  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.
