The nonlinear frequency response of graphene/piezoelectric sandwich nanoplates under the electric voltage exerted on piezoelectric layer and a shock force pulse is the objective of the present paper. First, using von Karman nonlinear relations, Kelvin–Voigt model, piezoelectric nonlocal elasticity theory and Hamilton's principle, the nonlinear governing differential equation of motion is derived. In the second step, to transform the partial differential equations to the ordinary differential ones, Galerkin technique is employed. Then, governing equation is solved based on multiple scale method. At the end, modulation equation of graphene/piezoelectric sandwich nanoplate is obtained. Emphasizing the effect of the electric voltage and the shock force pulse, the results for fundamental frequency response are obtained. The results emphasize that the exciting loads can cause multivaluedness in frequency response and intensify the jump phenomenon. Also, they can be effective to amplify the vibration behavior of the structure to make resonance. In addition, the thickness of piezoelectric layer plays a significant role in nonlinear frequency response.
EdaGFanchiniGChhowallaM.Large-area ultrathin films of reduced graphene oxide as a transparent and flexible electronic material. Nature Nanotech2008;
3: 270.
2.
LiDMüllerMBGiljeS, et al.
Processable aqueous dispersions of graphene nanosheets. Nature Nanotech2008;
3: 101.
3.
PotekinRKimSMcFarlandDM, et al.
A micromechanical mass sensing method based on amplitude tracking within an ultra-wide broadband resonance. Nonlinear Dyn2018;
92: 287–304.
4.
MahmoudMA.Validity and accuracy of resonance shift prediction formulas for microcantilevers: a review and comparative study. Crit Rev Solid State Mater Sci2016;
41: 386–429.
5.
JiYChoeMChoB, et al.
Organic nonvolatile memory devices with charge trapping multilayer graphene film. Nanotechnology2012;
23: 105202.
6.
ArashBWangQ.Detection of gas atoms with carbon nanotubes. Sci Rep2013;
3: 1782.
7.
BunchJSVan Der ZandeAMVerbridgeSS, et al.
Electromechanical resonators from graphene sheets. Science2007;
315: 490–493.
8.
KuillaTBhadraSYaoD, et al.
Recent advances in graphene based polymer composites. Prog Poly Sci2010;
35: 1350–1375.
9.
KaneCLMeleEJ.Size, shape, and low energy electronic structure of carbon nanotubes. Phys Rev Lett1997;
78: 1932.
10.
MaitiASvizhenkoAAnantramMP.Electronic transport through carbon nanotubes: effects of structural deformation and tube chirality. Phys Rev Lett2002;
88: 126805.
11.
JeongSChoJYSungTH, et al.
Electromechanical modeling and power performance analysis of a piezoelectric energy harvester having an attached mass and a segmented piezoelectric layer. Smart Mater Struct2017;
26: 035035.
12.
KhooSYRadeefZSOngZC, et al.
Structural dynamics effect on voltage generation from dual coupled cantilever based piezoelectric vibration energy harvester system. Measurement2017;
107: 41–52.
13.
ZhangJFangZShuC, et al.
A rotational piezoelectric energy harvester for efficient wind energy harvesting. Sens Actuators A Phys2017;
262: 123–129.
14.
SongJHuGTseKT, et al.
Performance of a circular cylinder piezoelectric wind energy harvester fitted with a splitter plate. Appl Phys Lett2017;
111: 223903.
LiCLiuJJChengM, et al.
Nonlocal vibrations and stabilities in parametric resonance of axially moving viscoelastic piezoelectric nanoplate subjected to thermo-electro-mechanical forces. Compos B Eng2017;
116: 153–169.
17.
ZhangDPLeiYJShenZB.Thermo-electro-mechanical vibration analysis of piezoelectric nanoplates resting on viscoelastic foundation with various boundary conditions. Int J Mech Sci2017;
131: 1001–1015.
18.
KundalwalSIMeguidSAWengGJ.Strain gradient polarization in graphene. Carbon2017;
117: 462–472.
19.
KundalwalSIShingareKBRathiA. Effect of flexoelectricity on the electromechanical response of graphene nanocomposite beam. Int J Mech Mater Des2018; 1–24.
20.
