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On the time decay in phase–lag thermoelasticity with two temperatures
1. | Departament de Matemàtiques, Universitat Politècnica de Catalunya, 08222 Terrassa, Barcelona, Spain |
2. | Laboratoire de Mathématiques et Applications, Université de Poitiers, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France |
The aim of this paper is to study the time decay of the solutions for two models of the one-dimensional phase-lag thermoelasticity with two temperatures. The first one is obtained when the heat flux vector and the inductive temperature are approximated by a second-order and first-order Taylor polynomial, respectively. In this case, the solutions decay in a slow way. The second model that we consider is obtained taking first-order Taylor approximations for the inductive thermal displacement, the inductive temperature and the heat flux. The decay is, therefore, of exponential type.
References:
[1] |
I. A. Abdallah, Dual phase lag heat conduction and thermoelastic properties of a semi-infinite medium induced by ultrashort pulsed laser, Progress in Physics, 3 (2009), 60-63. Google Scholar |
[2] |
S. Banik and M. Kanoria,
Effects of three-phase-lag on two temperatures generalized thermoelasticity for an infinite medium with a spherical cavity, Applied Mathematics and Mechanics, 33 (2012), 483-498.
doi: 10.1007/s10483-012-1565-8. |
[3] |
K. Borgmeyer, R. Quintanilla and R. Racke,
Phase-lag heat conduction: Decay rates for limit problems and well-posedness, J. Evolution Equations, 14 (2014), 863-884.
doi: 10.1007/s00028-014-0242-6. |
[4] |
P. J. Chen and M. E. Gurtin, On a theory of heat involving two temperatures, J. Applied Mathematics and Physics (ZAMP), 19 (1968), 614-627. Google Scholar |
[5] |
P. J. Chen, M. E. Gurtin and W. O. Williams,
A note on non-simple heat conduction, J. Applied Mathematics and Physics (ZAMP), 19 (1968), 969-970.
doi: 10.1007/BF01602278. |
[6] |
P. J. Chen, M. E. Gurtin and W. O. Williams, On the thermodynamics of non-simple materials with two temperatures, J. Applied Mathematics and Physics (ZAMP), 20 (1969), 107-112. Google Scholar |
[7] |
S. K. R. Choudhuri,
On a thermoelastic three-phase-lag model, J. Thermal Stresses, 30 (2007), 231-238.
doi: 10.1080/01495730601130919. |
[8] |
M. Dreher, R. Quintanilla and R. Racke,
Ill-posed problems in thermomechanics, Applied Mathematics Letters, 22 (2009), 1374-1379.
doi: 10.1016/j.aml.2009.03.010. |
[9] |
M. A. Ezzat, A. S. El-Karamany and S. M. Ezzat,
Two-temperature theory in magneto-thermoelasticity with fractional order dual-phase-lag heat transfer, Nuclear Engineering and Design, 252 (2012), 267-277.
doi: 10.1016/j.nucengdes.2012.06.012. |
[10] |
A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses, 15 (1992), 253–264.
doi: 10.1080/01495739208946136. |
[11] |
A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189–208.
doi: 10.1007/BF00044969. |
[12] |
M. A. Hader, M. A. Al-Nimr and B. A. Abu Nabah, The Dual-Phase-Lag heat conduction model in thin slabs under a fluctuating volumetric thermal disturbance, Int. J. Thermophysics, 23 (2002), 1669-1680. Google Scholar |
[13] |
F. L. Huang,
Strong asymptotic stability of linear dynamical systems in Banach spaces, J. Differential Equations, 104 (1993), 307-324.
doi: 10.1006/jdeq.1993.1074. |
[14] |
R. Quintanilla and P. M. Jordan,
A note on the two-temperature theory with dual-phase-lag decay: Some exact solutions, Mechanics Research Communications, 36 (2009), 796-803.
