# Analysis, Modeling and Experimental Study of the Normal Contact Stiffness of Rough Surfaces in Grinding

^{*}

## Abstract

**:**

^{2}+ mx + l of the cylindrical asperity model was established. After analyzing the rough surface data of the grinding process, the asperity distribution height was fitted with a Gaussian distribution function, which proved that asperity follows the Gaussian distribution law. The validity of this model was confirmed by the non-dimensional processing of the assumed model and the fitting of six plasticity indices. When the pressure is the same, the normal stiffness increases with the decrease in the roughness value of the joint surface. The experimental stiffness values are basically consistent with the fitting stiffness values of the newly established model, which verifies the reliability and effectiveness of the new model established for the grinding surface. In this paper, a new model for grinding joint surface is established, and an experimental platform is set up to verify the validity of the model.

## 1. Introduction

^{2}+ mx + l cylindrical asperity model. An experimental platform is set up to verify the effectiveness of the new model. In Figure 1, for ease of understanding, the technical route flow chart is presented.

## 2. Hypothetical Surface

^{2}+ mx + l is used to fit the profile data points of the asperity cross-section. The fitting results are shown in Figure 4, and the fitting data points of the profile of the asperity cross-section are basically consistent with the curve of the parabola function. This proved that the distribution of data points of a single asperity cross-section on the rough surface obtained by grinding is close to the parabolic function curve.

^{2}+ mx + l fitting, parabolic function y =cx

^{3}and trigonometric function gsin (x) were used to fit the profile data points of the asperity cross-section. A root mean square error (RMSE) analysis was performed on the fitting results, and the analysis results are shown in Table 1. It can be seen from the table that the root mean square error of fitting function y = nx

^{2}+ mx + l is the lowest, which proves that fitting function y = nx

^{2}+ mx + l is the most suitable for fitting data points of the asperity cross-section.

## 3. Normal Contact Stiffness Theoretical and Analytical Model

#### 3.1. Contact Model Assumptions

#### 3.2. Contact Stiffness of Mechanical Joint Surface

#### 3.3. Dimensionless Processing of the Model

_{n}E and (A

_{n}E)/σ, and all corresponding variables and length parameters in the formula are divided by σ for dimensionless processing.

^{*}= ω/σ is the normal deformation of a single asperity after non-dimensionalization. ω

_{1}

^{*}= ω

_{1}/σ is the normal critical deformation of the asperity after dimensionless transition from the stage of complete elastic deformation to the stage of elastoplastic deformation. ω

_{2}

^{*}= ω

_{2}/σ is the normal critical deformation of the asperity from the elastoplastic deformation stage to the complete plastic deformation stage after dimensionless deformation. z

^{*}= z/σ is the height of a single asperity without dimensionalization; y

^{*}= y/σ is the distance between the mean height plane of the asperity and the mean height plane of the surface after non-dimensionalization. h

^{*}= h/σ is the distance between the average height planes of the upper and lower bound surfaces after being dimensionless.

#### 3.4. Plasticity Index

#### 3.5. Dimensionless Stiffness Is Fitted by Simulation

## 4. Experiment

#### 4.1. Experimental Specimen

_{1}= E

_{2}= 195 GPa; Poisson’s ratio ν

_{1}= ν

_{2}= 0.29; hardness H = 1870 MPa. Figure 9a shows the processing and molding equipment of this experiment specimen. The model is an okamoto ultra-precision static press surface grinding machine (Okamoto, Tokyo, Honshu, Japan), upg series. Figure 9b shows the normal contact stiffness test specimen. The bonding surface is the convex surface obtained by grinding, and the height of the convex surface is 2 mm. The roughness of the joint surface was Sa 0.854 μm, Sa 1.173 μm, Sa 1.391 μm and Sa 1.524 μm, respectively.

#### 4.2. Experimental Apparatus

## 5. Results and Discussion

## 6. Conclusions

^{2}+ mx + l cylindrical asperity model was established. After analysis and research, the following conclusions were reached:

