Skip to main content

Advertisement

SpringerLink
Log in

Modeling and simulation of the curing process of epoxy resins using finite elements

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

Abstract

This article discusses several aspects of the curing process in polymers. First, we collect experimental data for an Araldite epoxy resin and calibrate the classical model of Kamal–Sourour. It is shown that there are strong correlations between the parameters in the material parameter identification process. Thus, a curing kinetics model with reduced number of parameters is proposed, which is calibrated to the experimental test data. Second, the model is implemented into a finite element program. Here, high-order, time-adaptive time integration schemes are chosen to treat the inherent instability resulting from the curing kinetics model. Since the curing variable determines the heat source in the heat equation, a particular finite element approach is applied. After the spatial discretization, we arrive at a large system of ordinary differential equations, where the diagonally implicit Runge–Kutta method in combination with the Multilevel-Newton method is chosen, which can be seen as an analogy to internal variable theories in nonlinear quasi-static finite element approaches. Temperature and curing-dependent heat capacity and heat conductivity are considered as well. Numerical examples and a new validation experiment conclude the investigations.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abliz, D., Artys, T., Ziegmann, G.: Influence of model parameter estimation methods and regression algorithms on curing kinetics and rheological modelling. J. Appl. Polym. Sci. 134(30), 45137 (2017)

    Article  Google Scholar 

  2. Alexander, R.: Diagonally implicit Runge–Kutta methods for stiff O.D.E’.s. SIAM J. Numer. Anal. 14, 1006–1021 (1977)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Balvers, J., Bersee, H., Beukers, A., Jansen, K.: Determination of cure dependent properties for curing simulation of thick-walled composites. In: 49th AIAA, Schaumburg (2008)

  4. Baur, H., Wunderlich, B.: About complex heat capacities and temperature-modulated calorimetry. J. Therm. Anal. Calorim. 54(2), 437–465 (1998)

    Article  Google Scholar 

  5. Beck, J.V., Arnold, K.J.: Parameter Estimation in Engineering and Science. Wiley, New York (1977)

    MATH  Google Scholar 

  6. Boey, F., Qiang, W.: Experimental modeling of the cure kinetics of an epoxy-hexaanhydro-4-methylphthalicanhydride (MHHPA) system. Polymer 41(6), 2081–2094 (2000)

    Article  Google Scholar 

  7. Bogetti, T.A., Gillespie Jr., J.W.: Two-dimensional cure simulation of thick thermosetting composites. J. Compos. Mater. 25(3), 239–273 (1991)

    Article  ADS  Google Scholar 

  8. Brauner, C., Bauer, S., Herrmann, A.S.: Analysing process-induced deformation and stresses using a simulated manufacturing process for composite multispar flaps. J. Compos. Mater. 49(4), 387–402 (2015)

    Article  ADS  Google Scholar 

  9. Chern, B.C., Moon, T.J., Howell, J.R., Tan, W.: New experimental data for enthalpy of reaction and temperature-and degree-of-cure-dependent specific heat and thermal conductivity of the Hercules 3501-6 epoxy system. J. Compos. Mater. 36(17), 2061–2072 (2002)

    Article  ADS  Google Scholar 

  10. Coleman, B.D., Gurtin, M.E.: Thermodynamics with internal state variables. J. Chem. Phys. 47, 597–613 (1967)

    Article  ADS  Google Scholar 

  11. Crane, L., Dynes, P., Kaelble, D.: Analysis of curing kinetics in polymer composites. J. Polym. Sci. Part C Polym. Lett. 11(8), 533–540 (1973)

    Article  Google Scholar 

  12. Diebels, S., Ellsiepen, P., Ehlers, W.: Error-controlled Runge–Kutta time integration of a viscoplastic hybrid two-phase model. Techn. Mech. 19, 19–27 (1999)

    Google Scholar 

  13. Draper, N.R., Smith, H.: Applied Regression Analysis, 3rd edn. Wiley, New York (1998)

    Book  MATH  Google Scholar 

  14. Droste, D., DiBenedetto, A.: The glass transition temperature of filled polymers and its effect on their physical properties. J. Appl. Polym. Sci. 13(10), 2149–2168 (1969)

    Article  Google Scholar 

  15. Ellsiepen, P.: Zeit- und ortsadaptive Verfahren angewandt auf Mehrphasenprobleme poröser Medien. Doctoral thesis, Institute of Mechanics II, University of Stuttgart, report no. II-3 (1999)

