Nonlocal Harmonically Varying Heat in a Magneto-Thermoelastic Thick Plate Using Simple and Refined Lord and Shulman Theories

Abstract

This work presents a solution to the nonlocal harmonically varying heat model in a magneto-thermoelastic thick plate. The classical, simple, and refined Lord and Shulman theories of thermoelasticity are applied. The medium is under a harmonic varying heat source with a constant strength and applied longitudinal magnetic field. Additionally, the nonlocal effect of thermoelastic materials is demonstrated using Eringen’s nonlocal theory. The Laplace transform technique is used to find the analytical solution. The numerical inversion approach of the Laplace transform is employed to determine the solution within the physical domain. The impacts of nonlocal, time parameters, and the angular frequency of thermal vibration on the field variables are presented graphically and analyzed in detail. The findings indicate that the responses of the magneto-thermoelastic thick plate to harmonically varying heat are significantly influenced by each one of the physical parameters. The refined Lord and Shulman model presents significant fluctuations in the results due to the theory’s additional terms.

1. Introduction

A subfield of continuum mechanics known as “nonlocal elasticity theory” considers the size effect and nonlocal behavior in materials by expanding classical elasticity theory. It acknowledges that the reaction of a substance at a specific point is influenced not only by its deformation but also by the deformation of neighboring points. In the context of classical elasticity, the stress experienced at a certain point is strictly determined by the strain occurring at the exact point, under the assumption that the size of the deformation does not influence the material’s response. Nonlocal elasticity theory incorporates a nonlocal or a small-scale parameter to describe nearby points’ impact on the material’s behavior. This nonlocal quantity shows the range across which stress at one point is influenced by the deformation at another. The size dependence and nonlocal interactions among material points are accurately captured. The nonlocal Eringen’s theory [1,2,3,4] helps investigate micro and nanostructure mechanical characteristics, where size effects are essential. It offers a more precise analysis of the deformation and distribution of stress in such structures of a smaller scale.

Bachher and Sarkar [5] used the Eringen’s nonlocal theory to investigate the transient wave propagation in infinite thermoelastic materials with voids. Xu et al. [6] reported the problem of a single-layer magneto-thermoviscoelastic plate. Mondal [7] applied Eringen’s nonlocal theory to investigate the memory effect of thermal distributions propagating through a magneto-thermoelastic rod. Zenkour and Abouelregal [8] utilized Eringen’s nonlocal elasticity theory to examine the magneto-thermoelastic interactions in a finite rod located in a magnetic field with a moving source.

In the classical theory of thermoelasticity, it is assumed that heat conduction and stress equilibrium happen immediately without considering any time delays or thermal inertia effects. Nevertheless, in actual situations, there usually exists a time delay between the change in temperature and the resulting response of a material to stress. The Lord and Shulman theory (LS) overcomes this limitation by integrating time-dependent factors into the thermoelastic equations. The Lord and Shulman theory (LS) is a modification of the classical theory (CTE) of thermoelasticity originally proposed by Lord and Shulman [9]. This theory aims to achieve more precise and realistic responses to materials’ thermal and mechanical properties, especially in scenarios that include fast temperature changes. The Lord and Shulman theory (LS) incorporates the thermal relaxation time as a singular relaxation time parameter. This parameter denotes the time scale in which the material achieves thermal equilibrium as a result of a temperature change. The LS theory incorporates the inertia of heat effects, giving an accurate prediction of temperature and stress distributions under transient thermal loading conditions.

The Lord and Shulman theory has been applied in numerous fields, such as heat conduction in materials, thermal stress analyses, and high-speed thermal processes. They are beneficial in transient thermal analyses, fast thermal cycling, and laser heating when the standard instantaneous heat conduction assumption is insufficient. This theory has enhanced our comprehension of the thermal behavior of materials and found practical use in areas such as thermal barrier coatings, electronic packaging, and high-speed manufacturing processes.

Karimi and Kiani [10] investigated the response of the nonlinear generalized thermoelasticity of a functionally graded material (FGM) layer subjected to thermal shock based on the Lord and Shulman theory. Xue et al. [11] used fractional Cattaneo–Vernotte models to examine the coupled thermoelastic fracture response of a fiber-reinforced composite hollow cylinder with a circumferential fracture. Mirzaei [12] studied the thermal shock effects on a linear viscoelastic strip. The Lord and Shulman theory of generalized thermoelasticity is utilized to determine the displacement, stress, and temperature within the strip. Alshaikh [13] conducted a study on the effects of thermal relaxation times on a porous medium based on the Lord and Shulman theory and revised multi-phase-lag models of generalized thermoelasticity. Oskouie et al. [14] investigated the propagation of nonlinear thermo-mechanical waves in a viscoelastic layer using the Lord and Shulman theory. Guha et al. [15] applied classical, dynamical, and coupled Lord and Shulman and Green–Lindsay thermoelasticity theories to examine thermoelastic damping and frequency shift in piezoelectro-magneto-thermoelastic composite micro-scale beams. Karimipour Dehkordi and Kiani [16] reported the magneto-thermoelastic behavior of a uniform and isotropic hollow cylinder using the Lord and Shulman theory and the Green–Lindsay theory. Many researchers consider the refined LS theory viable [17,18,19,20].

Magneto-thermoelasticity is an area of research that explores the interaction between magnetic fields and thermoelastic behaviors in materials. The field of magneto-thermoelasticity is widely used in several domains, particularly in biomedical engineering, nuclear devices, and geomagnetic research [20,21]. The generalized magneto-thermoelasticity equations were taken into consideration by many authors [22,23,24,25,26,27].

