Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

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Vatulyan A. O., Yurov V. O. Waves in a viscoelastic cylindrical waveguide with a defect. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, vol. 21, iss. 3, pp. 352-367. DOI: 10.18500/1816-9791-2021-21-3-352-367, EDN: JWCHHS

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Waves in a viscoelastic cylindrical waveguide with a defect

Vatulyan Alexander Ovanesovitsch, Southern Federal University
Yurov Victor Olegovych, Southern Federal University

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.

This work was supported by the Russian Foundation for Basic Research (project No. 19-31-90017).
  1. Vatulyan A. O., Yurov V. O. On the dispersion relations for an inhomogeneous waveguide with attenuation. Mechanics of Solids, 2016, vol. 51, iss. 5, pp. 576–582. https://doi.org/10.3103/S0025654416050101
  2. Vatul’yan A. O., Yurov V. O. Dispersion properties of an inhomogeneous piezoelectric waveguide with attenuation. Acoustical Physics. 2017, vol. 63, iss. 4, pp. 369–377. https://doi.org/10.1134/S1063771017040133
  3. Sohn H., Dutta D., Yang J. Y., Park H. J., DeSimio M., Olson S., Swenson E. Delamination detection in composites through guided wave field image processing. Composites Science and Technology, 2011, vol. 71, iss. 9, pp. 1250–1256. https://doi.org/10.1016/j.compscitech.2011.04.011
  4. Su Z., Ye L., Lu Y. Guided Lamb waves for identification of damage in composite structures: A review. Journal of Sound and Vibration, 2006, vol. 295, iss. 3–5, pp. 753–780. https://doi.org/10.1016/j.jsv.2006.01.020
  5. Eremin A. A., Golub M. V., Glushkov E. V., Glushkova N. V. Identification of delamination based on the Lamb wave scattering resonance frequencies. NDT& E International, 2019, vol. 103, pp. 145–153. https://doi.org/10.1016/j.ndteint.2019.03.001
  6. Golub M. V., Doroshenko O. V., Fomenko S. I., Wang Y., Zhang C. Elastic wave propagation, scattering and localization in layered phononic crystals with arrays of striplike cracks. International Journal of Solids and Structures, 2020, vol. 212, pp. 1–22. https://doi.org/10.1016/j.ijsolstr.2020.12.001
  7. Grinchenko V. T., Meleshko V. V. Garmonicheskie kolebaniya i volny v uprugikh telakh [Harmonic Vibrations and Waves in Elastic Bodies]. Moscow, Nauka, 1981. 282 p. (in Russian).
  8. Glushkov E. V., Glushkova N. V., Evdokimov A. A. Hybrid numerical-analytical scheme for calculating elastic wave diffraction in locally inhomogeneous waveguides. Acoustical Physics, 2018, vol. 64, iss. 1, pp. 1–9. https://doi.org/10.1134/S1063771018010086
  9. Alves C., Leitao V. Crack analysis using an enriched MFS domain decomposition technique. Engineering Analysis with Boundary Elements, 2006, vol. 30, iss. 3, pp. 160–166. https://doi.org/10.1016/j.enganabound.2005.08.012
  10. Gravenkamp H. Efficient simulation of elastic guided waves interacting with notches, adhesive joints, delaminations and inclined edges in plate structures. Ultrasonics, 2018, vol. 82, pp. 101–113. https://doi.org/10.1016/j.ultras.2017.07.019
  11. Aleksandrov V. M., Pozharskii D. A. To the problem of a crack on the elastic strip-halfplane interface. Mechanics of Solids, 2001, vol. 36, iss. 1, pp. 70–76.
  12. Lifanov I. K. Metod singuliarnykh integral’nykh uravneniy i chislennyi eksperiment [Method of Singular Integral Equations and Numerical Experiment]. Moscow, TOO “Janus”, 1995. 520 p. (in Russian).
