Advertisement

Mechanics of Time-Dependent Materials

, Volume 15, Issue 4, pp 389–406 | Cite as

A constitutive framework for modelling thin incompressible viscoelastic materials under plane stress in the finite strain regime

  • M. Kroon
Article

Abstract

Rubbers and soft biological tissues may undergo large deformations and are also viscoelastic. The formulation of constitutive models for these materials poses special challenges. In several applications, especially in biomechanics, these materials are also relatively thin, implying that in-plane stresses dominate and that plane stress may therefore be assumed. In the present paper, a constitutive model for viscoelastic materials in the finite strain regime and under the assumption of plane stress is proposed. It is assumed that the relaxation behaviour in the direction of plane stress can be treated separately, which makes it possible to formulate evolution laws for the plastic strains on explicit form at the same time as incompressibility is fulfilled. Experimental results from biomechanics (dynamic inflation of dog aorta) and rubber mechanics (biaxial stretching of rubber sheets) were used to assess the proposed model. The assessment clearly indicates that the model is fully able to predict the experimental outcome for these types of material.

Keywords

Viscoelastic Finite strain Large deformation Plane stress Rubber Incompressible 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Armentano, R.L., Barra, J.G., Levenson, J., Simon, A., Pichel, R.H.: Arterial wall mechanics in conscious dogs. Circ. Res. 76, 468–478 (1995) Google Scholar
  2. Arruda, E.M., Boyce, M.C.: A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J. Mech. Phys. Solids 41, 389–412 (1993) CrossRefGoogle Scholar
  3. Bonet, J.: Large strain viscoelastic constitutive models. Int. J. Solids Struct. 38, 2953–2968 (2001) MATHCrossRefGoogle Scholar
  4. Holzapfel, G.A.: Nonlinear Solid Mechanics. A Continuum Approach for Engineering. Wiley, Chichester (2000) MATHGoogle Scholar
  5. Holzapfel, G.A., Reiter, G.: Fully coupled thermomechanical behaviour of viscoelastic solids treated with finite elements. Int. J. Eng. Sci. 33, 1037–1058 (1995) MATHCrossRefGoogle Scholar
  6. Holzapfel, G.A., Simo, J.C.: A new viscoelastic constitutive model for continuous media at finite thermomechanical changes. Int. J. Solids Struct. 33, 3019–3034 (1996) MATHCrossRefGoogle Scholar
  7. Holzapfel, G.A., Gasser, T.C., Ogden, R.W.: A new constitutive framework for arterial wall mechanics and a comparative study of material models. J. Elast. 61, 1–48 (2000) MathSciNetMATHCrossRefGoogle Scholar
  8. Kawabata, S., Kawai, H.: Strain energy density functions of rubber vulcanizates from biaxial extension. Adv. Polym. Sci. 24, 89–124 (1977) Google Scholar
  9. Kroon, M.: An 8-chain model for rubber-like materials accounting for non-affine chain deformations and topological constraints. J. Elast. 102, 99–116 (2010a) MathSciNetCrossRefGoogle Scholar
  10. Kroon, M.: A constitutive model for smooth muscle including active tone and passive viscoelastic behaviour. Math. Med. Biol. 27, 129–155 (2010b) MathSciNetMATHCrossRefGoogle Scholar
  11. Lubliner, J.: A model of rubber viscoelasticity. Mech. Res. Commun. 12, 93–99 (1985) CrossRefGoogle Scholar
  12. Miehe, C., Göktepe, S., Lulei, F.: A micro-macro approach to rubber-like materials—part i: the non-affine micro-sphere model of rubber elasticity. J. Mech. Phys. Solids 52, 2617–2660 (2004) MathSciNetMATHCrossRefGoogle Scholar
  13. Mooney, M.: A theory of large elastic deformation. J. Appl. Phys. 11, 582–592 (1940) MATHCrossRefGoogle Scholar
  14. Ogden, R.W.: Large deformation isotropic elasticity—on the correlation of theory and experiment for incompressible rubberlike solids. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 326, 565–584 (1972) MATHCrossRefGoogle Scholar
  15. Ogden, R.W.: Non-linear Elastic Deformations. Dover, New York (1984) Google Scholar
  16. Rivlin, R.S.: Large elastic deformations of isotropic materials IV. Further developments of the general theory. Philos. Trans. R. Soc. Lond. Ser. A, Math. Phys. Sci. 241, 379–397 (1948) MathSciNetMATHCrossRefGoogle Scholar
  17. Simo, J.C.: On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects. Comput. Methods Appl. Mech. Eng. 60, 153–173 (1987) MathSciNetMATHCrossRefGoogle Scholar
  18. Simo, J.C., Hughes, T.J.R.: Computational Inelasticity. Springer, New York (1998) MATHGoogle Scholar
  19. Simo, J.C., Taylor, R.L.: Quasi-incompressible finite elasticity in principal stretches. Continuum basis and numerical algorithms. Comput. Methods Appl. Mech. Eng. 85, 273–310 (1991) MathSciNetMATHCrossRefGoogle Scholar
  20. Simo, J.C., Taylor, R.L., Pister, K.S.: Variational and projection methods for the volume constraint in finite deformation elastoplasticity. Comput. Methods Appl. Mech. Eng. 51, 177–208 (1985) MathSciNetMATHCrossRefGoogle Scholar
  21. Treloar, L.R.G., Riding, G.: A non-Gaussian theory of rubber in biaxial strain. I. Mechanical properties. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 369, 261–280 (1979) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, B. V. 2011

Authors and Affiliations

  1. 1.Department of Solid MechanicsRoyal Institute of TechnologyStockholmSweden

Personalised recommendations