Viscoelastic effects on nonlinear dynamics of microplates with fluid interaction based on consistent couple stress theory

Document Type : Research Paper

Authors

1 Department of Mechanical Engineering, Faculty of Engineering, University of Mohaghegh Ardabili, Ardabil, Iran

2 School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran

Abstract

In this study, the vibrations of a viscoelastic microplate in incompressible still fluid are investigated. The microplate is supposed simply supported in stationary fluid and made of viscoelastic material that follow the Leaderman viscoelastic model The fluid inertial effects as well as fluid damping on microplate vibration are also studied by applying three dimensional aerodynamic theory. The added mass values are obtained for various aspect ratios of the microplate and compared with available models. The non-classic stress and couple-stress tensors are calculated based on consistent couple stress theory and Leaderman integral. The resultant virtual works inserted in Hamilton’s principle and the governing equations of motion are derived. These coupled equation are simplified by Galerkin method and solved using harmonic balance analytical method. The results show that considering the added mass effects reduces the microsystem nonlinearity.

Keywords

[1]           M. M. Khoram, M. Hosseini, A. Hadi, M. J. I. J. o. A. M. Shishehsaz, Bending Analysis of Bidirectional FGM Timoshenko Nanobeam Subjected to Mechanical and Magnetic Forces and Resting on Winkler–Pasternak Foundation, Vol. 12, No. 08, pp. 2050093, 2020.
[2]           H. Gensler, R. Sheybani, P.-Y. Li, R. L. Mann, E. J. B. m. Meng, An implantable MEMS micropump system for drug delivery in small animals, Vol. 14, No. 3, pp. 483-496, 2012.
[3]           N.-T. J. J. Nguyen, et al.,“MEMS-Micropumps: A Review,” Transaction of ASME, Vol. 124, 2002.
[4]           A. Nisar, N. Afzulpurkar, B. Mahaisavariya, A. J. S. Tuantranont, A. B. Chemical, MEMS-based micropumps in drug delivery and biomedical applications, Vol. 130, No. 2, pp. 917-942, 2008.
[5]           S. Inaba, K. Akaishi, T. Mori, K. Hane, Analysis of the resonance characteristics of a cantilever vibrated photothermally in a liquid, Journal of applied physics, Vol. 73, No. 6, pp. 2654-2658, 1993.
[6]           M. M. Khoram, M. Hosseini, A. Hadi, M. Shishehsaz, Bending Analysis of Bidirectional FGM Timoshenko Nanobeam Subjected to Mechanical and Magnetic Forces and Resting on Winkler–Pasternak Foundation, International Journal of Applied Mechanics, Vol. 12, No. 08, pp. 2050093, 2020.
[7]           A. Soleimani, K. Dastani, A. Hadi, M. H. Naei, Effect of out-of-plane defects on the postbuckling behavior of graphene sheets based on nonlocal elasticity theory, Steel and Composite Structures, Vol. 30, No. 6, pp. 517-534, 2019.
[8]           A. Hadi, A. Rastgoo, N. Haghighipour, A. Bolhassani, Numerical modelling of a spheroid living cell membrane under hydrostatic pressure, Journal of Statistical Mechanics: Theory and Experiment, Vol. 2018, No. 8, pp. 083501, 2018.
[9]           A. Hadi, M. Z. Nejad, M. Hosseini, Vibrations of three-dimensionally graded nanobeams, International Journal of Engineering Science, Vol. 128, pp. 12-23, 2018.
[10]         M. Z. Nejad, A. J. I. J. o. E. S. Hadi, Non-local analysis of free vibration of bi-directional functionally graded Euler–Bernoulli nano-beams, Vol. 105, pp. 1-11, 2016.
[11]         A. Hadi, M. Z. Nejad, M. J. I. J. o. E. S. Hosseini, Vibrations of three-dimensionally graded nanobeams, Vol. 128, pp. 12-23, 2018.
[12]         M. M. Adeli, A. Hadi, M. Hosseini, H. H. J. T. E. P. J. P. Gorgani, Torsional vibration of nano-cone based on nonlocal strain gradient elasticity theory, Vol. 132, No. 9, pp. 1-10, 2017.
