# Fatigue Life Prediction for Injection-Molded Carbon Fiber-Reinforced Polyamide-6 Considering Anisotropy and Temperature Effects

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

^{2}= 0.9457 correlation coefficient. The present fatigue life prediction of CF-PA6 can be adopted when designers make suitable decisions considering the effects of temperature and anisotropy.

## 1. Introduction

^{2}= 0.9457. The results provide the possibility of predicting the fracture behavior considering the anisotropic behavior and temperature effect of CF-PA6 with the proposed model.

## 2. Fatigue Life Prediction Procedure

## 3. Material Characterization

#### 3.1. Static Mechanical Properties Characterization

^{3}/h heat flow in an oven dryer to remove the humidity before proceeding with experiments. All static experiments were repeated three times to increase experimental accuracy. The strain was obtained by means of a 20 mm gauge length extensometer in the case of the uniaxial tensile test.

#### 3.2. Cyclic Mechanical Properties Characterization

^{2}to 10

^{6}. The hysteresis loop moves in a positive direction of strain as the load amplitude increases, indicating that the degree of asymmetry intensifies. An example of a hysteresis loop of one low-cycle fatigue and one high-cycle fatigue is reported in Figure 4.

#### 3.3. Numerical Analysis

_{I}is the fiber interaction coefficient.

_{0}to λ

_{m,I}in Table 5. σ

_{0}is the stress level at which plastic strain becomes dominant. Using the experimental results by specimen direction and temperature, the constant with the highest agreement between the analysis and experimental result was derived through reverse engineering. Each constant was determined through the BFGS optimization technique to minimize the area difference between the load–displacement curve FEA result and the experimental result according to each coefficient combination. The orientation and elastic modulus of the fiber had a more significant influence on the calculation of the anisotropic behavior of the material, so the polymer matrix elastic modulus was derived relatively low.

_{0}to λ

_{m,I}. Therefore, the elastic modulus of fiber was obtained lower than the actual fiber value. Even though the elastic modulus was not directly evaluated and used, the coefficient obtained is meaningful because the stress–strain value utilized in predicting fatigue life can be derived through numerical analysis by minimizing the error between analysis and experiment. When an optimization technique is applied by experimentally deriving the elastic modulus of each matrix and the fiber, the constants other than the elastic modulus would have different values in Table 5, and a new combination constant would be derived.

## 4. Fatigue Life Prediction Model

_{f}, to predict fatigue life through the material coefficients A and c.

_{ref}) of CF-PA6, 20 °C, and the experimental temperature in Arrhenius law.

## 5. Results

^{2}) is slightly lower when considering various temperatures simultaneously compared to a single temperature; however, the proposed semi-empirical model confirms that fatigue life expectancy is well predicted. Based on a validated numerical analysis model, as presented in the previous section of the paper, each combination of investigated temperature and specimen direction, the von Mises stress, and strain at the minimum and maximum load associated with the fatigue experiment were exported from the simulation. The temperature was selected as the reference, room, and experimental temperature inside the heating chamber. The constants of the developed model Equation (5) were calculated by A = 5.6734 and c = −0.692 based on a total set of 54 data, as shown in Figure 9, and the correlation coefficient is R

^{2}= 0.9457. Thus, the reliability of the proposed function of the anisotropic fatigue test data integration obtained under various stress states and temperatures is demonstrated.

^{2}= 0.9729 and R

^{2}= 0.6548, respectively, which was further excellent in the performance of Equation (5). It was confirmed that the fatigue life prediction model developed using the principal stress worked well without considering the fiber orientation distribution, which is difficult to calculate, as an angle. The comparison of calculated results is shown in Figure 10.

## 6. Conclusions

- This study predicted the fatigue life expectancy of CF-PA6, a plastic reinforced with short fiber, through a strain-based semi-empirical model with a high correlation factor. A three-point bending test was performed to investigate various multi-axial stress states in actual components.
- A meaningful, intuitive fatigue life prediction model is proposed considering anisotropy as a stress term, which directly utilizes experimental results with a theoretical approach. It can be concluded that the fatigue life of materials with high temperature and anisotropy fiber orientation and polymers can be predicted with reasonable accuracy.
- SEM photography revealed that the higher the temperature and fatigue fracture cycle, the greater the deformation of the polymer matrix, and inversely, the more the deformation of the fiber. The higher the temperature, the more evenly the fiber’s fracture cross-section is.
- The developed numerical model and structural equation are highly consistent between experiments and FEA results. Furthermore, they could accurately export stress and strain as inputs to a semi-empirical model.
- The usefulness of the results proposed in this paper can be outlined in two parts. First, the paper summarizes the static and fatigue behavior considering the anisotropy and temperature of short fiber-reinforced plastic materials, which are increasingly utilized exponentially in the industry. Secondly, it provides insight into the availability of the developed semi-empirical model to predict the fatigue life of CF-PA6.
- The use of FRP affected by temperature and fiber orientation is a remaining challenge for research on much colder temperatures and compressive forces below 0 °C. In addition, using compressive force in testing and investigating the mechanical properties of FRP can accurately describe the complex stress states in industries. Therefore, it can be a better solution to predict the fatigue life and composite use of FRP considering low temperature and compression stress states in the future.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) The injection-molded plate with a runner. (

