Skip to main content
SpringerLink
Log in

Love-type wave in low-velocity piezoelectric-viscoelastic stratum with mass loading

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

An exact approach is used to investigate the propagation of a Love-type wave in a low-velocity piezoelectric-viscoelastic material (PV) stratum bonded to a functionally graded transversely isotropic viscoelastic (FGTIV) material substrate. The dispersion relations are obtained for electrically open (EO) and electrically short (ES) conditions on the upper free surface of the considered structure by using Valeev’s method of infinite determinants. The expression of group velocity of a Love-type wave in a low-velocity layered structure is also obtained in compact form. The obtained closed-form dispersion relations of a Love-type wave are matched with the standard result of a Love-type wave when reduced to the isotropic case. In order to observe the concealed characteristics of a Love-type wave, a detailed study is carried out graphically, to forefront the effect of various influencing parameters, viz. piezoelectric coupling parameter and magnifying gradient (MG) parameters. The profound impact of mass loading on the upper free surface of the considered layered structure is also studied.

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

Similar content being viewed by others

References

  1. Danoyan, Z.N., Piliposian, G.T.: Surface electro-elastic Love waves in a layered structure with a piezoelectric substrate and a dielectric layer. Int. J. Solids Struct. 44(18–19), 5829–5847 (2007)

    Article  Google Scholar 

  2. Piliposian, G.T., Danoyan, Z.N.: Surface electro-elastic Love waves in a layered structure with a piezoelectric substrate and two isotropic layers. Int. J. Solids Struct. 46(6), 1345–1353 (2009)

    Article  Google Scholar 

  3. Wang, X., Pan, E., Roy, A.K.: Scattering of antiplane shear wave by a piezoelectric circular cylinder with an imperfect interface. Acta Mech. 193(3–4), 177–195 (2007)

    Article  Google Scholar 

  4. Negi, A., Kumar Singh, A.: Analysis on scattering characteristics of Love-type wave due to surface irregularity in a piezoelectric structure. J. Acoust. Soc. Am. 145(6), 3756–3783 (2019)

    Article  Google Scholar 

  5. Vashishth, A.K., Gupta, V.: Wave propagation in transversely isotropic porous piezoelectric materials. Int. J. Solids Struct. 46(20), 3620–3632 (2009)

    Article  Google Scholar 

  6. Sharma, J.N., Sharma, K.K., Kumar, A.: Surface waves in a piezoelectric-semiconductor composite structure. Int. J. Solids Struct. 47(6), 816–826 (2010)

    Article  Google Scholar 

  7. Liu, J., Wang, Y., Wang, B.: Propagation of shear horizontal surface waves in a layered piezoelectric half-space with an imperfect interface. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 57(8), 1875–1879 (2010)

    Article  Google Scholar 

  8. Singh, B.M., Rokne, J.: Propagation of SH waves in layered functionally gradient piezoelectric-piezomagnetic structures. Philos. Mag. 93(14), 1690–1700 (2013)

    Article  Google Scholar 

  9. Li, P., Jin, F.: Excitation and propagation of shear horizontal waves in a piezoelectric layer imperfectly bonded to a metal or elastic substrate. Acta Mech. 226(2), 267–284 (2015)

    Article  MathSciNet  Google Scholar 

  10. Piliposyan, D.G., Piliposian, G.T., Ghazaryan, K.B.: Propagation and control of shear waves in piezoelectric composite waveguides with metallized interfaces. Int. J. Solids Struct. 106, 119–128 (2017)

    Article  Google Scholar 

  11. Guo, X., Wei, P., Li, L., Lan, M.: Effects of functionally graded interlayers on dispersion relations of shear horizontal waves in layered piezoelectric/piezomagnetic cylinders. Appl. Math. Modell. 55, 569–582 (2018)

    Article  MathSciNet  Google Scholar 

  12. Singh, A.K., Singh, S., Kumari, R., Ray, A.: Impact of point source and mass loading sensitivity on the propagation of an SH wave in an imperfectly bonded FGPPM layered structure. Acta Mech. 231(6), 2603–2627 (2020)

    Article  MathSciNet  Google Scholar 

  13. Othmani, C., Zhang, H., Lü, C.: Effects of initial stresses on guided wave propagation in multilayered PZT-4/PZT-5A composites: A polynomial expansion approach. Appl. Math. Modell. 78, 148–168 (2020)

