# Dilatant Nature of Sand Shear Strength
## 4 Shear Strength of Sand for the General Condition
The components of the dilatancy expressed by Equation (18) are:  
* For TXC: \[\left(\delta\varepsilon_{v}/\delta\varepsilon_{1}\right)_{c}=\left(\delta \varepsilon_{v}/\delta\varepsilon_{a}\right)_{c}=-0.3I_{R}\] (32) \[\left(\delta\varepsilon_{2}/\delta\varepsilon_{1}\right)_{c}=0.5\left( \delta\varepsilon_{v}/\delta\varepsilon_{1}\right)_{c}-0.5\] (33)
* For BXC: \[\left(\delta\varepsilon_{v}/\delta\varepsilon_{1}\right)_{b}=\left(\delta \varepsilon_{v}/\delta\varepsilon_{a}\right)_{b}=\left(\delta\varepsilon_{v} /\delta\varepsilon_{a}\right)_{c}\] (34) \[\left(\delta\varepsilon_{2}/\delta\varepsilon_{1}\right)_{b}=0\] (35)
* For TXE: \[\left(\delta\varepsilon_{2}/\delta\varepsilon_{1}\right)_{e}=1\] (36) \[\left(\delta\varepsilon_{v}/\delta\varepsilon_{1}\right)_{e}=2\{\left(\delta \varepsilon_{v}/\delta\varepsilon_{a}\right)_{e}/\left(\left(\delta \varepsilon_{v}/\delta\varepsilon_{a}\right)_{e}-1\right)\}\] (37) \[\left(\delta\varepsilon_{v}/\delta\varepsilon_{a}\right)_{e}=-0.5\left( \delta\varepsilon_{v}/\delta\varepsilon_{a}\right)_{c}\] (38)  
Assuming that the components of dilatancy, \(\left(\delta\varepsilon_{v}/\delta\varepsilon_{1}\right)\) and \(\left(\delta\varepsilon_{2}/\delta\varepsilon_{1}\right)\), are bilinear functions of the \(b\) parameter, we can write:  
\[\left(\delta\varepsilon_{v}/\delta\varepsilon_{1}\right)\ =\left(\delta \varepsilon_{v}/\delta\varepsilon_{a}\right)_{c}\ \text{ for }0\leq b\leq b_{b} \tag{39}\]  
\[\left(\delta\varepsilon_{v}/\delta\varepsilon_{1}\right)\ =\left(\frac{\delta \varepsilon_{v}}{\delta\varepsilon_{a}}\right)_{c}+\ \left\{\frac{2\left(\delta \varepsilon_{v}/\delta\varepsilon_{a}\right)_{c}}{2+\left(\delta \varepsilon_{v}/\delta\varepsilon_{a}\right)_{c}}-\left(\frac{\delta \varepsilon_{v}}{\delta\varepsilon_{a}}\right)_{c}\right\}\left(\frac{b-b_{b} }{1-b_{b}}\right)\text{ for }b_{b}<b\leq 1 \tag{40}\]  
and  
\[\left(\delta\varepsilon_{2}/\delta\varepsilon_{1}\right)\ =0.5\left\{\left(\frac{ \delta\varepsilon_{v}}{\delta\varepsilon_{a}}\right)_{c}-1\right\}\frac{b_{b}-b }{b_{b}}\text{ for }0\leq b\leq b_{b} \tag{41}\]  
\[\left(\delta\varepsilon_{2}/\delta\varepsilon_{1}\right)_{c}=0.5\left( \delta\varepsilon_{v}/\delta\varepsilon_{1}\right)_{c}-0.5\]0If the shear strength is not affected by the breakage and other undefined effects, the following can be assumed:  
\[\left(\delta\varepsilon_{2}/\delta\varepsilon_{1}\right)_{c}=0.5\left( \delta\varepsilon_{v}/\delta\varepsilon_{1}\right)_{c}-0.5\]1  
and  
\[\left(\delta\varepsilon_{2}/\delta\varepsilon_{1}\right)_{c}=0.5\left( \delta\varepsilon_{v}/\delta\varepsilon_{1}\right)_{c}-0.5\]2  
\[\left(\delta\varepsilon_{2}/\delta\varepsilon_{1}\right)_{c}=0.5\left( \delta\varepsilon_{v}/\delta\varepsilon_{1}\right)_{c}-0.5\]3  
The bilinear dependencies of \((\delta\varepsilon_{v}/\delta\varepsilon_{1})\), \((\delta\varepsilon_{2}/\delta\varepsilon_{1})\), and \(\beta_{f}\) on \(b\) are shown in Figures 12-14. Thus, the dilatancy at failure in the general state can be expressed simply by the strain increment ratio, \((\delta\varepsilon_{v}/\delta\varepsilon_{a})_{c}\), for TXC. The shear strength of sand can be calculated from Equation (16), for a known \(\phi^{\circ}=\phi^{\prime}_{cv}\), using Equations (13), (15), (18), and (39)-(45). Some failure criteria that are commonly known and used in engineering practice include the Mohr-Coulomb, Lade-Duncan [3], and Matsuoka-Nakai [2] criteria.