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Heat Conduction in Plates with Thin Two-Sided Multilayer Coatings Under the Conditions of Nonstationary Heating

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On the basis of the analytic solution of a one-dimensional problem of heat conduction for a plate with two-sided thin multilayer coating obtained with the help of generalized boundary conditions, we perform the investigation and establish the regularities of development of the thermal processes in the body with coating under the conditions of heating by an ambient medium with variable temperature.

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References

  1. N. M. Belyaev and A. A. Ryadno, Methods of Nonstationary Heat Conduction [in Russian], Vysshaya Shkola, Moscow (1978).

    MATH  Google Scholar 

  2. V. B. Veselovskii, Methods for the Numerical Analyses and Investigation of Thermal Processes in Industrial Installations and Technologies [in Russian], Dnipropetrovs’k University, Dnipropetrovs’k (2002).

    Google Scholar 

  3. V. B. Veselovskii, “Solution of the problems of nonstationary heat conduction for multilayer heat-protective coatings,” in: Applied Problems of Air-and-Gas Dynamics [in Russian], Naukova Dumka, Kiev (1987), pp. 95–100.

  4. V. M. Vigak, Optimal Control over Nonstationary Temperature Modes [in Russian], Naukova Dumka, Kiev (1979).

    Google Scholar 

  5. A. P. Gavris’, D. V. Ivashchuk, and P. R. Shevchuk, “Determination of residual stresses in a layered system during two-sided hightemperature spray-coating,” Mat. Met. Fiz.-Mekh. Polya, Issue 29, 8–12 (1989); English translation: J. Soviet Math., 65, No. 4, 1715–1719 (1993).

  6. N. O. Hembara, “Effect of a multilayer anticorrosion coating on the thermoelasticity of round plates,” Fiz.-Khim. Mekh. Mater., 49, No. 6, 50–54 (2013); English translation: Mater. Sci., 49, No. 6, 761–767 (2014).

  7. L. M. Dyakonyuk and Ya. G. Savula, “Computer modeling of heat transfer in a layer with a thin coating,” Visn. L’viv. Univ. Ser. Mekh.-Mat., Issue 50, 93–95 (1998).

  8. D. V. Evdokimov, D. N. Ivasishina, A. A. Kochubei, and N. V. Polyakov, “Analysis of heat conduction in a nonasymptotically thin layer,” in: Differential Equations and Their Application [in Russian], Dnipropetrovs’k National University, Dnipropetrovs’k (2006), pp. 141–156.

  9. R. Zh. Erzhanov, Yu. M. Matsevityi, U. M. Sultangazin, and V. P. Sheryshev, Concentrated Capacity in Problems of Thermal Physics and Microelectronics [in Russian], Naukova Dumka, Kiev (1992).

    Google Scholar 

  10. V. S. Zarubin, Numerical Analyses and Optimization of Thermal Insulation [in Russian], Énergoatomizdat, Moscow (1991).

    Google Scholar 

  11. O. T. Il’chenko, “Temperature field of a two-layered plate with time-varying heat-transfer conditions,” Inzh.-Fiz. Zh., 19, No. 6, 1094–1099 (1970); English translation: J. Eng. Phys. Thermophys., 19, No. 6, 1567–1570 (1970).

  12. É. M. Kartashov, Analytic Methods in the Theory of Heat Conduction in Solids [in Russian], Vysshaya Shkola, Moscow (1985).

    Google Scholar 

  13. Yu. A. Kirsanov, “Thermal state of coated solids in asymmetric cyclic heat exchange with ambient media,” Inzh.-Fiz. Zh., 69, No. 1, 123–128 (1996); English translation: J. Eng. Phys. Thermophys., 69, No. 1, 104–108 (1996).

  14. Yu. M. Kolyano and R. M. Kushnir, “Temperature stresses in plates heated by heat sources with two-sided coatings,” Mat. Met. Fiz.-Mekh. Polya, Issue 11, 72–75 (1980).

  15. H. M. Komarov, “Conditions of conjugation through a thermally thin layer in problems of heat conduction,” Dop. Nats. Akad. Nauk Ukr., No. 7, 26–32 (1996).

    MATH  Google Scholar 

  16. R. M. Kushnir, B. V. Protsyuk, and V. M. Synyuta, “Temperature stresses and displacements in a multilayer plate with nonlinear conditions of heat exchange,” Fiz.-Khim. Mekh. Mater., 38, No. 6, 31–38 (2002); English translation: Mater. Sci., 38, No. 6, 798–808 (2002).

  17. Yo. Yo. Luchko, V. M. Hembara, and N. O. Hembara, “Modeling of thermal conductivity of thin plates with multilayer coatings,” Diagnost. Dovhovich. Rekonstr. Mostiv Budiv. Konstruk., Issue 6, 65–70 (2004).

  18. A.V. Lykov, Analytical Heat Diffusion Theory, Academic Press, New York (1968).

    Google Scholar 

  19. P. A. Lyukshin, B. A. Lyukshin, N. Yu. Matolygina, and S. V. Panin, “Evaluation of temperature and temperature stresses in multilayer coatings,” Mekh. Kompoz. Mater. Konstruk., 16, No. 4, 563–574 (2010).

    Google Scholar 

  20. B. A. Lyashenko, V. A. Terletskii, N. A. Dolgov, and E. B. Soroka, “Distribution of temperature in a plate with a single-layer coating subjected to intense heating,” Probl. Prochn., No. 3, 128–133 (1998); English translation: Strength Mater., 30, No. 3, 340–344 (1998).

  21. Ya. S. Podstrigach and P. R. Shevchuk, “Temperature fields and stresses in bodies with thin coatings,” Tepl. Napryazh. Élem. Konstruk., Issue 7, 227–233 (1967).

  22. Ya. S. Podstrigach, P. R. Shevchuk, and D. V. Ivashchuk, “Effect of thin coatings on the stress-strain state of a layer with a twosided coating in diffusion impregnation,” Fiz.-Khim. Mekh. Mater., 10, No. 1, 74–80 (1974); English translation: Mater. Sci., 10, No. 1, 69–74 (1974).

  23. I. M. Prikhod’ko, “Thermal conductivity of a two-layer wall for a time-varying heat-transfer coefficient and ambient temperature,” Inzh.-Fiz. Zh., 18, No. 2, 323–327 (1970); English translation: J. Eng. Phys. Thermophys., 18, No. 2, 239–242 (1970).

  24. Ya. G. Savula, “Mathematical model of heat transfer through a three-dimensional body with thin plane coating,” Visn. L’viv. Univ. Ser. Mekh.-Mat., Issue 42, 3–7 (1995).

  25. M. Abramowitz and I. A. Stegun (editors), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, US Government Printing Office, Washington (1972).

    MATH  Google Scholar 

  26. G. N. Tret’yachenko and V. G. Barilo, “Thermal and stressed states of multilayered coatings,” Probl. Prochn., 25, No. 1, 41–49 (1993); English translation: Strength Mater., 25, No. 1, 34–41 (1993).

  27. G. N. Tret’yachenko, L. V. Kravchuk, R. I. Kuriat, et al., Thermal Fatigue of Materials under the Conditions of Nonuniform Thermal Stressed State [in Russian], Naukova Dumka, Kiev (1985).

    Google Scholar 

  28. N. P. Fleishman, “Mathematical models of thermal conjugation of media with thin foreign interlayers or coatings,” Visn. L’viv. Univ. Ser. Mekh.-Mat., Issue 39, 30–34 (1993).

  29. V. A. Shevchuk, “On the construction of the generalized boundary conditions of convective heat exchange with a medium through thin nonplanar coatings,” Dop. Nats. Akad. Nauk Ukr., No. 7, 76–82 (2011).

    Google Scholar 

  30. M. A. Al-Nimr and M. K. Alkam, “A generalized thermal boundary condition,” Heat Mass Transf., 33, No. 1-2, 157–161 (1997).

    Article  Google Scholar 

  31. F. Du, M. R. Lovell, and T. W. Wu, “Boundary-element-method analysis of temperature fields in coated cutting tools,” Int. J. Solids Struct., 38, No. 26-27, 4557–4570 (2001).

    Article  MATH  Google Scholar 

  32. T. Elperin and G. Rudin, “Temperature field in multilayer assembly affected by a local laser heating,” Int. J. Heat Mass Transf., 38, No. 17, 3143–3147 (1995).

    Article  Google Scholar 

  33. L. M. Heijnen, T. W. Kuijpers, and J. A. Klostermann, “Model description and experiments on carbon diffusion through protective layers,” High Temp.–High Press., 20, No. 3, 305–313 (1988).

  34. A. Mezin, J. Lepage, and P. B. Abel, “An analytical solution for nonsteady-state diffusion through thin films,” Thin Solid Films, 272, No. 1, 124–l31 (1996).

    Article  Google Scholar 

  35. O. Sarikaya, Y. Islamoglu, and E. Celik, “Finite-element modeling of the effect of the ceramic coatings on heat transfer characteristics in thermal barrier applications,” Mater. Design, 26, No. 4, 357–362 (2005).

    Article  Google Scholar 

  36. V. A. Shevchuk, “Calculation of thermal state of bodies with multilayer coatings,” in: Lecture Notes in Computer Science, Vol. 2330, Springer, Berlin (2002), pp. 500–509.

  37. V. A. Shevchuk, “Generalized boundary conditions to solving thermal stress problems for bodies with thin coatings,” in: R. B. Hetnarski (editor), Encyclopedia of Thermal Stresses, Vol. 4, Springer (2014), pp. 1942–1953.

  38. V. A. Shevchuk, “Modeling and computation of heat transfer in a system “body–multilayer coating,” Heat Transf. Res., 37, No. 5, 421–433 (2006).

  39. J. Zhao, Y. Li, and X. Ai, “Analysis of transient thermal stress in sandwich plate with functionally graded coatings,” Thin Solid Films, 516, No. 21, 7581–7587 (2008).

    Article  Google Scholar 

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Authors and Affiliations

  1. Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, Ukrainian National Academy of Sciences, Lviv, Ukraine

    V. А. Shevchuk

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  1. V. А. Shevchuk
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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 58, No. 2, pp. 148–157, April–June, 2015.

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Shevchuk, V.А. Heat Conduction in Plates with Thin Two-Sided Multilayer Coatings Under the Conditions of Nonstationary Heating. J Math Sci 223, 184–197 (2017). https://doi.org/10.1007/s10958-017-3347-y

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