Rotational and Thermodiffusion Waves in a Microelongated Thermoelastic Layer Due to Initial Stress and Thermal Heating with Electromagnetic Waves via the Moore-Gibson-Thompson Model

The behavior of elastic materials under stress is heavily influenced by both temperature variations and the movement of mass through diffusion. For structural components like aircraft engine blades and the electrodes found in lithium batteries, maintaining their integrity hinges on the complex relationships between heat transfer processes, the diffusion of materials at a microscopic level, the rates of chemical reactions, and the mechanical stresses they endure as [1]. Mass diffusion, a process fundamentally described by Fick's law, is the net movement of a substance from a region of high concentration to an area of low concentration. This principle is a crucial factor in understanding internal material transport, particularly within solid materials where atomic mobility dictates material properties and performance. Therefore, a thorough and detailed understanding of how temperature, diffusion phenomena, and the mechanical stresses interact within elastic materials is absolutely essential for predicting material behavior and designing reliable engineering components. Diffusion processes are not merely theoretical concepts; they are critically important in a wide variety of scientific and engineering fields. These areas include the study of the Earth's internal structure and dynamics in geophysics, numerous practical applications across various industrial sectors, enhanced oil recovery techniques for maximizing petroleum extraction, the precise and controlled fabrication processes used in semiconductor manufacturing, and the miniaturization and performance optimization of microelectronics. The traditional theory of thermoelastic diffusion, which was originally developed by Nowacki [2], uses Fourier's law as its mathematical foundation to describe both the flow of heat energy and the movement of mass due to diffusion. However, a significant limitation of these early models is that they assume instantaneous propagation of both heat and mass, an assumption that does not accurately reflect the physical reality of how these processes occur at a microscopic level. To address this unrealistic assumption and develop a more accurate representation of diffusion phenomena, a more sophisticated and refined theory of diffusion was later introduced by Aouadi et al. [3], incorporating the concepts of thermoelastic microtemperatures and microconcentrations to provide a more nuanced and physically plausible model. This advanced model considers the microstructural effects and provides a more realistic depiction of material behavior under combined thermal and mechanical loads.

Microstretch continua represent an extension of classical elasticity theory by incorporating additional degrees of freedom. In contrast to classical elasticity, each particle within this framework possesses seven degrees of freedom, encompassing three translational components, three rotational components at the micro-level, and one component accounting for internal stretching. Microstretch continua offer a more comprehensive approach compared to traditional elasticity theory, achieved through the introduction of supplementary degrees of freedom. Unlike the established principles of classical elasticity, this enhanced framework significantly expands the description of each constituent particle. Specifically, within the microstretch continuum, every particle is characterized by a richer set of descriptors, possessing a total of seven degrees of freedom. These encompass a broader range of possible motions and internal characteristics when compared to classical models. The seven degrees of freedom can be further delineated into distinct components that capture different aspects of particle behavior. Three of these degrees of freedom correspond to the translational movements of the particle, representing its ability to move in three-dimensional space. Additionally, three degrees of freedom are dedicated to rotational movements, but crucially, these rotations are defined at the micro-level, reflecting the internal structure and behavior of the particle. Finally, the seventh degree of freedom accounts for internal stretching within the particle, enabling the model to capture deformation modes beyond simple translation and rotation, thus providing a more accurate representation of material behavior under stress. This inclusion of micro-rotational and stretching components distinguishes microstretch continua from classical elasticity, allowing for the modeling of more complex material behaviors. Eringen [4], [5] introduced the theory behind microstretch materials, laying the foundation for this advanced field. Shaw and Mukhopadhyay [6], [7] explored how heat sources affect microelongated solids, while Ailawalia et al. [8] investigated the thermoelastic behavior in these materials under similar conditions. Marin and Stan [9] focused on weak solutions in the elasticity of dipolar bodies with stretch. Marin et al. [10] investigated a relaxed form of the Saint-Venant principle in the context of thermoelastic micropolar diffusion. Schoenberg and Censor [11] pointed out that two types of acceleration--Coriolis and centrifugal (or centripetal)--play a role. The impact of rotation on thermoelastic solids has been widely studied across numerous works [12], [13].

