# Multiaxial Fatigue Life Prediction of GH4169 Alloy Based on the Critical Plane Method

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## Abstract

**:**

## 1. Introduction

## 2. The Critical Plane Method

#### 2.1. Determination of the Critical Plane of a Smooth Specimen

#### 2.2. Determination of the Critical Plane of a Notched Specimen

^{T}is the transpose matrix of matrix M.

## 3. Multiaxial Fatigue Life Prediction Model

#### 3.1. Considering Additional Enhanced Multiaxial Damage Parameters

_{y}′, in order to compare the response level of materials to cyclic plasticity more directly. The greater the value of the cyclic yield stress σ

_{y}′, the stronger the ability to resist cyclic deformation, and σ

_{y}′ can be expressed as follows:

#### 3.2. Mean Strain Correction

## 4. Experimental Verification

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Brandão, P.; Infante, V.; Deus, A.M. Thermo-mechanical modeling of a high pressure turbine blade of an airplane gas turbine engine. Procedia Struct. Integrity
**2016**, 1, 189–196. [Google Scholar] [CrossRef] [Green Version] - Zhu, S.P.; Liu, Q.; Yu, Z.Y.; Liu, Y. Fatigue reliability analysis of a turbine disc under multi-source uncertainties. Procedia Struct. Integrity
**2017**, 5, 967–972. [Google Scholar] [CrossRef] - Zhu, S.P.; Liu, Q.; Zhou, J.; Yu, Z.Y. Fatigue reliability assessment of turbine discs under multi-source uncertainties. Fatigue Fract. Eng. Mater. Struct.
**2018**, 41, 1291–1305. [Google Scholar] [CrossRef] - Calvente, M.M.; Blason, S.; Canteli, A.F. A probabilistic approach for multiaxial fatigue criteria. Frattura ed Integrita Strutturale
**2017**, 11, 160–165. [Google Scholar] [CrossRef] - Correia, J.; Apetre, N.; Arcari, A.; Jesus, A.D.; Muñizcalvente, M.; Rui, C.; Berto, F.; Fernándezcanteli, A. Generalized probabilistic model allowing for various fatigue damage variables. Int. J. Fatigue
**2017**, 100, 187–194. [Google Scholar] [CrossRef] - Zhu, S.-P.; Liu, Q.; Lei, Q.; Wang, Q. Probabilistic fatigue life prediction and reliability assessment of a high pressure turbine disc considering load variations. Int. J. Damage Mech.
**2017**, 27, 1569–1588. [Google Scholar] [CrossRef] - Das, J.; Sivakumar, S.M. Multiaxial fatigue life prediction of a high temperature steam turbine rotor using a critical plane approach. Eng. Fail. Anal.
**2000**, 7, 347–358. [Google Scholar] [CrossRef] - Gates, N.R.; Fatemi, A. On the consideration of normal and shear stress interaction in multiaxial fatigue damage analysis. Int. J. Fatigue
**2017**, 100, 322–336. [Google Scholar] [CrossRef] - Jiang, C.; Liu, Z.C.; Wang, X.G.; Zhang, Z.; Long, X.Y. A structural stress-based critical plane method for multiaxial fatigue life estimation in welded joints. Fatigue Fract. Eng. Mater. Struct.
**2016**, 39, 372–383. [Google Scholar] [CrossRef] - Jin, D.; Tian, D.J.; Li, J.H.; Sakane, M. Low-cycle fatigue of 316l stainless steel under proportional and nonproportional loadings. Fatigue Fract. Eng. Mater. Struct.
**2016**, 39, 850–858. [Google Scholar] [CrossRef] - Kamal, M.; Rahman, M.M. Multiaxial fatigue life modelling using hybrid approach of critical plane and genetic algorithm. Fatigue Fract. Eng. Mater. Struct.
**2016**, 39, 479–490. [Google Scholar] [CrossRef] - Brnić, J.; Turkalj, G.; Čanađija, M.; Lanc, D.; Kršćanski, S.; Brčić, M.; Li, Q.; Niu, J. Mechanical properties, short time creep and fatigue of an austenitic steel. Materials
**2016**, 9, 298. [Google Scholar] [CrossRef] [PubMed] - Zhang, W.; Liu, H.; Wang, Q.; He, J. A fatigue life prediction method based on strain intensity factor. Materials
**2017**, 10, 689. [Google Scholar] [CrossRef] [PubMed] - Wei, H.; Liu, Y. A critical plane-energy model for multiaxial fatigue life prediction. Fatigue Fract. Eng. Mater. Struct.
**2017**, 40, 1973–1983. [Google Scholar] [CrossRef] - Aeran, A.; Siriwardane, S.C.; Mikkelsen, O.; Langen, I. A new nonlinear fatigue damage model based only on s-n curve parameters. Int. J. Fatigue
**2017**, 103, 327–341. [Google Scholar] [CrossRef] - Brown, M.W.; Miller, K.J. A theory for fatigue failure under multiaxial stress-strain conditions. Proc. Inst. Mech. Eng.
**2006**, 187, 745–755. [Google Scholar] [CrossRef] - Chu, C.C. Fatigue damage calculation using the critical plane approach. J. Eng. Mater. Technol.
**1995**, 117, 41–49. [Google Scholar] [CrossRef] - Liao, D.; Zhu, S.-P.; Correia, J.A.F.O.; Jesus, A.M.P.D.; Calçadac, R. Computational framework for multiaxial fatigue life prediction of compressor discs considering notch effects. Eng. Fract. Mech.
