Skip to main content
SpringerLink
Log in

Explicit transfer matrix for an incompressible orthotropic elastic layer and applications

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

In this paper, we establish transfer matrix for an incompressible orthotropic elastic layer. It is explicit and expressed compactly in terms of square brackets. This transfer matrix is a very convenient tool for solving various problems of wave propagation in layered elastic media including incompressible orthotropic layers. To prove this point, we apply it to investigate the reflection of SV-waves from an incompressible orthotropic layer overlying an incompressible orthotropic half-spaces and the propagation of Lamb waves in a composite plate consisting of two incompressible orthotropic layers. By using the obtained transfer matrix along with the effective boundary condition technique, we reduce the reflection of SV-waves from the layer to the reflection of SV-waves from the surface of half-space. The necessary and sufficient conditions for one or two reflected waves to exist have been established, and formulas for the reflection coefficients have been derived. Unlike the previously obtained formulas, the formulas derived in the present paper for the reflection coefficients are totally explicit. Employing the obtained transfer matrix, we arrive immediately at explicit dispersion equation of Lamb waves. Based on the obtained dispersion equation, it is shown numerically that for a two-layered plate with high-contrast material properties of the layers, the cutoff frequency of the first harmonic is close to zero. That means the low-frequency vibration spectrum of strongly inhomogeneous two-layered plates involves not only the fundamental bending mode, but also the first harmonic.

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. Rokhlin, S.I., Wang, L.: Stable recursive algorithm for elastic wave propagation in layered anisotropic media: Stiffness matrix method. J. Acoust. Soc. Am. 112, 822–834 (2002)

    Article  Google Scholar 

  2. Hosten, B., Castaings, M.: Surface impedance matrices to model the propagation in multilayered media. Ultrasonics 41, 501–507 (2003)

    Article  Google Scholar 

  3. Chen, X.: A systematic and efficient method of computing normal modes for multilayered half-space. Geophys. J. Int. 115, 391–409 (1993)

    Article  Google Scholar 

  4. Knopoff, L.: A matrix method for elastic waves problems. Bull. Seism. Soc. Am. 54, 431–438 (1964)

    Article  Google Scholar 

  5. Lowe, M.J.S.: Matrix techniques for modeling ultrasonic waves in multilayered media. IEEE Trans. 42, 525–542 (1995)

    Google Scholar 

  6. Thomson, W. T.: Transmission of elastic waves through a stratified solid medium. J. Appl. Phys. 21, 89–93 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  7. Haskell, N.A.: The dispersion of surface waves on multilayered media. Bull. Seism. Soc. Am. 43, 17–34 (1953)

    Article  Google Scholar 

  8. Green, W.A. and Green E.R.: Stress vibration due to an impact line load on a four-ply fibre composite plate. Int. J. Solids Struct. 28, 567–595 (1991)

    Article  Google Scholar 

  9. Solyanik, F. I.: Transmission of plane waves through a layered medium of anisotropic materials. Sov. Phys. Acoust. 23, 533–536 (1977)

    Google Scholar 

  10. Rokhlin, S.I. and Wang, Y.J.: Equivalent boundary conditions for thin orthotropic layer between, J. Acoust. Soc. Am. 91, 1875–1887 (1992)

    Article  Google Scholar 

  11. Vinh, P.C., Anh, V.T.N., Linh, N.T.K.: On a technique for deriving the explicit secular equation of Rayleigh waves in an orthotropic half-space coated by an orthotropic layer. Waves in Random and Complex media 26, 176–188 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Rogerson, G.A., Sandiford, K.J.: Some comments on the dispersion relation for periodically layered pre-stressed elastic media. Int. J. Eng. Sci. 40, 23–49 (2002)

    Article  Google Scholar 

  13. Wijeyewickrema, A.C., Leungvichcharoen, S.: Wave propagation in pre-stressed imperfectly bonded cimpressible elastic layered composites. Mech. Mater. 41, 1192–1203 (2009)

