Research Article  Open Access
Haiying Zhou, Hesheng Liu, Qingsong Jiang, Tangqing Kuang, Zhixin Chen, Weiping Li, "Effect of Process Parameters on Short Fiber Orientation along the Melt Flow Direction in WaterAssisted Injection Molded Part", Advances in Materials Science and Engineering, vol. 2019, Article ID 7201215, 10 pages, 2019. https://doi.org/10.1155/2019/7201215
Effect of Process Parameters on Short Fiber Orientation along the Melt Flow Direction in WaterAssisted Injection Molded Part
Abstract
The short fiber orientation (SFO) distribution in the waterassisted injection molding (WAIM) is more complicated than that in traditional injection molding due to the new process parameters. In this work, an improved fiber orientation tensor method was used to simulate the SFO in WAIM. The result was compared with the scanning electron micrograph, which was consistent with the experiments. The effect of six process parameters, including filling time, melt temperature, mold temperature, delay time, water pressure, and water temperature, on the SFO along the melt flow direction were studied through orthogonal experimental design, range analysis, and variance analysis. An artificial neural network was used to establish the nonlinear agent model between the process parameters and A_{11} representing the fiber orientation in melt flow direction. Results show that water pressure, melt temperature, and water temperature have significant effects on SFO. The threedimensional (3D) response surfaces and contour plots show that the values of A_{11} decrease with the increase in water pressure and melt temperature and increase as the water temperature rises.
1. Introduction
With the development of advanced, economical, and environmentally friendly society, higher requirements are placed on the performance of plastic products. The shortfiberreinforced polymer composite (SFRPC) is a material with a polymer as a matrix and short fibers as a dispersed phase. Its characteristics are lightweight, high specific strength and specific modulus, stable chemical properties, heat resistance, and good wear resistance [1]. In recent years, SFRPCs have gradually replaced metal materials in some fields, making them widely used in aviation, automotive, shipping, and medicine [2, 3].
Fluidassisted injection molding is an emerging process that includes gasassisted injection molding (GAIM) and waterassisted injection molding (WAIM) [4, 5]. The fluids used in GAIM and WAIM are nitrogen and water, respectively. Due to incompressibility, high heat capacity, and good thermal conductivity of water, the advantages of WAIM over GAIM are high product efficiency, more uniform and thinner residual wall thickness (RWT) [6]. Based on whether or not the cavity is completely filled with melt, WAIM can be categorized into two types: shortshot WAIM and overflow WAIM. In shortshot WAIM, the mold cavity is partially filled with melt, followed by the injection of water into the core of melt. In overflow WAIM, the mold cavity is completely filled with melt, followed by the injection of water that pushes the melt into the overflow cavity to form a part with a hollow cross section. Compared with the standard injection molding, WAIM offers significant advantages in the preparation of shaped hollow plastic parts with uniform RWT. However, due to the difficulty in controlling the water injection pressure and the turbulence characteristics of the injection water, the quality of the products is not stable [7, 8]. Present researches on WAIM focus on the distribution of RWT [9], the length of water column penetration [10, 11], and the defects of molded parts [12, 13].
The RWT of waterassisted injection molded parts is thin, and the mechanical properties of the parts can be greatly improved by using SFRPC as a raw material. Many studies reported that the distribution of short fiber orientation (SFO) determined the mechanical and physical properties of the plastic parts, while fiber orientation was affected by mold structure, molding process parameters, flow field distribution, initial state of fibers, fiber properties, and matrix properties [14–16]. The molding process parameters influence temperature, velocity distribution, melt viscosity gradient, and flow field, which ultimately determine the fiber orientation. Liu et al. [7, 12] found that the short fibers mostly aligned along the melt flow direction in WAIM. Huang et al. [17] suggested that highpressure water penetration significantly affected fiber orientation in WAIM, and increasing melt temperature decreased fiber orientation. Systematic studies on the influence of process parameters on the SFO help understand the fiber orientation mechanism, optimize the SFO distribution, and improve the overall performance of the parts in WAIM.
