface profile, i.e. quasi-planar plateaux separated by randomly spaced narrow grooves. The problem associated with the averaging effect can be rectified by the application of the RMS 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 can have the same height parameters their spatial arrangement and hence their wear and TEAM LRN FUNDAMENTALS OF CONTACT BETWEEN SOLIDS 453 T ABLE 10.1 Commonly used height parameters. Roughness average (CLA or R a ) R a = 1 L ⌠ ⌡ 0 L zdx x z L x z R a Root mean square roughness (RMS or R q ) R q = 1 L ⌠ ⌡ 0 L z 2 dx x z L x z L L L L L R max 1 R max 2 R max 3 R max 4 R max 5 Maximum peak-to- valley height ( R t ) Largest single peak-to- valley height in five adjoining sample lengths Ten-point height ( R z ) Average separation of the five highest peaks and the five lowest valleys within the sampling length Assessment length x z p 1 p 2 p 5 p 3 p 4 v 1 v 3 v 4 v 5 v 2 R z = p 1 + + p 5 + v 1 + + v 5 5 R t = 1 5 Σ R max i 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 in representing spatial variation. These functions are used to discriminate between the differing spatial surface characteristics by examining their decaying properties. Their limitation, however, is that they are not sensitive enough to be used to study changes in surface topography during wear. Wear usually occurs over almost all wavelengths and therefore changes 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 this function contains the same amount of information as the autocorrelation function it allows a much more accurate description of surface characteristics. The power spectral density TEAM LRN 454 ENGINEERING TRIBOLOGY TABLE 10.2 Statistical functions used to describe spatial characteristics of the real surfaces (adapted from [9]). Autocovariance function ( ACVF or R(τ) ) R(τ) = lim 1 L ⌠ ⌡ 0 L z(x)z(x + τ)dx x z τ L⇒ ∞ x z(x) z(x + τ) z x R(τ) z x τ τ R(τ) Autocorrelation function ( ACF or ρ(τ) ) ρ(τ) = R(τ) R q 2 often used in the form: ρ(τ) = e −τ/β* τ ρ(τ) 0 β* 2β* 2.3β* 0 1 0.1 Structure function ( SF or S(τ) ) S(τ) = lim 1 L ⌠ ⌡ 0 L [z(x) − z(x + τ)] 2 dx L⇒ ∞ τ S(τ) Worn Unworn Power spectral density function ( PSDF or G(ω) ) G(ω) = 2 π ⌠ ⌡ 0 ∞ R(τ)cos(ωτ)dτ G(ω) ω 2π Worn Unworn ρ(τ) 1 0 τ Worn Unworn where: 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]. TEAM LRN FUNDAMENTALS OF CONTACT BETWEEN SOLIDS 455 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 of the surfaces means that even a very simple structure needs a very broad spectrum to be well represented [9]. A detailed description of the height and spatial surface parameters can be found in, for example, [9,41]. Multi-Scale Characterization of Surface Topography A characteristic feature of the engineering surfaces is that they exhibit topographical details over a wide range of scales; from nano- to micro-scales. It has been shown that surface topography is a nonstationary random process for which the variance of height distribution (RMS 2 ) depends on the sampling length (i.e. the length over which the measurement is taken) [17]. Therefore the same surface can exhibit different values of the statistical parameters when a different sampling length or an instrument with a different resolution is used. 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 rough surface contains and the small number of particular length scales, i.e. sampling length and instrument resolution, that are used to define the surface parameters. Therefore traditional methods used in 3-D surface topography characterization provide functions or parameters that strongly depend on the scale at which they are calculated. This means that these parameters 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 the multi-scale nature of tribological surfaces new ‘multi-scale’ characterization methods still need to be developed. Recent developments in this area have been concentrated on three different approaches: · Fourier transform methods; · wavelet transformation methods, and · fractal methods. For the characterization of surfaces by wavelet and fractal methods the 3-D surface topography data is presented in the form of range images [45,46]. In these images the surface elevation data is encoded into a pixel brightness value, i.e. the brightest pixel, depicted by the grey level of ‘255’, represents the highest elevation point on the surface, while the darkest