Solution of a Problem of Generalized Thermoelasticity of an Annular Cylinder with Variable Material Properties by Finite Difference Method
Mukhopadhyay Santwana, Kumar Roushan
Department of Applied Mathematics
Institute of Technology
Banaras Hindu University, Varanasi-221005
e-mail: mukho_santwana@rediffmail.com
Received:
Received: 22 February 2009; accepted: 12 May; published online: 18 June 2009
DOI: 10.12921/cmst.2009.15.02.169-176
OAI: oai:lib.psnc.pl:672
Abstract:
The present work deals with a new problem of generalized thermoelasticity with one relaxation time for an infinitely long and isotropic annular cylinder of temperature dependent physical properties. The inner and outer curved surfaces of the cylinder are subjected to both the mechanical and thermal boundary conditions. A finite difference model is developed to derive the solution of the problem in which the governing equations are coupled non linear partial differential equations. The transient solution at any time can be evaluated directly from the model. In order to demonstrate the efficiency of the present model we consider a suitable material and obtain the numerical solution of displacement, temperature, and stresses inside the annulus for both the temperature-dependent and temperature-independent material properties of the medium. The results are analyzed with the help of different graphical plots.
Key words:
annular cylinder, finite difference method, generalized thermoelasticity, thermoelasticity with one relaxation parameter
References:
[1] M. Biot, Thermoelasticity and irreversible thermodynamics, J. Appl. Phys. 27, 240-253 (1956).
[2] H. Lord and Y. Shulman, Generalized dynamical theory of thermoelasticity, J. Mech. Phy. Solids 15, 299-309 (1967).
[3] R.S. Dhaliwal and H.H. Sherief, Generalized thermoelasticity for anisotropic media, Quart. Appl. Math. 33, 1-8 (1980).
[4] A.E. Green and K.A. Lindsay, Thermoelasticity J. Elasticity 2, 1-7 (1972).
[5] A.E. Green and P.M. Naghdi, A re-examination of the basic postulates of thermoemechanics, Proc. Roy. Soc. London A432, 171-194 (1991).
[6] A.E. Green and P.M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses 15, 253-264 (1992).
[7] A.E. Green and P.M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity 31, 189-209 (1993).
[8] W. Kaminski, Hyperbolic heat conduction equation for materials with a homogeneous inner structure, ASME J. Heat Transfer 112, 555-560 (1990).
[9] K. Mitra, S. Kumar, A. Vedavarz, M.A. Moallemi, Experimental evidence of hyperbolic heat conduction in processed meat, ASME J. Heat Transfer 117, 568-573 (1995).
[10] D.Y. Tzou, Experimental support for the lagging response in heat propagation, AIAA J. Thermophysics 9, 686-693 (1995).
[11] S.K. Roychoudhuri and S. Banerjee, Magneto thermoelastic waves induced by a thermal shock in a finitely conducting elastic half-space, Int. J. Math. Math. Sci. 19, 131-143 (1996).
[12] M.A. Ezzat, Fundamental solution in thermoelasticity with two relaxation times for cylindrical regions, Int. J. Eng. Sci. 33, 2011-2020 (1995).
[13] S. Banerjee and S.K. Roychoudhuri, Spherically symmetric thermo-viscoelastic waves in a visco-elastic medium with a spherical cavity, Computers Math. Applic. 30, 91-98 (1995).
[14] D.S. Chandrasekharaiah and H.R. Keshavan, Axisymmetric thermoelastic interactions in an unbounded body with cylindrical cavity, Acta Mechanica 92, 61-76 (1992).
[15] S. Mukhopadhyay and R. Kumar, A problem on thermoelastic interactions in an infinite medium with a cylindrical hole in generalized thermoelasticity III, J. Thermal Stresses 31, 455-475 (2008).
[16] P. Puri, P.M. Jordan, On the propagation of plane waves in type-III thermoelastic Media, Proc. Roy. Soc. London A 460, 3203-3221 (2004).