    Article  MathSciNet  Google Scholar 

  14. Biot, M.A.: Theory of deformation of a porous viscoelastic anisotropic solid. J. Appl. Phys. 27(5), 459–467 (1956)

    Article  MathSciNet  Google Scholar 

  15. Vashishth, A.K., Sharma, M.D.: Propagation of plane waves in poroviscoelastic anisotropic media. Appl. Math. Mech. 29(9), 1141 (2008)

    Article  Google Scholar 

  16. Yu, J.G., Ratolojanahary, F.E., Lefebvre, J.E.: Guided waves in functionally graded viscoelastic plates. Compos. Struct. 93(11), 2671–2677 (2011)

    Article  Google Scholar 

  17. Singh, A.K., Kumari, R., Ray, A., Chattopadhyay, A.: Love-type waves in a piezoelectric-viscoelastic bimaterial composite structure due to an impulsive point source. Int. J. Mech. Sci. 152, 613–629 (2019)

    Article  Google Scholar 

  18. Chaki, M.S., Singh, A.K.: Anti-plane wave in a piezoelectric viscoelastic composite medium: a semi-analytical finite element approach using PML. Int. J. Appl. Mech. 12(02), 2050020 (2020)

    Article  Google Scholar 

  19. Li, J., Dunn, M.L.: Viscoelectroelasticbehavior of heterogeneous piezoelectric solids. J. Appl. Phys. 89(5), 2893–2903 (2001)

    Article  Google Scholar 

  20. Kwok, K.W., Lee, T., Choy, S.H., Chan, H.L.: Lead-free piezoelectric transducers for microelectronic wirebonding applications. InTech (2010)

  21. Stovas, A., Ursin, B.: Reflection and transmission responses of layered transversely isotropic viscoelastic media. Geophys. Prospect. 51(5), 447–477 (2003)

    Article  Google Scholar 

  22. Liu, H., Wang, Z.K., Wang, T.J.: Effect of initial stress on the propagation behavior of Love waves in a layered piezoelectric structure. Int. J. Solids Struct. 38(1), 37–51 (2001)

    Article  Google Scholar 

Download references

Acknowledgements

Authors convey their sincere thanks to the Department of Science and Technology, Science and Engineering Research Board (DST-SERB), for their financial support to carry out this research work by sanction no. EMR/2016/003985/MS of the project entitled “Mathematical Study on Wave Propagation Aspects of Piezoelectric Composite Structures with Complexities.”

Author information

Authors and Affiliations

  1. Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad, Jharkhand, 826004, India

    Richa Kumari, Abhishek K. Singh & Anusree Ray

Authors
  1. Richa Kumari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abhishek K. Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Anusree Ray
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Richa Kumari.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