WAIM, including the melt and highpressure water filling stages, is more complicated than the standard injection molding process. Due to the difficultly in accurately controlling all the process parameters simultaneously, the research of SFO in laboratory is performed for qualitative analysis. With the development of computer technology, the threedimensional numerical simulation technology developed rapidly, enabling simulating complex injection molding. The process parameters can be accurately controlled in simulation, but the reliability of SFO simulation depends on the accuracy of the mathematical model. The fiber orientation distribution in injection molding is very complicated microscopically. In the past three decades, theoretical studies on fibers suspension rheology have achieved a great success. Based on the classic fiber orientation models, including Jeffery hydrodynamics model, Folgar–Tucker model [18], and ARDRSC model [19], Tseng et al. recently proposed an improved iARDRPR model [20, 21], which can provide good simulation results of SFO in standard injection molding.
In this work, based on the iARDRPR model, the SFO in WAIM was simulated, and the results were compared with the scanning electron micrographs (SEMs) to verify the applicability of this model for WAIM. The influences of process parameters on the fiber orientation along the melt flow direction were studied through the methods of orthogonal experimental design, range analysis, and variance analysis. The nonlinear proxy model between process parameters and fiber orientation along the melt flow direction was constructed by artificial neural network (ANN), and the interaction effect of significant process parameters was investigated using 3D response surfaces and contour plots.
2. Methods
2.1. Fluid Mechanics Governing Equations
The movement of short fibers in injection molding is a transient, nonNewtonian, and nonisothermal process. In numerical simulations, the melt is considered to be incompressible and laminar, and the inertial term is ignored. The governing equations for transient and nonisothermal fluid motion in WAIM processes are as follows:where is the velocity vector, is the shear viscosity, is the pressure, ρ is the melt density, is the total stress tensor, is the gravity, is the specific heat capacity, T is the temperature, k is the thermal conductivity, t is the time, and is the shear rate.
2.2. Rheological Model
The CrossWLF rheological model with seven parameters was used in the simulation, which can describe complex viscosity behaviors, including the viscosity varying with shear rate and the zeroshearrate viscosity, depending on temperature and pressure [20].where η is the viscosity of the melt, is the zero shear viscosity, is the shear rate, is the material constant, n is the power law index in the shear rate, T is the melt temperature, is the glass transition temperature, and , , , , and are the constants associated with the selected material.
2.3. Fiber Orientation Model
The fiber orientation is often described by two methods including orientation probability density distribution function and orientation tensor. The orientation probability density distribution function is used to calculate the ratio of short fibers to the total number of fibers in a certain orientation direction. It is intuitive and easy to understand, but its wide application is limited due to the large amount of calculation. In this work, the numerical simulation of SFO in WAIM is based on the fiber orientation tensor evolution equation. To succinctly represent the orientation of a large number of fibers, a secondorder orientation tensor is defined as follows [18]:where is the probability density distribution function of the whole orientation space, is the fiber’s unit vector, is a symmetric matrix, and when = /3, it represents the orientation state isotropic, where represents an identity matrix. The three components A_{11}, A_{22}, and A_{33} on the diagonal represent the fiber orientation in the melt flow direction, the crossflow direction, and the thickness direction, respectively.
Tseng recently proposed a new fiber orientation prediction model (iARDRPR) [21] as below:where is a Jeffery hydrodynamic model, is an improved anisotropic rotational diffusion model with two effective parameters: fiberfiber interaction factor and fibermatrix interaction influence factor , and is the retarding principal rate model that contains the parameter α (0 < α < 1), which is mean to slow the response rate of fiber orientation. The orientation tensor will be influenced by the parameters, and the default parameters ( = 0.005, = 0.1, and α = 0.7) of iARDRPR model for short fibers were used in the simulations:where is the vortex tensor, is the deformation rate tensor, is the particle shape factor, and the fourthorder orientation tensor A_{4} can be estimated from A by using the eigenvaluebased optimal fitting (EBOF) closure [22] or the invariantbased optimal fitting (IBOF) closure [23]. EBOF was chosen in this paper.where is a material derivative of a diagonal tensor and its superscript is the intrinsic orientation kinetic (IOK) assumption, is the ith diagonal component of , is a rotation matrix, is the eigenvalue of matrix and .
In the simulation, the melt flow problem was solved firstly, and then the resulting velocity field was applied to compute the short fiber orientation.
