Home Internal Variable Theory Formulated by One Extended Potential Function
Article
Licensed
Unlicensed Requires Authentication

Internal Variable Theory Formulated by One Extended Potential Function

  • Qiang Yang EMAIL logo , Zhuofu Tao and Yaoru Liu
Published/Copyright: June 3, 2020

Abstract

In the kinetic rate laws of internal variables, it is usually assumed that the rates of internal variables depend on the conjugate forces of the internal variables and the state variables. The dependence on the conjugate force has been fully addressed around flow potential functions. The kinetic rate laws can be formulated with two potential functions, the free energy function and the flow potential function. The dependence on the state variables has not been well addressed. Motivated by the previous study on the asymptotic stability of the internal variable theory by J. R. Rice, the thermodynamic significance of the dependence on the state variables is addressed in this paper. It is shown in this paper that the kinetic rate laws can be formulated by one extended potential function defined in an extended state space if the rates of internal variables do not depend explicitly on the internal variables. The extended state space is spanned by the state variables and the rate of internal variables. Furthermore, if the rates of internal variables do not depend explicitly on state variables, an extended Gibbs equation can be established based on the extended potential function, from which all constitutive equations can be recovered. This work may be considered as a certain Lagrangian formulation of the internal variable theory.

Award Identifier / Grant number: 51739006

Award Identifier / Grant number: 11572174

Funding statement: This work has been supported by the National Key R&D Program of China (2018YFC0407005) and the National Natural Science Foundation of China under projects 51739006 and 11572174.

References

[1] G. A. Maugin, The Thermodynamics of Nonlinear Irreversible Behaviors, World Scientific, Singapore, 1999.10.1142/3700Search in Google Scholar

[2] G. A. Maugin, The saga of internal variables of state in continuum thermo-mechanics (1893–2013), Mech. Res. Commun.69 (2015), 79–86.10.1016/j.mechrescom.2015.06.009Search in Google Scholar

[3] M. F. Horstemeyer and D. J. Bammann, Historical review of internal state variable theory for inelasticity, Int. J. Plast.26 (2010), 1310–1334.10.1016/j.ijplas.2010.06.005Search in Google Scholar

[4] P. W. Bridgman, The Nature of Thermodynamics, Harvard University Press, Cambridge, Mass, 1943.Search in Google Scholar

[5] M. A. Biot, The theory of elasticity and consolidation for a porous anisotropic solid, J. Appl. Phys.25 (1954), 1385–1391.10.1063/1.1721956Search in Google Scholar

[6] J. Meixner, The fundamental inequality in thermodynamics, Physica59 (1972), 305–313.10.1016/0031-8914(72)90085-7Search in Google Scholar

[7] J. Bataille and J. Kestin, Irreversible processes and physical interpretation of rational thermodynamics, J. Non-Equilib. Thermodyn.. 4 (1979), 229–258.10.1515/jnet.1979.4.4.229Search in Google Scholar

[8] J. Kestin, Note on the relation between the hypothesis of local equilibrium and the Clausius-Duhem Inequality, J. Non-Equilib. Thermodyn.. 15 (1990), 193–212.10.1515/jnet.1990.15.3.193Search in Google Scholar

[9] J. Kestin and J. R. Rice, Paradoxes in the application of thermodynamics to strained solids, in: A Critical Review of Thermodynamics, E. B. Stuart, et al. Eds., Mono Book, Baltimore (1970), 275–298.Search in Google Scholar

[10] J. R. Rice, Inelastic constitutive relations for solids: an internal variable theory and its application to metal plasticity, J. Mech. Phys. Solids19 (1971), 433–455.10.1016/0022-5096(71)90010-XSearch in Google Scholar

[11] J. R. Rice, Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms, in: Constitutive Equations in Plasticity, A. S. Argon, Ed., MIT Press, Cambridge, MA (1975), 23–79.Search in Google Scholar

[12] W. Muschik, Internal variables and nonequilibrium thermodynamics, J. Non-Equilib. Thermodyn.. 15 (1990), 127–137.10.1515/jnet.1990.15.2.127Search in Google Scholar

[13] G. A. Maugin and W. Muschik, Thermodynamics with internal variables. Part I. General concepts, J. Non-Equilib. Thermodyn.. 19 (1994), 217–249.10.1515/jnet.1994.19.3.217Search in Google Scholar

