On the solution of a problem of extended thermoelasticity theory (ETE) by using a complete finite element approach
Shivay Om Namha *, Mukhopadhyay Santwana
Department of Mathematical Sciences
Indian Institute of Technology (BHU)
Varanasi-221005, India
*E-mail: onsj79@gmail.com
Received:
Received: 08 December 2018; revised: 04 May 2019; accepted: 06 May 2019; published online: 12 June 2019
DOI: 10.12921/cmst.2018.0000062
Abstract:
This paper attempts to apply a complete finite element approach for the solution of problems on coupled dynamical thermoelasticity theory. Presently, we employ the extended thermoelasticity theory proposed by Lord and Shulman (1969) and consider a problem of linear thermoelasticity for the hollow disk with a thermal shock applied on its inner boundary. The thermoelastic equations have been solved using the complete finite element approach, where we have used discretization in the time domain as well as space domain and applied the Galerkin’s approach of the finite element for both time and space domain. We implement our scheme for a particular case and carry out computational work to obtain the numerical solution of the problem. Further, we compare the present results with the solutions obtained by FEM with Newmark time integration method and the solutions obtained by a trans-FEM method in which Laplace transform technique is used for the time domain. We show that, there is a perfect match in solutions of complete finite element approach with trans-finite element method and Newmark method. The efficiency of the method with respect to computation time is also compared with other two methods.
Key words:
coupled thermoelasticity, extended thermoelasticity theory, finite element method, Newmark method, trans-FEM
References:
[1] M.A. Biot, Thermoelasticity and irreversible thermodynamics, Journal of Applied Physics 27, 240–253 (1956).
[2] D.S. Chandrasekharaiah, Thermoelasticity with second sound: A review, Applied Mechanics Reviews 39(3), 355–376 (1986).
[3] D.S. Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature, Applied Mechanics Reviews 51(12), 705–729 (1998).
[4] C. Cattaneo, A form of heat conduction equation which eliminates the paradox of instantaneous propagation, Compte Rendus 247, 431–433 (1958).
[5] P. Vernotte, Les paradoxes de la theorie continue de l’equation de la chaleur, Compte Rendus 246, 3154–3155 (1958).
[6] P. Vernotte, Some possible complications in the phenomena of thermal conduction, Compte Rendus 252, 2190–2191 (1961).
[7] H.W. Lord, Y.A. Shulman, Generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids 15(5), 299–309 (1967).
[8] A.E. Green, K.A. Lindsay, Thermoelasticity, Journal of Elasticity 2, 1–7 (1972).
[9] A.E. Green, P.M. Naghdi, A re-examination of the base postulates of thermomechanics, Proceedings: Mathematical and Physical Sciences 432, 171–194 (1991).
[10] A.E. Green, P.M. Naghdi, On undamped heat waves in an elastic solid, Journal of Thermal Stresses 15, 253–264 (1992).
[11] A.E. Green, P.M. Naghdi, Thermoelasticity without energy dissipation, Journal of Elasticity 31, 189–208 (1993).
[12] R.B. Hetnarski, J. Ignaczak, Generalized thermoelasticity, Journal of Thermal Stresses 22, 451–476 (1999).
[13] R.B. Hetnarski, M.R. Eslami, Thermal stresses: Advanced theory and applications [In:] G.M.L. Gladwell, J.R. Barber, A. Klarbring (eds), Solid mechanics and its applications, 158, Dordrecht, The Netherlands, Springer (2010).
[14] G.S. Prakash, S.S. Reddy, S.K. Das, T. Sundararajan, K.N. Seetharamu, Numerical modeling of microscale effects in conduction for different thermal boundary conditions, Numerical Heat Transfer, Part A: Applications 38, 513–532 (2000).
[15] S.C. Mishra, T.B.P. Kumar, B. Mondal, Lattice Boltzmann method applied to the solution of energy equation of a radiation and non-Fourier heat conduction problem, Numerical Heat Transfer, Part A: Applications 54(8), 798–818 (2008).
