## 2 Viscoelastic Parameter Model of the Multi-Layered CGS
### Viscoelastic Parameter Model of One-Layered CGS  
Effective elastic parameters of the one-layered CGS can be calculated by those of the solid phase with consideration of the gas phase (pore). Figure 2 shows the calculation model of the solid phase for one-layered CGS, where \(\Omega_{\text{p}}\) is the region of pebble, \(\Omega_{\text{m}}\) is the region of sand-clay matrix and \(\Omega_{\text{eq}}\) is the region of equivalent media. The pebble is first simplified as isotropic sphere inclusion and completely surrounded by the isotropic sand-clay matrix. According to the generalized self-consistent model, the effective bulk modulus (\(K_{s}\)) and shear modulus (\(G_{s}\)) can be obtained [19; 36]. \[ \begin{array}{c}K_{s}=K_{m}+\frac{p_{0}(K_{p}-K_{m})(3K_{m}+4G_{m})}{3K_{m}+ 4G_{m}+3(1-p_{0})(K_{p}-K_{m})}\\ A_{1}\big{(}\frac{G_{s}}{G_{p}}\big{)}^{2}+2B_{1}\big{(}\frac{G_{s}}{G_{m}} \big{)}+C_{1}=0\end{array}  \tag{1}\]  
where \(K_{p}\) and \(K_{m}\), \(G_{p}\) and \(G_{m}\) are the bulk modulus and shear modulus of pebble and sand-clay matrix, \(p_{0}\) is the volume fraction of pebble, \(A_{1}\), \(B_{1}\), \(C_{1}\) are undetermined coefficients. It is worth mentioning that the pebble is actually an irregular polyhedron rather than the regular sphere. Under the same volume condition, the sphere has the smallest surface area while the tetrahedron (the simplest convex polyhedrons) has the largest surface area. Since the contacted surface area of pebble with sand-clay matrix affects the mechanical properties of the solid phase greatly, the volume of sphere pebble must be increased in order to match the same surface area of the actual irregular convex polyhedron. Define a shape factor \(\rho\) to describe the amplification effect of actual polyhedron on the  
Figure 1: Diagrammatic sketch model of multi-layered coarse-grained soil (CGS) structure. Figure 2: Calculation model of the solid phase for one-layered CGS. volume fraction of sphere pebble. For the same surface area, the sphere radius of \(r\) and the tetrahedron length of \(l\) have the relationship as Equation (2):  
\[\rho_{\text{max}}=\frac{V_{s}}{V_{p}}=\frac{\frac{4}{3}\pi r^{3}}{\frac{\sqrt{2} }{12}\beta}\approx 1.8 \tag{2}\]  
where \(\rho_{\text{max}}\) is the maximum value of shape factor \(\rho\). For the actually irregular convex polyhedron of pebble, \(\rho\) = 1-1.8. Substituting \(\rho\) into Equation (1) leads to the effective bulk and shear modulus of the solid phase, as Equation (3):  
\[ \begin{array}{l}K_{s}=K_{p}+\frac{\rho p_{0}(K_{p}-K_{m})(3K_{m}+4G_{m})}{3K_ {m}+4G_{m}+3(1-\rho p_{0})(K_{p}-K_{m})}\\ A_{1}\big{(}\frac{G_{m}}{G_{p}}\big{)}^{2}+2B_{1}\big{(}\frac{G_{m}}{G_{m}}\big{)} +C_{1}=0\end{array}  \tag{3}\]  
According to Equation (4):  
\[ \begin{array}{l}K=\frac{E}{3(1-2\
u)}\\ G=\frac{E}{2(1+\
u)}\end{array}  \tag{4}\]  
the effective elastic modulus (\(E_{s}\)) of the solid phase is shown as Equation (5):  
\[E_{s}=\frac{9K_{s}G_{s}}{3K_{s}+G_{s}} \tag{5}\]  
For calculating the effective elastic modulus of the one-layered CGS, gas phase (pore) must be considered. Figure 3 shows the calculation model of the one-layered CGS, i.e., the hollow sphere model, where \(\Omega_{\text{pl}}\) is the region of gas phase (pore) as inclusion, \(\Omega_{\text{m1}}\) is the region of solid phase (sand-clay-pebble matrix) and \(\Omega_{\text{eq1}}\) is the region of equivalent media. According to the hollow sphere model, the effective bulk modulus (\(K_{o}\)) and shear modulus (\(G_{o}\)) of one-layer CGS with two phases of solid and gas can be obtained [16; 36] as Equation (6):  
\[ \begin{array}{l}K_{o}=4K_{s}G_{s}(1-p_{1})/(4G_{s}+3K_{s}p_{1})\\ A_{2}\big{(}\frac{G_{o}}{G_{s}}\big{)}^{2}+2B_{2}\big{(}\frac{G_{o}}{G_{s}} \big{)}+C_{2}=0\end{array}  \tag{6}\]  
where \(p_{1}\) is pore volume fraction, \(A_{2}\), \(B_{2}\), \(C_{2}\) are undetermined coefficients. Similarly, the elastic modulus (\(E_{o}\)) is shown as Equation (7).