Research Article
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Thermoelastic Analysis For A Thick Plate Under The Radiation Boundary Conditions

Year 2023, , 35 - 44, 01.06.2023

G. Dhameja L. Khalsa Vinod Varghese

https://doi.org/10.5541/ijot.1170342

Abstract

A fractional Cattaneo model for studying the thermoelastic response for a finite thick circular plate with source function is considered. The thick plate is subjected to radiation-type boundary conditions on the upper and lower surfaces, and its curved surface is kept at zero temperature. The theory of integral transformations is used to solve the generalized fractional Cattaneo-type, classical Cattaneo-Vernotte and Fourier heat conduction model. The analytical expressions of displacement components using thermoelastic displacement potentials; and thermal-stress distribution are computed and depicted graphically. The effects of the fractional-order parameter and the relaxation time on the temperature fields and their thermal stresses are investigated. The findings show that the higher the fractional-order parameter, the higher the thermal response. The greater the relaxation period, the longer the heat flux propagates on thick structures.

Keywords

Fractional Cattaneo-type equation fractional calculus non-Fourier heat conduction thick plate thermal stress integral transform

References

  • M. Haskul, "Elastic state of functionally graded curved beam on the plane stress state subject to thermal load," Mech. Based Des. Struct. Mach., 48 (6), 739-754, 2020. DOI: 10.1080/15397734.2019.1660890.
  • E. Arslan, M. Haskul, "Generalized plane strain solution of a thick-walled cylindrical panel subjected to radial heating," Acta Mech, 226, 1213–1225, 2015. https://doi.org/10.1007/s00707-014-1248-4
  • M. Haskul, E. Arslan and W. Mack, "Radial heating of a thick-walled cylindrically curved FGM-panel," Z. Angew. Math. Mech., 97, 309-321, 2017. https://doi.org/10.1002/zamm.201500310
  • M. Haskul, "Yielding of functionally graded curved beam subjected to temperature," Pamukkale University Journal of Engineering Sciences, 26 (4), 587-593, 2020. DOI: 10.5505/pajes.2019.92331
  • E. Hoashi, T. Yokomine, A. Shimizu, and T. Kunugi, "Numerical analysis of wave-type heat transfer propagating in a thin foil irradiated by short-pulsed laser," Int. J. Heat Mass Transf., 46 (19), 4083–4095, 2003. DOI: 10.1016/S0017-9310(03)00225-4.
  • X. Ai and B. Q. Li, "Numerical simulation of thermal wave propagation during laser processing of thin films," J. Electron. Mater., 34 (5), 583–591, 2005. DOI: 10.1007/s11664-005-0069-6.
  • T. T. Lam and E. Fong, "Application of solution structure theorem to non-Fourier heat conduction problems: Analytical approach," Int. J. Heat Mass Transf., 54, 4796–4806, 2011. DOI: 10.1016/j.ijheatmasstransfer.2011.06.028.
  • T. T. Lam, "A unified solution of several heat conduction models," Int. J. Heat Mass Transf., 56 (1–2), 653–666, 2013. DOI: 10.1016/j.ijheatmasstransfer.2012.08.055.
  • C. Cattaneo, “Sur uneforme de l’équation de la chaleuréliminant le paradoxed’une propagation instantanée,” C. R. Acad. Sci., 247, 431–433, 1958.
  • P. Vernotte, “Les paradoxes de la théorie continue de l’équation de la chaleur,” C. R. Acad. Sci., 246, 3154-3155, 1958.
  • A. Compte and R. Metzler, "The generalized Cattaneo equation for the description of anomalous transport processes," J. Phys. A: Math. Gen., 30, 7277-7289, 1997.
  • F. M. Jiang, D. Y. Liu, and J. H. Zhou, "Non-Fourier heat conduction phenomena in porous material heated by microsecond laser pulse," Microscale Thermophys. Eng., 6 (4), 331–346, 2003. DOI: 10.1080/10893950290098386.
  • Y. Povstenko, Fractional thermoelasticty, Springer, New York, 2015.
  • Y. Povstenko, "Fractional heat conduction equation and associated thermal stress," J. Therm. Stresses, 28 (1), 83–102, 2005.
  • Y. Povstenko, "Fractional Cattaneo-type equations and generalized thermoelasticity," J. Therm. Stresses, 34 (2), 97-114, 2011. DOI: 10.1080/01495739.2010.511931.
  • T. N. Mishra and K. N. Rai, "Numerical solution of FSPL heat conduction equation for analysis of thermal propagation," Appl. Math. Comput., 273, 1006–1017, 2016. DOI: 10.1016/j.amc.2015.10.082.
  • H. Qi, H. Xu, and X. Guo, "The Cattaneo-type time fractional heat conduction equation for laser heating," Comput. Math. Appl., 66 (5), 824–831, 2013. DOI: 10.1016/j.camwa.2012.11.021.
