Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

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

For citation:

Prikazchikov D. A. Околорезонансные режимы в стационарной задаче о подвижной нагрузке в случае трансверсально изотропной упругой полуплоскости. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2015, vol. 15, iss. 2, pp. 215-221. DOI: 10.18500/1816-9791-2015-15-2-215-221, EDN: TXMFTV

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
Full text:
(downloads: 79)

Околорезонансные режимы в стационарной задаче о подвижной нагрузке в случае трансверсально изотропной упругой полуплоскости

Prikazchikov D. A., Keele University

A moving load problem on a transversely isotropic elastic half-plane is considered under steady-state assumption. The approach relies on the hyperbolic-elliptic asymptotic model for surface wave, allowing drastic simplifications. In particular, the formulation is reduced to a Dirichlet problem for a scaled Laplace equation having a straightforward solution in terms of elementary functions. The obtained approximate solutions are valid for loads travelling at speeds close to surface wave speed.

  1. Shamalta M., Metrikine A. V. Analytical study of the dynamic response of an embedded railway track to a moving load. Arch. Appl. Mech., 2003, vol. 73, pp. 131–146. DOI: 10.1007/s00419-002-0248-3.
  2. Cao Y., Xia H., Li Z. A semi-analytical/FEM model for predicting ground vibrations induced by highspeed train through continuous girder bridge. J. Mech. Sci. Technol., 2012, vol. 26, pp. 2485–2496. DOI: 10.1007s12206-012-0630-1.
  3. Cole J., Huth J. Stresses produced in a half plane by moving loads. J. Appl. Mech., 1958, vol. 25, pp. 433–436.
  4. Madshus C., Kaynia A. M. High-speed railway lines on soft ground : dynamic behaviour at critical train speed. J. Sound Vib., 2000, vol. 231, no. 3, pp. 689–701. DOI: 10.1006/jsvi.1999.2647.
  5. Mukherjee S. Stresses produced by a load moving over the rough boundary of a semiinfinite transversely isotropic solid. Pure Appl. Geophys., 1969, vol. 72, pp. 45–50. DOI:10.1007/BF00875691.
  6. De Hoop A. T. The moving-load problem in soil dynamics — the vertical displacement approximation. Wave Motion, 2002, vol. 36, pp. 335–346. DOI: 10.1016/S0165-2125(02)00028-8
  7. Kaplunov J., Nolde E., Prikazchikov D. A. A revisit to the moving load problem using an asymptotic model for the Rayleigh wave. Wave Motion, 2010, vol. 47, pp. 440–451. DOI:10.1016/j.wavemoti.2010.01.005.
  8. Kaplunov J., Prikazchikov D. A. Explicit models for surface, interfacial and edge waves in elastic solids. Dynamic localization phenomena in elasticity, acoustics and electromagnetism (Eds. R. Craster, J. Kaplunov), CISM Lecture Notes, Springer-Verlag, 2013, vol. 547, pp. 73–114.
  9. Kaplunov Yu. D., Kossovich L. Yu. Asymptotic model of Rayleigh waves in the far-field zone in an elastic half-plane. Doklady Physics, 2004, vol. 49, no. 4, pp. 234–236.
  10. Kaplunov J., Zakharov A., Prikazchikov D. A. Explicit models for elastic and piezoelastic surface waves. IMA J. Appl. Math., 2006, vol. 71, pp. 768–782. DOI: 10.1093/imamat/hxl012.
  11. Chadwick P. Surface and interfacial waves of arbitrary form in isotropic elastic media. J. Elast., 1976, vol. 6, pp. 73–80. DOI: 10.1007/BF00135177.
  12. Kaplunov J., Prikazchikov D. A., Erbas B., Sahin O. On a 3D moving load problem in an elastic half space. Wave Motion, 2013, vol. 50, pp. 1229–1238. DOI:10.1016/j.wavemoti.2012.12.008.
  13. Erbas B., Kaplunov J., Prikazchikov D. A., Sahin O. The near-resonant regimes of a moving load in a three-dimensional problem for a coated elastic half-space. Math. Mech. Solids, 2014. DOI:10.1177/1081286514555451.
  14. Mukhomodiarov R. R., Prikazchikov D. A. Asymptotic model for the Rayleigh wave in case of a transversely isotropic half-plane. XVIII Session of International School on the Models of Continuum Mechanics : Proc. Int. Conf., Saratov, Saratov Univ. Press, 2007, pp. 210–213 (in Russian).
  15. Prikazchikov D. A. Rayleigh waves of arbitrary profile in anisotropic media. Mech. Res. Comm., 2013, vol. 50, pp. 83–86. DOI:10.1016/j.mechrescom.2013.03.009.
  16. Buchwald V. T. Rayleigh waves in transversely isotropic media. Quart. J. Mech. Appl. Math., 1961, vol. 14, pp. 293–318. DOI:10.1093/qjmam/14.3.293.
  17. Ting T. C. T. Anisotropic elasticity. Oxford, Oxford Univ. Press, 1996, 570 p.
  18. Royer D., Dieulesaint E. Elastic Waves in Solids. II. Berlin, Springer, 2000, 446 p.
  19. Kiselev A. P., Parker D. F. Omni-directional Rayleigh, Stoneley and Scholte waves with general time dependence. Proc. Roy. Soc. London A, 2010, vol. 466, pp. 2241–2258. DOI:10.1098/rspa.2009.0595.
  20. Kaplunov J. D. Transient dynamics of an elastic half-plane subject to a moving load. Institute for Problems in Mechanics. Preprint / USSR Academy of Sciences, 1986, no. 277, 53 p. (in Russian).
  21. Courant R., Hilbert D. Methods of Mathematical Physics. Vol. 2. New York, Wiley, 1966, 811 p.
  22. Erbas B., Kaplunov J., Prikazchikov D. A. The Rayleigh wave field in mixed problems for a halfplane. IMA J. Appl. Math., 2013, vol. 78, pp. 1078–1086. DOI: 10.1093/imamat/ hxs010.