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

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Problem

- The equation of motion

- The heat conduction equation

- (i)
- Coupled thermoelasticity (CTE) model [1]: ${\tau}_{\vartheta}={\tau}_{\theta}={\tau}_{q}=0$, ${K}^{\ast}=0$, and $\varrho =1$,

- (ii)
- Lord and Shulman (L–S) model [2]: ${\tau}_{T}={\tau}_{\vartheta}=0$, ${\tau}_{q}={\tau}_{0}$, ${K}^{\ast}=0$, $N=1$, and $\varrho =1$,

- (iii)

- (iv)
- Simple generalized thermoelasticity theory with triple-phase-lag (Simple TPL-GN theory): ${\tau}_{q}\ge {\tau}_{\theta}>{\tau}_{\vartheta}>0$, $N=1$, and $\varrho =1$,

- (v)
- Refined generalized thermoelasticity theory with triple-phase-lag (Refined TPL-GN theory): ${\tau}_{q}\ge {\tau}_{T}\ge {\tau}_{\vartheta}>0$, $N>1$, and $\varrho =1$

## 3. Closed-Form Solution

- Continuous heat is applied to the spherical hole’s outer surface

- Due to the lack of traction on the hole’s surface, the mechanical boundary condition is met

## 4. Validation

#### 4.1. First Justification

- The RTPL models were developed with $N$ equal to 3, 4, and 5. Nevertheless, the STPL model was essentially provided when $N=1$.
- Using the RTPL model, incredibly accurate results were generated.
- The RTPL model yielded closed outcomes. All variables may be insensitive to larger values of $N$, particularly when $N$ exceeds 5.
- The magnetic field variables, which are the electric permittivity ${\epsilon}_{0}$ 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.

#### 4.2. Second Justification

**Figure 2.**The temperature $\Theta $ through the radial direction of the spherical hole for all models.

**Figure 3.**The volumetric strain $e$ through the radial direction of the spherical hole for all models.

**Figure 4.**The radial displacement $u$ through the radial direction of the spherical hole for all models.

**Figure 5.**The radial stress ${\sigma}_{rr}$ through the radial direction of the spherical hole for all models.

**Figure 6.**The circumferential stress ${\sigma}_{\theta \theta}$ through the radial direction of a spherical hole for all models.

#### 4.3. The Influence of Dimensionless Time

**Figure 7.**The influence of $t$ on temperature $\Theta $ through the radial direction of a spherical hole using the RTPL model.

**Figure 8.**The influence of $t$ on volumetric strain $e$ through the radial direction of the spherical hole using the RTPL model.

**Figure 9.**The influence of $t$ on radial displacement $u$ through the radial direction of the spherical hole using the RTPL model.

#### 4.4. The Influence of Dimensionless Magnetic Field Intensity

**Figure 10.**The influence of $t$ on radial stress ${\sigma}_{rr}$ through the radial direction of the spherical hole using the RTPL model.

**Figure 11.**The influence of $t$ on circumferential stress ${\sigma}_{\theta \theta}$ through the radial direction of the spherical hole using the RTPL model.

**Figure 12.**The influence of ${H}_{0}$ on temperature $\Theta $ through the radial direction of a spherical hole using the RTPL model.

**Figure 13.**The influence of ${H}_{0}$ on volumetric strain $e$ through the radial direction of the spherical hole using the RTPL model.

**Figure 14.**The influence of ${H}_{0}$ on radial displacement $u$ through the radial direction of the spherical hole using the RTPL model.

**Figure 15.**The influence of ${H}_{0}$ on radial stress ${\sigma}_{rr}$ through the radial direction of the spherical hole using the RTPL model.

#### 4.5. The Influence of Dimensionless Electric Permittivity

**Figure 16.**The influence of ${H}_{0}$ on circumferential stress ${\sigma}_{\theta \theta}$ through the radial direction of the spherical hole using the RTPL model.

**Figure 17.**The influence of ${\epsilon}_{0}$ on temperature $\Theta $ through the radial direction of a spherical hole using the RTPL model.

**Figure 18.**The influence of ${\epsilon}_{0}$ on volumetric strain $e$ through the radial direction of the spherical hole using the RTPL model.

**Figure 19.**The influence of ${\epsilon}_{0}$ on radial displacement $u$ through the radial direction of the spherical hole using the RTPL model.

**Figure 20.**The influence of ${\epsilon}_{0}$ on radial stress ${\sigma}_{rr}$ through the radial direction of the spherical hole using the RTPL model.

