# Generalized Thermoelastic Functionally Graded on a Thin Slim Strip Non-Gaussian Laser Beam

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Problem and Governing Equations

## 3. Theoretical Analysis

#### 3.1. The Non-Gaussian Laser Pulse

#### 3.2. Nondimensionalization

#### 3.3. The Solution to the Problem Considering LT Domain

#### 3.4. Exponential Variation of Nonhomogeneity

#### 3.5. Inversion of the LTs

## 4. Numerical Example and Discussion

_{0}= 10

^{11}J.m

^{−2}). Therefore, we adopted the Mathematica programming language for all numerical calculations.

## 5. Conclusions

- 1-
- The nonhomogeneity parameter significantly influences the solutions of displacement, temperature, stress and strain.
- 2-
- The impact of relaxation time has a considerable role in all distributions.
- 3-
- The Non-Gaussian laser pulse significantly affects all of the physical quantities (temperature, displacement, stress and strain).
- 4-
- The coupled thermoelasticity theory can be extracted as a special case.
- 5-
- The finite propagation speeds are manifested in all of the displayed graphs. This is expected because of the traveling of a thermal wave with a finite speed.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Ye, G.R.; Chen, W.Q.; Cai, J.B. A uniformly heated functionally graded cylindrical shell with transverse isotropy. Mech. Res. Commun.
**2001**, 28, 535–542. [Google Scholar] [CrossRef] - El-Naggar, A.M.; Abd-Alla, A.M.; Fahmy, M.A.; Ahmed, S.M. Thermal stresses in a rotating non-homogeneous orthotropic hollow cylinder. Heat Mass Transf.
**2002**, 39, 41–46. [Google Scholar] [CrossRef] - Wang, B.L.; Mai, Y.W. Transient one-dimensional heat conduction problems solved by finite element. Int. J. Mech. Sci.
**2005**, 47, 303–317. [Google Scholar] [CrossRef] - Ootao, Y.; Tanigawa, Y. Transient thermoelastic analysis for a functionally graded hollow cylinder. J. Therm. Stresses
**2006**, 29, 1031–1046. [Google Scholar] [CrossRef] - Shao, Z.S.; Wang, T.J.; Ang, K.K. Transient thermo-mechanical analysis of functionally graded hollow circular cylinders. J. Therm. Stresses
**2007**, 30, 81–104. [Google Scholar] [CrossRef] - 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. Solid
**1967**, 15, 299–309. [Google Scholar] [CrossRef] - Green, A.E.; Lindsay, K.A. Thermoelasticity. J. Elast.
**1972**, 2, 1–7. [Google Scholar] [CrossRef] - Kao, T.T. On Thermally Induced Non-Fourier Stress Waves in a Semi-Infinite Medium. AIAA J.
**1976**, 14, 818–820. [Google Scholar] [CrossRef] - McDonald, F.A. On the Precursor in Laser-Generated Ultrasound Waveforms in Metals. Appl. Phys. Lett.
**1990**, 56, 230–232. [Google Scholar] [CrossRef] - Enguehard, F.; Bertrand, L. Effects of Optical Penetration and Laser Pulse Duration on Laser Generated Longitudinal Acoustic Waves. Appl. Phys. Lett.
**1997**, 82, 1532–1538. [Google Scholar] [CrossRef] - Trajkovski, D.; Cukic, R. A coupled problem of thermoelastic vibrations of a circular plate with exact boundary conditions. Mech. Res. Commun.
**1999**, 26, 217–224. [Google Scholar] [CrossRef] - Sun, Y.; Fang, D.; Saka, M.; Soh, A.K. Laser-induced vibrations of micro-beams under different boundary conditions. Int. J. Solids Struct.
**2008**, 45, 1993–2013. [Google Scholar] [CrossRef] [Green Version] - Al-Huniti, N.S.; Al-Nimr, M.A.; Naji, M. Dynamic response of rod due to a moving heat source under the hyperbolic heat conduction model. J. Sound Vib.
**2001**, 242, 629–640. [Google Scholar] [CrossRef] - Abouelergal, E. Generalized thermoelasticity in an infinite nonhomogeneous solid having a spherical cavity using DPL model. Appl. Math.
**2011**, 2, 271–282. [Google Scholar] - Youssef, H.M.; Al-Felali, A.S. Generalized Thermoelasticity Problem of Material Subjected to Thermal Loading Due to Laser Pulse. Applied Math.
**2012**, 3, 142–146. [Google Scholar] [CrossRef] [Green Version] - Abo-Dahab, S.M.; Abouelregal, A.E. Investigation of the vibration of micro-beam resonators induced by a harmonically varying heat. J. Comput. Theor. Nanosci.
**2015**, 12, 924–933. [Google Scholar] [CrossRef] - Elsherbeny, K.Z.; Abouelregal, A.E.; Abo-Dahab, S.M.; Rashid, A.F. Thermoelastic analysis for an infinite solid cylinder due to harmonically varying heat with thermal conductivity variable. J. Comput. Theor. Nanosci.
**2016**, 13, 4493–4500. [Google Scholar] [CrossRef] - Marin, M.; Craciun, E.M.; Pop, N. Considerations on mixed initial-boundary value problems for micropolar porous bodies. Dyn. Syst. Appl.
**2016**, 25, 175–196. [Google Scholar] - Riaz, A.; Ellahi, R.; Bhatti, M.M.; Marin, M. Study of heat and mass transfer in the Eyring–Powell model of fluid propagating peristaltically through a rectangular compliant channel. Heat Transf. Res.
**2019**, 50, 1539–1560. [Google Scholar] [CrossRef] - Bhatti, M.M.; Ellahi, R.; Zeeshan, A.; Marin, M. Numerical study of heat transfer and Hall current impact on peristaltic propulsion of particle-fluid suspension with compliant wall properties. Mod. Phys. Lett. B
**2019**, 33, 1950439. [Google Scholar] [CrossRef] - Abo-Dahab, S.M.; Abouelregal, A.E. On a two-dimensional problem in thermoelasticity half-space with microstructure subjected to a uniform thermal shock. Phys. Waves Phemomena
**2019**, 27, 56–66. [Google Scholar] [CrossRef] - Abbas, I.A. The effects of relaxation times and a moving heat source on a two-temperature generalized thermoelastic thin slim strip. Can. J. Phys.
**2015**, 93, 585–590. [Google Scholar] [CrossRef] - Abbas, I.A.; Abo-Dahab, S.M. On the numerical solution of thermal shock problem for generalized magneto-thermoelasticity for an infinitely long annular cylinder with variable thermal conductivity. J. Comput. Theor. Nanosci.
**2014**, 11, 607–618. [Google Scholar] [CrossRef] - Othman, M.I.; Abouelregal, A.E. The effect of pulsed laser radiation on a thermoviscoelastic semi-infinite solid under two-temperature theory. Arch. Thermodyn.
**2017**, 38, 77–99. [Google Scholar] [CrossRef] [Green Version] - Abouelregal, A.E.; Zenkour, A.M. A generalized thermoelastic medium subjected to pulsed laser heating via a two-temperature model. J. Theor. Appl. Mech.
**2019**, 57, 631–639. [Google Scholar] [CrossRef] - Honig, G.; Hirdes, U. A method for the numerical inversion of the Laplace transform. J. Comput. Appl. Math.
**1984**, 10, 113–132. [Google Scholar] [CrossRef] [Green Version] - Fotuhi, A.R.; Fariborz, S.J. Anti-plane analysis of a functionally graded strip with multiple cracks. Int. J. Solids Struct.
**2006**, 43, 1239–1252. [Google Scholar] [CrossRef] [Green Version] - Zhou, Y.T.; Zhang, C.; Zhong, Z.; Wang, L. Transient thermo-electro-elastic contact analysis of a sliding punch acting on a functionally graded piezoelectric strip under non-Fourier heat conduction. Eur. J. Mech. A Solids
**2019**, 73, 90–100. [Google Scholar] [CrossRef] - Ross, T.S. An analysis of a non-Gaussian, Gaussian laser beam. In Proceedings: Laser Beam Control and Applications; Laser Beam Control and Applications: San Jose, CA, USA, 2006; Volume 6101, p. 610111. [Google Scholar]

