Propagation of Torsional Surface Waves under the Effect of Irregularity and Initial Stress

ABSTRACT

The present paper has been framed to study the influence of irregularity, initial stress and porosity on the propagation of torsional surface waves in an initially stressed anisotropic poro-elastic layer over a semi-infinite heterogeneous half space with linearly varying rigidity and density due to irregularity at the interface. The irregularity has been taken in the half-space in the form of a parabola. It is observed that torsional sur- face waves propagate in this assumed medium. In the absence of irregularity the velocity of torsional surface wave has been obtained. Further, it has been seen that for a layer over a homogeneous half space, the velo- city of torsional surface waves coincides with that of Love waves.

The present paper has been framed to study the influence of irregularity, initial stress and porosity on the propagation of torsional surface waves in an initially stressed anisotropic poro-elastic layer over a semi-infinite heterogeneous half space with linearly varying rigidity and density due to irregularity at the interface. The irregularity has been taken in the half-space in the form of a parabola. It is observed that torsional sur- face waves propagate in this assumed medium. In the absence of irregularity the velocity of torsional surface wave has been obtained. Further, it has been seen that for a layer over a homogeneous half space, the velo- city of torsional surface waves coincides with that of Love waves.

Cite this paper

nullS. Gupta, D. Majhi, S. Vishwakarma and S. Kundu, "Propagation of Torsional Surface Waves under the Effect of Irregularity and Initial Stress,"*Applied Mathematics*, Vol. 2 No. 12, 2011, pp. 1453-1461. doi: 10.4236/am.2011.212207.

nullS. Gupta, D. Majhi, S. Vishwakarma and S. Kundu, "Propagation of Torsional Surface Waves under the Effect of Irregularity and Initial Stress,"

References

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[2] W. M. Ewing, W. S. Jardetzky and F. Press, “Elastic Waves in LAYERED Media,” McgrawHill, New York, 1957.

[3] M. Bath, “Mathematical Aspects of Seismology,” Elsvier Publishing Comp., New York, 1968.

[4] L. Rayleigh, “On Waves Propagated along Plane Surface of an Elastic Solid,” Proceedings of the London Mathematical Society, Vol. 17, No. 3, 1885, pp. 4-11. doi:10.1112/plms/s1-17.1.4

[5] H. G. Georgiadis, I. Vardoulakis and G. Lykotrafitis “Torsional Surface Waves in a Gradient-Elastic Half Space,” Wave Motion, Vol. 31, No. 4, 2000, pp. 333-348. doi:10.1016/S0165-2125(99)00035-9

[6] E. Meissner, “Elastic Oberflachenwellen Mit Dispersion in Einem Inhomogeneous Medium,” Viertlagahrsschriftder Naturforschenden Gesellschaft, Zurich, Vol. 66, 1921, pp. 181-195 .

[7] I. Vardoulakis, “Torsional Surface Waves in Inhomogeneous Elastic Media,” International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 8, No. 3, 1984, pp. 287-296.

[8] S. Dey, A. K.Gupta and S. Gupta, “Propagation of Torsional Surface Waves in a Homogeneous Substratum over a Heterogeneous Half-Space,” International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 20, 1996, pp. 287-294. doi:10.1002/(SICI)1096-9853(199604)20:4<287::AID-NAG822>3.0.CO;2-2

[9] M. A. Biot, “Theory of Deformation of a Porous Viscoelastic Anisotropic Solid,” Journal of Applied Physics, Vol. 27, No. 5, 1956, pp. 459-467. doi:10.1063/1.1722402

[10] M. A. Biot, “Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid,” Journal of the Acoustical Society of America, Vol. 28, No. 2, 1956, pp. 168-178. doi:10.1121/1.1908239

[11] M. J. Buckingham, “Theory of Compressional and Shear Waves in Fluid-Like Marine Sediments,” Journal of the Acoustical Society of America, Vol. 103, No. 1, 1998, pp. 288-299. doi:10.1121/1.421091

[12] M. D. Sharma and M. L. Gogna, “Propagation of Love Waves in an Initially-Stressed Medium Consisting of sandy Layer Lying over a Liquid Saturated Porous Solid Half-Space,” Journal of the Acoustical Society of America, Vol. 89, No. 6, 1991, pp. 2584-2588. doi:10.1121/1.400697

[13] M. D. Sharma, R. Kumar and M. L. Gogna, “Surface Wave Propagation in a Transversely Isotropic Elastic Layer Overlying a Liquid-Saturated Porous Solid Half-Space and Lying under a Uniform Layer of Liquid,” Pure and Applied Geophysics, Vol. 133, No. 3, 1990, pp. 523-539. doi:10.1007/BF00878003

[14] M. D. Sharma, R. Kumar and M. L. Gogna, “Surface Wave Propagation in a Liquid-Saturated Porous Layer Over-Lying a Homogeneous Transversely Isotropic Half-Space and Lying under a Uniform Layer of Liquid,” International Journal of Solids and Structures, Vol. 27, 1991, pp. 1255-1267. doi:10.1016/0020-7683(91)90161-8

[15] M. A. Biot, “Influence of Initial Stress on Elastic Waves,” Journal of Applied Physics, Vol. 11, No. 8, 1940, pp. 522-530. doi:10.1063/1.1712807

[16] A. Chattopadhyay, S. Bose and M. Chakraborty, “Reflection of Elastic Waves under Initial Stress at a Free Surface,” Journal of the Acoustical Society of America, Vol. 72, No. 1, 1982, pp. 255-263. doi:10.1121/1.387987

[17] B. K. Kar and V. K. Kalyani, “Reflection and Refraction of SH-Waves Due to the Presence of a Sandwiched Initially Stressed Sandy Layer,” Geophysical research bulletin, Vol. 25, No. 3, 1987, pp. 117-124.

