## 2 Testing Equipment and Methods
### Resonant Column Test  
The RC test device is the most commonly used laboratory device for determining \(G\) and the material damping \(\xi\) of soils in the range of \(\gamma\) between \(10^{-6}\) and \(10^{-3}\) (e.g., [46; 47]). The RC technique was originally developed in the 1930s by the Japanese engineers [PERSON] and [PERSON] [48]. It was further developed in the 1960s (e.g., [49; 50; 51]) and has since become a popular technique for studying the stress-strain response of soil. It is based on oscillation of a solid or hollow cylindrical specimen in one of its vibration modes (torsional, flexural and normal modes) to determine the resonant frequency. In general, RC devices can be divided into two groups: free-free and fixed-free [52]. In the free-free RC device, the actuator is attached to either the upper or lower end of the sample, while the other end remains free to rotate. In a fixed-free RC device, one side of the specimen is restricted in its rotation, while the other end with the mounted actuator is free to rotate. RC devices are often limited in their performance, as they are usually unable to measure a very small strain (down to \(\gamma\approx 10^{-6}\)) without additional equipment [52]. In this study, the fixed-free torsional RC device from Wille Geotechnic was used (Figure 3); this device is capable of performing RC and BE tests simultaneously under isotropic and anisotropic stress conditions. The pneumatic system controls the cell and the back pressure up to 1000 kPa. To measure the axial deformation of the sample, the vertical displacement sensor is positioned in the confinement chamber. The volume changes during the saturation and consolidation stages are measured with a burette. The device works with sinusoidal torsional vibrations generated by two mini shakers attached to the top of the specimen, while the bottom of the specimen remains fixed. The force generated by the shakers is measured by a force sensor (excitation signal), while the movement of the sample is measured by an accelerometer (response signal) mounted on one side of the drive plate:  
\[amplitude=\frac{1}{\omega^{2}}\frac{V_{RMS,\bar{\mu}}}{V_{RMS,\bar{\mu}}}= \frac{1}{(2\cdot\pi\cdot f)^{2}}\frac{V_{RMS,\bar{\mu}}}{V_{RMS,F}} \tag{6}\]  
where \(V_{RMS,\bar{\mu}}\) is the root mean square \(RMS\) value of the response sensor voltage, \(V_{RMS,F}\) is the \(RMS\) value of the excitation signal voltage, \(\omega\) is the angular velocity, and \(f\) is the frequency. Normally, the resonant frequency is determined by finding the maximum amplitude (e.g., [53]). Theoretically, this method is not applicable to larger strains when the soil has significant damping \(\xi\), since the resonant frequency \(f_{r}\) is defined by a 90-degree phase shift between the response and the excitation (as shown in Figure 4 where the shift is applied to obtain a phase jump), and the frequency at maximum amplitude is:  
\[f=f_{r}\cdot\sqrt{(1-2\cdot\xi^{2})} \tag{7}\]  
Figure 3: (**a**) RC device setup. (**b**) Graphical representation of parameters estimating \(\overline{\gamma}_{max}\). The information required to calculate \(v_{s}\) in the soil under the given conditions results from the resonant frequency \(f_{r}\), specimen height \(h\) and boundary conditions defined by a factor \(\alpha\)[54]. Thus, \(G\) can be calculated as:  
\[G=\rho\cdot\left(\frac{2\cdot\pi\cdot h\cdot f_{r}}{\alpha}\right)^{2} \tag{8}\]  
The corresponding magnitude of engineering shear strain \(\overline{\gamma}_{max}\) induced in a specimen can be calculated from the peak acceleration sensor amplitude at resonance \(\ddot{u}\), the accelerometer relative position \(S\) and the specimen radius \(R\)[54]. Thus, \(\overline{\gamma}_{max}\) can be calculated as:  
\[\overline{\gamma}_{max}=\frac{\ddot{u}}{f_{r}^{2}}\ \frac{2\cdot R}{12\cdot\pi \cdot h\cdot S} \tag{9}\]  
Typically, RC devices adhere to a 2:1 ratio between specimen height and diameter.