# On the Dynamics of a Visco–Piezo–Flexoelectric Nanobeam

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Visco–Piezo–Flexoelectric Model

#### 2.1. Piezo–Flexoelectricity

_{i}(i = 1,3) represents the points’ displacements in the x and z directions, and w is the lateral displacement of the midplane. To show the thickness coordinate, the z parameter is used, and t is demonstrative of time.

#### 2.2. Size-Dependent Model

#### 2.3. Internal Viscoelasticity Coupling

## 3. The Solution Process

## 4. Result Validation

## 5. Frequency Analysis

_{0}a < 0.8 nm [72] and 0 < e

_{0}a ≤2 nm [73,74], and on the other side, the SGLP could be chosen arbitrarily. Regardless of time dependency and on the basis of Equation (45), the natural frequencies of the beam were extracted nondimensionally to be better plotted in an illustration as $\Omega =\omega \frac{{L}^{2}}{h}\sqrt{\frac{\rho}{{C}_{11}}}$ and $X=\frac{x}{L}$.

_{0}a produces larger frequencies compared to e

_{0}a > l. Moreover, by looking carefully at the figure, it can be seen that the further decreasing impact of the voltage is related to the case e

_{0}a > l.

_{0}a = 0 and piezo-e

_{0}a = 0.5 nm approach each other and are identical for large lengths. However, the most serious and marked result of this figure is the significance of flexoelectricity for larger lengths of the beam. Indeed, the results of piezo-e

_{0}a = 0 versus piezo–flexo-e

_{0}a = 0 and also piezo-e

_{0}a = 0.5 nm versus piezo–flexo-e

_{0}a = 0.5 nm become farther from one another after enlarging the length of the beam.

_{PF}/W

_{P}= 1.238, while for Figure 6d, W

_{PF}/W

_{P}= 1.293. In other words, when a material, in addition to piezoelectricity, also includes the flexoelectricity effect, it is further affected by the internal damping. In addition, this conclusion was observed to be clearer for higher mode shapes. Furthermore, it is worth noting that whenever the viscoelastic parameter’s value increased, the deflections decreased. Moreover, as flexoelectricity makes the material more flexible, the deflection by a piezo–flexoelectric nanomaterial would be higher than that by a piezoelectric one. It should be pointed out that all the mode shapes are symmetric. Consequently, these diagrams convey the noteworthy finding that a higher inner viscoelasticity parameter value strengthens the role of flexoelectricity in the nanomaterial.

## 6. Conclusions

- ∗
- The smaller the thickness, the larger the impact of flexoelectricity.
- ∗
- The lesser the SGLP values, the greater the flexoelectric effect.
- ∗
- The larger the length of the nanobeam, the larger the influence of flexoelectricity.
- ∗
- The greater the inner viscoelastic values, the greater the role of flexoelectricity.
- ∗
- The larger the SGLP values, the greater the inner viscoelastic impact.
- ∗
- The higher the mode number, the larger the influence of flexoelectricity.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**A dielectric hinged–hinged nanobeam’s configuration (designed in a CAD program (SolidWorks)).

**Figure 2.**Viscoelastic parameter vs. small-scale parameters and the effects of voltage on the nondimensional natural frequency (L = 10h, h = 2 nm).

**Figure 3.**Thickness parameter vs. flexoelectricity effect on the nondimensional natural frequency (e

_{0}a = 0.5 nm, l = 1 nm, V = 0.5 Volt, L = 10h, g = 100 N.s/m).

**Figure 4.**(

**a**) Strain gradient length scale parameter vs. flexoelectricity effect and viscoelastic parameter effect on the nondimensional natural frequency (e

_{0}a = 0.5 nm, h = 2 nm, L = 10h, V = 0.5 Volt). (

**b**) Strain gradient length scale parameter vs. flexoelectricity effect on the nondimensional natural frequency (e

_{0}a = 0.5 nm, h = 2 nm, L = 10h, V = 0.5 Volt, g = 100 N.s/m). (

**c**) Strain gradient length scale parameter vs. viscoelastic effect on the nondimensional natural frequency (e

_{0}a = 0.5 nm, h = 2 nm, L = 10h, V = 0.5 Volt).

**Figure 5.**Length-to-thickness ratio vs. flexoelectricity effect on the nondimensional natural frequency (l = 1 nm, h = 2 nm, V = 0.5 Volt, g = 0 N.s/m).

