Influence of Hall current and acoustic pressure on nanostructured DPL thermoelastic plates under ramp heating in a double-temperature model

Nomenclature

a

two-temperatures parameter

a n

material constant

C E

specific heat

e 0

material’s characteristic length or intrinsic length scale

e

cubical dilatation

e i j = 1 2 ( u i , j + u j , i )

components of the strain tensor

F i

Lorentz force

K

thermal conductivity

l

reference length scale of the system

P

acoustic pressure

T 0

absolute temperature

t o

pulse time rising

u i

components of the displacement vector

α t

coefficient of linear thermal expansion

σ i j

components of the stress tensor

φ

conductive heat temperature

ρ

density of the medium

ε 0

electric permittivity

δ i j

Kronecker delta function

λ , μ

Lame’s parameters

μ 0

magnetic permeability

ε 2 = a n e 0 / l

nonlocal parameter

ε i j r

permutation tensor

τ θ , τ q

thermal relaxation time

θ = T − T 0

thermodynamical temperature

γ = ( 3 λ + 2 μ ) α t

volume thermal expansion

1 Introduction

The interplay between the Hall current and acoustic pressure in nanostructured thermoelastic plates under ramp heating in a dual-temperature model is crucial for understanding the coupled electromechanical and thermal responses in advanced materials. The Hall current, induced by the interaction of electric and magnetic fields, modifies the charge carrier distribution and exerts a significant influence on the propagation of acoustic waves within the plate. This interaction becomes even more complex under ramp heating, which introduces a transient thermal gradient that drives thermal stresses and enhances the non-equilibrium effects described by the dual-temperature theory. In nanostructures, where size-dependent properties dominate, these effects are amplified due to the enhanced surface-to-volume ratio and quantum confinement. Investigating this problem is vital for applications in micro- and nanoscale devices, such as sensors and actuators, where precise control of thermal and electromechanical interactions is essential. Understanding these relations contributes to the design of materials and devices capable of withstanding high-frequency electromagnetic and thermal fields in emerging technologies.

The study of thermoelasticity, a fundamental area of continuum mechanics, is essential for understanding the complex interactions between thermal and mechanical processes in solid materials [1]. Classical thermoelasticity, which integrates heat conduction and elasticity, provides a foundational framework for examining how temperature changes influence mechanical deformations and vice versa [2]. However, traditional theories are often inadequate for capturing the nuances of heat transfer in scenarios involving rapid thermal transients, high-frequency oscillations, or nanoscale systems. Such limitations have driven the evolution of advanced thermoelastic theories that incorporate more realistic heat propagation mechanisms [3]. By introducing finite thermal wave speeds and addressing the temporal delay in thermal signal transmission, these extended models offer significant improvements in accurately describing non-equilibrium thermal processes [4]. This progression, marked by the transition from classical to hyperbolic and generalized thermoelasticity, has opened new avenues for studying phenomena like thermal shocks, second sound effects, and microstructural thermal inertia, which are critical in modern materials science and engineering applications [5]. The dual-phase-lag (DPL) heat conduction model, introduced by Tzou [6], revolutionized the understanding of transient thermal processes by incorporating two distinct delays: the phase lag of the heat flux and the phase lag of the temperature gradient. This approach addresses the limitations of classical Fourier heat conduction, which assumes infinite thermal propagation speed, making it unsuitable for applications involving rapid thermal responses or micro- and nanoscale systems [7]. By introducing these phase lags, the DPL model accounts for the finite relaxation times associated with heat flux and temperature gradient, providing a more accurate depiction of non-equilibrium thermal behavior.