The magnetic fields' interactions with thermoelastic materials have a vital effect in modern science and technology. It's especially important in biomedical engineering, where precise control over materials at microscopic scales is essential. These concepts are also critical in nuclear technology, where materials must perform reliably under extreme conditions to ensure both safety and efficiency. In thermoelastic materials, magnetic fields influence the equations of motion through a force known as the Lorentz force [14]. The foundational ideas of magnetoelasticity were first introduced by Knopoff [15]. Sherief and Helmy [16] expanded on this by exploring magneto-thermoelasticity with thermal relaxation time.

The hydrostatic initial stress effect in solid-state materials has a crucial role in shaping how the field descriptive functions are distributed. Geophysics studies the pre-stress effect caused by gravity beneath the Earth's surface significantly affects how elastic waves travel through the ground. The concept of initial stress first emerged when researchers began examining how mechanical loads impact various engineering components. Ames and Straughan [17] contributed to this field by establishing key results on how thermoelastic bodies behave under initial stress conditions. Montanaro [18] explored how hydrostatic initial stress influences thermoelastic materials. Wang and Slattery [19] developed the foundational concepts of thermoelastic bodies, focusing on cases in the context of the Green-Naghdi model with initial stress. The Moore-Gibson-Thompson (MGT) model was introduced to describe the thermal conductivity in a more accurate visualization. Building on this, Quintanilla [20] proposed a general thermoelastic theory that is considered an extension of the Green-Naghdi model. Abouelregal et al. [21] examined the behavior of dipolar bodies within the framework of MGT thermoelasticity. Several studies have extensively investigated further developments and complex problems in thermoelastic applications [22], [23], [24], [25], [26], [27], [28].

The present study explores the combined effects of magnetic fields and thermodiffusion in a two-dimensional thermoelastic microelongated layer within the frame of the MGT model. The proposed medium is assumed to be homogeneous, isotropic, rotating at a constant angular velocity, and initially stressed by uniform pressure. An analytical solution is obtained via the harmonic wave method to describe wave propagation behavior. Mechanical loading is modeled as a time-decaying wave at the surface of a half-space. Results are graphically presented using various aluminum-epoxy material parameters, showing finite solutions throughout. The observed damping effects stem from both the equation parameters and the imposed boundary conditions.

The descriptive and constitutive field equations of the prestressed linear, homogenous, and isotropic rotating thermoelastic microelongated medium without body forces in the magnetic field under the consideration of the thermodiffusion concept can be stated as Aouadi et al. [3], Eringen [4], Schoenberg and Censor [11], Montanaro [18], and Quintanilla [20].

The Maxwell equations in electrodynamics, for a slowly moving conducting medium, are

The Cartesian coordinate system $(x, y, z)$ has originated on the surface $z = 0$ with the $Z$-axis pointing vertically in the supposed medium, the plane configuration of the problem in the half-space, the positive $x-y$ plane. The displacement vector is $u_i \equiv(u, v, 0)$. The angular velocity will be $\Omega_T \equiv(0,0, \Omega)$. The electric intensity is $E_i \equiv\left(E_x, E_y, 0\right)$, the current density vector is $J_i \equiv\left(J_x, J_y, 0\right)$, the initial magnetic field will be $H_i \equiv\left(0,0, h_i(x, y, t)+H_0\right)$, and the Lorentz force is $F_i \equiv\left(F_x, F_y, 0\right)$ as in Figure 1.

