On the existence of weakly nonlocal Rayleigh waves with impedance boundary condition

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Authors

  • P.C. Vinh Faculty of Mathematics, Mechanics and Informatics, VNU University of Science, Vietnam
  • V.T.N. Anh Faculty of Mathematics, Mechanics and Informatics, VNU University of Science, Vietnam
  • T.T. Tuan Faculty of Mathematics, Mechanics and Informatics, VNU University of Science, Vietnam

Abstract

In this paper, the existence of Rayleigh waves propagating in weakly nonlocal incompressible isotropic elastic half-spaces subject to the tangential impedance boundary condition (TIBC) (at the surface of half-spaces, the tangential stress is proportional to the horizontal displacement and the normal stress is zero) is investigated. It is shown that for the negative values of the dimensionless tangential impedance parameter and the values of the dimensionless nonlocality parameter belong to the interval (0, 0.5), there exist exactly two Rayleigh waves. The first wave is the counterpart of the Rayleigh wave in local incompressible isotropic elastic half-spaces and the second is a new Rayleigh mode appearing due to the presence of nonlocality. Formulae for their velocities have been derived. Remarkably, the second wave can travel with very high velocity.

Keywords:

Rayleigh waves, weakly nonlocal elasticity, incompressible, impedance boundary condition, existence of Rayleigh waves

References


  1. L. Rayleigh, On waves propagating along the plane surface of an elastic solid, Proceedings of the Royal Mathematical London Society, A 17, 4–11, 1885.

  2. R.M. White, F.M. Voltmer, Direct piezoelectric coupling to surface elastic waves, Applied Physics Letters, 7, 314–316, 1965.

  3. A.C. Eringen, Nonlocal Continuum Field Theories, Springer, New York, 2002.

  4. S. Iijima, Helical microtubules of graphitic carbon, Nature, 354, 56–58, 1991.

  5. J.W. Yan, K.M. Liew, L.H. He, A higher-order gradient theory for modeling of the vibration behavior of single-wall carbon nanocones, Applied Mathematical Modelling, 38, 2946–2960, 2014.

  6. A.C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics, 54, 4703–4710, 1983.

  7. B. Singh, Propagation of waves in an incompressible rotating transversely isotropic nonlocal solid, Vietnam Journal of Mechanics, 43, 237–252, 2021.

  8. A.S. Pramanik, S. Biswas, Surface waves in nonlocal thermoelastic medium with state space approach, Journal of Thermal Stresses, 43, 667–686, 2020.

  9. A.M. Abd-Alla, S.M. Abo-Dahab, S.M. Ahmed, M.M. Rashid, Effect of magnetic field and voids on Rayleigh waves in a nonlocal thermoelastic half-space, The Journal of Strain Analysis for Engineering Design, 57, 61–72, 2022.

  10. A. Khurana, S.K. Tomar, Rayleigh-type waves in nonlocal micropolar solid half-space, Ultrasonics, 73, 162–168, 2017.

  11. K. Singh, S. Sawhney, Rayleigh waves with impedance boundary conditions in a nonlocal micropolar thermoelastic material, Journal of Physics: Conference Series, 1531, 012048, 2020, https://doi.org/10.1088/1742-6596/1531/1/012048.

  12. L.H. Tong, S.K. Lai, L.L. Zeng, C.J. Xu, J. Yang, Nonlocal scale effect on Rayleigh wave propagation in porous fluid-saturated materials, International Journal of Mechanical Sciences, 148, 459–466, 2018.

  13. G. Kaur, D. Singh, S.K. Tomar, Rayleigh-type wave in a nonlocal elastic solid with voids, European Journal of Mechanics A/Solids, 71, 134–150, 2018.

  14. B. Kaur, B. Singh, Rayleigh-type surface wave in nonlocal isotropic diffusive materials, Acta Mechanica, 232, 3407–3416, 2021.

  15. S. Biswas, Rayleigh waves in a nonlocal thermoelastic layer lying over a nonlocal thermoelastic half-space, Acta Mechanica, 231, 4129–4144, 2020.

  16. P. Lata, S. Singh, Rayleigh wave propagation in a nonlocal isotropic magneto-thermoelastic solid with multi-dual-phase lag heat transfer, GEM-International Journal on Geomathematics, 13, 5, 2022, https://doi.org/10.1007/s13137-022-00195-5.

  17. G. Romano, R. Barretta, M. Diaco, F.M. de Sciarra, Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams, International Journal of Mechanical Sciences, 121, 151–156, 2017.

  18. J. Kaplunov, D.A. Prikazchikov, L. Prikazchikova, On non-locally elastic Rayleigh wave, Philosophical Transactions of the Royal Society A, 380, 20210387, 2022.

  19. J. Kaplunov, D.A. Prikazchikov, L. Prikazchikova, On integral and differential formulations in nonlocal elasticity, European Journal of Mechanics A/Solids, 100, 104497, 2023.

  20. P.C. Vinh, V.T.N. Anh, On the well-posedness of Eringen’s nonlocal elasticity for harmonic plane wave problems, Proceedings of the Royal Society A, 480 (2293), 20230814, 2024.

  21. R. Chebakov, J. Kaplunov, G.A. Rogerson, Refined boundary conditions on the free surface of an elastic half-space taking into account non-local effects, Proceedings of the Royal Society A, 472 (2186), 20150800, 2016, https://doi.org/10.1098/rspa.2015.0800.

