# Prediction of Parameters of Equivalent Sum Rough Surfaces

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{0}, m

_{2}, and m

_{4}respectively, are the sum of spectral moments of two contacted surfaces. In this work, the selected parameters of the sum surfaces were predicted when the parameters of individual surfaces are known. During parameters selection, it was found that the pair of parameters: Sp/Sz (the emptiness coefficient) and Sq/Sa, better described the shape of the probability ordinate distribution of the analyzed textures than the frequently applied pair: the skewness Ssk and the kurtosis Sku. It was found that the RMS height Sq and the RMS slope Sdq were predicted with very high accuracy. The accuracy of prediction of the average summit curvature Ssc, the areal density of summits Sds, and parameters characterizing the shape of the ordinate distribution Sp/Sz and Sq/Sa was also good (the maximum relative errors were typically smaller than 10%).

## 1. Introduction

_{0}moment is the profile variance, the m

_{2}moment is the profile mean square slope, and the m

_{4}moment is the mean square curvature of the surface profile. Greenwood and Tripp [6] found that the contact of two surfaces can be replaced by the contact of the equivalent sum rough surface and the smooth flat surface. The elastic modulus of the equivalent surface can be obtained from the following equation:

_{i}and ν

_{i}(i = 1, 2) are Young’s moduli and Poisson’s ratios of the two contacting elements [6].

_{1}, R

_{2}, and R

_{3}are radii of the summits in contact. One can see that radius of summits, density, and height of summits are important parameters in contact mechanics.

_{0}moment is the profile variance, the m

_{2}moment is the profile mean square slope, and the m

_{4}moment is the mean square curvature of the surface profile. These moments are substantial in contact of two surfaces. This simplification can be used not only in statistical models of the elastic contact [7,9], but also in statistical models of the elastic–plastic contact [10,11,12,13]. The contact parameters such as the number of contacting asperities, the real area of contact, and the contact load, for any given separation between the equivalent sum rough surface and a smooth flat, can be also calculated by summing the contributions of all the contacting asperities using the summit identification model [14,15,16]. Combining two rough surfaces onto one equivalent rough surface and a smooth plane can also be helpful in deterministic contact models considering the Boussinesq problem [17,18,19]. In all cases, the ordinates of the equivalent sum rough surface should be the sum of the ordinates of two contacted surfaces. For the equivalent rough surface, the spectral moments are the sum of the spectral moments of two individual surfaces [20]. The parameters important in rough contact mechanics, such as the average curvature of summits, Ssc, and the areal summit density, can be computed from the spectral moments [2,21]:

## 2. Analyzed Textures

^{2}and contained 1024 × 1024 points. Spikes and isolated deep and narrow valleys were eliminated by truncation of the height corresponding to material ratios of 0.01–99.99%. Before calculations of parameters, flat surfaces were leveled, while forms of the curved surfaces were removed using the polynomials of the second degree. Digital filtration was not used. We tried to analyze surfaces of various types: isotropic, anisotropic, random, deterministic, or mixed, of symmetric and non-symmetric ordinate distribution, one-process, and two-process. Therefore, surfaces after vapor blasting, polishing, lapping, milling, one- and two-process honing (plateau honing), and vapor blasting followed by lapping were measured. In most cases, the measured surfaces were machined using typical techniques.

## 3. Selection of Surface Texture Parameters

## 4. Methods of Parameters Prediction

_{0}spectral moment, which is the profile variance. The Sdq parameter should be related to m

_{2}moment, which is a square of the profile RMS slope. The Ssc parameter can be related to m

_{4}moment which is the mean square curvature of the surface profile. For surface of Gaussian ordinate distribution, Equation (2) presents the dependence between m

_{4}profile spectral moment and the average curvature of summits. Because the spectral moment of the equivalent rough surface is the sum of spectral moments of both surfaces, these parameters of equivalent surfaces were predicted using the following formulae:

_{2}and m

_{4}spectral moments. Furthermore, this formula was proven only for a selected type of textures.

_{1}is a parameter of the first surface, P

_{2}is a parameter of the second surface, while P

_{sum}is predicted parameter of the sum surface. Equation (12) is an original conception of the authors of this paper.

## 5. Results of Parameters Prediction and Discussion

_{0}spectral moment is the Pq parameter of the profile or the Sq parameter of isotropic surface topography. When anisotropic one-directional surface texture is analyzed (for example, after grinding or milling), its Sq parameter is equal (or similar) to the mean value of the Pq parameter of the profile measured across the lay (main surface wavelength). In the perpendicular direction (along the lay), the mean value of the Pq parameter is smaller than the Sq parameter of areal surface texture. According to Equation (9), the square of the Sq parameter of the sum surface should be the sum of squares of the Sq parameters of both surfaces in the contact. This assumption was confirmed. For 50 sum surfaces, the maximum error was 0.99%, and the average error was 0.31%.

