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The importance of sampling interval for rough contact mechanics
Wear 276–277 (2012) 121–129
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The importance of sampling interval for rough contact mechanics P. Pawlus a,∗ , W. Zelasko b a b
Rzeszow University of Technology, Rzeszow, Poland Group of Technical Schools, Lezajsk, Poland
a r t i c l e
i n f o
Article history: Received 19 May 2011 Received in revised form 16 December 2011 Accepted 21 December 2011 Available online 28 December 2011 Keywords: Contact mechanics Sampling interval Surface topography
a b s t r a c t The contact of random surfaces is discussed in this paper. Commonly used elastic and elastic–plastic microcontact models were applied to modeled and measured isotropic surface topographies of a Gaussian ordinate distribution, assuming a constant and a distributed radius of surface summits, respectively. The calculation procedures allowed the mean surface separation and the real area of contact to be calculated as a function of contact load and sampling intervals. Results are presented for a wide range of contact load and sampling intervals, showing the difference obtained after using various models. The applicability of the profile spectral moment approach to areal random surface specification was checked. This technique relates geometrical properties of a surface important for rough contact mechanics to those of its constituent profiles. Suggestions for choosing the optimum sampling interval are provided. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The analysis of the deformation behavior of contacting surfaces is important to understand the mechanisms of friction, wear, lubrication, sealing, thermal and electrical contact as well as resistance. The pioneering contribution to this field was made by Greenwood and Williamson [1], who developed a basic contact model (GW model) of an isotropic surface. In this model all asperity summits have the same radius but their height varied randomly. This model has been extended to cover contact of curved surfaces [2], two rough surfaces with misaligned asperities [3], surfaces with non-uniform curvature of peaks [4] and strongly anisotropic rough surfaces [5]. McCool [6] suggested after comparison of the GW model with other models that the GW model gave good results. However, the GW contact model can only be used when the majority of contacting asperities deform elastically. A comparison among various contact models, often including the GW model was also given in [7–11]. The authors of Ref. [11] presented a slightly improved GW model in which all the spheres do not have the same radius; they found that all multi-asperity contact theories, taking into consideration the influence of the summit curvature variation as a function of summit height, gave similar results. However, it is difficult to find the direct numerical comparison of the same models for constant and distributed summits radius. A number of researchers studied contact of rough surfaces characterized by plastic deformation of asperities. Abbott and Firestone [12]
∗ Corresponding author. E-mail address: [email protected] (P. Pawlus). 0043-1648/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2011.12.015
developed the widely used model for fully plastic contact. It assumed that the deformation of a rough surface against a rigid smooth flat is equivalent to the truncation of the undeformed rough surface at its intersection with the flat. Pullen and Williamson [13] established a volume conservation model for the fully plastic contact of rough surface. The papers [7,14,15] present statistical models of elastic–plastic contact between rough surfaces. Chang et al. [14] put forward an elastic–plastic contact model (CEB model) for rough surfaces on the basis of volume conservation of plastically deformed asperities. Many publications appeared that are based on the CEB model or inspired by it; about 30 publications were cited in [16]. The paper [15] presented a different elastic–plastic contact model (ZMC model) for rough surfaces. Its main feature is the modeling of the long transitional regime from elastic deformation to the fully plastic flow. It is also widely used (see for example [17,18]). An elastic–plastic model for contacting rough surfaces that is based on Finite Element Analysis (FEA) of an elastic–plastic asperity contact was presented in paper [19]. Jackson and Green [7] developed a statistical elastic–plastic contact model using the results of a previous FEA of an elastic–plastic sphere in contact with a rigid flat [20]. It is widely known that the sampling interval has a significant influence on spatial and hybrid properties important in studying contact problems, particularly the summit density and curvature [21,22]. Several methods for determining the optimal sampling interval were suggested based on the principle that a tolerance between a parameter calculated from a sampled profiles and the true value for the real surface does not exceed an acceptable limit. Wu [23] presented a method of obtaining the minimum sampling interval for a random profile of normal ordinate distribution. The proposed sampling interval depends on the radius of the tip, the
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Nomenclature a A An d
contact radius true area of contact nominal contact area separation of the surfaces measured from the summits mean plane E1,2 Young’s moduli Hertz elastic modulus E h separation based on surface heights H hardness of softer material K maximum contact pressure factor m0 , m2 , m4 spectral moments of profile P contact load of individual asperity areal density of asperities Pds R mean radius of summits curvature SI sampling interval standard deviation of surface heights Sq () ys distance between the mean of asperity heights and that of surface ordinates height of summit z ˛ bandwidth parameter correlation between profile neighboring points Poisson’s ratios 1,2 s standard deviation of asperity height plasticity index ω interference critical interference at the inception of plastic deforωc mation
Fig. 1. Schematic of contact of two rough surfaces.
surfaces. However, there are surfaces for which the multi-scale technique will not converge. The fundamental aim of this paper is to analyze the effect of the sampling interval on contact mechanics of rough isotropic surfaces, while recommending the appropriate sampling interval. The second aim is to test the hypothesis that a constant peak radius affects the results of the contact mechanics study significantly. The present authors also checked if the areal (3D) surface topography parameters are important for rough contact mechanics can be estimated on the basis of (2D) profile analysis. 2. Theoretical background
correlation length and the standard deviation of the ordinate values. Lin [24], Dong et al. [25], Pawlus and Chetwynd [26] and Pawlus [27] recommended an optimum sampling interval based on the shape of the cumulative spectrum graph. It is also possible to select a sampling interval based on the autocorrelation function. The paper [28] focused on the evaluation of a general methodology to sampling in relation to the variation of the surface topography. The contact parameters, i.e., contact area versus contact load and separation versus contact load predicted by the GW and other statistical models are largely dependent on the sampling resolution (interval) used to characterize the surfaces [29]. Because long wavelengths exist on the surface after contact/wear, Whitehouse and Archard recommended a sampling interval equal to the distance at which the autocorrelation function decays to a value of 0.1 (correlation length) [30]. The significance of the shorter wavelength structures (revealed by the use of shorter sampling intervals) in the contact of surfaces may be questioned. McCool [31] determined the frequency interval over which purely elastic models are applicable. Thomas and Rosen [32] proposed a method for choosing a tribologically appropriate sampling interval, based on Archard’s observation that repetitive contact must be elastic. The sampling interval should be equal to half of the critical wavelength; all surface features smaller than this critical wavelength will disappear on the first encounter with another surface and so do not need to be taken into consideration with regard to the surface’s subsequent elastic behavior. Vallet et al. [33] analyzed the influence of the sampling interval on the determination of the roughness parameters, contact areas and flow through the aperture field resulting from the contact under load. Unfortunately, the approaches in [32,33] were restricted only to fractal surfaces. The problems with the sampling interval choice can be solved using the analysis of multi-scale rough surfaces. The works of Ciavarella et al. [34] can be the example of multi-scale analysis of fractal surfaces. The work [35] used a multiscale model based on the Fourier series obtained from measured
The contact between two rough surfaces is modeled by means of the contact of a single rough surface with a smooth plane [2]. The following fundamental assumptions (based on Greenwood and Williamson model [1]) are adopted: - the rough surface is isotropic, and has a Gaussian ordinate distribution, - asperities are spherical near their peaks (summits), - there is no interaction between asperities, and - only the asperities deform during contact. Fig. 1 shows the geometry model of the contacting rough surfaces. Two possibilities are investigated. First, all peaks have the same radius R (see Fig. 1a). Secondly, the peak radii vary randomly (Fig. 1b). In Fig. 1, z denotes the height of asperity, d separation of the surfaces measured from the summits mean plane, and h is the separation of the surfaces based on surface heights (ordinates). The interference is defined as: ω =z−d
(1)
During loading, the area of contact A and the contact load of each individual asperity depends only on its individual interference. For a sphere of radius R, elastically deflected by an amount ω, Hertz’s solution gives the contact area: A = Rω = R(z − d) = a2
(2)
where a is the contact radius. The corresponding asperity load is: P=
4 1/2 3/2 E R (z − d) 3
(3)
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where
E =
1 − 12 E1
1 − 22
+
123
−1 (4)
E2
Ei and i (i = 1, 2) are the Young’s moduli and the Poisson’s ratios for the two contacting elements [2], respectively. The critical interference ωc at the inception of plastic deformation is: ωc =
KH 2 2E
R
(5)
where H is the hardness of the softer material. A value of constant K = 0.6 is commonly selected. When ω < ωc the contact is elastic. Any summit, for which interference ω exceeds ωc will have some degree of plastic deformation. The CEB elastic–plastic contact model is based on volume conservation during plastic deformation [14]. The contact area of a plastically deformed asperity was calculated as:
A = a2 = Rω 2 −
ωc ω
(6)
The contact load for such an asperity is:
P = Rω 2 −
ωc ω
KH
(7)
3. Calculation procedure Isotropic surfaces with Gaussian ordinate distribution were generated, using the procedure developed by Wu [36]. Each surface was characterized by the correlation distance (in which the autocorrelation function decays to value of 0.1) and the standard deviation of height Sq. In addition, measured isotropic Gaussian surface topographies were analyzed. Talyscan 150 measuring equipment using a stylus tip with a nominal radius of 2 m was used. Modeled and measured surfaces contacted with a perfectly flat rigid surface. A surface point was selected to be a summit if its ordinate was higher than the ordinates of its eight nearest neighbors. This criterion was based on the work of Greenwood [37], Sayles and Thomas [38], as well as previous research of the present authors. This method does not take into consideration that a smaller asperity may become part of a bigger asperity. A newly developed n-point asperity model seems to be a good proposal for solving this problem. However, its application to areal surface topography without simplifying assumption would be computationally intensive [39]. For areal measurements, the radius of each summit was calculated as the reciprocal of its arithmetic average curvature in orthogonal directions. The summit curvature was calculated on the basis of a three-point formula [40]. The mean radius of the summits R, was considered to be the average of all the summits radii in the areal surface topography measurement. The same method was used in [41]. The areal density of asperities Pds, standard deviation of summits heights s , and distance between the mean of asperity heights and that of surface ordinates ys were calculated, too. In addition, the plasticity index defined by Greenwood and Williamson was computed, which in this case is expressed as: =
1/2 s
ωc
(8)
The total contact area A and total contact load P were calculated for various separations d in the following way. The vertical range of summit heights was divided into 100 intervals. For each separation, summits located above this height (plane) were identified. For each 8-neighbor summit in contact interference ω was computed (Eq. (1)). When it was smaller than the critical interference ωc , the contact area A and contact load P for elastic contact were calculated according to Eqs. (2) and (3), respectively. Else (when ω > ωc ) the contact area and contact load of a deformed asperity
Fig. 2. Three-dimensional views of surfaces A and C: for SI = 5 m (a), SI = 10 m (b), SI = 20 m (c), and SI = 40 m (d).