2.4. Geometric Model
The geometric model is shown in Figure 1. The length of the functional hollow duct is 245 mm, and the diameter is 20 mm. The overflow cavity was used to accommodate the melt pushed out by the highpressure water. The connecting pipe between the duct and the overflow cavity had a diameter of 10 mm. The 3D model was created using Pro/E software. Moldex3D, developed by CoreTech System Co. Ltd., was used to simulate the melt filling and highpressure water penetration process of WAIM. The combined model, imported into the Moldex3D R15.0, was meshed with Boundary Layer Mesh format. In the simulation, a shortfiberreinforced PP (Fiberfil J68/30/E with a short grass fiber mass fraction of 30% and an aspect ratio of 20) was selected as material, and its properties were available in the data bank of Moldex3D. The simulation of overflow WAIM process was carried out. First, the melt was injected into the mold cavity of the functional plastic part. Second, after a short delay time, the highpressure water was injected into the core of the melt and penetrated along the core position with the least flow resistance and pushed the core melt into the overflow cavity to form a plastic part with a hollow cross section.
2.5. Orthogonal Experimental Design
During the injection molding process, due to the differences in viscosity gradient and velocity field distribution, the melt was sheared and stretched by the surrounding fluid, the shearing action causes the short fibers to align along the melt flow direction, and the stretching action causes the short fibers to orient along the stretch direction. Many factors affect the melt flow field in WAIM. The process parameters considered in this study include filling time (A), melt temperature (B), mold temperature (C), delay time (D), water pressure (E), and water temperature (F). These factors and their corresponding levels are listed in Table 1.

The full factorial experimental design method demands 5^{6} experiments for 6 factors (A, B, C, D, E, and F) with 5 levels. To reduce the number of experiments and comprehensively examine the influence of process parameters on SFO, the orthogonal experimental design (L_{25} (5^{6})) was used to arrange the test plan. In the final stage, the short fibers were mainly oriented along the melt flow direction. The values of the component of orientation tensor representing the melt flow direction were got at twenty different points along the thickness direction at the position I of RWT, and objective A_{11} is the average value of these points (Table 2).

2.6. Range Analysis
Range analysis, a statistical method, was used to analyze the experimental results obtained in the orthogonal experimental design. Range is defined as the difference between the maximum value and the minimum value. The factor with the maximum range is regarded as the most sensitive process parameter.where i (i = 1, 2, 3, 4, 5) is the level of a factor, j (j = A, B, C, D, E, and F) is a certain factor, is the sum value of A_{11} of all level in each factor, is the value of the kth experimental result, is the test times with the jth factor and the ith level, and range represents the influence degree of the factor. The larger the range is, the stronger the influence of the factor is.
2.7. Analysis of Variance
Analysis of variance was employed to determine which factors were significant. In this method, the F value of each factor is used to indicate the degree of influence.where T is the sum of test indexes, n is the number of levels, m is the number of factors, is the total differences, is the differences of test results caused by the change in every level of factor j, is the differences of the test caused by the error, is the total test times, and are the degrees of freedom of each factor and test error, respectively, and other symbols are the same as above.
Given a significant level , the criterion can be found in the F distribution table. The influence of a factor with F value greater than value is significant.
2.8. Artificial Neural Network (ANN)
ANN inspired by the biologic neural system is a computing model used to map linear or nonlinear relationships between factors and responses. ANN can work as a human brain to establish a sample model from the perspective of information processing, without prior information or heuristic assumptions. An ANN model comprises three parts: one input layer, one or more hidden layers, and one output layer. The numbers of neurons in the input and output layer are determined by the numbers of factors and responses, respectively. In general, the number of neurons in hidden layer is determined by trial and error.
According to Kolmogorov’s theorem [24], an ANN model with a single hidden layer has the ability to map any complex nonlinear relationship between input and output. In this study, an ANN model with one hidden layer was used for modeling. Gradient descent algorithm was used for training model. The transfer functions of Tansig and Purelin were employed in the hidden and output layer, respectively. By changing the number of neurons in the hidden layer from 5 to 15, the ANN topology of 6131 with the minimum mean square error between the targets and the outputs was determined, indicating that there were six neurons in the input layer, thirteen neurons in the hidden layer, and one neuron in the output layer (Figure 2).
