Computational Mechanics

, Volume 60, Issue 4, pp 595–611 | Cite as

A fiber orientation-adapted integration scheme for computing the hyperelastic Tucker average for short fiber reinforced composites

  • Niels GoldbergEmail author
  • Felix Ospald
  • Matti Schneider
Original Paper


In this article we introduce a fiber orientation-adapted integration scheme for Tucker’s orientation averaging procedure applied to non-linear material laws, based on angular central Gaussian fiber orientation distributions. This method is stable w.r.t. fiber orientations degenerating into planar states and enables the construction of orthotropic hyperelastic energies for truly orthotropic fiber orientation states. We establish a reference scenario for fitting the Tucker average of a transversely isotropic hyperelastic energy, corresponding to a uni-directional fiber orientation, to microstructural simulations, obtained by FFT-based computational homogenization of neo-Hookean constituents. We carefully discuss ideas for accelerating the identification process, leading to a tremendous speed-up compared to a naive approach. The resulting hyperelastic material map turns out to be surprisingly accurate, simple to integrate in commercial finite element codes and fast in its execution. We demonstrate the capabilities of the extracted model by a finite element analysis of a fiber reinforced chain link.


Computational homogenization Hyperelasticity Short fibers Composite materials Orientation tensors Numerical integration Injection molding 



Niels Goldberg and Felix Ospald acknowledge support from the German Research Foundation (DFG) via the Federal Cluster of Excellence EXC 1075 “MERGE Technologies for Multifunctional Lightweight Structures”. We thank the professorship in Materials-Handling Technology (Fördertechnik) at TU Chemnitz for sharing the CAD geometry of the chain link.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.TU Chemnitz, Professur FestkörpermechanikChemnitzGermany
  2. 2.TU Chemnitz, Professur Numerische Mathematik (PDE)ChemnitzGermany
  3. 3.Fraunhofer ITWMKaiserslauternGermany

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