On attenuation of the seismic Rayleigh waves propagating in an elastic crustal layer over viscoelastic mantle

  • M NeginEmail author
  • S D Akbarov


This study investigates the attenuation of the seismic Rayleigh waves propagating in an elastic crustal layer of the Earth over its viscoelastic mantle. The exact equations of motion of the theory of linear viscoelasticity are used and the complex dispersion equation is obtained for an arbitrary type of hereditary operator of the viscoelastic materials. The viscoelasticity of the materials is described by the fractional-exponential operators of Rabotnov, and a solution algorithm is developed to obtain numerical results for the attenuation of the considered waves. Attenuation curves are obtained and discussed, and in particular, the influence of the rheological parameters of the materials on this attenuation is studied. It is established that a decrease in the creep time of the viscoelastic materials leads to an increase in the attenuation coefficient.


Seismic Rayleigh waves wave attenuation viscoelastic material rheological parameters fractional-exponential operator 



This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.


  1. Addy S K and Chakraborty N R 2005 Rayleigh waves in a viscoelastic half-space under initial hydrostatic stress in presence of the temperature field; Int. J. Math. Sci. 2005(24) 3883–3894.CrossRefGoogle Scholar
  2. Adolfson K, Enelund M and Olsson P 2005 On the fractional order model of viscoelasticity; Mech. Time-Depend Mater. 9 15–34.CrossRefGoogle Scholar
  3. Akbarov S D 2014 Axisymmetric time-harmonic Lamb’s problem for a system comprising a viscoelastic layer covering a viscoelastic half-space; Mech. Time-Depend Mater. 18(1) 153–178.CrossRefGoogle Scholar
  4. Akbarov S D 2015 Dynamics of pre-strained bi-material elastic systems: Linearized three-dimensional approach; Springer International, Switzerland, Scholar
  5. Akbarov S D and Kepceler T 2015 On the torsional wave dispersion in a hollow sandwich circular cylinder made from viscoelastic materials; Appl. Math. Model. 39(13) 3569–3587.CrossRefGoogle Scholar
  6. Akbarov S D and Negin M 2017a Near-surface waves in a system consisting of a covering layer and a half-space with imperfect interface under two-axial initial stresses; J. Vib. Control 23(1) 55–68.CrossRefGoogle Scholar
  7. Akbarov S D and Negin M 2017b Generalized Rayleigh wave dispersion in a covered half-space made of viscoelastic materials; CMC-Comput. Mater. Con. 53(4) 307–341.Google Scholar
  8. Akbarov S D, Kocal T and Kepceler T 2016a On the dispersion of the axisymmetric longitudinal wave propagating in a bi-layered hollow cylinder made of viscoelastic materials; Int. J. Solids Struct. 100 195–210.CrossRefGoogle Scholar
  9. Akbarov S D, Kocal T and Kepceler T 2016b Dispersion of axisymmetric longitudinal waves in a bi-material compound solid cylinder made of viscoelastic materials; CMC-Comput. Mater. Con. 51(2) 105–143.Google Scholar
  10. Aki K and Richards P G 2002 Quantitative seismology (2nd edn); University Science Books.Google Scholar
  11. Barshinger J N and Rose J L 2004 Guided wave propagation in an elastic hollow cylinder coated with a viscoelastic material; IEEE Trans. Ultrason. Ferroelectr. 51(11) 1547–1556.CrossRefGoogle Scholar
  12. Bosiakov S M 2014 On the application of a viscoelastic model with Rabotnov’s fractional exponential function for assessment of the stress-strain state of the periodontal ligament; Int. J. Mech. 8 353–358.Google Scholar
  13. Carcione J M 1992 Rayleigh waves in isotropic viscoelastic media; Geophys. J. Int. 108(2) 453–464.CrossRefGoogle Scholar
  14. Carcione J M 1995 Constitutive model and wave equations for linear, viscoelastic, anisotropic media; Geophysics 60(2) 537–548.CrossRefGoogle Scholar
