Skip to main content
Log in

Chemo-thermomechanical ageing of elastomers based on multiphase continuum mechanics

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

This article concerns the constitutive representation of one of the chemical ageing processes that occur in elastomers, chemo-thermomechanical ageing, which takes place as an irreversible, time-delayed chemical reaction when a medium diffuses into an unlike solid. This process is inhomogeneous in component parts of finite thickness, and as it can be thermally activated, ageing is accelerated on an increase in temperature. The application of multiphase continuum mechanics to these basic characteristics enables a thermodynamically coupled material model to be formulated, which is able to describe not only the viscoelasticity, but also the chemical decomposition and reformation processes that occur in the polymer network. The evolution principle of Liu-Müller is used to evaluate the thermomechanical consistency of the model obtained. Subsequent to this, the finite element method is applied to solve the resulting set of partial equations, which corresponds to a coupled multifield problem. The article closes with convincing simulations of illustrative examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Andrews R., Tobolsky A., Hanson E.: The theory of permanent set at elevated temperatures in natural and synthetic rubber vulcanizates. J. Appl. Phys. 17, 352–361 (1946)

    Article  ADS  Google Scholar 

  2. Atkin R.J., Craine R.E.: Continuum theories of mixtures: basic theory and historical developement. Q.J. Mechanics Appl. Math. 29(2), 209–244 (1976)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Audouin L., Langlois V., Verdu J., de Bruijn J.: Review: role of oxygen diffusion in polymer ageing: kinetic and mechanical aspects. J. Mater. Sci. 29, 569–583 (1994)

    Article  ADS  Google Scholar 

  4. Bathe K.J.: Finite-Elemente-Methoden. Springer, Berlin (1990)

    Google Scholar 

  5. Becker G., Braun D.: Kunststoff Handbuch 1, Die Kunststoffe-Chemie, Physik, Technologie. Carl Hanser Verlag, München (1996)

    Google Scholar 

  6. Blum G., Shelton J., Winn H.: Rubber oxidation and ageing studies. Ind. Eng. Chem. 43, 464–471 (1951)

    Article  Google Scholar 

  7. Bowen, R.M.: Theory of mixture. Continuum Physics III, 1–127 (1976)

  8. Bowen R.M.: Incompressible porous media models by use of the theory of mixtures. Int. J. Eng. Sci. 18, 1129–1148 (1980)

    Article  MATH  Google Scholar 

  9. Bowen R.M.: Compressible porous media models by use of the theory of mixtures. Int. J. Eng. Sci. 20, 697–735 (1982)

    Article  MATH  Google Scholar 

  10. Budzien J., Rottach D., Curro J., Lo C., Thompson A.: A new constitutive model for the chemical ageing of rubber networks in deformed states. Macromolecules 41, 9896–9903 (2008)

    Article  ADS  Google Scholar 

  11. de Boer R., Ehlers W.: On the problem of fluid- and gas-filled elasto-plastic solids. Int. J. Solids Struct. 22, 1231–1242 (1986)

    Article  MATH  Google Scholar 

  12. Duarte J., Achenbach M.: On the modelling of rubber ageing and performance changes in rubbery components. Kaut. Gummi Kunstst. 60, 172–175 (2007)

    Google Scholar 

  13. Dunn J., Scalan J., Watson W.: Stress relaxation during the thermal oxidation of vulcanized natural rubber. Trans. Faraday Soc. 55, 667–675 (1959)

    Article  Google Scholar 

  14. Dunwoody R.: A thermomechanical theory of diffusion in solid-fluid mixtures. Arch. Ration. Mech. Anal. 38, 348–371 (1970)

    MathSciNet  MATH  Google Scholar 

  15. Ehlers, W.: Compressible, incompressible and hybrid two-phase models in porous media theories. In: Angel, Y. (ed.) Anisotropy and Inhomogenity in Elasticity nad Plasticity, AMD(ASME), vol. 158, pp. 25–38. Springer, Berlin (1993)

  16. Ehlers W.: Foundations of multiphasic and porous materials. In: Ehlers, W., Bluhm, J. (eds.) Porous Media: Theory, Experiments and Numerical Applications, pp. 3–86. Springer, Berlin (2002)

    Google Scholar 

  17. Ehlers W., Ellsiepen P.: PANDAS: Ein FE-System zur Simulation von Sonderproblemen der Bodenmechanik. In: Wriggers, P., Meißner, U., Stein, E., Wunderlich, W. (eds.) Finite Elemente in der Baupraxis: Modellierung, Berechnung und Konstruktion, Beiträge zur Tagung FEM ’98 an der TU Darmstadt am 5. und 6. März 1998, pp. 400–431. Ernst & Sohn, Berlin (1998)

    Google Scholar 

  18. Ehlers W., Kubik J.: On finite dynamic equations for fluid-saturated porous media. Acta Mech. 105, 101–117 (1993)

    Article  MathSciNet  Google Scholar 

  19. Ehrenstein G., Pongratz S.: Beständigkeit von Kunststoffen. Carl Hanser Verlag, Munich (2007)

    Book  Google Scholar 

  20. Greiner R., Schwarzl F.: Thermal contraction and volume relaxation of amorphous polymers. Rheol. Acta 23, 378–395 (1984)

    Article  Google Scholar 

  21. Greve R.: Kontinuumsmechanik. Springer, Berlin (2003)

    Book  MATH  Google Scholar 

  22. Haupt P.: Continuum Mechanics and Theory of Materials. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  23. Hossain M., Possart G., Steinmann P.: A small-strain model to simulate the curing of thermosets. Comput. Mech. 43, 769–779 (2008)

