P ropagation of w aves in m icropolar generalized th erm o elastic

The aim of the present paper is to study the propagation of Lamb waves in micropolar generalized thermoelastic solids with two temperatures bordered with layers or half - spaces of inviscid liquid subjected to stress - free boundary conditions in the context o f Green and Lindsay (G - L) theory. The secular equations for go v-erning the symmetric and skew - symmetric leaky and nonleaky Lamb wave modes of propagation are derived. The computer simulated results with respect to phase velocity, attenuation coe f-ficient, am plitudes of dilatation, microrotation vector and heat flux in case of symmetric and skew - symmetric modes have been depicted graphically. Moreover, some particular cases of interest have also been discussed.


INTRODUCTION
Eringen (1966) developed the theory of micropolar elasticity which has aroused much interest in recent years because of its possible usefulness in investigating the deformation properties of solids for which the classical theory is inadequate.There are at least two different generalizations related to the classical theory of thermoelasticity.The first one given by Lord and Shulman (1967) admits only one relaxation time and the second one given by Green and Lindsay (1972) involves two relaxation times.
The linear theory of micropolar thermoelasticity has been developed by extending the theory of micropolar continua.Eringen (1970Eringen ( , 1999) ) and Nowacki (1986) have given detailed reviews on the subject.Boschi and Iesan (1973) extended the generalized theory of micropolar thermoelasticity which allows the transmission of heat as thermal waves of finite speed.The generalized ther-Latin American Journal of Solids and Structures 11 (2014)   moelasticity was presented by Dost and Taborrok (1978) by using Green and Lindsay theory.Chandrasekharaiah (1986) developed a heat flux dependent micropolar thermoelasticity.
Thermoelasticity with two temperatures is one of the non-classical theories of thermoelasticity.The thermal dependence is the main difference of this theory with respect to the classical one.A theory of heat conduction in deformable bodies depends on two distinct temperatures, the conductive temperature  and thermodynamic temperature T .The thermodynamic temperature T  has been introduced by Chen et al. (1968Chen et al. ( , 1969)).The difference between these two temperatures is proportional to the heat supply for time independent situations.For time dependent and for wave propagation problems, the two temperatures are in general different, regardless of the presence of heat supply.Warren and Chen (1973) investigated the wave propagation in the two temperatures theory of thermoelasticity.
The study related to the interaction of elastic waves with fluid loaded solids has been recognized as a viable means for deriving the non-destructive evaluation of solid structures.The reflected acoustic field from a fluid-solid interface provides details of many characteristics of solids.
These phenomena have been investigated for the simple isotropic semi-space as well as the complicated systems of multilayered anisotropic media.A detailed review in this respect is given by Nayfeh (1995).Qi (1994) investigated the influence of viscous fluid loading on the propagation of leaky Rayleigh waves in the presence of heat conduction effects.Wu and Zhu (1995) suggested an alternative approach to the treatment of Qi (1994).They presented solutions for the dispersion relations of leaky Rayleigh waves in the absence of heat conduction effects.Zhu and Wu (1995) used this technique to study Lamb waves in submerged and fluid coated plates.Nayfeh and Nagy (1997) formulated the exact characteristic equations for leaky waves propagating along the interfaces of systems which involve isotropic elastic solids loaded with viscous fluids, including halfspaces and finite thickness fluid layers.Youssef (2006) presented a new theory of generalized thermoelasticity by considering the interaction of heat conduction in deformable bodies.A uniqueness theorem for generalized linear thermoelasticity involving a homogeneous and isotropic body was also recorded in this study.Various authors, e.g.(Puri and Jordan (2007), Youssef and Al-Lehaibi (2007), Youssef and Al-Harby (2007), Magana and Quintanilla (2009), Mukhopadhyay and Kumar (2009), Kumar and Mukhopadhyay (2010), Kaushal et al. (2010Kaushal et al. ( , 2011) ) studied the problems of thermoelastic media with two temperatures.
Qingyong Sun et al. (2011) studied propagation characteristics of longitudinal displacement wave in micropolar fluid with micropolar elastic plate, Singh (2010) discussed propagation of thermoelastic waves in micropolar mixture of porous media.
In the present paper, the propagation of waves in an infinite homogeneous micropolar generalized thermoelastic plate with two temperatures bordered with layers or half-spaces of inviscid Latin American Journal of Solids and Structures 11 (2014) 1091-1113 liquid have been investigated.The secular equations have been derived.The phase velocity, attenuation coefficient, amplitudes of dilatation, microrotation vector and heat flux for the symmetric and skew-symmetric wave modes are computed numerically and presented graphically for G-L theory.

