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

TL;DR

This paper characterizes the phase space of fourth-order fiber-orientation tensors, establishing conditions for their realizability in 2D and 3D, which enhances modeling accuracy in fiber-reinforced composites.

Contribution

It proves that positive semidefiniteness and trace conditions are sufficient for realizing fourth-order fiber-orientation tensors in 2D and 3D, extending understanding beyond necessary conditions.

Findings

Necessary conditions are also sufficient in 2D and 3D.

Conditions are not sufficient in higher dimensions.

Implications for improved modeling of fiber-reinforced materials.

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 second-order fiber-orientation tensors implies artifacts. Closures based on the second-order fiber-orientation 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.