_{2}moment is the Pdq parameter, which is the RMS slope of the profile. Of course, this slope can be obtained using various methods (based on 2, 3, or 6 neighboring points). The method based on 2 points gave typically more correct value than the other methods [46].

_{4}spectral moment and the average summit curvature, was obtained for surfaces of Gaussian ordinate distribution. In addition, the m

_{4}spectral moment is the square of the mean curvature of the whole profile, not only peaks. For surfaces of non-Gaussian ordinate distributions (two-process textures, periodic surfaces), large errors of prediction of average summit curvature of the equivalent sum rough surface using Equation (11) are possible.

## 6. Conclusions

- In this work, we analyzed the relationships among the parameters of two contacted surfaces on parameters of equivalent surface, for which ordinates are sums of ordinates of both surfaces. Surfaces of various types (one- and two-process, isotropic and anisotropic, random or periodic) were studied.
- Selected parameters: Sq, Sdq, Ssc, Sds, Sp/Sz, and Sq/Sa, of sum surfaces were predicted precisely when the parameters of two individual surfaces are known. The other parameters: Sal, Str, Ssk, and Sku, were anticipated with lower accuracy.
- The parameters Sq, Sdq, and Ssc were predicted based on the changes of profile spectral moments during surfaces’ summation. The results revealed that RMS height Sq and RMS slope Sdq were predicted with very high accuracy. The maximum errors were smaller than one percent, and the average deviations were about 0.3%. In most cases, the relative errors of the Ssc parameter prediction were smaller than 8%, while the average errors were near 3%.
- The remaining parameters of equivalent sum surface were predicted on the base of parameters of both surfaces in contact, weighted by the values of the Sq parameter of these structures. The maximum errors of the summit density, Sds, and the Sq/Sa ratio predictions were near 7%, while the average errors were near 3% and 1.5%, respectively. In most cases, relative discrepancies between Sp/Sz parameter values of equivalent rough sum surfaces and predictions were smaller than 10% and the average errors were about 4%.
- During parameters selection, it was found that the pair of parameters, Sp/Sz (the emptiness coefficient) and Sq/Sa, better described the shape of the ordinate distribution of diversified surface textures than the typically applied set, the skewness, Ssk, and the kurtosis, Sku.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Sa (Ra) | arithmetical mean height |

Sal | fastest decay autocorrelation length |

Sds | areal density of summits |

Sdq | root mean square gradient |

Sku (Rku) | kurtosis |

Sp (Rp) | maximum peak height |

Spc | arithmetical mean peak curvature |

Spd | density of peaks |

Spq | plateau root mean square roughness |

Sq (Rq) | root mean square height |

Ssc | average summit curvature |

Ssk (Rsk) | skewness |

Str | texture aspect ratio |

Sv (Rv) | maximum valley depth |

Sz (Rz) | maximum height |

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**Figure 2.**Isometric view (

**a**), ordinate distribution (

**b**), and directionality plot (

**c**) of surface texture after vapor blasting.

**Figure 3.**Isometric view (

**a**), ordinate distribution (

**b**), and directionality plot (

**c**) of surface texture after polishing.

**Figure 4.**Isometric view (

**a**), ordinate distribution (

**b**), and directionality plot (

**c**) of surface texture after milling.

**Figure 5.**Isometric view (

**a**), ordinate distribution (

**b**), and directionality plot (

**c**) of surface texture after plateau honing.

**Figure 10.**Directionality plots of surface C (

**a**), of surface D (

**b**), and of equivalent sum surface (

**c**).

**Figure 12.**Directionality plots of surface E (

**a**), of surface F (

**b**), and of equivalent sum surface (

**c**).

**Figure 14.**Material ratio curves and ordinate distributions of surface G (

**a**), of surface H (

**b**), and of equivalent sum surface (

**c**).

Parameter | Ssk | Sku | Sq/Sa | Sp/Sz |
---|---|---|---|---|

Ssk | 1 | |||

Sku | −0.77 | 1 | ||

Sq/Sa | −0.82 | 0.91 | 1 | |

Sp/Sz | 0.71 | −0.48 | −0.45 | 1 |

Parameter | Average Error, % | Maximum Error, % |
---|---|---|

Sq | 0.31 | 0.99 |

Sdq | 0.27 | 0.97 |

Ssc | 3.1 | 12.1 |

Sds | 3.5 | 7.1 |

Sp/Sz | 3.8 | 12.3 |

Sq/Sa | 1.6 | 7.1 |

Ssk | 26.2 | 62.1 |

Sku | 13.1 | 45.2 |

Str | 16.5 | 89.9 |

Sal | 11.3 | 28.2 |

**Table 3.**Selected parameters of surface A, of surface B, of the sum surface, and of predicted parameters of the sum surface.

