Chen et al. In absence of heat supply, these two temperatures are identical for time independent situation. However, for time dependent cases, particularly for problems related to wave propagation, the two temperatures are in general different, regardless of heat supply. The two temperature thermoelasticity theory has gained much attention of the researchers in the recent years. The existence, structural stability, convergence and spatial behaviour of two temperature thermoelasticity have been provided by Quintanilla .
Roy Choudhuri and Debnath [24,25] and Othman [26,27]. Othman [28,29] studied the effect of rotation in a micropolar generalized thermoelastic and thermo-viscoelasticity half space under different theories. The propagation of plane harmonic waves in a rotating elastic medium without thermal field has been studied. It was shown there that the rotation causes the elastic medium to be dispersive and an isotropic.
These problems are based on more realistic elastic model since earth; moon and other plants have angular velocity. Owing to the mathematical difficulties encountered in two- and three dimensional multi-field coupled generalized heat conduction problems, the problems become too complicated to obtain an analytical solution. Instead of analytical methods, several authors have applied numerical methods such as finite difference method, finite element method and boundary value method etc. One can find several two-dimensional works based on the generalized thermoelasticity by using the normal mode analysis in the literatures Ezzat and Abd Elall , Othman and Lotfy , Lofty and Othman , Lofty  and Sarkar and Lahiri .
Using the normal mode analysis technique, we will get the solution in the Fourier transformed domain actually. To apply the normal mode analysis, we have to assume that all the relations are sufficiently smooth on the real axis such that the normal mode analysis of all these functions exist.
The normal mode analysis  was used to obtain the exact expression for the temperature distribution, thermal stresses, and the displacement components. The present article is concerned with a two-dimensional electro-magneto- thermoelastic coupled problem for a homogeneous, isotropic, thermally and electrically conducting half-space solid whose surface is subjected to a thermal shock with two temperature heat transfer. The normal mode analysis is used to obtain the exact analytical solutions of the problem.
Numerical results for the temperature, displacements and thermal stresses distribution are presented graphically and discussed. A comparison is made with the results obtained in the presence and absence of two temperature parameter and the magnetic field effect. Let us consider a perfectly conducting thermoelastic half-space x p 0 with two temperatures of a linear, homogeneous and isotropic medium whose state can be expressed in terms of the space variables x, y and the time variable t.
The initial conditions are assumed to be homogeneous. The dynamic displacement vector is actually measured from a steady state deformed position and the deformation is supposed to be small. The electric intensity vector is normal to both the magnetic intensity and the displacement vectors. Thus, it has the components. The heat conduction equation takes the form .
Now, we will suppose elastic and homogenous half-space x p 0 which obey Eqs. The displacement components for one dimension medium have the form. Using the summation convection. From 16 - For a two dimensional problem xy-plane all quantities depend only on space coordinates x, y and time t too. The field equations and constitutive relations in generalized linear thermoelasticity with the influence of magnetic field and without body forces and heat sources are.
Assuming the scalar potential functions P x,y, t and W x,y, t defined by the relations in the non-dimensional form:. The solution of the considered physical variable can be decomposed in terms of normal modes as the following form. Using Eqs. Substituting from Eqs. The normal mode analysis is, in fact, to look for the solution in Fourier transformed domain. Assuming that all the field quantities are sufficiently smooth on the real line such that normal mode analysis of these functions exists.
Analytic Solution for a Nonlinear Problem of Magneto-Thermoelasticity
In the physical problem, we should suppress the positive exponentials that are unbounded at infinity. Substituting the expressions of the variables considered into the above boundary conditions, we can obtain the following equations satisfied by the parameters. Hence, the expressions for the displacements, temperature distribution and other physical quantity of the medium can obtained. In order to analyze the above problem numerically, we now consider a numerical example for which computational results are given.
The results depict the variation of temperature, displacement and stress fields in the context of two theories. To study the effect of rotation and two temperature on wave propagation. The copper material was chosen for the purpose of numerical example. The numerical constants in SI unit of the problem were taken as:. The field quantities, temperature, displacement components and stress components depend not only on space x and time t but also on the thermal relaxation time t0.
Here all the variables are taken in non dimensional forms. The first group Fig. We notice that the results for the temperature, the displacement and stresses distribution when the relaxation time is including in the heat equation are distinctly different from those when the relaxation time is not mentioned in heat equation, because the thermal waves in the Fourier's theory of heat equation travel with an infinite speed of propagation as.
By comparing figures of solutions obtained under the three thermoelastic theories, important phenomena are observed:. It should be noted Fig. The variations of thermo-dynamical heat distribution Q with magnetic field are not similar in nature for the influence of magnetic field in the medium with difference in magnitude.
In other words, the temperature lines without magnetic field have the highest gradient when compared with that of with magnetic field in all range. In addition, all lines begin to coincide when the horizontal distance x increases to reach the reference temperature of the solid. These results obey physical reality for the behavior of copper as a polycrystalline solid.
It is also clear that the rate of change of u decreases with increasing the magnetic field. On the other hand, these variations are oscillatory in nature. In other words, the rxx component line for present the magnetic field take a exponential function has the lowest gradient when compared with that of absence take a wave function and satisfies the boundary.
That the variations of oxx for a thermoelastic medium without magnetic field are more oscillatory in nature ascompared with those variations obtained with the effect of Fig. Also the values force stress oyy for a thermoelastic medium with magnetic field satisfies the boundary condition. In other words, the oyy component line for magnetic has the lowest gradient when compared with that of absent the magnetic field. In addition, all lines begin to coincide when the horizontal distance x is increases to reach zero after their relaxations at infinity.
