Effects of Rotation and Gravity Field on Surface Waves in Fibre-Reinforced Thermoelastic Media under Four Theories

Estimation is done to investigate the gravitational and rotational parameters effects on surface waves in fibre-reinforced thermoelasticmedia.The theory of generalized surfacewaves has been firstly developed and then it has been employed to investigate particular cases of waves, namely, Stoneley waves, Rayleigh waves, and Love waves. The analytical expressions for surface waves velocity and attenuation coefficient are obtained in the physical domain by using the harmonic vibrations and four thermoelastic theories.The wave velocity equations have been obtained in different cases.The numerical results are given for equation of coupled thermoelastic theory (C-T), Lord-Shulman theory (L-S), Green-Lindsay theory (G-L), and the linearized (G-N) theory of type II. Comparisonwasmadewith the results obtained in the presence and absence of gravity, rotation, and parameters for fibre-reinforced of the material media. The results obtained are displayed by graphs to clear the phenomena physical meaning. The results indicate that the effect of gravity, rotation, relaxation times, and parameters of fibre-reinforced of the material medium is very pronounced.


Introduction
A reinforced concrete member will be designed for all conditions of stress that may occur accordance with the principle of mechanics.Fibre-reinforced composites are used in a variety of structures due to their low weight and high strength.The characteristic property of a reinforced composite is that its components act together as single anisotropic units as long as they remain in the elastic condition.Investigation of interaction between magnetic field, stress, and strain in a thermoelastic solid is very important due to its many applications in the field of geophysics, plasma physics, and related topics, especially in the nuclear field, where the extremely high temperature and temperature gradients, as well as the magnetic fields originating inside nuclear reactors, influence their design.Recently, more attention has been studied the dynamical problem of propagation of surface waves in a homogeneous and non-homogeneous elastic and thermoplastic media because of its utilitarian aspects in earthquake, engineering, and seismology on account of the occurrence of nonhomogeneities in the Earth's crust, as the Earth is made up of different layers.Abd-Alla et al. [1] investigated propagation of Rayleigh waves in generalized magnetothermoelastic orthotropic material under initial stress and gravity field.Thermal stresses in an infinite circular cylinder have been investigated by Abd-Alla et al. [2].Abd-Alla and Mahmoud [3] studied the magnetothermoelastic problem in rotational nonhomogeneous orthotropic hollow cylinder under the hyperbolic heat conduction model.Abd-Alla and Ahmed [4] investigated the Rayleigh waves in an orthotropic thermoelastic medium under gravity field and initial stress.Rayleigh waves in a magnetoelastic half-space of orthotropic material under the influence of initial stress and gravity field have been investigated by Abd-Alla et al. [5].Elnaggar and Abd-Alla [6] studied Rayleigh waves in magneto-thermomicroelastic halfspace under initial stress.Abd-Alla and Ahmed [7] discussed Rayleigh waves in an orthotropic thermoelastic medium under gravity field and initial stress.Effect of rotation and initial stress on an infinite generalized magnetothermoelastic diffusion body with a spherical cavity has been investigated by Abd-Alla and Abo-Dahab [8].Wu and Chai [9] studied propagation of surface waves in anisotropic solids: theoretical thermoelastic medium subjected to gravity field and relaxation times leading to particular cases such as Rayleigh waves, Love waves and Stoneley waves.The surface wave velocity and attenuation coefficient are obtained in the physical domain by using the harmonic vibrations.The effects of the gravity, relaxation times, and parameters for fibre-reinforced of the material medium on surface waves are studied simultaneously.The analytical expressions for surface waves velocity and attenuation coefficient are represented graphically.Numerical results for the surface waves velocity and attenuation coefficient are given and illustrated graphically in the presence and absence of the gravity field and rotation.

Formulation of the Problem
We consider homogeneous thermally conducting transversely two fibre-reinforced media.Let  1 and  2 be two fibre-reinforced elastic thermoelastic semi-infinite solid media.They are perfectly welded in contact to prevent any relative motion or sliding before and after the disturbances and that the continuity of displacement, stress, and so for the hold good across the common boundary surface.Further the mechanical properties of  1 are different from those of  2 .These media extend to an infinite great distance from the origin and are separated by a plane horizontal boundary and  2 is to be taken above  1 .Let  be a set of orthogonal Cartesian coordinates, let  be any point of the plane boundary, and let  points vertically downward to the medium  1 .We consider the possibility of a type of wave travelling in the direction  in such a manner that the disturbance is largely confined to the neighborhood of the boundary and, at any instant, all particles in any line parallel to -axis have equal displacements.These two assumptions conclude that the wave is a surface wave and all partial derivatives with respect to  are zero.Further let us assume that , V are the components of displacements at any point (, , ) at any time .It is also assumed that gravitational field produces a hydrostatic initial stress that is produced by a slow process of creep where the shearing stresses tend to become smaller or vanish after a long period of time.
The equilibrium equation of the initial stress is in the form The equations and constitutive relations for such medium in the absence of body forces and heat sources are

