# Development of a Model Material for Dynamic Geotechnical Model Tests
## 2 Selection of Similar Materials  
### Principle of Model Similarity  
Model testing involves scaling down a geological prototype according to the principle of similarity and then magnifying the physico-mechanical parameters of the model by observing the various physical phenomena of the model and using the same similarity principle to analyse the geological problems to be studied [23]. In the model test, it is difficult to realize complete similarity between prototype and model due to the limitations of test conditions, material preparation and other factors. In this study, dimensional analysis was used to deduce the similarity criterion; that is, the basic dimension was selected, and other physical quantities were derived from the basic dimension.  
In the process of dynamic testing of a geotechnical model, static physical quantities such as length (L), mass (m), density (\(\rho\)), elastic modulus (E\({}_{s}\)), Poisson's ratio (\(\mu_{s}\)), shear modulus (G\({}_{s}\)), cohesion (c), internal friction angle (\(\varphi\)), force (F), displacement (s), stress (\(\sigma\)), strain (\(\varepsilon\)), time (t), frequency (f), speed (\(v\)), gravitational acceleration (g) and acceleration (\(a\)) are considered. Dynamic physical parameters such as longitudinal wave velocity (\(v_{l}\)), horizontal wave velocity (\(v_{s}\)), dynamic elastic modulus (E\({}_{d}\)), dynamic Poisson's ratio (\(\mu_{d}\)) and dynamic shear modulus (G\({}_{d}\)) should also be considered. In this paper, C\({}_{a}\) = C\({}_{\rho}\) = 1 C\({}_{L}\) = \(n\) was selected as the fundamental dimension, the similarity relation was deduced by using the basic dimensional system [M][L][T] and the Buckingham \(\pi\) theorem, and the similarity criterion was determined [24,25], wherein [M] is the dimension of mass, [L] is the dimension of length and [T] is the dimension of time. The International System of Units (SI units) of the three are kg, m and s, respectively. In this dimensional system, [M][L][T] is independent of each other, and the dimensions of any other physical quantity can be derived from the power combination of these three dimensions.  
(1) Static parameter similarity ratio:  
\[\text{C}_{a}=\text{C}_{\rho}=\text{C}_{g}=\text{C}_{\varepsilon}=\text{C}_{ \phi}=\text{C}_{\mu_{s}}=1 \tag{1}\]  
\[\text{C}_{s}=\text{C}_{L}=n \tag{2}\]  
\[\text{C}_{r}=\text{C}_{\varepsilon}=\text{C}_{\varepsilon_{s}}=\text{C}_{ \varepsilon_{s}}=\text{C}_{\rho}\text{C}_{a}\text{C}_{L}=n \tag{3}\]  
\[\text{C}_{v}=\text{C}_{L}^{1/2}\text{C}_{a}^{1/2}=\sqrt{n} \tag{4}\]  
\[\text{C}_{t}=\text{C}_{L}^{1/2}\text{C}_{a}^{-1/2}=\sqrt{n} \tag{5}\]\[\mathrm{C}_{f}=\mathrm{C}_{L}^{-1/2}\mathrm{C}_{a}^{1/2}=1/\sqrt{n} \tag{6}\]  
\[\mathrm{C}_{F}=\mathrm{C}_{L}{}^{3}\mathrm{C}_{a}\mathrm{C}_{\rho}=n^{3} \tag{7}\]  
\[\mathrm{C}_{m}=\mathrm{C}_{L}{}^{3}\mathrm{C}_{\rho}=n^{3} \tag{8}\]  
(2) Dynamic parameter similarity ratio:  
\[\mathrm{C}_{\mu_{d}}=1 \tag{9}\]  
\[\mathrm{C}_{E_{d}}=\mathrm{C}_{G_{d}}=\mathrm{C}_{\rho}\mathrm{C}_{d}\mathrm{C }_{L}=n \tag{10}\]  
\[\mathrm{C}_{\
u_{l}}=\mathrm{C}_{\
u_{s}}=\mathrm{C}_{L}^{1/2}\mathrm{C}_{a}^ {1/2}=\sqrt{n} \tag{11}\]  
where \(\mathrm{C}_{x}\) is the similarity coefficient (that is, the proportions of prototype and model, and the subscript \(x\) is a related physical quantity), and \(n\) is the similarity multiple.