On thermoelastic problem based on four theories with the efficiency of the magnetic field and gravity

A.A. Kilany; A.N. Abd-alla; A.M. Abd-Alla; S.M. Abo-Dahab

doi:10.1016/j.joes.2022.02.007

Journal of Ocean Engineering and Science >

2024 , Vol. 9 >Issue 4: 338 - 347

DOI: https://doi.org/10.1016/j.joes.2022.02.007

Original article

On thermoelastic problem based on four theories with the efficiency of the magnetic field and gravity

A.A. Kilany ^,^a ,
A.N. Abd-alla ^a ,
A.M. Abd-Alla ^a ,
S.M. Abo-Dahab

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^a Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524 Egypt
^b Department of Computer Science, Faculty of Computers and Information, Luxor University, Egypt
^c Department of mathematics, Faculty of Science, South

Received date: 2021-11-14

Revised date: 2022-02-12

Accepted date: 2022-02-12

Online published: 2022-03-05

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Abstract

The fundamental objective of this paper is to study the effectiveness of magnetic field and gravity on an isotropic homogeneous thermoelastic structure based on four theories of generalized thermoelasticity. In another meaning, the models of coupled dynamic theory (CDT), Lord-Shulman (LS), Green-Lindsay (GL) as well as Green-Naghdi (GN II) will be taken in the consideration. Then, applying the harmonic method (normal mode technique), the solution of the governing equations and the expressions for the components of the displacement, temperature and (Mechanical and Maxwell's) stresses is taken into account and calculated numerically. The impacts of the gravity and magnetic field are illustrated graphically which are pronounced on the different physical quantities. Finally, the results of some research that others have previously obtained may be found some or all of them as special cases from this study.

Highlights

Investigate conducting thermoelectric materials as a new class of applicable thermoelectric solids. The result provides a motivation to investigate conducting thermoelectric materials as a new class of applicable thermoelectric solids. The results presented in this paper should prove useful for researchers in material science, designers of new materials, physicists as well as for those working on the development of magneto-thermo-elasticity and in practical situations as in geophysics, optics, acoustics, geomagnetic and oil prospecting etc. The used methods in the present article are applicable to a wide range of problems in thermodynamics and thermoelasticity.

Key words： Generalized thermoelasticity; Gravity; Magnetic field; Green and Naghdi Theory II; Normal mode approach

Cite this article

A.A. Kilany , A.N. Abd-alla , A.M. Abd-Alla , S.M. Abo-Dahab . On thermoelastic problem based on four theories with the efficiency of the magnetic field and gravity[J]. Journal of Ocean Engineering and Science, 2024 , 9(4) : 338 -347 . DOI: 10.1016/j.joes.2022.02.007

Nomenclature

\nabla^{2}

Laplacian operator

δ_{i j}

The Kronecker delta

γ

Material constant

λ, μ

Lame's constant

ρ

The density

ϑ_{0}, τ_{0}

Thermal relaxation time

e_{i j}

The strain tensor components

σ_{i j}

The stress tensor components

ζ_{i j}^{*}

The Maxwell stress tensor

μ_{0}

The magnetic permeability

a

The wave number

E

The induced electric permeability

g

The acceleration due to gravity

\vec{h}

The induced magnetic field vector

\vec{H}

The initial uniform magnetic intensity vector

\vec{J}

The current density vector

K

Thermal conductivity

r

The beam radius

T

The absolute temperature

u

The displacement vector

x, y, z

The coordinates of the system

C_{e}

The specific heat at constant strain

k^{*}

The material constant characteristic of the (GN) theory

G_{i}

The gravity force

I_{0}

The absorbed energy

T_{0}

The reference temperature chosen so that

| (T - T_{0}) / T_{0} | \leq 1

e_{k}

The unit vector

ε_{i j k}

The unit vector

1. Introduction

The topic of generalized thermoelasticity has attracted the attention of many scientists, experts and researchers because of its great importance in modern industrial technological applications. The expanded literature on this topic is usefully and currently available in abundance, but a few interesting recent investigations with various hypotheses may be mentioned (see Refs. Abd-alla et al. [1], Abo-Dahab et al. [2], Abd-alla et al. [3], Abd-Alla et al. [4], Abbas et al. [5], Abbas et al. [6], Abo-Dahab et al. [7]). The classical theory of thermoelasticity has been formulated by Boit [8] to overcome the paradox of the thermal disturbances propagated with infinite speeds. Furthermore, to remove this paradox generalized thermoelasticity models has been expanded. For examples, Lord and Shulman [9] founded the generalized thermoelasticity model that includes one relaxation time, which is producing the equation of heat conduction as hyperbolic in nature expects finite wave propagation.

Then, the temperature-rate-dependent model of generalized thermoelasticity has been given by Green and Lindsay [10] which has two thermal relaxation times. These two models are different and independent from each other, and neither can be derived as a special status of the other. Many contributions describing these models have been examined and exposed some motivating phenomenon. Shortened reviews of this point have been given in Chandrasekharaiah [11] after that, Green and Naghdi [12] and [13] suggested a new hypothesis in a generalized thermoelasticity model which foresees limited heat propagation speed and does not involve energy dissipation. This form was later known by the generalized model of thermoelasticity without the dissipation of energy. The basic idea of this theory is based on classical Fourier's law that is exchanged by a thermodynamic relation of heat flux rate of temperature.

