# Multiscale Study of the Effect of Fiber Twist Angle and Interface on the Viscoelasticity of 2D Woven Composites

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## Abstract

**:**

## 1. Introduction

## 2. The Microstructure, Multiscale Framework and RVEs of Woven Composites

#### 2.1. Multiscale Framework and RVEs

#### 2.2. Twist Angle and Coating of Fiber in the Yarn

## 3. Multiscale Viscoelastic Model of Woven Composites

#### 3.1. Viscoelastic Constitutive Model

#### 3.2. Multiscale Homogenization

#### 3.3. Periodic Boundary Conditions of RVEs

## 4. Validation

## 5. Numerical Investigations

#### 5.1. The Effect of Temperature

#### 5.2. The Effect of the Fiber Twist Angle

#### 5.3. The Effect of the Coating Thickness

## 6. Conclusions

- (1)
- The results using the multiscale method show that the fiber array considerably affects stiffness relaxation of the yarn. ${\tilde{C}}_{11}^{RVE1}$, ${\tilde{C}}_{12}^{RVE1}$, and ${\tilde{C}}_{22}^{RVE1}$ of the square array are higher than H-RVE1, while the H-RVE1’s tensors ${\tilde{C}}_{23}^{RVE1}$ and ${\tilde{C}}_{44}^{RVE1}$ are higher than that of S-RVE1. At the same temperature, the relaxation time and variation trend of S-RVE1 and H-RVE1 are almost identical.
- (2)
- The multiscale solutions show that the yarn surface twist angle has significant affection to the viscoelastic properties of the composites. The negative effect of high twist angle on moduli ${E}_{xx}^{RVE2}$ and ${E}_{yy}^{RVE2}$ are more important, while the improvement on other modulus is minor, and gradually disappears with the time. In addition, the lower twist angle has a more significant effect on the axial stiffness of the yarn, while the radial stiffness is more sensitive to the higher angle.
- (3)
- The coating, the material property, and the thickness can effectively improve the overall viscoelasticity of 2D woven composites, especially the in-plane relaxation moduli. When the stiffness of the coating is higher than that of the matrix, the coating will effectively improve the overall mechanical properties of the composite. Designing the coating is significant in exploiting the potentiality of 2D woven composites.
- (4)
- The multiscale method facilitates the calculation of the viscoelasticity of woven composites. Combining with the discrete theory and FEM, this paper provides an appropriate approach for analyzing non-isotropic composites. In addition, the effect of more microscopic parameters on the mesoscopic properties is considered and calculated.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

s | $\mathit{\infty}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|

${\lambda}_{s}(s)$ | - | 10^{2} | 10^{3} | 10^{4} | 10^{5} | 10^{6} | 10^{7} | 10^{8} |

${\tilde{C}}_{11}^{RVE1}$(MPa) | 150,185 | 152,861 | 152,790.3 | 152,719.9 | 152,558.5 | 152,310.8 | 152,068.5 | 151,699.5 |

${\tilde{C}}_{12}^{RVE1}$(MPa) | 1477 | 4049.66 | 4000.2 | 3951.22 | 3835.21 | 3648.91 | 3457.23 | 3149.23 |

${\tilde{C}}_{22}^{RVE1}$(MPa) | 4839.2 | 12,127.8 | 11,991. | 11,849.81 | 11,525.29 | 10,999.1 | 10,451.4 | 9559.33 |

${\tilde{C}}_{23}^{RVE1}$(MPa) | 1392.6 | 4582 | 4519.5 | 4454.74 | 4305.73 | 4063.47 | 3811.25 | 3401.7 |

${\tilde{C}}_{44}^{RVE1}$(MPa) | 1046 | 4634.2 | 4571.99 | 4507.85 | 4360.73 | 4123.69 | 3879.07 | 3485 |

${\tilde{C}}_{55}^{RVE1}$(MPa) | 1581 | 3335.45 | 3285.56 | 3234.27 | 3116.83 | 2928.93 | 2736.78 | 2430.8 |

s | $\mathit{\infty}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|

${\lambda}_{s}(s)$ | - | 10^{2} | 10^{3} | 10^{4} | 10^{5} | 10^{6} | 10^{7} | 10^{8} |

${\tilde{C}}_{11}^{RVE1}$(MPa) | 152,780.2 | 152,708.9 | 152,636.8 | 152,473.6 | 152,222.5 | 151,975.7 | 151,598 | |

${\tilde{C}}_{12}^{RVE1}$(MPa) | 4035.73 | 3984.31 | 3931.25 | 3809.39 | 3612.2 | 3407.64 | 3076.14 | |

${\tilde{C}}_{22}^{RVE1}$(MPa) | 11,860.1 | 11,713.22 | 11,561.45 | 11,212.65 | 10,646.66 | 10,057.74 | 9100.42 | |

${\tilde{C}}_{23}^{RVE1}$(MPa) | 4772.60 | 4714.71 | 4654.69 | 4516.45 | 4290.5 | 4053.69 | 3666.18 | |

${\tilde{C}}_{44}^{RVE1}$(MPa) | 4531.39 | 4466.61 | 4399.91 | 4247.02 | 4001.31 | 3748.72 | 3343.95 | |

${\tilde{C}}_{55}^{RVE1}$(MPa) | 3615.55 | 3573.31 | 3529.74 | 3429.71 | 3267.89 | 3099.82 | 2826.58 |

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**Figure 1.**SEM photographs: (

**a**) Natural fiber fabric [24]; (

**b**) The orientation of sisal fibers in the yarn [28]; (

**c**) The fiber rod end surface [29]; (

**d**) The fiber and coating in the yarn [30]. (Reproduced with permission ref. [24]. Copyright 2018 Elsevier; ref. [28]. Copyright 2016 Elsevier; ref. [29]. Copyright 2020 MDPI; ref. [30]. Copyright 2010 Elsevier).