$$\begin{aligned} \varepsilon= & {} \frac{\varepsilon _{1} C_{44}^{\prime }+\iota \omega \varepsilon _{2} C_{44}^{\prime \prime }}{C_{44}^{\prime }+\iota \omega C_{44}^{\prime \prime }}\,;\,\,\,\overline{\,C_{44}^{\left( 2 \right) } } =C_{44}^{\prime }+\iota \omega C_{44}^{\prime \prime };\\ Q11= & {} \left( {\iota s} \right) ^{-n}\left( {\left( {p+\iota s} \right) ^{2}-\omega _{1}^{2} } \right) ;\,\,\\ Q12= & {} \left( {\iota s} \right) ^{-n}\left( {A_{1}^{\prime }\left( {p+\iota s} \right) -A_{2} ^{\prime }} \right) ; \\ Q21= & {} \left( {-\frac{\varepsilon }{2}\left( {p+\iota s} \right) ^{2}+\frac{\varepsilon \iota s}{2}\left( {p+\iota s} \right) -\frac{k^{2}c^{2}\varepsilon _{3} {\rho }'}{2\overline{C_{44}^{\left( 2 \right) } } }} \right) ;\,\,\\ Q22= & {} \left( {A_{1}^{\prime }p+A_{2}^{\prime }} \right) ; \\ Q23= & {} \left( {-\frac{\varepsilon }{2}\left( {p-\iota s} \right) ^{2}-\frac{\varepsilon \iota s}{2}\left( {p-\iota s} \right) -\frac{k^{2}c^{2}\varepsilon _{3} {\rho }'}{2\overline{C_{44}^{\left( 2 \right) } } }} \right) ;\,\,\\ Q32= & {} \left( {-\iota s} \right) ^{-n}\left( {A_{1}^{\prime }\left( {p-\iota s} \right) +A_{2} ^{\prime }} \right) ;\\ Q33= & {} \left( {-\iota s} \right) ^{-n}\left( {\left( {p-\iota s} \right) ^{2}-\omega _{1}^{2} } \right) ;\,\,\\ Q1{2}'= & {} \left( {\iota s} \right) ^{-n}\left( {-\frac{\varepsilon }{2}p^{2}-\frac{\varepsilon \iota s}{2}p-\frac{k^{2}c^{2}\varepsilon _{3} {\rho }'}{2\overline{C_{44}^{\left( 2 \right) } } }} \right) ;\,\,\\ Q2{2}'= & {} \left( {p^{2}-\omega _{1}^{2} } \right) ;\\ Q3{2}'= & {} \left( {-\iota s} \right) ^{-n}\left( {-\frac{\varepsilon }{2}p^{2}+\frac{\varepsilon \iota s}{2}p-\frac{k^{2}c^{2}\varepsilon _{3} {\rho }'}{2\overline{C_{44}^{\left( 2 \right) } } }} \right) ;\,\,\\ M_{1}= & {} \left( {\frac{\left( {\cos \eta _{1} h-e^{-kh}} \right) \left( {e^{-kh}+e^{kh}} \right) }{\left( {e^{kh}-e^{-kh}} \right) }-e^{-kh}} \right) ;\\ M_{2}= & {} \frac{\sin \eta _{1} h\left( {e^{-kh}+e^{kh}} \right) }{\left( {e^{kh}-e^{-kh}} \right) };\,\,\\ A_{0}= & {} \left( {e^{kh}-e^{-kh}} \right) +\frac{\varepsilon _{11} }{\varepsilon _{0} }\left( {e^{kh}+e^{-kh}} \right) ; \\ A_{1}= & {} kC_{44}^{\left( 1 \right) } \left( {Kep} \right) ^{2}\left( {\frac{\left( {\cos \eta _{1} h-e^{-kh}} \right) \left( {e^{-kh}+e^{kh}} \right) }{\left( {e^{kh}-e^{-kh}} \right) }-e^{-kh}} \right) ;\,\,\\ A_{2}= & {} kC_{44}^{\left( 1 \right) } \left( {Kep} \right) ^{2}\left( {\frac{\sin \eta _{1} H\left( {e^{-kh}+e^{kh}} \right) }{\left( {e^{kh}-e^{-kh}} \right) }} \right) ; \\ A_{3}= & {} kC_{44}^{\left( 1 \right) } \left( {Kep} \right) ^{2}\left( {\frac{2\left( {\cos \eta _{1} h-e^{-kh}} \right) }{\left( {e^{kh}-e^{-kh}} \right) }-1} \right) \left( {-\overline{C_{44}^{\left( 1 \right) } } \eta _{1} \cos \eta _{1} h+A_{2} } \right) ; \\ A_{4}= & {} \left( {\overline{C_{44}^{\left( 1 \right) } } \eta _{1} -\frac{2kC_{44}^{\left( 1 \right) } \left( {Kep} \right) ^{2}\sin \eta _{1} h}{\left( {e^{kh}-e^{-kh}} \right) }} \right) \left( {\overline{C_{44}^{\left( 1 \right) } } \eta _{1} \sin \eta _{1} h+A_{1} } \right) ; \\ X_{1}= & {} \frac{kC_{44}^{\left( 1 \right) } \left( {Kep} \right) ^{2}}{A_{0} }\sin \eta _{1} h\left( {e^{-kh}+e^{kh}} \right) ;\,\,\\ X_{2}= & {} kC_{44}^{\left( 1 \right) } \left( {Kep} \right) ^{2}\left( {\frac{\left( {e^{-kh}+e^{kh}} \right) }{A_{0} }\left( {\cos \eta _{1} h-e^{-kh}+\frac{\varepsilon _{11} }{\varepsilon _{0} }e^{-kh}} \right) -e^{-kh}} \right) ;\\ Y_{1}= & {} kC_{44}^{\left( 1 \right) } \left( {Kep} \right) ^{2}\left( {\frac{2}{A_{0} }\left( {\cos \eta _{1} h-e^{-kh}+\frac{\varepsilon _{11} }{\varepsilon _{0} }e^{-kh}} \right) -1} \right) \left( {-\overline{C_{44}^{\left( 1 \right) } } \eta _{1} \cos \eta _{1} h+X_{1} } \right) ; \\ Y_{2}= & {} \left( {\overline{C_{44}^{\left( 1 \right) } } \eta _{1} +2kC_{44}^{\left( 1 \right) } \left( {Kep} \right) ^{2}\frac{\sin \eta _{1} h}{A_{0} }} \right) \left( {\overline{C_{44}^{\left( 1 \right) } } \eta _{1} \sin \eta _{1} h+X_{2} } \right) ; \\ L_{1}= & {} \overline{C_{44}^{\left( 1 \right) } } \eta _{1} \sin \eta _{1} h-k^{2}c^{2}\rho ^{\left( 0 \right) }{h}'\cos \eta _{1} h;\,\,\\ L_{2}= & {} kC_{44}^{\left( 1 \right) } \left( {Kep} \right) ^{2}\left( {\frac{\left( {\cos \eta _{1} h-e^{-kh}} \right) \left( {e^{-kh}+e^{kh}} \right) }{\left( {e^{kh}-e^{-kh}} \right) }-e^{-kh}} \right) ; \\ L_{3}= & {} \overline{C_{44}^{\left( 1 \right) } } \eta _{1} \cos \eta _{1} h-k^{2}c^{2}\rho ^{\left( 0 \right) }{h}'\sin \eta _{1} h;\,\,\\ L_{4}= & {} kC_{44}^{\left( 1 \right) } \left( {Kep} \right) ^{2}\left( {\frac{\sin \eta _{1} h\left( {e^{-kh}+e^{kh}} \right) }{\left( {e^{kh}-e^{-kh}} \right) }} \right) ;\\ L_{5}= & {} kC_{44}^{\left( 1 \right) } \left( {Kep} \right) ^{2}\left( {\frac{2\left( {\cos \eta _{1} h-e^{-kh}} \right) }{\left( {e^{kh}-e^{-kh}} \right) }-1} \right) \left( {-L_{3} +L_{4} } \right) ;\,\,\\ L_{6}= & {} \overline{C_{44}^{\left( 1 \right) } } \eta _{1} -2kC_{44}^{\left( 1 \right) } \left( {Kep} \right) ^{2}\frac{\sin \eta _{1} h}{\left( {e^{kh}-e^{-kh}} \right) }\left( {L_{1} +L_{2} } \right) ; \\ L_{7}= & {} -kC_{44}^{\left( 1 \right) } \left( {Kep} \right) ^{2}e^{-kh}+kC_{44}^{\left( 1 \right) } \frac{\left( {Kep} \right) ^{2}}{A_{0} }\left( {e^{kh}+e^{-kh}} \right) \left( {\cos \eta _{1} h-e^{-kh}+\frac{\varepsilon _{11} }{\varepsilon _{0} }e^{-kh}} \right) ; \\ L_{8}= & {} \overline{C_{44}^{\left( 1 \right) } } \eta _{1} -2kC_{44}^{\left( 1 \right) } \frac{\left( {Kep} \right) ^{2}}{A_{0} }\sin \eta _{1} h\left( {L_{1} +L_{7} } \right) ; \\ L_{9}= & {} k\left( {Kep} \right) ^{2}C_{44}^{\left( 1 \right) } \left( {\frac{2}{A_{0} }\left( {\cos \eta _{1} h-e^{-kh}+\frac{\varepsilon _{11} }{\varepsilon _{0} }e^{-kh}} \right) -1} \right) \left( {kC_{44}^{\left( 1 \right) } \frac{\left( {Kep} \right) ^{2}}{A_{0} }\sin \eta _{1} h\left( {e^{kh}+e^{-kh}} \right) -L_{3} } \right) ; \\ \overline{C_{44}^{\left( 