SobhyMZenkourAM.Magnetic field effect on thermomechanical buckling and vibration of viscoelastic sandwich nanobeams with CNT reinforced face sheets on a viscoelastic substrate. Compos B Eng2018;
154: 492–506.
21.
ZengSWangBLWangKF.Nonlinear vibration of piezoelectric sandwich nanoplates with functionally graded porous core with consideration of flexoelectric effect. Compos Struct2019;
207: 340–351.
22.
Ghorbanpour AraniAJamaliMGhorbanpour-AraniAH, et al.
Electro-magneto wave propagation analysis of viscoelastic sandwich nanoplates considering surface effects. Proc Inst Mech Eng C J Mech Eng Sci2017;
231: 387–403.
Ghorbanpour AraniAZamaniMH. Investigation of electric field effect on size-dependent bending analysis of functionally graded porous shear and normal deformable sandwich nanoplate on silica Aerogel foundation. J Sandw Struct Mater. Epub ahead of print 31 July 2017. DOI: 10.1177/1099636217721405.
25.
KundalwalSIKumarRSRayMC.Smart damping of laminated fuzzy fiber reinforced composite shells using 1–3 piezoelectric composites. Smart Mater Struct2013;
22: 105001.
26.
KundalwalSIRayMC.Smart damping of fuzzy fiber reinforced composite plates using 1–3 piezoelectric composites. J Vib Cont2016;
22: 1526–1546.
27.
ArefiMZenkourAM.Nonlocal electro-thermo-mechanical analysis of a sandwich nanoplate containing a Kelvin–Voigt viscoelastic nanoplate and two piezoelectric layers. Acta Mech2017;
228: 475–493.
28.
ArefiMZenkourAM.Thermo-electro-magneto-mechanical bending behavior of size-dependent sandwich piezomagnetic nanoplates. Mech Res Commun2017;
84: 27–42.
29.
ArefiMZenkourAM.Vibration and bending analysis of a sandwich microbeam with two integrated piezo-magnetic face-sheets. Compos Struct2017;
159: 479–490.
30.
ArefiMZamaniMHKianiM.Size-dependent free vibration analysis of three-layered exponentially graded nanoplate with piezomagnetic face-sheets resting on Pasternak’s foundation. J Intell Mater Syst Struct2018;
29: 774–786.
31.
ArefiMZenkourAM.Vibration and bending analyses of magneto–electro–thermo-elastic sandwich microplates resting on viscoelastic foundation. Appl Phys A2017;
123: 550.
32.
KundalwalSIMeguidSA.Effect of carbon nanotube waviness on active damping of laminated hybrid composite shells. Acta Mech2015;
226: 2035–2052.
33.
ArefiMZamaniMHKianiM. Smart electrical and magnetic stability analysis of exponentially graded shear deformable three-layered nanoplate based on nonlocal piezo-magneto-elasticity theory. J Sandw Struct Mater. Epub ahead of print 28 February 2018. DOI: 10.1177/1099636218760667.
34.
KumarRSKundalwalSIRayMC.Control of large amplitude vibrations of doubly curved sandwich shells composed of fuzzy fiber reinforced composite facings. Aerosp Sci Technol2017;
70: 10–28.
35.
KolahchiRZareiMSHajmohammadMH, et al.
Visco-nonlocal-refined Zigzag theories for dynamic buckling of laminated nanoplates using differential cubature-Bolotin methods. Thin-Walled Struct2017;
113: 162–169.
36.
KolahchiR.A comparative study on the bending, vibration and buckling of viscoelastic sandwich nano-plates based on different nonlocal theories using DC, HDQ and DQ methods. Aerosp Sci Technol2017;
66: 235–248.
37.
AsemiSRFarajpourAMohammadiM.Nonlinear vibration analysis of piezoelectric nanoelectromechanical resonators based on nonlocal elasticity theory. Compos Struct2014;
116: 703–712.
38.
Ghorbanpour-AraniAKolahdouzanFAbdollahianM.Nonlocal buckling of embedded magnetoelectroelastic sandwich nanoplate using refined zigzag theory. Appl Math Mech2018;
39: 529–546.
39.
KaramiBShahsavariDLiL, et al.
