Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Hybrid free energy approach for nearly incompressible behaviors at finite strain

  • 38 Accesses

  • 1 Citations


We explore the formulation of nearly incompressible behaviors at finite strain in the context of a hybrid or a mixed energy. Such an energy is a function of both an isochoric deformation and a pressure-like quantity that can be considered as an internal variable. From thermodynamical and physical considerations, new energy functions are developed to correctly describe both nearly incompressible elasticity and thermoelastic behaviors. We discuss the advantages of such a formulation; in particular, we show that this approach makes it possible to unify the variational and the thermodynamical formulations in the nearly incompressible context without using Lagrange multipliers or other specific variational principles.

This is a preview of subscription content, log in to check access.


  1. 1.

    Atluri, S.N., Reissner, E.: On the formulation of variational theorems involving volume constraints. Comput. Mech. 5(5), 337–344 (1989). https://doi.org/10.1007/BF01047050

  2. 2.

    Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1977)

  3. 3.

    Chadwick, P.: Thermo-mechanics of rubberlike materials. Phil. Trans. R. Soc. Lond. Ser A, Math. Phys. Sci. 276(1260), 371–403 (1974). http://www.jstor.org/stable/74231

  4. 4.

    Chadwick, P., Creasy, C.: Modified entropic elasticity of rubberlike materials. J. Mech. Phys. Solids 32(5), 337–357 (1984). https://doi.org/10.1016/0022-5096(84)90018-8. http://www.sciencedirect.com/science/article/pii/0022509684900188

  5. 5.

    Ciarlet, P.G.: Élasticité tridimensionnelle. Masson, Armand Colin (1986)

  6. 6.

    Ehlers, W., Eipper, G.: The simple tension problem at large volumetric strains computed from finite hyperelastic material laws. Acta Mech. 130, 17–27 (1998)

  7. 7.

    Flory, P.J.: Principles of Polymer Chemistry, first edn. Cornell University Press, Ithaca (1953)

  8. 8.

    Flory, R.J.: Thermodynamic relations for highly elastic materials. Trans. Faraday Soc. 57, 829–838 (1961)

  9. 9.

    Germain, P., Nguyen, Q., Suquet, P.: Continuum thermodynamics. J. Appl. Mech. 50, 1010–1020 (1983). https://doi.org/10.1115/1.3167184

  10. 10.

    Hartmann, S., Neff, P.: Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility. Int. J. Solids Struct. 40(11), 2767–2791 (2003). https://doi.org/10.1016/S0020-7683(03)00086-6. http://www.sciencedirect.com/science/article/pii/S0020768303000866

  11. 11.

    Holzapfel, G.: Nonlinear Solid Mechanics: A Continuum Approach for Engineering. Wiley, New York (2000)

  12. 12.

    Holzapfel, G., Simo, J.: Entropy elasticity of isotropic rubber-like solids at finite strains. Comput. Methods Appl. Mech. Eng. 132(12), 17–44 (1996). https://doi.org/10.1016/0045-7825(96)01001-8. http://www.sciencedirect.com/science/article/pii/0045782596010018

  13. 13.

    Kannan, K., Rajagopal, K.: A thermodynamical framework for chemically reacting systems. Zeitschrift fr Angewandte Mathematik und Physik (ZAMP) 62, 331–363 (2011). https://doi.org/10.1007/s00033-010-0104-1

  14. 14.

    Koprowski-Theiß, N., Johlitz, M., Diebels, S.: Compressible rubber materials: experiments and simulations. Arch. Appl. Mech. 82(8), 1117–1132 (2012). https://doi.org/10.1007/s00419-012-0616-6

  15. 15.

    Lejeunes, S., Eyheramendy, D., Boukamel, A., Delattre, A., Méo, S., Ahose, K.D.: A constitutive multiphysics modeling for nearly incompressible dissipative materials: application to thermo-chemo-mechanical aging of rubbers. Mech. Time-Depend. Mater. 22(1), 51–66 (2018). https://doi.org/10.1007/s11043-017-9351-2

  16. 16.

    Lion, A., Dippel, B., Liebl, C.: Thermomechanical material modelling based on a hybrid free energy density depending on pressure, isochoric deformation and temperature. Int. J. Solids Struct. 51(34), 729–739 (2014). https://doi.org/10.1016/j.ijsolstr.2013.10.036. http://www.sciencedirect.com/science/article/pii/S0020768313004319

  17. 17.

    Lion, A., Peters, J., Kolmeder, S.: Simulation of temperature history-dependent phenomena of glass-forming materials based on thermodynamics with internal state variables. Thermochim. Acta 522(1), 182–193 (2011). https://doi.org/10.1016/j.tca.2010.12.017. http://www.sciencedirect.com/science/article/pii/S0040603110004715. Special Issue: Interplay between Nucleation, Crystallization, and the Glass Transition

  18. 18.

    Liu, C., Hofstetter, G., Mang, H.: 3d finite element analysis of rubberlike materials at finite strains. Eng. Comput. 11, 111–128 (1994). https://doi.org/10.1108/02644409410799236

  19. 19.

    Miehe, C.: Aspects of the formulation and finite element implementation of large strain isotropic elasticity. Int. J. Numer. Meth. Eng. 37, 1981–2004 (1994)

  20. 20.

    Miehe, C.: Entropic thermoelasticity at finite strains. aspects of the formulation and numerical implementation. Computer Methods in Applied Mechanics and Engineering 120(3), 243–269 (1995). https://doi.org/10.1016/0045-7825(94)00057-T. http://www.sciencedirect.com/science/article/pii/004578259400057T

  21. 21.

    Ogden, R.: Elastic deformations of rubberlike solids. In: Hopkins, H., Sewell, M., (eds.) Mechanics of Solids, pp. 499–537. Pergamon, Oxford (1982). https://doi.org/10.1016/B978-0-08-025443-2.50021-5. https://www.sciencedirect.com/science/article/pii/B9780080254432500215

  22. 22.

    Reissner, E.: On a variational principle for elastic displacements and pressure. J. Appl. Mech. 51, 444–445 (1984). https://doi.org/10.1115/1.3167643

  23. 23.

    Rodriguez, E.L., Filisko, F.E.: Thermal effects in styrene-butadiene rubber at high hydrostatic pressures. Polymer 27, 1943–1947 (1986)

  24. 24.

    Simo, J., Taylor, R., Pister, K.: Variational and projection methods for the volume constraint in finite deformation elasto-plasticity. Comput. Methods Appl. Mech. Eng. 51(1), 177–208 (1985). https://doi.org/10.1016/0045-7825(85)90033-7. http://www.sciencedirect.com/science/article/pii/0045782585900337

  25. 25.

    Simo, J.C., Taylor, R.L.: Quasi-incompressible finite elasticity in principal stretches. continuum basis and numerical algorithms. Comput. Methods Appl. Mech. Eng. 85(3), 273–310 (1991). https://doi.org/10.1016/0045-7825(91)90100-K. http://www.sciencedirect.com/science/article/pii/004578259190100K

Download references

Author information

Correspondence to Stéphane Lejeunes.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Johlitz, Laiarinandrasana and Marco.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lejeunes, S., Eyheramendy, D. Hybrid free energy approach for nearly incompressible behaviors at finite strain. Continuum Mech. Thermodyn. 32, 387–401 (2020). https://doi.org/10.1007/s00161-018-0680-4

Download citation


  • Thermomechanical coupling
  • Entropic elasticity
  • Free energy
  • Volumetric behavior