Computational Mechanics

, Volume 53, Issue 4, pp 777–787 | Cite as

Degree of cure-dependent modelling for polymer curing processes at small-strain. Part I: consistent reformulation

  • M. Hossain
  • P. Steinmann
Original Paper


A physically-based small strain curing model has been developed and discussed in our previous contribution (Hossain et al. in Comput Mech 43:769–779, 2009a) which was extended later for finite strain elasticity and viscoelasticity including shrinkage in Hossain et al. (Comput Mech 44(5):621–630, 2009b) and in Hossain et al. (Comput Mech 46(3):363–375, 2010), respectively. The previously proposed constitutive models for curing processes are based on the temporal evolution of the material parameters, namely the shear modulus and the relaxation time (in the case of viscoelasticity). In the current paper, a thermodynamically consistent small strain constitutive model is formulated that is directly based on the degree of cure, a key parameter in the curing (reaction) kinetics. The new formulation is also in line with the earlier proposed hypoelastic approach. The curing process of polymers is a complex phenomenon involving a series of chemical reactions which transform a viscoelastic fluid into a viscoelastic solid during which the temperature, the chemistry and the mechanics are coupled. Part I of this work will deal with an isothermal viscoelastic formulation including shrinkage effects whereas the following Part II will give emphasis on the thermomechanical coupled approach. Some representative numerical examples conclude the paper and show the capability of the newly proposed constitutive formulation to capture major phenomena observed during the curing processes of polymers.


Curing Degree of cure Viscoelasticity Stiffness increase Cure-dependent model Volume shrinkage 


