Multi-Gaussian Stratified Modeling and Characterization of Multi-process Surfaces
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Surface serves as the fingerprint of a component. Most researchers understood surface topography from a single-stratum viewpoint, some ones have focused on a bi-Gaussian stratified model that respects formation mechanism, but few studies have considered a multi-Gaussian stratified possibility. A multi-Gaussian stratified model of surface topography is developed based on the existing bi-Gaussian stratified surface theory. The bi-Gaussian characterizing method, on the basis of linear regression, is revised to differentiate the mixed multiple Gaussian components. The model and the method are demonstrated on simulated multi-Gaussian stratified surfaces and real engineering surfaces manufactured by turning, rough lapping, and fine lapping. The results reveal the existence of multi-Gaussian stratified feature on multi-process surfaces in industry, and improve the stratified surface theory.
KeywordsSurface topography Stratified Characterization Simulation
Surface is regarded as the fingerprint of a component  because of its ability in evaluating process quality [2, 3], embodying functional performance [4, 5], and revealing intrinsic mechanism [6, 7]. Therefore, understanding the feature own by a surface is of primary importance before realizing the above capabilities. Researchers found that engineering surfaces can be categorized into two types based on the amplitude feature (Gaussian or non-Gaussian distribution). Most researchers mainly focused on the amplitude feature ranging from a Gaussian to a non-Gaussian distribution, while some ones further explored the stratified property, sometimes, owned by a non-Gaussian distribution. The stratified surface model was established in the characterization of a two-process (i.e., plateau honing operation) cylinder liner that consists of smooth wear-resistant and load-bearing plateau regions with intersecting deep valley regions working as oil reservoirs and debris traps [8, 9, 10]. In fact, such a surface is a large roughness-scale Gaussian surface superimposed with a small roughness-scale Gaussian truncating surface at a certain surface height, as shown in Fig. 1a, thus being independent from a non-Gaussian surface and being termed as a bi-Gaussian stratified surface . With respect to a plateau honing operation, the lower Gaussian component is related to the rough process; the upper Gaussian component is associated with the fine process. Because of the similarity, the bi-Gaussian stratified surface theory was extended to the modeling of worn surfaces [11, 12, 13, 14, 15, 16, 17]. By this, the lower Gaussian component is the unused surface; the upper Gaussian component is produced by a material wear.
To characterize a bi-Gaussian stratified surface, the probability material ratio curve (PMRC) is used to differentiate the components, as shown in Fig. 1b. The PMRC is obtained by scaling the probability coordinate of the material ratio curve (i.e., the Abbott curve) to a Gaussian standard deviation coordinate . The PMRC of a Gaussian surface exhibits as a straight line with its slope being the RMS of surface height and its intercept being the mean plane , achieving the deduction that the PMRC of a bi-Gaussian stratified surface should be bi-linear. A simple segmented separation method based on a segmented linear regression , as shown in Fig. 1b, was firstly proposed to obtain the bi-Gaussian surface parameters including the RMS of the upper component Spq, the mean plane of the upper component zmu, the RMS of the lower component Svq, the mean plane of the lower component zml, and the transition point (Smq, zk). Smq is the probability at the transition point, and zk is the corresponding surface height. Note that Smq in Fig. 1b has a transformation from a probability to a Gaussian standard deviation. Then, the simple segmented separation method was normalized and optimized by ISO 13565-3  by excluding non-linear regions (i.e., outlying upper region, outlying lower region, and the transition region) before the linear regression. Recently, Hu et al.  pointed out the drawback of the above two segmented separation methods in assuming a bi-linear PMRC with a knee point. Therefore, Hu et al.  developed a surface combination theory to obtain the continuous form of a bi-linear PMRC and proposed a continuous separation method. The continuous separation method was demonstrated to perfectly respect the unity-area demand on the probability density function. Furthermore, the continuous separation method was demonstrated to own high efficiency and perfect fluctuation resistance [14, 15]. Based on the bi-Gaussian characterization, researchers tried to assess or guide the manufacturing processes [20, 21, 22, 23]. In addition, bi-Gaussian stratified surface theory has also been applied to mechanism studies on lubrication , asperity contact [6, 24, 25, 26], wear and friction [16, 17], and acoustic emission , embodying its superiority compared with the common entire (or single-stratum) surface viewpoint.
