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Propagation of one-dimensional non-stationary waves in viscoelastic half space

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Abstract

The problem of one-dimensional non-stationary wave propagation in viscoelastic half space is considered. Relaxation kernel is considered exponential. Initial conditions are set to zero, displacement is determined on the half space boundary. The solution is represented in the form of perturbation and surface Green function resultant. For determination of Green function time Laplace transform is used. Its inverse transform is carried out both analytically expanding Green function in series and numerically. A good coincidence of analytical and numerical Green function calculation results is shown. The final solution is determined analytically. Examples of calculations are represented.

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References

  1. E. I. Shemyakin, “Non-stationary perturbations propagation in viscoelastic medium,” Dokl. Akad. Nauk SSSR 104, 34–37 (1955).

    MathSciNet  Google Scholar 

  2. A. A. Lokshin and Yu. V. Suvorova, Mathematical Theory of Wave Propagation in Memory Mediums (Mosk. Gos. Univ., Moscow, 1982) [in Russian]

    MATH  Google Scholar 

  3. I. G. Filippov and O. A. Egorychev, Wave Processes in Linear Viscoelastic Mediums (Mashinostroenie, Moscow, 1983) [in Russian]

    Google Scholar 

  4. F. B. Badalov, Hereditary Theory of Vicsoelasticity Integral and Integral-Differential Equations Solution Methods (Mekhnat, Tashkent, 1987) [in Russian]

    MATH  Google Scholar 

  5. S. A. Lychev, “Coupled dynamic problem of thermoviscoelasticity,” Mekh. Tverd. Tela, No. 5, 95–113 (2008).

    Google Scholar 

  6. M. Kh. Ilyasov, Non-Stationary Viscoelastic Waves (Azerbaijan Hava Yollary, Baku, 2011) [in Russian]

    MATH  Google Scholar 

  7. D. S. Berry and S. C. Hunter, “The propagation of dynamic stresses in visco-elastic rods,” J. Mech. Phys. Solids 4 (2), 72–95 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  8. O. W. Dillon, “Transient stresses in nongomogeneous viscoelastic (Maxwell) materials,” J. Aerospace Sci. 29, 284–288 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  9. H. Kolsky, “Stress waves in anelastic solids,” J. Geophys. Res. 68, 1193–1194 (1963).

    Article  MATH  Google Scholar 

  10. B. D. Coleman, M. E. Gurtin, and I. R. Herrera, “Waves in materials with memory,” Arch. Ration. Mech. Anal. 19, 1–19 (1965).

    MATH  Google Scholar 

  11. K. C. Valanis and S. Chang, “Stress wave propagation in a finite viscoelastic thin rod with a constitutive law of the hereditary type,” Develop. Theor. Appl.Mech. 4, 465–483 (1967).

    Article  Google Scholar 

  12. Lin Cong-mou and Yang Lin-de, “Analytic solution on propagating law of stress wave from explosion of extended charge in linear viscoelasticmedium,” J. Shandong Univ. Sci. Technol. Nat.Sci. 20 (3), 1–3 (2001).

    MathSciNet  Google Scholar 

  13. S. G. Pshenichnov, “Non-stationary dynamic problems of linear viscoelasticity,” Mekh. Tverd. Tela, №1, 84–96 (2013).

    Google Scholar 

  14. A. G. Gorshkov, A. L. Medvedskii, L. N. Rabinskii, and D. V. Tarlakovskii, Waves in Continious Media (Fizmatlit, Moscow, 2004) [in Russian]

    Google Scholar 

  15. Yu. N. Rabotnov, Mechanics of Deformed Solid (Nauka, Moscow, 1988) [in Russian].

    MATH  Google Scholar 

  16. V. A. Vestyak and D. V. Tarlakovskii, “Investigation of non-stationary radial vibrations of electromagnetoelastic thick-wall sphere using numerical inversion of Laplace transform,” Vestn. Tver. Univ., Ser.: Prikl.Mat. 1 (9), 51–64 (2014).

    Google Scholar 

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Correspondence to E. A. Korovaytseva.

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Submitted by A. V. Lapin

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Korovaytseva, E.A., Pshenichnov, S.G. & Tarlakovskii, D.V. Propagation of one-dimensional non-stationary waves in viscoelastic half space. Lobachevskii J Math 38, 827–832 (2017). https://doi.org/10.1134/S1995080217050237

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