# Constitutive Modeling on the Ti-6Al-4V Alloy during Air Cooling and Application

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## Abstract

**:**

^{−2}, 10

^{−3}, and 10

^{−4}/s. An Arrhenius model was employed to describe the yield stress at a strain of 0.1, and a simple generalized reduced gradient refinement was applied to optimize the parameters for a constitutive model. The mean error between the predicted and experimental flow stress was 65% and 16% before and after parameter optimization, respectively. The effects of strain on flow stress showed a linear relationship, so a strain compensation method was proposed. The modified Arrhenius model developed in this paper provided a good agreement between the predicted stresses and the experimental data. Finally, a finite element analysis (FEA) with a “UHARD” subroutine was employed, and the results indicated that the inner plate of the sandwich structure was the most vulnerable location during the air cooling process, and that the engineering strain due to a non-uniform temperature was calculated as 0.37%.

## 1. Introduction

## 2. Materials and Methods

^{−2}, 10

^{−3}, or 10

^{−4}/s. Due to the lengthy time requirements for the high temperature tensile test, the experiments at each designated temperature and strain rate were only performed once. The recorded force-displacement curves were employed to calculate the true strain and true stress data with the following equations [15]:

## 3. Flow Behavior of Ti-6Al-4V Alloy during Air Cooling

^{−2}, 10

^{−3}, 10

^{−4}/s, respectively. The main characteristics of the stress-strain curves for the Ti-6Al-4V alloy during cooling are listed as follows:

- During the initial stage of deformation, flow stress increased rapidly, which was caused by dislocation multiplication. Dislocations are mainly affected by dislocation multiplication, dynamic recovery (DRV), and dynamic recrystallization (DRX). The latter two are mechanisms of dislocation annihilation. As the deformation progresses, the annihilation mechanism gradually strengthens, gradually weakening the impact of dislocation multiplication.
- For large strains, a stress softening phenomenon appears after the stress reaches a peak. When the strain rate was 10
^{−2}/s, such softening behavior was continuous, which may be caused by the effects of voids at high strain rates. Previous research has demonstrated that obvious dynamic recrystallization can occur, resulting in significant void growth [9]. In contrast, the stress was able to maintain a stable state after softening at 10^{−3}/s, which is similar to the SPF process [16,17,18]. - For the low strain rate of 10
^{−4}/s, the flow behavior was directly related to the temperature. When T < 900 °C, stress softening also occurred. However, flow stress gradually increased with strain at high temperatures, indicating that the softening mechanism for titanium is not sufficient to eliminate the effects of dislocation multiplication.

^{−2}/s. Thus, it will be increasingly difficult for plastic deformation during air cooling. The strain-rate sensitivity parameter provides an estimate for the ability of a material to undergo continuous plastic deformation, which is given as $m=\partial \mathrm{ln}\sigma /\partial \mathrm{ln}\dot{\epsilon}$. The values for the strain-rate sensitivity parameter were lower at low temperatures, inferring the possibility that flow localization increased during the air cooling process.

## 4. Constitutive Modeling Based on an Arrhenius Model

#### 4.1. Arrhenius Model

#### 4.2. Determine the Constants of the Constitutive Model

^{−4}/s. The white area corresponds to β-phase grains, while the red, green, and blue colors represent the <0001>, <$\overline{1}$2$\overline{1}$0>, and <10$\overline{1}$0> crystallographic directions of the α-phase grains, respectively. There are many fine grains distributed around the initial grains, which is caused by dynamic recrystallization. Dynamic recrystallization occurs in the dislocation density gathering area with high energy, and the nucleation of recrystallized grains will reduce the dislocation density. When combined with the stress-strain curves from Figure 4a, dynamic recrystallization is the dominant mechanism for stress softening.