However, researchers have ignored the possibility of a multi-Gaussian (tri-Gaussian or higher) stratified feature. As we know, the bi-Gaussian stratified model was induced by a typical two-process operation, i.e., plateau honing operation; when it extended to a worn surface, the two Gaussian components can be understood as the unused and the wear-generated surfaces, respectively. Yet, in industry, an engineering surface usually undergoes processes more than two. It is interesting to answer a question that a multi-process surface will have a multi-Gaussian stratified feature or will only exhibit the bi-Gaussian stratified feature generated by the last two processes. This issue is also meaningful to a worn surface. Various wear modes (e.g., abrasive wear, adhesive wear, fatigue wear, etc.) will occur simultaneously during a wear procedure. The issue can benefit the individual wear-mode study.
The aim of the present study is to explore the possibility of multi-Gaussian stratified feature on a rough surface that undergoes multiple manufacturing processes. A multi-Gaussian stratified model of surface topography is developed on the basis of the existing bi-Gaussian stratified surface theory. The existing characterizing method of bi-Gaussian feature, on the basis of linear regression, is revised to differentiate the mixed multiple Gaussian components. Then, the newly established model and method are demonstrated on simulated and real surfaces, respectively. Herein, a surface simulation approach based on the selective superposition is performed to generate simulated pure multi-Gaussian stratified surfaces; and test samples are manufactured by turning, rough lapping and fine lapping, successively, to yield real multi-process surfaces.
2 Multi-Gaussian Stratified Model and Its Characterizing Method
2.1 Surface Modeling
Figure 2 illustrates the stratified model of surface topography from bi-Gaussian to multi-Gaussian components. In the existing bi-Gaussian stratified model , S1 is a large roughness-scale Gaussian surface, and S2 is another Gaussian surface with a small roughness-scale. Based on a selective superposition principle, a stratified surface S can be achieved by retaining the smaller surface height of the two components at a node, i.e., S = min (S1, S2). Herein, during the node selection, the surface height of S2 should involve the distance between the two mean planes Δzm1,2. Similarly, a tri-Gaussian stratified model consists of three Gaussian surfaces S1, S2 and S3. The surface heights of S2 and S3 are added by the two mean plane distances Δzm1,2 and Δzm1,3, respectively. A stratified surface S is then achieved by retaining the minimum surface height of the three components at each node. This assumption can be continued to model a multi-Gaussian stratified surface consisting of multiple components S1, S2,…, Sn. In a multi-Gaussian stratified surface, valley regions are produced by the initial Gaussian component with the largest roughness-scale, and plateau regions are the mixture of multiple high-order Gaussian components with different roughness scales.
The above multi-Gaussian stratified model has clearly embodied the difference between the multi-Gaussian stratified and the fractal features [29, 30]. A fractal feature emphasizes the self-affinity of different scales. Although a multi-Gaussian stratified surface also has multiple components owning different roughness scales, it mainly emphasizes the multi-component and the multi-stratum properties. Firstly, a multi-Gaussian stratified feature has multiple Gaussian components caused by different physical mechanisms. Secondly, these components are mixed based on a selective superposition principle . The selective superposition ensures that these components are not directly added at each node but are screened to keep a smaller surface-height value, thus rendering a multi-stratum property.
