Short glass or carbon fibers are commonly added to a polymer matrix to form engineering composites that may be used in injection molded products. The strength and stiffness of the composite is defined by the material properties of the fiber and matrix, as well as the geometry, volume fraction and orientation of the fibers. Randomly oriented fibers as shown in (A) exhibit an isotropic averaged response, whereas fibers highly aligned in the x1 direction as in (B) are closer to transversely isotropic in nature. The orientation of the short fibers is defined by the polymer melt flow during the molding process, so it is important to have accurate and efficient computational methods for predicting fiber orientation under general flow conditions.

The overwhelming task of following each individual fiber in actual injection molded products is avoided by simulating the orientation distribution function ψ for a collection of fibers. This analysis is further simplified by evaluating the evolution of the moments of the orientation distribution function which are written in terms of even-order tensors as

a_ij=∫p_ip_jψ(p)dp    a_ijkl=∫ p_i p_jp_ip_jψ(p)dp

where p is a unit vector that defines the fiber direction, and the integration is performed over the unit sphere. Unfortunately, the computation of each even-order tensor requires the next higher even-order tensor. Therefore, a closure approximation is introduced to write, for example, a_ijkl in terms of a_ij. Several closure approximations have been proposed in the literature, some of which are very accurate, but computationally expensive.

The goal of this undergraduate research project is to develop a neural network based calculation procedure for computing aijkl from values of aij as indicated in the figure below. Training is performed based on multiple homogeneous flow fields. Preliminary results indicate that this approach will likely provide a computationally efficient, accurate closure approximation.

A Neural Network Based Closure for Modeling Short-Fiber Suspensions. B.K. Schache, D.A. Jack., and D.E. Smith Manuscript under review in SPE ANTEC'2006.

A Neural Network Based Closure for Modeling Short-Fiber Suspensions. B.K. Schache. Honor's Undergraduate Thesis, University of Missouri, 2006, Under revision .

• U.S. National Science Foundation