[14] S. R. De Groot and P. Mazur, Non-equilibrium Thermodynamics, North-Holland, Amsterdam, 1962.Search in Google Scholar

[15] J. Bataille and J. Kestin, L’interprétation physique de la thermodynamique rationnelle, J. Méc.14 (1975), 365–384.Search in Google Scholar

[16] K. R. Rajagopal and A. R. Srinivasa, On thermomechanical restrictions of continua, Proc. R. Soc. Lond. A460 (2004), 631–651.10.1098/rspa.2002.1111Search in Google Scholar

[17] D. G. B. Edelen, A nonlinear Onsager theory of irreversibility, Int. J. Eng. Sci.10 (1972), 481–490.10.1016/0020-7225(72)90091-2Search in Google Scholar

[18] D. G. B. Edelen, Asymptotic stability, Onsager fluxes and reaction kinetics, Int. J. Eng. Sci.11 (1973), 819–839.10.1016/0020-7225(73)90032-3Search in Google Scholar

[19] Q. Yang, L. G. Tham and G. Swoboda, Normality structures with homogeneous kinetic rate laws, J. Appl. Mech.72 (2005), 322–329.10.1115/1.1867991Search in Google Scholar

[20] Q. Yang, R. K. Wang and L. J. Xue, Normality structures with thermodynamic equilibrium points, J. Appl. Mech.74 (2007), 965–971.10.1115/1.2722772Search in Google Scholar

[21] G. N. Hatsopoulos and J. H. Keenan, Principles of General Thermodynamics, John Wiley & Sons, New York, 1965.Search in Google Scholar

[22] E. P. Gyftopoulos and G. P. Beretta, Thermodynamics: Foundations and Applications, Dover, New York, 2005.Search in Google Scholar

[23] Q. Yang, J. Q. Bao and Y. R. Liu, Asymptotic stability in constrained configuration space for solids, J. Non-Equilib. Thermodyn.. 34 (2009), 155–170.10.1515/JNETDY.2009.009Search in Google Scholar

[24] V. Zorich, Mathematical Analysis of Problems in the Natural Science, Springer, Heidelberg, Dordrecht, London, New York, 2011.10.1007/978-3-642-14813-2Search in Google Scholar

[25] H. W. Haslach Jr., Maximum Dissipation Non-Equilibrium Thermodynamics and its Geometric Structure, Springer, Heidelberg, Dordrecht, London, New York, 2011.10.1007/978-1-4419-7765-6Search in Google Scholar

[26] K. H. Anthony, Hamilton’s action principle and thermodynamics of irreversible processes – A unifying procedure for reversible and irreversible processes, J. Non-Newton. Fluid Mech.96 (2001), 291–339.10.1016/S0377-0257(00)00187-7Search in Google Scholar

[27] K. S. Glavatskiy, Lagrangian formulation of irreversible thermodynamics and the second law of thermodynamics, J. Chem. Phys.142 (2015), 204106.10.1063/1.4921558Search in Google Scholar PubMed

[28] W. Kosinski and P. Perzyna, On consequences of the principle of stationary action for dissipative bodies, Arch. Mech.64 (2012), 1–12.Search in Google Scholar

[29] Q. Yang, Q. C. Lv and Y. R. Liu, Hamilton’s principle as inequality for inelastic bodies, Contin. Mech. Thermodyn.29 (2017), 747–756.10.1007/s00161-017-0557-ySearch in Google Scholar

[30] Q. Yang, Y. R. Liu and J. Q. Bao, Hamilton’s principle of entropy production for creep and relaxation processes, J. Eng. Mater. Technol.132 (2010), 011018.10.1115/1.4000302Search in Google Scholar

[31] H. Petryk, Thermodynamic conditions for stability in materials with rate-independent dissipation, Philos. Trans. R. Soc. Lond. A363 (2005), 2479–2515.10.1098/rsta.2005.1584Search in Google Scholar PubMed

Received: 2020-02-13
Revised: 2020-04-25
Accepted: 2020-05-15
Published Online: 2020-06-03
Published in Print: 2020-07-26

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 10.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/jnet-2020-0017/html
Scroll to top button