The magneto-thermoelastic phenomenon is involved by the interactive influences of use magnetic field on thermoelastic of deformable solids. This theory has a lot of interest of many researchers because it can be applied in many technological and industrial fields as well as in the study of smart materials. Moreover, investigation of such problems affects different applications in biomedical engineering, geophysical applications, and particular subjects in optics, acoustics and geomagnetic researches. The evolution of electromagnetic field interaction and thermoelastic phenomenon on a particular elastic material is given by a lot researchers Hetnarski and Ignazack [14]; Abo-Dahab et al. [15], Abd-alla [16], Abo-Dahab et al. [17], Abd-alla and Maugin [18], [19], [20], [21], Allam et al. [22], Abd-Alla et al. [23], Bayones et al. [24] and Abbas et al. [25] and [26].

The normal mode mechanism is utilized to obtain a set of ordinary differential equations as an alternative to a set of partial differential equations in a lot of problems as, Alshaikh [27], [28], [29], Othman and Eraki [30], Said [31], and Bayones et al. [32]. While Saeed and Abbas [33] applied the finite element technique for nonlinear thermoelastic model to investigate bioheat structure of tissues. Moreover, the cooperation between the motions of deformable structures with the existence of electromagnetic field has been studied by several researchers due to the potential of its implementation in many of the applications (see Refs [34,35]). Several investigators have studied the problems of propagation of electro-thermoelastic or thermo-magnetoelastic phenomena in smart materials [36], [37], [38], [39], [40], [41]. Furthermore, there is also a broad review of previous applications of these topics in [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56]. The gravity and inclined load effect in micropolar thermoelastic medium under G-N theory possessing cubic symmetry is investigated by Othman et al. [57]. Mohammad and Kumar [58] discussed the phenomenon of waves propagation over a rugged topography. Higazy et al. [59] investigated Computational solutions of wave considering some problems in nonlinear evolution equations. Varsoliwala and Singh [60] pointed out modeling of tsunami wave propagation at mid ocean and run-up on shore and its amplification. Authors [61], [62], [63] used the analytical solution to investigate from variables in this phenomenon.

This paper examines the effect of gravity and the primary magnetic field on an isotropic homogeneous solid in the framework of four models of generalized thermoelasticity. Using the normal mode approach, the components of temperature, displacement, stress are obtained. Then, the numerical calculations of these expressions have been performed and graphically represented to illustrate the impact of the new variables on the phenomena. The effect of the magnetic field and gravity are illustrated graphically which are pronounced on the different physical quantities. A comparison has been made between the present results obtained and the previous obtained by others when the new parameters neglected. Finally, the results of some research that others have previously obtained may be found some or all of them as special cases from this study. The results indicate to the applicable results on the waves propagation, especially that related to engineering and ocean.

2. Mathematical modeling and fundamental equations

The fundamental equations such as the equation of motion, Maxwell's equations and generalized heat conduction equation based on four theories of generalized magneto-thermoelasticity without heat sources and with the effect of the gravity are: Othman, and Eraki [30]

(1)

μ u_{i, j j} + (λ + μ) u_{j, i j} - γ (1 + θ_{0} \frac{\partial}{\partial t}) T_{, i} + f_{i} + G_{i} = ρ {\ddot{u}}_{i},

(2)

\vec{\nabla} \land \vec{H} = 0, \vec{\nabla} \land \vec{E} = - \vec{\dot{B}}, \vec{\nabla} \cdot \vec{B} = 0, \vec{B} = μ_{e} \vec{H},

(3)

\begin{matrix} k T_{, i i} + k^{*} {\dot{T}}_{, i i} & = & ρ C_{e} (n_{1} \frac{\partial}{\partial t} + τ_{0} \frac{\partial^{2}}{\partial t^{2}}) T \\ + γ T_{0} (n_{1} \frac{\partial}{\partial t} + n_{0} τ_{0} \frac{\partial^{2}}{\partial t^{2}}) (\vec{\nabla} \cdot \vec{u}), \end{matrix}

With

f_{i}

is Lorentz force that is written as: Salingaros [64]

(4)

f_{i} = {(\vec{J} \land \vec{B})}_{i}

The constitutive relations for an isotropic and homogeneous elastic structure may be written in sensorial form as:

(5)

e_{l k} = \frac{1}{2} (u_{l, k} + u_{k, l}), l, k = 1, 2, 3

(6)

σ_{l k} = [λ μ_{j, j} - (1 + θ_{0} \frac{\partial}{\partial t}) T] δ_{l k} + 2 μ

(7)

J_{i} = σ_{0} [E_{i} + ε_{i j k} {\dot{u}}_{i} B_{j} e_{k}]

The characteristics between four of generalized thermoelasticity theories (GTET) that will be study are:

(i) (CT) theory:

n_{0} = 0, n_{1} = 1, τ_{0} = 0, θ_{0} = 0,

(ii) (LS) theory:

n_{0} = 1, n_{1} = 1, τ_{0} = 0.2, θ_{0} = 0,

(iii) (GL) theory:

n_{0} = 0, n_{1} = 1, τ_{0} = 0.2, θ_{0} = 0.3,

(iv) (GN) theory:

n_{0} = 0, n_{1} = 1, τ_{0} = 1, θ_{0} = 0,

We assume the system of rectangular coordinates

(x, y, z)

that arose on y=0 surface. Also, it is supposed that the z -axis is vertically oriented to the center and constitutes a half space

(x \geq 0) .

