# Effect of Sinusoidal Corrugated Geometries on the Vibrational Response of Viscoelastic Nanoplates

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modeling Assumptions

#### 2.1. Problem Definition

#### 2.2. Non-Local Strain Gradient Theory

_{ij}and D

_{ij}denote the extensional and flexural equivalent stiffness of the waviness nanoplate, respectively.

#### 2.3. The Linear Viscoelastic Model

## 3. Analytical Approach

**w**is discretized by a double series function as follows [10]

## 4. Numerical Results and Discussion

^{2}and $\mu =1.34$ nm

^{2}. As visible in these two tables, the very good agreement between results reflects the accuracy of the proposed formulation to approach a similar problem, provided that non-local parameters are adequately selected. Note that few discrepancies between results can be mainly related to the different solution method of the governing equations, e.g., a Navier’s approach instead of a GDQ. These differences become even more negligible for increasing lengths ${L}_{x}={L}_{y}$ as visible in Table 1 and Table 2. Some possible effects can be also related to the selected plate theory, as evaluated comparatively in Table 3 according to a CPT, FSDT, or a Third-order Shear Deformation Theory (TSDT) [23], as well as according to a three-dimensional elasticity [37] and the S-FSDT [38].

^{2}or $g=0$ Ns/m

^{2}. The nanoplate features two sinusoidal corrugations with amplitude $F=0.5h$ (i.e., dimensionless amplitude ${F}^{*}=F/h=0.5$) and semi-length $c=0.25{L}_{x}$. Based on Figure 2, an increased non-local coefficient $\mu $ yields to a reduced natural frequency, whereas an increasing length scale parameter ${l}^{*}$ gets to an increased natural frequency of the nanostructure because of its increased stiffness, under the non-local strain gradient theory. This agrees with findings by Malikan and Nguyen [10]. Moreover, the natural frequency of the flat nanostructure is lower than the corrugated one, for the same fixed non-local coefficient $\mu $ (Figure 2a) and dimensionless length scale ${l}^{*}$ (Figure 2b). Remarkably, all the curves in Figure 2 are almost scalable with the varying geometry and damping parameter. A negligible increase in the natural frequency is also observed for an increasing viscoelastic damping parameter $g$. Note that the equations based on the nonlocal strain gradient theory revert to the classical mechanics ones, when the length scale and nonlocal parameter are exactly the same (i.e., when $l$ is equal to ${e}_{0}a$). Based on a comparative evaluation of Figure 2a,b, it is worth noting that the response of the nonlocal parameter $\mu $ compared to the length scale ${l}^{*}$, is more sensitive for each fixed geometry and damping parameters of the nanostructure.