- The rough surface data of grinding were analyzed, and the distribution height of asperity was fitted with the Gaussian distribution function to confirm that the height of asperity obeyed the Gaussian distribution law.
- The hypothesized model was treated without dimensions and fitted with six plasticity indices to confirm the validity of the model.
- The experimental stiffness values are basically consistent with the fitting stiffness values of the new model, which verifies the reliability and effectiveness of the new model established for the grinding surface.
- When the pressure is the same, the normal stiffness increases with the decrease in the roughness value of the joint surface. Roughness affects the contact stiffness between components. In practice, the stability of mechanical equipment can be effectively improved by improving the grinding precision and reducing the roughness of the joint surface.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Greenwood, J.A.; Williamson, J.B.P. Contact of nominally flat surfaces. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci.
**1966**, 295, 300–319. [Google Scholar] [CrossRef] - Chang, W.R.; Etsion, I.; Bogy, D.B. An Elastic-Plastic Model for the Contact of Rough Surfaces. J. Tribol.
**1987**, 109, 257–263. [Google Scholar] [CrossRef] - Kogut, L.; Etsion, I. Elastic-plastic contact analysis of a sphere and a rigid flat. J. Appl. Mech.-T Asme
**2002**, 69, 657–662. [Google Scholar] [CrossRef] - Xie, W.; Liu, C.; Huang, G.; Qin, Z.; Zong, K.; Jiang, D. Trans-scale rough surface contact model based on molecular dynamics method: Simulation, modeling and experimental verification. Eur. J. Mech. A/Solids
**2023**, 100, 105021. [Google Scholar] [CrossRef] - Jamshidi, H.; Tavakoli, E.; Ahmadian, H. Modeling polymer-metal frictional interface using multi-asperity contact theory. Mech. Syst. Signal Process.
**2022**, 164, 108227. [Google Scholar] [CrossRef] - Kang, H.; Li, Z.-M.; Liu, T.; Zhao, G.; Jing, J.; Yuan, W. A novel multiscale model for contact behavior analysis of rough surfaces with the statistical approach. Int. J. Mech. Sci.
**2021**, 212, 106808. [Google Scholar] [CrossRef] - Yuan, B.; Wang, Y.; Sun, W.; Mu, X.; Zhang, C.; Sun, Q. Theoretical and experimental study on interface stiffness measurement of rough surface using improved acoustic model. Mech. Syst. Signal Process.
**2023**, 186, 109839. [Google Scholar] [CrossRef] - Li, W.; Zhan, W.; Huang, P. A physics-based model of a dynamic tangential contact system of lap joints with non-Gaussian rough surfaces based on a new Iwan solution. AIP Adv.
**2020**, 10, 035207. [Google Scholar] [CrossRef] - Majumdar, A.; Bhushan, B. Role of Fractal Geometry in Roughness Characterization and Contact Mechanics of Surfaces. J. Tribol.
**1990**, 112, 205–216. [Google Scholar] [CrossRef] - Xie, Y.; Xiao, Y.; Lv, J.; Zhang, Z.; Zhou, Y.; Xue, Y. Influence of creep on preload relaxation of bolted composite joints: Modeling and numerical simulation. Compos. Struct.
**2020**, 245, 112332. [Google Scholar] [CrossRef] - Shen, F.; Li, Y.-H.; Ke, L.-L. On the size distribution of truncation areas for fractal surfaces. Int. J. Mech. Sci.
**2023**, 237, 107789. [Google Scholar] [CrossRef] - Yu, X.; Sun, Y.; Wu, S. Analytically decoupling of friction coefficient between mixed lubricated fractal surfaces. Int. J. Mech. Sci.
**2023**, 255, 108465. [Google Scholar] [CrossRef] - Chen, H.; Yin, Q.; Dong, G.; Xie, L.; Yin, G. Stiffness model of fixed joint considering self-affinity and elastoplasticity of asperities. Ind. Lubr. Tribol.
**2019**, 72, 128–135. [Google Scholar] [CrossRef] - Chen, J.; Liu, D.; Wang, C.; Zhang, W.; Zhu, L. A fractal contact model of rough surfaces considering detailed multi-scale effects. Tribol. Int.
**2022**, 176, 107920. [Google Scholar] [CrossRef] - Wang, H.; Jia, P.; Wang, L.; Yun, F.; Wang, G.; Liu, M.; Wang, X. Modeling of the Loading–Unloading Contact of Two Cylindrical Rough Surfaces with Friction. Appl. Sci.
**2020**, 10, 742. [Google Scholar] [CrossRef] - Yuan, Y.; Xu, K.; Zhao, K. The Loading–Unloading Model of Contact Between Fractal Rough Surfaces. Int. J. Precis. Eng. Manuf.
**2020**, 21, 1047–1063. [Google Scholar] [CrossRef] - Zhou, C.; Wang, H.; Wang, H.; Hu, B. Three-dimensional asperity model of rough surfaces based on valley–peak ratio of the maximum peak. Meccanica
**2021**, 56, 711–730. [Google Scholar] [CrossRef] - Shen, F.; Li, Y.-H.; Ke, L.-L. A novel fractal contact model based on size distribution law. Int. J. Mech. Sci.
**2023**, 249, 108255. [Google Scholar] [CrossRef] - Yu, Q.; Sun, J.; Ji, Z. Mechanics Analysis of Rough Surface Based on Shoulder-Shoulder Contact. Appl. Sci.
**2021**, 11, 8048. [Google Scholar] [CrossRef] - Zhu, L.; Chen, J.; Zhang, Z.; Hong, J. Normal contact stiffness model considering 3D surface topography and actual contact status. Mech. Sci.
**2021**, 12, 41–50. [Google Scholar] [CrossRef] - Xie, W.; Liu, C.; Huang, G.; Jiang, D. Numerical and Experimental Study on Rod-Fastened Rotor Dynamics Using Semi-Analytical Elastic-Plastic Model. J. Eng. Gas Turbines Power
**2022**, 144, 64501. [Google Scholar] [CrossRef] - Zhang, L.; Wen, J.; Liu, M.; Xing, G. A Revised Continuous Observation Length Model of Rough Contact without Adhesion. Mathematics
**2022**, 10, 3764. [Google Scholar] [CrossRef] - Yu, G.; Mao, H.; Jiang, L.; Liu, W.; Valerii, T. Fractal Contact Mechanics Model for the Rough Surface of a Beveloid Gear with Elliptical Asperities. Appl. Sci.
**2022**, 12, 4071. [Google Scholar] [CrossRef] - Lv, B.; Han, K.; Wang, Y.; Li, X. Analysis and Experimental Verification of the Sealing Performance of PEM Fuel Cell Based on Fractal Theory. Fractal Fract.
**2023**, 7, 401. [Google Scholar] [CrossRef] - Zhang, C.; Yu, W.; Yin, L.; Zeng, Q.; Chen, Z.; Shao, Y. Modeling of normal contact stiffness for surface with machining textures and analysis of its influencing factors. Int. J. Solids Struct.
**2023**, 262–263, 112042. [Google Scholar] [CrossRef] - An, Q.; Suo, S.; Lin, F.; Shi, J. A Novel Micro-Contact Stiffness Model for the Grinding Surfaces of Steel Materials Based on Cosine Curve-Shaped Asperities. Materials
**2019**, 12, 3561. [Google Scholar] [CrossRef] - Li, L.; Wang, J.; Pei, X.; Chu, W.; Cai, A. A modified elastic contact stiffness model considering the deformation of bulk substrate. J. Mech. Sci. Technol.
**2020**, 34, 777–790. [Google Scholar] [CrossRef] - Jiang, K.; Liu, Z.; Zhang, T. Research on contact mechanics of a 3D self-affine surface topography using mesh regeneration technology. In Proceedings of the 2022 2nd International Conference on Algorithms, High Performance Computing and Artificial Intelligence (AHPCAI), Guangzhou, China, 21–23 October 2022; pp. 312–318. [Google Scholar]
- Bai, Y.; An, Q.; Suo, S.; Wang, W.; Jia, X. An Analytical Model for the Normal Contact Stiffness of Mechanical Joint Surfaces Based on Parabolic Cylindrical Asperities. Materials
**2023**, 16, 1883. [Google Scholar] [CrossRef] - Nuri, K.A.; Halling, J. The normal approach between rough flat surfaces contact. Wear
**1975**, 32, 81–93. [Google Scholar] [CrossRef]

