# Numerical and Experimental Investigation on Bending Behavior for High-Performance Fiber Yarns Considering Probability Distribution of Fiber Strength

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## Abstract

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## 1. Introduction

_{yarn}, and realistic contact area Ar. An improved constitutive model with probability distribution constitutive laws was employed to establish the models and simulate the bending behavior of yarns. The probability distribution in terms of crimp strain is introduced to characterize the upper limit of crimp region. The model is demonstrated on carbon fiber yarns using the explicit solver Abaqus/Explicit. Meanwhile, to explore how the yarns’ architecture affects the bending behavior, the geometric deformation and bending behavior of yarns with various twists and plies were predicted on alumina and fiber quartz yarns, respectively. The model also provided the calculation approach for the bending stiffness of yarns at specific deflection.

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Experiment Method and Set Up

## 3. Simulation Details

#### 3.1. Constitutive Model Description

_{i}(i = 1~n) is defined as crimp length of the i-th fiber, which is arranged from small to large. Hence, the bending strain outside the neutral axis of the i-th fiber is obtained as:

_{0}is the initial yarn length. ε is the yarn strain. ${\epsilon}_{i}^{l}=\frac{{l}_{i}}{{l}_{0}}$ is defined as crimp strain of the i-th fiber and is used to characterize the crimp behavior of fibers within a yarn [21].

_{f}is Young’s modulus of the fiber, ${\epsilon}_{i}^{b}$ and ${\epsilon}_{i}^{l}$ are breaking strain and crimp strains of the i-th fiber.

_{f}is the apparent cross-section area of the fiber. The probability density function p(x) is the derivative of P(x). The stress of yarn with N fibers can be defined as:

#### 3.2. Simulation of Bending Test

## 4. Results and Discussion

#### 4.1. Verification of Numerical Simulation Model

_{yarn}and Ar of yarn were compared to experimental results using the present model. Here, EI

_{yarn}was calculated based on Equation (9) [12]:

_{yarn}and Ar at the deflection of 3 mm. It was found that the error of the experiment is larger than the simulation. This may be due to the fact that fiber rearrangement, and the accompanying cross-sectional change, is a dominant influence mechanism in the bending process of yarns in the experiment. Additionally, it is worth noting that the determined EI

_{yarn}by simulation falls within the range determined by the experiment whose changing rate is 1.7%. For contact area morphology, both showed a low center and high surrounding, the height changing rate was about 0.7 mm. The Ar of experiment and simulation were 17.5 and 15.8 mm

^{2}, respectively, whose changing rate for Ar was 12.6%.

_{yarn}and Ar) parameters, which validates the universality of the present model. Hence, it shows that the simulation model can be employed to explore the effect of more details on yarn bending properties.

#### 4.2. Influence of Twist Level on Bending Behavior

_{yarn}and Ar that are determined according to the methods described in the previous section for different twisted yarns at a deflection of 3 mm. Firstly, one can see that EI

_{yarn}is inversely proportional to twist level, especially 150 tpm to 200 tpm, which can be explained using the deformation mechanisms described. Yarns with a small twist result in weak cohesion and a great degree of freedom of fibers. Therefore, the EI

_{yarn}can be considered as the sum of each fiber. Conversely, yarns with a greater twist are susceptible to deformation. In addition, the results of bending simulation show that the effect of twist level on EI

_{yarn}is mainly concentrated in a finite range, whose conclusions are similar to the ones in the literature. Furthermore, the relationship between the realistic contact area and twist level was recorded at a deflection of 3 mm by the numerical simulation method in Figure 8b. It is shown that the Ar decreased non-linearly as the twist level increased, which is the same trend as the EI

_{yarn}’s. This can be explained by fiber rearrangement theory [7,37,38], that is, mainly due to the great degree of freedom between fibers within small twist yarns, a new contact surface of yarn is generated by rearrangement of inner layer fibers to the outer layer. As the twist level continues to increase, it is more difficult for the fibers to move with each other and the rate of change of Ar gradually decreases. It can be obtained that the rates of change of the realistic contact areas with different twists are 2.9, 2.5, 1.5, and 1.4%, respectively, further validating the above explanation from Figure 8b.