Thermal buckling of embedded sandwich piezoelectric nanoplates with functionally graded core by a nonlocal second-order shear deformation theory. Proc Inst Mech Eng C J Mech Eng Sci2018;
233: 287–301.
40.
EringenACEdelenDGB.On nonlocal elasticity. Int J Eng Sci1972;
10: 233–248.
41.
EringenAC.Theories of nonlocal plasticity. Int J Eng Sci1983;
21: 741–751.
42.
ZhuXLiL.Twisting statics of functionally graded nanotubes using Eringen’s nonlocal integral model. Compos Struct2017;
178: 87–96.
43.
ZhuXLiL.Longitudinal and torsional vibrations of size-dependent rods via nonlocal integral elasticity. Int J Mech Sci2017;
133: 639–650.
44.
ZhangYQLiuGRWangJS.Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression. Phys Rev B2004;
70: 205430.
45.
BarrettaRFeoLLucianoR, et al.
Variational formulations for functionally graded nonlocal Bernoulli–Euler nanobeams. Compos Struct2015;
129: 80–89.
46.
GhadiriMSafiM.Nonlinear vibration analysis of functionally graded nanobeam using homotopy perturbation method. Adv Appl Math Mech2017;
9: 144–156.
47.
EhyaeiJAkbarshahiAShafieiN.Influence of porosity and axial preload on vibration behavior of rotating FG nanobeam. Adv Nano Res2017;
5: 141–169.
48.
EbrahimiFHosseiniSHS.Thermal effects on nonlinear vibration behavior of viscoelastic nanosize plates. J Therm Stresses2016;
39: 606–625.
49.
AraniAGMaboudiMJKolahchiR.Nonlinear vibration analysis of visoelastically coupled DLAGS-systems. Eur J Mech A Solids2014;
45: 185–197.
50.
MurmuTAdhikariS.Nonlocal mass nanosensors based on vibrating monolayer graphene sheets. Sens Actuators B Chem2013;
188: 1319–1327.
51.
ShenLEShenHSZhangCL.Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments. Comput Mater Sci2010;
48: 680–685.
52.
SetoodehARMalekzadehPVosoughiAR.Nonlinear free vibration of orthotropic graphene sheets using nonlocal Mindlin plate theory. Proc Inst Mech Eng C J Mech Eng Sci2012;
226: 1896–1906.
53.
MalekzadehP.Nonlinear free vibration of tapered Mindlin plates with edges elastically restrained against rotation using DQM. Thin-Walled Struct2008;
46: 11–26.
54.
AraniAGKolahchiRBarzokiAAM, et al.
Elastic foundation effect on nonlinear thermo-vibration of embedded double-layered orthotropic graphene sheets using differential quadrature method. Proc Inst Mech Eng C J Mech Eng Sci2013;
227: 862–879.
55.
MurmuTSienzJAdhikariS, et al.
Nonlocal buckling of double-nanoplate-systems under biaxial compression. Compos B Eng2013;
44: 84–94.
56.
MurmuTAdhikariS.Nonlocal elasticity based vibration of initially pre-stressed coupled nanobeam systems. Eur J Mech A Solids2012;
34: 52–62.
57.
MurmuTMcCarthyMAAdhikariS.Vibration response of double-walled carbon nanotubes subjected to an externally applied longitudinal magnetic field: a nonlocal elasticity approach. J Sound Vib2012;
331: 5069–5086.
58.
JomehzadehESaidiAR.The small scale effect on nonlinear vibration of single layer graphene sheets. World Acad Sci Eng Technol2011;
5: 6–29.
59.
DyniewiczBPisarskiDBajerCI.Vibrations of a Mindlin plate subjected to a pair of inertial loads moving in opposite directions. J Sound Vib2017;
386: 265–282.
60.
ShenZBTangHLLiDK, et al.
Vibration of single-layered graphene sheet-based nanomechanical sensor via nonlocal Kirchhoff plate theory. Comput Mater Sci2012;
61: 200–205.
61.
ZhouSMShengLPShenZB.Transverse vibration of circular graphene sheet-based mass sensor via nonlocal Kirchhoff plate theory. Comput Mater Sci2014;
86: 73–78.
62.