**Figure 8.**Stiffness at different plasticity indices (

**a**) 0.5; (

**b**) 0.7; (

**c**) 1.0; (

**d**) 1.5; (

**e**) 2.0; (

**f**) 2.5.

Fitting function | y = nx^{2} + mx + l | y = cx^{3} | y = gsin(x) |

RMSE | 0.016 | 0.018 | 0.117 |

Ψ | β | σ/R |
---|---|---|

0.5 | 0.0302 | 8.75 × 10^{−5} |

0.7 | 0.0339 | 1.60 × 10^{−4} |

1.0 | 0.0414 | 3.02 × 10^{−4} |

1.5 | 0.0476 | 6.57 × 10^{−4} |

2.0 | 0.0541 | 1.144 × 10^{−3} |

2.5 | 0.0601 | 1.77 × 10^{−3} |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bai, Y.; Jia, X.; Guo, F.; Suo, S.
Analysis, Modeling and Experimental Study of the Normal Contact Stiffness of Rough Surfaces in Grinding. *Lubricants* **2023**, *11*, 508.
https://doi.org/10.3390/lubricants11120508

**AMA Style**

Bai Y, Jia X, Guo F, Suo S.
Analysis, Modeling and Experimental Study of the Normal Contact Stiffness of Rough Surfaces in Grinding. *Lubricants*. 2023; 11(12):508.
https://doi.org/10.3390/lubricants11120508

**Chicago/Turabian Style**

Bai, Yuzhu, Xiaohong Jia, Fei Guo, and Shuangfu Suo.
2023. "Analysis, Modeling and Experimental Study of the Normal Contact Stiffness of Rough Surfaces in Grinding" *Lubricants* 11, no. 12: 508.
https://doi.org/10.3390/lubricants11120508