# Compaction Characteristics and Minimum Void Ratio Prediction Model for Gap-Graded Soil-Rock Mixture
## 3 A New Model for Predicting Minimum Void Ratio of Gap-Graded SRMs Incorporating Particle Breakage
### A Model for Predicting Minimum Void Ratio of Gap-Graded SRMs Incorporating Particle Breakage
\(B_{g}=k_{1}f_{c}+b_{1}\), \(B_{g}\big{|}_{f=0}=b_{1}\) and thus:  
\[\Delta V_{B}=k_{2}(V_{f}+V_{C})k_{1}f_{c} \tag{8}\]  
Combining Equations (5) and (8):  
\[\Delta V=\Delta V_{A}+\Delta V_{B}=aV_{f}+k_{1}k_{2}(V_{f}+V_{C})f_{c} \tag{9}\]  
According to the phase diagrams of coarse-grained particles dominant structure, the minimum void ratio of the mixture can be expressed as Equation (10)  
\[e^{M}=\frac{V_{VC}-V_{f}+\Delta V}{V_{C}+V_{f}} \tag{10}\]  
According to the definition of \(f_{c}\):  
\[\Delta V_{A}=aV_{f} \tag{5}\]0  
where \(m_{c}\) is the mass of coarse-grained particles, \(m_{f}\) is the mass of fine-grained particles, \(\gamma_{c}\) is the unit weight of coarse-grained particles, \(\gamma_{f}\) is the unit weight of fine-grained particles. Because the coarse-grained particles and fine-grained particles used in this paper have the same lithology, their unit weights are almost the same, \(\gamma_{c}=\gamma_{f}\), and thus:  
\[\Delta V_{A}=aV_{f} \tag{5}\]1  
Substituting Equations (9) and (12) for Equation (10), and together with \(V_{VC}=e_{c}V_{C}\), the minimum void ratio of the mixture can be expressed as:  
\[\Delta V_{A}=aV_{f} \tag{5}\]2  
Let \(k=a+k_{1}k_{2}-1\), thus Equation (13) can be rewritten as:  
\[\Delta V_{A}=aV_{f} \tag{5}\]3  
where \(k\) is a material constant. It can be seen from Equation (14) that the minimum void ratio of gap-graded SRM is linearly related to the fine particles content, which is consistent with the compaction test results shown in \[\Delta V_{A}=aV_{f} \tag{5}\]4  
In the case of fine-grained particle dominant, the phase diagram is shown in \[\Delta V_{A}=aV_{f} \tag{5}\]5 The solid volume of fine-grained particles is \(V_{f}\), and the void among coarse-grained particles is \(V_{\textit{vf}}\). Then, as shown in Figure 14b, coarse-grained particles are added to the pure fine-grained particles. In a limiting case, all the added coarse-grained particles are embedded in the sea of fine-grained particles without altering the structure of fine-grained particles. The total void volume \(V_{\textit{vf}}\) remains constant. However, in a general sense, the total void volume will change due to the particle breakage and altering the structure of fine-grained particles. The change of total volume is \(\Delta V\), and it is caused by two parts: One is caused by the altering the structure of fine-grained particles \(\Delta V_{A}\), the other is caused by particle breakage \(\Delta V_{B}\), \(\Delta V=\Delta V_{A}+\Delta V_{B}\). \(\Delta V_{A}\) is proportional to the amount of coarse-grained particles added to the mixture. \(\Delta V_{B}\) is assumed to be the product of solid volume (\(V_{c}+V_{f}\)) and (\(B_{g}-B_{g}\Big{|}_{f_{c}=1}\)). Similar to the derivation of Equation (14), the minimum void ratio of gap-graded SRM with fine-grain dominant can be expressed as:  
\[\Delta V_{A}=aV_{f} \tag{5}\]6  
where \(k^{\prime}\) is material constant. For a given mass content in fine-grained group \(f_{c}\), two values of minimum void ratio \(e^{M}\) can be calculated: One is from Equation (14), the other is from Equation (15). The greater of the two void ratio values is likely to be achieved, because it requires less energy to reach the state. Therefore, the greater of the two values is considered to be the solution. The linear relationship between the minimum void ratio of gap-graded SRM and \(f_{c}\) is shown in \[\Delta V_{A}=aV_{f} \tag{5}\]7 The line represented by Equation (14) is shown in Figure 15 as the line AP. When \(f_{c}=0\), the minimum void ratio is \(e_{c}\). The line represented by Equation (15) is shown in Figure 15 as the line PB. When \(f_{c}=1\), the minimum void ratio is \(e_{f}\). The value of \(f_{c}\) corresponding to point P can be solved from Equations (14) and (15), and this value is the optimal mass content of fine-grained particles. \[\Delta V_{A}=aV_{f} \tag{5}\]8  
The model for estimating minimum void ratio of gap-graded SRM employs 4 parameters, that is, \(e_{c}\), \(e_{f}\), \(k\) and \(k^{\prime}\).