**1. Introduction**

The current needs for weight reduction and eco-friendly products (e.g., natural fiber composites) have fueled the use of polymers in various industries. They are of particular importance in the automotive and aerospace industries due to their low weight and durability. The polymers in most cases are further reinforced with fibers to enhance their mechanical properties. Those are strongly influenced by fiber volume fraction, fiber length distribution, and fiber orientation.

Fiber reinforced polymers show typically anisotropic material properties. The local orientation of fibers depends on the manufacturing process and the complexity of the part being produced. The prediction of the failure and lifetime of a part using simulative methods instead of experimental methods can significantly help to reduce the costs and speed up the development of new parts. The final fiber orientation induced by the manufacturing process is an important parameter for the structural simulation of the part. The current state-of-the-art fiber orientation models based on Jeffrey's equation [2] have been developed and implemented in different commercial software. These seek to emulate several effects like strain reduction [3–5] and anisotropic rotary diffusion [6–8] on fiber orientation. All these models require phenomenological parameters derived from experimental or microsimulation results to obtain a suitable prediction accuracy.

The experimental determination of the phenomenological parameter to describe the steady state fiber orientation by diffusion was for example conducted by Folgar and Tucker or Bay and Tucker [9,10]. In order to determine phenomenological parameters, which describe the fiber orientation evolution by the retarding rate, it is necessary to measure the transient fiber orientation development. Consequently, the methods are more demanding. Stover et al. [11] measured transient fiber orientations in semi-dilute solutions using a Couette device, a tracer fiber, and video cameras. For highly concentrated solutions, in situ methods with optical methods are not applicable yet. To overcome this shortcoming, experiments in homogeneous flows with a repeatable starting condition have been developed. Eberle et al. [12] measured fiber orientation evolution for a 30% wt. short fiber reinforced polybutylene terephthalate (PBT) in a cone and plate rheometer with specially shaped donut-like samples. Ortman et al. [13] used a sliding plate rheometer, where it is possible to control the initial conditions [14], to measure fiber orientation evolution. Kugler et al. [15] used the same setup to determine phenomenological parameters for a short fiber reinforced PBT-GF30. Recently, Perumal et al. [16] measured transient fiber orientation evolution for a 30% wt. glass fiber reinforced Nylon-6 in a parallel plate rheometer. All stated approaches determine fiber orientation parameters in shear flow. Lambert et al. [17,18] determined fiber orientation parameters for the first time in elongational flow.

Additionally to experimental determination, microscopic fiber simulation can be used to evaluate macroscopic fiber orientation parameters. Modeling approaches on the microscopic scale have the aim to approximate the physical behavior of the composite more accurately. Many authors determined the parameters based on microscopic simulation, for example Mezher et al. [19] and Perez [1]. For a detailed review, refer for example to [20].

In this work, we use a multi-scale simulation chain to enhance the fiber orientation prediction for complex industrial parts. The simulation workflow consists of three steps. Firstly, a virtual flow test on the particle scale (micro) is conducted. Secondly, macroscopic fiber orientation model parameters are determined based on the resulting fiber orientation evolution from the virtual flow test. Thirdly, the final fiber orientation in a part is obtained using continuum-based macro models with the material-dependent optimal parameters. The multi-scale process simulation is sketched in Figure 1.

**Figure 1.** Multi-scale simulation process for short fiber reinforced thermoplastics (SFRT). RVE, representative volume element.

Current fiber orientation models, on both scales, are mostly focused on the effects of shear flow. However, experiments in elongation flow [17,18] showed that fiber orientation develops differently under elongational flows. On the other hand, during injection molding, complex flow fields combining shear and elongational flow usually take place. Considering only shear-fitted fiber orientation models may lead to discrepancies in the final fiber orientation predictions in a part. Recently, Chen et al. [21,22] introduced a flow-dependent strain reduction factor for the different state-of-the-art macroscopic fiber orientation models. This last model approach is able to differentiate the orientation evolution speed between shear, elongational, and rotational flow.

In this work, we focus primarily on the influence of elongational flow on the fiber orientation phenomenon using a simulation at the particle level. Finally, a novel flow-dependent fiber orientation model, scaling between shear and elongational flows, is proposed and implemented in Autodesk Moldflow Insight Scandium® 2019. In comparison to the approach proposed by Chen et al. [21,22], we propose a more general model since it not only considers the change in strain reduction, but also in diffusion.