The two-temperatures theory (2TT), initially introduced by Chen et al. [8], provides a framework for heat transfer in deformable materials by distinguishing between two distinct temperature fields: The thermodynamic temperature and the conductive temperature. This distinction arises from the non-equilibrium heat conduction processes that occur at small scales or under rapid thermal variations. Many authors demonstrated that when external heat sources are present, the difference between these two temperatures is proportional to the heat supply, while in the absence of external heating, the temperatures converge to equilibrium [9,10,11]. Quintanilla [12] investigated the mathematical and physical properties of the 2TT, including stability and spatial resolution, reinforcing its applicability to advanced thermal systems. Building on this framework, Youssef [13] proposed a generalized thermoelasticity model based on the two-temperature concept, where the relationship between the temperature difference and the heat supply is defined through a non-negative constant. Over the years, the 2TT has gained significant attention in the context of generalized thermoelasticity and extended thermodynamics, providing critical insights into heat transfer processes in nano-engineered materials and complex thermal environments [14,15,16,17]. The use of 2TT (conductive and thermodynamic temperatures) in thermoelastic models addresses this shortcoming, offering greater accuracy in describing heat transfer dynamics. On the other hand, Awad [18] introduced a novel idea about the spatial decay estimates in semi-cylindrical with bounded domains, according to the non-classical linear thermoelasticity. Awad et al. [19] studied a thermal resonance according to the temperature amplitude attained and oscillations for an external exciting frequency source in the electron–phonon interaction process.

In thermoelasticity, the interaction between magnetic fields and acoustic pressure plays a critical role in analyzing the dynamic behavior of materials under external influences. Magnetic fields influence the elastic properties of materials, altering the propagation of acoustic waves through magnetoacoustic coupling, a phenomenon where the Lorentz force on charged particles induces or modifies acoustic wave behavior [20]. Lotfy et al. [21] explored the effects of magnetic fields on photothermal transport processes in semiconducting materials, incorporating Thomson effects and microtemperature phenomena. Their research also addressed Hall current impacts in rotating semiconducting media subjected to magnetic fields and ramp heating [22,23]. Mandelis [24] employed the photoacoustic effect to study thermal waves in elastic semiconductors, expanding the understanding of energy interactions in such systems. Additionally, Lotfy et al. [25] proposed a model for photoacoustic and plasmaelastic interactions in nonlocal thermoelastic media, emphasizing the role of laser-induced heating and variable thermal conductivity. These studies underscore the importance of magnetoacoustic interactions in advancing technologies like magnetoacoustic imaging and the development of magnetoelastic materials [26,27,28,29]. Some theoretical studies investigate the behavior of surface wave propagation in magneto-electro-elastic structures [30]. Tiwari et al. [31] studied the memory response according to the DPL thermoelasticity under the nonlocal effect. Singhal [32] examined the Love-type wave velocity during imperfect interface in the context of the piezo-structure of three distinct rheological models. Gupta et al. [33,34] studied the two temperature theories in the context of the photo-thermo-piezo-elastic theory with fractional order in a semiconductor material. Barak et al. [35] analyzed the nonlocal dual-phase-lag at distinct piezo-thermoelastic media to study the behavior of higher-order heat equations and wave propagations.

The novelty of the present study lies in its comprehensive investigation of the interplay between magnetic fields, acoustic pressure, and thermoelastic effects in nonlocal elastic media under ramp heating conditions. While previous research has explored these phenomena separately, this work introduces a coupled model that integrates the effects of Hall currents, two-temperature phenomena, and magnetic field-induced acoustic wave modulation in a nonlocal elastic medium. The methodology employs a combination of analytical and numerical techniques to solve the governing equations of magneto-acoustic coupling in the context of nonlocal thermoelasticity, with particular attention to ramp heating conditions. The motivation for this research stems from the need to improve our understanding of the behavior of advanced materials under extreme thermal and electromagnetic conditions, particularly for applications in magneto-acoustic imaging, sensor technologies, and laser-induced thermal processing. However, a limitation of this study is the complexity of the coupled system, which may restrict the applicability of the model to highly idealized scenarios, potentially overlooking real-world factors such as material heterogeneities or external noise. Despite this, the insights gained provide a significant step forward in modeling and understanding the dynamic behavior of semiconducting materials in the presence of magnetic fields and thermal gradients.