Special Cases of Thermoelasticity Theory
We get that the forms show the behavior under gravity field; the theories are as follows.
(1) The equation of the coupled thermoelasticity (C-T theory) is obtained when (2) Consider Lord-Shulman theory.
For the Lord-Shulman (L-S) theory (3) Consider the Green-Lindsay theory.
For the Green-Lindsay (G-L) theory (4) Consider the Green-Nagdhi theory.
For the Green-Nagdhi theory (G-N) of type 2, For plane strain deformation in the - plane, the components of stress take the form where By Helmholtz's theorem [35], the displacement vector  can be written in the form where the scalar  and the vector ⃗  represent irrotational and rotational parts of the displacement ⃗ .It is possible to take only one component of the vector ⃗  to be nonzero, as From (14a) and (14b) we obtain Substituting from (14c) into ( 2) and (3) we get From ( 16) and (18) we get where Also from (15) and (17) we get where The solution of ( 19) and ( 21) has the form Substituting from (23) in to (19) we get The solution for (24) as the form Equation (24) must have exponential solutions in order that  will describe surface waves; they must become vanishingly small as  → ∞: where Substituting from (23) in to (21) we get where From ( 26) and ( 27), we can obtain the temperature  and the potential function Ψ as follows: Substituting from ( 29) and (14c) into (4) for , Ψ, , and V we get where The solution of ( 30) is where After finding the values of  we get the following values: Substituting from ( 34), (35), and (36) into (14c), we get the components of displacement as Substituting from (36), (37), and (38a) in to ( 9)-( 12) we obtain

Stoneley Waves
It is the generalized form of Rayleigh waves in which we assume that the waves are propagated along the common boundary of two semi-infinite media  1 and  2 .Therefore (40) determines the wave velocity equation for Stoneley waves in anisotropic fibre-reinforced solid thermoelastic media under the influence of gravity.Clearly, from (40), it follows that wave velocity of the Stoneley waves depends upon the parameters for fibre-reinforced of the material medium, gravity, rotation, and the densities of both mediums, since the wave velocity equation (40) for Stoneley waves under the present circumstances depends on the particular value of  and creates a dispersion of a general wave form.Further, (40), of course, is in complete agreement with the corresponding classical result, when the effect of gravity, rotation, and parameters for fibre-reinforcement is ignored.

Love Waves
To investigate the possibility of love waves in a fibrereinforced thermoelastic solid media, we replace medium  2 that is obtained by two horizontal plane surfaces at a distance -apart, while  1 remains infinite.For medium  1 , the displacement component ] remains the same as in general case given by (38a), (38b), (38c), and (38d).

Numerical Results and Discussion
With a view to illustrating the analytical procedure presented earlier, we now consider a numerical example for which computational results are given.The results depict the variation of The output is plotted in Figures 1-30.   Figure 9 shows the variation of the Stoneley secular equation with respect to phase velocity , that has an oscillatory behavior of C-T theory, L-S theory, and G-L theory for different values of relaxation time  0 in the whole range of the phase velocity , while in G-N theory it decreases with increasing the constant  * and it increases with increasing phase velocity .This is due to the fact that the thermal waves in the coupled theory travel with an infinite speed of propagation as opposed to a finite speed in the generalized case.
Figure 10 displays the variation of the Stoneley wave velocity and attenuation coefficient with respect to phase velocity  that has oscillatory behavior in the whole range of  for L-S theory, which changes from the positive to the negative gradually.Figure 11 shows the variations of the Stoneley waves velocity and attenuation coefficient with respect to phase velocity  that has oscillatory behavior in the whole range of  for (G-L theory) that increases with increasing relaxation time  0 ,  0 .
Figure 12 shows the variations of the Stoneley waves velocity and attenuation coefficient with respect to phase velocity  for G-N theory in the whole range of  for different values of the constant  * which changes from the positive to the negative gradually.In both figures, it is clear that the Stoneley wave velocity and attenuation coefficient have a nonzero value only in a bounded region of space.It is observed that the Stoneley wave velocity decreases with increasing the phase velocity, while the attenuation coefficient increases with increasing the constant  * .
Figure 13 shows the variations of the Stoneley waves velocity and attenuation coefficient with respect to phase velocity  for G-N theory in the whole range of  for different values of constant  * .In both figureures, it is clear that the Stoneley wave velocity and attenuation coefficient have a nonzero value only in a bounded region of space.It is observed that the Stoneley wave velocity increases with increasing the phase velocity, while it decreases with increasing constant  * and attenuation coefficient increases with increasing the phase velocity, while it increases with increasing constant  * .Figures 14,15,16,17,18,19,20,and 21 show the variations of the Rayleigh velocity and attenuation coefficient with respect to gravity field  and rotation Ω, respectively, which has an oscillatory behavior in the whole range of  and Ω for C-T theory, L-S theory, G-L theory, and G-N theory.These figureures indicate that the medium of attenuation coefficient along  undergoes compressive deformation due to the thermal shock, except (C-T theory) it is a tension deformation.In both figureures, it is clear that the Rayleigh wave velocity and attenuation coefficient have a nonzero value only in a bounded region of space.Figure 22 shows the  In both figureures, it is clear that the Rayleigh wave velocity and attenuation coefficient have a nonzero value only in a bounded region of space.
Figure 26 shows the variations of the Rayleigh wave velocity and attenuation coefficient with respect to phase velocity  for G-N theory, the Rayleigh wave velocity increases with increasing phase velocity, while it decreases with increasing the constant  * , and the attenuation coefficient decreases with increasing phase velocity and the constant  * ; it is clear that the Rayleigh wave velocity and attenuation coefficient have a nonzero value only in a bounded region of space.
Figure 27 shows the variation of the Love waves secular equation with respect to phase velocity , which increases with increasing phase velocity for C-T theory, L-S theory, G-L theory, and G-N theory, while it decreases with increasing the constant  * ; it is clear that Love waves secular equation has a nonzero value only in a bounded region of space.