**Figure 21.**The influence of ${\epsilon}_{0}$ on circumferential stress ${\sigma}_{\theta \theta}$ through the radial direction of the spherical hole using the RTPL model.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Biot, M.A. Thermoelasticity and Irreversible Thermodynamics. J. Appl. Phys.
**1956**, 27, 240–253. [Google Scholar] [CrossRef] - Lord, H.; Shulman, Y. A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids
**1967**, 15, 299–309. [Google Scholar] [CrossRef] - Green, A.E.; Lindsay, K.A. Thermoelasticity. J. Elast.
**1972**, 2, 1–7. [Google Scholar] [CrossRef] - Green, A.E.; Naghdi, P.M. A re-examination of the basic postulates of thermomechanics. Proc. Roy. Soc. Lond. A
**1991**, 432, 171–194. [Google Scholar] [CrossRef] - Green, A.E.; Naghdi, P.M. On undamped heat waves in an elastic solid. J. Therm. Stress.
**1992**, 15, 253–264. [Google Scholar] [CrossRef] - Green, A.E.; Naghdi, P.M. Thermoelasticity without energy dissipation. J. Elast.
**1993**, 31, 189–208. [Google Scholar] [CrossRef] - Hetnarski, R.B.; Ignaczak, J. Generalized thermoelasticity. J. Therm. Stress.
**1999**, 22, 451–476. [Google Scholar] - Tzou, D.Y. A Unified Field Approach for Heat Conduction From Macro- to Micro-Scales. J. Heat Transf.
**1995**, 117, 8–16. [Google Scholar] [CrossRef] - Chandrasekharaiah, D.S. Thermoelasticity with Second Sound: A Review. Appl. Mech. Rev.
**1986**, 39, 355–376. [Google Scholar] [CrossRef] - Chandrasekharaiah, D.S. Hyperbolic Thermoelasticity: A Review of Recent Literature. Appl. Mech. Rev.
**1998**, 51, 705–729. [Google Scholar] [CrossRef] - Choudhuri, S.K.R. On A Thermoelastic Three-Phase-Lag Model. J. Therm. Stress.
**2007**, 30, 231–238. [Google Scholar] [CrossRef] - Quintanilla, R.; Racke, R. A note on stability in three-phase-lag heat conduction. Int. J. Heat Mass Transf.
**2008**, 51, 24–29. [Google Scholar] [CrossRef] - Quintanilla, R. A Well-Posed Problem for the Three-Dual-Phase-Lag Heat Conduction. J. Therm. Stress.
**2009**, 32, 1270–1278. [Google Scholar] [CrossRef] - El-Karamany, A.S.; Ezzat, M.A. On the phase-lag Green–Naghdi thermoelasticity theories. Appl. Math. Model.
**2016**, 40, 5643–5659. [Google Scholar] [CrossRef] - Zenkour, A.M. Refined two-temperature multi-phase-lags theory for thermomechanical response of microbeams using the modified couple stress analysis. Acta Mech.
**2018**, 229, 3671–3692. [Google Scholar] [CrossRef] - Zenkour, A.M. Refined microtemperatures multi-phase-lags theory for plane wave propagation in thermoelastic medium. Results Phys.
**2018**, 11, 929–937. [Google Scholar] [CrossRef] - Zenkour, A.M. Refined multi-phase-lags theory for photothermal waves of a gravitated semiconducting half-space. Compos. Struct.
**2019**, 212, 346–364. [Google Scholar] [CrossRef] - Zenkour, A.M. Effect of thermal activation and diffusion on a photothermal semiconducting half-space. J. Phys. Chem. Solids
**2019**, 132, 56–67. [Google Scholar] [CrossRef] - Zenkour, A.M. Wave propagation of a gravitated piezo-thermoelastic half-space via a refined multi-phase-lags theory. Mech. Adv. Mater. Struct.
**2020**, 27, 1923–1934. [Google Scholar] [CrossRef] - Zenkour, A.M. Magneto-thermal shock for a fiber-reinforced anisotropic half-space studied with a refined multi-dual-phase-lag model. J. Phys. Chem. Solids
**2020**, 137, 109213. [Google Scholar] [CrossRef] - Mukhopadhyay, S.; Kothari, S.; Kumar, R. On the representation of solutions for the theory of generalized thermoelasticity with three phase lags. Acta Mech.
**2010**, 214, 305–314. [Google Scholar] [CrossRef] - Kumar, R.; Mukhopadhyay, S. Effects of Three Phase Lags on Generalized Thermoelasticity for an Infinite Medium with a Cylindrical Cavity. J. Therm. Stress.
**2009**, 32, 1149–1165. [Google Scholar] [CrossRef] - Mukhopadhyay, S.; Kumar, R. Analysis of phase-lag effects on wave propagation in a thick plate under axisymmetric temperature distribution. Acta Mech.
**2010**, 210, 331–344. [Google Scholar] [CrossRef] - Knopoff, L. The interaction between elastic wave motions and a magnetic field in electrical conductors. J. Geophys. Res. Earth Surf.
**1955**, 60, 441–456. [Google Scholar] [CrossRef] - Chadwick, P. Elastic wave propagation in a magnetic field. In Proceedings of the 9th International Congress of Applied Mechanics, Brussels, Belgium, 5–13 September 1957; Volume 7, pp. 143–153. [Google Scholar]
- Kaliski, S.; Petykiewicz, J. Equation of motion coupled with the field of temperature in a magnetic field involving mechanical and electrical relaxation for anisotropic bodies. Proc. Vibr. Probl.
**1959**, 4, 17–28. [Google Scholar] - Chiriţă, S. High-order approximations of three-phase-lag heat conduction model: Some qualitative results. J. Therm. Stress.
**2018**, 41, 608–626. [Google Scholar] [CrossRef] - Sheikholeslami, M.; Ghasemi, A. Solidification heat transfer of nanofluid in existence of thermal radiation by means of FEM. Int. J. Heat Mass Transf.
**2018**, 123, 418–431. [Google Scholar] [CrossRef] - Sheikholeslami, M.; Jafaryar, M.; Li, Z. Nanofluid turbulent convective flow in a circular duct with helical turbulators considering CuO nanoparticles. Int. J. Heat Mass Transf.
**2018**, 124, 980–989. [Google Scholar] [CrossRef] - Sheikholeslami, M.; Haq, R.-U.; Shafee, A.; Li, Z. Heat transfer behavior of nanoparticle enhanced PCM solidification through an enclosure with V shaped fins. Int. J. Heat Mass Transf.
**2019**, 130, 1322–1342. [Google Scholar] [CrossRef] - Sheikholeslami, M.; Haq, R.-U.; Shafee, A.; Li, Z.; Elaraki, Y.G.; Tlili, I. Heat transfer simulation of heat storage unit with nanoparticles and fins through a heat exchanger. Int. J. Heat Mass Transf.
**2019**, 135, 470–478. [Google Scholar] [CrossRef] - Prasad, R.; Kumar, R.; Mukhopadhyay, S. Effects of phase lags on wave propagation in an infinite solid due to a continuous line heat source. Acta Mech.
**2011**, 217, 243–256. [Google Scholar] [CrossRef] - Zenkour, A.; El-Mekawy, H. On a multi-phase-lag model of coupled thermoelasticity. Int. Commun. Heat Mass Transf.
**2020**, 116, 104722. [Google Scholar] [CrossRef] - Biswas, S.; Mukhopadhyay, B.; Shaw, S. Effect of rotation in magne-to-thermoelastic transversely isotropic hollow cylinder with three-phase-lag model. Mech. Bas. Design Struct. Mach.
**2019**, 47, 234–254. [Google Scholar] [CrossRef] - Honig, G.; Hirdes, U. A method for the numerical inversion of Laplace transforms. J. Comput. Appl. Math.
**1984**, 10, 113–132. [Google Scholar] [CrossRef] - Zenkour, A.M. On Generalized Three-Phase-Lag Models in Photo-Thermoelasticity. Int. J. Appl. Mech.
**2022**, 14, 2250005. [Google Scholar] [CrossRef] - Mondal, S.; Sur, A. Photo-thermo-elastic wave propagation in an orthotropic semiconductor with a spherical cavity and memory responses. Waves Random Complex Media
**2021**, 31, 1835–1858. [Google Scholar] [CrossRef]

