Effect of Point Source, Self-Reinforcement and Heterogeneity on the Propagation of Magnetoelastic Shear Wave

ABSTRACT

This paper investigates the propagation of horizontally polarised shear waves due to a point source in a magnetoelastic self-reinforced layer lying over a heterogeneous self-reinforced half-space. The heterogeneity is caused by consideration of quadratic variation in rigidity. The methodology employed combines an efficient derivation for Green’s functions based on algebraic transformations with the perturbation approach. Dispersion equation has been obtained in the closed form. The dispersion curves are compared for different values of magnetoelastic coupling parameters and inhomogeneity parameters. Also, the comparative study is being made through graphs to find the effect of reinforcement over the reinforced-free case on the phase velocity. It is observed that the dispersion equation is in assertion with the classical Love-type wave equation in the absence of reinforcement, magnetic field and heterogeneity. Moreover, some important peculiarities have been observed in graphs.

This paper investigates the propagation of horizontally polarised shear waves due to a point source in a magnetoelastic self-reinforced layer lying over a heterogeneous self-reinforced half-space. The heterogeneity is caused by consideration of quadratic variation in rigidity. The methodology employed combines an efficient derivation for Green’s functions based on algebraic transformations with the perturbation approach. Dispersion equation has been obtained in the closed form. The dispersion curves are compared for different values of magnetoelastic coupling parameters and inhomogeneity parameters. Also, the comparative study is being made through graphs to find the effect of reinforcement over the reinforced-free case on the phase velocity. It is observed that the dispersion equation is in assertion with the classical Love-type wave equation in the absence of reinforcement, magnetic field and heterogeneity. Moreover, some important peculiarities have been observed in graphs.

Cite this paper

nullA. Chattopadhyay, S. Gupta, A. Singh and S. Sahu, "Effect of Point Source, Self-Reinforcement and Heterogeneity on the Propagation of Magnetoelastic Shear Wave,"*Applied Mathematics*, Vol. 2 No. 3, 2011, pp. 271-282. doi: 10.4236/am.2011.23032.

nullA. Chattopadhyay, S. Gupta, A. Singh and S. Sahu, "Effect of Point Source, Self-Reinforcement and Heterogeneity on the Propagation of Magnetoelastic Shear Wave,"

References

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[2] A. J. Belfield, T. G. Rogers and A. J. M. Spencer, “Stress in Elastic Plates Reinforced by Fibers Lying in Concentric Circles,” Journal of the Mechanics and Physics of Solids, Vol. 31, No. 1, 1983, pp. 25-54. doi:10.1016/0022-5096(83)90018-2

[3] P. D. S. Verma and O. H. Rana, “Rotation of a Circular Cylindrical Tube Reinforced by Fibres Lying along Helices,” Mechanics of Materials, Vol. 2, No. 4, 1983, pp. 353-359. doi:10.1016/0167-6636(83)90026-1

[4] P. D. S. Verma, “Magnetoelastic Shear Waves in Self- Reinforced Bodies,” International Journal of Engineering Science, Vol. 24, No. 7, 1986, pp. 1067-1073. doi:10.1016/0020-7225(86)90002-9

[5] P. D. S. Verma, O. H. Rana and M. Verma, “Magnetoelastic Transverse Surface Waves in Self-Reinforced Elastic Bodies,” Indian Journal of Pure and Applied Mathematics, Vol. 19, No. 7, 1988, pp. 713-716.

[6] A. Chattopadhyay and S. Chaudhury, “Propagation, Reflection and Transmission of Magnetoelastic Shear Waves in a Self Reinforced Medium,” International Journal of Engineering Science, Vol. 28, No. 6, 1990, pp. 485-495. doi:10.1016/0020-7225(90)90051-J

[7] A. Chattopadhyay and S. Chaudhury, “Magnetoelastic Shear Waves in an Infinite Self-Reinforced Plate,” International Journal of Numerical and Analytical Methods in Geomechanics, Vol. 19, No. 4, 1995, pp. 289-304. doi:10.1002/nag.1610190405

[8] A. Chattopadhyay and R. L. K. Venkateswarlu, “Stresses Produced in a Fibre-Reinforced Half Space Due to Moving Load,” Bulletin of Calcutta Mathematical Society, Vol. 90, 1998, pp. 337-342.

[9] S. Chaudhary, V. P. Kaushik and S. K. Tomar, “Transmission of Shear Waves through a Self-Reinforced Layer Sandwiched between Two Inhomogeneous Viscoelastic Half-Spaces,” International Journal of Mechanical Scien- ces, Vol. 47, No. 9, 2005, pp. 1455-1472. doi:10.1016/j.ijmecsci.2005.04.011

[10] S. Chaudhary, V. P. Kaushik and S. K. Tomar, “Plane SH-Wave Response from Elastic Slab Interposed between Two Different Self Reinforced Elastic Solids,” International Journal of Applied Mechanics and Engineering, Vol. 11, No. 4, 2006, pp. 787-801.

