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On the method of expansion on a variable interval for non-stationary problems in continuum mechanics

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Abstract

The present paper is devoted to the development of the method for searching an approximate solution of non-stationary problems on a variable interval. This method was first proposed by L.I. Slepyan. Here, the field of its applicability has been expanded to systems of equations, including equations of both hyperbolic and parabolic type. We describe the procedure of such approach in detail on a number of classical partial differential equations and compare the obtained results with exact analytical solution. Also, we consider a dynamic non-coupled thermoelastic problem. The method of expansion on a variable interval allows us to estimate the material response to a thermal disturbance at a large distance from the source.

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References

  1. Dech, G.: Guidance for practical application of Laplace transformation. M.: Nauka (1965) (in Russian)

  2. Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.: Tables of Integral Transforms, Vol. 1 (1954). Tables of Integral Transforms, 2 (1954)

  3. Debnath, L., Bhatta, D.: Integral Transforms and Their Applications. CRC Press, Boca Raton (2014)

    Book  Google Scholar 

  4. Slepyan, L.I.: Analysis of non-steady-state strain by means of series defined in a variable interval. Izv.Ac.Sci.USSR, Mekhanika, No. 4, 62–69 (1965). (in Russian)

  5. Slepyan, L.I.: Non-stationary Elastic Waves. Sudostroenie, Leningrad (1972). (in Russian)

    MATH  Google Scholar 

  6. Morozov, N.F., Indeitsev, D.A., Lukin, A.V., Popov, I.A., Privalova, O.V., Semenov, B.N., Shtukin, L.V.: Bernoulli–Euler beam under action of a moving thermal source: characteristics of the dynamic behavior. Dokl. Phys. Pleiades Publ. 64(4), 185–188 (2019)

    Article  Google Scholar 

  7. Cheng, C.W., Wang, S.Y., Chang, K.P., Chen, J.K.: Femtosecond laser ablation of copper at high laser fluence: modeling and experimental comparison. Appl. Surf. Sci. 361, 41–48 (2016)

    Article  Google Scholar 

  8. Wang, H., Dai, W., Melnik, R.: A finite difference method for studying thermal deformation in a double-layered thin film exposed to ultrashort pulsed lasers. Int. J. Therm. Sci. 45(12), 1179–1196 (2006)

    Article  Google Scholar 

  9. Danilovskaya, V.I.: On a dynamical problem of thermoelasticity. Prikl. Mat. Mekh 16(3), 341–344 (1952). (in Russian)

    MathSciNet  Google Scholar 

  10. Kovalenko, A.D.: Thermoelasticity. Basic Theory and Applications. Wolters-Noordhoff, Groningen (1970)

    MATH  Google Scholar 

  11. Nowacki, W.: Dynamic Problems of Thermoelasticity. Springer, Berlin (1975)

    MATH  Google Scholar 

  12. Jordan, P.M., Puri, P.: Revisiting the Danilovskaya problem. J. Therm. Stress. 29(9), 865–878 (2006)

    Article  Google Scholar 

  13. Vovnenko, N.V., Zimin, B.A., Sudenkov, Yu.V: Experimental investigations of thermoelastic stresses in heat and non-thermoconducting solids with submicroscopic durations of laser heating. J. Tech. Phys. 8(6), 57–62 (2011)

    Google Scholar 

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Authors and Affiliations

  1. Institute for Problems in Mechanical Engineering, V.O., Bolshoj pr., 61, St. Petersburg, Russia, 199178

    Dmitry A. Indeitsev, Boris N. Semenov & Dmitry S. Vavilov

  2. St. Petersburg University, Universitetskaya Emb., 13B, St. Petersburg, Russia, 199034

    Dmitry A. Indeitsev & Boris N. Semenov

  3. St. Petersburg Polytechnic University, Polytechnicheskaya, 29, St. Petersburg, Russia, 195251

    Dmitry A. Indeitsev

  4. Mozhaisky Military Space Academy, Jdanovskaya 13, St. Petersburg, Russia, 197198

    Dmitry S. Vavilov

Authors
  1. Dmitry A. Indeitsev
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  2. Boris N. Semenov
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  3. Dmitry S. Vavilov
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Correspondence to Dmitry S. Vavilov.

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Indeitsev, D.A., Semenov, B.N. & Vavilov, D.S. On the method of expansion on a variable interval for non-stationary problems in continuum mechanics. Acta Mech 232, 1961–1969 (2021). https://doi.org/10.1007/s00707-020-02890-6

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  • DOI: https://doi.org/10.1007/s00707-020-02890-6

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