Magneto-Thermoelastic Response in an Infinite Medium with a Spherical Hole in the Context of High Order Time-Derivatives and Triple-Phase-Lag Model

The article presents the interactions of magneto-thermoelastic effects in an isotropic material with a spherical cavity. The spherical cavity is expected to be tractionless and subjected to both heat and magnetic fields. The motion equation contains the Lorentz force. Laplace’s transformation methodology is used with a refined multi-time-derivative triple-phase-lag thermoelasticity theory to develop the generalized magneto-thermoelastic coupled solution. Many results were obtained to serve as benchmarks for future comparisons. The effects of time, magnetic field, and electric permittivity under the thermal environment were investigated.


Introduction
It is largely recognized that the theory of coupled thermoelasticity deals with the physical restriction that the thermal signal propagates at infinity. Biot [1] cultivated the theory of classical thermoelasticity based upon irreversible thermodynamic guidelines. The formula of motion within this idea is hyperbolic, while the formula of heat transmission is parabolic. As a result of the inclusion of a parabolic-type heat transmission formula, the theory still experiences the uncoupled theory flaw. Lord and Shulman [2] introduced the initial generalized thermoelasticity concept, through which the Fourier regulation of heat convection is substituted by the Maxwell-Cattaneo equation, which declares a single relaxation time in Fourier's regulation. Similarly, Green and Lindsay [3] (G-L) provided a temperature-dependent concept with dual relaxation times stated in the constitutive equations for stresses and entropy. Green and Naghdi [4][5][6] introduced three further models dubbed G-N types I, II, and III. The linearized form of the G-N type I model corresponds to the conventional thermoelasticity theory. The internal production rate of entropy is assumed to be zero in the G-N type II model, which does not include heat dissipation. The G-N type III model incorporates both prior models, as well as energy dissipation and the admission of damped thermoelastic waves. Hetnarski and Ignaczak [7] provided yet another generalized theory. This model is defined by a nonlinear field equation system based on the low-temperature thermoelasticity hypothesis. Tzou [8] and Chandrasekhariah [9,10] enhanced the dual-phase-lag (DPL) hypothesis. Tzou [8] stated that Fourier's regulation is substituted through an approximation to modify Fourier's law with a pair of distinct delay times for heat motion and temperature gradient. Thermoelasticity with three-phase-lag is one of the most recent advancements in thermoelasticity theory (Roy Choudhari [11]). Here, a phase lag for the thermic displacement gradient is added to the phase lags for the heat motion vector and the temperature gradient. Quintanilla and Racke [12] addressed some where Θ represents the temperature and u r , u Θ , and u φ are the displacement components. The non-vanishing strains can be stated as where e rr , e ΘΘ , and e φφ are the strain components and the dilatation e can be expressed as e = e rr + e ΘΘ + e φφ = ∂u ∂r The isotropic medium is considered to have a spherical hole with a radius and an introductory uniform temperature 0 . We assumed this environment to be a symmetric thermal space; therefore, the displacement components and the temperature are defined as = ( , ), = ( , ), where represents the temperature and , , and are the displacement components.
The non-vanishing strains can be stated as where , , and are the strain components and the dilatation can be expressed as .
The constitutive equations are given by the formula where represents stress tensor components, represents Kronecker's delta, = (3 + 2 ) represents the thermal modulus, is the thermal expansion coefficient, and and are Lame's constants.
Thus, by substituting Equation (2) into Equation (4), it gives The linear isotropic homogeneous thermoelastic body is governed by the following equation where volume forces are absent in this situation: • The equation of motion The constitutive equations are given by the formula where σ ij represents stress tensor components, δ ij represents Kronecker's delta, γ = (3λ + 2µ)α t represents the thermal modulus, α t is the thermal expansion coefficient, and λ and µ are Lame's constants. Thus, by substituting Equation (2) into Equation (4), it gives σ ΘΘ = σ φφ = λ ∂u ∂r The linear isotropic homogeneous thermoelastic body is governed by the following equation where volume forces are absent in this situation: • The equation of motion The equation of motion in spherical polar coordinates is introduced as a stress equation with the following form where µ 0 is electric permeability, H 0 is an initial magnetic field, ε 0 is electric permittivity, and ρ is density. Introducing the Laplacian operator in spherical coordinates by the form ∇ 2 ( * ) = ∂ 2 ( * ) ∂r 2 + 2 r ∂( * ) ∂r Hence, by applying Equations (3), (5), and (6) to Equation (7), one obtains • The heat conduction equation The heat conduction equation, in the context of the refined thermoelasticity form, is represented by a hyperbolic form as [36] (L * where L * T , L T , and L q are higher-order time-derivative operators given by and C ϑ is specific heat. Since N is an integer greater than zero; thus, Equation (10) is more generic. As a result, it gives some unique circumstances, such that (i) Coupled thermoelasticity (CTE) model [1]: τ ϑ = τ Θ = τ q = 0, K * = 0, and = 1, (ii) Lord and Shulman (L-S) model [2]: τ T = τ ϑ = 0, τ q = τ 0 , K * = 0, N = 1, and = 1, (iii) Green and Naghdi (G-N) model without energy dissipation [4][5][6]: τ T = τ ϑ = 0, τ q = 0, N = 1, and = 1, (iv) Simple generalized thermoelasticity theory with triple-phase-lag (Simple TPL-GN theory): τ q ≥ τ Θ > τ ϑ > 0, N = 1, and = 1, (v) Refined generalized thermoelasticity theory with triple-phase-lag (Refined TPL-GN theory): where K represents heat conductivity, K * represents the rate of thermal conductivity, τ q represents the phase lag of the heat flux, τ T represents the phase lag of the temperature gradient, and τ ϑ represents the phase lag of the thermal displacement.