  1. 1.
    Adolf DB, Martin JE (1990) Time-cure superposition during crosslinking. Macromolecules 23:3700–3704CrossRefGoogle Scholar
  2. 2.
    Adolf DB, Chambers RS (1997) Verification of the capability for quantitative stress prediction during epoxy cure. Polymer 38:5481–5490CrossRefGoogle Scholar
  3. 3.
    Adolf DB, Martin JE, Chambers RS, Burchett SN, Guess TN (1998) Stresses during thermoset cure. J Mater Res 13:530–550CrossRefGoogle Scholar
  4. 4.
    Ernst LJ, van’t Hof C, Yang DG, Kiasat M, Zhang GQ, Bressers HJL, Caers JFJ, den Boer AWJ, Janssen J (2002) Mechanical modeling and characterization of the curing process of underfill materials. J Electron Packag 124:97–105CrossRefGoogle Scholar
  5. 5.
    Haupt P, Lion A (2002) On finite linear viscoelasticity of incompressible isotropic materials. Acta Mechanica 159:87–124CrossRefzbMATHGoogle Scholar
  6. 6.
    Hojjati M, Johnston A, Hoa SV, Denault J (2004) Viscoelastic behaviour of Cytec FM73 adhesive during cure. J Appl Polym Sci 91:2548–2557CrossRefGoogle Scholar
  7. 7.
    Hossain M (2010) Modelling and computation of polymer curing, PhD Dissertation. University of Erlangen-Nuremberg, GermanyGoogle Scholar
  8. 8.
    Hossain M, Possart G, Steinmann P (2009a) A small-strain model to simulate the curing of thermosets. Comput Mech 43:769–779CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Hossain M, Possart G, Steinmann P (2009b) A finite strain framework for the simulation of polymer curing. Part I: elasticity. Comput Mech 44(5):621–630CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Hossain M, Possart G, Steinmann P (2010) A finite strain framework for the simulation of polymer curing. Part II: viscoelasticity and shrinkage. Comput Mech 46(3):363–375CrossRefzbMATHGoogle Scholar
  11. 11.
    Hossain M, Steinmann P (2011) Modelling and simulation of the curing process of polymers by a modified formulation of the Arruda–Boyce model. Arch Mech 63(5–6):621–633Google Scholar
  12. 12.
    Hubert P, Johnston A, Poursartip A, Nelson K (2001) Cure kinetics and viscosity models for Hexcel 8552 epoxy resin. In: Proceedings of the 46th international SAMPE symposium. Long Beach, CA, USA, pp 2341–2354Google Scholar
  13. 13.
    Jochum Ch, Grandidier JC (2004) Microbuckling elastic modelling approach of a single carbon fibre embedded in an epoxy matrix. Compos Sci Technol 64:2441–2449CrossRefGoogle Scholar
  14. 14.
    Kiasat M (2000) Curing shrinkage and residual stresses in viscoelastic thermosetting resins and composites, PhD Thesis. TU Delft, The NetherlandsGoogle Scholar
  15. 15.
    Kim YK, White SR (1996) Stress relaxation behaviour of 3501-6 epoxy resin during cure. Polym Eng Sci 36:2852–2862CrossRefGoogle Scholar
  16. 16.
    Klinge S, Bartels A, Steinmann P (2012a) Modeling of curing processes based on a multi-field potential: single and multi-scale aspects. Int J Solids Struct 49:2320–2333CrossRefGoogle Scholar
  17. 17.
    Klinge S, Bartels A, Steinmann P (2012b) The multi-scale approach to the curing of polymer incorporating viscous and shrinkage effects. Int J Solids Struct 49:3883–3990CrossRefGoogle Scholar
  18. 18.
    Kolmeder S, Lion A, Landgraf R, Ihlemann J (2011) Thermophysical properties and material modelling of acrylic bone cements used in vertebroplasty. J Therm Anal Calorim 105:705–718 Google Scholar
  19. 19.
    Lange J (1999) Viscoelastic properties and transitions during thermal and UV cure of a methacrylate resin. Polym Eng Sci 39:1651–1660CrossRefGoogle Scholar
  20. 20.
    Liebl C, Johlitz M, Yagimli B, Lion A (2012) Three-dimensional chemo-thermomechnically coupled simulation of curing adhesives including viscoplasticity and chemical shrinkage. Comput Mech 49(5):603–615CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Liebl C, Johlitz M, Yagimli B, Lion A (2012) Simulation of curing-induced viscoplastic deformation: a new approach considering chemo-thermomechanical coupling. Arch Appl Mech. doi: 10.1007/s00419-012-0639-z
  22. 22.
    Lion A, Höfer P (2007) On the phenomenological representation of curing phenomena in continuum mechanics. Arch Mech 59:59–89zbMATHGoogle Scholar
  23. 23.
    Lion A, Yagimli B, Baroud G, Goerke U (2008) Constitutive modelling of PMMA-based bone cement: a functional model of viscoelasticity and its approximation for time domain investigations. Arch Mech 60:197–218Google Scholar
  24. 24.
    Lion A, Yagimli B (2008) Differential scanning calorimetry-continuum mechanical considerations with focus to the polymerisation of adhesives. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 88:388–402CrossRefzbMATHGoogle Scholar
  25. 25.
    Lion A, Johlitz M (2012) On the representation of chemical ageing of rubber in continuum mechanics. Int J Solids Struct 49(10):1227–1240CrossRefMathSciNetGoogle Scholar
  26. 26.
    Matsuoka S, Quan X, Bair HE, Boyle DJ (1989) A model for the curing reaction of epoxy resins. Macromolecules 22:4093–4098CrossRefGoogle Scholar
  27. 27.
    Mergheim J, Possart G, Steinmann P (2012) Modelling and computation of curing and damage of thermosets. Comput Mater Sci 53:359–367CrossRefGoogle Scholar
  28. 28.
    O’Brien DJ, Mather PT, White SR (2001) Viscoelastic properties of an epoxy resin during cure. J Compos Mater 35:883–904CrossRefGoogle Scholar
  29. 29.
    Rabearison N, Jochum C, Grandidier JC (2009) A FEM coupling model for properties prediction during the curing of an epoxy matrix. Comput Mater Sci 45:715–724CrossRefGoogle Scholar
  30. 30.
    Retka J, Höfer P (2007) Numerische simulation aushärtender klebstoffe. Diploma Thesis. Universität der Bundeswehr, MünchenGoogle Scholar
  31. 31.
    Ruiz E, Trochu F (2005) Thermomechanical properties during cure of glass-polyester RTM composites: elastic and viscoelastic modeling. J Compos Mater 39:881–916CrossRefGoogle Scholar
  32. 32.
    Suzuki K, Miyano Y (1977) Change of viscoelastic properties of epoxy resin in the curing process. J Appl Polym Sci 21:3367–3379CrossRefGoogle Scholar
  33. 33.
    van’t Hof C (2006) Mechanical characterization and modeling of curing thermosets, PhD Thesis. TU Delft, The NetherlandsGoogle Scholar
  34. 34.
    Yagimli B, Lion A (2011) Experimental investigations and material modelling of curing processes under small deformations. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 91:342–359CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Chair of Applied MechanicsUniversity of Erlangen-NurembergErlangenGermany
  2. 2.University of Erlangen-NurembergErlangenGermany

Personalised recommendations