[11] A. Chattopadhyay, S. Gupta, S. K. Samal and V. K. Sharma, “Torsional Wave in Self-Reinforced Medium,” International Journal of Geomechanics, Vol. 9, No. 1, 2009, pp. 9-13. doi:10.1061/(ASCE)1532-3641(2009)9:1(9)

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[15] C. Vrettos, “Forced Anti-Plane Vibrations at the Surface of an Inhomogeneous Half-Space,” Soil Dynamics and Earthquake Engineering, Vol. 10, No. 5, 1991, pp. 230- 235. doi:10.1016/0267-7261(91)90016-S

[16] C. Vrettos, “The Boussinesq Problem for Soil with Bound Nonhomogeneity,” International Journal of Numerical and Analytical Methods in Geomechanics, Vol. 22, No. 8, 1998, pp. 655-669. doi:10.1002/(SICI)1096-9853(199808)22:8<655::AID-NAG938>3.0.CO;2-R

[17] K. Singh, “Love Waves Due to a Point Source in an Axially Symmetric Heterogeneous Layer between Two Homogeneous Half Spaces,” Pure and Applied Geophysics, Vol. 72, No. 1, 1969, pp. 35-44. doi:10.1007/BF00875690

[18] H. Deresiewich, “A Note on Love Waves in Homogeneous Crust Overlying an Inhomogeneous Substratum,” Bulletin of Seismological Society of America, Vol. 52, 1962, pp. 639-645.

[19] M. Ewing, W. S. Jardetzky and F. Press, “Elastic Waves in Layered Media,” McGraw-Hill, New York, 1957.

[20] K. Sezawa, “Love Waves Generated from a Source of a Certain Depth,” Bulletin of the Eathquake Research Institute, University of Tokyo, Vol. 13, 1935, pp. 1-17.

[21] Y. Sato, “Love Waves Propagated upon Heterogeneous Medium,” Bulletin of the Eathquake Research Institute, University of Tokyo, Vol. 30, 1952, pp. 1-12.

[22] M. L. Ghosh, “Love Wave Due to a Point Source in an Inhomogeneous Medium,” Gerlands Beitrage Zur Geophysik, Vol. 70, 1970, pp. 319-342.

[23] J. Bhattacharya, “The Possibility of the Propagation of Love Type Waves in an Intermediate Heterogeneous Layer Lying between Two Semi-Infinite Isotropic Homogeneous Elastic Layers,” Pure and Applied Geophysics, Vol. 72, No. 1, 1969, pp. 61-71. doi:10.1007/BF00875693

[24] A. Chattopadhyay and B. K. Kar, “Love Wave Due to a Point Source in an Isotropic Elastic Medium under Initial Stress,” International Journal of Non-Linear Mechanics, Vo. 16, No. 3-4, 1981, pp. 247-258. doi:10.1016/0020-7462(81)90038-X

[25] E. D. Covert, “Approximate Calculation of Green’s Function for Built-Up Bodies,” Journal of Mathematical Physics, Vol. 37, No. 1, 1958, pp. 58-65.

[26] A. Chattopadhyay, M. Chakraborty and V. Kaushwaha, “On the Dispersion Equation of Love Waves in a Porous Layer,” Acta Mechanica, Vol. 58, No. 3-4, 1986, pp. 125- 136. doi:10.1007/BF01176595

[27] K. Watanabe and R. G. Payton, “Green’s Function for SH-Wave in Cylindrically Monoclinic Material,” Journal of Mechanics and Physics, Vol. 50, No. 11, 2002, pp. 2425-2439. doi:10.1016/S0022-5096(02)00026-1

[28] G. D. Manolis and A. C. Bagtzoglou, “A Numerical Comparative Study of Wave Propagation in Inhomogeneous and Random Media,” Computational Mechanics, Vol. 10, No. 6, 1992, pp. 397-413. doi:10.1007/BF00363995

[29] A. O. Awojobi and O. A. Sobayo, “Ground Vibration Due to Seismic Detonation of a Buried Source,” Earthquake Engineering and Structural Dynamics, Vol. 5, No. 2, 2006, pp. 131-143. doi:10.1002/eqe.4290050203

[30] E. Kausel and J. Park, “Impulse Response of Elastic Half-Space in the Wave Number-Time Domain,” Journal of Engineering Mechanics ASCE, Vol. 130, No. 10, 2004, pp. 1211-1222. doi:10.1061/(ASCE)0733-9399(2004)130:10(1211)

[31] G. D. Manolis and R. P. Shaw, “Wave Motions in Stochastic Heterogenous Media,” Engineering Analysis with Boundary Element, Vol. 15, No. 3, 1995, pp. 225-234. doi:10.1016/0955-7997(95)00026-K

[32] M. F. Markham, “Measurements of Elastic Constants of Fibre Composite by Ultrasonics,” Composites, Vol. 1, 1970, pp. 145-149. doi:10.1016/0010-4361(70)90477-5

[33] G. A. Hool and W. S. Kinne, “Reinforced Concrete and Masonry Structure,” McGraw-Hill, New York, 1924.

