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Reflection and Transmission of Thermo-Viscoelastic Plane Waves at Liquid-Solid Interface
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Reflection and Transmission of Thermo-Viscoelastic Plane Waves at Liquid-Solid Interface

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The present paper is aimed at to study the reflection and transmission characteristics of plane waves at liquid-solid interface. The liquid is chosen to be inviscid and the solid …

The present paper is aimed at to study the reflection and transmission characteristics of plane waves at liquid-solid interface. The liquid is chosen to be inviscid and the solid
half-space is homogeneous isotropic, thermally conducting viscoelastic. Both classical (coupled) and non-classical (generalized) theories of linear thermo-viscoelasticity have been employed to investigate the characteristics of reflected and transmitted waves. Reflection and transmission coefficients are obtained for quasi-longitudinal ( qP ) wave. The numerical computations of reflection and transmission coefficients are carried out for water-copper structure with the help of Gauss-elimination by using MATLAB software and the results have been presented graphically.

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  • 1. AMAE Int. J. on Production and Industrial Engineering, Vol. 02, No. 01, April 2012 Reflection and Transmission of Thermo-Viscoelastic Plane Waves at Liquid-Solid Interface R Kaur and J N Sharma Department of Mathematics, National Institute of Technology, Hamirpur- 177005 India Email: kaur.rajbir22@gmail.com Email: jns@nitham.ac.inAbstract- The present paper is aimed at to study the reflection II. FORMULATION OF THE PROBLEMand transmission characteristics of plane waves at liquid-solidinterface. The liquid is chosen to be inviscid and the solid We consider a homogeneous isotropic, thermallyhalf-space is homogeneous isotropic, thermally conducting conducting, viscoelastic solid in the undeformed state initiallyviscoelastic. Both classical (coupled) and non-classical(generalized) theories of linear thermo-viscoelasticity have at uniform temperature T0 , underlying an inviscid liquid halfbeen employed to investigate the characteristics of reflected space. We take the origin of the co-ordinate systemand transmitted waves. Reflection and transmission (x, y, z) at any point on the plane surface (interface) andcoefficients are obtained for quasi-longitudinal ( qP ) wave. the z -axis pointing vertically downward into the solid halfThe numerical computations of reflection and transmission space which is thus represented by z  0 . We choose the x-coefficients are carried out for water-copper structure withthe help of Gauss-elimination by using MATLAB software axis along the direction of wave propagation in such a wayand the results have been presented graphically. that all the particles on the line parallel to the y-axis are equally displaced. Therefore, all the field quantities are independentKeywords- Reflection, Transmission, Viscoelastic Solid, of y -co-ordinate. Further, the disturbances are assumed toInviscid fluid, Critical angle. be confined to the neighborhood of the interface z  0 and I. INTRODUCTION hence vanish as z   . In the linear theory of homogeneous isotropic, the basic governing field equations The problems of reflection and transmission of waves at of motion and heat conduction for solid and liquid (inviscid)an interface between liquid and solid media has many medium, in the absence of heat sources and body forces, areapplications in under water acoustics and seismology. Ewing given byet al. [1], Hunter et al. [2] and Flugge [3] used mathematicalmodels to accommodate the energy dissipation due to viscouseffects in vibrating solids. Acharya and Mondal [4]investigated the propagation of Rayleigh surface waves in aVoigt-type [5] viscoelastic solid under the linear theory ofnon local elasticity. Schoenberg [6], Lockett [7], Cooper andReiss [8] and Cooper [9] have investigated the problems ofreflection and transmission of waves