An essential aspect of many studies is to examine the impact of harmonically varying heat. This is necessary to guarantee the strength and stability of structures, enhance their durability and resilience, optimize their design, and advance the development of materials and structural technologies to satisfy the demands of modern engineering applications. Abouelregal et al. [28] investigated the impact of the thermal conductivity parameter and dynamic load on the physical properties of rotating nonlocal nanobeams exposed to a varying heat source. Kaur and Singh [29] examined the induced forced flexural vibrations in a transversely isotropic thin rectangular plate generated by time-harmonic focused loads and considered the memory-dependent derivative. Sharma et al. [30] incorporated a thermoelastic model diffusion and elastic nonlocality to investigate the dynamic vibrations of a hollow sphere subjected to harmonically changing heat sources. Babaei and Chen [31] studied the transient thermopiezoelectric behavior of a functionally graded piezoelectric medium to a moving heat source. Abouelregal and Zenkour [32] examined a new nonlocal nanobeam model for thermoelastic exposed to harmonically varying heat.

Despite the valuable contributions of the existing research studies, numerous knowledge deficits require further investigation. Studying the interaction between harmonically varying heat, nonlocal effects, and magneto-thermoelastic coupling in thick plates is still largely unexplored. This research aims to determine a solution to the model of a nonlocal magneto-thermoelastic thick plate using the classical theory (CTE), simple Lord and Shulman theory (LS (s)), and refined Lord and Shulman theory (LS (r)). The plate is under a periodic varying heat source with a constant strength and applied longitudinal magnetic field. The Laplace transform technique is used to find the analytical solution. The impacts of nonlocal, time parameters and the angular frequency of thermal vibration on the field variables are presented graphically and analyzed in detail.

2. Modified Nonlocal Thermoelasticity

As indicated by the nonlocal elasticity theory of Eringen [1,2,3,4], the nonlocal stress tensor $\tilde{\tau }\left(\stackrel{⃑}{x}\right)$ at any point $\stackrel{⃑}{x}$ in a body can be communicated as in

$$\tilde{\tau }\left(\stackrel{⃑}{x}\right)={\int }_V\alpha \left(\left|{\stackrel{⃑}{x}}^{\prime }-\stackrel{⃑}{x}\right|,\xi \right)\tilde{\sigma }\left(\stackrel{⃑}{x}\right)\mathrm{d}V\left({\stackrel{⃑}{x}}^{\prime }\right),$$

(1)

where $\xi =e_{0}a/l$ is the nonlocal parameter, $\tilde{\sigma }\left(\stackrel{⃑}{x}\right)$ is the classical local stress tensor (Neumann–Duhamel thermoelasticity law) at two neighboring points ${\stackrel{⃑}{x}}^{\prime }$, and $\stackrel{⃑}{x}$, which are, respectively, given by

$$\tilde{\sigma }\left(\stackrel{⃑}{x}\right)=\lambda \left(\nabla ·\stackrel{⃑}{u}\right)\tilde{\mathrm{I}}+2\mu \tilde{\epsilon }-\gamma \theta \tilde{\mathrm{I}},$$

(2)

$$\tilde{\epsilon }={\displaystyle \frac{1}{2}}\left[\nabla \stackrel{⃑}{u}+\nabla \left({\stackrel{⃑}{u}}^{\mathrm{T}}\right)\right],$$

(3)

in which $\stackrel{⃑}{u}$ is the displacement vector, and $\tilde{\epsilon }$ is the linear strain tensor. Here, we present the temperature increment $\theta =T-T_0$ wherein $T_0$ denotes the reference temperature, and $T(x,t)$ is the absolute temperature.

Equation (1) may be simplified in an identical differential structure as [1,2,3,4]:

$$\left(1-{\xi }^2{\nabla }^2\right)\tilde{\tau }=\tilde{\sigma },$$

(4)

which considers the size impact on the reaction of nanostructures. The parity of linear momentum brings about the accompanying condition of motion as follows:

$$\nabla ·\tilde{\tau }+\stackrel{⃑}{F}=\rho {\displaystyle \frac{{\partial }^2\stackrel{⃑}{u}}{\partial t^2}},$$

(5)

where $\rho$ denotes the density of the material, and $\stackrel{⃑}{F}$ is the Lorentz force vector. After using Equation (4), the parity of linear momentum, Equation (5), brings about

$$\nabla ·\tilde{\sigma }+\left(1-{\xi }^2{\nabla }^2\right)\stackrel{⃑}{F}=\rho \left(1-{\xi }^2{\nabla }^2\right){\displaystyle \frac{{\partial }^2\stackrel{⃑}{u}}{\partial t^2}}.$$

(6)

At that point, the equations of motion can be obtained regarding the temperature and displacements as

$$\left(\lambda +\mu \right)\nabla \left(\nabla \stackrel{⃑}{u}\right)+\mu {\nabla }^2\stackrel{⃑}{u}-\gamma \nabla \theta +\left(1-{\xi }^2{\nabla }^2\right)\stackrel{⃑}{F}=\rho \left(1-{\xi }^2{\nabla }^2\right){\displaystyle \frac{{\partial }^2\stackrel{⃑}{u}}{\partial t^2}}.$$

(7)

One may observe that when the inward trademark length is ignored, i.e., the particles of a medium are viewed as constantly conveyed, $\xi$ is zero, and Equation (4) lessens to the constitutive condition of classical local thermoelasticity.

