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Computers, Materials & Continua
DOI:10.32604/cmc.2021.014255
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Article

Rayleigh Waves Propagation in an Infinite Rotating Thermoelastic Cylinder

A. M. Farhan1,2,*

1Department of Physics, Faculty of Science, Jazan University, Jazan, Saudi Arabia
2Department of Physics, Faculty of Science, Zagazig University, Zagazig, Egypt
*Corresponding Author: A. M. Farhan. Email: afarhan_afarhan@yahoo.com
Received: 09 September 2020; Accepted: 11 October 2020

Abstract: In this paper, we investigated the inuence of rotating half-space on the propagation of Rayleigh waves in a homogeneous isotropic, generalized thermo-elastic body, subject to the boundary conditions that the surface is traction free. In addition, it is subject to insulating thermal conduction. A general solution is obtained by using Lame’ potential’s and Hankel transform. The dispersion equations has been derived separately for two types of Rayleigh wave propagation properties by solving the equations of motion with appropriate boundary conditions. It is observed that the rotation, frequency and r exert some influence in the homogeneous isotropic medium due to propagation of Rayleigh waves. The frequency equation has been derived of homogeneous properties by solving the equations of motion with appropriate boundary conditions. It has been found that the frequency equation of waves contains a term involving the rotating. Therefore, the phase velocity of Rayleigh waves changes with respect to this rotating. When the rotating vanishes, the derived frequency equation reduces to that obtained in classical generalized thermo-elastic case which includes the relaxation time of heat conduction. In order to illustrate the analytical development, the numerical solution is carried out and computer simulated results in respect of Rayleigh wave velocity and attenuation coefficient are presented graphysically. A comparative and remarkable study has been carried out through various graphs to deliberate the consequences of different parameter on the frequency equation. The obtained results can be very useful in the design and optimization of Rayleigh wave.

Keywords: Rayleigh waves; wave propagation; generalized thermoelasticity; thermal stress; rotation

1  Introduction

There are many previous studies on generalized thermoelastic waves. Abd-Alla [1] found the phase velocity of propagation of Rayleigh waves in an elastic half-space of orthotropic material. Abd-Alla et al. [2] studied the effect of initial stress, gravity field, and rotation on the propagation of Rayleigh waves in an orthotropic material elastic half-space. Bagri et al. [3] discussed a unified generalized thermoelasticity solution for cylinders and spheres. El-Naggar et al. [4] investigated the effect of initial stress on the generalized thermoelastic problem in an infinite circular cylinder. Singh et al. [5] discussed the effect of rotation on the propagation of Rayleigh waves in an incompressible orthotropic elastic solid half-space with impedance boundary conditions. Singh [6] studied the Rayleigh waves in an incompressible fiber-reinforced elastic solid with impedance boundary conditions. Green et al. [7] introduced the theory of thermoelasticity. Lebon [8] introduced the generalized theory of thermoelasticity. Lord et al. [9] solved the dynamical theory of generalized thermoelasticity to take into account the time needed for the acceleration of the heat flow. Singh et al. [10] investigated the generalized thermoelastic waves in a transversely isotropic media. The generalized thermoelastic waves are discussed by Sharma et al. [11]. Sharma et al. [12] studied the effect of magnetic field on Rayleigh–Lamb waves in a thermoelastic homogeneous isotropic plate. Misra et al. [13] discussed the thermo-viscoelastic waves in an infinite aeolotropic body with a cylindrical cavity. Sharma [14] studied the Rayleigh waves at the surface of a general anisotropic poroelastic medium. Othman [15] investigated the effects of rotation on plane waves in generalized thermo-elasticity with two relaxation times. Othman [16] investigated the effect of rotation and relaxation time on the thermal shock problem for a half-space in generalized thermo-viscoelasticity. Sinha et al. [17] found the eigenvalue approach to study the effect of rotation and relaxation time in generalized thermoelasticity. Schoenberg et al. [18] studied the effect of rotation on propagation waves in an elastic media. Tanaka et al. [19] studied the application of the boundary element method to 3D problems of coupled thermoelasticity. Nowaki [20] introduced thermoelasticity.

This paper brings out the analytical study of generalized thermoelastic medium subjected to rotation. The generalized thermoelastic cylinder is assumed under the influence of rotation and the relaxation time. The main aim of the paper is to investigate the effects of involved parameters on the Rayleigh wave velocity and attenuation coefficient of the wave. Numerical computation has been accomplished to manifest the effect of rotation and relaxation time on the Rayleigh wave velocity and attenuation coefficient of the wave for different types of parameters. The numerical results have been obtained and presented graphically.

2  Formulations of the Problem and Boundary Conditions

Let us consider a homogeneous isotropic elastic solid with an infinite circular cylinder under initial temperature images. The elastic medium is rotating uniformly with an angular velocity images, where images is a unit vector representing the direction of the axis of rotation. The displacement equation of motion in the rotating frame has two additional terms, centripetal acceleration, images due to time-varying motion only where images is the displacement vector, and images. All quantities considered will be functions of the time variable t and the coordinates r and z.

The dynamic equation of motion is given by Sharma et al. [12] as follows:

images

images

The heat conduction equation is given by [4]:

images

Where images is the density of the material, k is thermal conductivity, cv is the specific heat of the material per unit mass, Trr, images, Tzz, and Trz are the stresses components, u and w are the displacement components, and images is the rotating component.

The stress–strain relations are given by Abd-Alla [12]:

images

images

images

images

images

where, images, images and images are thermal relaxation time.

