3.3.2. Elastic Modulus of Mechanical Stability
The yield limit and strength limit obtained in the tensile test reflect the bearing capacity of the material to the force, while the elongation δ or section shrinkage ψ reflect the plastic deformation capacity of the material. In order to express the difficulty of material resisting deformation in the elastic range, the meaning of material elastic modulus E is usually reflected by the rigidity of parts in practical engineering, because once the parts are designed according to the stress, the rigidity is judged by the deformation generated by the load in the service process in the elastic deformation range. Generally, the load-causing unit strain is the rigidity of the part. For example, for tension and compression parts, the rigidity is , and is the cross-section area of the part. It can be seen that in order to improve the stiffness of the parts, which is to reduce the elastic deformation of the parts, the materials with high elastic modulus can be selected, and the cross-sectional area of the bearing can be appropriately increased. The importance of stiffness is that it determines the stability of parts in service, especially for slender rod-shaped parts and thin-walled parts. Therefore, for the theoretical analysis and design calculation of components, the elastic modulus E is an important mechanical performance index that is often used.
Elastic modulus is defined as the ratio of stress to corresponding strain when the ideal material has small deformation. The nature of modulus depends on the nature of deformation. Elastic modulus (unit: GPa) includes the Young’s modulus (tensile stress/tensile strain, also called positive elastic modulus, tensile modulus), shear modulus (shear stress/shear strain angle), compression modulus (compressive stress/shrinkage strain), bulk modulus (bulk pressure/bulk strain)—the reciprocal of the modulus is called the flexibility expressed in . When it is not easy to cause confusion, the elastic modulus of general metal materials refers to the young’s modulus, that is, the positive elastic modulus.
From the macroscopic point of view, the modulus of elasticity is a measure of the ability of an object to resist elastic deformation. From the microscopic point of view, it is a reflection of the bonding strength among atoms, ions, or molecules. All the factors that affect the bonding strength can affect the elastic modulus of the material, such as bonding mode, crystal structure, chemical composition, microstructure, temperature, etc. The young’s modulus of metal materials will fluctuate by 5% or more due to different alloy compositions, heat treatment state and cold plastic deformation. However, generally speaking, the elastic modulus of metal material is a mechanical property index that is not sensitive to the structure. Alloying, heat treatment (fiber structure), cold plastic deformation, and other factors have little influence on the elastic modulus , and the external factors such as temperature and loading rate have little influence on it, so the elastic modulus is regarded as a constant in general engineering application.
Poisson’s ratio refers to the ratio of the absolute value of the transverse positive strain and the positive axial strain when the material is under tension or compression in a single direction, also known as the transverse deformation coefficient, which is the elastic constant reflecting the transverse deformation of the material. When the material extends (or shortens) along the load direction, it will also shorten (or lengthen) in the direction perpendicular to the load. The negative value of the ratio of the strain in the vertical direction to the strain in the load direction is called Poisson’s ratio of the material, i.e.; . Poisson’s ratio of materials is generally determined by test method. For traditional materials, is generally constant in the elastic working range, but beyond the elastic range, increases with the increase of stress until . For the material with large Poisson’s ratio, the amount of transverse deformation is larger than that of the longitudinal deformation after the material is stressed and before plastic deformation, otherwise, the amount of transverse deformation is smaller than that of the longitudinal deformation.
In
Table 6, we compare the results of calculation when there are vacancy point defects in different supercells of Mg
2X (X = Si, Ge) intermetallic compounds; it is found that for Mg
2Si type intermetallics, the
,
and
of Mg
2X with Mg vacancy (V
Mg) are higher than those of Mg
2X with X vacancy (V
X). The above results show that the vacancy defects in Mg position, compared with those in the Si position and Ge position, will lead to the stronger resistance to volume deformation, shear strain and elastic deformation of crystal materials, the greater the rigidity, the less deformation, the stronger the brittleness and the worse the plasticity.