To simplify the shown thermoelastic problem, use the dimensionless process in the form:

Consequently, Eqs. (1)–(4) in the ($x-y$) plane in a non-dimensional form, and by omitting the prime, will be:

where, $c_1=\frac{\lambda+\mu+\frac{p}{2}+\mu_0 H_0^2}{\mu-\frac{p}{2}}, c_2=\frac{\rho c_0^2}{\mu-\frac{p}{2}}, c_3=\frac{\rho \beta_1 c_0^2}{\mu-\frac{p}{2}}, c_4=\frac{c_0^2\left(\rho+\varepsilon_0 \mu_0 H_0^2\right)}{\mu-\frac{p}{2}}, c_5=\frac{\rho c_0^2 \Omega^2}{\mu-\frac{p}{2}}, c_6=\frac{2 \rho c_0^2 \Omega}{\mu-\frac{p}{2}}, c_7=\frac{\lambda_1 c_0^2}{\alpha_0 \omega^{* 2}}, c_8=\frac{\rho J_0 c_0^2}{2 \alpha_0}, c_9=\frac{\lambda_0^2}{\rho \alpha_0 \omega^{* 2}}, c_{10}=\frac{\lambda_0 \beta_1 c_0^2}{\beta_0 \alpha_0 \omega^{* 2}}, c_{11}=\frac{\lambda_0 \beta_1 c_0^2}{\beta_0 \alpha_0 \omega^*}, c_{12}=\frac{c_0^2}{b_1 \alpha_1 \omega^*}, c_{13}=\frac{\beta_2 \beta_0^2}{\rho c_0^2 b_1 \beta_1}, c_{14}=\frac{\alpha_1 \beta_2}{b_1 \beta_1}, c_{15}=\frac{k \omega^*}{k^*}, c_{16}=\frac{\beta_0 \beta_1 T_0 c_0^2}{\lambda_0}, c_{17}=\frac{\alpha 1 \beta_1 T_0 c_0^2}{\beta_2 k^*}, c_{18}=\frac{\rho C_E c_0^2}{k^*}, c_{19}=\frac{T_0 \beta_0^2}{\rho k^*}.$

Rewriting the descriptive equations in terms of the potential functions related to the displacement vectors as $\varphi_1=(x, y, t)$, and $\boldsymbol{\varphi}_2=(x, y, t)$, in the form:

Eqs. (13)–(17) will be reduced to

Harmonic wave approach solution of the studied system of the presented problem can be investigated in the following form:

where the amplitudes of the discussed functions are $\left[\varphi_1^{\dagger}, \varphi_2^{\dagger}, \psi^{\dagger}, C^{\dagger}, \theta^{\dagger}\right](x)$ the wave number of the propagated wave is $L$, $i$ is the imaginary unit, $\gamma$ is the phase velocity, with $\gamma=\frac{\omega}{L}$, while $\omega >0$, and is assigned to be the angular frequency; therefore, the descriptive equations are

with the variables $b_1=L^2-\frac{c_4 \gamma^2 L^2+c_5}{1+c_1}, \quad b_2=\frac{i L \gamma c_6}{1+c_1}, \quad b_3=\frac{c_2}{1+c_1}, \quad b_4=\frac{c_3}{1+c_1}, \quad b_5=-i L \gamma c_6, \quad b_6=L^2+c_4 \gamma^2 L^2+c_5, \quad b_7=c_7-c_8 \gamma^2 L^2, \quad b_8=i c_{11} \gamma L, \quad b_9=L^2+i L \gamma c_{12}, \quad b_{10}=\frac{c_{19} \gamma^2 L^2\left(1-i \gamma L \tau_0\right)}{1-i \gamma L c_{15}}, \quad b_{11}=\frac{i \gamma L c_{16}}{1-i \gamma L c_{15}}, \quad b_{12}=\frac{\gamma^2 L^2 c_{17}}{1-i \gamma L c_{15}}, \quad b_{13}=L^2-\frac{c_{18} \gamma^2 L^2\left(1-i \gamma L \tau_0\right)}{1-i \gamma L c_{15}}.$

System of Eqs. (25)–(29) gives the governing differential equations as:

where, $\pi_n(n=1,2,3,4,5)$ are the coefficients of the series obtained in MATLAB by eliminating the functions from Eqs. (25)–(29).

