# Lifetime Assessment for Multiaxial High-Cycle Fatigue Using Twin-Shear Unified Yield Criteria

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{4}cycles, which is a major part the crack initiation phase [2]. From the view of load path, fatigue can also be separated into uniaxial and multiaxial. Multiaxial high cycling is a kind of fatigue characterized by more than 10

^{4}cycles and a complex load path, for instance, bending and torsion stresses which are out of phase. For this fatigue, various approaches have been proposed and improved on the basis of experimental observations and mechanisms [3,4,5,6,7,8,9,10]. From these approaches, we may notice some yield criteria are in extensive application and have gained fairly good results, e.g., Tresca yield criteria and Huber-von Mises yield criteria. As one of outstanding delegates, the concept of critical plane associated with Tresca yield criteria have gained widespread usage [11]. It should be noted that critical plane approach is a product related to the generalized application of Tresca yield criteria. This conclusion can be obtained since maximum shear stress amplitude is constantly employed for critical plane approach [12,13]. However, the adaptability of the yield criterion determined by material properties (e.g., fragile materials and ductile materials) results in these approaches perhaps being more suitable for directional materials [14]. Detailly, Tresca yield criteria and Huber-von Mises yield criteria are experimentally suitable for fragile materials and ductile materials, respectively. Such experimental results may lead to adaptability of the critical plane approach in different kinds of materials. Consequently, it is very necessary to develop a new multiaxial high-cycle fatigue life prediction approach associated of yield criteria that is available for more materials. Twin-shear unified yield criteria, the clusters of yield criteria embodying Tresca and the linearization of Huber-von Mises, provides a reliable approach to develop the adaptability.

## 2. Fatigue Criteria Based on Twin-Shear Unified Yield Criterion

_{0}(O), R

_{1}(O

_{1}), and R

_{2}(O

_{2}) in Figure 1 reflect the evolution of local residual stress tensor of materials subjected to cyclic loading based on combining kinematic and isotropic hardening.

## 3. New Fatigue Criteria Applied to Tension and Torsion

#### 3.1. Stress State Analysis

_{1}(O

_{2}) (Figure 5) and its semi-axes C

_{a}and C

_{b}are given by [28]:

_{0}, the following equation can also be obtained:

_{0}. According to the theory above, we can also obtain the following equation:

_{1}.

#### 3.2. Evaluation of the Criteria

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 6.**Specimen made of aluminum alloy LY12CZ [29].

**Table 1.**Fatigue strength and material parameters of S–N curvilinear equation of aluminum alloy LY12CZ (${\sigma}_{-1}=168.73\mathrm{MPa}$, ${t}_{-1}=119.62\mathrm{MPa}$, ${{\tau}^{\prime}}_{f}=602.8\mathrm{MPa}$, ${b}_{0}=-0.1115$): experimental data [30] and predictions.

${\mathit{\sigma}}_{\mathbf{a}}\left(\mathbf{MPa}\right)$ | ${\mathit{\sigma}}_{\mathbf{m}}\left(\mathbf{MPa}\right)$ | ${\mathit{\tau}}_{\mathbf{a}}\left(\mathbf{MPa}\right)$ | ${\mathit{\tau}}_{\mathbf{m}}\left(\mathbf{MPa}\right)$ | $\mathit{\delta}$$\left({}^{\mathbf{o}}\right)$ | A | B | C | D | Exp. |
---|---|---|---|---|---|---|---|---|---|

126.491 | 0 | 91.571 | 0 | 0 | 281,670 | 1,082,449 | 243,628 | >10^{7} | 482,666 |

158.114 | 0 | 111.803 | 0 | 0 | 42,683 | 167,487 | 36,135 | 1,736,000 | 76,451 |

189.737 | 0 | 137.356 | 0 | 0 | 7421 | 28,518 | 6419 | 300,650 | 23,003 |

126.491 | 0 | 95.507 | 0 | 30 | 241,200 | 960,836 | 192,398 | 9,908,700 | 420,261 |

158.114 | 0 | 119.384 | 0 | 30 | 32,600 | 129,866 | 26,004 | 1,339,200 | 63,584 |

126.491 | 0 | 100 | 0 | 45 | 205,270 | 857,035 | 148,362 | 8,520,800 | 275,527 |

158.114 | 0 | 125 | 0 | 45 | 27,744 | 115,837 | 20,053 | 1,151,700 | 57,004 |

126.491 | 0 | 105.193 | 0 | 60 | 178,320 | 760,413 | 109,702 | 7,340,500 | 231,348 |

158.114 | 0 | 131.491 | 0 | 60 | 24,102 | 102,779 | 14,827 | 992,160 | 30,893 |

158.114 | 0 | 139.111 | 0 | 90 | 35,187 | 89,612 | 10,178 | 1,035,900 | 15,459 |

126.491 | 0 | 111.289 | 0 | 90 | 260,330 | 662,994 | 75,299 | 7,663,900 | 66,940 |

200 | 0 | 115.47 | 0 | 90 | 26,490 | 287,053 | 5303 | 874,420 | 14,296 |

250 | 0 | 144.34 | 0 | 90 | 3580 | 38,793 | 717 | 118,180 | 4634 |

200 | 0 | 100 | 0 | 90 | 40,853 | 837,823 | 7430 | 1,392,800 | 37,789 |

250 | 0 | 125 | 0 | 90 | 5522 | 113,241 | 1004 | 188,260 | 6811 |

**Table 2.**Fatigue strength and material parameters of S–N curvilinear equation of carbon structural steel SM45C (${\mathsf{\sigma}}_{-1}=442\mathrm{MPa}$, ${t}_{-1}=311\mathrm{MPa}$, $\eta =62.3\mathrm{MPa}$, ${b}_{1}=-0.53$): experimental data [29] and predictions.