pixel, 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 exponential functions of different frequencies. The Fourier methods were used to calculate the power spectrum and the autocorrelation function in order to obtain the surface topography parameters [e.g. 41,47-49]. However, the problem with the application of these methods to surfaces is that they provide results which strongly depend on the scale at which they are calculated, and hence they are not unique for a particular surface. This is because the Fourier transformation provides only the information whether a certain frequency component exists or not. As the result, the surface parameters calculated do not provide information about the scale at which the particular frequency component appears. TEAM LRN 456 ENGINEERING TRIBOLOGY · Characterization of Surface Topography by Wavelets Wavelet methods allow to decompose the surface data into different frequency components and then to characterize it at each individual scale. The wavelet methods were used to decompose 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. When applying wavelets the surfaces are usually first decomposed into roughness, waviness and form, and then the changes in surface peaks, pits and scratches together with their locations are obtained at different scales. However, there are still major difficulties in extracting the appropriate surface texture parameters from wavelets [44]. An example of the application of a wavelet transform to decompose a titanium alloy surface image at two different levels is shown in Figure 10.7. Each decomposition level contains a low resolution image and three images containing the vertical, horizontal and diagonal details of the original image at a particular scale. The low resolution images and the detail images were obtained by applying a combination of low-pass and/or high-pass filters along the rows and columns of the original image and a downsampling operator. The original image can be reconstructed back from these images obtained by using mirror filters and an upsampling operator [52]. Original image Level 1 wavelet decomposition Level 2 wavelet decomposition low resolution image horizontal detail image vertical detail image diagonal detail image low resolution image horizontal detail image vertical detail image diagonal detail image Figure 10.7 Example of application of wavelet transform to titanium alloy surface image. TEAM LRN FUNDAMENTALS OF CONTACT BETWEEN SOLIDS 457 · Characterization of Surface Topography by Fractals Fractal methods allow to characterize surface data in a scale-invariant manner. Usually fractal dimensions, since they are both ‘scale-invariant’ and closely related to self-similarity, are employed to characterize rough surfaces [19]. The basic difference between the characterization of real surfaces by statistical methods and fractals is that the statistical methods are used to characterize the disorder of the surface roughness while the fractals are used 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 be characterized by the Weierstrass-Mandelbrot function which has the fractal dimension ‘D’ and is given in the following form [19]: z(x) = G (D − 1) cos2 πγ n x γ (2 − D)n Σ n = n 1 ∞ for 1 < D < 2 and γ > 1 (10.1) where: 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 n 1 = 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]; γ n are the frequency modes corresponding to the reciprocal of roughness wavelength, i.e. γ n = 1/λ n [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 FIGURE 10.8 Example of a surface profile obtained by stylus or optical measurements. The Weierstrass-Mandelbrot function has the properties of generating a profile that does not appear 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 function closely 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 TEAM LRN 458 ENGINEERING TRIBOLOGY function. The fractal dimension and other parameters included in the Weierstrass- Mandelbrot function provide more consistent indicators of surface roughness than conventional parameters such as the standard deviation about a mean plane. This is because the fractal dimension is independent of the sampling length and the resolution of the instrument which otherwise directly affect the measured roughness [17]. 