**b**) 0°, 45°, and 90° directions-cut ASTM-D 638 specimens. (

**c**) The dimensions of 1.2 times ASTM-D 638 specimen.

**Figure 3.**(

**a**) Test jig dimensions for the upper part (

**left**), lower part (

**right**), and experimental setting for three-point bending test in an environmental chamber. (

**b**) Schematic dimensions of injection plates for extracting three-point bending test specimens and specimen shape and specifications with ASTM-D 790.

**Figure 4.**Test results of hysteresis loops for low- and high-cycle fatigues (90° and T = 60 °C). (

**a**) fatigue failure at 3800 cycles (0.6 kN load amplitude and 1.0 mean load) and (

**b**) fatigue failure at 346,900 cycles (0.2 kN load amplitude and 0.5 mean load).

**Figure 5.**(

**a**) Fiber orientation distribution from AME. (

**b**) Computational domain for structural analysis for the uniaxial specimen (

**left**) and the three-point bending specimen (

**right**).

**Figure 6.**Experimental results of true and engineering (eng.) stress–strain curves for 0°, 45°, and 90° specimens in uniaxial tensile tests.

**Figure 7.**Load–displacement comparison between experimental and FEA results for 0°, 45°, and 90° specimens in uniaxial tensile test conditions.

**Figure 8.**Load–displacement comparison between experimental and FEA results for 0°, 45°, and 90° specimens in three-point bending test conditions.

**Figure 10.**Fatigue experiments regression on (

**a**) modified Equation (5) model and (

**b**) Equation (4) model.

Temperature [°C] | Selected Mean Load for Each Specimen Direction [kN] | ||
---|---|---|---|

0° | 45° | 90° | |

40 | 2.000 | 1.250 | 1.125 |

60 | 1.750 | 1.125 | 1.000 |

100 | 1.500 | 1.000 | 0.750 |

Temperature [°C] | Specimen Angle [°] | Max. Load [kN] | Min. Load [kN] | Mean Load [kN] | Amplitude Load [kN] | N_{f}[Cycles] |
---|---|---|---|---|---|---|

