Top

Archive of Applied Mechanics

Published in:

08-02-2024 | Original

A size-dependent axisymmetric plate element: application to MEMS

Authors: Masoud Rahaeifard, Ali Karimzadeh

Published in: Archive of Applied Mechanics | Issue 3/2024

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

This paper presents the formulation of a novel axisymmetric plate element capable of capturing size effects observed in micro-scaled structures. To establish the element formulation, a size-dependent beam element is considered, and by axisymmetric expansion of the model, the stiffness and mass matrices and force vector for an axisymmetric plate element are derived. Comparing the results of this model with those from literature and the outcomes of COMSOL confirms that the present FE formulation can accurately predict the static and dynamic behavior of microplates as well as macro-scale plates. Furthermore, a convergence analysis is performed which indicates that this model can accurately predict the static deflection and natural frequency of circular plates utilizing very low number of elements and consequently with low values of computation cost. As an example of real-world application, the model is applied to analysis of microelectromechanical devices and its accuracy is confirmed.

previous article Dynamics of the perturbed restricted three-body problem with quantum correction and modified gravitational potential

next article Nanowire size-effect on the equilibrium positions of a dislocation dipole

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Karimzadeh, A., Rahaeifard, M.: Nonlinear vibrational analysis and linearization error investigation in size-dependent resonant pressure microsensors. Mech. Based Des. Struct (2023). https://doi.org/10.1080/15397734.2023.2272681CrossRef

Zhou, S.-M., Sheng, L.-P., Shen, Z.-B.: Transverse vibration of circular graphene sheet-based mass sensor via nonlocal Kirchhoff plate theory. Comput. Mater. Sci. 86, 73–78 (2014). https://doi.org/10.1016/j.commatsci.2014.01.031CrossRef

Asadi Dereshgi, H., Dal, H., Sayan, M.E.: Analytical analysis of a circular unimorph piezoelectric actuator in the range of low voltages and pressures. Microsyst. Technol. 26(8), 2453–2464 (2020). https://doi.org/10.1007/s00542-020-04786-wCrossRef

Kong, S.: A review on the size-dependent models of micro-beam and micro-plate based on the modified couple stress theory. Arch. Comput. Method E 29(1), 1–31 (2022). https://doi.org/10.1007/s11831-021-09567-wMathSciNetCrossRef

Ji, M., Wu, Y.-C., Ma, C.-C.: Theoretical analyses and numerical simulation of flexural vibration based on Reddy and modified higher-order plate theories for a transversely isotropic circular plate. Acta Mech. 232(7), 2825–2842 (2021). https://doi.org/10.1007/s00707-021-02973-yMathSciNetCrossRef

Wei, L., Qing, H.: Bending, buckling and vibration analysis of Bi-directional functionally graded circular/annular microplate based on MCST. Compos. Struct. 292, 115633 (2022). https://doi.org/10.1016/j.compstruct.2022.115633CrossRef

Jallouli, A., Kacem, N., Najar, F., et al.: Modeling and experimental characterization of squeeze film effects in nonlinear capacitive circular microplates. Mech. Syst. Signal Process. 127, 68–88 (2019). https://doi.org/10.1016/j.ymssp.2019.02.060ADSCrossRef

Caruntu, D.I., Oyervides, R.: Amplitude–frequency response of parametric resonance of electrostatically actuated MEMS clamped circular plate. Int. J. Non-Linear Mech. 149, 104310 (2023). https://doi.org/10.1016/j.ijnonlinmec.2022.104310ADSCrossRef

Das, M., Bhushan, A.: Investigation of an electrostatically actuated imperfect circular microplate under transverse pressure for pressure sensor applications. Eng. Res. Expr. 3(4), 045023 (2021). https://doi.org/10.1088/2631-8695/ac3771CrossRef

10.

Rai, A.K., Gupta, S.S.: Nonlinear vibrations of a thin isotropic circular plate subjected to moving point loads. J. Mech. Eng. Sci. 235(20), 4900–4912 (2021). https://doi.org/10.1177/0954406220976156CrossRef

11.

