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The aim of the present investigation is to study the effects of magnetic field, relaxation times, and rotation on the propagation of surface waves with imperfect boundary. The propagation between an isotropic elastic layer of finite thickness and a homogenous isotropic thermodiffusive elastic half-space with rotation in the context of Green-Lindsay (GL) model is studied. The secular equation for surface waves in compact form is derived after developing the mathematical model. The phase velocity and attenuation coefficient are obtained for stiffness, and then deduced for normal stiffness, tangential stiffness and welded contact. The amplitudes of displacements, temperature, and concentration are computed analytically at the free plane boundary. Some special cases are illustrated and compared with previous results obtained by other authors. The effects of rotation, magnetic field, and relaxation times on the speed, attenuation coefficient, and the amplitudes of displacements, temperature, and concentration are displayed graphically.

The foundations of magnetoelasticity were presented by Knopoff [

Recently, Sherief and Saleh [

In this paper, linear model is adopted to represent the imperfectly bonded interface conditions. The linear model is simplified and idealized situation of imperfectly bonded interface, where the discontinuities in displacements at interfaces have a linear relationship with the interface stresses. Taking these applications into account, the surface waves propagation at imperfect boundary between an isotropic elastic layer and isotropic thermodiffusive elastic half-space with rotation in the context of Green-Lindsay theory is investigated. The phase velocity and attenuation coefficients of wave propagation have been computed from the secular equations. The amplitudes of displacements, temperature, concentration, and specific loss are computed and depicted graphically to make clear the influence of magnetic field, rotation, stiffness, relaxation times, and diffusion on the phenomena and compare with the practical results.

The basic governing equations for homogenous generalized thermodiffusive solid in the absence of heat and mass diffusion sources are as follows (Singh [

Constitutive relations are

Equation of motion in the rotating frame of reference is

Equation of heat conduction is

Equation of mass diffusion is

Here, the medium is rotating with angular velocity

the centripetal acceleration

the Carioles acceleration

where

The symbols correspond to partial derivative and time derivative, respectively.

Following Bullen [

As shown in Figure

Schematic of the problem.

Upon introducing the quantities in (

For an isotropic elastic layer, we introduce potential functions

To solve (

The constants

Substituting the values of

In this paper, linear model is adopted to represent the imperfectly bonded interface conditions. The boundary conditions are the vanishing of the normal stress; Maxwell’s electromagnetic stress tensor

Mechanical conditions:

Temperature condition

Concentration condition

where

Substituting the value of

If we write

Hence, assume that

This shows that

The frequency equation (

The amplitude of surface displacements, temperature change, and concentration at the surface

The specific loss is the ratio of energy

Following Sherief and Saleh [

The concentration change, phase velocity and attenuation coefficient of wave propagation, displacement, stresses, temperature in the context of Green Lindsay (GL) theory of thermoelastic diffusion with variations of magnetic field, and rotation in 2D and 3D have been computed for various values of nondimensional wave number and calculated numerically and represented graphically in Figures

Variation of concentration with respect to the wave number with variation of rotation.

Variation of phase velocity with respect to the wave number with variation of rotation.

Variation of attenuation coefficient with respect to the wave number with variation of rotation.

Variation of the displacement

Variation of normal stress

Variation of shear stress

Variation of the temperature

Variation of the concentration charge with respect to the wave number with variation of the magnetic field.

Variation of phase velocity with respect to the wave number with variation of the magnetic field.

Variation of the attenuation coefficient with respect to the wave number with variation of the magnetic field.

Variation of the displacement

Variation of shear stress

Variation of shear stress

Variation of the temperature

Variation of the concentration charge with respect to the wave number and

Variation of the phase velocity with respect to the wave number and

Variation of the attenuation coefficient with respect to the wave number and

Variation of displacement

Variation of the stress

Variation of the stress

Variation of the temperature with respect to the wave number and

Figure

Figures

Figure

Figures

Totally, it is clear that the temperature decreases with increasing rotation values that physically indicates the negative influence of rotation on the temperature that takes into consideration engineering and structures.

Figures

It is obvious from Figure

From Figure

Finally, it appears that the temperature increases and decreases arrive to zero as the wave number tends to infinity but decreases with an increasing of the small

Surface waves at imperfect boundary between isotropic elastic layer of finite thickness and isotropic thermodiffusive elastic half-space with magnetic field, stiffness, and rotation with two thermal relaxation times (GL) model are illustrated. The secular equation in compact form has been derived. The concentration change, phase velocity, attenuation coefficient, displacement, stresses, and temperature are displayed graphically. The amplitudes of displacements, temperature, and concentration are computed at the free plane boundary and presented graphically. Specific loss of energy is obtained and depicted graphically.

The analysis to be carried will be useful in the design and construction of rotating sensors, engineering, structures, and surface acoustic waves devices and the following remarks have been concluded.

If there is no rotation, small wave number takes larger values than in the presence of rotation and large values of the wave number.

The phase velocity, attenuation coefficient, displacement, stresses components, and temperature begin from zero for zeros value of the wave number and tend to zero if the wave number tends to infinity; also, it is seen that they increase and decrease periodically with an increasing of the wave number and rotation and magnetic field.

The temperature decreases with increasing rotation values that physically indicates the negative influence of rotation on the temperature that takes into consideration engineering and structures and acoustic and rotating sensors.

Consider the following:

Consider the following:

Consider the following:

Consider the following:

The authors declare that there is no conflict of interests regarding the publication of this paper.