# Edge Pressures Obtained Using FEM and Half-Space: A Study of Truncated Contact Ellipses

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Finite Element Method

#### 2.2. Semi-Analytical Method

## 3. Model

#### 3.1. FEM Model

#### 3.2. SAM Model

## 4. Results and Discussion

#### 4.1. FEM

#### 4.2. SAM

#### 4.3. Comparison of FEM and SAM

## 5. Conclusions

- Very small undercut angles can be considered uncritical. The contact area is slightly limited, but not yet completely delimited by the edge due to deformations. Only moderate pressure peaks and plastic deformations occur.
- For very large angles, the contact area is sharply limited by the edge. Due to the steep edge, however, the local structural stiffness of the plane is reduced to such an extent that the entire contact area can deform elastically to a relatively high extent. The edge deflects. Therefore, only minor or no pressure peaks and plastic deformations occur. The contact is significantly characterized by elastic deformation. Thus, also very large angles seem to be uncritical.
- Medium angle ranges result in the highest pressure peaks and plastic deformations, as the contact area is significantly limited by the edge, but the edge still has a high structural stiffness. Elastic deflection of the edge is only marginally possible.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

a | contact radius given by Hertzian theory |

B, C, n | Swift isotropic hardening law parameters |

E | Young’s modulus |

f | undercut geometry |

F | applied load |

h | surface separation |

${h}_{0}$ | initial gap |

${h}_{0}^{\prime}$ | initial gap without undercut |

k, l | indices of the surface grid |

p | contact pressure |

${p}_{H}$ | maximum contact pressure given by Hertzian theory |

${p}_{max}$ | maximum contact pressure at the edge |

R | radius of the ball |

u | total surface deformation |

${u}_{el}$ | elastic surface deformation |

${u}_{el,x}$ | tangential elastic surface deformation |

${u}_{el,z}$ | normal elastic surface deformation |

${u}_{pl}$ | plastic surface deformation |

${u}_{pl,x}$ | tangential plastic surface deformation |

${u}_{pl,z}$ | normal plastic surface deformation |

x, y, z | space coordinates |

$\alpha $ | undercut angle |

$\Gamma $ | computational domain |

${\Gamma}_{c}$ | contact area |

${\Gamma}_{el}$ | elastic computational domain (2D) |

${\Gamma}_{pl}$ | plastic computational domain (3D) |

$\Delta $ | mesh size |

$\delta $ | rigid body displacement |

${\u03f5}_{\mathit{eff}}^{{\scriptscriptstyle p}}$ | effective plastic strain |

$\nu $ | Poisson’s ratio |

${\sigma}_{y}$ | yield stress |

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**Figure 1.**Schematic representation of the model studied: contact between rigid ball and elastic–plastic plane containing an undercut and resulting pressure distribution for the untruncated (dashed line) and truncated (solid line) contact.

**Figure 2.**Calculation schema of the elastic-plastic semi-analytical method, based on [11].

**Figure 6.**Pressure distribution (

**a**) and elastic deformation ${u}_{el,z}$ (

**b**) as benchmark comparison between FEM, SAM, and the analytical Hertzian solution for elastic material behavior.

**Figure 7.**Exemplary mesh convergence study for an undercut angle $\alpha ={3}^{\circ}$. Mesh density as well as pressure and plastic deformation at the edge are normalised with respect to the used discretisation.

**Figure 8.**Enlarged section of the mesh near the edge for meshing model 1 (Fem1) for the undercut angle $\alpha ={0}^{\circ}$ (

**a**) and $\alpha ={50}^{\circ}$ (

**b**).

**Figure 9.**Enlarged section of the mesh near the edge for meshing model 2 (Fem2) for the undercut angle $\alpha ={90}^{\circ}$ (

**a**) and $\alpha ={50}^{\circ}$ (

**b**).

**Figure 10.**Two-dimensional elastic computational domain ${\Gamma}_{el}$ and three-dimensional plastic computational domain ${\Gamma}_{pl}$ in the SAM model.

**Figure 11.**Pressure distributions for undercut angles $\alpha $ from ${0}^{\circ}$ to ${30}^{\circ}$ using FEM.

**Figure 12.**Profiles of the plastic deformation ${u}_{pl,z}$ (

**a**) and ${u}_{pl,x}$ (

**b**) for undercut angles $\alpha $ from ${0}^{\circ}$ to ${30}^{\circ}$ using FEM.

**Figure 13.**Profiles of the elastic deformation ${u}_{el,z}$ (

**a**) and ${u}_{el,x}$ (

**b**) for undercut angles $\alpha $ from ${0}^{\circ}$ to ${30}^{\circ}$ using FEM.

**Figure 14.**Profiles of the pressure for undercut angle $\alpha $ from ${30}^{\circ}$ to ${90}^{\circ}$ using FEM.

**Figure 15.**Profiles of the plastic deformation ${u}_{pl,z}$ (

**a**) and ${u}_{pl,x}$ (

**b**) for undercut angle $\alpha $ from ${30}^{\circ}$ to ${90}^{\circ}$ using FEM.

**Figure 16.**Profiles of the elastic deformation ${u}_{el,z}$ (

**a**) and ${u}_{el,x}$ (

**b**) for undercut angle $\alpha $ from ${30}^{\circ}$ to ${90}^{\circ}$ using FEM.

**Figure 17.**Pressure distributions for undercut angles $\alpha $ from ${0}^{\circ}$ to ${20}^{\circ}$ using SAM.

**Figure 18.**Profiles of the plastic deformation ${u}_{pl,z}$ for undercut angles $\alpha $ from ${0}^{\circ}$ to ${20}^{\circ}$ using SAM.

**Figure 19.**Profiles of the elastic deformation ${u}_{el,z}$ for undercut angles $\alpha $ from ${0}^{\circ}$ to ${20}^{\circ}$ using SAM.

**Figure 21.**Plastic deformation ${u}_{pl,z}$ at the edge ($x=0.5a$) plotted over the undercut angle $\alpha $.

**Figure 22.**Elastic deformation ${u}_{el,z}$ at the edge ($x=0.5a$) plotted over the undercut angle $\alpha $.

↓ Half-Plane | ← Undercut Angle $\mathit{\alpha}$ in ${}^{\circ}$ → | Quarter-Space ↓ | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 0.5 | 1 | 1.5 | 2 | 3 | 5 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 |

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**MDPI and ACS Style**

Juettner, M.; Bartz, M.; Tremmel, S.; Correns, M.; Wartzack, S. Edge Pressures Obtained Using FEM and Half-Space: A Study of Truncated Contact Ellipses. *Lubricants* **2022**, *10*, 107.
https://doi.org/10.3390/lubricants10060107

**AMA Style**

Juettner M, Bartz M, Tremmel S, Correns M, Wartzack S. Edge Pressures Obtained Using FEM and Half-Space: A Study of Truncated Contact Ellipses. *Lubricants*. 2022; 10(6):107.
https://doi.org/10.3390/lubricants10060107

**Chicago/Turabian Style**

Juettner, Michael, Marcel Bartz, Stephan Tremmel, Martin Correns, and Sandro Wartzack. 2022. "Edge Pressures Obtained Using FEM and Half-Space: A Study of Truncated Contact Ellipses" *Lubricants* 10, no. 6: 107.
https://doi.org/10.3390/lubricants10060107