## 4 Results
### \(G/G_{max}\)-\(\gamma\) Curves  
The normalized modulus reduction curve is used to represent the reduction of shear modulus with increasing shear strain, abbreviated as \(G/G_{\rm max}\). In CT tests, there are two methods for finding \(G_{\rm max}\). The first one is the extrapolation of the experimental data in the strain range of around \(10^{-4}\) to \(10^{-2}\), with the help of an empirical equation. The hyperbolic  
Figure 8: Percentage changes in \(G\) for two determination methods (\(D_{\rm r}=30\), \(50\), \(70\%\) and \(\sigma_{\rm c}=50\)–\(900\) kPa). model proposed by [PERSON] and [PERSON] [37] is used in this study. Equations (2) and (3) can be rewritten as follows:  
\[G=\frac{1}{a+b\gamma} \tag{16}\]  
where \(a\) and \(b\) are the fitting parameters, respectively. In general, the shear modulus at \(\gamma\) = 1 \(\times\) 10\({}^{-6}\) is assumed as the small strain shear modulus \(G_{\text{max}}\). This assumption is also utilized here. Revisiting the Equation (16), it can also be rewritten as follows:  
\[\frac{1}{G}=a+b\gamma \tag{17}\]  
Assuming \(\gamma\) = 0, \(G_{\text{max}}\) can thus be calculated using the following equation:  
\[G_{max}=\frac{1}{a} \tag{18}\]  
For the same set of measured data in CT tests, \(G_{\text{max}}\) can be found using Equations (16) and (18), respectively. As an example, Figure 9 schematically shows two methods to determine the \(G_{\text{max}}\), corresponding to \(G\)-\(\gamma\) curves for \(\sigma_{\text{c}}\) = 300 kPa and \(D_{\text{r}}\) = 70%. However, it should be noted that the values of \(a\) and \(b\) of Equations (16) and (17) show minor difference, since the hyperbolic and linear models are used to fit the measured data, respectively. Different coefficients of determination are also observed. Furthermore, the \(G_{\text{max}}\) is calculated based on 1/\(a\) assuming \(\gamma\) = 0, rather than \(\gamma\) = 10\({}^{-6}\). These two reasons which result in the calculated \(G_{\text{max}}\) values show certain difference. A systematic comparison between two estimation methods of \(G_{\text{max}}\) is carried out, and some discrepancies are observed in both the \(G_{\text{max}}\) and the corresponding \(G\)/\(G_{\text{max}}\) curves. This issue is discussed in detail in Section 5.1. The representative \(G\)/\(G_{\text{max}}\)-\(\gamma\) curves for three relative densities (\(D_{\text{r}}\) = 30, 50, 70%) and three effective confining pressures (\(\sigma_{\text{c}}\) = 50, 300, 900 kPa) are presented in Figure 10.  
These plots show that as the shear strain increases, the shear modulus decreases once a threshold strain is exceeded. The threshold strain ranges from 0.001% to 0.01% showing an increase in the threshold strain with increasing effective confining pressure. As noted in Figure 10, the normalized shear modulus curves determined by the hysteresis loop method and the arithmetical average method are almost identical. The \(G\)/\(G_{\text{max}}\)-\(\gamma\) curves measured by the other two methods (i.e., least square method and autocorrelation function method) shift on the downsides. The increased non-linearity can be attributed to the larger shear modulus measured at the medium strain while the smaller one at the higher strain leads to steeper shear modulus reduction curves (Figure 7). In addition, it can be observed  
Figure 9: Schematic representation of \(G_{\text{max}}\) = \(G_{\gamma=10}\)\(-\)6 and \(G_{\text{max}}\) = 1/a. that the \(G/G_{\rm max}\) curve varies in a narrow band for the high effective confining pressure (i.e., \(\sigma_{\rm c}\) = 900 kPa), indicating a decreased effect of determination methods. Available data of reference shear strain \(\gamma_{\rm r}\), corresponding to \(G/G_{\rm max}\) = 0.5, are reported in Figure 11 for different relative densities and effective confining pressures. A trend between \(\gamma_{\rm r}\) and \(\sigma_{\rm c}\), \(D_{\rm r}\) can be recognized. As \(\sigma_{\rm c}\) or \(D_{\rm r}\) increases, \(\gamma_{\rm r}\) also tends to increase linearly.