Applied Mathematics and Mechanics

, Volume 40, Issue 12, pp 1723–1740 | Cite as

An analytical study of vibration in functionally graded piezoelectric nanoplates: nonlocal strain gradient theory

  • Z. Sharifi
  • R. KhordadEmail author
  • A. Gharaati
  • G. Forozani
Open Access


In this paper, we analytically study vibration of functionally graded piezoelectric (FGP) nanoplates based on the nonlocal strain gradient theory. The top and bottom surfaces of the nanoplate are made of PZT-5H and PZT-4, respectively. We employ Hamilton’s principle and derive the governing differential equations. Then, we use Navier’s solution to obtain the natural frequencies of the FGP nanoplate. In the first step, we compare our results with the obtained results for the piezoelectric nanoplates in the previous studies. In the second step, we neglect the piezoelectric effect and compare our results with those obtained for the functionally graded (FG) nanoplates. Finally, the effects of the FG power index, the nonlocal parameter, the aspect ratio, and the side-tothickness ratio, and the nanoplate shape on natural frequencies are investigated.

Key words

nonlocal strain gradient nanoplate functionally graded piezoelectric (FGP) 

Chinese Library Classification


2010 Mathematics Subject Classification

70-08 74D05 


  1. [1]
    SLADEK, J., SLADEK, V., KASALA, J., and PAN, E. Nonlocal and gradient theories of piezoelectric nanoplates. Procedia Engineering, 190, 178–185 (2017)Google Scholar
  2. [2]
    LIANG, X., HU, S., and SHEN, S. Bernoulli-Euler dielectric beam model based on strain-gradient effect. Journal of Applied Mechanics, 80, 044502–044508 (2013)Google Scholar
  3. [3]
    GRAIGHEAD, H. G. Nanoelectromechanical systems. Science, 290, 1532–1535 (2000)Google Scholar
  4. [4]
    EKINCI, K. L. and ROUKES, M. L. Nanoelectromechanical systems. Review of Scientific Instruments, 76, 061101–061112 (2005)Google Scholar
  5. [5]
    DEQUESNES, M., ROTKIN, S. V., and ALURU, N. R. Calculation of pull-in voltages for carbon-nanotube-based nanoelectromechanical switches. Nanotechnology, 13, 120–131. (2002)Google Scholar
  6. [6]
    SAHMANI, S. and FATTAHI, A. M. Small scale effects on buckling and postbuckling behaviors of axially loaded FGM nanoshells based on nonlocal strain gradient elasticity theory. Applied Mathematics and Mechanics (English Edition), 39, 561–580. (2018) MathSciNetzbMATHGoogle Scholar
  7. [7]
    FLECK, N. A., MULLER, G. M., ASHBY, M. F., and HUTCHINSON, J. W. Strain gradient plasticity: theory and experiment. Acta Metallurgica et Materialia, 42, 475–487. (1994)Google Scholar
  8. [8]
    LI, X. F., WANG, B. L., and LEE, K. Y. Size effects of the bending stiffness of nanowires. Journal of Applied Physics, 105, 074306–074311. (2009)Google Scholar
  9. [9]
    LI, L., TANG, H., and HU, Y. The effect of thickness on the mechanics of nanobeams. International Journal of Engineering Science, 123, 81–91. (2018)MathSciNetGoogle Scholar
  10. [10]
    ZHU, X. and LI, L. On longitudinal dynamics of nanorods. International Journal of Engineering Science, 120, 129–145. (2017)Google Scholar
  11. [11]
    MINDLIN, R. D. and TIERSTEN, H. F. Effects of couple-stresses in linear elasticity. Archive for Rational Mechanics and Analysis, 11, 415–448. (1962)MathSciNetzbMATHGoogle Scholar
  12. [12]
    KOITER, W. T. Couple stresses in the theory of elasticity. Philosophical Transactions of the Royal Society of London B, 67, 17–44. (1964)MathSciNetzbMATHGoogle Scholar
  13. [13]
    BEVER, M. and DUWEZ, P. Gradients in composite materials. Materials Science and Engineering, 10, 1–8. (1972)Google Scholar
  14. [14]
    JHA, D., KANT, T., and SINGH, R. A critical review of recent research on functionally graded plates. Composite Structures, 96, 833–849. (2013)Google Scholar
  15. [15]
    KARAMI, B., SHAHSAVARI, D., and JANGHORBAN, M. Wave propagation analysis in functionally graded (FG) nanoplates under in-plane magnetic field based on nonlocal strain gradient theory and four variable refined plate theory. Mechanics of Advanced Materials and Structures, 25, 1047–1057. (2018)Google Scholar
