# 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
### Contact Surface Model Considering Initial Anisotropy and Roughness
#### 2.4.2 Quantitative Description of Roughness Effect  
The mechanical behavior of the contact surface when the structural surface is in a completely rough condition can be described by the hypoplasticity model in Equation (24). In order to extend the model to an arbitrary roughness case, the friction parameter (\(\lambda_{R}\)) and the relative roughness (\(R_{n}\)), as shown in Equation (28), referring to Gutjahr [23], are introduced, as they are widely used in various types of contact surface constitutive model such as elastic-plastic and hypoplasticity models:  
\[\lambda_{R}=0.25\text{lg}R_{n}+1.05 \tag{28}\]  
where \(R_{n}=R/d_{50}\), \(d_{50}\) is the average particle size, and \(R\) is the peak-to-valley distance of the structural surface, as shown in Figure 4.  
In Equation (24), the shear properties are mainly determined by the density factor, \(f_{d}\), and the critical friction angle, \(\varphi_{c}\), where \(\varphi_{c}\) determines the failure stress of loose sand and the residual stress of dense sand. Therefore, the friction parameter, \(\lambda_{R}\), is introduced into  
Figure 4: Diagram of peak-to-valley distance of the structural surface. Figure 3: Variations in peak stress coefficient with the deposition direction. the expression related to \(\varphi_{c}\) in order to consider the effect of structural surface roughness. And the expressions of \(c_{1}\) and \(c_{2}\) are modified to \(c_{1R}\) and \(c_{2R}\) as follows:  
\[c_{1R}=\sqrt{\frac{3}{8}}\times\frac{3-\sin(\lambda_{R}\varphi_{c})}{\sin( \lambda_{R}\varphi_{c})} \tag{29}\]  
\[c_{2R}=\frac{3}{8}\times\frac{3+\sin(\lambda_{R}\varphi_{c})}{\sin(\lambda_{ R}\varphi_{c})} \tag{30}\]  
the strength index, \(a_{1}\), is modified to \(a_{1R}\),  
\[a_{1R}=c_{1R}+c_{2R}\|\hat{\sigma}^{*}\|(1+\cos\theta)^{-1} \tag{31}\]  
Note that \(a_{1R}\) is the strength index considering only the effect of structural surface roughness. It is further rewritten to \(a_{1 IR}\) by substituting Equation (31) into Equation (27), which considers both the initial anisotropy and the effect of structural surface roughness,  
\[a_{1R}=\chi^{m}[c_{1R}+c_{2R}\|\hat{\sigma}^{*}\|(1+\cos\theta)]^{-1} \tag{32}\]  
The density factor, \(f_{d}\), describes the effect of the mean stress and the number of pores on shear properties and determines the response of the nonlinear part of the hypoplasticity model. The modeling experiments of [PERSON] [38] showed that the test results of loose sand on rougher structural surfaces are similar with that of dense sand on smoother surfaces. Therefore, the expression of \(f_{d}\) is rewritten to \(f_{dR}\) by taking into account the roughness effect:  
\[f_{dR}=\left(\frac{e-e_{d}}{e_{c}-e_{d}}\right)^{\alpha\lambda_{R}^{2}} \tag{33}\]  
Correspondingly, the parameter \(h_{i}\) is rewritten to \(h_{iR}\):  
\[h_{iR}=\frac{1}{c_{1R}^{2}}+\frac{1}{3}-\left(\frac{e_{i0}-e_{d0}}{e_{c0}-e_{d 0}}\right)^{\alpha\lambda_{R}^{2}}\frac{1}{c_{1R}\sqrt{3}} \tag{34}\]  
and the stiffness factor, \(f_{s}\), is rewritten to \(f_{sR}\):  
Figure 4.0  
By substituting Equations (32)-(35) into Equation (24), the hypoplasticity constitutive model of soil-structure contact surface considering structural surface roughness and initial anisotropy can be obtained as expressed in Equation (36):  
\[ \begin{bmatrix}\hat{\sigma}_{n}\\ \hat{\sigma}_{p}\\ \hat{\tau}_{x}\\ \hat{\tau}_{z}\end{bmatrix} =f_{sR}