## 3 Theoretical Models and Results
### Pressure Model  
The pressure model is supposed to describe the relationship between the pressure and the key factors including compression ratio and particle size distribution. The compression ratio \(\varepsilon\) is used to describe how much the rock material is compressed. Its value can be calculated using Equation (1). The size distribution index \(\sigma\) represents the particle size distribution of rock material. It can be calculated using Equations (2) and (3),  
\[\varepsilon=s/b, \tag{1}\]  
\[\overline{d}=\sum_{j=1}^{m}w_{j}\overline{d}_{j}, \tag{2}\]  
\[\sigma=\left[\sum_{j=1}^{m}w_{j}(\overline{d}_{j}-\overline{d})^{2}\right]^{ \frac{1}{2}}/\overline{d}, \tag{3}\]  
where \(s\) is the stroke of the RMT-150B rock mechanics testing system and its unit is mm, \(b\) is the height of rock material in the container and its unit is mm, \(w_{j}\) is the mass percentage of the material in size range \(j\), \(\overline{d}_{j}\) is the average size of the material in size range \(j\) and its unit is mm, \(m\) is the total number of size ranges, and \(\overline{d}\) is the average size of all material and its unit is mm. Generally, for cone crushers the value range of compression ratio is from 0 to 0.4 and the value range of size distribution index is from 0 to 1.  
Figure 2: (**a**) Typical flow sheet of crushing plant; (**b**) Typical flow sheet of two-stage aggregates production plant in China.  
The crushing pressure increases with the increase of compressive ratio in the compression experiment. When the compression ratio is constant, pressure value of basalt is the largest of the four materials. The largest compression strength of crushing materials is basalt in the construction industry, and the value of compression strength of basalt is less than 350 MPa. Thereby, the relative coefficient of material hardness is determined by Equation (4).  
\[\lambda=\frac{K}{350}, \tag{4}\]  
where \(K\) is compression strength and its unit is MPa, \(\lambda\) is a relative coefficient of material hardness.  
The scatter plots of pressure are obtained using the experimental data, and it is analyzed that the relationship between these parameters is nonlinear. The relationship between pressure and size distribution index is exponential change with the increase of compressive ratio. Therefore, based on nonlinear regression analysis of experimental data, the equation of pressure model was determined and the coefficients of the model were identified as shown in Equation (5).  
\[F=\left(a_{1}\lambda^{2}+a_{2}\right)\exp\ (a_{3}\varepsilon+a_{4}\sigma+a_{5}), \tag{5}\]  
where \(a_{1}=1.0378\), \(a_{2}=0.3071\), \(a_{3}=7.8752\), \(a_{4}=0.8891\), \(a_{5}=1.1117\).  
The evaluation index of the fitting model is R-square and Adjusted R-square. The values of R-square and Adjusted R-square are 0.9871 and 0.9851 respectively, which shows that the model can better fit the experimental data. Four kinds of material pressure model are drawn respectively as shown in Figure 4.  
Based on Figure 4, it can be concluded that the pressure will increase, while the size distribution index is increasing. That is because the smaller particles move into the interspace of bigger particles and protect them from getting crushed.