Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

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


For citation:

Wilde M. V., Sergeeva N. V. Development of Asymptotic Methods for the Analysis of Dispersion Relations for a Viscoelastic Solid Cylinder. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2017, vol. 17, iss. 2, pp. 183-195. DOI: 10.18500/1816-9791-2017-17-2-183-195

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
22.05.2017
Full text:
(downloads: 57)
Language: 
Russian
Heading: 
UDC: 
539.3

Development of Asymptotic Methods for the Analysis of Dispersion Relations for a Viscoelastic Solid Cylinder

Autors: 
Wilde Maria Vladimirovna, Saratov State University
Sergeeva Nadezhda Viktorovna, Saratov State University
Abstract: 

Propagation of time-harmonic waves in a viscoelastic solid cylinder is considered. Vibrations of the cylinder are described by three-dimensional viscoelasticity equations in  cylindrical coordinates. The stress-free surface boundary conditions are imposed. Viscoelastic properties are described by integral operators with a fractional-exponential kernel. For the case of a rational singularity parameter the method of asymptotic analysis of dispersion relations is proposed, which is based on the generalized power series expansion. For the axisymmetric waves the asymptotic expansions of the dispersion equation roots are obtained for low and high frequencies. The numerical results are presented to confirm the applicability of the proposed method.

