Variety of fiber orientation tensors

Abstract

Fiber orientation tensors are established descriptors of fiber orientation states in (thermo-)mechanical material models for fiber-reinforced composites. In this paper, the variety of fourth-order orientation tensors is analyzed and specified by parameterizations and admissible parameter ranges. The combination of parameterizations and admissible parameter ranges allows for studies on the mechanical response of different fiber architectures. Linear invariant decomposition with focus on index symmetry leads to a novel compact hierarchical parameterization, which highlights the central role of the isotropic state. Deviation from the isotropic state is given by a triclinic harmonic tensor with simplified structure in the orientation coordinate system, which is spanned by the second-order orientation tensor. Material symmetries reduce the number of independent parameters. The requirement of positive-semi-definiteness defines admissible ranges of independent parameters. Admissible parameter ranges for transversely isotropic and planar cases are given in a compact closed form and the orthotropic variety is visualized and discussed in detail. Sets of discrete unit vectors, leading to selected orientation states, are given.

On the phase space of fourth-order fiber-orientation tensors

“…However, the necessary condition of positive semidefiniteness of the completely symmetric tensor is assumed to be sufficient. In section 2.3 of this work, sufficiency of this condition for the cases inspected by Bauer and Böhlke [41] is proven. A scientific topic which is intimately connected to the question on the phase space of fourth-order fiberorientation tensors, is fiber-orientation closure approximations.…”

“…In particular, insights on the mathematical properties of fiber-orientation tensors might be transferred to diffusion tensors or fabric tensors. The phase space of second-order fiber-orientation tensors is known [36][37][38][39][40], see the recent review [41]. A spectral decomposition is typically used to separate the structural features of a second-order fiberorientation tensor, described by two independent eigenvalues with limited admissible parameter ranges, from the spatial alignment of this structural information in terms of a rotation or eigensystem.…”

“…Algebraic properties of symmetry and normalization are agreed upon and discussed in the literature [42][43][44][45][46]. Bauer and Böhlke [41] developed an eigensystem-based parameterization of fourth-order fiber-orientation tensors combining the framework of irreducible tensors [47][48][49][50][51][52][53] with the work of Kanatani [2]. This parameterization ensures normalization as well as symmetry conditions automatically and separates second-and fourth-order data.…”

“…Indeed, due to the characterization shown in the first part, we may identify a realizable fourth-oder fiber-orientation tensor with a positive semidefinite matrix in a suitable matrix representation (we use the Kelvin-Mandel notation). Extreme states of the phase space of fourth-order fiber-orientation tensors are studied based on the parameterizations introduced in previous work [41], which separate second-and fourth-order data and enable to incorporate constraints on material symmetry easily. For fixed second-order information, we identify fourth-order data maximizing the projection onto a specified direction in two and three dimensions.…”

Mechkit: A continuum mechanics toolkit in Python

“…To this end, the methods in mechkit are focussed on secondand fourth-order tensors. Main motivations can thus be found in the research concerning linear elasticity (Bertram & Glüge, 2015), (Mandel, 1965), (Fedorov, 1968), (Mehrabadi & Cowin, 1990), (Thomson, 1856), (Cowin & Mehrabadi, 1992), (Rychlewski, 2000), (Spencer, 1970), (Böhlke & Brüggemann, 2001), (Brannon, 2018) and the description of microstructures of fiber-reinforced composite materials (Bauer & Böhlke, 2021), (Ken-Ichi, 1984), (Advani & Tucker III, 1987).…”