Document Type : Research Paper
Authors
^{1} Mathematical and Theoretical Physics Department, Egyptian Atomic Energy Authority, EGYPT
^{2} Physics Department, Alexandria University Faculty of Science, EGYPT
Abstract
Keywords
Main Subjects
Influence of Electromagnetic Generalized Thermoelasticity Interactions with Nonlocal Effects under TemperatureDependent Properties in a Solid Cylinder
^{a} Mathematical and Theoretical Physics Department, Egyptian Atomic Energy Authority, EGYPT
^{b }Physics Department, Alexandria University Faculty of Science, EGYPT
KEYWORDS 

ABSTRACT 
Electromagnetic theory; Generalized thermoelasticity; Nonlocal thermoelasticity; Solid cylinder; Temperaturedependent properties. 
The temperaturedependent properties and the effect of nonlocal elasticity in the presence of a magnetic field have been studied in an infinitely long solid conductive circular cylinder. The issue arises in the setting of two relaxation times in extended magnetothermoelasticity theory. In the presence of a uniform magnetic field in the direction of the axis, the lateral surface is tractionfree and subjected to known temperatures. Techniques are employed to determine the answer in the Laplace transform domain. A numerical method based on Fourier series expansions is used to carry out the inversion operation. In addition, graphs depict comparisons to highlight the influence of various elements such as the difference in times and the effect of the nonlocal coefficient and Empirical material constant. 
The theory of thermoelasticity deals with the effect of thermal and mechanical disturbances on an elastic body. Earlier interest in it resulted in a large number of theoretical and experimental research works. Thermoelasticity is important because it has numerous applications in domains including aviation, nuclear reactors, modern propulsion system technology, plasma physics, and geophysics. Green and Lindsay [1] introduced the theory of generalized thermoelasticity, which is often referred to as the theory of generalized thermoelasticity with two relaxation times. The fundamental considerations of that theory have been the subject of numerous works. For example, Hany [2] has solved a model consisting of a halfspace governed by thermoelastic equations with two relaxation times. The action at the ambient level is carried out by a combination of thermal and mechanical shocks that act for a limited time. Khader et al. [3] applied Green and Lindsay's theory to a model of transient heat response in infinitely long annular cylinder surfaces with an internal heat source. Surendra and Elsibai [4] applied the two different theories of generalized thermal elasticity to a spherical cavity, and their results were compared with the classical dynamiccoupled theory. The interaction of magnetic fields and pressure in a hot solid material is gaining popularity due to its numerous applications in geophysics, plasma physics, and other domains. Extremely high temperatures and temperature gradients, as well as magnetic fields occurring within nuclear reactors, have an impact on their design and operation in the nuclear field [5]. The heat equation under discussion is usually the uncoupled or coupled equation, not the generalized one, in these investigations. In some cases, where the answers determined using either of these equations deviate slightly quantitatively, this attitude is valid. When considering shorttime impacts, however, the whole, generalized system of equations must be used; otherwise, a significant amount of accuracy would be lost. Among the authors who considered the generalized magnetothermoelastic equations are Sherief [69]. Ezzat and others [1014] have studied thermoelastic Bodies and viscoelastic materials with fractional. He [15] studied the generalized electromagneticthermoelastic problem for an infinitely long solid cylinder. AbdAlla et al. [16] have discussed the Rayleigh surface wave propagation in an orthotropic rotating magnetothermoelastic medium under the effect of gravity and initial stress. Othman and others [1721] have presented many papers in which they studied LordShulman theory, generalized thermoelasticity plane waves with two relaxation times, and temperaturedependent properties. Many authors contributed to this subject [2237]. As is widely known, the physical properties of engineering materials vary significantly with temperature. Because the modulus of elasticity drops dramatically with temperature, the material's ability to elastically resist thermal loads declines. As a result, plastic deformation occurs at far lower thermal stresses than systems with constant physical properties imply. Araki [38] studied a thermal stress analysis of a thermosviscoelastic hollow cylinder with temperaturedependent properties. Abbas [39] discussed the eigenvalue approach method in threedimensional generalized thermoelastic with temperaturedependent material properties. Abouelregal [40] solved the boundary value problem of a onedimensional semiinfinite piezoelectric medium. In several papers [4156] the temperaturedependent properties of materials have been studied as the nonlocal theory of elasticity. In this work, the temperaturedependent properties and the effect of nonlocal elasticity in the presence of a magnetic field have been studied numerically in an infinitely long solid conductive circular cylinder.
Let (r, φ, z) be cylindrical polar coordinates with the zaxis coinciding with the axis of a solid; infinitely long, elastic circular cylinder of a homogenous, isotropic material of radius r, . The cylinder is placed in a magnetic field, as shown in Figure 1. Due to Maxwell’s equations, electrodynamics equations are linearly simplified for a perfectly homogeneous, conducting elastic solid

