Investigation on Magneto-thermoelastic Disturbances Induced by Thermal Shock in an Elastic Half Space Having Finite Conductivity under Dual Phase-lag Heat Conduction
Tiwari Rakhi *, Kumar Anil, Mukhopadhyay Santwana
Department of Mathematical Sciences, Indian Institute of Technology
(Banaras Hindu University), Varanasi 221 005, India
∗E-mail: rakhibhu2117@gmail.com
Received:
Received: 20 April 2016; revised: 19 October 2016; accepted: 24 October 2016; published online: 16 November 2016
DOI: 10.12921/cmst.2016.0000019
Abstract:
The present work seeks to investigate the propagation of magneto-thermoelastic disturbances produced by a thermal shock in a finitely conducting elastic half-space in contact with vacuum. Normal load has been applied on the boundary of the existing media that is supposed to be permeated by a primary uniform magnetic field. We employ both the parabolic type (dual phase-lag magneto-thermoelasticity of type I (MTDPL-I)) and hyperbolic type (dual phase-lag magneto-thermoelasticity of type II (MTDPL-II)) dual phase-lag heat conduction models to account for the interactions among the magnetic, elastic and thermal fields. The integral transform technique is applied to solve the present problem and the analytical results of both cases have been obtained separately. A detailed analysis of results has been made in order to understand the nature of waves propagating inside the medium and the effects of the phase-lag parameters. The effect of the presence of magnetic field has been highlighted. Numerical results have also been obtained to analyze the effect of magnetic field on the behavior of the solution more clearly and a detailed analysis of the results predicted by two models has been presented. It has been noted that in some cases there are significant differences in the solution obtained in the contexts of MTDPL-I and MTDPL-II theory of magneto-thermoelasticity.
Key words:
dual phase-lag heat conduction theory-I, dual phase-lag heat conduction theory-II, finite conductivity, magneto-thermoelastic waves, thermal shock
References:
[1] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press, London 1959.
[2] A.N. Tikhonov, A.A. Samarskii, Equations of Mathematical Physics, Dover New York 1990.
[3] J.I. Frankel, B. Vick, M.N. Ozisik, General formulation and analysis of hyperbolic heat conduction in composite media, Int. J. Heat Mass Transfer 30(7), 1293-1305 (1987).
[4] D.D. Joseph, L. Preziosi, Heat waves, Rev. Mod. Phys. 61(1), 41-73 (1989).
[5] L. Wang, X. Zhou, X. Wei, Heat Conduction: Mathematical Models and Analytical Solutions, Springer-Verlag, Berlin Heidelberg 2008.
[6] M.A. Biot, Thermoelasticity and irreversible thermodynamics, J. Appl. Mech. Tech. Phys. 27, 240-253 (1956).
[7] C. Cattaneo, A form of heat conduction equation which eliminates the paradox of instantaneous propagation, Compt. Rend. 247, 431-433 (1948).
[8] P. Vernotte, Some possible complications in the phenomenon of thermal conduction, Compt. Rend. 252, 2190-2191 (1961).
[9] H.W. Lord, Y. Shulman, A Generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids 15, 299-309 (1967).
[10] A.E. Green, K.A. Lindsay, Thermoelasticity, J. Elasticity 2, 1-7 (1972).
[11] A.E. Green, P.M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity 31, 189-208 (1993).
[12] A.E. Green, P.M. Naghdi, A re-examination of the basic postulates of thermoemechanics, Proc. Roy. Soc. London A432, 171-194 (1991).
[13] D.Y. Tzou, Macro-to Microscale Heat Transfer: The Lagging Behavior, Taylor & Francis, New York 1997.
[14] D.Y. Tzou, On the thermal shock wave induced by a moving heat source, ASME J. Heat Transfer 111(2), 232-238 (1989).
