Circular Waves Propagation in Thermoelastic Solid-Liquid Interface

Vijayata Pathania

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

*Corresponding Author E-mail :vijayatapathania@yahoo.com

ABSTRACT:

The present investigation is to study the propagation of circular waves in a homogeneous isotropic, thermally conducting elastic solid half-space underlying a half-space of inviscid liquid at uniform temperature in the context of coupled theories of thermoelasticity. After developing the mathematical model, the secular equations for circular waves in compact form are derived. The results in the absence of fluid loading and uncoupled theories of thermoelasticity have been obtained as particular cases from the derived secular equations. 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. The MATLAB software tools have been employed for numerical computations. The theory and numerical computations are found to be in close agreement.

KEYWORD:

INTRODUCTION:

The study of dynamic properties of elastic solids is significant in various applications, especially in many engineering phenomena, including the response of soils, geological materials, the ultrasonic inspection of materials, in seismology and vibration of engineering structures. The structural components used in applications involving aerospace, off shore, submarine structures, pressure vessels, civil engineering structures, chemical pipes and even automotive suspension components are frequently exposed to thermal variations in different environments. In high temperature applications, thermal stresses which are induced from heat temperature build up and cooling processes may rise above the ultimate strength and lead to unexpected failures. The theory of thermoelasticity is well-established. The governing field equations in the classical dynamic coupled thermoelasticity CT are wave-type (hyperbolic) equations of motion and diffusion-type (parabolic) equation of heat conduction. A wave-like thermal disturbance is referred to as ‘‘second sound” by Chandrasekharaiah [1]. Nayfeh and Nasser [2] discussed the propagation of surface waves in homogeneous isotropic solids in the context of coupled and generalized thermoelastic bodies. Wu and Zhu [3] brought out detailed analysis of the characteristics of Lamb waves in electrokinetics. Sharma and Singh [4] investigated the circular crested thermoelastic waves in homogeneous isotropic plates, Sharma and Pathania [5] investigated the propagation of crested waves in thermoelastic solid plates immersed in liquid media maintained at uniform temperature. Sharma et al. [6] investigated the propagation of generalized Rayleigh waves in thermo elastic solids loaded with viscous fluid layer of varying temperature. Pathania et al. [7] studied the characteristics of propagation of circular waves in a homogeneous, transversely isotropic, thermally conducting elastic plate bordered with layers of conducting viscous fluid on both sides. The present investigation is concerned with the study of circular crested wave propagation in an infinite homogeneous, isotropic, thermoelastic half space underlying inviscid liquid half space. It is noticed that the motion for circular crested waves is also governed by Rayleigh type secular equations. More general dispersion equations of Rayleigh waves are derived and discussed in coupled and uncoupled theory of thermoelasticity. Numerical solution of the dispersion equations for Aluminium epoxy material has been carried out and presented graphically.

FORMULATION OF THE PROBLEM:

We consider a homogeneous isotropic, thermally conducting elastic solid half space 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 -axis points vertically downward into the solid half space (represented by). We choose -axis in the direction of wave propagation in such a way that all particles on a line parallel to -axis are equally displaced. Therefore, all the field quantities are independent of-coordinate. Further, it is assumed that the disturbances are small and are confined to the neighborhood of the interface and hence vanish as. The basic governing equations of coupled thermoelasticity in the absence of body forces and heat sources in non-dimensional form can be written as Noda et al. [8]

Fig 1: Variation of phase velocity with wave number

Fig.1 represents the plot of non-dimensional phase velocity of Rayleigh waves with non-dimensional wave number in case of coupled and uncoupled theories of thermoelasticity respectively. It is evident from Fig.1 that the phase velocity profile of Rayleigh waves shows decreasing trend with the increasing wave number in the considered cases before these become steady, stable and asymptotic afterwards to the reduced Rayleigh wave velocity. The phase velocity of the waves in coupled thermoelasticity is higher than the uncoupled thermoelasticity at the short wave numbers which is the distinctive feature of Rayleigh surface waves.

Fig 2 represents the variation of attenuation coefficients with respect to non-dimensional wave number for coupled and uncoupled 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 both the cases i.e. for coupled and uncoupled

Fig 2: Variation of attenuation coefficient with wave number

thermoelasticity. As the wave number ascends, it increases monotonically to attain maximal value at R=3 and 3.5 in case of uncoupled and coupled thermoelasticity respectively, and then it slashes down to zero with the increasing wave number. The maximum value of the attenuation coefficient decreases in case of the coupled thermoelasticity and is quite high for the uncoupled thermoelasticity. The positions of maximum amplitude are shifted towards the higher value of wave number as the wave progresses from uncoupled to coupled thermo elasticity.

REFERENCES:

1. Chandrasekharaiah, D.S. 1986: Thermo elasticity with second sound-a review, Appl. Mech. Rev. 39, 355–376.

2. Nayfeh, A. and Nasser, S. N., 1971: Thermoelastic waves in solid with thermal relaxation, Acta Mech., 12, 53-69.

3. Wu, J. R. and Zhu, Z. M., 1992: The propagation of Lamb waves in a plate bordered with layers of a liquid, J. Acous. Soc. Am., 91, 861-867.

4. Sharma, J.N. and Singh, D., 2002: Circular crested thermoelastic waves in homogeneous isotropic plates, Journal of Thermal stresses, 25, 1179-1193.

5. Sharma, J.N. and Pathania, V., 2005: Crested waves in thermoelastic plates immersed in liquid, J. Vib. Control, 11, 347–370.

6. Sharma, J.N., Sharma, R. and Sharma, Y.D., 2008, Generalized Rayleigh waves in thermoelastic solids under viscous fluid loading, 13, 217-238.

7. Pathania, S., Sharma P. K. and Sharma, J. N. 2012: Circular waves in thermoelastic plates sandwiched between liquid layers, Journal of International Academy of Physical Sciences, 16, 211-225.

8. Noda, N., Furukawa, T. and Ashida, F., 1989: Generalized thermoelasticity in an infinite solid with a hole. J. Thermal Stresses, 12, 385-402.

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Received on 20.11.2016 Modified on 25.11.2016

DOI: 10.5958/2349-2988.2017.00030.4

Research J. Science and Tech. 2017; 9(1):179-183.