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Implicit and explicit secular equations for Rayleigh waves in two-dimensional anisotropic media. (English) Zbl 1454.74071
Summary: This paper is concerned with the derivation of implicit and explicit secular equations for Rayleigh waves polarized in a plane of symmetry of an anisotropic linear elastic medium. It has been confirmed, in accord with T. C. T. Ting’s paper [in: Surface waves in anisotropic and laminated bodies and defects detection. Proceedings of the NATO advanced research workshop, 2002. Dordrecht: Kluwer Academic Publishers, 95–116 (2004; Zbl 1183.74121)], that the Rayleigh waves propagate with no geometric dispersion. Numerical evaluations of both the implicit and explicit equations give the same values of Rayleigh wave velocities. In the case of orthotropic material (thin composites) it has been found that Rayleigh wave velocity depends significantly (as with bulk waves) on the directions of principal material axes. For the same material model the analytical solutions, based on implicit and explicit secular equations, were compared against the finite element and experimental data that had been published by J. Cerv et al. in [“Influence of principal material directions of thin orthotropic structures on Rayleigh-edge wave velocity”, Compos. Struct. 92, No. 2, 568–577 (2010; doi:10.1016/j.compstruct.2009.09.001)]. It emerged that the theory was in accordance with the experiment.

MSC:
74J15 Surface waves in solid mechanics
74E10 Anisotropy in solid mechanics
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[1] Rayleigh, J. W.S., On waves propagated along the plane surface of an elastic solid, Proc. Lond. Math. Soc., 17, 4-11 (1885) · JFM 17.0962.01
[2] Ting, T. C.T., Explicit secular equations for surface waves in an anisotropic elastic half-space from Rayleigh to today, (Goldstein, R. W.; Maugin, G. A., Surface Waves in Anisotropic and Laminated Bodies and Defects Detection (2004), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 95-116 · Zbl 1183.74121
[3] Favretto-Cristini, N.; Komatitsch, D.; Carcione, J. M.; Cavallini, F., Elastic surface waves in crystals. Part 1: review of the physics, Ultrasonics, 51, 653-660 (2011)
[4] Cerv, J.; Kroupa, T.; Trnka, J., Influence of principal material directions of thin orthotropic structures on Rayleigh-edge wave velocity, Compos. Struct., 92, 568-577 (2010)
[5] Hess, P., Surface acoustic waves in material science, Phys. Today, 55, 42-47 (2002)
[6] Cerv, J., The influence of material properties on the behaviour of Rayleigh-edge waves in thin orthotropic media, Inter. Rev. Mech. Eng. (IREME), 2, 762-772 (2008)
[7] Stroh, A. N., Dislocations and cracks in anisotropic elasticity, Phil. Mag., 3, 625-646 (1958) · Zbl 0080.23505
[8] Synge, J. L., Elastic waves in anisotropic media, J. Math. Phys., 35, 323-334 (1956) · Zbl 0080.39003
[9] Stroh, A. N., Steady state problems in anisotropic elasticity, J. Math. Phys., 41, 77-103 (1962) · Zbl 0112.16804
[10] Stoneley, R., The seismological implications of aelotropy in continental structure, Mon. Not. R. Astron. Soc., Geophys. Suppl., 5, 343-353 (1949) · Zbl 0039.24004
[11] Alshits, V. I.; Lothe, J., On surface waves in hexagonal crystals, Sov. Phys. Crystallogr., 23, 509-515 (1978)
[12] Stoneley, R., The propagation of surface elastic waves in a cubic crystal, Proc. R. Soc. Lond. Ser. A, 232, 447-458 (1955) · Zbl 0067.23604
[13] Sveklo, V. A., Plane waves and Rayleigh waves in anisotropic media, Dokl. Akad. Nauk SSSR, 59, 871-874 (1948) · Zbl 0031.04302
[14] Stoneley, R., The propagation of surface waves in an elastic medium with orthorhombic symmetry, Geophys. J. R. Astron. Soc., 8, 176-186 (1963)
[15] Currie, P. K., The secular equation for Rayleigh waves on elastic crystal, Quart. J. Mech. Appl. Math., 32, 163-173 (1979) · Zbl 0417.73036
[16] Destrade, M., The explicit secular equation for surface acoustic waves in monoclinic elastic crystals, J. Acoust. Soc. Am., 109, 1398-1402 (2001)
[17] Ting, T. C.T., Explicit secular equations for surface waves in monoclinic materials with the symmetry plane \(x_1 = 0, x_2 = 0\) or \(x_3 = 0\), Proc. R. Soc. Lond. Ser. A, 458, 1017-1031 (2002) · Zbl 1031.74033
[18] Ting, T. C.T., An explicit secular equation for surface waves in an elastic material of general anisotropy, Quart. J. Mech. Appl. Math., 55, 297-311 (2002) · Zbl 1062.74023
[19] Ting, T. C.T., A unified formalism for elastostatics or steady state motion of compressible or incompressible anisotropic elastic materials, Int. J. Solids Struct., 39, 5427-5445 (2002) · Zbl 1044.74005
[20] Taziev, R. M., Dispersion relation for acoustic waves in an anisotropic elastic half-space, Sov. Phys.—Acoust., 35, 535-538 (1989)
[21] Destrade, M., Rayleigh waves in symmetry planes of crystals: explicit secular equations and some explicit wave speeds, Mech. Mater., 35, 931-939 (2003)
[22] Ting, T. C.T., The polarization vector and secular equation for surface waves in an anisotropic elastic half-space, Int. J. Solids Struct., 41, 2065-2083 (2004) · Zbl 1106.74358
[23] Ohyoshi, T., The propagation of Rayleigh waves along an obliquely cut surface in a directional fibre-reinforced composite, Compos. Sci. Technol., 60, 2191-2196 (2000)
[24] Ting, T. C.T., Anisotropic Elasticity (1996), Oxford University Press: Oxford University Press New-York · Zbl 0883.73001
[25] Tignol, J.-P., Galois’ Theory of Algebraic Equations (2001), World Scientific Publishing Co.
[26] Jones, R. M., Mechanics of Composite Materials (1975), Scripta Book Company: Scripta Book Company Washington, DC
[27] Destrade, M.; Fu, Y. B., The speed of interfacial waves polarized in a symmetry plane, Int. J. Eng. Sci., 44, 26-36 (2006) · Zbl 1213.74166
[28] Destrade, M., Elastic interface acoustic waves in twinned crystals, Int. J. Solids Struct., 40, 7375-7383 (2003) · Zbl 1063.74056
[29] Ingebrigtsen, K. A.; Tonning, A., Elastic surface waves in crystal, Phys. Rev., 184, 942-951 (1969)
[30] Schwarz, S., Foundations of a theory of equations (1958), Publish. House of Czechoslovak Academy of Sciences: Publish. House of Czechoslovak Academy of Sciences Prague, (in Slovak)
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