2.2 Characterizing Method
For a bi-Gaussian stratified surface, as displayed in Fig. 1b, the PMRC exhibits a bi-linear feature. The segmented or the continuous characterizing method can be applied to the PMRC to achieve the component surface parameters (including Spq, zmu, Svq, zml and the transition point which were illustrated in Sect. 1). For a multi-Gaussian stratified surface, the PMRC is multi-linear. However, in the use of the segmented or the continuous characterizing method, it will regard the PMRC as a bi-linear objective, and will differentiate the entire multi-Gaussian stratified surface into two target components. The characterizing method will mainly focus on the component with the largest roughness-scale, regarding it as one target component; the characterizing method will falsely mix high-order components, regarding them as the other target component. To further differentiate the high-order components, the component with the largest roughness-scale can be excluded based on the transition point [14, 15], and the remaining data can be characterized by the characterizing method again to obtain the component with the second largest roughness-scale. By repeating the above steps, each component can be characterized. The above procedure is illustrated in Fig. 3 in which some fundamental multi-Gaussian surface parameters are newly defined. Note that: for the 1-order component, its probability can be calculated by 1 − Smq1; for the 2-order component, its probability can be calculated by Smq1 × (1 − Smq2);…; for the n-order component, its probability can be calculated by Smq1 × Smq2 × … × Smq(n−1).
3 Simulation Research
3.1 Surface Simulation Approach
Pawlus  proposed a surface simulation approach based on the selective superposition to numerically generate a bi-Gaussian stratified surface. In his approach, two Gaussian components are generated based on their own sizes, nodes, autocorrelation function types, correlation lengths, and RMSs; the mean plane distance is determined; a new surface is generated by retaining the smaller surface value at each node; the new surface is translated so that its mean plane is at zero. Hu et al. [13, 14, 24, 27] have successfully used the Pawlus surface simulation approach to reconstruct real worn surfaces, finding that the Pawlus surface simulation approach is close to a real worn situation. Namely, it can generate a stratified feature, and can satisfy the target skewness and kurtosis simultaneously. Furthermore, Hu et al.  also improved the approach on the identification of component correlation lengths to generate flexible autocorrelation functions.
According to the multi-Gaussian stratified model, the Pawlus surface simulation approach can be revised to generate a multi-Gaussian stratified surface. In the present study, tri-Gaussian stratified feature is selected as a representative. The moving average model [31, 32, 33] with a fast Fourier transform solver [34, 35] is used to generate Gaussian components. They worked well in our previous works [13, 14, 24, 27]. The input parameters are listed in Table 1, and the corresponding simulation results are plotted in Fig. 4.
Input parameters of the simulated surfaces
Correlation length (µm)
6040 × 6040
1024 × 1024
Δzm1,2 = 0.5
Δzm1,3 = 0.5 + 0.05
Figure 4a shows a simulated Gaussian surface merely consisting of the first simulated Gaussian component. After mixing the first and the second Gaussian components, a simulated bi-Gaussian stratified surface is generated as plotted in Fig. 4b. Figure 4c shows the simulated tri-Gaussian stratified surface by mixing the three Gaussian components. It is clear that the bi-Gaussian and the tri-Gaussian stratified surfaces have an obvious stratified phenomenon compared with the Gaussian one. Furthermore, Fig. 4c has a smoother phenomenon in some plateau regions than Fig. 4b, indicating that the tri-Gaussian stratified surface owns a smaller roughness in some plateau regions than the bi-Gaussian stratified surface. The smaller roughness is introduced by the coupling of the third Gaussian component. However, this smoother phenomenon is a mixture, and is not obvious enough to directly offer a quantitative analysis. Therefore, a following surface characterization is required.
3.2 Surface Characterization
The PMRCs of the simulated surfaces are calculated, as shown in Fig. 5a. Note that the PMRCs should be transformed by the depth difference at a certain height reference . Namely, the surface heights of the bi-Gaussian and the tri-Gaussian stratified surfaces should be updated by translating their minimum heights to the minimum height of the Gaussian surface. In the present study, the height at 99.87% material ratio (its Gaussian standard deviation is 3) is adopted to exclude the measuring error.
The PMRCs embody linear, bi-linear and tri-linear features, validating the Gaussian, bi-Gaussian stratified and tri-Gaussian stratified features owned by the simulated surfaces. Then, the continuous separating method  is applied to the PMRCs, and the results are listed in Fig. 6. For the bi-Gaussian stratified surface, the resulting Spq, Svq, and the distance are 0.102 µm, 1.04 µm, and 0.466 µm (= 0.690 − 0.224 µm), which are close to the input values of 0.1 µm, 1 µm a,nd 0.5 µm, thus indicating that the continuous separation method has a great characterization. However, for the tri-Gaussian stratified surface, as mentioned in Sect. 2.2, the continuous separation method can only successfully identify the information of the first Gaussian component. The second and the third Gaussian components are regarded as a mixed component whose resulting Spq is 0.0667 µm.
