Effect of Varying Temperature on Waves in Anisotropic Thermoelastic Plate under Liquid Loading

An infinite speed of heat transference is predicted by the conventional theory of heat conduction which denies the physical proofs. To improve this ambiguity generalized theories are established. For the fixed speed in thermal signals, the generalized theory is formulated by including a flux-rate term in Fourier‘s law of heat conduction by Lord and Shulman [1]. If the considered body has a center of symmetry then Green and Lindsay [2] incorporated a temperature rate term amongst the fundamental relations to formulate a temperature-rate-dependent thermoelasticity that does not overrule the conventional Fourier‘s law of heat conduction and also this theory forecasts a limited speed of propagation of heat. By using these generalized theories, propagation of energy can be regarded as a wave-type instead of a diffusion-type. Dhaliwal and Sherief [3] elaborated on the findings of Lord and Shulman by deriving the equations for an anisotropic medium of generalized thermoelasticity and to these equations' uniqueness theorem is proved. Chandrasekharaiah [3] developed a thermal wave disturbance that he mentioned as ‗second sound‘. Ackerman et.al. [5] and Ackerman and Overtone [6] proved experimentally that the heat waves traveling with limited and high velocity for solid helium also exist, while the conforming frequency window i .e.the frequency range of thermal variations in which heatwaves are noticed is restricted for most of the solids. Firstly Lamb [7] studied the propagation of the waves in a uniform elastic plate. Since then disturbance traveling in an elastic solid material with free borders is referred to as ‗Lamb wave‘. The applications of Lamb waves are used in many fields like the checking of defects in thin-walled constituents and multi-sensors etc. The Lamb waves are constructed on the belief that when the solid plate is accompanied by the fluid it varies the amplitude and propagation velocity of the Lamb waves in the borders due to viscous and inertial properties of the fluid in the plate. The implications of the-viscous fluid on the Lamb waves propagation in a solid material of fixed-width inserted between homogeneous liquid half-space on its both sides investigated by Schoch [8] and found that some amount of energy in the plate is attached with the fluid in the form of radiation, while the most of energy remains in the solid. Chadwick and Seet [9] developed the time-harmonic plane wave propagation in a consistent anisotropic plate. Watkins et al. [10] used an acoustic impedance method to calculate the reduction in the magnitude of Lamb waves due to the existence of the non-viscous fluid. The growing attention in Lamb waves was moderately originated by its implementation in the multi-sensors [11]. Wu and Zhu [12] evaluated

Sharma and Pathania [13] investigated Lamb wave propagation in isotropic generalized thermoelastic homogeneous solid material fringe with layers or half-spaces of fluid on the surfaces of the plate. They showed that the shear horizontal component of waves decouples from the primary stream of wave motion. Sharma and Pathania [14] also derived the expressions for frequency equations, short-wavelength waves, and long-wavelength waves in a transversely isotropic homogeneous thermoelastic plate exposed to inviscid fluid loading. Propagation of wave in a liquid saturated porous solid with a micro-polar elastic skeleton at the boundary surface has been evaluated by [15, 16]. Sharma and Kumar [17] studied the effect of micropolarity on Lamb waves in thermally conducting elastic solid subjected to non-viscous fluid layers (or half-spaces) with varied temperatures and discussed the characteristic length and coupling factors impact on phase velocity. Pathania et. al. [18] investigated the Lamb waves in transversely isotropic thermoelastic plates immersed in viscous fluid layers in non-classical theories of thermo-elasticity. Pathania et. al. [19] analyzed the characteristics of the circular waves in a homogeneous anisotropic thermo-elastic material surrounded by conducting viscous fluid loading layers (or half-spaces) on the boundary surface of the plate. The effect of the loosely bonded interface on wave propagation between two half-space has been obtained by Barak and Kaliraman [20]. Recently, Pathania et. al. [21] investigated the impact of varying temperatures on the Lamb waves propagation in a homogeneous thermoelastic isotropic plate in the presence of ideal fluid in the frame of reference of the classical theory of thermoelasticity. In this paper, an effort has been done to discuss the propagation of plane waves in an anisotropic homogeneous thermo-elastic material having width 2d under the effect of non-viscous liquid with varying temperatures on borders of the plate, in the framework of coupled and generalized models of thermoelasticity. The liquid is supposed to be homogeneous and inviscid. The frequency equations are determined by using appropriate boundary conditions for Lamb waves. Also for Lamb waves, the frequency equation is solved analytically for the classical and non-classical models 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.