[15] J.R. Ho, C.P. Kuo, W.S. Jiaung, Study of heat transfer in multilayered structure within the framework of dual-phase-lag heat conduction model using lattice Boltzmann method, Int. J. Heat Mass Transfer
[16] K. Ramadan, Semi-analytical Solutions for the dual phase-lag heat conduction in multilayered media, Int. J. Therm. Sci. 48(1), 14-25 (2009).
[17] N.S. Al-Huniti, M.A. Al-Nimr, Thermoelastic behavior of a composite slab under a rapid dual-phase-lag heating, J. Therm. Stresses 27(7), 607-623 (2004).
[18] Y.M. Lee, T.W. Tsai, Ultra-fast pulse laser heating on a two-layered semi-infinite material with interfacial contact conductance, Int. Commun. Heat Mass Transfer 34(1), 45-51 (2007).
[19] K.C. Liu, Numerical analysis of dual-phase-lag heat transfer in a layered cylinder with nonlinear interface boundary conditions, Comput. Phys. Commun. 177(3), 307-314 (2007).
[20] D.S. Chandrasekharaiah, Hyperbolic thermoelasticity: a review of recent literature, Appl Mech Rev. 51(12), 705-729 (1998).
[21] W.S. Kim, L.G. Hector, M.N. Ozisik, Hyperbolic heat conduction due to axisymmetric continuous or pulsed surface heat sources, J. Applied Phys. 68, 5478-5485 (1990).
[22] M.N. Ozisik, D.Y. Tzou, On the wave theory in heat conduction, ASME Journal of Heat Transfer 116, 526-535 (1994).
[23] M.A. Al-Nimr, M. Naji, The Hyperbolic heat conduction equation in anisotropic material, Int. J. Thermophysics 21(1), 281-287 (2000).
[24] M.A. Al-Nimr, M. Naji, On the phase-lag effect on the non equilibrium entropy production, Microscale Thermophysical Engineering 4, 231-243 (2000).
[25] M.A. Al-Nimr, M. Naji„ V. Arpaci, Non equilibrium entropy production under the effect of dual-phase-lag heat conduction model, ASME J. Heat Transfer 122, 217-223 (2000).
[26] M.A. Al-Nimr, M. Hader, Melting and solidification under the effect of phase-lag concept in the hyperbolic conduction equation, Heat Transfer Engineering 22(2), 1-8 (2001).
[27] M.A. Al-Nimr, O.M. Haddad, The Phase-lag concept in the thermal behavior of lumped systems, Heat and Mass Transfer 37(2-3), 175-181 (2001).
[28] M.A. Al-Nimr, N. Al-Huniti, Transient thermal stresses in a thin elastic plate due to a rapid dual-phase-lag heating, J. Therm. Stresses 23, 731-746 (2000).
[29] J.L. Nowinski, Theory of Thermoelasticity with Applications, Sijthoff & Noordhoff, International Alphen Aan Den Rijn 1978.
[30] N.F. Jordan, A.C. Eringen, On the static non linear theory of electromagnetic thermoelastic solids Part I, Int. J. Eng. Sci. 2, 59-95 (1964).
[31] N.F. Jordan, A.C. Eringen, On the static non linear theory of electromagnetic thermoelastic solids Part II, Int. J. Eng. Sci. 2, 97-114 (1964).
[32] G.A. Maugin, The mechanical behaviour of electromagnetic solids, Phil. Trans. Roy. Soc. 302,189-215 (1984).
[33] E. Radzikowska, R. Kotowski, W. Muschik, A non equilibrium evolution criterion for electromagnetic bodies, J. Non Equilibrium Thermodynamics 26(31), 215-230 (2001).
[34] B.T. Maruszewski, A. Drzewiecki, R. Starosta, Anomalous features of the thermomagnetoelastic solid in a vortex array in a superconductor: Propagation of Love’s waves, J. Therm. Stresses 30(9-10), 1049-1065
[35] I-Shih Liu, Constitutive theory of anisotropic rigid heat conductors, Journal of Mathematical Physics 50(8), DOI: 10.1063/1.3190487 (2009).
