Abstract
A variational approach for fully coupled dynamic irreversible thermoelasticity is developed for continua, which considers both the conservative and dissipative character in terms of mixed variables. By introducing a consistent variational scheme for the spatial and temporal discretization of the governing equations, a mixed continuum element is established under the Hamiltonian-Lagrangian formalism. The proposed method leads to the development of minimum principles in discrete form with the proper selection of state variables and temporal action sum operators. Consequently, this novel mixed variational formulation can provide the basis for a class of optimization-based methods for irreversible thermomechanics. Several applications are considered to demonstrate the robustness of the proposed variational approach, including transient dynamic response of thermoelastic media due to surface heating caused by ramp- and step-type heat fluxes, and a sequence of laser pulses.
Similar content being viewed by others
References
Anthony K.H.: Hamilton’s action principle and thermodynamics of irreversible processes—a unifying procedure for reversible and irreversible processes. J. Non-Newtonian Fluid Mech. 96, 291–339 (2001)
Apostolakis G., Dargush G.F.: Mixed Lagrangian formulation for linear thermoelastic response of structures. J. Eng. Mech. 138, 508–518 (2012)
Apostolakis, G., Dargush, G.F.: Mixed variational principles for dynamic response of thermoelastic and poroelastic continua. Int. J. Solids Struct. 50, 642–650 (2013)
Askar Altay G., Cengiz Dokmeci M.: Some variational principles for linear coupled thermoelasticity. Int. J. Solids Struct. 33, 3937–3948 (1996)
Ben-Amoz M.: On a variational theorem in coupled thermoelasticity. J. Appl. Mech. 32, 943–945 (1965)
Biot M.A.: Theory of stress-strain relations in anisotropic viscoelasticity and relaxation phenomena. J. Appl. Phys. 25, 1385–1391 (1954)
Biot M.A.: Variational principles in irreversible thermodynamics with application to viscoelasticity. Phys. Rev. 97, 1463–1469 (1955)
Biot M.A.: Thermoelasticity and irreversible thermodynamics. J. Appl. Phys. 27, 240–253 (1956)
Boley B.A., Weiner J.H.: Theory of Thermal Stresses. Wiley, New York (1960)
Cadzow J.A.: Discrete calculus of variations. Int. J. Control 11, 393–407 (1970)
Cannarozzi A.A., Ubertini F.: A mixed variational method for linear coupled thermoelastic analysis. Int. J. Solids Struct. 38, 717–739 (2001)
Carter J.P., Booker J.R.: Finite element analysis of coupled thermoelasticity. Comput. Struct. 31, 73–80 (1989)
Chen J., Dargush G.F.: Boundary element method for dynamic poroelastic and thermoelastic analyses. Int. J. Solids Struct. 32, 2257–2278 (1995)
Chen T.C., Weng C.I.: Generalized coupled transient thermoelastic plane problems by Laplace transform/finite element method. J. Appl. Mech. ASME 55, 377–382 (1988)
Chester M.: Second sound in solids. Phys. Rev. 131, 2013–2015 (1963)
Danilovskaya V.I.: Thermal stresses in an elastic half-space due to a sudden heating of its boundary. Prikl. Mek. 14, 316–318 (1950)
Farhat C., Park K.C., Dubois-Pelerin Y.: An unconditionally stable staggered algorithm for transient finite element analysis of coupled thermoelastic problems. Comput. Methods Appl. Mech. Eng. 85, 349–365 (1991)
Fourier J.: Analytical Theory of Heat. English translation in 1878 by Alexander Freeman. Cambridge University Press, London (1822)
Gibbs J.: Fourier’s series. Nature 59, 200 (1898)
Gibbs J.: Fourier’s series. Nature 59, 606 (1899)
de Groot S.R.: Thermodynamics of Irreversible Processes. North-Holland Publishing Company, Amsterdam (1951)
Hamilton W.R.: On a general method in dynamics. Philos. Trans. R. Soc. Lond. 124, 247–308 (1834)
Hamilton W.R.: Second essay on a general method in dynamics. Philos. Trans. R. Soc. Lond. 125, 95–144 (1835)
Hector L.G. Jr, Kim W.S., Ozisik M.N.: Hyperbolic heat conduction due to a mode locked laser pulse train. Int. J. Eng. Sci. 30, 1731–1744 (1992)
Herrmann G.: On variational principles in thermoelasticity and heat conduction. Q. Appl. Maths 21(2), 151–155 (1963)
