# The Dynamic Response and Vibration of Functionally Graded Carbon Nanotube-Reinforced Composite (FG-CNTRC) Truncated Conical Shells Resting on Elastic Foundations

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

^{3}; the specific strength of up to 48,000 kN·m·kg

^{−1}is the best of known materials, compared to high-carbon steel’s 154 kN·m·kg

^{−1}[5]. About the electrical properties, because of CNTs’ nanoscale cross section, electrons propagate only along the tube’s axis. CNTs are one-dimensional conductors, or, in other words, metallic or semiconducting along the tubular axis. The maximum electrical conductance of a single-walled CNT is 4 e

^{2}/h, twice the conductivity of a single ballistic quantum channel [6]. As for the thermal properties, as far as we know, all nanotubes are expected to be very good thermal conductors along the tube; in fact, the temperature stability of CNTs is estimated to be up to 2800 °C in vacuum and about 750 °C in air [7]. Moreover, CNTs have optical properties such as useful absorption, photoluminescence (fluorescence), and Raman spectroscopy. The superior properties of CNTs are well established and have immediate applications in areas related to most industries, including aerospace, electronics, medicine, defense, automotive, energy, construction, and even fashion. For example, Aliahmad et al. presented the poly (vinylidene fluoride-hexafluoropropylene) porous membrane electrolyte enhanced with lithium bis(trifluoromethane sulphone)imide and lithium aluminum titanium phosphate with an ionic conductivity of 2.1 × 10

^{−3}S·cm

^{−1}for paper-based battery applications [8] and the paper–based lithium–ion batteries using carbon nanotube coated wood microfibers [9]. Agarwal et al. studied the conductive paper from lignocellulose wood microfibers coated with nanocomposite of carbon nanotubes and conductive polymers [10].

## 2. Formulation of the Problem

#### 2.1. Determination of the Elastic Modules of CNTRCs and FG-CNTRC

#### 2.2. Analytical Modeling of Elastic Medium

#### 2.3. Basic Formulation of the FG-CNTRC Truncated Conical Shells Surrounded by Elastic Foundations

#### 2.4. The Solution of Basic Equations

#### 2.5. Vibration Analysis

## 3. Numerical Results and Discussion

#### 3.1. Validation

#### 3.2. The Natural Frequency and Dynamic Response

## 4. Conclusions

- The value of the non-dimensional frequency negligible decreases when the values of semi-vertex angle $\gamma $ increase.
- The value of the amplitude and non-dimensional frequency of the shells are significantly affected by various types of CNT distributions. In the case of the FG-X type of CNT distribution, the amplitude value is the smallest and the non-dimensional frequency is the highest.
- The results obtained also demonstrate that the $t-f$ time–deflection curves are affected greatly by variations in parameters such as ratio ${R}_{1}/h$, length-to-radius ratio $L/{R}_{1}$, and amplitude $Q$.
- The elastic foundations strongly affect the dynamic response of FG-CNTRC truncated conical shells. The elastic foundations have positive effects on the amplitudes of FG-CNTRC truncated conical shells.
- The stress function, Galerkin method, Runge–Kutta method, and analytical approach are used to assess the dynamic responses of FG-CNTRC truncated conical shells resting on elastic foundations.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

## Appendix B

## Appendix C

## References

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**Figure 1.**(

**a**) The coordinate system and geometric characteristics of a FG-CNRTC truncated conical shell; (

**b**) the geometric characteristics of the truncated conical shell surrounded by elastic foundations.

**Figure 3.**Effect of CNT volume fraction of fibers on the dynamic response of the CNTRC truncated conical shells with uniform distribution type.

**Figure 4.**Effect of volume fraction of fibers on the dynamic response of the FG-CNTRC truncated conical shells.

**Figure 5.**Effect of semi-vertex angle $\gamma $ on the dynamic response of the FG-CNTRC truncated conical shells.

**Figure 6.**Effect of ratio $L/{R}_{1}$ on the dynamic response of the FG-CNTRC truncated conical shells.

**Figure 7.**Effect of ratio ${R}_{1}/h$ on the dynamic response of the FG-CNTRC truncated conical shells.

**Figure 8.**Effect of the Winkler modulus parameter ${K}_{w}$ on the dynamic response of the FG-CNTRC truncated conical shells.

**Figure 9.**Effect of the Pasternak modulus parameter ${K}_{p}$ on the dynamic response of the FG-CNTRC truncated conical shells.