  22. V.T.N. Anh, P.C. Vinh, Expressions of nonlocal quantities and application to Stoneley waves in weakly nonlocal orthotropic elastic half-spaces, Mathematics and Mechanics of Solids, 28, 2420–2435, 2023.

  23. V.T.N. Anh, P.C. Vinh, T.T. Tuan, L.T. Hue, Weakly nonlocal Rayleigh waves with impedance boundary conditions, Continuum Mechanics and Thermodynamics, 35, 2081–2094, 2023.

  24. V.T.N. Anh, P.C. Vinh, The incompressible limit method and Rayleigh waves in incompressible layered nonlocal orthotropic elastic media, Acta Mechanica, 234, 403–421, 2023.

  25. V.T.N. Anh, P.C. Vinh, T.T. Tuan, Transfer matrix for a weakly nonlocal elastic layer and Lamb waves in layered nonlocal composite plates, Mathematics and Mechanics of Solids, 2024, in press, https://doi.org/10.1177/10812865241258377.

  26. D.M. Barnett, J. Lothe, Free surface (Rayleigh) waves in anisotropic elastic half-spaces: the surface impedance method, Proceedings of the Royal Society of London A, 402, 135–152, 1985.

  27. A. Mielke, Y.B. Fu, Uniqueness of the surface-wave speed: a proof that is independent of the Stroh formalism, Mathematics and Mechanics of Solids, 9, 5–15, 2004.

  28. P.C. Vinh, V.T.N. Anh, Q.H. Dinh, The non-unique existence of Rayleigh waves in nonlocal elastic half-spaces, Zeitschrift für angewandte Mathematik und Physik, 74, 120, 2023.

  29. N.I. Muskhelishviili, Singular Intergral Equations, Noordhoff, Groningen, 1953.

  30. P. Henrici, Applied and Computational Complex Analysis, Vol. III, Wiley, New York, 1986.

  31. E. Godoy, M. Durn, J.-C. Ndlec, On the existence of surface waves in an elastic half-space with impedance boundary conditions, Wave Motion, 49, 585–594, 2012.

  32. P.C. Vinh, T.T.T. Hue, Rayleigh waves with impedance boundary conditions in anisotropic solids, Wave Motion, 51, 1082–1092, 2014.

  33. P.C. Vinh, T.T.T. Hue, Rayleigh waves with impedance boundary conditions in incompressible anisotropic half-spaces, International Journal of Engineering Science, 85, 175–185, 2014.

  34. B. Singh and B. Kaur, Propagation of Rayleigh waves in an incompressible rotating orthotropic elastic solid half-space with impedance boundary conditions, Journal of the Mathematical Behaviour of Biomedical Materials, 26, 73–78, 2017.

  35. B. Singh, B. Kaur, Rayleigh-type surface wave on a rotating orthotropic elastic half-space with impedance boundary conditions, Journal of Vibration and Control, 26, 1980–1987, 2020.

  36. B. Kaur, B. Singh, Rayleigh waves on the impedance boundary of a rotating monoclinic half-space, Acta Mechanica, 232, 2479–2491, 2021.

  37. B. Collet, M. Destrade, Explicit secular equations for piezoacoustic surface waves: Shear-horizontal modes, Journal of Acoustical Society of America, 116, 3432–3442, 2004.

  38. P.C. Vinh, N.Q. Xuan, Rayleigh waves with impedance boundary condition: Formula for the velocity, Existence and Uniqueness, European Journal of Mechanics A/Solids, 61, 180–185, 2017.

  39. P.T.H. Giang, P.C. Vinh, Existence and uniqueness of Rayleigh waves with normal impedance boundary conditions and formula for the wave velocity, Journal of Engineering Mathematics, 130, 13, 2021.

  40. D. Nkemzi, A new formula for the velocity of Rayleigh waves, Wave Motion, 26, 199–205, 1997.

  41. M. Romeo, Non-dispersive and dispersive electromagnetoacoustic SH surface modes in piezoelectric media, Wave Motion, 39, 93–110, 2004.

  42. P.C. Vinh, P.T.H. Giang, On formulas for the velocity of Stoneley waves propagating along the loosely bonded interface of two elastic half-spaces, Wave Motion, 48, 646–656, 2011.

  43. P.C. Vinh, P.G. Malischewsky, P.T.H. Giang, Formulas for the speed and slowness of Stoneley waves in bonded isotropic elastic half-spaces with the same bulk wave velocities, International Journal of Engineering Science, 60, 53–58, 2012.

  44. P.C. Vinh, Scholte-wave velocity formulae, Wave Motion, 50, 2, 180–190, 2013.

  45. P.T.H. Giang, P.C. Vinh, V.T.N. Anh, Formulas for the slowness of Stoneley waves with sliding contact, Archives of Mechanics, 72, 465–481, 2020.

  46. P.T.H. Giang, P.C. Vinh, T.T. Tuan, V.T.N. Anh, Electromagnetoacoustic SH waves: Formulas for the velocity, existence and uniqueness, Wave Motion, 105, 102757, 2021.

  47. P.C. Vinh, V.T.N. Anh, N.T.K. Linh, Exact secular equations of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer, International Journal of Solids and Structures, 83, 65–72, 2016.

  48. N.I. Muskhelishviili, Some Basic Problems of Mathematical Theory of Elasticity, Noordhoff, Netherlands, 1963.