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

Abstract

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.