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Peculiarities of the Boundary Conditions in the Analysis of Nonlinear Waves for Surface and Torsional Waves as Examples

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The peculiarities of using the boundary conditions in the nonlinear problems of wave propagation are analyzed for media that deform elastically and have an internal or external boundary. The analysis is carried out for two types of waves: a Rayleigh surface wave and a torsional wave. The statement and wave analysis of the specified two waves in the linear (classical) approach are briefly described since the linear solutions are used as a first approximation in the nonlinear approach. Common to both types of waves (and, apparently, to other waves in media with boundaries) is a significant complication in the nonlinear approach of the boundary conditions due to the difference between the shape of the boundary before and after deformation (in the linear approach, the shape of the boundary does not change). Also, a common feature is the significant complication of the mathematical representation of the boundary conditions due to the additional nonlinear terms. The example of the Rayleigh surface wave shows that the procedure for solving the problem using nonlinear boundary conditions is difficult but possible. The example of a torsional wave reveals the fact that the use of the condition of absence of stresses on the boundary surface (assumption of a free boundary) may not be completely correct.

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Authors and Affiliations

  1. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, 3 Nesterova St., Kyiv, 03057, Ukraine

    J. J. Rushchitsky, O. O. Khotenko & V. M. Yurchuk

  2. National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, 37 Beresteiskyi Av., Kyiv, 03056, Ukraine

    V. M. Yurchuk

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  1. J. J. Rushchitsky
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  2. O. O. Khotenko
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Correspondence to J. J. Rushchitsky.

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Translated from Prykladna Mekhanika, Vol. 59, No. 5, pp. 44–60, September–October 2023.

This study was sponsored by the budgetary program Support of Priority Areas of Research (KPKVK 6541230).

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Rushchitsky, J.J., Khotenko, O.O. & Yurchuk, V.M. Peculiarities of the Boundary Conditions in the Analysis of Nonlinear Waves for Surface and Torsional Waves as Examples. Int Appl Mech 59, 540–554 (2023). https://doi.org/10.1007/s10778-024-01240-6

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