# Bulk and Rayleigh Waves Propagation in Three-Phase Soil with Flow-Independent Viscosity
## 2 Viscoelastic Dynamic Model
The total stress of unsaturated soil is yielded by substituting Equation (7) into Equation (3), as  
\[\sigma_{ij}=\bigg{(}\lambda_{e}\epsilon_{v}+\lambda_{v}\frac{\partial \epsilon_{v}}{\partial t}\bigg{)}\delta_{ij}+2\bigg{(}\mu_{e}\epsilon_{ij}+ \mu_{v}\frac{\partial\epsilon_{ij}}{\partial t}\bigg{)}-\alpha_{e}p\delta_{ij} \tag{8}\]  
where \(\alpha_{e}\)= \(1~{}-~{}K_{b}/K_{s}\). The stress component acting on the soil skeleton, denoted by \(\sigma_{ij}^{s}\), can be assumed to  
\[\lambda_{v}=t_{s}\lambda_{e},\ \mu_{v}=t_{s}\mu_{e} \tag{1}\]0  
Substituting Equation (8) into Equation (9) and making algebraic operations, Equation (9) can be re-expressed as  
\[\lambda_{v}=t_{s}\lambda_{e},\ \mu_{v}=t_{s}\mu_{e} \tag{1}\]1The mass balance equation of each phase is constructed after ignoring phase transitions between components, as  
\[\lambda_{v}=t_{s}\lambda_{e},\ \mu_{v}=t_{s}\mu_{e} \tag{1}\]2  
Further expansion of Equation (11) yields the following formulas as  
\[\lambda_{v}=t_{s}\lambda_{e},\ \mu_{v}=t_{s}\mu_{e} \tag{1}\]3  
\[\lambda_{v}=t_{s}\lambda_{e},\ \mu_{v}=t_{s}\mu_{e} \tag{1}\]4  
\[\lambda_{v}=t_{s}\lambda_{e},\ \mu_{v}=t_{s}\mu_{e} \tag{1}\]5  
Following [PERSON] et al. [33], the derivative of each phase density with time is written as  
\[\lambda_{v}=t_{s}\lambda_{e},\ \mu_{v}=t_{s}\mu_{e} \tag{1}\]6  
According to the relationship between matrix suction and saturation of unsaturated soil, established by [PERSON] [34], the following formula is yielded  
\[\lambda_{v}=t_{s}\lambda_{e},\ \mu_{v}=t_{s}\mu_{e} \tag{1}\]7  
where \(S_{e}{=\left(S^{r}-S_{res}\right)/\left(1-S_{res}\right)}\). Substituting the first formula in Equation (15) into Equation (12) yields the derivative of porosity over time as  
\[\lambda_{v}=t_{s}\lambda_{e},\ \mu_{v}=t_{s}\mu_{e} \tag{1}\]8  
where \(\alpha_{v}{=\left(3\lambda_{v}{+}2\mu_{v}\right)/\left.3K_{s}\right.}\). Substituting Equations (15)-(17) into Equations (13) and (14), the following formulas are yielded  
\[\lambda_{v}=t_{s}\lambda_{e},\ \mu_{v}=t_{s}\mu_{e} \tag{1}\]9  
Figure 1:0  
in which  
Figure 1:1  
Figure 1:2  
Figure 1:3  
Figure 1:4  
Making some algebraic operations for Equations (18) and (19), the pore-fluid pressures are yielded as  
Figure 1:5  
Figure 1:6  
in which  
Figure 1:7  
Combination of Equations (20)-(23) and introduction of the relative displacement of liquid (\(u_{i}^{w}\)= \(n^{s}S^{r}(u_{i}^{l}-u_{i}^{s})\)) and gas (\(u_{i}^{g}=n^{s}S^{a}(u_{i}^{a}-u_{i}^{s})\)) phases yield the following formulas  
Figure 1:8  
Figure 1:9  
\[\sigma_{ij,j}=\overline{\rho}^{s}\dot{u}_{i}^{s}+\overline{\rho}^{l}\dot{u}_{i }^{l}+\overline{\rho}^{a}\dot{u}_{i}^{a} \tag{2}\]0  
with  
\[\sigma_{ij,j}=\overline{\rho}^{s}\dot{u}_{i}^{s}+\overline{\rho}^{l}\dot{u}_{i }^{l}+\overline{\rho}^{a}\dot{u}_{i}^{a} \tag{2}\]1  
\[\sigma_{ij,j}=\overline{\rho}^{s}\dot{u}_{i}^{s}+\overline{\rho}^{l}\dot{u}_{i }^{l}+\overline{\rho}^{a}\dot{u}_{i}^{a} \tag{2}\]2  
\[\sigma_{ij,j}=\overline{\rho}^{s}\dot{u}_{i}^{s}+\overline{\rho}^{l}\dot{u}_{i }^{l}+\overline{\rho}^{a}\dot{u}_{i}^{a} \tag{2}\]3  
\[\sigma_{ij,j}=\overline{\rho}^{s}\dot{u}_{i}^{s}+\overline{\rho}^{l}\dot{u}_{i }^{l}+\overline{\rho}^{a}\dot{u}_{i}^{a} \tag{2}\]4  
\[\sigma_{ij,j}=\overline{\rho}^{s}\dot{u}_{i}^{s}+\overline{\rho}^{l}\dot{u}_{i }^{l}+\overline{\rho}^{a}\dot{u}_{i}^{a} \tag{2}\]5  
Equations (25)-(27) represent the viscoelastic wave equations of triphase partially saturated soil. The advantages of the proposed viscoelastic dynamic model can describe the effect of both the flow-independent and flow-dependent viscosities on the soil dynamic behavior. The flow-independent viscosity is represented by the shear and dilatant constants \(\lambda_{v}\) and \(\mu_{v}\) in Equations (25)-(27). According to Equation (1), the soil skeleton viscosity can be represented by the relaxation time \(t_{s}\). Correspondingly, the fluid viscosity is characterized by the coefficients \(b_{w}\) and \(b_{g}\) in Equations (26) and (27).