New Class of Homogeneous Solutions for Plane Elastodynamic Problems

Abstract—

The paper presents a new type of functionally invariant Smirnov–Sobolev solutions for the wave equation, which can be used for solving many homogeneous problems of elastodynamics. The derived solution has a unique property: the double preimage of Laplace and Fourier transforms of this function with a new argument and coefficient coincides with the function itself. Thus, this property allows us to find an inversion formula for double integral transformations. The method for obtaining this formula is demonstrated by solving the Lamb problem for a half-plane under the transition from the image to the original.

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REFERENCES

  1. 1

    H. Lamb, “On the propagation of tremors over the surface of an elastic solid,” Phil. Trans. R. Soc. London A 203, 1–42 (1904).

    ADS  Article  Google Scholar 

  2. 2

    K. B. Broberg, Cracks and Fracture (Acad. Press, London, 1999).

    Google Scholar 

  3. 3

    V. B. Poruchikov, Methods of the Dynamic Theory of Elasticity (Nauka, Moscow, 1986) [in Russian].

    Google Scholar 

  4. 4

    A. V. Kravtsov, S. V. Kuznetsov, and S. Ya. Sekerzh-Zen’kovich, “Finite element models in Lamb’s problem,” Mech. Solids 46 (6), 952–959 (2011).

    ADS  Article  Google Scholar 

  5. 5

    Kh. Kh. Il’yasov, A. V. Kravtsov, S. V. Kuznetsov, and S. Ya. Sekerzh-Zen’kovich, “Exterior 3D Lamb problem: harmonic load distributed over a surface,” Mech. Solids 51 (1), 39–45 (2016).

    ADS  Article  Google Scholar 

  6. 6

    S. V. Kuznetsov and E. O. Terentjeva, “Planar internal Lamb problem: waves in the epicentral zone of a vertical power source,” Acoust. Phys. 61 (3), 356–367 (2015).

    ADS  Article  Google Scholar 

  7. 7

    V. A. Ditkin and A. P. Prudnikov, Integral Transformations and Operational Calculus (Nauka, Moscow, 1961) [in Russian].

    Google Scholar 

  8. 8

    V. I. Smirnov, The Course of Higher Mathematics (Nauka, Moscow, 1974) [in Russian].

    Google Scholar 

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Correspondence to N. B. Rasulova or M. B. Rasulov.

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Translated by I. Tselishcheva

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Rasulova, N.B., Rasulov, M.B. New Class of Homogeneous Solutions for Plane Elastodynamic Problems. Mech. Solids 55, 1057–1061 (2020). https://doi.org/10.3103/S0025654420070171

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Keywords:

  • homogeneous solutions
  • Laplace and Fourier transformations
  • Lamb’s problem
  • wave equation
  • Smirnov–Sobolev solutions