# 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
### Existing Model  
[PERSON] [7] measured porosities of sand-clay mixtures. He found that the minimum porosity of the mixture occurs when all the void space in the sand is completely filled by the bulk volume of clay. He also proposed an equation for estimating the minimum porosity of the binary mixtures, as Equation (2).  
\[n_{mix-\min}=n_{f}n_{c} \tag{2}\]  
where \(n_{mix-\min}\) is the minimum porosity of the sand-clay mixture, \(n_{c}\) is the porosity of pure sand, \(n_{f}\) is the porosity of pure clay.  
However, the ideal condition in which all the void space of coarse-grained particles is completely filled by the bulk volume of fine-grained particles can hardly be achieved. During the process of achieving minimum void ratio of the mixture, mutual interference between the coarse-grained particles and fine-grained particles will happen, that is, the so-called \"wall effect\" and \"loosening effect\" [30], as shown in Figure 10a. Wall effect means that when some isolated coarse-grained particles are immersed in a sea of fine-grained particles (which are dominant), there is a further amount of voids in the packing of fine-grained particles located in the interface vicinity. Loosening effect refers to when fine-grained particles are inserted in the porosity of a coarse-grain packing (coarse particles dominant), and if it is no longer able to fit in a void, there is locally a decrease of volume of the coarse-grained particles.  
[PERSON] et al. [23] considered the mutual inferences between fine-grained and coarse-grained particles when studying the minimum void ratio of sand-silt mixtures. They found that the added fine-grained particles would inevitably distort the structure of coarse-grained particles and cause a change of total void volume. They assumed that the change of void volume is proportional to the amount of fine-grained particles added to the mixture. Similarly, for a fine-grained particles dominant structure, the added coarse-grained particles will distort the structure of fine-grained particles and cause a change of total void volume, which is proportional to the amount of coarse particles added to the mixture. They gave the optimal \(f_{c}\) corresponding to the minimum void ratio as Equation (3).  
\[f_{c-optimum}=\frac{e_{c}-\widetilde{b}}{1+e_{c}+e_{f}-\widetilde{a}-\widetilde {b}} \tag{3}\]  
where \(e_{c}\) is the void ratio of pure coarse-grained particles, \(e_{f}\) is the void ratio of pure fine-grained particles, \(\widetilde{a}\) and \(\widetilde{b}\) are material constants.  
[PERSON] et al. [23] also proposed an equation for estimating the minimum porosity of the binary mixtures, as Equation (4).  
\[n_{mix-min}=e_{c}f_{c-optimum}+e_{f}(1-f_{c-optimum})-b_{12}e_{c}f_{c-optimum} \tag{4}\]  
where \(b_{12}\) is material constant.  
The minimum void ratio of gap-graded SRMs computed using [PERSON] and [PERSON] et al. models are shown in Figure 11 and compared with measured values from compaction tests. The comparison shows that the predictability of the model for sand-clay or sand-silt mixtures is not suitable for gap-graded SRMs. It is because, unlike sand-clay or sand-silt mixtures, gap-graded SRM will undergo more particle breakage during vibrating compaction, as the test results show in Section 2.2.1. The particle breakage during the process of compaction will generate some much finer particles, and these finer particles will further fill the voids of the mixture (shown in Figure 10b). Therefore, the effect of particle breakage should be taken into consideration when building models for predicting minimum void ratio of gap-graded SRMs.  
Figure 11: _Cont._  
Figure 10: The two parts constituting the change of void volume \(\Delta V\): (**a**) Mutual interference between coarse-grained and fine-grained particles \(\Delta V_{A}\), (**b**)Particle breakage \(\Delta V_{B}\).