## 4 Conclusions  
Molecular dynamic simulations were conducted to investigate the mechanical properties of montmorillonite with different compensation cations upon hydration with unconstrained system atoms, expanding further studies of nanoscale elastic properties. The conclusions are as follows:  
1. The basal spacing of all MMT systems increases from step to step in the initial crystalline swelling stage. While this step characteristic (plateau) disappears, the basal spacing shows approximately linear variation in the subsequent osmotic swelling stage. However, the quantitative values of the basal spacing for different compensation cationic MMTs are different, which is related to the ion radius of cations.
2. The nanoscale elastic properties of hydrated MMT show obvious anisotropy. The tensile and compressive strength (\(\sigma_{3}\)) of MMT orthogonal to the mineral plane is much lower than the in-plane strengths (\(\sigma_{1}\) and \(\sigma_{2}\)), and the hydration molecular layer mixed within MMT lamellae has certain shear strengths (\(\tau_{4}\) and \(\tau_{5}\)). The order of strength values from large to small is the compressive strength, the tensile strength, and the shear strength, respectively.
3. The components of \(C_{33}\), \(C_{44}\), and \(C_{55}\) all fluctuate with the increase of interlayer water content, while the variation of the in-plane stiffness tensor components \(C_{11}\), \(C_{22}\), and \(C_{12}\), all decrease nonlinearly with the increase of water content. These components are mainly controlled by the bonding strength of mineral atoms and the geometric conformation of the hydrated MMT system. The \(C_{66}\) component reflects the shear of hydrated MMT along the direction perpendicular to the mineral plane, which is mainly controlled by the relative torsion of bonded atoms. Due to the increase of interlayer water contribution in the weak shear zone, \(C_{66}\) decreases continuously with the increase of interlayer water content. Young's moduli all decrease nonlinearly with the increase of water content.  
Figure 10: Young modulus as a function of basal layer spacing.  
The simulation results reveal that the water molecules linked to the mineral surface will be arranged in an ordered structure (H-bond formation and thermodynamic dense stable state) and their shear strength are not zero. Preliminary discussion shows that this is related to the strength mechanism of clay at the microscale. Building a bottom-up procedure toward a constitutive law for soil mechanical still needs further research.  
Conceptualization, L.K. and Q.Z.; Methodology, X.S.; Validation, X.Z.; Formal analysis, L.K.; investigation, Q.Z.; writing--original draft preparation, L.K Writing-Original Draft Preparation, L.K.; Writing--review and editing, Q.Z. and X.Z.; Visualization, X.S.; Supervision, Q.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Fundamental Research Funds for the Central Universities, grant number 2020 ZDPYMS18.  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
The study presented in this article was supported by the Fundamental Research Funds for the Central Universities (2020 ZDPYMS18). This support is gratefully acknowledged.  
The authors declare no conflict of interest.