# Damping Ratio of Sand Containing Fine Particles in Cyclic Triaxial Liquefaction Tests
## 3 Results
However, the above phenomenon is contrary to the experimental data reported by [PERSON] [10] as discussed in the first section. A possible reason for this discrepancy might be the fact that different specimen preparation methods were employed creating different initial soil fabrics in which the influence of fines content might not necessarily be the same. As already known, the mechanical response of a soil element is highly nonlinear and strain-dependent. Subjected to cyclic loading, the strain-dependent property of sand can be manifested by the variation of energy loss per cycle \(\Delta W\) with the development of shear strain on the stress-strain curve. Figure 8 presents the stress-strain curves of HN31-C500 mixtures according to the shearing intensity and the fines content. The stress-strain curves at the initial loading stage primarily develop around the zero point, indicating a slight energy loss (i.e., quasi-elasticity) due to the external loading. However, after triggering the instability state with liquefaction failure, huge transient axial strains suddenly occur on the extension side, referring to a \"flow-type\" behaviour. Figure 8: Stress–strain curves for HN31-C500 specimens subjected to different levels of shearing with (**a**) \(F_{\rm c}\) = 5.0%, **(b**) \(F_{\rm c}\) = 10.0% and (**c**) \(F_{\rm c}\) = 15.0%. Figure 7: Cyclic shear resistance curves of HN31-C500 specimens with different fines contents. As a measure of dynamic characteristic, the damping ratio, defined as the energy loss per cycle \(\Delta W\) divided by the maximum stored energy \(W\), is more frequently employed as it is an indicative quantity describing the evolution of material states. A common manner of computing \(W\) is to assume that it is proportional to the area of the triangle bonded by a straight line defining the secant modulus on the stress-strain curve. However, this method is not suitable for this study as the axial deformations of the test specimens develops in an asymmetrical manner with an accumulative trend towards the extension side (\(\epsilon_{\text{a}}<0\)), as shown in Figures 6 and 8. From a microscopic viewpoint, this phenomenon can be explained by the following factors. Under gravity, the long axes of coarse sand grains preferentially orient along the horizontal direction to achieve a naturally stable state. Subjected to compressive shearing, sand grains tend to come into better contact with one another [30; 31]. In addition, near the liquefaction failure with the development of transient deformation, the inertia force that characterises the dynamic behaviour as distinguished from the static one appears in the form of enhancing the lateral confinement against external loading [23]. Therefore, for cyclic triaxial tests involving loading reversals (from triaxial compression to extension and vice versa), both the above reasons imply a higher axial resistance to deformation on the compression side than that on the extension side. In order to overcome the above asymmetrical property creating troubles in properly measuring the damping ratio \(D\), the maximum stored energy \(W\) is supposed to be proportional to the triangular area [32] (as shaded in Figure 8a, linking two extreme points belonging to a single hysteresis loop and one point sharing the same coordinates with them). Recently, such a definition has been successfully applied to establish the relationship between the damping ratio \(D\) and the amplitude of shear strain \(\gamma_{\text{a}}\) for the same HN31 sand for highlighting the influence of different waveforms on the liquefaction response [20]. Figure 9 presents the evolutions of \(D\) due to the progression of cycles for HN31-C500 mixtures with three different percentages.