Therefore, the displacement components ( u,0,w) and other physical variables become independent variables that relying on the variables t, x and y. Moreover, we presume that the small influence of temperature gradient on the conduction current

\vec{J}

is ignored. We assume that

\vec{H} = \vec{h} + {\vec{H}}_{0}

with

{\vec{H}}_{0} = (0, 0, H_{0})

acts parallel to z-axis. Due to the smallness of magnetic field

\vec{h}

, so that the product of

\vec{h}, \vec{u}

and their derivatives may be ignored during performing the process of linearization of the field equations. Therefore, Eqs. (1) And (3) may be expressed as:

(8)

\begin{matrix} μ \nabla^{2} u + (λ + μ) \frac{\partial e}{\partial x} - γ (1 + θ_{0} \frac{\partial}{\partial t}) \frac{\partial T}{\partial x} \\ + ρ g \frac{\partial w}{\partial x} - μ_{0} H_{0} \frac{\partial h}{\partial x} = ρ \frac{\partial^{2} u}{\partial t^{2}}, \end{matrix}

(9)

\begin{matrix} μ \nabla^{2} w + (λ + μ) \frac{\partial e}{\partial z} - γ (1 + θ_{0} \frac{\partial}{\partial t}) \frac{\partial T}{\partial z} \\ + ρ g \frac{\partial u}{\partial x} - μ_{0} H_{0} \frac{\partial h}{\partial z} = ρ \frac{\partial^{2} w}{\partial t^{2}}, \end{matrix}

(10)

\begin{matrix} k \nabla^{2} T + k^{*} \frac{\partial}{\partial t} \nabla^{2} T & = & ρ C_{e} (n_{1} \frac{\partial}{\partial t} + τ_{0} \frac{\partial^{2}}{\partial t^{2}}) T \\ + γ T_{0} (n_{1} \frac{\partial}{\partial t} + n_{0} τ_{0} \frac{\partial^{2}}{\partial t^{2}}) (\nabla . u) \end{matrix}

Applying a non-dimensional format as:

(11)

{x^{'}, z^{'}} = \frac{ω *}{c_{0}} {x, z}, θ_{o}^{'} = ω * θ_{0}, t^{'} = ω * t, τ_{0}^{'} = ω * τ,

(12)

\begin{matrix} {u^{'}, w^{'}} & = & \frac{ρ c_{0} ω^{*}}{υ T_{0}} {u, w}, T^{'} = \frac{T}{T_{0}}, δ_{i j}^{'} = \frac{δ_{i j}}{υ T_{0}}, \\ g^{'} & = & \frac{g}{c_{0} w^{*}} h^{'} = \frac{h}{H_{0}}, \end{matrix}

where

ω^{*} = \frac{ρ C_{E} {c_{0}}^{2}}{K}, ρ {c_{0}}^{2} = λ + 2 μ

The non-dimensional formats of Eqs. (8)-(10) (after omitting primes for suitability) become:

(13)

\nabla^{2} u + b_{1} \frac{\partial e}{\partial x} - b_{2} (1 + θ_{0} \frac{\partial}{\partial t}) \frac{\partial T}{\partial x} + b_{3} \frac{\partial w}{\partial x} - R_{h} \frac{\partial h}{\partial x} = b_{2} \frac{\partial^{2} u}{\partial t^{2}},

(14)

\nabla^{2} w + b_{1} \frac{\partial e}{\partial z} - b_{2} (1 + θ_{0} \frac{\partial}{\partial t}) \frac{\partial T}{\partial z} + b_{3} \frac{\partial u}{\partial x} - R_{h} \frac{\partial h}{\partial z} = b_{2} \frac{\partial^{2} w}{\partial t^{2}},

(15)

\begin{matrix} ε_{3} \nabla^{2} T + ε_{2} \frac{\partial}{\partial t} \nabla^{2} T & = & ε_{4} (n_{1} \frac{\partial}{\partial t} + τ_{0} ω^{*} \frac{\partial^{2}}{\partial t^{2}}) T \\ + ε_{1} (n_{1} \frac{\partial}{\partial t} + n_{0} τ_{0} \frac{\partial^{2}}{\partial t^{2}}) e \end{matrix}

where

\begin{matrix} ε_{1} & = & \frac{γ^{2} T_{0}}{w^{*} c_{0}^{2} ρ C_{e}}, ε_{2} = \frac{k^{*} w^{*}}{ρ c_{0}^{2} C_{e}}, ε_{3} = \frac{k}{ρ c_{0}^{2} C_{e}}, \\ ε_{4} & = & \frac{1}{ω^{*}}, b_{1} = \frac{λ + μ}{μ}, b_{2} = \frac{ρ c_{0}^{2}}{μ}, b_{3} = \frac{ρ g c_{0}^{2}}{μ}, R_{h} = \frac{μ_{0} H_{0}^{2}}{μ} . \end{matrix}

Here

ε_{k}, k = 1, 2, 3

is known as coupling parameter.

