Sixth-Order Fitted Closure Methods for Shot-Fiber Reinforced Polymer Composite Systems

Abstract

Fiber orientation tensors are widely used in injection molding simulations that include short-flbers suspended in the polymer melt. These tensors capture the stochastic nature of concentrated flber suspensions in a concise form suitable for numerical computation of short flber composites processing. Unfortunately, in the computation of the evolution equation for each even-order orientation tensor, the next higher even-order flber orientation tensor appears. It is, therefore, common to introduce a closure approximation where, for example, the fourth-order tensor is written in terms of the second-order tensor components. It has been shown that current fourth-order closures approach the fourth-order truncation limit in representing the true distribution of flbers, and to produce a substantial increase in accuracy it becomes necessary to investigate sixth-order closures. This paper presents a sixth-order fltted closure which assumes that the orthotropic planes of material symmetry of the sixth-order orientation tensor correspond to the principal directions of the second-order orientation tensor. The sixth-order closure is established using a fltting procedure which minimizes the difierence between the true and the fltted sixth-order orientation tensors over a range of orientations encompassing much of the eigenspace of the second-order orientation tensor. This new sixth-order closure is shown to better represent the distribution of flber orientations than previously obtained with any fourth-order closure.