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Nonlinear resonant behavior of thick multilayered nanoplates via nonlocal strain gradient elasticity theory

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Abstract

In this study, the nonlinear forced vibration of simply supported multilayered nanoplates in the framework of the first-order shear deformation of Mindlin plate theory based on the nonlocal strain gradient elasticity theory is investigated. The nonlinear von Karman strain–displacement relation is employed. The interaction of van der Waals forces between adjacent layers of a multilayered nanoplate is considered. The harmonic balance method is used to analyze the nonlinear resonance behavior of a thick nanoplate. The primary resonance and the secondary resonance which consist of superhamonic and subharmonic resonance are investigated. As a main result, by increasing the thickness of a thick nanoplate, the subharmonic resonance disappears and the frequency response curves imply non-resonant behavior, responses with finite amplitudes and different shapes for different thicknesses of thick nanoplates. At the end, the natural frequencies of thick multilayered nanoplates are obtained, and the effect of variation of nonlocal parameter, strain gradient parameter, number of layers and thickness of the nanoplate on the frequency response curves and stable and unstable regions is investigated.

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Notes

  1. Nanoelectromechanical system.

  2. Microelectromechanical system.

  3. Molecular Dynamics.

References

  1. Arsat, R., Breedon, M., Shafiei, M., Spizziri, P.G., Gilje, S., Kaner, R.B., Kalantar-zadeh, K., Wlodarski, W.: Graphene-like nano-sheets for surface acoustic wave gas sensor applications. Chem. Phys. Lett. 467, 344–347 (2009)

    Google Scholar 

  2. Schedin, F., Geim, A.K., Morozov, S.V., Hill, E.W., Blake, P., Katsnelson, M.I., Novoselov, K.S.: Detection of individual gas molecules adsorbed on graphene. Nat. Mater. 6, 652 (2007)

    Google Scholar 

  3. Rangel, N.L., Seminario, J.M.: Graphene terahertz generators for molecular circuits and sensors. J. Phys. Chem. A 112, 13699–13705 (2008)

    Google Scholar 

  4. Bai, S., Shen, X.: Graphene-inorganic nanocomposites. RSC Adv. 2, 64–98 (2012)

    Google Scholar 

  5. Bartolucci, S.F., Paras, J., Rafiee, M.A., Rafiee, J., Lee, S., Kapoor, D., Koratkar, N.: Graphene–aluminum nanocomposites. Mater. Sci. Eng. A 528, 7933–7937 (2011)

    Google Scholar 

  6. Kim, H., Abdala, A.A., Macosko, C.W.: Graphene/polymer nanocomposites. Macromolecules 43, 6515–6530 (2010)

    Google Scholar 

  7. Lin, Y.M., Dimitrakopoulos, C., Jenkins, K.A., Farmer, D.B., Chiu, H.Y., Grill, A., Avouris, P.: 100-GHz transistors from wafer-scale epitaxial graphene. Science 327, 662–662 (2010)

    Google Scholar 

  8. Das, A., Pisana, S., Chakraborty, B., Piscanec, S., Saha, S.K., Waghmare, U.V., Novoselov, K.S., Krishnamurthy, H.R., Geim, A.K., Ferrari, A.C., Sood, A.K.: Monitoring dopants by Raman scattering in an electrochemically top-gated graphene transistor. Nature 3, 210–215 (2008)

    Google Scholar 

  9. Xiang, Q., Yu, J., Jaroniec, M.: Graphene-based semiconductor photocatalysts. Chem. Soc. Rev. 41, 782–796 (2012)

    Google Scholar 

  10. Li, X., Wang, X., Zhang, L., Lee, S., Dai, H.: Chemically derived, ultrasmooth graphene nanoribbon semiconductors. Science 319, 1229–1232 (2008)

    Google Scholar 

  11. Zhang, Y.P., Challamel, N., Wang, C.M., Zhang, H.: Comparison of nano-plate bending behaviour by Eringen nonlocal plate, Hencky bar-net and continualised nonlocal plate models. Acta Mech. 230, 885–907 (2019)

