# Assessment of Analytical Orientation Prediction Models for Suspensions Containing Fibers and Spheres

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Measuring Fiber Orientation States by Orientation Tensors

#### 1.3. Orientation Prediction by Jeffery and Folgar Tucker

#### 1.4. Further Analytical Orientation Prediction Models

## 2. Method

#### 2.1. Suspension Simulations

#### 2.2. Computation of FT Parameters

## 3. Results

#### 3.1. Influence of the FT Model Formulation

#### 3.2. Influence of the Flow Type

#### 3.3. Influence of Disk Size

#### 3.4. Comparison of Existing Data

#### 3.5. Accuracy of FT Model in Contrast to Other Analytical Orientation Prediction Models

## 4. Discussion

#### 4.1. Formulation

#### 4.2. Underlying Flow Profile

#### 4.3. Suspension Composition

#### 4.4. Comparison with Literature

#### 4.5. Accuracy of the FT Model in Comparison to Other Prediction Models

#### 4.6. Limitation of This Work

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

2D | two-dimensional |

3D | three-dimensional |

ARD | Anisotropic Rotary Diffusion |

ARD-RSC | Anisotropic Rotary Diffusion Method with Reduced Strain Closure |

DNS | Direct Numerical Simulations |

FT | Folgar Tucker |

iARD | improved Anisotropic Rotary Diffusion |

IBOF | Invariant-based optimal fitting |

IRD | Isotropic Rotary Diffusion |

MRD | Moldflow Rotational Diffusion |

NAT | Natural Closure Approximiation |

OWR | Orthotropic Closure Approximation |

OWC | one-way coupling |

pARD | principle Anisotropic Rotary Diffusion |

RPR | Retarding Principal Rate |

RSC | Reduced Strain Closure |

RVE | Representative Volume Element |

SPH | Smoothed Particle Hydrodynamics |

TWC | two-way coupling |

## References

- Wang, Z.; Smith, D.E. Rheology Effects on Predicted Fiber Orientation and Elastic Properties in Large Scale Polymer Composite Additive Manufacturing. J. Compos. Sci.
**2018**, 2, 10. [Google Scholar] [CrossRef][Green Version] - Simon, S.A.; Bechara Senior, A.; Osswald, T. Experimental Validation of a Direct Fiber Model for Orientation Prediction. J. Compos. Sci.
**2020**, 4, 59. [Google Scholar] [CrossRef] - Fu, S.Y.; Hu, X.; Yue, C.Y. Effects of Fiber Length and Orientation Distribution On The Mechanical Properties of Short-Fiber-Reinforced Polymers. J. Soc. Mater. Sci. Jpn.
**1999**, 48, 74–83. [Google Scholar] [CrossRef][Green Version] - Terao, T.; Zhi, C.; Bando, Y.; Mitome, M.; Tang, C.; Golberg, D. Alignment of Boron Nitride Nanotubes in Polymeric Composite Films for Thermal Conductivity Improvement. J. Phys. Chem. C
**2010**, 114, 4340–4344. [Google Scholar] [CrossRef] - Yang, D.; Zhang, H.; Wu, J.; McCarthy, E.D. Fibre flow and void formation in 3D printing of short-fibre reinforced thermoplastic composites: An experimental benchmark exercise. Addit. Manuf.
**2020**, 101686. [Google Scholar] [CrossRef] - Hashemi, M.R.; Fatehi, R.; Manzari, M.T. SPH simulation of interacting solid bodies suspended in a shear flow of an Oldroyd-B fluid. J. Non–Newton. Fluid Mech.
**2011**, 166, 1239–1252. [Google Scholar] [CrossRef] - Meyer, N.; Saburow, O.; Hohberg, M.; Hrymak, A.N.; Henning, F.; Kärger, L. Parameter Identification of Fiber Orientation Models Based on Direct Fiber Simulation with Smoothed Particle Hydrodynamics. J. Compos. Sci.
**2020**, 4, 77. [Google Scholar] [CrossRef] - Do-Quang, M.; Amberg, G.; Brethouwer, G.; Johansson, A.V. Simulation of finite-size fibers in turbulent channel flows. Phys. Rev. E Stat. Nonlinear Soft Matter Phys.
**2014**, 89. [Google Scholar] [CrossRef] - Lu, J.; Das, S.; Peters, E.; Kuipers, J. Direct numerical simulation of fluid flow and mass transfer in dense fluid-particle systems with surface reactions. Chem. Eng. Sci.
**2018**, 176, 1–18. [Google Scholar] [CrossRef] - Zhang, J.; Wang, C. Numerical Study of Lateral Migration of Elliptical Magnetic Microparticles in Microchannels in Uniform Magnetic Fields. Magnetochemistry
**2018**, 4, 16. [Google Scholar] [CrossRef][Green Version] - Hwang, H.; Son, G. Direct numerical simulation of 3D particle motion in an evaporating liquid film. J. Mech. Sci. Technol.