The generalized heat conduction equation extended by the refined form of Lord and Shulman [9] theory is

$$k{\nabla }^2\theta =\left(1+{\sum }_{n=1}^N{\displaystyle \frac{t_0^n}{n!}}{\displaystyle \frac{{\partial }^n}{\partial t^n}}\right)\left[\rho c_{\nu }{\displaystyle \frac{\partial \theta }{\partial t}}+\gamma T_0{\displaystyle \frac{\partial }{\partial t}}\left(\nabla ·\stackrel{⃑}{u}\right)-Q\right],$$

(8)

where $k$ represents the thermal conductivity, $t_0$ is the first relaxation time, $c_{\nu }$ denotes the specific heat at a fixed strain, $Q(x,t)$ is the heat source, $\gamma =\left(3\lambda +2\mu \right){\alpha }_T$ represents the coupling parameter, $\lambda$, $\mu$ denote Lamé’s elastic coefficients, and ${\alpha }_T$ denotes the linear parameter of thermal expansion. However, in simple Lord and Shulman (LS) theory, $N=1$ is assumed [9]. The classical thermoelasticity theory is obtained by setting $t_0=0$ [33].

Maxwell’s electromagnetic field equations for a homogeneous and electrically conducting thermoelastic medium (neglecting the charge density) can be retrieved as [34]

$$\begin{gathered} \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\stackrel{⃑}{H}=\stackrel{⃑}{J}+{\epsilon }_0{\displaystyle \frac{\partial \stackrel{⃑}{E}}{\partial t}},\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\stackrel{⃑}{E}=-{\mu }_0{\displaystyle \frac{\partial \stackrel{⃑}{h}}{\partial t}},\stackrel{⃑}{h}=\nabla \times \left(\stackrel{⃑}{u}\times \stackrel{⃑}{H}\right), \\ \stackrel{⃑}{E}=-{\mu }_0\left({\displaystyle \frac{\partial \stackrel{⃑}{u}}{\partial t}}\times \stackrel{⃑}{H}\right), \nabla ·\stackrel{⃑}{h}=0, \end{gathered}$$

(9)

where $\stackrel{⃑}{J}$ is the electric current density vector, $\stackrel{⃑}{H}$ and $\stackrel{⃑}{E}$ are the magnetic and electric field intensities, ${\mu }_0$ is the magnetic permeability, and ${\epsilon }_0$ is the electric permeability. The electric current density vector $\stackrel{⃑}{J}$ according to Ohm’s law for a moving medium states that

$$\stackrel{⃑}{J}={\sigma }_0\left[\stackrel{⃑}{E}+{\mu }_0\left({\displaystyle \frac{\partial \stackrel{⃑}{u}}{\partial t}}\times \stackrel{⃑}{H}\right)\right],$$

(10)

where ${\sigma }_0$ is the electric conductivity of the medium (assumed to be infinite). Since $\stackrel{⃑}{J}$ is bounded and ${\sigma }_0$ is infinite, it follows that [35,36]:

$$\stackrel{⃑}{E}=-{\mu }_0\left({\displaystyle \frac{\partial \stackrel{⃑}{u}}{\partial t}}\times \stackrel{⃑}{H}\right).$$

(11)

3. Plan of the Problem

We will currently consider an infinite half-space at first at a constant temperature dispersion $T_0$ as shown in Figure 1. The dynamic issue of the half-space can be treated as a one-dimensional issue, at which point, the components of displacement can be expressed in the form

$$u_x=u\left(x,t\right), u_y=u_z=0.$$

(12)

The magneto-thermoelastic thick plate of perfect conductivity occupies the region $0\le x\le \mathcal{l}$ in an initial magnetic field ${\stackrel{⃑}{H}}_0$ in a $y$ direction at a uniform reference temperature $T_0$. This produces an induced magnetic field $\stackrel{⃑}{h}$ in the direction of the $y$-axis and an electric field E in the $z$-axis (perpendicular to $\stackrel{⃑}{H}$ and $\stackrel{⃑}{u}$).

We suppose the applied longitudinal magnetic field with uniform intensity acts perpendicular to the axial direction of the plate $\stackrel{⃑}{H}\equiv \left(0,H_0+h,0\right)$. The Lorentz force $\stackrel{⃑}{F}={\mu }_0\left(\stackrel{⃑}{J}\times \stackrel{⃑}{H}\right)$ induced by applying the longitudinal magnetic field $\stackrel{⃑}{H}$ appearing in the motion Equation (6) is represented as

$$\stackrel{⃑}{F}\equiv \left(f_x,f_y,f_z\right)=-{\epsilon }_0{\mu }_0^2{H}_0^2\left({\displaystyle \frac{{\partial }^{2}u}{\partial t^2}},\mathrm{0,0}\right).$$

(13)

For a one-dimensional case, Equation (4) may be expressed as:

$$\left(1-{\xi }^2{\displaystyle \frac{{\partial }^2}{\partial x^2}}\right){\tau }_x=\left(\lambda +2\mu \right){\displaystyle \frac{\partial u}{\partial x}}-\gamma \theta .$$

(14)

The assistance of Equations (13), (14), and (7) gives the following equation of motion:

$$\left(\rho +{\epsilon }_0{\mu }_0^2{H}_0^2\right)\left(1-{\xi }^2{\displaystyle \frac{{\partial }^2}{\partial x^2}}\right){\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}=\left(\lambda +2\mu \right){\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}-\gamma {\displaystyle \frac{\partial \theta }{\partial x}}.$$