${\mathit{\sigma}}_{\mathbf{a}}\left(\mathbf{MPa}\right)$ | ${\mathit{\sigma}}_{\mathbf{m}}\left(\mathbf{MPa}\right)$ | ${\mathit{\tau}}_{\mathbf{a}}\left(\mathbf{MPa}\right)$ | ${\mathit{\tau}}_{\mathbf{m}}\left(\mathbf{MPa}\right)$ | $\mathit{\delta}$$\left({}^{\mathbf{o}}\right)$ | A | B | C | D | Exp. |
---|---|---|---|---|---|---|---|---|---|

449 | 0 | 282 | 0 | 90 | 33,399 | 89,680 | 16,006 | >10^{7} | 29,900 |

354 | 0 | 334 | 0 | 90 | 37,667 | 41,856 | 20,236 | >10^{7} | 35,700 |

485 | 0 | 223 | 0 | 90 | 39,863 | >10^{7} | 25,852 | >10^{7} | 50,000 |

357 | 0 | 309 | 0 | 90 | 51,948 | 68,445 | 23,389 | >10^{7} | 73,800 |

449 | 0 | 217 | 0 | 90 | 63,825 | >10^{7} | 36,598 | >10^{7} | 106,000 |

370 | 0 | 285 | 0 | 90 | 67,675 | 133,212 | 25,257 | >10^{7} | 106,000 |

449 | 0 | 199 | 0 | 90 | 80,172 | >10^{7} | 41,222 | >10^{7} | 112,000 |

457 | 0 | 194 | 0 | 90 | 75,020 | >10^{7} | 38,516 | >10^{7} | 131,000 |

354 | 0 | 252 | 0 | 90 | 191,400 | 3,869,612 | 37,984 | >10^{7} | 333,000 |

437 | 0 | 154 | 0 | 90 | 236,950 | >10^{7} | 61,528 | >10^{7} | 431,000 |

286 | 0 | 143 | 0 | 90 | >10^{7} | >10^{7} | >10^{7} | >10^{7} | 1,660,000 |

354 | 0 | 165 | 0 | 90 | >10^{7} | >10^{7} | 2,980,185 | >10^{7} | 1,860,000 |

441 | 196 | 215 | 0 | 90 | 43,691 | >10^{7} | 40,658 | >10^{7} | 53,000 |

286 | 196 | 309 | 0 | 90 | 62,224 | 42,824 | 40,316 | >10^{7} | 59,200 |

464 | 196 | 155 | 0 | 90 | 56,322 | >10^{7} | 40,596 | >10^{7} | 70,100 |

473 | 196 | 136 | 0 | 90 | 57,808 | >10^{7} | 38,199 | >10^{7} | 86,300 |

173 | 196 | 334 | 0 | 90 | 83,584 | 40,007 | 81,343 | >10^{7} | 89,900 |

403 | 196 | 209 | 0 | 90 | 74,826 | 2,219,079 | 31,942 | >10^{7} | 92,100 |

437 | 196 | 177 | 0 | 90 | 66,929 | >10^{7} | 55,224 | >10^{7} | 102,000 |

167 | 196 | 321 | 0 | 90 | 130,430 | 59,441 | 128,539 | >10^{7} | 135,000 |

357 | 196 | 179 | 0 | 90 | 611,040 | 215,534 | 84,523 | >10^{7} | 351,000 |

182 | 196 | 274 | 0 | 90 | >10^{7} | 235,925 | 2,562,011 | >10^{7} | 394,000 |

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**MDPI and ACS Style**

Li, H.; Wang, J.; Wang, J.; Hu, M.; Peng, Y.
Lifetime Assessment for Multiaxial High-Cycle Fatigue Using Twin-Shear Unified Yield Criteria. *Metals* **2021**, *11*, 1178.
https://doi.org/10.3390/met11081178

**AMA Style**

Li H, Wang J, Wang J, Hu M, Peng Y.
Lifetime Assessment for Multiaxial High-Cycle Fatigue Using Twin-Shear Unified Yield Criteria. *Metals*. 2021; 11(8):1178.
https://doi.org/10.3390/met11081178

**Chicago/Turabian Style**

Li, Haoran, Jiadong Wang, Juncheng Wang, Ming Hu, and Yan Peng.
2021. "Lifetime Assessment for Multiaxial High-Cycle Fatigue Using Twin-Shear Unified Yield Criteria" *Metals* 11, no. 8: 1178.
https://doi.org/10.3390/met11081178