## 3 Results and Discussion
### Nanoscale Elastic Properties
The reason may lie in the difference between our simulation structure and the octahedral isomorphic substitutions of [29]. However, the measured range of UPV given by [17] (\(C_{11}\) = 13.8-46.1 GPa and \(C_{12}\) = 6.9\(\sim\)17.8 GPa) is much lower than the simulated results. This is because the MMT mineral lamellae are very thin, and the relative slip between lamellae easily occurs in the UPV test. However, the simulated results in [25] (\(C_{11}\) = 189\(\sim\)231 GPa) and test results by Brillouin scattering on mica minerals in [26] (\(C_{11}\) = 181 GPa and \(C_{22}\) = 178 GPa) both show that the results of this study are credible. For the shear of clay under high pressure, the most concerned is the shear resistance of limited interlayer water molecules in the hydrated MMT system, i.e., the in-plane shear stiffness tensor components \(C_{44}\) and \(C_{55}\). Their variations with basal spacing are shown in Figure 8; \(C_{44}\) and \(C_{55}\) fluctuate with the increase of water content which is like the variation of \(C_{33}\). Therefore, the mechanism of shear strength of interlayer water is closely related to the formation of H-bonds and atom-free volume. The shear strength between dry MMT lamellae is much larger than that of hydrated MMT lamellae, that is to say, the addition of water plays a lubricating role in interlayer friction, which makes \(C_{44}\) and \(C_{55}\) decrease sharply. However, these limited interlayer water molecules do not completely show the zero shear strength characteristics of free water. As water content increases, the in-plane shear strength increases to a local maximum when the system approaches the thermodynamic stable state and minimum atom free volume occurs. In the range of simulated water content, \(C_{44}\) and \(C_{55}\) eventually tend to be constant. Furthermore, the results in Figure 8 shows that the components \(C_{44}\) and \(C_{55}\) are in good agreement with those obtained by [29], and the in-plane shear strength between different MMT systems is close except for the dry MMT system. Meanwhile, the simulated results also fall well into the UPV measurement range of [17] (\(C_{44}\) = 2.9-8.9 GPa). Figure 7: The \(C_{11}\), \(C_{22}\), and \(C_{12}\) elastic constants as a function of basal layer spacing. The shear of hydrated MMT along the direction perpendicular to the mineral plane is mainly controlled by the relative torsion of bonded atoms. The variations of \(C_{66}\) with basal spacing are shown in Figure 9. With the increase of interlayer water content, \(C_{66}\) decreases continuously, which is mainly due to the increase of interlayer water contribution in the weak shear zone. Considering that the ratio of width to height of a true MMT lamella is very large (about 100:1 to 1000:1), shear along the direction of orthogonal to the mineral plane is almost impossible, so the variations of \(C_{66}\) will not be discussed here. Young's moduli of the different MMT systems under various hydration can be obtained by further conversion of the stiffness tensor, and the variation of Young's moduli with basal spacing is shown in Figure 10. The in-plane Young's moduli \(E_{1}\) and \(E_{2}\) are consistent with \(C_{11}\) and \(C_{12}\) that decrease nonlinearly with the increase of water content. For the Young's modulus of \(E_{3}\) which is perpendicular to the mineral plane, its value for the dry MMT system is 2-3 times that of the hydrated system for Na-MMT and Ca-MMT. \(E_{3}\) of the hydrated MMT system also decreases nonlinearly with the increase of water content. Interestingly, for Cs-MMT, the Young's modulus, E\({}_{3}\), of different hydrated MMT systems, even dry MMT systems, is almost the same. Preliminary analysis shows that this may be related to the larger radius of Cs\({}^{+}\) ions. For the dry MMT system, the interlayer spacing is large, which induces a low compression strength.