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Journal of Applied Mathematics
Volume 2010, Article ID 817680, 11 pages
http://dx.doi.org/10.1155/2010/817680
Research Article

Propagation of Elastic Waves in Prestressed Media

1Department of Mathematics, J. V. M. G. R. R. (P.G) College, Charkhi Dadri, Haryana 127306, India
2Department of Mathematics, The Technological Institute of Textile and Sciences, Bhiwani 127021, India
3Department of Electronic & Instumentation, The Technological Institute of Textile and Sciences, Bhiwani 127021, India

Received 13 June 2010; Accepted 29 October 2010

Academic Editor: George Jaiani

Copyright © 2010 Inder Singh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

3D solutions of the dynamical equations in the presence of external forces are derived for a homogeneous, prestressed medium. 2D plane waves solutions are obtained from general solutions and show that there exist two types of plane waves, namely, quasi-P waves and quasi-SV waves. Expressions for slowness surfaces and apparent velocities for these waves are derived analytically as well as numerically and represented graphically.

1. Introduction

In fact, the Earth is prestressed medium, due to many physical causes, that is, gravity variation, slow process of creep and variation of temperature, and so forth. Therefore, these problems are of much interest to seismologists due to its application in mineral prospecting and prediction of earthquakes.

For studying the propagation of elastic waves in prestressed solids of infinite extent, Sidhu and Singh [1, 2], Norris [3], and Day et al. [4] used a much simpler form of equation of motion of Biot [5]. The medium considered by earlier investigators is a homogenous prestressed with incremental elastic coefficients possessing orthotropic anisotropy due to normal components of initial stresses.

In the present paper, 3D problem of propagation of , SV and SH waves for a homogeneous prestressed medium is discussed. The model of prestressed medium used here is much more general than that used by earlier investigators.

The 2D plane waves solutions are obtained from general solutions for different conditions of initial stresses with or without external forces. Graphs of slowness surfaces are derived.

2. Basic Equations

Consider a homogenous prestressed solid. The state of prestresses is, therefore, defined by six components, that is, , and . Let all stress components be functions of . The state of initial stress introduces anisotropy so that even for an initially isotropic solid defined by two Lame’s constants , the number of the incremental elastic coefficients will be always larger than 2. Let be components of body forces along coordinate axes, respectively, where , , are all constant. The general form of dynamical equation for prestressed solids in the presence of external forces is given by Biot [5, page 52]: The two other equations are obtained by cyclic permutation of , , , 1, 2, 3 and , , . is the density and the displacement components along the axes. are the components of incremental body force, which are assumed to satisfy the equation The are the components of prestress, which are assumed to satisfy the equilibrium equation. and are related to the initial strain by Hooke’s law The incremental stresses are supposed to be linearly related to the incremental strains through the incremental elastic coefficients and We assume that Substituting (2.2), (2.5), and (2.6) in (2.1), we have The two other equations are obtained from (2.7) by cycle permutation of , , , 1, 2, 3, , , and , , .

Here

3. Propagation of Waves

For elastic waves propagating in a direction specified by direction cosines along the axes, we take are amplitude factors along the axes and is phase factor: where is phase velocity and is wave number.

Putting (3.1) and (3.2) in (2.7), we get where The explicit expressions for ,,  ,,, and are obtained from (3.4)–(3.9), by cyclic permutation of , , , 1, 2, 3, , , and , , , respectively.

Setting the determinant of the coefficients of , and of (3.3) equal to zero, on simplification, we get the cubic equation in : where

4. Special Cases

Two special cases may be dealt with immediately.

