Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Vibration and Stability Analysis Comparison for Nanoshell and Piezoelectric Nanoshell Subjected to Electrostatic Excitation

Discontinuity, Nonlinearity, and Complexity 9(4) (2020) 619--646 | DOI:10.5890/DNC.2020.12.014

Sayyid H. Hashemi Kachapi

Department of Mechanical Engineering, Babol Noshirvani University of Technology, P.O.Box484, Shariati Street, Babol, Mazandaran 47148-71167, Iran

Download Full Text PDF



In current study, vibration and stability analysis comparison of two nanostructures i.e. nanoshell (NS) and piezoelectric nanoshell (PENS) subjected to electrostatic excitation and Visco-Pasternak medium is investigated using the Gurtin--Murdoch surface/interface (S/I) theory. For this analysis, Hamilton's principles, the assumed mode method combined with Lagrange--Euler's and also Complex averaging method combined with Arc-length continuation are used. It can be seen that by changing the surface/interface densities and as a result, increasing or decreasing the system stiffness, the natural frequency can be less or greater than the state without taking into account the S/I effects. In both nanostructures (NS and PENS), considering the surface/interface effects increase the nonlinear behaviour compared with without S/I effects.


  1. [1]  Waggoner, P.S. and Craighead, H.G. (2007), Micro-and nanomechanical sensors for environmental, chemical, and biological detection, 7(10), 1238-1255
  2. [2]  Manbachi, A. and Cobbold, R.S.C. (2011), Development and application of piezoelectric materials for ultrasound generation and detection, Ultrasound, 11(4), 187-96.
  3. [3]  Jalili, N. (2010), Piezoelectric-Based Vibration Control: From Macro to Micro/Nano Scale Systems, Springer: New York.
  4. [4]  Gurtin, M.E. and Murdoch, A.I. (1975), A continuum theory of elastic material surface, Archive for Rational Mechanics and Analysis, 57(4), 291-323.
  5. [5]  Gurtin, M.E. and Murdoch, A.I. (1978), Surface stress in solids, International Journal of Solids and Structures, 14(6), 431-40.
  6. [6]  Nayfeh, A.H., Ouakad, H.M., Najar, F., Choura, S., and Abdel-Rahman, E.M. (2010), Nonlinear dynamics of a resonant gas sensor, Nonlinear Dynamics, 59, 607-618
  7. [7]  Fang, X.Q. and Zhu, C.S. (2017), Size-dependent nonlinear vibration of nonhomogeneous shell embedded with a piezoelectric layer based on surface/interface theory, Composite Structures, 160(15), 1191-1197.
  8. [8]  Fang, X.Q., Zhu, C.S., Liu, J.X., and Zhou, Z. (2018), Surface energy effect on nonlinear buckling and postbuckling behavior of functionally graded piezoelectric cylindrical nanoshells under lateral pressure, Materials Research Express, 5(4), DOI: 10.1088/2053-1591/aab914.
  9. [9]  Fang, X.Q., Zhu, C.S., and Liu, J.X. (2018), Surface energy effect on free vibration of nano-sized piezoelectric double-shell structures, Physica B: Condensed Matter, 529(15), 41-56.
  10. [10]  Sun, J., Wang, Z., Zhou, Z., Xu, X., and Lim, C.W. (2018), Surface effects on the buckling behaviors of piezoelectric cylindrical nanoshells using nonlocal continuum model, Applied Mathematical Modellin, 59, 341-356.
  11. [11]  Sahmani, S., Aghdam, M.M., and Bahrami, M. (2017), An efficient size-dependent shear deformable shell model and molecular dynamics simulation for axial instability analysis of silicon nanoshells, Journal of Molecular Graphics and Modelling, 77, 263-279.
  12. [12]  Sahmani, S. and Aghdam, M.M. (2017), Imperfection sensitivity of the size-dependent postbuckling response of pressurized FGM nanoshells in thermal environments, Archives of Civil and Mechanical Engineering, 17, 623-638.
