An invariant based fitted closure of the sixth-order orientation tensor for modeling short-fiber suspensions

Abstract

Orientation tensors are commonly used in short-fiber reinforced injection molding simulations of industrial polymer composite products. The evolution equation for each even-order orientation tensor is written in terms of the next higher even-order orientation tensor necessitating the use of a closure. It has been shown that current fourth-order closures approach the fourth-order truncation limit when representing the fiber orientation distribution function so that an increase in accuracy necessitates the development of a robust sixth-order closure. This paper presents a new fitted sixth-order closure formed from a general expression for a fully symmetric sixth-order tensor written as a function of a fourth-order orientation tensor. The components of this sixth-order closure are fit to a linear polynomial of the fourth-order orientation tensor invariants whose coefficients are computed by fitting the sixth-order components obtained from the closure to those computed from distribution function simulations for a variety of flow fields and interaction coefficients. The fitted sixth-order closure is shown to more accurately predict the second-order orientation tensor than simulations that employ existing fourth-order and sixth-order closures. Additionally, it is shown that the sixth-order closure more accurately represents the distribution function of fibers than current closure methods.