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.
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References
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)
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)
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)
Bathe K.J.: Finite-Elemente-Methoden. Springer, Berlin (1990)
Becker G., Braun D.: Kunststoff Handbuch 1, Die Kunststoffe-Chemie, Physik, Technologie. Carl Hanser Verlag, München (1996)
Blum G., Shelton J., Winn H.: Rubber oxidation and ageing studies. Ind. Eng. Chem. 43, 464–471 (1951)
Bowen, R.M.: Theory of mixture. Continuum Physics III, 1–127 (1976)
Bowen R.M.: Incompressible porous media models by use of the theory of mixtures. Int. J. Eng. Sci. 18, 1129–1148 (1980)
Bowen R.M.: Compressible porous media models by use of the theory of mixtures. Int. J. Eng. Sci. 20, 697–735 (1982)
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)
de Boer R., Ehlers W.: On the problem of fluid- and gas-filled elasto-plastic solids. Int. J. Solids Struct. 22, 1231–1242 (1986)
Duarte J., Achenbach M.: On the modelling of rubber ageing and performance changes in rubbery components. Kaut. Gummi Kunstst. 60, 172–175 (2007)
Dunn J., Scalan J., Watson W.: Stress relaxation during the thermal oxidation of vulcanized natural rubber. Trans. Faraday Soc. 55, 667–675 (1959)
Dunwoody R.: A thermomechanical theory of diffusion in solid-fluid mixtures. Arch. Ration. Mech. Anal. 38, 348–371 (1970)
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)
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)
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)
Ehlers W., Kubik J.: On finite dynamic equations for fluid-saturated porous media. Acta Mech. 105, 101–117 (1993)
Ehrenstein G., Pongratz S.: Beständigkeit von Kunststoffen. Carl Hanser Verlag, Munich (2007)
Greiner R., Schwarzl F.: Thermal contraction and volume relaxation of amorphous polymers. Rheol. Acta 23, 378–395 (1984)
Greve R.: Kontinuumsmechanik. Springer, Berlin (2003)
Haupt P.: Continuum Mechanics and Theory of Materials. Springer, Berlin (2000)
Hossain M., Possart G., Steinmann P.: A small-strain model to simulate the curing of thermosets. Comput. Mech. 43, 769–779 (2008)
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)
Hutchinson J.: Physical ageing of polymers. Prog. Polym. Sci. 20, 703–760 (1995)
Hutter K., Jöhnk K.: Continuum Methods of Physical Modeling. Springer, Berlin (2004)
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)
Kuhl D.: Modellierung und Simulation von Mehrfeldproblemen der Strukturmechanik. Shaker Verlag, Aachen (2005)
Lion A., Johlitz M.: On the representation of chemical ageing of rubber in continuum mechanics. Int. J. Solids Struct. 49(10), 1227–1240 (2012)
Liu I.S.: Method of Lagrangian multipliers for exploitation of the entropy principle. Arch. Ration. Mech. Anal. 46, 131–148 (1972)
Lustig S., Caruthers J., Peppas N.: Continuum thermodynamics and transport theory for polymer-fluid mixtures. Chem. Eng. Sci. 47, 3037–3057 (1992)
Müller I.: A thermodynamic theory of mixtures of fluids. Arch. Ration. Mech. Anal. 28, 1–39 (1968)
Müller I.: Thermodynamik, Die Grundlagen der Materialtheorie. Bertelsmann Universitätsverlag, Düsseldorf (1973)
Müller I.: Thermodynamics. Pitman, Boston (1985)
Ore S.: A modification of the method of intermittent stress relaxation measurements on rubber vulcanisates. J. Appl. Polym. Sci. 2, 318–321 (1959)
Pochiraju K., Tandon G.: Modeling thermo-oxidative layer growth in high-temperature resins. J. Eng. Mater. Technol. ASME 128, 107–116 (2006)
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)
Samohýl I., Šípek X.N.M.: Irreversible (rational) thermodynamics of fluid-solid mixtures. Collect. Czech. Chem. Commun. 50, 2346–2363 (1985)
Scalan J., Watson W.: The interpretation of stress relaxation measurements made on rubber during ageing. Trans. Faraday Soc. 54, 740–750 (1957)
Shaw J., Jones S., Wineman A.: Chemorheological response of elastomers at elevated temperatures: experiments and simulations. J. Mech. Phys. Solids 53, 2758–2793 (2005)
Smith L.: The Language of Rubber: An Introduction to the Specification and Testing of Elastomers. Butterworth-Heinemann publication house, Oxford (1993)
Steinke L., Spreckels J., Flamm M., Celina M.: Model for heterogeneous aging of rubber products. Plast. Rubber Compost. 40, 175–179 (2011)
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)
Tobolsky A.V.: Mechanische Eigenschaften und Struktur von Polymeren. Berliner Union, Stuttgart (1967)
Tobolsky A.V., Prettyman I.B., Dillon J.H.: Stress relaxation of natural and synthetic rubber stocks. J. Appl. Phys. 15, 380–395 (1944)
Truesdell, C.: Sulle basi delle termomeccanica. Rend. Lincei. 22, 33–38, 158–166 (1957)
Truesdell, C.A., Toupin, R.: The classical field theories. In: Flügge, S. (Herausgeber). Handbuch der Physik III/1, Springer, Berlin (1960)
Woltman R.: Beiträge zur hydraulischen Archtektur, vol. 3. Johann Christian Dietrich, Göttingen (1794)
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)
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Communicated by Andreas Öchsner.
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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
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DOI: https://doi.org/10.1007/s00161-012-0255-8