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Fiber orientation tensors are widely used to efficiently describe the fiber orientation evolution of anisotropic fiber suspensions during mold-filling. Real viscous fiber suspensions show fiber-induced anisotropic behavior. It is an open question how to correctly account for the anisotropic microstructure of fiber suspensions in general in the evolution equation of the fiber orientation tensors. In this study, a generalized evolution equation for the fiber orientation tensor of arbitrary even order is formulated, which takes into account the microstructural anisotropy of the fiber suspension. This formulation is based on a linear homogenization approach which allows the application of arbitrary mean-field models. The derived interaction term, representing the anisotropic environment of a single fiber, is discussed as a micromechanical convergence criterion of the underlying integral operator. It is shown and discussed in detail how the special cases of the Jeffery equation and the Folgar–Tucker equation follow from this general formulation. In addition, the generalized evolution equation is specified for selected mean-field models. In this context, an equation is presented to describe the evolution of the fiber orientation tensor depending on the spatial distribution of the fibers. For simple shear flow, all models describing the orientation dynamics are numerically investigated and compared. The results for the second-order orientation tensor as well as for a single fiber show that the well-known periodic behavior is present and strongly depends on the volume fraction and the chosen mean-field model. Considering the spatial distribution of the fibers has a significant effect on the orientation evolution and strongly affects the periodic reorientation. [Display omitted] • The evolution equation of orientation tensors of arbitrary even order is derived. • Fiber-induced anisotropic viscosity effects are covered in the orientation evolution. • Anisotropic effects are shown to be a micromechanical convergence criterion. • The chosen mean-field model greatly influences the fiber orientation dynamics. • The spatial fiber distribution strongly affects the periodic fiber reorientation. [ABSTRACT FROM AUTHOR]

A novel closure approximation method for fiber orientation tensors is proposed namely the fully symmetric implicit closure. Besides the full index symmetry, implicitly formulated closures based on the contraction condition fulfill the trace condition and the trace-preserving property of the Folgar–Tucker equation. As a first example, the fully symmetric implicit quadratic closure is considered as a simple modification of a recently proposed symmetric quadratic closure. It is shown that this closure can be realized by a fiber orientation distribution function. Secondly, the fully symmetric implicit hybrid closure is proposed as a counter-example of a closure not being based on a orientation distribution function. Both closures are compared against classical approximations in view of orientation evolution in a simple shear flow. Furthermore, the capability of predicting the effective viscous and elastic behavior of fiber suspensions and solid composites is investigated for a given fiber orientation state. The results show that the proposed implicit closures can be used to approximate the maximum entropy closure. Thereby, both the quadratic and the hybrid approach alleviate the high computational burden of the maximum entropy closure, as their evaluation requires solving a one-dimensional problem only. In addition, the predicted effective behavior based on the implicit closures shows an overall good agreement with predictions based on measured orientation data. • A novel implicit closure approach for fiber orientation tensors is proposed. • By considering full index symmetry, all necessary algebraic properties are fulfilled. • The considered closures can be reduced to an efficiently solvable 1D formulation. • The closures show a reliable prediction of effective viscous and elastic properties. • The closures considered are efficient maximum entropy approximations. [ABSTRACT FROM AUTHOR]