On Lamb Waves in a Thermoelastic Plate in the Presence of Ideal Fluid with Varied Temperature

The elastic waves are used to measure defects and elastic properties in solid materials have established great attention, and various important applications have been developed recently. In the non-destructive evaluation of hard materials, the coupling of elastic waves with liquid-loaded materials has appeared as an important field. The acoustic waves that are replicated from the solid-liquid interface have a lot of information regarding the solid structure properties together with the presence of the internal defect, interface quality etc. In deformable-body temperature varies from point to point and with time. This temperature variation is due to the deformation process and exchange of heat with the external medium in which mechanical energy is changed into heat energy. The degradation in thermo-elastic energy results in the damping of elastic body vibrations. Firstly, Lamb [1] developed the theory of waves known as Lamb waves. The Lamb waves are created on the belief that when the solid plate is cladded with the liquid it varies the amplitude and propagation velocity of the Lamb waves in the free boundaries due to viscous and inertial effects of the fluid in the plate. The consequence of the fluid on the Lamb waves propagation in a plate of fixed-width sandwiched between homogeneous liquidha lf-space on its both sides investigated by Schoch [2] and found that some amount of energyin the plate is attached with the fluid in the form of radiation, while the most of energy leftovers in the solid. For plane thermoelastic and magnetothermoelastic waves, an exact solution of the secular equation is derived by Puri [3] and obtained the solutions using approximate expansions for low and high frequencies and small coupling. Plona et. al. [4] investigated Lamb and Rayleigh waves at solid-liquid boundaries and derived that the simple Lamb or Rayleigh mode approach gives unexpected results when the solid and liquid densities are nearly equal. The influence of fluid layers on Lamb waves propagation in a solid plate is discussed by Wu and Zhu [6] and obtained the frequency equation for the same. Exposed to isothermal and insulated conditions, the thermally conducting elastic waves for a rigidly fixed and stress-free homogeneous and an isotropic material in the light of non-classical theories of thermo-elasticity are studied by Sharma et.al. [7]. The time-harmonic Lamb waves propagation in the thermoelastic material fringed with non-viscous fluid loading on the bottom and top of the plate is investigated by Sharma and Pathania [8]. Sharma and Pathania [9] studied the motion of Rayleigh and Lamb waves in thermally conducting elastic plate cladded with

shear horizontal component of waves decouples from the primary stream of wave motion and evaluated the frequency equations for non-leaky Lamb waves, leaky Rayleigh waves and leaky Lamb waves. Propagation of wave in a liquid saturated porous solid with micro-polar elastic skelton at the boundary surface has been evaluated by [10, 11]. Pathania et. al. [12] investigated the thermoelastic waves in anisotropic plates immersed in viscous liquid layers in non-classical theories of thermo-elasticity. Kumar et.al. [13] discussed the circular crested and straight wave motion in micro-stretch thermoelastic plate surrounded by non-viscous fluid coatingson both sides with varying temperatures. Pathania et. al. [14] also studied the characteristics of the circular waves in a homogeneous and transversely isotropic thermoelastic material surrounded by conducting viscous fluid loading layers (or halfspaces) on the top and bottom of the plate. Recently, Barak et al. [15] evaluated the reflection and refraction of wave in two welded contact infinite unbounded half-spaces and the effect of the loosely bounded interface on wave propagation between two half-space has been obtained by Barak and Kaliraman [16]. Here we analyze the motion of Lamb waves in athermally conducting elastic homogeneous and isotropic plate in the presence of an ideal fluid layer on its both sides at varying temperatures. The governing equations are solved in the x-z plane and it is found that there exist three coupled waves in the solid plate and one wave in each liquid layer. For Lamb waves, the frequency equation is solved analytically for the classical theory of thermo-elasticity and further deduced for uncoupled thermoelasticity. The equations for various regions have been deduced from the secular equation depending upon the type of characteristic roots. The aluminum-epoxy composite materialis selected for the solid plate and water is taken as a fluid to carry out the numerical results. The mathematical and graphical results are closely related to each other.

