Fractional Heat Conduction with Variable Thermal Conductivity of Infinite Annular Cylinder Under Thermoelasticity Theorem of Moore–Gibson–Thompson
Abstract
1. Introduction
2. Basic Formulations
3. The Problem Formulation
4. Numerical Results and Discussion
5. Conclusions and Future Work
- •
- The distributions of the temperature increment and stress components are significantly impacted by the fractional-order parameter, which has a considerable impact.
- •
- When the fractional-order parameter is increased, the temperature increment and stress component distributions also rise. This is because of the relationship between the two.
- •
- The values of the temperature increment and the absolute values of the stress components for the state of strong thermal conductivity are greater than their values in the state of normal thermal conductivity and than those in the state of weak thermal conductivity. This is because weak thermal conductivity has a lower thermal conductivity than strong thermal conductivity.
- •
- The distribution of the cubical deformation is not significantly affected by the fractional-order parameter in any way.
- •
- The thermal conductivity parameter has a significant influence on the distribution of the temperature rise as well as the stress components. When the thermal conductivity parameter is increased, the absolute value of the stress components and temperature increment distributions also rise. This is because the thermal conductivity parameter is increased.
- •
- In the strong thermal conductivity case, the values of the temperature increment and axial stress component for the three cases of thermal conductivity are closer to each other than in the usual thermal conductivity case and than in the weak thermal conductivity case.
- •
- The variation in the fractional-order parameter does not make any change in the differences between the three cases of thermal conductivity for the invariant average stress and the cubical deformation.
- •
- While the heat conductivity parameter does have some influence on the distribution of the cubical deformation, it is not very significant.
6. Future Work
Funding
Data Availability Statement
Conflicts of Interest
Nomenclature
The specific heat without deformation | |
The speed of the longitudinal wave | |
Hyperbolic two-temperature parameter | |
The cubical deformation | |
Diffusivity | |
Thermal conductivity | |
Thermal conductivity at room temperature | |
The rate of the thermal conductivity | |
The rate of the thermal conductivity at room temperature | |
The thermal conductivity parameter | |
Temperature | |
Reference temperature | |
Time | |
Thermal relaxation time | |
Lag-time of the temperature gradient | |
The radius | |
The displacement function | |
The z-axis | |
The linear thermal expansion coefficient | |
= | |
The mechanical coupling’s coefficient | |
The thermoelastic coupling’s coefficient | |
The coefficient of the thermal viscosity | |
Lamé’s parameters | |
The stress tensor | |
The density |
References
- Biot, M.A. Thermoelasticity and irreversible thermodynamics. J. Appl. Phys. 1956, 27, 240–253. [Google Scholar] [CrossRef]
- Sherief, H.; Naim Anwar, M.; Abd El-Latief, A.; Fayik, M.; Tawfik, A. A fully coupled system of generalized thermoelastic theory for semiconductor medium. Sci. Rep. 2024, 14, 13876. [Google Scholar] [CrossRef] [PubMed]
- Svanadze, M. On the coupled theory of thermoelastic double-porosity materials. J. Therm. Stress. 2022, 45, 576–596. [Google Scholar] [CrossRef]
- Youssef, H.M. Stat-Space Approach to Three-Dimensional Thermoelastic Half-Space Based on Fractional Order Heat Conduction and Variable Thermal Conductivity Under Moor–Gibson–Thompson Theorem. Fractal Fract. 2025, 9, 145. [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]
- Alihemmati, J.; Beni, Y.T. Generalized thermoelasticity of microstructures: Lord-Shulman theory with modified strain gradient theory. Mech. Mater. 2022, 172, 104412. [Google Scholar] [CrossRef]
- Hou, W.; Fu, L.-Y.; Carcione, J.M.; Wang, Z.; Wei, J. Simulation of thermoelastic waves based on the Lord-Shulman theory. Geophysics 2021, 86, T155–T164. [Google Scholar] [CrossRef]
- Bouslimi, J.; Omri, M.; Kilany, A.; Abo-Dahab, S.; Hatem, A. Mathematical model on a photothermal and voids in a semiconductor medium in the context of Lord-Shulman theory. Waves Random Complex Media 2024, 34, 5594–5611. [Google Scholar] [CrossRef]
