Abstract.
The nonlinear dynamic instability of reinforced piezoelectric nanoplates exposed to a parametric excitation and an electric voltage is the objective of the present paper. Firstly, a piezoelectric nanoplate reinforced with two graphene layers and resting on a visco-elastic foundation is modeled. Secondly, the piezoelectric nonlocal elasticity theory, the Kelvin-Voigt model, von Karman nonlinear relations and Hamilton’s principle, respectively, are used to derive the nonlinear governing differential equation of motion. In the next step, to transform partial differential equation to ordinary one and then, solve the equation, the Galerkin technique and multiple time scales method are employed respectively. At the end, the modulation equation of reinforced piezoelectric nanoplates is obtained. Emphasizing the effect of the electric voltage and parametric excitation on dynamic instability of the system, trivial and nontrivial steady-state solutions are discussed. The main results emphasize that the damping coefficient is responsible of the bifurcation point variation, while the amplitude response depends on the term of natural frequency. Therefore, damping can have a strong influence on the system.
Similar content being viewed by others
References
C.L. Kane, E.J. Mele, Phys. Rev. Lett. 78, 1932 (1997)
A. Maiti, A. Svizhenko, M.P. Anantram, Phys. Rev. Lett. 88, 126805 (2002)
B. Arash, Q. Wang, Sci. Rep. 3, 1782 (2013)
R. Potekin, S. Kim, D.M. McFarland, L.A. Bergman, H. Cho, A.F. Vakakis, Nonlinear Dyn. 92, 287 (2018)
M.A. Mahmoud, Crit. Rev. Solid State Mater. Sci. 41, 386 (2016)
Y. Ji, M. Choe, B. Cho, S. Song, J. Yoon, H.C. KO, T. Lee, Nanotechnology 23, 105202 (2012)
S. Jeong, J.Y. Cho, T.H. Sung, H.H. Yoo, Smart Mater. Struct. 26, 035035 (2017)
S.Y. Khoo, Z.S. Radeef, Z.C. Ong, Y.H. Huang, W.T. Chong, Z. Ismail, Measurement 107, 41 (2017)
J. Zhang, Z. Fang, C. Shu, J. Zhang, Q. Zhang, C. Li, Sensors Actuat. A: Phys. 262, 123 (2017)
J. Song, G. Hu, K.T. Tse, S.W. Li, K.C.S. Kwok, Appl. Phys. Lett. 111, 223903 (2017)
J. Zhang, C. Wang, J. Appl. Phys. 111, 094303 (2012)
C. Li, J.J. Liu, M. Cheng, X.L. Fan, Compos. Part B: Eng. 116, 153 (2017)
D.P. Zhang, Y.J. Lei, Z.B. Shen, Int. J. Mech. Sci. 131, 1001 (2017)
H.B. Li, Y.D. Li, X. Wang, X. Huang, J. Sound Vib. 358, 285 (2015)
A.G. Arani, M.H. Zamani, Arab. J. Sci. Eng. 43, 4675 (2018)
M. Sobhy, A.M. Zenkour, Compos. Part B: Eng. 154, 492 (2018)
S. Zeng, B.L. Wang, K.F. Wang, Compos. Struct. 207, 340 (2019)
A. Ghorbanpour Arani, M. Jamali, A.H. Ghorbanpour-Arani, R. Kolahchi, M. Mosayyebi, Proc. Inst. Mech. Eng., Part C 231, 387 (2017)
F. Ebrahimi, A. Dabbagh, J. Vib. Control 25, 933 (2018)
A. Ghorbanpour Arani, M.H. Zamani, J. Sandw. Struct. Mater. (2017) https://doi.org/10.1177/1099636217721405
M. Arefi, M. Kiani, M.H. Zamani, J. Sandw. Struct. Mater. (2018) https://doi.org/10.1177/1099636218795378
M. Arefi, A.M. Zenkour, Compos. Struct. 162, 108 (2017)
M. Arefi, A.M. Zenkour, Acta Mech. 228, 475 (2017)
M. Arefi, A.M. Zenkour, Mech. Res. Commun. 84, 27 (2017)
M. Arefi, A.M. Zenkour, Compos. Struct. 159, 479 (2017)
M. Arefi, M.H. Zamani, M. Kiani, J. Intell. Mater. Syst. Struct. 29, 774 (2018)
A.C. Eringen, D.G.B. Edelen, Int. J. Eng. Sci. 10, 233 (1972)
A.C. Eringen, Int. J. Eng. Sci. 21, 741 (1983)