GhadiriMRajabpourAAkbarshahiA.Non-linear forced vibration analysis of nanobeams subjected to moving concentrated load resting on a viscoelastic foundation considering thermal and surface effects. Appl Math Model2017;
50: 676–694.
63.
GhadiriMRajabpourAAkbarshahiA.Non-linear vibration and resonance analysis of graphene sheet subjected to moving load on a visco-Pasternak foundation under thermo-magnetic-mechanical loads: an analytical and simulation study. Measurement2018;
124: 103–119.
64.
EringenAC.Nonlocal polar elastic continua. Int J Eng Sci1972;
10: 1–16.
65.
EringenAC.On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys1983;
54: 4703–4710.
66.
ReddyJN.An introduction to continuum mechanics.
Cambridge:
Cambridge University Press, 2013.
67.
ReddyJN.Theory and analysis of elastic plates and shells.
Boca Raton:
CRC Press, 2006.
68.
KeLLWangYS.Thermoelectric-mechanical vibration of piezoelectric nanobeams based on the nonlocal theory. Smart Mater Struct2012;
21: 025018.
69.
LakesRLakesRS.Viscoelastic materials.
Cambridge:
Cambridge University Press, 2009.
70.
WangYLiFMWangYZ.Nonlinear vibration of double layered viscoelastic nanoplates based on nonlocal theory. Physica E Low Dimens Syst Nanostruct2015;
67: 65–76.
71.
AraniAGShiravandARahiM, et al.
Nonlocal vibration of coupled DLGS systems embedded on visco-Pasternak foundation. Physica B Condens Matter2012;
407: 4123–4131.
72.
KitipornchaiSHeXQLiewKM.Continuum model for the vibration of multilayered graphene sheets. Phys Rev B2005;
72: 075443.
73.
RajuISRaoGVRajuKK.Effect of longitudinal or inplane deformation and inertia on the large amplitude flexural vibrations of slender beams and thin plates. J Sound Vib1976;
49: 415–422.
74.
AlqasimiJEOuakadHM.Vibrational response of initially deformed bistable microbeams under the combined effect of mechanical shock loads and electrostatic forces. J Vib Acoust2018;
140: 021013.
75.
AskariARTahaniM.Size-dependent dynamic pull-in analysis of beam-type MEMS under mechanical shock based on the modified couple stress theory. Appl Math Model2015;
39: 934–946.
76.
OuakadHM.The response of a micro-electro-mechanical system (MEMS) cantilever-paddle gas sensor to mechanical shock loads. J Vib Cont2015;
21: 2739–2754.
77.
OuakadHMYounisMIAlsaleemF.Dynamic response of an electrostatically actuated microbeam to drop-table test. J Micromech Microeng2012;
22: 095003.
78.
YounisMIJordyDPitarresiJM.Computationally efficient approaches to characterize the dynamic response of microstructures under mechanical shock. J Microelectromech Syst2007;
16: 628–638.
79.
NayfehAHMookDT.Nonlinear oscillations.
Hoboken:
John Wiley & Sons, 2008.
80.
NourbakhshHMohammadzadehRRafieeM, et al.
Nonlinear effects on resonance behaviour of beams in micro scale. Appl Mech Mater2012;
110: 4178–4186.
81.
PradhanSCPhadikarJK.Nonlocal elasticity theory for vibration of nanoplates. J Sound Vib2009;
325: 206–223.
82.
ChuHN.Influence of large amplitudes on free flexural vibrations of rectangular elastic plates. J Appl Mech1956;
23: 532–540.
83.
WahT.Large amplitude flexural vibration of rectangular plates. Int J Mech Sci1963;
5: 425–438.
84.
MeiCDecha-UmphaiK.A finite element method for nonlinear forced vibrations of rectangular plates. AIAA J1985;
23: 1104–1110.
85.
ChiaCYPrabhakaraMK.A general mode approach to nonlinear flexural vibrations of laminated rectangular plates. J Appl Mech1978;
45: 623–628.
86.
ManojTAyyappanMKrishnanKS, et al.
Nonlinear vibration analysis of thin laminated rectangular plates on elastic foundations. Z Angew Math Mech2000;
80: 183–192.