(1) 
Ohm’s law for moving media

(2) 
z 


r 

H 

Z 



Y 
X 
Fig. 1. An infinitely long solid cylinder
From (1) and (2), we obtain:

(3) 

(4) 

(5) 
from equations (3) and (5), we get

(6) 
from equations (4) and (6), we obtain

(7) 
The strain components

(8) 
The cubic dilatation e is given by

(9) 
The stress tensor σij’s are given by

(10) 

(11) 

(12) 

(13) 
The equations of motion and equation of heat conduction [53]

(14) 

(15) 

(16) 
By applying the initial and boundary conditions, we can solve the problem.

(17) 
The induced magnetic and electric fields are continues.

(18) 
Tractionfree and thermal shock,

(19) 
from equations (4) and (18), substituting Fr into equation (14), we get

(20) 
to discuss the effect of the temperaturedependent material on the considered physical fields of the problem, following [39], we assume that
temperatureindependent properties, when we are put
Let us introduce the nondimension variables [20]
The equations (4), (7), (10), (16), and (20), can be written in the nondimensional form,

(21) 

(22) 

(23) 

(24) 

(25) 
where
Applying the Laplace transform with parameter s to both sides of equations (21)(25), we obtain

(26) 

(27) 

(28) 

(29) 

(30) 
Eliminating from equations (27), (29), and (30), we get

(31) 
where
and satisfy the equations,

(32) 

(33) 
The solutions of equations (31)(33), have the forms

(34) 

(35) 

(36) 
where, is the modified Bessel function of the first kind of order zero, and are the roots of the characteristic equation:

(37) 
from equations (26) and (36), we obtain

(38) 
The displacement, , can be found in equations (9) and (34)

(39) 
from equations (34), (35), (39), and 28), we obtain

(40) 
in the free space, around the cylinder, the induced fields E0 and h0 satisfy the following equations,

(41) 

(42) 
From equations (41) and (42) by eliminating , we get

(43) 
The solution of equation (43) is given by

(44) 
where is a modified Bessel function of the second kind of order zero, and is some parameter depending on s only.
from equation (44) and equation (42), we obtain

(45) 
The boundary conditions after taking the Laplace transform of both sides, we obtain

(46) 
by using equations (46), we can obtain the systems of linear equations for the unknown parameters

(47) 

(48) 

(49) 

(50) 
To get the inverse of Laplace transforms, we use the method proposed in [57]. A numerical analysis is performed to study the effects of nonlocal and temperaturedependent properties. For the purpose of illustration, the material properties are listed as [6], [39].

