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Stoneley waves with spring contact and evaluation of the quality of imperfect bonds

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Abstract

In this paper, the propagation of Stoneley waves along the interface between two orthotropic elastic half-spaces is investigated. The half-spaces are compressible or incompressible and they are in spring contact with each other. The main aim of the paper is to derive secular equations and formulas for the ratios of displacement components at the interface and to show that Stoneley waves are a good tool for evaluating the quality of imperfect bonds modeled as spring contacts. First, the secular equation and the formulas for the displacement-component ratios for the compressible case (two half-spaces are compressible) are derived. Then, those for the incompressible cases (one or both half-spaces are incompressible) are obtained by using the incompressible limit method. The obtained secular equations reveal that, while Stoneley waves are non-dispersive for the welded contact and the sliding contact, they are dispersive for the spring contact. Employing the obtained secular equations and formulas for the displacement-component ratios, two particular examples are carried out for determining the spring constants from the measured values of the displacement-component ratios and the wave velocity. It is shown that the ratios of displacement components at two sides of the interface are the best choice for evaluating the quality of imperfect bonds.

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References

  1. Lekesiz, H., Katsube, N., Rokhlin, S.I., Seghi, R.R.: Effective spring stiffness for a planar periodic array of collinear cracks at an interface between two dissimilar isotropic materials. Mech. Mater. 43, 87–98 (2011)

    Article  Google Scholar 

  2. Lekesiz, H., Katsube, N., Rokhlin, S.I., Seghi, R.R.: Effective spring stiffness for a periodic array of interacting coplanar penny-shaped cracks at an interface between two dissimilar isotropic materials. Int. J. Solids Struct. 50, 2817–2828 (2013)

    Article  Google Scholar 

  3. Bostrom, A., Bovik, P., Olsson, P.: A comparison of exact first order and spring boundary conditions for scattering by thin layers. J. Nondestruct. Eval. 11, 175–184 (1992)

    Article  Google Scholar 

  4. Martin, P.A.: Boundary integral equations for the scattering of elastic waves by elastic inclusions with thin interface layers. J. Nondestruct. Eval. 11, 167–174 (1992)

    Article  Google Scholar 

  5. Baltazar, A., Rokhlin, S.I., Pecorari, C.: On the relationship between ultrasonic and micromechanical properties of contacting rough surfaces. J. Mech. Phys. Solids. 50, 1397–1416 (2002)

    Article  Google Scholar 

  6. Sevostianov, I., Rodriguez-Ramos, R., Guinovart-Diaz, R., Bravo-Castillero, J., Sabina, F.J.: Connections between different models describing imperfect interfaces in periodic fiber-reinforced composites. Int. J. Solids Struct. 49, 1518–1525 (2012)

    Article  Google Scholar 

  7. Hess, P., Lomonosov, A.M., Mayer, A.P.: Laser-based linear and nonlinear guided elastic waves at surfaces (2D) and wedges (1D). Ultrasonics 54, 39–55 (2013)

    Article  Google Scholar 

  8. Every, A.G.: Measurement of the near-surface elastic properties of solids and thin supported films. Meas. Sci. Technol. 13, 21–39 (2002)

    Article  Google Scholar 

  9. Stoneley, R.: Elastic waves at the surface of seperation of two solids. Proc. R. Soc. Lond. A 106, 416–428 (1924)

    Article  Google Scholar 

  10. Sezawa, K., Kanai, K.: The range of possible existence of Stoneley waves, and some related problems. Bull. Earthq. Res. Inst. Tokyo Univ. 17, 1–8 (1939)

    Google Scholar 

  11. Scholte, J.G.: On the Stoneley wave equation. Proc. Kon. Acad. Sci. Amst. 45, 159–164 (1942)

    MathSciNet  MATH  Google Scholar 

  12. Scholte, J.G.: The range of existence of Rayleigh and Stoneley waves. Mon. Not. R. Astron. Soc. Geophys. Suppl. 5, 120–126 (1947)

    Article  MathSciNet  Google Scholar 

  13. Vinh, P.C., Malischewsky, P.G., Giang, P.T.H.: Formulas for the speed and slowness of Stoneley waves in bonded isotropic elastic half-spaces with the same bulk wave velocities. Int. J. Eng. Sci. 60, 53–58 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. Barnett, D.M., Lothe, J., Gavazza, S.D., Musgrave, M.J.P.: Consideration of the existence of interfacial (Stoneley) waves in bonded anisotropic elastic half-spaces. Proc. R. Soc. Lond. A. 412, 153–166 (1985)

    MathSciNet  MATH  Google Scholar 

  16. Chadwick, P., Jarvis, D.A.: Interfacial waves in a pre-strain neo-Hookean body I. Biaxial state of strain. Q. J. Mech. Appl. Math. 32, 387–399 (1979)

    Article  MathSciNet  Google Scholar 

  17. Chadwick, P., Jarvis, D.A.: Interfacial waves in a pre-strain neo-Hookean body II. Triaxial state of strain. Q. J. Mech. Appl. Math. 32, 401–418 (1979)