were computed using Eqs. (6) and (7). The total contact area A and contact load P were obtained by summing the contributions of all contacting asperities located above a given separation. This procedure was repeated for different surface separations. In addition, the contact area and contact load were calculated assuming only elastic contact; using Eqs. (2) and (3). The calculations of the contact area and contact load were repeated for various sampling intervals. The correlation between ordinates was calculated on the basis of the profile autocorrelation function, as the average value of 6 profiles. In addition, the mean values of zeroth m0 , second m2 and fourth m4 spectral moments of all profiles in perpendicular directions were computed. The m0 moment is the variance of profile height, i.e. Pq2 . The second moment m2 is equal to square of the profile r.m.s. slope i.e. Pq2 . The Pq parameter was calculated based on a 2-point formula. The fourth moment m4 being the second derivative of a profile was computed using the procedure described in [42]. For an isotropic areal surface some parameters connected with the peaks, such as the product R Pds s , depend on the parameter ˛, defined as: m0 m4 ˛= . (9) m22 This parameter was also calculated for each sampling interval used. The following material properties were selected (contact of steel-on-steel elements) E1 = E2 = 2.07 × 105 MPa, Brinell hardness H = 200 (1960 MPa), 1 = 2 = 0.29. These properties were also used in papers [14,15]. 4. Results and discussion Table 1 presents the surface topography parameters for various sampling intervals SI and correlation values between neighboring points . The standard deviation of surface ordinates Sq (), standard ordinates of peak heights s , areal peak density Pds, mean radius of peak R, product s PdsR, mean summit height ys , and the ˛ parameter are shown. The critical interference ωc and the plasticity index are also presented. Four surfaces were analyzed for four different sampling intervals. Surfaces A, B and C were computer generated. The correlation distances for surfaces A and B were 50 m, and 200 m for surface C. Surface D was experimentally determined and the correlation distance was found to be 72 m (the surface was finished using sand blasting). The dimensionless total load and area were also calculated using tabular values presented in [6] for given s , Pds, R, and ys . Fig. 2 shows an isometric view of surfaces A and B. Figs. 3–6 present the mean separation and real area of contact as a function of load and sampling interval
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Table 1 Surface topography parameters, critical interferences and plasticity indices. Surface A A A A B B B B C C C C D D D D
SI (m) 5 10 20 40 5 10 20 40 10 30 60 120 8 40 56 72
Sq (m)
˛
s (m)
Pds (m)−2
R (m)
s PdsR
ys (m)
ωc (m)
0.176 0.176 0.176 0.176 0.44 0.44 0.44 0.44 0.93 0.93 0.93 0.93 0.66 0.66 0.66 0.66
0.85 0.7 0.4 0.12 0.85 0.7 0.4 0.12 0.91 0.65 0.39 0.14 0.95 0.5 0.25 0.1
7.81 4.16 2.48 1.68 7.81 4.16 2.48 1.68 14.9 4.74 3.33 2.1 8.6 2.7 2.14 1.68
0.16 0.145 0.13 0.12 0.4 0.36 0.33 0.3 0.88 0.8 0.72 0.58 0.58 0.41 0.39 0.36
0.0012 0.0005 0.00019 0.00006 0.0012 0.0005 0.00019 0.00006 0.000232 0.000058 0.000021 0.000006 0.00035 0.000046 0.000026 0.000017
190.7 467.5 1172 3713.2 76.4 187.2 468.8 1483.2 212.8 830.9 2156.8 6900.2 347.2 1634.2 2457.2 3839.4
0.037 0.034 0.029 0.027 0.037 0.034 0.029 0.027 0.044 0.039 0.033 0.023 0.07 0.031 0.025 0.023
0.17 0.19 0.23 0.25 0.42 0.49 0.58 0.64 0.71 0.95 1.19 1.26 0.46 0.79 0.87 0.89
0.051 0.125 0.31 0.99 0.02 0.05 0.125 0.4 0.05 0.22 0.58 1.84 0.09 0.44 0.66 1.03
1.76 1.1 0.65 0.35 4.4 2.74 1.63 0.87 3.92 1.9 1.12 0.56 2.5 0.97 0.77 0.59
of surfaces A and D for various models: elastic and elastic–plastic for constant summit radius and elastic–plastic for variable summit radii. Table 2 shows the dimensionless separation h/ and contact area ratio A/An versus the dimensionless load P/An E for various values of the sampling interval of surface A and for various contact models. Fig. 3 presents the dimensionless separation h/ versus the dimensionless load P/An E for various sampling intervals of surface A. An increase in sampling interval caused a decrease in plasticity index. Elastic and elastic–plastic models were considered for constant value of the summit radius, but when this radius was distributed – only information about elastic–plastic model was graphically presented. The separation between contacting rough surfaces under a given load usually decreases with increasing magnitude of the sampling interval. For the same dimensionless load, the mean separation was smaller when the elastic model was used compared to the elastic–plastic model for sampling intervals of 5, 10 and 20 m. These differences were significant for the largest load, and the relative deviations increased with increasing magnitude of the sampling interval. For example, when the SI was set to 20 m, the relative difference was higher than 60%. For smaller loads than 0.001 the differences between the mean separations were found to be smaller than 10%. For the largest sampling interval (40 m) no differences between mean separations for elastic and elastic–plastic models were found for all analyzed loads, because the contact was elastic. When plastic contact occurs,
the elastic model underestimates the separation compared to the elastic–plastic model. Averaging of peak radius for elastic–plastic model caused increase in mean separation particularly for the highest sampling intervals corresponding to plasticity index much smaller than 1 in comparison to variable radius, the highest relative difference was bigger than 30%. Deviations decreased with decreasing load. A similar tendency was observed for the elastic model. However, the deviation in mean separation was negligible for the smallest sampling interval. Fig. 4 presents the true contact ratio A/An versus the dimensionless load P/An E for various sampling intervals and various contact models of surface A. In addition the plastic contact area ratio was computed. It was identical to that of total area ratio for the smallest sampling interval and almost identical for sampling interval of 10 m and the highest load. For smaller load the contribution of the plastic contact area to the total contact area was smaller. When the sampling interval was set to 20 m, most of the contact was elastic, and for the highest sampling interval only elastic contact occurred. In elastic and elastic–plastic models the contact area was higher for larger sampling interval and the same dimensionless load. This was mainly evident for elastic contact models. When the contact was plastic (sampling interval was 5 m), the contact area for a high load was larger after using the elastic–plastic model in comparison to the elastic model. When a model with variable radius was used, the contact area ratio for elastic–plastic contact was usually smaller than in the case of constant radius, particularly for a small load.
Table 2 Comparison of elastic–plastic model with constant summit radius EPRC, elastic model with constant summit radius ERC, elastic–plastic model with variable summit radii EPRV and elastic model with variable summit radii ERV for surface A. SI (m)
P/An E
h/
A/An
EPRC
ERC
EPRV
ERV
EPRC
ERC
EPRV
ERV
5
0.001 0.0001 0.00001 0.000001
0.68 2.02 2.83 3.37
0.48 1.82 2.73 3.37
0.55 1.96 2.74 3.37
0.48 1.82 2.69 3.3
0.096 0.0097 0.00113 0.000142
0.071 0.0096 0.00083 0.000117
0.102 0.0096 0.00111 0.000119
0.066 0.0083 0.00082 0.000116
10
0.001 0.0001 0.00001 0.000001
0.72 1.92 2.73 3.37
0.42 1.78 2.67 3.32
0.6 1.84 2.67 3.32
0.31 1.72 2.61 3.26
0.1 0.0126 0.0015 0.00016
0.103 0.012 0.0016 0.00019
0.104 0.0119 0.0013 0.00013
0.104 0.012 0.0021 0.00013
20
0.001 0.0001 0.00001 0.000001
0.31 1.65 2.46 3.03
0.12 1.6 2.46 3.03
0.26 1.6 2.31 2.98
0.022 1.55 2.37 2.98
0.14 0.021 0.0027 0.00039
0.135 0.022 0.0027 0.00039
0.136 0.018 0.002 0.00023
0.12 0.019 0.0017 0.00023
40
0.005 0.0001 0.00001 0.000001
0.31 1.39 2.37 2.98
0.31 1.39 2.37 2.98
0.23 1.28 2.23 2.91
0.23 1.28 2.23 2.91
0.14 0.036 0.0042 0.00073
0.14 0.036 0.0042 0.00073
0.131 0.03 0.0031 0.00043
0.131 0.03 0.0031 0.00043
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Fig. 4. Real area of contact of the surface A as a function of load and sampling interval: elastic plastic model, constant summit radius (a), elastic model, constant summit radius (b), and elastic–plastic model, variable summit radius (c).
Fig. 3. Mean separation of surface A as a function of load and sampling interval: elastic plastic model, constant summit radius (a), elastic model, constant summit radius (b), and elastic–plastic model, variable summit radius (c).
