# 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  
A gap-graded SRM consists of two grain size particles: Fine-grained and coarse-grained. The volume of the two grain size particles is denoted as \(V_{f}\) and \(V_{c}\). The void ratio of the pure two grain size groups is \(e_{f}\) and \(e_{c}\). Our objective is estimating the minimum void ratio of the gap-graded SRM. Firstly, we consider the coarse-grained particles as the dominant materials. The phase diagram of pure coarse-grained particles is shown in Figure 11(a). The solid volume of coarse particles is \(V_{c}\), and the void among coarse-grained particles is \(V_{vc}\). Then, as shown in Figure 11(b), fine-grained particles are added into the pure coarse-grained particles. In a limiting case, all the added fine-grained particles fill into the voids among coarse-grained particles without altering the structure of coarse-grained particles. Therefore, the solid volume of fine particles (\(V_{f}\)) occupies a space in the void volume (\(V_{vc}\)) and the total volume remains constant. However, as stated above, the limiting case can hardly be achieved, and the total void volume will change due to the particle breakage and altering of the structure of coarse-grained particles. The change of total volume is \(\Delta V\), and it is caused by two parts, one is caused by the altering of the structure of coarse-grained particles \(\Delta V_{A}\), the other is caused by particle breakage \(\Delta V_{B}\), \(\Delta V=\Delta V_{A}+\Delta V_{B}\). Figure 11: Comparison of measured minimum void ratios of gap-graded SRM and predicted void ratios using models from [PERSON] and [PERSON] et al. (1) Calculation of \(\Delta V_{A}\): According to [PERSON] et al. (2015), the change of void volume \(\Delta V_{A}\) caused by the added fine-grained particles is proportional to the amount of fine-grained particles added to the mixture. Therefore, \(\Delta V_{A}\) can be calculated using Equation (5). \[\Delta V_{A}=aV_{f} \tag{5}\]  
where \(a\) is material constant. (2) Calculation of \(\Delta V_{B}\): The data of particle breakage index \(B_{g}\), which is measured in the compaction tests, is plotted versus \(f_{c}\) in Figure 13. The particle breakage index \(B_{g}\) is found to be linearly related to \(f_{c}\). Here, we only show the cases of \(D=2.3\) and \(2.5\) for the limitation of the paper length. Therefore, \(B_{g}\) can be represented with Equation (6):  
\[B_{g}=k_{1}f_{c}+b_{1} \tag{6}\]  
where \(k_{1}\) and \(b_{1}\) are material constants. Figure 12: Phase diagrams for coarse-grained particles dominant: (**a**) pure coarse particles (before fine-grained particles added), (**b**) mixture (limiting case), (**c**) mixture (general case). Figure 13: Changes of particle breakage index \(B_{g}\) with \(f_{c}\) under fractal dimension \(D=2.3\) and \(2.5\). \(B_{g}\) reflects the degree of particle breakage of coarse-grained and fine-grained particles. The more particle breakage occurs, the larger \(B_{g}\) is, and thus the void volume change compared with the pure coarse-grained particles (Figure 10a) due to particle breakage \(\Delta V_{B}\) is larger. We assume \(\Delta V_{B}\) is proportional to the product of solid volume (\(V_{c}+V_{f}\)) and (\(B_{g}-B_{g}\big{|}_{f=0}\)), thus:  
\[\Delta V_{B}=k_{2}(V_{f}+V_{C})(B_{g}-B_{g}\big{|}_{f=0}) \tag{7}\]  
where \(k_{2}\) is material constant.