# Understanding Friction in Cam–Tappet Contacts—An Application-Oriented Time-Dependent Simulation Approach Considering Surface Asperities and Edge Effects

^{*}

## Abstract

**:**

## 1. Introduction

_{2}emissions could ideally be reduced by a factor of 4.5 [3]. However, this only applies if the electricity used is generated completely from renewable sources. Realistically, this might be possible only in a few countries, and even there, only within the next decades. As climate change is a global challenge, more efficient internal combustion engines (ICE) will be needed in the coming years [4,5]. Since a complete switch to electromobility is not feasible for resource reasons alone, alternative fuels, such as hydrogen, could become relevant in the combustion sector [6].Considering these aspects, the development of more efficient internal combustion engines will remain an important research focus in the upcoming years.

## 2. Materials and Methods

#### 2.1. Load Spectrum of the Cam–Tappet Contact

#### 2.2. Fluid Properites

#### 2.3. Numerical Modelling

#### 2.3.1. Dimensionless Scaling of the Contact Area

#### 2.3.2. Hydrodynamics

#### 2.3.3. Contact Mechanics

#### 2.3.4. Equilibrium of Forces

#### 2.3.5. Film Thickness Equation

#### 2.3.6. Cavitation

#### 2.3.7. Mixed Lubrication

#### 2.3.8. Numerical Implementation

#### 2.4. Target Values of the Simulation

## 3. Results

#### 3.1. Pressure and Lubriant Gap Distribution

#### 3.2. Friction in the Cam–Tappet Contact

#### 3.3. Influence of Surface Routhness on the Tribological Behaviour

## 4. Discussion

## 5. Conclusions

- Effects at the edges of the line contact appear to have an important influence on the tribological behavior. The narrower lubrication gap and the increased pressure at these areas suggest that the edge areas might contribute decisively to increased wear;
- The cam–tappet contact is in the mixed friction region, with the solid contact clearly dominating in the total friction force. The friction forces determined in the simulation agree well with those from experimental bench tests;
- The surface properties of cams and tappets have a considerable effect on the lubricant film structure and thus on the friction and wear behavior of the tribological system. The influence of roughness outweighs many other influencing factors, and thus deserves special attention;
- The selection of the simulation approach and the influencing variables should always be adapted to the aspect of the contact to be considered in order to find an optimal balance between accuracy and computational efficiency. The presented model is particularly suitable for the investigation of geometry adaptations and time-dependent friction force curves over the cycle.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${a}_{c}$ | Carreau parameter |