^{2}, $c=0.25{L}_{x}$, where the natural frequency of the structure increases proportionally with the viscoelastic damping. This increase is even more pronounced for increasing wave magnitudes ${F}^{*}$. This means that the geometry of corrugations can affect the response of the nanostructure significantly, especially for higher values of viscosity. Thus, the exact geometry of wrinkles and imperfections resulting from a manufacturing process must carefully account for the structural study of a hyper-viscoelastic nanoplate.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Sunthorn, S.; Kittiphat, R. The effect on rolling mill of waviness in hot rolled steel, World academy of sciences, Engineering and technology. Int. J. Metall. Mater. Eng.
**2014**, 8, 2077–2082. [Google Scholar] - Kamarian, S.; Salim, M.; Dimitri, R.; Tornabene, F. Free vibration analysis of conical shells reinforced with agglomerated carbon nanotubes. Int. J. Mech. Sci.
**2016**, 108–109, 157–165. [Google Scholar] [CrossRef] - Tornabene, F.; Fantuzzi, N.; Bacciocchi, M.; Viola, E. Effect of agglomeration on the natural frequencies of functionally graded carbon nanotube-reinforced laminated composite doubly-curved shells. Compos. Part B-Eng.
**2016**, 89, 187–218. [Google Scholar] [CrossRef] - Banić, D.; Bacciocchi, M.; Tornabene, F.; Ferreira, A.J.M. Influence of Winkler-Pasternak foundation on the vibrational behavior of plates and shells reinforced by agglomerated carbon nanotubes. Appl. Sci.
**2017**, 7, 1228. [Google Scholar] [CrossRef] - Nejati, M.; Dimitri, R.; Tornabene, F.; Hossein, Y.M. Thermal buckling of nanocomposite stiffened cylindrical shells reinforced by functionally graded wavy carbon nano-tubes with temperature-dependent properties. Appl. Sci.
**2017**, 7, 1223. [Google Scholar] [CrossRef] - Malikan, M.; Jabbarzadeh, M.; Dastjerdi, S. Non-linear Static stability of bi-layer carbon nanosheets resting on an elastic matrix under various types of in-plane shearing loads in thermo-elasticity using nonlocal continuum. Microsyst. Technol.
**2017**, 23, 2973–2991. [Google Scholar] [CrossRef] - Malikan, M. Electro-mechanical shear buckling of piezoelectric nanoplate using modified couple stress theory based on simplified first order shear deformation theory. Appl. Math. Model.
**2017**, 48, 196–207. [Google Scholar] [CrossRef] - Malikan, M. Analytical predictions for the buckling of a nanoplate subjected to nonuniform compression based on the four-variable plate theory. J. Appl. Comput. Mech.
**2017**, 3, 218–228. [Google Scholar] - Malikan, M. Buckling analysis of a micro composite plate with nano coating based on the modified couple stress theory. J. Appl. Comput. Mech.
**2018**, 4, 1–15. [Google Scholar] - Malikan, M.; Nguyen, V.B. Buckling analysis of piezo-magnetoelectric nanoplates in hygrothermal environment based on a novel one variable plate theory combining with higher-order nonlocal strain gradient theory. Phys. E Low-Dimens. Syst. Nanostruct.
**2018**, 102, 8–28. [Google Scholar] [CrossRef] - Malikan, M. Temperature influences on shear stability of a nanosize plate with piezoelectricity effect. Multidiscip. Model. Mater. Struct.
**2018**, 14, 125–142. [Google Scholar] [CrossRef] - Golmakani, M.E.; Malikan, M.; Far, M.N.S.; Majidi, H.R. Bending and buckling formulation of graphene sheets based on nonlocal simple first order shear deformation theory. Mater. Res. Express
**2018**, 5, 065010. [Google Scholar] [CrossRef] - Malikan, M.; Far, M.N.S. Differential quadrature method for dynamic buckling of graphene sheet coupled by a viscoelastic medium using neperian frequency based on nonlocal elasticity theory. J. Appl. Comput. Mech.
**2018**, 4, 147–160. [Google Scholar] - Malikan, M.; Dastjerdi, S. Analytical buckling of FG nanobeams on the basis of a new one variable first-order shear deformation beam theory. Int. J. Eng. Appl. Sci.
**2018**, 10, 21–34. [Google Scholar] [CrossRef] - Malikan, M. On the buckling response of axially pressurized nanotubes based on a novel nonlocal beam theory. J. Appl. Comput. Mech.
**2018**. [Google Scholar] [CrossRef] - Fantuzzi, N.; Tornabene, F.; Bacciocchi, M.; Dimitri, R. Free vibration analysis of arbitrarily shaped Functionally Graded Carbon Nanotube-reinforced plates. Compos. Part B-Eng.
**2017**, 115, 384–408. [Google Scholar] [CrossRef] - Nejati, M.; Asanjarani, A.; Dimitri, R.; Tornabene, F. Static and free vibration analysis of functionally graded conical shells reinforced by carbon nanotubes. Int. J. Mech. Sci.