(15)

Similarly, the heat conduction Equation (8) without the inclusion of heat source $Q$ is now rewritten as:

$$k{\displaystyle \frac{{\partial }^2\theta }{\partial x^2}}=\left(1+{\sum }_{n=1}^N{\displaystyle \frac{t_0^n}{n!}}{\displaystyle \frac{{\partial }^n}{\partial t^n}}\right){\displaystyle \frac{\partial }{\partial t}}\left(\rho c_{\nu }\theta +\gamma T_0{\displaystyle \frac{\partial u}{\partial x}}\right).$$

(16)

We consider the dimensionless quantities

$$\begin{gathered} \left\{x^{\prime },u^{\prime }\right\}=c_0\eta \left\{x,u\right\}, \left\{t^{\prime },t_0^{\prime }\right\}=c_0^2\eta \left\{t,t_0\right\}, {\xi }^{\prime }=c_0\eta \xi , \\ {\theta }^{\prime }={\displaystyle \frac{\theta }{T_0}}, \left\{{\sigma }_x^{\prime },{\tau }_x^{\prime }\right\}={\displaystyle \frac{1}{\lambda +2\mu }}\left\{{\sigma }_x,{\tau }_x\right\}, c_0^2={\displaystyle \frac{\lambda +2\mu }{\rho }}, \eta ={\displaystyle \frac{\rho c_{\nu }}{k}}, \end{gathered}$$

(17)

in the governing equations which may be finally written as (dropping the primes)

$$\left(1-{\xi }^2{\displaystyle \frac{{\partial }^2}{\partial x^2}}\right){\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}=c_1{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}-c_2{\displaystyle \frac{\partial \theta }{\partial x}},$$

(18)

$${\displaystyle \frac{{\partial }^2\theta }{\partial x^2}}=\left(1+{\sum }_{n=1}^N{\displaystyle \frac{t_0^n}{n!}}{\displaystyle \frac{{\partial }^n}{\partial t^n}}\right){\displaystyle \frac{\partial }{\partial t}}\left(\theta +c_3{\displaystyle \frac{\partial u}{\partial x}}\right),$$

(19)

$$\left(1-{\xi }^2{\displaystyle \frac{{\partial }^2}{\partial x^2}}\right){\tau }_x={\displaystyle \frac{\partial u}{\partial x}}-c_4\theta ,$$

(20)

where

$$c_1={\displaystyle \frac{\rho }{\rho +{\epsilon }_0{\mu }_0^2{H}_0^2}}, c_2={\displaystyle \frac{\gamma T_0}{c_0^2\left(\rho +{\epsilon }_0{\mu }_0^2{H}_0^2\right)}}, c_3={\displaystyle \frac{k\gamma }{\rho c_{\nu }{c}_0}}, c_4={\displaystyle \frac{T_0}{\lambda +2\mu }}.$$

(21)

These governing equations lead to the local definition if the nonlocal parameter $\xi$ is set to zero. The present work aims to determine the displacement, temperature, and nonlocal stress at the surface of the problem depicted by Equations (18)–(20).

The homogeneous initial conditions are assumed as

$$\theta \left(x,0\right)=u\left(x,0\right)={\displaystyle \frac{\partial \theta (x,0)}{\partial t}}={\displaystyle \frac{\partial u(x,0)}{\partial t}}=0.$$

(22)

4. Solution of the Problem in Laplace Transform Domain

By considering the Laplace transform characterized by the connection

$$\overline{f}\left(x,s\right)={\int }_0^{\mathrm{\infty }}f\left(x,t\right){\mathrm{e}}^{-st}\mathrm{d}t,$$

(23)

to the two sides of Equations (18)–(20) and utilizing the initial conditions (22), one obtains the field conditions in the Laplace transform domain as

$${\displaystyle \frac{{\mathrm{d}}^2\overline{u}}{\mathrm{d}{x}^2}}=c_5\overline{u}+c_6{\displaystyle \frac{\mathrm{d}\overline{\theta }}{\mathrm{d}x}},$$

(24)

$${\displaystyle \frac{{\mathrm{d}}^2\overline{\theta }}{\mathrm{d}{x}^2}}=\varpi \overline{\theta }+c_7{\displaystyle \frac{\mathrm{d}\overline{u}}{\mathrm{d}x}},$$

(25)

$$\left(1-{\xi }^2{\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}}\right){\overline{\tau }}_x={\displaystyle \frac{\mathrm{d}\overline{u}}{\mathrm{d}x}}-c_4\overline{\theta },$$

(26)

where

$$\left\{c_5,c_6\right\}={\displaystyle \frac{1}{c_1+{\xi }^2{s}^2}}\left\{s^2,c_2\right\}, c_7=\varpi c_3, \varpi =s\left(1+{\sum }_{n=1}^N{\displaystyle \frac{t_0^n}{n!}}{s}^n\right).$$

(27)

Equations (24) and (25) can be inscribed in a vector–matrix differential equation as [37,38]:

$${\displaystyle \frac{\mathrm{d}\mathit{V}\left(x,s\right)}{\mathrm{d}x}}=A\left(s\right)\mathit{V}\left(x,s\right),$$