To further differentiate the second and the third Gaussian components of the tri-Gaussian stratified surface, the first Gaussian component needs to be excluded, as mentioned in Sect. 2.2. According to Refs. [14, 15], based on the transition point obtained in the use of the continuous separating method, the bi-Gaussian and the tri-Gaussian stratified surfaces can be divided into two parts. By this, the lower part can be excluded. Then, the PMRCs can be recalculated from the retained part, as shown in Fig. 5b. For the bi-Gaussian stratified surface that excluded the first Gaussian component, its PMRC is not linear. The disturbance on the right hand is induced by the transition of the first and the second Gaussian components. Their transition is a smooth region rather than a knee point, and therefore the first Gaussian component cannot be completely excluded based on the transition point. Herein, the work interval can be adjusted to aviod this disturbance . In the present case, the work interval is set to [− 3, 1] in the second use of the continuous separating method to deal with Fig. 5b. The results are also listed in Fig. 6. The resulting Spq, Svq, and the distance are 0.0078 µm, 0.0342 µm, and − 0.0120 µm (= 0.759 − 0.771 µm). The corresponding input values are 0.01 µm, 0.1 µm, and 0.05 µm. It indicates that although the mixed two (i.e., second and third) Gaussian components are differentiated, the identified RMSs are smaller and the identified distance has an obvious deviation. These errors are caused by the node loss of components during the surface separation [14, 15].
4 Experimental Research
In the simulation research above, pure multi-Gaussian stratified surfaces have been discussed. This experimental section is used to explore the possibility of multi-Gaussian stratified feature owned by a multi-process surface topography. The conclusions drawn from the simulation research can be treated as a reference. Unlike simulated surfaces, the input parameters of real engineering surfaces are unknown, so the characterized parameters cannot be estimated. In addition, an obvious characterization error due to the component node loss has been observed in the simulation research. Therefore, only the PMRC is analyzed in this section.
4.1 Scheme and Facility
Three samples are formed from steel #45, and are named as Sample 1, Sample 2, and Sample 3. The size is illustrated in Fig. 7. Then, one end of each sample, successively, undergoes turning, rough lapping, and fine lapping, as shown in Fig. 8. During the rough lapping, samples are lapped by a cast-iron lapping machine (see Fig. 9) under a 5 kg normal load. Lubricating oil with boron-nitride particles is selected as the lapping lubricant. Samples 1, 2, and 3, respectively, take 1 min, 2 min, and 4 min at a 7.5 r/min revolving speed. During the fine lapping, samples are lapped by a brass lapping machine under a 5 kg normal load for 2 min at a 15 r/min revolving speed. Water with diamond paste W3.5 is adopted as the lapping lubricant. After each process, samples are cleaned in ethyl alcohol by an ultrasonic cleaner for 5 min, and then are measured at the same specified area by a Zygo white light interferometer. The circumferential position of the specified measured area can be ensured by marks, and the inner radius of the sample can be regarded as the reference for the radial position [16, 17, 37]. The ×2.5 objective is used to obtain a 6.04 × 6.04 mm2 area with 1024 × 1024 nodes. Fifty nodes near edges should be excluded to eliminate the edge effect of the measured area.
4.2 Results and Discussion
Figures 10, 11, and 12 are the visualizations of the measured surfaces and the corresponding shaded relief maps. In each sample, it is clear that after the rough lapping, the turned surface has a stratified feature. Then, after the fine lapping, the stratified feature becomes much more obvious. By this, the surface has smoother plateau regions than the surface before the fine lapping. It indicates that the three-process surface owns a smaller roughness in plateau regions than the two-process surface. The remaining issue is to answer a question: the above smaller roughness is a mixture of the last two processes (i.e., rough lapping and fine lapping) or is only the production of the last process (i.e., fine lapping). It equals to answer whether the three-process surface has a tri-Gaussian stratified feature or a bi-Gaussian one. Therefore, the PMRCs of the measured surfaces are calculated, and then plotted in Fig. 13.
