# Effects of Fiber Shape on Mechanical Properties of Fiber Assemblies

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Flexible Fiber Model

^{2}). The translational movement of the node sphere is driven by the normal contact force ${\mathit{F}}_{\mathrm{n}i}^{\mathrm{c}}$, tangential contact force ${\mathit{F}}_{\mathrm{t}i}^{\mathrm{c}}$, normal bond force ${\mathit{F}}_{\mathrm{n}i}^{\mathrm{b}}$, tangential bond force ${\mathit{F}}_{\mathrm{t}i}^{\mathrm{b}}$, contact damping forces ${\mathit{F}}_{\mathrm{n}i}^{\mathrm{cd}}$ and ${\mathit{F}}_{\mathrm{t}i}^{\mathrm{cd}}$, bond damping forces ${\mathit{F}}_{\mathrm{n}i}^{\mathrm{bd}}$ and ${\mathit{F}}_{\mathrm{t}i}^{\mathrm{bd}}$, and gravitational force ${m}_{i}\mathit{g}$. Rotational movement is induced by the moments ${\mathit{M}}_{i}^{\mathrm{c}}$, ${\mathit{M}}_{i}^{\mathrm{b}}$, ${\mathit{M}}_{i}^{\mathrm{cd}},$ and ${\mathit{M}}_{i}^{\mathrm{bd}}$ due to the contact forces, bond forces/moments, contact damping forces, and bond damping forces/moments, respectively. In the present fiber model, the basic element is a sphero-cylinder, as shown in Figure 1a. Thus, the contact detection between two fibers is based on the contact between two sphero-cylinders. As demonstrated in the previous work [10], three different contact types exist for a contact between two sphero-cylinders: hemisphere–hemisphere contact, hemisphere–cylinder contact, and cylinder–cylinder contact. The specific, geometry-dependent models of the normal and tangential contact forces, ${\mathit{F}}_{\mathrm{n}i}^{\mathrm{c}}$ and ${\mathit{F}}_{\mathrm{t}i}^{\mathrm{c}}$, which were proposed in the previous work [10] to calculate the contact forces for each contact type, are utilized in the present simulations. The forces $\mathit{F}$ on the right-hand side of Equation (1) have the unit of N, and the moments $\mathit{M}$ on the right-hand side of Equation (2) have the unit of N·m.

^{2}) and length (m), respectively, of the bond; $I=\pi {r}^{4}/4$ is the area moment of inertia (m

^{4}); ${I}_{\mathrm{p}}=\pi {r}^{4}/2$ is the polar area moment of inertia (m

^{4}); r is the radius of the fiber (m); and $\mathit{dt}$ is the time step (s). An illustration of bond forces and moments acting on a node sphere is shown in Figure 1b.

## 3. Compression Tests

#### 3.1. Fiber Rings

^{®}materials testing machine, moves down at a constant speed of 0.1 m/s into the container to compress the rubber ring bed, as shown in Figure 2b. The load exerted on the punch and the punch displacement are measured by the testing machine during the compression process.

#### 3.2. S-Shaped Fibers

^{3}. The other properties of the S-shaped fibers are the same to those of the above rubber rings (see Table 1). The curvature of an S-shaped $\kappa $ is defined as the reciprocal of the radius (i.e., $1/R$), as shown in Figure 5.

^{−1}and S-shaped fibers with $\kappa $ = 100 m

^{−1}are shown in Figure 7. The load–unload cycle 2 shifts to the right-hand side after the cycle 1 due to the fiber rearrangement and the consolidation of the fiber bed after the first load–unload cycle. The $P-\varphi $ loop of the load-unload cycle 3 is very close to that of the cycle 2. At a given solid volume fraction $\varphi $, a larger pressure $P$ is observed for the S-shaped fibers compared to the straight fibers, and a sharper increase in $P$ with $\varphi $ is obtained for the S-shaped fibers.

^{−1}and $\kappa $ = 100 m

^{−1}.

^{−1}compared to the crooked fibers with $\kappa $ = 100 m

^{−1}. However, larger than average fiber–fiber contact forces are obtained for the crooked fibers with $\kappa $ = 100 m

^{−1}(Figure 10), causing larger pressures in the compression of the fibers with $\kappa $ = 100 m

^{−1}. Thus, the straight fibers have more but weaker contacts and hence lower pressures in the compression than the crooked fibers.

