## 2 Theoretical Background  
### Relationship between Void Ratio and Dry Density in Gravel-Sand Mixtures  
[PERSON] et al. (2010) [20] conducted large-scale, one-dimensional compression tests on gravel-mixed decomposed granite soils (D-G-S) in an effort to analyze the influence of gravel on its compression characteristics. Before the application of compressive vertical loading of up to 10 MPa, the specimens with water content set at an optimum moisture content of 13% were compacted in a mold in 3 layers, with each layer given 21 blows of a 2.5 kg rammer falling through a height of 30 cm, resulting in constant compaction energy of \(E_{\text{c}}=187.469\) kJ/m\({}^{3}\). The optimum moisture content of 13% was obtained from the results of compaction tests on the soil components of the D-G-S, and the dry density of the mixture was obtained by taking the quotient between the wet density and the water content plus unity. Based on the results of compaction, they presented the limitation ofthe equation proposed by [PERSON] and [PERSON] (1951) [16] by comparing the predicted and measured relationship between gravel content and dry density, as shown in Figure 1. It was found that, for D-G-S, its initial dry density increased as the gravel content (\(P\)) increased up to a gravel content of 40% and decreased significantly at gravel contents (\(P\)) greater than 40%. This behavior is in disagreement with the values predicted by the [PERSON] and [PERSON] (1951) [16] equation, which assumes that an increase in the density of sand due to compaction increases the overall dry density of mixed material, and sand fills up the voids between gravels. In reality, the overall dry density reduces when the gravel content increases because there is less amount of sand that can fill up the voids between gravels, causing the overall void ratio to increase. [PERSON] et al. (2010) [20] employed the two-phase mixture theory suggested by [PERSON] (2005) [21] in an effort to study the influence of gravel content on initial dry density and compression index of D-G-S. In this theory, the overall void ratio \(e(P)\) of the mixed material is a function of the gravel content (\(P\)) by mass and is related to the void ratio of sand (\(e_{s}\)) and void ratio of gravel (\(e_{g}\)), as shown in Equation (1). \[e(P)=e_{s}(1-P)+e_{g}P \tag{1}\]  
where  
\[P=\frac{m_{S2}}{m_{S1}+m_{S2}} \tag{2}\]  
In Equation (2), \(m_{s1}\) is the dry mass of sand only, and \(m_{s2}\) is the dry mass of gravel only. Mixing sand with gravel during compaction would result in a change in the actual gravel void ratio. In that case, \(e_{g}\) in Equation (1) is replaced with the effective void ratio of gravel (\(e_{g}\)), and the overall void ratio of the gravel-sand mixture \(e(P)\) is calculated as  
\[e(P)=e_{s}(1-P)+e_{g}{}^{\prime}P \tag{3}\]  
where  
\[e_{g}{}^{\prime}=e_{g}-e_{g}R_{\rm m}(1-P) \tag{4}\]  
In Equation (4), the parameter \(R_{\rm m}\) indicates the degree of mixing between sand and gravel. If the sand and gravel are not mixed at all, \(R_{\rm m}=0\), but when they are mixed completely, \(R_{\rm m}=1\). When sand and gravel are partially mixed, \(0\leq R_{\rm m}\leq 1\). Figure 1: Limitation of [PERSON] and [PERSON] (1951) [16] equation as presented by [PERSON] et al. (2010) [20]. Substituting Equation (4) into Equation (3), the overall void ratio of the gravel-sand mixture \(e(P)\) can also be expressed as  
\[e(P)=e_{s}(1-P)+e_{g}P\{1-R_{m}(1-P)\} \tag{5}\]  
The specific gravity of the mixture \(G_{e}(P)\) is obtained by considering the ratio of the dry mass between sand and gravel, as shown in Equation (6), wherein \(G_{s}\) and \(G_{g}\) are the specific gravities of sand and gravel, respectively. To obtain the dry density of the mixture \(\rho_{d}(P)\), Equation (7) is used. Substituting Equations (5) and (6) into Equation (7), \(\rho_{d}(P)\) can be calculated using Equation (8), considering the effect of the change in actual gravel void ratio or degree of mixing between sand and gravel.