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Pressure Effect on Elastic Constants and Related Properties of Ti_{3}Al Intermetallic Compound: A First-Principles Study

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## Abstract

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## 1. Introduction

## 2. Theoretical Methods

#### 2.1. Computational Details

#### 2.2. Calculations of Elastic Constants

## 3. Results and Discussion

#### 3.1. Bulk Properties at Zero Pressure

#### 3.2. Pressure-Dependent Structure Parameters

#### 3.3. Pressure-Dependent Elastic Constants and Mechanical Properties

#### 3.4. Pressure-Dependent Elastic Anisotropy

#### 3.5. Pressure-Dependent Acoustic and Related Properties

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**B/G ratio (

**a**), Poisson’s ratio (

**b**), and Cauchy pressures ${C}_{12}-{C}_{66}$ (

**c**), ${C}_{13}-{C}_{44}$ (

**d**) as a function of pressure for Ti${}_{3}$Al.

**Figure 7.**Linear bulk modulus along the a and c principle axes (

**a**) and its anisotropy $\alpha $ (

**b**) as a function of pressure for Ti${}_{3}$Al.

**Figure 8.**Young’s modulus (

**a**), shear modulus (

**b**), and Poisson’s ratio (

**c**) on the $(2\overline{1}0)$, (010), and (001) planes as a function of pressure for Ti${}_{3}$Al.

**Figure 9.**Shear anisotropic factors ${A}_{\left\{100\right\}}$, ${A}_{\left\{010\right\}}$, and ${A}_{\left\{001\right\}}$ (

**a**), percentage anisotropy in compressibility ${A}^{B}$ and shear ${A}^{G}$ (

**b**), the universal anisotropy index ${A}^{U}$ and the log-Euclidean anisotropy index ${A}^{L}$) (

**c**) as a function of pressure for Ti${}_{3}$Al.

**Figure 10.**Mass density $\rho $ (

**a**), longitudinal and transverse sound velocities ${v}_{l}$, ${v}_{t1}$, and ${v}_{t2}$ in the [100] (

**b**) and [001] (

**c**) principle directions, and polycrystalline longitudinal, transverse, and average sound velocities ${V}_{L}$, ${V}_{T}$, and ${V}_{M}$ (

**d**) as a function of pressure for Ti${}_{3}$Al.

**Table 1.**The optimized lattice parameters a (in Å), c/a, and elastic constants (in GPa) of Ti${}_{3}$Al at zero pressure.

a | $\mathit{c}/\mathit{a}$ | ${\mathit{C}}_{11}$ | ${\mathit{C}}_{12}$ | ${\mathit{C}}_{13}$ | ${\mathit{C}}_{33}$ | ${\mathit{C}}_{44}$ | |
---|---|---|---|---|---|---|---|

Present | 5.758 | 0.808 | 186.18 | 85.57 | 62.76 | 233.03 | 59.75 |

Exp. [21,26] | 5.77 | 0.80 | 183.2 | 89.0 | 62.6 | 225.1 | 64.1 |

Theo. [22] | 5.74 | 0.81 | 184.67 | 82.37 | 63.41 | 225.08 | 53.97 |

Theo. [24] | 5.772 | 0.803 | 72 | ||||

Theo. [9] | 5.76 | 0.809 | 192.2 | 80.5 | 62.5 | 232.9 | 61.6 |

Theo. [23] | 5.72 | 0.81 | 185 | 83 | 63 | 231 | 57 |

Theo. [25] | 5.64 | 0.81 | 221 | 71 | 85 | 238 | 69 |

P | ${\mathit{C}}_{11}$ | ${\mathit{C}}_{12}$ | ${\mathit{C}}_{13}$ | ${\mathit{C}}_{33}$ | ${\mathit{C}}_{44}$ |
---|---|---|---|---|---|

0 | 186.18 | 85.57 | 62.76 | 233.03 | 59.75 |

10 | 224.30 | 108.15 | 75.51 | 286.13 | 66.15 |

20 | 261.44 | 124.83 | 89.61 | 323.41 | 70.22 |

30 | 291.09 | 142.23 | 105.72 | 354.83 | 72.90 |

40 | 318.32 | 157.86 | 122.34 | 380.17 | 74.16 |

50 | 343.06 | 174.27 | 135.46 | 402.06 | 74.06 |

60 | 366.48 | 191.39 | 149.52 | 430.75 | 73.29 |

70 | 389.37 | 208.75 | 161.36 | 460.66 | 71.69 |

80 | 409.54 | 228.60 | 172.83 | 492.29 | 69.81 |

**Table 3.**The elastic compliances ${S}_{ij}$ (in $\times {10}^{-3}$GPa${}^{-1}$) of Ti${}_{3}$Al under pressure up to 80 GPa.

P | ${\mathit{S}}_{11}$ | ${\mathit{S}}_{12}$ | ${\mathit{S}}_{13}$ | ${\mathit{S}}_{33}$ | ${\mathit{S}}_{44}$ |
---|---|---|---|---|---|

0 | 7.07 | −2.87 | −1.13 | 4.9 | 16.74 |

10 | 6.01 | −2.6 | −0.901945 | 3.97 | 15.12 |

20 | 5.15 | −2.17 | −0.823186 | 3.55 | 14.24 |

30 | 4.71 | −2.01 | −0.804515 | 3.3 | 13.72 |

40 | 4.37 | −1.86 | −0.809706 | 3.15 | 13.48 |

50 | 4.14 | −1.79 | −0.790763 | 3.02 | 13.5 |

60 | 3.96 | −1.75 | −0.764448 | 2.85 | 13.64 |

70 | 3.80 | −1.74 | −0.722140 | 2.68 | 13.95 |

80 | 3.73 | −1.8 | −0.679304 | 2.51 | 14.33 |

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**MDPI and ACS Style**

Zeng, X.; Peng, R.; Yu, Y.; Hu, Z.; Wen, Y.; Song, L.
Pressure Effect on Elastic Constants and Related Properties of Ti_{3}Al Intermetallic Compound: A First-Principles Study. *Materials* **2018**, *11*, 2015.
https://doi.org/10.3390/ma11102015

**AMA Style**

Zeng X, Peng R, Yu Y, Hu Z, Wen Y, Song L.
Pressure Effect on Elastic Constants and Related Properties of Ti_{3}Al Intermetallic Compound: A First-Principles Study. *Materials*. 2018; 11(10):2015.
https://doi.org/10.3390/ma11102015

**Chicago/Turabian Style**

Zeng, Xianshi, Rufang Peng, Yanlin Yu, Zuofu Hu, Yufeng Wen, and Lin Song.
2018. "Pressure Effect on Elastic Constants and Related Properties of Ti_{3}Al Intermetallic Compound: A First-Principles Study" *Materials* 11, no. 10: 2015.
https://doi.org/10.3390/ma11102015