# The Effect of Fractional Time Derivative on Two-Dimension Porous Materials Due to Pulse Heat Flux

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## Abstract

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## 1. Introduction

## 2. Basic Equations

## 3. Formulations of the Problem

## 4. Numerical Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B

## Appendix C

## References

- Biot, M.A. Thermoelasticity and irreversible thermodynamics. J. Appl. Phys.
**1956**, 27, 240–253. [Google Scholar] [CrossRef] - Rosencwaig, A.; Opsal, J.; Willenborg, D.L. Thin-film thickness measurements with thermal waves. Appl. Phys. Lett.
**1983**, 43, 166–168. [Google Scholar] [CrossRef] - Biot, M.A. General solutions of the equations of elasticity and consolidation for a porous material. J. Appl. Mech.
**1956**, 23, 91–96. [Google Scholar] - Biot, M.A. Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range. J. Acoust. Soc. Am.
**1956**, 28, 179–191. [Google Scholar] [CrossRef] - Lord, H.W.; Shulman, Y. A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids
**1967**, 15, 299–309. [Google Scholar] [CrossRef] - Green, A.E.; Naghdi, P.M. A re-examination of the basic postulates of thermomechanics. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci.
**1991**, 432, 171–194. [Google Scholar] - Green, A.; Naghdi, P. On undamped heat waves in an elastic solid. J. Therm. Stresses
**1992**, 15, 253–264. [Google Scholar] [CrossRef] - Green, A.; Naghdi, P. Thermoelasticity without energy dissipation. J. Elast.
**1993**, 31, 189–208. [Google Scholar] [CrossRef] - Youssef, H.M. Theory of fractional order generalized thermoelasticity. J. Heat Transf.
**2010**, 132, 061301. [Google Scholar] [CrossRef] - Youssef, H.M.; Al-Lehaibi, E.A. Variational principle of fractional order generalized thermoelasticity. Appl. Math. Lett.
**2010**, 23, 1183–1187. [Google Scholar] [CrossRef] [Green Version] - Sherief, H.H.; El-Sayed, A.M.A.; Abd El-Latief, A.M. Fractional order theory of thermoelasticity. Int. J. Solids Struct.
**2010**, 47, 269–275. [Google Scholar] [CrossRef] - Ezzat, M.A.; El Karamany, A.S. Theory of fractional order in electro-thermoelasticity. Eur. J. Mech. A Solids
**2011**, 30, 491–500. [Google Scholar] [CrossRef] - Ezzat, M.; El-Karamany, A.; El-Bary, A. Modeling of memory-dependent derivative in generalized thermoelasticity. Eur. Phys. J. Plus
**2016**, 131, 372. [Google Scholar] [CrossRef] - Marin, M. Some basic theorems in elastostatics of micropolar materials with voids. J. Comput. Appl. Math.
**1996**, 70, 115–126. [Google Scholar] [CrossRef] [Green Version] - Saeed, T.; Abbas, I.; Marin, M. A GL Model on Thermo-Elastic Interaction in a Poroelastic Material Using Finite Element Method. Symmetry
**2020**, 12, 488. [Google Scholar] [CrossRef] [Green Version] - Ouyang, X.L.; Xu, R.N.; Jiang, P.X. Three-equation local thermal non-equilibrium model for transient heat transfer in porous media: The internal thermal conduction effect in the solid phase. Int. J. Heat Mass Transf.
**2017**, 115, 1113–1124. [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] - Zenkour, A.M.; Abbas, I.A. Magneto-thermoelastic response of an infinite functionally graded cylinder using the finite element method. J. Vib. Control
**2014**, 20, 1907–1919. [Google Scholar] [CrossRef] - Abbas, I.A. Nonlinear transient thermal stress analysis of thick-walled FGM cylinder with temperature-dependent material properties. Meccanica
**2014**, 49, 1697–1708. [Google Scholar] [CrossRef] - El-Naggar, A.M.; Kishka, Z.; Abd-Alla, A.M.; Abbas, I.A.; Abo-Dahab, S.M.; Elsagheer, M. On the initial stress, magnetic field, voids and rotation effects on plane waves in generalized thermoelasticity. J. Comput. Theor. Nanosci.
**2013**, 10, 1408–1417. [Google Scholar] [CrossRef] - Sur, A.; Mondal, S. A generalized thermoelastic problem due to nonlocal effect in presence of mode I crack. J. Therm. Stresses
**2020**, 43, 1277–1299. [Google Scholar] [CrossRef] - Othman, M.I.; Mondal, S. Memory-dependent derivative effect on wave propagation of micropolar thermoelastic medium under pulsed laser heating with three theories. Int. J. Numer. Methods Heat Fluid Flow
**2019**, 30, 1025–1046. [Google Scholar] [CrossRef] - Abbas, I.A.; El-Amin, M.; Salama, A. Effect of thermal dispersion on free convection in a fluid saturated porous medium. Int. J. Heat Fluid Flow
**2009**, 30, 229–236. [Google Scholar] [CrossRef] - Hussein, E.M. Mathematical Model for Thermoelastic Porous Spherical Region Problems. Comput. Therm. Sci. Int. J.
