On Green and Naghdi Thermoelasticity Model without Energy Dissipation with Higher Order Time Differential and Phase-Lags

Document Type: Research Paper


1 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

2 Department of Mathematics, College of Science and Arts, Jouf University, Al-Qurayyat, Saudi Arabia


In the present work, a modified model of heat conduction including higher order of time derivative is derived by extending Green and Naghdi theory without energy dissipation. We introduce two phase lag times to include the thermal displacement gradient and the heat flux in the heat conduction and depict microscopic responses more precisely. The constructed model is applied to study thermoelastic waves in a homogeneous and isotropic perfect conducting unbounded solid body containing a spherical cavity. We use the Laplace transform method to analyze the problem. The solutions for the field functions are obtained numerically using the numerical Laplace inversion technique. The results are analyzed in different tables and graphs and compared with those obtained earlier in the contexts of some other theories of thermoelasticity.


Main Subjects

[1] Biot, M., Thermoelasticity and Irreversible Thermodynamics, J. Appl. Phys., 27, 1956, 240–253.

[2] Hetnarski, R.B., Ignaczak, J., Generalized Thermoelasticity, J. Thermal Stresses, 22, 1999, 451–470.

[3] Hetnarski, R.B. and Ignaczak, J., Nonclassical Dynamical Thermoelasticity, Inter. J. Solids Struct., 37, 2000, 215–224.

[4] Chandrasekharaiah, D.S., Hyperbolic Thermoelasticity, A Review of Recent Literature, Appl. Mech. Rev., 51, 1998, 705–729.

[5] Lord, H., Shulman, Y., A Generalized Dynamical Theory of Thermoelasticity, J. Mech. Phys. Solids, 15, 1967, 299.

[6] Green, A.E., Lindsay, K.A., Thermoelasticity, J. Elasticity, 2, 1972, 1–7.

[7] Ignaczak, J., Generalized Thermoelasticity and its Applications, in R. B. Hetnarski (ed.), Mechanical and Mathematical Methods, Thermal Stresses III, North Holland, 1989.

[8] Tzou, D.Y., A Unified Field Approach for Heat Conduction from Macro- to Micro-Scales, ASME J. Heat Transfer, 117, 1995, 8–16.

[9] Tzou, D.Y., The Generalized Lagging Response in Small-Scale and High-Rate Heating, Int. J. Heat Mass Transfer, 38, 1995, 3231–3240.

[10] Tzou, D.Y., Experimental Support for the Lagging Behavior in Heat Propagation, AIAA J. Thermophys. Heat Transfer, 9, 1995, 686–693.

[11] Chandrasekharaiah, D.S., Hyperbolic Thermoelasticity: A Review of Recent Literature, Appl. Mech. Rev., 51, 1998, 705–729.

[12] Quintanilla, R., Racke, R., A Note on Stability of Dual Phase-Lag Heat Conduction, Int. J. Heat. Mass Transf., 49, 2006, 1209–1213.

[13] Quintanilla, R., Racke, R., Qualitative Aspects in Dual Phase-Lag Heat Conduction, Proc. Royal Soc. A., 463, 659–674, 2007.

[14] Zenkour, A.M., Abouelregal, A.E., Effects of Phase-Lags in a Thermoviscoelastic Orthotropic Continuum with a Cylindrical Hole and Variable Thermal Conductivity, Arch. Mech., 67, 2015, 457–475.

[15] Zenkour, A.M., Mashat, D.S., Abouelregal, A.E., The Effect of Dual-Phase-Lag Model on Reflection of Thermoelastic Waves in a Solid Half Space with Variable Material Properties, Acta Mech. Solida Sinica, 26, 2013, 659–670.

[16] Prasad, R., Kumar, R., Mukhopadhyay, S., Propagation of Harmonic Plane Waves under Thermoelasticity with Dual-Phase-Lags, Int. J. Eng. Sci., 48(12), 2010, 2028–2043.

