Solution of the coupled nonstationary problem of thermoelasticity for a rigidly fixed multilayer circular plate by the finite integral transformations method

Cover Page


Cite item

Full Text

Abstract

A new closed solution of an axisymmetric non-stationary problem is constructed for a rigidly fixed round layered plate in the case of temperature changes on its upper front surface (boundary conditions of the 1st kind) and a given convective heat exchange of the lower front surface with the environment (boundary conditions of the 3rd kind).

The mathematical formulation of the problem under consideration includes linear equations of equilibrium and thermal conductivity (classical theory) in a spatial setting, under the assumption that their inertial elastic characteristics can be ignored when analyzing the operation of the structure under study.

When constructing a general solution to a non-stationary problem described by a system of linear coupled non-self-adjoint partial differential equations, the mathematical apparatus for separating variables in the form of finite integral Fourier–Bessel transformations and generalized biorthogonal transformation (CIP) is used. A special feature of the solution construction is the use of a CIP based on a multicomponent relation of eigenvector functions of two homogeneous boundary value problems, with the use of a conjugate operator that allows solving non-self-adjoint linear problems of mathematical physics. This transformation is the most effective method for studying such boundary value problems.

The calculated relations make it possible to determine the stress-strain state and the nature of the distribution of the temperature field in a rigid round multilayer plate at an arbitrary time and radial coordinate of external temperature influence. In addition, the numerical results of the calculation allow us to analyze the coupling effect of thermoelastic fields, which leads to a significant increase in normal stresses compared to solving similar problems in an unrelated setting.

About the authors

Dmitry A. Shlyakhin

Samara State Technical University

Email: d-612-mit2009@yandex.ru
ORCID iD: 0000-0003-0926-7388
SPIN-code: 7802-5059
Scopus Author ID: 26028953500
http://www.mathnet.ru/person52312

Dr. Techn. Sci.; Head of Dept.; Dept. of Strength of Materials and Structural Mechanics

244, Molodogvardeyskaya st., Samara, 443100, Russian Federation

Zhanslu M. Kusaeva

Samara State Technical University

Author for correspondence.
Email: zhkusaeva@mail.ru
ORCID iD: 0000-0001-7028-0130
SPIN-code: 6893-7012
Scopus Author ID: 57216585356
ResearcherId: AAQ-1159-2020
http://www.mathnet.ru/person157455

Postgraduate Student; Dept. of Structural Mechanics and Engineering Geology
Russian Federation, 244, Molodogvardeyskaya st., Samara, 443100, Russian Federation