Conclusion
Due to the complicated nature of the governing equations of the generalized thermoelasticity Fibre-reinforced theory, the work done in this field is unfortunately limited in number.The method used in this study provides a quite succesful in dealing with such problems.This method gives exact solutions in the generalized thermoelastic medium without any assumed restrictions on the actual physical quantities that appear in the governing equations of the problem considered.Important phenomena are observed in all these computations.
(i) It was found that for large values of time they give close results.The solutions obtained in the context of generalized thermoelasticity theory, however, exhibit the behavior of speed of surface wave propagation.(ii) By comparing Figures 1-30, it was found that the surface wave velocity has the same behavior in both media.But with the passage of rotation, relaxation times, and gravity, numerical values of surface wave velocity in the thermoelastic medium are large in comparison due to the influences of gravity and relaxation times in the elastic medium.
(iii) Special cases are considered as Rayleigh waves, Love wave, and Stoneley surface waves in anisotropic generalized thermoelastic medium, as well as in the isotropic case.
(iv) The results presented in this paper should prove useful for researchers in material science and designers of new materials.
(v) Study of the phenomenon of rotation and gravity is also used to improve the conditions of oil extractions.

Figure 1 :Figure 2 :Figure 3 :Figure 4 :
Figure 1: Variation of the Stoneley waves velocity and attenuation coefficient (CT) model with respect to the gravity.

Figure 5 :Figure 6 :Figure 7 :Figure 8 :
Figure 5: Variation of the Stoneley waves velocity and attenuation coefficient (CT) model with respect to the rotation.

Figure 9 :Figure 10 :Figure 11 :Figure 12 :Figure 13 :
Figure 9: Variation of the Stoneley secular equation for the (CT, LS, GL, and GN (type (II)) models with respect to the phase velocity.

Figures 1 ,
Figures 1, 2, 3, 4, 5, 6, 7, and 8 show the variation of the Stoneley waves and attenuation coefficient value with respect to gravity field  and rotation Ω, respectively, has oscillatory behavior in the whole range of gravity  and rotation for C-T theory, L-S theory, G-L theory, and G-N

Figure 14 :Figure 15 :Figure 16 :Figure 17 :Figure 18 :Figure 19 :
Figure 14: Variation of the Rayleigh waves velocity and attenuation coefficient (CT) model with respect to the gravity.

Figure 20 :Figure 21 :
Figure 20: Variation of the Rayleigh waves velocity and attenuation coefficient GL-model with respect to the rotation.

Figure 22 :Figure 23 :Figure 24 :Figure 25 :Figure 26 :
Figure 22: Variation of the Rayleigh secular equation for the (CT, LS, GL, and GN (type (II)) models with respect to the phase velocity.

Figure 29 :0Figure 30 :
Figure 29: Variation of the Love waves velocity and attenuation coefficient (GL) model with respect to the phase velocity.

Figures 28 ,
29, and 30  show the variations of the Love wave velocity and attenuation coefficient with respect to phase velocity , it has oscillatory behavior in the whole range of  for different values of relaxation times  0 ,  0 , and  * , respectively, for L-S theory, G-L theory, and G-N theory, which changes from the positive to the negative gradually.In both figures, it is clear that Love wave velocity and attenuation coefficient have a nonzero value only in a bounded region of space.