    Article  MathSciNet  Google Scholar 

  18. Dasgupta, A.: Effect of high initial stress on the propagation of Stoneley waves at the interface of two isotropic elastic incompressible media. Indian J. Pure Appl. Math. 12, 919–926 (1981)

    MATH  Google Scholar 

  19. Dunwoody, J.: Elastic interfacial standing waves. In: McCarthy, M.F., Hayes, M.A. (eds.) Elastic Waves Propagation, pp. 107–112. North-Holland, Amsterdam (1989)

    Google Scholar 

  20. Dowaikh, M.A., Ogden, R.W.: Interfacial waves and deformations in pre-stressed elastic media. Proc. R. Soc. Lond. A 433, 313–328 (1991)

    Article  MathSciNet  Google Scholar 

  21. Vinh, P.C., Giang, P.T.H.: Uniqueness of Stoneley waves in pre-stressed incompressible elastic media. Int. J. Non-Linear Mech. 47, 128–134 (2012)

    Article  Google Scholar 

  22. Murty, G.S.: A theoretical model for the attenuation and dispersion of Stoneley waves at the loosely bonded interface of elastichalf-spaces. Phys. Earth Planet. Inter. 11, 65–79 (1975)

    Article  Google Scholar 

  23. Vinh, P.C., Giang, P.T.H.: On formulas for the velocity of Stoneley waves propagating along the loosely bonded interface of two elastic half-spaces. Wave Motion 48, 647–657 (2011)

    Article  MathSciNet  Google Scholar 

  24. Barnett, D.M., Gavazza, S.D., Lothe, J.: Slip waves along the interface between two anisotropic elastic half-spaces in sliding contact. Proc. R. Soc. Lond. A 415, 389–419 (1988)

    Article  Google Scholar 

  25. Vinh, P.C., Anh, V.T.N., Linh, N.T.K.: Exact secular equations of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer. Int. J. Solids Struct. 83, 65–72 (2016)

    Article  Google Scholar 

  26. Vinh, P.C., Ogden, R.W.: Formulas for the Rayleigh wave speed in orthotropic elastic solids. Ach. Mech. 56, 247–265 (2004)

    MathSciNet  MATH  Google Scholar 

  27. Vinh, P.C.: Explicit secular equatins of Rayleigh waves in a non-homogeneous orthotropic elastic medium under the influence of gravity. Wave Motion 46, 427–434 (2009)

    Article  MathSciNet  Google Scholar 

  28. Sotiropoulos, D.A.: The effect of anisotropy on guided elastic waves in a layered half-space. Mech. Mater. 31, 215–223 (1999)

    Article  Google Scholar 

  29. Vinh, P.C., Tuan, T.T., Hue, L.T., Anh, V.T.N., Dung, T.T.T., Linh, N.T.K., Malischewsky, P.: Exact formula for the horizontal-to-vertical displacement ratio of Rayleigh waves in layered orthotropic half-spaces. J. Acoust. Soc. Am. 146(2), 1279–1289 (2019)

    Article  Google Scholar 

  30. Malischewsky, P.G., Scherbaum, F.: Love’s formula and H/V-ratio (ellipticity) of Rayleigh waves. Wave Motion 40, 57–67 (2004)

    Article  MathSciNet  Google Scholar 

  31. Vinh, P.C., Tuan, T.T., Hue, L.T.: Formulas for the H\(/\)V ratio (ellipticity) of Rayleigh waves in orthotropic elastic half-spaces. Wave Random Comlex. 29(4), 759–774 (2019)

    Article  MathSciNet  Google Scholar 

  32. Claus, R.O., Palmer, C.H.: Direct measurement of ultrasonic Stoneley waves. Appl. Phys. Lett. 31(8), 547–548 (1977)

    Article  Google Scholar 

  33. Claus, R.O., Palmer, C.H.: Optical measurements of ultrasonic waves on interfaces between bonded solids. IEEE Trans. Sonics Ultrason. 27, 97–103 (1980)

    Article  Google Scholar 

  34. Chevalier, Y., Louzar, M., Maugin, G.A.: Surface-wave characterization of the interface between two anisotropic media. J. Acoust. Soc. Am. 90(6), 3218–3227 (1991)

    Article  Google Scholar 

  35. Hashin, Z.: Thin interphase/imperfect interface in elasticity with application to coated fiber composites. J. Mech. Phys. Solids. 50(12), 2509–2537 (2002)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The work was supported by the Vietnam National Foundation For Science and Technology Development (NAFOSTED) under Grant No. 107.02-2016.15.

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Authors and Affiliations

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

    V. T. N. Anh, P. C. Vinh & T. T. Tuan

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

    L. T. Thang

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  1. V. T. N. Anh
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  2. L. T. Thang
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  3. P. C. Vinh
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  4. T. T. Tuan
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Correspondence to P. C. Vinh.

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Anh, V.T.N., Thang, L.T., Vinh, P.C. et al. Stoneley waves with spring contact and evaluation of the quality of imperfect bonds. Z. Angew. Math. Phys. 71, 36 (2020). https://doi.org/10.1007/s00033-020-1257-1

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