## 2 Materials and Methods
### Methods
#### 2.2.3 Method of Results Interpretation  
Standard RC test is performed to measure the propagation speed \(V_{S}\) of a transverse (shearing) wave in a soil specimen. For material with bulk density \(\rho\), the shear modulus is given by the following formula  
\[G=\rho\cdot{V_{S}}^{2}. \tag{1}\]  
Figure 7: Schematic drawing of RC/TS apparatus. And the propagation speed \(V_{S}\) of a transverse wave is  
\[V_{S}=\frac{2\pi\cdot f_{r}\cdot L}{\beta}, \tag{2}\]  
where \(f_{r}\)--resonant frequency of the system (specimen + drive system), \(L\)--specimen's height. The parameter \(\beta\) [rad] is obtained using the formula  
\[\frac{I}{I_{0}}=\beta\cdot\tan(\beta), \tag{3}\]  
where \(l\)--mass moment of inertia of the specimen, \(I_{0}\)--mass moment of inertia of the drive system, determined during the device calibration process. To find the resonant frequency \(f_{r}\) the specimen is subjected to torsional vibrations of gradually increasing frequency. The resonant frequency would be the frequency value corresponding to the maximum twist angle that was measured during the test. Equation (1) takes the following form  
\[G=\rho\frac{4\pi^{2}f_{r}^{2}L^{2}}{\beta^{2}}. \tag{4}\]  
The amplitude of applied torque and the measured corresponding twist angle can be converted to shear stress \(\tau\) and shear strain \(\gamma\). The shear modulus can then be calculated using the formula  
\[G=\frac{\tau(\gamma_{max})}{\gamma_{max}} \tag{5}\]  
considering the fact that the graph of the \(\tau(\gamma)\) function forms a characteristic hysteresis loop. Alternatively, the shear modulus of the specimen can be found using back analysis and numerically simulated TS test results. The goal of back analysis is to find a function that accurately describes relationship between the observed (known) effects and unknown causes. To avoid problems with solution ambiguity (there may be many sets of causes that lead to the same final result), the optimization approach is often used  
\[L=\underset{X}{Min}(|\Gamma(X)-Y|), \tag{6}\]  
where \(\Gamma(\cdot)\)--cause-and-effect relationship, \(X\)--set of unknown causes, \(Y\)--set of observed effects, \(L\)--objective function (or error function). The optimal \(X\) parameters for the \(\Gamma(\cdot)\) problem are found through minimization of the \(L\) function, which measures the difference between observed effects \(Y\) and the effects predicted by the \(\Gamma(X)\) model. The following objective function was used in the performed back analyses  
\[L=\underset{n}{\sum}w\Big{(}\tau_{\gamma}^{e}-\tau_{\gamma}^{e}\Big{)}^{2}, \tag{7}\]  
where \(\tau_{\gamma}^{e}\)--measured (experimentally, during the TS test) \(\tau\) for specific values of \(\gamma\), \(\tau_{\gamma}^{e}\)--numerically calculated \(\tau\) for specific values of \(\gamma\), \(w\)--weights of the computational nodes (set to 1), \(n\)--number of nodes. It is assumed that [PERSON]'s law applies, therefore  
\[\tau(\gamma)=G\gamma, \tag{8}\]  
where \(\tau\)--average shear stress, \(\gamma\)--average shear strain, \(G\)--shear modulus. This ensures that the problem solution is stable and unambiguous. However, considering the soil stiffness degradation in small strain range the following nonlinear relation is used  
\[\tau(\gamma)=G(\gamma)\gamma. \tag{9}\]The degree of nonlinearity can only be determined through repetitive \(G\) evaluation for different \(\gamma\) values. Therefore \(G(\gamma)\) can be obtained with iterative back analysis of a dataset from a single TS test. The \(\tau(\gamma)\) relation in numerical simulation is successively fitted to the \(\tau(\gamma)\) from the laboratory experiment. The numerical simulations were performed using the finite element method (FEM). Particular attention was paid to appropriate modeling of boundary conditions. It should be noted that the soil stiffness degradation function is not always monotonic, although it is usually a nonincreasing function.