Consider a homogeneous anisotropic solid plate of width ‗2d‘ consisted of a thermoelastic solid material. The solid plate is cladded with an ideal fluid of widthh on both borders i.e. on top and the bottom. The solid material is supposed at an undisturbed state and unvarying temperature T0 initially.

In the cartesian coordinate systemoxyz, the origin ―o‖ is chosen in the center of the plate and the z-axis is taken in a vertically downward direction along the width of the plate. The wave is traveled along thex-axis direction and the field parameters remain explicitly independent of y-coordinate but depend implicitly on y-coordinate such that the shear component of displacement is non-vanishing. The complete configuration of the problem is illustrated in Figure-1. The basic governing equations in non-dimensional form for homogenous transversely isotropic solid plate and liquid layers in the non-classical model of thermoelasticity, in the non-attendance of energy sources and body forces are given by [14] and [21] respectively as Where For Lord-Shulman (LS) theory the value of suffix is and for Green-Lindsay(GL) theory the value of suffix is k = 2 in Kronecker‘s delta is the velocity of the longitudinal wave in the thermoelastic half- space, and are the thermal relaxation times,is the thermomechanical coupling constant in the solid plate. For the top and the bottom liquid layers is the bulk modulus, is the velocity of sound, is the thermomechanical coupling constant, is the density of the liquid, is the specific heat at a constant volume, is the coefficient of volume thermal expansion. Here dot over a symbol represents the time derivative. In the liquid layers, we take where LjH is the vector velocity potential and is the scalar velocity potential element in the positive direction of z-axis. Since the shear motion is not supported by the inviscid liquid, therefore the shear motion of liquid vanishes, and thus we have

Here the comma in the subscript represents spatial derivatives. Plugging (6) in equations (4) and (5), we get equations in terms of the potential functions as

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

(i) For the solid plate, the dimensionless normal element of the stress tensor must be equal to the pressure of the fluid. This implies that tensor of the solid plate must be zero, implying that (iii) The dimensionless normal displacement element of the plate must be equal to the fluid. This yields to

(iv) The boundary condition for conduction-convection case is agreed by Where H is Biot‘s heat transfer coeffcient. The boundary condition (9.4) refers to the thermally insulated case if and corresponds to the isothermal case if , in the nonappearance of liquid i.e. for thermoelastic half-space.

4. SOLUTION OF THE PROBLEM

Analyzing the disturbances into the normal modes, the solutions of the considered problem are of the form where is the frequency,  is the wavenumber, and cis the dimensionless phase velocity. The inclination angle of wave normal with the z-axis (axis of symmetry) is denoted by  and m is an unidentified parameter. The displacements v, w, temperature T, and potentials concerning the displacementuhave the amplitude ratios V,W,S and respectively. Plugging the solutions (10.1) in equations (1)-(3), u, w, t, can be obtained as Where

The amplitude ratios qWandqSare given by where Upon using solutions (10.2) in equations (7)-(8) for the bottom liquid layer and top liquid layer , we have where Upon using the above formal solution (11), the stresses, displacements, and temperature change are obtained as Here

In this topic, the dispersal relation for Lamb waves propagation in the transversely isotropic plate under inviscid liquid loading is derived. An arrangement of eight linear equations in eight unknown amplitudes is obtained by applying thermal and mechanical boundary conditions at external plate surfaces We get the determinant of the coefficient of equal to zero when this system of equations has a non-zero, which gives the frequency equation for the propagation of Lamb waves in the thermoelastic plate. After straight forward algebraicsimplifications and reductions along with conditions is non zero and not odd multiple of , the distinctive equation for the thermoelastic plate yields to the frequency equations Here . The skew-symmetric mode corresponds to superscripted + sign and symmetric mode corresponds to superscripted – sign and For an inviscid fluid at a uniform temperature and homogeneous isotropic thermoelastic plate equation (21) get reduced to form which is the same as obtained and discussed by [13,14].

The complex transcendental frequency equation (21) gives evidence about the attenuation coefficient, phase velocity, and wavenumber of the Lamb waves. To solve these secular equations, we yield Implies and where R and Q are real quantities. Thus these complex quantities i.e. the phase velocity of the waves and wavenumber are attenuated in space. The exponential in the solutions of time-harmonic plane waves (10.1) and (10.2) respectively become Here Q is the attenuation coefficient and Vp is the phase velocity of plane waves respectively. The various modes of propagating wave for the attenuation coefficient Q and phase velocity Vp can be obtained by substituting relation (23) in equation (21).