We consider a thermally conducting elastic homogeneous and isotropic solid material having the thickness 2d. Initially, the solid plate is at an undisturbed state and unvarying temperature 0T. The solid plate is cladded with a homogeneous ideal fluid of thickness h on both borders i.e.on top and the bottom. It is supposed that from the interlayers of the liquid medium no reflection takes place. The origin of the cartesian coordinate system oxyz is taken at any point in the center of the plate. The wave propagates along thex- direction and the field extents remain

explicitly independent of y-coordinate which implies 0,y but depend implicitly on y-coordinate such that the shear component of displacement is non-zero. The z-axis points to a vertically downwards direction along the plate thicknessas illustrated in Figure-1. The constitutive relations and fundamental equations in the light of the classical theory of thermo-elasticity for the plate in the non-attendance ofbody forces and heat sources [8] are, The governing equations and temperature relation in the non-attendance of heat sources and body forces for the fluid medium [13], are given by Where Here the dot over a symbol represents the time derivative and the comma in the subscript denotes the spatial derivative.  , are Lame‘s parameters,  is the density of solid material, (,,)(,0,)uxztuw is the displacement vector, eC denotes the specific heat at a constant strain of the solid material, (,,)Txzt is temperature changeand K is the coefficient of thermal conductivity. Also 32,T where T is the linear thermal expansion. In the same manner, for the liquid, ,,,0,iiiLLLuxztuw is denoting the displacement vector in fluid, L is the density and L is the bulk modulus of liquid respectively. From the ambient temperature *0,TLT is the deviation in temperature of the fluid and *K is the thermal conductivity in fluid layers. Also **3,L where * is the coefficient of volume thermal expansion. The subscript irepresents that liquid takes the value 1 and 2 for the bottom and

Consider the dimensionless quantities as Making use of quantities (5), the dimensionless fundamental equations of motion and energyequation in the solid plate and liquid layers, after suppressing primes, are where Here *is the characteristic frequency of the material plate and *vC denotes the specific heat of the fluid at constant volume.Inthe solid plate 21 ,cc denote the longitudinal and shear wave velocities and Lc denotes the velocity of sound in the liquid,. Also Tand L are the thermomechanical coupling constants in the material plate and fluid layers respectively. To solve the above equations, we consider where ,ss represents the potential functions for the longitudinal and shear waves. In the inviscid fluid layers,the shear motion does not exist, thusin the absence of shear motioninthe fluid we get

where iL is the scalar velocity potential for the bottom and top fluid layers 1,2.i Plugging equations (11) and (12) in equations (6)-(9), we obtain the potential functions ,,,issL temperatures T and LT as

Since the waves are propagating along the x-axis in the positive direction ofthe thermo-elastic plate, we take the solutions as Here /,cc,  and  represent the dimensionless phase velocity, wavenumber andangular frequency of plane waves. Invoking the solutions (18) in equations (13)-(17), the expressions for 1,,,,ssLLTT and 2L are attained as Where Puri [3] firstly obtained the expressions for 2221,aa. Theacoustical pressure vanishes at external boundaries zdh to ensure a bounded solution due to the chosen potential functions 12 and LL. Here 1L and 2L are the solutions of the standing wave and they are solutions of traveling waves in leaky Lamb waves case.

The stress traction, displacement and heat flux at the solid-fluid interfaces dz may be written as:

(ii) The dimensionless shearelement of the stress tensor must be zero, i.e. (iii) The dimensionless normal displacement element of the plate must be equal to thefluid, i.e. (iv) The boundary condition for thermal case is specified by where H is the coefficient of heat transfer.

Plugging therequired interface conditions (24)-(27) at z=d, we obtain anarrangement of eight homogeneous linear equations with eight unknown amplitudes. For the existence ofa non-zero solution of the system of equations, the determinant of the coefficient matrix of these parameters is zero. Theprocedure used by Sharma and Pathania [8] aftersome arithmetical calculations of the determining factortogether with conditions 40m and 421, 1,2,3...2mhnn the dispersion relation for Lamb type waves with varying temperature yields where 44tan();1,2,3, taniiTmdiTmh. The skew-symmetric mode corresponds to by superscripted + sign and symmetric mode corresponds to superscripted – sign. If 0L, i.e. in the nonappearance of fluid layers, equation (28) becomes For a thermally insulated stress-free solid, the secular equation (29) can be obtained by setting 0H, thuswe have and for an isothermal stress-free plate H, thus we get The frequency equations (30) and (31) represent the thermally insulated and isothermal cases for a stress-free thermoelastic plate and matches exactly with already obtained results of Sharma et al. [7] and Sharma and Pathania [8].

The equations (22) can be written as Here depending on whether 2222212222,,,aa or ,1,1,,1222122aac we may have 123,,,mmm to be imaginary, zero, or real. Thus the dispersion equation (28) is transformed as follows. RegionI: For 12, , 1, 1,1caa and accordingly, we get ,i , 1,2,3iimimi. Therefore, equation (28) becomes

Region II: For 1c and the secular equation (28) yields

Region III: For 1c the dispersion equation remains the same as given by equation (28).

