Planar fiber orientation: Jeffery, non-orthotropic closures, and reconstructing distribution functions

Abstract

Many models for fiber suspensions use the second-order orientation tensor to represent the orientation state. This requires that the fourth-order orientation tensor be approximated using a closure formula. The prediction of non-linear properties also requires reconstructing an approximate orientation distribution function. All existing fourth-order closures and most reconstructions have orthotropic symmetry, which limits their accuracy. In this paper, closures and reconstruction are explored for planar fiber orientation, using a recent parameterization of fourth-order orientation tensors by Bauer and Böhlke (2021, 2022a, 2022b). We show that the orientation distribution for fibers that follow Jeffery’s equation, together with the associated natural closure, form the central axis of the space of fourth-order orientation tensors. A new family of non-orthotropic planar closure approximations is created by including the rate-of-deformation tensor as an additional independent variable. These closures are shown to be substantially more accurate than any planar orthotropic closure. We also explore the Jeffery, Bingham, ellipse radius, and fourth-order maximum entropy (ME4) functions for reconstructing the orientation distribution. When compared to distributions from the Folgar-Tucker model, ME4 using the exact fourth-order orientation tensor gives the most accurate reconstruction, while ME4 with the fourth-order tensor from the new non-orthotropic closure is nearly as good. • The Jeffery distribution is central to planar fourth-order orientation tensors. • The first-ever non-orthotropic closure approximations are developed. • Non-orthotropic closures are more accurate than any previous planar closure. • Fourth-order maximum entropy distributions give the most accurate reconstruction.