# Multiscale Wear Simulation in Textured, Lubricated Contacts

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Macro Model

_{def}(x,y) is necessary. Therefore, the lubrication gap geometry was drawn in a 3D CAD software and exported as a .stl file. The raycasting method can be used to determine the height field of the lubrication gap, which was implemented in python using the trimesh library [29]. On the bottom surface of the lubrication gap, the velocity u is assigned, while the top surface is stationary. This corresponds to a relative velocity of u. Furthermore, the ambient pressure p

_{0}is set at the vertical outer surfaces of the lubrication gap. The overall setup of the computational model is depicted in Figure 1.

_{cav}), the gap filling factor was set to ϴ = 1 in the cavitation region 0 < ϴ < 1. Thus, according to the implementation of Elrod, a two-phase flow can be modeled with the gap filling factor ϴ.

_{0}was adjusted in an iterative process until ${F}_{Hyd}={{\displaystyle \int}}_{A}^{}{p}_{Hyd}dA$ corresponded to ${F}_{Load}$. The frictional shear stress in the fluid can be calculated by Equation (3) [11].

_{0}decreases and the operating point shifts more and more into the mixed friction regime. Consequently, asperities come into contact and the force equilibrium is extended by the asperity contact force (Figure 1d). In the mixed friction regime, in addition to the hydrodynamic frictional force, the friction due to solid body contact ${F}_{Drag\_Asp}={\mu}_{0}{F}_{Asp}$ has to be considered. The total coefficient of friction in the mixed friction can be determined with Equation (4).

#### 2.2. Micro Model

_{6}for the lower moving surface, which corresponded to the bearing and shaft material in previous works [17,34].

#### 2.3. Wear

_{h}can be determined in dependence of the wear coefficient C, the local asperity contact pressure p

_{Asp}, the sliding velocity v

_{u}and the sliding time Δt (Equation (6)). The change in the macroscopic geometry due to the wear height w

_{h}was considered in the calculation of the height field in Equation (7).

_{Asp}(h,w

_{h}). Thus, in addition to the local lubrication gap height h, the local wear height w

_{h}was used as a coupling variable between the micro and macro models. For further information regarding the wear-dependent micro model, respectively, the multiscale wear simulation, please refer to [21,34]. The load and sliding velocity were defined in order to obtain the same nominal surface pressure and relative velocity as in [34]. In Table 2, the parameters for the calculations in mixed friction and the accompanying wear simulations are summarized.

_{t}. Texture heights of 5, 10, 20, and 50 µm were used.

## 3. Results

#### 3.1. Micro Model

#### 3.2. Hydrodynamic Simulation

_{0}and the hydrodynamic friction force F

_{Fric}were evaluated.

_{t}= 20 µm and 50 µm, the load-bearing capacity decreased significantly. In Figure 5b, it can be seen that for all texturing heights, the hydrodynamic friction force was lower than for the smooth wedge. For texture heights of 5 and 10 µm, the friction was reduced while the lubrication gap clearance remained the same, which corresponded to an improvement in the tribological performance for the considered operating condition in the hydrodynamic lubrication regime.

#### 3.3. Wear Simulation

_{L_Hydro}, solid contact force F

_{L_Solid}, friction coefficient µ, wear volume V

_{W}, minimum lubrication gap height h

_{min}, and the maximum wear height wh

_{max}, depending on the time in mixed friction of the smooth lubrication wedge, were compared with those of the textured variants with different texture heights. Selected textures exhibited beneficial tribological performance in the hydrodynamic friction regime (Figure 5). It was evident for all textured variants that in the initial condition, for the operating point in mixed friction (Table 2), lower hydrodynamic lifting forces and higher solid contact forces were determined, resulting in a higher coefficient of friction. Even the textured surfaces exhibited lower friction coefficients for the considered operating point in the hydrodynamic lubrication regime. In the case of the smooth wedge, the hydrodynamic lifting force decreased successively with increasing time in mixed friction, which was related to the wear in the lubrication gap and the associated change in the lubrication gap geometry toward a parallel gap. For the textured surfaces, an intermediate increase in the hydrodynamic lifting force was evident before it rapidly decreased with increasing wear. Furthermore, it was observed that the textured surfaces showed higher wear values, even if they showed better performance in the hydrodynamic lubrication regime. Neither the maximum wear height nor the wear volume showed an advantageous trend compared to the smooth lubrication gap.

_{film}in the initial state (t = 0). In the initial state, the smallest lubrication gap height h

_{min}was very localized at the outlet end of the lubrication gap. With an increasing time in mixed friction, the wedge became increasingly flattened, whereby the minimum lubrication gap height occurred in larger areas and reached a higher value (compared with Figure 7). This was also evident with the development of the asperity contact pressure. In the initial condition, high asperity contact pressures could be observed at the edge along the clear height h

_{0}. This area was rapidly worn due to the high asperity contact pressures (according to Equation (6)). The wear led to areas with the same lubricating gap heights. As the wedge effect decreased, it can be seen that the hydrodynamic pressure buildup was reduced. The pressure buildup due to the separate texturing can be seen in Figure 8. With increasing wear, the proportion of the pressure buildup of the texturing increased in relation to the pressure buildup of the main lubrication wedge. The plot of the wear height showed that the lubricating wedge mainly was subject to wear and the texture, due to its depth, remained almost unaffected.

