Modelling of Geometrical Microstructures and Mechanical Behaviour of Constituents

  • Heiko AndräEmail author
  • Dascha Dobrovolskij
  • Katja Schladitz
  • Sarah Staub
  • Ralf Müller
Part of the Mathematical Engineering book series (MATHENGIN)


In addition to the macroscopic component geometry, a morphological microstructure model and material models for all individual phases of the material are required as input data to apply multi-scale methods. However, the advantage is that complicated mechanical coupon tests on the composite material can be avoided. This chapter explains the computation of morphological and material parameters on the example of short glass fibre reinforced polymers. The fibre orientation is the most important geometrical micro-structural parameter which has to be computed from µCT scans, whereas other micro-structural parameters (e.g. fibre length distribution and diameter) are a priori known. State-of-the-art methods for estimating local fibre orientations based on 3D image data are used to determine this essential microstructure feature depending on the sample position w.r.t. the flow front. After that the generation of virtual microstructures with the same morphological parameters as the µCT scans is considered. In the second part of this chapter, the identification of the material parameters is described for the polymer polybutylene terephthalate (PBT). All necessary parameters of a rate-independent elastoplastic model with damage are computed from cyclic tensile tests with increasing load amplitudes. Finally, the validation of the morphological and material models are illustrated by using an FFT-accelerated pseudo-spectral method as micro-scale solver.


  1. 1.
    Baur, E., Osswald, T.A., Rudolph, N., Brinkmann, S., Schmachtenberg, E. (eds.): Saechtling Kunststoff Taschenbuch, 31st edn. Hanser (2013)Google Scholar
  2. 2.
    Fisher, N., Lewis, T., Embleton, B.: Statistical Analysis of Spherical Data. Cambridge University Press, Cambridge, UK (1987)CrossRefGoogle Scholar
  3. 3.
    Frangi, A., Niessen, W., Vincken, K., Viergever, M.: Multiscale vessel enhancement filtering. In: Proceedings of the Medical Image Computing and Computer-Assisted Intervention, pp. 130–137 (1998)CrossRefGoogle Scholar
  4. 4.
    Fraunhofer ITWM, Department of Image Processing: MAVI—modular algorithms for volume images. (2005)
  5. 5.
    GeoDict.: Accessed 16 Jan 2019
  6. 6.
    Ju, J.: On energy-based coupled elastoplastic damage theories: constitutive modeling and computational aspects. Int. J. Solids Struct. 25(7), 803–833 (1989)CrossRefGoogle Scholar
  7. 7.
    Kabel, M., Böhlke, T., Schneider, M.: Efficient fixed point and Newton-Krylov solvers for FFT-based homogenization of elasticity at large deformations. Comput. Mech. 54(6), 1497–1514 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kouznetsova, V., Brekelmans, W., Baaijens, F.: An approach to micro-macro modeling of heterogeneous materials. Comput. Mech. 27(1), 37–48 (2001)CrossRefGoogle Scholar
  9. 9.
    Kröner, E.: Bounds for effective elastic moduli of disordered materials. J. Mech. Phys. Solids 25(2), 137–155 (1977)CrossRefGoogle Scholar
  10. 10.
    Lippmann, B., Schwinger, J.: Variational principles for scattering processes. Phys. Rev. 79, 469–480 (1950)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Moulinec, H., Suquet, P.: A fast numerical method for computing the linear and nonlinear mechanical properties of composites. Comptes rendus de l’Académie des sciences. Série II, Mécanique, physique, chimie, astronomie 318(11), 1417–1423 (1994)zbMATHGoogle Scholar
  12. 12.
    Moulinec, H., Suquet, P.: A numerical method for computing the overall response of nonlinear composites with complex microstructure. Comput. Methods Appl. Mech. Eng. 157(1–2), 69–94 (1998)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mura, T.: Micromechanics of Defects in Solids, 2nd, revised edn. Mechanics of Elastic and Inelastic Solids. Martinus Nijhoff Publishers, Dordrecht (1987)Google Scholar
  14. 14.
    Niedziela, T., Strautins, U., Hosdez, V., Kech, A., Latz, A.: Improved multiscale fiber orientation modeling in injection molding of short fiber reinforced thermoplastics: simulation and Experiment. Int. J. Multiphys. Special Edition: Multiphys. Simul. Adv. Methods Ind. Eng. 357–366 (2011)Google Scholar
  15. 15.
    Ohser, J., Schladitz, K.: 3D Images of Materials Structures: Processing and Analysis. Wiley VCH (2009)Google Scholar
  16. 16.
    Onate, E. (ed.): Multiscale modeling of progressive damage in elasto-plastic composite materials (2014)Google Scholar
  17. 17.
    Otsu, N.: A threshold selection method from gray level histograms. IEEE Trans. Syst. Man Cybern. 9, 62–66 (1979)CrossRefGoogle Scholar
  18. 18.
    Schneider, R., Weil, W.: Stochastic and Integral Geometry. Probability and Its Applications. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  19. 19.
    Spahn, J., Andrä, H., Kabel, M., Müller, R.: A multiscale approach for modeling progressive damage of composite materials using fast Fourier transforms. Comput. Methods Appl. Mech. Eng. 268, 871–883 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Stoyan, D., Kendall, W., Mecke, J.: Stochastic Geometry and Its Applications, 2nd edn. Wiley, Chichester (1995)zbMATHGoogle Scholar
  21. 21.
    Wirjadi, O.: Models and algorithms for image-based analysis of microstructures. Ph.D. thesis. Technische Universität Kaiserslautern (2009)Google Scholar
  22. 22.
    Wirjadi, O., Schladitz, K., Easwaran, P., Ohser, J.: Estimating fibre direction distributions of reinforced composites from tomographic images. Image Anal. Stereol. 35(3), 167–179 (2016)CrossRefGoogle Scholar
  23. 23.
    Zeller, R., Dederichs, P.H.: Elastic constants of polycrystals. Phys. Status Solidi (b) 55(2), 831–842 (1973)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Heiko Andrä
    • 1
    Email author
  • Dascha Dobrovolskij
    • 1
    • 2
  • Katja Schladitz
    • 1
  • Sarah Staub
    • 1
  • Ralf Müller
    • 3
  1. 1.Fraunhofer Institute for Industrial MathematicsKaiserslauternGermany
  2. 2.University of Applied SciencesDarmstadtGermany
  3. 3.University of KaiserslauternKaiserslauternGermany

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