Dispersion of SH-Wave in a Heterogeneous Orthotropic Layer Sandwiched Between an Inhomogeneous Semi-Infinite Medium and a Heterogeneous Elastic Half-Space
الموضوعات :
R.M Prasad
1
(Department of Mathematics, S. N. Sinha College, Tekari Magadh University, Bodh-Gaya, India)
S Kundu
2
(Department of Mathematics & Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad, India)
الکلمات المفتاحية: Orthotropic medium, SH-wave, Heterogeneous half-space, Inhomogeneity,
ملخص المقالة :
The aim of this paper is to investigates the existence of the dispersion of SH-wave in a heterogeneous orthotropic layer lying over a heterogeneous elastic half-space and underlying an inhomogeneous semi-infinite medium. Hyperbolic variation in upper semi-infinite associated with directional rigidities and density has been considered while linear variation in the intermediate layer associated with initial stress, density, shear moduli and lower half-space associated with rigidity and density has been considered. The dispersion equation of SH-wave has been obtained in a closed form by using variable separation method. The effects of inhomogeneities of the assumed media are illustrated by figures using MATLAB programming. The Earth's composition is heterogeneous that incorporates extremely hard layers. The propagation of SH-wave across crustal layer of the Earth very much depends upon heterogeneity and orthotropic properties. In fact, the observation reveals that the phase velocity of SH-wave is directly proportionate to inhomogeneity parameter, orthotropic parameter and heterogeneity parameter. That means as inhomogeneity parameter and heterogeneity orthotropic parameter increases, the phase velocity of SH-wave increases proportionately. Moreover, the obtained dispersion equation of SH-wave coincides with the classical result of Love wave as initial stress, inhomogeneities, and the upper semi-infinite medium is neglected. This analysis may be helpful to expound the nature of the dispersion of seismic waves in elastic media.
[1] Shearer P.M., 2009, Introduction to Seismology, Cambridge University Press, Cambridge.
[2] Biot M.A., 1965, Mechanics of Incremental Deformation, John Wiley and Sons, New York.
[3] Gubbins D., 1990, Seismology and Plate Tectonics, Cambridge University Press, Cambridge,New york.
[4] Ewing W.M., Jardetzky W.S., Press F., 1957, Elastic Wave in Layered Media, Mcgraw-Hill Book Company, New York.
[5] Son M.S., Kang Y.J., 2012, Shear wave propagation in a layered poroelastic structure, Wave Motion 49: 490-500.
[6] Kalyani V.K., Sinha A., Pallavika, Chakraborty S.K., Mahanti N.C., 2008, Finite difference modeling of seismic wave propagation in monoclinic media, Acta Geophysica 56(4): 1074-1089.
[7] Pal P.C., Mandal D., 2014, Generation of SH-type waves due to shearing stress discontinuity in a sandy layer overlying an isotropic and inhomogeneous elastic half-space, Acta Geophysica 62(1): 44-58.
[8] Chattopadhyay A., Gupta S., Sharma V.K., Kumari P., 2008, Propagation of SH-waves in an irregular monoclinic crustal layer, Archive of Applied Mechanics 78(12): 989-999.
[9] Chattopadhyay A., Gupta S., Singh A.K., Sahu S.A., 2010, Propagation of SH-waves in an irregular non homogeneous monoclinic crustal layer over a semi-infinite monoclinic medium, Applied Mathematical Sciences 4(44): 2157-2170.
[10] Ding G., Darvinski M., 1998, Scattering of SH-waves in multi-layered media with irregular interfaces, Earthquake Engineering and Structural Dynamics 25(12): 1391-1404.
[11] Kaur J., Tomar S.K., Kaushik V.P., 2005, Reflection and refraction of SH-waves at a corrugated interface between two laterally and vertically heterogeneous viscoelastic solid half-space, International Journal of Solids and Structures 42(13): 3621-3643.
[12] Campillo M., 2006, Modeling of SH-wave propagation in an irregularly layered medium application to seismic profiles near a dome, Geophysical Prospecting 35(3): 236-249.
[13] Chaudhary S, Kaushik V.P., Tomar S.K., 2004, Reflection/transmission of plane SH wave through a self-reinforced elastic layer between two half-spaces, Acta Geophys Polon 52(2): 219-235.
[14] Du J., Jin X., Wang J., Zhou Y., 2007, SH wave propagation in a cylindrically layered piezoelectric structure with initial stress, Acta Mechanica 191(1-2): 59-74.
[15] Weiskopf W.H., 1945, Stresses in Soils Under Foundations, Franklin Inst.
[16] Abd-Alla A.M., Abo-Dahab S.M., Al-Thamali T.A., 2013, Love waves in a non-homogeneous orthotropic magneto-elastic layer under initial stress overlying a semi-infinite medium, Journal of Computational Theoretical Nanosciece 10(1): 10-18.
[17] Singh B., Kumar R., 1998, Reflection and refraction of plane waves at an interface between micropolar elastic solid and viscoelastic solid, International Journal of Engineering Science 36(2): 119-135.
[18] Abd-Alla A.M., Abo-Dahab S.M., Hammadc H.A.H., 2011, Propagation of Rayleigh waves in generalized magneto-thermoelastic orthotropic material under initial stress and gravity field, Applied Mathematics and Modelling 35: 2981-3000.
[19] Vardoulakis I., 1984, Torsional surface waves in inhomogeneous elastic media, International Journal for Numerical and Analytical Methods in Geomechanics 8: 287-296.
[20] Tomar S.K., Kaur J., 2007, Reflection and transmission of SH-waves at a corrugated interface between anisotropic elastic and viscoelastic solid half-spaces, Journal of Seismology 3: 235-258.
[21] Chattopadhyay A., Michel V., 2006, A model for spherical SH wave propagation in self-reinforced linearly elastic media, Archive of Applied Mechanics 76: 113-124.
[22] Prasad R.M., Kundu S. 2017, Torsional surface wave dispersion in pre-stressed dry sandy layer over a gravitating anisotropic porous half-space, Zeitschrift für Angewandte Mathematik und Mechanik 97(5): 550-560.
[23] Whittaker E.T., Watson G.N., 1991, A Course of Modern Analysis, Cambridge University Press, Cambridge.