3. Results and Discussion
3.1. Comparison between Simulation Result and Experimental Observation
Figure 3 shows the component of orientation tensor along the melt flow direction in a section taken from the position I (Figure 1) of the model used in the simulation with each process parameter set to level 3. The values of A_{11} first increase and then decrease from the region near the mold cavity to the region near the water channel, and the larger values of A_{11} indicate more short fibers orient in the flow direction in the corresponding region. Based on the SFO in the flow direction, the RWT can be divided into two regions, such as the ordered region with large A_{11} and disordered region with small A_{11}. To verify the simulation results, waterassisted injection molded tubes with shortfiberreinforced composite were observed by scanning electron microscopy (Nova NanoSEM 450) with an accelerating voltage of 5 kV. The tubes were produced by a laboratorydeveloped WAIM device improved from a 110 ton traditional injection molding machine. The material was PP containing 30% short grass fibers. The process parameters were set to the same as the simulation. A small sample was taken from the tube and immersed in liquid nitrogen for two hours and then cryofractured along the melt flow direction. The surface of the sample was subjected to goldsputtered treatment before observation. Figure 4 shows that the short fibers in the RWT mostly orient along the melt flow direction. Figures 4(a) and 4(b), the magnified micrographs taken from the ordered region and disordered region, indicate that more short fibers orient along the melt flow direction in the region near the mold cavity than that near the water channel. The experimental observations are consistent with the simulation results, indicating that the iARDRPR model can well predict the SFO distribution in overflow WAIM.
3.2. Analysis of Short Fiber Orientation Mechanism in WAIM
Figure 5 shows the values of A_{11} obtained from 100 points along the diameter in the simulation. The melt filling process was similar to the conventional injection molding process, in which the short fibers oriented with a typical shellcore structure. The short fibers in the shell layer mainly oriented along the melt flow direction due to the shear action, whereas the short fibers in the core layer weakly oriented due to stretch action. The values of A_{11} increased from about 0.7 to 0.88 within the thickness of 0 to 2 mm. This could be explained that the surface melt temperature decreases rapidly due to the cold mold cavity, and the short fibers are solidified without being sufficiently oriented; as the thickness increases, the cooling effect decreases and A_{11} increases.
In the water filling stage, the values of A_{11} in the region near the mold cavity are similar to that in the melt filling stage, which are caused by high shear action; the values of A_{11} decrease rapidly in the region near the water channel, indicating that the highpressure water filling has an important influence on the SFO in this region. To prevent the occurrence of water vaporization, the water filling process lasts very short, and the water column in a turbulent state exerts strong squeezing and friction at the interface between the melt and water. Furthermore, as the melt moves into the overflow cavity, the moving melt stretches the adjacent melt. The short fibers originally aligned along the flow direction readjust the posture due to the multiple disturbances, making short fibers in a disordered state in the region near water channel. In addition, the components of orientation tensors in all directions were assigned the values of 0.33 in the water channel with no fiber as shown in Figure 5.
3.3. Sensitivity Analysis
The results of range analysis are shown in Table 3. The rank of sensitivity of A_{11} to the six selected process parameters was determined and shown in the following based on the magnitude of the range R (water pressure, melt temperature, water temperature, mold temperature, filling rate, and delay time). As shown in Table 4, the results of the variance analysis indicate that the melt temperature and water pressure are highly significant factors for A_{11} with an F value of 7.346 and 8.612, respectively, and the water temperature with an F value of 4.102 is a significant factor for A_{11}. The filling rate, mold temperature, and delay time are not significant for A_{11} with the F values of 0.924, 1.163, and 0.913, respectively.

 
Significant. Highly significant. 
3.4. Artificial Neural Network Modeling
Experimental data (Table 2), obtained from orthogonal experimental design, were divided randomly into three data sets. 19 of overall 25 points were used to train the ANN model, and the others (3 + 3) were used to validate and test the model, respectively. The trained neural network was used to predict A_{11} in the orthogonal experimental scheme. As shown in Figure 6, the training sets and validation sets were almost on the diagonal line, and the test sets were very close to this line, indicating that the neural network has established the nonlinear relationship between the process parameters and A_{11} and could provide acceptable predictions.
3.5. Interaction Effect of Significant Process Parameter
Range and variance analysis results show that melt temperature, water pressure, and water temperature are significant for A_{11} of RWT in WAIM. The values of A_{11} were predicted by the trained neural network by changing two process parameters, while keeping the others constant. The 3D response surfaces and contour plots were used to describe the interaction effect of process parameters on A_{11}.
As shown in Figures 7 and 8, the value of A_{11} decreases as the melt temperature rises. The reasons come from two aspects. Firstly, the increasing melt temperature results in a decrease in viscosity. At the same shear rate, the shear stress decreases, and this is not conducive to the fiber orientation. Secondly, during the melt filling stage, the polymer molecular chains are also aligned, and the high melt temperature facilitates the deorientation behavior of the molecular chains after the completion of water injection process, which changes the orientation distribution of the short fibers and decreases A_{11}.
