# Nonlocal Free Vibration of Embedded Short-Fiber-Reinforced Nano-/Micro-Rods with Deformable Boundary Conditions

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Nonlocal Elasticity

_{0}specifies a material constant. The equation of motion for the longitudinal vibration of the composite micro- and nano-rods can be described by [44]:

## 3. Material Properties of Short-Fiber-Reinforced Composite

## 4. Fourier Infinite Series with Stokes’ Transformation

## 5. Frequency Determinants for the Short-Fiber-Reinforced Micro-/Nano-Rods

#### 5.1. General Case

#### 5.2. Without Elastic Medium Effect

#### 5.3. Without Nonlocal Effect

#### 5.4. Without Elastic Medium and Size-Effect

## 6. Discussion

_{0}a on the dimensionless frequency values of the short-fiber-reinforced nano-rod are investigated. For this purpose, the dimensionless axial frequency values for nonlocal parameter values ranging from 0 nm to 0.5 nm are plotted for the first seven modes in Figure 3. The following properties are used for this figure: ${\rho}_{f}/{\rho}_{m}=4$, ${E}_{f}/{E}_{m}=10$, $l/d=4$, ${V}_{f}=0.5$ and $L=20$ nm. Also, the elastic medium effect is omitted in this example. When we look at the changes in the dimensionless axial frequencies of the short-fiber-reinforced composite nano-rod with the help of the figure, we can say that a general decrease has occurred. In the first mode, when the nonlocal parameter is 0.0 nm, 0.1 nm, 0.2 nm and 0.3 nm, or when the nonlocal parameter is 0.0 nm and 0.1 nm in the second mode, there is no change in the dimensionless axial frequencies. This can be easily explained by the impact of the nonlocal parameter on the vibrational modes. It should be noted that the amount of the decrease in the non-dimensional frequencies increases with the increase in the vibration mode number. In the first and second modes the changes are especially negligible, while in the higher modes the differences become more pronounced. Thus, it can be concluded that the impact of the nonlocal parameter on the axial frequencies of the short-fiber-reinforced composite nano-rod in higher modes is more significant.

_{f}/E

_{m}ratio on the first dimensionless frequencies of the embedded short-fiber-reinforced composite nano-rod. In this figure, the change in the first mode dimensionless frequencies of the composite nano-rod versus E

_{f}/E

_{m}for various foundation parameters K is plotted. The dimensionless foundation parameter and E

_{f}/E

_{m}are changed from zero to six and from 5 to 30, respectively. Furthermore, the following properties are assumed for this figure: ${\rho}_{f}/{\rho}_{m}=4$, $l/d=2$, ${e}_{0}a=0.2\mathrm{nm}$, ${V}_{f}=0.5$, $L=20\mathrm{nm}$. It can be understood from this figure that an increment in the E

_{f}/E

_{m}value is accompanied by an increase in the first-mode axial frequencies.

_{f}/E

_{m}ratios on the dimensionless frequency values of the short-fiber-reinforced nano-rod are examined in Figure 9. For this purpose, the non-dimensional axial frequency values for E

_{f}/E

_{m}values ranging from 5 to 30 are demonstrated for the first seven modes. The following properties are considered for this study: ${\rho}_{f}/{\rho}_{m}=4$, ${e}_{0}a=0.2nm$, ${V}_{f}=0.5$, $K=0$, $l/d=2$, and $L=20$ nm. It is clearly seen here that with increasing E

_{f}/E

_{m}values, the dimensionless frequencies of the composite nano-sized rod increase. It should be emphasized here that at low E

_{f}/E

_{m}values, the change in the dimensionless axial frequencies of the short-fiber-reinforced nano-rod is more prominent. As the E

_{f}/E

_{m}values increase, the change in the dimensionless axial frequencies of the composite nano-rod decreases.

_{f}/ρ

_{m}ratios on the dimensionless frequencies of the composite nano-sized rod are examined in Figure 10. For this purpose, non-dimensional axial frequency values for ρ

_{f}/ρ

_{m}values ranging from 2 to 12 are shown for the first seven modes. The following properties are considered for this study: ${E}_{f}/{E}_{m}=10$, ${e}_{0}a=0.2nm$, ${V}_{f}=0.5$, $K=0$, $l/d=2$, and $L=20$ nm. It is observed here that with increasing ρ

_{f}/ρ

_{m}values, the non-dimensional frequency values of the short-fiber-reinforced composite nano-rod decrease. This decrement in the frequencies is valid for all modes examined. It should be highlighted here that at low ρ

_{f}/ρ

_{m}values, the variation in the dimensionless axial frequencies of the short-fiber-reinforced nano-rod is more conspicuous.