${b}_{Hertz}$ | Herzian contact half-wide |

$C$ | Elasticity tensor |

${C}_{u}$ | Speed correction factor |

${C}_{F}$ | Force correction factor |

${C}_{R}$ | Radius correction factor |

${E}_{g}$ | Function of the cam edge geometry |

${F}_{n}$ | Contact normal force |

${F}_{R}$ | Friction force |

${F}_{R,solid}$ | Solid friction force |

${F}_{R,fluid}$ | Fluid friction force |

${G}_{c}$ | Critical shear stress |

$h$ | Lubricant gap height |

${h}_{\mathrm{liq}}$ | Gap height with fluid |

${h}_{min}$ | Minimum lubricant gap height |

$H$ | Dimensionless lubricant gap height |

${H}_{0}$ | Dimensionless distance between undeformed bodies |

$L$ | Dimensionless contact length in y-direction |

${n}_{c}$ | Carreau parameter |

$p$ | Hydrodynamic pressure |

${p}_{solid}$ | Solid contact pressure |

${p}_{cav}$ | Cavitation pressure |

${p}_{\mathrm{max}}$ | Maximum pressure |

$P$ | Dimensionless hydrodynamic pressure |

${P}_{solid}$ | Dimensionless solid contact pressure |

${P}_{tot}$ | Dimensionless total contact pressure |

${p}_{Hertz}$ | Hertzian contact pressure |

$R$ | Cam radius |

$S$ | y-axis scaling factor |

$t$ | Time |

$T$ | Dimensionless time |

${u}_{m}$ | Mean entrainment velocity |

$x,y,z$ | Coordinates |

$X,Y,Z$ | Dimensionless coordinates |

${\alpha}_{\eta}$ | Pressure viscosity coefficient |

$\gamma $ | Penalty function |

$\gamma $ | Ratio of the x and y correlation lengths |

$\dot{\gamma}$ | Shear rate |

$\overline{\delta}$ | Dimensionless elastic deformation |

$\epsilon $ | Strain tensor |

$\eta $ | Viscosity |

${\eta}_{0}$ | Base viscosity |

$\overline{\eta}$ | Dimensionless viscosity |

${\eta}_{\infty}$ | Second plateau viscosity |

${\eta}_{liq}$ | Viscosity of the liquid phase |

$\mathsf{\theta}$ | Fractional film content |

$\lambda $ | Lubricant gap height ratio |

$\mu $ | Coefficient of friction |

$\xi $ | Penalty factor |

$\rho $ | Density |

${\rho}_{0}$ | Base density |

${\rho}_{liq}$ | Density of the liquid phase |

$\overline{\rho}$ | Dimensionless density |

$\sigma $ | Stress tensor |

$\tau $ | Shear stress |

$\phi $ | Cam angle |

$\psi $ | Term of the Reynolds equation |

$\mathsf{\Omega}$ | Calculation area |

${\mathsf{\Omega}}_{\mathrm{c}}$ | Central calculation area |

$\nabla $ | Nabla operator |

## References

- Holmberg, K.; Erdemir, A. Influence of Tribology on Global Energy Consumption, Costs and Emissions. Friction
**2017**, 5, 263–284. [Google Scholar] [CrossRef] - Holmberg, K.; Andersson, P.; Erdemir, A. Global Energy Consumption Due to Friction in Passenger Cars. Tribol. Int.
**2012**, 47, 221–234. [Google Scholar] [CrossRef] - Holmberg, K.; Erdemir, A. The Impact of Tribology on Energy Use and CO
_{2}Emission Globally and in Combustion Engine and Electric Cars. Tribol. Int.**2019**, 135, 389–396. [Google Scholar] [CrossRef] - Serrano, J.R.; Novella, R.; Piqueras, P. Why the Development of Internal Combustion Engines Is Still Necessary to Fight against Global Climate Change from the Perspective of Transportation. Appl. Sci.
