# Dilatant Nature of Sand Shear Strength
## 3 Shear Strength of Sand for TXC, BXC, and TXE  
### 3.1. Trainal Triassic Compression  
For drained triaxial compression (TXC), \(\sigma_{2}^{\prime}=\sigma_{3}^{\prime}\) and \(\epsilon_{2}^{p}=\delta\epsilon_{3}^{p}\ (\theta=\theta_{\epsilon}=\pi/6)\). At failure, \(\delta p^{\prime}=0\) elastic parts of the strain increments vanish; therefore:  
\[\left(\delta\epsilon_{2}/\delta\epsilon_{1}\right)_{c}=\left(\delta\epsilon_ {3}/\delta\epsilon_{1}\right)_{c}=\frac{1}{2}\{\left(\delta\epsilon_{p}/ \delta\epsilon_{a}\right)_{c}-1\} \tag{19}\]  
\[D_{c}=\frac{3(\delta\epsilon_{p}/\delta\epsilon_{a})_{c}}{3-\left(\delta \epsilon_{p}/\delta\epsilon_{a}\right)_{c}} \tag{20}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Sand**} & \multirow{2}{*}{\(\mathbf{d_{50}}\)} & \multirow{2}{*}{\(\mathbf{C_{U}}\)} & \multirow{2}{*}{\(\mathbf{e_{max}}\)} & \multirow{2}{*}{\(\mathbf{e_{min}}\)} & \multirow{2}{*}{\(\boldsymbol{\phi_{cv}^{{}^{\prime}}}\)} & \\ \cline{2-2} \cline{5-6}  & [mm] & [-] & [-] & [-] & [\({}^{\circ}\)] & \\ \hline Sydney & 0.30 & 1.50 & 0.856 & 0.565 & 31.0 & [PERSON] and [PERSON] [43] \\ \hline Ticino & 0.53 & 1.30 & 0.923 & 0.574 & 34 & Prearo [44] \\ \hline Portaway & 0.40 & 2.05 & 0.790 & 0.460 & 29.8 & [PERSON] et al. [45] \\ \hline Firoozkuh & 0.26 & 1.90 & 0.943 & 0.568 & 30.5 & [PERSON] et al. [37] \\ \hline Fuji River & 0.22 & 2.21 & 0.990 & 0.550 & 36.6 & [PERSON] et al. [37] \\ \hline Nevada & 0.17 & 2.00 & 0.887 & 0.511 & 33.4 & [PERSON] and [PERSON] [46] \\ \hline Hokksund & & & 0.910 & 0.550 & 32.0 & [PERSON] [47] \\ \hline Kogyuk 350/2 & & & 0.830 & 0.470 & 31.0 & [PERSON] [47] \\ \hline Reid Bedford & & & 0.870 & 0.550 & 32.0 & [PERSON] [47] \\ \hline Till & 0.11 & & 0.835 & 0.363 & 34.6 & [PERSON] and [PERSON] [36] \\ \hline Brasted & 0.28 & 2.14 & 0.792 & 0.475 & 32.7 & [PERSON] [8] \\ \hline Berlin & 0.38 & 3.00 & 0.550 & 0.390 & 31.0 & [PERSON] et al. [48] \\ \hline Karlsruhe & 0.42 & 1.85 & 0.840 & 0.530 & 34.0 & Hettler and Vardoulakis [49] \\ \hline Changi & 0.30 & 2.00 & 0.916 & 0.533 & 33.4 & [PERSON] and [PERSON] [50] \\ \hline Ham River & 0.27 & 1.67 & 0.849 & 0.547 & 33.7 & Green and Reades [51] \\ \hline River Walland & & & 0.980 & 0.621 & 30.0 & Barden and Khayatt [52] \\ \hline Chiba & & 2.10 & 0.946 & 0.500 & 33.0 & [PERSON] et al. [53] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._ \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Values**} & \multicolumn{3}{c}{**Deformation Mode**} \\ \cline{2-4}  & **Triaxial Compression** & **Plane Strain** & **Triaxial Extension** \\ \hline \(\theta\) & \(\pi/6\) & \(\theta_{b}\) & \(-\pi/6\) \\ \hline \(\theta_{\epsilon}\) & \(\pi/6\) & \(-sin^{-1}(D_{b}/3)\) & \(-\pi/6\) \\ \hline \(M^{\circ}\) & \(\frac{6 sin\phi^{\circ}}{3\ -\ sin\phi^{\circ}}\) & \(\left(3 sin\phi^{\circ}\right)/\left(\sqrt{3}cos\theta_{b}\ -\ sin\phi^{\circ}\ sin \theta_{b}\right)\) & \(\frac{6 sin\phi^{\circ}}{3\ +\ sin\phi^{\circ}}\) \\ \hline \(A^{\circ}\) & \(1-\ \left(M^{\circ}_{c}/3\right)\) & \(\frac{1}{\cos(\theta_{b}\ -\ \theta_{db})}\left\{1-\frac{2}{3}M^{\circ}_{b}\ sin \left(\theta_{b}+\frac{2}{3}\pi\right)\right\}\) & \(1-\ \left(2M^{\circ}_{e}/3\right)\) \\ \hline \(D\) & \(\frac{3\left(\delta\epsilon_{p}/\delta\epsilon_{a}\right)_{c}}{3-\ \left(\delta\epsilon_{p}/\delta\epsilon_{a}\right)_{c}}\) & \(\frac{3\left(\delta\epsilon_{p}/\delta\epsilon_{a}\right)_{b}}{2\sqrt{3\ +\ \left(\delta \epsilon_{p}/\delta\epsilon_{a}\right)_{b}}^{2}\ -\ 3\left(\delta\epsilon_{p}/\delta\epsilon_{a}\right)_{b}}\) & \(-\frac{3\left(\delta\epsilon_{p}/\delta\epsilon_{a}\right)_{e}}{3\ -\ \left(\delta \epsilon_{p}/\delta\epsilon_{a}\right)_{e}}\) \\ \hline \(\alpha_{f}\) & & 0 \\ \hline \(\beta_{f}\) & 1 & 1.4 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Characteristic values of stress–dilatancy relationship for drained conditions at a constant \(\sigma_{3}^{\prime}\) stress path. where, without the breakage effects, \(\alpha_{f}=0\) and \(\beta_{f}=1\)[22].