# New Geometrical Modelling for 2D Fabric and 2.5D Interlock Composites

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Objective

#### 1.2. State of the Art

## 2. Geometrical Modelling

#### 2.1. 2D Fabric

#### 2.1.1. Plain Weave Fabric [1/1]

#### 2.1.2. Twill [2/1, 3/1]

#### 2.1.3. Sateen/Satin [5-end, 8-end]

#### 2.1.4. Basket Weave [2/2, 4/4]

#### 2.1.5. 2D Fabric Hybrid Composites

#### 2.2. Interlock Composite 2.5D

- 1
- Layer to Layer Angle Interlock
- 2
- Through the Thickness Angle Interlock

#### 2.3. Construction Testing

- -
- Follow up test.
- -
- Pattern test.
- -
- Symmetry test (not obligatory).

#### 2.4. Analytical Modelling

_{i},G

_{i}) and fiber volume fraction (Vf) as shown in Figure 14 as a user interface to the analytical modelling.

_{11}= V

^{f}E

^{f}

_{11}+ V

^{m}E

^{m}

_{22}= E

_{33}= E

^{m}/(1 − √V

^{f}(1 − E

^{m}/(E

^{f}

_{22})))

_{23}= G

^{m}/(1 − √V

^{f}(1 − G

^{m}/(G

^{f}

_{23})))

_{1}

_{2}= G

_{1}

_{3}= G

^{m}/(1 − √V

^{f}(1 − G

^{m}/(G

^{f}

_{22})))

_{23}= V

^{f}ν

^{f}

_{23}+ V

^{m}(2ν

^{m}− ν

_{1}

_{2}(E

_{22}/E

_{11}))

_{1}

_{2}= ν

_{1}

_{3}= ν

^{m}+ ν

^{f}(ν

^{f}

_{1}

_{2}− ν

^{m})

^{f}is the fiber volume fraction, E

^{f}

_{11}is the Young’s elastic modulus of the fiber in principle axis 1, E

^{f}

_{22}is the Young’s elastic modulus of the fiber in principle axis 2, G

^{f}

_{1}

_{2}is the longitudinal shear modulus of the fiber, G

^{f}

_{23}is the transverse shear modulus of the fiber, ν

^{f}

_{1}

_{2}is the primary Poisson’s ratio of the fiber, and E

^{m}, ν

^{m}, and G

^{m}represent the Young’s elastic modulus, Poisson’s ratio, and shear modulus of the matrix, respectively.

^{b}] = [T]

_{k}

^{T}[C]

_{k}[T]

_{k}

^{b}

^{K}ij + (1 − p) C

^{K}mij

^{b}] and [Cm] are the bounder and resin stiffness matrix in the global coordinate system, respectively, p is the percentage in volume of the bounder and resin in the composite structure, and k is the number of element in the whole structure.

_{i}] are the inverse of matrices found in Equation (14).

^{1}

_{i}] are the matrices found in Equation (15).

## 3. Materials

#### 3.1. 2D Fabric

- E-glass/epoxy—I;
- E-glass/epoxy—II;
- T300/epoxy;
- EW220/5284.

#### 3.2. 2.5D Interlock

## 4. Results and Discussion

#### 4.1. 2D Fabric

#### 4.2. 5D Interlock

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 9.**Shapes of the puzzle: Shape 1: the filler element; Shape 2: upper change in direction (horizontal); Shape 3: the reflected interlock; Shape 4: the incident interlock; Shape 5: lower change in direction (horizontal).

Geometry Parameters | E-Glass/Epoxy—I | E-Glass/Epoxy—II | T300/Epoxy | EW220/5284 |
---|---|---|---|---|

V^{f} | 0.42 | 0.25 | 0.44 | 0.55 |

E_{1} (GPa) | 51.5 | 51.1 | 148.8 | 65.1 |

E_{2} (GPa) | 17.5 | 16 | 12.2 | 22.9 |

G_{1}_{2} (GPa) | 5.8 | 5.77 | 4.81 | 8.4 |

ν_{1}_{2} | 0.31 | 0.31 | 0.29 | 0.24 |

E^{m} (GPa) | 3.5 | 3.5 | 3.5 | 3.2 |

G^{m} (GPa) | 1.3 | 1.3 | 1.3 | 1.1 |

ν^{m} (GPa) | 0.35 | 0.35 | 0.35 | 0.42 |

**Table 2.**Mechanical properties of carbon fibers and matrix [58].

Carbon Fibers | Ef_{11} (GPa) | Ef_{22} (GPa) | Gf_{1}_{2} (GPa) | vf_{12} | vf_{23} |
---|---|---|---|---|---|

T300-J | 230 | 15 | 50 | 0.278 | 0.3 |

Resin RTM6 | E^{m} (GPa) | — | — | vm | — |

— | 2.89 | — | — | 0.35 | — |

E_{1} (GPa) [57] | E_{1} (GPa)—Author | Percentage of Error (%) | |
---|---|---|---|

E-glass/epoxy—I | 14.5 | 14.38 | 0.82 |

E-glass/epoxy—II | 60.3 | 60.57 | 0.44 |

T300/epoxy | 58.91 | 59.39 | 0.814 |

EW220/5284 | 19.3 | 19.67 | 1.91 |

**Table 4.**Analytical results of the iso-strain model compared to numerical results for the composites H2 [58].

Effective Elastic Properties | E_{x} (GPa) | E_{y} (GPa) | G_{xy} (GPa) |
---|---|---|---|

Results by 3SHM | 25.93 | 54.82 | 3.22 |

Results by Author | 28.8 | 55.3 | 2.86 |

Percentage of error (%) | 10 | 1 | 1.125 |

**Table 5.**Analytical results of the iso-strain model compared to numerical results for the composites 71 [58].

Effective Elastic Properties | E_{x} (GPa) | E_{y} (GPa) | G_{xy} (GPa) |
---|---|---|---|

Results by 3SHM | 40.7 | 31.21 | 3.14 |

Results by Author | 40.95 | 32.5 | 3.102 |

Percentage of error (%) | 0.614 | 4.13 | 1.21 |

**Table 6.**Analytical results of the iso-strain model compared to numerical results for the composites 69 [58].

Effective Elastic Properties | E_{x} (GPa) | E_{y} (GPa) | G_{xy} (GPa) |
---|---|---|---|

Results by 3SHM | 28.98 | 37.23 | 3.41 |

Results by Author | 28.69 | 39.12 | 3.365 |

Percentage of error (%) | 1 | 5.07 | 1.31 |

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**MDPI and ACS Style**

Kaddaha, M.A.; Younes, R.; Lafon, P.
New Geometrical Modelling for 2D Fabric and 2.5D Interlock Composites. *Textiles* **2022**, *2*, 142-161.
https://doi.org/10.3390/textiles2010008

**AMA Style**

Kaddaha MA, Younes R, Lafon P.
New Geometrical Modelling for 2D Fabric and 2.5D Interlock Composites. *Textiles*. 2022; 2(1):142-161.
https://doi.org/10.3390/textiles2010008

**Chicago/Turabian Style**

Kaddaha, Mohamad Abbas, Rafic Younes, and Pascal Lafon.
2022. "New Geometrical Modelling for 2D Fabric and 2.5D Interlock Composites" *Textiles* 2, no. 1: 142-161.
https://doi.org/10.3390/textiles2010008