[16] B. Xu, B.Q. Li, Finite element solution of non-Fourier thermal wave problems, Numerical Heat Transfer: Part B: Fundamentals 44, 45–60 (2003).
[17] A. Bagri, M.R. Eslami, Generalized coupled thermoelasticity of functionally graded annular disk considering the Lord-Shulman theory, Composite Structures 83, 168–179 (2008).
[18] S. Kothari, S. Mukhopadhyay, Study of a problem of functionally graded hollow disk under different thermoelasticity theories. An analysis of phase-lag effects, Computers & Mathematics with Applications 66, 1306-1321 (2013).
[17] M.A. Rincon, B.S. Santos, J. Limaco, Numerical method, existence and uniqueness for thermoelasticity system with moving boundary, Computational & Applied Mathematics 24(3), 439-60 (2005).
[18] I.A. Abbas, S.F. Alzahrani, A Green-Naghdi model in a 2D problem of a mode I crack in an isotropic thermoelastic plate, Physical Mesomechanics 21(2), 99-103 (2018).
[19] F.L. Stasa, Applied finite element formulation for thermal stress analysis, CBS Publishing, New York (1985).
[20] M. Balla, Formulation of coupled problems of thermoelasticity by finite elements, Periodica Polytechnica Mechanical Engineering 33(1-2), 59 (1989).
[23] H.H. Sherief, H.A. Salah, A half space problem in the theory of generalized thermoelastic diffusion, International Journal of Solids and Structures 42(15), 4484–4493 (2005).
[24] G. Honig, U. Hirdes, A method for the numerical inversion of Laplace transforms, Journal of Computational and Applied Mathematics 10, 113–132 (1984).
[25] R.E. Bellman, R.E. Kalaba, J.A. Lockett, Numerical inversion of the Laplace transform: Applications to biology, economics, engineering and physics, American Elsevier (1966).
[26] H. Stehfest, Numerical inversion of Laplace transform, Communications of the ACM 13(1), 47–49 (1970).
[21] N.M. Newmark, A method of computation for structural dynamics, J. Eng. Mech. Div. ASCE 85, 67–94 (1959).
This paper attempts to apply a complete finite element approach for the solution of problems on coupled dynamical thermoelasticity theory. Presently, we employ the extended thermoelasticity theory proposed by Lord and Shulman (1969) and consider a problem of linear thermoelasticity for the hollow disk with a thermal shock applied on its inner boundary. The thermoelastic equations have been solved using the complete finite element approach, where we have used discretization in the time domain as well as space domain and applied the Galerkin’s approach of the finite element for both time and space domain. We implement our scheme for a particular case and carry out computational work to obtain the numerical solution of the problem. Further, we compare the present results with the solutions obtained by FEM with Newmark time integration method and the solutions obtained by a trans-FEM method in which Laplace transform technique is used for the time domain. We show that, there is a perfect match in solutions of complete finite element approach with trans-finite element method and Newmark method. The efficiency of the method with respect to computation time is also compared with other two methods.
Key words:
coupled thermoelasticity, extended thermoelasticity theory, finite element method, Newmark method, trans-FEM
References:
[1] M.A. Biot, Thermoelasticity and irreversible thermodynamics, Journal of Applied Physics 27, 240–253 (1956).
[2] D.S. Chandrasekharaiah, Thermoelasticity with second sound: A review, Applied Mechanics Reviews 39(3), 355–376 (1986).
[3] D.S. Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature, Applied Mechanics Reviews 51(12), 705–729 (1998).
[4] C. Cattaneo, A form of heat conduction equation which eliminates the paradox of instantaneous propagation, Compte Rendus 247, 431–433 (1958).
[5] P. Vernotte, Les paradoxes de la theorie continue de l’equation de la chaleur, Compte Rendus 246, 3154–3155 (1958).
