Effect of Rigid Boundary, Initial Stress and Inhomogeneity on the Propagation of Torsional Surface Waves

Amares Chattopadhyay*, Sudarshan Dhua, Sanjeev A. Sahu

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

*Corresponding Author: amares.c@gmail.com, dhuasudarshan@gmail.com; ism.sanjeev@gmail.com

ABSTRACT:

This paper is developed to investigate the effect of presence of rigid boundary on the propagation of torsional surface waves in non homogeneous anisotropic half-space under initial stress. It has been assumed that the rigidity, initial compressive stress and density vary linearly with depth. The medium with assumed conditions allow the propagation of torsional surface waves. The dispersion relation has been obtained in a closed form and presented by means of graphs. It has been observed that when the initial stress increases, the velocity of torsional wave decreases and the increase of anisotropy factor increase the velocity. Also the velocity of torsional waves increases with the increases nonhomogeneity parameters.

KEY WORDS: Rigid boundary, Initial stress, Inhomogeneity, Torsional Surface Waves, Dispersion relation.

INTRODUCTION:

The study of surface waves in a half-space is important to seismologists due to its possible applications in Geophysical prospecting and in understanding the cause and estimation of damage due to earthquakes. Apart from the well known Love and Rayleigh waves, another type of surface wave may be available in non-homogeneous earth, known as torsional surface waves. These waves are horizontally polarized but give a twist to the medium when they propagate. The existence of torsional waves in heterogeneous elastic half-space considering the quadratic and linear variation for shear modulus and density has been discussed by Meissner [10]. Vardoulakis [13] has studied that torsional surface waves also propagate in Gibson’s half-space where the shear modulus varies linearly with depth but the density remains constant. Georgiadis et al. [7] have demonstrated that torsional surface waves do exist in a gradient elastic half-space. Selim [11] has discussed the propagation of torsional surface waves in heterogeneous half-space with irregular free surface. The propagation of torsional surface waves in an elastic half-space with void pores has been studied by Dey et al. [4]. The propagation of torsional surface wave in an initially stressed cylinder has been discussed by Dey and Dutta [5]. Chattopadhyay et al. [2] investigated the propagation of torsional waves in heterogeneous layer over a heterogeneous half-space, while the propagation of torsional surface waves in heterogeneous anisotropic medium with constant density and variable rigidity.

The development of initial stresses in the medium is due to many reasons, for example resulting from the difference of temperature, process of quenching, slow process of creep, differential external forces, gravity variations etc. These stresses have a pronounced influence on the propagation of waves as shown by Biot [1]. The study of surface waves in an initially stressed medium is of interest not for theoretical taste only but for practical purposes too. Based on the pioneering work of Biot [1] on pre-stressed solids, various studies of body and surface wave propagation in the pre-stressed solids have been carried out by many researchers such as Chattopadhyay et al. [3], Kar and Kalyani [9], Dey and Addy [6] and Roy [12].

Inside the Earth, a very hard layer (also known as “rigid”) is present. Since the composition of the Earth is heterogeneous including a very hard layer, the rigid interface plays significant roles in the propagation of the seismic waves. Gupta et al. [8] discussed the effect of rigid boundary on propagation of torsional surface waves in porous elastic layer by.

This paper investigates the possibility of propagation of torsional surface waves in anisotropic and non-homogeneous elastic half-space with rigid boundary. The medium has been considered pre-stressed and the variation in rigidity and density has been taken linear with the depth. As a result, we conclude that the increase of initial stress parameter decreases the velocity of torsional surface waves and the increase of anisotropy factor increases the velocity of torsional waves significantly. Also the velocity of torsional waves increases with the increases nonhomogeneity parameters. This study may have some possible applications in geophysics and seismology.

Note: A₁, A_2, A₃ are dimension less.

Numerical Calculations and Discussion:

To study the effect of the anisotropy, non homogeneity and initial stress on the propagation of torsional surface wave under rigid boundary, numerical calculations have been performed with different values of parameters representing the above characteristic from equation (12). Equation (12) is a quadratic equation in . For the calculation we have taken the following data and

The results have been presented graphically in the figs. 2, 3, 4 for three different types of materials namely (1) Isotropic elastic non homogeneous material (2) Anisotropic Sandstone and (3) Anisotropic Quartz respectively. The values of are given in the table.