## 5. Effect of the Strain on Flow Stress

#### 5.1. Strain Hardening Law

#### 5.2. Constitutive Model with Strain Compensation

## 6. Application of the Constitutive Model to a Sandwich Structure

## 7. Conclusions

- (1)
- Owing to a high cooling rate and a small thickness value, the inner plate might be the first location where damage occurs during the air cooling process;
- (2)
- The engineering strain for the sandwich structure due to the temperature gradient was 0.37%, and distortions in the sandwich structure were caused by the temperature gradient;
- (3)
- The strain stress ratio n
_{h}was shown to have a linear relationship with strain, so a strain compensation method based on a linear function is proposed; - (4)
- Parameter optimization for the Arrhenius model for flow stress was indispensable, since it reduced the mean error from 65% to 16% for the difference between the predicted results and the experimental data.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**The flow behavior of a Ti-6Al-4V alloy during air cooling at the following temperatures: (

**a**) 700 °C; (

**b**) 800 °C; (

**c**) 900 °C; (

**d**) 930 °C.

**Figure 5.**The flow behavior of a Ti-6Al-4V alloy at ε = 0.2 during the air cooling process after superplastic deformation: (

**a**) true stress; (

**b**) strain-rate sensitivity parameter.

**Figure 6.**Relationship between (

**a**) ln$\dot{\epsilon}$ and ln$\sigma $, and (

**b**) ln$\dot{\epsilon}$ and$\sigma $.

**Figure 7.**Relationship between $ln\dot{\epsilon}$ and$\text{}ln\left[\mathrm{sin}\mathrm{h}\left(\alpha \sigma \right)\right]$.

**Figure 8.**Relationship between $ln\left[\mathrm{sin}\mathrm{h}\left(\alpha \sigma \right)\right]$ and 1/T.

**Figure 9.**Dynamic recrystallization during air cooling deformation as measured by EBSD at 700 °C and 10

^{−4}/s.

**Figure 10.**Comparison of the predicted stress with the experimental data at ε = 0.1 before (dotted lines) and after (solid lines) optimization.

**Figure 11.**The hardening law for a Ti-6Al-4V alloy at a strain rate of: (

**a**) 10

^{−2}/s; (

**b**) 10

^{−3}/s; (

**c**) 10

^{−4}/s.

**Figure 13.**Comparison between the experimental data and the predicted stresses from the constitutive model with strain compensation.

**Figure 14.**The results of an FEM analysis for the air cooling process with respect to: (

**a**) temperature; (

**b**) equivalent strain; (

**c**) displacement.

**Figure 15.**The evolution of selected points by FEM analysis: (

**a**) temperature at points A–C; (

**b**) displacement at points A and B.

Nomenclature | Descriptions | Nomenclature | Descriptions |
---|---|---|---|

SPF/DB | Superplastic forming/diffusion bonding | $\sigma $ | True stress |

$\dot{\epsilon}$ | Strain rate | $\epsilon $ | True strain |

$F$ | Loading force | ${A}_{0}$ | Initial cross-sectional area |

${L}_{0}$ | Initial gauge length | $\mathsf{\Delta}L$ | Elongation |

$m$ | Strain-rate sensitivity parameters | $Z$ | Zener–Hollomon parameter |

$Q$ | Deformation active energy | $T$ | Temperature |

R | Gas constant | ${n}_{h}$ | Strain stress ratio |

Chemistry (wt.%) | Ti | Al | V | C |
---|---|---|---|---|

Ti-6Al-4V | 88.31 | 5.59 | 4.85 | 1.25 |

Temperature (°C) | $\dot{\mathit{\epsilon}}\left({\mathit{s}}^{-1}\right)$ | ||
---|---|---|---|

${10}^{-2}$ | ${10}^{-3}$ | ${10}^{-4}$ | |

930 | 56.92 | 26.84 | 11.91 |

900 | 76.17 | 36.78 | 13.35 |

800 | 171.49 | 109.86 | 47.96 |

700 | 297.58 | 230.84 | 136.6 |

Parameter | 930 °C | 900 °C | 800 °C | 700 °C | Mean Value |
---|---|---|---|---|---|