Fig. 2. Temperature distribution 

Fig. 3. Displacement distribution 

Fig. 4. Radial stress distribution 

Fig. 5. Electric field distribution 

Fig. 6. Induced magnetic field distribution 

Fig. 7. Temperature distribution 

Fig. 8. Displacement distribution 

Fig. 9. Radial stress distribution 

Fig. 10. Electric field distribution 

Fig. 11. Induced magnetic field distribution 

Fig. 12. Temperature distribution 

Fig. 13. Displacement distribution 

Fig. 14. Radial stress distribution 

Fig. 15. Electric field distribution 

Fig. 16. Induced magnetic field distribution 
Figures (26) represent the graphs for the induced electric and magnetic field, stress, displacement, and temperature, in the case of absent nonlocal ( ) and temperatureindependent properties ( ). We can solve the problem by using three values of times, namely for , , and . Solid lines represent the case when t = 0.12, Dotted lines represent the solution for t = 0.05, and dashed lines represent the solution for t = 0.08. We first evaluate the derivation in the last section and the numerical inverse of the Laplace transform algorithm. The temperature on the cylinder surface is constant at 1. From all the graphs, it is easy to see that all functions satisfy the boundary conditions. We observe that the value of the temperature increases when the value of time increases. The position of the maximum points of stress and displacementinduced electric and magnetic fields with respect to radius increases when the time increases. The finite speed of wave propagation is seen in all graphs, which is the major distinguishing feature of the expanded theory with two relaxation times. That is, unlike in the coupled theory, the effects of thermal shock on the boundary do not immediately fill the entire body of the cylinder.
Figures 711 represent nonlocal and temperatureindependent properties ( ). The problem was solved for t = 0.08 and three values of nonlocal parameter, namely for , , and dotted lines represent the solution for , the dashed lines represent the solution for , and solid lines represent the case when .
It clearly shows the effect of the nonlocal parameter on the temperature and stress (Figures 7 and 9). In figure 7, when ε = 0 (absent of nonlocal), the impacts of the thermal shock arrive at namely r = 0.4, but in present nonlocal the impacts of the thermal shock arrive namely
r = 0.35. In figure 9, the effect of nonlocal is appearing at the point of discontinuity. The effect of the nonlocal parameter is very small in the displacement, induced electric and magnetic fields, figures (8, 10, and 11).
Figures 1216, represent the case of absent nonlocal and temperaturedependent properties. The problem was solved for t = 0.08 and three values of Empirical material constant namely for , , and . Dotted lines represent the solution for , the dashed lines represent the solution for , and solid lines represent the case when . In Figure (12), we note that by increasing the Empirical material constant α*, the speed of heat wave propagation decreases; this is due to the change in the physical properties of matter. At the impacts of the thermal shock have arrived at namely r = 0.44, at the impacts of the thermal shock, have arrived at namely r = 0.47, at the impacts of the thermal shock, have arrived at namely r = 0.51. In all the figures (1216), it should be noted that there is a significant difference in the values of the considered function. This may be due to the fact that the temperature in thermoelasticity is an infinitesimal deviation from the reference temperature. Thus, the dependence of the modulus of elasticity on reference temperature has a significant effect on the thermal and mechanical interactions. And we show that by increasing the Empirical material constant α*, increases the values of the functions, which indicates that the physical reasons for elastic behavior can be quite different for different materials. In metals, lattice changes size and shape when forces are applied (energy is added to the system). This is evident from the figures (1314).
In this work, the temperaturedependent properties and the effect of nonlocal elasticity in the presence of a magnetic field have been studied numerically in an infinitely long solid conductive circular cylinder. We can draw the following conclusions in light of the above analysis: The physical quantities satisfy the boundary conditions, the finite speed of wave propagation is apparent, and all the results are in concurrence with the generalized theory of thermoelasticity. The studied physical quantities are affected by the presence of a nonlocal parameter, as they are strongly affected by some physical quantities such as temperature and stress, and have a negligible effect on some physical quantities such as displacement, and electric and magnetic fields. The dependence of the modulus of elasticity on the reference temperature has a significant effect on thermal and mechanical interactions. This is very clear from the graphs. The magnitude of all studied physical quantities increased in the presence of temperaturedependent properties. This study is useful for these problems. The study mentioned above is important in the study of structural components and mechanical elements such as pressure vessels and pipes in nuclear reactors, chemical plants, and highspeed aircraft. It is subject to thermal loads due to high temperature, hightemperatures gradients, and periodic temperature changes.
Nomenclature
J 
electric current density 
ε_{0} 
electric permeabilities 
μ_{0} 
magnetic permeabilities 
D 
electric induction vectors 
σ_{0} 
electric conductivity 
u 
displacement vector 
λ, μ 
Lamé’s modulii 
γ 
material constant 
T 
absolute temperature 
ρ 
density 
θ_{0} 
constant 
H_{0} 
magnetic field 

Laplace’s operator 
α_{t} 
coefficient of linear thermal expansion 
T_{0} 
reference temperature 
k 
thermal conductivity 
c_{E} 
specific heat at constant strain 
τ_{1}, τ_{2} 
the relaxation time 
H(t) 
Heaviside unit step function 
E_{0} 
Induced electric fields in the free space 
h_{0} 
Induced magnetic fields in the free space 
F 
Lorentz force 

Constants 

Empirical material constant 
B 
magnetic induction vectors 

nonlocal parameter 
Acknowledgments
This computational facility support from the Department of Mathematical and Theoretical Physics at the nuclear research center is gratefully acknowledged
Conflicts of Interest
The author declares that there is no conflict of interest regarding the publication of this manuscript. In addition, the authors have entirely observed the ethical issues, including plagiarism, informed consent, misconduct, data fabrication and/or falsification, double publication and/or submission, and redundancy.
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