The differences increased for increasing magnitude of the sampling interval. As a result of elastic contact no differences between the area ratios for the same load after elastic and elastic–plastic models use were found for the largest sampling interval. Spatial properties of surface B are identical with those of surface A, only the height of surface B is higher, which resulted in higher values of the plasticity index. The separation between contacting rough surfaces, under a given load usually decreases with the increasing magnitude of the sampling interval (this is mainly evident after elastic model use). Averaging the peak radius caused an increase of the mean separation in comparison to the case of a variable summit radius. Relative differences were larger for elastic contact than for elastic–plastic contact, and they were found to be the biggest for the largest sampling interval. Deviations between the mean separations after application of different models were lower for smaller load. After application of the elastic model the contact area was higher for a larger sampling interval and the same load. When contact was mostly plastic (sampling intervals were 5, 10 and 20 m), the contact area for high load was larger after using the elastic–plastic model in comparison to the elastic model. Application of the elastic–plastic contact model especially for variable summits radii caused smaller differences between contact areas for different sampling intervals used in comparison to the elastic model. Surface C was characterized by the highest height and the largest main wavelengths. The contact parameters were similar to those from surface B. For the same load the mean separation was smaller when the elastic model was used for the sampling intervals of 30, 60
and 120 m, the relative difference was higher for a larger sampling interval. The elastic model underestimates the separation compared to the elastic–plastic model, but case with a high plasticity index value formed exception. For surfaces with a higher plasticity index (sampling intervals were 10, 30 and 60 m), the contact area for a high load was larger after using the elastic–plastic model in comparison to the elastic model. The plasticity index of the measured surface D was comparatively high only for the smallest sampling interval 8 m. The separation between contacting rough surfaces, under a given load decreases with the higher sampling interval for comparatively high loads when all models were used (see Fig. 5); for a small load the mean separation was the smallest for a sampling interval of 40 m. For the same load the mean separation was smaller when the elastic model was used in comparison to the elastic–plastic model – relative differences (in the range 14–47% for dimensionless load 0.001 and a constant peak radius) were the largest for the highest sampling interval. For smaller loads differences were found to be lower. Similarly to surfaces A, B and C, averaging the peak radius caused an increase in the mean separation compared to using a variable radius. The plastic contact area was almost identical to that of the total area for the smallest sampling interval used. For the surface with the highest plasticity index, the contact area for the high load was larger after using the elastic–plastic model in comparison to the elastic model (see Fig. 6) – the relative difference was found to be 25% for a dimensionless load of 0.001 and for a constant summit radius. In other cases the contact area was rather similar after applying these models. For all the models, increasing the sampling interval increased the contact area for the same load. Differences between curves corresponding to various sampling intervals were small when the elastic–plastic model with variable summit radius
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Fig. 5. Mean separation of surface D as a function of load and sampling interval: elastic plastic model, constant summit radius (a), elastic model, constant summit radius (b), and elastic–plastic model, variable summit radius (c).
was used. The contact area was smaller than in the case of a constant radius, particularly for a small load. Generally, the separation between contacting rough surfaces, under a given load, decreases with increasing sampling interval. This is mainly evident after applying the elastic model. When contact was mainly plastic (plasticity index higher than 2.5), the elastic model overestimated the separation compared to the elastic–plastic model. In the other cases the elastic model underestimated the separation. Relative differences were larger for larger sampling intervals and for higher loads. No differences between the elastic and the elastic–plastic models were found for a plasticity index smaller than 0.4. Averaging the peak radius caused an increase of the mean separation in comparison to using a variable peak radius. The contact area was higher for larger a sampling interval and constant dimensional load. However, the differences in contact areas between the different sampling intervals were smaller after applying the elastic–plastic model compared applying the elastic model. Those differences were particularly small when the elastic–plastic model with variable radius was used. When the plasticity index was higher than 1.6, the contact area for a high load was larger after using the elastic–plastic contact model in comparison to the elastic model. For smaller plasticity index values, the differences between the area ratio were negligible. For the elastic–plastic contact when the variable summit radius was used, the contact area was smaller than in the case of a constant radius, particularly for a small load and comparatively large sampling intervals. According to [4] for a given load, the true area of contact should be smaller after consideration of the peak curvatures distribution.
Fig. 6. Real area of contact of the surface D as a function of load and sampling interval: elastic plastic model, constant summit radius (a), elastic model, constant summit radius (b), and elastic–plastic model, variable summit radius (c).
This finding was confirmed. When a variable radius was used, the contact area was smaller than in the case of a constant radius, particularly for a small load and comparatively large sampling intervals (in the case of elastic contact). The calculation of the paper [4] was concerned with the main structure of the surface topography ( = 0.1), so the sampling interval was also high. It is evident from the analysis of Table 1 that an increase in the sampling interval SI caused a decrease in the plasticity index and the product s PdsR. When increasing the SI, the summit density Pds decreases and the summit radius R increases, while the standard deviation of asperity heights s decreases. The plasticity index decreases because the decrease in s is smaller than the increase in R. The product s PdsR decreases because in the decrease in both Pds and s is higher than the increase in R. High correlations between neighboring points correspond to large values of the bandwidth parameter ˛. Measured and simulated surface topographies were analyzed. It was found that when was near 0.1, the bandwidth parameter value ˛ was in the range 1.5–1.8 (Nayak [43] found it should be higher than 1.5). When ˛ is larger, the product s PdsR is also larger. For a correlation smaller than 0.5 this product is usually smaller than 0.04. High ˛ values (higher than 8) can be obtained for highly correlated points ( ≥ 0.9). However the large product s PdsR value often found in the technical literature (for example 0.06 [14,15]) proves a high correlation between neighboring points ( ≥ 0.9). An increase in the magnitude of the sampling interval (and decrease in and ˛) causes not only a decrease in the standard deviation of asperity heights s but also an increase in the distance between the mean of asperity heights and that of surface ordinates ys . These changes were anticipated by Whitehouse and Archard for
P. Pawlus, W. Zelasko / Wear 276–277 (2012) 121–129
profiles [30]. Because oftentimes contact parameters are obtained on the basis of profile analysis it is necessary to consider correlations among parameters applied in contact mechanics in 2D and 3D systems. Dependencies between profile spectral moments of anisotropic Gaussian surfaces were developed by Nayak [43] and Bush et al. [44]. They were presented by McCool [6]. The surface summit density is given by: Pds =
1 √ 6 3
m 2 4
m2
(10)
The mean summit curvature averaged over all summit heights is: √ 8 m4 (11) Spc = √ 3 The variance of the summit height distribution is: s2 =
1−
0.8968 ˛
m0
(12)
The distance between the mean of the summit height distribution and the surface mean plane is: √ 4 m0 (13) ys = √ ˛ Surface and profile measurement and their resultant statistics were compared and their interrelationship examined for several common engineering surfaces [8]. Good agreement was found between theory and measurement over a large range of sampling intervals. Yu and Polycarpou [41] compared summit density and summit radius obtained from a numerically generated isotropic Gaussian surface, and predicted these parameters by means of the spectrum moment approach. It was found that at small ˛ values, the summit density and summit radius based on their definition were higher than prediction from the spectrum moment approach. Sets of parallel profiles were obtained from modeled surfaces and average profile spectral moments m0 , m2 and m4 were calculated. It was evident that high values of the ˛ parameter (larger than 6) correspond to large errors of summit density anticipation. Mean summit density predicted by the spectral moment approach were much bigger than value obtained from its definition. For ˛ values from range 1.7–5 the difference was smaller than 5%. For non-correlated neighboring points ( between 0.1 and 0.12) application of the spectral moments method caused underestimation of the summit density (the errors were smaller than 15%). The mean summit curvature was accurately predicted by the spectral moments approach, independently of the ˛ value. Errors were usually smaller than 8%. The relative difference between the standard deviation of summit heights was usually smaller than 5% but not higher than 10%. For ˛ values smaller than 2 predicted ys distance was higher than value obtained from the analyses of simulated and measured surfaces, however for ˛ values larger than 7.5 it was usually smaller but differences were not higher than 15%. Generally good agreement was found between the two methods except for the summit density, which could be overestimated by the theory for comparatively high ˛ values. The obtained results were roughly confirmed after the analysis of the measured surfaces. Here, measurements were in good agreement again with the theory. However, sometimes the differences were found to be larger than in case of modeled surfaces. Some deviations in the parameter estimation can be caused by the calculation of mean radius of asperity curvature. Two methods can be used. In the first of them, the summit curvature is averaged over the entire surface, and the summit radius is calculated as the reciprocal value of the curvature. This method was applied in [4,6]. In the second method, the summit curvature for each summit is calculated, and the summit radius is then obtained as the reciprocal of its curvature, and this radius is averaged over the surface. This
127