**2020**, 12, 233–248. [Google Scholar] [CrossRef] - Hobiny, A.; Abbas, I. Generalized thermoelastic interaction in a two-dimensional porous medium under dual phase lag model. Int. J. Numer. Methods Heat Fluid Flow
**2020**, 30, 4865–4881. [Google Scholar] [CrossRef] - Biswas, S. Surface waves in porous nonlocal thermoelastic orthotropic medium. Acta Mech.
**2020**, 231, 2741–2760. [Google Scholar] [CrossRef] - Carini, M.; Zampoli, V. On porous matrices with three delay times: A study in linear thermoelasticity. Mathematics
**2020**, 8, 371. [Google Scholar] [CrossRef] [Green Version] - Abbas, I.A.; Marin, M. Analytical Solutions of a Two-Dimensional Generalized Thermoelastic Diffusions Problem Due to Laser Pulse. Iran. J. Sci. Technol. Trans. Mech. Eng.
**2018**, 42, 57–71. [Google Scholar] [CrossRef] - Shekhar, S. Study of deformation due to thermal shock in porous thermoelastic material with reference temperature dependent properties. Mech. Based Des. Struct. Mach.
**2020**. [Google Scholar] [CrossRef] - Itu, C.; Öchsner, A.; Vlase, S.; Marin, M.I. Improved rigidity of composite circular plates through radial ribs. Proc. Inst. Mech. Eng. Part L J. Mater. Des. Appl.
**2019**, 233, 1585–1593. [Google Scholar] [CrossRef] - Abd-Elaziz, E.M.; Marin, M.; Othman, M.I. On the effect of Thomson and initial stress in a thermo-porous elastic solid under GN electromagnetic theory. Symmetry
**2019**, 11, 413. [Google Scholar] [CrossRef] [Green Version] - Abbas, I.A.; Kumar, R. Deformation due to thermal source in micropolar generalized thermoelastic half-space by finite element method. J. Comput. Theor. Nanosci.
**2014**, 11, 185–190. [Google Scholar] [CrossRef] - Sheikholeslami, M.; Ellahi, R.; Shafee, A.; Li, Z. Numerical investigation for second law analysis of ferrofluid inside a porous semi annulus: An application of entropy generation and exergy loss. Int. J. Numer. Methods Heat Fluid Flow
**2019**, 29, 1079–1102. [Google Scholar] [CrossRef] - Ellahi, R.; Sait, S.M.; Shehzad, N.; Ayaz, Z. A hybrid investigation on numerical and analytical solutions of electro-magnetohydrodynamics flow of nanofluid through porous media with entropy generation. Int. J. Numer. Methods Heat Fluid Flow
**2019**, 30, 834–854. [Google Scholar] [CrossRef] - Singh, B. Wave propagation in a generalized thermoelastic material with voids. Appl. Math. Comput.
**2007**, 189, 698–709. [Google Scholar] [CrossRef] - Palani, G.; Abbas, I. Free convection MHD flow with thermal radiation from an impulsively started vertical plate. Nonlinear Anal. Model. Control
**2009**, 14, 73–84. [Google Scholar] [CrossRef] [Green Version] - Villatoro, F.R.; Pérez, J.; Santander, J.L.G.; Borovsky, M.A.; Ratis, Y.L.; Izzheurov, E.A.; de Córdoba, P.F. Perturbation analysis of the heat transfer in porous media with small thermal conductivity. J. Math. Anal. Appl.
**2011**, 374, 57–70. [Google Scholar] [CrossRef] [Green Version] - Abbas, I. Natural frequencies of a poroelastic hollow cylinder. Acta Mech.
**2006**, 186, 229–237. [Google Scholar] [CrossRef] - Alzahrani, F.; Hobiny, A.; Abbas, I.; Marin, M. An Eigenvalues Approach for a Two-Dimensional Porous Medium Based Upon Weak, Normal and Strong Thermal Conductivities. Symmetry
**2020**, 12, 848. [Google Scholar] [CrossRef] - Hobiny, A.; Alzahrani, F.; Abbas, A.; Marin, M. The Effect of fractional time derivative of bioheat model in skin tissue induced to laser irradiation. Symmetry
**2020**, 12, 602. [Google Scholar] [CrossRef] - Das, N.C.; Lahiri, A.; Giri, R.R. Eigenvalue approach to generalized thermoelasticity. Indian J. Pure Appl. Math.
**1997**, 28, 1573–1594. [Google Scholar] - Abbas, I.A.; Alzahrani, F.S.; Elaiw, A. A DPL model of photothermal interaction in a semiconductor material. Waves Random Complex Media
**2019**, 29, 328–343. [Google Scholar] [CrossRef] - Saeed, T.; Abbas, I. Thermomechanical response in a two-dimension porous medium subjected to thermal loading. Int. J. Numer. Methods Heat Fluid Flow
**2019**, 30, 4103–4117. [Google Scholar] [CrossRef] - Stehfest, H. Algorithm 368: Numerical inversion of Laplace transforms [D5]. Commun. ACM
**1970**, 13, 47–49. [Google Scholar] [CrossRef] - Othman, M.I.; Marin, M. Effect of thermal loading due to laser pulse on thermoelastic porous medium under GN theory. Results Phys.
**2017**, 7, 3863–3872. [Google Scholar] [CrossRef]