**Table 1.**Physical constants of the copper material [23].

Parameter | Value | Parameter | Value | Parameter | Value |
---|---|---|---|---|---|

$\lambda $ | $1.628\times {10}^{11}\mathrm{N}{\mathrm{m}}^{-2}$ | $K$ | $3.86\times {10}^{2}\mathrm{kg}\mathrm{m}{\mathrm{K}}^{-1}$ | ${R}_{a}$ | 0.5 |

$\mu $ | $0.362\times {10}^{11}\mathrm{N}{\mathrm{m}}^{-2}$ | ${C}_{v}$ | $0.3831\times {10}^{3}{\mathrm{m}}^{2}{\mathrm{K}}^{-1}{\mathrm{s}}^{-2}$ | b | $1.13849\times {10}^{10}$ |

$\rho $ | $8.954\times {10}^{3}\mathrm{Kg}{\mathrm{m}}^{-3}$ | ${\alpha}_{t}$ | $1.78\times {10}^{-5}{\mathrm{k}}^{-1}$ | ${\epsilon}_{1}$ | $0.0168$ |

${T}_{0}$ | $298\mathrm{K}$ | ${\delta}_{0}$ | $0.01$ | $h$ | $0.01$ |

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Abo-Dahab, S.M.; Abouelregal, A.E.; Marin, M.
Generalized Thermoelastic Functionally Graded on a Thin Slim Strip Non-Gaussian Laser Beam. *Symmetry* **2020**, *12*, 1094.
https://doi.org/10.3390/sym12071094

**AMA Style**

Abo-Dahab SM, Abouelregal AE, Marin M.
Generalized Thermoelastic Functionally Graded on a Thin Slim Strip Non-Gaussian Laser Beam. *Symmetry*. 2020; 12(7):1094.
https://doi.org/10.3390/sym12071094

**Chicago/Turabian Style**

Abo-Dahab, Sayed M., Ahmed E. Abouelregal, and Marin Marin.
2020. "Generalized Thermoelastic Functionally Graded on a Thin Slim Strip Non-Gaussian Laser Beam" *Symmetry* 12, no. 7: 1094.
https://doi.org/10.3390/sym12071094