Acta Mechanica

, Volume 225, Issue 6, pp 1685–1698 | Cite as

Permanent set and stress-softening constitutive equation applied to rubber-like materials and soft tissues

  • M. Rebouah
  • G. Chagnon


Many rubber-like materials present a phenomenon known as Mullins effect. It is characterized by a difference of behavior between the first and second loadings and by a permanent set after a first loading. Moreover, this phenomenon induces anisotropy in an initially isotropic material. A new constitutive equation is proposed in this paper. It relies on the decomposition of the macromolecular network into two parts: chains related together and chains related to fillers. The first part is modeled by a simple hyperelastic constitutive equation, whereas the second one is described by an evolution function introduced in the hyperelastic strain energy. It contributes to describe both the anisotropic stress softening and the permanent set. The model is finally extended to soft tissues’ mechanical behavior that present also stress softening but with an initially anisotropic behavior. The two models are successfully fitted and compared to experimental data.


Constitutive Equation Constitutive Model Silicone Rubber Strain Energy Density Evolution Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Mullins L.: Effect of stretching on the properties of rubber. Rubber Chem. Technol. 21, 281–300 (1948)CrossRefGoogle Scholar
  2. 2.
    Mullins L.: Softening of rubber by deformation. Rubber Chem. Technol. 42, 339–362 (1969)CrossRefGoogle Scholar
  3. 3.
    Gurtin M.E., Francis E.C.: Simple rate-independent model for damage. J. Spacecraft 18, 285–286 (1981)CrossRefGoogle Scholar
  4. 4.
    Simo J.C.: On a fully three-dimensional finite-strain viscoelastic damage model:Formulation and computational aspects. Comput. Meth. Appl. Mech. Eng. 60, 153–173 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Miehe C.: Discontinuous and continuous damage evolution in Ogden type large strain elastic materials. Eur. J. Mech. A/Solids 14, 697–720 (1995)zbMATHGoogle Scholar
  6. 6.
    Ogden, R.W. and Roxburgh, D.G.: An energy based model of the Mullins effect. In: Dorfmann, Muhr, (ed.) Constitutive Models for Rubber. I. A. A. Balkema (1999)Google Scholar
  7. 7.
    Ogden, R.W.: Mechanics of Rubberlike Solids. In: XXI ICTAM, Warsaw, Poland (2004)Google Scholar
  8. 8.
    Diani J., Brieu M., Vacherand J.M.: A damage directional constitutive model for the Mullins effect with permanentset and induced anisotropy. Eur. J. Mech. A/Solids 25, 483–496 (2006)CrossRefzbMATHGoogle Scholar
  9. 9.
    Merckel Y., Diani J., Roux S., Brieu M.: A simple framework for full-network hyperelasticity and anisotropic damage. J. Mech. Phys. Solids 59, 75–88 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Laraba-Abbes F., Ienny P., Piques R.: A new Taylor-made methodology for the mechanical behaviour analysis of rubber like materials: II. Application of the hyperelastic behaviour characterization of a carbon-black filled natural rubber vulcanizate. Polymer 44, 821–840 (2003)CrossRefGoogle Scholar
  11. 11.
    Itskov M., Haberstroh E., Ehret A.E., Vohringer M.C.: Experimental observation of the deformation induced anisotropy of the Mullins effect in rubber. KGK-Kautschuk Gummi Kunststoffe 59, 93–96 (2006)Google Scholar
  12. 12.
    Machado G., Favier D., Chagnon G.: Determination of membrane stress-strain full fields of bulge tests from SDIC measurements. Theory, validation and experimental results on a silicone elastomer. Exp. Mech. 52, 865–880 (2012)CrossRefGoogle Scholar
  13. 13.
    Merckel Y., Brieu M., Diani J., Caillard J.: A Mullins softening criterion for general loading conditions. J. Mech. Phys. Solids 60, 1257–1264 (2012)CrossRefGoogle Scholar
  14. 14.
    Dorfmann A., Pancheri F.: A constitutive model for the Mullins effect with changes in material symmetry. Int. J. Nonlinear Mech 47, 874–887 (2012)CrossRefGoogle Scholar
  15. 15.
    Mooney M.: A theory of large elastic deformation. J. Appl. Phys. 11, 582–592 (1940)CrossRefzbMATHGoogle Scholar
  16. 16.
    Treloar, L.R.G.: The elasticity of a network of long chain molecules (I and II). Trans. Faraday Soc. 39:36–64; 241–246 (1943)Google Scholar
  17. 17.
    Ogden R.W.: Large deformation isotropic elasticity—on the correlation of theory and experiment for incompressible rubber like solids. Proc. R. Soc. Lond. A 326, 565–584 (1972)CrossRefzbMATHGoogle Scholar
  18. 18.
