American Institute of Mathematical Sciences

Advanced
  2019, 27: 7-19. doi: 10.3934/era.2019007

On the time decay in phase–lag thermoelasticity with two temperatures

Antonio Magaña 1, , Alain Miranville 2,, and  Ramón Quintanilla 1,

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

* Corresponding author: Alain Miranville

Received  June 2019 Revised  November 2019 Published  September 2019

Fund Project: The first and the third authors are supported by the project "Análisis Matemático de Problemas de la Termomecánica" (MTM2016-74934-P), (AEI/FEDER, UE) of the Spanish Ministry of Economy and Competitiveness.

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.

Keywords: Phase-lag thermoelasticity, exponential stability, slow decay, spectral analysis.
Mathematics Subject Classification: Primary: 74F05, 74H40; Secondary: 74H20, 34B35, 35P20.
Citation: Antonio Magaña, Alain Miranville, Ramón Quintanilla. On the time decay in phase–lag thermoelasticity with two temperatures. Electronic Research Archive, 2019, 27: 7-19. doi: 10.3934/era.2019007
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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[11]

A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189–208. doi: 10.1007/BF00044969.  Google Scholar

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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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

J. Prüss, On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189–208. doi: 10.1007/BF00044969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

J. Prüss, On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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