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

Journal of Elasticity https://doi.org/10.1007/s10659-022-09977-2 On the Phase Space of Fourth-Order Fiber-Orientation Tensors Julian Karl Bauer1 · Matti Schneider1 · Thomas Böhlke1 Received: 15 November 2022 / Accepted: 20 December 2022 © The Author(s) 2023 Abstract Fiber-orientation tensors describe the relevant features of the fiber-orientation distribution compactly and are thus ubiquitous in injection-molding simulations and subsequent mechanical analyses. In engineering applications to date, the second-order fiber-orientation tensor is the basic quantity of interest, and the fourth-order fiber-orientation tensor is obtained via a closure approximation. Unfortunately, such a description limits the predictive capabilities of the modeling process significantly, because the wealth of possible fourth-order fiber-orientation tensors is not exploited by such closures, and the restriction to secondorder fiber-orientation tensors implies artifacts. Closures based on the second-order fiberorientation tensor face a fundamental problem – which fourth-order fiber-orientation tensors can be realized? In the literature, only necessary conditions for a fiber-orientation tensor to be connected to a fiber-orientation distribution are found. In this article, we show that the typically considered necessary conditions, positive semidefiniteness and a trace condition, are also sufficient for being a fourth-order fiber-orientation tensor in the physically relevant case of two and three spatial dimensions. Moreover, we show that these conditions are not sufficient in higher dimensions. The argument is based on convex duality and a celebrated theorem of D. Hilbert (1888) on the decomposability of positive and homogeneous polynomials of degree four. The result has numerous implications for modeling the flow and the resulting microstructures of fiber-reinforced composites, in particular for the effective elastic constants of such materials. Based on our findings, we show how to connect optimization problems on fourth-order fiber-orientation tensors to semi-definite programming. The proposed formulation permits to encode symmetries of the fiber-orientation tensor naturally. As an application, we look at the differences between orthotropic and general, i.e., triclinic, fiber-orientation tensors of fourth order in two and three spatial dimensions, revealing the severe limitations inherent to orthotropic closure approximations. Keywords Fiber-orientation tensor · Injection molding · Closure approximation · Effective elastic stiffness · Fiber-reinforced composite  M. Schneider J.K. Bauer T. Böhlke 1 Karlsruhe Institute of Technology (KIT), Institute of Engineering Mechanics, Karlsruhe, Germany J.K. Bauer et al. Mathematics Subject Classification 46N60 · 62P30 · 74A40 · 74A60 · 74B05 · 74Q05 1 Introduction 1.1 State of the Art Fiber-orientation tensors [1] date back to the far-reaching works [2, 3] and describe the relevant features of the fiber-orientation distribution of discontinuous fiber-reinforced composites. Within the virtual development and design process of such composites [1, 4–6], fiberorientation tensors appear in material modeling [7–12], microstructure generation [13–19], mold filling or flow simulations [20–25] and the experimental computer tomography analysis [26–28]. This wide field of application motivates a detailed understanding of the mathematical properties of fiber-orientation tensors. Actually, the motivation and field of application is much more general, as fabric tensors and diffusion tensors share structural properties with fiber-orientation tensors. Fabric tensors [7, 29] share all characteristics with fiberorientation tensors, except for a normalization constraint and are used in the field of porous materials. Diffusion tensors [30–32] also differ from fiber-orientation tensors only by a missing trace constraint and are used in medicine to describe the orientation of body tissues based on the diffusion motion of water molecules. This procedure is called diffusion-weighted magnetic resonance imaging (DW-MRI) [33, 34] and is, e.g., used on brain tissue to prevent strokes [35]. 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 [36–40], see the recent review [41]. A spectral decomposition is typically used to separate the structural features of a second-order fiber-orientation 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. The limited structural variability of secondorder fiber-orientation tensors is a critical ingredient for applications in process simulations [5, 9–11]. In contrast to the second-order case, the phase space of fourth-order fiber-orientation tensors is less well understood, a circumstance which motivated the work at hand. Algebraic properties of symmetry and normalization are agreed upon and discussed in the literature [42–46]. Bauer and Böhlke [41] developed an eigensystem-based parametrization of fourth-order fiber-orientation tensors combining the framework of irreducible tensors [47–53] with the work of Kanatani [2]. This parametrization ensures normalization as well as symmetry conditions automatically and separates second- and fourth-order data. Moreover, additional material-symmetry constraints may be taken into account in a natural way. Bauer and Böhlke [41] assume that positive semidefiniteness of the tensor is a sufficient condition to derive admissible parameter ranges, specifying the variety 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. In Sect. 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 fiber-orientation tensors, is fiber-orientation closure approximations. Such closure approximations [22, 37, 56–61] are tensor-valued functions which postulate a functional relationship between a given second-order fiber-orientation tensor and an unknown On the Phase Space of Fourth-Order Fiber-Orientation Tensors fourth-order fiber-orientation tensor [19]. Identifying the phase space of fourth-order fiberorientation tensors is essential to solve a fundamental problem of closure approximations – which fourth- order fiber-orientation tensors can be realized? 1.2 Contributions This work is divided into a basic and an applied part. In the first part, we provide a proof (...truncated)