## 4. Discussion

_{e}number is below a defined limit, and the texture aspect ratio λ is above a threshold, the Reynolds equation is applicable. In the context of this study, for the considered operating conditions, a maximum Reynolds number of ${R}_{{e}_{max}}=0.076$ and a minimum texture aspect ratio of ${\lambda}_{min}=20$ were obtained, hence the applicability of the Reynolds equation is given.

_{t}decreased. In the initial condition, it can be seen that for larger texture heights, the hydrodynamic force for the operating condition in the mixed friction regime was more reduced. With increasing wear, an intermediate increase in the hydrodynamic lifting force was evident for the textured surfaces before the hydrodynamic force decreased rapidly. The reversal point depends on the texture depth (Figure 10) and ranged between 24 s and 35 s. Up to the reversal point, the hydrodynamic lifting force for the textured surfaces increased, which can be interpreted as a ‘recovery’ of the hydrodynamic performance.

_{w}= 4.4–5.1 mm for all texture heights. This means that up to the time of a half worn lubrication wedge, the hydrodynamic pressure buildup is increased compared to the initial condition. For better clarification, the surface plots at the times of the reversal points for the textured surfaces are shown in Figure 11.

_{inlet}corresponds to the lubrication gap height at the inlet and h

_{outlet}to the lubrication gap height at the outlet). In scope of the present study, it was evident that in the first stadium of the wear, up to the reversal point, the performance improved (F

_{L_Hydro}was increased, µ was reduced). With increasing wear, the relative position of the texture was more at the outlet side and the convergence ratio also decreased. At the reversal point of the characteristic curves, a convergence ratio of approx. K ~1.5 was observed for all texture heights h

_{t}. Up to this point, an intermediate improvement was evident before the tribological performance dropped again. This corresponds to the trend of the friction coefficient in [42].

## 5. Conclusions

- Surface textures show the potential to positively affect the performance of lubricated contacts in the hydrodynamic lubrication regime in terms of reduced friction and increased lifting force.
- For a given texture shape, the texture height h
_{t}has a significant influence on the operating performance. - A specific texture geometry, which has a positive effect on tribological performance in a specific operating point in the hydrodynamic lubrication regime, is not inevitably associated with lower wear values. Hence, it can be inferred that a certain texture geometry only improved the tribological performance within the range of a certain operating point.
- With increasing wear, an intermediate improvement in the tribological performance for the textured surfaces could be observed. From a reversal point, which occurred for all considered textured heights at a maximum wear height of ~1.6 µm, respectively, a convergence ratio of K ~1.5, the tribological performance deteriorated again, which corresponded to the observations regarding the coefficient of friction for different convergence ratios of another author [42].
- In the present study, the ability to trap wear particles in the texturing dimples was not modeled. This mechanism can improve the wear behavior, therefore, the obtained simulation results should be interpreted as relative values. The presented simulation model is intended to provide a better understanding of the contact conditions, particularly the asperity contact pressure, in the lubricated and textured contacts and considering the wear dependent surfaces.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Dimensions and velocity boundary conditions of the lubrication gap, (

**b**) pressure boundary condition, (

**c**) force equilibrium for hydrodynamic lubrication, (

**d**) force equilibrium for mixed lubrication.

**Figure 4.**Results of the wear dependent micro model (

**a**) asperity contact pressure, (

**b**) flow factors, (

**c**) shear factors.

**Figure 5.**Results of the hydrodynamic simulation: (

**a**) relative lubrication gap clearance, (

**b**) relative hydrodynamic friction force.

**Figure 7.**Development of the selected tribosystem parameters as a function of time in mixed friction and the associated wear.

**Figure 8.**Plot of the time dependent lubrication gap height, asperity contact pressure, hydrodynamic pressure, and wear height for a texture height of h

_{t}= 10 µm.

**Figure 9.**Plot of the time dependent lubrication gap height, asperity contact pressure, hydrodynamic pressure, and wear height for the smooth lubrication wedge.

**Figure 11.**Surface plots for the wear dependent lubrication gap height, asperity contact pressure, hydrodynamic pressure, and wear height at the reversal point.

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

Load | F_{Load} = 200 N |

Velocity | u = 5 m/s |

Density | ρ = 819.63 kg/m^{3} |

Dynamic viscosity | η = 0.01021 Pas |

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

Load | F_{load} = 200 N |

Velocity | u = 0.124 m/s |

Density | ρ = 819.63 kg/m^{3} |

Dynamic viscosity | η = 0.01021 Pas |

Wear coefficient | C = 3.72 × 10^{−14} m^{3}/Nm |

Static friction coefficient | µ_{0} = 0.132 |

E-Modulus upper body | E_{upper} = 77,590 N/mm^{2} |

E-Modulus lower body | E_{lower} = 210,000 N/mm^{2} |

Poisson’s ratio upper body | ν_{upper} = 0.3 |

Poisson’s ratio lower body | ν_{lower} = 0.3 |

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**MDPI and ACS Style**

Maier, M.; Pusterhofer, M.; Grün, F.
Multiscale Wear Simulation in Textured, Lubricated Contacts. *Coatings* **2023**, *13*, 697.
https://doi.org/10.3390/coatings13040697

**AMA Style**

Maier M, Pusterhofer M, Grün F.
Multiscale Wear Simulation in Textured, Lubricated Contacts. *Coatings*. 2023; 13(4):697.
https://doi.org/10.3390/coatings13040697

**Chicago/Turabian Style**

Maier, Michael, Michael Pusterhofer, and Florian Grün.
2023. "Multiscale Wear Simulation in Textured, Lubricated Contacts" *Coatings* 13, no. 4: 697.
https://doi.org/10.3390/coatings13040697