# 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

- Eitner, U. Thermomechanics of Photovoltaic Modules. Ph.D. Thesis, Zentrum für Ingenieurwissenschaften der Martin-Luther-Universität Halle-Wittemberg, Halle, Germany, 2011. [Google Scholar]
- Eitner, U.; Kajari-schröder, S.; Köntges, M.; Brendel, R. Non-linear mechanical properties of ethylene-vynil acetate (EVA) and its relevance to thermomechanics photovoltaic modules. In Proceedings of the 25th European Conference on Photovoltaic Solar Energy, Valencia, Spain, 6–10 September 2010; pp. 4366–4368. [Google Scholar]
- Paggi, M.; Kajari-Schröder, S.; Eitner, U. Thermomechanical deformations in photovoltaic laminates. J. Strain Anal. Eng. Des.
**2011**, 46, 772–782. [Google Scholar] [CrossRef][Green Version] - Paggi, M.; Sapora, A. An Accurate Thermoviscoelastic Rheological Model for Ethylene Vinyl Acetate Based on Fractional Calculus. Int. J. Photoenergy
**2015**, 2015. [Google Scholar] [CrossRef] - Gagliardi, M.; Lenarda, P.; Paggi, M. A reaction-diffusion formulation to simulate EVA polymer degradation in environmental and accelerated ageing conditions. Sol. Energy Mater. Sol. Cells
**2017**, 164, 93–106. [Google Scholar] [CrossRef] - Paggi, M.; Reinoso, J. A variational approach with embedded roughness for adhesive contact problems. Mech. Adv. Mater. Struct.
**2020**, 27, 1731–1747. [Google Scholar] [CrossRef][Green Version] - Paggi, M.; Bemporad, A.; Reinoso, J. Computational Methods for Contact Problems with Roughness. In Modeling and Simulation of Tribological Problems in Technology; Paggi, M., Hills, D., Eds.; Springer International Publishing: Cham, Switzerland, 2020; pp. 131–178. [Google Scholar] [CrossRef]
- Bonari, J.; Paggi, M.; Reinoso, J. A framework for the analysis of fully coupled normal and tangential contact problems with complex interfaces.
**2020**. submitted. [Google Scholar] - Wriggers, P. Computational Contact Mechanics; Springer: Berlin/Heidelberg, Germany, 2006. [Google Scholar] [CrossRef]
- Zienkiewicz, O.; Taylor, R. The Finite Element Method for Solid and Structural Mechanics; Butterworth-Heinemann: Oxford, UK, 2000; Volume 2. [Google Scholar]
- Hills, D.; Nowell, D. Mechanics of Fretting Fatigue; Springer-Science + Business Media: Berlin/Heidelberg, Germany, 2020; pp. 20–24. [Google Scholar]
- Jäger, J. A New Principle in Contact Mechanics. J. Tribol.
**1998**, 120, 677–684. [Google Scholar] [CrossRef]

**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