Closed-Form Solution
It is suitable to present the following dimensionless quantities: where All of the governing equations are reformulated using the aforesaid dimensionless variables (removing the dashes for the sake of simplicity) and where Equations (22) and (23) are resolved to obtain temperature Θ and radial displacement u, which are the first two variables needed to complete the set of complete solutions. Then, the subsequent volumetric strain (dilatation) e and thermal stresses are possible to display as expressions in Θ and u. To achieve this objective, the following initial conditions are applied The thermomechanical boundary conditions were also employed in conjunction with the homogeneous initial conditions. The surface of the spherical cavity is considered to be constantly heated and traction-free and the current unbounded body will be investigated as such. It is possible to describe these conditions as:

•
Continuous heat is applied to the spherical hole's outer surface where Θ 0 is thermal constant, and H(t) represents the Heaviside unit step function.
• Due to the lack of traction on the hole's surface, the mechanical boundary condition is met In addition, the following regularity requirements are taken into account: With the homogeneous initial conditions mentioned in Equation (24), the Laplace transform is used on Equations (19)- (23), and the results are as follows: and in which s represents the Laplace parameter.
The coupled system of Equations (33) and (34) can easily give the displacement u in terms of the higher derivatives of Θ as In addition, the differential equation of temperature is expressed as where the coefficients β i are given by Equation (39), which is presented in a polar coordinate system, is difficult to solve because of this. It can be stated as follows: where ζ 2 j are the roots of These roots are given, respectively, by Equation (41) tends to the next modified Bessel equation of zero-order which can be solved with the regularity condition: u, Θ → 0 as r → ∞ . Consequently, the general solution of Equation (41), which is bounded at infinity, is provided by By substituting it into Equation (37), hence u, taking into account the regularity condition: u → 0 as r → ∞ , is given by where A j are integration parameters and Using the relation between u and e to obtain The problem has been solved up to this point. The boundary conditions from Equations (25) and (26) are sufficient to determine the two parameters A j . If the stresses are expressed in terms of radial displacement and temperature, then it is simple to give these values as Laplace-domain analytical solutions for the modified generalized G-N theory already exist. To obtain the solutions in the physical realm, the function ψ(t) is viewed as an inversion of the Laplace transform ψ(s) using the formula where L is a sufficiently large integer, i denotes the imaginary number unit, Re represents the real part, and q is an arbitrary constant. To speed up computations, there have been numerous numerical studies demonstrating that the estimate of q that satisfies the relation is qt ≈ 4.7 [37]. An inversion of the expressions of temperature Θ, radial displacement u, dilatation e, radial stress σ rr , and hoop stress σ ΘΘ can be achieved using the numerical approach mentioned.