40 | 0 | 3.20 | 0.80 | 2.00 | 1.20 | 1512 |

3.40 | 0.60 | 2.00 | 1.40 | 980 | ||

2.80 | 1.20 | 2.00 | 0.80 | 3400 | ||

1.40 | 0.60 | 1.00 | 0.40 | 106,263 | ||

2.40 | 2.20 | 2.30 | 0.10 | 674,785 | ||

45 | 1.90 | 0.60 | 1.25 | 0.65 | 7500 | |

2.10 | 0.40 | 1.25 | 0.85 | 2420 | ||

1.70 | 0.80 | 1.25 | 0.45 | 46,700 | ||

0.90 | 0.40 | 0.65 | 0.25 | 268,410 | ||

1.60 | 1.40 | 1.50 | 0.10 | 340,485 | ||

90 | 1.75 | 0.50 | 1.12 | 0.62 | 1774 | |

1.95 | 0.30 | 1.12 | 0.82 | 1670 | ||

1.55 | 0.70 | 1.12 | 0.42 | 9800 | ||

0.90 | 0.30 | 0.60 | 0.30 | 76,000 | ||

1.50 | 1.30 | 1.40 | 0.10 | 361,807 | ||

60 | 0 | 2.90 | 0.60 | 1.75 | 1.15 | 1820 |

3.25 | 0.25 | 1.75 | 1.50 | 770 | ||

2.60 | 0.90 | 1.75 | 0.85 | 8700 | ||

1.30 | 0.50 | 0.90 | 0.40 | 79,805 | ||

2.10 | 1.90 | 2.00 | 0.10 | 892,500 | ||

45 | 1.75 | 0.50 | 1.12 | 0.62 | 5840 | |

1.95 | 0.30 | 1.12 | 0.82 | 642 | ||

1.55 | 0.70 | 1.12 | 0.42 | 10,400 | ||

0.80 | 0.30 | 0.55 | 0.25 | 89,600 | ||

1.50 | 1.30 | 1.40 | 0.10 | 428,500 | ||

90 | 1.50 | 0.50 | 1.00 | 0.50 | 11,650 | |

1.60 | 0.40 | 1.00 | 0.600 | 3800 | ||

1.30 | 0.70 | 1.00 | 0.30 | 51,007 | ||

0.70 | 0.30 | 0.50 | 0.20 | 346,900 | ||

1.40 | 1.20 | 1.30 | 0.10 | 276,000 | ||

100 | 0 | 2.25 | 0.75 | 1.50 | 0.75 | 4753 |

2.50 | 0.50 | 1.50 | 1.00 | 1670 | ||

2.00 | 1.00 | 1.50 | 0.50 | 23,060 | ||

1.00 | 0.40 | 0.70 | 0.30 | 660,650 | ||

1.80 | 1.60 | 1.70 | 0.10 | 475,439 | ||

45 | 1.50 | 0.50 | 1.00 | 0.50 | 4460 | |

1.60 | 0.40 | 1.00 | 0.60 | 1790 | ||

1.30 | 0.70 | 1.00 | 0.30 | 29,800 | ||

0.70 | 0.30 | 0.50 | 0.20 | 108,699 | ||

1.40 | 1.20 | 1.30 | 0.10 | 90,335 | ||

90 | 1.15 | 0.35 | 0.75 | 0.40 | 5700 | |

1.35 | 0.15 | 0.75 | 0.60 | 1213 | ||

1.00 | 0.50 | 0.75 | 0.25 | 39,355 | ||

0.50 | 0.20 | 0.35 | 0.15 | 270,883 | ||

1.10 | 0.90 | 1.00 | 0.10 | 166,068 |

Temperature [°C] | Specimen Angle [°] | Max. Load [kN] | Min. Load [kN] | Mean Load [kN] | Amplitude Load [kN] | N_{f}[Cycles] |
---|---|---|---|---|---|---|

40 | 0 | 0.13 | 0.11 | 0.12 | 0.01 | 2109 |

45 | 0.10 | 0.80 | 0.90 | 0.01 | 905 | |

90 | 0.08 | 0.06 | 0.07 | 0.01 | 2160 | |

60 | 0 | 0.11 | 0.09 | 0.10 | 0.01 | 15,041 |

45 | 0.08 | 0.06 | 0.07 | 0.01 | 9800 | |

90 | 0.08 | 0.06 | 0.07 | 0.01 | 5061 | |

100 | 0 | 0.09 | 0.07 | 0.08 | 0.01 | 1200 |

45 | 0.06 | 0.04 | 0.05 | 0.01 | 671 | |

90 | 0.05 | 0.03 | 0.04 | 0.01 | 2780 |

Symbol | Definition |
---|---|

ε_{p,eff} | Effective plastic strain |

T | Temperature in the environmental chamber in °C |

K | Strength coefficient |

n | Hardening exponent |

α_{m} | Weight factor for the fiber direction |

β_{m} | Weight factor for the direction normal to the fibers |

E_{m} | Polymer matrix elastic modulus |

E_{f} | Fiber’s elastic modulus |

λ_{m}_{, I} | The first eigenvalue of the fiber orientation matrix in the region with strong fiber alignment with the polymer flow |

**Table 5.**Ramberg–Osgood model constants for each experimental temperature in injection molding process analysis by AME.

T [°C] | σ_{0} [MPa] | n | α_{m} | β_{m} | E_{m} [GPa] | E_{f} [GPa] | λ_{m}_{,I} |
---|---|---|---|---|---|---|---|

40 | 250.10 | 3.34 | 18.85 | 11.33 | 0.78 | 60.02 | 0.85 |

60 | 215.76 | 4.14 | 7.13 | 8.40 | 0.56 | 30.37 | 0.85 |

100 | 227.12 | 4.46 | 16.71 | 17.65 | 0.40 | 30.48 | 0.85 |

Temperature [°C] | Tensile Strength of Each Specimen Direction [MPa] | ||
---|---|---|---|

0° | 45° | 90° | |

40 | 130.3 | 101.3 | 77.2 |

60 | 122.6 | 79.6 | 60.8 |

100 | 98.8 | 62.8 | 51.2 |

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**MDPI and ACS Style**

Choi, J.; Andrian, Y.O.; Lee, H.; Lee, H.; Kim, N.
Fatigue Life Prediction for Injection-Molded Carbon Fiber-Reinforced Polyamide-6 Considering Anisotropy and Temperature Effects. *Materials* **2024**, *17*, 315.
https://doi.org/10.3390/ma17020315

**AMA Style**

Choi J, Andrian YO, Lee H, Lee H, Kim N.
Fatigue Life Prediction for Injection-Molded Carbon Fiber-Reinforced Polyamide-6 Considering Anisotropy and Temperature Effects. *Materials*. 2024; 17(2):315.
https://doi.org/10.3390/ma17020315

**Chicago/Turabian Style**

Choi, Joeun, Yohanes Oscar Andrian, Hyungtak Lee, Hyungyil Lee, and Naksoo Kim.
2024. "Fatigue Life Prediction for Injection-Molded Carbon Fiber-Reinforced Polyamide-6 Considering Anisotropy and Temperature Effects" *Materials* 17, no. 2: 315.
https://doi.org/10.3390/ma17020315