Propagation of plane waves in a rotating magneto-thermoelastic fiber-reinforced medium under GN theory

The article is concerned with the possibility of plane wave propagation in a rotating elastic medium under the action of magnetic and thermal fields. The material is assumed to be fibre-reinforced with increased stiffness, strength and load bearing capacity. Green and Nagdhi’s concepts of generalized thermoelastic models II and III have been followed in the governing equations expressed in tensor notation. The effects of various parameters of the applied fields on the plane wave velocity have been shown graphically. c © 2017 University of West Bohemia. All rights reserved.


Introduction
The concept of fibre-reinforced solids stems from the necessity of generating more strength, stiffness and load bearing capacity of solids through the process of suitably mixing certain kind of materials with such solids.In situations where light material with high strength and stiffness is essential, as for example, in manufacturing aircraft components, fibre-reinforced material can be a good option.Fibre-reinforcing is not any new idea.In fact, fibre-reinforcing in its crude form was a common practice followed in hardening soil by mixing straw dust or rice husk with the soil in early stages of civilization and is still followed in some rural areas.Another very common instance is the use of iron wire/rods in manufacturing concrete structures.With the advancement of technology and innovation of modern methods different kinds of materials are manufactured as per need using fibre-reinforcing concept.Different methods are adopted in fibre-reinforcing process, viz., long fibres are arranged in parallel in the matrix, short fibres are randomly arranged in the matrix, or fibres in the form of cloth are set in the matrix.Obviously, the strength and the load bearing capacity of the fibre-reinforced material will depend upon the fibre arrangement in the matrix.
In the study of deformation in solids great care and attention are necessary for bodies subjected to flow or generation of heat in it due to the fact that thermal loading causes deformation different from that caused by mechanical loading.Thermoelasticity takes care of the deformations and stresses produced by thermal loadings as well as those produced by mechanical loadings.The equations governing thermoelastic behavior consist of the modified stress strain relations, the stress equations of motion and the heat conduction equation.But the parabolic type of heat conduction equation as was used initially in the study of thermoelastic behavior was found to yield some unrealistic situation in the sense that the velocity of heat wave propagation was infinite.Modified version of the Fourier law of heat conduction as was suggested by Lord and Shulman [15] in 1967 yielded a finite velocity of heat wave propagation by introducing relaxation time into the energy equation, rendering the new thermoelastic field equations fully hyperbolic.In 1972 Green and Lindsay [12] proposed a theory of thermoelasticity keeping the Fourier law of heat conduction unchanged and modifying the classical energy equation and stress-strain-temperature relations.Investigation of various problems characterizing the two theories based on totally independent ideas, have been discussed by Chandrasekharaiah [5,6].Modifications of the constitutive equations of thermoelasticity done afterwards by Green and Nagdhi [12,13] paved the way to accommodate a wider class of heat flow problems.In G-N model II constitutive equations are derived using thermal displacement gradient among the constitutive variables and reduced energy equation.The model considers no energy dissipation.G-N model III is a modification of G-N model II in the sense that the temperature gradient has also been considered in the constitutive equation in addition to those considered in model II.This theory allows energy dissipation and permits the propagation of thermal signals at both finite and infinite speeds.
Study of the effects of magnetic field on thermoelastic medium is another area of interest.The field of study known as magneto thermoelasticity has applications in several areas, particularly in nuclear devices, biomedical engineering and geomagnetic investigations.Some of the works related to this field are Abd-Alla and Al-Dawy [1,2], Ezzat and Othman [10], Ezzat [9], Ezzat et al. [11], Wang et al. [25], Othman and Song [22], Othman [19], Othman and Said [23].A number of discussions relating wave propagation in rotating isotropic or anisotropic media were reported in literature, some of which are the works of RoyChoudhuri [7], Gupta and Gupta [14], Singh [24], Maity et al. [16,17].Roychoudhuri and Banerjee [8] studied the propagation of time-harmonic coupled electromagnetoelastic dilatational thermal shear waves using the thermoelasticity theory of type II (G-N model) [13].Thermoelastic plane waves in a rotating isotropic medium has been studied by Ahmad and Khan [3].A number of discussion relating to fibre-reinforced materials were discussed by Belfield [4], Othman [20], Othman [21], Markham [18], Zorammuana [26].
The present discussion aims at the study of the propagation of plane waves in a rotating thermoelastic fiber-reinforced medium with and without energy dissipation under Green-Nagdhi model.A magnetic field of uniform magnitude is supposed to be acting on the medium but there is no body force.Fiber-reinforcing of general type has been considered and the governing equations of motion are framed taking into account of the thermoelastic characteristics of the material, rotational effects and the applied magnetic field.Equations have been presented using tensor notations.Possibilities of plane wave propagation in the medium have been studied in this discussion.Effects of rotation, applied magnetic field, and temperature of the material on plane wave propagation have been examined severally and jointly.Finally, some graphical presentations have been made to assess the effects of various parameters in the plane wave propagation in fiber-reinforced media of different nature.