**Figure 1.**An infinite environment with constant heat and magnetic field that affect externally on the spherical hole.

**Table 1.**Different thermoelasticity theories with a range of r values show the effects of dimensionless time t on volumetric strain e.

r | t | CTE | G–N | L–S | STPL | RTPL | ||
---|---|---|---|---|---|---|---|---|

N = 1 | N = 3 | N = 4 | N = 5 | |||||

1.001 | 0.3 | −0.1600507 | −0.1103681 | −0.1550835 | −0.1590269 | −0.1588199 | −0.1596750 | −0.1596267 |

0.5 | −0.4421455 | −0.4275114 | −0.4390305 | −0.4434662 | −0.4443045 | −0.4443921 | −0.4443562 | |

0.8 | −0.7204259 | −0.7270958 | −0.7191311 | −0.7256527 | −0.7265913 | −0.7265860 | −0.7265783 | |

1.0108 | 0.3 | 0.7907324 | 0.9283637 | 0.8158773 | 0.7976376 | 0.8076523 | 0.8034979 | 0.8027793 |

0.5 | 0.1513132 | 0.1735829 | 0.1667559 | 0.1553263 | 0.1569277 | 0.1563401 | 0.1563651 | |

0.8 | −0.3411962 | −0.3475165 | −0.3323632 | −0.3422239 | −0.3425316 | −0.3425963 | −0.3425864 | |

1.035 | 0.3 | 2.2338375 | 2.8512535 | 2.2802879 | 2.2427386 | 2.2771291 | 2.2703739 | 2.2670290 |

0.5 | 1.3241083 | 1.4192143 | 1.3607036 | 1.3359762 | 1.3456487 | 1.3440881 | 1.3439886 | |

0.8 | 0.4858891 | 0.4793232 | 0.5099324 | 0.4931183 | 0.4951689 | 0.4949403 | 0.4949480 |

**Table 2.**Different thermoelasticity theories with a range of r values show the effects of dimensionless time t on radial displacement u.

r | t | CTE | G–N | L–S | STPL | RTPL | ||
---|---|---|---|---|---|---|---|---|

N = 1 | N = 3 | N = 4 | N = 5 | |||||

1.02 | 0.3 | −0.4446010 | −0.4587127 | −0.4416027 | −0.4442389 | −0.4446845 | −0.4452007 | −0.4451150 |

0.5 | −0.6879661 | −0.6641208 | −0.6859852 | −0.6897589 | −0.6907317 | −0.6907757 | −0.6907412 | |

0.8 | −0.94270823 | −0.9609906 | −0.9420769 | −0.9482101 | −0.9491696 | −0.9491588 | −0.9491515 | |

1.2 | 0.3 | −0.1990386 | −0.1923100 | −0.1999577 | −0.1995811 | −0.1997097 | −0.1990796 | −0.1991871 |

0.5 | −0.3095016 | −0.3218320 | −0.3080032 | −0.3111561 | −0.3096710 | −0.3097699 | −0.3098091 | |

0.8 | −0.4649046 | −0.4700720 | −0.4608963 | −0.4675405 | −0.4668571 | −0.4669065 | −0.4669071 | |

1.4 | 0.3 | −0.0421606 | −0.0347817 | −0.0412343 | −0.0414258 | −0.0426042 | −0.0424287 | −0.0422553 |

0.5 | −0.1976639 | −0.1892863 | −0.1990575 | −0.1983774 | −0.1989738 | −0.1988528 | −0.1988503 | |

0.8 | −0.3399756 | −0.3397367 | −0.3428069 | −0.3434390 | −0.3428693 | −0.3428636 | −0.3428703 |

**Table 3.**Different thermoelasticity theories with a range of r values show the effects of dimensionless time t on temperature $\Theta $.

r | t | CTE | G–N | L–S | STPL | RTPL | ||
---|---|---|---|---|---|---|---|---|

N = 1 | N = 3 | N = 4 | N = 5 | |||||

1.02 | 0.3 | 0.5197161 | 0.5231952 | 0.8158523 | 0.6321797 | 0.7257061 | 0.6687072 | 0.6619560 |

