The Dirichlet–Neumann conditions and orthogonality of three-dimensional natural waves in layered solids

D. D. Zakharov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 9, pp. 1598-1612

Résumé

Orthogonality relations for homogeneous waves in layered plates are obtained, and they are generalized to the case of a contact with fluid layers. For layers of infinite thickness, it is shown that the homogeneous waves of the discrete spectrum are orthogonal to each other and to the waves of the continuous spectrum. For finite-size sources, exact formulas are derived for the coefficients multiplying the modes. Based on the orthogonality relations, a nonlocal radiation principle is proposed such that the infinite domain in the numerical solution of diffraction problems for layered plates can be replaced by a virtual cylinder.

Bibliographie

[1] Bobrovnitskii Yu. I., “Sootnosheniya ortogonalnosti dlya voln Lemba”, Akustich. zhurnal, 18:4 (1972), 513–515

[2] Fedoryuk M. V., “Sootnosheniya tipa ortogonalnosti v tverdykh volnovodakh”, Akustich. zhurnal, 20:2 (1974), 310–314

[3] Fraser W. B., “Orthogonality relations for Rayleigh–Lamb modes of vibration of a plate”, J. Acoust. Soc. Amer., 59 (1976), 215–216 | DOI | Zbl

[4] Prakash B. G., “Generalized orthogonality relations for rectangular strips in elastodynamics”, Mech. Res. Communs., 5:4 (1978), 251–255 | DOI | Zbl

[5] Zil'bergleit A. S., Nuller B. M., “Obobschennaya ortogonalnost odnorodnykh reshenii v dinamicheskikh zadachakh teorii uprugosti”, Dokl. AN SSSR, 234:2 (1977), 333–335 | MR

[6] Slepyan L. I., “Teorema Betti i sootnosheniya ortogonalnosti dlya sobstvennykh funktsii”, Izv. AN SSSR. Mekhan. tverdogo tela, 1979, no. 1, 83–87 | MR

[7] Folk R. T., Herszynski A., “Solutions of elastodynamic slab problems using a new orthogonality condition”, J. Acoust. Soc. Amer., 80:4 (1986), 1103–1110 | DOI | MR

[8] Zakharov D. D., “Sootnosheniya obobschennoi ortogonalnosti dlya sobstvennykh funktsii v prostranstvennykh zadachakh dinamiki uprugogo sloya”, Izv. AN SSSR. Mekhan. tverdogo tela, 1988, no. 6, 62–68

[9] Achenbach J. D., “Explicit solution for carrier waves supporting surface waves and plane waves”, Wave Motion, 28 (1998), 89–97 | DOI | MR | Zbl

[10] Achenbach J. D., Xu Y., “Use of elastodynamic reciprocity to analyze point-load generated axisymmetric waves in a plate”, Wave Motion, 30 (1998), 57–67 | DOI | MR

[11] Achenbach J. D., Xu Y., “Wave motion in an isotropic elastic layer generated by time harmonic load of arbitrary direction”, J. Acoust. Soc. Amer., 106 (1999), 83–90 | DOI

[12] Achenbach J. D., “Calculation of wave fields using elastodynamic reciprocity”, Int. J. Solids and Struct., 37 (2000), 7043–7053 | DOI | MR | Zbl

[13] Lawrie J. B., Abrahams I. D., “An orthogonality relation for a class of problems with high order boundary conditions; Applications in sound-structure interaction”, Quart. J. Mechs Appl. Maths., 52:2 (1999), 161–181 | DOI | MR | Zbl

[14] Pelts S. P., Shikhman V. M., “Rasseyanie volny Releya na uprugoi polupolose, stseplennoi na tortse s uprugoi poluploskostyu”, Dokl. AN SSSR, 292:2 (1987), 299–303

[15] Pelts S. P., Shikhman V. M., “O skhodimosti metoda odnorodnykh reshenii v dinamicheskoi smeshannoi zadache dlya polupolosy”, Dokl. AN SSSR, 295:4 (1987), 821–824 | MR

[16] Pelts S. P., Shikhman V. M., “Rasprostranenie voln v krestoobraznom soedinenii beskonechnykh uprugikh polos”, Prikl. matem. i mekhan., 51:1 (1987), 54–59

[17] Kasatkin B. A., “Difraktsiya normalnykh voln na myagkom ekrane tipa rassloeniya v plastine”, Akustich. zhurnal, 27:2 (1981), 135–137 | MR

[18] Shkerdin G., Glorieux G., “Lamb mode conversion in a plate with a delamination”, J. Acoust. Soc. Amer., 116 (2004), 2089–2100 | DOI

[19] Shkerdin G., Glorieux G., “Lamb mode conversion in an absorptive bi-layer with a delamination”, J. Acoust. Soc. Amer., 118 (2006), 2253–2264 | DOI