[36] J. Casas-Vazquez, M. Criado-Sancho, D. Jou, Extended thermodynamics of polymers and superfluids, Journal of Non-newtonian Fluid Mechanics 152(1-3), 36-44 (2008).
[37] G. Lebon, D. Jou, Early history of extended irreversible thermodynamics (1953-1983): An exploration beyond local equilibrium and classical transport theory, European Physical Journal H 40(2), 205-240
[38] D. Jou, J. Casas-Vazquez, Extended irreversible thermodynamics and its relation with other continuum approaches, Journal of Non Newtonian Fluid Mechanics 96(1-2), 77-104 (2001).
[39] B.T. Maruszewski, A. Drzewiecki, R. Starosta, Thermodynamics of unconventional thermoelastic damping in auxetic media, Phys. Status Solidi B 250(10), 2044-2050 (2013).
[40] S. Kaliski, W. Nowacki, Excitation of mechanical-electromagnetic waves induced by a thermal shock, BUll. Acad. Polon. Sci. Series. Sci. Tech. 1(10), (1962).
[41] C. Massalas, A. Dalamangas, Coupled magneto-thermoelastic problem in elastic half-space, Lett. Appl. Eng. Sci. 21(8) (1983).
[42] S.K. Roychoudhuri, G. Chatterjee, A coupled magneto-thermoelastic problem in a perfectly conducting half-space with thermal relaxation, Int. J. Math. and Math. Sci. 13(3), (1990).
[43] S.K. Roychoudhuri, G. Chatterjee, Temperature-rate dependent magneto-thermoelastic waves in a finitely conducting elastic half-space, Computers Math. Applic. 19(5), (1990).
[44] J.N. Sharma, S. Dayal Chand, Transient generalized magneto-thermoelastic waves in a half space, Int. J. Eng. Sci. 26 (9), (1988).
[45] S.K. Roychoudhuri, S. Banerjee (Mukhopadhyay), Magneto-Thermoelastic waves induced by a thermal shock in a finitely conducting elastic half space, Int. J. Math., Math. Sci. 19(1), 131-144 (1996).
[46] J.C. Maxwell, A dynamical theory of the electromagnetic field, Phil. Trans. Roy. Soc. 155, 459-512 (1865).
[47] R. Bellman, R.E. Kolaba, J.A. Lockette, Numerical Inversion of the Laplace Transform, American Elsevier Pub. Co., New York 1966.
The present work seeks to investigate the propagation of magneto-thermoelastic disturbances produced by a thermal shock in a finitely conducting elastic half-space in contact with vacuum. Normal load has been applied on the boundary of the existing media that is supposed to be permeated by a primary uniform magnetic field. We employ both the parabolic type (dual phase-lag magneto-thermoelasticity of type I (MTDPL-I)) and hyperbolic type (dual phase-lag magneto-thermoelasticity of type II (MTDPL-II)) dual phase-lag heat conduction models to account for the interactions among the magnetic, elastic and thermal fields. The integral transform technique is applied to solve the present problem and the analytical results of both cases have been obtained separately. A detailed analysis of results has been made in order to understand the nature of waves propagating inside the medium and the effects of the phase-lag parameters. The effect of the presence of magnetic field has been highlighted. Numerical results have also been obtained to analyze the effect of magnetic field on the behavior of the solution more clearly and a detailed analysis of the results predicted by two models has been presented. It has been noted that in some cases there are significant differences in the solution obtained in the contexts of MTDPL-I and MTDPL-II theory of magneto-thermoelasticity.
Key words:
dual phase-lag heat conduction theory-I, dual phase-lag heat conduction theory-II, finite conductivity, magneto-thermoelastic waves, thermal shock
References:
[1] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press, London 1959.
[2] A.N. Tikhonov, A.A. Samarskii, Equations of Mathematical Physics, Dover New York 1990.
[3] J.I. Frankel, B. Vick, M.N. Ozisik, General formulation and analysis of hyperbolic heat conduction in composite media, Int. J. Heat Mass Transfer 30(7), 1293-1305 (1987).
