Document Type: Research Paper

**Authors**

^{1}
Department of Mathematics, Kurukshetra University , Kurukshetra, Haryana, India

^{2}
Department of Mathematics, MM University, Mullana, Ambala, Haryana, India

^{3}
Department of Basic and Applied Sciences, Punjabi University, Patiala, Punjab, India

**Abstract**

The present paper is concerned with the investigation of disturbances in a homogeneous transversely isotropic thermoelastic rotating medium with two temperatures, in the presence of the combined effects of Hall currents and magnetic field due to thermomechanical sources. The formulation is applied to the thermoelasticity theories developed by Green-Naghdi Theories of Type-II and Type-III. Laplace and Fourier transform technique is applied to solve the problem. As an application, the bounding surface is subjected to concentrated and distributed sources (mechanical and thermal sources). The analytical expressions of displacement, stress components, temperature change and current density components are obtained in the transformed domain. Numerical inversion technique has been applied to obtain the results in the physical domain. Numerical simulated results are depicted graphically to show a comparison of effect of Hall current on the two theories GN-II and GN-III on resulting quantities. Some special cases are also deduced from the present investigation.

**Keywords**

Abbas I.A., Kumar R., Reen L,S., 2014, Response of thermal sources in transversely isotropic thermoelastic materials without energy dissipation and with two temperatures, Canadian Journal of Physics 92(11): 1305-1311.

[2] Abbas I.A., 2011, A two dimensional problem for a fibre- reinforced anisotropic thermoelastic half-space with energy dissipation, Sadhana(c) Indian academy of sciences 36(3): 411-423.

[3] Attia H.A., 2009, Effect of Hall current on the velocity and temperature distributions of couette flow with variable properties and uniform suction and injection, Computational and Applied Mathematics 28(2): 195-212.

[4] Atwa S.Y., Jahangir A., 2014, Two temperature effects on plane waves in generalized thermo-microstretch elastic solid, International Journal of Thermophysics 35: 175-193.

[5] Boley B.A., Tolins I.S., 1962, Transient coupled thermoelastic boundary value problem in the half space, Journal of Applied Mechanics 29: 637-646.

[6] Chandrasekharaiah D. S., 1998, Hyperbolic thermoelasticity: A review of recent literature, Applied Mechanics Reviews 51: 705-729.

[7] Chen P.J., Gurtin M.E., 1968, On a theory of heat conduction involving two parameters, Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 19: 614-627.

[8] Chen P.J., Gurtin M.E., WilliamsW.O., 1968, A note on simple heat conduction, Journal of Applied Mathematics and Physics 19: 969-970.

[9] Chen P.J., Gurtin M.E., Williams W.O., 1969, On the thermodynamics of non simple elastic materials with two temperatures, Journal of Applied Mathematics and Physics 20: 107-112.

[10] Das P., Kanoria M., 2014, Study of finite thermal waves in a magnetothermoelastic rotating medium, Journal of Thermal Stresses 37(4): 405-428.

[11] Dhaliwal R.S., Singh A., 1980, Dynamic Coupled Thermoelasticity, Hindustance Publisher Corp, New Delhi, India.

[12] Ezzat M.A, Awad E.S.,2010, Constitutive relations, uniqueness of solutionand thermal shock application in the linear theory of micropolar generalized thermoelasticity involving two temperatures, Journal of Thermal Stresses 33(3): 225-250.

[13] Green A.E., Naghdi P.M., 1991, A re-examination of the basic postulates of thermomechanics, Proceedings of the Royal Society of London A 432: 171-194.

[14] Green A.E., Naghdi P.M., 1992, On undamped heat waves in an elastic solid, Journal of Thermal Stresses 15: 253-264.

[15] Green A.E., Naghdi P.M.,1993, Thermoelasticity without energy dissipation, Journal of Elasticity 31:189-208.

[16] Honig G., Hirdes U., 1984, A method for the inversion of Laplace transform, Journal of Computational and Applied Mathematics 10:113-132.

[17] Kaushal S., Kumar R., Miglani A., 2011, Wave propagation in temperature rate dependent thermoelasticity with two temperatures, Mathematical Sciences 5:125-146.

