## 3 Simulation of Sand Plane Strain Test
### Simulation Result
#### 3.3.1 Influence of Confining Pressure on Strength and Deformation
Moreover, the value of \(\sigma_{1}\)/\(\sigma_{3}\) decreases compared with the dense sand, and different rotation angles of the principal stress axis, which are from 0 to 90 degrees, bring about a certain degree of strength anisotropy, although  
Figure 10: The \(\sigma_{1}\)/\(\sigma_{3}\)-\(\varepsilon_{\rm a}\) and \(\varepsilon_{\rm v}\)-\(\varepsilon_{\rm a}\) relationship simulated by the proposed model corresponding to (**a**,**b**) dense sand and (**c**,**d**) loose sand, when \(\sigma_{\rm c}\) = 0.05 kgf/cm\({}^{2}\). the degree of anisotropy is slightly weaker than that of dense sand. The dilation tendency is similarly weaker in loose sand, as indicated by Figure 10d. Simulation results of the \(\sigma_{1}/\sigma_{3}\)-\(\epsilon_{\mathrm{a}}\) and \(\epsilon_{\mathrm{v}}\)-\(\epsilon_{\mathrm{a}}\) under \(\sigma_{\mathrm{c}}\) of 4.0 kgf/cm\({}^{2}\) (400 kPa) are presented in Figure 11. As the confining pressure increases to 400 kPa, both loose and dense sand stress-strain curves reflect a decreasing trend in the overall stress ratio and volumetric strain, while exhibiting similar behavior to that observed at 0.05 kgf/cm\({}^{2}\). In order to explore the differences and connections in simulating anisotropic capabilities between the anisotropic hypoplastic model proposed in this paper and the existing mature models, the present study compares the simulation results with the transversely isotropic hypoplastic model developed by [PERSON] (1998) [PERSON] (1998). [PERSON]'s model considers the effect of variation in the principal stress loading direction (deposition angle), and the constitutive equation is expressed in Equations (9)-(12). Both models consider the impact of principal stress axis rotation. [PERSON]'s model takes the anisotropic parameters as \(\alpha=0.9\), \(\beta=1.0\), and \(\gamma=1.04\), with a maximum void ratio of 0.85, a minimum void ratio of 0.3, and confining pressure of \(\sigma_{\mathrm{c}}=4.0\) kgf/cm\({}^{2}\) (400 kPa), and \(\mathbf{A}\) represents the second-order tensor related to the angle between the principal stress and the principal fabric direction. The remaining model parameters are listed in Table 2.  
\[\overset{o}{\sigma}=\mathbf{L}(\mathbf{\sigma},\dot{\mathbf{\epsilon}})+ \mathbf{M}(\mathbf{A},\mathbf{N})\left\|\dot{\mathbf{\epsilon}}\right\| \tag{9}\]  
\[\mathbf{M}(\mathbf{A},\mathbf{N})\mathbf{=}\left(\alpha+\gamma-2\beta\right) \text{tr}(\mathbf{A}\mathbf{\cdot}\mathbf{N})+\gamma\mathbf{N}+(\beta-\gamma )(\mathbf{A}\mathbf{\cdot}\mathbf{N}+\mathbf{N}\mathbf{\cdot}\mathbf{A}) \tag{10}\]  
Figure 11: The \(\sigma_{1}/\sigma_{3}\)-\(\epsilon_{\mathrm{a}}\) and \(\epsilon_{\mathrm{v}}\)-\(\epsilon_{\mathrm{a}}\) relationship simulated by the proposed model corresponding to (**a**,**b**) dense sand and (**c**,**d**) loose sand, when \(\sigma_{\mathrm{c}}=4.0\) kgf/cm\({}^{2}\). The linear term \(\mathbf{L}\) and the nonlinear term \(\mathbf{N}\) are defined as follows, and the form of \(f(e)\) is shown in Formula (4):  
\[\mathbf{L}=C_{1}\text{tr}(\sigma)\dot{e}+C_{2}\text{tr}(\dot{e})\sigma+C_{3} \frac{\text{tr}\sigma\dot{e}}{\text{tr}\sigma}\sigma \tag{11}\]  
Figure 7.0  
The simulation results of [PERSON]'s anisotropic model are presented in Figure 7.1 The strength of the model for dense sand shows a trend of decreasing first and then increasing as \(\delta\) decreases, and the minimum shear strength corresponds to \(\delta\) of approximately 34\({}^{\circ}\) and the maximum strength to 90\({}^{\circ}\), which is consistent with the experimental results mentioned above. The strength of loose sand shows a trend of decreasing first and then increasing as \(\delta\) advances from 90\({}^{\circ}\) to 0\({}^{\circ}\), but the trend is less significant, and the contraction deformation increases with the changing of \(\delta\). Figures 13 and 14 illustrate the relationship between peak friction angle \(\varphi\) and \(\delta\) at a confining stress of \(\sigma_{\text{c}}=1.0\) kgf/cm\({}^{2}\) (100 kPa).