# Mechanical Homogenization of Transversely Isotropic CNT/GNP Reinforced Biocomposite for Wind Turbine Blades: Numerical and Analytical Study

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## Abstract

**:**

## 1. Introduction

## 2. First Homogenization

#### 2.1. Mathematical Model

#### 2.1.1. CNT Inclusion

_{1}, x

_{2}, x

_{3}), where the x

_{3}-axis lies along its longitudinal direction and the (x

_{1}, x

_{2}) plane coincides with its transversely isotropic plane. The carbon nanotube CNT is considered as a rolled graphene sheet and is only strong in its length direction as seen in Figure 2a. Comparably, a single-layered graphene sheet GNP is strong along all its in-plane directions, as seen in Figure 2b, and offers a large surface/volume ratio [26]. When the nanofillers are perfectly randomly distributed in the matrix, the overall elastic property of the reinforced composite is isotropic as an average effect of orientation.

#### 2.1.2. GNP Inclusion

#### 2.2. Numerical Model

#### 2.2.1. RVE Geometry

- -
- The CNT inclusions are considered to have an ellipsoidal (prolate) shape. GNP inclusions are considered as disc-shaped spheroidal inclusions with aligned and random orientations.
- -
- The phases are considered as perfectly bounded.

#### 2.2.2. Materials Properties

^{3}and 10

^{−4}as aspect ratios for CNT and GNP, respectively.

#### 2.3. First Homogenization Results

## 3. Second Homogenization

#### 3.1. Mathematical Model

#### 3.1.1. Isotropic Fiber

- a.
- Chamis approach

- b.
- Hashin–Rosen model

- c.
- Halpin–Tsai model

- ζ = 2 L/d for calculation of the longitudinal modulus.
- ζ = 2 for calculation of the transversal modulus.

#### 3.1.2. Transversely Isotropic Fiber

- a.
- Hahn model

- b.
- Halpin-Tsai model

#### 3.2. Numerical Model

#### 3.3. Second Homogenization Results

## 4. Effect of Aspect Ratio on the Mechanical Properties

^{−4}.

## 5. Effect of Agglomeration on the Mechanical Properties

#### 5.1. Agglomeration of CNTs

#### 5.1.1. Method Comparison

^{4}, and the agglomeration to have a spherical shape. The volume fraction of CNTs was assumed to be equal to 1%.

#### 5.1.2. Results and Discussions

#### 5.2. Agglomeration of GNPs

#### Results and Discussions

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Conflicts of Interest

## References

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**Figure 3.**RVE of random CNT and GNP nanofillers: (

**a**) CNT cylindrical inclusion; (

**b**) CNT ellipsoidal inclusion; (

**c**) GNP platelet inclusion; and (

**d**) GNP ellipsoidal inclusion.

**Figure 4.**Effective elastic moduli of a matrix reinforced by randomly oriented nanofillers, a comparison between Digimat MF and Mori–Tanaka scheme: (

**a**) effective Young’s modulus${E}_{m}^{*}$; and (

**b**) effective Poisson’s ratio ${\nu}_{m}^{*}$.

**Figure 5.**Comparison of the effective elastic moduli of a matrix reinforced by randomly oriented nanofillers, by using numerical method with Digimat MF: (

**a**) effective Young’s modulus${E}_{m}^{*}$; and (

**b**) effective Poisson’s ratio ${\nu}_{m}^{*}$.

**Figure 7.**The transversely elastic moduli of CNT-PP matrix reinforced by alfa-aligned fiber, using numerical and analytical methods: (

**a**) ${E}_{11}$ axial Young’s modulus; (

**b**) ${E}_{22}$ in-plane Young’s modulus; (

**c**) ${\mathsf{\nu}}_{23}$ in-plane Poisson’s ratio; and (

**d**) ${\nu}_{12}$ axial Poisson’s ratio.

**Figure 8.**The transversely elastic moduli of CNT-PP matrix reinforced by carbon-(T-300) aligned fiber, using numerical and analytical methods: (

**a**) ${E}_{11}$ axial Young’s modulus; (

**b**) ${E}_{22}$ in-plane Young’s modulus; (

**c**) ${\mathsf{\nu}}_{23}$ in-plane Poisson’s ratio; and (

**d**) ${\nu}_{12}$ axial Poisson’s ratio.

**Figure 9.**Comparison of the transversely elastic moduli of an effective matrix reinforced by aligned fibers, by using the numerical method with Digimat MF: (