[18] Kaushal S.,Sharma N., Kumar R., 2010, Propagation of waves in generalized thermoelastic continua with two temperature, International Journal of Applied Mechanics and Engineering 15:1111-1127.

[19] Kumar R., Devi S., 2010, Magnetothermoelastic (Type-II AND III) half-space in contact with vacuum, Applied Mathematical Sciences 69(4): 3413-3424.

[20] Kumar R., Kansal T., 2010, Effect of rotation on rayleigh lamb waves in an isotropic generalized thermoelastic diffusive plate, Journal of Applied Mechanics and Technical Physics 51(5):751-761.

[21] Kumar R., Mukhopdhyay S., 2010, Effects of thermal relaxation times on plane wave propagation under two temperature thermoelasticity, International Journal of Engineering Sciences 48(2): 128-139.

[22] Kumar R., Rupender., 2009, Effect of rotation in magneto-micropolar thermoelastic medium due to mechanical and thermal sources, Solitons and Fractals 41:1619-1633.

[23] Kumar R., Sharma K.D., Garg S.K., 2014, Effect of two temperature on reflection coefficient in micropolar thermoelastic media with and without energy dissipation, Advances in Acoustics and Vibrations ID846721.

[24] Mahmoud S.R., 2013, An analytical solution for effect of magnetic field and initial stress on an infinite generalized thermoelastic rotating non homogeneous diffusion medium, Abstract and Applied Analysis ID 284646.

[25] Press W.H., Teukolshy S.A., Vellerling W.T., Flannery B.P., 1986, Numerical Recipes in Fortran, Cambridge University Press, Cambridge.

[26] Quintanilla R., 2002, Thermoelasticity without energy dissipation of materials with microstructure, Journal of Applied Mathematical Modeling 26:1125-1137.

[27] Salem A.M., 2007, Hall current effects on MHD flow of a Power-Law fluid over a rotating disk, Journal of the Korean Physical Society 50(1): 28-33.

[28] Sarkar N., Lahiri A., 2012, Temperature rate dependent generalized thermoelasticity with modified Ohm's law, International Journal of Computational Materials Science and Engineering 1(4): 1250031.

[29] Sharma K., Bhargava R.R., 2014, Propagation of thermoelastic plane waves at an imperfect boundary of thermal conducting viscous liquid/generalized thermolastic solid, Afrika Matematika 25: 81-102.

[30] Sharma K., Marin M., 2013, Effect of distinct conductive and thermodynamic temperatures on the reflection of plane waves in micropolar elastic half-space, UPB Scientific Bulletin 75(2):121-132.

[31] Sharma K., Kumar P., 2013, Propagation of plane waves and fundamental solution in thermoviscoelastic medium with voids, Journal of Thermal Stresses 36: 94-111.

[32] Sharma N., Kumar R., 2012, Elastodynamics of an axi-symmetric problem in generalised thermoelastic diffusion , International Journal of Advanced Scientific and Technical Research 2(3): 478-492.

[33] Sharma N., Kumar R., Ram P., 2012, Interactions of generalised thermoelastic diffusion due to inclined load, International Journal of Emerging Trends in Engineering and Development 5(2): 583-600.

[34] Sharma S., Sharma K., Bhargava R.R., 2013, Effect of viscousity on wave propagation in anisotropic thermoelastic with Green-Naghdi theory Type-II and Type-III, Materials Physics and Mechanics 16:144-158.

[35] Slaughter W.S., 2002, The Linearised Theory of Elasticity, Birkhausar.

[36] Warren W.E., Chen P.J., 1973, Wave propagation in the two temperature theory of thermoelasticity, Journal of Acta Mechanica 16: 21-33.

[37] Youssef H.M., 2006, Theory of two temperature generalized thermoelasticity, IMA Journal of Applied Mathematics 71(3): 383-390.

[2] Abbas I.A., 2011, A two dimensional problem for a fibre- reinforced anisotropic thermoelastic half-space with energy dissipation, Sadhana(c) Indian academy of sciences 36(3): 411-423.

[3] Attia H.A., 2009, Effect of Hall current on the velocity and temperature distributions of couette flow with variable properties and uniform suction and injection, Computational and Applied Mathematics 28(2): 195-212.