  16. [16]
    LI, X., LI, L., HU, Y., DING, Z., and DENG, W. Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory. Composite Structures, 165, 250–265. (2017)Google Scholar
  17. [17]
    LYU, C. F., CHEN, W. Q., and LIM, C. W. Elastic mechanical behavior of nano-scaled FGM films incorporating surface energies. Composites Science and Technology, 69, 1124–1130. (2009)Google Scholar
  18. [18]
    SEDIGHI, H. M., DANESHMAND, F., and ABADYAN, M. Modified model for instability analysis of symmetric FGM double-sided nano-bridge: corrections due to surface layer, finite conductivity and size effect. Composite Structures, 132, 545–557. (2015)Google Scholar
  19. [19]
    SEDIGHI, H. M., KEIVANI, M., and ABADYAN, M. Modified continuum model for stability analysis of asymmetric FGM double-sided NEMS: corrections due to finite conductivity, surface energy and nonlocal effect. Composites Part B: Engineering, 83, 117–133. (2015)Google Scholar
  20. [20]
    EBRAHIMI, F., BARATI, M. R., and DABBAGH, A. A nonlocal strain gradient theory for wave propagation analysis in temperature-dependent inhomogeneous nanoplates. International Journal of Engineering Science, 107, 169–182. (2016)Google Scholar
  21. [21]
    CIVALEK, Ö. Buckling analysis of composite panels and shells with different material properties by discrete singular convolution (DSC) method. Composite Structures, 161, 93–110. (2017)Google Scholar
  22. [22]
    AKGÖZ, B. and CIVALEK, Ö. Longitudinal vibration analysis of strain gradient bars made of functionally graded materials (FGM). Composites Part B: Engineering, 55, 263–268. (2013)Google Scholar
  23. [23]
    LEE, C. Y. and KIM, J. H. Hygrothermal postbuckling behavior of functionally graded plates. Composite Structures, 95, 278–282. (2013)Google Scholar
  24. [24]
    EBRAHIMI, F. and BARATI, M. R. Vibration analysis of piezoelectrically actuated curved nano-size FG beams via a nonlocal strain-electric field gradient theory. Mechanics of Advanced Materials and Structures, 47, 350–359. (2018)Google Scholar
  25. [25]
    SOBHY, M. An accurate shear deformation theory for vibration and buckling of FGM sandwich plates in hygrothermal environment. International Journal of Mechanical Sciences, 110, 62–77. (2016)Google Scholar
  26. [26]
    JAFARI, A. A., JANDAGHIAN, A. A., and RAHMANI, O. Transient bending analysis of a functionally graded circular plate with integrated surface piezoelectric layers. International Journal of Mechanics and Material Engineering, 9, 8–21. (2014)Google Scholar
  27. [27]
    JANDAGHIAN, A. A., JAFARI, A. A., and RAHMANI, O. Vibrational response of functionally graded circular plate integrated with piezoelectric layers: an exact solution. Engineering Solid Mechanics, 2, 119–130. (2014)Google Scholar
  28. [28]
    JANDAGHIAN, A. A., JAFARI, A. A., and RAHMANI, O. Exact solution for transient bending of a circular plate integrated with piezoelectric layers. Applied Mathematics Modelling, 37, 7154–7163. (2013)MathSciNetzbMATHGoogle Scholar
  29. [29]
    ZHANG, S., XIA, R., LEBRUN, L., ANDERSON, D., and SHROUT, T. R. Piezoelectric materials for high power, high temperature applications. Materials Letters, 59, 3471–3475. (2005)Google Scholar
  30. [30]
    MAHINZARE, M., RANJBARPUR, H., and GHADIRI, M. Free vibration analysis of a rotary smart two directional functionally graded piezoelectric material in axial symmetry circular nanoplate. Mechanical Systems and Signal Processing, 100, 188–207. (2018)Google Scholar
  31. [31]
    MAHINZARE, M., ALIPOUR, M. J., SADATSAKKAK, S. A., and GHADIRI, M. A nonlocal strain gradient theory for dynamic modeling of a rotary thermo piezoelectrically actuated nano FG circular plate. Mechanical Systems and Signal Processing, 115, 323–337. (2019)Google Scholar
  32. [32]
    QIU, J., TANI, J., UENO, T., MORITA, T., TAKAHASHI, H., and DU, H. Fabrication and high durability of functionally graded piezoelectric bending actuators. Smart Materials and Structures, 12, 115–121. (2003)Google Scholar