doi: 10.1016/j.mechrescom.2009.05.002. |
[15] |
M. C. Leseduarte, R. Quintanilla and R. Racke,
On (non-)exponential decay in generalized thermoelasticity with two temperatures, Applied Mathematics Letters, 70 (2017), 18-25.
doi: 10.1016/j.aml.2017.02.020. |
[16] |
Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC Research Notes in Mathematics, vol. 398, Chapman & Hall/CRC, Boca Raton, FL, 1999. |
[17] |
A. Magaña, A. Miranville and R. Quintanilla,
On the stability in phase-lag heat conduction with two temperatures, J. of Evolution Equations, 18 (2018), 1697-1712.
doi: 10.1007/s00028-018-0457-z. |
[18] |
J. E. Marsden and T. J. R. Hughes, Topics in the mathematical foundations of elasticity, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. II, 30-285, Res. Notes in Math., 27, Pitman, Boston, Mass.-London, 1978. |
[19] |
A. Miranville and R. Quintanilla,
A phase-field model based on a three-phase-lag heat conduction, Applied Mathematics and Optimization, 63 (2011), 133-150.
doi: 10.1007/s00245-010-9114-9. |
[20] |
S. Mukhopadhyay, R Prasad and R. Kumar, On the theory of Two-Temperature Thermoelaticity with Two Phase-Lags, J. Thermal Stresses, 34 (2011), 352-365. Google Scholar |
[21] |
M. A. Othman, W. M. Hasona and E. M. Abd-Elaziz,
Effect of rotation on micropolar generalized thermoelasticity with two temperatures using a dual-phase-lag model, Canadian J. Physics, 92 (2014), 149-158.
doi: 10.1139/cjp-2013-0398. |
[22] |
J. Prüss,
On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.
doi: 10.2307/1999112. |
[23] |
R. Quintanilla,
Exponential stability in the dual-phase-lag heat conduction theory, J. Non-Equilibrium Thermodynamics, 27 (2002), 217-227.
doi: 10.1515/JNETDY.2002.012. |
[24] |
R. Quintanilla, A well-posed problem for the Dual-Phase-Lag heat conduction, J. Thermal Stresses, 31 (2008), 260-269. Google Scholar |
[25] |
R. Quintanilla,
A well-posed problem for the three-dual-phase-lag heat conduction, J. Thermal Stresses, 32 (2009), 1270-1278.
doi: 10.1080/01495730903310599. |
[26] |
R. Quintanilla and R. Racke,
Qualitative aspects in dual-phase-lag thermoelasticity, SIAM J. Appl. Math., 66 (2006), 977-1001.
doi: 10.1137/05062860X. |
[27] |
R. Quintanilla and R. Racke,
A note on stability of dual-phase-lag heat conduction, Int. J. Heat Mass Transfer, 49 (2006), 1209-1213.
doi: 10.1016/j.ijheatmasstransfer.2005.10.016. |
[28] |
R. Quintanilla and R. Racke,
Qualitative aspects in dual-phase-lag heat conduction, Proc. Royal Society London A, 463 (2007), 659-674.
doi: 10.1098/rspa.2006.1784. |
[29] |
R. Quintanilla and R. Racke, A note on stability in three-phase-lag heat conduction, Int. J. Heat Mass Transfer, 51 (2008), 24-29. Google Scholar |
[30] |
R. Quintanilla and R. Racke,
Spatial behavior in phase-lag heat conduction, Differential and Integral Equations, 28 (2015), 291-308.
|
[31] |
S. A. Rukolaine,
Unphysical effects of the dual-phase-lag model of heat conduction, Int. J. Heat and Mass Transfer, 78 (2014), 58-63.
doi: 10.1016/j.ijheatmasstransfer.2014.06.066. |
[32] |
D. Y. Tzou,
A unified approach for heat conduction from macro to micro-scales, ASME J. Heat Transfer, 117 (1995), 8-16.