[4] Atwa S.Y., Jahangir A., 2014, Two temperature effects on plane waves in generalized thermo-microstretch elastic solid, International Journal of Thermophysics 35: 175-193.

[5] Boley B.A., Tolins I.S., 1962, Transient coupled thermoelastic boundary value problem in the half space, Journal of Applied Mechanics 29: 637-646.

[6] Chandrasekharaiah D. S., 1998, Hyperbolic thermoelasticity: A review of recent literature, Applied Mechanics Reviews 51: 705-729.

[7] Chen P.J., Gurtin M.E., 1968, On a theory of heat conduction involving two parameters, Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 19: 614-627.

[8] Chen P.J., Gurtin M.E., WilliamsW.O., 1968, A note on simple heat conduction, Journal of Applied Mathematics and Physics 19: 969-970.

[9] Chen P.J., Gurtin M.E., Williams W.O., 1969, On the thermodynamics of non simple elastic materials with two temperatures, Journal of Applied Mathematics and Physics 20: 107-112.

[10] Das P., Kanoria M., 2014, Study of finite thermal waves in a magnetothermoelastic rotating medium, Journal of Thermal Stresses 37(4): 405-428.

[11] Dhaliwal R.S., Singh A., 1980, Dynamic Coupled Thermoelasticity, Hindustance Publisher Corp, New Delhi, India.

[12] Ezzat M.A, Awad E.S.,2010, Constitutive relations, uniqueness of solutionand thermal shock application in the linear theory of micropolar generalized thermoelasticity involving two temperatures, Journal of Thermal Stresses 33(3): 225-250.

[13] Green A.E., Naghdi P.M., 1991, A re-examination of the basic postulates of thermomechanics, Proceedings of the Royal Society of London A 432: 171-194.

[14] Green A.E., Naghdi P.M., 1992, On undamped heat waves in an elastic solid, Journal of Thermal Stresses 15: 253-264.

[15] Green A.E., Naghdi P.M.,1993, Thermoelasticity without energy dissipation, Journal of Elasticity 31:189-208.

[16] Honig G., Hirdes U., 1984, A method for the inversion of Laplace transform, Journal of Computational and Applied Mathematics 10:113-132.

[17] Kaushal S., Kumar R., Miglani A., 2011, Wave propagation in temperature rate dependent thermoelasticity with two temperatures, Mathematical Sciences 5:125-146.

[18] Kaushal S.,Sharma N., Kumar R., 2010, Propagation of waves in generalized thermoelastic continua with two temperature, International Journal of Applied Mechanics and Engineering 15:1111-1127.

[19] Kumar R., Devi S., 2010, Magnetothermoelastic (Type-II AND III) half-space in contact with vacuum, Applied Mathematical Sciences 69(4): 3413-3424.

[20] Kumar R., Kansal T., 2010, Effect of rotation on rayleigh lamb waves in an isotropic generalized thermoelastic diffusive plate, Journal of Applied Mechanics and Technical Physics 51(5):751-761.

[21] Kumar R., Mukhopdhyay S., 2010, Effects of thermal relaxation times on plane wave propagation under two temperature thermoelasticity, International Journal of Engineering Sciences 48(2): 128-139.

[22] Kumar R., Rupender., 2009, Effect of rotation in magneto-micropolar thermoelastic medium due to mechanical and thermal sources, Solitons and Fractals 41:1619-1633.

[23] Kumar R., Sharma K.D., Garg S.K., 2014, Effect of two temperature on reflection coefficient in micropolar thermoelastic media with and without energy dissipation, Advances in Acoustics and Vibrations ID846721.

[24] Mahmoud S.R., 2013, An analytical solution for effect of magnetic field and initial stress on an infinite generalized thermoelastic rotating non homogeneous diffusion medium, Abstract and Applied Analysis ID 284646.

[25] Press W.H., Teukolshy S.A., Vellerling W.T., Flannery B.P., 1986, Numerical Recipes in Fortran, Cambridge University Press, Cambridge.

[26] Quintanilla R., 2002, Thermoelasticity without energy dissipation of materials with microstructure, Journal of Applied Mathematical Modeling 26:1125-1137.

