# Wave Dispersion Behaviors of Multi-Scale CNT/Glass Fiber/Polymer Nanocomposite Laminated Plates

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## Abstract

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## 1. Introduction

## 2. Theory and Formulation

#### 2.1. Problem Definition

#### 2.2. Homogenization Procedure

#### 2.3. Refined Higher-Order Plate Theory

#### 2.4. Motion Equations

#### 2.5. Constitutive Equations

#### 2.6. Governing Equations

## 3. Analytical Solution

## 4. Numerical Results and Discussion

#### 4.1. Validation

#### 4.2. Influence of Different Compositions on the Wave Propagation Response of the Continua

#### 4.3. Influence of Number of Plies on the Wave Propagation Response of the Continua

#### 4.4. Influence of Volume Fraction of the GFs on the Wave Propagation Response of the Continua for Different Lay-Ups

_{s}lay-up which corresponds to the case that GFs are parallel to the x-axis and y-axis, respectively. The reason for this can be explained throughout Equation (33) that each parameter has its maximum value while [0, 90, –90, 0]

_{s}lay-up is used. The more the deviation from this lay-up, the more the wave frequency will be reduced. The ultimate effect of this kind of stiffness reduction can be observed in [45, 45, –45, –45]

_{s}lay-up where wave frequency has plateaued on a little above 1.2 kHz for all of the considered volume fractions of GFs.

#### 4.5. Influence of Weight Fraction of the CNTs on the Wave Propagation Response of the Continua for Various Length-to-Diameter Ratios of CNTs

## 5. Concluding Remarks

- Since an increase in the number of plies has a downside effect on the wave frequency, it is best to find the optimal number to satisfy the design criteria.
- The MSH nanocomposite plate can achieve its greatest dynamic response once the ${\left[0,90,-90,0\right]}_{\mathrm{s}}$ lay-up is employed.
- Adding either the weight fraction of the CNTs or the volume fraction of the GFs amplifies the natural frequency of the propagated waves.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Conflicts of Interest

## Appendix A

## Appendix B

## Appendix C

## References

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**Figure 1.**Schematic view of the analyzed multi-layered plate (

**left**figure) and the order by which the ingredients must be combined to manufacture a single ply of hybrid nanocomposite material (

**right**figure).

**Figure 2.**Comparison of the wave frequency versus wave number curve of the acoustic branch of waves dispersed in FG plates. The solid line is related to the response provided in [39] and the diamonds are the responses of present study.

**Figure 3.**Comparison of acoustic wave frequency responses of sandwich plates fabricated from polymer, GF/polymer, CNT/polymer, and GF/CNT/polymer. α = 100 is employed in the cases which CNTs exist in the composition of the material. In this figure, the ${\left[0,30,45,60,90\right]}_{\mathrm{s}}$ lay-up is used for the laminated composite.

**Figure 4.**The variation in the wave frequency versus number of the plies for different amounts of the fibers’ orientation angle (β = 2).

**Figure 5.**The variation in the wave frequency versus volume fraction of the GFs for various lay-ups of eight-layered laminates (β = 2).

**Figure 6.**Variation in the wave frequency versus weight fraction of the CNTs for various CNTs’ length-to-diameter ratio. In this diagram, [0, 90, −90, 0]

_{s}lay-up is used (V

_{f}= 0.2, β = 2).

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**MDPI and ACS Style**

Ebrahimi, F.; Enferadi, A.; Dabbagh, A.
Wave Dispersion Behaviors of Multi-Scale CNT/Glass Fiber/Polymer Nanocomposite Laminated Plates. *Polymers* **2022**, *14*, 5448.
https://doi.org/10.3390/polym14245448

**AMA Style**

Ebrahimi F, Enferadi A, Dabbagh A.
Wave Dispersion Behaviors of Multi-Scale CNT/Glass Fiber/Polymer Nanocomposite Laminated Plates. *Polymers*. 2022; 14(24):5448.
https://doi.org/10.3390/polym14245448

**Chicago/Turabian Style**

Ebrahimi, Farzad, Alireza Enferadi, and Ali Dabbagh.
2022. "Wave Dispersion Behaviors of Multi-Scale CNT/Glass Fiber/Polymer Nanocomposite Laminated Plates" *Polymers* 14, no. 24: 5448.
https://doi.org/10.3390/polym14245448