Variation of magnetic field has a serious effect on both magnitudes of mechanical stresses. These trends obey elastic and thermoelastic properties of the solid under investigation. The lines for. The conductive temperature begins from the positive values and then decreases to arrive the minimum amplitudes in two cases of variation magnetic field, also move in the wave propagation when the magnetic field is absent and exponential function when presented, beyond it falls again to try to retain zero at infinity.
The second group Fig. It should be noted fig. The displacements u show the different behaviors at different values of b. The stress component, rxx reach coincidence with zero value fig. We can say, significant difference in the stress component rxx is noticed for different values of the non-dimensional two-temperature parameter.
In addition, all lines begin to coincide when the horizontal distance x increases to reach zero at infinity. The stress component distribution is continuous, smooth and moves in the exponential function. These trends obey elastic and thermoelastic properties of the solid. The stress component, ryy start from the negative value fig. The two cases are different the exponential function. Finally, Figs. It is concluded that all the values decreases with an increasing of smaller values of y-axis and with the larger values increase. The curves of the physical quantities with CD theory in most of figures are lower in comparison with those under LS theory and GL theory, due to the relaxation times.
Analytical solutions based upon normal mode analysis for themoelastic problem in solids have been developed and utilized. The theory of two-temperature generalize thermoelasticity describes the behavior of the particles of the elastic body more real than the theory of one-temperature generalized thermoelasticity. The context of the theory of two-temperature. The value of all the physical quantities converges to zero with an increase in distance x and All functions are continuous.
Deformation of a body depends on the nature of forced applied as well as the type of boundary conditions. It is clear from all the figures that all the distributions considered have a non-zero value at the beginning except in a bounded region of the half-space. Outside of this region, the values vanish identically and this means that the region has not felt thermal disturbance yet. From the temperature distributions, we have found a wave and exponential type heat propagation with finite speeds in the medium. The heat wave front moves forward with a finite speed in the medium with the passage of time which proves that the generalized thermoelasticity theory with two temperature heat transfer is much closed to the behavior to the elastic materials.
This is not the case for the CD theory where an infinite speed of thermal propagation can be found and hence all the considered physical quantities have a non-zero possibly very small value for any point in the medium. This indicates that the generalized Fourier's heat conduction mechanism is completely different from the classical Fourier's law. The effect of magnetic field in the medium plays significant role in the study of deformation of a body.
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Hetnarski, in: Proc. Hereafter, for simplicity, the following non-dimensional variables will be considered throughout this study. Using the above non-dimensional variables, and by dropping the primes for convenience, the governing equations and expressions for stress components take the following form:. Eliminating u and v between Eq. The Fourier cosine and sine transforms are defined respectively by the following relations  :.
Substituting from Eq. The solution of Eq. Applying the Fourier cosine transform with respect to the variable x on both sides of Eq. Appling Fourier cosine transforms to both sides of Eq. Inverting the Fourier transform for Eq. Using the boundary conditions 44a and 44c together with Eq.
The boundary conditions Eq. In Eq. Equations 50 and 51 are a set of dual integral equations whose solution gives the unknown parameter A. The following substitution is now taken in order to solve these equations:. Interchanging the order of the integration in the previous equation and using the following relation:. Equation 56 is called the Able integral equation and its solution is given by  :. Assuming that. Regarding the above equation and Eq. To obtain B q , the dual integral Eq.
Using Eq. Considering Eq. The above evaluations are applied to copper material, whose constants are shown in Table 1. A Fortran computer program, compiled with Visual Fortran v. Romberg technique of numerical integration with variable step size was employed to calculate the involved definite integrals.
It should be noted Fig. Graph lines for both values of y show different slopes at crack ends according to y-values. In addition, all lines begin to coincide when the horizontal distance x is beyond the double of the crack size to reach the reference temperature of the solid. These results obey physical reality for the behaviour of copper as a polycrystalline solid. The horizontal displacement, u, begins with a parabolic decrease then increases again to reach its maximum magnitude just at the crack end.
Beyond it, u falls again to try to retain zero at infinity. It is also clear that the rate of change of u decreases with increasing y as we go farther apart from the crack Fig. On the other hand, Fig. The displacements u and v show different behaviours, because of the elasticity of the solid tends to resist vertical displacements in the problem under investigation.
Both of the components show different behaviours; the former tends to increase to maximum just before the end of the crack. Then it falls to a minimum with a highly negative gradient. Variation of y has a serious effect on both magnitudes of mechanical stresses Fig.
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These trends obey elastic and thermoelastic properties of the solid under investigation. All functions are continuous, especially that of u and v which prove that the crack will not propagate because of the fact that both of the mechanical and thermal stresses are not sufficient to propagate such a crack, and to propagate it, the solid need to be subject to an external stress tensile, shear,. Subscribe Today.
Science Alert. All Rights Reserved. Research Article. Elfalaky and A. Abdel- Halim. Similar Articles in this Journal. Search in Google Scholar. How to cite this article: A. Abdel- Halim, Journal of Applied Sciences, 6: DOI: Analytical solutions based upon Fourier transforms for thermoelastic problem in solids have been developed and utilized. Temperature, radial and axial distributions were estimated at different distances from the crack edge. Horizontal and vertical stress distributions were evaluated as functions of the distance from the crack edge.
It can be concluded that a change of volume is attended by a change of the temperature while the effect of the deformation upon the temperature distribution is the subject of the theory of thermoelasticity. Anwar, M. Sherief, State space approach to generalized thermoelasticity. Thermal Stresses, Thermoelasticity and irreversible thermodynamics. Applied Phys. Operational Mathematics. Danilovskaya, V. Thermal stresses in an elastic half-space due to a sudden heating of its boundary.
Dhaliwal, R. Generalized thermoelasticity for anisotropic media, Q.
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