2 Formulation of the problem

Through the elastic medium, the main equations that describe the interactions between acoustic, thermal, electromagnetic field, and mechanical waves can be derived by coupling the heat conduction, thermoelasticity, and wave propagation equations. The general form of the dual-phase-lag heat conduction equation with two temperatures in the presence of thermal waves is expressed as follows [23,36]:

For a transversely isotropic (or transversely thermoelastic) material under the influence of the Lorentz force ( F i = μ 0 ( J ⇀ × H ⇀ ) ), the equation of motion in the general form can be obtained by considering both the mechanical and thermal effects, as well as the interaction between the material’s deformation and the magnetic field. In such materials, the stress–strain relationships and the thermal field are coupled with the mechanical wave propagation, and the Lorentz force modifies the equations of motion by influencing the charge carriers. For the mechanical wave interactions, the displacement field u i in the elastic medium is governed by the following nonlocal equation of motion under the effect of the Lorentz force [32,34]:

In the context of the 2TT, the interaction between φ and T is described by a set of coupled differential equations that account for the heat flow and temperature gradients in the material. These equations capture the transient heat conduction process, where the difference between φ and T arises due to the lag in thermal propagation between the two temperatures. The general form of the 2TT equations is given by [13]

The acoustic diffusivity in the elastic medium can be characterized by the acoustic pressure or the thermoacoustic effects resulting from the heating processes. The acoustic pressure equation, which describes the interaction between acoustic pressure and temperature in a thermoelastic medium, can be derived from the fundamental principles of thermodynamics and wave propagation in the medium. In the presence of a temperature field, acoustic waves are influenced by thermal effects, and the coupling between the acoustic pressure and temperature can be modeled as follows [24,25,26,37]:

The speed of sound in the medium is c s = 8,430 m/s. The specific heat ratio is ℑ = 1.666 and the coefficient β = 2.56 × 10⁻⁶°C is the volumetric thermal expansion [30].

For an elastic material, the nonlocal stress–strain relation is given by [38,39]

In generalized thermoelasticity theory, the phase-lags associated with the temperature gradient and heat flux are key concepts that describe the delay in the response of thermal variables to thermal disturbances. Specifically, τ θ and τ q ( 0 ≤ τ θ < τ q ) represent the phase-lags of the temperature gradient and heat flux. The general form of the equations incorporating these phase-lags can be derived from the extended versions of the thermoelasticity theory, such as the Lord and Shulman (LS, τ θ = 0 ) theory, which introduces a relaxation time for the thermal response (no phase-lags are present). The phase-lag model reflects the fact that changes in temperature and heat flux do not propagate instantaneously but rather with a delay determined by the material’s thermal properties [34].

To incorporate the Hall current impact and the finite conductivity of the material into the system of equations, we need to include the generalized form of Ohm’s law and the Lorentz force. The generalized Ohm’s law accounts for the current density in a conducting medium, which can be influenced by both the electric field and the magnetic field (in the presence of the Hall effect). In the presence of a magnetic field, in a more general tensor form, the components of the current density tensor J r and Lorentz force can be reformulated as follows [21,23,30]:

When an elastic medium is subjected to an initial magnetic field of strength H r = ( H 1 , H 2 , H 3 ) = ( 0 , H 0 , 0 ) along the y-axis, the electrical conductivity coefficient is denoted by σ 0 = n e t e e n 2 / m e , the electron charge by e n , and the electron mass by m e . The density of charge carriers is represented by n e , and t e signifies the time between electron collisions. In this scenario, the induced electric field can be neglected to isolate the Hall effect, which arises solely due to the magnetic force. Additionally, by considering the system in two dimensions (2D), specifically within the x z -plane of the rectangular Cartesian coordinate system, the displacement vector is expressed as u ⇀ = u i = ( u , 0 , w ) . In this setup, the current density and Lorentz force components are reformulated as follows [23]:

The influence of the Hall current, represented by the Hall parameter m = t e ω e , can be determined using the electron frequency ω e = e μ 0 H 0 / m e , which is related to the parameter m.