(28)

where

$$\mathit{V}\left(x,s\right)=\left\{\begin{array}{c}\overline{\theta }\\ \overline{u}\\ {\displaystyle \frac{\mathrm{d}\overline{\theta }}{\mathrm{d}x}}\\ {\displaystyle \frac{\mathrm{d}\overline{u}}{\mathrm{d}x}}\end{array}\right\}, A\left(s\right)=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ \varpi & 0& 0& c_7\\ 0& c_5& c_6& 0\end{array}\right].$$

(29)

Using the solution procedure through the eigenvalue approach presented in [38,39,40], the solutions in the Laplace transform space for $\overline{u}\left(x\right)$ and $\overline{\theta }\left(x\right)$ are

$$\overline{\theta }\left(x\right)={\sum }_{j=1}^2\left(A_j{\mathrm{e}}^{m_{j}x}+A_{j+2}{\mathrm{e}}^{-m_{j}x}\right),$$

(30)

$$\overline{u}\left(x\right)={\sum }_{j=1}^2{\widehat{m}}_j\left(A_j{\mathrm{e}}^{m_{j}x}-A_{j+2}{\mathrm{e}}^{-m_{j}x}\right),$$

(31)

where $A_j$ ($j=1,2,3,4$) are integration parameters and

$${\widehat{m}}_j={\displaystyle \frac{m_j\left(c_6{c}_7-m_j^2+\varpi \right)}{c_5{c}_7}}.$$

(32)

Moreover, the local stress is given by

$$\overline{\sigma }\left(x\right)={\sum }_{j=1}^2\left(m_j{\widehat{m}}_j-c_4\right)\left(A_j{\mathrm{e}}^{m_{j}x}+A_{j+2}{\mathrm{e}}^{-m_{j}x}\right),$$

(33)

where

$${\tilde{m}}_j=m_j{\widehat{m}}_j-c_4.$$

(34)

Using Equations (30) and (31) in Equation (20), the nonlocal stress ${\overline{\tau }}_{xx}$ can be resolved as

$$\overline{\tau }\left(x\right)={\sum }_{j=1}^2{\check{m}}_j\left(A_j{\mathrm{e}}^{m_{j}x}+A_{j+2}{\mathrm{e}}^{-m_{j}x}\right)+A_5{\mathrm{e}}^{x/\xi }+A_6{\mathrm{e}}^{-x/\xi },$$

(35)

where $A_5$ and $A_6$ are additional integration parameters and

$${\check{m}}_j={\displaystyle \frac{{\tilde{m}}_j}{1-m_j^2{\xi }^2}}.$$

(36)

Now, let us consider that the thick plate is thermally loaded by harmonically varying heat incidents into the edge $x=0$. Therefore, this condition and other boundary conditions are considered as

$$\theta \left(0,t\right)=f\left(t\right)={\theta }_0\sin \omega t, {\displaystyle \frac{\partial \theta (\mathcal{l},t)}{\partial x}}=0, \sigma \left(0,t\right)=\sigma \left(\mathcal{l},t\right)=0.$$

(37)

where ${\theta }_0$ is a constant and $\omega$ is the angular frequency of the thermal vibration.

The above conditions as well as Equations (30) and (33) give

$$\begin{gathered} {\sum }_{j=1}^2\left(A_j+A_{j+2}\right)=\overline{f}\left(s\right), \\ {\sum }_{j=1}^2{m}_j\left(A_j{\mathrm{e}}^{m_j\mathcal{l}}-A_{j+2}{\mathrm{e}}^{-m_j\mathcal{l}}\right)=0, \\ {\sum }_{j=1}^2{\tilde{m}}_j\left(A_j+A_{j+2}\right)=0, \\ {\sum }_{j=1}^2{\tilde{m}}_j\left(A_j{\mathrm{e}}^{m_j\mathcal{l}}+A_{j+2}{\mathrm{e}}^{-m_j\mathcal{l}}\right)=0, \end{gathered}$$

(38)

where $\overline{f}\left(s\right)$ is the Laplace transform of $f\left(t\right)$ given by

$$\overline{f}\left(s\right)={\displaystyle \frac{{\theta }_{0}s}{s^2-{\omega }^2}}.$$

(39)

The solution of the above system of linear equations is given in the form

$$\begin{gathered} A_1={\displaystyle \frac{{\tilde{m}}_2\overline{f}\left(s\right)}{∆}}\left[{\mu }_1^{+}{\mathrm{e}}^{\left(m_1+m_2\right)\mathcal{l}}-{\mu }_1^{-}{\mathrm{e}}^{\left(m_1-m_2\right)\mathcal{l}}-2m_2{\tilde{m}}_1{\mathrm{e}}^{2m_1\mathcal{l}}\right], \\ {A}_2={\displaystyle \frac{{\tilde{m}}_1\overline{f}\left(s\right)}{∆}}\left[{\mu }_1^{+}{\mathrm{e}}^{\left(3m_1-m_2\right)\mathcal{l}}+{\mu }_1^{-}{\mathrm{e}}^{\left(m_1-m_2\right)\mathcal{l}}-2m_1{\tilde{m}}_2{\mathrm{e}}^{2m_1\mathcal{l}}\right], \\ {A}_3={\displaystyle \frac{{\tilde{m}}_2\overline{f}\left(s\right)}{∆}}\left[2m_2{\tilde{m}}_1{\mathrm{e}}^{2m_1\mathcal{l}}+{\mu }_1^{-}{\mathrm{e}}^{\left(3m_1+m_2\right)\mathcal{l}}-{\mu }_1^{+}{\mathrm{e}}^{\left(3m_1-m_2\right)\mathcal{l}}\right], \\ {A}_4={\displaystyle \frac{{\tilde{m}}_1\overline{f}\left(s\right)}{∆}}\left[2m_1{\tilde{m}}_2{\mathrm{e}}^{2m_1\mathcal{l}}-{\mu }_1^{-}{\mathrm{e}}^{\left(3m_1+m_2\right)\mathcal{l}}-{\mu }_1^{+}{\mathrm{e}}^{\left(m_1+m_2\right)\mathcal{l}}\right], \end{gathered}$$