    Article  MATH  Google Scholar 

  14. Vinh, P.C., Anh, V.T.N., Merodio, J., Hue, L.T.: Explicit tramsfer matrices of pre-stressed elastic layers. Int. J. Non-linear Mech. 106, 288–296 (2018)

    Article  Google Scholar 

  15. Chadwick, P.: Wave propagation in incmpressible transversely isotropic elastic media II. Proc. R. Ir. Acad. 94A, 85–104 (1994)

    MATH  Google Scholar 

  16. Kaplunov, J., Prikazchikov, D. A., Prikazchikova, L. A.: Dispersion of elastic waves in a strongly inhomogeneous three-layered plate. Int. J. Solids Struct. 113–114, 169–179 (2017)

    Article  Google Scholar 

  17. Keith, C. M., Crampin, S.: Seismic body waves in anisotropic media: reflection and refraction at a plane interface. Geophys. J. Int. 49, 181–208 (1977)

    Article  Google Scholar 

  18. Rokhlin, S. I., Belland, T. K. and Adler, L.: Reflection and refraction of elastic waves on a plane interface between two generally anisotropic media. J. Acoust. Soc. Am. 79, 906–918 (1986)

    Article  Google Scholar 

  19. Nair, S. and Sotiropoulos, D. A.: Elastic waves in orthotropic incompressible materials and reflection from an interface. J. Acoust. Soc. Am. 102 102–109 (1997)

    Article  Google Scholar 

  20. Chattopadhyay, A.: Reflection of three-dimensional plane waves in triclinic crystalline medium. Appl. Math. Mech. 28, 1309–1316 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chatterjee, M., Dhua, S., Sahu, S. A., Chattopadhyay, A.: Reflection in a highly anisotropic medium for three-dimensional plane waves under initial stresses. Int. J. Eng. Sci. 85, 136–149 (2014)

    Article  Google Scholar 

  22. Chattopadhyay, A., Kumari, P., Sharma, V.K.: Reflection and refraction at the interface between distinct generally anisotropic half spaces for three-dimensional plane quasi-P waves. J. Vibration and Control 21, 493–508 (2015)

    Article  Google Scholar 

  23. Paswan, B., Sahu, S.A., Chattopadhyay, A.: Reflection and transmission of plane wave through fluid layer of finite width sandwiched between two monoclinic elastic half-spaces. Acta Mech. 227, 3687–3701 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Modi, C., Kumari, P., Sharma, V.K.: Reflection/refraction of qP/qSV wave in layered self-reinforced media. Appl. Math. Modelling 40, 8737–8749 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kumari, P., Sharma, V.K., Modi, C.: Reflection/refraction pattern of quasi- (P/SV) waves in dissimilar monoclinic media separated with finite isotropic layer. J. Vibr. Control 22, 2745–2758 (2016)

    Article  MathSciNet  Google Scholar 

  26. Sahu, S.A., Paswan, B., Chattopadhyay, A.: Reflection and transmission of plane waves through isotropic medium sandwiched between two highly anisotropic half-spaces. Waves in Random and Complex Media 26, 42–67 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kumari, P., Sharma, V.K.: Dynamics of seismic waves in highly ansotropic triclinic media with intermedite monoclinic layer. Appl. Math. Modelling 71, 375–393 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  28. Chadwick, P.: Wave propagation in incmpressible transversely isotropic elastic media I. Proc. R. Ir. Acad. 93A, 231–253 (1993)

    MATH  Google Scholar 

  29. Lamb, H.: On waves in an elastic plate. Proc. R. Soc. Lond. A 93, 114–128 (1917)

    Article  MATH  Google Scholar 

  30. Mindlin, R.D.: An Introduction to the Mathematical Theory of Vibrations of Elastic Plates, editted by J. Yang. World Scientific Publishing Co. Pte. Ltd, Singapore (2006)

    Book  MATH  Google Scholar 

  31. Gazis, D.C.: Exact analysis of the plane-strain vibrations of thick-walled hollow cylinders. J. Acoust. Soc. Am. 30, 786–794 (1958)

