Vibration Analysis of Circular Nanoplates under Nonlinear Electrostatic Excitation Considering the Surface Energy and Size Effects

Authors

1 Department of Engineering Sciences, Faculty of Advanced Technologies, University of Mohaghegh Ardabili, Ardabil, Iran

2 Department of Engineering Sciences, Faculty of Advanced Technologies, University of of Mohaghegh Ardabili, Ardabil, Iran.

3 Department of Bio Information, Faculty of Advanced Technologies, University of Mohaghegh Ardabili, Ardabil, Iran.

Abstract

This article investigates the primary resonant behavior and static pull-in instability of a circular nanoplate under nonlinear electrostatic actuation. The consistent couple stress theory, Gurtin-Murdoch surface elasticity theory and Hamilton principle were utilized to derive the governing differential equation of transverse vibration Kirchhoff nanoplate by considering the fluid damping and Casimir forces. The governing equation were solved for small amplitude vibrations. To this end, it is assumed that the elastic nanoplate is deflected using a DC bias voltage and then driven to vibrate around its deflected position by a harmonic AC load. The weighted residual method of Galerkin was used to obtain a reduced order model. The method of multiple scales is used to solve the nonlinear equation of motion and, the primary resonance mode frequency response equation is derived. The obtained numerical results were compared to those of previous research works, and a good agreement observed between them. The numerical results revealed that electrostatic actuation and Casmier force have softening effects; but the surface energy can has hardening or softening effect depending on the surface mechanical properties, dimensions and boundary condtions of the nanoplate.

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References

 [1] Lam D, Yang F, Chong A, Wang J, and Tong P, (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51(8): 1477-1508.
[2] Miller R E and Shenoy V B, (2000) Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11(3): 139.
[3] Arash B and Wang Q, (2012) A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Comp Mater Sci 51(1): 303-313.
[4] Wang K, Wang B, and Kitamura T, (2016) A review on the application of modified continuum models in modeling and simulation of nanostructures. Acta Mech Sinica 32(1): 83-100.
[5] Toupin R A, (1962) Elastic materials with couple-stresses. Arch Ration Mech An 11(1): 385-414.
[6] Mindlin R and Tiersten H, (1962) Effects of couple-stresses in linear elasticity. Arch Ration Mech An 11(1): 415-448.
[7] Koiter W, (1964) Couple-stresses in the theory of elasticity, I and II, Prec. Roy. Netherlands Acad. Sci. B 67.
[8] Hadjesfandiari A R and Dargush G F, (2011) Couple stress theory for solids. Int J Solids Struct 48(18): 2496-2510.
[9] Aghababaie Beni M, Ghazavi M-R, and Rezazadeh G, (2017) A study of fluid media and size effect on dynamic response of microplate. Modares Mech Eng 17(9): 153-164.
[10] Akbari Alashti R and Abolghasemi A H, (2014) A size-dependent Bernoulli-Euler beam formulation based on a new model of couple stress theory. Int J Eng 27(6): 951-960.
[11] Yang F, Chong A C M, Lam D C C, and Tong P, (2002) Couple stress based strain gradient theory for elasticity. Int J of Solids Struct 39(10): 2731-2743.
[12] Park S and Gao X, (2006) Bernoulli–Euler beam model based on a modified couple stress theory. J Micromech Microeng 16(11): 2355.
[13] Asghari M, Rahaeifard M, Kahrobaiyan M, and Ahmadian M, (2011) The modified couple stress functionally graded Timoshenko beam formulation. Mater Design 32(3): 1435-1443.
[14] Jomehzadeh E, Noori H, and Saidi A, (2011) The size-dependent vibration analysis of micro-plates based on a modified couple stress theory. Physica E Low Dimens Syst Nanostruct 43(4): 877-883.
[15] Ke L-L, Wang Y-S, Yang J, and Kitipornchai S, (2012) Free vibration of size-dependent Mindlin microplates based on the modified couple stress theory. J Sound Vib 331(1): 94-106.
[16] Reddy J and Kim J, (2012) A nonlinear modified couple stress-based third-order theory of functionally graded plates. Compos Struct 94(3): 1128-1143.
[17] Dingreville R, Qu J, and Cherkaoui M, (2005) Surface free energy and its effect on the elastic behavior of nano-sized particles, wires and films. J Mech Phys Solids 53(8): 1827-1854.
[18] Gurtin M E and Murdoch A I, (1975) A continuum theory of elastic material surfaces. Arch Ration Mech An 57(4): 291-323.
[19] Gurtin M E, ME G, and AI M, (1978) Surface stress in solids.
[20] Ansari R and Sahmani S, (2011) Surface stress effects on the free vibration behavior of nanoplates. Int J Eng Sci 49(11): 1204-1215.
[21] Wang K and Wang B, (2012) Effects of residual surface stress and surface elasticity on the nonlinear free vibration of nanoscale plates. J Appl Phys 112(1): 013520.
[22] Ansari R, Gholami R, Faghih Shojaei M, Mohammadi V, and Sahmani S, (2013) Surface stress effect on the vibrational response of circular nanoplates with various edge supports. J Appl Mech 80(2).
[23] Wang K and Wang B L, (2014) Influence of surface energy on the non-linear pull-in instability of nano-switches. Int J Nonlin Mech 59: 69-75.
[24] Wang K, Wang B, and Zhang C, (2017) Surface energy and thermal stress effect on nonlinear vibration of electrostatically actuated circular micro-/nanoplates based on modified couple stress theory. Acta Mech 228(1): 129-140.
[25] Hamidi B A, Hosseini S A, Hassannejad R, and Khosravi F, (2020) Theoretical analysis of thermoelastic damping of silver nanobeam resonators based on Green–Naghdi via nonlocal elasticity with surface energy effects. Eur Phys J Plus 135(1): 1-20.
[26] Hosseini S H S and Ghadiri M, (2021) Nonlinear dynamics of fluid conveying double-walled nanotubes incorporating surface effect: A bifurcation analysis. Appl Math Model 92: 594-611.
[27] Abdelrahman A A, Mohamed N A, and Eltaher M A, (2020) Static bending of perforated nanobeams including surface energy and microstructure effects. Eng Comput: 1-21.
[28] Puers R and Lapadatu D, (1996) Electrostatic forces and their effects on capacitive mechanical sensors. Sens Actuator A Phys 56(3): 203-210.
[29] Nguyen C-C, Katehi L P, and Rebeiz G M, (1998) Micromachined devices for wireless communications. Proc IEEE 86(8): 1756-1768.
[30] Sarafraz A, Sahmani S, and Aghdam M M, (2019) Nonlinear secondary resonance of nanobeams under subharmonic and superharmonic excitations including surface free energy effects. Appl Math Model 66: 195-226.
[31] Mamandi A and Mirzaei ghaleh M, (2020) Nonlinear Vibration of a Microbeam on a Winkler Foundation and Subjected to an Axial Load using Modified Couple StressTheory. J Solid Fluid Mech 10(4): 181-194.
[32] Sarafraz A, Sahmani S, and Aghdam M, (2020) Nonlinear primary resonance analysis of nanoshells including vibrational mode interactions based on the surface elasticity theory. Appl Math Mech 41(2): 233-260.
[33] Sahmani S, Fattahi A, and Ahmed N, (2020) Surface elastic shell model for nonlinear primary resonant dynamics of FG porous nanoshells incorporating modal interactions. Int J Mech Sci 165: 105203.
[34] Xie B, Sahmani S, Safaei B, and Xu B, (2021) Nonlinear secondary resonance of FG porous silicon nanobeams under periodic hard excitations based on surface elasticity theory. Eng Comput 37(2): 1611-1634.
 