[17] N. Noda, Thermal stress in material with temperaturedependent properties, In: Thermal stresses, R. B. Hetnarski (ed), Elsevier Science, North Holland, Amsterdam, 391-483 (1986).
[18] N. Noda, Thermal stresses in functionally graded materials, J. Thermal Stresses 22, 27-40 (1999).
[19] H. Argeso and A.N. Eraslan, On the use of temperaturedependent physical properties in thermomechanical calculations for solid and hollow cylinder, Int. J. Thermal Sciences 47, 136-146 (2008).
[20] A.N. Eraslan and Y. Kartal, A nonlinear shooting method applied to solid mechanics: Part 1, Numerical solution of a plane stress model, Int. J. Nonlinear Analysis and Phenomena 1, 27-40 (2004).
[21] H.M. Youssef, Generalized thermoelasticity of an infinite medium with cylindrical cavity and variable material properties, J. Thermal Stresses 5, 521-532 (2005).
[22] M.A. Ezzat, M. Zakaria and A. Abdel-Bary, Generalized thermoelasticity with temperature dependent modulus of elasticity under three theories, J. Applied Math. & Computing 14, 193-212 (2004).
[23] H.M. Youssef and I.A. Abbas, Thermal shock problem of generalized thermoelasticity for an annular cylinder with variable thermal conductivity, Computational Methods in Science and Technology 13(2), 95-100 (2007).
[24] M.A. Ezzat, M.I. Othman and A.S. EI-Karamany, The dependence of the modulus of elasticity on the reference temperature in generalized thermoelasticity, J. Thermal Streses 24, 1159-1176 (2001).
[25] H.H. Sherief and M. Anwar, A problem in generalized thermoelasticity for aninfinitely long annular cylinder composed of two different material, Acta Mechanica 80, 137-149 (1989).
[26] H.H. Sherief and M. Anwar, A problem in generalized thermoelasticity for an infinitely long annular cylinder, Int. J. Eng. Sci. 26, 301-306 (1988).
[27] D.A. Anderson, J.C. Tannehill and R.H. Pletcher, Computational fluid mechanics and heat transfer, Hemisphere Publishing Corporation, McGraw Hill Book Company (1997).
The present work deals with a new problem of generalized thermoelasticity with one relaxation time for an infinitely long and isotropic annular cylinder of temperature dependent physical properties. The inner and outer curved surfaces of the cylinder are subjected to both the mechanical and thermal boundary conditions. A finite difference model is developed to derive the solution of the problem in which the governing equations are coupled non linear partial differential equations. The transient solution at any time can be evaluated directly from the model. In order to demonstrate the efficiency of the present model we consider a suitable material and obtain the numerical solution of displacement, temperature, and stresses inside the annulus for both the temperature-dependent and temperature-independent material properties of the medium. The results are analyzed with the help of different graphical plots.
Key words:
annular cylinder, finite difference method, generalized thermoelasticity, thermoelasticity with one relaxation parameter
References:
[1] M. Biot, Thermoelasticity and irreversible thermodynamics, J. Appl. Phys. 27, 240-253 (1956).
[2] H. Lord and Y. Shulman, Generalized dynamical theory of thermoelasticity, J. Mech. Phy. Solids 15, 299-309 (1967).
[3] R.S. Dhaliwal and H.H. Sherief, Generalized thermoelasticity for anisotropic media, Quart. Appl. Math. 33, 1-8 (1980).
[4] A.E. Green and K.A. Lindsay, Thermoelasticity J. Elasticity 2, 1-7 (1972).
[5] A.E. Green and P.M. Naghdi, A re-examination of the basic postulates of thermoemechanics, Proc. Roy. Soc. London A432, 171-194 (1991).
[6] A.E. Green and P.M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses 15, 253-264 (1992).
[7] A.E. Green and P.M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity 31, 189-209 (1993).
