# Efficient Sub-Modeling for Adhesive Wear in Elastic–Plastic Spherical Contacts

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

## 3. FE Model

## 4. Results and Discussion

#### 4.1. Verification of the Improved Sub-Model

#### 4.2. The Effect of the Normal Load ${P}^{*}$

#### 4.3. Wear Characteristics

^{−7}, and the relative difference between the simulation result and the predicted results from Equation (16) was approx. 2.5%.

#### 4.4. Parametric Study for the Material Properties

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$a$ | mesh size |

$c$ | characteristic width of crack |

$d$ | asperity radius |

$E$ | Young’s modulus |

${G}_{F}$ | size-independent fracture energy |

${G}_{f}$ | fracture energy |

$H$ | hardness |

$K$ | Archard’s wear coefficient |

${L}_{c}$ | critical load at yield inception under full stick contact condition |

$L$ | sliding distance for Archard’s wear model |

${l}_{p}$ | length of wear particle |

$P$ | normal load |

${P}^{*}$ | dimensionless normal load, ${P}^{*}=P/{L}_{c}$ |

$R$ | sphere radius |

$r$ | section I radius |

$s$ | sliding distance |

$t$ | section I thickness |

${t}_{p}$ | thickness of wear particle |

${u}_{p}$ | tangential displacement when wear particle is formed |

${u}_{x}$ | tangential displacement |

${V}_{0}$ | volume of original hemisphere, ${V}_{0}=2/3\pi {R}^{3}$ |

${V}_{p}$ | volume of wear particle |

$W$ | Archard’s wear volume |

${W}_{p}$ | wear particle width |

$w$ | wear rate |

$Y$ | yield strength |

$\mathsf{\upsilon}$ | Poisson’s ratio |

$\mathsf{\omega}$ | interference |

${\omega}_{p}$ | interference when the wear particle is formed |

Subscripts | |

$0$ | at normal preloading |

$p$ | wear particle |

$s$ | sub-model |

$1$ | global model |

Superscript | |

$*$ | dimensionless |

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**Figure 1.**Adhesive wear of the spherical contact with a rigid flat (

**a**) schematic and (

**b**) cross section (plane y = 0) of the wear particle, taken from [13].

**Figure 2.**Global model with a mesh design, the improved sub-model assembly and boundary condition: (

**a**) global model with a mesh design, (

**b**) location of the improved sub-model with sub-model parameters, (

**c**) boundary conditions.

**Figure 3.**Evolution of the fracture, from instants ${I}_{A}$ to ${I}_{E}$ for $R=10\left[\mathrm{m}\mathrm{m}\right]$, ${P}^{\mathrm{*}}=100$.

**Figure 4.**Dimensionless tangential force $Q/P$ and dimensionless interference $\omega /{\omega}_{0}$ vs. dimensionless tangential displacement ${u}_{x}/{\omega}_{0}$ for R = 10 [mm], ${P}^{\mathrm{*}}=100$, comparing with Ref. [13].

**Figure 5.**Dimensionless tangential load $Q/P$ and dimensionless interference $\omega /{\omega}_{0}$ vs. dimensionless tangential displacement ${u}_{x}/{\omega}_{0}$ for different normal loads and radii R, (

**a**) R = 5 [mm], (

**b**) R = 10 [mm], (

**c**) R = 50 [mm].

**Figure 6.**Static friction $\mu $ vs. dimensionless normal load ${P}^{\mathrm{*}}$ for the different radii and comparison with Ref. [17].

**Figure 7.**Maximum tangential force vs. normal load for the different radii and different normal loads.

**Figure 8.**Wear particle geometrical characteristics: (

**a**) wear particle length, (

**b**) wear particle thickness, (

**c**) wear particle half width, presented in both dimensional and dimensionless manners, comparing with Ref [15].

**Figure 9.**(

**a**) Wear volume ${V}_{P}$ and dimensionless wear volume ${V}_{p}^{\mathrm{*}}={V}_{p}/{V}_{0}$; (

**b**) wear rate $w={V}_{p}/s$ and dimensionless wear rate ${w}^{\mathrm{*}}={V}_{p}^{\mathrm{*}}/\left(({u}_{p}+{l}_{p})/{\omega}_{0}\right)$.

**Figure 10.**Dimensionless tangential load $Q/P$ and dimensionless interference $\mathsf{\omega}/{\mathsf{\omega}}_{0}$ vs. dimensionless tangential displacement ${u}_{x}/{\mathsf{\omega}}_{0}$ for the different fracture energies.

${\mathit{P}}^{\mathit{*}}$ | ${\mathit{r}}_{1}^{\mathit{*}}({\mathit{r}}_{1}/\mathit{R})$ | ${\mathit{t}}_{1}^{\mathit{*}}({\mathit{t}}_{1}/\mathit{R})$ | ${\mathit{a}}_{1}^{\mathit{*}}({\mathit{a}}_{1}/\mathit{R})$ | ${\mathit{r}}_{\mathit{s}}^{\mathit{*}}({\mathit{r}}_{\mathit{s}}/\mathit{R})$ | ${\mathit{t}}_{\mathit{s}}^{\mathit{*}}({\mathit{t}}_{\mathit{s}}/\mathit{R})$ | ${\mathit{a}}_{\mathit{s}}^{\mathit{*}}({\mathit{a}}_{\mathit{s}}/\mathit{R})$ |
---|---|---|---|---|---|---|

15 | 0.1 | 0.015 | 0.0015 | 0.07 | 0.008 | 5 × 10^{−4} |

20 to 30 | 0.1 | 0.018 | 0.002 | 0.08 | 0.012 | 6 × 10^{−4} |

50 | 0.13 | 0.022 | 0.002 | 0.1 | 0.02 | 7 × 10^{−4} |

75 | 0.2 | 0.05 | 0.003 | 0.14 | 0.026 | 0.001 |

100 | 0.2 | 0.05 | 0.004 | 0.15 | 0.038 | 0.001 |

150 | 0.25 | 0.07 | 0.004 | 0.19 | 0.05 | 0.002 |

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**MDPI and ACS Style**

Li, M.; Xiang, G.; Goltsberg, R.
Efficient Sub-Modeling for Adhesive Wear in Elastic–Plastic Spherical Contacts. *Lubricants* **2023**, *11*, 228.
https://doi.org/10.3390/lubricants11050228

**AMA Style**

Li M, Xiang G, Goltsberg R.
Efficient Sub-Modeling for Adhesive Wear in Elastic–Plastic Spherical Contacts. *Lubricants*. 2023; 11(5):228.
https://doi.org/10.3390/lubricants11050228

**Chicago/Turabian Style**

Li, Minsi, Guo Xiang, and Roman Goltsberg.
2023. "Efficient Sub-Modeling for Adhesive Wear in Elastic–Plastic Spherical Contacts" *Lubricants* 11, no. 5: 228.
https://doi.org/10.3390/lubricants11050228