[34] G. A. Maugin, “Review Article: Wave Motion in Magnetizable Deformable Solids,” International Journal of Engineering Science, Vol. 19, No. 3, 1981, pp. 321-388. doi:10.1016/0020-7225(81)90059-8

[1] J. D. Achenbach, “Wave Propagation in Elastic Solids,” North Holland Publication Company, New York, 1975.

[2] A. J. Belfield, T. G. Rogers and A. J. M. Spencer, “Stress in Elastic Plates Reinforced by Fibers Lying in Concentric Circles,” Journal of the Mechanics and Physics of Solids, Vol. 31, No. 1, 1983, pp. 25-54. doi:10.1016/0022-5096(83)90018-2

[3] P. D. S. Verma and O. H. Rana, “Rotation of a Circular Cylindrical Tube Reinforced by Fibres Lying along Helices,” Mechanics of Materials, Vol. 2, No. 4, 1983, pp. 353-359. doi:10.1016/0167-6636(83)90026-1

[4] P. D. S. Verma, “Magnetoelastic Shear Waves in Self- Reinforced Bodies,” International Journal of Engineering Science, Vol. 24, No. 7, 1986, pp. 1067-1073. doi:10.1016/0020-7225(86)90002-9

[5] P. D. S. Verma, O. H. Rana and M. Verma, “Magnetoelastic Transverse Surface Waves in Self-Reinforced Elastic Bodies,” Indian Journal of Pure and Applied Mathematics, Vol. 19, No. 7, 1988, pp. 713-716.

[6] A. Chattopadhyay and S. Chaudhury, “Propagation, Reflection and Transmission of Magnetoelastic Shear Waves in a Self Reinforced Medium,” International Journal of Engineering Science, Vol. 28, No. 6, 1990, pp. 485-495. doi:10.1016/0020-7225(90)90051-J

[7] A. Chattopadhyay and S. Chaudhury, “Magnetoelastic Shear Waves in an Infinite Self-Reinforced Plate,” International Journal of Numerical and Analytical Methods in Geomechanics, Vol. 19, No. 4, 1995, pp. 289-304. doi:10.1002/nag.1610190405

[8] A. Chattopadhyay and R. L. K. Venkateswarlu, “Stresses Produced in a Fibre-Reinforced Half Space Due to Moving Load,” Bulletin of Calcutta Mathematical Society, Vol. 90, 1998, pp. 337-342.

[9] S. Chaudhary, V. P. Kaushik and S. K. Tomar, “Transmission of Shear Waves through a Self-Reinforced Layer Sandwiched between Two Inhomogeneous Viscoelastic Half-Spaces,” International Journal of Mechanical Scien- ces, Vol. 47, No. 9, 2005, pp. 1455-1472. doi:10.1016/j.ijmecsci.2005.04.011

[10] S. Chaudhary, V. P. Kaushik and S. K. Tomar, “Plane SH-Wave Response from Elastic Slab Interposed between Two Different Self Reinforced Elastic Solids,” International Journal of Applied Mechanics and Engineering, Vol. 11, No. 4, 2006, pp. 787-801.

[11] A. Chattopadhyay, S. Gupta, S. K. Samal and V. K. Sharma, “Torsional Wave in Self-Reinforced Medium,” International Journal of Geomechanics, Vol. 9, No. 1, 2009, pp. 9-13. doi:10.1061/(ASCE)1532-3641(2009)9:1(9)

[12] K. Aki and P. G. Richards, “Quantitative Seismology: Theory and Methods,” W. H. Freeman & Co., New York, 1980.

[13] A. T. De Hoop, “Handbook of Radiation and Scattering of Waves: Acoustic Waves in Fluids, Elastic Waves in Solids, Electromagnetic Waves,” Academic Press, London, 1995.

[14] L. M. Brekhovskikh and O. A. Godin, “Acoustics of Layered Media,” Springer-Verlag, Berlin, 1992.

