Thermoelastic Waves Propagation in Layered Plates in Anisotropic Media

 

Vijayata Pathania1, S. Pathania2

1Department of Mathematics, H.P.U.R.C.Dharamshala-176215 India

2Department of Mathematics, National Institute of Technology, Hamirpur-177005 India

*Corresponding Author: vijayatapathania@yahoo.com, shwetanithmr@gmail.com

   

ABSTRACT:

Analysis for the propagation of thermoelastic waves in transversely isotropic plates is investigated, commencing with a formal analysis of waves in a layered plate of transversely isotropic media with viscous fluid on both sides, the dispersion relations of thermoelastic waves are obtained by invoking continuity at the interface and boundary of conditions on the surfaces of layered plate. The secular equations for governing the symmetric and antisymmetric wave motion of the plate, in completely separate terms, are derived. Finally, in order to illustrate the analytical results, the numerical solution is carried out for transversely isotropic plate of zinc material bordered with water.  The results have been deduced and compared with the existing one in relevant publications available in the literature at various stages of this work.

 

KEYWORDS: Lamb waves, viscous, relaxation time, Biot’s constant, anisotropic.

 

 

INTRODUCTION:

Anisotropy is exhibited in its purest form in single crystals but also occurs on collections of crystals or minerals which have crystallized or have been deposited with a preferred orientation, or have been subjected to non uniform forces after formation. A layered medium by its very nature is anisotropic in the large, but the individual layers may also be anisotropic in a manner which cannot be handled by a further subdivision into finer layers. We shall consider materials that possess an axis of symmetry in the sense that all rays at right angles to the axis are equivalent. Such media are called transversely isotropic. Many authors such as Wu and Zhu (1991); Zhu and Wu (1995); 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). Recently, Sharma and Sharma (2010) investigated the propagation of thermoelastic Rayleigh waves in a solid half-space and Lamb waves in thermoviscoelastic plate loaded with viscous fluid layer of varying temperature. Deshmukh et al. (2010) studied the thermal stresses in a simply supported plate with thermal bending moments. Lal et al. (2010) investigated the transverse vibrations of nonhomogeneous rectangular plates of uniform thickness using boundary characteristic orthogonal polynomials.     

 

In the present paper, we have discussed 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. More general dispersion equations of Lamb type waves are derived and discussed. The analytical results so obtained have been verified numerically and are illustrated graphically in case of zinc material and water.

 

Conclusion:

It is observed that increasing viscosity of liquid loading magnifies the value of phase velocity of both symmetric and asymmetric modes. The profiles of attenuation coefficient and specific loss factor of acoustic modes are noticed to be highly dispersive. Significant effect of liquid temperature has been observed on specific loss factor of energy dissipation profiles in the considered material plate. This information may also be useful and utilized in ultrasonic applications. 

 

REFERENCE:

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

2.       J. Wu and Z. Zhu (1991). The propagation of Lamb waves in a plate bordered with layers of a liquid. Journal of Acousical  Society of  America 91, pp 861 –867.

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

4.       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), pp. 1-20.

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

6.       K. C.  Deshmukh , M. V.  Khandait , S. D. Warbhe , and V. S.  Kulkarni  (2010). Thermal stresses in a simply supported plate with thermal bending moments. International Journal of Applied Mathematics and Mechanics 6(18), pp. 1 – 12.

7.       R. Lal , Y Kumar, and U S Gupta (2010). Transverse vibrations of nonhomogeneous rectangular plates of uniform thickness using boundary characteristic orthogonal polynomials. International Journal of Applied Mathematics and Mechanics 6, pp. 93 – 109.

8.       S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover , New York, 1961.

9.       S. Kaliski (1963). Absorption of magneto viscoelastic surface waves in a real conductor in magnetic field. Proceedings of Vibration Problems 4, pp. 319–329.

10.    Z. Zhu  and J.  Wu (1995). The propagation of Lamb waves in a plate bordered with a viscous liquid. Journal of Acoustical Society of America 98, pp. 1057-1067.

 

 

Received on 30.01.2013                                                 Accepted on 08.02.2013        

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Research J. Science and Tech 5(1): Jan.-Mar.2013 page 123-129