**Figure 2.**Multiscale framework of the 2D woven composites considering the twisted angle and interface in RVE. The H-RVE is fiber arrayed in hexagonal and the S-RVE is fiber arrayed in square.

**Figure 3.**Definitions of the fiber orientation angles. (

**a**) The twisting angle of fiber from inner periphery to outer periphery in the yarn; (

**b**) The relation between the fiber orientation with the RVE1 coordinate system.

Parameter | Young’s Modulus (GPa) | Shear Modulus (GPa) | Poisson’s Ratio | |||
---|---|---|---|---|---|---|

${\mathit{E}}_{\mathit{f}1}$ | ${\mathit{E}}_{\mathit{f}2}={\mathit{E}}_{\mathit{f}3}$ | ${\mathit{G}}_{\mathit{f}12}={\mathit{G}}_{\mathit{f}13}$ | ${\mathit{G}}_{\mathit{f}23}$ | ${\mathit{\upsilon}}_{\mathit{f}12}={\mathit{\upsilon}}_{\mathit{f}13}$ | ${\mathit{\upsilon}}_{\mathit{f}23}$ | |

Value | 233 | 15 | 8.963 | 5.639 | 0.2 | 0.33 |

$\mathit{i}$ | $\mathit{\infty}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|

${E}_{i}(\mathrm{MPa})$ | 1000 | 224.1 | 450.8 | 406.1 | 392.7 | 810.4 | 203.7 | 1486.0 |

${\rho}_{i}(s)$ | - | 1.0 × 10^{3} | 1.0 × 10^{5} | 1.0 × 10^{6} | 1.0 × 10^{7} | 1.0 × 10^{8} | 1.0 × 10^{9} | 1.0 × 10^{10} |

Parameter | Twist Angle $\mathit{\theta}(\xb0)$ | Young’s Modulus (GPa) | Shear Modulus (GPa) | Poisson’s Ratio | |||
---|---|---|---|---|---|---|---|

${\mathit{E}}_{\mathit{f}1}$ | ${\mathit{E}}_{\mathit{f}2}={\mathit{E}}_{\mathit{f}3}$ | ${\mathit{G}}_{\mathit{f}12}={\mathit{G}}_{\mathit{f}13}$ | ${\mathit{G}}_{\mathit{f}23}$ | ${\mathit{\upsilon}}_{\mathit{f}12}={\mathit{\upsilon}}_{\mathit{f}13}$ | ${\mathit{\upsilon}}_{\mathit{f}23}$ | ||

Value | 0° | 233 | 15 | 8.963 | 5.639 | 0.2 | 0.33 |

30° | 179.9583 | 17.67307 | 22.116 | 6.8 | 0.65 | 0.299 | |

60° | 53.07674 | 37.777 | 33.202 | 17.056 | 0.47 | 0.107 |

Parameter | Young’s Modulus (GPa) | Shear Modulus (GPa) | Poisson’s Ratio | |||
---|---|---|---|---|---|---|

${\mathit{E}}_{1}^{\mathbf{int}}$ | ${\mathit{E}}_{2}^{\mathbf{int}}={\mathit{E}}_{3}^{\mathbf{int}}$ | ${\mathit{G}}_{\mathit{y}12}^{\mathbf{int}}={\mathit{G}}_{\mathit{y}13}^{\mathbf{int}}$ | ${\mathit{G}}_{\mathit{y}23}^{\mathbf{int}}$ | ${\mathit{\upsilon}}_{12}^{\mathbf{int}}={\mathit{\upsilon}}_{13}^{\mathbf{int}}$ | ${\mathit{\upsilon}}_{23}^{\mathbf{int}}$ | |

Pyrolytic carbon | 30 | 12 | 2 | 4.3 | 0.12 | 0.4 |

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## Share and Cite

**MDPI and ACS Style**

Li, B.; Liu, C.; Zhao, X.; Ye, J.; Guo, F. Multiscale Study of the Effect of Fiber Twist Angle and Interface on the Viscoelasticity of 2D Woven Composites. *Materials* **2023**, *16*, 2689.
https://doi.org/10.3390/ma16072689

**AMA Style**

Li B, Liu C, Zhao X, Ye J, Guo F. Multiscale Study of the Effect of Fiber Twist Angle and Interface on the Viscoelasticity of 2D Woven Composites. *Materials*. 2023; 16(7):2689.
https://doi.org/10.3390/ma16072689

**Chicago/Turabian Style**

Li, Beibei, Cheng Liu, Xiaoyu Zhao, Jinrui Ye, and Fei Guo. 2023. "Multiscale Study of the Effect of Fiber Twist Angle and Interface on the Viscoelasticity of 2D Woven Composites" *Materials* 16, no. 7: 2689.
https://doi.org/10.3390/ma16072689