# 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
### Hypoplasticity Constitutive Expression of Soil-Structure Contact Surfaces  
According to the stress tensor and strain rate tensor in Equations (15) and (16), the trace of the stress tensor can be obtained as follows:  
\[\text{tr}\sigma=\sigma_{n}+2\sigma_{p} \tag{17}\]  
Then, the stress ratio tensor can be given as follows:  
\[\hat{\sigma}=\frac{\sigma}{\text{tr}\sigma}= \begin{bmatrix}\frac{\sigma_{n}}{ \sigma_{n}+2\sigma_{p}}\\ \frac{\sigma_{p}}{\sigma_{n}+2\sigma_{p}}\\ \frac{\tau_{p}}{\sigma_{n}+2\sigma_{p}}\\ \frac{\tau_{p}}{\sigma_{n}+2\sigma_{p}}\end{bmatrix}  \tag{18}\]  
And the deviatoric stress ratio is expressed as follows:  
\[\hat{\sigma}^{*}= \begin{bmatrix}\frac{2}{3}\cdot\frac{\sigma_{n}-\sigma_{p}}{ \sigma_{n}+2\sigma_{p}}\\ \frac{1}{3}\cdot\frac{\sigma_{p}-\sigma_{n}}{\sigma_{n}+2\sigma_{p}}\\ \frac{\tau_{p}}{\sigma_{n}+2\sigma_{p}}\\ \frac{\tau_{p}}{\sigma_{n}+2\sigma_{p}}\end{bmatrix}  \tag{19}\]  
By substituting the stress rate tensor into the formula of norm, we can obtain the following:  
\[\left\|\hat{\epsilon}\right\|=\sqrt{\hat{\epsilon}\cdot\hat{\epsilon}}=\sqrt{ \hat{\epsilon}_{n}^{2}+2\left(\frac{\hat{\gamma}_{x}}{2}\right)^{2}+2\left( \frac{\hat{\gamma}_{z}}{2}\right)^{2}}=\sqrt{\hat{\epsilon}_{n}^{2}+\frac{1} {2}\hat{\gamma}_{x}^{2}+\frac{1}{2}\hat{\gamma}_{z}^{2}} \tag{20}\]  
Similarly, substituting the deviatoric stress ratio tensor into the norm formula yields the following:  
\[\left\|\hat{\sigma}^{*}\right\|=\sqrt{\left(\frac{2}{3}\cdot\frac{\sigma_{n}- \sigma_{p}}{\sigma_{n}+2\sigma_{p}}\right)^{2}+2\left(\frac{1}{3}\cdot\frac{ \sigma_{p}-\sigma_{n}}{\sigma_{n}+2\sigma_{p}}\right)^{2}+2\left(\frac{\tau_ {x}}{\sigma_{n}+2\sigma_{p}}\right)^{2}+2\left(\frac{\tau_{z}}{\sigma_{n}+2 \sigma_{p}}\right)^{2}} \tag{21}\]  
Then, by substituting Equations (15)-(21) into the Gudehus-Bauer eight-parameter equation, the linear and nonlinear terms of this equation can be calculated as follows:  
\[\text{L}\left(\hat{\sigma},\hat{\epsilon}\right)=a_{1}^{2}\hat{\epsilon}+ \hat{\sigma}\text{tr}\left(\hat{\sigma}\hat{\epsilon}\right)=a_{1}^{2}\hat{ \epsilon}+\frac{\sigma_{n}\hat{\epsilon}_{n}+\frac{3}{2}\tau_{x}\hat{\gamma}_ {x}+\frac{1}{2}\tau_{z}\hat{\gamma}_{z}}{\left(\sigma_{n}+2\sigma_{p}\right) ^{2}}\sigma \tag{22}\]  
\[\text{N}(\hat{\sigma})=a_{1}(\hat{\sigma}+\hat{\sigma}^{*})=a_{1} \begin{bmatrix} \frac{2\sigma_{p}}{\sigma_{n}+2\sigma_{p}}-\frac{1}{3}\\ \frac{2\sigma_{p}}{\sigma_{n}+2\sigma_{p}}-\frac{1}{3}\\ \frac{2\tau_{p}}{\sigma_{n}+2\sigma_{p}}\\ \frac{2\tau_{z}}{\sigma_{n}+2\sigma_{p}}\end{bmatrix}  \tag{23}\]  
The final constitutive equation of contact surface can be obtained in vector form as follows:\[ \begin{bmatrix}\dot{\sigma}_{n}\\ \dot{\sigma}_{p}\\ \dot{\tau}_{x}\\ \dot{\tau}_{z}\end{bmatrix} =f_{s} \begin{bmatrix}{\dot{e}}_{n}\\ 0\\ \frac{\dot{\gamma}_{z}}{2}\\ \frac{\dot{\gamma}_{z}}{2}\end{bmatrix} +\frac{\frac{\sigma_{n}\dot{e}_{n}+ \frac{3}{2}\tau_{z}\dot{\gamma}_{x}+\frac{1}{2}\tau_{z}\dot{\gamma}_{z}}{\left( \sigma_{n}+2\sigma_{p}\right)^{2}}}{\left(\sigma_{n}+2\sigma_{p}\right)^{2}} \begin{bmatrix}\sigma_{n}\\ \sigma_{p}\\ \tau_{x}\\ \tau_{z}\end{bmatrix} \\ +f_{d}a_{1}\frac{\sqrt{\frac{\dot{\gamma}_{z}}{2}+\frac{1}{2}\dot{\gamma}_{z} +\frac{1}{2}\dot{\gamma}_{z}}^{2}}{\sigma_{n}+2\sigma_{p}} \begin{bmatrix}\frac{ 2\sigma_{n}}{\sigma_{n}+2\sigma_{p}}-\frac{1}{3}\\ \frac{2\sigma_{p}}{\sigma_{n}+2\sigma_{p}}-\frac{1}{3}\\ \frac{2\tau_{z}}{\sigma_{n}+2\sigma_{p}}\end{bmatrix} \\ \frac{2\tau_{z}}{\sigma_{n}+2\sigma_{p}}\end{bmatrix} \tag{24}\]  
The basic parameters of the above model are the same as those of the Gudehus-Bauer model, but with modified stress and strain rate tensors, which should be calculated using the simplified vector expressions.