Magneto-Thermoelastic Response in an Unbounded Medium Containing a Spherical Hole via Multi-Time-Derivative Thermoelasticity Theories

This article introduces magneto-thermoelastic exchanges in an unbounded medium with a spherical cavity. A refined multi-time-derivative dual-phase-lag thermoelasticity model is applied for this reason. The surface of the spherical hole is considered traction-free and under both constant heating and external magnetic field. A generalized magneto-thermoelastic coupled solution is developed utilizing Laplace’s transform. The field variables are shown graphically and examined to demonstrate the impacts of the magnetic field, phase-lags, and other parameters on the field quantities. The present theory is examined to assess its validity including comparison with the existing literature.


Introduction
The thermoelastic responses of different structures with spherical cavities have received much attention because of their usefulness in many industrial applications. In the following, we restrict our attention to the application of continuums with spherical cavities. All the problems discussed are concerned with thermoelastic exchanges within the framework of several generalized thermoelasticity theories.
Generalized thermoelasticity models, with one or more relaxation times, have been proposed to modify the heat conduction equation. One of the original forms of the heat conduction equation, associated with gases theory, was introduced by Maxwell [1]. Another form was proposed within the framework of heat conduction in rigid structures by Cattaneo [2]. A third form was introduced by Dhaliwal and Sherief [3] by extension to the case of an anisotropic medium. To overcome the contradiction of an endless velocity of thermal waves intrinsic to classical coupled thermoelasticity (CTE) theory [4], attempts have been made by various investigators, for a range of reasons, to modify coupled thermoelasticity to entail a wave-type heat conduction equation.
Lord and Shulman (L-S) [5] developed generalized thermoelasticity theory presenting one relaxation time in Fourier's law of heat conduction equation and therefore converting it into a hyperbolic type. Banerjee and Roychoudhuri [6] discussed the generalized theory of thermo-elasticity suggested by L-S [5] to examine thermo-visco-elastic wave propagation in an unlimited viscoelastic body of Kelvin-Voight type with a spherical hole. Sinha and Elsibai [7] discussed thermoelastic exchanges in an unlimited solid with a spherical inclusion considering L-S and G-L theories. Rakshit Kundu and Mukhopadhyay [8] described field variables in a viscoelastic body with a spherical hole. Youssef [9] described a problem of thermoelastic exchanges in a limitless body including a spherical hole subjected to a moving heat source according to L-S theory. Elhagary [10] described the problem of a thermoelastic unbounded solid including a spherical hole in the framework of L-S diffusion theory. Karmakar et al. [11] determined the temperatures, stress, displacement, and strain in an unbounded solid including a spherical hole in the framework of processes addressed by two-temperature theory (2TT).
Later, Green-Naghdi (G-N) [12][13][14] created three versions for generalized thermoelasticity that were identified as I, II, and III types. Mukhopadhyay [15,16] presented thermoelastic exchanges in an unbounded solid including a spherical hole in the framework of G-N theory. Mukhopadhyay and Kumar [17] considered thermoelastic exchanges in an infinite solid with a spherical hole in the framework of several theories. Allam et al. [18] investigated electro-magneto-thermoelastic exchanges in an infinite solid with a spherical hole in the framework of G-N theory. Banik and Kanoria [19] determined the thermoelastic quantities in an infinite solid with a spherical inclusion in the framework of the 2TT. Abbas [20] investigated a general solution to the field equations of 2TT in an unbounded medium with a spherical hole in the framework of the G-N model. Bera et al. [21] investigated the waves arising from the boundary of a spherical cavity in an infinite medium. Biswas [22] examined the thermoelastic exchange in a limitless body including a spherical cavity in the context of the G-N model. Chandrasekharaiah and Narasimha Murthy [23] considered thermoelastic exchanges in an infinite body including a spherical inclusion.
Green and Lindsay [24] pioneered an additional theory, known as the G-L model, that included two relaxation times. Roy Choudhuri and Chatterjee [25] studied spherically symmetric thermoelastic waves in an unbounded body containing a spherical hole. Sherief and Darwish [26] presented a problem of a thermoelastic unbounded solid containing a spherical hole in the framework of thermoelasticity theory with two relaxation times. Mukhopadhyay [27] discussed thermally induced vibrations of an unbounded viscoelastic body including a spherical hole in the framework of G-L theory. Ghosh and Kanoria [28] determined thermoelastic quantities in a functionally graded (FG) spherically unbonded body including a spherical hole in the framework of G-L theory. Kanoria and Ghosh [29] examined thermoelastic exchanges in an FG hollow sphere in the framework of the G-L model. Das and Lahiri [30] considered a thermoelastic problem for an unbounded FG and temperature-dependent spherical inclusion in the framework of G-L theory.
Many investigators have used dual/triple-phase-lag (D/TPL) heat transfer theory to examine thermoelastic exchanges in unbounded mediums including spherical cavities. DPL theory was originally presented by Tzou [31,32] to describe some problems at a macroscopic scale. Abouelregal and Abo-Dahab [33] presented thermal quantities in an unbounded solid with a spherical hole in the framework of DPL theory. Hobiny and Abbas [34] applied DPL theory in the examination of photo-thermal exchanges in an infinite solid containing a spherical cavity. Mondal and Sur [35] studied a coupled problem in an infinite solid with a spherical hole in the framework of a photothermal transport process in relation to 2TT. Singh and Sarkar [36] examined thermoelastic exchange in a 2TT unbounded isotropic body containing a spherical cavity in the framework of a memory-dependent derivative (MDD). Comparisons were made graphically between the 2T TPL theory and 2T L-S theory with MDD. Many researchers have dealt with one-dimensional (1D) problems in generalized thermoelasticity in unbounded mediums with spherical cavities [37][38][39][40][41][42][43][44].
In the current article, magneto-thermoelastic exchanges in an infinite solid with a spherical hole are studied with respect to multi-time-derivative thermoelasticity theories [45][46][47][48][49][50][51][52][53]. A refined DPL model is used for this purpose. The technique of Laplace transforms in the time domain is applied to obtain the governing equations analytically. The derived equations are solved and then Laplace inversion is carried out to obtain the field quantities numerically. For verification proposes, the outcomes are compared with those obtained previously. Additional results are presented graphically and others are reported for future comparison.