(40)

where

$$∆=\left({\tilde{m}}_1-{\tilde{m}}_2\right)\left\{{\mu }_1^{+}\left[{\mathrm{e}}^{\left({3m}_1-m_2\right)\mathcal{l}}-{\mathrm{e}}^{\left(m_1+m_2\right)\mathcal{l}}\right]+{\mu }_1^{-}\left[{\mathrm{e}}^{\left(m_1-m_2\right)\mathcal{l}}-{\mathrm{e}}^{\left(3m_1+m_2\right)\mathcal{l}}\right]\right\},$$

(41)

in which

$${\mu }_1^{+}=m_1{\tilde{m}}_2+m_2{\tilde{m}}_1, {\mu }_1^{-}=m_1{\tilde{m}}_2-m_2{\tilde{m}}_1.$$

(42)

Finally, to get the nonlocal stress $\overline{\tau }\left(x\right)$ in Equation (35), we need two additional boundary conditions in the Laplace domain in the form

$$\overline{\tau }\left(0\right)=\overline{\tau }\left(\mathcal{l}\right)=0.$$

(43)

Therefore, the forms of $A_5$ and $A_6$ are given in terms of other parameters by

$$\begin{gathered} A_5=-{\displaystyle \frac{{\check{m}}_1{A}_1\left({\mathrm{e}}^{m_1\mathcal{l}}-{\mathrm{e}}^{-m_3\mathcal{l}}\right)+{\check{m}}_2{A}_2\left({\mathrm{e}}^{m_2\mathcal{l}}-{\mathrm{e}}^{-m_3\mathcal{l}}\right)+{\check{m}}_1{A}_3\left({\mathrm{e}}^{-m_1\mathcal{l}}-{\mathrm{e}}^{-m_3\mathcal{l}}\right)+{\check{m}}_2{A}_4\left({\mathrm{e}}^{-m_2\mathcal{l}}-{\mathrm{e}}^{-m_3\mathcal{l}}\right)}{{\mathrm{e}}^{m_3\mathcal{l}}-{\mathrm{e}}^{-m_3\mathcal{l}}}}, \\ {A}_6={\displaystyle \frac{{\check{m}}_1{A}_1\left({\mathrm{e}}^{m_1\mathcal{l}}-{\mathrm{e}}^{m_3\mathcal{l}}\right)+{\check{m}}_2{A}_2\left({\mathrm{e}}^{m_2\mathcal{l}}-{\mathrm{e}}^{m_3\mathcal{l}}\right)+{\check{m}}_1{A}_3\left({\mathrm{e}}^{-m_1\mathcal{l}}-{\mathrm{e}}^{m_3\mathcal{l}}\right)+{\check{m}}_2{A}_4\left({\mathrm{e}}^{-m_2\mathcal{l}}-{\mathrm{e}}^{m_3\mathcal{l}}\right)}{{\mathrm{e}}^{m_3\mathcal{l}}-{\mathrm{e}}^{-m_3\mathcal{l}}}}. \end{gathered}$$

(44)

The problem in the Laplace transform domain has been solved. Due to the complexity of Equations (30) and (31), it is challenging to perform the inverse transform analytically in the time domain. Therefore, the numerical inversion Laplace transform approach is employed to determine the actual time domain responses of temperature, displacement, and local and nonlocal stresses.

5. Numerical Inversion of Laplace Transform

We obtained the solutions to two problems in the Laplace transform domain $(x,s)$ in the previous section. This section will provide an overview of the numerical inversion approach employed to determine the solution within the physical domain. Consider $\bar{f}\left(s\right)$ to be the Laplace transform of $f\left(t\right)$. The inversion formula for the Laplace transform can be expressed as stated by Churchill [40] as follows:

$$f(t)={\displaystyle \frac{1}{2\pi l}}{\int }_{v-i\mathrm{\infty }}^{v+i\mathrm{\infty }} {\mathrm{e}}^{st}\bar{f}(s)\mathrm{d}s,$$

(45)

in which $v$ represents an arbitrary real number greater than all the real parts of the singularities of $\bar{f}\left(s\right)$. By utilizing the equation $s=v+iw$ and applying the Fourier series in $[0,2T]$, an approximation formula may be derived, as demonstrated in the work of Honig and Hirdes [41]:

$$f(t)\cong f_N(t)={\displaystyle \frac{1}{2}}{c}_0+{\sum }_{k=1}^N c_k \mathrm{for} 0\le t\le 2T,$$

(46)

where

$$c_k={\displaystyle \frac{e^{vt}}{T}}\mathrm{R}\mathrm{e}\left[e^{ik\pi t/T}\bar{f}\left(v+{\displaystyle \frac{ık\pi }{T}}\right)\right].$$

(47)

Two approaches are employed to minimize the total error. The “Korrecktur” approach is used to minimize the discretization error. Following that, the ε-algorithm is applied to minimize the error caused by truncation and enhance the convergence rate. Honig and Hirdes [41] provide more information about these techniques and provide the criteria by which the values of $v$ and $T$ are selected.