**2017**, 130, 383–398. [Google Scholar] [CrossRef] - Tornabene, F.; Fantuzzi, N.; Bacciocchi, M. Linear static response of nanocomposite plates and shells reinforced by agglomerated carbon nanotubes. Compos. Part B-Eng.
**2017**, 115, 449–476. [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.
**2017**, in press. [Google Scholar] [CrossRef] - Hashemi, S.H.; Mehrabani, H.; Ahmadi-Savadkoohi, A. Forced vibration of nanoplate on viscoelastic substrate with consideration of structural damping: An analytical solution. Compos. Struct.
**2015**, 133, 8–15. [Google Scholar] [CrossRef] - Wang, Y.; Li, F.; Wang, Y. Nonlinear vibration of double layered viscoelastic nanoplates based on nonlocal theory. Physica E
**2015**, 67, 65–76. [Google Scholar] [CrossRef] - Zenkour, A.M.; Sobhy, M. Nonlocal piezo-hygrothermal analysis for vibration characteristics of a piezoelectric Kelvin–Voigt viscoelastic nanoplate embedded in a viscoelastic medium. Acta Mech.
**2018**, 229, 3–19. [Google Scholar] [CrossRef] - Aghababaei, R.; Reddy, J.N. Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates. J. Sound Vib.
**2009**, 326, 277–289. [Google Scholar] [CrossRef] - Malikan, M.; Tornabene, F.; Dimitri, R. Nonlocal three-dimensional theory of elasticity for buckling behavior of functionally graded porous nanoplates using volume integrals. Mater. Res. Express
**2018**, 5, 095006. [Google Scholar] [CrossRef] - Malikan, M.; Nguyen, V.B.; Tornabene, F. Damped forced vibration analysis of single-walled carbon nanotubes resting on viscoelastic foundation in thermal environment using nonlocal strain gradient theory. Eng. Sci. Technol. Int. J.
**2018**, 21, 778–786. [Google Scholar] [CrossRef] - Malikan, M.; Nguyen, V.B.; Tornabene, F. Electromagnetic forced vibrations of composite nanoplates using nonlocal strain gradient theory. Mater. Res. Express
**2018**, 5, 075031. [Google Scholar] [CrossRef] - Malikan, M. Electro-thermal buckling of elastically supported double-layered piezoelectric nanoplates affected by an external electric voltage. Multidiscip. Model. Mater. Struct.
**2018**. [Google Scholar] [CrossRef] - She, G.-L.; Yuan, F.-G.; Ren, Y.-R.; Xiao, W.-S. On buckling and postbuckling behavior of nanotubes. Int. J. Eng. Sci.
**2017**, 121, 130–142. [Google Scholar] [CrossRef] - She, G.-L.; Yuan, F.-G.; Ren, Y.-R. Thermal buckling and post-buckling analysis of functionally graded beams based on a general higher-order shear deformation theory. Appl. Math. Model.
**2017**, 47, 340–357. [Google Scholar] [CrossRef] - She, G.-L.; Yuan, F.-G.; Ren, Y.-R.; Liu, H.-B.; Xiao, W.-S. Nonlinear bending and vibration analysis of functionally graded porous tubes via a nonlocal strain gradient theory. Compos. Struct.
**2018**, 203, 614–623. [Google Scholar] [CrossRef] - Lim, C.W.; Zhang, G.; Reddy, J.N. A Higher-order nonlocal elasticity and strain gradient theory and Its Applications in wave propagation. J. Mech. Phys. Solids
**2015**, 78, 298–313. [Google Scholar] [CrossRef] - Barati, M.R.; Shahverdi, H. Hygro-thermal vibration analysis of graded double-refined-nanoplate systems using hybrid nonlocal stress-strain gradient theory. Compos. Struct.
**2017**, 176, 982–995. [Google Scholar] [CrossRef] - Briassoulis, D. Equivalent orthotropic properties of corrugated sheets. Comput. Struct.
**1986**, 23, 129–138. [Google Scholar] [CrossRef] - Liew, K.M.; Peng, L.X.; Kitipornchai, S. Buckling analysis of corrugated plates using a mesh-free Galerkin method based on the first-order shear deformation theory. Comput. Mech.
**2006**, 38, 61–75. [Google Scholar] [CrossRef] - Liew, K.M.; Peng, L.X.; Kitipornchai, S. Vibration analysis of corrugated Reissner–Mindlin plates using a mesh-free Galerkin method. Int. J. Mech. Sci.
**2009**, 51, 642–652. [Google Scholar] [CrossRef] - Ansari, R.; Sahmani, S.; Arash, B. Nonlocal plate model for free vibrations of single-layered graphene sheets. Phys. Lett. A
**2010**, 375, 53–62. [Google Scholar] [CrossRef] - Wu, C.; Li, W. Free vibration analysis of embedded single-layered nanoplates and graphene sheets by using the multiple time scale method. Comput. Math. Appl.
**2017**, 73, 838–854. [Google Scholar] [CrossRef] - Malekzadeh, P.; Shojaee, M. Free vibration of nanoplates based on a nonlocal two-variable refined plate theory. Compos. Struct.
**2013**, 95, 443–452. [Google Scholar] [CrossRef]

