## 3 Test Results and Discussion
### Small Strain Modulus and Modulus Reduction  
In light of the high damping and energy absorbing characteristics of the rubber chips [30; 31], from an engineering perspective, understanding of the dynamic properties of rubber-calcareous sand mixtures is also important. In Figure 7, the small strain shear modulus (G\({}_{0}\)) of the mixtures determined from the resonant column test are presented. To facilitate the comparison, the initial void ratios of the specimens remained very close (e\({}_{0}\) = 1.37\(\sim\)1.39). It can be seen in this plot that, under otherwise similar conditions, a reduction of the G\({}_{0}\) values was obtained for the specimens with higher rubber contents. Moreover, the trend between the effective confining pressure and the G\({}_{0}\) value can be characterized using a power relationship, such as G\({}_{0}\) = \(A\) (\(\sigma^{\prime}\))\({}^{n}\). The reduction of G\({}_{0}\) was mainly reflected by the coefficient \(A\) in the way that it decreased with increasing rubber content. More specifically, the \(A\) value decreased by approximately 55% with a 10% rubber content. Accordingly, the coefficient \(n\) was stress-dependent and it increased with the rubber content. Although the G\({}_{0}\) gradually decreased with the add-in rubber chips, the reduction was less remarkable when the rubber content was beyond 5%. For instance, at \(\sigma^{\prime}\) = 100 kPa, the difference in G\({}_{0}\) value was approximately 20 kPa between the specimens of sieved (0) and sieved (5), whereas further reduction of the G\({}_{0}\) value was only 12 kPa for the specimen with a 10% rubber content.  
Moreover, the modulus reduction curves and the associated damping curves of the specimens at \(\sigma^{\prime}\) = 200 kPa are presented in Figures 8 and 9, respectively, along with the literature values of the Fujian sand [32]. As a matter of fact, the ground response analysis during earthquakes depends on an accurate estimation of the soil modulus and damping properties at working strain (in the order of 10\({}^{-3}\) or greater) [33; 34]. Nevertheless, it is impractical to conduct a field test with this high level of ground shaking. Hence, the normalized modulus reduction curve and the damping curve are often used in design. Providing a small strain shear stiffness of soil (G\({}_{0}\)), the shear stiffness (G) at any given shear strain level can be estimated. Unlike the observed discrepancies in Figure 7, little difference for the specimens of varying rubber contents can be found in Figure 8. However, the modulus reduction curves of the rubber-calcareous sand mixtures are steeper than that of the Fujian sand. Accordingly, in Figure 9, the damping curves of the mixtures appear to locate above that of the Fujian sand. Again, no significant difference was yielded by varying the rubber content.  
Figure 7: Small strain shear modulus (G\({}_{0}\)) of the rubber–calcareous sand mixtures.  
In the above context, although the inclusion of rubber chips lowered the shear stiffness of the calcareous sand (e.g., at 400 kPa, the small strain shear stiffness of calcareous sand decreased by approximately 63% with a 10% rubber content), it is clear that the most detrimental impact of grain crushing on the calcareous sand was well minimized. In the current state of practice, on the other hand, the soil stiffness can be reinforced and improved by using a geomembrane or geogrid [8]. Of course, further studies along this line are worthwhile.