# Hypoplastic Modeling of Soil-Structure Contact Surface Considering Initial Anisotropy and Roughness
## 2. Fundamental of the Hypoplastic Constitutive Model for the Soil-Structure Contact Surface
### Model Parameters Determination
Index \(\alpha\) Parametric equations for two scalars can be obtained from Equation (24):  
\[\dot{\sigma}_{n}=f_{s}\frac{\left(\sigma_{n}+2\sigma_{p}\right)^{2}}{{\sigma_{n }}^{2}+2{\sigma_{p}}^{2}}\left[\dot{\epsilon}_{n}+a^{2}\frac{\sigma_{n}\dot{ \epsilon}_{n}}{\left(\sigma_{n}+2\sigma_{p}\right)^{2}}\sigma_{n}+f_{d}\frac{a }{3}\frac{5\sigma_{n}-2\sigma_{p}}{\sigma_{n}+2\sigma_{p}}\dot{\epsilon}_{n}\right] \tag{39}\]  
\[\dot{\sigma}_{p}=f_{s}\frac{\left(\sigma_{n}+2\sigma_{p}\right)^{2}}{{\sigma_{ n}}^{2}+2{\sigma_{p}}^{2}}\left[a^{2}\frac{\sigma_{n}\dot{\epsilon}_{n}}{ \left(\sigma_{n}+2\sigma_{p}\right)^{2}}\sigma_{p}+f_{d}\frac{a}{3}\frac{4 \sigma_{p}-\sigma_{n}}{\sigma_{n}+2\sigma_{p}}\dot{\epsilon}_{n}\right] \tag{40}\]  
When the soil reaches the peak stress state, \(\dot{\sigma}_{n}=0\), so \(\alpha\) can be derived from Equation (23) as follows:  
\[\alpha=\frac{\ln\left[6\frac{\left(2+K_{p}\right)^{2}+a^{2}K_{p}\left(K_{p}-1 -\tan v_{p}\right)}{a\left(2+K_{p}\right)\left(5K_{p}-2\right)\sqrt{4+2\left(1 +\tan v_{p}\right)^{2}}}\right]}{\ln\left(\frac{e-\epsilon_{d}}{\dot{\epsilon }_{c}-\dot{\epsilon}_{d}}\right)} \tag{41}\]  
where \(K_{p}\) is the peak stress ratio:  
\[K_{p}=\frac{\sigma_{n}}{\sigma_{p}}=\frac{1+\sin\varphi_{p}}{1-\sin\varphi_{p}} \tag{42}\]  
\[\tan v_{p}=-\frac{\dot{\epsilon}_{p}}{\dot{\epsilon}_{n}} \tag{43}\]  
\[\sin\varphi_{p}=\left(\frac{\sigma_{n}-\sigma_{p}}{\sigma_{n}+\sigma_{p}} \right)_{p} \tag{44}\]  
Since the model uses a reduced component of the stress tensor, \(\dot{\epsilon}_{22}=\dot{\epsilon}_{33}=0\), and \(\dot{\epsilon}_{v}=\dot{\epsilon}_{n}\), where \(\dot{\epsilon}_{v}\) is the volume strain rate, and \(\dot{\epsilon}_{n}\) is axial strain rate. Hence, \(\tan v_{p}=-1\). The natural logarithm function in the denominator of Equation (41) can be regarded as the relative void ratio related to pressure and can be denoted as \(r_{e}\); \(r_{e}\) is a function of the void ratio and average stress. Integrate \(\dot{\sigma}_{n}=0\) into Equation (40) at the peak stress condition, and the peak dilatancy angle can be obtained:  
\[\tan v_{p}=2\frac{K_{p}-4+5 AK_{p}{}^{2}-2 AK_{p}}{\left(5K_{p}-2\right)\left(1 +2A\right)}-1 \tag{45}\]  
\[A=\frac{a^{2}}{\left(2+K_{p}\right)^{2}}\left[1-\frac{K_{p}\left(4-K_{p} \right)}{5K_{p}-2}\right] \tag{46}\]  
In summary, when the critical friction angle, peak friction angle, and relative void ratio are known, the index \(\alpha\) can be calculated according to the above formula. 7. Index \(\beta\)  
The index \(\beta\) only plays a role when \(e\) is significantly smaller than \(e_{i}\) (dense sand), and the beta has little effect on the model response in the rest of the cases. A large number of test results show that the value of \(\beta\) is roughly 1, so for natural soil, the general method is to assume that \(\beta=1\). 8. Anisotropic parameters: \(b_{1}\), \(b_{2}\), \(b_{3}\), and The parameters \(b_{1}\), \(b_{2}\), and \(b_{3}\) can be determined through direct shear tests conducted on the contact surface under normal stress conditions. By performing direct shear tests on the contact surface, we can obtain the shear stress-shear displacement relationship curves at various initial deposition angles. This process then enables the determination of peak shear stress under these different conditions. The peak shear strength coefficient's relationship with the initial deposition angle can be charted by normalizing the peak stress values obtained under isotropic conditions. This process yields a regular variation relationship, as expressed in Equation (26), where \(b_{1}\), \(b_{2}\), and \(b_{3}\) serve as fitting parameters. The exponent \(m\) in this context is a dimensionless constant. 9. Roughness, \(R\)  
For the regular structural planes, \(R\) represents the peak-to-valley distance (or depth) of the structural plane, as depicted in Figure 4. In the case of irregular structural planes, an arithmetic mean value derived from multiple measurements can be employed for \(R\).