## 3 Simulation of Sand Plane Strain Test
### Simulation Result
#### 3.3.1 Influence of Confining Pressure on Strength and Deformation
The peak stress ratio and residual stress ratio from the constitutive model simulations were approximately 6.3 and 4.0, respectively, closely matching the physical experiment results of 6.0 and 4.5, respectively. At \(\epsilon_{\rm a}\) of 10%, the corresponding \(\epsilon_{\rm v}\) was 4%, which was slightly greater than that of the physical sample. When \(\sigma_{\rm c}\) was 4.0 kgf/cm\({}^{2}\), the constitutive model simulations yielded peak stress ratio and residual stress ratio values of 4.5 and 3.8, respectively; whereas the physical experiment showed values of 5.1 and 4.0, respectively. At \(\epsilon_{\rm a}\) of 10%, the corresponding \(\epsilon_{\rm v}\) was 3%. The results indicate relatively fine agreement between the hypoplastic constitutive model and physical testing trends, which can broadly reflect the strength and deformation characteristics of soil under different confining pressures, especially stress ratio. The curve of axial strain versus volumetric strain can also reflect the overall evolution law corresponding to physical testing. Figure 6: The stress–strain relationship measured by the test [31] under different confining pressures at \(\delta=90^{\circ}\). The test results demonstrate that dense and loose sand exhibit prominent strength and deformation anisotropy at different confining pressure levels. When the rotation angle of the major principal stress axis is 90\({}^{\circ}\)--that is, perpendicular to the principal fabric direction--the sand sample exhibits the maximum peak shear strength, and then the peak shear strength shows a decreasing trend with the decreasing rotation angle of the major principal stress axis, reaching the minimum value at 23\({}^{\circ}\), and subsequently gradually increases, indicating a trend of first decreasing and then increasing overall, but the strength at a rotation angle of 0\({}^{\circ}\) is not as robust compared to 90\({}^{\circ}\). On the other hand, the deformation (\(\varepsilon_{\rm v}\)) of the sand under a confining pressure of \(\sigma_{\rm c}\) = 4.0 kgf/cm\({}^{2}\) is insensitive to the initial fabric; under larger deformations, which correspond to an axial strain of 7.5-10%, the d\(\varepsilon_{\rm v}\) is extremely small, while the stress ratio at different deposition angles gradually reaches a nearly identical state. Figure 9(a),b depicts the \(\sigma_{1}\)/\(\sigma_{3}\)-\(\varepsilon_{\rm a}\) and \(\varepsilon_{\rm v}\)-\(\varepsilon_{\rm a}\) curves of dense sand simulated by the hypoplastic model at \(\sigma_{\rm c}\) = 0.05 kgf/cm\({}^{2}\) (5 kPa). Simulation results demonstrate that dense sand undergoes stress softening after its \(\sigma_{1}\)/\(\sigma_{3}\) reaches its peak value at \(\varepsilon_{\rm a}\) = 4%. The \(\sigma_{1}\)/\(\sigma_{3}\) value gradually reaches a fixed value, consistent with experimental results, as \(\varepsilon_{\rm a}\) approaches 10% after experiencing a significant amount of deformation. The maximum stress ratio of 6.3 occurs at \(\delta\) of 90\({}^{\circ}\), followed by a monotonically decreasing trend in the peak stress ratio as \(\delta\) decreases and reaches its minimum value of 5.0 at \(\delta\) of 0\({}^{\circ}\). A discrepancy compared to the experimental result is observed with the minimum stress ratio occurring at 23\({}^{\circ}\). As shown in Figure 9(b), as \(\delta\) decreases, \(\varepsilon_{\rm v}\) also decreases, indicating weakened dilatancy. Loose sand undergoes strain hardening during shearing. As shown in Figure 9(c), there is no decrease in the stress ratio when \(\varepsilon_{\rm a}\) reaches about 4%, and the stress ratio remains constant during the subsequent shearing process.