[35] Talebian S, Rezazadeh G, Fathalilou M, and Toosi B, (2010) Effect of temperature on pull-in voltage and natural frequency of an electrostatically actuated microplate. Mechatronics 20(6): 666-673.
[36] Gies H and Klingmüller K, (2006) Casimir effect for curved geometries: Proximity-Force-Approximation validity limits. Phys rev lett 96(22): 220401.
[37] Bao M and Yang H, (2007) Squeeze film air damping in MEMS. Sens. Actuator A Phys 136(1): 3-27.
[38] Rezazadeh G, Tahmasebi A, and Zubstov M, (2006) Application of piezoelectric layers in electrostatic MEM actuators: controlling of pull-in voltage. Microsyst technol 12(12): 1163-1170.
[39] Nayfeh A H and Mook D T,(1979) Nonlinear oscillations. Willey ,. New York.
[40] Kroeger F and Swenson C, (1977) Absolute linear thermal expansion measurements on copper and aluminum from 5 to 320 K. J Appl Phys 48(3): 853-864.
[41] Al-Damook A, Summers J, Kapur N, and Thompson H,(2016) Effect of temperature-dependent air properties on the accuracy of numerical simulations of thermal airflows over pinned heat sinks. Int Commun Heat Mass Transfer 78:163-167.
[42] Raback P and Pursula A,(2004) Finite Element Simulation of the Electro-Mechanical Pull-In Phenomenon, in European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS: Jyväskylä, Finland.
[43] Osterberg P,(1995) Electrostatically Actuated Microelectromechanical Test Structures for Material Property Measurement, in Department of Electrical Engineering and Computer Science. MIT.
[44] Caruntu D I, Martinez I, and Taylor K N, (2013) Voltage–amplitude response of alternating current near half natural frequency electrostatically actuated MEMS resonators. Mech Res Commun 52: 25-31.
[45] Gheshlaghi B and Hasheminejad S M, (2011) Surface effects on nonlinear free vibration of nanobeams. Compos B Eng 42(4): 934-937.
Journal of Solid and Fluid Mechanics
Volume 12, Issue 5
November and December 2022
Pages 133-146

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