Numerical Solution of Electro-magneto-thermo-mechanical Shock Problem
Department of Basic and Applied Science
Arab Academy for Science and Technology
P.O. Box 1029 Alexandria, Egypt
e-mail: aaelbary@aast.edu
Received:
Rec. 20 April 2006
DOI: 10.12921/cmst.2006.12.02.101-108
OAI: oai:lib.psnc.pl:617
Abstract:
A conducting half-space, permeated by an initial magnetic field governed by the generalized equations of thermoelasticity is considered. The bounding plane is acted upon by a combination of thermal and mechanical shock. The formulation is applied to both generalizations, Lord-Shulman theory and the Green-Lindsay theory, as well as to the coupled theory. Laplace transform techniques together with the method of potentials are used. The inversion of the Laplace is carried out using a numerical approach. Numerical results for the
temperature, the stress and the induced magnetic and electric field distributions are obtained and illustrated graphically for a particular case. Comparisons are made with the results obtained in the case of the absence of the magnetic field.
Key words:
generalized thermoelasticity, Laplace transforms, magneto-thermoelasicity
References:
[1] J. H. Duhamel, Second memoir, sur les phenomenes thermomechanique, J. de L’ Ecole Polytechnique 15 (1837).
[2] M. Biot, Thermoelasticity and irreversible thermodynamics, J. Appl. Phys. 27 (1956).
[3] C. Cattaneo, Sullacondizione del calore, Atti. Sem. Mat. Fis. Univ. Modena 3 (1948).
[4] C. Truesdell and R. G. Muncaster, Fundamental of Maxwell’s kinetic theory of a simple monatomic gas, Acad. Press, New York (1980).
[5] D. E. Glass and B. Vick, Hyperbolic heat conduction with surface radiation, Int. J. Heat Mass Transfer 28, 1823 (1985).
[6] D. D. Joseph and L. Preziosi, Heat waves, Rev. Modern Phys. 61, 41 (1989).
[7] D. D. Joseph and L. Preziosi, Addendum to the paper: heat waves, Rev. Modern Phys. 62, 375 (1989).
[8] W. Dreyer and H. Struchtrup, Heat pulse experiments revisited, Count. Mech. Thermodyn. 5, 3 (1993).
[9] P. Puri and P. K. Kythe, Non-classical thermal effects in Stoke’s second problem, Acta Mech. 112, 1 (1995).
[10] D. S. Chandrasekharaiah, Hyperbolic thermoelasticity. A review of recent literature, Appl. Mech. Rev. 51, 705 (1998).
[11] H. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, Mech. Phys. Solid. 15, 299 (1967).
[12] I. Müller, The coldness, a universal function in thermoelastic solids, Arch. Rat. Mech. Anal. 41, 319 (1971).
[13] A. Green and N. Laws, On the entropy production inequality, Arch. Rat. Anal. 54, 7 (1972).
[14] A. Green and K. Lindsay, Thermoelasticity, J. Elast. 2, 7 (1972).
[15] E. Şuhubi, Thermoelastic solids, in: A. C. Eringen (ED), Cont. Phys. II, Academic Press, New York (1975) Ch. 2.
[16] M. Ezzat, Fundamental solution in thermoelasticity with two relaxation times for cylindrical regions, Int. J. Engng. Sci. 33, 2011 (1995).
[17] A. Nayfeh and S. Nemat-Nasser, Electromagneto-Thermoelastic Plane Waves in Solids with Thermal Relaxation, J. Appl. Mech. Series E 39, 108 (1972).
[18] H. Sherief and M. Ezzat, A thermal-shock problem in magneto-thermoelasticity with thermal relaxation, Int. J. Solids and Structures 33, 4449 (1996).
[19] M. Ezzat, Generation of generalized magneto-thermoelasticity waves by thermal shock in a perfectly conducting halfspace, J. Thermal Stresses 20, 617 (1997).
[20] H. Sherief, A thermal-mechanical shock problem for thermoelasticity with two relaxation time, Int. J. Engng. Sci. 32, 313 (1994).
[21] M. Ezzat and A. El-Karamany, Magnetothermoelasticity with two relaxation times in conducting medium with variable electrical and thermal conductivity, J. App. Math and
Computaions 142, 449 (2003).
[22] M. Ezzat and A. El-Karamani, The uniqueness and reciprocity theorems for generalized thermoviscoelasticity for anisotropic media, J. Thermal Stresses 25, 507 (2002).