**2016**, 30, 3929–3934. [Google Scholar] [CrossRef] - Polfer, P.; Kraft, T.; Bierwisch, C. Suspension modeling using smoothed particle hydrodynamics: Accuracy of the viscosity formulation and the suspended body dynamics. Appl. Math. Model.
**2016**, 40, 2606–2618. [Google Scholar] [CrossRef] - Gudžulić, V.; Dang, T.S.; Meschke, G. Computational modeling of fiber flow during casting of fresh concrete. Comput. Mech.
**2018**, 31, 751. [Google Scholar] [CrossRef] - Li, M.; Zhang, Y.; Zhang, S.; Hou, B.; Zhou, H. Experimental investigation and modeling study of the fiber orientation behavior. Eng. Comput.
**2020**, 367. [Google Scholar] [CrossRef] - Advani, S.G.; Tucker, C.L. The Use of Tensors to Describe and Predict Fiber Orientation in Short Fiber Composites. J. Rheol.
**1987**, 31, 751–784. [Google Scholar] [CrossRef] - Jeffery, G.B. The Motion of Ellipsoidal Particles Immersed in a Viscous Fluid. Proc. R. Soc. A: Math. Phys. Eng. Sci.
**1922**, 102, 161–179. [Google Scholar] [CrossRef][Green Version] - Folgar, F.; Tucker, C.L. Orientation Behavior of Fibers in Concentrated Suspensions. J. Reinf. Plast. Compos.
**1984**, 3, 98–119. [Google Scholar] [CrossRef] - Advani, S.G.; Tucker, C.L. A numerical simulation of short fiber orientation in compression molding. Polym. Compos.
**1990**, 11, 164–173. [Google Scholar] [CrossRef] - Hand, G.L. A theory of anisotropic fluids. J. Fluid Mech.
**1962**, 13, 33–46. [Google Scholar] [CrossRef] - Doi, M. Molecular dynamics and rheological properties of concentrated solutions of rodlike polymers in isotropic and liquid crystalline phases. J. Polym. Sci. Polym. Phys. Ed.
**1981**, 19, 229–243. [Google Scholar] [CrossRef] - Lipscomb, G.G.; Denn, M.M.; Hur, D.U.; Boger, D.V. The flow of fiber suspensions in complex geometries. J. Non–Newton. Fluid Mech.
**1988**, 26, 297–325. [Google Scholar] [CrossRef] - Du Chung, H.; Kwon, T.H. Invariant-based optimal fitting closure approximation for the numerical prediction of flow-induced fiber orientation. J. Rheol.
**2002**, 46, 169–194. [Google Scholar] [CrossRef][Green Version] - Verleye, V.; Couniot, A.; Dupret, F. Numerical prediction of fibre orientation in complex injection-moulded parts. Eng. Sci.
**1994**, 4. [Google Scholar] [CrossRef] - Bay, R.S. Fiber Orientation in Injection-Molded Composites: A Comparison of Theory and Experiment. Ph.D. Thesis, University of Illinois, Champaign, IL, USA, 1991. [Google Scholar]
- Mezher, R.; Perez, M.; Scheuer, A.; Abisset-Chavanne, E.; Chinesta, F.; Keunings, R. Analysis of the Folgar & Tucker model for concentrated fibre suspensions in unconfined and confined shear flows via direct numerical simulation. Compos. Part A Appl. Sci. Manuf.
**2016**, 91, 388–397. [Google Scholar] [CrossRef] - Phan-Thien, N.; Fan, X.J.; Tanner, R.I.; Zheng, R. Folgar–Tucker constant for a fibre suspension in a Newtonian fluid. J. Non–Newton. Fluid Mech.
**2002**, 103, 251–260. [Google Scholar] [CrossRef] - Fan, X.; Phan-Thien, N.; Zheng, R. A direct simulation of fibre suspensions. J. Non–Newton. Fluid Mech.
**1998**, 74, 113–135. [Google Scholar] [CrossRef] - Ranganathan, S.; Advani, S.G. Fiber–fiber interactions in homogeneous flows of nondilute suspensions. J. Rheol.
**1991**, 35, 1499–1522. [Google Scholar] [CrossRef] - Advani, S.G. Flow and Rheology in Polymer Composites Manufacturing: Processing Short-Fiber Systems, 1st ed.; Composite Materials Series; Elsevier Science: Amsterdam, The Netherlands, 1994; Volume 10. [Google Scholar]
- Wang, J.; O’Gara, J.F.; Tucker, C.L. An objective model for slow orientation kinetics in concentrated fiber suspensions: Theory and rheological evidence. J. Rheol.
