# 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
### Establishment for the Hypoplastic Constitutive Model of the Soil-Structure Contact Surface  
The shear behavior and control mechanisms of the contact surface are fundamentally in line with those of soil. The contact surface can be regarded as a miniature continuous soil mass, and its deformation pattern is generally akin to that of soil. Consequently, the contact surface model can be extrapolated from soil constitutive models.  
[PERSON] and [PERSON] (2016) established the contact surface model using the von Wolffersdorff eight-parameter hypoplastic model for sand by assuming an isotropic stress state at the contact surface. They also assumed that only in-plane shear stress exists at the contact surface, with all shear stresses outside the contact surface being zero, i.e., \(\varphi_{23}=\varphi_{32}=0\). The coordinate system for the contact surface is shown in Figure 1.  
The stress tensor for the Arnold-Herle contact surface model is given by the following:  
\[\sigma=\begin{bmatrix}\sigma_{11}&\sigma_{12}&\sigma_{13}\\ \sigma_{21}&\sigma_{22}&\sigma_{23}\\ \sigma_{31}&\sigma_{32}&\sigma_{33}\end{bmatrix}=\begin{bmatrix}\sigma_{n}& \tau_{x}&\tau_{z}\\ \tau_{x}&\sigma_{n}&0\\ \tau_{z}&0&\sigma_{n}\end{bmatrix} \tag{13}\]  
Similarly, the strain rate tensor is given by the following:  
\[\dot{\epsilon}=\begin{bmatrix}\dot{\epsilon}_{11}&\dot{\epsilon}_{12}&\dot{ \epsilon}_{13}\\ \dot{\epsilon}_{21}&\dot{\epsilon}_{22}&\dot{\epsilon}_{23}\\ \dot{\epsilon}_{31}&\dot{\epsilon}_{32}&\dot{\epsilon}_{33}\end{bmatrix}= \begin{bmatrix}\dot{\epsilon}_{n}&\frac{\dot{\gamma}_{x}}{2}&\frac{\dot{ \gamma}_{z}}{2}\\ \frac{\dot{\gamma}_{x}}{2}&\dot{\epsilon}_{n}&0\\ \frac{\dot{\gamma}_{z}}{2}&0&\dot{\epsilon}_{n}\end{bmatrix} \tag{14}\]  
where \(\dot{\epsilon}_{n}\) is the normal strain rate at the contact surface; and \(\dot{\gamma}_{x}/2\) and \(\dot{\gamma}_{z}/2\) are the shear strain rates in the \(x\) and \(z\) directions, respectively.  
However, the stresses within the contact surface are autonomous and do not equate to the normal stress, as depicted in Figure 2. To address this, [PERSON] [25] revised the assumptions of the Arnold-Herle contact surface model, proposing that an isotropic stress state exists within the contact surface, i.e., \(\sigma_{22}=\sigma_{33}\
eq\sigma_{11}\), which develops independently of the normal stress during shear. The in-plane stress is denoted by \(\sigma_{p}\). The modified stress tensor and strain rate tensor can be expressed in vector form as follows:  
\[\sigma=\begin{bmatrix}\sigma_{n}&\sigma_{p}&\tau_{x}&\tau_{z}\end{bmatrix}^{T} \tag{15}\]  
\[\dot{\epsilon}=\begin{bmatrix}\dot{\epsilon}_{n}&0&\frac{\dot{\gamma}_{x}}{2}& \frac{\dot{\gamma}_{z}}{2}\end{bmatrix}^{T} \tag{16}\]  
The three-dimensional hypoplastic constitutive model for the soil-structure contact surface developed by [PERSON] [25] can better predict the behavior of the contact surface and offers higher computational efficiency in numerical simulations. Therefore, this paper will adopt the assumptions of [PERSON] [25] to transform the Gudehus-Bauer hypoplastic model into a hypoplastic constitutive model for the soil contact surface.  
Figure 1: Schematic diagram of the contact surface coordinate system.  
Figure 2: Schematic diagram of contact surface stress and strain components.