# Low Cycle Fatigue Life Prediction Using Unified Mechanics Theory in Ti-6Al-4V Alloys

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## Abstract

**:**

## 1. Introduction

## 2. Unified Mechanics Theory-Based Life Prediction Model

#### 2.1. Unified Mechanics Theory

#### 2.1.1. Second Law of Unified Mechanics Theory

**F**is defined by Newton’s second universal law of motion. However, Newton’s laws do not account for energy loss after the initial momentum. Energy loss takes place according to the first and second laws of thermodynamics. As a result, a marriage of laws of second law of Newton and laws of thermodynamic is given by:

**P**represents the momentum and $v$ represents the velocity. Assuming a constant mass system,

#### 2.1.2. Third Law of Unified Mechanics Theory

#### 2.1.3. Thermodynamic State Index (TSI) for Damage in Low Cycle Fatigue of Materials

#### 2.2. Analytical Approach for the Prediction of Damage and Fatigue Life

#### 2.3. Computational 3-D Model for the Prediction of Damage

#### 2.3.1. Derivation of the Computational Model

#### 2.3.2. Algorithm for the Computational Model

## 3. Validation of the Computational Model for Monotonic Loading

#### 3.1. Validation of the Numerical Model for Monotonic Tensile Loading

#### 3.2. Validation of the 3-D Numerical Model for Monotonic Compressive Loading

## 4. Model Predictions for Low Cycle Fatigue Life

^{−3}s

^{−1}at room temperature. Similar quasi-static loading condition is established in our numerical loading by controlling the step time of the numerical model in ABAQUS. The material model used in developing the 3-Dimensional numerical model is independent of the strain rate and the temperature and hence the strain rate hardening behavior and temperature effects, including the thermal dissipation are not considered in our study. Unified mechanics theory-based approach for damage calculation, described in Section 2, is used to predict the low cycle fatigue life of Ti-6Al-4V alloys. Details of the one-dimensional analytical model as well as the three-dimensional numerical model to predict fatigue life or Ti-6Al-4V are given in Section 4.1 below.

#### 4.1. Analytical Approach for Fatigue Life Prediction

#### 4.2. Computational Procedure for Fatigue Life Prediction

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Schematic representation of computing plastic dissipation from the engineering stress-plastic strain graph.

**Figure 3.**Schematics of numerical model for displacement controlled monotonic tensile loading in ABAQUS.

**Figure 4.**Comparison between monotonic tensile stress-strain graphs obtained from the test data [76] and numerical model.

**Figure 6.**Schematics of numerical model for displacement controlled monotonic compressive. loading in ABAQUS.

**Figure 7.**Comparison between monotonic compressive stress-strain graphs obtained from the test data [2] and numerical model.

**Figure 10.**Low cycle fatigue life (Nf) prediction at different strain amplitudes of cyclic loading in comparison with the test data [76].

**Figure 11.**Numerical results on engineering stress-strain hysteresis loops for 1.2% strain amplitude of cyclic loading. (

**a**) hysteresis loops at 1.2% strain amplitude for 50 cycles of loading; (

**b**) comparative hysteresis plot for first cycle and 50th cycle of loading.

Material Parameter | Value | Unit |
---|---|---|

Young’s modulus, E | 106 | GPa |

Poisson’s ratio, ν | 0.31 | |

Density,$\mathbf{\rho}$ | 4540 | kg/m^{3} |

Critical TSI,${\mathit{\Phi}}_{\mathit{c}}$ | 1 | |

Hardening parameter,$\mathit{K}$ | 968.00 | MPa |

Hardening exponent,$\mathit{r}$ | 0.64 | |

Yield strength,${\mathbf{\sigma}}_{{\mathit{y}}_{0}}$ | 992.00 | MPa |

Molar mass,${\mathit{m}}_{\mathit{s}}$ | 0.047867 | kg/mol |

Reference temperature, T | 298 | K |

**Table 2.**Material parameters used in the numerical model for compressive loading in Ti-6Al-4V alloy.

Material Parameter | Value | Unit |
---|---|---|

Young’s modulus, E | 118 | GPa |

Poisson’s ratio, ν | 0.31 | |

Density,$\mathbf{\rho}$ | 4540 | kg/m^{3} |

Critical TSI,${\mathit{\Phi}}_{\mathit{c}}$ | 1 | |

Hardening parameter,$\mathit{K}$ | 550.00 | MPa |

Hardening exponent,$\mathit{r}$ | 0.65 | |

Yield strength,${\mathbf{\sigma}}_{{\mathit{y}}_{0}}$ | 1047.00 | MPa |

Molar mass,${\mathit{m}}_{\mathit{s}}$ | 0.047867 | kg/mol |

Reference temperature, T | 298 | K |

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**MDPI and ACS Style**

Bin Jamal M, N.; Kumar, A.; Lakshmana Rao, C.; Basaran, C. Low Cycle Fatigue Life Prediction Using Unified Mechanics Theory in Ti-6Al-4V Alloys. *Entropy* **2020**, *22*, 24.
https://doi.org/10.3390/e22010024

**AMA Style**

Bin Jamal M N, Kumar A, Lakshmana Rao C, Basaran C. Low Cycle Fatigue Life Prediction Using Unified Mechanics Theory in Ti-6Al-4V Alloys. *Entropy*. 2020; 22(1):24.
https://doi.org/10.3390/e22010024

**Chicago/Turabian Style**

Bin Jamal M, Noushad, Aman Kumar, Chebolu Lakshmana Rao, and Cemal Basaran. 2020. "Low Cycle Fatigue Life Prediction Using Unified Mechanics Theory in Ti-6Al-4V Alloys" *Entropy* 22, no. 1: 24.
https://doi.org/10.3390/e22010024