**2008**, 52, 1179–1200. [Google Scholar] [CrossRef] - Tseng, H.C.; Chang, R.Y.; Hsu, C.H. Phenomenological improvements to predictive models of fiber orientation in concentrated suspensions. J. Rheol.
**2013**, 57, 1597–1631. [Google Scholar] [CrossRef] - Phelps, J.H.; Tucker, C.L. An anisotropic rotary diffusion model for fiber orientation in short- and long-fiber thermoplastics. J. Non–Newton. Fluid Mech.
**2009**, 156, 165–176. [Google Scholar] [CrossRef] - Tseng, H.C.; Chang, R.Y.; Hsu, C.H. Comparison of recent fiber orientation models in injection molding simulation of fiber-reinforced composites. J. Thermoplast. Compos. Mater.
**2020**, 33, 35–52. [Google Scholar] [CrossRef] - Favaloro, A.J.; Tucker, C.L. Analysis of anisotropic rotary diffusion models for fiber orientation. Compos. Part A Appl. Sci. Manuf.
**2019**, 126, 105605. [Google Scholar] [CrossRef] - Tseng, H.C.; Chang, R.Y.; Hsu, C.H. The use of principal spatial tensor to predict anisotropic fiber orientation in concentrated fiber suspensions. J. Rheol.
**2018**, 62, 313–320. [Google Scholar] [CrossRef] - Bakharev, A. (Ed.) Using New Anisotropic Rotational Diffusion Model to Improve Prediction of Short Fibers in Thermoplastic Injection Molding; Society of Plastics Engineers: Lubbock, TX, USA, 2018. [Google Scholar]
- Dietemann, B.; Bosna, F.; Kraft, T.; Kruggel-Emden, H.; Bierwisch, C. Folgar Tucker constants fitted against simulations of various suspensions under shear and/or elongation. Fraunhofer Fordatis
**2021**. [Google Scholar] [CrossRef] - Dietemann, B.; Bosna, F.; Lorenz, M.; Travitzky, N.; Kruggel-Emden, H.; Kraft, T.; Bierwisch, C. Modeling Robocasting with Smoothed Particle Hydrodynamics: Printing gap-spanning filaments. Addit. Manuf.
**2020**, 36, 101488. [Google Scholar] [CrossRef] - Dietemann, B.; Bosna, F.; Lorenz, M.; Travitzky, N.; Kruggel-Emden, H.; Kraft, T.; Bierwisch, C. Numerical study of texture in material extrusion: Orientation in a multicomponent system of spheres and ellipsoids. J. Non–Newton. Fluid Mech.
**2021**, 291, 104532. [Google Scholar] [CrossRef] - Cintra, J.S.; Tucker, C.L. Orthotropic closure approximations for flow–induced fiber orientation. J. Rheol.
**1995**, 39, 1095–1122. [Google Scholar] [CrossRef] - Du Chung, H.; Kwon, T.H. Improved model of orthotropic closure approximation for flow induced fiber orientation. Polym. Compos.
**2001**, 22, 636–649. [Google Scholar] [CrossRef] - Takano, M. Viscosity Effect on Flow Orientation of Short Fibers; Monsanto Research Corporation: Springfield, MO, USA, 1973. [Google Scholar]
- Yamane, Y.; Kaneda, Y.; Dio, M. Numerical simulation of semi-dilute suspensions of rodlike particles in shear flow. J. Non–Newton. Fluid Mech.
**1994**, 54, 405–421. [Google Scholar] [CrossRef] - Férec, J.; Ausias, G.; Heuzey, M.C.; Carreau, P.J. Modeling fiber interactions in semiconcentrated fiber suspensions. J. Rheol.
**2009**, 53, 49–72. [Google Scholar] [CrossRef][Green Version] - Bertevas, E.; Férec, J.; Khoo, B.C.; Ausias, G.; Phan-Thien, N. Smoothed particle hydrodynamics (SPH) modeling of fiber orientation in a 3D printing process. Phys. Fluids
**2018**, 30, 103103. [Google Scholar] [CrossRef] - Jezek, J.; Schulmann, K.; Paterson, S. Modified Jeffery model: Influence of particle concentration on mineral fabric in moderately concentrated suspensions. J. Geophys. Res. Solid Earth
**2013**, 118, 852–861. [Google Scholar] [CrossRef] - Cieslinski, M.J.; Wapperom, P.; Baird, D.G. Fiber orientation evolution in simple shear flow from a repeatable initial fiber orientation. J. Non–Newton. Fluid Mech.
**2016**, 237, 65–75. [Google Scholar] [CrossRef] - Willems, F.; Reitinger, P.; Bonten, C. Calibration of Fiber Orientation Simulations for LFT—A New Approach. J. Compos. Sci.
**2020**, 4, 163. [Google Scholar] [CrossRef] - Favaloro, A.J.; Tseng, H.C.; Pipes, R.B. A new anisotropic viscous constitutive model for composites molding simulation. Compos. Part A Appl. Sci. Manuf.
**2018**, 115, 112–122. [Google Scholar] [CrossRef] - Tseng, H.C.; Favaloro, A.J. The use of informed isotropic constitutive equation to simulate anisotropic rheological behaviors in fiber suspensions. J. Rheol.
**2019**, 63, 263–274. [Google Scholar] [CrossRef] - Li, T.; Luyé, J.F. Flow-Fiber Coupled Viscosity in Injection Molding Simulations of Short Fiber Reinforced Thermoplastics. Int. Polym. Process.
**2019**, 34, 158–171. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**Illustration of an extrusion process with a zoom into the suspension showing round and elongated particles within a homogeneous matrix material.