For convenience the components of displacement may be rewritten by the functions

ψ_{1} (x, z, t)

and

ψ_{2} (x, z, t)

in non-dimensional formats as:

(16)

u = \frac{\partial ψ_{1}}{\partial x} - \frac{\partial ψ_{2}}{\partial z}, w = \frac{\partial ψ_{1}}{\partial z} + \frac{\partial ψ_{2}}{\partial x},

Gives

(17)

\nabla^{2} ψ_{1} = e and \nabla^{2} ψ_{2} = (\frac{\partial u}{\partial z} - \frac{\partial w}{\partial x}) .

Using Eqs. (16) and (17) into Eqs. (13)-) 15) yields

(18)

[(1 + b_{1} - R_{h}) \nabla^{2} - b_{2} \frac{\partial^{2}}{\partial t^{2}}] ψ_{1} + b_{3} \frac{\partial}{\partial x} ψ_{2} - b_{2} (1 + θ_{0} \frac{\partial}{\partial t}) T = 0,

(19)

- b_{3} \frac{\partial}{\partial x} ψ_{1} + [\nabla^{2} - b_{2} \frac{\partial^{2}}{\partial t^{2}}] ψ_{2} = 0,

(20)

\begin{matrix} - ε_{1} (\frac{n_{1}}{ω^{*}} \frac{\partial}{\partial t} + n_{0} τ_{0} \frac{\partial^{2}}{\partial t^{2}}) \nabla^{2} ψ_{1} + (ε_{3} + ε_{2} \frac{\partial}{\partial t}) \nabla^{2} T \\ - ε_{4} (n_{1} \frac{\partial}{\partial t} + τ_{0} ω^{*} \frac{\partial^{2}}{\partial t^{2}}) T = 0 \end{matrix}

The components of the stress as functions of the displacement and temperature may be expressed as:

(21)

σ_{x x} = \frac{\partial u}{\partial x} + L \frac{\partial w}{\partial z} - (1 + θ_{0} \frac{\partial}{\partial t}) T,

(22)

σ_{y y} = L e - (1 + θ_{0} \frac{\partial}{\partial t}) T,

(23)

σ_{z z} = \frac{\partial w}{\partial z} + L \frac{\partial u}{\partial x} - (1 + θ_{0} \frac{\partial}{\partial t}) T,

(24)

σ_{x z} = \frac{1}{b_{2}} (\frac{\partial u}{\partial z} + \frac{\partial w}{\partial x}), and σ_{x y} = σ_{y z} = 0 .

(25)

τ_{z z} = G (\frac{\partial u}{\partial x} + \frac{\partial w}{\partial z})

Where we have used:

L = \frac{λ}{λ + 2 μ}, G = \frac{μ_{e} H_{0}^{2}}{λ + 2 μ} .

3. Normal mode technique

The normal mode mechanism is applied to solve the governing Eqs.(18)-(20) by transferring them from partial differential equations to ordinary differential equations. This process assists us to get analytical, accurate solutions of fundamental physical fields (Othman et al. [38])

Therefore, the normal mode mechanism may be written as:

(26)

[ψ_{1}, ψ_{2}, T] (x, z, t) = [ψ_{1}^{*}, ψ_{2}^{*}, T^{*}] (x) \exp [i (ω t + a z)],

With

[ψ_{1}^{*}, ψ_{2}^{*}, T^{*}] (x)

represent the amplitudes,

a and ω

are the wave number and the angular frequency, respectively. Applying equation (26), into equations (18)-(20) tend to:

(27)

[D^{2} - B_{1}] ψ_{1}^{*} + B_{2} D ψ_{2}^{*} - B_{3} (1 + i ω θ_{0}) T^{*} = 0,

(28)

B_{3} D ψ_{1}^{*} - [D^{2} - B_{2}] ψ_{2}^{*} = 0,

(29)

B_{5} [D^{2} - a^{2}] ψ_{1}^{*} + [D^{2} - a^{2}] T^{*} - B_{6} T^{*} = 0

where

\begin{matrix} B_{1} & = & a^{2} - \frac{b_{2} ω^{2}}{1 + b_{1} - R_{H}}, B_{2} = \frac{b_{3}}{1 + b_{1} - R_{H}}, \\ B_{3} & = & \frac{b_{2}}{1 + b_{1} - R_{H}}, B_{4} = a^{2} - b_{2} ω^{2}, \end{matrix}

\begin{matrix} B_{5} & = & \frac{ε_{1} ω (n_{1} i - n_{0} τ_{0} ω^{*} ω)}{(ε_{3} + ε_{2} i ω)} B_{6} = \frac{- ε_{4} ω (- n_{1} i + τ_{0} ω^{*} ω)}{(ε_{3} + ε_{2} i ω)}, and \\ D & = & \frac{d}{d x} . \end{matrix}

Eliminating

ψ_{1}^{*}, ψ_{2}^{*}, and T^{*}

from Eqs. (27)-(29), yields:

(30)

[D^{6} - B_{7} D^{4} + B_{8} D^{2} - B_{9}] (ψ_{1}^{*}, ψ_{2}^{*}, T^{*}) = 0 .

