Abstract
In this paper, we study the propagation of Rayleigh waves in a nonlocal orthotropic elastic half-space subject to impedance boundary conditions using the weakly nonlocal elasticity model which is introduced recently by the first two authors of the paper. The half-space may be compressible or incompressible. Our main aim is to derive explicit secular equations of Rayleigh waves and based on them examine the effect of nonlocality, impedance boundary conditions and incompressibility on the Rayleigh wave characteristics, such as the velocity and the displacement components at the half-space surface. The secular equation for the compressible half-space is derived using the surface impedance matrix method. The secular equation for the incompressible half-space is then obtained by employing the incompressible limit method. It is shown numerically that: (i) While the Rayleigh wave velocity decreases when increasing the nonlocal and impedance parameters, the incompressibility makes it increasing. (ii) The nonlocality affects very strongly the displacement amplitudes at the half-space’s surface, the incompressibility and the impedance boundary conditions affect them considerably.
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Abd-Alla, A.M., Abo-Dahab, S.M., Ahmed, S.M., Rashid, M.M.: Effect of magnetic field and voids on Rayleigh waves in a nonlocal thermoelastic half-space. J. Strain Anal. Eng. Des. (2021). https://doi.org/10.1177/03093247211001243
Anh, V.T.N., Vinh, P.C.: Expressions of nonlocal quantities and application to Stoneley waves in weakly nonlocal orthotropic elastic half-spaces. Math. Mech. Solids (2023). https://doi.org/10.1177/10812865231164332
Biswas, B.: Rayleigh waves in a nonlocal thermoelastic layer lying over a nonlocal thermoelastic half-space. Acta Mech. 231, 4129–4144 (2020)
Chadwick, P.: The existence of pure surface modes in elastic materials with orthohombic symmetry. J. Sound. Vib. 47, 39–52 (1976)
Chebakov, R., Kaplunov, J., Rogerson, G.A.: Refined boundary conditions on the free surface of an elastic half-space taking into account non-local effects. Proc. R. Soc. A-Math. Phys. Eng. Sci. 472(2186), 20150800 (2016)
Collet, B., Destrade, M.: Explicit secular equations for Piezoacoustic surface waves: shear-horizontal modes. J. Acoust. Soc. Am. 116, 3432–3442 (2020)
Destrade, M.: The explicit secular equation for surface waves in monoclinic elastic crystals. J. Acoust. Soc. Am. 109, 1398–1402 (2001)
Duffy, D.G.: Green’s functions with applications. CRC Press,Tayor & Francis Group, NW (2015)
Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)
Godoy, G., Durn, M., Ndlec, J.-C.: On the existence of surface waves in an 432 elastic half-space with impedance boundary conditions. Wave Motion 49, 585–594 (2012)
Iijima, S.: Helical microtubules of graphitic carbon. Nature 354, 56–58 (1991)
Kaplunov, J., Prikazchikov, D.A., Prikazchikova, L.: On non-locally elastic Rayleigh wave. Phil. Trans. R. Soc. A 380(2231), 20210387 (2022)
Kaplunov, J., Prikazchikov, D.A., Prikazchikova, L.: On integral and differential formulations in nonlocal elasticity. Eur. J. Mech. A Solids 104497 (2022)
Kaur, G., Singh, D., Tomer, S.K.: Rayleigh-type wave in a nonlocal elastic solid with voids. Eur. J. Mech. A Solids 71, 134–150 (2018)
Kaur, B., Singh, B.: Rayleigh-type surface wave in nonlocal isotropic diffusive materials. Acta Mech. 232, 3407–3416 (2021)
Khurana, A., Tomar, S.K.: Rayleigh-type waves in nonlocal micropolar solid half-space. Ultrasonics 73, 162–168 (2017)
Lata, P., Singh, S.: Rayleigh wave propagation in a nonlocal isotropic magneto-thermoelastic solid with multi-dual-phase lag heat transfer. GEM-Int. J. Geomath. 13, 5 (2022). https://doi.org/10.1007/s13137-022-00195-5