    Article  Google Scholar 

  32. Viktorov, I.A.: Rayleigh and Lamb Waves. Plenum Press, New York (1967)

    Book  Google Scholar 

  33. Kaplunov, J.D., Kossovich, L.Y., Nolde, E.V.: Dynamics of Thin Walled Elastic Bodies. Academic Press (1998)

    MATH  Google Scholar 

  34. Abubakar, I.: Free Vibrations of aT ransversely Isotropic Plate. Q. Jl. Mech. Appl. Math. 15, 129–136 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  35. Baylis, E.R., Green, W.A.: Flexural waves in fiber reinforced laminated plates. J. Sound Vibr. 110, 1–26 (1986)

    Article  Google Scholar 

  36. Kaplunov, J.D., Kossovich, L.Y., Rogerson, G.A.: Direct Asyptotic integration of the equations of transversely isotropic elasticity for a plate near cut-off frequencies. Q. Jl. Mech. Appl. Math. 53, 323–341 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  37. Kossovich, L. Y., Moukhomodiarov, R. R. and Rogerson, G. A.: Analysis of the dispersion relation for an incompressible transversely isotropic elastic plate, Acta Mech. 153, 89–111 (2002)

    Article  MATH  Google Scholar 

  38. Solie, L.P., and Auld, B.A.: Elastic waves in free anisotropic plates. J. Acoust. Soc. Am. 54, 50–65 (1973)

    Article  Google Scholar 

  39. Baid, H.K.: Detection of damage in a Composite Structure using Guided Waves. University of California, Los Angeles (2012).. (Ph.D. thesis)

    Google Scholar 

  40. Baid H., Schaal C., Samajder H., Mail A.: Dispersion of Lamb waves in a honeycomb composite sandwich panel. Ultrasonics 56, 409–416 (2015)

    Article  Google Scholar 

  41. Nayfeh, A. H., Chimenti, D. E.: Free wave propagation in plates of general anisotropic media. ASME J. Appl. Mech. 56, 881–886 (1989)

    Article  MATH  Google Scholar 

  42. Shuvalov, A. L.: On the theory of wave propagation in anisotropic plates. Proc. R. Soc. Lond. A 456, 2197–2222 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  43. Kuznetsov, S. V.: Lamb waves in anisotropic plates (review). Acoust. Phys. 60, 95–103 (2014)

    Article  Google Scholar 

  44. Ogen, R. W., Roxburgh, D. G.: The effect of pre-stress on the vibration and stability of elastic plates. Int. J. Eng. Sci. 31, 1611–1639 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  45. Ogen, R. W., Roxburgh, D. G.: Stability and vibration of pre-stressed compressible elastic plates. Int. J. Eng. Sci. 32, 427–454 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  46. Rogerson, G. A.: Some asymptotic expansions of the dispersion relation for an incompressible elastic plate. Int. J. Solids Struct. 34, 2785–2802 (1997)

    Article  MATH  Google Scholar 

  47. Kaplunov, J.D., Nolde, E.V. and Rogerson, G.A.: A low-frequency model for dynamic motion in pre-stressed incompressible elastic structures. Proc. R. Soc. Lond. A 456, 2589–2610 (2000)

    Article  MATH  Google Scholar 

  48. Kaplunov, J.D., Nolde, E.V., Rogerson, G.A.: An Asymptotically consistent model for long-wave hight-frequency motion in a pre-stressed elastic plate. Math. Mech. Solids 7, 581–606 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  49. Kaplunov, J.D., Nolde, E.V. and Rogerson, G.A.: Short wave motion in a pre-stressed incompressible elastic plate. IMA J. Appl. Mech. 67, 383–399 (2002)

    MathSciNet  MATH  Google Scholar 

  50. Kaplunov, J.D., Nolde, E.V.: Long-wave vibrations of a nearly incompressible isotropic plate with fixed faces. Q. Jl. Mech. Appl. Math. 55, 345–356 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  51. Jones, J. P.: Wave propagation in a two-layered medium. ASME J. Appl. Mech. 31, 213–222 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  52. Yao-Jun, W., Wei, N., Xian-Hua, O.: Lamb wave modes in a two-layered solid medium with weak interface. Acta Physica Sinica 3, 561–566 (1994)

