# 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  
### Hypoplastic Model for Soil  
Traditional geotechnical plasticity mechanics theories are primarily constrained by two significant limitations [8]: (1) the deformation of the soil is continuous and cannot objectively be decomposed into elastic and plastic components; and (2) the theory does not adequately describe the verified incremental nonlinear phenomena. Therefore, [PERSON] and [PERSON] [36] proposed a tensor function to describe the relationship between stress rate and strain rate by considering the stress rate as a function of stress and strain rate. The fundamental equation is as follows:  
\[\overset{\circ}{\sigma}=\mathrm{H}(\sigma,\dot{\varepsilon}) \tag{1}\]  
where \(\overset{\circ}{\sigma}\) is the Jaumann stress rate tensor, defined as follows:  
\[\overset{\circ}{\sigma}=\dot{\sigma}+\sigma\dot{\omega}-\dot{\omega}\sigma \tag{2}\]  
[PERSON] [12] and [PERSON] [13] introduced the critical state (critical stress surface) into the hypoplastic model and proposed a widely used eight-parameter hypoplastic model (referred to as the Gudehus-Bauer model), with the basic formula as follows:  
\[\overset{\circ}{\sigma}=\mathrm{L}(\sigma,\dot{\varepsilon})-\mathrm{N}( \sigma,\dot{\varepsilon}) \tag{3}\]  
In Equation (3), the expressions for tensor functions \(\mathrm{L}\big{(}\hat{\sigma},\dot{\varepsilon}\big{)}\) and \(\mathrm{N}(\hat{\sigma})\) are as follows:  
\[\mathrm{L}\big{(}\hat{\sigma},\dot{\varepsilon}\big{)}={a_{1}}^{2}\dot{ \varepsilon}+\hat{\sigma}\mathrm{tr}\big{(}\hat{\sigma}\dot{\varepsilon}\big{)} \tag{4}\]  
\[\mathrm{N}(\hat{\sigma})={a_{1}}(\hat{\sigma}+\hat{\sigma}^{*}) \tag{5}\]where \(a_{1}\) is a dimensionless scalar factor representing the radius of the critical stress point, and its expression is as follows:  
\[a_{1}=\left[c_{1}+c_{2}\|\hat{\sigma}^{*}\|(1+\cos 3\theta)\right]^{-1} \tag{6}\]  
\[c_{1}=\sqrt{\frac{3}{8}}\times\frac{3-\sin\varphi_{c}}{\sin\varphi_{c}} \tag{7}\]  
\[c_{2}=\frac{3}{8}\times\frac{3+\sin\varphi_{c}}{\sin\varphi_{c}} \tag{8}\]  
where \(\varphi_{c}\) is the critical friction angle.  
In addition, the parameters \(f_{s}\) and, in Equation (3), \(f_{d}\) are the stiffness factor and the density factor, respectively. The stiffness factor, \(f_{s}\), is introduced to reflect the influence of stress level and void ratio:  
\[f_{s}=\frac{h_{s}}{nh_{i}}\left(\frac{e_{i}}{e}\right)^{\beta}\frac{1+e_{i}}{e _{i}}\left(\frac{3p_{s}}{h_{s}}\right)^{1-n} \tag{9}\]  
where  
\[h_{i}=\frac{1}{c_{1}^{2}}+\frac{1}{3}-\left(\frac{e_{i0}-e_{i0}}{e_{c0}-e_{i0 }}\right)^{\alpha}\frac{1}{\sqrt{3}c_{1}} \tag{10}\]  
The density factor, \(f_{d}\), is related to the material density:  
\[f_{d}=\left(\frac{e-e_{d}}{e_{c}-e_{d}}\right)^{\alpha} \tag{11}\]  
In Equations (9)-(11), the exponents \(\alpha\) and \(\beta\) are dimensionless constants, while \(h_{s}\) and \(n\) are two compression constants (with \(h_{s}\) defined as the particle hardness). Moreover, \(e_{i}\), \(e_{c}\), and \(e_{d}\) represent the upper void ratio, critical void ratio, and lower void ratio under the current stress state, respectively. They are related to the stress level and can be determined by the following equation:  
\[\frac{e_{i}}{e_{i0}}=\frac{e_{c}}{e_{c0}}=\frac{e_{d}}{e_{d0}}=\exp\left[- \left(\frac{3p_{s}}{h_{s}}\right)^{n}\right] \tag{12}\]  
where \(e_{i0}\), \(e_{c0}\), and \(e_{d0}\) represent the upper, critical, and lower void ratios of the initial state, respectively.  
Totally, the Gudehus-Bauer hypoplastic model includes 8 parameters: \(\alpha\), \(\beta\), \(e_{i0}\), \(e_{c0}\), \(e_{d0}\), \(h_{s}\), \(n\), and \(\varphi_{c}\). This model effectively simulates the nonlinear constitutive relationships of granular materials while considering the effects of stress level, void ratio, and density.