References: 
  1. Ainola L. J., Nigul U. K. Stress waves in elastic plates and shells. Trans. Estonian SSR Acad. Sci., Ser. of Phis.-Math. Tech. Sci., 1965, vol. 14, no. 1, pp. 3–63 (in Russian).
  2. Kaplunov J. D., Kovalev V. A., Wilde M. V. Matching of asymptotic models in scattering of a plane acoustic waves by an elastic cylindrical shell. Journal of Sound and Vibration, 2003, vol. 264, iss. 3, pp. 639–655. DOI: https://doi.org/10.1016/S0022-460X(02)01212-9.
  3. Wilde М. V., Kaplunov Yu. D., Kossovich L. Yu. Kraevye i interfejsnye rezonansnye javlenija v uprugih telah [Edge and interfacial resonance phenomena in elastic bodies]. Мoscow, Fizmatlit, 2010. 280 p. (in Russian).
  4. Rzhanicin А. R. Teorija polzuchesti [Creep theory]. Moscow, Stroyizdat, 1968. 418 p. (in Russian).
  5. Rabotnov Yu. N. Polzuchest jelementov konstrukcij [Creep of structural elements]. Moscow, Nauka, 1966. 752 p. (in Russian).
  6. Chervinko O. P., Senchenkov I. K. Garmonicheskie volny v sloe i beskonechnom cilindre [Harmonic waves in a layer and in an infinite cylinder]. Prikladnaya mekhanika [Applied Mechanics], 1986, vol. 22, no. 12, pp. 31–37 (in Russian).
  7. Tanaka K., Kon-No A. Harmonic Waves in Lenear Viscoelastic Plate. Bull. JSME, 1980, vol. 23, no. 176, pp. 185–193. DOI: https://doi.org/10.1299/jsme1958.23.185.
  8. Voropaev G. A., Popkov V. I. Rasprostranenie osesimmetrichnyh voln v vjazkouprugom polom cilindre [Propagation of axisymmetric waves in a viscoelastic hollow cylinder]. Prikladnaja mehanika [Applied Mechanics], 1989, vol. 25, no. 10, pp. 19–23 (in Russian).
  9. Anofrikova N. S., Sergeeva N. V. Chislennyj analiz dispersionnyh uravnenij v sluchae nasledstvenno-uprugogo sploshnogo cilindra [Numerical analysis of dispersion equations for viscoelastic solid cylinder]. Sovremennye problemy mehaniki sploshnoj sredy: trudy XVII Mezhdunarodnoj konferencii [Proc. XVII Intern. Conf. „Modern Problems of continuum mechanics“]. Rostov-on-Don, South. Fed. Univ. Press, 2014, vol. 1, pp. 44–48 (in Russian).
  10. Vatul’yan A. O., Yurlov V. O. On the dispersion relations for an inhomogeneous waveguide with attenuation. Mech. Solids, 2016, iss. 5, pp. 576–582. DOI: https://doi.org/10.3103/S0025654416050101.
  11. Anofrikova N. S., Kossovich L. Yu., Chernenko V. P. Asymptotic methods for constructing solutions in the neighborhood of wave fronts in a viscoelastic rod for large values of time. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2005, vol. 5, iss. 1, pp. 82–88 (in Russian).
  12. Anofrikova N. S., Sergeeva N. V. Investigation of harmonic waves in the viscoelastic layer. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2014, vol. 14, iss. 3, pp. 321–328 (in Russian).
  13. Wilde M. V., Sergeeva N. V. Asymptotic analysis of the viscoelastic material properties effect on the dispersion of harmonic waves in a solid cylinder. Sovremennye problemy mehaniki sploshnoj sredy: trudy XVIII Mezhdunarodnoj konferencii [Proc. XVIII Intern. Conf „Modern Problems of continuum mechanics“]. Rostov-on-Don, South. Fed. Univ. Press, 2016, vol. 1, pp. 135–139 (in Russian).
  14. Rabotnov Yu. N. Elements of Hereditary Solid Mechanics. Moscow, Nauka, 1977. 384 p. (Engl. transl. by Mir Publ., Moscow, 1980.)
  15. Man’kovskii V. A., Sapunov V. T. Nomographic properties of the exponential fractional e-function used for the description of linear viscoelasticity. Industrial laboratory, 2000, vol. 66, iss. 3, pp. 188–191.
  16. Badalov F. B., Abdukarimov A., Khudayarov B. A. A numerical investigation of the influence of rheological parameters on the character of vibrations in heredity-defortable systems. Computational Technologies, 2007, vol. 12, no. 4, pp. 17–26 (in Russian).
  17. Erokhin S. V. Modelling of creep and relaxation with the use of fractional derivatives. Internet-Vestnik VolgGASU, 2015, iss. 4(40). Available at: http://vestnik.vgasu.ru/attachments/8Erokhin.pdf (accessed 03, March, 2017) (in Russian).
  18. Grinchenko V. T, Meleshko V. V. Garmonicheskie kolebaniya i volny v uprugih telah [Harmonic oscillations and waves in elastic bodies]. Kiev, Nauk. Dumka, 1981. 284 p. (in Russian).
  19. Nayfe A. H. Metody vozmushhenij [Perturbation methods]. Moscow, Mir, 1976. 454 p. (in Russian).
  20. Kozhanova T. V., Kossovich L. Yu. Dispersionnye uravneniya Releya–Lemba [Rayleigh–Lamb dispersion equations]. Saratov, Saratov. Univ. Press, 1990. 21 p. (in Russian).
  21. Lavrentev М. А., Shabat B. V. Metody teorii funkcij kompleksnogo peremennogo [Methods of the complex variable functions theory]. Мoscow, Nauka, 1973. 736 p. (in Russian).
  22. Rabotnov Yu. I. Mehanika deformiruemogo tverdogo tela : Ucheb. posobie dlja vuzov [Mechanics of deformable solids]. Moscow, Nauka, 1988. 712 p. (in Russian).
  23. Rossikhin Yu. A., Shitikova M.V. Application of fractional calculus for dynamic problems of solid mechanics: Novel trends and recent results. Applied Mechanics Reviews, 2010, vol. 63, no. 1, 010801. DOI: https://doi.org/10.1115/1.4000563.
  24. Rossikhin Yu. A., Shitikova M.V. Comparative analysis of viscoelastic models involving fractional derivatives of different orders. Fractional Calculus and Applied Analysis, 2007, vol. 10, iss. 2, pp. 111–121.
  25. Zhuravkov M. A., Romanova N. S. Review of methods and approaches for mechanical problem solutions based on fractional calculus. Math. Mech. Solids, 2016, vol. 21, iss. 5, pp. 595–620. DOI: https://doi.org/10.1177/1081286514532934.