The Extended Non-affine Tube Model for Crosslinked Polymer Networks: Physical Basics, Implementation, and Application to Thermomechanical Finite Element Analyses

  • Ronny Behnke
  • Michael Kaliske
Part of the Advances in Polymer Science book series (POLYMER, volume 275)


This chapter is devoted to a summary of the so-called extended non-affine tube model. First, general model approaches for representation of the behavior of elastomers within numerical simulations are discussed. Second, the extended non-affine tube model is considered in the context of hyperelastic material models. Starting from molecular-statistical considerations, a Helmholtz free energy function is derived and formulated in terms of continuum mechanical quantities of the macroscale. Furthermore, combination with a model approach to represent continuum damage and time-dependent effects is addressed. The free energy function of the model approach is further set into the context of thermomechanics to account for temperature-dependent behavior of elastomers within numerical simulations. Finally, finite element implementation of the extended non-affine tube model and its application to uniaxial and biaxial tension tests performed on elastomer specimens are presented.


Continuum damage Elastomers Finite element analysis Hyperelasticity Temperature dependency Tube model Viscoelasticity 



Some of the research results reported herein are partially supported by the Deutsche Forschungsgemeinschaft (DFG) under grant KA 1163/16. The financial support is gratefully acknowledged. The authors thank J.-B. Le Cam (IFMA, France) and his research group for their experimental-numerical cooperation.


  1. 1.
    Behnke R, Dal H, Geißler G, Näser B, Netzker C, Kaliske M (2013) Macroscopical modeling and numerical simulation for the characterization of crack and durability properties of particle-reinforced elastomers. In: Grellmann W, Heinrich G, Kaliske M, Klüppel M, Schneider K, Vilgis T (eds) Fracture mechanics and statistical mechanics of reinforced elastomeric blends. Lecture notes in applied and computational mechanics, vol 70. Springer, Berlin, pp 167–226Google Scholar
  2. 2.
    Kaliske M, Behnke R (2015) Material laws of rubbers. In: Kobayashi S, Müllen K (eds) Encyclopedia of polymeric nanomaterials. Springer, Berlin, pp 1187–1197CrossRefGoogle Scholar
  3. 3.
    Grambow A (2002) Determination of material parameters for filled rubber depending on time, temperature and loading condition. Ph.D. Thesis, Rheinisch-Westfälische Technische Hochschule Aachen, GermanyGoogle Scholar
  4. 4.
    Heinrich G, Straube E, Helmis G (1988) Rubber elasticity of polymer networks: theories. Adv Polym Sci 85:33–87CrossRefGoogle Scholar
  5. 5.
    Edwards S, Vilgis T (1988) The tube model theory of rubber elasticity. Rep Prog Phys 51:243–297CrossRefGoogle Scholar
  6. 6.
    Valanis K, Landel R (1967) The strain-energy function of a hyperelastic material in terms of the extension ratios. J Appl Phys 38:2997–3002CrossRefGoogle Scholar
  7. 7.
    Heinrich G, Kaliske M (1997) Theoretical and numerical formulation of a molecular based constitutive tube-model of rubber elasticity. Comput Theor Polym Sci 7:227–241CrossRefGoogle Scholar
  8. 8.
    Kaliske M, Heinrich G (1999) An extended tube-model for rubber elasticity: statistical-mechanical theory and finite element implementation. Rubber Chem Technol 72:602–632CrossRefGoogle Scholar
  9. 9.
    Bergström J (2015) Mechanics of solid polymers. Theory and computational modeling. Elsevier, San DiegoGoogle Scholar
  10. 10.
    Klüppel M, Schramm J (2000) A generalized tube model of rubber elasticity and stress softening of filler reinforced elastomer systems. Macromol Theory Simul 9:742–754CrossRefGoogle Scholar
  11. 11.
    Klüppel M (2003) Hyperelasticity and stress softening of filler reinforced polymer networks. Macromol Symp Funct Netw Gels 200:31–44CrossRefGoogle Scholar
  12. 12.
    Heinrich G, Straube E (1987) A theory of topological constraints in polymer networks. Polym Bull 17:247–253Google Scholar
  13. 13.
    Heinrich G, Straube E (1984) On the strength and deformation dependence of tube-like topological constraints in polymer networks, melts and concentrated solutions I. The polymer network case. Acta Polym 34:589–594CrossRefGoogle Scholar
  14. 14.
