# 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  
The contact surface model presented in this study encompasses 8 fundamental parameters (\(\alpha\), \(\beta\), \(e_{i0}\), \(e_{c0}\), \(e_{d0}\), \(h_{s}\), \(n\), and \(\varphi_{c}\)), 4 initial anisotropy parameters (\(b_{1}\), \(b_{2}\), \(b_{3}\), and \(m\)), and 1 roughness parameter (\(R\)). The methods for determining these parameters are detailed as follows. Extensive testing has demonstrated that the method of estimating the critical friction angle using the angle of repose is relatively precise, with accuracy sufficient for the majority of sandy soils. Table 2[39] presents the angle of repose and critical friction angle values obtained from shear tests on various types of sand. The data reveal a minimal discrepancy between these two parameters, indicating that it is entirely viable to approximate the critical friction angle using the angle of repose. However, it is important to note that for particles with a size d \(\leq\) 0.1 mm, the error in predicting the critical friction angle using the angle of repose becomes significant, necessitating the use of tests to accurately determine the critical friction angle. Particle hardness (\(h_{s}\)) and index \(n\)  
The values of \(h_{s}\) and \(n\) are ascertained through an optimization method that employs local search to identify the best solution, thereby circumventing the issue of excessive error that may stem from the selection of disparate stress state points based on test outcomes. Upper void ratio, \(e_{i0}\), at zero pressure  
During the experimental procedures, it is nearly impossible to replicate the upper void ratio; hence, dry sand is commonly poured into a standard mold. During the pouring process, a critical value is derived which exceeds the theoretical maximum void ratio, \(e_{\text{max}}\), and it is widely accepted that \(e_{i0}\approx 1.2e_{\text{max}}\). Lower bound void ratio, \(e_{d0}\), at zero pressure  
According to the compression characteristics of sand, the void ratio, \(e_{d}\), diminishes as stress increases. Typically, \(e_{d}\) is ascertained from the \(e_{d}-C_{u}\) relationship curve, which is suggested by [PERSON][40]. The \(e_{d}-C_{u}\) relationship is conducted under conditions where the pressure is \(p_{s}=55\) kPa, \(K_{0}=0.40\), and the vertical stress is maintained at 96 kPa. Once the value of \(e_{d}\) is determined, the value of \(e_{d0}\) can be deduced by applying the formula that is retroactively derived from Equation (25):  
\[e_{d0}=e_{d}\exp\left[\left(\frac{\sigma_{n}+2\sigma_{p}}{h_{s}}\right)^{n}\right] \tag{37}\]  
As a general guideline, it is often more favorable to set \(e_{d0}\approx e_{\text{min}}\). Critical void ratio, \(e_{c0}\), at zero pressure  
Similar to the upper void ratio, the critical void ratio, \(e_{c}\), also diminishes with increasing stress. The value can likewise be extracted from the expression given in Equation (25):  
\[e_{c0}=e_{c}\exp\left[\left(\frac{\sigma_{n}+2\sigma_{p}}{h_{s}}\right)^{n}\right] \tag{38}\] \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Materials**} & **Angle of** & **Critical Friction** & \multirow{2}{*}{**Test**} \\  & **Repose/\({}^{c}\)** & **Angle/\({}^{c}\)** & \\ \hline Hochstetten gravel & 35.7 & 36.5 & Direct shear \\ Hochstetten sand & 34.0 & 34.0 & Consolidation drained \\ Hostun sand & 32.0 & 32.0 & Consolidation drained \\ Karlsruhe sand & 29.1 & 30.0 & Consolidation drained \\ Lausitz sand & 33.0 & 32.2 & Consolidation drained \\ Toyoura sand & 30.4 & 30.9 & Consolidation undrained \\ Zbraslav sand & 30.8 & 29.7 & Direct shear \\ \hline \end{tabular}
\end{table} 
Table 2: Angle of repose and critical friction angle from shear tests. The results of undrained triaxial test on Sacramento River sand and Monterey sand are summarized, and \(e_{c0}\approx e_{\max}\) is generally used. 6.