[13]         G.-L. She, H.-B. Liu, B. J. T.-W. S. Karami, Resonance analysis of composite curved microbeams reinforced with graphene nanoplatelets, Vol. 160, pp. 107407, 2021.
[14]         J. K. Sinha, S. Singh, A. Rama Rao, Added mass and damping of submerged perforated plates, Journal of Sound Vibration, Vol. 260, pp. 549-564, 2003.
[15]         D. I. Caruntu, I. Martinez, Reduced order model of parametric resonance of electrostatically actuated MEMS cantilever resonators, International Journal of Non-Linear Mechanics, Vol. 66, pp. 28-32, 2014.
[16]         S. D. Vishwakarma, A. K. Pandey, J. M. Parpia, D. R. Southworth, H. G. Craighead, R. Pratap, Evaluation of mode dependent fluid damping in a high frequency drumhead microresonator, Journal of Microelectromechanical Systems, Vol. 23, No. 2, pp. 334-346, 2014.
[17]         Y. Yadykin, V. Tenetov, D. Levin, The added mass of a flexible plate oscillating in a fluid, Journal of Fluids Structures, Vol. 17, No. 1, pp. 115-123, 2003.
[18]         D. G. Gorman, I. Trendafilova, A. J. Mulholland, J. Horáček, Analytical modelling and extraction of the modal behaviour of a cantilever beam in fluid interaction, Journal of Sound Vibration, Vol. 308, No. 1-2, pp. 231-245, 2007.
[19]         A. J. Pretlove, Note on the virtual mass for a panel in an infinite baffle, Journal of the Acoustical Society of America, Vol. 38, pp. 266–270, 1965.
[20]         H. Dai, L. Wang, Q. Ni, Dynamics and pull-in instability of electrostatically actuated microbeams conveying fluid, Microfluidics Nanofluidics, Vol. 18, No. 1, pp. 49-55, 2015.
[21]         S. JK, S. S, R. R. A, Added mass and damping of submerged perforated plates, Journal of Sound and Vibration, Vol. 260, No. 3, pp. 549–564, 2003.
[22]         A. Amiri, R. Shabani, G. Rezazadeh, Coupled vibrations of a magneto-electro-elastic micro-diaphragm in micro-pumps, Microfluidics Nanofluidics, Vol. 20, No. 1, pp. 18, 2016.
[23]         G. Jabbari, R. Shabani, G. Rezazadeh, Nonlinear vibrations of an electrostatically actuated microresonator in an incompressible fluid cavity based on the modified couple stress theory, Journal of Computational Nonlinear Dynamics, Vol. 11, No. 4, pp. 041029, 2016.
[24]         M. Elwenspoek, H. V. Jansen, 2004, Silicon micromachining, Cambridge University Press,
[25]         K. S. Teh, L. Lin, Time-dependent buckling phenomena of polysilicon micro beams, Microelectronics journal, Vol. 30, No. 11, pp. 1169-1172, 1999.
[26]         J. Liu, Y. Zhang, L. Fan, Nonlocal vibration and biaxial buckling of double-viscoelastic-FGM-nanoplate system with viscoelastic Pasternak medium in between, Physics letters A, Vol. 381, No. 14, pp. 1228-1235, 2017.
[27]         S. Pouresmaeeli, E. Ghavanloo, S. Fazelzadeh, Vibration analysis of viscoelastic orthotropic nanoplates resting on viscoelastic medium, Composite Structures, Vol. 96, pp. 405-410, 2013.
[28]         D. Karličić, P. Kozić, R. Pavlović, Free transverse vibration of nonlocal viscoelastic orthotropic multi-nanoplate system (MNPS) embedded in a viscoelastic medium, Composite Structures, Vol. 115, pp. 89-99, 2014.
[29]         H. Farokhi, M. H. Ghayesh, Viscoelasticity effects on resonant response of a shear deformable extensible microbeam, Nonlinear Dynamics, Vol. 87, No. 1, pp. 391-406, 2017.
[30]         M. Ajri, M. M. S. Fakhrabadi, Nonlinear free vibration of viscoelastic nanoplates based on modified couple stress theory, Journal of Computational Applied Mechanics, Vol. 49, No. 1, 2018.
[31]         M. Ajri, A. Rastgoo, M. M. S. J. P. S. Fakhrabadi, How does flexoelectricity affect static bending and nonlinear dynamic response of nanoscale lipid bilayers?, Vol. 95, No. 2, pp. 025001, 2019.