[4] D.D. Joseph, L. Preziosi, Heat waves, Rev. Mod. Phys. 61(1), 41-73 (1989).
[5] L. Wang, X. Zhou, X. Wei, Heat Conduction: Mathematical Models and Analytical Solutions, Springer-Verlag, Berlin Heidelberg 2008.
[6] M.A. Biot, Thermoelasticity and irreversible thermodynamics, J. Appl. Mech. Tech. Phys. 27, 240-253 (1956).
[7] C. Cattaneo, A form of heat conduction equation which eliminates the paradox of instantaneous propagation, Compt. Rend. 247, 431-433 (1948).
[8] P. Vernotte, Some possible complications in the phenomenon of thermal conduction, Compt. Rend. 252, 2190-2191 (1961).
[9] H.W. Lord, Y. Shulman, A Generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids 15, 299-309 (1967).
[10] A.E. Green, K.A. Lindsay, Thermoelasticity, J. Elasticity 2, 1-7 (1972).
[11] A.E. Green, P.M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity 31, 189-208 (1993).
[12] A.E. Green, P.M. Naghdi, A re-examination of the basic postulates of thermoemechanics, Proc. Roy. Soc. London A432, 171-194 (1991).
[13] D.Y. Tzou, Macro-to Microscale Heat Transfer: The Lagging Behavior, Taylor & Francis, New York 1997.
[14] D.Y. Tzou, On the thermal shock wave induced by a moving heat source, ASME J. Heat Transfer 111(2), 232-238 (1989).
[15] J.R. Ho, C.P. Kuo, W.S. Jiaung, Study of heat transfer in multilayered structure within the framework of dual-phase-lag heat conduction model using lattice Boltzmann method, Int. J. Heat Mass Transfer
[16] K. Ramadan, Semi-analytical Solutions for the dual phase-lag heat conduction in multilayered media, Int. J. Therm. Sci. 48(1), 14-25 (2009).
[17] N.S. Al-Huniti, M.A. Al-Nimr, Thermoelastic behavior of a composite slab under a rapid dual-phase-lag heating, J. Therm. Stresses 27(7), 607-623 (2004).
[18] Y.M. Lee, T.W. Tsai, Ultra-fast pulse laser heating on a two-layered semi-infinite material with interfacial contact conductance, Int. Commun. Heat Mass Transfer 34(1), 45-51 (2007).
[19] K.C. Liu, Numerical analysis of dual-phase-lag heat transfer in a layered cylinder with nonlinear interface boundary conditions, Comput. Phys. Commun. 177(3), 307-314 (2007).
[20] D.S. Chandrasekharaiah, Hyperbolic thermoelasticity: a review of recent literature, Appl Mech Rev. 51(12), 705-729 (1998).
[21] W.S. Kim, L.G. Hector, M.N. Ozisik, Hyperbolic heat conduction due to axisymmetric continuous or pulsed surface heat sources, J. Applied Phys. 68, 5478-5485 (1990).
[22] M.N. Ozisik, D.Y. Tzou, On the wave theory in heat conduction, ASME Journal of Heat Transfer 116, 526-535 (1994).
[23] M.A. Al-Nimr, M. Naji, The Hyperbolic heat conduction equation in anisotropic material, Int. J. Thermophysics 21(1), 281-287 (2000).
[24] M.A. Al-Nimr, M. Naji, On the phase-lag effect on the non equilibrium entropy production, Microscale Thermophysical Engineering 4, 231-243 (2000).
[25] M.A. Al-Nimr, M. Naji„ V. Arpaci, Non equilibrium entropy production under the effect of dual-phase-lag heat conduction model, ASME J. Heat Transfer 122, 217-223 (2000).
[26] M.A. Al-Nimr, M. Hader, Melting and solidification under the effect of phase-lag concept in the hyperbolic conduction equation, Heat Transfer Engineering 22(2), 1-8 (2001).
[27] M.A. Al-Nimr, O.M. Haddad, The Phase-lag concept in the thermal behavior of lumped systems, Heat and Mass Transfer 37(2-3), 175-181 (2001).