  16. Ellsiepen, P., Hartmann, S.: Remarks on the interpretation of current non-linear finite-element-analyses as differential–algebraic equations. Int. J. Numer. Methods Eng. 51, 679–707 (2001)

    Article  MATH  Google Scholar 

  17. Forrest, J., Dalnoki-Veress, K., Stevens, J., Dutcher, J.: Effect of free surfaces on the glass transition temperature of thin polymer films. Phys. Rev. Lett. 77(10), 2002 (1996)

    Article  ADS  Google Scholar 

  18. Fournier, J., Williams, G., Duch, C., Aldridge, G.A.: Changes in molecular dynamics during bulk polymerization of an epoxide- amine system as studied by dielectric relaxation spectroscopy. Macromolecules 29(22), 7097–7107 (1996)

    Article  ADS  Google Scholar 

  19. Fox, T.G., Loshaek, S.: Influence of molecular weight and degree of crosslinking on the specific volume and glass temperature of polymers. J. Polym. Sci. Part A Polym. Chem. 15(80), 371–390 (1955)

    ADS  Google Scholar 

  20. Fritzen, P.: Numerische Behandlung nichtlinearer Probleme der Elastizitäts- und Plastizitätstheorie. Doctoral thesis, Department of Mathematics, University of Darmstadt (1997)

  21. Gill, P., Sauerbrunn, S., Reading, M.: Modulated differential scanning calorimetry. J. Therm. Anal. 40(3), 931–939 (1993)

    Article  Google Scholar 

  22. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II, 2nd edn. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

  23. Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I, 2nd edn. Springer, Berlin (1993)

    MATH  Google Scholar 

  24. Hamkar, A.W.: Eine iterationsfreie Finite-Elemente Methode im Rahmen der finiten Thermoviskoelastizität. Phd-thesis, report no. 1/2013, Institute of Applied Mechanics, Clausthal University of Technology, Clausthal-Zellerfeld (2013)

  25. Hamkar, A.W., Hartmann, S.: Theoretical and numerical aspects in weak-compressible finite strain thermo-elasticity. J. Theor. Appl. Mech. 50, 3–22 (2012)

    Google Scholar 

  26. Hamkar, A.W., Hartmann, S., Rang, J.: A stiffly accurate Rosenbrock-type method of order 2 applied to FE-analyses in finite strain viscoelasticity. Appl. Numer. Math. 62(12), 1837–1848 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hartmann, S.: Computation in finite strain viscoelasticity: finite elements based on the interpretation as differential–algebraic equations. Comput. Methods Appl. Mech. Eng. 191(13–14), 1439–1470 (2002)

    Article  ADS  MATH  Google Scholar 

  28. Hartmann, S.: A remark on the application of the Newton–Raphson method in non-linear finite element analysis. Comput. Mech. 36(2), 100–116 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hartmann, S.: Comparison of the multiplicative decompositions \({\bf F}={\bf F}_\varTheta {\bf F}_{M}\) and \({\bf F}={\bf F}_{M} {\bf F}_\varTheta \) in finite strain thermo-elasticity. Technical Report Series Fac3-12-01, Faculty of Mathematics/Computer Sciences and Mechanical Engineering, Clausthal University of Technology (Germany) (2012)

  30. Hartmann, S., Hamkar, A.W.: Rosenbrock-type methods applied to finite element computations within finite strain viscoelasticity. Comput. Methods Appl. Mech. Eng. 199(23–24), 1455–1470 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Hartmann, S., Rothe, S.: A rigorous application of the method of vertical lines to coupled systems in finite element analysis. In: Ansorge, R., Bijl, H., Meister, A., Sonar, T. (eds.) Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws, Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 120, pp. 161–175. Springer, Berlin (2013)

    Chapter  Google Scholar 

  32. Hartmann, S., Duintjer Tebbens, J., Quint, K.J., Meister, A.: Iterative solvers within sequences of large linear systems in non-linear structural mechanics. ZAMM J. Appl. Math. Mech. 89(9), 711–728 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Haupt, P.: Continuum Mechanics and Theory of Materials, 2nd edn. Springer, Berlin (2002)

    Book  MATH  Google Scholar 

  34. Heuser, H.: Lehrbuch der Analysis Teil II, 7th edn. Teubner, Stuttgart (1992)

    MATH  Google Scholar 

  35. Hossain, M., Steinmann, P.: Continuum physics of materials with time-dependent properties: reviewing the case of polymer curing. Adv. Appl. Mech. 48, 141–259 (2015)

