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

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## 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

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

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

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**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