Fiber networks at different length scales represent a prevalent microstructure of highly deformable materials and biological matter. At the microscale, these fiber networks are key for the function of biological systems, while at the macroscale they endow materials with striking characteristics, such as unusual kinematic behavior and high defect tolerance. Resolving the microstructure in discrete network models has helped understanding the mechanisms responsible for these outstanding characteristics, and computational homogenization can be used to simulate the macroscopic response. Notwithstanding, nonlinear continuum mechanics, by definition only applicable at the larger length scales, has likewise proved suitable to capture many of these special features in dedicated approaches.

After discussing some recent examples of special characteristics and their implications in network materials, this seminar will focus on analytical methods to model the transition from the single fiber to the homogenized network scale in continuum mechanical models. The commonly used micro-macro approaches used to this end are based on establishing relations between the macroscopic deformation field and the deformation of vectorial line elements, which represent referential fiber directions and are defined on the unit sphere. For non-affine networks, this concept reaches its limit, and an alternative concept will be presented instead. The latter is based on a new type of constitutive relation between the distribution of fiber stretch and the macroscopic deformation gradient. This new approach, albeit not free of challenges, opens up new routes for constitutive modelling of network materials, able to capture both the macroscale behavior and features of their distinct microscopic kinematics. Finally, the approach allows reformulating the classical concepts, and thus not only provides alternative strategies for their numerical implementation but also new perspectives that reveal inherent and potentially limiting assumptions behind these theories.