# S-N Curve Characterisation for Composite Materials and Prediction of Remaining Fatigue Life Using Damage Function

^{*}

## Abstract

**:**

## 1. Introduction

_{max}) at the lowest number of loading cycles. However, no one seems to have paid attention to how valid such adopted ultimate strength values are when used for the fatigue characterisation of polymeric matrix materials for composites.

## 2. The Theory

#### 2.1. S-N Curve Model

_{f}) in the model with the half cycle (N = N

_{0}) is given as a function of applied peak stress (σ

_{max}):

_{uT}= ultimate tensile strength, and α, β = damage parameters.

_{f}is the fatigue damage at tensile fatigue failure [8] defined as,

#### 2.2. Prediction of Remaining Fatigue Life

_{0}= 0.5 cycles. Note the value of 0.3 is from −log(N

_{0}).

#### 2.3. Determination of the Exponent n

_{fB}) and A (= D

_{fA}) (Figure 1a) and for ${\sigma}_{Hmax}$ and ${\sigma}_{Lmax}$ respectively using Equation (4);

_{f}at point B (= N

_{HB}) and point C (= N

_{HC}) using Equation (1);

_{fB}and N

_{fC}are N

_{f}(see Equation (1)) at ${\sigma}_{Hmax}$ and ${\sigma}_{Lmax}$ respectively;

## 3. Experimental

#### 3.1. Material and Specimens

#### 3.2. Mechanical Tests

## 4. Development of Method for Data Point at the Lowest Number of Loading Cycles

^{2}) of 0.99. Accordingly, the extrapolated value using the least square line for the first 0.5 cycle stress (σ) at log(−1) or 0.1 sec is found to be 52 MPa. It should be noted that the extrapolated stress (52 MPa) and highest experimental stress (50.85 MPa) obtained at a crosshead speed of 1000 mm/min are 14% and 11% higher, respectively, than the lowest (45.63 MPa) experimental ultimate stress obtained at 1 mm/min.

## 5. Fatigue Results and Discussion

_{u}= 52 MPa obtained from extrapolation, and the dashed S-N curve for σ

_{u}= 45.6 MPa obtained at a crosshead speed of 1 mm/min. The solid S-N curve appears to represent experimental data adequately, whereas the other appears to significantly deviate from the experimental data. The deviation of the dashed line is obviously caused by the introduction of the data point for σ

_{u}= 45.6 MPa, which is the violation of the fatigue damage axiom [8]. The parameters α and β in Equation (1) were obtained to be logα = −7.016 (or α = 9.65 × 10

^{−8}) and β = 1.031 for the solid S–N curve, and logα = −6.7072 (or α = 1.96 × 10

^{−7}and β = 0.8598 for the dashed S-N curve. The parameter values for the latter would not have been possible to obtained without removing an invalid data point for log$\left(\partial {D}_{f}/\partial {N}_{f}\right)$ versus log${\sigma}_{max}$ (see Equation (3)) from the violation resulted. Accordingly, as the results indicated, the data point for the ultimate strength corresponding to the lowest number of loading cycles should be obtained using the adequately verified method rather than at an arbitrary loading rate. Otherwise, the result is erroneous outcomes not only for the S–N characterisation but also for various predictions.

_{max}= 50 MPa displays no visible cracks outside the fracture surface, indicating that fatigue damage is not much spread across the whole specimen prior to the breaking point. This is not unexpected at a high σ

_{max}, because the fatigue damage tends to accumulate less at the high σ

_{max}due to the small number of loading cycles. However, all other fatigue specimens display multiple cracks in the form of damage (indicated with solid arrows) except at σ

_{max}= 35 MPa. Additionally, some whitening is seen at σ

_{max}= 43 MPa in another form of fatigue damage. It is also seen at σ

_{max}= 50 MPa that there is a tendency of the fracture angle towards 45°, resembling the broken tensile specimens (Figure 7), and a significant permanent deformation along the fracture path occurred whereas all other specimens display approximately horizontal cracking paths. The damage mechanism of the fatigued specimens mainly involves crack initiation and then propagation. The observations here would represent the final stage of the mechanism involved.

_{u}= 52 MPa obtained from extrapolation, experimental results, and loading paths. Each of the triangle symbols in Figure 10a,b represent the final breaking point at N = N

_{f2}of each fatigue specimen (notation is given using one of the data sets in the figure.) The accuracies of the prediction depend on how close the experimental results are to the S-N curve. Figure 10a shows the high–low loading where the first maximum stress is denoted by σ

_{max}

_{1}and the second maximum stress is denoted by σ

_{max}

_{2}. Each location of point b was found according to Equation (7) with a validated exponent n = 10.1 using the Matlab script in the Appendix A. Additionally, detailed numerical values are listed in Table 1 with accuracies calculated using (logN

_{f}−logN

_{f2})/logN

_{f}. The prediction of the remaining fatigue lives appear to be in good agreement with experimental results.