[18] S. Dey and S. K. Addy, “Reflection of Plane Waves under Initial Stress at a Free Surface,” International Journal of Non-Linear Mechanics, Vol. 12, No. 6, 1977, pp. 371-381. doi:10.1016/0020-7462(77)90038-5

[19] M. A. Biot, “Mechanics of Incremental Deformation,” Wiley, New York, 1965

[20] W.H. Weiskopf, “Stresses in Soils under a Foundation,” Journal of the Franklin Institute, Vol. 239, No. 6, 1945, pp. 445-465. doi:10.1016/0016-0032(45)90189-X

[1] J. D. Achenbach, “Wave Propagation in Elastic Solids,” North Holland Publishing Comp., New York, 1973

[2] W. M. Ewing, W. S. Jardetzky and F. Press, “Elastic Waves in LAYERED Media,” McgrawHill, New York, 1957.

[3] M. Bath, “Mathematical Aspects of Seismology,” Elsvier Publishing Comp., New York, 1968.

[4] L. Rayleigh, “On Waves Propagated along Plane Surface of an Elastic Solid,” Proceedings of the London Mathematical Society, Vol. 17, No. 3, 1885, pp. 4-11. doi:10.1112/plms/s1-17.1.4

[5] H. G. Georgiadis, I. Vardoulakis and G. Lykotrafitis “Torsional Surface Waves in a Gradient-Elastic Half Space,” Wave Motion, Vol. 31, No. 4, 2000, pp. 333-348. doi:10.1016/S0165-2125(99)00035-9

[6] E. Meissner, “Elastic Oberflachenwellen Mit Dispersion in Einem Inhomogeneous Medium,” Viertlagahrsschriftder Naturforschenden Gesellschaft, Zurich, Vol. 66, 1921, pp. 181-195 .

[7] I. Vardoulakis, “Torsional Surface Waves in Inhomogeneous Elastic Media,” International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 8, No. 3, 1984, pp. 287-296.

[8] S. Dey, A. K.Gupta and S. Gupta, “Propagation of Torsional Surface Waves in a Homogeneous Substratum over a Heterogeneous Half-Space,” International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 20, 1996, pp. 287-294. doi:10.1002/(SICI)1096-9853(199604)20:4<287::AID-NAG822>3.0.CO;2-2

[9] M. A. Biot, “Theory of Deformation of a Porous Viscoelastic Anisotropic Solid,” Journal of Applied Physics, Vol. 27, No. 5, 1956, pp. 459-467. doi:10.1063/1.1722402

[10] M. A. Biot, “Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid,” Journal of the Acoustical Society of America, Vol. 28, No. 2, 1956, pp. 168-178. doi:10.1121/1.1908239

[11] M. J. Buckingham, “Theory of Compressional and Shear Waves in Fluid-Like Marine Sediments,” Journal of the Acoustical Society of America, Vol. 103, No. 1, 1998, pp. 288-299. doi:10.1121/1.421091

[12] M. D. Sharma and M. L. Gogna, “Propagation of Love Waves in an Initially-Stressed Medium Consisting of sandy Layer Lying over a Liquid Saturated Porous Solid Half-Space,” Journal of the Acoustical Society of America, Vol. 89, No. 6, 1991, pp. 2584-2588. doi:10.1121/1.400697

[13] M. D. Sharma, R. Kumar and M. L. Gogna, “Surface Wave Propagation in a Transversely Isotropic Elastic Layer Overlying a Liquid-Saturated Porous Solid Half-Space and Lying under a Uniform Layer of Liquid,” Pure and Applied Geophysics, Vol. 133, No. 3, 1990, pp. 523-539. doi:10.1007/BF00878003

[14] M. D. Sharma, R. Kumar and M. L. Gogna, “Surface Wave Propagation in a Liquid-Saturated Porous Layer Over-Lying a Homogeneous Transversely Isotropic Half-Space and Lying under a Uniform Layer of Liquid,” International Journal of Solids and Structures, Vol. 27, 1991, pp. 1255-1267. doi:10.1016/0020-7683(91)90161-8

[15] M. A. Biot, “Influence of Initial Stress on Elastic Waves,” Journal of Applied Physics, Vol. 11, No. 8, 1940, pp. 522-530. doi:10.1063/1.1712807

[16] A. Chattopadhyay, S. Bose and M. Chakraborty, “Reflection of Elastic Waves under Initial Stress at a Free Surface,” Journal of the Acoustical Society of America, Vol. 72, No. 1, 1982, pp. 255-263. doi:10.1121/1.387987

[17] B. K. Kar and V. K. Kalyani, “Reflection and Refraction of SH-Waves Due to the Presence of a Sandwiched Initially Stressed Sandy Layer,” Geophysical research bulletin, Vol. 25, No. 3, 1987, pp. 117-124.

[18] S. Dey and S. K. Addy, “Reflection of Plane Waves under Initial Stress at a Free Surface,” International Journal of Non-Linear Mechanics, Vol. 12, No. 6, 1977, pp. 371-381. doi:10.1016/0020-7462(77)90038-5

[19] M. A. Biot, “Mechanics of Incremental Deformation,” Wiley, New York, 1965

[20] W.H. Weiskopf, “Stresses in Soils under a Foundation,” Journal of the Franklin Institute, Vol. 239, No. 6, 1945, pp. 445-465. doi:10.1016/0016-0032(45)90189-X