# Meso-Macro Simulations of the Forming of 3D Non-Crimp Woven Fabrics

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## Abstract

**:**

## 1. Introduction

## 2. Macroscopic Analysis of Woven Reinforcements

## 3. Mesoscopic Analysis by Macro-Meso Embedded Approach

## 4. Mesoscopic Analysis Enriched by Local Mesoscopic Simulation

#### 4.1. Local Mesoscopic Simulation

#### 4.2. Comparison with Experimental Results

#### 4.3. Influence of the Number of RVEs

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

_{k}described in Section 2 is defined by the classical invariants (Equation (A1)), and the strain energy density w

_{k}can be expressed in polynomial form (Equation (A2)). In order to take into consideration the curvature of the fibers, an independent bending stiffness of fibers χ is added to calculate the bending moment M

_{bend}by Equation A3. The material parameters ki, D

_{0}, and D

_{1}are characterized by the corresponding mechanical tests (Table A1).

**Table A1.**Material parameters identified in the macroscopic simulations (k

_{i}in MPa; D

_{0}in Nmm; D

_{1}in Nmm

^{2}).

w_{elong1/2} | k_{1} | k_{2} | k_{3} | k_{4} | k_{5} | k_{6} |

8.692 | 4816 | −1.002 × 10^{6} | 8.275 × 10^{7} | −2.419 × 10^{9} | 2.442 × 10^{10} | |

k_{7} | k_{8} | k_{9} | k_{10} | k_{11} | k_{12} | |

8.692 | −505.6 | 1.073 × 10^{5} | −2.687 × 10^{6} | 0 | 0 | |

w_{comp} | k_{1}/k_{7} | k_{2}/k_{8} | k_{3}/k_{9} | k_{4}/k_{10} | k_{5}/k_{11} | k_{6}/k_{12} |

9.69 × 10^{−2} | 0.7009 | 6.426 × 10^{−2} | −1.339 | 0.8777 | 2.855 × 10^{−2} | |

w_{sh} | k_{1}/k_{7} | k_{2}/k_{8} | k_{3}/k_{9} | k_{4}/k_{10} | k_{5}/k_{11} | k_{6}/k_{12} |

0.3441 | −2.019 | 7.416 | −14.01 | 14.01 | −5.444 | |

w_{shT1} | k_{1}/k_{7} | k_{2}/k_{8} | k_{3}/k_{9} | k_{4}/k_{10} | k_{5}/k_{11} | k_{6}/k_{12} |

3.721 × 10^{−2} | 3.844 × 10^{−2} | −0.521 | 1.877 | −2.78 | 1.572 | |

w_{shT2} | k_{1}/k_{7} | k_{2}/k_{8} | k_{3}/k_{9} | k_{4}/k_{10} | k_{5}/k_{11} | k_{6}/k_{12} |

347.3 | −5.674 × 10^{−2} | −1.46 × 10^{−2} | 0.5835 | −1.201 | 0.8084 | |

D_{0} (warp) | D_{1} (warp) | D_{0} (weft) | D_{1} (weft) | |||

3.6 | 117.9 | 3.748 | 123.6 |

## Appendix B

_{iso}, concerning the longitudinal behavior of a yarn, and the potential w

_{trans}, concerning the mechanical behavior in the transverse plane of a yarn (Equation (A4)).

_{A}, Poisson’s ratio ν, and longitudinal shear modulus G

_{A}, respectively. The material parameters used in the mesoscopic simulation are given in Table A2 (details in [29]).

E | E_{A_}Elongation | E_{A_}Compression | $\mathit{\nu}$ | G_{A} |
---|---|---|---|---|

3200 | 45,516 | 100 | 0 | 1600 |

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**Figure 2.**Comparisons between the numerical and experimental results for three-point bending (

**a**) deformed geometry and (

**b**) the deformation of the middle line of the reinforcement.

**Figure 3.**Comparisons between the experimental and numerical results for hemispherical forming. (

**a**,

**b**) In-plane shear angle in the same position. (

**c**) Deformed geometry.

**Figure 5.**Three-point bending. (

**a**) Deformed geometry in experimental result. (

**b**) Elongation of yarns in mesoscopic analysis. (

**c**) Elongation of an extracted yarn.

**Figure 6.**Hemispherical stamping. (

**a**) Deformed geometry in experimental result. (

**b**) Compaction of yarns in mesoscopic analysis.

**Figure 7.**Influence of the enriched local mesoscopic simulation on the elongation of yarns in three-point bending. (

**a**) Macro-meso embedded analysis. (

**b**) Macro-meso embedded method with local mesoscopic simulation analysis. (

**c**) Change in elongation for the same elements.

**Figure 8.**Influence of the enriched local mesoscopic simulation on the elongation of yarns in hemispherical forming. (

**a**) Macro-meso embedded analysis. (

**b**) Macro-meso embedded method with local mesoscopic simulation analysis. (

**c**) Change in elongation for the same elements.

**Figure 9.**Deformed geometry of RVE in the same position obtained by: (

**a**) simulation, (

**b**) experiment, and (

**c**) comparison of the outlines of binder yarns and warp yarns.

**Figure 11.**Comparisons of yarn elongation for ten elements in the same positions between the meso results of one RVE and threes RVEs: (

**a**) with the result of Macro-Meso embedded approach, (

**b**) without the result of Macro-Meso embedded approach.

Size of Model RVE: 4.78 × 4.64 × 3.25 mm ^{3} | Number of Elements (RVE: 39,888 Elements) | Calculation Time | |||
---|---|---|---|---|---|

Macro Simulation | Macro-Meso Embedded Analysis | Meso Local Simulation | |||

Three-point bending | 41 RVEs | 1.6 millions | 6 h | 15 min | 30 min |

Hemispherical forming | 625 RVEs | 25 millions | 1 day | 1 h | 2 h |

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**MDPI and ACS Style**

Wang, J.; Wang, P.; Hamila, N.; Boisse, P. Meso-Macro Simulations of the Forming of 3D Non-Crimp Woven Fabrics. *Textiles* **2022**, *2*, 112-123.
https://doi.org/10.3390/textiles2010006

**AMA Style**

Wang J, Wang P, Hamila N, Boisse P. Meso-Macro Simulations of the Forming of 3D Non-Crimp Woven Fabrics. *Textiles*. 2022; 2(1):112-123.
https://doi.org/10.3390/textiles2010006

**Chicago/Turabian Style**

Wang, Jie, Peng Wang, Nahiene Hamila, and Philippe Boisse. 2022. "Meso-Macro Simulations of the Forming of 3D Non-Crimp Woven Fabrics" *Textiles* 2, no. 1: 112-123.
https://doi.org/10.3390/textiles2010006