6. Numerical Results

A discussion of the numerical results of the field variables under study is offered to evaluate the validity of the suggested model in this study. The properties of material required in numerical calculations are provided by applying the copper material constants as [35,36]

$$\begin{gathered} \lambda =7.76\times {10}^{10} \left(\mathrm{k}\mathrm{g} {\mathrm{m}}^{-1} {\mathrm{s}}^{-2}\right), \mu =3.86\times {10}^{10}\left(\mathrm{k}\mathrm{g} {\mathrm{m}}^{-1} {\mathrm{s}}^{-2}\right), \\ \rho =8954 \left(\mathrm{k}\mathrm{g} {\mathrm{m}}^{-3}\right), c_{\nu }=383.1 \left(\mathrm{J} {\mathrm{k}\mathrm{g}}^{-1} {\mathrm{K}}^{-1}\right), {\alpha }_t=1.78\times {10}^{-5}\left({\mathrm{K}}^{-1}\right), \\ K=386 \left(\mathrm{W} {\mathrm{m}}^{-1} {\mathrm{K}}^{-1}\right), T_0=293 \mathrm{K}, {\epsilon }_0=5.7\times {10}^7 {\left(\mathsf{\Omega } \mathrm{m}\right)}^{-1}, \\ {\mu }_0=4\pi \times {10}^{-7}\left(\mathrm{H} {\mathrm{m}}^{-1}\right), H_0={\displaystyle \frac{1}{36\pi }}\times {10}^{-7}\left(\mathrm{H} {\mathrm{m}}^{-1}\right). \end{gathered}$$

(48)

6.1. The Study of CTE, Simple, and Refined LS Models

Figure 2, Figure 3, Figure 4 and Figure 5 depict the field quantities resulting from the CTE, LS (s), and LS (r) theories ($\xi =0.3$, $\omega =5$). Figure 2 illustrates the temperature distributions across the $x$-axis of the thick plate based on several theories. Across the $x$-axis, the temperature gradually drops in the CTE and LS generalized thermoelastic models. This is due to the heat source varying harmonically over a brief period. The behavior of $\theta$ under the refined LS generalized thermoelastic model appears distinct from the other theories. The temperature $\theta$ fluctuates across the $x$-axis of the thick plate.

The distribution of displacement $u$ across the $x$-axis of the thick plate is shown in Figure 3 using various theories. The displacement amplitude increases across the $x$-axis of the plate thickness. The material exhibits reduced rigidity in the load direction, which leads to increased displacement as the load increases across the plate. The LS (s) generalized thermoelastic model produces the lowest displacement amplitude, whereas the LS (r) model has the highest.

Figure 4 shows the local stress $\sigma$ across the $x$-axis of the thick plates using several theories $(\xi =0,\mathsf{\omega }=5)$. The CTE and LS (s) models exhibit similar behavior. The LS (r) model produces the most stress at $x=0.75$. The LS (r) model, with additional terms, captures the variations in stress across the plate. This may result in fluctuations since the model considers the impact of several factors.

The nonlocal stress $\tau$ across the $x$-axis of the thick plates according to different theories is presented in Figure 5. The LS (r) model shows a higher value of nonlocal stress $\tau$, and reaches its maximum value of about $x=0.75$. The CTE thermoelastic model gives the lowest nonlocal stress $\tau$ compared to the other models.

6.2. The Effect of the Angular Frequency of Thermal Vibration $\omega$

Figure 6, Figure 7, Figure 8 and Figure 9 illustrate the effect of the angular frequency of thermal vibration on different fields $(\xi =0.3)$. The current results of the fields studied are presented with several angular frequency values $\omega =1,2,3,4$. Figure 6 presents the temperature $\theta$ distribution of the thick plate for different $\omega$ via the CTE, LS (s), and LS (r) theories. The increase in the value of $\omega$ induces an increase in the temperature distribution in the three models. Higher Angular frequencies of thermal vibration $\omega$ produce more significant thermal gradients resulting from faster thermal energy oscillations. However, the LS (r) model behaves differently with the value of $\omega$ rising. Along the $x$-axis, the value of the temperature distribution decreases, then all zero about $x=0.75$. After $x=0.75$, the increase of $\omega$ raises the temperature.

Figure 7 illustrates the displacement $u$ distributions of the thick plate for different $\omega (\xi =1)$ via the CTE, LS (s), and LS (r) theories. The displacement amplitude decreases as the angular frequency of thermal vibration increases. As $\omega$ increases, the material’s effective rigidity can impact the response, resulting in lower displacements. In the LS (r) model, the response of increases $\omega$ increases $u$ where $0.8\le x\le 1$.

The local stress distribution responses of the thick plate for different $\omega$ via the CTE, LS (s), and LS (r) theories are presented in Figure 8. From these figures, it is evident that the amplitude of the investigated field decreases as the thermal vibration angular frequency parameter is increased. The influence on the local stress distributions is reversed in the LS (r) model on $x\in \left[0,0.3\right]\cup [0.6,1]$. This could occur from wave interactions or other dynamical effects considered by the LS (r) theory, which are not included in the CTE or LS (s) model.