For each sample in Fig. 13, after the rough lapping, the PMRC of the turned surface transfers from linear to bi-linear. The lower parts (on the right hand of the sub-images) of the two PMRCs are overlapped, indicating that the valley regions produced by the turning are retained during the rough lapping. Then, after the fine lapping, the non-linear feature of the PMRC becomes much more obvious while still keeps the valley regions (produced by the turning) unaltered.
Then, the multi-Gaussian revised continuous separating method (see Sect. 2.2) is carried out. Namely, the continuous separating method is firstly used to yield the transition points; secondly, based on the resulting transition points, the rough lapped surfaces and the fine lapped surfaces are, respectively, divided into two parts; thirdly, after excluding the lower part, the PMRCs are recalculated within the retained part, as displayed in Fig. 14. Indeed, as similar to the simulated cases in Fig. 5b, there is a disturbance induced by the smooth component transition. However, in the undisturbed region, the bi-linear feature can be observed for Samples 2 and 3. It demonstrates that the tri-Gaussian stratified feature does exist on a three-process surface. Yet, in the undisturbed region of Sample 1, the PMRCs before and after the fine lapping are two lines with different slopes, indicating the feature generated by the fine lapping has totally covered the feature generated by the rough lapping. The fine lapped surface only has a bi-Gaussian stratified feature arising from the turning and the fine lapping processes. It is because that Sample 1 has the shortest period for the rough lapping, and therefore has the smallest ratio for the component caused by the rough lapping. It can be concluded that the occurrence of tri-Gaussian stratified feature depends on the relative time and intensity of processes.
5 Additional Discussion
In the above simulation and experimental analyses, the existence of multi-Gaussian stratified feature has been observed. And the established multi-Gaussian stratified model and its characterizing method have been demonstrated. In this section, another two issues are further emphasized.
Before the characterization of a multi-Gaussian stratified feature, it is necessary to predetermine the number of processes to ensure the physical significance of each component.
One may argue the functional performances owned by higher-order Gaussian components because of their small roughness scales. In fact, before the introduction of the bi-Gaussian stratified model into mechanism studies, researchers treated the surface topography as a single-stratum body, and can, to some extent, reveal the intrinsic mechanism. However, after the introduction, the bi-Gaussian stratified model has adequately embodied its superiority in the investigations on lubrication , asperity contact [6, 24, 25, 26], wear and friction [16, 17], and acoustic emission . Therefore, it is meaningful to research the multi-Gaussian stratified feature. The following work is to gauge the functional performances of higher-order Gaussian components, and to determine the cutoff order. In addition, the full characterization of components is the basis to evaluate or guide manufacturing processes. it again emphasizes the significance of exploring the multi-Gaussian stratified feature.
Surface works as the fingerprint of a component. Most researchers modeled the surface topography from a classical single-stratum viewpoint, and some works have focused on a bi-Gaussian stratified perspective that respects the surface formation mechanism. However, few studies have taken into account the factor of multiple manufacturing processes. A multi-Gaussian stratified model of surface topography is developed on the basis of the existing bi-Gaussian stratified surface theory. The bi-Gaussian characterizing method, on the basis of linear regression, is revised to differentiate the mixed multiple Gaussian components. The newly established model and the method are demonstrated on simulated multi-Gaussian stratified surfaces and real engineering surfaces manufactured by turning, rough lapping, and fine lapping. The results reveal the existence of multi-Gaussian stratified feature on multi-process surfaces in industry.
This work was supported by the China Postdoctoral Science Foundation (Grant No. 2017M621458), the National Natural Science Foundation of China (Grant No. 11572192), the National Natural Science Foundation of China (Grant No. 11632011), and the National Science and Technology Support Plan Project (Grant No. 2015BAA08B02).
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