Assessing the use of Tensor Closure Methods with Orientation Distribution Reconstruction Functions

Abstract

Orientation tensors are widely used to describe fiber orientations in mold filling simulations of short-fiber-reinforced composite systems. In these flow calculations, a closure is employed that approximates the fourth-order orientation tensor as a function of the second-order orientation tensor. Sixth-order closures have also been proposed. This paper assesses the effect of using closure approximations in fiber orientation predictions by reconstructing the fiber orientation distribution function from successively higher order orientation tensors in a Fourier series representation. This approach recognizes that the orientation tensors are related to the series expansion coefficients of the distribution function. An error metric is introduced and applied that makes it possible to compare closures of varying order. Errors associated with several fourth-order closures and a sixth-order closure are investigated and compared with the truncation error that results from a reconstruction of the exact second-, fourth-, and sixth-order orientation tensors. Examples are provided over a range of interaction coefficients and flow fields typical of injection molded fiber-reinforced composites to illustrate the proposed closure assessment method.