**a**) ${E}_{11}$ axial Young’s modulus; (

**b**) ${E}_{22}$ in-plane Young’s modulus; (

**c**) ${\mathsf{\nu}}_{23}$ in-plane Poisson’s ratio; and (

**d**) ${\nu}_{12}$ axial Poisson’s ratio.

**Figure 10.**The effect of CNT/GNP aspect ratios on the effective elastic moduli of matrix using Digimat MF: (

**a**) ${E}_{m}^{*}$ Young’s modulus for CNT-PP and ${\nu}_{m}^{*}$ Poisson’s ratio for CNT-PP; (

**b**) ${E}_{m}^{*}$ Young’s modulus for GNP-PP, and ${\nu}_{m}^{*}$ Poisson’s ratio for GNP-PP.

**Figure 11.**Agglomeration model of CNTs for the hybrid matrix: (

**a**) partial agglomeration; (

**b**) null agglomeration; and (

**c**) complete agglomeration.

**Figure 12.**Effective elastic moduli ${E}_{m}^{*}$ of a matrix reinforced by agglomerated randomly oriented nanofillers, a comparison between Digimat MF and Mori–Tanaka scheme.

**Figure 13.**Effective elastic moduli of composite reinforced by randomly CNT inclusion in PP matrix, in function agglomeration parameter $\mu $ at small volume fraction (1–4%): (

**a**) effective Young’s modulus${E}_{m}^{*}$; and (

**b**) effective Poisson’s ratio ${\nu}_{m}^{*}$.

**Figure 14.**Effective elastic moduli of composite reinforced by random CNT inclusion in PP matrix, in function of η for $\mu $ = 0.2 with a maximum of Young’s modulus ${E}_{m}^{*}$ and minimum of Poisson’s ratio ${\nu}_{m}^{*}$: (

**a**) effective Young’s modulus${E}_{m}^{*}$; and (

**b**) effective Poisson’s ratio ${\nu}_{m}^{*}$.

**Figure 15.**Effective elastic moduli of composite reinforced by randomly CNT inclusion in PP matrix, in function of $\eta $ for $\mu $ = 0.35 with a maximum of Young’s modulus ${E}_{m}^{*}$ and minimum of Poisson’s ratio ${\nu}_{m}^{*}$: (

**a**) effective Young’s modulus${E}_{m}^{*}$; and (

**b**) effective Poisson’s ratio ${\nu}_{m}^{*}$.

**Figure 16.**Effective elastic moduli of composite reinforced by randomly GNP inclusion in PP matrix, in function agglomeration (parameter $\mu $ at small volume fraction (1–4%): (

**a**) effective Young’s modulus${E}_{m}^{*}$; and (

**b**) effective Poisson’s ratio ${\nu}_{m}^{*}$.

**Figure 17.**Effective elastic moduli of composite reinforced by randomly GNP inclusion in PP matrix, in function of $\eta $ for $\mu $ = 0.2 with a maximum of Young’s modulus ${E}_{m}^{*}$ and minimum of Poisson’s ratio ${\nu}_{m}^{*}$: (

**a**) effective Young’s modulus${E}_{m}^{*}$; and (

**b**) effective Poisson’s ratio ${\nu}_{m}^{*}$.

**Figure 18.**Effective elastic moduli of composite reinforced by randomly GNP inclusion in PP matrix, in function of $\eta $ for $\mu $ = 0.35 with a maximum of Young’s modulus ${E}_{m}^{*}$ and minimum of Poisson ratio ${\nu}_{m}^{*}$: (

**a**) effective Young’s modulus${E}_{m}^{*}$; and (

**b**) effective Poisson’s ratio ${\nu}_{m}^{*}$.

${\mathit{k}}_{\mathit{r}}\left(\mathrm{GPa}\right)$ | ${\mathit{l}}_{\mathit{r}}\left(\mathrm{GPa}\right)$ | ${\mathit{m}}_{\mathit{r}}\left(\mathrm{GPa}\right)$ | ${\mathit{n}}_{\mathit{r}}\left(\mathrm{GPa}\right)$ | ${\mathit{p}}_{\mathit{r}}\left(\mathrm{GPa}\right)$ | |
---|---|---|---|---|---|

