Abstract
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.
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-Newton. Fluid Mech. 96, 291–339 (2001)
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)
De Groot, S.R., Mazur, P.: Non-equilibrium Thermodynamics. North-Holland, Amsterdam (1962)
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)
Glavatskiy, K.S.: Lagrangian formulation of irreversible thermodynamics and the second law of thermodynamics. J. Chem. Phys. 142, 204106 (2015)
Il’yushin, A.A.: On a postulate of plasticity. J. Appl. Math. Mech. 25, 746–750 (1961)
Fung, Y.C., Tong, P.: Classical and Computational Solid Mechanics. World Scientific, Singapore (2001)
Kim, J., Dargush, G.F., Ju, Y.K.: Extended framework of Hamiltons principle for continuum dynamics. Int. J. Solids Struct. 50, 3418–3429 (2013)
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)
Kosinski, W., Perzyna, P.: On consequences of the principle of stationary action for dissipative bodies. Arch. Mech. 64, 1–12 (2012)
Leng, K.D., Yang, Q.: Generalized Hamilton’s principle for inelastic bodies within non-equilibrium thermodynamics. Entropy 13, 1904–1915 (2011)
Maugin, G.A.: Internal variables and dissipative structures. J. Non-equilib. Thermodyn. 15, 173–192 (1990)
Maugin, G.A., Muschik, W.: Thermodynamics with internal variables. Part I. General concepts. J. Non-Equilib. Thermodyn. 19, 217–249 (1994)
Maugin, G.A.: The Thermodynamics of Nonlinear Irreversible Behaviors. World Scientific, Singapore (1999)
Maugin, G.A.: The saga of internal variables of state in continuum thermo-mechanics (1893–2013). Mech. Res. Commun. 69, 79–86 (2015)
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)
Petryk, H.: Thermodynamic conditions for stability in materials with rate-independent dissipation. Philos. Trans. R. Soc. Lond. A 363, 2479–2515 (2005)
Reddy, J.N.: An Introduction to Continuum Mechanics. Cambridge University Press, Cambridge (2008)
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)
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)
Van, P., Muschik, W.: Structure of variational principles in nonequilibrium thermodynamics. Phys. Rev. E 52, 3584–3590 (1995)
Van, P., Berezovski, A., Engelbrecht, J.: Internal variables and dynamic degrees of freedom. J. Non-equilib. Thermodyn. 33, 235–254 (2008)
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)
Vujanovic, B.: On one variational principle for irreversible phenomena. Acta Mech. 19, 259–275 (1974)
Vujanovic, B.: A variational principle for non-conservative dynamical systems. ZAMM 55, 321–331 (1975)
Yang, Q., Tham, L.G., Swoboda, G.: Normality structures with homogeneous kinetic rate laws. ASME J. Appl. Mech. 72, 322–329 (2005)
Yang, Q., Wang, R.K., Xue, L.J.: Normality structures with thermodynamic equilibrium points. ASME J. Appl. Mech. 74, 965–971 (2007)
Yang, Q., Bao, J.Q., Liu, Y.R.: Asymptotic stability in constrained configuration space for solids. J. Non-equilib. Thermodyn. 34, 155–170 (2009)
Yang, Q., Guan, F.H., Liu, Y.R.: Hamilton’s principle for Green-inelastic bodies. Mech. Res. Commun. 37, 696–699 (2010)
Ziegler, H.: An Introduction to Thermomechanics. North-Holland, Amsterdam (1977)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Andreas Öchsner.
Rights and permissions
About this article
Cite this article
Yang, Q., Lv, Q.C. & Liu, Y.R. Hamilton’s principle as inequality for inelastic bodies. Continuum Mech. Thermodyn. 29, 747–756 (2017). https://doi.org/10.1007/s00161-017-0557-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00161-017-0557-y