## 3 Results and Discussion
### Nanoscale Elastic Properties  
To fully demonstrate the yield behavior of the MMT mineral, the maximum strain of each direction is expanded to 10% and the corresponding calculation step is increased to 100 steps. The stress-strain curves of the typical MMT system obtained from simulation are shown in Figure 4, where the interlayer compensation cation is Ca\({}^{2+}\) ion, and the interlayer water number \(n\) is 80, corresponding to bilayer hydration. Figure 4: Typical stress–strain behavior for hydrated Ca-montmorillonite. Figure 3: Hydration energy and immersion energy of montmorillonite as a function of increasing interlayer water molecule number. The micromechanical properties of hydrated MMT show obvious anisotropy. The in-plane uniaxial tensile and compressive strength of MMT is approximately equal (for \(\sigma_{1}\) and \(\sigma_{2}\)), while the tensile strength is slightly less than the compressive strength. When the tensile strain reaches about 7%, the material yields. Meanwhile, the compressive stress-strain relationship shows a good linear relationship within the maximum compressive strain (10%). The tensile and compressive strength (\(\sigma_{3}\)) of MMT orthogonal to the mineral plane is much lower than that in-plane strength (\(\sigma_{1}\) and \(\sigma_{2}\)). This is mainly due to the appearance of the interlayer water in MMT lamellae, and the bonding strength between the MMT crystal and water molecule through non-bonding is much lower than that of covalent or ionic bonding in the mineral plane. In addition, the compression of MMT in the \(Z\) direction shows obvious non-linear characteristics and hardening under high pressure, which is different from the linear characteristics in the crystal plane. The hydration molecular layer that mixed within MMT lamellae has a certain shear strength (\(\tau 4\) and \(\tau 5\)) and also shows anisotropy along with different directions of the MMT crystal plane. The shear stiffness of the interlayer water along the Y direction (\(\tau 5\)) is slightly larger than that in the X direction (\(\tau 4\)), while its shear strength is slightly lower. The shear capability of MMT lamellae is very high (\(\tau 6\)) and shows good linear characteristics in the simulated strain ranges. By comparing the quantitative values of in-plane strength, it was found that the in-plane compressive strength of hydrated MMT is the largest, followed by tensile strength and shear strength. Similarly, this variation also applies to the strengths that are perpendicular to the mineral plane. All of the stress responses show good linearity in the range of 1% small strain, and this is illustrated in the lower right of each block in Figure 4. This also shows that the linear elastic constants calculated by the small strain of 1% are reliable. The stiffness tensor and linear elastic constants of typical MMT in the 1% strain range corresponding to Figure 4 are listed in Table 1.  
The simulated stiffness tensor components in Table 1 are in good agreement with simulations and measurements in the literature. The in-plane components \(C_{11}\) and \(C_{22}\) of the MMT with water bilayer at 300 K are 189 GPa and 191 GPa, respectively, from [25], which are almost equal to the results of this study that \(C_{11}=189\) GPa and \(C_{22}=212\) GPa. Furthermore, there are differences in the in-plane Young's moduli \(E_{1}\) and \(E_{2}\) which are derived from the stiffness tensor, and these differences are induced by the different topologies of trivalent cations in dioctahedral structure of MMT, as shown in Figure 5. However, the in-plane Young's moduli (\(E_{1}\) and \(E_{2}\)) are all much higher than the Young's modulus perpendicular to the plane (\(E_{3}\)). On the contrary, the in-plane shear moduli (\(G_{23}\) and \(G_{31}\)) are lower than the shear modulus perpendicular to the plane (\(G_{12}\)). The larger difference of [PERSON]'s ratio in different directions once again shows the anisotropy of the MMT structure. The b