0.5 | 0.4551331 | 0.4591944 | 0.6034615 | 0.5250257 | 0.5347617 | 0.5288343 | 0.5292230 | |

0.8 | 0.4065923 | 0.4113274 | 0.4823044 | 0.4480926 | 0.4475421 | 0.4469535 | 0.4470211 | |

1.2 | 0.3 | 1.1328376 | 1.1551850 | 1.5158575 | 1.2732250 | 1.8693517 | 1.8697841 | 1.7797076 |

0.5 | 1.1441023 | 1.1762359 | 1.6486669 | 1.3572497 | 1.5595642 | 1.5331112 | 1.5294970 | |

0.8 | 1.0991674 | 1.1409540 | 1.4633732 | 1.2932336 | 1.3452712 | 1.3412060 | 1.3411794 | |

1.4 | 0.3 | 0.6166223 | 0.6403425 | 0.2553162 | 0.5498148 | 0.4213529 | 0.6197291 | 0.6189961 |

0.5 | 0.7944132 | 0.8302729 | 0.7959542 | 0.8370789 | 0.9866523 | 0.9826821 | 0.9777945 | |

0.8 | 0.9145463 | 0.9654823 | 1.1009292 | 1.0413519 | 1.1155064 | 1.112294 | 1.1119570 |

**Table 4.**Different thermoelasticity theories with a range of $r$ values show the effects of dimensionless time $t$ on radial stress ${\sigma}_{rr}$.

r | t | CTE | G–N | L–S | STPL | RTPL | ||
---|---|---|---|---|---|---|---|---|

N = 1 | N = 3 | N = 4 | N = 5 | |||||

1.02 | 0.3 | 1.4250082 | 1.7169427 | 1.1644764 | 1.3218307 | 1.2488604 | 1.3002398 | 1.3052216 |

0.5 | 0.8697574 | 0.8919956 | 0.7447060 | 0.8095222 | 0.8052005 | 0.8101600 | 0.8097287 | |

0.8 | 0.5084246 | 0.5016690 | 0.4473304 | 0.4747826 | 0.4767497 | 0.4771983 | 0.4771340 | |

1.2 | 0.3 | −0.3476186 | −0.2060406 | −0.7381048 | −0.4810923 | −1.1013084 | −1.0979774 | −1.0048155 |

0.5 | −1.0358693 | −0.4342299 | −1.5929489 | −1.2634836 | −1.4691980 | −1.4406952 | −1.4372923 | |

0.8 | 0.4730670 | 0.4169757 | 0.0996810 | 0.2792902 | 0.2373867 | 0.2412938 | 0.2412457 | |

1.4 | 0.3 | 0.0720812 | −0.0097442 | 0.4509219 | 0.1469006 | 0.2787176 | 0.0756354 | 0.0768349 |

0.5 | 0.0635672 | 0.0801358 | 0.0794198 | 0.0336441 | −0.1175433 | −0.1139438 | −0.1088892 | |

0.8 | −1.1153279 | −1.1761286 | −1.3292837 | −1.2426448 | −1.3245219 | −1.3209060 | −1.3205495 |

**Table 5.**Different thermoelasticity theories with a range of $r$ values show the effects of dimensionless time $t$ on circumferential stress ${\sigma}_{\theta \theta}$.

r | t | CTE | G–N | L–S | STPL | RTPL | ||
---|---|---|---|---|---|---|---|---|

N = 1 | N = 3 | N = 4 | N = 5 | |||||

1.02 | 0.3 | 0.0198399 | 0.1669378 | −0.2555124 | −0.0876141 | −0.1712722 | −0.1175958 | −0.1116476 |

0.5 | −0.4645790 | −0.4416541 | −0.5992993 | −0.5313858 | −0.5393601 | −0.5339610 | −0.5343373 | |

0.8 | −0.8709283 | −0.8892813 | −0.9386944 | −0.9138763 | −0.9135551 | −0.9130260 | −0.9130849 | |

1.2 | 0.3 | −0.9048642 | −0.8304659 | −1.2923918 | −1.0422374 | −1.6505469 | −1.6485684 | −1.5570346 |

0.5 | −1.3474299 | −1.0602676 | −1.8770728 | −1.5692064 | −1.7719893 | −1.7445910 | −1.7411154 | |

0.8 | −0.6979375 | −0.7436193 | −1.0634094 | −0.8940525 | −0.9404411 | −0.9364963 | −0.9365076 | |

1.4 | 0.3 | −0.3014565 | −0.3536854 | 0.0693004 | −0.2301085 | −0.1008053 | −0.3014154 | −0.3003248 |

0.5 | −0.5053201 | −0.5032281 | −0.4991360 | −0.5421070 | −0.6929150 | −0.6890443 | −0.6840712 | |

0.8 | −1.2577217 | −1.3109739 | −1.4599464 | −1.3872542 | −1.4648736 | −1.4614548 | −1.4611128 |

**Table 6.**Different thermoelasticity theories with a range of $r$ values show the effects of dimensionless magnetic field intensity ${H}_{0}$ on volumetric strain $e$.

r | H_{0} | CTE | G–N | L–S | STPL | RTPL | ||
---|---|---|---|---|---|---|---|---|

N = 1 | N = 3 | N = 4 | N = 5 | |||||

1.001 | 0 | 0.1782137 | 0.5045876 | 0.1876537 | 0.1803101 | 0.1815128 | 0.1801641 | 0.1801591 |

$5\times {10}^{8}$ | −0.1600507 | −0.1260185 | −0.1550835 | −0.1590269 | −0.1588199 | −0.1596750 | −0.1596267 | |