^{−1}) exhibit the largest average inclination angle.

## 4. Tensile Tests

^{−1}, and rapid decreases in ${\sigma}_{yy}$ are observed for the fibers with $\kappa \ge $ 58 m

^{−1}. In addition, the yield tensile stress ${\sigma}_{yy}^{0}$ generally decreases with increasing fiber curvature $\kappa $ due to reduction in the interlocking of the fibers.

## 5. Shear Tests

^{−1}) with various sample lengths. The shear stress increases with the shear strain at the early stage and then fluctuates around a constant level, which is referred to as the steady state. At a given shear strain, the shear stress generally increases as the normalized sample length ${L}_{s}/d$ is reduced. In the shear process, the coordination number also increases with the shear strain at the early stage and then achieves a constant level, as shown in Figure 16b. Nevertheless, the coordination number, unlike the shear stress, shows a nonmonotonic and irregular dependence on the sample length.

^{−1}, U-shaped fibers with $\theta $ = 0° and 90°, and Z-shaped fibers. Similar to the yield tensile stress (see Figure 14), the yield shear stress ${\sigma}_{xy}^{0}$ decreases as the sample length ${L}_{s}/d$ increases. As shown in Figure 17a, for a given sample length, ${\sigma}_{xy}^{0}$ decreases as the fiber curvature $\kappa $ increases. The S-shaped fibers with a larger curvature $\kappa $ = 100 m

^{−1}, U-shaped fibers, and Z-shaped fibers have the smallest yield shear stresses. The average coordination numbers are insensitive to the normalized sample length, as shown in Figure 17b. In general, the coordination number decreases with increasing fiber curvature $\kappa $, and smaller coordination numbers are obtained for the U-shaped and Z-shaped fibers.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) A sketch of the flexible fiber model, (

**b**) bond forces, and moments exerted on a node sphere, as well as (

**c**) contact forces distributed to the two node spheres.

**Figure 4.**Comparison of experimental and simulation results with rubber rings: (

**a**) bed height versus loading force and (

**b**) pressure versus solid volume fraction.

**Figure 5.**Sketches of S-shaped, U-shaped, and Z-shaped fibers. For the U-shaped and Z-shaped fibers, the length of the middle part is twice the length of each tip.

**Figure 7.**$P-\varphi $ curves for the first three load–unload cycles of (

**a**) straight fibers ($\kappa $ = 0 m

^{−1}) and (

**b**) S-shaped fibers ($\kappa $ = 100 m

^{−1}).

**Figure 8.**Pressure $P$ versus solid volume fraction $\varphi $ for (

**a**) the first loading path and (

**b**) the second loading path. The pressure $P$ versus fiber curvature $\kappa $ at $\varphi $ = 0.4 is plotted in (

**c**).

**Figure 9.**Variation of coordination number with (

**a**) solid volume fraction and (

**b**) pressure in the load–unload cycle 1.

**Figure 10.**Variation of average fiber–fiber contact force with (

**a**) solid volume fraction and (

**b**) pressure in the load–unload cycle 1.

**Figure 11.**Probability density distributions of fiber inclination angles for the packings (

**a**) before and (

**b**) after the first load–unload cycle. Average inclination angle as a function of fiber curvature is plotted in (

**c**).

**Figure 13.**(

**a**) Tensile stress ${\sigma}_{yy}$ versus tensile strain ${\u03f5}_{yy}$ for the S-shaped fibers with various curvatures $\kappa $ and (

**b**) dependence of yield tensile stress ${\sigma}_{yy}^{0}$ on the fiber curvature. The normalized fiber assembly sample length is specified as ${L}_{s}/d$ = 4.17.

**Figure 14.**Yield tensile stress versus sample length with various cross-sectional areas of the samples of straight fibers ($\kappa $ = 0).

**Figure 16.**Variations of (

**a**) shear stress and (

**b**) coordination number with shear strain for the straight fibers with various sample lengths.

**Figure 17.**Variation of (

**a**) yield shear stress and (

**b**) average coordination number with normalized sample size for the fibers of various shapes. The half-length of an error bar represents a standard deviation from the average value at the steady state of shearing deformation.