[23] G. Honig and Hirdes, A Method for the Numerical Inversion of the Laplace Transform, J. Comp. Appl. Math. 10, 113(1984).
[24] B. A. Boley and J. H. Weiner, Theory of thermal stresses, Wiley, New York, 1960.
A conducting half-space, permeated by an initial magnetic field governed by the generalized equations of thermoelasticity is considered. The bounding plane is acted upon by a combination of thermal and mechanical shock. The formulation is applied to both generalizations, Lord-Shulman theory and the Green-Lindsay theory, as well as to the coupled theory. Laplace transform techniques together with the method of potentials are used. The inversion of the Laplace is carried out using a numerical approach. Numerical results for the
temperature, the stress and the induced magnetic and electric field distributions are obtained and illustrated graphically for a particular case. Comparisons are made with the results obtained in the case of the absence of the magnetic field.
Key words:
generalized thermoelasticity, Laplace transforms, magneto-thermoelasicity
References:
[1] J. H. Duhamel, Second memoir, sur les phenomenes thermomechanique, J. de L’ Ecole Polytechnique 15 (1837).
[2] M. Biot, Thermoelasticity and irreversible thermodynamics, J. Appl. Phys. 27 (1956).
[3] C. Cattaneo, Sullacondizione del calore, Atti. Sem. Mat. Fis. Univ. Modena 3 (1948).
[4] C. Truesdell and R. G. Muncaster, Fundamental of Maxwell’s kinetic theory of a simple monatomic gas, Acad. Press, New York (1980).
[5] D. E. Glass and B. Vick, Hyperbolic heat conduction with surface radiation, Int. J. Heat Mass Transfer 28, 1823 (1985).
[6] D. D. Joseph and L. Preziosi, Heat waves, Rev. Modern Phys. 61, 41 (1989).
[7] D. D. Joseph and L. Preziosi, Addendum to the paper: heat waves, Rev. Modern Phys. 62, 375 (1989).
[8] W. Dreyer and H. Struchtrup, Heat pulse experiments revisited, Count. Mech. Thermodyn. 5, 3 (1993).
[9] P. Puri and P. K. Kythe, Non-classical thermal effects in Stoke’s second problem, Acta Mech. 112, 1 (1995).
[10] D. S. Chandrasekharaiah, Hyperbolic thermoelasticity. A review of recent literature, Appl. Mech. Rev. 51, 705 (1998).
[11] H. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, Mech. Phys. Solid. 15, 299 (1967).
[12] I. Müller, The coldness, a universal function in thermoelastic solids, Arch. Rat. Mech. Anal. 41, 319 (1971).
[13] A. Green and N. Laws, On the entropy production inequality, Arch. Rat. Anal. 54, 7 (1972).
[14] A. Green and K. Lindsay, Thermoelasticity, J. Elast. 2, 7 (1972).
[15] E. Şuhubi, Thermoelastic solids, in: A. C. Eringen (ED), Cont. Phys. II, Academic Press, New York (1975) Ch. 2.
[16] M. Ezzat, Fundamental solution in thermoelasticity with two relaxation times for cylindrical regions, Int. J. Engng. Sci. 33, 2011 (1995).
[17] A. Nayfeh and S. Nemat-Nasser, Electromagneto-Thermoelastic Plane Waves in Solids with Thermal Relaxation, J. Appl. Mech. Series E 39, 108 (1972).
[18] H. Sherief and M. Ezzat, A thermal-shock problem in magneto-thermoelasticity with thermal relaxation, Int. J. Solids and Structures 33, 4449 (1996).
[19] M. Ezzat, Generation of generalized magneto-thermoelasticity waves by thermal shock in a perfectly conducting halfspace, J. Thermal Stresses 20, 617 (1997).
[20] H. Sherief, A thermal-mechanical shock problem for thermoelasticity with two relaxation time, Int. J. Engng. Sci. 32, 313 (1994).
[21] M. Ezzat and A. El-Karamany, Magnetothermoelasticity with two relaxation times in conducting medium with variable electrical and thermal conductivity, J. App. Math and
Computaions 142, 449 (2003).
[22] M. Ezzat and A. El-Karamani, The uniqueness and reciprocity theorems for generalized thermoviscoelasticity for anisotropic media, J. Thermal Stresses 25, 507 (2002).
[23] G. Honig and Hirdes, A Method for the Numerical Inversion of the Laplace Transform, J. Comp. Appl. Math. 10, 113(1984).
[24] B. A. Boley and J. H. Weiner, Theory of thermal stresses, Wiley, New York, 1960.
