## 3 Mechanical Properties of Layered Rock Mass
### Comparative Analysis of Elastic Modulus  
Elastic modulus is the ratio of stress to strain change value of the object in the elastic deformation stage, that is, the stress required for the unit elastic deformation of the material under the action of external force. The elastic modulus characterizes the ability of the object to resist elastic deformation. Therefore, it is of great significance to clarify the influence of combination characteristics on the elastic modulus for understanding the deformation law of rock in deep combined strata. To compare the elastic modulus, Figure 8 is drawn according to the order of the elastic modulus and the component content and layered characteristics of the sample. The values of the elastic modulus was closely related to the component content. The higher the shale content is, the lower the sandstone content is, and the higher the elastic modulus of the rock mass is. In addition, the number of layers and the location of the rock layer also affect the elastic modulus. When the component content is the same, the elastic modulus of the hard sandwich is smaller than that of the soft sandwich. The elastic modulus of the interbedded rock mass with the same component content is smaller than that of the sandwich rock mass. Figure 9 shows the stress diagram of superimposed, sandwiched and interbedded rock masses. Assuming that the stress between layers is equal, according to [PERSON]'s law, the shape variable of the assemblage can be expressed by Equation (1), and that of each rock layer can be expressed by Equation (2). \[\Delta h=\frac{Fh}{EA} \tag{1}\]  
\[\Delta h_{i}=\frac{Fh_{i}}{E_{i}A} \tag{2}\]  
where \(h\) represents the height of rock strata, \(F\) represents the pressure of the rock mass, \(E\) represents the elastic modulus and \(A\) represents the upper and lower area of the rock mass. Figure 8: Analysis diagram of elastic modulus. Figure 9: Stress analysis diagram of layered rock mass. The total deformation is the sum of the deformation of the rock layer, as shown in Equation (3). Thus, the elastic modulus of the combination can be expressed by Equation (4). \[\Delta h=\sum h_{i} \tag{3}\]  
\[\frac{1}{E}=\frac{1}{h}\bigg{(}\frac{h_{1}}{E_{1}}+\frac{h_{2}}{E_{2}}\bigg{)} \tag{4}\]  
For Figure 9b,c, the elastic modulus of the combination can be expressed by Equations (5) and (6) as  
\[\frac{1}{E}=\frac{1}{h}\bigg{(}\frac{h_{1}}{E_{1}}+\frac{h_{2}}{E_{2}}+\frac{h _{3}}{E_{3}}\bigg{)} \tag{5}\]  
\[\frac{1}{E}=\frac{1}{h}\bigg{(}\frac{h_{1}}{E_{1}}+\frac{h_{2}}{E_{2}}+\frac{h _{3}}{E_{3}}+\frac{h_{4}}{E_{4}}+\frac{h_{5}}{E_{5}}\bigg{)} \tag{6}\]  
The elastic model of the combination is only related to the elastic modulus of each component and the height of the rock layer. For the combination containing only two materials, the elastic modulus can be expressed by Equation (7). \[\frac{1}{E}=\frac{\alpha}{E_{1}}+\frac{\beta}{E_{2}} \tag{7}\]  
The percentage of the height of the material with a surface elastic modulus of \(E_{1}\) to the total height of the combination. The percentage of the height of the material whose modulus of elasticity is \(E_{2}\) in the total height of the combination. Substitute \(\beta=1-\alpha\) into Equation (8) to obtain:  
\[E=\frac{E_{1}E_{2}}{\alpha(E_{2}-E_{1})+E_{1}} \tag{8}\]  
If \(E_{2}>E_{1}\), the larger \(\alpha\) (the higher the total height of the material whose elastic modulus is \(E_{1}\)), the lower the elastic modulus of the combination. If \(E_{2}<E_{1}\), the larger \(\alpha\) (the higher the total height of the material whose elastic modulus is \(E_{1}\)), the higher the elastic modulus of the combination will be. Therefore, for the elastic modulus of a combination containing only two materials, when the component with a large elastic modulus accounts for a large number, the elastic modulus of the combination will be larger. On the contrary, when the component with a small elastic modulus accounts for a large number, the elastic modulus of the combination will be smaller.