4.1. Three Dimensional Prestressed Medium
4.1.1. Propagation of Waves along the Unique Axis

Putting , in (3.3)–(3.8) and in the expressions of , ,  ,,, and , we get Equation (3.10) takes the form where

4.1.2. In the Absence of Body Forces

When and ,, , and so forth are all constant, then (4.3) becomes, with the help of (4.2) and (4.4),

4.1.3. Prestressed Is Defined by Normal Components

There are only normal components of prestress present in the medium, that is putting in (4.1), (4.5) takes the form on simplification, we get three values of :

4.2. Two Dimensional Prestressed  Medium
4.2.1. Plane Waves Solution

Here, we consider the behaviour of plane waves in -plane perpendicular to the -axis; putting in (3.4)–(3.9) and in the expressions of and so forth, we get the set of equations from (3.3): where Equation (4.8) has nontrivial solution when It is quadratic equation in , and it has two values of corresponding to quasi-SV waves and quasi- waves.

In the absence of body forces and when , , and so forth are constants, then from (4.8) and (4.9), we get If , then (4.11) takes the form where The set of homogeneous (4.12) in , has a nontrivial solution when This quadratic equation in may be solved to obtain Thus, in general, in this two-dimensional model of the prestressed medium, there exist two types of plane waves, namely, quasi- waves and quasi-SV waves whose phase velocities correspond to upper and lower signs of (4.15).

4.2.2. Propagation of Plane Waves in Orthotropic Medium

Consider a homogenous prestressed elastic solid. The material is either isotropic in finite strain or anisotropic with orthotropic symmetry. The principal directions of initial stress are chosen to coincide with the directions of elastic symmetry and the coordinate axes. Let the state of uniform initial stresses have principal stresses ,, and . We further assume that and and are constant. The principal stress does not enter explicitly into the equations of motion. Its influence is, however, included indirectly in the values of the incremental elastic coefficients. We put and ; (4.15) can be written as where , where Let and be the values of associated with upper and lower signs in (4.16), corresponding to the velocities for quasi- waves and quasi-SV waves, respectively. Hence these expressions coincide with the expressions obtained by Sidhu and Singh [1] for velocities of quasi- waves and quasi-SV waves.

5. Numerical Calculation and Discussions

Here we consider the model for an initially stressed medium: Using (5.1) in (4.17), we get a nondimensional form of velocity equations as where The apparent velocities for quasi- waves and quasi-SV waves can be obtained from (5.2) as The numerical values of the dimensionless slowness have been calculated from (5.2) assuming that for different values of vary from and for different values of vary from . Figures 1 and 2 show the variation of the dimensionless slowness for the quasi- and quasi-SV waves with the angle of incidence −0.8, −0.6, 0.2, 0.4, 0.6, and 0.8.

817680.fig.001
Figure 1
817680.fig.002
Figure 2

6. Conclusion

The study shows that the phase velocities of quasi- waves and quasi-SV waves are highly affected by the initial stresses present in the medium and also the direction of propagation.

Acknowledgment

The authors are thankful to Dr. George Jaiani, Tbilisi State University, Georgia for his valuable comments and encouragement to prepare the paper.

References

  1. R. S. Sidhu and S. J. Singh, “Comments on “Reflection of elastic waves under initial stress at a free surface: P and SV motion,” Journal of the Acoustical Society of America, vol. 74, pp. 1640–1644, 1983. View at Google Scholar
  2. R. S. Sidhu and S. J. Singh, “Reflection of P and SV waves at the free surface of a prestressed elastic half-space,” Journal of Acoustical Society of America, vol. 76, no. 2, pp. 594–598, 1984. View at Google Scholar
  3. A. N. Norries, “Propagation of plane waves in a prestressed elastic medium,” Journal of the Acoustical Society of America, vol. 47, pp. 1642–1643, 1983. View at Google Scholar
  4. S. Day, N. Ray, and A. Datta, “Propagation of P and S waves and reflection of P in a medium under normal initial stresses,” Geophysical Research Bulletin, vol. 23, pp. 1–28, 1985. View at Google Scholar
  5. M. A. Biot, Mechanics of Incremental Deformations, John Wiley & Sons, New York, NY, USA, 1965.