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

  • Received: 01 June 2019 Revised: 01 November 2019
  • Primary: 74F05, 74H40; Secondary: 74H20, 34B35, 35P20

  • 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.

    Citation: Antonio Magaña, Alain Miranville, Ramón Quintanilla. On the time decay in phase–lag thermoelasticity with two temperatures[J]. Electronic Research Archive, 2019, 27: 7-19. doi: 10.3934/era.2019007

    Related Papers:

  • 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.



    加载中


    [1] Abdallah I. A. (2009) Dual phase lag heat conduction and thermoelastic properties of a semi-infinite medium induced by ultrashort pulsed laser. Progress in Physics 3: 60-63.
    [2] Banik S., Kanoria M. (2012) Effects of three-phase-lag on two temperatures generalized thermoelasticity for an infinite medium with a spherical cavity. Applied Mathematics and Mechanics 33: 483-498.
    [3] Borgmeyer K., Quintanilla R., Racke R. (2014) Phase-lag heat conduction: Decay rates for limit problems and well-posedness. J. Evolution Equations 14: 863-884.
    [4] Chen P. J., Gurtin M. E. (1968) On a theory of heat involving two temperatures. J. Applied Mathematics and Physics (ZAMP) 19: 614-627.
    [5] Chen P. J., Gurtin M. E., Williams W. O. (1968) A note on non-simple heat conduction. J. Applied Mathematics and Physics (ZAMP) 19: 969-970.
    [6] Chen P. J., Gurtin M. E., Williams W. O. (1969) On the thermodynamics of non-simple materials with two temperatures. J. Applied Mathematics and Physics (ZAMP) 20: 107-112.
    [7] Choudhuri S. K. R. (2007) On a thermoelastic three-phase-lag model. J. Thermal Stresses 30: 231-238.
    [8] Dreher M., Quintanilla R., Racke R. (2009) Ill-posed problems in thermomechanics. Applied Mathematics Letters 22: 1374-1379.
    [9] Ezzat M. A., El-Karamany A. S., Ezzat S. M. (2012) Two-temperature theory in magneto-thermoelasticity with fractional order dual-phase-lag heat transfer. Nuclear Engineering and Design 252: 267-277.
    [10] A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses, 15 (1992), 253–264.

    10.1080/01495739208946136

    MR1175235

    [11] A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189–208.

    10.1007/BF00044969

    MR1236373

    [12] Hader M. A., Al-Nimr M. A., Abu Nabah B. A. (2002) The Dual-Phase-Lag heat conduction model in thin slabs under a fluctuating volumetric thermal disturbance. Int. J. Thermophysics 23: 1669-1680.
    [13] Huang F. L. (1993) Strong asymptotic stability of linear dynamical systems in Banach spaces. J. Differential Equations 104: 307-324.
    [14] Quintanilla R., Jordan P. M. (2009) A note on the two-temperature theory with dual-phase-lag decay: Some exact solutions. Mechanics Research Communications 36: 796-803.
    [15] Leseduarte M. C., Quintanilla R., Racke R. (2017) On (non-)exponential decay in generalized thermoelasticity with two temperatures. Applied Mathematics Letters 70: 18-25.
    [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.

    MR1681343

    [17] Magaña A., Miranville A., Quintanilla R. (2018) On the stability in phase-lag heat conduction with two temperatures. J. of Evolution Equations 18: 1697-1712.
    [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.

    MR576233

    [19] Miranville A., Quintanilla R. (2011) A phase-field model based on a three-phase-lag heat conduction. Applied Mathematics and Optimization 63: 133-150.
    [20] Mukhopadhyay S., Prasad R, Kumar R. (2011) On the theory of Two-Temperature Thermoelaticity with Two Phase-Lags. J. Thermal Stresses 34: 352-365.
    [21] Othman M. A., Hasona W. M., Abd-Elaziz E. M. (2014) Effect of rotation on micropolar generalized thermoelasticity with two temperatures using a dual-phase-lag model. Canadian J. Physics 92: 149-158.
    [22] Prüss J. (1984) On the spectrum of C0-semigroups. Trans. Amer. Math. Soc. 284: 847-857.
    [23] Quintanilla R. (2002) Exponential stability in the dual-phase-lag heat conduction theory. J. Non-Equilibrium Thermodynamics 27: 217-227.
    [24] Quintanilla R. (2008) A well-posed problem for the Dual-Phase-Lag heat conduction. J. Thermal Stresses 31: 260-269.
    [25] Quintanilla R. (2009) A well-posed problem for the three-dual-phase-lag heat conduction. J. Thermal Stresses 32: 1270-1278.
    [26] Quintanilla R., Racke R. (2006) Qualitative aspects in dual-phase-lag thermoelasticity. SIAM J. Appl. Math. 66: 977-1001.
    [27] Quintanilla R., Racke R. (2006) A note on stability of dual-phase-lag heat conduction. Int. J. Heat Mass Transfer 49: 1209-1213.
    [28] Quintanilla R., Racke R. (2007) Qualitative aspects in dual-phase-lag heat conduction. Proc. Royal Society London A 463: 659-674.
    [29] Quintanilla R., Racke R. (2008) A note on stability in three-phase-lag heat conduction. Int. J. Heat Mass Transfer 51: 24-29.
    [30] Quintanilla R., Racke R. (2015) Spatial behavior in phase-lag heat conduction. Differential and Integral Equations 28: 291-308.
    [31] Rukolaine S. A. (2014) Unphysical effects of the dual-phase-lag model of heat conduction. Int. J. Heat and Mass Transfer 78: 58-63.
    [32] Tzou D. Y. (1995) A unified approach for heat conduction from macro to micro-scales. ASME J. Heat Transfer 117: 8-16.
    [33] Warren W. E., Chen P. J. (1973) Wave propagation in two temperatures theory of thermoelaticity. Acta Mechanica 16: 83-117.
    [34] Zhang Y. (2009) Generalized dual-phase lag bioheat equations based on nonequilibrium heat transfer in living biological tissues. Int. J. of Heat and Mass Transfer 52: 4829-4834.
  • Reader Comments
  • © 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2657) PDF downloads(291) Cited by(4)

Article outline

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog