# Damping Ratio of Sand Containing Fine Particles in Cyclic Triaxial Liquefaction Tests
## 3 Results  
Regarding cyclic liquefaction tests, it is customary to take either (i) the excess pore water pressure ratio \(r_{\text{u}}\) (\(r_{\text{u}}=\Delta u/\sigma^{\prime}_{\text{c}}\)) or (ii) axial strain \(\varepsilon_{\text{a}}\) as the standard to recognise the outset of sand liquefaction [7; 23; 24]. Between these two criteria, the latter [15] is more preferred since the excess pore water pressure \(\Delta u\) of sandy specimens containing fine particles was experimentally found not to completely develop till the consolidation stress but rather stop building up when it reached a value equal to approximately 90-95% of the consolidation stress. For this reason, the achievement of 5% axial strain in double amplitude (D.A.) during one cycle was thought to estimate the attainment of liquefaction failure. Figure 5 presents the experimental results of HN31-C500 mixtures with the fines content \(F_{\text{c}}\) equal to 5.0%. For better viewing, two partial enlarged views are provided on the right. The excess pore water pressure \(\Delta u\) displayed in Figure 5a suggests that the initial increment in \(CSR\) from 0.075 to 0.090 could significantly accelerate the development of sand liquefaction. Figure 5b displays the relationship between the axial strain \(\varepsilon_{\text{a}}\) and the number of cycles \(N_{\text{cyc}}\). In the initial loading stage, \(\varepsilon_{\text{a}}\) accumulated in a granular manner. After \(r_{\text{u}}\) attained a higher value of about 80%, the accumulation rate of \(\varepsilon_{\text{a}}\) was sharply amplified and the specimen finally liquefied with the transient axial strain on the extension side. With the increase in \(CSR\) from 0.075 to 0.090, the number of cycles to liquefaction (i.e., 5% in D.A.) \(N_{\text{L}}\) decreased from 269 to 56. In contrast, the next increment in \(CSR\) was relatively inefficient  
Figure 4: (**a**) Schematic layout of the reaction frame, (**b**) test specimen and (**c**) auto compensation system [21]. to favour the build-up of liquefaction. The effective stress paths shown in Figure 5c were mainly governed by the instability line [25] referring to the flow liquefaction during which the test specimens passed from the solid to liquid state and behaved almost like a liquid. Close to the failure stage, a tiny dilatant response (as outlined by a blue arrow in Figure 5c) was indeed observed for \(CSR\) = 0.075. This puzzling phenomenon can be attributed to the fact that the coupled hydraulic actuator was incapable of perfectly managing the transient axial displacement during the liquefaction stage. Given the limited space, only the evolutions of axial strain against the number of cycles are provided with partial enlarged views in Figure 6 for \(F_{\text{c}}\) = 10.0% and 15.0%, respectively. In the graph, the axial strain first developed in a gradual manner followed by a rapid increase with the attainment of liquefaction triggering, very similar to what has been previously observed in Figure 5. These curves are two key ingredients in constructing the subsequent cyclic shear resistance curves for further analysis. Figure 5: Evolutions of (**a**) excess pore water pressure, (**b**) axial strain and (**c**) effective stress path for HN31-C500 mixtures with \(F_{\text{c}}\) = 5.0% [26]. Figure 7 displays all cyclic shear resistance curves for HN31-C500 mixtures as a function of fines content \(F_{\rm c}\). With the increase in \(F_{\rm c}\), the curve clearly moves towards the left. This indicates that the gradual addition of fine particles provides a quite negative contribution to the global soil structure because \(N_{\rm cyc}\) required for the attainment of the given failure criterion (5.0% in D.A.) decreases to be much smaller for specimens containing more inclusions. This adverse effect due to the addition of fine particles is consistent with many experimental investigations documented in the published literature [27; 28; 29].