## 4 Results
### G-\(\gamma\) Curves  
For a given \(\gamma\) amplitude, the hysteresis loops of the specimens subjected to five numbers of loading cyclic (\(N\)) are measured with a loading frequency of 1.0 Hz, for \(D_{\text{r}}\) = 30, 50, 70% and \(\sigma_{\text{c}}\) = 50, 100, 300, 500, 700, 900 kPa. The value of \(G\) can then be calculated from the measured cyclic hysteresis loop. As an example, Figure 6 depicts the representative hysteresis loops (\(N\) = 2-4) for increasing \(\gamma\) values in the range of 0.045-1.5% (the specific strain amplitudes are given in the subplots), which corresponds to the \(D_{\text{r}}\) = 50%, and \(\sigma_{\text{c}}\) = 100 kPa. As can be observed, the shapes of the stress-strain loops measured are consistent with the expected shape up to a shear strain of 0.74% whereas the hysteresis loop becomes progressively asymmetric with increasing \(\gamma\). At high strains, the \"banana shape\" of the hysteresis loop appears. It is often assumed that the \"banana shape\" is due to the dilation of the test specimen at higher shear strains. This trend is consistent with the findings in the literature [29]. Moreover, it appears that the loading cyclic number \(N\) has a pronounced influence on the shape of hysteresis loops with the influence becoming larger as the \(\gamma\) increases. From Figure 6, the shear modulus shows a tendency of decrease with increasing \(N\)[7; 15; 19; 34]. In this study, the second cyclic loading and the corresponding hysteresis loop has been utilized for determining the value of \(G\). The influence of the hysteresis loop selection on the measured \(G\) will be discussed in the next section. Figure 7 presents the variation of shear modulus with changes in shear strain for three different relative densities (\(D_{\text{r}}\) = 30, 50, 70%) and three representative confining pressures (\(\sigma_{\text{c}}\) = 50, 300, 900 kPa). Figure 7a,c,e present the results for the variation of shear modulus  
in the entire shear strain range while the variations of shear modulus at medium shear strains (\(10^{-4}\)-\(10^{-3}\)), corresponding to \(\sigma_{\rm c}=50\) kPa are shown in Figure 7b,d,f. As expected, the shear modulus is strongly dependent on the shear strain amplitude and the effective confining pressure. The increase in the confining pressure leads to an increase in the shear modulus while the shear modulus decreases as the shear strain increases. Concerning the influence of determination methods, four methods capture similar trends. More specifically, at medium shear strain (\(10^{-4}\)-\(10^{-3}\)), the variation of shear modulus is not significant. The least square method and the autocorrelation function method predict similar \(G\) values, both of them are slightly larger than the ones of the conventional method (i.e., hysteresis loop method). At higher shear strain (\(10^{-3}\)-\(10^{-2}\)), an opposite trend is observed. Both the autocorrelation function method and least square method predict the smaller \(G\) in comparison to the hysteresis loop method, with the minimum value appearing for the autocorrelation  
Figure 7: Representative G-\(\gamma\) curves for four determination methods (**a**) \(D_{\rm r}=30\%\), (**c**) \(D_{\rm r}=50\%\), (**e**) \(D_{\rm r}=70\%\), and (**b**,d,f**) are the corresponding subplots for \(\sigma_{\rm c}=50\) kPa. function method. It appears that the difference in shear modulus at higher shear strain is more obvious for lower \(\sigma_{\rm c}\). In addition, the arithmetical average method and hysteresis loop method match very well for all the strain amplitudes tested here. It indicates that the arithmetical average method proposed by [PERSON] et al. [29] does not improve the accuracy of the hysteresis loop method on the estimation of shear modulus. This observation agrees well with the findings in [PERSON] et al. [29]. In their study, the comparison of two methods was conducted in the strain range of around \(4\times 10^{-4}\)-\(2\times 10^{-2}\), corresponding to \(D_{\rm r}=30\%\), and \(\sigma_{\rm c}=100\) kPa.