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Illustrations of composite micro-/nano-rods with elastic springs: (

**a**) aligned composite micro-/nano-rod; (

**b**) randomly oriented composite micro-/nano-rod.

**Figure 3.**The variations of non-dimensional frequencies of short-fiber-reinforced composite nano-rods versus nonlocal parameter: (

**a**) 1st mode (

**b**) 2nd mode (

**c**) 3rd mode (

**d**) 4th mode (

**e**) 5th mode (

**f**) 6th mode (

**g**) 7th mode.

**Figure 4.**The variations of non-dimensional frequencies of short-fiber-reinforced composite nano-rods versus l/d: (

**a**) 1st mode (

**b**) 2nd mode (

**c**) 3rd mode (

**d**) 4th mode (

**e**) 5th mode (

**f**) 6th mode (

**g**) 7th mode.

**Figure 5.**The variations of non-dimensional frequencies of short-fiber-reinforced composite nano-rods versus length: (

**a**) 1st mode (

**b**) 2nd mode (

**c**) 3rd mode (

**d**) 4th mode (

**e**) 5th mode (

**f**) 6th mode (

**g**) 7th mode.

**Figure 6.**The variations of non-dimensional frequencies of short-fiber-reinforced composite nano-rods versus the foundation parameter: (

**a**) 1st mode (

**b**) 2nd mode (

**c**) 3rd mode (

**d**) 4th mode (

**e**) 5th mode (

**f**) 6th mode (

**g**) 7th mode.

**Figure 7.**The variations of non-dimensional frequencies of short-fiber-reinforced composite nano-rods versus l/d: (

**a**) K = 0 (

**b**) K = 1 (

**c**) K = 2 (

**d**) K = 3 (

**e**) K = 4 (

**f**) K = 5 (

**g**) K = 6.

**Figure 8.**The variations of non-dimensional frequencies of short-fiber-reinforced composite nano-rods versus E

_{f}/E

_{m}: (

**a**) K = 0 (

**b**) K = 1 (

**c**) K = 2 (

**d**) K = 3 (

**e**) K = 4 (

**f**) K = 5 (

**g**) K = 6.

**Figure 9.**The variations of non-dimensional frequencies of short-fiber-reinforced composite nano-rods versus E

_{f}/E

_{m}: (

**a**) 1st mode (

**b**) 2nd mode (

**c**) 3rd mode (

**d**) 4th mode (

**e**) 5th mode (

**f**) 6th mode (

**g**) 7th mode.

**Figure 10.**The variations of non-dimensional frequencies of short-fiber-reinforced composite nano-rods versus

**ρ**

_{f}/

**ρ**

_{m}: (

**a**) 1st mode (

**b**) 2nd mode (

**c**) 3rd mode (

**d**) 4th mode (

**e**) 5th mode (

**f**) 6th mode (

**g**) 7th mode.

**Table 1.**Comparison of the first three non-dimensional axial frequencies of randomly oriented short-fiber-reinforced composite nano-rods for the clamped–clamped boundary condition.

Mode Number | Analytical Solution [1] | Present $\overline{({\mathit{\Omega}}_{0}}=\overline{{\mathit{\Omega}}_{\mathit{L}}}=\mathit{\infty})$ |
---|---|---|

e_{0}a = 0 nm | ||

1 | 3.5819 | 3.5819 |

2 | 7.1639 | 7.1639 |

3 | 10.7459 | 10.7459 |

e_{0}a = 0.5 nm | ||

1 | 3.5709 | 3.5709 |

2 | 7.0771 | 7.0771 |

3 | 10.4594 | 10.4594 |

e_{0}a = 1 nm | ||

1 | 3.5385 | 3.5385 |

2 | 6.8345 | 6.8345 |

3 | 9.7206 | 9.7206 |

**Table 2.**Comparison of the first three non-dimensional axial frequencies of randomly oriented short-fiber-reinforced composite nano-rods for the clamped–free boundary condition.

Mode Number | Analytical Solution [1] | Present $\overline{({\mathit{\Omega}}_{0}}=\mathit{\infty},\overline{{\mathit{\Omega}}_{\mathit{L}}}=0)$ |
---|---|---|

e_{0}a = 0 nm | ||

1 | 1.7909 | 1.7909 |

2 | 5.3729 | 5.3729 |

3 | 8.9549 | 8.9549 |

e_{0}a = 0.5 nm | ||

1 | 1.7896 | 1.7896 |

2 | 5.3360 | 5.3360 |

3 | 8.7871 | 8.7871 |

e_{0}a = 1 nm | ||

1 | 1.7854 | 1.7854 |

2 | 5.2297 | 5.2297 |

3 | 8.3352 | 8.3352 |

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**MDPI and ACS Style**

Civalek, Ö.; Uzun, B.; Yaylı, M.Ö.
Nonlocal Free Vibration of Embedded Short-Fiber-Reinforced Nano-/Micro-Rods with Deformable Boundary Conditions. *Materials* **2022**, *15*, 6803.
https://doi.org/10.3390/ma15196803

**AMA Style**

Civalek Ö, Uzun B, Yaylı MÖ.
Nonlocal Free Vibration of Embedded Short-Fiber-Reinforced Nano-/Micro-Rods with Deformable Boundary Conditions. *Materials*. 2022; 15(19):6803.
https://doi.org/10.3390/ma15196803

**Chicago/Turabian Style**

Civalek, Ömer, Büşra Uzun, and Mustafa Özgür Yaylı.
2022. "Nonlocal Free Vibration of Embedded Short-Fiber-Reinforced Nano-/Micro-Rods with Deformable Boundary Conditions" *Materials* 15, no. 19: 6803.
https://doi.org/10.3390/ma15196803