  • H. Qi, and X. Guo, "Transient fractional heat conduction with generalized Cattaneo model," Int. J. Heat Mass Transf., 76, 535–539, 2014.
  • H. Xu, H. Qi, and X. Jiang, "Fractional Cattaneo heat equation on a semi-infinite medium," Chin. Phys. B, 22 (1), 014401, 2013. DOI: 10.1088/1674-1056/22/1/014401.
  • G. Xu, J. Wang, and Z. Han, "Study on the transient temperature field based on the fractional heat conduction equation for laser heating," Appl. Math. Mech., 36, 844–849, 2015.
  • G. Xu and J. Wang, "Analytical solution of time fractional Cattaneo heat equation for finite slab under pulse heat flux," Appl. Math. Mech., 39 (10), 1465–1476, 2018. DOI: 10.1007/s10483-018-2375-8.
  • G. Xu, J. Wang, and Z. Han, "Notes on 'The Cattaneo-type time fractional heat conduction equation for laser heating' [Comput. Math. Appl. 66 (2013) 824–831]," Comput. Math. Appl., 71 (10), 2132–2137, 2016. DOI: 10.1016/j.camwa.2016.03.011.
  • C. Cattaneo, “Sulla conduzione del calore,” Atti Sem. Mat. Fis. Univ. Modena, 3, 83–101, 1948.
  • H. R. Ghazizadeh, M. Maerefat, and A. Azimi, "Explicit and implicit finite difference schemes for fractional Cattaneo equation," J. Comput. Phys., 229 (16), 7042–7057, 2010. DOI: 10.1016/j.jcp.2010.05.039.
  • Z. M. Odibat, N. T. Shawagfeh, "Generalized Taylor's formula," Appl. Math. Comput., 186, 286–293, 2007.
  • I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
  • Z. Zhang and D.Y. Liu, "Advanced in the study of non-Fourier heat conduction," Advance Mechanics, 30, 446-456, 2000.
  • R. Gorenflo and F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order, A. Carpinteri and F. Mainardi (Editors): Fractals and Fractional Calculus in Continuum Mechanics), 223-276, Springer Verlag, Wien and New York, 1997.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, 204, Elsevier Science, Amsterdam, 2006.
  • Y. Povstenko, "Axisymmetric Solutions to Time-fractional heat conduction equation in a half-space under Robin boundary conditions," Int. J. Differ. Equ., 1–13, 2012. DOI: 10.1155/2012/154085.
  • Y. Povstenko, "Axisymmetric solutions to fractional diffusion-wave equation in a cylinder under Robin boundary condition," Eur. Phys. J. Spec. Top., 222, 1767–1777, 2013. DOI: 10.1140/epjst/e2013-01962-4.
  • Y. Povstenko, "Fundamental solutions to the fractional heat conduction equation in a ball under Robin boundary condition," Centr. Eur. J. Math., 12 (4), 611–622, 2014. DOI: 10.2478/s11533-013-0368-8.
  • H. S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, 2nd ed., Oxford University Press, Oxford, 1959.
  • G. M. L. Gladwell, J. R. Barber, and Z. Olesiak, "Thermal problems with radiation boundary conditions," Q. J. Mech. Appl. Math., 36 (3), 387–401, 1983. DOI: 10.1093/qjmam/36.3.387.
  • E. Marchi and G. Zgrablich, "Heat conduction in hollow cylinders with radiation," Proc. Edimburgh Math. Soc., 14(11), 159-164, 1964.
  • E. Marchi and A. Fasulo, "Heat conduction in sector of hollow cylinder with radiation," Atti, della Acc. Sci. di. Torino, 101, 373-382, 1967.
  • R. Kumar, N. K. Lamba, and V. Varghese, "Analysis of thermoelastic disc with radiation conditions on the curved surfaces," Mater. Phys. Mech., 16 (2), 175-186, 2013.
  • N. Noda, R. B. Hetnarski, Y. Tanigawa, Thermal stresses, 2nd ed., Taylor and Francis, New York, 2003.
  • A. E. H. Love, A Treatise on the mathematical theory of elasticity, 4th ed., Dover publications, New York, 1944.
  • W. Nowacki, Thermoelasticity, 2nd ed., PWN-Polish Scientific Publishers, Warsaw and Pergamon Press, Oxford, 1986.
  • J. J. Tripathi, K. C. Deshmukh and J. Verma, Fractional Order Generalized Thermoelastic Problem in a Thick Circular Plate with Periodically Varying Heat Source, Int. J. Thermodyn., 20 (3), 132-138, 2017. DOI: 10.5541/ijot.5000190819.
  • K. C. Deshmukh, S. D. Warbhe, and V. S. Kulkarni, "Brief Note on Heat Flow With Arbitrary Heating Rates in a Hollow Cylinder," Therm. Sci., 15 (1), 275–280, 2011. DOI: 10.2298/TSCI100817063D.
  • S. N. Li, B. Y. Cao, "Fractional Boltzmann transport equation for anomalous heat transport and divergent thermal conductivity," Int. J. Heat Mass Transf., 137, 84-89, 2019. DOI: 10.1016/j.ijheatmasstransfer.2019.03.120.
  • S. N. Li, B. Y. Cao, "Fractional-order heat conduction models from generalized Boltzmann transport equation," Philos. Trans. R. Soc. A, 378, 20190280, 2020. DOI: 10.1098/rsta.2019.0280.

Year 2023, , 35 - 44, 01.06.2023

G. Dhameja L. Khalsa Vinod Varghese

https://doi.org/10.5541/ijot.1170342

Abstract

References

  • M. Haskul, "Elastic state of functionally graded curved beam on the plane stress state subject to thermal load," Mech. Based Des. Struct. Mach., 48 (6), 739-754, 2020. DOI: 10.1080/15397734.2019.1660890.
  • E. Arslan, M. Haskul, "Generalized plane strain solution of a thick-walled cylindrical panel subjected to radial heating," Acta Mech, 226, 1213–1225, 2015. https://doi.org/10.1007/s00707-014-1248-4
  • M. Haskul, E. Arslan and W. Mack, "Radial heating of a thick-walled cylindrically curved FGM-panel," Z. Angew. Math. Mech., 97, 309-321, 2017. https://doi.org/10.1002/zamm.201500310
  • M. Haskul, "Yielding of functionally graded curved beam subjected to temperature," Pamukkale University Journal of Engineering Sciences, 26 (4), 587-593, 2020. DOI: 10.5505/pajes.2019.92331
  • E. Hoashi, T. Yokomine, A. Shimizu, and T. Kunugi, "Numerical analysis of wave-type heat transfer propagating in a thin foil irradiated by short-pulsed laser," Int. J. Heat Mass Transf., 46 (19), 4083–4095, 2003. DOI: 10.1016/S0017-9310(03)00225-4.
  • X. Ai and B. Q. Li, "Numerical simulation of thermal wave propagation during laser processing of thin films," J. Electron. Mater., 34 (5), 583–591, 2005. DOI: 10.1007/s11664-005-0069-6.
  • T. T. Lam and E. Fong, "Application of solution structure theorem to non-Fourier heat conduction problems: Analytical approach," Int. J. Heat Mass Transf., 54, 4796–4806, 2011. DOI: 10.1016/j.ijheatmasstransfer.2011.06.028.
  • T. T. Lam, "A unified solution of several heat conduction models," Int. J. Heat Mass Transf., 56 (1–2), 653–666, 2013. DOI: 10.1016/j.ijheatmasstransfer.2012.08.055.
  • C. Cattaneo, “Sur uneforme de l’équation de la chaleuréliminant le paradoxed’une propagation instantanée,” C. R. Acad. Sci., 247, 431–433, 1958.
  • P. Vernotte, “Les paradoxes de la théorie continue de l’équation de la chaleur,” C. R. Acad. Sci., 246, 3154-3155, 1958.
  • A. Compte and R. Metzler, "The generalized Cattaneo equation for the description of anomalous transport processes," J. Phys. A: Math. Gen., 30, 7277-7289, 1997.
  • F. M. Jiang, D. Y. Liu, and J. H. Zhou, "Non-Fourier heat conduction phenomena in porous material heated by microsecond laser pulse," Microscale Thermophys. Eng., 6 (4), 331–346, 2003. DOI: 10.1080/10893950290098386.
  • Y. Povstenko, Fractional thermoelasticty, Springer, New York, 2015.
  • Y. Povstenko, "Fractional heat conduction equation and associated thermal stress," J. Therm. Stresses, 28 (1), 83–102, 2005.
  • Y. Povstenko, "Fractional Cattaneo-type equations and generalized thermoelasticity," J. Therm. Stresses, 34 (2), 97-114, 2011. DOI: 10.1080/01495739.2010.511931.
  • T. N. Mishra and K. N. Rai, "Numerical solution of FSPL heat conduction equation for analysis of thermal propagation," Appl. Math. Comput., 273, 1006–1017, 2016. DOI: 10.1016/j.amc.2015.10.082.
  • H. Qi, H. Xu, and X. Guo, "The Cattaneo-type time fractional heat conduction equation for laser heating," Comput. Math. Appl., 66 (5), 824–831, 2013. DOI: 10.1016/j.camwa.2012.11.021.
  • H. Qi, and X. Guo, "Transient fractional heat conduction with generalized Cattaneo model," Int. J. Heat Mass Transf., 76, 535–539, 2014.
  • H. Xu, H. Qi, and X. Jiang, "Fractional Cattaneo heat equation on a semi-infinite medium," Chin. Phys. B, 22 (1), 014401, 2013. DOI: 10.1088/1674-1056/22/1/014401.
  • G. Xu, J. Wang, and Z. Han, "Study on the transient temperature field based on the fractional heat conduction equation for laser heating," Appl. Math. Mech., 36, 844–849, 2015.
  • G. Xu and J. Wang, "Analytical solution of time fractional Cattaneo heat equation for finite slab under pulse heat flux," Appl. Math. Mech., 39 (10), 1465–1476, 2018. DOI: 10.1007/s10483-018-2375-8.
