# 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

- Rouway, M.; Nachtane, M.; Tarfaoui, M.; Chakhchaoui, N.; Omari, L.E.H.; Fraija, F.; Cherkaoui, O. Mechanical Properties of a Biocomposite Based on Carbon Nanotube and Graphene Nanoplatelet Reinforced Polymers: Analytical and Numerical Study. J. Compos. Sci.
**2021**, 5, 234. [Google Scholar] [CrossRef] - Nachtane, M.; Tarfaoui, M.; Goda, I.; Rouway, M. A review on the technologies, design considerations and numerical models of tidal current turbines. Renew Energy
**2020**, 157, 1274–1288. [Google Scholar] [CrossRef] - Lamhour, K.; Rouway, M.; Tizliouine, A.; El Hachemi Omari, L.; Salhi, H.; Cherkaoui, O. Experimental study on the properties of Alfa/wool woven fabrics reinforced epoxy composite as an application in wind turbine blades. J. Compos. Mater.
**2022**, 56, 3253–3268. [Google Scholar] [CrossRef] - Mahmoud Zaghloul, M.Y.; Yousry Zaghloul, M.M.; Yousry Zaghloul, M.M. Developments in polyester composite materials—An in-depth review on natural fibres and nano fillers. Compos. Struct.
**2021**, 278, 114698. [Google Scholar] [CrossRef] - Nachtane, M.; Meraghni, F.; Chatzigeorgiou, G.; Harper, L.T.; Pelascini, F. Multiscale viscoplastic modeling of recycled glass fiber-reinforced thermoplastic composites: Experimental and numerical investigations. Compos. Part B Eng.
**2022**, 242, 110087. [Google Scholar] [CrossRef] - Nunna, S.; Blanchard, P.; Buckmaster, D.; Davis, S.; Naebe, M. Development of a cost model for the production of carbon fibres. Heliyon
**2019**, 5, e02698. [Google Scholar] [CrossRef] [Green Version] - Zaghloul, M.M.Y.; Zaghloul, M.Y.M.; Zaghloul, M.M.Y. Experimental and modeling analysis of mechanical-electrical behaviors of polypropylene composites filled with graphite and MWCNT fillers. Polym. Test.
**2017**, 63, 467–474. [Google Scholar] [CrossRef] - García-Macías, E.; Castro-Triguero, R. Coupled effect of CNT waviness and agglomeration: A case study of vibrational analysis of CNT/polymer skew plates. Compos. Struct.
**2018**, 193, 87–102. [Google Scholar] [CrossRef] - Maghsoudlou, M.A.; Barbaz Isfahani, R.; Saber-Samandari, S.; Sadighi, M. Effect of interphase, curvature and agglomeration of SWCNTs on mechanical properties of polymer-based nanocomposites: Experimental and numerical investigations. Compos. Part B Eng.
**2019**, 175, 107119. [Google Scholar] [CrossRef] - Pan, Z.-Z.; Chen, X.; Zhang, L.-W. Modeling large amplitude vibration of pretwisted hybrid composite blades containing CNTRC layers and matrix cracked FRC layers. Appl. Math. Model.
**2020**, 83, 640–659. [Google Scholar] [CrossRef] - Iijima, S. Helical microtubules of graphitic carbon. Nature
**1991**, 354, 56–58. [Google Scholar] [CrossRef] - Hassanzadeh-Aghdam, M.K.; Mahmoodi, M.J.; Ansari, R. Creep performance of CNT polymer nanocomposites -An emphasis on viscoelastic interphase and CNT agglomeration. Compos. Part B Eng.
**2019**, 168, 274–281. [Google Scholar] [CrossRef] - Thostenson, E.T.; Chou, T.-W. On the elastic properties of carbon nanotube-based composites: Modelling and characterization. J. Phys. Appl. Phys.
**2003**, 36, 573–582. [Google Scholar] [CrossRef] - Bonnet, P.; Sireude, D.; Garnier, B.; Chauvet, O. Thermal properties and percolation in carbon nanotube-polymer composites. Appl. Phys. Lett.
**2007**, 91, 201910. [Google Scholar] [CrossRef] - Fidelus, J.D.; Wiesel, E.; Gojny, F.H.; Schulte, K.; Wagner, H.D. Thermo-mechanical properties of randomly oriented carbon/epoxy nanocomposites. Compos. Part Appl. Sci. Manuf.
