# Viscoelastic Effects during Tangential Contact Analyzed by a Novel Finite Element Approach with Embedded Interface Profiles

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Proposed Solution Scheme for the Contact Problem

#### 2.2. MPJR Formulation

#### 2.3. Rheological Model

#### 2.4. Problem Set Up

## 3. Results

#### 3.1. Bulk Stresses

#### 3.2. Interface Tractions

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Model Validation

#### Appendix A.1. Evaluation of Normal Reaction Forces

#### Appendix A.2. Evaluation of Tangential Reaction Forces

## References

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**Figure 1.**Profile discretization and equivalent interface definition. The interface element $\Gamma $ is defined with the lower two nodes that belong to the deformable bulk, and the others placed at a given offset normal to the lower boundary. An abscissa s can be defined along the boundary to map the indenter’s elevation field, which is stored inside the element and it is used to correct the normal gap.

**Figure 2.**Four-nodes, zero-thickness eMbedded Profiles for Joint Roughness (MPJR) interface finite element.

**Figure 5.**Model predictions: bulk stresses during the normal approach, (

**a**,

**b**), and during full sliding, (

**c**,

**d**), all scaled by a reference elastic modulus ${E}_{f,0}=8.147\times {10}^{2}\phantom{\rule{0.166667em}{0ex}}\mathrm{Pa}$.

**Figure 6.**Selected distributions of normal and tangential contact tractions during the different stages of loading.

**Figure 7.**Time evolution of the resultant tangential force ${Q}_{x}$ for different rheological models.

**Table 1.**Rheological parameters for Ethylene Vynil Acetate (EVA), where n is the number of Prony series’ arms.

n | ${\mathit{G}}^{\mathit{\infty}}$ | ${\mathit{\mu}}_{\mathbf{0}}$ | ${\mathit{\mu}}_{\mathit{n}}$ | ${\mathit{\tau}}_{\mathit{n}}$ |
---|---|---|---|---|

[–] | [Pa] | [–] | [–] | [s] |

1 | 568.498 | 0.421 | 0.579 | 0.817 |

2 | 674.606 | 0.306 | 0.398 | 0.212 |

0.296 | 2.458 | |||

3 | 749.386 | 0.254 | 0.310 | 0.102 |

0.226 | 0.545 | |||

0.210 | 4.104 |

v |
---|

$\mathbf{[}\mathbf{m}\mathbf{/}\mathbf{s}\mathbf{]}$ |

$1.000\times {10}^{-03}$ |

$2.154\times {10}^{-03}$ |

$4.642\times {10}^{-03}$ |

$1.000\times {10}^{-02}$ |

$2.154\times {10}^{-02}$ |

$4.642\times {10}^{-02}$ |

$1.000\times {10}^{-01}$ |

$2.154\times {10}^{-01}$ |

$4.642\times {10}^{-01}$ |

$1.000\times {10}^{+00}$ |

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**MDPI and ACS Style**

Bonari, J.; Paggi, M.
Viscoelastic Effects during Tangential Contact Analyzed by a Novel Finite Element Approach with Embedded Interface Profiles. *Lubricants* **2020**, *8*, 107.
https://doi.org/10.3390/lubricants8120107

**AMA Style**

Bonari J, Paggi M.
Viscoelastic Effects during Tangential Contact Analyzed by a Novel Finite Element Approach with Embedded Interface Profiles. *Lubricants*. 2020; 8(12):107.
https://doi.org/10.3390/lubricants8120107

**Chicago/Turabian Style**

Bonari, Jacopo, and Marco Paggi.
2020. "Viscoelastic Effects during Tangential Contact Analyzed by a Novel Finite Element Approach with Embedded Interface Profiles" *Lubricants* 8, no. 12: 107.
https://doi.org/10.3390/lubricants8120107