Analytical solution for thermally stressed two-layer elastic strip


Аuthors

Hoa V. D.*, Zveryaev E. M., Pykhtin A. V.

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

*e-mail: dong.hoavan@yandex.ru

Abstract

The article considers a thin rectangular strip response to the mechanical load impact (in the plane of the object) and a temperature field in the formulation of a plane problem of the theory of elasticity. The basis of the solution is application of the Saint-Venant–Picard–Banach method for integrating the equations of the theory of elasticity of thin-walled systems (SVPB). The method combines iterative and asymptotic approaches and is of greater freedom from assumptions limiting the solution.

The first feature is transition to the sequential integration of the original equations. The relationships are being lined-up in such a way that the result of the previous one is used in the subsequent expression as a known value. Introduction of the initial approximation allows considering such sequence as an iterative operator of the successive approximations method. The unknown functions selection as the initial ones, being determined (refined) in the solution process, corresponds to the idea of the semi-inverse Saint-Venant method, expanding its interpretation to an iterative one.

Elimination of differentiation operators with respect to the thickness coordinate in equations by integration includes in the iterative operator the integration operators correlated with the Picard operators of the method for solving first-order differential equations resolved with respect to the derivative (which is also iterative).

Consistent application of the iterative operator gives integrals (the form of solution for the unknowns of the problem) in the form of asymptotic rads with respect to the small thin-wall parameter. The solution error is being estimated by the degree of the small parameter (which is an arbitrarily small value) of the leading term of the discarded part of the series. The solution existence and uniqueness are determined by the principle of compressed mappings (Banach's fixed point theorem).

The resulting integrals (iterative approximations for the functions of the stress-strain state) are used to satisfy the boundary conditions of the problem. As the result of this, the main unknown problems are determined (the arbitrary rules of integration, which include initial approximation functions).

SVPB is an analytical method, and the asymptotic approach is also usually used to isolate from the equations the available relations characterizing the components of the solution with certain properties (in particular, quickly and slowly changing components responsible for the edge effect and the main solution). When solving the problem under consideration, the type of solution for the main unknowns is obtained by direct transformations without the use of asymptotic hypotheses. For the first iteration, a comparison with the asymptotic solution was performed. The solution is supplemented with the results obtained from the relations of the next iteration.

Keywords:

plane problem, rectangular strip, thermally stressed state, iterations, Saint-Venant–Picard–Banach method

References

  1. Vol'mir A.S. Gibkie plastiny i obolochki (Flexible plates and shells), Moscow, Gosudarstvennoe izdatel'stvo tekhniko-teoreticheskoi literatury, 1956, 420 p.

  2. Vasil'ev V.V. Izvestiya RAN. Mekhanika tverdogo tela, 1992, no. 3, pp. 26–47.

  3. Pikul' V.V. Izvestiya Rossiiskoi akademii nauk. Mekhanika tverdogo tela, 2000, no. 2, pp. 153–168.

  4. Zveryaev E.M., Tupikova E.M. Stroitel'naya mekhanika inzhenernykh konstruktsii i sooruzhenii, 2021, vol. 17, no. 6, pp. 588–607.

  5. Gol'denveizer A.L. Prikladnaya matematika i mekhanika, 1962, vol. 26, no. 4, pp. 668–686.

  6. Zveryaev E.M. Postroenie osnovnogo napryazhennogo sostoyaniya tonkoi uprugoi obolochki metodom prostykh iteratsii: Detormirovanie i razrushenie elementov konstruktsii letatel'nykh apparatov (Construction of the main stress state of a thin elastic shell using the method of simple iterations: Detortion and destruction of aircraft structural elements. ny devices), Moscow, MATI, 1989, pp. 56-63.

  7. Zveryaev E.M. Prikladnaya matematika i mekhanika, 2019, vol. 83, no. 5–6, pp. 823–833. DOI: 10.1134/S0032823519050126

  8. Zveryaev E.M., Pykhtin A.V., Khoa V.D. Stroitel'naya mekhanika i raschet sooruzhenii, 2021, no. 4, pp. 2–11. DOI: 10.37538/0039-2383.2021.4.2.11

  9. Lyav A. Matematicheskaya teoriya uprugosti (Mathematical theory of elasticity), Moscow-Leningrad, Ob"edinennoe nauchno-tekhnicheskoe izdanie NKTP SSSR, 1935, 674 p.

  10. Kolmogorov A.N., Fomin S.V. Elementy teorii funktsii i funktsional'nogo analiza (Elements of the theory of functions and functional analysis), Moscow, Nauka, 1976, 543 p.

  11. Kamke E. Spravochnik po obyknovennym differentsial'nym uravneniyam (Handbook of ordinary differential equations), Moscow, Nauka. Glavnaya redaktsiya fiziko-matematicheskoi literatury, 1976, 576 p.

  12. Lebedev N.N. Temperaturnye napryazheniya v teorii uprugosti (Temperature stresses in the theory of elasticity), Moscow-Leningrad, ONTI. Glavnaya redaktsiya tekhniko-teoreticheskoi literatury, 1937, 110 p.

  13. Zveryaev E.M. Vestnik Rossiiskogo universiteta druzhby narodov. Seriya: Inzhenernye issledovaniya, 2021, vol. 22, no. 3, pp. 293–304.

  14. Zveryaev E.M., Pykhtin A.V. Materialy XXII Mezhdunarodnoi konferentsii po vychislitel'noi mekhanike i sovremennym prikladnym programmnym sistemam, VMSPPS’2021 (Alushta, 4–13 sentyabrya 2021), Moscow, Izd-vo MAI, 2021, pp. 214-215.

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

  16. Boli B., Ueiner D. Teoriya temperaturnykh napryazhenii (Theory of temperature stress), Moscow, Mir, 1964, 512 p.

  17. Aleksandrov A.V., Alfutov N.A., Astanin V.V. et al. Dinamika i prochnost' mashin. Teoriya mekhanizmov i mashin. Mashinostroenie: Entsiklopediya. Kn. 2. (Dynamics and strength of machines. Theory of mechanisms and machines. Mechanical engineering. Encyclopedia), Moscow, Mashinostroenie, 1995, 624 p.

  18. Sementsova A.N. Trudy MAI, 2013, no. 65. URL: https://trudymai.ru/eng/published.php?ID=35951

  19. Lur'e S.A., Solyaev Yu.O., Nguen D.K., Medvedskii A.L., Rabinskii L.N. Trudy MAI, 2013, no. 71. URL: https://trudymai.ru/eng/published.php?ID=47084

  20. Lur'e S.A., Dudchenko A.A., Nguen D.K. Trudy MAI, 2014, no. 75. URL: https://trudymai.ru/eng/published.php?ID=49674

  21. Chigrinets E.G., Rodriges S.B., Zabolotnii D.I., Chotchaeva S. K. Trudy MAI, 2021, no. 116. URL: https://trudymai.ru/eng/published.php?ID=121111

  22. Firsanov V.V., Fam V.T. Trudy MAI, 2019, no. 105. URL: http://trudymai.ru/eng/published.php?ID=104174


Download

mai.ru — informational site MAI

Copyright © 2000-2024 by MAI

 Вход