CNT | 536 | 184 | 132 | 2143 | 791 |

GNP | 850 | 6.8 | 369 | 102,000 | 102,000 |

${\mathbf{E}}_{11}^{\mathbf{r}}$ $\left(\mathbf{GPa}\right)$ | ${\mathbf{E}}_{22}^{\mathbf{r}}={\mathbf{E}}_{33}^{\mathbf{r}}$ $\left(\mathbf{GPa}\right)$ | ${\mathsf{\nu}}_{23}^{\mathbf{r}}$ | ${\mathsf{\nu}}_{12}^{\mathbf{r}}={\mathsf{\nu}}_{13}^{\mathbf{r}}$ | ${\mathbf{G}}_{23}^{\mathbf{r}}$ $\left(\mathbf{GPa}\right)$ | ${\mathbf{G}}_{12}^{\mathbf{r}}={\mathbf{G}}_{13}^{\mathbf{r}}$ $\left(\mathbf{GPa}\right)$ | |
---|---|---|---|---|---|---|

CNT | 421.14 | 2079.8 | 0.17164 | 0.59522 | 132 | 791 |

GNP | 1029.204 | 102,000 | 0.004 | 0.4 | 369 | 102,000 |

Materials | PP | UP |
---|---|---|

Model | Elastic | Elastic |

Symmetry | Isotropic | Isotropic |

Density ρ (g/cm^{3}) | 0.9 | 1.3 |

Young’s modulus E (GPa) | 1.4 | 3.8 |

Poisson’s ratio v | 0.45 | 0.42 |

Elastic Properties | CNT | GNP |
---|---|---|

Axial Young’s modulus (GPa) | 2079.8 | 102,000 |

In plane Young’s modulus (GPa) | 421.14 | 1029.204 |

In plane Poisson’s ratio | 0.59522 | 0.4 |

Transverse Poisson’s ratio | 0.17164 | 0.004 |

Transverse shear modulus (GPa) | 791 | 102,000 |

In plane shear modulus (GPa) | 132 | 369 |

Density (g/cm^{3}) | 1.2 | 2.2 |

Materials | Alfa | E-Glass |
---|---|---|

Model | Elastic | Elastic |

Symmetry | Isotropic | Isotropic |

Density ρ (g/cm^{3}) | 1.52 | 2.54 |

Young’s modulus E (GPa) | 19.4 | 73 |

Poisson’s ratio v | 0.34 | 0.23 |

CNT | GNP | |
---|---|---|

Axial Young’s modulus (GPa) | 2079.8 | 102,000 |

In plane Young’s modulus (GPa) | 421.14 | 1029.204 |

In plane Poisson’s ratio | 0.59522 | 0.4 |

Transverse Poisson’s ratio | 0.17164 | 0.004 |

Transverse shear modulus (GPa) | 791 | 102,000 |

In plane shear modulus (GPa) | 132 | 369 |

Density (g/cm^{3}) | 1.2 | 2.2 |

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**MDPI and ACS Style**

El Bahi, A.; Rouway, M.; Tarfaoui, M.; Moumen, A.E.; Chakhchaoui, N.; Cherkaoui, O.; Omari, L.E.H. Mechanical Homogenization of Transversely Isotropic CNT/GNP Reinforced Biocomposite for Wind Turbine Blades: Numerical and Analytical Study. *J. Compos. Sci.* **2023**, *7*, 29.
https://doi.org/10.3390/jcs7010029

**AMA Style**

El Bahi A, Rouway M, Tarfaoui M, Moumen AE, Chakhchaoui N, Cherkaoui O, Omari LEH. Mechanical Homogenization of Transversely Isotropic CNT/GNP Reinforced Biocomposite for Wind Turbine Blades: Numerical and Analytical Study. *Journal of Composites Science*. 2023; 7(1):29.
https://doi.org/10.3390/jcs7010029

**Chicago/Turabian Style**

El Bahi, Amine, Marwane Rouway, Mostapha Tarfaoui, Ahmed El Moumen, Nabil Chakhchaoui, Omar Cherkaoui, and Lhaj El Hachemi Omari. 2023. "Mechanical Homogenization of Transversely Isotropic CNT/GNP Reinforced Biocomposite for Wind Turbine Blades: Numerical and Analytical Study" *Journal of Composites Science* 7, no. 1: 29.
https://doi.org/10.3390/jcs7010029