    Article  Google Scholar 

  36. Hoyer, W., Schmidt, J.W.: Newton-type decomposition methods for equations arising in network analysis. ZAMM Z. Angew. Math. Mech. 64, 397–405 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  37. Hughes, T.J.R.: The Finite Element Method. Prentice-Hall, Englewood Cliffs (1987)

    MATH  Google Scholar 

  38. Huntsman: Advanced Materials Araldite LY 556 / Aradur 917 / Accelerator DY 070. Huntsman Advanced Materials GmbH, Basel, Switzerland (2007)

  39. Johnston, A.A.: An integrated model of the development of process-induced deformation in autoclave processing of composite structures. Ph.D. thesis, University of British Columbia (1997)

  40. Kamal, M.R.: Thermoset characterization for moldability analysis. Polym. Eng. Sci. 14(3), 231–239 (1974)

    Article  Google Scholar 

  41. Karkanas, P.I., Partridge, I.K.: Cure modeling and monitoring of epoxy/amine resin systems. I. Cure kinetics modeling. J. Appl. Polym. Sci. 77(7), 1419–1431 (2000)

    Article  Google Scholar 

  42. Klan, H., Thess, A.: Wärmeübertragung bei freier Konvektion: Außenströmungen. In: VDI-Wärmeatlas, Springer, chap F2, pp. 753–764 (2006)

  43. Krämer, S., Rothe, S., Hartmann, S.: Homogeneous stress–strain states computed by 3D-stress algorithms of FE-codes: application to material parameter identification. Eng. Comput. 31, 141–159 (2015)

    Article  Google Scholar 

  44. Krantz, S.G., Parks, H.R.: The Implicit Function Theorem, 1st edn. Birkhäuser, Boston (2003)

    Book  Google Scholar 

  45. Kreisselmeier, G., Steinhauser, R.: Systematische Auslegung von Reglern durch Optimierung eines vektoriellen Gütekriteriums/Systematic controller design by optimization of a vector performance index. at-Automatisierungstechnik 27(1–12):76–79 (1979)

  46. Laidler, K.J.: The development of the Arrhenius equation. J. Chem. Educ. 61(6), 494 (1984)

    Article  Google Scholar 

  47. Landgraf, R.: Modellierung und Simulation der Aushärtung polymerer Werkstoffe. Ph.D. thesis, Technische Universität Chemnitz. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-187720 (2015)

  48. Landgraf, R., Ihlemann, J., Kolmeder, S., Lion, A., Lebsack, H., Kober, C.: Modelling and simulation of acrylic bone cement injection and curing within the framework of vertebroplasty. ZAMM-J. Appl. Math. Mech./Z. Angew. Math. Mech. 95(12), 1530–1547 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  49. Lang, J., Verwer, J.: ROS3P—an accurate third-order Rosenbrock solver designed for parabolic problems. BIT 41, 731–738 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  50. Leistner, C., Hartmann, S., Wittrock, J., Bode, K.: Shrinkage behavior of araldite epoxy resin using Archimedes’ principle. Polym. Test. 67, 409–416 (2018)

    Article  Google Scholar 

  51. Liebl, C., Johlitz, M., Yagimli, B., Lion, A.: Three-dimensional chemo-thermomechanically coupled simulation of curing adhesives including viscoplasticity and chemical shrinkage. Comput. Mech. 49(5), 603–615 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  52. Lion, A., Dippel, B., Liebl, C.: Thermomechanical material modelling based on a hybrid free energy density depending on pressure, isochoric deformation and temperature. Int. J. Solids Struct. 51(3), 729–739 (2014)

    Article  Google Scholar 

  53. Loshaek, S., Fox, T.: Cross-linked polymers. I. Factors influencing the efficiency of cross-linking in copolymers of methyl methacrylate and glycol dimethacrylates. J. Am. Chem. Soc. 75(14), 3544–3550 (1953)

    Article  Google Scholar 

  54. Lu, S., Pister, K.: Decomposition of deformation and representation of the free energy function for isotropic thermoelastic solids. Int. J. Solids Struct. 11, 927–934 (1975)

    Article  MATH  Google Scholar 

  55. Mijovic, J., Wang, H.T.: Modeling of processing of composites. Part II—temperature distribution during cure. SAMPE J. 24(2), 42–55 (1988)