$1\times {10}^{9}$ | −0.5963013 | −0.5838559 | −0.5942398 | −0.5959014 | −0.5961448 | −0.5966622 | −0.5965978 | |

1.0108 | 0 | 1.6863479 | 3.1787268 | 1.7322308 | 1.6976456 | 1.7199434 | 1.7128199 | 1.7110496 |

$5\times {10}^{8}$ | 0.7907324 | 0.9125334 | 0.8158773 | 0.7976376 | 0.8076523 | 0.8034979 | 0.8027793 | |

$1\times {10}^{9}$ | −0.0216654 | −0.0121308 | −0.0097624 | −0.0180902 | −0.0141696 | −0.0163392 | −0.0165856 | |

1.035 | 0 | 2.4534254 | 7.7106569 | 2.4972843 | 2.4545775 | 2.5100973 | 2.5053423 | 2.4991835 |

$5\times {10}^{8}$ | 2.2338375 | 2.8364917 | 2.2802879 | 2.2427386 | 2.2771291 | 2.2703739 | 2.2670290 | |

$1\times {10}^{9}$ | 1.1346513 | 1.1417442 | 1.1625329 | 1.1418607 | 1.1575618 | 1.1530407 | 1.1516357 |

**Table 7.**Different thermoelasticity theories with a range of $r$ values show the effects of dimensionless magnetic field intensity ${H}_{0}$ on radial displacement $u$.

$\mathit{r}$ | ${\mathit{H}}_{0}$ | CTE | G–N | L–S | STPL | RTPL | ||
---|---|---|---|---|---|---|---|---|

$\mathit{N}=1$ | $\mathit{N}=3$ | $\mathit{N}=4$ | $\mathit{N}=5$ | |||||

1.02 | 0 | −0.1767007 | −0.2365661 | −0.1712572 | −0.1759073 | −0.1761001 | −0.1767683 | −0.1766927 |

$5\times {10}^{8}$ | −0.4446010 | −0.4556610 | −0.4416027 | −0.4442389 | −0.4446845 | −0.4452007 | −0.4451150 | |

$1\times {10}^{9}$ | −0.8377249 | −0.8346584 | −0.8365695 | −0.8376432 | −0.8381650 | −0.8385221 | −0.8384422 | |

1.2 | 0 | 0.0006626 | 0.0211928 | 0.0018281 | 0.0009625 | 0.0010703 | 0.0015785 | 0.0014699 |

$5\times {10}^{8}$ | −0.1990386 | −0.1927053 | −0.1999577 | −0.1995811 | −0.1997097 | −0.1990796 | −0.1991871 | |

$1\times {10}^{9}$ | −0.4200627 | −0.4180902 | −0.4204997 | −0.4207381 | −0.4193394 | −0.4190292 | −0.4192868 | |

1.4 | 0 | 0.0415410 | 0.0465105 | 0.0418797 | 0.0420817 | 0.0412667 | 0.0413940 | 0.0415588 |

$5\times {10}^{8}$ | −0.0421606 | −0.0352435 | −0.0412343 | −0.0414258 | −0.0426042 | −0.0424287 | −0.0422553 | |

$1\times {10}^{9}$ | −0.2954475 | −0.2962609 | −0.2956049 | −0.2953487 | −0.2970595 | −0.2968102 | −0.2965584 |

**Table 8.**Different thermoelasticity theories with a range of $r$ values show the effects of dimensionless magnetic field intensity ${H}_{0}$ on temperature $\Theta $.

$\mathit{r}$ | ${\mathit{H}}_{0}$ | CTE | G–N | L–S | STPL | RTPL | ||
---|---|---|---|---|---|---|---|---|

$\mathit{N}=1$ | $\mathit{N}=3$ | $\mathit{N}=4$ | $\mathit{N}=5$ | |||||

1.02 | 0 | 0.5195277 | 0.5231952 | 0.8149122 | 0.6317500 | 0.7249938 | 0.6681243 | 0.6613913 |

$5\times {10}^{8}$ | 0.5197161 | 0.5231952 | 0.8158523 | 0.6321797 | 0.7257061 | 0.6687072 | 0.6619559 | |

$1\times {10}^{9}$ | 0.5202053 | 0.5231952 | 0.8171815 | 0.6329834 | 0.7267366 | 0.6695850 | 0.6628219 | |

1.2 | 0 | 1.1328485 | 1.1551850 | 1.5160141 | 1.2730366 | 1.8699681 | 1.8702848 | 1.7801007 |

$5\times {10}^{8}$ | 1.1328376 | 1.1551850 | 1.5158575 | 1.2732250 | 1.8693517 | 1.8697841 | 1.7797076 | |

$1\times {10}^{9}$ | 1.1302030 | 1.1551850 | 1.5072709 | 1.2688359 | 1.8634498 | 1.8652445 | 1.7751423 | |

1.4 | 0 | 0.6178070 | 0.6403425 | 0.2557158 | 0.5506273 | 0.4221051 | 0.6206186 | 0.6198641 |

$5\times {10}^{8}$ | 0.6166223 | 0.6403425 | 0.2553162 | 0.5498148 | 0.4213529 | 0.6197291 | 0.6189961 | |

$1\times {10}^{9}$ | 0.6177501 | 0.6403425 | 0.2597756 | 0.5516250 | 0.4242953 | 0.6217558 | 0.6210694 |

**Table 9.**Different thermoelasticity theories with a range of $r$ values show the effects of dimensionless magnetic field intensity ${H}_{0}$ on radial stress ${\sigma}_{rr}$.