1 \right) } }^{\dag }= & {} C_{44}^{\left( R \right) } +\frac{\left( {e_{15}^{\left( R \right) } } \right) ^{2}}{\varepsilon _{11}^{\left( R \right) \,} };\,\,\\ \eta _{1}^{\dag }= & {} k\sqrt{\frac{c^{2}}{\beta _{1}^{2} }-1} ;\,\,\,\,A_{1}^{\dag }=kC_{44}^{\left( R \right) } \left( {Kep} \right) ^{2}\left( {\frac{\left( {\cos \eta _{1}^{\dag }h-e^{-kh}} \right) \left( {e^{-kh}+e^{kh}} \right) }{\left( {e^{kH}-e^{-kH}} \right) }-e^{-kh}} \right) ;\, \\ A_{2}^{\dag }= & {} kC_{44}^{\left( R \right) } \left( {Kep} \right) ^{2}\left( {\frac{\sin \eta _{1}^{\dag }h\left( {e^{-kh}+e^{kh}} \right) }{\left( {e^{kh}-e^{-kh}} \right) }} \right) ;\,\,\\ \overline{\,C_{44}^{\left( 2 \right) } }^{\dag }= & {} C_{44}^{\prime };\,\,\\ A_{0}^{\dag }= & {} \left( {e^{kh}-e^{-kh}} \right) +\frac{\varepsilon _{11}^{\left( R \right) } }{\varepsilon _{0} }\left( {e^{kh}+e^{-kh}} \right) ; \\ A_{3}^{\dag }= & {} kC_{44}^{\left( R \right) } \left( {Kep} \right) ^{2}\left( {\frac{2\left( {\cos \eta _{1}^{\dag }h-e^{-kh}} \right) }{\left( {e^{kh}-e^{-kh}} \right) }-1} \right) \left( {-\overline{C_{44}^{\left( 1 \right) } }^{\dag }\eta _{1}^{\dag }\cos \eta _{1}^{\dag }h+A_{2}^{\dag }} \right) ; \\ A_{4}^{\dag }= & {} \left( {\overline{C_{44}^{\left( 1 \right) } }^{\dag }\eta _{1} -\frac{2kC_{44}^{\left( R \right) } \left( {Kep} \right) ^{2}\sin \eta _{1}^{\dag }h}{\left( {e^{kh}-e^{-kh}} \right) }} \right) \left( {\overline{C_{44}^{\left( 1 \right) } }^{\dag }\eta _{1}^{\dag }\sin \eta _{1}^{\dag }h+A_{4}^{\dag }} \right) ;\\ X_{1}^{\dag }= & {} \frac{kC_{44}^{\left( R \right) } \left( {Kep} \right) ^{2}}{A_{0}^{\dag }}\sin \eta _{1}^{\dag }h\left( {e^{-kh}+e^{kh}} \right) ;\, \\ X_{2}^{\dag }= & {} kC_{44}^{\left( R \right) } \left( {Kep} \right) ^{2}\left( {\frac{\left( {e^{-kh}+e^{kh}} \right) }{A_{0}^{\dag }}\left( {\cos \eta _{1}^{\dag }h-e^{-kh}+\frac{\varepsilon _{11}^{\left( R \right) } }{\varepsilon _{0} }e^{-kh}} \right) -e^{-kh}} \right) ; \\ Y_{1}^{\dag }= & {} kC_{44}^{\left( R \right) } \left( {Kep} \right) ^{2}\left( {\frac{2}{A_{0}^{\dag }}\left( {\cos \eta _{1}^{\dag }h-e^{-kh}+\frac{\varepsilon _{11} }{\varepsilon _{0} }e^{-kh}} \right) -1} \right) \left( {-\overline{C_{44}^{\left( 1 \right) } } \eta _{1}^{\dag }\cos \eta _{1}^{\dag }h+X_{1}^{\dag }} \right) ; \\ Y_{2}^{\dag }= & {} \left( {\overline{C_{44}^{\left( 1 \right) } }^{\dag }\eta _{1}^{\dag }+2kC_{44}^{\left( R \right) } \left( {Kep} \right) ^{2}\frac{\sin \eta _{1}^{\dag }h}{A_{0}^{\dag }}} \right) \left( {\overline{C_{44}^{\left( 1 \right) } }^{\dag }\eta _{1}^{\dag }\sin \eta _{1}^{\dag }h+X_{2}^{\dag }} \right) . \\ \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kumari, R., Singh, A.K. & Ray, A. Love-type wave in low-velocity piezoelectric-viscoelastic stratum with mass loading. Acta Mech 232, 1253–1271 (2021). https://doi.org/10.1007/s00707-020-02831-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-020-02831-3

Search

Navigation