## 5 Discussion
### Definition of \(G_{\rm max}\)  
In the laboratory, the small strain shear modulus \(G_{\rm max}\) is more practical to be determined using the bender element or resonant column tests [12; 39; 40; 41]. In the present study, it is determined by the extrapolation of the experimental data in the strain range of around \(10^{-4}\) to \(10^{-2}\), with the help of an empirical equation [37]. As mentioned above, there are two definitions of the small strain shear modulus \(G_{\rm max}\): \(G_{\gamma=10}{}^{-6}\) and \(1/a\). Both of the two definitions have been utilized to estimate approximately the small strain shear modulus  
Figure 11: Relationship between \(\gamma_{T}\) and \(\sigma_{\rm c}\) for different determination methods. Figure 12: Percentage changes in \(G/G_{\rm max}\) for two determination methods (\(D_{\rm r}\) = 30, 50, 70% and \(\sigma_{\rm c}\) = 50–900 kPa). in the literature. Therefore, the uncertainties of \(G/G_{\rm max}\) associated with the definitions of \(G_{\rm max}\) from \(G_{\gamma=10}^{-6}\) and \(1/a\) may be different when compared to each other. In this section, a systematic comparison between two definitions of \(G_{\rm max}\) is carried out, corresponding to \(D_{\rm r}=70\%\), \(\sigma_{\rm c}=50\)-\(900\) kPa. Figure 13 shows the comparison of the calculated \(G_{\rm max}\) for the four \(G\) determination methods. As can be observed, the \(G_{\rm max}\) calculated by \(1/a\) is larger than that of \(G_{\gamma=10}^{-6}\). For instance, the autocorrelation function method predicts the maximum \(G_{\rm max}\), with a relative difference of more than 20% for \(\sigma_{\rm c}=900\) kPa. The hysteresis loop method and arithmetical average method compute the same value of \(G_{\rm max}\), but slightly smaller than that of the least square method, with the relative difference generally around 10%. Nevertheless, two definitions of \(G_{\rm max}\) capture the same trend in terms of the influence of four determination methods. This phenomenon is in agreement with the findings shown in Figure 8.  
The previous discussion has highlighted the effects of different definitions on \(G_{\rm max}\). The discrepancy observed in the \(G_{\rm max}\) will influence the corresponding \(G/G_{\rm max}\) curves. Figure 14 presents the comparison of the normalized shear modulus curves for different determination methods. The figure clearly shows the relationship between the \(G_{\rm max}\) and the corresponding \(G/G_{\rm max}\) curves for various effective confining pressures and determination methods. In fact, it can be observed that a slope with the \(G_{\rm max}\) that has been determined by the \(1/a\) will have a steeper gradient than with one that has been determined by the \(G_{\gamma=10}^{-6}\). This phenomenon is valid for various determination methods considered in this study. Additionally, for high \(\sigma_{\rm c}\) (i.e., 500-\(900\) kPa), the \(G/G_{\rm max}\) curves normalized by \(G_{\rm max}=1/a\) varies generally in a narrow band, and are almost identical for the autocorrelation function method in particular. Oppositely, the confining effect (i.e., the higher \(\sigma_{\rm c}\), the larger \(G/G_{\rm max}\)) seems to be well captured when \(G_{\rm max}\) is estimated at \(\gamma=10^{-6}\). The previous discussion has clearly illustrated that the \(G/G_{\rm max}\) curves obtained using the autocorrelation function method and the least square method are generally low when \(G\) is normalized by \(G_{\gamma=10}^{-6}\). This phenomenon is also observed when \(G_{\rm max}\) is calculated using \(1/a\), as shown in Figure 15a. As noted in Figure 15a, for the same \(G\) determination method, a significant difference in the \(G/G_{\rm max}\) appears for two definitions of \(G_{\rm max}\).