Melnikov, Y.A., Melnikov, M.Y.: Green’s Functions: Construction and Applications, p. 10785. Walter de Gruyter GmbH & Co KG, Berlin/Boston (2012)
Nobili, A., Prikazchikov, D.A.: Explicit formulation for the Rayleigh wave field induced by surface stresses in an orthorhombic half-plane. Eur. J. Mech. A Solids 70, 86–94 (2018)
Ogden, R.W., Vinh, P.C.: On Rayleigh waves in incompressible orthotropic elastic solids. J. Acoust. Soc. Am. 115, 530–533 (2004)
Pramanik, A.S., Biswas, S.: Surface waves in nonlocal thermoelastic medium with state space approach. J. Therm. Stresses 43, 667–686 (2020)
Prikazchikov, D.A.: Rayleigh waves of arbitrary profile in anisotropic media. Mech. Res. Commun. 50, 83–86 (2013)
Rayleigh, L.: On waves propagated along the plane surface of an elastic solid. Proc. R. Soc. Lond. A 17, 4–11 (1885)
Romano, G., Barretta, R., Diaco, M., de Sciarra, F.M.: Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams. Int. J. Mech. Sci. 121, 151–156 (2017)
Singh, K.: Rayleigh waves with impedance boundary conditions in a nonlocal micropolar thermoelastic material. J. Phys. Conf. Ser. 1531, 012048 (2020). https://doi.org/10.1088/1742-6596/1531/1/012048
Singh, B.: Propagation of waves in an incompressible rotating transversely isotropic nonlocal solid. Vietnam J. Mech. 43, 237–252 (2021)
Stoneley, R.: The propagation of surface waves in an elastic medium with orthohombic symmetry. Geophys. J. Int. 8, 176–186 (1963)
Taziev, R.M.: Dispersion relation for acoustic waves in an anisotropic elastic half-space. Sov. Phys. Acoust. 35, 535–538 (1989)
Ting, T.C.T.: Anisotropic Elasticity: Theory and Applications. Oxford University Press, New York (1996)
Ting, T.C.T.: Explicit secular equations for surface waves in monoclinic materials with the symmetry plane at \(x_1=0, x_2=0\) or \(x_3=0\). Proc. R. Soc. Lond. A 458, 1017–1031 (2002)
Tung, D.X.: Wave propagation in nonlocal orthotropic micropolar elastic solids. Arch. Mech. 73, 237–251 (2021)
Tung, D.X.: Surface waves in nonlocal transversely isotropic liquid-saturated porous solid. Arch. Appl. Mech. 91, 2881–2892 (2021)
Vinh, P.C.: Explicit secular equations of Rayleigh waves in elastic media under the influence of gravity and initial stress. Appl. Math. Compt. 215, 395–404 (2009)
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)
Vinh, P.C., Hue, T.T.T.: Rayleigh waves with impedance boundary conditions in anisotropic solids. Wave Motion 51, 1082–1092 (2014)
Vinh, P.C., Hue, T.T.T.: Rayleigh waves with impedance boundary conditions in incompressible anisotropic half-spaces. Int. J. Eng. Sci. 85, 175–185 (2014)
Vinh, P.C., Seriani, G.: Explicit secular equations of Stoneley waves in a non-homogeneous orthotropic elastic medium under the influence of gravity. Appl. Math. Compt. 215, 3515–3525 (2010)
White, R.M., Voltmer, F.M.: Direct piezoelectric coupling to surface elastic waves. Appl. Phys. Lett. 7, 314–316 (1965)
Yan, J.W., Liew, K.M., He, L.H.: A higher-order gradient theory for modeling of the vibration behavior of single-wall carbon nanocones. Appl. Math. Model. 38, 2946–2960 (2014)
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Vu Thi Ngoc Anh was funded by Vingroup JSC and supported by the Postdoctoral Scholarship Programme of Vingroup Innovation Foundation (VINIF), Institute of Big Data, code VINIF.2022.STS.42.
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Anh, V.T.N., Vinh, P.C., Tuan, T.T. et al. Weakly nonlocal Rayleigh waves with impedance boundary conditions. Continuum Mech. Thermodyn. 35, 2081–2094 (2023). https://doi.org/10.1007/s00161-023-01235-7
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DOI: https://doi.org/10.1007/s00161-023-01235-7