    Haines D.W., Wilson D.W.: Strain energy density function for rubber like materials. J. Mech. Phys. Solids 27, 345–360 (1979)CrossRefzbMATHGoogle Scholar
  19. 19.
    Gent A.N.: A new constitutive relation for rubber. Rubber Chem. Technol. 69, 59–61 (1996)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Dorfmann A., Ogden R.W.: A constitutive model for the Mullins effect with permanent set in particule-reinforced rubber. Int. J. Solids Struct. 41, 1855–1878 (2004)CrossRefzbMATHGoogle Scholar
  21. 21.
    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
  22. 22.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Miehe C., Göktepe S.: A micro-macro approach to rubber-like materials. Part II: The micro-sphere model of finite rubber viscoelasticity. J. Mech. Phys. Solids 53, 2231–2258 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Göktepe S., Miehe C.: A micro-macro approach to rubber-like materials. Part III: The micro-sphere model of anisotropic mullins-type damage. J. Mech. Phys. Solids 53, 2259–2283 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Shariff M.H.B.M.: An anisotropic model of the Mullins effect. J. Eng. Math. 56, 415–435 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Rebouah M., Machado G., Chagnon G., Favier D.: Anisotropic Mullins stress softening of a deformed silicone holey plate. Mech. Res. Commun. 49, 36–43 (2013)CrossRefGoogle Scholar
  27. 27.
    Gillibert J., Brieu M., Diani J.: Anisotropy of direction-based constitutive models for rubber-like materials. Int. J. Solids Struct. 47, 640–646 (2010)CrossRefzbMATHGoogle Scholar
  28. 28.
    Ehret A.E., Itskov M., Schmid H.: Numerical integration on the sphere and its effect on the material symmetry of constitutive equations- a comparative study. Int. J. Numer. Meth. Eng. 81, 189–206 (2010)zbMATHGoogle Scholar
  29. 29.
    Rickaby S.R., Scott N.H.: A model for the Mullins effect during multicyclic equibiaxial loading. Acta mech. 224,1887–1900 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Itskov M., Ehret A., Kazakeviciute-Makovska R., Weinhold G.: A thermodynamically consistent phenomenological model of the anisotropic Mullins effect. ZAMM J. Appl. Math. Mech. 90, 370–386 (2010)CrossRefzbMATHGoogle Scholar
  31. 31.
    Merckel Y., Diani J., Brieu M., Caillard J.: Constitutive modeling of the anisotropic behavior of Mullins softened filled rubbers. Mech. Mater. 57, 30–41 (2013)CrossRefGoogle Scholar
  32. 32.
    Lanir Y.: A structural theory for the homogeneous biaxial stress-strain relationshipin flat collagenous tissues. J. Biomech. 12, 423–436 (1979)CrossRefGoogle Scholar
  33. 33.
    Lanir Y.: Constitutive equations for fibous connective tissues. J. Biomech. 16, 1–12 (1983)CrossRefGoogle Scholar
  34. 34.
    Fung Y.C.: Biomechanics, Mechanical Properties of Living Tissues. Springer, New York (1993)Google Scholar
  35. 35.
    Holzapfel G.A.: Nonlinear Solid Mechanics—A Continuum Approach for Engineering. Wiley, NY (1993)Google Scholar
  36. 36.
    Vande Geest J.P., Sacks M.S., Vorp D.A.: The effects of aneurysm on the biaxial mechanical behavior of human abdominal aorta. J. Biomech. 39, 1324–1334 (2006)CrossRefGoogle Scholar
  37. 37.
    Maher E., Creane A., Lally C., Kelly D.J.: An anisotropic inelastic constitutive model to describe stress softening and permanent deformation in arterial tissue. J. Mech. Behav. Biomed. Mater. 12, 9–19 (2012)CrossRefGoogle Scholar
  38. 38.
    Alastrué V., Peña E., Martinez M.A., Doblaré M.: Experimental study and constitutive modelling of the passive mechanical properties of the ovine infrarenal vena cava tissue. J. Biomech. 41, 3038–3045 (2008)CrossRefGoogle Scholar
  39. 39.
    Peña E., Calvo B., Martinez M.A., Martins P., Mascarenhas T., Jorge R.M.N., Ferreira A., Doblaré M.: Experimental study and constitutive modeling of the viscoelastic mechanical properties of the human prolapsed vaginal tissue. Biomech. Model. Mechanobiol. 9, 35–44 (2010)CrossRefGoogle Scholar
  40. 40.
    Natali A.N., Carniel E.L., Gregersen H.: Biomechanical behaviour of oesophageal tissues: Material and structural configuration, experimental data and constitutive analysis. Med. Eng. Phys. 31, 1056–1062 (2009)CrossRefGoogle Scholar
  41. 41.
    Franceschini G., Bigoni D., Regitnig P., Holzapfel G.A.: Brain tissue deforms similarly to filled elastomers and follows consolidation theory. J. Mech. Phys. Solids 54, 2592–2620 (2006)CrossRefzbMATHGoogle Scholar
  42. 42.
