# Further Validation of a Simple Mathematical Description of Wear and Contact Pressure Evolution in Sliding Contacts

^{*}

## Abstract

**:**

## 1. Introduction

^{®}or similar codes. Obtained trends of the maximum pressure and wear depth evolution are in very good agreement with FE solutions described in the literature [7,13] by the same authors. However, some discrepancies between the two approaches, FE and analytical, were observed at the borders of the contact regions.

## 2. Mathematical Procedure

_{M}is the maximum pressure, p

_{a}the pressure at the border of the contact area, a the contact half-width, as explained in Figure 2. During wear/time the three quantities p

_{M}p

_{a}and a varied continuously.

^{®}for t > 0 or in a discrete form implementing a simple cycle, and the whole procedure required only four simple lines in Matlab

^{®}, in addition to input data.

## 3. Test Cases

#### 3.1. Line Contact

_{w}), vertical displacement (d

_{v}) and modified diameter. We used the worn mass to estimate the wear coefficient, given the density of the worn material ($\rho =1.34$ g/cm

^{3}); thus,

_{v}to the predicted wear depths.

#### 3.2. Frictional Line Contact

#### 3.3. Point Contact

_{M}over the test, which was continuously measured capacitively within a resolution of ±1 μm. Since the wear of the disc was below 200 nm, we could consider it negligible with respect to the pins so we could use of our model.

## 4. Results and Discussion

#### 4.1. Line Contact

#### 4.2. Frictional Line Contact

#### 4.3. Point Contact

^{−9}mm

^{2}N to fit the experimental data. It must be observed that it was not clear whether the experimental measure actually provided the maximum depth or a kind of average value over the contact area. According to the parabolic assumption, the ratio between the average and maximum wear depth was about 0.78; taking this into account, we should rescale all k values.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Definition of the geometry in the x-y plane. Schemes of the test cases: (

**b**) Line contact: pin-on-plate test, (

**c**) Point contact: pin-on-plate/disc test.

**Figure 3.**Comparison of the predicted maximum wear depth (in red) and the experimental vertical displacement (in blue) from Samyin et al. 2008 [16] at three load levels (50, 100, 150 N).

**Figure 4.**Comparison of the maximum wear depth (

**a**) and maximum pressure (

**b**) obtained from the proposed approach and from FE simulations (Mattei et al. 2022) [15].

**Figure 5.**Comparison of the pressure profiles obtained from the proposed procedure and from FE simulations [15] for two travelled distances d = 5 m (left) and d = 15 m (right). First row: FE model with a rather fine mesh; second row: FE model with mesh refined at the borders. (Legend as in Figure 4: red solid line = present study; dashed blue line = FE solution from [15]).

**Figure 6.**Predicted trends of the maximum wear depth (

**a**) and diameter (

**b**) of the worn area (i.e., 2 a) compared to the experimental values (the blue *) at the end of the test 1 for F = 10 N from (Bose et al., 2019) [6].

**Figure 7.**Average pressure obtained with the proposed procedure and identification of a threshold value (17.88 MPa); that in (Bose et al. 2019) [6] is found at cycle 110, (F = 10 N).

**Figure 8.**(

**a**): Comparison of maximum wear depth: experimental data vs. analytical model, Average pressure obtained with the proposed procedure and identification of a threshold value (17.88 MPa); that in [6] is found at cycle 110, (F = 10 N). (

**b**): Maximum (p

_{M}) and minimum (p

_{a}) pressure in the first part of the test for F = 0.8 N.

**Table 1.**Main data for the three experimental test cases [16].

Ref. | r_{0}(mm) | L (mm) | E_{1}(MPa) | ν_{1} | E_{2}(GPa) | ν_{2} | F_{N}(N) | k (mm^{2}/N) | v_{s}(m/s) | d (m) |
---|---|---|---|---|---|---|---|---|---|---|

[16] | 2.5 | 15 | 2480 | 0.4 | 200 | 0.3 | 50, 100, 150 | 10^{−8} | 0.3 | 15,000 |

**Table 2.**Wear results and wear coefficient for three tested conditions [16].

Tests | F_{N} (N) | m_{w} (g) | k (mm^{2}/N) |
---|---|---|---|

Test 1 | 50 | 0.0164 | 1.63·10^{−8} |

Test 2 | 100 | 0.0392 | 1.95·10^{−8} |

Test 3 | 150 | 0.0367 | 2.11·10^{−8} |

**Table 3.**Main data of the test case in [15].

Ref. | r_{0}(mm) | E (GPa) | ν | F (N/mm) | k (mm ^{2}/N) | v_{s}(m/s) | d (m) |
---|---|---|---|---|---|---|---|

[15] | 10 | 200 | 0.3 | 100 | 10^{−8} | 0.01 | 20 |

**Table 4.**Main data of the test case in [6].

Ref. | r_{0}(mm) | E_{p}(GPa) | ν_{p} | E_{d}(GPa) | ν_{d} | F (N) | v_{s}(m/s) | d (m) |
---|---|---|---|---|---|---|---|---|

[6] | 5 | 100 | 0.33 | 200 | 0.3 | 10, 20, 30 | 0.4 | 500 |

**Table 5.**Wear volume and wear coefficient for three tested conditions [6].

Tests | F (N) | V_{w} (mm^{3}) | k (mm^{2}/N) |
---|---|---|---|

Test 1 | 10 | 2.144 | 42.288·10^{−8} |

Test 2 | 20 | 3.67 | 36.7·10^{−8} |

Test 3 | 30 | 5.501 | 36.66·10^{−8} |

**Table 6.**Main data of the test case in [3].

Ref. | r_{0}(mm) | E_{p} = E_{d} (GPa) | ${\mathit{\nu}}_{\mathbf{p}}={\mathit{\nu}}_{\mathbf{d}}$ | F (N) | v_{s}(m/s) | d (m) | k (mm ^{2} N) |
---|---|---|---|---|---|---|---|

[3] | 0.794 | 304 | 0.24 | 0.2, 0.4, 0.8 | 0.4 | 500 | 13.5·10^{−9} |

**Table 7.**Comparison of experimental and estimated wear depths and contact area diameters [6].

Test | Experimental Wear Depth (mm) | Estimated Wear Depth (mm) | Error (%) | Experimental Worn Area Diameter (mm) | Estimated Worn Area Diameter (mm) | Error (%) |
---|---|---|---|---|---|---|

Test 1 | 0.378 | 0.369 | 2.4 | 3.805 | 3.84 | −0.92 |

Test 2 | 0.493 | 0.4834 | 1.9 | 4.329 | 4.397 | −1.57 |

Test 3 | 0.602 | 0.5918 | 1.7 | 4.758 | 4.8654 | −2.25 |

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**MDPI and ACS Style**

Di Puccio, F.; Mattei, L.
Further Validation of a Simple Mathematical Description of Wear and Contact Pressure Evolution in Sliding Contacts. *Lubricants* **2023**, *11*, 230.
https://doi.org/10.3390/lubricants11050230

**AMA Style**

Di Puccio F, Mattei L.
Further Validation of a Simple Mathematical Description of Wear and Contact Pressure Evolution in Sliding Contacts. *Lubricants*. 2023; 11(5):230.
https://doi.org/10.3390/lubricants11050230

**Chicago/Turabian Style**

Di Puccio, Francesca, and Lorenza Mattei.
2023. "Further Validation of a Simple Mathematical Description of Wear and Contact Pressure Evolution in Sliding Contacts" *Lubricants* 11, no. 5: 230.
https://doi.org/10.3390/lubricants11050230