The strain components and the rotation are given by Bagri et al. [13] as follows:

images

Using Eqs. (4)(9) into Eqs. (1) and (2), the following can be written as:

images

images

By Helmholtz’s theorem [14], the displacement vector images can be written in the following form:

images

Where the scalar images and the vector images represent irrational and rotational parts of the displacement images. The component of the vector images to be non-zero, as

images

From Eqs. (12a) and (12b), we obtain

images

Substituting from (12a)–(12c) into Eqs. (3), (10), and (11), we get two equations for images and images as follows:

images

images

images

Where,

images

Eq. (14) represents the longitudinal wave in the direction of r with velocity images and Eq. (15) represents the velocity of the shear wave in the direction of r with velocity images.

3  Boundary Conditions

Let us consider a homogeneous and isotropic elastic solid with an infinite cylinder of the radius a. The axis of the cylinder is taken along the z direction, subjected to the boundary conditions are given as follows:

images

The thermal boundary condition is

images

4  Solution of the Problem

Assume that the temperature and potential functions of solid satisfy

images

Eqs. (13)(15) yield a set of differential equations:

images

images

images

where images is the Laplace operator. images can be eliminated from Eq. (20) by substituting it in Eq. (19), giving

images

Where

images

The general solution of Eqs. (21) and (22) can be found. If we introduce the inversion of the Hankel transform which is defined by

images

We obtain

images

By putting images, the indicial equation governing (24) is

images

where images are the roots of Eq. (24) and images, j = 1, 2.

If images, then the roots of Eq. (25) become

images

The above roots correspond to the case in which the elastic wave and generalized heat condition equations are not coupled. For small images, i.e., only the first order is taken. The roots of Eq. (25) take the form

images

Then the solution of Eq. (24) is

images

which leads to

images

The solution of Eq. (21) is

images

Where images and images.

Substituting Eq. (29) into Eq. (20) yields

images

The stress components Trr and Trz are given by

images

Substituting Eqs. (29)(31) into Eqs. (12c) and (32), we get

images

images

images

images

5  Frequency Equation

In this section, we are going to obtain the frequency equation for the boundary conditions which specify that the outer surface of the cylinder is traction free and the thermal boundary conditions are illustrated. Substituting Eqs. (31) and (32) into the boundary conditions (17) and (18), we get

images

images

images

By eliminating constraints images and images, the frequency equation is given in a form of third order determinant as follows:

images

where

images

The frequency Eq. (40) has a complex root. The real part (Re) gives the velocity of Rayleigh waves and the imaginary part (I’m) gives the attenuation coefficient.

6  Numerical Results and Discussion

The numerical calculation was carried out of the Rayleigh waves velocity and attenuation coefficient. To illustrate the theoretical results obtained in the preceding section, we now present some numerical results. The material chosen for this purpose was carbon steel, the physical data for which are given below [16]:

images

Fig. 1 shows the variations of Rayleigh waves velocity and attenuation coefficient with respect to frequency images for different values of rotation images for the Lord-Schulman theory. The Rayleigh wave velocity decreases with the increase of rotation and frequency, while the attenuation coefficient increases with the increase of rotation and frequency.

images

Figure 1: Variations of Rayleigh waves velocity and attenuation coefficient with respect to frequency images for different values of images at images, r = 2

Fig. 2 shows the variations of Rayleigh wave velocity and attenuation coefficient with respect to frequency images for different values of relaxation time images for the Lord-Schulman theory. The Rayleigh wave velocity decreases with the increase of relaxation time and frequency, while the attenuation coefficient increases with the increase of relaxation time and frequency.

images

Figure 2: Variations of Rayleigh waves velocity and attenuation coefficient with respect to frequency images for different values of relaxation time of images at images, and r = 2

(II) GL-model (images, images), images.

Fig. 3 shows the variations of Rayleigh wave velocity and attenuation coefficient with respect to frequency images for different values of rotation images for the Green-Linsay theory. The Rayleigh wave velocity decreases with the increase of rotation and frequency, while the attenuation coefficient increases with the increase of rotation and frequency; the attenuation coefficient shifts from positive to negative in the range of frequency.

images

Figure 3: Variations of Rayleigh wave velocity and attenuation coefficient with respect to frequency images for different values of rotation images, at images, images

Fig. 4 shows the variations of Rayleigh wave velocity and attenuation coefficient with respect to frequency images for different values of relaxation time images for the Lord-Schulman theory. The Rayleigh wave velocity decreases with the increase of relaxation time and frequency, while the attenuation coefficient increases with the increase of relaxation time and frequency.

images

Figure 4: Variations of Rayleigh waves velocity and attenuation coefficient with respect to frequency images for different values of relaxation time images at images and images

7  Conclusions

The governing field equations for linear homogeneous and isotropic thermoelastic materials with rotation are solved to work out appropriate surface wave solutions in an infinite cylinder. The frequency equation for the Rayleigh surface wave is obtained. The numerical results are illustrated graphically against frequency for different values of rotation and relaxation time. Some concluding remarks are given as follows

1.    The rotation and relaxation time significantly influence the variations of the Rayleigh wave.

2.    Analysis of Rayleigh wave developed into a body due to rotation and relaxation time.

3.    The rotation and relaxation time of an infinite cylinder give the same effect in the problem as mentioned above in the results.

4.    The present theoretical results may provide interesting information for experimental scientists, researchers, and seismologists working on this subject.

Funding Statement: The author received no specific funding for this study.

Conflicts of Interest: The author declares that there are no conflicts of interest to report regarding the present study.

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