## 2 Model Establishing
### Boundary and Initial Conditions of the Pile-Soil-Cap System  
#### 2.3.1 Boundary Conditions of the Soil Layer in Four Cases  
In the previous section, permafrost and seasonally frozen soil are divided into four cases according to the freezing/thawing processes caused by seasonal factors or artificial activities, in which the upper part is the active layer, and the lower part is the inactive layer, as shown in Figure 1.  
(1) The boundary conditions of the inactive layer  
The boundary condition of the soil layer is expressed as follows in an axisymmetric coordinate system:  
\[E_{s1}^{(av)}\frac{\partial u_{s1}^{(s)}}{\partial z}(r,H_{0})+k_{s 1}u_{z1}^{(s)}(r,H_{0})=0 \tag{4}\] \[-E_{s1}^{(av)}\frac{\partial u_{s1}^{(s)}}{\partial z}(r,H_{1})+k_{ s1}u_{z1}^{(s)}(r,H_{1})=0 \tag{5}\]  
where \(E_{s1}^{(av)}\) represents the elastic modulus of solid in the inactive layer. Subscript 1 indicates the lower layer of the soil. For seasonally frozen soil, the inactive layer is thawed soil. \(E_{s1}^{(av)}\)= \(\mu\) (\(\lambda\)\(+\)\(2\mu\))/(\(\lambda\)\(+\)\(\mu\)), and \(\lambda\) and \(\mu\) are Lame constants. For permafrost, the inactive layer is frozen soil; \(E_{s1}^{(av)}\)= \(R_{11}\). \(u_{z1}^{(s)}\) represents the vertical displacement of solid phase in the inactive layer. The radial displacement of the soil layer at infinity is zero; there is no lateral displacement, and the pile shaft is impermeable. For seasonally frozen soil,  
\[u_{r1}^{(s)}(a,z)=0,\ u_{r1}^{(f)}(a,z)=0 \tag{6}\]  
where \(u_{r1}^{(s)}\) and \(u_{r1}^{(s)}\) are the radial displacements of the solid and liquid in the inactive layer, and subscript 1 represents the inactive layer (i.e., the lower layer). Figure 1: Schematic diagram of model of a pile–soil–cap dynamic interaction system. (**a**) Pile–soil–cap coupling system and (**b**) pile cap and pile–soil system substructure. For permafrost,  
\[u_{r1}^{(s)}(a,z)=0,\ u_{r1}^{(i)}(a,z)=0,\ u_{r1}^{(f)}(a,z)=0 \tag{7}\]  
where \(u_{r1}^{(i)}\) is the radial displacements of the ice phases in the inactive layer, and subscript 1 represents the inactive layer (i.e., the lower layer). The shear stresses at the interface between the pile and soil remain equal. A slippage interface with a spring and damper is used to simulate the dynamic stiffness and damping of the pile-soil interface. It is assumed that the dynamic stiffness is independent of the frequency, and the damping ratio is proportional to the frequency within a certain range. The corresponding pile-soil contact conditions of seasonally frozen soil are expressed as follows:  
\[\tau_{zr1}^{(s)}(a,z)=-f_{s1}(z)/2\pi a,\ f_{s1}=k_{s1}\Delta w_{1}+D_{s1} \Delta\dot{w}_{1} \tag{8}\]  
The pile-soil contact conditions of permafrost are as follows:  
\[\tau_{zr1}^{(s)}(a,z)+\tau_{zr1}^{(i)}(a,z)=-[f_{s1}(z)+f_{1}(z)]/2\pi a,\ f_{s1}=k_{s1}\Delta w_{1}+D_{s1}\Delta\dot{w}_{1}, \tag{9}\]  
where \(\tau_{zr1}^{(s)}\) and \(\tau_{zr1}^{(i)}\) are the solid and ice phase shear stress acting on a plane perpendicular to the \(z\) axis but along the \(r\) axis, respectively. \(\Delta w_{i}\) = (\(w_{bi}-u_{z}^{(i)}\)) represents the relative movement between the pile and soil; the overdot denotes the derivative with respect to time. \(k_{si}\) (or \(k_{ii}\)) and \(D_{si}\) (or \(D_{ii}\)) represent the dynamic stiffness and damping coefficient of the pile-soil interface, respectively. Their first subscript indicates the solid or ice phase, and the second subscript indicates the label of the soil layer, with the subsoil being labeled as 1 and the upper layer as 2. The damping coefficient is directly proportional to frequency, and the dot symbol represents the derivative of time. The boundary conditions of the active layer  
The boundary condition of the soil layer is expressed as follows:  
\[E_{s2}^{(av)}\frac{\partial u_{s2}^{(s)}}{\partial z}(r,H_{1})+k_{s02}u_{s2}^ {(s)}(r,H_{1})=0 \tag{10}\]  
where \(E_{s2}^{(av)}\) represents the elastic modulus of solid in the active layer.