and Computational Methods

doi: 10.18698/2309-3684-2020-4-84110

An asymptotic theory of thermoelasticity of multilayer composite shells is proposed, the derivation of the basic equations of which is based on the asymptotic expansion in terms of a small geometric parameter of three-dimensional thermoelasticity equations. This method was previously developed by the authors for thin composite plates, and in this article it is applied to thin-walled shells of an arbitrary frame. According to the developed method, the original three-dimensional problem of thermoelasticity decomposes into a recurrent successor of one-dimensional local problems of thermoelasticity and an averaged two-dimensional problem of thin shells. For local problems of thermoelasticity, analytical solutions are obtained, which make it possible to close the averaged formulation of the problem of the theory of shells with respect to 5 unknown functions: longitudinal displacements, deflection, and two shear forces. It is shown that the averaged problem for multilayer shells coincides with the classical system of equations for Kirchhoff–Love shells, however, it is more substantiated, since the asymptotic theory does not contain any assumptions regarding the pattern of the distribution of permutations and stresses over thickness. In addition, the asymptotic theory makes it possible to calculate all the stresses in the shell, without solving any additional problems, but only by differentiating the averaged displacements.

[1] Vaglio-Laurin R. Laminar heat transfer on three-dimensional blunt nosed bodies in hypersonic flow. ARS Journal, 1959, vol. 29, iss. 1, pp. 123–129.

[2] Avduevsky V.S. Raschet trekhmernogo laminarnogo pogranichnogo sloya na liniyah rastekaniya [Calculation of a three-dimensional laminar boundary layer on spreading lines]. Izvestiya Akademii nauk SSSR. Otdelenie tekhniche-skih nauk. Mekhanika i mashinostroenie [Proceedings of the USSR Academy of Sciences. Department of Technical Sciences. Mechanics and Mechanical Engineering], 1962, no. 1, pp. 123–130.

[3] Bashkin V.A.Raschetnye sootnosheniya i programmy dlya chislennogo integrirovaniya uravnenii prostranstvennogo pogranichnogo sloya na konicheskih telah [Calculation relations and programs for numerical integration of spatial boundary layer equations on conic bodies]. Trudy TsAGI [Proceedings of TsAGI], 1968, iss. 106, pp. 97–118.

[4] Dorodnicyn A.A. Ob odnom metode resheniya uravnenii laminarnogo pogranichnogo sloya [On a method for solving the equations of a laminar boundary layer]. Journal of Applied Mechanics and Technical Physics, 1960, no. 3, pp. 111–118.

[5] Bashkin V.A., Kolina N.P. Raschet soprotivleniya treniya i teplovogo poto-ka na sfericheski zatuplennyh krugovyh konusah v sverhzvukovyh potokah [Calculation of friction resistance and heat flow on spherically blunted circular cones in supersonic flows]. Trudy TsAGI [Proceedings of TsAGI], 1968, iss. 106, pp. 119–181.

[6] Shevelev Yu.D. Trekhmernye zadachi teorii laminarnogo pogranichnogo sloya [Three-dimensional problems of the theory of a laminar boundary layer]. Moscow, Nauka Publ., 1977, 224 p.

[7] Dimitrienko Yu.I., Zakharov A.A., Koryakov M.N., Syzdykov E.K. Modeling of coupled aerogasdynamics and heat transfer processes on the thermal protection surface of a future hypersonic aircraft. BMSTU Journal of Mechanical Engineering, 2014, no. 3, pp. 23–34.

[8] Dimitrienko Yu.I., Kotenev V.P., Zakharov A.A. Metod lentochnyh adaptivnyh setok dlya chislennogo modelirovaniya v gazovoi dinamike [The method of ribbon adaptive grids for numerical modeling in gas dynamics]. Moscow, Fizmatlit Publ., 2011, 280 p.

[9] Dimitrienko Y.I., Koryakov M.N., Zakharov A.A. Application of RKDG method for computational solution of three-dimensional gasdynamic equations with non-structured grids. Mathematical Modeling and Computational Methods, 2015, no. 4, pp. 75–91.

[10] Aleksin V.A. Modelirovanie turbulentnyh szhimaemyh turbulentnyh techenii [Modeling of turbulent compressible turbulent flows]. Giperzvukovaya aerodinamika i teplomassoobmen spuskaemyh kosmicheskih apparatov i planetnyh zondov [Hypersonic aerodynamics and heat and mass transfer of descent spacecraft and planetary probes], Moscow, 2011, pp. 433–462.

[11] Zemlyansky B.A., Lunev V.V., Vlasov V.I., Gorshkov A.B., Zalogin G.N. Konvektivnyi teploobmen letatel'nyh apparatov [Convective heat exchange of aircraft]. Moscow, Fizmatlit Publ., 2014, 377 p.

[12] Avduevsky V.S., Galitseysky B.M., Glebov G.A., etc. Osnovy teploperedachi v aviacionnoi i raketno-kosmicheskoi tekhnike [Fundamentals of heat transfer in aviation and rocket and space technology]. Moscow, Mashinostroenie Publ., 1975, 623 p.

[13] Gorsky V.V. Teoreticheskie osnovy rascheta ablyacionnoi teplovoi zashchity [Theoretical bases of calculation of ablative thermal protection]. Moscow, Nauchnyi mir Publ., 2015, 688 p.

[14] Samarskiy A.A. Vvedenie v teoriyu raznostnyh skhem [Introduction to the theory of difference schemes]. Moscow, Nauka Publ., 1971, 522 p.

Димитриенко Ю.И., Губарева Е.А., Пичугина А.Е. Моделирование термона-пряжений в композитных оболочках на основе асимптотической теории. Часть 1. Общая теория оболочек. Математическое моделирование и численные методы, 2020, № 4, с. 84–110.

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