For intermetallic compounds with anti-site point defects, the , and E of Mg2X (X = Si, Ge) with the XMg anti-site are higher than those of Mg2X with the MgX anti-site, indicating that when the Si or Ge atom replace the Mg atom to form anti-vacancy defects, compared with Mg atom replacing Si or Ge atom to form anti vacancy defect, the material has a strong resistance to volume deformation, shear strain and elastic deformation, but poor plasticity. Besides, for Mg2X (X = Si, Ge) with same point defect structure, the , and E of 1 × 1 × 2 supercells are smaller than those of 2 × 2 × 2 supercells, which indicates that the Mg2X (X = Si, Ge) with point defects are easier to form in 2 × 2 × 2 supercells. Therefore, the point defects generated on the Mg position of 2 × 2 × 2 supercells can enhance the mechanical stability of the material, while the plasticity is poor.
Table 6 also shows that the Poisson’s ratio
(0.168 and 0.164) of
Mg16X8 without point defects is the smallest, and the Poisson’s ratio
with defects is larger (greater than 0.18), indicating that the formation of defects makes the transverse deformation of the material larger and the transverse elastic mechanical instability increased. It was also found that the Poisson’s ratio of Mg
2X (X = Si, Ge) with X vacancy is higher than that of Mg
2X with the Mg
X anti-site. Similarly, the Poisson’s ratio of Mg
2X with Mg vacancy was slightly higher than that of Mg
2X with X
Mg anti-site. Thus, the above results show that when there are defects in the Mg, Si, and Ge position, the void point defects increase the instability of transverse deformation more than the anti-void point defects. Note that for Mg
2X (X = Si, Ge) with the same point defect structure (except Mg
Si), the Poisson’s ratio
of the 2 × 2 × 2 supercells is smaller than that of the 1 × 1 × 2 supercells, indicating that the point defects in larger supercells can only increase the instability of transverse deformation slightly, compared with 1 × 1 × 2 supercells.
According to Pugh’s report [
44], G/B can predict the ductility or brittleness of materials, and materials with G/B value higher than the critical value of 0.57 belong to brittle materials; it can be seen from
Table 6 that both Mg
2X (X = Si, Ge) with and without point defects are brittle materials. The maximum G/B value of Mg
2X without point defects indicates that the formation of defects will improve the ductility of materials; the G/B value of vacancy point defects is smaller than that of anti-site defects, indicating that vacancy point defects can improve the plasticity of materials more than anti-site defects. Compared with 2 × 2 × 2 supercells, the G/B of Mg
2X with the same point defect structure (except
) in 1× 1 × 2 supercells can enhance the plasticity of materials.
In theory, only two of the three elastic constants , and of isotropic materials are independent, because they have the following relationship: This is consistent with the essence of the definition of Poisson’s ratio. For anisotropic materials, the elastic modulus in x, y and z directions, shear modulus and Poisson’s ratio in xy, xz, and yz planes are required
3.3.3. Elastic Anisotropy
Anisotropy exists widely in all kinds of materials. In short, crystals have different properties in different directions. It is easy to confuse anisotropy and inhomogeneity, which are two completely different concepts. Anisotropy means that the properties of materials are related to directions, while inhomogeneity means that the properties of materials are related to parts. In the case of single crystals, at any point in the interior, the structures and properties are the same, but it has different performance in different directions.
Crystal is anisotropic, with different properties in different directions, and has strict symmetry. Here, the symmetry is a very remarkable property, which is widely used in mathematics and physics, and also commonly used in the analysis of material properties. Polycrystalline materials have preferred orientation and certain anisotropy.
All kinds of materials are elastic, and the elastic properties of most materials are anisotropic [
45,
46]. For example, in cubic crystals, the [111] direction is usually more difficult to compress than the [100] direction. The three-dimensional surface Young’’s modulus
of intermetallics is calculated as follows [
47]
where
represent the flexibility coefficient,
indicate direction cosines relative to the x, y, and z directions, respectively.
Figure 4 and
Figure 5 show the three-dimensional Young’s modulus
surface of each intermetallic compound Mg
2X (X = Si, Ge) at 0 GPa with 1 × 1 × 2 supercells and 2 × 2 × 2 supercells, respectively. The near sphere shows that these intermetallic compounds are elastic and anisotropic [
48]. Compared with other point defects of Mg
2X, the Mg
2X with X vacancy exhibits a relatively obvious deviation from spherical shape, which indicates that the Mg
2X with X vacancy has a relatively strong anisotropy. The degree of spherical deviation of Mg
2X decreases with the increase of supercell size, implying that Mg
2X with point defects show a relatively weak anisotropy in the 2 × 2 × 2 supercells.