Variety of Planar Fourth‐Order Fiber Orientation Tensors and Implications on Effective Elastic Stiffnesses

In this contribution, selected results from [1–3] are presented in a compact and simplified way. In addition, the variety of fiber orientation tensors is used to determine a maximum deviation of the direction-dependent Young's modulus, which can arise if only second-order directional information is included in a specific meanfield homogenization. Focusing on the special case of planar fiber distributions, the variety of fiber orientation tensors identified in [1] is considered as a design space. This design space is completely explored for the orientation-averaging homogenization following [4], fixed material parameters and fixed fiber volume content. The possible directional dependence of the resulting effective stiffnesses is graphically presented using polar plots of the direction-dependent Young's modulus. These polar plots are arranged on two-dimensional slices within the parameter space of planar fourth-order fiber orientation tensors. This gives a complete representation of the influence of the orientation tensor on the anisotropic stiffness tensor. Consequences of closure approximations, i.e., restriction to second-order directional information, are demonstrated and motivate measurement of fourth-order fiber orientation tensors.