  33. [33]
    HE, J. and LILLEY, C. M. Surface effect on the elastic behavior of static bending nanowires. Nano Letters, 8, 1798–1802. (2008)Google Scholar
  34. [34]
    ASGHARI, M., RAHAEIFARD, M., KAHROBAIYAN, M., and AHMADIAN, M. The modified couple stress functionally graded Timoshenko beam formulation. Material Design, 32, 1435–1443. (2011)Google Scholar
  35. [35]
    ANSARI, R., GHOLAMI, R., and SAHMANI, S. Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory. Composite Structures, 94, 221–228. (2011)Google Scholar
  36. [36]
    RAHMANI, O. and PEDRAM, O. Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory. International Journal of Engineering Sciences, 77, 55–70. (2014)MathSciNetzbMATHGoogle Scholar
  37. [37]
    THAI, H. T. A nonlocal beam theory for bending, buckling, and vibration of nanobeams. International Journal of Engineering Sciences, 52, 56–64. (2012)MathSciNetGoogle Scholar
  38. [38]
    TOUNSI, A., BENGUEDIAB, S., ADDA, B., SEMMAH, A., and ZIDOUR, M. Nonlocal effects on thermal buckling properties of double-walled carbon nanotubes. Advances in Nano Research, 1, 1–11. (2013)Google Scholar
  39. [39]
    BENGUEDIAB, S., HEIRECHE, H., BOUSAHLA, A. A., TOUNSI, A., and BENZAIR, A. Nonlinear vibration properties of a zigzag single-walled carbon nanotube embedded in a polymer matrix. Advances in Nano Research, 3, 29–37. (2015)Google Scholar
  40. [40]
    BENGUEDIAB, S., TOUNSI, A., ZIDOUR, M., and SEMMAH, A. Chirality and scale effects on mechanical buckling properties of zigzag double-walled carbon nanotubes. Composite Part B: Engineering, 57, 21–24. (2014)Google Scholar
  41. [41]
    ERINGEN, A. C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54, 4703–4710. (1983)Google Scholar
  42. [42]
    ERINGEN, A. C. Nonlocal Continuum Field Theories, Springer, Berlin (2002)zbMATHGoogle Scholar
  43. [43]
    ERINGEN, A. C. and EDELEN, D. On nonlocal elasticity. International Journal of Engineering Sciences, 10, 233–248. (1972)MathSciNetzbMATHGoogle Scholar
  44. [44]
    JANDAGHIAN, A. A. and RAHMANI, O. Vibration analysis of functionally graded piezoelectric nanoscale plates by nonlocal elasticity theory: an analytical solution. Superlattices and Microstructures, 100, 57–75. (2016)Google Scholar
  45. [45]
    EZZIN, H., MKAOIR, M., and AMOR, M. B. Rayleigh wave behavior in functionally graded magneto-electro-elastic material. Superlattices and Microstructures, 112, 455–469. (2017)Google Scholar
  46. [46]
    ARANI, A. G., KOLAHCHI, R., and VOSSOUGH, H. Buckling analysis and smart control of SLGS using elastically coupled PVDF nanoplate based on the nonlocal Mindlin plate theory. Physica B, 407, 4458–4469. (2012)Google Scholar
  47. [47]
    KE, L. L., LIU, C., and WANG, Y.S. Free vibration of nonlocal piezoelectric nanoplates under various boundary conditions. Physica E, 66, 93–106. (2015)Google Scholar
  48. [48]
    TANG, H., LI, L., and HU, Y. Coupling effect of thickness and shear deformation on size-dependent bending of micro/nano-scale porous beams. Applied Mathematical Modelling, 66, 527–547. (2019)MathSciNetGoogle Scholar
  49. [49]
    TANG, H., LI, L., HU, Y., MENG, W., and DUAN, K. Vibration of nonlocal strain gradient beams incorporating Poisson’s ratio and thickness effects. Thin-Walled Structures, 137, 377–391. (2019)Google Scholar
  50. [50]
    NATARAJAN, S., CHAKRABORTY, S., THANGAVEL, M., BORDAS, S., and RABCZUK, T. Size-dependent free flexural vibration behavior of functionally graded nanoplates. Computational Material Sciences, 65, 74–80. (2012)Google Scholar

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Authors and Affiliations

  • Z. Sharifi
    • 1
  • R. Khordad
    • 2
    Email author
  • A. Gharaati
    • 1
  • G. Forozani
    • 1
  1. 1.Department of PhysicsPayame Noor UniversityTehranIran
  2. 2.Department of Physics, College of ScienceYasouj UniversityYasoujIran

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