# Generalized MGT Heat Transfer Model for an Electro-Thermal Microbeam Lying on a Viscous-Pasternak Foundation with a Laser Excitation Heat Source

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Analysis and Basic Equations

## 3. Model Description

## 4. Solution of the Governing Equations

^{2}), $d$ symbolizes the laser penetration depth, ${t}_{p}$ denotes the length of time that the laser pulse is active, and the reflectance of the surface denoted by $R$ and $\beta $ is a constant parameter.

## 5. Initial and Boundary Conditions

## 6. Laplace Transform Solution

## 7. Numerical Inversion Technique

## 8. Numerical Results and Analysis

#### 8.1. The Influence of Visco-Pasternak’s Basis Factors

#### 8.2. Effect of Laser Pulse Duration ${t}_{p}$

#### 8.3. The Effect of Electrothermal Voltage

## 9. Conclusions

- The results show that increasing the Winkler and foundation shear moduli reduces microbeam deflection and first-side axial deformations. This reduction is a direct consequence of the higher beam stiffness. However, as the viscosity index is raised, the deflection and deformation significantly increase.
- As Pasternak’s modulus and Winkler’s viscosity factor increase, flexural moment and axial stress also increase. Increasing the coefficient of the visco-Pasternak base reduces the dynamic behavior of the microbeam, facilitating a much faster attempt at equilibrium. Increasing the viscous damping coefficients results in a significant decrease in the amplitudes of the studied work areas and the response time.
- Theoretical analysis and experimental results showed that a solid base is necessary for the stability of the building when it is in a state of oscillation. Every building needs a solid foundation on which to rest, as this helps prevent any kind of mechanical failure from occurring.
- The results show that the MGTE model has a good theoretical and experimental agreement between the mechanical properties and the physical aspects.
- The magnitudes of deflection increase with increasing electrical voltages. This is due to the increase in the thermos-electrical load (electrothermal voltage) produced inside the microbeam.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Zenkour, A.M.; Abouelregal, A.E. Effect of harmonically varying heat on FG nanobeams in the context of a nonlocal two-temperature thermoelasticity theory. Euro. J. Comp. Mech.
**2014**, 23, 1–14. [Google Scholar] [CrossRef] - Craighead, H.G. Nanoelectromechanical Systems. Science
**2000**, 290, 1532–1535. [Google Scholar] [CrossRef][Green Version] - Li, C.; Zhu, C.X.; Lim, C.W.; Li, S. Nonlinear in-plane thermal buckling of rotationally restrained functionally graded carbon nanotube reinforced composite shallow arches under uniform radial loading. Appl. Math. Mech.
**2022**, 43, 1821–1840. [Google Scholar] [CrossRef] - LI, C. Size-dependent thermal behaviors of axially traveling nanobeams based on a strain gradient theory. Struc. Eng. Mech.
**2013**, 48, 415–434. [Google Scholar] [CrossRef] - Li, C.; Zhu, C.X.; Zhang, N.; Sui, S.H.; Zhao, J.B. Free vibration of self-powered nanoribbons subjected to thermal-mechanical-electrical fields based on a nonlocal strain gradient theory. Appl. Math. Model.
**2022**, 110, 583–602. [Google Scholar] [CrossRef] - Sui, S.; Zhu, C.; Li, C.; Lei, Z. Free Vibration of Axially Traveling Moderately Thick FG Plates Resting on Elastic Foundations. J. Vib. Eng. Technol.
**2023**, 11, 329–341. [Google Scholar] [CrossRef] - Zemskov, A.; Hao, L.; Tarlakovskii, D. Bernoulli-Euler beam unsteady bending model with consideration of heat and mass transfer. J. Appl. Comp. Mech.
**2023**, 9, 168–180. [Google Scholar] - Ashraf, M.W.; Tayyaba, S.; Afzulpurkar, N. Micro electromechanical systems (MEMS) based microfluidic devices for biomedical applications. Int. J. Mol. Sci.
