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We present theoretical studies for the third-order elastic constants of Mg, Be, Ti, Zn, Zr, and Cd with a hexagonal-close-packed (HCP) structure. The method of homogeneous deformation combined with first-principles total-energy calculations is employed. The deformation gradient

In the theory of linear elasticity, infinitesimal deformation strains are assumed, and using finite deformation on materials is very significant for practical application. The second-order elastic constants (SOECs) are sufficient to describe the elastic stress-strain response and wave propagation in solids [

In the present study, we describe the method combined homogeneous deformation with first-principles total-energy calculations for determining TOECs of HCP metals (Mg, Be, Ti, Zn, Zr, and Cd). It is well known that the properties in HCP materials have been studied for a long time, especially for Mg. Mg has good ductility, better characteristics suppressing vibration and noise than aluminium, and excellent castability [

In this work, we will recall some basic facts from the nonlinear theory of elasticity [

In the nonlinear elasticity theory, the Lagrangian strain tensor is related to the internal energy, and the free energy is related to the Lagrangian strain tensor. The elastic constants can be obtained by expanding the internal energy as a Taylor series in terms of the Lagrangian strain tensor at constant entropy by Bruuger [

And the isothermal elastic constants are

Only six of each set of the nine Lagrangian strain tensors are independent due to its symmetry, and it is convenient to introduce the Voigt notation (11

Equations (

Besides, the Lagrangian stress

In the work presented here, we have determined the second-order elastic constants and third-order elastic constants for Mg, Be, Ti, Zn, Zr, and Cd combined with the first-principles calculations based on density functional theory (DFT). The deformation gradient

To carry out the different deformation modes in our work, the deformation gradient matrix

In general, the deformation gradient

For hexagonal structure, there are five independent SOECs

The coefficients

Strain type | ||
---|---|---|

2 | ||

Coefficients

In this work, we implement first-principles total-energy calculations on the basis of the density functional theory (DFT), which is embodied in the Vienna ab simulation package (VASP) [

Taking Mg as an example, Figure

In the calculated internal energy of Mg as a function of k-points grid size, the cutoff energy, and lattice constants, respectively. (a) The analogous dependence on the density of k-points mesh (energy cutoff of 600 eV is used for all points). (b) The dependence of internal energy on the cutoff energy (Monk-Pack sampling

Calculated values for the lattice parameters

Mg | Be | Ti | Zn | Zr | Cd | |
---|---|---|---|---|---|---|

3.193 | 2.262 | 2.925 | 2.648 | 3.234 | 3.032 | |

3. | 2. | 2. | 2. | 3. | 3. | |

1.627 | 1.578 | 1.580 | 1.910 | 1.600 | 1.910 | |

1. | 1. | 1. | 1. | 1. | 1. | |

63.96 | 390.06 | 175.26 | 167.71 | 142.46 | 92.18 | |

67.27 | 367.95 | 191.25 | 61.79 | 164.98 | 40.04 | |

17.02 | 160.46 | 36.35 | 26.30 | 22.25 | 17.38 | |

46. | 33. | |||||

28.30 | 23.93 | 96.14 | 37.86 | 71.66 | 44.96 | |

20.51 | 13.823 | 81.21 | 48.46 | 63.19 | 31.89 | |

^{a}Reference [^{b}Reference [^{c}Reference [^{d}Reference [^{e}Reference [^{f}Reference [^{g}Reference [56] obtianed from an ultrasonic wave interference technique.

We list our calculated values of second-order elastic constants (SOECs)

Predictions for the third-order elastic constants (TOECs) of Mg, Be, Ti, Zn, Zr, and Cd. The unit of all data is GPa.

Mg | Be | Ti | Zn | Zr | Cd | |
---|---|---|---|---|---|---|

−879.14 | −3865.41 | −1591.00 | −3007.54 | −856.89 | −1978.36 | |

−801.04 | −2931.07 | −1457.22 | −2988.01 | −835.38 | −2340.52 | |

−670.98 | −2932.89 | −1635.42 | −528.17 | −1303.69 | −414.87 | |

−801.58 | 250.623 | 309.761 | −289.74 | −224.99 | −368.93 | |

−74.08 | −21.59 | −750.07 | −125.04 | −220.521 | −134.95 | |

−53.82 | −176.83 | −1112.21 | −81.77 | −556.40 | −42.00 | |

−193.31 | 81.44 | 419.96 | −265.92 | −94.83 | −185.52 | |

−28.16 | −256.83 | 8.22 | −33.88 | 28.61 | 14.33 | |

−36.03 | 111.16 | −345.29 | −101.28 | 22.94 | −131.33 | |

−160.09 | −930.03 | −109.11 | −139.51 | −77.46 | 56.53 |

The strain-energy relations for Mg. The hollow circle denotes the results of DFT results; solid lines represent the curves obtained from nonlinear elasticity theory; dashed lines indicate the curves obtained from linear elasticity. (a–j) Lagrangian strains

The determination of

Next, our concern is to investigate for which range of strains

In the case of materials under larger hydrostatic pressure, it is of value to describe the nonlinear elastic properties employing the concept of effective elastic constants

The Lagrangian parameters

The calculated hydrostatic pressure derivatives of SOEC

Predictions for the pressure derivatives of second-order elastic constants for Mg, Be, Ti, Zn, Zr, and Cd.

Mg | Be | Ti | Zn | Zr | Cd | |
---|---|---|---|---|---|---|

12.21 | 7.51 | 3.25 | 6.91 | 2.13 | 7.92 | |

14.94 | 1.53 | 1.47 | 2.50 | 3.24 | 2.52 | |

1.41 | 0.34 | 3.88 | 3.30 | 2.99 | 3.68 | |

5.62 | 4.23 | -0.32 | 3.74 | 2.53 | 5.32 | |

0.27 | 1.35 | 0.16 | 0.76 | −0.84 | −2.29 |

In this section, elastic anisotropy of Mg, Be, Ti, Zn, Zr, and Cd has been investigated. It is well known that the acoustic velocities are related to the elastic constants by the Christoffel equation

Elastic anisotropy factors varied with the applied pressures. (a–d) describe

In order to investigate the elastic anisotropy of HCP metals systematically, the universal anisotropy

A universal ductile-to-brittle criterion: to evaluate material ductility or brittleness, the classical criteria of pressure and of Pugh’s modulus ratio

Brittleness and ductility as functions of pressure. The material is ductile when the ratio is below the critical value of 0.5, and the material is considered brittle when the ratio G/B is larger than 0.571. The five-pointed star, triangle-right, circle, triangle-up, diamond, and square represent Mg, Be, Ti, Zn, Zr, and Cd, respectively.

Vickers hardness

Vickers hardness

In this work, we have described a systematic scheme to calculate the SOECs and TOECs for Mg, Be, Ti, Zn, Zr, and Cd using the homogeneous deformation method and first-principles calculations. Our calculated values of SOECs and TOECs for Mg, Be, Ti, Zn, Zr, and Cd are listed in Tables

The data used to support the findings of this study are included in the article.

The authors declare that they have no conflicts of interest.

This research was funded by the Fundamental Research Funds for the Central Universities (Grant no. 2019CDXYTX0023); Chongqing Natural Science Foundation (Grant no. cstc2018jscx-msyb1002); and the Doctoral Startup Fund of Chongqing University of Posts and Telecommunications (Grant no. E012A2021224).