Free vibration response of functionally graded carbon nanotube double curved shells and panels with piezoelectric layers in a thermal environment

Document Type : Article


1 Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran

2 New Technologies Research Center, Amirkabir University of Technology, Tehran, Iran


This paper presents free vibration of the double-curved shells and panels with piezoelectric layers in a thermal environment. Vibration characteristics of elliptical, spherical, cycloidal, and toro circular shells of revolution are studied in detail. Vibration behavior of carbon nanotubes (CNTs) reinforced composite shells embedded with piezoelectric layers at the upper and lower surfaces is scrutinized. It is supposed that temperature changes linearly through-thickness direction. Reissner- Mindlin and the first order shear deformation (FSDT) theories are implemented to derive the governing equations of the considered structures. The distribution of nanotubes is assumed to be linear along the thickness direction. For solving the equation, the General Differential Quadrature (GDQ) method is used to obtain a numerical analysis for the dynamics of the objective structures. Finally, the effects of boundary conditions, the thickness of piezoelectric layers, functional distribution of CNTs, thermal environment and kinds of the circuit (opened-circuit and closed-circuit) are analyzed. Eigenvalue system is solved to obtain natural frequencies. It is delineated that the obtained fundamental frequency by the closed -circuit is smaller than those obtained by the opened-circuit. Another interesting result is that the natural frequency is decreased by increasing temperature.


