Continuum Mechanics and Thermodynamics

, Volume 31, Issue 3, pp 775–806 | Cite as

A solution set-based entropy principle for constitutive modeling in mechanics

  • Julian HeßEmail author
  • Alexei F. Cheviakov
Original Article


Entropy principles based on thermodynamic consistency requirements are widely used for constitutive modeling in continuum mechanics, providing physical constraints on a priori unknown constitutive functions. The well-known Müller–Liu procedure is based on Liu’s lemma for linear systems. While the Müller–Liu algorithm works well for basic models with simple constitutive dependencies, it cannot take into account nonlinear relationships that exist between higher derivatives of the fields in the cases of more complex constitutive dependencies. The current contribution presents a general solution set-based procedure, which, for a model system of differential equations, respects the geometry of the solution manifold, and yields a set of constraint equations on the unknown constitutive functions, which are necessary and sufficient conditions for the entropy production to stay nonnegative for any solution of the model. Similarly to the Müller–Liu procedure, the solution set approach is algorithmic, its output being a set of constraint equations and a residual entropy inequality. The solution set method is applicable to virtually any physical model, allows for arbitrary initially postulated forms of the constitutive dependencies, and does not use artificial constructs like Lagrange multipliers. A Maple implementation makes the solution set method computationally straightforward and useful for the constitutive modeling of complex systems. Several computational examples are considered, in particular models of gas, anisotropic fluid, and granular flow dynamics. The resulting constitutive function forms are analyzed, and comparisons are provided. It is shown how the solution set entropy principle can yield classification problems, leading to several complementary sets of admissible constitutive functions; such classification problems have not previously appeared in the constitutive modeling literature.


Constitutive modeling Continuum mechanics Entropy principle Solution set Symbolic computations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


Supplementary material (309 kb)
Supplementary material 1 (zip 308 KB)


