Orthotropic Fitted Closure

A family of closure approximations called orthotropic closures was developed by Cintra and Tucker (1995). Since the fourth-order orientation tensor aiju must take 

5 Row-Induced Alignment in Short-Fiber Reinforced Polymers 

The closure using the above coefficient matrix is labeled as ORF closure, which provides an excellent match to the probability distribution function calculations for C/ = 0.01-0.1. However, it exhibits non-physical oscillation for C[ — 0.001 in a simple shear flow. Cintra and Tucker (1995) introduced another coefficient matrix with the label ORL for C/ = 0.001-0.01, which is given by 

Further improvements of the orthotropic fitted closure and application to injection molding problems have been reported by VerWeyst (1998) and VerWeyst et al. (1999). 

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The coefhcients pi to jSg are polynomial expansions of the second and third invariants of ay, and chosen to satisfy ay k — ay. The polynomial coefficients can be found by matching particular solutions for the exact solutions of the probability distribution function. The idea is similar to the orthotropic fitted closure. However, the natural closure uses analytical solutions of the probability distribution function the analytical solution is only available for Cj—O and ar oo. 

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