# Dynamic Behavior of an Embedded Foundation under Horizontal Vibration in a Poroelastic Half-Space
## 2 Problem Definition and Governing Equations
### Governing Equations of Porelastic Soil  
The motion of vibration \(P\)(t) is assumed to be time-harmonic with a time factor of \(e^{i\omega t}\), where \(\omega\) is the circular frequency and \(i=\sqrt{-1}\). The time factor of \(e^{i\omega t}\) is canceled henceforth from all expressions for simplicity. The equilibrium governing equation of foundation motion is:  
\[m\ddot{u}(t)+F_{b}(t)+F_{s}(t)=P(t) \tag{1}\]  
where \(m\) is the mass of the cylindrical foundation, \(u(t)=ue^{i\omega t}\) is the time-harmonic horizontal displacement of the foundation. When \(\ddot{u}(t)\) equals to 0, means the dynamic acceleration of foundation is zero, the dynamic motion equation can degenerate to static equilibrium function. \(P(t)=Pe^{i\omega t}\) is the excitation force. \(F_{b}(t)=F_{b}e^{i\omega t}\) is the horizontal dynamic reaction force at the foundation base, \(F_{s}(t)=F_{s}e^{i\omega t}\) is the horizontal dynamic reaction force along the foundation side. The governing differential equations for the poroelastic medium are expressed under cylindrical coordinates system \(o\) (\(r\), \(\theta\), \(z\)), shown as follows [7,8],  
\[G\
abla^{2}u_{r}+(\lambda+G)\frac{\partial e}{\partial r}-\frac{G}{r}(2\frac{ \partial u_{\theta}}{r\partial\theta}+\frac{u_{r}}{r})-\alpha\frac{\partial p _{f}}{\partial r}+G\frac{\partial^{2}u_{r}}{\partial z^{2}}-\rho_{s}\ddot{u}_ {r}-\rho_{f}\ddot{w}_{r}=0 \tag{2}\]  
\[G\
abla^{2}u_{\theta}+(\lambda+G)\frac{\partial e}{\partial\theta}-\frac{G}{r }(\frac{u_{\theta}}{r}-2\frac{\partial u_{r}}{r\partial\theta})-\alpha\frac{1 }{r}\frac{\partial p_{f}}{\partial r}+G\frac{\partial^{2}u_{\theta}}{\partial z ^{2}}-\rho_{s}\ddot{u}_{\theta}-\rho_{f}\ddot{w}_{\theta}=0 \tag{3}\]  
\[G\
abla^{2}u_{z}+(\lambda+G)\frac{\partial e}{\partial z}-\alpha\frac{1}{r} \frac{\partial p_{f}}{\partial z}-\rho_{s}\ddot{u}_{z}-\rho_{f}\ddot{w}_{z}=0 \tag{4}\]  
\[-\frac{\partial p_{f}}{\partial r}=\rho_{f}\ddot{u}_{r}+a\ddot{w}_{r}+b\dot{w} _{r} \tag{5}\]  
\[-\frac{1}{r}\frac{\partial p_{f}}{\partial\theta}=\rho_{f}\ddot{u}_{\theta}+a \ddot{w}_{\theta}+b\dot{w}_{\theta} \tag{6}\]  
Figure 1: Rigid cylindrical foundation embedded in a poroelastic half-space. The seepage function under cylindrical coordinates is expressed as follows under the assumption of soil grain, and the fluid is incompressible. \[M\bigg{(}\frac{\partial\dot{w}_{r}}{\partial r}+\frac{\dot{w}_{r}}{r}+\frac{1}{r} \frac{\partial\dot{w}_{\theta}}{\partial\theta}\bigg{)}+\alpha M\bigg{(}\frac{ \partial\dot{u}_{r}}{\partial r}+\frac{\dot{u}_{r}}{r}+\frac{1}{r}\frac{ \partial\dot{u}_{r\theta}}{\partial\theta}\bigg{)}=-\dot{p}_{f} \tag{7}\]  
where \(u_{x}\), \(u_{r}\), \(u_{\theta}\) is the vertical, radial, and tangential displacement of the solid matrix, \(w_{x}\), \(w_{r}\), \(w_{\theta}\) is the average fluid displacement of the solid matrix in the vertical radial and tangential direction. \(u_{r}(t)=u_{r}e^{i\omega t}\), \(w_{r}(t)=w_{r}e^{i\omega t}\), \(u_{\theta}(t)=u_{\theta}e^{i\omega t}\), \(w_{\theta}(t)=w_{\theta}e^{i\omega t}\). \(\rho\) is the mass density of the bulk material, and \(\rho=(1-n)\rho_{s}+n\rho_{f}\), where \(\rho_{f}\) is the density of the fluid, \(\rho_{s}\) is the density of solid mass. \(b\) is defined as the ratio between the fluid viscosity and the intrinsic permeability of the porous medium (\(\text{s}\cdot\text{Pa}/\text{m}^{2}\)). The ratio of fluid density \(\rho_{f}\) over the porosity \(n\) is defined as \(a\). \(\lambda\), \(G\) are the Lame parameter \(g\) is the acceleration of gravity. \(\alpha\), \(M\) are the Biot's parameters accounting for compressibility of the poroelastic material. \(p_{\text{f}}\) is the pore water pressure.