  • G. Xu, J. Wang, and Z. Han, "Notes on 'The Cattaneo-type time fractional heat conduction equation for laser heating' [Comput. Math. Appl. 66 (2013) 824–831]," Comput. Math. Appl., 71 (10), 2132–2137, 2016. DOI: 10.1016/j.camwa.2016.03.011.
  • C. Cattaneo, “Sulla conduzione del calore,” Atti Sem. Mat. Fis. Univ. Modena, 3, 83–101, 1948.
  • H. R. Ghazizadeh, M. Maerefat, and A. Azimi, "Explicit and implicit finite difference schemes for fractional Cattaneo equation," J. Comput. Phys., 229 (16), 7042–7057, 2010. DOI: 10.1016/j.jcp.2010.05.039.
  • Z. M. Odibat, N. T. Shawagfeh, "Generalized Taylor's formula," Appl. Math. Comput., 186, 286–293, 2007.
  • I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
  • Z. Zhang and D.Y. Liu, "Advanced in the study of non-Fourier heat conduction," Advance Mechanics, 30, 446-456, 2000.
  • R. Gorenflo and F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order, A. Carpinteri and F. Mainardi (Editors): Fractals and Fractional Calculus in Continuum Mechanics), 223-276, Springer Verlag, Wien and New York, 1997.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, 204, Elsevier Science, Amsterdam, 2006.
  • Y. Povstenko, "Axisymmetric Solutions to Time-fractional heat conduction equation in a half-space under Robin boundary conditions," Int. J. Differ. Equ., 1–13, 2012. DOI: 10.1155/2012/154085.
  • Y. Povstenko, "Axisymmetric solutions to fractional diffusion-wave equation in a cylinder under Robin boundary condition," Eur. Phys. J. Spec. Top., 222, 1767–1777, 2013. DOI: 10.1140/epjst/e2013-01962-4.
  • Y. Povstenko, "Fundamental solutions to the fractional heat conduction equation in a ball under Robin boundary condition," Centr. Eur. J. Math., 12 (4), 611–622, 2014. DOI: 10.2478/s11533-013-0368-8.
  • H. S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, 2nd ed., Oxford University Press, Oxford, 1959.
  • G. M. L. Gladwell, J. R. Barber, and Z. Olesiak, "Thermal problems with radiation boundary conditions," Q. J. Mech. Appl. Math., 36 (3), 387–401, 1983. DOI: 10.1093/qjmam/36.3.387.
  • E. Marchi and G. Zgrablich, "Heat conduction in hollow cylinders with radiation," Proc. Edimburgh Math. Soc., 14(11), 159-164, 1964.
  • E. Marchi and A. Fasulo, "Heat conduction in sector of hollow cylinder with radiation," Atti, della Acc. Sci. di. Torino, 101, 373-382, 1967.
  • R. Kumar, N. K. Lamba, and V. Varghese, "Analysis of thermoelastic disc with radiation conditions on the curved surfaces," Mater. Phys. Mech., 16 (2), 175-186, 2013.
  • N. Noda, R. B. Hetnarski, Y. Tanigawa, Thermal stresses, 2nd ed., Taylor and Francis, New York, 2003.
  • A. E. H. Love, A Treatise on the mathematical theory of elasticity, 4th ed., Dover publications, New York, 1944.
  • W. Nowacki, Thermoelasticity, 2nd ed., PWN-Polish Scientific Publishers, Warsaw and Pergamon Press, Oxford, 1986.
  • J. J. Tripathi, K. C. Deshmukh and J. Verma, Fractional Order Generalized Thermoelastic Problem in a Thick Circular Plate with Periodically Varying Heat Source, Int. J. Thermodyn., 20 (3), 132-138, 2017. DOI: 10.5541/ijot.5000190819.
  • K. C. Deshmukh, S. D. Warbhe, and V. S. Kulkarni, "Brief Note on Heat Flow With Arbitrary Heating Rates in a Hollow Cylinder," Therm. Sci., 15 (1), 275–280, 2011. DOI: 10.2298/TSCI100817063D.
  • S. N. Li, B. Y. Cao, "Fractional Boltzmann transport equation for anomalous heat transport and divergent thermal conductivity," Int. J. Heat Mass Transf., 137, 84-89, 2019. DOI: 10.1016/j.ijheatmasstransfer.2019.03.120.
  • S. N. Li, B. Y. Cao, "Fractional-order heat conduction models from generalized Boltzmann transport equation," Philos. Trans. R. Soc. A, 378, 20190280, 2020. DOI: 10.1098/rsta.2019.0280.
There are 44 citations in total.

