Studying electroelastic state of cylindrical shells from piezomaterials based on refined theory


Firsanov V. V.1*, Nguyen L. H.1**, Tran N. D.2***

1. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia
2. Le Quy Don Technical University, 236, Hoang Quoc Viet, Ha Noi, Viet Nam



Piezoelectric effect was discovered by the Pierre and Jacques Curie brothers. The essence of the piezoelectric effect consists in the fact that electric charges originate on the surface of crystals of certain classes while their deformation, and vice versa, mechanical strains occur in the crystal under electric field impact. Functional structural elements based on piezoelectrics application are widely employed in various fields of engineering, automation, computer engineering, and especially in the aerospace industry. Such structures are rather technological and allow effectively control their deformations. Thus, studying and computing the stress-strain state of structural elements from piezoelectric materials is an up-to-date task.

The article suggests a version of a refined theory for electroelastic state calculating of cylindrical shells from piezoelectric material. Mathematical model is built based on the three-dimensional equations of theory of elasticity. The problem of reducing three-dimensional equations to two-dimensional ones is realized through representing the required displacements by polynomials along the normal coordinate two degrees higher with respect to the classical theory.

A system of differential equations of equilibrium in displacements and potentials was obtained using Lagrange variation principle. Operational method, based on Laplace transform, and Maple software are employed while solving the formulated boundary value problem. Shell deformations are obtained by geometric relationships, tangential stresses are determined from the Hooke law relationships, and transverse stresses are determined from the equilibrium equations of three-dimensional theory of elasticity. Electric potential distribution on the shell surface is obtained by direct integration of the electric field intensity expressions.

A cylindrical shell from PZT piezo material is being considered as an example. Let us apply an electric potential on the upper and lower surfaces of the shell with hinged-fixed edges on the boundaries. Based on the computational result, the article demonstrates that with the absence of mechanical loads, and electric potential action only on the shell surface the stress-strain state existed inside the shell.

Comparing the results obtained in the presented work with the classical theory data allowed establishing that while studying electroelastic state of the cylindrical piezo shell near the zones of the stressed state distortion, for example, near hinged edges, the refined theory should be employed, since maximum stresses in this zone were being substantially refined.

In the boundary zone, transverse normal and tangential stresses, which are neglected in classical theory, appear to be of the same order with maximum stress values corresponding to the classical theory. Such high levels of additional stresses should be accounted for when evaluating the strength and durability of shell structures.


  1. Parton V.Z., Kudryavtsev B.A. Elektromagnitouprugost’ p’ezoelektricheskikh i elektroprovodnykh tel (Electro-magneto-elasticity of piezoelectric and electrically conductive bodies), Moscow, Nauka, 1998, 470 p.

  2. Grishanina T.V., Shklyarchuk F.N. Dinamika upravlyaemykh konstruktsii (Dynamics of controlled structures), Moscow, Izd-vo MAI, 2007, 326 p.

  3. Shklyarchuk F.N., Grishanina T.V. Dinamika upravlyaemykh konstruktsii (Dynamics of controlled structures), Moscow, Izd-vo MAI, 1999, 54 p.

  4. Timoshenko C.P., Voinovskii-Kriger S. Plastinki i obolochki (Plates and shells), Moscow, Nauka, 1966, 636 p.

  5. Gol’denveizer A.L. Teoriya uprugikh tonkikh obolochek (Theory of elastic thin shells), Moscow, Nauka, 1976, 512 p.

  6. Vasil’ev V.V., Lur’e S.A. Izvestiya AN SSSR. Mekhanika tverdogo tela, 1990, no. 6, pp. 139-146.

  7. Vlasov V.Z. Obshchaya teoriya obolochek i ee prilozheniya v tekhnike. Izbrannye Trudy (General theory of shells and its applications in engineering. Selected Works.), Moscow, AN SSSR, 1962, vol. 1, 528 p.

  8. Lur’e A.I. Teoriya uprugosti (Theory of elasticity), Moscow, Nauka, 1970, 940 p.

  9. Firsanov V.V., Chan N.D. Problemy mashinostroeniya i nadezhnosti mashin, 2011, no. 6, pp. 49-54. (V.V. Firsanov and Ch.N.Doan. Energy-consistent theory of cylindrical shells, Journal of machinery, manufacture and reliability, 2011, vol. 40, no. 6, pp.543 – 548.)

  10. Firsanov V.V. Problemy mashinostroeniya i nadezhnosti mashin, 2016, no. 6, pp. 35 – 43. (Study of stress-deformed state of rectangular plates based on non-classical theory, Journal of Machinery Manufacture and Reliability, 2016, vol. 45, no. 6, pp.515 – 522).

  11. Firsanov V.V., Vo A.Kh. Trudy MAI, 2018, no. 102, available at:

  12. Firsanov V.V., Fam V.T. Trudy MAI, 2019, no. 105, available at:

  13. Nasedkin A.V., Shevtsova M.S. Vestnik Donskogo gosudarstvennogo universiteta, 2013, no. 3-4 (72-73), pp. 16 – 26.

  14. Lyav A. Matematicheskaya teoriya uprugosti (Mathematical Theory of Elasticity) Moscow-Leningrad, ONTI, 1935, 674 p.

  15. Reissner E. Nekotorye problemy teorii obolochek. Uprugie obolochki (Some problems of theory of shells. Elastic shells), Moscow, Izd-vo inostrannoi literatury, 1962, 151 p.

  16. Zveryaev E.M., Olekhova L.V. Trudy MAI, 2015, no. 79, available at:

  17. Gol’denveizer A.L. Prikladnaya matematika i mekhanika, 1963, vol. 27, no. 4, pp. 593 – 608.

  18. H.S Tzou. Piezoelectric Shells, Distributed Sensing and Control of Continua. ISBN 978-94-010-4784-5, 1993.

  19. Reddy J.N. Mechanics of laminated composite plates and shells: theory and analysis, CRC Press, 2004, 831 p.

  20. Xiao-Hong Wu, Changqing Chen, Ya-Peng Shen, Xiao-Geng Tian. A high order theory for functionally graded piezoelectric shells, International Journal of Solids and Structures, 2002, no. 39, pp. 5325 – 5344. DOI: 10.1016/S0020-7683(02)00418-3.

Download — informational site MAI

Copyright © 2000-2020 by MAI