# Study of Thermoelectric Responses of a Conductive Semi-Solid Surface to Variable Thermal Shock in the Context of the Moore–Gibson–Thompson Thermoelasticity

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## Abstract

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## 1. Introduction

## 2. Mathematical Model and Basic Equations

## 3. Problem Formulation

## 4. Solution Methodology

## 5. Normal Mode Solution

## 6. Boundary Conditions

## 7. The Calculated Values and Explanation

#### 7.1. Comparison of Thermoelastic Models

**Figure 2.**The temperature change $\theta $ versus $x$ for various thermoelastic models when $t=0.2$, $\mathsf{\Omega}=0.7$, and $y=0.1$.

**Figure 3.**The displacement component $u$ versus $x$ for various thermoelastic models when $t=0.2$, $\mathsf{\Omega}=0.7$, and $y=0.1$.

**Figure 4.**The stress component ${\sigma}_{xx}$ versus $x$ for various thermoelastic models when $t=0.2$, $\mathsf{\Omega}=0.7$, and $y=0.1$.

**Figure 5.**The stress component ${\sigma}_{xy}$ versus $x$ for various thermoelastic models when $t=0.2$, $\mathsf{\Omega}=0.7$, and $y=0.1$.

**Figure 6.**The induced magnetic field $h$ versus $x$ for various thermoelastic models when $t=0.2$, $\mathsf{\Omega}=0.7$, and $y=0.1$.

**Figure 7.**The induced electric component ${E}_{2}$ versus $x$ for various thermoelastic models when $t=0.2$, $\mathsf{\Omega}=0.7$, and $y=0.1$.

- It has been shown that the behavior of the field quantum distributions is quite sensitive to changing the values of the thermodynamic parameters, ${\tau}_{0}$ and ${K}^{*}$
- Compared to other extended theories (MGT, LS, GN-II, and GN-III), the coupled theory curves (CTE) are found to be higher than their counterparts are.
- Although Biot’s theory (CTE) was applied to many thermoelasticity problems, it produces unacceptable results in situations involving short-range temperatures, such as thermal shocks and laser material interactions.
- Extended models contradict the coupled theory, which suggests that heat waves may move at an infinite speed.
- The results converged for the two modified generalized theories LS and MGTE. According to the two overarching ideas, a thermoelastic reaction has a “cool-down” period. When the two generalized theories were modified, heat diffusion assumed the appearance of a wave phenomenon rather than the diffusion phenomenon usually associated with the second acoustic effect. The propagation velocity of the heat wave is found to be constrained after modifying the Fourier formula for thermal conductivity in the two theories.
- The results demonstrate that the behavior of the various distributions is more pronounced in the case of thermoelastic theory, GN-III, than it is in the case of thermoelastic type, GN-II. One possible explanation is that there is no energy loss in the second form.
- The numerical findings distinguish between the Green and Naghdi GN-III theories and the novel MGTE model, as well as other generalized thermal models. The GN-III model relies more on the classic theory of thermoelasticity and shows a greater temperature change than the MGTE model does. This result confirms the validity of the current model, which suggests that the GN-III model follows the behavior of the traditional theory that predicts infinite velocities for heat waves.
- Since the relaxation time, ${\tau}_{0},$ is factored into the heat equation, the solutions converge in both the MGTE and LS models. This means that the presence of relaxation time leads to a reduction in the propagation of thermal and mechanical waves in accordance with the experimental results.
- In reality, the strain and stress fields are affected by the fluctuating core body temperature, and the reverse is also true. Force loads, as well as temperature stresses, are common conditions for many structural parts. The material might crack under these forces or under the combined mechanical and thermal stresses resulting from external loads. The amount and distribution of thermal stresses must be determined for a comprehensive strength study of structures. Because of this, professionals from a wide range of areas focus their attention on problems related to calculating temperature fields and thermal stresses.

#### 7.2. Effect of the Angular Velocity on Variables of the Problem

**Figure 9.**Impact of rotational speed, $\mathsf{\Omega},$ on the displacement component, $u,$ when $t=0.2$, $\mathsf{\Omega}=0.7$, and ${k}_{0}=0.25\times {10}^{3}$.

**Figure 10.**Impact of rotational speed, $\mathsf{\Omega},$ on the stress component, ${\sigma}_{xx},$ when $t=0.2$, $\mathsf{\Omega}=0.7$, and ${k}_{0}=0.25\times {10}^{3}$.

**Figure 11.**Impact of rotational speed, $\mathsf{\Omega},$ on the stress component, ${\sigma}_{xy},$ when $t=0.2$, $\mathsf{\Omega}=0.7$, and ${k}_{0}=0.25\times {10}^{3}$.

**Figure 12.**Impact of rotational speed, $\mathsf{\Omega},$ on the induced magnetic field, $h$, when $t=0.2$, $\mathsf{\Omega}=0.7$, and ${k}_{0}=0.25\times {10}^{3}$.

