Abstract
Theoretical aspects of applying Lamb waves to nondestructive testing of layered anisotropic media are considered. Propagation of Lamb waves is analyzed using a six-dimensional complex Cauchy formalism that allows one to obtain a closed-form equation for determining the dispersion of Lamb waves in media with arbitrary elastic anisotropy. The possibility for applying the higher modes of Lamb waves to nondestructive testing is noted.
Similar content being viewed by others
References
Rayleigh, J.W., On the free vibrations of an infinite plate of homogeneous isotropic elastic matter, Proc. Math. Soc., London, 1889, vol. 20, p. 225–234.
Lamb, H., On the flexure of an elastic plate, Proc. Math. Soc., London, 1889, vol. 21, p. 70–90.
Lamb, H., On waves in an elastic plate, Proc. Roy. Soc, 1917, vol. A93, pp. 114–128.
Rayleigh, J.W., On wave propagating along the plane surface of an elastic solid, Proc. London Math. Soc., 1885, vol. 17, pp. 4–11.
Mindlin, R.D., Waves and vibrations in isotropic elastic plates, in: The First Symp. Nav. Struct. Mech., 1958, Oxford: Pergamon Press, 1960.
Viktorov, I.A., Fizicheskie osnovy primeneniya ul’trazvukovykh voln Releya i Lemba v tekhnike (Physical Foundations of Technical Applications of Ultrasonic Rayleigh and Lamb Waves), Moscow: Nauka, 1966.
Jeffreys, H., The surface waves of earthquakes, Geophys. J. Int., 1935, vol. 3, pp. 253–261.
Gogoladze, V.G., Dispersion of Rayleigh waves in a layer, Tr. Seism. Inst. Akad. Nauk SSSR, 1947, vol. 119, pp. 27–38.
Wilson, J.T. and Baykal, O., On the North Atlantic basin as determined from Rayleigh wave dispersion, Bull. Seism. Soc. Am., 1948, vol. 38, pp. 41–53.
Dobrin, M.B., Dispersion in seismic surface waves, Geophys., 1951, vol. 16, pp. 63–80.
Freedman, A., The variation, with the Poisson ratio, of Lamb modes in a free plate. I: General spectra, J. Sound Vib., 1990, vol. 137, pp. 209–230.
Freedman, A., The variation, with the Poisson ratio, of Lamb modes in a free plate. II: At transitions and coincidence values, J. Sound Vib., 1990, vol. 137, pp. 231–247.
Freedman, A., The variation, with the Poisson ratio, of Lamb modes in a free plate. III: Behavior of individual modes, J. Sound Vib., 1990, vol. 137, pp. 249–266.
Zhu, Q. and Mayer, W.G., On the crossing points of Lamb wave velocity dispersion curves, J. Acoust. Soc. Am, 1993, vol. 93, no. 4, pp. 1893–1895.
Stokes, G.G., On the theory of oscillatory waves, Trans. Cambridge Philos. Soc., 1847, vol. 8, pp. 441–455.
Rayleigh, J.W., On progressive waves, Proc. London Math. Soc., 1877, vol. 9, pp. 21–26.
Rayleigh, J.W., Theory of Sound, Ulan Press, 2011.
Reynolds, O., On the rate of progression of groups of waves and the rate at which energy is transmitted by waves, Nature, 1877, vol. 16, pp. 343–344.
Sezawa, K. and Kanai, K., On the propagation of Rayleigh waves in dispersive elastic media, Bull. Earthquake Res. Inst., Univ. Tokyo, 1941, vol. 19, pp. 549–553.
Sato, Y., Study on surface waves. II: Velocity of surface waves propagated upon elastic plates, Bull. Earthquake Res. Inst., Univ. Tokyo, 1951, vol. 29, pp. 223–261.
Press, F. and Ewing, M., Theory of air-coupled flexural waves, J. Appl. Phys., 1951, vol. 22, pp. 892–899.
Press, F. and Oliver, J., Model study of air-coupled surface waves, J. Acoust. Soc. Am., 1955, vol. 27, pp. 43–46.
Tolstoy, I., Dispersion and simple harmonic sources in wave ducts, J. Acoust. Soc. Am., 1955, vol. 27, pp. 897–910.
Tolstoy, I. and Usdin, E., Wave propagation in elastic plates: low and high mode dispersion, J. Acoust. Soc. Am., 1957, vol. 29, pp. 37–42.
Negishi, K., Existence of negative group velocities for Lamb waves, Jpn. J. Appl. Phys., 1987, vol. 26, nos. Suppl. 26-1, pp. 171–173.
Biot, M.A., General theorems on the equivalence of group velocity and energy transport, Phys. Rev., 1957, vol. 105, pp. 1129–1137.