method was used in [22,41]. The present authors used the second method because the mean summit radius was similar to the variable summit radius. However, it was found that the summit radius estimated using the second method exceeds that obtained by the first method by 1.1–1.2 times. It is also possible to estimate parameters characterizing the summits from the parameters characterizing peaks (summits are local maxima on the surface, as distinct from peaks, which are local maxima on a profile). A peak was identified using a three-point method. However, significant errors occurred. The mean summit height was about two times higher than the mean peak height. The mean summit curvature was about 1.3–1.4 times higher than mean peak curvature. The differences between the standard deviations of summit and peak heights were small for highly correlated ordinates (for > 0.85 relative difference was about 5%). The standard deviation of summits height was smaller than that of peak heights. The summit density was close to the peak density square (according to Nayak [43] summit density should be 1.2 of peak density square). The results obtained for correlation between neighboring points > 0.85 were different. In those cases the square of peak density was similar to twice of the summit density. Similar results were obtained after the analysis of measured surfaces. On the basis of the profile peak analysis, the radius of summits and the standard deviation of the summit heights can be overestimated. Thus, the errors of the calculation of the product s PdsR after the profile analysis can be high, especially for small correlation between measuring points. However, the errors of computing the plasticity index on the basis of profile analysis can be small. In general, applying profile spectral moments is better than profile peak analysis. The question arises how the sampling intervals should be chosen? Whitehouse and Archard [30] as well as Onions and Archard [4] recommend sampling interval corresponding to correlation of 0.1. For this spacing events reach the condition where they can be regarded as independent. Smaller scale structures are characterized by a high value of the plasticity index. Greenwood and Williamson stated that most surfaces have a plasticity index higher than 1.0 and, thus, except for especially smooth surfaces, the asperities will flow plastically under the lightest load [1]. The sampling interval should be not higher than that proposed by Whitehouse and Archard [30] ( = 0.1) in order to avoid of aliasing. A correlation of 0.1 corresponds to the ˛ bandwidth parameter about 1.5. However, in this case only a narrow range of wavelengths is present [43]. When ˛ is small, the distribution of summit heights of a random Gaussian surface is not Gaussian, so the errors can occur after using a Gaussian summit heights distribution in statistical models [41]. There are opinions that an adequate representation of a surface can be obtained by taking ˛ = 5 [37,45]. It was found that for conventionally machined surfaces a sampling interval for a correlation = 0.1 is rather large (usually much larger than 30 m). So maybe the correlation can be higher (sampling interval smaller) than the authors of Refs. [4,30] suggested. However, in our opinion, the data should not be highly correlated. A correlation higher than 0.85 should be avoided. In this case the data are redundant, and false information about the summits can be obtained from the analysis of profiles. The higher frequencies, which are more susceptible to instrumental idiosyncrasies, are often not as significant tribologically as the lower frequencies. An optimal sampling interval should be obtained after measurement with initial SI as small as possible. For minimising the cost, it can be done by measuring of a few profiles from the surface. After that, the surface should be remeasured on a scale appropriate for a larger measured area with the optimum sampling interval, or if the number of points is not too small by increasing the sampling interval for initially measured surface area. The autocorrelation function calculated by most of the commercial instrument can be helpful. The high value of ˛ (larger
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than 8) or product s PdsR (larger than 0.045) prove that correlation between neighboring points is too high. It is known that [5] the real contact area at moderate separations (h/ > 1) must be equal to half of bearing area at that separation. For the analyzed surfaces the dimensionless elastic contact area was compared to half of the material ratio with the same separation h/ from the range 1–3. It was found that the smallest differences were obtained for values between 0.5 and 0.7. For example for surface D the mean error in this case was 7%. The criterion for the mechanism of deformation of surface asperities should be taken into consideration when selecting the sampling interval. As smaller and sharper asperities are observed, a size will be reached below which they will deform plastically during the first cycle of contact and thus disappear. During the subsequent lifetime of the contact, it will behave elastically as if the corresponding range of surface wavelength does not exist [32]. As we insert the limiting value of at which the deformation will be plastic at any load, we can select an optimal sampling interval. The plasticity index should be calculated while increasing the sampling interval, and the value of sampling interval should be selected for which is not much higher than this limiting value. Because of small errors, the plasticity index can be calculated on the basis of profile analysis, especially for small correlation between ordinates. Great care should be taken for surfaces having large value of Sq/R [7]. For the elastic contact model, when the interference ω is equal to the standard deviation of the surface heights, the ratio of the contact radius to summit radius is: a = R
Sq R
(14)
It was found that the resulting deformation after application of experimental surface topography parameters can be very large, so the models may calculate the contact area and the contact force for deformation outside of their intended range and this calculation can produce meaningless or misleading results [7]. However increasing the sampling interval can cause an increase in the summit radius and a decrease in the a/R ratio (in our calculation, the highest a/R value, for the smallest sampling interval was not higher than 7%). Other sources of error in the contact parameter computation are important as well. The described method of summit radius calculation is the first of them. The assumption often made by researchers that the heights of peaks and summits are the same is the second of them. It can cause serious errors especially for small correlation between ordinates. This assumption leads to overestimation of standard deviation of summit heights. The other more serious error is the assumption that standard deviation of summit heights is equal to the standard deviation of ordinate heights. Only for very high correlation between neighboring points s is close to .
5. Conclusions The sampling interval selection is important for rough contact mechanics. An increase in the sampling interval causes a decrease in the plasticity index. The separation between contacting rough surfaces, under a given load decreases with the higher sampling interval used. The contact area was higher for a larger sampling interval and the same dimensionless load. Differences among separations and contact areas calculated with different sampling interval were smaller after applying the elastic–plastic model than after using the elastic model. Deviations between contact areas for different sampling intervals were particularly small when the elastic–plastic model with distributed radius was used.
Applicability of the profile spectral moments’ approach to areal random surface specification was checked. Good agreement between the analysis of modeled and measured surfaces and theory was found, except for the summit density which could be overestimated by theory for comparatively high ˛ and values. The errors of calculation of the different parameters important for contact mechanics after the analysis of the profile peaks are higher than after using the profile spectral moments technique. However, the discrepancies of computing the plasticity index on the basis of profile analysis can be small, especially for small correlation between ordinates. Problems related to the sampling interval can be resolved by determination of the optimal sampling interval after initial an measurement. The surface should then be re-measured or the roughness parameters re-calculated with the optimal sampling interval. For the selected sampling interval, the correlation between neighboring ordinates should be smaller than 0.85. For such high values of the correlation, the data are redundant, and false information about surface summits can be obtained during the analysis of profiles. The higher frequencies are not often as significant tribologically as the lower frequencies. The correlation between 0.5 and 0.7 is suggested by the present authors. The sampling interval should not exceed the value that corresponds to a correlation of 0.1 in order to avoid aliasing. When selecting a sampling interval, the mechanism of surface deformation should be taken into consideration. Averaging of summit radius caused an increase of the mean separation in comparison to distributed radii. For a given load, the true area of contact was smaller after considering the peak curvature distribution, particularly for a small load and comparatively large sampling intervals. Method of summit radius calculation can be source of uncertainty in contact mechanics. Assumption that standard deviation of summit heights is equal to standard deviation of ordinate heights can be the other source of errors. The statistical models in contact mechanics may not be valid for a certain set of surface topography parameters. References [1] J.A. Greenwood, J.B.P. Williamson, Contact of nominally flat surfaces, Proc. Roy. Soc. Lond. A 295 (1966) 300–319. [2] J.A. Greenwood, J.H. Tripp, The elastic contact of rough spheres, ASME J. Appl. Mech. 34 (1967) 153–159. [3] J.A. Greenwood, J.H. Tripp, The contact of two nominally flat rough surfaces, Proc. Inst. Mech. Eng. 185 (1970–71) 625–633. [4] R.A. Onions, J.F. Archard, The contact of surfaces having a random structure, J. Phys. D: Appl. Phys. 6 (1973) 289–305. [5] A.W. Bush, R.D. Gibson, G.P. Keogh, Strongly anisotropic rough surface, ASME J. Tribol. 1010 (1979) 15–20. [6] J.I. McCool, Comparison of models for the contact of rough surfaces, Wear 107 (1986) 37–60. [7] R.L. Jackson, I. Green, A statistical model of elasto-plastic asperity contact between rough surfaces, Tribol. Int. 39 (2006) 906–914. [8] M. Paggi, M. Ciavarella, The coefficient of proportionality k between real contract are and load, with new asperity models, Wear 268 (2010) 1020–1029. [9] G. Zavarise, M. Borri-Brunetto, M. Paggi, On the reliability of microscopical contact models, Wear 257 (2004) 229–245. [10] G. Pugliese, S.M.O. Tavares, E. Ciulli, L.A. Ferreira, Rough contact between actual engineering surfaces. Part II. Contact mechanics, Wear 264 (2008) 1116–1128. [11] G. Carbone, F. Bottiglione, Asperity contact theories: do they predict linearity between contact area and load, J. Mech. Phys. Solids 56 (2008) 2555–2572. [12] E.J. Abbott, F.A. Firestone, Specifying surface quality – a method based on accurate measurement and comparison, Mech. Eng. 55 (1933) 569. [13] J. Pullen, J.B.P. Williamson, On the plastic contact of rough surfaces, Proc. Roy. Soc. Lond. A 327 (1972) 159–173. [14] W.R. Chang, I. Etsion, D.B. Bogy, An elastic–plastic model for the contact of rough surfaces, ASME J. Tribol. 109 (1987) 257–263. [15] Y. Zhao, D.M. Maietta, L. Chang, An asperity microcontact model incorporating the transition from elastic deformation to fully plastic flow, ASME J. Tribol. 20 (2000) 86–93. [16] A. Sepehri, K. Farhang, Closed-form equations for three dimensional elastic–plastic contact of nominally flat rough surfaces, ASME J. Tribol. 131 (2009), 041402-1–8.
P. Pawlus, W. Zelasko / Wear 276–277 (2012) 121–129 [17] Y.-R. Jeng, S.-R. Peng, Contact behavior of surfaces containing elliptical asperities with Gaussian and non-Gaussian height distributions, ASME J. Tribol. 129 (2007) 743–753. [18] Z. Dimkovski, Surfaces of honed cylinder liners, PhD Dissertation, Chalmers University of Technology, Sweden, 2011. [19] L. Kogut, I. Etsion, A finite element based elastic–plastic model for the contact of rough surfaces, Tribol. Trans. 46 (2003) 383–390. [20] R.L. Jackson, I. Green, A finite element study of elasto-plastic hemispheral contact, ASME J. Tribol. 127 (2005) 343–354. [21] A.Y. Suh, A.A. Polycarpou, Digital filtering methodology used to reduce scale of measurement effects in roughness parameters for magnetic storage supersmooth hard disks, Wear 260 (2006) 538–548. [22] E. Tomanik, Modelling the asperity contact area on actual 3D surfaces, SAE Paper 2005-01-1864. [23] J.-J. Wu, Spectral analysis for the effect of stylus tip curvature on measuring rough surfaces, Wear 230 (1999) 194–200. [24] T.Y. Lin, L. Blunt, K.J. Stout, Determination of the proper frequency bandwidth for 3-D topography measurement using spectral analysis. Part I: isotropic surface, Wear 166 (1993) 221–232. [25] W.P. Dong, E. Mainsah, K.J. Stout, Determination of appropriate sampling conditions for three-dimensional microtopography measurement, Int. J. Mach. Tools Manuf. 36/12 (1996) 1347–1362. [26] P. Pawlus, D.G. Chetwynd, Efficient characterization of surface topography in cylinder bores, Precis. Eng. 19 (1996) 164–174. [27] P. Pawlus, Digitisation of surface topography measurement results, Measurement 40 (2007) 672–686. [28] B.-G. Rosen, Towards optimised sampling in surface roughness, in: XII International Colloquium on Surfaces, Chemnitz, Germany, 2008, pp. 383–392. [29] L. Kogut, R.L. Jackson, A comparison of contact modeling utilizing statistical and fractal approaches, ASME J. Tribol. 128 (2006) 213–217. [30] D.J. Whitehouse, J.F. Archard, The properties of random surface of significance in their contact, Proc. Roy. Soc. Lond. A 316 (1970) 97–121.