**Figure 13.**Impact of rotational speed, $\mathsf{\Omega},$ on the induced electric component, ${E}_{2,}$ when $t=0.2$, $\mathsf{\Omega}=0.7$, and ${k}_{0}=0.25\times {10}^{3}$.

- It has been shown that rotation has a significant effect on the variance of all physical fields considered in the studied problem.
- The initial value of some system variables is the same and may be zero according to the boundary criteria.
- The insights from the current theoretical conclusions may be useful to experimental investigators, engineers, and seismologists who study the field of spinning bodies.
- Temperature, $\theta ,$ travels through the medium in a wave pattern at finite speeds. With increasing values of the angular velocity of rotation, $\mathsf{\Omega}$, the temperature distributions increase.
- There is a significant difference in deformation in the presence and absence of rotation. Both the deformation and the magnetic induction field increase significantly with the increase in the constant angular velocity of rotation, $\mathsf{\Omega}$.
- The change in the angular velocity of the body’s rotation has a significant impact on the thermal stresses inside the medium. The amount of compressive behavior of the stresses increases with the increase in the amount of rotation.
- The proposed results may have significant technological applications, including designing and constructing gyroscopes and other rotating sensors, because the rotational change affects different domains in various ways.
- Investigating issues of thermoelasticity in rotating media makes more sense because of the angular velocity of small objects that go into designing machines as well as massive bodies such as the Earth, Moon, and other planets.

#### 7.3. Effect of the Modified Ohm’s Law Coefficient on Variables of the Problem

**Figure 14.**Influence of the coefficient, ${k}_{0},$ on the temperature change, $\theta ,$ when $t=0.2$, $\mathsf{\Omega}=0.7$, and $y=0.1$.

**Figure 15.**Influence of the coefficient, ${k}_{0},$ the displacement component, $u,$ when $t=0.2$, $\mathsf{\Omega}=0.7$, and $y=0.1$.

**Figure 16.**Influence of the coefficient, ${k}_{0},$ on the stress component, ${\sigma}_{xx},$ when $t=0.2$, $\mathsf{\Omega}=0.7$, and $y=0.1$.

**Figure 17.**Influence of the coefficient, ${k}_{0,}$ on the stress component, ${\sigma}_{xy,}$ when $t=0.2$, $\mathsf{\Omega}=0.7$, and $y=0.1$.

**Figure 18.**Influence of the coefficient, ${k}_{0,}$ on the induced magnetic field, $h,$ when $t=0.2$, $\mathsf{\Omega}=0.7$, and $y=0.1$.

**Figure 19.**Influence of the coefficient, ${k}_{0,}$ on the induced electric component, ${E}_{2},$ when $t=0.2$, $\mathsf{\Omega}=0.7$, and $y=0.1$.

- To a large extent, the electromagnetic field affects the distributions of all studied physical quantities, both with and without the electromagnetic field.
- Due to the influence of the magnetic field, the values of all physical variables become zero as x increases, and all the functions examined have continuous curves.
- When the temperature gradient coefficient, ${k}_{0},$ is increased, the modified Ohm’s law leads to an increase in temperature change, $\theta $. It is noted that the thermal diffusion is much greater than it is in the case of neglecting this factor, ${k}_{0}$. For this reason, this effect must be taken into account in the design of some thermoelectric devices.
- It is seen that the Seebeck modulus has a prominent effect on deformation. One of the observations that must be taken into account is that deformation behavior in the case of neglecting this parameter is the opposite of the behavior in the case of taking it into account.
- The Seebeck coefficient has a compressive effect on the behavior of thermal stresses and the induced electric field component. The higher the value of the temperature gradient coefficient in the modified Ohm’s law is, the greater the magnitudes of these quantities are.
- One of the many crucial elements in the effective operation of thermoelectric generators and thermoelectric coolers is the use of materials with a high Ohm’s law coefficient (Seebeck coefficient).

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Megahid, S.F.; Abouelregal, A.E.; Askar, S.S.; Marin, M.
Study of Thermoelectric Responses of a Conductive Semi-Solid Surface to Variable Thermal Shock in the Context of the Moore–Gibson–Thompson Thermoelasticity. *Axioms* **2023**, *12*, 659.
https://doi.org/10.3390/axioms12070659

**AMA Style**

Megahid SF, Abouelregal AE, Askar SS, Marin M.
Study of Thermoelectric Responses of a Conductive Semi-Solid Surface to Variable Thermal Shock in the Context of the Moore–Gibson–Thompson Thermoelasticity. *Axioms*. 2023; 12(7):659.
https://doi.org/10.3390/axioms12070659

**Chicago/Turabian Style**

Megahid, Sami F., Ahmed E. Abouelregal, Sameh S. Askar, and Marin Marin.
2023. "Study of Thermoelectric Responses of a Conductive Semi-Solid Surface to Variable Thermal Shock in the Context of the Moore–Gibson–Thompson Thermoelasticity" *Axioms* 12, no. 7: 659.
https://doi.org/10.3390/axioms12070659