## 3 Results and Discussion
This hypothesis is predicated on the notion that in loose and low-stiffness sands, the dynamic elastic modulus is substantially greater than the static modulus; however, this disparity diminishes as soil stiffness increases. The experimental data acquired in this study predominantly exceed the Alpan correlation curve, particularly at lower E\({}_{stat}\) values. The correlation between E\({}_{dyn}\)/E\({}_{0}\) and E\({}_{0}\) demonstrates a closer alignment with the Alpan trend, whereas the E\({}_{dyn}\)/E\({}_{50}\) and E\({}_{50}\) relationship exhibits greater variability. The results reveal that Poisson's ratio significantly affects E\({}_{dyn}\) and hence E\({}_{dyn}\)/E\({}_{stat}\) ratio. The data obtained in this study were analyzed in comparison with the experimental results presented by [8], who conducted experiments on four different sands (Sands 1, 2, 3 and 4) with varying grain distributions, utilizing the [3] correlation (Figure 8). Ref. [8] measured [PERSON]'s ratio (\(\upsilon\)) during the tests about 0.2. Therefore, comparisons are made for \(\upsilon\) = 0.2 and compared for different R\({}_{sec}\) ratios (R\({}_{sec}\) = 1, 2, 3, and 4). A high level of agreement was observed between the sands tested in this study (S1-S55) and the data reported by [8] in terms of E\({}_{dyn}\)/R\({}_{sec}\)E\({}_{50}\) ratios. This consistency supports the validity and reliability of the experimental methods and analytical approaches employed in both studies. However, the curve proposed by [3] underestimates the experimentally obtained E\({}_{dyn}\)/R\({}_{sec}\)E\({}_{50}\) (R\({}_{sec}\) = 1) values by approximately 1.5 to 2.5 times (Figure 8a). Nonetheless, when Figure 8b,c are  
Figure 6: (**a**) Scheme of a curve q(\(\varepsilon\)) in a monotonic triaxial test with an unloading–reloading cycle, definition of E\({}_{0}\), E\({}_{50}\) and E\({}_{ur}\), (**b**) definition of the tangent elastic modulus E\({}_{i}\) according to [3]. considered, the assumption of R\({}_{\rm sec}\) = 2-3 indicates that the experimental data are in good agreement with the curve proposed by Alpan. However, for higher R\({}_{\rm sec}\) values (R\({}_{\rm sec}\) = 4), this relationship deteriorates slightly (Figure 8d). Furthermore, the distribution observed among the specimens varies according to the fines content of the sands. The clean sand specimens (S5) exhibit lower E\({}_{\rm dyn}\)/E\({}_{\rm stat}\) ratios, indicating a more rigid internal structure and a smaller discrepancy between dynamic and static stiffness. Conversely, specimens with higher fines content (S1-S4) display a broader range of variation. Consequently, the position of the data relative to the Alpan curve also varies with fines content: as the fines content decreases, the data tend to align more closely with the sand region defined by [3]. In order check the reliability of the experimental results, statistical error analyses were conducted on the case discussed in Figure 8. In relation to this, Figure 9 presents the variation in the differences (residuals) between the stiffness ratios predicted according to the [3] equation and the experimental data with respect to the R\({}_{\text{sec}}\) parameter. As R\({}_{\text{sec}}\) increases, the model tends to converge to the measured values; especially for R\({}_{\text{sec}}\) = 2, the residuals decrease to the lowest level. This shows that the assumption that E\({}_{\text{stat}}\) equals twice the secant elastic modulus (E\({}_{50}\)) definition is the most compatible with the Alpan approach. On the other hand, a wider distribution of residuals in both the positive and negative directions is observed for R\({}_{\text{sec}}\) = 1 and R\({}_{\text{sec}}\) = 4, which proves that the model tends to deviate in these extreme cases. Figure 10 illustrates the residual behavior of the Alpan correlation across various [PERSON]'s ratios (\(\upsilon\)). It is evident that for \(\upsilon\) = 0.4, the residual distribution is most constrained, yielding predictions that are closest to the model.