Pazhouheshgar, A., Haghighatfar, Y., Moghanian, A.: Finite element method and analytical analysis of static and dynamic pull-in instability of a functionally graded microplate. J. Vib. Control 28(3–4), 425–438 (2022). https://doi.org/10.1177/1077546320980208MathSciNetCrossRef

12.

Raeisi Estabragh, E., Baradaran, G.H.: Analysis of large deflection of nanobeams based on the modified couple stress theory by using finite element method. Arch. Appl. Mech. 91(12), 4717–4734 (2021). https://doi.org/10.1007/s00419-021-02029-6ADSCrossRef

13.

Belounar, A., Boussem, F., Houhou, M.N., et al.: Strain-based finite element formulation for the analysis of functionally graded plates. Arch. Appl. Mech. 92(7), 2061–2079 (2022). https://doi.org/10.1007/s00419-022-02160-yADSCrossRef

14.

Gupta, N.K.: Response simulations of clamped circular steel plates under uniform impulse and effects of axisymmetric stiffener configurations. Int. J. Impact Eng 159, 104049 (2022). https://doi.org/10.1016/j.ijimpeng.2021.104049CrossRef

15.

Esen, I.: A new finite element for transverse vibration of rectangular thin plates under a moving mass. Finite Elem. Anal. Des. 66, 26–35 (2013). https://doi.org/10.1016/j.finel.2012.11.005MathSciNetCrossRef

16.

Moayeri, M., Darabi, B., Sianaki, A.H., et al.: Third order nonlinear vibration of viscoelastic circular microplate based on softening and hardening nonlinear viscoelastic foundation under thermal loading. Eur. J. Mech. A-Solid 95, 104644 (2022). https://doi.org/10.1016/j.euromechsol.2022.104644ADSMathSciNetCrossRef

17.

Moayeri, M., Darabi, B., Hoseini Sianaki, A., et al.: Effect of small scale on nonlinear vibrations of viscoelastic circular microplate rested on nonlinear viscoelastic foundation. J. Aeronaut. Eng. 24(2), 152–166 (2022)

18.

Fleck, N.A., Muller, G.M., Ashby, M.F., et al.: Strain gradient plasticity: theory and experiment. Acta Metal. Mater. 42(2), 475–487 (1994). https://doi.org/10.1016/0956-7151(94)90502-9CrossRef

19.

McFarland, A.W., Colton, J.S.: Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J. Micromech. Microeng. 15(5), 1060 (2005). https://doi.org/10.1088/0960-1317/15/5/024ADSCrossRef

20.

Lam, D.C.C., Yang, F., Chong, A.C.M., et al.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51(8), 1477–1508 (2003). https://doi.org/10.1016/S0022-5096(03)00053-XADSCrossRef

21.

Esen, I.: Response of a micro-capillary system exposed to a moving mass in magnetic field using nonlocal strain gradient theory. Int. J. Mech. Sci. 188, 105937 (2020). https://doi.org/10.1016/j.ijmecsci.2020.105937CrossRef

22.

Esen, I., Özmen, R.: Thermal vibration and buckling of magneto-electro-elastic functionally graded porous nanoplates using nonlocal strain gradient elasticity. Compos. Struct. 296, 115878 (2022). https://doi.org/10.1016/j.compstruct.2022.115878CrossRef

23.

Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11(1), 415–448 (1962). https://doi.org/10.1007/BF00253946MathSciNetCrossRef

24.

Yang, F., Chong, A.C.M., Lam, D.C.C., et al.: Couple stress based strain gradient theory for elasticity. Int. J. Solid. Struct. 39(10), 2731–2743 (2002). https://doi.org/10.1016/S0020-7683(02)00152-XCrossRef

25.

Özarpa, C., Esen, I.: Modelling the dynamics of a nanocapillary system with a moving mass using the non-local strain gradient theory. Math. Method Appl. Sci. (2020). https://doi.org/10.1002/mma.6812CrossRef

26.

Koç, M.A., Esen, İ, Eroğlu, M.: Thermomechanical vibration response of nanoplates with magneto-electro-elastic face layers and functionally graded porous core using nonlocal strain gradient elasticity. Mech. Adv. Mater. Struct. (2023). https://doi.org/10.1080/15376494.2023.2199412CrossRef

27.