The 2D heat Eq. (1) is reformulated as follows:

The 2D stress–strain relations are

The 2D motion equations are

The 2D of 2TT is

The 2D acoustic pressure equation can be reformulated as follows:

By introducing dimensionless field variables, the governing equations can be expressed in their dimensionless forms. The main variables are then defined through their corresponding dimensionless quantities as follows [40,41]:

where P 0 represents the initial acoustic pressure. For simplicity, the dashes can be omitted. The core 2D equations in this case can be expressed in their dimensionless form as follows:

where ξ = γ ρ C E and β = a η 2 c 0 2 , a 1 ⁎ = μ ρ C 0 2 , a 2 = μ + λ ρ C 0 2 , a 0 = γ T 0 ρ C 0 2 , ε p = c 0 2 c s 2 , η p = ℑ β T 0 c 0 2 P 0 , Γ = σ 0 C T μ 0 2 H 0 2 ρ 0 , and ∇ 2 = ∂ 2 ∂ x 2 + ∂ 2 ∂ z 2 .

Due to the various field variables that have been analyzed, we can introduce the potential functions Π ( x , z , t ) and ψ ( x , z , t ) . These functions allow us to describe the acoustic waves more effectively, and the corresponding partial differential equations can be expressed as follows [42,43]:

By transforming Eq. (22) into Eqs. (17), (19), and (20), the corresponding system of equations can be formulated as follows:

where a 3 = 1 a 1 ⁎ + a 2 and a 4 = 1 a 1 ⁎ .

Eqs. (24) and (25) describe the dynamics of transverse waves in the system. These equations specifically describe transverse waves, which are independent of the overall motion and do not rely on thermal or mechanical forces. These elastic waves propagate through space, maintaining their strength throughout their journey.

3 Solution to the problem

To solve the system of governing equations, we will assume that the domain variables are evaluated using the following forms based on normal mode analysis [44,45]:

where ω is the angular frequency of the wave, i = − 1 , the propagation constant (wave number) is b , and the spatial-dependent part of each field variable, evaluated at position x are ( Π ⁎ , ψ ⁎ , φ ⁎ , P ⁎ , θ ⁎ ) ( x ) , and σ i j ⁎ ( x ) . Eq. (26) can be applied for the main Eqs. (18), (21) and (23)–(25), yields:

where A = ω ( 1 + ω τ q ) ( 1 + ω τ θ ) , B = ξ ω ( 1 + ω τ q ) ( 1 + ω τ θ ) , Ω 1 = 1 + a 3 ε 2 ϑ 1 , A 1 = b 2 + a 3 Ω 2 ϑ 1 , Ω 2 = 1 + ε 2 b 2 , ϑ 1 = ω 2 + ω Γ 1 + m 2 , A 2 = a 0 a 3 ( 1 + τ θ ω ) , B 1 = ω a 3 m Γ 1 + m 2 , B 2 = ( b 2 ( 1 + ε 2 ) − 1 ) Ω 3 , Ω 4 = 1 + ε 2 Ω 3 , Ω 3 = a 4 ω 2 + Γ ω a 1 ⁎ ( 1 + m 2 ) , B 3 = Γ m ω a 1 ⁎ ( 1 + m 2 ) A 3 = ( β b 2 + 1 ) / β , α 1 = b 2 + ε p ω 2 , α 2 = η p ω 2 , β ⁎ = 1 β , and D = d d x .

The eighth-order differential equation is obtained after the elimination method is applied for Eqs. (27)–(31), one can obtain

where

where A 4 = β ⁎ + A , A 5 = β ⁎ b 2 + A A 3 , and Θ 5 = Ω 4 ( A 4 Ω 1 + A 2 B ) + A 4 B 1 B 3 ε 4 .

Here is the structured presentation of the homogeneous factorization Eq. (32).

