Constitutive modelling of fibre networks with stretch distributions, Part II: Alternative representation, affine distribution and anisotropy

In this paper, the concept for modelling materials with fibre network microstructures introduced in Part I of this series is reconsidered. At first, an alternative representation of the theory is provided that entails advantages in terms of numerical implementation. Next, the new constitutive approach is applied to affine networks, whose fibre stretches are distributed according to the affine distribution. Despite the tremendously widespread use of this model, it seems that the general form of the corresponding distribution has remained largely unexplored to date. The thus obtained reformulation of the affine full network model provides deep insight into this concept, and may help overcoming well-known numerical problems with the associated spherical integration, e.g. when highly non-linear or piece-wise defined fibre laws are used. The latter case is typical for applications in biomechanics, where fibres are frequently assumed to have negligible compressive resistance. While the developments of our theory thus far had focused on isotropic networks, we here showcase for the affine case how the anisotropy caused by non-uniform directional distributions of the fibres can be incorporated in the novel approach. Finally, it is shown that several earlier approaches to model networks of affinely deforming fibres or polymer chains result as special cases of our theory.