Rayleigh-type wave propagation in incompressible visco-elastic media under initial stress

doi:10.1007/s10483-018-2306-9

Abstract

Abstract:

Propagation of Rayleigh-type surface waves in an incompressible visco-elastic material over incompressible visco-elastic semi-infinite media under the effect of initial stresses is discussed. The dispersion equation is determined to study the effect of different types of parameters such as inhomogeneity, initial stress, wave number, phase velocity, damping factor, visco-elasticity, and incompressibility on the Rayleigh-type wave propagation. It is found that the affecting parameters have a significant effect on the wave propagation. Cardano's and Ferrari's methods are deployed to estimate the roots of differential equations associated with layer and semi-infinite media. The MATHEMATICA software is applied to explicate the effect of these parameters graphically.

Key words: Rayleigh-type wave, incompressible, inhomogeneity, normal form theory, singularity theory, universal unfolding, transition set, strongly nonlinear Duffing system, visco-elasticity, initial stress

2010 MSC Number:

74J15

P. SINGH, A. CHATTOPADHYAY, A. K. SINGH. Rayleigh-type wave propagation in incompressible visco-elastic media under initial stress. Applied Mathematics and Mechanics (English Edition), 2018, 39(3): 317-334.

TrendMD

References

[1] Biot, M. A. Mechanics of Incremental Deformations, John Wiley, New York (1965)
[2] Achenbach, J. Wave Propagation in Elastic Solids, Elsevier, New York (1973)
[3] Stoneley, R. The transmission of Rayleigh waves in a heterogeneous medium. Geophysical Journal International, 3, 222-232(1934)
[4] Dutta, S. Rayleigh wave propagation in a two-layer anisotropic media. Pure and Applied Geophysics, 60, 51-60(1965)
[5] Chattopadhyay, A. Propagation of SH waves in a viscoelastic medium due to irregularity in the crustal layer. Bulletin of Calcutta Mathematical Society, 70, 303-316(1978)
[6] Dey, S., Chattopadhyay, A., and Chakraborty, M. Effect of initial stresses on reflection on transmission of seismic-wave at the Earth core-mantle boundary. Archive of Mechanics, 34, 61-72(1982)
[7] Pal, A. K. and Chattopadhyay, A. The reflection phenomena of plane waves at a free boundary in a prestressed elastic half-space. Journal of Acoustical Society of America, 76, 924-925(1984)
[8] Chattopadhyay, A., Mahata, N. P., and Keshri, A. Rayleigh waves in a medium under initial stresses. Acta Geophysica Polonica, 34, 57-62(1986)
[9] Sharma, M. D. and Gogna, M. L. Seismic wave propagation in a viscoelastic porous solid saturated by viscous liquid. Pure and Applied Geophysics, 135(3), 383-400(1991)
[10] Fu, Y. and Rogerson, G. A. A nonlinear analysis of instability of a pre-stressed incompressible elastic plate. In Proceedings of the Royal Society of London A:Mathematical, Physical and Engineering Sciences, 446, 233-254(1994)
[11] Rogerson, G. A. and Fu, Y. B. An asymptotic analysis of the dispersion relation of a pre-stressed incompressible elastic plate. Acta Mechanica, 111, 59-74(1995)
[12] Abd-Alla, A. M., Hammad, H. A. H., and Abo-Dahab, S. M. Rayleigh waves in a magneto elastic half-space of orthotropic material under influence of initial stress and gravity field. Applied Mathematics and Computation, 154, 583-597(2004)
[13] Abd-Alla, A. M., Abo-Dahab, S. M., Hammad, H. A., and Mahmoud, S. R. On generalized magneto-thermoelastic Rayleigh waves in a granular medium under the influence of a gravity field and initial stress. Journal of Vibration and Control, 17, 115-128(2011)
[14] Abd-Alla, A. M., Abo-Dahab, S. M., and Al-Thamali, T. A. Propagation of Rayleigh waves in a rotating orthotropic material elastic half-space under initial stress and gravity. Journal of Mechanical Science and Technology, 26, 2815-2823(2012)
[15] Sharma, M. D. Rayleigh waves in dissipative poro-viscoelastic media. Bulletin of the Seismological Society of America, 102, 2468-2483(2012)
[16] Sharma, M. D. Propagation and attenuation of Rayleigh waves in generalized thermo elastic media. Journal of Seismology, 18, 61-79(2014)
[17] Ahmed, S. M. and Abo-Dahab, S. M. Influence of initial stress and gravity field on propagation of Rayleigh and Stoneley waves in a thermoelastic orthotropic granular medium. Mathematical Problems in Engineering, 2012, 245965(2012)
[18] Ogden, R. W. and Singh, B. The effect of rotation and initial stress on the propagation of waves in a transversely isotropic elastic solid. Wave Motion, 51, 1108-1126(2014)
[19] Wang, Y. Z., Li, F. M., Kishimoto, K., Wang, Y. S., and Huang, W. H. Band gaps of elastic waves in three-dimensional piezoelectric phononic crystals with initial stress. European Journal of Mechanics-A/Solids, 29, 182-189(2010)
[20] Wang, Y., Li, F., Kishimoto, K., Wang, Y., and Huang, W. Wave localization in randomly disordered periodic piezoelectric rods with initial stress. Acta Mechanica Solida Sinica, 21, 529-535(2008)
[21] Chatterjee, M., Dhua, S., and Chattopadhyay, A. Response of moving load due to irregularity in slightly compressible, finitely deformed elastic media. Mechanics Research Communications, 66, 49-59(2015)
[22] Chatterjee, M., Dhua, S., and Chattopadhyay, A. Propagation of shear waves in viscoelastic heterogeneous layer overlying an initially stressed half space. Journal of Physics:Conference Series, 662, 012001(2015)
[23] Dhua, S. and Chattopadhyay, A. Wave propagation in heterogeneous layers of the Earth. Waves in Random and Complex Media, 26, 626-641(2016)
[24] Kumari, P., Modi, C., and Sharma, V. K. Torsional waves in a magneto-viscoelastic layer over an inhomogeneous substratum. The European Physical Journal Plus, 131, 263(2016)
[25] Kumari, P., Sharma, V. K., and Modi, C. Modeling of magnetoelastic shear waves due to point source in a viscoelastic crustal layer over an inhomogeneous viscoelastic half space. Waves in Random and Complex Media, 26, 101-120(2016)
[26] Khurana, A. and Tomar, S. K. Rayleigh-type waves in nonlocal micropolar solid half-space. Ultrasonics, 73, 162-168(2017)
[27] Chatterjee, M. and Chattopadhyay, A. Propagation, reflection and transmission of SH-waves in slightly compressible, finitely deformed elastic media. Applied Mathematics and Mechanics (English Edition), 36(8), 1045-1056(2015) https://doi.org/10.1007/s10483-015-1961-7
[28] Gubbins, D. Seismology and Plate Tectonics, Cambridge University Press, Cambridge (1990)
[29] Caloi, P. Comportement des ondes de Rayleigh dans unmilieu firmoelastique indefini (in French). Publications du Bureau Central Séismoloqique International:Travaux Scientifiques Série A, 17, 89-108(1948)