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[31] J.I. McCool, Limits of applicability of elastic contact models of rough surfaces, Wear 86 (1983) 105–118. [32] T.R. Thomas, B.-G. Rosen, Determination of the optimum sampling interval for rough contact mechanics, Tribol. Int. 33 (2000) 601–610. [33] C. Vallet, D. Lasseux, H. Zahouani, P. Sainsot, Sampling effects on contact and transport properties between fractal surfaces, Tribol. Int. 42 (2009) 1132–1145. [34] M. Ciavarella, G. Murolo, G. Demelio, On the elastic contact of rough surfaces: numerical experiments and comparisons with recent theories, Wear 261 (2006) 1102–1113. [35] W.E. Wilson, S.V. Angadi, R.L. Jackson, Surface separation and contact resistance considering sinusoidal elastic–plastic multi-scale rough surface contact, Wear 268 (2010) 190–201. [36] J.J. Wu, Simulation of rough surfaces with FFT, Tribol. Int. 33 (2000) 47–58. [37] J.A. Greenwood, A unified theory of surface roughness, Proc. Roy. Soc. Lond. A 393 (1984) 133–157. [38] R.S. Sayles, T.R. Thomas, Measurements of the statistical properties of engineering surfaces, ASME J. Lubr. Technol. 101 (1979) 409–417. [39] A. Hariri, J.W. Zu, R. Ben Mrad, Modeling of elastic/plastic contact between nominally flat rough surfaces using an n-point asperity model, ASME J. Tribol. 128 (2006) 876–885. [40] D.J. Whitehouse, The digital measurement of peak parameters on surface profiles, J. Mech. Eng. Sci. IMechE 20/4 (1978) 221–226. [41] N. Yu, A.A. Polycarpou, Extracting summit roughness parameters from random surfaces accounting for asymmetry of the summit heights, ASME J. Tribol. 126 (2004) 761–766. [42] J.I. McCool, Finite difference spectral moments estimation for profiles: the effect of sample spacing and quantization error, Precis. Eng. 4/4 (1982) 181–184. [43] P.R. Nayak, Random process model of rough surfaces, ASME J. Lubr. Technol. 93 (1971) 398–407. [44] A.W. Bush, R.D. Gibson, G.P. Keogh, The limit of elastic deformation in the contact of rough surfaces, Mech. Res. Commun. 3 (1976) 169–174. [45] J.A. Greenwood, A simplified elliptic model of rough surface contact, Wear 261 (2006) 191–200.
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Wear journal homepage: www.elsevier.com/locate/wear
The importance of sampling interval for rough contact mechanics P. Pawlus a,∗ , W. Zelasko b a b
Rzeszow University of Technology, Rzeszow, Poland Group of Technical Schools, Lezajsk, Poland
a r t i c l e
i n f o
Article history: Received 19 May 2011 Received in revised form 16 December 2011 Accepted 21 December 2011 Available online 28 December 2011 Keywords: Contact mechanics Sampling interval Surface topography
a b s t r a c t The contact of random surfaces is discussed in this paper. Commonly used elastic and elastic–plastic microcontact models were applied to modeled and measured isotropic surface topographies of a Gaussian ordinate distribution, assuming a constant and a distributed radius of surface summits, respectively. The calculation procedures allowed the mean surface separation and the real area of contact to be calculated as a function of contact load and sampling intervals. Results are presented for a wide range of contact load and sampling intervals, showing the difference obtained after using various models. The applicability of the profile spectral moment approach to areal random surface specification was checked. This technique relates geometrical properties of a surface important for rough contact mechanics to those of its constituent profiles. Suggestions for choosing the optimum sampling interval are provided. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The analysis of the deformation behavior of contacting surfaces is important to understand the mechanisms of friction, wear, lubrication, sealing, thermal and electrical contact as well as resistance. The pioneering contribution to this field was made by Greenwood and Williamson [1], who developed a basic contact model (GW model) of an isotropic surface. In this model all asperity summits have the same radius but their height varied randomly. This model has been extended to cover contact of curved surfaces [2], two rough surfaces with misaligned asperities [3], surfaces with non-uniform curvature of peaks [4] and strongly anisotropic rough surfaces [5]. McCool [6] suggested after comparison of the GW model with other models that the GW model gave good results. However, the GW contact model can only be used when the majority of contacting asperities deform elastically. A comparison among various contact models, often including the GW model was also given in [7–11]. The authors of Ref. [11] presented a slightly improved GW model in which all the spheres do not have the same radius; they found that all multi-asperity contact theories, taking into consideration the influence of the summit curvature variation as a function of summit height, gave similar results. However, it is difficult to find the direct numerical comparison of the same models for constant and distributed summits radius. A number of researchers studied contact of rough surfaces characterized by plastic deformation of asperities. Abbott and Firestone [12]
∗ Corresponding author. E-mail address: [email protected] (P. Pawlus). 0043-1648/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2011.12.015
developed the widely used model for fully plastic contact. It assumed that the deformation of a rough surface against a rigid smooth flat is equivalent to the truncation of the undeformed rough surface at its intersection with the flat. Pullen and Williamson [13] established a volume conservation model for the fully plastic contact of rough surface. The papers [7,14,15] present statistical models of elastic–plastic contact between rough surfaces. Chang et al. [14] put forward an elastic–plastic contact model (CEB model) for rough surfaces on the basis of volume conservation of plastically deformed asperities. Many publications appeared that are based on the CEB model or inspired by it; about 30 publications were cited in [16]. The paper [15] presented a different elastic–plastic contact model (ZMC model) for rough surfaces. Its main feature is the modeling of the long transitional regime from elastic deformation to the fully plastic flow. It is also widely used (see for example [17,18]). An elastic–plastic model for contacting rough surfaces that is based on Finite Element Analysis (FEA) of an elastic–plastic asperity contact was presented in paper [19]. Jackson and Green [7] developed a statistical elastic–plastic contact model using the results of a previous FEA of an elastic–plastic sphere in contact with a rigid flat [20]. It is widely known that the sampling interval has a significant influence on spatial and hybrid properties important in studying contact problems, particularly the summit density and curvature [21,22]. Several methods for determining the optimal sampling interval were suggested based on the principle that a tolerance between a parameter calculated from a sampled profiles and the true value for the real surface does not exceed an acceptable limit. Wu [23] presented a method of obtaining the minimum sampling interval for a random profile of normal ordinate distribution. The proposed sampling interval depends on the radius of the tip, the
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Nomenclature a A An d
contact radius true area of contact nominal contact area separation of the surfaces measured from the summits mean plane E1,2 Young’s moduli Hertz elastic modulus E h separation based on surface heights H hardness of softer material K maximum contact pressure factor m0 , m2 , m4 spectral moments of profile P contact load of individual asperity areal density of asperities Pds R mean radius of summits curvature SI sampling interval standard deviation of surface heights Sq () ys distance between the mean of asperity heights and that of surface ordinates height of summit z ˛ bandwidth parameter correlation between profile neighboring points Poisson’s ratios 1,2 s standard deviation of asperity height plasticity index ω interference critical interference at the inception of plastic deforωc mation
Fig. 1. Schematic of contact of two rough surfaces.
surfaces. However, there are surfaces for which the multi-scale technique will not converge. The fundamental aim of this paper is to analyze the effect of the sampling interval on contact mechanics of rough isotropic surfaces, while recommending the appropriate sampling interval. The second aim is to test the hypothesis that a constant peak radius affects the results of the contact mechanics study significantly. The present authors also checked if the areal (3D) surface topography parameters are important for rough contact mechanics can be estimated on the basis of (2D) profile analysis. 2. Theoretical background
correlation length and the standard deviation of the ordinate values. Lin [24], Dong et al. [25], Pawlus and Chetwynd [26] and Pawlus [27] recommended an optimum sampling interval based on the shape of the cumulative spectrum graph. It is also possible to select a sampling interval based on the autocorrelation function. The paper [28] focused on the evaluation of a general methodology to sampling in relation to the variation of the surface topography. The contact parameters, i.e., contact area versus contact load and separation versus contact load predicted by the GW and other statistical models are largely dependent on the sampling resolution (interval) used to characterize the surfaces [29]. Because long wavelengths exist on the surface after contact/wear, Whitehouse and Archard recommended a sampling interval equal to the distance at which the autocorrelation function decays to a value of 0.1 (correlation length) [30]. The significance of the shorter wavelength structures (revealed by the use of shorter sampling intervals) in the contact of surfaces may be questioned. McCool [31] determined the frequency interval over which purely elastic models are applicable. Thomas and Rosen [32] proposed a method for choosing a tribologically appropriate sampling interval, based on Archard’s observation that repetitive contact must be elastic. The sampling interval should be equal to half of the critical wavelength; all surface features smaller than this critical wavelength will disappear on the first encounter with another surface and so do not need to be taken into consideration with regard to the surface’s subsequent elastic behavior. Vallet et al. [33] analyzed the influence of the sampling interval on the determination of the roughness parameters, contact areas and flow through the aperture field resulting from the contact under load. Unfortunately, the approaches in [32,33] were restricted only to fractal surfaces. The problems with the sampling interval choice can be solved using the analysis of multi-scale rough surfaces. The works of Ciavarella et al. [34] can be the example of multi-scale analysis of fractal surfaces. The work [35] used a multiscale model based on the Fourier series obtained from measured