**2011**, 12, 3648–3704. [Google Scholar] [CrossRef][Green Version] - Shi, D.; Zhang, H.; Wang, Q.; Zha, S. Free and forced vibration of the moderately thick laminated composite rectangular plate on various elastic Winkler and Pasternak foundations. Shock Vibr.
**2017**, 2017, 7820130. [Google Scholar] [CrossRef][Green Version] - Zemskov, A.V.; Tarlakovskii, D.V. Unsteady elastic diffusion bending model for a Timoshenko beam on a Winkler foundation. Arch. Appl. Mech.
**2022**, 92, 1355–1366. [Google Scholar] [CrossRef] - Zemskov, A.V.; Okonechnikov, A.S.; Tarlakovskii, D.V. Unsteady Elastic–Diffusion Vibrations of a Simply Supported Euler–Bernoulli Beam Under the Distributed Transverse Load. In Multiscale Solid Mechanics. Advanced Structured Materials; Altenbach, H., Eremeyev, V.A., Igumnov, L.A., Eds.; Springer: Cham, Switzerland, 2021; Volume 141. [Google Scholar]
- Togun, N.; Bağdatli, S.M. The vibration of nanobeam resting on elastic foundation using modified couple stress theory. Teh. Glas.
**2018**, 12, 221–225. [Google Scholar] [CrossRef][Green Version] - Saadatnia, Z.; Askari, H.; Esmailzadeh, E. Multi-frequency excitation of microbeams supported by Winkler and Pasternak foundations. J. Vibr. Control
**2017**, 24, 2894–2911. [Google Scholar] [CrossRef] - Hajjaj, A.Z.; Alcheikh, N.; Ramini, A.; Al Hafiz, M.A.; Younis, M.I. Highly Tunable Electrothermally and Electrostatically Actuated Resonators. J. Microelectromechanical Syst.
**2016**, 25, 440–449. [Google Scholar] [CrossRef][Green Version] - Liu, Y.; Zhou, S.; Wu, K.; Qi, L. Size-dependent electromechanical responses of a bilayer piezoelectric microbeam. Int. J. Mech. Mater. Design.
**2020**, 16, 443–460. [Google Scholar] [CrossRef] - Pimpare, S.B.; Sutar, C.S. Hollow cylinder with thermoelastic modelling by reduced differential transform. J. Indones. Math. Soc.
**2022**, 28, 8–18. [Google Scholar] [CrossRef] - Biot, M. Thermoelsticity and Irreversible Thermodynamics. J. Appl. Phys.
**1956**, 27, 240–253. [Google Scholar] [CrossRef] - Lord, H.; Shulman, Y. A generalized dynamic 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. R. Soc. Lond. Ser. A
**1991**, 432, 171–194. [Google Scholar] - 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] - Quintanilla, R. Moore–Gibson–Thompson thermoelasticity. Math. Mech. Solids.
**2019**, 24, 4020–4031. [Google Scholar] [CrossRef] - Quintanilla, R. Moore-Gibson-Thompson thermoelasticity with two temperatures. Appl. Eng. Sci.
**2020**, 1, 100006. [Google Scholar] [CrossRef] - Abouelregal, A.E.; Sedighi, H.M.; Eremeyev, V.A. Thermomagnetic behavior of a semiconductor material heated by pulsed excitation based on the fourth-order MGT photothermal model. Continuum Mech. Thermodyn.
**2023**, 35, 81–102. [Google Scholar] [CrossRef] - Abouelregal, A.E.; Rayan, A.; Mostafa, D.M. Transient responses to an infinite solid with a spherical cavity according to the MGT thermo-diffusion model with fractional derivatives without nonsingular kernels. Waves Random Complex Media
**2022**. [Google Scholar] [CrossRef] - Moaaz, O.; Abouelregal, A.E.; Alsharari, F. Analysis of a transversely isotropic annular circular cylinder immersed in a magnetic field using the Moore–Gibson–Thompson thermoelastic model and generalized Ohm’s law. Mathematics
**2022**, 10, 3816. [Google Scholar] [CrossRef] - Conti, M.; Pata, V.; Quintanilla, R. Thermoelasticity of Moore–Gibson–Thompson type with history dependence in the temperature. Asymptot. Anal.
**2020**, 120, 1–21. [Google Scholar] [CrossRef] - Gupta, S.; Dutta, R.; Das, S.; Pandit, D.K. Hall current effect in double poro-thermoelastic material with fractional-order Moore–Gibson–Thompson heat equation subjected to Eringen’s nonlocal theory. Waves Random Complex Media
**2022**. [Google Scholar] [CrossRef] - Marin, M. On existence and uniqueness in thermoelasticity of micropolar bodies, Comptes rendus de l’Académie des Sciences Paris. Série II B
**1995**, 321, 375–480. [Google Scholar] - Marin, M.; Marinescu, C. Thermoelasticity of initially stressed bodies. Asymptotic equipartition of energies. Int. J. Eng. Sci.
**1998**, 36, 73–86. [Google Scholar] [CrossRef] - Hetényi, M. Beams on Elastic Foundation: Theory with Applications in the Fields of Civil and Mechanical Engineering; University of Michigan: Ann Arbor, MI, USA, 1971. [Google Scholar]
- Younesian, D.; Hosseinkhani, A.; Askari, H.; Esmailzadeh, E. Elastic and viscoelastic foundations: A review on linear and nonlinear vibration modeling and applications. Nonlinear Dyn.