Where

B_{7} = B_{1} + B_{4} + B_{6} + B_{2} b_{3} - B_{3} B_{5} - B_{3} B_{5} ϑ_{0} i + ω + a^{2},

\begin{matrix} B_{8} & = & a^{2} B_{1} + a^{2} B_{4} + a^{2} B_{2} b_{3} - a^{2} B_{3} B_{5} - B_{3} B_{5} B_{4} + B_{1} B_{4} \\ + B_{6} B_{4} + B_{1} B_{6} + b_{3} B_{2} B_{6} + B_{3} B_{5} a^{2} i ω ϑ_{0} + B_{3} B_{4} B_{5} i ω ϑ_{0}, \end{matrix}

B_{9} = a^{2} B_{1} B_{4} - a^{2} B_{3} B_{5} B_{4} + B_{1} B_{4} B_{6} - B_{3} B_{4} B_{5} a^{2} i ω ϑ_{0},

Factoring the non-homogeneous Eq. (30) gives:

(31)

(D^{2} - k_{1}^{2}) (D^{2} - k_{2}^{2}) (D^{2} - k_{3}^{2}) (ψ_{1}^{*}, ψ_{2}^{*}, T^{*}) = 0 .

With

k_{j}^{2} (j = 1, 2, 3)

define the roots of Eq. (30). Therefore, utilizing the bounded condition,

z \to \infty,

hence Eq. (26), has the following solutions:

(32)

Ψ_{1} (x, z, t) = \sum_{j = 1}^{3} R_{j} \exp (- k_{j} x + i ω t + i a z) .

(33)

Ψ_{2} (x, z, t) = \sum_{j = 1}^{3} H_{1 j} R_{j} \exp (- k_{j} x + i ω t + i a z) .

(34)

T (x, z, t) = \sum_{j = 1}^{3} H_{2 j} R_{j} \exp (- k_{j} x + i ω t + i a z) .

Here

(35)

H_{1 j} = \frac{- b_{3} k_{j}}{(k_{j}^{2} - B_{4})}, H_{2 j} = \frac{(k_{j}^{2} - B_{1}) - B_{2} H_{1 j} k_{j}}{B_{3}}, j = 1, 2, 3 .

where,

R_{j} (j = 1, 2, 3)

are undefined parameters.

The displacement components may be obtained by using Eqs. (32) and (33) into Eq. (16) as:

(36)

u (x, z, t) = \sum_{j = 1}^{3} M_{1 j} R_{j} \exp (- k_{j} x + i ω t + i a z) .

(37)

w (x, z, t) = \sum_{j = 1}^{3} M_{2 j} R_{j} \exp (- k_{j} x + i ω t + i a z) .

where

M_{1 j} = - k_{j} - i a H_{1 j}, M_{2 j} = i a - k_{j} H_{1 j}, j = 1, 2, 3 .

The stress components may be given by applying Eqs. (32)-(34) into Eqs. (21)-(25) as:

(38)

σ_{x x} (x, z, t) = \sum_{j = 1}^{3} H_{3 j} R_{j} \exp (- k_{j} x + i ω t + i a z),

(39)

σ_{y y} (x, z, t) = \sum_{j = 1}^{3} H_{4 j} R_{j} \exp (- k_{j} x + i ω t + i a z),

(40)

σ_{z z} (x, z, t) = \sum_{j = 1}^{3} H_{5 j} R_{j} \exp (- k_{j} x + i ω t + i a z),

(41)

σ_{x z} (x, z, t) = \sum_{j = 1}^{3} H_{6 j} R_{j} \exp (- k_{j} x + i ω t + i a z),

(42)

τ_{z z} = \sum_{j = 1}^{3} H_{7 j} \exp (- k_{j} x + i ω t + i a z),

Here

\begin{matrix} H_{3 j} & = & - M_{1 j} k_{j} + L M_{2 j} i a - H_{2 j} - i ω θ_{0} H_{2 j}, \\ H_{4 j} & = & - k_{j} M_{1 j} L + i a M_{2 j} L - H_{2 j} - i ω θ_{0} H_{2 j}, \\ H_{5 j} & = & i a M_{2 j} - L M_{1 j} k_{j} - H_{2 j} - i ω θ_{0} H_{2 j}, \\ H_{6 j} & = & \frac{1}{b_{2}} (i a M_{1 j} - M_{2 j} k_{j}), H_{7 j} = G (i a M_{2 j} - M_{1 j} k_{j}) . \end{matrix}

4. Boundary conditions

When choosing boundary conditions, they must achieve an important characteristic, which is that solutions must be limited at infinity, and they should, also help in setting the values of the optional constants that arise in the problem such as

R_{j} (j = 1, 2, 3) .

So, at

x = 0

one may apply the following boundary conditions:

(43)

ζ_{z z}^{*} = σ_{z z} + τ_{z z} = - p_{1} \exp (ω t + i a z), σ_{x z} = 0, \frac{\partial T}{\partial x} = 0 .

where,

p_{1}

defines the mechanical power and

ζ_{z z}^{*}

is known as Maxwell stress component.

Applying Eqs. (43) into the Eqs. (34), (40), (41) and (42), one may get the following equations:

(44)

\sum_{j = 1}^{3} (H_{5 j} + H_{7 j}) R_{j} = - p_{1},

(45)

\sum_{j = 1}^{3} H_{6 j} R_{j} = 0,

(46)

\sum_{j = 1}^{3} - k_{j} H_{2 j} R_{j} = 0 .

