# Bulk and Rayleigh Waves Propagation in Three-Phase Soil with Flow-Independent Viscosity
## 2 Viscoelastic Dynamic Model  
Generally, the deformation of viscoelastic materials is strictly associated with the loading force and relaxation time of stress and strain. Usually, the damping of three-phase unsaturated soil to the elastic wave propagation mainly comes from two aspects: ovine is the viscosity from the solid skeleton, called flow-independent viscosity, and the other is the viscosity from the liquid and gas in pores, called flow-dependent viscosity. To express the flow-independent viscosity in three-phase unsaturated soil, the generalized Kelvin-Voigt model [27; 28] is introduced in this study, as illustrated in Figure 1. The dashpot depicted in Figure 1 is utilized to represent the negative effect of the solid skeleton on elastic wave propagation. In Figure 1, the spring represents the linear-elastic response of the solid skeleton in unsaturated soil under load, while the dashpot represents the damping behaviors of the solid skeleton in unsaturated soil. The dashpot avoids the spring directly reaching the load. The correlation between elastic constants and viscous constants of viscoelastic soils can be expressed as [29]  
\[\lambda_{v}=t_{s}\lambda_{e},\ \mu_{v}=t_{s}\mu_{e} \tag{1}\]  
where the two sets of symbols \(\lambda_{e}\), \(\mu_{e}\) and \(\lambda_{v}\), \(\mu_{v}\) stand for the elastic and viscosity constants of soil, respectively. \(t_{s}\) stands for the relaxation time, representing the normalized viscosity without loss of generality [24; 30]. In this article, \(t_{s}\) is utilized to evaluate the viscosity of the solid skeleton, that is, the magnitude of flow-independent viscosity. Figure 1: Generalized Kelvin–Voigt model [27; 28]. In this work, the unsaturated soil is assumed to be homogeneous, isotropic, and viscoelastic material with multiple pores, composed of liquid and gas occupying pores as well as solid particles. The liquid (water) saturation and gas (air) saturation are represented by \(S^{r}\) and \(S^{a}\), respectively, and the constraint \(S^{r}+S^{a}=1\) is satisfied in this work. According to Biot porous medium theory, the motion equation of the partially saturated soil, when ignoring the body forces and dissipation, can be expressed as [31]  
\[\sigma_{ij,j}=\overline{\rho}^{s}\dot{u}_{i}^{s}+\overline{\rho}^{l}\dot{u}_{i }^{l}+\overline{\rho}^{a}\dot{u}_{i}^{a} \tag{2}\]  
where \(\overline{\rho}^{s}=(1-n^{s})\rho^{s}\), \(\overline{\rho}^{l}=n^{s}S^{r}\rho^{l}\), and \(\overline{\rho}^{a}=n^{s}S^{a}\rho^{a}\). The nomenclature of all the Roman and Greek symbols is stated in Nomenclature. To save space, the nomenclature (physical meaning) of the symbols in the formulas used below is listed in Nomenclature, and this point will not be repeated. According to [PERSON] and [PERSON] [32], the following formula is yielded  
\[\sigma^{\prime}{}_{ij}=\sigma_{ij}+p\delta_{ij} \tag{3}\]  
where \(p=S^{r}p_{l}+(1~{}-~{}S^{r})p_{a}\) represents the pore fluid pressure formed by liquid and gas in pores. The stress-strain relationship of the solid skeleton is described as [33]  
\[\sigma^{\prime}{}_{ij}=c_{ij}^{s}\epsilon_{ij}-c_{ij}^{e}\epsilon_{ij}^{p} \tag{4}\]  
The strain tensors under the general state and the pore pressure are written, respectively, as [33]  
\[\epsilon_{ij}=\frac{1}{2}\Big{(}u_{i,j}^{s}+u_{j,i}^{s}\Big{)} \tag{5}\]  
\[\epsilon_{11}^{p}=\epsilon_{22}^{p}=\epsilon_{33}^{p}=-\frac{1}{3K_{s}}p. \tag{6}\]  
The effective stress tensor of unsaturated soil is yielded through combining Equations (4)-(6), as  
\[\sigma^{\prime}{}_{ij}=\bigg{(}\lambda_{e}\epsilon_{v}+\lambda_{v}\frac{ \partial\epsilon_{v}}{\partial t}\bigg{)}\delta_{ij}+2\bigg{(}\mu_{e} \epsilon_{ij}+\mu_{v}\frac{\partial\epsilon_{ij}}{\partial t}\bigg{)}+\frac{ K_{b}}{K_{s}}p\delta_{ij} \tag{7}\]  
where \(\epsilon_{v}=u_{i,i^{\prime}}^{s}\) and \(K_{b}\)= \(\lambda_{e}+2\mu_{e}/3\).