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 arithmetic progression as multiples of a basic frequency, while in a Weierstrass-Mandelbrot function they increase in a geometric progression [19]. In Fourier series the phases of some frequencies coincide at certain nodes which make the function appear non-random. With the application 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 provides both 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) = G 2(D − 1) 2ln γ 1 ω (5 − 2D) (10.2) where: S(ω) is the power spectrum [m 3 ]; 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 high frequency limit corresponds to the Nyquist frequency which is related to the resolution of the instrument [19]. The fractal dimension ‘D’ is obtained from the slope ‘m’ of the log-log plot of ‘S(ω)’ versus ‘ω’, i.e.: m 2 m 1 logS(ω) [m 3 ] logω [m −1 ] Slope m = −(5 − 2D) 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 of the 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 of scale independent parameters which characterize an isotropic rough surface [20]. When they are known then the surface roughness at any length scale can be determined from the Weierstrass-Mandelbrot function [20]. It should also be mentioned that in the fractal model of roughness, as developed so far, the scale 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. In practice, engineering surfaces contain a limit to roughness, i.e. the surfaces are machined ‘smooth’ and in this respect, the fractal model diverges from reality. TEAM LRN FUNDAMENTALS OF CONTACT BETWEEN SOLIDS 459 There have been various techniques developed to evaluate the fractal dimension from a profile, 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 some fundamental 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 distinguish between the two worn surfaces [60]. Also tests conducted on artificially generated profiles demonstrated that the problem of choosing any particular algorithm for the calculation of fractal 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 surface topographies. 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 ground surfaces [62], engineering surfaces measured with different resolutions [63], etc. The most popular 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] and variation method [67], generalized fractal analysis based on a Ganti-Bhushan model [63] and the patchwork method [68]. The basic limitation of these methods is that they work well only with isotropic surfaces, i.e. with surfaces which exhibit the same statistical characteristics in all directions [45]. Majority of surfaces, however, are anisotropic, i.e. they exhibit different surface patterns along different directions. In order to overcome this limitation and characterize the surface in all directions a modified Hurst Orientation Transform (HOT) method was developed [69]. The HOT method allows calculation 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 of orientation, reveal surface anisotropy [45,69]. The problem is that none of the methods mentioned provide a full description of surface topography since they were designed to characterize only particular morphological surface features such as surface roughness and surface directionality. Even though a modified HOT method allows for a characterization of surface anisotropy it still does not provide a full description of the surface topography. It seems that fractal methods currently used only work well 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 in nature 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 a certain degree of ‘self-transformability’, i.e. one part of the image can be transformed into another part of the image reproducing itself almost exactly [72]. In other words, a surface image is composed of image parts which can be converted to fit approximately other parts located elsewhere in the image [45]. This is illustrated in Figure 10.9 which shows a mild steel surface with the ‘self-transformable’ parts marked by the squares. PIFS method is based on these affine transformations and allows to encapsulate the whole information about the surface in a set of mathematical formulae [44,45]. These formulae when iteratively applied into any initial image result in a sequence of images which converge to the original surface image. This is illustrated in Figure 10.10 where the sets of TEAM LRN [...]