**2019**, 9, 4597. [Google Scholar] [CrossRef] [Green Version] - Kalghatgi, G. Is It Really the End of Internal Combustion Engines and Petroleum in Transport? Appl. Energy
**2018**, 225, 965–974. [Google Scholar] [CrossRef] - Bae, C.; Kim, J. Alternative Fuels for Internal Combustion Engines. Proc. Combust. Inst.
**2017**, 36, 3389–3413. [Google Scholar] [CrossRef] - Lee, P.; Zhmud, B. Low Friction Powertrains: Current Advances in Lubricants and Coatings. Lubricants
**2021**, 9, 74. [Google Scholar] [CrossRef] - Ciulli, E.; Fazzolari, F.; Pugliese, G. Contact Force Measurements in Cam and Follower Lubricated Contacts. Front. Mech. Eng.
**2020**, 6, 601410. [Google Scholar] [CrossRef] - Van Helden, A.K.; van der Meer, R.J.; van Staaden, J.J.; van Gelderen, E. Dynamic Friction in Cam/Tappet Lubrication. SAE Trans.
**1985**, 94, 224–231. [Google Scholar] - Kano, M. Super Low Friction of DLC Applied to Engine Cam Follower Lubricated with Ester-Containing Oil. Tribol. Int.
**2006**, 39, 1682–1685. [Google Scholar] [CrossRef] - Dobrenizki, L.; Tremmel, S.; Wartzack, S.; Hoffmann, D.C.; Brögelmann, T.; Bobzin, K.; Bagcivan, N.; Musayev, Y.; Hosenfeldt, T. Efficiency Improvement in Automobile Bucket Tappet/Camshaft Contacts by DLC Coatings—Influence of Engine Oil, Temperature and Camshaft Speed. Surf. Coat. Technol.
**2016**, 308, 360–373. [Google Scholar] [CrossRef] - Marian, M.; Weikert, T.; Tremmel, S. On Friction Reduction by Surface Modifications in the TEHL Cam/Tappet-Contact-Experimental and Numerical Studies. Coatings
**2019**, 9, 843. [Google Scholar] [CrossRef] [Green Version] - Mabuchi, Y.; Yamashita, T.; Izumi, H.; Sekikawa, T.; Nishimura, K.; Hirano, S.; Moriguchi, Y. Examination of the Axial Shape of the Automotive Valvetrain Cam for Engine Friction Reduction. Tribol. Trans.
**2017**, 60, 1088–1098. [Google Scholar] [CrossRef] - Ai, X.; Yu, H. A Numerical Analysis for the Transient EHL Process of a Cam-Tappet Pair in I. C. Engine. J. Tribol.
**1989**, 111, 413–417. [Google Scholar] [CrossRef] - Dowson, D.; Taylor, C.M.; Zhu, G. A Transient Elastohydrodynamic Lubrication Analysis of a Cam and Follower. J. Phys. D Appl. Phys.
**1992**, 25, A313–A320. [Google Scholar] [CrossRef] - Messé, S.; Lubrecht, A.A. Transient Elastohydrodynamic Analysis of an Overhead Cam/Tappet Contact. Proc. Inst. Mech. Eng. Part J J. Eng. Tribol.
**2000**, 214, 415–425. [Google Scholar] [CrossRef] - Lubrecht, A.A.; Venner, C.H. Elastohydrodynamic Lubrication of Rough Surfaces. Proc. Inst. Mech. Eng. Part J J. Eng. Tribol.
**1999**, 213, 397–404. [Google Scholar] [CrossRef] - Teodorescu, M.; Taraza, D.; Henein, N.A.; Bryzik, W. Simplified Elasto-Hydrodynamic Friction Model of the Cam-Tappet Contact. SAE Trans.
**2003**, 112, 1271–1282. [Google Scholar] - Dowson, D.; Higginson, G.R. Elasto-Hydrodynamic Lubrication: International Series on Materials Science and Technology; Dowson, D., Higginson, G.R., Eds.; Pergamon Press: Oxford, UK; Elsevier: New York, NY, USA, 1977; ISBN 978-0-08-021302-6. [Google Scholar]
- Wang, J.; Yang, P. A Numerical Analysis for TEHL of Eccentric-Tappet Pair Subjected to Transient Load. J. Tribol.
**2003**, 125, 770–779. [Google Scholar] [CrossRef] - Wang, J.; Venner, C.H.; Lubrecht, A.A. Influence of Surface Waviness on the Thermal Elastohydrodynamic Lubrication of an Eccentric-Tappet Pair. J. Tribol.
**2013**, 135, 021001. [Google Scholar] [CrossRef] - Chong, W.; Teodorescu, M.; Rahnejat, H. Mixed Thermo-Elastohydrodynamic Cam-Tappet Power Loss in Low-Speed Emission Cycles. Int. J. Engine Res.
**2014**, 15, 153–164. [Google Scholar] [CrossRef] [Green Version] - Raisin, J.; Fillot, N.; Vergne, P.; Dureisseix, D.; Lacour, V. Transient Thermal Elastohydrodynamic Modeling of Cam–Follower Systems: Understanding Performance. Tribol. Trans.
**2016**, 59, 720–732. [Google Scholar] [CrossRef] - Habchi, W. Finite Element Modeling of Elastohydrodynamic Lubrication Problems; John Wiley & Sons: Hoboken, NJ, USA, 2018; ISBN 978-1-119-22514-0. [Google Scholar]
- Habchi, W.; Eyheramendy, D.; Vergne, P.; Morales-Espejel, G. A Full-System Approach of the Elastohydrodynamic Line/Point Contact Problem. J. Tribol.