**Figure 1.**The change in volume fraction field of void variations $\phi $ via $x$ and $y=0.4$ for strong normal and weak conductivities.

**Figure 2.**The variation of temperature $\mathsf{\Theta}$ via $x$ and $y=0.4$ for strong normal and weak conductivities.

**Figure 3.**The variations of vertical displacement $v$ via $x$ with $y=0.4$ for strong normal and weak conductivities.

**Figure 4.**The variations of horizontal displacement $u$ via $x$ with $y=0.4$ for strong normal and weak conductivities.

**Figure 5.**The variation of stress ${\sigma}_{xy}$ via $x$ with $y=0.4$ for strong normal and weak conductivities.

**Figure 6.**The variation of stress ${\sigma}_{xx}$ via $x$ and $y=0.4$ for strong normal and weak conductivities.

**Figure 7.**The change in volume fraction field of void variations $\phi $ via $y$ and $x=0.4$ for strong normal and weak conductivities.

**Figure 8.**The variations of temperature $\mathsf{\Theta}$ via $y$ and $x=0.4$ for strong normal and weak conductivities.

**Figure 9.**The variations of vertical displacement $v$ via $y$ with $x=0.4$ for strong normal and weak conductivities.

**Figure 10.**The variations of horizontal displacement $u$ via $y$ with $x=0.4$ for strong normal and weak conductivities.

**Figure 11.**The variation of stresses component ${\sigma}_{xy}$ versus $y$ when $x=0.4$ for strong normal and weak conductivities.

**Figure 12.**The variation of stresses components ${\sigma}_{xx}$ versus $y$ when $x=0.4$ for strong normal and weak conductivities.

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Saeed, T.; A. Abbas, I.
The Effect of Fractional Time Derivative on Two-Dimension Porous Materials Due to Pulse Heat Flux. *Mathematics* **2021**, *9*, 207.
https://doi.org/10.3390/math9030207

**AMA Style**

Saeed T, A. Abbas I.
The Effect of Fractional Time Derivative on Two-Dimension Porous Materials Due to Pulse Heat Flux. *Mathematics*. 2021; 9(3):207.
https://doi.org/10.3390/math9030207

**Chicago/Turabian Style**

Saeed, Tareq, and Ibrahim A. Abbas.
2021. "The Effect of Fractional Time Derivative on Two-Dimension Porous Materials Due to Pulse Heat Flux" *Mathematics* 9, no. 3: 207.
https://doi.org/10.3390/math9030207