**Figure 2.**Variation of the natural frequency with the nonlocal parameter $\mu $ (

**a**) and the length scale ${l}^{*}$ (

**b**).

**Figure 3.**Natural frequency vs. number of corrugations. ${l}^{*}=0.5$, $\mu =1.41$ nm

^{2}, $g=0.5$ Ns/m

^{2}.

**Figure 4.**Natural frequency vs. viscoelastic damping. ${l}^{*}=0.5$, $\mu =1.41$ nm

^{2}, $c=0.25{L}_{x}$.

**Figure 5.**Natural frequency vs. dimensionless wave amplitude ${F}^{*}$. $\mu =1.41$ nm

^{2}, $g=2$ Ns/m

^{2}, $c=0.1{L}_{x}$.

**Table 1.**Numerical results based on the one-variable first-order shear deformation plate theory (OVFSDT), molecular dynamics, and non-local FSDT. $E=1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{TPa}$, $\nu =0.16$, $h=0.34\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$, $\mu =1.41$ nm

^{2}.

Natural Frequency (THz) | |||
---|---|---|---|

Present-Non-Local, Navier | Non-Local-FSDT, GDQ [36] | Zigzag Graphene, MD [36] | ${\mathit{L}}_{\mathit{x}}={\mathit{L}}_{\mathit{y}}\text{}\left(\mathbf{nm}\right)$ |

OVFSDT | |||

0.0592339 | 0.0584221 | 0.0587725 | 10 |

0.0287035 | 0.0282888 | 0.0273881 | 15 |

0.0153208 | 0.0164593 | 0.0157524 | 20 |

0.0103514 | 0.0107085 | 0.0099480 | 25 |

0.0074258 | 0.0075049 | 0.0070655 | 30 |

0.0055703 | 0.0055447 | 0.0052982 | 35 |

0.0043250 | 0.0042608 | 0.0040985 | 40 |

0.0034512 | 0.0033751 | 0.0032609 | 45 |

0.0028157 | 0.0027388 | 0.0026194 | 50 |

**Table 2.**Numerical results based on the OVFSDT, molecular dynamics, and non-local FSDT. $E=1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{TPa}$, $\upsilon =0.16$, $h=0.34\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$, $\mu =1.34$ nm

^{2}.

Natural Frequency (THz) | |||
---|---|---|---|

Present-Non-Local, Navier | Non-Local-FSDT, GDQ [36] | Armchair Graphene, MD [36] | ${\mathit{L}}_{\mathit{x}}={\mathit{L}}_{\mathit{y}}\text{}\left(\mathbf{nm}\right)$ |

OVFSDT | |||

0.0603135 | 0.0592359 | 0.0595014 | 10 |

0.0290542 | 0.0284945 | 0.0277928 | 15 |

0.0154374 | 0.0165309 | 0.0158141 | 20 |

0.0104078 | 0.0107393 | 0.0099975 | 25 |

0.0074558 | 0.0075201 | 0.0070712 | 30 |

0.0055876 | 0.0055531 | 0.0052993 | 35 |

0.0043356 | 0.0042657 | 0.0041017 | 40 |

0.0034580 | 0.0033782 | 0.0032614 | 45 |

0.0028202 | 0.0027408 | 0.0026197 | 50 |

**Table 3.**Comparative evaluation between OVFSDT and other non-local plate models from literature. $E=1.02\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{TPa}$, $\upsilon =0.3$, $h=0.34\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$.

Theories | Non-Dimensional Natural Frequency ($\overline{\mathit{\omega}}=\mathit{\omega}\mathit{h}\sqrt{\frac{\mathit{\rho}}{\mathit{G}}}$) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{L}}_{\mathit{x}}/{\mathit{L}}_{\mathit{y}}=1\text{}$ | ${\mathit{L}}_{\mathit{x}}/{\mathit{L}}_{\mathit{y}}=2\text{}$ | |||||||||||

${\mathit{L}}_{\mathit{x}}/\mathit{h}=10\text{}$ | ${\mathit{L}}_{\mathit{x}}/\mathit{h}=20\text{}$ | ${\mathit{L}}_{\mathit{x}}/\mathit{h}=10\text{}$ | ${\mathit{L}}_{\mathit{x}}/\mathit{h}=20\text{}$ | |||||||||