Lowe, M.J.S., Plate waves for the NDT of diffusion bonded titanium, PhD thesis, Imperial College, UK, 1992.
Lyon, R.H., Response of an elastic plate to localized driving forces, J. Acoust. Soc. Am., 1955, vol. 27, pp. 259–265.
Bobrovnitskii, Y.I., Orthogonality relations for Lamb waves, Sov. Acoust. Phys., 1973, vol. 18, no. 4, pp. 432–433.
Auld, B.A., Acoustic Fields and Waves in Solids, Malabar. Florida: Robert E. Krieger Publ. Co., 1990, vol. 2.
Vovk, A.E. and Tyutekin, V.V., Excitation of normal waves in a planar elastic duct by forces given in its cross section, Tr. Akust. Inst., 1969, no. IX, pp. 5–26.
Duquenne, L., Moulin, E., Assaad, J., and Grondel, S., Transient modeling of Lamb waves generated in viscoelastic materials by surface bonded piezoelectric transducers, J. Acoust. Soc. Am., 2004, vol. 116, pp. 133–141.
Kirrmann, P., On the completeness of Lamb modes, J. Elasticity, 1995, vol. 37, pp. 39–69.
Zakharov, D.D., Orthogonality of 3D guided waves in viscoelastic laminates and far filed evaluation to a local acoustic source, Int. J. Solids Struct., 2008, vol. 45, p. 1788–1803.
Pagneux, V. and Maurel, A., Determination of Lamb mode eigenvalues, J. Acoust. Soc. Am., 2001, vol. 110, pp. 1307–1314.
Malischewsky, W., Orthonormalization of plane surface and body waves (in German), Gerlands Beitr. Geophys., 1970, vol. 79, pp. 468–474.
Stange, S. and Friederich, W., Guided wave propagation across sharp lateral heterogeneities: the complete wavefield at plane vertical discontinuities, Geophys. J. Int, 1992, vol. 109, pp. 183–190.
Keldysh, M.V., On the completeness of eigenfunctions of some classes of non self-adjoint linear operators, Russ. Math. Surv., 1971, vol. 26, no. 4, pp. 15–44.
Agmon, S., On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Commun. Pure Appl. Math., 1962, vol. 15, pp. 119–147.
Lowe, M.J.S. and Diligent, O., Low-frequency refection characteristics of the s 0 Lamb wave from a rectangular notch in a plate, J. Acoust. Soc. Am., 2002, vol. 111, pp. 64–74.
Brevdo, L., On the stability of Lamb modes, Acta Mech., 1996, vol. 117, pp. 71–80.
Stoneley, R., The seismological implications of aelotropy in continental structures, Mon. Not. R. Astron. Soc., Geophys. Suppl., 1949, vol. 5, pp. 343–353.
Musgrave, M.J.P., On the propagation of elastic waves in aeolotropic media. I. General principles, Proc. R. Soc. London A, 1954, vol. 226, pp. 339–355.
Musgrave, M.J.P., On the propagation of elastic waves in aeolotropic media. II. Media of hexagonal symmetry, Proc. R. Soc. London A, 1954, vol. 226, pp. 356–366.
Buchwald, V.T., Rayleigh waves in transversely isotropic media, Quart. J. Mech. Appl. Math., 1961, vol. 14, pp. 293–317.
Stoneley, R., The propagation of surface elastic waves in a cubic crystal, Proc. R. Soc. London A, 1955, vol. 232, pp. 447–458.
Miller, G.F. and Musgrave, M.J.P., On the propagation of elastic waves in aeolotropic media. III. Media of cubic symmetry, Proc. R. Soc. London A, 1956, vol. 236, pp. 352–383.
Mitra, M., Propagation of elastic waves in an infnite plate of cylindrically aelotropic material, Z. Angew. Math. Phys., 1959, vol. 10, pp. 579–583.
Anderson, D.L., Elastic wave propagation in layered anisotropic media, J. Geophys. Res., 1961, vol. 66, pp. 2953–2963.
Solie, L.P. and Auld, B.A., Elastic waves in free anisotropic plates, J. Acoust. Soc. Am., 1973, vol. 54, pp. 50–65.
Nayfeh, A.H. and Chimenti, D.E., Free wave propagation in plates of general anisotropic media, J. Appl. Mech., 1989, vol. 56, no. 4, pp. 881–886.
Dayal, V. and Kinra, V.K., Leaky Lamb waves in an anisotropic plate. I: An exact solution and experiments, J. Acoust. Soc. Am., 1989, vol. 85, pp. 2268–2276.
