Analytical solution for thermally stressed two-layer elastic strip

Аuthors

Hoa V. D.^*, Zveryaev E. M.^**

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

*e-mail: dong.hoavan@yandex.ru
**e-mail: zveriaev@gmail.com

Abstract

The authors consider the problem of the stress-strain state determining of a two-layer elastic strip staying in a temperature field under the action of a transverse bending load. Certain conditions may be imposed on the displacement or stress at the short sides of the strip.

A method called in [1] the Saint-Venant–Picard–Banach (SVPB) method is used to construct a solution. Being based on a generalization of the ideas of the Saint-Venant semi-inverse method [2] and the Picard method of successive approximations of [3], it allows finding all the unknowns for a system of equations of the elasticity theory by sequential calculations without any preliminary hypotheses. The dimensionless partial differential equations with a small thin-walled parameter for a thin strip are being written as integral ones with respect to the transverse coordinate, similar to what is done in the Picard method. Next, the equations and transformed elasticity relationships are being arranged in a sequence that, accordingly, allows sequentially calculate the unknown problems, expressing them through the four arbitrary functions of the longitudinal coordinate obtained by integration along the transverse coordinate and integral coefficients from the function of changing the stiffness of the layers in the transverse direction. Having these formulas for all the unknowns of the problem being searched for, the expressions for the boundary conditions on the long sides to determine four arbitrary integration functions according to Picard [4] may be written. These expressions represent ordinary differential equations with small parameters for derivatives with respect to the longitudinal coordinate. Their solutions are slowly changing functions of the main stress-strain state and rapidly changing functions such as the edge effect. The integration constants for both solutions are being determined from the boundary conditions on the short sides of the strip. For slowly changing ones, the conditions coincide with the classical ones for a beam. The conditions unsatisfied by the classical solution are being satisfied by the integration constants of rapidly changing solutions, adding to the slowly changing solution the edge effect and features in the corners of the strip [5]. Substituting these solutions of the main unknowns into the formulas previously obtained after the first iteration, we obtain formulas for all the unknown unknowns of the problem of the theory of elasticity of a layered strip, valid for each point of the strip.

Keywords:

planar problem, layered material, thermally stressed state, asymptotic approach, iterations, Saint-Venant–Picard–Banach method

References

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

2. Lyav A. Matematicheskaya teoriya uprugosti (A treatise on the mathematical theory of elasticity), Moscow-Leningrad, ONTI, 1935, 674 p.

3. Kamke E. Spravochnik po obyknovennym differentsial’nym uravneniyam (Handbook of ordinary differential equations), Moscow, Mir, 1984, 584 p.

4. 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.

5. Friedrichs K.O. Bulletin of the American Mathematical Society, 1955, vol. 61, no. 6, pp. 485‒504. DOI: 10.1090/S0002-9904-1955-09976-2

6. Milton Gr. W. The theory of composites, Cambridge University Press, 2004, 719 p.

7. Vasil’ev V.V., Protasov V.D., Bolotin V.V. et al. Kompozitsionnye materialy: Spravochnik. (Composite materials: Handbook), Moscow, Mashinostroenie, 1990, 512 p.

8. Baker A., Dutton S., Kelly D. Composite materials for Aircraft structures, American Institute of Aeronautics and Astronautics, Inc, 2004, 603 p.

9. Zveryaev E.M., Olekhova L.V. Preprinty IPM im. M.V.Keldysha, 2014, no. 95. DOI: 10.13140/RG.2.2.21968.00000

10. Zveryaev E.M., Olekhova L.V. Trudy MAI, 2015, no. 79. URL: https://trudymai.ru/eng/published.php?ID=55762

11. Chakrabarti A., Sheikh A.H., Griffith M., Oehlers D.J. Analysis of composite beams with partial shear interactions using a higher order beam theory, Engineering Structures, 2012, no. 36, pp. 283–291. DOI: 10.12989/sem.2015.53.4.625

12. Firsanov V.V., Fam V.T, Chan N.D. Trudy MAI, 2020, no. 114. URL: http://trudymai.ru/eng/published.php?ID=118893. DOI: 10.34759/trd-2020-114-07

13. Grishchenko S.V., Popov Yu.I. Trudy MAI, 2013, no. 65. URL: https://trudymai.ru/eng/published.php?ID=35854

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

15. 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. DOI: 10.34759/trd-2021-116-17

16. 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

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

18. Zveryaev E.M. IV Vserossiiskii simpozium «Mekhanika kompozitsionnykh materialov i konstruktsii»: sbornik trudov. Vol. 2. Moscow, IPRIM RAN, 2012, pp. 226-234.

19. Zveryaev E.M. Vestnik Rossiiskogo universiteta druzhby narodov. Seriya: Inzhenernye issledovaniya, 2021, vol. 22, no. 2, pp. 293–304.

20. 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.

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

22. Feodos’ev V.I. Soprotivlenie materialov (Strength of materials), Moscow, Nauka. Glavnaya redaktsiya fiziko-matematicheskoi lituratury, 1986, 512 p.

23. Zveryaev E.M. Preprinty IPM im. M.V. Keldysha, 2018, no. 83, 19 p. DOI: 10.20948/prepr-2018-83

Download