Three Dimensional Thermal Shock Problem in Magneto-Thermoelastic Orthotropic Medium

Document Type : Research Paper


1 Department of Mathematics, University of North Bengal, Darjeeling, India

2 Department of Mathematics, Faculty of Science, Taif University, Saudi Arabia---- Department of Mathematics, Faculty of Science, South Valley University, Egypt



The paper is concerned with the study of magneto-thermoelastic interactions in three dimensional thermoelastic medium under the purview of three-phase-lag model of generalized thermoelasticity. The medium under consideration is assumed to be homogeneous orthotropic medium. The fundamental equations of the three-dimensional problem of generalized thermoelasticity are obtained as a vector-matrix differential equation form by employing normal mode analysis which is then solved by eigenvalue approach. Stresses and displacements are presented graphically for different thermoelastic models.


[1] Lord H.W., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids 15: 299-309.
[2] Green A.E., Lindsay K.A., Thermoelasticity, Journal of Elasticity 2: 1-7.
[3] Green A.E., Naghdi P.M., 1991, A re-examination of the basic properties of thermomechanics, Proceedings of Royal Society London Series A 432: 171-194.
[4] Green A.E., Naghdi P.M., 1992, On damped heat waves in an elastic solid, Journal of Thermal Stresses 15: 252-264.
[5] Green A.E., Naghdi P.M., 1993, Thermoelasticity without energy dissipation, Journal of Elasticity 31: 189-208.
[6] Chandrasekharaih D.S., 1986, Thermoelasticity with second sound: A review, Applied Mechanics Reviews 39(3): 355-376.
[7] Chandrasekharaih D.S., 1998, Hyperbolic thermoelasticity: A review of recent literature, Applied Mechanics Reviews 51(12): 705-729.
[8] Ignaczak J., Hetnarski R.B., 2014, Generalized Thermoelasticity: Mathematical Formulation, Encyclopedia of Thermal Stresses 2014: 1974-1986.
[9] Tzou D.Y., 1995, A unique field approach for heat conduction from macro to micro scales, Journal of Heat Transfer 117: 8-16.
[10] Roy Choudhuri S.K., 2007, On a thermoelastic three phase lag model, Journal of Thermal Stresses 30: 231-238.
[11] Biswas S., Mukhopadhyay B., Shaw S., 2017, Thermal shock response in magneto-thermoelastic orthotropic medium with three-phase-lag model, Journal of Electromagnetic waves and Applications 31(9): 879-897.
[12] Othman M.I.A., Hasona W.M., Mansour N.T., 2015, The effect of magnetic field on generalized thermoelastic medium with two temperature under three phase lag model, Multidiscipline Modeling in Materials and Structures 11(4): 544-557.
[13] Said S.M., 2016, Influence of gravity on generalized magneto-thermoelastic medium for three-phase -lag model, Journal of Computational and Applied Mathematics 291:142-157.
[14] Othman M.I.A., Said S.M., 2014, 2-D problem of magneto-thermoelasticity fiber reinforced medium under temperature-dependent properties with three-phase-lag theory, Meccanica 49(5): 1225-1243.
[15] El-Karamany A.S., Ezzat M.A., 2004, Thermal shock problem in generalized thermoelasticity under four theories, International Journal of Engineering Science 42: 649-671.
[16] Sherief H.H., El-Maghraby N.M., Allam A.A., 2013, Stochastic thermal shock in generalized thermoelasticity, Applied Mathematical Modelling 37: 762-775.
[17] Ezzat M.A., Youssef H.M., 2010, Three dimensional thermal shock problem of generalized thermoelastic half-space, Applied Mathematical Modelling 34: 3608-3622.
[18] Kalkal K.K., Deswal S., 2014, Effects of phase lags on three dimensional wave propagation with temperature dependent properties, International Journal of Thermophysics 35(5): 952-969.
[19] El-Karamany A.S., Ezzat M.A., 2013, On the three-phase-lag linear micropolar thermoelasticity theory, European Journal of Mechanics A/ Solids 40: 198-208.
[20] Ezzat M. A., El-Karamany A.S., Fayik M.A., 2012, Fractional order theory in thermoelastic solid with three-phase-lag heat transfer, Archive of Applied Mechanics 82(4): 557-572.
[21] Said S.M., Othman M.I.A., 2016, Effects of gravitational and hydrostatic initial stress on a two-temperature fiber-reinforced thermoelastic medium for three-phase-lag, Journal of Solid Mechanics 8(4): 806-822.
[22] Lofty K.h., 2014, Two temperature generalized magneto-thermoelastic interactions in an elastic medium under three theories, Applied Mathematics and Computation 227: 871-888.
[23] Sarkar N., Lahiri A., 2012, Electromagneto-thermoelastic interactions in an orthotropic slab with two thermal relaxation times, Computational Mathematics and Modelling 23(4): 461-477.
[24] Das N.C., Bhakta P.C., 1985, Eigen function expansion method to the solution of simultaneous equations and its application in mechanics, Mechanics Research Communications 12: 19-29.
[25] Ezzat M.A., 2006, The relaxation effects of the volume properties of electrically conducting viscoelastic material, Materials Science and Engineering B: Solid-State Materials for Advanced Technology 130: 11-23.
[26] Ezzat M.A., 2004, Fundamental solution in generalized magneto-thermoelasticity with two relaxation times for perfect conductor cylindrical region, International Journal of Engineering Science 42: 1503-1519.
[27] Ezzat M.A., El-Karamany A.S., El-Bary A.A., 2016, Electro-thermoelasticity theory with memory-dependent heat transfer, International Journal of Engineering Science 99: 22-38.