Variational Constituive Models for Soft Biological Tissues

  • Jakson Manfredini VassolerEmail author
  • Eduardo Alberto Fancello
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 49)


Biological soft tissues are heterogeneous composite materials made of cells and molecules of the extracellular matrix. These tissues are frequently classified into four basic categories: muscle, epithelial, nervous and connective, each one with its own mechanical and functional properties. Their mechanical response to external forces (excluding those mechanisms associated with time scales typical of tissue remodeling), are characterized by anisotropy, high nonlinearity, strain rate dependency, permanent deformation and eventually, damage. Despite a wide set of constitutive models that have already been proposed in the specialized literature to represent the macroscopic behavior of these materials, this work focuses attention on a particular group, coined as variational in the sense that the incremental internal variable updates are found as minimizers of a pseudo strain-energy potential, called Incremental Potential evaluated at each time-step. General cases of models for viscoelastic, viscoplastic and fiber reinforced soft materials are discussed with the aid of numerical examples exploring the features of the corresponding approach.


Internal Variable Soft Biological Tissue Viscous Deformation Ogden Model Incremental Potential 
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  1. 1.
    Junqueira, L.C., Carneiro, J.: Histologia Basica, 10th edn. Guanabara Koogan S.A, Rio de Janeiro (2004)Google Scholar
  2. 2.
    Fung, Y.: Biomechanics—Mechanical Properties of Living Tissues. Springer, New York (1993)Google Scholar
  3. 3.
    Shergold, O.A., Fleck, N.A., Radford, D.: The uniaxial stress versus strain response of pig skin and silicone rubber at low and high strain rates. Int. J. Impact Eng. 32, 1384–1402 (2006)CrossRefGoogle Scholar
  4. 4.
    Giles, J.M., Black, A.E., Bischoff, J.E.: Anomalous rate dependence of the preconditioned response of soft tissue during load controlled deformation. J. Biomech. 40, 777–785 (2007)CrossRefGoogle Scholar
  5. 5.
    Munõz, M.J., Bea, J.A., Rodríguez, J.F., Ochoa, I., Grasa, J., Pérez Del Palomar, A., Zaragoza, P., Osta, R., Doblaré, M.: An experimental study of the mouse skin behaviour: damage and inelastic aspects. J. Biomech. 41(1), 93–99 (2008)CrossRefGoogle Scholar
  6. 6.
    Miller, K.: Constitutive model of brain tissue suitable for finite element analysis of surgical procedures. J. Biomech. 32(5), 531537 (1999)CrossRefGoogle Scholar
  7. 7.
    Limbert, G., Taylor, M., Middleton, J.: Three-dimensional finite element modelling of the human ACL: simulation of passive knee flexion with a stressed and stress-free ACL. J. Biomech. 11(41), 1723–1731 (2004)CrossRefGoogle Scholar
  8. 8.
    Holzapfel, G.A., Weizscker, H.W.: Biomechanical behavior of the arterial wall and its numerical characterization. Comput. Biol. Med. 28(4), 37792 (1998)CrossRefGoogle Scholar
  9. 9.
    Lanir, Y.: Constitutive equations for fibrous connective tissues. J. Biomech. 16(1), 1–12 (1983)CrossRefGoogle Scholar
  10. 10.
    Pioletti, D.P., Rakotomanana, L.R.: Viscoelastic constitutive law in large deformations: application to human knee ligaments and tendons. J. Biomech. 31(8), 753–757 (1998)CrossRefGoogle Scholar
  11. 11.
    Pioletti, D.P., Rakotomanana, L.R.: Non-linear viscoelastic laws for soft biological tissues. Eur. J. Mech. A/Solids 19, 749–759 (2000)CrossRefzbMATHGoogle Scholar
  12. 12.
    Limbert, G., Taylor, M.: On the constitutive modeling of biological soft connective tissues: a general theoretical framework and explicit forms of the tensors of elasticity for strongly anisotropic continuum fiber-reinforced composites at finite strain. Int. J. Solids Struct. 39(8), 2343–2358 (2002)CrossRefzbMATHGoogle Scholar
  13. 13.
    El Sayed, T., Mota, A., Fraternali, F., Ortiz, M.: A variational constitutive model for soft biological tissues. J. Biomech. 41, 1458–1466 (2008)CrossRefGoogle Scholar
  14. 14.
    Ehret, A.E., Itskov, M.: Modeling of anisotropic softening phenomena: application to soft biological tissues. Int. J. Plast. 25(5), 901–919 (2009)CrossRefzbMATHGoogle Scholar
  15. 15.
    Ortiz, M., Stainier, L.: The variational formulation of viscoplastic constitutive updates. Comput. Methods Appl. Mech. Eng. 171(31), 419–444 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Radovitzky, R., Ortiz, M.: Error estimation and adaptive meshing in strongly nonlinear dynamic problems. Comput. Methods Appl. Mech. Eng. 172, 203–240 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fancello, E., Ponthot, J.P., Stainier, L.: A variational formulation of constitutive models and updates in nonlinear finite viscoelasticity. Int. J. Numer. Methods Eng. 65(13), 1831–1864 (2006)CrossRefzbMATHGoogle Scholar
  18. 18.
    Fancello, E.A., Vigneron, L., Ponthot, J., Stainier, L.: A viscoelastic formulation for finite strains: application to brain soft tissues. In: XXVII CILAMCE—Iberian Latin American Congress on Computational Methods in Engineering (2006)Google Scholar
  19. 19.
    Fancello, E., Vassoler, J., Stainier, L.: Comput. Methods Appl. Mech. Eng. A variational constitutive update algorithm for a set of isotropic hyperelastic viscoplastic material models 197, 4132–4148 (2008)Google Scholar
  20. 20.
    Vassoler, J.M., Reips, L., Fancello, E.A.: A variational framework for fiber-reinforced viscoelastic soft tissues. Int. J. Numer. Methods Eng. 89(13), 16911706 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Holzapfel, G.A.: Nonlinear Solid Mechanics: A Continuum Approach for Engineering. Wiley, New York (2000)Google Scholar
  22. 22.
    Neto, E.D.S., Peric, D., Owen, D.R.J.: Computational Methods for Plasticity. Wiley, Chichester (2008)Google Scholar
  23. 23.
    Anand, L., Weber, G.: Finite deformations constitutive equations and a time integration procedure for isotropic hyperelastic-viscoplastic solids. Comput. Methods Appl. Mech. Eng. 79(14), 173–202 (1990)zbMATHGoogle Scholar
  24. 24.
    Holzapfel, G., Gasser, T.C.: A viscoelastic model for fiber-reinforced composites at finite strains: continuum basis, computational aspects and applications. Comput. Methods Appl. Mech. Eng. 190(21), 4379–4403 (2001)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Jakson Manfredini Vassoler
    • 1
    Email author
  • Eduardo Alberto Fancello
    • 2
  1. 1.Department of Mechanical EngineeringUniversidade Federal do Rio Grande do SulPorto AlegreBrazil
  2. 2.Department of Mechanical EngineeringUniversidade Federal de Santa CatarinaFlorianópolisBrazil

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