  15. Carcione J M 2007 Wave fields in real media: Wave propagation in anisotropic, anelastic, porous and electromagnetic media; Vol. 38, Elsevier, Amsterdam.Google Scholar
  16. Carcione J M, Poletto F and Gei D 2004 3-D wave simulation in anelastic media using the Kelvin–Voigt constitutive equation; J. Comput. Phys. 196(1) 282–297.CrossRefGoogle Scholar
  17. Castaings M and Hosten B 2003 Guided waves propagating in sandwich structures made of anisotropic, viscoelastic, composite materials; J. Acoust. Soc. Am. 113(5) 2622–2634.CrossRefGoogle Scholar
  18. Chen Z J, He Y and Gao J 2015 On the comparison of properties of Rayleigh waves in elastic and viscoelastic media; Int. J. Numer. Anal. Mod. 12(2) 254–267.Google Scholar
  19. Chiriţă S, Ciarletta M and Tibullo V 2014 Rayleigh surface waves on a Kelvin–Voigt viscoelastic half-space; J. Elast. 115(1) 61–76.CrossRefGoogle Scholar
  20. Eldred L B, Baker W P and Palazotto A N 1995 Kelvin–Voigt versus fractional derivative model as constitutive relations for viscoelastic materials; AIAA J. 33(3) 547–550.CrossRefGoogle Scholar
  21. Ely G P, Day S M and Minster J B 2008 A support-operator method for viscoelastic wave modelling in 3-D heterogeneous media; Geophys. J. Int. 172(1) 331–344.CrossRefGoogle Scholar
  22. Ewing W M, Jardetzky W S, Press F and Beiser A 1957 Elastic waves in layered media; Phys. Today 10 27.CrossRefGoogle Scholar
  23. Fan J 2004 Surface seismic Rayleigh wave with nonlinear damping; Appl. Math. Model. 28(2) 163–171.CrossRefGoogle Scholar
  24. Garg N 2007 Effect of initial stress on harmonic plane homogeneous waves in viscoelastic anisotropic media; J. Sound. Vib. 303(3) 515–525.CrossRefGoogle Scholar
  25. Golub V P, Fernati P V and Lyashenko Y G 2008 Determining the parameters of the fractional exponential heredity kernels of linear viscoelastic materials; Int. Appl. Mech. 44(9) 963–974.CrossRefGoogle Scholar
  26. Ivanov T P and Savova R 2014 Motion of the particles due to viscoelastic surface waves of an assigned frequency; Math. Mech. Solids 19(6) 725–731.CrossRefGoogle Scholar
  27. Jiangong Y 2011 Viscoelastic shear horizontal wave in graded and layered plates; Int. J. Solids Struct. 48(16) 2361–2372.CrossRefGoogle Scholar
  28. Jousset P, Neuberg J and Jolly A 2004 Modelling low-frequency volcanic earthquakes in a viscoelastic medium with topography; Geophys. J. Int. 159(2) 776–802.CrossRefGoogle Scholar
  29. Kaminskii A A and Selivanov M F 2005 An approach to the determination of the deformation characteristics of viscoelastic materials; Int. Appl. Mech. 41(8) 867–875.CrossRefGoogle Scholar
  30. Kielczyriski P and Cheeke J D N 1997 Love waves propagation in viscoelastic media [and NDT application]; In: Proceedings of the IEEE ultrasonics symposium, 1997, Vol. 1, pp. 437–440.Google Scholar
  31. Kocal T and Akbarov S D 2017 On the attenuation of the axisymmetric longitudinal waves propagating in the bi-layered hollow cylinder made of viscoelastic materials; Struct. Eng. Mech. 61(2) 145–165.Google Scholar
  32. Kolsky H 1963 Stress waves in solids; Vol. 1098, Courier Corporation, North Chelmsford.Google Scholar
  33. Kumar R and Parter G 2009 Analysis of free vibrations for Rayleigh-Lamb waves in a microstretch thermoelastic plate with two relaxation times; J. Eng. Phys. Thermophys. 82 35–46.CrossRefGoogle Scholar
  34. Lai C G and Rix G J 2002 Solution of the Rayleigh eigen problem in viscoelastic media; Bull. Seismol. Soc. Am. 92(6) 2297–2309.CrossRefGoogle Scholar
  35. Manconi E and Sorokin S 2013 On the effect of damping on dispersion curves in plates; Int. J. Solids. Struct. 50(11) 1966–1973.CrossRefGoogle Scholar
  36. Meral F C, Royston T J and Magin R L 2009 Surface response of a fractional order viscoelastic halfspace to surface and subsurface sources; J. Acoust. Soc. Am. 126(6) 3278–3285.CrossRefGoogle Scholar
