## 2 Background of Scale Model Similitude Theories
### The Concept of Application of Scale Model Similitude to Soil Mechanics  
The scale modelling of geotechnical problems was initially described by [PERSON] (1958) [14] who differentiated between total and effective stress and also developed independent similarity relationships for each situation. For scale modelling to be employed in different types of stress system in a 1g scale model, this approach assumes a linear stress-strain relation between the model and prototype. Accordingly, the soil constitutive model can be scaled. This hypothesis is shown in Figure 2, where \(\alpha\) and \(\beta\) represent the stress and strain scaling factors, respectively. The scaling of strain conflicts with the underlying philosophy of the dimensional analysis method. However, the restriction of derivations within elastic deformations is justified by the analysis becoming insurmountably complicated once the nonlinear response is considered. [PERSON] (1968), [15], examined the complexity of replicating the constitutive behaviour of a prototype in scale models for soils whose response hinges on the confining pressure  
Figure 2: Constitutive behaviour of a scale model defined by stress and strain scaling factors. Modified from, [PERSON] (1958), [14]. loading condition, i.e., soil self-weight. The assumptions adopted by [PERSON] (1958) were extended such that the strain behaviours of the scaled and prototype elements of a soil are assumed to be identical only if these components are subjected to two geometrically comparable stress paths (provided that their initial states in the \(\mathrm{e}-\ln\sigma^{\prime}\) relation are equidistant from the critical state line). This hypothesis is demonstrated in Figure 3 and has been validated by a limited number of physical tests. [PERSON] also asserted that a centrifuge programme is an applicable approach for this type of scaling method. In further developments, the \"Buckingham pi theorem\" was developed in 1986 to solve the scale-modelling problem of the dynamic interaction of a pile in clay soil [16]. A nondimensional equation was developed to define this theory; with reference to Figure 4, \(D,EI,\ \mathrm{and}\ M\) are reference parameters in accordance with Equation (14):  
\[\frac{x}{D}=f\left(\frac{y}{D},\ \frac{M_{c}}{DM_{p}},\ \frac{J_{c}}{D^{3}M_{p}}, \frac{M_{p}}{\rho D^{2}},\frac{EI}{E_{r}D^{4}},\ \frac{EI}{E_{l}D^{4}},\frac{FD^{2}}{EI},\frac{M_{p}D^{4} \omega^{2}}{EI},\omega T_{O},\frac{\omega^{2}D}{g}\right) \tag{14}\]  
where \(x\) is the pile lateral displacement,\(y\) is the y-coordinate of maximum lateral deflection, \(D\) is the pile diameter, \(M_{c}\) is the pile cap mass, \(M_{p}\) is the pile mass per unit length, \(J_{c}\) is the moment of inertia of the pile cap, \(\rho\) is the soil mass density, \(E\) is Young's modulus of the pile, \(I\) is the moment of inertia of the pile section, \(E_{r}\) is the soil storage modulus, \(E_{l}\) is the soil loss modulus, \(F\) is the applied lateral load, \(\omega\) is frequency of vibration, \(T_{o}\) is the linear frequency sweep period, and \(g\) is the acceleration of gravity, which indicates the necessity for significant gravity in the last term. Figure 4: Formulation of tangent modulus of the constitutive behaviour of scaled soil. modified from [PERSON] (1989) [17]. Figure 3: Critical state of soil with identical stress trajectories for prototype and model. Modified from [PERSON] (1968), [15]. [PERSON] et al. (1986) [16] concluded that the gravity effects for the lateral pile response for overconsolidated clay were negligible, so the experiments were performed within this scaling system and in a 1g environment. The results demonstrated that the gravitational effects for these particular test conditions were insignificant. The frequency response was predominantly dependent on the material properties of the soil and piles.