## 3 Results and Discussion
### Nanoscale Elastic Properties
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(C_{ij}\) & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline
1 & \(189.6710\pm 1.0140\) & \(81.7433\pm 0.8172\) & \(10.1290\pm 0.4701\) & \(1.2430\pm 0.3777\) & \(-10.4597\pm 0.2771\) & \(0.4964\pm 0.4787\) \\
2 & & \(212.5464\pm 1.7814\) & \(8.4054\pm 0.4993\) & \(3.5123\pm 0.4520\) & \(-3.4307\pm 0.3618\) & \(1.1526\pm 1.0715\) \\
3 & & & \(28.4335\pm 0.5057\) & \(0.3278\pm 0.2660\) & \(1.3468\pm 0.1654\) & \(0.1060\pm 0.2955\) \\
4 & & & & \(4.2630\pm 0.1881\) & \(-0.6221\pm 0.1371\) & \(-2.7480\pm 0.2758\) \\
5 & & & Symmetric & & \(5.4343\pm 0.1049\) & \(0.3864\pm 0.2236\) \\
6 & & & & & & 45.4441 \(\pm\) 0.6942 \\ \hline Young’s & \(E_{1}\) & \(139.1605\) & \(E_{2}\) & \(174.2115\) & \(E_{3}\) & \(27.0289\) \\ Abdulus & \(G_{23}\) & \(2.1315\) & \(G_{31}\) & \(2.7172\) & \(G_{12}\) & \(22.7221\) \\ Poisson’s ratios & \(\mu_{12}\) & \(\mu_{21}\) & \(\mu_{13}\) & \(\mu_{23}\) & \(\mu_{32}\) \\
0.3463 & 0.4335 & 0.3436 & 0.0667 & 0.1199 & 0.0186 \\ Bulk modulus & & & & 70.1340 & & \\ Compressibility & & & & 38.6319 & & \\ (1/TPa) & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Typical stiffness coefficients and elastic constants of hydrated Ca-MM (GPa). strength in the mineral plane is much higher than that of non-bonding strength between mineral lamellae. The evolution of elastic properties of hydrated MMT under different stable state water content will be further illustrated by analyzing the changes of some components in the stiffness tensor. To validate the study, the simulation results from [29] and UPV test results from [17] are adopted for comparison due to the high correlation with the work in this study. The authors in [29] gave the variation of stiffness components during the hydration process of Na-Wyoming MMT with a water content of 0-40%. The authors in [17] obtained the range of elastic stiffness constants of smectite minerals on the microscale based on UPV test results of clay composition in shale. For compression of clay under high pressure, the most concerning component of the stiffness tensor is \(C_{33}\), that is, the ability to resist deformation in the direction of perpendicular to the mineral plane. Figure 6 shows the variation of \(C_{33}\) obtained upon hydration by [29], Na-, Cs-, Ca- MMT in this study under different stable state conditions, and the test range results of [17]. Since the stiffness tensor elements in this study are obtained by a statistical average of 20 frame trajectory files, the statistical standard deviation of data is also given in Figure 6.  
[PERSON] et al. (2012) revealed that \(C_{33}\) decreased sharply from dry MMT to the formation of the first hydrated layer between lamellae, then increased to a local maximum with the formation of the stable hydrated layer, and finally maintained a constant until  
Figure 5: Different topologies of Al atom in the dioctahedral structure of montmorillonite. Figure 6: The \(C_{33}\) elastic constant as a function of basal layer spacing. the basic spacing d \(>\) 15.6 A [29]. The values of \(C_{33}\) obtained in this study are in good agreement with the results of [29], apart from the first hydrated layer. The \(C_{33}\) of Ca-MMT is slightly higher than that of Na-MMT, while the \(C_{33}\) of Cs-MMT is the lowest among all the same hydrated states. Apart from the dry MMT, all of the simulated \(C_{33}\) fall well into the given range of UPV measured by [17] (\(C_{33}\) = 11.5\(\sim\)30.4 GPa). The in-plane stiffness tensor components \(C_{11}\), \(C_{22}\), and \(C_{12}\), which reflect the Poisson effect in the plane, are shown in Figure 7 with basal spacing. \(C_{11}\), \(C_{22}\), and \(C_{12}\) all decrease nonlinearly with the increase of water content. These components are mainly controlled by the ionic or covalent bonding strength of mineral atoms and the geometric conformation of the hydrated MMT system. In addition, the simulation results of \(C_{11}\) and \(C_{12}\) are in good agreement with those of [29], but \(C_{12}\) is slightly lower.