  13. Vatulyan A. O., Yurov V. O. Numerical andasymptotic solution of the problem of oscillations of an inhomogeneous waveguide with an annular crack of finite width. Acoustical Physics. 2020, vol. 66, iss. 5, pp. 441–448. https://doi.org/10.1134/S1063771020050140
  14. Antonenko N. N. The problem of a longitudinal crack with a filler in a strip. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2015, vol. 15, iss. 3, pp. 315– 322 (in Russian). https://doi.org/10.18500/1816-9791-2015-15-3-315-322
  15. Doroshenko O. V., Kirillova E. V., Fomenko S. I. An asymptotic solution of the hypersingular boundary integral equation simulating wave scattering by the interface strip-like crack. PNRPU Mechanics Bulletin, 2019, iss. 2, pp. 86–99 (in Russian). https://doi.org/10.15593/perm.mech/2019.2.07
  16. Glushkov E. V., Glushkova N. V., Golub M. V. Blocking of traveling waves and energy localization due to the elastodynamic diffraction by a crack. Acoustical Physics, 2006, vol. 52, iss. 3, pp. 259–269. https://doi.org/10.1134/S1063771006030043
  17. Vatul’yan A., Yavruyan O. An asymptotic approach in problems of crack identification. Journal of Applied Mathematics and Mechanics, 2006, vol. 70, iss. 4, pp. 647–656. https://doi.org/10.1016/j.jappmathmech.2006.09.015
  18. Bostrom A., Golub M. Elastic SH wave propagation in a layered anisotropic plate with interface damage modeled by spring boundary conditions. The Quarterly Journal of Mechanics and Applied Mathematics, 2009, vol. 62, iss. 1, pp. 39–52. https://doi.org/10.1093/qjmam/hbn025
  19. Ma L., Wu L., Zhou Z., Guo L. Scattering of the harmonic anti-plane shear waves by a crack in functionally graded piezoelectric materials. Composite Structures, 2005, vol. 69, iss. 4, pp. 436–441. https://doi.org/10.1016/j.compstruct.2004.08.001
  20. Vatul’yan A. O., Baranov I. V. Determination of the crack configuration in an anisotropic elastic medium. Acoustical Physics, 2005, vol. 51, iss. 4, pp. 385–391. https://doi.org/10.1134/1.1983599
  21. Ijjeh A. A., Ullah S., Kudela P. Full wavefield processing by using FCN for delamination detection. Mechanical Systems and Signal Processing, 2021, vol. 153, 107537. https://doi.org/10.1016/j.ymssp.2020.107537
  22. Christensen R. M. Theory of Viscoelasticity: An Introduction. New York, Academic Press, 1971. 245 p. (Russ. ed.: Moscow, Mir, 1974. 338 p.).
  23. Vorovich I. I., Babeshko V. A. Dinamicheskie smeshannye zadachi teorii uprugosti dlia neklassicheskikh oblastei [Dynamic Mixed Problems of Elasticity for Non-classical Domains]. Moscow, Nauka, 1979. 320 p. (in Russian).
  24. Kecs W., Teodorescu P. Introduction to the Theory of Generalized Functions with Applications to Engineering. Bucuresti, Editura Tehnica, 1975. 412 p. (Russ. ed.: Moscow, Mir, 1978. 518 p.).
  25. Belotserkovsky S. M., Lifanov I. K. Chislennye metody v singul’arnykh integral’nykh uravneniyakh i ikh primeneniya v aerodinamike, teorii uprugosti i elektrodinamike [Numerical Methods in Singular Integral Equations and Their Application in Aerodynamics, Elasticity Theory, Electrodynamics]. Moscow, Nauka, 1985. 253 p. (in Russian).
  26. Vatul’yan A. O., Yurov V. O. Analysis of forced vibrations in a functionally gradient cylindrical waveguide. Acoustical Physics, 2018, vol. 64, iss. 6, pp. 649–656. https:// doi.org/10.1134/S1063771019010147
  27. Vatulyan A. O. Koeffitsyentnye obratnye zadachi mekhaniki [Coefficient Inverse Problems of Mechanics]. Moscow, Fizmatlit, 2019. 272 p. (in Russian).