[17] Borgmeyer, K., Quintanilla, R., Racke, R., Phase-Lag Heat Condition: Decay Rates for Limit Problems and Well-Posedness, J. Evol. Equ., 14, 2014, 863-884.

[18] Liu, Z., Quintanilla, R., Time Decay in Dual-Phase-Lag Thermoelasticity: Critical Case, Comm. Pure Appl. Analy., 17(1), 2018, 177-190.

[19] Guo, F.L., Wang, G.Q., Rogerson, G.A., Analysis of Thermoelastic Damping in Micro- and Nanomechanical Resonators based on Dual-Phase-Lag Generalized Thermoelasticity Theory, Int. J. Eng. Sci., 60, 2012, 59-65.

[20] Abbas, I.A., A Dual Phase Lag Model on Thermoelastic Interaction in an Infinite Fiber-Reinforced Anisotropic Medium with a Circular Hole, Mech. Based Des. Struct. Machines, 43, 2015, 501–513.

[21] Green, A.E., Naghdi, P.M., A Re-examination of the Basic Postulates of Thermomechanics, Proc. Roy. Soc. Lond. A, 432, 1991, 171–194.

[22] Green, A.E., Naghdi, P.M., On Undamped Heat Waves in an Elastic Solid, J. Therm. Stress., 15, 1992, 253–264.

[23] Green, A.E., Naghdi, P.M., Thermoelasticity without Energy Dissipation, J. Elasticity, 31, 1993, 189–208.

[24] Chandrasekharaiah, D.S., A Note on the Uniqueness of Solution in the Linear Theory of Thermoelasticity without Energy Dissipation, J. Elasticity, 43(3), 1996, 279–283.

[25] Chandrasekharaiah, D.S., A Uniqueness Theorem in the Theory of Thermoelasticity without Energy Dissipation, J. Therm. Stresses, 19(3), 1996, 267–272.

[26] Choudhuri, S.R., On a Thermoelastic Three-Phase-Lag Model, J. Therm. Stresses, 30(3), 2007, 231–238.

[27] El-Karamany, A.S., Ezzat, M.A., On the Phase-Lag Green-Naghdi Thermoelasticity Theories, Appl. Math. Model., 40, 2016, 5643–5659.

[28] Ciarletta, M.A., Theory of Micropolar Thermoelasticity without Energy Dissipation, J. Therm. Stresses, 22, 2009, 581–594.

[29] Chiriţă, S., Ciarletta, M., Reciprocal and Variational Principles in Linear Thermoelasticity without Energy Dissipation, Mech. Res. Commun., 37, 2010, 271–275.

[30] Ieşan, D., On a Theory of Thermoelasticity without Energy Dissipation for Solids with Microtemperatures, Z. Angew. Math. Mech., 98(6), 2018, 870-885.

[31] Quintanilla, R., On Existence in Thermoelasticity without Energy Dissipation, J. Therm. Stresses, 25, 2002, 195-202.

[32] Allam, M.N., Elsibai K.A., Abouelregal, A.E., Electromagneto-Thermoelastic Problem in a Thick Plate using Green and Naghdi Theory, Int. J. Eng. Sci., 47, 2009, 680-690.

[33] Allam, M.N., Elsibai K.A., Abouelregal, A.E., Electromagneto-Thermoelastic Plane Waves without Energy Dissipation for an Infinitely Long Annular Cylinder in a Harmonic Field, J. Therm. Stresses, 30, 2007,195–210.

[34] Marin, M., Baleanu, D., On Vibrations in Thermoelasticity without Energy Dissipation for Micropolar Bodies, Bound. Val. Prob., 2016, 2016, 111.

[35] Khedr M.El., Khader, S.A., A Problem in Thermoelasticity with and without Energy Dissipation, J. Phys. Math., 8(3), 2017, 1000243.