doi: 10.1115/1.2822329. |
[33] |
W. E. Warren and P. J. Chen, Wave propagation in two temperatures theory of thermoelaticity, Acta Mechanica, 16 (1973), 83-117. Google Scholar |
[34] |
Y. Zhang,
Generalized dual-phase lag bioheat equations based on nonequilibrium heat transfer in living biological tissues, Int. J. of Heat and Mass Transfer, 52 (2009), 4829-4834.
doi: 10.1016/j.ijheatmasstransfer.2009.06.007. |
show all references
References:
[1] |
I. A. Abdallah, Dual phase lag heat conduction and thermoelastic properties of a semi-infinite medium induced by ultrashort pulsed laser, Progress in Physics, 3 (2009), 60-63. Google Scholar |
[2] |
S. Banik and M. Kanoria,
Effects of three-phase-lag on two temperatures generalized thermoelasticity for an infinite medium with a spherical cavity, Applied Mathematics and Mechanics, 33 (2012), 483-498.
doi: 10.1007/s10483-012-1565-8. |
[3] |
K. Borgmeyer, R. Quintanilla and R. Racke,
Phase-lag heat conduction: Decay rates for limit problems and well-posedness, J. Evolution Equations, 14 (2014), 863-884.
doi: 10.1007/s00028-014-0242-6. |
[4] |
P. J. Chen and M. E. Gurtin, On a theory of heat involving two temperatures, J. Applied Mathematics and Physics (ZAMP), 19 (1968), 614-627. Google Scholar |
[5] |
P. J. Chen, M. E. Gurtin and W. O. Williams,
A note on non-simple heat conduction, J. Applied Mathematics and Physics (ZAMP), 19 (1968), 969-970.
doi: 10.1007/BF01602278. |
[6] |
P. J. Chen, M. E. Gurtin and W. O. Williams, On the thermodynamics of non-simple materials with two temperatures, J. Applied Mathematics and Physics (ZAMP), 20 (1969), 107-112. Google Scholar |
[7] |
S. K. R. Choudhuri,
On a thermoelastic three-phase-lag model, J. Thermal Stresses, 30 (2007), 231-238.
doi: 10.1080/01495730601130919. |
[8] |
M. Dreher, R. Quintanilla and R. Racke,
Ill-posed problems in thermomechanics, Applied Mathematics Letters, 22 (2009), 1374-1379.
doi: 10.1016/j.aml.2009.03.010. |
[9] |
M. A. Ezzat, A. S. El-Karamany and S. M. Ezzat,
Two-temperature theory in magneto-thermoelasticity with fractional order dual-phase-lag heat transfer, Nuclear Engineering and Design, 252 (2012), 267-277.
doi: 10.1016/j.nucengdes.2012.06.012. |
[10] |
A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses, 15 (1992), 253–264.
doi: 10.1080/01495739208946136. |
[11] |
A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189–208.
doi: 10.1007/BF00044969. |
[12] |
M. A. Hader, M. A. Al-Nimr and B. A. Abu Nabah, The Dual-Phase-Lag heat conduction model in thin slabs under a fluctuating volumetric thermal disturbance, Int. J. Thermophysics, 23 (2002), 1669-1680. Google Scholar |
[13] |
F. L. Huang,
Strong asymptotic stability of linear dynamical systems in Banach spaces, J. Differential Equations, 104 (1993), 307-324.
doi: 10.1006/jdeq.1993.1074. |
[14] |
R. Quintanilla and P. M. Jordan,
A note on the two-temperature theory with dual-phase-lag decay: Some exact solutions, Mechanics Research Communications, 36 (2009), 796-803.
doi: 10.1016/j.mechrescom.2009.05.002. |
[15] |
M. C. Leseduarte, R. Quintanilla and R. Racke,
On (non-)exponential decay in generalized thermoelasticity with two temperatures, Applied Mathematics Letters, 70 (2017), 18-25.
doi: 10.1016/j.aml.2017.02.020. |
[16] |
Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC Research Notes in Mathematics, vol. 398, Chapman & Hall/CRC, Boca Raton, FL, 1999. |
[17] |
A. Magaña, A. Miranville and R. Quintanilla,
On the stability in phase-lag heat conduction with two temperatures, J. of Evolution Equations, 18 (2018), 1697-1712.