- Green, A.; Naghdi, P. On undamped heat waves in an elastic solid. J. Therm. Stress. 1992, 15, 253–264. [Google Scholar] [CrossRef]
- Green, A.; Naghdi, P. Thermoelasticity without energy dissipation. J. Elast. 1993, 31, 189–208. [Google Scholar] [CrossRef]
- Hendy, M.M.; Ezzat, M.A. A modified Green-Naghdi fractional order model for analyzing thermoelectric MHD. Int. J. Numer. Methods Heat Fluid Flow 2024, 34, 2376–2398. [Google Scholar] [CrossRef]
- Jagtap, A.D.; Mitsotakis, D.; Karniadakis, G.E. Deep learning of inverse water waves problems using multi-fidelity data: Application to Serre–Green–Naghdi equations. Ocean Eng. 2022, 248, 110775. [Google Scholar] [CrossRef]
- Green, A.; Lindsay, K. Thermoelasticity. J. Elast. 1972, 2, 1–7. [Google Scholar] [CrossRef]
- Sharifi, H. Dynamic response of an orthotropic hollow cylinder under thermal shock based on Green–Lindsay theory. Thin-Walled Struct. 2023, 182, 110221. [Google Scholar] [CrossRef]
- Karimipour Dehkordi, M.; Kiani, Y. Lord–Shulman and Green–Lindsay-based magneto-thermoelasticity of hollow cylinder. Acta Mech. 2024, 235, 51–72. [Google Scholar] [CrossRef]
- Danyluk, M.; Geubelle, P.; Hilton, H. Two-dimensional dynamic and three-dimensional fracture in viscoelastic materials. Int. J. Solids Struct. 1998, 35, 3831–3853. [Google Scholar] [CrossRef]
- Ezzat, M.A.; Youssef, H.M. Three-dimensional thermal shock problem of generalized thermoelastic half-space. Appl. Math. Model. 2010, 34, 3608–3622. [Google Scholar] [CrossRef]
- Ezzat, M.A.; Youssef, H.M. Three-dimensional thermo-viscoelastic material. Mech. Adv. Mater. Struct. 2016, 23, 108–116. [Google Scholar] [CrossRef]
- Youssef, H.; Abbas, I. Thermal shock problem of generalized thermoelasticity for an infinitely long annular cylinder with variable thermal conductivity. Comput. Methods Sci. Technol. 2007, 13, 95–100. [Google Scholar] [CrossRef]
- Youssef, H.M. Two-dimensional thermal shock problem of fractional order generalized thermoelasticity. Acta Mech. 2012, 223, 1219–1231. [Google Scholar] [CrossRef]
- Hafed, Z.S.; Zenkour, A.M. Refined generalized theory for thermoelastic waves in a hollow sphere due to maintained constant temperature and radial stress. Case Stud. Therm. Eng. 2025, 68, 105905. [Google Scholar] [CrossRef]
- Khader, S.; Marrouf, A.; Khedr, M. A model for elastic half space under a visco-elastic layer in generalized thermoelasticity. Contin. Mech. Thermodyn. 2025, 37, 25. [Google Scholar] [CrossRef]
- Yahya, A.; Saidi, A. Response of Generalized Thermoelastic for Free Vibration of a Solid Cylinder with Voids Under a Dual-Phase Lag Model. Iran. J. Sci. Technol. Trans. Mech. Eng. 2025, 1–11. [Google Scholar] [CrossRef]
- Li, S.; Sun, J.; Zhu, J. Analytical solution of dual-phase-lagging generalized thermoelastic damping in Levinson micro/nano rectangular plates. J. Therm. Stress. 2025, 1–19. [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]
- Singh, B.; Mukhopadhyay, S. Galerkin-type solution for the Moore–Gibson–Thompson thermoelasticity theory. Acta Mech. 2021, 232, 1273–1283. [Google Scholar] [CrossRef]
- Bazarra, N.; Fernández, J.R.; Quintanilla, R. Analysis of a Moore–Gibson–Thompson thermoelastic problem. J. Comput. Appl. Math. 2021, 382, 113058. [Google Scholar] [CrossRef]
- Fernández Sare, H.D.; Quintanilla, R. Moore Gibson Thompson thermoelastic plates: Comparisons. J. Evol. Equ. 2023, 23, 70. [Google Scholar] [CrossRef]
- Mondal, S.; Srivastava, A.; Mukhopadhyay, S. Thermoelastic wave propagation and reflection in biological tissue under nonlocal elasticity and Moore–Gibson–Thompson heat conduction: Modeling and analysis. Z. Angew. Math. Phys. 2025, 76, 30. [Google Scholar] [CrossRef]
- Kadian, P.; Kumar, S.; Sangwan, M. Influence of initial stress and gravity on fiber-reinforced thermoelastic solid using Moore–Gibson–Thompson generalized theory of thermoelasticity. Multidiscip. Model. Mater. Struct. 2025, 21, 217–238. [Google Scholar] [CrossRef]
- Ahmed, Y.; Zakria, A.; Osman, O.A.A.; Suhail, M.; Rabih, M.N.A. Fractional Moore–Gibson–Thompson Heat Conduction for Vibration Analysis of Non-Local Thermoelastic Micro-Beams on a Viscoelastic Pasternak Foundation. Fractal Fract. 2025, 9, 118. [Google Scholar] [CrossRef]
- Heymans, N.; Podlubny, I. Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheol. Acta 2006, 45, 765–771. [Google Scholar] [CrossRef]
