# Parameter Identification of Fiber Orientation Models Based on Direct Fiber Simulation with Smoothed Particle Hydrodynamics

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## Abstract

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## 1. Introduction

#### 1.1. Point-Wise Interaction Methods

#### 1.2. Resolved Methods

## 2. Theory

#### 2.1. Fluid Model

#### 2.2. Interaction between Fluid and Solid Particles

#### 2.3. Basic Model for Flexible Fibers

#### 2.4. Viscous Traction at Fiber Surface

#### 2.5. Fiber Interactions

**Surface-Surface:**- If both interacting particles are located at the center of the fiber (e.g., they have two neighbor particles each, compare Figure 4 at ${t}_{0}$), the normal direction of contact pair $(i,j)$ can be computed using the cross product$${\mathbf{n}}_{ij}=\left[\phantom{\rule{-0.166667em}{0ex}}[{\mathbf{p}}_{i}\times {\mathbf{p}}_{j}]\phantom{\rule{-0.166667em}{0ex}}\right]$$$$[{\mathbf{p}}_{i},{\mathbf{n}}_{ij},-{\mathbf{p}}_{j}]{[{\mathcal{P}}_{ij},{D}_{ij},{\mathcal{P}}_{ji}]}^{\top}={\mathbf{x}}_{ij}$$
**Surface-End****and****End-Surface:**- If a particle of a fiber end interacts with a central particle of another fiber, the vector between these two particles ${\mathbf{x}}_{ij}$ can be used to obtain the normal direction by projection. It is assumed that ${\mathbf{p}}_{i}$ describes a unit vector in fiber direction at one fiber particle at position $\mathcal{A}$. Let ${\mathbf{x}}_{ij}$ be the vector from another fibers’ end particle $\mathcal{B}$ to the point $\mathcal{A}$. The closest point to $\mathcal{B}$ on a line with direction ${\mathbf{p}}_{i}$ is denoted as $\mathcal{C}$ and can be used to define the normal direction as$${\mathbf{n}}_{ij}=\left[\phantom{\rule{-0.166667em}{0ex}}\left[\overrightarrow{\mathcal{B}\mathcal{C}}\right]\phantom{\rule{-0.166667em}{0ex}}\right]=\left[\phantom{\rule{-0.166667em}{0ex}}[\overrightarrow{\mathcal{A}\mathcal{C}}-\overrightarrow{\mathcal{A}B}]\phantom{\rule{-0.166667em}{0ex}}\right].$$Using the definitions above and the fact that $\mathcal{C}$ is the projection of $\mathcal{B}$ to the line with direction ${\mathbf{p}}_{i}$, Equation (28) can be rewritten as$${\mathbf{n}}_{ij}=\left[\phantom{\rule{-0.166667em}{0ex}}[{\mathcal{P}}_{ij}{\mathbf{p}}_{i}-(-{\mathbf{x}}_{ij})]\phantom{\rule{-0.166667em}{0ex}}\right]=\left[\phantom{\rule{-0.166667em}{0ex}}[{\mathbf{x}}_{ij}+{\mathcal{P}}_{ij}{\mathbf{p}}_{i}]\phantom{\rule{-0.166667em}{0ex}}\right].$$The projection on the destination fiber is given as ${\mathcal{P}}_{ij}=-{\mathbf{p}}_{i}\xb7{\mathbf{x}}_{ij}$ and the contact distance is computed as ${D}_{ij}=\parallel {\mathbf{x}}_{ij}+{\mathcal{P}}_{ij}{\mathbf{p}}_{i}\parallel $.
**End-End:**- The simplest case is the interaction of two fiber ends. Here, the vector between those two particles can be simply determined by$${\mathbf{n}}_{ij}=\left[\phantom{\rule{-0.166667em}{0ex}}\left[{\mathbf{x}}_{ij}\right]\phantom{\rule{-0.166667em}{0ex}}\right]$$

#### 2.6. Time Integration and Implementation

## 3. Results

#### 3.1. Rotation and Bending in a Simple Shear Flow

#### 3.2. Parameter Identification for the Orientation Evolution in a Non-Dilute Short Fiber Suspensions

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Evaluation of the Surface Traction Integral

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**Figure 2.**A fiber is placed horizontally with $\varphi =\pi /2$ in a shear flow. The top and bottom walls have a prescribed velocity and a no-slip condition, the boundary conditions in ${x}_{1}$ direction are periodical.