For uncoupled thermo-elasticity 210,1,a 122ia thus,221m 121222cim. Therefore, the dispersion equations (28) yields In the absence of varying temperature equation (35) reduces to If 0H i.e. for a thermally insulated plate, the equation (36) reduces to If H i.e. for a isothermal plate, the equation (36) becomes If 0L i.e. in the non-appearance of liquid, equations (37) and (38) respectively reduces to The equations (39) and (40) are the same as obtained by Sharma and Pathania [8], Sharma et. al. [7] and in elastokinetics for stress-free boundary conditions conferred in detail by Graff [5].

The complex transcendental secular equations (28) contain a plethora of information like wave number, phase velocity, attenuation coefficient, etc. To solve these secular equations, we take Here V and Q represent phase velocity and attenuation coefficientof plane waves respectively. Also RiQ, R and Q are real numbers and RV. The exponential ()ixcte in the time-harmonic plane wave solution (18) turns out to be iRxtQx. The various modes of propagating waveforthe attenuation coefficient Q and phase velocity V can be obtained by substitutingequation (41) in dispersion equation (28). The values of the attenuation coefficient Q and phase velocity V are computed by using relation (41).

In this section, we present some numerical results for the aluminum-epoxy composite material with the opinion of demonstrating the theoretical results obtained in the previous sections. The numerical values for the material are given as [8,11]: For numerical calculations, the fluid taken is water. The density of the fluid is 31000LKgm and the sound velocity in the fluid is sec/105.13mcL.

Figure 2 and Figure 3 represent the profiles of phase velocity )(V with respect to the wavenumber R of symmetric and asymmetric modes of wave propagation with ideal (inviscid) liquid loading respectively. Itis noticed from Figure 2 that for the bottommost symmetric mode 0n the phase velocity declines approximately from unity at a long wavelength and with the increasing wave number it moves towards the thermoelastic Rayleigh wave velocity. For other symmetrical modes 1,2,3nnn, the phase velocity accomplishes moderately high values at vanishing wavenumber and at short wavelength, it decreases asymptotically and becomes closer to shear wave velocity. In the case of Figure 3, the value of phase velocity of the lowermost skew-symmetric mode rises from zero at starting wavenumber and the n-remains almost constant and approaches to thermoelastic

wavenumber that sharply decreases and attains steady and asymptotic Rayleigh wave velocity with advanced wavenumber. For skew-symmetric optical modes the profiles of phase velocity with respect to wavenumber follow the same movements as that observed in the case of symmetric one. In both cases, for higher modes the value of phase velocity is found to progress at an amount nearly n-times the phase velocity magnitude of the initial mode. Figure 4 indicates the profiles of phase velocities with respect to liquid loaded temperature change and it is noticed that the propagating phase velocity of non-viscous fluid almost remains constant with fluid loaded temperature change in both symmetric and asymmetric modes except variation in values of the phase velocity and magnitude of symmetrical mode is higher than the magnitude of asymmetric mode. It is also found that the symmetric and asymmetric modes of phase velocity profile have non-dispersive nature with respect to the liquid loaded temperatures i.e. there is no effect of different temperature loading. Figure 5 indicates the variations of attenuation coefficients with respect to the liquid loaded temperature of wave propagation for symmetrical and skew-symmetrical modes. Here we noticed that the attenuation coefficient profile with respect to fluid loaded temperature follows the same movement except for variations in the magnitude of the In both Figures 4 and 5, it is observed that liquid loading temperature has the opposite effect in symmetric as well as the asymmetric mode in case of phase velocity and attenuation coefficient.

The Lamb waves propagation in a thermally conducting elastic homogeneous isotropic plate in the presence of non-viscous fluid layers on its both sides, with varying temperature is studied in the frame of reference of coupled thermo-elasticity and following conclusions are obtained. 1. There exists a coupled system of three types of waves viz. longitudinal waves, the vertical component of transverse waves and waves due to thermal variation in the solid plate. 2. It is found that apart from this coupled system of waves, there is a horizontal component of transverse waves that keeps itself isolated from the rest of the coupled motion and is not affected by the mechanical and thermal load.

4. With the variation in the parameter wave number, we find three different regions of secular equations. 5. For asymmetric mode, the plots of the attenuation coefficient with varied liquid temperatures show a decreasingtrend. Thus, it is inferred that the skew-symmetric mode is more sensitive and quite useful in ultrasonic applications. 6. The present analysis is very much useful in the field of earthquake engineering, soil dynamics, seismology, hydrology and geophysics.

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Department of Mathematics, Indira Gandhi University, Meerpur, Rewari ms_barak@igu.ac.in