# Modeling the Thermoforming Process of a Complex Geometry Based on a Thermo-Visco-Hyperelastic Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Thermo-Visco-Hyperelastic Material Model

#### 2.1. Hyperelastic Model

- Mooney–Rivlin model [20,21]: this model is highly recommended in previous existing studies for the simulation of material-forming processes. There exist many versions of this model based on the order of level, for instance: two-parameter, three-parameter, etc. The form of the strain energy for the two-parameter model is$$W={C}_{10}({\overline{I}}_{1}-3)+{C}_{01}({\overline{I}}_{2}-3)+\frac{1}{{D}_{1}}{(J-1)}^{2}$$
- Ogden model [20,21]: this model is also one of the most adopted hyperelastic models in the literature. The strain energy function is calculated as follows:$$\sum _{i=1}^{N}\frac{{\mu}_{i}}{{\alpha}_{i}}({\lambda}_{1}^{{\alpha}_{i}}+{\lambda}_{2}^{{\alpha}_{i}}+{\lambda}_{3}^{{\alpha}_{i}}-3)+\sum _{k=1}^{N}\frac{1}{D}{(J-1)}^{2k}$$
- Marlow model [21]: this is a particular model, since it does not require any parameter calibration. Rather, a simple test is enough to ensure that the model follows the given data [22]. The form of the Marlow strain energy potential is$$W={W}_{dev}\left(\overline{{I}_{1}}\right)+{W}_{vol}\left({J}_{el}\right)$$

#### 2.2. Limitations of Hyperelasticity

## 3. Experimental and Numerical Tests

#### 3.1. DMA Tests

- Temperature sweep: A frequency of 1 Hz was used, and a temperature ramp of 1 °C per minute was applied;
- Frequency sweep: Frequencies ranging from 0.1 Hz to 10 Hz were used, with 10 points per decade.

#### 3.2. Tensile Tests

#### 3.3. The Constitutive Model Used in ABAQUS

## 4. Experimental–Numerical Modeling of the Industrial Demonstrator of the Thermoforming Process

## 5. Results and Discussion

#### 5.1. Baseline Simulation

#### 5.2. Temperature Distribution

#### 5.3. Stress Distribution

#### 5.4. Thickness Distribution

## 6. Conclusions

- The type of mesh used in this study does not allow for using automatic selective mesh refinement to improve the deformation of a complex shape.
- The influence of the friction coefficient on the evolution of the thickness was not studied.
- Another important factor that was not studied is the conduction coefficient.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Thermoforming process: (

**a**) heating: the sheet is heated using radiant heat until it reaches the optimal temperature for thermoforming; (

**b**) pre-stretching: the sheet is mechanically stretched before molding, ensuring uniform thickness; (

**c**) mold up: the mold goes up until it touches the film; (

**d**) vacuum on: vacuum pressure is used to remove air from under the sheet and ensure close contact with the mold contours.