Details

Primary Language English
Subjects Thermodynamics and Statistical Physics
Journal Section Research Articles
Authors

G. Dhameja This is me

L. Khalsa This is me

Vinod Varghese 0000-0002-9660-7610

Early Pub Date April 27, 2023
Publication Date June 1, 2023
Published in Issue Year 2023

Cite

APA Dhameja, G., Khalsa, L., & Varghese, V. (2023). Thermoelastic Analysis For A Thick Plate Under The Radiation Boundary Conditions. International Journal of Thermodynamics, 26(2), 35-44. https://doi.org/10.5541/ijot.1170342
AMA Dhameja G, Khalsa L, Varghese V. Thermoelastic Analysis For A Thick Plate Under The Radiation Boundary Conditions. International Journal of Thermodynamics. June 2023;26(2):35-44. doi:10.5541/ijot.1170342
Chicago Dhameja, G., L. Khalsa, and Vinod Varghese. “Thermoelastic Analysis For A Thick Plate Under The Radiation Boundary Conditions”. International Journal of Thermodynamics 26, no. 2 (June 2023): 35-44. https://doi.org/10.5541/ijot.1170342.
EndNote Dhameja G, Khalsa L, Varghese V (June 1, 2023) Thermoelastic Analysis For A Thick Plate Under The Radiation Boundary Conditions. International Journal of Thermodynamics 26 2 35–44.
IEEE G. Dhameja, L. Khalsa, and V. Varghese, “Thermoelastic Analysis For A Thick Plate Under The Radiation Boundary Conditions”, International Journal of Thermodynamics, vol. 26, no. 2, pp. 35–44, 2023, doi: 10.5541/ijot.1170342.
ISNAD Dhameja, G. et al. “Thermoelastic Analysis For A Thick Plate Under The Radiation Boundary Conditions”. International Journal of Thermodynamics 26/2 (June 2023), 35-44. https://doi.org/10.5541/ijot.1170342.
JAMA Dhameja G, Khalsa L, Varghese V. Thermoelastic Analysis For A Thick Plate Under The Radiation Boundary Conditions. International Journal of Thermodynamics. 2023;26:35–44.
MLA Dhameja, G. et al. “Thermoelastic Analysis For A Thick Plate Under The Radiation Boundary Conditions”. International Journal of Thermodynamics, vol. 26, no. 2, 2023, pp. 35-44, doi:10.5541/ijot.1170342.
Vancouver Dhameja G, Khalsa L, Varghese V. Thermoelastic Analysis For A Thick Plate Under The Radiation Boundary Conditions. International Journal of Thermodynamics. 2023;26(2):35-44.