First Justification
Tables 1-15 provide the results for all variables using various thermoelasticity models of triplephase-lag in various locations. Each model's field values were impacted by the magnetic field H 0 and electric permittivity ε 0 at a time-independent dimensionless rate of t = 0.3. Figures 2-21 show further results from an unbounded medium with a spherical hole in the radial direction. Except when otherwise noted, the numerical data in these tables were acquired for t = 0.3, µ 0 = 1.256629 × 10 −6 , H 0 = 5 × 10 8 , and ε 0 = 8.85418782 × 10 −12 .              Tables 1-5. As t increased, the dilatation e, radial displacement u, and circumferential stress σ ΘΘ decreased at all positions. Distinct observations were noticed for the radial stress σ rr . However, the temperature Θ decreased as t increased at the first position (r = 1.02), while Θ increased as t increased at the third position (r = 1.4).
The inclusion of the dimensionless magnetic field intensity H 0 at some positions in the medium are shown in Tables 6-10. As H 0 increased, the dilatation e, radial displacement u, and circumferential stress σ ΘΘ rapidly decreased at all positions. This may not occur for σ ΘΘ when r = 1.4. At r = 1.02, the radial stress σ rr decreased as H 0 increased while it increased at r = 1.4. However, σ rr no longer increased at r = 1.2 and decreased again when H 0 = 10 9 . However, the temperature Θ slowly increased as H 0 increased at r = 1.02 for different theories except for the G-N theory, in which Θ is still constant. At r = 1.2, the temperature Θ due to CTE, L-S, and RTPL theories slightly decreased as H 0 increased Otherwise, and at r = 1.2, Θ no longer decreased and increased again when H 0 = 10 9 . Generally, the temperature Θ due to the G-N theory was still constant at all discussed positions.
Finally, in this respect, the inclusion of the dimensionless electric permittivity ε 0 at distinct positions in the medium are shown in Tables 11-15. It is clear that all variables were sensitive to the inclusion of ε 0 .
According to the above-presented data, it is clear that: 1.
The RTPL models were developed with N equal to 3, 4, and 5. Nevertheless, the STPL model was essentially provided when N = 1.
The RTPL model yielded closed outcomes. All variables may be insensitive to larger values of N, particularly when N exceeds 5.

4.
The magnetic field variables, which are the electric permittivity ε 0 and magnetic field intensity H 0 , were taken into account to show their effects via all thermoelasticity theorems with various values and in different positions.

Second Justification
A time t = 0.3 was used in Figures 2-6 to demonstrate the effects of each model on the variables. Accordingly, all of the following graphs are shown about the refined triple-phase-lag (RTPL) model with N = 5 and H 0 = 50, so that all field variables are examined. 0 = 10 9 . Generally, the temperature due to the G-N theory was still constant at all discussed positions.
Finally, in this respect, the inclusion of the dimensionless electric permittivity 0 at distinct positions in the medium are shown in Tables 11-15. It is clear that all variables were sensitive to the inclusion of 0 .
According to the above-presented data, it is clear that: 1. The RTPL models were developed with equal to 3, 4, and 5. Nevertheless, the STPL model was essentially provided when = 1. 2. Using the RTPL model, incredibly accurate results were generated. 3. The RTPL model yielded closed outcomes. All variables may be insensitive to larger values of , particularly when exceeds 5. 4. The magnetic field variables, which are the electric permittivity 0 and magnetic field intensity 0 , were taken into account to show their effects via all thermoelasticity theorems with various values and in different positions.