Basic Equations
Let us be concerned with thermoelastic analysis of an isotropic body including a spherical cavity of radius R based on unified multi-phase-lag theory. It is assumed that the outer edge of the spherical cavity is traction-free and subjected to harmonically varying heat (See Figure 1). The spherical cavity coordinate system (r,θ,φ) is used to address the present problem.

Basic Equations
Let us be concerned with thermoelastic analysis of an isotropic body including a spherical cavity of radius based on unified multi-phase-lag theory. It is assumed that the outer edge of the spherical cavity is traction-free and subjected to harmonically varying heat (See Figure 1). The spherical cavity coordinate system ( , , ) is used to address the present problem. The governing equations for a linear isotropic homogeneous thermoelastic body in the absence of volume forces are given by:

•
The equations of motion: • The constitutive equations: where and are the stresses and strains and denotes Kronecker's delta tensor.
• The heat conduction equation: is considered in the context of the refined thermoelasticity form in which and denote the following higher-order time-derivative operators: Equation (3) with the aid of Equation (2) are the more general ones when has numerous integers more than zero. Some specific cases may be achieved as  The governing equations for a linear isotropic homogeneous thermoelastic body in the absence of volume forces are given by:

•
The equations of motion: • The constitutive equations: where σ ij and e ij are the stresses and strains and δ ij denotes Kronecker's delta tensor.

