Skip to main content
SpringerLink
Log in

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

  • Published:
Journal of Earth System Science Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  • 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.

    Article  Google Scholar 

  • Adolfson K, Enelund M and Olsson P 2005 On the fractional order model of viscoelasticity; Mech. Time-Depend Mater. 9 15–34.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Akbarov S D 2015 Dynamics of pre-strained bi-material elastic systems: Linearized three-dimensional approach; Springer International, Switzerland, https://doi.org/10.1007/978-3-319-14460-3.

    Book  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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 

  • Aki K and Richards P G 2002 Quantitative seismology (2nd edn); University Science Books.

  • 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.

    Article  Google Scholar 

  • 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 

  • Carcione J M 1992 Rayleigh waves in isotropic viscoelastic media; Geophys. J. Int. 108(2) 453–464.

    Article  Google Scholar 

  • Carcione J M 1995 Constitutive model and wave equations for linear, viscoelastic, anisotropic media; Geophysics 60(2) 537–548.

    Article  Google Scholar 

  • Carcione J M 2007 Wave fields in real media: Wave propagation in anisotropic, anelastic, porous and electromagnetic media; Vol. 38, Elsevier, Amsterdam.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • Chiriţă S, Ciarletta M and Tibullo V 2014 Rayleigh surface waves on a Kelvin–Voigt viscoelastic half-space; J. Elast. 115(1) 61–76.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Ewing W M, Jardetzky W S, Press F and Beiser A 1957 Elastic waves in layered media; Phys. Today 10 27.

    Article  Google Scholar 

  • Fan J 2004 Surface seismic Rayleigh wave with nonlinear damping; Appl. Math. Model. 28(2) 163–171.

    Article  Google Scholar 

  • Garg N 2007 Effect of initial stress on harmonic plane homogeneous waves in viscoelastic anisotropic media; J. Sound. Vib. 303(3) 515–525.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Jiangong Y 2011 Viscoelastic shear horizontal wave in graded and layered plates; Int. J. Solids Struct. 48(16) 2361–2372.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

  • 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 

  • Kolsky H 1963 Stress waves in solids; Vol. 1098, Courier Corporation, North Chelmsford.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Manconi E and Sorokin S 2013 On the effect of damping on dispersion curves in plates; Int. J. Solids. Struct. 50(11) 1966–1973.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Negin M 2015 Generalized Rayleigh wave propagation in a covered half-space with liquid upper layer; Struct. Eng. Mech. 56(3) 491–506.

    Article  Google Scholar 

  • 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 

  • Pasternak M 2008 New approach to Rayleigh wave propagation in the elastic halfspace-viscoelastic layer interface; Acta Phys. Pol. A 114(6A).

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Rabotnov Y N 1980 Elements of hereditary solid mechanics; Mir, Moscow.

    Google Scholar 

  • Romeo M 2001 Rayleigh waves on a viscoelastic solid half-space; J. Acoust. Soc. Am. 110(1) 59–67.

    Article  Google Scholar 

  • 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 

  • 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 

  • 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.

    Article  Google Scholar 

  • Sharma J N 2005 Some considerations on the Rayleigh Lamb waves in viscoelastic plates; J. Vib. Control 11 1311–1335.

    Article  Google Scholar 

  • 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 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Zhou Y 2009 Surface-wave sensitivity to 3-D anelasticity; Geophys. J. Int. 178(3) 1403–1410

    Article  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Bahcesehir University, Istanbul, Turkey

    M Negin

  2. Department of Mechanical Engineering, Yildiz Technical University, Istanbul, Turkey

    S D Akbarov

  3. Institute of Mathematics and Mechanics of the National Academy of Sciences of Azerbaijan, Baku, Azerbaijan

    S D Akbarov

Authors
  1. M Negin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S D Akbarov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M Negin.

Additional information

Corresponding Editor: Munukutla Radhakrishna

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Negin, M., Akbarov, S.D. On attenuation of the seismic Rayleigh waves propagating in an elastic crustal layer over viscoelastic mantle. J Earth Syst Sci 128, 181 (2019). https://doi.org/10.1007/s12040-019-1202-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12040-019-1202-x

Keywords

Search

Navigation