Generalized Thermoelastic Waves in a Plate Sandwiched with Layers of Viscid Liquid

Vijayata Pathania

Department of Mathematics, H.P.U.R.C.Dharamshala-176215 India

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

ABSTRACT:

Analysis for the propagation of thermoelastic waves in a homogeneous, transversely isotropic, thermally conducting plate bordered with layers of viscous liquid on both sides is investigated in the context of generalized theories of thermoelasticity. Secular equations for homogeneous transversely isotropic plate in closed form and isolated mathematical conditions for symmetric and anti-symmetric wave modes in completely separate terms are derived. The amplitudes of displacement components and temperature change have been computed and studied. Finally, the numerical solution is carried out for transversely isotropic plate of zinc material bordered with water. The amplitudes of displacement and temperature change in case of fundamental symmetric and skew symmetric modes are presented in order to illustrate and compare the theoretical results. The theory and numerical computations are found to be in close agreement.

KEYWORDS: Transversely isotropic, Viscous, Biot’s constant, Displacement, Amplitude

1 INTRODUCTION:

The plates are frequently used as structural components, and their vibration characteristics are obviously important for practical design. 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. Wu and Zhu 1991; Graff 1991 brought out detailed analysis of the characteristics of Lamb waves in elastokinetics The investigations on the propagation of Lamb waves in thermoelastic solid plates bordered by liquid media maintained at uniform temperature have been carried out by Sharma and Pathania 2004. Sharma et al. (2009) investigated the propagation of Lamb waves in viscothermoelastic plate under inviscid fluid layer. Sharma and Sharma 2010 investigated the propagation of thermoelastic Rayleigh waves in a solid half-space and Lamb waves in thermo-viscoelastic plate loaded with viscous fluid layer of varying temperature.

The analysis of Lamb type wave propagation in an infinite homogeneous, transversely isotropic, thermoelastic plate bordered with an viscous liquid layers or half spaces on both sides in the context of generalized (Lord-Shulman (LS) and Green-Lindsay (GL)) theories of thermoelasticity have been discussed. The dispersion equations of Lamb type waves, displacement and amplitude are derived and discussed. The analytical results so obtained have been verified numerically and are illustrated graphically in case of zinc material and water.

2 FORMULATION OF THE PROBLEM:

We consider an infinite homogeneous, transversely isotropic, thermally conducting elastic plate of thicknessinitially at temperature, having a viscous liquid layer of thicknesson both sides. We take origin of the co-ordinate system o-xyz on the middle surface of the plate. The x-y plane is chosen to coincide with the middle surface and the z-axis normal to it along the thickness and is taken as axis of symmetry. We take x-z plane as the plane of incidence. The basic governing equations for homogeneous transversely isotropic solid plate in generalized thermoelasticity, in the absence of body forces and heat sources and for viscous fluid (homogeneous liquid) medium, are given by (Sharma and Pathania 2004; Chandrasekhar 1961) respectively.

8. CONCLUSIONS:

1. The comparison of Fig. 1 and 2 reveals that skew symmetric x-component of displacement (u) is dominant as compared to the symmetric one in this case.

2. It is noticed that the thermal relaxations time has negligibly small effect on the displacement and temperature change.

3. It is also noticed that the displacements u and w has significantly higher magnitude in both symmetric and skew symmetric modes of wave propagation as compared to temperature change in the present situation.

4. This information may also be useful and utilized in ultrasonic applications.

9.REFERENCES:

H. W. Lord and Y. Shulman (1967). The generalized dynamical theory of Thermoelasticity. Journal of the Mechanics and Physics of Solids 15, 299-309.

J. Wu and Z. Zhu (1991). The propagation of Lamb waves in a plate bordered with layers of a liquid. J. Acoust. Soc. Am. 91, 861 –867.

K. F. Graff (1991). Wave Motion in Elastic Solids. Dover, New York.

J. N. Sharma and V. Pathania (2004). Generalized thermoelastic waves in anisotropic plates sandwiched between liquid layers. Journal of Sound and Vibration 278, 383-411.

J.N. Sharma, Rattan Chand, Mohamed I.A. Othman (2009). On the propagation of Lamb waves in viscothermoelastic plates under fluid loadings. International Journal of Engineering Science 47, 391–404.

J. N. Sharma and R. Sharma (2010). Propagation characteristics of Lamb waves in a thermo-viscoelastic plate loaded with viscous fluid layers. International Journal of Applied Mathematics and Mechanics 6(3), 1-20.

Received on 13.01.2014 Accepted on 01.02.2014

Int. J. Tech. 4(1): Jan.-June. 2014; Page 148-154