... D ≠ 1.5 is calculated from the following expression [10] : W 4 π = G*(D − 1) g1 (D)A* D /2 r Aa E' 3 [( ) * (2 − D)Ar D (3 − 2D) /2 ] − a*(3 − 2D) /2 c (10. 6) + K φg2 (D)A* D /2 a* (2 − D) /2 r c and for D = 1.5 from the relation [10] : ( ) ( ) A* W = π G*1/ 2 r 3 Aa E' 3/ 4 ln ( ) A* 3K φ A* r r + 3a* 4 3 c 3/4 ac 1/4 * TEAM LRN (10. 7) 466 ENGINEERING TRIBOLOGY where: W is the total load [N]; E' is the composite... area, i.e Aa = L2 [m2]; * Ar is the nondimensional real contact area, i.e A r/A a; Ar is the real contact area [m2], i.e.: D a 2 D l Ar = al is the contact area of the largest spot [m2]; ac is the critical contact area demarcating elastic and plastic regimes [m2], i.e.: ac = φ G2 πKφ 2 ( ) 2/ (D − 1) is a material property parameter, i.e φ = σy /E' The first part of equations (10. 6) and (10. 7) represents... modulus (eq 7.35) [Pa]; D is the fractal dimension, i.e 1 < D < 2; K is the factor relating hardness ‘H’ to the yield strength ‘σ y ’ of the material, i.e H = Kσy and ‘K’ is in the range 0.5 - 2 [33 ,10] ; g 1 , g2 are parameters expressed in terms of the fractal dimension ‘D’, i.e.: g1(D) = ( ) D 2 D (3 − 2 D) D D/ 2 and g2(D) = ( ) D 2 D (2 − D) /2 G* is the roughness parameter, i.e G* = G/ A a; G is the... skewed Gaussian profile results TEAM LRN FUNDAMENTALS OF CONTACT BETWEEN SOLIDS 461 Rq [µm] 1.0 2. 5 N 0.5 10 N 50 N Unsafe 0 .2 0.1 Safe 0.05 0. 02 0.01 1 2 5 10 20 50 β* [µm] FIGURE 10. 11 Relationship between safe and unsafe operating regions in terms of the height and spatial surface characteristics [22 ] 10. 3 CONTACT BETWEEN SOLIDS Surface roughness limits the contact between solid bodies to a very... possible Although this view of ‘stick-slip’ motion is only hypothetical, it serves to illustrate the complex nature of the phenomenon in the absence of more detailed research 400 300 10 ms 20 0 3 2 100 1 0 a) Time 0 300 3 20 0 2 100 10ms 0 b) Time 1 0 Displacement [µm] 400 Friction force [N] 500 Displacement [µm] Friction force [N] 500 FIGURE 10. 19 Effect of the rate of friction force application on friction... transfer particles to form larger transfer particles 1 1 ⇒ 2 Exit 2 Prolong persistance of central transfer particles Irreversible boundary of contact (approximates to contact area) Irreversible boundary 3 2 ⇒ 3 Return Reversible boundary FIGURE 10 .21 Schematic illustration of the loss of transfer particles at the edge of contact TEAM LRN FUNDAMENTALS OF CONTACT BETWEEN SOLIDS 473 Irreversibility of wear particle... LRN 464 ENGINEERING TRIBOLOGY 2 0.5 (2. 3β*) r= 9σ 2 The expression for a total load is given in a form [26 ]: ∞ W= ∞ 4 ⌠ ⌠ f*(z*,C) dCdz* Aσ E' (2. 3 β*) (z* − d)1.5 3 ⌡ ⌡ N C d 0 (10. 4) where: W is the total load [N]; E' is the composite Young's modulus (eq 7.35) [Pa] The ratio of load to real contact area, i.e the mean contact pressure, is therefore given by: ∞ pmean = W 4 σ E' = Ar 3 π n (2. 3 β*) ⌠... Once this level of force is reached, the TEAM LRN 470 ENGINEERING TRIBOLOGY coefficient of friction maintains a steady value over the range of velocities, declining gradually after reaching a critical velocity level as shown in Figure 10. 17 µ 2. 0 Indium block 1.5 Lead block 1.0 0.5 0 Vertical load ≈ 3 N 10 9 10 6 10 3 103 1 Sliding speed [mm/s] FIGURE 10. 17 Variation of friction coefficient for indium...460 ENGINEERING TRIBOLOGY 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 a) b) c) d) 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,... the work of Whitehouse and Archard [25 ], Onions and Archard [26 ], Pullen and Williamson [28 ], Nayak [29 ] and others [e.g 30- 32] In these models statistical methods are applied to describe the complex nature of the contact between two rough surfaces For example, in the Onions and Archard model [26 ], which is based on the Whitehouse and Archard statistical model [25 ], the true contact area is given by . range 0.5 - 2 [33 ,10] ; g 1 , g 2 are parameters expressed in terms of the fractal dimension ‘D’, i.e.: g 1 (D) = () D 2 − D D /2 (3 − 2D) D and g 2 (D) = () 2 − D D (2 − D) /2 G* is the. equation (10 .2) . The contact load for D ≠ 1.5 is calculated from the following expression [10] : = () A a E' W 3 4 π G* (D − 1) g 1 (D)A r * D /2 [ D (2 − D)A r * − a c * (3 − 2D) /2 (3 − 2D) /2 ] +. LRN FUNDAMENTALS OF CONTACT BETWEEN SOLIDS 461 0.01 0. 02 0.05 0.1 0 .2 0.5 1.0 R q [µm] 12 51 020 50 β* [µm] 50 N 10 N 2. 5 N Safe Unsafe FIGURE 10. 11 Relationship between safe and unsafe operating