The Eqs. (43)-(45) consists of a set of 3 non-homogeneous equations that may be expressed as:

(47)

(\begin{matrix} R_{1} \\ R_{2} \\ R_{3} \end{matrix}) = {(\begin{matrix} (H_{51} + H_{71}) & (H_{52} + H_{72}) & (H_{53} + H_{73)} \\ H_{61} & H_{62} & H_{63} \\ - k_{1} H_{21} & - k_{2} H_{22} & - k_{3} H_{23} \end{matrix})}^{- 1} (\begin{matrix} - p_{1} \\ 0 \\ 0 \end{matrix})

Therefore, we have obtained full solutions for all required physical parameter fields.

5. Analysis of the numerical simulations

The magnesium material is selected for numerical calculations and its thermal and mechanical data have been given in reference (Abo-Dahab et al. [17]), where all the variables utilized in the computations has been considered in SI units, as:

\begin{matrix} ρ = 1.74 \times 10^{3} k g / m^{3} & λ = 9.4 \times 10^{10} N / m^{2} & μ = 4 \times 10^{10} N / m^{2}, \\ K = 1.7 \times 10^{2} W / m K, & C_{e} = 1.04 \times 10^{3} j / k g K, & ω^{*} = 3.58 \times 10^{11} / s, \\ μ_{0} = 4 \times π \times 10^{- 3}, & T_{0} = 298 \end{matrix}

In addition to the comparisons, we applied the following:

\begin{matrix} p_{1} = 0.25 N / m^{2}, k^{*} = 100 W / m K, a = 0.5, ω_{0} = 5, ξ = 0.6, \\ t = 0.9 s, ω = ω_{0} + i ξ, 0 \leq x \leq 5 m \end{matrix}

The numerical calculations of the real part values of the field of physical quantities in two dimensions as a function of distance

x

have been divided into three categories:

(i) The first group is to study the effect of alteration the four generalized thermoelastic theories (GTET) on physical quantities when both the gravitational field (

g

) and the initial magnetic field

(H_{0})

are constants. In more details, in this case, it considers the following:

H_{0} = 9 \times 10^{5}, g = 9.8,

the other values for variables of different (GTET) given as, for (CT) theory

n_{0} = 0, n_{1} = 0, τ_{0} = 1, and θ_{0} = 0 .

While, for (LS) theory,

n_{0} = 1, n_{1} = 1, τ_{0} = 0.2, and θ_{0} = 0 .

But, for (GL) theory,

n_{0} = 0, n_{1} = 1, τ_{0} = 0.2, and θ_{0} = 0.3 .

Finally, for (GN) theory,

n_{0} = 0, n_{1} = 1, τ_{0} = 1, and θ_{0} = 0 .

This group is shown in Figures (1)-(7).

(ii) The second group is to examine the impact of variation the gravitational field to be as

g = 0, 5.7, 9.8

on physical quantities when both the initial magnetic field to be

H_{0} = 9 \times 10^{5},

and the parameters of (GTET) are chosen to be fixed as:

n_{0} = 0, n_{1} = 0, τ_{0} = 0.1, and θ_{0} = 0 .

This group is displayed in Figures (8)-(14).

(iii) The third group is to illustrate the influence of change the initial magnetic field

(H_{0})

to become as

H_{0} = (0, 2, 3, 5) \times 10^{5}

on physical quantities when both the gravitational field

g = 9.8

and the variables of (GTET) are constants such as:

n_{0} = 0, n_{1} = 0, τ_{0} = 0.1, and θ_{0} = 0 .

This group is displayed in Figures (15)-(21).

In more details, one may remark the following:

Fig. 1 considers the changes on the displacement

u

versus distance

x .

It is noted that in this case when the thermal relaxation times change, the contrast of the G-L model (green-colored curve) is completely different from the rest of the other three models. While the remaining three curves of the models (CT, L-S and G-N) exhibit similar oscillatory behavior with small variations.

View original graphic|Download|PPT slide

Fig.1 Displacement distribution $u$ versus $x$ with four theories.

Fig. 2 examines the variation on the displacement component

w

as function of

x .

It is observed that the behavior of the curves in this case is very similar. For example, the curves begin with a maximum value when

x = 0

, then as the distance grows, these values decay to reach the value almost zero in the interval

x = (3.3 - 5) .

But the effect of relaxation times is the greatest possible for the G-L model, then the C-T model, after that the L-S model and finally the G-N model.

View original graphic|Download|PPT slide

Fig.2 Displacement distribution $w$ versus $x$ with four theories.

Fig. 3 illustrates the changes on the normal component of stress

σ_{x x}

against off

x .

In this case, all the curves begin with negative values for all the models, due to the variation of the thermal relaxation times, and they are gradually enlarged linearly with similar behavior for all the curves. So, they are reaching almost a maximum value at

x = 1.5

. After that, the curves begin to decrease so that they all coincide at

x = 5

View original graphic|Download|PPT slide

Fig.3 Stress distribution $σ_{x x}$ versus $x$ with four theories.

Figs. 4 and 6 display the modifications on the components of stresses

σ_{z z}

and

ζ_{z z}^{*}

versus

x .