The descriptive differential Eq. (30) has its characteristic equation:

Eq. (31) gives the roots of the descriptive differential equation, which can accept only the negative roots to avoid the unbounded solution. Really, we attain the net analytical solution of the presented problem:

where,

$\begin{aligned}M_{1 n} & =\frac{-b_5}{k_n^2-b_6}, M_{2 n}=\frac{\delta_1}{\delta_2}, M_{3 n}=\frac{-\left(k_n^2-L^2\right)\left(c_{13}\left(k_n^2-L^2\right)\right)+c_{14} M_{4 n}}{k_n^2-b_9}, \\M_{4 n} & =\frac{-\left(k_n^2-L^2\right)\left(c_9 b_{12}+b_8 b_{10}+M_{2 n}\left(b_{12}\left(k_n^2-b_7\right)-b_8 b_{11}\right)\right.}{k_n^2-b_9}, \\M_{5 n}&=-k_n+i L M_{1 n}, M_{6 n} =i L+k_n M_{1 n}, \\M_{7 n} & =c_{20}\left(-k_n M_{5 n}+i L M_{6 n}\right)-c_{21} k_n M_{5 n}+M_{2 n}-M_{3 n}-M_{4 n}-p^*, \\M_{8 n} & =c_{20}\left(-k_n M_{5 n}+i L M_{6 n}\right)-i L c_{21} M_{6 n}+M_{2 n}-M_{3 n}-M_{4 n}-p^*, \\M_{10 n} & =c_{23}\left(i L M_{5 n}-k_n M_{6 n}\right)+c_{24}\left(i L M_{5 n}+k_n M_{6 n}\right), \\M_{11 n} & =-\alpha_0 k_n M_{2 n}, M_{12 n}=i L \alpha_0 M_{2 n}, \\M_{13 n} & =c_{25}\left(i L M_{5 n}-k_n M_{6 n}\right)+c_{26} M_{3 n}+c_{27} M_{4 n},\end{aligned}$

$\begin{aligned} & \delta_1=\left(b_3 b_8-b_4 c_{10}\right)\left(k_n^2-L^2\right)\left(b_8 b_{10}+b_{12} c_9\right)-\left(b_8\left(k_n^2-b_{13}\right)-c_{10} b_{12}\right)\left(b_8\left(k_n^2-b_1\right)\right. \\ & \left. -b_2 b_8 M_{1 n}-c_9 b_4\left(k_n^2-L^2\right)\right), \\ & \delta_2=\left(b_3 b_8+b_4 c_{10}\right)\left(b_{12}\left(k_n^2-b_7\right)-b_8 b_{11}\right)+\left(b_8\left(k_n^2-b_{13}\right)-c_{10} b_{12}\right)\left(b_4\left(k_n^2-b_7\right)+b_3 b_8\right),\end{aligned}$

$\begin{aligned} & c_{20}=\frac{\lambda}{\rho c_0^2}, c_{21}=\frac{2 \mu}{\rho c_0^2}, c_{22}=\frac{\beta_1}{\rho c_0^2}, c_{23}=\frac{\mu}{\rho c_0^2}, \\ & c_{24}=\frac{p^*}{2 \rho c_0^2}, c_{25}=\frac{\beta_0 T_0}{\rho c_0^2}, c_{26}=\frac{b_1 \beta_1 T_0}{\beta_2^2}, c_{27}=\frac{a_1 T_0}{\beta_2} .\end{aligned}$

Boundary conditions on the half-space were used to evaluate the coefficients $B_n$ will be:

Since the half-space is subjected to a constant mechanical load $p_1$. Using the boundary conditions Eq. (33) in Eq. (32) to get the coefficients.

