Characterization of Constitutive Parameters for Hyperelastic Models Considering the Baker-Ericksen Inequalities

  • Felipe Tempel Stumpf
  • Rogério José MarczakEmail author
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 49)


Hyperelastic models are used to simulate the mechanical behavior of rubber-like materials ranging from elastomers, such as natural rubber and silicon, to biologic materials, such as muscles and skin tissue. Once the desired hyperelastic model has its parameters fitted to the available experimental results, these hyperelastic parameters have to fulfill the requirements imposed by the Baker-Ericksen inequalities in order to guarantee a plausible physical behavior to the material, although seldom used. When applied to an incompressible isotropic hyperelastic model, these inequalities state that the first derivative of the strain energy density function with respect to the first strain invariant must be positive and the first derivative of the strain energy density function with respect to the second strain invariant must be non-negative. The aim of this work is to study which improvements the requirement of the Baker-Ericksen inequalities can bring when fitting hyperelastic models to experimental data. This is accomplished through a constrained optimization procedure. Results obtained for natural rubber and silicon samples considering classical and newly developed hyperelastic models are shown and discussed.


Hyperelasticity Optimization Constitutive parameters 


  1. 1.
    Baker, M., Ericksen, J.L.: Inequalities Restricting the Form of the Stress- Deformation Relation for Isotropic Elastic Solids and Reiner-Rivlin Fluids. J. Wash. Acad. Sci. 44, 33–35 (1954)Google Scholar
  2. 2.
    Ball, J.M.: Convexity Conditions and Existence Theorems in Nonlinear Elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1977) Google Scholar
  3. 3.
    Balzani, D., Neff, P., Schröder, J., Holzapfel, G.A.: A polyconvex framework for soft biological tissues. Adjustment to experimental data. Int. J. Solids Struct. 43, 6052–6070 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bilgili, E.: Restricting the Hyperelastic Models for Elastomers Based on Some Thermodynamical, Mechanical and Empirical Criteria. J. Elastom. Plast. 36, 159–175 (2004)Google Scholar
  5. 5.
    Fung, Y.C.B.: Elasticity of soft tissues in sample elongation. Am. J. Physiol. 213, 1532–1544 (1967)Google Scholar
  6. 6.
    Hartmann, S., Neff, P.: Polyconvexity of Generalized Polynomial-Type Hyperelastic Strain Energy Functions for Near-Incompressibility. Int. J. Solids and Struct. 40, 2767–2791 (2003) Google Scholar
  7. 7.
    Hoss, L., Stumpf, F.T., de Bortoli, D., Marczak, R.J.: Constitutive models for rubber-like materials: comparative analysis, goodness of fitting measures, and proposal of a new model. In: International Rubber Conference, IRCO (2011)Google Scholar
  8. 8.
    Jones, D., Treloar, L.: The properties of rubber in pure homogeneous strain. J. Phys. D Appl. Phys. 8, 1285–1304 (1975)CrossRefGoogle Scholar
  9. 9.
    Marczak, R.J., Gheller, J.J., Hoss, L.: Characterization of Elastomers for Numerical Simulation. Centro Tecnológico de Polímeros SENAI, São Leopoldo (2006) (in Portuguese)Google Scholar
  10. 10.
    Meunier, L., Chagnon, G., Favier, D., Orgeás, L., Vacher, P.: Mechanical experimental characterization and numerical modelling of an unfilled silicone rubber. Polym. Testing 27, 765–777 (2008)CrossRefGoogle Scholar
  11. 11.
    Nelder, J.A., Mead, R.: A simplex method for function minimization. Comput. J. 7, 308–313 (1965)CrossRefzbMATHGoogle Scholar
  12. 12.
    Ogden, R.W.: Non-Linear Elastic Deformations. Dover Publications, New York (1984) Google Scholar
  13. 13.
    Pucci, E., Saccomandi, G.: A note on the Gent model for rubber-like materials. Rubber Chem. Technol. 75, 839–851 (2002)CrossRefGoogle Scholar
  14. 14.
    Stumpf, F.T.: Assessment of a hyperelastic model for incompressible materials: analysis of restrictions, numerical implementation and optimization of the constitutive parameters. M.Sc. Dissertation, UFRGS, Porto Alegre (2009) (in Portuguese)Google Scholar
  15. 15.
    Truesdell, C.: The main unsolved problem in finite elasticity theory. Z. Angew. Math. Phys. 36, 97–103 (1956) (in German)Google Scholar
  16. 16.
    Truesdell, C., Noll, W.: The nonlinear Field Theories of Mechanics Flügge’s Handbuch der Physik, vol. III/3. Springer, Berlin/Heidelberg (1965)Google Scholar
  17. 17.
    Venkataraman, P.: Applied Optimization with MATLAB Programming. Willey (2009)Google Scholar
  18. 18.
    Yeoh, O.H.: Characterization of Elastic Properties of Carbon Black Filled Rubber Vulcanizates. Rubber Chem. Tech. 63, 792–805 (1990) Google Scholar

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Felipe Tempel Stumpf
    • 1
  • Rogério José Marczak
    • 2
    Email author
  1. 1.School of EngineeringFURGRio GrandeBrazil
  2. 2.Mechanical Engineering DepartmentUFRGSPorto AlegreBrazil

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