References

  1. Podstrigach Ya. S., Lomakin V. A., Kolyano Yu. M. Termouprugost’ tel neodnorodnoi struktury [Thermoelasticity of Bodies with Inhomogeneous Structure]. Moscow, Nauka, 1984, 368 pp. (In Russian)
  2. Boley B., Weiner J. Theory of Thermal Stresses. New York, Wiley, 1960, xvi+586 pp.
  3. Nowacki W. Dinamicheskie zadachi termouprugosti [Dynamic Problems of Thermoelasticity]. Moscow, Mir, 1970, 256 pp. (In Russian)
  4. Kovalenko A. D. Vvedenie v termouprugost’ [Introduction to Thermoelasticity]. Kiev, Nauk. dumka, 1965, 202 pp. (In Russian)
  5. Radayev Yu. N., Taranova M. V. Wavenumbers of type III thermoelastic waves in a long waveguide under sidewall heat interchanging, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2011, no. 2(23), pp. 53–61 (In Russian). https://doi.org/10.14498/vsgtu965.
  6. Shashkov A. G., Bubnov V. A., Ianovsky S. Yu. Volnovye iavleniia teploprovodnosti. Sistemno-strukturnyi podkhod [Wave Phenomena of Heat Conductivity. System and Structural Approach]. Moscow, Editorial URSS, 2004, 296 pp. (In Russian)
  7. Kudinov V. A., Kartashev E. M., Kalashnikov V. V. Analiticheskie resheniia zadach teplomassoperenosa i termouprugosti dlia mnogosloinykh konstruktsii [Analytical Solutions of Heat and Mass Transfer and Thermoelasticity Problems for Multilayer Structures]. Moscow, Vyssh. Shk., 2005, 430 pp. (In Russian)
  8. Kudinov V. A., Klebleev R. M., Kuklova E. A. Obtaining exact analytical solutions for nonstationary heat conduction problems using orthogonal methods, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2017, vol. 21, no. 1, pp. 197–206 (In Russian). https://doi.org/10.14498/vsgtu1521.
  9. Kartashov E. M. Analiticheskie metody v teploprovodnosti tverdykh tel [Analytical Methods in the Theory of the Thermal Conductivity of Solids]. Moscow, Vyssh. shk., 1985, 480 pp. (In Russian)
  10. Filatov V. N. Calculation of the temperature effects of flexible gently sloping shells supported by an orthogonal grid of edges, In: Nelineinye zadachi rascheta tonkostennykh konstruktsii [Nonlinear Problems of Calculating Thin-Walled Structures]. Saratov, Saratov State Univ., 1989, pp. 108–110 (In Russian).
  11. Kudinov V. A., Kuznetsova A. E., Eremin A. V., Kotova E. V. Analytical solutions of the quasistatic thermoelasticity task with variable physical properties of a medium, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2014, no. 2(35), pp. 130–135 (In Russian). https://doi.org/10.14498/vsgtu1219.
  12. Kobzar’ V. N., Fil’shtinskii L. A. The plane dynamic problem of coupled thermoelasticity, J. Appl. Math. Mech., 2008, vol. 72, no. 5, pp. 611–618. https://doi.org/10.1016/j.jappmathmech.2008.11.002.
  13. Sargsyan S. H. Mathematical model of micropolar thermo-elasticity of thin shells, J. Thermal Stresses, 2013, vol. 36, no. 11, pp. 1200–1216. https://doi.org/10.1080/01495739.2013.819265.
  14. Zhornik A. I., Zhornik V. A., Savochka P. A. On a problem of thermoelasticity for a solid cylinder, Izv. YuFU. Tekhn. Nauki, 2012, no. 6(131), pp. 63–69 (In Russian).
  15. Zhukov P. V. Calculation of temperature fields and thermal stresses in thick-walled cylinder under impulse heat supply, Vestnik IGEU, 2013, no. 3, pp. 54–57 (In Russian).
  16. Makarova I. S. The solution of uncoupled thermoelastic problem with first kind boundary conditions, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2012, no. 3(28), pp. 191–195 (In Russian). https://doi.org/10.14498/vsgtu1088.
  17. Harmatij H., Król M., Popovycz V. Quasi-static problem of thermoelasticity for thermosensitive infinite circular cylinder of complex heat exchange, Adv. Pure Math., 2013, vol. 3, no. 4, pp. 430–437. https://doi.org/10.4236/apm.2013.34061.
  18. Lee Z.-Y. Coupled problem of thermoelasticity for multilayered spheres with time-dependent boundary conditions, J. Mar. Sci. Tech., 2004, vol. 12, no. 2, pp. 93–101.
  19. Lord H. W., Shulman Y. A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids, 1967, pp. 299–309. https://doi.org/10.1016/0022-5096(67)90024-5.
  20. Kovalev V. A., Radayev Yu. N., Semenov D. A. Coupled dynamic problems of hyperbolic thermoelasticity, Izv. Saratov Univ. Math. Mech. Inform., 2009, vol. 9, no. 4(2), pp. 94–127 (In Russian). https://doi.org/10.18500/1816-9791-2009-9-4-2-94-127.
  21. Kovalev V. A., Radayev Yu. N., Revinsky R. A. Generalized cross-coupled type-III thermoelastic waves propagating via a waveguide under sidewall heat interchange, Izv. Saratov Univ. Math. Mech. Inform., 2011, vol. 11, no. 1, pp. 59–70 (In Russian). https://doi.org/10.18500/1816-9791-2011-11-1-59-70.
  22. Senitskii Yu. E. Solution of coupled dynamic thermoelasticity problem for an infinite cylinder and sphere, Sov. Appl. Mech., 1982, vol. 18, no. 6, pp. 514–520. https://doi.org/10.1007/BF00883341.
  23. Shlyakhin D. A., Kalmova M. A. A coupled unsteady thermoelasticity problem for a long hollow cylinder, Engineering Journal of Don, 2020, no. 3 (In Russian). http://ivdon.ru/ru/magazine/archive/n3y2020/6361.
  24. Lychev S. A. A coupled dynamic problem of thermoelasticity for a finite cylinder, Vestn. Samar. Gos. Univ., Estestvennonauchn. Ser., 2003, no. 4(30), pp. 112–124 (In Russian).
  25. Lychev S. A., Manzhirov A. V., Joubert S. V. Closed solutions of boundary-value problems of coupled thermoelasticity, Mech. Solids, 2010, vol. 45, no. 4, pp. 610–623. https://doi.org/10.3103/S0025654410040102.
  26. Shlyakhin D. A., Dauletmuratova Zh. M. Non-stationary coupled axisymmetric thermoelasticity problem for a rigidly fixed round plate, PNRPU Mechanics Bulletin, 2019, no. 4, pp. 191–200 (In Russian). https://doi.org/10.15593/perm.mech/2019.4.18.
  27. Fu J. W., Chen Z. T., Qian L. F. Coupled thermoelastic analysis of a multi-layered hollow cylinder based on the C–T theory and its application on functionally graded materials, Compos. Struct., 2015, vol. 131, no. 1, pp. 139–150. https://doi.org/10.1016/j.compstruct.2015.04.053.
  28. Vitucci G., Mishuris G. Analysis of residual stresses in thermoelastic multilayer cylinders, J. Eur. Ceram. Soc., 2016, vol. 36, no. 9, pp. 2411–2417, arXiv: 1511.06562 [cond-mat.mtrl-sci]. https://doi.org/10.1016/j.jeurceramsoc.2015.12.003.
  29. Shliakhin D. A., Dauletmuratova Zh. M. Nonstationary axisymmetric thermoelasticity problem for a rigidly fixed circular plate, Engineering Journal: Science and Innovation, 2018, no. 5(77), 1761, 18 pp (In Russian). https://doi.org/10.18698/2308-6033-2018-5-1761
  30. Sneddon I. N. Fourier Transforms. New York, McGraw-Hill, 1950, xii+542 pp.
  31. Senitskij Yu. È. Biorthogonal multicomponent finite integral transformation and its application to boundary value problems of mechanics, Russian Math. (Iz. VUZ), 1996, vol. 40, no. 8, pp. 69–79.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2021 Authors; Samara State Technical University (Compilation, Design, and Layout)

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies