Skip to main content
Log in

A Generalized Function in Heat Conductivity Problems for Multilayer Structures with Heat Sources

  • Reliability, Strength, and Wear Resistance of Machines and Structures
  • Published:
Journal of Machinery Manufacture and Reliability Aims and scope Submit manuscript

Abstract

An accurate analytical solution of the stationary problem of heat conductivity for a multilayer structure with heat sources under asymmetric boundary conditions of the third kind is obtained based on the use of the theory of generalized functions. Using the asymmetric unit function (Heaviside function), the problem for a multilayer structure is reduced to a single-layer with discontinuous (piecewise-continuous) medium properties. By introducing a new independent variable, the nonlinear differential equation (nonlinearity of the second kind—the thermal-physical properties depend on the spatial variable) is reduced to a linear one, direct integration of which allows obtaining an accurate analytical solution of the problem considered. Examples of investigations of the temperature fields for some certain multilayer structures are given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kudinov, V.A., Averin, B.V., and Stefanyuk, E.V., Teploprovodnost’ i termouprugost' v mnogosloinykh konstruktsiyakh (Heat Conductivity and Thermoelasticity and Multilayered Constructions), Moscow: Vysshaya Shkola, 2008.

    Google Scholar 

  2. Averin, B.V., Kolotilkin, D.I., and Kudinov, V.A., Sturm-Liouville problem for a differential equation of second order with discontinuous coefficients, J. Eng. Phys. Thermophys., 2000, vol. 73, no. 4, pp. 735–740.

    Google Scholar 

  3. Vladimirov, V.S., Obobshchennye funktsii v matematicheskoi fizike (Generalized Functions in Mathematical Physics), Moscow: Nauka, 1979.

    MATH  Google Scholar 

  4. Gel’fand, I.M. and Shilov, G.E., Obobshchennye funktsii i deistviya nad nimi (Generalized Functions and Operations over them), Moscow: Fizmatgiz, 1959.

    MATH  Google Scholar 

  5. Ketch, V. and Teodorescu, P., Introduction to the Theory of Generalized Functions with Applications in Engineering, New York: Wiley, 1978.

    Google Scholar 

  6. Vigak, V.M., On the construction of the solution of the heat equation for a piecewise homogeneous body, Dokl. AN USSR, Ser. A, 1980, no. 1, p. 30.

    MathSciNet  Google Scholar 

  7. Kolyano, Yu.M., Application of generalized functions in thermal mechanics of piecewise homogeneous bodies, in Matematicheskie metody i fiziko-mekhanicheskie polya (Mathematical Methods and Physical-Mechanical Fields), Kiev: Naukova Dumka, 1978, no. 7, pp. 7–11.

    MathSciNet  Google Scholar 

  8. Obraztsov, I.F. and Onanov, G.G., Stroitel’naya mekhanika skoshennykh tonkostennykh sistem (Structural Mechanics of Tapered Thin Walled Systems), Moscow: Mashinostroenie, 1973.

    Google Scholar 

  9. Onanov, G.G., Equations with singular coefficients of the type of the delta function and its derivatives, Dokl. Akad. Nauk SSSR, 1970, vol. 191, no. 5, pp. 997–1000.

    MathSciNet  MATH  Google Scholar 

  10. Protsyuk, B.V., On solution of heat conductivity and thermoelasticity problems for multilayer bodies, Dokl. Akad. Nauk USSR, Ser. A, 1977, no. 11, pp. 1019–1021.

    Google Scholar 

  11. Kudinov, V.A., Kudinov, I.V., and Skvortsova, M.P., Generalized functions and additional boundary conditions in heat conduction problems for multilayered bodies, Comput. Math. Math. Phys., 2015, vol. 55, no. 4, pp. 666–676.

    Article  MathSciNet  MATH  Google Scholar 

  12. Volkov, A.G., Radio-transparent shelters for powerful transmitting phased antenna arrays, Extended Abstract of Cand. Sci. Dissertation, Moscow: NIRF, 1984.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. V. Eremin.

Additional information

Original Russian Text © A.V. Eremin, E.V. Stefanyuk, O.Yu. Kurganova, V.K. Tkachev, M.P. Skvortsova, 2018, published in Problemy Mashinostroeniya i Nadezhnosti Mashin, 2018, No. 3, pp. 52–58.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Eremin, A.V., Stefanyuk, E.V., Kurganova, O.Y. et al. A Generalized Function in Heat Conductivity Problems for Multilayer Structures with Heat Sources. J. Mach. Manuf. Reliab. 47, 249–255 (2018). https://doi.org/10.3103/S1052618818030056

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1052618818030056

Navigation