[20] Gregory R. D., Gladwell I., “The reflection of a symmetric Rayleigh–Lamb wave at a fixed or free edge of a plate”, J. Elasticity, 13 (1983), 185–206 | DOI | MR | Zbl

[21] Flores-Lopez M. A., Gregory R. D., “Scattering of Rayleigh–Lamb waves by a surface breaking crack in an elastic plate”, J. Acoust. Soc. Amer., 119:4 (2006), 2041–2049 | DOI

[22] Givoli D., “High-order local non-reflecting boundary conditions: a review”, Wave Motion, 39 (2004), 319–326 | DOI | MR | Zbl

[23] Berenger J.-P., “A perfectly matched layer for the absorption of electromagnetic waves”, J. Comput. Phys., 114 (1994), 185–200 | DOI | MR | Zbl

[24] Skelton E. A., Adams S. D.M., Craster R. V., “Guided elastic waves and perfectly matched layers”, Wave Motion, 44 (2007), 573–592 | DOI | MR

[25] Astrakhantsev G. P., “Metod fiktivnykh oblastei dlya ellipticheskikh uravnenii vtorogo poryadka s estestvennymi granichnymi usloviyami”, Zh. vychisl. matem. i matem. fiz., 18:1 (1978), 118–125 | MR | Zbl

[26] Vabischevich P. N., Metody fiktivnykh oblastei v kraevykh zadachakh matematicheskoi fiziki, MGU, M., 1993

[27] Burnett D. S., “A three-dimensional acoustic infinite element based on a prolate spheroidal multiple expansion”, J. Acoust. Soc. Amer., 96 (1994), 2798–2816 | DOI | MR

[28] Keller J. B., Givoli D., “Exact non-reflecting boundary conditions”, J. Comput. Phys., 82 (1989), 172–192 | DOI | MR | Zbl

[29] Givoli D., Keller J. B., “Non-reflecting boundary conditions for elastic waves”, Wave Motion, 12 (1990), 261–279 | DOI | MR | Zbl

[30] Rayben'kii V. S., Tsynkov S. V., “Artificial boundary conditions for the numerical solution of external viscous flow problems”, SIAM J. Numer. Anal., 32 (1995), 1355–1389 | DOI | MR

[31] Baronian V., Bonnet-Ben Dhia A.-S., Luneville E., “Transparent boundary conditions for the harmonic diffraction problem in an elastic waveguide”, J. Comput. Appl. Math., 234:6 (2010), 1945–1952 | DOI | MR | Zbl

[32] Jezzine K., Lhemery A., “Simulation of guided wave inspection based on the reciprocity principle and the semianalytical finite element method”, Rev. Progress in QNDE 26, AIP Conf. Proc., 894, American Inst. Phys., Melville, N.Y., 2007, 39–46

[33] Kostyuchenko A. G., Orazov M. B., “O polnote kornevykh vektorov nekotorykh samosopryazhennykh kvadratichnykh puchkov”, Funkts. analiz, 11:4 (1977), 85–87 | MR | Zbl

[34] Kirrmann P., “On the completeness of Lamb's modes”, J. Elasticity, 37 (1995), 39–69 | DOI | MR | Zbl

[35] Kostyuchenko A. G., Orazov M. B., “Problem of oscillations of an elastic half-cylinder and related self-adjoint quadratic pencil”, J. Math. Sciences, 33:3 (1986), 1025–1065 | DOI | Zbl

[36] Auld B. A., Acoustic fields and waves in solids, v. 2, Wyley, N. Y., 1973

[37] M. Abramovitz, I. A. Stegun (ed.), Handbook of mathematical functions, Nat. Bur. Standards, Washington, 1972

[38] Biot V. A., “General theorems of the equivalence of group velocity and energy transport”, Phys. Rev., 105:4 (1957), 1129–1137 | DOI | Zbl

[39] Lighthill M. J., “Group velocity”, J. Inst. Math. Appl., 1:1 (1965), 1–28 | DOI | MR | Zbl

[40] Zilbergleit A. S., Kopilevich Yu. I., “O skorosti perenosa energii volnami. I; II”, Zh. tekhn. fiz., 50:2 (1980), 241–251; 3, 449–460

[41] Achenbach J. D., Wave propagation in elastic solids, North Holland Publ. Co., Amsterdam, 1973 | Zbl

[42] Graff K. F., Wave motion in elastic solids, Dover Publ., New York, 1975

[43] Poruchikov V. B., Metody dinamicheskoi teorii uprugosti, Nauka, M., 1986 | MR

[44] Sretenskii L. N., Teoriya volnovykh dvizhenii zhidkosti, Nauka, M., 1977

[45] Landau L. D., Lifshits E. M., Gidrodinamika, Nauka, M., 1986 | MR