  13. [13]  Sahmani, S., Aghdam, M.M., and Bahrami, M. (2017), Nonlinear buckling and postbuckling behavior of cylindrical shear deformable nanoshells subjected to radial compression including surface free energy effects, Acta Mechanica Solida Sinica, 30(2), 209-222.
  14. [14]  Farokhi, H., Pa\"{\i}doussis, M.P., and Misra, A. (2016), A new nonlinear model for analyzing the behaviour of carbon nanotube-based resonators, Journal of Sound and Vibration, 378, 56-75
  15. [15]  Sun, J., Wang, Z., Zhou, Z., Xu, X., and Lim, C.W. (2018), Surface effects on the buckling behaviors of piezoelectric cylindrical nanoshells using nonlocal continuum model, Applied Mathematical Modelling, 59, 341-356.
  16. [16]  Oskouie, M.F., Ansari, R., and Sadeghi, F. (2017), Nonlinear vibration analysis of fractional viscoelastic Euler--Bernoulli nanobeams based on the surface stress theory, Acta Mechanica Solida Sinica, 30, 416-424.
  17. [17]  Arani, A.G., Fereidoon, A., and Kolahchi, R. (2015), Nonlinear surface and nonlocal piezoelasticity theories for vibration of embedded single-layer boron nitride sheet using harmonic differential quadrature and differential cubature methods, Journal of Intelligent Materials Systems and Structures, 26, 1150-1163.
  18. [18]  Fereidoon, A., Andalib, E., and Mirafzal, A. (2016), Nonlinear Vibration of Viscoelastic Embedded-DWCNTs Integrated with Piezoelectric Layers-Conveying Viscous Fluid Considering Surface Effects, Physica E: Low-Dimensional Systems and Nanostructures, 81, 205-218.
  19. [19]  Rouhi, H., Ansari, R., and Darvizeh, M. (2015), Exact solution for the vibrations of cylindrical nanoshells considering surface energy effect, Journal of Ultrafine Grained and Nanostructured Materials, 48(2), 113-124.
  20. [20]  Sarafraz, A., Sahmani, S., and Aghdam, M.M. (2019), Nonlinear secondary resonance of nanobeams under subharmonic and superharmonic excitations including surface free energy effects, Applied Mathematical Modelling, 66, 195-226.
  21. [21]  Amabili, M. (2008), Nonlinear Vibrations and Stability of Shells and Plates, Cambridge University Press: New York.
  22. [22]  Donnell, L.H. (1976), Beam, Plates and Shells, McGraw-Hill: New York.
  23. [23] Boroujerdy, M.S. and Eslami, M.R. (2014), Axisymmetric snap-through behavior of Piezo-FGM shallow clamped spherical shells under thermo-electro-mechanical loading, International Journal of Pressure Vessels and Piping, 120-121 19-26.
  24. [24]  Kirchhoff, G. (1850), \"{U}ber das Gleichgewicht und die Bewegung einer elastischen Scheibe, Journal fur reine und angewandte Mathematik, 40, 51-88.
  25. [25]  Lu, P., He, L.H., Lee, H.P., and Lu, C. (2006), Thin plate theory including surface effects, International Journal Solids and Structures, 43(16), 4631-47.
  26. [26]  Ke, L.L., Wang, Y.S., and Reddy, J.N. (2014), Thermo-electro-mechanical vibration of size-dependent piezoelectric cylindrical nanoshells under various boundary conditions, Composite Structures, 116, 626-636.
  27. [27]  Ghorbanpour Arani, A., Kolahchi, R., and Hashemian, M. (2014), Nonlocal surface piezoelasticity theory for dynamic stability of double-walled boron nitride nanotube conveying viscose fluid based on different theories, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, DOI: 10.1177/0954406214527270.
  28. [28]  Manevitch, A.I. and Manevitch, L.I. (2005), Themechanics of Nonlinear Systems with Internal Resonance, Imperial College Press: London.
  29. [29]  Parseh, M., Dardel, M., Ghasemi, M.H., and Pashaei, M.H. (2016), Steady state dynamics of a non-linear beam coupled to a non-linear energy sink, International Journal of Non-Linear Mechanics, 79, 48-65.
  30. [30]  Ansari, R., Gholami, R., Norouzzadeh, A., and Darabi, M.A. (2015), Surface Stress Effect on the Vibration and Instability of Nanoscale Pipes Conveying Fluid Based on a Size-Dependent Timoshenko Beam Model Acta Mechanica Sinica, 31, 708-719.