Ben-Menahem, A. and Sena, A.G., Seismic source theory in stratifed anisotropic media, J. Geophys. Res., 1990, vol. 95, no. B10, pp. 15.395–15.427.
Liu, G.R., Tani, J., Watanabe, K., and Ohyoshi, T., Lamb wave propagation in anisotropic laminates, J. Appl. Mech., 1990, vol. 57, pp. 923–929.
Lin, W. and Keer, L.M., A study of Lamb waves in anisotropic plates, J. Acoust. Soc. Am., 1992, vol. 92, pp. 888–894.
Neau, G., Lamb waves in anisotropic viscoelastic plates. Study of the wave fronts and attenuation, PhD thesis, L’Universite de Bordeaux, 2003.
Neau, G., Deschamps, M., and Lowe, M.J.S., Group velocity of lamb waves in anisotropic plates: comparison between theory and experiment, In: Review of Progress in Quantitative NDED, New York: AIP, 2001, vol. 20.
Wang, L. and Yuan, F.G., Group velocity and characteristic wave curves of Lamb waves in composites: modeling and experiments, Compos. Sci. Technol., 2007, vol. 67, pp. 1370–1384.
Kuznetsov, S.V., Surface waves of non-Rayleigh type, Quart. Appl. Math., 2003, vol. 61, p. 575–582.
Kuznetsov, S.V., Subsonic Lamb waves in anisotropic plates, Quart. Appl. Math., 2002, vol. 60, pp. 577–587.
Markus, S.A., Low-frequency approximations for the zero modes of normal waves in anisotropic plates, Akust. Zh., 1987, vol. 33, pp. 1091–1095.
Al'shits, V.I. and Lyubimov, V.N., Abnormal dispersion of surface elastic waves in an anisotropic plate, Kristallografiya, 1988, vol. 33, pp. 279–285.
Stroh, A.N., Steady state problems in anisotropic elasticity, J. Math. Phys., 1962, vol. 41, pp. 77–103.
Ingebrigtsen, K.A. and Tonning, A., Elastic surface waves in crystals, Phys. Rev, 1969, vol. 184, pp. 942–951.
Barnett, D.M. and Lothe, J., Synthesis of the sextic and the integral formalism for dislocations, Greens functions, and surface waves in anisotropic elastic solids, Phys. Norv., 1973, vol. 7, pp. 13–19.
Chadwick, P. and Smith, G.D., Foundations of the theory of surface waves in anisotropic elastic materials, In; Advances in Applied Mechanics, New York: Academic Press, 1977, vol. 17.
Ting, T.C.T. and Barnett, D.M., Classifcations of surface waves in anisotropic elastic materials, Wave Motion, 1997, vol. 26, pp. 207–218.
Ting, T.C.T., On extraordinary semisimple matrix N(ν) for anisotropic elastic materials, Quart. Appl. Math., 1997, vol. 55, pp. 723–738.
Fu, Y.B., Hamiltonian interpretation of the Stroh formalism in anisotropic elasticity, Proc. R. Soc. A, 2007, vol. 463, pp. 3073–3087.
Tanuma, K., Stroh Formalism and Rayleigh Waves, Springer, 2010.
Shuvalov, A.L., On the theory of wave propagation in anisotropic plates, Proc. R. Soc. London. Ser. A, 2000, vol. 456, pp. 2197–2222.
Alshits, V., Deschamps, M., and Maugin, G., Elastic waves in anisotropic plates: short-wavelength asymptotics of the dispersion branches Vn(k), Wave Motion, 2003, vol. 37, no. 3, pp. 273–292.
Ting, T.C.T., Anisotropic Elasticity: Theory and Applications, New York: Oxford Univ. Press, 1996.
Kuznetsov, S.V., Love waves in stratifed monoclinic media, Quart. Appl. Math., 2004, vol. 62, pp. 749–766.
Kuznetsov, S.V., SH-waves in laminated plates, Quart. Appl. Math., 2006, vol. 64, pp. 153–165.
Kuznetsov, S.V., Love waves in nondestructive diagnostics of layered composites, Acoust. Phys., 2010, vol. 56, pp. 877–892.
Gregory, R.D. and Gladwell, I., The refection of a symmetric Rayleigh-Lamb wave at the fixed or free edge of a plate, J. Elasticity, 1983, vol. 13, pp. 185–206.
Aki, K. and Richards, P.G., Quantitative Seismology. Theory and Methods, New York: W.H. Freeman and Co., 1980.
Ben-Menahem, A. and Singh, S.J., Seismic Waves and Sources, Dover Publications, 2012.
Chapman, C., Fundamentals of Seismic Wave Propagation, New York: Cambridge Univ. Press, 2010.
Sheriff, R.E. and Geldart, R.P., Exploration Seismology, New York: Cambridge Univ. Press, 1995.