In figs. 2, 3 and 4 has been plotted against Fig. 1 shows that two wave front increases when increases. Also the curves show that the increase of initial stress parameter Q decreases the velocity of torsional surface waves and same happens for the other two Figures. Also from the figs. 2, 3 and 4 it has been seen that the increase of (the anisotropy factor) increases the velocity of propagation. Also velocity of torsional surface wave is always remains more than the velocity of shear wave of the medium.

Table 1: For isotropic elastic non homogeneous material.

Curve no.

(dynes/cm²)

For

the root

R₁

R₂

0.4

R₁

R₂

0.8

R₁

R₂

Fig. 2: Torsional surface wave dispersion curve for linear variation of rigidity, initial compressive stress and density for isotropic elastic non homogeneous material with rigid boundary

Table 2: For anisotropic sandstone

Curve no.

(dynes/cm²)

For the root

R₁

R₂

0.4

R₁

R₂

0.8

R₁

R₂

Fig.3: Torsional surface wave dispersion curve for linear variation of rigidity, initial compressive stress and density for anisotropic sandstone material with rigid boundary

Table 3: For Anisotropic Quartz material.

Curve no.

(dynes/cm²)

For the root

R₁

R₂

0.4

R₁

R₂

0.8

R₁

R₂

Fig.4: Torsional surface wave dispersion curve for linear variation of rigidity, initial compressive stress and density for anisotropic quartz material with rigid boundary

REFERENCE:

1. Biot, M.A.:The influence of initial stress on elastic wave, J. Appl. Phys. 11 (1940) 522–530.

2. Chattopadhyay, A., Gupta, S., Kumari, P. Sharma, V.K.: Propagation of torsional waves in an inhomogeneous layer over an inhomogeneous half space. Meccanica. 46(4), 671-680 (2011).

3. Chattopadhyay, A., Bose, S., Chakraborty, M. : Reflection of elastic waves under initial stress at a free surface, J. Acoust Soc. Am. 72,255–263 (1982).

4. Dey, S., Gupta, S., Gupta, A.K. : Torsional surface waves in an elastic half-space with void pores, International Journal for Numerical and Analysis Methods in Geomechanics 17 (3), 97–204(1993).

5. Dey, S., Dutta, A.: Torsional wave propagation in an initially stressed cylinder, Proceedings of the Indian National Science Academy 58A (5) 425–429(1992).

6. Dey, S., Addy, S.K. : Reflection of plane waves under initial stress at a free surface, Int. J. Non-linear Mech. 12 371–381(1977).

7. Georgiadis, H.G., Vardoulakis, I., Lykotrafitis, G. : Torsional surface waves in a gradient-elastic half-space, Wave Motion, 31 (4) ,333–348(2000).

8. Gupta, S., Chattopadhyay, A., Kundu, S.: Effect of rigid boundary on propagation of torsional surface waves in porous elastic layer, Appl. Math. Mech.,32(3), 327–338 (2011).

9. Kar, B.K., Kalyani, V.K.: Reflection and refraction of SH-waves due to the presence of a sandwiched initially stressed sandy layer, Geophys. Res. Bulletin 25, 117–124(1987).

10. Meissner, E.: Elastic oberflachenwellen mit dispersion in einem inhomogeneous medium. Viertelgahrsschriftden Naturforschender Ge-sellschaft in Zurich. 66, 181-185 (1921).

11. Selim, M.M.: Propagation of torsional surface waves in heterogeneous half-space with irregular free surface, Applied Mathematical Sciences 1 (29) 1429–1437(2007).

12. Roy, P.P. :Wave propagation in a thinly two layered medium with stress couples under initial stresses, Acta Mech. 54 ,1–21(1984).

13. Vardoulakis, I.: Torsional surface wave in inhomogeneous elastic media. Int. J. Numer. Anal. Methods Geomech. 8, 287-296 (1984).

14. Whittaker, E.T., Watson, G.N.: A course of modern analysis. Cambridge University Press. (1991).

Received on 06.01.2013 Accepted on 05.02.2013

Research J. Science and Tech 5(1): Jan.-Mar.2013 page 160-164