A comparative study between fiber orientation closure approximations and a new orthotropic closure

In injection and compression molding simulation, orientation tensors provide an efficient way with less computational effort to calculate flow-induced fiber orientations. In these flow calculations, the solution of any even-ordered orientation tensor needs the following even higher-ordered orientation tensor. Therefore, a closure is used in order to approximate the higher-ordered orientation tensor as a function of the components of the lower-ordered orientation tensors. There exist many closures of the fourth-order orientation tensor in terms of the second-order orientation. This paper gives a review with mathematical details for different closure approximations including simple closures, composite closures, eigenvalue- based orthotropic closures, invariant-based closures, and neural-network- based closures. Moreover, a new closure approximation that assumes an orthotropic stiffness tensor was suggested in this work. The results of the closure approximations were validated through the comparison of Young moduli between the closure approximations and the experimental results for both short- and long-fiber-reinforced composites.