Shear horizontal waves in a nonlinear elastic layer overlying a rigid substratum

Abstract

In this work, the propagation of shear horizontal (SH) waves in a homogeneous, isotropic and compressible nonlinear hyper-elastic layer having
finite thickness is studied. The upper surface of the layer is assumed to be free from traction and the lower boundary is rigidly fixed. These waves are dispersive like the Love waves. The problem is examined by a perturbation method that balances the nonlinearity and dispersion in the analysis. A nonlinear Schrödinger equation is derived describing the nonlinear self modulation of the waves. Then, the effect of nonlinear
properties of the material on the propagation characteristics and on the existence of solitary waves are discussed.

Keywords

• Dispersive nonlinear shear horizontal waves,Multiple scale method,Bright solitary waves,Dark solitary waves

References

1. Ewing, W.M., Jardetsky, W.S., Press, F. Elastic Waves in Layered Media, McGraw-Hill, New York, 1957.
2. Love, A. E. H. Some Problems of Geodynamics, Cambridge University Press, Cambridge, 1911. of seismology. Cambridge University Press, Cambridge, 1963.
3. Achenbach, J.D. Wave Propagation in Elastic Solids, North Holland Publishing Co., Ams- terdam, 1973.
4. Graf,K.F. Wave Motion in Elastic Solids, Clarendon Press, Oxford, 1975.
5. Farnell, G.W. Types and properties of surface waves, In: A.A. Oliner (Ed.), Acoustic Surface Waves, pp. 13-60, Springer, Berlin, 1978.
6. Maugin, G.A. Elastic surface waves with transverse horizontal polarization, In: Hutchinson, J.W. (Ed.) Advances in Applied Mechanics. 23, pp. 373-434. Academic Press, New York, 1983.
7. Parker, D.F., Maugin, G.A. (Eds.). Recent Developments in Surface Acoustic Waves, Springer Series on Wave Phenomena, Vol. 7., Springer-Verlag, Berlin, 1988.
8. Maugin, G.A. Physical and mathematical models of nonlinear waves in solids, In: Jerey, A., Engelbrecht, J. (Eds.) Nonlinear Waves in Solids, International Centre for Mechanical Sciences, Course and Lectures-No.341, pp. 109-233, Springer-Verlag, New York, 1994.

9. Parker, D.F. Nonlinear surface acoustic waves and waves on stratied media, In: Jerey, A., Engelbrecht, J. (Eds.) Nonlinear Waves in Solids, International Centre for Mechanical Sciences, Course and Lectures-No.341, pp. 289-347. Springer-Verlag, New York, 1994.
10. Mayer, A.P. Surface Acoustic waves in nonlinear elastic media, Physics Reports, 256, 4-5, 1995.
11. Norris, A. Finite Amplitude Waves in Solids, In: M.F. Hamilton and D.T. Blackstock (Eds.), Nonlinear Acoustics, Chap. 9, pp.263-277. Academic Press, San Diego, 1998.
12. Porubov, A.V. Amplication of Nonlinear Strain Waves in Solids, World Scientic, Singa- pore, 2003.
13. Bataille K., Lund, F. Nonlinear waves in elastic media, Physica D 6, pp. 95-104, 1982.
14. Teymur, M. Nonlinear modulation of Love waves in a compressible hyperelastic layered half space, Int. J. Eng. Sci. 26, 907-927, 1988.
15. Maugin, G.A., Hadouaj, H. Solitary surface transverse waves on an elastic substrate coated with a thin lm, Phy. Rev. B. 44 (3), 1266-1280, 1991.
16. Teymur, M., Demirci, A., Ahmetolan, S. Propagation of surface SH waves on a half space covered by a nonlinear thin layer, Int. J. Eng. Sci. 85, 150-162, 2014.
17. Ahmetolan, S., Teymur, M. Non-linear modulation of SH waves in a two-layered plate and formation of surface SH waves, Int. J. Non Linear Mech. 38, 1237-1250, 2003.
18. Ahmetolan, S., Teymur, M. Nonlinear modulation of SH waves in an incompressible hyperelastic plate, Z. angew. Math. Phy. 58, 457-474, 2007.
19. Teymur, M. Small but nite amplitude waves in a two-layered incompressible elastic medium, Int. J. Eng. Sci., 34, 227-241, 1996.
20. Ferreira, E.R., Boulanger, Ph. Large Amplitude Love Waves, Q. Jl. Mech. Appl. Math. 61 (3), 353-371, 2008.
21. Pucci, E., Saccomandi, G. Secondary Motions Associated With Anti-Plane Shear in Nonlinear Isotropic Elasticity, Q. Jl. Mech. Appl. Math. 66, 221-239, 2013.
22. Kramer, S.L. Geotechnical Earthquake Engineering, Prentice Hall, New Jersey, 1996.
23. Eringen, A.C., Suhubi, E.S. Elastodynamics, Vol. I. Academic Press, New York, 1974.
24. Saccomandi,G., Ogden, R.W. (Eds.)Mechanics and Thermomechanics of Rubberlike Solids, CISM Lecture Notes 452, Springer, New York, 2004.
25. Carroll, M.M. Some results on nite amplitude elastic waves, Acta Mech., 3, 167-181, 1967.
26. Pucci,E., Saccomandi,G. The Anti-Plane Shear Problem in Nonlinear Elasticity-Revisited, J. Elast., 113, 167-177, 2013.
27. Rogers, C., Saccomandi,G., Vergori, L. Carroll-type deformations in nonlinear elastodynamics, J. Phys. A: Math. Theor. 47, 205204, (17pp), 2014.
28. Jerey, A., Kawahara, T. Asymptotic Methods in Nonlinear Wave Theory, Pitman, Boston, 1981.
29. Hudson, J.A. Love waves in a heterogeneous medium, R. Astr. Soc. The Geophys. Jou. 6, 131-147, 1962
30. Dodd, R.K, Eilbeck J.C., Gibbon, J.D., Morris, H.C. Solitons and Nonlinear Wave Equa- tions, Academic Press, London, 1982.
31. Ablowitz, M.J., Clarkson, P.A. Solitons, Nonlinear Evolution Equations and Inverse Scat- tering, Cambridge University Press, Cambridge, 1991.
32. Peregrine, D.H. Water Waves, Nonlinear Schrödinger Equations and Their Solutions, J. Aust. Math. Soc. Ser. B. 25, 16-43, 1983.
33. Engin, H., Askar A. Cakmak, A.S. An Analytic Method for Strong motion Studies in Layered Media, Int. J.Non Linear Mech. 16, 165-186, 1981.
34. Dimitriu, P.P. Experiments on Nonlinear Eects in Ground-Surface Vibration, In Parker, D.F., Maugin, G.A. (Eds.), Recent Developments in Surface Acoustic Waves, pp.72-78, Springer Series on Wave Phenomena, Vol. 7. Springer-Verlag, Berlin, 1988.