    Article  Google Scholar 

  24. Hossain M., Possart G., Steinmann P.: A finite strain framework for the simulation of polymer curing part I: elasticity. Comput. Mech. 44, 621–630 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hutchinson J.: Physical ageing of polymers. Prog. Polym. Sci. 20, 703–760 (1995)

    Article  Google Scholar 

  26. Hutter K., Jöhnk K.: Continuum Methods of Physical Modeling. Springer, Berlin (2004)

    MATH  Google Scholar 

  27. Johlitz, M., Retka, J., Lion, A.: Chemical ageing of elastomers: experiments and modelling. In: Jerrams, S., Murphy, N. (eds.) Constitutive Models for Rubber VII, pp. 113–118 (2011)

  28. Kuhl D.: Modellierung und Simulation von Mehrfeldproblemen der Strukturmechanik. Shaker Verlag, Aachen (2005)

    Google Scholar 

  29. Lion A., Johlitz M.: On the representation of chemical ageing of rubber in continuum mechanics. Int. J. Solids Struct. 49(10), 1227–1240 (2012)

    Article  MathSciNet  Google Scholar 

  30. Liu I.S.: Method of Lagrangian multipliers for exploitation of the entropy principle. Arch. Ration. Mech. Anal. 46, 131–148 (1972)

    MATH  Google Scholar 

  31. Lustig S., Caruthers J., Peppas N.: Continuum thermodynamics and transport theory for polymer-fluid mixtures. Chem. Eng. Sci. 47, 3037–3057 (1992)

    Article  Google Scholar 

  32. Müller I.: A thermodynamic theory of mixtures of fluids. Arch. Ration. Mech. Anal. 28, 1–39 (1968)

    Article  MATH  Google Scholar 

  33. Müller I.: Thermodynamik, Die Grundlagen der Materialtheorie. Bertelsmann Universitätsverlag, Düsseldorf (1973)

    MATH  Google Scholar 

  34. Müller I.: Thermodynamics. Pitman, Boston (1985)

    MATH  Google Scholar 

  35. Ore S.: A modification of the method of intermittent stress relaxation measurements on rubber vulcanisates. J. Appl. Polym. Sci. 2, 318–321 (1959)

    Article  Google Scholar 

  36. Pochiraju K., Tandon G.: Modeling thermo-oxidative layer growth in high-temperature resins. J. Eng. Mater. Technol. ASME 128, 107–116 (2006)

    Article  Google Scholar 

  37. Quang N., Samohýl I., Thoang H.: Irreversible (rational) thermodynamics of mixtures of a solid substance with chemical reacting fluids. Collect. Czech. Chem. Commun. 53, 1620–1635 (1988)

    Article  Google Scholar 

  38. Samohýl I., Šípek X.N.M.: Irreversible (rational) thermodynamics of fluid-solid mixtures. Collect. Czech. Chem. Commun. 50, 2346–2363 (1985)

    Article  Google Scholar 

  39. Scalan J., Watson W.: The interpretation of stress relaxation measurements made on rubber during ageing. Trans. Faraday Soc. 54, 740–750 (1957)

    Article  Google Scholar 

  40. Shaw J., Jones S., Wineman A.: Chemorheological response of elastomers at elevated temperatures: experiments and simulations. J. Mech. Phys. Solids 53, 2758–2793 (2005)

    Article  ADS  MATH  Google Scholar 

  41. Smith L.: The Language of Rubber: An Introduction to the Specification and Testing of Elastomers. Butterworth-Heinemann publication house, Oxford (1993)

    Google Scholar 

  42. Steinke L., Spreckels J., Flamm M., Celina M.: Model for heterogeneous aging of rubber products. Plast. Rubber Compost. 40, 175–179 (2011)

    Article  Google Scholar 

  43. Steinke, L., Veltin, U., Flamm, M., Lion, A., Celina, M.: Numerical analysis of the heterogeneous ageing of rubber products. In: Jerrams, S., Murphy, N. (eds.) Constitutive Models for Rubber VII, pp. 155–160 (2011b)

  44. Tobolsky A.V.: Mechanische Eigenschaften und Struktur von Polymeren. Berliner Union, Stuttgart (1967)

    Google Scholar 

  45. Tobolsky A.V., Prettyman I.B., Dillon J.H.: Stress relaxation of natural and synthetic rubber stocks. J. Appl. Phys. 15, 380–395 (1944)

    Article  ADS  Google Scholar 

  46. Truesdell, C.: Sulle basi delle termomeccanica. Rend. Lincei. 22, 33–38, 158–166 (1957)

    Google Scholar 

  47. Truesdell, C.A., Toupin, R.: The classical field theories. In: Flügge, S. (Herausgeber). Handbuch der Physik III/1, Springer, Berlin (1960)

  48. Woltman R.: Beiträge zur hydraulischen Archtektur, vol. 3. Johann Christian Dietrich, Göttingen (1794)

    Google Scholar 

  49. Ziegler, C., Mehling, V., Baaser, H., Häusler, O.: Simulation of relaxation and set in elastomer components using the multi-axial freudenberg-ageing model. In: Proceedings of the International Rubber Conference Nürnberg (2009)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Michael Johlitz.

Additional information

Communicated by Andreas Öchsner.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Johlitz, M., Lion, A. Chemo-thermomechanical ageing of elastomers based on multiphase continuum mechanics. Continuum Mech. Thermodyn. 25, 605–624 (2013). https://doi.org/10.1007/s00161-012-0255-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-012-0255-8

Keywords

Navigation