    Heinrich G, Straube E (1984) On the strength and deformation dependence of tube-like topological constraints in polymer networks, melts and concentrated solutions II. Polymer melts and concentrated solutions. Acta Polym 35:115–119CrossRefGoogle Scholar
  15. 15.
    Edwards S (1965) The statistical mechanics of polymers with excluded volume. Proc Phys Soc 85:613–624CrossRefGoogle Scholar
  16. 16.
    Edwards S, Vilgis T (1986) The effect of entanglements in rubber elasticity. Polymer 27:483–492CrossRefGoogle Scholar
  17. 17.
    Freed K (1972) Functional integrals and polymer statistics. Adv Chem Phys 22:1–128Google Scholar
  18. 18.
    Treloar L (1975) The physics of rubber elasticity. Clarendon, OxfordGoogle Scholar
  19. 19.
    Arruda E, Boyce M (1993) A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J Mech Phys Solids 41:389–412CrossRefGoogle Scholar
  20. 20.
    Marckmann G, Verron E (2006) Comparison of hyperelastic models for rubberlike materials. Rubber Chem Technol 79:835–858CrossRefGoogle Scholar
  21. 21.
    Ogden R (1972) Large deformation isotropic elasticity – on the correlation of theory and experiment for incompressible rubberlike solids. Proc R Soc A 326:565–584CrossRefGoogle Scholar
  22. 22.
    Miehe C, Göktepe S, Lulei F (2004) A micro-macro approach to rubber-like materials. Part I: The non-affine micro-sphere model of rubber elasticity. J Mech Phys Solids 52:2617–2660CrossRefGoogle Scholar
  23. 23.
    Drozdov A, Dorfmann A (2002) Finite viscoelasticity of filled rubbers: the effects of preloading and thermal recovery. Continuum Mech Thermodyn 14:337–361CrossRefGoogle Scholar
  24. 24.
    Green M, Tobolsky A (1946) A new approach to the theory of relaxing polymeric media. J Chem Phys 14:80–92CrossRefGoogle Scholar
  25. 25.
    Tanaka F, Edwards S (1992) Viscoelastic properties of physically crosslinked networks. Part 1. Non-linear stationary viscoelasticity. J Non-Newton Fluid Mech 43:247–271CrossRefGoogle Scholar
  26. 26.
    Tanaka F, Edwards S (1992) Viscoelastic properties of physically crosslinked networks. Part 2. Dynamic mechanical moduli. J Non-Newton Fluid Mech 43:273–288CrossRefGoogle Scholar
  27. 27.
    De Gennes P (1971) Reptation of a polymer chain in the presence of fixed obstacles. J Chem Phys 55:572–579CrossRefGoogle Scholar
  28. 28.
    Doi M, Edwards S (1986) The theory of polymer dynamics. Clarendon, OxfordGoogle Scholar
  29. 29.
    Bergström J, Boyce M (1998) Constitutive modeling of the large strain time-dependent behavior of elastomers. J Mech Phys Solids 46:931–954CrossRefGoogle Scholar
  30. 30.
    Dal H, Kaliske M (2009) Bergström-Boyce model for nonlinear finite rubber viscoelasticity: theoretical aspects and algorithmic treatment for the FE method. Comput Mech 44:809–823CrossRefGoogle Scholar
  31. 31.
    Areias P, Matouš K (2008) Finite element formulation for modeling nonlinear viscoelastic elastomers. Comput Methods Appl Mech Eng 197:4702–4717CrossRefGoogle Scholar
  32. 32.
    Miehe C, Göktepe S (2005) A micro-macro approach to rubber-like materials. Part II: The micro-sphere model of finite rubber viscoelasticity. J Mech Phys Solids 53:2231–2258CrossRefGoogle Scholar
  33. 33.
    Wineman A (2009) Nonlinear viscoelastic solids – a review. Math Mech Solids 14:300–366CrossRefGoogle Scholar
  34. 34.
    Freund M, Lorenz H, Juhre D, Ihlemann J, Klüppel M (2011) Finite element implementation of a microstructure-based model for filled elastomers. Int J Plast 27:902–919CrossRefGoogle Scholar
  35. 35.
    Kaliske M (2000) A formulation of elasticity and viscoelasticity for fibre reinforced material at small and finite strains. Comput Methods Appl Mech Eng 185:225–243CrossRefGoogle Scholar
  36. 36.
