Fiber orientation distributions based on planar fiber orientation tensors of fourth order

Abstract

Fiber orientation tensors represent averaged measures of fiber orientations inside a microstructure. Although, orientation-dependent material models are commonly used to describe the mechanical properties of representative microstructure, the influence of changing or differing microstructure on the material response is rarely investigated systematically for directional measures which are more precise than second-order fiber orientation tensors. For the special case of planar orientation distributions, a set of admissible fiber orientation tensors of fourth-order is known. Fiber orientation distributions reconstructed from given orientation tensors are of interest both for numerical averaging schemes in material models and visualization of the directional information itself. Focusing on the special case of planar orientations, this paper draws the geometric picture of fiber orientation distribution functions reconstructed from fourth-order fiber orientation tensors. The developed methodology can be adopted to study the dependence of material models on planar fourth-order fiber orientation tensors. Within the set of admissible fiber orientation tensors, a subset of distinct tensors is identified. Advantages and disadvantages of the description of planar orientation states in two- or three-dimensional tensor frameworks are demonstrated. Reconstruction of fiber orientation distributions is performed by truncated Fourier series and additionally by deploying a maximum entropy method. The combination of the set of admissible and distinct fiber orientation tensors and reconstruction methods leads to the variety of reconstructed fiber orientation distributions. This variety is visualized by arrangements of polar plots within the parameter space of fiber orientation tensors. This visualization shows the influence of averaged orientation measures on reconstructed orientation distributions and can be used to study any simulation method or quantity which is defined as a function of planar fourth-order fiber orientation tensors.

Multiscale mechanics and molecular dynamics simulations of the durability of fiber-reinforced polymer composites

“…The distribution of fiber orientations is critical to the mechanical performance of FRP composites, and the distribution functions can serve as accurate microstructural descriptors of the axisymmetric fiber orientation within a specific volume of a fiber-reinforced composite 112,113 . Fiber orientation tensors serve as a measure of the average fiber orientations within a microstructure 113,114 . The anisotropic behavior of the material is a result of the reorientation effect of the fibers during the extrusion process.…”

On the Phase Space of Fourth-Order Fiber-Orientation Tensors

“…This variety is presented for special cases motivated by material symmetry. The case of planar fourth-order fiber-orientation tensors and derived quantities is studied in successive papers [54,55]. However, the necessary condition of positive semidefiniteness of the completely symmetric tensor is assumed to be sufficient.…”

Maximum Entropy Closure Relation for Higher Order Alignment and Orientation Tensors Compared to Quadratic and Hybrid Closure

“…Calculating the elasticity tensor or the stiffness tensor of the composite from a simulation of the second order orientation tensor requires again a closure relation. For planar orientation states, as they occur in sheet molding compounds [30], a minimal invariant set of structurally differing planar fourth-order fiber orientation tensors and orientation distribution functions derived from them have been constructed [30,31].…”