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In this work, we study an analytical procedure for evaluation of the displacement and stresses in fibre-reinforced anisotropic elastic media under effects of rotation and initial magnetic field, and due to the application of the rotation and initial magnetic field. Effects of rotation and initial magnetic field are analyzed theoretically and computed numerically. Numerical results have been given and illustrated graphically. Comparison was made with the results obtained in the presence of rotation and initial magnetic field in fibre-reinforced anisotropic and isotropic elastic media. The results indicate the effect of rotation and initial magnetic field.

The linear theory of elasticity of paramount importance in the stress analysis of steel is the commonest engineering structural material. To a lesser extent, linear elasticity describes the mechanical behavior of the other common solid materials, e.g. concrete, wood and coal. The problem of rotating disks or cylinders has its application in high-speed cameras, steam and gas turbines, planetary landings and in many other domains. Various authors have formulated these generalized theories on different grounds. Lord and Shulman [

infinite generalized magneto-thermoelastic diffusion body with a spherical cavity. Effects of rotation and initial stress on generalized-thermoelastic problem in an infinite circular cylinder are due to Abd-Alla et al. [

Fibre-reinforced composites are used in a variety of structures due to their low weight and high strength. The mechanical behavior of many fibre-reinforced composite materials is adequately modeled by the theory of linear elasticity for transversely isotropic materials, with the preferred direction coinciding with the fibre direction. In such composites, the fibres are usually arranged in parallel straight lines. The characteristic property of a reinforced composite is that its components act together as a single anisotropic unit as long as they remain in the elastic condition.

The idea of introducing a continuous self-reinforcement at every point of an elastic solid was discussed by Belfied et al. [

In this paper, we studied an analytical procedure for evaluation of the displacement, and stresses in fibre-reinforced anisotropic elastic media under effect of rotation and initial magnetic field. Using the harmonic vibrations, we found the general solution, determining the displacements and stress components. The special case was studied in isotropic generalized elastic medium with rotation and initial magnetic field. Finally, we represented this case graphically.

The propagation of general surface waves is examined here for a fiber-reinforced elastic solid semi-infinite medium

Both media are under the primary magnetic field

The electromagnetic field is governed by Maxwell equations , under the consideration that the medium is a perfect electric conductor taking into account the absence of the displacement current

where

where

The constitutive equation for the fiber reinforced linearly elastic anisotropic medium with respect to preferred direction

where are

Are the components of strain,

The equations of motion are:

where,

where

To examine dilatational and rotational disturbances, we introduce two displacement potentials

The component

And for medium

The boundary conditions for the titled problem are:

a) The component of displacement at the boundary surface between the media

b) The stress components

where

where

Similar relations in

We seek harmonic solutions for (11), (13) and (14) in the form (see Bullen [

where is a complex frequency. In M and similar relations in M_{1} with the factions

with speed C. It is convenient to introduce

And similar expressions

Now substituting from (23) into (11), (13) and (14), we obtain for the medium

Equation (25) has solutions:

And for the medium

In the above, for the effect to be essentially a surface one ,each expression must diminish indefinitely with increasing distance from the boundary this with be the case if each expression contains an exponential factor in with the exponent is teal and negative. Hence,

Similar relations in

By using the boundary conditions a and b, we can determined the constants

We can study the components of displacement and stresses in fibre-reinforced anisotropic elastic media under effect of rotation and initial magnetic field from Equations (28)-(35) by using Maple program, is clear up from Figures 2-9.

In this case, substituting

To study the surface waves in fibre-reinforced we use the following physical constants for anisotropic elastic media under the in influence of rotation and initial magnetic field, are considered [

The numerical technique outlined above was used to obtain of the displacement, stresses in fibre-reinforced anisotropic and isotropic elastic media under effect of rotation and initial magnetic field. These distributions are shown in Figures 2-17. For the sake of brevity some computational results are being presented here.

rotation, we find in medium

In the light of the above analysis, the following conclusions may be made:

- Effects of rotation and initial magnetic field are cleared on the components of displacement and stresses;

- Effect of complex frequency is cleared on the components of displacement and stresses;

- There is a clear difference in the two cases, anisotropic and isotropic elastic media;

- Deformation of a body depends on the nature of the forces applied as well as the type of boundary conditions.

F. S. Bayones, (2015) Effect of Rotation and Initial Magnetic Field in Fibre-Reinforced Anisotropic Elastic Media. Applied Mathematics,06,877-898. doi: 10.4236/am.2015.65081