**2005**, 36, 1555–1561. [Google Scholar] [CrossRef] - Hassanzadeh-Aghdam, M.K. Evaluating the effective creep properties of graphene-reinforced polymer nanocomposites by a homogenization approach. Compos. Sci. Technol.
**2021**, 209, 108791. [Google Scholar] [CrossRef] - Gao, C.; Zhan, B.; Chen, L.; Li, X. A micromechanical model of graphene-reinforced metal matrix nanocomposites with consideration of graphene orientations. Compos. Sci. Technol.
**2017**, 152, 120–128. [Google Scholar] [CrossRef] - Rafiee, M.A.; Rafiee, J.; Wang, Z.; Song, H.; Yu, Z.-Z.; Koratkar, N. Enhanced Mechanical Properties of Nanocomposites at Low Graphene Content. ACS Nano
**2009**, 3, 3884–3890. [Google Scholar] [CrossRef] - Narh, K.A.; Jallo, L.; Rhee, K.Y. The effect of carbon nanotube agglomeration on the thermal and mechanical properties of polyethylene oxide. Polym. Compos.
**2008**, 29, 809–817. [Google Scholar] [CrossRef] - Alian, A.R.; El-Borgi, S.; Meguid, S.A. Multiscale modeling of the effect of waviness and agglomeration of CNTs on the elastic properties of nanocomposites. Comput. Mater. Sci.
**2016**, 117, 195–204. [Google Scholar] [CrossRef] - Daghigh, H.; Daghigh, V.; Milani, A.; Tannant, D.; Lacy, T.E.; Reddy, J.N. Nonlocal bending and buckling of agglomerated CNT-Reinforced composite nanoplates. Compos. Part B Eng.
**2020**, 183, 107716. [Google Scholar] [CrossRef] - Shi, D.-L.; Feng, X.-Q.; Huang, Y.Y.; Hwang, K.-C.; Gao, H. The Effect of Nanotube Waviness and Agglomeration on the Elastic Property of Carbon Nanotube-Reinforced Composites. J. Eng. Mater. Technol.
**2004**, 126, 250–257. [Google Scholar] [CrossRef] - Ji, X.-Y.; Cao, Y.-P.; Feng, X.-Q. Micromechanics prediction of the effective elastic moduli of graphene sheet-reinforced polymer nanocomposites. Model. Simul. Mater. Sci. Eng.
**2010**, 18, 045005. [Google Scholar] [CrossRef] - Yun, G.J.; Zhu, F.-Y.; Lim, H.J.; Choi, H. A damage plasticity constitutive model for wavy CNT nanocomposites by incremental Mori-Tanaka approach. Compos. Struct.
**2021**, 258, 113178. [Google Scholar] [CrossRef] - Fazilati, J.; Khalafi, V.; Jalalvand, M. Free vibration analysis of three-phase CNT/polymer/fiber laminated tow-steered quadrilateral plates considering agglomeration effects. Thin-Walled Struct.
**2022**, 179, 109638. [Google Scholar] [CrossRef] - Mori, T.; Tanaka, K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall.
**1973**, 21, 571–574. [Google Scholar] [CrossRef] - Tornabene, F.; Bacciocchi, M.; Fantuzzi, N.; Reddy, J.N. Multiscale approach for three-phase CNT/polymer/fiber laminated nanocomposite structures. Polym. Compos.
**2019**, 40, E102–E126. [Google Scholar] [CrossRef] - Chamis, C.C. Simplified Composite Micromechanics Equations For Hygral, Thermal And Mechanical Properties. In Proceedings of the Ann. Conf. of the Society of the Plastics Industry (SPI) Reinforced Plastics/Composites Inst., Houston, TX, USA, 1 January 1983. [Google Scholar]
- Hashin, Z.; Rosen, B.W. The Elastic Moduli of Fiber-Reinforced Materials. J. Appl. Mech.
**1964**, 31, 223–232. [Google Scholar] [CrossRef] - Halpin, J.C. Stiffness and Expansion Estimates for Oriented Short Fiber Composites. J. Compos. Mater.
**1969**, 3, 732–734. [Google Scholar] [CrossRef]

**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