VIBRATION OF PLATE ON FOUNDATION AND INTEGRAL TRANSFORM
The field equations following Eringen (1966), Ezzat and Awad (2010) and Green and Lindsay (1967) in a homogeneous, isotropic, micropolar elastic medium in the context of generalized theory of thermoelasticity with two temperatures, without body forces, body couples and heat sources, are as follows ( ) and the constitutive relations are λ and µ are Lame's constants.K , α , β and γ are micropolar constants.t ij and m ij are the components of stress tensor and couple stress tensor respectively.u i and ϕ i are the dis- placement and microrotation vectors, ρ is the density , j is the microinertia, K * is the thermal conductivity, c * is the specific heat at constant strain, T is the thermodynamic temperature, Φ is the conductive temperature, T 0 is the reference temperature, ν = 3λ + 2µ + K ( )α T , where α T is the coefficient of linear thermal expansion, τ 0 and τ 1 are the thermal relaxation time δ ij is the Kronecker delta, ε ijr is the alternating tensor.The relation connecting T and Φ is given by T = (1− a∇ 2 )Φ , where a is a two temperature parameter.
Following Achenbach (1976), the field equations can be expressed in terms of velocity potential for inviscid fluid as Latin American Journal of Solids and Structures 11 (2014) 1091-1113 where c L 2 = λ L ρ L is the velocity of acoustic fluid, λ L is the bulk modulus, ρ L is the density of the fluid, p L is the acoustic pressure in the fluid, ϕ L is the velocity potential of the fluid, u L is the velocity vector, ∇ is gradient operator, ∇ 2 is the Laplacian operator.

NUMERICAL RESULTS
An infinite homogeneous isotropic, thermally conducting micropolar thermoelastic plate of thickness 2d initially undisturbed and at uniform temperature 0 T is considered.We consider that two infinitely large homogeneous inviscid liquid half-spaces or layers of thickness h bordered the plate on both sides as shown in figures 1(a) and 1(b) respectively.The origin of the coordinate system ) , , ( is taken on the middle of the plate and the axis normal to the solid plate along the thickness is taken as 3 x -axis.We consider the propagation of plane waves in the − 3 1 x x plane with a wavefront parallel to the 2 x -axis so that field components are independent of 2 x -coordinates.
The displacement and microrotation vectors for two dimensional problem are taken as For inviscid fluid, we take ( ) ( ) ( ) The physical quantities can be made dimensionless by defining the following Latin American Journal of Solids and Structures 11 (2014) 1091-1113 such that * ω is the characteristic frequency of the medium.
The displacement components u !, u ! and 3 u are related to the potential functions ϕ, ψ and ψ in dimensionless form as In the liquid layers, we have φ are the scalar velocity potential components for the top liquid layer ( 1 = i ) and for the bottom liquid layer ( x and 3 x components of the particle velocity for the top liquid layer and the bottom liquid layer respectively. Using eqs.( 8)-( 11) in eqs.( 1)-( 3) and ( 7) and after suppressing the primes, we obtain the following equations Latin American Journal of Solids and Structures 11 (2014) 1091-1113 where The solutions of eqs.( 12)-( 16) are assumed as is the phase velocity, ω is the frequency and ξ is the wave number.

BOUNDARY CONDITIONS
We consider the following boundary conditions at the solid-liquid interfaces where H is the surface heat transfer coefficient.Here H → 0 and H → ∞ corresponds to thermally insulated and isothermal boundaries, respectively.

Leaky Lamb Waves
The solutions for solid media of finite thickness d 2 sandwiched between two liquid half-spaces is given by eqs.( 20)-( 23) together with

Nonleaky Lamb Waves
The corresponding solutions for a solid media of finite thickness d 2 sandwiched between two finite liquid layers of thickness h are given by eqs.( 20)-( 23) and

DISPERSION EQUATIONS
We apply the formal solutions of previous section to study the specific situations with inviscid fluid.

