Stress-strain state of arbitrary shells with account for thermoelectric impact based on refined theory


Аuthors

Firsanov V. V.*, Nguyen L. H.**

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: k906@mai.ru
**e-mail: lehung.mai@mail.ru

Abstract

The article regards the study of the stress-strain state of thin-walled elements such as plates and shells of arbitrary geometric shapes. The shell stress-strain state depends on various exter- nal factors. Besides mechanical loads, shells can be affected by thermal process. For certain classes of materials with the polarization property and piezoelectric effect, the electric field im- pact substantially affects the stress-strain state of the shell. The presented article considers the stress-strain state of arbitrary shells under the action of mechanical and electrical loads, and temperatures. A refined theory, based on the equations of the three-dimensional elasticity theo- ry, is employed for mathematical modelling of the joint problem of thermo-electro-elasticity of shells. The sought shells displacements are being represented as polynomials on the normal to the median surface coordinate two degrees higher, compared to the classical theory of the Kirchhoff-Love type. Heating and electrical impacts will result in extra deformations occur- rence, stipulated by all-around thermal expansion and field of intensity. These deformations are being superimposed on the elastic ones and accounted for while solving the problem of stress- strain state of shells. The system of basic equations of equilibrium of the shells' thermo-electro- elasticity and corresponding boundary conditions are obtained by minimizing the total energy of the shell, based on Lagrange variation principle. The article presents the example of the stress- strain state computing of cylindrical shells, rigidly clamped at the edges under various loading options. The solution of the boundary value problem is based on the Laplace transform and the matrix sweep method.

Keywords:

stress-strain state, thermoelectroelasticity, electric action, shell of rotation, tem- perature effect, variational Lagrange principle, Laplace transform.

References

  1. Goldenveizer A.L. Teoriya uprugikh tonkikh obolochek [Theory of elastic thin shells]. Moscow: Nauka, 1976. 512 p. In Russ.

  2. Vasiliev V.V., Lurie S.A. K probleme utochneniya teorii pologikh obolochek [On the problem of refinement of the theory of flat shells]. Izvestiya АN SSSR. Mekhanika tver- dogo tela — Proceedings of the USSR Academy of Sciences. Mechanics of solid, 1990, no. 6, pp. 139–146. In Russ.

  3. Vlasov V.Z. Izbrannye trudy. Obshhaya teoriya obolochek [Se- lected Works. General theory of shells]. Moscow: Publ. house of Academy of Sciences of the USSR, 1962. 530 p. In Russ.

  4. Lurie A.I. Teoriya uprugosti [Theory of elasticity]. Mos- cow, Nauka, 1970. 939 p. In Russ.

  5. Love A.E.H. A treatise on the mathematical theory of elas- ticity. Cambridge [Eng.]: University Press, 1927. 643 p. (Russ. ed. Love A. Matematicheskaya teoriya uprugosti. Moscow-Leningrad: ONTI, 1935. 674 p.)

  6. Gol’denveizer A.L. Derivation of an approximate theory of shells by means of asymptotic integration of the equations of the theory of elasticity. Journal of Applied Mathematics and Mechanics. 1963, vol. 27, no. 4, pp. 903–924. https://doi.org/10.1016/0021-8928(63)90177-1

  7. Kovalenko A.D. Osnovy termouprugosti [Fundamentals of thermo-elasticity]. Kiev: Publishing house Naukova Dumka, 1970. 309 p. In Russ.

  8. Samsonenko G.I. General method of solving problems of thermoelastic bending of thin rectangular plates made of anisotropic material — resisting materials. Materialy 6 Mezhdu- narodnoj konferentsii po problemam gornoj promyshlennosti, stroitel’stva i ehnergetiki «Sotsiаl’no-ehkonomicheskie i ehko- logicheskie problemy gornoj promyshlennosti, stroitel’stvа i ehnergetiki» [Proceedings of the 6th Int. Conf. on Mining, Construction and Energy "Socio-economic and environmen- tal problems of mining industry, construction and energy]. Tula, 2010, vol. 2, pp. 84–88. In Russ.

  9. Parton V.Z., Kudryavtsev B.A. Elektromagnitouprugost’ p’ezoelektricheskikh i elektroprovodnykh tel [Electromagne- toelasticity of piezoelectric and electrically conductive bo- dies]. Moscow: Nauka, 1998. 470 p. In Russ.

  10. Grishanina T.V., Shklyarchuk F.N. Dinamika upravlyae- mykh konstruktsij [Dynamics of controlled structures]. Mos- cow: Publishing House of the Moscow Aviation Institute, 2007. 326 p. In Russ.

  11. Grinchenko V.T., Ulitko A.F., Shulga N.A. Elektroupru- gost’ [Electroelasticity]. Kiev: Naukova Dumka, 1989. 280 p. In Russ.

  12. Firsanov V.V., Doan Ch.N. Energy-cousistent theory of cylindrical shells. Journal of machinery, manufacture and reliabitity, 2011, vol. 40, no. 6, pp. 543–548.

  13. Firsanov V.V. Study of stress-deformed state of rectangular plates based on nonclassical theory. Journal of Machinery Manufacture and Reliability, 2016, vol. 45, no. 6, pp. 515–522.

  14. Firsanov V.V., Doan T.N. Investigation of the statics and free vibrations of cylindrical shells on the basis of a nonclassical theory. Composites: Mechanics, Computations, Applications: An International Journal, 2015, vol. 6, no. 2. pp. 135–166.

  15. Tran Ngoc Doan, Do Van Thom, Nguyen Truong Thanh, Phan Van Chuong, Nguyen Chi Tho, Nguyen Tri Ta, Hoang Nam Nguyen. Analysis of stress concentration phe- nomenon of cylinder laminated shells using higher-order shear deformation Quasi-3D theory. Composite Structures, 2020, vol. 232, article 111526. https://doi.org/10.1016/j.compstruct.2019.111526struct.2019.111526

mai.ru — informational site of MAI

Copyright © 2009-2024 by MAI