[27] Salem A.M., 2007, Hall current effects on MHD flow of a Power-Law fluid over a rotating disk, Journal of the Korean Physical Society 50(1): 28-33.

[28] Sarkar N., Lahiri A., 2012, Temperature rate dependent generalized thermoelasticity with modified Ohm's law, International Journal of Computational Materials Science and Engineering 1(4): 1250031.

[29] Sharma K., Bhargava R.R., 2014, Propagation of thermoelastic plane waves at an imperfect boundary of thermal conducting viscous liquid/generalized thermolastic solid, Afrika Matematika 25: 81-102.

[30] Sharma K., Marin M., 2013, Effect of distinct conductive and thermodynamic temperatures on the reflection of plane waves in micropolar elastic half-space, UPB Scientific Bulletin 75(2):121-132.

[31] Sharma K., Kumar P., 2013, Propagation of plane waves and fundamental solution in thermoviscoelastic medium with voids, Journal of Thermal Stresses 36: 94-111.

[32] Sharma N., Kumar R., 2012, Elastodynamics of an axi-symmetric problem in generalised thermoelastic diffusion , International Journal of Advanced Scientific and Technical Research 2(3): 478-492.

[33] Sharma N., Kumar R., Ram P., 2012, Interactions of generalised thermoelastic diffusion due to inclined load, International Journal of Emerging Trends in Engineering and Development 5(2): 583-600.

[34] Sharma S., Sharma K., Bhargava R.R., 2013, Effect of viscousity on wave propagation in anisotropic thermoelastic with Green-Naghdi theory Type-II and Type-III, Materials Physics and Mechanics 16:144-158.

[35] Slaughter W.S., 2002, The Linearised Theory of Elasticity, Birkhausar.

[36] Warren W.E., Chen P.J., 1973, Wave propagation in the two temperature theory of thermoelasticity, Journal of Acta Mechanica 16: 21-33.

[37] Youssef H.M., 2006, Theory of two temperature generalized thermoelasticity, IMA Journal of Applied Mathematics 71(3): 383-390.

[38] Youssef H.M., AI-Lehaibi E.A.,2007, State space approach of two temperature generalized thermoelasticity of one dimensional problem, International Journal of Solids and Structures 44:1550-1562.

[39] Youssef H.M., AI-Harby A.H., 2007, State space approach of two temperature generalized thermoelasticity of infinite body with a spherical cavity subjected to different types of thermal loading, Journal of Archives of Applied Mechanics 77(9): 675-687.

[40] Youssef H.M., 2011, Theory of two - temperature thermoelasticity without energy dissipation, Journal of Thermal Stresses 34:138-146.

[41] Youssef H.M.,2013, Variational principle of two - temperature thermoelasticity without energy dissipation, Journal of Thermoelasticity 1(1): 42-44.

[42] Zakaria M., 2012, Effects of hall current and rotation on magneto-micropolar generalized thermoelasticity due to ramp-type heating, International Journal of Electromagnetics and Applications 2(3): 24-32.

[43] Zakaria M., 2014, Effect of hall current on generalized magneto-thermoelasticity micropolar solid subjected to ramp-type heating, International Applied Mechanics 50(1):92-104.

[39] Youssef H.M., AI-Harby A.H., 2007, State space approach of two temperature generalized thermoelasticity of infinite body with a spherical cavity subjected to different types of thermal loading, Journal of Archives of Applied Mechanics 77(9): 675-687.

[40] Youssef H.M., 2011, Theory of two - temperature thermoelasticity without energy dissipation, Journal of Thermal Stresses 34:138-146.

[41] Youssef H.M.,2013, Variational principle of two - temperature thermoelasticity without energy dissipation, Journal of Thermoelasticity 1(1): 42-44.

[42] Zakaria M., 2012, Effects of hall current and rotation on magneto-micropolar generalized thermoelasticity due to ramp-type heating, International Journal of Electromagnetics and Applications 2(3): 24-32.

[43] Zakaria M., 2014, Effect of hall current on generalized magneto-thermoelasticity micropolar solid subjected to ramp-type heating, International Applied Mechanics 50(1):92-104.

Volume 8, Issue 4

Autumn 2016

Pages 840-858