The behavior of the curves in these two cases is very close. So, they start for all models from one point at

x = 0

, the stresses values become (-0.14). After that, they start to grow non-linearly until it reaches

x = 3

, then the changes become almost constant and linear for all curves until they all coincide when

x = 5

View original graphic|Download|PPT slide

Fig.4 Stress distribution $σ_{z z}$ versus $x$ with four theories.

Fig. 5 discusses the alterations on the component of stress

σ_{x z}

with respect to

x .

It may be seen that all curves start at

x = 0

with a value of

σ_{x z} = 0

.Then, these values gradually decrease linearly to reach a minimum value in the period

x = (0.5 - 1) .

After that, its value increases non-linearly until they become almost constant values in the period

x = (3.5 - 5) .

View original graphic|Download|PPT slide

Fig.5 Stress distribution $σ_{x z}$ versus $x$ with four theories.

Fig. 7 presents the changes in

T

versus

x

It is remarked that when

x = 0

, the values of

T

varies according to different models, but almost all curves have similar behavior with a small difference in the values. It is remarked that the curves of the variable

T

leads to zero when the distance is maximized.

From the second group of Figures (8-14). It is easy to see that Figs. 8 and 10 examine the variations on

w

as and

σ_{x x}

function of

x

at different values of

g

. It is observed that the curves start with different values at

x = 0

, then as the distance enlarges, these values change to reach the value almost zero at

x = 1.75

for u curves but at

x = 2.75

for

σ_{x x}

curves, After that, the curves for both of

u

and

σ_{x x}

coincide until they reach the maximum value of

x

3 m

. It may easily be noted that

u

inversely proportional to the gravitational field

g

, while

σ_{x x}

is directly proportional to it.

Figs. 9 and 14 give the variations on the variables

w

and

T

against of

x

. It is found that all the curves in the two variables start with different values at

x = 0

. Then they reduce non-linearly with increasing

x

to reach to the following values at

x = 0.4, 0.7 and 0.4

, respectively. After that, the values of the

w

curves diminish to coincide at

x = 3

. While curves of the

T

increase almost linearly in small amounts to become stable until reach

x = 3

Figs. 11, 12 and 13 demonstrate the variety of

σ_{x x}, σ_{x z}, and ζ_{z z}^{*}

versus

x

at various values of

g

. It is easy to notice that the behavior of the curves in these three cases has the same behavior as the curves in Figs 4, 5 and 7 from in the first group.

Finally, from the third group of Figures (15-21). It is remarked that Fig. 15 shows the changes on

u

as a function of

x

at several values of

H_{o}

. The attitude of the curves in this case is similar. So, they start for all curves from one point at

x = 0

, then, they start to grow non-linearly until it reaches to its maximum at the interval

x = (2.5 - 4.2) .

After that, They reduce nonlinearly until becoming minimum almost in the period

x = (1 - 1.35) .

Next, they increase linearly again until they coincide at about

x = 3

. It may also be seen that the effect of

H_{o}

u

is an inverse proportion.

Fig. 16 considers the alteration of

w

as a function of

x

. It is noticed that the demeanor of the curves in this case is analogous. Therefore, the curves commencement with a maximum value when

x = 0

, then as the distance expands, these values reduce to reach the value almost zero at

x = 3

. Also, it is found that the more value of

H_{o}

leads to increase

w .

Fig. 17 display the change of the normal stresses

σ_{x x}

as a function of

x

. By increasing the value of

H_{o}

, the values of curves

σ_{x x}

goes up nonlinearly in the period

0 \leq x \leq 1.5

, while after this period the changes become linear until they reach at

x = 3 .

Also, it is noted that the more value of

H_{o}

leads to increase

σ_{x x}

Figs. 18 and 20 display the variety of

σ_{x x}

and

ζ_{z z}^{*}

against

x

at various values of

H_{o}

. It is clear that the demeanor of the curves in these two cases has the same character as the curves in Figs 4 and 6 from in the first group and Fig. 12, Fig. 13 in the second group.

View original graphic|Download|PPT slide

Fig.18 Stress distribution $σ_{z z}$ versus $x$ with magnetic field.

Fig. 19 illustrates the alterations on the component of stress

σ_{x z}

with respect to

x

at many values of

H_{o} .

It may be seen that all curves start at

x = 0

with a value of

σ_{x z} .

Then, these values gradually decrease linearly to reach a minimum value about

x = 0.5

. Then, its value increases non-linearly until they become almost constant values in the period

x = (2 - 3) .

Also, increasing the value of

H_{o}

clearly reduces

σ_{x z} .

View original graphic|Download|PPT slide

Fig.19 Stress distribution $σ_{x z}$ versus $x$ with magnetic field.

View original graphic|Download|PPT slide

**Fig.20 Stress distribution $ζ_{z z}^{*}$ versus $x$ with magnetic field.**

Fig. 21 considers the change on the curves of

T

versus

x

at different values of

H_{o}

. By increasing the value of

H_{o}

, the values of curves

T

goes down nonlinearly until they reach to its minimum in the interval

x = (0.4 - 0.6)

, After that the changes become almost linear until they reach at

x = 3 .

Also, it is noticed that the more value of

H_{o}

leads to decrease

T

View original graphic|Download|PPT slide

Fig.21 Temperature distribution $T$ versus $x$ with magnetic field.