**Figure 3.**Micro-structures examples where in each column we show one variation of the quantity being denoted on top of the column.

**Figure 4.**Schematic drawing of RVEs under deformation: (

**a**) planar elongation, (

**b**) simple shear, (

**c**) combined shear and elongation.

**Figure 5.**Exemplary Folger Tucker (FT) curves (dashed lines) being fitted to the simulation curves (solid lines).

**Figure 6.**Influence of the dimensionality of the FT formulation for the case of shear and elongation for various FT parameters and two aspect ratios. The black line denotes the solution according to the Jeffery model.

**Figure 7.**Distribution of Folgar Tucker parameters among all simulations for the 2D and 3D formulation of the Folgar Tucker model.

**Figure 9.**${C}_{\mathrm{I}}$ over area ratio of disks to fibers ${A}_{\mathrm{d}}/{A}_{\mathrm{f}}$.

**Figure 10.**Literature data of Figure 2 shown in gray with an additional blue layer showing the FT parameters of this work.

**Figure 11.**Total deviation between the numerical data and the Jeffery model (no parameter), the FT model (1 parameter), the Reduced Strain Closure (RSC) model (2 parameters), the MOldflow Rotational Diffusion (MRD) model (4 parameter) and the Anisotropic Rotary Diffusion (ARD) model (5 parameters). The total deviation is normalized by the deviation of the Jeffery model.

Model Name | Abbreviation | Parameters |
---|---|---|

Jeffery [16] | - | 0 |

Folgar Tucker [17] | FT | 1 |

Reduced Strain Closure [30] | RSC | 2 |

improved Anisotropic Rotary Diffusion Retarding Principal Rate [31] | iARD-RPR | 3 |

principle Anisotropic Rotary Diffusion [35] | pARD | 3 |

Principal Anisotropic Rotary Diffusion Retarding Principal Rate [35] | pARD-RPR | 3 |

Moldflow Rotational Diffusion [36] | MRD | 4 |

Anisotropic Rotary Diffusion [32] | ARD | 5 |

Anisotropic Rotary Diffusion Reduced Strain Closure [32] | ARD-RSC | 6 |

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**MDPI and ACS Style**

Dietemann, B.; Bosna, F.; Kruggel-Emden, H.; Kraft, T.; Bierwisch, C. Assessment of Analytical Orientation Prediction Models for Suspensions Containing Fibers and Spheres. *J. Compos. Sci.* **2021**, *5*, 107.
https://doi.org/10.3390/jcs5040107

**AMA Style**

Dietemann B, Bosna F, Kruggel-Emden H, Kraft T, Bierwisch C. Assessment of Analytical Orientation Prediction Models for Suspensions Containing Fibers and Spheres. *Journal of Composites Science*. 2021; 5(4):107.
https://doi.org/10.3390/jcs5040107

**Chicago/Turabian Style**

Dietemann, Bastien, Fatih Bosna, Harald Kruggel-Emden, Torsten Kraft, and Claas Bierwisch. 2021. "Assessment of Analytical Orientation Prediction Models for Suspensions Containing Fibers and Spheres" *Journal of Composites Science* 5, no. 4: 107.
https://doi.org/10.3390/jcs5040107