Continuum Mechanics and Thermodynamics

, Volume 29, Issue 3, pp 747–756 | Cite as

Hamilton’s principle as inequality for inelastic bodies

  • Q. Yang
  • Q. C. Lv
  • Y. R. Liu
Original Article


This paper is concerned with Hamilton’s principle for inelastic bodies with conservative external forces. Inelasticity is described by internal variable theory by Rice (J Mech Phys Solids 19:433–455, 1971), and the influence of strain change on the temperature field is ignored. Unlike Hamilton’s principle for elastic bodies which has an explicit Lagrangian, Hamilton’s principle for inelastic bodies generally has no an explicit Lagrangian. Based on the entropy inequality, a quasi Hamilton’s principle for inelastic bodies is established in the form of inequality and with an explicit Lagrangian, which is just the Lagrangian for elastic bodies by replacing the strain energy with free energy. The quasi Hamilton’s principle for inelastic bodies states that the actual motion is distinguished by making the action an maximum. The evolution equations of internal variables can not be recovered from the quasi Hamilton’s principle.


Hamilton’s principle Inelastic body Internal variables Inequality Entropy inequality 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anthony, K.H.: Hamilton’s action principle and thermodynamics of irreversible processes—a unifying procedure for reversible and irreversible processes. J. Non-Newton. Fluid Mech. 96, 291–339 (2001)CrossRefGoogle Scholar
  2. 2.
    Batra, G.: On Hamilton’s principle for thermo-elastic fluids and solids, and internal constraints in thermo-elasticity. J. Ration. Mech. Anal. 99, 37–59 (1987)MathSciNetCrossRefGoogle Scholar
  3. 3.
    De Groot, S.R., Mazur, P.: Non-equilibrium Thermodynamics. North-Holland, Amsterdam (1962)zbMATHGoogle Scholar
  4. 4.
    Drucker, D.C.: A more fundamental approach to stress-strain relations. In: Proceedings of First U.S. National Congress of Applied Mechanics, ASME, pp. 487–497 (1951)Google Scholar
  5. 5.
    Glavatskiy, K.S.: Lagrangian formulation of irreversible thermodynamics and the second law of thermodynamics. J. Chem. Phys. 142, 204106 (2015)ADSCrossRefGoogle Scholar
  6. 6.
    Il’yushin, A.A.: On a postulate of plasticity. J. Appl. Math. Mech. 25, 746–750 (1961)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fung, Y.C., Tong, P.: Classical and Computational Solid Mechanics. World Scientific, Singapore (2001)CrossRefGoogle Scholar
  8. 8.
    Kim, J., Dargush, G.F., Ju, Y.K.: Extended framework of Hamiltons principle for continuum dynamics. Int. J. Solids Struct. 50, 3418–3429 (2013)CrossRefGoogle Scholar
  9. 9.
    Kestin, J., Rice, J.R.: Paradoxes in the application of thermodynamics to strained solids. In: Stuart, E.B., et al. (eds.) A Critical Review of Thermodynamics, pp. 275–298. Mono Book, Baltimore (1970)Google Scholar
  10. 10.
    Kosinski, W., Perzyna, P.: On consequences of the principle of stationary action for dissipative bodies. Arch. Mech. 64, 1–12 (2012)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Leng, K.D., Yang, Q.: Generalized Hamilton’s principle for inelastic bodies within non-equilibrium thermodynamics. Entropy 13, 1904–1915 (2011)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Maugin, G.A.: Internal variables and dissipative structures. J. Non-equilib. Thermodyn. 15, 173–192 (1990)ADSCrossRefGoogle Scholar
  13. 13.
    Maugin, G.A., Muschik, W.: Thermodynamics with internal variables. Part I. General concepts. J. Non-Equilib. Thermodyn. 19, 217–249 (1994)ADSzbMATHGoogle Scholar
  14. 14.
    Maugin, G.A.: The Thermodynamics of Nonlinear Irreversible Behaviors. World Scientific, Singapore (1999)CrossRefGoogle Scholar
  15. 15.
    Maugin, G.A.: The saga of internal variables of state in continuum thermo-mechanics (1893–2013). Mech. Res. Commun. 69, 79–86 (2015)CrossRefGoogle Scholar
  16. 16.
    Truesdell, C., Toupin, R.A.: The classical field theories. In: Flügge, S. (ed.) Handbuch der Physik, Band III/1, pp. 226–793. Springer, Berlin (1960)Google Scholar
  17. 17.
    Petryk, H.: Thermodynamic conditions for stability in materials with rate-independent dissipation. Philos. Trans. R. Soc. Lond. A 363, 2479–2515 (2005)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Reddy, J.N.: An Introduction to Continuum Mechanics. Cambridge University Press, Cambridge (2008)Google Scholar
  19. 19.
    Rice, J.R.: Inelastic constitutive relations for solids: an internal variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433–455 (1971)ADSCrossRefGoogle Scholar
  20. 20.
    Rice, J.R.: Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms. In: Argon, A.S. (ed.) Constitutive Equations in Plasticity, pp. 23–79. MIT Press, Cambridge (1975)Google Scholar
  21. 21.
    Van, P., Muschik, W.: Structure of variational principles in nonequilibrium thermodynamics. Phys. Rev. E 52, 3584–3590 (1995)ADSCrossRefGoogle Scholar
  22. 22.
    Van, P., Berezovski, A., Engelbrecht, J.: Internal variables and dynamic degrees of freedom. J. Non-equilib. Thermodyn. 33, 235–254 (2008)ADSCrossRefGoogle Scholar
  23. 23.
    Van, P.: Weakly nonlocal non-equilibrium thermodynamics-variational principles and second law. In: Soomere, T., Quak, E. (eds.) Applied Wave Mathematics, pp. 153–186. Springer, New York (2009)CrossRefGoogle Scholar
  24. 24.
    Vujanovic, B.: On one variational principle for irreversible phenomena. Acta Mech. 19, 259–275 (1974)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Vujanovic, B.: A variational principle for non-conservative dynamical systems. ZAMM 55, 321–331 (1975)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Yang, Q., Tham, L.G., Swoboda, G.: Normality structures with homogeneous kinetic rate laws. ASME J. Appl. Mech. 72, 322–329 (2005)ADSCrossRefGoogle Scholar
  27. 27.
    Yang, Q., Wang, R.K., Xue, L.J.: Normality structures with thermodynamic equilibrium points. ASME J. Appl. Mech. 74, 965–971 (2007)ADSCrossRefGoogle Scholar
  28. 28.
    Yang, Q., Bao, J.Q., Liu, Y.R.: Asymptotic stability in constrained configuration space for solids. J. Non-equilib. Thermodyn. 34, 155–170 (2009)ADSzbMATHGoogle Scholar
  29. 29.
    Yang, Q., Guan, F.H., Liu, Y.R.: Hamilton’s principle for Green-inelastic bodies. Mech. Res. Commun. 37, 696–699 (2010)CrossRefGoogle Scholar
  30. 30.
    Ziegler, H.: An Introduction to Thermomechanics. North-Holland, Amsterdam (1977)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.State Key Laboratory of Hydroscience and EngineeringTsinghua UniversityBeijingPeople’s Republic of China

Personalised recommendations