# Numerical and Experimental Assessment of the Effect of Residual Stresses on the Fatigue Strength of an Aircraft Blade

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Examination

#### 2.1. Experimental Object

^{3}. EI-961 alloy steel has the following strength properties (at 20 °C):

- Ultimate Tensile Strength: UTS =1050 MPa;
- Yield Strength: YS = 850 MPa;
- Young Modulus: E = 200 GPa;
- Poisson ratio: υ = 0.3.

#### 2.2. Residual Stress Measurements

^{®}(Pulstec Industrial Co., Ltd., Hamamatsu-City, Japan) (Figure 3).

_{res}= −464 MPa) was used in the numerical tests as a value of residual-initial stresses.

## 3. Fatigue Life Assessment

_{eqv}= 987 MPa. The maximum principal stresses σ

_{1max}occurred at the same location and were 184 MPa higher [30]. The value of maximum principal stress σ

_{1max}= 1171 MPa was used in later calculations as the basic quantity describing the blade load. The equations used in the calculations were presented in Section 3.2.

#### 3.1. Numerical Models of the Material Properties

_{f}), fatigue ductility coefficient (ε’

_{f}), fatigue strength exponent (b), and fatigue ductility exponent (c), were estimated based on material data and static tensile test. The calculated values are summarized in Table 3 and presented in Figure 7.

- σ′
_{f}–fatigue strength coefficient, MPa - ε′
_{f}–fatigue ductility coefficient - b–fatigue strength exponent
- c–fatigue ductility exponent
- $Nf$–number of cycles to failure
- ${\epsilon}_{C}$–total strain amplitude
- $\sigma $–stress, MPa
- $E$–Young Modulus, GPa
- ${K}^{\prime}$–cyclic strength coefficient, MPa
- ${n}^{\prime}$–cyclic strain hardening exponent

_{res}= −464 MPa). The second value used in the fatigue calculations was the maximum value of the principal stresses σ

_{1max}= 1171 MPa, which was taken from the numerical strength analysis presented in the literature [30].

#### 3.2. Algorithm of the Fatigue Life Assessment

^{®}software (R2021a).

^{3}), and the lowest for the 4-point Manson model. For the average model, the estimated durability was nearly 1000 load cycles. In turn, in the case of Ong’s fatigue model (and Fatiemi’s hardening model), the number of cycles to crack initiation was as high as 401 × 10

^{3}. These results are much higher than all the others calculations (more than 4 times greater than the second-highest result for a given hardening model-the Mitchell fatigue model). Interestingly, the smallest scatter in the results was observed for the model of cyclic hardening according to Xianxin. The highest value (1.61 × 10

^{3}) occurred for the Manson fatigue model and the lowest for the Mitchell model (0.13 × 10

^{3}). The smallest result was only 12 times smaller than the largest one. The value for the average model, in the case of the Xianxin cyclic hardening model, was 0.81 × 10

^{3}and was slightly lower than the same fatigue model with hardening according to the Manson model.

#### 3.3. Comparison of Numerical and Experimental Results

^{3}. The aforementioned value was used to evaluate the obtained numerical results. A group summary of all the results related to the various material configurations is presented in Table 6.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

- E = 200 GPa–Young Modulus
- UTS = 1050 MPa–Ulitmate Tensile Strength
- ${\sigma}_{f}^{}=980\mathrm{MPa}$–fracture stress
- ${\epsilon}_{f}^{}$ = 0.12 mm/mm–strain during fracture (static tensile test)
- RA = 50%–percentage reduction (from the tensile test)
- HB = 302–Brinell Hardness

_{f}), and the remaining one quantity (${\mathsf{\sigma}}_{f}{}^{\prime}$)) is determined directly from the tensile strength.

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**Figure 1.**Photograph of a damaged compressor of a turbine engine (

**a**) and a blade damaged by a collision with a foreign object (

**b**).

**Figure 4.**Measurement of residual/initial stresses on the blade surface, at a distance of h = 3 mm (

**a**) and h = 1 mm (

**b**) from the blade root.

Distance from the Blade Root h, mm | Convex (Outer) Side of the Blade | Concave (Inner) Side of the Blade |
---|---|---|

- | Residual Stress σ_{res}, MPa | |

1 | - | −713 |

3 | −230 | −659 |

25 | −260 | −600 |

50 | −459 | −478 |

- | Inner (Concave) Side of the Blade | Outer (Convex) Side of the Blade | ||||
---|---|---|---|---|---|---|

Measurement point | 1 | 2 | 3 | 4 | 5 | 6 |

Residual stress σ_{res}, MPa | −478 | −659 | −255 | −197 | −230 | −213 |

Name of the Model | Short-Name | σ ′_{f}, MPa | ε ′_{f}, mm/mm | b | c |
---|---|---|---|---|---|