    MathSciNet  MATH  Google Scholar 

  12. Arefi, M.: Analysis of wave in a functionally graded magneto-electro-elastic nano-rod using nonlocal elasticity model subjected to electric and magnetic potentials. Acta Mech. 227(9), 2529–2542 (2016)

    MathSciNet  MATH  Google Scholar 

  13. Karimipour, I., Fotuhi, A.R.: Anti-plane analysis of an infinite plane with multiple cracks based on strain gradient theory. Acta Mech. 228(5), 1793–1817 (2017)

    MathSciNet  MATH  Google Scholar 

  14. Asghari, M., Kahrobaiyan, M.H., Nikfar, M., Ahmadian, M.T.: A size-dependent nonlinear Timoshenko microbeam model based on the strain gradient theory. Acta Mech. 223(6), 1233–1249 (2012)

    MathSciNet  MATH  Google Scholar 

  15. Sahmani, S., Safaei, B.: Nonlinear free vibrations of bi-directional functionally graded micro/nano-beams including nonlocal stress and microstructural strain gradient size effects. Thin-Walled Struct. 140, 342–356 (2019)

    Google Scholar 

  16. Barretta, R., Faghidian, S.A., Luciano, R., Medaglia, C.M., Penna, R.: Free vibrations of FG elastic Timoshenko nano-beams by strain gradient and stress-driven nonlocal models. Compos. Part B Eng. 154, 20–32 (2018)

    Google Scholar 

  17. Sahmani, S., Aghdam, M.M.: Nonlinear instability of hydrostatic pressurized microtubules surrounded by cytoplasm of a living cell including nonlocality and strain gradient microsize dependency. Acta Mech. 229(1), 403–420 (2018)

    MathSciNet  MATH  Google Scholar 

  18. Sahmani, S., Aghdam, M.M.: Nonlinear primary resonance of micro/nano-beams made of nanoporous biomaterials incorporating nonlocality and strain gradient size dependency. Results Phys. 8, 879–892 (2018)

    Google Scholar 

  19. Ebrahimi, F., Barati, M.R.: Vibration analysis of viscoelastic inhomogeneous nanobeams resting on a viscoelastic foundation based on nonlocal strain gradient theory incorporating surface and thermal effects. Acta Mech. 228(3), 1197–1210 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Mahmoudpour, E., Hosseini-Hashemi, S.H., Faghidian, S.A.: A nonlocal strain gradient theory for nonlinear free and forced vibration of embedded thick FG double layered nanoplates. Struct. Eng. Mech. 68, 103–119 (2018)

    Google Scholar 

  21. Mahmoudpour, E., Hosseini-Hashemi, S.H., Faghidian, S.A.: Nonlinear resonant behaviors of embedded thick FG double layered nanoplates via nonlocal strain gradient theory. Microsyst. Technol. 25, 951–964 (2019)

    Google Scholar 

  22. Farajpour, A., Yazdi, M.H., Rastgoo, A., Mohammadi, M.: A higher-order nonlocal strain gradient plate model for buckling of orthotropic nanoplates in thermal environment. Acta Mech. 227(7), 1849–1867 (2016)

    MathSciNet  MATH  Google Scholar 

  23. Barati, M.R.: Vibration analysis of porous FG nanoshells with even and uneven porosity distributions using nonlocal strain gradient elasticity. Acta Mech. 229(3), 1183–1196 (2018)

    MathSciNet  MATH  Google Scholar 

  24. Zhen, Y.X., Wen, S.L., Tang, Y.: Free vibration analysis of viscoelastic nanotubes under longitudinal magnetic field based on nonlocal strain gradient Timoshenko beam model. Phys. E Low-dimensional Syst. Nanostruct. 105, 116–124 (2019)