**2008**, 130, 021501. [Google Scholar] [CrossRef] - Wassim, H. Une Approche Éléments Finis avec Couplage fort des Problèmes de Lubrification Élastohydrodynamique: Application aux Fluides de très Faible Viscosité. Ph.D. Thesis, L’Institut National des Sciences Appliquées de Lyon, Lyon, France, 2008. Available online: http://docinsa.insa-lyon.fr/these/2008/habchi/these.pdf (accessed on 25 October 2021).
- Wu, W.; Wang, J.; Venner, C.H. Thermal Elastohydrodynamic Lubrication of an Optimized Cam–Tappet Pair in Smooth Contact. J. Tribol.
**2016**, 138, 021501. [Google Scholar] [CrossRef] - Tsuha, N.A.H.; Nonato, F.; Cavalca, K.L. Formulation of a Reduced Order Model for the Stiffness on Elastohydrodynamic Line Contacts Applied to Cam-Follower Mechanism. Mech. Mach. Theory
**2017**, 113, 22–39. [Google Scholar] [CrossRef] - Shirzadegan, M.; Almqvist, A.; Larsson, R. Fully Coupled EHL Model for Simulation of Finite Length Line Cam-Roller Follower Contacts. Tribol. Int.
**2016**, 103, 584–598. [Google Scholar] [CrossRef] [Green Version] - Yu, C.; Meng, X.; Xie, Y. Numerical Simulation of the Effects of Coating on Thermal Elastohydrodynamic Lubrication in Cam/Tappet Contact. Proc. Inst. Mech. Eng. Part J J. Eng. Tribol.
**2017**, 231, 221–239. [Google Scholar] [CrossRef] - Meng, X.; Yu, C.; Xie, Y.; Mei, B. Thermal Insulation Effect on EHL of Coated Cam/Tappet Contact during Start Up. Ind. Lubr. Tribol.
**2018**, 70, 917–926. [Google Scholar] [CrossRef] - Lyu, B.; Meng, X.; Zhang, R.; Cui, Y. A Comprehensive Numerical Study on Friction Reduction and Wear Resistance by Surface Coating on Cam/Tappet Pairs under Different Conditions. Coatings
**2020**, 10, 485. [Google Scholar] [CrossRef] - Marian, M.; Tremmel, S.; Wartzack, S. Microtextured Surfaces in Higher Loaded Rolling-Sliding EHL Line-Contacts. Tribol. Int.
**2018**, 127, 420–432. [Google Scholar] [CrossRef] - Torabi, A.; Akbarzadeh, S.; Salimpour, M.; Khonsari, M.M. On the Running-in Behavior of Cam-Follower Mechanism. Tribol. Int.
**2018**, 118, 301–313. [Google Scholar] [CrossRef] - Tang, H.; Wang, J.; Sun, N.; Zhu, J. Effect of Angular Speed of Cam on Oil Film Variation in the Line Contact Thermal EHL of a Cam-Tappet Pair. Ind. Lubr. Tribol.
**2020**, 72, 713–722. [Google Scholar] [CrossRef] - Marian, M.; Orgeldinger, C.; Rothammer, B.; Nečas, D.; Vrbka, M.; Křupka, I.; Hartl, M.; Wimmer, M.A.; Tremmel, S.; Wartzack, S. Towards the Understanding of Lubrication Mechanisms in Total Knee Replacements—Part II: Numerical Modeling. Tribol. Int.
**2021**, 156, 106809. [Google Scholar] [CrossRef] - Marian, M. Numerische Auslegung von Oberflächenmikrotexturen für geschmierte tribologische Kontakte; Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU): Erlangen, Germany, 2021. [Google Scholar]
- Weschta, M. Untersuchungen zur Wirkungsweise von Mikrotexturen in elastohydrodynamischen Gleit/Wälz-Kontakten. Ph.D. Thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Erlangen, Germany, 2017. [Google Scholar]
- Carreau, P.J. Rheological Equations from Molecular Network Theories. Trans. Soc. Rheol.
**1972**, 16, 99–127. [Google Scholar] [CrossRef] - Bair, S. A Rough Shear-Thinning Correction for EHD Film Thickness. Tribol. Trans.
**2004**, 47, 361–365. [Google Scholar] [CrossRef] - Roelands, C.J.A. Correlational Aspects of the Viscosity-Temperature-Pressure Relationship of Lubricating Oils. Ph.D. Thesis, TU Delft, Delft, The Netherlands, 1966. [Google Scholar]
- Lohner, T.; Ziegltrum, A.; Stemplinger, J.-P.; Stahl, K. Engineering Software Solution for Thermal Elastohydrodynamic Lubrication Using Multiphysics Software. Adv. Tribol.