The values of A_{11} decrease with the increase in the water pressure (Figures 7 and 9). After melt filling stage, the short fibers in the shell layer highly orient along the melt flow direction. During the highpressure water filling stage, the inner layers of RWT bear the pressing and friction of water column as well as the stretching of the moving melt. The short fibers originally oriented along the melt flow direction are postureadjusted and arranged in a disordered state. The greater the water pressure is, the stronger the disturbance to the RWT is, and the smaller A_{11} in the melt flow direction is.
While the water injection temperature rises, the values of A_{11} of RWT in WAIM increase (Figures 8 and 9). The polymer melt has a viscoelastic property and can return to the original state after the external force is removed. The multiple disturbances cause the short fibers to be disordered in the inner layers of RWT during the water filling stage, and the higher water temperature favors the elastic recovery of the short fiber posture, thereby resulting in an increase of A_{11}.
4. Conclusions
The iARDRPR model was used to simulate the orientation of short fibers in WAIM. Compared with the SEM micrographs, the simulation results indicated that this model was suitable for the SFO prediction in overflow WAIM. The short fibers in the region near the mold cavity mostly oriented in the melt flow direction, while those in the region near the water channel were disordered. The melt temperature, water pressure, and water temperature have significant effects on the SFO along the melt flow direction. It was found that the values of A_{11} decreased with increasing water injection pressure and melt temperature and increased with increasing water temperature. Finally, this study is beneficial for the subsequent optimization of SFO in WAIM.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was funded by the National Natural Science Foundation of China (Grant nos. 21664002 and 51563010) and the Jiangxi Provincial Key Technology R&D Program (Grant no. 20161BBE50073).
References
 S. Mortazavian and A. Fatemi, “Fatigue behavior and modeling of short fiber reinforced polymer composites: a literature review,” International Journal of Fatigue, vol. 70, pp. 297–321, 2015. View at: Publisher Site  Google Scholar
 C. Röhrig, T. Scheffer, and S. Diebels, “Mechanical characterization of a short fiberreinforced polymer at room temperature: experimental setups evaluated by an optical measurement system,” Continuum Mechanics and Thermodynamics, vol. 29, no. 5, pp. 1093–1111, 2017. View at: Publisher Site  Google Scholar
 B. M. Rudresh and B. N. Ravi Kumar, “Influence of experimental parameters on friction and wear mechanisms of PA66/PTFE blend reinforced with glass fiber,” Transactions of the Indian Institute of Metals, vol. 71, no. 2, pp. 339–349, 2018. View at: Publisher Site  Google Scholar
 S.J. Liu and Y.C. Wu, “Dynamic visualization of cavityfilling process in fluidassisted injection moldinggas versus water,” Polymer Testing, vol. 26, no. 2, pp. 232–242, 2007. View at: Publisher Site  Google Scholar
 L. Li, Y. Peng, and W. Wei, “Recent advances on fluid assisted injection molding technique,” Recent Patents on Mechanical Engineering, vol. 7, no. 1, pp. 82–91, 2014. View at: Publisher Site  Google Scholar
 H. P. Park, B.S. Cha, S.B. Park et al., “A study on the void formation in residual wall thickness of fluidassisted injection molding parts,” Advances in Materials Science and Engineering, vol. 2014, Article ID 238251, 6 pages, 2014. View at: Publisher Site  Google Scholar
 S. Liu, M. Lin, and Y. Wu, “An experimental study of the waterassisted injection molding of glass fiber filled polybutyleneterephthalate (PBT) composites,” Composites Science and Technology, vol. 67, no. 78, pp. 1415–1424, 2007. View at: Publisher Site  Google Scholar
 T. Q. Kuang, B. P. Xu, G. F. Zhou, and L. S. Turng, “Numerical simulation on residual thickness of pipes with curved sections in waterassisted coinjection molding,” Journal of Applied Polymer Science, vol. 132, no. 34, pp. 1–10, 2015. View at: Publisher Site  Google Scholar