•
The heat conduction equation: is considered in the context of the refined thermoelasticity form in which L T and L q denote the following higher-order time-derivative operators: Equation (3) with the aid of Equation (2) are the more general ones when N has numerous integers more than zero. Some specific cases may be achieved as 4 of 18 (i) Dynamical coupled thermoelasticity (CTE) model [4]: τ T = τ q = 0 and = 1, (ii) Lord and Shulman (L-S) model [5]: τ T = 0, τ q = τ 0 and = 1, (iii) Green and Naghdi (G-N) model without energy dissipation [12][13][14]: (iv) The simple dual-phase-lag (SDPL) model [50][51][52]: τ q ≥ τ T > 0, = 1 and N = 1, (v) The refined with dual-phase-lag (RDPL) model [50][51][52]: N > 1, τ q ≥ τ T > 0, and = 1, The displacements of the present, axially symmetric spherical medium are summarized as The non-vanishing strains and volumetric strain can be expressed as Thus, the volumetric strain e has the form e = e rr + e θθ + e φφ = ∂u ∂r The constitutive equations for the spherical symmetric system can be stated as Applying the operator (∂/∂r + 2/r) to both sides of Equation (15), one gets in which ∇ 2 denotes the Laplacian operator in spherical coordinates. It meets the formulation

Formulation of the Problem
It is proper to establish the non-dimensional variables in the following parts: The whole governing equations, with the above dimensionless variables, are diminished to (throwing down the dash for convenience) where

Closed-Form Solution
The comprehensive solutions are provided by resolving Equations (21) and (22) to obtain, firstly, temperature Θ and volumetric strain (dilatation) e. Then, the subsequent radial displacement and thermal stresses may be presented as functions of Θ and e. For this objective, we will first employ the next initial conditions: In adding together to the above homogenous initial conditions, we also used the thermomechanical boundary conditions. The current unbounded body will be studied as quiescent and the surface of the spherical cavity is assumed to be exposed to constant heat and traction free. Such conditions can be explained as

•
The surface of the spherical hole is subjected to a constant heat • The mechanical boundary condition is respected as the surface of the spherical hole is traction free Moreover, we take into consideration the following regularity conditions The Laplace transform is carried out for Equations (19)- (22), and, with the homogeneous initial conditions that appeared in Equation (24), one gets: The system of equations provided in Equations (30) and (31) can be indicated in the differential equation where the coefficients β i are given by and the temperature Θ is reformed as follows Equation (33) is very complicated since it is presented in a polar coordinate system. It can be expressed as where ζ 2 j are the roots of These roots ζ j are given, respectively, by Equation (36) tends to the next modified Bessel's equation of zero-order which has a solution under the regularity conditions: u, Θ → 0 as r → ∞ . Therefore, the general solution of Equations (35) and (39), that is bounded at infinity, is provided by where B j are integration parameters and Using the relation between u and e e(r) = Du(r), D = d dr one can pick up the solution for the dimensionless form of radial displacement pretending that u disappears at infinity as: whereζ Up to here, the solution is finished. It is as much as needed to establish the two parameters B j with the aid of the boundary conditions given in Equations (25) and (26). So, one gets Therefore, the current analytical solution is already provided for the modified formulations in Laplace space. To achieve the solution in the basic time-space one can consider a function ψ(t) as an inversion of the Laplace function ψ(s) in the form where p is an arbitrary constant, Re is the real part, i suggests the imagined number unit and L denotes a sufficiently big integer. For faster combination, various numerical analyses have shown that the approximation of p fulfills the connection pt ≈ 4.7 [35]. The numerical procedure cited is used to invert the terms of temperature Θ, radial displacement u, volumetric strain e, radial stress σ 1 , and circumferential stress σ 2 .

First Justification
The outcomes for all variables using various thermoelasticity models of dual-phaselag are presented in Tables 1-5    The outcomes described in Tables 1-5 are offered as benchmarks for other researchers. It is evident from the tabulated results that:

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The G-N model provides the lowest absolute value of all variables. It may vanish at some positions.

