**3. Results**

In this Section, the influences of different material and process settings are evaluated. Based on the experimental data, the optimization results for the simulation are shown, and the two different optimization methods are compared.

## *3.1. Experimental Results*

#### 3.1.1. Fiber Orientation at Different Positions

The following Figure 17a shows the fiber orientation distribution along the plate thickness at three different positions (according to Figure 13, Pos. A, B, and C) along the flow path. In addition, the experimental results were smoothed and symmetrized according to Section 3.2 in order to be compared with the simulated results, Figure 17b.

**Figure 17.** Experimentally determined fiber orientations along the flow path for PP-LGF20 v30.

The analysis of the fiber orientation distribution along the flow path (position A–C) shows an increasing core layer, which also means that more fibers are oriented perpendicular to the flow direction. This also leads to enhancement of the mechanical properties in this direction, while the properties in the flow direction are diminished. However, the increase in shear layer thickness along the flow path has often been observed and is not a particular phenomenon of long fiber reinforced plastics.

#### 3.1.2. Fiber Orientation of Different Materials and Process Conditions

In the following section, the influence of fiber concentration and injection speed is analyzed. In general, it can be observed (Figure 18) that all fiber concentrations and processing conditions of the injection-molded long glass fiber reinforced polypropylene lead to a low degree of fiber orientation in the shear layers, and a wide core layer with a high degree of orientation.

All the analyzed specimens show a higher degree of orientation perpendicular to the flow (*A*22) than parallel to the flow direction (*A*11) in the core layer region, whereas in the shear layers, the degree of orientation is nearly the same in both directions (parallel *A*11 and perpendicular to the flow *A*22). This effect changes only imperceptibly, even with a variation in fiber concentration and injection speed. In contrast to short fiber reinforced plastics, the wide core layer with its high degree of fiber orientation in the long fiber reinforced material also leads to higher mechanical properties (e.g., stiffness, strengthm and impact resistance as shown in Figure 1) perpendicular to the flow. A unique detail of the analyzed parts can be observed in the shifted core layers, which are not completely centered. This effect is caused by a different cooling performance (comparable to [101]) of the mold halves and has no influence on the interpretation of the results.

The analysis of the three different fiber concentrations (fiber mass fractions 20%, 40%, and 60%) of the long glass fiber reinforced polypropylene also show that an increase in fiber concentration results in a more distinct core layer Figure 18a,c,e. This means that a more concentrated suspension leads to an increased number of fibers that are oriented perpendicular to the flow direction *A*22. The same effect can also be seen in the variation of the injection speed. With increasing flow velocity, a wider core layer can be observed for all analyzed fiber concentrations.

**Figure 18.** Resulting fiber orientation along the thickness of different fiber concentrations (20, 40 and 60% mass fraction long glass fiber) and different injection speeds (low = 30 cm<sup>3</sup>/s, high = 100 cm<sup>3</sup>/s).

In order to compare the skin core layer ratio systematically, two different methods were developed and used in this study. As a first method, the *A*11 entry of the fiber orientation tensor is approximated by a function, and the transition from skin to core layer (green line) is determined by the mean value of the two inflection points of its derivative as shown in Figure 19a. The determined threshold lines are also depicted in the grinded pattern of the cross section as shown in Figure 19b. Subsequently, the skin layer ratio can be calculated by the ratio of the associated thickness values. As a second method, the two intersections between the fiber orientation curves *A*11 and *A*22 are determined and their mean value is defined as the transition from skin to core.

**Figure19.** Methodtodeterminetheskincorelayer ratioofareinforcedmaterialbyobjectivecriteria.

For the different materials and injection speeds the resulting skin core layer ratios showed only slight differences between the two described methods. Therefore, only the results of the intersection method of *A*11 and *A*22 were chosen and shown in Table 2.

**Table 2.** Skin core layer ratio for different fiber concentrations and injection speeds of PP-LGF.


The results for the calculated skin-to-core layer ratios show an increase in the core-layer thickness with increasing fiber content. Thus, the previously recognized tendency of increase in core layer thickness can be proven objectively. The effect can be seen in general for low as well as for high injection speeds, except the outlying value for a very thin core layer thickness with 36% for PP-LGF40 at a low injection speed. No reason has been found to explain this effect.