**Figure 3.**Cylindrical fiber segment with length $\Delta L$, orientation $\mathbf{p}$ and an arbitrary surface normal $\mathbf{n}$.

**Figure 4.**Snapshots of two fibers in contact. At contact initiation ${t}_{0}$, ${\mathbf{p}}_{i}$ and ${\mathbf{p}}_{j}$ denote unit vectors for the fiber directions and ${\mathbf{x}}_{ij}$ is the vector between particles of one active contact pair $(i,j)$. The contact forces are indicated by arrows which scale with contact force magnitude and rotate in the subsequent time steps (${t}_{1},{t}_{2},{t}_{3}$).

**Figure 5.**Orientation angle ϕ for fiber in a shear flow. The fiber length ${L}_{\mathrm{f}}$ is expressed in multiples of the particle spacing Δx. The dashed gray line represents the solution to Jeffery’s equation with Zhang’s [53] fit. The solid black line represents the solution obtained with this SPH implementation. If the viscous surface traction term (Equation (24)) is neglected, the fiber stops rotating after one half rotation, as shown with the black dotted line.

**Figure 6.**Setup for the 3D shear with fibers in a cube of edge length $L=15\Delta x$. Lees-Edwards boundary conditions [54] are employed to induce a shear rate G. Therefore dummy particles (light gray) and particles leaving the domain in ${x}_{1}$-direction are shifted periodically in ${x}_{2}$ during each domain update. Conventional periodic boundary conditions apply to all other sides of the cube. The initial state is generated from fibers aligned in ${x}_{1}$-direction at unique random positions in the entire volume. (Some fibers appear longer in this figure due to other overlapping fibers behind it.)

**Figure 7.**Ensemble average of fiber orientation tensor components for fibers with aspect ratio r

_{p}= 4.43 (L

_{f}= 5) in a 3D shear flow and its comparison to the reduced strain model with optimal parameters. Each simulation result was obtained from five independently sampled initial configurations. The standard deviation is indicated by a light green filled area for each volume fraction.

**Figure 8.**Snapshot of fibers at 10% fiber volume fraction after 30 strains. The colors are introduced to distinguish fiber particles and represent the particle ID.

**Table 1.**Bending modes of a fiber with length ${L}_{\mathrm{f}}=11\Delta x$ for varied dimensionless stiffness. One example with $S=10$ and critical fiber bending angle ${\theta}_{\mathrm{c}}=\frac{\pi}{4}$ is shown to demonstrate fiber fracture.

Strain | 0 | $\frac{1}{4}\mathit{TG}$ | $\frac{1}{2}\mathit{TG}$ | $\frac{3}{4}\mathit{TG}$ | $\mathit{TG}$ |
---|---|---|---|---|---|

$S=5,{\theta}_{\mathrm{c}}=\infty $ | |||||

$S=10,{\theta}_{\mathrm{c}}=\infty $ | |||||

$S=10,{\theta}_{\mathrm{c}}=\frac{\pi}{4}$ | |||||

$S=20,{\theta}_{\mathrm{c}}=\infty $ | |||||

$S=100,{\theta}_{\mathrm{c}}=\infty $ |

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**MDPI and ACS Style**

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.
https://doi.org/10.3390/jcs4020077

**AMA Style**

Meyer N, Saburow O, Hohberg M, Hrymak AN, Henning F, Kärger L. Parameter Identification of Fiber Orientation Models Based on Direct Fiber Simulation with Smoothed Particle Hydrodynamics. *Journal of Composites Science*. 2020; 4(2):77.
https://doi.org/10.3390/jcs4020077

**Chicago/Turabian Style**

Meyer, Nils, Oleg Saburow, Martin Hohberg, Andrew N. Hrymak, Frank Henning, and Luise Kärger. 2020. "Parameter Identification of Fiber Orientation Models Based on Direct Fiber Simulation with Smoothed Particle Hydrodynamics" *Journal of Composites Science* 4, no. 2: 77.
https://doi.org/10.3390/jcs4020077