[28] M.A. Al-Nimr, N. Al-Huniti, Transient thermal stresses in a thin elastic plate due to a rapid dual-phase-lag heating, J. Therm. Stresses 23, 731-746 (2000).
[29] J.L. Nowinski, Theory of Thermoelasticity with Applications, Sijthoff & Noordhoff, International Alphen Aan Den Rijn 1978.
[30] N.F. Jordan, A.C. Eringen, On the static non linear theory of electromagnetic thermoelastic solids Part I, Int. J. Eng. Sci. 2, 59-95 (1964).
[31] N.F. Jordan, A.C. Eringen, On the static non linear theory of electromagnetic thermoelastic solids Part II, Int. J. Eng. Sci. 2, 97-114 (1964).
[32] G.A. Maugin, The mechanical behaviour of electromagnetic solids, Phil. Trans. Roy. Soc. 302,189-215 (1984).
[33] E. Radzikowska, R. Kotowski, W. Muschik, A non equilibrium evolution criterion for electromagnetic bodies, J. Non Equilibrium Thermodynamics 26(31), 215-230 (2001).
[34] B.T. Maruszewski, A. Drzewiecki, R. Starosta, Anomalous features of the thermomagnetoelastic solid in a vortex array in a superconductor: Propagation of Love’s waves, J. Therm. Stresses 30(9-10), 1049-1065
[35] I-Shih Liu, Constitutive theory of anisotropic rigid heat conductors, Journal of Mathematical Physics 50(8), DOI: 10.1063/1.3190487 (2009).
[36] J. Casas-Vazquez, M. Criado-Sancho, D. Jou, Extended thermodynamics of polymers and superfluids, Journal of Non-newtonian Fluid Mechanics 152(1-3), 36-44 (2008).
[37] G. Lebon, D. Jou, Early history of extended irreversible thermodynamics (1953-1983): An exploration beyond local equilibrium and classical transport theory, European Physical Journal H 40(2), 205-240
[38] D. Jou, J. Casas-Vazquez, Extended irreversible thermodynamics and its relation with other continuum approaches, Journal of Non Newtonian Fluid Mechanics 96(1-2), 77-104 (2001).
[39] B.T. Maruszewski, A. Drzewiecki, R. Starosta, Thermodynamics of unconventional thermoelastic damping in auxetic media, Phys. Status Solidi B 250(10), 2044-2050 (2013).
[40] S. Kaliski, W. Nowacki, Excitation of mechanical-electromagnetic waves induced by a thermal shock, BUll. Acad. Polon. Sci. Series. Sci. Tech. 1(10), (1962).
[41] C. Massalas, A. Dalamangas, Coupled magneto-thermoelastic problem in elastic half-space, Lett. Appl. Eng. Sci. 21(8) (1983).
[42] S.K. Roychoudhuri, G. Chatterjee, A coupled magneto-thermoelastic problem in a perfectly conducting half-space with thermal relaxation, Int. J. Math. and Math. Sci. 13(3), (1990).
[43] S.K. Roychoudhuri, G. Chatterjee, Temperature-rate dependent magneto-thermoelastic waves in a finitely conducting elastic half-space, Computers Math. Applic. 19(5), (1990).
[44] J.N. Sharma, S. Dayal Chand, Transient generalized magneto-thermoelastic waves in a half space, Int. J. Eng. Sci. 26 (9), (1988).
[45] S.K. Roychoudhuri, S. Banerjee (Mukhopadhyay), Magneto-Thermoelastic waves induced by a thermal shock in a finitely conducting elastic half space, Int. J. Math., Math. Sci. 19(1), 131-144 (1996).
[46] J.C. Maxwell, A dynamical theory of the electromagnetic field, Phil. Trans. Roy. Soc. 155, 459-512 (1865).
[47] R. Bellman, R.E. Kolaba, J.A. Lockette, Numerical Inversion of the Laplace Transform, American Elsevier Pub. Co., New York 1966.
