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The purpose of this paper is to study the effect of rotation on the general three-dimensional model of the equations of the generalized thermoelasticity for a homogeneous isotropic elastic half-space solid. The problem is studied in the context of the Green-Naghdi theory of type II (without energy dissipation). The normal mode analysis is used to obtain the expressions for the temperature, thermal stress, strain and displacement. The distributions of variables considered are represented graphically.

The propagation of waves in thermoelastic materials has many applications in various fields of science and technology, namely, atomic physics, industrial engineering, thermal power plants, submarine structures, pressure vessel, aerospace, chemical pipe and metallurgy. Thermoelasticity theories, which admit a finite speed for thermal signals, have received a lot of attention for the past four decades. In contrast to the conventional coupled thermoelasticity theory based on a parabolic heat equation by Biot [

The first generalization, for isotropic bodies, is due to Lord and Shulman [

The second generalization is known as the theory of thermoelasticity with two relaxation times, or the theory of temperature-rate-dependent thermoelasticity, and is proposed by Green and Lindsay [

The theory of thermoelasticity without energy dissipation is another generalized theory and is formulated by Green and Naghdi [

The problems for rotating media have also been investigated. Chand et al. [

Recently, Othman et al. [

In the present work, we studied the effect of rotation on the general three-dimensional model of the equations of the generalized thermoelasticity for a homogeneous isotropic elastic half-space solid in the context of Green- Naghdi theory of type II without any body forces or heat sources. The effect of rotation on different characteristics is shown graphically for generalized thermoelasticity.

We consider a homogeneous thermoelastic half-space, rotating uniformly with an angular velocity

The governing equations of the medium in the context of the generalized thermoelasticity of the Green-Naghdi theory of type IIin the absence of body forces and heat sources are:

Equation of motion:

Heat conduction equation:

Stress-displacement-temperature relation:

In the preceding equations,

We can rewrite the equation of motion as

and the conduction equation takes the form

and the stress-displacement-temperature relation as:

where

For convenience, we will transform the above equations in non-dimensional forms, so the following non-di- mensional variables are used:

where C_{T} represents the non-dimensional thermal wave speed and

Equations (4)-(13) in the non-dimensional forms (after suppressing the primes) reduce to

where

From Equations (20)-(22) by addition, we get

where,

We consider plane waves propagating in the plane such that at any instant all the particles in a line parallel to the y axis have equal displacements, i.e. all partial derivatives with respect to y vanish.

We may separate out the purely dilatation and purely rotational disturbances associated with the components

and

By using Equation (27) in Equations (16)-(19), we obtain

where,

The solution of the considered physical variables can be decomposed in terms of normal modes as in the following form

where

Using Equation (32), then Equations (28)-(31) take the form

where

Eliminating

In a similar manner, we can show that

where

Equation (37) can be factored as

where

The solution of Equation (40), which is bounded as

similarly

where

from Equations (41), (42) and (27) then we obtain

from Equations (44) and (45) in (14) we get

from Equations (26), (32), (43) and (46), then we obtain

In order to complete the solution we have to know the parameters

a) The thermal boundary condition is

where

From Equations (48), (49) and (29), we get

b) Mechanical boundary condition:

It is assumed that at

Using the boundary conditions (50) and (51) in Equations (43)-(45) respectively, we get

Solving the system of Equations (52)-(54), we get the parameters

where

In order to illustrate the theoretical results obtained in the preceding section, we now present some numerical results. In the calculation process, we take the case of copper material. Since

Figures 1-4 represented 2D curves for the change of behavior of the values of the real part of the displacement component

the boundary condition at

Figures 5-8 are representing 3D surface for curves for distribution of the values of the real part of thedisplacement component

Figures 9-12 are showing 3D surface for curves for distribution of all physical quantities for a wide range of

All these Figures 5-12 are in the presence of rotation

1) The values of the distributions of all physical quantities converge to zero with increasing distance x. Using these results; it is possible to investigate the disturbance caused by more general sources for practical applications.

2) It is clearly observed from Figures 1-4 that the rotation Ω plays a significant role in all physical quantities.

3) It is clear from Figures 5-8 that the changes in the values of the time cause significant changes on all the studied fields.

4) It is observed from Figures 9-12 that the changes in the values of the dimensions cause significant changes on all the studied fields.

5) The speed of wave propagation of the thermoelastic field variables is finite and coincides with the physical behavior of the elastic materials.