**2018**, 202, 423–435. [Google Scholar] [CrossRef] - Gates, N.; Fatemi, A. Notch deformation and stress gradient effects in multiaxial fatigue. Theor. Appl. Fract. Mech.
**2016**, 84, 3–25. [Google Scholar] [CrossRef] - Gates, N.R.; Fatemi, A. A simplified cyclic plasticity model for calculating stress-strain response under multiaxial non-proportional loadings. Eur. J. Mech. A/Solid
**2016**, 59, 344–355. [Google Scholar] [CrossRef] - Lou, Y.; Yoon, J.W.; Huh, H.; Chao, Q.; Song, J.H. Correlation of the maximum shear stress with micro-mechanisms of ductile fracture for metals with high strength-to-weight ratio. Intl. J. Mech. Sci.
**2018**, 146, 583–601. [Google Scholar] [CrossRef] - Luo, P.; Yao, W.; Susmel, L.; Wang, Y.; Ma, X. A survey on multiaxial fatigue damage parameters under non-proportional loadings. Fatigue Fract. Eng. Mater. Struct.
**2017**, 40, 1323–1342. [Google Scholar] [CrossRef] - Ma, S.; Markert, B.; Yuan, H. Multiaxial fatigue life assessment of sintered porous iron under proportional and non-proportional loadings. Int. J. Fatigue
**2017**, 97, 214–226. [Google Scholar] [CrossRef] - Romanowicz, P. Numerical assessment of fatigue load capacity of cylindrical crane wheel using multiaxial high-cycle fatigue criteria. Arch. Appl. Mech.
**2017**, 87, 1–20. [Google Scholar] [CrossRef] - Shang, D.G.; Sun, G.Q.; Chen, J.H.; Cai, N.; Yan, C.L. Multiaxial fatigue behavior of ni-based superalloy gh4169 at 650 c. Mat. Sci. Eng. A Struct.
**2006**, 432, 231–238. [Google Scholar] [CrossRef] - Pan, J.; Nicholas, T. Effects of mean stresses on multiaxial fatigue life prediction based. Int. J. Fatigue
**2001**, 23, 87–92. [Google Scholar] [CrossRef] - Brown, M.W.; Surer, D.K.; Wang, C.H. An analysis of mean stress in multiaxial random fatigue. Fatigue Fract. Eng. Mater. Struct.
**2010**, 19, 323–333. [Google Scholar] [CrossRef] - Ge, J.; Sun, Y.; Zhou, S. Fatigue life estimation under multiaxial random loading by means of the equivalent lemaitre stress and multiaxial s–n curve methods. Int. J. Fatigue
**2015**, 79, 65–74. [Google Scholar] [CrossRef] - Zhang, J.; Xiao, Q.; Shi, X.; Fei, B. Effect of mean shear stress on torsion fatigue failure behavior of 2a12-t4 aluminum alloy. Int. J. Fatigue
**2014**, 67, 173–182. [Google Scholar] [CrossRef] - Marciniak, Z.; Rozumek, D.; Macha, E. Verification of fatigue critical plane position according to variance and damage accumulation methods under multiaxial loading. Int. J. Fatigue
**2014**, 58, 84–93. [Google Scholar] [CrossRef] - Spear, A.D.; Hochhalter, J.D.; Cerrone, A.R.; Li, S.F.; Lind, J.F.; Suter, R.M.; Ingraffea, A.R. A method to generate conformal finite-element meshes from 3d measurements of microstructurally small fatigue-crack propagation. Fatigue Fract. Eng. Mater. Struct.
**2016**, 39, 737–751. [Google Scholar] [CrossRef] - Susmel, L.; Taylor, D. A critical distance/plane method to estimate finite life of notched components under variable amplitude uniaxial/multiaxial fatigue loading. Int. J. Fatigue
**2012**, 38, 7–24. [Google Scholar] [CrossRef] - Fatemi, A.; Socie, D.F. A critical plane approach to multiaxial fatigue damage including out-of-phase loading. Fatigue Fract. Eng. Mater. Struct.
**2010**, 11, 149–165. [Google Scholar] [CrossRef] - Wang, C.H.; Brown, M.W. A path-independent parameter for fatigue under proportional and non-proportional loading. Fatigue Fract. Eng. Mater. Struct.
**2010**, 16, 1285–1297. [Google Scholar] [CrossRef] - Smith, K.N. A stresss train function for the fatigue metals. J. Mater.
**1970**, 5, 767–778. [Google Scholar] - Wu, Z.R.; Li, X.; Fang, L.; Song, Y.D. Multiaxial fatigue life prediction based on nonlinear continuum damage mechanics and critical plane method. J. Mater. Eng. Perform.
**2018**, 1–9. [Google Scholar] [CrossRef] - Xu, S.; Zhu, S.-P.; Hao, Y.-Z.; Liao, D. Critical plane–based multiaxial fatigue life prediction of turbine disk alloys by refining normal stress sensitivity. J. Strain Anal. Eng.
**2018**, 53, 719–729. [Google Scholar] [CrossRef] - Ranganathan, R.M. An improved, automated finite element analysis for fatigue life predictions of notched components. Int. J. Mater. Prod. Technol.
**2004**, 21, 539–554. [Google Scholar] [CrossRef] - Xu, S.; Zhu, S.; Hao, Y.; Liao, D. Multiaxial fatigue life prediction of an hpt disk based on critical plane-damage parameters. Acta Aeronautica et Astronautica Sinica
**2018**, 39, 221930. (In Chinese) [Google Scholar]