Parameter | Surface A | Surface B | Sum Surface | Predicted Sum Surface | Unit |
---|---|---|---|---|---|

Sq | 0.047 | 1.89 | 1.89 | 1.89 | µm |

Sdq | 0.016 | 0.46 | 0.46 | 0.46 | - |

Ssc | 7.28 | 139.5 | 139.5 | 139.7 | 1/mm |

Sds | 6094 | 3593 | 3612 | 3654 | 1/mm^{2} |

Sp/Sz | 0.42 | 0.39 | 0.39 | 0.39 | - |

Sq/Sa | 1.31 | 1.31 | 1.31 | 1.31 | - |

Ssk | −0.3 | −0.65 | −0.64 | −0.64 | - |

Sku | 3.04 | 4.73 | 4.68 | 4.71 | - |

Str | 0.033 | 0.8 | 0.83 | 0.78 | - |

Sal | 0.0078 | 0.013 | 0.014 | 0.013 | mm |

**Table 4.**Selected parameters of surface C, of surface D, of the sum surface, and of predicted parameters of the sum surface.

Parameter | Surface C | Surface D | Sum Surface | Predicted Sum Surface | Unit |
---|---|---|---|---|---|

Sq | 0.7 | 0.79 | 1.05 | 1.05 | µm |

Sdq | 0.13 | 0.16 | 0.21 | 0.21 | - |

Ssc | 41.5 | 57.4 | 73.2 | 70.9 | 1/mm |

Sds | 5199 | 6028 | 5388 | 5639 | 1/mm^{2} |

Sp/Sz | 0.2 | 0.31 | 0.26 | 0.26 | - |

Sq/Sa | 1.391 | 1.41 | 1.34 | 1.4 | - |

Ssk | −2.15 | −1.9 | −1.44 | −2.02 | - |

Sku | 9.51 | 7.68 | 5.71 | 8.54 | - |

Str | 0.028 | 0.019 | 0.29 | 0.023 | - |

Sal | 0.016 | 0.016 | 0.023 | 0.016 | mm |

**Table 5.**Selected parameters of surface E, of surface F, of the sum surface, and of predicted parameters of the sum surface.

Parameter | Surface E | Surface F | Sum Surface | Predicted Sum Surface | Unit |
---|---|---|---|---|---|

Sq | 2.06 | 2.9 | 3.57 | 3.56 | µm |

Sdq | 0.23 | 0.53 | 0.58 | 0.58 | - |

Ssc | 62.4 | 157.9 | 177.5 | 169.8 | 1/mm |

Sds | 3239 | 3056 | 3098 | 3132 | 1/mm^{2} |

Sp/Sz | 0.56 | 0.44 | 0.47 | 0.49 | - |

Sq/Sa | 1.32 | 1.33 | 1.3 | 1.32 | - |

Ssk | −0.32 | −0.44 | −0.3 | −0.39 | - |

Sku | 4.39 | 4.62 | 3.9 | 4.52 | - |

Str | 0.024 | 0.86 | 0.62 | 0.51 | - |

Sal | 0.031 | 0.02 | 0.024 | 0.025 | mm |

**Table 6.**Selected parameters of surface G, of surface H, of the sum surface, and of predicted parameters of the sum surface.

Parameter | Surface G | Surface H | Sum Surface | Predicted Sum Surface | Unit |
---|---|---|---|---|---|

Sq | 1.37 | 1.26 | 1.86 | 1.86 | µm |

Sdq | 0.38 | 0.19 | 0.42 | 0.42 | - |

Ssc | 120.3 | 37.2 | 131.4 | 125.9 | 1/mm |

Sds | 4137 | 4136 | 3978 | 4136 | 1/mm^{2} |

Sp/Sz | 0.48 | 0.143 | 0.36 | 0.32 | - |

Sq/Sa | 1.39 | 1.62 | 1.41 | 1.5 | - |

Ssk | −0.52 | −3.26 | −1.23 | −1.84 | - |

Sku | 5.86 | 16.67 | 6.82 | 11.05 | - |

Str | 0.72 | 0.85 | 0.79 | 0.78 | - |

Sal | 0.0091 | 0.025 | 0.017 | 0.017 | mm |

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Pawlus, P.; Reizer, R.; Zelasko, W.
Prediction of Parameters of Equivalent Sum Rough Surfaces. *Materials* **2020**, *13*, 4898.
https://doi.org/10.3390/ma13214898

**AMA Style**

Pawlus P, Reizer R, Zelasko W.
Prediction of Parameters of Equivalent Sum Rough Surfaces. *Materials*. 2020; 13(21):4898.
https://doi.org/10.3390/ma13214898

**Chicago/Turabian Style**

Pawlus, Pawel, Rafal Reizer, and Wieslaw Zelasko.
2020. "Prediction of Parameters of Equivalent Sum Rough Surfaces" *Materials* 13, no. 21: 4898.
https://doi.org/10.3390/ma13214898