The contact between two rough surfaces is modeled by means of the contact of a single rough surface with a smooth plane [2]. The following fundamental assumptions (based on Greenwood and Williamson model [1]) are adopted: - the rough surface is isotropic, and has a Gaussian ordinate distribution, - asperities are spherical near their peaks (summits), - there is no interaction between asperities, and - only the asperities deform during contact. Fig. 1 shows the geometry model of the contacting rough surfaces. Two possibilities are investigated. First, all peaks have the same radius R (see Fig. 1a). Secondly, the peak radii vary randomly (Fig. 1b). In Fig. 1, z denotes the height of asperity, d separation of the surfaces measured from the summits mean plane, and h is the separation of the surfaces based on surface heights (ordinates). The interference is defined as: ω =z−d
(1)
During loading, the area of contact A and the contact load of each individual asperity depends only on its individual interference. For a sphere of radius R, elastically deflected by an amount ω, Hertz’s solution gives the contact area: A = Rω = R(z − d) = a2
(2)
where a is the contact radius. The corresponding asperity load is: P=
4 1/2 3/2 E R (z − d) 3
(3)
P. Pawlus, W. Zelasko / Wear 276–277 (2012) 121–129
where
E =
1 − 12 E1
1 − 22
+
123
−1 (4)
E2
Ei and i (i = 1, 2) are the Young’s moduli and the Poisson’s ratios for the two contacting elements [2], respectively. The critical interference ωc at the inception of plastic deformation is: ωc =
KH 2 2E
R
(5)
where H is the hardness of the softer material. A value of constant K = 0.6 is commonly selected. When ω < ωc the contact is elastic. Any summit, for which interference ω exceeds ωc will have some degree of plastic deformation. The CEB elastic–plastic contact model is based on volume conservation during plastic deformation [14]. The contact area of a plastically deformed asperity was calculated as:
A = a2 = Rω 2 −
ωc ω
(6)
The contact load for such an asperity is:
P = Rω 2 −
ωc ω
KH
(7)
3. Calculation procedure Isotropic surfaces with Gaussian ordinate distribution were generated, using the procedure developed by Wu [36]. Each surface was characterized by the correlation distance (in which the autocorrelation function decays to value of 0.1) and the standard deviation of height Sq. In addition, measured isotropic Gaussian surface topographies were analyzed. Talyscan 150 measuring equipment using a stylus tip with a nominal radius of 2 m was used. Modeled and measured surfaces contacted with a perfectly flat rigid surface. A surface point was selected to be a summit if its ordinate was higher than the ordinates of its eight nearest neighbors. This criterion was based on the work of Greenwood [37], Sayles and Thomas [38], as well as previous research of the present authors. This method does not take into consideration that a smaller asperity may become part of a bigger asperity. A newly developed n-point asperity model seems to be a good proposal for solving this problem. However, its application to areal surface topography without simplifying assumption would be computationally intensive [39]. For areal measurements, the radius of each summit was calculated as the reciprocal of its arithmetic average curvature in orthogonal directions. The summit curvature was calculated on the basis of a three-point formula [40]. The mean radius of the summits R, was considered to be the average of all the summits radii in the areal surface topography measurement. The same method was used in [41]. The areal density of asperities Pds, standard deviation of summits heights s , and distance between the mean of asperity heights and that of surface ordinates ys were calculated, too. In addition, the plasticity index defined by Greenwood and Williamson was computed, which in this case is expressed as: =
1/2 s
ωc
(8)
The total contact area A and total contact load P were calculated for various separations d in the following way. The vertical range of summit heights was divided into 100 intervals. For each separation, summits located above this height (plane) were identified. For each 8-neighbor summit in contact interference ω was computed (Eq. (1)). When it was smaller than the critical interference ωc , the contact area A and contact load P for elastic contact were calculated according to Eqs. (2) and (3), respectively. Else (when ω > ωc ) the contact area and contact load of a deformed asperity
Fig. 2. Three-dimensional views of surfaces A and C: for SI = 5 m (a), SI = 10 m (b), SI = 20 m (c), and SI = 40 m (d).
were computed using Eqs. (6) and (7). The total contact area A and contact load P were obtained by summing the contributions of all contacting asperities located above a given separation. This procedure was repeated for different surface separations. In addition, the contact area and contact load were calculated assuming only elastic contact; using Eqs. (2) and (3). The calculations of the contact area and contact load were repeated for various sampling intervals. The correlation between ordinates was calculated on the basis of the profile autocorrelation function, as the average value of 6 profiles. In addition, the mean values of zeroth m0 , second m2 and fourth m4 spectral moments of all profiles in perpendicular directions were computed. The m0 moment is the variance of profile height, i.e. Pq2 . The second moment m2 is equal to square of the profile r.m.s. slope i.e. Pq2 . The Pq parameter was calculated based on a 2-point formula. The fourth moment m4 being the second derivative of a profile was computed using the procedure described in [42]. For an isotropic areal surface some parameters connected with the peaks, such as the product R Pds s , depend on the parameter ˛, defined as: m0 m4 ˛= . (9) m22 This parameter was also calculated for each sampling interval used. The following material properties were selected (contact of steel-on-steel elements) E1 = E2 = 2.07 × 105 MPa, Brinell hardness H = 200 (1960 MPa), 1 = 2 = 0.29. These properties were also used in papers [14,15]. 4. Results and discussion Table 1 presents the surface topography parameters for various sampling intervals SI and correlation values between neighboring points . The standard deviation of surface ordinates Sq (), standard ordinates of peak heights s , areal peak density Pds, mean radius of peak R, product s PdsR, mean summit height ys , and the ˛ parameter are shown. The critical interference ωc and the plasticity index are also presented. Four surfaces were analyzed for four different sampling intervals. Surfaces A, B and C were computer generated. The correlation distances for surfaces A and B were 50 m, and 200 m for surface C. Surface D was experimentally determined and the correlation distance was found to be 72 m (the surface was finished using sand blasting). The dimensionless total load and area were also calculated using tabular values presented in [6] for given s , Pds, R, and ys . Fig. 2 shows an isometric view of surfaces A and B. Figs. 3–6 present the mean separation and real area of contact as a function of load and sampling interval
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Table 1 Surface topography parameters, critical interferences and plasticity indices. Surface A A A A B B B B C C C C D D D D
SI (m) 5 10 20 40 5 10 20 40 10 30 60 120 8 40 56 72
Sq (m)
˛
s (m)
Pds (m)−2
R (m)
s PdsR
ys (m)
ωc (m)
0.176 0.176 0.176 0.176 0.44 0.44 0.44 0.44 0.93 0.93 0.93 0.93 0.66 0.66 0.66 0.66
0.85 0.7 0.4 0.12 0.85 0.7 0.4 0.12 0.91 0.65 0.39 0.14 0.95 0.5 0.25 0.1
7.81 4.16 2.48 1.68 7.81 4.16 2.48 1.68 14.9 4.74 3.33 2.1 8.6 2.7 2.14 1.68
0.16 0.145 0.13 0.12 0.4 0.36 0.33 0.3 0.88 0.8 0.72 0.58 0.58 0.41 0.39 0.36
0.0012 0.0005 0.00019 0.00006 0.0012 0.0005 0.00019 0.00006 0.000232 0.000058 0.000021 0.000006 0.00035 0.000046 0.000026 0.000017
190.7 467.5 1172 3713.2 76.4 187.2 468.8 1483.2 212.8 830.9 2156.8 6900.2 347.2 1634.2 2457.2 3839.4
0.037 0.034 0.029 0.027 0.037 0.034 0.029 0.027 0.044 0.039 0.033 0.023 0.07 0.031 0.025 0.023
0.17 0.19 0.23 0.25 0.42 0.49 0.58 0.64 0.71 0.95 1.19 1.26 0.46 0.79 0.87 0.89
0.051 0.125 0.31 0.99 0.02 0.05 0.125 0.4 0.05 0.22 0.58 1.84 0.09 0.44 0.66 1.03
1.76 1.1 0.65 0.35 4.4 2.74 1.63 0.87 3.92 1.9 1.12 0.56 2.5 0.97 0.77 0.59
of surfaces A and D for various models: elastic and elastic–plastic for constant summit radius and elastic–plastic for variable summit radii. Table 2 shows the dimensionless separation h/ and contact area ratio A/An versus the dimensionless load P/An E for various values of the sampling interval of surface A and for various contact models. Fig. 3 presents the dimensionless separation h/ versus the dimensionless load P/An E for various sampling intervals of surface A. An increase in sampling interval caused a decrease in plasticity index. Elastic and elastic–plastic models were considered for constant value of the summit radius, but when this radius was distributed – only information about elastic–plastic model was graphically presented. The separation between contacting rough surfaces under a given load usually decreases with increasing magnitude of the sampling interval. For the same dimensionless load, the mean separation was smaller when the elastic model was used compared to the elastic–plastic model for sampling intervals of 5, 10 and 20 m. These differences were significant for the largest load, and the relative deviations increased with increasing magnitude of the sampling interval. For example, when the SI was set to 20 m, the relative difference was higher than 60%. For smaller loads than 0.001 the differences between the mean separations were found to be smaller than 10%. For the largest sampling interval (40 m) no differences between mean separations for elastic and elastic–plastic models were found for all analyzed loads, because the contact was elastic. When plastic contact occurs,
the elastic model underestimates the separation compared to the elastic–plastic model. Averaging of peak radius for elastic–plastic model caused increase in mean separation particularly for the highest sampling intervals corresponding to plasticity index much smaller than 1 in comparison to variable radius, the highest relative difference was bigger than 30%. Deviations decreased with decreasing load. A similar tendency was observed for the elastic model. However, the deviation in mean separation was negligible for the smallest sampling interval. Fig. 4 presents the true contact ratio A/An versus the dimensionless load P/An E for various sampling intervals and various contact models of surface A. In addition the plastic contact area ratio was computed. It was identical to that of total area ratio for the smallest sampling interval and almost identical for sampling interval of 10 m and the highest load. For smaller load the contribution of the plastic contact area to the total contact area was smaller. When the sampling interval was set to 20 m, most of the contact was elastic, and for the highest sampling interval only elastic contact occurred. In elastic and elastic–plastic models the contact area was higher for larger sampling interval and the same dimensionless load. This was mainly evident for elastic contact models. When the contact was plastic (sampling interval was 5 m), the contact area for a high load was larger after using the elastic–plastic model in comparison to the elastic model. When a model with variable radius was used, the contact area ratio for elastic–plastic contact was usually smaller than in the case of constant radius, particularly for a small load.