Second Justification
A time = 0.3 was used in Figures 2-6 to demonstrate the effects of each model on the variables. Accordingly, all of the following graphs are shown about the refined triplephase-lag (RTPL) model with = 5 and 0 = 50, so that all field variables are examined.    Figure 2, the temperature oscillated along the trajectory of the RTPL model. After = 2.25, the temperature remained stable in all models, and the temperature values were near to one another. According to the STPL, L-S, and CTE models in Figure 3, the values follow the RTPL theory's vibrational trajectory exactly. This model's trajectory varies concerning the G-N model's values of . There was a noticeable shift in the values after = 1.68, after According to the STPL, L-S, and CTE models in Figure 3, the e values follow the RTPL theory's vibrational trajectory exactly. This model's trajectory varies concerning the G-N model's values of e. There was a noticeable shift in the e values after r = 1.68, after which the values were similar to each other. In Figure 4, the CTE and STPL models have radial displacements u potentially identical to that of the RTPL model, which vanished in the direction of the radial motion. To put it another way, the G-N and L-S models' displacements u may represent upper or lower limitations on the values produced by the RTPL model. According to the STPL, L-S, and CTE models in Figure 3, the values follow the RTPL theory's vibrational trajectory exactly. This model's trajectory varies concerning the G-N model's values of . There was a noticeable shift in the values after = 1.68, after which the values were similar to each other. In Figure 4, the CTE and STPL models have radial displacements potentially identical to that of the RTPL model, which vanished in the direction of the radial motion. To put it another way, the G-N and L-S models' displacements may represent upper or lower limitations on the values produced by the RTPL model.    Figure 5 shows that the RTPL model radial stress vibrated with a large amplitude around the rest of the models from = 1.06 to = 1.8. Around this location, the radial stress levels oscillated around those predicted by the RTPL theory, although with a modest amplitude, until they cease after = 2.6.
Finally, Figure 6 shows similar behaviors of the circumferential stress as those of the radial stress. It shows that the RTPL model's circumferential stress vibrated with a large amplitude around the rest of the models from = 1.06 to = 1.8. Around this location, the radial stress levels oscillated around those predicted by the RTPL theory, although with a modest amplitude, until they cease after = 2.6.  shows that the RTPL model radial stress σ rr vibrated with a large amplitude around the rest of the models from r = 1.06 to r = 1.8. Around this location, the radial stress levels oscillated around those predicted by the RTPL theory, although with a modest amplitude, until they cease after r = 2.6.
Finally, Figure 6 shows similar behaviors of the circumferential stress as those of the radial stress. It shows that the RTPL model's circumferential stress σ ΘΘ vibrated with a large amplitude around the rest of the models from r = 1.06 to r = 1.8. Around this location, the radial stress levels oscillated around those predicted by the RTPL theory, although with a modest amplitude, until they cease after r = 2.6.
In brief, based on the data shown above, it can be determined that the RTPL model produces the most straightforward results. When it comes to this problem, we will use the RTPL theory to see how various parameters affect the field variables. Figure 5. The radial stress through the radial direction of the spherical hole for all models. Figure 5 shows that the RTPL model radial stress vibrated with a large amplitude around the rest of the models from = 1.06 to = 1.8. Around this location, the radial stress levels oscillated around those predicted by the RTPL theory, although with a modest amplitude, until they cease after = 2.6. Finally, Figure 6 shows similar behaviors of the circumferential stress as those of the radial stress. It shows that the RTPL model's circumferential stress vibrated with a large amplitude around the rest of the models from = 1.06 to = 1.8. Around this location, the radial stress levels oscillated around those predicted by the RTPL theory, although with a modest amplitude, until they cease after = 2.6. Figure 6. The circumferential stress through the radial direction of a spherical hole for all models.