The angular frequency of thermal vibration $\omega$ effects on the nonlocal stress distributions via the CTE, LS (s), and LS (r) theories are displayed in Figure 9. The nonlocal stress $\tau$ decreases as the value of $\omega$ increases for CTE and LS (s) models. In the case of LS (r), the value of $\tau$ rises as $\omega$ increases, and the difference in the values of τ seems to be magnified when $x=0.75$. The LS (r) theory incorporates additional factors, including higher-order effects that enhance the nonlocal stress $\tau$ response. Figure 6, Figure 7, Figure 8 and Figure 9 show that the thermal vibration angular frequency parameter significantly impacts all fields.

6.3. The Effect of Nonlocal Parameter $\xi$

To investigate the impacts of nonlocal parameters on the distribution of temperature, displacement, and nonlocal stress for thick plates, Figure 10, Figure 11 and Figure 12 present the findings based on CTE, simple, and refined LS theories. The local thermoelasticity model is represented by the nonlocal parameter $\xi =0$, whereas the following values $(\xi =0.2,0.3,$ and $0.5)$ represent nonlocal models of thermoelasticity. In this case, $\omega =5$ was considered.

In Figure 10, we can see the response of the temperature distribution $\theta$ for different nonlocal parameters. It can be seen that the temperature distribution increases as the nonlocal parameter increases.

The effects of the nonlocal parameter $\xi$ on the displacement $u$ distributions of the thick plate via the CTE, LS (s), and LS (r) theories are presented in Figure 11. The displacement amplitude decreases as the nonlocal parameter increases till it reaches $x=0.75$ for the CTE model, $x=0.8$ for the LS (s) model, and $x=0.7$ for the LS (r) model, then $u$ increases as the nonlocal parameter value rises.

Figure 12 shows the nonlocal stress $\tau$ distributions of the thick plate for different $\xi$ via the CTE, LS (s), and LS (r) theories. By increasing the value of the nonlocal parameter, the nonlocal stress $\tau$ increases. Additionally, the local model has the lowest stress in these distributions. Sharp edges in the local case ($\xi =0$) result from localized stress response, causing concentrated stress and singularities at particular points. Conversely, the nonlocal examples ($\xi =0.1, 0.15, 0.2$) illustrate smooth curves as the nonlocal effects minimize the large gradients, leading to a more uniform stress distribution. This emphasizes the significance of incorporating nonlocal effects in materials when stress distribution is affected by neighboring material points, resulting in more realistic and physically precise models.

6.4. The Effect of Time Parameter $t$

Figure 13, Figure 14, Figure 15 and Figure 16 illustrate the effect of the time parameter $t$ on different fields based on the CTE, LS (s), and LS (r) theories ($\omega =5$). The current results of the fields studied are presented for time parameter values $t=0.5,0.55,0.6,0.65$. In Figure 13, the impact of time parameter $t$ on temperature $\theta$ distributions of the thick plate is shown. Based on the CTE and LS (s) theories, the distribution of temperature increases as $t$ increases. In the LS (r) model, the response of $\theta$ exhibits different fluctuations in behavior for various values of $t$. The fluctuations arise from the LS (r) theory’s ability to represent thermal waves and microstructural interactions, resulting in a more complex and dynamic temperature distribution.

Figure 14 shows the impact of the time parameter $t$ on displacement $u$ distributions of the thick plate. For the CTE, LS (s), and LS (r) theories, displacement distribution decreases as $t$ increases. However, the effect of $t$ changes as $x=0.9$ for the CTE theory, $x=0.95$ for the LS (s) theory, and $x=0.65$ for the LS (r) theory.

The local stress $\sigma$ distributions of the thick plate for different $t$ $(\omega =5)$ via the CTE, LS (s), and LS (r) theories are displayed in Figure 15. By considering the CTE and LS (s) theories, the value of $\sigma$ gradually decreases with increasing time $t$. The disparity in the values of $\sigma$ seems to be increased at $x=0.45$. In the case of the LS (r) model, the local stress solution exhibits different oscillating patterns over the range of the $x$-axis for various values of time $t$.

According to the CTE, LS (s), and LS (r) theories, Figure 16 presents the nonlocal stress $\tau$ distributions of the thick plate for various $t$ ($\omega =5$). Based on the theories of CTE and LS, the value of nonlocal stress $\tau$ progressively goes down as time t increases. The discrepancy in the values of $\tau$ appears to be amplified when $x=0.47$. For the LS (r) model, the nonlocal stress solution $\tau$ displays distinct oscillating shapes along the $x$-axis at various time $t$ values.

In general, the additional terms in the LS (r) theory can cause a variation in the results across the plate. This may cause significant fluctuations in the responses of some physical fields, especially under transitory circumstances.

7. Conclusions

The main goal of this work is to study the effects of nonlocal harmonically varying heat in a magneto-thermoelastic thick plate, using the CTE, simple, and refined LS theories. Eringen’s nonlocal elasticity theory is incorporated into the analysis to capture the nonlocal effect of thermoelastic materials. The medium is subjected to a longitudinal magnetic field and a harmonic varying heat source of a constant intensity. The governing equations are transformed into the Laplace–Fourier domain. The numerical inversion approach of the Laplace transform is applied to find the solution within the physical domain. Solutions are provided for the displacement, temperature, stress, nonlocal, time parameters, and angular frequency of thermal vibration. The following conclusions can be made based on the findings:

• Thermal vibration angular frequency has a substantial impact on all fields studied.
• Increasing the nonlocal parameter value escalates the temperature and nonlocal stress $\tau$, while reducing the displacement.
• The LS (r) theory can cause a variation in the results across the plate due to the theory’s additional terms. This may cause significant fluctuations in the responses of some physical fields.