$\mathit{r}$ | ${\mathit{H}}_{0}$ | CTE | G–N | L–S | STPL | RTPL | ||
---|---|---|---|---|---|---|---|---|

$\mathit{N}=1$ | $\mathit{N}=3$ | $\mathit{N}=4$ | $\mathit{N}=5$ | |||||

1.02 | 0 | 2.1413094 | 5.0084304 | 1.9012246 | 2.0400614 | 1.9882193 | 2.0368781 | 2.0397306 |

$5\times {10}^{8}$ | 1.4250082 | 1.7098567 | 1.1644764 | 1.3218307 | 1.2488604 | 1.3002398 | 1.3052216 | |

$1\times {10}^{9}$ | 0.7663735 | 0.7619035 | 0.4877997 | 0.6591606 | 0.5742749 | 0.6284021 | 0.6344444 | |

1.2 | 0 | −0.6773446 | 0.6615173 | −1.0776863 | −0.8119883 | −1.4290790 | −1.4239547 | −1.3317608 |

$5\times {10}^{8}$ | −0.3476186 | −0.2009060 | −0.7381048 | −0.4810923 | −1.1013084 | −1.0979774 | −1.0048155 | |

$1\times {10}^{9}$ | −0.8607713 | −0.8969050 | −1.2863322 | −1.0105215 | −1.6267403 | −1.6169840 | −1.5257144 | |

1.4 | 0 | −0.7407481 | −0.7747852 | −0.3827128 | −0.6759330 | −0.5439698 | −0.7468172 | −0.7458980 |

$5\times {10}^{8}$ | 0.0720812 | −0.0016496 | 0.4509219 | 0.1469006 | 0.2787176 | 0.0756354 | 0.0768349 | |

$1\times {10}^{9}$ | 0.2771069 | 0.2558551 | 0.6657660 | 0.3510599 | 0.4892780 | 0.2833072 | 0.2842716 |

**Table 10.**Different thermoelasticity theories with a range of $r$ values show the effects of dimensionless magnetic field intensity ${H}_{0}$ on circumferential stress ${\sigma}_{\theta \theta}$.

$\mathit{r}$ | ${\mathit{H}}_{0}$ | CTE | G–N | L–S | STPL | RTPL | ||
---|---|---|---|---|---|---|---|---|

$\mathit{N}=1$ | $\mathit{N}=3$ | $\mathit{N}=4$ | $\mathit{N}=5$ | |||||

1.02 | 0 | 0.6413170 | 1.9878850 | 0.3789836 | 0.5353730 | 0.4626946 | 0.5147940 | 0.5196557 |

$5\times {10}^{8}$ | 0.0198399 | 0.1372332 | −0.2555124 | −0.0876141 | −0.1712722 | −0.1175958 | −0.1116476 | |

$1\times {10}^{9}$ | −0.6954887 | −0.6792086 | −0.9821086 | −0.8053970 | −0.8952159 | −0.8399301 | −0.8334501 | |

1.2 | 0 | −0.9039566 | −0.2244379 | −1.2947625 | −1.0411157 | −1.6480632 | −1.6452294 | −1.5541281 |

$5\times {10}^{8}$ | −0.9048642 | −0.8273155 | −1.2923918 | −1.0422374 | −1.6505469 | −1.6485684 | −1.5570346 | |

$1\times {10}^{9}$ | −1.3447378 | −1.3652770 | −1.7464786 | −1.4895059 | −2.0937861 | −2.0895322 | −1.9990591 | |

1.4 | 0 | −0.6498027 | −0.6794221 | −0.2895031 | −0.5834226 | −0.4537569 | −0.6543521 | −0.6533975 |

$5\times {10}^{8}$ | −0.3014565 | −0.3479797 | 0.0693004 | −0.2301085 | −0.1008053 | −0.3014154 | −0.3003248 | |

$1\times {10}^{9}$ | −0.3799260 | −0.4038305 | −0.0066819 | −0.3098064 | −0.1782388 | −0.3797877 | −0.3787822 |

**Table 11.**Different thermoelasticity theories with a range of $r$ values show the effects of dimensionless electric permittivity ${\epsilon}_{0}$ on volumetric strain $e$.

$\mathit{r}$ | ${\mathit{\epsilon}}_{0}$ | CTE | G–N | L–S | STPL | RTPL | ||
---|---|---|---|---|---|---|---|---|

$\mathit{N}=1$ | $\mathit{N}=3$ | $\mathit{N}=4$ | $\mathit{N}=5$ | |||||

1.001 | 0.00 | −0.1600507 | −0.1817118 | −0.1550835 | −0.1590269 | −0.1588199 | −0.1596750 | −0.1596267 |

0.01 | −0.0728988 | −0.2495945 | −0.0685479 | −0.0719793 | −0.0716577 | −0.0723496 | −0.0723247 | |

0.05 | 0.1069584 | 0.5070073 | 0.1099905 | 0.1076340 | 0.1080314 | 0.1076008 | 0.1075979 | |

0.20 | 0.3598124 | −0.1230445 | 0.3614521 | 0.3602013 | 0.3605292 | 0.3603088 | 0.3602955 | |

1.0108 | 0.00 | 0.7907324 | 0.7422693 | 0.8158773 | 0.7976376 | 0.8076523 | 0.8034979 | 0.8027793 |

0.01 | 1.0444167 | 1.1310202 | 1.0660570 | 1.0501743 | 1.0593502 | 1.0558573 | 1.0551759 | |

0.05 | 1.6586577 | 2.2876971 | 1.6733614 | 1.6622524 | 1.6694769 | 1.6672005 | 1.6666240 | |

0.20 | 2.5373701 | −0.1390619 | 2.5446030 | 2.5388002 | 2.5434033 | 2.5423482 | 2.5419391 | |

1.035 | 0.00 | 2.2338375 | 2.0604843 | 2.2802879 | 2.2427386 | 2.2771291 | 2.2703739 | 2.2670290 |

0.01 | 2.4541706 | 3.6055484 | 2.4889375 | 2.4596437 | 2.4888941 | 2.4840987 | 2.4811503 | |

0.05 | 2.3896894 | 3.7194494 | 2.4028985 | 2.3898009 | 2.4073189 | 2.4059421 | 2.4039812 | |

0.20 | 0.5555053 | −3.5966438 | 0.5537322 | 0.5538753 | 0.5579398 | 0.5586249 | 0.5580035 |

**Table 12.**Different thermoelasticity theories with a range of $r$ values show the effects of dimensionless electric permittivity ${\epsilon}_{0}$ on radial displacement $u$.