**2019**, 97, 853–895. [Google Scholar] [CrossRef] - Miao, Y.; He, H.; Yang, Q.; Shi, Y. Analytical solution considering the tangential effect for an infinite beam on a viscoelastic Pasternak foundation. Appl. Math. Model.
**2020**, 85, 231–243. [Google Scholar] [CrossRef] - Demir, C.; Civalek, Ö. On the analysis of microbeams. Int. J. Eng. Sci.
**2017**, 121, 14–33. [Google Scholar] [CrossRef] - Darban, H.; Fabbrocino, F.; Feo, L.; Luciano, R. Size-dependent buckling analysis of nanobeams resting on two-parameter elastic foundation through stress-driven nonlocal elasticity model. Mech. Adv. Mater. Struct.
**2021**, 28, 2408–2416. [Google Scholar] [CrossRef] - Younesian, D.; Kargarnovin, M.H. Response of the beams on random Pasternak foundations subjected to harmonic moving loads. J. Mech. Sci. Technol.
**2009**, 23, 3013–3023. [Google Scholar] [CrossRef] - Ramadan, K.; Tyfour, W.R.; Al-Nimr, M.A. On the analysis of short-pulse laser heating of metals using the dual phase lag heat conduction model. ASME J. Heat Transf.
**2009**, 131, 111301. [Google Scholar] [CrossRef] - Lee, H.-L.; Chen, W.-L.; Chang, W.-J.; Yang, Y.-C. Estimation of energy absorption rate and temperature distributions in short-pulse laser heating of metals with a dual-phase-lag model. Appl. Therm. Eng.
**2014**, 65, 352–360. [Google Scholar] [CrossRef] - Campos, R.G.; Huet, A. Numerical inversion of the Laplace transform and its application to fractional diffusion. Appl. Math. Comput.
**2018**, 327, 70–78. [Google Scholar] [CrossRef] - Davies, B.; Martin, B. Numerical inversion of the Laplace transform: A survey and comparison of methods. J. Comput. Phys.
**1979**, 33, 1–32. [Google Scholar] [CrossRef] - Gzyl, H.; Tagliani, A.; Milev, M. Laplace transform inversion on the real line is truly ill-conditioned, Appl. Math. Comput.
**2013**, 219, 9805–9809. [Google Scholar] - Honig, G.; Hirdes, U. A method for the numerical inversion of Laplace transforms. J. Comput. Appl. Math.
**1984**, 10, 113–132. [Google Scholar] [CrossRef][Green Version] - Stehfest, H. Algorithm 368: Numerical inversion of Laplace transforms [D5]. Commun. ACM
**1970**, 13, 47–49. [Google Scholar] [CrossRef] - Stehfest, H. Remark on algorithm 368: Numerical inversion of Laplace transforms. Commun. ACM
**1970**, 13, 624. [Google Scholar] [CrossRef] - Wójcik, M.; Szukiewicz, M.; Kowalik, P. Application of Numerical Laplace Inversion Methods in Chemical Engineering with Maple
^{®}. J. Appl. Comput. Sci. Methods**2015**, 7, 5–15. [Google Scholar] [CrossRef] - Marin, M.; Öchsner, A. The effect of a dipolar structure on the Hölder stability in Green–Naghdi thermoelasticity, Contin. Mech. Thermodyn.
**2017**, 29, 1365–1374. [Google Scholar] [CrossRef] - Luminiţa, S.M.; Sorin, V.; Marin, M.; Arina, M. New analytical method based on dynamic response of planar mechanical elastic systems. Bound. Val. Probl.
**2020**, 2020, 104. [Google Scholar] - Abouelregal, A.E.; Marin, M. The response of nanobeams with temperature-dependent properties using state-space method via modified couple stress theory. Symmetry
**2020**, 12, 1276. [Google Scholar] [CrossRef] - Marin, M.; Ellahi, R.; Vlase, S.; Bhatti, M.M. On the decay of exponential type for the solutions in a dipolar elastic body. J. Taibah Univ. Sci.
**2020**, 14, 534–540. [Google Scholar] [CrossRef][Green Version]

