# Mechanical Properties of Al–Mg–Si Alloys (6xxx Series): A DFT-Based Study

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{3}, a bulk modulus of 83.3 GPa, a shear modulus of 34.4 GPa, a Vickers hardness of 2.79 GPa, a Poisson’s ratio of 0.413, a Pugh’s ratio of 5.42, and a yield strength of 8.38 GPa. The optimum Si/Mg ratio was found to be 4.5 for most of the mechanical properties. The study successfully established that the Si/Mg ratio is a critical factor when dealing with the mechanical properties of the Al–Mg–Si alloys. The alloys with the optimum Si/Mg ratio can be used for industrial applications such as plane skins and mining equipment where these properties are required.

## 1. Introduction

_{5}Si

_{6}phase, also known as the β” phase), both existing as needle-like structures [7]. The mechanical properties of alloys can be significantly altered with the addition of small amounts of alloying elements and by suitable heat treatments [8]. Specifically, the mechanical strength of the alloys can be improved through cold working and alloying. However, both processes tend to diminish their resistance to corrosion. Moreover, their applicability faces the limitation of a low melting point (660 °C). The principal alloying elements are copper, Si, Mg, manganese, and zinc [9].

## 2. Materials and Methods

#### 2.1. Density Functional Theory

#### 2.2. Modeling the Structures of the Alloys

#### 2.3. Structural Optimization

^{−4}Ry. K_points varied from 2 to 9 in steps of 1. To obtain the equilibrium lattice parameters of the crystal, calculations on the total energy were carried out for a range of unit cell volumes by varying the lattice parameters in steps of 0.2 a.u from 19.3 to 22.1, producing 15 data points. The equilibrium lattice parameters were obtained by fitting the resulting total energies versus volumes data into the third-order Birch–Murnaghan equation of state, given by Equation (2) [10]:

_{o}is the reference volume, V is the deformed volume, ${\mathrm{B}}_{\mathrm{o}}$ is the bulk modulus, and ${\mathrm{B}}_{\mathrm{o}}^{\u2019}$ is the derivative of the bulk modulus with respect to pressure. From Equation (2), the minimum equilibrium volumes were obtained. The values of the equilibrium volumes were then fitted into the equation for finding the volume of a simple cubic cell, given by Equation (3):

^{−4}Ry/Å.

#### 2.4. Calculation of Mechanical Properties

_{i}in a material are directly proportional to the corresponding applied strain δ

_{i}within the linear regime of the crystal:

_{ij}is the elastic stiffness constant corresponding to the spring constant in Hooke’s law.

## 3. Results and Discussion

#### 3.1. Structural Properties

^{−3}Ry. Figure 3 presents the total energy per atom against k_points for samples A_00, A_19, and all the alloy samples combined, which was found to stabilize at the 5 × 5 × 5 mesh. The energy difference corresponding to this mesh was found to be 3.5 × 10

^{−4}Ry. Thus, the 5 × 5 × 5 k_point mesh was chosen for all the other supercells. Although higher values for the ecut and k_point mesh would have been chosen so as to improve the accuracy of the calculation, it was noted that the higher values would have been more computationally expensive, considering the large number of atoms (108 atoms) in each supercell that were modeled in this study. However, the 50 Ry ecut and 5 × 5 × 5 Ry k_points are sufficient to give accurate results.

_{o}) of the alloys were obtained with the help of Equation (2). By applying the formula for finding the volume of a cube (Equation (3)), the volumes of the cells were calculated. The normalized volumes (v/v

_{o}) were also obtained and the graphs of total energy versus normalized plotted in order to optimize the equilibrium volumes as shown in Figure 4. It is evident from Table 2 that the optimum lattice parameters of the supercells decrease from A_19 to A_91. Since the trend from A_19 to A_91 is accompanied by an increase in the Si/Mg ratio, it implies that as the ratio increases, the lattice parameters of the alloys decrease (Table 2). This shows that the unit cells shrink with an increase in the Si/Mg ratio and is in agreement with the corresponding consistent increase in the densities of the alloy samples.