$\mathit{r}$ | ${\mathit{\epsilon}}_{0}$ | CTE | G–N | L–S | STPL | RTPL | ||
---|---|---|---|---|---|---|---|---|

$\mathit{N}=1$ | $\mathit{N}=3$ | $\mathit{N}=4$ | $\mathit{N}=5$ | |||||

1.02 | 0.00 | −0.4446010 | −0.4702098 | −0.4416027 | −0.4442389 | −0.4446845 | −0.4452007 | −0.4451150 |

0.01 | −0.3761698 | −0.3938417 | −0.3735556 | −0.3758274 | −0.3760925 | −0.3764870 | −0.3764284 | |

0.05 | −0.2514169 | −0.2002410 | −0.2496761 | −0.2511617 | −0.2512163 | −0.2514271 | −0.2514038 | |

0.20 | −0.1285389 | −0.1350768 | −0.1277263 | −0.1284124 | −0.1283882 | −0.1284682 | −0.1284637 | |

1.2 | 0.00 | −0.1990386 | −0.1982898 | −0.1999577 | −0.1995811 | −0.1997097 | −0.1990796 | −0.1991871 |

0.01 | −0.1388393 | −0.1242475 | −0.1387854 | −0.1389520 | −0.1390226 | −0.1386521 | −0.1387124 | |

0.05 | −0.0514207 | −0.0617681 | −0.0510383 | −0.0513124 | −0.0512864 | −0.0511259 | −0.0511593 | |

0.20 | 0.0085746 | 0.0088881 | 0.0087642 | 0.0086495 | 0.0086748 | 0.0087255 | 0.0087156 | |

1.4 | 0.00 | −0.0421606 | −0.0399129 | −0.0412343 | −0.0414258 | −0.0426042 | −0.0424287 | −0.0422553 |

0.01 | 0.0104176 | 0.0181003 | 0.0115143 | 0.0111229 | 0.0104788 | 0.0105647 | 0.0106848 | |

0.05 | 0.0128624 | 0.0116589 | 0.0129452 | 0.0130233 | 0.0127674 | 0.0128069 | 0.0128582 | |

0.20 | 0.0039736 | 0.0026442 | 0.0039234 | 0.0039991 | 0.0039099 | 0.0039224 | 0.0039389 |

**Table 13.**Different thermoelasticity theories with a range of $r$ values show the effects of dimensionless electric permittivity ${\epsilon}_{0}$ on temperature $\Theta $.

$\mathit{r}$ | ${\mathit{\epsilon}}_{0}$ | CTE | G–N | L–S | STPL | RTPL | ||
---|---|---|---|---|---|---|---|---|

$\mathit{N}=1$ | $\mathit{N}=3$ | $\mathit{N}=4$ | $\mathit{N}=5$ | |||||

1.02 | 0.00 | 0.5197161 | 0.5231952 | 0.8158523 | 0.6321797 | 0.7257061 | 0.6687072 | 0.6619559 |

0.01 | 0.5196226 | 0.5231952 | 0.8155291 | 0.6320045 | 0.7254572 | 0.6684988 | 0.6617520 | |

0.05 | 0.5195045 | 0.5231952 | 0.8149457 | 0.6317351 | 0.7250108 | 0.6681320 | 0.6613971 | |

0.20 | 0.5194992 | 0.5231952 | 0.8144369 | 0.6315960 | 0.7246144 | 0.6678177 | 0.6611030 | |

1.2 | 0.00 | 1.1328376 | 1.1551850 | 1.5158575 | 1.2732250 | 1.8693517 | 1.8697841 | 1.7797076 |

0.01 | 1.1325365 | 1.1551850 | 1.5158537 | 1.2728365 | 1.8696724 | 1.8700076 | 1.7798489 | |

0.05 | 1.1326225 | 1.1551850 | 1.5155747 | 1.2727896 | 1.8697239 | 1.8700990 | 1.7799023 | |

0.20 | 1.1330736 | 1.1551850 | 1.5160089 | 1.2731337 | 1.8702705 | 1.8706068 | 1.7803877 | |

1.4 | 0.00 | 0.6166223 | 0.6403425 | 0.2553162 | 0.5498148 | 0.4213529 | 0.6197291 | 0.6189961 |

0.01 | 0.6163464 | 0.6403425 | 0.2546471 | 0.5492825 | 0.4209161 | 0.6193444 | 0.6186026 | |

0.05 | 0.6175008 | 0.6403425 | 0.2554513 | 0.5503468 | 0.4216919 | 0.6202147 | 0.6194853 | |

0.20 | 0.6182038 | 0.6403425 | 0.2561957 | 0.5510730 | 0.4223512 | 0.6209149 | 0.6201905 |

**Table 14.**Different thermoelasticity theories with a range of $r$ values show the effects of dimensionless electric permittivity ${\epsilon}_{0}$ on radial stress ${\sigma}_{rr}$.