**2016**, 2016, 6507203. [Google Scholar] [CrossRef] [Green Version] - Hertz, H. Ueber die Berührung fester elastischer Körper. Journal für die reine und angewandte Mathematik
**1882**, 1882, 156–171. [Google Scholar] [CrossRef] - Tan, X.; Goodyer, C.E.; Jimack, P.K.; Taylor, R.I.; Walkley, M.A. Computational Approaches for Modelling Elastohydrodynamic Lubrication Using Multiphysics Software. Proc. Inst. Mech. Eng. Part J J. Eng. Tribol.
**2012**, 226, 463–480. [Google Scholar] [CrossRef] - Winkler, A.; Marian, M.; Tremmel, S.; Wartzack, S. Numerical Modeling of Wear in a Thrust Roller Bearing under Mixed Elastohydrodynamic Lubrication. Lubricants
**2020**, 8, 58. [Google Scholar] [CrossRef] - Reynolds, O. On the Theory of Lubrication and Its Application to Mr. Beauchamp Tower’s Experiments, Including an Experimental Determination of the Viscosity of Olive Oil. Philos. Trans. R. Soc. Lond.
**1886**, 177, 157–234. [Google Scholar] [CrossRef] - Marian, M.; Weschta, M.; Tremmel, S.; Wartzack, S. Simulation of Microtextured Surfaces in Starved EHL Contacts Using Commercial FE Software. Matls. Perf. Charact.
**2017**, 6, 165–181. [Google Scholar] [CrossRef] - Zhao, Y.; Maietta, D.M.; Chang, L. An Asperity Microcontact Model Incorporating the Transition from Elastic Deformation to Fully Plastic Flow. J. Tribol.
**2000**, 122, 86–93. [Google Scholar] [CrossRef] - Marian, M.; Grützmacher, P.; Rosenkranz, A.; Tremmel, S.; Mücklich, F.; Wartzack, S. Designing Surface Textures for EHL Point-Contacts—Transient 3D Simulations, Meta-Modeling and Experimental Validation. Tribol. Int.
**2019**, 137, 152–163. [Google Scholar] [CrossRef] - Masjedi, M.; Khonsari, M.M. Film Thickness and Asperity Load Formulas for Line-Contact Elastohydrodynamic Lubrication With Provision for Surface Roughness. J. Tribol.
**2012**, 134, 011503. [Google Scholar] [CrossRef] - Patir, N.; Cheng, H.S. An Average Flow Model for Determining Effects of Three-Dimensional Roughness on Partial Hydrodynamic Lubrication. J. Lubr. Technol.
**1978**, 100, 12–17. [Google Scholar] [CrossRef] - Patir, N.; Cheng, H. Application of Average Flow Model to Lubrication between Rough Sliding Surfaces. J. Lubr. Technol.
**1979**, 101, 220–229. [Google Scholar] [CrossRef] - The Finite Element Method for Fluid Dynamics, 7th ed.; Zienkiewicz, O.C.; Taylor, R.L.; Nithiarasu, P. (Eds.) Butterworth-Heinemann: Oxford, UK, 2014; ISBN 978-1-85617-635-4. [Google Scholar]
- Tribologie-Handbuch; Czichos, H.; Habig, K.-H. (Eds.) Springer Fachmedien Wiesbaden: Wiesbaden, Germany, 2015; ISBN 978-3-8348-1810-2. [Google Scholar]
- Habchi, W.; Vergne, P.; Bair, S.; Andersson, O.; Eyheramendy, D.; Morales-Espejel, G.E. Influence of Pressure and Temperature Dependence of Thermal Properties of a Lubricant on the Behaviour of Circular TEHD Contacts. Tribol. Int.
**2010**, 43, 1842–1850. [Google Scholar] [CrossRef] - Liu, H.C.; Zhang, B.B.; Bader, N.; Venner, C.H.; Poll, G. Simplified Traction Prediction for Highly Loaded Rolling/Sliding EHL Contacts. Tribol. Int.
**2020**, 148, 106335. [Google Scholar] [CrossRef] - Björling, M.; Habchi, W.; Bair, S.; Larsson, R.; Marklund, P. Towards the True Prediction of EHL Friction. Tribol. Int.
**2013**, 66, 19–26. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Simplified structure of the cam–tappet contact (