  1. 1.
    Hutter, K., Jöhnk, K.: Continuum Methods of Physical Modeling. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  2. 2.
    Magnenet, V., Rahouadj, R., Ganghoffer, J., Cunat, C.: On the lie symmetry groups with application to a non linear viscoelastic behaviour. J. Mech. Behav. Mater. 16(4–5), 241–248 (2005)zbMATHGoogle Scholar
  3. 3.
    Magnenet, V., Rahouadj, R., Ganghoffer, J.-F.: A new methodology for determining the mechanical behavior of polymers exploiting lie symmetries: application to a stick-like material. Mech. Mater. 41(9), 1017–1024 (2009)CrossRefGoogle Scholar
  4. 4.
    Magnenet, V., Rahouadj, R., Ganghoffer, J.-F.: Symmetry analysis and invariance relations in creep. Math. Mech. Solids 19(8), 988–1010 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Magnenet, V., Rahouadj, R., Ganghoffer, J.-F., Cunat, C.: Continuous symmetry analysis of a dissipative constitutive law: application to the time-temperature superposition. Eur. J. Mech. A/Solids 28(4), 744–751 (2009)ADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Rahouadj, R., Ganghoffer, J.-F., Cunat, C.: A thermodynamic approach with internal variables using lagrange formalism. Part II. Continuous symmetries in the case of the time-temperature equivalence. Mech. Res. Commun. 30(2), 119–123 (2003)CrossRefzbMATHGoogle Scholar
  7. 7.
    Goda, I., Assidi, M., Belouettar, S., Ganghoffer, J.: A micropolar anisotropic constitutive model of cancellous bone from discrete homogenization. J. Mech. Behav. Biomed. Mater. 16, 87–108 (2012)CrossRefGoogle Scholar
  8. 8.
    Goda, I., Assidi, M., Ganghoffer, J.-F.: A 3d elastic micropolar model of vertebral trabecular bone from lattice homogenization of the bone microstructure. Biomech. Model. Mechanobiol. 13(1), 53–83 (2014)CrossRefGoogle Scholar
  9. 9.
    Coleman, B.D., Noll, W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Ration Mech. Anal. 13(1), 167–178 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kremer, G.: Extended thermodynamics of ideal gases with 14 fields. Ann. l’IHP Phys. théor. 45, 419–440 (1986)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Liu, I.-S.: Method of Lagrange multipliers for exploitation of the entropy principle. Arch. Ration. Mech. Anal. 46(2), 131–148 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Müller, I.: A thermodynamic theory of mixtures of fluids. Arch. Ration. Mech. Anal. 28(1), 1–39 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Müller, I., Liu, I.-S.: Thermodynamics of mixtures of fluids. In: Truesdell, C. (ed.) Rational Thermodynamics. Springer, Berlin (1984)Google Scholar
  14. 14.
    Müller, I.: On the entropy inequality. Arch. Ration. Mech. Anal. 26(2), 118–141 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Truesdell, C.: Sulle basi della termomeccanica. Rend. Lincei 22(8), 33–38 (1957)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Truesdell, C.: Mechanical basis of diffusion. J. Chem. Phys. 37(10), 2336–2344 (1962)ADSCrossRefGoogle Scholar
  17. 17.
    Liu, I.-S.: On irreversible thermodynamics. PhD thesis, Johns Hopkins University, Baltimore, (1972)Google Scholar
  18. 18.
    Hauser, R., Kirchner, N.: A historical note on the entropy principle of Müller and Liu. Contin. Mech. Thermodyn. 14(2), 223–226 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Reis, M.C., Wang, Y.: A two-fluid model for reactive dilute solid-liquid mixtures with phase changes. Contin. Mech. Thermodyn. 29(2), 1–26 (2016)MathSciNetGoogle Scholar
  20. 20.
    Liu, I.-S.: Entropy flux relation for viscoelastic bodies. J. Elast. 90(3), 259–270 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Liu, I.-S., Müller, I.: On the thermodynamics and thermostatics of fluids in electromagnetic fields. Arch. Ration. Mech. Anal. 46(2), 149–176 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Svendsen, B., Chanda, T.: Continuum thermodynamic formulation of models for electromagnetic thermoinelastic solids with application in electromagnetic metal forming. Contin. Mech. Thermodyn. 17(1), 1–16 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Heß, J., Wang, Y., Hutter, K.: Thermodynamically consistent modeling of granular-fluid mixtures incorporating pore pressure evolution and hypoplastic behavior. Contin. Mech. Thermodyn. 29(1), 311–343 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Schneider, L., Hutter, K.: Solid-Fluid Mixtures of Frictional Materials in Geophysical and Geotechnical Context: Based on a Concise Thermodynamic Analysis. Springer, Berlin (2009)CrossRefGoogle Scholar
  25. 25.
    Cheviakov, A., Heß, J.: A symbolic computation framework for constitutive modelling based on entropy principles. Appl. Math. Comput. 324, 105–118 (2018)MathSciNetGoogle Scholar
  26. 26.
    Olver, P.J.: Applications of Lie Groups to Differential Equations, vol. 107. Springer, Berlin (2000)zbMATHGoogle Scholar
  27. 27.
    Müller, I.: A new systematic approach to non-equilibrium thermodynamics. Pure Appl. Chem. 22(3–4), 335–342 (1970)CrossRefGoogle Scholar
  28. 28.
    Müller, I.: Die Kältefunktion, eine universelle Funktion in der Thermodynamik viskoser wärmeleitender Flüssigkeiten. Arch. Ration. Mech. Anal. 40(1), 1–36 (1971)CrossRefzbMATHGoogle Scholar
  29. 29.
    Müller, I.: The coldness, a universal function in thermoelastic bodies. Arch. Ration. Mech. Anal. 41(5), 319–332 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Cheviakov, A.F.: Gem software package for computation of symmetries and conservation laws of differential equations. Comput. Phys. Commun. 176(1), 48–61 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Cheviakov, A.F.: Computation of fluxes of conservation laws. J. Eng. Math. 66(1–3), 153–173 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Cheviakov, A.F.: Symbolic computation of local symmetries of nonlinear and linear partial and ordinary differential equations. Math. Comput. Sci. 4(2–3), 203–222 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Reid, G.J., Wittkopf, A.D., Boulton, A.: Reduction of systems of nonlinear partial differential equations to simplified involutive forms. Eur. J. Appl. Math. 7(6), 635–666 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Wang, Y., Hutter, K.: Comparison of two entropy principles and their applications in granular flows with/without fluid. Arch. Mech. 51(5), 605–632 (1999)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Wang, Y., Hutter, K.: Shearing flows in a Goodman–Cowin type material—theory and numerical results. Part. Sci. Technol. 17(1), 97–124 (1999)ADSCrossRefGoogle Scholar
  36. 36.
    Svendsen, B., Hutter, K.: On the thermodynamics of a mixture of isotropic materials with constraints. Int. J. Eng. Sci. 33(14), 2021–2054 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Liu, I.-S.: Continuum Mechanics. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  38. 38.
    Müller, I., Ruggeri, T.: Rational Extended Thermodynamics. Volume 37 of Springer Tracts in Natural Philosophy. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  39. 39.
    GeM software package, examples, and description (2007).
  40. 40.
    Goodman, M., Cowin, S.: A continuum theory for granular materials. Arch. Ration. Mech. Anal. 44(4), 249–266 (1972)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Chair of Fluid DynamicsTechnical University of DarmstadtDarmstadtGermany
  2. 2.Department of Mathematics and StatisticsUniversity of SaskatchewanSaskatoonCanada

Personalised recommendations