Keramidas G.A., Ting E.C.: A finite element formulation for thermal stress analysis. Part I: Variational formulation. Nucl. Eng. Des. 39, 267–275 (1976)
Keramidas G.A., Ting E.C.: A finite element formulation for thermal stress analysis. Part II: Finite element formulation. Nucl. Eng. Des. 39, 277–287 (1976)
Lee T.W., Sim W.J.: Efficient time-domain finite element analysis for dynamic coupled thermoelasticity. Comput. Struct. 45, 785–793 (1992)
Manolis G.D., Beskos D.E.: Integral formulation and fundamental solutions of dynamic poroelasticity and thermoelasticity. Acta Mech. 76, 89–104 (1989)
Matlab Computer Software: The Language of technical computing. The MathWorks Inc., Natick, MA (2006)
Maugin G.A., Kalpakides V.K.: A Hamiltonian formulation for elasticity and thermoelasticity. J. Phys. A Math. Gen. 35, 10775–10788 (2002)
Maxwell J.C.: On the dynamical theory of gases. Philos. Trans. R. Soc. Lond. 157, 49–88 (1867)
Nayfeh A., Nemat-Nasser S.: Thermoelastic waves in solids with thermal relaxation. Acta Mech. 12, 53–69 (1971)
Nickell R.E., Sackman J.L.: Approximate solutions in linear, coupled thermoelasticity. J. Appl. Mech. 35, 255–266 (1968)
Nickell R.E., Sackman J.L.: Variational principles for linear coupled thermoelasticity. Q. Appl. Math. 26, 11–26 (1968)
Nowacki W.: Thermo-Elasticity. 2nd edn. Pergamon Press, New York (1986)
Oden J.T.: Finite element analysis of nonlinear problems in dynamical theory of coupled thermoelasticity. Nucl. Eng. Des. 10, 465–475 (1969)
Onsager L.: Reciprocal relations in irreversible processes I. Phys. Rev. 37, 405–426 (1931)
Onsager L.: Reciprocal relations in irreversible processes II. Phys. Rev. 38, 2265–2279 (1931)
Prathap G., Naganarayana B.P.: Consistent thermal stress evaluation in finite elements. Comput. Struct. 54, 415–426 (1995)
Prevost J.H., Tao D.: Dynamic coupled thermoelasticity problems with relaxation times. J. Appl. Mech. ASME 50, 817–822 (1983)
Prigogine I.: Introduction to Thermodynamics of Irreversible Processes. 3rd edn. Interscience Publishers, New York (1967)
Rafalski P.: A variational principle for the coupled thermoelastic problem. Int. J. Eng. Sci. 6, 465–471 (1968)
Sanderson T., Ume C., Jarzynski J.: Hyperbolic heat conduction effects caused by temporally modulated laser pulses. Ultrasonics 33, 423–427 (1995)
Serra E., Bonaldi M.: A finite element formulation for thermoelastic damping analysis. Int. J. Numer. Methods Eng. 78, 671–691 (2009)
Sivaselvan M.V., Reinhorn A.M.: Lagrangian approach to structural collapse simulation. J. Eng. Mech. ASCE 132, 795–805 (2006)
Sivaselvan M.V., Lavan O., Dargush G.F., Kurino H., Hyodo Y., Fukuda R., Sato K., Apostolakis G., Reinhorn A.M.: Numerical collapse simulation of large-scale structural systems using an optimization-based algorithm. Earthq. Eng. Struct. Dyn. 38, 655–677 (2009)
Sladek J., Sladek V., Solek P., Tan C.L., Zhang C.: Two- and three-dimensional transient thermoelastic analysis by the MLPG method. Comput. Model. Eng. Sci. 47, 61–95 (2009)
Sternberg E., Chakravorty J.G.: On inertia effects in a transient thermoelastic problem. J. Appl. Mech. ASME 26, 503–509 (1959)
Tamma K.K., Railkar S.B.: A generalized hybrid transfinite element computational approach for nonlinear/linear unified thermal-structural analysis. Comput. Struct. 26, 655–665 (1987)
Tamma K.K., Railkar S.B.: On heat displacement based hybrid transfinite element formulations for uncoupled/coupled thermally induced stress wave propagation. Comput. Struct. 30, 1025–1036 (1988)
Vujanovic B., Djukic D.: On one variational principle of Hamilton’s type for nonlinear heat transfer problem. Int. J. Heat Mass Trans. 15, 1111–1123 (1972)
Wriggers P., Reese S.: Thermoelastic stability of trusses with temperature-dependent constitutive relations. Int. J. Numer. Methods Eng. 35, 1891–1906 (1992)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Apostolakis, G., Dargush, G.F. Variational methods in irreversible thermoelasticity: theoretical developments and minimum principles for the discrete form. Acta Mech 224, 2065–2088 (2013). https://doi.org/10.1007/s00707-013-0843-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-013-0843-0