    Horgan C.O., Saccomandi G.: A new constitutive theory for fiber-reinforced incompressible nonlinearly elastic solids. J. Mech. Phys. Solids 53, 1985–2015 (2005)CrossRefMathSciNetGoogle Scholar
  43. 43.
    Alastrué V., Martinez M.A., Doblaré M., Menzel A.: Anisotropic microsphere-based finite elasticity applied to blood vessel modelling. J. Mech. Phys. Solids 57, 178–203 (2009)CrossRefzbMATHGoogle Scholar
  44. 44.
    Balzani D., Neff P., Schroder J., Holzapfel G.A.: A polyconvex framework for soft biological tissues. Adjustement to experimental data. Int. J. Solids Struct. 43, 6052–6070 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  45. 45.
    Nerurkar N.L., Mauck R.L., Elliott D.M.: Modeling interlamellar interactions in angle-ply biologic laminates for annulus fibrosus tissue engineering. Biomech. Model. Mechanobiol. 10, 973–984 (2011)CrossRefGoogle Scholar
  46. 46.
    Calvo B., Peña E., Martinez M.A., Doblaré M.: An uncoupled directional damage model for fibred biological soft tissues. formulation and computational aspects. Int. J. Numer. Methods Eng. 69, 2036–2057 (2007)CrossRefzbMATHGoogle Scholar
  47. 47.
    Caner F.C., Carol I.: Microplane constitutive model and computational framework for blood vessel tissue. J. Biomech. Eng. 128, 419–427 (2006)CrossRefGoogle Scholar
  48. 48.
    Driessen, N.J.B., B.C.V.C. and Baaiens, F.T.ens, F.P.T.: A structural constitutive model for collagenous cardiovascular tissues incorporating the angular fiber distribution. J. Biomech. Eng. 127, 494–503 (2005)Google Scholar
  49. 49.
    Peña E., Martins P., Mascarenhasd T., Natal Jorge R.M., Ferreirae A., Doblaré M., Calvo B.: Mechanical characterization of the softening behavior of human vaginal tissue. J. Mech. Beh. Biomed. Mater. 4, 275–283 (2011)CrossRefGoogle Scholar
  50. 50.
    Peña E., Doblaré M.: An anisotropic pseudo-elastic approach for modelling Mullins effect in fibrous biological materials. Mech. Res. Comm. 36, 784–790 (2009)CrossRefzbMATHGoogle Scholar
  51. 51.
    Machado G., Chagnon G., Favier D.: Induced anisotropy by the Mullins effect in filled silicone rubber. Mech. Mater. 50, 70–80 (2012)CrossRefGoogle Scholar
  52. 52.
    Kaliske M.: A formulation of elasticity and viscoelasticity for fibre reinforced material at small and finite strains. Comput. Methods Appl. Mech. Eng. 185, 225–243 (2000)CrossRefzbMATHGoogle Scholar
  53. 53.
    Govindjee S., Simo J.C.: Mullins’ effect and the strain amplitude dependence of the storage modulus. Int. J. Solids. Struct. 29, 1737–1751 (1992)CrossRefzbMATHGoogle Scholar
  54. 54.
    Bazant Z.P., Oh B.H.: Efficient numerical integration on the surface of a sphere. Z. Angew. Math. Mech. 66, 37–49 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  55. 55.
    Zuñiga A.E., Beatty M.F.: A new phenomenological model for stress-softening in elastomers. Z. Angew. Math. Mech. 53, 794–814 (2002)CrossRefzbMATHGoogle Scholar
  56. 56.
    Coleman B.D., Gurtin M.E.: Thermodynamics with internal state variables. J. Chem. Phys. 47, 597–613 (1967)CrossRefGoogle Scholar
  57. 57.
    Schröder J., Neff P., Balzani D.: A variational approach for materially stable anisotropic hyperelasticity. Int. J. Solids. Struct. 42, 4352–4371 (2005)CrossRefzbMATHGoogle Scholar
  58. 58.
    Li D., Robertson A.M.: A structural multi-mechanism constitutive equation for cerebral arterial tissue. Int. J. Solids Struct. 46, 2920–2928 (2009)CrossRefzbMATHGoogle Scholar
  59. 59.
    Ehret A.E., Itskov M.: Modeling of anisotropic softening phenomena: Application to soft biological tissues. Int. J. Plast. 25, 901–919 (2009)CrossRefzbMATHGoogle Scholar
  60. 60.
    Peña E.: Prediction of the softening and damage effects with permanent set in fibrous biological materials. J. Mech. Phys. Solids 59, 1808–1822 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  61. 61.
    Demiray H.: A note on the elasticity of soft biological tissues. J. Biomech. 5, 309–311 (1972)CrossRefGoogle Scholar
  62. 62.
    Delfino A., Stergiopulos N., Moore J.E. Jr, Meister J.J.: Residual strain effects on the stress field in a thick wall finite element model of the human carotid bifurcation. J. Biomech. 30, 777–786 (1997)CrossRefGoogle Scholar
  63. 63.
    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)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Wien 2013

Authors and Affiliations

  1. 1.Universite de Grenoble/CNRS/TIMC-IMAG UMR 5525GrenobleFrance

Personalised recommendations