## 3 Simulation of Sand Plane Strain Test
### Simulation Result  
#### 3.3.1 Influence of Confining Pressure on Strength and Deformation  
Figure 6 displays the relationship between the stress ratio (\(\sigma_{1}/\sigma_{3}\)) and volume strain (\(\varepsilon_{\text{v}}\)), with axial strain (\(\varepsilon_{\text{a}}\)) obtained from physical experiments (reference [31]). The experiments were conducted under different pressure levels, where the void ratio \(e\) was approximately 0.75 and \(\delta\) was set at 90\({}^{\circ}\) to investigate the effect of confining pressure (\(\sigma_{\text{c}}\)) on deformation characteristics. Three conclusions can be drawn from the results: First, prior to reaching peak stress, the ratio of the increment in stress ratio divided by the increment \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sand Sample** & **Confining Pressure (kgf/cm\({}^{2}\))** & **Void Ratio \(e\)** & **Deposition Angle (\({}^{\circ}\))** \\ \hline  & 0.05 & & 90 \\  & 0.1 & \(e\approx 0.7\) & 45 \\ Toyoura sand & 0.5 & & 34 \\  & 1.0 & \(e\approx 0.8\) & 23 \\  & 4.0 & & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Test scheme for plane strain. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**\(\boldsymbol{E(\text{Mpa})}\)/\(\sigma_{\text{c}}\)** & \(\boldsymbol{v}_{1}\) & \(\boldsymbol{\varphi}\)(\({}^{\circ}\)) & \(\boldsymbol{\psi}\)(\({}^{\circ}\)) \\ \hline
400 & 0.2 & 45 & 0 \\ \hline \(C_{1}\) & \(C_{2}\) & \(C_{3}\) & \(C_{4}\) \\ \(-111.11\) &in axial strain (d(\(\sigma_{1}/\sigma_{3}\))/d\(\epsilon_{\rm a}\) values) declines with the rise of \(\sigma_{\rm c}\) under constant \(\sigma_{1}/\sigma_{3}\) conditions. The axial strain (\(\epsilon_{\rm a}\)) at peak stress increases with the rise of confining pressure (\(\sigma_{\rm c}\)), and this trend becomes more prominent when \(\sigma_{3}\) exceeds 0.5 kgf/cm\({}^{2}\). In addition, after reaching peak stress, \(-\)d(\(\sigma_{1}/\sigma_{3}\))/d\(\epsilon_{\rm a}\) increases with the increase in \(\sigma_{\rm c}\). Finally, both \(\sigma_{1}/\sigma_{3}\) and \(\epsilon_{\rm v}\) decrease with the increase in \(\sigma_{\rm c}\) when the samples reach a higher strain state (5-10%). When the confining pressure \(\sigma_{\rm c}\) is 0.5, 1.0, 4.0 (kgf/cm\({}^{2}\)), the increment of volumetric strain d\(\epsilon_{\rm v}\) corresponding to axial strain of 5-10% is nearly zero, and the corresponding \(\sigma_{1}/\sigma_{3}\) is about 3.5. This finding indicates that some of the samples have reached the residual stress state under the existing confining pressure. In contrast, when the \(\sigma_{\rm c}\) is 0.05 kgf/cm\({}^{2}\) and \(\epsilon_{\rm a}\) reaches 5-10%, the corresponding d\(\epsilon_{\rm v}\) is not zero and the stress ratio \(\sigma_{1}/\sigma_{3}\) is greater than 3.5, indicating that under this confining pressure level, the specimen has not yet entered the residual stress state and will still undergo further deformation. The results of the hypoplastic constitutive model simulation are presented in Figure 7. First, it is important to clarify that this article focuses on qualitatively elucidating the capability of the proposed model to simulate anisotropy. The objective is to establish a correlation between the simulation results and the trends observed in experimental data. However, a thorough quantitative comparison with experimental results is not performed. The void ratio in the hypoplastic constitutive simulation was set at \(e=0.35\) to emphasize the stress softening aspect, while keeping all the other conditions the same as in the experiment. The simulation and experimental results exhibit similar trends, with reductions in stress ratio and \(\epsilon_{\rm v}\) as the confining pressure increases. The maximum values of \(\sigma_{1}/\sigma_{3}\) and \(\epsilon_{\rm v}\) were obtained at the minimum confining pressure condition, \(\sigma_{\rm c}=0.05\) kgf/cm\({}^{2}\).