This transforms the equation into the characteristic equation with roots m n 2 ( n = 1 , 2 , 3 , 4 )

The linear solutions are formulated from exponential solutions. The general solution is obtained when x → ∞ as follows:

where D n is a constant determined by boundary conditions and

Γ 1 n = ( m n 2 − α 1 ) α 2 , Γ 2 n = − Γ 1 n β ⁎ m n 2 − A 3 , Γ 3 n = − ( m n 2 − α 1 ) ( A 4 m n 2 − A 5 ) α 2 B ( m n 2 − A 3 ) ( m n 2 − b 2 ) , Γ 4 n = − B 3 ( m n 2 − α 1 ) ( ε 2 m n 2 − Ω 2 ) ( A 4 m n 2 − A 5 ) α 2 B ( m n 2 − A 3 ) ( Ω 4 m n 2 − B 2 ) ( m n 2 − b 2 ) .

The strain–displacement components are generally derived from Eq. (22) as follows:

Utilizing Eqs. (39), (41), and (42) to obtain the stress components from the constitutive relations

where

4 Boundary conditions

To solve the presented system of differential equations and determine the unknown parameters M n ( n = 1 , 2 , 3 , 4 ) discussed in Section 2, certain boundary conditions for the problem under investigation can be provided. In this study, we examine a thermally elastic quasi-space subjected to mechanical stress and acoustic pressure on its surface, influenced by a temperature gradient of the ramp type, all under the effects of a magnetic field and Hall current at x = 0 for a nonlocal elastic medium. Consequently, the following boundary conditions are considered [46,47,48]:

1. Ramp-type heating (varies sinusoidally with time) is applied based on thermal conditions with an arbitrary constant temperature θ 1 , determined by the heating rise time or the sinusoidal pulse width t 0 , which is considered on the outer surface [46].

   (48) θ ( 0 , z , t ) = θ 1 sin π t t 0 exp ( ω t + i b z ) .

   Sinusoidal time variations allow for studying the transient heat transfer behavior in materials or structures, where thermal effects are not constant but vary periodically. This helps to analyze the thermal responses which means how the material or structure responds to oscillating heat inputs, which can lead to phenomena like thermal resonance, cyclic expansion, and contraction.

2. The Mechanical boundary condition which describes how the material interacts with its surroundings can be taken with constant magnitude force load Ω 1 for normal stress as [1,2]

   (49) σ x x ( 0 , z , t ) = Ω 1 ( z , t ) = − Ω 1 ⁎ exp ( ω t + i b z ) .

3. The tangent mechanical stress can be chosen as zero traction (free surface) as follows:

   (50) σ x y ( 0 , z , t ) = 0 .

4. Acoustic pressure refers to the fluctuating pressure within a medium caused by the propagation of acoustic waves, which result from disturbances such as sound, vibrations, or thermal gradients. At the free surface of an elastic medium, the acoustic pressure is typically related to external disturbances P 0 ⁎ which can be chosen in the form of sinusoidal acoustic pressure and is commonly used in cases where the pressure is oscillatory [31].

By applying the normal mode method to Eqs. (48)–(51), the following system of equations was obtained:

The parameters D n ( n = 1 , 2 , 3 , 4 ) can be determined once the equations mentioned above are solved. As a result, the various physical fields involved have been obtained, and the entire system of equations has been resolved.

5 Validation

5.2 Influence of Hall current parameter

The proposed set of equations addresses the DPL thermoelasticity theory, capturing the interaction between elastic, thermal, and mechanical waves while neglecting the effects of external magnetic fields. To validate the model and its implementation, by neglecting the Lorentz force F ⇀ , the equations were reduced to the classical formulation of the nonlocal DPL thermoelasticity. This reduction was verified numerically and analytically, yielding results consistent with established literature in conventional thermoelastic models (Figure 1). In this case, the motion equation takes the form [34]

(57) ρ ( 1 − ε 2 ∇ 2 ) u ̈ i = σ i j , j .

Figure 1

Variation in the primary dimensionless nonlocal physical fields for the vertical distance for two different values of the Hall current parameter, based on the DPL model, with varying sinusoidal heating parameters.