$\mathit{r}$ | ${\mathit{\epsilon}}_{0}$ | CTE | G–N | L–S | STPL | RTPL | ||
---|---|---|---|---|---|---|---|---|

$\mathit{N}=1$ | $\mathit{N}=3$ | $\mathit{N}=4$ | $\mathit{N}=5$ | |||||

1.02 | 0.00 | 1.4250082 | 1.4417132 | 1.1644764 | 1.3218307 | 1.2488604 | 1.3002398 | 1.3052216 |

0.01 | 1.6799725 | 2.3462039 | 1.4135395 | 1.5747718 | 1.4996322 | 1.5520136 | 1.5571447 | |

0.05 | 2.1955956 | 2.8931530 | 1.9176461 | 2.0867605 | 2.0068048 | 2.0610981 | 2.0665780 | |

0.20 | 2.1293602 | −1.9570943 | 1.8395231 | 2.0174381 | 1.9310497 | 1.9872334 | 1.9932340 | |

1.2 | 0.00 | −0.3476186 | −0.7002070 | −0.7381048 | −0.4810923 | −1.1013084 | −1.0979774 | −1.0048155 |

0.01 | −0.0084140 | 0.7900171 | −0.3912182 | −0.1439469 | −0.7489823 | −0.7479519 | −0.6564771 | |

0.05 | −0.5214154 | 0.2873304 | −0.9091673 | −0.6595318 | −1.2628424 | −1.2615757 | −1.1707216 | |

0.20 | −1.1843148 | −3.7578587 | −1.5710519 | −1.3248846 | −1.9236769 | −1.9235961 | −1.8331840 | |

1.4 | 0.00 | 0.0720812 | −0.0630823 | 0.4509219 | 0.1469006 | 0.2787176 | 0.0756354 | 0.0768349 |

0.01 | −0.0733074 | −0.1260033 | 0.2946380 | −0.0050862 | 0.1296550 | −0.0721929 | −0.0714200 | |

0.05 | −0.6513165 | −0.4203154 | −0.2904336 | −0.5848771 | −0.4551246 | −0.6549940 | −0.6542153 | |

0.20 | −0.6302124 | −1.3114760 | −0.2682290 | −0.5632011 | −0.4341115 | −0.6331041 | −0.6323633 |

**Table 15.**Different thermoelasticity theories with a range of $r$ values show the effects of dimensionless electric permittivity ${\epsilon}_{0}$ on circumferential stress ${\sigma}_{\theta \theta}$.

$\mathit{r}$ | ${\mathit{\epsilon}}_{0}$ | CTE | G–N | L–S | STPL | RTPL | ||
---|---|---|---|---|---|---|---|---|

$\mathit{N}=1$ | $\mathit{N}=3$ | $\mathit{N}=4$ | $\mathit{N}=5$ | |||||

1.02 | 0.00 | 0.0198399 | −0.1007279 | −0.2555124 | −0.0876141 | −0.1712722 | −0.1707365 | −0.1706840 |

0.01 | 0.2147006 | 0.3924400 | −0.0638714 | 0.1062539 | 0.0217218 | 0.0222527 | 0.0223048 | |

0.05 | 0.5953855 | 1.0740053 | 0.3104173 | 0.4851070 | 0.3984551 | 0.3989805 | 0.3990321 | |

0.20 | 0.6824975 | −1.5054263 | 0.3909123 | 0.5706121 | 0.4809410 | 0.4814663 | 0.4815179 | |

1.2 | 0.00 | −0.9048642 | −1.1056940 | −1.2923918 | −1.0422374 | −1.6505469 | −1.6485537 | −1.6483581 |

0.01 | −0.6845724 | −0.3075093 | −1.0675876 | −0.8225765 | −1.4235815 | −1.4215722 | −1.4213750 | |

0.05 | −0.8690243 | −0.4528431 | −1.2540643 | −1.0080733 | −1.6081824 | −1.6061455 | −1.6059456 | |

0.20 | −1.1516241 | −2.2032886 | −1.5363075 | −1.2918775 | −1.8898230 | −1.8877660 | −1.8875641 | |

1.4 | 0.00 | −0.3014565 | −0.3782174 | 0.0693004 | −0.2301085 | −0.1008053 | −0.0993506 | −0.0992079 |

0.01 | −0.3366938 | −0.3633738 | 0.0289190 | −0.2685467 | −0.1374440 | −0.1359819 | −0.1358384 | |

0.05 | −0.6252768 | −0.5124963 | −0.2637530 | −0.5583662 | −0.4293436 | −0.4278723 | −0.4277280 | |

0.20 | −0.6213890 | −0.9706441 | −0.2594291 | −0.5542999 | −0.4254574 | −0.4239819 | −0.4238371 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Allehaibi, A.M.; Zenkour, A.M.
Magneto-Thermoelastic Response in an Infinite Medium with a Spherical Hole in the Context of High Order Time-Derivatives and Triple-Phase-Lag Model. *Materials* **2022**, *15*, 6256.
https://doi.org/10.3390/ma15186256

**AMA Style**

Allehaibi AM, Zenkour AM.
Magneto-Thermoelastic Response in an Infinite Medium with a Spherical Hole in the Context of High Order Time-Derivatives and Triple-Phase-Lag Model. *Materials*. 2022; 15(18):6256.
https://doi.org/10.3390/ma15186256

**Chicago/Turabian Style**

Allehaibi, Ashraf M., and Ashraf M. Zenkour.
2022. "Magneto-Thermoelastic Response in an Infinite Medium with a Spherical Hole in the Context of High Order Time-Derivatives and Triple-Phase-Lag Model" *Materials* 15, no. 18: 6256.
https://doi.org/10.3390/ma15186256