→ H
) of the plate where Latin American Journal of Solids and Structures 11 (2014) 1091-1113 for stress-free isothermal boundaries (

∞ → H
) of the plate.where

Nonleaky Lamb Waves
We consider an isotropic thermoelastic micropolar plate with two temperatures bordered with layers of inviscid liquid on both sides as shown in Fig. 1(b).
for stress-free thermally insulated boundaries (

→ H
) of the plate.
Latin American Journal of Solids and Structures 11 (2014) 1091-1113 for stress-free isothermal boundaries (

∞ → H
) of the plate.
Here the superscript +1 refers to skew-symmetric and -1 refers to symmetric modes.All the coefficients and other quantities are recorded in Appendix.
Eqs. ( 35) and ( 38) are the general dispersion relations involving wave number and phase velocity of various modes of propagation in a micropolar thermoelastic plate bordered with layers of inviscid liquid or half-spaces on both sides.

Special cases
The removal of the liquid layers or half-spaces on the both sides, provide the wave propagation in micropolar thermoelastic solid with two temperatures.Analytically, if we take 0 = L ρ in eqs.( 35) and ( 37) then the secular equations for stress-free thermally insulated boundaries ( 0 → H ) for the said case reduce to If we take 0 = a in eq. ( 30), then we obtain the secular equations in micropolar generalized thermoelastic plate.

AMPLITUDES OF DILATATION, MICROROTATION AND HEAT FLUX
The amplitudes of dilatation, microrotation and heat flux for symmetric and skew-symmetric modes have been computed for micropolar thermoelastic plate.By using eq.( 17) in ( 12)-( 16) and then using eqs.( 20)-( 25), we obtain Latin American Journal of Solids and Structures 11 (2014)

NUMERICAL RESULTS AND DISCUSSION
For numerical computation we select Magnesium crystal (micropolar thermoelastic solid).The physical data for this medium is given below: (i) The values of micropolar constants are taken from Eringen (1984) (ii) and thermal parameters are taken from Dhaliwal and Singh (1980): For numerical calculations, water is taken as liquid and the speed of sound in water is given by sec In general, wave number and phase velocity of the waves are complex quantities, therefore, the waves are attenuated in space.If we write V and Q are real numbers.This shows that V is the propaga- tion speed and Q is the attenuation coefficient of waves.Using eq. ( 45) in secular eqs. ( 35 and (37), the value of propagation speed V and attenuation coefficient Q for different modes of propagation can be obtained.
In figures 2 to 9, GLS and GNLS refer to leaky and nonleaky symmetric waves in micropolar thermoelastic solid with two temperatures, GLSK and GNLSK refer to leaky and nonleaky skewsymmetric waves in micropolar thermoelastic solid with two temperatures, GALS and GANLS refer to leaky and nonleaky symmetric waves in micropolar thermoelastic solid, GALSK and GANLSK refer to leaky and nonleaky skew-symmetric waves in micropolar thermoelastic solid.In figures 10 to 15, GT represents the amplitude for micropolar thermoelastic solid with two temperatures and TS represents the amplitude for micropolar thermoelastic solid.It is depicted from fig. 2 that the phase velocity for lowest symmetric mode of propagation for GLS and GALS coincide.The magnitude of phase velocity for GALS is greater than GLS for (n=1) symmetric mode of propagation for ξd = 1 , 3 ≤ ξd ≤ 6 and in the remaining range the behavior is reversed.It is noticed that for (n=2) mode, the phase velocities for GALS remain more than the values for GLS for 1 ≤ ξd ≤ 3, ξd = 6 and in the remaining region, the behavior is reversed.Fig. 4 shows that for (n=0) symmetric mode of propagation, the velocities for GANLS are greater than GNLS for ξd = 2, 4 ≤ ξd ≤ 6, 8 ≤ ξd ≤ 10 and in the remaining region, the behavior is op- posite.It is also noted that for (n=1) mode, the phase velocities for GANLS are greater than the phase velocities for GNLS in the whole region.The values for GANLS are greater than the values for GNLS in the whole region, except for ξd = 1.Fig. 6 depicts that for symmetric leaky Lamb wave mode (n=0), the attenuation coefficient for GLS remain more than the attenuation coefficient for GALS when ξd = 1 and 3 ≤ ξd ≤ 6 , while in the remaining region, the behavior is reversed.For (n=1), the values for GLS oscillate and attain maximum value at ξd = 10 .It is noticed that for (n=2), the attenuation coefficient for GALS remain more than the values for GLS in the whole region, except in the region 7 ≤ ξd ≤ 10 , where the behavior is reversed.Fig. 7 shows that for (n=0) mode, the magnitude of attenuation for GLSK and GALSK attain maximum value at ξd = 1 .For (n=1) skew-symmetric mode, the values for GLSK decrease with increase in wave number.It is noticed that for (n=2) mode, the magnitude of attenuation coefficient for GALSK remain more than in case of GLSK in the whole region, except for ξd = 1 , where the values coincide.It is evident from fig. 8 that for symmetric nonleaky Lamb wave mode (n=0), the attenuation coefficient for GNLS and GANLS attain maximum value at ξd = 1 .It is noticed that the mag- nitude of attenuation coefficient for GNLS and GANLS attain maximum value 0.01212 and 0.00756 at ξd = 3 respectively for (n=1) mode.For (n=2) mode, the values for GNLS and GANLS attain maximum value at ξd = 5.Fig. 9 depicts that for (n=0) skew-symmetric nonleaky Lamb wave mode of propagation, the magnitude of attenuation coefficient for GNLSK and GANLSK decrease with increase in wave number.It is depicted that for (n=1) mode, the magnitude for GNLSK and GANLSK attain maximum value at ξd = 1 .For (n=2) mode, the values for GANLSK are greater than GNLSK in the whole region except for ξd = 2,10.

