Abstract
This paper uses the finite element method to simulate the mechanical, electric, and polarization behaviors of piezoelectric nanoplates resting on elastic foundations subjected to static loads, in which the flexoelectric effect is taken into consideration. The finite element formulations are established by employing a new type of shear deformation theory, which does not need any shear correction factors, but still accurately describes the stress field of the plate. The numerical results show clearly that the flexoelectric effect has a strong influence on the mechanical responses of the nanoplates. In particular, the normal stress distribution in the thickness direction is no longer linear when the flexoelectric coefficient is large enough, and this phenomenon differs completely from that of conventional plates. In addition, the distribution of the electric field and the polarization also depend on boundary conditions, which were not investigated in the published works.
Similar content being viewed by others
Data availability
Data used to support the findings of this study are included in the article.
References
Yan, Z.: Size-dependent bending and vibration behaviors of piezoelectric circular nanoplates. Smart Mater. Struct. 25(3), 13 (2016). https://doi.org/10.1088/0964-1726/25/3/035017
Yang, W., Liang, X., Shen, S.: Electromechanical responses of piezoelectric nanoplates with flexoelectricity. Acta Mech. 226(9), 3097–3110 (2015). https://doi.org/10.1007/s00707-015-1373-8
Li, A., Zhou, S., Qi, L.: Size-dependent electromechanical coupling behaviors of circular micro-plate due to flexoelectricity. Appl. Phys. A 122(10), 918 (2016). https://doi.org/10.1007/s00339-016-0455-3
Wang, X., Zhang, R., Jiang, L.: A study of the flexoelectric effect on the electroelastic fields of a cantilevered piezoelectric nanoplate. Int. J. Appl. Mech. 09(04), 1750056 (2017). https://doi.org/10.1142/S1758825117500569
He, L., Lou, J., Zhang, A., Wu, H., Du, J., Wang, J.: On the coupling effects of piezoelectricity and flexoelectricity in piezoelectric nanostructures. AIP Adv. 7(10), 105106 (2017). https://doi.org/10.1063/1.4994021
Ebrahimi, F., Barati, M.: Static stability analysis of embedded flexoelectric nanoplates considering surface effects. Appl. Phys. A (2017). https://doi.org/10.1007/s00339-017-1265-y
Ghobadi, A., Beni, Y.T., Golestanian, H.: Nonlinear thermo-electromechanical vibration analysis of size-dependent functionally graded flexoelectric nano-plate exposed magnetic field. Arch. Appl. Mech. 90(9), 2025–2070 (2020). https://doi.org/10.1007/s00419-020-01708-0
Amir, S., BabaAkbar-Zarei, H., Khorasani, M.: Flexoelectric vibration analysis of nanocomposite sandwich plates. Mech. Based Des. Struct. Mach. 48(2), 146–163 (2020). https://doi.org/10.1080/15397734.2019.1624175
Ghobadi, A., Beni, Y.T., Golestanian, H.: Size dependent thermo-electro-mechanical nonlinear bending analysis of flexoelectric nano-plate in the presence of magnetic field. Int. J. Mech. Sci. 152, 118–137 (2019). https://doi.org/10.1016/j.ijmecsci.2018.12.049
Ghobadi, A., Tadi Beni, Y., Golestanian, H.: Size dependent nonlinear bending analysis of a flexoelectric functionally graded nano-plate under thermo-electro-mechanical loads. J. Solid Mech. 12(1), 33–56 (2020). https://doi.org/10.22034/jsm.2019.569280.1296
Giannakopoulos, A.E., Zisis, T.: Steady-state antiplane crack considering the flexoelectrics effect: surface waves and flexoelectric metamaterials. Arch. Appl. Mech. 91(2), 713–738 (2021). https://doi.org/10.1007/s00419-020-01815-y
Qu, Y., Jin, F., Yang, J.: Torsion of a flexoelectric semiconductor rod with a rectangular cross section. Arch. Appl. Mech. 91(5), 2027–2038 (2021). https://doi.org/10.1007/s00419-020-01867-0
Yang, W., Hu, T., Liang, X., Shen, S.: On band structures of layered phononic crystals with flexoelectricity. Arch. Appl. Mech. 88(5), 629–644 (2018). https://doi.org/10.1007/s00419-017-1332-z
Shimpi, R.: Refined plate theory and its variants. AIAA J. 40(1), 137–146 (2002)
Thai, T., Park, T., Choi, D.-H.: An efficient shear deformation theory for vibration of functionally graded plates. Arch. Appl. Mech. 83, 137–149 (2013)
Thai, H.-T., Vo, T.P.: A new sinusoidal shear deformation theory for bending, buckling, and vibration of functionally graded plates. Appl. Math. Model. 37(5), 3269–3281 (2013). https://doi.org/10.1016/j.apm.2012.08.008
El Meiche, N., Tounsi, A., Ziane, N., Mechab, I., Adda.Bedia, E.A.: A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate. Int. J. Mech. Sci. 53(4), 237–247 (2011). https://doi.org/10.1016/j.ijmecsci.2011.01.004
Thai, H.-T., Choi, D.-H.: Finite element formulation of various four unknown shear deformation theories for functionally graded plates. Finite Elem. Anal. Des. 75, 50–61 (2013). https://doi.org/10.1016/j.finel.2013.07.003
Touratier, M.: An efficient standard plate theory. Int. J. Eng. Sci. 29(8), 901–916 (1991). https://doi.org/10.1016/0020-7225(91)90165-Y
Shu, L., Wei, X., Pang, T., Yao, X., Wang, C.: Symmetry of flexoelectric coefficients in crystalline medium. J. Appl. Phys. 110(10), 104106 (2011). https://doi.org/10.1063/1.3662196
Han, J.-B., Liew, K.M.: Numerical differential quadrature method for Reissner/Mindlin plates on two-parameter foundations. Int. J. Mech. Sci. 39(9), 977–989 (1997). https://doi.org/10.1016/S0020-7403(97)00001-5
Thai, H.-T., Park, M., Choi, D.-H.: A simple refined theory for bending, buckling, and vibration of thick plates resting on elastic foundation. Int. J. Mech. Sci. 73, 40–52 (2013). https://doi.org/10.1016/j.ijmecsci.2013.03.017
Matsunaga, H.: Stress analysis of functionally graded plates subjected to thermal and mechanical loadings. Compos. Struct. 87(4), 344–357 (2009). https://doi.org/10.1016/j.compstruct.2008.02.002
Acknowledgements
This work gratefully acknowledges the support of Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 107.02-2020.18.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
All authors declare that there is no conflict of interest regarding the publication of this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Thai, L.M., Luat, D.T., Phung, V.B. et al. Finite element modeling of mechanical behaviors of piezoelectric nanoplates with flexoelectric effects. Arch Appl Mech 92, 163–182 (2022). https://doi.org/10.1007/s00419-021-02048-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-021-02048-3