Bergmann, L., Der Ultraschall, Zurich: S. Hirzel Verlag, 1949.
Viktorov, I.A., Ultrasonic Lamb waves: a review, Akust. Zh., 1965, vol. 11, pp. 1–18.
Doxbeck, M.A., Hussain, M.A., Rama, J., Abate, A., and Frankel, J., An algorithm for the determination of coating properties from laser generated and detected Rayleigh waves using wavelet analysis, Rev. Prog. QNDE, 2002, vol. 21, pp. 292–299.
Golubev, E.V., Gurevich, S.Yu., and Petrov, Yu.V., On the theory of Lamb wave excitation in metals by pulsed laser radiation, Acoust. Phys., 2011, vol. 57, pp. 620–626.
Royer, D. and Dieulesaint, E., Elastic Waves in Solids 1. Free and Guided Propagation, New York: Springer, 2009.
Moler, C. and Loan, Ch.V., Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Review, 2003, vol. 45, pp. 1–46.
Hartman, P., Ordinary Differential Equations, New York: SIAM, 2002.
Pease, M.C., Methods of Matrix Algebra, London: Academic Press, 1965.
Dunkin, J.W., Computation of modal solutions in layered elastic media at high frequencies, Bull. Seism. Soc. Am., 1965, vol. 55, p. 335–358.
Lévesque, D. and Piche, L., A robust transfer matrix formulation for the ultrasonic response of multilayered absorbing media, J. Acoust. Soc., 1992, vol. 92, pp. 452–467.
Castaings, M. and Hosten, B., Delta operator technique to improve the Thomson–Haskell method stability for propagation in multilayered anisotropic absorbing plate, J. Acoust. Soc. Am., 1994, vol. 95, pp. 1931–1941.
Lowe, M.J.S., Matrix techniques for modeling ultrasonic waves in multilayered media. IEEE Trans. Ultrason. Eng., 1995, vol. 42, pp. 525–542.
Wang, L. and Rokhlin, S.I., Stable reformulation of transfer matrix method for wave propagation in layered anisotropic media, Ultrason., 2001, vol. 39, pp. 413–424.
Bailey, D.H., A portable high performance multiprecision package, NASA RNR Tech. Rep., 1993.
Bailey, D.H., Automatic translation of Fortran programs to multiprecision, NASA RNR Tech. Rep., 1993.
Guo, C.Y. and Wheeler, L., Extreme Poisson’s ratios and related elastic crystal properties, J. Mech. Phys. Solids, 2006, vol. 54, pp. 690–707.
Schecklman, S., Zurk, L.M., Henry, S., and Kniffin, G.P., http://dx.doi.org/ doi 10.1063/1.3561806
Kulesh, M.A., Matveenko, V.P., and Shardakov, I.N., Propagation of surface elastic waves in the Cosserat medium, Acoust. Phys., 2006, vol. 52, no. 2, pp. 186–193.
Kulesh, M.A., Grekova, E.F., and Shardakov, I.N., The problem of surface wave propagation in a reduced Cosserat medium, Acouts. Phys., 2009, vol. 55, no. 2, pp. 218–226.
Markov, M.G., Rayleigh wave propagation along the boundary of a non-Newtonian fluid-saturated porous medium, Acoust. Phys., 2006, vol. 52, no. 4, pp. 429–434.
Bobrovnitskii, Yu.I., A Rayleigh-type wave at the plane interface of two homogeneous fluid half-spaces, Acouts. Phys., 2011, vol. 57, no. 5, p. 595–597.
Kuznetsov, S.V. and Terentjeva, E.O., Planar internal Lamb problem: waves in the epicentral zone of a vertical power source, Acoust. Phys., 2015, vol. 61, no. 3, pp. 356–367.
Kuznetsov, S.V. and Terentjeva, E.O., Wave felds and domination regions for the interior Lamb problem, Mech. Solids, 2015, vol. 5, pp. 508–520.
Sekerzh-Zenkovich, S.J., Ilyasov, H.H., Kravtsov, A.V., and Kuznetsov, S.V., Outer Lamb problem. Distributed harmonic surface loading, Mech. Solids, 2016, vol. 1, pp. 50–56.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.V. Ilyashenko, S.V. Kuznetsov, 2017, published in Defektoskopiya, 2017, No. 4, pp. 3–21.
Rights and permissions
About this article
Cite this article
Ilyashenko, A.V., Kuznetsov, S.V. Theoretical Aspects of Applying Lamb Waves in Nondestructive Testing of Anisotropic Media. Russ J Nondestruct Test 53, 243–259 (2017). https://doi.org/10.1134/S1061830917040039
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1061830917040039