## 2 Methodology
### Physical Simulation
#### 2.3.2 Similar Materials  
In experimental research, the different scales between the prototype and the physical model are very common. Since the size of the physical model is extremely smaller compared to the practical engineering, the similarity of the simulated material should be considered [38]. The principle of similitude was first systematically proposed by [PERSON] in 1915 when discussing the dimensional analysis [39; 40]. It was generally used in fluid mechanics. Recently, with the development of experimental rock mechanics, more researchers started to investigate the scale effects on the interpretation of experimental results and the validity of the small-scale tests [41; 42; 43]. According to the principle of similitude, we have the following ratios of the geometric and material properties between the prototype and the practical case. The geometric similarity of the model experiment can be determined by \(C_{L}\). In the present study, the physical model is scaled at 1:200.  
\[C_{L}=\frac{L_{p}}{L_{m}}=200 \tag{1}\]  
where \(L_{p}\) is the geometric parameter of the prototype and \(L_{m}\) is the geometric parameter of the model.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Layer**} & **Simulated** & **Young’s Modulus** & **Sand/Lime/Gypsum** & **Cohesion** & **Internal Friction** & **UCS** \\  & **Material** & **(MPa)** & **(kPa)** & **Angle (\({}^{\circ}\))** & **(MPa)** \\ \hline \(1\) & Material I & 43 & 26:1:1 & 16 & 30 & 55 \\ \hline \(2\) & Material II & 52 & 26:1:2 & 19 & 32 & 68 \\ \hline \(3\) & Material III & 65 & 26:2:1 & 24 & 33 & 88 \\ \hline \(4\) & Material IV & 72 & 26:2:2 & 28 & 35 & 107 \\ \hline \(5\) & Material V & 73 & 26:2:2.5 & 29 & 35 & 111 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The mixing ratios and mechanical properties of the simulated materials.  
Since the bulk density of the simulated material and the rock strata is very close, the similarity ratio of the bulk density can be seen as one:  
\[C_{\gamma}=\frac{\gamma_{p}}{\gamma_{m}}=1 \tag{2}\]  
where \(\gamma_{p}\) is the bulk density of the prototype, and \(\gamma_{m}\) is the bulk density of the model.  
With the aid of the above two ratios, the similarity ratios about stress, strain, frictional coefficient, cohesion, and elastic modulus can then be determined.  
\[C_{e}=C_{\gamma}C_{\gamma}=200 \tag{3}\]  
\[C_{e}=C_{f}=1 \tag{4}\]  
\[C_{E}=C_{\sigma}=200 \tag{5}\]  
where \(C_{e}\) is the similarity ratio of strain, \(C_{f}\) is the similarity ratio of the friction coefficient, and \(C_{E}\) is the similarity ratio of elastic modulus.  
The key to the accuracy of the physical simulation in rock mechanics is the selection of similar materials [43]. The compounds of the aggregate and cement or gypsum are commonly used in rock mechanics. Due to the scale effects, the mechanical properties of the simulated material should be reduced accordingly based on the principle of similitude. For example, when the Young's modulus of the strata is about 10 GPa, the Young's modulus of the simulated material should be around 50 MPa. As listed in Table 3, the Young's modulus of the simulated materials is much lower than the practical value, which ranges from 43 to 72, depending on the proportions of the three main components. The values are reasonable for the principle of similitude. However, there is a limitation about the simulated material, in that it is hard to find a perfect material with a much lower Young's modulus but a similar strength to the prototype one. Though the internal frictional angle of the present simulated material is a little lower, fortunately, the ratios between them are not big. Therefore, they can still be used as a similar material.