Journal of Materials Science

, Volume 40, Issue 22, pp 5875–5881 | Cite as

Numerical prediction of the foam structure of polymeric materials by direct 3D simulation of their expansion by chemical reaction based on a multidomain method

  • J. Bikard
  • J. Bruchon
  • T. Coupez
  • B. Vergnes
Mechanical Behavior of Cellular Solids


The quality of thermosetting polymer foams (like polyurethane foam, used for example in automotive industry) mainly depends on the manufacturing process. At a mesoscopic scale, the foam can be modelled by the expansion of gas bubbles in a polymer matrix with evolutionary rheological behaviour. The initial bubbles correspond to germs, which are supposed quasi-homogeneously distributed in the polymer. An elementary foam volume (∼1 mm3) is phenomenologically modelled by a diphasic medium (polymer and immiscible gas bubbles). The evolution of each component is governed by equations resulting from thermodynamics of irreversible processes: the relevant state variables in gas, resulting from chemical reaction creating carbon dioxide (assimilated then to a perfect gas), are pressure, temperature and conversion rate of the reaction. The number of gas moles in each bubble depends on this conversion rate. The foam is considered as a shear-thinning viscous fluid, whose rheological parameters evolve with the curing reaction, depending on the process conditions (temperature, pressure). A mixed finite element method with multidomain approach is developed to simulate the average growth rate of the foam during its manufacture and to characterize the influence of the manufacturing conditions (or initial rheological behaviour of the components) on macroscopic parameters of the foam (cell size, heterogeneity of porosity, wall thickness).


Foam Polyurethane Rheological Behaviour Polyurethane Foam Mixed Finite Element 
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.


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  1. 1.
    L. Lefebvre and R. Keunings, Mathematical Modelling and Computer Simulation of the Flow of Chemically-Reacting Polymeric Foams, in “Mathematical Modelling for Materials Processing,” edited by M. Cross, J. F. T. Pittman, R. D. Wood, (Clarendon Press, Oxford, 1993) p 417.Google Scholar
  2. 2.
    G. Oertel, in “Polyurethane Handbook” (Hanser Publishers, Munich, 1985).Google Scholar
  3. 3.
    D. Weaire and S. Hutzler, “The Physics of Foams” (Oxford University Press, Oxford, 1999).Google Scholar
  4. 4.
    E. Mora, L. D. Artavia and C. W. Macosko, Modulus development during reactive polyurethane foaming, J. Rheol. 35 (1991) 921.CrossRefGoogle Scholar
  5. 5.
    S. L. Everitt, O. G. Harlen, H. J. Wilson and D. J. Read, Bubble dynamics in viscoelastic fluids with application to reacting and non-reacting polymer foams, J. Non Newt. Fluid Mech. 114 (2003) 83.CrossRefGoogle Scholar
  6. 6.
    P. Perzyna, Fundamental problems in viscoelasticity, Adv Appl. Mech. 9 (1966) 243.Google Scholar
  7. 7.
    M. Amon and D. C. Denson, A study of the dynamics of foam growth: analysis of the growth of closely spaced spherical bubbles, Polym. Eng. Sci. 24 (1984) 1026.CrossRefGoogle Scholar
  8. 8.
    M. I. Aranguren and R. J. J. Williams, Kinetic and statistical aspects of the formation of polyurethanes from toluene diisocyanate, Polymer 27 (1986) 425.CrossRefGoogle Scholar
  9. 9.
    J. Bikard, T. Coupez and B. Vergnes, Modèlisation numérique multidomaines de l’expansion réactive d’une mousse polymère par création de gaz, Actes du 38ème Colloque du Groupe Français de Rhéologie, CD ROM (2003), Brest, France.Google Scholar
  10. 10.
    G. O. Piloyan, I. D. Ryabchikov and O. S. Novikora, Determination of activation energies of chemical reactions by differential thermal analysis, Nature, 5067 (1966) 1229.Google Scholar
  11. 11.
    R. B. Kellog and B. Liu, A finite element method for the compressible Navier-Stokes equations. SIAM J. Num. Anal., 33 (1996) 788.Google Scholar
  12. 12.
    D. N. Arnold, F. Brezzi and M. Fortin, A stable finite element for Stokes equations. Calcolo 21 (1984) 344.Google Scholar
  13. 13.
    S. Batkam, J. Bruchon and T. Coupez, A space-time discontinuous Galerkin method for convection and diffusion in injection moulding. Intern. J. Form. Proc. 7 (2003) 11.Google Scholar
  14. 14.
    E. Pichelin and T. Coupez, Finite element solution of the 3D mold filling problem for viscous incompressible fluid. Comput. Meth. Appl. Mech. Eng. 163 (1999) 371.Google Scholar
  15. 15.
    E. Bigot and T. Coupez, Capture of 3D moving free surfaces and material interfaces by mesh deformation. in Proceedings ECCOMAS 2000, Barcelona, CD Rom (2000).Google Scholar
  16. 16.
    J. Bruchon and T. Coupez, Étude 3D de la formation d'une structure de mousse polymère par simulation de l'expansion anisotherme de bulles de gaz, Mécanique & Industries 4 (4)(2003) pp. 331.Google Scholar
  17. 17.
    Z. H. Tu, V. P. W. Shim and C. T. Lim, Plastic deformation modes in rigid polyurethane foam under static loading, Int. J. Solids Struct., 38 (2001) 9267.CrossRefGoogle Scholar
  18. 18.
    J. M. Castro and C. W. Macosko, Kinetics and rheology of typical polyurethane reaction injection molding systems, SPE ANTEC Tech. Papers (1980) 434.Google Scholar
  19. 19.
    F. Dimier, N. Sbirrazzuoli, B. Vergnes and M. Vincent, Curing kinetics and chemorheological analysis of polyurethane formation, Polym. Eng. Sci., 44 (2004) 518.CrossRefGoogle Scholar
  20. 20.
    G. A. Campbell, Polyurethane foam process development. A systems engineering approach. J. Appl. Polym. Sci. 16 (1972) 1387.Google Scholar

Copyright information

© Springer Science + Business Media, Inc 2005

Authors and Affiliations

  1. 1.Centre de Mise en Forme des Matériaux (CEMEF), Ecole des Mines de ParisSophia AntipolisFrance

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