## 3 Simulation of Sand Plane Strain Test
### Simulation Result
#### 3.3.1 Influence of Confining Pressure on Strength and Deformation
Figure 13 represents the correlation between \(\varphi\) and \(\delta\), and Figure 14a illustrates the curves of \(\varphi(\delta)/\varphi(\delta=90^{\circ})\)-\(\delta\) under plane strain and triaxial compression tests. Figure 14b exhibits the curves of \(\varphi(\delta)/\varphi(\delta=90^{\circ})\)-\(\delta\) simulated by the model. The results in Figure 13a,b depict the simulated response of the model presented in this paper. It can be observed that the variation of peak friction angle \(\varphi\) with the principal stress axis rotation angle \(\delta\) have similar patterns for both dense sand (\(\epsilon\) = 0.35) and loose sand (\(\epsilon\) = 0.8). The maximum value of \(\varphi\) corresponds to a rotation angle of \(\delta\) = 90\({}^{\circ}\), followed by a decreasing trend as \(\delta\) decreases. The slope of the \(\varphi\)-\(\delta\) curve in the range of 90\({}^{\circ}\) to 30\({}^{\circ}\) is relatively steep, indicating that the influence of \(\delta\) on the strength is significant in this range. In contrast, the slope of the \(\varphi\)-\(\delta\) curve in the range of 30\({}^{\circ}\) to 0\({}^{\circ}\) is relatively gentle, implying that \(\delta\) has a minor impact on strength in this range. The simulated results of the proposed model are compared with [PERSON]'s anisotropic model in Figure 13c,d. [PERSON]'s model shows that \(\varphi\) first decreases and then increases with \(\delta\)  
Figure 7.2 The relationship between \(\varphi(\delta)/\varphi(\delta=90^{\circ})\) and principal stress rotation angle: (**a**) test results [31]; (**b**) simulation results. Figure 7.3 The relationship between peak friction angle and principal stress rotation angle: (**a**) \(\epsilon\) = 0.35; (**b**) \(\epsilon\) = 0.8; (**c**,d**) comparison with [PERSON]’s model. within the range of 90\({}^{\circ}\) to 0\({}^{\circ}\), and the strength at 0\({}^{\circ}\) is lower than that of 90\({}^{\circ}\). The minimum strength is achieved at a rotation angle of approximately 30\({}^{\circ}\). Figure 14a illustrates the variation of the internal friction angle \(\varphi\) with \(\delta\) (from 90\({}^{\circ}\) to 0\({}^{\circ}\)) showing a tendency to decrease initially and then subsequently increasing. The minimum value of \(\varphi\) is observed at \(\delta=34^{\circ}\), and it gradually increases as \(\delta\) approaches 0\({}^{\circ}\). As \(\delta\) approaches 0\({}^{\circ}\), \(\varphi(\delta)/\varphi(\delta=90^{\circ})\) varies from 0.88 to 0.90. The variation of \(\varphi\) under triaxial compression conditions is similar to that under plane strain conditions, but the degree of anisotropy is lower, and the tendency of an initial decrease followed by an increase is less pronounced. Figure 14b presents the simulation results obtained under plane strain conditions using the hypoplastic constitutive model derived in this paper. \(\varphi\) changes considerably within the range of 90\({}^{\circ}\) to 30\({}^{\circ}\) as \(\delta\) varies, but the decreasing trend of \(\varphi\) becomes slower when \(\delta\) exceeds 30\({}^{\circ}\). At a rotation angle of 0\({}^{\circ}\), \(\varphi(\delta)/\varphi(\delta=90^{\circ})\) falls within the range of 0.9 to 0.94, which is roughly consistent with experimental values. Figure 15 compares the variation of strength with \(\delta\) between the proposed model and [PERSON]'s anisotropic model.