5.3 Non-local impact

The formulated fundamental equations incorporate nonlocal effects to study their impact on the main physical fields in the context of thermoelastic wave propagation. To validate the robustness of the model, the following methodologies were employed:

Reduction to the DPL thermoelasticity model, by setting the nonlocal parameter to zero ( ε = 0 ), the equations transform into the DPL thermoelasticity model under the influence of Hall’s current theory. Numerical results obtained from this limiting case were benchmarked against previously established results for the DPL framework, demonstrating a high degree of consistency and confirming the accuracy of the model reduction (Figure 2). In this case, the motion equation takes the form

(58) ρ u ̈ i = σ i j , j + F i ( i , j = 1 , 2 , 3 ) .

Figure 2

Variation in the primary dimensionless physical fields for the vertical distance for local and nonlocal media under the impact of Hall current, based on the DPL model, with varying sinusoidal heating parameters.

6 Numerical results

This section provides a quantitative assessment of the analytical solutions to the various problem areas discussed earlier. This section presents numerical results for several dimensionless physical quantities associated with thermoelastic behavior, emphasizing temperature, acoustic pressure, and stress. The numerical results are presented graphically and analyzed theoretically. To highlight the influence of each factor in facilitating wave propagation within an elastic medium, a detailed discussion and analysis of its effect on the system of equations is conducted. While additional variables were chosen individually, all studies were completed within a relatively short time period (t = 0.001 s) when z = − 1 . To improve the comprehension of the thermoacoustic mathematical model and its theoretical outcomes, the actual values of the physical variables were considered. MATLAB (2022a) software was used to perform the numerical computations. For reference, the physical properties of copper crystals were utilized, with the material properties of copper (Cu) measured in International System (SI) units and reported as follows [31,32,33,34]:

λ = 7.59 × 10 9 N/m 2 , μ = 3.86 × 10 10 kg / ms 2 , ρ = 8 , 960 kg / m 3 , τ 0 = 0.02 s , α = − 1.28 × 10 9 N / m 2 , β = 0.32 × 10 9 N / m 2 , η = 8886.73 m / s 2 , ε = 0.0168 , α t = 1.78 × 10 − 5 K − 1 , k = 386 W m − 1 K − 1 , b = 1 , C E = 383.1 J / ( kgk ) , T 0 = 298 K , y = − 1 , Ω 1 ⁎ = 0.5 , θ 1 = 300 K , P 0 ⁎ = 2 , ω 0 = 2 , and ξ = 1 , where ω = ω 0 + i ξ and e ω t = e ω 0 t ( cos ξ t + i sin ξ t ) . When the time is very small, the value of ω is given in real [35,36].

6.3 Impact of DPL thermal times

The relaxation times in the LS model and the DPL model play a crucial role in the propagation of waves in a nonlocal medium, especially under the influence of the Hall current which is obtained in Figure 3. These models introduce memory effects or time lags into the material’s response to mechanical and thermal disturbances, which are essential for accurately modeling complex phenomena in materials with thermal and elastic relaxation processes. The LS model introduces thermal relaxation time ( τ θ = 0 ​), which governs the delay in the heat conduction process. As the relaxation time increases, the material’s thermal response becomes more sluggish, with delayed heat conduction and diffusion. The Hall current, which induces an additional electromagnetic force on charge carriers, will affect the efficiency of heat generation and transport in the medium. Thus, a longer relaxation time would cause the temperature wave to propagate more slowly and with greater damping, as the material lags in responding to thermal changes. In the DPL model ( 0 ≤ τ θ < τ q ), there are two relaxation times: one for the thermal flux (related to heat conduction) and another for the temperature gradient (related to the rate of change of temperature). Both relaxation times induce a phase lag in the thermal response, which causes the temperature wave to propagate with delayed changes in both the temperature and heat flux. Relaxation times in the thermal field represent the material’s ability to respond to temperature changes. Longer relaxation times imply a slower response to thermal gradients, meaning heat propagation is delayed and dampened. When Hall current is present, charge carriers are driven by electromagnetic forces, further complicating the thermal behavior due to altered energy dissipation and conduction. Relaxation times control how quickly the material responds to thermal and mechanical disturbances, with larger times causing delayed, damped, or oscillatory wave propagation in all fields.