[32]         M. Ajri, A. Rastgoo, M. M. S. J. I. J. o. A. M. Fakhrabadi, Primary and secondary resonance analyses of viscoelastic nanoplates based on strain gradient theory, Vol. 10, No. 10, pp. 1850109, 2018.
[33]         M. Mousavi Khoram, M. Hosseini, M. J. J. o. C. A. M. Shishesaz, A concise review of nano-plates, Vol. 50, No. 2, pp. 420-429, 2019.
[34]         M. Ajri, M. M. S. Fakhrabadi, A. J. L. A. J. o. S. Rastgoo, Structures, Analytical solution for nonlinear dynamic behavior of viscoelastic nano-plates modeled by consistent couple stress theory, Vol. 15, No. 9, 2018.
[35]         A. R. Hadjesfandiari, G. F. Dargush, Couple stress theory for solids, International Journal of Solids and Structures, Vol. 48, No. 18, pp. 2496-2510, 2011.
[36]         A. R. Hadjesfandiari, G. F. Dargush, Fundamental solutions for isotropic size-dependent couple stress elasticity, International Journal of Solids and Structures, Vol. 50, No. 9, pp. 1253-1265, 2013.
[37]         A. R. Hadjesfandiari, G. F. Dargush, A. Hajesfandiari, Consistent skew-symmetric couple stress theory for size-dependent creeping flow, Journal of Non-Newtonian Fluid Mechanics, Vol. 196, pp. 83-94, 2013.
[38]         M. Z. Nejad, A. Hadi, A. J. S. e. Farajpour, m. A. i. journal, Consistent couple-stress theory for free vibration analysis of Euler-Bernoulli nano-beams made of arbitrary bi-directional functionally graded materials, Vol. 63, No. 2, pp. 161-169, 2017.
[39]         A. Hadi, M. Z. Nejad, A. Rastgoo, M. J. S. Hosseini, C. Structures, Buckling analysis of FGM Euler-Bernoulli nano-beams with 3D-varying properties based on consistent couple-stress theory, Vol. 26, No. 6, pp. 663-672, 2018.
[40]         L. JE., 1989, Boundary stabilization of thin plates, SIAM, Philadelphia
[41]         J. N. Reddy, 2006, Theory and analysis of elastic plates and shells, CRC press,
[42]         H. Ma, X.-L. Gao, J. Reddy, A non-classical Mindlin plate model based on a modified couple stress theory, Acta mechanica, Vol. 220, No. 1-4, pp. 217-235, 2011.
[43]         A. M. Esfahani, M. Bahrami, S. R. G. Anbarani, Forced vibration analysis of a viscoelastic polymeric piezoelectric microplate with fluid interaction, Micro & Nano Letters, Vol. 11, No. 7, pp. 395-401, 2016.
[44]         S. Kirstein, M. Mertesdorf, M. Schönhoff, The influence of a viscous fluid on the vibration dynamics of scanning near-field optical microscopy fiber probes and atomic force microscopy cantilevers, Journal of Applied Physics, Vol. 84, No. 4, pp. 1782-1790, 1998.
[45]         H. Lamb, 1932,  Hydrodynamics, Cambridge   University   Press, Cambridge
[46]         F. M. White, I. Corfield, 2006, Viscous fluid flow, McGraw-Hill New York,
[47]         H. J. J. o. F. Minami, Added mass of a membrane vibrating at finite amplitude, Journal of Fluids Structures, Vol. 12, No. 7, pp. 919-932, 1998.
[48]         A. Lucey, P. Carpenter, The hydroelastic stability of three-dimensional disturbances of a finite compliant wall, Journal of Sound Vibration, Vol. 165, No. 3, pp. 527-552, 1993.
[49]         A. Niyogi, Nonlinear bending of rectangular orthotropic plates, International Journal of Solids and Structures, Vol. 9, No. 9, pp. 1133-1139, 1973.
[50]         R. E. Mickens, 2010, Truly nonlinear oscillations: harmonic balance, parameter expansions, iteration, and averaging methods, World Scientific,
[51]         T. Y.-T. J. J. o. F. M. Wu, Hydromechanics of swimming propulsion. Part 1. Swimming of a two-dimensional flexible plate at variable forward speeds in an inviscid fluid, Vol. 46, No. 2, pp. 337-355, 1971.
Volume 52, Issue 3
September 2021
Pages 394-407
  • Receive Date: 06 March 2021
  • Revise Date: 02 September 2021
  • Accept Date: 03 September 2021