Thermal Shock Problem of Generalized Thermoelasticity for an Infinitely Long Annular Cylinder with Variable Thermal Conductivity
Youssef Hamdy M. 1*, Abbas Ibrahim A. 2
1Mathematical Department, Faculty of Education
Alexandria University, Alexandria, Egypt
e-mail: yousefanne@yahoo.com
2Department of Mathematics, Faculty of Science
Sohag University 82524, Egypt
e-mail: ibrabbas7@yahoo.com
Received:
Rec. May 17, 2007
DOI: 10.12921/cmst.2007.13.02.95-100
OAI: oai:lib.psnc.pl:639
Abstract:
In this paper, a general finite element model is proposed to analyze transient phenomena in thermoelastic model in the context of the theory of generalized thermoelasticity with one relaxation time with variable thermal conductivity. An application of an infinitely long annular cylinder was studied, where the inner surface is traction free and subjected to thermal shock, while the outer surface is traction free and thermally isolated. The results for the temperature increment, the stress components and the displacement component are illustrated graphically.
Key words:
annular cylinder, finite element, generalized thermoelasticity, thermoelasticity
References:
[1] M. Biot, Thermoelasticity and irreversible thermo-dynamics, J. Appl. Phys. 127, 240-253 (1956).
[2] H. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, Mech. Phys. Solid 15, 299-309 (1967).
[3] A. E. Green and K. A. Lindsay, Thermoelasticity, J. Elast. 2, 1-7 (1972).
[4] S. Erbay and E. Ôuhubi, Longitudinal wave propagation in a generalized thermo-elastic cylinder, J. Thermal. Stresses, 9, 279-295 (1986).
[5] J. Ignaczak, A strong discontinuity wave in thermoelastic with relaxation times, J. Thermal Stresses 8, 25-40 (1985).
[6] J. Ignaczak, Decomposition theorem for thermoelasticity with finite wave speeds, J. Thermal Stresses 1, 41-52 (1978).
[7] M. Ezzat, Fundamental solution in thermoelasticity with two relaxation times for cylindrical regions, Int. J. Eng. Sci. 33, 2011-2020 (1995).
[8] N. M. El-Maghraby and H. M. Youssef, State space approach to generalized thermoelastic problem with thermomechanical shock, J. Applied Mathematics and Computation, 156(2), 577-586 (2004).
[9] N. M El-Maghraby and H. M. Youssef, A two-dimensional thermoelasticity problem for thermo. mechanical shock with two relaxation times, J. Applied Mathematics and Computation (AMC) 120, 172-184 (2005).
[10] H. M. Youssef, N. M. El-Maghraby and A. A. El-Bary, State space approach to thermoelastic problem with vibrational stresses, Computational Mathematics and Modeling 17, 243-253 (2006).
[11] H. Youssef, The Dependence of The Modulus of Elasticity And The Thermal Conductivity on The Reference Temperature in Generalized Thermoelasticity For An Infinite Material With A Spherical Cavity, Journal of Applied Mathematics and Mechanics, ISSN0253-4827, 26(4) (2005).
[12] H. Youssef, Generalized Thermoelasticity of an Infinite Body with a Cylindrical Cavity and Variable Material Properties, J. Thermal Stresses 28(5), 521-532 (2005).
[13] H. Youssef, Problem of Generalized Thermoelastic Infinite Medium with Cylindrical Cavity Subjected to a Ramp-Type Heating and Loading, J. Archive of App. Mech. 75, 553-565 (2006).
[14] N. Noda, Thermal Stresses in Materials with Temperature-Dependent Properties, Thermal Stresses I, Richard B. Hetnarski (Editor), North-Holland, Amsterdam, 1986.
[15] D. P. H. Hasselman and R. A. Heller, Thermal Stresses in Sever Environments, Plenum Press, New York, 1980.
[16] R. B. Hetnaraski, Thermal Stresses, North-Holland, Amsterdam, 1, 391-396 (1986).
[17] O. C. Zienkiewicz and R. L. Taylor, The finite element method. Fluid dynamics, 5th ed. Butterworth-Heinemann, 2000.