_{1}, and U

_{2}precipitates by Froseth’s group [19], which represents a 0.7463% deviation. The computational lattice parameter by Nakashima [20] also compares favorably with the result of this study, representing a 0.2488% deviation. The known trend of overestimation of the lattice parameters by general gradient approximation (GGA) was not witnessed in this work. However, this can be attributed to the improvement of the GGA over time [27].

^{3}) and the volume of the whole crystal, and, hence, its density, since density is affected by volume. At lower Si/Mg ratios, there is a sharp decrease in the unit cell parameters with a corresponding increase in the densities of the alloy samples. At higher values of the Si/Mg ratios, however, both curves tend to be constant. This shows that as the Si/Mg ratio increases, the distance between atoms is becoming smaller, which implies that as you move from A_19 towards A_91, the interatomic distances decrease. A_19, therefore, having the lowest density, will be the lightest and more appropriate for use in making aircraft parts. It is also worth noting that the difference between the highest (2.821 g/cm

^{3}) and the lowest (2.762 g/cm

^{3}) density is very small and hence insignificant, since the difference between the masses of 1 cm

^{3}of A_19 and A_91 is just 0.059 g.

#### 3.2. Mechanical Properties

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Three-dimensional structures of (

**a**) an aluminium unit cell and (

**b**) a supercell of the A_19 (a 3 × 3 × 3 modeled Al–Mg–Si) alloy as visualized in Burai. The grey spheres represent the aluminium atoms; the brown spheres represent the silicon atoms; and the green spheres represent the magnesium atoms.

**Figure 2.**Graphs of total energy against kinetic energy cut-off for (

**a**) sample A_00, (

**b**) sample A_19, and (

**c**) all 10 samples combined.

**Figure 3.**Graphs of total energy against K_points for (

**a**) sample A_00, (

**b**) sample A_19, and (

**c**) all 10 samples combined.

**Figure 5.**Graphs of equilibrium unit cell parameters and the densities of the alloy samples against the Si/Mg ratio.

**Figure 6.**Calculated stress–strain curves for elastic stiffness constants (

**a**) c

_{11}, (

**b**) c

_{12}, and (

**c**) c

_{44}for samples A_00 and A_19.

**Figure 7.**Variations of the mechanical properties of the Al–Mg–Si alloys: (

**a**) bulk modulus, (

**b**) shear modulus, (

**c**) Young’s modulus, (

**d**) Poisson’s ratio, (

**e**) Pugh’s ratio, and (

**f**) Vickers hardness as a function of Si/Mg ratio.

**Table 1.**Concentrations of various Al–Mg–Si alloys (6xxx series) and the number of atoms of each element.

Sample ID | Silicon | Magnesium | Aluminium | Si/Mg Ratio | Si/(Mg+Si) | |||
---|---|---|---|---|---|---|---|---|

Conc. (%) | Atoms | Conc. (%) | Atoms | Conc. (%) | Atoms | |||

A_00 | 0 | 0 | 0 | 0 | 100 | 108 | - | - |

A_19 | 1 | 1 | 9 | 10 | 90 | 97 | 0.100 | 0.091 |

A_28 | 2 | 2 | 8 | 9 | 90 | 97 | 0.222 | 0.182 |

A_37 | 3 | 3 | 7 | 8 | 90 | 97 | 0.375 | 0.270 |

A_46 | 4 | 4 | 6 | 7 | 90 | 97 | 0.571 | 0.364 |

A_55 | 5 | 5 | 5 | 5 | 90 | 97 | 1.000 | 0.455 |

A_64 | 6 | 7 | 4 | 4 | 90 | 97 | 1.750 | 0.545 |

A_73 | 7 | 8 | 3 | 3 | 90 | 97 | 2.667 | 0.636 |

A_82 | 8 | 9 | 2 | 2 | 90 | 97 | 4.500 | 0.727 |

A_91 | 9 | 10 | 1 | 1 | 90 | 97 | 10.00 | 0.818 |

Alloy Sample | a (Å) | ρ (g/cm^{3}) |
---|---|---|

A_00 | 4.020 (4.050 ^{a}) (4.032 ^{b}) | 2.756 (2.700 ^{c}) (2.700 ^{d}) |

A_19 | 4.005 | 2.762 |

A_28 | 4.003 | 2.770 |

A_37 | 4.002 | 2.779 |

A_46 | 4.000 | 2.783 |

A_55 | 3.999 | 2.790 |

A_64 | 3.997 | 2.801 |

A_73 | 3.995 | 2.808 |

A_82 | 3.994 | 2.815 |

A_91 | 3.993 | 2.821 |

**Table 3.**Calculated elastic stiffness constants (${c}_{11}$, ${c}_{12}$ and ${c}_{44}$) of the alloy samples.

Alloy Sample | Si/Mg Ratio | ${\mathit{c}}_{11}$ (GPa) | ${\mathit{c}}_{12}$ (GPa) | ${\mathit{c}}_{44}$ (GPa) |
---|---|---|---|---|