Generally, from Figures (1-7), it is noticed through the graphs and results which come from numerical calculations that the effect of temperature, displacement compounds and stress components when the thermal relaxation time is implicitly located in the heat equation is clearly dissimilar from that case when the thermal relaxation times are not present in the thermal equation, while they exist in the constitutive equations of the system being studied.

Finally, the results obtained in this work are more general, so one may obtain from them some results of others' research as special cases from them. For example, Othman and Lotfy [5], Othman et al. [36], Abo-Dahab [41], Lotfy [37] studied a problem of generalized magneto-thermo-elasticity without considering more theories of GTET and with different assumptions.

6. Conclusions

One may conclude the following based on the previous analysis and outcomes:

i) Using the normal mode analysis, it is possible to convert a group of partial differential equations into a structure of ordinary differential equations which are easy to solve analytically and then one may get the numerical simulations that represent graphically for all the physical variables of the problem.

ii) It is found that the primary magnetic field, gravity, and thermal relaxation times have a significant impact on physical variables according to the four theories of GTET. Moreover, all variables that have been studied are continuous functions in addition to achieving a property that leads to zero when the distance is maximized.

iii) The four theories of generalized thermoelasticity subjected to primary magnetic field characterize the attitude of the particles of an elastic structure is more reasonable than applying one or more of thermal relaxation times of the theories of generalized magnetothermo-elasticity.

iv) The results of this research can be used in many physical, engineering and industrial applications, for example analysis and design of heat-resistant materials. In addition, there are many uses such as semiconductors and reactions during the process of photovoltaic cells and electronic devices, transistors, as well as medical instruments. Furthermore, This problem may be extended as a future work to study and solve the phenomena of reflection and refraction of wave propagation at free surfaces in anisotropic piezoelectric or in thermo-piezoelectric smart materials.

Declaration of Competing Interest

None.

References

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Abstract

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模态框（Modal）标题

Abstract

Cite this article

Nomenclature

1. Introduction

2. Mathematical modeling and fundamental equations

3. Normal mode technique

4. Boundary conditions

5. Analysis of the numerical simulations

Fig.1 Displacement distribution u versus x with four theories.

Fig.2 Displacement distribution w versus x with four theories.

Fig.3 Stress distribution σxx versus x with four theories.

Fig.4 Stress distribution σzz versus x with four theories.

Fig.5 Stress distribution σxz versus x with four theories.

Fig.6 Stress distributionζzz* versus x with four theories.

Fig.7 Temperature distribution T versus x with four theories.

Fig.8 Displacement distribution u versus x with gravity.

Fig.9 Displacement distribution w versus x with gravity.

Fig.10 Stress distribution σxx versus x with gravity.

Fig.11 Stress distribution σzz versus x with gravity.

Fig.12 Stress distribution σxz versus x with gravity.

Fig.13 Stress distribution ζzz* versus x with gravity.

Fig.14 Temperature distribution T versus x with gravity.

Fig.15 Displacement distribution u versus x with magnetic field.

Fig.16 Displacement distribution w versus x with magnetic field.

Fig.17 Stress distribution σxx versus x with magnetic field.

Fig.18 Stress distribution σzz versus x with magnetic field.

Fig.19 Stress distribution σxz versus x with magnetic field.

Fig.20 Stress distribution ζzz* versus x with magnetic field.

Fig.21 Temperature distribution T versus x with magnetic field.

6. Conclusions

References

Links

Fig.1 Displacement distribution $u$ versus $x$ with four theories.

Fig.2 Displacement distribution $w$ versus $x$ with four theories.

Fig.3 Stress distribution $σ_{x x}$ versus $x$ with four theories.

Fig.4 Stress distribution $σ_{z z}$ versus $x$ with four theories.

Fig.5 Stress distribution $σ_{x z}$ versus $x$ with four theories.

**Fig.6 Stress distribution $ζ_{z z}^{*}$ versus $x$ with four theories.**

Fig.7 Temperature distribution $T$ versus $x$ with four theories.

Fig.8 Displacement distribution $u$ versus $x$ with gravity.

Fig.9 Displacement distribution $w$ versus $x$ with gravity.

Fig.10 Stress distribution $σ_{x x}$ versus $x$ with gravity.

Fig.11 Stress distribution $σ_{z z}$ versus $x$ with gravity.

Fig.12 Stress distribution $σ_{x z}$ versus $x$ with gravity.

**Fig.13 Stress distribution $ζ_{z z}^{*}$ versus $x$ with gravity.**

Fig.14 Temperature distribution $T$ versus $x$ with gravity.

Fig.15 Displacement distribution $u$ versus $x$ with magnetic field.

Fig.16 Displacement distribution $w$ versus $x$ with magnetic field.

Fig.17 Stress distribution $σ_{x x}$ versus $x$ with magnetic field.

Fig.18 Stress distribution $σ_{z z}$ versus $x$ with magnetic field.

Fig.19 Stress distribution $σ_{x z}$ versus $x$ with magnetic field.

**Fig.20 Stress distribution $ζ_{z z}^{*}$ versus $x$ with magnetic field.**

Fig.21 Temperature distribution $T$ versus $x$ with magnetic field.