[8] W. Kaminski, Hyperbolic heat conduction equation for materials with a homogeneous inner structure, ASME J. Heat Transfer 112, 555-560 (1990).
[9] K. Mitra, S. Kumar, A. Vedavarz, M.A. Moallemi, Experimental evidence of hyperbolic heat conduction in processed meat, ASME J. Heat Transfer 117, 568-573 (1995).
[10] D.Y. Tzou, Experimental support for the lagging response in heat propagation, AIAA J. Thermophysics 9, 686-693 (1995).
[11] S.K. Roychoudhuri and S. Banerjee, Magneto thermoelastic waves induced by a thermal shock in a finitely conducting elastic half-space, Int. J. Math. Math. Sci. 19, 131-143 (1996).
[12] M.A. Ezzat, Fundamental solution in thermoelasticity with two relaxation times for cylindrical regions, Int. J. Eng. Sci. 33, 2011-2020 (1995).
[13] S. Banerjee and S.K. Roychoudhuri, Spherically symmetric thermo-viscoelastic waves in a visco-elastic medium with a spherical cavity, Computers Math. Applic. 30, 91-98 (1995).
[14] D.S. Chandrasekharaiah and H.R. Keshavan, Axisymmetric thermoelastic interactions in an unbounded body with cylindrical cavity, Acta Mechanica 92, 61-76 (1992).
[15] S. Mukhopadhyay and R. Kumar, A problem on thermoelastic interactions in an infinite medium with a cylindrical hole in generalized thermoelasticity III, J. Thermal Stresses 31, 455-475 (2008).
[16] P. Puri, P.M. Jordan, On the propagation of plane waves in type-III thermoelastic Media, Proc. Roy. Soc. London A 460, 3203-3221 (2004).
[17] N. Noda, Thermal stress in material with temperaturedependent properties, In: Thermal stresses, R. B. Hetnarski (ed), Elsevier Science, North Holland, Amsterdam, 391-483 (1986).
[18] N. Noda, Thermal stresses in functionally graded materials, J. Thermal Stresses 22, 27-40 (1999).
[19] H. Argeso and A.N. Eraslan, On the use of temperaturedependent physical properties in thermomechanical calculations for solid and hollow cylinder, Int. J. Thermal Sciences 47, 136-146 (2008).
[20] A.N. Eraslan and Y. Kartal, A nonlinear shooting method applied to solid mechanics: Part 1, Numerical solution of a plane stress model, Int. J. Nonlinear Analysis and Phenomena 1, 27-40 (2004).
[21] H.M. Youssef, Generalized thermoelasticity of an infinite medium with cylindrical cavity and variable material properties, J. Thermal Stresses 5, 521-532 (2005).
[22] M.A. Ezzat, M. Zakaria and A. Abdel-Bary, Generalized thermoelasticity with temperature dependent modulus of elasticity under three theories, J. Applied Math. & Computing 14, 193-212 (2004).
[23] H.M. Youssef and I.A. Abbas, Thermal shock problem of generalized thermoelasticity for an annular cylinder with variable thermal conductivity, Computational Methods in Science and Technology 13(2), 95-100 (2007).
[24] M.A. Ezzat, M.I. Othman and A.S. EI-Karamany, The dependence of the modulus of elasticity on the reference temperature in generalized thermoelasticity, J. Thermal Streses 24, 1159-1176 (2001).
[25] H.H. Sherief and M. Anwar, A problem in generalized thermoelasticity for aninfinitely long annular cylinder composed of two different material, Acta Mechanica 80, 137-149 (1989).
[26] H.H. Sherief and M. Anwar, A problem in generalized thermoelasticity for an infinitely long annular cylinder, Int. J. Eng. Sci. 26, 301-306 (1988).
[27] D.A. Anderson, J.C. Tannehill and R.H. Pletcher, Computational fluid mechanics and heat transfer, Hemisphere Publishing Corporation, McGraw Hill Book Company (1997).