Manson (1965) | Man | 2280 | 0.61 | −0.12 | −0.6 |

4 point Manson (1965) | Man4 | 1172.2 | 0.528 | −0.064 | −0.57 |

Mitchell (1977) | Mit | 1545 | 0.12 | −0.068 | −0.6 |

Muralidharan Manson (1988) | MM | 1765.8 | 0.279 | −0.09 | −0.56 |

Baumel Seeger (1990) | BS | 1800 | 0.368 | −0.087 | −0.58 |

Ong (1993) | Ong | 1344 | 0.12 | −0.047 | −0.48 |

Roessle Fatemi (2000) | RF | 1508.5 | 0.14 | −0.09 | −0.56 |

Median (2002) | Med | 1800 | 0.45 | −0.09 | −0.59 |

Average | Ave | 1651.9 | 0.327 | −0.082 | −0.57 |

- | K′_{1}, MPa | n′_{1} | K′_{2}, MPa | n′_{2} | K′_{3}, MPa | n′_{3} |
---|---|---|---|---|---|---|

Man | 2516.9 | 0.2 | 2516.9 | 0.2 | 1775.5 | 0.17 |

Man4 | 1332 | 1258.3 | 0.11 | |||

Mit | 2361 | 1967.9 | 0.11 | |||

MM | 2279.8 | 2168.3 | 0.16 | |||

BS | 2198.4 | 2091.2 | 0.15 | |||

Ong | 2053.8 | 1656.1 | 0.10 | |||

RF | 2236.7 | 2070.2 | 0.16 | |||

Med | 2111.7 | 2033.2 | 0.15 | |||

Ave | 2136.3 | 1970.3 | 0.14 | |||

Manson′s model | Fatemi′s model | Xianxin′s model |

**Table 5.**Results of numerical fatigue life tests for the compressor blade with a notch tested in resonance conditions.

- | Cyclic hardening model | |||
---|---|---|---|---|

- | - | Manson′s | Fatemi′s | Xianxin′s |

ε-N Fatigue material model | Man | $7.4\times {10}^{3}$ | $7.4\times {10}^{3}$ | $1.61\times {10}^{3}$ |

Man4 | $0.03\times {10}^{3}$ | $1.08\times {10}^{3}$ | $1.47\times {10}^{3}$ | |

Mit | $0.46\times {10}^{3}$ | $89.12\times {10}^{3}$ | $0.13\times {10}^{3}$ | |

MM | $1.62\times {10}^{3}$ | $11.98\times {10}^{3}$ | $0.7\times {10}^{3}$ | |

BS | $1.49\times {10}^{3}$ | $21.77\times {10}^{3}$ | $0.92\times {10}^{3}$ | |

Ong | $0.31\times {10}^{3}$ | $401\times {10}^{3}$ | $0.37\times {10}^{3}$ | |

RF | $0.37\times {10}^{3}$ | $1.96\times {10}^{3}$ | $0.2\times {10}^{3}$ | |

Med | $1.22\times {10}^{3}$ | $17.17\times {10}^{3}$ | $1.11\times {10}^{3}$ |

**Table 6.**The results of the numerical fatigue analysis related to the actual blade life (geometrical V-shape notch, resonance A = 1.8 mm), determined experimentally N

_{in}= 12.9 × 10

^{3}) [27].

- | Cyclic Hardening Model | |||
---|---|---|---|---|

- | - | Manson′s | Fatemi′s | Xianxin′s |

ε-N Fatigue material model | Man | 57.37% | 57.37% | 12.51% |

Man4 | 0.22% | 8.42% | 11.37% | |

Mit | 3.54% | 690.88% | 1.01% | |

MM | 12.52% | 92.86% | 5.43% | |

BS | 11.52% | 168.79% | 7.13% | |

Ong | 2.38% | 3108.91% | 2.90% | |

RF | 2.87% | 15.16% | 1.52% | |

Med | 9.48% | 133.07% | 8.59% |

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**MDPI and ACS Style**

Bednarz, A.; Misiolek, W.Z. Numerical and Experimental Assessment of the Effect of Residual Stresses on the Fatigue Strength of an Aircraft Blade. *Materials* **2021**, *14*, 5279.
https://doi.org/10.3390/ma14185279

**AMA Style**

Bednarz A, Misiolek WZ. Numerical and Experimental Assessment of the Effect of Residual Stresses on the Fatigue Strength of an Aircraft Blade. *Materials*. 2021; 14(18):5279.
https://doi.org/10.3390/ma14185279

**Chicago/Turabian Style**

Bednarz, Arkadiusz, and Wojciech Zbigniew Misiolek. 2021. "Numerical and Experimental Assessment of the Effect of Residual Stresses on the Fatigue Strength of an Aircraft Blade" *Materials* 14, no. 18: 5279.
https://doi.org/10.3390/ma14185279