## 3 Results and Discussion
### Mechanical Property Analysis  
This study employed a stress fluctuation method to calculate the mechanical properties of the system. Under a constant temperature, T, the relationship between the elastic stiffness coefficient (\(C_{ijkl}\)) and the stress (\(\sigma_{ij}\), N \(\cdot\) m\({}^{-2}\)) and strain (\(\varepsilon_{kl}\)) can be obtained using Equation (5) [52]:  
\[C_{ijkl}=\left.\frac{\partial\sigma_{ij}}{\partial\sigma_{kl}}\right|_{T, \varepsilon_{kl}}=\frac{1}{V_{0}}\left.\frac{\partial^{2}A}{\partial \varepsilon_{ij}\partial\varepsilon_{kl}}\right|_{T,\varepsilon_{ij}, \varepsilon_{kl}} \tag{5}\]  
where \(V_{0}\) represents the initial volume of the simulation unit in its undeformed state, m\({}^{3}\), and \(A\) denotes the Helmholtz free energy in J. By utilizing Equations (6) to (9), the computation of two distinct coefficients, namely the bulk modulus (\(K\), GPa) and the shear modulus (\(G\), GPa), enables the characterization of the elastic properties of isotropic materials. \[G_{V}=[C_{11}+C_{22}+C_{33}+2(C_{44}+C_{55}+C_{66})-(C_{12}+C_{13}+C_{23})]/15 \tag{6}\]  
\[G_{R}=15/[4(S_{11}+S_{22}+S_{33}-S_{12}-S_{13}-S_{23})+3(S_{44}+S_{55}+S_{66})] \tag{7}\]  
\[K_{V}=[C_{11}+C_{33}+2(C_{21}+C_{13}+C_{23})]/9 \tag{8}\]  
\[K_{R}=[S_{11}+S_{22}+S_{33}+2(S_{12}+S_{13}+S_{23})]^{-1} \tag{9}\]  
where the subscripts \(R\) and \(V\) refer to the Reuss and Voigt average values, respectively. \(S_{ij}\) represents the components of the elastic compliance matrix derived from the inverse of the elastic stiffness matrix. Hill represents the actual values of the bulk and shear moduli, which typically fall within the range of the Reuss and Voigt values. This approximation is commonly known as the Voigt-Reuss-Hill (VRH) approximation. The bulk modulus and shear modulus of the system can be determined using Equations (10) and (11) [53]:  
\[K_{VRH}=(K_{V}+K_{R})/2 \tag{10}\]  
\[G_{VRH}=(G_{V}+G_{R})/2 \tag{11}\]  
Figure 7: The adsorption energies of the three epoxy resins on the sand surface. Utilizing the average values from the aforementioned two methods, the calculation of [PERSON]'s modulus (\(E\), GPa) and [PERSON]'s ratio (\(v\)) can be achieved through Equations (12) and (13):  
\[E=9G_{VRH}/(3+G_{VRH}/K_{VRH}) \tag{12}\]  
\[v=(3-2G_{VRH}/K_{VRH})/(6+2G_{VRH}/K_{VRH}) \tag{13}\]  
The mechanical properties of the three epoxy resins did not differ significantly, as shown in \[G_{V}=[C_{11}+C_{22}+C_{33}+2(C_{44}+C_{55}+C_{66})-(C_{12}+C_{13}+C_{23})]/15 \tag{6}\]0 After adsorption with sand, DGEBA exhibited a slightly higher shear modulus and Young's modulus compared to the other two epoxy resins, indicating a stronger resistance to shear and tensile deformation. Furthermore, its Poisson's ratio was slightly higher in the XY direction than those of the other epoxy resins, suggesting increased susceptibility to transverse deformation in that direction. The overall mechanical properties of DGEBDS after adsorption with sand were similar to those of DGEBA. However, the [PERSON]'s modulus of DGEBDS displayed greater variation across different directions, and its Poisson's ratio also showed notable discrepancies among directions. This can be attributed to the presence of more polar groups within the DGEBDS molecule, which likely enhanced intermolecular forces and, consequently, its mechanical properties. Conversely, the [PERSON]'s modulus of AEOR after adsorption with sand varied considerably across different directions compared to the other two epoxy resins. It exhibited relatively lower shear modulus and bulk modulus values, suggesting weaker resistance to shear and volumetric deformation. Nevertheless, its Poisson's ratio was slightly higher in the ZY direction than those of the other epoxy resins, indicating increased susceptibility to transverse deformation in that direction. The Poisson's ratios for the three epoxy-sand composites exhibit slight variations across different directions.