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The other CTE and L-S models provide appropriate outcomes for all variables. For the RDPL model the temperature, displacement, and circumferential stress slightly increase as the value of N increases, while volumetric strain, radial stress, and circumferential stress slightly decrease. All variables may be insensitive to the higher values of N especially when N ≥ 5. The discrepancy of the temperature through radial direction of a spheri corresponding to all models is produced in Figure 2. Similar figures for the rem variables are presented in Figures 3-6. Figure 2 reveals that the temperature du CTE, L-S, and SDPL models vibrates across the trajectory of the RDPL model, wh perature due to the G-N model vibrates below the trajectory of the RDPL mod temperature according to the G-N model may vanish earlier than the temperature ing to other models. Figure 3 reveals that the values of ̅ of SDPL, L-S, and CTE models vibrate cally to the trajectory of the RDPL theory. While the value of ̅ for the G-N model across and below the trajectory of the RDPL model.  The discrepancy of the temperature Θ through radial direction of a spherical hole corresponding to all models is produced in Figure 2. Similar figures for the remaining variables are presented in Figures 3-6. Figure 2 reveals that the temperature due to the CTE, L-S, and SDPL models vibrates across the trajectory of the RDPL model, while temperature due to the G-N model vibrates below the trajectory of the RDPL model. The temperature according to the G-N model may vanish earlier than the temperature according to other models. The discrepancy of the temperature through radial direction of a spher corresponding to all models is produced in Figure 2. Similar figures for the re variables are presented in Figures 3-6. Figure 2 reveals that the temperature du CTE, L-S, and SDPL models vibrates across the trajectory of the RDPL model, wh perature due to the G-N model vibrates below the trajectory of the RDPL mo temperature according to the G-N model may vanish earlier than the temperature ing to other models. Figure 3 reveals that the values of ̅ of SDPL, L-S, and CTE models vibrat cally to the trajectory of the RDPL theory. While the value of ̅ for the G-N model across and below the trajectory of the RDPL model.       Figure 5 reveals that the radial stress σ 1 of the G-N model may rapidly vanish through the radial direction when r > 1.1. The radial stress of the L-S model vibrates around the RDPL model with wide amplitude, then it also vanishes when r > 1.24. The other CTE and SDPL theories give radial stresses that vibrate around those of the RDPL theory but with small amplitude. Finally, Figure 6 shows similar behaviors of circumferential stress as those of the radial stress. It shows that the circumferential stress σ 2 of the G-N model may rapidly vanish through the radial direction when r > 1.1. The radial stress of the L-S model vibrates around the RDPL theory with wide amplitude, then it also vanishes when r > 1.24. The other CTE and SDPL theories give radial stresses that vibrate around those of the RDPL theory but with small amplitude.

Second Justification
It is concluded from the above figures that the outcomes of the RDPL model are the most straightforward. So, we restrict our attention to using this theory for yielding the outcomes of this problem considering the effect of various parameters on the field variables.

