^{1}Department of Applied Mathematics, Indian School of Mines, Dhanbad-826004, India

^{2}Department, Qena Faculty of Science, Egypt

Abstract

The present paper studies the possibility of propagation of torsional surface waves in an inhomogeneous anisotropic layer lying between two heterogeneous half-spaces (upper and lower half-space). Both the half-spaces are assumed to be under compressive initial stress. The study reveals that under the assumed conditions, a torsional surface wave propagates in the medium. The dispersion relation of torsional surface wave has been obtained in the presence of heterogeneity, initial stress and anisotropic, and it is observed that the inhomogeneity factor due to quadratic and hyperbolic variations in rigidity, density and initial stress of the medium decreases the phase velocity as it increases. The result also shows that the initial stresses have a pronounced influence on the propagation of torsional surface waves. In the absence of anisotropy, Initial stress, inhomogeneity and rigidity of the upper half-space, then the dispersion relation coincide with the classical dispersion relation of Love wave.

Abo-Dahab S.M., 2011, Reflection of P and SV waves from stress-free surface elastic half-space under influence of magnetic field and hydro-static initial stress without energy dissipation, Journal of Vibration and Control 17: 2213-2221. [2] Abd-Alla A.M., Ahmed S.M., 1999, Propagation of love waves in a nonhomogeneous orthotropic elastic layer under initial stress overlying semi-infinite medium, Applied Mathematics and Computation 106: 265-275. [3] 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. [4] Ahmed S.M., Abo-Dahab S.M., 2010, Propagation of Love waves in an orthotropic granular layer under initial stress overlying a semi-infinite granular medium, Journal of Vibration and Control 16: 1845-1858. [5] Chattopadhyay A., Gupta S., Sharma V.K., Kumari P., 2013a, Torsional wave propagation in harmonically inhomogeneous media, International Journal for Numerical and Analytical Methods in Geomechanics 1280-1291. [6] Chattopadhyay A., Gupta S., Sahu S.A., Dhua S., 2013b, Torsional surface waves in heterogeneous anisotropic half-space under initial stress, Archive of Applied Mechanics 83: 357-366. [7] Gupta S., Chattopadhyay A., Kundu S., Gupta A.K., 2010, Effect of rigid boundary on the propagation of torsional waves in a homogeneous layer over a heterogeneous half-space, Archive of Applied Mechanics 80: 143-150. [8] Gupta S., Chattopadhyay A., Majhi D.K., 2011, Effect of rigid boundary on propagation of torsional surface waves in porous elastic layer, Applied Mathematics and Mechanics 32: 327-338. [9] Biot M.A., 1940, The influence of initial stress on elastic wave, Journal of Applied Physics 11: 522-530. [10] Kepceler T., 2010, Torsional wave dispersion relation in a pre-stressed bi-material compounded cylinder with an imperfect interface, Applied Mathematical Modelling 34: 4058-4073. [11] Ozturk A., Akbbarov S.D., 2009, Torsional wave propagation in a pre-stressed circular cylinder embedded in a pre-stressed elastic medium, Applied Mathematical Modelling 33: 3636-3649. [12] Love A.E.H., 1927, The Mathematical Theory of Elasticity, Cambridge University Press, Cambridge. [13] Love A.E.H., 1911, Some Problems of Geodynamics, Cambridge University Press, Cambridge. [14] Ewing W.M., Jardetzky W.S., Press F., 1957, Elastic Waves in Layered Media, Mcgraw-Hill, New York. [15] Biot M.A., 1965, Mechanics of Incremental Deformation, John Willey and Sons, New York. [16] Khaled A., Gepreel, Abo-Dahab S.M., Nofal T.A., 2012, Homotopy perturbation method and variational iteration method for harmonic waves propagation in nonlinear magneto-thermoelasticity with rotation, Mathematical Problems in Engineering 2012: 1-30. [17] Biot M.A., 1956, Theory of propagation of elastic waves in a fluid saturated porous solid. I. Low frequency range, The Journal of the Acoustical Society of America 28: 168-178. [18] Kumari P., Sharma V.K., Modi C., 2015, Propagation of torsional waves in an inhomogeneous layer sandwiched between inhomogeneous semi infinite strata, Journal of Engineering Mathematics 90: 1-11. [19] Selim M.M., 2007, Propagation of torsional surface waves in heterogeneous half-space with irregular free surface, Applied Mathematical Sciences 1: 1429-1437. [20] Shearer T., Abrahams I.D., Parnell W.J., Daros C.H., 2013, Torsional wave propagation in a pre-stressed hyperelastic annular circular cylinder, The Guarterly Journal of Mechanics and Applied Mathematics 66: 465-487. [21] Chattopadhyay A., Singh A.K., 2012, Propagation of magnetoelastic shear waves in an irregular self-reinforced layer, Journal of Engineering Mathematics 75: 139-155. [22] Vishwakarma S.K., Gupta S., 2013, Existence of torsional surface waves in an earth’s crustal layer lying over a sandy mantle, Journal of Earth System Science 122: 1411-1421. [23] Georgiadis H.G., Vardoulakis I., Lykotrafitis G., 2000, Torsional surface waves in a gradient-elastic half-space, Wave Motion 31: 333-348. [24] Dey S., Dutta D., 1992, Torsional wave propagation in an initially stressed cylinder, Proceedings of the National Academy of Sciences 58: 425-429.