Rayleigh Waves in Isotropic Medium Loaded with Inviscid Fluid of Varying Temperature

V. Pathania

H.P.U. R. C. Khaniyara, Dharamshala-176218

*Corresponding Author Email: vijayatapathania@yahoo.com

ABSTRACT:

The paper focuses on the propagation of Rayleigh waves in a homogeneous isotropic, thermally conducting elastic solid half-space underlying a half-space of inviscid liquid at varying temperature in the context of coupled theory of thermoelasticity. The fundamental equations for coupled thermoelasticity with boundary conditions are derived. After developing mathematical model, the secular equations for Rayleigh waves in compact form are derived. Finally, in order to illustrate the analytical results, the numerical calculations have been carried out for aluminum-epoxy composite material solid half-space underlying an inviscid liquid half-space and presented graphically.

KEYWORDS:

1 INTRODUCTION:

Surface acoustic waves which propagate along the surface are known as Rayleigh waves. Rayleigh waves, also called ground roll, are surface waves that travel as ripples with motions that are similar to those of waves on the surface of water. Ground roll is a particular type of Rayleigh wave that travels along or near the ground surface and is usually characterized by relatively low velocity, low frequency, and high amplitude. Rayleigh wave are distinct from other types of seismic waves, such as P-waves and S-waves or Love waves and travel the surface of a relatively thick solid material penetrating to a depth of one wavelength approximately. These waves combine both longitudinal and transverse motion to create an elliptic orbital motion. Rayleigh waves are useful because they are very sensitive to surface defects. These waves can be used to inspect areas that other waves might have difficulty in reaching. The existence of Rayleigh waves was predicted by Lord Rayleigh [1]. Nayfeh and Nasser [2] studied the propagation of plane harmonic and Rayleigh waves in thermoelastic solids in the context of generalized thermoelasticity. The influence of viscous fluid loading on the propagation of leaky Rayleigh wave in the presence of heat conduction effects was studied by Qi [3]. Love [4] gave the first comprehensive treatment of dispersion of Rayleigh and Love waves in the case of an elastic solid half space covered by a single solid layer. According to Andle and Vetelino [5], the development of micro-acoustic wave sensors in biosensing have created the need for further investigation into surface wave propagation in a fluid loaded layered medium. Plona et al. [6] studied Rayleigh and Lamb waves at liquid-solid boundaries. Sharma and Pathania [7] studied the propagation of Rayleigh-Lamb waves in homogeneous isotropic plates bordered with layers of inviscid liquid on both sides in the context of different generalized theories of thermoelasticity. Sharma et. al. [8] investigated the propagation characteristics of Rayleigh surface waves in microstretch thermoelastic continua under inviscid fluid loadings. Pathania et. al. [9] studied the generalized thermoelastic waves in anisotropic plates sandwiched between viscous liquid layers.

In the present paper, the Rayleigh surface wave propagation in thermoelastic solids under inviscid fluid loading with varying temperature has been discussed. The coupled theory of thermoelasticity is employed to understand the effect of thermomechanical coupling. More general dispersion equations of thermoelastic Rayleigh type waves are derived. Numerical solutions of the dispersion equations for an aluminium epoxy composite have also been presented.

2 FORMULATION OF THE PROBLEM:

We consider a homogeneous isotropic, thermally conducting elastic solid in the undeformed state initially at uniform temperature , underlying an inviscid liquid half space. We take O as the origin of the coordinate system on the plane surface (interface) and points vertically downward into the solid half space (represented by ). We choose in the direction of wave propagation in such a way that all particles on a line parallel to be equally displaced. Therefore, all the field quantities are independent of .

Fig. 2 represents the plot of non-dimensional phase velocity of Rayleigh waves with non-dimensional wave number (R) in case of high temperature, low temperature and uniform temperature respectively, in coupled thermoelasticity (CT) It is evident from Fig. 2 that the phase velocity profiles of Rayleigh waves shows decreasing trend with the increasing wave number in all the three considered cases before these become steady, stable and asymptotic afterwards to the reduced Rayleigh wave velocity. The phase velocity of waves increases with an increase in temperature at short wave numbers which is the distinctive feature of Rayleigh surface waves.

Fig 3 represents the variation of attenuation coefficients with respect to non-dimensional wave number for coupled thermoelastic Rayleigh waves propagation in an infinite half-space solid underlying a homogeneous infinite half-space liquid. The attenuation coefficient profile has zero value at vanishing wave number in case of high temperature, uniform temperature and low temperature. After that it increases monotonically to attain maximal value at R=1, 3 and 3.5 in case of high temperature, uniform temperature and low temperature respectively, and then slashes down to zero with increasing wave number. The maximum value of attenuation coefficient decreases in case of low and high temperature and is quite high at uniform temperature. The positions of maximum amplitude are shifted towards the higher value of wave number as the wave progresses in high temperature, uniform temperature and low temperature respectively.

8 REFERENCES:

1. Rayleigh, L., On waves propagated along the plane surface of an elastic solid, Proc. London Math. Soc., 17 (1885) 4-11.

2. Nayfeh A.H., Nasser S.N., Thermoelastic waves in solids with thermal relaxation, Acta Mechanica 12 (1971) 53–69.

3. Qi Q., Attenuated leaky Lamb waves, Journal of Acoustical Society of America, 95 (1994) 3222–3231.

4. Love, A.E.H, Some problems of Geodynamics, Cambridge University Press London (1926).

5. Andle, J.C., Vetelino, J.F., Acoustic wave biosensors. Sens. and Actuat., A, 44 (1994) 167-176

6. Plona, T.J., Behravesh, M. and Mayer, W.G., Rayleigh and Lamb waves at liquid-solid boundaries, Ultrasonics, 13 (1975) 171-174.

7. Sharma, J. N. and Pathania, V., Propagation of leaky surface waves in thermoelastic solids due to inviscid fluid loadings, J. Thermal Stresses, 28 (2005) 485-519.

8. Sharma, J.N., Kumar, S. and Sharma, Y.D., Propagation characteristics of Rayleigh surface waves in microstretch thermoelastic continua under inviscid fluid loadings, J. Thermal Stresses, 31 (2008) 18-39.

9. Pathania, S., Sharma, J. N., Pathania, V. and Sharma, P.K., Generalized thermoelastic waves in anisotropic plates sandwiched between viscous liquid layers, Int. J. of Appl. Math and Mech., 7 (5) (2011) 62-88.

Received on 28.08.2016 Accepted on 16.09.2016

Int. J. Tech. 2016; 6(2): 253-257

DOI: 10.5958/2231-3915.2016.00039.0