    Simo J (1987) On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects. Comput Methods Appl Mech Eng 60:153–173CrossRefGoogle Scholar
  37. 37.
    Govindjee S, Simo J (1992) Mullins’ effect and the strain amplitude dependence of the storage modulus. Int J Solids Struct 29:1737–1751CrossRefGoogle Scholar
  38. 38.
    Kaliske M, Rothert H (1997) Formulation and implementation of three-dimensional viscoelasticity at small and finite strains. Comput Mech 19:228–239CrossRefGoogle Scholar
  39. 39.
    Lee E (1969) Elastic-plastic deformation at finite strains. J Appl Mech 36:1–6CrossRefGoogle Scholar
  40. 40.
    Sidoroff F (1974) Un modèle viscoélastique nonlinéaire avec configuration intermédiaire. Journal de Mécanique 13:697–713Google Scholar
  41. 41.
    Lubliner J (1985) A model of rubber viscoelasticity. Mech Res Commun 12:93–99CrossRefGoogle Scholar
  42. 42.
    Simo J (1992) Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory. Comput Methods Appl Mech Eng 99:61–112CrossRefGoogle Scholar
  43. 43.
    Reese S, Govindjee S (1998) A theory of finite viscoelasticity and numerical aspects. Int J Solids Struct 35:3455–3482CrossRefGoogle Scholar
  44. 44.
    Kachanov L (1986) Introduction to continuum damage mechanics. Martinus Nijhoff, DordrechtCrossRefGoogle Scholar
  45. 45.
    Chaboche JL (1981) Continuous damage mechanics – a tool to describe phenomena before crack initiation. Nucl Eng Des 64:233–247CrossRefGoogle Scholar
  46. 46.
    Chaboche JL (1988) Continuum damage mechanics: Part I – General concepts. Part II – Damage growth, crack initiation and crack growth. J Appl Mech 55:59–72CrossRefGoogle Scholar
  47. 47.
    Lemaitre J (1984) How to use damage mechanics. Nucl Eng Des 80:233–245CrossRefGoogle Scholar
  48. 48.
    Chagnon G, Verron E, Gornet L, Marckmann G, Charrier P (2004) On the relevance of continuum damage mechanics as applied to the Mullins effect in elastomers. J Mech Phys Solids 52:1627–1650CrossRefGoogle Scholar
  49. 49.
    Simo J, Ju J (1987) Strain- and stress-based continuum damage models – II. Computational aspects. Int J Solids Struct 23:841–869CrossRefGoogle Scholar
  50. 50.
    Simo J, Ju J (1987) Strain- and stress-based continuum damage models – I. Formulation. Int J Solids Struct 23:821–840CrossRefGoogle Scholar
  51. 51.
    Kaliske M, Nasdala L, Rothert H (2001) On damage modelling for elastic and viscoelastic materials at large strain. Comput Struct 79:2133–2141CrossRefGoogle Scholar
  52. 52.
    Nasdala L, Kaliske M, Rothert H, Becker A (1999) A realistic elastic damage model for rubber. In: Dorfmann A, Muhr A (eds) Constitutive models for rubber. Balkema, Rotterdam, pp 151–158Google Scholar
  53. 53.
    Göktepe S, Miehe C (2005) A micro-macro approach to rubber-like materials. Part III: The micro-sphere model of anisotropic Mullins-type damage. J Mech Phys Solids 53:2259–2283CrossRefGoogle Scholar
  54. 54.
    Flory P (1961) Thermodynamic relations for high elastic materials. Trans Faraday Soc 57:829–838CrossRefGoogle Scholar
  55. 55.
    Simo J, Taylor R, Pister K (1985) Variational and projection methods for the volume constraint in finite deformation elasto-plasticity. Comput Methods Appl Mech Eng 51:177–208CrossRefGoogle Scholar
  56. 56.
    Miehe C (1994) Aspects of the formulation and finite element implementation of large strain isotropic elasticity. Int J Numer Methods Eng 37:1981–2004CrossRefGoogle Scholar
  57. 57.
    Simo J, Taylor R (1982) Penalty function formulations for incompressible nonlinear elastostatics. Comput Methods Appl Mech Eng 35:107–118CrossRefGoogle Scholar
  58. 58.
    Dal H (2011) Approaches to the modeling of inelasticity and failure of rubberlike materials. Theory and numerics. Ph.D. Thesis, TU Dresden, GermanyGoogle Scholar
  59. 59.