The Influence of Dimensionless Time
The outcomes of dimensionless time t on all variables due to the RDPL model are presented in Figures 7-11. Figure 7 reveals the effects of t on Θ through radial direction of a spherical hole. Similar figures for the remaining variables are presented in Figures 8-11. It is clear in Figure 7 that Θ vibrates through the radial direction for various values of t with different wavelengths. The temperature Θ no longer increases and has its highest values when r = 1.04. The temperature vanishes as r increases, irrespective of the values of t. Figure 8 reveals that the volumetric strain e vibrates through the radial direction of a spherical hole with different amplitudes and different wavelengths. The wavelength increases as t increases. For t = 0.02 the volumetric strain e firstly vanishes when r > 1.024, while for t = 0.05, the volumetric strain e finally vanishes when r > 1.06. In Figure 9, the radial displacement u rapidly increases through the radial direction of the spherical hole when t = 0.02, while u slowly increases when t = 0.03. u is slowly decreasing when t = 0.05 It is obvious that the radial displacement u increases with increase in dimensionless time t at fixed positions. Figure 4. The radial displacement through radial direction of spherical hole p models. Figure 5 reveals that the radial stress of the G-N model may r through the radial direction when > 1.1. The radial stress of the L-S m around the RDPL model with wide amplitude, then it also vanishes when other CTE and SDPL theories give radial stresses that vibrate around thos theory but with small amplitude. Finally, Figure 6 shows similar behaviors of circumferential stress as t dial stress. It shows that the circumferential stress of the G-N model ma ish through the radial direction when > 1.1. The radial stress of the L-S m around the RDPL theory with wide amplitude, then it also vanishes when other CTE and SDPL theories give radial stresses that vibrate around thos theory but with small amplitude. It is concluded from the above figures that the outcomes of the RDPL most straightforward. So, we restrict our attention to using this theory fo of a spherical hole. Similar figures for the remaining variables are presented in Fig  11. It is clear in Figure 7 that vibrates through the radial direction for various v with different wavelengths. The temperature no longer increases and has its values when = 1.04. The temperature vanishes as increases, irrespective of the of .    The radial stress σ 1 through the radial direction of spherical hole due to the RDPL model is described in Figure 10 for various values of t. The radial stress σ 1 oscillates on a very small scale, then increases when t = 0.03 and 0.05, while it decreases when t = 0.02. At any fixed position, the radial stress σ 1 increases with increase in the dimensionless time t. The circumferential stress σ 2 is drawn through the radial direction of the spherical cavity utilizing the RDPL model in Figure 11 for distinctive values of t. It vibrates over a very small range, then it increases for t = 0.02, but decreases when t = 0.03 and 0.05. At any fixed position, the circumferential stress σ 2 increases with increase in the dimensionless time t. Figure 8. The influence of on volumetric strain ̅ through radial direction of sphe RDPL model. Figure 9. The influence of on radial displacement ̅ through radial direction o using RDPL model.
The radial stress ̅ 1 through the radial direction of spherical hole du model is described in Figure 10 for various values of . The radial stress ̅ a very small scale, then increases when = 0.03 and 0.05, while it decrea 0.02. At any fixed position, the radial stress ̅ 1 increases with increase in t less time . The circumferential stress ̅ 2 is drawn through the radial di  Figure 11 for distinctive valu brates over a very small range, then it increases for = 0.02, but decreases w and 0.05. At any fixed position, the circumferential stress increases wi the dimensionless time .

Conclusions
The present refined dual-phase-lag model is innovative and produce sults for variables such as temperature, volumetric strain, displacement, and multi-time derivatives heat equation was explained. The constitutive relatio coordinates were considered to examine the thermoelastic coupling behavio medium with a spherical cavity due to a uniform heat. To create a unified m combine other models, including the coupled dynamical thermoelasticity m and Shulman model, the Green and Naghdi model without energy dissipat Figure 11. The influence of t on circumferential stress σ 2 through radial direction of spherical hole using RDPL model.

Conclusions
The present refined dual-phase-lag model is innovative and produces accurate results for variables such as temperature, volumetric strain, displacement, and stresses. The multi-time derivatives heat equation was explained. The constitutive relations of spherical coordinates were considered to examine the thermoelastic coupling behavior of an infinite medium with a spherical cavity due to a uniform heat. To create a unified model, one can combine other models, including the coupled dynamical thermoelasticity model, the Lord and Shulman model, the Green and Naghdi model without energy dissipation, as well as a simple dual-phase-lag model. The system of two high-time-derivative differential coupled equations was solved, and all field variables were developed for the thermoelastic coupling response of an infinite medium with a spherical hole. Various confirmation examples and applications were offered to compare the outcomes due to all models with the refined ones. A sample set of graphs were presented to demonstrate relationships of variables through radial direction of a spherical hole. Some tables have been provided as confirmation examples to provide benchmark outcomes for future comparisons by other researchers. The described and demonstrated outcomes revealed different behaviors of all field variables and dimensionless time parameters. The present dual-phase-lag theory diminished the magnitudes of the examined variables, which may be significant in some practical applications. The G-N model provided appropriate outcomes over a small range. However, the refined model produced improved and exact outcomes.