[15] C. Vrettos, “Forced Anti-Plane Vibrations at the Surface of an Inhomogeneous Half-Space,” Soil Dynamics and Earthquake Engineering, Vol. 10, No. 5, 1991, pp. 230- 235. doi:10.1016/0267-7261(91)90016-S

[16] C. Vrettos, “The Boussinesq Problem for Soil with Bound Nonhomogeneity,” International Journal of Numerical and Analytical Methods in Geomechanics, Vol. 22, No. 8, 1998, pp. 655-669. doi:10.1002/(SICI)1096-9853(199808)22:8<655::AID-NAG938>3.0.CO;2-R

[17] K. Singh, “Love Waves Due to a Point Source in an Axially Symmetric Heterogeneous Layer between Two Homogeneous Half Spaces,” Pure and Applied Geophysics, Vol. 72, No. 1, 1969, pp. 35-44. doi:10.1007/BF00875690

[18] H. Deresiewich, “A Note on Love Waves in Homogeneous Crust Overlying an Inhomogeneous Substratum,” Bulletin of Seismological Society of America, Vol. 52, 1962, pp. 639-645.

[19] M. Ewing, W. S. Jardetzky and F. Press, “Elastic Waves in Layered Media,” McGraw-Hill, New York, 1957.

[20] K. Sezawa, “Love Waves Generated from a Source of a Certain Depth,” Bulletin of the Eathquake Research Institute, University of Tokyo, Vol. 13, 1935, pp. 1-17.

[21] Y. Sato, “Love Waves Propagated upon Heterogeneous Medium,” Bulletin of the Eathquake Research Institute, University of Tokyo, Vol. 30, 1952, pp. 1-12.

[22] M. L. Ghosh, “Love Wave Due to a Point Source in an Inhomogeneous Medium,” Gerlands Beitrage Zur Geophysik, Vol. 70, 1970, pp. 319-342.

[23] J. Bhattacharya, “The Possibility of the Propagation of Love Type Waves in an Intermediate Heterogeneous Layer Lying between Two Semi-Infinite Isotropic Homogeneous Elastic Layers,” Pure and Applied Geophysics, Vol. 72, No. 1, 1969, pp. 61-71. doi:10.1007/BF00875693

[24] A. Chattopadhyay and B. K. Kar, “Love Wave Due to a Point Source in an Isotropic Elastic Medium under Initial Stress,” International Journal of Non-Linear Mechanics, Vo. 16, No. 3-4, 1981, pp. 247-258. doi:10.1016/0020-7462(81)90038-X

[25] E. D. Covert, “Approximate Calculation of Green’s Function for Built-Up Bodies,” Journal of Mathematical Physics, Vol. 37, No. 1, 1958, pp. 58-65.

[26] A. Chattopadhyay, M. Chakraborty and V. Kaushwaha, “On the Dispersion Equation of Love Waves in a Porous Layer,” Acta Mechanica, Vol. 58, No. 3-4, 1986, pp. 125- 136. doi:10.1007/BF01176595

[27] K. Watanabe and R. G. Payton, “Green’s Function for SH-Wave in Cylindrically Monoclinic Material,” Journal of Mechanics and Physics, Vol. 50, No. 11, 2002, pp. 2425-2439. doi:10.1016/S0022-5096(02)00026-1

[28] G. D. Manolis and A. C. Bagtzoglou, “A Numerical Comparative Study of Wave Propagation in Inhomogeneous and Random Media,” Computational Mechanics, Vol. 10, No. 6, 1992, pp. 397-413. doi:10.1007/BF00363995

[29] A. O. Awojobi and O. A. Sobayo, “Ground Vibration Due to Seismic Detonation of a Buried Source,” Earthquake Engineering and Structural Dynamics, Vol. 5, No. 2, 2006, pp. 131-143. doi:10.1002/eqe.4290050203

[30] E. Kausel and J. Park, “Impulse Response of Elastic Half-Space in the Wave Number-Time Domain,” Journal of Engineering Mechanics ASCE, Vol. 130, No. 10, 2004, pp. 1211-1222. doi:10.1061/(ASCE)0733-9399(2004)130:10(1211)

[31] G. D. Manolis and R. P. Shaw, “Wave Motions in Stochastic Heterogenous Media,” Engineering Analysis with Boundary Element, Vol. 15, No. 3, 1995, pp. 225-234. doi:10.1016/0955-7997(95)00026-K

[32] M. F. Markham, “Measurements of Elastic Constants of Fibre Composite by Ultrasonics,” Composites, Vol. 1, 1970, pp. 145-149. doi:10.1016/0010-4361(70)90477-5

[33] G. A. Hool and W. S. Kinne, “Reinforced Concrete and Masonry Structure,” McGraw-Hill, New York, 1924.

[34] G. A. Maugin, “Review Article: Wave Motion in Magnetizable Deformable Solids,” International Journal of Engineering Science, Vol. 19, No. 3, 1981, pp. 321-388. doi:10.1016/0020-7225(81)90059-8