**Figure 6.**(

**a**) First mode shape for two cases: the presence of flexoelectricity and its absence without inner damping (e

_{0}a = 0.5 nm, l = 1 nm, h = 2 nm, V = 0.5 Volt, g = 0 N.s/m). (

**b**) First mode shape for two cases: the presence of flexoelectricity and its absence including internal viscoelasticity (e

_{0}a = 0.5 nm, l = 1 nm, h = 2 nm, V = 0.5 Volt, g = 100 N.s/m). (

**c**) Third mode shape for two cases: the presence of flexoelectricity and its absence without inner damping (e

_{0}a = 0.5 nm, l = 1 nm, h = 2 nm, V = 0.5 Volt, g = 0 N.s/m). (

**d**) Third mode shape for two cases: the presence of flexoelectricity and its absence with inner damping (e

_{0}a = 0.5 nm, l = 1 nm, h = 2 nm, V = 0.5 Volt, g = 100 N.s/m). (

**e**) Fifth mode shape for two cases: the presence of flexoelectricity and its absence without inner damping (e

_{0}a = 0.5 nm, l = 1 nm, h = 2 nm, V = 0.5 Volt, g = 0 N.s/m). (

**f**) Fifth mode shape for two cases: the presence of flexoelectricity and its absence with inner damping (e

_{0}a = 0.5 nm, l = 1 nm, h = 2 nm, V = 0.5 Volt, g = 100 N.s/m).

Configuration | Conditions |
---|---|

S | w(0, L) = 0, M_{x}(0, L) = 0, N_{x}(0, L) = 0 |

L/h | (e_{0}a)^{2} | Present | Euler–Bernoulli [68] |
---|---|---|---|

5 | 0 | 9.7112 | 9.7112 |

1 | 9.2647 | 9.2647 | |

2 | 8.8747 | 8.8747 | |

3 | 8.5301 | 8.5301 | |

4 | 8.2228 | 8.2228 | |

10 | 0 | 9.8293 | 9.8293 |

1 | 9.3774 | 9.3774 | |

2 | 8.9826 | 8.9826 | |

3 | 8.6338 | 8.6338 | |

4 | 8.3228 | 8.3228 | |

20 | 0 | 9.8595 | 9.8595 |

1 | 9.4062 | 9.4062 | |

2 | 9.0102 | 9.0102 | |

3 | 8.6604 | 8.6604 | |

4 | 8.3483 | 8.3483 |

L/D | [69] (MD-Armchair CNT) | Nonlocal Strain Gradient Theory | ||
---|---|---|---|---|

[69] (FSDT, Navier) | [70] (OVFSDT, Navier) | Present | ||

4.86 | 1.138 | 1.209 | 1.25535 | 1.23117 |

8.47 | 0.466 | 0.448 | 0.43207 | 0.49103 |

13.89 | 0.190 | 0.192 | 0.19004 | 0.21306 |

17.47 | 0.122 | 0.126 | 0.12431 | 0.13213 |

Pb(Zr, Ti)O_{3} or PZT-5H |
---|

C_{11} = 102 GPa, ρ = 7500 kg/m^{3},f _{31} = 10^{−7} C/m, d_{31} = 17.05 C/m^{2},a _{33} = 1.76 × 10^{−8} C/V.m, b_{33} = 10^{−9} J.m^{3}/C, |

κ_{1} = 6.62 C/V.m, κ_{2} = 8.85 × 10^{−12} C/V.m |

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Malikan, M.; Eremeyev, V.A.
On the Dynamics of a Visco–Piezo–Flexoelectric Nanobeam. *Symmetry* **2020**, *12*, 643.
https://doi.org/10.3390/sym12040643

**AMA Style**

Malikan M, Eremeyev VA.
On the Dynamics of a Visco–Piezo–Flexoelectric Nanobeam. *Symmetry*. 2020; 12(4):643.
https://doi.org/10.3390/sym12040643

**Chicago/Turabian Style**

Malikan, Mohammad, and Victor A. Eremeyev.
2020. "On the Dynamics of a Visco–Piezo–Flexoelectric Nanobeam" *Symmetry* 12, no. 4: 643.
https://doi.org/10.3390/sym12040643