On the Rayleigh surface waves on an anisotropic homogeneous thermoelastic half-space. (English) Zbl 1401.74149

Summary: In this paper, we consider the propagation of surface waves in half-spaces made of anisotropic homogeneous thermoelastic materials. When the thermal dissipative properties of a half-space are taken into consideration, the undamped characteristic features of Rayleigh waves do not remain valid. Then, the process is irreversible and the Rayleigh waves are damped in time and dispersive. Here, we show that the Stroh formulation of the problem leads to a first-order linear partial differential system with constant coefficients. The associated characteristic equation (the propagation condition) is an eight degree equation with complex coefficients and, therefore, its solutions are complex numbers. Consequently, the secular equation results to be with complex coefficients, and therefore, the surface wave is damped in time and dispersed. The results are illustrated for the case of an orthotropic homogeneous thermoelastic half-space, when an explicit bicubic form of the characteristic equation with complex coefficients is obtained. The analysis of these Rayleigh waves in a homogeneous orthotropic half-space is numerically exemplified. Further, in the case of an isotropic homogeneous thermoelastic material, the characteristic equation is solved exactly, and the general solution of the first-order differential system follows. On this basis, the Rayleigh-type surface waves are studied, and the dispersion condition is found.


74J15 Surface waves in solid mechanics
74F05 Thermal effects in solid mechanics
74E10 Anisotropy in solid mechanics
Full Text: DOI


