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Modelling of rectangular Kirchhoff plate oscillations under unsteady elastodiffusive perturbations

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Abstract

We study unsteady elastic-diffusion vibrations of a rectangular isotropic Kirchhoff plate. The coupled elastic-diffusion multicomponent continuum model is used to formulate the problem. We are using the d’Alembert variational principle to obtain from the model the equations of longitudinal and transverse vibrations of a rectangular isotropic elastodiffusive Kirchhoff plate. The problem solution is sought in integral form. The kernels of integral representations are the Green’s functions. For their determination, the Laplace transform and expansion in double Fourier series are used. As an example, we consider the bending of a simply supported plate. The effect of bending on the diffusion processes in the plate is investigated.

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References

  1. Gorskij, V.S.: Issledovanie uprugogo posledejstviya v splave Si–Au s uporyadochennoj reshetkoj. Zhurnal e’ksperimental’noj i teoreticheskoj fiziki 6(3), 272–276 (1936). [in Russian]

    Google Scholar 

  2. Nachtrieb, N.H., Handler, G.S.: A relaxed vacancy model for diffusion in crystalline metals. Acta Metall. 2(6), 797–802 (1954)

    Article  Google Scholar 

  3. Petit, J., Nachtrieb, N.H.: Self-diffusion in liquid gallium. J. Chem. Phys. 24, 1027 (1956)

    Article  Google Scholar 

  4. Shvets, R.N., Flyachok, V.M.: The equations of mechanothermodiffusion of anisotropic shells taking account of transverse strains. Mat. Met. Fiz.-Mekh. Polya 20, 54–61 (1984)

    MATH  Google Scholar 

  5. Aouadi, M., Copetti, M.I.M.: Analytical and numerical results for a dynamic contact problem with two stops in thermoelastic diffusion theory. ZAMM Z. Angew. Math. Mech. (2015). https://doi.org/10.1002/zamm.201400285

    Article  Google Scholar 

  6. Copetti, M.I.M., Aouadi, M.: A quasi-static contact problem in thermoviscoelastic diffusion theory. Appl. Numer. Math. 109, 157183 (2016)

    Article  MathSciNet  Google Scholar 

  7. Aouadi, M., Miranville, A.: Smooth attractor for a nonlinear thermoelastic diffusion thin plate based on GurtinPipkins model. Asympt. Anal. 95, 129160 (2015)

    MATH  Google Scholar 

  8. Aouadi, M.: On thermoelastic diffusion thin plate theory. Appl. Math. Mech. Engl. Ed. 36(5), 619632 (2015)

    Article  MathSciNet  Google Scholar 

  9. Aouadi, M., Miranville, A.: Quasi-stability and global attractor in nonlinear thermoelastic diffusion plate with memory. Evol. Equ. Control Theory 4(3), 241–263 (2015)

    Article  MathSciNet  Google Scholar 

  10. Bhattacharya, D., Kanoria, M.: The influence of two temperature generalized thermoelastic diffusion inside a spherical shell. Int. J. Eng. Tech. Res. (IJETR) 2(5), 151–159 (2014)

    Google Scholar 

  11. Eremeev, V.S.: Diffuziya i napryazheniya, p. 182. Energoatomizdat, Moscow (1985) (in Russian)

  12. Knyazeva A.G.: Introduction to the thermodynamics of irreversible processes. In: Lectures About Models, p. 172. Ivan Fedorov Publishing House, Tomsk (2014) (in Russian)

  13. Deswal, S., Kalkal, K.: A two-dimensional generalized electro-magneto-thermoviscoelastic problem for a half-space with diffusion. Int. J. Therm. Sci. 50(5), 749–759 (2011)

    Article  Google Scholar 

  14. Elhagary, M.A.: A two-dimensional generalized thermoelastic diffusion problem for a half-space subjected to harmonically varying heating. Acta Mech. 224, 3057–3069 (2013)

    Article  MathSciNet  Google Scholar 

  15. Kumar, R., Kothari, S., Mukhopadhyay, S.: Some theorems on generalized thermoelastic diffusion. Acta Mech. 217, 287–296 (2011)

    Article  Google Scholar 

  16. Sherief, H.H., El-Maghraby, N.M.: A thick plate problem in the theory of generalized thermoelastic diffusion. Int. J. Thermophys. 30, 2044–2057 (2009)

    Article  Google Scholar 

  17. Igumnov, L.A., Tarlakovskii, D.V., Zemskov, A.V.: A two-dimensional nonstationary problem of elastic diffusion for an orthotropic one-component layer. Lobachevskii J. Math. 38(5), 808817 (2017). https://doi.org/10.1134/S1995080217050146

    Article  MathSciNet  MATH  Google Scholar 

  18. Zemskov, A.V., Tarlakovskiy, D.V.: Two-dimensional nonstationary problem elastic for diffusion in an isotropic one-component layer. J. Appl. Mech. Tech. Phys. 56(6), 1023–1030 (2015). https://doi.org/10.1134/S0021894415060127

    Article  MathSciNet  MATH  Google Scholar 

  19. Le, K.C.: Vibrations of Shells and Rods, vol. 425. Springer, Berlin (1999)

    Book  Google Scholar 

  20. Le, K.C., Yi, J.-H.: An asymptotically exact theory of smart sandwich shells. Int. J. Eng. Sci. 106, 179–198 (2016)

    Article  MathSciNet  Google Scholar 

  21. Le, K.C.: An asymptotically exact theory of functionally graded piezoelectric shells. Int. J. Eng. Sci. 112, 42–62 (2017)

    Article  MathSciNet  Google Scholar 

  22. Mikhailova, E.Y., Tarlakovskii, D.V., Fedotenkov, G.V.: Obshchaya teoriya uprugikh obolochek. MAI, Moscow (2018) (in Russian)

  23. Tarlakovskii, D.V., Zemskov, A.V.: An elastodiffusive orthotropic Euler–Bernoulli beam with considering diffusion flux relaxation. Math. Comput. Appl. 24, 23 (2019). https://doi.org/10.3390/mca24010023

    Article  MathSciNet  Google Scholar 

  24. Larikov, L.I., Fel‘chenko, B.M., Mazarenko, V.F., Gurevich, C.M., Xapchenko, G.K.: Anomal‘noe uskorenie diffuzii pri impul‘snom razrushenii metallov, Doklady AN SSSR. Texnicheskaya Fizika 221(5), 1073–1075 (1975) (in Russian)

  25. Nirano, K., Cohen, M., Averbach, V., Ujiiye, N.: Self-diffusion in alpha iron during compressive plastic flow. Trans. Metall. Soc. AIME 227, 950 (1963)

    Google Scholar 

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Acknowledgements

This work was funded by the subsidy from RFBR (Project No. 20-08-00589 A).

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Authors and Affiliations

  1. Moscow Aviation Institute (National Research University), Moscow, Russia

    A. V. Zemskov & D. V. Tarlakovskii

  2. Institute of Mechanics Lomonosov Moscow State University, Moscow, Russia

    A. V. Zemskov & D. V. Tarlakovskii

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  1. A. V. Zemskov
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  2. D. V. Tarlakovskii
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Correspondence to A. V. Zemskov.

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Zemskov, A.V., Tarlakovskii, D.V. Modelling of rectangular Kirchhoff plate oscillations under unsteady elastodiffusive perturbations. Acta Mech 232, 1785–1796 (2021). https://doi.org/10.1007/s00707-020-02879-1

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  • DOI: https://doi.org/10.1007/s00707-020-02879-1

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