    Google Scholar 

  56. Nielsen, L.E.: Cross-linking-effect on physical properties of polymers. J. Macromol. Sci. Part C 3(1), 69–103 (1969)

    Article  Google Scholar 

  57. Parker, W., Jenkins, R., Butler, C., Abbott, G.: Flash method of determining thermal diffusivity, heat capacity, and thermal conductivity. J. Appl. Phys. 32(9), 1679–1684 (1961)

    Article  ADS  Google Scholar 

  58. Quint, K.J., Hartmann, S., Rothe, S., Saba, N., Steinhoff, K.: Experimental validation of high-order time-integration for non-linear heat transfer problems. Comput. Mech. 48, 81–96 (2011)

    Article  MATH  Google Scholar 

  59. Rabbat, N.B.G., Sangiovanni-Vincentelli, A.L., Hsieh, H.Y.: A multilevel Newton algorithm with macromodeling and latency for the analysis of large-scale nonlinear circuits in the time domain. IEEE Trans. Circuits Syst. 26, 733–740 (1979)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  60. Ramos, J., Pagani, N., Riccardi, C., Borrajo, J., Goyanes, S., Mondragon, I.: Cure kinetics and shrinkage model for epoxy-amine systems. Polymer 46(10), 3323–3328 (2005)

    Article  Google Scholar 

  61. Reading, M., Elliott, D., Hill, V.: A new approach to the calorimetric investigation of physical and chemical transitions. J. Therm. Anal. Calorim. 40(3), 949–955 (1993)

    Article  Google Scholar 

  62. Rothe, S., Hamkar, A.W., Quint, K.J., Hartmann, S.: Comparison of diagonal-implicit, linear-implicit and half-explicit Runge–Kutta methods in non-linear finite element analyses. Arch. Appl. Mech. 82(8), 1057–1074 (2012)

    Article  ADS  MATH  Google Scholar 

  63. Ruiz, E., Trochu, F.: Numerical analysis of cure temperature and internal stresses in thin and thick RTM parts. Compos. Part A Appl. Sci. Manuf. 36(6), 806–826 (2005)

    Article  Google Scholar 

  64. Schawe, J.: A description of chemical and diffusion control in isothermal kinetics of cure kinetics. Thermochim. Acta 388(1), 299–312 (2002)

    Article  Google Scholar 

  65. Schwarz, H.R., Köckler, N.: Numerische Mathematik. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  66. Simo, J.C., Taylor, R.L.: Consistent tangent operators for rate-independent elastoplasticity. Comput. Methods Appl. Mech. Eng. 48, 101–118 (1985)

    Article  ADS  MATH  Google Scholar 

  67. Urbaniak, M.: A relationship between the glass transition temperature and the conversion degree in the curing reaction of the EPY epoxy system. Polimery 56(3), 240–243 (2011)

    Article  Google Scholar 

  68. Van Assche, G., Van Hemelrijck, A., Rahier, H., Van Mele, B.: Modulated temperature differential scanning calorimetry: cure, vitrification, and devitrification of thermosetting systems. Thermochim. Acta 304, 317–334 (1997)

    Article  Google Scholar 

  69. Wriggers, P.: Nichtlineare Finite-Elemente Methoden. Springer, Berlin (2001)

    Book  MATH  Google Scholar 

  70. Zhang, J., Xu, Y., Huang, P.: Effect of cure cycle on curing process and hardness for epoxy resin. Express Polym. Lett. 3(9), 534–541 (2009)

    Article  Google Scholar 

Download references

Acknowledgements

We gratefully acknowledge our colleagues at Clausthal University of Technology Dr. J. Wittrock and K. Bode for their support in the shrinkage experiments. Furthermore, we would like to thank Marco Löffelholz for his support in performing some validation experiments

Author information

Authors and Affiliations

  1. Institute of Applied Mechanics, Clausthal University of Technology, Adolph-Roemer-Str. 2a, 38678, Clausthal-Zellerfeld, Germany

    C. Leistner & S. Hartmann

  2. Institute of Polymer Materials and Plastics Engineering, Clausthal University of Technology, Agricolastraße 6, 38678, Clausthal-Zellerfeld, Germany

    D. Abliz & G. Ziegmann

Authors
  1. C. Leistner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Hartmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. Abliz
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. G. Ziegmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Hartmann.

Additional information

Communicated by Johlitz, Laiarinandrasana and Marco.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Leistner, C., Hartmann, S., Abliz, D. et al. Modeling and simulation of the curing process of epoxy resins using finite elements. Continuum Mech. Thermodyn. 32, 327–350 (2020). https://doi.org/10.1007/s00161-018-0708-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-018-0708-9

Keywords

Search

Navigation