Abstract
We analyze the existence and number of solutions of the boundary-value problem describing time-harmonic surface shear waves on the free boundary of a functionally graded half-space with density and shear modulus depending continuously on the semi-infinite depth coordinate. The mathematical formulation amounts to the parametric Sturm-Liouville equation on a half-line with frequency and wave number as the parameters, which is subjected to the Neumann (traction-free) condition at the origin and the condition of vanishing at infinity. The solvability requirement determines the wave number versus frequency dispersion dependence of the surface waves. We establish the criteria for non-existence of surface waves and for the existence of a finite number of surface-wave solutions, which grows and tends to infinity with growing wave number. An interesting result is a possibility of the existence of infinite number of solutions for any given wave number. These three options are conditioned by the asymptotic behavior of the shear modulus and density close to the infinite depth. The paper ends up with an overview of possible extensions of the present study in the physical and mathematical context.
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References
Achenbach, J.D., Balogun, O.: Antiplane surface waves on a half-space with depth-dependent properties. Wave Motion 47(1), 59–65 (2010)
Arnold, V.I.: The Sturm theorems and symplectic geometry. Funct. Anal. Appl. 19, 251–259 (1985)
Atkinson, F.V., Mingarelli, A.B.: Multiparameter Eigenvalue Problems. Sturm-Liouville Theory. CRC Press, Boca Raton (2011)
Babich, V.M., Kiselev, A.P.: Elastic Waves. High Frequency Theory. CRC Press, Boca Raton (2018)
Bellman, R.: Stability Theory of Differential Equations. McGraw-Hill, New York (1953)
Birman, V., Byrd, L.W.: Modeling and analysis of functionally graded materials and structures. Appl. Mech. Rev. 60(5), 195–216 (2007)
Biryukov, S.V., Gulyaev, Y.V., Krylov, V.V., Plessky, V.P.: Surface Acoustic Waves in Inhomogeneous Media. Springer, Berlin (1995)
Brekhovskikh, L.M., Godin, O.: Acoustics of Layered Media II. Springer Series on Wave Phenomena. Springer, Belin (1999)
Cerveny, V.: Seismic Ray Theory. Cambridge University Press, Cambridge (2010)
Collet, B., Destrade, M., Maugin, G.A.: Bleustein-Gulyaev waves in some functionally graded materials. Eur. J. Mech A/Solid 25(5), 695–706 (2006)
Darinskii, A.N., Shuvalov, A.L.: Surface electromagnetic waves in anisotropic superlattices. Phys. Rev. A 102, 033515 (2020)
Destrade, M.: Seismic Rayleigh waves on an exponentially graded, orthotropic half-space. Proc. Roy. Soc. A. 463(2078), 495–502 (2007)
Fedoryuk, M.V.: Asymptotic Analysis. Springer, Berlin (1993)
Hartman, Ph.: Ordinary Differential Equations. SIAM, Philadelphia (2002)
Kennett, B.L.N.: Seismic Wave Propagation in Stratified Media. Cambridge University Press, Cambridge (1983)
Shuvalov, A.L.: The high-frequency dispersion coefficient for the Rayleigh velocity in a vertically inhomogeneous anisotropic halfspace. J. Acoust. Soc. Am. 123(5), 2484–2487 (2008)
Shuvalov, A.L., Poncelet, O., Deschamps, M.: General formalism for plane guided waves in transversely inhomogeneous anisotropic plates. Wave Motion 40(4), 413–426 (2004)
Shuvalov, A.L., Poncelet, O., Golkin, S.V.: Existence and spectral properties of shear horizontal surface acoustic waves in vertically periodic half-spaces. Proc. R. Soc. A 465(2105), 1489–1511 (2009)
Ting, T.C.T.: Existence of anti-plane shear surface waves in anisotropic elastic half-space with depth-dependent material properties. Wave Motion 47(6), 350–357 (2010)
**aoshan, C., Feng, J., Kishimoto, K.: Transverse shear surface wave in a functionally graded material infinite half space. Philos. Mag. Lett. 92(5), 245–253 (2012)
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Sarychev, A., Shuvalov, A., Spadini, M. (2021). Surface Shear Waves in a Functionally Graded Half-Space. In: Mariano, P.M. (eds) Variational Views in Mechanics. Advances in Mechanics and Mathematics(), vol 46. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-90051-9_2
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DOI: https://doi.org/10.1007/978-3-030-90051-9_2
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