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 V_ijkl 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. The example below of simple shear flow with C_I=10^-2 is provided to demonstrate the difference between the exact distribution function ψ and the reconstructed distribution functions of second-, fourth-, and sixth-order reconstructions. Observe that the sixth-order reconstruction is nearly visually indistinguishable than the exact distribution function.

Assessing the Use of Tensor Closure Methods with Orientation Distribution Reconstruction Functions. D.A. Jack and D.E. Smith. Journal of Composite Materials 38:1851-1872, 2004.

Assessing the Use of Tensor Closure Methods with Orientation Distribution Reconstruction Functions. D.A. Jack and D.E. Smith. Proceedings of ASME IMECE'03, Washington D.C., November, 2003.

Assessing the Use of Lower Order Tensors in Numerical Predictions of Flow-Induced Fiber Orientation. D.A. Jack and D.E. Smith. SPE ANTEC'2003, Nashville, Tennessee, May, 2003.

• U.S. National Science Foundation
• Missouri Research Board.