[18] J. N. Reddy, An Introduction to the finite element method, second ed., McGraw-Hill, New York, 1993.
[19] R. D. Cook, D. S. Malkus and M. E. Plesha, Concepts and applications of finite element analysis, third ed., John Wiley, New York, 1989.
In this paper, a general finite element model is proposed to analyze transient phenomena in thermoelastic model in the context of the theory of generalized thermoelasticity with one relaxation time with variable thermal conductivity. An application of an infinitely long annular cylinder was studied, where the inner surface is traction free and subjected to thermal shock, while the outer surface is traction free and thermally isolated. The results for the temperature increment, the stress components and the displacement component are illustrated graphically.
Key words:
annular cylinder, finite element, generalized thermoelasticity, thermoelasticity
References:
[1] M. Biot, Thermoelasticity and irreversible thermo-dynamics, J. Appl. Phys. 127, 240-253 (1956).
[2] H. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, Mech. Phys. Solid 15, 299-309 (1967).
[3] A. E. Green and K. A. Lindsay, Thermoelasticity, J. Elast. 2, 1-7 (1972).
[4] S. Erbay and E. Ôuhubi, Longitudinal wave propagation in a generalized thermo-elastic cylinder, J. Thermal. Stresses, 9, 279-295 (1986).
[5] J. Ignaczak, A strong discontinuity wave in thermoelastic with relaxation times, J. Thermal Stresses 8, 25-40 (1985).
[6] J. Ignaczak, Decomposition theorem for thermoelasticity with finite wave speeds, J. Thermal Stresses 1, 41-52 (1978).
[7] M. Ezzat, Fundamental solution in thermoelasticity with two relaxation times for cylindrical regions, Int. J. Eng. Sci. 33, 2011-2020 (1995).
[8] N. M. El-Maghraby and H. M. Youssef, State space approach to generalized thermoelastic problem with thermomechanical shock, J. Applied Mathematics and Computation, 156(2), 577-586 (2004).
[9] N. M El-Maghraby and H. M. Youssef, A two-dimensional thermoelasticity problem for thermo. mechanical shock with two relaxation times, J. Applied Mathematics and Computation (AMC) 120, 172-184 (2005).
[10] H. M. Youssef, N. M. El-Maghraby and A. A. El-Bary, State space approach to thermoelastic problem with vibrational stresses, Computational Mathematics and Modeling 17, 243-253 (2006).
[11] H. Youssef, The Dependence of The Modulus of Elasticity And The Thermal Conductivity on The Reference Temperature in Generalized Thermoelasticity For An Infinite Material With A Spherical Cavity, Journal of Applied Mathematics and Mechanics, ISSN0253-4827, 26(4) (2005).
[12] H. Youssef, Generalized Thermoelasticity of an Infinite Body with a Cylindrical Cavity and Variable Material Properties, J. Thermal Stresses 28(5), 521-532 (2005).
[13] H. Youssef, Problem of Generalized Thermoelastic Infinite Medium with Cylindrical Cavity Subjected to a Ramp-Type Heating and Loading, J. Archive of App. Mech. 75, 553-565 (2006).
[14] N. Noda, Thermal Stresses in Materials with Temperature-Dependent Properties, Thermal Stresses I, Richard B. Hetnarski (Editor), North-Holland, Amsterdam, 1986.
[15] D. P. H. Hasselman and R. A. Heller, Thermal Stresses in Sever Environments, Plenum Press, New York, 1980.
[16] R. B. Hetnaraski, Thermal Stresses, North-Holland, Amsterdam, 1, 391-396 (1986).
[17] O. C. Zienkiewicz and R. L. Taylor, The finite element method. Fluid dynamics, 5th ed. Butterworth-Heinemann, 2000.
[18] J. N. Reddy, An Introduction to the finite element method, second ed., McGraw-Hill, New York, 1993.
[19] R. D. Cook, D. S. Malkus and M. E. Plesha, Concepts and applications of finite element analysis, third ed., John Wiley, New York, 1989.