**Figure 4.**Experimental tensile curves of HIPS at different temperatures and a velocity of 0.15 mm/s.

**Figure 5.**Experimental tensile curves of HIPS at different velocities and temperature tests of 60 °C.

**Figure 6.**Comparison of experimental and numerical tensile curves at different temperatures above Tg.

**Figure 14.**Distribution of experimental temperature values of the plate obtained through an IR camera.

**Figure 15.**Steps of the numerical simulation. (

**a**): the sheet is mechanically stretched before molding, ensuring uniform thickness. (

**b**): the mold goes up until it touches the film. (

**c**): vacuum pressure is used to remove air from under the sheet and ensure close contact with the mold contours.

**Figure 16.**(

**a**) Distribution of sheet temperature during the pre-stretching step, (

**b**) distribution of sheet temperature during the mold up step, and (

**c**) distribution of sheet temperature during the vacuum on step.

**Figure 17.**(

**a**) Distribution of von Mises stress during the pre-stretching step, (

**b**) distribution of von Mises stress during the mold-up step, and (

**c**) distribution of von Mises stress during the vacuum on the step.

**Figure 19.**(

**a**) Thickness distribution on side 1. (

**b**) Experimental measurement points of the thickness on side 1.

**Figure 20.**(

**a**,

**b**) Thickness distribution on side 2. (

**c**) Experimental measurement points of the thickness on side 2.

**Figure 21.**(

**a**) Thickness distribution on side 3. (

**b**) Experimental measurement points of the thickness on side 3.

Temperature [°C] | Test Specimens | Loading Speeds [mm/s] |
---|---|---|

60 | 3 | 0.015–0.15 |

90 | 3 | 0.015–0.15 |

110 | 3 | 0.15–1.5–15 |

120 | 3 | 0.15–1.5–15 |

130 | 3 | 0.15–1.5–15 |

140 | 3 | 0.15–1.5–15 |

Temperature [°C] | Input Data | Curve Used |
---|---|---|

60 | Experimental test data | Nominal stress–strain curve |

90 | Experimental test data | Nominal stress–strain curve |

110 | Experimental test data | Nominal stress–strain curve |

120 | Experimental test data | Nominal stress–strain curve |

130 | Experimental test data | Nominal stress–strain curve |

140 | Experimental test data | Nominal stress–strain curve |

Term Number | ${\mathit{g}}_{\mathit{i}}$ | ${\mathit{\tau}}_{\mathit{i}}$ [s] |
---|---|---|

1 | $31\times {10}^{-5}$ | $104\times {10}^{4}$ |

2 | $32\times {10}^{-5}$ | $94\times {10}^{4}$ |

3 | $37\times {10}^{-5}$ | ${10}^{4}$ |

4 | ${10}^{-3}$ | ${10}^{3}$ |

5 | $3\times {10}^{-3}$ | 97 |

6 | $18\times {10}^{-3}$ | 9.9 |

7 | $17\times {10}^{-2}$ | $94\times {10}^{-2}$ |

8 | $27\times {10}^{-2}$ | ${10}^{-1}$ |

9 | $14\times {10}^{-2}$ | ${10}^{-2}$ |

10 | $17\times {10}^{-2}$ | ${10}^{-4}$ |

Step | Pressure [Bar] | Displacement [mm] | Duration [s] |
---|---|---|---|

Pre-stretching | 0.01 | - | 0.5 |

Mould up | - | 102 | 2 |

Vacuum on | 1 | - | 2 |

Contact Type | Parts | Properties |
---|---|---|

Mechanical contact | Sheet and mold | Hard contact and friction coefficient (f = 0.3) |

Thermal contact | Sheet and mold | Gap conductance ^{1} |

^{1}The data are given below.

Conductance [$\frac{\mathbf{mW}}{{\mathbf{mm}}^{2}{\xb7}^{\circ}\mathbf{C}}$] | Clearance [mm] |
---|---|

10 | 0 |

5 | 0.005 |

2.5 | 0.01 |

1.66 | 0.015 |

1.25 | 0.02 |

1 | 0.025 |

0 | 0.05 |

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**MDPI and ACS Style**

Ragoubi, A.; Ducloud, G.; Agazzi, A.; Dewailly, P.; Le Goff, R.
Modeling the Thermoforming Process of a Complex Geometry Based on a Thermo-Visco-Hyperelastic Model. *J. Manuf. Mater. Process.* **2024**, *8*, 33.
https://doi.org/10.3390/jmmp8010033

**AMA Style**

Ragoubi A, Ducloud G, Agazzi A, Dewailly P, Le Goff R.
Modeling the Thermoforming Process of a Complex Geometry Based on a Thermo-Visco-Hyperelastic Model. *Journal of Manufacturing and Materials Processing*. 2024; 8(1):33.
https://doi.org/10.3390/jmmp8010033

**Chicago/Turabian Style**

Ragoubi, Ameni, Guillaume Ducloud, Alban Agazzi, Patrick Dewailly, and Ronan Le Goff.
2024. "Modeling the Thermoforming Process of a Complex Geometry Based on a Thermo-Visco-Hyperelastic Model" *Journal of Manufacturing and Materials Processing* 8, no. 1: 33.
https://doi.org/10.3390/jmmp8010033