- Meral, F.; Royston, T.; Magin, R. Fractional calculus in viscoelasticity: An experimental study. Commun. Nonlinear Sci. Numer. Simul. 2010, 15, 939–945. [Google Scholar] [CrossRef]
- Youssef, H.M. Theory of fractional order generalized thermoelasticity. J. Heat Transf. 2010, 132, 061301. [Google Scholar] [CrossRef]
- Youssef, H.M. State-space approach to fractional order two-temperature generalized thermoelastic medium subjected to moving heat source. Mech. Adv. Mater. Struct. 2013, 20, 47–60. [Google Scholar] [CrossRef]
- Youssef, H.M.; Al-Lehaibi, E.A. Fractional order generalized thermoelastic half-space subjected to ramp-type heating. Mech. Res. Commun. 2010, 37, 448–452. [Google Scholar] [CrossRef]
- Hasselman, D.P. Thermal Stresses in Severe Environments; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Hetnarski, R.B.; Eslami, M.R.; Gladwell, G. Thermal Stresses: Advanced Theory and Applications; Springer: Berlin/Heidelberg, Germany, 2009; Volume 158. [Google Scholar]
- Bahar, L.Y.; Hetnarski, R.B. State space afproach to thermoelasticity. J. Therm. Stress. 1978, 1, 135–145. [Google Scholar] [CrossRef]
- Youssef, H.M. Generalized thermoelasticity of an infinite body with a cylindrical cavity and variable material properties. J. Therm. Stress. 2005, 28, 521–532. [Google Scholar] [CrossRef]
- Youssef, H.M. Two-Temperature Generalized Thermoelastic Infinite Medium with Cylindrical Cavity Subjected to Different Types of Thermal Loading. WSEAS Trans. Heat Mass Transf. 2006, 1, 769. [Google Scholar]
- Youssef, H.M. Problem of generalized thermoelastic infinite medium with cylindrical cavity subjected to a ramp-type heating and loading. Arch. Appl. Mech. 2006, 75, 553–565. [Google Scholar] [CrossRef]
- Youssef, H.M. Generalized thermoelastic infinite medium with cylindrical cavity subjected to moving heat source. Mech. Res. Commun. 2009, 36, 487–496. [Google Scholar] [CrossRef]
- Youssef, H.M. Two-temperature generalized thermoelastic infinite medium with cylindrical cavity subjected to moving heat source. Arch. Appl. Mech. 2010, 80, 1213–1224. [Google Scholar] [CrossRef]
- Ezzat, M.A.; Youssef, H.M. Generalized Magneto-Thermoelasticity For An Infinite Perfect Conducting Body with A Cylindrical Cavity. Mater. Phys. Mech. 2013, 18, 156–170. [Google Scholar]
- Ezzat, M.; El-Bary, A. Effects of variable thermal conductivity and fractional order of heat transfer on a perfect conducting infinitely long hollow cylinder. Int. J. Therm. Sci. 2016, 108, 62–69. [Google Scholar] [CrossRef]
- Youssef, H.M. Theory of generalized thermoelasticity with fractional order strain. J. Vib. Control 2016, 22, 3840–3857. [Google Scholar] [CrossRef]
- El-Bary, A.A.; Youssef, H.M.; Nasr, M.A.E. Hyperbolic two-temperature generalized thermoelastic infinite medium with cylindrical cavity subjected to the non-Gaussian laser beam. J. Umm Al-Qura Univ. Eng. Archit. 2022, 13, 62–69. [Google Scholar] [CrossRef]
- Abbas, I.; Saeed, T.; Alhothuali, M. Hyperbolic two-temperature photo-thermal interaction in a semiconductor medium with a cylindrical cavity. Silicon 2021, 13, 1871–1878. [Google Scholar] [CrossRef]
- Tzou, D.Y. A unified field approach for heat conduction from macro-to micro-scales. J. Heat Transf. 1995, 117, 8–16. [Google Scholar] [CrossRef]
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. |
© 2025 by the author. 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
Al-Lehaibi, E.A.N. Fractional Heat Conduction with Variable Thermal Conductivity of Infinite Annular Cylinder Under Thermoelasticity Theorem of Moore–Gibson–Thompson. Fractal Fract. 2025, 9, 272. https://doi.org/10.3390/fractalfract9050272
Al-Lehaibi EAN. Fractional Heat Conduction with Variable Thermal Conductivity of Infinite Annular Cylinder Under Thermoelasticity Theorem of Moore–Gibson–Thompson. Fractal and Fractional. 2025; 9(5):272. https://doi.org/10.3390/fractalfract9050272
Chicago/Turabian StyleAl-Lehaibi, Eman A. N. 2025. "Fractional Heat Conduction with Variable Thermal Conductivity of Infinite Annular Cylinder Under Thermoelasticity Theorem of Moore–Gibson–Thompson" Fractal and Fractional 9, no. 5: 272. https://doi.org/10.3390/fractalfract9050272
APA StyleAl-Lehaibi, E. A. N. (2025). Fractional Heat Conduction with Variable Thermal Conductivity of Infinite Annular Cylinder Under Thermoelasticity Theorem of Moore–Gibson–Thompson. Fractal and Fractional, 9(5), 272. https://doi.org/10.3390/fractalfract9050272