at an interface betweenviscoelastic isotropic media. where In order to eliminate the paradox of infinite velocity ofthermal signals in classical (coupled) thermoelasticity, Lordand Shulman [11] and Green and Lindsay [12] proposednonclassical (generalized) theories of thermoelasticity whichpredict a finite speed for heat propagation. Sharma, et al. [13]studied the reflection of piezothermoelastic waves from thecharge free and stress free boundary of transversely isotropichalf space. Here  ,  are Lame’s parameters,  0 and  1 are thermo- In this paper, we discuss the reflection and transmission viscoelastic relaxation times and  t is the coefficient of lin-of plane waves at the interface between inviscid liquid half-space and thermo-viscoelastic solid half-space. The effects ear thermal expansion.  is the density of the solid,of incident angles and fluid loading on reflection and T ( x, z , t ) is the temperature change andtransmission coefficients are considered. The analyticalresults so obtained have been verified numerically and are is the displacement vector; K isillustrated graphically. the thermal conductivity; C e is the specific heat at constant© 2012 AMAE 8DOI: 01.IJPIE.02.01.44
  • 2. AMAE Int. J. on Production and Industrial Engineering, Vol. 02, No. 01, April 2012strain of the solid; t 0 and III. BOUNDARY CONDITIONS t1 are thermal relaxation times;  L is the bulk modulus, The boundary conditions at z  0 can be expressed  L and   are the density and  as  zz   p ,  xz  0 , w  wL , T, z  H (T  TL )  0coefficient of volume thermal expansion, u L is the velocity where H is the Biot’s heat transfer constant. (13)vector and TL is the temperature deviation in the liquid tem- IV. SOLUTION OF THE PROBLEMperature from ambient temperature T0 ;  jk is the We assume wave solutions of the formKronecker’s delta with k  1 for LS theory and k  2 forGL theory. The superposed dot notation is used for timedifferentiation. To facilitate the solution we define thefollowing dimensionless quantities.  where c  is the non dimensional phase velocity,  is k the frequency and k is the wave number. . Upon using solution (14) in equations (8)-(12), we obtain a system of algebraic equations in unknowns A, B, C and D. The condition for the existence of non-trivial solution of this system of equations upon solving provide uswhere where Here   is the characteristic frequency of the solid plate; is the thermomechanical coupling constant and c1 , c2are respectively, the longitudinal and shear wave velocitiesin the thermoelastic solid half-space; L is the In the absence of viscous effects (  0  0   1 ) andthermomechanical coupling and c L is the velocity of sound thermal field (T  0   , TL  0   L ) , we havein the fluid. Upon using quantities (5) alongwith the relations A. qP -WAVE INCIDENCE UPON A PLANE SURFACE Let the suffix i and r represent incident and reflected waves,in equations (1)-(4), we get respectively. Omitting the term exp(it ) , we can write© 2012 AMAE 9DOI: 01.IJPIE.02.01.44
  • 3. AMAE Int. J. on Production and Industrial Engineering, Vol. 02, No. 01, April 2012where L L  (29)  2 V. NUMERICAL RESULTS AND DISCUSSIONS Upon using equations (18)-(22) in the boundary conditions In this section the reflection and transmission coefficients(14) alongwith the fact that all the waves, incident, reflectedand transmitted must be in phase at the interface z  0 for for qP wave incidence at an interface between thermo- viscoelastic solid and inviscid fluid have been computedall values of x and t , we get numerically. The material chosen for this purpose is Copper, the physical data for which is given by Sharma, et al. [14]This with the help of equation (15) implies that   0.00265,   8.2  1010 Nm 2 ,   4.2  1010 Nm 2 ,   8.950  10 3 kg m 3 , The equation (25) is modified Snell’s law in this situation.In the absence of thermal field, viscous effect and liquid, (25) K  1.13102 Cal m1s 1 K 1 ,  T  1.0  10 8 K 1 ,becomes  0   1  6.8831 10 13 K , T 0 300 K The liquid chosen for the purpose of numerical calculations is water, the velocity of sound in which is givenThe