# On the Effect of Thomson and Initial Stress in a Thermo-Porous Elastic Solid under G-N Electromagnetic Theory

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Problem

**u**, with components $\mathit{u}=(u,v,0)$. The functions that are considered in this context are dependent on the time variable t and of the spatial variables x and y.

**h**, and these satisfy the electromagnetism equations, in the linearized form. We will use the Maxwell’s equations [24] in order to characterize the evolution of the electric field and for variation of the magnetic field, as follows:

## 3. The Solution of the Problem

#### 3.1. Decomposition by Normal Mode Analysis

#### 3.2. Boundary Conditions

_{1}, R

_{2}, R

_{3}and R

_{4}.

#### 3.2.1. The mechanical boundary condition

#### 3.2.2. The Boundary Restriction of Heat

#### 3.2.3. Voids Conditions

#### 3.2.4. The Boundary Restriction for Electromagnetic Field

_{1}, R

_{2}, R

_{3}and R

_{4}, we will use the dimensionless size ${{\theta}^{\prime}}_{0}=\frac{{\theta}_{0}}{{T}_{0}}\hspace{0.17em}$ and the expressions of the variables into the boundary restrictions imposed above. Additionally, we will use the normal mode analysis in order to obtain the system of equations

## 4. Special Cases

#### 4.1. Pores Neglect

#### 4.2. Neglecting the Initial Stress

## 5. Numerical Results and Discussion

^{2}, $\mu =3.278\times {10}^{10}$ N/m

^{2},$k=1.7\times {10}^{2}$ W/m·deg, $\rho =1.74\times {10}^{3}$ Kg/m

^{3}, $\beta =2.68\hspace{0.17em}\times \hspace{0.17em}{10}^{6}$ N/m

^{2}·deg, ${C}_{e}=1.04\hspace{0.17em}\times \hspace{0.17em}{10}^{3}$ J/Kg·deg, ${\omega}_{1}^{*}\hspace{0.17em}=3.58\times {10}^{11}$ /s, ${\alpha}_{t}=1.78\times {10}^{-5}$ N/m

^{2}and T

_{0}= 298 K.

^{2}, $\alpha =3.688\times {10}^{-5}$ N, ${\xi}_{1}=1.475\times \hspace{0.17em}{10}^{10}$ N/m

^{2}, ${\lambda}_{0}=1.13849\times {10}^{10}$ N/m

^{2}, $m=2\times \hspace{0.17em}{10}^{6}$ N/m

^{2}·deg and ${\omega}_{0}=0.0787\times \hspace{0.17em}{10}^{-3}$ N/m

^{2}s.

^{2}/Cal·cm·sec.

- (i)
- whether we have an initial stress or not [L
^{*}=0 and 10^{5}at M_{0}=0.5 and H_{0}=10^{5}]; - (ii)
- whether we have a Thomson effect or not [M
_{0}=0 and 0.5 at H_{0}=10^{5}and L^{*}=10^{5}]; - (iii)
- whether we have some void parameters or not $[{M}_{0}=\hspace{0.17em}0.5,\hspace{0.17em}{H}_{0}={10}^{5}\hspace{0.17em}and\hspace{0.17em}{L}^{\ast}={10}^{5}].$

**,**and we have a comparison between the values of the strain in the case of the presence of the pores to those in the case of neglecting the voids, in the range $0\le y\le 1.7$; while, the values are the same for two cases at $y\ge 1.7$. Figure 10 illustrates the repartition of the temperature $\theta ,$ and a comparison between the temperature in the case of presence of pores to those in the case of neglecting the voids, for y in the range 0 < y < 1.9. In the case y > 1.9, the values are the same for two cases. Figure 11 depicts the repartition of the magnetic field h, an the values of the magnetic field h in the case of the presence of pores are compared to those in the case of neglecting the voids, for y in the range $0\le y\le 3$; in the case $y\ge 3$. The values are the same for the two cases.

## 6. Conclusions

- (1)
- We have derived the field equations of homogeneous, isotropic, electro-magneto-thermo- porous elastic half-plane with the Thomson effect and initial stress.
- (2)
- The analytical solutions that are based upon normal mode analysis for the thermoelastic problem in solids have been developed and utilized.
- (3)
- The presence of initial stress, void parameters, and Thomson effect play significant roles in all of the physical quantities.
- (4)
- The value of all physical quantities converges to zero with the increase in distance $y$ and all of the functions are continuous.
- (5)
- The deformation of a body depends on the nature of the applied forces and Thomson effect, as well as the type of boundary conditions.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**The change in the volume fraction field $\varphi $ distribution at ${M}_{0}=0.5$ and ${H}_{0}={10}^{5}$.

**Figure 8.**The change in the volume fraction field $\varphi $ distribution at ${H}_{0}={10}^{5}$ and ${L}^{*}={10}^{5}$.

**Figure 10.**The temperature $\theta $ distribution at ${H}_{0}={10}^{5}$, ${M}_{0}=0.5$, and ${L}^{*}={10}^{5}$.

**Figure 11.**The induced magnetic field $h$ distribution at ${H}_{0}={10}^{5}$, ${M}_{0}=0.5$, and ${L}^{*}={10}^{5}$.

**Figure 12.**Three-dimensional (3D) curve distribution of the strain $e$ versus the distances at: ${M}_{0}=0.5$, ${H}_{0}={10}^{5}$, and ${L}^{*}={10}^{5}$.

**Figure 13.**3D Curve distribution of the temperature $\theta $ versus the distances, at: ${M}_{0}=0.5$, ${H}_{0}={10}^{5}$ and ${L}^{*}={10}^{5}$.

**Figure 14.**3D Curve distribution of the induced magnetic field $h$ versus the distances, at: ${M}_{0}=0.5$, ${H}_{0}={10}^{5}$ and ${L}^{*}={10}^{5}$.

**Figure 15.**3D Curve distribution of the change in the volume fraction field $\varphi $ versus the distances, at: ${M}_{0}=0.5$, ${H}_{0}={10}^{5}$ and ${L}^{*}={10}^{5}$.

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**MDPI and ACS Style**

Abd-Elaziz, E.M.; Marin, M.; Othman, M.I.A.
On the Effect of Thomson and Initial Stress in a Thermo-Porous Elastic Solid under G-N Electromagnetic Theory. *Symmetry* **2019**, *11*, 413.
https://doi.org/10.3390/sym11030413

**AMA Style**

Abd-Elaziz EM, Marin M, Othman MIA.
On the Effect of Thomson and Initial Stress in a Thermo-Porous Elastic Solid under G-N Electromagnetic Theory. *Symmetry*. 2019; 11(3):413.
https://doi.org/10.3390/sym11030413

**Chicago/Turabian Style**

Abd-Elaziz, Elsayed M., Marin Marin, and Mohamed I. A. Othman.
2019. "On the Effect of Thomson and Initial Stress in a Thermo-Porous Elastic Solid under G-N Electromagnetic Theory" *Symmetry* 11, no. 3: 413.
https://doi.org/10.3390/sym11030413