# Modelling Transitions in Regimes of Lubrication for Rough Surface Contact

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## Abstract

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## 1. Introduction

## 2. Mathematical Approach

#### 2.1. Hydrodynamic Pressure

#### 2.2. Interacting Asperity Pressure

#### 2.3. Frictional Conjunction

#### 2.4. Numerical Method

## 3. Experimental Approach

#### 3.1. Friction Testing

#### 3.2. Lubricant Viscosity-Pressure Correlation

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

A | Apparent contact area (m^{2}) |

a | Hertzian contact radius in the x-direction (m) |

b | Hertzian contact radius in the y-direction (m) |

D | Influence coefficient (m) |

E | Modulus of elasticity (m) |

${E}^{*}$ | Effective modulus of elasticity (m) |

${f}_{b}$ | Boundary friction force (N) |

${f}_{n}$ | Rough surface model statistical function, $n=2$ and $5/2$ (−) |

${f}_{v}$ | Viscous friction force (N) |

${f}_{tot}$ | Total friction force (N) |

$\overline{G}$ | Non-dimensional material parameter (−) |

${g}_{E}$ | Non-dimensional elasticity parameter (−) |

${g}_{V}$ | Non-dimensional viscosity parameter (−) |

H | Non-dimensional elastic lubricant film profile (−) |

h | Elastic lubricant film profile (m) |

${h}_{0}$ | Minimum clearance (m) |

${h}_{s}$ | Local gap along conjunction (m) |

m | Pressure coefficient of boundary shear strength (−) |

${m}^{\prime}$ | Slope of the logarithmic linear relationship between the lubricant (−) |

Dynamic viscosity and temperature (−) | |

${N}^{\prime}$ | Interception of the logarithmic linear relationship between |

The lubricant dynamic viscosity and temperature (−) | |

P | Non-dimensional hydrodynamic pressure (−) |

${p}_{a}$ | Asperity interacting pressure (Pa) |

${p}_{h}$ | Hydrodynamic pressure (Pa) |

${p}_{hz}$ | Maximum Hertzian pressure (Pa) |

R | Pin curvature radius (m) |

T | Temperature (°C) |

t | Time (s) |

U | Non-dimensional average contact surface sliding speed in the x-direction (m/s) |

$\overline{U}$ | Non-dimensional sliding speed parameter (−) |

u | Contact surface sliding speed in the x-direction (m/s) |

${u}_{avg}$ | Average contact surface sliding speed in the x-direction (m/s) |

V | Non-dimensional average contact surface sliding speed in the y-direction (m/s) |