**Table 1.**Material property parameters [39].

Material | K′/MPa | n′ | E/GPa | σ_{y}/MPa | ${\mathit{\sigma}}_{\mathit{f}}^{\prime}/\mathbf{MPa}$ | ${\mathit{\epsilon}}_{\mathit{f}}^{\prime}$ | b | c |
---|---|---|---|---|---|---|---|---|

GH4169 at 650° | 1933 | 0.1483 | 182 | 626.4 | 1476 | 0.162 | −0.086 | −0.58 |

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**MDPI and ACS Style**

Liu, J.; Zhang, Z.; Li, B.; Lang, S.
Multiaxial Fatigue Life Prediction of GH4169 Alloy Based on the Critical Plane Method. *Metals* **2019**, *9*, 255.
https://doi.org/10.3390/met9020255

**AMA Style**

Liu J, Zhang Z, Li B, Lang S.
Multiaxial Fatigue Life Prediction of GH4169 Alloy Based on the Critical Plane Method. *Metals*. 2019; 9(2):255.
https://doi.org/10.3390/met9020255

**Chicago/Turabian Style**

Liu, Jianhui, Zhen Zhang, Bin Li, and Shanshan Lang.
2019. "Multiaxial Fatigue Life Prediction of GH4169 Alloy Based on the Critical Plane Method" *Metals* 9, no. 2: 255.
https://doi.org/10.3390/met9020255