    Google Scholar 

  53. Demcenko, A., Mazeika, L.: Calculation of Lamb waves dispersion curves in multi-layered planar structures. Ultragarsas 44 (3), 15–17 (2002)

    Google Scholar 

  54. Liu, G. R., Tani, J., Watanabe, K., Ohyoshi, T.: Lamb wave propagation in anisotropic laminates. ASME J. Appl. Mech. 57, 923–929 (1990)

    Article  Google Scholar 

  55. Nayfeh, A. H.: The general problem of elastic wave propagation in multilayered anisotropic media. J. Acout. Soc. Am. 60, 1521–1531 (1991)

    Article  Google Scholar 

  56. Verma, K.L.: On the wave propagation in layered plates of general anisotropic media. World Acad. Sci. Eng. Tech. 13, 487–493 (2008)

    Google Scholar 

  57. Nafchi, A.M., Zangeneh, V., Moradi, A., Mohamadirad, H.: Modeling of ultrasonic wave propagation in two-layer plate and drawing the dispersion curves. In: International Conference on Mechanical Engineering and Advanced Technology-ICMEAT2012 1012 October, 2012, Abbasi International Hotel, Isfahan, Iran, ICMEAT2012-1106

  58. Bratton, R., Datta, S.: Analysis of guided waves in a bilayered plate. In: D.O. Thompson and D.E. Chimenti (eds) Review of Progress in Quantitative Nondestructive Evaluation, Vol. 11, pp. 193–200. Plenum Press, New York (1992)

    Chapter  Google Scholar 

  59. Wang, Z., Jen, C-K., Cheeke, J. D, : Analytical solutions for sagittal plane waves in three-layer composites. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 4, 293–301 (1993)

    Article  Google Scholar 

  60. Lutianov, M., Rogerson, G.A.: Long wave motion in layered elastic media. Int. J. Eng. Sci. 48, 1856–1871 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  61. Ogden, R.W.: Elastic deformations of rubberlike solids. In: H.G. Hopkins and M.J. Sewell (eds) Mechanics of Solids The Rodney Hill 60th Anniversary Volume. Pergamon Press, Oxford, pp. 499-537 (1982)

    Google Scholar 

  62. Amabil, M., Breslavsky, I.D., Reddy, J.N.: Nonlinear higher-order shell theory for incompressible biological hyperelastic materials. Comput. Methods Appl. Mech. Engrg 346, 841–861 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  63. Stroh, A. N.: Steady state problems in anisotropic elasticity. J. Math. Phys. 41, 77–103 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  64. Vinh, P.C., Anh, V.T.N.: Effective boundary condition method and approximate secular equations of Rayleigh waves in orthotropic half-spaces coated by a thin layer. J. Mech. Mater. Struct. 11, 259–277 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The work was supported by the Vietnam National Foundation For Science and Technology Development (NAFOSTED) under Grant No. 107.02-2019.314.

Author information

Authors and Affiliations

  1. Faculty of Mathematics, Mechanics and Informatics, VNU University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam

    V. T. N. Anh & P. C. Vinh

  2. Department of Engineering Mechanics, Thuyloi University of Vietnam, 175 Tay Son Str., Hanoi, Vietnam

    N. T. K. Linh

  3. Department of Mathematics, Vietnam National University of Forestry, Xuan Mai-Chuong My, Hanoi, Vietnam

    L. T. Thang

Authors
  1. V. T. N. Anh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. C. Vinh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. T. K. Linh
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. L. T. Thang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. C. Vinh.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Anh, V.T.N., Vinh, P.C., Linh, N.T.K. et al. Explicit transfer matrix for an incompressible orthotropic elastic layer and applications. Z. Angew. Math. Phys. 72, 145 (2021). https://doi.org/10.1007/s00033-021-01579-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-021-01579-7

Keywords

Mathematics Subject Classification

Search

Navigation