Fraunhofer Institute for Industrial Mathematics ITWM
Fraunhofer Institute for Industrial Mathematics ITWM
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© fotalia / Fraunhofer ITWM
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© fotalia / Fraunhofer ITWM

Modelling of Microstructures: Fiber Systems

Stochastic geometry models are fit to real structures based on geometric characteristics measured from 3D image data. Varying model parameters creates virtual fiber systems with different microstructures. Consequences on the macroscopic behavior can be studied by numerical simulations

Widely used stochastic models for fibrous structures are Boolean models of cylinders and Poisson processes of dilated lines. For both models, the orientation distribution of the cylinders can be chosen to match the one observed in the materials – isotropic or anisotropic, e. g. with a preferred direction or directions isotropically distributed in a plane.

Boolean Model

Dilatiertes Poissonsches Geradenfeld
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Dilated Poisson line system, orientations in z-direction preferred
Boolsches Modell trilobaler Zylinder
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Boolean model of trilobal cylinders, isotropic
Boolsches Modell trilobaler Zylinder
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Boolean model of trilobal cylinders, orientations in xy-plane preferred

Modeling dense systems of non overlapping fibers with a controllable level of bending requires a different typ of model. The Altendorf-Jeulin model operates in two steps.

  1. First, a random walk with a multivariate von Mises-Fisher orientation distribution defines bent fibers.
  2. Second, a force-biased packing approach arranges them in a non overlapping configuration.

With a fiber length to fiber thickness ratio of up to 15, isotropic fiber systems with fiber volume fractions above 60 percent can be generated.

Altendorf-Jeulin Model

Z-Richtung bevorzugt
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Z-direction preferred
Isotropy
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Isotropy
Isotropy in the x-y plane
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Isotropy in the x-y plane

The original Altendorf-Jeulin generates fiber systems with fibers as separate as possible. For nonwovens, Easwaran's variation allowing for fiber bundles is more suitable.

Easwaran-Altendorf-Jeulin Model

Z-Richtung bevorzugt
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Z-direction preferred
Isotropie
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Isotropy
Isotropie in der x-y-Ebene
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Isotropy in the x-y plane

Publications

  • Easwaran, P.;Lehmann, M. J.; Wirjadi, O.; Prill, T. ; Didas, S.; Redenbach, C.:
    Automatic Fiber Thickness Measurement in SEM Images Validated Using Synthetic Data.
    Proceedings of FILTECH 2015.
    Filtech Exhibitions Germany: Cologne
  • Redenbach, C.; Schladitz, K.; Vecchio, I.; Wirjadi, O. :
    Image Analysis for Microstructures Based on Stochastic Models.
    GAMM-Mitteilungen. 37, (2), 281-305, (2014).
  • Altendorf H.; Jeulin D.: 
    Random-walk-based stochastic modeling of three-dimensional fiber systems.
    Phys. Rev. E 83, 041804, 2011.

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