    Miehe C (1995) Discontinuous and continuous damage evolution in Ogden-type large-strain elastic materials. Eur J Mech A/Solids 14:697–720Google Scholar
  60. 60.
    Coleman B, Gurtin M (1967) Thermodynamics with internal state variables. J Chem Phys 47:597–613CrossRefGoogle Scholar
  61. 61.
    Bonet J, Wood R (1997) Nonlinear continuum mechanics for finite element analysis. Cambridge University Press, CambridgeGoogle Scholar
  62. 62.
    Weber G, Anand L (1990) Finite deformation constitutive equations and a time integration procedure for isotropic, hyperelastic-viscoplastic solids. Comput Methods Appl Mech Eng 79:173–202CrossRefGoogle Scholar
  63. 63.
    Wriggers P (2001) Nichtlineare Finite-Element-Methoden. Springer, BerlinCrossRefGoogle Scholar
  64. 64.
    Reese S, Govindjee S (1998) Theoretical and numerical aspects in the thermoviscoelastic material behaviour of rubber-like polymers. Mech Time Dependent Mater 1:357–396CrossRefGoogle Scholar
  65. 65.
    Heinrich G (1987) Thermoelasticity of tube-like constrained polymer networks. Acta Polym 38:637–638CrossRefGoogle Scholar
  66. 66.
    Miehe C (1988) Zur numerischen Behandlung thermomechanischer Prozesse. Ph.D. Thesis, Universität Hannover, GermanyGoogle Scholar
  67. 67.
    Miehe C (1995) Entropic thermoelasticity at finite strains. Aspects of the formulation and numerical implementation. Comput Methods Appl Mech Eng 120:243–269CrossRefGoogle Scholar
  68. 68.
    Lion A (1997) On the large deformation behaviour of reinforced rubber at different temperatures. J Mech Phys Solids 45:1805–1834CrossRefGoogle Scholar
  69. 69.
    Lion A (1997) A physically based method to represent the thermo-mechanical behaviour of elastomers. Acta Mech 123:1–25CrossRefGoogle Scholar
  70. 70.
    Bérardi G, Jaeger M, Martin R, Carpentier C (1996) Modelling of a thermoviscoelastic coupling for large deformations through finite element analysis. Int J Heat Mass Transfer 39:3911–3924CrossRefGoogle Scholar
  71. 71.
    Allen G, Bianchi U, Price C (1963) Thermodynamics of elasticity of natural rubber. Trans Faraday Soc 59:2493–2502CrossRefGoogle Scholar
  72. 72.
    Allen G, Kirkham M, Padget J, Price C (1971) Thermodynamics of rubber elasticity at constant volume. Trans Faraday Soc 67:1278–1292CrossRefGoogle Scholar
  73. 73.
    Chadwick P (1974) Thermo-mechanics of rubberlike materials. Philos Trans R Soc Lond Ser A Math Phys Sci 276:371–403CrossRefGoogle Scholar
  74. 74.
    Chadwick P, Creasy C (1984) Modified entropic elasticity of rubberlike materials. J Mech Phys Solids 32:337–357CrossRefGoogle Scholar
  75. 75.
    Haupt P (1993) On the mathematical modelling of material behavior in continuum mechanics. Acta Mech 100:129–154CrossRefGoogle Scholar
  76. 76.
    Morland L, Lee E (1960) Stress analysis for linear viscoelastic materials with temperature variation. Trans Soc Rheol 4:233–263CrossRefGoogle Scholar
  77. 77.
    Holzapfel G, Simo J (1996) A new viscoelastic constitutive model for continuous media at finite thermomechanical changes. Int J Solids Struct 33:3019–3034CrossRefGoogle Scholar
  78. 78.
    Reese S (2003) A micromechanically motivated material model for the thermoviscoelastic material behaviour of rubber-like polymers. Int J Plast 19:909–940CrossRefGoogle Scholar
  79. 79.
    Dippel B, Johlitz M, Lion A (2015) Thermo-mechanical couplings in elastomers – experiments and modelling. Zeitschrift für Angewandte Mathematik und Mechanik 95:1117–1128CrossRefGoogle Scholar
  80. 80.
    Simo J, Miehe C (1992) Associative coupled thermoplasticity at finite strains: formulation, numerical analysis and implementation. Comput Methods Appl Mech Eng 98:41–104CrossRefGoogle Scholar
  81. 81.