In brief, based on the data shown above, it can be determined that the RTPL model produces the most straightforward results. When it comes to this problem, we will use the RTPL theory to see how various parameters affect the field variables. Figure 6. The circumferential stress σ ΘΘ through the radial direction of a spherical hole for all models.  Figure 7 clearly shows that for various values of t and different wavelengths, Θ vibrated in the radial direction. When r = 1.14, the temperature Θ no longer increased and reached its maximum value. Regardless of the value of t, the temperature decreased as r increased. With time, its immense worth has begun to diminish from its former heights. Figure 7 clearly shows that for various values of and different wavelengths, vibrated in the radial direction. When = 1.14, the temperature no longer increased and reached its maximum value. Regardless of the value of , the temperature decreased as increased. It is shown in Figure 8 that the volumetric strain vibrated in the radial direction of the spherical hole with distinct frequencies and varying amplitudes. An increase in time led to an increase in the wavelength. When = 0.3, the volumetric strain initially disappeared when > 1.55 and = 0.80 . The volumetric strain finally disappeared when > 3 . When = 0.3 in Figure 9, the radial displacement increased swiftly across the radial direction of the spherical hole but increased slowly when = 0.80. It is shown in Figure 8 that the volumetric strain e vibrated in the radial direction of the spherical hole with distinct frequencies and varying amplitudes. An increase in time t led to an increase in the wavelength. When t = 0.3, the volumetric strain e initially disappeared when r > 1.55 and t = 0.80. The volumetric strain e finally disappeared when r > 3. When t = 0.3 in Figure 9, the radial displacement u increased swiftly across the radial direction of the spherical hole but increased slowly when t = 0.80. Figure 8 that the volumetric strain vibrated in the radial direction of the spherical hole with distinct frequencies and varying amplitudes. An increase in time led to an increase in the wavelength. When = 0.3, the volumetric strain initially disappeared when > 1.55 and = 0.80 . The volumetric strain finally disappeared when > 3 . When = 0.3 in Figure 9, the radial displacement increased swiftly across the radial direction of the spherical hole but increased slowly when = 0.80.  Using various values, Figure 10 shows how the RTPL model influences the radial stress in a spherical hole's radial direction. Different wavelengths were used to describe the stress , which increased in frequency as the time increased. Figure 11 shows the circumferential stress in the radial direction of the spherical cavity using the RTPL model for various values. Depending on the value of , the circumferential stress vibrated at various wavelengths and became smoother as increased. The radial and circumferential stresses, and , disappeared as time progressed, as shown in the figures. Figures 12-16 show the effects of the dimensionless magnetic field strength 0 on all variables using the RTPL model. Figure 12 depicts the effects of 0 on along the radial direction of a spherical hole. Figures 13-16 show comparable figures for the remaining variables. Figure 12 shows that the vibrated in the radial direction for various values of 0 and produced the same wavelengths with very slight differences, making them quite comparable. Figure 9. The influence of t on radial displacement u through the radial direction of the spherical hole using the RTPL model.

