# Dilatant Nature of Sand Shear Strength
## 4 Shear Strength of Sand for the General Condition
The differences calculated by using the FSC and Bolton's relation are similar (Figure 16). The differences, \((\phi^{\prime}_{e}-\phi^{\prime}_{c})\), which were calculated using different failure criteria and the FSC, are shown in \[\left(\delta\varepsilon_{v}/\delta\varepsilon_{1}\right)_{b}=\left(\delta \varepsilon_{v}/\delta\varepsilon_{a}\right)_{b}=\left(\delta\varepsilon_{v} /\delta\varepsilon_{a}\right)_{c}\]6  
The Mohr-Coulomb and Matsuoka-Nakai criteria assume that \(\phi_{e}^{\prime}=\phi_{c}^{\prime}\). The difference, \(\left(\phi_{e}^{\prime}-\phi_{c}^{\prime}\right)\), which is calculated using the Lade-Duncan criterion, is higher than about \(4^{\circ}\) for non-dilative behavior, and increases for more dilative behavior, reaching around \(7^{\circ}\) for \(\left(\delta\varepsilon_{v}/\delta\varepsilon_{a}\right)_{c}=-1.2\), \(\left(I_{R}=4\right)\). Using the FSC for not very dilative behavior, \(\left(\delta\varepsilon_{v}/\delta\varepsilon_{a}\right)_{c}>-0.6\), \(\left(I_{R}<2\right)\), the difference \(\left(\phi_{e}^{\prime}-\phi_{c}^{\prime}\right)<0\); and for very dilative behavior, \(\left(\delta\varepsilon_{v}/\delta\varepsilon_{a}\right)_{c}<-0.6\), \(\left(I_{R}>2\right)\), the difference \(\left(\phi_{e}^{\prime}-\phi_{c}^{\prime}\right)>0\), reaching little more than \(5^{\circ}\) for \(\left(\delta\varepsilon_{v}/\delta\varepsilon_{a}\right)_{c}=-1.2\), \(\left(I_{R}=4\right)\). The difference, \(\left(\phi_{e}^{\prime}-\phi_{c}^{\prime}\right)\approx 4^{\circ}\), is obtained from the Lade-Duncan criterion for the non-dilative behavior and it assumes that the critical state angle of friction for the TXE is much greater than that for the TXC. However, the experiments do not confirm this. [PERSON] et al. [18] and [PERSON] [12] collected many true triaxial test results for sand. [PERSON] [12] analyzed the influence of the shear banding, the specimen slenderness ratio, the cross anisotropy, and the stability of the experimental technique on the sand strength. The analyzed data do not conform to one common failure criterion. [PERSON] and [PERSON] [19] calculated the ratio, \(\left(\phi^{\prime}/\phi_{tc}^{\prime}\right)\), on the basis of the test results collected by [PERSON] et al. [18]. The results are shown in \[\left(\delta\varepsilon_{v}/\delta\varepsilon_{1}\right)_{b}=\left(\delta \varepsilon_{v}/\delta\varepsilon_{a}\right)_{b}=\left(\delta\varepsilon_{v} /\delta\varepsilon_{a}\right)_{c}\]7  
\[\left(\delta\varepsilon_{v}/\delta\varepsilon_{1}\right)_{b}=\left(\delta \varepsilon_{v}/\delta\varepsilon_{a}\right)_{b}=\left(\delta\varepsilon_{v} /\delta\varepsilon_{a}\right)_{c}\]8 _Cont._  
The ratios, \((\phi^{\prime}/\phi^{\prime}_{lc})\) were calculated using the Matsuoka-Nakai and Lade-Duncan criteria and the FSC for \(\phi^{o}=\phi^{\prime}_{cv}=30^{\circ}\) and \(34^{\circ}\), for non-dilative \(\left(\delta\varepsilon_{v}/\delta\varepsilon_{a}\right)_{c}=0\), \((I_{R}=0)\), for moderate-dilative \(\left(\delta\varepsilon_{v}/\delta\varepsilon_{a}\right)_{c}=-0.6\), \((I_{R}=2)\), and for high-dilative, \(\left(\delta\varepsilon_{v}/\delta\varepsilon_{a}\right)_{c}=-1.2\), \((I_{R}=4)\), and the behaviors are also shown in \[\left(\delta\varepsilon_{v}/\delta\varepsilon_{1}\right)_{b}=\left(\delta \varepsilon_{v}/\delta\varepsilon_{a}\right)_{b}=\left(\delta\varepsilon_{v} /\delta\varepsilon_{a}\right)_{c}\]9 For the Mohr-Coulomb criterion, the ratio, \(\phi^{\prime}/\phi^{\prime}_{c}\)=1, is independent of the dilatancy. The effect of \(\phi^{\prime}_{cv}\), \(\left(\delta\varepsilon_{v}/\delta\varepsilon_{a}\right)_{c}\), and \(b\) on \((\phi^{\prime}/\phi^{\prime}_{c})\) is small for the Matsuoka-Nakai and Lade-Duncan criteria. In contrast, the ratio, \(\left(\phi^{\prime}/\phi^{\prime}_{c}\right)\), which is calculated using the FSC, is very sensitive to the strain increment ratio, \(\left(\delta\varepsilon_{v}/\delta\varepsilon_{a}\right)_{c}\) (dilatancy), and the critical friction angle (\(\phi^{\prime}_{cv}=\phi^{o}\)).