M. Ghadiri, N. Shafiei, A. Akbarshahi, Appl. Phys. A 122, 673 (2016)
S. Sahmani, A.M. Fattahi, Eur. Phys. J. Plus 132, 231 (2017)
Y.Q. Zhang, G.R. Liu, J.S. Wang, Phys. Rev. B 70, 205430 (2004)
R. Barretta, L. Feo, R. Luciano, F.M. de Sciarra, Compos. Struct. 129, 80 (2015)
F. Ebrahimi, S.H.S. Hosseini, J. Therm. Stresses 39, 606 (2016)
R. Kolahchi, M.S. Zarei, M.H. Hajmohammad, A.N. Oskouei, Thin-Walled Struct. 113, 162 (2017)
R. Kolahchi, Aerospace Sci. Technol. 66, 235 (2017)
S.R. Asemi, A. Farajpour, M. Mohammadi, Compos. Struct. 116, 703 (2014)
A. Ghorbanpour-Arani, F. Kolahdouzan, M. Abdollahian, Appl. Math. Mech. 39, 529 (2018)
B. Karami, D. Shahsavari, L. Li, M. Karami, M. Janghorban, Proc. Inst. Mech. Eng., Part C 233, 287 (2019)
C. Li, C.W. Lim, J.L. Yu, Smart Mater. Struct. 20, 015023 (2010)
I.M. Fokou, C.N.D. Buckjohn, M.S. Siewe, C. Tchawoua, Eur. Phys. J. Plus 132, 344 (2017)
Y. Huang, J. Fu, A. Liu, Compos. Part B: Eng. 164, 226 (2019)
Y.Z. Wang, Y.S. Wang, L.L. Ke, Physica E 83, 195 (2016)
C. Li, C.W. Lim, J.L. Yu, Smart Mater. Struct. 20, 015023 (2010)
P. Alevras, S. Theodossiades, H. Rahnejat, Appl. Phys. Lett. 110, 233901 (2017)
Y.A. Amer, A.T. El-Sayed, A.A. Kotb, Nonlinear Dyn. 85, 2497 (2016)
R.V. Bobryk, D. Yurchenko, J. Sound Vib. 366, 407 (2016)
M. Darabi, R. Ganesan, Compos. Struct. 176, 82 (2017)
Y.Z. Wang, Appl. Math. Model. 48, 621 (2017)
S. Krylov, I. Harari, Y. Cohen, J. Micromech. Microeng. 15, 1188 (2005)
F. Ebrahimi, S.H.S. Hosseini, Eur. Phys. J. Plus 132, 172 (2017)
A.C. Eringen, Int. J. Eng. Sci. 10, 1 (1972)
A.C. Eringen, J. Appl. Phys. 54, 4703 (1983)
J.N. Reddy, An Introduction to Continuum Mechanics (Cambridge University Press, 2013)
J.N. Reddy, Theory and Analysis of Elastic Plates and Shells (CRC Press, 2006)
L.L. Ke, Y.S. Wang, Smart Mater. Struct. 21, 025018 (2012)
R. Lakes, R.S. Lakes, Viscoelastic Materials (Cambridge University Press, 2009)
Y. Wang, F.M. Li, Y.Z. Wang, Physica E 67, 65 (2015)
J.N. Reddy, Theory and Analysis of Elastic Plates and Shells (CRC Press, 2006)
A.G. Arani, A. Shiravand, M. Rahi, R. Kolahchi, Physica B 407, 4123 (2012)
I.S. Raju, G.V. Rao, K.K. Raju, J. Sound Vib. 49, 415 (1976)
L.E. Shen, H.S. Shen, C.L. Zhang, Comput. Mater. Sci. 48, 680 (2010)
A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations (John Wiley & Sons, 2008)
H. Nourbakhsh, R. Mohammadzadeh, M. Rafiee, R. Rafiee, Appl. Mech. Mater. 110, 4178 (2012)
S.C. Pradhan, J.K. Phadikar, J. Sound Vib. 325, 206 (2009)
A.G. Arani, R. Kolahchi, A.A.M. Barzoki, M.R. Mozdianfard, S.M.N. Farahani, Proc. Inst. Mech. Eng., Part C 227, 862 (2013)
H.N. Chu, J. Appl. Mech. 23, 532 (1956)
T. Wah, Int. J. Mech. Sci. 5, 425 (1963)
C. Mei, K.A.M.O.L.P.H.A.N. Decha-Umphai, AIAA J. 23, 1104 (1985)
C.Y. Chia, M.K. Prabhakara, J. Appl. Mech. 45, 623 (1978)
T. Manoj, M. Ayyappan, K.S. Krishnan, B. Nageswara Rao, Z. Angew. Math. Mech. 80, 183 (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
The EPJ Publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ghadiri, M., Hosseini, S.H.S. Parametrically excited nonlinear dynamic instability of reinforced piezoelectric nanoplates. Eur. Phys. J. Plus 134, 413 (2019). https://doi.org/10.1140/epjp/i2019-12784-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/i2019-12784-9