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Frontiers of Mechanical Engineering

ISSN 2095-0233

ISSN 2095-0241(Online)

CN 11-5984/TH

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Front. Mech. Eng.    2015, Vol. 10 Issue (2) : 187-197    https://doi.org/10.1007/s11465-015-0339-9
RESEARCH ARTICLE
Vibration analysis of nano-structure multilayered graphene sheets using modified strain gradient theory
Amir ALLAHBAKHSHI1,Masih ALLAHBAKHSHI2,*()
1. Department of Mechanical Engineering, Science and Research Branch, Islamic Azad University, Arak, Iran
2. Department of Civil Engineering, Mazandaran University of Technolo- gy, Babol, Iran
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Abstract

In this paper, for the first time, the modified strain gradient theory is used as a new size-dependent Kirchhoff micro-plate model to study the effect of interlayer van der Waals (vdW) force for the vibration analysis of multilayered graphene sheets (MLGSs). The model contains three material length scale parameters, which may effectively capture the size effect. The model can also degenerate into the modified couple stress plate model or the classical plate model, if two or all of the material length scale parameters are taken to be zero. After obtaining the governing equations based on modified strain gradient theory via principle of minimum potential energy, as only infinitesimal vibration is considered, the net pressure due to the vdW interaction is assumed to be linearly proportional to the deflection between two layers. To solve the governing equation subjected to the boundary conditions, the Fourier series is assumed for w=w(x, y). To show the accuracy of the formulations, present results in specific cases are compared with available results in literature and a good agreement can be seen. The results indicate that the present model can predict prominent natural frequency with the reduction of structural size, especially when the plate thickness is on the same order of the material length scale parameter.
Keywords graphene      van der Waals (vdW) force      modi- fied strain gradient elasticity theory      size effect parameter     
Corresponding Author(s): Masih ALLAHBAKHSHI   
Online First Date: 30 June 2015    Issue Date: 14 July 2015
 Cite this article:   
Amir ALLAHBAKHSHI,Masih ALLAHBAKHSHI. Vibration analysis of nano-structure multilayered graphene sheets using modified strain gradient theory[J]. Front. Mech. Eng., 2015, 10(2): 187-197.
 URL:  
https://academic.hep.com.cn/fme/EN/10.1007/s11465-015-0339-9
https://academic.hep.com.cn/fme/EN/Y2015/V10/I2/187
Fig.1  Rectangular multilayered nano-graphene sheet
m n ω 1 M S G T /THz ω 2 M S G T /THz ω 1 M C S T /THz ω 2 M C S T /THz ω 1 CPT /THz ω 2 CPT /THz
1 1 0.1895 2.6888 0.1165 2.6846 0.0690 2.6830
1 2 0.4742 2.7237 0.2911 2.6979 0.1725 2.6877
1 3 0.9499 2.8454 0.5823 2.7446 0.3450 2.7042
2 1 0.4742 2.7237 0.2911 2.6979 0.1725 2.6877
2 2 0.7594 2.7876 0.4658 2.7223 0.2760 2.6963
2 3 1.2361 2.9532 0.7570 2.7869 0.4485 2.7194
3 1 0.9499 2.8454 0.5823 2.7446 0.3450 2.7042
3 2 1.2361 2.9532 0.7570 2.7869 0.4485 2.7194
3 3 1.7143 3.1832 1.0481 2.8796 0.6210 2.7531
Tab.1  Classical natural frequencies and resonant natural frequencies for l / h = 0.1 of a square double-layered GS with width b = 10 nm for the MSGT, MCST and CPT in the first nine modes
m n ω 1 /THz ω 2 /THz ω 3 /THz ω 4 /THz ω 5 /THz
1 1 0.1895 1.1486 2.2001 3.0571 3.6083
1 2 0.4742 1.2281 2.2426 3.0878 3.6344
1 3 0.9499 1.4784 2.3889 3.1957 3.7264
2 1 0.4742 1.2281 2.2426 3.0878 3.6344
2 2 0.7594 1.3639 2.3197 3.1443 3.6825
2 3 1.2361 1.6767 2.5164 3.2921 3.8095
3 1 0.9499 1.4784 2.3889 3.1957 3.7264
3 2 1.2361 1.6767 2.5164 3.2921 3.8095
3 3 1.7143 2.0548 2.7827 3.4998 3.9903
Tab.2  Classical natural frequencies and resonant frequencies for l = 0.1 h of a square five-layered GS with width b =10nm for MSGT
m n ω 1 /THz ω 2 /THz ω 3 /THz ω 4 /THz ω 5 /THz
1 1 0.1165 1.1389 2.1950 3.0534 3.6052
1 2 0.2911 1.1697 2.2112 3.0651 3.6151
1 3 0.5823 1.2738 2.2679 3.1063 3.6501
2 1 0.2911 1.1697 2.2112 3.0651 3.6151
2 2 0.4658 1.2249 2.2409 3.0866 3.6333
2 3 0.7570 1.3625 2.3189 3.1437 3.6820
3 1 0.5823 1.2738 2.2679 3.1063 3.6501
3 2 0.7570 1.3625 2.3189 3.1437 3.6820
3 3 1.0481 1.5434 2.4296 3.2262 3.7527
Tab.3  Classical natural frequencies and resonant frequencies for l = 0.1 h of a square five-layered GS with width b =10 nm for the MCST
Fig.2   ω 1 of m = n = 1 with a variable size scale h / l for the three models of MSGT, MCST and CPT
Fig.3   ω 2 of m = n = 1 with a variable size scale h / l for the three models of MSGT, MCST and CPT
Fig.4   ω 1 and ω 2 of a square double-layered GS with m = 1 and different n of MSGT for l = 0.1 h
Fig.5   ω 1 , ω 2 , ω 3 , ω 4 and ω 5 of a square five-layered GS with m = 1 and different n of MSGT for l = 0.1 h
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