A_00 | - | 100.5 | 62.5 | 34.6 |

A_19 | 0.100 | 90.5 | 68.0 | 33.2 |

A_28 | 0.222 | 87.8 | 69.5 | 29.5 |

A_37 | 0.375 | 87.0 | 69.9 | 29.5 |

A_46 | 0.571 | 82.5 | 71.0 | 24.5 |

A_55 | 1.000 | 88.6 | 71.5 | 29.6 |

A_64 | 1.750 | 114.3 | 61.8 | 34.4 |

A_73 | 2.667 | 118.2 | 59.9 | 34.2 |

A_82 | 4.500 | 126.3 | 58.7 | 34.8 |

A_91 | 10.00 | 123.3 | 63.3 | 33.2 |

**Table 4.**Calculated elastic constants (bulk modulus (B), shear modulus (G), Young’s modulus (E), Poisson’s ratio (μ), Pugh’s ratio (n), and Vickers hardness (Hv) of the alloy samples. μ and n do not have units.

Alloy Sample | Si/Mg Ratio | B (GPa) | G (GPa) | E (GPa) | µ | n | Hv (GPa) Chen | Hv (GPa) Tian |
---|---|---|---|---|---|---|---|---|

A_00 | - | 74.0 | 27.6 | 73.1 | 0.335 | 2.70 | 1.35 | 3.14 |

A_19 | 0.100 | 75.5 | 21.5 | 59.0 | 0.370 | 3.51 | −0.22 | 1.94 |

A_28 | 0.222 | 75.6 | 18.5 | 51.3 | 0.387 | 4.09 | −0.88 | 1.46 |

A_37 | 0.375 | 75.6 | 18.0 | 50.1 | 0.390 | 4.20 | −0.97 | 1.39 |

A_46 | 0.571 | 74.8 | 13.8 | 39.0 | 0.413 | 5.42 | −1.71 | 0.86 |

A_55 | 1.000 | 77.2 | 18.0 | 50.2 | 0.392 | 4.28 | −1.02 | 1.37 |

A_64 | 1.750 | 79.3 | 30.9 | 82.0 | 0.328 | 2.57 | 1.93 | 3.57 |

A_73 | 2.667 | 79.3 | 32.1 | 84.6 | 0.322 | 2.48 | 2.27 | 3.82 |

A_82 | 4.500 | 81.2 | 34.4 | 90.4 | 0.315 | 2.36 | 2.79 | 4.24 |

A_91 | 10.00 | 83.3 | 31.9 | 84.8 | 0.330 | 2.61 | 1.93 | 3.58 |

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**MDPI and ACS Style**

Pius, K.K.; Ongwen, N.O.; Mageto, M.; Odari, V.; Gaitho, F.M.
Mechanical Properties of Al–Mg–Si Alloys (6xxx Series): A DFT-Based Study. *Alloys* **2023**, *2*, 213-226.
https://doi.org/10.3390/alloys2030015

**AMA Style**

Pius KK, Ongwen NO, Mageto M, Odari V, Gaitho FM.
Mechanical Properties of Al–Mg–Si Alloys (6xxx Series): A DFT-Based Study. *Alloys*. 2023; 2(3):213-226.
https://doi.org/10.3390/alloys2030015

**Chicago/Turabian Style**

Pius, Kipkorir Kirui, Nicholas O. Ongwen, Maxwell Mageto, Victor Odari, and Francis Magiri Gaitho.
2023. "Mechanical Properties of Al–Mg–Si Alloys (6xxx Series): A DFT-Based Study" *Alloys* 2, no. 3: 213-226.
https://doi.org/10.3390/alloys2030015