    Google Scholar 

  25. She, G.L., Yuan, F.G., Ren, Y.R., Liu, H.B., Xiao, W.S.: Nonlinear bending and vibration analysis of functionally graded porous tubes via a nonlocal strain gradient theory. Compos. Struct. 203, 614–623 (2018)

    Google Scholar 

  26. Lu, L., Guo, X., Zhao, J.: Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory. Int. J. Eng. Sci. 116, 12–24 (2017)

    MathSciNet  MATH  Google Scholar 

  27. Wang, K.F., Wang, B., Zhang, C.: 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 (2017)

    MathSciNet  MATH  Google Scholar 

  28. Tao, C., Fu, Y.: Thermal buckling and postbuckling analysis of size-dependent composite laminated microbeams based on a new modified couple stress theory. Acta Mech. 228(5), 1711–1724 (2017)

    MathSciNet  MATH  Google Scholar 

  29. Mahmoudpour, E., Hosseini-Hashemi, S.H., Faghidian, S.A.: Nonlinear vibration analysis of FG nano-beams resting on elastic foundation in thermal environment using stress-driven nonlocal integral model. Appl. Math. Model. 57, 302–315 (2018)

    MathSciNet  MATH  Google Scholar 

  30. Mehralian, F., Beni, Y.T., Zeverdejani, M.K.: Nonlocal strain gradient theory calibration using molecular dynamics simulation based on small scale vibration of nanotubes. Phys. B Condens. Matter. 514, 61–69 (2017)

    Google Scholar 

  31. Kitipornchai, S., He, X.Q., Liew, K.M.: Continuum model for the vibration of multilayered graphene sheets. Phys. Rev. B 72, 075443 (2005)

    Google Scholar 

  32. Ansari, R., Arash, B., Rouhi, H.: Nanoscale vibration analysis of embedded multi-layered graphene sheets under various boundary conditions. Comput. Mater. Sci. 50, 3091–3100 (2011)

    Google Scholar 

  33. Ansari, R., Rajabiehfard, R., Arash, B.: Nonlocal finite element model for vibrations of embedded multi-layered graphene sheets. Comput. Mater. Sci. 49, 831–838 (2010)

    Google Scholar 

  34. Jomehzadeh, E., Saidi, A.R., Pugno, N.M.: Large amplitude vibration of a bilayer graphene embedded in a nonlinear polymer matrix. Phys. E Low-Dimensional Syst. Nanostruct. 44, 1973–1982 (2012)

    Google Scholar 

  35. He, X.Q., Wang, J.B., Liu, B., Liew, K.M.: Analysis of nonlinear forced vibration of multi-layered graphene sheets. Comput. Mater. Sci. 61, 194–199 (2012)

    Google Scholar 

  36. Ma’en, S.S.: Superharmonic resonance analysis of nonlocal nano beam subjected to axial thermal and magnetic forces and resting on a nonlinear elastic foundation. Microsyst. Technol. 23, 3319–3330 (2017)

    Google Scholar 

  37. Hosseini, S.M., Shooshtari, A., Kalhori, H., Mahmoodi, S.N.: Nonlinear-forced vibrations of piezoelectrically actuated viscoelastic cantilevers. Nonlinear Dyn. 78, 571–583 (2014)

    MathSciNet  MATH  Google Scholar 

  38. Mahmoodi, S.N., Jalili, N., Ahmadian, M.: January. Subharmonic Resonances of Piezoelectrically Actuated Microcantilevers. In: ASME 2009 Dynamic Systems and Control Conference. American Society of Mechanical Engineers, pp. 533–540(2009)

  39. Mahmoodi, S.N., Jalili, N., Ahmadian, M.: Subharmonics analysis of nonlinear flexural vibrations of piezoelectrically actuated microcantilevers. Nonlinear Dyn. 59, 397–409 (2010)

    MATH  Google Scholar 

  40. Alijani, F., Bakhtiari-Nejad, F., Amabili, M.: Nonlinear vibrations of FGM rectangular plates in thermal environments. Nonlinear Dyn. 66, 251 (2011)