Table 2 Comparison of elastic–plastic model with constant summit radius EPRC, elastic model with constant summit radius ERC, elastic–plastic model with variable summit radii EPRV and elastic model with variable summit radii ERV for surface A. SI (m)
P/An E
h/
A/An
EPRC
ERC
EPRV
ERV
EPRC
ERC
EPRV
ERV
5
0.001 0.0001 0.00001 0.000001
0.68 2.02 2.83 3.37
0.48 1.82 2.73 3.37
0.55 1.96 2.74 3.37
0.48 1.82 2.69 3.3
0.096 0.0097 0.00113 0.000142
0.071 0.0096 0.00083 0.000117
0.102 0.0096 0.00111 0.000119
0.066 0.0083 0.00082 0.000116
10
0.001 0.0001 0.00001 0.000001
0.72 1.92 2.73 3.37
0.42 1.78 2.67 3.32
0.6 1.84 2.67 3.32
0.31 1.72 2.61 3.26
0.1 0.0126 0.0015 0.00016
0.103 0.012 0.0016 0.00019
0.104 0.0119 0.0013 0.00013
0.104 0.012 0.0021 0.00013
20
0.001 0.0001 0.00001 0.000001
0.31 1.65 2.46 3.03
0.12 1.6 2.46 3.03
0.26 1.6 2.31 2.98
0.022 1.55 2.37 2.98
0.14 0.021 0.0027 0.00039
0.135 0.022 0.0027 0.00039
0.136 0.018 0.002 0.00023
0.12 0.019 0.0017 0.00023
40
0.005 0.0001 0.00001 0.000001
0.31 1.39 2.37 2.98
0.31 1.39 2.37 2.98
0.23 1.28 2.23 2.91
0.23 1.28 2.23 2.91
0.14 0.036 0.0042 0.00073
0.14 0.036 0.0042 0.00073
0.131 0.03 0.0031 0.00043
0.131 0.03 0.0031 0.00043
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Fig. 4. Real area of contact of the surface A as a function of load and sampling interval: elastic plastic model, constant summit radius (a), elastic model, constant summit radius (b), and elastic–plastic model, variable summit radius (c).
Fig. 3. Mean separation of surface A as a function of load and sampling interval: elastic plastic model, constant summit radius (a), elastic model, constant summit radius (b), and elastic–plastic model, variable summit radius (c).
The differences increased for increasing magnitude of the sampling interval. As a result of elastic contact no differences between the area ratios for the same load after elastic and elastic–plastic models use were found for the largest sampling interval. Spatial properties of surface B are identical with those of surface A, only the height of surface B is higher, which resulted in higher values of the plasticity index. The separation between contacting rough surfaces, under a given load usually decreases with the increasing magnitude of the sampling interval (this is mainly evident after elastic model use). Averaging the peak radius caused an increase of the mean separation in comparison to the case of a variable summit radius. Relative differences were larger for elastic contact than for elastic–plastic contact, and they were found to be the biggest for the largest sampling interval. Deviations between the mean separations after application of different models were lower for smaller load. After application of the elastic model the contact area was higher for a larger sampling interval and the same load. When contact was mostly plastic (sampling intervals were 5, 10 and 20 m), the contact area for high load was larger after using the elastic–plastic model in comparison to the elastic model. Application of the elastic–plastic contact model especially for variable summits radii caused smaller differences between contact areas for different sampling intervals used in comparison to the elastic model. Surface C was characterized by the highest height and the largest main wavelengths. The contact parameters were similar to those from surface B. For the same load the mean separation was smaller when the elastic model was used for the sampling intervals of 30, 60
and 120 m, the relative difference was higher for a larger sampling interval. The elastic model underestimates the separation compared to the elastic–plastic model, but case with a high plasticity index value formed exception. For surfaces with a higher plasticity index (sampling intervals were 10, 30 and 60 m), the contact area for a high load was larger after using the elastic–plastic model in comparison to the elastic model. The plasticity index of the measured surface D was comparatively high only for the smallest sampling interval 8 m. The separation between contacting rough surfaces, under a given load decreases with the higher sampling interval for comparatively high loads when all models were used (see Fig. 5); for a small load the mean separation was the smallest for a sampling interval of 40 m. For the same load the mean separation was smaller when the elastic model was used in comparison to the elastic–plastic model – relative differences (in the range 14–47% for dimensionless load 0.001 and a constant peak radius) were the largest for the highest sampling interval. For smaller loads differences were found to be lower. Similarly to surfaces A, B and C, averaging the peak radius caused an increase in the mean separation compared to using a variable radius. The plastic contact area was almost identical to that of the total area for the smallest sampling interval used. For the surface with the highest plasticity index, the contact area for the high load was larger after using the elastic–plastic model in comparison to the elastic model (see Fig. 6) – the relative difference was found to be 25% for a dimensionless load of 0.001 and for a constant summit radius. In other cases the contact area was rather similar after applying these models. For all the models, increasing the sampling interval increased the contact area for the same load. Differences between curves corresponding to various sampling intervals were small when the elastic–plastic model with variable summit radius
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Fig. 5. Mean separation of surface D as a function of load and sampling interval: elastic plastic model, constant summit radius (a), elastic model, constant summit radius (b), and elastic–plastic model, variable summit radius (c).
was used. The contact area was smaller than in the case of a constant radius, particularly for a small load. Generally, the separation between contacting rough surfaces, under a given load, decreases with increasing sampling interval. This is mainly evident after applying the elastic model. When contact was mainly plastic (plasticity index higher than 2.5), the elastic model overestimated the separation compared to the elastic–plastic model. In the other cases the elastic model underestimated the separation. Relative differences were larger for larger sampling intervals and for higher loads. No differences between the elastic and the elastic–plastic models were found for a plasticity index smaller than 0.4. Averaging the peak radius caused an increase of the mean separation in comparison to using a variable peak radius. The contact area was higher for larger a sampling interval and constant dimensional load. However, the differences in contact areas between the different sampling intervals were smaller after applying the elastic–plastic model compared applying the elastic model. Those differences were particularly small when the elastic–plastic model with variable radius was used. When the plasticity index was higher than 1.6, the contact area for a high load was larger after using the elastic–plastic contact model in comparison to the elastic model. For smaller plasticity index values, the differences between the area ratio were negligible. For the elastic–plastic contact when the variable summit radius was used, the contact area was smaller than in the case of a constant radius, particularly for a small load and comparatively large sampling intervals. According to [4] for a given load, the true area of contact should be smaller after consideration of the peak curvatures distribution.
Fig. 6. Real area of contact of the surface D as a function of load and sampling interval: elastic plastic model, constant summit radius (a), elastic model, constant summit radius (b), and elastic–plastic model, variable summit radius (c).
This finding was confirmed. When a variable radius was used, the contact area was smaller than in the case of a constant radius, particularly for a small load and comparatively large sampling intervals (in the case of elastic contact). The calculation of the paper [4] was concerned with the main structure of the surface topography ( = 0.1), so the sampling interval was also high. It is evident from the analysis of Table 1 that an increase in the sampling interval SI caused a decrease in the plasticity index and the product s PdsR. When increasing the SI, the summit density Pds decreases and the summit radius R increases, while the standard deviation of asperity heights s decreases. The plasticity index decreases because the decrease in s is smaller than the increase in R. The product s PdsR decreases because in the decrease in both Pds and s is higher than the increase in R. High correlations between neighboring points correspond to large values of the bandwidth parameter ˛. Measured and simulated surface topographies were analyzed. It was found that when was near 0.1, the bandwidth parameter value ˛ was in the range 1.5–1.8 (Nayak [43] found it should be higher than 1.5). When ˛ is larger, the product s PdsR is also larger. For a correlation smaller than 0.5 this product is usually smaller than 0.04. High ˛ values (higher than 8) can be obtained for highly correlated points ( ≥ 0.9). However the large product s PdsR value often found in the technical literature (for example 0.06 [14,15]) proves a high correlation between neighboring points ( ≥ 0.9). An increase in the magnitude of the sampling interval (and decrease in and ˛) causes not only a decrease in the standard deviation of asperity heights s but also an increase in the distance between the mean of asperity heights and that of surface ordinates ys . These changes were anticipated by Whitehouse and Archard for
P. Pawlus, W. Zelasko / Wear 276–277 (2012) 121–129
profiles [30]. Because oftentimes contact parameters are obtained on the basis of profile analysis it is necessary to consider correlations among parameters applied in contact mechanics in 2D and 3D systems. Dependencies between profile spectral moments of anisotropic Gaussian surfaces were developed by Nayak [43] and Bush et al. [44]. They were presented by McCool [6]. The surface summit density is given by: Pds =
1 √ 6 3
m 2 4
m2
(10)
The mean summit curvature averaged over all summit heights is: √ 8 m4 (11) Spc = √ 3 The variance of the summit height distribution is: s2 =
1−
0.8968 ˛
m0
(12)
The distance between the mean of the summit height distribution and the surface mean plane is: √ 4 m0 (13) ys = √ ˛ Surface and profile measurement and their resultant statistics were compared and their interrelationship examined for several common engineering surfaces [8]. Good agreement was found between theory and measurement over a large range of sampling intervals. Yu and Polycarpou [41] compared summit density and summit radius obtained from a numerically generated isotropic Gaussian surface, and predicted these parameters by means of the spectrum moment approach. It was found that at small ˛ values, the summit density and summit radius based on their definition were higher than prediction from the spectrum moment approach. Sets of parallel profiles were obtained from modeled surfaces and average profile spectral moments m0 , m2 and m4 were calculated. It was evident that high values of the ˛ parameter (larger than 6) correspond to large errors of summit density anticipation. Mean summit density predicted by the spectral moment approach were much bigger than value obtained from its definition. For ˛ values from range 1.7–5 the difference was smaller than 5%. For non-correlated neighboring points ( between 0.1 and 0.12) application of the spectral moments method caused underestimation of the summit density (the errors were smaller than 15%). The mean summit curvature was accurately predicted by the spectral moments approach, independently of the ˛ value. Errors were usually smaller than 8%. The relative difference between the standard deviation of summit heights was usually smaller than 5% but not higher than 10%. For ˛ values smaller than 2 predicted ys distance was higher than value obtained from the analyses of simulated and measured surfaces, however for ˛ values larger than 7.5 it was usually smaller but differences were not higher than 15%. Generally good agreement was found between the two methods except for the summit density, which could be overestimated by the theory for comparatively high ˛ values. The obtained results were roughly confirmed after the analysis of the measured surfaces. Here, measurements were in good agreement again with the theory. However, sometimes the differences were found to be larger than in case of modeled surfaces. Some deviations in the parameter estimation can be caused by the calculation of mean radius of asperity curvature. Two methods can be used. In the first of them, the summit curvature is averaged over the entire surface, and the summit radius is calculated as the reciprocal value of the curvature. This method was applied in [4,6]. In the second method, the summit curvature for each summit is calculated, and the summit radius is then obtained as the reciprocal of its curvature, and this radius is averaged over the surface. This
