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


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.

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  1. 1

    Mullins L.: Effect of stretching on the properties of rubber. Rubber Chem. Technol. 21, 281–300 (1948)

    Article  Google Scholar 

  2. 2

    Mullins L.: Softening of rubber by deformation. Rubber Chem. Technol. 42, 339–362 (1969)

    Article  Google Scholar 

  3. 3

    Gurtin M.E., Francis E.C.: Simple rate-independent model for damage. J. Spacecraft 18, 285–286 (1981)

    Article  Google 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)

    Article  MATH  MathSciNet  Google 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)

    MATH  Google 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)

  7. 7

    Ogden, R.W.: Mechanics of Rubberlike Solids. In: XXI ICTAM, Warsaw, Poland (2004)

  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)

    Article  MATH  Google 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)

    Article  MATH  MathSciNet  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google Scholar 

  15. 15

    Mooney M.: A theory of large elastic deformation. J. Appl. Phys. 11, 582–592 (1940)

    Article  MATH  Google 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)

    Article  MATH  Google 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)

    Article  MATH  Google Scholar 

  19. 19

    Gent A.N.: A new constitutive relation for rubber. Rubber Chem. Technol. 69, 59–61 (1996)

    Article  MathSciNet  Google 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)

    Article  MATH  Google 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)

    Article  Google 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)

    Article  MATH  MathSciNet  Google 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)

    Article  MATH  MathSciNet  Google 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)

    Article  MATH  MathSciNet  Google Scholar 

  25. 25

    Shariff M.H.B.M.: An anisotropic model of the Mullins effect. J. Eng. Math. 56, 415–435 (2006)

    Article  MATH  MathSciNet  Google 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)

    Article  Google 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)

    Article  MATH  Google 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)

    MATH  Google 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)

    Article  MATH  MathSciNet  Google 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)

    Article  MATH  Google 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)

    Article  Google Scholar 

  32. 32

    Lanir Y.: A structural theory for the homogeneous biaxial stress-strain relationshipin flat collagenous tissues. J. Biomech. 12, 423–436 (1979)

    Article  Google Scholar 

  33. 33

    Lanir Y.: Constitutive equations for fibous connective tissues. J. Biomech. 16, 1–12 (1983)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  MATH  Google 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)

    Article  MathSciNet  Google 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)

    Article  MATH  Google 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)

    Article  MATH  MathSciNet  Google 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)

    Article  Google 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)

    Article  MATH  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  MATH  Google 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)

    Article  Google 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)

    Article  MATH  Google 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)

    Article  MATH  Google 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)

    Article  MATH  MathSciNet  Google 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)

    Article  MATH  Google Scholar 

  56. 56

    Coleman B.D., Gurtin M.E.: Thermodynamics with internal state variables. J. Chem. Phys. 47, 597–613 (1967)

    Article  Google 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)

    Article  MATH  Google 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)

    Article  MATH  Google 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)

    Article  MATH  Google 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)

    Article  MATH  MathSciNet  Google Scholar 

  61. 61

    Demiray H.: A note on the elasticity of soft biological tissues. J. Biomech. 5, 309–311 (1972)

    Article  Google 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)

    Article  Google 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)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to G. Chagnon.

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Rebouah, M., Chagnon, G. Permanent set and stress-softening constitutive equation applied to rubber-like materials and soft tissues. Acta Mech 225, 1685–1698 (2014).

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  • Constitutive Equation
  • Constitutive Model
  • Silicone Rubber
  • Strain Energy Density
  • Evolution Function