Numerical scheme employed according to [4-5] took the physical parameters for aluminum-epoxy material, that used as $\rho=2330 \mathrm{~kg} \cdot \mathrm{m}^{-3}, \quad \lambda=3.64 \times 10^{10} \mathrm{~N} \cdot \mathrm{m}^{-2}, \quad \mu=5.64 \mathrm{~N} \cdot \mathrm{m}^{-2}, \quad k=150 \mathrm{~N} \mathrm{~s}^{-1} \cdot \mathrm{K}^{-1}, \quad \alpha_{t 1}=3 \times 10^{-6} \mathrm{~K}^{-1}, \quad \alpha_{t 2}=3 \times 10^{-6} \mathrm{~K}^{-1}, \quad C_E=695 \mathrm{~J} \cdot \mathrm{kg} \cdot \mathrm{K}^{-1}, \quad H_0=10^4 \mathrm{~A} \cdot \mathrm{m}^{-1}, \quad \mu_0=4 \pi \times 10^{-7} \mathrm{H} \cdot \mathrm{m}^{-1}, \quad \varepsilon_0=10^{-9} / 36 \pi \mathrm{~F} \mathrm{~m}^{-1}, \quad t=5 \times 10^{-2} \mathrm{~s}, \quad \lambda_0=\lambda_1=0.3 \times 10^{10} \mathrm{~N} \cdot \mathrm{m}^{-2}, \quad \beta_0=\beta_1=0.05 \times 10^5 \mathrm{~N} \mathrm{k}^{-1} \mathrm{~m}^{-2}, \quad \alpha_0=0.61 \times 10^{-10} \mathrm{~N}, \quad \alpha_c=2.65 \times 10^{-4} / \mathrm{m}^2 \cdot \mathrm{kg}, \quad J_0=0.196 \times 10^{-4} \mathrm{~m}^2, \quad \alpha_1=0.85 \times 10^{-8} \mathrm{~kg} / \mathrm{m}^3, b_1=32 \times 10^5 \mathrm{~m}^5 / \mathrm{kg} \cdot \mathrm{s}^2, \quad \tau_0=0.08 \mathrm{~s}, \quad \mathrm{k}=252 \mathrm{~J} / \mathrm{m} \cdot \mathrm{sk}, \quad k^*=170 \mathrm{~J} / \mathrm{m} \cdot \mathrm{sk}.$

The clarified figures show the effect of rotation and the wave number in the descriptive functions of the medium with two values for both the rotation and the wave number.

Figure 2 depicts that the rotation causes the displacement component $u(x, y, t)$ is increasing for $0 < x < 0.75$, then a decrease in the domain $0.75 < x < 8$, the same behavior of the displacement component $v(x,y,t)$ is increasing in the domain $0 < x < 1.5$, then decreasing for $x > 1.5$. Figure 3 clarifies that the increasing of the rotation is responsible for the increasing of the temperature $\theta(x,y,t)$ for $x > 0$, since the behavior of $\psi(x, y, t)$ is increasing in the domains $0 < x < 0.5$, and $15 < x < 8$, and decreasing in $0.5 < x <1.5$.

Figure 4 reports that the increase in the rotation causes a decrease of the stress component $\sigma_{x x}(x, y, t)$ is decreasing for $0 < x < 0.75$, then decreasing in the domain $0.75 < x < 8$, while the stress component $\sigma_{x y}(x, y, t)$ is decreasing for $x > 0$. Figure 5 states that the increase in the rotation is responsible for the increase in the chemical concentration $C(x, y, t)$ in the domain $0 < x < 5$, then it decreases for $x >5$, while the behavior of $m_x(x, y, t)$ is decreasing for $x > 0$.

Figure 6 depicts that the increase in the wave number causes the displacement component $u(x, y, t)$ is increasing for $x > 0$, while the behavior of the displacement component $v(x, y, t)$ is increasing in the domain $0 < x < 1.7$, then decreasing for $x > 1.7$. Figure 7 clarifies that the increase in the wave number is responsible for the increase in the temperature $\theta(x, y, t)$ for $x > 0$. since the behavior of $\psi(x, y, t)$ is increasing in the domains $0 < x < 0.5$, and $2.5 < x < 8$ while it decreased in $0.5 < x < 2.5$.

Figure 8 reports that the increase in the wave number causes an increase in the stress component $\sigma_{x x}(x, y, t)$ is increasing for the domains $0 < x < 0.5$, and $1.3 < x < 3$, and it is decreasing in domains $0.3 < x < 1.3$, and $3 < x < 8$, while the stress component $\sigma_{x y}(x, y, t)$ is increasing for $x > 0$.