Main Subjects

1. Qatu, M., Sullivan, R.W., and Wanga, W. Recent research advances on the dynamic analysis of composite shells 2000{2009", Compos. Struct., 93, pp. 14{31 (2010). 2. Alijani, F. and Amabili, M. Non-linear vibrations of shells: A literature review from 2003 to 2013", Int. J. Nonlin Mech., 58, pp. 233{257 (2014). 3. Kulkarni, H., Khusru, K.Z., and Shravan Aiyappa, K. Application of piezoelectric technology in automotive systems", Mater Today Proc., 5, pp. 21299{21304 (2018). 4. Tornabene, F. 2-D GDQ solution for free vibrations of anisotropic doubly-curved shells and panels of revolution", Compos. Struct., 93, pp. 1854{1876 (2011). 5. Viola, E., Tornabene, F., and Fantuzzi, N. Static analysis of completely doubly-curved laminated shells and panels using general higher-order shear deformation theories", Compose. Struc., 101, pp. 59{93 (2013). 6. Shooshtari, A. and Razavi, S. Large-amplitude free vibration of magneto-electro-elastic curved panels", Sci. Iran., 23, pp. 2606{2615 (2016). 7. Pang, F., Li, H., Cui, J., Du, Y., and Gao, C. Application of ugge thin shell theory to the solution of free vibration behaviors for spherical-cylindricalspherical shell: A uni_ed formulation", Eur. J. Mech A-Solid, 74, pp. 381{393 (2019). 8. Rout, M., Hota, S., and Karmakar, A. Thermoelastic free vibration response of graphene reinforced laminated composite shells", Eng. Struct., 178, pp. 179{ 190 (2019). 9. Pang, F., Li, H., Wang, X., Miao, X., and Li, S. A semi analytical method for the free vibration of doublycurved shells of revolution", Comput. Math Appl., 75, pp. 3249{3268 (2018). 10. Awrejcewicz, J., Kurpa, L., and Shmatko, T. Linear and nonlinear free vibration analysis of laminated functionally graded shallow shells with complex plan form and di_erent boundary conditions", Int. J. Nonlin Mech., 107, pp. 161{169 (2018). 11. Fang, X., Zhu, C., Liu, J., and Liu, X. Surface energy e_ect on free vibration of nano-sized piezoelectric double-shell structures", Physica B Condens Matter., 529, pp. 41{56 (2018). 12. Zhoua, Z., Nia, Y., Zhua, S., Tonga, Z., Sunb, J., and Xua, X. An accurate and straightforward approach to thermo-electro-mechanical vibration of piezoelectric _ber-reinforced composite cylindrical shells", Compose. Struc., 207, pp. 292{303 (2019). 13. Mallek, H., Jrad, H., Wali, M., and Dammak, F. Piezoelastic response of smart functionally graded structure with integrated piezoelectric layers using discrete double directors shell element", Compose. Struc., 210, pp. 354{366 (2019). 14. Akbari Alashti, A., Khorsand, M., and Tarahhomi, M.H. Thermo-elastic analysis of a functionally graded spherical shell with piezoelectric layers by di_erential quadrature method", Sci. Iran., 20, pp. 109{119 (2013). 15. Behjat, B., Salehi, M., Armin, A., Sadighi, M., Abbasi, M. Static and dynamic analysis of functionally graded piezoelectric plates under mechanical and electrical loading", Sci. Iran., 20, pp. 986{994 (2011). 16. Wang, Q., Shi, Q., Liang, D., and Pang, F. Free vibration of four-parameter functionally graded moderately thick doubly-curved panels and shells of revolution with general boundary conditions", Appl .Math. Model, 4, pp. 705{734 (2017). 17. Reddy, I.N., Mechanics of Laminated Composite Plates and Shells, Washington, D.C, CRC Press (2004). 18. Ventsel, E. and Kravthammer, T. Thin plates and shells", New York, NY10016 (2001). 19. Sayyaadi, H. and Askari Farsangi, M.A. An analytical solution for dynamic behavior of thick doubly curved functionally graded smart panels", Compose. Struct., 107, pp. 88{102 (2014). 20. Kiani, Y. Free vibration of functionally graded carbon nanotube reinforced composite plates integrated with piezoelectric layers", Comput. Math Appl., 72, pp. 2433{2449 (2016). 21. Jafari, A.A., Khalili, S.M.R., and Tavakolian, M. Nonlinear vibration of functionally graded cylindrical shells embedded with a piezoelectric layer", Thin Walled Struct., 79, pp. 8{15 (2014). 2408 S. Khorshidi et al./Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 2391{2408 22. Song, Z.G., Zhang, L.W., and Liew, K.M. Vibration analysis of CNT-reinforced functionally graded composite cylindrical shells in thermal environments", Int. J. Mech Sci., 115-116, pp. 339{347 (2016). 23. Zghal, S., Farikha, A., and dammak, F. Free vibration analysis of carbon nanotube-reinforced functionally graded composite shell structures", Appl. Math. Model, 53, pp. 132{155 (2018). 24. Arshid, E. and Khorshidvand, A.R. Free vibration analysis of saturated porous FG circular plates integrated with piezoelectric actuators via di_erential quadrature method", Thin-Walled Struct, 125, pp. 220{233 (2018). 25. Bodaghi, M. and Shakeri, M. An analytical approach for free vibration and transient response of functionally graded piezoelectric cylindrical panels subjected to impulsive loads". Compose. Struct., 94, pp. 1721{1735 (2012). 26. Shu, C. Free vibration analysis of composite laminated conical shells by generalized di_erential quadrature", J. Sound Vib., 194, pp. 587{604 (1996). 27. Behjat, B., Salehi, M., Sadighi, M., Armin, A., and Abbasi, M. Static, dynamic, and free vibration analysis of functionally graded piezoelectric panels using _nite element method", J. Intell Mater Syst. Struct., 20, pp. 1635{1646 (2009). 28. Ansari, R., Torabi, J., and Faghih Shojaei, M. Vibrational analysis of functionally graded carbon nanotube reinforced composite", Eur. J. Mech A-Solid, 60, pp. 166{182 (2016). 29. Xie, X., Zheng, H., and Jin, G. Free vibration of four-parameter functionally graded spherical and parabolic shells of revolution with arbitrary boundary conditions", Compos Part B: Eng., 77, pp. 59{73 (2015). 30. Zhang, H., Shi, D., Wang, Q., and Qin, B. Free vibration of functionally graded parabolic and circular panels with general boundary conditions", Curved and Layer. Struct., 4, pp. 52{84 (2017). 31. Shen, H.S. and Xiang, Y. Nonlinear analysis of nanotube-reinforced composite beams resting on elastic foundations in thermal environments", Eng. Struct., 56, pp. 698{708 (2013). 32. Han, Y. and Elliott, J. Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites", Comput. Mater. Sci, 39, pp. 315{323 (2007). 33. Shariyat, M. Dynamic buckling of imperfect laminated plates with piezoelectric sensors and actuators subjected to thermo-electro-mechanical loadings, considering the temperature-dependency of the material properties", Compose. Struct., 88, pp. 228{239 (2016).