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Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, Volume 21, Issue 3, Pages 352–367
DOI: https://doi.org/10.18500/1816-9791-2021-21-3-352-367
(Mi isu901)
 

This article is cited in 1 scientific paper (total in 1 paper)

Scientific Part
Mechanics

Waves in a viscoelastic cylindrical waveguide with a defect

A. O. Vatulyanab, V. O. Yurova

a Southern Federal University, 8a Mil’chakova St., Rostov-on-Don 344090, Russia
b Southern Mathematical Institute -- the Affiliate of Vladikavkaz Scientific Center of Russian Academy of Sciences, 22 Markusa St., Vladikavkaz 362027, Russia
Full-text PDF (351 kB) Citations (1)
References:
Abstract: In this paper, we consider a direct problem on waves in a viscoelastic inhomogeneous cylindrical waveguide with annular delamination and investigate an inverse problem on the identification of the delamination parameters on the basis of the additional data on the displacement field at the outer boundary of the waveguide. In order to account rheological properties within the framework of the complex modules concept, we use a model of a standard viscoelastic body. After applying the integral Fourier transform along the axial coordinate in the transform space, the problem is reduced to solving a canonical system of first-order differential equations with two spectral parameters. The corresponding boundary-value problems are solved numerically by using the shooting method. To satisfy the boundary conditions on the delamination, a system of two hypersingular integral equations for the opening functions (radial and axial displacements jumps) are compiled and solved on the basis of the boundary element method. To construct the displacement field on the outer boundary of the waveguide, the techniques of direct numerical integration by quadrature formulas and the residue theorem are used. When using the theorem on residues, the calculations are performed considering the three smallest complex poles in the absolute value, which corresponds to the retention of three non-uniform vibration modes. We carry out a series of computational experiments allowing to construct the wave field at the waveguide’s outer boundary. We perform the analysis of the effect of the delamination width and geometric characteristics of loading on the wave fields. On the basis of the asymptotic formula for the field at the outer boundary of the waveguide and additional data on the radial and axial displacements at one given point, a system of transcendental equations is compiled to find the delamination width and distance to the loading region. A series of computational experiments on the reconstruction of the axial position of the defect and its width are also carried out. We also perform the analysis of the damping effect on the inverse problem equations and estimate the error. Finally, we reveal the area of applicability of the proposed reconstruction method.
Key words: inhomogeneous cylindrical waveguide, viscoelasticity, delamination, inverse problem.
Funding agency Grant number
Russian Foundation for Basic Research 19-31-90017
This work was supported by the Russian Foundation for Basic Research (project No. 19-31-90017).
Received: 25.03.2021
Accepted: 29.04.2021
Bibliographic databases:
Document Type: Article
UDC: 539.3
Language: Russian
Citation: A. O. Vatulyan, V. O. Yurov, “Waves in a viscoelastic cylindrical waveguide with a defect”, Izv. Saratov Univ. Math. Mech. Inform., 21:3 (2021), 352–367
Citation in format AMSBIB
\Bibitem{VatYur21}
\by A.~O.~Vatulyan, V.~O.~Yurov
\paper Waves in a viscoelastic cylindrical waveguide with a defect
\jour Izv. Saratov Univ. Math. Mech. Inform.
\yr 2021
\vol 21
\issue 3
\pages 352--367
\mathnet{http://mi.mathnet.ru/isu901}
\crossref{https://doi.org/10.18500/1816-9791-2021-21-3-352-367}
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  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Известия Саратовского университета. Новая серия. Серия Математика. Механика. Информатика
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    Full-text PDF :37
    References:14
     
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