127
method was used in [22,41]. The present authors used the second method because the mean summit radius was similar to the variable summit radius. However, it was found that the summit radius estimated using the second method exceeds that obtained by the first method by 1.1–1.2 times. It is also possible to estimate parameters characterizing the summits from the parameters characterizing peaks (summits are local maxima on the surface, as distinct from peaks, which are local maxima on a profile). A peak was identified using a three-point method. However, significant errors occurred. The mean summit height was about two times higher than the mean peak height. The mean summit curvature was about 1.3–1.4 times higher than mean peak curvature. The differences between the standard deviations of summit and peak heights were small for highly correlated ordinates (for > 0.85 relative difference was about 5%). The standard deviation of summits height was smaller than that of peak heights. The summit density was close to the peak density square (according to Nayak [43] summit density should be 1.2 of peak density square). The results obtained for correlation between neighboring points > 0.85 were different. In those cases the square of peak density was similar to twice of the summit density. Similar results were obtained after the analysis of measured surfaces. On the basis of the profile peak analysis, the radius of summits and the standard deviation of the summit heights can be overestimated. Thus, the errors of the calculation of the product s PdsR after the profile analysis can be high, especially for small correlation between measuring points. However, the errors of computing the plasticity index on the basis of profile analysis can be small. In general, applying profile spectral moments is better than profile peak analysis. The question arises how the sampling intervals should be chosen? Whitehouse and Archard [30] as well as Onions and Archard [4] recommend sampling interval corresponding to correlation of 0.1. For this spacing events reach the condition where they can be regarded as independent. Smaller scale structures are characterized by a high value of the plasticity index. Greenwood and Williamson stated that most surfaces have a plasticity index higher than 1.0 and, thus, except for especially smooth surfaces, the asperities will flow plastically under the lightest load [1]. The sampling interval should be not higher than that proposed by Whitehouse and Archard [30] ( = 0.1) in order to avoid of aliasing. A correlation of 0.1 corresponds to the ˛ bandwidth parameter about 1.5. However, in this case only a narrow range of wavelengths is present [43]. When ˛ is small, the distribution of summit heights of a random Gaussian surface is not Gaussian, so the errors can occur after using a Gaussian summit heights distribution in statistical models [41]. There are opinions that an adequate representation of a surface can be obtained by taking ˛ = 5 [37,45]. It was found that for conventionally machined surfaces a sampling interval for a correlation = 0.1 is rather large (usually much larger than 30 m). So maybe the correlation can be higher (sampling interval smaller) than the authors of Refs. [4,30] suggested. However, in our opinion, the data should not be highly correlated. A correlation higher than 0.85 should be avoided. In this case the data are redundant, and false information about the summits can be obtained from the analysis of profiles. The higher frequencies, which are more susceptible to instrumental idiosyncrasies, are often not as significant tribologically as the lower frequencies. An optimal sampling interval should be obtained after measurement with initial SI as small as possible. For minimising the cost, it can be done by measuring of a few profiles from the surface. After that, the surface should be remeasured on a scale appropriate for a larger measured area with the optimum sampling interval, or if the number of points is not too small by increasing the sampling interval for initially measured surface area. The autocorrelation function calculated by most of the commercial instrument can be helpful. The high value of ˛ (larger
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than 8) or product s PdsR (larger than 0.045) prove that correlation between neighboring points is too high. It is known that [5] the real contact area at moderate separations (h/ > 1) must be equal to half of bearing area at that separation. For the analyzed surfaces the dimensionless elastic contact area was compared to half of the material ratio with the same separation h/ from the range 1–3. It was found that the smallest differences were obtained for values between 0.5 and 0.7. For example for surface D the mean error in this case was 7%. The criterion for the mechanism of deformation of surface asperities should be taken into consideration when selecting the sampling interval. As smaller and sharper asperities are observed, a size will be reached below which they will deform plastically during the first cycle of contact and thus disappear. During the subsequent lifetime of the contact, it will behave elastically as if the corresponding range of surface wavelength does not exist [32]. As we insert the limiting value of at which the deformation will be plastic at any load, we can select an optimal sampling interval. The plasticity index should be calculated while increasing the sampling interval, and the value of sampling interval should be selected for which is not much higher than this limiting value. Because of small errors, the plasticity index can be calculated on the basis of profile analysis, especially for small correlation between ordinates. Great care should be taken for surfaces having large value of Sq/R [7]. For the elastic contact model, when the interference ω is equal to the standard deviation of the surface heights, the ratio of the contact radius to summit radius is: a = R
Sq R
(14)
It was found that the resulting deformation after application of experimental surface topography parameters can be very large, so the models may calculate the contact area and the contact force for deformation outside of their intended range and this calculation can produce meaningless or misleading results [7]. However increasing the sampling interval can cause an increase in the summit radius and a decrease in the a/R ratio (in our calculation, the highest a/R value, for the smallest sampling interval was not higher than 7%). Other sources of error in the contact parameter computation are important as well. The described method of summit radius calculation is the first of them. The assumption often made by researchers that the heights of peaks and summits are the same is the second of them. It can cause serious errors especially for small correlation between ordinates. This assumption leads to overestimation of standard deviation of summit heights. The other more serious error is the assumption that standard deviation of summit heights is equal to the standard deviation of ordinate heights. Only for very high correlation between neighboring points s is close to .
5. Conclusions The sampling interval selection is important for rough contact mechanics. An increase in the sampling interval causes a decrease in the plasticity index. The separation between contacting rough surfaces, under a given load decreases with the higher sampling interval used. The contact area was higher for a larger sampling interval and the same dimensionless load. Differences among separations and contact areas calculated with different sampling interval were smaller after applying the elastic–plastic model than after using the elastic model. Deviations between contact areas for different sampling intervals were particularly small when the elastic–plastic model with distributed radius was used.
Applicability of the profile spectral moments’ approach to areal random surface specification was checked. Good agreement between the analysis of modeled and measured surfaces and theory was found, except for the summit density which could be overestimated by theory for comparatively high ˛ and values. The errors of calculation of the different parameters important for contact mechanics after the analysis of the profile peaks are higher than after using the profile spectral moments technique. However, the discrepancies of computing the plasticity index on the basis of profile analysis can be small, especially for small correlation between ordinates. Problems related to the sampling interval can be resolved by determination of the optimal sampling interval after initial an measurement. The surface should then be re-measured or the roughness parameters re-calculated with the optimal sampling interval. For the selected sampling interval, the correlation between neighboring ordinates should be smaller than 0.85. For such high values of the correlation, the data are redundant, and false information about surface summits can be obtained during the analysis of profiles. The higher frequencies are not often as significant tribologically as the lower frequencies. The correlation between 0.5 and 0.7 is suggested by the present authors. The sampling interval should not exceed the value that corresponds to a correlation of 0.1 in order to avoid aliasing. When selecting a sampling interval, the mechanism of surface deformation should be taken into consideration. Averaging of summit radius caused an increase of the mean separation in comparison to distributed radii. For a given load, the true area of contact was smaller after considering the peak curvature distribution, particularly for a small load and comparatively large sampling intervals. Method of summit radius calculation can be source of uncertainty in contact mechanics. Assumption that standard deviation of summit heights is equal to standard deviation of ordinate heights can be the other source of errors. The statistical models in contact mechanics may not be valid for a certain set of surface topography parameters. References [1] J.A. Greenwood, J.B.P. Williamson, Contact of nominally flat surfaces, Proc. Roy. Soc. Lond. A 295 (1966) 300–319. [2] J.A. Greenwood, J.H. Tripp, The elastic contact of rough spheres, ASME J. Appl. Mech. 34 (1967) 153–159. [3] J.A. Greenwood, J.H. Tripp, The contact of two nominally flat rough surfaces, Proc. Inst. Mech. Eng. 185 (1970–71) 625–633. [4] R.A. Onions, J.F. Archard, The contact of surfaces having a random structure, J. Phys. D: Appl. Phys. 6 (1973) 289–305. [5] A.W. Bush, R.D. Gibson, G.P. Keogh, Strongly anisotropic rough surface, ASME J. Tribol. 1010 (1979) 15–20. [6] J.I. McCool, Comparison of models for the contact of rough surfaces, Wear 107 (1986) 37–60. [7] R.L. Jackson, I. Green, A statistical model of elasto-plastic asperity contact between rough surfaces, Tribol. Int. 39 (2006) 906–914. [8] M. Paggi, M. Ciavarella, The coefficient of proportionality k between real contract are and load, with new asperity models, Wear 268 (2010) 1020–1029. [9] G. Zavarise, M. Borri-Brunetto, M. Paggi, On the reliability of microscopical contact models, Wear 257 (2004) 229–245. [10] G. Pugliese, S.M.O. Tavares, E. Ciulli, L.A. Ferreira, Rough contact between actual engineering surfaces. Part II. Contact mechanics, Wear 264 (2008) 1116–1128. [11] G. Carbone, F. Bottiglione, Asperity contact theories: do they predict linearity between contact area and load, J. Mech. Phys. Solids 56 (2008) 2555–2572. [12] E.J. Abbott, F.A. Firestone, Specifying surface quality – a method based on accurate measurement and comparison, Mech. Eng. 55 (1933) 569. [13] J. Pullen, J.B.P. Williamson, On the plastic contact of rough surfaces, Proc. Roy. Soc. Lond. A 327 (1972) 159–173. [14] W.R. Chang, I. Etsion, D.B. Bogy, An elastic–plastic model for the contact of rough surfaces, ASME J. Tribol. 109 (1987) 257–263. [15] Y. Zhao, D.M. Maietta, L. Chang, An asperity microcontact model incorporating the transition from elastic deformation to fully plastic flow, ASME J. Tribol. 20 (2000) 86–93. [16] A. Sepehri, K. Farhang, Closed-form equations for three dimensional elastic–plastic contact of nominally flat rough surfaces, ASME J. Tribol. 131 (2009), 041402-1–8.
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