    MathSciNet  MATH  Google Scholar 

  41. Vatankhah, R., Kahrobaiyan, M.H., Alasty, A., Ahmadian, M.T.: Nonlinear forced vibration of strain gradient microbeams. Appl. Math. Model. 37, 8363–8382 (2013)

    MathSciNet  MATH  Google Scholar 

  42. Wang, Y., Li, F., Wang, Y., Jing, X.: Nonlinear responses and stability analysis of viscoelastic nanoplate resting on elastic matrix under 3:1 internal resonances. Int. J. Mech. Sci. 128, 94–104 (2017)

    Google Scholar 

  43. Permoon, M.R., Haddadpour, H., Javadi, M.: Nonlinear vibration of fractional viscoelastic plate: primary, subharmonic, and superharmonic response. Int. J. Non-Linear Mech. 99, 154–164 (2018)

    Google Scholar 

  44. Jomehzadeh, E., Saidi, A.R., Jomehzadeh, Z., Bonaccorso, F., Palermo, V., Galiotis, C., Pugno, N.M.: Nonlinear subharmonic oscillation of orthotropic graphene-matrix composite. Comput. Mater. Sci. 99, 164–172 (2015)

    Google Scholar 

  45. Cong, P.H., Duc, N.D.: New approach to investigate the nonlinear dynamic response and vibration of a functionally graded multilayer graphene nanocomposite plate on a viscoelastic Pasternak medium in a thermal environment. Acta Mech. 229(9), 3651–3670 (2018)

    MathSciNet  MATH  Google Scholar 

  46. Sahmani, S., Fotouhi, M., Aghdam, M.: Size-dependent nonlinear secondary resonance of micro-/nano-beams made of nano-porous biomaterials including truncated cube cells. Acta Mech. 230, 1077–1103 (2019)

    MathSciNet  MATH  Google Scholar 

  47. Asadi, H., Bodaghi, M., Shakeri, M., Aghdam, M.M.: Nonlinear dynamics of SMA-fiber-reinforced composite beams subjected to a primary/secondary-resonance excitation. Acta Mech. 226(2), 437–455 (2015)

    MathSciNet  MATH  Google Scholar 

  48. Shooshtari, A., Rafiee, M.: Nonlinear forced vibration analysis of clamped functionally graded beams. Acta Mech. 221(1–2), 23 (2011)

    MATH  Google Scholar 

  49. Sarafraz, A., Sahmani, S., Aghdam, M.M.: Nonlinear secondary resonance of nanobeams under subharmonic and superharmonic excitations including surface free energy effects. Appl. Math. Model. 66, 195–226 (2019)

    MathSciNet  MATH  Google Scholar 

  50. Sahmani, S., Aghdam, M.M.: Size-dependent nonlinear mechanics of biological nanoporous microbeams. In: Rahmandoust, M., Ayatollahi, M. (eds.) Nanomaterials for Advanced Biological Applications, Vol. 104, pp. 181–207. Springer, Cham (2019)

  51. Sahmani, S., Mohammadi Aghdam, M., Akbarzadeh, A.: Surface stress effect on nonlinear instability of imperfect piezoelectric nanoshells under combination of hydrostatic pressure and lateral electric field. AUT J. Mech. Eng. 2(2), 177–190 (2018)

    Google Scholar 

  52. Razzak, M.A.: A simple harmonic balance method for solving strongly nonlinear oscillators. J. Assoc. Arab Univ. Basic Appl. Sci. 21, 68–76 (2016)

    Google Scholar 

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  1. Department of Mechanical Engineering, Borujerd Branch, Islamic Azad University, Borujerd, Iran

    E. Mahmoudpour

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Mahmoudpour, E. Nonlinear resonant behavior of thick multilayered nanoplates via nonlocal strain gradient elasticity theory. Acta Mech 231, 2651–2667 (2020). https://doi.org/10.1007/s00707-020-02636-4

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  • DOI: https://doi.org/10.1007/s00707-020-02636-4

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