Field equations
Following Belfield et al. [4] the stress-strain relations for linearly fibre-reinforced elastic medium may be expressed in tensor form as where τ ij are the cartesian components of the stress tensor; are the strain components, related to the displacement vector u i ; λ, μ T are elastic constants; α * , β * , (μ L − μ T ) are reinforcing parameters; ν is a temperature parameter and a = (a 1 , a 2 , a 3 ) such that a 2 1 + a 2 2 + a 2 3 = 1.For a rotating elastic medium the equation of motion, in absence of body force, can be written as In ( 2), ρ denotes the material density, Ω is the angular velocity vector, overhead dot represents differentiation with respect to time and the suffix i after second bracket represent the ith component of the vector inside.
If, in addition, the solid is under the action of magnetic field H, then the governing field equations involving the displacement u = u i (x, t) and the temperature T (x, t), for a fiberreinforced material with rotation, in absence of body force, may be written as where α = 1 and α = 0 respectively correspond to heat conduction with and without energy dissipation.The term J × B in (3) arises from the presence of the applied magnetic field.Due to the application of the initially applied magnetic field H 0 , an induced magnetic field h, an induced electric field E and a current density J are developed.For a slowly moving homogeneous electrically conducting medium, the simplified system of linear equations of electrodynamics are where 0 is the electrical conductivity and μ 0 is the magnetic permeability so that B = μ 0 H is the magnetic field in the medium due to total magnetic field H = H 0 + h, arising from applied field H 0 and induced field h.
If we assume that H 0 = (H 01 , H 02 , H 03 ) and Ω = (Ω 1 , Ω 2 , Ω 3 ), then utilizing relations (5) and neglecting products of h with u and its derivatives, the governing equations of motion (3) and (4) for a medium in thermoelasticity with and without energy dissipation under the action of applying magnetic field and rotation may be written in tensor notation as where T * is the reference temperature, ν = (3λ + 2μ)α t , K is the coefficient of thermal conductivity, K * is the additional material constant, ρ is the mass density, c e is the specific heat of the solid at constant strain, α t is the coefficient of linear thermal expansion, λ and μ are Lame' constants.In (6), ijk represents the Levi-civita tensor which has a non-zero value only if i, j, k are all distinct and has a value of 1 if i, j, k are in cyclic order, whereas it has a value of −1 when they are acyclic.

Plane wave propagation
In order to examine the possibility of a plane wave propagation in the medium under consideration we shall assume a solution of governing equations ( 6) and ( 7) in the form The speed of the wave is The direction of plane wave propagation is represented by the unit vector n = (n 1 , n 2 , n 3 ), while the direction of particle displacement is denoted by the unit vector p = (p 1 , p 2 , p 3 ).A and B appearing in (8) are constants.Substituting ( 8) into ( 6) and ( 7), using τ 2 = −1, n k n k = 1 and setting B/A = D, we get Eliminating D from (10) and (11) we get Equation ( 12) is a set of three linear homogeneous equations in p k of the form Also where Rewriting (13) as and noting that not all p k s are zero, it follows that Equation ( 17) with a third order determinant on its left hand side yields an algebraic equation in q 2 with complex coefficients which will determine the wave speed c n in (9).It is clear that the velocity of the plane wave propagation depends on the terms F ik arising from the elastic behavior of the material and the direction n i of propagation of the wave, the terms M ik arising from the applied magnetic field, the terms R ik arising from the rotation of the medium and the terms T ik arising from the thermal character of the material.

