Mixed Lagrangian Formulation for Linear Thermoelastic Response of Structures
Publication: Journal of Engineering Mechanics
Volume 138, Issue 5
Abstract
Although a complete unified theory for elasticity, plasticity, and damage does not yet exist, an approach on the basis of thermomechanical principles may be able to serve as the foundation for such a theory. With this in mind, as a first step, a mixed formulation is developed for fully coupled, spatially discretized linear thermoelasticity under the Lagrangian formalism by using Hamilton’s principle. A variational integration scheme is then proposed for the temporal discretization of the resulting Euler-Lagrange equations. With this discrete numerical time-step solution, it becomes possible, for proper choices of state variables, to restate the problem in the form of an optimization. Ultimately, this allows the formulation of a principle of minimum generalized complementary potential energy for the discrete-time thermoelastic system.
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References
Apostolakis, G. (2010). “A Lagrangian approach for thermomechanics towards damage and deterioration of structures.” Ph.D dissertation, State Univ. of NY, Buffalo, NY.
Biot, M. A. (1954). “Theory of stress-strain relations in anisotropic viscoelasticity and relaxation phenomena.” J. Appl. Phys., 25(11), 1385–1391.JAPIAU
Biot, M. A. (1955). “Variational principles in irreversible thermodynamics with application to viscoelasticity.” Phys. Rev., 97(6), 1463–1469.PHRVAO
Biot, M. A. (1956). “Thermoelasticity and irreversible thermodynamics.” J. Appl. Phys., 27(3), 240–253.JAPIAU
Boley, B. A., and Tolins, I. S. (1962). “Transient coupled thermoelastic boundary value problems in the half-space.” J. Appl. Mech., 29(4), 637–646.JAMCAV
Cadzow, J. A. (1970). “Discrete calculus of variations.” Int. J. Control, 11(3), 393–407.IJCOAZ
Calvo, M. P., López-Marcos, M. A., and Sanz-Serna, J. M. (1998). “Variable step implementation of geometric integrators.” Appl. Numer. Math., 28(1), 1–16.ANMAEL
Chen, J. B. (2005). “Variational formulation for the multisymplectic hamiltonian systems.” Lett. Math. Phys., 71(3), 243–253.LMPHDY
Chester, M. (1963). “Second sound in solids.” Phys. Rev., 131(5), 2013–2015.PHRVAO
De León, M., and Martín de Diego, D. (2002). “Variational integrators and time-dependent lagrangian systems.” Rep. Math. Phys.RMHPBE, 49(2–3), 183–192.
De León, M., Martín de Diego, D., and Santamaría-Merino, A. (2004). “Geometric integrators and nonholonomic mechanics.” J. Math. Phys. (N.Y.)JMAPAQ, 45(3), 1042–1064.
de Groot, S. R. (1951). Thermodynamics of irreversible processes, North-Holland Amsterdam, Holland.
Fraeijs de Veubeke, B. M. (1951). “Diffusion des inconnues hyperstatiques dans les voilures à longeron couplés.” Bull. Serv. Technique de L’Aéronautique, No. 24, Imprimerie Marcel Hayez, Bruxelles.
Fraeijs de Veubeke, B. M. (1965). “Displacement and equilibrium models.” stress analysis, Zienkiewicz, O. C., and Hollister, G., eds., Wiley, London, 145–197.
Gonzalez, O., and Simo, J. C. (1996). “On the stability of symplectic and energy-momentum algorithms for non-linear hamiltonian systems with symmetry.” Comput. Methods Appl. Mech. Eng., 134(3–4), 197–222.CMMECC
Green, A. E., and Naghdi, P. M. (1993). “Thermoelasticity without energy-dissipation.” J. Elast., 31(3), 189–208.JELSAY
Hamilton, W. R. (1834). “On a general method in dynamics: by which the study of the motions of all free systems of attracting or repelling points is reduced to the search and differentiation of one central relation, or characteristic function.” Philos. Trans. R. Soc. LondonPTRSAV, 124, 247–308.
Hamilton, W. R. (1835). “Second essay on a general method in dynamics.” Philos. Trans. R. Soc. London, 125, 95–144.PTRSAV
Hu, H. C. (1955). “On some variational principles in the theory of elasticity and the theory of plasticity.” Scientia Sinica, 4(1), 33–54.SSINAV
Kane, C., Marsden, J. E., and Ortiz, M. (1999). “Symplectic-energy-momentum preserving variational integrators.” J. Math. Phys. (N.Y.)JMAPAQ, 40(7), 3353–3371.
Lord, H. W., and Shulman, Y. (1967). “A generalized dynamical theory of thermoelasticity.” J. Mech. Phys. Solids, 15(5), 299–309.JMPSA8
Maxwell, J. C. (1867). “On the dynamical theory of gases.” Philos. Trans. R. Soc. London, 157, 49–88.PTRSAV
Onsager, L. (1931a). “Reciprocal relations in irreversible processes I.” Phys. Rev., 37(4), 405–426.PHRVAO
Onsager, L. (1931b). “Reciprocal relations in irreversible processes II.” Phys. Rev., 38(12), 2265–2279.PHRVAO
Oravas, G. O. (1966). “Historical review of extremum principles in elastomechanics.” The theory of equilibrium of elastic systems and its applications, CAP Castigliano, Dover, NY, 20–32.
Petrehus, V. (2007). “Variational integrators for higher order lagrangians.” Bulletin mathematique de la Societe des sciences mathematiques de Roumanie, 50(2), 169–183.
Prigogine, I. (1967). Introduction to thermodynamics of irreversible processes, 3rd Ed., Interscience Publishers, New York.
Rayleigh, J. W. S. (1877). The theory of sound—Volume I, Chapter 4 § 81, 2nd Ed., reprint in 1945 Dover Publications, New York.
Reissner, E. (1950). “On a variational theorem in elasticity.” J. Math. Phys.ZAMPA8, 29, 90–95.
Sanz-Serna, J. M., and Vadillo, F. (1987). “Studies in numerical nonlinear instability III: Augmented hamiltonian systems.” SIAM J. Appl. Math., 47(1), 92–108.SMJMAP
Sivaselvan, M. V. et al. (2009). “Numerical collapse simulation of large-scale structural systems using an optimization-based algorithm.” Earthquake Eng. Struct. Dyn.IJEEBG, 38(5), 655–677.
Sivaselvan, M. V., and Reinhorn, A. M. (2004).“Nonlinear structural analysis towards collapse simulation: A dynamical systems approach.” Rep. MCEER-05-0004, MCEER/SUNY at Buffalo, Buffalo, NY.
Sivaselvan, M. V., and Reinhorn, A. M. (2006). “Lagrangian approach to structural collapse simulation.” J. Eng. Mech.JENMDT, 132(8), 795–805.
Washizu, K. (1955). “On the variational principles of elasticity and plasticity.” Technical Rep. 25-18 MIT, Aeroelastic and Structures Research Laboratory, Cambridge, MA.
Washizu, K. (1968). Variational methods in elasticity and plasticity, Pergamon, New York.
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Published In
Journal of Engineering Mechanics
Volume 138 • Issue 5 • May 2012
Pages: 508 - 518
Copyright
© 2012. American Society of Civil Engineers.
History
Received: Mar 14, 2011
Accepted: Oct 11, 2011
Published online: Oct 12, 2011
Published in print: May 1, 2012
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