#
A Response Surface Methodology Approach to Develop a Multiphysics Simulation Model of a Tensile Friction Test^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

_{2}emissions. However, the formability and shape accuracy of HSS decrease with increasing strength. An approach to reduce the springback behavior at cold forming and to improve the formability is press hardening, which allows adjusting the desired properties of the final product in terms of the microstructure, surface condition, and strength. However, to optimize this process, knowledge of the interfacial phenomena and material behavior at high temperatures is required.

## 2. Materials and Methods

_{T}is the tensile force applied to the specimen; F

_{P}is the compressive force between the jaws. The testing device used to apply the pressure on the die and the tensile load on the steel strip is the Instron Structure Testing System Portal 100 kN. This testing machine is equipped to apply vertical compression and horizontal tension forces. Both are driven by pneumatic pistons, capable of applying a maximum force of 100 kN. Before passing through the die, the strip was heated by a custom-shaped inductor powered by an Eldec HFG 50 generator, which has a maximum output of 50 kW. An Optris Quotient pyrometer is used to evaluate the temperature value reached by the samples in the heating phase. The specimens were pulled through a pair of jaws applied to an area where the specimen faced the test pressure. This experimental setup was created in the laboratories of the Fraunhofer IWU (Dresden, Germany) with equipment provided by the Chemnitz University of Technology (Chemnitz, Germany).

- A factorial (or fractional factorial) design in the factors studied; each has two levels;
- A set of center points, experimental runs whose values of each factor are the medians of the values used in the factorial portion. This point is often replicated to improve the precision of the experiment;
- A set of axial points, experimental runs equal to the center points except for one factor, which will assume values below and above the median of the two factorial levels. These allow us to estimate curvature.

^{k}+ 2k + n

_{1}, …, X

_{k}= independent variables, b

_{0}= overall mean response, b

_{1}, …, b

_{k}= regression model coefficients, K = number of independent variables, ϵ = error. Equation (3) is a mathematical model to approximate the observed values of the dependent variables y, that considers the main effects for factor (X

_{1}, …, X

_{k}), their interactions (X

_{1}X

_{2}, X

_{1}X

_{3}, …, X

_{k−1}X

_{k}) and their quadratic components (X

_{1}

^{2}, ……, X

_{k}

^{2}).

_{c}= number of corner points; n

_{s}= number of star points. The main factors that influence the press hardening process, according to the literature, are:

- Metal sheet temperature at which the semi-finished product enters the tool before being formed;
- Sliding speed, namely the relative speed between the die and the metal sheet during the forming phase of the piece;
- Mold pressure; that is, the pressure value with which the metal sheet is formed.

- Metal sheet temperature: 400–720 °C;
- Sliding speed: 10–80 mm/s;
- Pressure: 10–40 MPa.

^{3}= 8 combinations of parameters.

#### FEA Model

## 3. Results

_{1}the strip speed, X

_{2}the strip temperature, and X

_{3}the jaws pressure:

#### FEA Results

- Metal sheet temperature: 550 °C;
- Sliding speed: 45 mm/s;
- Mould pressure: 25 MPa.

_{strip}= 550 °C − 470 °C = 80 °C

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Circumscribed Central Composite Design (CCCD) for 3 factors: Temperature, Speed, and Pressure.

**Figure 3.**Contour Plot of: (

**a**) Friction Coefficient vs. Temperature and Speed for a Hold Value of Pressure of 25 MPa; (

**b**) Friction Coefficient vs. Pressure and Speed for a Hold Value of Temperature of 560 °C.

**Figure 4.**Contour Plot of Friction Coefficient vs. Temperature and Pressure for a Hold Value of Speed of 45 mm/s.

Level − | Level + | |
---|---|---|

Temperature [°C] | 465 | 655 |

Speed [mm/s] | 24 | 66 |

Pressure [MPa] | 16 | 34 |

S | R-sq (${\mathit{R}}^{2}$) | Adjusted R-sq (${\mathit{R}}_{\mathit{a}\mathit{d}\mathit{j}}^{2}$) | Predicted R-sq (${\mathit{R}}_{\mathit{p}\mathit{r}\mathit{e}\mathit{d}}^{2}$) |
---|---|---|---|

0.0212 | 95% | 90% | 79% |

Model Components | p-Value |
---|---|

Speed | 0.000 |

Temperature | 0.000 |

Pressure | 0.001 |

Speed × Speed | 0.088 |

Temperature × Temperature | 0.011 |

Pressure × Pressure | 0.000 |

Speed × Temperature | 0.225 |

Speed × Pressure | 0.027 |

Temperature × Pressure | 0.722 |

S | ${\mathit{R}}^{2}$ | ${\mathit{R}}_{\mathit{a}\mathit{d}\mathit{j}}^{2}$ | ${\mathit{R}}_{\mathit{p}\mathit{r}\mathit{e}\mathit{d}}^{2}$ |
---|---|---|---|

0.0233 | 92% | 88% | 79% |

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**MDPI and ACS Style**

Adamo, L.; Birnbaum, P.; Kräusel, V.; Penta, F.; Lanzotti, A. A Response Surface Methodology Approach to Develop a Multiphysics Simulation Model of a Tensile Friction Test. *Eng. Proc.* **2022**, *26*, 22.
https://doi.org/10.3390/engproc2022026022

**AMA Style**

Adamo L, Birnbaum P, Kräusel V, Penta F, Lanzotti A. A Response Surface Methodology Approach to Develop a Multiphysics Simulation Model of a Tensile Friction Test. *Engineering Proceedings*. 2022; 26(1):22.
https://doi.org/10.3390/engproc2022026022

**Chicago/Turabian Style**

Adamo, Luca, Peter Birnbaum, Verena Kräusel, Francesco Penta, and Antonio Lanzotti. 2022. "A Response Surface Methodology Approach to Develop a Multiphysics Simulation Model of a Tensile Friction Test" *Engineering Proceedings* 26, no. 1: 22.
https://doi.org/10.3390/engproc2022026022