# 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

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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 |

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**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