## 3 Undrained Shear Behavior for Binary Mixtures
### Sand Modification and Shear Strength  
For the NS range of 0-15% investigated, greater NS content generally produced an increase in failure shear strength, \(q^{\prime}{}_{\text{f}}\). Treated sand isotropically consolidated at \(p^{\prime}{}_{\text{c}}=200\) kPa experienced the greatest rise in \(q^{\prime}{}_{\text{f}}\), which is consistent with earlier discussions on the significance of \(p^{\prime}{}_{\text{c}}=200\) kPa as the threshold confining pressure. By adding 15% NS, the \(q^{\prime}{}_{\text{f}}\) for confining pressures of 100, 200, 300 and 400 kPa increase by 95%, 201%, 115% and 69%, respectively, compared to the untreated sand. At particle level, this is mainly due to improvements in first stage bonding (enhanced cohesion) that produces the enhanced deviatoric stress of the treated specimens. This observation is supported by computed relative improvements in failure apparent cohesion and effective friction angle (\(c^{\prime}{}_{\text{f}}\), \(\phi^{\prime}{}_{\text{f}}\)). In other words, compared to the base sand, the 1%, 5%, 10% and 15% NS contents mobilized greater \(c^{\prime}{}_{\text{f}}\) of 73%, 250%, 450%, and 750%, respectively. Whereas, the corresponding improvements in \(\phi^{\prime}{}_{\text{f}}\) were limited to 5%. To better understand the relationship between mean isotropic effective confining pressure and peak strength, the normalized peak shear strength (\(q^{\prime}{}_{\text{peak}}/p^{\prime}{}_{\text{c}}\)) is plotted against NS content in Figure 9.  
Figure 8: Variation of flow potential with nano-silica content. Referring to Figure 9, over the NS range of 0-15%, greater NS content produced an increase in \(q^{\prime}_{\text{peak}}/p^{\prime}_{\text{c}}\), with maximum levels appearing at 15% NS for all confining pressure levels investigated. Similar findings have been reported by [44; 45]. Note that the 15% NS content is lower than the threshold fines content beyond which the overall behavior of the binary mixture would be controlled by the fines. For \(<\) 15% NS content, however, the rounded silica flocs locate in between the sand particles to build an open packing on the 600 (loose) to 402 (fairly loose) packing transition route described earlier in the paper. The chains of sand and rounded silica flocs enjoy the generally enhanced coordination number (i.e., contact number per particle) and transmit greater levels of skeletal stresses. In Figure 10, axial strain at failure deviatoric stress (\(\varepsilon_{\text{f}}\)) is plotted against NS content and indicates that strain at failure state decreased with increasing NS content for all confinement levels investigated. This confirms a more brittle dilative response for the treated sand compared to the base sand. Failure strain also decreased with increasing mean effective stress (\(p^{\prime}_{\text{c}}\)), with the maximum decrease occurring for 15% NS, while the largest recorded drop was for \(p^{\prime}_{\text{c}}\) = 200 kPa. That is, compared to the base sand, \(\varepsilon_{\text{f}}\) for the 15% NS-treated sand decreased by 24%, 36%, 35%, and 27% for \(p^{\prime}_{\text{c}}\) = 100, 200, 300 and 400 kPa, respectively. This lends further evidence to the significance of micro-structural events that take place at the 'threshold' 200 kPa mean effective stress. In Figure 11, residual deviatoric stress (\(q^{\prime}_{\text{res}}\)) is plotted against NS content for the four mean effective confinement stress levels investigated. This figure indicates a gentle increase in \(q^{\prime}_{\text{res}}\) for up to 10% NS, followed by a more marked increase for increasing NS content from 10% to 15%. As expected,  
Figure 10: Variation in axial strain at peak deviatoric stress with nano-silica content. Figure 9: Variation of normalized peak shear strength with nano-silica content. confinement pressure has a clear direct impact on \(q^{\prime}\) res. Once the specimens are sheared beyond the PT point, silica flocs migrate and'stuff' the macro-pore spaces, thereby providing support to direct sand particle-to-particle contact points.