v | Contact surface sliding speed in the y-direction (m/s) |

${v}_{avg}$ | Average contact surface sliding speed in the y-direction (m/s) |

W | Contact load (N) |

${W}_{ref}$ | Reference contact load (N) |

$\overline{W}$ | Non-dimensional load parameter (−) |

X | Non-dimensional coordinate along the x-direction (−) |

$x,{x}^{\prime}$ | Coordinate along the x-direction (m) |

Y | Non-dimensional coordinate along the y-direction (−) |

$y,{y}^{\prime}$ | Coordinate along the y-direction (m) |

$\alpha $ | Lubricant viscosity-pressure coefficient (Pa^{−1}) |

${\beta}_{a}$ | Curvature radius at the asperity peak (m) |

$\gamma $ | Slope of the limiting shear stress-pressure relation (−) |

$\delta $ | Contact elastic deformation (m) |

$\zeta $ | Surface density of asperity peaks (−) |

$\eta $ | Lubricant dynamic viscosity (Pa.s) |

${\eta}_{0}$ | Bulk lubricant dynamic viscosity at $p=0$ (Pa.s) |

$\overline{\eta}$ | Non-dimensional lubricant dynamic viscosity (−) |

$\lambda $ | Separation parameter (−) |

$\nu $ | Poisson’s ratio (−) |

$\rho $ | Lubricant density (kg/m^{3}) |

${\rho}_{0}$ | Bulk lubricant density at $p=0$ (kg/m^{3}) |

$\overline{\rho}$ | Non-dimensional lubricant density (−) |

$\sigma $ | Composite surface roughness (m) |

${\tau}_{b}$ | Boundary shear (Pa) |

${\tau}_{0}$ | Eyring shear stress (Pa) |

${\tau}_{v}$ | Viscous shear (Pa) |

$\mathsf{\Omega}$ | Relaxation factor for pressure convergence loop (−) |

${\mathsf{\Omega}}_{w}$ | Relaxation factor for load balance loop (−) |

## Appendix A

Parameters | Non-Dimensional | Relation |
---|---|---|

x | X | $X=x/b$ |

y | Y | $Y=y/a$ |

$\rho $ | $\overline{\rho}$ | $\overline{\rho}=\rho /{\rho}_{0}$ |

$\eta $ | $\overline{\eta}$ | $\overline{\eta}=\eta /{\eta}_{0}$ |

h | H | $H=hR/{b}^{2}$ |

p | P | $P=p/{p}_{h}$ |

${u}_{avg}$ | U | $U=u/{u}_{avg}$ |

${v}_{avg}$ | V | $V=v/{u}_{avg}$ |

## Appendix B. Finite Difference Scheme

## Appendix C

Parameter | Value | Unit |
---|---|---|

Pin curvature radius, R | 5 | mm |

Wear track radius | 20 | mm |

Young’s modulus (disk) | 210.0 | GPa |

Young’s modulus (pin) | 110.0 | GPa |

Poisson’s ratio (disk) | 0.27 | - |

Poisson’s ratio (pin) | 0.21 | - |

Eyring shear stress | 2 | MPa |

## Appendix D

Lubricant Type | T_{supply} (°C) | $\mathit{\gamma}$ (-) | m (-) |
---|---|---|---|

SAE5W40 | 35 | 0.048 | 0.107 |

SAE10W40 | 30 | 0.043 | 0.106 |

SAE15W40 | 23 | 0.051 | 0.115 |

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**Figure 4.**Rheological properties at different temperatures for commercially-available engine lubricants.

**Figure 5.**Lubrication regime map for the SAE10W40-grade engine lubricant with an applied normal load of 20 N at disk rotating speeds between 60 rpm and 2000 rpm.

**Figure 6.**Simulated tribological properties for a point contact lubricated with SAE10W40-grade engine lubricant at 4 m/s.

**Figure 7.**Contact pressure distribution and lubricant film profile for the point contact lubricated with SAE10W40-grade engine lubricant.

**Figure 8.**Comparison between the simulated and measured coefficient of friction for SAE10W40-grade engine lubricant at different sliding velocities.

**Figure 9.**Viscous and boundary shear properties for the point contact lubricated with SAE10W40-grade engine lubricant.

**Figure 10.**Comparison between the simulated and measured coefficient of friction for selected SAE-grade engine lubricants at different sliding velocities.

**Table 1.**Measured surface roughness parameters for Greenwood and Tripp’s rough surface contact model.

Parameter | Value | Unit |
---|---|---|

Composite surface roughness, $\sigma $ | 0.105 | μm |

$\zeta \beta \sigma $ | 0.4 | - |

$\sigma /\beta $ | 0.055 | - |

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**MDPI and ACS Style**

Chong, W.W.F.; Hamdan, S.H.; Wong, K.J.; Yusup, S.
Modelling Transitions in Regimes of Lubrication for Rough Surface Contact. *Lubricants* **2019**, *7*, 77.
https://doi.org/10.3390/lubricants7090077

**AMA Style**

Chong WWF, Hamdan SH, Wong KJ, Yusup S.
Modelling Transitions in Regimes of Lubrication for Rough Surface Contact. *Lubricants*. 2019; 7(9):77.
https://doi.org/10.3390/lubricants7090077

**Chicago/Turabian Style**

Chong, William Woei Fong, Siti Hartini Hamdan, King Jye Wong, and Suzana Yusup.
2019. "Modelling Transitions in Regimes of Lubrication for Rough Surface Contact" *Lubricants* 7, no. 9: 77.
https://doi.org/10.3390/lubricants7090077