    Eterovic A, Bathe KJ (1990) A hyperelastic-based large strain elasto-plastic constitutive formulation with combined isotropic-kinematic hardening using the logarithmic stress and strain measures. Int J Numer Methods Eng 30:1099–1114CrossRefGoogle Scholar
  82. 82.
    Behnke R (2015) Thermo-mechanical modeling and durability analysis of elastomer components under dynamic loading. Ph.D. Thesis, TU Dresden, GermanyGoogle Scholar
  83. 83.
    Felippa C, Park K (1980) Staggered transient analysis procedures for coupled mechanical systems: Formulation. Comput Methods Appl Mech Eng 24:61–111CrossRefGoogle Scholar
  84. 84.
    Armero F, Simo J (1992) A new unconditionally stable fractional step method for non-linear coupled thermomechanical problems. Int J Numer Methods Eng 35:737–766CrossRefGoogle Scholar
  85. 85.
    Bathe KJ (1996) Finite element procedures. Prentice-Hall, New JerseyGoogle Scholar
  86. 86.
    Dal H, Kaliske M (2005) Q1P0-brick element for thermomechanical analysis. Leipzig Annu Civil Eng Rep 10:105–115Google Scholar
  87. 87.
    Zienkiewicz O, Taylor R (2000) The finite element method, vol 2; Solid mechanics, 5 edn. Butterworth-Heinmann, OxfordGoogle Scholar
  88. 88.
    Simo J, Armero F, Taylor R (1993) Improved versions of assumed enhanced strain tri-linear elements for 3D finite deformation problems. Comput Methods Appl Mech Eng 110:359–386CrossRefGoogle Scholar
  89. 89.
    Behnke R, Kaliske M, Klüppel M (2016) Thermo-mechanical analysis of cyclically loaded particle-reinforced elastomer components: Experiment and finite element simulation. Rubber Chem Technol 89:154–176CrossRefGoogle Scholar
  90. 90.
    Behnke R, Dal H, Kaliske M (2011) An extended tube model for thermoviscoelasticity of rubberlike materials: theory and numerical implementation. In: Jerrams S, Murphy N (eds) Constitutive models for rubber VII. CRC, London, pp 87–92CrossRefGoogle Scholar
  91. 91.
    Pottier T, Moutrille MP, Le Cam JB, Balandraud X, Grédiac M (2009) Study on the use of motion compensation techniques to determine heat sources. Application to large deformations on cracked rubber specimens. Exp Mech 49:561–574CrossRefGoogle Scholar
  92. 92.
    Balandraud X, Toussaint E, Le Cam J, Grédiac M, Behnke R, Kaliske M (2011) Application of full-field measurements and numerical simulations to analyze the thermo-mechanical response of a three-branch rubber specimen. In: Jerrams S, Murphy N (eds) Constitutive models for rubber VII. CRC, London, pp 45–50CrossRefGoogle Scholar
  93. 93.
    Guélon T, Toussaint E, Le Cam JB, Promma N, Grédiac M (2009) A new characterisation method for rubber. Polym Test 28:715–723CrossRefGoogle Scholar
  94. 94.
    Sutton M, Wolters W, Peters W, Ranson W, McNeil S (1983) Determination of displacements using an improved digital correlation method. Image Vis Computating 1:133–139CrossRefGoogle Scholar
  95. 95.
    Sasso M, Palmieri G, Chiappini G, Amodio D (2008) Characterization of hyper-elastic rubber-like materials by biaxial and uniaxial stretching tests based on optical methods. Polym Test 27:995–1004CrossRefGoogle Scholar
  96. 96.
    Chevalier L, Calloch S, Hild F, Marco Y (2001) Digital image correlation used to analyze the multiaxial behavior of rubber-like materials. Eur J Mech A/Solids 2:169–187CrossRefGoogle Scholar
  97. 97.
    Behnke R, Kaliske M (2014) Thermo-mechanical finite element analysis of steady state rolling off-the-road tires with respect to thermal damage. In: Proceedings international rubber conference, Beijing, 16–18 Sept 2014, pp 1455–1460Google Scholar
  98. 98.
    Behnke R, Kaliske M (2015) Thermo-mechanically coupled investigation of steady state rolling tires by numerical simulation and experiment. Int J Non Linear Mech 68:101–131CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institut für Statik und Dynamik der TragwerkeTechnische Universität DresdenDresdenGermany

Personalised recommendations