Particular Case
Let us consider a fibre-reinforced elastic body with fibre-reinforcing direction a = (a 1 , a 2 , a 3 ) is rotating with uniform angular velocity Ω = Ω(0, 0, 1) (Fig. 1).Let us suppose that a uniform magnetic field H 0 = H 0 (0, 1, 0) is applied to the body.We investigate propagation of a plane wave in the medium in a direction specified by the unit vector n = (0, n 2 , n 3 ).In this case we rewrite (16) in the form where D ik = η ik − θδ ik .
Writing x = q/ω, we find that, in this particular case , where Since all the values of p k are not zero, the equation |D ik | = 0 obtained from ( 18) yields two algebraic equations and where A closer look at equations ( 19) and (20) will reveal that (19) is the same for both the models II and III, whereas ( 20) is an additional equation for the model III.A real and positive root of ( 19) and (20) will lead to the determination of the speed of plane wave propagation in the medium.The speed of wave propagation is actually 1/|x|.Furthermore if the wave is propagating in the direction of the axis of symmetry, i.e., if n 2 = 0, n 3 = 1 and a = (0, 0, a 3 ), then equation (17) becomes where The equation |D ik | = 0 yields four different velocities of plane wave propagations (V com ) i , i = 1, 2, 3, 4 common to models II and III and an additional velocity V 2 for model III only, given by which clearly depends upon the reinforcing direction, applied magnetic field and temperature parameters of the medium which depends upon the applied magnetic field, rotation of the medium but not the direction of reinforcing parameter and temperature parameters and which depends upon the reinforcing direction, applied magnetic field and rotation parameter of the medium, where and

Numerical results and discussions
The present study focuses on the effects of fibre-reinforcing, rotation, magnetic field and temperature on the propagation of plane waves in a solid.To observe the effects numerically we have adopted three sets of values of relevant parameters from the works of Othman et al. [19], Markham [18], Zorammuana [26] as given below Although our analysis is quite general in respect of fibre-reinforcing direction, direction of the axis of rotation and the direction of the applied magnetic field, for simplicity of calculations and for a quick view on the trend of behavior of the plane wave velocity under the actions of rotation, thermal and magnetic effects, we shall limit our numerical computation for the fibre-reinforced medium which (a) has fibre-reinforcing in the direction (0, 0, 1), (b) is under the applied magnetic field in the direction (0, 1, 0), (c) is rotating about an axis having direction cosines (0, 0, 1).
Furthermore, we shall assume that a plane wave is propagating in the direction (0, 0, 1).Under such conditions, the wave velocity values are given by equations ( 22), ( 23) and (24).Adopting the parameter as listed above in our numerical computation, we have tried to analyse the trends of the behavior of plane wave propagating in the medium under different media conditions and load conditions with respect to two G-N models.In Figs.2-9 we have plotted the velocity of plane wave propagation against applied magnetic field H 0 under different states of rotation, reinforced and thermal conditions.One interesting observation from the figures indicate that in absence of applied magnetic field the wave velocity has its maximum value under thermal, rotational and fibre-reinforced conditions.In other words, increase of magnetic field under any of these conditions depresses the wave velocity.
Figs. 2-4 show that for a fixed applied magnetic field, the velocity of propagation decreases with the increase of reinforcing parameter values μ L − μ T .Fig. 2. Effect of reinforcing parameter on wave velocity (V com ) 1 Figs.5-6 display the rotational effects on the wave velocity.It is seen that for a fixed value of applied magnetic field, the wave velocity decreases as rotation parameter increases.Fig. 9 shows the thermal effects on the wave velocity indicating that the velocity decreases with the increase of temperature.This figure also shows that at a fixed temperature the increase of a magnetic field decreases the wave velocity.

Conclusion
The present study aiming for examining the possibility of plane wave propagation in a fibrereinforced medium under the action of rotation, magnetic and thermal loadings, shows that the plane wave propagation is significantly influenced by the parameter values of the medium as well as other conditions imposed on the medium.Study shows that fibre-reinforcing direction in the medium plays an important role on the wave velocity.Effects of rotation, magnetic and thermal fields applied jointly or severally are also very significant.Numerical evaluation of plane wave velocity based on available parameter values of fibre-reinforced materials shows that keeping the other parameter values unchanged, the increase of applied magnetic field in a certain direction on the medium depresses the wave velocity.Also, for a fixed value of applied magnetic field, wave velocity decreases with the increased values of rotation and fibre-reinforcing parameters.It is also found that G-N model III predicts two more possible velocities of plane wave propagation in the medium in comparison with those in G-N model II.