Effect of Body Forces on the Propagation of Love Wave in a semi-infinite Orthotropic Medium

P. C. Pal, S. Kumar and D. Mandal

Department of Applied Mathematics, Indian School of Mines, Dhanbad-826004, India

*Corresponding Author: pcpal_ism@yahoo.co.in, santosh453@gmail.com, dinbandhumandal@gmail.com

ABSTRACT:

The effect of time dependent body forces on the propagation of Love wave motion in an orthotropic elastic half space is considered in this paper. The surface displacement component is obtained in a closed form for special type of crystals. Numerical results are obtained for a particular case of time dependent body force at different distances from the source for the different values of the non-dimensional time. The results are shown graphically.

KEY WORDS: Body Forces, Love wave, Orthotropic Media, Cagniard de-Hoop Technique

INTRODUCTION:

The study of elastic waves propagation in layered media has been a subject to extensive investigation by geophysicists, seismologist, Applied Mechanician and Applied Mathematicians in the last four decades. It is of considerable important in a variety of applications ranging from earth geophysics and earthquake engineering to laminated composites used in aerospace and mechanical engineering applications and to non-destructive evaluation of structural elements composed of laminated composites.

The propagation of SH-type wave in a viscoelastic medium consisting of a layer over a half-space, focusing on the reflected and transmitted waves due to incidence of a point torque SH source at the surface. This is done in a manner to accommodated most qualitative theories such a plane wave methods (Boarchedt, 1977), asymptotic ray theory method, (Krebes and Hron, 1980), and integral transform methods, Le et al. (1994), Hearn and Krebes (1990), in addition to the related line source problem for viscoelastic media. The problem of evaluating the displacement produced in an isotropic semi-infinite medium has been studied by Cagniard (1939), Garvin (1956), Garvin used a modified version of cagniard’s method. Mitra and Maiti (1979) have considered the effect of sources (asymmetric and symmetric) on the generation of different pulses and waves in an isotropic elastic half-space. They have adopted Cagniard’s modified method to solve the problem. Abd-Alla and Ahmed (1999) have discussed the Love waves in a non-homogeneous orthotropic elastic medium under changeable initial stress.

Liu and Liu (2004) have studied the influence of anisotropy of the solid skeleton on the propagation of characteristic of Rayleigh waves in orthotropic fluid-saturated porous media. Rangelov et al. (2010) have considered a special case of wave propagation problem in a restricted class of orthotropic inhomogeneous half-space.

The present problem is concerned with orthotropic half space of Love waves. Unlike the isotropic problem solved by the different authors, the orientation of the crystallographic axes relative to the half-space surface does not make a considerable difference in the complexity of the analysis for the corresponding orthotropic material.

1. Formulation of Problem

Geometry of the problem is depicted in Fig. 1, where x and y axes are taken along the free surface of the semi-infinite orthotropic elastic medium and z-axis perpendicular to the free surface. Our main concern here is to evaluate displacement at the surface due to body forces; the displacement being assumed to tend to zero as . For SH-type of waves the displacement and body forces do not depend on y and if be the displacement at any point into the medium then . The two equations of motions are identically satisfied.

Fig. 1 Geometry of the Problem.

REFERENCE:

1. Abd-Alla, A.M. and Ahmed, S.M.: Propagation of Love waves in a non-homogeneous orthotropic elastic layer under initial stress overlying semi-infinite medium. Appl. Math. Comp. 106 (2-3), 265-275 (1999)

2. Borcherdt, R. D.: Reflection and refraction of type-II S-waves in elastic and anelastic media. Bull. Seism. Soc. Am. 67, 43-67 (1977)

3. Cagniard, L.: Reflection et refraction des ondes seismiques progressive, Paris: Gauthier-villars (1939)

4. Garvin, W. W.: Exact transient solution of the Buried line source problem. Proceeding of Royal Society London A, 234, 1199, 528-541 (1956)

5. Hearn, D. J. and Krebes, E. S.: Complex rays applied to wave propagation in a viscoelastic medium. Pageoph 132, 401-415 (1990a)

6. Hearn, D. J. and Krebes, E. S.: On computing ray-synthetic seismograms for anelastic media using complex rays. Geophysics 55, 422-432 (1990b)

7. Krebes, E. S. and Hron, F.: Ray-synthetic seismograms for SH waves in anelastic media. Bull. Seism. Soc. Am. 70, 29-46 (1980a)

8. Krebes, E. S. and Hron, F.: Synthetic seismograms for SH waves in layered anelastic media by asymptotic ray theory. Bull. Seism. Soc. Am. 70, 2005-2020 (1980b)

9. Le, L.H.T., Krebes, E.S. and Quiroga-Goode, G.E.: Synthetic seismograms for SH waves in anelastic transversely isotropic media, Geophys. J. Int. 116, 598-604 (1994)

10. Liu, K. and Liu, Y.: Propagation of characteristic of Rayleigh waves in orthotropic fluid-saturated porous media. J. Sound Vib. 271 (1-2), 1-13 (2004)

11. Maiti, N. C. and Mitra, M.: Wave propagation from extended, asymmetric surface sources in an elastic half-space. Bull. Seism. Soc. Am. 69 (3), 713-735 (1979)

12. Rangelov, T.V., Manolis, G.D. and Dineva, P.S.: Wave Propagation in a restricted class of orthotropic inhomogeneous half-planes. Acta Mech. 210, 169-182 (2010)

Received on 02.01.2013 Accepted on 13.02.2013

Research J. Science and Tech 5(1): Jan.-Mar.2013 page 216-219