Therefore the secular equation (21) becomes For an inviscid fluid at a uniform temperature and homogeneous isotropic thermoelastic plate equation (25) get reduced to form which is the same as obtained and discussed by [13,14].

Based on the values of characteristic roots 531,,mmm to be imaginary, zero, or real, the dispersion equation (21) is converted as given below.

If the frequency equation has characteristic roots in the form then which is purely imaginary. Therefore the partial wave superposition confirms that they have the ―exponential decay‖ property. Therefore the secular equation (21) can be changed to hyperbolic tangent functions by

where Herecorresponding expressions can be found by substituting.

If two roots of the equation are in the form , then the secular equation (21) can be altered with hyperbolic tangent functions in by substituting circular tangent

where

Herecorresponding expressions can be found by substituting .

when the roots of the characteristic equation 5,3,1 ,2kmk are complex numbers then the dispersion equation remains the same as given by equation (21).

We now present some numerical solutions to the theoretical results obtained in the earlier sections. The following values of material constants are taken for solid zinc plate and liquid water.

Figure 2 presents the profiles of phase velocity (Vp) with wavenumber ® with liquid loaded temperature for the angle of inclination 0=750, 900 for symmetric and significantly decrease to become nearer to the shear wave velocity asymptotically but the lowest (acoustic) asymmetric mode (A0) approaches the Rayleigh wave speed at higher wavenumbers. The trembling energy is mostly spread by the surfaces (interfaces) of the solid material in the situations when a plate of finite thickness seems to be half-space. The free sides possess Rayleigh-type waves having complex phase velocity and wavenumber. The acoustic symmetric mode (S0) turns in to a non-scattered phase velocity concerning the wavenumber. 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 (n=1). The lowest asymmetric mode(A0) is detected most sensitive and affected.

Figure 3 indicates the variations of attenuation coefficients concerning the liquid loaded temperature of wave propagation for symmetrical and asymmetrical manners for the angle of inclination 0=750, 900 and it is observed that for n=0 and n=1, the attenuation coefficient has an insignificant variation concerning wavenumber along the direction of wave propagation in the framework of generalized theories of thermo-elasticity. For the other optical modes, the magnitude of the attenuation coefficient shoots up to achieve maximum value and then decreases monotonically to zero with the increasing wavenumber. Figure 4 depicts the variation of phase velocity with liquid temperature for symmetrical and asymmetrical modes for the angle of inclination 45O. It is evident from the figure that for both symmetrical and asymmetrical modes, the phase velocity of the liquid has constant values with changing liquid loading temperature. For higher modes, the value of phase velocity is found to progress at an amount nearly n-times the values of phase velocity of the initial mode (n=1). The behavior and trend of asymmetric and symmetric modes be similar to each other the difference in their values or magnitudes. For symmetric modes, the value of the phase velocity is found to be high to that of the asymmetric one at different liquid temperatures.

In Figure 5, the phase velocity of the asymmetric mode of wave propagation achieves quite high values at a small value of the plate thickness that significantly decreases with the increasing values of the thickness of the plate and remains constant. This happens because the coupling effect of various interacting fields increases as the thickness of the plate increases, resulting in a decrease in the values phase velocity. Also, it is found that with the increasing values of the plate thickness the Rayleigh wave velocity is attained as the heat transmission primarily occurs in the vicinity of the free sides of the plate.

Figure 6 shows the variation of the attenuation coefficient with the thickness of the plate for asymmetric mode for the angle of inclination 0=750, 900. It is evident from the figure that the values of the attenuation coefficient increases moderately to achieve a maximum value in case of 0=900 and then decreases with the increase in the thickness of the plate while for 0=750, the profile of attenuation coefficient increases linearly with plate thickness. For 0=900 the magnitude of the attenuation coefficient found to be higher than that of 0=750 increasing values of the plate thickness.

The Lamb waves propagation in a thermally conducting elastic homogeneous, transversely isotropic plate in the presence of non-viscous fluid layers on its both sides, with varying temperature has been studied in the frame of reference of non-classical theories of thermoelasticity. It is found that there exists a coupling of three types of waves in the solid plate and two mechanical waves in each liquid layer due to mechanical stresses. It is noticed that for the symmetrical and skew-symmetrical modes, the profiles of the phase velocity show distinguishable results at the long-wavelengths and attains the Rayleigh waves velocity with increasing wavenumbers. The effect of the liquid temperature and thickness of the fluid layers has also been detected significantly on attenuation coefficient and distribution curves in the considered material plate.

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Department of Mathematics, Indira Gandhi University, Meerpur, Rewari (HR)-123401, India ms_barak@igu.ac.in