The Influence of Dimensionless Magnetic Field Intensity
Using various t values, Figure 10 shows how the RTPL model influences the radial stress σ rr in a spherical hole's radial direction. Different wavelengths were used to describe the stress σ rr , which increased in frequency as the time t increased. Figure 11 shows the circumferential stress σ ΘΘ in the radial direction of the spherical cavity using the RTPL model for various t values. Depending on the value of t, the circumferential stress vibrated at various wavelengths and became smoother as t increased. The radial and circumferential stresses, σ rr and σ ΘΘ , disappeared as time progressed, as shown in the figures. Figures 12-16 show the effects of the dimensionless magnetic field strength H 0 on all variables using the RTPL model. Figure 12 depicts the effects of H 0 on Θ along the radial direction of a spherical hole. Figures 13-16 show comparable figures for the remaining variables. Figure 12 shows that the Θ vibrated in the radial direction for various values of H 0 and produced the same wavelengths with very slight differences, making them quite comparable. Figures 12-16 show the effects of the dimensionless magnetic field strength all variables using the RTPL model. Figure 12 depicts the effects of 0 on al radial direction of a spherical hole. Figures 13-16 show comparable figures for the ing variables. Figure 12 shows that the vibrated in the radial direction for vari ues of 0 and produced the same wavelengths with very slight differences, maki quite comparable.   Figure 11. The influence of on circumferential stress through the radial directio spherical hole using the RTPL model. Figure 11. The influence of t on circumferential stress σ ΘΘ through the radial direction of the spherical hole using the RTPL model. Figure 13 shows that the volumetric strain e vibrated with multiple frequencies and varied amplitudes along the radial direction of the spherical hole. An increase in H 0 led to an increase in wavelength. When H 0 = 0, the volumetric strain e first dissipated when r > 1. 35 and H 0 = 10 9 , and eventually disappeared when r > 2. In Figure 14, when H 0 = 0, the radial displacement u increased rapidly throughout the radial direction of the spherical hole, but it increased slowly when H 0 = 10 9 . As H 0 increased, the radial displacement u may have vanished. Figure 11. The influence of on circumferential stress through the radial direction of the spherical hole using the RTPL model. Figure 12. The influence of 0 on temperature through the radial direction of a spherical hole using the RTPL model. Figure 13 shows that the volumetric strain vibrated with multiple frequencies and varied amplitudes along the radial direction of the spherical hole. An increase in 0 led to an increase in wavelength. When 0 = 0, the volumetric strain first dissipated when > 1.35 and 0 = 10 9 , and eventually disappeared when > 2. In Figure 14, when 0 = 0, the radial displacement increased rapidly throughout the radial direction of the spherical hole, but it increased slowly when 0 = 10 9 . As 0 increased, the radial dis placement may have vanished.  Figure 15 illustrates the impact of the RTPL model on the radial stress in a spher ical hole's radial direction for a variety of 0 values. Different wavelengths were em ployed to characterize the radial stress , which increased in frequency as 0 increased Figure 16 depicts the circumferential stress in the radial direction of the spherica cavity as calculated by the RTPL model for a range of 0 values. Depending on the value, the circumferential stress vibrated at different wavelengths and became increas ingly smooth as 0 increased. As shown by the graphs, the radial and circumferentia stresses, denoted by and , vanished as 0 advanced. Figure 13. The influence of H 0 on volumetric strain e through the radial direction of the spherical hole using the RTPL model. Figure 15 illustrates the impact of the RTPL model on the radial stress σ rr in a spherical hole's radial direction for a variety of H 0 values. Different wavelengths were employed to characterize the radial stress σ rr , which increased in frequency as H 0 increased. Figure 16 depicts the circumferential stress σ ΘΘ in the radial direction of the spherical cavity as calculated by the RTPL model for a range of H 0 values. Depending on the H 0 value, the circumferential stress vibrated at different wavelengths and became increasingly smooth as H 0 increased. As shown by the graphs, the radial and circumferential stresses, denoted by σ rr and σ ΘΘ , vanished as H 0 advanced. ployed to characterize the radial stress , which increased in frequency as 0 increased Figure 16 depicts the circumferential stress in the radial direction of the spherica cavity as calculated by the RTPL model for a range of 0 values. Depending on the 0 value, the circumferential stress vibrated at different wavelengths and became increasingly smooth as 0 increased. As shown by the graphs, the radial and circumferentia stresses, denoted by and , vanished as 0 advanced. Figure 14. The influence of 0 on radial displacement through the radial direction of the spher ical hole using the RTPL model.