$\mathit{\mu}=0$ nm^{2} | $\mathit{\mu}=1$ nm^{2} | $\mathit{\mu}=2$ nm^{2} | $\mathit{\mu}=0$ nm^{2} | $\mathit{\mu}=1$ nm^{2} | $\mathit{\mu}=2$ nm^{2} | $\mathit{\mu}=0$ nm^{2} | $\mathit{\mu}=1$ nm^{2} | $\mathit{\mu}=2$ nm^{2} | $\mathit{\mu}=0$ nm^{2} | $\mathit{\mu}=1$ nm^{2} | $\mathit{\mu}=2$ nm^{2} | |

[37]-case 1 | 0.0955 | 0.0873 | 0.0809 | 0.0240 | 0.0220 | 0.0203 | 0.0599 | 0.0565 | 0.0536 | 0.0150 | 0.0142 | 0.0135 |

[37]-case 2 | 0.0931 | 0.0850 | 0.0788 | 0.0239 | 0.0218 | 0.0202 | 0.0589 | 0.0556 | 0.0528 | 0.0150 | 0.0141 | 0.0134 |

[37]-case 3 | 0.0931 | 0.0851 | 0.0789 | 0.0239 | 0.0218 | 0.0202 | 0.0589 | 0.0556 | 0.0528 | 0.0150 | 0.0141 | 0.0134 |

[37]-case 4 | 0.0931 | 0.0851 | 0.0789 | 0.0239 | 0.0218 | 0.0202 | 0.0589 | 0.0556 | 0.0528 | 0.0150 | 0.0141 | 0.0134 |

Non-local-CPT [23] | 0.0963 | 0.0880 | 0.0816 | 0.0241 | 0.0220 | 0.0204 | 0.0602 | 0.0568 | 0.0539 | 0.0150 | 0.0142 | 0.0135 |

Non-local-FSDT [23] | 0.0930 | 0.0850 | 0.0788 | 0.0239 | 0.0218 | 0.0202 | 0.0589 | 0.0556 | 0.0527 | 0.0150 | 0.0141 | 0.0134 |

Non-local-TSDT [23] | 0.0935 | 0.0854 | 0.0791 | 0.0239 | 0.0218 | 0.0202 | 0.0591 | 0.0557 | 0.0529 | 0.0150 | 0.0141 | 0.0134 |

Non-local-S-FSDT [38] | 0.0930 | 0.0850 | 0.0787 | 0.02386 | 0.0218 | 0.0202 | 0.0588 | 0.0555 | 0.0527 | 0.0149 | 0.0141 | 0.0134 |

Non-local-Present, OVFSDT | 0.0928 | 0.0849 | 0.0780 | 0.02337 | 0.0217 | 0.02019 | 0.0589 | 0.0552 | 0.0521 | 0.0145 | 0.0140 | 0.0134 |

**Table 4.**Material properties of the viscoelastic nanoplate [36].

Elastic properties |

$E=1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{TPa}$, $\upsilon =0.16$ |

Density |

$\rho =2250$ kg/m^{3} |

Dimensional values |

$h=0.34\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$, ${L}_{x}={L}_{y}=10\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ |

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**MDPI and ACS Style**

Malikan, M.; Dimitri, R.; Tornabene, F.
Effect of Sinusoidal Corrugated Geometries on the Vibrational Response of Viscoelastic Nanoplates. *Appl. Sci.* **2018**, *8*, 1432.
https://doi.org/10.3390/app8091432

**AMA Style**

Malikan M, Dimitri R, Tornabene F.
Effect of Sinusoidal Corrugated Geometries on the Vibrational Response of Viscoelastic Nanoplates. *Applied Sciences*. 2018; 8(9):1432.
https://doi.org/10.3390/app8091432

**Chicago/Turabian Style**

Malikan, Mohammad, Rossana Dimitri, and Francesco Tornabene.
2018. "Effect of Sinusoidal Corrugated Geometries on the Vibrational Response of Viscoelastic Nanoplates" *Applied Sciences* 8, no. 9: 1432.
https://doi.org/10.3390/app8091432