Sixth-order Fitted Closures for Short-fiber Reinforced Polymer Composites

Abstract

Orientation tensors are widely used in short-fiber reinforced injection molding simulations of polymer composite products. In these computations, the evolution of an orientation tensor is given as a function of higher-order tensors, necessitating the use of a closure approximation. Current fourth-order fitted closures enable accurate and efficient computation of second-order tensor evolutions, but have been shown to be limited in their ability to accurately represent the fiber orientation distribution function. This article presents two sixth-order fitted closures that may be used to improve the accuracy of fiber orientation distribution function reconstructions. These new closures are written in terms of the second-order orientation tensor where it is assumed that the orthogonal planes of material symmetry of the sixth-order tensor are defined by the principal directions of the second-order tensor. Sixth-order tensor components are fit to either eigenvalues or invariants of the second-order tensor over numerous flow conditions and interaction coefficients. Example calculations illustrate that the new closures are capable of surpassing accuracy limits experienced when using any fourth-order closure.