$n$ | 2.94 | 2.62 | 3.51 | 5.69 | 3.69 |

$\beta $ | 0.099 | 0.072 | 0.037 | 0.028 | 0.059 |

$\alpha =\beta /n$ | 0.016 |

Parameter | 930 °C | 900 °C | 800 °C | 700 °C | Mean Value |
---|---|---|---|---|---|

$\eta $ | 2.72 | 2.33 | 2.07 | 1.76 | 2.22 |

Parameter | 10^{−2} s^{−1} | 10^{−3} s^{−1} | 10^{−4} s^{−1} | Mean Value |
---|---|---|---|---|

$K$ | 20,534.89 | 19,483.77 | 16,370.95 | 18,796.54 |

Strain Rate (s^{−1}) | 930 °C | 900 °C | 800 °C | 700 °C |
---|---|---|---|---|

10^{−2} | 30.03 | 30.05 | 29.79 | 29.30 |

10^{−3} | 29.63 | 29.76 | 29.73 | 29.37 |

10^{−4} | 29.19 | 29.82 | 30.10 | 30.44 |

Mean value | 29.77 |

$\mathit{n}$ | $\mathit{\beta}$ | $\mathit{\alpha}$ | $\mathit{\eta}$ | $\mathit{K}$ | $\mathit{Q}$ | $\mathit{l}\mathit{n}{\mathit{A}}_{3}$ | ${\mathit{A}}_{3}$ |
---|---|---|---|---|---|---|---|

3.69 | 0.059 | 0.016 | 2.22 | 18,796.53 | 347.39 | 29.77 | 8.48 × 10^{12} |

$\mathit{\alpha}$ | $\mathit{\eta}$ | $\mathit{Q}$ | ${\mathit{A}}_{3}$ | |
---|---|---|---|---|

Before optimization | 0.016 | 2.22 | 347.39 | 8.48 × 10^{12} |

After optimization | 0.036 | 0.44 | 351.93 | 1.11 × 10^{13} |

Strain Rate (s^{−1}) | 700 °C | 800 °C | 900 °C | 930 °C | Mean Values |
---|---|---|---|---|---|

10^{−2} | −0.85 | −0.84 | −0.86 | −0.83 | −0.85 |

10^{−3} | −0.99 | −0.76 | −0.61 | −0.48 | −0.71 |

10^{−4} | −0.71 | −0.59 | 0.12 | 0.54 | −0.16 |

Strain Rate (s^{−1}) | 700 °C | 800 °C | 900 °C | 930 °C | Mean Values |
---|---|---|---|---|---|

10^{−2} | 1.10 | 1.07 | 1.04 | 1.03 | 1.06 |

10^{−3} | 1.10 | 1.03 | 1.02 | 1.03 | 1.04 |

10^{−4} | 1.04 | 1.01 | 1.02 | 1.00 | 1.02 |

${k}_{k}$ | ${b}_{k}$ | ${k}_{b}$ | ${b}_{b}$ | |

Values | −0.15 | −1.59 | 0.0092 | 1.10 |

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**MDPI and ACS Style**

Han, X.; Yang, J.; Li, J.; Wu, J.
Constitutive Modeling on the Ti-6Al-4V Alloy during Air Cooling and Application. *Metals* **2022**, *12*, 513.
https://doi.org/10.3390/met12030513

**AMA Style**

Han X, Yang J, Li J, Wu J.
Constitutive Modeling on the Ti-6Al-4V Alloy during Air Cooling and Application. *Metals*. 2022; 12(3):513.
https://doi.org/10.3390/met12030513

**Chicago/Turabian Style**

Han, Xiaoning, Junzhou Yang, Jinshan Li, and Jianjun Wu.
2022. "Constitutive Modeling on the Ti-6Al-4V Alloy during Air Cooling and Application" *Metals* 12, no. 3: 513.
https://doi.org/10.3390/met12030513