doi: 10.1007/s00028-018-0457-z. |
[18] |
J. E. Marsden and T. J. R. Hughes, Topics in the mathematical foundations of elasticity, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. II, 30-285, Res. Notes in Math., 27, Pitman, Boston, Mass.-London, 1978. |
[19] |
A. Miranville and R. Quintanilla,
A phase-field model based on a three-phase-lag heat conduction, Applied Mathematics and Optimization, 63 (2011), 133-150.
doi: 10.1007/s00245-010-9114-9. |
[20] |
S. Mukhopadhyay, R Prasad and R. Kumar, On the theory of Two-Temperature Thermoelaticity with Two Phase-Lags, J. Thermal Stresses, 34 (2011), 352-365. Google Scholar |
[21] |
M. A. Othman, W. M. Hasona and E. M. Abd-Elaziz,
Effect of rotation on micropolar generalized thermoelasticity with two temperatures using a dual-phase-lag model, Canadian J. Physics, 92 (2014), 149-158.
doi: 10.1139/cjp-2013-0398. |
[22] |
J. Prüss,
On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.
doi: 10.2307/1999112. |
[23] |
R. Quintanilla,
Exponential stability in the dual-phase-lag heat conduction theory, J. Non-Equilibrium Thermodynamics, 27 (2002), 217-227.
doi: 10.1515/JNETDY.2002.012. |
[24] |
R. Quintanilla, A well-posed problem for the Dual-Phase-Lag heat conduction, J. Thermal Stresses, 31 (2008), 260-269. Google Scholar |
[25] |
R. Quintanilla,
A well-posed problem for the three-dual-phase-lag heat conduction, J. Thermal Stresses, 32 (2009), 1270-1278.
doi: 10.1080/01495730903310599. |
[26] |
R. Quintanilla and R. Racke,
Qualitative aspects in dual-phase-lag thermoelasticity, SIAM J. Appl. Math., 66 (2006), 977-1001.
doi: 10.1137/05062860X. |
[27] |
R. Quintanilla and R. Racke,
A note on stability of dual-phase-lag heat conduction, Int. J. Heat Mass Transfer, 49 (2006), 1209-1213.
doi: 10.1016/j.ijheatmasstransfer.2005.10.016. |
[28] |
R. Quintanilla and R. Racke,
Qualitative aspects in dual-phase-lag heat conduction, Proc. Royal Society London A, 463 (2007), 659-674.
doi: 10.1098/rspa.2006.1784. |
[29] |
R. Quintanilla and R. Racke, A note on stability in three-phase-lag heat conduction, Int. J. Heat Mass Transfer, 51 (2008), 24-29. Google Scholar |
[30] |
R. Quintanilla and R. Racke,
Spatial behavior in phase-lag heat conduction, Differential and Integral Equations, 28 (2015), 291-308.
|
[31] |
S. A. Rukolaine,
Unphysical effects of the dual-phase-lag model of heat conduction, Int. J. Heat and Mass Transfer, 78 (2014), 58-63.
doi: 10.1016/j.ijheatmasstransfer.2014.06.066. |
[32] |
D. Y. Tzou,
A unified approach for heat conduction from macro to micro-scales, ASME J. Heat Transfer, 117 (1995), 8-16.
doi: 10.1115/1.2822329. |
[33] |
W. E. Warren and P. J. Chen, Wave propagation in two temperatures theory of thermoelaticity, Acta Mechanica, 16 (1973), 83-117. Google Scholar |
[34] |
Y. Zhang,
Generalized dual-phase lag bioheat equations based on nonequilibrium heat transfer in living biological tissues, Int. J. of Heat and Mass Transfer, 52 (2009), 4829-4834.
doi: 10.1016/j.ijheatmasstransfer.2009.06.007. |
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