Esen, I., Özmen, R.: Free and forced thermomechanical vibration and buckling responses of functionally graded magneto-electro-elastic porous nanoplates. Mech. Based Des. Struct. (2022). https://doi.org/10.1080/15397734.2022.2152045CrossRef

28.

Li, Z., He, Y., Lei, J., et al.: A standard experimental method for determining the material length scale based on modified couple stress theory. Int. J. Mech. Sci. 141, 198–205 (2018). https://doi.org/10.1016/j.ijmecsci.2018.03.035CrossRef

29.

Rahaeifard, M., Kahrobaiyan, M., Asghari, M., et al.: Static pull-in analysis of microcantilevers based on the modified couple stress theory. Sensor Actuators A-Phys. 171(2), 370–374 (2011)CrossRef

30.

Romanoff, J., Reddy, J.: Experimental validation of the modified couple stress Timoshenko beam theory for web-core sandwich panels. Compos. Struct. 111, 130–137 (2014)CrossRef

31.

Al-Furjan, M.S.H., Samimi-Sohrforozani, E., Habibi, M., et al.: Vibrational characteristics of a higher-order laminated composite viscoelastic annular microplate via modified couple stress theory. Compos. Struct. 257, 113152 (2021). https://doi.org/10.1016/j.compstruct.2020.113152CrossRef

32.

Rahi, A.: Vibration analysis of multiple-layer microbeams based on the modified couple stress theory: analytical approach. Arch. Appl. Mech. 91(1), 23–32 (2021). https://doi.org/10.1007/s00419-020-01795-zADSCrossRef

33.

Zhou, H., Jiang, H., Li, P., et al.: Thermoelastic damping in the size-dependent micro/nanobeam resonator with nonlocal dual-phase-lag heat conduction. Thin Wall. Struct. 169, 108437 (2021). https://doi.org/10.1016/j.tws.2021.108437CrossRef

34.

Karimzadeh, A., Ahmadian, M.T., Rahaeifard, M.: Effect of size dependency on in-plane vibration of circular micro-rings. Sci. Iran 24(4), 1996–2008 (2017). https://doi.org/10.24200/SCI.2017.4289CrossRef

35.

Karimzadeh, A., Roohi, R., Akbari, M.: Piezoelectric wind energy harvesting from vortex induced vibrations of an elastic beam. Sci. Iran (2022). https://doi.org/10.24200/SCI.2022.59718.6393CrossRef

36.

Kahrobaiyan, M.H., Asghari, M., Ahmadian, M.T.: A Timoshenko beam element based on the modified couple stress theory. Int. J. Mech. Sci. 79, 75–83 (2014). https://doi.org/10.1016/j.ijmecsci.2013.11.014CrossRef

37.

Wu, C.P., Lyu, Y.S.: An asymptotic consistent couple stress theory for the three-dimensional free vibration analysis of functionally graded microplates resting on an elastic medium. Math. Method Appl. Sci. 46(4), 4891–4919 (2023). https://doi.org/10.1002/mma.8810ADSMathSciNetCrossRef

38.

Sahrawat, R.K., Duhan, A., Kumar, K.: Study of vibrations in micro-scale piezothermoelastic beam resonator utilising modified couple stress theory. Acta Mech. (2023). https://doi.org/10.1007/s00707-023-03575-6MathSciNetCrossRef

39.

Kahrobaiyan, M.H., Asghari, M., Ahmadian, M.T.: Strain gradient beam element. Finite Elem. Anal. Des. 68, 63–75 (2013). https://doi.org/10.1016/j.finel.2012.12.006MathSciNetCrossRef

40.

Karimzadeh, A., Ahmadian, M.T.: Vibrational characteristics of size-dependent vibrating ring gyroscope. Sci. Iran 25(6 Special Issue Dedicated to Professor Goodarz Ahmadi), 3151–3160 (2018). https://doi.org/10.24200/SCI.2018.20495CrossRef

41.