The Influence of Dimensionless Electric Permittivity
The effects of dimensionless electric permittivity 0 on all variables, as predicted by the RTPL model ( 0 = 50), are depicted in Figures 17-21. The effects of 0 on in the radial direction of a spherical hole are shown in Figure 17. Figures 18-21 provide analogous data for the other variables. Figure 17 demonstrates that the vibrated in the radia direction for a variety of 0 values and exhibited wavelengths that are almost identica with minor variations, making them similar.

The Influence of Dimensionless Electric Permittivity
The effects of dimensionless electric permittivity ε 0 on all variables, as predicted by the RTPL model (H 0 = 50), are depicted in Figures 17-21. The effects of ε 0 on Θ in the radial direction of a spherical hole are shown in Figure 17. Figures 18-21 provide analogous data for the other variables. Figure 17 demonstrates that the Θ vibrated in the radial direction for a variety of ε 0 values and exhibited wavelengths that are almost identical with minor variations, making them similar.
The effects of dimensionless electric permittivity 0 on all variables, as predicted by the RTPL model ( 0 = 50), are depicted in Figures 17-21. The effects of 0 on in the radial direction of a spherical hole are shown in Figure 17. Figures 18-21 provide analogous data for the other variables. Figure 17 demonstrates that the vibrated in the radia direction for a variety of 0 values and exhibited wavelengths that are almost identica with minor variations, making them similar. Figure 16. The influence of 0 on circumferential stress through the radial direction of the spherical hole using the RTPL model. According to Figure 18, the spherical hole's volumetric strain vibrated in the radia direction with different frequencies and varying amplitudes. The wavelength decrease as 0 increased. In both cases, when 0 = 0, the volumetric strain initially reduced a > 1.7 and 0 = 0.2, and finally vanished at > 1.22. 0 increased the maximum valu of and became too close to the boundary layer, making 0 a more desirable parameter On the spherical hole in Figure 19, the radial displacement increased slowly when 0 = 0 and quickly when 0 = 0.2. As 0 grew, vanished. According to Figure 18, the spherical hole's volumetric strain e vibrated in the radial direction with different frequencies and varying amplitudes. The wavelength decreased as ε 0 increased. In both cases, when ε 0 = 0, the volumetric strain e initially reduced at r > 1.7 and ε 0 = 0.2, and finally vanished at r > 1. 22. ε 0 increased the maximum value of e and became too close to the boundary layer, making ε 0 a more desirable parameter. On the spherical hole in Figure 19, the radial displacement u increased slowly when ε 0 = 0 and quickly when ε 0 = 0.2. As ε 0 grew, u vanished. direction with different frequencies and varying amplitudes. The wavelength de as 0 increased. In both cases, when 0 = 0, the volumetric strain initially red > 1.7 and 0 = 0.2, and finally vanished at > 1.22. 0 increased the maximu of and became too close to the boundary layer, making 0 a more desirable par On the spherical hole in Figure 19, the radial displacement increased slowly wh 0 and quickly when 0 = 0.2. As 0 grew, vanished.   Figure 20 to illustrate the effect of the RTPL on the radial stress in a spherical hole's radial direction. A wide range of wave was used to study how radial stress changed with increasing values of 0 . Fi shows the circumferential stress , as determined by an iterative application RTPL model, in a spherical hole's radial direction with varied 0 values. The cir ential stress vibrated at different frequencies depending on the value of 0 . It can from the figures that as 0 increased, the radial and circumferential stresses, a were diminished, but their maximum values increased.  Figure 20 to illustrate the effect of the RTPL model on the radial stress σ rr in a spherical hole's radial direction. A wide range of wavelengths was used to study how radial stress σ rr changed with increasing values of ε 0 . Figure 21 shows the circumferential stress σ ΘΘ , as determined by an iterative application of the RTPL model, in a spherical hole's radial direction with varied ε 0 values. The circumferential stress vibrated at different frequencies depending on the value of ε 0 . It can be seen from the figures that as ε 0 increased, the radial and circumferential stresses, σ rr and σ ΘΘ , were diminished, but their maximum values increased. was used to study how radial stress changed with increasing values of 0 . Figure 21 shows the circumferential stress , as determined by an iterative application of the RTPL model, in a spherical hole's radial direction with varied 0 values. The circumfer ential stress vibrated at different frequencies depending on the value of 0 . It can be seen from the figures that as 0 increased, the radial and circumferential stresses, and were diminished, but their maximum values increased.

Conclusions
The new revised triple-phase-lag model is innovative and precise in terms of temper ature, volumetric strain, displacement, and stresses. The heat equation with multi-tim derivatives was explained. The thermoelastic coupling behavior of an infinite medium with a spherical cavity due to uniform heat was studied using spherical coordinates. Th paired dynamical thermoelasticity model, the Lord and Shulman model, the Green and Naghdi model without energy dissipation, and a simple triple-phase-lag model were used to create a unified model. As a result of solving the two high-order time-derivative differ ential coupled equations, the thermoelastic coupling response of an infinite material with a spherical hole was produced. Several examples and applications were provided to com pare the results of all models, regardless of whether or not they were subject to Lorent Forces effects. As an example, a spherical hole was used to show the correlations between several factors. Tables have been supplied to serve as benchmarks for future comparison by other scholars, as shown in the following instances. The disclosed and verified out Figure 21. The influence of ε 0 on circumferential stress σ ΘΘ through the radial direction of the spherical hole using the RTPL model.

Conclusions
The new revised triple-phase-lag model is innovative and precise in terms of temperature, volumetric strain, displacement, and stresses. The heat equation with multi-time derivatives was explained. The thermoelastic coupling behavior of an infinite medium with a spherical cavity due to uniform heat was studied using spherical coordinates. The paired dynamical thermoelasticity model, the Lord and Shulman model, the Green and Naghdi model without energy dissipation, and a simple triple-phase-lag model were used to create a unified model. As a result of solving the two high-order time-derivative differential coupled equations, the thermoelastic coupling response of an infinite material with a spherical hole was produced. Several examples and applications were provided to compare the results of all models, regardless of whether or not they were subject to Lorentz Forces effects. As an example, a spherical hole was used to show the correlations between several factors. Tables have been supplied to serve as benchmarks for future comparisons by other scholars, as shown in the following instances. The disclosed and verified outcomes revealed that all field variables and dimensionless temporal parameters behaved differently from what had been assumed beforehand. Because of the triple-phase-lag theory, the magnitudes of several variables may be reduced in practical implementations. As long as the L-S model was used, the results were accurate. In contrast, the updated model yielded more precise results.