 H. Park, B. S. Cha, and B. Rhee, “Experimental and numerical investigation of the effect of process conditions on residual wall thickness and cooling and surface characteristics of waterassisted injection molded hollow products,” Advances in Materials Science and Engineering, vol. 2015, Article ID 161938, 11 pages, 2015. View at: Publisher Site  Google Scholar
 J. G. Yang, X. H. Zhou, and Q. Niu, “Model and simulation of water penetration in waterassisted injection molding,” The International Journal of Advanced Manufacturing Technology, vol. 67, no. 1–4, pp. 367–375, 2013. View at: Publisher Site  Google Scholar
 T. Q. Kuang, K. Zhou, L. X. Wu, G. F. Zhou, and L. S. Turng, “Experimental study on the penetration interfaces of pipes with different crosssections in overflow waterassisted coinjection molding,” Journal of Applied Polymer Science, vol. 133, no. 1, pp. 1–8, 2016. View at: Publisher Site  Google Scholar
 S.J. Liu and Y.S. Chen, “The manufacturing of thermoplastic composite parts by waterassisted injectionmolding technology,” Composites Part A: Applied Science and Manufacturing, vol. 35, no. 2, pp. 171–180, 2004. View at: Publisher Site  Google Scholar
 S.J. Liu and S.P. Lin, “Study of “fingering” in water assisted injection molded composites,” Composites Part A: Applied Science and Manufacturing, vol. 36, no. 11, pp. 1507–1517, 2005. View at: Publisher Site  Google Scholar
 I. Fiebig and V. Schoeppner, “Factors influencing the fiber orientation in welding of fiberreinforced thermoplastics,” Welding in the World, vol. 62, no. 5, pp. 997–1012, 2018. View at: Publisher Site  Google Scholar
 Q.S. Jiang, H.S. Liu, Q.W. Xiao, S.F. Chou, A.H. Xiong, and H.R. Nie, “Threedimensional numerical simulation of total warpage deformation for shortglassfiberreinforced polypropylene composite injectionmolded parts using coupled FEM,” Journal of Polymer Engineering, vol. 38, no. 5, pp. 493–502, 2018. View at: Publisher Site  Google Scholar
 S. C. Lee, D. Y. Yang, J. Ko, and J. R. Youn, “Effect of compressibility on flow field and fiber orientation during the filling stage of injection molding,” Journal of Materials Processing Technology, vol. 70, no. 13, pp. 83–92, 1997. View at: Publisher Site  Google Scholar
 H.X. Huang, R.H. Zhou, and C. Yang, “Fiber orientation propelled by highpressure water penetration in waterassisted injection molded fiberreinforced thermoplastics part,” Journal of Composite Materials, vol. 47, no. 2, pp. 183–190, 2013. View at: Publisher Site  Google Scholar
 S. G. Advani and C. L. Tucker, “The use of tensors to describe and predict fiber orientation in short fiber composites,” Journal of Rheology, vol. 31, no. 8, pp. 751–784, 1987. View at: Publisher Site  Google Scholar
 B. O. Agboola, D. A. Jack, and S. MontgomerySmith, “Effectiveness of recent fiberinteraction diffusion models for orientation and the part stiffness predictions in injection molded shortfiber reinforced composites,” Composites Part A: Applied Science and Manufacturing, vol. 43, no. 11, pp. 1959–1970, 2012. View at: Publisher Site  Google Scholar
 H.C. Tseng, R.Y. Chang, and C.H. Hsu, “Numerical prediction of fiber orientation and mechanical performance for short/long glass and carbon fiberreinforced composites,” Composites Science and Technology, vol. 144, pp. 51–56, 2017. View at: Publisher Site  Google Scholar
 H.C. Tseng, R.Y. Chang, and C.H. Hsu, “The use of shearratedependent parameters to improve fiber orientation predictions for injection molded fiber composites,” Composites Part A: Applied Science and Manufacturing, vol. 104, pp. 81–88, 2018. View at: Publisher Site  Google Scholar
 J. S. Cintra Jr. and C. L. Tucker III, “Orthotropic closure approximations for flowinduced fiber orientation,” Journal of Rheology, vol. 39, no. 6, pp. 1095–1122, 1995. View at: Publisher Site  Google Scholar
 D. H. Chung and T. H. Kwon, “Invariantbased optimal fitting closure approximation for the numerical prediction of flowinduced fiber orientation,” Journal of Rheology, vol. 46, no. 1, pp. 169–194, 2002. View at: Publisher Site  Google Scholar
 V. Kůrková, “Kolmogorov’s theorem and multilayer neural networks,” Neural Networks, vol. 5, no. 3, pp. 501–506, 1992. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2019 Haiying Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.