[6] P. Vernotte, Some possible complications in the phenomena of thermal conduction, Compte Rendus 252, 2190–2191 (1961).
[7] H.W. Lord, Y.A. Shulman, Generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids 15(5), 299–309 (1967).
[8] A.E. Green, K.A. Lindsay, Thermoelasticity, Journal of Elasticity 2, 1–7 (1972).
[9] A.E. Green, P.M. Naghdi, A re-examination of the base postulates of thermomechanics, Proceedings: Mathematical and Physical Sciences 432, 171–194 (1991).
[10] A.E. Green, P.M. Naghdi, On undamped heat waves in an elastic solid, Journal of Thermal Stresses 15, 253–264 (1992).
[11] A.E. Green, P.M. Naghdi, Thermoelasticity without energy dissipation, Journal of Elasticity 31, 189–208 (1993).
[12] R.B. Hetnarski, J. Ignaczak, Generalized thermoelasticity, Journal of Thermal Stresses 22, 451–476 (1999).
[13] R.B. Hetnarski, M.R. Eslami, Thermal stresses: Advanced theory and applications [In:] G.M.L. Gladwell, J.R. Barber, A. Klarbring (eds), Solid mechanics and its applications, 158, Dordrecht, The Netherlands, Springer (2010).
[14] G.S. Prakash, S.S. Reddy, S.K. Das, T. Sundararajan, K.N. Seetharamu, Numerical modeling of microscale effects in conduction for different thermal boundary conditions, Numerical Heat Transfer, Part A: Applications 38, 513–532 (2000).
[15] S.C. Mishra, T.B.P. Kumar, B. Mondal, Lattice Boltzmann method applied to the solution of energy equation of a radiation and non-Fourier heat conduction problem, Numerical Heat Transfer, Part A: Applications 54(8), 798–818 (2008).
[16] B. Xu, B.Q. Li, Finite element solution of non-Fourier thermal wave problems, Numerical Heat Transfer: Part B: Fundamentals 44, 45–60 (2003).
[17] A. Bagri, M.R. Eslami, Generalized coupled thermoelasticity of functionally graded annular disk considering the Lord-Shulman theory, Composite Structures 83, 168–179 (2008).
[18] S. Kothari, S. Mukhopadhyay, Study of a problem of functionally graded hollow disk under different thermoelasticity theories. An analysis of phase-lag effects, Computers & Mathematics with Applications 66, 1306-1321 (2013).
[17] M.A. Rincon, B.S. Santos, J. Limaco, Numerical method, existence and uniqueness for thermoelasticity system with moving boundary, Computational & Applied Mathematics 24(3), 439-60 (2005).
[18] I.A. Abbas, S.F. Alzahrani, A Green-Naghdi model in a 2D problem of a mode I crack in an isotropic thermoelastic plate, Physical Mesomechanics 21(2), 99-103 (2018).
[19] F.L. Stasa, Applied finite element formulation for thermal stress analysis, CBS Publishing, New York (1985).
[20] M. Balla, Formulation of coupled problems of thermoelasticity by finite elements, Periodica Polytechnica Mechanical Engineering 33(1-2), 59 (1989).
[23] H.H. Sherief, H.A. Salah, A half space problem in the theory of generalized thermoelastic diffusion, International Journal of Solids and Structures 42(15), 4484–4493 (2005).
[24] G. Honig, U. Hirdes, A method for the numerical inversion of Laplace transforms, Journal of Computational and Applied Mathematics 10, 113–132 (1984).
[25] R.E. Bellman, R.E. Kalaba, J.A. Lockett, Numerical inversion of the Laplace transform: Applications to biology, economics, engineering and physics, American Elsevier (1966).
[26] H. Stehfest, Numerical inversion of Laplace transform, Communications of the ACM 13(1), 47–49 (1970).
[21] N.M. Newmark, A method of computation for structural dynamics, J. Eng. Mech. Div. ASCE 85, 67–94 (1959).