Figure 3

Variation in the primary dimensionless nonlocal physical fields for the vertical distance for different thermal memories according to the LS model and DPL model, based on the Hall current model, with varying sinusoidal heating parameters.

6.4 Surface wave propagation

The surface wave propagation of various dimensionless nonlocal physical fields (temperature, acoustic pressure, displacement, normal stress, tangent stress, and conductive temperature) is influenced by several factors. In the context of Figure 4, the DPL model and the Hall current model are used to describe the effects of sinusoidal heating on wave propagation. The figure likely shows the behavior of these fields as a function of both vertical (x) and horizontal (z) distances, while varying sinusoidal heating parameters (amplitude, frequency, or other characteristics). Let us break down the effects in detail. Sinusoidal heating typically involves a periodic temperature variation that can significantly affect the propagation of waves. In this case, the sinusoidal heating parameters influence the material’s response, modifying the characteristics of wave propagation, especially in terms of amplitude, frequency, and damping. Overall, the combined effects of sinusoidal heating, Hall current, and the DPL model lead to more complex and slower wave propagation, with significant changes in the amplitude, phase, and damping of the waves.

Figure 4

Variation in the primary dimensionless nonlocal physical fields for the horizontal and vertical distances according to the DPL model, based on the Hall current model, with varying sinusoidal heating parameters.

7 Life applications

• The DPL thermoelasticity theory and its extension to nonlocal effects provide insights into heat transfer and stress distribution in nanostructures. These results are critical for developing efficient thermal management strategies in microelectronics and photonics.

• The study of elastic and thermal wave propagation with nonlocal effects can advance acoustic wave applications, such as surface acoustic wave devices, which are used in sensors, filters, and signal processing systems in communication technology.

• The framework developed in this work provides a foundation for studying mechanical and thermal behaviors in advanced materials, including nanocomposites and metamaterials. The incorporation of nonlocal effects is particularly relevant for understanding and designing materials with tailored properties at the nanoscale.

• In aerospace engineering, where materials are exposed to extreme thermal and electromagnetic environments, this study provides essential tools for predicting material behavior and ensuring reliability in advanced structural applications.

• The coupling between thermal, elastic, and electromagnetic fields can be exploited in piezoelectric and thermoelectric energy harvesting systems. Insights from this investigation could contribute to the efficiency and miniaturization of such systems.

8 Conclusion

This work presents an advanced model of magnetothermal elasticity, incorporating acoustic pressure and extending the heat transfer concept to include both dynamic and conductive temperatures, along with relaxation times. The governing equations were derived from the nonlocal thermoelastic model influenced by acoustic pressure in the presence of a strong magnetic field. The model was solved using the harmonic wave method, with specific boundary conditions applied to the free surface. The study focused on the response of nonlocal thermoelastic materials to thermal and acoustic stresses, temperature variations, and deformations, while also considering the impact of Hall effect. The role of the Hall current is emphasized, as the variations in conductivity caused by a strong magnetic field make it a significant factor. This current is generated when ions and electrons collide and move perpendicular to the electromagnetic fields along the magnetic lines of force. The current DPL model is found to offer more accurate predictions compared to the LS models, highlighting the importance of incorporating different relaxation times for a better understanding of heat and stress propagation dynamics within materials. Additionally, the varied heating sinusoidally (ramp heating) with time parameter is identified as a crucial factor in determining how heat is applied to the thermoelastic medium over time, significantly affecting the behavior of the studied thermophysical domains. Integrating the Hall current effect, varying relaxation times, and ramp-type heating parameters into thermoelastic models not only deepens our understanding of wave propagation phenomena but also facilitates the development of advanced materials and technologies. This enhanced understanding has wide-ranging implications across fields such as science, engineering, and medicine, particularly in the design of magnetostrictive materials for sensors and actuators. Furthermore, it aids in optimizing material processing techniques to improve performance and efficiency, especially in the context of tailored materials with specialized functionalities. Understanding the complex interactions between magnetic fields, thermal properties, and mechanical responses is essential for the advancement of next-generation materials.