analytical expression of reflection and transmission by c L  1.5  10 3 m / s and density iscoefficients R1qP  A4 / Ai1 a nd  L  1000 kg m 3 . T0  298K .T kqP  Ark / Ai1 ( k  1, 2 , 3) in the presence of thermal Figs. 1 and 2 yields the behaviour of reflection / transmission coefficients for the angles of incidence offield for incident qP wave are obtained as longitudinal wave propagates from fluid into solid. It is 1 3 4 observed that for longitudinal wave incidence, the reflectedT1qP  , qP qP  2 , T3   , R1   (27) longitudinal wave passes through a minimum at critical angles  T2  qPwhere    50 0 for elastic case which is known as Rayleigh-wave angle. At this angle a wave with large surface components is generated. These results parallel those obtained by Mott [10] in the analysis of incidence at a water-stainless steel interface, under the influence of dissipation.and  1 ,  2 ,  3 ,  4 can be obtained from  by replacingfirst, second ,third and fourth column by  a14 a 24 a34 0 respectively. .here a11  cos 2 3 , a12  cos 2 3a13   sin 2 3 , a14   2 La 21   1 2 a12 sin 21 , a 22   2  2 a 2 sin 2 2  2a 23   0 cos 2 3 , a31  a1 cos 1a32  a 2 cos 2 , a33  a3 sin  3 , ia34  a 4 cos 4 , a 41  i  a1 cos 1  H , Figure 1. qP -wave incidence at the interface (TVE/E) 0 ia 42  i a2 cos 2  H , a  0 , a   HS ,  0 43 44 L© 2012 AMAE 10DOI: 01.IJPIE.02.01.44
  • 4. AMAE Int. J. on Production and Industrial Engineering, Vol. 02, No. 01, April 2012 [2] C. Hunter, I. Sneddon and R. Hill, Viscoelastic Waves: Progress in Solid Mechanics, North Interscience, Amsterdam, New York, 1960. [3] W. Flugge, Viscoelasticity, Blasdell, London, 1967. [4] D. P. Acharya and A. Mondal, “Propagation of Rayleigh surface waves with small wave-lengths non-local viscoelastic solids,” Sadhana, vol. 27, pp. 605-612, 2002. [5] W. Voigt, “Theortische student uberdie elasticitats verhalinisse krystalle,” Abhandlungen der Gesellschaft der Wissenschaften zu Goettingen vol. 34, 1887. [6] M. Schoenberg, “Transmission and reflection of plane waves at an elastic-viscoelastic interface,” Geophys. J. Royal Astron. Soc., vol. 25, pp. 35-47, 1971. [7] F. J. Lockett, “The reflection and refraction of waves at an interface between viscoelastic materials,” J. Mech. Phys. Solids, vol. 10, pp. 53-64, 1962. [8] H. F. Cooper and E. L. Reiss, “Reflection of plane viscoelastic waves from plane boundaries,” J. Acoust. Soc. Am., vol. 39, pp. 1133-1138, 1966. Figure2. qP -wave incidence at the interface (VE/E) in the [9] H. F. Cooper, “Reflection and transmission of oblique plane absence of thermal field waves at a plane interface between viscoelastic media,” J. Acoust. Soc. Am., vol. 42, pp. 1064-1069, 1967. CONCLUSIONS [10] G Mott, “Reflection and refraction coefficients at a Fluid- Solid interface,”. J. Acoust. Soc. Am., pp. 819-829, 1970. The reflection and transmission of plane waves at inviscid [11] H.W. Lord and Y. Shulman, “A generalized dynamical theoryliquid- thermoviscoelastic solid interface has been analyzed of thermoelasticity,” J. Mech. Phys. Solids, vol. 15, pp. 299-309,theoretically. The significant effect of incident angle, thermal, 1967.viscosity and presence of liquid on the amplitude ratios of [12] A. E. Green and K. A. Lindsay, “Thermoelasticity,” J. Elast.,reflected and transmitted waves have been observed. vol. 2, pp. 1-7, 1972.Rayleigh angle phenomenon is explained. It is shown that [13] J. N. Sharma, V. Walia and S. K. Gupta, “Reflection ofreflected surface wave exist for incidence angles greater than piezothermoelastic waves from the charge and stress free boundaryRayleigh wave angle as explained in [8]. of a transversely isotropic half-space,” Int. J. Engng. Sci., vol. 46, pp. 131-146, 2008. [14] J. N. Sharma and R. Sharma, “Propagation characteristics of REFERENCES Lamb waves in a thermo-viscoelastic plate loaded with viscous[1] M. Ewing, W.S. Jardetzky and F. Press, Elastic Waves in Layered fluid layers,” Int. J. of Appl. Math and Mech., vol. 6, pp. 1-20,Media, McGraw, New York, 1957. 2010.© 2012 AMAE 11DOI: 01.IJPIE.02.01.44

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