_{max}

_{2}= 45 MPa may be noticed. The scatter may be due to the instability from the high stress (σ

_{max}

_{2}= 45 MPa) which is close to the ultimate strength of 45.65 at the crosshead speed of 1 mm/min. A combined plot for both high-low and low–high loadings is given in Figure 10c, showing the clustered data points along the S-N curve as an overall view of the experimental data points.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{Hmax}) and low (σ

_{Lmax}) stresses

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**Figure 1.**Calculation and notation: (

**a**) sequence for finding exponent n value at given high (${\sigma}_{Hmax}$) and low (${\sigma}_{Lmax}$) stresses; and (

**b**) notation on schematic S–N plane.

**Figure 4.**Partial results following the initial fatigue loading set-up for σ

_{max}= 50 MPa and σ

_{min}= 20 MPa.

**Figure 5.**Tensile test results for extrapolation at 0.1 s (or log(−1)) to find an equivalent σ

_{max}at 0.5 cycle failure.

**Figure 7.**Failure modes of the specimens following tensile testing for a crosshead speed range of 1–1000 mm/min.

**Figure 8.**S-N curves fitted and experimental data: log α = −7.016 and β = 1.031 for solid line with σ

_{u}= 52.0 MPa obtained from the extrapolation; and log α = −6.707 and β = 0.856 for dashed line with σ

_{u}= 45.6 MPa obtained at a crosshead speed of 1 mm/min.

**Figure 9.**Broken fatigue specimens after fatigue testing: (

**a**) σ

_{max}= 50 MPa—no visible cracks outside fracture surface; (

**b**) σ

_{max}= 43 MPa—multiple small cracks with whitening along the edges; (

**c**) σ

_{max}= 35 MPa—one crack is clearly visible; (

**d**) σ

_{max}= 25 MPa—many cracks are seen; (

**e**) σ

_{max}= 20 MPa—multiple crack are visible; and (

**f**) untested.

**Figure 10.**Remaining fatigue life prediction represented by an S-N curve and experimental data points represented by triangular data points: (

**a**) high-low loading paths; (

**b**) low-high loading paths; and (

**c**) combined experimental results.

σ_{max}_{1} (MPa) | logN_{1} (Cycles) | σ_{max}_{2} (MPa) | At Point b log (Cycles) | Experimental Remaining Loading Cycles at σ_{max}_{2} | ΔN_{2} | N_{f2} | Accuracy (%) |
---|---|---|---|---|---|---|---|

45 | 4.000 | 30 | 4.015 | 116,159 | 1.087 | 5.102 | 2.23 |

45 | 3.699 | 30 | 3.712 | 84,013 | 1.238 | 4.950 | −0.81 |

45 | 4.000 | 35 | 4.006 | 83,331 | 0.965 | 4.971 | 2.56 |

44 | 4.000 | 40 | 4.001 | 34,640 | 0.649 | 4.650 | −0.37 |

40 | 3.477 | 35 | 3.481 | 67,458 | 1.367 | 4.848 | 0.03 |

40 | 3.477 | 30 | 3.489 | 127,098 | 1.626 | 5.115 | 2.48 |

40 | 4.301 | 27 | 4.323 | 105,463 | 0.779 | 5.102 | 0.68 |

35 | 3.845 | 23 | 3.874 | 149,425 | 1.322 | 5.196 | 0.62 |

σ_{max}_{1} (MPa) | logN_{1} (Cycles) | σ_{max}_{2} (MPa) | At Point b log (Cycles) | Experimental Remaining Loading Cycles at σ_{max}_{2} | ΔN_{2} | N_{f2} | Accuracy (%) |
---|---|---|---|---|---|---|---|

35 | 4.000 | 45 | 3.994 | 31,978 | 0.628 | 4.622 | 4.85 |

35 | 3.301 | 45 | 3.296 | 10,440 | 0.798 | 4.094 | −7.11 |

30 | 4.000 | 45 | 3.986 | 20,251 | 0.490 | 4.476 | 1.56 |

27 | 4.477 | 40 | 4.454 | 42,731 | 0.398 | 4.852 | 3.97 |

20 | 3.699 | 38 | 3.657 | 53,186 | 1.104 | 4.761 | 0.34 |

23 | 3.845 | 33 | 3.820 | 80,589 | 1.120 | 4.940 | 0.68 |

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**MDPI and ACS Style**

Kim, H.S.; Huang, S.
S-N Curve Characterisation for Composite Materials and Prediction of Remaining Fatigue Life Using Damage Function. *J. Compos. Sci.* **2021**, *5*, 76.
https://doi.org/10.3390/jcs5030076

**AMA Style**

Kim HS, Huang S.
S-N Curve Characterisation for Composite Materials and Prediction of Remaining Fatigue Life Using Damage Function. *Journal of Composites Science*. 2021; 5(3):76.
https://doi.org/10.3390/jcs5030076

**Chicago/Turabian Style**

Kim, Ho Sung, and Saijie Huang.
2021. "S-N Curve Characterisation for Composite Materials and Prediction of Remaining Fatigue Life Using Damage Function" *Journal of Composites Science* 5, no. 3: 76.
https://doi.org/10.3390/jcs5030076