Mohammadi, V., Ansari, R., Faghih Shojaei, M., et al.: Size-dependent dynamic pull-in instability of hydrostatically and electrostatically actuated circular microplates. Nonlinear Dynam. 73(3), 1515–1526 (2013). https://doi.org/10.1007/s11071-013-0882-zMathSciNetCrossRef

42.

Esen, I.: Dynamics of size-dependant Timoshenko micro beams subjected to moving loads. Int. J. Mech. Sci. 175, 105501 (2020). https://doi.org/10.1016/j.ijmecsci.2020.105501CrossRef

43.

Zhang, B., Li, H., Kong, L., et al.: Size-dependent static and dynamic analysis of Reddy-type micro-beams by strain gradient differential quadrature finite element method. Thin Wall. Struct. 148, 106496 (2020). https://doi.org/10.1016/j.tws.2019.106496CrossRef

44.

Ma, Y., Gao, Y., Yang, W., et al.: Free vibration of a micro-scale composite laminated Reddy plate using a finite element method based on the new modified couple stress theory. Results Phys. 16, 102903 (2020). https://doi.org/10.1016/j.rinp.2019.102903CrossRef

45.

Yang, L., Li, P., Fang, Y., et al.: A generalized methodology for thermoelastic damping in axisymmetric vibration of circular plate resonators covered by multiple partial coatings. Thin Wall. Struct. 162, 107576 (2021). https://doi.org/10.1016/j.tws.2021.107576CrossRef

46.

Qing, H.: Well-posedness of two-phase local/nonlocal integral polar models for consistent axisymmetric bending of circular microplates. Appl. Math. Mech. 43(5), 637–652 (2022). https://doi.org/10.1007/s10483-022-2843-9MathSciNetCrossRef

47.

Lai, W.M., Rubin, D., Krempl, E.: Introduction to continuum mechanics. Butterworth-Heinemann (2009)

48.

Logan, D.L.: A first course in the finite element method. Cengage Learning (2016)

49.

Reddy, J.N.: Introduction to the finite element method. McGraw-Hill Education (2019)

50.

Huebner, K.H., Dewhirst, D.L., Smith, D.E., et al.: The finite element method for engineers. Wiley (2001)

51.

Rahaeifard, M., Ahmadian, M.T., Firoozbakhsh, K.: Vibration analysis of electrostatically actuated nonlinear microbridges based on the modified couple stress theory. Appl. Math. Model. 39(21), 6694–6704 (2015). https://doi.org/10.1016/j.apm.2015.02.020MathSciNetCrossRef

52.

Reddy, J.N.: Theory and analysis of elastic plates and shells. CRC Press (2006)CrossRef

53.

Leissa, A.W.: Vibration of plates. Vol. 160. Scientific and Technical Information Division, National Aeronautics and Space Administration (1969)

54.

Li, Z., Zhao, L., Jiang, Z., et al.: Mechanical behavior analysis on electrostatically actuated rectangular microplates. J. Micromech. Microeng. 25(3), 035007 (2015). https://doi.org/10.1088/0960-1317/25/3/035007ADSCrossRef

Title: A size-dependent axisymmetric plate element: application to MEMS
Authors: Masoud Rahaeifard
Ali Karimzadeh
Publication date: 08-02-2024
Publisher: Springer Berlin Heidelberg
Published in: Archive of Applied Mechanics / Issue 3/2024
Print ISSN: 0939-1533
Electronic ISSN: 1432-0681
DOI: https://doi.org/10.1007/s00419-024-02544-2

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Other articles of this Issue 3/2024

A parametric study on the elastic limit stresses of rotating variable thickness orthotropic disk

Penetration mechanism of grouting by using the cement-based slurry with time-dependent viscosity

Non-Fourier thermoelastic damping in small-sized ring resonators with circular cross section according to Moore–Gibson–Thompson generalized thermoelasticity theory

Dynamics of the perturbed restricted three-body problem with quantum correction and modified gravitational potential

Sun–Venus CR3BP, part 2: resonance investigation and planar periodic orbit family generation

Stacked multi-wedge acoustic black holes for low-frequency attenuation of flexible beams

Premium Partners