#### *3.2. Results of the Parameter Optimization*

As shown in Figure 15, the resulting fiber orientation distribution varies very strongly, depending on the chosen fiber orientation model parameters. Figure 20 shows the results of direct parameter optimization, as explained in Section 2.5.1, by coupling MATLAB® and Moldex3D® with different fiber orientation models, (a) basic iARD and (b) iARD with shear rate dependency.

**Figure 20.** Results of the direct parameter optimization by coupling MATLAB® and Moldex3D®.

To prove the functionality of the optimization routine, deviant values were chosen as initial parameters for the corresponding fiber orientation model (Run 1), which did not match the experiments. The experimentally determined fiber orientation is depicted as blue line with triangular markers. For a clearer presentation of the optimization progress, only some selected optimization steps are shown. However, the run numbers still correspond to the respective iteration step. The results obviously show a better agreemen<sup>t</sup> to the experiment with increasing iterations. In the skin layer, the optimization result of the basic iARD model shows very good accordance with the experimental data, whereas in the core layer, the simulation result does not represent the low degree of orientation determined by the experiment. However, this cannot be further improved by optimizing the model parameters, as there is no parameter set that can represent the entire orientation distribution from the experiment. This means that the best possible parameter set was found after 37 optimization steps for the basic iARD fiber orientation model.

For the optimization of the iARD model with shear rate dependency in version R17, the adjustment of the parameters has only a very one-dimensional influence on the change of the fiber orientation. Because the change from core to shear layer is very abrupt, no real transition zone is formed. Therefore, the degree of orientation in the skin layers can only be adjusted by varying the parameters *Ci* and *Cm*. All the other areas remain largely constant despite the parameter variation.

#### *3.3. Influence of Parameters and Viscosity*

In addition to the model parameters, the defined shear rate-dependent viscosity also has a significant influence on the resulting velocity profile as already explained in Section 1.1.2 as well as the results of the fiber orientation model. For this reason, the parameters of the defined viscosity model (Herschel–Bulkley) were varied and the influence on the fiber orientation distribution was investigated. Figure 21 shows the analyzed viscosity curves, which are moved slightly towards higher and lower viscosities (deviating from experimental data).

**Figure 21.** Shear rate and temperature dependent viscosity and analyzed viscosity curves.

In Figure 22 the influence of the viscosity on the resulting fiber orientation distribution is investigated for di fferent fiber orientation model versions and parameters (comparable to Figure 15). On the left side of Figure 22a,c,e, the fiber orientation was calculated with the experimentally determined viscosity for PP-LGF20 (blue curve in Figure 21), and on the right side of the Figure 22b,d,f, a higher viscosity, especially in the area of low shear rates (grey curve in Figure 21), is used. In the first row (a, b) the basic iARD model in Moldex3D ® version R13, and in the second row (c, d) the iARD model with shear rate dependency in version R16, and the last row (e, f) the iARD model with shear rate dependency and fiber coupling in version R17 is used. For the comparison of the three fiber orientation model versions, the same variation of fiber orientation model parameters was used. The parameter α was varied in the range of 0–0.99 for high and also low *Ci* and *Cm* values.

(**e**) iARD with shear rate dependency and fiber coupling in R17, normal viscosity 

(**f**) iARD with shear rate dependency and fiber coupling in R17, high viscosity 

**Figure 22.** Influence of the viscosity for different fiber orientation model versions for PP-LGF20v30.

First of all, all results show a change of the fiber orientation distribution within the core layer due to the change in viscosity. However, the fiber orientation distribution in the shear layers remains almost unchanged for all settings (viscosity models, fiber orientation models and parameters) due to the same viscosity in the range of high shear rates (blue and grey curve in Figure 21). The viscosity change mainly only affects the thickness of the core layer. The core layer is widened with a higher viscosity curve, which corresponds more closely to the experimental determined fiber orientation distribution. However, even if the adjustment of the viscosity curve improves the thickness of the core

layer, a deviation between the predicted and measured fiber orientation still remains for all analyzed fiber orientation models, as well as for the viscosity models and their respective parameters.

