## 4 CSSM Framework for Binary Mixtures
### Critical State Locus in \(q^{\prime}-p^{\prime}\) Plane  
Referring to Table 6 and Figure 12, addition of NS to the base sand produced substantial increases in \(M_{\rm cs}\), although the variation of \(M_{\rm cs}\) for 1-15% NS content was marginal. Compared to the base sand, \(\phi^{\prime}{}_{\rm cs}\) increased by 3.7\({}^{\circ}\) (i.e., increasing from 29.1\({}^{\circ}\) to 32.8\({}^{\circ}\)) for 15% NS addition, with a 2.9\({}^{\circ}\) increase achieved for NS content increasing from 1% to 15%. However, NS appears to be a relatively more effective proppant than silica silt (that is, of similar mineralogy but with larger \(D_{50}\)). As a means of comparison, our experimental results are contrasted with similar recent findings for two quarzitic angular to sub-rounded uniformly graded Toyoura and Fujian sands mixed with 0-15% angular silica silt. For these silty sand mixtures, Yang and Wei [4] showed a very modest (<2%) increase in \(\phi^{\prime}{}_{\rm cs}\). This example also demonstrates the significance of the particle shape and size comprising the fines content. The non-intrinsic component of friction angle in granular materials (\(\phi^{\prime}{}_{\rm g}\)) is dependent on particle interlocking and comprises of mobilized particle dilation (\(\phi^{\prime}{}_{\rm d}\)) and particle re-arrangement/damage (\(\phi^{\prime}{}_{\rm p}\)) contributions, with the former a function of confinement level and particle texture. As such, \(\phi^{\prime}{}_{\rm d}\) is inversely proportional to void ratio and adopts a nominal zero value when void ratio reaches the critical (or constant volume) state. At critical state, the friction angle comprises the intrinsic mineral-controlled interparticle friction angle (\(\phi^{\prime}{}_{\rm u}\)) and a limited \(\phi^{\prime}{}_{\rm g}\) (<6\({}^{\circ}\)) that largely depends on the levels of particle rearrangement and damage. Unlike silica silt, rounded NS flocs minimize the shearing-induced particle damage to sharp sand particle asperities by allowing neighboring sand particles to move over one another, thereby generating a dilative material response. Further, the relatively smaller mean size of the NS flocs leads to the development of higher levels of specific surface and a tendency of the sand particles to be finely coated. This provides additional support to sand particle edges and increases \(\phi^{\prime}{}_{\rm p}\). Collectively, these factors limit the 'edge chipping' effect, thereby increasing the magnitude of \(\phi^{\prime}{}_{\rm cs}\). To better understand the effect of fines content, \(\phi^{\prime}{}_{\rm cs}\) is plotted against NC content in Figure 13a. For consistency and comparison purposes, the general equation offered in Yang and Wei [4] for linear variation of \(\phi^{\prime}{}_{\rm cs}\) with fines content (Equation (5)) is employed here,  
\[\phi^{\prime}{}_{\rm cs}\left({}_{F_{\rm c}}\right)=k\,F_{\rm c}+\phi^{\prime} {}_{\rm cs}\left({}_{F_{\rm c}=0}\right) \tag{5}\]  
where \(F_{\rm c}\) is the fines content and \(k\) is the gradient of the of \(\phi^{\prime}{}_{\rm cs}\left({}_{F_{\rm c}}\right)\) against \(F_{\rm c}\) regression line. For the F161 test sand, a \(k\) = 0.27 was determined (for a regression coefficient of 0.95). This is greater than reported \(k\) values of 0.05 and 0.12 for the Toyoura and Fujian sands, respectively, when mixed with crushed silica in the investigations by [4]. The mineralogical composition of the base F161, Toyoura and Fujian sands and their fines contents are similar. Variations in \(D_{50}\) and roundness (\(R\)) of these sands are marginal (refer to values listed in Figure 13a). As such, the main difference between these testing materials lies in the particle shapes and sizes of their fines content.