  37. Meral F C, Royston T J and Magin R L 2011 Rayleigh–lamb wave propagation on a fractional order viscoelastic plate; J. Acoust. Soc. Am. 129(2) 1036–1045.CrossRefGoogle Scholar
  38. Negin M 2015 Generalized Rayleigh wave propagation in a covered half-space with liquid upper layer; Struct. Eng. Mech. 56(3) 491–506.CrossRefGoogle Scholar
  39. Negin M, Akbarov S D and Erguven M E 2014 Generalized Rayleigh wave dispersion analysis in a pre-stressed elastic stratified half-space with imperfectly bonded interfaces; CMC-Comput. Mater. Con. 42(1) 25–61.Google Scholar
  40. Pasternak M 2008 New approach to Rayleigh wave propagation in the elastic halfspace-viscoelastic layer interface; Acta Phys. Pol. A 114(6A).CrossRefGoogle Scholar
  41. Quintanilla F H, Fan Z, Lowe M J S and Craster R V 2015 Guided waves’ dispersion curves in anisotropic viscoelastic single-and multi-layered media; Int. Proc. R. Soc. A 471(2183) 20150268.CrossRefGoogle Scholar
  42. Rabotnov Y N 1980 Elements of hereditary solid mechanics; Mir, Moscow.Google Scholar
  43. Romeo M 2001 Rayleigh waves on a viscoelastic solid half-space; J. Acoust. Soc. Am. 110(1) 59–67.CrossRefGoogle Scholar
  44. Rossikhin Y A 2010 Reflections on two parallel ways in the progress of fractional calculus in mechanics of solids; Appl. Mech. Rev. 63(1) 010701-1-12.Google Scholar
  45. Rossikhin Y A and Shitikova M V 2014 The simplest models of viscoelasticity involving fractional derivatives and their connectedness with the Rabotnov fractional order operators; Int. J. Mech. 8 326–331.Google Scholar
  46. Sawicki J T and Padovan J 1999 Frequency driven phasic shifting and elastic-hysteretic portioning properties of fractional mechanical system representation schemes; J. Franklin Inst. 336 423–433.CrossRefGoogle Scholar
  47. Sharma J N 2005 Some considerations on the Rayleigh Lamb waves in viscoelastic plates; J. Vib. Control 11 1311–1335.CrossRefGoogle Scholar
  48. Sharma M D 2011 Phase velocity and attenuation of plane waves in dissipative elastic media: Solving complex transcendental equation using functional iteration method; Int. J. Eng. Sci. Technol. 3(2) 130–136.Google Scholar
  49. Sharma J N and Kumar S 2009 Lamb waves in micropolar thermoelastic solid plates immersed in liquid with varying temperature; Mechanics 44 305–319.Google Scholar
  50. Sharma J N and Othman M I A 2007 Effect of rotation on generalized thermo-viscoelastic Rayleigh-Lamb waves; Int. J. Solids Struct. 44 4243–4255.CrossRefGoogle Scholar
  51. Sharma J N, Sharma R and Sharma P K 2009 Rayleigh waves in a thermo-viscoelastic solid loaded with viscous fluid of varying temperature; Int. J. Theor. Appl. Sci. 1(2) 60–70.Google Scholar
  52. Simonetti F and Cawley P 2003 A guided wave technique for the characterization of highly attenuative viscoelastic materials; J. Acoust. Soc. Am. 114(1) 158–165.CrossRefGoogle Scholar
  53. Vishwakarma S K and Gupta S 2012 Torsional surface wave in a homogeneous crustal layer over a viscoelastic mantle; Int. J. Appl. Math. Mech. 8(16) 38–50.Google Scholar
  54. Yuan S, Song X, Cai W and Hu Y 2018 Analysis of attenuation and dispersion of Rayleigh waves in viscoelastic media by finite-difference modeling; J. Appl. Geophys. 148 115–126.CrossRefGoogle Scholar
  55. Zhang K, Luo Y, Xia J and Chen C 2011 Pseudospectral modeling and dispersion analysis of Rayleigh waves in viscoelastic media; Soil Dyn. Earthq. Eng. 31(10) 1332–1337.CrossRefGoogle Scholar
  56. Zhou Y 2009 Surface-wave sensitivity to 3-D anelasticity; Geophys. J. Int. 178(3) 1403–1410CrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.Department of Civil EngineeringBahcesehir UniversityIstanbulTurkey
  2. 2.Department of Mechanical EngineeringYildiz Technical UniversityIstanbulTurkey
  3. 3.Institute of Mathematics and Mechanics of the National Academy of Sciences of AzerbaijanBakuAzerbaijan

Personalised recommendations