#### *3.4. Validation and Results of the New Calibration Approach*

In order to ensure the correct implementation of the new optimization routine, a test case was chosen from literature [87]. Thus, the different parts of the implemented orientation model were tested for performance and verified by reference values [87]. The investigated test case is a simple shear flow within a single element. Two different versions of the fiber orientation models explained in Section 1.1.8, the basic iARD model (Equation (9)) and the iARD-RPR model (Equation (12)), were tested. The parameters of the fiber orientation model are chosen to make the iARD model (*Cm* = 0, *Ci* = 0.01 and α = 0.9) correspond to the standard model of Folgar-Tucker (FT) [86]. In Figure 23 the reference values are shown as points and the calculated fiber orientation as lines.

**Figure 23.** Validation of the implemented fiber orientation calculation within the calibration tool.

The results of the fiber orientation calculation with the new calibration tool in Figure 23 show quite good agreemen<sup>t</sup> with the results from literature. Thus, a correct implementation of the fiber orientation model is assumed. A further validation can be obtained by comparing the calculation of the fiber orientation by Moldex3D® with the fiber orientation calculated by the new calibration tool in MATLAB® on the basis of the same flow field exported from Moldex3D®. The results of the calculated fiber orientation distribution are almost identical, as shown in Figure 24a.

**Figure 24.** Calculation and calibration results of the fiber orientation calibration tool.

Based on this initial calculation, a parameter optimization was performed with the calibration tool in MATLAB ®. The calibration results of the parameter optimization are shown in Figure 24b for the experimentally measured viscosity, and in (c) for the increased viscosity according to the gray curve in Figure 21. Even if the fiber orientation model does not represent the entire orientation distribution, the adjusted viscosity curve (high viscosity) leads to a significant improvement in the resulting fiber orientation distribution in the core layer.

Compared with direct optimization by Moldex3D ® (Section 2.5.1), the new calibration approach (Section 2.5.2) fully implemented in MATLAB ® is significantly faster, as the entire calculation is based on only one initial calculation of the flow field without any necessary recalculation. The following Figure 25 clearly demonstrates that a significant amount (approximately factor 15) of the required calculation time can be saved by the new approach. Because the calculation of the fiber orientation is performed as an independent post-processing step, the calculation of the flow field is independent of the fiber orientation calculation and remains the same for each optimization step.

**Figure 25.** Comparison of calculation time of direct optimization method and the new optimization approach implemented in the fiber orientation optimization tool.

#### **4. Conclusions and Outlook**

A novel method for the objective comparison of experimentally determined and by process simulation predicted fiber orientation was developed. As a first step, a new standardized method for the objective comparison of experimentally determined and simulation predicted fiber orientation based on the error deviation was defined. Furthermore, a corresponding validation of the method showed that the fiber orientation model has successfully been implemented in the novel calibration tool, and that the developed problem-specific optimization algorithm can adjust the model parameters simultaneously in order to minimize the objective function.

After only a few iteration loops and within a few minutes, the automated method shows an optimized fiber orientation distribution for long fiber reinforced materials based on the calculated flow field. Thus, the model parameters for the fiber orientation model are determined in the most accurate way and can be used for further predictions. This also allows for calculating the structural mechanics of fiber reinforced parts with high prediction accuracy. These results clearly confirm the scientific hypotheses stated at the beginning of this study.

The investigation also showed that there is further optimization potential which should be investigated and considered in the future. The parameter optimization is currently only implemented for the fiber orientation model, but can easily be extended to the prediction of fiber length distribution and fiber concentration. Further development of this method also allows a transfer of the calculated properties to structural mechanics with high quality predictions.

**Author Contributions:** Conceptualization, F.W. and P.R.; methodology, F.W. and P.R.; software, F.W. and P.R.; validation, F.W. and P.R.; formal analysis, F.W. and P.R.; investigation, F.W. and P.R.; resources, F.W. and P.R.; data curation, F.W. and P.R.; writing—original draft preparation, F.W. and P.R.; writing—review and editing, F.W., P.R. and C.B.; visualization, F.W. and P.R.; supervision, C.B.; project administration, F.W.; funding acquisition, C.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the German Federal Ministry of Economics and Energy (BMWi) within the Central Innovation Program for SMEs (ZIM). Grant number [ZF4041122BL7].

**Conflicts of Interest:** The authors declare no conflict of interest.