**Figure 10.**Effect of amplitude $Q$ on the dynamic response of the FG-CNTRC truncated conical shells.

**Figure 11.**The dynamic response of CNTRC truncated conical shells with various types of CNT reinforcements.

**Table 1.**Comparisons of non-dimensional frequency parameter of the isotropic truncated conical shells.

$\mathit{\gamma}$ | $\mathit{n}$ | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

${30}^{\xb0}$ | Li et al. [59] | 0.8431 | 0.7416 | 0.6419 | 0.5590 | 0.5008 |

Lam and Hua [60] | 0.8429 | 0.7376 | 0.6362 | 0.5528 | 0.4950 | |

Present | 0.8700 | 0.7934 | 0.6831 | 0.5491 | 0.3976 | |

${45}^{\xb0}$ | Li et al. [59] | 0.7642 | 0.7211 | 0.6747 | 0.6336 | 0.6049 |

Lam and Hua [60] | 0.7655 | 0.7212 | 0.6739 | 0.6323 | 0.6035 | |

Present | 0.7205 | 0.7023 | 0.6689 | 0.6149 | 0.5350 |

**Table 2.**Influences of CNT volume fraction, various types of FG-CNTRC and ratio ${R}_{1}/h$ on the non-dimensional frequency of the FG-CNTRC truncated conical shells.

${\mathit{R}}_{1}/\mathit{h}$ | ${\mathit{V}}_{\mathit{C}\mathit{N}\mathit{T}}^{*}=0.12$ | ${\mathit{V}}_{\mathit{C}\mathit{N}\mathit{T}}^{*}=0.17$ | ${\mathit{V}}_{\mathit{C}\mathit{N}\mathit{T}}^{*}=0.28$ | ||||||
---|---|---|---|---|---|---|---|---|---|

FG-O | FG-V | FG-X | FG-O | FG-V | FG-X | FG-O | FG-V | FG-X | |

50 | 1.039 | 1.236 | 1.905 | 1.214 | 1.455 | 2.275 | 1.586 | 1.878 | 2.905 |

60 | 0.816 | 0.989 | 1.561 | 0.940 | 1.153 | 1.859 | 1.260 | 1.513 | 2.387 |

70 | 0.646 | 0.804 | 1.310 | 0.726 | 0.925 | 1.555 | 1.013 | 1.242 | 2.012 |

80 | 0.505 | 0.656 | 1.118 | 0.544 | 0.740 | 1.320 | 0.814 | 1.029 | 1.725 |

100 | 0.255 | 0.421 | 0.838 | 0.154 | 0.430 | 0.974 | 0.485 | 0.699 | 1.310 |

**Table 3.**Influences of semi-vertex angle $\gamma $ on the non-dimensional frequency of the CNTRC truncated conical shells.

$\mathit{\gamma}$ | UD | FG-O | FG-V | FG-X |
---|---|---|---|---|

15° | 1.3672 | 0.8341 | 1.0526 | 1.7414 |

30° | 1.2463 | 0.8160 | 0.9889 | 1.5606 |

45° | 1.1507 | 0.7866 | 0.9311 | 1.4238 |

60° | 1.0898 | 0.7658 | 0.8935 | 1.3371 |

75° | 1.0567 | 0.7544 | 0.8731 | 1.2900 |

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**MDPI and ACS Style**

Nguyen Dinh, D.; Nguyen, P.D.
The Dynamic Response and Vibration of Functionally Graded Carbon Nanotube-Reinforced Composite (FG-CNTRC) Truncated Conical Shells Resting on Elastic Foundations. *Materials* **2017**, *10*, 1194.
https://doi.org/10.3390/ma10101194

**AMA Style**

Nguyen Dinh D, Nguyen PD.
The Dynamic Response and Vibration of Functionally Graded Carbon Nanotube-Reinforced Composite (FG-CNTRC) Truncated Conical Shells Resting on Elastic Foundations. *Materials*. 2017; 10(10):1194.
https://doi.org/10.3390/ma10101194

**Chicago/Turabian Style**

Nguyen Dinh, Duc, and Pham Dinh Nguyen.
2017. "The Dynamic Response and Vibration of Functionally Graded Carbon Nanotube-Reinforced Composite (FG-CNTRC) Truncated Conical Shells Resting on Elastic Foundations" *Materials* 10, no. 10: 1194.
https://doi.org/10.3390/ma10101194