Amplitudes
In figures 10 to 15, GT represents the amplitude for micropolar thermoelastic solid with two temperatures and TS represents the amplitude for micropolar thermoelastic solid.Figs. 10 and 11 depict the variations of symmetric and skew-symmetric amplitudes of dilatation for G-L theory for stress-free thermally insulated boundary.The dilatation is having minimum value at the centre and maximum value between the centre and the surfaces for symmetric mode and maximum value at the centre for skew-symmetric mode.It is observed that the dilatation for TS remain more than the dilatation for GT in the whole region for both symmetric and skew-symmetric modes.It is evident from figs. 12 and 13 that the amplitude of symmetric and skew-symmetric microrotation is minimum at the centre and obtain maximum value at the surfaces.
The amplitude of symmetric and skew-symmetric heat flux is oscillatory.For symmetric mode and skew-symmetric mode, the amplitude of heat flux attain maximum value at the sufaces and minimum value at the centre, as shown in figures 14 and 15.

CONCLUSION
It is observed that the variation of phase velocities of lowest symmetric and skew-symmetric mode for leaky and nonleaky Lamb waves almost coincide with increase in wave number.The phase velocities for higher symmetric and skew-symmetric mode attain maximum value at vanishing wave number and as wave number increase the phase velocities get decreased sharply.For (n=2) symmetric mode, the attenuation coefficient for GLS is greater than the values for GALS in the whole region .It is noticed that the values of attenuation coefficient for lowest symmetric and skew-symmetric mode for leaky and nonleaky Lamb waves are very small as compared to the values for highest mode.The values of symmetric and skew-symmetric dilatation for TS are higher in comparison to GT that reveals the effect of two temperatures.

Figure 1
Figure 1(a) Geometry of leaky Lamb waves the characteristic dispersion equations as

Figure 1
Figure 1(b) Geometry of nonleaky Lamb waves

Figure 2
Figure 2 Variations of phase velocity for symmetric leaky Lamb waves

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Figure 3 Variations of phase velocity for skew-symmetric leaky Lamb waves

Figure 4
Figure 4 Variations of phase velocity for symmetric nonleaky Lamb waves

Figure 5
Figure 5 Variations of phase velocity for skew-symmetric nonleaky Lamb waves

Fig. 5
Fig.5depicts that for (n=0) skew-symmetric mode of propagation, the velocities for GNLSK and GANLSK coincide.It is evident that for (n=1) mode, the phase velocities for GANLSK and the values for GNLSK differ near the vanishing wave number and with increase in wave number they coincide.For (n=2) skew-symmetric mode, the phase velocities for NLSK and ANLSK coincide in the whole region, except for ξd = 3.

Figure 7
Figure 7 Variations of attenuation coefficient for skew-symmetric leaky Lamb waves

Figure 8
Figure 8 Variations of attenuation coefficient for symmetric nonleaky Lamb waves

Figure 9
Figure 9 Variations of attenuation coefficient for skew-symmetric nonleaky Lamb waves

Figure 10
Figure 10 Amplitude of symmetric dilatation Figure 11 Amplitude of skew-symmetric dilatation

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Figure 12 Amplitude of symmetric microrotation Figure 13 Amplitude of skew-symmetric microrotation