Figure 9 states that the increase in the wave number is responsible for the decrease in the chemical concentration $C(x, y, t)$ for $x > 0$, while the behavior of $m_x(x, y, t)$ is increasing in the domain $0 < x < 3.2$, then it decreases for $3.2 < x < 8$. It can be seen that the studied functions propagate as the wave behavior.

Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, and Figure 16 were plotted to describe the represented constitutive functions in 2-D to show the homogeneity between the 2-D and 3-D presented figures in the case of the variation of the spatial coordinates. Figures show the attenuated propagation of the physical properties of the medium.

This investigation delves into the combined effects of magnetic fields and thermodiffusion phenomena within a rotating thermoelastic microelongated half-space. The material under consideration is initially stressed and characterized as linear, homogeneous, and isotropic. To model the thermoelastic behavior, the MGT model is employed, providing a framework for understanding the interactions between thermal, elastic, and microstructural effects. The analysis focuses on a rotating system to capture the influence of Coriolis forces on the material's response. A harmonic wave method serves as the mathematical foundation for deriving the medium's descriptive function. This approach allows for the study of wave propagation within the material and the determination of how various parameters affect wave characteristics. The analytical solution obtained through this method is carefully examined to ensure its consistency with the imposed boundary conditions. The agreement between the analytical solution and boundary conditions is a critical step in validating the accuracy of the model. The study explores the influence of both rotation and wave number on the behavior of field functions. These field functions represent physical quantities such as displacement, temperature, and stress within the material. The analysis reveals that rotation exerts a significant impact on the variation of these field functions. The Coriolis forces induced by rotation lead to changes in the distribution of stress and temperature throughout the microelongated half-space. In contrast, the wave number plays a crucial role in ensuring the solutions remain bounded. The wave number effectively controls the spatial oscillation of the field functions, preventing them from growing indefinitely. To visualize the results, both three-dimensional and two-dimensional representations are employed. These visualizations provide a comprehensive understanding of the spatial distribution of the field functions. The consistency between the three-dimensional and two-dimensional representations further validates the accuracy of the boundary conditions and the overall model. The findings obtained from this analysis, along with the discussed scenarios for this microelongated layer, suggest potential relevance in a wide range of manufacturing applications. Understanding the behavior of such materials under combined magnetic, thermal, and rotational effects is crucial for optimizing manufacturing processes and designing new materials with tailored properties.

This research presents an original analytical study on coupled wave propagation and damping in aluminum-epoxy composites. The investigation uniquely incorporates magnetic, thermos-diffusive, rotational, and initial stress considerations in a microelongated thermoelastic half-space, employing the MGT model.

Further research is warranted to extend the current framework to encompass three-dimensional geometries and nonlinear regimes, thereby facilitating a thorough understanding of stability in microelongated thermoelastic media. Investigation of alternative composite systems, dynamic boundary conditions, and additional coupled fields, such as piezoelectric or electromagnetic interactions, is also recommended to enhance the model's versatility. To accurately represent complex geometries and heterogeneous structures, and to validate theoretical predictions, numerical simulations and experimental verification are essential. Ultimately, adapting the analysis to applications including vibration control, damping, and wave filtering will translate theoretical advancements into practical engineering solutions.

The present study is limited by its reliance on a two-dimensional half-space model, which simplifies the geometry and may not fully capture three-dimensional effects encountered in practical applications. The analysis is conducted within a linear framework, thereby neglecting potential nonlinearities in stress–strain and thermodiffusion behavior under extreme loading conditions. Furthermore, the choice of aluminum epoxy as the illustrative composite restricts the generalizability of the findings to other advanced materials with distinct microstructural properties. The boundary conditions employed are idealized, whereas real engineering systems often involve dynamic or irregular constraints. Finally, the investigation remains purely theoretical, with no experimental validation to substantiate the predicted stability and damping characteristics.