**a**) and the corresponding sections of the cam (

**b**).

**Figure 2.**Load collective of the cam–tappet contact at different camshaft speeds. Mean entrainment speed ${u}_{m}$ (

**a**) and contact normal force ${F}_{n}$ (

**b**) are shown over cam cycle from pre-cam (approx. ±90°…±55°) via rising flank (approx. ±55°…±40°) to cam tip (approx. ±40°…0°).

**Figure 3.**Scaled FE calculation area $\mathsf{\Omega}$ and representation of the mesh in the central contact area ${\mathsf{\Omega}}_{\mathrm{c}}$.

**Figure 4.**Solid-state contact pressure curve (

**a**) and flux factors (

**b**) depending on the height ratio $\lambda =\frac{h}{\sigma}$.

**Figure 5.**Lubricant gap $h$ (

**a**,

**b**) and total pressure p (

**c**,

**d**) for 500 rpm (

**a**,

**c**) and 2000 rpm (

**b**,

**d**) camshaft speed at pre-cam, rising flank and cam tip. In the center of the contact area, uniform distributions typical for line contacts were formed; at the edges, these deviated due to the edge geometry. The color scales of the enlargements are adjusted to show the edge effects. It should be noted that the length of the simulation area was extended to $10{b}_{Hertz}$ for the 2000 rpm load case.

**Figure 6.**Maximum pressure ${p}_{max}$ (

**a**) and minimum lubricant gap ${h}_{min}$ (

**b**) over one cam cycle for different load cases. The pressure curves were similar for all load cases, and the minimum lubricant film height increased considerably with increasing speed.

**Figure 7.**Fluid (

**a**) and solid (

**b**) friction forces over one cam cycle for different load cases using a combined surface roughness of $\sigma =0.1\mathsf{\mu}\mathrm{m}$. Solid-state friction was dominant in all load cases and occurred primarily in cam tip contact. It decreased with increasing speed. Fluid friction occurred mainly in the area of the rising flank. The solid friction was additionally evaluated for a lower coefficient of friction. Note different axis scaling.

**Figure 8.**Fluid (

**a**) and solid (

**b**) friction forces dependent on the surface roughness at 2000 rpm. With increasing roughness, the solid friction increased considerably, whereas the proportion of fluid friction decreased. Note different axis scaling.

Base density ${\rho}_{0}$ | $805\frac{\mathrm{k}\mathrm{g}}{{\mathrm{m}}^{3}}$ |

Base viscosity ${\eta}_{0}$ | $0.03\mathrm{P}\mathrm{a}\cdot \mathrm{s}$ |

Pressure viscosity coefficient ${\alpha}_{\eta}$ | $1.31\cdot {10}^{-8}\mathrm{P}{\mathrm{a}}^{-1}$ |

Critical shear stress ${G}_{c}$ | $6\mathrm{M}\mathrm{P}\mathrm{a}$ |

Second plateau viscosity ${\eta}_{\infty}$ | $0.2{\eta}_{0}$ |

Carreau parameter ${a}_{c}$ | 2.2 |

Carreau parameter ${n}_{c}$ | 0.8 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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**

Orgeldinger, C.; Tremmel, S.
Understanding Friction in Cam–Tappet Contacts—An Application-Oriented Time-Dependent Simulation Approach Considering Surface Asperities and Edge Effects. *Lubricants* **2021**, *9*, 106.
https://doi.org/10.3390/lubricants9110106

**AMA Style**

Orgeldinger C, Tremmel S.
Understanding Friction in Cam–Tappet Contacts—An Application-Oriented Time-Dependent Simulation Approach Considering Surface Asperities and Edge Effects. *Lubricants*. 2021; 9(11):106.
https://doi.org/10.3390/lubricants9110106

**Chicago/Turabian Style**

Orgeldinger, Christian, and Stephan Tremmel.
2021. "Understanding Friction in Cam–Tappet Contacts—An Application-Oriented Time-Dependent Simulation Approach Considering Surface Asperities and Edge Effects" *Lubricants* 9, no. 11: 106.
https://doi.org/10.3390/lubricants9110106