Quantity | Symbol | Value |
---|---|---|

Density | $\rho $ | $2332$$(\mathrm{kg}/{\mathrm{m}}^{3})$ |

Young’s Modulus | $E$ | $120$ GPa |

Length | $L$ | $600$ μm |

Width | $b$ | $20$ μm |

Thickness | $h$ | $10$ μm |

Thermal expansion coefficient | ${\alpha}_{t}$ | $2.59\times {10}^{-6}$ (1/K) |

Thermal conductivity | $K$ | $165$ W/(m K) |

Specific heat | ${C}_{E}$ | $130$ J/(kg K) |

Electrical conductivity | ${\sigma}_{e}$ | $0.78\times {10}^{4}$ (S/m) |

Poisson’s ratio | $\nu $ | 0.22 |

Laser intensity | ${I}_{0}$ | $13.4$$(\mathrm{W}/{\mathrm{m}}^{2}$) |

Laser penetration depth | $d$ | $15.3\times {10}^{-9}$ m |

Constant parameter | $\beta $ | 1.992 |

Ambient temperature | ${T}_{0}$ | $300$ K |

Laser pulse duration | ${t}_{p}$ | $100\times {10}^{-15}$ s |

Relaxation time | ${\tau}_{0}$ | $8.5\times {10}^{-12}$ s |

Surface reflectivity | $R$ | $0.93$ |

**Table 2.**The influence of visco-Pasternak parameters ${K}_{p}$, ${K}_{w}$, and ${K}_{v}$ on the studied fields.

${\mathit{K}}_{\mathit{v}}$ | ${\mathit{K}}_{\mathit{w}}$ | ${\mathit{K}}_{\mathit{p}}$ | Thermo-Physical Fields | ||||
---|---|---|---|---|---|---|---|

$\mathit{w}$ | $\mathit{\theta}$ | $\mathit{u}$ | ${\mathit{\sigma}}_{\mathit{x}}$ | $\mathit{M}$ | |||

0.2 | 0.0 | 0.0 | −0.0609143 | 2.85630 | 0.101218 | −0.564227 | −0.406249 |

0.1 | 0.2 | −0.0597582 | 2.85663 | 0.101314 | −0.558596 | −0.406141 | |

0.2 | 0.3 | −0.0591818 | 2.85680 | 0.101359 | −0.555789 | −0.405988 | |

0.3 | 0.4 | −0.0586066 | 2.85696 | 0.101402 | −0.552988 | −0.405758 | |

0.4 | 0.0 | 0.0 | −0.0608022 | 2.85631 | 0.101190 | −0.563906 | −0.406248 |

0.1 | 0.2 | −0.0601662 | 2.85649 | 0.101243 | −0.560807 | −0.406133 | |

0.2 | 0.3 | −0.0590141 | 2.85682 | 0.101335 | −0.555195 | −0.405971 | |

0.3 | 0.4 | −0.0584398 | 2.85699 | 0.101378 | −0.552398 | −0.405727 | |

0.6 | 0.0 | 0.0 | −0.060650 | 2.85632 | 0.101152 | −0.563469 | −0.406246 |

0.1 | 0.2 | −0.059497 | 2.85665 | 0.101247 | −0.557849 | −0.406136 | |

0.2 | 0.3 | −0.0589221 | 2.85682 | 0.101291 | −0.555049 | −0.405982 | |

0.3 | 0.4 | −0.0583485 | 2.85698 | 0.101334 | −0.552255 | −0.405750 | |

0.8 | 0.0 | 0.0 | −0.0604983 | 2.85633 | 0.101114 | −0.563033 | −0.406244 |

0.1 | 0.2 | −0.0593470 | 2.85667 | 0.101208 | −0.557421 | −0.406133 | |

0.2 | 0.3 | −0.0587731 | 2.85683 | 0.101252 | −0.554624 | −0.405978 | |

0.3 | 0.4 | −0.0582004 | 2.85700 | 0.101295 | −0.551833 | −0.405747 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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**

Abouelregal, A.E.; Marin, M.; Askar, S.S. Generalized MGT Heat Transfer Model for an Electro-Thermal Microbeam Lying on a Viscous-Pasternak Foundation with a Laser Excitation Heat Source. *Symmetry* **2023**, *15*, 814.
https://doi.org/10.3390/sym15040814

**AMA Style**

Abouelregal AE, Marin M, Askar SS. Generalized MGT Heat Transfer Model for an Electro-Thermal Microbeam Lying on a Viscous-Pasternak Foundation with a Laser Excitation Heat Source. *Symmetry*. 2023; 15(4):814.
https://doi.org/10.3390/sym15040814

**Chicago/Turabian Style**

Abouelregal, Ahmed E., Marin Marin, and Sameh S. Askar. 2023. "Generalized MGT Heat Transfer Model for an Electro-Thermal Microbeam Lying on a Viscous-Pasternak Foundation with a Laser Excitation Heat Source" *Symmetry* 15, no. 4: 814.
https://doi.org/10.3390/sym15040814