#### 4.3. Influence of Ply on Bending Behavior

_{yarn}and Ar were employed to characterize the bending behavior of plied yarn. Figure 10 shows the values for the EI

_{yarn}and Ar after the bending of plied yarns, which illustrates that the effect of ply number on the bending behavior is significant. The bending load of plied yarn showed a non-linear increasing trend with increasing deflection during the bending, which can be divided into two phases, that is, the deformation and bending phase in Figure 10a. The bending load of the defection of 3 mm was proportional to the number of plies. However, it is seen that as the number of plies increased, the non-linear variation trend of the bending load at 3 mm gradually became obvious, especially in 3-ply and 6-ply. This means that the effect of inter-fiber friction force on the bending load of multi-plies yarns is significant. Figure 10b represents the relationships of EI

_{yarn}and Ar to the number of plies. There was an increase in EI

_{yarn}of yarns with the increasing number of plies at the deflection of 3 mm. This is a result of the increase in the number of fibers in the cross-section of yarn with the increase in the number of plies. Furthermore, the trend described is explained by the following equation:

_{yarn}is the bending stiffness of fiber.

^{2}of 1-ply to 96.2 mm

^{2}of 10-ply. The overall behavior was similar to Ar in Figure 8b, but with a few notable differences: somewhat greater range of variation in the results, and finally, a more marked transition was evident at around 3-ply where the rate of change of Ar changed. The rates of change were 31%, 30%, 47%, and 31%, respectively, within the scope of the current research, for which the arrangement of fibers may be responsible.

## 5. Conclusions

_{yarn}, and realistic contact area Ar after the comparison between the experimental measurements and the simulation. Furthermore, the predictions of bending behavior for twisted and plied yarns were carried out by the described model. It is shown that the bending load decreases gradually as the twist level increases and increases as the number of plies increases at the deflection of 3 mm. The EI

_{yarn}and Ar are inversely proportional to twist level of yarns though, which varies on a minor interval. However, mainly influenced by the number of fibers, the EI

_{yarn}and Ar increase with the increasing number of strands, which varies on a large interval. In the future, more detailed characterization parameters that cannot be experimentally obtained need to be explored to analyze the mechanical behavior of yarns through simulation models.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**The 3D segmented clockwise twisted yarns. (Note: the gray is a cross-section of yarn, and the green is a cross-section of fiber).

**Figure 6.**Comparison of experimental and simulation results of yarn bending behavior: (

**a**) bending load–deflection curve, (

**b**) resulting values of EI

_{yarn}and Ar (taking carbon fiber yarn as an example).

**Figure 7.**Prediction of the geometric model with different twist levels: (

**a**) 50 tpm; (

**b**) 80 tpm; (

**c**) 100 tpm; (

**d**) 150 tpm; and (

**e**) 200 tpm.

**Figure 8.**Prediction of bending behavior of twisted yarn: (

**a**) curve of deflection and bending load, (

**b**) realistic contact area and bending stiffness for different twisted yarns at a deflection of 3 mm.

**Figure 9.**The prediction of the geometric model with the different number of plies: (

**a**) 1 ply; (

**b**) 2 plies; (

**c**) 3 plies; (

**d**) 6 plies; and (

**e**) 10 plies.

**Figure 10.**Prediction of bending behavior of plied yarn: (

**a**) curve of deflection and bending load, (

**b**) realistic contact area and bending stiffness at a deflection of 3 mm.

Yarn and Fiber Type | Density (g/cm^{3}) | Tex (g/1000 m) | Twist Level (tpm) | Radius of Fiber (μm) |
---|---|---|---|---|

Carbon fiber yarn | 1.91 | 218 | 0 | 5.0 |

Alumina fiber yarn | 2.88 | 330 | 170 | 7.0 |

Quartz fiber yarn | 2.20 | 190 | 80 | 7.5 |

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**MDPI and ACS Style**

Wang, Y.; Li, X.; Xie, J.; Wu, N.; Jiao, Y.; Wang, P.
Numerical and Experimental Investigation on Bending Behavior for High-Performance Fiber Yarns Considering Probability Distribution of Fiber Strength. *Textiles* **2023**, *3*, 129-141.
https://doi.org/10.3390/textiles3010010

**AMA Style**

Wang Y, Li X, Xie J, Wu N, Jiao Y, Wang P.
Numerical and Experimental Investigation on Bending Behavior for High-Performance Fiber Yarns Considering Probability Distribution of Fiber Strength. *Textiles*. 2023; 3(1):129-141.
https://doi.org/10.3390/textiles3010010

**Chicago/Turabian Style**

Wang, Yu, Xuejiao Li, Junbo Xie, Ning Wu, Yanan Jiao, and Peng Wang.
2023. "Numerical and Experimental Investigation on Bending Behavior for High-Performance Fiber Yarns Considering Probability Distribution of Fiber Strength" *Textiles* 3, no. 1: 129-141.
https://doi.org/10.3390/textiles3010010