[36] Marin, M., Cesaro means in thermoelasticity of dipolar bodies, Acta Mech., 122(1-4), 1997, 155-168.

[37] Hassan, M., Marin, M., Ellahi, R., Alamri, S.Z., Exploration of Convective Heat Transfer and Flow Characteristics Synthesis by Cu–Ag/Water Hybrid-Nanofluids, Heat Transfer Research, 49(18), 2018, 1837-1848.

[38] Chiriţă, S., Ciarletta, M., Tibullo, V., On the Wave Propagation in the Time Differential Dual-Phase-Lag Thermoelastic Model, Proc. Royal Soc. A, 471, 2015, 20150400.

[39] Chiriţă, S., High-Order Effects of Thermal Lagging in Deformable Conductors, Int. J. Heat Mass Trans., 127, 2018, 965–974.

[40] Cattaneo, C., A Form of Heat Conduction Equation which Eliminates the Paradox of Instantaneous Propagation, Comp. Rend., 247, 1958, 431-433.

[41] Vernotte, P., Les paradoxes de la Theorie Continue de l’Equation de la Chaleur, Comp. Rend., 246, 1958, 3154-3155.

[42] Chiriţă, S., On the Time Differential Dual-Phase-Lag Thermoelastic Model, Meccanica, 52, 2017, 349–361.

[43] Zenkour, A.M., Abouelregal, A.E., Alnefaie, K.A., Abu-Hamdeh, N.H., Seebeck Effect on a Magneto-Thermoelastic Long Solid Cylinder with Temperature-Dependent Thermal Conductivity, European J. Pure Appl. Math., 10(4), 2017, 786-808.

[44] Chandrasekharaiah, D.S., Srinath, K.S., Thermoelastic Interactions without Energy Dissipation due to a Point Heat Source, J. Elasticity, 50, 1998, 97–108.

[45] Morse, P., Feshbach, H., Methods of Theoretical Physics, 1st ed., McGraw-Hill, New York, 1953.

[46] Honig, G., Hirdes, U., A Method for the Numerical Inversion of the Laplace Transform, J. Comput. Appl. Math., 10, 1984, 113–132.

[47] Mashat, D.S., Zenkour, A.M., Abouelregal, A.E., Fractional Order Thermoelasticity Theory for a Half-Space Subjected to an Axisymmetric Heat Distribution, Mech. Adv. Mater. Struct., 22(11), 2015, 925–932.

[48] Zenkour, A.M. and Abouelregal, A.E., State-Space Approach for an Infinite Medium with a Spherical Cavity Based Upon Two-Temperature Generalized Thermoelasticity Theory and Fractional Heat Conduction, Zeitsch. Angewandte Math. Phys., 65(1), 2014, 149–164.

[49] Quintanilla, R., Exponential Stability in the Dual-Phase-Lag Heat Conduction Theory, J. Non-Equil. Thermod., 27, 2002, 217–227.

[50] El-Karamany, A.S., Ezzat, M.A., On the Phase-Lag Green-Naghdi Thermoelasticity Theories, Appl Math. Model., 40, 2016, 5643–5659.

[51] Ezzat, M.A., El-Karamany, A.S., Fractional Order Heat conduction Law in Magneto-Thermoelasticity Involving Two Temperatures, Zeitsch. Angewandte Math. Phys., 62, 2011, 937–952.

[52] Ezzat, M.A., El-Karamany, A.S., Theory of Fractional Order in Electro-thermoelasticity, Eur. J. Mech. A/Solid, 30, 2011, 491–500.

[53] Ezzat, M.A., Fayik, M., Fractional Order Theory of Thermoelastic Diffusion, J. Therm. Stresses, 34, 2013, 851–872.

[54] Xu, H.Y., Jiang, X.Y., Time Fractional Dual-Phase-Lag Heat Conduction Equation, Chin. Phys. B, 24(3), 2015, 034401.