**Figure 18.**(

**a**) An illustration of Feret diameter of a fiber. (

**b**) Yield shear stress versus normalized maximum Feret diameter of a fiber, ${D}_{F}^{\mathit{max}}/{l}_{f}$, for various sample lengths. The half-length of an error bar represents a standard deviation from the average value at the steady state of shearing deformation.

Parameters | Values |
---|---|

Fiber shape | Ring |

Outer diameter of a fiber ring, ${D}_{\mathit{out}}$ (mm) | 23 |

Diameter of fiber line, ${d}_{s}$ (mm) | 2.4 |

Number of fiber rings, (-) | 300 |

Material density, ${\rho}_{\mathrm{f}}$ (kg/m^{3}) | 1340 |

Material Poisson’s ratio, $\zeta $ (-) | 0.5 |

Elastic modulus for fiber–fiber contact, ${E}_{\mathrm{c}}$ (Pa) | 7.5 $\times $10^{5} |

Elastic modulus for fiber bond, ${E}_{\mathrm{b}}$ (Pa) | 7.5 $\times $10^{5} |

Shear modulus for fiber bond, ${G}_{\mathrm{b}}$ (Pa) | ${G}_{\mathrm{b}}=0.5{E}_{\mathrm{b}}/\left(1+\zeta \right)$ |

Fiber–fiber friction coefficient, ${\mu}_{\mathrm{f}\mathrm{f}}$ (-) | 1.4 |

Contact damping coefficient, ${\beta}_{\mathrm{c}}$ (-) | 0.113 |

Bond damping coefficient, ${\beta}_{\mathrm{b}}$ (-) | 3.35 $\times $10^{−2} |

Diameter of cylindrical container, (mm) | 80 |

Fiber-cylindrical wall friction coefficient, (-) | 0.6 |

Fiber-flat wall friction coefficient, (-) | 0.6 |

Loading speed, $v$ (m/s) | 0.1 |

Parameters | Values |
---|---|

Dimensions of domain, ${l}_{x}\times {l}_{y}\times {l}_{z}$ (mm^{3}) | 60 $\times $300 $\times $60 |

Fiber shape | S-shape, U-shape, Z-shape |

Linear length of a fiber, ${l}_{f}$ (mm) | 62.83 |

Diameter of fiber line, ${d}_{s}$ (mm) | 2.4 |

Number of fibers, (-) | 300, 500 |

Material density, ${\rho}_{\mathrm{f}}$ (kg/m^{3}) | 1340 |

Material Poisson’s ratio, $\zeta $ (-) | 0.5 |

Elastic modulus for fiber–fiber contact, ${E}_{\mathrm{c}}$ (Pa) | 7.5 $\times $10^{5} |

Elastic modulus for fiber bond, ${E}_{\mathrm{b}}$ (Pa) | 7.5 $\times $10^{5} |

Shear modulus for fiber bond, ${G}_{\mathrm{b}}$ (Pa) | ${G}_{\mathrm{b}}=0.5{E}_{\mathrm{b}}/\left(1+\zeta \right)$ |

Fiber-fiber friction coefficient, ${\mu}_{\mathrm{f}\mathrm{f}}$ (-) | 1.4 |

Contact damping coefficient, ${\beta}_{\mathrm{c}}$ (-) | 0.113 |

Bond damping coefficient, ${\beta}_{\mathrm{b}}$ (-) | 3.35 $\times $10^{−2} |

Fiber-flat wall friction coefficient | 0.6 |

Loading speed, $v$ (m/s) | 0.1 |

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**MDPI and ACS Style**

Xu, D.; Ma, H.; Guo, Y. Effects of Fiber Shape on Mechanical Properties of Fiber Assemblies. *Materials* **2023**, *16*, 2712.
https://doi.org/10.3390/ma16072712

**AMA Style**

Xu D, Ma H, Guo Y. Effects of Fiber Shape on Mechanical Properties of Fiber Assemblies. *Materials*. 2023; 16(7):2712.
https://doi.org/10.3390/ma16072712

**Chicago/Turabian Style**

Xu, Dandan, Huibin Ma, and Yu Guo. 2023. "Effects of Fiber Shape on Mechanical Properties of Fiber Assemblies" *Materials* 16, no. 7: 2712.
https://doi.org/10.3390/ma16072712