[1] Lord Rayleigh: On waves propagated along the plane surface of an elastic solid. Proc. Lond. Math. Soc. 17, 4–11 (1885)
[2] Stroh A.N.: Steady state problems in anisotropic elasticity. J. Math. Phys. 41, 77–103 (1962) · Zbl 0112.16804
[3] Destrade M.: Seismic Rayleigh waves on an exponentially graded, orthotropic half space. Proc. R. Soc. A 463, 495–502 (2007) · Zbl 1127.86003 · doi:10.1098/rspa.2006.1774
[4] Tanuma K.: Stroh formalism and Rayleigh waves. J. Elast. 89, 5–154 (2007) · Zbl 1127.74004 · doi:10.1007/s10659-007-9117-1
[5] Ting T.C.T.: Secular equations for Rayleigh and Stoneley waves in exponentially graded elastic materials of general anisotropy under the influence of gravity. J. Elast. 105, 331–347 (2011) · Zbl 1320.74061 · doi:10.1007/s10659-011-9314-9
[6] Ting T.C.T.: Surface waves in an exponentially graded, general anisotropic elastic material under the influence of gravity. Wave Motion 48, 335–344 (2011) · Zbl 1283.74029 · doi:10.1016/j.wavemoti.2010.12.001
[7] Norris A.N., Shuvalov A.L.: Elastodynamics of radially inhomogeneous spherically anisotropic elastic materials in the Stroh formalism. Proc. R. Soc. A 468, 467–484 (2012) · Zbl 1364.74018 · doi:10.1098/rspa.2011.0463
[8] Ting T.C.T.: Anisotropic Elasticity: Theory and Applications. Oxford University Press, New York (1996) · Zbl 0883.73001
[9] Lockett F.J.: Effect of thermal properties of a solid on the velocity of Rayleigh waves. J. Mech. Phys. Solids 7, 71–75 (1958) · Zbl 0088.17301 · doi:10.1016/0022-5096(58)90040-1
[10] Lockett F.J., Sneddon I.N.: Propagation of thermal stresses in an infinite medium. Proc. Edinb. Math. Soc. 11, 237–244 (1959) · Zbl 0094.37301 · doi:10.1017/S0013091500021970
[11] Nowacki W., Sokoł owski M.: Propagation of thermoelastic waves in plates. Arch. Mech. (Arch. Mech. Stos.) 11, 715–727 (1959) · Zbl 0092.18901
[12] Deresiewicz H.: Plane waves in a thermoelastic solid. J. Acoust. Soc. Am. 29, 204–209 (1957) · doi:10.1121/1.1908832
[13] Deresiewicz H.: Effect of boundaries on waves in a thermoelastic solid. J. Mech. Phys. Solids 8, 164–172 (1960) · Zbl 0122.19406 · doi:10.1016/0022-5096(60)90035-1
[14] Chadwick P., Sneddon I.N.: Plane waves in an elastic solid conducting heat. J. Mech. Phys. Solids 6, 223–230 (1958) · Zbl 0088.17204 · doi:10.1016/0022-5096(58)90027-9
[15] Lessen M.: Thermoelastic waves and thermal shock. J. Mech. Phys. Solids 7, 77–84 (1959) · doi:10.1016/0022-5096(59)90012-2
[16] Chadwick P.: Thermoelasticity. The dynamical theory. In: Sneddon, I.N, Hill, R. (eds) Progress in Solid Mechanics, vol. 1, pp. 263–328. North-Holland, Amsterdam (1960)
[17] Nowacki, W.: On some dynamical problems of thermoelasticity. In: Problems of Continuum Mechanics, Contributions in honor of the Seventieth Birthday of Academician N. I. Muskhelishvili, pp. 291–3305. Published by SIAM Philadelphia, Pennsylvania (1961)
[18] Achenbach J.D.: The propagation of stress discontinuities according to the coupled equations of thermoelasticity. Acta Mech. 3, 342–351 (1967) · doi:10.1007/BF01181493
[19] Puri P.: Plane waves in thermoelasticity and magnetothermoelasticity. Int. J. Eng. Sci. 10, 467–477 (1972) · Zbl 0235.73011 · doi:10.1016/0020-7225(72)90052-3
[20] Ivanov T.P.: On the propagation of thermoelastic Rayleigh waves. Wave Motion 10, 73–82 (1988) · Zbl 0626.73006 · doi:10.1016/0165-2125(88)90007-8
[21] Flavin J.N., Green A.E.: Plane thermoelastic waves in an initially stressed medium. J. Mech. Phys. Solids 9, 179–190 (1961) · Zbl 0117.19501 · doi:10.1016/0022-5096(61)90016-3
[22] Flavin J.N.: Thermoelastic Rayleigh waves in a prestressed medium. Proc. Camb. Philos. Soc. 58, 532–538 (1962) · doi:10.1017/S0305004100036811
[23] Ivanov T.P., Savova R.: Viscoelastic surface waves of an assigned wavelength. Eur. J. Mech. A/Solids 24, 305–310 (2005) · Zbl 1079.74560 · doi:10.1016/j.euromechsol.2004.11.002
[24] Ivanov T.P., Savova R.: Thermoviscoelastic surface waves of an assigned frequency. IMA J. Appl. Math. 74, 250–263 (2009) · Zbl 1169.76345 · doi:10.1093/imamat/hxn028
[25] Ivanov T.P., Savova R.: Thermoviscoelastic surface waves of an assigned wavelength. Int. J. Solids Struct. 47, 1972–1978 (2010) · Zbl 1194.74164 · doi:10.1016/j.ijsolstr.2010.04.001
[26] Savova R., Ivanov T.P.: On viscous effects in surface wave propagation. J. Theor. Appl. Mech. 37, 25–34 (2007)
[27] Chadwick P., Windle D.W.: Propagation of Rayleigh waves along isothermal and insulated boundaries. Proc. R. Soc. Lond. A 280, 47–71 (1964) · Zbl 0128.19501 · doi:10.1098/rspa.1964.0130
[28] Chakraborty S.K., Pal R.P.: Thermo-elastic Rayleigh waves in transversely isotropic solids. Pure Appl. Geophys. 76, 79–86 (1969) · doi:10.1007/BF00877839
[29] Chadwick P., Seet L.T.C.: Wave propagation in a transversely isotropic heat-conducting elastic material. Mathematika 17, 255–274 (1970) · Zbl 0248.73013 · doi:10.1112/S002557930000293X
[30] Abd-Alla A.M., Ahmed S.M.: Rayleigh waves in an orthotropic thermoelastic medium under gravity field and initial stress. Earth Moon Planets 75, 185–197 (1996) · Zbl 0882.73021 · doi:10.1007/BF02592996
[31] Lazutkin D.F., Rossikhin Y.A.: Plane harmonic waves in anisotropic thermoelastic media. Int. Appl. Mech. 10, 46–53 (1974)
[32] Banerjee D.K., Pao Y.H.: Thermoelastic waves in anisotropic solids. J. Acoust. Soc. Am. 56, 1444–1454 (1974) · Zbl 0297.73011 · doi:10.1121/1.1903463
[33] Sharma J.N., Singh H.: Thermoelastic surface waves in a transversely isotropic half-space with thermal relaxation. Indian J. Pure Appl. Math. 16, 1202–1219 (1985) · Zbl 0577.73031
[34] Abouelregal A.E.: Rayleigh waves in a thermoelastic solid half space using dual-phase-lag model. Int. J. Eng. Sci. 49, 781–791 (2011) · Zbl 1231.74202 · doi:10.1016/j.ijengsci.2011.03.007
[35] Straughan, B.: Heat Waves. In: Applied Mathematical Sciences, vol. 177, Springer, New York, Dordrecht, Heidelberg, London (2011) · Zbl 1232.80001
[36] Carlson D.E.: Linear thermoelasticity. In: Truesdell, C.A. (ed) Handbuch der Physik, vol. VIa/2, pp. 297–345. Springer, Berlin (1972)
[37] Gurtin M.E.: The linear theory of elasticity. In: Truesdell, C.A. (ed) Handbuch der Physik, vol. VIa/2, pp. 1–295. Springer, Berlin (1972)
[38] Hawwa M.A., Nayfeh A.H.: The general problem of thermoelastic waves in anisotropic periodically laminated composites. Compos. Eng. 5, 1499–1517 (1995) · doi:10.1016/0961-9526(95)00087-4
[39] Verma K.L., Hasebe N.: On the flexural and extensional thermoelastic waves in orthotropic plates with two thermal relaxation times. J. Appl. Math. 1, 69–83 (2004) · Zbl 1121.74356 · doi:10.1155/S1110757X04308041
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